یک مدل برنامه ریزی خطی فازی برای بهینه سازی شبکه های زنجیره تامین چندمرحله ای با توابع مثلثی و عضویت ذوزنقه ای

Supply chain management (SCM) is concerned with a complex business relations network that contains interrelationships between various entities, such as suppliers, manufacturers, distribution centers and customers. SCM integrates these entities and manages their interrelationships through the use of information technology to meet customer expectations (i.e., higher product variety and quality, lower costs and faster responses) effectively along the entire value chain. Thus, one of the vital issues in supply chain management is the design of the value chain network. In this paper, a fuzzy linear programming model for the optimization of the multi-stage supply chain model with triangular and trapezoidal membership functions is presented. The model determines the fuzzy capacities of the facilities (plants or distribution centers (DCs)) and the design of the network configuration with a minimum total cost. The total cost involves the shipping cost from suppliers; transportation costs between plants and DCs; distribution costs between DCs and customer zones; and opportunity costs from not having the material at the right time. The developed model is solved by a professional software package (LINDO), and the computational results are discussed.

In the current age of globalization, enterprises struggle against their competitors and survive by cooperating with their strategic partners. These cooperative or collective efforts evolve a new type of relationship, the supply chain relationship, among these companies, and furthermore, they foster a new concept in management: the supply chain management (SCM) [9], [7], [4], [3], [12], [8], [19] and [6]. In the last several years, many studies have been proposed and much research has been performed on the design and optimization of supply chain networks. In one study, Pirkul and Jayaraman [21] studied a multi-commodity, multi-plant, capacitated facility location problem and proposed an efficient heuristic solution to the problem. In the capacitated plant and warehouse location model, customers typically demand multiple units of different products that are distributed to customer outlets from open warehouses that receive these products from several manufacturing plants. The objective function of the model minimizes the sum of the fixed cost of establishing and operating the plants and the warehouses plus the variable cost of transporting units of products from the plants to the warehouses and distributing the products from the warehouses to the customer, to satisfy the multiple demands of the customers. Timpe and Kallrath [26] considered a multi-site, multi-product production network and presented a general mixed integer linear programming model that combines aspects related to production, distribution and marketing and involves production sites (plants) and sales points. Lakhal et al. [17] proposed a mathematical programming model of an extended enterprise, which can be used to investigate strategic networking. A number of general network modeling constructs are proposed. A model to optimize the supply chain structure under specific assumptions on the nature of the production, cost and value functions in typical production/distribution companies is then derived. A heuristic to obtain solutions from the model is also presented. Finally, an example based on a refrigerator company is used to illustrate the usefulness of the approach. Cakravastia et al. [5] developed an analytical model of the supplier selection process in designing a supply chain network. The constraints on the capacity of each potential supplier are considered in the process. The 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. The operational level concerns decisions related to optimizing the manufacturing and logistical activities of each potential supplier, to meet the customer's requirements. At the chain level, all of the bids from potential suppliers are evaluated, and the final configuration of the supply chain is determined. The structure of the chain depends on the product specifications and on the customer's order size. An optimal solution in terms of the models for the two levels can be obtained using a mixed-integer programming technique [5] and [25] presented a multi-phase mathematical programming approach for effective supply chain design. More specifically, the methodology develops and applies a combination of multi-criteria efficiency models based on game theory concepts and linear and integer programming methods. Korpela et al. [15] proposed a framework with which the risks were related to a customer–supplier relationship; the service requirements by the customers and the strategies of the supplier company can be included in the production capacity allocation and the supply chain design. Essentially, the target is to prepare a sales plan where the limited production capacity is allocated to the customers based on their strategic importance and the risk involved. Furthermore, the supply chain is designed on the basis of the customers' strategic importance and service requirements. The framework is demonstrated with a numerical example, and it is based on integrating the analytic hierarchy process (AHP) and mixed integer programming (MIP). Syarif et al. [24] considered the logistic chain network problem formulated by the 0–1 mixed integer linear programming problem. The design of the problem involves the choice of the facilities (plants and distribution centers) to be opened and the distribution network design, with the goal of satisfying the demand with minimum cost. For the solution method, the spanning tree-based genetic algorithm using Prüfer number representation is proposed. Gnoni et al. [11] studied the lot sizing and scheduling problem (LSSP) of a multi-site manufacturing system with capacity constraints and uncertain multi-product and multi-period demands. LSSP is solved by a hybrid model resulting from the integration of a mixed-integer linear programming model and a simulation model. The hybrid modeling approach is adopted to test local and global production strategies in solving the LSSP concerned. The proposed model is applied to a supply chain of a multi-site manufacturing system of braking equipment for the automotive industry. Yan et al. [29] proposed a strategic production–distribution model for supply chain design with consideration of bills of materials (BOM). Logical constraints are used to represent BOMs and the associated relationships among the main entities of a supply chain, such as suppliers, producers and distribution centers. Chen and Lee [6] studied the multi-objective optimization of multi-echelon supply chain networks with uncertain product demands and prices. A multi-product, multi-stage and multi-period scheduling model is proposed to deal with multiple incommensurable goals for a multi-echelon supply chain network with uncertain market demands and product prices. The uncertain market demands are modeled as a number of discrete scenarios with known probabilities, and the fuzzy sets are used for describing the sellers’ and buyers’ incompatible preferences on product prices. The supply chain scheduling model is constructed as a mixed-integer nonlinear programming problem. Schulz et al. [23] formulated a supply chain model for a petrochemical complex as a multi-period mixed integer nonlinear program (MINLP). The model includes production, product delivery, inventory management and decisions such as individual production levels for each product, as well as operating conditions for each plant in the complex. Two multi-periods MINLP are formulated, to compare models with different levels of rigorousness. Also, although its volume is less than the non-fuzzy papers mentioned above, some papers concerning fuzziness in the design and optimization of supply chain networks were proposed recently. Kumar et al. [16] proposed a fuzzy programming approach for the vendor selection problem (VSP) in a supply chain. VSP has been treated as a “fuzzy multi-objective integer programming VSP” (f-MIP_VSP) formulation that incorporates the three important goals: cost-minimization, quality-maximization and maximization of on-time-delivery, with realistic constraints, such as meeting the buyers’ demands, the vendors’ capacities, and the vendors’ quota flexibilities. In the proposed model, various input parameters were treated as being vague, with a linear membership function of the fuzzy type. The model was tested on a data set that was adopted from a case study at a company. Tsai et al. [27] introduced a channel allocation problem that includes decisions of channel mix and capacity allocation for each distribution channel. They formulate the problem as a fuzzy mixed integer multiple goal programming problem, which includes objective functions such as maximizing net profits, minimizing the rate of end user claims, and minimizing the rate of late lading, and is subject to constraints regarding manufacturing capacity, customer demands, channel capacity, channel quota flexibility and budget limitations. Xu et al. [28] proposed a random fuzzy multi-objective mixed-integer nonlinear programming model that includes two objective functions: the minimization of the total cost and the maximization of the customer services, under the conditions of random fuzzy customer demands and transportation costs between facilities. Liang [18] developed a fuzzy multi-objective linear programming model that includes fuzzy objectives, simultaneously minimizing total costs and total delivery time in relation to inventory levels, available machine capacity and labor levels at each source. The model forecasts demands and available warehouse space at each destination and predicts the total budget. Sanayeia et al. [22] proposed an integrated approach of multi-attribute utility theory (MAUT) and linear programming (LP) for rating and choosing the best suppliers and defining the optimum order quantities among selected ones in order to maximize total additive utility. Javadi et al. [14] developed a fuzzy multi-objective linear programming (FMOLP) model for solving the multi-objective no-wait flow shop scheduling problem in a fuzzy environment. The proposed model attempted to simultaneously minimize the weighted mean completion time and the weighted mean earliness. A numerical example demonstrated the feasibility of applying the proposed model to no-wait flow shop scheduling problem. The proposed model yielded a compromised solution and the decision maker’s overall levels of satisfaction. Mula et al. [20] presented a review of mathematical programming models for supply chain production and transport planning. The purpose of their study was to identify current and future research in this field and to propose a taxonomy framework based on the elements of supply chain structure, decision level, modeling approach, purpose, shared information, limitations, novelty and application. Al-Aomar [1] described a simulation-based approach for developing a lean production system of multi-lean measures. Three lean measures are defined to characterize the leanness of the underlying production system: productivity, cycle time and work-in-process inventory. An optimized setting to certain operational parameters was determined so that a best tradeoff of the three lean measures is reached. The problem formulation resulted in a multi-objective optimization problem with no closed-form definition of problem objective functions and constraints. Ignaciuk and Bartoszewicz [13] considered the problem of designing an efficient supply strategy for logistic systems with perishable goods. In the analyzed systems, the stock at a distribution center was used to fulfill an unknown, time-varying market demand. The stock deteriorated exponentially, and was replenished with delay from a remote supply source. The objective was to specify a supply strategy such that high level of demand satisfaction was obtained despite unknown pattern of demand variations. As opposed to the previous approaches based mainly on heuristics and static optimization, they applied formal design methodology of sliding-mode control and discrete-time dynamical optimization. In this paper, a multi-echelon supply chain model with uncertain facility capacities is addressed with a fuzzy linear programming approach. The model allows decision-makers to reflect their supply policies and also includes the choice of the facilities (plants or distribution centers (DCs)) to be opened and the design of the network configuration with the minimum total cost. The total cost involves the following: the supply cost (a combination of the purchasing cost and the shipping cost from suppliers); the transportation costs between plants and DCs; the distribution cost between DCs and customer zones; the fixed operating costs; and the opportunity cost of not having the material at the right time. This paper is organized into four sections. After the introduction, in which some supply chain models are described, the remainder of the paper is structured as follows. In Section 2, the proposed multi-stage supply chain model is introduced. This model is solved by a professional software package and the results of the implementation are discussed in Section 3. Conclusions are presented in Section 4.

Effective supply chain network design and optimization of the network are tasks that provide a competitive advantage to firms and organizations in today’s highly intractable global business environment. In this study, we address the design and optimization of the strategic production-distribution problem, which consists of choosing the location of fuzzy capacitated plants and distribution centers and determining the optimal physical flow of goods from supply sources to consumption points. The proposed fuzzy model includes the choice of the facilities (the plants or distribution centers to be opened) and the design of the network configuration with a minimum total cost under the fuzzy capacity constraints with triangular and trapezoidal membership functions. The total cost involves the following: the supply cost (a combination of the purchasing cost and shipping cost from the suppliers); the transportation costs between plants and DCs; the distribution cost between DCs and customer zones; and the opportunity cost associated with not having the material at the right time. This study differs from studies in the literature in terms of introducing the opportunity cost (the value of the time for the transportation of materials and products) and the fuzziness concept in the supply chain design problem. Finally, we make a sensitivity analysis to show the correlation between the objective function value and the structure of the membership function forms. Results of sensitivity analysis on hypothetical instances yielded the following significant results: • Trapezoidal membership functions are better than the triangular membership functions with respect to representing the problem more realistically. • While the capacities of plant 1 and DC 1 should be increased an average of 16.43% and 15.69%, the capacities of plant 2 and DC 2 should be decreased an average of 38.95% and 9.98%. In addition, there is no need for any capacity adjustments in DC 3. • Finally, experiments show that increasing the interval width of the trapezoidal membership functions decreases the total cost in the supply chain network. However, this type of problem is regarded inherently as difficult, because the solution requires significant resources regardless of the algorithm used [24]. The time complexity of the supply chain design problems increase in parallel with the size of the actual data in the problem data [2]. Thus, real world applications need improved efficient heuristics for solving these problems algorithmically. Further research should consider the following goals: (i) to study robust fuzzy programming in which both vagueness and ambiguity are considered simultaneously; (ii) to apply other fuzzy mathematical programming-based approaches; (iii) to design an expert system that works according to a decision maker’s aspirations, experiences and business; (iv) to apply a simulation model as an interesting option to integrate the best capacities of the facilities for supply chain planning problems; and (v) to apply fuzzy multi-objective linear programming models due to the multi-objective nature of the problem. Consequently, the design and management of supply chain systems are closely related to the success of a company. This perspective indicates the importance of effective logistics systems and their management for organizational effectiveness and competitiveness. If the managers have fuzzy data to be analyzed, they can produce more flexible scenarios to improve and understand their decision models better. Here, we showed that modeling supply chains using fuzzy techniques provides more helpful data for supply chain managers than crisp models.