The LR-Type Fuzzy Multi-Objective Vendor Selection Problem in Supply Chain Management

Vendor selection is an established problem in supply chain management. It is regarded as a strategic resource by manufacturers, which must be managed efficiently. Any inappropriate selection of the vendors may lead to severe issues in the supply chain network. Hence, the desire to develop a model that minimizes the combination of transportation, deliveries, and ordering costs under uncertainty situation. In this paper, a multi-objective vendor selection problem under fuzzy environment is solved using a fuzzy goal programming approach. The vendor selection problem was modeled as a multi-objective problem, including three primary objectives of minimizing the transportation cost; the late deliveries; and the net ordering cost subject to constraints related to aggregate demand; vendor capacity; budget allocation; purchasing value; vendors’ quota; and quantity rejected. The proposed model input parameters are considered to be LR fuzzy numbers. The effectiveness of the model is illustrated with simulated data using R statistical package based on a real-life case study which was analyzed using LINGO 16.0 optimization software. The decision on the vendor’s quota allocation and selection under different degree of vagueness in the information was provided. The proposed model can address realistic vendor selection problem in the fuzzy environment and can serve as a useful tool for multi-criteria decision-making in supply chain management.


Introduction
One of the most vexing problems facing purchasing managers in organizational business decision-making is a vendor selection problem (VSP). A supply chain is a complex network in both manufacturing and service industries with interconnected components such as the suppliers of raw materials, enterprise, manufacturers, retailers, distributors, storage facilities and transporters of goods and services to desired customers in its simplest form. Thus, decision-making in a supply chain encompasses all the involved partners and activities in satisfying the demand of customers. Vendor selection plays a vital role in supply chain management; this is because of the pressing need to incorporate alliance strategies with the vendors. The types of equipment and materials supplied from vendors play a significant role in the effective management of the supply network. A location of a vendor, for instance, has a substantial impact on the firm's logistic decision for planning transportation

Literature Review
Research on supply chain and supplier selection has a long tradition since the 1960s, the criteria for vendor selection and vendor rating is a main central area of research in supply chain management (SCM). Studies related to this include but not limited to Ravindran [10], Wind et al. [11], Stewart [12], Hinkle et al. [13], Miller [14], McMillan [15], Lucas and Moore [16], Wieters and Ostrom [17], Manzer et al. [18]. More recently, fuzzy set methods were introduced in the study of supply chains, in order to cope with robustness, uncertainty and vagueness, see for instance [19][20][21][22]. Three quantitative techniques used for decision-making in supplier selection are mathematical programming, multiple attribute, and intelligent approaches. Of the three methods, mathematical programming models are used extensively in the vendor selection problem (VSP) [see Tables 1 and 2]. The nature of the vendor selection in the supply chain was discussed as a multi-criterion decision-making problem by Kumar et al. [23]. Weber and Current [24] gave the concept of multi-objective programming technique in selecting multiple conflicting criteria of vendors alongside their order quantities. Several authors such as Xia and Wu [3], Dahel [25], Pokharel [26], Tsai and Wang [27], Rezaei and Davoodi [28] worked on the multi-objective vendor selection problem (MOVSP). Kumar et al. [23,29] formulated a fuzzy mixed integer goal programming model for a multiple sourcing supplier selection problems (MSSP) with cost, quality, and delivery as fuzzy goals, subject to the constraints of buyer's demand, the suppliers' capacity, and others. The max-min technique developed by Zimmermann [8] employed in solving the multi-objective problem in their study. A systematic vendor selection process was developed by Lamberson et al. [30] to identify and prioritize relevant criteria, evaluate the trade-offs between the economic, technical, and performance criteria. Kumar et al. [31] used a lexicographic goal programming approach for solving a piecewise linear VSP of quantity discounts.
In real-life situations, the criteria for purchasing department may have various weights related to their strategies. To cope with these problems, Amid et al. [32,33] formulated a fuzzy-based model for MSSP, including three fuzzy goals: cost, quality and delivery, which are subject to capacity restriction and market demand. They used the additive model developed by Tiwari et al. [34] in solving their multi-criterion model. A weighted max-min (WMM) model was subsequently proposed by Lin [35] for solving fuzzy multi-criterion model of supplier selection. The approach was applied later by Amid et al. [36] to a fuzzy multi-objective supplier selection problem (MOSSP). Liao and Kao [37] combined the Taguchi loss function, AHP, and multi-choice goal programming model to solve the supplier selection problem. Liao and Kao [38] also gave a two-stage model for selecting suppliers in a company which engaged in the watch manufacturing sector by using a fuzzy technique for order preference by similarity to an ideal solution (TOPSIS) with fuzzy triangular numbers and multi-choice goal programming (MCGP) to optimize the problem. Wu et al. [39] used a trapezoidal membership function and solved the probabilistic multi-criteria vendor selection model by using sequential programming, considering risk factors. Ozkok and Tiryaki [40] established a compensatory fuzzy approach for solving a multi-objective linear supplier selection problem with multiple items by using cost, service, and quality as objectives. Some latest research work in the vendor selection problem using fuzzy goal programming includes: Pandey et al. [41], Mirzaee et al. [42], Abbas [43], Sari [44], Wadhwa [45], Kamal et al. [46], Shaw [47], Aggarwal et al. [48], Islam and Deb [49], Ho [50], Torres-Ruiz and Ravindran [51], Mahmudul Hassan et al. [52], Alizadeh and Yousefi [53], Ozkan and Aydin [54], Jia et al. [55], Krishankumar et al. [56]. Arikan [57] formulated a multi-objective VSP based on price, quality, customer service and delivery criteria, by using Lai and Hwang [58,59], gave fuzzy augmented approach to solve the fuzzy MOVSP. Shirkouhi et al. [60] developed an interactive two-phase fuzzy multi-objective linear programming model for the supplier selection under multi-price level and multi-products. Kilic [61] developed an integrated approach, including a fuzzy technique for selecting the best supplier in a multi-item/multi-supplier environment. Rouyendegh and Saputro [62] described an optimum decision-making method for selecting a supplier and allocating order by applying fuzzy TOPSIS and MCGP. Jadidi et al. [63] developed a normalized goal programming model with predetermined goals and predetermined weights for solving a multi-objective supplier selection problem. Chang et al. [64] considered multiple aspiration levels and vague goal relations to help the decision-makers for choosing better suppliers by using multi-choice goal programming (MCGP) with the fuzzy approach. Karimi and Rezaeinia [65] adopted a revised multi-segment goal programming model for selecting the suppliers. Sivrikaya et al. [66] adopted a fuzzy-AHP goal programming approach with linguistic variables that are express in trapezoidal fuzzy numbers, and applied to assess weights and ratings of supplier selection criteria.
The single objective programming problem can be easily solved. However, the "multi-criteria decision-making" (MCDM) problem cannot be directly solved using any algorithm, and therefore some other algorithms are used to address this issue. The algorithms which are frequently use for solving the MCDM problems are "fuzzy goal programming, goal programming and interactive fuzzy-goal programming respectively." These algorithms are used to obtain the compromise solution of the MCDM problems. Presently, several methods are available to solve MCDM problems; one of the simplest methods is weighted criterion method. In weighting method we can use the following criterion: point allocation, pairwise comparison, ranking or rating methods, and trade-off analysis. Each criterion differs according to its accuracy, ease of use, complexity for users, and theoretical foundations and produces different sets of weight criteria. Selecting a proper method of weighting is a crucial step for solving an MCDM problem. The main purpose of any criteria weighting method is to attach ordinal values to different standards indicating their relative importance in an MCDM problem. Weighted criterion works well if the objectives behaved well, and trade-offs between the objectives allow the weights to be certainly determined. The weights can be changed both during and after the optimization. This change allows the decision maker to optimize only for one objective, change the weights and optimize to another objective. It can accomplish by setting one weight to 1 and all the others to 0 (or some small positive number). Also, the decision maker can make his priority level for the objective functions and optimize the problem accordingly to his preference. The choices of weights assigned to the objective functions depend on the choices of the decision maker. The flexibility in the weights allows to the decision maker to generate new solutions accordingly to his special ordering of the objective functions. Therefore, there is always an opportunity to solve the problem for setting the better-improved solutions.

Fuzzy Sets Preliminaries
This section is devoted to an introduction to our fuzzy set notations and definitions, which will be used in the next section for the problem formulation. Starting with several basic definitions involving fuzzy sets, fuzzy numbers and its types, especially LR fuzzy numbers, are outlined. A brief discussion of α-cuts for Gaussian and exponential fuzzy numbers is also given. For more details, we refer to [7,9,80,81].
Let X be a set, called universe of discourse. A mapping µ : ForÃ := (X, µ) being a fuzzy set and α ∈ [0, 1] the following (non-fuzzy, classical, or crisp) sets are defined: The α-cut or α-level set is A≥ α := A α := {x ∈ X : µ(x) ≥ α} and the strong α-cut or strong α-level set is A> α := A α := {x ∈ X : µx > α}. Based on these two, we define the support S(A) = Supp(A) as the strong 0-cut and the core or kernel C(A) = Core(A) as the 1-cut ofÃ.
The height of a fuzzy setÃ is defined by Let X ⊆ R n . Then the fuzzy setÃ : A fuzzy setÃ = (X, µ) for X ⊆ R is called a fuzzy number ifÃ is convex and normal, µ is piecewise continuous, A α is a closed interval for all α ∈ (0, 1], and S(A) is bounded.
Some special cases of fuzzy numbers with their α-cuts are: • A trapezoidal fuzzy numberÃ = (R, µ) is a fuzzy number whose membership function µ is defined as: Please note that the membership function is piecewise continuous andÃ is normal, hence it is a well-defined fuzzy number. Further, note that the quadruple (a, b, c, d) is sufficient to describeÃ, so we writeÃ = (a, b, c, d) as abbreviation for a trapezoidal fuzzy number. In case of b = c, the trapezoidal fuzzy number is called a triangular fuzzy number, and one can writeÃ = (a, b, d). The α-cut of the trapezoidal fuzzy numberÃ = (a, b, c, d) is the closed intervalÃ The α-cut of a triangular fuzzy numberÃ = (a, b, d) is the closed interval • A trapezoidal fuzzy number is a special case of an LR fuzzy numberÃ = (R, µ), where in addition to (a, b, c, d), two continuous, strictly monotone function L, R : [0, 1] → [0, 1] with L(0) = R(0) = 0 and L(1) = R(1) = 1 are specified, and µ is defined as Please note that the core ofÃ is the interval [b, c]. The left spread and the right spread are defined as β := b − a and γ := d − c, respectively. The α-cut is is called a Gaussian fuzzy number, if its membership function µ is defined as where m ∈ R. Here σ L , σ R are called the left spread and right spread, respectively. As abbreviation, we writeÃ = (m, σ L , σ R ), and if σ L = σ R , we write as abbreviationÃ = (m, σ). Please note that {m} is the core of the Gaussian fuzzy number. Please note that the Gaussian fuzzy number is a special case of the LR fuzzy number. The α-cut of a Gaussian fuzzy numberÃ = (m, σ) is • An exponential fuzzy number is a fuzzy number which has a membership function µ given by where {m} is the core, for m ∈ R. Here σ L , σ R are called the left spread and right spread, respectively. As abbreviation, we writeÃ = (m, σ L , σ R ), and if σ L = σ R , we write as abbreviationÃ = (m, σ). Please note that the exponential fuzzy number is a special case. The α-cut of an exponential fuzzy numberÃ = (m, σ) isÃ

Multi-Objective Vendor Selection Problem (MOVSP)
The MOVSP is an extension of the problem discussed by [23], where a system of n vendors with a deterministic parameter is considered. The list of symbols used in the model formulation is given in Table 3. per unit item price for the quantity ordered q j to the j th vendor t j unit item transportation cost from the j th vendor l j percentage for the late delivered units by the j th vendor U j upper limit for the j th vendor's quantity available B j each vendor's budget constraint allocation s j percentage for the j th vendor rejected units r j rating value for the j th vendor P vendor's minimum total purchasing value f j j th vendor quota flexibility F vendor's minimum value flexible in supply quota R maximum affordable rejection by a purchaser Decision Variable q j order quantity is given to the vendor j The MOVSP is formulated as follows: Optimize j=1 l j q j , related to late delivered items Subject to set o f constraints : ∑ n j=1 q j = D, constraint related to aggregate demand q j ≤ U j , for all j = 1, 2, ..., n., constraint related to vendor capacity p j q j ≤ B j for all j = 1, 2, ..., n., constraint related to budget allocation ∑ n j=1 r j q j ≥ P. constraint related to Purchasing value ∑ n j=1 f j q j ≤ F. constraint related to vendor s quota ∑ n j=1 s j q j ≤ R constraint related to rejected items q j ≥ 0, ∀ j = 1, 2, ..., n. constraint related to non − negative vendor The assumptions contained in [23] is hold, we therefore, formulate the MOVSP with n decision variables and m constraints under fuzzy logic set theory. The fuzzy mathematical formulation is stated as follows: In problem (2), we assume that all the vague parameters are LR type fuzzy number.

The Conversion Procedure of Fuzzy MOVSP into Crisp MOVSP
In this section, the MOVSP formulated under three different types of LR fuzzy numbers, enlightening their applications in real-world problems. (2) is defined by the generalized exponential LR normalize fuzzy numberÃ = (a, b, σ L , σ R ), where a, b are left and right extreme (lower and upper values) endpoints, and σ L , σ R are left and right spread. Let L(α),R(α) are two monotonic functions respectively.

Case 1. The vague input parameters of the problem
We now define the ranking function value as Problem (2) is transformed into the equivalent crisp form using Equation (3): where λ ∈ [0, 1] is an arbitrary chosen fixed value. Case 2. The vague input parameters of the problem (2) is defined by the Gaussian LR normalize fuzzy numberÃ = (a, b, σ L , σ R ).
We now define the ranking function value as Problem (2) is transformed into the equivalent crisp form using Equation (5) as where λ ∈ [0, 1] is an arbitrary chosen fixed value. Case 3. The vague input parameters of the problem (2) is defined by the Gaussian LR normalize fuzzy numberÃ = (a, b, σ L , σ R ). and We now define the ranking function value as Problem (2) is transformed into the equivalent crisp form using Equation (5) as where λ ∈ [0, 1] is an arbitrary chosen fixed value.

Fuzzy Goal Programming Procedure
To solve the formulated problems (4), (6) and (8), we use Zimmermann's fuzzy programming approach, see [8,82]. This approach was applied to solve various multi-objective optimization problems (such as linear programming, non-linear programming, or stochastic programming). In this section, we present a brief fuzzy programming method for solving the crisp MOVSP (4), (6) and (8), which in an abstract way can be written as: where x ∈ X is used to represent all set of the feasible constraint of the MOVSP (4), (6) and (8), respectively. We assume that the problem is feasible and that an optimal compromise solution exists.
Step 1. Firstly, we solve the problem to obtain the ideal solutions (g 1 , g 2 , . . . , g k ) for the respective objective functions. Using these ideal solutions, we formulate a pay-off matrix. Then lower and upper bound of each of the objective functions is estimated from the pay-off matrix as: g k ≤ [(Z k (Q)) λ ] ≤ u k , k = 1, 2, . . . K Where g k are imprecise aspiration levels of fuzzy goals (Z k (Q)) λ for which we define linear membership functions in the next step.
Step 2. In this step, we define a fuzzy membership function for the k-th objective function (Z k (Q)) λ :

1.
For first minimize type objective function, the membership function is constructed as: For second minimize type objective function, the membership function is constructed as: For third minimize type objective function, the membership function is constructed as: Here u 1 , u 2 and u 3 are upper tolerance limit for the fuzzy goal (Z 1 (Q)) λ , (Z 1 (Q)) λ and (Z 3 (Q)) λ , respectively.
Step 3. Next, we consider the conversion of the objective functions into fuzzy goals using aspiration level to the corresponding objective function. Thus, problems (4), (6) and (8) are approximated as a fuzzy goal program by taking certain aspiration levels and introducing an auxiliary variable for the deviation from below to the objective function.
The minimization type objective function is approximated as The above membership function is used to formulate the multi-objective optimization problem as a single-objective optimization problem. Now we can use the FGP approach for attaining the highest degree of membership and thus the above problem is transformed into a goal programming problem as Here, Z represents the achievement function of the problems (4), (6) and (8), and w − k = 1 u k −g k ∀ k = 1, 2, . . . K attached to each of the objective functions measures the relative importance of a deviation from their respective target.

Computational Procedure
The conversion method presented in Section 3 can be organized into the following steps as: Step 1. Formulate the real world problem as a mathematical programming problem with fuzzy parameter. In our case, we formulated the MOVSP with LR fuzzy parameters.
Step 2. Define the membership function for fuzzy parameters in the mathematical formulation of the stated problem.
Step 3. Compute the ranking function for these fuzzy parameters to convert them into an equivalent crisp parameter.
Step 4. Determine the solution of multi-objective programming problem with objectives (Z 1 (Q)) λ , (Z 1 (Q)) λ and (Z 3 (Q)) λ respectively with crisp parameter as obtained in Step 3 for an individual optimal solution by any optimization software (Lingo, R, Matlab and any other appropriate software).
Step 5. Set the optimal solution obtained in Step 4 as a goal, and compute the appropriate aspiration level for each goal. Step 6. Repeat the Steps 4 and 5 for the various choices of values of lambda set by the decision maker.
Step 9. Solve the deterministic goal programming problem for various values of alpha-cut set by decision maker by any optimization software.
Step 10. Choose the optimal solution from the obtained solution set and the corresponding decision variables.

Case Study
In order to illustrate the proposed work, we considered a simulated case study for the vendor Selection Problem in continuation of [31] with some extensions. The instance consists of four vendors. Their profiles with uncertainty are giving in Table 4. All the parameters of the problem are considered to be randomly distributed and are simulated using uniform distribution through the R software package. If the purchasers follow a 95% (2σ limits) for the accepted policy, then the rejection maximum limit should not be more than 5% of the demand. Hence, the tolerance for the maximum rejection for a purchaser should be 25,000 × 0.05 = 1250. The minimum value for the vendors' quota flexibility and that of the total purchase for supplied items are the policy decisions, which depend on the demand. The minimum flexible value in the quota of the suppliers is F = f · D, and that of total purchase value for the items supplied is given as P = r · D. The other information is fuzzy. We assumed that fuzzy numbersÃ = (a, b, σ L , σ R ) are LR type fuzzy numbers. The instance data for the vendors with LR type fuzzy numbers are given in Table 4. Furthermore, we studied the following pattern of the fuzzy number, i.e., Exponential, Normal, and Gaussian-Normal. Parameters in the form of LR-fuzzy numbers (a, b, σ L , σ R ) are given below: Using the ranking function approaches (3), (5), and (7) as defined above in the preliminary section, the LR generalized exponential, the LR normalized, and the LR Gaussian-Normal fuzzy number, respectively, are transformed into an equivalent deterministic form. Tables 5-7 provide the parameters equivalent to the deterministic or crisp value for cases 1, 2, 3 at the distinct value of λ = 0, 0.5 and 1, respectively. When demand is fixed to D = 25,000 units. Using values of Table 5, problem (4) of case 1 was solved using the FGP approach as defined in Section 4.2. The results obtained at the distinct value of λ = 0, 0.5 and 1 are given in Table 8. Similarly, problems (6) and (8) of case 2 and 3 respectively were solved for the distinct value of λ = 0, 0.5 and 1 using the FGP approach, and the result is summarized below in Tables 9 and 10.  The optimal result for various discrete choices for λ is presented in Tables 8-10. They show that vendor 4 has lost most of his quota due to inferior performance on the criterion set up by the manufacturer, i.e., percentage of flexibility, purchase rating, percentage of rejection, purchasing cost, and budget allocation. However, vendor 2 received more quota allocation as he performed best among the other vendors on the different performance criteria. The quota for vendor 3 is higher than the vendor 1 due to their inferior capacity. The result also indicates that if the value of λ increases or decreases, then the quota allocation to different vendors also starts changing consistently. When the demand is not fixed (D = 22,000 or 23,000 or 2000 or 4000) units. Since the parameters are in uncertain form; we generated the optimistic, pessimistic and most likely value of the fuzzy parameters, and their corresponding solutions. The compromise solution values for Z 1 , Z 2 and Z 3 and optimum allocation of order quantities for the vendors for the same discrete values for λ are summarized below in Tables 8-10.
By using the values in Table 5, problem (4) of case 1 was solved using the FGP approach as defined in Section 4.2. The results obtained at the distinct value of λ = 0, 0.5 and 1 are given in Table 11. Similarly, problem (6) and (8) of case 2 and 3 respectively were solved for the distinct value of λ = 0, 0.5 and 1 using the FGP approach, and the result is summarized in Tables 12 and 13. The optimal result for various discrete choices λ is presented in Tables 11-13. They show that vendor 4 lost most of their quota due to inferior performance on the criterion set up by the manufacturer, i.e., percentage of flexibility, purchase rating, percentage of rejection, purchasing cost, and budget allocation. However, the second vendor received more quota allocation as he performed best among the other vendors on the different performance criteria. The quota for vendor 3 is higher than the vendor 1 due to their inferior capacity. The result also indicates that if the value of λ increases or decreases, then the quota allocation to different vendors also starts changing consistently.

Result Analysis and Discussion
The efficiency of the solution at a different value λ was compared within the different methods by calculating the Relative Effectiveness (R.E.).
Here T Approach 1 represents the trace value obtained when demand is fixed and T Approach 2 represents the trace value when demand is not fixed. Approach 1, and Approach 2 are based on the use of different types of ranking function and are evaluated on the different scales of λ, i.e., 0, 0.5, and 1, because of this value of the objective function and the quota allocation to the vendor's also changes. However, among all the methods used, case 3 provides the best optimum result under the given restrictions, and the solution found, is very close to the aspiration level set by the decision-maker. The solution to the problem instance given above was solved by LINGO 16.0. The following results were generated, which indicate that most of the goals are attainable either with some minor and significant improvement in the set of targets. The result analysis of vendor selection for discrete choices of λ is given in Figures 1 and 2.   Table 6 is in the case of vagueness, where the input information is present in the form of LR fuzzy numbers. The suitable fuzzy set theory concept was applied to obtain the equivalent crisp form. Moreover, we can consider the mean method of extreme values to handle the vagueness in the input parameters. Therefore, Table 4 is revised accordingly to the mean values method, and resultant Table 14 is given below: Also, we used some well-known existing multi-objective approaches, i.e., goal programming, Chebyshev goal programming, and also the two different types of fuzzy goal programming for solving the proposed MOVSP model and found that the solution obtained by these approaches are ranging within the solution range provided by the proposed approach. The different approaches for solving the proposed MOVSP model are shortly defined below:

Mean value Method. Since
The Goal Programming. The data given in Table 15 was used to solve the MOVSP by the goal programming approach.
The Chebyshev Goal Programming. The data given in Table 15 was used to solve the MOVSP by the Chebyshev goal programming approach.
After using the Chebyshev goal programming approach, the optimal compromise solution is obtained as (6,439,980, 292,074.2, 1313.05) with order quantities (5750, 11, 527, 6863, 860). The Fuzzy Goal Programming. The data given in Table 15 was used to solve the MOVSP by the fuzzy goal programming approach.
After using the fuzzy goal programming approach, the optimal compromise solution is obtained as (6,925,348, 297,672, 1140.43) with order quantities (4063, 16,602, 3819, 516). Also, by introducing the auxiliary variables for each objective function as λ K , the above model is reformulated as: After using the fuzzy goal programming approach with auxiliary variables, the optimal compromise solution is obtained as (6,731,794, 299,191.8, 1088.83) with order quantities (5078, 16,590, 3332, 0).

Conclusions
A Multi-Objective Vendor Selection Problem (MOVSP) incorporating vagueness in the parameters was presented in this paper, which is useful to select vendors' quota allocation in supply chain management. Two approaches, certainty and uncertainty in demand along with different types of defuzzification methods of fuzzy numbers are considered. Various techniques such as mean value method, goal programming with some of its variants are incorporated in this work Under both cases of fixed and uncertainty in demand considered, three values of the fuzzy parameters are generated. That is, the optimistic, the pessimistic, and the most likely as shown in Tables 5-7. The trace values of each case are represented with an approach. Different ranking functions were used in evaluating the approaches at varying values of λ. The relative effectiveness of these approaches is compared to ascertain the efficiency of the solution (Table 15). The optimal compromise multi-objective solution for various λ values is presented in Tables 8-13. It can be seen that the forth vendor lost most of their quota as a result of inferior performance on the criteria such as flexibility percentage, purchase rating, rejection percentage, purchasing cost, and budget allocation set by the decision-maker. However, the second vendor received more quota allocation due to his best performance among the vendors on the different criteria. The quota for the third vendor is higher than the first vendor due to their low capacity. The result also indicates quota allocation to vendors changes with the value of λ consistently. Fuzzy multi-objective programming is used to solve the entire MOVSP. Since in multiple objective optimization problems, a single satisfying solution for all the objectives is not possible, the concept of Pareto multi-objective solution is considered. In this regard, we solved the formulated MOVSP to obtain the compromise multi-objective solution of the problem to enable manufacturers to make a viable and profitable decision in their manufacturing process. The Pareto solution for the various vendors is analyzed above. The technique can handle the complexity of quota allocation problems of vendors in the supply chain even if their capacities are vague, which may be due to the information gap between suppliers and buyers. The proposed MOVSP under fuzzy environment has advantages over the deterministic techniques in the sense that precise information is not readily available in supply chain management. The inherent uncertainty in the input information such as transportation cost; the demand, the price, late delivery cost of an item as well as budget and vendors capacity are considered to be fuzzy. In addition, this will enable decision-makers to address VSPs under uncertainty. Another advantage of this proposed model is that, any available commercial packages such as LINDO/LINGO, GAMS and the like could be useful in solving such problems. A case study with simulated data concerning vendor selection is used to demonstrate the effectiveness of the proposed MOVSP model. This research opens to researchers a new methodology which can be applied to solve other similar supplier selection problems with slight modifications. For instance, in the future, this study can be further examined under an extended fuzzy environment, or where an astronomical data on the real-life situation is available, a probabilistic approach to solving the problem can be employed.