An Enhanced TwoPhase Fuzzy Programming Model for MultiObjective Supplier Selection Problem
 Author: Fatrias Dicky, Shimizu Yoshiaki
 Organization: Fatrias Dicky; Shimizu Yoshiaki
 Publish: Industrial Engineering and Management Systems Volume 11, Issue1, p1~10, 01 March 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 companies 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 assess suppliers is not known precisely and typically fuzzy in nature. In this paper, therefore, fuzzy multiobjective linear programming (fuzzy MOLP) is presented under fuzzy goals: cost minimization, service level maximization and purchasing risk. To solve the problem, we introduce an enhanced twophase 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.

KEYWORD
Supplier Selection , TwoPhase Fuzzy Programming , Supply Chain

1. INTRODUCTION
In global and competitive market, the need for establishing a longerterm 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 chain 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 twophase 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 maxmin 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 twophase fuzzy programming for multiobjective 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 multiobjective programming (MOP) and Data Envelope Analysis (DEA) to deal with noncooperative supplier negotiation strategies where the selection of one supplier results in another being left out of the solution. Dahel (2003) studied multiobjective mixed integer programming to select supplier and allocate product to them in multiproduct environment. Xia and Wu (2007) improved AHP using rough set theory and multiobjective 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 nonlinear 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. Chamodrakazet al. (2010) provided new approach of twostage 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 multiobjective mathematical programming model.This paper focuses on fuzzy multiobjective linear programming (fuzzy MOLP) to deal with supplier selection problem. Kumar
et al . (2006) developed a fuzzy multiobjective integer programming approach for vendor selection problem subject to constraints including buyer’s demand, vendors’ capacity, and derived an optimal solution using maxmin 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. Amidet 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, αcut approach is then proposed to improve the resulted achievement level. Later on, Amidet al. (2010) applied weighted maxmin approach in supplier selection problem and compared the performance of the proposed approach with maxmin 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 supertransitive approximation to determine the weights of objectives and constraint in formulating fuzzy MOLP model in supplier selection and solved the model using maxmin 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 maxmin 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 twophase approach to compute efficient solutions of fuzzy MOLP problems as the improvement of compromise approach of Wuet al. (2001). Liet 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 maxmin operator. In this sense, the performance of compromise approach can be improved and, on the other hand, the disadvantage of maxmin operator can be overcome. However, the twophase approach will face the same obstacle if maxmin operator outputs the result closer or equal to the most optimistic value, and hence, cannot provide the improvement. To release the abovementioned shortcomings and to help obtain a more reasonable compromise solution, therefore, this paper proposes an enhanced twophase 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.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 Multiobjective Linear Programming, the proposed “an enhanced twophase 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 subcriteria 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 twophase 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” (twosided
equal specification or twosidedunequal specification), “smaller is better” and “larger is better.” IfL (y ) is the loss associated with a particular value of qualityy ,m is the target value of the specification, andk 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 besttwo sided equal specification” type, “target is besttwo sided unequal specification” type, “smaller is better” type, and “l arger is better” type, the formulation ofL (y ) are given is eq.(1)(4), respectively.3.2 Analytical Hierarchy Process
The analytic hierarchy process (AHP) was developed to provide a simple but theoretically multiplecriteria 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, pairwise 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 pairwise 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 Multiobjective Linear Programming
A linear multiobjective problem can be stated as: find vector
x in the transformed formx ^{T} = x _{1},x _{2},…,X_{n}  which minimize objective functionZ _{k} and maximize objective functionZ _{l} withsubject to:
where
X_{d} is the set of feasible solution that satisfy the set of system constraints.Zimmermann (1978) first adopted fuzzy programming model proposed by Bellman and Zadeh (1970) into conventional LP problems. The fuzzy formulation for eq. (5)(7) can be stated as
subject to:
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 asand linear membership function for fuzzy constraints is given as
d_{r} is subjectively chosen tolerance interval expressing the limit of the violation of the rth inequalities constraints. In the above formulation,Z _{k} ^{max},Z _{l} ^{max},Z _{k} ^{min} andZ_{l} ^{min} mean the maximum value (worst solution) and the minimum value (best solution) ofZ _{k} and ,Z _{l} respectively. They are obtained through solving a single objective optimization problem respectively under each objective function (Lai and Hwang, 1994).Zimmermann (1978) proposed a maxmin operator approach to solve the above fuzzy MOLP. The Eq. (5)~(7) can be transformed into the following crisp formulation by introducing additional variable
λ which represent an overall achievement level for both fuzzy objectives and constraints.3.4 Enhanced twoPhase Fuzzy Programming
Li
et al. (2005) proposed a twophase approach to compute efficient solutions of fuzzy MOLP as the improvement of compromise approach of Wuet al. (2001). The steps of twophase approach are as follow:Step 1: Solve the maxmin operator problem and output the optimal value, sayx ^{0}.Step 2: Set the lower boundfor objective function and
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 abovementioned shortcomings and to help obtain a more reasonable compromise solution, therefore, this research proposes an enhanced twophase approach of fuzzy MOLP. Namely, we propose to solve the following model to get the final solution
x :where
ε_{j} andδ_{j} are augmented variables to relax the overall achievement level resulted from the foregoing maxmin operator problem, respectively, andp 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 notations are defined in order to describe the model.
i = index for supplier (i = 1, 2, …, N)
D = demand of buyer (unit)
B = total budget of buyer to purchase product ($)
xi = order quantity to supplier i (unit)
pi = unit price of supplier i ($)
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:
subject to:
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 nonnegativity 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 problemStep 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 subcriteria
b. Measure the relative important of subcriteria using AHP
c. For each subcriteria, define a target value, calculate loss coefficient and Taguchi loss
d. Find weighted Taguchi loss by employing the output of AHP. This value is used in MOLP model as the coefficient of objective of minimizing purchasing risk
Step 3: Find membership function for each criteria and constraint.a. Determine a lower bound of each objective by solving MOLP as a single objective supplier selection problem using each time only one objective.
b. As in a), determine an upper bound of each objective by solving MOLP as a single objective supplier selection problem using each time only one objective.
Step 4: Calculate relative importance of criteria and constraints using AHP.Step 5: Reformulate the MOLP supplier selection into equivalent crisp model using the enhanced twophase 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 of100 and 150 unit.
Purchasing risk is measured from four subcriteria: quality, order fulfillment, ontime delivery, and distance/ proximity. Concerning product quality, DM sets the target value of defect products at zero and the upper specification 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 quantity (100%) and the total loss will occur if supplier can only satisfy 80% of total order. For ontime 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 zeroloss will occur at the closest supplier and the specification 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 subcriterion 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 ontime delivery,k_{1} = 4 andk_{2} = 1 (since an unequal two side specification is considered for ontime delivery, there exists two loses coefficients,k_{1} andk_{2} ).The actual values (Table 3), together with the value of loss subcriterion
k previously calculated for these four subcriteria, 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 pairwise comparison matrix and local weight for each of these four subcriteria 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 subcriterion. 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:
MinZ1 = 10x1 + 12x2 + 9x3 ≤~Z10
MaxZ2 = 0.75x1 + 0.9x2 + 0.85x3 ≥~Z20
MinZ3 = 0.284x1 + 0.363x2 + 0.353x3 ≤~Z30
subject to:
x1 + x2 + x3 ？ 1400
10x1 + 12x2 + 9x3 ≤ 20000
x1 ≤ 500, x2 ≤ 600, x3 ≤ 550
xi ≥ 0
The criteria and constraint can be considered equally important and added together for comparison. However, such a comparison is generally unfair due to certain criteria that that may be more important than others. In this model, the weight of cost, service level, purchasing risk and demand are derived from AHP. Table 5 shows the pairwise comparison matrix and local weights 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 _{z1}(x ^{0}),μ _{z2}(x ^{0}), andμ _{z3}(x ^{0}), respectively. The crisp formulation of the above fuzzy MOLP using the enhanced twophase approach according to Eq. (18) is given asMax (1 ？ ρ) (0.447λ1 + 0.282λ2 + 0.164λ3 + 0.106γ1) ? p(ε1 + ε2 + ε3 + δ1)
subject to:
12x1 + 16x2 + 12x3 ≤ 20000
x1 ≤ 500, x2 ≤ 600, x3 ≤ 550
ω1 + ω2 + ω3 + β1 = 1
ε1 ≤ 0.566, ε2 ≤ 0.566, ε3 ≤ 0.566, δ1 ≤ 0.847
λ1, λ2, λ3, γ 1 ∈ [0, 1]
x1, x2, x3, ω1, ω2, ω3, β1 ≥ 0
In this problem, the original twophase approach fails to yield the feasible solution. The constraint associated with
λ _{1} cannot be satisfied because the value ofλ _{1} equal to 0.526 which is lower than the designated value of its lower bound (λ _{1}^{1} = 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 ofp is lower than 0.5. Whenp is equal or greater than 0.5, the overall achievement level decreases. A lowerp 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 (
Z _{2} ) and the purchasing risk (Z _{3} ) are a critical objectives as the corresponding constraints are relaxed for almost anyp 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 whenp is 0.10. The achievement level ofZ _{2} is totally relaxed (μ _{z}_{2} = 0) to achieve a better achievement level forZ _{1} followed byZ _{3}. The achievement level ofZ _{2} reaches the best possible value for the entire value ofp whenZ _{3} is relaxed forp is equal to0.54 and 0.6. Moreover,
Z _{1} is free from relaxation as it is the most important objective, whose assigned weight is the highest, according to the DM’s preference ( ω_{1} >> ω_{2} > ω_{3} ).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,μ _{z1} andμ _{z3} improve to 0.991 and 0.980, respectively, which results in the best value ofZ _{1} andZ _{3}. However, the DMs should carefully notice that the achievement level of service level, the second most important criteria, declines toward the worst performance (μ _{z2} = 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.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 twophase fuzzy programming model has been developed. Through numerical experiment, we demonstrate the promising advantage of our proposed approach over the maxmin 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 multiobjective optimization problem that should be concerned in future studies.
> APPENDIX

[Table 1.] Suppliers’ Quantitative Information.

[Table 2.] The Specification Limit and Range Value of Four Subcriteria.

[Table 3.] Actual Value of the Four Subcriteria.

[Table 4.] Taguchi Loss.

[Table 5.] Pairwise Comparison Matrix

[Table 6.] Comparison of Maxmin Operator and the Proposed Approach.