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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.11 No.1 pp.1-10

# An Enhanced Two-Phase Fuzzy Programming Model for Multi-Objective Supplier Selection Problem

Dicky Fatrias*, 1Yoshiaki Shimizu
*Department of Electronic and Information Engineering, Toyohashi University of Technology
1Department of Electronic and Information Engineering, Toyohashi University of Technology
Received: November 23, 2011 / Revised: February 12, 2012 / Accepted: February 13, 2012

### Abstract

Supplier selection is an essential task within the purchasing function of supply chain management because it provides companies with opportunities to reduce various costs and realize stable and reliable production. However, many com-panies find it difficult to determine which suppliers should be targeted as each of them has varying strengths and weaknesses in performance which require careful screening by the purchaser. Moreover, information required to as-sess suppliers is not known precisely and typically fuzzy in nature. In this paper, therefore, fuzzy multi-objective line-ar programming (fuzzy MOLP) is presented under fuzzy goals: cost minimization, service level maximization and purchasing risk. To solve the problem, we introduce an enhanced two-phase approach of fuzzy linear programming for the supplier selection. In formulated problem, Analytical Hierarchy Process (AHP) is used to determine the weights of criteria, and Taguchi Loss Function is employed to quantify purchasing risk. Finally, we provide a set of alternative solution which enables decision maker (DM) to select the best compromise solution based on his/her preference. Numerical experiment is provided to demonstrate our approach.

11-1-01.pdf331.8KB

### 1. INTRODUCTION

In global and competitive market, the need for es­tablishing a longer-term relationship that fosters coop­eration among suppliers and their customers has been highlighted. However, many purchasers find it difficult to determine which suppliers should be targeted as each of them has varying strengths and weaknesses in per­formance. Moreover, the importance of each criterion tends to vary from one purchaser to others. This prob­lem becomes more complicated as the simultaneous evaluation is required in terms of qualitative and quanti­tative criteria. So, every decision needs to be integrated by trading off performances of different suppliers at each supply cha
in stage.
One of the main characteristic of supplier selection is that this task is characterized by an imprecision and incomplete of data which results in vagueness of infor­mation related to decision criteria. Stochastic models are usually based on representation of existing uncertainty by probability concepts and are, consequently, limited to tackling the uncertainties captured (Aliev et al., 2007). Moreover, the estimation of probability distribution is difficult to carry out in a fuzzy environment because of the imprecision of the data. This is why, Fuzzy set the­ory (FST) is applied as an appropriate tool to handle this problem effectively.
Li et al. (2005) proposed a two-phase approach to compute efficient solutions of fuzzy programming as an improvement of compromise approach. In their model, they proposed that minimum acceptable achievement level of fuzzy objectives and constraints is set to the solution of max-min operator (Zimmermann approach). However, this method may not necessarily yield a feasi­ble solution when the minimum acceptable achievement level is closer or equal to the most optimistic value (closer to 1). In this research, an enhanced two-phase fuzzy programming for multi-objective supplier selec­tion problem has been developed as decision support tool for DMs in order to select the best compromise so­lution based on his/her preference regarding the assign­ment of order quantity to the selected supplier(s).

### 2. LITERATURE REVIEW

A number of studies have been devoted to examin­ing supplier selection methods. Quantitative techniques have become increasingly applied recently. A compre­hensive review of numerous quantitative techniques used for supplier selection has been done by Weber et al. (1991). They found that linear weighting models, mathematical programming models and statistical/probabilistic appro­aches have been most applied approaches.
Some researches used single objective, such as cost, to evaluate suppliers. Kaslingam and Lee (1996) devel­oped an integer programming model to select suppliers and to determine order quantities with the objective of minimizing total supplying costs which include purchas­ing and transportation costs. Caudhry at al. (1993) used linear and mixed binary integer programming to mini­mize aggregate price considering both all unit and in­cremental quantity discount.
As an extension of single objective techniques, mul­ti-objective mathematical programming has been pro­posed to solve a more complex supplier selection prob­lem. Weber et al. (1998) combined multi-objective pro­gramming (MOP) and Data Envelope Analysis (DEA) to deal with non-cooperative supplier negotiation strate­gies where the selection of one supplier results in an­other being left out of the solution. Dahel (2003) studied multi-objective mixed integer programming to select supplier and allocate product to them in multi-product environment. Xia and Wu (2007) improved AHP using rough set theory and multi-objective mixed integer pro­gramming to determine the best suppliers and optimal quantity allocated to each of them in the case of multiple sourcing, multiple product with multiple criteria. Ko­kangul and Susuz (2009) proposed an integration of ana­lytical hierarchy process (AHP) and non-linear integer MOP to determine the best supplier and optimal order quantity among them that simultaneously maximize total value of purchase and minimize total cost of purchase. Chamodrakaz et al. (2010) provided new approach of two-stage supplier selection problem. At the first stage, an initial screening is performed through the enforce­ment of hard constraint on the selection criteria, and in the second stage, final selection is performed using a modified variant of fuzzy preference programming (FPP). Eroll and Farell (2003) used qualitative and quantitative factor in supplier selection. A fuzzy QFD (Quality func­tion Deployment) is used to translate linguistic input into qualitative data and then combine it with other quan­titative data to develop a multi-objective mathematical programming
model.
This paper focuses on fuzzy multi-objective linear programming (fuzzy MOLP) to deal with supplier selec­tion problem. Kumar et al. (2006) developed a fuzzy multi-objective integer programming approach for ven­dor selection problem subject to constraints including buyer’s demand, vendors’ capacity, and derived an op­timal solution using max-min operator (Zimmermann’s approach). To evaluate the performance of the model, they perform sensitivity analysis on the order allocation and objective function by changing the degree of uncer­tainty in vendor capacity. Amid et al. (2006) solved fuzzy MOLP supplier selection problem by applying weighted additive method to facilitate an asymmetric fuzzy deci­sion making technique. Since they found the perform­ance of such a method is not adequate to support deci­sion making process, α-cut approach is then proposed to improve the resulted achievement level. Later on, Amid et al. (2010) applied weighted max-min approach in sup­plier selection problem and compared the performance of the proposed approach with max-min operator and weighted additive model. They found that the ratio of achievement level of objectives matches the ratio of the objectives weight.
Although there were a number of publications adop­ting fuzzy programming model in supplier selection problem in the literature, most of them rely on the appli­cation of the existing method and very few researches have concerned with the improvement in the methodo­logical process of deriving optimal solution. Kagnicio­glu (2006) proposed super-transitive approximation to determine the weights of objectives and constraint in formulating fuzzy MOLP model in supplier selection and solved the model using max-min operator and weighted additive model. Yucel and Guneri (2010) proposed a new method of weights calculation in fuzzy MOLP sup­plier selection. Both researches mentioned above only focus on the process for weights calculation for fuzzy objective and constraints.
It has been approved that solving fuzzy MOLP us­ing max-min operator may not result in a optimal solu­tion (Tseng and Chen, 1998; Dubois and Fortemps, 1999; Lin, 2004). Such a lack has been resolved by Li et al. (2005) who proposed two-phase approach to compute efficient solutions of fuzzy MOLP problems as the im­provement of compromise approach of Wu et al. (2001). Li et al. (2005) found that the performance of compro­mise approach decreases when the DM prefers to choose the minimum acceptable achievement level closer or equal to the most optimistic value. In their proposed me­thod, minimum acceptable achievement level is set to the solution of max-min operator. In this sense, the per­formance of compromise approach can be improved and, on the other hand, the disadvantage of max-min operator can be overcome. However, the two-phase approach will face the same obstacle if max-min operator outputs the result closer or equal to the most optimistic value, and hence, cannot provide the improvement. To release the above-mentioned shortcomings and to help obtain a more reasonable compromise solution, therefore, this paper proposes an enhanced two-phase approach of fuzzy MOLP by introducing additional variables which con­trol the relaxation of resulted overall achievement level and apply it to solve supplier selection problem.
In the proposed supplier selection model, net cost minimization, service level maximization and purchas­ing risk minimization are incorporated as fuzzy goals. The first two criteria are cited most often in ordering decision (Ghodsypour and O’Brien, 1998). Purchasing risk is included as one objective to measure the risk of potential loss incurred if purchaser allocates a certain amount of product to purchase to a certain supplier. To this end, Taguchi loss function (TLF) is used to quantify this risk. AHP is employed to determine relative impor­tant between fuzzy goals and constraints.
The rest of the paper is organized as follows. The comprehensive description of the proposed model is described in section 3. It includes the theoretic descrip­tions for Taguchi Loss Function, AHP, Fuzzy Multi-objective Linear Programming, the proposed “an en­hanced two-phase approach” to solve fuzzy MOLP. This section is closed with the “solution prodedures” which describes step by step procedures to solve fuzzy MOLP supplier selection problem. Then a numerical experi­ment is presented in Section 4. Finally, the paper is con­cluded in section 5.

### 3. THE PROPOSED INTEGERATED METHOD

This section presents all methods involved in our fuzzy MOLP model. First, Taguchi loss function is de­scribed to quantify the risk associated with purchasing decision, followed by AHP to calculate a relative impor­tance of sub-criteria used to measure risk as well as the relative importance between objectives and constraints in the final formulation. Next, fuzzy MOLP supplier
selection model and an enhanced two-phase approach are presented

#### 3.1 Taguchi Loss Function

In traditional system, the product is accepted if the quality measurement falls within the specification limit.
Otherwise, the product is rejected. The quality losses occur only when the product deviates beyond the speci­fication limits, therefore becoming unacceptable (Pi and Low, 2005). Taguchi suggests a narrower view of qual­ity acceptability by indicating that any deviation from quality’s target value results in a loss. If the quality measurement is the same as the target value, the loss is zero. Otherwise, the loss can be measured using a quad­ratic function (Kathley and Waler, 2002).
There are three types of Taguchi loss functions: “target is best” (two-sided equal specification or two-sided unequal specification), “smaller is better” and “lar­ger is better.” If L(y) is the loss associated with a par­ticular value of quality y, m is the target value of the specification, and k is the loss coefficient whose value is constant depending on the cost at the specification limits and the width of the specification, then for “target is best-two sided equal

#### 3.2 Analytical Hierarchy Process

The analytic hierarchy process (AHP) was devel­oped to provide a simple but theoretically multiple-criteria methodology for evaluating alternatives (Saaty, 1980). The major reasons for applying AHP are because it can handle both qualitative and quantitative criteria and because it can be easily understood and applied by the DMs. AHP involves the principles of decomposition, pair-wise comparisons, and priority vector generation and synthesis.
The procedures of AHP to solve a complex prob­lem involve six essential steps (Lee, 1999): define the unstructured problem and state clearly the objectives and outcomes; decompose the problem into a hierarchi­cal structure with decision elements (e.g., criteria and alternatives); employ pair-wise comparisons among de­cision elements and form comparison matrices; use the eigen value method to estimate the relative weights of the decision elements; check the consistency property of matrices to ensure that the judgments of decision makers are consistent; and aggregate the relative weights of decision elements to obtain an overall rating for the al­ternatives.

#### 3.3 Fuzzy Multi-objective Linear Programming

A linear multi-objective problem can be stated as: find vector x in the transformed form xT=|x , x ,L,
X| which minimize objective function Zk and maxi-mize objective function Zl with

where Xd is the set of feasible solution that satisfy the set of system constraints.
Zimmermann (1978) first adopted fuzzy program¬ming model proposed by Bellman and Zadeh (1970) into conventional LP problems. The fuzzy formulation for eq. (5)-(7) can be stated as

The above fuzzy MOLP is characterized by linear membership function whose value changes between 0 and 1. The membership function (μ)
for fuzzy objectives are given as

and linear membership function for fuzzy constraints is given as

d is subjectively chosen tolerance interval expressing the limit of the violation of the rth inequalities con-straints. In the above formulation, ,

and  mean the maximum value (worst solution) and the mi¬nimum value (best solution) of Zk and Zl respectively.
kl , They are obtained through solving a single objective optimization problem respectively under each objective function (Lai and Hwang, 1994).
Zimmermann (1978) proposed a max-min operator approach to solve the above fuzzy MOLP. The Eq. (5)~
(7) can be transformed into the following crisp formula¬tion by introducing additional variable λwhich repre¬sent an overall achievement level for both fuzzy objec¬tives and constraints.

#### 3.4 Enhanced two-Phase Fuzzy Programming

Li et al. (2005) proposed a two-phase approach to compute efficient solutions of fuzzy MOLP as the im­provement of compromise approach of Wu et al. (2001). The steps of two-phase approach are as follow:

##### Step 1:

Solve the max-min operator problem and output the optimal value, say x0.

##### Step 2:

Set the lower bound λj zj (x0 ) lfor objective function and γr gr (x0 ) for fuzzy constraints and solve the following model to get a final solution x.

It should be noted that the value of minimum ac­ceptable achievement level is a compromise preference value of decision maker. However, this method may not necessarily yield a feasible solution when the minimum acceptable achievement level is closer or equal to the most optimistic value. Moreover, due to the problem structure of supplier selection under consideration, for­mulating linear programming requires a careful parame­ter setting because selection criteria are quantified using wide range of numerical input. Inappropriate parameter setting may also result in infeasible solution. To release the above-mentioned shortcomings and to help obtain a more reasonable compromise solution, therefore, this research proposes an enhanced two-phase approach of fuzzy MOLP. Namely, we propose to solve the follow­ing model to get the final solution x:

where εj and δj are augmented variables to relax the overall achievement level resulted from the foregoing max-min operator problem, respectively, and p is a wei­ghting factor which control the original objective func­tion value and the relaxation value. Apparently, it is desirable such relaxation is as small as possible as long as the feasibility is hold

#### 3.5 Supplier Selection Problem

In this section, we formulate a mathematical model of fuzzy MOLP supplier selection. The following nota¬tions are defined in order to describe the model.
i = index for supplier (i = 1, 2, , N)
D = demand of buye (unit)
B = total budget of buyer to purchase product($) xi= order quantity to supplier i(unit) pi = Iunit price of supplier i ($)
fi= service level of supplier i(% fulfillment)
ri= purchasing risks of supplier i (% risk)
Ci= capacity of supplier i (unit)

The MOLP model for supplier selection is as fol­low:

Eq. (19) minimizes the net cost for ordering prod­uct to satisfy demand. Eq. (20) maximizes the service level of suppliers. Eq. (21) minimizes the purchasing risk when the firm allocates a certain amount of product to purchase to a certain supplier. Eq. (22) puts restriction that order quantity assigned to suppliers must satisfy the total demand. Eq. (23) ensures that the total cost of pur­chasing does not exceed the amount of budget allocated by the firm. Eq. (24) guarantees that the order quantity assigned to each supplier will not exceed supplier capac­ity limit. Eq. (25) is non-negativity constraint.

#### 3.6 Solution Procedures

The proposed fuzzy MOLP supplier selection prob­lem is constructed through the following steps:

##### Step 1:

Define the criteria for supplier selection prob­lem

##### Step 2:

Construct the MOLP supplier selection problem according to defined criteria (minimize purchasing cost, maximize service level, and minimize purchasing risk) and constraint of the buyer and suppliers. The purchas­ing risk is quantified as followed:
a. Define sub-criteria
b. Measure the relative important of sub-criteria using AHP
c. For each sub-criteria, define a target value, calcu¬late loss coefficient and Taguchi loss
d. Find weighted Taguchi loss by employing the out¬put of AHP. This value is used in MOLP model as the coefficient of objective of minimizing purchas¬ing risk

##### Step 3:

Find membership function for each criteria and constraint.
a. Determine a lower bound of each objective by solv¬ing MOLP as a single objective supplier selection problem using each time only one objective.
b. As in a), determine an upper bound of each objec¬tive by solving MOLP as a single objective sup¬plier selection problem using each time only one objective.

##### Step 4:

relative importance of criteria and con­straints using AHP.

##### Step 5:

Reformulate the MOLP supplier selection into equivalent crisp model using the enhanced two-phase fuzzy MOLP and find the set of feasible solution.

### 4. NUMERICAL EXAMPLE

Suppose that one firm should manage three suppli­ers for one product. Management wants to improve the efficiency of the purchasing process by evaluating their suppliers. The management considers three objective functions i.e. minimizing net cost, maximizing service level and minimizing purchasing risk subject to con­straint regarding demand of product, supplier capacity limitation, firm’s budget allocation, etc. The estimated value of suppliers’ net price, service level and suppliers’ capacity are given in Table 1. An allocated budget of the firm to purchase the product is \$20,000. The demand is a fuzzy number and is predicted to be about 1400 unit with refraction of-100 and 150 unit

Purchasing risk is measured from four sub-criteria: quality, order fulfillment, on-time delivery, and distance/ proximity. Concerning product quality, DM sets the target value of defect products at zero and the upper specifica¬tion limit at 3% to indicate the allowable deviation from the target value. Zero loss will occur for 0% defective parts and 100% loss will occur at the specification limit of 3% defective parts. For order fulfillment rate, the loss will be zero for the supplier who fulfills all order quan-tity (100%) and the total loss will occur if supplier can only satisfy 80% of total order. For on-time delivery, the specification limit of delivery is 10 days and 5 days for early and delay shipment, respectively. The DM will tolerate the shipment for maximum 5 days delay and 10 early. In this case, manufacturer will incur 100% loss if shipment is 5 days delayed or 10 days earlier from scheduled shipment, and on contrary, no loss incurred if the shipment is on time. For distance/proximity, a zero-loss will occur at the closest supplier and the specifica¬tion limit is up to 40% of the closest supplier. It means that the manufacturer will incur 100% loss if there is other suppliers in consideration whose distance reaches the specification limit. The specification limit and range value of each sub-criterion are presented in Table 2.
Calculating the value of k from Eq. (1)~(4) gives 1111.11, 0.64, and 6.25 for quality, order fulfillment, distance/proximity, respectively. For on-time delivery, k1 = 4 and k2 = 1 (since an unequal two side specification is considered for on-time delivery, there exists two loses coefficients, k1 and k2 ).
The actual values (Table 3), together with the value of loss sub-criterion k previously calculated for these four sub-criteria, are used to calculate the individual Taguchi Loss for each supplier for each criterion using Eq. (1)~(4). For example, the actual quality value of supplier A is 1.0% defective rate, which means 1.0% deviation from the target value. Individual Taguchi loss is then calculated by entering this value into Eq. (1)~(4). The result is shown in Table 4.
Suppose the pair-wise comparison matrix and local weight for each of these four sub-criteria using AHP is that shown in Table 5. The consistency Ratio (CR) of table 5 is 0.0971 (less than 0.1). The weighted Taguchi loss is then calculated using Taguchi losses and the local weight of sub-criterion. Table 4 shows the weighted Taguchi loss and the normalized Taguchi loss for each supplier. The normalized Taguchi loss is then used as a coefficient of purchasing risk in fuzzy MOLP. Based on suppliers’ data in Table 1 and the normalized Taguchi Loss in Table 4, the fuzzy MOLP supplier selection of the presented problem is constructed according to Eq. (8)-(12) as follow:

The criteria and constraint can be considered equ¬ally important and added together for comparison. How¬ever, such a comparison is generally unfair due to cer¬tain criteria that that may be more important than others. In this model, the weight of cost, service level, purchas¬ing risk and demand are derived from AHP. Table 5 shows the pair-wise comparison matrix and local wei¬ghts for criteria and constraint. The consistency Ratio (CR) is 0.026 (less than 0.1).
Calculating the membership function using max­min operator (Eq. (16)) gives 0.566, 0.566 and 0.566 for μ(x), μ(x), and μ(x), respectively. The crisp for-mulation of the above fuzzy MOLP using the enhanced two-phase approach according to Eq. (18) is given as

In this problem, the original two-phase approach fails to yield the feasible solution. The constraint associ¬ated with λ1cannot be satisfied because the value of λ1equal to 0.526 which is lower than the designated value of its lower bound ( λ1 l= 0.566).
Table 6 provides a set of the feasible solutions re­sulted by utilizing the proposed method which includes the overall achievement level, individual achievement level, ordering plan and the objective value of the equivalent crisp model along with the upper and lower bounds of fuzzy objectives and constraint. As shown in the Table 6, the overall achievement level of the pro­posed approach is known to be better than that of Max­min operator ( λ = 0.566) when value of p is lower than 0.5. When p is equal or greater than 0.5, the overall achievement level decreases. A lower p value indicates the model attempts to find a solution by relaxing more the critical objective related to the corresponding con­straint to achieve a better achievement level of the other objective
In this model, the service level ( Z2 and the pur¬chasing risk ( Z3 are a critical objectives as the corre¬sponding constraints are relaxed for almost any p value (critical constraint). This implies that the model tends to sacrifice the performance of these objectives because it is at less of cost decreasing the performance of these objectives rather than decreasing other. The greatest relaxation is occurred when p is 0.10. The achievement level of Z is totally relaxed ( μ = 0)  to achieve a better
achievement level for Z1followed by Z3. The achieve¬ment level of Z2 reaches the best possible value for the entire value of p when Z3 is relaxed for p is equal to 0.54 and 0.6. Moreover, Z1is free from relaxation as it is the most important objective, whose assigned weight is the highest, according to the DM’s preference ( ω1 >> ω2> ω3).

In this fuzzy formulation, all suppliers are selected to supply product to the firm. Moreover, upon more careful observation, it is revealed that ordering to Sup­plier 1 and Supplier 3 is more preferable. It is indicated from the order quantity assigned to these suppliers as they receive the biggest amount of order quantity which is equal/closer to their full capacity. In this case, it is not profitable to order more quantity to supplier 2 because it offers the most expensive price and the highest purchas­ing risk among others. As mentioned above, the price (net cost) is put as the main concern of the DM (the highest weight). Thus, placing a smaller order quantity to Supplier 2 is the best decision. Without loss of generality, suppose that the DM wants to select p equals 0.10. In this solution, μz1 and μz 3 improve to 0.991 and 0.980, respectively, which results in the best value of Z1 and Z3. However, the DMs should carefully notice that the achievement level of ser­vice level, the second most important criteria, declines toward the worst performance ( μz 2 = 0). Eventually, final decision should be made by the DM to choose the most favorable decision among the feasible alternative solu­tions according to his/her preference.

Table 1. Suppliers’ Quantitative Information.

Table 2. SThe Specification Limit and Range Value of Four Sub-criteria.

Table 3. Actual Value of the Four Sub-criteria.

Table 4. Taguchi Loss.

Table 5. Pair-wise Comparison Matrix

Table 6. Comparison of Max-min Operator and the Proposed Approach.

### 5. CONCLUSION

Supplier selection is an essential task within the purchasing function that needs careful screening under some qualitative and quantitative criteria. Moreover, most information required to assess supplier is usually not known precisely and typically fuzzy in nature over the planning horizon. Concerning such characteristics, this research proposes integrated methodology for FMOLP model for supplier selection. In formulated problem, the most common fuzzy objectives and parameter in practi­cal ordering decision have been presented. AHP is used to avoid the subjective judgment on qualitative/quan­titative criteria and TLF is employed to quantify the purchasing risk. For the purpose of solving the FMOLP problem, the enhanced two-phase fuzzy programming model has been developed. Through numerical experi­ment, we demonstrate the promising advantage of our proposed approach over the max-min operator (Zimmer­mann’s approach). Finally, this integrated approach pro­vides a set of potential feasible solutions which guide DMs to select the best solution according to their prefer­ence. This also refers to a multi-objective optimization problem that should be concerned in future studies.

### APPENDIX

The membership functions for objective functions and demand constraint.

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