The Pennsylvania State University The Graduate School College of Engineering

RAW MATERIALS PROCUREMENT PLANNING MODELS WITH APPLICATIONS

A Thesis in Industrial Engineering and Operations Research by Moises Eidelman Ghelman

© 2010 Moises Eidelman Ghelman

Submitted in Partial Fulfillment of the Requirements for the Degree of

Master of Science

May 2010

The thesis of Moises Eidelman was reviewed and approved* by the following:

A. Ravi Ravindran Professor of Industrial Engineering and Affiliate Professor of IST School Thesis Adviser

Felisa Preciado Clinical Assistant Professor of Supply Chain Management

Paul Griffin Professor of Industrial Engineering Head of the Department of Industrial and Manufacturing Engineering

*Signatures are on file in the Graduate School.

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Abstract The main focus of the thesis is to develop an optimal procurement policy for raw materials, that is, reorder points and optimal order quantities, for a multinational company. We develop a multiple period mixed integer programming model that takes into account the variation in demand that occurs in the forecasted demands for raw materials from one month to another. Later, we allow for backorders and introduce a service level criterion for each of the raw material. The model also includes the cost of holding inventory and a fixed cost of ordering whenever an order is placed during a given period. The model is illustrated with real data collected from a health and hygiene company that operates in more than 35 countries around the globe. For this case study, inventory, financial information, leadtimes and demand information are collected. Raw materials demand requirements in turn relates to the demand of finished goods through the Bill of Materials (BOM) structure. Thus, finished goods demand is characterized so raw materials demand can be accurately estimated. Financial information includes procurement, transportation, administrative and holding costs, as well as suppliers’ discounts policies or any other costs related to stocking up raw materials. Order lead-times, which account for supplier processing time, shipping and administrative times due to orders consolidation and handling, have also been taken into account. The model was run monthly, on a rolling horizon basis, over the entire planning horizon, and only the results obtained for the period in which the model was run (namely, the first month) was implemented each time. By doing this, we were able to capture the changes in the forecasted demand from month to month and the variations in the raw materials delivery schedule. Finally, recommendations regarding the implementation of the proposed policy, as well as future issues, such as demand and lead time variations, have been discussed.

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TABLE OF CONTENTS LIST OF FIGURES ................................................................................................................................. v LIST OF TABLES .................................................................................................................................. vi ACKNOWLEDGEMENTS .................................................................................................................. viii Chapter 1 INTRODUCTION ................................................................................................................. 1 1.1 Introduction to Supply Chain Management ................................................................................... 1 1.2 Problem Description ...................................................................................................................... 3 Chapter 2 LITERATURE REVIEW ...................................................................................................... 6 2.1 2.2 2.3 2.4

Inventory Control ........................................................................................................................... 6 Raw Material Inventory Management ......................................................................................... 11 Customer Responsiveness ............................................................................................................ 12 Contribution of Thesis ................................................................................................................. 13

Chapter 3 MODEL FORMULATION ................................................................................................. 14 3.1 Deterministic (Base) Multi-Period Model ................................................................................... 14 3.1.1 Notation.............................................................................................................................. 14 3.1.2 Objective Function ............................................................................................................. 16 3.1.3 Constraints ......................................................................................................................... 17 3.1.4 Illustrative Example ........................................................................................................... 18 Chapter 4 IMPLEMENTATION AND RESULTS ............................................................................. 28 4.1 Base Model with Real Data ......................................................................................................... 28 4.1.1 Given Data ......................................................................................................................... 28 4.1.2 Results ................................................................................................................................ 33 4.2 Extension to the Base Model ....................................................................................................... 40 4.2.1 Notation.............................................................................................................................. 40 4.2.2 Objective Function ............................................................................................................. 41 4.2.3 Constraints ......................................................................................................................... 41 4.2.4 Given Data ......................................................................................................................... 42 4.2.5 Results of the extension to the Base Model with the Real Data ........................................ 42 Chapter 5 CONCLUSIONS AND FUTURE RESEARCH ................................................................. 52 5.1 Conclusion ................................................................................................................................... 52 5.2 Future Research ........................................................................................................................... 53 References .............................................................................................................................................. 54 Appendix A Results from Extended Model at s = 4 and s = 5 ............................................................. 57 Appendix B GAMS Code .................................................................................................................... 60

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LIST OF FIGURES Figure 2-1: Inventory Model Classifications .................................................................................................. 7

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LIST OF TABLES Table 3-1: Illustrative Example: Estimated demands for raw materials .............................................. 22 Table 3-2: Illustrative Example: Unit costs for raw materials ................................................................ 23 Table 3-3: Illustrative Example: Results from model run at s = 4......................................................... 24 Table 3-4: Illustrative Example: Results from model run at s = 5......................................................... 25 Table 3-5: Illustrative Example: Results from model run at s = 6......................................................... 25 Table 3-6: Illustrative Example: Results from model run at s = 7......................................................... 26 Table 3-7: Illustrative Example: Results from model run at s = 8......................................................... 26 Table 3-8: Illustrative Example: Consolidated results implemented from using all model runs .. 27 Table 4-1: Forecasted Demand ........................................................................................................................ 29 Table 4-2: Raw Materials Lead Times .......................................................................................................... 30 Table 4-3: Raw Material Cost ($/unit)........................................................................................................... 31 Table 4-4: Raw Materials Storage Information .......................................................................................... 32 Table 4-5: Real Data: Results from the model run at s = 1 (month 1) ................................................. 34 Table 4-6: Real Data: Results from the model run at s = 2 (month 2) ................................................. 35 Table 4-7: Real Data: Results from the model run at s = 3 (month 3) ................................................. 36 Table 4-8: Real Data: Results from the model run at s = 4 (month 4) ................................................. 37 Table 4-9: Real Data: Results from the model run at s = 5 (month 5) ................................................. 38 Table 4-10: Real Data: Consolidated results implemented from using all model runs ................... 39 Table 4-11: Extended Model: Orders from the model run at s = 1 ....................................................... 43 Table 4-12: Extended Model: Inventory from the model run at s = 1 .................................................. 44 Table 4-13: Extended Model: Backorders from the model run at s = 1 ............................................... 44 Table 4-14: Extended Model: Orders from the model run at s = 2 ....................................................... 45

vii Table 4-15: Extended Model: Inventory from the model run at s = 2 .................................................. 46 Table 4-16: Extended Model: Backorders from the model run at s = 2 ............................................... 46 Table 4-17: Extended Model: Orders from the model run at s = 3 ....................................................... 47 Table 4-18: Extended Model: Inventory from the model run at s = 3 .................................................. 48 Table 4-19: Extended Model: Backorders from the model run at s = 3 ............................................... 48 Table 4-20: Extended Model: Consolidated results implemented from using all model runs ...... 50 Table A-1: Extended Model: Orders from the model run at s = 4 ......................................................... 57 Table A-2: Extended Model: Inventory from the model run at s = 4 ................................................... 57 Table A-3: Extended Model: Backorders from the model run at s = 4 ................................................ 58 Table A-4: Extended Model: Orders from the model run at s = 5 ......................................................... 58 Table A-5: Extended Model: Inventory from the model run at s = 5 ................................................... 59 Table A-6: Extended Model: Backorders from the model run at s = 5 ................................................ 59

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ACKNOWLEDGEMENTS I would like to thank my thesis advisor Dr. Ravindran for giving me the opportunity to work with him in this research. He provided me the necessary guidance throughout the entire research and writing of the thesis. At the same time, I benefitted from his invaluable knowledge and expertise in the topic. I express my gratitude to my thesis reader Dr. Presiado for the all the guidance and for pointing out areas of improvements in the thesis. I’d like to take this time to thank all of you who in one way or another made this work possible; specially to my family and friends who were there for me at all time. My mother Betty, my father Isak, my sister Rachel and my brother Miguel, thanks for giving me your love and support. I would like to thank my girlfriend Raquel who helped me from the beginning and was there to push me trough the hard times. Also there were people from the company who dedicated time for this project, especially Rodrigo who was always there to answer any of my questions. Finally I’d like to dedicate this work to my grandparents, who are no longer here to appreciate this work but, they were the once who lighted my intellectual capabilities and for that they will always remain my inspiration.

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Chapter 1 INTRODUCTION This chapter is an introduction to supply chain management with the most relevant concepts in the area. Then we will describe the problem presented in the thesis and an overview of the following chapters.

1.1 Introduction to Supply Chain Management With globalization, many companies have changed the way they operate and do business; from the way they buy raw material from suppliers to the way they market their products in order to reach their final customer. These changes have forced many companies to consider supply chain management an important area. In the literature, there are several definitions for supply chain management. One of the comprehensive ones is given by Simchi-Levi (2003), “Supply chain management is a set of approaches utilized to efficiently integrate suppliers, manufacturers, warehouses and stores so that merchandise is produced and distributed at the right locations at the right time in order to minimize the system wide costs while satisfying service level requirements.” To measure the supply chain performance, there are six drivers that have been identified in the literature as the most important ones. The drivers indicated by Chopra and Meindl (2007) are facilities, inventory, transportation, information, sourcing and pricing. Facilities are the place where raw material is stored and distributed so that products can be later manufactured or assembled. Inventory is the goods and materials held available in stock by a business. Transportation refers to the movement of goods and material from one place to another. It can be performed by various modes, such as air, sea, roads, pipelines and electronic

2 form. Information in a supply chain refers to all the data and analysis regarding facility, inventory, transportation, cost, prices and customers shared throughout the supply chain. Sourcing is when we decide on who will perform a particular supply chain activity such as the production, storage, transportation or the management of information. Pricing is how much the firm will charge for their goods and services to the customers. This research focuses mainly on inventory, which is also known as an idle asset for companies because it doesn’t generate any revenue until it is sold. Moreover, it implies having capital invested in inventory leaving out other potential investment opportunities. A company’s asset in inventory can vary between 2% for an online retail up to 50% for a grocery store. Besides the regular cycle inventory, companies maintain an extra amount of inventory to buffer against a stock out caused by an unexpected change in demand. This extra amount of inventory is best known as the safety stock. Different types of inventory held throughout the supply chain. These include the inventory of raw material, work in process (WIP), finished goods and spare parts. In this research we give attention to the inventory of raw material, as it is a challenging and important task that represents the beginning of the manufacturing process and is essential to achieve a certain service level. It is well known that inventory plays a big role in the financial and operational performance of a company, as it considers both investments and logistics to adequately meet demand at the lowest possible cost. By holding inventory, the inevitable variability in demand and supply can be softened. Consequently this will help to satisfy the customer demand faster and reduce the number of stock-outs, which in the case of raw material, will lead to unexpected costly production stops. Another reason is to take advantage of the economies of scale. Typically

3 the ordering cost is a fixed cost and is independent of the number of items in the order. Hence by having more frequent orders, the cost will be higher. Also economies of scale plays a role in the transportation cost where there are discounts when buying in large quantities.

1.2 Problem Description In this thesis we consider the problem of determining an optimal inventory policy for the raw materials of a company. This company is a leading global health and hygiene company employing nearly 53,000 people worldwide and posting sales of 19.4 billion in 2008. It has operations in 35 countries and their products are sold in more than 150 countries everyday. For the study, we will consider only one county and do the analysis with their real data. The current process of ordering the raw material from the suppliers starts when the buyer of the raw material receives the Master Production Schedule (MPS), which contains information about the future demand estimates for finished goods. Raw materials demand information in turn relates to the demand of finished goods through the Bill of Material (BOM) structure. Thus, finished goods demand will be characterized so raw materials demand can be accurately estimated. In order to ensure that variations in the MPS from month to month are considered in the raw material planning process, only the first month’s decisions in the multi-period optimization model are implemented. After an order is placed, there are several things that can happen. Most of the raw material that is ordered comes from another country, which means that it will arrive by sea and has a long lead-time. Because of the long lead-time, the orders need to be placed well ahead of time to make sure that they will arrive when needed. This means that by the time we run the optimization model, we don’t have the most accurate forecast of deliveries for the planning

4 horizon. Keep in mind that the forecast will be more accurate as we get closer to the month for which the forecast is made. This can cause some differences in the amount ordered and the amount really needed. During the hurricane season (between the months of June and November) there can be some delays in the delivery of the shipments. This will affect the time at which the planning has to be done or otherwise the material will not arrive on time. Also we will have to make the necessary adjustments in the model formulation to be sure that we are considering all the possible scenarios and thus make the model more realistic. When dealing with procurement of raw material cross-national there are other important considerations in the process. At what time the materials arrive at the country where they will be used, they need to be nationalized and it can take a couple of extra days until all the paperwork is ready. Then they are transported to the warehouse where they will be stored or sent directly to the production plant if they will be used immediately. Different modeling methodologies will be studied to determine the optimal order quantities as well as the reorder points. In doing so, inventory, financial information, lead-times and demand information will be collected. Financial information will include procurement, transportation, administrative and holding costs, as well as suppliers’ discounts policies or any other costs related to stocking up raw materials. Order lead-times, which account for supplier production, shipping and administrative times due to orders consolidation and handling, will be also taken into account. Finally the current process will be compared to the proposed model using several performance metrics. We feel that the work done for one country can be extended to other countries or to the entire region.

5 The overview of the thesis is as follows. Chapter 2 will include a literature review that forms the background to the problem. Inventory control models as well as supply chain management studies will be reviewed in detail. In Chapter 3 the model formulation will be introduced, starting with the base model where both the demand and lead-times are deterministic. We will then use a case study illustrating the results of the model. In Chapter 4 the model introduced earlier will be used to find the optimal ordering policy and inventory levels for a real world scenario. Also we will extend the base model by relaxing some of the assumptions and allow for shortages; and thus the possibility of having backorders. Finally, Chapter 5 will present the conclusions of the thesis and suggestions for future research work. It will be followed by the list of references used for the research work.

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Chapter 2 LITERATURE REVIEW In this section we will present a review of the literature related to inventory control, raw material inventory management and customer responsiveness.

2.1 Inventory Control Research in inventory control deals with the control of production or ordering quantities so as to minimize the total cost to a firm. This cost can include the production cost, inventory holding cost, ordering cost, transportation cost and others. The inventory models started in 1915 when Harris introduced the first deterministic inventory model for a single product, the Economic Order Quantity (best known as the EOQ). Since the 1950s, there have been many advances in this area. Arrow et al (1958), Zipkin (2000) and Azadirav and Rangarajan (2008) are several good sources providing an overview of the most common inventory policies. Inventory models can be classified using different criteria depending on their characteristics. Figure 2.1shows a detailed classification of the inventory models. When we are classifying the models based on demand, they can be broken into two categories; deterministic and stochastic. When the demand is known it is said to be deterministic and when there is a known probability distribution for the demand it is said to be stochastic. Also with the stochastic demand, the parameters that define the stochastic process can be either constant in time (stationary), or changing over time (nonstationary). If we classify the inventory models based on supply, we encounter several scenarios. Starting with the instantaneous supply, where it is assumed that when an order is placed it is

7 immediately received and available to use (lead time equal to zero). The lead time can also be a positive value; either a constant or it can vary with time, or it can be stochastic. Inventory models may be used to manage the inventory for a single product or multipleproduct simultaneously. The planning horizon can be just a single period or it can be multipleperiods. For multiple period analyses there are differences in the models for the finite period to the infinite period planning horizon.

Deterministic Stationary and Non-stationary

Based on Demand Stochastic Instantaneous Constant Lead Time Based on Supply Variable Lead Time Stochastic Lead Time Based on Planning Horizon

Single-Period Multiple-Period

Finite Based on the number of items

Infinite

One Item Multiple-Items Figure 2-1: Inventory Model Classifications

Researchers have tried to review the literature to provide a common terminology among the models, with the goal to reduce the effort that it takes to track the different models. Having a common terminology also provides the ease of understanding in using these models. Aggrawal

8 (1974) looked at the inventory-related models published during the last ten years from a system point of view. In the paper he provides a chart showing six categories based on similarities of approaches, the categories are: (1) models for determining optimal inventory policies, (2) lot size optimization, (3) optimization of various specific management objectives, (4) models for optimizing highly specialized inventory situations, (5) application of advanced mathematical theories to inventory problems, and (6) models bridging the gap between theory and practice. There are two types of policies that can be used to determine the optimal order quantity; the continuous review policy and the periodic review policy. In the continuous review policy, the inventory is reviewed on a continuous basis. This constant monitoring of inventory allows placing a new order of size Q as soon as the inventory reaches the reorder point (ROP). Here the order quantity Q is constant but the time between orders can change depending on the demand. It is evident that in order to have a continuous review policy in place, it is necessary to have a good information technology system to monitor the inventory levels at all times. On the other hand, in the periodic review policy (also known as the Base Stock policy) the inventory levels are reviewed at designated times separated by equal intervals (e.g., monthly) and the orders are placed to raise the inventory level to a specified threshold. Here the time between orders remains constant but the size of each order vary depending on the inventory on hand. The ABC classification is commonly used in practice to determine which inventory items need what type of review policy. Usually class-A items are assigned a continuous review policy, class-B items can be assigned either a continuous or periodic review policy and class-C items are controlled using a periodic review policy. Ng (2007) proposed a simple model for multiple criteria ABC classification. By converting the criteria measures into a scalar score, they can obtain a score for each inventory

9 item. Chu, Liang and Liao (2008) proposed a new inventory control approach called ABC-fuzzy classification (ABC-FC). As opposed to the regular ABC classification, this method is not limited to considering only one criterion. It can handle nominal or non-nominal attributes, can incorporate manager’s experience and can be easily implemented. In supply chain management Warsing (2008), there are two broad categories in which a supply chain can be classified, centralized and decentralized. The centralized supply chain is one where one company owns all the stages of the supply chain including the transportation. A decentralized supply chains is the most common, and is when there are multiple owners of the different stages of the supply chain. The main difference between them is that in a centralized supply chain the information is shared from end to end and the inventory policies can be set for all the stages, whereas in a decentralized supply chain there is limited information sharing and each stage has its own inventory policies. Clark and Scarf's (1960) paper on centralized control is the basis of most subsequent work. They were able to show that under a periodic review policy with no set up cost, an order up to policy (also called base stock policy) in all stages is optimal. They were also the first ones to introduce the concept of echelon inventory position, which is the sum of the inventory position at the considered stage and at all the downstream stages. Later DeBolt and Graves (1985) extended this model for a continuous review policy. Lee and Billington (1993), motivated by the real situation at Hewlett-Packard (HP) Deskjet printer supply, developed a model where the supply chain is not under complete centralized control. Also they identified important topics to be considered in future research, such as the modeling of correlated demands, modeling the material delay, developing build-to-order models and others.

10 Hackman and Leachman (1989) introduced a general framework that guides the management scientist’s formulation of deterministic models. Giri and Chaudhuri (1998) derived an approximate optimal solution to an extended EOQ-type model for perishable products where demand rate is a function of the on-hand inventory. Oh and Hwang (2006) discussed the inventory control for a recycling system assuming that demand is deterministic. Having a deterministic demand is not always the case in real world situations. For this reason there are inventory models that assume a stochastic demand. Hopp et al. (1997) developed three heuristic algorithms based on simplified representations of the inventory and service expressions. With these algorithms they found a closed form solution to the control parameters without the need to re-optimize the rest of the system. Kijima and Takimoto (1999) considered a single-item, single location (T,S) inventory/production model with uncertain demand in which they had a limited capacity per period. They were able to express the time-dependent distribution of the leadtime and the customer waiting time by using a Markov Chain model. Other researchers have used robust optimization to deal with inventory models that have a stochastic demand. Bertsimas and Thiele (2006) developed an approach that incorporates randomness using robust optimization. The robust problem is of the same class as the nominal problem, and they showed that it lead to a high-quality solution and often outperforming dynamic programming based policies in single stations problems.

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2.2 Raw Material Inventory Management

There are many types of inventory that can be held throughout the supply chain. They include raw material, work in process (WIP), finished goods inventory and spare parts. In this thesis, we are more concerned with managing the inventory of raw material. Hopp and Spearman (2000) define raw material as the components, subassemblies, or materials that are purchased from outside the plant and used in the fabrication/assembly process inside the plant. The idea of managing the raw material is to have it available when it is needed by the production system. Since most of the time we cannot receive the raw material immediately (in a just-in-time fashion), carrying inventory stock of raw material seems appropriate. The size of the stock can be influenced by several factors; quantity discount received from the supplier, variability in either the supply or demand, or the materials can become obsolete. Goyal (1977) considered an inventory model that unified the inventory problem of raw material and finished products for a single product manufacturing system. He pointed out that the problem of determining the optimal procurement policy for raw material cannot be treated in isolation; it depends on the production batch size of the products that required the raw material. Later Kim and Chandra (1987) developed a heuristic procedure to classify the raw material into fewer groups so that it could be used in Goyal’s model. They assumed that the unit replenishment cost in a group is a decreasing function of the number of raw material in that group. Hong and Hayya (1992) developed an exact solution procedure for finding the optimal inventory policy for raw materials and grouping them into fewer groups. They presented a numerical example and used Kim and Chandra’s procedure to compare their result. Their total

12 cost was about 20% lower than Kim and Chandra’s model but there were some differences in the assumptions for the demand rate for the raw material. Sarker and Parija (1996) developed an ordering policy for raw material for a single manufacturing batch. The objective was to minimize the total cost while at the same time meeting the demand of the production facility. Using an integer approximation they were able to reach the optimal solution.

2.3 Customer Responsiveness Having a service level constraint in the inventory models is very important because it measures the probability that all orders will be delivered with the stock on hand for a specific time interval. If this constraint is omitted, then we might not have any incentive to carry inventory at all, and our customers orders will be backordered until we have accumulated a number of them, and at that point we will place the order. Another possibility is to assign a cost for backorders; that way the model itself will try to minimize the number of backorders occurring in a period of time. Tijms and Groenevelt (1984 present a useful approximation for the reorder point, s, such that the required service level is achieved. In the paper they considered both scenarios where the review policy can be either periodic or continuous. Chen and Krass (2001) investigated inventory models in which the cost of running out of stock is replaced by a minimal service level constraint (SLC) that requires a certain level of service to be met every period. They found that above a “critical” service level, the optimal (s,S) policy “collapses” to a simple base-stock or order-up-to policy. Here the service level refers to the availability of stock in a probabilistic or expected sense. Tarim and Kingsman (2004)

13 extended on the topic by addressing a stochastic dynamic lot-sizing problem where they consider a specified customer service level. Using a mixed integer programming model they found the optimal solution allowing them to find both the number and timing of the replenishment orders at the same time. Another common measure of service used for inventory models is the Fill Rate. Fill Rate is the fraction of demand satisfied from stock. Thomas (2005) investigated several factors like contract horizon, demand distribution and the cost of failing to meet the target and evaluated their impact on the Fill Rate. Axsäter (2006) studied a single-echelon continuous review system with normally distributed lead-time demand, under a Fill Rate constraint. They show an alternative approximate technique to find order quantities with computational efforts similar to the EOQ formula.

2.4 Contribution of Thesis This thesis is an extension of the work previously mentioned in the area of inventory control models. We consider deterministic and then stochastic models. The thesis represents an application of existing inventory models to a real world situation. In this work, we focused mainly in managing the raw material inventory for a large consumer product company in the area of health and hygiene. After running the optimization models we will provide useful information to the company regarding optimal inventory policies for the different raw materials. This will translate into savings for the company due to better inventory management and reordering policies.

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Chapter 3 MODEL FORMULATION In this chapter we will present the mathematical programming model that will be used to find optimal inventory policies for raw materials procurement. The first model, called the Base Model, assumes that both demand and lead time are deterministic.

3.1 Deterministic (Base) Multi-period Model The model determines the amount of each raw material to be ordered each period assuming that we have a forecast of demands and known lead-times. The objective is to minimize the purchasing cost, inventory holding cost and the fixed ordering cost for a planning horizon, while maintaining a desired service level for each product and not exceeding the total capacity available each period. In order to ensure that variations in the demand forecast are considered in the procurement planning process, only the decisions made for the first period of the planning cycle will be implemented. The model will be run on a rolling horizon basis; that is, the end of the planning cycle will be shifted one period ahead each time. This will ensure that updated forecasts of raw material needs and supplier deliveries in the pipe line are included on a continuous basis.

3.1.1 Notation Indices: i = raw material (i = 1,…N) t = time period

15 s = time period at which the model is run for the next T periods T = length of planning horizon for each model run

Parameters: dit = estimate of demand for raw material i in period t,

i = 1,…,N and t = s, s+1,…, s+T-1

cit = unit cost of raw material i in period t Li = lead time for raw material i hi = inventory holding cost per unit of raw material i per period Capt = total normal storage capacity for all raw material in period t (Note: Additional capacity can be leased at a cost) ai = constant representing the space occupied by each unit of raw material i et = cost per unit of extra storage space needed in period t Ft = fixed cost of ordering one or more raw materials in period t M = very large number (e.g. M = 999999)

Variables: xit = amount of raw material i ordered in period t (will arrive at t + Li),

t = s, s + 1, …, s + T -1

Iit = inventory of raw material i at the end of period t 𝐶𝑎𝑝𝑡 − = unused normal storage capacity in period t 𝐶𝑎𝑝𝑡 + = additional capacity leased in period t δt 𝜖{0,1}, where δt = 1 if any raw material is ordered in period t, and δt = 0 otherwise Note: total capacity available in period 𝑡 = 𝐶𝑎𝑝𝑡 + 𝐶𝑎𝑝𝑡 +

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3.1.2 Objective Function The objective of the model is to minimize the total cost of the system over the planning horizon. In this case, we will be considering four costs that are relevant to the company, purchasing cost, inventory holding cost, the cost of increasing the capacity (acquiring extra storage space) and the fixed cost of ordering in a given period. The purchasing cost is the cost of the raw material itself. To obtain this cost we multiply the amount of each raw material ordered over the planning horizon (xit), by the cost of each raw material (cit). The inventory holding cost is the cost of carrying inventory from one period to another. It is calculated by multiplying the amount of raw material carried in inventory in each period (Iit), by the cost of holding that particular material (hi). The cost of acquiring extra capacity is the cost to rent another warehouse; this can be calculated by multiplying the cost per unit (et) of extra space by the amount of extra capacity leased (Capt+) at period t. The fixed ordering cost (Ft) is only incurred when an order is placed in that particular period, independent of how many raw materials are ordered. For this we use a binary variable (δt) that can takes the value of 1 when an order is placed or 0 when no orders are placed in a given period. Minimize Total Cost = Purchasing Cost + Inventory Holding Cost + Extra Capacity Cost + Fixed Ordering Cost 𝑠+𝑇−1 𝑁

𝑠+𝑇−1 +

𝑀𝑖𝑛 𝑍 =

[(𝑐𝑖𝑡 𝑥𝑖𝑡 + ℎ𝑖 𝐼𝑖𝑡 ) + 𝑒𝑡 𝐶𝑎𝑝𝑡 ] + 𝑡=𝑠

𝑖=1

𝐹𝑡 𝛿𝑡 𝑡=𝑠

[3.1]

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3.1.3 Constraints Demand Constraint: This constraint is to make sure that in every period t we have balanced the inflows with the outflows (quantity coming in with quantity going out in every period t). 𝐼 𝑜 𝑖,𝑠−1 + 𝑥 𝑜 𝑖,𝑠−𝐿𝑖 = 𝑑𝑖𝑠 + 𝐼𝑖𝑠 𝐼𝑖,𝑡−1 + 𝑥 𝑜 𝑖,𝑡−𝐿𝑖 = 𝑑𝑖𝑡 + 𝐼𝑖𝑡 𝐼𝑖,𝑡−1 + 𝑥𝑖,𝑡−𝐿𝑖 = 𝑑𝑖𝑡 + 𝐼𝑖𝑡

for all 𝑖 = 1, … , N for all 𝑖 = 1, … , N and 𝑡 = 𝑠 + 1, … , 𝑠 + 𝐿𝑖 − 1 for all 𝑖 = 1, … , N and 𝑡 = 𝑠 + 𝐿𝑖 , … , 𝑠 + 𝑇 − 1

[3.2]

xoit, Ioit refers to the values of the decision variables that we already know from the previous model runs at t = s -1, s -2, etc. Note: 𝑥𝑖,𝑡−𝐿𝑖 for t ≤ s+Li-1 

are known from previous model runs for t ≤ s -1

𝑥𝑖,𝑡−𝐿𝑖 for t ≥ s+Li



are unknown decision variables

Ii,s-1



are known ∀ i

Capacity Constraint: This constraint is written for every t as a goal constraint since additional capacity can be obtained for a price. 𝑁

𝑎𝑖 𝐼𝑖,𝑡−1 + 𝑥𝑖,𝑡−𝐿𝑖 + 𝐶𝑎𝑝𝑡 − − 𝐶𝑎𝑝𝑡 + = 𝐶𝑎𝑝𝑡

for all t = 𝑠, 𝑠 + 1 … , 𝑠 + T − 1

[3.3]

𝑖=1

Binary Constraint: This constraint will make sure that the orders quantities have a relationship with the actual ordering in every period. If no orders are placed in a period, these order quantities have to be zero for that period. 𝑁

𝑥𝑖,𝑡 ≤ 𝑀𝛿𝑡 𝑖=1

for all t = 𝑠, 𝑠 + 1 … , 𝑠 + T − 1

[3.4]

18

3.1.4 Illustrative Example In order to show a numerical example of the model, we will use the following information: i =1, 2 (two raw materials), s = 4 (means the model is run in the month of April), T = 6 months, L1 = 2 months , L2 = 3 months Parameters: c14 – unit cost of raw material 1 in period 4 … c19 – unit cost of raw material 1 in period 9 c24 – unit cost of raw material 2 in period 4 … c29 – unit cost of raw material 2 in period 9 h1 – inventory holding cost for raw material 1 per period h2 – inventory holding cost for raw material 2 per period e1 – cost per unit of extra storage space needed in period 1 … e9 – cost per unit of extra storage space needed in period 9 a1 – constant representing the space occupied by each unit of raw material 1 a2 – constant representing the space occupied by each unit of raw material 2 Cap4 – total normal storage capacity for all raw materials in period 4 … Cap9 – total normal storage capacity for all raw materials in period 9 d14 – demand estimate for raw material 1 in period 4 …

19 d19 – demand estimate for raw material 1 in period 9 d24 – demand estimate for raw material 2 in period 4 … d29 – demand estimate for raw material 2 in period 9 F4 – fixed ordering cost in period 4 … F9 – fixed ordering cost in period 9 M – very large number (e.g. M = 99999)

Variables: x14 – amount of raw material 1 ordered in period 4 … x19 – amount of raw material 1 ordered in period 9 x24 – amount of raw material 2 ordered in period 4 … x29 – amount of raw material 2 ordered in period 9 Note: xit for t < 4 are known quantities and are denoted by xoit for i = 1, 2 and t = 1, 2, 3

I14 – inventory of raw material 1 at the end of period 4 … I19 – Inventory of raw material 1 at the end of period 9 I24 – Inventory of raw material 2 at the end of period 4 …

20 I29 – Inventory of raw material 2 at the end of period 9

Note: Initial Inventories I13 and I23, are known quantities and are denoted by I13o and I23o

Cap4– – unused normal storage capacity in period 4 … Cap9– – unused normal storage capacity in period 9 Cap4+ – additional capacity leased in period 4 … Cap9+ – additional capacity leased in period 9 δ4 – binary variable indicating if ordering in period 4 … δ9 – binary variable indicating if ordering in period 9

Objective Function: Min Z = (c14x14+c15x15+c16x16 +c17x17+c18x18 +c19x19) + (c24x24+c25x25+c26x26+c27x27+c28x28+c29x29) + h1(I14 + I15 + I16 + I17 + I18 + I19) + h2(I24 + I25 + I26 + I27 + I28 +I29) + (e4Cap4+ + e5Cap5+ + e6Cap6+ + e7Cap7+ + e8Cap8+ + e9Cap9+) + F4 δ4 + F5 δ5 + F6 δ6 + F7 δ7 + F8 δ8 + F9 δ9

21 Constraints:

Demand Constraints 

For raw material 1:

At t=4: Io13 + xo12 = d14 + I14 At t=5: I14 + xo13 = d15 + I15 At t=6: I15 + x14 = d16 + I16 At t=7: I16 + x15 = d17 + I17 At t=8: I17 + x16 = d18 + I18 At t=9: I18 + x17 = d19 + I19



For raw material 2:

At t=4: Io23 + xo21 = d24 + I24 At t=5: I24 + xo22 = d25 + I25 At t=6: I25 + xo23 = d26 + I26 At t=7: I26 + x24 = d27 + I27 At t=8: I27 + x25 = d28 + I28 At t=9: I28 + x26 = d29 + I29 xoit, Ioit are known constants from t ≤ 3 .

Capacity Constraints At t=4: a1 (Io13 + xo12) + a2 (Io23 + xo21) + Cap4- - Cap4+ = Cap4 At t=5: a1 (I14 + xo13) + a2 (I24 + xo22) + Cap5- - Cap5+ = Cap5

22 At t=6: a1 (I15 + x14) + a2 (I25 + xo23) + Cap6- - Cap6+ = Cap6 At t=7: a1 (I16 + x15) + a2 (I26 + x24) + Cap7- - Cap7+ = Cap7 At t=8: a1 (I17 + x16) + a2 (I27 + x25) + Cap8- - Cap8+ = Cap8 At t=9: a1 (I18 + x17) + a2 (I28 + x26) + Cap9- - Cap9+ = Cap9 xoit, Ioit refers to the values that we already know from the previous model runs.

Binary Constraints At t=4: x14 + x24 ≤ M δ4 At t=5: x15 + x25 ≤ M δ5 At t=6: x16 + x26 ≤M δ6 At t=7: x17 + x27 ≤ M δ7 At t=8: x18 + x28 ≤M δ8 At t=9: x19 + x29 ≤ M δ9

The model was run using the following information: Table 3-1: Illustrative Example: Estimated demands for raw materials

period 4 5 6 7 8 9 10 11 12 13 14

Demand (dit) raw material 1 raw material 2 100 55 90 40 110 50 120 50 90 45 100 55 110 40 100 55 90 50 100 55 120 50

23

Table 3-2: Illustrative Example: Unit costs for raw materials

period 4 5 6 7 8 9 10 11 12 13 14

Unit Costs (cit) in $ raw material 1 raw material 2 2.00 1.00 2.00 1.00 4.00 1.50 4.00 1.50 3.00 1.70 3.00 1.70 2.00 1.00 2.00 1.00 4.00 1.50 4.00 1.50 3.00 1.70

h1 = $0.30/unit/month h2 = $0.14/unit/month F4 = F5 = F6 = F7 = F8 = F9 = F10 = F11 = F12 = F13 = F14 = $3.00 e4 = e5 = e6 = e7 = e8 = e9 = e10 = e11 = e12 = e13 = e14 = $0.50/m2 Cap4 = Cap5 = Cap6 = Cap7 = Cap8 = Cap9 = Cap10 = Cap11 = Cap12 = Cap13 = Cap14 = 160 m2 a4 = a5 = a6 = a7 = a8 = a9 = a10 = a11 = a12 = a13 = a14 = 1 xo12 = 100; xo13 = 90 xo21 = 55; xo22 = 40; xo23 = 50 Io13 = 0 Io23 = 0

The results from running the mixed integer programming model at s = 4 for the planning horizon of 6 months (t = 4, 5, 6, 7, 8, 9) is given in Table 3-3:

24

Table 3-3: Illustrative Example: Results from model run at s = 4

x14 = 110 x15 = 310 x16 = 0 x17 = 0 x18 = 0 x19 = 0

x24 = 50 x25 = 45 x26 = 55 x27 = 0 x28 = 0 x29 = 0

I14 = 0 I15= 0 I16 = 0 I17 = 190 I18 = 100 I19 = 0

I24 = 0 I25 = 0 I26 = 0 I27 = 0 I28 = 0 I29 = 0

Results Cap+4 = 0 Cap+5 = 0 Cap+6 = 0 Cap+7 = 200 Cap+8 = 75 Cap+9 = 0 δ4 = 1 δ5 = 1 δ6 = 1 δ7 = 0 δ8 = 0 δ9 = 0

Cap-4 = 5 Cap-5 = 30 Cap-6 = 0 Cap-7 = 0 Cap-8 = 0 Cap-9 = 5 Z = 1251

Looking at the results for s = 4 (Table 3-3), we note that optimal solution to the model gives a total minimum cost of $1251. The model suggests ordering only in the first 2 months for raw material 1. The total demand of 610 units for 6 months is met by incoming orders of 100 and 90 at t = 4 and 5 respectively, plus new orders of 110 and 310 in months 4 and 5. Because of the lead time of 2 months, no orders are placed for raw material 1 in the last 2 months of the planning horizon. Similar results are exhibited for raw material 2 also. It is to be noted here that the only decision that will be made after running the model for s = 4 is to order 110 units of raw material 1 (x14) and 50 units of raw material 2 (x24). Decisions on x15, x25, x26 will not be implemented at this time. The mixed integer programming (MIP) model was solved using GAMS. The complete model can be found in Appendix B. The model had 42 variables (including the 6 binary variables) and 30 constraints and it took approximately 2 seconds to solve.

25 In the optimal solution we obtained (Table 3.3), the values of x18, x19, x27, x28, and x29 are all zero. This is always going to be the case in the optimal solution because at those periods (t = 7, 8 and 9) the lead times are larger than the remaining periods in the planning horizon. The same thing happens to the ending inventories, I19, I29 which are also zero in the optimal solution. This is not going to be a problem since we are implementing only what the model suggest for that particular period, namely s = 4. For the future periods, the model will be run again and the optimal solutions will be recalculated. To illustrate this, we have rerun the model at s = 5, 6, 7 and 8. Their results are shown in Tables 3-4, 3-5, 3-6 and 3-7 respectively.

Table 3-4: Illustrative Example: Results from model run at s = 5

x15 = 310 x16 = 0 x17 = 0 x18 = 110 x19 = 0 x110 = 0

x25 = 45 x26 = 55 x27 = 40 x28 = 0 x29 = 0 x210 = 0

I15= 0 I16 = 0 I17 = 190 I18 = 100 I19 = 0 I110 = 0

I25 = 0 I26 = 0 I27 = 0 I28 = 0 I29 = 0 I210 = 0

Results Cap+5 = 0 Cap+6 = 0 Cap+7 = 200 Cap+8 = 75 Cap+9 = 0 Cap+10 = 0 δ5 = 1 δ6 = 1 δ7 = 1 δ8 = 1 δ9 = 0 δ10 = 0

Cap-5 = 30 Cap-6 = 0 Cap-7 = 0 Cap-8 = 0 Cap-9 = 5 Cap-10 =10 Z = 1644

Table 3-5: Illustrative Example: Results from model run at s = 6

x16 = 0 x17 = 0 x18 = 110 x19 = 100 x110 = 0 x111 = 0

x26 = 55 x27 = 50 x28 = 45 x29 = 0 x210 = 0 x211 = 0

Results Cap+6 = 0 Cap+7 = 200 Cap+8 = 75 Cap+9 = 0 Cap+10 = 0 Cap+11 = 0

Cap-6 = 0 Cap-7 = 0 Cap-8 = 0 Cap-9 = 5 Cap-10 = 0 Cap-11 = 5

26 I16 = 0 I17 = 190 I18 = 100 I19 = 0 I110 = 0 I111 = 0

I26 = 0 I27 = 0 I28 = 0 I29 = 0 I210 =10 I211 = 0

δ6 = 1 δ7 = 1 δ8 = 1 δ9 = 1 δ10 = 0 δ11 = 0

Z = 2036

Table 3-6: Illustrative Example: Results from model run at s = 7

x17 = 0 x18 = 110 x19 = 100 x110 = 90 x111 = 0 x112 = 0

x27 = 50 x28 = 45 x29 = 50 x210 = 0 x211 = 0 x212 = 0

I17 = 190 I18 = 100 I19 = 0 I110 = 0 I111 = 0 I112 = 0

I27 = 0 I28 = 0 I29 = 0 I210 = 10 I211 = 0 I212 = 0

Results Cap+7 = 200 Cap+8 = 75 Cap+9 = 0 Cap+10 = 0 Cap+11 = 0 Cap+12 = 0 δ7 = 1 δ8 = 1 δ9 = 1 δ10 = 1 δ11 = 0 δ12 = 0

Cap-7 = 0 Cap-8 = 0 Cap-9 = 5 Cap-10 = 0 Cap-11 = 5 Cap-12=20 Z = 2301

Table 3-7: Illustrative Example: Results from model run at s = 8

x18 = 110 x19 = 100 x110 = 90 x111 = 100 x112 = 0 x113 = 0

x28 = 45 x29 = 50 x210 = 55 x211 = 0 x212 = 0 x213 = 0

I18 = 100 I19 = 0 I110 = 0 I111 = 0 I112 = 0 I113 = 0

I28 = 0 I29 = 0 I210 = 10 I211 = 0 I212 = 0 I213 = 0

Results Cap+8 = 75 Cap+9 = 0 Cap+10 = 0 Cap+11 = 0 Cap+12 = 0 Cap+ 13= 75 δ8 = 1 δ9 = 1 δ10 = 1 δ11 = 1 δ12 = 0 δ13 = 0

Cap-8 = 0 Cap-9 = 5 Cap-10 = 0 Cap-11 = 5 Cap-12=20 Cap-13 = 5 Z = 2456

27 Since we are only implementing the results obtained in the period in which the model is run, Table 3-8 shows the consolidated results that will used for decision making from each model run: Table 3-8: Illustrative Example: Consolidated results implemented from using all model runs

x14 = 110 x15 = 310 x16 = 0 x17 = 0 x18 = 110

x24 = 50 x25 = 45 x26 = 55 x27 = 50 x28 = 45

I14 = 0 I15= 0 I16 = 0 I17 =190 I18 = 100

I24 = 0 I25 = 0 I26 = 0 I27 = 0 I28 = 0

Results Cap+4 = 0 Cap+5 = 0 Cap+6 = 0 Cap+7 = 200 Cap+8 = 75

Cap-4 = 5 Cap-5 = 30 Cap-6 = 0 Cap-7 = 0 Cap-8 = 0

δ4 = 1 δ5 = 1 δ6 = 1 δ7 = 1 δ8 = 1

In real world situations, the forecast of demands are known for 6 months only and are subject to change each month the model is run. Hence the optimal value of x15 and x25 in Table 3-4 might be different than those shown in Table 3-3. The same applies to results shown in Tables 3-5 to 3-7. Thus, the consolidated results that will be implemented could be very different than those shown in Table 3-8.

28

Chapter 4 IMPLEMENTATION AND RESULTS In this chapter we will use the optimization model described in Chapter 3 to find the optimal ordering policy as well as the inventory levels for a real world situation. The information of the different types of raw materials as well as all the demand and cost data was collected from a global health and hygiene company that operates in more than 35 countries around the world.

4.1 Base Model with Real Data The list of raw materials used by the company to produce all their products has more than 100 items. In order for us to keep track of the policies as well as the inventory levels of each raw material, we selected a representative sample to conduct the analysis.

4.1.1 Given Data Number of raw materials, i = 1, 2, …, 15. Planning horizon length, T = 6 months. Table 4-1 shows the forecasted demand requirements for each of the 15 raw material. To maintain the confidentiality, we will show the materials with an ID number created for the analysis instead of the SKU (Stock Keeping Unit) used by the company. Also information regarding provider’s exact location, or any other case sensitive information will not be revealed. In the case of demand and cost related information, the numbers will be multiplied by a factor in order to maintain confidentiality.

29 It is important to remember that the demand for each raw material was obtained using the demand for finished goods and the bill of materials (BOM). The demand of raw material is related to the demand of finished goods through the BOM. The BOM shows the different raw materials as well as quantities needed to produce each type of finished good. The forecasted demand changes every month when the new MPS (Master Production Schedule) is generated. As we move closer to the planning month, the forecasted demand becomes more accurate; but we need to keep in mind that because of the long lead times, we have to plan well in advance to have the materials ready when they are needed. Table 4-1: Forecasted Demand Raw Material ID

1

2

3

4

5

6

7

8

9

10

1

1782108

1812187

1938653

1757225

1769502

1534347

1849860

1999840

1849305

1787060

2

760949

859092

801449

919081

815382

661795

854690

930451

856772

806698

3

1344402

1657734

1334956

1587953

1576087

1262282

1637898

1743257

1607078

1495376

4

15423

20249

16875

19526

17614

14463

18526

20368

18442

17382

5

559020

625499

708920

719850

1286176

1320088

826565

503251

495262

575513

6

139163

198017

148416

150274

201188

194257

180560

112819

114664

99816

7

842503

920171

829868

798879

804656

961897

875438

921717

798333

809726

8

3477356

3826667

3350962

3024762

2518163

2509015

3190379

4290076

3974999

3040726

9

1514561

1185797

1283457

1094641

1264099

796192

1362816

1676779

1541613

1448846

10

281997

402473

285751

400388

456189

459068

421455

291324

257162

244237

11

12243

5882

1705

12890

0

18381

13920

9709

1280

16636

12

819971

199888

635505

743537

147583

997399

348595

877327

60393

982819

13

0

78488

0

89983

0

96567

0

105610

0

103528

14

21373

18930

32445

23412

32678

62749

14006

2287

14589

36993

15

42017

50671

55185

61000

37258

34301

35638

35837

86333

90632

Monthly Demand

In Table 4-1 there are some months for which the demand is 0. This could be due to several reasons; for example, it may be a product that is scheduled to be produced every other month (not on a monthly basis), or a situation where there is a promotion and a certain product is only produced for that particular month and then is discontinued. Also certain products can

30 transition to a different product either by changing its package or its presentation, which represents a change in the SKU and also this information is reflected in the demand data. Table 4-2 shows the information about the lead times of the raw materials. The lead times were estimated by looking at past data. These lead times are going to depend directly on the location of the provider as well as the mode of transportation. Usually all the raw materials come by sea; this implies that there must be a good coordination with the shipping companies to receive all the materials on time. Also during the hurricane season the lead times may vary and these changes can be easily implemented in the model. Table 4-2: Raw Materials Lead Times

ID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Lead Time (months) 2 2 3 2 3 3 2 3 2 2 3 3 3 2 2

In Table 4-3 the unit cost of each raw material for each month are shown. These costs are revised and updated every two months according to changes in price for the raw materials. Usually the price changes occur because of the supply and demand conditions, and sometimes

31 because natural events might create scarcity or excess surplus of a raw material and thus affecting dramatically the price. Table 4-3: Raw Material Cost ($/unit)

ID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

4 0.65 1.64 0.05 7.43 0.08 0.25 0.14 0.06 0.23 0.16 0.25 0.03 0.04 0.14 0.13

5 0.65 1.64 0.05 7.43 0.08 0.25 0.14 0.06 0.23 0.16 0.25 0.03 0.04 0.14 0.13

6 0.71 1.77 0.44 8.17 0.08 0.23 0.15 0.06 0.25 0.13 0.27 0.04 0.04 0.14 0.11

7 0.71 1.77 0.44 8.17 0.08 0.23 0.15 0.06 0.25 0.13 0.27 0.04 0.04 0.14 0.11

8 0.80 1.95 0.06 9.24 0.10 0.29 0.17 0.07 0.27 0.19 0.30 0.04 0.05 0.16 0.15

Month 9 10 11 12 13 14 15 0.80 0.95 0.95 0.87 0.87 0.77 0.77 1.95 2.22 2.22 1.95 1.95 1.78 1.78 0.06 0.07 0.07 0.06 0.06 0.06 0.06 9.24 10.71 10.71 9.75 9.75 8.68 8.68 0.10 0.11 0.11 0.10 0.10 0.09 0.09 0.29 0.33 0.33 0.30 0.30 0.27 0.27 0.17 0.18 0.18 0.16 0.16 0.15 0.15 0.07 0.08 0.08 0.07 0.07 0.06 0.06 0.27 0.29 0.29 0.26 0.26 0.23 0.23 0.19 0.20 0.20 0.18 0.18 0.16 0.16 0.30 0.31 0.31 0.29 0.29 0.27 0.27 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.05 0.05 0.05 0.04 0.04 0.04 0.04 0.16 0.17 0.17 0.15 0.15 0.14 0.14 0.15 0.16 0.16 0.15 0.15 0.15 0.15

To calculate the inventory holding cost for each raw material, we use the information provided by the finance department of the company. In industry, the inventory holding cost is estimated to be approximately 5 – 35% of the material’s cost. In our particular example we will let the holding cost to be 10% of the average raw material cost. The inventory holding cost is measured in cost (in dollars) per unit (measurement units can be either Kg or m2) per month. The storage capacity for the raw materials is not a fixed constraint because there is the possibility of acquiring extra storage capacity at a cost. This extra cost is introduced in the model to penalize the objective function every time extra storage capacity is needed. The monthly base storage capacity available is 2000 m2 and the cost to acquire extra storage capacity is $8.5 per m2 per month. Also the fixed cost of ordering in any given month is $300,000. This cost is incurred

32 in every month that an order is placed and is independent of the number of items ordered (it will be the same cost whether we order 1 raw material or 15 raw materials). The cost was estimated taking various factors into account, such as salary of people involved, number of hours spent in planning the orders, cost associated with the contract with suppliers and so on. Since not all the raw material is measured in the same units (some are measured in Kg and others in m2) we need to find a conversion factor to relate the amount of raw material that we have with the available storage capacity. This must be done keeping in mind that some of the materials can be stored on top of each other forming different levels, and others must go by themselves. The number of levels in which the materials can be stored will depend on the weight of the materials as well as its volume. In the raw materials warehouse, they have classified the materials into families according to their dimensions and number of levels that the material can be stored. Having the materials classified in families also helps in the allocation of space among the materials in the warehouse. Table 4-4 summarizes the information mentioned above. Table 4-4: Raw Materials Storage Information

ID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Measurement Unit KG KG M2 KG M2 M2 M2 M2 M2 M2 M2 M2 M2 M2 M2

Family 11 13 8 1 5 6 7 8 10 10 6 8 8 7 1

Number of levels 6 2 1 2 1 4 4 1 4 4 4 1 1 4 2

Area Occupied by measurement unit (m2) 0.00026889 0.00064000 0.00003919 0.00194529 0.00004952 0.00009076 0.00006304 0.00004732 0.00008289 0.00006826 0.00009104 0.00002462 0.00004976 0.00005404 0.00004183

33

4.1.2 Results The optimization software GAMS is used to generate the mixed integer programming (MIP) model and then to solve it. The model had 394 variables (including 12 binary variables) and 370 constraints and it took approximately 3 seconds to solve. The model is run each month and updated with the most current information; this includes the output from the previous month’s model, any change in the unit price of the materials or lead times, and delivery times of raw materials on order. In the model for month 1, we have adjusted the initial conditions in order to maintain feasibility. The orders placed on the previous month are going to be exactly the same as the forecasted demand (xit = dit) for the first month for which the model is run initially, otherwise, the orders placed at t = 1 will not arrive on time to meet the demand of month 1. Also we assumed that the initial inventory is going to be zero for all the raw materials. Tables 4-5 shows the results from running the mixed integer programming model at s = 1 for the planning horizon of 6 months (t = 1, 2, 3, 4, 5, 6). The optimal solution to the model gives a minimum cost of $14,912,378. The model results indicate that no inventory is kept for any of the raw materials in any month; Iit = 0 for all i and t. In the optimal solution (Table 4-5), xit = 0 for t = 5 and 6 for all i. This is due to the fact that the minimum lead time of 2 months is larger than the remaining months in the planning horizon. Also, for some months, there is unused warehouse capacity (i.e. Cap-t >0) and for others, additional capacity is needed (i.e. Cap+t > 0). In those months, where additional capacity is needed, the cost of extra storage capacity is incurred.

34

Table 4-5: Real Data: Results from the model run at s = 1 (month 1)

x11 = 1938653 x21 = 801449 x31 = 1587953 x41 = 16875 x51 = 719850 x61 = 150274 x71 = 829868 x81 = 3024762 x91 = 1283457 x10,1 = 285751 x11,1 = 12890 x12,1 = 743537 x13,1 = 89983 x14,1 = 32445 x15,1 =55185 Cap+1 = 0 Cap+2 = 85.09 Cap+3 = 27.66 Cap+4 = 60.88 Cap+5 = 45.34 Cap+6 = 0

Results x12 = 1757225 x13 = 1769502 x22 = 919081 x23 = 815382 x32 = 2838369 x33 = 0 x42 = 19526 x43 = 17614 x52 = 1286176 x53 = 1320088 x62 = 201188 x63 = 194257 x72 = 798879 x73 = 804656 x82 = 2518163 x83 = 2509015 x92 = 1094641 x93 = 1264099 x10,2 = 400388 x10,3 = 456189 x11,2 = 0 x11,3 = 18381 x12,2 = 147583 x12,3 = 997399 x13,2 = 0 x13,3 = 96567 x14,2 = 23412 x14,3 = 32678 x15,2 = 61000 x15,3 = 37258 Cap-1 = 37.12 Cap-2 = 0 Cap-3 = 0 Cap-4 = 0 Cap-5 = 5 Cap-6 = 258.67

δ1 = 1 δ2 = 1 δ3 = 1 δ4 = 1 δ5 = 0 δ6 = 0

x14 = 1534347 x24 = 661795 x34 = 0 x44 = 14463 x54 = 0 x64 = 0 x74 = 961897 x84 = 0 x94 = 796192 x10,4 = 459068 x11,4 = 0 x12,4 = 0 x13,4 = 0 x14,4 = 62749 x15,4 = 34301 Z = 14912378

After running the model at month 1, we will have results for the next four months but will only be implementing the results that appear in the first column which correspond to the orders placed at month 1. The following month (s = 2), we will rerun the model for the next 6 months and at that point, we will implement the results obtained for the second month. This allows us to use the most accurate demand forecast available and to take into account any delays in the receipt of the orders placed at previous months. Similar to the Illustrative Example in Chapter 3, the model for the future months (s = 2, 3, 4 and 5) are rerun and the optimal solutions recalculated. The results can be found in Tables 4-6, 4-7, 4-8 and 4-9 respectively.

35

Table 4-6: Real Data: Results from the model run at s = 2 (month 2)

x12 = 1757225 x22 = 919081 x32 = 4476267 x42 = 19526 x52 = 1286176 x62 = 201188 x72 = 798879 x82 = 2518163 x92 = 1094641 x10,2 = 0 x11,2 = 0 x12,2 = 147583 x13,2 = 0 x14,2 = 23412 x15,2 = 61000 Cap+2 = 85.09 Cap+3 = 177.72 Cap+4 = 174.59 Cap+5 = 203 Cap+6 = 0 Cap+7 = 77.37

Results x13 = 1769502 x14 = 1769572 x23 = 815382 x24 = 661795 x33 = 0 x34 = 0 x43 = 17614 x44 = 32989 x53 = 1320088 x54 = 826565 x63 = 194257 x64 = 180560 x73 = 804656 x74 = 961897 x83 = 2509015 x84 = 3190379 x93 = 1264099 x94 = 796192 x10,3 = 456189 x10,4 = 84198 x11,3 = 18381 x11,4 = 13920 x12,3 = 997399 x12,4 = 348595 x13,3 = 96567 x13,4 = 0 x14,3 = 32678 x14,4 = 76755 x15,3 = 37258 x15,4 = 69939 Cap-2 = 0 Cap-3 = 0 Cap-4 = 0 Cap-5 = 0 Cap-6 = 0 Cap-7 = 0

δ2 = 1 δ3 = 1 δ4 = 1 δ5 = 1 δ6 = 0 δ7 = 0

x15 = 1614635 x25 = 854690 x35 = 0 x45 = 0 x55 = 0 x65 = 0 x75 = 875438 x85 = 0 x95 = 1362816 x10,5 = 0 x11,5 = 0 x12,5 = 0 x13,5 = 0 x14,5 = 0 x15,5 = 0 Z = 19371438

In the model run at s = 2, we assumed that the forecasted demand for the 6 months of the planning horizon did not change. Thus, the demand forecast for month 2 through 6 were the same as in the model run at s = 1 and forecast of demand for month 7 was added.

36

Table 4-7: Real Data: Results from the model run at s = 3 (month 3)

x13 = 1769502 x23 = 815382 x33 = 0 x43 = 17614 x53 = 1320088 x63 = 194257 x73 = 804656 x83 = 2509015 x93 = 1264099 x10,3 = 456189 x11,3 = 18381 x12,3 = 997399 x13,3 = 96567 x14,3 = 32678 x15,3 = 37258 Cap+3 = 177.72 Cap+4 = 174.59 Cap+5 = 203 Cap+6 = 0 Cap+7 = 0 Cap+8 = 0

Results x14 = 1695613 x15 = 1688594 x24 = 661795 x25 = 854690 x34 = 0 x35 = 1743257 x44 = 32989 x45 = 0 x54 = 826565 x55 = 503251 x64 = 293379 x65 = 0 x74 = 961897 x75 = 875438 x84 = 3190379 x85 = 4290076 x94 = 796192 x95 = 1362816 x10,4 = 375522 x10,5 = 0 x11,4 = 13920 x11,5 = 9709 x12,4 = 348595 x12,5 = 877327 x13,4 = 0 x13,5 = 105610 x14,4 = 76755 x14,5 = 0 x15,4 = 69939 x15,5 = 0 Cap-3 = 0 Cap-4 = 0 Cap-5 = 0 Cap-6 = 0 Cap-7 = 0 Cap-8 = 0

δ3 = 1 δ4 = 1 δ5 = 1 δ6 = 1 δ7 = 0 δ8 = 0

x16 = 1999840 x26 = 930451 x36 = 0 x46 = 20368 x56 = 0 x66 = 0 x76 = 921717 x86 = 0 x96 = 1676779 x10,6 = 0 x11,6 = 0 x12,6 = 0 x13,6 = 0 x14,6 = 2287 x15,6 = 35837 Z = 24167963

I10,3 = 1652902; I10,4 = 1252514; I35 = 2900180, I10,5 = 796325; I16 = 161266, I36 = 1637898, I46 = 18526, I10,6 = 712779, I14,6 = 14006, I15,6 = 35638;

37

Table 4-8: Real Data: Results from the model run at s = 4 (month 4)

x14 = 1630326 x24 = 661795 x34 = 0 x44 = 32989 x54 = 826565 x64 = 293379 x74 = 961897 x84 = 3190379 x94 = 796192 x10,4 = 632684 x11,4 = 13920 x12,4 = 348595 x13,4 = 0 x14,4 = 76755 x15,4 = 69939 Cap+4 = 174.59 Cap+5 = 203 Cap+6 = 0 Cap+7 = 140.78 Cap+8 = 1756.82 Cap+9 = 86.95

Results x15 = 1753881 x16 = 3849145 x25 = 854690 x26 = 1787223 x35 = 1743257 x36 = 1607078 x45 = 0 x46 = 38810 x55 = 503251 x56 = 495262 x65 = 0 x66 = 114664 x75 = 875438 x76 = 921717 x85 = 4290076 x86 = 3974999 x95 = 1362816 x96 = 1676779 x10,5 = 0 x10,6 = 0 x11,5 = 9709 x11,6 = 1280 x12,5 = 877327 x12,6 = 60393 x13,5 = 105610 x13,6 = 0 x14,5 = 0 x14,6 = 2287 x15,5 = 0 x15,6 = 35837 Cap-4 = 0 Cap-5 = 0 Cap-6 = 0 Cap-7 = 0 Cap-8 = 0 Cap-9 = 0

δ4 = 1 δ5 = 1 δ6 = 1 δ7 = 0 δ8 = 0 δ9 = 0

x17 = 0 x27 = 0 x37 = 0 x47 = 0 x57 = 0 x67 = 0 x77 = 798333 x87 = 0 x97 = 1541613 x10,7 = 459068 x11,7 = 0 x12,7 = 0 x13,7 = 0 x14,7 = 14589 x15,7 = 86333 Z = 29000537

I10,4 = 1252514; I35 = 2900180, I10,5 = 796325; I16 = 95979, I36 = 1637898, I46 = 18526, I10,6 = 969941, I14,6 = 14006, I15,6 = 35638; I67 = 112819, I10,7 = 548486;

38

Table 4-9: Real Data: Results from the model run at s = 5 (month 5)

x15 = 1753881 x25 = 854690 x35 = 1743257 x45 = 0 x55 = 503251 x65 = 0 x75 = 875438 x85 = 4290076 x95 = 1362816 x10,5 = 0 x11,5 = 9709 x12,5 = 877327 x13,5 = 105610 x14,5 = 0 x15,5 = 0 Cap+5 = 203 Cap+6 = 0 Cap+7 = 140.78 Cap+8 = 1756.82 Cap+9 = 202.79 Cap+10 = 0

Results x16 = 3849145 x17 = 0 x26 = 1787223 x27 = 0 x36 = 3102454 x37 = 0 x46 = 38810 x47 = 0 x56 = 1070775 x57 = 0 x66 = 114664 x67 = 99816 x76 = 921717 x77 = 798333 x86 = 3974999 x87 = 3040726 x96 = 1676779 x97 = 1541613 x10,6 = 0 x10,7 = 0 x11,6 = 1280 x11,7 = 16636 x12,6 = 60393 x12,7 = 982819 x13,6 = 0 x13,7 = 103528 x14,6 = 2287 x14,7 = 14589 x15,6 = 35837 x15,7 = 86333 Cap-5 = 0 Cap-6 = 0 Cap-7 = 0 Cap-8 = 0 Cap-9 = 0 Cap-10 = 11.49

δ5 = 1 δ6 = 1 δ7 = 1 δ8 = 1 δ9 = 0 δ10 = 0

x18 = 1787060 x28 = 806698 x38 = 0 x48 = 17382 x58 = 0 x68 = 0 x78 = 809726 x88 = 0 x98 = 1448846 x10,8 = 244237 x11,8= 0 x12,8 = 0 x13,8 = 0 x14,8 = 36993 x15,8 = 90632 Z = 33788165

I35 = 2900180, I10,5 = 796325; I16 = 95979, I36 = 1637898, I46 = 18526, I10,6 = 969941, I14,6 = 14006, I15,6 = 35638; I67 = 112819, I10,7 = 548486; I18 = 1849305, I28 = 856772, I48 = 18442, I10,8 = 257162; Since we are only implementing the results obtained in the period in which the model is run, Table 4-10 shows the consolidated results that will used for decision making from each model run. The results from month 1 were obtained from the model run at s = 1, the results for month 2 were obtained from the model run at s = 2, and so on with the other months.

39 Table 4-10: Real Data: Consolidated results implemented from using all model runs

x11 = 1938653 x21 = 801449 x31 = 1587953 x41 = 16875 x51 = 719850 x61 = 150274 x71 = 829868 x81 = 3024762 x91 = 1283457 x10,1 = 285751 x11,1 = 12890 x12,1 = 743537 x13,1 = 89983 x14,1 = 32445 x15,1 =55185

x12 = 1757225 x22 = 919081 x32 = 4476267 x42 = 19526 x52 = 1286176 x62 = 201188 x72 = 798879 x82 = 2518163 x92 = 1094641 x10,2 = 0 x11,2 = 0 x12,2 = 147583 x13,2 = 0 x14,2 = 23412 x15,2 = 61000

Results x13 = 1769502 x23 = 815382 x33 = 0 x43 = 17614 x53 = 1320088 x63 = 194257 x73 = 804656 x83 = 2509015 x93 = 1264099 x10,3 = 456189 x11,3 = 18381 x12,3 = 997399 x13,3 = 96567 x14,3 = 32678 x15,3 = 37258

x14 = 1630326 x24 = 661795 x34 = 0 x44 = 32989 x54 = 826565 x64 = 293379 x74 = 961897 x84 = 3190379 x94 = 796192 x10,4 = 632684 x11,4 = 13920 x12,4 = 348595 x13,4 = 0 x14,4 = 76755 x15,4 = 69939

Cap+1 = 0 Cap+2 = 85.09 Cap+3 = 177.72 Cap+4 = 174.59 Cap+5 = 203

Cap-1 = 37.12 Cap-2 = 0 Cap-3 = 0 Cap-4 = 0 Cap-5 = 0

δ1 = 1 δ2 = 1 δ3 = 1 δ4 = 1 δ5 = 1

x15 = 1753881 x25 = 854690 x35 = 1743257 x45 = 0 x55 = 503251 x65 = 0 x75 = 875438 x85 = 4290076 x95 = 1362816 x10,5 = 0 x11,5 = 9709 x12,5 = 877327 x13,5 = 105610 x14,5 = 0 x15,5 = 0

In these models we assumed the forecasted demand for the raw materials (Table 4-1) did not change on a monthly basis. This is also true for the delivery schedules that remained without change. In real world situations, the forecast of demands are known for 6 months only, and are subject to change each month the model is run, thus we need to modify some things in the Base Model in order to capture these changes. The next section will deal with the extension of the Base Model.

40

4.2 Extension to the Base Model In the base model we assumed that there were no shortages allowed. This extension will relax that assumption and thus includes the possibility of having backorders. As we know, backorders are undesirable, especially, for raw materials. Not having enough raw materials will directly affect the production at the plant, and the company’s revenue. However, shortage can occur during a month because of sudden increases in forecasted demand or delays in expected deliveries.

4.2.1 Notation The following notation remains very similar to the Base Model but there are some changes that are needed in this model. Only those parameters, variables and equations that would change will be shown and, all the others will remain the same.

Parameters: These parameters will be added to those in the Base Model. SLi = desired service level for each raw material i The service level measure as fill rate is the fraction of demand satisfied from inventory. bi = back order cost per unit of raw material i per period

Variables: we are changing the inventory variable into two distinct variables, positive variable indicates an inventory and negative variable indicates a shortage. 𝐼 +𝑖𝑡 = amount of inventory of raw material i at the end of period t 𝐼 −𝑖𝑡 = amount of shortage of raw material i at the end of period t

41

4.2.2 Objective Function The objective function in this problem remains similar to the base model; we just add a backorder cost. This backorder cost is used to achieve the desired service level for each product. Gallego (1990) showed that the implicit optimal backorder cost can be represented using the service level (Fill Rate) by the critical ratio shown in equation [4.2]. Minimize Total cost = Purchasing Cost + Inventory Holding Cost + Back Order Cost + Extra Capacity Cost + Fixed Ordering Cost, 𝑠+𝑇−1 𝑁

𝑠+𝑇−1 +

+ − [(𝑐𝑖𝑡 𝑥𝑖𝑡 + ℎ 𝑖 𝐼 𝑖𝑡 + 𝑏𝑖 𝐼 𝑖𝑡 ) + 𝑒𝑡 𝐶𝑎𝑝𝑡 ] +

𝑀𝑖𝑛 𝑍 = 𝑡=𝑠

𝑖=1

𝐹𝑡 𝛿𝑡

[4.1]

𝑡=𝑠

Where, 𝑆𝐿𝑖 =

𝑏𝑖 𝑏 𝑖 +ℎ 𝑖

[4.2]

4.2.3 Constraints Demand Constraint: In this constraint we are adding the cumulative backorder term to both sides of the equation. 𝑜

− 𝑜 + 𝐼 + 𝑖,𝑠−1 − 𝐼 𝑜 − 𝑖,𝑠−1 + 𝑥 𝑖,𝑠−𝐿𝑖 = 𝑑𝑖𝑠 + 𝐼 𝑖𝑠 − 𝐼𝑖𝑠 for all 𝑖 = 1, … , N − 𝐼 + 𝑖,𝑡−1 − 𝐼𝑖,𝑡−1 + 𝑥 𝑜 𝑖,𝑡−𝐿𝑖 = 𝑑𝑖𝑡 + 𝐼 + 𝑖𝑡 − 𝐼𝑖𝑡− for all 𝑖 = 1, … , N and 𝑡 = 𝑠 + 1, … , 𝑠 + 𝐿𝑖 − 1 [4.3] − 𝐼 + 𝑖,𝑡−1 − 𝐼𝑖,𝑡−1 + 𝑥𝑖,𝑡−𝐿𝑖 = 𝑑𝑖𝑡 + 𝐼 + 𝑖𝑡 − 𝐼𝑖𝑡− for all 𝑖 = 1, … , N and 𝑡 = 𝑠 + 𝐿𝑖 , … , 𝑠 + 𝑇 − 1

xoit, Ioit refers to the values of the decision variables that we already know from the previous model runs at t = s -1, s -2, etc.

42 Capacity Constraint: In this constraint, we have to make sure that the capacity is measured only with respect to the inventory and not the backorders. 𝑁

𝑎𝑖 𝐼 +𝑖,𝑡−1 + 𝑥𝑖,𝑡−𝐿𝑖 + 𝐶𝑎𝑝𝑡 − − 𝐶𝑎𝑝𝑡 + = 𝐶𝑎𝑝𝑡 for all t = 𝑠, 𝑠 + 1 … , 𝑠 + T − 1

[4.4]

𝑖=1

Now that we have the adjusted model for shortages, we can proceed to rerun the model and optimize the ordering quantities for the raw materials. At this time we are also going to take into account the changes that occur between the forecasted demands from one period to another.

4.2.4 Given Data We are going to use the same data that we used for the Base Model. Each month the demand forecast will be revised and updated. Also, we are adding a new parameter for the Fill Rate for each raw material (SLi). This fill rate will be the same for all the materials because when selecting the materials from the list, we chose the most relevant ones. We want to attain a 98% service level/fill rate for all the raw materials.

4.2.5 Results of the Extension to the Base Model with the Real Data The optimization software GAMS is used to generate the mixed integer programming (MIP) model and then to solve it. The model had 394 variables (including 12 binary variables) and 370 constraints and it took approximately 3 seconds to solve. The model is run each month and updated with the most current information; this includes the output from the previous month’s models, the forecasted demand and any change in the unit price of the materials or lead times. Model results from month 1 (s =1) are shown in Table 4-11, 4-12 and 4-13.

43 Unlike the models run in Chapter 4.1, here we are using real data collected from the company for the initial conditions; both for the orders in transit and the initial inventory. Also the forecasted demand changes according to the data provided by the company. In the first run of models, we may experience some strange pattern in the orders indicated by the models. This is due to the fact that we are taking a snapshot of the current levels of inventory of the company at certain point in time, and they may not have the right amount of inventory on hand to satisfy the desired service level.

Table 4-11: Extended Model: Orders from the model run at s = 1

x11 = 284224 x21 = 287554 x31 = 0 x41 = 0 x51 = 0 x61 = 0 x71 = 92550 x81 = 0 x91 = 266252 x10,1 = 0 x11,1 = 0 x12,1 = 191695 x13,1 = 64310 x14,1 = 0 x15,1 = 0

x12 = 251879 x22 = 93865 x32 = 0 x42 = 0 x52 = 0 x62 = 0 x72 = 124894 x82 = 0 x92 = 113387 x10,2 = 56999 x11,2 = 0 x12,2 = 182372 x13,2 = 34326 x14,2 = 0 x15,2 = 0

x13 = 655631 x23 = 260340 x33 = 0 x43 = 4361 x53 = 0 x63 = 0 x73 = 369614 x83 = 0 x93 = 362344 x10,3 = 146162 x11,3 = 0 x12,3 = 3112 x13,3 = 2449 x14,3 = 0 x15,3 = 0

x14 = 0 x24 = 0 x34 = 0 x44 = 0 x54 = 0 x64 = 0 x74 = 0 x84 = 0 x94 = 0 x10,4 = 0 x11,4 = 0 x12,4 = 0 x13,4 = 0 x14,4 = 0 x15,4 = 0

44 Table 4-12: Extended Model: Inventory from the model run at s = 1

I+11 = 414831 I+21 = 0 I+31 = 1265717 I+41 = 0 I+51 = 0 I+61 = 173134 I+71 = 229027 I+81 = 1221799 I+91 = 33302 I+10,1 = 33765 I+11,1 = 22652 I+12,1 = 448 I+13,1 = 0 I+14,1 = 17011 I+15,1 = 62392

I+12 = 72836 I+22 = 0 I+32 = 983957 I+42 = 6408 I+52 = 0 I+62 = 285331 I+72 = 120860 I+82 = 639829 I+92 = 0 I+10,2 = 141353 I+11,2 = 19900 I+12,2 = 0 I+13,2 = 0 I+14,2 = 58717 I+15,2 = 74862

I+13 = 0 I+23 = 0 I+33 = 2579223 I+43 = 3473 I+53 = 2183858 I+63 = 398520 I+73 = 0 I+83 = 4998836 I+93 = 0 I+10,3 = 7857 I+11,3 = 15632 I+12,3 = 0 I+13,3 = 0 I+14,3 = 47279 I+15,3 = 66787

I+14 = 0 I+24 = 0 I+34 = 2395405 I+44 = 1391 I+54 = 1977186 I+64 = 362451 I+74 = 0 I+84 = 4550207 I+94 = 0 I+10,4 = 0 I+11,4 = 15223 I+12,4 = 0 I+13,4 = 0 I+14,4 = 47279 I+15,4 = 59670

Table 4-13: Extended Model: Backorders from the model run at s = 1

I-11 = 0 I-21 = 109878 I-31 = 0 I-41 = 2840 I-51 = 62597 I-61 = 0 I-71 = 0 I-81 = 0 I-91 = 0 I-10,1 = 0 I-11,1 = 0 I-12,1 = 0 I-13,1 = 12679 I-14,1 = 0 I-15,1 = 0

I-12 = 0 I-22 = 153966 I-32 = 0 I-42 = 0 I-52 = 446670 I-62 = 0 I-72 = 0 I-82 = 0 I-92 = 124549 I-10,2 = 0 I-11,2 = 0 I-12,2 = 34517 I-13,2 = 12679 I-14,2 = 0 I-15,2 = 0

I-13 = 0 I-23 = 0 I-33 = 0 I-43 = 0 I-53 = 0 I-63 = 0 I-73 = 0 I-83 = 0 I-93 = 0 I-10,3 = 0 I-11,3 = 0 I-12,3 = 186501 I-13,3 = 60820 I-14,3 = 0 I-15,3 = 0

I-14 = 0 I-24 =0 I-34 = 0 I-44 = 0 I-54 = 0 I-64 = 0 I-74 = 0 I-84 = 0 I-94 = 0 I-10,4 = 0 I-11,4 = 0 I-12,4 = 0 I-13,4 = 0 I-14,4 = 0 I-15,4 = 0

Similar to the models in Chapter 4.1, we implement only the results obtained for the month that the model is run. In this case, from the results obtained in the model at s = 1, we will only order what the model results indicate for month 1. The other order quantities, for the months 2, 3, 4 and 5 will be determined later when we rerun the model at s = 2, 3, 4 and 5.

45 For the future months, the model will be run again and the optimal solutions will be recalculated. To illustrate this, we have rerun the model at s = 2, 3, 4 and 5. The results for s = 2 and 3 are shown in Tables 4-14 to 4-19 respectively, and for s = 4 and 5 are shown in Tables A-1 to A-6 respectively (located in Appendix A). In these models, the forecasted demand changed every month according to new information received from the customers. Because of this reason, it is possible that we experience a shortage of raw material for some months, due to an increase in the demand for a month, for which the raw materials were ordered in the previous months. The opposite situation can happen also, we can experience excess inventory when there is a decrease in the demand for a month, early in the planning cycle.

Table 4-14: Extended Model: Orders from the model run at s = 2

x12 = 3729191 x22 = 1689035 x32 = 4312818 x42 = 33226 x52 = 1715113 x62 = 201542 x72 = 2042525 x82 = 5135028 x92 = 2382497 x10,2 = 991258 x11,2 = 16646 x12,2 = 2070846 x13,2 = 195430 x14,2 = 36429 x15,2 = 11198

x13 = 1610451 x23 = 705415 x33 = 0 x43 = 15477 x53 = 1139977 x63 = 177846 x73 = 1016729 x83 = 2926061 x93 = 1028314 x10,3 = 331850 x11,3 = 5104 x12,3 = 64842 x13,3 = 0 x14,3 = 31064 x15,3 = 30827

x14 = 2138167 x24 = 600325 x34 = 0 x44 = 27554 x54 = 1071182 x64 = 146054 x74 = 826541 x84 = 3408352 x94 = 766156 x10,4 = 711591 x11,4 = 17954 x12,4 = 1012999 x13,4 = 121686 x14,4 = 58842 x15,4 = 61976

x15 = 759688 x25 = 647785 x35 = 0 x45 = 0 x55 = 0 x65 = 0 x75 = 944626 x85 = 0 x95 = 925001 x10,5 = 0 x11,5 = 0 x12,5 = 0 x13,5 = 0 x14,5 = 0 x15,5 = 0

46

Table 4-15: Extended Model: Inventory from the model run at s = 2

I+12 = 0 I+22 = 0 I+32 = 0 I+42 = 0 I+52 = 0 I+62 = 141946 I+72 = 0 I+82 = 0 I+92 = 0 I+10,2 = 0 I+11,2 = 22652 I+12,2 = 0 I+13,2 = 0 I+14,2 = 34333 I+15,2 = 45134

I+13 = 0 I+23 = 0 I+33 = 428365 I+43 = 0 I+53 = 493123 I+63 = 147494 I+73 = 0 I+83 = 1027069 I+93 = 0 I+10,3 = 0 I+11,3 = 4895 I+12,3 = 0 I+13,3 = 0 I+14,3 = 0 I+15,3 = 20143

I+14 = 0 I+24 = 0 I+34 = 0 I+44 = 0 I+54 = 0 I+64 = 0 I+74 = 0 I+84 = 0 I+94 = 0 I+10,4 = 0 I+11,4 = 1308 I+12,4 = 0 I+13,4 = 0 I+14,4 = 0 I+15,4 = 0

I+15 = 0 I+25 = 0 I+35 = 2367682 I+45 = 0 I+55 = 0 I+65 = 0 I+75 = 0 I+85 = 0 I+95 = 0 I+10,5 = 0 I+11,5 = 0 I+12,5 = 0 I+13,5 = 0 I+14,5 = 0 I+15,5 = 0

Table 4-16: Extended Model: Backorders from the model run at s = 2

I-12 = 1354671 I-22 = 825260 I-32 = 310370 I-42 = 8031 I-52 = 1348773 I-62 = 0 I-72 = 475629 I-82 = 1296364 I-92 = 1230797 I-10,2 = 183924 I-11,2 = 0 I-12,2 = 147135 I-13,2 = 12679 I-14,2 = 0 I-15,2 = 0

I-13 = 2414161 I-23 = 1124664 I-33 = 0 I-43 = 20968 I-53 = 0 I-63 = 0 I-73 = 1242371 I-83 = 0 I-93 = 1707403 I-10,3 = 551225 I-11,3 = 0 I-12,3 = 1029806 I-13,3 = 129758 I-14,3 = 8651 I-15,3 = 0

I-14 = 0 I-24 = 0 I-34 = 648202 I-44 = 0 I-54 = 643931 I-64 = 57376 I-74 = 0 I-84 = 1398197 I-94 = 0 I-10,4 = 0 I-11,4 = 0 I-12,4 = 905937 I-13,4 = 65448 I-14,4 = 0 I-15,4 = 0

I-15 = 0 I-25 = 0 I-35 = 0 I-45 = 0 I-55 = 0 I-65 = 0 I-75 = 0 I-85 = 0 I-95 = 0 I-10,5 = 0 I-11,5 = 0 I-12,5 = 0 I-13,5 = 0 I-14,5 = 0 I-15,5 = 0

From this result in Table 4-14, we will only order what the model results indicate to order in month 2, which are the orders shown in the first column of the table. The other order

47 quantities for the later months (3, 4 and 5) will be determined when we rerun the model at s = 3, 4 and 5. Here the forecasted demand also changed from the previous month according to the updates made to the forecast. For example, the forecasted demand for raw material 1 in period 3 used at s = 1 was 357060, the forecast for the same raw material in the same period made at s = 2 was 1343714 and at s = 3 was 1534347. The forecasted demand for raw material 1 in period 3 at s = 1 was more than 3 times smaller than the one forecasted at s = 2. Also the forecast at s = 3 is 14% larger than the forecast at s = 2. This will impact both the inventory and the shortage for the raw materials.

It is clear that the forecast of demands can change dramatically from month to month. Table 4-15 and 4-16 show that for raw materials 6, 11, 14 and 15 there is inventory available at the end of month 2, but for all the other raw materials we will have shortage.

Table 4-17: Extended Model: Orders from the model run at s = 3

x13 = 2251240 x23 = 1022690 x33 = 0 x43 = 22388 x53 = 710051 x63 = 212186 x73 = 1152567 x83 = 4223416 x93 = 1738940 x10,3 = 421235 x11,3 = 2909 x12,3 = 81428 x13,3 = 0 x14,3 = 33476 x15,3 = 49033

x14 = 1686960 x24 = 647040 x34 = 0 x44 = 27693 x54 = 1832880 x64 = 347473 x74 = 878801 x84 = 6269357 x94 = 889229 x10,4 = 1199001 x11,4 = 23826 x12,4 = 1176438 x13,4 = 58305 x14,4 = 41565 x15,4 = 136459

x15 = 1208012 x25 = 603675 x35 = 1826328 x45 = 0 x55 = 0 x65 = 0 x75 = 814989 x85 = 0 x95 = 878658 x10,5 = 0 x11,5 = 0 x12,5 = 0 x13,5 = 0 x14,5 = 0 x15,5 = 0

x16 = 1460368 x26 = 644901 x36 = 0 x46 = 13729 x56 = 0 x66 = 0 x76 = 819556 x86 = 0 x96 = 864456 x10,6 = 0 x11,6 = 0 x12,6 = 0 x13,6 = 0 x14,6 = 36620 x15,6 = 74396

48

Table 4-18: Extended Model: Inventory from the model run at s = 3

I+13 = 0 I+23 = 0 I+33 = 278348 I+43 = 0 I+53 = 331139 I+63 = 118689 I+73 = 0 I+83 = 985117 I+93 = 0 I+10,3 = 0 I+11,3 = 4271 I+12,3 = 0 I+13,3 = 0 I+14,3 = 0 I+15,3 = 10833

I+14 = 0 I+24 = 0 I+34 = 0 I+44 = 0 I+54 = 0 I+64 = 0 I+74 = 0 I+84 = 0 I+94 = 0 I+10,4 = 0 I+11,4 = 4271 I+12,4 = 0 I+13,4 = 0 I+14,4 = 0 I+15,4 = 0

I+15 = 0 I+25 = 0 I+35 = 1742381 I+45 = 0 I+55 = 251421 I+65 = 0 I+75 = 0 I+85 = 0 I+95 = 0 I+10,5 = 0 I+11,5 = 0 I+12,5 = 0 I+13,5 = 38806 I+14,5 = 0 I+15,5 = 0

I+16 = 206636 I+26 = 0 I+36 = 519976 I+46 = 13540 I+56 = 0 I+66 = 0 I+76 = 0 I+86 = 0 I+96 = 0 I+10,6 = 778662 I+11,6 = 0 I+12,6 = 0 I+13,6 = 38806 I+14,6 = 13585 I+15,6 = 63685

Table 4-19: Extended Model: Backorders from the model run at s = 3

I-13 = 2604794 I-23 = 1199501 I-33 = 0 I-43 = 22494 I-53 = 0 I-63 = 0 I-73 = 1344976 I-83 = 0 I-93 = 1760737 I-10,3 = 642992 I-11,3 = 0 I-12,3 = 1144534 I-13,3 = 109246 I-14,3 = 28416 I-15,3 = 0

I-14 = 753744 I-24 = 349716 I-34 = 1373999 I-44 = 7528 I-54 = 1027852 I-64 = 145848 I-74 = 132781 I-84 = 1484296 I-94 = 585413 I-10,4 = 209665 I-11,4 = 0 I-12,4 = 952839 I-13,4 = 44936 I-14,4 = 7961 I-15,4 = 23254

I-15 = 0 I-25 = 0 I-35 = 0 I-45 = 0 I-55 = 0 I-65 = 21404 I-75 = 0 I-85 = 761378 I-95 = 0 I-10,5 = 0 I-11,5 = 2909 I-12,5 = 81428 I-13,5 = 0 I-14,5 = 0 I-15,5 = 0

I-16 = 0 I-26 = 0 I-36 = 0 I-46 = 0 I-56 = 0 I-66 = 0 I-76 = 0 I-86 = 0 I-96 = 0 I-10,6 =0 I-11,6 = 0 I-12,6 = 0 I-13,6 = 0 I-14,6 = 0 I-15,6 = 0

The model was run up to period s = 5 and the tables with the results for the last two months can be found in Appendix A.

49 Table 4-20 illustrates the consolidated result that will be implemented from running the model on a monthly basis. We can see how the inventory and backorders change due to changes in demand that occur from updating the forecast every month. When the forecasted demand is lower than the actual demand, there is going to be a shortage, and thus we will have backorders; raw material 1 is a good example of this. On the other hand, when the forecasted demand is higher than the actual demand, there is going to be inventory; raw material 6 is a good example of this situation, but only until period 3, because after that there are backorders on the raw material. Implementing only the results obtained in the period in which the model is run has some benefits; one can make decisions with the most recent information and can easily adjust the future month order quantities whenever necessary. From Table 4-20 it is clear that there are orders placed every month; these orders do not necessarily include all the raw materials. Some raw materials are used only for a specific product and hence, not ordered every month. Also if the amount of inventory held is sufficiently large to cover the forecasted demand then no orders will be placed for that raw material. It is important to point out that the large amount of backorders seen in Table 4-20 is due to the fact that the forecasted demand changed dramatically from month to month. Since the changes occur only after the order has been placed, in some cases it leaves no time to react. The model adjust itself to minimize the number of backorders whenever possible by having the cost of running out of stock (bi) higher than the cost of holding inventory (hi). To reduce the number of backorders, there are several things that can be done. The backorder cost can be raised so that there is a higher penalty in the objective function. Another option is to carry at least one month of demand in inventory to compensate for the sudden

50 changes in the forecast. Lastly, we can put restrictions on how much the forecasted demand can be changed from month to month. This will help reduce the number of backorders by not creating these peaks in demand.

Table 4-20: Extended Model: Consolidated results implemented from using all model runs

x11 = 284224 x21 = 287554 x31 = 0 x41 = 0 x51 = 0 x61 = 0 x71 = 92550 x81 = 0 x91 = 266252 x10,1 = 0 x11,1 = 0 x12,1 = 191695 x13,1 = 64310 x14,1 = 0 x15,1 = 0

x12 = 3729191 x22 = 1689035 x32 = 4312818 x42 = 33226 x52 = 1715113 x62 = 201542 x72 = 2042525 x82 = 5135028 x92 = 2382497 x10,2 = 991258 x11,2 = 16646 x12,2 = 2070846 x13,2 = 195430 x14,2 = 36429 x15,2 = 11198

x13 = 2251240 x23 = 1022690 x33 = 0 x43 = 22388 x53 = 710051 x63 = 212186 x73 = 1152567 x83 = 4223416 x93 = 1738940 x10,3 = 421235 x11,3 = 2909 x12,3 = 81428 x13,3 = 0 x14,3 = 33476 x15,3 = 49033

I+11 = 414831 I+21 = 0 I+31 = 1265717 I+41 = 0 I+51 = 0 I+61 = 173134 I+71 = 229027 I+81 = 1221799 I+91 = 33302 I+10,1 = 33765 I+11,1 = 22652 I+12,1 = 448 I+13,1 = 0 I+14,1 = 17011 I+15,1 = 62392

I+12 = 0 I+22 = 0 I+32 = 0 I+42 = 0 I+52 = 0 I+62 = 141946 I+72 = 0 I+82 = 0 I+92 = 0 I+10,2 = 0 I+11,2 = 22652 I+12,2 = 0 I+13,2 = 0 I+14,2 = 34333 I+15,2 = 45134

I+13 = 0 I+23 = 0 I+33 = 278348 I+43 = 0 I+53 = 331139 I+63 = 118689 I+73 = 0 I+83 = 985117 I+93 = 0 I+10,3 = 0 I+11,3 = 4271 I+12,3 = 0 I+13,3 = 0 I+14,3 = 0 I+15,3 = 10833

x14 = 1552302 x24 = 551432 x34 = 0 x44 = 27223 x54 = 1604132 x64 = 299337 x74 = 696490 x84 = 4054220 x94 = 647292 x10,4 = 1432762 x11,4 = 29869 x12,4 = 1617478 x13,4 = 63030 x14,4 = 26120 x15,4 = 150374 I+14 = 0 I+24 = 0 I+34 = 0 I+44 = 0 I+54 = 0 I+64 = 0 I+74 = 0 I+84 = 0 I+94 = 0 I+10,4 = 0 I+11,4 = 0 I+12,4 = 0 I+13,4 = 0 I+14,4 = 0 I+15,4 = 0

x15 = 2233341 x25 = 1162518 x35 = 2758245 x45 = 10505 x55 = 0 x65 = 11719 x75 = 1298332 x85 = 5480717 x95 = 2191687 x10,5 = 0 x11,5 = 0 x12,5 = 0 x13,5 = 40629 x14,5 = 0 x15,5 = 8701 I+15 = 0 I+25 = 0 I+35 = 1210011 I+45 = 0 I+55 = 716436 I+65 = 26852 I+75 = 52961 I+85 = 0 I+95 = 0 I+10,5 = 56722 I+11,5 = 0 I+12,5 = 0 I+13,5 = 44884 I+14,5 = 25196 I+15,5 = 0

51 I-11 = 0 I-21 = 109878 I-31 = 0 I-41 = 2840 I-51 = 62597 I-61 = 0 I-71 = 0 I-81 = 0 I-91 = 0 I-10,1 = 0 I-11,1 = 0 I-12,1 = 0 I-13,1 = 12679 I-14,1 = 0 I-15,1 = 0

I-12 = 1354671 I-22 = 825260 I-32 = 310370 I-42 = 8031 I-52 = 1348773 I-62 = 0 I-72 = 475629 I-82 = 1296364 I-92 = 1230797 I-10,2 = 183924 I-11,2 = 0 I-12,2 = 147135 I-13,2 = 12679 I-14,2 = 0 I-15,2 = 0

I-13 = 2604794 I-23 = 1199501 I-33 = 0 I-43 = 22494 I-53 = 0 I-63 = 0 I-73 = 1344976 I-83 = 0 I-93 = 1760737 I-10,3 = 642992 I-11,3 = 0 I-12,3 = 1144534 I-13,3 = 109246 I-14,3 = 28416 I-15,3 = 0

I-14 = 725463 I-24 = 365156 I-34 = 359550 I-44 = 7794 I-54 = 495426 I-64 = 61871 I-74 = 177889 I-84 = 205262 I-94 = 741056 I-10,4 = 3189 I-11,4 = 9649 I-12,4 = 301434 I-13,4 = 44936 I-14,4 = 5993 I-15,4 = 13607

I-15 = 474063 I-25 = 272917 I-35 = 0 I-45 = 5774 I-55 = 0 I-65 = 0 I-75 = 0 I-85 = 1360310 I-95 = 678895 I-10,5 = 0 I-11,5 = 2712 I-12,5 = 107915 I-13,5 = 0 I-14,5 = 0 I-15,5 = 411

52

Chapter 5 CONCLUSIONS AND FUTURE RESEARCH In this chapter we will give a conclusion to the thesis and also point out different areas where there can be a focus for future research.

5.1 Conclusion In this thesis we presented a mixed integer programming model that took into account the variation in demand that occurs in the forecasted demands for raw materials from one month to another. Later, we allowed for backorders and introduced a service level for each of the raw material. There was also a cost for holding inventory and a fixed cost for ordering whenever an order was placed. We started with a small example to illustrate how the model works, and then used the same model with real data collected from a global health and hygiene company that operates in more than 35 countries around the globe. The model was run on a monthly basis over the entire planning horizon, and only the results obtained for the period in which the model was run (namely, the first month) were implemented. By doing this we were able to capture the changes in the forecasted demand from month to month and the variations in the raw materials deliveries schedule. From the results of the models, it is clear that an order was placed every month but not all the materials were included in every order. Because of this, the fixed cost of ordering was incurred every month. There was also backorders in some of the months, which were found to be the consequence of the big changes in the forecasted demands from month to month, which left no time to react.

53

5.2 Future Research In the area of inventory control models and supply chain management, there are always different ways to approach a problem. In the model presented in the thesis, it was always assumed that the lead time was very stable and thus never considered its variability. An interesting topic for future research is the relaxation of this assumption, and to represent the lead time as a random variable and study its effect by combining simulation with optimization. The forecasted demand, which we changed on a monthly basis, can be approached differently. We can develop different demand scenarios based on some interval of uncertainty. Also this will help in showing the impact of multiple sources of uncertainty in the objective function; in our example, this could be demand uncertainty and lead time uncertainty. Another modeling approach for this particular situation could be to vary the frequency of the orders. By making orders more frequently we can react quicker to sudden changes in the demand of the raw materials and thus reducing the number of stock outs. These conditions have to be arranged with the supplier, who can also set restrictions on the size of the order. A supplier can decide to deliver only one raw material per container, which will then change the order quantities. We would no longer order half of a container; it would have to be either one container or no containers. Finally an important point that we can possibly consider is the risk of the supply chain disruption due to natural (e.g. hurricanes, floods, etc) and manmade events (e.g. riot at the port, strike at the supplier’s plant, etc). Also when receiving the orders, we can encounter defects in the raw materials and some of them may need to be discarded. A probability of occurring would be assigned to each of scenarios depending on the threat level they represent. Then we can plan accordingly taking into account these possibilities.

54

REFERENCES Aggrawal, S.C., (1974), “A review of current inventory theory and its applications,” International Journal of Production Research, vol.12, pp.443-482. Arrow, K., S.Karling, H.Scarf, (1958), “Studies in Mathematical Theory of Inventory and Production,” Stanford University Press, Stanford, CA. Axsäter,S., (2006), “A simple procedure for determining order quantities under fill rate constraint and normally distributed lead-time demand,” European Journal of Operations Research, vol.174, pp.480-491. Bertsimas,D. and A.Thiele, (2006), “A robust optimization approach to inventory theory,” Operations Research, vol.54, pp.150-168. Chen,F.Y. and D.Krass, (2001), “Inventory models with minimal service level constraints,” European Journal of Operations Research, vol.134, pp.120-140. Chu,C.W., G.S.Liang and C.T.Liao, (2008), “Controlling inventory by combining ABC analysis and fuzzy classification,” Computer & Industrial Engineering, vol.55, pp.841-851. Clark, A. and H. Scarf, (1960), “Optimal policies for a multi-echelon inventory problem,” Management Science, vol.6, pp. 474-490. Chopra.S. and P.Meindl,(2007), “ Supply Chain Management: Strategy, Planning and Operation,” 3rd ed, Prentice Hall, Upper Saddle River, N.J. DeBodt, M. and S.C.Graves, (1985), “Continuous review policies for a multi-echelon inventory problem with Stochastic demand,” Management Science, vol.31, pp.1286-1299. Gallego,G., (1990), “Scheduling the production of several items with random demands in a single facility,” Management Science, vol.36, pp.1579-1592. Giri.B.C. and K.S.Chaudhuri, (1998), “Deterministic models of perishable inventory with stockdependent demand rate and nonlinear holding cost,” European Journal of Operational Research, vol.105, pp.467-474. Goyal,S.K., (1977), “An integrated inventory model for a single production system,” Operational Research Quarterly, vol.28, pp.539-545.

55 Hackman,S.T., R.Leachman, (1989), “A general framework for modeling production,” Management Science, vol.35, pp.478-495 Hong,J. and J.C.Hayya, (1992), “An optimal algorithm for integrating raw materials inventory in a single-product manufacturing system,” European Journal of Operational Research, vol.59, pp.313-318. Hopp,W.J. and M.L.Spearman, (2000), “Factory Physics: foundations of manufacturing management,” 2nd ed, McGraw-Hill, New York. Hopp,W.J., M.L.Spearman and R.Q.Zhang, (1997), “Easiliy implementable inventory control policies,” Operations Research, vol.45, pp.327-340. Kijima,M. and T.Takimoto, (1999), “A (T,S) inventory/production system with limited production capacity and uncertain demands,” Operations Research Letters, vol.15, pp.67-79. Kim,S.H. and J.Chandra, (1987), “An integrated inventory model for a single product and its raw material,” International Journal of Production Research, vol.25, pp.627-634. Lee,H.L. and C.Billington, (1993), “Material management in decentralized supply chain,” Operations Research, vol.41, pp.835-847. Ng,W.L., (2007), “A simple classifier for multiple criteria ABC analysis,” European Journal of Operational Research, vol.177, pp.344-353. Oh, Y.H. and H.Hwang (2006), “Determinisic inventory model for recycling system,” Journal of Intelligent Manufacturing, vol.17, pp.423-428. Sarker,B.R and G.R.Parija, (1996), “Optimal batch size and raw material ordering policy for a production system with a fixed-interval lumpy demand delivery system,” European Journal of Operational Research, vol.89, pp.593-608. Simchi-Levi, D., P.Kaminsky and E.Simchi-Levi, (2003), “ Designing & Managing the Supply Chain: Concepts, Strategies & Case Studies,”2nd ed, McGraw Hill, Boston. Tarim,S.A and B.G.Kingsman, (2004), “The stochastic dynamic production/inventory lot sizing problem with service-level constraints,” International Journal of Production Economics, vol.88, pp.105-119.

56 Thomas,D., (2005), “Measuring items fill-rate performance in a finite horizon,” Manufacturing & Service Operations Management, vol.7, pp.74-80. Tijms,H.C and H.Groenevelt, (1984), “Simple approximations for the reorder point in periodic and continuous review (s,S) inventory systems with service level constraints,” European Journal of Operations Research, vol.17, pp.175-190. Zipkin, P (2000), “Foundations of Inventory Management,” McGraw-Hill, New York.

57

APPENDIX A Results from Extended Model at s = 4 and s = 5 Table A-1: Extended Model: Orders from the model run at s = 4

x14 = 1552302 x24 = 551432 x34 = 0 x44 = 27223 x54 = 1604132 x64 = 299337 x74 = 696490 x84 = 4054220 x94 = 647292 x10,4 = 1432762 x11,4 = 29869 x12,4 = 1617478 x13,4 = 63030 x14,4 = 26120 x15,4 = 150374

x15 = 1333853 x25 = 668876 x35 = 1713255 x45 = 0 x55 = 0 x65 = 0 x75 = 956038 x85 = 2178081 x95 = 1042484 x10,5 = 0 x11,5 = 0 x12,5 = 0 x13,5 = 0 x14,5 = 0 x15,5 = 0

x16 = 2715650 x26 = 1199389 x36 = 1099040 x46 = 26161 x56 = 730741 x66 = 131005 x76 = 760174 x86 = 3538397 x96 = 820012 x10,6 = 0 x11,6 = 9390 x12,6 = 841357 x13,6 = 54796 x14,6 = 19926 x15,6 = 70609

x17 = 0 x27 = 0 x37 = 0 x47 = 0 x57 = 0 x67 = 0 x77 = 833071 x87 = 0 x97 = 933987 x10,7 = 0 x11,7 = 0 x12,7 = 0 x13,7 = 0 x14,7 = 17339 x15,7 = 54498

Table A-2: Extended Model: Inventory from the model run at s = 4

I+14 = 0 I+24 = 0 I+34 = 0 I+44 = 0 I+54 = 0 I+64 = 0 I+74 = 0 I+84 = 0 I+94 = 0 I+10,4 = 0 I+11,4 = 0 I+12,4 = 0 I+13,4 = 0 I+14,4 = 0 I+15,4 = 0

I+15 = 115883 I+25 = 70634 I+35 = 1852427 I+45 = 1492 I+55 = 308724 I+65 = 0 I+75 = 145948 I+85 = 0 I+95 = 167205 I+10,5 = 0 I+11,5 = 0 I+12,5 = 0 I+13,5 = 44526 I+14,5 = 14574 I+15,5 = 0

I+16 = 182145 I+26 = 0 I+36 = 656664 I+46 = 14958 I+56 = 0 I+66 = 842 I+76 = 0 I+86 = 710938 I+96 = 0 I+10,6 = 965907 I+11,6 = 0 I+12,6 = 0 I+13,6 = 44526 I+14,6 = 20730 I+15,6 = 58140

I+17 = 0 I+27 = 0 I+37 = 0 I+47 = 0 I+57 = 820804 I+67 = 169174 I+77 = 0 I+87 = 878534.3 I+97 = 0 I+10,7 = 662993 I+11,7 = 0 I+12,7 = 0 I+13,7 = 0 I+14,7 = 0 I+15,7 = 0

58

Table A-3: Extended Model: Backorders from the model run at s = 4

I-14 = 725463 I-24 = 365156 I-34 = 1359550 I-44 = 7794 I-54 = 495426 I-64 = 61871 I-74 = 177889 I-84 = 2205262 I-94 = 741056 I-10,4 = 73189 I-11,4 = 9649 I-12,4 = 1301434 I-13,4 = 44936 I-14,4 = 5993 I-15,4 = 13607

I-15 = 0 I-25 = 0 I-35 = 0 I-45 = 0 I-55 = 0 I-65 = 16422 I-75 = 0 I-85 = 259940 I-95 = 0 I-10,5 = 17944 I-11,5 = 10829 I-12,5 = 298186 I-13,5 = 0 I-14,5 = 0 I-15,5 = 2956

I-16 = 0 I-26 = 0 I-36 = 0 I-46 = 0 I-56 = 822 I-66 = 0 I-76 = 0 I-86 = 0 I-96 = 0 I-10,6 = 0 I-11,6 = 7920 I-12,6 = 322171 I-13,6 = 0 I-14,6 = 0 I-15,6 = 0

I-17 = 0 I-27 = 0 I-37 = 578754 I-47 = 0 I-57 = 0 I-67 = 0 I-77 = 0 I-87 = 0 I-97 = 0 I-10,7 = 0 I-11,7 = 0 I-12,7 = 0 I-13,7 = 0 I-14,7 = 0 I-15,7 = 0

Table A-4: Extended Model: Orders from the model run at s = 5

x15 = 2233341 x25 = 1162518 x35 = 2758245 x45 = 10505 x55 = 0 x65 = 11719 x75 = 1298332 x85 = 5480717 x95 = 2191687 x10,5 = 0 x11,5 = 0 x12,5 = 0 x13,5 = 40629 x14,5 = 0 x15,5 = 8701

x16 = 3454395 x26 = 1534503 x36 = 2904049 x46 = 33855 x56 = 2076603 x66 = 181938 x76 = 868260 x86 = 5148054 x96 = 892890 x10,6 = 0 x11,6 = 11694 x12,6 = 1065126 x13,6 = 105049 x14,6 = 4632 x15,6 = 81042

x17 = 0 x27 = 0 x37 = 0 x47 = 0 x57 = 0 x67 = 188157 x77 = 1217657 x87 = 4061858 x97 = 1341806 x10,7 = 139465 x11,7 = 0 x12,7 = 66561 x13,7 = 0 x14,7 = 11615 x15,7 = 65756

x18 = 1573113 x28 = 708422 x38 = 0 x48 = 15467 x58 = 0 x68 = 0 x78 = 961126 x88 = 0 x98 = 1009544 x10,8 = 410287 x11,8 = 0 x12,8 = 0 x13,8 = 0 x14,8 = 15187 x15,8 = 68186

59 Table A-5: Extended Model: Inventory from the model run at s = 5

I+15 = 0 I+25 = 0 I+35 = 1210011 I+45 = 0 I+55 = 716436 I+65 = 26852 I+75 = 52961 I+85 = 0 I+95 = 0 I+10,5 = 56722 I+11,5 = 0 I+12,5 = 0 I+13,5 = 44884 I+14,5 = 25196 I+15,5 = 0

I+16 = 0 I+26 = 0 I+36 = 0 I+46 = 5890 I+56 = 497483 I+66 = 72027 I+76 = 0 I+86 = 0 I+96 = 0 I+10,6 = 1142562 I+11,6 = 197 I+12,6 = 0 I+13,6 = 44884 I+14,6 = 33585 I+15,6 = 56835

I+17 = 0 I+27 = 0 I+37 = 0 I+47 = 0 I+57 = 1124326 I+67 = 190190 I+77 = 0 I+87 = 0 I+97 = 0 I+10,7 = 734881 I+11,7 = 9186 I+12,7 = 373966 I+13,7 = 0 I+14,7 = 22677 I+15,7 = 0

I+18 = 1899072 I+28 = 858543 I+38 = 0 I+48 = 19117 I+58 = 48262 I+68 = 0 I+78 = 0 I+88 = 0 I+98 = 0 I+10,8 = 263915 I+11,8 = 9186 I+12,8 = 373966 I+13,8 = 0 I+14,8 = 0 I+15,8 = 0

Table A-6: Extended Model: Backorders from the model run at s = 5

I-15 = 474063 I-25 = 272917 I-35 = 0 I-45 = 5774 I-55 = 0 I-65 = 0 I-75 = 0 I-85 = 1360310 I-95 = 678895 I-10,5 = 0 I-11,5 = 2712 I-12,5 = 107915 I-13,5 = 0 I-14,5 = 0 I-15,5 = 411

I-16 = 547052 I-26 = 431709 I-36 = 111085 I-46 = 0 I-56 = 0 I-66 = 0 I-76 = 197151 I-86 = 1798731 I-96 = 1110751 I-10,6 = 0 I-11,6 = 0 I-12,6 = 26487 I-13,6 = 0 I-14,6 = 0 I-15,6 = 0

I-17 = 0 I-27 = 0 I-37 = 1467194 I-47 = 0 I-57 = 0 I-67 = 0 I-77 = 0 I-87 = 1989679 I-97 = 0 I-10,7 = 0 I-11,7 = 0 I-12,7 = 0 I-13,7 = 40629 I-14,7 = 0 I-15,7 = 0

I-18 = 0 I-28 = 0 I-38 = 0 I-48 = 0 I-58 = 0 I-68 = 0 I-78 = 0 I-88 = 0 I-98 = 0 I-10,8 = 0 I-11,8 = 0 I-12,8 = 0 I-13,8 = 0 I-14,8 = 0 I-15,8 = 0

60

APPENDIX B GAMS Code (the code is from the Illustrative Example run in Chapter 3) Sets i raw material /1*2/ t periods /1*12/ ; Parameters h(i) inventory holding cost of raw material i in dollars /1 2

0.30 0.14 /

CAP(t) total capacity of raw material in period t /1 2 3 4 5 6 7 8 9 10 11 12

160 160 160 160 160 160 160 160 160 160 160 160 /

e(t) cost of additional storage capacity in period t /1 2 3 4 5 6 7 8

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

61 9 10 11 12

0.5 0.5 0.5 0.5 /

F(t) Fixed ordering cost in period t /1 2 3 4 5 6 7 8 9 10 11 12

3 3 3 3 3 3 3 3 3 3 3 3/

a(i) area occupied by each raw material i (in m2 per measurement unit) /1 2

1 1/

Table c(i,t) unit cost for raw material i in period t

1 2

1 0 0

2 0 0

3 0 0

4 2 1

5 2 1

6 4 1.5

7 4 1.5

8 3 1.7

9 3 1.7

10 0 0

11 0 0

12 0 0

8 90 45

9 100 55

10 0 0

11 0 0

12 0 0

; Table d(i,t) demand estimates for raw material i in period t

1 2

1 0 0

2 0 0

3 0 0

4 100 55

5 90 40

6 110 50

7 120 50

; Table L(i,t) lead time for raw material i in period t 1 1 2 2 3

2 2 3

3 2 3

4 2 3

5 2 3

6 2 3

7 2 3

8 2 3

9 2 3

10 2 3

11 2 3

12 2 3

62 ; Table inix(i,t) initial condition for x to the model for the periods before s

1 2

1 0 55

2 100 40

3 90 50

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

9 0 0

10 0 0

11 0 0

12 0 0

10 0 0

11 0 0

12 0 0

; Table iniINVP(i,t) initial condition for x to the model for the periods before s

1 2

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

9 0 0

; Scalar s period at which the model is run /4/ ; Scalar M very large number /9999/ ;

Variables X(i,t)

Amount of raw material i ordered in period t (will arrive at t+L)

INVP(i,t) Positive inventory of raw material i at the end of period t ACAP(t)

Additional capacity needed in period t

UCAP(t)

Underused capacity in period t

delta(t)

Binary variable indicating if ordering in period t

z

Total inventory cost in dollars

; Positive variable X(i,t),INVP(i,t),ACAP(t),UCAP(t), IP(i,t) ; Binary variable delta(t)

63 ; Equations cost ordering cost

minimize sum of purchasing cost plus inventory holding cost plus ordering cost and fixed

demand(i,t)

ensures that the demand for every product i is met at every period t

capacity(t)

restrict total capacity in period t

ordering(i,t)

makes sure that the orders are tied up with the binary variable

inieqx(i,t)

defines the initial conditions of x to the model for the periods before s

inieqINVP(i,t) defines the initial conditions of INVP to the model for the periods before s ; cost.. z =e= sum((i,t), c(i,t)*X(i,t) + h(i)*INVP(i,t)) + sum(t, e(t)*ACAP(t)) + sum(t, F(t)*delta(t)) ; demand(i,t)$(ord(t) ge s).. INVP(i,t-1) + (X(i,t-L(i,t))) =e= d(i,t) + INVP(i,t) ; capacity(t)$(ord(t) ge s).. sum(i, a(i)*(INVP(i,t-1) + X(i,t-L(i,t)))) + UCAP(t) - ACAP(t) =e= CAP(t) ; ordering(i,t)$(ord(t) ge s).. X(i,t) =l= M*delta(t) ; inieqx(i,t)$(ord(t) lt s).. X(i,t) =e= inix(i,t) ; inieqINVP(i,t)$(ord(t) lt s).. INVP(i,t) =e= iniINVP(i,t) ;

Model transport /all/; solve transport using mip minimizing z ; display X.l, INVP.l, delta.l, UCAP.l, ACAP.l;