Vendor selection and order allocation using an integrated fuzzy mathematical programming model

Article history: Received March 29, 2015 Received in revised format: May 12, 2015 Accepted May 12, 2015 Available online May 18 2015 In the context of supply chain management, supplier selection plays a key role in reaching desirable production planning. In today's competitive world, many enterprises have focused on selecting the appropriate suppliers in an attempt to reduce purchasing costs and improve quality products and services. Supplier selection is a multi-criteria decision problem, which includes different qualitative and quantitative criteria such as purchase cost, on time delivery, quality of service, etc. In this study, a fuzzy multi-objective mathematical programming model is presented to select appropriate supplier and assign desirable order to different supplies. The proposed model was implemented for an organization by considering 16 different scenarios and the results are compared with two other existing methods. Growing Science Ltd. All rights reserved. 5 © 201


Introduction
In the context of supply chain management, supplier selection plays a key role in reaching desirable production planning (Chopra & Meindl, 2007;Chai et al., 2013).In today's competitive world, many enterprises have focused on selecting the appropriate suppliers in an attempt to reduce purchasing costs and improve quality products and services (Dickson, 1966).Supplier selection is a multi-criteria decision problem, which includes different qualitative and quantitative criteria such as purchase cost, on time delivery, quality of service, etc. (Ghodsypour & O'Brien, 2001).Ahi and Searcy (2013) presented a comparative literature analysis of definitions for green and sustainable supply chain management (Werners, 1988).They discussed the concept of green supply chain management (GSCM) and sustainable supply chain management (SSCM).They reported that definitions for GSCM were generally more narrowly concentrated than those for SSCM and put an emphasis on the characteristics of environmental, flow, and coordination focuses.Amin et al. (2011) quantified SWOT (Strengths, Weaknesses, Opportunities and Threats) method in the context of supplier selection.SWOT is one of well-known methods for conducting strategic studies.Besides, they used the fuzzy logic and triangular fuzzy numbers with SWOT analysis to deal with vagueness of human thought.SWOT analysis may take into account both qualitative and quantitative criteria.The managers may understand the position of suppliers in a competitive environment using SWOT matrix.In addition, they used a fuzzy linear programming model to determine how much purchased should be accomplished from each supplier.Amin and Zhang (2012) proposed an integrated model in two phases where, they first proposed a framework for supplier selection criteria and then used a fuzzy method to evaluate suppliers based on qualitative criteria.They proposed a multi objective mixed-integer linear programming model to determine which suppliers and refurbishing sites should be selected, and reported the optimal number of parts and products in closed loop supply chain (CLSC) network.Arikan (2013) presented a multiple sourcing supplier selection problem as a multi objective linear programming problem.They proposed a fuzzy mathematical model and to satisfy the decision maker's aspirations for fuzzy goals.Awasthi et al. (2009) considered a supplier selection problem for a single manufacturer/retailer who encounters a random demand.All the available suppliers may provide various prices and may have restrictions on minimum and maximum order sizes.They determined a low-cost assortment of suppliers which is capable of satisfying the demand.Nazari-Shirkouhi et al. ( 2013) solved a supplier selection problem under multi-price level and multi-product using interactive two-phase fuzzy multi-objective linear programming (FMOLP) model.The model minimized total purchasing and ordering costs, a number of defective units, and late delivered units ordered from suppliers.Ozkok and Tiryaki (2011) proposed a compensatory fuzzy technique to solve multi-objective linear supplier selection problem with multiple-item (MLSSP-MI) by using Werners' "fuzzy and" (μand) operator.Shaw et al. (2012) proposed a method for supplier selection using fuzzy (Zadeh, 1965) analytical hierarchy process (AHP) and fuzzy multi-objective linear programming for developing low carbon supply chain.Yu et al. (2013) presented a mathematical model for optimal selection of retailers for a manufacturing vendor in a vendor managed inventory system.

The proposed study
In this paper, a fuzzy multi-objective mathematical programming model is presented to select supplier and assign order to different supplies.The proposed model is implemented for an organization by considering 16 different scenarios and the results are compared with two other existing methods.• Discount conditions are specified in contracts.

Definitions
• No shortage is permitted.
• It is possible to order different parts from each supplier.
• One part can be purchased from different suppliers.
• Demand, budget and maximum acceptable failure rate are defined in fuzzy.

Mathematical model
The following presents the mathematical model and integer, , , , The first objective function (z1) minimizes total cost of purchased, the second objective function minimizes (z2) the expected failure ratio and the last objective function (z3) minimizes the delivery of all parts.The first constraint (Eq.( 4)) is associated with the budget limitation, the second constraint (Eq.( 5)) is related to capacity of suppliers while Eq. ( 6) specifies the maximum failure ratio.Eq. ( 7) determines demand for each part, Eq. ( 8) determines the minimum amount of purchased from each supplier and finally, Eq. ( 9) determines the type of variables.

Fuzzy approach
The proposed study of this paper uses fuzzy approach to handle uncertainty associated with different parameters.Let w1 to w3 be the weights associated with three objective functions, respectively.In addition, let 1 ω to 3 ω be the weights associated with budget, demand and failure constraints, respectively (Lai & Hwang, 1993;Wang et al., 2009).The proposed study of this paper uses the following fuzzy model to handle uncertainty.
( ) Moreover, the proposed study uses pair-wise comparison to find the relative importance of each weight based on the following mathematical model (Tiwari et al., 1987), max λ (18) subject to Table 1 demonstrates the summary of fuzzy numbers and Table 2 shows the results of numbers assigned for various parameters.

The results
For the proposed study of this paper, we use 16 different scenarios and Table 3 presents the summary of the data used for this study.We have solved the resulted problem under different scenarios and Fig. 1 shows the overall satisfaction under all possible scenarios.

Fig. 1. The overall customer satisfaction under various scenarios
As we can observe from the results of Fig. 1, among different scenarios, six scenarios, 11, 12, 14, 15 and 16, appear to perform better than others although the purchase cost seems to be higher as shown in Fig. 2 as follows, This could be because of lower delivery time and failure, which are summarized in Fig. 3 and Fig. 4 as follows,

Conclusion
In this paper, we have presented a fuzzy mathematical model for supplier selection and order allocation.
The proposed study has considered three objective functions namely total purchase cost, delivery time and failure rate and using fuzzy analytical hierarchy process, the study has assigned weights for various objective functions as well as constraints.The study has implemented an efficient technique to convert the fuzzy mathematical problem into a crisp one and using 16 scenarios, the study analyzed the results.
The preliminary results of this survey have conducted that the model could be practically used for realworld decision making issues.

Table 1
The summary of fuzzy numbers

Table 2
The summary of fuzzy numbers

Table 4
The summary of data for various scenarios calculated using FAHP