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J. of Mult.-Valued Logic & Soft Computing, Vol. 24, pp. 475–500 Reprints available directly from the publisher Photocopying permitted by license only

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A Group Decision Making Approach Using Interval Type-2 Fuzzy AHP for Enterprise Information Systems Project Selection Basar Oztaysi Istanbul Technical University, Industrial Engineering Department, 34367, Istanbul, Turkey E-mail: [email protected] Received: February 26, 2013. Accepted: July 2, 2013.

As a result of emerging computation technologies, information systems are developed to support various processes of enterprises. ERP (Enterprise Resource Planning) has been a leading EIS (Enterprise Information Systems) application and companies are allocating resources to such new projects. However, studies report that most of the projects fail or cannot reach the desired outcomes selecting inappropriate software is represented as one of the major sources for project failures. This paper focuses on EIS selection problem which is a multicriteria decision making (MCDM) problem since it contains various and conflicting criteria. In this study, a group decision making approach using Analytic Hierarchy Process (AHP) and Interval type-2 fuzzy sets is used on a real world ERP selection problem with six criteria and four alternatives. The originality of the study comes from using interval type-2 fuzzy sets in the decision problem. By using interval type-2 fuzzy sets for handling fuzzy group decision-making problems more flexible decisions can be given since interval type-2 fuzzy sets are more suitable to represent uncertainties than type-1 fuzzy set. Keywords:  Enterprise Information Systems, Enterprise Resource Planning, Type-2 Fuzzy Sets, Fuzzy AHP, Fuzzy Numbers, Multicriteria, Uncertainty

1 Introduction Today, with the emerging information technologies, business processes are widely supported by information systems. Enterprise information systems (EIS) can be defined as any kind of computing system that works across the

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enterprise, such as Enterprise Resource Planning (ERP), Customer Relationship Management (CRM), Knowledge Management Systems (KMS), and Supply Chain Management (SCM). An EIS provides a central system to the organization, which ensures integration of business process and information sharing across all levels of the organization. ERP is a leading EIS application which automates business processes and supports all other business functions such as; finance, human resources, operations and logistic, sale and marketing. The integration of business processes, increases the effectiveness of management, improves decision making and enables organization wide visibility. Because of ERP systems’ numerous promises, companies have started many project initiatives and made high amounts of investments on these projects. ERP projects start with the selection of the software system and continues with the implementation which includes changes in business processes and information technology all over the company. As a result of this complex nature, ERP project may fail and cause no return on the investments. ERP projects, just like other EIS projects, need considerable financial investment and have potential risks and benefits. Thus the selection of the proper software is a very important issue for companies. The selection process can be formulated as a multi criteria decision making (MCDM) problem which focuses on problems with discrete decision space and predetermined decision alternatives. MCDM techniques can handle various different and conflicting criteria such as cost, functionality and the project durations. In the EIS selection literature WSM (Perez and Rojas, 2000; Collier et al., 1999), TOPSIS (Mao et al. 2009), AHP (Wei et al., 2005; Ngai and Chan, 2005), and ANP (Sarkis and Sundarraj, 2006; Yazgan et al., 2009) are widely used MDCM methods. In this study, a group decision making based on AHP is used to select the best project because of its flexibility, ability to decompose a decision problem into its constituent parts and support for group decision making through consensus. In the traditional formulation of MCDM problems, human’s judgments are represented as exact numbers. However, in many practical cases, usage of exact numbers may not be possible, the data may be imprecise, or the decision makers might be unable to assign exact numerical values to the evaluation. The fuzzy set theory (Zadeh, 1965) is developed to mathematically represent uncertainty and provide formalized tools for dealing with the imprecision in decision-making problems. In fuzzy sets, each element has a degree of membership which is described with a membership function valued in the interval [0, 1] (Zadeh, 1965). In the literature, traditional MCDM methods are extended using fuzzy numbers to handle uncertainties. The ordinary fuzzy sets are capable of handling uncertainties and vagueness, however, due to some shortcoming; the concept of a type-2 fuzzy set was introduced by Zadeh (1975) as an extension of ordinary fuzzy sets. Type-2 fuzzy sets can be defined as the fuzzy sets whose membership grades themselves are type-1



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fuzzy sets and they are useful to model vagueness and incorporating linguistic uncertainties (Mendel 2000; Karnik and Mendel, 2001). In this paper, the steps of interval type-2 fuzzy AHP are introduced and the method is applied to an EIS selection problem which contains both tangible and intangible criteria. The originality of the paper comes from (i) extending the current studies (Kahraman et al. 2012; Sari et al. 2013) on interval type-2 fuzzy AHP to group decision making and (ii) applying the proposed approach to enterprise information selection problem. Using interval type-2 fuzzy sets for group decision-making problems a more flexible and more intelligent results are determined due to the fact that interval type-2 fuzzy sets can represent uncertainity better than traditional type-1 fuzzy sets (Chen and Lee; 2010a, 2010b). The rest of the paper is organized as follows. Section 2 presents the literature review on enterprise information systems selection problem. In Section 3 methodology is introduced, starting with interval type-2 fuzzy sets first, and then the proposed AHP method is presented. ERP selection decision model, which consists of six criteria and four alternative is given in Section 4. In Section 5 a numerical case study is given and interval type-2 fuzzy AHP is applied to a case study. Finally Section 6 gives the conclusions.

2  Literature Review In recent years, selection of information systems has been the focus of academic studies. The current studies in this field can be classified into four main groups (Jadhav and Sonar, 2011): (i) evaluation and selection of specific products (Ngai and Chan, 2005; Colombo and Francalanci, 2004) (ii) methodology for selection (Kunda, 2003; Mehrez et al., 1993; Williams, 1992; Hlupic and Paul, 1996) (iii) criteria for selection (Chau, 1995; Stylianou et al., 1992; Arditi and Singh, 1991) (iv) automated systems/tools that assist decision makers (Mohamed et al., 2004; Grau et al., 2004). Evaluation and selection of enterprise information systems can be formulated multicriteria decision making (MCDM) problem. MCDM is an important branch of decision making that focuses on problems with discrete decision space in which the decision alternatives are predetermined (Triantaphyllou, 2000). In an EIS selection problem it is necessary to take into consideration various points of view such as costs, benefits, and usability. In MCDM each point of view is represented with a different criterion and the alternatives are evaluated according to these criteria. By doing so different aspects of decision makers can be involved into the decision process (Roy 2005). Though there are various techniques that fall into the same group, the main steps of MCDM can be given follows (Opricovic and Tzeng, 2004): (1) Establishing a system of evaluation criteria. (2) Developing the alternatives. (3) Evaluating alternatives in terms of criteria. (4)

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Applying a multi criteria analysis method. (5) Selecting one alternative as the optimal one. The techniques used for information systems selection include MCDM techniques such as weighted sum method (WSM) and Analytic Hierarchy Process (AHP), Technique for Order Performance by Similarity to Ideal Solution (TOPSIS) , fuzzy MCDM approaches, Data Envelopment Analysis (DEA), real options and feature analyses. The Weighted Sum Method (WSM) assumes that any decision-maker attempts implicitly to maximize a function by aggregating different points of view that are taken into account (Triantaphyllou, 2000). For each criterion, weights and rating scales are assigned. Then, the alternatives are evaluated and numerical ratings are assigned to each alternative for each criterion. Finally, these numerical ratings are multiplied by the criterion’s weight, and these scores are summed to produce an overall score. Perez and Rojas (2000) used WSM in a case study that evaluates workflow software. Collier et al. (1999) proposed a methodology for the evaluation and selection of data mining software for Northern Arizona University’s data center. Blanc and Jelassi (1989) proposed a three step framework for evaluating DSS software. Morisio and Tsoukis (1997) presented a formal methodology to evaluate any software packages; Williams (1992) suggested a WSM based method to evaluate the suitability of software products against the total requirements of a project. TOPSIS method has been also used in the information systems literature selcetion problem. TOPSIS is a useful technique for ranking and selecting the best alternative based on positive ideal and negative ideal solutions (Hwang and Yoon 1981). In TOPSIS technique, first the positive ideal solution and negative ideal solutions are determined for each criteria, then the relative closeness of each alternative with these ideals are obtained. Finally and a preference order of alternaties is ranked according to a combitaniton of these two closeness measures. There is a limited number of TOPSIS applications in the field of IS selection. Mao et al. (2009) suggested using TOPSIS and balanced scorecard for information systems selection. DEA is a nonparametric method for the estimation of efficiency frontiers. It is used to empirically measure productive efficiency of decision making units by the comparison of inputs and outputs. In the field of IS selection DEA has a limited number of applications. Fisher et al. (2004) utilized DEA to compare the performance of ERP systems. Lall and Teyarachakul (2006) also suggested using DEA for ERP performance evaluation. Bernroider and Stix (2006) proposed a hybrid method for software selection integrating utility ranking method and DEA. Asosheh et al. (2010) presented an approach using DEA and balanced scorecard for IT project evaluation and selection. Real Options approach is also used for the selection of information systems. Wu and Ong (2008) proposed a framework for evaluating the IT investments based on real options. Chen et al. (2009) suggested using real options for evaluation the information technology projects under multiple risk situations. Tolga (2012), integrated



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ELECTRE with real options approach for the evaluation of software development projects. You et al.(2012) proposed using real options in fuzzy environment for the evaluation of ERP investments. An other method to be mentioned is the Feature Analysis. The method is not a formal MCDM method but can be evaluated as a primitive application used for IS selection. In its simplest form, feature analysis provides a list of “yes/no” responses to the existence of a particular property in a product. First the decision maker decides on the features that are required from the product, then all the alternatives are check whether they fulfill the requirements or not. At the end of the process for each alternative, the number of fulfilled features is determined and the alternative with the highest value is selected (Kitchenham, 1996). The application case gets complicated when the applications are about software packages because it becomes hard to respond with true/false kind of answers. One of the most popular methods in the field of IS selection is Analytic Hierarch Process (AHP). AHP, developed by Saaty (1980), enables decisionmakers to structure a multi-criteria decision making problem into a hierarchy. The technique is based on pair-wise comparison between the alternatives. The weights of the criteria and alternatives are found by calculating the eigenvector from the comparison matrix’s largest eigenvalue. There are various studies that use AHP for IS selection. Teltumbde (2000) proposed a methodology for the selection of ERP software using AHP. Morera (2002) presented a combined methodology using DESMET and AHP for the evaluation of COTS. Kunda (2003) proposed an AHP based social-technical approach for commercial of the shelf (COTS) software components. Sarkis and Talluri (2004), used AHP to evaluate and select e-commerce software and communication systems for supply chain. Phillips-Wren et al. (2004) proposed a framework for evaluating decision support systems using AHP. Colombo and Francalanci (2004) focused on the selection of CRM packages covering both functional and architectural quality using AHP. Wei et al. (2005) proposed an AHP based methodology for ERP selection. Ngai and Chan (2005) applied AHP for the evaluation of knowledge management tools. Analytic Network Process (ANP) is an extension of AHP which can handle the decision problems that has dependence and feedback between the criteria. Sarkis and Sundarraj (2003) suggested a methodology which uses ANP, for componentized enterprise information technology evaluation and selection. Sarkis and Sundarraj (2006) proposed a two-stage methodology which involves a combination of the ANP and integer programming in order to deal with interdependent set of IS evaluation. Yazgan et al. (2009) used ANP and artificial neural networks to make an ERP software selection. Kop et. al (2011) investigated the risk of ERP projects using ANP in fuzzy environment from different risk perspectives such as; executive risks, organizational risks, project management risks and technical risks. Another group of MCDM approach can be given as the fuzzy approaches. The studies in this group generally enable decision makers to use linguistic

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terms to evaluate alternatives. Using linguistic terms for evaluations improve decision making procedure by accommodating the vagueness and ambiguity in human decision making. Chen (2002) proposed an algorithm based on the fuzzy measure and fuzzy integral to deal with the IS project selection problems. Bozdag et al. (2003) focused on selection of computer integrated manufacturing systems and used four different fuzzy multi-attribute group decision-making methods including; Blin’s model, fuzzy synthetic evaluation, Yager’s weighted goals method, and fuzzy analytic hierarchy process. Mikhailov and Masizana (2004), proposed a decision support system for IS selection using linguistic terms and fuzzy numbers. Cochran and Chen (2005) applied fuzzy multicriteria decision making on object oriented simulation software selection using linguistic terms and rule based systems. Lin et al. (2006) proposed group decision making with linguistic terms in the context of data warehouse selection. Liao et al. (2007) proposed a group decision making model using linguistic terms and linear programming for the process of ERP systems selection. Şen et al. (2009) proposed a fuzzy multi attribute decision making model that can handle qualitative and quantitative objectives for enterprise software selection. Karsak and Özogul (2009) proposed a novel approach that combines quality function deployment, fuzzy linear regression and zero-one goal programming for the selection of ERP systems. Chen and Cheng (2009) presented a MCDM based on the fuzzy measure and the fuzzy integral for selecting an information system. As an extension of regular fuzzy sets, Zadeh (1975) introduced type-2 fuzzy sets whose membership grades are themselves type-1 fuzzy sets. Type-2 fuzzy sets are very useful in circumstances where it is difficult to determine an exact membership function for a fuzzy set, thus it can model uncertainties associated with usage of words (Mendel, 2000; Mendel 2007). Due to the computational difficulties, interval type-2 fuzzy sets, as sub set of type-2 fuzzy sets are used in the literature. Although interval type-2 fuzzy sets are not directly used in IS selection problems, the approach is recently used in other decision making problems. Chen and Lee (2010a) proposed a group decision making approach based interval type-2 fuzzy sets. In the study, mathematical operators are introduced and the proposed ranking approach is compared with other methods. Chen and Lee (2010b) proposed an extended TOPSIS method and presented numerical examples using interval type-2 fuzzy sets. Wang et al. (2012) presented a group decision making approach where the information provided by the decision makers are expressed as interval type-2 fuzzy decision matrices, and criteria weights are partially known. Chen et al. (2012) proposed a new multi attribute group decision making method using interval type-2 fuzzy sets including a novel ranking technique. Kahraman et al. (2012), integrated type-2 fuzzy sets with analytic hierarchy process (AHP). Sari et al. (2013) extended AHP method using trapezoidal interval type-2 fuzzy sets and applied it to warehouse selection problem. Zhang and Zhang (2013) poposed trapezoidal interval type-2



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fuzzy soft sets and use it to represent linguistic variables. The authors also proposed a novel approach to multi attribute group decision making under interval type-2 fuzzy environment. Chen and Wang (2013) proposed using a-cut approach to rank and evaluate interval type-2 fuzzy sets and developed a new decision making method based on this approach. Ngan (2013) extended probabilistic linguistic framework to develop methodologies such as taking union and intersection and performing arithmetic operations on type-2 fuzzy numbers and apply these on a multi criteria decision making problem. The presented literature review shows that MCDM methods are widely used for the selection of information systems. In order to model uncertainties in linguistic evaluations, fuzzy extensions of the methods are proposed and used. However type-2 fuzzy sets, which can handle the uncertainties in decision models in a better way, do not have any applications in this area. Thus this study aims to contribute to the literature by proposing a group decision making approach based on interval type-2 fuzzy numbers and AHP.

3 Methodology In this study, a group decision making approach based on interval type-2 fuzzy sets and AHP is proposed by extending the current studies of Kahraman et al. sets (2012) and Sari et al. (2013). In this section, in order to clarify the method, first the interval type-2 fuzzy sets are explained and then the steps of the method is introduced. 3.1  Interval type-2 Fuzzy Sets The fuzzy set theory (Zadeh, 1965) was specifically designed to mathematically represent uncertainty and provide formalized tools for dealing with the imprecision intrinsic to many problems. Fuzzy sets enable the use of approximate information and uncertainty in the decision process. Since knowledge can be expressed in a more natural way by fuzzy sets, many engineering and decision problems can be greatly simplified and enhanced. Thus many crisp analysis methods are extended using fuzzy sets to solve real-world problems, which contain imprecision in the variables and parameters measured and processed. Various MCDM techniques are extended to use linguistic variables to achieve this benefit. Linguistic variables are variables that can take words in natural languages as its value such as ‘‘good’’, ‘‘medium’’, and ‘‘bad’’. These linguistic values are characterized by fuzzy sets in the universe of discourse in which the variable is defined and integrated into the techniques. Several geometric mapping functions functions such as, triangular, trapezoidal and S-shaped membership functions are being used in the literature to mathematically represent linguistic variables (Kahraman and Kaya, 2010). In the classical set theory, an element either belongs or does not belong to a set. However in type-1 fuzzy sets (Zadeh 1965), each element has a degree

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of membership which is described by a membership function valued in the interval [0, 1]. This property enable classical (type-1) fuzzy set to handle uncertainties; however shortcomings of classical fuzzy set (Mendel and John, 2002) are reported such as usage of words, difficulties in aggregation of experts opinions and working with noisy data. Type-2 fuzzy sets are proposed by Zadeh (1975) to handle these kinds of uncertainties. The membership values of type-2 fuzzy sets are also fuzzy numbers themselves. While, the membership functions of type-1 fuzzy sets are two-dimensional, membership functions of type-2 fuzzy sets are three-dimensional. It is the new third dimension that provides additional degrees of freedom that make it possible to directly model uncertainties (Mendel and John, 2002). A type-2 fuzzy sets  A in the universe of discourse X can be represented by a type-2 membership function µ A ( x, u) = 1 where x ∈X and u ∈ J x ⊆ [ 0,1] as follows (Zadeh, 1975):

 A=

{((x, u) , µ

 A

( x, u)) | ∀x ∈ X, ∀u ∈ J x ⊆ [0,1] , 0 ≤ µ A ( x, u) ≤ 1} (1)

where Jx denotes an interval [0,1]. The type-2 fuzzy set  A also can be represented as follows (Mendel et. al, 2006):  A=∫



∫

x ∈X u ∈J x

µ A ( x, u ) / ( x, u ) J x ⊆ [ 0,1]

(2)

where J x ⊆ [ 0,1] and ∫ ∫ denote union over all admissible x and u. With this definition, if all µ  ( x, u) = 1, then  A is called an interval type-2 A fuzzy set (Buckley, 1985). An interval type-2 fuzzy set  A can be regarded as a special case of a type-2 fuzzy set, represented as follows (Mendel et. al, 2006):

，

 A=



1

∫ ∫ ( x, u) , (3) x∈X

u∈J x

where J x ⊆ [ 0,1] . In accordance with the given definitions, a trapezoidal interval type-2 fuzzy set can be repreented as

(

) (

( ) ( )) (

( ) ( ))

L L U L U U Ai = Ai ; Ai =  aiU1 , aiU2 , aiU3 , aiU4 ; H1 Ai , H 2 Ai , aiL1 , aiL2 , aiL3 , aiL4 ; H1 Ai , H 2 Ai   

U L U U U U L L L L  Ai and  where ， Ai are type-1 fuzzy sets; ai1 , ai 2 , ai 3 , ai 4 , ai1 , ai 2 , ai 3 and ai 4 are

( )

U the reference points of the interval type-2 fuzzy set  Ai , H j  Ai ;, shows the

membership value of the element aUj ( j +1) in the upper trapezoidal membership



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( )

U L function  Ai ,1 ≤ j ≤ 2, H j  Ai denotes the membership value of the element

( )

U ≤ j ≤ 2, H1  Ai

( )

L U U in the lower trapezoidal membership function  Ai ,1 ≤ j ≤ 2, H1  Ai ∈[0,1], H 2 (  Ai ) U L L ∈[0,1], H 2 (  Ai ) ∈[0,1], H1  Ai ∈[0,1], H 2  Ai ∈[0,1] and 1 ≤ i ≤ n (Chen

a

L j ( j +1)

( )

( )

and Lee, 2010a). Figure 1 represents a sample trapezoidal interval type-2 fuzzy set.   Let A1 and A2 be interval type-2 fuzzy sets and k be a crisp number,

(

( ) ( )) (

( ) ( ))

(

( ) ( ))

( ) ( ))

L   U U U U  1U , H A  1U , a L , a L , a L , a L , H A  1L , H   , a12 , a13 , a14 ; H1 A A1 =   a11  13 14 1 2 A1 2 11 12 

L U U U U U U L L L  2L  A2 =   a21 , a22 , a23 , a24 ; H1  , a22 , a2L3 , a24 A2 , H 2  A2 , ( a21 , H1  A2 , H 2 A  

the arithmetic operations with these numbers are shown in the following (Chen and Lee, 2010a). Addition :

( ( ) ( ) ) ( ( ) ( ))

((

1 ⊕ A  2 = aU + aU , aU + aU , aU + aU , aU + aU ; min H A  1U ; H A  U2 , min H A  1U ; H A  U2  , A 11 21 12 22 13 23 14 24 1 1 2 2 

(a

L 11

 ); H ( A  2 )), min( H ( A  1 ); H ( A  2 )))) + a , a + a , a + a , a + a ; min( H1 ( A 1 2 2 L 21

L 12

L 22

L 13

L 23

L 14

L 1

L 24

L

L

L

(4)

((

  U U U U U U U A1  A2 = a11 − a24 , a12 − a23 , a13 − a22 , a14 − a2U1 ;

Subtraction :

( ( ) ( ) ) ( ( ) ( ))

U U U U min H1  A1 ; H1  A2 , min H 2  A1 ; H 2  A2  , 

(a

L 11

L L L L − a24 , a12L − a23 , a13L − a22 , a14L − a21 ;

L L L L A1 ); H 2 (  A2 )))) min( H1 (  A1 ); H1 (  A2 )), min( H 2 ( 

Figure 1 Interval Type-2 Fuzzy sets.

(5)

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((

U U U U U U U U 1 ⊗  A A2 ≅ a11 × a21 , a12 × a22 , a13 × a23 , a14 × a24 ;

Multiplication :

( ( ) ( ) ) ( ( ) ( ))

1 ; H  min H1 A , min H 2  A1 ; H 2  A2  , 1 A2  (6) L L L L a11L × a21 , a12L × a22 , a13L × a23 , a14L × a24 ; U

U

U

U

((

L  1L ); H ( A  2L )))) min( H1 (  A1 ); H1 ( A ?2L )), min( H 2 ( A 2

Multiplication with a crisp number :

((

( ) ( ) (k × a , k × a , k × a , k × a ; H ( A ), H ( A )) (7) )

U  U U U  1U , ; H1  A1 , H 2 A k A1 = k × a11 , k × a12 , k × a1U3 , k × a14

L 11

L 12

L 13

L 14

1

L 1

2

L 1

Division by a crisp number :

 A1   1 U 1 U 1 U 1 U U U =   × a11 , × a12 , × a13 , × a14  ; H1 A1 , H 2 A1 ,   k k k k k 1 L 1 L 1 L 1 L L L   × a11 , × a12 , × a13 , × a14 ; H1 A1 , H 2 A1    k k k k (8)

( ) ( )

( ) ( )

where k > 0. Based on these arithmetic operations the interval type-2 fuzzy AHP is introduced in the next section. 3.2  Interval Type-2 Fuzzy AHP AHP, developed by Saaty (1980), is a structured approach used for decision making. AHP aims quantifying relative priorities for a given set of alternatives based on the decision makers’ judgements, It also stresses the consistency of the comparison of alternatives and has the ability to detect and incorporate inconsistencies inherent in the decision making process. The original scale used for decision maker’s evaluations is composed of crisp numbers. However in real world, a decision maker’s judgements may not be crisp and it can be relatively difficult for him/her to provide exact numerical



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values. In these cases, using fuzzy logic provides a mathematical strength to capture the uncertainties associated with human cognitive process (Kahraman et al. 2010). From this point of view, fuzzy extensions of classical AHP have been proposed. Literature provides several fuzzy AHP applications which use fuzzy set theory to identify the linguistic variables. Laarhoven and Pedrycz (1983) proposed the first algorithm in fuzzy AHP by using triangular fuzzy membership functions and Lootsma’s logarithmic least square method. Buckley (1985) extended the method with trapezoidal fuzzy numbers and used the geometric mean method to derive fuzzy weights and performance scores. Chang (1996) proposed a methodology based on triangular fuzzy numbers for pairwise comparisons and used the extent analysis method for the synthetic extent values of the pairwise comparisons. In one of the recent studies Zeng et al. (2007), proposed using arithmetic averaging method to get performance scores and extend the method with different scales contains triangular, trapezoidal, and crisp numbers. In order to better handle the uncertainty and vagueness, type-2 fuzzy sets can be integrated with the AHP method. Kahraman et al. (2012) and Sari et al. (2013) are the initial studies that focus on using interval type-2 fuzzy sets with AHP. Their proposed method is based on Buckley’s (1985) fuzzy AHP method and use trapezoidal interval type-2 fuzzy sets. In this study these initial studies are extended to handle group decision making. The extended procedure of the interval type-2 fuzzy AHP method is given in the following: Step 1: Define the problem and establish its goal. Step 2: Structure the hierarchy from the top through the intermediate levels by determining the criteria and finally at the lowest level, list of the alternatives. Step 3: Construct fuzzy pairwise comparison matrices among all the criteria in the dimensions of the hierarchy system. The experts use linguistic variables to evaluate the pairwise comparisons. The linguistic variables and their trapezoidal interval type-2 fuzzy scales used in the study are given in Table 1 (Sari et al., 2013). The resulting comparison matrices are constructed as type-2 interval fuzzy  sets A as given in the following;

()



1   =  a 21 A      a n1

a12 1  a n2

  a1n     a 2 n   =        1   

1 1 a12

a12

 1 a1n



1

1

a 2 n

 a1n    a 2 n        1  

(9)

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Linguistic variables

Trapezoidal Interval Type-2 fuzzy scales

Absolutely Strong (AS)

(7,8,9,9;1,1) (7.2,8.2,8.8,9;0.8,0.8)

Very Strong (VS)

(5,6,8,9;1,1) (5.2,6.2,7.8,8.8;0.8,0.8)

Fairly Strong (FS)

(3,4,6,7;1,1) (3.2,4.2,5.8,6.8;0.8,0.8)

Slightly Strong (SS)

(1,2,4,5;1,1) (1.2,2.2,3.8,4.8;0.8,0.8)

Exactly Equal (E)

(1,1,1,1;1,1) (1,1,1,1;1,1) Reciprocals of above

If factor i has one of the above linguistic variables assigned to it when compared with factor j, then j has the reciprocal value when compared with i.

Table 1 Definition and interval type 2 fuzzy scales of the linguistic variables

where  1 1 1 1  U U    aU , aU , aU , aU ; H1 a12 , H 2 a13  ,  1 =  14 13 12 11  a   1 1 1 1  L L   L , L , L , L ; H1 a22 , H 2 a23     a24 a23 a22 a21 



( )

( )

( )

( )

Step 4: Examine the consistency of the fuzzy pairwise comparison matrices. In this manner, the matrices are defuzzified and checked for consistencies. If there are inconsistencies detected, the experts are asked to reevaluate the comparison matrices. Step 5: Aggregate the expert evaluations using geometric mean. 1

n  n  1 a ij =  a ij ⊗ …⊗ a ij  (11)  

Where



n

( (

( ) ( ))  (12) ( ) ( ))

 n aU , n aU , n aU , n aU ; H u a , H u a 1 2 ij1 ij 2 ij 3 ij 4 ij ij a ij =   n L n L n L n L  , aij1 , aij 2 , aij 3 , aij 4 ; H1L aij , H 2L aaij

Step 6: Calculate the fuzzy weights for each criterion. In this manner first the geometric mean of each row ri is calculated using Eq. 11 and Eq. 12;



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In order to find the fuzzy weights, the fuzzy priorities are computed by normalization. The fuzzy weight pi of the ith criterion is calculated as follows; −1 p = r i ⊗ r1 ⊕ …⊕ r i ⊕ …⊕ r n  (13) i  

Where



  a ij  a1u a2u a3u a4u =  u , u , u , u , min H1u ( a ) , H1u ( b ) , min H 2u ( a ) , H 2u ( b )    bij  b4 b3 b2 b1

(

)

(

)

 a1L a2L a3L a4L  L L L L  b L , b L , b L , b L , min H1 ( a ) , H1 (b) , min H 2 (a ), H 2 (b)  4 3 2 1

(

)

(

)

and Eq. 4 is used for the summation, Step 7: Calculate the fuzzy performance scores of each alternative using the Equation 14. n    U i = ∑ j =1 w j s j , ∀i.



(14)

 is the fuzzy utility of alternative i,  is the weight of the criterion where U wj i  j, and s j is the score of the alternative with respect to criterion j. Recall that  j and s j are computed from different pairwise comparison matrices using w  represents the fuzzy priority of the the same formulas in Step 6. While w j related pairwise comparison of the criteria, s j represents the fuzzy priority calculated from the related pairwise comparison of the alternatives with respect to the related criterion. Step 8: Defuzzify type-2 interval fuzzy sets in order to determine the importance ranking of the alternatives are defuzzified. The DTtrT method (Kahraman et. al 2012) is used for defuzzification in this step.

(uU − lU ) + (βU .m1U − lU ) + (αU .m2U − lU ) + l

U

4

 (u − l ) + (β L .m1L − lL ) + (α L .m2 L − lL )  + lL  + L L

DTtrT =

4



2



(15)

488

Basar Oztaysi

Step 9: Determine the best alternative using the defuzzified utility values of the alternatives. The alternative with the highest utility is selected. 4  A Numerical Case Study 4.1 Background In this section, a numerical application of interval type-2 fuzzy AHP in a real world case study is presented. The case study is from a workwear manufacturing company which has many shops allover and in some European countries in Europe. The company’s head office is located in Istanbul but the manufacturing facility operates in Adapazari. Since the workwear industry includes many products and materials, the company has its own production as well as some products are being outsourced. The management of the company seems to be complex since the head office, shops and manufacturing facility is located in different geographical areas. In the initial status, the information systems are based on accounting software and local production planning software customized for textile sector. In order to maintain an improved management and visibility, the company considers making an investment on the information systems. In order to make a comprehensive decision, the company builds a decision making team which contains a high level manager from the company, a consultant from academia and a consultant from a IS company. As a result of the initial studies, the decision model is constructed as given in Section 4.2 and a short list of alternatives is determined. The four alternatives are determined as follows; Alternative1 (A1) No Investment, Alternative2 (A2) A Local Midsized ERP Software. Alternative3 (A3) an International Midsized ERP, Alternative4 (A4) An Internal ERP Software. 4.2  Decision Model The evaluation criteria for information system selection problem are determined with respect to relevant literature (Kahraman et al. 2010; Sen, et al. 2009; Lall and Liao et al. 2007; Teyarachakul, 2006; Fish et al. 2004; Teltumbde, 2000) and the comments of the decision making team. As a result a three level AHP model is constructed as shown in Figure 2. The criteria selected for the Enterprise Resource Planning (ERP) system selection are given as follows: (1) Technology (TEC): This criterion represents the accordance between technological infrastructure of the company (such as; databases, operation systems, hardware and security infrastructure) and the ERP system alternatives. If there is a lack of accordance, many additional costs can emerge and unexpected problems may occur. (2) Schedule (SCH): Schedule refers to the duration of the project from the starting of the project to the establishment of the system. The criterion



Group Decision Making

489

Figure 2 Decision model for ERP System Selection.

(3)

(4)

(5)

(6)

schedule contains the database and software installations, system training, conceptual design, graphical design, data migration from the legacy systems, tests and going live with the system. Economic (ECO): This criterion refers to the total cost of ownership. Not only the initial cost of the software and hardware, but also training, consultancy, implementation and supporting costs must be evaluated in order to calculate the total cost of ownership. Ease of Use (EU): Users are among the most important factors of an information system. Ease of use is the most important issue for a technology to be accepted by the users. If the users experience difficulties in using software, their attitude towards the software may change. Userspecific home pages, role-based functionality, familiar user interface can be defined as some of the marks of ease of use. Functionality (FUN): Functionality refers to capabilities of the software. It is mostly maintained by built in applications and the software modules. The functionality of the software should be evaluated based on the current requirements of the company. Generally a pilot project is given to the vendors to evaluate the functionality of the software in real time. Vendor (VEN): Besides the functionality of the software, the vendor that take place in the implementation has a vital role in the success of the projects. That’s why the reputation of the vendor is very important for the selection process. The current references and capabilities of the vendors should be examined.

4.3  Application of the methodology In this section the steps of interval type-2 fuzzy AHP which is defined in Section 3 are applied in the following. The goal of the decision process is defined as selecting the best IS project among the four alternatives with a decision

490

Basar Oztaysi

model containing six criteria; namely technology, schedule, economic, ease of use, functionality, and vendor. and four alternatives. After the decision model is structured in the first two stage, in the third stage of the application the pairwise comparison matrices for criteria and alternatives are constructed. One matrix for comparison of the criteria and six matrices for the comparison of alternatives with respect to check, each criterion are formed; and three experts evaluated the pairwise comparisons using linguistic variables given in Table 1. Table 2 shows the linguistic evaluations of the six criteria and Table 3 shows comparison of alternatives with respect to each criterion. After the consistency check the priority weights are calculated following the steps 5, 6, and 7. In this part of the study sample calculations are given for a single matrix which contains the expert evaluations about the technology criteria. In this matrix, the pair wise comparison of A4 and A2, with respect to Technology criterion is evaluated by the three different experts as FS, SS and VS. To aggregate the evaluations Equations 11 and 12 are used. 1

 3 (3.4.6.7,1.1)(3.2, 4.2, 5.8, 6.8; 0.8, 0.8) ⊗ (1, 2, 4, 5; 1,1) a 12 =   (1.2, 2.2, 3.8, 4.8; 0.8, 0.8) ⊗ (5, 6, 8, 9; 1,1)(5.2, 6.2, 7.8, 8.8; 0.8, 0.8) = ( 2.46, 3.63, 5.76, 6.80;1,1) ( 2.71, 3.85, 5.56, 6..59; 0.8, 0.8) Using the same calculations the aggregated decision matrix is obtained as follows. The geometric mean of each row is calculated to find the ri values. As an example r1 is calculated as : 1



 4 (1,1,1,1;1,1)(1,1,1,1;1,1) ⊗ (0.30, 0.34, 0.5, 0.69; 1,1) (0.31, 0.35, 0.47, 0.63; 0.8, 0.8) ⊗ (0.15, 0.19, 0.31, 0.48; 1,1)    r1 =  (0.16, 0.19, 0.29, 0.43; 0.8, 0.8) ⊗ (0.15, 0.19, 0.31, 0.48; 1,1)    (0.16, 0.19, 0.29, 0.43; 0.8, 0.8) = ( 0.29, 0.33, 0.47, 0.63;; 1,1)( 0.30, 0.34, 0.45, 0.58; 0.8, 0.8)

After all other ri values are calculated in a similar way, the fuzzy weight of each criterion, pi, is calculated by normalization denoted in Equation 13.

−1 p = r1 ⊗ r1 ⊕ r 2 ⊕ r 3 ⊕ r 4  1  

FS

FS

VEN

FUN

FS

SS

SS

SS

FS

FS

SS

SS

E

Exp3

FS

1/SS

FS

SS

E

1/SS

Exp1

Table 2 The pairwise comparison for the criteria

FS

EU

FS

E

SS

SS

SCH

ECO

E

E

TEC

Exp2

Exp1

TEC

 

 

FS

1/FS

SS

FS

E

E

Exp2

SCH

SS

1/SS

SS

SS

E

1/SS

Exp3

SS

1/SS

E

E

1/SS

1/SS

Exp1

FS

E

SS

E

1/FS

1/FS

Exp2

ECO

E

1/SS

E

E

1/SS

1/SS

Exp3

SS

1/FS

E

1/E

1/FS

1/FS

Exp1

SS

E

E

1/SS

1/SS

1/SS

Exp2

EU

FS

1/FS

E

1/E

1/SS

1/FS

Exp3

VS

E

FS

SS

1/SS

1/FS

Exp1

FS

E

E

E

1/FS

1/SS

Exp2

VEN

VS

E

FS

SS

1/SS

1/FS

Exp3

E

1/VS

1/SS

1/SS

1/FS

1/FS

Exp1

E

1/FS

1/SS

1/FS

1/FS

1/FS

Exp2

FUN

E

1/VS

1/FS

E

1/SS

1/SS

Exp3

Group Decision Making 491

492

Basar Oztaysi

 

 

w.r.t. TEC

A1

 

 

A2

 

 

A3

 

 

A4

 

Exp1 Exp2 Exp3 Exp1 Exp2 Exp3 Exp1 Exp2 Exp3 Exp1 Exp2 Exp3

A1

E

E

E

E

1/FS

1/SS

1/FS

1/FS

1/SS

1/FS

1/SS

1/FS

A2

E

FS

SS

E

E

E

1/FS

1/SS

1/FS

1/FS

1/SS

1/VS

A3

FS

FS

SS

FS

SS

FS

E

E

E

1/SS

E

E

A4

FS

SS

FS

FS

SS

VS

SS

E

E

E

E

E

 

 

A1

 

 

A2

 

 

A3

 

 

A4

 

w.r.1/. Exp1 Exp2 Exp3 Exp1 Exp2 Exp3 Exp1 Exp2 Exp3 Exp1 Exp2 Exp3 SCH A1

E

E

A2

1/FS

1/VS 1/FS

A3

E

FS

VS

FS

A

VS

VS

A

VS

A

E

E

E

E

E

SS

FS

VS

VS

1/A

1/VS 1/VS E

E

1/SS

E

E

E

FS

VS

FS

A4

FS

FS

VS

1/FS

1/VS 1/VS 1/FS

1/VS 1/FS

E

E

E

 

 

A1

 

 

A2

A3

 

A4

 

 

 

 

w.r.1/. Exp1 Exp2 Exp3 Exp1 Exp2 Exp3 Exp1 Exp2 Exp3 Exp1 Exp2 Exp3 ECO A1

E

E

E

E

SS

E

FS

SS

SS

VS

A

VS

A2

E

1/SS

E

E

E

E

FS

SS

SS

VS

VS

VS

A3

1/FS

1/SS

1/SS

1/VS 1/VS 1/VS E

E

E

FS

SS

FS

A4

1/VS 1/A

1/VS 1/FS

1/SS

1/FS

1/FS

1/SS

1/FS

E

E

E

 

 

 

A2

 

 

A3

 

 

A4

 

A1

 

w.r.1/. Exp1 Exp2 Exp3 Exp1 Exp2 Exp3 Exp1 Exp2 Exp3 Exp1 Exp2 Exp3 EU A1

E

E

E

E

E

1/SS

1/E

1/SS

1/SS

1/SS

1/SS

1/FS

A2

1/E

1/E

SS

E

E

E

1/E

1/SS

1/SS

1/FS

1/SS

1/SS

A3

E

SS

SS

E

SS

SS

E

E

E

1/E

1/SS

1/SS

A4

SS

SS

FS

FS

SS

SS

E

SS

SS

E

E

E

 

 

A1

 

 

A2

 

 

A3

 

 

A4

 

w.r.1/. Exp1 Exp2 Exp3 Exp1 Exp2 Exp3 Exp1 Exp2 Exp3 Exp1 Exp2 Exp3 VEN A1

E

E

E

E

1/SS

1/SS

1/E

1/SS

1/FS

1/SS

1/FS

1/SS

A2

1/E

SS

SS

E

E

E

1/E

1/SS

1/FS

E

1/SS

1/FS

A3

E

SS

FS

E

SS

FS

E

E

E

E

E

SS

A4

SS

FS

SS

E

SS

FS

E

E

1/SS

E

E

E

 

 

A1

 

 

A2

 

 

A3

 

 

A4

 

w.r.1/. Exp1 Exp2 Exp3 Exp1 Exp2 Exp3 Exp1 Exp2 Exp3 Exp1 Exp2 Exp3 FUN A1

E

E

E

1/E

1/SS

1/SS

1/FS

1/SS

1/FS

1/FS

A2

E

SS

SS

E

E

E

1/FS

1/FS

1/SS

1/VS 1/E

1/SS

A3

FS

SS

FS

FS

FS

SS

E

E

E

E

1/SS

E

A4

FS

SS

FS

VS

E

SS

E

SS

E

E

E

E

Table 3 The pairwise comparison for the alternatives

1/SS

1/FS

 p1 = ( 0.29, 0.33, 0.47, 0.63; 1,1)( 0.30, 0.34, 0.45, 0.58; 0.8, 0.8) ecision  ( 0.29, 0.33, 0.47, 0G.roup 63;1,D1)( 0.30,M 0.aking 34, 0. 45, 0.58; 0.8, 0.8) ⊕   (0.42, 0.50, 0.70, 0.89;1,1)(0.44, 0.52, 0.68, 0.84; 0.8, 0.8) ⊕ ⊗  (1.26,1.58, 2.16, 2.50; 1,1)(1.33,1.64, 2.10, 2.42; 0.8, 0.8) ⊕     (1.50,1.95, 2.63, 2.92;1,1)(1.60, 2.02, 2.57, 2.86; 0.8, 0.8) 





−1

493



p = ( 0.042, 0.056, 0.10, 0.18; 1,1) ( 0.045, 0.059, 0.0999, 0.15; 0.8, 0.8) 1

The geometric means and fuzzy weights of the alternatives with respect to technology criterion are given in Table 5. IN a similar way the aggregated local scores of the alternatives are calculated and presented in Table 6. Table 7 represents the interval type-2 weights of each criteria with respect to the goal. Later, Eq. 14 is used to find the global scores of alternatives with respect to each criteria and the results are represented in Table 8. The fuzzy global scores are then summed up to find the overall score of each alternative. Table 9 shows the Interval type-2 fuzzy scores of alternatives. The type-2 fuzzy score are defuzzified using DTtrT Table 9 shows the result of the defuzzification. The normalized crisp score value of Alternative 1 is calculated as an example in the following.

DTtrT =

A1

(0.91 − 0.045) + (0.37 − 0.045) + ( 0.098 − 0.045) + 0.045 4

2

 ( 0.74 − 0.053) + ( 0.8x 0.32 − 0.053)    +(0.8x 0.11 − 0.053)  + 0.053   4   A1  A2 A3 A4  + = 0.324 (0.30,0.34,0.5, 2 (0.15,0.19,0.31, (0.15,0.19,0.31, (1,1,1,1;1,1) (1,1,1,1;1,1)

0.69;1,1)(0.31,0.35, 0.47,0.63;0.8,0.8)

0.48;1,1)(0.16,0.19, 0.29,0.43;0.8,0.8)

0.48;1,1)(0.16,0.19, 0.29,0.43;0.8,0.8)

A2

(1.44,2,2.88, 3.27;1,1)(1.56,2.09, 2.80,3.19;0.8,0.8)

(1,1,1,1;1,1) (1,1,1,1;1,1)

(0.15,0.19,0.31, 0.48;1,1)(0.16,0.19, 0.29,0.43;0.8,0.8)

(0.14,0.17,0.27, 0.40;1,1)(0.15,0.17, 0.25,0.36;0.8,0.8)

A3

(2.08,3.17,5.24, 6.25;1,1)(2.30,3.38, 5.03,6.05;0.8,0.8)

(2.08,3.17,5.24, 6.25;1,1)(2.30,3.38, 5.03,6.05;0.8,0.8)

(1,1,1,1;1,1) (1,1,1,1;1,1)

(0.58,0.62,0.79, 1;1,1)(0.59,0.64, 0.76,0.94;0.8,0.8)

A4

(2.08,3.17,5.24, 6.25;1,1)(2.30,3.38, 5.03,6.05;0.8,0.8)

(2.46,3.63,5.76, 6.80;1,1)(2.71,3.85, 5.56,6.59;0.8,0.8)

(1,1.25,1.58, 1.70;1,1)(1.06,1.30, 1.56,1.68;0.8,0.8)

(1,1,1,1;1,1) (1,1,1,1;1,1)

Table 4 Aggregated evaluations of the experts with respect to technology

494

Basar Oztaysi values

values

A1

(0.29,0.33,0.47,0.63;1,1) (0.30,0.34,0.45,0.58;0.8,0.8)

(0.042,0.056,0.10,0.18;1,1) (0.045,0.059,0.099,0.15;0.8,0.8)

A2

(0.42,0.50,0.70,0.89;1,1) (0.44,0.52,0.68,0.84;0.8,0.8)

(0.061,0.084,0.16,0.25;1,1) (0.066,0.090,0.14,0.22;0.8,0.8)

A3

(1.26,1.58,2.16,2.50;1,1) (1.33,1.64,2.10,2.42;0.8,0.8)

(0.18,0.26,0.49,0.71;1,1) (0.19,0.28,0.46,0.65;0.8,0.8)

A4

(1.50,1.95,2.63,2.92;1,1) (1.60,2.02,2.57,2.86;0.8,0.8)

(0.21,0.32,0.60,0.83;1,1) (0.23,0.34,0.56,0.77;0.8,0.8)

Table 5  ri and pi value of the decision matrix for the technology criterion

Scores w.r.t.TEC

Scores w.r.t. SCH

Scores w.r.t. ECO

A1

(0.042,0.056,0.10, 0.18;1,1)(0.045,0.059, 0.099,0.15;0.8,0.8)

(0.40,0.50,0.75, 0.92;1,1)(0.42,0.53, 0.72,0.88;0.8,0.8)

(0.23,0.34,0.61, 0.85;1,1)(0.25, 0.36,0.58,0.79;0.8,0.8)

A2

(0.061,0.084,0.16, 0.25;1,1) (0.066,0.090, 0.14,0.22;0.8,0.8)

(0.10,0.13,0.20, 0.26;1,1)(0.10,0.13, 0.19,0.24;0.8,0.8)

(0.15,0.24,0.46, 0.68;1,1)0.17,0.25, 0.43,0.62;0.8,0.8)

A3

(0.18,0.26,0.49, 0.71;1,1)(0.19,0.28, 0.46,0.65;0.8,0.8)

(0.083,0.10,0.15, 0.20;1,1)(0.086, 0.10,0.14,0.18;0.8,0.8)

(0.070,0.10,0.21, 0.36;1,1)0.075,0.10, 0.19,0.32;0.8,0.8)

A4

(0.21,0.32,0.60, 0.83;1,1)(0.23,0.34, 0.56,0.77;0.8,0.8)

(0.058,0.07,0.11, 0.15;1,1)(0.061,0.076, 0.11,0.14;0.8,0.8)

(0.031,0.041,0.077, 0.12;1,1)(0.033,0.043, 0.071,0.11;0.8,0.8)

Scores w.r.t. EU

Scores w.r.t. VEN

Scores w.r.t. FUN

A1

(0.052,0.069,0.13, 0.23;1,1)(0.055,0.073, 0.12,0.20;0.8,0.8)

(0.061,0.080,0.15, 0.26;1,1)(0.064,0.084, 0.14,0.22;0.8,0.8)

(0.046,0.060,0.11, 0.20;1,1)(0.048,0.064, 0.10,0.17;0.8,0.8)

A2

(0.06,0.082,0.16, 0.26;1,1)(0.064,0.087, 0.14,0.23;0.8,0.8)

(0.091,0.12,0.22, 0.34;1,1)(0.098,0.13, 0.21,0.31;0.8,0.8)

(0.069,0.09,0.18, 0.28;1,1)(0.075,0.10, 0.16,0.25;0.8,0.8)

A3

(0.18,0.27,0.52, 0.77;1,1)(0.20,0.29, 0.48,0.70;0.8,0.8)

(0.19,0.28,0.48, 0.65;1,1)(0.21,0.30, 0.46,0.60;0.8,0.8)

(0.19,0.27,0.51, 0.73;1,1)(0.20,0.29, 0.48,0.67;0.8,0.8)

A4

(0.17,0.29,0.57, 0.83;1,1)(0.19,0.31, 0.54,0.77;0.8,0.8)

(0.17,0.25,0.46, 0.65;1,1)(0.19,0.27, 0.43,0.60;0.8,0.8)

(0.20,0.30,0.53, 0.73;1,1)(0.22,0.32, 0.50,0.68;0.8,0.8)

Table 6 Aggregated local scores of alternatives with respect to the criteria



Group Decision Making Criteria

495

Interval Type-2 Weights

TEC

(0.022,0.03,0.072,0.13;1,1)(0.024,0.034,0.06,0.11;0.8,0.8)

SCH

(0.02,0.042,0.098,0.18;1,1)(0.030,0.045,0.08,0.16;0.8,0.8)

ECO

(0.077,0.12,0.26,0.42;1,1)(0.08,0.13,0.24,0.38;0.8,0.8)

EU

(0.092,0.14,0.31,0.49;1,1)(0.10,0.15,0.28,0.44;0.8,0.8)

VEN

(0.036,0.052,0.11,0.19;1,1)(0.039,0.056,0.10,0.16;0.8,0.8)

FUN

(0.16,0.28,0.60,0.91;1,1)(0.18,0.30,0.56,0.83;0.8,0.8)

Table 7 Criteria weights with respect to the goal

A1

A2

A3

A4

TEC

(0.009,0.001, 0.007,0.025;1,1) (0.001,0.002,0.006, 0.018;0.8,0.8)

(0.001,0.002, 0.011,0.03;1,1) (0.001,0.003,0.009, 0.027;0.8,0.8)

(0.00,0.008, 0.035,0.10;1,1) (0.004,0.009,0.030, 0.078;0.8,0.8)

(0.004,0.010, 0.043,0.11;1,1) (0.005,0.012,0.037, 0.092;0.8,0.8)

SCH

(0.011,0.021, 0.074,0.17;1,1) (0.013,0.024, 0.064,0.14;0.8,0.8)

(0.002,0.005, (0.002,0.004, (0.001,0.003, 0.02,0.049;1,1) 0.015,0.037;1,1) 0.011,0.029;1,1) (0.003,0.006, (0.002,0.004, (0.001,0.003, 0.017,0.040;0.8,0.8) 0.012,0.030;0.8,0.8) 0.009,0.023;0.8,0.8)

ECO

(0.017,0.043, 0.16,0.36;1,1) (0.022,0.050, 0.14,0.30;0.8,0.8)

(0.012,0.030, 0.12,0.29;1,1) (0.015,0.035, 0.10,0.23;0.8,0.8)

(0.005,0.012, 0.056,0.15;1,1) (0.006,0.014, 0.048,0.12;0.8,0.8)

(0.002,0.005, 0.020,0.054;1,1) (0.002,0.006, 0.017,0.043;0.8,0.8)

(0.004,0.010, 0.042,0.11;1,1) (0.005,0.011, 0.035,0.091;0.8,0.8)

(0.005,0.012, 0.050,0.13;1,1) (0.006,0.013, 0.042,0.10;0.8,0.8)

(0.017,0.040, 0.16,0.38;1,1) (0.02,0.046, 0.14,0.31;0.8,0.8)

(0.016,0.042, 0.18,0.41;1,1) (0.020,0.050, 0.15,0.34;0.8,0.8)

(0.002,0.004, (0.003,0.006, 0.016,0.05;1,1) 0.025,0.065;1,1) VEN (0.002,0.004, (0.003,0.007, 0.014,0.038;0.8,0.8) 0.021,0.051;0.8,0.8)

(0.007,0.015, 0.053,0.12;1,1) (0.008,0.017, 0.046,0.10;0.8,0.8)

(0.006,0.013, 0.050,0.12;1,1) (0.007,0.015, 0.043,0.10;0.8,0.8)

(0.031,0.078, 0.30,0.67;1,1) (0.039,0.090, 0.26,0.56;0.8,0.8)

(0.034,0.085, 0.32,0.66;1,1) (0.042,0.099, 0.28,0.56;0.8,0.8)

EU

FUN

(0.007,0.017, 0.071,0.18;1,1) (0.009,0.01, 0.061,0.14;0.8,0.8)

(0.011,0.027, 0.10,0.25;1,1) (0.014,0.031, 0.094,0.21;0.8,0.8)

Table 8 Global scores of alternatives with respect to criteria

496

Basar Oztaysi Interval Type 2 Fuzzy Global Scores

Crisp Scores

Normalized Crisp score

A1

(0.045,0.098,0.37,0.91;1,1) (0.053,0.11,0.32,0.74;0.8,0.8)

0.324

0,196

A2

(0.03,0.085,0.34,0.83;1,1) (0.045,0.098,0.29,0.67;0.8,0.8)

0.291

0,176

A3

(0.06,0.15,0.63,1.47;1,1) (0.082,0.18,0.54,1.21;0.8,0.8)

0.527

0,318

A4

(0.066,0.16,0.63,1.40;1,1) (0.081,0.18,0.54,1.16;0.8,0.8)

0.513

0,310

Table 9 Fuzzy and normalized crisp score values of the alternatives

According to the results of Table 9 the overall utility score of the alternatives are found as Alt. 3 > Alt.4 > Alt.1 > Alt.2. The calculation results showed that the third alternative is the best one and the second alternative is the worst ERP alternative for the company. As a result, Alternative 3 is selected by the company.

5 Conclusion Selection of the right IS project constituted a major component of effective management of technology. Since the selection process contains various conflicting criteria, the EIS selection can be modeled as a MCDM problem. In this study, interval type-2 fuzzy AHP, which is based on Buckley’s fuzzy AHP method, is applied for a real world EIS selection problem. In the case study six criteria are determined as a result of literature survey and evaluations of the decision making team. As a result of the proposed technique, the most important factors are determined as functionality and ease of use. Four alternative ERP projects are compared including the current system, a local solution, a midsized international solution and an industry leader software company. When the pairwise comparison matrices are analyzed, it can be seen that Alternative 1 and Alternative 2 are very good at technology and project schedule. But since Alternative 3 and Alternative 4 provide better functionality and ease of use they outperformed the other alternatives. Both classical AHP and fuzzy AHP methods propose that the linguistic evaluations can be handled by the methods. In classical AHP crisp numbers are used, for pair wise comparison evaluations. However in fuzzy AHP, the linguistic variables are represented as fuzzy numbers instead of crisp. In fuzzy AHP triangular fuzzy numbers are used to represent linguistic variables. The originality of this paper comes from using trapezoidal interval type-2 fuzzy sets to represent linguistic evaluations and extend the AHP



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method to handle arithmetic operations based on these numbers. By doing this, more reliable results can be maintained since interval type-2 fuzzy sets can better represent uncertainties. For the future studies, the same linguistic evaluations can be used for classical and fuzzy AHP and the result can be compared with the result of this study. Also, the represented method can also be compared with the results of other MCDM techniques such as TOPSIS, VIKOR and COPRAS. Finally the other AHP methods such as Chang’s (1996) or Laarhoven and Pedrycz’s (1983) can be modified using type-2 fuzzy sets.

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