# Evaluating Suppliers Using AHP in a Fuzzy Environment and Allocating Order Quantities to Each Supplier in a Supply Chain

Authors: C. Lavanpriya; V. Muthukumaran; P. Manoj Kumar
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The most challenging task in the supply chain is selecting the appropriate supplier, and it has a great influence on industry productivity growth. Because of the large budget allotted to raw materials, supplier selection is a critical issue for the industry’s financial status. The right supplier in the industry drastically reduces problems throughout the supply chain. The several managers of a company were interviewed, and the most essential criteria considered by the managers when choosing their supplier firms were identified. In this study, Fuzzy-AHP and goal programming are combined to identify suitable suppliers and allocate orders in the supply chain. The mathematical models developed the goals of minimizing periodic budget, minimizing defect rate, maximizing the total value of purchase, and maximizing demand. To reach the desired values of periodic goals, the multiobjective multi-integer linear programming model is presented. A mathematical example is offered to show the effectiveness of suggested method. The Archimedean goal programming method is used to solve these multiple periods of single products within the Lingo 18.0 software package. Multiple period demand allocations to suppliers are calculated asX11 = 9000, X12 = 7800, X13 = 7900, X21 = 3434.9, X22 = 5098, X23 = 6002.56, X31 = 100, X32 = 100, and X33 = 100. All the target values of budget, defective rate, and demand are achieved for all periods except the total values of purchasing. The results are subjected to sensitivity analysis. It will help a manufacturer for selecting a suitable supplier to allocate the order for maximizing the profit, improving the quality of product, enhancing the supplier buyer relationship, strengthening the supply chain, reducing the risk of purchasing decision, and improving customer satisfaction. Therefore, the supply chain is effective to supply the continuity of products.


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## Body

## 1. Introduction

Globalization of the economy has resulted in procurement of products and services becoming increasingly significant among various supply chain-related operations. Purchasing costs can make 60% or more of a company’s income. Selection of the vendor is an important decision in many situations, with long-term consequences for a company’s profitability and efficiency.Coordination between a manufacturer and suppliers is often a tough and critical link in supply chains. When a supplier adopts a well-managed and well-established supply chain, the relationship has a long-term impact on the supply chain’s competitiveness. As a result, the most essential concern for developing an effective supply chain system is selecting the appropriate supplier. Furthermore, supplier selection is a difficult procedure because several variables must be examined during the decision-making process. Poor decision-making will have greater effects on the business as it becomes more reliant on its suppliers. This type of decision is frequently difficult and unstructured. Supplier selection decision-making requires trade-offs among various criteria including both quantitative and qualitative variables that may conflict [1].As a result, if a company’s buyer wishes to effectively manage the purchasing function, it should select qualified suppliers based on the proper model and criteria. Selecting the supplier is an MCDM issue with tangible and intangible criteria. When choosing vendors, numerous variables are taken into account.Decision-makers determine ten criteria for this research work to select potential suppliers during the determination of supplier selection criteria. The ten criteria for selecting a supplier for this research work are listed below. Cost (C1), quality (C2), lead time (C3), communication system (C4), warranties and claims (C5), risk factor (C6), performance history (C7), carbon reduction initiatives (C8), payment flexibility (C9), and product personalization (C10) are all significant factors.In the research, primary goal of choosing the vendor’s procedure is to maximize total profit for the buyer, reduce the purchasing risk, establish long-term relationships between firms and suppliers, and increase the quality of product. The order allocation to each suitable supplier will enhance the flow of supply products to all the selected supplier, customer satisfaction will improve, and the supply chain will get strengthen.Choosing the right supplier and assigning order quantities are two of the most important functions that purchasing decision-makers must perform, as they affect the company’s long-term viability. The main goal is to obtain the proper product quality in the appropriate amount from the appropriate supplier at the exact time and at an affordable rate. In this research, paper buyers consider multiple input data from several decision-makers.Additionally, in supplier selection, order allocation decisions perform a vital role in determining the cost-effectiveness of the industry. It consists of identifying various quantities of goods that are purchased from different vendors because an organization’s requirements may exceed the capacity of a single supplier [2]. As a result, this research attempts to determine the order quantity to be acquired from every vendor over a certain planning horizon to optimize the allocation of order quantities.Purchasing is a strategic activity because it increases profits and reduces costs. As a result, the author considers the issue of choosing the appropriate supplier and allocating order quantities, both of which have an impact on the supply chain. When a buyer places an order, the supplier lowers the unit purchase price through goal programming.In AHP, buyer can consolidate the output of the response of several decision maker. However, in ANP, buyer cannot consider the opinion of many decision-makers. Therefore, the AHP model is used to select the potential supplier. To make decision-makers opinion effective, fuzzy is incorporated with AHP [3, 4].The focus of this research is to develop mathematical modelling that can be used to solve the issue of integrating supplier selection and demand allocation. A single product, multiple periods, multiple suppliers, and multiple objective mathematical models are all built in this research. One of the major contributions in this research is the increased profit of the buyer, increased quality of the product, and the continuous flow of products for suppliers. As a result, buyer will allocate the order with the four goals. Minimizing periodic budget, minimizing defect rate, maximizing the total value of purchase (TVP), and maximizing demand are the four goals of the proposed MOMILP model. In the first phase, FAHP employs criteria for supplier evaluation. Based on these criteria, each supplier is assigned a weight for each item, and suppliers are rated to find the most suitable suppliers. In the second phase, the AGP method is used to determine a recommended compromise solution for assigning order quantities to selected vendors in each period for each item. Finally, FAHP weights are fed into a LINGO-solved optimization model. The implementation of a developed mathematical model is also evaluated using a numerical example.The rest of the paper is organized as follows: Section2 will focus on both a review of the literature on the selection of the supplier and order allocations as well as the research gap. Section 3 explains the case study as well as the methodology for solving it. Finally, there are results and conclusions in Section 4.


## 2. Literature Review

The studies in the literature are grouped into two categories: supplier selection approaches are the first group and order allocation procedures are the second. The following section provides a brief overview of acceptable alternative approaches in terms of general application, unique features, and significant limitations.


### 2.1. Supplier Selection

The author concentrated on Taguchi’s loss function, TOPSIS, and multiple-criteria goal programming as integrative methodologies. There are three phases in the model. Taguchi’s loss function is used to determine the quality losses in the first phase. In the second step, suitable factors are found using TOPSIS weights, and in the third phase, a goal programming model is created using weights and loss associates to select the best-performing vendor. To choose a vendor who performs better than average, the author used an integrated method with different criteria [5].For supply chain success, the author emphasized the importance of choosing the appropriate supplier and assigning the order quantity of products in uncertain conditions. To deal with market changes, the author used a strategy for handling a group MCDM problem that combines selecting a supplier with assigning order quantities for highly dynamic supply chains. The proposed technique imitates decision-makers’ data collection and manipulation in the purchasing sector. However, precise data are insufficient to make realistic circumstances in many cases, and fuzzy logic can be employed to deal with the ambiguity of human judgment [6]. The fuzzy TOPSIS method is utilized for supplier evaluation by employing four qualitative metrics: overall performance strategy, quality of service, technological innovation, and financial risk. The criterion application is then investigated systematically for order distribution only among the shortlisted vendors, using a simulation-based fuzzy TOPSIS technique [7].According to the author, one of the most difficult aspects of waste management is determining a proper location for the disposal of infectious trash. The selection of highly contagious waste disposal centers is a challenging procedure that involves complicated social and environmental criteria with cost criteria including the allocation of resources. The model offered by the author can result in the selection of new acceptable locations for highly contagious waste disposal while taking into account both the overall price and ultimate prioritized weight objectives [8].For supplier evaluation, the author used the integrated FUCOM-Rough SAW model. The evaluation was based on 21 factors that were organized into a two-level hierarchical structure. The weight of the factors was calculated using the FUCOM approach. First, key criteria values were determined: economic, social, and environmental. The weights over all subfactors for each major group of factors were then determined. To rank and evaluate suppliers from a probable set, the rough SAW approach was used [9].In the dairy business, the author utilized the fuzzy TOPSIS approach to solve a vendor selection issue. Business is a freshly constructed plant. The company makes pasteurized and packaged milk, as well as yogurt and other dairy products. As a result, a daily milk supply is required to stay competitive in the business, and the company wishes to choose the optimal vendors and boost its efficiency. The fuzzy TOPSIS approach was used to choose providers for the purpose [10].The author offered a new way to solving a vendor selection issue in the iron and steel sector by combining rough statistics with the MABAC approach and DOE to create a metamodel. The system begins with the aggregate of five competing vendors’ relative efficiency scores using rough numbers to allow for the uncertainty considered in the decision-making process [11].The author used fuzzy multicriteria analysis to investigate supplier evaluation and selection. The FAHP was investigated in the study to determine the best vendor for the procurement of raw material required for the fabrication of preinsulated pipes. On the basis of nine factors, decision-makers chose among five vendors. The total efficiency of the business is influenced by the proper execution of procurement. In the scenario, procurement of raw materials is required for the production system [12].


### 2.2. Order Allocation

The author highlighted the elements of a successful supplier selection approach. The usage of MCDM methodologies and mathematical models is then compared in terms of those parameters. As an outcome, a hybrid solution based on fuzzy TOPSIS and goal programming has been proposed [13].To successfully manage uncertainty and vagueness in supplier evaluation situations, a new weighted additive fuzzy programming paradigm is proposed. Then, utilizing the constraints, goals, and weights of the parameters, a fuzzy multiobjective linear model is established to resolve the vendor selection dilemma and distribute the optimal level of order quantities to each vendor [14, 15].After capturing the vagueness of the problem, the model applies the linguistic variable to assess the weight of each factor. Finally, the model has a less computational procedure and can deal with the rating of factors effectively. This rating of factor can be easily applied to other management decision problems.The author proposed a combined methodology of a rule-based weighted fuzzy method, a FAHP, and multiobjective mathematical programming for long-term vendor evaluation and placed on order allotment with the multiple periods and multiple product lot-sizing issues [16].The authors studied a scenario in which purchasers must estimate the number of products to order of each supplier in every period to achieve the production plan’s requirements. However, they must also adhere to the given constraints [17, 18].The researcher has highlighted the two independent optimization techniques for selecting the appropriate supplier and allocating the optimal order quantity. These two techniques are combined to assess the impact of market demand on the procurement of products. For more efficient procurement, the author suggested the MOILP model for integrating highly dynamic evaluation of the supplier and demand allotment with market conditions [2].According to the author, reverse logistics encompasses all operations relating to product reuse in a closed-loop supply chain network that includes the manufacturer, disassembly, refurbishment, and disposal locations. Meanwhile, the manufacturer is responsible for it. The author presents a multiphased integrated model. The first part proposes a framework for reverse logistics supplier selection criteria. Furthermore, a fuzzy technique is used to assess providers using qualitative criteria. The outcome of this process is the weight of each provider according to each part. In the second phase, author offers a MOMILP model to identify suitable providers and the appropriate amount of goods in the CLSC network [19].The author proposed a decision-making tool to handle various periods for green selection of supplier and allocate appropriate demand allotment issue. The tool is made up of three parts that are all linked together. First, fuzzy TOPSIS is utilized to allocate two preferred weight ratings to each possible supplier, one is for traditional and other for green criteria. Second, top executives employ an AHP to give a global priority weight to both sets of criteria, depending on the corporate goals and independent of possible suppliers. Third, the fuzzy TOPSIS preference weights for traditional and green criteria are multiplied by the set’s overall importance weight for each supplier [20].A new multiobjective mathematical programming method is suggested to manage the long-term selection of the supplier and allocate the demand quantities simultaneously in various periods, multiple goods, and multiple supplier supply chains. Quantity discounts are taken into account in two scenarios: all-unit and incremental discounts [21].


### 2.3. Research Gap

According to a review of the literature, there are a variety of methodologies and techniques for selecting the suppliers. After reviewing a number of contributions to supply chain concepts [22], it was recognized that more work was needed to identify specific ways that manufacturers could use. Although much research has been done on the implementation of supplier selection and order allotment models, more information on how they will be implemented at the manufacturing level is still needed. Different analytical approaches and strategies for selecting suitable suppliers and demand allocation are described in the literature, with the most of them adopting MCDM systems, such as TOPSIS, FUZZY TOPSIS, ANP method, and Linear programming. Despite the fact that various studies have been done on these strategies, little study has been done on the trade-offs of integrating the FAHP and goal programming methodologies in supply chain. No previous research, according to the authors, has looked into allocating orders among acceptable suppliers for single product and time periods, as well as integrating FAHP with goal programming. Many studies have just focused on the buyer’s profit. No study has focused on product quality, product flow to suppliers, and productivity. To achieve this purpose, the research focuses on decreasing defect rate, maximizing demand, and maximizing total value of products. This paper describes a method for identifying possible suppliers, emphasizing the importance of getting the right one and allocating demand to the relevant supplier.


## 2.1. Supplier Selection

The author concentrated on Taguchi’s loss function, TOPSIS, and multiple-criteria goal programming as integrative methodologies. There are three phases in the model. Taguchi’s loss function is used to determine the quality losses in the first phase. In the second step, suitable factors are found using TOPSIS weights, and in the third phase, a goal programming model is created using weights and loss associates to select the best-performing vendor. To choose a vendor who performs better than average, the author used an integrated method with different criteria [5].For supply chain success, the author emphasized the importance of choosing the appropriate supplier and assigning the order quantity of products in uncertain conditions. To deal with market changes, the author used a strategy for handling a group MCDM problem that combines selecting a supplier with assigning order quantities for highly dynamic supply chains. The proposed technique imitates decision-makers’ data collection and manipulation in the purchasing sector. However, precise data are insufficient to make realistic circumstances in many cases, and fuzzy logic can be employed to deal with the ambiguity of human judgment [6]. The fuzzy TOPSIS method is utilized for supplier evaluation by employing four qualitative metrics: overall performance strategy, quality of service, technological innovation, and financial risk. The criterion application is then investigated systematically for order distribution only among the shortlisted vendors, using a simulation-based fuzzy TOPSIS technique [7].According to the author, one of the most difficult aspects of waste management is determining a proper location for the disposal of infectious trash. The selection of highly contagious waste disposal centers is a challenging procedure that involves complicated social and environmental criteria with cost criteria including the allocation of resources. The model offered by the author can result in the selection of new acceptable locations for highly contagious waste disposal while taking into account both the overall price and ultimate prioritized weight objectives [8].For supplier evaluation, the author used the integrated FUCOM-Rough SAW model. The evaluation was based on 21 factors that were organized into a two-level hierarchical structure. The weight of the factors was calculated using the FUCOM approach. First, key criteria values were determined: economic, social, and environmental. The weights over all subfactors for each major group of factors were then determined. To rank and evaluate suppliers from a probable set, the rough SAW approach was used [9].In the dairy business, the author utilized the fuzzy TOPSIS approach to solve a vendor selection issue. Business is a freshly constructed plant. The company makes pasteurized and packaged milk, as well as yogurt and other dairy products. As a result, a daily milk supply is required to stay competitive in the business, and the company wishes to choose the optimal vendors and boost its efficiency. The fuzzy TOPSIS approach was used to choose providers for the purpose [10].The author offered a new way to solving a vendor selection issue in the iron and steel sector by combining rough statistics with the MABAC approach and DOE to create a metamodel. The system begins with the aggregate of five competing vendors’ relative efficiency scores using rough numbers to allow for the uncertainty considered in the decision-making process [11].The author used fuzzy multicriteria analysis to investigate supplier evaluation and selection. The FAHP was investigated in the study to determine the best vendor for the procurement of raw material required for the fabrication of preinsulated pipes. On the basis of nine factors, decision-makers chose among five vendors. The total efficiency of the business is influenced by the proper execution of procurement. In the scenario, procurement of raw materials is required for the production system [12].


## 2.2. Order Allocation

The author highlighted the elements of a successful supplier selection approach. The usage of MCDM methodologies and mathematical models is then compared in terms of those parameters. As an outcome, a hybrid solution based on fuzzy TOPSIS and goal programming has been proposed [13].To successfully manage uncertainty and vagueness in supplier evaluation situations, a new weighted additive fuzzy programming paradigm is proposed. Then, utilizing the constraints, goals, and weights of the parameters, a fuzzy multiobjective linear model is established to resolve the vendor selection dilemma and distribute the optimal level of order quantities to each vendor [14, 15].After capturing the vagueness of the problem, the model applies the linguistic variable to assess the weight of each factor. Finally, the model has a less computational procedure and can deal with the rating of factors effectively. This rating of factor can be easily applied to other management decision problems.The author proposed a combined methodology of a rule-based weighted fuzzy method, a FAHP, and multiobjective mathematical programming for long-term vendor evaluation and placed on order allotment with the multiple periods and multiple product lot-sizing issues [16].The authors studied a scenario in which purchasers must estimate the number of products to order of each supplier in every period to achieve the production plan’s requirements. However, they must also adhere to the given constraints [17, 18].The researcher has highlighted the two independent optimization techniques for selecting the appropriate supplier and allocating the optimal order quantity. These two techniques are combined to assess the impact of market demand on the procurement of products. For more efficient procurement, the author suggested the MOILP model for integrating highly dynamic evaluation of the supplier and demand allotment with market conditions [2].According to the author, reverse logistics encompasses all operations relating to product reuse in a closed-loop supply chain network that includes the manufacturer, disassembly, refurbishment, and disposal locations. Meanwhile, the manufacturer is responsible for it. The author presents a multiphased integrated model. The first part proposes a framework for reverse logistics supplier selection criteria. Furthermore, a fuzzy technique is used to assess providers using qualitative criteria. The outcome of this process is the weight of each provider according to each part. In the second phase, author offers a MOMILP model to identify suitable providers and the appropriate amount of goods in the CLSC network [19].The author proposed a decision-making tool to handle various periods for green selection of supplier and allocate appropriate demand allotment issue. The tool is made up of three parts that are all linked together. First, fuzzy TOPSIS is utilized to allocate two preferred weight ratings to each possible supplier, one is for traditional and other for green criteria. Second, top executives employ an AHP to give a global priority weight to both sets of criteria, depending on the corporate goals and independent of possible suppliers. Third, the fuzzy TOPSIS preference weights for traditional and green criteria are multiplied by the set’s overall importance weight for each supplier [20].A new multiobjective mathematical programming method is suggested to manage the long-term selection of the supplier and allocate the demand quantities simultaneously in various periods, multiple goods, and multiple supplier supply chains. Quantity discounts are taken into account in two scenarios: all-unit and incremental discounts [21].


## 2.3. Research Gap

According to a review of the literature, there are a variety of methodologies and techniques for selecting the suppliers. After reviewing a number of contributions to supply chain concepts [22], it was recognized that more work was needed to identify specific ways that manufacturers could use. Although much research has been done on the implementation of supplier selection and order allotment models, more information on how they will be implemented at the manufacturing level is still needed. Different analytical approaches and strategies for selecting suitable suppliers and demand allocation are described in the literature, with the most of them adopting MCDM systems, such as TOPSIS, FUZZY TOPSIS, ANP method, and Linear programming. Despite the fact that various studies have been done on these strategies, little study has been done on the trade-offs of integrating the FAHP and goal programming methodologies in supply chain. No previous research, according to the authors, has looked into allocating orders among acceptable suppliers for single product and time periods, as well as integrating FAHP with goal programming. Many studies have just focused on the buyer’s profit. No study has focused on product quality, product flow to suppliers, and productivity. To achieve this purpose, the research focuses on decreasing defect rate, maximizing demand, and maximizing total value of products. This paper describes a method for identifying possible suppliers, emphasizing the importance of getting the right one and allocating demand to the relevant supplier.


## 3. Methodologies

### 3.1. FAHP

Chatterjee proposed a FAHP approach based on the extent analysis method. The method uses linguistic variables to express the comparative judgments given by decision-makers. LetX=x1,xj…xm represent an object set and G=g1,gj…gm a goal set [22]. In the method proposed by Chatterjee, each object xi is taken and extent analysis is performed for each goal gi. As a result, the following signs represented in equation (1) can be used to get m extent analysis values for each object. Equation (2) denotes fuzzy triangular numbers.(1)Mgi1,Mgij,…,Mgim,i=1,2,…,n,(2)where allMgijj=1,2,…,nare triangular fuzzy numbers.The procedure follows the steps outlined in the text.Step 1: find the value of the fuzzy synthetic extent for theith object using equation (3). Equation (4) demonstrates how to do a fuzzy addition. Equations (5) and (6) are used to calculate the value of fuzzy synthetic extent:(3)si=∑j=1mMgii⊗∑i=1n∑j=1mMgii−1,(4)Where∑j=1mMgii.Equation (4) is obtained by performing addition operation of m extent analysis values for a particular matrix such that (5)∑j=1mMgii=∑j=1mlj,∑j=1mmj,∑j=1muj,(6)∑i=1n∑j=1mMgii−1=1∑i=1n∑j=1mui,1∑i=1n∑j=1mmi,1∑i=1n∑j=1mli.Step 2: calculate the probability ofs2l2,m2,u2≥s1=l1,m1,u1, where s2 and s1 are provided by siin equation (3). Equation (7) defines the degree of possibility between two fuzzy synthetic extents. (7)Vs2≥s1=supy≥xminμs2y,μs1x.This can be equivalently expressed as in equations (8) and (9):(8)vs2≥s1=hgts1∩s2=μs2d,(9)μs2d=1ifm2≥m10ifl1≥u2l1−u2m2−u2−m1−l1otherwise.The comparison betweenM1 and M2 requires the values ofVs2≥s1andVs1≥s2.Step 3: compute the degree of possibility for a convex fuzzy number to be greater thank convex fuzzy numbers sii=1,…,k. The degree of possibility is calculated in the following equation:(10)Vs≥s1,s2…sk=Vs≥s1ands≥s2and…ands≥sk=minVs≥si,i=1,2,…k.Step 4: the following equation is used to calculate vectorW′:(11)W′=d′A1,d′A2,…,d′AkT.Assuming that(12)d′Ai=minVsi≥sj,fori=1,2,…,k,j=1,2,…,k,k≠j.The normalized vector is given by the following equation:(13)W=dA1,dA2,…,dAk.W represents the nonfuzzy number computed for each comparison matrix.


### 3.2. Goal Programing

A model for allocating order quantities is presented for three suppliers, multiperiods, and single products, using AGP. In the goal programming model, the achievement function measures the degree to which the unwanted deviation variables of the goals are minimized. The AGP approach has been used to solve the MOMILP model by taking into consideration of the model. The following is a list of the model’s notation.


#### 3.2.1. Notations

Dt = demand of the item in period t;ht = holding cost of the item during period t;Oit = ordering cost for supplier i in period t;qit = rate of defective goods from supplier i in period t;Q = maximum defect rate the buyer will tolerate;Bt = budget of the buyer in period t;Vit = capacity of the supplier i in period t;Cit = price of the item for supplier i in period t;L = minimum order quantity;Wi = overall score of suppliers i obtained from the FAHP model Pi=∑i=1nWiVit.


#### 3.2.2. Decision Variables

Xit = number of products ordered from supplier i in period t;Yit = 1 if an order is placed on supplier i in time period t, 0 otherwise.


#### 3.2.3. Intermediate Variables

It = inventory of the product, carried over from t to period t + 1;zjt+ = positive deviation from the target value of ith goal in period t;zjt− = negative deviation from the target value of ith goal in period.Goal programming formulation for single product, multiple suppliers, and multiple time periods is shown in Table1.Table 1
Goal programming formulation for single product and multiple suppliers.
ModelGoal and constraintU1=∑t=1Tz1t+/∑t=1TBt,U2=∑t=1Tz2t+/∑t=1tQDt,Minimize the achievement functionU3=∑t=1Tz3t−/∑t=1TPt,U4=∑t=1Tz4t−/∑t=1TDtMinU=w1U1+w2U2+w3U3+w4U4MinU=w1∑t=1Tz1t+/∑i−1TBt+w2∑t=1Tz2t+/∑i−1TQDt+w3∑t=1Tz3t−/∑i−1TPt+w4∑t=1tz4t−/∑i−1TDtBudget∑i=1nCitXit+∑j=1nOitYit+htIt+z1t−−z1t+=Bt∀tt=1,2…T∑i=1nqitXit+z2t−−z2t+=QDt−It−1∀tExpected number of defect rate product∑i=1nWiXit+z3t−−z3t+=Pt∀tTotal value of purchasingIt−1+∑i=1nXit+z4t−−z4t+=Dt∀Demand constraintsIt=∑i=1nXit−Dt+It−1∀Material balance equationXit⩽VitYit∀i,tNonnegativity and binary constraintXit≥LYit∀i,tXit⩾0∀i,tYit−0or1integer∀i,t


### 3.3. Case Study

The industry is a well-known large manufacturer that sells the products that make up its supply chain network. In order to gain a competitive advantage in the market, its board of directors wishes to identify suitable suppliers to purchase raw materials for the development of new goods. To choose a supplier, a decision committee of three decision-makers was constituted. For this scenario, the industry selects ten criteria. Cost (C1), quality (C2), lead time (C3), communication system (C4), warranties and claims (C5), risk factor (C6), performance history (C7), carbon reduction programs (C8), payment flexibility (C9), and product personalization (C10) are the factors to consider (C10). Top management has set four goals based on the company’s sales history and sales forecast: minimizing periodic budget, decreasing defect rate, maximizing total value of purchase, and maximizing demand. The following is a summary of the integrated FAHP and goal programming method used to solve this problem, as well as the numerical procedure. In terms of supplier selection and allotment of demand, the integrated fuzzy TOPSIS and MCGP are seen in Figure 1.Figure 1
Framework for the supplier selection and order allocation problem.The linguistic scale for weighing criteria and evaluating alternatives is shown in Table2. Table 3 compares the weights given by decision-makers to various criteria. Table 4 depicts the supplier rating’s fuzzy numbers in relation to criterion C1. Tables 5–13 show the fuzzy number supplier ratings for all criteria. Table 14 depicts weight vector matrix that has been normalized. Table 15 shows the weight and ranking of each supplier.Table 2
Linguistic scale for weighing criteria and evaluating alternatives.
Linguistic termsFuzzy triangular numberInverse fuzzy triangular fuzzy numberEqually important (EI)(1, 1, 1)(1, 1, 1)Equally preferable (EQ)(1, 1, 3)(0.33, 1, 1)Slightly preferable (SP)(1, 3, 5)(0.20, 0.33, 1)Fairly preferable (FP)(3, 5, 7)(0.14, 0.20, 0.33)Extremely preferable (XP)(5, 7, 9)(0.11, 0.14, 0.20)Absolutely preferable (AP)(7, 9, 9)(0.11, 0.11, 0.14)Table 3
Comparison of the weights given to different criteria by decision-makers.
DMC1C2C3C4C5C6C7C8C9C10C1DM1EIEQSPAPXPEQSPEQFPXPDM2EISPFPXPSPXPFPSPXPXPDM3EIEQFPFPFPAPXPFPAPEQC2DM11/EQEIAPXPSPEQXPEQXPEQDM21/SPEIAPFPFPSPAPSPXPFPDM31/EQEIAPFPEQFPAPFPAPAPC3DM11/SP1/APEIEQSPXPEQXPSPEQDM21/FP1/APEIEQFPFPSPFPFPSPDM31/FP1/APEIEQXPAPFPAPXPFPC4DM11/AP1/XP1/EQEIEQFPXPEQFPAPDM21/XP1/FP1/EQEISPFPFPSPXPAPDM31/FP1/FP1/EQEIFPFPAPSPAPAPC5DM11/XP1/SP1/SP1/EQEIEQXPXPXPSPDM21/SP1/FP1/FP1/SPEISPXPAPSPFPDM31/FP1/EQ1/FP1/FPEIFPXPAPFPFPC6DM11/EQ1/EQ1/XP1/FP1/EQEIXPAPEQXPDM21/XP1/SP1/FP1/FP1/SPEIXPAPXPXPDM31/AP1/FP1/AP1/FP1/FPEIEQAPAPEQC7DM11/SP1/XP1/EQ1/XP1/XP1/XPEIXPXPXPDM21/FP1/AP1/SP1/FP1/XP1/XPEISPAPXPDM31/XP1/AP1/FP1/AP1/XP1/XPEIFPAPXPC8DM11/EQ1/EQ1/EQ1/EQ1/XP1/AP1/XPEISPXPDM21/SP1/SP1/SP1/SP1/AP1/AP1/SPEIFPAPDM31/FP1/AP1/SP1/SP1/AP1/AP1/FPEIFPAPC9DM11/FP1/EQ1/SP1/FP1/XP1/XP1/XP1/XPEIEQDM21/XP1/FP1/FP1/XP1/SP1/XP1/XP1/APEISPDM31/AP1/AP1/XP1/AP1/FP1/EQ1/XP1/APEIFPC10DM11/XP1/EQ1/EQ1/AP1/SP1/XP1/XP1/XP1/EQEIDM21/XP1/FP1/SP1/AP1/FP1/XP1/XP1/AP1/SPEIDM31/EQ1/AP1/AP1/AP1/FP1/EQ1/XP1/AP1/FPEITable 4
Fuzzy numbers associated with the supplier rating for criterionC1.
DM1A1A2A3A1(1, 1, 1)(1, 1, 3)(1, 3, 5)A2(0.33, 1, 1)(1, 1, 1)(7, 9, 9)A3(0.20, 0.33, 1)(0.11, 0.11, 0.14)(1, 1, 1)DM2A1A2A3A1(1, 1, 1)(5, 7, 9)(3, 5, 7)A2(0.11, 0.14, 0.20)(1, 1, 1)(7, 9, 9)A3(0.14, 0.20, 0.33)(0.11, 0.11, 0.14)(1, 1, 1)DM3A1A2A3A1(1, 1, 1)(1, 1, 3)(1, 3, 5)A2(0.33, 1, 1)(1, 1, 1)(3, 5, 7)A3(0.20, 0.33, 1)(0.11, 0.11, 0.14)(1, 1, 1)A1A2A3A1(1, 1, 1)(2.33, 3, 5)(1.66, 3.66, 5.66)A2(0.25, 0.17, 0.73)(1, 1, 1)(5.66, 7.66, 8.33)A3(0.18, 0.28, 0.77)(0.11, 0.11, 0.14)(1, 1, 1)Table 5
Fuzzy numbers associated with the supplier rating for criterionC2.
A1A2A3A1(1, 1, 1)(3, 4.3, 5.6)(3, 4.3, 5.6.3)A2(0.21, 0.48, 0.71)(1, 1, 1)(3, 4.3, 5.6.3)A3(0.19, 0.44, 1.53)(0.19, 0.44, 0.51)(1, 1, 1)Table 6
Fuzzy numbers associated with the supplier rating for criterionC3.
A1A2A3A1(1, 1, 1)(3.66, 5, 6.33)(3.66, 5.66, 7.66)A2(0.19, 0.43, 0.49)(1, 1, 1)(4.33, 6.33, 7.66)A3(0.14, 0.20, 0.46)(0.13, 0.17, 0.26)(1, 1, 1)Table 7
Fuzzy numbers associated with the supplier rating for criterionC4.
A1A2A3A1(1, 1, 1)(3.66, 5, 6.33)(3, 5, 7)A2(0.19, 0.43, 0.49)(1, 1, 1)(3.66, 5.66, 7.66)A3(0.15, 0.22, 0.51)(0.13, 0.18, 0.28)(1, 1, 1)Table 8
Fuzzy numbers associated with the supplier rating for criterionC5.
A1A2A3A1(1, 1, 1)(3, 4.33, 5.66)(3.66, 5.66, 7.66)A2(0.21, 0.48, 0.71)(1, 1, 1)(3, 3.66, 5)A3(0.13, 0.18, 0.28)(0.25, 0.70, 0.71)(1, 1, 1)Table 9
Fuzzy numbers associated with the supplier rating for criterionC6.
A1A2A3A1(1, 1, 1)(3, 4.33, 5.66)(5.66, 7.66, 9)A2(0.21, 0.48, 0.71)(1, 1, 1)(3.66, 5, 6.33)A3(0.11, 0.13, 0.18)(0.19, 0.43, 0.49)(1, 1, 1)Table 10
Fuzzy numbers associated with the supplier rating for criterionC7.
A1A2A3A1(1, 1, 1)(1.66, 3.66, 5.66)(3, 3.66, 5)A2(0.18, 0.28, 0.77)(1, 1, 1)(2.33, 3.66, 5.66)A3(0.25, 0.70, 0.71)(0.21, 0.49, 0.73)(1, 1, 1)Table 11
Fuzzy numbers associated with the supplier rating for criterionC8.
A1A2A3A1(1, 1, 1)(3, 5, 7)(2.33, 3.66, 5.66)A2(0.15, 0.22, 0.51)(1, 1, 1)(4.33, 5.66, 7)A3(0.20, 0.46, 0.55)(0.18, 0.41, 0.44)(1, 1, 1)Table 12
Fuzzy numbers associated with the supplier rating for criterionC9.
A1A2A3A1(1, 1, 1)(4.33, 5.66, 7)(3.66, 5.66, 7)A2(0.18, 0.41, 0.44)(1, 1, 1)(2.66, 5.66, 7.66)A3(0.15, 0.21, 0.49)(0.13, 0.18, 0.28)(1, 1, 1)Table 13
Fuzzy numbers associated with the supplier rating for criterionC10.
A1A2A3A1(1, 1, 1)(1.66, 3, 5)(4.33, 6.33, 7.66)A2(0.22, 0.51, 0.77)(1, 1, 1)(3, 5, 7)A3(0.14, 0.19, 0.44)(0.15, 0.22, 0.51)(1, 1, 1)Table 14
Normalized weight vector.
C1C2C3C4C5C6C7C8C9C10A10.280.470.450.4400.550.440.410.440.43A20.390.200.2060.210.230.090.210.240.210.23A30.320.320.340.330.760.340.330.340.340.33Weight of the criteria0.100.100.100.090.100.100.100.100.100.10Table 15
Performance of the supplier.
SupplierPrioritizing weightRanking of the supplierA10.391A20.223A30.382The supplier’s criteria value∑j=1mMgii is calculated as(14)C1=29.92,44.65,60.1,C2=32.51,45.7,57.93,C3=21.59,32.35,45.35,C4=24.24,40.66,44.99,C5=20.06,30.12,39.89,C6=20.47,27.38,33.53,C7=16.15,22.59,29.03,C8=10.83,16.28,19.9,C9=3.84,6.38,9.52,C10=2.40,3.59,7.23,∑i=1n∑j=1mMgii=182.01,269.7,347.02,∑i=1n∑j=1mMgii−1=0.0028,0.0037,0.0054.The value of the fuzzy synthetic extent for the criteria matrix is(15)Sc1=29.92,44.65,60.15∗0.0028,0.0037,0.0054=0.083,0.16,0.32,Sc2=0.091,0.16,0.31,Sc3=0.06,0.11,0.24,Sc4=0.06,0.15,0.24,Sc5=0.05,0.11,0.21,Sc6=0.05,0.10,0.18,Sc7=0.04,0.08,0.15,Sc8=0.03,0.06,0.10,Sc9=0.01,0.02,0.05,Sc10=0.006,0.01,0.03.Weight vectorW′ = (1,1,1,0.93,1,1,1,1,1,1).After normalization, the weight vector is (0.10,0.10,0.10,0.09,0.10,0.10,0.10,0.10,0,10).The weight vector for the alternative supplier is calculated as follows.A1 = (4.99, 7.66, 11.66), A2 = (6.91, 9.37, 10.06), and A3 = (1.29, 1.39, 1.91)∑i=1n∑j=1mMgii is calculated as (13.19, 18.42, 23.63)∑i=1n∑j=1mMgii−1 is calculated as (0.042, 0.054, 0.075)The value of the fuzzy extent for the criteria matrix is(16)SA1=4.99,7.66,11.66∗0.042,0.054,0.075=0.20,0.41,0.87,SA2=6.91,9.37,10.06∗0.042,0.054,0.075=0.29,0.50,0.75,SA3=1.29,1.39,1.91∗0.042,0.054,0.075=0.054,0.07,0.14.The degree of possibility of these fuzzy values is computed as(17)VSA1≥SA2=0.86,VSA1≥SA3=1.70,VSA2≥SA1=1.19,VSA2≥SA3=2.61,VSA3≥SA1=1,VSA3≥SA2=1,W′=0.86,1.19,1.The buyer will not rely on a single supplier because they used multiple sourcing strategies in the purchasing activity. Single sourcing cannot satisfy the demand of the buyer. Therefore, the buyer selects all three suppliers for allocating the order quantities of a single product, multiple suppliers, and multiple time periods using the goal programming approach.The AGP is constructed on the Lingo environment for this mathematical model with three suppliers and three periods. The order is assigned to each supplier and each of the time periods in using Lingo software, as shown in Table16.(18)Cit=26,27,28Cit=24,23,27Cit=24,23,24,qit=0.00250.00250.00250.00350.00250.00250.00400.00400.0040,Oit=150120150210190195180160160,Vit=900078007900800060007900850070008500,Dt=12500,13000,14000,Bt=510000,500000,550000Wi=0.39,0.22,0.38,Q=0.0050,ht=1,1.5,1,L=100.Table 16
Periodic order quantity for three time periods.
XitTP1TP2TP3S1900078007900S23434.9250986002.56S3100100100


#### 3.3.1. Sensitivity Analysis for Budget

For budget,Bt′=Bt+k∗ΔB, where ΔB = 5000, t=1,2,…3, and k=−3,−2,−1,0. Budget values are negatively deviated from target values, as shown in Table 17. The K values of −3, −2, −1, and 0 are used to reduce the budget values in sensitivity analysis. It is observed that undesirable deviations from goal 1, goal 2, and goal 4 are zero for all K values. The undesirable deviations from goal3 and total unwanted deviations are constant over all K values. Target values of budget are achieved. The sensitivity analysis for budget results is summarized in Table 18 and shown in Figure 2.Table 17
Deviations from the desired values.
DeviationsPeriod 1Period 2Period 3z1t−179815192349.3219436.7z1t+000z2t−27.532.1834.63z2t+000z3t−4196.312820.503609.43z3t+000z4t−000z4t+34.9232.6435.2Table 18
Sensitivity analysis for budget.
Objective functionK = −3K = −2K = −1K = 0U10000U20000U30.4500.4500.4500.450U40000U0.1120.1120.1120.112Figure 2
Sensitivity analysis for budget.


#### 3.3.2. Sensitivity Analysis for Defective Rate

For defective rate,Qt′=Qt+k∗ΔQ, where ΔQ = 0.0005, t=1,2,…3, and k=−3,−2,−1,0. Defective rate values are negatively deviated from target values as shown in Table 17. As a result, the target values of defective rate are achieved. The K values of −3, −2, −1, and 0 are used to reduce the number of defective products in sensitivity analysis. It is observed that undesirable deviation from the target values of goal 1, goal 2, and goal 4 is decreased to zero over K values whereas undesirable deviations from target values of goal 3 and total unwanted deviation are constant over all K values. The sensitivity analysis for defective rate results is summarized in Table 19 and shown in Figure 3.Table 19
Sensitivity analysis for defective rate.
Objective functionK = −3K = −2K = −1K = −0U10000U20000U30.450.450.450.45U40000U0.1120.1120.1120.112Figure 3
Sensitivity analysis for defective rate.


#### 3.3.3. Sensitivity Analysis for TVP

For TVP,Pt′=Pt+k∗ΔP, where ΔP = 300, t=1,2,…6, and k=−3,−2,−1,0. TVP values are negatively deviated from target values, as shown in Table 17. The K values of −3, −2, −1, and 0 are used to reduce the TVP values in sensitivity analysis. As a result, the target values of TVP are not achieved. Therefore, undesirable deviations from the target values of goal 1, goal 2, goal 4 are zero for all K values. The undesirable deviations from target values of goal 3 and total unwanted deviations appear to increase overall K values. The sensitivity analysis for TVP results is summarized in Table 20 and shown in Figure 4.Table 20
Sensitivity analysis for TVP.
Objective functionK = −3K = −2K = −1K = 0U10000U20000U30.3780.4050.4290.450U40000U0.949E − 010.1010.1070.112Figure 4
Sensitivity analysis for TVP.


#### 3.3.4. Sensitivity Analysis for Demand

For demand,Dt′=Dt+k∗ΔD, where ΔD = 1000, t=1,2,…3, and k=0,1,2,3. Demand values are positively deviated from target values, as shown in Table 17. The K values in the range of 0, 1, 2, and 3 are used for sensitivity analysis for increasing the demand. By increasing the demand, the results show that for all K values, undesirable deviation from target values of goal 1, goal 2, and goal 4 is zero, but total unwanted deviation is declined. Over all K values, the undesirable deviation from goal 3 is decreasing. As a result, the target values of demand are achieved. The sensitivity analysis for budget results is summarized in Table 21 and shown in Figure 5.Table 21
Sensitivity analysis for demand.
Objective functionK = 0K = 1K = 2K = 3U10000U20000U30.4500.4200.3860.343U40000U0.1120.1050.9665E−010.851E−01Figure 5
Sensitivity analysis for demand.


## 3.1. FAHP

Chatterjee proposed a FAHP approach based on the extent analysis method. The method uses linguistic variables to express the comparative judgments given by decision-makers. LetX=x1,xj…xm represent an object set and G=g1,gj…gm a goal set [22]. In the method proposed by Chatterjee, each object xi is taken and extent analysis is performed for each goal gi. As a result, the following signs represented in equation (1) can be used to get m extent analysis values for each object. Equation (2) denotes fuzzy triangular numbers.(1)Mgi1,Mgij,…,Mgim,i=1,2,…,n,(2)where allMgijj=1,2,…,nare triangular fuzzy numbers.The procedure follows the steps outlined in the text.Step 1: find the value of the fuzzy synthetic extent for theith object using equation (3). Equation (4) demonstrates how to do a fuzzy addition. Equations (5) and (6) are used to calculate the value of fuzzy synthetic extent:(3)si=∑j=1mMgii⊗∑i=1n∑j=1mMgii−1,(4)Where∑j=1mMgii.Equation (4) is obtained by performing addition operation of m extent analysis values for a particular matrix such that (5)∑j=1mMgii=∑j=1mlj,∑j=1mmj,∑j=1muj,(6)∑i=1n∑j=1mMgii−1=1∑i=1n∑j=1mui,1∑i=1n∑j=1mmi,1∑i=1n∑j=1mli.Step 2: calculate the probability ofs2l2,m2,u2≥s1=l1,m1,u1, where s2 and s1 are provided by siin equation (3). Equation (7) defines the degree of possibility between two fuzzy synthetic extents. (7)Vs2≥s1=supy≥xminμs2y,μs1x.This can be equivalently expressed as in equations (8) and (9):(8)vs2≥s1=hgts1∩s2=μs2d,(9)μs2d=1ifm2≥m10ifl1≥u2l1−u2m2−u2−m1−l1otherwise.The comparison betweenM1 and M2 requires the values ofVs2≥s1andVs1≥s2.Step 3: compute the degree of possibility for a convex fuzzy number to be greater thank convex fuzzy numbers sii=1,…,k. The degree of possibility is calculated in the following equation:(10)Vs≥s1,s2…sk=Vs≥s1ands≥s2and…ands≥sk=minVs≥si,i=1,2,…k.Step 4: the following equation is used to calculate vectorW′:(11)W′=d′A1,d′A2,…,d′AkT.Assuming that(12)d′Ai=minVsi≥sj,fori=1,2,…,k,j=1,2,…,k,k≠j.The normalized vector is given by the following equation:(13)W=dA1,dA2,…,dAk.W represents the nonfuzzy number computed for each comparison matrix.


## 3.2. Goal Programing

A model for allocating order quantities is presented for three suppliers, multiperiods, and single products, using AGP. In the goal programming model, the achievement function measures the degree to which the unwanted deviation variables of the goals are minimized. The AGP approach has been used to solve the MOMILP model by taking into consideration of the model. The following is a list of the model’s notation.


### 3.2.1. Notations

Dt = demand of the item in period t;ht = holding cost of the item during period t;Oit = ordering cost for supplier i in period t;qit = rate of defective goods from supplier i in period t;Q = maximum defect rate the buyer will tolerate;Bt = budget of the buyer in period t;Vit = capacity of the supplier i in period t;Cit = price of the item for supplier i in period t;L = minimum order quantity;Wi = overall score of suppliers i obtained from the FAHP model Pi=∑i=1nWiVit.


### 3.2.2. Decision Variables

Xit = number of products ordered from supplier i in period t;Yit = 1 if an order is placed on supplier i in time period t, 0 otherwise.


### 3.2.3. Intermediate Variables

It = inventory of the product, carried over from t to period t + 1;zjt+ = positive deviation from the target value of ith goal in period t;zjt− = negative deviation from the target value of ith goal in period.Goal programming formulation for single product, multiple suppliers, and multiple time periods is shown in Table1.Table 1
Goal programming formulation for single product and multiple suppliers.
ModelGoal and constraintU1=∑t=1Tz1t+/∑t=1TBt,U2=∑t=1Tz2t+/∑t=1tQDt,Minimize the achievement functionU3=∑t=1Tz3t−/∑t=1TPt,U4=∑t=1Tz4t−/∑t=1TDtMinU=w1U1+w2U2+w3U3+w4U4MinU=w1∑t=1Tz1t+/∑i−1TBt+w2∑t=1Tz2t+/∑i−1TQDt+w3∑t=1Tz3t−/∑i−1TPt+w4∑t=1tz4t−/∑i−1TDtBudget∑i=1nCitXit+∑j=1nOitYit+htIt+z1t−−z1t+=Bt∀tt=1,2…T∑i=1nqitXit+z2t−−z2t+=QDt−It−1∀tExpected number of defect rate product∑i=1nWiXit+z3t−−z3t+=Pt∀tTotal value of purchasingIt−1+∑i=1nXit+z4t−−z4t+=Dt∀Demand constraintsIt=∑i=1nXit−Dt+It−1∀Material balance equationXit⩽VitYit∀i,tNonnegativity and binary constraintXit≥LYit∀i,tXit⩾0∀i,tYit−0or1integer∀i,t


## 3.2.1. Notations

Dt = demand of the item in period t;ht = holding cost of the item during period t;Oit = ordering cost for supplier i in period t;qit = rate of defective goods from supplier i in period t;Q = maximum defect rate the buyer will tolerate;Bt = budget of the buyer in period t;Vit = capacity of the supplier i in period t;Cit = price of the item for supplier i in period t;L = minimum order quantity;Wi = overall score of suppliers i obtained from the FAHP model Pi=∑i=1nWiVit.


## 3.2.2. Decision Variables

Xit = number of products ordered from supplier i in period t;Yit = 1 if an order is placed on supplier i in time period t, 0 otherwise.


## 3.2.3. Intermediate Variables

It = inventory of the product, carried over from t to period t + 1;zjt+ = positive deviation from the target value of ith goal in period t;zjt− = negative deviation from the target value of ith goal in period.Goal programming formulation for single product, multiple suppliers, and multiple time periods is shown in Table1.Table 1
Goal programming formulation for single product and multiple suppliers.
ModelGoal and constraintU1=∑t=1Tz1t+/∑t=1TBt,U2=∑t=1Tz2t+/∑t=1tQDt,Minimize the achievement functionU3=∑t=1Tz3t−/∑t=1TPt,U4=∑t=1Tz4t−/∑t=1TDtMinU=w1U1+w2U2+w3U3+w4U4MinU=w1∑t=1Tz1t+/∑i−1TBt+w2∑t=1Tz2t+/∑i−1TQDt+w3∑t=1Tz3t−/∑i−1TPt+w4∑t=1tz4t−/∑i−1TDtBudget∑i=1nCitXit+∑j=1nOitYit+htIt+z1t−−z1t+=Bt∀tt=1,2…T∑i=1nqitXit+z2t−−z2t+=QDt−It−1∀tExpected number of defect rate product∑i=1nWiXit+z3t−−z3t+=Pt∀tTotal value of purchasingIt−1+∑i=1nXit+z4t−−z4t+=Dt∀Demand constraintsIt=∑i=1nXit−Dt+It−1∀Material balance equationXit⩽VitYit∀i,tNonnegativity and binary constraintXit≥LYit∀i,tXit⩾0∀i,tYit−0or1integer∀i,t


## 3.3. Case Study

The industry is a well-known large manufacturer that sells the products that make up its supply chain network. In order to gain a competitive advantage in the market, its board of directors wishes to identify suitable suppliers to purchase raw materials for the development of new goods. To choose a supplier, a decision committee of three decision-makers was constituted. For this scenario, the industry selects ten criteria. Cost (C1), quality (C2), lead time (C3), communication system (C4), warranties and claims (C5), risk factor (C6), performance history (C7), carbon reduction programs (C8), payment flexibility (C9), and product personalization (C10) are the factors to consider (C10). Top management has set four goals based on the company’s sales history and sales forecast: minimizing periodic budget, decreasing defect rate, maximizing total value of purchase, and maximizing demand. The following is a summary of the integrated FAHP and goal programming method used to solve this problem, as well as the numerical procedure. In terms of supplier selection and allotment of demand, the integrated fuzzy TOPSIS and MCGP are seen in Figure 1.Figure 1
Framework for the supplier selection and order allocation problem.The linguistic scale for weighing criteria and evaluating alternatives is shown in Table2. Table 3 compares the weights given by decision-makers to various criteria. Table 4 depicts the supplier rating’s fuzzy numbers in relation to criterion C1. Tables 5–13 show the fuzzy number supplier ratings for all criteria. Table 14 depicts weight vector matrix that has been normalized. Table 15 shows the weight and ranking of each supplier.Table 2
Linguistic scale for weighing criteria and evaluating alternatives.
Linguistic termsFuzzy triangular numberInverse fuzzy triangular fuzzy numberEqually important (EI)(1, 1, 1)(1, 1, 1)Equally preferable (EQ)(1, 1, 3)(0.33, 1, 1)Slightly preferable (SP)(1, 3, 5)(0.20, 0.33, 1)Fairly preferable (FP)(3, 5, 7)(0.14, 0.20, 0.33)Extremely preferable (XP)(5, 7, 9)(0.11, 0.14, 0.20)Absolutely preferable (AP)(7, 9, 9)(0.11, 0.11, 0.14)Table 3
Comparison of the weights given to different criteria by decision-makers.
DMC1C2C3C4C5C6C7C8C9C10C1DM1EIEQSPAPXPEQSPEQFPXPDM2EISPFPXPSPXPFPSPXPXPDM3EIEQFPFPFPAPXPFPAPEQC2DM11/EQEIAPXPSPEQXPEQXPEQDM21/SPEIAPFPFPSPAPSPXPFPDM31/EQEIAPFPEQFPAPFPAPAPC3DM11/SP1/APEIEQSPXPEQXPSPEQDM21/FP1/APEIEQFPFPSPFPFPSPDM31/FP1/APEIEQXPAPFPAPXPFPC4DM11/AP1/XP1/EQEIEQFPXPEQFPAPDM21/XP1/FP1/EQEISPFPFPSPXPAPDM31/FP1/FP1/EQEIFPFPAPSPAPAPC5DM11/XP1/SP1/SP1/EQEIEQXPXPXPSPDM21/SP1/FP1/FP1/SPEISPXPAPSPFPDM31/FP1/EQ1/FP1/FPEIFPXPAPFPFPC6DM11/EQ1/EQ1/XP1/FP1/EQEIXPAPEQXPDM21/XP1/SP1/FP1/FP1/SPEIXPAPXPXPDM31/AP1/FP1/AP1/FP1/FPEIEQAPAPEQC7DM11/SP1/XP1/EQ1/XP1/XP1/XPEIXPXPXPDM21/FP1/AP1/SP1/FP1/XP1/XPEISPAPXPDM31/XP1/AP1/FP1/AP1/XP1/XPEIFPAPXPC8DM11/EQ1/EQ1/EQ1/EQ1/XP1/AP1/XPEISPXPDM21/SP1/SP1/SP1/SP1/AP1/AP1/SPEIFPAPDM31/FP1/AP1/SP1/SP1/AP1/AP1/FPEIFPAPC9DM11/FP1/EQ1/SP1/FP1/XP1/XP1/XP1/XPEIEQDM21/XP1/FP1/FP1/XP1/SP1/XP1/XP1/APEISPDM31/AP1/AP1/XP1/AP1/FP1/EQ1/XP1/APEIFPC10DM11/XP1/EQ1/EQ1/AP1/SP1/XP1/XP1/XP1/EQEIDM21/XP1/FP1/SP1/AP1/FP1/XP1/XP1/AP1/SPEIDM31/EQ1/AP1/AP1/AP1/FP1/EQ1/XP1/AP1/FPEITable 4
Fuzzy numbers associated with the supplier rating for criterionC1.
DM1A1A2A3A1(1, 1, 1)(1, 1, 3)(1, 3, 5)A2(0.33, 1, 1)(1, 1, 1)(7, 9, 9)A3(0.20, 0.33, 1)(0.11, 0.11, 0.14)(1, 1, 1)DM2A1A2A3A1(1, 1, 1)(5, 7, 9)(3, 5, 7)A2(0.11, 0.14, 0.20)(1, 1, 1)(7, 9, 9)A3(0.14, 0.20, 0.33)(0.11, 0.11, 0.14)(1, 1, 1)DM3A1A2A3A1(1, 1, 1)(1, 1, 3)(1, 3, 5)A2(0.33, 1, 1)(1, 1, 1)(3, 5, 7)A3(0.20, 0.33, 1)(0.11, 0.11, 0.14)(1, 1, 1)A1A2A3A1(1, 1, 1)(2.33, 3, 5)(1.66, 3.66, 5.66)A2(0.25, 0.17, 0.73)(1, 1, 1)(5.66, 7.66, 8.33)A3(0.18, 0.28, 0.77)(0.11, 0.11, 0.14)(1, 1, 1)Table 5
Fuzzy numbers associated with the supplier rating for criterionC2.
A1A2A3A1(1, 1, 1)(3, 4.3, 5.6)(3, 4.3, 5.6.3)A2(0.21, 0.48, 0.71)(1, 1, 1)(3, 4.3, 5.6.3)A3(0.19, 0.44, 1.53)(0.19, 0.44, 0.51)(1, 1, 1)Table 6
Fuzzy numbers associated with the supplier rating for criterionC3.
A1A2A3A1(1, 1, 1)(3.66, 5, 6.33)(3.66, 5.66, 7.66)A2(0.19, 0.43, 0.49)(1, 1, 1)(4.33, 6.33, 7.66)A3(0.14, 0.20, 0.46)(0.13, 0.17, 0.26)(1, 1, 1)Table 7
Fuzzy numbers associated with the supplier rating for criterionC4.
A1A2A3A1(1, 1, 1)(3.66, 5, 6.33)(3, 5, 7)A2(0.19, 0.43, 0.49)(1, 1, 1)(3.66, 5.66, 7.66)A3(0.15, 0.22, 0.51)(0.13, 0.18, 0.28)(1, 1, 1)Table 8
Fuzzy numbers associated with the supplier rating for criterionC5.
A1A2A3A1(1, 1, 1)(3, 4.33, 5.66)(3.66, 5.66, 7.66)A2(0.21, 0.48, 0.71)(1, 1, 1)(3, 3.66, 5)A3(0.13, 0.18, 0.28)(0.25, 0.70, 0.71)(1, 1, 1)Table 9
Fuzzy numbers associated with the supplier rating for criterionC6.
A1A2A3A1(1, 1, 1)(3, 4.33, 5.66)(5.66, 7.66, 9)A2(0.21, 0.48, 0.71)(1, 1, 1)(3.66, 5, 6.33)A3(0.11, 0.13, 0.18)(0.19, 0.43, 0.49)(1, 1, 1)Table 10
Fuzzy numbers associated with the supplier rating for criterionC7.
A1A2A3A1(1, 1, 1)(1.66, 3.66, 5.66)(3, 3.66, 5)A2(0.18, 0.28, 0.77)(1, 1, 1)(2.33, 3.66, 5.66)A3(0.25, 0.70, 0.71)(0.21, 0.49, 0.73)(1, 1, 1)Table 11
Fuzzy numbers associated with the supplier rating for criterionC8.
A1A2A3A1(1, 1, 1)(3, 5, 7)(2.33, 3.66, 5.66)A2(0.15, 0.22, 0.51)(1, 1, 1)(4.33, 5.66, 7)A3(0.20, 0.46, 0.55)(0.18, 0.41, 0.44)(1, 1, 1)Table 12
Fuzzy numbers associated with the supplier rating for criterionC9.
A1A2A3A1(1, 1, 1)(4.33, 5.66, 7)(3.66, 5.66, 7)A2(0.18, 0.41, 0.44)(1, 1, 1)(2.66, 5.66, 7.66)A3(0.15, 0.21, 0.49)(0.13, 0.18, 0.28)(1, 1, 1)Table 13
Fuzzy numbers associated with the supplier rating for criterionC10.
A1A2A3A1(1, 1, 1)(1.66, 3, 5)(4.33, 6.33, 7.66)A2(0.22, 0.51, 0.77)(1, 1, 1)(3, 5, 7)A3(0.14, 0.19, 0.44)(0.15, 0.22, 0.51)(1, 1, 1)Table 14
Normalized weight vector.
C1C2C3C4C5C6C7C8C9C10A10.280.470.450.4400.550.440.410.440.43A20.390.200.2060.210.230.090.210.240.210.23A30.320.320.340.330.760.340.330.340.340.33Weight of the criteria0.100.100.100.090.100.100.100.100.100.10Table 15
Performance of the supplier.
SupplierPrioritizing weightRanking of the supplierA10.391A20.223A30.382The supplier’s criteria value∑j=1mMgii is calculated as(14)C1=29.92,44.65,60.1,C2=32.51,45.7,57.93,C3=21.59,32.35,45.35,C4=24.24,40.66,44.99,C5=20.06,30.12,39.89,C6=20.47,27.38,33.53,C7=16.15,22.59,29.03,C8=10.83,16.28,19.9,C9=3.84,6.38,9.52,C10=2.40,3.59,7.23,∑i=1n∑j=1mMgii=182.01,269.7,347.02,∑i=1n∑j=1mMgii−1=0.0028,0.0037,0.0054.The value of the fuzzy synthetic extent for the criteria matrix is(15)Sc1=29.92,44.65,60.15∗0.0028,0.0037,0.0054=0.083,0.16,0.32,Sc2=0.091,0.16,0.31,Sc3=0.06,0.11,0.24,Sc4=0.06,0.15,0.24,Sc5=0.05,0.11,0.21,Sc6=0.05,0.10,0.18,Sc7=0.04,0.08,0.15,Sc8=0.03,0.06,0.10,Sc9=0.01,0.02,0.05,Sc10=0.006,0.01,0.03.Weight vectorW′ = (1,1,1,0.93,1,1,1,1,1,1).After normalization, the weight vector is (0.10,0.10,0.10,0.09,0.10,0.10,0.10,0.10,0,10).The weight vector for the alternative supplier is calculated as follows.A1 = (4.99, 7.66, 11.66), A2 = (6.91, 9.37, 10.06), and A3 = (1.29, 1.39, 1.91)∑i=1n∑j=1mMgii is calculated as (13.19, 18.42, 23.63)∑i=1n∑j=1mMgii−1 is calculated as (0.042, 0.054, 0.075)The value of the fuzzy extent for the criteria matrix is(16)SA1=4.99,7.66,11.66∗0.042,0.054,0.075=0.20,0.41,0.87,SA2=6.91,9.37,10.06∗0.042,0.054,0.075=0.29,0.50,0.75,SA3=1.29,1.39,1.91∗0.042,0.054,0.075=0.054,0.07,0.14.The degree of possibility of these fuzzy values is computed as(17)VSA1≥SA2=0.86,VSA1≥SA3=1.70,VSA2≥SA1=1.19,VSA2≥SA3=2.61,VSA3≥SA1=1,VSA3≥SA2=1,W′=0.86,1.19,1.The buyer will not rely on a single supplier because they used multiple sourcing strategies in the purchasing activity. Single sourcing cannot satisfy the demand of the buyer. Therefore, the buyer selects all three suppliers for allocating the order quantities of a single product, multiple suppliers, and multiple time periods using the goal programming approach.The AGP is constructed on the Lingo environment for this mathematical model with three suppliers and three periods. The order is assigned to each supplier and each of the time periods in using Lingo software, as shown in Table16.(18)Cit=26,27,28Cit=24,23,27Cit=24,23,24,qit=0.00250.00250.00250.00350.00250.00250.00400.00400.0040,Oit=150120150210190195180160160,Vit=900078007900800060007900850070008500,Dt=12500,13000,14000,Bt=510000,500000,550000Wi=0.39,0.22,0.38,Q=0.0050,ht=1,1.5,1,L=100.Table 16
Periodic order quantity for three time periods.
XitTP1TP2TP3S1900078007900S23434.9250986002.56S3100100100


### 3.3.1. Sensitivity Analysis for Budget

For budget,Bt′=Bt+k∗ΔB, where ΔB = 5000, t=1,2,…3, and k=−3,−2,−1,0. Budget values are negatively deviated from target values, as shown in Table 17. The K values of −3, −2, −1, and 0 are used to reduce the budget values in sensitivity analysis. It is observed that undesirable deviations from goal 1, goal 2, and goal 4 are zero for all K values. The undesirable deviations from goal3 and total unwanted deviations are constant over all K values. Target values of budget are achieved. The sensitivity analysis for budget results is summarized in Table 18 and shown in Figure 2.Table 17
Deviations from the desired values.
DeviationsPeriod 1Period 2Period 3z1t−179815192349.3219436.7z1t+000z2t−27.532.1834.63z2t+000z3t−4196.312820.503609.43z3t+000z4t−000z4t+34.9232.6435.2Table 18
Sensitivity analysis for budget.
Objective functionK = −3K = −2K = −1K = 0U10000U20000U30.4500.4500.4500.450U40000U0.1120.1120.1120.112Figure 2
Sensitivity analysis for budget.


### 3.3.2. Sensitivity Analysis for Defective Rate

For defective rate,Qt′=Qt+k∗ΔQ, where ΔQ = 0.0005, t=1,2,…3, and k=−3,−2,−1,0. Defective rate values are negatively deviated from target values as shown in Table 17. As a result, the target values of defective rate are achieved. The K values of −3, −2, −1, and 0 are used to reduce the number of defective products in sensitivity analysis. It is observed that undesirable deviation from the target values of goal 1, goal 2, and goal 4 is decreased to zero over K values whereas undesirable deviations from target values of goal 3 and total unwanted deviation are constant over all K values. The sensitivity analysis for defective rate results is summarized in Table 19 and shown in Figure 3.Table 19
Sensitivity analysis for defective rate.
Objective functionK = −3K = −2K = −1K = −0U10000U20000U30.450.450.450.45U40000U0.1120.1120.1120.112Figure 3
Sensitivity analysis for defective rate.


### 3.3.3. Sensitivity Analysis for TVP

For TVP,Pt′=Pt+k∗ΔP, where ΔP = 300, t=1,2,…6, and k=−3,−2,−1,0. TVP values are negatively deviated from target values, as shown in Table 17. The K values of −3, −2, −1, and 0 are used to reduce the TVP values in sensitivity analysis. As a result, the target values of TVP are not achieved. Therefore, undesirable deviations from the target values of goal 1, goal 2, goal 4 are zero for all K values. The undesirable deviations from target values of goal 3 and total unwanted deviations appear to increase overall K values. The sensitivity analysis for TVP results is summarized in Table 20 and shown in Figure 4.Table 20
Sensitivity analysis for TVP.
Objective functionK = −3K = −2K = −1K = 0U10000U20000U30.3780.4050.4290.450U40000U0.949E − 010.1010.1070.112Figure 4
Sensitivity analysis for TVP.


### 3.3.4. Sensitivity Analysis for Demand

For demand,Dt′=Dt+k∗ΔD, where ΔD = 1000, t=1,2,…3, and k=0,1,2,3. Demand values are positively deviated from target values, as shown in Table 17. The K values in the range of 0, 1, 2, and 3 are used for sensitivity analysis for increasing the demand. By increasing the demand, the results show that for all K values, undesirable deviation from target values of goal 1, goal 2, and goal 4 is zero, but total unwanted deviation is declined. Over all K values, the undesirable deviation from goal 3 is decreasing. As a result, the target values of demand are achieved. The sensitivity analysis for budget results is summarized in Table 21 and shown in Figure 5.Table 21
Sensitivity analysis for demand.
Objective functionK = 0K = 1K = 2K = 3U10000U20000U30.4500.4200.3860.343U40000U0.1120.1050.9665E−010.851E−01Figure 5
Sensitivity analysis for demand.


## 3.3.1. Sensitivity Analysis for Budget

For budget,Bt′=Bt+k∗ΔB, where ΔB = 5000, t=1,2,…3, and k=−3,−2,−1,0. Budget values are negatively deviated from target values, as shown in Table 17. The K values of −3, −2, −1, and 0 are used to reduce the budget values in sensitivity analysis. It is observed that undesirable deviations from goal 1, goal 2, and goal 4 are zero for all K values. The undesirable deviations from goal3 and total unwanted deviations are constant over all K values. Target values of budget are achieved. The sensitivity analysis for budget results is summarized in Table 18 and shown in Figure 2.Table 17
Deviations from the desired values.
DeviationsPeriod 1Period 2Period 3z1t−179815192349.3219436.7z1t+000z2t−27.532.1834.63z2t+000z3t−4196.312820.503609.43z3t+000z4t−000z4t+34.9232.6435.2Table 18
Sensitivity analysis for budget.
Objective functionK = −3K = −2K = −1K = 0U10000U20000U30.4500.4500.4500.450U40000U0.1120.1120.1120.112Figure 2
Sensitivity analysis for budget.


## 3.3.2. Sensitivity Analysis for Defective Rate

For defective rate,Qt′=Qt+k∗ΔQ, where ΔQ = 0.0005, t=1,2,…3, and k=−3,−2,−1,0. Defective rate values are negatively deviated from target values as shown in Table 17. As a result, the target values of defective rate are achieved. The K values of −3, −2, −1, and 0 are used to reduce the number of defective products in sensitivity analysis. It is observed that undesirable deviation from the target values of goal 1, goal 2, and goal 4 is decreased to zero over K values whereas undesirable deviations from target values of goal 3 and total unwanted deviation are constant over all K values. The sensitivity analysis for defective rate results is summarized in Table 19 and shown in Figure 3.Table 19
Sensitivity analysis for defective rate.
Objective functionK = −3K = −2K = −1K = −0U10000U20000U30.450.450.450.45U40000U0.1120.1120.1120.112Figure 3
Sensitivity analysis for defective rate.


## 3.3.3. Sensitivity Analysis for TVP

For TVP,Pt′=Pt+k∗ΔP, where ΔP = 300, t=1,2,…6, and k=−3,−2,−1,0. TVP values are negatively deviated from target values, as shown in Table 17. The K values of −3, −2, −1, and 0 are used to reduce the TVP values in sensitivity analysis. As a result, the target values of TVP are not achieved. Therefore, undesirable deviations from the target values of goal 1, goal 2, goal 4 are zero for all K values. The undesirable deviations from target values of goal 3 and total unwanted deviations appear to increase overall K values. The sensitivity analysis for TVP results is summarized in Table 20 and shown in Figure 4.Table 20
Sensitivity analysis for TVP.
Objective functionK = −3K = −2K = −1K = 0U10000U20000U30.3780.4050.4290.450U40000U0.949E − 010.1010.1070.112Figure 4
Sensitivity analysis for TVP.


## 3.3.4. Sensitivity Analysis for Demand

For demand,Dt′=Dt+k∗ΔD, where ΔD = 1000, t=1,2,…3, and k=0,1,2,3. Demand values are positively deviated from target values, as shown in Table 17. The K values in the range of 0, 1, 2, and 3 are used for sensitivity analysis for increasing the demand. By increasing the demand, the results show that for all K values, undesirable deviation from target values of goal 1, goal 2, and goal 4 is zero, but total unwanted deviation is declined. Over all K values, the undesirable deviation from goal 3 is decreasing. As a result, the target values of demand are achieved. The sensitivity analysis for budget results is summarized in Table 21 and shown in Figure 5.Table 21
Sensitivity analysis for demand.
Objective functionK = 0K = 1K = 2K = 3U10000U20000U30.4500.4200.3860.343U40000U0.1120.1050.9665E−010.851E−01Figure 5
Sensitivity analysis for demand.


## 4. Result and Conclusion

A FAHP is presented in this paper as a method for selecting suppliers for vagueness and uncertainty. By integrating fuzzy logic into MCDM methods, uncertainty can be avoided and decision-makers can determine the criteria in an uncertain environment. To select suppliers and allocate order quantities, the study integrated FAHP with goal programming. When a supplier offers multiple time periods, finding the right supplier and splitting the orders with the appropriate suppliers have become a major challenge for companies. Therefore, it is challenging to choose a supplier that can procure materials according to predetermined criteria and organizational needs. This study was done where three suppliers were ranked by three decision-makers based on ten criteria, and then the most appropriate supplier was selected. This paper considers a case in which the total demand for a single product over multiple periods is known. In this issue, the MOMILP model is presented to determine the optimum quantities among selected suppliers. The objectives of using the mentioned model were enhancing the TVP, minimizing the budget, diminishing the defective rate, and maximizing the demand. According to the suggested model, by applying the supplier’s weight as a coefficient of TVP, the model determines the appropriate supplier. It also assigns optimal order quantities to the determined suppliers under model constraints. This research paper presented a numerical example and a sensitivity model to show the efficiency of the model as well as their impacts on the values of supplier order quantity.FAHP determines that A1 is the optimal supplier based on its relative closeness coefficient. Multiple period demand allocations to FAHP determines that A1 is the optimal supplier based on its relative closeness coefficient. Multiple period demand allocations to suppliers are calculated by Lingo software asX11 = 9000, X12 = 7800, X13 = 7900, X21 = 3434.9, X22 = 5098, X23 = 6002.56, X31 = 100, X32 = 100, and X33 = 100. All the target values of goals are achieved for all periods except the total values of purchasing. Therefore, the profit of the buyer is maximized, quality of product is maximized, the supply chain is strengthened, customer satisfaction is increased, and continuous flow of products is achieved. The numerical example shows that the single item, multiobjective model can provide buyers with an effective way to select the optimal suppliers and allocate order quantities among them.The industry’s buyer was not reliant on a single supplier. When a supplier refuses to deliver materials, the suggested model helps managers by allowing them to change the values of the variables in the model to their preferences and determining the appropriate order quantity among the suppliers. If a buying manager is ready to reduce costs, lower faulty rates, and increase demand, this model could be useful.Various extension strategies can be used to extend the research presented in this paper on multiobjective evaluation among suppliers. The first is to consider quantity discounts during demand allocation among suppliers. A second possibility is to consider the uncertainty of demand instead of fixed demand in supplier selection problems. Finally, the research can be extended by combining strategies to allocate demand among suppliers that are uncertain and multiobjective. Modifying the demand valve while assigning orders to suppliers is not possible, which is one of the study’s limitations. Customer needs might cause demand to fluctuate in real time. The model supports purchasing managers in assessing the performance of suppliers and in making a decision about satisfactory demand allocation among suppliers based on the evaluations of different suppliers.



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*Source: 8695983-2022-07-13.xml*