Selection of internet assessment vendor using TOPSIS method in fuzzy environment
Saurav Datta*, Chitrasen Samantra and Siba Sankar Mahapatra Department of Mechanical Engineering, National Institute of Technology, Rourkela-769008, India E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] *Corresponding author
Partha Sarathi Chakraborty Department of Adult and Continuing Education and Extension, Jadavpur University, Kolkata-700032, India E-mail: [email protected]
Gautam Majumdar Department of Mechanical Engineering, Jadavpur University, Kolkata-700032, India E-mail: [email protected]
Abstract: In recent days an explosion in demand is being resulted for internet-based assessment solutions. The number of internet assessment vendors has increased dramatically, and they differ widely in offerings, capabilities, technology, assessment expertise, customer support and other factors. Choosing an appropriate vendor capable of fulfilling various needs and become a value-added partner can seem somewhat daunting. Conventional approaches for selection and evaluation of vendors separately tend to be less effective in dealing with the imprecise of attributes assessment individually. Hence, this study aims to propose an effective integrated fuzzy multi-attribute group decision making (FMAGDM) approach to deal both qualitative and quantitative attributes associated with internet assessment vendor selection criteria’s. In this paper, a technique for order preference by similarity to ideal
2 S. Datta et al.
solution (TOPSIS) method has been adopted in a fuzzy environment for selection and ranking of internet assessment vendor based on decision makers’ linguistic viewpoint on various criteria attributes. Generalised trapezoidal fuzzy numbers are used to express linguistic variables that consider making a fuzzy decision matrix. A numerical illustration has been reported to illustrate the procedural hierarchy of fuzzy-TOPSIS method in relation to a vendor selection problem.
Keywords: internet assessment vendor; vendor selection; fuzzy set; TOPSIS.
Reference to this paper should be made as follows: Datta, S., Samantra, C., Mahapatra, S.S., Mondal, G., Chakraborty, P.S. and Majumdar, G. (2013) ‘Selection of internet assessment vendor using TOPSIS method in fuzzy environment’, Int. J. Business Performance and Supply Chain Modelling, Vol. 5, No. 1, pp.1–27.
Biographical notes: Saurav Datta is an Assistant Professor in the Department of Mechanical Engineering, National Institute of Technology, Rourkela, India. His current area of research includes weld quality optimisation, modelling and simulation of production processes, and multi-criteria decision-making. He has published a number of journal papers in national/international repute and presented a number of papers in various conferences/symposia in India and abroad. He is currently guiding a number of research scholars for MTech/PhD.
Chitrasen Samantra is a Research Scholar (Production Specialisation) in the Department of Mechanical Engineering, National Institute of Technology, Rourkela, India. He is currently pursuing his MTech in the area of MCDM in agile manufacturing.
Siba Sankar Mahapatra is a Professor in the Department of Mechanical Engineering, National Institute of Technology, Rourkela, India. He has more than 20 years of experience in teaching and research. His current area of research includes multi-criteria decision-making, quality engineering, assembly line balancing, group technology, neural networks, and non-traditional optimisation and simulation. He has published more than 40 journal papers. He has written few books related to his research work. He is currently dealing with few sponsored projects.
Goutam Mondal is currently working as Section Engineering in Tata Motors Limited, Jamshedpur, Jharkhand, India. He is currently pursuing his PhD (Production Specialisation) in the Department of Mechanical Engineering, Jadavpur University, Kolkata, India. His area of research is organisation behaviour and performance evaluation in supply chain management.
Partha Sarathi Chakraborty is an Assistant Professor in the Department of Adult and Continuing Education and Extension, Jadavpur University, Kolkata, India. His current area of research includes operations research and supply chain management.
Gautam Majumdar is a Professor in the Department of Mechanical Engineering, Jadavpur University, Kolkata, India. He has vast experience in teaching and research. His current area of research includes surface coatings, operations research and supply chain management.
Selection of internet assessment vendor using TOPSIS method 3
1 Introduction: internet assessment vendor evaluation
Application service providers (ASPs) are firms that provide online applications and other services to their clients. Instead of purchasing or developing software and installing it in an organisation, the ASP hosts and manages the application. Advantages of using an ASP include the following:
a Lower cost than developing and maintaining internal systems, and much faster deployment.
b Access to the latest technology. Good ASPs, with their need to remain competitive, offer leading edge technology.
c More powerful and robust administration and reporting capabilities than most internal systems would provide.
d Expertise in leveraging assessment information to improve organisational effectiveness.
e Allows clients to focus on core functions (e.g., selection, training, management development) instead of IT issues.
Depending on the ASP and a client’s particular needs, disadvantages could include things like dependence on an outside party, inability to customise the application in very specific ways, uncertainty about customer support, concerns about data security, and the possibility of the ASP going out of business and leaving stranded. Before going to the process of choosing an assessment vendor, one has to first decide whether it is needed at all. It gets down to a build or buys decision. The following conditions would tend to increase the desirability of finding an ASP to partner.
a having many assessment applications versus just one or two, assessments used for multiple purposes (e.g., selection and development), and a relatively high volume of assessments annually
b limited internal assessment and/or IT expertise (or capacity) to build and maintain applications
c need for assessment content that may already be developed (e.g., a validated selection test offered by a vendor) versus an assessment associated with an internal training program that may have to develop anyway
d desire to implement a solution quickly, with low start-up costs.
Assuming an internet assessment vendor makes sense for the business, the first step is to clarify (and document) the specific needs. This includes who’s being assessed and why, requirements for assessment content (e.g., validation evidence for selection tests), the number of users and functionality needed, individual and group reporting desired, company access to the database, and a variety of technology issues. It is to be checked whether various departments in the organisation also have online assessment needs. There are strong advantages to having different functions working with the same vendor, and preferably that vendor having an integrated database and the ability to transfer information to the client’s HRIS or LMS in a seamless fashion.
4 S. Datta et al.
The criteria (Censeo Corporation INSIGHT, 2007) for evaluating the content, psychometric properties, validity, etc. are more important and should be carefully considered if to seek vendor assessments in addition to administration, scoring and reporting capabilities. An assessment that does not accurately measure what it purports to measure does no good, and may cause harm, no matter how well it is delivered. The following points are to be noted:
a In most cases, it is strongly preferable to partner with a vendor who has both assessment and technology expertise in-house. This allows the vendor to offer a full range of services in a timely fashion, and to proactively look for ways to add value rather than merely respond to requests.
b Where the assessment program is intended for development purposes, it is to be ensured that the vendor can easily make the assessment information available to line managers at various levels. Managers are to hold accountable for developing their people; a good assessment platform can help them to do that.
c A good vendor should be able to provide with two important documents – a clear description of all the features and customisable options available, and a technology document that describes the architecture, performance and reliability, security, backup procedures, etc. in detail.
d It is to be ensured that the vendor has both the expertise and the capacity to provide good customer support, including technical support, and that the service will be available when it is needed.
2 Literature review and prior state of art on vendor selection problem
In recent days business marketplace has become highly competitive, at the same time tremendously challenging to satisfy wide variety of customers’ needs. In order to secure a competitive position in the global marketplace, enterprises have become more concerned to follow certain strategies to achieve shorter lead times, reduced costs and higher quality. Vendors play a key role in achieving corporate competitiveness. Consequently, selection of appropriate vendors is a major issue of these new strategies. Several conflicting quantitative and qualitative factors or criteria attributes seem to affect supplier selection problem; therefore, it is treated as multi-attribute decision making (MADM) problem.
Kumar et al. (2004) adopted a fuzzy goal programming approach for solving vendor selection problem with multiple objectives which were fuzzy in nature. The vendor selection problem was formulated as a fuzzy mixed integer goal programming that included three primary goals: minimising the net cost, minimising the net rejections, and minimising the net late deliveries subject to realistic constraints regarding buyer’s demand, vendors’ capacity, vendors’ quota flexibility, purchase value of items, budget allocation to individual vendor, etc. The proposed approach had capability to handle realistic situations in a fuzzy environment and provided a better decision tool for the vendor selection decision in a supply chain. Bayazit (2006) provided a good insight into the use of analytic network process (ANP) that is a multiple criteria decision-making methodology in evaluating supplier selection problems. ANP is a complex methodology and requires more comparisons than the traditional AHP and it increases the effort. The
Selection of internet assessment vendor using TOPSIS method 5
study provided an effective framework to guide managers for evaluating suppliers. Nukala and Gupta (2007) developed an integrated multi-criteria decision making methodology using Taguchi loss functions, AHP and Fuzzy Programming that evaluated the suppliers and determined the order quantities under different degrees of information vagueness in the decision parameters in a closed-loop supply chain network. While the Taguchi loss functions quantified the suppliers attributes to quality loss, the AHP transformed these quality losses into a variable for decision making that could be used in formulating the fuzzy programming objective function to determine the order quantities. Jadidi et al. (2008a) proposed a new approach based on the concepts of technique for order preference by similarity to ideal solution (TOPSIS) to evaluate and select the best supplier. It was demonstrated that the improved method, which was used to solve the MADM problems for selecting the best supplier, was a good means of evaluation, and it appeared to be more appropriate. Amid and Ghodsypour (2008) developed an additive weighted model for fuzzy multi-objective supplier selection problem with fuzzy weights. In a real situation, the proposed model could be implemented as a vector optimisation problem; the basic concept was to use a single utility function to express the preference of decision makers (DMs), in which the values of criteria and constraints were expressed in vague terms and were not equally important. Guo et al. (2009) presented a support vector machine technology, potential support vector machine, combined with decision tree to address issues on supplier selection including feature selection, multiclass classification etc.
Kaur et al. (2009) presented a fuzzy-statistical comparative case study on reinforcing impact factor of a vendor as an indirect measure of quality in vendor selection problem. It was found that the vendor who got the highest allocation was the one who had the highest impact factor. Shu and Wub (2009) proposed supplier selection and evaluation on the basis of the quality criterion under fuzzy environment since fuzzy quality data are ubiquitous in the real world. A numerical example was illustrated to present the possible application by incorporating fuzzy data into the quality-based supplier selection and evaluation. Thanaraksakul and Phruksaphanrat (2009) developed a supplier evaluation framework based on balanced scorecard (BSC) with integrated corporate social responsibility (CSR). It was found that quality, delivery, and cost were the most significant criteria. Moreover, some criteria were seemed to be changed according to shorten product life cycle, technologies, improvement of service, evolution of production system, and emergence of supply chain management (SCM). Evaluating supplier using the proposed framework could be helpful for DMs to qualify the most eligible supplier who could meet qualifications and buyer’s strategies as well as environmental and social responsibility issues. Azadeh et al. (2009) introduced a framework for decision making about the vendor selection problem which was capable to cover the crisp, stochastic, and uncertainty conditions in the supply chain environment. The proposed framework included of Monte Carlo simulation analysis of three types of vendor selection models [data envelopment analysis (DEA), fuzzy DEA (FDEA), and chance constraint DEA (CCDEA)] in the supply chain and presented a decision making plan for choosing the appropriate model for the supplier selection under special conditions as stated above. Enyinda et al. (2010) presented a case study on solving the supplier selection process problem in a generic pharmaceutical firm using the analytic hierarchy process (AHP) model and implemented with the support of the expert choice software. The AHP was considered a reliable methodology for developing a generic pharmaceutical firm strategic supplier selection and evaluation framework. Based on the research findings, the
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regulatory compliance selection criterion was most favoured, followed by quality, risk, cost, supplier profile, and service.
Songhori et al. (2011) provided a comprehensive and systematic framework that embraced both quantitative and qualitative criteria for supplier selection. The study addressed the need in the supplier evaluation literature for methods that considered different (transportation alternatives) TAs in the supplier selection and order allocation decisions encompassing multiple discrete time periods. Shah et al. (2010) developed an integrated inventory policy for single vendor and multiple buyers. The demand of a product was assumed to be quadratic. It was established numerically that the integrated approach resulted in significant decrease in the total cost compared with the independent decision approach by all the buyers. Kumar et al. (2011) proposed use of AHP based on fuzzy simulation (FSAHP) towards supplier selection.
Khorasani and Bafruei (2011) studied on supplier selection in pharmaceutical industry in Iran to select the best supplier of maize starch. They considered the most important criteria for supplier selection like price, quality, service, organisation, and technical issues appropriate for pharmaceutical industry. Fuzzy analytical hierarchical process (FAHP) was used for selecting the best supplier. Shirouyehzad et al. (2011) applied a fuzzy logic controller as a robust and easy understanding approach to transform the quantitative variable to linguistic terms in order to measure the vendors’ performance. They considered four criteria: service quality, price, lateness deliveries and rate of rejected parts which could influence vendors’ performance. Meena et al. (2012) studied to identify the factors that affect the suppliers’ satisfaction in buyer-supplier relationships and explore their relationships with suppliers’ satisfaction. Safari et al. (2012) applied PROMETHEE method based on entropy weight for Supplier Selection. The outcome of this research was ranking and selecting supplier with the help of Shannon’s Entropy and PROMETHEE techniques. Huang et al. (2012) proposed cooperative game-theoretic approach for optimal supplier selection, pricing and inventory decisions in a multi-level supply chain.
In supply chain management, supplier selection is one of the most important activities. The importance is enhanced even more by new strategies in a supply chain, because of the key role suppliers perform in terms of quality, costs and services which affect the outcome in the buyer’s company. Supplier selection can be viewed as a multiple criteria group decision making problem in which the objectives merely equally important. In practice, uncertainty, vagueness and imprecision of the goals, constraints and parameters in this problem make the decision making process more complicated. The vagueness of the input data and varying priority weight reflecting importance of criteria are essentially to be considered as well. In real cases, where DMs face up to uncertain data and situations, fuzzy logic model can help DMs to find out the appropriate ordering from each supplier, and allows purchasing manager(s) to manage supply chain performance on cost, quality, on time delivery, etc. Such supplier selection problems were attempted by previous researchers using a variety of MCDM tools in fuzzy context. Recently fuzzy-based TOPSIS approach (Chen et al., 2006; Jadidi et al., 2008a; Taghavifard and Mirheydari, 2008; Shahanaghi and Yazdian, 2009; Vahdania et al., 2009; Sreekumar and Mahapatra, 2009; Verma et al., 2009; Jadidi et al., 2010; Anisseh and Yusuff, 2011; Izadikhah, 2012) has been found efficient in this regard. Inspiring by the wide application of the aforesaid approach in supply chain management, the present
Selection of internet assessment vendor using TOPSIS method 7
study attempts to solve a MADM problem in internet assessment vendor selection by applying TOPSIS method combined with fuzzy linguistic evaluation. Numerical illustration has been reported here to confirm application feasibility of this method in the prescribed area.
3 Fuzzy linguistic modelling
In this section, some basic definitions of fuzzy sets, fuzzy numbers and linguistic variables are reviewed from Zadeh (1975), Buckley (1985), Negi (1989) and Kaufmann and Gupta (1991). The basic definitions and notations below will be used throughout this paper until otherwise stated.
Definition 2.1. A fuzzy set A in a universe of discourse X is characterised by a membership function ( )Aμ x which associates with each element x in X a real number in the interval [0, 1]. The function value ( )Aμ x is termed the grade of membership of x in
A (Kaufmann and Gupta, 1991).
Definition 2.2. A fuzzy set A in a universe of discourse X is convex if and only if
( ) ( ) ( )( )1 2 1 2(1 ) min ,A A Aμ λx λ x μ x μ x+ − ≥ (1)
For all x1, x2 in X and all λ∈[0,1], where min denotes the minimum operator (Klir and Yuan, 1995).
Definition 2.3. The height of a fuzzy set is the largest membership grade attained by any element in that set. A fuzzy set A in the universe of discourse X is called normalised when the height of A is equal to 1 (Klir and Yuan, 1995).
Definition 2.4. A fuzzy number is a fuzzy subset in the universe of discourse X that is both convex and normal. Figure shows a fuzzy number n in the universe of discourse X that conforms to this definition (Kaufmann and Gupta, 1991).
Definition 2.5. The α-cut of fuzzy number n is defined as:
( ){ }: , ,i n i in x μ x x X= ≥ ∈α α (2)
where α∈[0,1].
The symbol nα represents a non-empty bounded interval contained in X, which can be denoted by , ,ul ln n n n= ⎡ ⎤⎣ ⎦α α α α and unα are the lower and upper bounds of the closed interval, respectively (Kaufmann and Gupta, 1991; Zimmermann, 1991). For a fuzzy number ,n if 0ln >α and 1un ≤α for all α∈[0,1], then n is called a standardised (normalised) positive fuzzy number (Negi, 1989).
Definition 2.6. A positive trapezoidal fuzzy number (PTFN) n can be defined as (n1, n2, n3, n4), shown in Figure 2. The membership function ( )nμ x is defined as (Kaufmann and Gupta, 1991)
8 S. Datta et al.
1
11 2
2 1
2 3
43 4
3 4
4
0, ,
, ,
( ) 1, ,
, ,
0, .
n
x nx n n x n
n nμ x n x n
x n n x nn n
x n
<⎧⎪ −⎪ ≤ ≤⎪ −⎪= ≤ ≤⎨⎪ −⎪ ≤ ≤
−⎪⎪ >⎩
(3)
For a trapezoidal fuzzy number ( )1 2 3 4, , , ,n n n n n= if n2 = n3, then n is called a triangular fuzzy number. A non-fuzzy number r can be expressed as (r, r, r, r). By the extension principle (Dubois and Prade, 1980), the fuzzy sum ⊕ and fuzzy subtraction Θ of any two trapezoidal numbers are also trapezoidal fuzzy numbers; but the multiplication ⊗ of any two trapezoidal numbers is only an approximate trapezoidal fuzzy number. Given any two PTFNs, ( )1 2 3 4, , ,m m m m m= and ( )1 2 3 4, , ,n n n n n= and a positive real number r, some main operations of fuzzy numbers m and n can be expressed as follows:
[ ]1 1 2 2 3 3 4 4, , ,m n m n m n m n m n⊕ = + + + + (4)
[ ]1 1 2 2 3 3 4 4, , ,m n m n m n m n m nΘ = − − − − (5)
[ ]1 1 2 2 3 3 4 4, , ,m n m n m n m n m n⊗ ≅ (6)
[ ]1 2 3 4, , ,m r m r m r m r m r⊗ ≅ (7)
Definition 2.7. A matrix D is called a fuzzy matrix if at least one element is a fuzzy number (Buckley, 1985).
Definition 2.8. A linguistic variable is a variable whose values are expressed in linguistic terms (Zimmermann, 1991).
The concept of a linguistic variable is very useful in dealing with situations, which are too complex or not well defined to be reasonably described in conventional quantitative expressions (Zimmermann, 1991). Fuzzy numbers can also represent these linguistic values.
Let ( )1 2 3 4, , ,m m m m m= and ( )1 2 3 4, , ,n n n n n= be two trapezoidal fuzzy numbers. Then the distance between them can be calculated by using the vertex method as (Chen, 2000)
( ) ( ) ( ) ( ) ( )2 2 221 1 2 2 3 3 4 4
1,4vd m n m n m n m n m n⎡ ⎤= − + − + − + −⎣ ⎦ (8)
Let ( )1 2 3, ,m m m m= and ( )1 2 3 4, , ,n n n n n= be two triangular fuzzy numbers. Then the distance between them can be calculated by using the vertex method as (Chen, 2000)
( ) ( ) ( ) ( )2 2 21 1 2 2 3 3
1,3vd m n m n m n m n⎡ ⎤= − + − + −⎣ ⎦ (9)
Selection of internet assessment vendor using TOPSIS method 9
The vertex method is an effective and simple method to calculate the distance between two trapezoidal fuzzy numbers. According to the vertex method, two trapezoidal fuzzy numbers m and n are identical if and only if ( ), 0.vd m n = Let ,m n and p be three trapezoidal fuzzy numbers. Fuzzy number n is closer to fuzzy number m than the other fuzzy number p if and only if ( ) ( ), ,v vd m n d m p< (Chen, 2000).
Figure 1 A fuzzy number n
Figure 2 A trapezoidal fuzzy number n
( )n xμ
0
1
x1n 2n 3n 4n
4 Methodology adopted for vendor selection
A systematic approach to extend the TOPSIS has been adopted to solve the vendor selection problem under fuzzy environment. The importance weights of various criteria
0
1
x
( )n xμ
10 S. Datta et al.
and the ratings of qualitative criteria have been considered as linguistic variables. Because linguistic assessments merely approximate the subjective judgment of DMs, linear trapezoidal membership functions have been considered and seemed adequate for capturing the vagueness of these linguistic assessments (Delgado et al., 1998; Herrera et al., 1996; Herrera and Herrera-Viedma, 2000). These linguistic variables can be expressed in PTFNs (Figure 3 and Figure 4). The importance weight of each criterion can be set by directly assigning or indirectly using pairwise comparison (Cook, 1992).
It has been suggested that the DMs use the linguistic variables shown in Figure 3 and Figure 4 to evaluate the importance of the criteria and the ratings of alternatives with respect to qualitative criteria.
Figure 3 Linguistic variables for importance weight of each criterion (see online version for colours)
In fact, vendor selection in supply chain system is a group multiple-criteria decision-making (GMCDM) problem, which may be described by means of the following sets:
Set of K DMs called E = {D1, D2, …, DK};
Selection of internet assessment vendor using TOPSIS method 11
1 a set of m possible suppliers called A = {A1, A2, …, Am}
2 a set of n criteria, C = {C1, C2, …, Cn}, with which supplier performances are measured
3 a set of performance ratings of Ai(i = 1, 2, …, m) with respect to criteria Cj(j = 1, 2, …, n), called X = {xij, i = 1, 2, …, m; j = 1, 2, …, n}.
Assume that a decision group has K DMs, and the fuzzy rating of each DM Dk(k = 1, 2, …, K) can be represented as a PTFN ( 1, 2, , )kR k K= … with membership function ( ).
kRμ x A good aggregation method should be considered the range of fuzzy rating of each DM. It means that the range of aggregated fuzzy rating must include the ranges of all DMs’ fuzzy ratings. Let the fuzzy ratings of all DMs be trapezoidal fuzzy numbers ( , , , ), 1, 2, .k k k k kR a b c d k K= = … Then the aggregated fuzzy rating can be defined as:
( ), , , , 1, 2, ,R a b c d k K= = … (10)
Here,
{ }min ,kk
a a=
1
1 ,K
kk
b bK =
= ∑
1
1 K
kk
c cK =
= ∑
and
{ }max kk
d d=
Let the fuzzy rating and importance weight of the kth DM be ( , , , )ijk ijk ijk ijk ijkx a b c d= and 1 2 3 4( , , , ); 1, 2, , ; 1, 2, , ,jk jk jk jk jkw w w w w i m j n= = =… … respectively. Hence, the aggregated fuzzy ratings ( )ijx of alternatives with respect to each criterion can be calculated as:
( ), , ,ij ij ij ij ijx a b c d= (11)
Here,
{ }min ,ij ijkk
a a=
1
1 ,K
ij ijkk
b bK =
= ∑
1
1 K
ij ijkk
c cK =
= ∑
12 S. Datta et al.
and
{ }maxij ijkk
d d=
The aggregated fuzzy weights jw of each criterion can be calculated as:
( )1 2 3 4, , , ,j j j j jw w w w w= (12)
Here,
{ }1 1min ,j jkk
w w=
2 21
1 ,K
j jkk
w wK =
= ∑
3 31
1 K
j jkk
w wK =
= ∑
and
{ }4 4max .j jkk
w w=
As stated above, a supplier-selection problem can be concisely expressed in matrix format as follows:
11 12 1
21 22 2
1 2
. . .
. . .. . . . . .
,. . . . . .. . . . . .
. . .
n
n
m m mn
x x xx x x
x x x
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥
= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
D
[ ]1 2, ,..., ,nw w w=W
Here,
( ), , ,ij ij ij ij ijx a b c d=
and
( )1 2 3 4, , , ;j j j j jw w w w w=
1, 2, , ; 1, 2, ,i m j n= =… …
can be approximated by PTFNs. To avoid complexity of mathematical operations in a decision process, the linear
scale transformation is used here to transform various criteria scales into comparable scales. The set of criteria can be divided into benefit criteria (the larger the rating, the
Selection of internet assessment vendor using TOPSIS method 13
greater the preference) and cost criteria (the smaller the rating, the greater is the preference). Therefore, the normalised fuzzy decision matrix can be expressed as:
[ ] ,ij m nr×
=R (13)
where B and C are the sets of benefit criteria and cost criteria, respectively, and
* * * *, , , , ,ij ij ij ij
ijj j j j
a b c dr j B
d d d d⎛ ⎞= ∈⎜ ⎟⎝ ⎠
, , , , ,j j j jij
ij ij ij ij
a a a ar j C
d c b a
− − − −⎛ ⎞= ∈⎜ ⎟⎝ ⎠
* max , ,j iji
d d j B= ∈
min , .j iji
a a j C− = ∈
The normalisation method mentioned above is designed to preserve the property in which the elements , ,ijr i j∀ are standardised (normalised) trapezoidal fuzzy numbers.
Considering the different importance of each criterion, the weighted normalised fuzzy decision matrix is constructed as:
[ ] , 1, 2, , ; 1, 2, , ,ij m nv i m j n
×= = =V … … (14)
where
.~(.)~~jijij wrv =
According to the weighted normalised fuzzy decision matrix, normalised OTFNs can also approximate the elements , .ijv i j= ∀ Then, the fuzzy positive-ideal solution (FPIS, A*) and fuzzy negative-ideal solution (FNIS, A–) can be defined as:
( )* * * *1 2, , , ,nA v v v= … (15)
( )1 2, , , nA v v v− − − −= … (16)
Here,
( )* maxj iji
v v=
and
( )min ; 1, 2, , ; 1, 2, , .j iji
v v i m j n− = = =… …
The distance of each alternative (supplier) from A* and A– can be currently calculated as:
( )* *
1
, , 1,2, , ,n
i v ij jj
d d v v i m=
= =∑ … (17)
14 S. Datta et al.
( )1
, , 1, 2, , ,n
i v ij jj
d d v v i m− −
=
= =∑ … (18)
Here, dv(.,.) is the distance measurement between two fuzzy numbers. A closeness coefficient is defined to determine the ranking order of all possible
candidate vendors once *id and id − of each supplier ( )1,2, ,iA i m= … has been
calculated. The closeness coefficient represents the distances to the fuzzy positive-ideal solution (A*) and the fuzzy negative-ideal solution (A–) simultaneously by taking the relative closeness to the fuzzy positive-ideal solution. The closeness coefficient (CCi) of each alternative (supplier) is calculated as:
*, 1, 2, , .i
ii i
dCC i md d
−
−= =
+… (19)
It is clear that CCi = 1 if Ai = A* and CCi = 0 if Ai = A–. In other words, vendor Ai is closer to the FPIS (A*) and farther from FNIS (A–) as CCi approaches to 1. According to the descending order of CCi, the ranking order of all vendors can be determined and the best one from among a set of feasible vendors can be selected. In order to determine the ranking order off all feasible vendors, a realistic approach given by Chen et al. (2006) is summarised in Table 1. Table 1 Recommended approval status
Closeness coefficient (CCi) Assessment status
CCi ∈ [0, 0.2] Do not recommend
CCi ∈ [0.2, 0.4] Recommend with high risk
CCi ∈ [0.4, 0.6] Recommend with low risk
CCi ∈ [0.6, 0.8] Approved
CCi ∈ [0.8, 1.0] Approved and preferred
In summation, an algorithm of the fuzzy decision-making method for dealing with the supplier selection is given as follows.
Step 1 from a committee of DMs, and then identify the evaluation criteria
Step 2 choose the appropriate linguistic variables for the importance weight of the criteria and the linguistic ratings for alternative vendors
Step 3 aggregate the weight of criteria to get the aggregated fuzzy weight jw of criterion Cj, and pull the DMs’ ratings to get the aggregated fuzzy rating ijx of vendor Ai under criterion Cj
Step 4 construct the fuzzy-decision matrix and subsequently the normalised fuzzy-decision matrix
Step 7 determine the distance of each supplier from FPIS and FNIS respectively
Selection of internet assessment vendor using TOPSIS method 15
Step 8 calculate the closeness coefficient of each supplier
Step 9 according to the closeness coefficient, the assessment status of each supplier can be understood and finally to determine the ranking order of all suppliers.
5 Numerical illustrations
The model discussed above has been applied to an internet assessment vendor selection process carried out by an industry in eastern part of India has been explored. After preliminary screening, five candidate vendors (V1, V2, V3, V4, and V5) remain for further evaluation. A committee of four DMs, DM1; DM2; DM3 and DM4, has been formed to select the most suitable vendor. Eleven criteria (C1, C2, …, C11) have been considered as illustrated as follows. Other than C10 (cost criteria) all have been considered as benefit criteria.
1 Assessment expertise (C1) a in-house talent to develop and validate tests and other assessments b experience with a variety of assessments (knowledge and ability tests,
behavioural skills, personality and work preferences, 360 surveys, etc.) c ability to provide counsel on leveraging assessment data and process to improve
organisational effectiveness d main focus is assessments versus it being a side business, but also breadth in
related HR areas (e.g., selection, training, management development, performance management, employee research)
2 Assessment platform (C2) a easy to understand and use, clear navigation, online help, etc. b good user functionality for intended purposes (e.g., save and bookmark, back
navigation) c timers available for items, sections and total assessment d item randomisation and randomly drawing items from a larger pool e ability to turn on/off functionality by assessment (e.g., whether report is
viewable to user when assessment is completed, number of times assessment can be taken)
f ability to administer multiple assessments on the same platform, with an integrated database
g control over which users have access to which assessments h user access to assessment history i detailed configuration document that explains the various options available
(Note: the points above are only a few of many possible features) j sound code management and SQA practices for managing upgrades before code
is promoted to production servers.
16 S. Datta et al.
3 Customisation (C3) a company branding (logo, colours, etc.) b assessment content, including flexibility on types of items, response scales, etc.;
can handle multiple choice, multiple select, matching, ranking, fill-in-the-blank, various rating scales, etc.
c flexibility on scoring for items, sections and total assessment d flexibility on individual and group reports e foreign language support, including double-byte characters f fast turnaround and low cost for customising.
4 Reporting (C4) a immediate scoring and reporting b individual feedback report can show scores for total assessment and
competencies/sections; developmental feedback can be linked to scores at any level, from total down to item responses
c ability to enable managers at any level to access individual and group reports so they can effectively use assessment information to increase the knowledge, skills and performance of their people
d ease of viewing reports online and printing those desired as nicely formatted documents.
5 Company administrative functionality (C5) a ability to create new users and manage current users b flexibility in how users are registered (e.g., admin can do it individually, self
registration, file transfer to vendor for import) c ability to view assessments online (without actually taking them) d ease with which individual and group reports can be accessed e company admin access to powerful and robust group reporting capabilities,
which provide answers to key questions they will have f export features so data can be used in other applications g ability to control what functionality admins have access to by role and individual
paths to internet, etc.) b state of the art hardware and software c fast performance; scalable as more clients and assessments are added (i.e., you
should have NO concerns about capacity for at least 10x the volume you will bring to the vendor)
d reliable – uptime > 99%; redundant systems and automatic failover protection; immediate/automatic alerts of problems to vendor
e sound backup procedures
Selection of internet assessment vendor using TOPSIS method 17
f in-house technology expertise g detailed technology document that explains the infrastructure and security.
7 Security (C7) a company codes, user IDs, passwords, etc. to protect confidential information;
strong confidentiality statement and real execution b hierarchical permission structure for client access c firewalls and sound practices to protect the servers d full encryption available (for user data and assessment content) without any
degradation in performance
8 Customer support (C8) a fast set up, posting new assessments, etc. (< 24 hours) b 24 × 7 coverage; immediate response to any technical problems c access via e-mail and phone d adequate staffing for total volume (including yours, if it is large!) e expertise and capacity to help implement the assessment program in a way that
ensures success.
9 Implementation process (C9) a little or no software to install; very easy to implement b vendor conducts technology audit to ensure all will work properly c availability of implementation support.
10 Costs (C10) a company setup; posting assessments b per assessment costs or annual license c other costs.
11 Vendor reputation (C11) a stability (no chance of going out of business) – several years in the assessment
business, solid financially b good customer base of well-known companies c references give high marks in all areas above
18 S. Datta et al.
Table 2 Importance weight of criteria from four DMs
Decision-makers (DMs) Criteria
DM1 DM2 DM3 DM4 C1 VH VH VH H C2 VH H VH H C3 VH H VH H C4 MH MH M MH C5 MH MH M M C6 MH MH M M C7 VH VH VH VH C8 H H H H C9 VH H VH H C10 M M ML M C11 MH H H ML
Table 3 Ratings of the five candidate vendors by DMs under various criteria
Decision makers (DMs) Criteria Vendors
DM1 DM2 DM3 DM4 V1 G G G VG V2 VG G G VG V3 F MF G G V4 G F G G
C1
V5 VG G G VG V1 F G F F V2 G G F MF V3 MF MF MF MF V4 G VG VG G
C2
V5 VG VG VG VG V1 F MF G G V2 G F G G V3 VG G G VG V4 F G F F
C3
V5 G G F MF V1 G VG G VG V2 VG VG VG VG V3 F G F G V4 G G G G
C4
V5 VG VG VG VG
Selection of internet assessment vendor using TOPSIS method 19
Table 3 Ratings of the five candidate vendors by DMs under various criteria (continued)
Decision makers (DMs) Criteria Vendors
DM1 DM2 DM3 DM4 V1 F F F F V2 G MF G MF V3 MF MF MF MF V4 G G G G
C5
V5 VG VG VG VG V1 VG VG G MF V2 VG VG G G V3 G G F VG V4 G G F F
C6
V5 VG VG MF G V1 F MF G G V2 G F G G V3 VG G G VG V4 F G F F
C7
V5 G G F MF V1 VG MF VG VG V2 G G VG VG V3 G VG G G V4 VG F G G
C8
V5 F G VG VG V1 G VG G VG V2 G VG G G V3 F G VG G V4 F G F VG
C9
V5 MF VG VG F V1 G G G VG V2 VG G G VG V3 F MF G G V4 G F G G
C10
V5 VG G G VG V1 F G F F V2 G G F MF V3 MF MF MF MF V4 G VG VG G
C11
V5 VG VG VG VG
20 S. Datta et al.
Table 4 Fuzzy-decision matrix and fuzzy weights of five candidates
Cri
teri
a
C1
C2
C3
C4
C5
C6
Wei
ght
(0.7
0, 0
.87,
0.9
5, 1
.00)
(0
.70,
0.8
0, 0
.80,
0.9
0)
(0.7
0, 0
.85,
0.9
0, 1
.00)
(0.4
0, 0
.57,
0.6
5, 0
.80)
(0.4
0, 0
.55,
0.6
0, 0
.80)
(0
.40,
0.5
5, 0
.60,
0.8
0)V
1 (0
.70,
0.8
2, 0
.85,
1.0
0)
(0.4
0, 0
.57,
0.5
7, 0
.90)
(0
.40,
0.6
7, 0
.70,
0.9
0)(0
.70,
0.8
5, 0
.90,
1.0
0)(0
.40,
0.5
0, 0
.50,
0.6
0)
(0.5
0, 0
.80,
0.8
7, 1
.00)
V2
(0.7
0, 0
.85,
0.9
0, 1
.00)
(0
.40,
0.6
7, 0
.70,
0.9
0)
(0.4
0, 0
.72,
0.7
2, 0
.90)
(0.8
0, 0
.90,
1.0
0, 1
.00)
(0.5
0, 0
.75,
0.7
7, 0
.90)
(0
.70,
0.8
5, 0
.90,
1.0
0)V
3 (0
.40,
0.6
7, 0
.70,
0.9
0)
(0.5
0, 0
.60,
0.7
0, 0
.80)
(0
.70,
0.8
5, 0
.90,
1.0
0)(0
.40,
0.6
5, 0
.65,
0.9
0)(0
.50,
0.6
0, 0
.70,
0.8
0)
(0.4
0, 0
.75,
0.7
7, 1
.00)
V4
(0.4
0, 0
.72,
0.7
2, 0
.90)
(0
.70,
0.8
5, 0
.90,
1.0
0)
(0.4
0, 0
.57,
0.5
7, 0
.90)
(0.7
0, 0
.80,
0.8
0, 0
.90)
(0.7
0, 0
.80,
0.8
0, 0
.90)
(0
.40,
0.6
5, 0
.65,
0.9
0)V
5 (0
.70,
0.8
5, 0
.90,
1.0
0)
(0.8
0, 0
.90,
1.0
0, 1
.00)
(0
.40,
0.6
7, 0
.70,
0.9
0)(0
.80,
0.9
0, 1
.00,
1.0
0)(0
.80,
0.9
0, 1
.00,
1.0
0)
(0.5
0, 0
.80,
0.8
0, 1
.00)
C
7 C
8 C
9 C
10
C11
Wei
ght
(0.8
0, 0
.90,
1.0
0, 1
.00)
(0
.70,
0.8
0, 0
.80,
0.9
0)
(0.7
5, 0
.85,
0.8
7, 1
.00)
(0.2
0, 0
.45,
0.4
7, 0
.60)
(0.2
0, 0
.62,
0.6
7, 0
.77)
V1
(0.4
0, 0
.67,
0.7
0, 0
.90)
(0
.50,
0.8
2, 0
.92,
1.0
0)
(0.7
0, 0
.85,
0.9
0, 1
.00)
(0.7
0, 0
.82,
0.8
5, 1
.00)
(0.4
0, 0
.57,
0.5
7, 0
.90)
V2
(0.4
0, 0
.65,
0.6
5, 0
.90)
(0
.40,
0.7
7, 0
.82,
1.0
0)
(0.7
0, 0
.82,
0.8
5, 1
.00)
(0.7
0, 0
.85,
0.9
0, 1
.00)
(0.4
0, 0
.67,
0.7
0, 0
.90)
V3
(0.7
0, 0
.85,
0.8
7, 1
.00)
(0
.40,
0.8
2, 0
.85,
1.0
0)
(0.4
0, 0
.75,
0.7
7, 1
.00)
(0.4
0, 0
.67,
0.7
0, 0
.90)
(0.5
0, 0
.60,
0.7
0, 0
.80)
V4
(0.4
0, 0
.57,
0.5
7, 0
.90)
(0
.40,
0.7
5, 0
.77,
1.0
0)
(0.4
0, 0
.67,
0.7
0, 1
.00)
(0.4
0, 0
.72,
0.7
2, 0
.90)
(0.7
0, 0
.85,
0.9
0, 1
.00)
V5
(0.4
0, 0
.67,
0.7
0, 0
.90)
(0
.40,
0.7
7, 0
.82,
1.0
0)
(0.4
0, 0
.72,
0.8
0, 1
.00)
(0.7
0, 0
.85,
0.9
0, 1
.00)
(0.8
0, 0
.90,
1.0
0, 1
.00)
Selection of internet assessment vendor using TOPSIS method 21
Table 5 Normalised fuzzy-decision matrix
Cri
teri
a
C1
C2
C3
C4
C5
C6
V1
(0.7
0, 0
.82,
0.8
5, 1
.00)
(0
.40,
0.5
7, 0
.57,
0.9
0)
(0.4
0, 0
.67,
0.7
0, 0
.90)
(0.7
0, 0
.85,
0.9
0, 1
.00)
(0.4
0, 0
.50,
0.5
0, 0
.60)
(0
.50,
0.8
0, 0
.87,
1.0
0)V
2 (0
.70,
0.8
5, 0
.90,
1.0
0)
(0.4
0, 0
.67,
0.7
0, 0
.90)
(0
.40,
0.7
2, 0
.72,
0.9
0)(0
.80,
0.9
0, 1
.00,
1.0
0)(0
.50,
0.7
5, 0
.77,
0.9
0)
(0.7
0, 0
.85,
0.9
0, 1
.00)
V3
(0.4
0, 0
.67,
0.7
0, 0
.90)
(0
.50,
0.6
0, 0
.70,
0.8
0)
(0.7
0, 0
.85,
0.9
0, 1
.00)
(0.4
0, 0
.65,
0.6
5, 0
.90)
(0.5
0, 0
.60,
0.7
0, 0
.80)
(0
.40,
0.7
5, 0
.77,
1.0
0)V
4 (0
.40,
0.7
2, 0
.72,
0.9
0)
(0.7
0, 0
.85,
0.9
0, 1
.00)
(0
.40,
0.5
7, 0
.57,
0.9
0)(0
.70,
0.8
0, 0
.80,
0.9
0)(0
.70,
0.8
0, 0
.80,
0.9
0)
(0.4
0, 0
.65,
0.6
5, 0
.90)
V5
(0.7
0, 0
.85,
0.9
0, 1
.00)
(0
.80,
0.9
0, 1
.00,
1.0
0)
(0.4
0, 0
.67,
0.7
0, 0
.90)
(0.8
0, 0
.90,
1.0
0, 1
.00)
(0.8
0, 0
.90,
1.0
0, 1
.00)
(0
.50,
0.8
0, 0
.80,
1.0
0)
C
7 C
8 C
9 C
10
C11
V1
(0.4
0, 0
.67,
0.7
0, 0
.90)
(0
.50,
0.8
2, 0
.92,
1.0
0)
(0.7
0, 0
.85,
0.9
0, 1
.00)
(0.4
0, 0
.47,
0.4
9, 0
.57)
(0.4
0, 0
.57,
0.5
7, 0
.90)
V2
(0.4
0, 0
.65,
0.6
5, 0
.90)
(0
.40,
0.7
7, 0
.82,
1.0
0)
(0.7
0, 0
.82,
0.8
5, 1
.00)
(0.4
0, 0
.44,
0.4
7, 0
.57)
(0.4
0, 0
.67,
0.7
0, 0
.90)
V3
(0.7
0, 0
.85,
0.8
7, 1
.00)
(0
.40,
0.8
2, 0
.85,
1.0
0)
(0.4
0, 0
.75,
0.7
7, 1
.00)
(0.4
4, 0
.57,
0.6
0, 1
.00)
(0.5
0, 0
.60,
0.7
0, 0
.80)
V4
(0.4
0, 0
.57,
0.5
7, 0
.90)
(0
.40,
0.7
5, 0
.77,
1.0
0)
(0.4
0, 0
.67,
0.7
0, 1
.00)
(0.4
4, 0
.56,
0.5
6, 1
.00)
(0.7
0, 0
.85,
0.9
0, 1
.00)
V5
(0.4
0, 0
.67,
0.7
0, 0
.90)
(0
.40,
0.7
7, 0
.82,
1.0
0)
(0.4
0, 0
.72,
0.8
0, 1
.00)
(0.4
0, 0
.44,
0.4
7, 0
.57)
(0.8
0, 0
.90,
1.0
0, 1
.00)
22 S. Datta et al.
Table 6 Weighted normalised fuzzy-decision matrix
Cri
teri
a
C1
C2
C3
C4
C5
C6
V1
(0.4
9, 0
.71,
0.8
1, 1
.00)
(0
.28,
0.4
6, 0
.46,
0.8
1)
(0.2
8, 0
.57,
0.6
3, 0
.90)
(0.2
8, 0
.48,
0.5
8, 0
.80)
(0.1
6, 0
.27,
0.3
0, 0
.48)
(0
.20,
0.4
4, 0
.52,
0.8
0)V
2 (0
.49,
0.7
4, 0
.85,
1.0
0)
(0.2
8, 0
.54,
0.5
6, 0
.81)
(0
.28,
0.6
1, 0
.65,
0.9
0)(0
.32,
0.5
1, 0
.65,
0.8
0)(0
.20,
0.4
1, 0
.46,
0.7
2)
(0.2
8, 0
.47,
0.5
4, 0
.80)
V3
(0.2
8, 0
.58,
0.6
6, 0
.90)
(0
.35,
0.4
8, 0
.56,
0.7
2)
(0.4
9, 0
.72,
0.8
1, 1
.00)
(0.1
6, 0
.37,
0.4
2, 0
.72)
(0.2
0, 0
.33,
0.4
2, 0
.64)
(0
.16,
0.4
1, 0
.46,
0.8
0)V
4 (0
.28,
0.6
3, 0
.68,
0.9
0)
(0.4
9, 0
.68,
0.7
2, 0
.90)
(0
.28,
0.4
8, 0
.51,
0.9
0)(0
.28,
0.4
6, 0
.52,
0.7
2)(0
.28,
0.4
4, 0
.48,
0.7
2)
(0.1
6, 0
.36,
0.3
9, 0
.72)
V5
(0.4
9, 0
.74,
0.8
5, 1
.00)
(0
.56,
0.7
2, 0
.80,
0.9
0)
(0.2
8, 0
.57,
0.6
3, 0
.90)
(0.3
2, 0
.51,
0.6
5, 0
.80)
(0.4
8, 0
.49,
0.6
0, 0
.80)
(0
.20,
0.4
4, 0
.48,
0.8
0)
C
7 C
8 C
9 C
10
C11
V1
(0.3
2, 0
.60,
0.7
0, 0
.90)
(0
.35,
0.6
6, 0
.74,
0.9
0)
(0.5
2, 0
.72,
0.7
8, 1
.00)
(0.0
8, 0
.21,
0.2
3, 0
.34)
(0.0
8, 0
.35,
0.3
8, 0
.69)
V2
(0.3
2, 0
.58,
0.6
5, 0
.90)
(0
.28,
0.6
2, 0
.66,
0.9
0)
(0.5
2, 0
.70,
0.7
2, 1
.00)
(0.0
8, 0
.20,
0.2
2, 0
.34)
(0.0
8, 0
.42,
0.4
7, 0
.69)
V3
(0.5
6, 0
.76,
0.8
7, 1
.00)
(0
.28,
0.6
6, 0
.68,
0.9
0)
(0.3
0, 0
.64,
0.6
7, 1
.00)
(0.0
9, 0
.26,
0.2
8, 0
.60)
(0.1
0, 0
.37,
0.4
7, 0
.62)
V4
(0.3
2, 0
.51,
0.5
7, 0
.90)
(0
.28,
0.6
0, 0
.62,
0.9
0)
(0.3
0, 0
.57,
0.6
1, 1
.00)
(0.0
9, 0
.25,
0.2
6, 0
.60)
(0.1
4, 0
.53,
0.6
0, 0
.77)
V5
(0.3
2, 0
.60,
0.7
0, 0
.90)
(0
.28,
0.6
2, 0
.66,
0.9
0)
(0.3
0, 0
.61,
0.7
0, 1
.00)
(0.0
8, 0
.20,
0.2
2, 0
.34)
(0.1
6, 0
.56,
0.6
7, 0
.77)
Selection of internet assessment vendor using TOPSIS method 23
Table 7 Distances between Ai(i = 1, 2, …, 5) and A* with respect to each criterion
The fuzzy-based TOPSIS method has been applied to solve this decision making problem, the procedural steps of which have been summarised as follows:
Step 1 Four DMs use the linguistic weighting variables shown in Figure 3 to assess the importance (priority weight) of each of the individual criterion. The importance weights of the criteria determined by these three DMs have been shown in Table 2.
Step 2 The four DMs have been instructed to assign the linguistic rating variables for subjective judgment shown in Figure 4 to evaluate the ratings of candidate vendors with respect to each criterion. The ratings of the five candidates by the DMs under the various criteria have been furnished in Table 3.
Step 3 Then the linguistic evaluations shown in Table 2 and Table 3 have been converted into trapezoidal fuzzy numbers to construct the fuzzy-decision matrix and determine the fuzzy weight of each criterion, as in Table 4.
Step 4 The normalised fuzzy-decision matrix has been constructed as in Table 5.
Step 5 Weighted normalised fuzzy-decision matrix has been constructed as in Table 6.
Step 7 Calculate the distance of each vendor from FPIS and FNIS with respect to each criterion, respectively, as Table 7 and Table 8.
Step 8 Calculate *id and id − of five possible vendor Ai(i = 1, 2, …, 5) as Table 9.
Step 9 Calculate the closeness coefficient of each vendor as:
1
2
3
4
5
0.490.510.480.500.55
CCCCCCCCCC
=====
Step 10 According to the closeness coefficients of five vendors, it has been observed that all belong to group III and the assessment status is ‘recommend with low risk’ (Table 1).
However, closeness coefficient of V5 > V2 > V4 > V1 > V3. Therefore, vendor V5 has been selected.
6 Conclusions
The foregoing study highlights application feasibility of fuzzy-set theory in vendor selection problem which is involved with uncertain and imprecise data (DMs or respondents’ opinions). In this scenario of a multi-criteria decision making process, the use of linguistic variables on subjective judgment is highly advantageous when criteria estimates cannot be expressed by means of numerical scores. Therefore, in assessing of feasible vendors with respect to criteria and corresponding priority weights, it is suggested as well as recommended to assign linguistic variables instead of numerical estimates. Due to the DMs’ perception, subjective estimates often appear in the process of vendor selection problem, an extension version of TOPSIS in a fuzzy environment has been adopted in this paper. The fuzzy TOPSIS method can efficiently manage with the ratings of both quantitative as well as qualitative criteria and select the most appropriate alternative vendor effectively. In has been experienced that the fuzzy TOPSIS method is very flexible. With respect to the closeness coefficient, not only the ranking order can be determined but also the assessment status of all possible vendors can be examined. The systematic decision making hierarchy in a fuzzy environment has been presented in this
Selection of internet assessment vendor using TOPSIS method 25
paper through a case study of internet assessment vendor selection. The said approach can also be extended to the analysis of other managerial decision making problems.
Acknowledgements
The authors sincerely express their heartiest thanks to Prof. Kannan Govindan, Chief Editor, International Journal of Business Performance and Supply Chain Modelling (IJBPSCM) and the anonymous reviewers for their valuable comments and suggestions to make the paper a good contributor.
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