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Dynamic pricing strategies for new products with extended warranty contracts

Zhang, Shengqiu; 张盛球 Zhang, S. [张盛球]. (2015). Dynamic pricing strategies for new products with extended warranty contracts. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5435661 2015

http://hdl.handle.net/10722/209470

The author retains all proprietary rights, (such as patent rights) and the right to use in future works.

Dynamic Pricing Strategies for New Products with Extended Warranty Contracts

Submitted by

Zhang Shengqiu

A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at The University of Hong Kong

Jan. 2015

Abstract of thesis entitled

Dynamic Pricing Strategies for New Products with Extended Warranty Contracts

Submitted by Zhang Shengqiu for the degree of Doctor of Philosophy at The University of Hong Kong Jan. 2015

An extended warranty provides consumers the opportunity to rectify product failures at little or no cost after the expiry of the base warranty. Empirical evidence has shown that the selling of extended warranty contract (EWC) has become a profitable business in many manufacturing and retail industries. This thesis investigates the dynamic pricing problem for a new product with the option of purchasing an extended warranty contract (EWC). The research simultaneously determines the pricing strategies for both the product and the associated EWC, and the production rate to maximize the manufacturer’s long-term total profit. The results show that the provision of EWC will significantly affect the optimal pricing strategy for the new product, and may also affect its optimal production plan.

The research establishes three mathematical models for making optimal pricing decisions under different operational settings. The first model considers a centralized selling system in which the manufacturer sells the product and offers the associated EWC to the consumer directly. The second model extends the first one by incorporating the production and inventory decisions in the analysis. The third model considers a decentralized system in which the manufacturer sells

the new product to consumers through an independent retailer. The EWC can be offered either by the manufacturer or by the retailer. It is shown that each scenario leads to a differential Stackelberg game in which the manufacturer and the retailer are players.

For the first model, the Pontryagin maximum principle is used to derive the necessary condition for the optimal pricing strategies for both the new product and the associated EWC. Some properties for pricing the new product optimally are then studied. Apart from analysing the characteristics of the optimal pricing strategy under general demand conditions, closed-form solutions for the problem are also derived for some specific demand functions. In cases where closed-form solutions cannot be found, a gradient algorithm is applied to solve the problem numerically. In the second model, the production rate becomes a decision variable because the unit production cost depends on the chosen production rate. Results of the analysis show that the optimal selling price for the EWC remains the same as that in the first model, while the optimal selling price for the new product are affected by the production rate. The results also show that the gradient algorithm fails to converge, thus is no longer suitable for the second model due to the complexity caused by the boundary conditions. A more robust control vector parameterization method is then developed to compute the numerical solution. Analysing the third model theoretically indicates that some necessary conditions related to the optimal wholesale price and the optimal retail price must be satisfied for the existence of an open-loop Stackelberg equilibrium. Some important managerial insights are derived on the basis of the properties characterizing the optimal solution. The control vector parameterization method is then further developed to solve the differential game problem. Numerical experiments are then carried out to demonstrate which distribution channel results in the largest profit. (498 words)

Declaration

I declare that this thesis represents my own work, except where due acknowledgement is made, and that is has not been previously included in a thesis, dissertation or report submitted to this University or to any other institution for a degree, diploma or other qualifications.

Signed________________________________________ Zhang Shengqiu Jan. 2015

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Acknowledgements

I would like to express my sincere gratitude to my supervisor, Prof. K.L. Mak, for his supervision and guidance in my research in the past few years. His feedback has helped me refine my thinking and writing. I also want to thank Dr. Chu for his valuable advice during the last leg of my revisions. Special thanks go to Dr. M. Song for her invaluable advice and patience. Thanks for all the great things she said and did, for her kindness and graciousness. Dr. Song has always enlightened me when I had difficulties to continue my research.

I am also especially grateful to my lab-mates who have accompanied me throughout this meaningful journey. Acknowledgments also go to my friend Dr. R.B. King for going through my dissertation and providing language advice.

I would also like The University of Hong Kong and the Department of Industrial and Manufacturing Systems Engineering for offering me the opportunity to study in this great PhD programme.

This thesis is dedicated to my entire family, who has always supported me in my academic undertakings. My heartfelt thanks is given to my parents for their endless love and emotional support. Without their encouragement and understanding, I can never go this far in my study.

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Table of Contents Declaration ............................................................................................................ i Acknowledgements .............................................................................................. ii Table of Contents ................................................................................................ iii List of Figures .................................................................................................... vii List of Tables ....................................................................................................... ix List of Symbols ..................................................................................................... x Chapter 1 Introduction ........................................................................................ 1 1.1 Research Background ........................................................................................................1 1.2 Research Motivation and Objectives ..................................................................................4 1.2.1 Research Motivation ....................................................................................................4 1.2.2 Research Objectives ....................................................................................................5 1.3 Research Contributions ......................................................................................................6 1.4 Thesis Organization ...........................................................................................................7

Chapter 2 Literature Review .............................................................................. 9 2.1 The Role of Warranty Policies in Marketing ......................................................................9 2.2 Dynamic Pricing Strategies for New Products in Marketing and Management ................. 10 2.2.1 Dynamic Effects in the Market .................................................................................. 11 2.2.2 Diffusion Models (Market Growth Models) ............................................................... 13 2.2.3 Estimating the Parameters in Diffusion Models ......................................................... 16 2.3 The Diffusion of New Products with Warranty Policies ................................................... 17 2.3.1 The Diffusion of New Products with BW .................................................................. 17 2.3.2 Research on Consumer Demand for the EWC ........................................................... 21 iii

2.4 Channel Design ............................................................................................................... 23 2.4.1 Pricing in Marketing Channels .................................................................................. 23 2.4.2 Dynamic Stackelberg (Leader-Follower) Game in Marketing Channels ..................... 24 2.4.3 Channel Design for New Product with EWC ............................................................. 25 2.5 Chapter Summary ............................................................................................................ 26

Chapter 3 Dynamic Pricing of a Product and Its EWC ....................................27 3.1 Introduction ..................................................................................................................... 27 3.2 Mathematical Formulation ............................................................................................... 28 3.2.1 Costs ......................................................................................................................... 30 3.2.2 Demand Functions..................................................................................................... 30 3.2.3 Mathematical formulation for the dynamic optimization problem .............................. 32 3.3 Theoretical Analysis ........................................................................................................ 34 3.3.1 Necessary and Sufficient Conditions for the Optimal Solution ................................... 34 3.3.2 Properties of the Optimal Solution ............................................................................. 47 3.4 Analysis of Warranty Contracts ....................................................................................... 53 3.5 Case Study ....................................................................................................................... 64 3.5.1 Static Market ............................................................................................................. 64 3.5.2 Dynamic Market........................................................................................................ 69 3.6 Numerical Study .............................................................................................................. 82 3.6.1 Gradient Method ....................................................................................................... 82 3.6.2 Numerical Experiments for Example 3.3 and 3.4 ....................................................... 84 3.7 Chapter Summary ............................................................................................................ 91

Chapter 4 Dynamic Pricing with Production Rate Decision ............................92 4.1 Introduction ..................................................................................................................... 92

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4.2 Mathematical Formulation ............................................................................................... 93 4.3 Theoretical Analysis ........................................................................................................ 97 4.4 Case Study ..................................................................................................................... 117 4.4.1 Static Market ........................................................................................................... 118 4.4.2 Dynamic Market...................................................................................................... 124 4.5 Numerical Study ............................................................................................................ 126 4.5.1 Control Vector Parameterization.............................................................................. 126 4.5.2 Numerical Results ................................................................................................... 130 4.6 Chapter Summary .......................................................................................................... 139

Chapter 5 Dynamic Pricing and Distribution Channel Design ......................141 5.1 Introduction ................................................................................................................... 141 5.2 Model R: Retailer Offers the EWC ................................................................................ 143 5.2.1 Mathematical Formulation of Model R .................................................................... 143 5.2.2 Theoretical Analysis ................................................................................................ 144 5.2.3 Case Study .............................................................................................................. 155 5.3 Model M: Manufacturer Offers the EWC ....................................................................... 160 5.3.1 Mathematical Formulation of Model M ................................................................... 160 5.3.2 Theoretical Analysis ................................................................................................ 161 5.3.3 Case Study .............................................................................................................. 164 5.4 Comparisons Among Distribution Channels................................................................... 167 5.5 Chapter Summary .......................................................................................................... 169

Chapter 6 Conclusions and Future Work .......................................................170 6.1 Conclusions ................................................................................................................... 170 6.2 Recommendations for Future Work ............................................................................... 173

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Reference ...........................................................................................................173 Appendix A........................................................................................................183

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List of Figures Figure 3.1 The instantaneous profit flow at time t ...................................................................... 32 Figure 3.2 Simulation of the warranty claims when the product life follows a Weibull distribution ................................................................................................................................ 55 Figure 3.3 The state-price phase diagram .................................................................................. 78 Figure 3.4 The vector field for the state-price differential equations .......................................... 80 Figure 3.5 The state-costate phase diagram ............................................................................... 81 Figure 3.6 Comparison of analytical and numerical results under different values of the tolerance 𝛿 in the gradient method............................................................................................................ 85 Figure 3.7 (𝑋(𝑡), 𝑝1 (𝑡), 𝜃1 (𝑡)) under different BW lengths ...................................................... 86 Figure 3.8 Comparison of the total profit and the profit from selling EWC under different 𝐸(𝑝2 , 𝜂) .................................................................................................................................... 87 Figure 3.9 (𝑋(𝑡), 𝑝1 (𝑡), 𝜃1 (𝑡)) under different 𝑞1 and 𝑞2 ........................................................... 89 Figure 3.10 The total profit and the profit from selling EWC under different 𝐸(𝑝2 , 𝜂) when 𝑞1 = 3, 𝑞2 = 4.......................................................................................................................... 90 Figure 3.11 The total profit and the profit from selling EWC under different 𝐸(𝑝2 , 𝜂) when 𝑞1 = 1, 𝑞2 = 5.......................................................................................................................... 91

Figure 4.1 Sample inventory / backlogging cost function........................................................... 94 Figure 4.2 The instantaneous profit flow at time t ...................................................................... 95 Figure 4.3 Diagrammatic sketch in the proof of Proposition 4.1 .............................................. 103 Figure 4.4 Trajectories of the functions 𝜁1 (𝜃3 ) and

𝑎1 2

𝜁2 (𝜃3 ) ................................................. 122

Figure 4.5 The (𝐼, 𝜃3 ) space phase diagram .............................................................................. 123 Figure 4.6 Control parameterization ........................................................................................ 127

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Figure 4.7 Comparison of the analytical result and the numerical results obtained by control vector parameterization ........................................................................................................... 131 Figure 4.8 The optimal price 𝑝1 (t) under different diffusion effects ........................................ 135 Figure 4.9 The optimal production rate 𝑞(𝑡) under different diffusion effects .......................... 135 Figure 4.10 The trajectories of (𝑋(𝑡), 𝐼(𝑡), 𝜃1 (𝑡), 𝜃3 (𝑡)) under different diffusion effect ......... 136 Figure 4.11 The inventory level under different backlogging costs .......................................... 137 Figure 4.12 The inventory level under different inventory costs given large backlogging cost . 138 Figure 4.13 The inventory level under different 𝑐2 .................................................................. 139

Figure 5.1 Two decentralized channels .................................................................................... 142 Figure 5.2 The approximated optimal wholesale and retail prices for Model R ........................ 159 Figure 5.3 The approximated optimal wholesale and retail prices for Model M ....................... 166 Figure 5.4 The wholesale and retail prices for Model R and Model M ..................................... 168

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List of Tables Table 2.1 Bass diffusion model with marketing variables .......................................................... 15

Table 3.1 The notations used in chapter 3 .................................................................................. 29 Table 3.2 Parameters for the numerical analysis ........................................................................ 84 Table 3.3 Total profit under different BW lengths ..................................................................... 85 Table 3.4 Comparison of the total profit and the profit from selling EWC under different 𝐸(𝑝2 , 𝜂) .................................................................................................................................... 87 Table 3.5 Total profit under different BW lengths when 𝑞1 = 3, 𝑞2 = 4 .................................... 89 Table 3.6 Total profit under different BW lengths when 𝑞1 = 1, 𝑞2 = 5 .................................... 90

Table 4.1 Gradient computation procedure .............................................................................. 129 Table 4.2 Control vector parameterization algorithm outline ................................................... 130 Table 4.3 Parameters for numerical experiment 1 .................................................................... 131 Table 4.4 Optimal BW length 𝜏 under different BW length elasticity b and expected unit repair cost per time 𝐶𝑟 ....................................................................................................................... 133 Table 4.5 Parameters for numerical experiment 2 .................................................................... 134

Table 5.1 Parameter values ...................................................................................................... 158 Table 5.2 Profit analysis under different 𝐸(𝑝2 , 𝜂) for Model R ................................................ 160 Table 5.3 Profit analysis under different 𝐸(𝑝2 , 𝜂) for Model M ............................................... 166

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List of Symbols BW

Basic warranty

EWC

Extended warranty contract

𝑝1 (𝑡)

The selling price of the new product at time t

𝑝2 (𝑡)

The selling price of the EWC at time t

𝑈1

The domain of variable of 𝑝1 (𝑡)

𝑈2

The domain of variable of 𝑝2 (𝑡)

T

The life cycle of the new product which starts with the launch of the product onto the market and ends when the manufacturer stops selling

𝜏

The length of the BW

𝜂

The length of the EWC

𝐶𝑚

The unit production cost of the product

𝐶𝐵𝑊

The expected repair cost of the BW

𝐶𝐸𝑊𝐶

The expected repair cost of the EWC

𝑋(𝑡)

The cumulative sales volume of the product in the time interval [0, 𝑡]

𝑌(𝑡)

The cumulative sales volume of EWC in the time interval [0, 𝑡]

𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏)

The demand rate function of the new product at time t.

𝜌(𝑝2 (𝑡), 𝜂)

The probability that consumers are willing to pay for EWC

𝑃𝐸𝐷𝑝1

The price elasticity of demand for the new product

𝑃𝐸𝐷𝑝2

The price elasticity of demand for the EWC

MR

The marginal revenue for selling the new product

𝐌𝐑 𝐄𝐖𝐂

The marginal revenue for selling the EWC

𝐶𝑟

The expected repair cost for each product failure

x

𝑁(𝑡)

The number of failures in (0, 𝑡]

𝑇𝑖 (i = 1,2 … )

The time of the i’th failure

𝑊𝑖 (i = 1,2 … )

The repair cost of the i’th failure

𝑅 (𝑡 )

The total repair cost during (0, 𝑡]

𝜆(𝑡)

The failure rate (or intensity function) of the new product

NHPP

Non-Homogeneous Poisson Process

𝑞(𝑡)

The production rate at time t

𝑈3

The domain of variable of production rate 𝑞(𝑡)

𝐶𝑚 (𝑞(𝑡))

The unit production cost function when this cost depends on the production rate

𝑐0

The cost parameter which is independent of production rate

𝑐1

The cost parameter linearly depends on production rate

𝑐2

The cost parameter nonlinearly depends on production rate

𝛽

The nonlinear degree parameter

𝑄(𝑡)

The cumulative production volume in the time interval [0, 𝑡]

𝐼(𝑡)

The inventory level at time t

𝑆(𝐼(𝑡))

The inventory / backlogging cost function at time t

𝛿+

The cost parameter for holding positive inventory

𝛿−

The cost parameter for backlog demand (i.e., negative inventory)

NLP

Nonlinear optimization problem

𝑝1𝑀 (𝑡)

The wholesale price for the new product in decentralized channels

𝑈1𝑀

The domain of variable of 𝑝1𝑀 (𝑡)

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Chapter 1 Introduction 1.1 Research Background “Would you like to buy an extended warranty contract?” This is the most frequently asked question when a consumer buys a new product, such as a digital camera, a computer, an automobile and a wide range of other audio/visual equipment. Most new products include a Basic Warranty (BW), offered by the manufacturer, which guarantees to replace or repair the product once it fails to perform its normal function during the BW’s valid period, e.g., one year or two years, starting from the time of initial purchase. Compared to the lifetime of the product, the BW is rather short, so the seller usually recommends consumers to buy an Extended Warranty Contract (EWC), which is a prolonged warranty offered to consumers after the expiration of the BW. Unlike the BW whose price is included in the price of the product (Murthy and Jack 2003), the EWC is purchased separately at the time of initial purchase. It essentially works like an insurance contract offered by a manufacturer, a retailer or a third-party provider. The warranty policies, the BW and the EWC, are the most widely used after-sales services in marketing. The BW usually serves as a persuasive marketing tool, since a satisfactory BW policy implies good quality which certainly enhances consumers' willingness to purchase the product. The EWC policy has become very prevalent among consumers and manufacturers during the last two decades, and it is one of key revenue generators for the manufacturers. According to the citing sources (such as the PC World, Wall Street Journal and Automotive News) of Desai and Padmanabhan (2004), nearly 20% of automobile buyers and over 75% of home electronics and appliances buyers have chosen to purchase an EWC. The 2009 statistics from the USA National Automobile Dealers Association showed that 34.4% of new car buyers bought an EWC in the first half of that year, which was up from 23.5% in 1999. With average car dealers now losing money on each car sale, selling the EWC is an important source of dealer profits. Another example comes from PC companies: the profit of selling EWC accounted for nearly 30% of Dell’s and nearly 43% of Apple’s net income (Warranty Week, 2004). It has been estimated that

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the average profit margin for selling EWC can reach 50%-60%, much higher than the profit margin for selling new products alone. Traditionally, pricing strategies and production decisions (such as production rate and inventory level) for new products are two important factors for maximizing a manufacturer’s profit. Optimal pricing strategies can improve selling profit, while optimal production decisions can reduce cost. Nowadays, both BW and EWC have become crucial generators of revenue and profit for a manufacturer. As pointed out by Cohen et al. (2006), this is the golden age of services. In order to survive and prosper, every manufacturer (or retailer) must transform itself into a service business with the aim of extending the profit chains and boosting the market share. Then, the following questions arise when the manufacturer tries to maximize its total profit (including selling new products as well as selling EWC): 

Should pricing strategies and production decisions be adjusted when the company considers offering warranty policies?



How to find the optimal trade-off balance among pricing strategies, production decisions and warranty policies?

To better understand these questions, it might be helpful to justify how the pricing strategies, production decisions, as well as warranty policies interactively affect the manufacturer’s profit (Sterman, 2002). The product pricing strategies play a central role in sustaining a manufacturer’s profitability. Generally speaking, a lower product price usually increases the sales volume, but reduces the profit per unit of product. On the other hand, a higher price can increase the profit per unit of product. However, it tends to dissuade consumers from purchasing. Therefore, the manufacturer must set a rational selling price for its product. In addition, the unit production cost of a new product, which is usually affected by the production rate, influences the pricing strategies. Although factors such as the cost of raw material and the labour cost of each product are independent from the production rate, empirical observations show that factors such as rent and fixed investment are inversely proportion to the production rate, and factors such as tool wearout cost and rework cost are proportion to the production rate. In reality, the production rate must

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be adjusted to match the demand, because high production rate may lead to excessive inventory, while low production rate may lead to shortage. Hence, the manufacturer should consider pricing strategies and production rate simultaneously because the unit production cost of the product is significantly affected by the production rate. As for warranty policies, BW is usually used by a manufacturer as a means of advertising product quality in order to improve the sales volume (Ladany and Shore, 2007) while EWC is essentially analogous to an insurance. In general, EWC has little impact on improving the demand of the regular product because it is usually sold as a separate contract after consumers have purchased the new product. However, selling the EWC creates additional benefits that are not provided by the original BW (Pingle, 2010). Obviously, when the sales volume of the new product increases, potential buyers of the EWC will also increase, which leads to an increase in revenue contributed by selling the EWC. However, offering warranty polices results in extra maintenance cost to the manufacturer. The manufacturer therefore should accurately estimate the extra cost incurred by the warranty programme (the BW and the EWC) so as to assess its impact on profitability (Chattopadhyay and Rahman, 2008; Kleyner and Sandborn, 2008; Wu and Xie, 2008), before formulating comprehensive plans via linking warranty polices to pricing strategies and production decisions (Murthy and Blischke 2000; Huang et al. 2007). Manufacturers often aim to maximize total profit from a long-term perspective. In situations where the manufacturer’s decisions in the current period are likely to impact future outcomes (e.g., sales), designing appropriate pricing and production strategies for a new product with warranty policies is a difficult task, because of the complex dynamic effects associated with the diffusion of the new product over the planning horizon (Chatterjee, 2009). To assist the manufacturer in making optimal decisions, this study focuses on a rather general type of demand function for the new product which is affected by the following important factors (Chenavaz et al., 2011; Ruiz-Conde et al., 2006): (a) Diffusion effect (network effect) - Consumers who have purchased the regular product will inform other people who have not bought the product. This is a result of demand-side learning through, for example, word-of-mouth (Bass, 1969).

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(b) Saturation effect - As the number of potential adopters in the market is usually limited, the more cumulative adopters, the fewer remaining potential adopters (Bass, 1969). (c) The selling price of the new product - The demand for the new product is inversely proportional to the selling price. (d) The length of the BW contract – As longer BW length usually implies better product quality and better after-sales service to consumers, it can promote the demand. The market is dynamic if the diffusion and saturation effects have an important influence on the new product diffusion process. Otherwise, the market is static if it does not exhibit any diffusion and saturation (Fan, 2005; Wu, 2009; Faridimehr, 2013). This study investigates the pricing strategies, warranty polices, and production decisions for launching a new product into both market environments for a manufacturer targeting profit maximization.

1.2 Research Motivation and Objectives 1.2.1 Research Motivation Pricing strategies, warranty policies (BW and EWC) and production decisions are the essential issues that affect a manufacturer’s profit in a long-term planning horizon. Lin and Shue (2005), Wu et al. (2006), Huang et al. (2007) and Zhou et al. (2009) tried to examine the influence of the BW policy on the pricing strategy, but the influence of the EWC policy was not considered. Lin and Shue (2005), Wu et al. (2006) and Zhou et al. (2009) were based on static markets. Although the work of Huang et al. (2007) was based on a dynamic market, it did not consider the effect of production rate. The works of Li et al. (2012) and Desai et al. (2004) incorporated the EWC policy in the distribution channel design. They considered a simple supply chain consisting of a manufacturer and a retailer under two channel configurations, i.e., either the manufacturer or the retailer offers the EWC. The optimal pricing strategies for a static demand with additive demand function were studied under each channel configuration. Although pricing models for new products have been extensively studied by researchers, most of these models are developed for the single-period situation and do not account for the effect of current decisions on future

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outcomes. In addition, research on simultaneous optimization of pricing strategies, warranty policies, and production rate under dynamic market condition is very limited. Indeed, earlier research studies have indicated the following three research gaps: 1. When a manufacturer offers both BW and EWC to customers, how do these two warranty polices influence the pricing strategy of the new product? 2. When the unit production cost of the new product is significantly affected by production rate, how do pricing strategies, warranty policies and production rate affect one another? 3. How to extend the work of Li et al. (2012) and Desai et al. (2004) to include dynamic market?

1.2.2 Research Objectives This study aims at developing optimal pricing strategies of a new product and its EWC for maximizing a manufacturer’s total profit over the planning horizon under the dynamic market condition. Optimal decisions related to the production rate and the distribution channel design are also formulated. Indeed, the study covers the following aspects: (a) A basic mathematical model is developed to derive the optimal pricing strategies of the new

product and its EWC for both static and dynamic markets. The properties of these optimal pricing strategies are studied to provide some managerial insights, and to highlight the impact of warranty policies on pricing strategy via both analytical and numerical methods. (b) If the unit production cost depends on the production rate, the basic mathematical model is

extended to include the production rate as a decision variable. Based on the structure of the extended mathematical model and the properties of the optimal decisions, some managerial insights are obtained, and an efficient algorithm is developed to establish the pricing strategies of the new product and its EWC, and the production rate for maximizing the manufacturer’s profit. (c) When the new product is sold through a retailer, two dynamic Stackelberg games are

proposed for developing the optimal pricing strategies of the new product and its EWC for

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different distribution channel designs (i.e., either the manufacturer or the retailer offers the EWC). The properties of the corresponding Stackelberg equilibria are analysed for each game, and an efficient solution algorithm is developed to help the manufacturer choose the distribution channel which leads to larger profit.

1.3 Research Contributions Three novel mathematical models have been developed in this study to formulate dynamic pricing strategies and production decisions for a manufacturer targeting profit maximization. These models respectively generate the optimal results for each of the following scenarios: (1) the manufacturer sells the product then directly offers the customer the associated EWC; (2) the manufacturer sells the product then directly offers the customer the associated EWC, and the production rate has significant impact on the production cost; and (3) the manufacturer sells the product to the consumer through an independent retailer. Either the manufacturer or the retailer may offer the associate EWC to the customer. Based on the structure of the mathematical models, this research has reached the following conclusion. In the presence of EWC, the diffusion / saturation effect on the demand side, and the effect of the production rate on the supply side, the manufacturer has to trade-off among the profit from selling the new product, the profit from selling the EWC, and the marginal cost affected by the production rate. Therefore, the economic rule that marginal revenue is equal to marginal cost in the single-period monopoly setting does not apply in the above three scenarios. To maximize the manufacturer’s total profit, an efficient algorithm has been developed for the first scenario. The algorithm is based on the gradient method which relies on the optimality conditions derived by invoking the Pontryagin maximum principle. However, when production and inventory decisions are included in the model (i.e., the second scenario), the complexity of the model increases and the algorithm based on the gradient method cannot effectively handle problems with complex boundary conditions and / or path constraints. Another algorithm is then developed based on the control vector parameterization method, by approximating the control 6

decision variables with linear combination of piecewise-constant functions, and by converting the optimal control problem into a finite-dimensional non-linear optimization problem. Past studies on dynamic pricing problems have mainly focused on theoretical analysis, such as deriving optimality conditions of the solutions and their related managerial properties. The three mathematical models and the algorithms developed in this study, however, provide not only theoretical closed-form solutions for some specific cases, but also numerical solutions that can validate managerial properties of the optimal solution. This is a methodological contribution to the field of dynamic pricing problems.

1.4 Thesis Organization This thesis is organized as follows: Chapter 2 presents a review of previous research. First, the role of warranty policies (both the BW and the EWC) in marketing is discussed. Second, the review introduces the basic concepts of dynamic pricing for new products in marketing and management as well as some important preceding research. Third, some important dynamic demand functions widely adopted in preceding research are summarised. Finally, the state-of-the-art research related to channel design problems is elucidated. Chapter 3 proposes a dynamic optimization model to jointly optimize the pricing strategies of the new product and its EWC. The model reflects the relationship between the warranty policies and the pricing strategies when the production rate has little effect on the unit production cost of the new product. The important properties of the optimal selling prices of the new product and the optional EWC are derived by invoking the Pontryagin maximum principle. Optimal closed-form solutions are derived for some special demand functions, and a gradient algorithm is developed to obtain numerical solutions for general demand functions. Chapter 4 extends the model in Chapter 3 to simultaneously consider the pricing strategies of the new product and its EWC, and the production decisions (i.e., production rate and inventory). The important properties of the optimal selling prices of the new product and its EWC, and the

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optimal production rate are studied by invoking the Pontryagin maximum principle to provide some managerial insights. Since the model is very complex, the technique of robust control vector parameterization is used to derive the optimal decisions in the forms of piecewiseconstant functions. Chapter 5 focuses on the distribution channel design problem. Two differential Stackelberg game models are formulated based on whether the manufacturer or the retailer offers the EWC. The necessary conditions to derive the open-loop Stackelberg equilibria (i.e., the optimal pricing strategies for the manufacturer and the retailer) and some properties characterizing the optimal solutions are studied. In addition, the control vector parameterization method is amended to compute approximated solutions for the differential game problems. Based on these numerical results, the manufacturer can compare which distribution channel is more profitable. Chapter 6 presents a summary of the work done in this study, the limitations of this study and the recommendations for future research.

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Chapter 2 Literature Review This chapter provides a thorough survey of the relevant literature. Section 2.1 discusses the role of two widely accepted warranty policies (the BW and the EWC) in marketing. An overview of the previous studies related to dynamic pricing of new products is presented in Section 2.2. Section 2.3 narrows the scope to the profit optimization problems for new products with BW and / or EWC. Particularly, the demand functions for new products with BW and the consumer demand for EWC are also summarised in this section. The last Section 2.4 introduces the channel design problems and the related research incorporating the EWC.

2.1 The Role of Warranty Policies in Marketing Consumers usually have no idea about the performance of a new product over its useful life. Therefore, manufacturers need to promote the product brand and provide information to help consumers make the purchasing decision. In this process, warranty policies serve as persuasive marketing tools (Murthy et al., 2006) in two aspects. First, as a promotional tool, warranty policies promote the quality and reliability of a new product with longer and better warranty terms. Second, as a protection tool, warranty policies reduce the risks taken by consumers by guaranteeing to repair defective products that fail to perform normally during the warranty period. Therefore, warranty policies are important strategies for manufacturers to create and manage the market for the new product. Many different types of warranty policies for new products have been proposed both in industrial practices and theoretical research (Murthy et al., 2002; 2006). This study only focuses on the most common ones: the basic warranty (BW) and the extended warranty contract (EWC). As pointed out in Chapter 1, a BW is an integral part of a new product and is offered by the manufacturer at no additional cost to consumers. While an EWC provides additional coverage beyond the BW, consumers have to pay extra premium to obtain it. In addition, the EWC is an optional warranty which is not tied to the sale of a new product and may be offered either by the manufacturer or by a third party.

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According to marketing signal theory (Murthy et al. 2006), the BW provides consumers information about product quality and reliability. In general, longer BW period must result in higher maintenance cost for the manufacturer. If the new product has a high reliability and better quality, the cost will decrease. Therefore, consumers reckon that the BW is positively related to product quality. In contrast, the EWC has little impact on the sales volume of the new product, but it can bring extra profit for the manufacturer because the selling price of the EWC is well in excess of the fair maintenance cost incurred by the EWC (Murthy et al. 2006). Selling the EWC can generate huge profits. For example, in 2003, 50% of Best Buy’s profit and almost 100% of Circuit City’s profits stemmed from selling the EWC, although they only accounted for 3%-4% of the total revenue (Chen et al., 2009). In economic theory, consumer choice problems are usually modelled as an optimization problem with the aim of maximizing consumers’ expected utility function through taking into account consumers’ attitude to risk and information at the time of making the purchase decision. Many studies applied these models to analyse the choice problem for the optional EWC. They do provide some insights for managers in warranty management. For example, when consumers vary in their degree of risk aversion, manufacturers can segment the market by offering EWC to more risk-averse consumers and offering no warranty for less risk-averse consumers (Padmanabhan, 1996). When consumers vary in usage intensity, low-usage consumers prefer less EWC, while high-usage prefer an EWC with a high price because these consumers face a higher risk of failure (Murthy and Blischke, 2006).

2.2 Dynamic Pricing Strategies for New Products in Marketing and Management Dynamic pricing refers to a pricing strategy using a long-term perspective, which recognizes the future implications of the current price (Rao, 2009). This section reviews previous research about designing an appropriate pricing strategy for a new product over its whole planning horizon (sometimes called the life cycle of the new product, i.e., from the time the product is launched until it exits the market). In this situation, the appropriate pricing strategy should be dynamic

10

because it usually involves many complicated dynamic effects (e.g., learning effects, diffusion and saturation effects) associated with the diffusion of the new product in a given market. Since the seminal work of Robinson and Lakhani (1975), this topic has been a central research theme in marketing. Chatterjee (2009) gave a detailed and comprehensive review. Most models related to dynamic pricing strategies for new products seek to determine the optimal price trajectory over the planning horizon to optimize certain objectives (e.g., the profit stream), given the demand function, the initial and terminal times, and / or the boundary conditions. These models can all be formulated as follows:

𝑇

𝑀𝑎𝑥 ∫ [𝑝(𝑡) − 𝑐(𝑁(𝑡))] ( 𝑝(𝑡)

0

𝑑𝑁(𝑡) ) 𝑑𝑡, 𝑑𝑡

(2.1)

subject to

𝑑𝑁 (𝑡) = 𝑓(𝑁(𝑡), 𝑝(𝑡)); 𝑁(0) = 0, 𝑑𝑡

(2.2)

where 𝑁(𝑡) represents the cumulative sales (or market penetration), 𝑓(𝑁 (𝑡), 𝑝(𝑡)) is the demand function, and 𝑐(𝑁(𝑡)) represents the marginal cost, which may decline in cumulative sales due to cost learning effects. Note that both current demand and marginal cost depend on the cumulative sales, which is determined by the past price. Therefore, the optimal pricing strategies in these models must be dynamic. Typically, these dynamic optimization problems are solved by applying calculus of variations or optimal control theory (Sethi et al., 2000). In the following Section 2.2.1, some of the effects that lead to dynamic pricing are summarised.

2.2.1 Dynamic Effects in the Market Learning effects

11

One important learning effect is the cost learning effect, i.e., the unit production cost decreases with the cumulative output. Empirical studies in marketing show that current productivity is linked to past production (Arrow, 1962; Dockner et al., 2000), i.e., productivity depends on past experience in the production of a particular product. This observation can be modelled by using past production quantities as a proxy for accumulated experience (Arrow, 1962). The learningby-doing effect implies that productivity increases with cumulative production. This increase in productivity results in a decrease in the unit production cost (Dockner et al., 2000).

Diffusion effects Diffusion effects refer to all the phenomena related to a higher market penetration (i.e., cumulative sales) that affect the probability of purchase, which include greater knowledge of the market through word-of-mouth, self-advertisement of the product, and uncertainty reduction due to a better reputation of the product among others (Mahajan et al., 1990; Rao et al., 2009). Word-of-mouth has an internal impact on sales. It is linked to social interactions of previous buyers who contribute to the product's reputation. Obviously, the more a product sells the stronger the results are. For a good-quality product, this effect is often viewed as a positive aspect and thus has a positive impact on sales. However, this effect can also be negative as is the case when buyers discover that the product is not suitable for them or the quality is poor (Kalish, 1983).

Saturation effects While diffusion effects mainly imply that past sales could stimulate future sales, saturation effects occur when current sales reduce the market for the product in the future. The number of the potential adopters, i.e., individuals who are willing to purchase the product, is usually limited. For a “perfectly durable product” (i.e., a product that can last for a long time), the number of products each potential adopter needs is also limited. Therefore, the potential market is finite. As total sales increase, there is less and less unsatisfied demand, which reduces future sales (Kalish, 1983; Levinthal et al., 1989). In other words, each new sale reduces the future market.

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As mentioned in the above discussion, cost learning effects mainly influence the marginal cost, while the diffusion and saturation effects have a significant impact on the demand function. The demand function always plays an important role in marketing. Clearly, the dynamic pricing models for new products require specifying the demand functions of the new products. Recently, marketing researchers have turned their attention to investigating the diffusion of a new product within a social system (Ruiz-Conde et al., 2006). (Note: the concept “innovation” is widely used in these literatures, and the new product is viewed as a type of innovation.). A number of diffusion models have been developed to capture the typical sales pattern (i.e., demand function) associated with the diffusion process of new products. More concrete descriptions of these diffusion models are summarised in Section 2.2.2.

2.2.2 Diffusion Models (Market Growth Models) Fundamental diffusion model The fundamental idea behind these diffusion models is that the sales rate of a new product at any point in time depends on the cumulative sales, that is 𝑑𝑁(𝑡) = 𝑓(𝑁(𝑡)), 𝑑𝑡 where the notations are the same as those in (2.2). Among these models, the Bass diffusion model is the most famous one and can forecast the long-term sales pattern of new technologies as well as new durable products. It has been widely accepted and applied in the marketing literature (Mahajan et al., 2000; Parker, 1994). The original Bass diffusion model has the form:

𝑑𝑁(𝑡) 𝑁 (𝑡 ) ] [𝑀 − 𝑁(𝑡)], = [𝑝 + 𝑞 𝑑𝑡 𝑀

13

(2.3)

where 𝑀 is the population size of the potential adopters, while 𝑝 and 𝑞 are the coefficients of innovation (external influence) and imitation (internal influence) respectively. The underlying demand dynamics are driven by the saturation effect and the diffusion effect. The saturation effect is the second term on the right hand side of (2.3), which is decreasing in cumulative sales. The diffusion effect is the first term and it is increasing in cumulative sales. Moreover, the diffusion effect drives the dynamics in the early stage of the life cycle, while the saturation effect dominates in the end. Thus, the demand should be increasing in cumulative sales at the beginning, but decreasing later in the life cycle.

Models with marketing variables The parameters in the classical Bass diffusion model have clear economic interpretations. However, this model only captures the impact of marketing variables (such as price, advertising, warranty policy, etc.) on the diffusion process of a new product implicitly. These variables, if incorporated explicitly in diffusion models, provide more insight and directions for influencing the diffusion process through marketing instruments (Peres et al., 2010; Ruiz-Conde et al., 2006). The importance of including marketing mix variables in diffusion models has been widely recognized in the literature (Josrgensen,1983; Mahajan et al.,1990; Mahajan et al., 2000; Bass et al., 2000). As pointed out by Ruiz-Conde et al. (2006), marketing variables can be incorporated in the Bass diffusion model via a separable or a non-separable function. Here, a separable function means that the marketing variables have direct effects on sales, which is separated from the part that describes the diffusion process. In a non-separable specification, the marketing variables are assumed to have impacts on the diffusion process, so that both parts cannot be separated in the model. Table 2.1 summarises the ways of including marketing variables in the Bass diffusion model. The rest of this subsection reviews some early studies that explicitly incorporate price into the Bass diffusion model so as to derive the optimal dynamic pricing strategies for new products.

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Table 2.1 Bass diffusion model with marketing variables Bass diffusion model: 𝑑𝑁(𝑡) 𝑁(𝑡) ] [𝑀 − 𝑁(𝑡)] = [𝑝 + 𝑞 𝑑𝑡 𝑀 Four extensions: 1) External influence which has a non-separable form: 𝑝 = 𝑝(𝑡) = 𝑓(marketing variables (𝑡)) 2) Internal influence which has a non-separable form: 𝑞 = 𝑞(𝑡) = 𝑓(marketing variables (𝑡)) 3a) Both external and internal influence in a non-separable form: 𝑝 = 𝑝(𝑡) = 𝑓(marketing variables (𝑡)) 𝑞 = 𝑞(𝑡) = 𝑓(marketing variables (𝑡)) 3b) Both external and internal influence in a separable form: Multiply [𝑝 + 𝑞

𝑁(𝑡) 𝑀

] [𝑀 − 𝑁(𝑡)] with 𝑓(marketing variables (𝑡))

4) Potential market (non-separable function): 𝑀 = 𝑀(𝑡) = 𝑓(marketing variables (𝑡))

The seminal work of Robinson and Lakhani (1975) extended the Bass diffusion model using the option 3b) in Table 2.1, and proposed the following demand function 𝑓(𝑁(𝑡), 𝑃(𝑡)) = [𝑝 + 𝑞

𝑁 (𝑡 ) ] [𝑀 − 𝑁(𝑡)]𝑒 −𝛿1 𝑃(𝑡) , 𝑀

where 𝑃(𝑡) is the price at time t. They used this model to study the diffusion process of a semiconductor device, and determined the optimal pricing strategy for this product. Kalish (1983) used the same demand function to derive the optimal pricing strategies for new products, and the simulation results showed that penetration pricing is suitable for both durables and non-durables. The work of Dockner and Jorgensen (1988) extended the model of Kalish (1983) to oligopolistic 15

markets. However, as the problem becomes more challenging when considering oligopolistic competition, they only studied a linear price specification, i.e., 𝑒 −𝛿1 𝑃(𝑡) is replaced by a linear price function. They found that the optimal oligopolistic pricing strategy should increase (decrease) whenever there are positive (negative) word-of-month effects. In addition, Bass (1980) extended the Bass diffusion model via multiplying the original model by 𝑃(𝑡)−𝛿1 , that is 𝑓(𝑁(𝑡), 𝑃 (𝑡)) = [𝑝 + 𝑞

𝑁 (𝑡 ) ] [𝑀 − 𝑁(𝑡)]𝑃(𝑡)−𝛿1 , 𝑀

where 𝑃(𝑡) is the price at time t, and 𝛿1 is the price parameter. The study calibrated the extended diffusion model using data on consumer durables, and then investigated the optimal pricing strategies of new products.

2.2.3 Estimating the Parameters in Diffusion Models In practice, the new product diffusion models are used to forecast the sale / demand of a new product. However, the parameters in these diffusion models are not known in advance and they must be estimated according to empirical or experimental means. There are three parameters, 𝑝, 𝑞 and 𝑀, that must be estimated in the Bass model. With scarce diffusion data, the market potential parameter (M) is usually estimated via external sources of information such as surveys of long-term purchase intentions or previous sales data. The parameters 𝑝 and 𝑞 are estimated via expert judgments or algebraic estimation procedures. The latter one has to guess some key parameters. For example, Lawrence and Lawton (1981) proposed a procedure which is based on the guesses of 𝑀, 𝑞/𝑝 and 𝑝 + 𝑞 . The algebraic estimation procedure suggested by Mahajan and Sharma (1986) requires expert judgments to guess three pieces of information: market size, the peak time of the adoption rate and the adoption rate at the peak. With sufficient diffusion data, the parameters in Bass model can be estimated by more sophisticated mathematical techniques, which can be divided into two categories: the time 16

invariant and the time variant. The time invariant estimation techniques include ordinary least squares (OLS) proposed by Bass (1969), maximum likelihood estimation (MLE) proposed by Schmittlein and Mahajan (1982), and nonlinear least squares (NLLS) proposed by Srinivasan and Mason (1986). Mahajan et al. (1986) compared the performance of the above time invariant estimation techniques based on real data for seven products. Their study results showed that the NLLS technique provides better predictions. Therefore, the NLLS technique has gradually become the standard in diffusion model parameter estimation problems and it has been extended to estimate parameters in general diffusion models (see e.g., Mahajan et al., 2000; Putsis and Srinfvasan, 2000). The time variant estimation techniques include Bayesian estimation, feedback filters, etc. All these estimation techniques are summarised by Putsis and Srinfvasan (2000). For the general diffusion model, many methods are proposed in the literature to estimate the unknown parameters (see, e.g., Putsis Jr and Srinfvasan, 2000; Scitovski and Meler, 2002). One prevalent technique is the least squares (LS) method whose numerical method is based on solving the nonlinear LS (NLLS) problem depicted by Gill et al. (1981) as well as Dennis and Schnabel (1996). To apply the LS method, the existence of LS estimations needs to be considered, and this question is hard to answer when facing NLLS problems. Although Srinivasan and Mason (1986) propose the NLLS estimation method for the Bass model, the problem of existence of NLLS for Bass model is considered by Jukić (2011, 2013). For other special classes of functions, one can refer to Demidenko (1996, 2006), Jukić and Marković (2010), Jukić et al. (2008) and Jukić (2009).

2.3 The Diffusion of New Products with Warranty Policies As discussed in Section 2.1, warranty polices have become a powerful marketing tool for manufacturers to promote their new products and generate extra profit in recent years. Offering BW may promote the sales volume of the new product, while selling EWC can generate extra profit. However, these two types of warranty policies also result in maintenance costs. Therefore, there are trade-offs when offering warranty policies. Suitable warranty policies can raise the manufacturer’s total profit.

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2.3.1 The Diffusion of New Products with BW During the last two decades, the topic of deriving optimal price for new products with BW has become prevalent. These studies mainly focus on optimizing two major marketing strategies, price and BW, which influence the sales volume and ultimately the profit level. To maximize the profit of a product sold under a BW, Glickman et al. (1976) proposed one of the earliest models which use a displaced log-linear function to represent the sales rate as a function of price and BW length. Mesak (1996) considered a monopolist selling new products. Based on the assumption that the demand function depends on the length of BW, the product price, and the cumulative sales volume, the optimal trajectories for both the price and the BW length are derived over a given planning horizon. Teng and Thompson (1996) developed a general framework to determine the optimal price and quality strategies of new products for a monopolistic manufacturer over a fixed planning horizon. In the proposed framework, they considered the learning effects on the supply (manufacturer) side and the diffusion and saturation effects on the demand (customers) side. Lin and Shue (2005), Wu et al. (2006) and Huang et al. (2007) extended the model proposed by Teng and Thompson (1996) through the consideration of the BW length instead of the quality. These studies assumed that the demand function is determined by product price and BW length. The optimal price and BW length under the free replacement policy were derived in a pre-determined planning horizon. Particularly, the models developed by Huang et al. (2007), Lin et al. (2009) and Wu et al. (2009) include the BW length, cost learning effect, diffusion and saturation effects. There are two types of market environments that can be gleaned from the literature reviewed above. They are the dynamic market and the static market - depending on whether the demand function of the new product is affected by the cumulative sales volume (i.e., diffusion and saturation effects) or not (Faridimehr et al., 2013; Lin et al., 2009; Wu et al., 2009).

Static market The static market, which is characterized by a “static” demand function, considers the diffusion process of new products depending on price and BW length, but independent from the

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cumulative sales. The static demand function characterizes the market where word-of-mouth is not important and where the demand function exhibits no learning or saturation (Wu et al., 2009). It means that diffusion and saturation effects are not important in the development of the product and that most of the whole life cycle of the product lies in the maturity phase. Some typical specific static demand functions widely accepted in the literature include: (a) Additive price-warranty demand function The additive price-warranty demand function attracts a lot of attention due to the simplicity of its mathematical formulation and ease of understanding and applications (Desai et al., 2004; Kim et al., 2008; Li et al., 2012). It has the following form: ℎ(𝑝, 𝜏) = 𝑎0 − 𝑎1 𝑝 + 𝑎2 𝜏, where 𝑎0 (> 0) is the potential demand when the price 𝑝1 and the BW length 𝜏 are zeros; 𝑎1 (≥ 0) measures the price sensitivity in the demand function; 𝑎2 (≥ 0) measures the BW sensitivity in the demand function. (b) Glickman-Berger demand function The Glickman-Berger demand function has been widely accepted by researchers (Glickman et al., 1976; Lin et al., 2009; Manna, 2008; Wu et al., 2009). It has the following mathematical form: ℎ(𝑝, 𝜏) = 𝑘1 [𝑘2 + 𝜏]𝑏 𝑝−𝑎 , where 𝑝 ∈ (0, +∞), 𝑘1 > 0, 𝑘2 ≥ 0, 𝑎 > 1 and 0 < 𝑏 < 1 . The constant 𝑘1 is an amplitude factor and 𝑘2 is a constant of time displacement which ensures nonnegative demand even when 𝜏 is zero. Parameters 𝑎 and 𝑏 can be interpreted as the price elasticity and basic warranty period elasticity, respectively. This function possesses the sensible properties of decreasing exponentially with respect of price and increasing exponentially with protection period length.

Dynamic market

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The dynamic market is characterized by a “dynamic” demand function, which refers to a demand function depending on the price, BW length and the cumulative sales. The dynamic market is important to model the diffusion process of durable products, in which diffusion effects and / or saturation effects are significant (Wu et al., 2006). Indeed, word-of-mouth is an important factor in the diffusion of products in the dynamic market. Therefore, current demand affects the quantity of future demand. Various dynamic demand functions have been developed. This section only focuses on the limited-growth demand function and the Bass-type demand functions in the dynamic market. (a) Limited-growth demand function This type of demand function describes situations wherein the saturation effect dominates the diffusion process of the new product over the whole planning horizon (Aoki et al., 2002; Lin et al., 2009). It can be expressed as: 𝑓 (𝑁(𝑡), 𝑝(𝑡), 𝜏) = 𝑘1 [𝑘2 + 𝜏]𝑏 𝑝−𝑎 (𝑡)[𝑀 − 𝑁(𝑡)], where 𝑀 − 𝑁(𝑡) reflects the saturation effect, whereas the rest of the expression captures the effect of price and warranty on the sales rate and coincides with the Glickman-Berger demand function depicted above. (b) Bass-type demand function This type of demand function is fairly standard in the literature (Huang et al., 2007; Ruiz-Conde et al., 2006). Ruiz-Conde et al. (2006) considered the following form: 𝑓 (𝑁(𝑡), 𝑝(𝑡), 𝜏) = 𝑘1 [𝑘2 + 𝜏]𝑏 𝑝−𝑎 (𝑡)[𝑀 − 𝑁(𝑡)] [𝑞1 +

where [𝑞1 +

𝑞2 𝑁(𝑡) 𝑀

𝑞2 𝑁(𝑡) ], 𝑀

] and [𝑀 − 𝑁(𝑡)] model the diffusion and satuation effects, respectively; the

rest terms capture the effect of price and warranty on the sales rate and coincide with the Glickman-Berger demand function.

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Huang et al. (2007) assumed that the useful life of the product is much longer than the product life cycle so that there are no repeat purchase sales. They confined attention to the first purchase sales, the sales rate, 𝑓 (𝑁(𝑡), 𝑝(𝑡), 𝜏), is modelled as follows: 𝑓 (𝑁(𝑡), 𝑝(𝑡), 𝜏) = 𝐾𝜏 𝛼 𝑒 −𝛽𝑝(𝑡) [1 −

𝑁(𝑡) 𝑁 (𝑡 ) ]. ] [𝜓 + 𝑀 𝑀

The salient features of the model are as follows: (1) The terms [1 −

𝑁(𝑡) 𝑀

] [𝜓 +

𝑁(𝑡) 𝑀

] represent the sales as a diffusion process involving

innovators and imitators as in the Bass model, where the parameter  reflects the relative influence of innovators. (2) The term 𝐾𝜏 𝛼 𝑒 −𝛽𝑝(𝑡) captures the effect of price and warranty on the sales rate. The parameter 𝛼 > 0 is the warranty elasticity parameter and the parameter 𝛽 > 0 reflects the influence of the price 𝑝(𝑡). Note that the warranty length has a positive effect (with the sale rate increasing in the warranty length) and the selling price has a negative effect (with the sales rate decreasing in the price). (3) The parameter K is a scale factor, which reflects the influence of the competitors and other environmental influences, such as the number of potential consumers, the consumer purchasing power, etc.

2.3.3 Research on Consumer Demand for the EWC Although the issue of optimal pricing strategies for new products with BW policy has received much attention in the literature as summarised in section 2.3.1, few studies have examined the effects of selling the EWC especially from a long-term perspective. The EWC can essentially be viewed as an insurance product. However, predicting consumers' perceptions about the EWC is more complex. According to the insurance literature, the major determinants of purchasing insurance products include the extent of loss, the probability of loss, the insurance premium charged and the consumers’ risk aversion (Padmanabhan and Rao, 1993; Schlesinger, 2000).

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Empirical studies indicate that these determinants objectively influence the purchase of life, disaster and health insurance with few exceptions. From the consumers’ purchasing perspective, these determinants, however, may be subjective when the insurance product is the EWC. For instance, the failure rate of the same model of a computer should be the same for all consumers. However, the real situation is that different consumers may have different perceptions of the failure rate due to their different ideas and estimations. Therefore, research on consumer demand for the EWC needs to consider consumer behaviour under uncertainty and needs to highlight the variables that are the likely determinants of consumer demand for warranty. Intuitively, decreasing the EWC price and providing longer coverage of the EWC should encourage consumer demand, because it is cheaper to hedge risk. Previous studies pertaining to the consumer demand for the EWC are formulated involving consumer heterogeneity in risk aversion and usage rates. The work of Day and Fox (1985) is the first qualitative study aimed at understanding consumer perception and decision making about the EWC. They suggested that the demand for the EWC is likely to be influenced by factors such as risk preference, product experience and usage habits. Padmanabhan and Rao (1993) used the agency theory framework to analyse the optimal warranty policy for the manufacturer in a market based on consumer variation in risk preference and consumer moral hazard. They tested their theory with empirical data collected from a mail questionnaire of new car buyer in the Dallas-Fort Worth Metroplex area. Hartman and Laksana (2009) proposed a dynamic programming method to compute the optimal pricing policy for the provider given the consumers’ risk characterization, and the population of consumers has different risk tolerances as opposed to targeting a specific risk category. Recently, Chen et al. (2009) considered the EWC as an insurance product, and investigated the impact of consumer characteristics (income, gender, and prior usage) on insurance purchase determinants: probability of loss, extent of loss, risk aversion, and amount of insurance premium. They also used the panel data of electronic purchases across several product categories to test their predictions. In their demand forecasting model, the consumer decision of purchasing the EWC is modelled using a discrete choice framework. The first step is to derive consumer’s perceived product failure rate, and then calculate the expected benefit of purchasing the EWC, the last step is to make purchase decision on the basis of their expected utilities. This discrete choice framework is one of the most popular

22

methodologies for the EWC demand forecasting. For similar work, one can refer to Chen and Sun (2007) as well as Moore (2002,2011).

2.4 Channel Design 2.4.1 Pricing in Marketing Channels Two elements of the marketing mix that managers use to control their marketing strategies are price and channel, yet they differ fundamentally in the ways managers use them to influence market demand. Price is the most flexible in the marketing mix, and is usually used to influence short-run demand. Channel, or distribution channel, however, is the least flexible and perhaps the costliest to change in a short term. As such, channel design is often seen as part of a company’s long-term policy. Most importantly, in the presence of a typically decentralized distribution channel, an upstream price change by a manufacturer does not affect consumer demand directly, but only through how this upstream price change affects the retail price set downstream in the channel. Rao (1984) reviewed the pricing literature and stated that “the issues of pricing along the distribution channel have not received much attention in the literature.” Over the last several decades, this research gap has been remedied substantially. Game theory has revolutionized the theoretical analysis of pricing within the distribution channel and clarified many problems about how prices are determined. In addition, these studies have yielded deep insights into the optimal long-term channel design. Based on the short-term nature of price and the long-term nature of the distribution channel, McGuire and Staelin (1983) laid the foundation for game theoretic analysis of distribution channels in marketing. The heart of the channel pricing literature is the concept of double marginalization (Spengler, 1950). Double marginalization refers to different decision makers having their respective market powers but applying their own markup in prices at different vertical levels. In the subsequent section, the applications of the dynamic Stackelberg game in marketing channels are firstly summarised, and then channel design problems for new products with an optional EWC are discussed. 23

2.4.2 Dynamic Stackelberg (Leader-Follower) Game in Marketing Channels The field of marketing distribution channel has received much attention over the last two decades. Most early studies in this field are based on a static market, which only examine the one-shot interactions between the channel members (such as the manufacturer and the retailer). In practice, the channel members, however, often interact with each other over a rather long time horizon. Therefore, the decisions of the channel members evolve over time. For this reason, it is necessary to adopt the framework of the differential game. The study of differential games originated from Isaacs (1965) who studied the pursuit-evasion problems. Since then, differential game models have been applied extensively to study problems of conflict arising in supply chain management, marketing, economics, etc. More recently, dynamic Stackelberg games have been formulated to study hierarchical or sequential decision making processes. There are two types of information structure in a dynamic Stackelberg game: open loop and closed-loop. The open loop structure assumes a pre-announcement of the strategy by the leader at the initial time of the game. The closed-loop structure is also called the feedback information structure. In a closed-loop equilibrium, the leader and the follower need to make their decisions at any time based on the current state at that particular time. For the open-loop differential game in marketing, Jørgensen et al. (2001) studied joint pricing and advertising policies in dynamic channel. Later on, He et al. (2008) considered a supply chain in which a manufacturer sells an innovative durable product to an independent retailer over a planning horizon. They assumed that the product demand follows a Bass-type diffusion process and is determined by market influences, retail price of the product, and shelf space allocated to it. Two profit optimization strategies were considered for the retailer: one is the myopic strategy which maximizes the current-period profit, and the other is the far-sighted strategy which maximizes the life-cycle profit. They characterized the optimal dynamic shelf-space allocation and retail pricing policies for the retailer as well as the optimal wholesale pricing policy for the manufacturer. More recently, Gutierrez et al. (2011) constructed a dynamic Stackelberg game to

24

derive the open loop equilibrium pricing strategy under vertical competition. The demand function in this model is affected by the diffusion and saturation effects. As shown in He et al. (2007), majority of the papers apply the differential Stackelberg game to study the open loop equilibria due to their mathematical tractability. Studies related to the closed-loop solution are very limited, because it is extremely difficult to drive the feedback Stackelberg equilibria.

2.4.3 Channel Design for New Product with EWC This part reviews previous studies that consider the EWC in the channel design problem. Desai and Padmanabhan (2004) derived explicit demand functions for a durable product and its EWC based on the consumer’s utility, which explicitly captures the complementary property of the EWC. They investigated the impact of different distributional arrangements for market outcomes and manufacturer profitability. The key driving forces of the results are shown to be the double marginalization effect. Different channel arrangements for marketing of the EWC cause these effects to occur at different levels within a distribution channel, which have significant implications for the optimal warranty policy. Li et al. (2012) studied the design of extended warranties in a supply chain consisting of a manufacturer and an independent retailer. Their results indicated that the retailer, with a higher rate of profit improvement, benefits more from being the provider of EWC, unless he or she has a substantial repair cost disadvantage compared to the manufacturer. They also provided guidelines to retailers and manufacturers in terms of how to make appropriate choices with regard to being a provider or a re-seller of the EWC. Furthermore, their research implies that the EWC is not only a source of revenue, but also a strategic tool in channel choices to improve system profits and to make product pricing decisions. These papers contribute to the channel design problems with due consideration of the EWC. However, one fundamental assumption of these studies is that the demand is a simple linear demand function and no dynamic effects are considered in the demand function. Extending the

25

distribution channel design problem of EWC to a dynamic market setting would be difficult but meaningful.

2.5 Chapter Summary This chapter provides a comprehensive review of the previous studies developed over the past few decades related to the pricing problems for the diffusion of new product with warranty policies. The purpose of this review is twofold. The first is to understand the state-of-the-art research in related areas. The second is to identify the research gaps which can overcome the shortcomings in these previous studies. It is worth noting that previous studies have focused on the dynamic pricing problems for new products with BW. There is still a lack of research on dynamic pricing problem for new products with EWC in a general framework. Furthermore, although the cost learning effect has been widely considered in dynamic pricing, little attention has been paid to the effect of production rate. As actual price data indicates that production rate has a much greater effect on manufacturing cost than learning (Markell, 1993), the problem of simultaneously optimizing dynamic price and production rate should be carefully studied. Lastly, as pointed out in Section 2.4.3, the works of Li et al. (2012) and Desai et al. (2004) can be extended to a dynamic market. This study, therefore, contribute to the literature by addressing these research gaps.

26

Chapter 3 Dynamic Pricing of a Product and Its EWC 3.1 Introduction Designing appropriate pricing strategies for new products play a crucial role in sustaining a manufacturer’s profitability and have become a challenging issue in today’s marketplace, because it involves complex dynamic effects (such as diffusion and saturation effects) with regard to the diffusion of the new product (Lin et al., 2005). By modelling these dynamic effects, previous research seeks the optimal dynamic pricing strategy over time to maximize the total profit over the planning horizon (Bhargava and Natsuyama, 2005). The determination of dynamic pricing for new product planning has become a central theme in marketing and management, and more studies related to this research theme are summarised in Section 2.2. In addition to the traditional product pricing strategies, warranty policies have also become visible and important marketing strategies. Recall from Chapter 2 that the basic warranty (BW) and the extended warranty contract (EWC) are the two most widely accepted warranty policies for new products such as computers, electric appliances, and automobiles. In general, a BW with a longer period can increase a manufacturer’s competitive advantage. However, it also results in higher costs for the manufacturer (Murthy et al., 2002). In addition to the free BW which comes bundled with the product, consumers usually have the option to buy an EWC which extends the period of the BW. The EWC is being purchased by a large number of consumers (Consumers' CHECKBOOK, 2007). In modern business, selling the EWC is regarded as a key revenue generator for the manufacturer. There has been increasing research attention on the joint determination of pricing strategy and BW policy. Teng and Thompson (1996) developed a general mathematical model to derive the optimal price and quality policy of a new product for a monopolistic manufacturer given a planning horizon. In the proposed model, they considered the unit production cost learning curve on the supply (manufacturer) side and the diffusion and saturation effects on the demand (customers) side. This model was extended by Lin and Shue (2005), Wu et al. (2006), Huang et al. (2007) and Zhou et al. (2009). These new models considered the BW instead of product quality, assuming that the demand for the new product is determined by the price of the product

27

and the length of the BW. The optimal trajectories of price and warranty length of free replacement policy were derived in a pre-determined life cycle of the product. The model proposed in this chapter is based on the aforementioned studies. However, the monopolist pricing problem addressed in this model extends previous models in four aspects: 1) The demand function is more general. It is related not only to the selling price of new product and the BW length but also to the cumulative sales volume, i.e., it also considers the diffusion and saturation effects. 2) The proposed model considers the EWC as a separate product and examines its impact on the dynamic pricing strategy for the new product. 3) When considering the warranty cost, previous studies mainly considered the free replacement policy. The new model estimates the expected warranty cost based on the minimum repair policy, which is more common in reality. 4) Previous studies mainly focused on analysing the properties of the optimal solution. This chapter also develops a gradient algorithm to compute the numerical results when the demand function is too complex to find the closed-form solution. The rest of this chapter is organized as follows: Section 3.2 describes the formulation of the mathematical model. Section 3.3 details the results of the theoretical analysis. Section 3.4 develops a model for estimating the expected warranty cost incurred in the BW and the EWC. Section 3.5 presents some case studies when specifying the demand functions of the new product. The last section 3.6 introduces the gradient algorithm and implements it to analyse the Bass-type demand function, for which the closed-form solution cannot be derived.

3.2 Mathematical Formulation To develop the diffusion model of a new product with warranty policies (including both the BW and the EWC), the following notations are used:

28

Table 3.1 The notations used in chapter 3 Decision variables 𝑝1 (𝑡)

The selling price of the new product at time t.

𝑝2 (𝑡)

The selling price of the EWC at time t.

Parameters 𝑇

The life cycle of the new product which starts with the launch of a product onto the market and ends when the manufacturer stops selling

𝜏

The length of the BW

𝜂

The length of the EWC

𝐶𝑚

The unit production cost of the product

𝐶𝐵𝑊

The expected repair cost of the BW

𝐶𝐸𝑊𝐶

The expected repair cost of the EWC

𝑃𝐸𝐷𝑝1

The price elasticity of demand for the new product

𝑃𝐸𝐷𝑝2

The price elasticity of demand for the EWC

𝐶𝑟

The expected repair cost for each product failure

𝜆 (𝑡 )

The failure rate (or intensity function) of the product

Consider a manufacturer who manufactures and sells a new repairable product in the planning horizon [0, 𝑇]. The production lead-time is sufficiently short (i.e., inventory is neglected). During the planning horizon, the manufacturer sells the product to consumers at a unit price 𝑝1 (𝑡) at time t. After each deal, two types of warranty services, the BW and the EWC, are offered to the consumers: The BW of length 𝜏 is bundled with the product. When a consumer purchases one unit of the product at time t, the manufacturer guarantees to repair the product free of charge whenever it fails to perform normal functions in the period [𝑡, 𝑡 + 𝜏]. On the other hand, the EWC is optional. If the consumer accepts the EWC, he or she has to pay an extra charge of 𝑝2 (𝑡). In return, the manufacturer guarantees to repair the failed product at no charge for a prolonged period 𝜂.

29

Before formulating the diffusion model, the cost and demand functions must first be specified.

3.2.1 Costs The costs borne by the manufacturer include the manufacturing cost and the expected repair cost when the product breaks down during the period covered by the BW and / or the EWC. The manufacturing cost of the unit product is assumed to be a constant 𝐶𝑚 . The repair costs incurred by the BW and the EWC are designated as random variables, because both the number of failures and the repair cost for each failure are random variables. However, this model considers the expected repair costs rather than the random variables. Let 𝐶𝐵𝑊 and 𝐶𝐸𝑊𝐶 denote the expected repair costs incurred by the BW and the EWC respectively. Obviously, 𝐶𝐵𝑊 and 𝐶𝐸𝑊𝐶 are independent from the price 𝑝1 (𝑡) and 𝑝2 (𝑡). In addition, a longer warranty length (𝜏 and 𝜂) leads to a higher expected repair cost, so 𝐶𝐵𝑊 and 𝐶𝐸𝑊𝐶 must be functions of 𝜏 and 𝜂, respectively, and satisfy: 𝑑𝐶𝐵𝑊 (𝜏)⁄𝑑𝜏 > 0, 𝐶𝐵𝑊 (0) = 0;

(3.1)

𝑑𝐶𝐸𝑊𝐶 (𝜂)⁄𝑑𝜂 > 0, 𝐶𝐸𝑊𝐶 (0) = 0.

(3.2)

The precise estimation method about the expected warranty cost function 𝐶𝐵𝑊 (𝜏) and 𝐶𝐸𝑊𝐶 (𝜂) can be found in Section 3.4. For notation simplification, the arguments 𝜏 and 𝜂 are often omitted in the two cost functions.

3.2.2 Demand Functions Demand Function for the New Product Let 𝑋(𝑡) be the cumulative sales volume of the product in time interval [0, 𝑡]. The demand rate, 𝑋̇ (𝑡)(= 𝑑𝑋(𝑡)/𝑑𝑡), at time t is assumed to be a function of 𝑋(𝑡), 𝑝1 (𝑡) and 𝜏: 𝑋̇ (𝑡) = 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏),

30

(3.3)

where the function 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏) is positive and differentiable with respect to 𝑋(𝑡), 𝑝1 (𝑡) and 𝜏. Any feasible price 𝑝1 (𝑡) should be a piecewise continuous function and takes positive values in a fixed open interval 𝑈1 in 𝑅+ , i.e., 𝑈1 = (0, 𝑙 ) for some 𝑙 > 0 or 𝑈1 = (0, +∞) . Moreover, this demand function satisfies 𝑓𝑝1 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏) < 0, which means that the demand rate decreases when the selling price increases; 𝑓𝜏 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏) > 0, which means that the demand rate increases as the BW length increases. (Note: Please refer to Appendix A for the mathematical explanation of expressions 𝑓𝑝1 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏) and 𝑓𝜏 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏) as well as other similar mathematical notations used in this study.)

Demand function for the EWC Note that only consumers who have bought the new product are potential consumers for the EWC, i.e., the maximum demand for the EWC is limited by the new product demand 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏) ( Mahajan, 1991; Li, et al. 2012). More specifically, demand rate of the EWC = (demand rate of new product) × (probability that consumers will purchase the EWC) This formula captures the fact that the EWC is an optional choice after the purchase of the new product (Lam, 2001). Consumers, who are faced with the option of purchasing an EWC, are supposed to make the decision based on two important factors: one is the price of the EWC 𝑝2 (𝑡), and the other is the length of the EWC 𝜂. A higher EWC price 𝑝2 (𝑡) leads to a lower demand, and a longer EWC length attracts more consumers to buy the EWC. Similarly, the price 𝑝2 (𝑡) considered in this chapter is a piecewise continuous function and takes positive values in a fixed open interval 𝑈2 in 𝑅+ . Define 31

𝜌(𝑝2 (𝑡), 𝜂):

The probability that consumers are willing to pay for EWC.

𝑌(𝑡):

The cumulative sales volume of EWC in the time interval [0, 𝑡].

The partial derivatives of 𝜌(𝑝2 (𝑡), 𝜂) must satisfy 𝜌𝑝2 (𝑝2 (𝑡), 𝜂) < 0 and 𝜌𝜂 (𝑝2 (𝑡), 𝜂) > 0. The demand rate of EWC, 𝑌̇ (𝑡)(= 𝑑𝑌(𝑡)/𝑑𝑡), should naturally be expressed as: 𝑌̇ (𝑡) = 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)𝜌(𝑝2 (𝑡), 𝜂).

(3.4)

3.2.3 Mathematical formulation for the dynamic optimization problem

Unit expected BW cost

Length of BW

Revenue from unit

𝜏

product

Selling price of product 𝑝1 (𝑡)

Demand rate

Instantaneous revenue from selling product

of product Profit

EWC price

Demand rate

𝑝2 (𝑡)

of EWC Instantaneous revenue from selling EWC

Length of

Revenue from unit

EWC 𝜂

EWC

Unit expected EWC cost

Figure 3.1 The instantaneous profit flow at time t

32

Given the value of 𝜏, 𝜂, 𝑝1 (𝑡) and 𝑝2 (𝑡), Figure 3.1 schematically describes the instantaneous expected profit flow of the manufacturer at time t. As shown in Figure 3.1, the total profit for the manufacturer is generated by selling the new product and selling the EWC. The direction of each arrow indicates that the variable on the left would influence the one on the right. For example, the selling price of the new product, 𝑝1 (𝑡), influences the unit revenue of the new product as well as the demand rate for both the new product and the EWC. Therefore, the instantaneous expected profit rate function for selling the new product is 𝑓(𝑋 (𝑡), 𝑝1 (𝑡), 𝜏)[𝑝1 (𝑡) − 𝐶𝑚 − 𝐶𝐵𝑊 (𝜏)]. The instantaneous expected profit rate function for selling the EWC is 𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏)𝜌(𝑝2 , 𝜂)[𝑝2 (𝑡) − 𝐶𝐸𝑊𝐶 (𝜂)]. In this model, 𝜏, and 𝜂 are constants, because the length of BW and the length of EWC are predetermined by the manufacturer. To maximize the total expected profit in the planning horizon, the manufacturer obviously needs to optimize the dynamic pricing strategies: 𝑝1 (𝑡) and 𝑝2 (𝑡). Therefore, the diffusion model for this dynamic optimization problem can be formulated as an optimal control problem (Seierstada and Sydsaeter, 1987; Liberzon, 2012; Sethi and Thompson, 2000; Troutman, 1996; Kamien and Schwartz, 2012): 𝑀𝑎𝑥 𝐽[𝑝1 (𝑡), 𝑝2 (𝑡)]

𝑝1 (𝑡),𝑝2 (𝑡)

𝑇

=

𝑀𝑎𝑥 ∫ 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)[𝑝1 (𝑡) − 𝐶𝑚 − 𝐶𝐵𝑊 ] 𝑑𝑡

𝑝1 (𝑡),𝑝2 (𝑡) 0

𝑇

+ ∫ 𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏)𝜌(𝑝2 (𝑡), 𝜂)[𝑝2 (𝑡) − 𝐶𝐸𝑊𝐶 ] 𝑑𝑡 0

𝑇

=

𝑀𝑎𝑥 ∫ 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)[𝑝1 (𝑡) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝜌(𝑝2 (𝑡), 𝜂)[𝑝2 (𝑡) − 𝐶𝐸𝑊𝐶 ]] 𝑑𝑡

𝑝1 (𝑡),𝑝2 (𝑡) 0

(3.5) Subject to 𝑋̇(𝑡) = 𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏)

33

(3.6)

𝑌̇(𝑡) = 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)𝜌(𝑝2 (𝑡), 𝜂)

(3.7)

𝑋(0) = 0, 𝑌(0) = 0

(3.8)

𝑋(𝑇) and 𝑌(𝑇) are free

(3.9)

(𝑝1 (𝑡), 𝑝2 (𝑡)) ∈ 𝑈1 × 𝑈2 ⊆ 𝑅+2

(3.10)

Here, 𝑝1 (𝑡) and 𝑝2 (𝑡) are the control variables representing the pricing strategies of the new product and the EWC; (3.5) is the objective function, i.e., the total profit of the manufacturer; 𝑋(𝑡) and 𝑌(𝑡) are the state variables; (3.6)-(3.9) are the state equations and corresponding boundary conditions; (3.10) is the control region.

3.3 Theoretical Analysis 3.3.1 Necessary and Sufficient Conditions for the Optimal Solution Before proceeding to derive the necessary conditions of the dynamic optimization problem (3.5)(3.10), define 𝐸 (𝑝2 (𝑡), 𝜂) = 𝜌(𝑝2 (𝑡), 𝜂)[𝑝2 (𝑡) − 𝐶𝐸𝑊𝐶 (𝜂)] for all 𝑡 ∈ [0, 𝑇].

(3.11)

Note that 𝜌(𝑝2 (𝑡), 𝜂) is the probability that consumers are willing to pay for the EWC given the EWC price 𝑝2 (𝑡) and the length 𝜂 , while 𝑝2 (𝑡) − 𝐶𝐸𝑊𝐶 (𝜂) is the profit accrued from successfully selling EWC. Therefore, 𝐸 (𝑝2 (𝑡), 𝜂) can be viewed as the expected profit from the EWC after selling one unit new product. Let (𝑝1 (𝑡), 𝑝2 (𝑡)) be a pair of pricing strategies satisfying condition (3.10). State equations (3.6) -(3.9) yield the unique state variables (𝑋(𝑡|𝑝1 (𝑡)), 𝑌(𝑡|𝑝1 (𝑡), 𝑝2 (𝑡))), which, for simplicity, are also denoted by (𝑋 (𝑡), 𝑌(𝑡)) . Thus, (𝑋(𝑡), 𝑌 (𝑡), 𝑝1 (𝑡), 𝑝2 (𝑡)) is a feasible solution for the problem (3.5)-(3.10). With these preparations, the following lemma asserts that the total profit for the manufacturer increases with 𝐸 (𝑝2 (𝑡), 𝜂).

34

Lemma 3.1 Let (𝑋 (𝑡), 𝑌 𝑖 (𝑡), 𝑝1 (𝑡), 𝑝2𝑖 (𝑡)) (where 𝑖 = 1,2) be two pairs of feasible solutions for the dynamic optimization problem (3.5)-(3.10). If 𝐸 (𝑝21 (𝑡), 𝜂) ≤ 𝐸 (𝑝22 (𝑡), 𝜂) (∀𝑡 ∈ [0, 𝑇]), then 𝐽[𝑝1 (𝑡), 𝑝21 (𝑡)] ≤ 𝐽[𝑝1 (𝑡), 𝑝22 (𝑡)]. Proof: Consider 𝐽[𝑝1 (𝑡), 𝑝21 (𝑡)] − 𝐽[𝑝1 (𝑡), 𝑝22 (𝑡)] 𝑇

= ∫ 𝑓(𝑋 (𝑡), 𝑝1 (𝑡), 𝜏)[𝐸(𝑝21 (𝑡), 𝜂) − 𝐸 (𝑝22 (𝑡), 𝜂)]𝑑𝑡. 0

For any feasible pricing strategy 𝑝1 (𝑡), the demand function 𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏) > 0 ∀𝑡 ∈ [0, 𝑇]. As (𝑝21 (𝑡), 𝜂) ≤ 𝐸 (𝑝22 (𝑡), 𝜂) ∀𝑡 ∈ [0, 𝑇], the integrand 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)[𝐸(𝑝21 (𝑡), 𝜂) − 𝐸 (𝑝22 (𝑡), 𝜂)] ≤ 0 yields 𝐽[𝑝1 (𝑡), 𝑝21 (𝑡)] ≤ 𝐽[𝑝1 (𝑡), 𝑝22 (𝑡)]. ∎ The definition of 𝐸 (𝑝2 (𝑡), 𝜂) in (3.11) implies that 𝐸 (𝑝2 (𝑡), 𝜂) does not depend on t explicitly. Thus, 𝐴𝑟𝑔𝑚𝑎𝑥𝑝2(𝑡)∈𝑈2 𝐸(𝑝2 (𝑡), 𝜂 ) must be a constant. Let 𝑝̅2 = 𝐴𝑟𝑔𝑚𝑎𝑥𝑝2∈𝑈2 𝐸 (𝑝2 , 𝜂 ) for all 𝑡 ∈ [0, 𝑇].

(3.12)

As 𝑝̅2 maximizes 𝐸 (𝑝2 (𝑡), 𝜂) for all 𝑡 ∈ [0, 𝑇], Lemma 3.1 implies that the corresponding total profit 𝐽 is also higher than that of any other EWC price 𝑝2 (𝑡), i.e., the manufacturer is better to set the selling price of EWC to 𝑝2 (𝑡) ≡ 𝑝̅2 , and 𝐸(𝑝̅2 , 𝜂 ) is the maximum expected profit from EWC after selling a new product. The following Theorem 3.1 describes the necessary conditions which correspond to the optimal solution for the dynamic optimization problem (3.5)-(3.10). It further asserts that the optimal

35

selling price of the EWC must be 𝑝̅2 , which also affects the optimal selling price of the new product.

Theorem 3.1 Suppose (𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝑝1∗ (𝑡), 𝑝2∗ (𝑡)) is the optimal solution for the dynamic optimization problem (3.5)-(3.10). There must exist a piecewise continuously differentiable function 𝜃1 (𝑡), such that the optimal solution and 𝜃1 (𝑡) satisfy the following relations: 𝑝2∗ (𝑡) ≡ 𝑝̅2 = 𝐴𝑟𝑔𝑚𝑎𝑥 𝐸 (𝑝2 , 𝜂) , ∀𝑡 ∈ [0, 𝑇];

(3.13)

𝑝2 ∈𝑈2

𝑝1∗ (𝑡) = 𝐴𝑟𝑔𝑚𝑎𝑥 𝑓 (𝑋 ∗ (𝑡), 𝑝1 , 𝜏)[𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 (𝑡)], 𝑝1 ∈𝑈1

(3.14) ∀𝑡 ∈ [0, 𝑇];

𝑋̇ ∗ (𝑡) = 𝑓 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏), 𝑋 ∗ (0) = 0;

(3.15)

𝑌̇ ∗ (𝑡) = 𝑓(𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)𝜌(𝑝̅2 , 𝜂), 𝑌 ∗ (0) = 0;

(3.16)

𝜃̇1 (𝑡) = −𝑓𝑋 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)[𝑝1∗ (𝑡) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 (𝑡)], 𝜃1 (𝑇) = 0.

(3.17)

Furthermore, (3.17) is equivalent to the following integral form: 𝑇𝜔 ̅ (𝑠)

𝜃1 (𝑡) = ∫𝑡

𝑓 (𝑋 𝜔 ̅ (𝑡) 𝑋

∗

(𝑠), 𝑝1∗ (𝑠), 𝜏)[𝑝1∗ (𝑠) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂)]𝑑𝑠,

(3.18)

𝑇

where 𝜔 ̅ (𝑡) = exp [− ∫𝑡 𝑓𝑋 (𝑋 ∗ (𝑠), 𝑝1∗ (𝑠), 𝜏)𝑑𝑠]. Proof: Formulate the Hamiltonian function for the dynamic optimization problem (3.5)-(3.10) as follow: 𝐻 = 𝐻(𝑋 (𝑡), 𝑌(𝑡), 𝑝1 (𝑡), 𝑝2 (𝑡), 𝜃1 (𝑡), 𝜃2 (𝑡)) = 𝜃0 𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏)[𝑝1 (𝑡) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝2 (𝑡), 𝜂)] + 𝜃1 (𝑡)(𝑋(𝑡), 𝑝1 (𝑡), 𝜏) + 𝜃2 (𝑡)𝑓(𝑋 (𝑡), 𝑝1 (𝑡), 𝜏)𝜌(𝑝2 (𝑡), 𝜂),

(3.19)

36

where 𝜃0 the constant multiplier associated with the integrand of the objective function; 𝜃1 (𝑡) and 𝜃2 (𝑡) are the costate functions associate with the state equations (3.6) and (3.7). Suppose (𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝑝1∗ (𝑡), 𝑝2∗ (𝑡)) is the optimal solution to dynamic optimization problem (3.5)-(3.10). The Pontryagin maximum principle asserts: (1) The constant multiplier 𝜃0 (= 0 or 1) and the costate function 𝜃1 (𝑡) and 𝜃2 (𝑡) must satisfy (𝜃0 , 𝜃1 (𝑡), 𝜃2 (𝑡)) ≠ (0,0,0) for all 𝑡 ∈ [0, 𝑇].

(3.20)

(2) (𝑝1∗ (𝑡), 𝑝2∗ (𝑡)) maximizes 𝐻(𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝑝1 , 𝑝2 , 𝜃1 (𝑡), 𝜃2 (𝑡)) for all 𝑡 ∈ [0, 𝑇] and (𝑝1 , 𝑝2 ) ∈ 𝑈1 × 𝑈2 , i.e., 𝐻 ∗ ≜ 𝐻(𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝑝1∗ (𝑡), 𝑝2∗ (𝑡), 𝜃1 (𝑡), 𝜃2 (𝑡)) ≥ 𝐻(𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝑝1 , 𝑝2 , 𝜃1 (𝑡), 𝜃2 (𝑡)).

(3.21)

(3) For any time 𝑡 at which the functions (𝑝1∗ (𝑡), 𝑝2∗ (𝑡)) are continuous, 𝜃̇1 (𝑡) = −

𝜕𝐻 ∗ 𝜕𝑋

= −𝜃0 𝑓𝑋 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)[𝑝1∗ (𝑡) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑃2∗ (𝑡), 𝜂)] − 𝜃1 (𝑡)𝑓𝑋 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏) − 𝜃2 (𝑡)𝑓𝑋 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)𝜌(𝑝2∗ (𝑡), 𝜂); 𝜃̇2 (𝑡) = −

𝜕𝐻 ∗ = 0. 𝜕𝑌

(3.22)

(3.23)

(4) The transversality conditions corresponding to the boundary conditions (3.9) are (𝜃1 (𝑇), 𝜃2 (𝑇)) = (0,0). At the end time 𝑡 = 𝑇, both (3.20) and (3.24) must hold, so 𝜃0 = 1. By (3.24) and (3.23), it is obvious that 𝜃2 (𝑡) ≡ 0. Substituting 𝜃2 (𝑡) ≡ 0 into (3.21) yields

37

(3.24)

𝐻(𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝑝1∗ (𝑡), 𝑝2∗ (𝑡), 𝜃1 (𝑡), 𝜃2 (𝑡)) = 𝑓 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)[𝑝1∗ (𝑡) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝜃1 (𝑡) + 𝐸 (𝑃2∗ (𝑡), 𝜂)] ≥ 𝑓 (𝑋 ∗ (𝑡), 𝑝1 , 𝜏)[𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝜃1 (𝑡) + 𝐸 (𝑝2 , 𝜂)] = 𝐻 (𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝑝1 , 𝑝2 , 𝜃1 (𝑡), 𝜃2 (𝑡)),

(3.25)

for all 𝑡 ∈ [0, 𝑇] and (𝑝1 , 𝑝2 ) ∈ 𝑈1 × 𝑈2 . Define 𝑝̅1 (𝑡) = 𝐴𝑟𝑔max 𝑓 (𝑋 ∗ (𝑡), 𝑝1 , 𝜏)[𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝜃1 (𝑡) + 𝐸 (𝑝̅2 , 𝜂)] ; ∀𝑡 ∈ [0, 𝑇]. 𝑝1 ∈𝑈1

(3.26) We assert that 𝑝1∗ (𝑡) = 𝑝̅1 (𝑡) ∀𝑡 ∈ [0, 𝑇]. Suppose for the sake of contradiction that ∃𝑡̅ ∈ [0, 𝑇] such that 𝑝1∗ (𝑡̅) ≠ 𝑝̅1 (𝑡̅). Then 𝐻(𝑋 ∗ (𝑡̅), 𝑌 ∗ (𝑡̅), 𝑝1∗ (𝑡̅), 𝑝2∗ (𝑡̅), 𝜃1 (𝑡̅), 𝜃2 (𝑡̅)) = 𝑓 (𝑋 ∗ (𝑡̅), 𝑝1∗ (𝑡̅), 𝜏)[𝑝1 (𝑡̅) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝜃1 (𝑡̅) + 𝐸 (𝑝2∗ (𝑡̅), 𝜂)] ≤ 𝑓 (𝑋 ∗ (𝑡̅), 𝑝1∗ (𝑡̅), 𝜏)[𝑝1 (𝑡̅) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝜃1 (𝑡̅) + 𝐸 (𝑝̅2 , 𝜂)] < 𝑓 (𝑋 ∗ (𝑡̅), 𝑝̅1 (𝑡̅), 𝜏)[𝑝̅1 (𝑡̅) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝜃1 (𝑡̅) + 𝐸 (𝑝̅2 , 𝜂)] = 𝐻(𝑋 ∗ (𝑡̅), 𝑌 ∗ (𝑡̅), 𝑝̅1 (𝑡̅), 𝑝̅2 , 𝜃1 (𝑡̅), 𝜃2 (𝑡̅)).

(3.27)

The first inequality is due to 𝑓 (𝑋 ∗ (𝑡̅), 𝑝1∗ (𝑡̅), 𝜏) > 0 and 𝐸 (𝑝2∗ (𝑡̅), 𝜂) ≤ 𝐸 (𝑝̅2 , 𝜂) . Particularly, 𝐸 (𝑝2∗ (𝑡̅), 𝜂) ≤ 𝐸 (𝑝̅2 , 𝜂) is a direct conclusion of (3.12). The second strictly inequality follows from 𝑝1∗ (𝑡̅) ≠ 𝑝̅1 (𝑡̅) and the definition of (3.26). As (3.27) contradicts (3.25), the hypothesis that ∃𝑡̅ ∈ [0, 𝑇] such that 𝑝1∗ (𝑡̅) ≠ 𝑝̅1 (𝑡̅) is not true, i.e., 𝑝1∗ (𝑡) = 𝑝̅1 (𝑡) ∀𝑡 ∈ [0, 𝑇]. Next, consider 𝑝2∗ (𝑡). (3.25) asserts that 𝐻(𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝑝1∗ (𝑡), 𝑝2∗ (𝑡), 𝜃1 (𝑡), 𝜃2 (𝑡)) = 𝑓 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)[𝑝1∗ (𝑡) − 𝐶𝑚 − 𝐶𝐵𝑊 (𝜏) + 𝜃1 (𝑡) + 𝐸 (𝑝2∗ (𝑡), 𝜂)] ≥ 𝑓 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)[𝑝1∗ (𝑡) − 𝐶𝑚 − 𝐶𝐵𝑊 (𝜏) + 𝜃1 (𝑡) + 𝐸 (𝑝̅2 , 𝜂)]

38

= 𝐻(𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝑝1∗ (𝑡), 𝑝̅2 , 𝜃1 (𝑡), 𝜃2 (𝑡)),

(3.28)

for any 𝑡 ∈ [0, 𝑇]. Suppose for the sake of contradiction that 𝑝2∗ (𝑡) ≠ 𝑝̅2 = 𝐴𝑟𝑔max 𝐸 (𝑝2 , 𝜂), 𝑝2 ∈𝑈2

which implies 𝐸 (𝑝2∗ (𝑡), 𝜂) < 𝐸(𝑝̅2 , 𝜂). As 𝑓(𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏) > 0, then 𝑓 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)[𝑝1∗ (𝑡) − 𝑐𝑚 − 𝐶𝐵𝑊 (𝜏) + 𝜃1 (𝑡) + 𝐸 (𝑃2∗ (𝑡), 𝜂)] < 𝑓(𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)[𝑝1∗ (𝑡) − 𝑐𝑚 − 𝐶𝐵𝑊 (𝜏) + 𝜃1 (𝑡) + 𝐸 (𝑝̅2 , 𝜂)], which is in contradiction with (3.28). Therefore, 𝑝2∗ (𝑡) = 𝑝̅2

(3.29)

Substitute 𝜃0 = 1 and 𝜃2 (𝑡) ≡ 0 into (3.22) yields 𝜃̇1 (𝑡) + 𝜃1 (𝑡)𝑓𝑋 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏) = −𝑓𝑋 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)[𝑝1∗ (𝑡) − 𝑐𝑚 − 𝐶𝐵𝑊 (𝜆) + 𝐸 (𝑝̅2 , 𝜂)]. Define the factor 𝑇

𝜛(𝑡) = exp [− ∫ 𝑓𝑋 (𝑋 ∗ (𝑠), 𝑝1∗ (𝑠), 𝜏)𝑑𝑠]. 𝑡

Then 𝜛̇(𝑡) =

𝑑𝜛(𝑡) = 𝑓𝑋 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)𝜛(𝑡). 𝑑𝑡

Multiplying the factor  (t ) on both sides of the costate equation (3.30) yields 𝑑(𝜃1 (𝑡)𝜛(𝑡)) = 𝜃̇1 (𝑡)𝜛(𝑡) + 𝜃1 (𝑡)𝜛̇(𝑡) 𝑑𝑡 = −𝑓𝑋 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)[𝑝1∗ (𝑡) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂)]𝜛(𝑡).

39

(3.30)

Integrating the above differential equation from 𝑡 < 𝑇 to 𝑇 yields 𝑇

𝜃1 (𝑡)𝜛(𝑡) = ∫ 𝑓𝑋 (𝑋 ∗ (𝑠), 𝑝1∗ (𝑠), 𝜏)[𝑝1∗ (𝑠) − 𝑐𝑚 − 𝐶𝐵𝑊 (𝜆) + 𝐸(𝑝̅2 , 𝜂)]𝜛(𝑠)𝑑𝑠 + 𝐴, 𝑡

where A is a constant. The transversality condition (3.24) yields A  0 . Therefore, 𝑇

𝜃1 (𝑡) = ∫ 𝑡

𝜛(𝑠) 𝑓 (𝑋 ∗ (𝑠), 𝑝1∗ (𝑠), 𝜏)[𝑝1∗ (𝑠) − 𝑐𝑚 − 𝐶𝐵𝑊 (𝜆) + 𝐸(𝑝̅2 , 𝜂)]𝑑𝑠. 𝜛(𝑡) 𝑋 ∎

Remark 3.1 Economic explanations of the Hamiltonian function Consider the Hamiltonian function defined in the proof of Theorem 3.1, that is 𝐻 = 𝜃0 𝑓 (𝑋 (𝑡), 𝑝1 (𝑡), 𝜏)[𝑝1 (𝑡) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝2 (𝑡), 𝜂)] +𝜃1 (𝑡)(𝑋(𝑡), 𝑝1 (𝑡), 𝜏) + 𝜃2 (𝑡)𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏)𝜌(𝑝2 (𝑡), 𝜂). The constant multiplier 𝜃0 = 1 implies the problem is not degenerate. Furthermore, the first term in 𝐻 degenerates into 𝐿 = 𝑓 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)[𝑝1∗ (𝑡) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂)] , which can be interpreted as the instantaneous profit flow at time t. Then 𝐿𝑋 = 𝑓𝑋 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)[𝑝1∗ (𝑡) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂)] is the marginal change in the instantaneous profit with respect to the sales volume 𝑋(𝑡) of the new product at time t. In other words, if the sales volume 𝑋(𝑡) of the new product is increased by 𝛿, then the instantaneous profit flow will be approximately increased by 𝐿𝑋 𝛿. At time t, notice that the costate function 𝜃1 (𝑡) has the integral form (3.18): 𝑇

𝜃1 (𝑡) = ∫ 𝑡

𝜔 ̅(𝑠) 𝑓 (𝑋 ∗ (𝑠), 𝑝1∗ (𝑠), 𝜏)[𝑝1∗ (𝑠) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂)]𝑑𝑠. 𝜔 ̅(𝑡) 𝑋

If one interprets 𝜔 ̅(𝑠)/𝜔 ̅(𝑡) as the discount factor that discounts a cash flow at time 𝑠 to time 𝑡, where 𝑡 ≤ 𝑠, 𝜃1 (𝑡) can be viewed as the marginal change of the profit from time 𝑡 to 𝑇 with respect to the cumulative sales volume 𝑋(𝑡) of the new product at time t. In other words, if the cumulative sales volume 𝑋(𝑡) of the new product at time 𝑡 is increased by 𝛿, then the profit from time 𝑡 to 𝑇 will be approximately increased by 𝜃1 (𝑡)𝛿 . The above formula reinforces the 40

forward-looking nature of the optimal solution, and 𝜃1 (𝑡) is sometimes named as the shadow value of the sales volume 𝑋(𝑡) at time t ( Dorfman, 1969; Sethi et al., 2000). Formula (3.18) yields the following observations: (a) If 𝑓𝑋 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏) > 0 ∀𝑡 ∈ [0, 𝑇] , then 𝜃1 (𝑡) > 0 ∀𝑡 ∈ [0, 𝑇) . It implies that if the demand rate increases with the increase of the sales volume (𝑡) , the marginal profit 𝜃1 (𝑡) is always positive. That is, a small increase in the sales volume 𝑋(𝑡) will increase the manufacturer’s total profit in the remaining life cycle of the product. (b) If 𝑓𝑋 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏) < 0 ∀𝑡 ∈ [0, 𝑇], then 𝜃1 (𝑡) < 0 ∀𝑡 ∈ [0, 𝑇). Similarly, a small increase in the sales volume 𝑋(𝑡) will decrease the manufacturer’s total profit in the remaining life cycle of the product. (c) If 𝑓𝑋 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏) = 0 ∀𝑡 ∈ [0, 𝑇], then 𝜃1 (𝑡) = 0 ∀𝑡 ∈ [0, 𝑇). It implies that if the sales volume 𝑋(𝑡) does not affect the demand rate, then a small change in the sales volume 𝑋(𝑡) has no impact on the manufacturer’s total profit in the remaining life cycle of the product. Similarly, 𝜃2 (𝑡) is the marginal change in the total profit from time 𝑡 to 𝑇 with respect to the cumulative sales volume of the EWC 𝑌(𝑡) at time t. As indicated in the proof of Theorem 3.1, the sales volume Y does not affect the demand rate of the EWC, and hence 𝜃2 (𝑡) = 0 ∀𝑡 ∈ [0, 𝑇] which coincides with case (c) in the interpretation of 𝜃1 (𝑡). Based on the interpretations of 𝐿, 𝜃1 (𝑡) and 𝜃2 (𝑡), there is a conclusion that the Hamiltonian H measures the total profit contribution accumulated in a short time interval. It includes the instantaneous profit flow 𝐿 and the future effect 𝜃1 (𝑡)𝑓(𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏) of the sales volume X, whereas the future effect of the sales volume Y is zero. Obviously, the Hamiltonian H only depends on (𝑋(𝑡), 𝑝1 (𝑡), 𝑝2 (𝑡), 𝜃1 (𝑡)). To simplify notations, the Hamiltonian will be denoted as 𝐻(𝑋 (𝑡), 𝑝1 (𝑡), 𝑝2 (𝑡), 𝜃1 (𝑡)) in the subsequent sections.

41

Remark 3.2 According to the conclusion of Theorem 3.1, finding the optimal price (𝑝1∗ (𝑡), 𝑝2∗ (t)) for the dynamic optimization problem (3.5)-(3.10) can be divided into two steps: Step 1: Find the maximum point of the non-linear function: 𝑝2∗ (𝑡) ≡ 𝑝̅2 = 𝐴𝑟𝑔𝑚𝑎𝑥 𝐸 (𝑝2 , 𝜂 ) , ∀ 𝑡 ∈ [0, 𝑇]. 𝑝2 ∈𝑈2

Step 2: Find the optimal 𝑝1∗ (𝑡) via (3.14)-(3.15) and (3.17). More specifically, find the triple (𝑋 ∗ (𝑡), 𝑝1 (𝑡), 𝜃1 (𝑡)) which satisfies:

𝑝1∗ (𝑡) = 𝐴𝑟𝑔𝑚𝑎𝑥 𝐻(𝑋 ∗ (𝑡), 𝑝1 , 𝑝̅2 , 𝜃1 (𝑡)) ; 𝑝1 ∈𝑈1

𝑋̇ ∗ (𝑡) = 𝑓(𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏), 𝑋(0) = 0;

(3.31)

{𝜃̇1 (𝑡) = −𝐻𝑋 (𝑋 ∗ (𝑡), 𝑝1 (𝑡), 𝑝̅2 , 𝜃1 (𝑡)), 𝜃1 (𝑇) = 0.

Given the specific form of function 𝜌(𝑝2 (𝑡), 𝜂), the optimization problem in Step 1 is a typical non-linear optimization problem. The optimal pricing strategy 𝑝2∗ (𝑡) ≡ 𝑝̅2 is constant over the planning horizon [0, 𝑇] , 𝑝̅2 can be derived by standard non-linear optimization procedures. Hence, the remaining task will focus on studying the property of 𝑝1∗ (𝑡) and the related solution procedure. In the following, the time argument t will be omitted when no confusion arises.

Remark 3.3 (3.14) implies that at any time 𝑡 ∈ [0, 𝑇] , given the value of (𝑋 ∗ , 𝑝̅2 , 𝜃1 ), 𝑝1∗ maximizes the Hamiltonian function 𝐻 (𝑋 ∗ , 𝑝1 , 𝑝̅2 , 𝜃1 ) among all 𝑝1 ∈ 𝑈1 . Note that the domain 𝑈1 ⊆ 𝑅+ is an open interval. The first-order necessary condition for maximum point yields 𝐻𝑝1 (𝑋 ∗ , 𝑝1 , 𝑝̅2 , 𝜃1 ) = 𝑓𝑝1 (𝑋 ∗ , 𝑝1∗ , 𝜏)[𝑝1∗ − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 ] + 𝑓 (𝑋 ∗ , 𝑝1∗ , 𝜏) = 0, which is equivalent to

42

𝑝1∗

𝑓 (𝑋 ∗ , 𝑝1∗ , 𝜏) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 = − . 𝑓𝑝1 (𝑋 ∗ , 𝑝1∗ , 𝜏)

(3.32)

Therefore, (3.32) is the first-order condition of (3.14). It not only reveals the relationship between 𝑋 ∗ , 𝑌 ∗ , 𝑝1∗ , 𝑝̅2 and 𝜃1 , but helps to derive further properties of the optimal solution. For example, after substituting this first-order condition into (3.18), the costate function 𝜃1 can be simplified into a more compact integral form: 𝑇

𝜃1 (𝑡) = ∫ −𝑓(𝑋 𝑡

∗

𝑓 (𝑋 ∗ , 𝑝1∗ , 𝜏) ∗ ) 𝑋 , 𝑝1 , 𝜏 𝑑𝑠. 𝑓𝑝1 (𝑋 ∗ , 𝑝1∗ , 𝜏)

Further assume that 𝑓(𝑋, 𝑝1 , 𝜏) has a second-order partial derivative with respect to 𝑝1 . The second-order optimality condition for maximum point yields 𝐻𝑝1 𝑝1 (𝑋 ∗ , 𝑝1 , 𝑝̅2 , 𝜃1 ) = 𝑓𝑝1 𝑝1 (𝑋 ∗ , 𝑝1∗ , 𝜏)[𝑝1∗ − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 ] + 2𝑓𝑝1 (𝑋 ∗ , 𝑝1∗ , 𝜏) ≤ 0. Substituting (3.32) into the above inequality yields 𝑓(𝑋 ∗ , 𝑝1∗ , 𝜏) − 𝑓 (𝑋 ∗ , 𝑝1∗ , 𝜏) + 2𝑓𝑝1 (𝑋 ∗ , 𝑝1∗ , 𝜏) ≤ 0. 𝑓𝑝1 (𝑋 ∗ , 𝑝1∗ , 𝜏) 𝑝1 𝑝1 Multiplying 𝑓𝑝1 (𝑋 ∗ , 𝑝1∗ , 𝜏)(< 0) on both sides of the above inequality yields 2𝑓𝑝21 (𝑋 ∗ , 𝑝1∗ , 𝜏) ≥ 𝑓(𝑋 ∗ , 𝑝1∗ , 𝜏)𝑓𝑝1 𝑝1 (𝑋 ∗ , 𝑝1∗ , 𝜏).

(3.33)

Note that the instantaneous revenue for selling the new product is 𝑝1 𝑓 (𝑋, 𝑝1 , 𝜏) . And the corresponding marginal revenue (MR) can be described as the change in the instantaneous revenue divided by the change in number of units sold (Barnett et al., 2002). More formally, MR is the derivative of the instantaneous revenue with respect to the demand 𝑓(𝑋, 𝑝1 , 𝜏): 𝐌𝐑 =

∂(𝑝1 𝑓) . ∂𝑓

43

Recall that the instantaneous demand rate 𝑓 = 𝑓(𝑋, 𝑝1 , 𝜏) is a decreasing function with respect to 𝑝1 when 𝑋 and 𝜏 are given. Thus, there is an inverse function 𝑝1 = 𝑝1 (𝑓, 𝑋, 𝜏), and

∂𝑝1 ∂𝑓

1

= 𝜕𝑓/𝜕𝑝 . 1

Therefore, the expression of MR can be expanded as 𝐌𝐑 =

∂(𝑝1 𝑓) d𝑝1 1 = 𝑝1 + 𝑓 = 𝑝1 + 𝑓 , ∂𝑓 d𝑓 𝑓𝑝1

Furthermore, take the derivative of MR with respect to 𝑝1 𝑓𝑝1 𝑝1 2𝑓𝑝21 − 𝑓𝑓𝑝1 𝑝1 ∂𝐌𝐑 ∂ 1 1 = (𝑝 + 𝑓 ) = 1 + 𝑓𝑝1 −𝑓 2 = . ∂𝑝1 ∂𝑝1 1 𝑓𝑝1 𝑓𝑝1 𝑓𝑝1 𝑓𝑝21 Combine the expression of ∂𝐌𝐑/ ∂𝑝1 and the condition (3.33), it is not difficult to notice that the optimal solution 𝑝1∗ must obey the following economic rule: the marginal revenue is an increasing function of the product price 𝑝1 , i.e., ∂𝐌𝐑/ ∂𝑝1 |𝑝1 =𝑝1∗ ≥ 0. Similarly, (3.13) implies that 𝐸 (𝑝2 , 𝜂) attains its maximum at 𝑝̅2 in the domain 𝑈2 , i.e., 𝑝̅2 = 𝐴𝑟𝑔𝑚𝑎𝑥 𝐸 (𝑝2 , 𝜂). Suppose that 𝜌(𝑝2 , 𝜂) has a first-order partial derivative with respect to 𝑝2 , 𝑝2 ∈𝑈2

then the first-order condition for the maximum point 𝑝̅2 is 𝐸𝑝2 (𝑝̅2 , 𝜂) = (𝑝̅2 − 𝐶𝐸𝑊𝐶 )𝜌𝑝2 (𝑝̅2 , 𝜂) + 𝜌(𝑝̅2 , 𝜂) = 0. Further assume 𝜌(𝑝2 , 𝜂) has a second-order partial derivative with respect to 𝑝2 , then 𝐸𝑝2 𝑝2 (𝑝̅2 , 𝜂) =

2𝜌𝑝22 (𝑝̅2 , 𝜂) − 𝜌(𝑝̅2 , 𝜂)𝜌𝑝2 𝑝2 (𝑝̅2 , 𝜂) ≤ 0. 𝜌𝑝2 (𝑝̅2 , 𝜂)

𝑓 (𝑋, 𝑝1 , 𝜏)𝜌(𝑝2 , 𝜂) is the demand rate of EWC. Denote 𝐌𝐑 𝐄𝐖𝐂 as the marginal change in revenue with respect to the demand rate of EWC. Then we have ∂𝐌𝐑 𝐄𝐖𝐂 2𝜌𝑝22 (𝑝2 , 𝜂) − 𝜌(𝑝2 , 𝜂)𝜌𝑝2 (𝑝2 , 𝜂) = . ∂𝑝2 𝜌𝑝22 (𝑝2 , 𝜂) Therefore, 𝐸𝑝2 𝑝2 (𝑝̅2 , 𝜂) ≤ 0 implies the marginal revenue with respect to the demand rate of EWC is an increasing function of the EWC price 𝑝2 , i.e., ∂𝐌𝐑 𝐄𝐖𝐂 / ∂𝑝2 |𝑝2 =𝑝̅2 ≥ 0. 44

Suppose that (𝑋 ∗ , 𝑌 ∗ , 𝑝1∗ , 𝑝̅2 ) satisfies the necessary optimality conditions for the dynamic optimization problem (3.5)-(3.10) in Theorem 3.1. To make sure 𝐽 attains its global maximum at (𝑋 ∗ , 𝑌 ∗ , 𝑝1∗ , 𝑝̅2 ), the following concavity condition of 𝐻 (𝑋, 𝑝1 , 𝑝2 , 𝜃1 ) is sufficient.

Theorem 3.2 (Mangasarian-type sufficient condition) Let (𝑋 ∗ , 𝑌 ∗ , 𝑝1∗ , 𝑝̅2 ) be a feasible solution for the dynamic optimization problem (3.5)-(3.10). Suppose that (𝑋 ∗ , 𝑌 ∗ , 𝑝1∗ , 𝑝̅2 ) satisfies the conditions in Theorem 3.1 with the costate function 𝜃1 . If the Hamiltonian 𝐻 (𝑋, 𝑝1 , 𝑝2 , 𝜃1 ) is concave in (𝑋, 𝑝1 ) for any given 𝑝2 and 𝜃1 , then (𝑋 ∗ , 𝑌 ∗ , 𝑝1∗ , 𝑝̅2 ) attains the global maximum of 𝐽. If 𝐻 (𝑋, 𝑝1 , 𝑝2 , 𝜃1 ) is strictly concave in (𝑋, 𝑝1 ), then (𝑋 ∗ , 𝑌 ∗ , 𝑝1∗ , 𝑝̅2 ) is the unique optimal solution. Proof: Let (𝑋(𝑡), 𝑌 (𝑡), 𝑝1 (𝑡), 𝑝2 (𝑡)) be an arbitrary feasible solution for the problem (3.5)-(3.10). Then (𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝑝1∗ (𝑡), 𝑝̅2 ) is optimal provided the following inequality is always true ∆= 𝐽[𝑝1∗ (𝑡), 𝑝̅2 ] − 𝐽[𝑝1 (𝑡), 𝑝2 (𝑡)] ≥ 0. Note that 𝑋̇ = 𝑓 (𝑋, 𝑝1 , 𝜏), and the integrand of the objective (3.5) can be rewritten as 𝑓(𝑋, 𝑝1 , 𝜏)[𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝2 , 𝜂)] = 𝐻(𝑋, 𝑝1 , 𝑝2 , 𝜃1 ) − 𝜃1 𝑓(𝑋, 𝑝1 , 𝜏) = 𝐻(𝑋, 𝑝1 , 𝑝2 , 𝜃1 ) − 𝜃1 𝑋̇. Then, 𝑻

∆= ∫ {[𝐻(𝑋 ∗ , 𝑝1∗ , 𝑝̅2 , 𝜃1 ) − 𝜃1 𝑋̇ ∗ ] − [𝐻 (𝑋, 𝑝1 , 𝑝2 , 𝜃1 ) − 𝜃1 𝑋̇ ]}𝑑𝑡 𝟎

𝑻

𝑻

= ∫ [𝐻(𝑋 ∗ , 𝑝1∗ , 𝑝̅2 , 𝜃1 ) − 𝐻 (𝑋, 𝑝1 , 𝑝2 , 𝜃1 )] 𝑑𝑡 + ∫ 𝜃1 (𝑋̇ − 𝑋̇ ∗ ) 𝑑𝑡. 𝟎

𝟎

Note that 𝐸 (𝑝2 , 𝜂) ≤ 𝐸 (𝑝̅2 , 𝜂) ∀ 𝑝2 ∈ 𝑈2 and 𝑓(𝑋, 𝑝1 , 𝜏) > 0 ∀ 𝑝1 ∈ 𝑈1 . Then

45

𝐻 (𝑋 ∗ , 𝑝1∗ , 𝑝̅2 , 𝜃1 ) − 𝐻 (𝑋, 𝑝1 , 𝑝2 , 𝜃1 ) ≥ 𝐻 (𝑋 ∗ , 𝑝1∗ , 𝑝̅2 , 𝜃1 ) − 𝐻 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ). If the Hamiltonian 𝐻 (𝑋, 𝑝1 , 𝑝2 , 𝜃1 ) is concave in (𝑋, 𝑝1 ) for all t, then 𝐻 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ) ≤ 𝐻 (𝑋 ∗ , 𝑝1∗ , 𝑝̅2 , 𝜃1 ) + 𝐻𝑋 (𝑋 ∗ , 𝑝1∗ , 𝑝̅2 , 𝜃1 )[𝑋 − 𝑋 ∗ ] + 𝐻𝑝1 (𝑋 ∗ , 𝑝1∗ , 𝑝̅2 , 𝜃1 )[𝑝1 − 𝑝1∗ ].

(3.34)

The first-order necessary condition implies 𝐻𝑝1 (𝑋 ∗ , 𝑝1∗ , 𝑝̅2 , 𝜃1 ) ≡ 0 and 𝜃̇1 = −𝐻𝑋 (𝑋 ∗ , 𝑝1∗ , 𝑝̅2 , 𝜃1 ). Therefore, 𝑻

∆≥ ∫ [𝐻(𝑋 𝟎

∗

, 𝑝1∗ , 𝑝̅2 , 𝜃1 )

𝑻

𝑻

− 𝐻 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 )] 𝑑𝑡 + ∫ 𝜃1 (𝑋̇ − 𝑋̇ ∗ ) 𝑑𝑡 𝟎

𝑻

≥ ∫ 𝜃̇1 (𝑋 − 𝑋 ∗ ) 𝑑𝑡 + ∫ 𝜃1 (𝑋̇ − 𝑋̇ ∗ ) 𝑑𝑡 𝟎

𝟎

𝑻

= ∫ 𝑑(𝜃1 (𝑋 − 𝑋 ∗ )) 𝟎

𝑇 = (𝜃1 (𝑋 − 𝑋 ∗ )) | 0 = 𝜃1 (𝑇)(𝑋(𝑇) − 𝑋 ∗ (𝑇)) − 𝜃1 (0)(𝑋(0) − 𝑋 ∗ (0)) = 0 (by the boundary conditions 𝑋(0) = 𝑋 ∗ (0) = 0 and 𝜃1 (𝑇) = 0). Consequently, (𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝑝1∗ (𝑡), 𝑝̅2 ) is optimal. If 𝐻 (𝑋, 𝑝1 , 𝑝2 , 𝜃1 ) is strictly concave in (𝑋, 𝑝1 ), then the inequality (3.34) becomes a strict inequality if either 𝑋 ≠ 𝑋 ∗ or 𝑝1 ≠ 𝑝1∗ . This strict inequality leads to the conclusion that 𝐽[𝑝1∗ (𝑡), 𝑝̅2 ] > 𝐽[𝑝1 (𝑡), 𝑝2 (𝑡)] , i.e., any feasible solution (𝑋(𝑡), 𝑝1 (𝑡)) that is not identically equal to (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡)) is nonoptimal. ∎ The Mangasarian-type sufficient condition is applicable to the general demand rate function defined in (3.3), but it requires relatively strong concavity of the Hamiltonian function, 𝐻 (𝑋, 𝑝1 , 𝑝2 , 𝜃1 ). In Section 3.5, two important and widely studied types of demand rate functions (corresponding to two marketing environments) will be discussed, and the sufficient condition corresponding to the two special types of demand rate function require a weaker concavity

46

assumption. Prior to these detailed discussions, let us focus on the properties of the optimal solution (𝑋 ∗ , 𝑌 ∗ , 𝑝1∗ , 𝑝̅2 ). In the remaining sections of this chapter, the superscript “*” in the notation of the optimal solution will be omitted unless otherwise stated.

3.3.2 Properties of the Optimal Solution In economics, elasticity is an important concept which refers to the ratio of the proportional change in one variable with respect to the proportional change in another variable (Tirole, 1988). It is usually expressed as a negative number but shown as a positive value. According to this definition, the price elasticities of demand for the new product and its EWC measure how sensitive the demand rates of the product and its EWC are to the changes in their prices, respectively. For dynamic monopolists, the formula to calculate the price elasticity of demand (𝑃𝐸𝐷𝑝1 ) for the new product (Sarikavanij et al., 2011) is: proportional change in the demand rate of the new product 𝑃𝐸𝐷𝑝1 = | |, proportional change in price of the new product where proportional change in price of the new product =

𝑑𝑝1 ; 𝑝1

proportional change in the demand rate of the new product = 𝑑𝑓 𝑝1 𝑝1 𝑃𝐸𝐷𝑝1 = | ∙ | = −𝑓𝑝1 . 𝑑𝑝1 𝑓 𝑓 Similarly, the price elasticity of demand (𝑃𝐸𝐷𝑃2 ) for the EWC is: 𝑃𝐸𝐷𝑝2 = |

𝑑(𝑓𝜌) /(𝑓𝜌) 𝑝̅2 | = −𝜌𝑝2 . 𝑑𝑝̅2 /𝑝̅2 𝜌

Notice that the revenue function for selling the new product is: 𝑅 = price × demand rate = 𝑝1 × 𝑓. 47

𝑑𝑓 (𝑋, 𝑝1 , 𝜏) ; 𝑓 (𝑋, 𝑝1 , 𝜏)

Then the marginal revenue for selling the new product is: 𝐌𝐑 =

𝑑𝑅 𝑑𝑝1 1 𝑓 = 𝑝1 + 𝑓 = 𝑝1 + 𝑓 = 𝑝1 + . 𝑑𝑓 𝑑𝑓 𝑑𝑓/𝑑𝑝1 𝑓𝑝1

Similarly, the marginal revenue for selling the EWC is: 𝐌𝐑 𝐄𝐖𝐂 = 𝑝2 +

𝜌 . 𝜌𝑝2

With the expressions of 𝑃𝐸𝐷𝑝1 and 𝑃𝐸𝐷𝑃2 , the marginal revenues with respect to the demand rates of the product and the EWC can be expressed as: 𝐌𝐑 = 𝑝1 +

𝑓 𝑓 1 = 𝑝1 (1 + ) = 𝑝1 (1 − ); 𝑓𝑝1 𝑝1 𝑓𝑝1 𝑃𝐸𝐷𝑝1 𝐌𝐑 𝐄𝐖𝐂 = 𝑝̅2 (1 −

1 ). 𝑃𝐸𝐷𝑝2

Review the standard single period monopolist problem. The problem is static, and the optimal price, which will generate maximum profit for the monopolist, must obey the following equation (Tirole, 1988) marginal cost = marginal revenue.

(3.34)

Note that marginal revenue = price × (1 −

1 ). price elasticity of demand

Then (3.34) is equivalent to price − marginal cost 1 = , price price elasticity of demand

48

(3.35)

which indicates that the inverse price elasticity of demand equals the ratio between the profit margin and the price. (3.35) implies that the higher price elasticity of demand, the lower pricing power (left part of (3.35)) of the monopolist. The dynamic optimization problem (3.5)-(3.10) formulated in this chapter is different from the standard single period monopolist problem in two aspects. Firstly, problem (3.5)-(3.10) is dynamic, i.e., the current price may influence the future decision. Secondly, the manufacturer sells not only the new product but also the optional EWC. One question arises: should the optimal prices of the new product and the EWC also obey the basic rule (i.e., (3.34) or (3.35)) in the standard single period monopoly problem? The following proposition gives the answer.

Proposition 3.1 Suppose (𝑝1 , 𝑝̅2 ) is the optimal price for the dynamic optimization problem (3.5) -(3.10). The marginal revenues with respect to the demand rates of the new product and the EWC can be expressed as

𝐌𝐑 = (1 −

1 ) 𝑝 = 𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂) − 𝜃1 ; 𝑃𝐸𝐷𝑝1 1

𝐌𝐑 𝐄𝐖𝐂 = (1 −

1 ) 𝑝̅ = 𝐶𝐸𝑊𝐶 . 𝑃𝐸𝐷𝑝2 2

(3.36)

(3.37)

Proof: As discussed in Remark 3.3, the optimal price 𝑝1 and 𝑝̅2 must satisfy the first-order condition for optimality, that is 𝐻𝑝1 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ) = [𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 ]𝑓𝑝1 + 𝑓 = 0; 𝐸𝑝2 (𝑝̅2 , 𝜂) = (𝑝̅2 − 𝐶𝐸𝑊𝐶 )𝜌𝑝2 + 𝜌 = 0. After simple algebraic operation,

49

(1 +

𝑓 ) 𝑝 = 𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂) − 𝜃1 ; 𝑝1 𝑓𝑝1 1 (1 +

𝜌 ) 𝑝̅ = 𝐶𝐸𝑊𝐶 . 𝑝̅2 𝜌𝑝2 2

Insert in the expressions of 𝑃𝐸𝐷𝑝1 and 𝑃𝐸𝐷𝑝2 , we have (1 −

1 ) 𝑝 = 𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂) − 𝜃1 ; 𝑃𝐸𝐷𝑝1 1 (1 −

1 ) 𝑝̅ = 𝐶𝐸𝑊𝐶 . 𝑃𝐸𝐷𝑝2 2 ∎

The manufacturer, who monopolizes in a long-term planning horizon, sells both the new product and the EWC. 𝐶𝑚 + 𝐶𝐵𝑊 can be viewed as the marginal cost of the new product. (3.36) indicates that the optimal price 𝑝1 for the new product changes the standard pricing rule of marginal revenue = marginal cost. The standard pricing rule fails in this situation, because the manufacturer’s revenue is affected not only by the selling price 𝑝1 but also the shadow price 𝜃1 and the expected profit 𝐸 (𝑝̅2 , 𝜂) from the EWC. The next proposition will quantify the relationship between the revenue, the elasticity 𝑃𝐸𝐷𝑝1 and the Hamiltonian H.

Proposition 3.2 The Hamiltonian function H is constant over time, and the instantaneous revenue of selling the new product, 𝑝1 𝑓, is linearly proportional to 𝑃𝐸𝐷𝑝1 with the rate H, i.e., 𝑝1 𝑓 = 𝐻 ⋅ 𝑃𝐸𝐷𝑝1 . Proof: The derivative of the Hamiltonian function H with respect to t is 𝑑𝐻(𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ) = 𝐻𝑋 𝑋̇ + 𝐻𝑝1 𝑝1̇ + 𝐻𝑝2 𝑝̅2̇ + 𝐻𝜃1 𝜃1̇ . 𝑑𝑡

50

Note that 𝑋̇ = 𝐻𝜃1 , 𝐻𝑝1 = 𝐻𝑝2 = 𝑝̅2̇ = 0, and 𝜃1̇ = −𝐻𝑋 . Thus, 𝑑𝐻(𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ) = 𝐻𝑋 𝐻𝜃1 + 0 + 0 + 𝐻𝜃1 (−𝐻𝑋 ) = 0. 𝑑𝑡 Therefore, H remains constant over time. Multiplying 𝑝1 𝑓𝑝1 on both sides of (3.32) yields 𝑝1 𝑓 = −𝑝1 𝑓𝑝1 [𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 ] =−

𝑝1 𝑓𝑝1 𝑓[𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 ]. 𝑓

Inserting in the expression of 𝑃𝐸𝐷𝑝1 , 𝐻 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ) and the above equation yields 𝑝1 𝑓 = 𝐻 ∙ 𝑃𝐸𝐷𝑝1 . ∎ When the demand rate function 𝑓(𝑋, 𝑝1 , 𝜏) satisfies the second-order condition, the following proposition can characterize how the optimal price 𝑝1 changes over time.

Proposition 3.3 Assume 𝑓 (𝑋, 𝑝1 , 𝜏) has a second-order partial derivative with respect to (𝑋, 𝑝1 ). Then the optimal price path 𝑝1 is smooth over time, and the derivative satisfies (2𝑓𝑝21 − 𝑓𝑓𝑝1 𝑝1 )𝑝̇1 = (−2𝑓𝑋 ∙ 𝑓𝑝1 + 𝑓𝑓𝑝1 𝑋 )𝑓, as well as the end time boundary condition

𝑝1 (𝑇) +

𝑓(𝑋(𝑇), 𝑝1 (𝑇), 𝜏) = 𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂). 𝑓𝑝1 (𝑋(𝑇), 𝑝1 (𝑇), 𝜏)

Proof: As discussed in Remark 3.3, the optimal price 𝑝1 satisfies the equation (3.32):

51

(3.38)

𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 = −

𝑓 . 𝑓𝑝1

At 𝑡 = 𝑇, combining (3.32) with the boundary condition of 𝜃1 (𝑇) = 0 in the costate equation (3.17) yields 𝑝1 (𝑇) +

𝑓 (𝑋(𝑇), 𝑝1 (𝑇), 𝜏) = 𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂). 𝑓𝑝1 (𝑋(𝑇), 𝑝1 (𝑇), 𝜏)

Note that 𝐶𝑚 , 𝐶𝐵𝑊 and 𝐸 (𝑝̅2 , 𝜂) are constants. Taking the derivative with respect to t on both side of the equation 𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 = −

𝑓 . 𝑓𝑝1

Then we have 𝑝̇1 + 𝜃̇1 = −

𝑓𝑝1 ∙ 𝑝̇1 + 𝑓𝑋 ∙ 𝑋̇ 𝑓(𝑓𝑝1 𝑋 ∙ 𝑋̇ + 𝑓𝑝1 𝑝1 ∙ 𝑝̇1 ) + . 𝑓𝑝1 𝑓𝑝21

Substitute the costate equation (3.17) into the above equation yields 𝑝̇1 − 𝑓𝑋 (𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 ) = −

𝑓𝑝1 ∙ 𝑝̇1 + 𝑓𝑋 ∙ 𝑓 𝑓(𝑓𝑝1 𝑋 ∙ 𝑓 + 𝑓𝑝1 𝑝1 ∙ 𝑝̇1 ) + . 𝑓𝑝1 𝑓𝑝21

Applying (3.32) yields 𝑝̇1 +

𝑓𝑓𝑋 −(𝑓𝑝1 ∙ 𝑝̇1 + 𝑓𝑋 ∙ 𝑓)𝑓𝑝1 + 𝑓(𝑓𝑝1 𝑋 ∙ 𝑓 + 𝑓𝑝1 𝑝1 ∙ 𝑝̇1 ) = , 𝑓𝑝1 𝑓𝑝21

i.e., 𝑝̇1 (1 +

𝑓𝑝21 − 𝑓𝑓𝑝1 𝑝1 −2𝑓𝑋 ∙ 𝑓𝑓𝑝1 + 𝑓 2 𝑓𝑝1 𝑋 ) = , 𝑓𝑝21 𝑓𝑝21

i.e., (2𝑓𝑝21 − 𝑓𝑓𝑝1 𝑝1 )𝑝̇1 = (−2𝑓𝑋 ∙ 𝑓𝑝1 + 𝑓𝑓𝑝1 𝑋 )𝑓. 52

Because 2𝑓𝑝21 − 𝑓𝑓𝑝1 𝑝1 ≥ 0 and 𝑓 > 0, the sign of 𝑝̇1 is totally determined by −2𝑓𝑋 ∙ 𝑓𝑝1 + 𝑓𝑓𝑝1 𝑋 . More specifically, 𝑝1 (𝑡) is increasing, if −2𝑓𝑋 ∙ 𝑓𝑝1 + 𝑓𝑓𝑝1 𝑋 > 0; {𝑝1 (𝑡) is constant,

if −2𝑓𝑋 ∙ 𝑓𝑝1 + 𝑓𝑓𝑝1 𝑋 = 0;

𝑝1 (𝑡) is decreasing, if −2𝑓𝑋 ∙ 𝑓𝑝1 + 𝑓𝑓𝑝1 𝑋 < 0. ∎

3.4 Analysis of Warranty Contracts Assume a repairable product is new at time 0. Let 𝑁(𝑡) denote the number of failures in (0, 𝑡]. Let 𝑇𝑖 be the time of the 𝑖’th failure, where 𝑇0 = 0. Let 𝑍𝑖 = 𝑇𝑖 − 𝑇𝑖−1 be the time between the (𝑖 − 1)’th failure and the 𝑖’th failure. Then the observed sequence {𝑇𝑖 , 𝑖 = 1, 2, … } forms a point process, and {𝑁(𝑡), 𝑡 ≥ 0} is the corresponding counting process (Lindqvist, 2008). To estimate the expected repair cost in (0, 𝑡], the first task is to find the number of failures during (0, 𝑡], i.e., the distribution of 𝑁(𝑡). In the failure-repair model, it is assumed that the time to repair the product is negligible, and the distribution of 𝑁(𝑡) depends on the failure rate of the product and the effects of successive repair actions. The failure rate is determined by the product design and so is fixed; the repair actions are usually divided into two categories, namely, perfect repair and minimal repair (Lindqvist, 2008). 1) Perfect repair: the product is repaired to a condition as good as new. 2) Minimal repair: the product is repaired only to the state it had immediately before the failure. In other words, after the repair, the product is exactly as if no failure had ever occurred. In most pricing models considering the warranty policy, the warranty cost functions were based on perfect repair (Lin et al., 2005; Huang et al., 2007). However, it is well known that minimal repair is more common in real life than perfect repair. To fill this research gap, the study in this section aims to conduct the warranty cost analysis based on minimal repair.

53

Let 𝜆(𝑡), 𝑡 ≥ 0 denote the failure rate (or the intensity function) of the product. Suppose that {𝑁(𝑡), 𝑡 ≥ 0} is a Non-Homogeneous Poisson Process (NHPP) with the failure rate function 𝜆(𝑡), 𝑡 ≥ 0 under the minimal repair action (Lindqvist, 2008; Karbasian et al., 2010). The number of failures in the interval (𝑠, 𝑡], 𝑡 > 𝑠 , follows a Poisson distribution with the 𝑡

parameter ∫𝑠 𝜆(𝑢)𝑑𝑢, i.e., 𝑡

𝑃 (𝑁 (𝑡 ) − 𝑁 ( 𝑠 ) = 𝑘 ) =

(∫𝑠 𝜆(𝑢)𝑑𝑢)

𝑘

𝑘!

𝑡

exp (− ∫ 𝜆(𝑢)𝑑𝑢). 𝑠

𝑡

Define 𝑚(𝑡) = ∫0 𝜆(𝑢)𝑑𝑢. The expected number of failures can be obtained as follows: ∞

∞

𝑘

𝐸[𝑁(𝑡)] = ∑ 𝑘𝑃(𝑁 (𝑡) = 𝑘) = ∑ 𝑘 𝑘=0

𝑘=0

(𝑚(𝑡)) exp(−𝑚(𝑡)) 𝑘!

∞

(3.39)

𝑘−1

(𝑚(𝑡)) = 𝑚(𝑡) exp(−𝑚(𝑡)) ∑ = 𝑚(𝑡). (𝑘 − 1)! 𝑘=1

The variance of the number of failures is 𝑉𝑎𝑟[𝑁 (𝑡)] = 𝐸[𝑁(𝑡)2 ] − (𝐸[𝑁(𝑡)])2 ∞

= ∑ 𝑘 2 𝑃(𝑁(𝑡) = 𝑘) − (𝑚(𝑡))

2

𝑘=0 ∞

𝑘

(𝑚(𝑡)) 2 = ∑𝑘 exp(−𝑚(𝑡)) − (𝑚(𝑡)) 𝑘! 2

(3.40)

𝑘=0

∞

𝑘−1

(𝑚(𝑡)) 2 = 𝑚(𝑡) exp(−𝑚(𝑡)) ∑ 𝑘 − (𝑚(𝑡)) (𝑘 − 1)! 𝑘=1

= 𝑚(𝑡) exp(−𝑚(𝑡)) (𝑚(𝑡) + 1) exp(𝑚(𝑡)) − (𝑚(𝑡))

2

= 𝑚(𝑡).

To better illustrate the concept of NHPP, Figure 3.2 presents the simulation results given that the product life follows the Weibull distribution with parameters 𝛼 and 𝛽. Note that the Weibull distribution is widely used to model product life and the corresponding NHPP is called the

54

Power Law Process (PLP). Figure 3.2 (a) plots the failure rate functions under three pairs of the parameters 𝛼 and 𝛽, while the dashed lines and the bold lines in Figure 3.2 (b) correspond to the cumulative failure rate function and sample paths, respectively. The blue lines illustrates a PLP with 𝛽 < 1. In this case, the failure rate decreases over time, which implies that the defective product improves its performance after each repair. The green lines correspond to the case when 𝛽 = 1. The failure rate is constant over time and the interval between two consecutive failures follows an exponential distribution with parameter 𝛼. The red lines show the case when 𝛽 > 1. Note that the failure rate increases with time, i.e., the product is more likely to fail as time goes on. (a) Failure rate functions  (t;  ,) of the PLP( ,) for three chosen pair of  and 

(b) Cumulative failure rate function and sample paths of the PLP( ,)

20

40

15

(t;20,0.3) (t;4,1)

35

(t;0.5,1.6)

30 25

10

20 15

5

10 5

0

0

1

2

3

4

5

6

0

0

1

2

3

4

5

6

Figure 3.2 Simulation of the warranty claims when the product life follows a Weibull distribution

When an item is returned for repair under warranty (BW or EWC), the manufacturer incurs various costs, including labour cost, material cost, transportation costs, etc. All these costs are aggregated into a single cost termed the “repair cost” for each claim of the failure. Because some of the costs are uncertain, this cost should be a random variable (Blischke and Murthy, 1996). Let 𝑊𝑖 (i = 1,2 … . ) denote the repair cost of the 𝑖’th failure, then the total repair cost during (0, 𝑡] can be represented as 55

𝑁(𝑡)

𝑅 (𝑡) = ∑ 𝑊𝑖 . 𝑖=1

The continuous-time stochastic process {𝑅(𝑡), 𝑡 ≥ 0} is a reward process associated with NHPP. Obviously, the expected total repair costs during BW and EWC can be represented as 𝐶𝐵𝑊 (𝜏) = 𝐸 [𝑅(𝜏)] and 𝐶𝐸𝑊𝐶 (𝜏, 𝜂) = 𝐸 [𝑅(𝜏 + 𝜂) − 𝑅(𝜏)], respectively. And the calculation of these two cost functions is based on the following property of {𝑅 (𝑡), 𝑡 ≥ 0}.

Lemma 3.2 Consider a NHPP {𝑁 (𝑡), 𝑡 > 0} with intensity function 𝜆(𝑡), 𝑡 ≥ 0. Suppose that (a) 𝑊𝑖 (i = 1,2 … . ) is a sequence of independent and identically distributed (i.i.d.) nonnegative random variables; (b) 𝑊𝑖 (i = 1,2 … . ) is independent of 𝑁(𝑡) . 𝑡

( ) Denote 𝑚(𝑡) = ∫0 𝜆(𝑢)𝑑𝑢 and 𝑅(𝑡) = ∑𝑁(𝑡) 𝑖=1 𝑊𝑖 . Then the expectation of 𝑅 𝑡 is 𝐸 [𝑅(𝑡)] = 𝐸 [𝑊1 ]𝐸 [𝑁(𝑡)] = 𝐸 [𝑊1 ]𝑚(𝑡),

(3.41)

which is the Wald’s equation of NHPP, and the variance of 𝑅(𝑡) is 𝑉𝑎𝑟[𝑅(𝑡)] = 𝐸 [𝑊12 ]𝑚(𝑡). Proof: (1) For the equation (3.41) 𝑁(𝑡)

𝐸 [𝑅(𝑡)] = E [∑ 𝑊𝑖 ]. 𝑖=1

According to the law of total expectation,

56

(3.42)

∞

𝑁(𝑡)=𝑘

𝐸 [𝑅(𝑡)] = ∑ 𝐸 [ ∑ 𝑊𝑖 |𝑁(𝑡) = 𝑘] ∙ 𝑃(𝑁(𝑡) = 𝑘). 𝑘=1

𝑖=1

Because 𝑊𝑖 (i = 1,2 … . ) is independent of 𝑁(𝑡), ∑𝑁(𝑡)=𝑘 𝑊𝑖 is also independent of 𝑁(𝑡) and so 𝑖=1 𝑁(𝑡)=𝑘

∞

𝐸[𝑅(𝑡)] = ∑ 𝐸 [ ∑ 𝑊𝑖 ] ∙ 𝑃(𝑁(𝑡) = 𝑘). 𝑘=1

𝑖=1

Since 𝑊𝑖 (i = 1,2 … . ) is a sequence of i.i.d. non-negative random variables, we have ∞

𝐸 [𝑅(𝑡)] = ∑(𝑘 ∙ 𝐸[𝑊1 ] ∙ 𝑃(𝑁(𝑡) = 𝑘) 𝑘=1 ∞

= 𝐸 [𝑊1 ] ∙ ∑(𝑘 ∙ 𝑃(𝑁(𝑡) = 𝑘) 𝑘=1

= 𝐸 [𝑊1 ]𝐸 [𝑁(𝑡)]. (2) The proof for (3.42) is similar. Note that 𝑁(𝑡)

2

𝑉𝑎𝑟[𝑅(𝑡)] = 𝐸 [(∑ 𝑊𝑖 ) ] − (𝐸[𝑅(𝑡)])2 ,

(3.43)

𝑖=1

where 2

𝑁(𝑡)

2

𝑁(𝑡)=𝑘

∞

𝐸 [(∑ 𝑊𝑖 ) ] = ∑ 𝐸 [( ∑ 𝑊𝑖 ) | 𝑁(𝑡) = 𝑘] ∙ 𝑃(𝑁(𝑡) = 𝑘). 𝑖=1

𝑘=1

𝑖=1

𝑊𝑖 (i = 1,2 … . ) is independent of 𝑁(𝑡) and so 𝑁(𝑡)

2

∞

𝑘

𝑘

𝐸 [(∑ 𝑊𝑖 ) ] = ∑ (∑ ∑ 𝐸[𝑊𝑖 𝑊𝑗 ]) ∙ 𝑃(𝑁(𝑡) = 𝑘). 𝑖=1

𝑘=1

𝑖=1 𝑗=1

Furthermore, as 𝑊𝑖 (i = 1,2 … . ) is a sequence of i.i.d. non-negative random variables, we obtain

57

𝑁(𝑡)

2

∞

𝐸 [(∑ 𝑊𝑖 ) ] = ∑ (𝑘𝐸[𝑊12 ] + 𝑘(𝑘 − 1)(𝐸[𝑊1 ])2 ) ∙ 𝑃(𝑁(𝑡) = 𝑘) 𝑖=1

𝑘=1 ∞

= ∑ (𝑘𝑉𝑎𝑟[𝑊1 ] + 𝑘 2 (𝐸 [𝑊1 ])2 ) ∙ 𝑃(𝑁(𝑡) = 𝑘) 𝑘=1 ∞

∞

= 𝑉𝑎𝑟[𝑊1 ] ∑ 𝑘 ∙ 𝑃(𝑁(𝑡) = 𝑘) + (𝐸 [𝑊1

])2

𝑘=1

∑ 𝑘 2 ∙ 𝑃 (𝑁 (𝑡 ) = 𝑘 ) 𝑘=1

= 𝑉𝑎𝑟[𝑊1 ]𝐸 [𝑁(𝑡)] + (𝐸 [𝑊1 ])2 𝐸 [𝑁(𝑡)2 ] = 𝑉𝑎𝑟[𝑊1 ]𝐸 [𝑁(𝑡)] + (𝐸 [𝑊1 ])2 (𝑉𝑎𝑟[𝑁(𝑡)] + (𝐸 [𝑁(𝑡)])2 ). Substituting the above equation and (3.41) into (3.43) yields 𝑉𝑎𝑟[𝑅(𝑡)] = 𝑉𝑎𝑟[𝑊1 ]𝐸 [𝑁(𝑡)] + (𝐸 [𝑊1 ])2 (𝑉𝑎𝑟[𝑁(𝑡)] + (𝐸 [𝑁(𝑡)])2 ) − (𝐸 [𝑊1 ]𝐸 [𝑁(𝑡)])2 = 𝑉𝑎𝑟[𝑊1 ]𝐸 [𝑁(𝑡)] + (𝐸 [𝑊1 ])2 𝑉𝑎𝑟[𝑁(𝑡)]. Note that {𝑁(𝑡), 𝑡 > 0} is NHPP. (3.39) and (3.40) imply 𝑉𝑎𝑟[𝑅(𝑡)] = (𝑉𝑎𝑟[𝑊1 ] + (𝐸[𝑊1 ])2 )𝑚(𝑡) = 𝐸 [𝑊12 ]𝑚(𝑡). ∎

Remark 3.4 In Lemma 3.2, the i.i.d. non-negative random variables 𝑊𝑖 (i = 1,2 … . ) correspond to the repair cost of the i’th failure in the aforementioned cost model. The conditions (a) and (b) only require that the repair costs are non-negative and independent of each other as well as the failure counting process {𝑁(𝑡), 𝑡 ≥ 0}. These are fairly loose restrictions and can usually be satisfied. If one applies Lemma 3.2 to compute the warranty cost, it is straightforward that the expected repair costs during the BW and the EWC are 𝐶𝐵𝑊 (𝜏) = 𝐸 [𝑊1 ]𝑚(𝜏);

58

(3.44)

𝐶𝐸𝑊𝐶 (𝜏, 𝜂) = 𝐸 [𝑊1 ](𝑚(𝜏 + 𝜂) − 𝑚(𝜏)).

(3.45)

respectively, and the corresponding variances are 𝑉𝐵𝑊 (𝜏) = 𝐸 [𝑊12 ]𝑚(𝜏);

(3.46)

𝑉𝐸𝑊𝐶 (𝜏, 𝜂) = 𝐸 [𝑊12 ](𝑚(𝜏 + 𝜂) − 𝑚(𝜏)).

(3.47)

Remark 3.5 The equations (3.39) and (3.44)-(3.45) provide the number of the expected warranty claims and the expected warranty cost, which are good measures of the overall cost of warranty and directly affect the expected total profit and the pricing strategy presented in Section 3.3. However, (3.40) and (3.46)-(3.47) are supplementary results which provide information about the risk contained in the warranty programmes. The risk measures are helpful in warranty management but these are beyond the research scope of this study.

Remark 3.6 The remaining part of this study focuses on the constant failure rate, i.e., 𝜆(𝑡) = 𝜆, ∀𝑡 ≥ 0. The results for the general intensity function can be derived with a similar procedure. In addition, let 𝐸 [𝑊1 ] = 𝐶𝑟 simplify the notation, then 𝐶𝐵𝑊 = 𝜆𝐶𝑟 𝜏 and 𝐶𝐸𝑊𝐶 = 𝜆𝐶𝑟 𝜂.

As described in Section 3.2.2, longer warranty length (𝜏 and 𝜂) can promote the sales volume (both of the new product and the EWC). However, longer warranty length always increases the expected warranty cost as shown in the above analysis. The remaining part of this section studies how the time-independent parameters 𝜏 and 𝜂 affect the total profit 𝑇

𝐽 = ∫ 𝑓 (𝑋, 𝑝1 , 𝜏)[𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂)] 𝑑𝑡. 0

Note that the optimal solution (𝑋, 𝑌, 𝑝1 , 𝑝̅2 ), the costate function 𝜃1 and the corresponding total profit 𝐽 all depend implicitly on the parameters 𝜏 and 𝜂. In order to study the impact of 𝜏 and 𝜂 59

on these quantities, they are clearly denoted as 𝑋(𝑡; 𝜏, 𝜂), 𝑌(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝑝̅2 (𝜏, 𝜂), 𝜃1 (𝑡; 𝜏, 𝜂) and 𝐽(𝜏, 𝜂) . The result of the parameter analysis is summarised in the following proposition.

Proposition 3.4 Let (𝑋(𝑡; 𝜏, 𝜂), 𝑌(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝑝̅2 (𝜏, 𝜂)) be the optimal solution for the problem (3.5)-(3.10), and 𝜃1 (𝑡; 𝜏, 𝜂) is the corresponding costate function. If (𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝑝̅2 (𝜏, 𝜂)) are continuously differentiable functions with respect to the parameters 𝜏 and 𝜂, then 𝑇

𝐽𝜏 (𝜏, 𝜂) = − ∫ 𝑓 (𝑋 (𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏) [𝜆𝐶𝑟 + 0

𝑓𝜏 (𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏) ] 𝑑𝑡 ; 𝑓𝑝1 (𝑋 (𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏)

𝐽𝜂 (𝜏, 𝜂) = 𝐸𝜂 (𝑝̅2 (𝜏, 𝜂), 𝜂)𝑋 (𝑇; 𝜏, 𝜂). Proof: The derivative of 𝐽(𝜏, 𝜂) with respect to 𝜏 is 𝑇

𝐽𝜏 (𝜏, 𝜂) = ∫ {(𝑓𝜏 (𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏) + 𝑓𝑋 (𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏)𝑋𝜏 (𝑡; 𝜏, 𝜂) 0

+ 𝑓𝑝1 (𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏)𝑝1 𝜏 (𝑡; 𝜏, 𝜂)) [𝑝1 (𝑡; 𝜏, 𝜂) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 (𝜏, 𝜂), 𝜂)] + 𝑓 (𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏)[𝑝1 𝜏 (𝑡; 𝜏, 𝜂) − 𝜆𝐶𝑟 ]} 𝑑𝑡. Because (𝑋(𝑡; 𝜏, 𝜂), 𝑌(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝑝̅2 (𝜏, 𝜂)) is the optimal solution, it must satisfy the state equations and boundary conditions for all (𝜏, 𝜂), thereby implying the equations 𝑓 (𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏) − 𝑋̇(𝑡; 𝜏, 𝜂) = 0; 𝑋 (0; 𝜏, 𝜂) = 0. Differentiating the two equations with respect to 𝜏 yields 𝑓𝑋 (𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏)𝑋𝜏 (𝑡; 𝜏, 𝜂) + 𝑓𝑝1 (𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏)𝑝1 𝜏 (𝑡; 𝜏, 𝜂) + 𝑓𝜏 (𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏) − 𝑋̇𝜏 (𝑡; 𝜏, 𝜂) = 0;

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𝑋𝜏 (0; 𝜏, 𝜂) = 0. Multiply the first equation by 𝜃1 (𝑡; 𝜏, 𝜂), and integrate it 𝑇

∫ [𝑓𝑋 (𝑋 (𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏)𝑋𝜏 (𝑡; 𝜏, 𝜂) + 𝑓𝑝1 (𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏)𝑝1 𝜏 (𝑡; 𝜏, 𝜂) 0

+ 𝑓𝜏 (𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏) − 𝑋̇𝜏 (𝑡; 𝜏, 𝜂)]𝜃1 (𝑡; 𝜏, 𝜂)𝑑𝑡 = 0. Adding it to 𝐽𝜏 (𝜏, 𝜂) yields 𝑇

𝐽𝜏 (𝜏, 𝜂) = ∫ {(𝑓𝜏 (𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏) + 𝑓𝑋 (𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏)𝑋𝜏 (𝑡; 𝜏, 𝜂) 0

+ 𝑓𝑝1 (𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏)𝑝1 𝜏 (𝑡; 𝜏, 𝜂)) [𝑝1 (𝑡; 𝜏, 𝜂) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 (𝜏, 𝜂), 𝜂) + 𝜃1 (𝑡; 𝜏, 𝜂)] − 𝑋̇𝜏 (𝑡; 𝜏, 𝜂)𝜃1 (𝑡; 𝜏, 𝜂) + 𝑓 (𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏)[𝑝1 𝜏 (𝑡; 𝜏, 𝜂) − 𝜆𝐶𝑟 ]} 𝑑𝑡. Denote 𝐻(𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ; 𝜏, 𝜂) = 𝑓 (𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏)[𝑝1 (𝑡; 𝜏, 𝜂) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 (𝜏, 𝜂), 𝜂) + 𝜃1 (𝑡; 𝜏, 𝜂)]. Then 𝑇

𝐽𝜏 (𝜏, 𝜂) = ∫ [𝐻𝜏 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ; 𝜏, 𝜂) + 𝐻𝑋 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ; 𝜏, 𝜂)𝑋𝜏 (𝑡; 𝜏, 𝜂) 0

+ 𝐻𝑝1 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ; 𝜏, 𝜂)𝑝1 𝜏 (𝑡; 𝜏, 𝜂) − 𝑋̇𝜏 (𝑡; 𝜏, 𝜂)𝜃1 (𝑡; 𝜏, 𝜂)] 𝑑𝑡 𝑇

= ∫ [𝐻𝜏 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ; 𝜏, 𝜂) + (𝐻𝑋 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ; 𝜏, 𝜂)𝑋𝜏 − 𝑋̇𝜏 (𝑡; 𝜏, 𝜂)𝜃1 (𝑡; 𝜏, 𝜂)) 0

+ 𝐻𝑝1 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ; 𝜏, 𝜂)𝑝1 𝜏 (𝑡; 𝜏, 𝜂)] 𝑑𝑡. In addition, 𝑇

∫ 𝑋̇𝜏 (𝑡; 𝜏, 𝜂)𝜃1 (𝑡; 𝜏, 𝜂)𝑑𝑡 0

61

𝑇

𝑇 = 𝑋𝜏 (𝑡; 𝜏, 𝜂)𝜃1 (𝑡; 𝜏, 𝜂) | − ∫ 𝑋𝜏 (𝑡; 𝜏, 𝜂)𝜃̇1 (𝑡; 𝜏, 𝜂)𝑑𝑡 0 0 𝑇

= 𝑋𝜏 (𝑇; 𝜏, 𝜂)𝜃1 (𝑇; 𝜏, 𝜂) − 𝑋𝜏 (0; 𝜏, 𝜂)𝜃1 (0; 𝜏, 𝜂) − ∫ 𝑋𝜏 (𝑡; 𝜏, 𝜂)𝜃̇1 (𝑡; 𝜏, 𝜂)𝑑𝑡 0

𝑇

= − ∫ 𝑋𝜏 (𝑡; 𝜏, 𝜂)𝜃̇1 (𝑡; 𝜏, 𝜂)𝑑𝑡, 0

where the last equality follows from the boundary conditions 𝜃1 (𝑇; 𝜏, 𝜂) = 𝑋𝜏 (0; 𝜏, 𝜂) = 0 . Insert 𝜃̇1 (𝑡; 𝜏, 𝜂) = −𝐻𝑋 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ; 𝜏, 𝜂, 𝐻𝑝1 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ; 𝜏, 𝜂) = 0 and the above expression into 𝐽𝜏 (𝜏, 𝜂), we have 𝐽𝜏 (𝜏, 𝜂) 𝑇

= ∫ [𝐻𝜏 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ; 𝜏, 𝜂) + (𝐻𝑋 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ; 𝜏, 𝜂) + 𝜃̇1 (𝑡; 𝜏, 𝜂)) 𝑋𝜏 (𝑡; 𝜏, 𝜂) 0

+ 𝐻𝑝1 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ; 𝜏, 𝜂)𝑝1 𝜏 (𝑡; 𝜏, 𝜂)] 𝑑𝑡 𝑇

= ∫ 𝐻𝜏 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ; 𝜏, 𝜂)𝑑𝑡 0

𝑇

= ∫ {𝑓𝜏 (𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏)[𝑝1 (𝑡; 𝜏, 𝜂) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 (𝜏, 𝜂), 𝜂) + 𝜃1 (𝑡; 𝜏, 𝜂)] 0

− 𝑓(𝑋 (𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏)𝜆𝐶𝑟 }𝑑𝑡. According to (3.32), the first-order condition of 𝐻(𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ; 𝜏, 𝜂) implies 𝑝1 (𝑡; 𝜏, 𝜂) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 (𝜏, 𝜂), 𝜂) + 𝜃1 (𝑡; 𝜏, 𝜂) = −

𝑓(𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏) , 𝑓𝑝1 (𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏)

and so 𝑇

𝐽𝜏 (𝜏, 𝜂) = ∫ {𝑓𝜏 (𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏) (− 0

𝑓(𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏) ) 𝑓𝑝1 (𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏)

− 𝑓 (𝑋 (𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏)𝜆𝐶𝑟 } 𝑑𝑡 𝑇

= − ∫ 𝑓 (𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏) [𝜆𝐶𝑟 + 0

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𝑓𝜏 (𝑋 (𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏) ] 𝑑𝑡. 𝑓𝑝1 (𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏)

Similarly, 𝑇

𝐽𝜂 (𝜏, 𝜂) = ∫ 𝐻𝜂 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ; 𝜏, 𝜂)𝑑𝑡 0

𝑇

= ∫ 𝑓 (𝑋(𝑡; 𝜏, 𝜂), 𝑝1 (𝑡; 𝜏, 𝜂), 𝜏)𝐸𝜂 (𝑝̅2 (𝜏, 𝜂), 𝜂)𝑑𝑡 0

𝑇

= 𝐸𝜂 (𝑝̅2 (𝜏, 𝜂), 𝜂) ∫ 𝑋̇(𝑡; 𝜏, 𝜂)𝑑𝑡 0

= 𝐸𝜂 (𝑝̅2 (𝜏, 𝜂), 𝜂)𝑋 (𝑇; 𝜏, 𝜂). ∎ The observations of Proposition 3.4 are as follows: 

If 𝐽𝜏 (𝜏, 𝜂) > 0, the manufacturer can increase the total profit by extending the BW length 𝜏.



If 𝐽𝜏 (𝜏, 𝜂) < 0, the manufacturer can increase the total profit via cutting down the BW length 𝜏.



If 𝐽𝜏 (𝜏, 𝜂) = 0, the total profit has reached a local maximum or a local minimum. If it is a local maximum, a small adjustment of the BW length only decreases the total profit. If it is a local minimum, it is better to change the BW length so as to increase the total profit.

For the manufacturer, the best value of the BW length 𝜏 is the one which achieves the maximum value of 𝐽(𝜏, 𝜂). This task, however, is usually difficult as we do not know the closed-form of 𝑋(𝑡; 𝜏, 𝜂) and 𝑝1 (𝑡; 𝜏, 𝜂). In practice, the manufacturer can set the BW length 𝜏 to be half year, one year, one and a half year, etc. After that, the expected total profits under different value of 𝜏 can be compared to find the best BW length. In the following numerical experiments, details of this method will be discussed. The discussion of 𝐽𝜂 (𝜏, 𝜂) is analogous and thus omitted.

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3.5 Case Study As discussed in Remark 3.2, the optimal price for the EWC is a constant function 𝑝̅2 . The challenge is to derive the optimal solution 𝑝1 (𝑡) for the new product, i.e., to find the triple (𝑋(𝑡), 𝑝1 (𝑡), 𝜃1 (𝑡)) satisfying (3.31). This section tackles this challenge under two special types of the demand rate function 𝑓 (𝑋, 𝑝1 , 𝜏).

3.5.1 Static Market In a static market, the demand rate is independent of the cumulative sales volume. In this case, the demand is just a function of 𝑝1 and 𝜏, i.e., 𝑓𝑋 (𝑋, 𝑝1 , 𝜏) ≡ 0 (or 𝑓(𝑋, 𝑝1 , 𝜏) = ℎ(𝑝1 , 𝜏)). This assumption is reasonable when the market is sufficiently large and the consumers know the product very well, i.e., when both the diffusion and saturation effects are not important. Suppose (𝑝1 , 𝑝̅2 ) is the optimal price for the dynamic optimization problem (3.5)-(3.10). Through carefully reviewing Theorem 3.1, it is not difficult to notice that the costate function 𝜃1 would degenerate into 𝜃1 ≡ 0, ∀𝑡 ∈ [0, 𝑇] and the state constraint (3.48) has no influence on the objective function. Therefore, the Hamiltonian function H is degenerate and equals the integrand of the objective function: 𝐻(𝑝1 , 𝑝2 ) = ℎ(𝑝1 , 𝜏)[𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝2 , 𝜂)]. Then the optimal price 𝑝1 must satisfy: 𝑝1 ≡ 𝐴𝑟𝑔𝑚𝑎𝑥 𝐻 (𝑝1 , 𝑝̅2 ) 𝑝1 ∈𝑈1

= 𝐴𝑟𝑔𝑚𝑎𝑥 ℎ(𝑝1 , 𝜏)[𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂)]; ∀𝑡 ∈ [0, 𝑇]. 𝑝1 ∈𝑈1

Obviously, this is also the sufficient condition for 𝑝1 being optimal.

Example 3.1 Additive price-warranty demand function (Desai et al., 2004; Kim et al., 2008; Li et al., 2012)

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ℎ(𝑝1 , 𝜏) = 𝑎0 − 𝑎1 𝑝1 + 𝑎2 𝜏. where 𝑎0 (> 0) is the potential demand when the price 𝑝1 and the BW length 𝜏 are zero; 𝑎1 (≥ 0) measures the price sensitivity in the demand function; 𝑎2 (≥ 0) measures the BW sensitivity in the demand function; 𝑈1 = (0,

𝑎0 +𝑎2𝜏 𝑎1

) is the admissible region for 𝑝1 .

Solution: As discussed in the beginning of Section 3.5.1, note that 𝐻(𝑝1 , 𝑝̅2 ) = (𝑎0 − 𝑎1 𝑝1 + 𝑎2 𝜏)[𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂)] = −𝑎1 𝑝12 + [(𝑎0 + 𝑎2 𝜏) + 𝑎1 (𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂))]𝑝1 − (𝑎0 + 𝑎2 𝜏)(𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂)). This is a quadratic function and the feasible region is the open interval 𝑈1 = (0,

𝑎0 +𝑎2𝜏 𝑎1

).

Therefore, the optimal solution can be obtained as follows: If (𝑎0 + 𝑎2 𝜏) + 𝑎1 (𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂)) > 0, the optimal price 𝑝1 𝑝1 =

(𝑎0 + 𝑎2 𝜏) + 𝑎1 (𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂)) . 2𝑎1

Otherwise, (𝑎0 + 𝑎2 𝜏) + 𝑎1 (𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂)) ≤ 0 and 𝐻(𝑝1 , 𝑝̅2 ) can never reach the maximum if 𝑝1 ∈ (0,

𝑎0 +𝑎2 𝜏 𝑎1

). Therefore, no optimal price 𝑝1 exists at this time. (Note: The

discussion in the beginning of Section 3.5.1 is based on the existence of the optimal solution, but it does not provide the condition under which an optimal solution exists)

Parameter analysis: (a) When the expected profit 𝐸 (𝑝̅2 , 𝜂) from selling the EWC is high, the manufacturer lowers the price of the new product to increase the sales volume, because the profit from selling the EWC can cover the loss of selling the new product, and the optimal price 𝑝1 maximizes the

65

total profit. The situation 𝐸 (𝑝̅2 , 𝜂) ≥

𝑎0 +𝑎2 𝜏 𝑎1

+ (𝐶𝑚 + 𝐶𝐵𝑊 ) means that the expected profit

from the EWC is much greater than the marginal cost 𝐶𝑚 + 𝐶𝐵𝑊 , so it is unlikely to have such a situation in practice. However, the situation of 𝑝1 ≤ 𝐶𝑚 + 𝐶𝐵𝑊 (the selling price of new product is lower than the marginal cost) may happen, and the condition is (𝑎0 + 𝑎2 𝜏) + 𝑎1 (𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂)) ≤ 𝐶𝑚 + 𝐶𝐵𝑊 , 2𝑎1 i.e., 𝐸 (𝑝̅2 , 𝜂) ≥

(b) Substituting the optimal price 𝑝1 =

𝑎0 + 𝑎2 𝜏 − (𝐶𝑚 + 𝐶𝐵𝑊 ). 𝑎1

(𝑎0 +𝑎2 𝜏)+𝑎1 (𝐶𝑚 +𝐶𝐵𝑊 −𝐸(𝑝̅2,𝜂)) 2𝑎1

into the objective function

yields: 2

[(𝑎0 + 𝑎2 𝜏) − 𝑎1 (𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂))] 𝐽(𝜏, 𝜂) = 𝑇. 4𝑎1 𝐽(𝜏, 𝜂) is concave in 𝜏, and 𝐽𝜏 (𝜏, 𝜂) = − [𝜆𝐶𝑟 −

𝑎2 ] 𝑋 (𝑇 ). 𝑎1

These results are consistent with Proposition 3.4. As 𝑋 (𝑇) > 0, we have 𝑎



If 𝜆𝐶𝑟 > 𝑎2 , then 𝐽𝜏 (𝜏, 𝜂) < 0, i.e., the shorter 𝜏, the higher 𝐽(𝜏, 𝜂).



If 𝜆𝐶𝑟 < 𝑎2 , then 𝐽𝜏 (𝜏, 𝜂) > 0, i.e., the longer 𝜏, the higher 𝐽(𝜏, 𝜂).



If 𝜆𝐶𝑟 = 𝑎2 , then 𝐽𝜏 (𝜏, 𝜂) = 0, i.e., 𝐽(𝜏, 𝜂) is independent of 𝜏.

1

𝑎

1

𝑎

1

(c) Usually, increasing the BW length 𝜏 can promote the demand, but the warranty cost also increases. Therefore, the optimal price increases in 𝜏, and the increasing rate is a constant

66

𝜕𝑝1 𝑎2 + 𝑎1 𝜆𝐶𝑟 = > 0. 𝜕𝜏 2𝑎1

Example 3.2 Glickman-Berger demand function (Glickman et al., 1976; Lin et al., 2009; Manna, 2008; Wu et al., 2009) ℎ(𝑝1 , 𝜏) = 𝑘1 [𝑘2 + 𝜏]𝑏 𝑝1−𝑎 , where 𝑝1 ∈ (0, +∞), 𝑘1 > 0, 𝑘2 ≥ 0, 𝑎 > 1 and 0 < 𝑏 < 1 . The constant 𝑘1 is an amplitude factor and 𝑘2 is a constant of time displacement which ensures nonzero demand when 𝜏 is zero. The parameters 𝑎 and 𝑏 can be interpreted as the price elasticity and basic warranty length elasticity, respectively. Solution: Consider 𝐻 (𝑝1 , 𝑝̅2 ) = 𝑘1 [𝑘2 + 𝜏]𝑏 𝑝1−𝑎 [𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂)]. An optimal price p1 exists only when Cm + 𝐶𝐵𝑊 > 𝐸 (𝑝̅2 , 𝜂). Otherwise, lim 𝐻(𝑝1 , 𝑝̅2 ) = +∞.

𝑝1 ↓0

Suppose that 𝐶𝑚 + 𝐶𝐵𝑊 > 𝐸 (𝑝̅2 , 𝜂). Given 𝑝̅2 , the Hamiltonian H as a function of 𝑝1 achieve its maximum at 𝐻𝑝1 (𝑝1 , 𝑝̅2 ) = 𝑘1 (𝑘2 + 𝜏)𝑏 𝑝1−𝑎 [

𝑎−1 𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂)] = 0. 𝑎

Therefore, the optimal price 𝑝1 must be 𝑝1 =

𝑎 [𝐶 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂)]. 𝑎−1 𝑚

Obviously, the optimal price increases with respect to 𝜏, and the increasing rate is a constant

67

𝜕𝑝1 𝑎𝜆𝐶𝑟 = > 0. 𝜕𝜏 𝑎−1 Parameter analysis Substituting 𝑝1 into the objective function yields the total profit −𝑎+1 𝑘1 [𝑘2 + 𝜏]𝑏 𝑎 [ 𝐽(𝜏, 𝜂) = [𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂)]] 𝑇. 𝑎 𝑎−1

The derivative of 𝐽(𝜏, 𝜂) with respect to 𝜏 is 𝐽𝜏 (𝜏, 𝜂) = [ =

𝑏 [𝐶 + 𝜆𝐶𝑟 𝜏 − 𝐸 (𝑝̅2 , 𝜂)] − 𝜆𝐶𝑟 ] 𝑋 (𝑇) (𝑘2 + 𝜏)(𝑎 − 1) 𝑚

(𝑏 + 1 − 𝑎)𝜆𝐶𝑟 𝜏 + (𝐶𝑚 − 𝐸 (𝑝̅2 , 𝜂))𝑏 − (𝑎 − 1)𝜆𝐶𝑟 𝑘2 𝑋 (𝑇 ), (𝑘2 + 𝜏)(𝑎 − 1)

which is coincident with Proposition 3.4. Let 𝑀1 = (𝑏 + 1 − 𝑎)𝜆𝐶𝑟 and 𝑀2 = (𝐶𝑚 − 𝐸 (𝑝̅2 , 𝜂))𝑏 − (𝑎 − 1)𝜆𝐶𝑟 𝑘2 . Obviously, the signs of 𝑀1 and 𝑀2 determine the sign of 𝐽𝜏 (𝜏, 𝜂): 

If 𝑀1 > 0 and 𝑀2 > 0, then 𝐽𝜏 (𝜏, 𝜂) > 0 for all 𝜏 ≥ 0, i.e., the manufacturer is better to set a longer BW length.



𝑀

If 𝑀1 < 0 and 𝑀2 > 0, then 𝐽𝜏 (𝜏, 𝜂) = 0 at 𝜏 = − 𝑀2 > 0. In addition, 𝐽𝜏 (𝜏, 𝜂) > 0 when 1

𝑀

𝑀

𝑀

0 ≤ 𝜏 < − 𝑀2 ; 𝐽𝜏 (𝜏, 𝜂) < 0 when 𝜏 > − 𝑀2 . It implies that 𝜏 = − 𝑀2 is the optimal choice for 1

1

1

the manufacturer. 

If 𝑀1 < 0 and 𝑀2 < 0, then 𝐽𝜏 (𝜏, 𝜂) < 0 for all 𝜏 ≥ 0, i.e., the manufacturer is better to set the BW length is zero.



𝑀

If 𝑀1 > 0 and 𝑀2 < 0, then 𝐽𝜏 (𝜏, 𝜂) = 0 at 𝜏 = − 𝑀2 > 0. In addition, 𝐽𝜏 (𝜏, 𝜂) < 0 when 1

𝑀

𝑀

𝑀

0 ≤ 𝜏 < − 𝑀2 ; 𝐽𝜏 (𝜏, 𝜂) > 0 when 𝜏 > − 𝑀2 . It implies that 𝜏 = − 𝑀2 is the minimum point of 1

1

1

𝐽𝜏 (𝜏, 𝜂). The manufacturer is better to set the BW length far from the value 𝜏 = −

68

𝑀2 𝑀1

.

The other cases, 𝑀1 = 0 or 𝑀2 = 0, are similar and thus will not be explicitly discussed here.

3.5.2 Dynamic Market The pricing of durable products is usually studied under a dynamic market environment, in which the demand exhibits diffusion and saturation effects. In this case, current demand affects future demand, and hence the demand is a function of price, warranty length, and cumulative sales (Faridimehr et al., 2013). This study mainly focuses on a dynamic market where the demand rate function is multiplicatively separable in 𝑋 and 𝑝1 , that is 𝑓(𝑋, 𝑝1 , 𝜏) = 𝑔(𝑋)ℎ(𝑝1 , 𝜏). This type of demand function is fairly standard in the literature (Huang et al., 2007; Jørgensen et al., 1999; Jørgensen and Zaccour, 2007). Here 𝑔(𝑋) represents the diffusion process of how the new product gets adopted in the market. Similar to the Bass model, it reflects the behaviours of consumers who are classified as innovators or imitators. The other term ℎ(𝑝1 , 𝜏) captures the effect of price and the BW on the sales rate, and it is assumed to satisfy ℎ(𝑝1 , 𝜏) > 0, ℎ𝑝1 (𝑝1 , 𝜏) < 0, ℎ𝜏 (𝑝1 , 𝜏) > 0 for any 𝑝1 ∈ 𝑈1 and 𝑡 ∈ [0, 𝑇]. Section 3.3 has derived some conclusions about the optimal pricing strategy 𝑝1 under general demand function 𝑓 (𝑋, 𝑝1 , 𝜏) . While these conclusions are useful in gaining insights on the properties of 𝑝1 , there are stronger results when the demand rate function is multiplicatively separable in 𝑋 and 𝑝1 . One can refer to the following Corollary 3.1 and 3.2.

Corollary 3.1 Consider the demand rate function 𝑓 (𝑋, 𝑝1 , 𝜏) = 𝑔(𝑋)ℎ(𝑝1 , 𝜏) . Suppose that (𝑋 ∗ , 𝑌 ∗ , 𝑝1∗ , 𝑝̅2 ) satisfy the necessary conditions in Theorem 3.1 for the dynamic optimization problem (3.5)-(3.10) with the corresponding costate function 𝜃1 . If 𝑈1 = (0, +∞) and 𝑔(𝑋) is concave function of X in [0, +∞), then (𝑋 ∗ , 𝑌 ∗ , 𝑝1∗ , 𝑝̅2 ) yields the global maximum of 𝐽.

69

Proof: Similar to the proof of Theorem 3.2, consider an arbitrary feasible solution (𝑋, 𝑝1 , 𝑝2 , 𝜃1 ) for the dynamic problem (3.5)-(3.10). Then (𝑋 ∗ , 𝑌 ∗ , 𝑝1∗ , 𝑝̅2 ) is optimal provided the following inequality is always true 𝑻

∆ = ∫ {[𝐻(𝑋 ∗ , 𝑝1∗ , 𝑝̅2 , 𝜃1 ) − 𝜃1 𝑋̇ ∗ ] − [𝐻 (𝑋, 𝑝1 , 𝑝2 , 𝜃1 ) − 𝜃1 𝑋̇ ]}𝑑𝑡 𝟎

𝑻

𝑻

= ∫ [𝐻 (𝑋 ∗ , 𝑝1∗ , 𝑝̅2 , 𝜃1 ) − 𝐻 (𝑋, 𝑝1 , 𝑝2 , 𝜃1 )] 𝑑𝑡 + ∫ (𝑋̇ − 𝑋̇ ∗ )𝜃1 𝑑𝑡 𝟎

𝟎

≥ 0. For any feasible pricing strategy {𝑝1 ∈ 𝑈1 , 𝑡 ∈ [0, 𝑇]} , 𝑓(𝑋, 𝑝1 , 𝜏) = 𝑔(𝑋)ℎ(𝑝1 , 𝜏) > 0 and ℎ(𝑝1 , 𝜏) > 0 imply 𝑔(𝑋) > 0 , where 𝑋 is the solution of the state equations 𝑋̇(𝑡) = 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏) and 𝑋(0) = 0. Then 𝑝1∗ = Argmax 𝐻 (𝑋 ∗ , 𝑝1 , 𝑝̅2 , 𝜃1 ) 𝑝1 ∈𝑈1

= Argmax 𝑔(𝑋 ∗ )ℎ(𝑝1 , 𝜏)[𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 ] 𝑝1 ∈𝑈1

= Argmax ℎ(𝑝1 , 𝜏)[𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 ], 𝑝1 ∈𝑈1

and 𝐻 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ) = 𝑔(𝑋)ℎ(𝑝1 , 𝜏)[𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 ] ≤ 𝑔(𝑋)ℎ(𝑝1∗ , 𝜏)[𝑝1∗ − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 ] = 𝐻 (𝑋, 𝑝1∗ , 𝑝̅2 , 𝜃1 ). 𝜃1 is a finite piecewise differentiable function on [0, 𝑇] . If 𝑝1 is sufficiently large, then max𝑝1 ∈𝑈1 ℎ(𝑝1 , 𝜏)[𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 ] must be positive. Moreover, 𝑔(𝑋) is concave in 𝑋 and so

70

𝐻(𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ) − 𝐻(𝑋 ∗ , 𝑝1∗ , 𝑝̅2 , 𝜃1 ) ≤ 𝐻(𝑋, 𝑝1∗ , 𝑝̅2 , 𝜃1 ) − 𝐻(𝑋 ∗ , 𝑝1 , 𝑝̅2 , 𝜃1 ) = [𝑔(𝑋) − 𝑔(𝑋 ∗ )]ℎ(𝑝1∗ , 𝜏)[𝑝1∗ − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸(𝑝̅2 , 𝜂) + 𝜃1 ] ≤ 𝑔′ (𝑋 ∗ )(𝑋 − 𝑋 ∗ )ℎ(𝑝1∗ , 𝜏)[𝑝1∗ − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸(𝑝̅2 , 𝜂) + 𝜃1 ] = 𝐻𝑋 (𝑋 ∗ , 𝑝1∗ , 𝑝̅2 , 𝜃1 )(𝑋 − 𝑋 ∗ ) = −𝜃̇1 (𝑋 − 𝑋 ∗ ).

(3.49)

Substituting (3.49) into ∆ yields 𝑻

𝑻

𝑇 ∆≥ ∫ −𝜃̇1 (𝑋 − 𝑋 ∗ ) 𝑑𝑡 + ∫ 𝜃1 (𝑋̇ − 𝑋̇ ∗ ) 𝑑𝑡 = 𝜃1 (𝑋 − 𝑋 ∗ ) | = 0. 0 𝟎 𝟎 The last equality can be derived by the boundary conditions 𝜃1 (𝑇) = 0 and 𝑋 (0) = 𝑋 ∗ (0) = 0. ∎

Corollary 3.2 Consider the demand rate function 𝑓 (𝑋, 𝑝1 , 𝜏) = 𝑔(𝑋)ℎ(𝑝1 , 𝜏). If ℎ(𝑝1 , 𝜏) has a second-order partial derivative with respect to 𝑝1 , then the optimal price path 𝑝1 is smooth over time and its derivative satisfies (2ℎ𝑝21 − ℎℎ𝑝1 𝑝1 )𝑝̇1 = −𝑔̇ (𝑋)ℎℎ𝑝1 , with the following boundary condition at time 𝑇 𝑝1 (𝑇) +

ℎ(𝑝1 (𝑇), 𝜏) = 𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂). ℎ𝑝1 (𝑝1 (𝑇), 𝜏)

Because −ℎℎ𝑝1 > 0 and 2ℎ𝑝21 − ℎℎ𝑝1 𝑝1 ≥ 0 , the sign of 𝑝̇1 is determined by 𝑔̇ (𝑋) . More specifically,

71

𝑝1 is increasing, if 𝑔̇ (𝑋) > 0; {𝑝1 is constant,

if 𝑔̇ (𝑋) = 0;

𝑝1 is decreasing, if 𝑔̇ (𝑋) < 0. Proof: Follow the proof procedure of Proposition 3.3 ∎ Note that Corollary 3.1 provides the sufficient condition for the optimal solution, and this condition is easier to check comparing with the condition of Theorem 3.2. In addition, Corollary 3.2 describes the pattern of the optimal price 𝑝1 over the planning horizon [0, 𝑇]. To illustrate these mathematical results, the following subsection discusses two examples.

Example 3.3 Limited-growth demand function (Aoki et al., 2002; Lin et al., 2009) 𝑓 (𝑋, 𝑝1 , 𝜏) = 𝑘1 [𝑘2 + 𝜏]𝑏 𝑝1−𝑎 [𝑀 − 𝑋], where 𝑀 is the population size of potential consumers; 𝑀 − 𝑋(𝑡) represents the saturation effect; the term 𝑘1 [𝑘2 + 𝜏]𝑏 𝑝1−𝑎 captures the effect of price and warranty on the sales rate and coincides with the Glickman-Berger demand function introduced in Example 3.2. Solution: The Hamiltonian for this example is 𝐻 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ) = [𝑀 − 𝑋]𝑘1 (𝑘2 + 𝜏)𝑏 𝑝1−𝑎 [𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 ]. The first equation in (3.31) indicates that the optimal price 𝑝1 ∈ (0, +∞), ∀𝑡 ∈ [0, 𝑇] must maximize the Hamiltonian 𝐻 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ) for any given (𝑋, 𝑝̅2 , 𝜃1 ) at any time 𝑡 ∈ [0, 𝑇] . Besides, the state equation (3.6) implies 𝑋 (𝑡) ∈ [0, 𝑀), ∀𝑡 ∈ [0, 𝑇]. Therefore, 𝑝1 maximizes the Hamiltonian 𝐻 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ) for any given (𝑋, 𝑝̅2 , 𝜃1 ) if and only if 𝑝1 maximizes 𝑘1 (𝑘2 + 𝜏)𝑏 𝑝1−𝑎 [𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 ] for the given (𝑝̅2 , 𝜃1 ).

72

Similar to Example 3.2, the optimal price 𝑝1 satisfies the first-order condition of the Hamiltonian H, that is 𝐻𝑝1 = [𝑀 − 𝑋]𝑘1 (𝑘2 + 𝜏)𝑏 𝑝1−𝑎 [

𝑎−1 𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 ] = 0. 𝑎

The state equations 𝑋̇ = 𝑓 (𝑋, 𝑝1 , 𝜏) and 𝑋 (0) = 0 imply 𝑀 > 𝑋 for any feasible price 𝑝1 . Therefore, 𝐻𝑝1 = 0 yields 𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 =

1 𝑝. 𝑎 1

Replacing the first equation in (3.31) with this first-order condition of the Hamiltonian H yields 𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 = 𝑋̇ = [𝑀 − 𝑋]𝑘1 (𝑘2 + 𝜏)𝑏 𝑝1−𝑎 ,

1 𝑝 ; 𝑎 1

𝑋 (0) = 0;

{𝜃1̇ = 𝑘1 (𝑘2 + 𝜏)𝑏 𝑝1−𝑎 [𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 ],

𝜃 (𝑇) = 0.

Substituting the first-order condition of the Hamiltonian H into the costate equation of 𝜃1 yields 𝜃1̇ =

𝑘1 (𝑘2 + 𝜏)𝑏 𝑝1−𝑎+1 . 𝑎

Take the derivative with respect to t on both sides of the first-order condition 𝜃1̇ = −

𝑎−1 𝑝̇1 . 𝑎

Combining the above two differentials yields 𝑝̇1 =

𝑘1 (𝑘2 + 𝜏)𝑏 −𝑎+1 𝑝1 . 1−𝑎

Integrating this differential equation yields 1

𝑎 𝑎𝑘1 (𝑘2 + 𝜏)𝑏 𝑝1 (𝑡) = [ (𝐶1 − 𝑡)] , 𝑎−1

73

where 𝐶1 is a constant. Next, obtain the value of 𝐶1 from the end time constraint of 𝜃1 . Consider the first-order condition at time t = T: 𝑝1 (𝑇) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 (𝑇) =

1 𝑝 (𝑇 ). 𝑎 1

Substituting 𝜃 (𝑇) = 0 and the closed-form 𝑝1 (𝑡) into the above equation yields 1

𝑎 𝑎𝑘1 (𝑘2 + 𝜏)𝑏 𝑎 [ (𝐶 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂)), (𝐶1 − 𝑇)] = 𝑎−1 𝑎−1 𝑚

i.e., 𝑎 − 1 1−𝑎 [𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂)]𝑎 𝐶1 = ( ) + 𝑇. 𝑎 𝑘1 (𝑘2 + 𝜏)𝑏 The closed-form of costate function 𝜃1 (𝑡) follows directly from the first-order condition: 𝜃1 (𝑡) =

1−𝑎 𝑝1 (𝑡) + 𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂). 𝑎

To derive the sales volume 𝑋(𝑡), notice that 𝑋̇ = [𝑀 − 𝑋]𝑘1 (𝑘2 + 𝜏)𝑏 𝑝1−𝑎 , i.e., 𝑑𝑋 = [𝑀 − 𝑋]𝑘1 (𝑘2 + 𝜏)𝑏 𝑝1−𝑎 , 𝑑𝑡 which is equivalent to 𝑑[𝑀 − 𝑋] = 𝑑𝑙𝑛(𝑀 − 𝑋) = −𝑘1 (𝑘2 + 𝜏)𝑏 𝑝1−𝑎 𝑑𝑡. [𝑀 − 𝑋] Hence, 𝑀 − 𝑋(𝑡) = 𝐶2 𝑒 −𝑘1 (𝑘2 +𝜏) 74

𝑏 𝑡 𝑝−𝑎 (𝑠)𝑑𝑠 ∫0 1

= 𝐶2 𝑒

𝑘1(𝑘2 +𝜏)𝑏

= 𝐶2 (1 −

𝑎−1 [ln(𝐶1 −𝑡)−𝑙𝑛𝐶1 ] 𝑎𝑘1 (𝑘2 +𝜏)𝑏

𝑡 𝑎−1 ) 𝑎 . 𝐶1

𝐶2 is a constant and can be obtained from the initial constrain 𝑋(0) = 0, that is 𝐶2 = 𝑀 − 𝑋 (0) = 𝑀. As a result, 𝑡 𝑋 (𝑡) = 𝑀 [1 − (1 − ) 𝐶1

𝑎−1 𝑎

].

Parameter analysis (a) Take the derivative of 𝑝1 (𝑡) with respect to t: 1−𝑎 𝑎

𝑘1 (𝑘2 + 𝜏)𝑏 𝑎𝑘1 (𝑘2 + 𝜏)𝑏 ( ) (𝐶1 − 𝑡)] [ 𝑝̇1 𝑡 = − 𝑎−1 𝑎−1

< 0.

It implies the optimal price 𝑝1 (𝑡) is always decreasing in the planning horizon [0, 𝑇] . This observation agrees with Corollary 3.2.

(b) Take the derivative of 𝐽(𝜏, 𝜂) with respect to 𝜏, that is 𝑇

𝐽𝜏 (𝜏, 𝜂) = − ∫ 𝑘1 [𝑘2 + 0

𝜏]𝑏 𝑝1−𝑎

𝑡 (1 − ) 𝐶1

𝑎−1 𝑎

[𝜆𝐶𝑟 −

𝑏 𝑝 ] 𝑑𝑡. 𝑎 (𝑘 2 + 𝜏 ) 1

This expression coincides with Proposition 3.4. It is possible to substitute the expression of 𝑝1 (𝑡) into the above integration, but no further property can be derived due to the complexity of the integrand. In addition, we can only determine the optimal 𝜏 numerically, which will be discussed in Section 3.6.

75

Example 3.4 Bass-type demand function (Huang et al., 2007; Ruiz-Conde et al., 2006). 𝑓 (𝑋, 𝑝1 , 𝜏) = 𝑘1 [𝑘2 + 𝜏]𝑏 𝑝1−𝑎 [𝑀 − 𝑋] [𝑞1 +

𝑞2 𝑋 ]. 𝑀

Obviously, the limited-growth demand function is a special case of the Bass-type demand function with coefficients of innovation 𝑞1 = 1 and coefficients of imitation 𝑞2 = 0. For more details of this demand function, please refer to the literature review (Section 2.2.2 and Section 2.3.1). Similar to Example 3.3, define the Hamiltonian function 𝐻 = [𝑀 − 𝑋] [𝑞1 +

𝑞2 𝑋 ] 𝑘 (𝑘 + 𝜏)𝑏 𝑝1−𝑎 [𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 ]. 𝑀 1 2

The optimal price 𝑝1 (𝑡) should be a solution to the following equation system (with boundary constraints): 𝑞2 𝑋 𝑎−1 ] 𝑘1 (𝑘2 + 𝜏)𝑏 𝑝1−𝑎 [ 𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 ] = 0; 𝑀 𝑎 𝑞2 𝑋 𝑋̇ = 𝑘1 (𝑘2 + 𝜏)𝑏 𝑝1−𝑎 [𝑀 − 𝑋] [𝑞1 + ], 𝑋(0) = 0; 𝑀 2𝑞 ̇1 = − [𝑞2 − 𝑞1 − 2 𝑋] 𝑘1 (𝑘2 + 𝜏)𝑏 𝑝1−𝑎 [𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 ], 𝜃(𝑇) = 0. 𝜃 { 𝑀 𝐻𝑝1 = [𝑀 − 𝑋] [𝑞1 +

Note that the demand function is rather complex. It is hard to derive the closed-form solution as we did in Example 3.3. However, it is possible to compute the numerical solution via the gradient method, which will be discussed in Section 3.6. In the following subsection, another approach, the phase diagram analysis method, is applied to qualitatively characterize the optimal solution of Example 3.4. This approach projects the solution to a phase space and partitions the space into regions. In each region, we can analyse whether the variables increase or decrease over time, which yields restrictions on the trajectory shape of the candidates for the optimal solution. For more details about the general phase diagram analysis, one can refer to Weber (2011).

76

Phase diagram analysis Similar to the procedure in Example 3.3 or the proof of Proposition 3.3, let us combine the firstorder condition 𝐻𝑝1 = 0 and the differential equation for the costate function 𝜃1 , then eliminate 𝜃1 and derive the following differential equation for 𝑝1 : 𝑘1 (𝑘2 + 𝜏)𝑏 𝑝1−𝑎+1 2𝑞2 𝑝̇1 = − [𝑞2 − 𝑞1 − 𝑋] , 1−𝑎 𝑀

𝑝1 (𝑇) =

𝑎(𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂)) . 𝑎−1

Focus on the following pair of state-price differential equations, which only involves 𝑋 and 𝑝1 :

𝑋̇ = [𝑀 − 𝑋] [𝑞1 +

𝑞2 𝑋 ] 𝑘 (𝑘 + 𝜏)𝑏 𝑝1−𝑎 , 𝑀 1 2

𝑋(0) = 0;

𝑘1 (𝑘2 + 𝜏)𝑏 𝑝1−𝑎+1 2𝑞2 [𝑞2 − 𝑞1 − 𝑋] , {𝑝̇1 = − 1−𝑎 𝑀

𝑝1 (𝑇) =

𝑎(𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂)) . 𝑎−1

Since the price 𝑝1 is positive and the corresponding state variable 𝑋 is nonnegative, it is suffices to consider the positive quadrant of the (𝑋, 𝑝1 ) space. The state-price differential equations can be used to partition the (𝑋, 𝑝1 ) space into regions in which the respective signs of 𝑋̇ and 𝑝̇1 are known. The first step is to find the points satisfying 𝑋̇ = 0 and 𝑝̇1 = 0 . Suppose 𝑞2 > 𝑞1 , in the positive quadrant, the curve for 𝑋̇ = 0 is the straight line 𝑋 = 𝑀, while the curve for 𝑝̇1 = 0 is the straight line 𝑋 =

𝑞2 −𝑞1 2𝑞2

𝑀 . As shown in

Figure 3.3, the two lines divide the positive quadrant of the (𝑋, 𝑝1 ) space into three regions, which are labeled Ⅰ-Ⅲ. In region Ⅰ, [𝑀 − 𝑋] [𝑞1 +

𝑞2 𝑋

In region Ⅱ, [𝑀 − 𝑋] [𝑞1 +

𝑀

] > 0 and 𝑞2 − 𝑞1 −

𝑞2 𝑋 𝑀

] > 0 and 𝑞2 − 𝑞1 −

2𝑞2 𝑀 2𝑞2 𝑀

𝑋 < 0 imply that 𝑋̇ > 0 and 𝑝̇1 > 0. 𝑋 > 0 imply that 𝑋̇ > 0 and 𝑝̇1 < 0 .

According to the state equation, the state variable 𝑋 has an upper bound 𝑀, hence region Ⅲ can be neglected.

77

Figure 3.3 The state-price phase diagram

In Figure 3.3, the signs of 𝑋̇ and 𝑝̇1 in each region are represented by two perpendicular arrows. The vertical arrow indicates an increase or a decrease in 𝑝1 , while the horizontal one refers to a change in 𝑋. Since the optimal solution must obey the pair of state-price differential equations and the boundary constraints, it must follow the direction indicated by the pair of arrows in each region. The curve between the two arrows indicates the direction of admissible trajectories in each region, with the arrows on the curve denoting the movement over time. The slope of the trajectories in the (𝑋, 𝑝1 ) space can be obtained from 𝑑𝑝1 /𝑑𝑋 = (𝑑𝑝1 /𝑑𝑡)( 𝑑𝑡/ 𝑑𝑋) = 𝑝̇1 /𝑋̇. Hence, when a trajectory goes through a locus where 𝑝̇1 = 0, its slope is zero. This information, along with the direction of trajectories in each region, allows us to obtain the shape of the trajectories. The trajectories go up and to the right in region Ⅰ. Once a trajectory reaches 𝑝̇1 = 0, i.e., it crosses the line 𝑋 =

𝑞2 −𝑞1 2𝑞2

𝑀 and enters region Ⅱ from region Ⅰ, it begins to turn

down and to the right. When the trajectories approach the line 𝑋̇ = 0, 𝑋̇ becomes arbitrarily small, so the trajectories never cross the line 𝑋̇ = 0.

78

𝑎

Next, let us focus on the boundary constraints: 𝑋(0) = 0 and 𝑝1 (𝑇) = 𝑎−1 (𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂)). As indicated in Figure 3.3, any optimal trajectory must begin at some point (initial time price 𝑝1 (0)) on the line 𝑋 = 0 and end at some point (end time sales volume 𝑋(𝑇)) on the 𝑎

line 𝑝1 = 𝑎−1 (𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂)). The course of travelling must take the specified amount of time 𝑇, so the optimal trajectory changes with time 𝑇. In Figure 3.3, trajectory (1) takes 𝑇1 to finish its course, when the time changes to 𝑇2 (< 𝑇1 ) , trajectory (1) cannot reach the line 𝑎

𝑝1 = 𝑎−1 (𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂)) at 𝑇2 . To satisfy the end time constraint, the initial price 𝑝1 (0) should be shifted down so as to increase the rate of 𝑋̇ and shorten the travel time, which can be reflected in trajectory (2). All the above discussion is based on 𝑞2 > 𝑞1 . If 𝑞2 ≤ 𝑞1 , the discussion is similar except that the region Ⅰ disappears. Furthermore, the state-price phase diagram analysis can be illustrated via plotting the vector field of the state-price equations when all the parameters in the state-price equations are given. An example is shown in Figure 3.4 (For the values of the relevant parameters, please refer to Table 3.2 in Section 3.6).

The above analysis focuses on the state-price phase diagram. It is also possible to conduct the phase diagram analysis in the state-costate plane. According to the first-order condition 𝐻𝑝1 = 0, the optimal price 𝑝1 can be expressed as an explicit function of 𝑋 and 𝜃1 : 𝑝1 =

𝑎 (𝐶 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂) − 𝜃1 ). 𝑎−1 𝑚

Substituting this expression of 𝑝1 into the state and costate function yields: 𝑋̇ = [𝑀 − 𝑋] [𝑞1 +

𝑞2 𝑋 𝑘1 (𝑘2 + 𝜏)𝑏 𝑎−𝑎 (𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂) − 𝜃1 )−𝑎 , 𝑋 (0) = 0; ] (𝑎 − 1)−𝑎 𝑀

2𝑞2 𝑘1 (𝑘2 + 𝜏)𝑏 𝑎−𝑎 ̇ (𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂) − 𝜃1 )−𝑎+1 , 𝜃1 (𝑇) = 0. 𝜃 = − [𝑞2 − 𝑞1 − 𝑋] (𝑎 − 1)−𝑎+1 { 1 𝑀

79

The phase diagram in the (𝑋, 𝜃1 ) space can be constructed in the same way as the preceding one

p

1

in the (𝑋, 𝑝1 ) space.

30 25 20 15 10 5 0

10

20

30

40

50

60

70

80

90

70

80

90

X

p

1

(a) Vector field for q1=1,q2=5

25 20 15 10 5 0

10

20

30

40

50

60

(b) Vector field for q1=1, q2=0

X

Figure 3.4 The vector field for the state-price differential equations (The straight line in (a) is 𝑋 =

𝑞2 −𝑞1 2𝑞2

𝑀, it indicates that the price 𝑝1 changes from increasing to

decreasing; the red curves in both (a) and (b) are the sample optimal trajectories given the initial price 𝑝1 (0))

80

𝑎

Note that 𝑝1 = 𝑎−1 (𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂) − 𝜃1 ) is positive, i.e., 𝜃1 falls in (−∞, 𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂)). In addition, the possible range of 𝑋 is [0, 𝑀). The next step is to partition the possible range of the (𝑋, 𝜃1 ) space into regions. Then the respective signs of 𝑋̇ and 𝜃̇1 are known in each regions. The procedure is similar to the discussion of (𝑋, 𝑝1 ) space and thus need not be explicitly discussed. When 𝑞2 > 𝑞1 , Figure 3.5 shows the resulting regions, which are labeled Ⅰ -Ⅱ.

Figure 3.5 The state-costate phase diagram

The state-costate phase diagram provides important information regarding the trajectory shape. Any trajectory should go down and to the right in region Ⅰ. When it reaches 𝜃̇1 = 0, it begins to turn up and to the right. Besides, the boundary constraints 𝑋 (0) = 0 and 𝜃1 (𝑇) = 0 imply that the optimal trajectories must begin at some point on the line 𝑋 (0) = 0 and then end at some point of the line 𝜃1 (𝑇) = 0. The course of travelling takes the exact amount of time 𝑇.

81

3.6 Numerical Study 3.6.1 Gradient Method In Section 3.3.1, the Pontryagin maximum principle was used to derive the necessary conditions of the optimal solution for the problem (3.5)-(3.10). Remark 3.2 has pointed out that the most challenging task is to find the triple (𝑋, 𝑝1 , 𝜃1 ) via (3.31): 𝑝1 = 𝐴𝑟𝑔𝑚𝑎𝑥 𝐻 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ) ; 𝑝1 ∈𝑈1

𝑋̇ = 𝑓(𝑋, 𝑝1 , 𝜏), 𝑋 (0) = 0; {𝜃̇1 = −𝐻𝑋 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ), 𝜃1 (𝑇) = 0. It is desirable to obtain an explicit solution of the triple (𝑋, 𝑝1 , 𝜃1 ) as illustrated in Example 3.3. However, it might be impossible to do so for the general demand rate function. The difficulty arises from the complex Hamiltonian function H, the split boundary condition of (𝑋, 𝜃1 ), and the nonlinear differential equations.

Here, an iterative gradient algorithm has been designed to solve (3.31). The algorithm is based on the observation that the price trajectory 𝑝1 maximizes the Hamiltonian function H and also maximizes the objective function J. Thus, the algorithm determines the optimal price trajectory 𝑝1 by making the first-order derivative of the Hamiltonian function H with respect to 𝑝1 equal to zero: 𝐻𝑝1 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ) = 0. The main idea of this algorithm is to update the price 𝑝1 in the steepest descent direction of the Hamiltonian function H, more specific descriptions are presented as follows: 1. The manufacturer chooses an initial price 𝑝1 randomly. 2. Use 𝑝1 to solve the state equation 𝑋̇ = 𝑓(𝑋, 𝑝1 , 𝜏) forward by the Runge-Kutta method. 3. According to 𝑝1 and 𝑋 obtained in the second step, proceed to get the costate function 𝜃1 by solving the costate equation 𝜃̇1 (𝑡) = −𝐻𝑋 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ) backward by the Runge-Kutta method. 82

4. Use the values of (𝑋, 𝑝1 , 𝜃1 ) to compute 𝐻𝑝1 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ). If 𝐻𝑝1 (𝑋, 𝑝1 , 𝑝̅2 , 𝜃1 ) = 0, 𝑝1 is the optimal solution; otherwise, use the steepest descent algorithm to get a new price 𝑝1 and go to the second step. As for the convergence of the above iterative procedure, one can refer to Kirk (2004).

Based on the above iterative procedure, the gradient algorithm can be summarised as follows: 1. Generate randomly a discrete approximation to the control 𝑝1 (𝑡), 𝑡 ∈ [0, 𝑇], that is 𝑝1 (𝑡) = 𝑝1 (𝑡𝑘 ),

𝑡 ∈ [𝑡𝑘 , 𝑡𝑘+1 ), 𝑘 = 1, 2, … , 𝑁

2. Calculate 𝑋(𝑡) by integrating the state equation (3.6) from 0 to T using the value 𝑝1 (𝑡) with the initial condition 𝑋 (0) = 0 (the 4-step Runge-Kutta method). 3. Calculate 𝜃1 (𝑡) by integrating the costate equation (3.17) backward using 𝑝1 (𝑡) and 𝑋(𝑡) (the 4-step Runge-Kutta method). 4. Use the discrete values of 𝑋(𝑡) and 𝜃1 (𝑡) to evaluate

𝜕𝐻 𝜕𝑝1

.

𝜕𝐻

5. If ‖𝜕𝑝 ‖ < 𝛿 (a given small value), then terminate the iterative procedure and output the 1

optimal solution (𝑋(𝑡), 𝑝1 (𝑡)). If the stopping criterion is not satisfied, generate a new piecewise constant control given by 𝑝1 (𝑡𝑘 ) = 𝑝1 (𝑡𝑘 ) + 𝑙 ×

𝜕𝐻 (𝑡 ), 𝑘 = 1, 2, … , 𝑁. 𝜕𝑝1 𝑘

The step length 𝑙 is determined by binary search so that the new price 𝑝1 (𝑡) yields a smaller H. Then go back to Step 2.

83

3.6.2 Numerical Experiments for Example 3.3 and 3.4 This section conducts a numerical analysis to learn more about the triple (𝑋(𝑡), 𝑝1 (𝑡), 𝜃1 (𝑡)) under the demand functions introduced in Example 3.3 and 3.4. The first experiment considers the limited-growth demand function in Example 3.3, which is a special case of Example 3.4 with 𝑞1 = 1 and 𝑞2 = 0 . Note that the closed-form optimal solution under such a demand rate function has been derived. This experiment can compare the numerical solutions obtained by the gradient method with the analytical results so as to evaluate the performance of the gradient method. The parameter values used in the numerical experiment are listed in Table 3.2: the unit manufacturing cost 𝐶𝑚 = 6 × 103 HKD; the expected unit repair cost per time 𝐶𝑟 = 250 HKD; the intensity function (damage rate) 𝜆 = 1 time/year; the BW length 𝜏 = 1 year; the planning horizon 𝑇 = 3 years; the maximum expected profit from selling an EWC 𝐸 (𝑝̅2 , 𝜂) = 600 HKD; the amplitude factor 𝑘1 = 6 × 1010 and the time displacement parameter 𝑘2 =5; the price elasticity 𝑎 = 3 and the BW length elasticity 𝑏 = 0.5; the population size of potential consumers 𝑀 = 105 ; the coefficient of innovation 𝑞1 = 1 and the coefficient of imitation 𝑞2 = 0.

Table 3.2 Parameters for the numerical analysis 𝐶𝑚

𝐶𝑟

λ

𝜏

T

𝐸(𝑝̅2 , 𝜂)

𝑘1

𝑘2

𝑎

b

M

𝑞1

𝑞2

6 × 103

250

1

1

3

600

6 × 1010

5

3

0.5

105

1

0

Given the parameters in Table 3.2, Figure 3.6 displays the numerical results obtained via the gradient method under different values of the tolerance 𝛿 in the gradient method. As shown in Figure 3.6, when 𝛿 < 10−5 , the numerical solution almost coincides with the closed-form solution, i.e., the numerical accuracy is sufficient for the experiment. Thus, the remaining numerical experiments are based on the tolerance value 𝛿 < 10−5 .

84

Price (x103)

11

 0; ̇ ( ) ( ) 𝑆 0 = 0 and 𝑆 𝐼 {= 0 when 𝐼 = 0; < 0 when 𝐼 < 0. Note that 𝑆 (0) = 0 means the inventory / backlogging cost is zero if there is no inventory or backlog. The description of 𝑆̇ (𝐼 ) implies that the inventory / backlogging cost is increasing with the absolute value of inventory level |𝐼|. Figure 4.1 presents an example of the inventory / backlogging cost function.

Figure 4.1 Sample inventory / backlogging cost function

A typical inventory / backlogging cost function is 𝑆 (𝐼 ) = {

𝛿+ 𝐼 2 ,

𝐼 ≥ 0;

𝛿− 𝐼 2 , 𝐼 < 0.

Where the parameters 𝛿+ and 𝛿− determine the costs to hold inventory or backlog demand, respectively.

94

In Chapter 3, the cost of the unit product is assumed to be a constant 𝐶𝑚 . The production plan is based on the demand rate at time t. Therefore, the instantaneous expected profit at time 𝑡 for the manufacturer would be Total profit = Profit from selling new product + Profit from selling EWC = Demand of new product × (Unit price of new product − 𝐶𝑚 − 𝐶𝐵𝑊 ) + Demand of EWC × (Unit price EWC − 𝐶𝐸𝑊𝐶 ) , which has been indicated directly in Figure 3.1. In the extended model considered in this chapter, the cost of the unit product depends on the production rate 𝑞(𝑡), and the production plan is directly based on the production rate. Therefore, inventory / backlogging cost would accrue because the demand rate and production rate are usually different. Also the instantaneous expected profit at time 𝑡 for the manufacturer would be changed into Total profit = Revenue from selling new product + Revenue from selling EWC −Production cost − Inventory / backlogging cost.

Selling price of new product 𝑝1 (𝑡)

Inventory level 𝐼(𝑡)

Cumulative production 𝑄(𝑡)

Revenue from selling new products Inventory / backlogging cost Production cost

EWC price 𝑝2 (𝑡)

Demand rate of EWC

Cumulative sales 𝑌(𝑡)

Revenue from selling EWC

Figure 4.2 The instantaneous profit flow at time t

95

Profit

Production rate 𝑞(𝑡)

Cumulative sales 𝑋(𝑡)

Demand rate of new product

The following Figure 4.2 schematically depicts the instantaneous expected profit flow at time t for the extended model. For each arrow in Figure 4.2, it implies that the variable at the beginning would affect the one at the end. With the notations introduced in Chapter 3 Section 3.2, it is not difficult to notice that the revenue from selling the new product would be: 𝑓 (𝑋 (𝑡), 𝑝1 (𝑡), 𝜏)[𝑝1 (𝑡) − 𝐶𝐵𝑊 ]. The revenue from selling the EWC would be: 𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏)𝜌(𝑝2 (𝑡), 𝜂)(𝑝2 (𝑡) − 𝐶𝐸𝑊𝐶 ). The production cost would be: 𝐶(𝑞 (𝑡)) = 𝑐0 + 𝑐1 𝑞 (𝑡) + 𝑐2 𝑞(𝑡)𝛽+1 . The inventory / backlogging cost would be: 𝑆(𝐼 (𝑡)). In the end, the integrated pricing and production problem can be formulated as follows: 𝑀𝑎𝑥

𝑝1 (𝑡),𝑝2 (𝑡),𝑞(𝑡)

𝐽[𝑝1 (𝑡), 𝑝2 (𝑡), 𝑞 (𝑡)] 𝑇

=

𝑀𝑎𝑥

∫ {𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏)[𝑝1 (𝑡) − 𝐶𝐵𝑊

𝑝1 (𝑡),𝑝2 (𝑡),𝑞(𝑡) 0

+ 𝜌(𝑝2 (𝑡), 𝜂)(𝑝2 (𝑡) − 𝐶𝐸𝑊𝐶 )] − 𝐶(𝑞 (𝑡)) − 𝑆(𝐼(𝑡))}𝑑𝑡

(4.2)

Subject to 𝑋̇(𝑡) = 𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏)

(4.3)

𝑌̇(𝑡) = 𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏)𝜌(𝑝2 (𝑡), 𝜂)

(4.4)

𝐼 ̇(𝑡) = 𝑞(𝑡) − 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)

(4.5)

𝑋(0) = 0, 𝑌(0) = 0, 𝐼 (0) = 0, 𝑋(𝑇), 𝑋(𝑇) free, but 𝐼(𝑇) ≥ 0

96

(4.6)

(𝑝1 (𝑡), 𝑝2 (𝑡), 𝑞(𝑡)) ∈ 𝑈1 × 𝑈2 × 𝑈3 ⊆ 𝑅+3

(4.7)

4.3 Theoretical Analysis Similar to Section 3.3, define 𝐸 (𝑝2 (𝑡), 𝜂) = 𝜌(𝑝2 (𝑡), 𝜂)[𝑝2 (𝑡) − 𝐶𝐸𝑊𝐶 (𝜂)] for all 𝑡 ∈ [0, 𝑇], and 𝑝̅2 = 𝐴𝑟𝑔𝑚𝑎𝑥𝑝2∈𝑈2 𝐸 (𝑝2 , 𝜂 ) for all 𝑡 ∈ [0, 𝑇]. Following the proof procedure of Lemma 3.1, it is not difficult to derive a similar result: Lemma 4.1 Let (𝑋 (𝑡), 𝑌 𝑖 (𝑡), 𝐼(𝑡), 𝑝1 (𝑡), 𝑝2𝑖 (𝑡), 𝑞(𝑡)) (where 𝑖 = 1,2) be two feasible solutions for the dynamic optimization problem (4.2)-(4.7). If 𝐸 (𝑝21 (𝑡), 𝜂) ≤ 𝐸 (𝑝22 (𝑡), 𝜂) (∀𝑡 ∈ [0, 𝑇]), then 𝐽[𝑝1 (𝑡), 𝑝21 (𝑡), 𝑞 (𝑡)] ≤ 𝐽[𝑝1 (𝑡), 𝑝22 (𝑡), 𝑞(𝑡)]

In order to derive the necessary conditions for the optimal solution of problem (4.2)-(4.7), define the following Hamiltonian: 𝐻 = 𝐻(𝑋 (𝑡), 𝑌(𝑡), 𝐼 (𝑡), 𝑝1 (𝑡), 𝑝2 (𝑡), 𝑞(𝑡), 𝜃1 (𝑡), 𝜃2 (𝑡), 𝜃3 (𝑡)) = 𝜃0 {𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏)[𝑝1 (𝑡) − 𝐶𝐵𝑊 + 𝐸 (𝑝2 (𝑡), 𝜂)] − 𝐶(𝑞 (𝑡)) − 𝑆 (𝐼(𝑡))} + 𝜃1 (𝑡)𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏) + 𝜃2 (𝑡)𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏)𝜌(𝑝2 (𝑡), 𝜂) + 𝜃3 (𝑡)(𝑞(𝑡) − 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)), where 𝜃0 is the constant multiplier associated with the integrand of the objective function; 𝜃1 (𝑡), 𝜃2 (𝑡) and 𝜃3 (𝑡) are the costate functions associated with the state equations (4.3), (4.4) and (4.5), respectively.

97

Suppose (𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝐼 ∗ (𝑡), 𝑝1∗ (𝑡), 𝑝2∗ (𝑡), 𝑞 ∗ (𝑡)) is the optimal solution for the dynamic optimization problem (4.2)-(4.7). According to the Pontryagin maximum principle, we assert that (1) The constant multiplier 𝜃0 (= 0 or 1) and the costate functions 𝜃1 (𝑡), 𝜃2 (𝑡) and 𝜃3 (𝑡) satisfy (𝜃0 , 𝜃1 (𝑡), 𝜃2 (𝑡), 𝜃3 (𝑡)) ≠ (0,0,0,0) for all 𝑡 ∈ [0, 𝑇]. (2) For all 𝑡 ∈ [0, 𝑇] and (𝑝1 , 𝑝2 , 𝑞 ) ∈ 𝑈1 × 𝑈2 × 𝑈3 , (𝑝1∗ (𝑡), 𝑝2∗ (𝑡), 𝑞 ∗ (𝑡)) maximizes 𝐻(𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝐼 ∗ (𝑡), 𝑝1 , 𝑝2 , 𝑞, 𝜃1 (𝑡), 𝜃2 (𝑡), 𝜃3 (𝑡)), that is 𝐻 ∗ ≜ 𝐻(𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝐼 ∗ (𝑡), 𝑝1∗ (𝑡), 𝑝2∗ (𝑡), 𝑞 ∗ (𝑡), 𝜃1 (𝑡), 𝜃2 (𝑡), 𝜃3 (𝑡)) ≥ 𝐻(𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝐼 ∗ (𝑡), 𝑝1 , 𝑝2 , 𝑞, 𝜃1 (𝑡), 𝜃2 (𝑡), 𝜃3 (𝑡)).

(4.8)

(3) For any time 𝑡 at which the functions (𝑝1∗ (𝑡), 𝑝2∗ (𝑡), 𝑞 ∗ (𝑡)) are continuous, the costate functions satisfy the following differential equations: 𝜕𝐻∗ 𝜕𝑋 = −𝜃0 {𝑓𝑋 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)[𝑝1∗ (𝑡) − 𝐶𝐵𝑊 + 𝐸 (𝑝2∗ (𝑡), 𝜂)]} + 𝜃1 (𝑡)𝑓𝑋 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)

𝜃̇1 (𝑡) = −

+ 𝜃2 (𝑡)𝑓𝑋 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)𝜌(𝑝2∗ (𝑡), 𝜂) − 𝜃3 (𝑡)𝑓𝑋 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏); 𝜕𝐻 ∗ = 0; 𝜕𝑌 𝜕𝐻 ∗ ̇𝜃3 (𝑡) = − = 𝜃0 𝑆̇(𝐼 ∗ ). 𝜕𝐼 𝜃̇2 (𝑡) = −

(4) The transversality conditions corresponding to boundary conditions (4.6) (also called the end time constraints of the costate functions) are (𝜃1 (𝑇), 𝜃2 (𝑇)) = (0, 0) and 𝜃3 (𝑇) ≥ 0 (= 0 if 𝐼 ∗ (𝑇) > 0).

First, derive the following properties regarding the constant multiplier 𝜃0 and the costate function 𝜃2 (𝑡).

98

Lemma 4.2 𝜃0 = 1 and 𝜃2 (𝑡) ≡ 0. Proof: Note that the costate equation of 𝜃2 (𝑡) implies that 𝜃2 (𝑡) is constant over [0, 𝑇]. Hence, the end time boundary condition 𝜃2 (𝑇) = 0 implies 𝜃2 (𝑡) ≡ 0. Next, prove the conclusion of 𝜃0 = 1 by contradiction. Suppose on the contrary that 𝜃0 = 0. Substituting 𝜃0 = 0 and 𝜃2 (𝑡) ≡ 0 into the Hamiltonian function yields 𝐻 ∗ = [𝜃1 (𝑡) − 𝜃3 (𝑡)]𝑓(𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏) + 𝜃3 (𝑡)𝑞 ∗ (𝑡). Then the costate equation of 𝜃2 (𝑡) becomes 𝜃̇3 (𝑡) = 0. This obviously lead to the conclusion that 𝜃3 (𝑡) is constant over [0, 𝑇]. As the end time boundary condition of 𝜃3 (𝑡) is 𝜃3 (𝑇) ≥ 0, then 𝜃3 (𝑡) ≡ 𝜃3 (𝑇) ≥ 0. Consider the following two cases: 

Suppose that 𝜃3 (𝑇) = 0, i.e., 𝜃3 (𝑡) ≡ 0. Substituting it into the costate function of 𝜃1 (𝑡) yields 𝜃̇1 (𝑡) = 𝜃1 (𝑡)𝑓𝑋 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏). Notice that the end time boundary condition 𝜃1 (𝑇) = 0 implies 𝜃1 (𝑡) ≡ 0 over [0, 𝑇]. That is (𝜃0 , 𝜃1 (𝑇), 𝜃2 (𝑇), 𝜃3 (𝑇)) = (0,0,0,0), which contradicts with the preceding conclusion: (𝜃0 , 𝜃1 (𝑡), 𝜃2 (𝑡), 𝜃3 (𝑡)) ≠ (0,0,0,0) for all 𝑡 ∈ [0, 𝑇].



Suppose that 𝜃3 (𝑡) ≡ 𝜃3 (𝑇) > 0. Substituting 𝜃0 = 0, 𝜃2 (𝑡) ≡ 0 and 𝜃3 (𝑡) ≡ 𝜃3 (𝑇) > 0

99

into (4.8), we obtain (𝜃1 (𝑡) − 𝜃3 (𝑇))𝑓 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏) + 𝜃3 (𝑇)𝑞∗ (𝑡) ≥ (𝜃1 (𝑡) − 𝜃3 (𝑇))𝑓(𝑋 ∗ (𝑡), 𝑝1 , 𝜏) + 𝜃3 (𝑇)𝑞 for all 𝑡 ∈ [0, 𝑇] and any (𝑝1 , 𝑝2 , 𝑞) ∈ 𝑈1 × 𝑈2 × 𝑈3 . However, because 𝜃3 (𝑇) > 0 and the admissible region of 𝑞 is 𝑈3 = [0, +∞) . If set 𝑝1 = 𝑝1∗ (𝑡) for all 𝑡 ∈ [0, 𝑇], then (𝜃1 (𝑡) − 𝜃3 (𝑇))𝑓 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏) + 𝜃3 (𝑇)𝑞∗ (𝑡) < (𝜃1 (𝑡) − 𝜃3 (𝑇))𝑓(𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏) + 𝜃3 (𝑇)𝑞 for any 𝑞 > 𝑞 ∗ (𝑡), which leads to contradiction. Therefore, the constant multiplier satisfies 𝜃0 = 1. ∎ Based on Lemma 4.2, the next conclusion asserts that the optimal production rate 𝑞 ∗ (𝑡) can be explicitly expressed as a function of 𝜃3 (𝑡).

Lemma 4.3 If the parameter 𝑐2 > 0, the optimal production rate 𝑞 ∗ (𝑡) satisfies ,

if 𝜃3 (𝑡) < 𝑐1 ;

𝑞 ∗ (𝑡) = { 𝜃3 (𝑡) − 𝑐1 ( ) , (𝛽 + 1)𝑐2

if 𝜃3 (𝑡) ≥ 𝑐1 .

0 1 𝛽

(4.9)

Proof: As 𝜃0 = 1 and 𝜃2 (𝑡) ≡ 0, the Hamiltonian would be simplified into 𝐻 = 𝑓(𝑋 (𝑡), 𝑝1 (𝑡), 𝜏)[𝑝1 (𝑡) − 𝐶𝐵𝑊 + 𝐸 (𝑝2 (𝑡), 𝜂) + 𝜃1 (𝑡) − 𝜃3 (𝑡)] − 𝐶(𝑞 (𝑡)) − 𝑆(𝐼 (𝑡)) + 𝜃3 (𝑡)𝑞(𝑡). 100

Note that the optimal solution (𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝐼 ∗ (𝑡), 𝑝1∗ (𝑡), 𝑝2∗ (𝑡), 𝑞 ∗ (𝑡)) satisfies (4.8): 𝐻∗ ≜ 𝐻(𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝐼 ∗ (𝑡), 𝑝1∗ (𝑡), 𝑝2∗ (𝑡), 𝑞 ∗ (𝑡), 𝜃1 (𝑡), 𝜃2 (𝑡), 𝜃3 (𝑡)) ≥ 𝐻(𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝐼 ∗ (𝑡), 𝑝1 , 𝑝2 , 𝑞, 𝜃1 (𝑡), 𝜃2 (𝑡), 𝜃3 (𝑡)), i.e., 𝑓 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)[𝑝1∗ (𝑡) − 𝐶𝐵𝑊 + 𝐸 (𝑝2∗ (𝑡), 𝜂) + 𝜃1 (𝑡) − 𝜃3 (𝑡)] − 𝐶(𝑞 ∗ (𝑡)) − 𝑆(𝐼 ∗ (𝑡)) + 𝜃3 (𝑡)𝑞 ∗ (𝑡) ≥ 𝑓 (𝑋 ∗ (𝑡), 𝑝1 , 𝜏)[𝑝1 (𝑡) − 𝐶𝐵𝑊 + 𝐸 (𝑝2 , 𝜂) + 𝜃1 (𝑡) − 𝜃3 (𝑡)] − 𝐶 (𝑞 ) − 𝑆(𝐼 ∗ (𝑡)) + 𝜃3 (𝑡)𝑞

(4.10)

for all 𝑡 ∈ [0, 𝑇] and (𝑝1 , 𝑝2 , 𝑞) ∈ 𝑈1 × 𝑈2 × 𝑈3 . According to the optimality of (𝑝1∗ (𝑡), 𝑝2∗ (𝑡), 𝑞 ∗ (𝑡)), (4.10) is equivalent to 𝑓 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)[𝑝1∗ (𝑡) − 𝐶𝐵𝑊 + 𝐸 (𝑝2∗ (𝑡), 𝜂) + 𝜃1 (𝑡) − 𝜃3 (𝑡)] − 𝐶(𝑞 ∗ (𝑡)) − 𝑆(𝐼 ∗ (𝑡)) + 𝜃3 (𝑡)𝑞 ∗ (𝑡) =

𝑀𝑎𝑥

(𝑝1 ,𝑝2 ,𝑞)∈𝑈1 ×𝑈2 ×𝑈3

{𝑓 (𝑋 ∗ (𝑡), 𝑝1 , 𝜏)[𝑝1 (𝑡) − 𝐶𝐵𝑊 + 𝐸 (𝑝2 , 𝜂) + 𝜃1 (𝑡) − 𝜃3 (𝑡)] − 𝐶 (𝑞 )

− 𝑆(𝐼 ∗ (𝑡)) + 𝜃3 (𝑡)𝑞 } =

𝑀𝑎𝑥

(𝑝1 ,𝑝2 )∈𝑈1 ×𝑈2

{𝑓 (𝑋 ∗ (𝑡), 𝑝1 , 𝜏)[𝑝1 (𝑡) − 𝐶𝐵𝑊 + 𝐸 (𝑝2 , 𝜂) + 𝜃1 (𝑡) − 𝜃3 (𝑡)]} − 𝑆(𝐼 ∗ (𝑡)) + 𝑀𝑎𝑥[−𝐶 (𝑞 ) + 𝜃3 (𝑡)𝑞 ]. 𝑞∈𝑈3

Hence, (𝑝1∗ (𝑡), 𝑝2∗ (𝑡)) = 𝐴𝑟𝑔𝑚𝑎𝑥 {𝑓 (𝑋 ∗ (𝑡), 𝑝1 , 𝜏)[𝑝1 (𝑡) − 𝐶𝐵𝑊 + 𝐸 (𝑝2 , 𝜂) + 𝜃1 (𝑡) − 𝜃3 (𝑡)]}, (𝑝1 ,𝑝2 )∈𝑈1 ×𝑈2

∀𝑡 ∈ [0, 𝑇];

101

(4.11)

𝑞 ∗ (𝑡) = 𝐴𝑟𝑔𝑚𝑎𝑥[−𝐶 (𝑞 ) + 𝜃3 (𝑡)𝑞] , 𝑞∈𝑈3

𝑡 ∈ [0, 𝑇].

(4.12)

Similar to the proof of Theorem 3.1, Lemma 4.1 and (4.11) yield 𝑃2∗ (𝑡) ≡ 𝑝̅2 = 𝐴𝑟𝑔𝑚𝑎𝑥 𝐸 (𝑝2 , 𝜂) , ∀𝑡 ∈ [0, 𝑇]; 𝑝2 ∈𝑈2

𝑝1∗ (𝑡) = 𝐴𝑟𝑔𝑚𝑎𝑥 𝑓 (𝑋 ∗ (𝑡), 𝑝1 , 𝜏)[𝑝1 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 (𝑡) − 𝜃3 (𝑡)] , ∀𝑡 ∈ [0, 𝑇]. 𝑝1 ∈𝑈1

Besides, substituting the expression of 𝐶 (𝑞(𝑡)) into (4.12) yields 𝑞 ∗ (𝑡) = 𝐴𝑟𝑔𝑚𝑎𝑥 [−𝑐0 + (𝜃3 (𝑡) − 𝑐1 )𝑞 − 𝑐2 𝑞 𝛽+1 ] , 𝑡 ∈ [0, 𝑇]. 𝑞∈𝑈3

Define the following function 𝑅(𝑞 ) = −𝑐0 + (𝜃3 (𝑡) − 𝑐1 )𝑞 − 𝑐2 𝑞 𝛽+1 , where 𝑞 ∈ [0, +∞). Obviously, the derivative of 𝑅 (𝑞 ) with respected to 𝑞 is 𝑅̇(𝑞 ) = (𝜃3 (𝑡) − 𝑐1 ) − (𝛽 + 1)𝑐2 𝑞 𝛽 . If 𝜃3 (𝑡) < 𝑐1 at time t, 𝑅̇(𝑞 ) < 0 for any 𝑞 ∈ [0, +∞), which means 𝑅(𝑞 ) is decreasing is [0, +∞) and 𝑅(𝑞 ) reaches the maximum at 𝑞 ∗ = 0. If 𝜃3 (𝑡) ≥ 𝑐1 at time t, then 𝑅′ (𝑞 ) = 0 at 1

𝜃3 (𝑡) − 𝑐1 𝛽 𝑞∗ = ( ) . (𝛽 + 1)𝑐2 Note that 𝑅̇(𝑞 ) > 0 for any 𝑞 ∈ [0, 𝑞 ∗ ) and 𝑅̇(𝑞 ) < 0 for any 𝑞 ∈ (𝑞 ∗ , +∞) , i.e., 𝑅 (𝑞 ) is increasing in [0, 𝑞 ∗ ) and decreasing in (𝑞 ∗ , +∞) . Therefore, 𝑅(𝑞 ) reaches the maximum at 1

∗

𝑞 =

𝜃3 (𝑡)−𝑐1 𝛽 ( (𝛽+1)𝑐 ) . 2

(4.9) follows immediately from the above two cases.

102

∎

Proposition 4.1 𝜃3 (𝑡) ≥ 𝑐1 for any 𝑡 ∈ [0, 𝑇]. Proof: Suppose for the sake of contradiction that there is at least one point 𝑡0 ∈ (0, 𝑇) such that 𝜃3 (𝑡) < 𝑐1 . Obviously, one of the following two cases must hold: Case (a): 𝐼 ∗ (𝑡0 ) ≥ 0; Case (b): 𝐼 ∗ (𝑡0 ) < 0.

Figure 4.3 Diagrammatic sketch in the proof of Proposition 4.1

For case (a), let us focus on the trajectories of 𝜃3 (𝑡) and 𝐼 ∗ (𝑡) in the interval [0, 𝑡0 ]. Suppose that there exists some 𝑡 ∈ [0, 𝑡0 ] such that 𝜃3 (𝑡) ≥ 𝑐1 . Denote 𝑡 = 𝑚𝑎 𝑥 {𝑡 ∈ [0, 𝑡0 )|𝜃3 (𝑡) = 𝑐1 },

103

(4.13)

i.e., 𝑡 is the first time that the trajectory of 𝜃3 (𝑡) touches 𝑐1 when 𝜃3 (𝑡) moves backward from 𝑡0 (as illustrated in the case (a) region in Figure 4.3). According to (4.9) in Lemma 4.3, there must be 𝑞 ∗ (𝑡) = 0 for any 𝑡 ∈ [𝑡, 𝑡0 ]. It implies that 𝐼 ∗ (𝑡) is decreasing in [𝑡, 𝑡0 ] and hence 𝐼 ∗ (𝑡) ≥ 0 for any 𝑡 ∈ [𝑡, 𝑡0 ]. On the other hand, the costate equation of 𝜃3 (𝑡) and the property of S ( I ) imply > 0 when 𝐼∗ (𝑡) > 0; 𝜃̇3 (𝑡) = 𝑆̇(𝐼 ∗ (𝑡)) {= 0 when 𝐼∗ (𝑡) = 0; < 0 when 𝐼 ∗ (𝑡) < 0.

(4.14)

Because 𝐼 ∗ (𝑡) ≥ 0 for any 𝑡 ∈ [𝑡, 𝑡0 ], 𝜃3 (𝑡) is increasing in [𝑡, 𝑡0 ]. Therefore, 𝜃3 (𝑡) ≤ 𝜃3 (𝑡) < 𝑐1 , which contradicts with the definition (4.13). Now, we have 𝜃3 (𝑡) < 𝑐1 for any 𝑡 ∈ [0, 𝑡0 ]. (4.9) implies 𝑞 ∗ (𝑡) = 0 for any 𝑡 ∈ [0, 𝑡0 ], which implies 𝐼 ∗ (𝑡) is strictly decreasing in [0, 𝑡0 ], i.e., 𝐼 ∗ (0) > 0, i.e., case (a) will lead to a contradiction with the initial condition of 𝐼 ∗ (0) = 0 in (4.6)). For case (b), focus on the trajectories of 𝜃3 (𝑡) and 𝐼 ∗ (𝑡) in the interval [𝑡0 , 𝑇]. Similarly, suppose that there exists some 𝑡 ∈ [𝑡0 , 𝑇] such that 𝜃3 (𝑡) ≥ 𝑐1 . Then define 𝑡̅ = 𝑚𝑖 𝑛{𝑡 ∈ (𝑡0 , 𝑇]|𝜃3 (𝑡) = 𝑐1 }.

(4.15)

i.e., 𝑡̅ is the first time that the trajectory of 𝜃3 (𝑡) touches 𝑐1 when 𝜃3 (𝑡) moves forward from 𝑡0 (as illustrated in the case (b) region of Figure 4.3). According to (4.9), because 𝜃3 (𝑡) < 𝑐1 for any 𝑡 ∈ [𝑡0 , 𝑡̅ ], there must be 𝑞 ∗ (𝑡) = 0 for any 𝑡 ∈ [𝑡0 , 𝑡̅ ]. Therefore, 𝐼 ∗ (𝑡) is decreasing in [𝑡0 , 𝑡̅ ] , which also implies 𝐼 ∗ (𝑡) < 0 for any 𝑡 ∈ [𝑡0 , 𝑡̅ ] . Therefore, (4.14) yields 𝜃3 (𝑡) is decreasing in [𝑡0 , 𝑡̅ ] and so 𝜃3 (𝑡̅) ≤ 𝜃3 (𝑡) < 𝑐1 , which contradicts to the definition (4.15). As a result, we obtain that 𝜃3 (𝑡) < 𝑐1 for any 𝑡 ∈ [𝑡0 , 𝑇]. Again, (4.9) implies 𝑞 ∗ (𝑡) = 0 for any 𝑡 ∈ [𝑡0 , 𝑇], i.e., 𝐼 ∗ (𝑡) is strictly decreasing in [𝑡0 , 𝑇] and 𝐼 ∗ (𝑇) < 0. It is contradict with the end time boundary condition of 𝐼 ∗ (𝑇) ≥ 0 in (4.6). The analysis for cases (a) and (b) shows that there exists no 𝑡0 ∈ (0, 𝑇) such that 𝜃3 (𝑡) < 𝑐1 , i.e., 𝜃3 (𝑡) ≥ 𝑐1 for any 𝑡 ∈ (0, 𝑇). Note that 𝜃3 (𝑡) is piecewise differentiable, then 𝜃3 (𝑡) ≥ 𝑐1 for any 𝑡 ∈ [0, 𝑇]. 104

∎ Lemma 4.3 and Proposition 4.1 assert that the optimal production rate can be simplified to 1

𝜃3 (𝑡) − 𝑐1 𝛽 𝑞 ∗ (𝑡 ) = ( ) , (𝛽 + 1)𝑐2

𝑡 ∈ [0, 𝑇].

The next proposition indicates that the optimal inventory level at the end time 𝑇 should be zero.

Proposition 4.2 𝐼 ∗ (𝑇) = 0 Proof: If 𝐼 ∗ (𝑇) > 0, then the transversality condition implies that 𝜃3 (𝑇) = 0 < 𝑐1 , which contradicts Proposition 4.1. Therefore, 𝐼 ∗ (𝑇) = 0. ∎ Based on Lemmas 4.2 and 4.3 as well as Propositions 4.1 and 4.2, the following theorem 4.1 summarises the necessary conditions of the optimal solution for the problem (4.2)-(4.7).

Theorem 4.1 Suppose (𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝐼 ∗ (𝑡), 𝑝1∗ (𝑡), 𝑝2∗ (𝑡), 𝑞 ∗ (𝑡)) is the optimal solution for the dynamic optimization problem (4.2)-(4.7) and the parameter 𝑐2 > 0. There must exist piecewise continuously differentiable functions 𝜃1 (𝑡) and 𝜃3 (𝑡), such that the optimal solution, 𝜃1 (𝑡) and 𝜃3 (𝑡) must satisfy the following relations: 𝑃2∗ (𝑡) ≡ 𝑝̅2 = 𝐴𝑟𝑔𝑚𝑎𝑥 𝐸 (𝑝2 , 𝜂) , ∀𝑡 ∈ [0, 𝑇]; 𝑝2 ∈𝑈2

(4.16)

1

𝜃3 (𝑡) − 𝑐1 𝛽 𝑞 ∗ (𝑡 ) = ( ) , (𝛽 + 1)𝑐2

105

𝑡 ∈ [0, 𝑇];

(4.17)

𝑝1∗ (𝑡) = 𝐴𝑟𝑔𝑚𝑎𝑥 𝑓 (𝑋 ∗ (𝑡), 𝑝1 , 𝜏)[𝑝1 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 (𝑡) − 𝜃3 (𝑡)] , 𝑝1 ∈𝑈1

∀𝑡 ∈ [0, 𝑇];

(4.18)

𝑋̇ ∗ (𝑡) = 𝑓 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏), 𝑋 ∗ (0) = 0;

(4.19)

𝑌̇ ∗ (𝑡) = 𝑓 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)𝜌(𝑝̅2 ), 𝜂), 𝑌 ∗ (0) = 0;

(4.20)

𝐼 ∗̇ (𝑡) = 𝑞 ∗ (𝑡) − 𝑓 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏), 𝐼 ∗ (0) = 𝐼 ∗ (𝑇) = 0;

(4.21)

𝜃̇1 (𝑡) = −𝑓𝑋 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)[𝑝1∗ (𝑡) − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 (𝑡) − 𝜃3 (𝑡)], 𝜃1 (𝑇) = 0; 𝜃̇3 (𝑡) = 𝑆̇ (𝐼 ∗ (𝑡)).

(4.22) (4.23)

Proof: (4.16) and (4.18) are derived in the proof of Lemma 4.3. (4.17) is an immediately result of Lemma 4.3 and Proposition 4.1. (4.19)-(4.21) are the state equations as well as the boundary conditions in the dynamic optimization problem (4.2)-(4.7). The only difference is the end time boundary condition 𝐼 ∗ (𝑇) = 0 indicated in Proposition 4.2. (4.22)-(4.23) are the costate equations and their boundary conditions. ∎

Remark 4.1 Economic explanations (a) Recall the definition the Hamiltonian function H for dynamic problem (4.2)-(4.7) 𝐻 = 𝜃0 {𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏)[𝑝1 (𝑡) − 𝐶𝐵𝑊 + 𝐸 (𝑝2 (𝑡), 𝜂)] − 𝐶(𝑞 (𝑡)) − 𝑆 (𝐼(𝑡))} + 𝜃1 (𝑡)𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏) + 𝜃2 (𝑡)𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏)𝜌(𝑝2 (𝑡), 𝜂) + 𝜃3 (𝑡)(𝑞(𝑡) − 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)). The constant multiplier 𝜃0 = 1 implies the problem is not degenerate.

106

𝜃3 (𝑡) can be regarded as the marginal production cost at time t. The value of 𝜃3 (𝑡) is jointly determined by the production rate and the inventory cost. As indicated in (4.1), the unit production cost is at least 𝑐1 for any production rate, which explains the result 𝜃3 (𝑡) ≥ 𝑐1 in Proposition 4.1. Similar to Remark 3.1, 𝜃1 (𝑡) can be interpreted as the discounted value of the marginal change in the profit from time 𝑡 to 𝑇 with respect to the sales volume X. The main difference is that the production cost 𝐶𝑚 in Remark 3.1 should be replaced by the marginal cost 𝜃3 (𝑡). Besides, 𝜃2 (𝑡) is the marginal change in the total profit from time 𝑡 to 𝑇 with respect to the sales volume Y. Lemma 4.2, however, asserts that 𝜃2 (𝑡) ≡ 0. Obviously, the Hamiltonian function depends only on the variables (𝑋(𝑡), 𝐼(𝑡), 𝑝1 (𝑡), 𝑝2 (𝑡), 𝑞 (𝑡), 𝜃1 (𝑡), 𝜃1 (𝑡)). Therefore, it is usually denoted as 𝐻(𝑋 (𝑡), 𝐼 (𝑡), 𝑝1 (𝑡), 𝑝2 (𝑡), 𝑞(𝑡), 𝜃1 (𝑡), 𝜃1 (𝑡)). (b) As for Proposition 4.2, if 𝐼 ∗ (𝑇) > 0, the manufacturer has produced more units of the new product than the demand from the market. Obviously, the production and inventory costs could be reduced and the demand can still be satisfied as long as the production quantity equals the demand, which yields 𝐼 ∗ (𝑇) = 0.

Remark 4.2 Similar to Remark 3.2, Theorem 4.1 also suggest the following procedure to obtain the optimal solution (𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝐼 ∗ (𝑡), 𝑝1∗ (𝑡), 𝑝2∗ (𝑡), 𝑞 ∗ (𝑡)): Step 1: Find the maximum point of the non-linear function: 𝑝2∗ (𝑡) ≡ 𝑝̅2 = 𝐴𝑟𝑔𝑚𝑎𝑥 𝐸 (𝑝2 , 𝜂 ). 𝑝2 ∈𝑈2

Step 2: Substitute (4.17) into (4.21), and then obtain the quintuple (𝑋 ∗ (𝑡), 𝐼 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜃1 (𝑡), 𝜃3 (𝑡)) by solving (4.18)-(4.19) and (4.21)-(4.23), i.e.,

107

𝑝1∗ (𝑡) = 𝐴𝑟𝑔𝑚𝑎𝑥 𝑓 (𝑋 ∗ (𝑡), 𝑝1 , 𝜏)[𝑝1 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 (𝑡) − 𝜃3 (𝑡)] , ∀𝑡 ∈ [0, 𝑇]; 𝑝1 ∈𝑈1

𝑋̇ ∗ (𝑡) = 𝑓 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏), 𝑋 ∗ (0) = 0; 1

𝜃3 (𝑡) − 𝑐1 𝛽 𝐼 ̇∗ (𝑡) = ( ) − 𝑓(𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏), 𝐼 ∗ (0) = 𝐼 ∗ (𝑇) = 0; (𝛽 + 1)𝑐2 𝜃̇1 (𝑡) = −𝑓𝑋 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)[𝑝1∗ (𝑡) − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 (𝑡) − 𝜃3 (𝑡)], 𝜃1 (𝑇) = 0; 𝜃̇3 (𝑡) = 𝑆̇ (𝐼 ∗ (𝑡)).

{

(4.24) Step 3: Based on (𝑋 ∗ (𝑡), 𝐼 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜃1 (𝑡), 𝜃3 (𝑡)), obtain 𝑞 ∗ (𝑡) and 𝑌 ∗ (𝑡) by 1

𝜃3 (𝑡) − 𝑐1 𝛽 𝑞 ∗ (𝑡 ) = ( ) ; (𝛽 + 1)𝑐2 𝑌̇ ∗ (𝑡) = 𝑓 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)𝜌(𝑝̅2 ), 𝜂), 𝑌 ∗ (0) = 0.

Remark 4.3 In the following, the time argument t will be omitted when no confusion arises. (4.18) indicates that 𝑝1∗ = 𝐴𝑟𝑔𝑚𝑎𝑥 𝑓(𝑋 ∗ , 𝑝1 , 𝜏)[𝑝1 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 − 𝜃3 ] 𝑝1 ∈𝑈1

= 𝐴𝑟𝑔𝑚𝑎𝑥 { 𝑓 (𝑋 ∗ , 𝑝1 , 𝜏)[𝑝1 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 − 𝜃3 ] − 𝑆 (𝐼 ∗ ) + 𝜃3 𝑞 ∗ − 𝐶 (𝑞 ∗ )} 𝑝1 ∈𝑈1

= 𝐴𝑟𝑔𝑚𝑎𝑥 𝐻(𝑋 ∗ , 𝐼 ∗ , 𝑝1 , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 ), ∀𝑡 ∈ [0, 𝑇]. 𝑝1 ∈𝑈1

Similar to Remark 3.3, the first-order condition of (4.18) is 𝐻𝑝1 (𝑋 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 ) = 𝑓𝑝1 (𝑋 ∗ , 𝑝1∗ , 𝜏)[𝑝1∗ − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 − 𝜃3 ] + 𝑓(𝑋 ∗ , 𝑝1∗ , 𝜏) = 0, i.e.,

108

𝑝1∗

𝑓 (𝑋 ∗ , 𝑝1∗ , 𝜏) − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 − 𝜃3 = − . 𝑓𝑝1 (𝑋 ∗ , 𝑝1∗ , 𝜏)

(4.25)

In addition, if 𝑝1∗ can be derived as a function of (𝑋 ∗ , 𝜃1 , 𝜃3 ) via (4.25), substitute 𝑝1∗ (𝑋 ∗ , 𝜃1 , 𝜃3 ) into (4.24) to get a group of differential equations for the quadruple (𝑋 ∗ , 𝐼 ∗ , 𝜃1 , 𝜃3 ). Further assume that 𝑓(𝑋, 𝑝1 , 𝜏) has a second-order partial derivative with respect to 𝑝1 . The second-order condition of (4.18) is 𝐻𝑝1 𝑝1 (𝑋 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 ) = 𝑓𝑝1 𝑝1 (𝑋 ∗ , 𝑝1∗ , 𝜏)[𝑝1∗ − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 − 𝜃3 ] + 2𝑓𝑝1 (𝑋 ∗ , 𝑝1∗ , 𝜏) ≤ 0. Substituting (4.25) into the about inequality yields 2𝑓𝑝21 (𝑋 ∗ , 𝑝1∗ , 𝜏) ≥ 𝑓(𝑋 ∗ , 𝑝1∗ , 𝜏)𝑓𝑝1 𝑝1 (𝑋 ∗ , 𝑝1∗ , 𝜏),

(4.26)

which can be interpreted in the same manner as (3.33) in Remark 3.3.

As described in the basic model in Chapter 3, the manufacturer is not necessary to keep inventory (neither positive nor negative) when the unit production is constant, because keeping inventory must give rise of extra inventory or backlogging cost. However, if the unit production cost is related with production rate as depicted by expression (4.1), the appropriate production plan through the planning horizon can reduce the production cost, and there is usually a trade-off between the production cost and the inventory / backlogging cost. In this situation, the inventory level is usually not euqal to zero unless one degenerated case which will be specified in the following corollary.

Corollary 4.1 Suppose (𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝐼 ∗ (𝑡), 𝑝1∗ (𝑡), 𝑝2∗ (𝑡), 𝑞 ∗ (𝑡)) is an optimal solution to the dynamic problem (4.2)-(4.7) with the costate functions 𝜃1 (𝑡) and 𝜃3 (𝑡) , and the parameter

109

𝑐2 = 0, then the inventory level 𝐼(𝑡) is zero (i.e., the production rate is equal to demand rate) over the planning horizon [0, 𝑇]. Proof: Carefully review the proof of Lemma 4.2 and Proposition 4.1, these conclusions does not rely on the condition of 𝑐2 , so they are also true when 𝑐2 = 0. However, the conclusion of Lemma 4.3 needs the condition 𝑐2 > 0. If 𝑐2 = 0, then 𝑞 ∗ (𝑡) = 𝐴𝑟𝑔𝑚𝑎𝑥[−𝑐0 + (𝜃3 (𝑡) − 𝑐1 )𝑞 − 𝑐2 𝑞 𝛽+1 ], ∀𝑡 ∈ [0, 𝑇] 𝑞∈𝑈3

= 𝐴𝑟𝑔𝑚𝑎𝑥[−𝑐0 + (𝜃3 (𝑡) − 𝑐1 )𝑞 ]. 𝑞∈𝑈3

According to Proposition 4.1, 𝜃3 (𝑡) ≥ 𝑐1 for any 𝑡 ∈ [0, 𝑇]. Suppose 𝜃3 (𝑡) > 𝑐1 for some 𝑡0 ∈ [0, 𝑇]. Note that 𝑞 ∗ (𝑡0 ) = 𝐴𝑟𝑔𝑚𝑎𝑥𝑞∈𝑈3 [−𝑐0 + (𝜃3 (𝑡0 ) − 𝑐1 )𝑞 ] , but the domain is 𝑈3 = [0, +∞). If take 𝑞0 ∈ 𝑈3 such that 𝑞0 > 𝑞 ∗ (𝑡0 ), then −𝑐0 + (𝜃3 (𝑡) − 𝑐1 )𝑞0 > −𝑐0 + (𝜃3 (𝑡0 ) − 𝑐1 )𝑞 ∗ (𝑡0 ) contradict with the expression of 𝑞 ∗ (𝑡) = 𝐴𝑟𝑔𝑚𝑎𝑥𝑞∈𝑈3 [−𝑐0 + (𝜃3 (𝑡) − 𝑐1 )𝑞 ] , ∀𝑡 ∈ [0, 𝑇]. Therefore, the only possible situation is 𝜃3 (𝑡) = 𝑐1 for any 𝑡 ∈ [0, 𝑇]. In this situation, 𝜃̇3 (𝑡) = 0 for any 𝑡 ∈ [0, 𝑇], but (4.23) also implies that 𝜃̇3 (𝑡) = 𝑆̇(𝐼 (𝑡)) = 0. According to the definition of the inventory / backlogging cost function described in Section 4.2, 𝑆̇(𝐼 (𝑡)) = 0 implies 𝐼 (𝑡) = 0 for any 𝑡 ∈ [0, 𝑇]. Note that 𝐼 ̇(𝑡) = 𝑞 (𝑡) − 𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏) = 0 for any 𝑡 ∈ [0, 𝑇] i.e., 𝑞 (𝑡) = 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏) for any 𝑡 ∈ [0, 𝑇] 110

i.e., the production rate is equal to the demand rate. ∎

Remark 4.4 It is not difficult to notice that if parameters 𝑐0 = 𝑐1 = 0, the unit production cost will always be constant 𝑐1 , and this model will degenerate to the basic model in Chapter 3. What’s more, if we rewite the objective function (4.2) as 𝑇

𝑀𝑎𝑥

∫ {𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏)[𝑝1 (𝑡) − 𝐶𝐵𝑊 + 𝜌(𝑝2 (𝑡), 𝜂)(𝑝2 (𝑡) − 𝐶𝐸𝑊𝐶 )] − 𝐶(𝑞 (𝑡))

𝑝1 (𝑡),𝑝2 (𝑡),𝑞(𝑡) 0

− 𝑆(𝐼 (𝑡))}𝑑𝑡 𝑇

= −𝑐0 𝑇 +

𝑀𝑎𝑥

∫ {𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏)[𝑝1 (𝑡) − 𝐶𝐵𝑊 + 𝜌(𝑝2 (𝑡), 𝜂)(𝑝2 (𝑡) − 𝐶𝐸𝑊𝐶 )]

𝑝1 (𝑡),𝑝2 (𝑡),𝑞(𝑡) 0

− [𝑐1 + 𝑐2 𝑞 (𝑡)𝛽 ]𝑞(𝑡) − 𝑆(𝐼 (𝑡))}𝑑𝑡, then it is not difficult to see that the parameter 𝑐0 do not influence the decision of pricing strategy and production strategy. That is why the inventory level may be equal to zero even when 𝑐0 ≠ 0 but 𝑐2 = 0. Similar to Theorem 3.2, under certain conditions, the necessary conditions in Theorem 4.1 are also sufficient conditions for optimality.

Theorem 4.2 (Mangasarian-type sufficient condition) Let (𝑋 ∗ , 𝑌 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ ) be a feasible solution for the dynamic optimization problem (4.2)-(4.7). Suppose that (𝑋 ∗ , 𝑌 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ ) satisfies the conditions in Theorem 4.1 with the costate functions 𝜃1 and 𝜃3 . (a) If the Hamiltonian 𝐻 (𝑋, 𝐼, 𝑝1 , 𝑝2 , 𝑞, 𝜃1 , 𝜃3 ) is concave in (𝑋, 𝑝1 ) for any given (𝐼, 𝑝2 , 𝑞, 𝜃1 , 𝜃3 ), then (𝑋 ∗ , 𝑌 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ ) attains the global maximum of 𝐽. (b) Suppose that the demand function 𝑓 (𝑋, 𝑝1 , 𝜏) is multiplicatively separable in 𝑋 and 𝑝1 , that is

111

𝑓(𝑋, 𝑝1 , 𝜏) = 𝑔(𝑋)ℎ(𝑝1 , 𝜏) If 𝑈1 = (0, +∞) , and 𝑔(𝑋) is concave in 𝑋 ∈ [0, +∞), then (𝑋 ∗ , 𝑌 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ ) attains

the global maximum of 𝐽. Proof: The proof procedure is similar to Theorem 3.2 and Corollary 3.1. Let (𝑋, 𝑌, 𝐼, 𝑝1 , 𝑝2 , 𝑞, ) be an arbitrary feasible solution of the problem (4.2)-(4.7). Then (𝑋 ∗ , 𝑌 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ ) is optimal provided that the following inequality is always true ∆= 𝐽[𝑝1∗ , 𝑝̅2 , 𝑞 ∗ ] − 𝐽[𝑝1 , 𝑝2 , 𝑞] ≥ 0. Note that 𝑋̇ = 𝑓(𝑋, 𝑝1 , 𝜏) and 𝐼 ̇ = 𝑞 − 𝑓(𝑋, 𝑝1 , 𝜏). The integrand of the objective function (4.2) can be rewritten as 𝑓(𝑋, 𝑝1 , 𝜏)[𝑝1 − 𝐶𝐵𝑊 + 𝐸 (𝑝2 , 𝜂)] − 𝐶 (𝑞 ) − 𝑆(𝐼 ) = 𝐻 (𝑋, 𝐼, 𝑝1 , 𝑝2 , 𝑞, 𝜃1 , 𝜃3 ) − 𝜃1 𝑓(𝑋, 𝑝1 , 𝜏) − (𝑞 − 𝑓(𝑋, 𝑝1 , 𝜏))𝜃3 = 𝐻 (𝑋, 𝑝1 , 𝑝2 , 𝜃1 ) − 𝜃1 𝑋̇ − 𝜃3 𝐼 ̇. Then, 𝑻

∆= ∫ {[𝐻(𝑋 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 ) − 𝜃1 𝑋̇ ∗ − 𝜃3 𝐼 ̇∗ ] − [𝐻(𝑋, 𝐼, 𝑝1 , 𝑝2 , 𝑞, 𝜃1 , 𝜃3 ) − 𝜃1 𝑋̇ − 𝜃3 𝐼 ̇]}𝑑𝑡 𝟎

𝑻

𝑻

= ∫ [𝐻(𝑋 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 ) − 𝐻(𝑋, 𝐼, 𝑝1 , 𝑝2 , 𝑞, 𝜃1 , 𝜃3 )] 𝑑𝑡 + ∫ 𝜃1 (𝑋̇ − 𝑋̇ ∗ ) 𝑑𝑡 𝟎

𝟎

𝑻

+ ∫ 𝜃3 (𝐼 ̇ − 𝐼 ∗̇ ) 𝑑𝑡. 𝟎

Note that 𝐸 (𝑝2 , 𝜂) ≤ 𝐸 (𝑝̅2 , 𝜂) ∀ 𝑝2 ∈ 𝑈2 , and 𝑓(𝑋, 𝑝1 , 𝜏) > 0 ∀ 𝑝1 ∈ 𝑈1 . Hence 𝐻(𝑋 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 ) − 𝐻 (𝑋, 𝐼, 𝑝1 , 𝑝2 , 𝑞, 𝜃1 , 𝜃3 ) ≥ 𝐻 (𝑋 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 ) − 𝐻 (𝑋, 𝐼, 𝑝1 , 𝑝̅2 , 𝑞, 𝜃1 , 𝜃3 ).

112

(4.27)

Proof for (a) According to the assumption, the Hamiltonian 𝐻 (𝑋, 𝐼, 𝑝1 , 𝑝2 , 𝑞, 𝜃1 , 𝜃3 ) is concave in (𝑋, 𝑝1 ) for any given (𝐼, 𝑝2 , 𝑞, 𝜃1 , 𝜃3 ). Note that 𝐻 (𝑋, 𝐼, 𝑝1 , 𝑝2 , 𝑞, 𝜃1 , 𝜃3 ) = 𝑓 (𝑋, 𝑝1 , 𝜏)[𝑝1 − 𝐶𝐵𝑊 + 𝐸 (𝑝2 , 𝜂) + 𝜃1 − 𝜃3 ] − 𝑆(𝐼 ) + 𝜃3 𝑞 − 𝐶 (𝑞 ). In addition, −𝑆 (𝐼 ) is concave in 𝐼 and 𝜃3 𝑞 − 𝐶 (𝑞 ) is concave in 𝑞. Therefore, 𝐻 (𝑋, 𝐼, 𝑝1 , 𝑝2 , 𝑞, 𝜃1 , 𝜃3 ) must be concave in (𝑋, 𝐼, 𝑝1 , 𝑞) for any given (𝑝2 , 𝜃1 , 𝜃3 ). Then 𝐻 (𝑋, 𝐼, 𝑝1 , 𝑝̅2 , 𝑞, 𝜃1 , 𝜃3 ) − 𝐻 (𝑋 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 ) ≤ 𝐻𝑋 (𝑋 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 )[𝑋 − 𝑋 ∗ ] + 𝐻𝑝1 (𝑋 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 )[𝑝1 − 𝑝1∗ ] +𝐻𝐼 (𝑋 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 )[𝐼 − 𝐼 ∗ ] + 𝐻𝑞 (𝑋 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 )[𝑞 − 𝑞 ∗ ].

(4.28)

Theorem 4.1 implies that 𝐻𝑝1 (𝑋 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 ) = 0 , 𝐻𝑞 (𝑋 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 ) = 0 , 𝜃̇1 = −𝐻𝑋 (𝑋 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 ) , and 𝜃̇3 = −𝐻𝐼 (𝑋 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 ) . Substituting (4.27) and (4.28) into the expression of ∆ yields 𝑻

∗

∆≥ ∫ [𝐻 (𝑋 , 𝐼 𝟎

∗

, 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 )

𝑻

− 𝐻 (𝑋, 𝐼, 𝑝1 , 𝑝̅2 , 𝑞, 𝜃1 , 𝜃3 )] 𝑑𝑡 + ∫ 𝜃1 (𝑋̇ − 𝑋̇ ∗ ) 𝑑𝑡 𝟎

𝑻

+ ∫ 𝜃3 (𝐼 ̇ − 𝐼 ̇∗ )𝑑𝑡 𝟎

𝑻

𝑻

𝑻

𝑻

≥ ∫ 𝜃̇1 (𝑋 − 𝑋 ∗ ) 𝑑𝑡 + ∫ 𝜃1 (𝑋̇ − 𝑋̇ ∗ ) 𝑑𝑡 + ∫ 𝜃3 (𝐼 ̇ − 𝐼 ∗̇ )𝑑𝑡 + ∫ 𝜃3̇ (𝐼 − 𝐼 ∗ )𝑑𝑡 𝟎

𝟎

𝟎

𝟎

𝑇 𝑇 = (𝜃1 (𝑋 − 𝑋 ∗ )) | + 𝜃3 (𝐼 − 𝐼 ∗ ) | . 0 0 Note that 𝜃3 (𝑡) ≥ 𝑐1 for all 𝑡 ∈ [0, 𝑇] and the boundary conditions: 𝑋(0) = 𝑋 ∗ (0) = 0; 𝜃1 (𝑇) = 0; 𝑋 (0) = 𝑋 ∗ (0) = 0; 𝐼 (𝑇) ≥ 0; 𝐼 ∗ (𝑇) = 0. Then ∆≥ 0 + 𝜃3 (𝑇)𝐼(𝑇) ≥ 0. Consequently, (𝑋 ∗ , 𝑌 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ ) is optimal. Proof for (b) 113

Note that both 𝑔(𝑋) and ℎ(𝑝1 , 𝜏) are positive. Then 𝑝1∗ = 𝐴𝑟𝑔𝑚𝑎𝑥 𝐻(𝑋 ∗ , 𝐼 ∗ , 𝑝1 , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 ) 𝑝1 ∈𝑈1

= 𝐴𝑟𝑔𝑚𝑎𝑥 𝑓 (𝑋 ∗ , 𝑝1 , 𝜏)[𝑝1 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 − 𝜃3 ] − 𝑆 (𝐼 ∗ ) + 𝜃3 𝑞 ∗ − 𝐶 (𝑞 ∗ ) 𝑝1 ∈𝑈1

= 𝐴𝑟𝑔𝑚𝑎𝑥 𝑔(𝑋 ∗ )ℎ(𝑝1 , 𝜏)[𝑝1 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 − 𝜃3 ] 𝑝1 ∈𝑈1

= 𝐴𝑟𝑔𝑚𝑎𝑥 ℎ(𝑝1 , 𝜏)[𝑝1 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 − 𝜃3 ]. 𝑝1 ∈𝑈1

If 𝑝1 is sufficiently large, then max𝑝1 ∈𝑈1 ℎ(𝑝1 , 𝜏)[𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 ] must be positive, and 𝐻 (𝑋, 𝐼, 𝑝1 , 𝑝̅2 , 𝑞, 𝜃1 , 𝜃3 ) = 𝑔(𝑋)ℎ(𝑝1 , 𝜏)[𝑝1 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 − 𝜃3 ] − 𝑆 (𝐼 ) + 𝜃3 𝑞 − 𝐶 (𝑞 ) ≤ 𝑔(𝑋)ℎ(𝑝1∗ , 𝜏)[𝑝1∗ − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 − 𝜃3 ] − 𝑆 (𝐼 ) + 𝜃3 𝑞 − 𝐶 (𝑞 ) = 𝐻 (𝑋, 𝐼, 𝑝1∗ , 𝑝̅2 , 𝑞, 𝜃1 , 𝜃3 ). Substituting the above inequality into (4.27) yields 𝐻 (𝑋 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 ) − 𝐻 (𝑋, 𝐼, 𝑝1 , 𝑝2 , 𝑞, 𝜃1 , 𝜃3 ) ≥ 𝐻 (𝑋 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 ) − 𝐻 (𝑋, 𝐼, 𝑝1∗ , 𝑝̅2 , 𝑞, 𝜃1 , 𝜃3 ). Note that 𝑔(𝑋), −𝑆 (𝐼 ) and 𝜃3 𝑞 − 𝐶 (𝑞 ) are concave. Since 𝐻𝑞 (𝑋 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 ) = 0, 𝜃̇1 = −𝐻𝑋 (𝑋 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 ) and 𝜃̇3 = −𝐻𝐼 (𝑋 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 ), then 𝐻 (𝑋 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 ) − 𝐻 (𝑋, 𝐼, 𝑝1∗ , 𝑝̅2 , 𝑞, 𝜃1 , 𝜃3 ) ≥ −𝐻𝑋 (𝑋 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 )[𝑋 − 𝑋 ∗ ] − 𝐻𝐼 (𝑋 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 )[𝐼 − 𝐼 ∗ ] − 𝐻𝑞 (𝑋 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 )[𝑞 − 𝑞 ∗ ] = 𝜃̇1 [𝑋 − 𝑋 ∗ ] + 𝜃̇3 [𝐼 − 𝐼 ∗ ]. Therefore,

114

𝑻

∗

∆≥ ∫ [𝐻(𝑋 , 𝐼 𝟎

∗

, 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ , 𝜃1 , 𝜃3 ) −

𝐻 (𝑋, 𝐼, 𝑝1∗ , 𝑝̅2 , 𝑞, 𝜃1 , 𝜃3 )] 𝑑𝑡

𝑻

+ ∫ 𝜃1 (𝑋̇ − 𝑋̇ ∗ ) 𝑑𝑡 𝟎

𝑻

+ ∫ 𝜃3 (𝐼 ̇ − 𝐼 ̇∗ )𝑑𝑡 𝟎

𝑻

𝑻

≥ ∫ [𝜃̇1 (𝑋 − 𝑋 ∗ ) + 𝜃1 (𝑋̇ − 𝑋̇ ∗ )] 𝑑𝑡 + ∫ [𝜃3 (𝐼 ̇ − 𝐼 ∗̇ ) + 𝜃3̇ (𝐼 − 𝐼 ∗ )]𝑑𝑡 ≥ 0. 𝟎

𝟎

Consequently, (𝑋 ∗ , 𝑌 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ ) is optimal. ∎ Consider the price elasticity of demand for the new product (𝑃𝐸𝐷𝑝1 ) and the price elasticity of demand for the EWC ( 𝑃𝐸𝐷𝑃2 ) introduced in Sections 3.3. The following Proposition 4.3 summaries some interesting properties of the optimal solution (𝑋 ∗ , 𝑌 ∗ , 𝐼 ∗ , 𝑝1∗ , 𝑝̅2 , 𝑞 ∗ ) for the problem (4.2)-(4.7). In the remaining section of this chapter, the superscript “*” in the notation of the optimal solution will be omitted unless otherwise stated and the arguments in a multivariable function will be suppressed, e.g., 𝑓 = 𝑓 (𝑋, 𝑝1 , 𝜏).

Proposition 4.3 Suppose (𝑋, 𝑌, 𝐼, 𝑝1 , 𝑝2 , 𝑞 ) is an optimal solution for the dynamic problem (4.2)(4.7) with the costate functions 𝜃1 and 𝜃3 . (a) The marginal revenues with respect to the demand rates of the new product and the EWC can be expressed as (1 −

1 ) 𝑝 = 𝜃3 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂) − 𝜃1 ; 𝑃𝐸𝐷𝑝1 1 (1 −

1 ) 𝑝̅ = 𝐶𝐸𝑊𝐶 . 𝑃𝐸𝐷𝑝2 2

(b) If 𝑓(𝑋, 𝑝1 , 𝜏) has a second-order partial derivative with respect to (𝑋, 𝑝1 ), then the optimal price path 𝑝1 is smooth over time, and the derivative satisfies 115

(2𝑓𝑝21

𝑆̇(𝐼 )𝑓𝑝21 − 𝑓𝑓𝑝1 𝑝1 )𝑝̇1 = (−2𝑓𝑋 𝑓𝑝1 + 𝑓𝑓𝑝1 𝑋 + ) 𝑓, 𝑓

as well as the end time boundary condition 𝑝1 (𝑇) +

𝑓 (𝑋 (𝑇), 𝑝1 (𝑇), 𝜏) = 𝜃3 (𝑇) + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂). 𝑓𝑝1 (𝑋(𝑇), 𝑝1 (𝑇), 𝜏)

(c) When the demand function 𝑓 (𝑋, 𝑝1 , 𝜏) is multiplicatively separable in 𝑋 and 𝑝1 , that is 𝑓(𝑋, 𝑝1 , 𝜏) = 𝑔(𝑋)ℎ(𝑝1 , 𝜏). The result in (b) can be simplified to (2ℎ𝑝21 − ℎℎ𝑝1 𝑝1 )𝑝̇1 = −𝑔̇ (𝑋)ℎℎ𝑝1 + 𝑆̇ (𝐼 )ℎ2𝑝1 , as well as the end time boundary condition 𝑝1 (𝑇) +

ℎ (𝑝1 (𝑇), 𝜏) = 𝜃3 (𝑇) + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂). ℎ𝑝1 (𝑝1 (𝑇), 𝜏)

Proof: Proof for (a). The optimal prices 𝑝1 and 𝑝̅2 must satisfy (4.18) and (4.16) which imply 𝐻𝑝1 = (𝑝1 − 𝜃3 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 )𝑓𝑝1 + 𝑓 = 0; 𝐸𝑝2 (𝑝̅2 , 𝜂) = (𝑝̅2 − 𝐶𝐸𝑊𝐶 )𝜌𝑝2 + 𝜌 = 0. Then (1 −

1 𝑓 ) 𝑝1 = (1 + ) 𝑝 = 𝜃3 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂) − 𝜃1 ; 𝑃𝐸𝐷𝑝1 𝑝1 𝑓𝑝1 1 (1 −

1 𝜌 ) 𝑝̅2 = 𝑝̅2 (1 + ) = 𝐶𝐸𝑊𝐶 . 𝑃𝐸𝐷𝑝2 𝑝̅2 𝜌𝑝2

116

Proof for (b) Consider (4.25) at the end time 𝑡 = 𝑇, that is 𝑝1 (𝑇) − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 (𝑇) − 𝜃3 (𝑇) = −

𝑓 (𝑋(𝑇), 𝑝1 (𝑇), 𝜏) . 𝑓𝑝1 (𝑋(𝑇), 𝑝1 (𝑇), 𝜏)

Note that 𝜃1 (𝑇) = 0. Hence,

𝑝1 (𝑇) +

𝑓(𝑋(𝑇), 𝑝1 (𝑇), 𝜏) = 𝜃3 (𝑇) + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂). 𝑓𝑝1 (𝑋 (𝑇), 𝑝1 (𝑇), 𝜏)

Next, take the derivative with respect to t on both sides of (4.25) 𝑝̇1 + 𝜃̇1 − 𝜃̇3 = −

𝑓𝑝1 ∙ 𝑝̇1 + 𝑓𝑋 ∙ 𝑋̇ (𝑓𝑝1 𝑋 ∙ 𝑋̇ + 𝑓𝑝1 𝑝1 ∙ 𝑝̇1 )𝑓 + . 𝑓𝑝1 𝑓𝑝21

Substituting 𝜃̇1 = −(𝑝1 − 𝜃3 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 )𝑓𝑋 = 𝑓𝑋

𝑓 𝑓𝑝1

and 𝜃̇3 = 𝑆̇(𝐼 ) into the above

equation yields (2𝑓𝑝21 − 𝑓𝑓𝑝1 𝑝1 )𝑝̇1 = (−2𝑓𝑋 𝑓𝑝1 + 𝑓𝑓𝑝1 𝑋 +

𝑆̇ (𝐼 )𝑓𝑝21 ) 𝑓. 𝑓

Notice that 𝑓 > 0 and 2𝑓𝑝21 − 𝑓𝑓𝑝1 𝑝1 ≥ 0 obtained from (4.26). The sign of 𝑝̇1 is totally determined by −2𝑓𝑋 𝑓𝑝1 + 𝑓𝑓𝑝1 𝑋 + 𝑆̇ (𝐼)𝑓𝑝21 /𝑓. Proof for (c) It follows directly form (b) by substituting 𝑓(𝑋, 𝑝1 , 𝜏) = 𝑔(𝑋)ℎ(𝑝1 , 𝜏) into the results of (b). ∎

4.4 Case Study As discussed in Remark 4.2, the optimal price for the EWC would be a constant function 𝑝̅2 . Hence, the key problem is to derive the optimal price 𝑝1 and the optimal production rate 𝑞 for the

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new product. This section examines this problem when specifying two special type of demand functions 𝑓(𝑋, 𝑝1 , 𝜏).

4.4.1 Static Market Consider a demand function 𝑓 which depends on 𝑝1 and 𝜏, i.e., 𝑓 = ℎ(𝑝1 , 𝜏). In this case, the costate functions 𝜃1 degenerates to 𝜃1 = 0 ∀𝑡 ∈ [0, 𝑇] due to 𝑓𝑋 = 0. The state constraint (4.19) has no influence on the objective function, so the Hamiltonian function H can be simplified as: 𝐻 (𝐼, 𝑝1 , 𝑝̅2 , 𝑞, 𝜃3 ) = ℎ(𝑝1 , 𝜏)[𝑝1 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) − 𝜃3 ] − 𝑆(𝐼 ) + 𝜃3 𝑞 − 𝐶 (𝑞 ). Given the specific form of the demand function 𝑓 = ℎ(𝑝1 , 𝜏) , it is possible to derive more concrete conclusions about the optimal price 𝑝1 and the optimal production rate 𝑞. To illustrate the general analysis procedure which may vary from case to case, one example will be studied in the remaining subsection.

Example 4.1 Additive price-warranty demand function ℎ(𝑝1 , 𝜏) = 𝑎0 − 𝑎1 𝑝1 + 𝑎2 𝜏. Solution: The Hamiltonian for the additive price-warranty demand function is 𝐻 (𝐼, 𝑝1 , 𝑝̅2 , 𝑞, 𝜃3 ) = (𝑎0 − 𝑎1 𝑝1 + 𝑎2 𝜏)[𝑝1 − 𝐶𝐵𝑊 − 𝜃3 + 𝐸 (𝑝̅2 , 𝜂)] − (𝑐0 + (𝑐1 − 𝜃3 )𝑞 + 𝑐2 𝑞 2 ) − 𝑆 (𝐼 ) = −𝑎1 𝑝12 + [(𝑎0 + 𝑎2 𝜏) + 𝑎1 (𝐶𝐵𝑊 + 𝜃3 − 𝐸 (𝑝̅2 , 𝜂))]𝑝1 − (𝑎0 + 𝑎2 𝜏)(𝐶𝐵𝑊 + 𝜃3 − 𝐸 ) − (𝑐0 + (𝑐1 − 𝜃3 )𝑞 + 𝑐2 𝑞 𝛽+1 ) − 𝑆 (𝐼 ). Based on Theorem 4.1 as well as Remarks 4.2 and 4.3, the optima price 𝑝1 and optimal production rate 𝑞 can be explicitly expression as function of 𝜃3 : 𝑝1 =

𝑎1 (𝐶𝐵𝑊 + 𝜃3 − 𝐸 (𝑝̅2 , 𝜂)) + (𝑎0 + 𝑎2 𝜏) ; 2𝑎1

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1

𝜃3 − 𝑐1 𝛽 𝑞=( ) . (𝛽 + 1)𝑐2 Then, the remaining task is to solve the following equation system

{

𝐼 ̇ = 𝑞 − (𝑎0 − 𝑎1 𝑝1 + 𝑎2 𝜏),

𝐼 (0) = 𝐼 (𝑇) = 0;

𝜃3̇ = 𝑆̇(𝐼 ).

Substituting 𝑝1 and 𝑞 into the above equations yields 1

𝜃3 − 𝑐1 𝛽 𝑎1 (𝐶𝐵𝑊 + 𝜃3 − 𝐸 (𝑝̅2 , 𝜂)) − (𝑎0 + 𝑎2 𝜏) 𝐼̇ = ( ) + , 𝐼 (0) = 𝐼(𝑇) = 0; { (𝛽 + 1)𝑐2 2 𝜃3̇ = 𝑆̇ (𝐼 ). It is difficult to directly find the closed-form solution of the differential equations about (𝐼, 𝜃3 ). Therefore, one can first consider a simplified case with 𝛽 = 1 and 𝑆 (𝐼 ) = 𝛿𝐼 2 . The equation system degenerates into 𝐼̇ = {

𝜃3 − 𝑐1 𝑎1 (𝐶𝐵𝑊 + 𝜃3 − 𝐸 (𝑝̅2 , 𝜂)) − (𝑎0 + 𝑎2 𝜏) + , 𝑐2 2 2 𝜃3̇ = 2𝛿𝐼.

𝐼 (0) = 𝐼(𝑇) = 0;

Taking the derivative of 𝐼 ̇ with respect to t yields 𝐼̈ =

𝜃̇3 𝑎1 𝜃̇3 (𝑎1 𝑐2 + 1)𝜃̇3 (𝑎1 𝑐2 + 1)𝛿 + = = 𝐼, 𝑐2 2 2 𝑐2 2 2𝑐2

𝐼(0) = 𝐼(𝑇) = 0.

Set 𝑅 = ((𝑎1 𝑐2 + 1)𝛿)/2𝑐2 , the general solution for the above second order differential equation can be expressed as 𝐼(𝑡) = 𝑘1 𝑒 √𝑅𝑡 + 𝑘2 𝑒 −√𝑅𝑡 . where 𝑘1 and 𝑘2 are two constant coefficients.

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Because the solution of 𝐼(𝑡) must satisfy the initial time and end time boundary conditions 𝐼 (0) = 𝐼(𝑇), that is

{

𝐼 (0) = 𝑘1 + 𝑘2 = 0; 𝐼 (𝑇) = 𝑘1 𝑒 √𝑅𝑇 + 𝑘2 𝑒 −√𝑅𝑇 = 0.

Obviously, 𝑘1 = 𝑘2 = 0. Therefore, 𝐼 (𝑡) ≡ 0 ∀ 𝑡 ∈ [0, 𝑇] , i.e., there is no backlog or positive inventory during the planning horizon. In addition, 𝐼 (𝑡) ≡ 0 ∀ 𝑡 ∈ [0, 𝑇] also implies that 𝐼̇ =

𝜃3 − 𝑐1 𝑎1 (𝐶𝐵𝑊 + 𝜃3 − 𝐸 (𝑝̅2 , 𝜂)) − (𝑎0 + 𝑎2 𝜏) + = 0, 𝑐2 2 2

i.e., 𝜃3 =

𝑎1 𝑐2 (𝐸 (𝑝̅2 , 𝜂) − 𝐶𝐵𝑊 − 𝑐1 ) + (𝑎0 + 𝑎2 𝜏)𝑐2 + 𝑐1 . 1 + 𝑎1 𝑐2

Thus 𝑝1 =

−𝑎1 (𝐸 (𝑝̅2 , 𝜂) − 𝐶𝐵𝑊 − 𝑐1 ) + (𝑎0 + 𝑎2 𝜏)(1 + 2𝑎1 𝑐2 ) ; 2(1 + 𝑎1 𝑐2 )𝑎1 𝑞=

Because 𝑝1 ∈ (0,

𝑎0 +𝑎2 𝜏 𝑎1

𝑎1 (𝐸(𝑝̅2 , 𝜂) − 𝐶𝐵𝑊 − 𝑐1 ) + (𝑎0 + 𝑎2 𝜏) . 2(1 + 𝑎1 𝑐2 )

), i.e., ( 𝑎 0 + 𝑎2 𝜏 ) > 𝐶𝐵𝑊 + 𝑐1 − 𝐸 (𝑝̅2 , 𝜂), 𝑎1

which implies 𝜃3 > 𝑐1, which agrees with Proposition 4.1.

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So far, we have proved that 𝐼 (𝑡) ≡ 0 when 𝛽 = 1 and 𝑆 (𝐼 ) = 𝛿𝐼 2 . What is the conclusion if 𝛽 > 0 and 𝑆(𝐼 ) is an arbitrary inventory / backlogging cost function depicted in Section 4.2? The following phase diagram analysis in (𝐼, 𝜃3 ) space asserts that 𝐼 (𝑡) ≡ 0. Start with identifying the points where 𝐼 ̇ = 0 and 𝜃3̇ = 0, i.e., 1

𝜃3 − 𝑐1 𝛽 𝑎1 (𝐶𝐵𝑊 + 𝜃3 − 𝐸 (𝑝̅2 , 𝜂)) − (𝑎0 + 𝑎2 𝜏) ( ) + = 0; { (𝛽 + 1)𝑐2 2 𝑆̇(𝐼 ) = 0. Obviously, the straight line 𝐼 = 0 corresponds to the set of the points satisfying 𝜃3̇ = 0. Let 1

𝜃3 − 𝑐1 𝛽 𝑎1 (𝐶𝐵𝑊 + 𝜃3 − 𝐸 (𝑝̅2 , 𝜂)) − (𝑎0 + 𝑎2 𝜏) 𝜁(𝜃3 ) = ( ) + . (𝛽 + 1)𝑐2 2 The roots of 𝜁 (𝜃3 ) = 0 satisfy 𝐼 ̇ = 0. Lemma 4.4 Suppose

(𝑎0+𝑎2 𝜏) 𝑎1

> 𝐶𝐵𝑊 + 𝑐1 − 𝐸 (𝑝̅2 , 𝜂). There exists a unique 𝜃3° ∈ (𝑐1 , +∞) such

that 𝜁(𝜃3° ) = 0. Proof: Let 1

𝜃3 − 𝑐1 𝛽 𝜁1 (𝜃3 ) = ( ) ; (𝛽 + 1)𝑐2 𝜁2 (𝜃3 ) = −𝜃3 + Because 𝜁 (𝜃3 ) = 𝜁1 (𝜃3 ) − satisfying 𝜁1 (𝜃3 ) =

𝑎1 2

𝑎1 2

𝑎0 + 𝑎2 𝜏 − (𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂)). 𝑎1

𝜁2 (𝜃3 ) , the point satisfying 𝜁(𝜃3 ) = 0 is equivalent to the one

𝜁2 (𝜃3 ).

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Note that

(𝑎0 +𝑎2 𝜏) 𝑎1

− 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) > 𝑐1 . It implies the two functions 𝜁1 (𝜃3 ) and

𝑎1 2

𝜁2 (𝜃3 )

must intersect at some point of 𝜃3° ∈ (𝑐1 , +∞) and then diverge again. Figure 4.4 illustrates this property and validates the result in Lemma 4.4.

Figure 4.4 Trajectories of the functions 𝜁1 (𝜃3 ) and

𝑎1 2

𝜁2 (𝜃3 ) ∎

Although the proof of Lemma 4.2 is based on a geometric approach, it can also be proved through algebraic derivation, which is rather cumbersome and hence omitted here. Lemma 4.2 indicates that the straight line 𝜃3 = 𝜃3° satisfies 𝐼 ̇ = 0. Therefore, the two lines, 𝐼 = 0 and 𝜃3 = 𝜃3° , divide the (𝐼, 𝜃3 ) space into four regions, which are labeled Ⅰ-Ⅳ as shown in Figure 4.5. Note that the region 𝜃3 ≤ 𝑐1 can be ignored as 𝜃3 > 𝑐1. Since the inventory level 𝐼 satisfies the initial time and end time boundary conditions 𝐼(0) = 𝐼 (𝑇) = 0, the trajectory starts at some point on the line 𝐼 = 0, then it travels for a time period of T, and returns to the line 𝐼 = 0 at the end time T. Obviously, the initial point at the line 𝐼 = 0 has only three possible locations: 

𝜃3 > 𝜃3° which corresponds to the trajectory (a) in Figure 4.5;

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

𝑐1 < 𝜃3 < 𝜃3° which corresponds to the trajectory (b) in Figure 4.5;



𝜃3 = 𝜃3° .

As indicated in the (𝐼, 𝜃3 ) space phase diagram, if the trajectory begins at some point 𝜃3 > 𝜃3° , it will go up and to the right and is impossible to travel back to the line 𝐼 = 0 at the end time T. Similarly, the trajectory can never begin at some point 𝑐1 < 𝜃3 < 𝜃3° either. Therefore, the trajectory can only begin at the point 𝜃3 = 𝜃3° and remain there over the whole time period [0, T]. Therefore, 𝐼 (𝑡) ≡ 0, 𝑡 ∈ [0, 𝑇];

𝑝1 =

𝑎1 (𝐶𝐵𝑊 + 𝜃3° − 𝐸 (𝑝̅2 , 𝜂)) + (𝑎0 + 𝑎2 𝜏) 2𝑎1 1

𝜃3° − 𝑐1 𝛽 𝑞=( ) . (𝛽 + 1)𝑐2

Figure 4.5 The (𝐼, 𝜃3 ) space phase diagram

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;

4.4.2 Dynamic Market Consider the dynamic market introduced in Section 3.5.2. Theorem 4.2 (b) and Proposition 4.3 (c) have provided some important results regarding a typical type of such demand rate function which is multiplicatively separable in 𝑋 and 𝑝1 , that is 𝑓(𝑋, 𝑝1 , 𝜏) = 𝑔(𝑋)ℎ(𝑝1 , 𝜏). However, it is difficult to derive the closed-form solution even when the demand function has been specified. The following Example 4.2 illustrates that the phase diagram method is not applicable in this situation and the most feasible approach is the numerical method.

Example 4.2 Bass-type demand function 𝑓 (𝑋, 𝑝1 , 𝜏) = 𝑘1 [𝑘2 + 𝜏]𝑏 𝑝1−𝑎 [𝑀 − 𝑋] [𝑞1 +

𝑞2 𝑋 ]. 𝑀

(Note: 𝑞1 = 1, 𝑞2 = 0 is the limited-growth demand function) Solution: The Hamiltonian for the Bass-type demand function is 𝐻 (𝑋, 𝐼, 𝑝1 , 𝑝2 , 𝑞, 𝜃1 , 𝜃3 ) = 𝑘1 [𝑘2 + 𝜏]𝑏 𝑝1−𝑎 [𝑀 − 𝑋] [𝑞1 +

𝑞2 𝑋 ] [𝑝1 − 𝐶𝐵𝑊 + 𝐸 (𝑝2 , 𝜂) + 𝜃1 − 𝜃3 ] − 𝑆(𝐼 ) + 𝜃3 𝑞 − 𝐶 (𝑞 ). 𝑀 1 𝛽

𝜃3 −𝑐1

According to Remark 4.2, 𝑝2 = 𝑝̅2 and 𝑞 = ((𝛽+1)𝑐 ) . Hence, the challenge is to obtain the 2

quintuple (𝑋, 𝐼, 𝑝1 , 𝜃1 , 𝜃3 ) by solving (4.24), that is

124

𝑞2 𝑋 ] 𝑘 (𝑘 + 𝜏)𝑏 𝑝1−𝑎 [𝑝1 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 − 𝜃3 ] ; 𝑀 1 2 𝑞2 𝑋 𝑋̇ = [𝑀 − 𝑋] [𝑞1 + ] 𝑘 (𝑘 + 𝜏)𝑏 𝑝1−𝑎 , 𝑋(0) = 0; 𝑀 1 2

𝑝1 = 𝐴𝑟𝑔𝑚𝑎𝑥[𝑀 − 𝑋] [𝑞1 + 𝑝1 ∈𝑈1

1

𝜃3 − 𝑐1 𝛽 𝑞2 𝑋 𝐼̇ = ( ) − [𝑀 − 𝑋] [𝑞1 + ] 𝑘1 (𝑘2 + 𝜏)𝑏 𝑝1−𝑎 , 𝐼 (0) = 𝐼 (𝑇) = 0; (𝛽 + 1)𝑐2 𝑀 2𝑞2 𝜃̇1 = − [𝑞2 − 𝑞1 − 𝑋] 𝑘1 (𝑘2 + 𝜏)𝑏 𝑝1−𝑎 [𝑝1 − 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂) + 𝜃1 − 𝜃3 ], 𝜃1 (𝑇) = 0; 𝑀 { 𝜃̇3 = 𝑆̇ (𝐼 ). It is impossible to derive an explicit closed-form solution for the quintuple (𝑋, 𝐼, 𝑝1 , 𝜃1 , 𝜃3 ) due to the complexity of this equation system. Moreover, the interrelationship among the quintuple (𝑋, 𝐼, 𝑝1 , 𝜃1 , 𝜃3 ) is too complicated to separate them into pairs with only two variables, e.g., (𝐼, 𝜃3 ) in Example 4.1. Therefore, the phase diagram technique is not applicable for studying the qualitative properties. A possible approach is to generate numerical solutions through numerical methods. Unfortunately, the gradient algorithm developed in Chapter 3 is no longer applicable for the dynamic problem (4.2)-(4.7) formulated in this chapter, because the boundary conditions are more complicated. As indicated in Theorem 4.1 and Proposition 4.2, the inventory must satisfy both the starting time and end time conditions I(t) = I(T) = 0. However, the costate function 𝜃3 has no boundary condition. Therefore, it is difficult to apply Steps 2-3 in the gradient algorithm to problem (4.2)-(4.7), and there is no guarantee that the gradient algorithm can converge to an optimal solution. Therefore, a more robust algorithm based on the idea of control vector parameterization has been developed in this chapter. The basic idea of control vector parameterization is to approximate the control decision variables by the linear combination of piecewise-constant functions. This approximating scheme discretizes the original optimal control problem and yields a finite-dimensional non-linear optimization problem, where the decision variables correspond to the coefficients in the linear combination. Section 4.5 presents the details of this approach.

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4.5 Numerical Study 4.5.1 Control Vector Parameterization To illustrate the idea of control vector parameterization, consider a general but typical optimal control problem: 𝑇

𝑀𝑎𝑥 𝐽[𝑢(𝑡)] = ∫ 𝐿(𝑥(𝑡), 𝑢(𝑡), 𝑡)𝑑𝑡.

𝑢(𝑡)∈𝑈

(4.29)

0

with the following state equation and the boundary conditions: 𝑥̇ (𝑡) = 𝑓 (𝑥(𝑡), 𝑢(𝑡), 𝑡);

(4.30)

𝑥 (0) = 𝑥 0 ,

(4.31)

𝑥 (𝑇 ) = 𝑥 𝑇 .

To simplify the problem, 𝑥 and 𝑢 are assumed to be scalars. It is straightforward to generalize the approach to the case where 𝑥 and 𝑢 are vectors. As shown in Figure 4.6, any piecewise control 𝑢(𝑡) can be approximated by a piecewiseconstant basis function 𝑢𝑝 (𝑡|𝑦⃗):

𝑁−1

𝑢𝑝 (𝑡|𝑦⃗) ≔ ∑ 𝑦𝑖 𝜒[𝑡𝑖 ,𝑡𝑖+1) (𝑡) , 𝑡 ∈ [0, 𝑇], 𝑘=0

where 𝑦⃗ = [𝑦0 , 𝑦1 , … , 𝑦𝑁−1 ]𝑇 ∈ 𝑈 𝑁 and 𝜒[𝑡𝑖 ,𝑡𝑖+1 ) (𝑡) ≔ {

126

1, if 𝑡 ∈ [𝑡𝑖 , 𝑡𝑖+1 ) . 0, if 𝑡 ∈ [𝑡𝑖 , 𝑡𝑖+1 )

(4.32)

Figure 4.6 Control parameterization

Substituting (4.32) and the initial condition 𝑥(0) = 𝑥0 into the state equation (4.30) yields the corresponding state variable 𝑥 𝑝 (𝑡|𝑦⃗), which is uniquely defined by 𝑢𝑝 (𝑡|𝑦⃗), because (4.32) and the initial condition 𝑥(0) = 𝑥0 form a standard initial value problem (IVP). Next, substituting 𝑢𝑝 (𝑡|𝑦⃗) and 𝑥 𝑝 (𝑡|𝑦⃗) into the objective function (4.29) yields 𝑇

𝑀𝑎𝑥 𝐽(𝑦⃗) = ∫ 𝐿(𝑥 𝑝 (𝑡|𝑦⃗), 𝑢𝑝 (𝑡|𝑦⃗), 𝑡)𝑑𝑡. 𝑁

𝑦⃗⃗∈𝑈

(4.33)

0

Now, define the following approximation problem: choose a vector 𝑦⃗ ∈ 𝑈 𝑁 that maximizes the objective function (4.33) subject to the end time constraint ℱ (𝑦⃗) = 𝑥 𝑝 (𝑇|𝑦⃗) − 𝑥 𝑇 = 0.

(4.34)

The above approximation problem is a nonlinear optimization problem (NLP) with N decision variables. Thus, it is much easier to solve than the original dynamic optimal control problem, because the optimal control problem needs to determine the value of a function 𝑢(𝑡) at an infinite number of time points (i.e., every point in the time interval [0, 𝑇] ). After these preliminary works, an important question is how to solve the approximation NLP problem.

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For the approximation problem (4.33)-(4.34), the objective and constraint functions, i.e., 𝐽(𝑦⃗) and ℱ (𝑦⃗), are implicit rather than explicit functions of the decision vector 𝑦⃗. The standard algorithms for NLP (e.g., sequential quadratic programming and interior point method) exploit the gradients of the objective function and the constraints to generate search directions. In order to apply these approaches, we have to compute the gradients of 𝐽(𝑦⃗) and ℱ (𝑦⃗) with respect to each component 𝑦𝑘 of the control vector 𝑦⃗, which are denoted by 𝐽𝑦𝑘 (𝑦⃗) and ℱ𝑦𝑘 (𝑦⃗), 𝑘 = 0, 1, … ,𝑁 − 1. To compute the gradients of 𝐽(𝑦⃗) and ℱ (𝑦⃗), consider the IVP for the vector 𝑦⃗ = [𝑦0 , 𝑦1 , … , 𝑦𝑁−1 ]𝑇 , that is 𝑥̇ 𝑝 (𝑡|𝑦⃗) = 𝑓(𝑥 𝑝 (𝑡|𝑦⃗), 𝑢𝑝 (𝑡|𝑦⃗), 𝑡),

𝑥(0) = 0.

Take the derivative on both sides of the above equation with respect to 𝑦𝑘 , 𝑘 = 0,1, … , 𝑁 − 1, then 𝑝

𝑝

𝑥̇ 𝑦𝑘 (𝑡|𝑦⃗) = 𝑓𝑥 (𝑥 𝑝 (𝑡|𝑦⃗), 𝑢𝑝 (𝑡|𝑦⃗), 𝑡)𝑥𝑦𝑘 (𝑡|𝑦⃗) + 𝑓𝑦𝑘 (𝑥 𝑝 (𝑡|𝑦⃗), 𝑢𝑝 (𝑡|𝑦⃗), 𝑡), 𝑥𝑦𝑘 (0) = 0.

(4.35)

𝑝

𝑥𝑦𝑘 (𝑡|𝑦⃗) is called the sensitivity variables (Errico, 1997; Bea, 1998; Chachuat, 2009) and (4.35) the sensitivity equations with respect to 𝑦𝑘 , 𝑘 = 0,1, … , 𝑁 − 1 . Obviously, the sensitivity equations are also IVPs, and so numerical methods, e.g., Runge-Kutta, can be applied to obtain 𝑝

𝑥𝑦𝑘 (𝑡|𝑦⃗). The gradients of the objective and constraint functions in (4.33) and (4.34) can then be computed using the chain rule of differentiation: 𝑇

𝑝 𝐽𝑦𝑘 (⃗𝑦⃗) = ∫ [𝑓𝑥 (𝑥𝑝 (𝑡|⃗𝑦⃗), 𝑢𝑝 (𝑡|⃗𝑦⃗), 𝑡)𝑥𝑦 (𝑡|⃗𝑦⃗) + 𝑓𝑦 (𝑥𝑝 (𝑡|⃗𝑦⃗), 𝑢𝑝 (𝑡|⃗𝑦⃗), 𝑡)]𝑑𝑡 ; 0

𝑘

𝑘

𝑝

ℱ𝑦𝑘 (𝑦⃗) = 𝑥𝑦𝑘 (𝑇|𝑦⃗). where 𝑘 = 0,1, … , 𝑁 − 1.

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In summary, given the value of 𝑢𝑝 (𝑡|𝑦⃗), the procedure to compute both the values and the derivatives of the objective and constraint functions is as follows:

Table 4.1 Gradient computation procedure (a) Integrate the state and sensitivity variables from 0 to 𝑇: 𝑥̇ 𝑝 (𝑡|𝑦⃗) = 𝑓 (𝑥 𝑝 (𝑡|𝑦⃗), 𝑢𝑝 (𝑡|𝑦⃗), 𝑡), 𝑝

𝑥(0) = 0;

𝑝

𝑥̇ 𝑦𝑘 (𝑡|𝑦⃗) = 𝑓𝑥 (𝑥 𝑝 (𝑡|𝑦⃗), 𝑢𝑝 (𝑡|𝑦⃗), 𝑡)𝑥𝑦𝑘 (𝑡|𝑦⃗) + 𝑓𝑦𝑘 (𝑥 𝑝 (𝑡|𝑦⃗), 𝑢𝑝 (𝑡|𝑦⃗), 𝑡),

𝑥𝑦𝑘 (0) = 0

for all 𝑘 = 0,1, … , 𝑁 − 1 𝑝

(b) Given 𝑥 𝑝 (𝑡|𝑦⃗) and 𝑥𝑦𝑘 (𝑡|𝑦⃗), compute 𝑇

𝐽(𝑦⃗) = ∫ 𝐿(𝑥 𝑝 (𝑡|𝑦⃗), 𝑢𝑝 (𝑡|𝑦⃗), 𝑡)𝑑𝑡 ; 0

ℱ (𝑦⃗) = 𝑥 𝑝 (𝑇|𝑦⃗) − 𝑥 𝑇 ; 𝑇

𝑝

𝐽𝑦𝑘 (𝑦⃗) = ∫ [𝑓𝑥 (𝑥 𝑝 (𝑡|𝑦⃗), 𝑢𝑝 (𝑡|𝑦⃗), 𝑡)𝑥𝑦𝑘 (𝑡|𝑦⃗) + 𝑓𝑦𝑘 (𝑥 𝑝 (𝑡|𝑦⃗), 𝑢𝑝 (𝑡|𝑦⃗), 𝑡)]𝑑𝑡 ; 0

𝑝

ℱ𝑦𝑘 (𝑦⃗) = 𝑥𝑦𝑘 (𝑇|𝑦⃗) for all 𝑘 = 0,1, … , 𝑁 − 1.

So far, the gradient computation idea has been illustrated. And the next task is to achieve the gradient computation procedure with numerical calculation method. Obviously, the state and sensitivity variables in Table 4.1 step (a) form a standard IVP. In addition, the integrated form of 𝐽(𝑦⃗) and 𝐽𝑦𝑘 (𝑦⃗) are also essentially IVPs. Thus, the numerical method to solve IVPs can be 𝑝

applied to compute 𝑥 𝑝 (𝑡|𝑦⃗) , 𝑥𝑦𝑘 (𝑡|𝑦⃗) , 𝐽(𝑦⃗) and 𝐽𝑦𝑘 (𝑦⃗) , such as the Runge-Kutta method adopted in the gradient algorithm proposed in Chapter 3. Finally, the last task is to update the value of 𝑦⃗ and attain the maximum of the objective function. This can be achieved via the state-of-the-art NLP algorithm, and then the approximation problem

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(4.33)-(4.34) can be solved efficiently. Based on these discussions, Table 4.2 summarises the control vector parameterization algorithm as follows:

Table 4.2 Control vector parameterization algorithm outline 1. Initialization of the decision variables 𝑦⃗ = 𝑦⃗0 . 2. Forward integration of the state and sensitivity variables with the initial conditions 𝑥(0) = 𝑥0 and 𝑥𝑦𝑘 (0) = 0, c.f. Table 4.1 (a). (the 4-step Runge-Kutta method) 3. Calculation of the values and derivatives of the objective and constraint functions, c.f. Table 4.1 (b). (the 4-step Runge-Kutta method) 4. Iterative solution of the NLP (i.e., the NLP algorithm): (a) If the optimum is achieved (the optimality conditions are satisfied), the algorithm terminates and returns the solution 𝑦⃗. (b) Otherwise, go to Step 2 with new values of the decision variables 𝑦⃗: = 𝑦⃗𝑛𝑒𝑤 , which are determined based on the gradients obtained in Step 3. Note: The NLP algorithm applied in the numerical examples of this study is the interior point algorithm (Wächter, et al. 2006), which is implemented by the fmincon function in Matlab’s Optimization Toolbox.

4.5.2 Numerical Results Numerical experiment 1 The framework of the optimal control problem (4.29)-(4.31) includes both the initial time and end time boundary conditions. If one relaxes the end time boundary condition, the approximation problem for (4.29)-(4.31) degenerates into an unconstrained NLP (4.33). Review the optimal control problem (3.5)-(3.10) in Chapter 3. Note that the problem (3.5)-(3.10) only considers the initial time boundary conditions. This means that problem (3.5)-(3.10) can be

130

viewed as a special case of problem (4.29)-(4.31). Control vector parameterization, therefore, is applicable to solve the problem (3.5)-(3.10). In order to evaluate the performance of the control vector parameterization method, the first numerical experiment applies the control vector parameterization to solve the problem (3.5)-(3.10) based on the demand function in Example 3.3, and compares the numerical results with the analytical solutions. (Note: As the optimal price for the EWC has been proved to be the constant 𝑝̅2 , we can substitute 𝑝̅2 into (4.29)-(4.31) and optimize the price of the new product by applying the control vector parameterization method.) Table 4.3 Parameters for numerical experiment 1 𝐶𝑟

𝜆

𝜏

T

𝐸(𝑝̅2 , 𝜂)

𝑘1

𝑘2

𝑎

b

M

6 × 103

250

1

1.5

3

600

6 × 1010

5

3

0.5

105

1

p (t) (x103)

𝐶𝑚

11.5 N=20 analytical result N=6

11

10.5

10

9.5

9

8.5

0

0.5

1

1.5

2

2.5

3

Figure 4.7 Comparison of the analytical result and the numerical results obtained by control vector parameterization

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The parameters used in this experiment are displayed in Table 4.3 (c.f. Section 3.6.2 for the interpretations of the parameter). Set the NLP solver tolerance 𝜀 < 10−7 . The analytical and numerical solutions of the optimal product price 𝑝1 are plotted in Figure 4.7. Note that N is the number of time intervals used in the control vector parameterization model. As N increases, the approximate solution does approach the analytical result.

Table 3.3 in Section 3.6.2 presents the total profit under five different BW lengths obtained by the gradient algorithm given the values of the other parameters in Table 4.3. Among the five values of the BW length 𝜏 = 0.5, 1, 1.5, 2, 2.5, , the total profit is maximized (157.10156 × 106 ) when the BW length is 𝜏 = 1.5 years. In practice, it is sufficient for the manufacturer to consider five possible lengths of the BW. However, from a theoretical perspective, we cannot assert that 𝜏 = 1.5 is optimal when 𝜏 can take any value within a region, e.g., [0, 3]. The gradient algorithm can only be applied when the value of 𝜏 is given. The control vector parameterization method, however, can also optimize this constant parameter 𝜏 simultaneously. More specifically, suppose that the general optimal control problem (4.29)-(4.31) includes a constant parameter ν, which also needs to be optimized. Define 𝑦⃗ = [𝑦0 , 𝑦1 , … , 𝑦𝑁−1 , ν]𝑇 . Obviously, this definition simply adds one more decision variable into the approximation problem (4.33)-(4.34), and the remaining procedure to solve the approximation problem (4.33)-(4.34) would not be changed. This argument demonstrates how to incorporate additional decision variables in the control vector parameterization method. From this perspective, the control vector parameterization method is more robust than the gradient method. Given parameters other than 𝜏, 𝑏 and 𝐶𝑟 in Table 4.3, set different combinations of the BW length elasticity 𝑏 and the expected unit repair cost per time 𝐶𝑟 , the control vector parameterization method can be applied to optimize the BW length 𝜏 over the interval [0, 3] as well as the corresponding optimal profit. The results are summarised in Table 4.4. Table 3.3 indicates that the total profit can reach 157.10156 × 106 when 𝜏 = 1.5, 𝑏 = 0.25 and 𝐶𝑟 = 0.25 × 103 . If we optimize 𝜏 in the admissible region [0, 3], the optimal BW length is 𝜏 = 1.62 years and the corresponding total profit is 157.19197 × 106 , which are highlighted in bold italic fonts in Table 4.4.

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Table 4.4 Optimal BW length 𝜏 under different BW length elasticity b and expected unit repair cost per time 𝐶𝑟 b

𝐶𝑟 (× 103 )

Optimal 𝜏

Total Profit (× 106 )

0.4

0.25

0

136.60888

0.45

0.25

0.71

146.11876

0.5

0.25

1.62

157.19197

0.55

0.25

2.64

170.01953

0.4

0.2

1.40

137.63018

0.45

0.2

2.51

148.92758

0.5

0.2

3

161.89377

0.55

0.2

3

175.87123

Note: The NLP solver tolerance is set to be 𝜀 < 10−7 and the number of time intervals in the control vector parameterization method is 𝑁 = 20 . And the two parameter values remain unchanged in the following Numerical experiment 2 and 3.

Numerical experiment 2 The second numerical experiment corresponds to the properties of the optimal solution of Example 4.2. Consider the following inventory / backlogging cost function: 𝑆 (𝐼 ) = {

𝛿+ 𝐼 2 ,

𝐼≥0

𝛿− 𝐼 2 , 𝐼 < 0

where 𝛿+ is the unit inventory cost per year; 𝛿− is the unit backlogging cost per year. The input parameters are specified in Table 4.5.

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Table 4.5 Parameters for numerical experiment 2 𝐶𝑟

𝜆

𝜏

T

𝐸(𝑝̅2 , 𝜂)

250

1

1

3

600

𝑘1 6× 10

10

𝑘2

𝑎

b

M

5

3

0.5

105

𝑐0

𝑐1

𝑐2

2×

4×

0.1 ×

3

3

10

10

103

𝛿+

𝛿−

200

600

The first result focuses on how the diffusion effect affects the optimal solution. As indicated in Example 3.4, the term 𝑀 − 𝑋 in the Bass-type demand function captures the saturation effect and 𝑞1 + 𝑞2 𝑋/𝑀 captures the diffusion effect. Particularly, 𝑞1 is the coefficient of innovation and 𝑞2 is the coefficient of imitation. Consider the three pairs of (𝑞1 , 𝑞2 ), that is

Pair 1

Pair 2

Pair 3

𝑞1

3

1

1

𝑞2

4

5

0

Figures 4.8-4.10 plot the optimal solution trajectories computed by the control vector parameterization method. Obviously, when the parameters (𝑞1 , 𝑞2 ) are set to be 1 and 0, respectively, the Bass-type demand function degenerates into a limited-growth demand function. The numerical results in Figure 4.8-Figure 4.10 show that when the imitation effect is zero, i.e., 𝑞2 = 0, the optimal selling price for the new product decreases over the planning horizon; the production rate increases over the planning horizon; the inventory level is always positive. When the imitation effect is significant, i.e., 𝑞2 > 0, the demand rate of the new product ought to be greater than the case when 𝑞2 = 0, as shown in Figure 4.9 (d), the green line and red line represent the production trajectories for the case of 𝑞2 > 0, and they are obviously higher than the production trajectories for the case of 𝑞2 = 0. In addition, the inventory level is not always positive when 𝑞2 > 0. The optimal price is increasing at the beginning of the life cycle, i.e., when the diffusion effect is strong, and it decreases when the saturation effect dominates.

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q1=3,q2=4 q1=1,q2=5

13

1

Price p (t) (x 103 )

13.5

q1=1,q2=0 12.5 12 11.5 11 10.5 10 9.5 9 8.5

0

0.5

1

1.5

2

2.5

3

Figure 4.8 The optimal price 𝑝1 (𝑡) under different diffusion effects

Production q(t) (x 103)

21 20 19 18 17 q1=3,q2=4

16

q1=1,q2=5

15

q1=1,q2=0

14 13 12 11

0

0.5

1

1.5

2

2.5

3

Figure 4.9 The optimal production rate 𝑞(𝑡) under different diffusion effects 135

q1=3,q2=4 q1=1,q2=5

50

0.3

I(t) (x 103)

X(t) (x 103)

60

0.2

q1=1,q2=0

0.1

40

0 30 -0.1 20

-0.2

10 0

-0.3 -0.4 0

0.5

1

1.5

2

2.5

3

0

1

1.5

2

2.5

3

(b) Inventory level trajectories I(t)

(a) Cumulative sale trajectories X(t) 2

3(t) (x 103)

3 1(t) (x 10 )

0.5

1.5

8.5

8

1 7.5

0.5 0

7

-0.5 6.5 -1 -1.5

0

0.5 1 1.5 2 2.5 (c) Costate function trajectories 1(t)

6

3

0

0.5 1 1.5 2 2.5 (d) Costate function trajectories 3(t)

3

Figure 4.10 The trajectories of (𝑋(𝑡), 𝐼(𝑡), 𝜃1 (𝑡), 𝜃3 (t)) under different diffusion effect

Numerical experiment 3 This experiment investigates how the inventory level changes with different parameters given the Bass-type demand function and its corresponding innovation coefficients 𝑞1 = 1 and imitation coefficient 𝑞2 = 5. Case 1: Inventory level change with diffenent backlogging cost 𝛿−

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Given parameters other than the backlogging cost 𝛿− in Table 4.5, Figure 4.11 shows that the backlogging situration shrinks when the backlogging cost increases, while there is little change in the positive inventory. This numerical result is consistent with intuitive observations. Note that there is a trade-off between production cost and inventory / backlooging cost. When the backlogging cost is very large, the manufacturer has to sacrifice the benefits associated with appropriate production plans in order to reduce the extremely large backlogging cost. On the other hand, if the inventory cost is not very large, then it is possible to keep the positive

I(t) (x 103)

inventory. 0.1

0.05

0

-0.05

-0.1

-=3+=600

-0.15

-=10+=2000 -0.2

-0.25

-=30+=6000 -=100+=20000 0

0.5

1

1.5

2

2.5

3

Figure 4.11 The inventory level under different backlogging costs

Case 2: Inventory level change with diffenent inventory cost 𝛿+ given large backlogging cost In this case, we set the backlogging cost to be 200000 so as to make the backlogging situation nearly disappear. Then we keep the inventory with different values of the cost parameter 𝛿+ as shown in the following Figure 4.12. Obviously, the numerical result implies that the positive inventory level shrinks as the inventory carrying cost increases. In addition, if one combines case

137

1 and case 2, there is an intuitive conclusion that the inventory level will approch zero as the backloging cost and inventory cost approch infinity.

I(t) (x103)

0.035

+=200 +=1000

0.03

+=5000 +=20000

0.025

0.02

0.015

0.01

0.005

0

-0.005

0

0.5

1

1.5

2

2.5

3

Figure 4.12 The inventory level under different inventory costs given large backlogging cost

Case 3: Inventory level change with parameter 𝑐2 Given parameters other than the backlogging cost 𝑐2 in Table 4.5, Figure 4.13 indicates that: 

When 𝑐2 is relatively large, a high production rate will give rise to a high unit production cost. Therefore, the manufacturer perfers to pay the backlogging cost to reduce the production rate and thereby reduce the total cost. This situration is be reflected by the blue inventory curve.



When 𝑐2 is neither very large nor small, the unit production cost will be moderately influenced by the production rate. There will be a trade-off between the production cost and the backlogging / inventory cost. The manufacturer allows a backlog situation when the

138

demand is very large and keeps a positive inventory when the demand is relatively small. This situation is reflected by the red, yellow, and green inventory curves. 

When 𝑐2 is relatively small, the influence of the production rate on the unit production cost will be very small. Therefore, the production rate will approach the demand rate so as to decrease the backlogging cost and inventory cost. This situation is reflected by the purple and black inventory curves. Obviously, when 𝑐2 approaches zero, the inventory level will

I(t)(x103)

also approach zero, this numerical result is consistent with the conclusion of Corollary 4.1.

0.5 c 2 =1000 c 2 =100

0.4

c 2 =10 c 2 =1

0.3

c 2 =0.1 c 2 =0.01

0.2

0.1

0

-0.1

-0.2

-0.3

0

0.5

1

1.5

2

2.5

time 3

Figure 4.13 The inventory level under different 𝑐2

4.6 Chapter Summary This chapter extends the model in Chapter 3 to situations wherein the unit production cost is significantly affected by the production rate. The necessary conditions of the optimal solution for the extended model have been derived through the Pontryagin maximum principle. Properties of

139

the optimal pricing strategies for the new product and its optional EWC have been analysed. This chapter further considers the problem under two demand functions corresponding to the static market and dynamic markets, respectively. It is possible to derive a closed-form solution under a static market. However, this becomes extremely difficult to do so in dynamic market. A robust numerical algorithm based on the control vector parameterization idea has been developed to compute the numerical solution. The numerical experiments also reveal some additional properties regarding the optimal solution.

140

Chapter 5 Dynamic Pricing and Distribution Channel Design 5.1 Introduction ‘Price’ and ‘channel’ are two of the four elements in marketing (Sudhir et al., 2008). They are the most crucial considerations when launching a new product into the market. Determining price is a

tactical decision as it can be easily changed to influence the short-run demand. On the other hand, the channel structure refers to the way through which the manufacturer reaches its end consumers. In general, it is very costly to change the distribution channel. Therefore, the channel design problem can be viewed as a long-run strategic decision for the manufacturer. The most fundamental decision in channel design is to choose between a centralized channel (sell the products directly to the end consumer) and a decentralized channel (sell the products through intermediaries such as a retailer) (Xu, et al. 2011; Zaccour, 2012; Zaccour, 2004; Zuo, 2011). Obviously, the two problems discussed in Chapters 3 and 4 are based on the centralized channel. The primary focus of this chapter is to study the pricing strategies for the new product and its optional EWC in decentralized systems. In such systems, the manufacturer produces a single repairable product and sells it exclusively to the retailer at a wholesale price. The retailer then sells the product to the end customers at a retail price. Either the retailer or the manufacturer can offer the EWC, which leads to two decentralized channels. Models R and M refer to cases wherein the EWC is offered by the retailer and the manufacturer, respectively. As the manufacturer and the independent retailer form a simple supply chain, both of them strive to optimize their own decisions to maximize their respective profits over the planning horizon (i.e., in a long-term perspective). More specifically, in Model R, the manufacturer determines the wholesale price whereas the retailer decides the retail price and the EWC price. In Model M, the manufacturer determines the wholesale price and the EWC price, whereas the retailer controls the retail price. Figure 5.1 schematically describes the two models with the corresponding decision variables.

141

Manufacturer

Manufacturer

Wholesale price

Wholesale price

Retailer

Retailer

Retail price

EWC price Retail price

EWC price

Consumers

Consumers

Model R

Model M

Figure 5.1 Two decentralized channels

The research problem in this chapter is related to the studies of Li et al. (2012) and Desai et al. (2004), which were based on the static market. However, this chapter considers the problem in a rather general framework, which is applicable to both static and dynamic markets. The research questions addressed in this chapter include: (1) What are the optimal pricing strategies in the two decentralized channels? (2) Which decentralized channel leads to a higher supply chain profit? Is it Model M or Model R? Does the centralized system (Chapter 3) provide the largest profit? The rest of this chapter is organized as follows. Section 5.2 formulates the mathematical model for Model R, and then derives the necessary conditions for the optimal pricing strategies. The last part of this section is the case study and numerical analysis. Section 5.3 discusses Model M, whose structure is similar to Model R. Section 5.4 focuses on the numerical comparison among the Centralization channel, Model R and Model M.

142

5.2 Model R: Retailer Offers the EWC 5.2.1 Mathematical Formulation of Model R Consider a manufacturer who sells a new repairable product to consumers through an independent retailer at a wholesale price. The retailer resells the product and its EWC to the end consumers. The Nikon - Fortress partnership is an example of this channel, where Nikon sells its new digital single-lens reflex camera (DSLR) through its authorized retailer Fortress, while Fortress also offers end consumers the optional “An Xin Bao” extended warranty programme, which prolongs the Nikon’s BW. In this chapter, the manufacturer’s and retailer’s problems will be formulated as a dynamic Stackelberg (Leader-Follower) game, in which the manufacturer acts as the game leader and the retailer acts as the follower. Let 𝑝1𝑀 (𝑡), 𝑝1 (𝑡), and 𝑝2 (𝑡) denote the wholesale price of the product, the retail price of the product, and the price of the EWC at time t, respectively. Consider the same cost and demand functions introduced in Chapter 3 (c.f. Sections 3.2.1 and 3.2.2 for details). Note that the demand functions for the new product and the EWC are only determined by the retail price 𝑝1 (𝑡) and the EWC price 𝑝2 (𝑡), but independent of the wholesale price 𝑝1𝑀 (𝑡). With these preliminaries, the manufacturer’s and retailer’s instantaneous profit rate functions can be expressed as 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)[𝑝1𝑀 (𝑡) − 𝐶𝑚 − 𝐶𝐵𝑊 ], and 𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏)[𝑝1 (𝑡) − 𝑝1𝑀 (𝑡) + 𝜌(𝑝2 (𝑡), 𝜂)(𝑝2 (𝑡) − 𝐶𝐸𝑊𝐶 )]. Then, the dynamic Stackelberg game can be formulated as follows: The goal of the manufacturer is to maximize the following objective function:

143

𝐽𝑀 = 𝑀𝑎𝑥 𝐽𝑀 [𝑝1𝑀 (𝑡)] 𝑀 𝑝1 (𝑡)

𝑇

∫ 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)[𝑝1𝑀 (𝑡) − 𝐶𝑚 − 𝐶𝐵𝑊 ]𝑑𝑡. = 𝑀𝑎𝑥 𝑀 𝑝1 (𝑡), 0

(5.1)

The goal of the retailer is to maximize his or her own profit given by 𝐽𝑅 =

𝑀𝑎𝑥

𝑝1 (𝑡),𝑝2 (𝑡)

𝐽𝑅 [𝑝1 (𝑡), 𝑝2 (𝑡)] 𝑇

=

𝑀𝑎𝑥 ∫ 𝑓(𝑋 (𝑡), 𝑝1 (𝑡), 𝜏)[𝑝1 (𝑡) − 𝑝1𝑀 (𝑡)

𝑝1 (𝑡),𝑝2 (𝑡) 0

+ 𝜌(𝑝2 (𝑡), 𝜂)(𝑝2 (𝑡) − 𝐶𝐸𝑊𝐶 )]𝑑𝑡.

(5.2)

Both the manufacturer and retailer must obey the dynamic demand rate constraints 𝑋̇(𝑡) = 𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏)

(5.3)

𝑌̇(𝑡) = 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)𝜌(𝑝2 (𝑡), 𝜂)

(5.4)

𝑋(0) = 𝑌(0) = 0; 𝑋(𝑇) and 𝑌(𝑇) are free

(5.5)

(𝑝1𝑀 (𝑡), 𝑝1 (𝑡), 𝑝2 (𝑡)) ∈ 𝑈1𝑀 × 𝑈1 × 𝑈2 ⊆ 𝑅+3

(5.6)

5.2.2 Theoretical Analysis The game proceeds in two stages. In stage 1, the manufacturer announces the wholesale price {𝑝1𝑀 (𝑡): 𝑡 ∈ [0, 𝑇]} for the whole planning horizon [0, 𝑇]. In stage 2, after observing {𝑝1𝑀 (𝑡): 𝑡 ∈ [0, 𝑇]}, the retailer decides the retail price and the EWC price {𝑝1 (𝑡), 𝑝2 (𝑡), 𝑡 ∈ [0, 𝑇]}. To find the optimal pricing strategies (Stackelberg equilibrium) of the differential game, the solution procedure should be in the reverse order: solve the retailer’s problem first, and then use the retailer’s best response to formulate the manufacturer’s problem. Given the manufacturer’s wholesale price {𝑝1𝑀 (𝑡): 𝑡 ∈ [0, 𝑇]}, (5.2)-(5.6) formulates the retailer’s problem which is an optimal control problem with the retail and EWC price {𝑝1 (𝑡), 𝑝2 (𝑡), 𝑡 ∈

144

[0, 𝑇]} and the cumulative sales {𝑋(𝑡), 𝑌 (𝑡): 𝑡 ∈ [0, 𝑇]} as control and state variables, respectively. Note that the optimal control problem (5.2)-(5.6) has the same structure as the problem (3.5)-(3.10) in Chapter 3. Define 𝐸 (𝑝2 (𝑡), 𝜂) = 𝜌(𝑝2 (𝑡), 𝜂)[𝑝2 (𝑡) − 𝐶𝐸𝑊𝐶 (𝜂)] for all 𝑡 ∈ [0, 𝑇], which can be viewed as the expected profit from the EWC after one unit of the new product has been sold. The following Lemma 5.1 is similar to Lemma 3.1.

Lemma 5.1 Let (𝑋 (𝑡), 𝑌 𝑖 (𝑡), 𝑝1 (𝑡), 𝑝2𝑖 (𝑡)) (where 𝑖 = 1,2) be two feasible solutions for the retailer’s problem (5.2)-(5.6). If 𝐸 (𝑝21 (𝑡), 𝜂) ≤ 𝐸 (𝑝22 (𝑡), 𝜂) (∀𝑡 ∈ [0, 𝑇]), then 𝐽𝑅 [𝑝1 (𝑡), 𝑝21 (𝑡)] ≤ 𝐽𝑅 [𝑝1 (𝑡), 𝑝22 (𝑡)].

Combining Lemma 5.1 and the proof procedure of Theorem 3.1, it is not difficult to derive the necessary optimality conditions for the retailer’s problem.

Lemma 5.2 Given the manufacturer’s wholesale price {𝑝1𝑀 (𝑡): 𝑡 ∈ [0, 𝑇]}, suppose {𝑋(𝑡), 𝑌 (𝑡), 𝑝1° (𝑡), 𝑝2° (𝑡)} is the optimal solution for the retailer’s problem (5.2)-(5.6). There must exist a piecewise continuously differentiable function 𝜃 𝑅 (𝑡) satisfying the following relations: 𝑝2° (𝑡) ≡ 𝑝2 = 𝐴𝑟𝑔𝑚𝑎𝑥 𝐸 (𝑝2 , 𝜂) , ∀𝑡 ∈ [0, 𝑇];

(5.7)

𝑝1° (𝑡) = 𝐴𝑟𝑔𝑚𝑎𝑥 𝐻𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1 , 𝑝2 ), ∀𝑡 ∈ [0, 𝑇];

(5.8)

𝑋̇ (𝑡) = 𝑓(𝑋 (𝑡), 𝑝1° (𝑡), 𝜏), 𝑋(0) = 0;

(5.9)

𝑌̇ (𝑡) = 𝑓(𝑋 (𝑡), 𝑝1° (𝑡), 𝜏)𝜌(𝑝2 , 𝜂), 𝑌(0) = 0;

(5.10)

𝑝2 ∈𝑈2

𝑝1 ∈𝑈1

𝜃̇ 𝑅 (𝑡) = −𝐻𝑋𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1° (𝑡), 𝑝2 ),

145

𝜃 𝑅 (𝑇) = 0,

(5.11)

where 𝐻𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1 (𝑡), 𝑝2 (𝑡)) is the Hamiltonian function for retailer’s problem, and this function is defined as 𝐻𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1 (𝑡), 𝑝2 (𝑡)) = 𝑓 (𝑋 (𝑡), 𝑝1 (𝑡), 𝜏)(𝑝1 (𝑡) − 𝑝1𝑀 (𝑡) + 𝜃 𝑅 (𝑡) + 𝐸 (𝑝2 (𝑡), 𝜂)).

(5.12)

Similar to Theorem 3.1, the optimal selling price of the EWC is constant over the planning horizon. The retail price of the new product is more complex. It depends on the wholesale price 𝑝1𝑀 (𝑡) , the profit from selling EWC 𝐸(𝑝2 , 𝜂) and the future marginal profit 𝜃 𝑅 (𝑡) . The relationship can be reflected in condition (5.8). In addition, condition (5.8) can be interpreted as follows: Given the time 𝑡 ∈ [0, 𝑇] and the value of (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝2 ), the Hamiltonian function 𝐻𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1 , 𝑝2 ) can be viewed as a function of the argument 𝑝1 with domain 𝑈1 = (0, +∞). (5.8) implies that 𝐻𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1 , 𝑝2 ) attains its maximum at 𝑝1° (𝑡). Note that 𝑈1 is an open interval. Hence, 𝑝1° (𝑡), as the maximum point of the function 𝐻𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1 , 𝑝2 ) in domain 𝑈1 , must satisfy the first-order necessary condition, that is 𝐻𝑝𝑅1 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1° (𝑡), 𝑝2 ) = 𝑓𝑝1 (𝑋(𝑡), 𝑝1° (𝑡), 𝜏) (𝑝1° (𝑡) − 𝑝1𝑀 (𝑡) + 𝜃 𝑅 (𝑡) + 𝐸(𝑝2 , 𝜂)) + 𝑓(𝑋 (𝑡), 𝑝1° (𝑡), 𝜏) = 0.

(5.13)

Obviously, condition (5.8) implies (5.13). However, the inverse implication is true in some special situations, one of which will be stated and proved in the following Lemma 5.3.

146

Lemma 5.3 If the demand function 𝑓 (𝑋, 𝑝1 , 𝜏) satisfies 2𝑓𝑝21 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏) − 𝑓𝑝1 𝑝1 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏) > 0 for all 𝑝1 (𝑡) ∈ 𝑈1 .

(5.14)

Then condition (5.8) can be replaced by (5.13) and the optimal retail price 𝑝1° (𝑡) with respect to any given wholesale price 𝑝1𝑀 (𝑡) is unique. Proof: Given any time 𝑡 ∈ [0, 𝑇] and the value of (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝2 ), let 𝑝1° (𝑡) be a retail price satisfying (5.13), which implies that 𝑝1° (𝑡) must be a local maximum or local minimum of the Hamiltonian function𝐻𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1 , 𝑝2 ). The second-order derivative of 𝐻𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1 , 𝑝2 ) at 𝑝1° (𝑡) would be 𝐻𝑝𝑅1 𝑝1 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1° (𝑡), 𝑝2 ) = 𝑓𝑝1 𝑝1 (𝑋(𝑡), 𝑝1° (𝑡), 𝜏) (𝑝1° (𝑡) − 𝑝1𝑀 (𝑡) + 𝜃 𝑅 (𝑡) + 𝐸(𝑝2 , 𝜂)) + 2𝑓𝑝1 (𝑋(𝑡), 𝑝1° (𝑡), 𝜏). (5.15) Rearranging the first-order condition (5.13) yields

𝑝1° (𝑡) − 𝑝1𝑀 (𝑡) + 𝜃 𝑅 (𝑡) + 𝐸(𝑝2 , 𝜂) = −

𝑓(𝑋 (𝑡), 𝑝1° (𝑡), 𝜏) . 𝑓𝑝1 (𝑋(𝑡), 𝑝1° (𝑡), 𝜏)

Substituting (5.16) into (5.15) yields 𝐻𝑝𝑅1 𝑝1 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1° (𝑡), 𝑝2 ) = −𝑓𝑝1 𝑝1 (𝑋(𝑡), 𝑝1° (𝑡), 𝜏)

𝑓(𝑋 (𝑡), 𝑝1° (𝑡), 𝜏) + 2𝑓𝑝1 (𝑋(𝑡), 𝑝1° (𝑡), 𝜏) 𝑓𝑝1 (𝑋(𝑡), 𝑝1° (𝑡), 𝜏)

2𝑓𝑝21 (𝑋(𝑡), 𝑝1° (𝑡), 𝜏) − 𝑓𝑝1 𝑝1 (𝑋(𝑡), 𝑝1° (𝑡), 𝜏)𝑓(𝑋 (𝑡), 𝑝1° (𝑡), 𝜏) = 𝑓𝑝1 (𝑋(𝑡), 𝑝1° (𝑡), 𝜏) < 0.

147

(5.16)

This inequality implies 𝑝1° (𝑡) must be a strict local maximum of 𝐻𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1 , 𝑝2 ), and the function 𝐻𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1 , 𝑝2 ) has no local minimum in the admissible region 𝑈1 . If not, suppose 𝑝1 (t) is a local minimum, then (5.16) must be true at 𝑝1 (𝑡). However, (5.14) also implies 𝐻𝑝𝑅1 𝑝1 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1 (𝑡), 𝑝2 ) < 0, which is not the second-order necessary condition for a minimum. Therefore, the hypothesis that 𝑝1 (t) is a local minimum would lead to a contradiction. Furthermore, 𝑝1° (𝑡) must be the unique global maximum. If not, there exists 𝑝1°° (𝑡) ∈ 𝑈1 and 𝑝1°° (𝑡) ≠ 𝑝1° (𝑡) such that 𝐻𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1° (𝑡), 𝑝2 ) ≤ 𝐻𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1°° (𝑡), 𝑝2 ). Consider the Hamiltonian function 𝐻𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1 , 𝑝2 ) on the closed interval [𝑝1°° (𝑡), 𝑝1° (𝑡)] if 𝑝1°° (𝑡) < 𝑝1° (𝑡) (or [ 𝑝1° (𝑡), 𝑝1°° (𝑡)] if 𝑝1°° (𝑡) > 𝑝1° (𝑡) ). According to the continuity of 𝐻𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1 , 𝑝2 ), there must be a minimum point 𝑝1̀ (𝑡) on the closed interval [𝑝1°° (𝑡), 𝑝1° (𝑡)]. Notice that 𝑝1° (𝑡) and 𝑝1°° (𝑡) are strict local maximum points, then 𝑝1̀ (𝑡) ∈ (𝑝1°° (𝑡), 𝑝1° (𝑡)), then 𝑝1̀ (𝑡) will also be a local minimum. This leads to a contradiction with the preceding conclusion that the function 𝐻𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1 , 𝑝2 ) has no local minimum in the domain 𝑈1 . Therefore, 𝑝1° (𝑡) must be the unique global maximum of 𝐻𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1 , 𝑝2 ), i.e., 𝑝1° (𝑡) satisfies condition (5.8). ∎ Remark 5.1: Condition (5.14) just states that the marginal revenue is an increasing function of the retail price when the retailer sells the new product. For more detailed economic explanation, one can refer to Remark 3.3 in Chapter 3. Based on condition (5.14), the next proposition reveals an important managerial insight.

148

Proposition 5.1 Suppose that the demand function 𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏) satisfies condition (5.14). Whenever the manufacturer decides to raise or lower the wholesale price, the optimal retail price should follow the movement of the wholesale price. Proof: Given the manufacturer’s wholesale price 𝑝1𝑀 (𝑡), the retailer’s optimal retail price 𝑝1° (𝑡) satisfies the first-order condition (5.13) which can be rearranged as

𝑝1𝑀 (𝑡)

=

𝑝1° (𝑡) +

𝑓(𝑋 (𝑡), 𝑝1° (𝑡), 𝜏) + 𝜃 𝑅 (𝑡) + 𝐸(𝑝2 , 𝜂). °( ) 𝑓𝑝1 (𝑋(𝑡), 𝑝1 𝑡 , 𝜏)

(5.17)

To simplify the notations, suppress the time argument t and denote 𝑝1𝑀 = 𝑇(𝑝1° ) = 𝑝1° +

𝑓(𝑋, 𝑝1° , 𝜏) + 𝜃 𝑅 + 𝐸(𝑝2 , 𝜂). ° 𝑓𝑝1 (𝑋, 𝑝1 , 𝜏)

Then 𝑑𝑝1𝑀 𝑑𝑇(𝑝1° ) = 𝑑𝑝1° 𝑑𝑝1° = 1+

𝑓𝑝1 (𝑋, 𝑝1° , 𝜏) 𝑓(𝑋, 𝑝1° , 𝜏)𝑓𝑝1 𝑝1 (𝑋, 𝑝1° , 𝜏) + 𝑓𝑝1 (𝑋, 𝑝1° , 𝜏) 𝑓𝑝21 (𝑋, 𝑝1° , 𝜏)

2𝑓𝑝21 (𝑋, 𝑝1° , 𝜏) − 𝑓(𝑋, 𝑝1° , 𝜏)𝑓𝑝1 𝑝1 (𝑋, 𝑝1° , 𝜏) = 𝑓𝑝21 (𝑋, 𝑝1° , 𝜏) > 0. Thus, 𝑝1𝑀 = 𝑇(𝑝1° ) is monotone increasing in 𝑝1° , and hence the inverse function of 𝑝1𝑀 = 𝑇(𝑝1° ) must exist, that is 𝑝1° = 𝑇 −1 (𝑝1𝑀 ). Therefore, the optimal retail price 𝑝1° can be viewed as an implicit function of the wholesale price 𝑝1𝑀 . In addition,

149

𝑑𝑝1° 𝑑𝑇 −1 (𝑝1𝑀 ) 1 = = > 0, 𝑑𝑝1𝑀 𝑑𝑝1𝑀 𝑑𝑇(𝑝1° ) 𝑑𝑝1° which implies the result in Proposition 5.1. ∎ So far, this section has analysed the retailer’s problem. Lemma 5.3 and Proposition 5.1 imply that if the demand function 𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏) satisfies (5.14), the retailer has the unique best response retail price 𝑝1° (𝑡) = 𝑇 −1 (𝑝1𝑀 (𝑡)). The manufacturer takes the retailer’s best response into consideration when maximizing his or her own total profit. Similar to the optimization problem studied in Chapter 3, in most cases, it is impossible to derive a closed-form solution for this best response retail price. Therefore, condition (5.17) must be imposed as an equality constraint on the manufacturer’s problem (Basar et al., 1995; Weber, 2011), that is

𝑀

𝐽 =

𝑇

𝑀𝑎𝑥

𝑝1𝑀 (𝑡),𝑝1 (𝑡)

∫ 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)[𝑝1𝑀 (𝑡) − 𝐶𝑚 − 𝐶𝐵𝑊 ]𝑑𝑡

(5.18)

0

subject to 𝑋̇ (𝑡) = 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏); 𝑋 (0) = 0

(5.19)

𝑌̇(𝑡) = 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)𝜌(𝑝2 , 𝜂); 𝑌(0) = 0

(5.20)

𝜃̇ 𝑅 (𝑡) = −𝐻𝑋𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1 (𝑡), 𝑝2 ); 𝑝1𝑀 (𝑡) = 𝑝1 (𝑡) +

𝜃 𝑅 (𝑇 ) = 0

𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏) + 𝜃 𝑅 (𝑡) + 𝐸(𝑝2 , 𝜂) 𝑓𝑝1 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)

(𝑝1𝑀 (𝑡), 𝑝1 (𝑡)) ∈ 𝑈1𝑀 × 𝑈1

(5.21) (5.22) (5.23)

Section 5.2.2 will focus on how to find the best wholesale price 𝑝1𝑀 (𝑡) and the corresponding best response retail price 𝑝1 (𝑡) through the manufacturer’s problem.

150

∗

∗

Theorem 5.1 Suppose (𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1∗ (𝑡)) is the optimal solution for the manufacturer’s problem. There must exist piecewise continuously differential functions 𝜃1𝑀 (𝑡) and 𝜃3𝑀 (𝑡) , such that the optimal solution as well as 𝜃1𝑀 (𝑡) and 𝜃3𝑀 (𝑡) satisfy the following relations: ∗

𝑝1∗ (𝑡) = 𝐴𝑟𝑔𝑚𝑎𝑥 𝐻𝑀 (𝑋 ∗ (𝑡), 𝜃 𝑅 (𝑡), 𝑝1 , 𝜃1𝑀 (𝑡), 𝜃3𝑀 (𝑡)), ∀𝑡 ∈ [0, 𝑇]; 𝑝1 ∈𝑈1

∗

𝑝1𝑀 (𝑡) = 𝑇(𝑝1∗ (𝑡)) = 𝑝1∗ (𝑡) +

𝑓 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏) ∗ + 𝜃 𝑅 (𝑡) + 𝐸(𝑝2 , 𝜂); 𝑓𝑝1 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)

𝑋̇ ∗ (𝑡) = 𝑓 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏), 𝑋 ∗ (0) = 0; 𝑌̇ ∗ (𝑡) = 𝑓(𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)𝜌(𝑝2 , 𝜂), 𝑌 ∗ (0) = 0; ∗ ∗ 𝜃̇ 𝑅 (𝑡) = −𝐻𝑋𝑅 (𝑋 ∗ (𝑡), 𝜃 𝑅 (𝑡), 𝑇(𝑝1∗ (𝑡)), 𝑝1∗ (𝑡), 𝑝2 ),

∗

𝜃 𝑅 (𝑇) = 0;

∗ 𝜃̇1𝑀 (𝑡) = −𝐻𝑋𝑀 (𝑋 ∗ (𝑡), 𝜃 𝑅 (𝑡), 𝑝1∗ (𝑡), 𝜃1𝑀 (𝑡), 𝜃3𝑀 (𝑡)), 𝜃1𝑀 (𝑇) = 0; ∗ 𝜃̇3𝑀 (𝑡) = −𝐻𝜃𝑀𝑅 (𝑋 ∗ (𝑡), 𝜃 𝑅 (𝑡), 𝑝1∗ (𝑡), 𝜃1𝑀 (𝑡), 𝜃3𝑀 (𝑡)), 𝜃3𝑀 (0) = 0,

where ∗

𝐻𝑀 (𝑋 ∗ (𝑡), 𝜃 𝑅 (𝑡), 𝑝1∗ (𝑡), 𝜃1𝑀 (𝑡), 𝜃3𝑀 (𝑡)) = 𝑓(𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏) [𝑇(𝑝1∗ (𝑡)) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝜃1𝑀 (𝑡) + 𝜃3𝑀 (𝑡)

𝑓𝑋 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏) ]. 𝑓𝑝1 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)

Proof: The result is proved by the following four steps. Step 1 Note that the wholesale price 𝑝1𝑀 (𝑡) can be explicitly expressed by the retail price, that is 𝑇 (𝑝1 (𝑡)) = 𝑝1 (𝑡) +

𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏) + 𝜃 𝑅 (𝑡) + 𝐸(𝑝2 , 𝜂). ( ) ( ) ( ) 𝑓𝑝1 𝑋 𝑡 , 𝑝1 𝑡 , 𝜏

151

The manufacturer’s problem can be simplified to an optimal control problem with one decision variable 𝑝1 (𝑡), that is 𝑇

𝑀

𝐽 = 𝑀𝑎𝑥 ∫ 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)[𝑇(𝑝1 (𝑡)) − 𝐶𝑚 − 𝐶𝐵𝑊 ]𝑑𝑡

(5.24)

𝑋̇ (𝑡) = 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏); 𝑋 (0) = 0

(5.25)

𝑌̇(𝑡) = 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)𝜌(𝑝2 , 𝜂); 𝑌(0) = 0

(5.26)

𝑝1 (𝑡)

0

subject to

𝜃̇ 𝑅 (𝑡) = −𝐻𝑋𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑇(𝑝1 (𝑡)), 𝑝1 (𝑡), 𝑝2 );

𝜃 𝑅 (𝑇 ) = 0

𝑝1 (𝑡) ∈ 𝑈1

(5.27) (5.28)

Step 2: Formulate the Hamiltonian function for the problem (5.24)-(5.28) as follows 𝐻𝑀 = 𝐻𝑀 (𝑋(𝑡), 𝑌(𝑡), 𝜃 𝑅 (𝑡), 𝑝1 (𝑡), 𝜃1𝑀 (𝑡), 𝜃2𝑀 (𝑡), 𝜃3𝑀 (𝑡)) = 𝜃0 {𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏) [𝑇 (𝑝1 (𝑡)) − 𝐶𝑚 − 𝐶𝐵𝑊 ]} + 𝜃1𝑀 (𝑡) 𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏) + 𝜃2𝑀 (𝑡)𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)𝜌(𝑝2 , 𝜂) − 𝜃3𝑀 (𝑡)𝐻𝑋𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑇 (𝑝1 (𝑡)) , 𝑝1 (𝑡), 𝑝2 ), where 𝜃0 is the constant multiplier associated with the integrand of the objective function; 𝜃1𝑀 (𝑡), 𝜃2𝑀 (𝑡) and 𝜃3𝑀 (𝑡) are the costate functions associated with the state equations (5.25), (5.26) and (5.28). ∗

Suppose (𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝜃 𝑅 (𝑡), 𝑝1∗ (𝑡)) is the optimal solution to the problem (5.24)-(5.28). The Pontryagin maximum principle asserts that (1) The constant multiplier 𝜃0 (= 0 or 1) and the costate function 𝜃1 (𝑡), 𝜃2 (𝑡) and 𝜃3 (𝑡) satisfy (𝜃0 , 𝜃1𝑀 (𝑡), 𝜃2𝑀 (𝑡), 𝜃3𝑀 (𝑡)) ≠ (0,0,0,0) for all 𝑡 ∈ [0, 𝑇]. (2) For any 𝑡 ∈ [0, 𝑇] and 𝑝1 ∈ 𝑈1

152

∗

𝑝1∗ (𝑡) = 𝐴𝑟𝑔𝑚𝑎𝑥 𝐻𝑀 (𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝜃 𝑅 (𝑡), 𝑝1 , 𝜃1𝑀 (𝑡), 𝜃2𝑀 (𝑡), 𝜃3𝑀 (𝑡)). 𝑝1 ∈𝑈1

(3) For any time 𝑡 at which the function 𝑝1∗ (𝑡) is continuous, the costate functions satisfy the following differential equations as well as the corresponding boundary conditions: ∗ 𝜃̇1𝑀 (𝑡) = −𝐻𝑋𝑀 (𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝜃 𝑅 (𝑡), 𝑝1∗ (𝑡), 𝜃1𝑀 (𝑡), 𝜃2𝑀 (𝑡), 𝜃3𝑀 (𝑡)), 𝜃1𝑀 (𝑇) = 0;

∗ 𝜃̇2𝑀 (𝑡) = −𝐻𝑌𝑀 (𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝜃 𝑅 (𝑡), 𝑝1∗ (𝑡), 𝜃1𝑀 (𝑡), 𝜃2𝑀 (𝑡), 𝜃3𝑀 (𝑡)) = 0, 𝜃2𝑀 (𝑇) = 0;

∗ 𝜃̇3𝑀 (𝑡) = −𝐻𝜃𝑀𝑅 (𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝜃 𝑅 (𝑡), 𝑝1∗ (𝑡), 𝜃1𝑀 (𝑡), 𝜃2𝑀 (𝑡), 𝜃3𝑀 (𝑡)), 𝜃3𝑀 (0) = 0;

Step 3: 𝜃̇2𝑀 (𝑡) = 0 and 𝜃2𝑀 (𝑇) = 0 obviously imply that 𝜃2𝑀 (𝑡) ≡ 0 over [0, 𝑇] . In addition, we can assert that the constant multiplier 𝜃0 = 1 with the method of proof by contradiction. Suppose in contrary that 𝜃0 = 0. Note that 𝐻𝑋𝑅 (𝑋 ∗ (𝑡), 𝜃 𝑅 (𝑡), 𝑇(𝑝1∗ (𝑡)), 𝑝1∗ (𝑡), 𝑝2 ) = 𝑓𝑋 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏) (𝑝1∗ (𝑡) − 𝑇(𝑝1∗ (𝑡)) + 𝜃 𝑅 (𝑡) + 𝐸 (𝑝2 (𝑡), 𝜂)) =−

𝑓 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)𝑓𝑋 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏) . 𝑓𝑝1 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)

Then ∗ 𝜃̇3𝑀 (𝑡) = −𝐻𝜃𝑀𝑅 (𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝜃 𝑅 (𝑡), 𝑝1∗ (𝑡), 𝜃1𝑀 (𝑡), 𝜃2𝑀 (𝑡), 𝜃3𝑀 (𝑡))

=

𝜃3𝑀 (𝑡)

𝜕𝐻𝑋𝑅 (𝑋 ∗ (𝑡), 𝜃 𝑅 (𝑡), 𝑇(𝑝1∗ (𝑡)), 𝑝1∗ (𝑡), 𝑝2 ) 𝜕𝜃𝑅

= 0. Due to the initial time condition 𝜃3𝑀 (0) = 0, the differential equation 𝜃̇3𝑀 (𝑡) = 0 has only one solution 𝜃3𝑀 (𝑡) ≡ 0 over [0, 𝑇]. Therefore, (𝜃0 , 𝜃1𝑀 (𝑇), 𝜃2𝑀 (𝑇), 𝜃3𝑀 (𝑇)) = (0,0,0,0).

153

This contradicts the statement (1) in step 1, that is (𝜃0 , 𝜃1𝑀 (𝑡), 𝜃2𝑀 (𝑡), 𝜃3𝑀 (𝑡)) ≠ (0,0,0,0) for all 𝑡 ∈ [0, 𝑇]. Therefore, the constant multiplier 𝜃0 satisfies 𝜃0 = 1. Step 4: Simplify the Hamiltonian function H by applying 𝜃0 = 1 and 𝜃2𝑀 (𝑡) ≡ 0: ∗

𝐻𝑀 (𝑋 ∗ (𝑡), 𝜃 𝑅 (𝑡), 𝑝1∗ (𝑡), 𝜃1𝑀 (𝑡), 𝜃3𝑀 (𝑡)) = 𝑓 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏) [𝑇(𝑝1∗ (𝑡)) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝜃1𝑀 (𝑡) + 𝜃3𝑀 (𝑡)

𝑓𝑋 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏) ]. 𝑓𝑝1 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)

∗

Then the optimal solution (𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝜃 𝑅 (𝑡), 𝑝1∗ (𝑡)) to the problem (5.24)-(5.28) can be derived via the following necessary conditions: ∗

𝑝1∗ (𝑡) = 𝐴𝑟𝑔𝑚𝑎𝑥 𝐻𝑀 (𝑋 ∗ (𝑡), 𝜃 𝑅 (𝑡), 𝑝1 , 𝜃1𝑀 (𝑡), 𝜃3𝑀 (𝑡)), ∀𝑡 ∈ [0, 𝑇]; 𝑝1 ∈𝑈1

∗ 𝑋̇ ∗ (𝑡) = 𝑓(𝑋 (𝑡), 𝑝∗1 (𝑡), 𝜏), 𝑋 ∗ (0) = 0;

𝑌̇ ∗ (𝑡) = 𝑓(𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)𝜌(𝑝2 , 𝜂), 𝑌 ∗ (0) = 0; ∗

∗ 𝑅 ∗ 𝑅 𝜃̇ 𝑅 (𝑡) = −𝐻𝑋 (𝑋 (𝑡), 𝜃 (𝑡), 𝑇 (𝑝∗1 (𝑡)) , 𝑝∗1 (𝑡), 𝑝2 ),

∗

𝜃 𝑅 (𝑇) = 0;

∗ 𝜃̇1𝑀 (𝑡) = −𝐻𝑋𝑀 (𝑋 ∗ (𝑡), 𝜃 𝑅 (𝑡), 𝑝1∗ (𝑡), 𝜃1𝑀 (𝑡), 𝜃3𝑀 (𝑡)), 𝜃1𝑀 (𝑇) = 0;

∗ 𝜃̇3𝑀 (𝑡) = −𝐻𝜃𝑀𝑅 (𝑋 ∗ (𝑡), 𝜃 𝑅 (𝑡), 𝑝1∗ (𝑡), 𝜃1𝑀 (𝑡), 𝜃3𝑀 (𝑡)), 𝜃3𝑀 (0) = 0.

Once obtaining the optimal retail price 𝑝1∗ (𝑡), the optimal wholesale price for the manufacturer is ∗ 𝑝1𝑀 (𝑡)

=

𝑇(𝑝1∗ (𝑡))

=

𝑝1∗ (𝑡) +

𝑓 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏) ∗ + 𝜃 𝑅 (𝑡) + 𝐸(𝑝2 , 𝜂). ∗ ∗ 𝑓𝑝1 (𝑋 (𝑡), 𝑝1 (𝑡), 𝜏) ∎

154

5.2.3 Case Study 5.2.3.1 Static Market In a static market, the demand function is independent of the cumulative sales volume 𝑋(𝑡) and degenerates into 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏) = ℎ(𝑝1 (𝑡), 𝜏). Obviously, the selling price of the EWC must be 𝑝2 = 𝐴𝑟𝑔𝑚𝑎𝑥𝑝2∈𝑈2 𝐸 (𝑝2 , 𝜂) . Furthermore, the static market assumption implies that the differential constraints (5.3)-(5.5) do not affect the manufacturer’s and the retailer’s objective function, so the optimal wholesale and retail price will be constant over the planning horizon. In this case, it is sufficient to optimize the instantaneous profit rate functions for both the manufacturer and the retailer, that is For the manufacturer: For the retailer:

𝐿𝑀 = ℎ(𝑝1 , 𝜏)[𝑝1𝑀 − 𝐶𝑚 − 𝐶𝐵𝑊 ]; 𝐿𝑅 = ℎ(𝑝1 , 𝜏)[𝑝1 − 𝑝1𝑀 + 𝐸(𝑝2 , 𝜂)].

Example 5.1 Additive price-warranty demand function ℎ(𝑝1 , 𝜏) = 𝑎0 − 𝑎1 𝑝1 + 𝑎2 𝜏. Given the wholesale price 𝑝1𝑀 , retailer’s instantaneous profit rate is 𝐿𝑅 = (𝑎0 − 𝑎1 𝑝1 + 𝑎2 𝜏)[𝑝1 − 𝑝1𝑀 + 𝐸 (𝑝̅2 , 𝜂)] = −𝑎1 𝑝12 + [(𝑎0 + 𝑎2 𝜏) + 𝑎1 (𝑝1𝑀 − 𝐸 (𝑝̅2 , 𝜂))]𝑝1 − (𝑎0 + 𝑎2 𝜏)(𝑝1𝑀 − 𝐸 (𝑝̅2 , 𝜂)), which is a quadratic function. Obviously, the best response retail price 𝑝1 is (𝑎0 + 𝑎2 𝜏) + 𝑎1 (𝑝1𝑀 − 𝐸 (𝑝̅2 , 𝜂)) 𝑝1 = . 2𝑎1 Next, substituting (5.29) into the manufacturer’s problem yields 𝐿𝑀 = (𝑎0 − 𝑎1 𝑝1 + 𝑎2 𝜏)[𝑝1𝑀 − 𝐶𝑚 − 𝐶𝐵𝑊 ]

155

(5.29)

(𝑎0 + 𝑎2 𝜏) − 𝑎1 (𝑝1𝑀 − 𝐸 (𝑝̅2 , 𝜂)) 𝑀 [𝑝1 − 𝐶𝑚 − 𝐶𝐵𝑊 ]. = 2 Hence, the optimal wholesale price is

𝑝1𝑀 =

𝑎0 + 𝑎1 (𝐶𝑚 + 𝐶𝐵𝑊 + 𝐸 (𝑝̅2 , 𝜂)) + 𝑎2 𝜏 . 2𝑎1

(5.30)

Substituting this optimal wholesale price into (5.29) yields the corresponding optimal retail price

𝑝1 =

3(𝑎0 + 𝑎2 𝜏) + 𝑎1 (𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂)) . 4𝑎1

(5.31)

Then the total profits for the manufacturer and retailer during the planning horizon are 2

𝐽𝑀 =

̅2 , 𝜂))] [(𝑎0 + 𝑎2 𝜏) − 𝑎1 (𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸(𝑝

𝑇;

8𝑎1 2

[(𝑎0 + 𝑎2 𝜏) − 𝑎1 (𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂))] 𝐽 = 𝑇. 16𝑎1 𝑅

and the total profit in the whole supply chain is 2

3[(𝑎0 + 𝑎2 𝜏) − 𝑎1 (𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂))] 𝐽 +𝐽 = 𝑇. 16𝑎1 𝑅

𝑀

Also note that 𝐽𝑅 𝐽

𝑀

=

8𝑎1 1 = , 16𝑎1 2

which indicates that the manufacturer’s profit is twice that of the retailer’s.

156

5.2.3.2 Dynamic Market Lemma 5.2 asserts that the optimal price for EWC should be 𝑝2 = 𝐴𝑟𝑔𝑚𝑎𝑥𝑝2∈𝑈2 𝐸 (𝑝2 , 𝜂) . Therefore, the challenge is to find the optimal wholesale and retail prices of the new product. Although Theorem 5.1 provides the necessary conditions of these optimal prices, it is usually impossible to derive a closed-form solution due to the complexity of the necessary conditions. The only possible way is to generate a numerical solution via numerical methods. Consider the control vector parameterization method introduced in Chapter 4. The basic idea is to approximate the continuous control variables by piecewise-constant functions and then the original dynamic optimization problem can be transformed into a standard nonlinear optimization problem (NLP). Note that 𝑝1𝑀 (𝑡) and 𝑝1 (𝑡) are the continuous decision variables for the manufacturer’s problem (5.18)-(5.23). Following the procedure of control vector parameterization, they are approximated by piecewise-constant basis functions 𝑝1𝑀 (𝑡|𝑦⃗ 𝑀 ) and 𝑝1 (𝑡|𝑦⃗): 𝑁−1

𝑝1𝑀 (𝑡|𝑦⃗ 𝑀 ) = ∑ 𝑦𝑖𝑀 𝜒[𝑡𝑖 ,𝑡𝑖+1 ) (𝑡) , 𝑡 ∈ [0, 𝑇]; 𝑘=0 𝑁−1

𝑝1 (𝑡|𝑦⃗) = ∑ 𝑦𝑖 𝜒[𝑡𝑖 ,𝑡𝑖+1 ) (𝑡) , 𝑡 ∈ [0, 𝑇], 𝑘=0 𝑀 ]𝑇 where 𝑦⃗ 𝑀 = [𝑦0𝑀 , 𝑦1𝑀 , … , 𝑦𝑁−1 , 𝑦⃗ = [𝑦0 , 𝑦1 , … , 𝑦𝑁−1 ]𝑇 and

𝜒[𝑡𝑖 ,𝑡𝑖+1 ) (𝑡) ≔ {

1, 0,

if 𝑡 ∈ [𝑡𝑖 , 𝑡𝑖+1 ) . if 𝑡 ∈ [𝑡𝑖 , 𝑡𝑖+1 )

However, unlike the problem in Chapter 4, the above approximation cannot simplify the manufacturer’s problem (5.18)-(5.23) into a NLP, because the path constraint (5.22) must hold at each time 𝑡 ∈ [0, 𝑇], which gives rise to an infinite number of constraints. To overcome this challenge, this chapter introduces the following integral constraint (Becerra, 2004):

157

2

𝑇

𝑓 (𝑋(𝑡|𝑦⃗), 𝑝1 (𝑡|𝑦⃗), 𝜏) 𝜈(𝑦⃗ 𝑀 , 𝑦⃗) = ∫ (𝑝1 (𝑡|𝑦⃗) − 𝑝1𝑀 (𝑡|𝑦⃗ 𝑀 ) + + 𝜃 𝑅 (𝑡|𝑦⃗) + 𝐸(𝑝2 , 𝜂)) 𝑑𝑡. ( ( ) ( | ) ) 𝑓 𝑋 𝑡|𝑦 ⃗ , 𝑝 𝑡 𝑦 ⃗ , 𝜏 𝑝 1 0 1 Note that 𝜈(𝑦⃗ 𝑀 , 𝑦⃗) measures the overall violation of the path constraint (5.22). Hence, a relaxation of the manufacturer’s problem can be obtained by replacing the path constraint (5.22) with 𝜈 (𝑦⃗ 𝑀 , 𝑦⃗) ≤ 𝜀1 . The control vector parameterization method can be then applied to the relaxation, which yields an approximation of the optimal wholesale and prices.

Example 5.2 Limited-growth demand function 𝑓 (𝑋, 𝑝1 , 𝜏) = 𝑘1 [𝑘2 + 𝜏]𝑏 𝑝1−𝑎 [𝑀 − 𝑋]. Two numerical experiments will be conducted through the modified control vector parameterization method. It firstly derives the approximate solution for the manufacturer’s problem under this demand function, and then compares the profits in Model R with that in the centralized channel, i.e., the optimal profit of the model presented in Chapter 3. Numerical experiment 1 The parameter values for this numerical experiment are summarised in the following table:

Table 5.1 Parameter values 𝐶𝑚

𝐶𝑟

𝜆

𝜏

T

𝐸(𝑝̅2 , 𝜂)

𝑘1

𝑘2

𝑎

B

M

6 × 103

250

1

1

3

600

6 × 1010

5

3

0.5

105

158

Price (x 103)

16

15

14

13

retail price N=6 wholesale price N=6 retail price N=20 wholesale price N=20

12

11

10

0

0.5

1

1.5

2

2.5

Figure 5.2 The approximated optimal wholesale and retail prices for Model R

In addition, the experiment sets the relax parameter 𝜀1 < 0.5 and the NLP solver tolerance 𝜀 < 10−7 , Figure 5.1 plots the approximated optimal wholesale and retail prices obtained by the control vector parameterization method for different values of 𝑁 , i.e., the number of time intervals that the control vector parameterization method divides the planning horizon [0, 𝑇] into. Note that the wholesale price decreases over the planning horizon and so does the retail price. However, the decreasing rate of the retail price is much higher than that of the wholesale price.

Numerical experiment 2 This experiment examines how the profits of the manufacturer and the retailer change with respect to the value of 𝐸 (𝑝̅2 , 𝜂), i.e., the maximum expected profit from selling EWC after one unit of new product has been sold. The values of parameters other than 𝐸 (𝑝̅2 , 𝜂) are given in

159

Table 5.1. The number of time intervals in control vector parameterization is N = 6, the relax parameter 𝜀1 < 0.5, and the NLP solver tolerance 𝜀 < 10−7 . Table 5.2 summarises the profits under three different 𝐸 (𝑝̅2 , 𝜂), which listed in the first row. The 2nd, 3rd, and the 4th rows display the profits of the manufacturer, the retailer, and the entire supply chain under the decentralized Model R, while the last row report the total profit under the centralized channel, which can be obtained from Table 3.4. Table 5.2 indicates that the total profits for both the manufacturer and the retailer increase if 𝐸 (𝑝̅2 , 𝜂) increases, and the retailer’s increment is larger than the manufacturer’s. Furthermore, as expected, the total profit under the centralized channel is always higher than the supply chain profit in Model R.

Table 5.2 Profit analysis under different 𝐸 (𝑝̅2 , 𝜂) for Model R 𝐸(𝑝̅2 , 𝜂) (× 103 HKD)

0.4

0.6

1

Manufacturer’s profit (Model R) × 106 HKD

49.1942

49.4176

54.9981

Retailer’s profit (Model R) × 106 HKD

65.9354

69.1505

76.1358

Total profit (Model R) × 106 HKD

112.8310

118.5681

131.1339

Total profit (Centralized channel) × 106 HKD

149.6894

157.1016

173.2603

5.3 Model M: Manufacturer Offers the EWC 5.3.1 Mathematical Formulation of Model M Model M considers the case that the manufacturer sells the new product to end consumers through a retailer whereas the EWC is sold directly to the consumer by the manufacturer. Sony’s “Partnership programme” adopts such a decentralized channel.

160

The structure of the dynamic Stackelberg game for Model M is slightly different from that for Model R. In stage 1, the manufacturer announces the wholesale price of the new product 𝑝1𝑀 (𝑡) as well as the selling price of the optional EWC 𝑝2 (𝑡). In stage 2, the retailer sets the retail price 𝑝1 (𝑡) according to the manufacturer’s wholesale price. Similar to Section 5.2.1, the dynamic Stackelberg game for Model M can be formulated as follows: The goal of the manufacturer is to maximize the following objective function:

𝐽𝑀 =

𝑀𝑎𝑥 𝐽𝑀 [𝑝1𝑀 (𝑡), 𝑝2 (𝑡)]

𝑝1𝑀 (𝑡),𝑝2 (𝑡)

𝑇

=

𝑀𝑎𝑥 𝑀

∫ 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)[𝑝1𝑀 (𝑡) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝2 (𝑡), 𝜂)]𝑑𝑡.

𝑝1 (𝑡),𝑝2 (𝑡) 0

The goal of the retailer is to maximize his or her profit given by

𝑇

𝐽 = 𝑀𝑎𝑥 𝐽 𝑝1 (𝑡)] = 𝑀𝑎𝑥 ∫ 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)[𝑝1 (𝑡) − 𝑝1𝑀 (𝑡)]𝑑𝑡. 𝑅

𝑝1 (𝑡)

𝑅[

𝑝1 (𝑡)

0

Both the manufacturer and retailer must obey the dynamic demand rate constraints 𝑋̇(𝑡) = 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏) 𝑌̇(𝑡) = 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)𝜌(𝑝2 (𝑡), 𝜂) 𝑋(0) = 𝑌(0) = 0; 𝑋(𝑇) and 𝑌(𝑇) are free (𝑝1𝑀 (𝑡), 𝑝1 (𝑡), 𝑝2 (𝑡)) ∈ 𝑈1𝑀 × 𝑈1 × 𝑈2 ⊆ 𝑅+3

5.3.2 Theoretical Analysis Following the analysis procedure in Section 5.2.2, it is not difficult to derive similar theoretical results for Model M. Therefore, this section only states the results without providing proofs and explanations.

161

Lemma 5.2’ Given the manufacturer’s wholesale price {𝑝1𝑀 (𝑡): 𝑡 ∈ [0, 𝑇]} , suppose {𝑋(𝑡), 𝑌(𝑡), 𝑝1° (𝑡)} is the optimal solution to the retailer’s problem for Model R. Then there must exist a piecewise continuously differentiable function 𝜃 𝑅 (𝑡) satisfying the following relations: 𝑝1° (𝑡) = 𝐴𝑟𝑔𝑚𝑎𝑥 𝐻𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1 ), ∀𝑡 ∈ [0, 𝑇]; 𝑝1 ∈𝑈1

𝑋̇ (𝑡) = 𝑓(𝑋 (𝑡), 𝑝1° (𝑡), 𝜏), 𝑋(0) = 0; 𝑌̇ (𝑡) = 𝑓(𝑋 (𝑡), 𝑝1° (𝑡), 𝜏)𝜌(𝑝2 , 𝜂), 𝑌(0) = 0; 𝜃̇ 𝑅 (𝑡) = −𝐻𝑋𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1° (𝑡)), 𝜃 𝑅 (𝑇) = 0, where 𝐻𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1 (𝑡)) is the Hamiltonian function for retailer’s problem defined as 𝐻𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1 (𝑡)) = 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)(𝑝1 (𝑡) − 𝑝1𝑀 (𝑡) + 𝜃 𝑅 (𝑡))

Lemma 5.3’ If the demand function 𝑓(𝑋, 𝑝1 , 𝜏) satisfies 2𝑓𝑝21 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏) − 𝑓𝑝1 𝑝1 (𝑋 (𝑡), 𝑝1 (𝑡), 𝜏)𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏) > 0 ; for all 𝑝1 (𝑡) ∈ 𝑈1 . Then the condition 𝑝1° (𝑡) = 𝐴𝑟𝑔𝑚𝑎𝑥 𝐻𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1 ), ∀𝑡 ∈ [0, 𝑇], 𝑝1 ∈𝑈1

is equivalent to 𝐻𝑝𝑅1 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1° (𝑡)) = 𝑓𝑝1 (𝑋(𝑡), 𝑝1° (𝑡), 𝜏) (𝑝1° (𝑡) − 𝑝1𝑀 (𝑡) + 𝜃 𝑅 (𝑡)) + 𝑓(𝑋 (𝑡), 𝑝1° (𝑡), 𝜏) = 0, and the optimal retail price 𝑝1° (𝑡) with respect to any given wholesale price is unique.

162

Proposition 5.1’ Suppose that the demand function 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏) satisfies 2𝑓𝑝21 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏) − 𝑓𝑝1 𝑝1 (𝑋 (𝑡), 𝑝1 (𝑡), 𝜏)𝑓(𝑋(𝑡), 𝑝1 (𝑡), 𝜏) > 0 , for all 𝑝1 (𝑡) ∈ 𝑈1 . Whenever the manufacturer decides to raise or lower the wholesale price, the optimal retailer price should follow the movement of the wholesale price.

Note that the manufacturer takes the retailer’s best response into consideration when optimizing his or her own total profit. Based on the optimality conditions for the retailer’s problem, the manufacturer’s optimization problem can be formulated as:

𝐽𝑀 =

𝑇

𝑀𝑎𝑥

∫ 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)[𝑝1𝑀 (𝑡) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝2 (𝑡), 𝜂)]𝑑𝑡

𝑝1𝑀 (𝑡),𝑝1 (𝑡),𝑝2 (𝑡) 0

subject to 𝑋̇ (𝑡) = 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏); 𝑋 (0) = 0 𝑌̇(𝑡) = 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)𝜌(𝑝2 , 𝜂); 𝑌(0) = 0 𝜃̇ 𝑅 (𝑡) = −𝐻𝑋𝑅 (𝑋(𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1 (𝑡)); 𝑝1𝑀 (𝑡) = 𝑝1 (𝑡) +

𝜃 𝑅 (𝑇 ) = 0

𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏) + 𝜃 𝑅 (𝑡 ) 𝑓𝑝1 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)

(𝑝1𝑀 (𝑡), 𝑝1 (𝑡), 𝑝2 (𝑡)) ∈ 𝑈1𝑀 × 𝑈1 × 𝑈2 ⊆ 𝑅+3 Then Pontryagin maximum principle would be applied to derive the necessary conditions of the optimal solution to the manufacturer’s problem:

∗

∗

Theorem 5.1’ Suppose (𝑋 ∗ (𝑡), 𝑌 ∗ (𝑡), 𝜃 𝑅 (𝑡), 𝑝1𝑀 (𝑡), 𝑝1∗ (𝑡), 𝑝2∗ (𝑡)) is the optimal solution for the manufacturer’s problem. There must exist piecewise continuously differential functions 𝜃1𝑀 (𝑡) and 𝜃3𝑀 (𝑡) , such that the optimal solution as well as 𝜃1𝑀 (𝑡) and 𝜃3𝑀 (𝑡) satisfy the following relations: 163

𝑝2∗ (𝑡) ≡ 𝑝2 = 𝐴𝑟𝑔𝑚𝑎𝑥 𝐸 (𝑝2 , 𝜂) , ∀𝑡 ∈ [0, 𝑇]; 𝑝2 ∈𝑈2 ∗

𝑝1∗ (𝑡) = 𝐴𝑟𝑔𝑚𝑎𝑥 𝐻𝑀 (𝑋 ∗ (𝑡), 𝜃 𝑅 (𝑡), 𝑝1 , 𝑝2 , 𝜃1𝑀 (𝑡), 𝜃3𝑀 (𝑡)), ∀𝑡 ∈ [0, 𝑇]; 𝑝1 ∈𝑈1

∗

𝑝1𝑀 (𝑡) = 𝑇(𝑝1∗ (𝑡)) = 𝑝1∗ (𝑡) +

𝑓 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏) ∗ + 𝜃 𝑅 (𝑡 ); 𝑓𝑝1 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)

∗ 𝑋̇ ∗ (𝑡) = 𝑓(𝑋 (𝑡), 𝑝∗1 (𝑡), 𝜏), 𝑋 ∗ (0) = 0;

𝑌̇ ∗ (𝑡) = 𝑓(𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)𝜌(𝑝2 , 𝜂), 𝑌 ∗ (0) = 0; ∗

∗ 𝑅 ∗ 𝑅 𝜃̇ 𝑅 (𝑡) = −𝐻𝑋 (𝑋 (𝑡), 𝜃 (𝑡), 𝑇 (𝑝∗1 (𝑡)) , 𝑝∗1 (𝑡)),

∗

𝜃 𝑅 (𝑇) = 0;

∗ 𝜃̇1𝑀 (𝑡) = −𝐻𝑋𝑀 (𝑋 ∗ (𝑡), 𝜃 𝑅 (𝑡), 𝑝1∗ (𝑡), 𝑝2 , 𝜃1𝑀 (𝑡), 𝜃3𝑀 (𝑡)), 𝜃1𝑀 (𝑇) = 0; ∗ 𝜃̇3𝑀 (𝑡) = −𝐻𝜃𝑀𝑅 (𝑋 ∗ (𝑡), 𝜃 𝑅 (𝑡), 𝑝1∗ (𝑡), 𝑝2 , 𝜃1𝑀 (𝑡), 𝜃3𝑀 (𝑡)), 𝜃3𝑀 (0) = 0,

where ∗

𝐻𝑀 (𝑋 ∗ (𝑡), 𝜃 𝑅 (𝑡), 𝑝1∗ (𝑡), 𝑝2 , 𝜃1𝑀 (𝑡), 𝜃3𝑀 (𝑡)) = 𝑓(𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏) [𝑇(𝑝1∗ (𝑡)) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸(𝑝2 , 𝜂) + 𝜃1𝑀 (𝑡) + 𝜃3𝑀 (𝑡)

𝑓𝑋 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏) ]. 𝑓𝑝1 (𝑋 ∗ (𝑡), 𝑝1∗ (𝑡), 𝜏)

5.3.3 Case Study This section considers the two specific demand functions introduced in Section 5.2.3 for Model M. In particular, the demand function in Example 5.1’ is the same as that in Example 5.1. It corresponds to a case in a static market and the closed-form optimal solution is derived. Example 5.2’ has the same demand function as Example 5.2, which is under a dynamic market. The modified control vector parameterization method can be applied to obtain the numerical solution. Note that the derivation and the algorithm in this section are the same as those in Section 5.2.3. Therefore, this section just presents the results but omit the detailed analysis.

164

Example 5.1’ Additive price-warranty demand function ℎ(𝑝1 , 𝜏) = 𝑎0 − 𝑎1 𝑝1 + 𝑎2 𝜏. Solution: The optimal wholesale price is 𝑎0 + 𝑎2 𝜏 + 𝑎1 (𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂)) . 2𝑎1

(5.32)

3(𝑎0 + 𝑎2 𝜏) + 𝑎1 (𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂)) . 4𝑎1

(5.33)

𝑝1𝑀 = The optimal retail price is

𝑝1 =

The total profits for the manufacturer and the retailer during the planning horizon are

𝐽𝑀 =

̅2 , 𝜂))] [(𝑎0 + 𝑎2 𝜏) − 𝑎1 (𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸(𝑝

2

𝑇;

8𝑎1 2

[(𝑎0 + 𝑎2 𝜏) − 𝑎1 (𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂))] 𝐽 = 𝑇. 16𝑎1 𝑅

While the total profit in the whole supply chain is 2

3[(𝑎0 + 𝑎2 𝜏) − 𝑎1 (𝐶𝑚 + 𝐶𝐵𝑊 − 𝐸 (𝑝̅2 , 𝜂))] 𝐽 +𝐽 = 𝑇. 16𝑎1 𝑅

𝑀

Example 5.2’ Limited-growth demand function 𝑓 (𝑋, 𝑝1 , 𝜏) = 𝑘1 [𝑘2 + 𝜏]𝑏 𝑝1−𝑎 [𝑀 − 𝑋]. The two numerical experiments in Example 5.2 are repeated in Model M. The results are presented in Figure 5.2 and Table 5.3, which are the counterparts of Figure 5.1 and Table 5.2 for Model M, respectively.

165

Price (x 103)

16

15

14

13 retail price N=6 wholesale price N=6 retail price N=20 wholesale price N=20

12

11

10

9 0

0.5

1

1.5

2

2.5

3

Figure 5.3 The approximated optimal wholesale and retail prices for Model M

Table 5.3 Profit analysis under different 𝐸 (𝑝̅2 , 𝜂) for Model M 𝐸(𝑝̅2 , 𝜂) (× 103 HKD)

0.4

0.6

1

Manufacturer’s profit (Model M) × 106 HKD

46.8372

49.4000

55.0784

Retailer’s profit (Model M) × 106 HKD

65.7825

69.0404

76.1250

Total profit (Model M) × 106 HKD

112.6197

118.4402

131.2034

Total profit (Centralized channel) × 106 HKD

149.6894

157.1016

173.2603

166

5.4 Comparisons Among Distribution Channels This section compares the optimal profits and prices in the centralized channel (Chapter 3) and the two decentralized channels (Model R and Model M), which is based on the case study results in Sections 5.2.3 and 5.3.3. Comparison 1: The first part compares the total profit under the centralized channel (Chapter 3) with those under the decentralized channels (Model R and Model M). The literature of supply chain management suggests that the profit of a centralized supply chain is always as much as, if not higher than, that of a decentralized counterpart. This property also holds for this problem. Given the same demand function and input parameters, the centralized channel generates the largest total profit. Take Model R as an example, the total profit in the whole supply chain (or the whole channel) is: 𝑇

𝐽 + 𝐽 = ∫ 𝑓(𝑋 (𝑡), 𝑝1 (𝑡), 𝜏)[𝑝1𝑀 (𝑡) − 𝐶𝑚 − 𝐶𝐵𝑊 ]𝑑𝑡 𝑀

𝑅

0

𝑇

+ ∫ 𝑓 (𝑋(𝑡), 𝑝1 (𝑡), 𝜏)[𝑝1 (𝑡) − 𝑝1𝑀 (𝑡) + 𝐸 (𝑝2 (𝑡), 𝜂)]𝑑𝑡 0

𝑇

= ∫ 𝑓(𝑋 (𝑡), 𝑝1 (𝑡), 𝜏)[𝑝1 (𝑡) − 𝐶𝑚 − 𝐶𝐵𝑊 + 𝐸 (𝑝2 (𝑡), 𝜂)]𝑑𝑡, 0

(5.34)

which is exactly the objective function for the centralized channel. Similarly, this observation can be verified for Model M. Also note that the optimal retail price in either Model R or Model M is a feasible price trajectory in the problem under the centralized channel. Therefore, the centralized channel must generate the largest total profit. When the demand function is the additive price-warranty demand function, the total profit in Model R (or Model M) only accounts for 3/4 of the total profit in the centralized channel (c.f. Examples 3.1, 5.1 and 5.1’).

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When the demand function is the limited-growth demand function, the numerical results in Tables 3.4, 5.2 and 5.3 also confirm that the centralized channel has significantly higher profits than both Model R and Model M (c.f. Examples 3.3, 5.2 and 5.2’). Comparison 2: The second part compares the two decentralized channels, i.e., Model R and Model M. As pointed out in Comparison 1, both Model R and Model M have the same total profit function (5.34). Therefore, if the optimal retail prices for Model R and Model M are the same, then the total profits for Model R and Model M must be the same. When the demand function is the additive price-warranty demand function, the closed-form solutions in Examples 5.1 and 5.1’ assert that the retail prices in Model R and Model M are the

price (x 103)

same and the total profits for Model R and Model M are also the same.

16

15

14

13

retail price for Model R wholesale price for Model R wholesale price for Model M retail price for Model M

12

11

10

9 0

0.5

1

1.5

2

2.5

Figure 5.4 The wholesale and retail prices for Model R and Model M

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3

Consider a case wherein the demand function is the limited-growth demand function. Given the parameter values in Table 5.1, the optimal wholesale and retail prices for Model R and Model M are illustrated in Figure 5.4, which is obtained by combining Figures 5.2 and 5.3. Along with the total profits displayed in Example 5.2 and Example 5.2’, Figure 5.4 also confirms that the retail prices in Model R and Model M are the same and the total profits for Model R and Model M are also the same (Note: In Figure 5.4, the retail prices for Model R and Model M are not the same, but the discrepancy may be caused by the approximation errors of the control vector parameterization method). Therefore, for a general demand function, one can conjecture that the optimal retail prices in Model R and Model M are the same. In addition, the manufacturer’s profit may remain the same in either Model R or Model M. An intuitive explanation is that the manufacturer will set a relatively higher wholesale price when the retailer offers the EWC so as to cover the loss of not selling the EWC. We leave the rigorous proof of these results to future work.

5.5 Chapter Summary This chapter studies the optimal dynamic pricing strategies under decentralized distribution channels. In order to incorporate the imitation (e.g., word-of-mouth) and saturation effects in a dynamic market, the demand (state equation) is assumed to depend on both the retail price of the new product and past cumulative sales. Both the manufacturer and the retailer want to maximize their own total profit over the planning horizon. Therefore, the appropriate solution approach is the open loop Stackelberg game. In particular, two dynamic Stackelberg games have been formulated according to whether the retailer or the manufacturer offers the EWC. This chapter analyses the necessary optimality conditions and studies the properties of the optimal solutions. In most cases, the optimal solution for each game is analytically intractable. Therefore, the control vector parameterization method has been modified to compute an approximate solution.

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Chapter 6 Conclusions and Future Work 6.1 Conclusions This thesis has studied a complex product diffusion problem in which a monopoly manufacturer plans to launch a new product into a dynamic market along with a free basic warranty contract (BW) and an optional extended warranty contract (EWC). The objective is to formulate the optimal pricing strategies for the product and its EWC, and the optimal production decisions to maximize the manufacturer’s long-term profit. Three mathematical models have been developed to describe the interrelationships among the EWC, the production rate and the distribution channel design, and the demand function of the new product which is affected by factors such as the diffusion effect and the saturation effect. There are three major assumptions made in the first model: (1) the new product is launched into the market directly by the manufacturer along with a free BW and an optional EWC which is modelled as an “insurance contract” with a price and a coverage period; (2) the unit production cost for the new product is constant (i.e., the effect of production rate can be neglected); and (3) the demand function for the new product depends on the selling price, the length of the period covered by the BW and the cumulative sales volume. This model has been formulated in the form of an optimal control problem for answering the following questions: (a) How to set the price of the new product and its optional EWC, the “insurance contract”? (b) What are the properties of the optimal pricing strategies for the new product and the EWC? (c) How does the length of the period covered by BW affect the total profit? (d) How does the optimal price of the new product behave for some specified demand functions? By invoking the Pontryagin maximum principle, this research has shown that the optimal price of the EWC solely depends on the length of the period covered by the EWC and the expected maintenance cost incurred during the period, and is constant over time. The optimal price of the new product, however, is more complex because it is influenced by both the price of the EWC and the dynamic characteristics of the demand function. According to classical economic theory, 170

the optimal price of the new product for a single period monopolist should satisfy the principle that marginal revenue is equal to marginal cost. However, this principle is not applicable in this case because the manufacturer is a multi-period monopolist whose aim is to maximize the total profit over the whole planning horizon. Indeed, it has been shown theoretically that the marginal cost should be equal to the sum of the instantaneous marginal revenue, the future marginal profit, and the expected profit from selling the EWC, and that the manufacturer needs to lower the selling price of the new product when the expected profit from selling the EWC increases. Although the profit margin of selling the new product may decrease due to lowering the selling price, the profit loss incurred in selling the new product at a lower price can be covered by selling the EWC. These results have been validated by the case study in Chapter 3. In addition, a longer BW coverage period promotes the sales volume of the new product but incurs higher maintenance cost. Based on the structure of the model, a parameter analyzing method has been used to study how the optimal total profit changes with the length of the BW coverage period, and the changing rate is measured by an explicit integral expression. The phase diagram analysis has been used to provide a qualitative characterization of the behaviour of the optimal solution, and the gradient method has been adopted to develop an efficient algorithm to compute the optimal pricing strategies. If the unit production cost of the new product is significantly affected by the production rate, the first model has been extended to include the production rate as a decision variable. In this case, a high production rate may lead to extra inventory, while a low production rate may lead to shortage (backlogging). The problem then becomes determining the optimal pricing strategies for the new product and the EWC, and the optimal production rate to maximize the manufacturer’s total profit. The extended model has also been formulated in the form of an optimal control problem. The Pontryagin maximum principle has been applied to derive the necessary conditions for the optimal solution. Similar to the first model, the optimal selling price for the EWC is constant over time. However, the optimal price for the new product in the extended model needs to satisfy a modified economic principle, i.e., the marginal cost should be equal to the sum of the instantaneous marginal revenue, the future marginal profit, and the expected profit from selling the EWC. Unlike the first model where the marginal cost is a constant, the marginal cost in the

171

extended model is jointly determined by the production rate, and the inventory cost and / or backlogging cost. In this respect, the optimal selling price for the new product and the optimal production rate affect each other. A more robust algorithm based on the control vector parameterization method has also been developed to provide the optimal solution numerically. As the boundary condition in the extended model is much more complex than that in the first model, the gradient method developed for the first model fails to converge. The control vector parameterization method, which uses the piecewise-constant basic functions to approximate the optimal solution, has less restrictions when dealing with optimal control problems compared with the gradient method. Therefore, it has been successfully applied to find the approximate solution for the extended model. Although the control vector parameterization method has not received much attention in the dynamic pricing literature, the numerical experiments conducted in Chapters 4 and 5 demonstrate its robustness and effectiveness. The third model focuses on decentralized distribution channels where the new product must be sold through an independent retailer to end consumers. As the EWC can be offered by either the manufacturer or the retailer, two different decentralized channels have been investigated. For each decentralized channel, a dynamic Stackelberg game has been formulated to study how the EWC affects the optimal wholesale price and retail price chosen by the manufacturer and the retailer, respectively. Previous studies considering the EWC in channel design problems were limited to examining one-shot interactions between the manufacturer and retailer. The dynamic Stackelberg game has explored how their pricing strategies (i.e., interactions) evolve over time. The necessary conditions for the optimal wholesale and retail prices have been derived. In addition, control vector parameterization method has been applied to design a modified numerical algorithm and generate the approximate optimal wholesale and retail prices. The research has also compared the two decentralized channels. The case study results indicate that the optimal retail prices seem to be the same in the two decentralized channels. However, this observation has to be rigorously proved under a general framework in future research.

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6.2 Recommendations for Future Work Research on the formulation of pricing strategies for launching a new product into a dynamic market along with a free BW and an optional EWC is in its infancy, and its outcomes can have significant economic importance. This thesis has only presented the solutions to some key problems. There are still enormous opportunities for contribution to the advancement of this field. To advance the accomplishments of this thesis, the following research problems are recommended for further investigation: The three models presented in this thesis assume that the manufacturer is a monopolist in the market (i.e., no competitors producing similar products). This assumption is a good approximation of reality in many situations, e.g., when the manufacturer holds the patent to a certain technology, or has monopoly over a certain natural resource. In these situations, the demand function depends on the monopolist’s price. However, the monopoly assumption is no longer suitable if new manufacturers producing similar products are allowed to enter the market. In this situation, the demand function of the original manufacturer depends on the competitors’ price as well. Future research should include competition among manufacturers when modelling the diffusion problem of launching the new product with an optional EWC. This problem can be modelled as another dynamic game. The optimal pricing strategies for the manufacturers would be determined by the Nash equilibrium of this game. In the distribution channel design problem, the two proposed dynamic Stackelberg games have assumed that there is no information asymmetry between the manufacturer and the retailer. However, in real situations, the retailer may have private information about the consumers’ demand for the new product and the EWC, and may use this information to improve his or her own profit in the distribution channel. Future research can consider the existence of information asymmetry between the manufacturer and the retailer. The problem can again be formulated as a dynamic Stackelberg game. The difficulty is in modelling the information asymmetry. Furthermore, the two proposed dynamic games have also assumed that the unit production cost is constant. Further work can be done to extend these two dynamic games to include a more general unit production cost function as well as an inventory / backlog cost function. However, the extended dynamic games with general cost functions are not always concave under certain

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specific conditions, and can only be optimized via more general bi-level programming methods or numerical optimization methods. Finally, the three proposed model have assumed that the demand for the new product is deterministic. Although this assumption is valid as shown in many diffusion models which have been formulated to depict accurately the new product diffusion process, it may be appropriate in some cases to assume consumer demand to be stochastic. Therefore, a relevant extension of this study is to discuss the impact of demand uncertainty on the optimal pricing strategies for both the new product and the EWC. Stochastic control theory could be applied in this cases to solve the problems.

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Reference Aoki, Masanao, & Yoshikawa, Hiroshi. (2002). Demand saturation-creation and economic growth. Journal of Economic Behavior & Organization, 48(2), 127-154. Arrow, Kenneth J. (1962). The economic implications of learning by doing. The review of economic studies, 155-173. Barnett, R.A., Ziegler, M.R., & Byleen, K.E. (2002). Applied Mathematics for Business, Economics, Life Sciences, and Social Sciences: Prentice Hall PTR. Bass, Frank M. (1969). A new product growth model for consumer durables. Management Science. 15 215-227. Bass, Frank M. (1980). The relationship between diffusion rates, experience curves, and demand elasticities for consumer durable technological innovations. The Journal of Business, 53(3), S5167. Bass, Frank M, Jain, Dipak, & Krishnan, Trichy. (2000). Modeling the marketing-mix influence in newproduct diffusion: Kluwer Academic Publishers: Boston, MA. Bea, Randal W. (1998). Successive Galerkin approximation algorithms for nonlinear optimal and robust control. International Journal of Control, 71(5), 717-743. doi: 10.1080/002071798221542 Becerra, Victor M. (2004). Solving optimal control problems with state constraints using nonlinear programming and simulation tools. Education, IEEE Transactions on, 47(3), 377-384. Blischke, WR, & Murthy, DNP. Product Warranty Handbook. 1996. Miircel Dekker. Chattopadhyay & Rahman. (2008), Development of lifetime warranty policies and models for estimating costs, Reliability Engineering and System Safety, 93, 522–529. Chachuat, B.C. (2009). Nonlinear and Dynamic Optimization: From Theory to Practice: EPFL. Chatterjee, Rabikar. (2009). Strategic pricing of new product and services. Handbook of Pricing Research in Marketing, 169. Chen, Tao, Kalra, Ajay, & Sun, Baohong. (2009). Why do consumers buy Extended Service contracts? Journal of Consumer Research, 36(4), 611-623. Chen, T., & Sun, B. (2007). An Empirical Investigation of Consumer Purchases and Intertemporal Pricing of Retailers’ Extended Service Contracts. Gumegie: Carnegie Mellon University.

175

Chenavaz, Régis, Carrier, Louis-Philippe, Etienne, Lydie, & Paraschiv, Corina. (2011). Dynamic Pricing in Management Science. Journal of Economic Studies and Research DOI, 10(2011.283). Cohen, Morris A, Agrawal, Narendra, & Agrawal, Vipul. (2006). Winning in the aftermarket. Harvard business review, 84(5), 129. Consumers' CHECKBOOK. (2007). Automobile Extended Service Contracts. From http://www.checkbook.org/auto/ExtendedService.cfm Day, E., & Fox, R. J. (1985). Extended Warranties, Service Contracts, and Maintenance Agreements—A Marketing Opportunity? Journal of Consumer Marketing, 2(4), 77-86. Demidenko, E. (1996). On the existence of the least squares estimate in nonlinear growth curve models of exponential type. Communications in Statistics-Theory and Methods, 25(1), 159-182. Demidenko, E. (2006). Criteria for global minimum of sum of squares in nonlinear regression. Computational statistics & data analysis, 51(3), 1739-1753. Dennis, J. E., & Schnabel, R. B. (1996). Numerical methods for unconstrained optimization and nonlinear equations (Vol. 16): Siam. Desai, Preyas S, & Padmanabhan, Paddy. (2004). Durable good, extended warranty and channel coordination. Review of Marketing Science, 2(1). Dockner, & Engelbert. (2000). Differential games in economics and management science: Cambridge University Press. Dockner, Engelbert, & Jørgensen, Steffen. (1988). Optimal pricing strategies for new products in dynamic oligopolies. Marketing Science, 7(4), 315-334. Dorfman, Robert. (1969). An economic interpretation of optimal control theory. The American Economic Review, 59(5), 817-831. Errico, Ronald M. (1997). What is an adjoint model? Bulletin of the American Meteorological Society, 78(11), 2577-2591. Fan, YY, Bhargava, Hemant K, & Natsuyama, HH. (2005). Dynamic Pricing via Dynamic Programming1. Journal of optimization theory and applications, 127(3), 565-577. Faridimehr, Sina, & Niaki, STA. (2013). Optimal strategies for price, warranty length, and production rate of a new product with learning production cost. Transactions E: Industrial Engineering, 20. Gill, P. E., Murray, W., & Wright, M. H. (1981). Practical optimization.

176

Giri, BC, & Moon, I. (2004). Accounting for idle capacity cost in the scheduling of economic lot sizes. International journal of production research, 42(4), 677-691. Glickman, Theodore S, & Berger, Paul D. (1976). Optimal price and protection period decisions for a product under warranty. Management Science, 22(12), 1381-1390. Gutierrez, Genaro J, & He, Xiuli. (2011). Life‐Cycle Channel Coordination Issues in Launching an Innovative Durable Product. Production and Operations Management, 20(2), 268-279. Hartman, J. C., & Laksana, K. (2009). Designing and pricing menus of extended warranty contracts. Naval Research Logistics, 56(3). He, Xiuli. (2007). Stackelberg differential game models in supply chain management. (PhD). He, Xiuli, Prasad, Ashutosh, Sethi, Suresh P, & Gutierrez, Genaro J. (2007). A survey of Stackelberg differential game models in supply and marketing channels. Journal of Systems Science and Systems Engineering, 16(4), 385-413. He, Xiuli, & Sethi, SP. (2008). Dynamic slotting and pricing decisions in a durable product supply chain. Journal of Optimization Theory and Applications, 137(2), 363-379. Huang, Hong-Zhong, Liu, Zhi-Jie, & Murthy, DNP. (2007). Optimal reliability, warranty and price for new products. IIE Transactions, 39(8), 819-827. Isaacs, RUFUS. (1965). Differential Games: Wiley, New York. Jørgensen, Steffen, Kort, Peter M, & Zaccour, Georges. (1999). Production, inventory, and pricing under cost and demand learning effects. European Journal of Operational Research, 117(2), 382-395. Jørgensen, Steffen, Sigue, Simon-Pierre, & Zaccour, Georges. (2001). Stackelberg leadership in a marketing channel. International Game Theory Review, 3(01), 13-26. Jørgensen, Steffen, & Zaccour, Georges. (2007). Developments in differential game theory and numerical methods: economic and management applications. Computational Management Science, 4(2), 159-181. Jukić, D. (2009). On the existence of the best discrete approximation in lp norm by reciprocals of real polynomials. Journal of Approximation Theory, 156(2), 212-222. Josrgensen, Steffen. (1983). Optimal control of a diffusion model of new product acceptance with pricedependent total market potential. Optimal Control Applications and Methods, 4(3), 269-276.

177

Jukić, D. (2011). Total least squares fitting Bass diffusion model. Mathematical and computer modelling, 53(9), 1756-1770. Jukić, D. (2013). On nonlinear weighted least squares estimation of Bass diffusion model. Applied mathematics and computation, 219(14), 7891-7900. Jukić, D., Benšić, M., & Scitovski, R. (2008). On the existence of the nonlinear weighted least squares estimate for a three-parameter Weibull distribution. Computational statistics & data analysis, 52(9), 4502-4511. Jukić, D., & Marković, D. (2010). On nonlinear weighted errors-in-variables parameter estimation problem in the three-parameter Weibull model. Applied mathematics and computation, 215(10), 3599-3609. Kalish, Shlomo. (1983). Monopolist pricing with dynamic demand and production cost. Marketing Science, 2(2), 135-159. Kamien, Morton I, & Schwartz, Nancy Lou. (2012). Dynamic optimization: the calculus of variations and optimal control in economics and management: DoverPublications. com. Kar, S, Roy, TK, & Maiti, M. (2008). Multi-objective Inventory Model of Deteriorating Items with Space Constraint in a Fuzzy Environment. Tamsui Oxford Journal of Mathematical Sciences, 24(1), 37. Kleyner. A & Sandborn.P. (2008), Minimizing life cycle cost by managing product reliability via validation plan and warranty return cost, International Journal of Production Economics, 112 (2) 796–807. Khouja, Moutaz. (1995). The economic production lot size model under volume flexibility. Computers & operations research, 22(5), 515-523. Kim, Bowon, & Park, Sangsun. (2008). Optimal pricing, EOL (end of life) warranty, and spare parts manufacturing strategy amid product transition. European Journal of Operational Research, 188(3), 723-745. Kirk, D.E. (2004). Optimal Control Theory: An Introduction: Dover Publications. Lam, Yeh, & Kwok Wai Lam, Peggo. (2001). An extended warranty policy with options open to consumers. European Journal of Operational Research, 131(3), 514-529. Ladany, S.P., Shore, H., (2007). Profit maximizing warranty period with sales expressed by a demand function. Quality and Reliability Engineering International, 23, 291–301.

178

Levinthal, Daniel A, & Purohit, Devavrat. (1989). Durable goods and product obsolescence. Marketing Science, 8(1), 35-56. Li, Kunpeng, Mallik, Suman, & Chhajed, Dilip. (2012). Design of extended warranties in supply chains under additive demand. Production and Operations Management, 21(4), 730-746. Liberzon, Daniel. (2012). Calculus of variations and optimal control theory: a concise introduction: Princeton University Press. Lin, Pei-Chun. (2008). Optimal pricing, production rate, and quality under learning effects. Journal of Business Research, 61(11), 1152-1159. Lin, Pei-Chun, & Shue, Li-Yen. (2005). Application of optimal control theory to product pricing and warranty with free replacement under the influence of basic lifetime distributions. Computers & Industrial Engineering, 48(1), 69-82. Lin, Pei-Chun, Wang, Jenhung, & Chin, Shiao-Shan. (2009). Dynamic optimisation of price, warranty length and production rate. International Journal of Systems Science, 40(4), 411-420. Lindqvist, Bo Henry. (2008). Maintenance of Repairable Systems Complex System Maintenance Handbook (pp. 235-261): Springer. Mahajan, Vijay, & Muller, Eitan. (1991). Pricing and diffusion of primary and contingent products. Technological Forecasting and Social Change, 39(3), 291-307. Mahajan, V., Mason, C. H., & Srinivasan, V. (1986). An Evaluation of Estimation Procedures for New Product Diffusion Models. In V. Mahajan & Y. Wind (Eds.), Innovation Diffusion Models of New Product Acceptance (pp. pp. 203–232): Ballinger Publishing Company, Cambridge. Mahajan, Vijay, Muller, Eitan, & Bass, Frank M. (1990). New product diffusion models in marketing: A review and directions for research. The Journal of Marketing, 1-26. Mahajan, Vijay, Muller, Eitan, & Wind, Yoram. (2000). New-product diffusion models: Springer. Manna, Dipak K. (2008). Price-warranty length decision with Glickman-Berger model. International Journal of Reliability and Safety, 2(3), 221-233. Markell, Alan G. (1993). Learning and Production Rate in Cost Estimating: DTIC Document. McGuire, Timothy W, & Staelin, Richard. (1983). An industry equilibrium analysis of downstream vertical integration. Marketing Science, 2(2), 161-191. Mesak, Hani I. (1996). Modelling monopolist pricing and protection period decisions for new products under warranty. Optimal Control Applications and Methods, 17(4), 231-252.

179

Moore, Evan C. (2002). An investigation into the demand for service contracts. Citeseer. Moore, Evan C. (2011). Investigating the demand for new vehicle service contracts. Southwestern Economic Review, 35, 37-52. Murthy, D.N.P., & Blischke Wallace R., (2000), Strategic warranty management: a life-cycle approach, IEEE Transactions on Engineering Management, 47, 40–54. Murthy, D.N.P., & Blischke, W.R. (2006). Warranty management and product manufacture: Springer. Murthy, D.N.P., & Djamaludin, I. (2002). New product warranty: A literature review. International Journal of Production Economics, 79(3), 231-260. Murthy, D.N.P., & Jack, N. (2003). Warranty and maintenance, Handbook of reliability engineering (pp. 305-316): Springer. Karbasian, M, & Ibrahim, Z. (2010). Estimation of Parameters of the Power-Law-Non-Homogenous Poisson Process in the Case of Exact Failures Data. International Journal of Industrial Engineering, 21(2) Padmanabhan, V., & Rao, R. C. (1993). Warranty policy and extended service contracts: Theory and an application to automobiles. Marketing science, 12(3), 230-247. Padmanabhan,V.(1996). Extended Warranties, Product Warranty Handbook, New York, 439–452. Parker, Philip M. (1994). Aggregate diffusion forecasting models in marketing: A critical review. International journal of forecasting, 10(2), 353-380. Peres, Renana, Muller, Eitan, & Mahajan, Vijay. (2010). Innovation diffusion and new product growth models: A critical review and research directions. International Journal of Research in Marketing, 27(2), 91-106. Pingle, Mark. (2010). Using gambling to teach insurance principles. Putsis Jr, W. P., & Srinfvasan, V. (2000). 11. ESTIMATION TECHNIQUES FOR MACRO DIFFUSION MODELS. New-product diffusion models. Rao, Vithala R. (1984). Pricing research in marketing: The state of the art. Journal of Business, S39-S60. Rao, Vithala R. (2009). Handbook of pricing research in marketing: Edward Elgar Publishing. Rao, Vithala R, & Kartono, Benjamin. (2009). Pricing objectives and strategies: A cross-country survey. Handbook of pricing research in marketing. Elgar, Cheltenham, 9-36.

180

Robinson, Bruce, & Lakhani, Chet. (1975). Dynamic price models for new-product planning. Management Science, 21(10), 1113-1122. Ruiz-Conde, Enar, Leeflang, Peter SH, & Wieringa, Jaap E. (2006). Marketing variables in macro-level diffusion models. Journal für Betriebswirtschaft, 56(3), 155-183. Sarikavanij, Sukuman, Pongpullponsak, Adisak, & Jiarasuksakun, Thiradet. (2011). Decision and Computing Program of Optimal Price and Warranty Length for a Gamma Lifetime Distributed Product. Schlesinger, H. (2000). The theory of insurance demand, Handbook of insurance (pp. 131-151): Springer. Schmittlein, D. C., & Mahajan, V. (1982). Maximum likelihood estimation for an innovation diffusion model of new product acceptance. Marketing science, 1(1), 57-78. Scitovski, R., & Meler, M. (2002). Solving parameter estimation problem in new product diffusion models. Applied mathematics and computation, 127(1), 45-63. Seierstad, Atle, & Sydsaeter, Knut. (1987). Optimal control theory with economic applications (Vol. 20): North-Holland Amsterdam. Sethi, Suresh P, & Thompson, Gerald L. (2000). Optimal control theory: applications to management science and economics: Springer. Spengler, Joseph J. (1950). Vertical integration and antitrust policy. The Journal of Political Economy, 347-352. Srinivasan, V., & Mason, C. H. (1986). Technical Note-Nonlinear Least Squares Estimation of New Product Diffusion Models. Marketing science, 5(2), 169-178. Sterman, John D. (2002). System dynamics: systems thinking and modeling for a complex world. Paper presented at the Proceedings of the ESD Internal Symposium. Sudhir, K, & Datta, Sumon. (2008). Pricing in Marketing Channels Handbook of Pricing (V. Rao Ed.), Edward Elgar Publishers. Teng, Jinn-Tsair, & Thompson, Gerald L. (1996). Optimal strategies for general price-quality decision models of new products with learning production costs. European Journal of Operational Research, 93(3), 476-489. Tirole, Jean. (1988). The theory of industrial organization: MIT press. Troutman, J.L. (1996). Variational Calculus and Optimal Control: Optimization With Elementary Convexity: Springer-Verlag.

181

Wächter, Andreas, & Biegler, Lorenz T. (2006). On the implementation of an interior-point filter linesearch algorithm for large-scale nonlinear programming. Mathematical programming, 106(1), 2557. Warranty Week. 2004. Published on 10/26. From http://www.warrantyweek.com/archive/ww20041026.html Weber, Thomas A. (2011). Optimal control theory with applications in economics (Vol. 1). Wu, Chin-Chun, Chou, Chao-Yu, & Huang, Chikong. (2009). Optimal price, warranty length and production rate for free replacement policy in the static demand market. Omega, 37(1), 29-39. Wu, Chin-Chun, Lin, Pei-Chun, & Chou, Chao-Yu. (2006). Determination of price and warranty length for a normal lifetime distributed product. International Journal of Production Economics, 102(1), 95-107. Xu, Ke, Chiang, Wei‐Yu Kevin, & Liang, Liang. (2011). Dynamic pricing and channel efficiency in the presence of the cost learning effect. International Transactions in Operational Research, 18(5), 579-604. Wu.S, & Xie, M.(2008), Warranty cost analysis for non-repairable services products, International Journal of Systems Science, 39 (3), 279–288 Yano, Candace A, & Gilbert, Stephen M. (2005). Coordinated pricing and production/procurement decisions: A review Managing Business Interfaces (pp. 65-103): Springer. Zaccour, G. (2012). Optional-Contingent-Products Pricing in Marketing Channels P. Kort, S. Taboubi. Zaccour, G. (2004). Differential games in marketing (Vol. 15): Springer. Zhou, Zhefang, Li, Yanjun, & Tang, Kwei. (2009). Dynamic pricing and warranty policies for products with fixed lifetime. European Journal of Operational Research, 196(3), 940-948. Zuo, Xu. (2011). Dynamic Pricing Policies in a Two-echelon Supply Chain. City University of Hong Kong. Retrieved from http://books.google.com.hk/books?id=aj9FMwEACAAJ

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Appendix A Mathematical notation: Uses in this study In this study, we use the following abbreviations which are now standard in mathematical literature: ∈

for “is an element of” or “is in” or “in”

∉ or ∈

for “is not an element of” or “is not in”

∀

for “for each” or “for every”

R𝑑+

for “positive part of d-dimensional Euclidean space”

∎

for “end of proof”

In addition, we shall need notations which are not standard in order to handle effectively the following situations: (1) For the univariate function 𝐹 (𝑥 ) with x as the argument, its first-order derivative would be denoted simply as 𝐹̇ (𝑥) =

𝑑𝐹(𝑥) . 𝑑𝑥

(2) For the multivariate function 𝐹(𝑥1 , 𝑥2 , … , 𝑥𝑛 ), its first-order derivative with respect to 𝑥𝑖 , 𝑖 = 1,2, … , 𝑛 (or called partial derivative with respect to 𝑥𝑖 ) are as usual denoted 𝐹𝑥𝑖 (𝑥1 , 𝑥2 , … , 𝑥𝑛 ) =

𝜕𝐹(𝑥1 , 𝑥2 , … , 𝑥𝑛 ) ; 𝜕𝑥𝑖

second-order derivatives are denoted as: 𝐹𝑥𝑖𝑥𝑗 (𝑥1 , 𝑥2 , … , 𝑥𝑛 ) =

𝜕 2 𝐹(𝑥1 , 𝑥2 , … , 𝑥𝑛 ) . 𝜕𝑥𝑖 𝜕𝑥𝑗

(3) For the multivariate composite function 𝐹 (𝑥1 (𝑡), 𝑥2 (𝑡), … , 𝑥𝑛 (𝑡)), the following expression 𝐹𝑥𝑖 (𝑥1 (𝑡), 𝑥2 (𝑡), … , 𝑥𝑛 (𝑡)); can be explained as: 183

𝑖 = 1,2, … , 𝑛

First, we write 𝐹 = 𝐹 (𝑥1 , 𝑥2 , … , 𝑥𝑛 ) for which the partial derivatives are usually denoted as 𝐹𝑥𝑖 (𝑥1 , 𝑥2 , … , 𝑥𝑛 ). Then, we evaluate at (𝑥1 (𝑡), 𝑥2 (𝑡), … , 𝑥𝑛 (𝑡)), and denote the respective results by 𝐹𝑥𝑖 (𝑥1 (𝑡), 𝑥2 (𝑡), … , 𝑥𝑛 (𝑡));

𝑖 = 1,2, … , 𝑛.

Observe that with this notation, the chain rule takes the following forms 𝑑𝐹 = 𝐹𝑥1 (𝑥1 (𝑡), 𝑥2 (𝑡), … , 𝑥𝑛 (𝑡))𝑥̇ 1 (𝑡) + ⋯ + 𝐹𝑥𝑛 (𝑥1 (𝑡), 𝑥2 (𝑡), … , 𝑥𝑛 (𝑡))𝑥̇ 𝑛 (𝑡). 𝑑(𝑡) (4) For the multivariate composite function 𝐹 (𝛼, 𝑥1 (𝑡), 𝑥2 (𝑡), … , 𝑥𝑛 (𝑡)) which accompanies parameter 𝛼, introduce the 𝑛 + 1′𝑡ℎ variate 𝑥0 (𝑡) ≡ 0, then 𝐹𝛼 (𝛼, 𝑥1 (𝑡), 𝑥2 (𝑡), … , 𝑥𝑛 (𝑡)) = 𝐹𝑥0 (𝑥0 (𝑡), 𝑥1 (𝑡), 𝑥2 (𝑡), … , 𝑥𝑛 (𝑡)); 𝐹𝑥𝑖 (𝛼, 𝑥1 (𝑡), 𝑥2 (𝑡), … , 𝑥𝑛 (𝑡)) = 𝐹𝑥𝑖 (𝑥0 (𝑡), 𝑥1 (𝑡), 𝑥2 (𝑡), … , 𝑥𝑛 (𝑡)).

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