Int J Adv Manuf Technol (2008) 36:401–418 DOI 10.1007/s00170-006-0842-6

ORIGINAL ARTICLE

A supply chain distribution network design model: An interactive fuzzy goal programming-based solution approach Hasan Selim & Irem Ozkarahan

Received: 20 April 2006 / Accepted: 12 October 2006 / Published online: 12 December 2006 # Springer-Verlag London Limited 2006

Abstract A supply chain (SC) distribution network design model is developed in this paper. The goal of the model is to select the optimum numbers, locations and capacity levels of plants and warehouses to deliver products to retailers at the least cost while satisfying desired service level to retailers. A maximal covering approach is used in statement of the service level. The model distinguishes itself from other models in this field in the modeling approach used. Because of somewhat imprecise nature of retailers’ demands and decision makers’ (DM) aspiration levels for the goals, a fuzzy modeling approach is used. Additionally, a novel and generic interactive fuzzy goal programming (IFGP)-based solution approach is proposed to determine the preferred compromise solution. To explore the viability of the proposed model and the solution approach, computational experiments are performed on realistic scale case problems. Keywords Supply chain . Distribution network design . Interactive fuzzy goal programming . Maximal covering Abbreviations DM decision maker FGP fuzzy goal programming FST fuzzy set theory GP goal programming IFGP interactive fuzzy goal programming SC supply chain TCOST total cost INV investment TSERVL total service level H. Selim (*) : I. Ozkarahan Department of Industrial Engineering, Dokuz Eylul University, 35100 Izmir, Turkey e-mail: [email protected] I. Ozkarahan e-mail: [email protected]

1 Introduction The network design problem is one of the most comprehensive strategic decision issues that need to be optimized for the long-term efficient operation of whole supply chain (SC). The decisions made for network design determine the number and locations of raw material suppliers, manufacturing plants, intermediate warehouses and distribution centers as well as select the distribution channel from suppliers to customers and identify the transportation volume among distributed facilities for an extended time horizon. Effective design and management of SC networks assists in production and delivery of a variety of products at low cost, high quality, and short lead times. It is obvious that an effective SC network structure is crucial in gaining a competitive operational performance. To cope with the complexity of the SC network design problem, the network has been divided into several stages in many previous studies. The number of stages is determined based on the tradeoffs between fragmented network problem complexity and the network integrality [1]. For the SC network design problem, Jang et al. [2] decompose the entire network into three sub-networks, inbound network, distribution network and outbound network. A common objective in designing a distribution network is to determine the least cost system design such that the retailers’ demand is satisfied without exceeding the capacities of the warehouses and plants. This usually involves making tradeoffs inherent among the cost components of the system that include cost of opening and operating the plants and warehouses and the inbound and outbound transportation costs. There are two key decisions when designing a distribution network [3]: 1) Will products be delivered to the customer location or picked up from a pre-ordained site? 2) Will products flow through an intermediary? Based on the

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choices for the two decisions, there are various types of distribution network designs that may be used to move products from manufacturing plants to customer. The readers may refer to Chopra and Meindl [3] for the detailed description of each distribution option and the discussion on its strengths and weaknesses. In this paper, we deal with a SC distribution network comprised of a set of manufacturing plants, warehouses and retailers. The network is illustrated in Fig. 1. In the design option considered, inventory is stored locally at retail stores and distributor warehouses. The main advantage of such a network structure is that it can lower the delivery cost and provide a faster response than other networks. The major disadvantage is the increased inventory and facility costs. As emphasized by Chopra and Meindl [3], such a network is best suited for fast-moving items where customers value the rapid response. Cost or profit-based optimization is the most widely used method for SC distribution network design problems. However, more customer oriented approaches are required in order to provide a sustainable competitive advantage in today’s business environment. Nowadays, there is a trend to consider customer service level as more critical. Customer service level can be measured by various measures such as customer response time, consistency of order cycle time, accuracy of order fulfillment rate, delivery lead time, flexibility in order quantity. In this paper, we develop a multiobjective SC distribution network design model. The goal is to select the optimum numbers, locations and capacity levels of plants and warehouses to deliver the products to the retailers at the Fig. 1 Structure of the SC distribution network

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least cost while satisfying the desired service level to the retailers. Since the maximal covering approach [4] has proved to be one of the most useful facility location models from both theoretical and practical points of view, we use it in statement of the service level. In the proposed model, a coverage function which may differ among the retailers according to their service standard request is defined for each retailer. Much of the decision making in the real world takes place in an environment in which the goals, the constraints, and the consequences of the possible actions are not known precisely. As in the same manner, SCs operate in a somehow uncertain environment. Uncertainty may be associated with target values of objectives, external supply and customer demand etc. Supply chain distribution network design models developed so far either ignored uncertainty or consider it approximately through the use of probability concepts. However, when there is lack of evidence available or lack of certainty in evidence, the standard probabilistic reasoning methods are not appropriate. In this case, uncertain parameters can be specified based on the experience and managerial subjective judgment. Fuzzy set theory (FST) [5] provides the appropriate framework to describe and treat uncertainty [6]. In decision sciences, fuzzy sets have had a great impact in preference modeling and multi-criteria evaluation and have helped bringing optimization techniques closer to the users needs. As emphasized previously, SC distribution network design problem is a strategic decision problem, the optimization of which is crucial for the long-term efficient operation of whole SC. The decisions to be made have

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long-lasting effects and are costly, or sometimes impossible to reverse. Because data are often incomplete and imprecise, SC distribution network design decisions generally use forecasts based on aggregated data. As the environment or system changes, such impreciseness also propagates and can exert serious effects on the operation and management of the system. Another source of impreciseness may be due to vagueness in DMs’ interpretation or intent. This situation occurs when exactness is not required or is not possible to specify, or when exactness would unnecessarily limit options. As stated by Reznik and Pham [7], exact or crisp models when used in such cases would not be able to faithfully simulate the richness and subtlety of the real work space. Furthermore, in the worst case, misleading or incorrect outcomes may result. Thus, there is a need to articulate the problems arising from such impreciseness with the view to construct appropriate models for their representation, as well as suitable methods for processing and manipulating them within the environment. Considering this need, in this study, we use FST in handling SC distribution network design problem. More specifically, two kind of impreciseness that may be faced in the problem, i.e., imprecision in retailers’ demand and DMs’ aspiration levels for the goals, are treated. Additionally, to provide the DMs with a flexible and robust multi-objective decision making technique, we propose a novel and generic interactive fuzzy goal programming (IFGP)-based solution approach. Through the solution approach, DMs determine the preferred compromise solution. The paper is further organized as follows: A review of the related literature is presented in the next section. Thereafter, basic concepts and the framework of fuzzy goal programming (FGP), and the proposed IFGP-based solution approach are presented in Sects. 3 and 4, respectively. Section 5 is devoted to the presentation of the crisp formulation of the proposed SC distribution network design model. Computational experiments are presented in Sect. 6. Finally, conclusions are presented in Sect. 7.

2 Literature review Numerous researchers have extensively studied facility and demand allocation problems. Interested readers may refer to detailed survey results, e.g., Francis et al. [8], Aikens [9], Brandeau and Chiu [10], Beamon [11], Avella et al. [12] and Pontrandolfo and Okogbaa [13]. Beamon [11] provides a focused review of literature in the area of multi-stage SC design and analysis and classify the models in the area into four categories: deterministic analytic models, stochastic analytic models, economic models, and simulation models. Avella et al. [12] present their views on the state of the art and the future trends in

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location analysis. Pontrandolfo and Okogbaa [13] review the literature on the configuration as well as the coordination of the network of global facilities problems. They define a framework that systematically addresses the global manufacturing planning problem by identifying and classifying the variables involved therein. The last decades of the twentieth century witnessed a considerable expansion of SCs into international locations. This growth in globalization, and the additional management challenges it brings, has motivated both practitioner and academic interest in global SC management [14]. In this paper, we do not consider the global issues in the proposed model. After mentioning some important reviews on SC network design problem, we present a review of the several relevant papers in the following. In our review, we focus especially on the papers that develop or consider linear deterministic SC network design models. Geoffrion and Graves [15] presents a new method for the solution of the problem addresses the optimal location of distribution centers between plants and customers. They develop an algorithm based on Benders’ decomposition for solving multi-commodity distribution network design problem. Brown et al. [16] present a mixed integer model for a multi-commodity production-distribution system. The objective of the model is to minimize the variable production and shipping cost, fixed cost of equipment assignment and plant operations. They apply to the model a primal decomposition technique similar to Geoffrion and Graves’ [15] algorithm. Cohen and Lee [17] present a strategic model structure and a hierarchical decomposition approach. The scope of their work is to analyze interactions between functions in a complete SC network. To model these interactions they consider four sub modules where each represents a part of the overall SC: (1) material control, (2) production control, (3) finished goods stockpile, and (4) distribution network control. Pirkul and Jayaraman [18] consider a tri-echelon, multicommodity system concerning production, distribution and transportation planning. The authors use a Lagrangean relaxation-based heuristic to provide effective an effective feasible solution. Jayaraman [19] studies the capacitated warehouse location problem that involves locating a given number of warehouses to satisfy customer demands for different products. Pirkul and Jayaraman [1] extend the previous problem by considering locating also a given number of plants. They present a model for multicommodity, multi-plant, capacitated facility location problem, and develop a Lagrangean-based heuristic solution procedure. Dogan and Goetschalckx [20] develop a mixed integer linear programming model for the integrated design of multi-period production-distribution systems. Their paper contributes to the literature by developing an

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integrated design methodology for strategic production and distribution systems using primal decomposition theory. It also provides an acceleration methodology for solving the problem. Tragantalerngsak et al. [21] consider a twoechelon facility location problem in which the facilities in the first echelon are incapacitated and the facilities in the second echelon are capacitated. The goal in their model is to determine the number and locations of facilities in both echelons in order to satisfy customer demand of the product. They develop a Lagrangean relaxation-based branch and bound algorithm to solve the problem. Lee et al. [22] develop a multi-product mixed integer nonlinear programming model to develop a capacity expansion of an integrated production and distribution system. The system comprises the multi-site batch plants and warehouses. Melachrinodis and Min [23] design a multi-objective, multi-period mixed integer programming model that determine the optimal relocation site and phase out schedule of a combined manufacturing and distribution facility from SC perspectives. Their research differentiates from the literature by considering both dynamic aspects and multiechelon network design. Sabri and Beamon [24] develop a SC model that considers simultaneous strategic and operational SC planning. The main contribution of the work is the incorporation of production, delivery, and demand uncertainty into one model. Pirkul and Jayaraman [25] present an integrated logistic model, and develop an efficient solution procedure for multi-commodity production-distribution problem. Tsiakis et al. [26] develop a strategic planning model for SC networks. The paper takes into consideration flexible production facilities in which a number of products are produced making use of shared resources, the economies of scale in transportation, and uncertainty in product demand. Jang et al. [2] propose a supply network with a global bill of material. They model design and planning problems of a supply network in a single combined system. Talluri and Baker [27] propose a multi-phase mathematical programming approach for effective SC network design. Cakravastia et al. [28] develop a mixed integer programming model of the supplier (any manufacturer playing a lower-level supporting role) selection process in designing a SC network. The assumed objective of the SC is to minimize the level of customer dissatisfaction, which is evaluated by two criteria: (i) price and (ii) delivery lead time. More recently, Beamon and Fernandes [29] study a closed-loop SC in which manufacturers produce new products and remanufacture used products. The multiperiod integer programming model uses the present worth method to jointly analyze investment and operational costs. Yeh [30] presents a hybrid heuristic algorithm to solve the multi-stage SC network design problem. The algorithm combines a greedy method, the linear programming

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technique and three local search methods, the pair-wise exchange procedure, the insert procedure and the remove procedure. Eskigun et al. [31] deal with the design of a SC distribution network considering lead time, location of distribution facilities and choice of transportation mode. They present a Lagrangian heuristic that gives good solution quality in reasonable computational time. In a recent paper, Amiri [32] addresses the distribution network design problem in a SC system. The author develops a mixed integer programming model and provides a heuristic solution procedure. The contribution of our current paper to the literature is twofold: First, a fuzzy multi-objective model has been developed for SC distribution network design problem. Second, a novel and generic IFGP-based solution approach is proposed to determine the preferred compromise solution.

3 Fuzzy goal programming Goal programming (GP) is one of the most powerful, multiobjective decision making approaches in practical decision making. In a standard GP formulation, goals are defined precisely. However, application of GP to the real life problems may be faced with two important difficulties. One of which is expressing the DMs’ vague goals mathematically and the second is the need to optimize all goals simultaneously. In such situations, the use of FST comes in handy. Applying FST into goal programming (GP) has the advantage of allowing for the vague aspirations of a DM, which can then be qualified by some natural language terms. The FST in GP was first considered by Narasimhan [33]. Goal programming in fuzzy environment is further developed by Hannan [34], Ignizio [35], Narasimhan and Rubin [36], Tiwari et al. [37, 38] and others. A fuzzy set A can be characterized by a membership function, usually denoted by μ, which assigns to each object of a domain its grade of membership in A. The nearer the value of membership function to unity, the higher the grade of membership of element or object in a fuzzy set A. Various types of membership functions can be used to represent the fuzzy set. A typical FGP problem formulation can be stated as follows: Find

xi ;

i ¼ 1; :::; n

9 > > > > > > =

Zm ðxi Þ
Z m

m ¼ 1; :::; M

Zk ðxi Þ  Z k gj ð x i Þ  bj

k ¼ M þ 1; :::; K > > > > j ¼ 1; :::; J > > ; i ¼ 1; :::; n

xi  0

ð1Þ

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405

where, Zm(xi) is the mth goal constraint, Zk (xi) is the kth goal constraint, Z m ðxi Þ is the target value of the mth goal, Z k ðxi Þ is the target value of the kth goal, gj (xi) is the jth inequality constraint and bj is the available resource of inequality constraint j. In formulation (1), the symbols “≺ and ≻” denote the fuzzified versions of “≤ and ≥” and can be read as approximately less / greater than or equal to. These two types of linguistic terms have different meanings. Under approximately less than or equal to situation, the goal m is allowed to be spread to the right-hand-side of Z m (Z m ¼ lm where lm denote the lower bound for the mth objective) with a certain range of rm(Z m þ rm ¼ um , where um denote the upper bound for the mth objective). Similarly, with approximately
greater than or equal to, pk is the allowed left side of Z k Z k  pk ¼ lk ; and Z k ¼ uk . As can be seen, GP and FGP have some similarities. Both of them need an aspiration level for each objective. These aspiration levels are determined by DMs. In addition to the aspiration levels of the goals, FGP needs max-min limits (uk,lk) for each goal. While the DMs decide the maxmin limits, the linear programming results are starting points and the intervals are covered by these results. Generally, the DMs find estimates of the upper (u) and lower (l) values for each goal using payoff table (see Table 1). Therefore, the feasibility of each fuzzy goal is guaranteed. Here, Zm(X) denotes the mth objective function, and X(m) is the optimal solution of the mth single objective problem. Solving the problem with X(m) (m=1,..., M)
for each objective, a payoff matrix with entries Zpm ¼ Zm X ðpÞ , m, p=1,..., M can be formulated as presented in Table 1. Here, um ¼ max ðZ1m ; Z2m ; . . . ; ZMm Þ and lm =Zmm, m=1,..., M. After constructing fuzzified aspiration levels with respect to the linguistic terms of approximately less than or equal to, and approximately greater than or equal to, the membership functions can be developed for each goal. Using Belman & Zadeh’s [39] min operator approach, one can obtain the feasible fuzzy solution set by the intersection of all membership functions representing the fuzzy goals. This solution set is then characterized by its membership μF(x) which is: mF ð xÞ ¼ mZ1 ð xÞ \ mZ2 ð xÞ:::: \ mZk ð xÞ   ¼ min mZ1 ð xÞ; mZ2 ð xÞ; ::::; mZk ð xÞ

ð2Þ

Then the optimum decision can be determined to be the maximum degree of membership for the fuzzy decision:   max mF ð xÞ ¼ max min mZ1 ð xÞ; mZ2 ð xÞ; :::; mZk ð xÞ x2F

x2F

ð3Þ

By introducing the auxiliary variable λ, which is the overall satisfactory level of compromise, formulation (2)

Table 1 The payoff table

X (1) X (2) .. . X (M)

Z1(X)

Z2(X)

...

ZM(X)

Z11 Z21 ... ZM1

Z12 Z22 ... ZM2

... ... ... ...

Z1M Z2M ... ZMM

can be transformed to the following conventional linear programming problem [40]: 9 maximize 1 > > > > > > subject to > > > = 1  μ Zk k ¼ 1; :::; K ð4Þ gj ðxi Þ  bj i ¼ 1; :::; n; j ¼ 1; :::; J > > > > > > xi  0 i ¼ 1; :::; n > > > ; 1 2 ½0; 1:

4 The proposed IFGP-based solution approach By use of the interactive paradigm, interactive fuzzy multiobjective decision making approaches have been investigated to improve the flexibility and robustness of multiobjective decision making techniques. They provide learning process about the system, whereby the DM can learn to recognize good solutions and relative importance of factors in the system [41]. The main advantage of interactive approaches is that the DM controls the search direction during the solution procedure and, as a result, the efficient solution achieves his/her preferences [42]. Literature in the class of fuzzy interactive programming includes Werners [43, 44], Leung [45], Fabian et al. [46], Sasaki et al. [47] and Baptistella and Ollero [48]. Belman and Zadeh’s [39] min operator focuses only on the maximization of the minimum membership grade. It is not a compensatory operator. That is, goals with a high degree of membership are not traded off against goals with a low degree of membership. Therefore, some computationally efficient compensatory operators (see [41]) can be used in setting the objective function in fuzzy programming to investigate better results. One criterion used to evaluate the performance of compensatory operators in fuzzy optimization is monotonicity. Among the compensatory operators which are well suited in solving multiobjective programming problems, Werners’ [49] ‘fuzzy and’ operator has an advantage of being a strongly monotonically increasing function. That is, it is positively related with the compensation rate. Furthermore, it is easy to handle, and has generated reasonable consistent results in applications. For those reasons, we employ Werners’s ‘fuzzy and’ operator in the proposed IFGP-based solution approach.

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Werners [49] formulates the ‘fuzzy and’ operator as follows: ( ) X μD ð xÞ ¼ Max + min ðμk ð xÞÞ þ ð1  + Þð1=K Þ μk ð xÞ k

k

ð5Þ where K is the total number of fuzzy objectives and constraints, μk(x) is the membership function of fuzzy goal k, and + is the coefficient of compensation defined within the interval [0,1]. By adopting min operator into Eq. (5), the following linear programming problem can be formed: maximize

+ 1 þ ð1  + Þð1=K Þ

subject to

μ k ð xÞ  1 þ 1 k ;

X k

1 ; 1 k ; + 2 ½0; 1:

1k

9 > > > =

8k 2 K; 8x 2 X > > > ; ð6Þ

In real life decision problems, the relative importance of the objectives assigned by the DMs may not be equal, and may change over time. Different from ‘fuzzy and’ operator, the proposed IFGP approach consider the relative importance of the objectives, and consequently provides a more realistic structure. To reflect the relative importance of λks to the objective function, we develop and use a modified version of the ‘fuzzy and’ operator in the proposed solution approach. More specifically, we use the following formulation: P maximize gl þ ð1  g Þ wk lk ð7Þ k

subject to the constraints in formulation (6), where P wk ¼ 1. k In order to determine the weights, there are some good approaches in the literature, such as analytic hierarchy process, weighted least square method and the entropy method etc. Also, there are some fuzzy approaches for finding crisp weights in fuzzy environment. However, determination of the weights is not the focus of this study. We think that the coefficient of compensation (γ) can be treated as the degree of willingness of the SC partners to sacrifice the aspiration levels for their goals to some extent in the short run to provide the loyalty of their partners and/ or to strengthen their competitive position in the long run. We also think that the coefficient of compensation can be determined through a consensus decision making process. In this process, complete unanimity is not the goal - that is rarely possible. However, it is possible for each SC member to have had the opportunity to express their opinion, be listened to, and accept a group decision based on its logic and feasibility considering all relevant factors. This requires the mutual trust and respect of each member.

Fig. 2 Flowchart of the proposed IFGP-based solution approach

We assert that uncertainties of the input data and the DMs’ aspiration levels for the goals in multiobjective linear programming problems can be treated through the proposed IFGP-based solution approach, and consequently, the preferred compromise solution can be determined. Before introducing the proposed solution approach, we think that presentation of some definitions and theoretic explanations related to the topic may be useful for clarity. Definitions of the compromise solution and the preferred compromise solution are presented in the following [42].

mij

LBi Fig. 3 Illustration of linear coverage function

UBi

dtij

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407

Table 2 Formulation of the objective functions

Compromise solution: A feasible vector X* Z S is called a compromise solution of the problem iff X* Z E and Z ðX  Þ  ^X 2S Z ð X Þ where Z(X) is the objective function, S is the feasible region, $ stands for “minimum” and E is the set of efficient solutions. This definition imposes two conditions on the solution for it to be a compromise solution. First, the solution should be efficient. Second, the compromise solution is the closest solution to the ideal one that maximizes the underlying utility function of the DM. In real-world cases, knowledge of the set of efficient solutions E is not always necessary. On the other hand, the DMs’ preferences are to be considered in determining the final compromise solution. Preferred compromise solution: If the compromise solution satisfies the DMs’ preferences, then it is called the preferred compromise solution. The proposed IFGP-based solution approach can be summarized in the following steps: –

Step 1: Develop the conventional (crisp) linear programming formulation of the problem.

Table 3 Expected demand of the retailers Demand for product I (ai l) 992 423 659 806 107 500 741 257 956 129

408 573 873 222 377 439 783 936 317 887

556 388 467 501 777 759 516 327 664 929

831 683 653 669 701 264 507 174 878 609

Demand for product II (ai2) 352 378 649 483 709 449 455 528 484 514

229 898 844 306 845 758 508 916 171 403

282 968 960 789 195 736 198 630 955 186

894 845 935 968 610 976 496 813 424 505

976 885 847 334 776 923 927 320 658 700

880 988 629 971 743 339 811 466 720 724

–

– –

– –

Step 2: Obtain efficient extreme solutions (payoff values) used for constructing the membership functions of the objectives. If the DM selects one of them as a preferred compromise solution go to Step 7. Otherwise go to Step 3. Step 3: Define the membership function of each fuzzy objective using upper and lower bounds of the objectives. Step 4: Considering the membership functions defined in Step 3 and γ (fix the value of γ to 1 in the first iteration) develop the formulation of the problem using proposed ‘modified fuzzy and’ operator. Step 5: Obtain a compromise solution and present the solution to the DM. If the DM accepts it, go to Step 7. Otherwise, go to Step 6. Step 6: Ask the DM if he want to modify the coefficient of compensation (γ), and membership functions of the objectives, and go to Step 3. Actually, definition of a unique rule, e.g., selection of the initial value, change direction or rate of variation in each step, by which the value of γ is varied is difficult since it depends on DMs’ preferences. For instance, if the problem under concern is so sensitive to γ, the rate of variation should be sufficiently slow. When the DM tries to modify membership functions of the objectives and constraints, only the following variations are acceptable [43]: a) the increase of lower bound (lk) for the maximization objectives, b) the decrease of upper bound (um) for the minimization objective, c) the

Table 4 Lower and upper bounds for the distance Retailers

Lower bound (LBi)

Upper bound (UBi)

1–12 13–27 28–50

500 600 700

650 750 850

408

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Table 5 The payoff table

TCOST INV TSERVL

–

Table 7 Compromise solution results with min operator

TCOST

INV

TSERVL

γ λ

2,720,667 3,250,000 3,250,000

2,663,710 2,229,724 2,994,668

52,046 50,000 61,532

1 0.5317 2,968,563 2,583,509 56,131

decrease of maximum tolerance (dj) is an acceptable modification which can guarantee an efficient solution in the recalculated compromise solution step. In order to avoid the possibility of getting into infeasible solution sets because of excess increase of lk or excess decrease of um, we should increase lk and decrease um as few requirements as possible in each iteration. Step 7: Stop. The flow chart of these steps is shown in Fig. 2.

In this section, we present the crisp formulation of the proposed SC distribution network design model. The goal of the model is to select the optimum numbers, locations and capacity levels of plants and warehouses to deliver the products to the retailers at the least cost while satisfying the desired service level to the retailers. Maximal covering approach is used in statement of the service level, and a coverage function which may differ among the retailers according to their service standard request is defined for each retailer. The proposed model distinguishes itself from other models in this field in the modeling approach used. Because of somehow imprecise nature of retailers’ demand and DMs’ aspiration levels for the goals, fuzzy modeling approach is used. Additionally, a novel and generic IFGPbased solution approach is proposed to determine the preferred compromise solution. The mathematical model is developed on the basis of the following assumptions:

–

– –

INV

TSERVL μTCOST μINV

μTSERVL

0.5317 0.5317 0.5317

The retailers have demand for a multitude of products, and the warehouses are responsible for right-time delivery of a right amount of products. Decision makers of the plants, warehouses and retailers share information and collaborate with each other to design an effective distribution network. Decisions are made within a single period.

5.1 Notations and definitions Definitions of sets, parameters and decision variables of the proposed model are presented in the following.

5 Crisp formulation of the proposed model

–

–

TCOST

The network considered encompasses a set of retailers with known locations, and possible discrete set of location zones/sites where warehouses and plants are located. Different capacity levels are available to both the potential plants and warehouses.

Table 6 Lower and upper bounds for the objectives

–

Sets

I J K L R H

set of zones where retailers are located, potential warehouse locations, potential plant locations, set of products, set of capacity levels available for warehouses, set of capacity levels available for plants.

–

Parameters

Tjkl

Cijl

fkh gjr

OPkh OWjr ail sl Wjr

Objectives

Lower bound

Upper bound

TCOST INV TSERVL

2,720,667 2,229,724 50,000

3,250,000 2,994,668 61,532

ql Dkh dtij

variable cost to transport one unit of product l from the plant in zone k to the warehouse in zone j variable cost to transport one unit of product l from the warehouse in zone j to the retailer in zone i fixed portion of the operating cost of the plant in zone k with capacity level h fixed portion of the annual possession and operating costs of the warehouse in zone j with capacity level r opening cost of the plant in zone k with capacity level h opening cost of the warehouse in zone j with capacity level r demand for product l by the retailer in zone i required throughput capacity of a warehouses for product l throughput capacity of the warehouse in zone j with capacity level r required production capacity of a plant for product l capacity of the plant in zone k with capacity level h distance between zone i and zone j

Int J Adv Manuf Technol (2008) 36:401–418 Table 8 Solution results of the model by the proposed solution approach

a

409

γ

λ

TCOST

INV

TSERVL

μTCOST

μINV

μTSERVL

1a 0.9 0.8-0.7 0.6-0.5 0.4-0.3 0.2-0

0.5317 0.5317 0.5280 0.5234 0.4374 0.4231

2,968,563 2,947,188 2,924,110 2,909,366 2,827,727 2,823,297

2,583,509 2,583,509 2,586,257 2,589,729 2,654,768 2,665,540

56,131 56,131 56,229 56,185 57,216 57,310

0.5317 0.5721 0.6157 0.6435 0.7977 0.8061

0.5317 0.5317 0.5280 0.5234 0.4374 0.4231

0.5317 0.5317 0.5402 0.5364 0.6257 0.6339

min operator

mij LBi, UBi

– Yjkl Xijl Zjr Pkh

coverage parameter that denotes the coverage level of the retailer in zone i by the warehouse in zone j the parameters that are used in defining the coverage parameter of the retailer in zone i, LBi and UBi denote lower and upper bound for the distance, respectively (see Fig. 3). Decision variables amount of product l transported to the warehouse in zone j from the plant in zone k, amount of product l transported to the retailer in zone i from the warehouse in zone j, binary variable that indicates whether a warehouse with capacity level r is constructed in zone j, binary variable that indicates whether a plant with capacity level h is constructed in zone k.

5.2 The objective functions

houses, the third one maximizes the total service level provided to the retailers. 5.3 The constraints The constraints of the proposed model and their definitions are presented in the following. X Xijl ¼ ail for all i 2 I and l 2 L; ð11Þ j

Constraint set (11) ensures that all demand from retailers is satisfied by warehouses. XX X sl Xijl  Wjr Zjr for all j 2 J ; ð12Þ i

r

l

Constraint set (12) limits the distribution quantities that are shipped from warehouses to retailers to the throughput limits of warehouses. X Zjr  1 for all j 2 J ; ð13Þ r

Objective functions of the model are formulated as follows: As can be seen from Table 2, the first objective function minimizes total cost made of: the transportation costs of products from plants to warehouses and from warehouses to retailers, and the fixed costs associated with the plants and the warehouses. While the second objective function minimizes the investment in opening plants and wareFig. 4 Solution results by the proposed solution approach

Constraint set (13) ensures that a warehouse can be assigned at most one capacity level. X X Xijl  Yjkl for all j 2 J and l 2 L; ð14Þ i

k

Constraint set (14) guarantees that all demand from retailer in zone i for product l is balanced by the total units

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Table 9 Solution results obtained by the proposed solution approach in terms of utilization of the plants and warehouses γ

1 0.9 0.8-0.7 0.6-0.5 0.4-0.3 0.2-0

Utilization of the plants

Utilization of the warehouses

Min

Max

Average

Min

Max

Average

0.87 0.87 0.89 0.87 0.87 0.88

1 1 1 1 1 1

0.97 0.97 0.97 0.97 0.97 0.96

0.90 0.91 0.82 0.90 0.86 0.75

1.00 1.00 1.00 1.00 1.00 1.00

0.97 0.97 0.97 0.97 0.97 0.93

of product l available at warehouse in zone j that has been supplied from open plants. XX X qkl Yjkl  Dkh Pkh for all k 2 K; ð15Þ j

l

h

Constraints in set (15) represent the capacity restrictions of the plants in terms of their total shipments to the warehouses. X Pkh  1 for all k 2 K; ð16Þ h

Constraint set (16) ensures that a plant can be assigned at most one capacity level. Zjr 2 f0; 1g for all j 2 J ; r 2 R; Pkh 2 f0; 1g for all k 2 K; h 2 H

ð17Þ

Finally, constraint set (17) enforces the binary and nonnegativity restrictions on the decision variables.

6 Computational experiments 6.1 The problem and parameter structuring To explore the viability of the proposed model and the IFGP-based solution approach, computational experiments are presented in this section. The experiments are classified into two categories. Imprecision in the DMs’ aspiration levels for the goals is treated in the first category, while imprecision in both the retailers’ demand and DMs’ aspiration levels for the goals is treated simultaneously in the second category. Solutions of the proposed model are

performed using Werners’ (1988) ‘fuzzy and’ operator and the proposed solution approach, and the results are compared. A hypothetically constructed SC distribution network design problem with 50 retailer zones, 20 potential warehouse sites and 15 potential plant sites is considered in the computational experiments. It is assumed that two different types of product are demanded by the retailers. Coordinates of the retailer zones, potential warehouses and plant sites are generated from a uniform distribution over a square with side 3000. Euclidean distances are used in defining the coverage parameters. CPLEX 9.1 optimization software is used at the solution stage. Before presenting the computational experiments, let us explain the parameter structuring of the hypothetical SC distribution network design problem under concern. Expected demand of the retailers for two different products is drawn from a uniform distribution between 100 and 1000 as given in Table 3. Five capacity levels are used for the capacities available to both the potential plants and warehouses. The opening cost of the warehouse in zone j with capacity level 3 (OWj3) are drawn from a uniform distribution between 90,000 and 120,000. The opening costs of the warehouses for the other capacity levels are computed as follows: OWj1 =0.75*OWj3, OWj2 =0.85*OWj3, OWj4 =1.15*OWj3, OWj5 =1.25*OWj3. Cost coefficients of OPkh are computed in terms of the warehouses costs as OPkh =4*OWkh. Fixed portion of the annual possession and operating costs of the warehouse in zone j with capacity level 3 (gj3) and the plant in zone k with capacity level 3 (fk3) are drawn from a uniform distribution between 18,000 and 25,000 and 75,000 and 100,000, respectively. Fixed portion of the annual possession and operating costs of warehouses and plants for the other capacity levels are computed as follows: gj1 =0.75*gj3, gj2 = 0.85*gj3, gj4 =1.15*gj3, gj5 =1.25*gj3 and fk1 =0.75*fk3, fk2 = 0.85*fk3, fk4 =1.15*fk3, fk5 =1.25*fk3. Required throughput capacity of a warehouse for product l and required production capacity of a plant for product l are given as follows: s1 =1, s2 =1 and q1 =1, q2 =2. The cost coefficients Cijl and Tjkl are computed as being proportional to the Euclidean distance among the locations of warehouses and retailers, and plants and warehouses, respectively.

Table 10 Solution results of the model by Werners’ ‘fuzzy and’ operator γ

λ

TCOST

INV

TSERVL

μTCOST

μINV

μTSERVL

1 0.9 0.8-0.6 0.5 0.4-0.1 0

0.5317 0.5317 0.5280 0.5234 0.4448 0.0000

2,968,563 2,947,407 2,924,274 2,909,366 2,879,877 2,879,877

2,583,509 2,583,509 2,586,257 2,589,729 2,649,159 2,649,159

56,131 56,135 56,233 56,185 58,338 58,338

0.5317 0.5716 0.6154 0.6435 0.6992 0.6992

0.5317 0.5317 0.5280 0.5234 0.4448 0.4448

0.5317 0.5320 0.5405 0.5364 0.7231 0.7231

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411

Fig. 5 Solution results of the model by Werners’ ‘fuzzy and’ operator

Specifically, Cijl and Tjkl are drawn from a uniform distribution between 0.025*dtij and 0.035*dtij and 0.045*dtjk and 0.055*dtjk, respectively. The parameters that are used in defining the coverage parameter of retailer in zone i (LBi, UBi), are given in Table 4. Throughput limit of warehouse in zone j with capacity level r (Wjr) and capacity of the plant in zone k with capacity level h (Dkh) are taken as follows. Wjr =4000, 6000, 8000, 10000, 12000, Dkh =15000, 20000, 30000, 35000, 40000. 6.2 Implementation category I: treatment of fuzzy aspiration levels for the goals In this category of experiments, the imprecision in goal achievement is allowed through the specification of an interval of acceptable achievement rather than a crisp value.

µa

il

6.2.1 Solution by the proposed approach Solution of the SC distribution network design problem by the proposed approach is presented step by step in the following. –

Step 1: The crisp formulation of the problem has been developed using Eqs. (8 to 17) in Sect. 5. – Step 2: Efficient extreme solutions of the problem are presented in Table 5. It is assumed here that the DMs don’t choose any of the efficient extreme solutions as the preferred compromise solution and proceed to Step 3. – Step 3: Considering the efficient extreme solutions given in Table 5, the lower and upper bounds of the objectives can be determined. In our case, the corresponding minimum and maximum values of the efficient extreme solutions are determined as the lower and upper bounds, respectively, as presented in Table 6.

Table 11 The payoff table

0.8 a il

ail

1.2 a il

Fig. 6 Membership function of the retailers’ demand

ail

TCOST INV TSERVL TCOST INV TSERVL

TCOST

INV

TSERVL

2,236,907 8,405,707 21,173,751 2,720,667 3,250,000 3,250,000

2,401,231 1,480,163 8,076,997 2,663,710 2,229,724 2,994,668

41,180 11,072 73,839 52,046 50,000 61,532

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Membership functions of fuzzy objectives can be formulated now using the upper and lower bounds as follows: 8 0 if TCOST > 3; 250; 000 > > > < 3; 250; 000  TCOST if 2; 720; 667 < TCOST  3; 250; 000 μTCOST ¼ > 3; 250; 000  2; 720; 667 > > : 1 if TCOST  2; 720; 667: ð18Þ

μINV ¼

8 0 > > >
2; 994; 668  2; 229; 724 > > : 1

if

INV > 2; 994; 668

if

2; 229; 724 < INV  2; 994; 668

if

INV  2; 229; 724: ð19Þ

μTSERVL ¼

8 1 > < TSERVL  50; 000 > : 61; 532  50; 000 0

if

TSERVL > 61; 532

if

50; 000 < TSERVL  61; 532

if

TSERVL  50; 000: ð20Þ

–

Step 4: Considering the membership functions in Step 3, and using the ‘modified fuzzy and’ operator, mathematical formulation of the problem can be developed as follows: Table 12 Lower and upper bounds for the objectives Objectives

Lower bound

Upper bound

TCOST INV TSERVL

2,236,907 1,480,163 50,000

3,250,000 2,994,668 73,839

9 maximize γλ þ ð1  γ Þ½0:45λ1 þ 0:20λ2 þ 0:35λ3  > > > > subject to > > = μTCOST λ þ λ1 ; μINV λ þ λ2 ; > > > > μTSERVL λ þ λ3 ; > > ; μTCOST ; μINV ; μTSERVL ; λ; λk ðk ¼ 1; 2; 3Þ; γ 2 ½0; 1 ð21Þ and other system constraints (11 to 17). As we stated previously, the relative weights for the membership functions of the objectives can be determined by the DMs using various methods. It is assumed here that, the weights are determined by the DMs as presented in the objective function of the model.

Table 13 Solution results of the model by the proposed solution approach γ

λ

TCOST

INV

TSERVL

μTCOST

μINV

μTSERVL

1 0.9-0.6 0.5 0.4-0.3 0.2 0.1 0

0.3327 0.3327 0.2938 0.2669 0.2390 0.0932 0.0000

2,912,471 2,911,738 2,758,226 2,680,598 2,628,604 2,511,505 2,428,348

2,484,436 2,484,436 2,542,970 2,583,538 2,625,496 2,500,082 2,486,397

57,947 57,931 57,004 56,362 55,697 52,221 50,161

0.3327 0.3339 0.4854 0.5620 0.6134 0.7290 0.8110

0.3327 0.3327 0.2938 0.2669 0.2390 0.3223 0.3314

0.3327 0.3327 0.2938 0.2669 0.2390 0.0932 0.0068

Int J Adv Manuf Technol (2008) 36:401–418

–

Step 5: By fixing the value of γ to 1, the solution given in Table 7 is obtained. It is assumed here that the DMs need more improvement in the results, and want to consider the solution results of the model with different coefficient of compensation (γ) to make the final decision. – Step 6: In this step, a set of solutions corresponding to the different values of γ are obtained and presented to the DMs. The results are given in Table 8. The results presented in Table 8 are illustrated graphically in Fig. 4. Besides the cost, investment and service level, utilizations of the plants and warehouses are important indicators which should be considered in designing a SC distribution network. Therefore, we compare the solution results in terms of utilizations of the plants and warehouses in Table 9. If the results of the model with γ=0.4 (or 0.3) are compared to those of the model with γ=1, it can be realized that a substantial increase (50.03%) can be provided in achievement level of the membership function of total cost objective (μTCOST) with a decrease by 17.74% in that of the second (μINV) objective. Furthermore, achievement level of the total service level objective is also improved by 17.68%. It can also be concluded that all solution results seem reasonable from the utilization rates of the plants and warehouses point of view. The average utilization rates are considerably high in all of the solution alternatives. Considering the results given in Tables 8 and 9 together, let’s suppose that the DMs accept the results of the model with γ=0.4, and that they consider this solution the preferred compromise solution. Then, the procedure is terminated at Step 7. Fig. 7 Solution results of the model with the proposed solution approach

413 Table 14 Solution results of the proposed model in terms of utilization of the plants and warehouses γ

1 0.9-0.6 0.5 0.4-0.3 0.2 0.1 0

Utilization of the plants

Utilization of the warehouses

Min

Max

Average

Min

Max

Average

0.97 0.92 1 0.97 0.95 0.97 0.87

1 1 1 1 1 1 1

0.99 0.97 1 0.99 0.98 0.99 0.97

0.67 0.61 0.79 0.63 0.56 0.67 0.67

1 1 1 1 1 1 1

0.95 0.92 0.97 0.91 0.93 0.95 0.95

6.2.2 Solution by Werners’ ‘fuzzy and’ operator Using Werners’ ‘fuzzy and’ operator we can formulate the SC network design problem under concern as follows: 9 maximize + 1 þ ð1  + Þ½ð1 1 þ 1 2 þ 1 3 Þ=3 > > > > subject to > > = μTCOST  1 þ 1 1 μINV  1 þ 1 2 > > > > μTSERVL  1 þ 1 3 > > ; μTCOST ; μINV ; μTSERVL ; 1 ; 1 k ðk ¼ 1; 2; 3Þ; + 2 ½0; 1 ð22Þ and other system constraints (11 to 17). In the same manner as the previous application, a set of solutions corresponding to the different values of γ are obtained in this application. The results are presented in Table 10 and illustrated graphically in Fig. 5.

414

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Table 15 Solution results of the problem with 'fuzzy and' operator γ

λ

TCOST

INV

TSERVL

μTCOST

μINV

μTSERV

1 0.9-0.6 0.5 0.4 0.3 0.2 0.1 0

0.3327 0.3327 0.3313 0.3308 0.2948 0.2762 0.2443 0.0000

2,912,471 2,911,738 2,914,338 2,914,819 2,763,686 2,706,503 2,654,303 2,500,376

2,484,436 2,484,436 2,470,430 2,467,130 2,535,344 2,569,532 2,573,163 2,430,673

57,947 57,931 57,898 57,887 57,028 56,584 55,823 51,679

0.3327 0.3339 0.3313 0.3308 0.4800 0.5365 0.5880 0.7399

0.3327 0.3327 0.3420 0.3442 0.2989 0.2762 0.2738 0.3684

0.3327 0.3327 0.3313 0.3308 0.2948 0.2762 0.2443 0.0704

Considering the results given in Table 10, let’s suppose that the DMs accept again the results of the model with γ=0.4. 6.3 Implementation category II: treatment of fuzzy demand and fuzzy aspiration levels for the goals We assume here that the retailers’ demand pattern can be affected to some extent using demand management techniques. That is, the retailers’ demand can be altered within a range. Under this assumption, we state retailers’ demand as fuzzy parameters using triangular membership function as illustrated in Fig. 6. The lower and upper bounds for each of the retailer’s demand are assumed 80% and 120% of their corresponding expected demand (ail) values, respectively. 6.3.1 Solution by the proposed approach Efficient extreme solutions of the problem are presented in Table 11. The upper part of the table is constructed by solving the problem considering the individual objective functions subject to fuzzy constraint set while the crisp constraint set is considered in the lower part. As in the same manner of our preceding experiment with the proposed solution approach, it is assumed here that the Fig. 8 Solution results of the problem with Werners’ ‘fuzzy and’ operator

DMs don’t select any of the efficient extreme solutions as the preferred compromise solution and proceed to the next step. Considering the efficient extreme solutions given in Table 11, the lower and upper bounds of the objectives are determined as documented in Table 12. Using the ‘modified fuzzy and’ operator, mathematical formulation of the problem can be developed as follows: max imize γλ þ ð1  γ Þ½0:45λ1 þ 0:20λ2 þ 0:35λ3  subject to μTCOST λ þ λ1 ; μINV λ þ λ2 ; μTSERVL λ þ λ3 ; μDEMANDil λ; μTCOST ; μINV ; μTSERVL ; μDEMANDil ; λ; λk ; γ 2 ½0; 1

9 > > > > > > > > > > > = > > > > > > > > > > > ;

ð23Þ

and other system constraints (11 to 17). A set of solutions for the above problem with different values of γ are obtained and presented in Table 13. The solution results presented in Table 13 are illustrated graphically in Fig. 7. The solution results are also compared in terms of utilization of the plants and warehouses in Table 14.

Int J Adv Manuf Technol (2008) 36:401–418

415

Table 16 Comparison of the solution results

Solution approach Implementation category

Objective function

Proposed approacha

Werners’ 'fuzzy and' operatorb

Category I: Fuzzy aspiration levels for the goals

TCOST INV TSERVL Utilization of plants

2,827,727 2,654,768 57,216 0.87 1 0.97 0.86 1 0.97 2,758,226 2,542,970 57,004 1 1 1 0.79 1 0.97

2,879,877 2,649,159 58,338 0.84 1 0.95 0.81 1 0.97 2,763,686 2,535,344 57,028 0.98 1 1 1 1 1

min max aver. min max aver.

Utilization of warehouses

Category II: Fuzzy demand and fuzzy aspiration levels for the goals

TCOST INV TSERVL Utilization of plants

γ=0.4 and γ=0.5 for the first and second category, respectively. b γ=0.4 and γ=0.3 for the first and second category, respectively.

min max aver. min max aver.

a

Utilization of warehouses

6.3.2 Solution by Werners’ ‘fuzzy and’ operator

Comparing the results given in Table 13 together with consideration of the utilization of the plants and warehouses, let’s suppose that the DMs accept the results of the model with γ=0.5, and that they consider this solution the preferred compromise solution. Then, the procedure is terminated at Step 7. If the results of the model with γ=0.5 are compared to those of the model with γ=1, it can be realized that substantial increase (45.9%) can be provided in achievement level of the total cost objective with a decrease by 11.7% in those of the second and third objective function. In terms of utilization rates of the plants and warehouses, the model with γ=0.5 provides better results compared to the other solutions.

Fig. 9 Comparison of the results in terms of TCOST and INV objectives

Using Werners’ ‘fuzzy and’ operator, we obtain a set of solutions corresponding to different values of γ. The results are presented in Table 15 and illustrated graphically in Fig. 8. 6.4 Comparison of the results The solution results obtained by the proposed solution approach and Werners’ ‘fuzzy and’ operator for the two different implementation categories are compared in Table 16. The results presented in Table 16 are illustrated graphically in Figs. 9, 10 and 11. If the results presented in Table 16 are analyzed, it can be seen that, for the first category of experiments, the

2,900,000 2,850,000 2,800,000 2,750,000 2,700,000 2,650,000 2,600,000 2,550,000 2,500,000 2,450,000 2,400,000

Proposed approach Werners' approach

TCOST

INV

Category I

TCOST

INV

Category II

416

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Fig. 10 Comparison of the results in terms of TSERVL

60,000

TSERVL

58,000 Proposed approach

56,000

Werners' approach 54,000 52,000 50,000 Category I

relative importance levels assigned to the membership functions of the objectives can be reflected to the results to some extent by the proposed solution approach. More specifically, by using the proposed approach with given weighting structure, total cost objective can be improved by 1.81% while total service level objective is deteriorated by 1.92%, and investment objective remains almost the same compared to the results by Werners’ approach. On the other hand, the results obtained by the proposed and Werners’ approaches are almost the same for the second category of experiments. That is, the relative importance of the objectives doesn’t influence on the results of the problem instance. It can be concluded here that, the proposed solution approach may provide different and even more preferable results compared to the Werners’ ‘fuzzy and’ approach. But, it should be emphasized that the weight structure and the structure of the problem, e.g., fuzzy vs. crisp parameters, influences the level of differences between the two approaches. The proposed IFGP-based solution approach can be employed by the DMs as a flexible and robust multi-objective decision making approach to determine the preferred compromise solution. Fig. 11 Comparison of the results in terms of utilization of the plants and warehouses

Category II

7 Conclusions A multi-objective linear programming model is developed in this paper to address the SC distribution network design problem. The goal of the model is to select the optimum numbers, locations and capacity levels of plants and warehouses to deliver the products to the retailers at the least cost while satisfying the desired service level. The model distinguishes itself from other models in this field in the modeling approach used. Decision makers’ imprecise aspiration levels for the goals and retailers’ imprecise demand are incorporated into the model using fuzzy modeling approach, which is otherwise not possible by conventional mathematical programming methods. This paper also contributes to the literature by proposing a novel and generic IFGP-based solution approach which determines the preferred compromise solution for multiobjective decision problems. Results of the computational experiments performed on realistic scale case problems explore the viability of the proposed solution approach and the SC distribution network design model. The results also point out that SC

1 0.9 0.8 0.7 0.6 Proposed approach

0.5

Werners' approach 0.4 0.3 min

aver.

Util.of plants

min

aver.

Util. of warehouses

Category I

min

aver.

Util.of plants

min

aver.

Util. of warehouses

Category II

Int J Adv Manuf Technol (2008) 36:401–418

distribution network design problem can be handled in a more flexible, robust and realistic way through the proposed model and the IFGP-based solution approach. The proposed model can be used in real life industrial application in restructuring, i.e., expanding or narrowing, of an existing SC distribution network besides the design of a new network. Such a strategic model can be a part of a decision support system developed for the collaborative SC management practices.

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