روش برنامه ریزی خطی دو مرحله ای برای مشکلات ارزیابی تامین کننده چند هدفه و تخصیص سفارش

In this study, an integration of the analytic hierarchy process (AHP) and a multi-objective possibilistic linear programming (MOPLP) technique is developed to account for all tangible, intangible, quantitative, and qualitative factors which are used to evaluate and select suppliers and to define the optimum order quantities assigned to each. A multi-objective linear programming technique is first employed to solve the problem. To model the uncertainties encountered in the integrated supplier evaluation and order allocation methodology, fuzzy theory is adopted. Hence, possibilistic linear programming (PLP) is proposed for solving the problem, as it is believed to be the best approach for absorbing the imprecise nature of the real world. In the supplier evaluation phase, environmental criteria are also considered.

Supply chain management (SCM) is often defined as optimization of the network, which comprises a number of systems that are responsible for the procurement, manufacturing, warehousing, and transportation activities. Management of each of these systems involves a series of complex trade-offs between various business function costs. SCM is generally considered as an effective means to help companies reduce costs, improve responsiveness, and increase competitiveness. In order to remain competitive in the global market, the entire supply chain network should be designed as an integrated system, where the focus is on planning and management of the flow of materials from suppliers to end users. Suppliers are highly responsive and flexible in responding to end user orders. Hence, many issues in the supply network are influenced by the proper selection of suppliers. Supplier evaluation and selection affect almost every subsequent decision to be made in the management of supply network. Selecting the right suppliers and quota allocations significantly reduces purchasing costs, improves competitiveness, and enhances end user satisfaction by eliminating waste, improving quality and flexibility to meet the requirements of the end users, and reducing lead-time at different stages of the network [28]. Supplier evaluation and selection is a complex and multiple-criteria decision-making (MCDM) problem that is affected by different tangible and intangible variables including, but not limited to, price, quality, performance, technical capability, delivery. Basically, there are two kinds of supplier selection and sourcing problems: single-sourcing and multiple-sourcing. In the former, one supplier can satisfy all requirements of the buyer, whereas in the latter, more than one supplier is required to satisfy all needs of the buyer. In the first case, management needs to make only one decision – “which supplier is the best” – whereas in the second case, management allocates order quantities among suppliers to create a competitive environment [15]. Order allocation or sourcing decisions incorporate operational metrics such as cost, quality and delivery, and strategic capabilities of suppliers such as quality management practices, management practices, and design development, process and cost reduction capabilities. The problem of how to allocate orders among suppliers properly in multiple supplier environments is more complicated than the supplier evaluation and selection problem in the single-sourcing case, because the activities of order allocation, or sourcing problems, are co-dependent. However, both supplier evaluation and order allocation problems are selection problems [42] and [25]. Supplier evaluation and selection has received considerable attention in the SCM literature. Also, many methods have been developed for solving single- and multiple-sourcing problems since the 1960s. The most commonly used methods for these kinds of problems are mathematical programming models, multiple attribute decision aid methods, cost-based methods, statistical and probabilistic methods and combined methodologies. Mathematical programming approaches have been widely used for modeling selection and allocation problems. They include linear programming [36], [2] and [46], mixed integer programming [5], [37], [12], [34], [7], [11] and [39], goal programming [6], [26] and [8], and multi-objective programming [49], [50], [51], [52] and [53]. Multiple attribute decision aid methods such as linear weighting methods [13], [16], [44] and [45] and the analytic hierarchy process (AHP) [34], [3], [35] and [43] have also been applied to problems. Several studies have been presented, which use cost-based methods and statistical/probabilistic methods. Additionally, there are some combined methods such as a combination of the AHP and linear programming [15], the AHP and mixed integer programming [27], the analytic network process (ANP) and multiple objective programming [14], ANP, TOPSIS and the nominal group technique [41]. Several papers on order allocation or sourcing have been published recently. Some of them have been combined with supplier evaluation and selection problems. The supplier evaluation and selection method can be adopted to model the order allocation problem by considering the dependence between the activities, especially in the form of precedence constraints. Hammami et al. [18] proposed a mathematical model to solve the supplier selection and order allocation problem. They then applied a long-term optimization based on the AHP and weighted point methods. Hazra et al. [19] analyzed capacity allocation among multiple suppliers and presented a basis for allocating a buyer’s requirements among multiple suppliers in order to minimize the cost. Talluri and Narasimhan [42] proposed a methodology for effective supplier sourcing by using data envelopment analysis (DEA) models and non-parametric statistical techniques. Choi and Chang [10] developed a two-phased semantic optimization modeling approach for strategic supplier selection and allocation problems. Demirtas and Ustun [14] proposed an integrated ANP and multi-objective mixed integer programming methodology for choosing the most suitable suppliers and defining optimum quantities among the selected suppliers. The supplier evaluation and selection problem is a complex, multi-criterion decision-making problem which includes both tangible and intangible criteria, some of which may conflict. To handle tangible and intangible criteria, certain methods were suggested by some authors. The most important point is that all methods make some kind of trade-off between tangible and intangible factors to find the best supplier. When considering complex supplier evaluation and order allocation problems together, a multi-stage multi-object programming approach that includes both quantitative and qualitative aspects should be used to find a solution. However, much of the information discovered in this kind of process is not known with certainty. Due to the imprecision and fuzziness of the information related to parameters, deterministic models are not suitable to obtain an effective solution for supplier evaluation and order allocation problems. To overcome the natural difficulties of these problems, fuzzy set theory provides a way to obtain precise answers. Zadeh [54] suggested a fuzzy set theory to describe systems of imprecise nature. Bellman and Zadeh [4] presented a fuzzy programming model for decisions in fuzzy environments. Based on this theory, Zimmermann [58] developed a fuzzy linear programming (FLP) method with single and multiple objectives. Zadeh [56], in his pioneering work, used the fuzzy sets as a basis to derive the theory of possibility. After his initial study, possibility theory has found considerable acceptance. Essentially, fuzzy mathematical programming treats decision-making problems under fuzzy goals and constraints. The fuzzy goals and constraints assure the flexibility of the target values of the objective functions and the elasticity of constraints. Another type of fuzzy mathematical programming treats uncertain coefficients of objective functions and constraints. According to possibility theory [56], linear equations with uncertain coefficients can be treated as a possibilistic application to fuzzy mathematical programming, and the fuzzy coefficients can be regarded as possibility distributions on coefficient values; this is called a possibilistic optimization approach [20] and [38]. This approach is applied to various areas. For example, in one recent study, Zhang et al. [57] proposed lower and upper possibilistic mean–variance models for portfolio selection by utilizing the lower and upper possibilistic means and variances of fuzzy numbers. A number of authors have applied fuzzy single or multi-objective mathematical programming techniques in various areas. But there are few papers published recently about handling imprecise information and uncertainty in supplier evaluation, and specifically, in order allocation models. Li et al. [32] evaluated both qualitative and quantitative criteria for supplier performance using the fuzzy bag method. Kumar et al. [28] developed a fuzzy mixed integer goal programming supplier evaluation and selection model. They also developed a fuzzy multi-objective integer programming model to deal with the same problem [29]. Chen et al. [9] proposed a MCDM model based on fuzzy set theory to deal with supplier evaluation and selection problems. Humpreys et al. [21] presented a methodology for assessing environmental performance in the supplier evaluation and selection process based on fuzzy logic. Amid et al. [1] developed a fuzzy multi-objective model to assign different weights to various criteria. This model also considered the calculation of order quantities assigned to each supplier. According to the above-mentioned literature, there are few researchers who have paid attention to order allocation, and there are only a few studies that integrate the supplier evaluation and order allocation concepts in the same methodology. Although several effective techniques and models have been utilized for evaluating supplier performance, there is very little work in incorporating vagueness and impreciseness of the information into the evaluation and allocation problem from the multi-objective point of the view. In this paper, an integration of the AHP and a multi-objective possibilistic linear programming (MOPLP) technique is developed to consider tangible, intangible, quantitative, and qualitative factors for evaluation of suppliers and to define the optimum order quantities assigned to each supplier. The major objective of this study is to model the uncertainty problems faced by the integrated supplier evaluation and order allocation methodology. A multi-objective linear programming technique is first employed to solve the problem. Fuzzy theory is adopted for dealing with the uncertainties in cost, quality, demand, etc. Consequently, possibilistic linear programming (PLP) is proposed for solving the problem because it is apparently the best approach for absorbing the imprecise nature of the real world. This paper is further organized as follows. Section 2 explains the theoretical background of the MOPLP methodology. In Section 3, the MOPLP-order allocation model to cope with the supplier evaluation and order allocation problem is defined. This section also introduces the basic problem descriptions and general assumptions of the model. In Section 4, the PLP model is defined with auxiliary crisp objective functions and crisp constraints. Theoretical information is given on how to solve and improve the multi-objective linear programming (MOLP) model using a two-phase approach in the second part of Section 4. The proposed method is illustrated with an example in Section 5. Finally, results are discussed and some conclusions are pointed out in Sections 6 and 7, respectively.

Supplier evaluation decisions are strategic and extremely critical to the success of companies due to their direct impacts on system effectiveness. As a decision-making problem, the supplier evaluation has multiple objectives that conflict with each other. In addition, depending on the increasing level of environmental problems, companies should also consider environmental criteria in their evaluation processes. Considering this reality environmental criteria added SCOR level 1 performance metrics are used as evaluation criteria in this study. This process is unique in the literature of supplier evaluation–order allocation problems. Additionally, in real-life problems, some data cannot be obtained precisely. In this case, data for the price, the rate of rejected units, and aggregate demand of the company are imprecise. Traditional models are insufficient to express the vagueness of this related data. However, a multi-objective possibilistic linear programming (MOPLP) model provides the ability to express imprecise data in a logical way. It must be noted that this study also utilized a two-phased approach for MOPLP. The two-phased approach provides some advantages to the DMs. First, the satisfaction degree can be improved with the use of MOPLP. Secondly, various types of membership functions, such as hyperbolic and piece-wise linear functions, can be used. MOPLP provides more computational efficiency and flexibility in an uncertain environment. Consequently, MOPLP with two-phases is suitable for a supplier evaluation and order allocation problem including uncertainties.