ترکیب برنامه ریزی خطی احتمالاتی و تحلیل سلسله مراتبی فازی برای حل مشکل موقعیت چند تاسیسات چندهدفه

The capacitated multi-facility location problem is a complex and imprecise decision-making problem which contains both quantitative and qualitative factors. In the literature, many objectives for optimizing many types of logistics networks are described: (i) minimization objectives such as cost, inventory, transportation time, environmental impact, financial risk and (ii) maximization objectives such as profit, customer satisfaction, and flexibility and robustness. However, only a few papers have considered quantitative and qualitative factors together with imprecise methodologies. Unlike traditional cost-based optimization techniques, the approach proposed here evaluates these factors together while considering various viewpoints. Decision-makers must deal both factors together to model complex structure of real-world applications. In this paper, a two-phase possibilistic linear programming approach and a fuzzy analytical hierarchical process approach have been combined to optimize two objective functions (“minimum cost” and “maximum qualitative factors benefit”) in a four-stage (suppliers, plants, distribution centers, customers) supply chain network in the presence of vagueness. The results and findings of this method are illustrated with a numerical example, and the advantages of this methodology are discussed in the conclusion.

Today’s global market competition and high customer expectations have forced enterprises to consider their supply chains (SC) more carefully. Supply chain decisions are important strategic decisions which affect every member of the chain because the various functions performed by these members are integrated with each other. Among these functions are marketing, distribution, planning, manufacturing, and purchasing. The capacitated multi-facility location and SC network design problem is one of the most comprehensive strategic decision problems that need to be optimized for long-term efficient operation of the whole supply chain. This problem determines the number, location, capacity, and type of the plants, warehouses, and distribution centers to be used. It also establishes distribution channels and the quantities of materials and items to consume, produce, and ship from suppliers to customers [1]. Location-allocation decisions involve substantial capital investment and result in long-term constraints on the production and distribution of goods. These problems are complex and, like most real-world problems, depend on a number of tangible and intangible factors which are unique to each problem. The complexity of these systems arises from a multitude of quantitative and qualitative factors which influence location choices as well as from the intrinsic difficulty of making numerous tradeoffs among those factors [5]. Over and above this complexity, global SC management is difficult because multiple sources of uncertainty and complex interrelationships at various levels between diverse entities exist in the SC, and therefore it is very difficult to determine simultaneously the supply chain configuration and the SC total cost. Fast-changing transportation and facilities costs, facility capacities, and customer demands are some of the SC parameters which are difficult to predict accurately because of imprecision in the environment. Supply chain network (SCN) design problems reviewed in the literature have been examined for situations ranging from a single product type to complex multi-product systems; the models developed range from linear deterministic models to complex nonlinear stochastic ones. The number of objective functions also depends on the degree of complexity of the problem. Generally, these problems involve multiple and conflicting objectives such as cost, service level, and resource utilization. To deal with multiple objectives and to enable the decision-maker to evaluate a greater number of alternative solutions, various numbers of supply chain levels or stages and various solution approaches and methodologies have been used. Supply chain network design levels are determined according to the components of the supply chain network problem being considered. In this research, papers in the literature have been categorized based on the number of SCN levels. The criteria considered in the objective functions and the solution methods and methodologies used in the literature are also reviewed. Vercellis [40] presented a capacitated master production planning and capacity allocation problem for a multi-plant manufacturing system with two serial stages in each plant. The objective of the problem is to minimize the sum of the various cost factors, namely the production cost in stages 1 and 2, inventory, lost demand, transportation, and overtime. The resulting mixed {0, 1} linear programming model is solved by means of LP-based heuristic algorithms. Zhou and Liu [47] proposed a mathematical model and an efficient solution procedure for a bi-criteria allocation problem involving multiple warehouses with different capacities. They also considered two conflicting objectives, transit time and shipping cost, with respect to the warehouse allocation problem. Their proposed solution procedure used a genetic algorithm that is designed to find Pareto optimal solutions for this problem in a short period of time. Romeijn et al. [32] considered a traditional deterministic single-DC multi-retailer (SDMR) model. They tried to minimize the location and transportation costs and the two-level inventory costs. An additional cost term that represents costs related to safety stocks or capacity issues was also proposed. They formulated the problem as a set covering model. Cakravastia et al. [9] aimed to develop an analytical model for the supplier selection process when designing a supply chain network. The assumed objective of the supply chain is to minimize the level of customer dissatisfaction, which is evaluated by two performance criteria: (i) price and (ii) delivery lead time. The overall model operates at two levels of decision-making: the operational level and the chain level. An optimal solution in terms of the models for the two levels can be obtained using a mixed-integer programming technique. Syam [36] extended traditional facility location models by introducing several logistical cost components such as holding, ordering, and transportation costs in a multi-commodity, multi-location framework. Their paper provided an integrated model and sought to minimize total physical distribution costs by simultaneously determining optimal locations, flows, shipment compositions, and shipment cycle times. Two sophisticated heuristic methodologies, based on Lagrangean relaxation and simulated annealing respectively, were provided and compared in an extensive computational experiment. Yan et al. [43] proposed a strategic production–distribution model for supply chain design with consideration of bills of materials (BOM). Logical constraints were used to represent BOM and the associated relationships among the main entities of a supply chain such as suppliers, producers, and distribution centers. Moreover, these relationships were formulated as logical constraints in a mixed integer programming (MIP) model, thus capturing the role of BOM in supplier selection in the strategic design of a supply chain. The total cost of the supply chain included purchasing cost, production cost, transportation and distribution cost, and fixed costs such as the fixed ordering cost, the fixed cost to open and operate a producer, and the fixed cost to open and operate a DC. Chen and Lee [11] proposed a multi-product, multi-stage, and multi-period scheduling model to deal with multiple incommensurable goals for a multi-echelon supply chain network with uncertain market demands and product prices. The supply chain scheduling model is constructed as a mixed integer nonlinear programming problem to satisfy several conflicting objectives, including fair profit distribution among all participants, safe inventory levels, maximum customer service levels, and robustness of decisions to uncertain product demands. For the solution, a two-phase fuzzy decision-making method was presented. Amiri [3] developed a mixed integer programming model and presented a Lagrangean-based solution procedure for the problem. The model minimizes total costs, including the costs to serve the demands of customers from the warehouses, the costs of shipments from the plants to the warehouses, and the costs associated with opening and operating the warehouses and the plants. Yilmaz and Çatay [44] addressed a strategic planning problem for a three-stage production–distribution network. The problem consisted of a single-item, multi-supplier, multi-producer, and multi-distributor production–distribution network with deterministic demand. The objective was to minimize the costs associated with production, transportation, and inventory as well as capacity expansion costs over a given time horizon. The problem was formulated as a 0–1 mixed integer programming model. Efficient relaxation-based heuristics were considered to obtain a good feasible solution. Tsiakis and Papageorgiou [39] proposed a mixed integer linear programming (MILP) model to assist senior operations management in making decisions about production allocation, production capacity per site, purchase of raw materials, and network configuration, while taking into account financial aspects (exchange rates, duties, etc.) and costs. The objective function included fixed infrastructure costs, production costs, material-handling costs at distribution centers, transportation costs, and duties. Pirkul and Jayaraman [30] presented a Lagrangian relaxation of this model and developed a heuristic solution procedure which uses the information provided by this relaxation to generate good feasible solutions. Their model minimized the sum of the costs to distribute products from open warehouses to customers, the costs for transporting units of different commodities from plants to warehouses, and the fixed costs associated with locating and operating manufacturing plants and warehouses. Jayaraman and Pirkul [18] studied an integrated logistics model for locating production and distribution facilities in a multi-echelon environment. The objective function minimized the total cost of the supply chain, including the fixed costs of operating and opening plants and warehouses, the variable costs of production and distribution, and the costs of transportation of raw materials from vendors to plants and of transportation of the finished products from plants to customer outlets through warehouses. A mixed integer programming approach was formulated, and a Lagrangean relaxation scheme was applied to the resulting model. Syarif et al. [37] considered a logistic chain network problem formulated by a 0–1 mixed integer linear programming model. This problem involved the choice of facilities to be opened and the design of the distribution network to satisfy the demand at minimum cost. To solve this problem, a spanning-tree-based genetic algorithm using a Prüfer number representation was used. Results were compared with those from a traditional matrix-based genetic algorithm and from the LINDO professional software package. Braun et al. [8] first described a six-node network and a model predictive control (MPC)-based management policy. The objective function for an MPC controller was constructed using three terms: penalized predicted setpoint tracking error, excess movement of the manipulated variable, and deviation of the manipulated variable from a target value. The optimization problem can be readily solved using standard quadratic programming (QP) algorithms. Melo et al. [27] proposed a mathematical modeling framework that captures many practical aspects of supply chains, such as dynamic planning horizon, generic supply chain network structure, external supply of materials, inventory opportunities for goods, distribution of commodities, facility configuration, availability of capital for investments, and storage limitations. A mixed integer linear programming model (MILP) for the dynamic relocation problem was formulated. Altiparmak et al. [1] considered three objectives: (1) minimization of total cost, including fixed costs of plants and distribution centers and inbound and outbound distribution costs, (2) maximization of customer services that can be rendered to customers in terms of acceptable delivery time, and (3) maximization of capacity utilization balance for distribution centers. They used a new solution procedure based on genetic algorithms to find the set of Pareto-optimal solutions for the multi-objective SCN design problem. In recent years, many people have brought fuzzy theory into facility location and design to deal with the supply-chain network problem. Zhou et al. [48] proposed three types of fuzzy programming models to model capacitated location-allocation problem with fuzzy demands. Kahraman et al. [19] used fuzzy theory for the select a facility location among alternative locations. Bilgen [7] presented a fuzzy model consisting of multiple manufacturers, multiple production lines and multiple distribution centers for application in the consumer goods industry. Liang [26] presented an interactive fuzzy multi-objective linear programming (f-MOLP) model for solving integrated production and transportation problem with multiple fuzzy goals in fuzzy environments. Roghanian et al. [31] considered a “probabilistic bi-level linear multi-objective programming problem” and its application to an enterprise-wide supply-chain planning problem. Sakawa et al. [34] are introduced fuzzy goals into the formulated fuzzy random noncooperative bilevel linear program by taking into account the vagueness of decision makers’ judgements. Selim and Ozkarahan [35] developed an interactive fuzzy goal program for supply-chain distribution network design. Chen and Chang [13] developed an approach for deriving the membership function of the fuzzy minimum total cost in a multi-product, multi-echelon, multi-period supply chain with fuzzy parameters. Ghatee and Hashemi [15] in their work dealt with fuzzy quantities and relations in multi-objective minimum-cost flow problem in a supply-chain network. Torabi and Hassini [38] proposed a new multi-objective possibilistic mixed integer linear programming model (MOPMILP) for integrating procurement, production and distribution planning considering various conflicting objectives simultaneously as well as the imprecise nature of certain critical parameters such as market demands, cost/time coefficients and capacity levels. According to the literature described above, few researchers have considered the inclusion of qualitative factors in multi-objective problems. Although several effective techniques and models have been used to design the best supply chain network and to optimize various objectives, little work has been done on incorporating vagueness and imprecision of information into the capacitated multi-facility location problem. In this paper, an integrated approach using possibilistic linear programming and fuzzy AHP is developed to consider both quantitative and qualitative factors. The major objective of this study is to model the uncertainty problems faced by decision-makers and supply chain managers. A multi-objective linear programming technique is first used to solve the problem. Fuzzy theory is used to deal with transportation costs between the stages of the supply chain, fixed costs of facilities, and expert opinions in AHP. Consequently, possibilistic linear programming (PLP) is proposed for solving the problem because it appears a convenient approach for incorporating the imprecise nature of the real world [28]. Alternatively, bi-level programming can be used to describe model for decision-making situations where a hierarchy exists. The bi-level programming model consists of two sub-models, which is defined as an upper-level problem and the other as a lower-level problem. The choice of dominant level limits or strongly affects the choice of strategy on the lower level. The rest of the paper is organized as follows: in Section 2, the theoretical background of the possibilistic linear programming and fuzzy AHP methodologies is given. In Section 3, the problem assumptions and the mathematical model are defined. In Section 4, the model is developed as a crisp model with the aid of the discussions in Section 2. The proposed method is illustrated with an example in Section 5. The results and findings are also discussed in this section. Finally, Section 6 concludes the study.

The multi-facility location problem is one of the most important strategic decision problems which can affect the future of companies. This importance is increased even more when the supply chain is considered globally. Multi-facility location problems are multi-criteria decision-making problems which contain both imprecise quantitative and qualitative factors. Estimating the customer requirements and expert opinions for facility location problems is not easy due to the scarcity and volatility of data. To cope with ambiguity and vaguness problems, fuzzy set theory has been used in this research. In this paper, interactive integrated “two-phase PLP” and “fuzzy AHP” approaches were used for solving the multi-objective multi-facility location problem. Combining two approaches can effectively handle the imprecision of input data. The auxiliary multiple-objective linear programming model attempts to minimize total SC transportation and facilities costs and maximize the qualitative-factor benefits. The proposed model tries to minimize as much as possible the imprecise total cost, to maximize the possibility of obtaining lower total cost, to minimize the risk of obtaining higher total cost, and to maximize qualitative-factor benefits. It must be noted that this study also used a two-phase approach to MOPLP. The two-phase approach provides some advantages to DMs. First, the degree of satisfaction can be improved with the use of MOPLP. Moreover, various types of interactive solutions achieved by use of this approach could help decision-makers to formulate decisions under variable conditions.