A theorical model design for ERP software selection process under the constraints of cost and quality: A fuzzy approach

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1 Journal of Intelligent & Fuzzy Systems 21 (2010) DOI: /IFS IOS Press 365 A theorical model design for ERP software selection process under the constraints of cost and quality: A fuzzy approach Semih Onut and Tugba Efendigil Department of Industrial Engineering, Yildiz Technical University, Yildiz, Istanbul, Turkey Abstract. Enterprise Resource Planning (ERP) software selection is one of the most important decision making issues covering both qualitative and quantitative factors for organizations. Multiple criteria decision making (MCDM) has been found to be a useful approach to analyze these conflicting factors. Qualitative criteria are often accompanied by ambiguities and vagueness. This makes fuzzy logic a more natural approach to this kind of problems. This study presents a beneficial structure to the managers for use in ERP software vendor selection process. In order to evaluate ERP vendors methodologically, a hierarchical framework is also proposed. As a MCDM tool, we used analytic hierarchy process (AHP) and its fuzzy extension to obtain more decisive judgments by prioritizing criteria and assigning weights to the alternatives. The objective of this paper is to select the most appropriate alternative that meets the customer s requirements with respect to cost and quality constraints. In the end of this study, a real-world case study from Turkey is also presented to illustrate efficiency of the methodology and its applicability in practice. Keywords: AHP, fuzzy AHP, multiple criteria decision making, software selection 1. Introduction Technological opportunities, demands for innovation, and changes in global markets force organizations to be more outward looking, market-oriented and knowledge driven with a flexible infrastructure. This development affects organizations business practices and procedures to remain in a competitive environment. At this point, information technology has an important role in increasing the competitiveness of organizations. Enterprise resource planning (ERP) systems were introduced into companies to solve various organizational problems and to provide an integrated frame as an information technology. Corresponding author. Tel.: ; Fax: ; onut@yildiz.edu.tr (Semih Onut). An ERP system is an integrated software package composed by a set of standard functional modules (production, sales, human resources, finance, etc.) developed or integrated by the vendor that can be adapted to the specific needs of each customer [14]. A formal definition of ERP given by APICS (American Production and Inventory Control Society) dictionary is An accounting-oriented information system for identifying and planning the enterprise-wide resources needed to take, make, ship and account for customer orders. A successful ERP project involves selecting an ERP software system and vendor, implementing this system, managing business processes change and examining the practicality of the system [38]. ERP implementation process involves macro-implementation at the strategic level and microimplementation at the operational level [3] /10/$ IOS Press and the authors. All rights reserved

2 366 S. Onut and T. Efendigil / A theorical model design for ERP software selection process under the constraints of cost and quality Shang and Seddon [32] classify ERP benefits into five groups as follows: Operational, relating to cost reduction, cycle time reduction, productivity improvement, quality improvement, and customer services improvement. Managerial, relating to better resource management, improved decision making and planning, and performance improvement. Strategic, concerning supporting business growth, supporting business alliance, building business innovations, building cost leadership, generating product differentiation, and building external linkages. IT infrastructure, involving building business flexibility, IT cost reduction, and increased IT infrastructure capability. Organizational, relating to supporting organizational changes, facilitating business learning, empowering, and building common visions. ERP is a key ingredient for gaining competitive advantage, streamlining operations and having lean manufacturing. Also, ERP systems implementations are considered costly and time and resource consuming processes. Especially, the package selection includes important decisions regarding budgets, timeframes, goals that will affect the entire project. Choosing the right ERP software package that best meets the organizational needs and processes is crucial to ensure minimal modification and successful implementation and use. A better fit between software vendor and user organization is beneficially associated with packaged software implementation success. This relationship provides maximizing the compatibility of organizations with their vendors. The other important issue is vendor support with extended technical assistance, emergency maintenance, updates and special user training. Selecting the wrong ERP software may cause a misfit between the package and organizational goals or business processes. Obviously, choosing an ERP software package and vendor, and implementing and maintaining this system is a very critical process since this decision will affect the organization positively or negatively. Therefore, the objective of this paper is to propose a framework in order to select an appropriate ERP software vendor considering both company demands and ERP system characteristics under the constraints of cost and quality criteria. ERP software selection issue can be viewed as a multiple criteria decision making (MCDM) problem in the presence of many quantitative and qualitative criteria that should be considered in the selection procedure including a set of possible vendor alternatives. A decision maker (DM) is required to choose among quantifiable or non-quantifiable and multiple criteria. The DM s evaluations on qualitative criteria are always subjective and thus imprecise. The objectives are usually conflicting and therefore the solution is highly dependent on the preferences of the DM. Besides, it is very difficult to develop a selection criterion that can precisely describe the preference of one alternative over another. The evaluation data of ERP software alternatives suitability for various subjective criteria, and the weights of the criteria are usually expressed in linguistic terms. This makes fuzzy logic a more natural approach to this kind of problems. In this paper, we used analytic hierarchy process (AHP) and its fuzzy extension to obtain more decisive judgments by prioritizing criteria and assigning weights to the software alternatives. The paper briefly reviews the software packages approaches, the concepts of applied decision-making methodologies (AHP and Fuzzy AHP) through their applications, and the comparison of the methodologies which gives the best result. The rest of the paper is organized as follows: Section 2 presents a review of previous works on the selection of ERP systems via Commercial Off-the- Shelf (COTS) selection techniques followed by detailed AHP and Fuzzy AHP approaches. In Section 3, a decision framework is proposed to show an ERP software vendor selection problem under the cost and quality constraints. Finally, some conclusions and future work directions are given in Section Literature review 2.1. ERP software selection methods ERP is a kind of application software package that encompasses many functions, such as planning production, purchasing materials, maintaining inventories, interacting with suppliers, providing customer service, and tracking orders. Application software packages are defined by a vendor to provide a set of standard functions that are usable for different kinds of companies. COTS term refers to application software packages. By considering the explanation above, we intended to search for the selection techniques of ERP systems. Current literature provides a number of methods for the evaluation of COTS components. Each of these methods emphasizes one or more critical aspects of COTS

3 S. Onut and T. Efendigil / A theorical model design for ERP software selection process under the constraints of cost and quality 367 software evaluation. The primary goal is to identify those aspects of the methodologies that might be useful in developing an integrated approach to software evaluation. Jeanrenaud and Romanazzi [18] proposed Checklist Driven Software Evaluation Methodology (CDSEM) to evaluate software that employs checklists, which they use to determine a quality metric for each item in the checklist. Kontio [22] suggested Off-The-Shelf- Option (OTSO) which is a multi-phased (search, screening and evaluation, analysis phases) approach and is concerned with the actual selection process, not with implementation. The central theme to the OTSO method is the construction of a product evaluation criteria hierarchy. Maiden and Ncube [27] presented Procurement-Oriented Requirements Engineering (PORE) Method which is a template approach to requirements definition that depends on evaluating COTS products. Obendorf [30] investigated COTS Acquisition Process (CAP). This approach has three phases: initialization, execution, and reuse. Tran and Liu [33] proposed, COTS-Based Integrated System Development (CISD) Method that has a two-stage selection process. The first stage is product identification, where candidates are identified and classified. The second is evaluation, where the final candidates are chosen (and unsuitable candidates eliminated). Morisio and Tsoukias [29] prepared IusWare methodology which is based on the multi-criteria decision aid approach and consists of two main phases: design of an evaluation model and application. Scenario Based COTS Selection approach, which was proposed by Feblowitz and Greenspan [15], is an impact analysis of COTS. This technique considers system scenarios that can be modified under the hypothesis of using different COTS candidates; a new scenario set is produce together with a list of issued encountered during COTS adaptation. Lawlis et al. [26] presented Requirements-driven COTS Product Evaluation Process (RCPEP) that requires trade study and hand-on evaluation. Sedigh-Ali et al. [31] applied Risk Management Metrics method, which is related to the Cost of Software Quality (CoSQ) and to CMM maturity levels in the selection process of COTS. Umble et al. [36] identified success factors, software selection steps, and implementation procedures critical to a successful ERP software implementation. A case study of a largely ERP implementation was also presented and discussed in terms of these key factors. By combining the basic concept of the utility ranking technique with data envelopment analysis, Bernroider and Stix [5] proposed a conceptual approach, named profile distance method, to support ERP selection problems. Ayağ and Özdemir [4] presented a fuzzy analytic network process based approach to ERP software selection problem. Karsak and Ozogul [20] developed a decision framework for ERP software selection based on quality function deployment, fuzzy linear regression and zero-one goal programming The AHP MCDM is one of the most well known branches of decision making. The weighted sum model (WSM) is the earliest and probably the most widely used method. The weighted product model (WPM) can be considered as a modification of the WSM and has been proposed in order to overcome some of its weakness. AHP, ELECTRE and TOPSIS methods are some other widely used methods [35]. In this paper, we used the AHP since it has been very popular in practical areas. The AHP approach also has some advantages. One of the most important advantages of the AHP is that it is based on pair-wise comparison. Besides, the AHP calculates the inconsistency index which is the ratio of the DM s inconsistency. The AHP has been proposed in recent literature as an emerging solution approach to large, dynamic, and complex real world multi criteria decision making problems, such as the strategic planning of organizational resources and the justification of new manufacturing technology. In many industrial applications, the final decision is based on the evaluation of a number of alternatives in terms of a number of criteria. This problem may become a very difficult when the criteria are expressed in different units or the related data are difficult to be quantified. The AHP is an effective approach in dealing with this kind of decision problems [34]. The AHP treats the decision as a system, which is difficult for many DMs to do and makes complex decision processes more rational by synthesizing all available information about the decision in a system-wide and systematic manner. Fundamentally, the AHP works by developing priorities for alternatives and the criteria used to judge the alternatives. The AHP has been applied to a large variety of decision-making processes in the different application areas. Some related studies utilizing this technique can be summarized as follows. Min [28] reported that the AHP can be used in three ways in COTS evaluation; (a) to define weights, priorities or criteria at all levels of a criteria hierarchy, (b) as an aggregation tool for

4 368 S. Onut and T. Efendigil / A theorical model design for ERP software selection process under the constraints of cost and quality computing a score to any level of criteria hierarchy, and (c) as an assessment tool for expressing preferences for products against any particular criterion. Yang and Lee [39] presented an AHP decision model for facility location selection from the view of organizations that contemplate locations of a new facility or a relocation of existing facilities. Bevilacqua and Braglia [6] described the application of the AHP for selecting the best maintenance strategy for an important Italian oil refinery. Al-Harbi [2] presented this technique as a potential group decision making method for use in project management. Khalil [21] applied the AHP process to select the most appropriate project delivery method. Chan [9] used this method to choose the most favourable material equipment type. Handfield et al. [16] illustrated the use of the AHP as a decision support model to demonstrate how the AHP can be used to evaluate the relative importance of various environmental criteria and to assess the relative performance of several suppliers with these criteria. As a software selection case, Lai et al. [25] evaluated a multi-media authorizing system in a group decision environment by using the AHP technique. Abdi and Labib [1] presented this approach for structuring the decision making process for the selection of a manufacturing system among feasible alternatives based on the reconfigurable manufacturing system. Recently, AHP methodology has been applied to several decision problems such as project and strategy selection, vendor rating, software selection, location selection, etc. In the conventional AHP, the pairwise comparison of each level is carried out using a nine-point scale that expresses preferences between options, such as equally, moderately, strongly, very strongly, or extremely. The essential steps in the AHP application contain (1) decomposing a general decision problem in a hierarchical form into sub-problems that can be easily realized and evaluated, (2) determining the priorities of the elements at each level of the decision hierarchy, and (3) synthesizing the priorities to determine the overall priorities of the decision alternatives. In this study, we will not explain the procedure of the AHP methodology Fuzzy AHP Generally, some decision data of real world problems cannot be gathered accurately. At this point, it is important to notice that making quantitative and qualitative predictions are difficult since the ability of every DM is different and contingent upon his/her experience, knowledge, evaluation, etc. In order to obtain a reasonable result, DMs are forced to employ numerical values instead of linguistic expressions. For this reason, fuzzy decision making models carry out this process more accurately by allowing the translation the linguistic expressions into numerical ones. Fuzziness is a kind of uncertainty. Basically, fuzzy logic is a precise logic of imprecision and approximate reasoning. It is natural to employ fuzzy logic as a modeling language when the objects of modeling are not well defined [40]. Fuzzy logic is derived from fuzzy set theory that are based on vague definitions of sets, not randomness. Hence, the advantage of fuzzy approach is to be able to express relative importance of the alternatives and the criteria with fuzzy numbers instead of complex values because it is more confident to give interval judgment than fixed value judgment. Conventional AHP methodology helps to solve problems in which all decision data are supposed to be known and must be represented by crisp numbers with a crisp aggregation score when the fuzzy AHP methodology causes to have difficulty in judging the preferred alternatives because all values and aggregation score are fuzzy data. However, the fuzzy AHP approach allows a more accurate description of the decision making process. In the fuzzy AHP procedure, the pairwise comparison in the judgment matrix is fuzzy numbers that are modified by the designer s emphasis. Using fuzzy arithmetic and α-cuts the procedure calculates a sequence of weight vectors that will be used to combine the scores on each attribute. The procedure calculates a corresponding set of scores and determines one composite score that is the average of these fuzzy scores. Also there are many fuzzy AHP methodologies presented by various authors. There are a number of studies related with the fuzzy AHP technique in the literature. Some past works are as follows. Van Laarhoven and Pedrycz [37] proposed a method that compared fuzzy judgments by triangular fuzzy number. Buckley [8] assigned trapezoidal fuzzy number to fuzzy priorities for comparison. Chang [11] introduced a new approach for tackling the fuzzy AHP, with the use of triangular fuzzy numbers and the use of the extent analysis method for the synthetic extent values of the pairwise comparisons. Zhu et al. [41] improved the formulation of comparing the triangular fuzzy number s size to discuss on extent analysis method and applications of the fuzzy AHP. Deng [13] presented a fuzzy approach for tackling qualitative multi-criteria analysis problems in a simple and straightforward manner. Chan et al. [10] proposed a

5 S. Onut and T. Efendigil / A theorical model design for ERP software selection process under the constraints of cost and quality 369 technology selection algorithm to quantify both tangible and intangible benefits in fuzzy environment. They described an application of the theory of fuzzy sets from the economic evaluation perspective; a fuzzy cash flow analysis is employed. Kuo et al. [23] developed a decision support system for locating a new convenience store. Kwong and Bai [24] presented a fuzzy AHP approach to determine the importance weights of customer requirements and to prioritize them. Bozdag et al. [7] applied the fuzzy AHP to select the best computer integrated manufacturing system by taking into account both intangible and tangible factors. Kahraman et al. [19] used the fuzzy AHP process to select the best supplier firm providing the most satisfaction for the criteria determined Fuzzy numbers of pairwise comparison In this study, we utilize triangular fuzzy numbers represented as 1 to 9 that are used to express the relative strength of each pair of elements in the same hierarchy. A fuzzy number is a special fuzzy set F = {(x, µ F (x), x R}, where x takes its values on the real line, R: x and µ F (x) is a continuous mapping from R to the closed interval [0, 1]. A triangular fuzzy number denoted as M =(a, b, c), where a b c, has the following triangular type membership function: 0 x<a x a a x b (x) = b a µ M c x b x c c b 0 x>c (1) Alternatively by defining the interval of confidence level α, the triangular fuzzy number can be characterized as: M α = [ a α,c α] = [(b a)α + a, (c b)α + c] α [0, 1] (2) Here a α and c α denote the left side representation and right side representation of a fuzzy number respectively in Fig. 1. By using triangular fuzzy numbers via pairwise comparison the fuzzy judgment matrix Ã(a ij ) is constructed as follow: µ M ~ 1 0 à = a α c α a b c Fig. 1. A representation of a fuzzy number M. 1 ã 12 ã ã 1(n 1) ã 1n ã 21 1 ã ã 2(n 1) ã 2n ã (n 1)1 ã (n 1)2 ã (n 1) ã (n 1)n ã n1 ã n2 ã n3... ã n(n 1) 1 The judgment matrix à is an n n fuzzy matrix containing fuzzy numbers ã ij. { 1, i = j ã ij = 1, 3, 5, 7, 9or 1 1, 3 1, 5 1, 7 1, 9 1, i /= j (3) In the fuzzy AHP, triangular fuzzy numbers and interval arithmetic are utilized to improve the scaling scheme in the judgment matrices. In this study, we assume that the interval of confidence level α = 0.50 to obtain the α-cuts fuzzy comparison matrices Fuzzy AHP methodology Once the triangular fuzzy numbers are assigned to indicate the relationship between each pair we used the method of Chang s extent analysis [12]. Let X = {x 1, x 2,..., x n } be an object set, and U = {u 1, u 2,..., u n } be a goal set. According to this method, each object is taken and an extent analysis for each goal is performed respectively. Therefore, extent analysis values m for each object can be obtained with the following symbols: Mgi 1,M2 gi,...,mm gi, i = 1, 2,...,n (4) where all the M j gi (j =1,2,..., 3) are triangular fuzzy numbers.. M.

6 370 S. Onut and T. Efendigil / A theorical model design for ERP software selection process under the constraints of cost and quality The value of fuzzy synthetic extent with respect to the ith object is defined as: 1 m m S i = M j gi (5) M j gi i=1 The degree of possibility of M 1 M 2 is defined as: V (M 1 M 2 ) = sup min(µm1 (x),µ M2 (y) (6) x y When a pair (x, y) exists such that x y and µ M1 (x) = µ M2 (y), then we have V(M 1 M 2 ) = 1. Since M 1 and M 2 are convex fuzzy numbers we can have: V (M 1 M 2 ) = 1 if m 1 m 2, (7) V (M 1 M 2 ) = hgt(m 1 M 2 ) = µ M1 (d) (8) where d is the ordinate of the highest intersection point D between u M1 and u M2 (see Fig. 2). When M 1 =(l 1, m 1, u 1 ) and M 2 =(l 2, m 2, u 2 ), the ordinate of D is given by the following equation: V (M 2 M 1 ) = hgt(m 1 M 2 ) l 1 u 2 = (9) (m 2 u 2 ) (m 1 l 1 ) To compare M 1 and M 2, we need both values of V(M 1 M 2 ) and V(M 2 M 1 ). The degree possibility for a convex fuzzy number to be greater than k convex fuzzy numbers M i (i =1,2,..., k) can be defined by: V (M M 1,M 2,...,M k ) = V [(M M 1 ) (M M 2 and...and (M M k )] and = min V (M M i )i = 1, 2, 3,...,k. (10) Assume that: d (A i ) = min V (S i S k ) (11) 1 V (M 2 M 1 ) l 2 M 2 M 1 m 2 l 1 d u 2 m 1 u1 Fig. 2. Intersection point d between two fuzzy numbers M 1 and M 2. For k =1, 2,..., n; k /= i. Then, the weight vector is given by: W = [d (A 1 ),d (A 2 ),...,d (A n )] T, (12) where A i (i =1,2,..., n) are n elements. The normalized weight vectors are: W = [d(a 1 ),d(a 2 ),..., d(a n )] T (13) W is a nonfuzzy number. 3. A numerical example An estimated 50 to 75 percent of U.S. firms experience some degree of failure in implementing advanced manufacturing technology. Most problems occur when the new technology s capabilities are incompatible with the organization s existing business processes and procedures. Since an ERP system, by its very nature, will impose its own logic on a company s strategy, organization, and culture, it is imperative that the system selection decision be a wise one [36]. Hence in this study an ERP software vendor selection methodology is proposed (see Fig. 3). The methodology is applied to decide on which ERP software package that will best meet the needs of a company. A real world case problem is selected to illustrate the application of the proposed methodology. The selected company is a medium sized manufacturing enterprise, which is active in chemical industry located in Turkey. This company has a variety of integration problems between the functional departments. In order to resolve this issue, function managers would like to make a decision on purchasing an ERP package for their different departments to increase the communication between them, to implement new technology, to lower MIS cost, to improve customer service, to strengthen supplier partnerships, to enhance organizational flexibility, to reduce inventory, and to increase market share, etc. First of all, managers formed a project team which was included personnel chosen from different departments and was supported by top management to select an ERP software. This team created a vision to define the corporate mission, objectives, and strategy. Then, this team conducted the business process reengineering with a function list which was created to define what the requirements were. After collecting all possible information about the current system and establishing the evaluation criteria, the project team evaluated all software vendors characteristics in the

7 S. Onut and T. Efendigil / A theorical model design for ERP software selection process under the constraints of cost and quality 371 Determine the requirement for an ERP software system in the company Form a project team Discuss the goals, strategy and characteristics of the ideal ERP software system Conduct the business process reengineering with an identified requirement function list Establish the ERP vendors evaluation criteria and technical features of the software Evaluate all vendor alternatives in the market Filter out unqualified vendors Evaluate the rest due to the RFP lists and the demonstrations Establish the hierarchical structure of selection criteria Express the opinions of team members about the criteria, subcriteria and alternatives in linguistic form via a questionnaire by comparing pairwises Convert the linguistics variables into the crisp numbers Convert the linguistics variables into the fuzzy numbers Apply the AHP calculation steps Apply the Fuzzy AHP calculation steps Defuzzification process Transform the fuzzified values of the weights to crisp ones Compare the results of AHP and Fuzzy AHP approaches Select the best ERP vendor Fig. 3. The proposed methodology for the selection of ERP software vendor. market. Later on, they made a preliminary analysis of the strengths and weaknesses of each vendors and the goodness of fit of the software. Finally, they filtered out unqualified vendors and selected three software vendors. A request for proposal (RFP) list was prepared for the selected vendors to be filled in to figure out the

8 372 S. Onut and T. Efendigil / A theorical model design for ERP software selection process under the constraints of cost and quality capability of them. The company requested the qualified vendors to demonstrate their packages. Although this step was applied, the project team did not reach a consensus. To help the project team make a decision, we offer to use AHP and fuzzy AHP decision making methodologies to decide on the best vendor to select. The AHP and fuzzy AHP approaches allow team members to use their experience, values and knowledge to decompose a problem into smaller sets by solving them with their own procedures in making a decision. Team members expressed their opinions on the importance and the strengths of the relationships between selection criteria pairwises in the form of linguistic variables such as very strong, strong, medium, weak, and none to build the structure of comparison frame. Here, Delphi technique, a group decisionmaking approach by reaching consensus of relevant information was utilized in expressing the opinion of team members for selection criteria. This paper aims to compare these two techniques to obtain a sensitive and a accurate solution for ERP software selection problem in order to assist the managers in a fluctuating business environment under the cost and quality constraints. Although the decision making methods mentioned above perform well, they suffer from some limitations such as; lack of expertise, inconsistency among semantic descriptions or scoring assignments to the attributes or alternatives. Thus, during the evaluation and development process, consistency checks were conducted and in some cases DMs were wanted to explain the reasons of their assessments in a detailed way. Selecting a suitable ERP project involves various factors. In this study, the cost criterion is based on purchasing price. This price contains licensing arrangement cost, product and technology cost and consulting cost, which involves adapting and integrating cost, supporting cost, training cost, maintenance (upgrades) cost. The quality criteria is derived from the international norm ISO/IEC 9126 [17]. Currently, ISO/IEC 9126 standard is widely used by researchers and practitioners to evaluate software quality. Standards for the software quality model defines software quality characteristics as composed of six external attributes of interest, namely functionality, reliability, efficiency, usability, maintainability, and portability. In turn, each of these qualities is refined into sub-attributes in following Table 1. In addition to system cost and quality criteria we considered vendors factors such as vendor s condition and vendor s ability. We gathered these factors based on vendor s reputation. By vendor s abil- Table 1 The ISO 9126 software quality model Characteristic Functionality Reliability Usability Efficiency Maintability Portability Sub-characteristic Suitability Accuracy Interoperability Compliance Maturity Fault tolerance Recoverability Understandability Learnability Operability Time behavior Resource behavior Analysability Changeability Stability Testability Adaptability Installability Replaceability Conformance ity criteria, we implied vendor s technology level, implementation and service ability, consulting service, training support. As far as vendor s condition we considered vendor s financial condition, certifications and credentials. To assign the ability weights, it is required to develop an evaluation form including linguistic expressions. In our model, the project team members evaluated the vendors according to the linguistic scale in Table 2. As shown in Fig. 4. our model includes four hierarchy levels. Finally, with the weights of importance we attempted to find best ERP vendor among all alternatives. In this study, we do not discuss about the calculations of the AHP, but fuzzy AHP. In the end of paper, a comparison is given between AHP and fuzzy AHP. The project team compared the sub-attributes with respect to main attributes in the hierarchical approach by utilizing fuzzy triangular numbers in fuzzy AHP procedure. A detailed questionnaire related with the data regarding the qualitative criteria for ERP software selection model was prepared for the paired comparisons (see Appendix A) to tackle the ambiguities involved in the process of the linguistic assessment of the data. The following tables show the pairwise comparisons and weight vector of each matrix (see Tables 3 8). Instead of giving all pairwise comparisons of the model, some basic matrix structures are depicted in tables as examples to avoid the repetitive processes. Table 4 indi-

9 S. Onut and T. Efendigil / A theorical model design for ERP software selection process under the constraints of cost and quality 373 Table 2 Linguistic variables describing weights of attributes and values of ratings Linguistic scale Numerical ratings Fuzzy numbers Membership function Domain Triangular fuzzy scale for importance for AHP for fuzzy AHP (l, m, u) Just equal (1.0, 1.0, 1.0) Equal importance 1 1 µ M (x) = (3 x)/(3 1) 1 x 3 (1.0, 1.0, 3.0) Weak importance of one 3 3 µ M (x) = (x 1)/(3 1) 1 x 3 (1.0, 3.0, 5.0) over another µ M (x) = (5 x)/(5 3) 3 x 5 Essential or strong 5 5 µ M (x) = (x 3)/(5 3) 3 x 5 (3.0, 5.0, 7.0) importance µ M (x) = (7 x)/(7 5) 5 x 7 Very strong importance 7 7 µ M (x) = (x 5)/(7 5) 5 x 7 (5.0, 7.0, 9.0) µ M (x) = (9 x)/(9 7) 7 x 9 Extremely preferred 9 9 µ M (x) = (x 7)/(9 7) 7 x 9 (7.0, 9.0, 9.0) Intermediate values between 2, 4, 6, 8 the two adjacent judgments If factor i has one of the Reciprocals Reciprocals of above above numbers assigned to it when compared to factor j, then j has the reciprocal value when compared with i of above M 1 1 (1/u 1, 1/m 1, 1/l 1 ) Level 1: GOAL Level 2: CRITERIA Level 3: SUB-CRITERIA Level 4: ALTERNATIVES Functionality Reliability Usability Efficiency System A Quality Maintability Portability System B Evaluation of an ERP software system Cost Purchasing Consulting System C Reputation Vendor s ability Vendor s condition Fig. 4. The hierarchy of the proposed model.

10 374 S. Onut and T. Efendigil / A theorical model design for ERP software selection process under the constraints of cost and quality Table 3 The fuzzy evaluation matrix regarding object function Quality Cost Reputation Quality (1, 1, 1) (4, 5, 6) (2, 3, 4) Cost (1/6, 1/5, 1/4) (1, 1, 1) (1/4, 1/3, 1/2) Reputation (1/4, 1/3, 1/2) (2, 3,4) (1, 1, 1) via AHP formulation we have S QUALITY = (0.38, 0.61, 0.94), S COST = (0.08, 0.10, 0.15), S REPUTATION = (0.18, 0.29, 0.47). Using these vectors V(S QUALITY S COST ) = 1.00, V(S COST S QUALITY ) = 0.00, V(S QUALITY S REPUTATION ) = 1.00, V(S REPUTATION S QUALITY ) = 0.22, V(S COST S QREPUTATION )= 0.00, V(S REPUTATION S COST ) = 1.00 are obtained. The weight vector is calculated as W O = (0.82, 0.00, 0.18) T. cates the sub-attributes evaluation of quality criteria. Table 5 shows the examples for comparison of the software vendors regarding various sub-criteria of quality, cost and reputation criteria. Table 6 summarized the combination of priority weights of quality, cost and reputation criteria. System A has the highest priority weight and it is selected as the best candidate vendor according to all utilized approaches. When we solve this problem by the AHP technique (Table 8) we obtain value of 0.44; while we use fuzzy AHP (Table 7) we obtain the value of 0.54 as priority weights. Clearly it is observed that Table 4 Evaluation of sub-attribute s quality criteria Quality Functionality Reliability Usability Efficiency Maintability Portability Functionality (1, 1, 1) (1/4, 1/3, 1/2) (1, 1, 2) (4, 5, 6) (2, 3, 4) (2, 3, 4) Reliability (2, 3, 4) (1, 1, 1) (2, 3, 4) (4, 5, 6) (2, 3, 4) (2, 3, 4) Usability (1/2, 1, 1) (1/4, 1/3, 1/2) (1, 1, 1) (2, 3, 4) (2, 3, 4) (1, 1, 2) Efficiency (1/6, 1/5, 1/4) (1/6, 1/5, 1/4) (1/4, 1/3, 1/2) (1, 1, 1) (1/4, 1/3, 1/2) (1/4, 1/3, 1/2) Maintability (1/4, 1/3, 1/2) (1/4, 1/3, 1/2) (1/4, 1/3, 1/2) (2, 3, 4) (1, 1, 1) (1/4, 1/3, 1/2) Portability (1/4, 1/3, 1/2) (1/4, 1/3, 1/2) (1/2, 1, 1) (2, 3, 4) (2, 3, 4) (1, 1, 1) S FUNCTIONALTY = (0.14, 0.23, 0.42), S RELIABILITY = (0.18, 0.32, 0.55), S USABILITY = (0.09, 0.16, 0.30), S EFFICIENCY = (0.03, 0.04, 0.07), S MAINTABILITY = (0.05, 0.09, 0.17), S PORTABILITY = (0.08, 0.15, 0.26). V(S FUNCTIONALITY S RELIABILITY ) = 0.75, V(S RELIABILITY S FUNCTIONALTY ) = 1.00, V(S FUNCTIONALITY S USABILITY ) = 1.00, V(S USABILITY S FUNCTIONALTY ) = 0.69, V(S FUNCTINALITYY S EFFICIENCY ) = 1.00, V(S EFFICIENCY S FUNCTIONALITY ) = 0.00, V(S FUNCTIONALTY S MAINTABILITY ) = 1.00, V(S MAINTABILITY S FUNCTIONALITY ) = 0.17, V(S FUNCTIONALTY S PORTABILITY ) = 1.00, V(S PORTABILITY S FUNCTIONALITY ) = 0.60, V(S RELIABILITY S USABILITY ) = 1.00, V(S USABILITY S RELIABILITY ) = 0.44, V(S RELIABILITY S EFFICIENCY ) = 1.00, V(S EFFICIENCY S RELIABILITY ) = 0.00, V(S RELIABILITY S MAINTABILITY ) = 1.00, V(S MAINTABILITY S RELIABILITY ) = 0.00, V(S RELIABILITY S PORTABILITY ) = 1.00, V(S PORTABILITY S RELIABILITY ) = 0.34, V(S USABILITY S EFFICIENCY ) = 1.00, V(S EFFICIENCY S USABILITY ) = 0.00, V(S USABILITY S MAINTABILITY ) = 1.00, V(S MAINTABILITY S USABILITY ) = 0.52, V(S USABILITY S PORTABILITY ) = 1.00, V(S PORTABILITY S USABILITY ) = 0.94, V(S EFFICIENCY S MAINTABILITY ) = 0.25, V(S MAINTABILITY S EFFICINECY ) = 1.00, V(S EFFICIENCY S PORTABILITY ) = 0.00, V(S PORTABILITY S EFFICINECY ) = 1.00, V(S MAINTABILITY S PORTABILITY ) = 0.59, V(S PORTABILITY S MAINTABILITY ) = W Q = (0.29, 0.40, 0.17, 0.00, 0.00, 0.13) T. Cost Purchasing Consulting Reputation Vendor s ability Vendor s condition Purchasing (1, 1, 1) (1/4, 1/3, 1/2) Vendor s ability (1, 1, 1) (4, 5, 6) Consulting (2, 3,4) (1, 1, 1) Vendor s condition (1/6, 1/5, 1/4) (1, 1, 1) W C = (0.00, 1.00) T. W R = (1.00, 0.00) T. Table 5 Examples for evaluation of the software vendors regarding various sub-criteria of quality, cost and reputation criteria System Sub-attribute functionality of quality Sub-attribute purchasing of cost Sub-attribute vendor s ability of criteria a criteria b reputation criteria c A B C A B C A B C A (1, 1, 1) (1, 1, 2) (4, 5, 6) (1, 1, 1) (2, 3, 4) (6, 7, 8) (1, 1, 1) (1/6, 1/5, 1/4) (1/8, 1/7, 1/6) B (1/2, 1, 1) (1, 1, 1) (4, 5, 6) (1/4, 1/3, 1/2) (1, 1, 1) (4, 5, 6) (4, 5, 6) (1, 1, 1) (1/4, 1/3, 1/2) C (1/6, 1/5, 1/4) (1/6, 1/5, 1/4) (1, 1, 1) (1/8, 1/7, 1/6) (1/6, 1/5, 1/4) (1, 1, 1) (6, 7, 8) (2, 3, 4) (1, 1, 1) a W F = (0.50, 0.50, 0.00) T. b W PC = (0.82, 0.18, 0.00) T. c W VA = (0.00, 0.18, 0.82) T.

11 S. Onut and T. Efendigil / A theorical model design for ERP software selection process under the constraints of cost and quality 375 Table 6 Summarized combination of priority weights Sub-attributes of quality Functionality Reliability Usability Efficiency Maintability Portability Alternative priority weights Weight Alternatives System A System B System C Sub-attributes of cost Sub-attributes of reputation Purchasing Consulting Alternative Vendor s Vendor s Alternative priority weights ability condition priority weighst Weight Alternatives System A System B System C Table 7 The solution with fuzzy AHP Main-attributes of the objective Quality Cost Reputation Alternative priority weights Weight Alternatives System A System B System C Table 8 The solution with conventional AHP Main-attributes of the objective Quality Cost Reputation Alternative priority weights Weight Alternatives System A System B System C these two approaches helps managers to make a strategic decision. However, fuzzy AHP approach allows the users get more accurately values to model the vagueness which changes according subjective ideas in the decision-making environment for ERP software package selection problem. Therefore, fuzzy AHP method is proposed to use in order to obtain firmly decision and noticeably solution. 4. Conclusion Although a number of methods have been applied to selection problems including scoring, ranking, mathematical optimization and multi-criteria decision making, in ERP software selection literature very few studies have been considered using decision making techniques in fuzzy environment for this area. This

12 376 S. Onut and T. Efendigil / A theorical model design for ERP software selection process under the constraints of cost and quality paper intends to show how effective is fuzzy AHP as a decision-making tool in software vendor selection problem by a comparison of the traditional AHP and fuzzy AHP approaches. Even with the complete accurate information, different decision making methods may lead to totally different results. Thus, the proposed methodology demonstrates the selection of the best ERP vendor under the cost and quality restrictions in the presence of vagueness. It is seen that fuzzy AHP is a useful decision-making methodology to make more precise selection-decisions that may help the company to achieve a competitive edge in a complexity environment. Fuzzy AHP approach incorporates quantitative data of the criteria, which have to be evaluated by qualitative measures. The proposed selection methodology is flexible to incorporate new or extra criteria or DMs for the evaluation process. Besides, this methodology gives the opportunity to the project team to decompose such a complicated selection problem into smaller parts. Thus, the project team can understand the relationships between different criteria and assign easily their own ideas to the hierarchical structure. In the future we offer to apply other decision-making methods using fuzzy concept to capture the uncertainty in complex approaches. Appendix A The following questions are utilized for a questionnaire structure (see Tables 9 and 10) to determine the importance of the criteria and the weights of alternatives by putting check marks on the pairwise comparison matrices. Question 1. How important is cost when it is compared with quality? Question 2. How important is functionality when it is compared with reliability? Question 3. How sufficient is system A respect to cost when it is compared with other alternatives? Table 9 Questionnaire form used to evaluate an ERP software selection in fuzzy AHP stage Criteria (C) Compared Extremely Very strong Strong Weak Equal Just equal criteria (CoC) preferred importance importance importance importance Con. I Con. II Con. I Con. II Con. I Con. II Con. I Con. II Con. I Con. II Con. I Con. II Comparison of sub-criteria pairwises Quality Functionality Reliability Usability Efficiency Reliability Usability Efficiency Maintability Portability Usability Efficiency Maintability Portability Efficiency Maintability Portability Maintability Portability Portability Maintability Cost Purchasing Consulting Reputation Vendor s ability Vendor s condition Comparison of main-criteria pairwises Quality Cost Reputation Cost Reputation Condition I (Con. I): C is more important than CoC. Condition II (Con. II): CoC is more important than C.

13 S. Onut and T. Efendigil / A theorical model design for ERP software selection process under the constraints of cost and quality 377 Table 10 Questionnaire form used to evaluate the alternatives of ERP vendors according to evaluation criteria Criteria Alternative Extremely Very strong Strong Weak Equal Just equal preferred importance importance importance importance A1 A2 Con. I Con. II Con. I Con. II Con. I Con. II Con. I Con. II Con. I Con. II Con. I Con. II Quality System A System B System C System B System C Functionality System A System B System C System B System C Reliability System A System B System C System B System C... Cost System A System B System C System B System C Purchasing System A System B System C System B System C... Reputation System A System B System C System B System C Vendor s ability System A System B System C System B System C... Condition I (Con. I): A1 is more important than A2. Condition II (Con. II): A1 is more important than A2. References [1] M.R. Abdi and A.W. Labib, A design strategy for reconfigurable manufacturing systems (RMSs) using analytical hierarchical process (AHP): A case study, International Journal of Production Research 41(10) (2003), [2] K.M.A.-S. Al-Harbi, Application of the AHP in project management, International Journal of Project Management 19(1) (2001), [3] A. Al-Mudimigh, M. Zairi and M. Al-Mashari, ERP software implementation: An integrative framework, European Journal of Information Systems 10(4) (2001), [4] Z. Ayağ and R.G. Özdemir, An intelligent approach to ERP software selection through fuzzy ANP, International Journal of Production Research 45(10) (2007), [5] E.W.N. Bernroider and V. Stix, Profile distance methoda multi-attribute decision making approach for information system investments, Decision Support Systems 42 (2006), [6] M. Bevilacqua and M. Braglia, The analytic hierarchy process applied to maintenance strategy selection, Reliability Engineering and System Safety 70(1) (2000), [7] C.E. Bozdag, C. Kahraman and D. Ruan, Fuzzy group decision making for selection among computer integrated manufacturing systems, Computers in Industry 51(1) (2003), [8] J.J. Buckley, Fuzzy hierarchical analysis, Fuzzy Sets and Systems 17 (1985), [9] F.T.S. Chan, Design of material handling equipment selection system: An integration of expert system with analytic hierarchy process approach, Integrated Manufacturing Systems 13(1) (2002), [10] F.T.S. Chan, M.H. Chan and N.K.H. Tang, Evaluation methodologies for technology selection, Industrial and Manufacturing Systems Engineering, Journal of Materials Processing Technology 107(1 3) (2000), [11] D.-Y. Chang, Application of the extent analysis method on fuzzy AHP, European Journal of Operational Research 95(3) (1996), [12] D.-Y. Chang, Extent analysis and synthetic decision, Optimization Techniques and Applications 1, World Scientific, Singapore, 1992, [13] H. Deng, Multicriteria analysis with fuzzy pairwise comparison, International Journal of Approximate Reasoning 21(3) (1999), [14] J. Esteves and J. Pastor, Towards the unification of critical success factors for ERP implementations, Published in 10th Annual Business Information Technology (BIT) 2000 Conference, Manchester, 2000.

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15 Journal of Intelligent & Fuzzy Systems 21 (2010) DOI: /IFS IOS Press An input relaxation measure of efficiency in fuzzy data envelopment analysis (FDEA) M. Khodabakhshi* and Mona Hejrizadeh Department of Mathematics, Faculty of Science, Lorestan University, Khorram Abad, Iran Abstract. While data envelopment analysis (DEA) models are very sensitive to possible data errors, in many real applications, the data of production processes cannot be precisely measured. Fuzzy DEA is a successful method that allows to deal with imprecise data in DEA. In this paper, we develop fuzzy version of the input relaxation model introduced in Jahahshahloo and Khodabakhshi [15] by using some ranking methods based on the comparison of α-cuts. The resulting auxiliary crisp problems can be solved by the usual DEA software. It is shown that, using a numerical example, how the proposed model become specially useful for detecting sensitive decision-making units. Keywords: Data envelopment analysis, input relaxation, fuzzy mathematical programming, fuzzy intervals, possibilistic programming 1. Introduction Data envelopment analysis (DEA) was originated in 1978 by Charnes et al. [4], and later developed by Banker et al. [1] in 1984 to evaluate the relative efficiency of a set of decision-making units (DMUs) involved in a production process. DEA models provide efficiency scores that assess the performance of the different DMUs in terms of either the use of several inputs or the production of certain outputs (or even simultaneously). Traditionally, the coefficients of DEA models, i.e., the data of inputs and outputs of the different DMUs, are assumed to be measured with precision. However, as some authors point out (see e.g., [11]), this is not always possible. In these cases it may be more appropriate to interpret the experts understanding of the parameters as fuzzy numerical data which can be represented by means of fuzzy numbers or fuzzy intervals. Fuzzy mathematical programming provides us with a tool to deal with the natural uncertainty inherent to some production processes. Alternatively, Corresponding author. mkhbakhshi@yahoo.com (Mohammad Khodabakhshi). other authors propose chance constrained programming formulations of DEA as stochastic approaches to deal with variations in data (see e.g., [5 7, 19 25, 35]). Several fuzzy approaches have been provided to the assessment of efficiency in the DEA literature. Sengupta [36] considers fuzzy both objective and constraints and analyzes the resulting fuzzy DEA model by using Zimmermann s method [44]. Kao and Liu [17] develop a method to find the membership functions of the fuzzy efficiency scores when some observations are fuzzy numbers. The idea is based on the α-cuts and Zadeh s extension principle [42]. Hougaard s approach [12] allows the decision makers to use scores of technical efficiency in combination with other sources of information as expert opinions for instance. Entani et al. in [10] propose a DEA model with an interval efficiency consisting of efficiencies obtained from the pessimistic and the optimistic viewpoints. Their model, which is able to deal with fuzzy data, also consider inefficiency intervals. See also [2, 9, 28, 34, 36, 39, 43] for further discussion on fuzzy programming. Here, we are particularly interested in the approach by Guo and Tanaka [11], which uses the possibilistic programming. Some /10/$ IOS Press and the authors. All rights reserved

16 396 M. Khodabakhshi and M. Hejrizadeh / An input relaxation measure of efficiency in fuzzy data envelopment analysis (FDEA) gains are obtained with respect to both computational and interpretative aspects. The remainder of this paper is presented as follows: Section 2 contains some results on fuzzy interval analysis that will be used in the paper. In section 3, we develop fuzzy input relaxation model in DEA. We also include an example that shows how the use of the proposed possibilistic DEA model may provide some very useful information which would remain unnoticed with a crisp approach. Section 4 contains our final conclusions. 2. Preliminaries As it is pointed out in [37] there is not a total agreement in the literature with respect to the classification of fuzzy mathematical programming approaches and some authors prefer to call such problems linear programs with fuzzy coefficient problems. Inuiguchi et al. [13] and Lai and Hwang [26] refer to linear programming with imprecise coefficients restricted by possibilistic distributions as possibilistic programming. Different classifications can be find in Zimmermann [45], Leung [29] or Lujandjula [31], for instance. Dubois et al. [8] present a complete survey of the state of the art about fuzzy interval analysis. In this section, following Leon et al. [27] we are simply recalling how to perform the basic operations of arithmetics and the comparison of fuzzy intervals for ranking purposes. To be more precise, we deal with LR-fuzzy numbers whose definition is as follows. These definitions are taken from Leon et al. [27]. Definition 1. A fuzzy number M is said to be a LRfuzzy number, M = (m L,m R,α L,α R ) L,R if its membership function has the following form: ( ) L m L r, r6 m L, α L µ M (r) = 1, m L 6 r 6 m R, ( R r m R, r> m R α R ) where L and R are reference functions, i.e., L, R: [0, + ] [0, 1] are strictly decreasing in supp( M) = {r : µ M (r) > 0} and upper semi-continuous functions such that L(0) = R(0) = 1 If supp(( M)) is a bounded set, L and R are defined on [0, 1] and satisfy L(1) = R(1) = 0. In the context of FLP, the min T-norm is the most applied to evaluate a linear combination of fuzzy quantities, ã 1 x 1 ã 2 x 2... ã n x n when the fuzzy numbers are noninteractive [31]. In particular, for a given set of LR-fuzzy numbers ã j = (aj L,aR j,αl j,αr j ) L,R,j = 1,...,n and some scalars x j > 0,j = 1,...,n,we have that ã j x j = a L j x j, a R j x j α L j x j, α R j x j, where L and R are the common left and right reference functions, and n ã j x j denotes the combination ã 1 x 1 ã 2 x 2... ã n x n. Due to important of fuzzy quantities in practice, they have received attention by many researchers (see for instance [3, 40, 41]). The ranking process depends heavily on the environment or the framework of the problem at hand. Based on Chang and Lee s classification [3], if fuzzy numbers are ranked by comparing their α-cuts the ranking method belongs to what they call approach by using α-cuts. Although this kind of methods does not use all the information stored in the fuzzy sets and also appears to be restrictive to apply, they provide quick results. This is probably the reason why they are so widely used in the FLP framework. Let us recall the definition of maximum of two fuzzy numbers. Definition 2. Let M and Ñ be two fuzzy numbers. Then M Ñ represents the fuzzy number having the following membership function: µ M Ñ (r) = sup {µ M (s) µ Ñ (t)}. (2) r=s t Based on fuzzy max operator, Dubois and Prade [9] define the following ordering relation. Definition 3. Then, L,R (1) Let M and Ñ be two fuzzy numbers. M & Ñ M Ñ = M. (3) Tanaka et al. [38] and Ramik and Rimanek [33] have formulated FLP problems by using this order. In fact, Rimak and Rimanek provided an operative characterization of (3) in terms of the α-level sets:

17 M. Khodabakhshi and M. Hejrizadeh / An input relaxation measure of efficiency in fuzzy data envelopment analysis (FDEA) 397 Lemma 1. (Ramik and Rimanek [33]). Let M and Ñ be two fuzzy numbers. Then, M Ñ = M if, and only if, h [0, 1] the two statements below hold: inf{s : µ M (s)> h}> inf{t : µ Ñ (t)> h}, sup{s : µ M (s)> h}> sup{t : µ Ñ (t)> h}. (4) In particular, for two LR-fuzzy numbers, M = (m L,m R,α L,α R ) L,R and Ñ = (n L,n R,β L,β R ) L,R, (4) holds if, and only if, m L L (h)α L > n L L (h)β L h [0, 1] m R + R (h)α R > n R + R (h)β R h [0, 1] (5) where L (h) = sup{z : L(z)> h},l (h) = sup{z : L (z)> h}, R (h) = sup{z : R(z)> h},r (h) = sup{z : R (z)> h}, Moreover, if M = (m L,m R,α L,α R ) L,R and Ñ = (n L,n R,β L,β R ) L,R have bounded support and both L = L and R = R, then (5) becomes m L > n L, m L α L > n L β L, m R > n R, m R + α R > n R + β R. (6) In spite of its well-foundedness, this order could provoke situations of undecisiveness, where, as Dubois et al. [8] state, intuitively one would expect that M should be declared greater than Ñ because they are very different. To provide a solution to this issue, we have considered Tanaka et al. [38] proposal. Definition 4. Let M and Ñ be two fuzzy numbers and h a real number h [0, 1]. Then M h Ñ if, and only if, k [h, 1] the following two statements hold: inf{s : µ M (s)> k}> inf{t : µ Ñ (t)> k}, sup{s : µ M (s)> k}> sup{t : µ Ñ (t)> k}, (7) For LR-fuzzy numbers with bounded support, and using this ranking method, for a given h, expression (7) becomes m L L (k)α L > n L L (k)β L k [h, 1] m R + R (k)α R > n R + R (k)β R k [h, 1] (8) Notice that (8) is less restrictive than (5). Indeed when comparing M and Ñ at a given possibility level h,itmay happen that M h Ñ although M Ñ does not hold. As we show in the next section, this ranking method allows us to provide the efficiency scores for different possibility levels to the decision maker. 3. Fuzzy input relaxation model In this section, input relaxation model introduced in [15, 16, 18, 19] is developed in fuzzy data envelopment analysis DEA models with fuzzy data Although data envelopment analysis methodology has many advantages, such as no requirement for a priori weights or explicit specification of functional relations among the multiple inputs and outputs, there is a weakness in conventional DEA models. DEA doesn t allow incorporating uncertainty in the model formulation. Furthermore, an efficient DMU which is sensitive to variation in input-output data can not be identified by crisp DEA models. In other words, a DMU which is measured as efficient relative to other DMUs, may turn inefficient if variations such as data entry errors are considered. Some authors propose chance constrained programming formulations of DEA as stochastic approaches to deal with variations in data (see e.g., [5 7, 19 25, 35]). Fuzzy linear programming also allows us to deal with the natural uncertainty inherent to some production processes. In recent years, fuzzy set theory has been proposed as a way to quantify imprecise and vague data in DEA models. Efficiency measurement in DEA leads to comparision of fuzzy quantities. Many researchers provided different approaches to comparision and ranking fuzzy quantities. Dubois et al. [8] present a complete survey of the state of the art about fuzzy interval analysis. Different classifications in Zimmermann [45], Leung [29] or Lujandjula [31] can be find, for instance. See also, Sengupta [35], Inuiguchi et al. [13, 14], Lai and Hwang [26], Guo and Tanaka [11], Lertworasirikul et al. [28], Lai et al. [26], and Leon et al. [27] among others. In what follows, fuzzy version of input relaxation model is develop based on comparing α cuts Fuzzy input relaxation model Suppose that we are going to evaluate the relative efficiency of n DMUs which use m inputs to produce

18 398 M. Khodabakhshi and M. Hejrizadeh / An input relaxation measure of efficiency in fuzzy data envelopment analysis (FDEA) s outputs. Assume that the data of inputs and outputs cannot be precisely measured and, also, that they can be expressed as LR-fuzzy numbers with bounded support x ij = (xij L,xR ij,αl ij,αr ij ) L ij,r ij, i = 1,...,m, j = 1,...,n, ỹ rj =(yrj L,yR rj,βl rj,βr rj ) L, r=1,...,s, rj,r rj j = 1,...,n, and s + i2 = (s i2 +L,s+R i2,δl i2,δr i2 ) L j i2,rj i2 satisfying which means that DMU 0 produces maximum possible output. Otherwise, if φ 0 > 1 it means that DMU 0 could produce φ 0 y r0, r = 1,..., s, therefore, it is inefficient in its current state. Efficiency score from fuzzy point of view is defined in Definition 5. Since inputs and outputs are LR numbers, the constraints in (10) (except that of convexity) can be regarded as inequalities between LR numbers. If, in particular, is interpreted as in (3) and the linear combinations as in (1), then (10) can be transformed in L i1 = =L in := L i, L r1 = =L rn := L r, L j i2 = =Ln i2 := L" i, i = 1,...,m, r = 1,...,s, i = 1,...,s max φ 0 Subject to: R i1 = =R in := R i, i = 1,...,m, R r1 = =R rn := R r, r = 1,...,s. (9) R 1 i2 = =Rn i2 := R" i, i = 1,...,s. Note that (9) is not too restrictive, as we are simply requiring that, for any variable (both inputs and outputs), we can use LR-fuzzy numbers with the same type for the corresponding n data. For instance, if these are trapezoid or triangular fuzzy numbers then (9) holds. Let us also assume that the input relaxation model introduced in Jahanshahloo and Khodabakhshi [15] is used to evaluate the relative efficiency of this set of DMUs. Then, the extended input relaxation model can be expressed as the following fuzzy LP problem: max φ 0 Subject to: x i0 λ j x ij s i2 + φ 0 ỹ r0 λ j ỹ rj λ j = 1 (10) λ j > 0, s + i2 0 Note that in the optimal solution of the crisp input relaxation model, φ 0 is equal to unity for efficient DMU 0 x L i0 > x R i0 > x L i0 αl i0 > x R i0 + αr i0 > φ 0 y L r0 6 φ 0 y R r0 6 λ j x L ij s+l i2 λ j x R ij s+r i2 λ j x L ij s+l i2 λ j xij R s+r λ j yrj L λ j yrj R φ 0 y L r0 φ 0β L r0 6 φ 0 y R r0 + φ 0β R r0 6 λ j = 1 λ j α L ij δl i2 i2 + λ j α R ij + δr i2 λ j yrj L λ j βrj L λ j yrj R + λ j βrj R (11) λ j > 0, s i2 +L > 0 L,s i2 +R > 0 R,s i2 +L δ L i2 > 0, s i2 +R + δ R i2 > 0. Hence, the optimal value of (11) provides an evaluation of the efficiency of DMU 0 in which all the possible

19 M. Khodabakhshi and M. Hejrizadeh / An input relaxation measure of efficiency in fuzzy data envelopment analysis (FDEA) 399 values of the different variables for all the DMUs at all the possibility levels are considered. On the other hand, the analyst may want to have available the efficiency results with respect to a given possibility level, or they may even be interested in knowing how the efficiency changes for different possibility levels. In this case we can use h for ranking (see (7)), then model (10) can be expressed as the following linear programming problem: (P h ): max φ 0 Subject to: x L i0 > x R i0 > λ j x L ij s+l i2 λ j xij R sr i2 x L i0 L i (h)αl i0 > ( s i2 +L x R i0 + R i (h)αr i0 > ( s i2 +R φ 0 y L r0 6 λ j xij L L i (h) λ j α L ij ) L i (h)δ L i2 λ j xij R + R i (h) λ j α R ij ) + R i (h)δ R i2 λ j yrj L λ j = 1 λ j > 0, s i2 +L > 0 L,s i2 +R > 0 R,s i2 +L L i (h)δ L i2 > 0, s +R i2 + R i (h)δ R i2 > 0. The optimal value of (12), φ0 (h), provides the efficiency score of DMU 0 at the h possibility level. As said before, in practice we can solve this model for different values of h to observe how the efficiency scores of the DMUs change when the possibility level h varies. For instance, the decision maker could shown a table displaying φ0 (h) for values of h from 0 to 1 by 0.1 for each DMU. Notice that if h = 0 then (12) coincides with (11). One of the advantages of such a table is in allowing the analyst to identify the sensitive units. By this we mean a DMU for which small modifications in some input or output would lead us to change our mind about its efficiency. Remark 1. Thus, the efficiency score of a given DMU 0 can be regarded as a fuzzy set whose membership function is defined as µ 0 (φ) = sup{h : φ is an optimal value of (P h )}. (13) In Guo and Tanaka s approach the efficiency score of DMU 0 for each possibility level h is a fuzzy number. However, in ours this efficiency score is a real number in (0,1]. The following result shows that the h-possibilistic efficiency score of a given unit might improve for lower values of the possibility level. φ 0 y R r0 6 λ j yrj R Proposition 1. The h-possiblistic efficiency score is a non-decreasing function of the possibility level h. φ 0 y L r0 L r (h)φ 0β L r0 6 φ 0 y R r0 + R r (h)φ 0β R r0 6 λ j yrj L L r (h) λ j βrj L (12) λ j yrj R + R r (h) λ j βrj R Proof. Let (λ 1,...,λ n,φ 0 ) be an optimal solution of (10) when is interpreted as h. Then, n λ j x ij s i2 + α x i0,i= 1,...,m, n λ j ỹ rj α φ 0 ỹ r0,r= 1,...,s, α [h, 1]. Therefore, (λ 1,...,λ n,φ 0 ) is a feasible solution of (10) for all α such that h α 1. Consequently, the optimal value of any of these problems will be less than or equal to φ 0.

20 400 M. Khodabakhshi and M. Hejrizadeh / An input relaxation measure of efficiency in fuzzy data envelopment analysis (FDEA) The previous result reflects the idea that the efficiency may improve if we are willing to consider more production scenarios as plausible. In particular, a DMU evaluated as inefficient for h =1 may become efficient for a lower possibility level. We can define now what the fuzzy set of efficient units is. Definition 5. For a given set {DMU i } n i=1 of n DMUs we define the fuzzy set of efficient units as Ẽf ={(DMU i,µẽf (DMU i )), i= 1,...,n}, where the membership function is given by is redundant as a consequence of the symmetry. Therefore, in this situation (12) becomes max φ 0 Subject to: x i0 (1 h)α i0 λ j x ij (1 h) (s + i2 (1 h)δ i2) x i0 + (1 h)α i0 λ j x ij + (1 h) (s + i2 + (1 h)δ i2) λ j α ij λ j α ij µẽf (DMU i ) = { 0 if φi (h) 1 for all h [0, 1] sup{h : φi (h) = 1} if φ i (h) = 1 for some h [0, 1]. Remark 2. (1) As stated by Proposition 1, if φ i (h) = 1 then φ (u) = 1 for all u6 h. (2) As the exact computation of µẽf (DMU) requires to obtain a supremum, if it is shown that the decision maker a table displaying φ (h) for values of h from 0 to 1 by 0.1 for each DMU, we can only compute µẽf (DMU) approximately. In case that more precision is required only a few extra computational effort is necessary. φ 0 y r0 (1 h)β r0 φ 0 y r0 + (1 h)β r0 λ j y rj (1 h) λ j β rj λ j y rj + (1 h) λ j β rj λ j = 1 (14) λ j 0, s + i Particular case: Triangular fuzzy numbers If inputs and outputs are now assumed to be symmetrical triangular fuzzy numbers, denoted by the pairs consisting of the corresponding centers and spreads, x ij = (x ij,α ij ), i = 1,...,m, j = 1,...,n and ỹ rj = (y rj,β rj ), r = 1,...,s, j = 1,...,n, and s i2 + = (s + i2,δ i2) then (12) can be substantially simplified. Note that for triangular fuzzy numbers L i (h) = R i (h) = L i (h) = R i (h) = L i (h) = R i (h) = 1 h Also, the two constraints associated with the main values reduce to only one, and can be eliminated since it By a direct substitution in (14) we can simplify (P h T ) in the following two situations. Proposition 2. If for some output r, there exist two scalars c r and β r such that β rj = c r y rj + β r, j = 1,...,n,then n λ j ỹ rj h ỹ ro will be equivalent to n λ j y rj > y r0. (2) If for some input i, there exists a scalar c i such that α ij = c i x ij j = 1,..., n, then n λ j x ij h x i0 will be equivalent to n λ j x ij 6 x io Numerical example We are evaluating the efficiency of data in Table 1 with model (14) to illustrate the use of the methodology developed in this paper. Results are presented in Table 2.