Logistics Distribution Centers Location Problem under Fuzzy Environment

Transcription

1 Chapter 79 Logistics Distribution Centers Location Problem under Fuzzy Environment Muhammad Hashim, Liming Yao, Abid Hussain Nadeem, Muhammad Nazim and Muhammad Nazam Abstract Logistics distribution centers location problem is concerned with how to select distribution centers from the potential set for minimizing cost and fulfill the demand. This paper aims at multi-objective optimization for three-echelon supply chain architecture consisting of manufacturer, distribution centers (DCs) and customers. The key design decisions considered are: the number and location of distribution centers, the quantity of products to be shipped from manufacturer to DCs and then from DCs to customers. The present study mainly investigates the proposed problem under fuzzy environment and the uncertain model is converted into a deterministic form by the expected value measure. The approximate best solution of the model is provided using fuzzy simulation. A numerical example is used to illustrate the effectiveness of the proposed model and solution approach. Keywords Distribution center location problem Multi-objective model Expected value measure Fuzzy programming method 79.1 Introduction Manufacturer, customer and supplier are three important members of a supply chain. To some extent, the success of a manufacturer depends on its ability to connect these members seamlessly. In the real logistic system, distribution centers required to connect the manufacturer and their customers to facilitate the movements of goods. The term logistics refers to the science of ensuring the movements of goods in supply chain are carried out efficiently. It involves the location of plants and distribution centers, and selecting the best strategy for the allocation of product from plants to distribution centers and then from distribution centers to customers. The purpose M. Hashim (B) L. Yao A. Nadeem M. Nazim M. Nazam Business School, Sichuan University, Chengdu , P. R. China hashimscu@gmail.com. Xu et al. (eds.), Proceedings of the Seventh International Conference on Management Science and Engineering Management (Volume 2), Lecture Notes in Electrical Engineering 242, DOI: / _79, Ó Springer-Verlag Berlin Heidelberg

2 928 M. Hashim & L. Yao & et al of logistics distribution centers location is to minimize the cost and improve the service quality. Therefore, the distribution centers location is a long-term strategic problem, which not only determines where a new set of facilities should be located to facilitate the customers, but also how much capacities should be allocated to new facilities. The optimal location of distribution centers has been an area of considerable research. Determining good locations for distribution centers in order to meet growing demands has continued to be of significant interest to researchers. A good location planning gives a clear strategic advantage to a firm over its competitors in the market. Consequently, a number of studies have been conducted for the development of mathematical models for distribution centers location planning [1, 2, 11, 13], [15], such as network location models, continuous location models and mixed integer programming models. Perl et al [1] presented a mixed integer programming formulation for warehouse location-routing problem and proposed a heuristic solution method. Aksen et al [2] reported the static conversion from brick-and-mortar retailing to the hybrid click-and-mortar business model from the perspective of distribution logistics. Amiri [3] developed a mixed integer programming model for the distribution network design problem in supply chain system that involves the location of plants and distribution centers, and determining the best strategy for distributing the products from plants to distribution centers and from distribution centers to customers. Syam [4] investigated an integrated model and methodologies for the location problem with logistical components. Chen et al [5] proposed a multiple criteria decision-making method to solve the distribution center location selection problem under fuzzy environment. Yang et al [6] employed chance-constrained programming model for logistics distribution centers location problem under fuzzy environment. Moreno et al [7] have investigated fuzzy location problems on networks with fuzzy values. Klose et al [10] reviewed some of the contributions to the current state of facility location models for distribution systems. Shankar et al [14] presented a multi-objective optimization of single-product for four echelon supply chain architecture consisting of suppliers, production plants, distribution centers and customer zones. Keshteli [21] presented a model for the selection of potential places as distribution centers in order to supply demands of all customers. So, a significant research has been done in this field on fuzzy and random parameters [8, 9]. In traditional supply chain, minimizing cost or maximizing profit is the primitive objective in most of the supply chain network design models [16, 17]. To minimize the cost or maximize the profit is not the only objective in supply chain network, but satisfying customer demand is also equally important. In fact, the success of any manufacturer depends entirely on the quality of its order fulfillment and its level of customer satisfaction. Usually, the customer satisfaction level and the cost control are two key factors to measure the level of a logistics system. However, although distribution location problem has been studied widely in the last decade. Most of the research, addresses deterministic model with single objective. Thus, constructing a multi-objective programming model is necessary for improving the service. In real decision making, the situation is often not deterministic, and some factors

3 79 Logistics Distribution Centers Location Problem 929 such as customer demand and transportation cost are usually changing, therefore distribution center location problem should be consider under fuzzy environment. This paper investigates the logistics distribution centers location problem under fuzzy environment from another point of view, in which transportation cost, setup cost and demand are supposed to be fuzzy variables. The remainder of the paper is organized as follows: Sect explains the research problem and distribution center location problem under uncertain environment, Sect describes mathematical programming model under fuzzy environment and expected value operator to deal with fuzzy parameters, Sect states the solution method for solving the model, Sect presents a numerical example, results and sensitivity analysis, and finally, the concluding remarks are given in Sect Problem Statement Generally, distribution centers location problem involves how to select the location of distribution centers from a potential set and how to transport the products from manufacturer to customers via distribution centers so that the total relevant costs should be minimized and customer service level should be maximized. The choice of locations for distribution centers is an important question in an efficient logistics systems. A well-designed distribution centers location not only reduces the transportation and operational cost but also increases the customer service level and profits. The cost of distribution centers and customer service level provided by the design system are directly effected by the locations, size and number of distribution centers. So there is a need to find a best plan to design the solution of this problem. It is not easy to change the locations of distributions centers, because it is not only directly related to the operating expenses but also has a great influence on working and the control of whole logistics system. It is a complicated problem and determining the best locations for new facilities is thus an important strategic challenge for decision makers [22]. For solving the above problem this paper considers four kinds of logistics costs: (1) Distribution centers setup cost. (2) Transportation cost from manufacturer to distribution centers. (3) Transportation cost from distribution centers to customers. (4) Operational cost of distribution centers. Decision makers have the following two tasks: (i) select the locations of distribution centers from the potential set. (ii) determine the transported quantity of products from manufacturer to distribution centers and also from selected distribution centers to customers. So, manufacturer should supply the material to selected distribution centers and then from distribution centers to customers. For the understanding of this problem see Fig In real life, decision making takes place in an environment where the objectives, constraints or parameters are not known precisely. It is hard to describe the problem parameters as known due to the complexity of social and economic environment

4 930 M. Hashim & L. Yao & et al Manufacturer Potential Distribution Centers Customers x j y jk... j... k Fig Supply chain layout of logistics distribution centers location problem as well as some unpredictable factors such as bad weather and vehicle breakdowns. These challenges increased the importance of stochastic and fuzzy programming for solving the real world problems where data are not known precisely. Many scholars have been presented stochastic distribution location models that are closer to real situations. Most of them have modeled the uncertainty (e.g., demand and transportation cost) by using probability distribution. Although probability theory has been proved to be a useful tool for dealing with uncertainty, some times it may not be suitable due to the lack of historical data. For example, if a decision maker makes the annual plan for the next year, it is not easy for him to obtain the exact values of some parameters indeed, e.g., distribution setup cost, transportation cost and demand. So many parameters like customer demand and transportation costs are usually uncertain rather than deterministic and it is very difficult to determine exact costs due to the fluctuation in the values. In fact, the decision maker cannot obtain perfect information for each parameter in the decision system. As different people have different feeling about the uncertain demand and cost and since there is no clear definition of this change. Thus in such kind of problem, it can be characterized by uncertainty of fuzziness and stochastic models may not be suitable. Some scholars have observed this uncertainty and imprecision and dealt with them using fuzzy theory [18, 19]. However, distribution centers location problem often faced with uncertain environment where fuzziness is exists in a decision making process. Therefore, fuzzy variables that can take into account fuzziness are favored by decision-makers to describe the uncertainty. In this paper, uncertain parameters (i.e demand, transportation cost and setup cost) are represented by fuzzy numbers that can be further characterized by triangular fuzzy numbers. The decision makers can not get the exact information for each uncertain variable. So in this situation, the decision maker can describe the parameters into triangular fuzzy numbers that are more suitable to explain the uncertainty such as demand is about D m but definitely not less than D l and grater than D u (see Fig. 79.2).

5 79 Logistics Distribution Centers Location Problem 931 Fig Triangular fuzzy variable (x) 1 0 D D m D u 79.3 Model Formulation In this paper the problem is formulated as a multi-objectives programming problem with fuzzy coefficients, in which the decision makers have two objectives first minimize the relevant cost and second is to maximize the fill rate (service quality) Notations Suppose that there is one manufacturer, j distribution centers and k customer markets. The task is to transfer products from plant to DCs and then from DCs to customers. Indices Ω : set of potential distribution centers, j is an index, j Ω = {1, 2, 3,, }; Φ : set of customer markets, k is an index, k Φ = {1,2,3,,K}. Parameters d jk : demand of customer markets k at distribution centers j; tr j : unit transportation cost of transported product from manufacture to distribution centers j; tr jk : unit transportation cost of transported product from distribution centers j to customer market k; a j : distribution centers j capacity; C : setup cost of the distribution centers j; t j : operational cost of the distribution centers j. Decision variables X j : this variable denotes the transported quantity from manufacturer to distributer centers j; Y jk : this variable denotes the transported quantity from distribution centers j to customer market k.

6 932 M. Hashim & L. Yao & et al For the proposed problem, there is a need to select the distribution centers from the potential set 1,2,3,,. In this paper binary variable w j is used to denote whether the distribution center j is selected or not, that is, { 1, if distribution center j is open, w j = 0, otherwise Modelling Formulation The mathematical formulations of objectives are as follows. In this model, the first objective function is to minimize the cost, which typically consists of its product transporting, operational and setup costs. minf 1 = E Me [ tr j ].X j + (t j ).X j + K k=1 Second objective function maximizes the fill rate. max F 2 = K k=1 Y jk / k=1 E Me [ tr jk ].Y jk + K E Me [ C j ]w j. (79.1) E Me [ d jk ]. (79.2) Generally speaking, some mandatory conditions must be satisfied when the decision maker makes the decision and these are listed below: The first constraint states that manufacturer supply should be greater than or equal to DCs supply. K k=1 Y jk X j. (79.3) Second constraint states that manufacturer supply should be less than DCs capacity. X j a j w j. (79.4) Third constraint states that DCs should not exceed from the given numbers. w j N. (79.5) Fourth constraint states that fill rate can vary from 80% to 100%. If it exceed the define maximum limit it shows that supply is greater than demand which will be not acceptable by customers. In other case, if it is less than the define limit then it shows that supply is less than the customer need. So it should be within the defined limit.

7 79 Logistics Distribution Centers Location Problem K K jk/ k=1y k=1 E Me [ d k ] 1. (79.6) If a distribution center j is opened then X j and Y jk will be greater than zero and if not then X j and Y jk will be equal to zero. X j,y jk 0, w j = 0or1, j Ω, k Φ. (79.7) From the above discussion, by integration of Equations (79.1) to (79.7), a fuzzy multi-objective programming model can be formulated as follows: minf 1 = E Me [ tr j ].X j + (t j ).X j + / maxf 2 = K K Y jk E Me [ d jk ], s.t. k=1 K k=1 Y jk X j, X j a j w j, k=1 K k=1 w j N, / 0.8 K K Y jk E Me [ d k ] 1, k=1 k=1 X j,y jk 0, w j = 0or1, j Ω, k Φ. E Me [ tr jk ].Y jk + E Me [ C j ]w j, It is very difficult to handle a multi-objective problem when it involves uncertain variables. So it is necessary to convert the fuzzy model into a deterministic one. In this paper expected value model is used to solve the proposed problem. For calculating the expected values of the triangular variables, a new fuzzy measure with an optimistic-pessimistic adjusting index is applied to characterize the problem. The detail information of this fuzzy measure Me can be found in [20]. It is a convex combination of possibility (Pos) and necessity (Nec) and the basic knowledge for measure (Pos) and necessity (Nec) can be found in [26]. Let F= (γ 1,γ 2,γ 3 ) denotes a triangular fuzzy variable. Based on the definition and properties of expected value operator of fuzzy variable using measure Me [20], if the fuzzy variable F= (γ 1,γ 2,γ 3 ), where γ 1,γ 2,γ 3 > 0, then the expected value of F should be: E Me [ F]= 1 λ 2 (γ 1 + γ 2 )+ λ 2 (γ 2 + γ 3 ), λ [0,1]. (79.8)

8 934 M. Hashim & L. Yao & et al 79.4 Solution Approach The fuzzy programming [23, 24] is very useful and efficient tool to overcome problems under uncertainty. A classical and stochastic programming method may cost a lot to obtain the exact coefficient value, while fuzzy programming method does not cost too much [25]. From this fact, fuzzy programming method is very suitable when the coefficients are not known precisely. It offers a powerful mean of handling optimization problems with fuzzy parameters and has been used in different ways in the past. Zimmermann [27] introduced this method for multi-objective programming problems and later has been advanced by Sakawa et al [28]. In this method, the fuzziness in the decision making process is represented by using the fuzzy concept which has been studied widely and many results have been published [29, 30]. This method can be used for both linear and nonlinear multi-objective programming. Under the fuzzy programming method a multi-objective problem is equivalent to: { max { F1 (x),f 2 (x)}, s.t. x X. Considering the imprecise nature of the decision makers judgement for each objective function of problem, the fuzzy goals such as make F 1 (x) and F 2 (x) approximately larger than a certain values are introduced and then problem is converted into: { max {μ1 ( F 1 (x)), μ 2 (F 2 (x))}, s.t. x X. The fuzzy goal is characterized by the following linear membership function: 1, F i (x) > Fi 1, F μ i (F i (x)) = i (x) Fi 0, F F i 1 F0 i 0 F i (x) Fi 1, i 0, F i (x) < Fi 0 and Fi 1 and Fi 0 denote the maximum and minimum values of the objective functions F i (x), which could be determined as follows: F 1 i = max F i(x), Fi 0 = min F i(x), i = 1,2. (79.9) x X x X For each objective function μ i (F i (x)), assume that the decision maker can specify the so-called reference membership function value μ i which reflects the desired membership function value of μ i (F i (x)). The corresponding Pareto optimal solution, which is nearest to the requirements in the minimax sense or better than that if the reference membership function value is attainable, is obtained by solving the following minimax problem: { min max { μi μ i (F i (x))}, i = 1,2, s.t. x X.

9 79 Logistics Distribution Centers Location Problem 935 By introducing auxiliary variable y, the above problem will be equivalent to: min y, μ i μ i (F i (x)) y, i = 1,2, s.t. 0 y 1, x X. It shows that above problem is a convex programming problem and the global optimal solution can be easily obtained from this method Numerical Example In this section, a numerical example is used to show the application of the model. In this paper, we supposed that a decision maker needs to select the 3 distribution centers from 5 potential distribution centers to serve 5 customers. The data is shown in the Tables 79.1, 79.2 and 79.3: Table 79.1 Customer demand Customer (k) Demand ( d jk ) (55,60,65) (95,100,105) (155,160,165) (120,125,130) (60,65,70) Table 79.2 Transportation cost of unit product from distribution centers to customers (Y jk ) k \ j (0.31,0.35,0.39) (0.80,0.84,0.88) (0.19,0.23,0.27) (0.32,0.36,0.40) (0.39,0.43,0.47) 2 (0.42,0.46,0.50) (0.41,0.45,0.49) (0.30,0.34,0.37) (0.18,0.22,0.26) (0.20,0.24,0.28) 3 (0.19,0.23,0.27) (0.30,0.34,0.37) (0.28,0.32,0.36) (0.63,0.67,0.71) (0.19,0.23,0.27) 4 (0.32,0.36,0.40) (0.18,0.22,0.26) (0.42,0.46,0.50) (0.48,0.52,0.57) (0.21,0.25,0.29) 5 (0.39,0.43,0.47) (0.20,0.24,0.28) (0.83,0.87,0.91) (0.59,0.63,0.67) (0.32,0.36,0.40) Table 79.3 Distribution centers operational cost, transportation cost, setup cost and capacity Distribution centers j t j (0.47) (0.59) (0.52) (0.28) (0.50) tr j (0.51,0.56,0.60) (0.63,0.67,0.71) (0.39,0.43,0.47) (0.61,0.65,0.69) (0.94,0.98,0.102) C j (16,19,21) (26,29,31) (12,15,17) (15,17,21) (16,20,21) a j (124) (185) (294) (102) (290)

10 936 M. Hashim & L. Yao & et al The Result of the Numerical Example The optimal solution is obtained on based fuzzy simulation and set optimisticpessimistic index λ = 0.5. The approximate best object values are F 1 = , F 2 = 1, and corresponding optimal solution is W 1 = 1, W 3 = 1, W 4 = 1, X 1 = 124, X 3 = 286, X 4 = 100, Y 13 = 59, Y 31 = 60, Y 42 = 100, Y 15 = 65, Y 33 = 101, Y 34 = 125 and the values of other decision variables are equal to zero. The distribution center 1 serves the customers 3,5; the distribution center 3 serves the customers 1,3,4; the distribution center 4 serves the customer 2. In order to understand this approximate optimal solution by a straightforward way, this paper refer to Fig Manufacturer Distribution centers Customers x j 124 y jk x j Supply from manufacturer to DC y jk Supply from DC to customer Fig Optimal solution layout Sensitivity Analysis The optimal objective is dependent on the value of parameter λ, thus it is meaningful to investigate the sensitivity of the optimal objectives with respect to λ, with the increase in λ values the objectives values are increased and vice versa. Fig shows that the optimal objective will increase with increasing value of λ. Actually, the sensitivity of the result can be tested for other values of parameter λ (see Table 79.4). The decision maker wants to minimize the cost and maximize the customer satisfaction. In real life, it is very difficult for a decision maker to achieve the de-

11 79 Logistics Distribution Centers Location Problem 937 sirable level of objective function due to uncertain parameters. Since there exists operational cost in each distribution center, it is suitable that all quantity in each distribution center could be transported to satisfy the demands of the customers. For minimizing the total cost, it is reasonable to suppose that the total quantity transported from manufacturer is just the total demands of the customers. So, we could see the impact of increase or decrease in supply from demand in Table Fig Sensitivity of optimal objective with different λ Table 79.4 Sensitivity with respect to parameter λ λ Objective value (F 1 ) Table 79.5 Objective function values Actual demand Demand satisfied Total cost (F 1 ) Fill rate (F 2 ) Conclusion This paper mainly investigates a multi-objective programming model with fuzzy coefficients for solving the logistics distribution center location problem. As a result, fuzzy expected value programming is constructed as a decision model to convert the uncertain model into a deterministic one. Fuzzy programming method is proposed to seek the approximate best transportation plan and customers assignment plan for the distribution centers. At last, the effectiveness of the proposed model is tested by a numerical example. In addition, sensitivity of the optimal objective value with respect to the parameter is also discussed in this example. The proposed model and

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