38 Int. J. Integrated Supply Management, Vol. 10, No. 1, 2016

Copyright © 2016 Inderscience Enterprises Ltd.

A goal programming approach for supplier evaluation and demand allocation among suppliers

Amol Singh Faculty of Operations Management, Indian Institute of Management, Rohtak Rohtak, Haryana, 124001, India Email: amol.singh@iimrohtak.ac.in

Abstract: This paper presents a hybrid algorithm for supplier evaluation and demand allocation among the suppliers. The objective here is to minimise the inventory and transportation costs and simultaneously to maximise the total purchase value of the items taking into consideration demand condition, supplier capacity, budget and delivery lead-time constraints. Since the problem is multi-objective decision making, we solve this problem by converting all mixed integer programming objectives in to single objective with the help of goal programming approach. The customer demand is allocated among the suppliers by using a hybrid algorithm based on the technique for order preference by similarity to ideal solution (TOPSIS), fuzzy set theory, MILP, and goal programming approaches. The results are validated by computational experiment and prove the efficacy of the hybrid algorithm.

Keywords: supplier evaluation; supplier selection; demand allocation; goal programming; TOPSIS; technique for order preference by similarity to ideal solution.

Reference to this paper should be made as follows: Singh, A. (2016) ‘A goal programming approach for supplier evaluation and demand allocation among suppliers’, Int. J. Integrated Supply Management, Vol. 10, No. 1, pp.38–62.

Biographical notes: Amol Singh received his Masters in Industrial Engineering from the Moti Lal Nehru National Institute of Technology Allahabad and a PhD in Industrial Engineering from the Indian Institute of Technology Roorkee. He received his MHRD Fellowship for full time PhD Research Work at the IIT Roorkee. Currently, he is a Faculty in Operations Management at the Indian Institute of Management Rohtak, India. He has published several research papers in international journals and conferences. His main research interests include operations management, simulation modelling and analysis, and applications of operations research.

1 Introduction

Supplier selection is defined as the process of selecting the right suppliers, at the right place, at the right time, in the right quantities and with the right quality (Ayhan, 2013). A typical manufacturer spends approximately 60% of total income from sales on procurement of material such as raw material, intermediate parts and components (Krajewski et al., 2007). It has also been reported that procurement of goods and services

A goal programing approach for supplier evaluation 39

constitutes up to 70% of product cost (Ghodsypour and O’Brien, 1998). These stylised facts indicate that selecting the right supplier will result in reducing operational costs, increasing profitability and quality of products, improving competitiveness in the market and responding to customers’ demands rapidly (Abdollahi et al., 2015; Onut et al., 2009). Moreover, customer satisfaction is also enhanced by determining the best supplier (Amin and Razmi, 2009). In the supplier-selection problem, various criteria are specified and evaluated with respect to different suppliers’ attributes. Therefore, this problem may be considered as a multiple criteria decision-making (MCDM) problem and can enhance the purchase value in terms of cost, quality and on-time delivery of the items purchased (Singh, 2014). Additionally, companies are also facing tough competition from their rivals. To overcome this competitive pressure, companies are paying more attention to core competencies. They have increased their level of outsourcing, and are relying predominantly on their supply chains as the source of competitive advantage.

In supplier-selection problem there is a set of multiple and usually conflicting criteria, which are supposed to be ranked to select the best favourable supplier by evaluating the criteria among various candidates (Yu et al., 2013). Moreover, supplier-selection criteria are qualitative or quantitative in nature. These criteria can be defined variously as buyers take into account numerous conflicting factors. Illustratively, low price can offset poor quality or delivery lead time. Dickson (1966) identified 23 criteria in his study of various supplier-selection problems. He reported that quality, delivery and performance history are the three most important criteria. Similarly, Weber et al. (1991) in a review of 74 papers obtained similar results pertaining to the multi-criteria nature of supplier-selection problem. In the following years, several studies have been reported to specify the factors affecting the supplier performance. Ellram (1990), Roa and Kiser (1980), Stamm and Golhar (1993) identified 60, 18 and 13 criteria for supplier selection, respectively (Ghodsypour and O’Brien, 1998). Similarly, Wang et al. (2009) adopted 12 performance metrics to assess the supplier’s performance. In a review of 42 research papers, Singh (2014) identified 40 criteria for various supplier-selection problems. Nudurupati et al. (2015) discussed the role of stakeholders in the supplier-selection problem. Dey et al. (2015) reported quality, delivery, costing, organisational capability, environmental practices, social practice and risk management practice as the important criteria for measuring the performance of the suppliers. Although there are many different criteria used in various studies, Weber et al. (1991) determined that net price, delivery and quality are the most important criteria in the performance evaluation of the suppliers.

After selecting the performance metric for supplier evaluation, stakeholders assign scores to each supplier against each criterion (Singh, 2014), which forms multi-criteria group decision-making problem for ranking the suppliers under the scores assigned by different stakeholders and is solved by using various solution approaches. A literature review of 74 research papers carried out by Weber et al. (1991) classified the solution approach into three categories: linear weighting models, mathematical programming approaches and probabilistic approaches. In the following years, several methodologies have been used for the supplier-selection problem but many of them only discuss the case of traditional supplier-selection problems (TSSPs) (Ware et al., 2014). In the TSSP, suppliers are ranked and the top ranked supplier is supposed to be selected for the entire planning horizon unless it is re-ranked or re-assessed (Ware et al., 2014). In the literature, a variety of approaches have been suggested to solve the problem of supplier selection, which include analytic hierarchy process (AHP) (Deng et al., 2014; Shaw et al., 2012),

40 A. Singh

grey relational analysis (Rajesh and Ravi, 2015), technique for order preference by similarity to ideal solution (TOPSIS) (Lima et al., 2014), data envelopment analysis (DEA) (Kumar et al., 2014; Toloo and Nalchigar, 2011) and some hybrid methods (Abdollahi et al., 2015; Singh, 2014; Vahdani et al., 2013). In addition, a detailed review and classification of the supplier-selection methods can be found in Chai et al. (2013) and Ho et al. (2010). In the supplier-selection problem, exact data are inadequate to model real-life situations, hence, fuzzy set theory (Chen et al., 2006) was incorporated to deal with the vagueness and ambiguity in the real decision-making process. Kannan et al. (2014) proposed a framework using fuzzy TOPSIS to evaluate green suppliers for a Brazilian electronics company based on the criteria of green supply chain management practices. Roshandel et al. (2013) presented the fuzzy hierarchical TOPSIS for supplier selection and evaluation in a detergent production industry. Moreover, several researchers used fuzzy set theory in their study to capture the uncertainties of input data (Liou et al., 2014; Lee et al., 2014; Karsak and Dursun, 2014; Buyukozkan and Çifçi, 2012; Deng and Chan, 2011).

Plethora of literatures on supplier-selection or vendor-selection problems are available, whereas the dynamic supplier-selection problem considering multi-period multi-parts multi-source is not adequately addressed in the past (Ware et al., 2014). Most of the reported literature on supplier selection focuses on the selection of a single supplier for the entire planning horizon (Ware et al., 2014; Singh, 2014). However, there are instances when a single supplier is not capable to meet the demand of the buyer. Furthermore, the aspect of full and low truckload, inventory cost and transportation cost is not addressed adequately in the context of supplier-selection problems.

Hence, to find the best suppliers under suppliers and buyer limitations environment, a two-stage novel integrated approach including fuzzy TOPSIS (to determine the relative weights of different criteria and rating of suppliers with respect to given criteria), mixed integer linear programming (MILP) and goal programming model is proposed in this study. The contributions of the present study differ from the other related studies in the literature. First, qualitative and quantitative criteria apply in the selection of a pool of suppliers. Second, fuzzy TOPSIS algorithm tackles the risk of supplier-selection environment. It defuzzifies the fuzzy data in the final step of the ranking process. Third, integration is provided between the outputs of fuzzy TOPSIS, MILP and the goal programming model. The model uses the weights of criteria and supplier rating which are derived from fuzzy TOPSIS as an input and finally minimises the transportation cost, inventory cost and decides the suppliers along with the number of full and low truckload. Unlike the existing MILP models in the literature, the model differs from them as it considers many conditions simultaneously.

The remainder of the paper is planned as follows. Section 2 provides the review of solution methodology used in supplier-selection problems. Section 3 identifies the research issues, which form the basis for problem formulation, and then presents the objectives and framework of the study. Section 4 suggests a mathematical model to solve the supplier evaluation and multi-objective demand allocation problem. Section 5 presents the methodology used in the study. Section 6 reports the computational experiment and the findings of the computational experiments. Section 7 presents the managerial implications, conclusion and future research directions are presented in Section 8.

A goal programing approach for supplier evaluation 41

2 Review of the literature

Supplier selection and demand allocation problem is a multi-objective decision-making problem involving both qualitative and quantitative performance measures. Usually, several conflicting criteria make the supplier-selection problem a complex problem. It is often desirable to make a compromise among the conflicting criteria. In the supplier-selection problem, quality, delivery and performance history are the three most important criteria (Dickson, 1966; Weber et al., 1991). Although most buyers still consider cost to be their primary concern, new more interactive and interdependent selection criteria are increasingly being used. Being a multi-criteria decision-making problem, several approaches have been reported in the literature. Bhutta (2003) provided a review of 154 supplier-selection research papers and alternative methods/techniques adopted. Supplier evaluation has assumed a strategic role in determining competitiveness of large manufacturing companies. An increasing number of researches have been devoted to the development of different kinds of methodologies to cope with this problem. Bruno et al. (2012) reported that mathematics, statistics, artificial intelligence, qualitative models, mathematical combined models, artificial intelligence combined models and hybrid combined models were used by 70, 21, 35, 28, 27, 3 and 34 researchers, respectively, in a supplier-selection problem. Moreover, Bruno et al. (2012) emphasised the use of fuzzy set theory and a need for the development of hybrid-combined models for supplier-selection problems.

The literature shows a variety of methodologies and approaches used for the supplier-selection problem. Traditionally, linear weighting models, total cost approach, multiple attribute utility theory and total cost ownership were used for supplier selection. In the last one decade researchers focused on optimisation techniques, multi-objective programming, AHP, data envelop analysis, artificial intelligence and hybrid approaches. A brief description of alternative approaches in terms of general application, features and limitations is discussed in the following sections.

2.1 Optimisation techniques

Linear programming (Ghodsypour and O’Brien, 1998), dynamic programming (Masella and Rangone, 2000) and multi-objective programming (Weber and Ellram, 1993) are the popular optimisation techniques. Zhang and Zhang (2011) used the MILP approach to solve the supplier-selection problem under stochastic demand. They selected the suppliers and allocated the ordering quantity properly among the selected suppliers to minimise the total cost including selection, purchasing, holding and shortage costs. Swaik (2011) also applied the MILP approach to study the problem of order allocation of parts among the suppliers in a customer-driven supply chain. The study suggested that future research could consider supplier selection in a customer-driven supply chain, taking into account risk and dynamic multi-period demand. Osman and Demirli (2010) addressed the supplier-selection problem related to an aerospace company and tried to optimise its outsourcing strategies to meet the expected demand and customer satisfaction requirements under delivery dates and approved budget. They used the goal-programming approach to achieve the company’s objectives.

42 A. Singh

2.2 Multi-objective programming

Multi-objective programming approach considers several criteria simultaneously and makes the tradeoff among the key supplier-selection criteria. This approach is especially suitable to just-in-time scenarios (Weber and Ellram, 1993). An additional flexibility of this approach is that it allows a varying number of suppliers into consideration and offers suggested volume allocation to the supplier. Ho et al. (2010) provide a literature review on multi-criteria decision-making approaches for supplier evaluation and selection. The objective of their survey was to determine which approaches were used widely in the literature, which evaluating criteria received more attention and whether there was any inadequacy of the approaches. They found the individual approach (58.97%) more popular than the integrated approach (41.03%). In the individual approach, the most popular approach is DEA followed by mathematical programming, AHP, CBR, analytic network process (ANP), fuzzy set theory and genetic algorithm. DEA attracted researchers mainly because of its robustness. The wide applicability of individual approach was due to its simplicity, ease of use and flexibility (Ho et al., 2010). In the supplier-selection problem, besides the rating of suppliers, the decision-makers also need to consider the resource limitations (budget of buyer and capability of supplier). In this situation, goal programming can be preferred to AHP/ANP/TOPSIS techniques. The decision-making process facilitates when AHP/ANP/TOPSIS and GP are integrated. They suggested that the voice of stakeholders should be considered and the evaluating criteria should be derived from the requirements of stakeholders using a series of house of quality. As far as the evaluating criteria for supplier selection is concerned, they reported that 87.18% of buyers considered quality in the supplier selection. The second most popular criterion (82.05%) was delivery and the third most popular criterion (80.77%) was price or cost.

Choi and Hartley (1996) studied the supplier-selection practice in US auto industries and concluded that quality, delivery and consistency were the key factors. However, price turned out to be the least important factor in supplier selection. These facts indicate that the traditional single criterion approach based on lowest cost bidding is no longer supportive in supply chain management. In the last decade, numerous studies have focused on the supplier-selection process using alternative approaches such as single objective technique, i.e., cost ratio method, linear or mixed integer programming and multi-objective techniques, i.e., goal programming (Ghodsypour and O’Brien, 1998; Yan et al., 2003; Oliveria and Lourenco, 2002). Despite their usefulness, the optimisation methods suffer from certain drawbacks associated with their implementation. Of particular interest, the major shortcoming is the exclusion of qualitative criteria, considered important in supplier-selection problem in both the single and multi-objective programming.

2.3 Analytic hierarchy process

AHP developed by Saaty (1980) is a mathematical procedure to assign weights to several alternatives using a scheme of pairwise comparison. The model has witnessed applications to a wide variety of decision-making areas including research and development, project selection, supplier selection, evaluating alternatives, etc. This method allows the decision-maker to convert the complex problems in the form of a hierarchy or a set of integrated levels. The advantage of the hierarchical structure is that it

A goal programing approach for supplier evaluation 43

allows the systematic decomposition of the overall problem into its fundamental components and interdependencies with a large degree of flexibility. This is the main reason for choosing the AHP for tackling the supplier-selection problem, which involves many intangible factors (Nydick and Hill, 1992). Generally, the hierarchy has at least three levels, namely the goal, the criteria and the alternatives. For the supplier-selection problem, the goal is to select the best overall supplier, the criteria could be quality, on-time delivery, price, etc., and the alternatives are the different proposals supplied by the suppliers. Bruno et al. (2012) reported that in a corporate environment, AHP is one of the most prominent methodologies for supplier evaluation. They also reported the strength and weakness of AHP methodology in the supplier evaluation process. The suitability of AHP to the supplier-selection problem derives from four distinctive characteristics:

• ability to handle both tangible and intangible attributes

• ability to structure the problems in a hierarchical manner

• ability to monitor the consistency with which a decision-maker makes a judgement

• ability to provide a synthetic score for each supplier.

As regards the drawbacks, its use is not straightforward for practitioners. A consensus is required for aggregating individual judgements for the pair-wise comparison matrices. There cannot be a single hierarchy for most of the supplier-selection problems. The reliability of the outcome depends not only on the quality of the data but also on the knowledge and judgements of decision-makers (Chan and Chan, 2004).

2.4 Data envelopment analysis

DEA postulates the concept of the efficiency of a decision alternative. The benefit criterion (output) and the cost criterion (input) determine the decision alternative, i.e., the supplier. The ratio of the weighted sum of outputs (i.e., the performance of the supplier) to the weighted sum of inputs (i.e., the costs of using the supplier) determines the efficiency of a supplier. For each supplier, the DEA method finds the most favourable set of weights, i.e., the set of weights that maximises the supplier's efficiency without making its own or any other supplier's efficiency greater than one. In this way, DEA helps the buyer in classifying the suppliers into two categories: efficient suppliers (efficient frontier) and inefficient suppliers. Weber (1996) reported the efficacy of the DEA approach in supplier-selection problems, especially when multiple and conflicting criteria are considered. Toloo and Nalchigar (2011) proposed an integrated DEA model to evaluate the overall performance of the suppliers in the presence of cardinal as well as ordinal data. They considered three evaluating factors including total cost of shipment, supplier reputation and bill received from the supplier without error. They emphasised on the simultaneous consideration of cardinal and ordinal data in a supplier-selection process neglected by previous studies.

2.5 Artificial intelligence

The literature provides evidence of researchers using various advanced techniques: neural network (Luo et al., 2009; Aksoy and Ozturk, 2011), genetic algorithm, ANP, fuzzy set

44 A. Singh

theory (Ozkok and Tiryaki, 2011; Yucel and Guneri, 2011) and hybrid approach for supplier selection. In the real supplier-selection environment, the modelling of many situations may not be sufficient or exact, as the available data are inexact, vague, imprecise and uncertain by nature (Ozkok and Tiryaki, 2011; Yucel and Guneri, 2011). The decision-making processes that take place in such situations have to contend with uncertain or imprecise information. For managing the vagueness and uncertainty of the problems, the fuzzy set theory can prove an effective method (Chen et al., 2006). To model such situations, the fuzzy set theory represents the uncertainty in terms of linguistic variable converted into fuzzy numbers. Most of the fuzzy multi-criteria supplier-selection models defuzzify into a crisp one in the initial stage, thereby defeating the very purpose of collecting fuzzy data pertaining to different opinions of decision-makers.

2.6 Hybrid approaches

Recognising the fact no particular technique can provide a generalised perspective on the supplier-selection problems, researchers have been encouraged to develop the hybrid approaches. Within the framework of hybrid approaches, alternative supplier-selection models are integrated for achieving a richer model based on a combination of advantages of different techniques. Kumaraswamy et al. (2011) used TOPSIS and quality function deployment (QFD) techniques for supplier selection in an SME. Amid et al. (2009) used the fuzzy set theory and MILP techniques for demand allocation to suppliers. During their investigation, they considered three objective functions encompassing minimisation of the cost, the rejected items and the late deliveries, under the capacity and demand requirement constraints. Bhattacharya et al. (2010) used AHP and QFD in combination with cost factor measure to rank the suppliers. For evaluating the suppliers, they considered eight criteria, namely delivery, quality, responsiveness, management discipline, financial position, facility and technical capability. Liao and Kao (2011) developed a hybrid methodology using fuzzy set theory, TOPSIS and goal programming to measure the comparative rating of the suppliers. This integrated approach allows the decision-makers to set multiple aspiration levels for supplier-selection problems. Zouggari and Benyoucef (2012) proposed a hybrid approach combining fuzzy set theory, ANP and TOPSIS for prioritising the suppliers. Lee et al. (2014) integrated AHP and TOPSIS based on the fuzzy theory to determine the prior weights of multiple criteria and select the best-fit suppliers taking the subjective and vague preferences of decision-makers into consideration. Karsak and Dursun (2014) proposed a novel fuzzy MCDM framework for supplier selection by integrating QFD and DEA which considers the impacts of inner dependence among supplier assessment criteria through constructing a house of quality. Buyukozkan and Çifçi (2012) proposed a hybrid MCDM model to evaluate green suppliers by combining fuzzy DEMATEL, fuzzy ANP and fuzzy TOPSIS. Deng and Chan (2011) developed an MCDM methodology by using fuzzy set and TOPSIS to deal with supplier selection under uncertain environments. Chen (2011) proposed a structured methodology for supplier selection and evaluation based on a combination of DEA and TOPSIS approaches. The major drawback of this model is that it does not capture the uncertainties of supplier-selection environment. Vinod et al. (2011) used a hybrid model, which

A goal programing approach for supplier evaluation 45

integrated AHP with fuzzy set theory. The AHP methodology enabled crisp value of numerical judgements at the initial stage rather than at the final stage, which could have been beneficial for addressing the problem of fuzzification in the supplier-selection model.

From a critical perspective, most of the reported studies share some common features. Firstly, a single supplier-selection problem constitutes the focus for most studies. However, there may be instances when the single supplier does not have the capacity to fulfil the demand of the buyer company. The supplier may also be incapable to supply the quantity of desirable items due to some unexpected events. In these circumstances, a pool of suppliers could satisfy the demand of the buyer company. Secondly, in a real supplier-selection process, most input information can be known imprecisely. In this context, researchers rely on hybrid models involving fuzzy sets theory for handling uncertainty. However, hybrid models also suffer from drawbacks. Moreover, hybrid approaches are scarce (Sevkli et al., 2008; Bruno et al., 2012). Research in this area can be useful to develop efficient hybrid methodologies, which could improve the efficacy of the input information and could select a pool of suppliers along with the order size and number of full and low truckload.

3 The proposed approach

A critical review of the literature brings to the fore some crucial issues for research. Firstly, in the real supplier-selection environment, companies often have a pool of suppliers for meeting the demand of the items. This situation creates a new scenario in the supplier-selection problem. Secondly, supplier selection is a multi-criteria decision-making problem, encompassing risk and uncertainties. Thirdly, supplier-selection models based on fuzzy set theory adopt the practice of data defuzzified into a crisp one in the initial stage. This approach undermines the advantage of collecting the fuzzy data (opinion of all decision-makers). Fourthly, in the literature, simultaneous consideration of procurement value (PV), transportation cost, inventory cost, and the concept of full and low truckload has not received much attention in the context of supplier-selection process.

In view of the above, the main objective of the present study is to develop a hybrid model to address all the above issues. Hence, the study aims at addressing the following tasks. Firstly, a complex problem of multi-objective demand allocation among suppliers is considered and the rating of the suppliers is integrated with the MILP model to minimise the inventory and transportation costs and simultaneously to maximise the value of procurement. The model selects a pool of suppliers along with the number of full and low truckload. Secondly, qualitative and quantitative criteria apply in the computation of the supplier rating. Thirdly, fuzzy TOPSIS algorithm tackles the risk of supplier-selection environment. It defuzzifies the fuzzy data in the final step of the ranking process. Fourthly, the hybrid fuzzy TOPSIS algorithm is integrated with the MILP model and is solved by using goal programming approach.

The conceptual framework of the hybrid model is shown in Figure 1. The hybrid model integrates fuzzy set theory, TOPSIS, MILP and goal programming methodologies to solve the problem of multi-objective demand allocation among candidate suppliers

46 A. Singh

under demand, budget, delivery lead-time and capacity constraints. ‘n’ number of stakeholders are involved in the selection process of the suppliers, they assign their assessment to each candidate suppliers based on ‘m’ criteria, namely reliability, quality, price, consistency, on time delivery, etc. Each stakeholder assigns his assessment to each candidate supplier. The assessment of each stakeholder is converted into fuzzy score by applying the fuzzy set theory. Further, the rating of candidate suppliers is computed by using the fuzzy TOPSIS algorithm. The available capacity and delivery lead-time of the ith supplier are Ci and LTi, respectively. The demand of the buyer company is D and the maximum/minimum budget for procurement of the items is Bmax and Bmin, respectively. Under these constraints of buyer/suppliers limitations, the MILP model is developed with the objective of PV and cost including transportation and inventory cost. Finally, these two objectives are converted into single objective and the multi-objective model is solved by using the goal programming approach. These steps are discussed briefly in Sections 4 and 5.

Figure 1 The structure of the proposed model

4 Model formulation

Decision-makers assign their assessment to each candidate suppliers based on a set of criteria and in terms of linguistic variables. These linguistic variables are converted into fuzzy number as shown in Tables 1 and 2. Each decision-maker assigns fuzzy score to the supplier against a criterion and similarly assigns the fuzzy score to each criterion.

A goal programing approach for supplier evaluation 47

These fuzzy scores are expressed in the fuzzy matrix format. X = [xij]m×n is a fuzzy decision matrix in which A1, A2, …, Am are ‘m’ possible suppliers and C1, C2, …, Cn are ‘n’ possible criteria. The performance of supplier Ai with respect to criteria Cj is expressed by xij and the weight of criterion Cj is expressed by wj. Where xij and wj are represented by triangular fuzzy scores. A triangular fuzzy score is represented as

1 1 2 3 3 [( , ); ; ( )].x x x x x x′ ′= Technique to order preference by similarity to ideal solution (TOPSIS) owes to Hwang and Yoon (1981). The steps of hybrid fuzzy TOPSIS methodology are explained below:

Step 1: A group of ‘k’ decision-makers is identified and this group defines a set of relevant criteria for supplier evaluation. It may be noted that the interval value allows the decision-maker to define the lower bound and upper bound values for matrix element and for the weight of each criterion.

Step 2: Determine the score of the considered suppliers with respect to each criterion and similarly the score of each chosen criterion.

Step 3: Compute average score of each supplier with respect to a criterion and average weight of each criterion. For instance, if in the decision-making group there are ‘K’ decision-makers and each assigns their own score to each supplier with respect to a criterion and similarly to each criterion. The average scores of each supplier with respect to a criterion and importance of each criterion are computed by using the following relations:

1 2 1/[ ]k kij ij ij ijx x x x= × × (1)

1 2 1/[ ] .k kij ij ij ijw w w w= × × (2)

Table 1 Linguistic variables for the rating of the suppliers

Very poor (VP) [(0, 0); 0; (1, 1.5)] Poor (P) [(0, 0.5); 1; (2.5, 3.5)] Moderately poor (MP) [(0, 1.5); 3; (4.5, 5.5)] Fair (F) [(2.5, 3.5); 5; (6.5, 7.5)] Moderately good (MG) [(4.5, 5.5); 7; (8, 9.5)] Good (G) [(5.5, 7.5); 9; (9.5, 10)] Very good (VG) [(8.5, 9.5); 10; (10, 10)]

Table 2 Linguistic variables for the importance of the criterion

Very low (VL) [(0, 0); 0; (.1, .15)] Low (L) [(0, 0.05); 0.1; (0.25, 0.35)] Medium low (ML) [(0, 0.15); 0.3; (0.45, 0.55)] Medium (M) [(0.25, 0.35); 0.5; (0.65, 0.75)] Medium high (MH) [(0.45, 0.55); 0.7; (0.8, 0.95)] High (H) [(0.55, 0.75); 0.9; (0.95, 1)] Very high (VH) [(0.85, 0.95); 1; (1, 1)]

48 A. Singh

Step 4: The fuzzy scores computed in Step 3 are in different units, hence these scores are normalised by converting them into non-dimensional ratio for comparison. Fuzzy scores xij = [(aij, aij

’); bij; (cij cij’)] computed in step 3 are converted into

non-dimensional ratio by using the following equations:

' '

, ; ; , , 1, 2, , , ij ij ij ij ijij

j j j j j

a a b c cr i m j

c c c c c+ + + + +

= = ∈ Ω

… (3)

' ', ; ; , , 1, 2, , , ,j j j j jij

ij ij ijjj ij

a a a a ar i m j

a b ca c

− − − − − = = ∈ Ω

… (4)

where

max( ),j ij bc c j+ = ∈ Ω

min( ), .j ij ca a j− ′= ∈ Ω

Step 5: The normalised fuzzy decision matrix computed in Step 4 into weightage normalised fuzzy decision matrix (v = [vij]n×m, where vij = rij × wj) by using the following relation:

( ) ( )( ) ( )

1 1 1 1 2 2 3 3 3 3 , ; ; , .

, ; ; , .

ij ij j ij j j j ij j ij j

ij ij ij ij ij

v r w r w r w r w r w

g g h l l

′ ′= × × × × × ′ ′=

(5)

Step 6: Determine positive ideal (A+) and negative ideal (A–) solutions by using the following equations:

A+ = [(1, 1); 1; (1, 1)] for j ∈ Ωb (6)

A– = [(0, 0); 0; (0, 0)] for j ∈ Ωc. (7)

Step 7: Compute the separation measures (Euclidean distances) of each supplier from the positive ideal (A+) and negative ideal (A-) solutions by using the following relations:

32

1

1( , ) [( ) ]3

i

xi yii

D N M N M=

− − −

=

= −∑ (8)

32

1

1( , ) [( ) ],3

i

xi yii

D N M N M=

+ + +

=

= −∑ (9)

where D– (N, M) and D+ (N, M) are the primary and secondary separation measures, hence the separation measure of each supplier 1 2,I ID D+ + from the positive ideal solution is computed by using the following equations:

2 2 21

1

1 [( 1) ( 1) ( 1)3

n

i ij ij ijj

D g h l+

== − + − + −∑ (10)

A goal programing approach for supplier evaluation 49

2 2 22

1

1[( 1) ( 1) ( 1) .3

n

i ij ij ijj

D g h l+

=

′ ′ ′= − + − + −∑ (11)

Similarly, the separation measure of each supplier 1 2( , )i iD D− − from the negative ideal solution is computed by using the following relations:

2 2 21

1

1[( 0) ( 0) ( 0)3

n

i ij ij ijj

D g h l−

== − + − + −∑ (12)

2 2 22

1

1[( 0) ( 0) ( 0) .3

n

i ij ij ijj

D g h l−

=

′ ′ ′= − + − + −∑ (13)

Step 8: The fuzzy relative closeness of each supplier by using each pair of separation measure as computed in Step 7. The fuzzy relative closeness of supplier Ai is computed by using the following equations:

2 11 2

2 2 1 1

, and .i i

i i i i

D DRC RC

D D D D

− −

+ − + −= =+ +

(14)

Step 9: Compute crisp value of the fuzzy relative closeness as computed in step 8 of each supplier by using the following relation:

* 1 2 .2i

RC RCRC

+=

(15)

The above relation determines the ranking of the suppliers. A buyer company makes a choice among several suppliers and splits the order quantity among the suppliers. In this context, we propose a hybrid model based on TOPSIS, fuzzy set theory and goal programming for demand allocation in terms of full truckload and low truckload among the candidate suppliers. The proposed model minimises the inventory and transportation costs and simultaneously maximises the PV of goods subjected to budget, demand, capacity and lead-time constraints. The symbols used in the proposed model are defined as follows:

n: Number of suppliers D: Demand of the buyer Ci: Capacity of ith supplier Q: Order quantity to all suppliers Qi: Order quantity to ith supplier in each Qfi: Number of units allocated to ith supplier for full truck load Qli: Number of units allocated to ith supplier for low truck load. Co: Ordering cost per order of ith supplier Pi: Sales price of ith supplier

50 A. Singh

Sfi: Fixed shipping cost of ith supplier for full truck load Sli: Fixed shipping cost of ith supplier for low truck load Svfi: Variable shipping cost of ith supplier for full truck load Svli: Variable shipping cost of ith supplier for low truck load fi: Number of full truck load assigned to ith supplier li: Number of low truck load assigned to ith supplier Oi: Ordering cost for ith supplier Ci: Production capacity of ith supplier Li: Delivery lead time required by ith supplier LT: Lead time imposed by the buyer company Cc: Inventory carrying cost of buyer per unit per period di: Distance of ith supplier from the buyer company Bmax: Maximum budget limit Bmin: Minimum budget limit RCi: Relative closeness index of ith supplier FTc: Capacity of full truck load LTc: Capacity of low truck load.

The following relation denotes the total cost, which includes transportation, ordering, carrying and purchasing cost:

total1

( ) ( )

( ) ( ) .2

i n

i fi i i li i i fi vfi i li vli ii

i i c i fi li

DC f S X l S Y d Q S X Q S YQ

D QO Z C P Q QQ

=

=

= + + +

+ + + +

∑

The first term of the expression is the fixed shipping cost for full truckload (Sfi) and low truckload (Sli). The fixed shipping cost includes the cost per unit distance and is independent of the load. The different shipping costs are considered for full truckload and low truckload. Illustratively, the fuel consumption of a large capacity vehicle is more than a small capacity vehicle. The second term represents the variable shipping cost for full truckload (Svfi) and low truckload (Svli). The variable shipping cost is a cost per load and it is independent of the distance covered. The third term represents the total ordering cost (non-transportation) and includes the ordering and inspection cost. The fourth term represents the inventory holding cost by the buyer. The last terms represents the total purchasing cost of goods.

Q is the optimal order quantity; it can be calculated by using the derivative of Ctotal.

total 1

( ) ( ) ( )0 2 .

n

i fi i i li i i fi vfi i li vli i i ii

c

f S X l S Y d Q S X Q S Y O ZC

Q DQ C

=

+ + + +∂

= ⇒ =∂

∑

A goal programing approach for supplier evaluation 51

By substituting the value of Q in Ctotal, it becomes

total1

1

1

( ) ( )

( ) ( ) ( )2

( ) ( ( )2 (

2

i ni fi i i li i i fi vfi i li vli i i i

ni

i fi i i li i i fi vfi i li vli i i ii

c

n

i fi i i li i i fi vfi i li vli i i ic i

i fic

D f S X l S Y d Q S X Q S Y O ZC

f S X l S Y d Q S X Q S Y O ZD

C

f S X l S Y d Q S X Q S Y O ZC

D P QC

=

=

=

=

+ + + +

= + + + +

+ + + ++ +

∑∑

∑) .liQ

+

(16a)

The PV of the goods is computed by using the following equation:

1( ).

n

i fi lii

PV RC Q Q=

= × +∑ (16b)

The multi-objective model of demand allocation among candidate suppliers is formulated by using the objective functions and constraints as given below:

max total min( ) , ( ) .Z PV C= (17)

Subject to:

for 1, ,i c i c if FT l LT C i n+ ≤ = … (18)

for 1, ,fi li iQ Q C i n+ ≤ = … (19)

for 1, ,c fi cLT Q FT i n≤ ≤ = … (20)

for 1, ,vi cQ LT i n≤ = … (21)

for 1, ,fi liQ Q D i n+ ≥ = … (22)

max1

( )i n

i ii

P Q B=

=

× ≤∑ (23)

min1

( )i n

i ii

P Q B=

=

× ≥∑ (24)

1

i n

i ii

L Q QLT=

=

≤∑ (25)

1

i n

ii

Q Q=

=

=∑ (26)

Xi, Yi = (0, 1) (27)

Zi = 1 if Xi + Yi > 0 otherwise 0. (28)

52 A. Singh

Equations (16a) and (16b) represent the objective functions of the supplier selection and demand allocation problem. The first and second constraints (equations (18) and (19)) ensure the supplier capacity restriction in terms of number of full truckload and number of low truck load. Third and fourth constraints (equations (20) and (21)) ensure the units restriction to full and low truckload. The fifth constraint (equation (22)) ensures the demand restriction of the buyer. The sixth constraint (equation (23)) represents the minimum budget restriction. The seventh constraint (equation (24)) represents the maximum budget restriction. The eighth constraint (equation (25)) represents the lead-time restriction. The ninth constraint (equation (26)) ensures the order quantity to each supplier. The last two constraints represent the binary requirement on variables Xi, Yi and Zi.

5 Methodology

Most commonly used multi-objective programming approaches are hierarchical, utility (weighted sum method), Pareto, interactive and goal programming. The multi-objective programming approach determines a compromise solution, which satisfies a number of considered criteria simultaneously. In this study, we use goal-programming approach to solve the multi-objective demand allocation problem under a set of constraints. In this approach, all the objectives are taken into consideration as constraints, which denote some satisfying levels (or goals) and try to find out a solution that is as close as possible to the predefined goals. Goals G1 and G2 of the objective functions are decided by solving the model described in the previous section by considering Total cost (Ctotal) and PV as the single objective function. Since the PV and total cost have different order of magnitudes. The deviations of total cost and PV are normalised by using the optimum values of total cost (G1) and PV (G2). In this context, a goal-programming model is formulated and the objective function and additional constraints of the model are as follows:

1 2

1 2

Minimise (1 ) .d d

Z w wG G

− +

= + − (29)

Subject to:

1

1

1

( ) ( )

( ) ( ) ( )2

( ) ( ( )2 ( )

2

i ni fi i i li i i fi vfi i li vli i i i

ni

i fi i i li i i fi vfi i li vli i i ii

c

n

i fi i i li i i fi vfi i li vli i i ic i

i fi lic

D f S X l S Y d Q S X Q S Y O Z

f S X l S Y d Q S X Q S Y O ZD

C

f S X l S Y d Q S X Q S Y O ZC

D P Q QC

=

=

=

=

+ + + + + + + +

+ + + +

+ + +

∑∑

∑

1 1 1d d G+ −

+ − =

(30)

A goal programing approach for supplier evaluation 53

2 2 21

( ) .n

i fi lii

RC Q Q d d G+ −

=

× + + − =∑ (31)

The goal programming model converts the single objective functions into constraints by adding over-achievement and under-achievement deviation factors as shown by relations (30) and (31). The objective function of goal-programming model is denoted in terms of deviation factors as shown by equation (29). The first deviation factor (d–) represents the over-achievement of cost. Here, the objective is to minimise the over-achievement of cost, since we are interested in the minimisation of cost. The second deviation factor (d+) denotes the under-achievement of PV. In this case, the objective is to minimise the under-achievement of PV, since we are interested in the maximisation of PV. The objective function of the multi-objective problem is solved as a single objective optimisation problem subject to constraints (18)–(28), (30) and (31). The experimental results obtained by the model are discussed in the following section.

6 Computational experiment

A manufacturing firm decides the demand allocation among five candidate suppliers, namely S1, S2, S3, S4 and S5. Four decision-makers, namely M1, M2, M3 and M4 are involved in the selection process. Decision-makers assign their assessment to each candidate suppliers based on five criteria, namely reliability (C1), quality (C2), on time delivery (C3), consistency (C4) and price (C5). Each decision-maker assigns his or her assessment to each candidate supplier in terms of linguistic variables as shown in Tables 3 and 4, respectively.

Table 3 Rating of decision makers on importance of criterion

Criteria M1 M2 M3 M4

C1 ML ML ML M C2 VH H H VH C3 H H MH MH C4 MH H H MH C5 MH MH M M

Let the expected demand, available capacity of suppliers, their lead times and cost for the items are as follows: company wants to purchase 20,000 units. The unit material cost for suppliers S1, S2, S3 and S4 are $4, $5, $5, $6 and $7, respectively, and the capacity of candidate suppliers are 12,000, 14,000, 10,000, 16,000 and 10,000 units, respectively. The annual budget for purchasing the items is at least $140,000 but less than $160,000. The delivery lead times of suppliers are 3, 4, 6, 3, 2 days, respectively. According to company’s policy, average delivery lead time should not be more than 4 days. The distance of the suppliers from the company is 40, 60, 70, 50, 30 km. The maximum capacity of the full truckload and low truckload is 4000 and 2000 units. The fixed shipping cost and variable shipping cost for full truckload and low truckload are $4, $3 per km and per truck and $1.5, $2.0 per unit, respectively, for all suppliers. Ordering cost is $100 per order and annual inventory carrying cost is $1 per unit.

54 A. Singh

Table 4 Assessment of suppliers based on each criterion

Criterion

Decision makers Suppliers M1 M2 M3 M4

C1 S1 G VG MG MG S2 F F MP MP S3 MP MP F F S4 F F MP MG S5 VG G G VG C2 S1 MP MP F F S2 MG F MG F S3 F F MG MG

S4 G VG G MG S5 VG VG G VG C3 S1 MG MG G MG S2 F F MG MP S3 MP MP P P S4 MG MG F MG S5 VG VG G G C4 S1 G VG G MG S2 F F MG MG S3 MG G G MG S4 VG VG G VG S5 MP MP P P C5 S1 VG VG G VG S2 G G MG MG S3 MG MG F F S4 F MP F F S5 P P MP P

All the linguistic variables given in Tables 3 and 4 are converted into fuzzy score by using the values given in Tables 1 and 2, and further these scores are averaged out by using equations (1) and (2). The calculated fuzzy scores are given in Table 5. The computed fuzzy scores given in Table 5 are in different units. These scores are normalised by using equations (3) and (4). The normalised values are given in Table 6. Each criterion has linguistics weights, these weights are applied to the normalised values, and weighted normalised values are computed by using equation (5). The weighted normalised values are shown in Table 7. Ideal and negative ideal solutions are identified by using equations (6) and (7). Separation measures from ideal and negative ideal solution are computed by using equations (8)–(13), respectively. These separation measures are presented in Table 8. The relative closeness from ideal and negative ideal solution would be as an interval and this interval is computed by using equation (14). Finally, with the help of equation (15), the relative closeness of each supplier is

A goal programing approach for supplier evaluation 55

computed. The interval value of relative closeness and final relative closeness is given in Table 9.

The suppliers rating shown in Table 9 is integrated with the MILP approach to maximise the total purchase value of the items. The objective function of the model is shown by equation (16b) that maximises the total purchase value of the items subjected to the constraints shown by equations (17)–(28) as given in Section 4. The model is solved and the results are summarised in Table 10.

Table 5 Interval value of supplier assessment and weights

Suppliers C1 C2 C3 C4 C5

S1 [(5.55, 6.81); 8.14; (8.83,9.74)]

[(0, 2.29); 3.87; (5.4, 6.42)]

[(4.73, 5.94); 7.45; (8.35, 9.6)]

[(5.8, 7.4); 8.67; (9.21,9.87)]

[(7.62, 8.95); 9.74; (9.87, 10)]

S2 [(0, 2.29); 3.87; (5.4, 6.42)]

[(3.35, 4.38); 5.9; (7.21, 8.4)]

[(0, 3.17); 4.78; (6.25,7.36)]

[(3.35, 4.38); 5.9; (7.2, 8.4)]

[(4.97, 6.4); 7.9; (8.7, 9.74)]

S3 [(0, 2.3); 3.87; (5.4, 6.42)]

[(3.35, 4.38); 5.9; (7.2, 8.44)]

[(0, .86); 1.73; (3.35, 4.38)]

[(4.97, 6.4); 7.93; (8.7, 9.7)]

[(3.35, 4.38); 5.9; (7.2, 8.4)]

S4 [(0, 3.17); 4.78; (6.24,7.36)]

[(5.8, 7.4); 8.67; (9.2, 9.87)]

[(3.88, 4.9); 6.43; (7.59, 8.9)]

[(7.62, 8.95); 9.7; (9.8, 10)]

[(0, 2.83); 4.4; (5.92, 6.94)]

S5 [(6.83, 8.4); 9.48; (9.74,10)]

[(7.83, 8.95); 9.74; (9.87, 10)]

[(6.84, 8.4); 9.48; (9.74, 10)]

[(0, .86); 1.73; (3.35, 4.38)]

[(0, .65); 1.31; (2.89, 3.91)]

Weight [(0, 0.17); 0.35; (0.49, 0.59)]

[(0.68, 0.84); 0.94; (0.97, 1)]

[(0.5, 0.64); 0.79; (0.87, 0.94)]

[(0.5, 0.64); 0.8; (0.87, 0.97)]

[(0.33, 0.43); 0.6; (0.71, 0.84)]

Table 6 Normalised decision matrix

Suppliers C1 C2 C3 C4 C5

S1 [(0.56, 0.68); 0.82; (0.88, 0.98)]

[(0, 0.23); 0.39; (0.54, 0.64)]

[(0.48, 0.6); 0.75; (0.83, 0.96)]

[(0.58, 0.74); 0.87; (0..92, 0.98)]

[(0.78, 0.9); 0.98; (0.99, 1)]

S2 [(0, 0.23); 0.38; (0.54, 0.64)]

[(0.34, 0.44); 0.59; (0.72, 0.84)]

[(0.24, 0.35); 0.5; (0.64, 0.75)]

[(0.34, 0.44); 0.6; (0.72, 0.84)]

[(0.5, 0.65); 0.8; (0.88, 0.98)]

S3 [(0, 0.23); 0.39; (0.54, 0.64)]

[(0.34, 0.44); 0.59; (0.72, 0.84)]

[(0, 0.1); 0.2; (0.35, 0.45)]

[(0.49, 0.64); 0.8; (0.88, 0.97)]

[(0.35, 0.45); 0.6; (0.73, 0.85)]

S4 [(0, 0.32); 0.48; (0.63, 0.74)]

[(0.58, 0.74); 0.87; (0.92, 0.99)]

[(0.4, 0.5); .65; (0.76, 0.9)]

[(0.76, 0.9); 0.97; (0.98, 1)]

[(0.19, 0.3); 0.45; (0.6, 0.7)]

S5 [(0.68, 0.84); 0.95; (0.97, 1)]

[(0.78, 0.89); 0.97; (0.98, 1)]

[(0.7, 0.85); 0.95; (0.98, 1)]

[(0, 0.09); 0.17; (0.34, 0.44)]

[(0, .08); 0.15; (0.3, 0.4)]

The second model minimises the total cost subjected to a set of constraints. The objective function of the model is shown by equation (16a) that minimises the inventory and transportation costs subjected to the constraints shown by equations (17)–(28) as given in Section 4. The model is solved and the results are summarised in Table 11.

The solutions of the above two models are further used to set the goals for total cost (G1) and PV (G2). A goal-programming model is developed for multi-objective demand allocation among candidate suppliers. The objective function of the goal-programming model is shown by equation (29) and the constraints are shown by equations (18)–(28), (30) and (31). The model is solved and the results obtained by the model are summarised in Table 12.

56 A. Singh

Table 7 Weightage normalised decision matrix

Suppliers C1 C2 C3 C4 C5

S1 [(0.04, 0.12), 0.29, (0.44, 0.59)]

[(0.09, 0.21), 0.38, (0.53, 0.65)]

[(0.24, 0.39), 0.6, (0.72, 0.93)]

[(0.3, 0.49), 0.71, (0.72, 0.96)]

[(0.27, 0.41), 0.59, (0.71, 0.85)]

S2 [(0.01, 0.043), 0.14, (0.28, 0.39)]

[(0.25, 0.38), 0.57, (0.71, 0.85)]

[(0.12, 0.23), 0.4, (0.56, 0.73)]

[(0.12, 0.29), 0.48, (0.64, 0.83)]

[(0.18, 0.29), 0.48, (0.71, 0.83)]

S3 [(0.01, 0.04), 0.14, (0.28, 0.39)]

[(0.25, 0.38), 0.57, (0.71, 0.85)]

[(0, 0.07), 0.16, (0.31, 0.44)]

[(0.25, 0.42), 0.64, (0.77, 0.95)]

[(0.12, 0.20), 0.36, (0.71, 0.72)]

S4 [(0.02, 0.06), 0.18, (0.32, 0.45)]

[(0.42, 0.64), 0.84, (0.90, 0.98)]

[(0.2, 0.33), 0.52, (0.66, 0.87)]

[(0.39, 0.59), 0.78, (0.86, 0.97)]

[(0.07, 0.14), 0.27, (0.71, 0.59)]

S5 [(0.04, 0.14), 0.33, (0.49, 0.6)]

[(0.55, 0.77), 0.93, (0.96, 1)]

[(0.35, 0.55), 0.76, (0.85, 0.97)]

[(0, 0.07), 0.16, (0.31, 0.44)]

[(0, 0.04), 0.09, (0.71, 0.34)]

Table 8 Separation measures from ideal and negative ideal solutions

Suppliers ( )1 2, i iD D+ + ( )1 2, i iD D− −

S1 (2.95, 2.5) (2.41, 2.9) S2 (3.25, 2.88) (2.1, 2.52) S3 (3.4, 3.07) (1.98, 2.33) S4 (2.85, 2.47) (2.61, 2.91) S5 (3.06, 2.75) (2.48, 2.56)

Table 9 Interval value of relative closeness and the final relative closeness of each supplier

Suppliers Interval of relative closeness Final score of suppliers (Rc*)

S1 (0.45, 0.54) 0.49 S2 (0.39, 0.47) 0.43 S3 (0.37, 0.43) 0.40 S4 (0.48, 0.54) 0.51 S5 (0.45, 0.48) 0.47

Table 10 Demand allocation to suppliers by using the single objective function as procurement value

Suppliers Units allocated No. of full truckload Number of half truckload S4 10,000 2 1 S5 10,000 2 1 G2 (procurement value) 9800

The demand allocation to the suppliers with the number of full truckload and low truckload, over-achievement of cost and under-achievement of PV computed by the goal programming model is Qf1 = 4000, Ql1 = 2000, Qf4 = 4000, Ql4 = 2000, Qf5 = 8000, f1 = 1, l1 = 1, f4 = 1, l4 = 1, f5 = 2, 2 40d + = and 1 5349.d − = Total PV = 9670 and total cost is $152,425. According to the relative closeness computed by single approach (i.e.,

A goal programing approach for supplier evaluation 57

TOPSIS), suppliers S4 and S1 are the best suppliers. However, a total of 6000, 6000 and 8000 units are assigned to suppliers S1, S4 and S5 by using the hybrid algorithm based on TOPSIS and goal programming approaches. The maximum units are assigned to supplier S5. This allocation of the units maximises the value of procurement and at the same time minimises the total cost, which includes inventory and transportation costs. On the contrary, the individual approach (i.e., TOPSIS) assigns the low rating to the supplier S5. In this context, it may be concluded that the hybrid approach improves the efficacy of the individual approaches. However, the wide applicability of the individual approach in the literature is due to its simplicity, ease of use and great flexibility. These observations support the literature findings to develop more hybrid approaches for the supplier-selection problems.

Table 11 Demand allocation to suppliers by using the single objective as total cost

Suppliers Units allocated No. of full truckload Number of half truckload

S1 8000 2 0 S4 4000 1 0 S5 8000 2 G1 (Total cost) 147,076

Table 12 Demand allocation to suppliers by using goal programming model

Suppliers Units allocated No. of full truckload Number of half truckload S1 6000 1 1 S4 6000 1 1 S5 8000 2

Procurement value = 9760, Total Cost = 52,425, 2 40,d + = 1 5349d − =

7 Managerial implications

The present research work focuses on the supplier evaluation and multi-objective demand allocation among the candidate suppliers. Three models were developed based on three objective functions. The first model focuses on the maximisation of PV of the items being purchased. The second model focuses on the minimisation of inventory and transportation costs. The third multi-objective model is a combination of PV and inventory and transportation costs. Ratings of the suppliers are computed by the fuzzy TOPSIS algorithm and these ratings are used during the model formulation. In the first model, supplier ratings are integrated with the PV of the items and the demand of the buyer company is allocated optimally among the suppliers, and this allocation maximises the PV of the items. This model could be beneficial for the purchasing managers if they are willing to maximise the value of procurement.

In the second model, demand of the buyer company is allocated optimally among the suppliers and this allocation minimises the inventory and transportation costs. Nowadays more and more companies are looking at transportation optimisation in the face of rising

58 A. Singh

fuel costs. This model would be helpful for the purchasing managers if inventory and transportation costs are the most important criteria for them.

The third multi-objective model minimises the inventory and transportation costs and simultaneously maximises the PV of the items during the demand allocation among the suppliers. This model is supportive for the managers as they can assign their importance to the total cost and PV as per their choices and can determine the optimal quantity for allocation among the suppliers. The model provides flexibility to the managers for evaluation of the different available alternatives to take a decision about optimal demand allocation among the suppliers. As fuel prices are increasing day by day, purchasing managers may assign 100% importance to the transportation cost and may allocate the demand among the suppliers.

The multi-objective demand allocation model helps out the purchasing managers to decide the number of full and low truckload and it also helps out to decide the optimal order size. This feature of the proposed model would definitely assist the purchasing managers to figure out the balance between the inventory and transportation costs.

Further, in the multi-objective model, the buyer company did not depend upon the single supplier. Whenever a supplier declines to supply the items, then the proposed model is supportive for the managers as they can adjust the value of variables in the model as per their choices and determine the optimal quantity for allocation among the rest of the suppliers. Further, a rated group of suppliers can help the managers for operating their supply chain smoothly without any breakdown due to the non-availability of the items not supplied by one of the supplier. Further, the model would be helpful for the managers who are interested to reconfigure their supply chain under a changing business environment.

An important feature of the proposed multi-criteria demand allocation model is its ability to capture qualitative as well as quantitative criteria consistent with the real-world situations. In the model, TOPSIS is used to decide the rating of the supplier and handle qualitative and quantitative criteria. The rating of the suppliers is decided by the closeness index, which is computed by TOPSIS. According to the closeness index, the managers can determine not only the rating of the suppliers but also they can assess the status of all possible suppliers. Hence, this feature of the model would help the managers in their decisions pertaining to supplier evaluation as well as multi-objective demand allocation among the suppliers. In real supplier-selection problems, the modelling of many situations may not be sufficient or exact, as the available data are inexact, vague, imprecise and uncertain by nature. In these situations, managers usually face a high degree of uncertainties and fuzzy-set theory is the most effective method for managing the vagueness and uncertainties of the problems. In this situation, the fuzzy TOPSIS method is beneficial for the managers and it can capture their subjective estimates in terms of linguistic variables. In fact, the fuzzy TOPSIS method is very flexible and it can handle both tangible and intangible attributes and evaluate the suppliers effectively. Further, integration of fuzzy TOPSIS with goal programming provides an opportunity to the managers to optimise their decision about multi-objective demand allocation among suppliers. Significantly, the proposed hybrid (fuzzy, TOPSIS and goal programming) model provides more objective information for supplier evaluation and demand allocation among suppliers in supply chain. The model provides flexibility to the managers for evaluation of the different available alternatives to take a multi-objective decision about demand allocation among the suppliers.

A goal programing approach for supplier evaluation 59

8 Conclusions and future research directions

In an increasingly competitive environment, firms are paying more attention to selecting the right suppliers for procurement of raw materials and component parts for their products. Supplier evaluation and selection together has an important role in the supply chain process and is crucial to the success of a manufacturing firm (Choi and Hartley 1996). Therefore, it is widely studied in the literature. Although different aspects of this problem are analysed previously, the present study provides a new approach to solve the problem of supplier selection and multi-objective demand allocation among candidate suppliers. This problem is solved in two stages: in the first stage the rating of suppliers are evaluated by using the fuzzy TOPSIS approach. In the second stage, mixed integer-programming model is developed by considering the PV, which is based on supplier rating evaluated in the first stage, and transportation and inventory cost including the concept of full and low truckload.

The study on supplier evaluation and multi-objective demand allocation among suppliers presented in this research paper could be extended in various ways. First, more case studies on manufacturing systems engaged in diversified operations could underline the practical usefulness of the hybrid methodology as derived from the experimental results. Second, future research could consider the multi-period demand during demand allocation among suppliers. Third, uncertainty of demand instead of fixed demand in supplier-selection problem could be taken up. Finally, research can be extended by developing more hybrid approaches for uncertain multi-objective demand allocation among suppliers.

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