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: ; [email protected] (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.

14 378 S. Onut and T. Efendigil / A theorical model design for ERP software selection process under the constraints of cost and quality [15] M.D. Feblowitz and S.J. Greenspan, SceneraioScenario-based analysis of COTS acquisition impacts, in: Requirements Engineering 3(3/4) (1998), [16] R. Handfield, S.V. Walton, R. Sroufe and S.A. Melnyk, Applying environmental criteria to supplier assessment: A study in the application of the Analytical Hierarchy Process, European Journal of Operational Research 141(1) (2002), [17] IEEE Std , IEEE standard for a software quality metrics methodology, [18] J. Jeanrenaud and P. Romanazzi, Software product evaluation: A methodological approach, in: Software Quality Management II: Building SoftwareSoftware into Qualityo Quality (1994), [19] C. Kahraman, U. Cebeci and Z. Ulukan, Multi-criteria supplier selection using fuzzy AHP, Logistics Information Management 16(6) (2003), [20] E. Karsak and C.O. Ozogul, An integrated decision making approach for ERP system selection, Expert Systems with Applications (691) 36(1) (2009), [21] M.I.A. Khalil, Selecting the appropriate project delivery method using AHP, International Journal of Project Management 20(6) (2002), [22] J. Kontio, A case study in applying a systematic method for COTS selection, in: 18th International Conference on Software Engineering (1996), [23] R.J. Kuo, S.C. Chi and S.S. Kao, A decision support system for selecting convenience store location through integration of fuzzy AHP and artificial neural network, Computers in Industry 47(2) (2002), [24] C.K. Kwong and H. Bai, A fuzzy AHP approach to the determination of importance weights of customer requirements in quality function deployment, Journal of Intelligent Manufacturing 13(5) (2002), [25] V.S. Lai, B.K. Wong and W. Cheung, Group decision making in a multiple criteria environment: A case using the AHP in software selection, European Journal of Operational Research 137(1) (2002), [26] P.K. Lawlis, K.E. Mark, D.A. Thomas and T. Courtheyn, A formal process for evaluating COTS software products, IEEE Computer 34(5) (2001), [27] N.A. Maiden and C. Ncube, Acquiring COTS software selection requirements, IEEE Software 15(2) (1998), [28] H. Min, Selection of software: The analytic hierarchy process, International Journal of Physical Distribution and Logistics Management 22(1) (1992), [29] M. Morisio and A. Tsoukias, IusWare: A methodology for the evaluation and selection of software products, IEEE Proceedings Software Engineering 144(3) (1997), [30] T. Oberndorf, COTS and open systems An overview, 2008, Available at cots body.html [31] S. Sedigh-Ali, A. Ghafoor and R. Paul, Software engineering metrics for COTS-based systems, IEEE Computer 34(5) (2001), [32] S. Shang and P. Seddon, A comprehensive framework for classifying the benefits of ERP systems, in: Proceedings of AMCIS (2000), [33] V. Tran and D.-B. Liu, A risk-mitigating model for the development of reliable and maintainable large-scale commercialoff-the-shelf integrated software systems, in: Proceedings of Annual Reliability and Maintainability Symposium, 1997, [34] E. Triantaphyllou and S.H. Mann, Using the analytic hierarchy process for decision making in engineering applications: Some challenges, International Journal of Industrial Engineering: Applications and Practice 2(1) (1995), [35] E. Triantaphyllou, Multi-criteria Decision Making Methods: A Comparative Study, Kluwer Academic Publishers, [36] E.J. Umble, R.R. Haft and M.M. Umble, Enterprise resource planning: Implementation procedures and critical success factors, European Journal of Operational Research 146(2) (2003), [37] P.J.M. Van Laarhoven and W. Pedrycz, A fuzzy extension of Saaty s priority theory, Fuzzy Sets and Systems 11 (1983), [38] C.C. Wei and M.J. Wang, A comprehensive framework for selecting an ERP system, International Journal of Project Management 22(2) (2004), [39] J. Yang and H. Lee, An AHP decision model for facility location selection, Facilities 15(9 10) (1997), MCB University Press, [40] L.A. Zadeh, Is there a need for fuzzy logic? Information Sciences 178 (2008), [41] K.-J. Zhu, Y. Jing and D.-Y. Chang, A discussion on Extent Analysis Method and applications of fuzzy AHP, European Journal of Operational Research 116(2) (1999),

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. [email protected] (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.

21 M. Khodabakhshi and M. Hejrizadeh / An input relaxation measure of efficiency in fuzzy data envelopment analysis (FDEA) 401 Table 1 Data of illustrative example DMU X Y A 3 = (3, 2) 3 = (3, 1) B 4 = (4, 0.5) 2.5 = (2.5, 1) C 4.5 = (4.5, 1.5) 6 = (6, 1) D 6.5 = (6.5, 0.5) 4 = (4, 1.25) E 7 = (7, 2) 5 = (5, 0.5) F 8 = (8, 0.5) 3.5 = (3.5, 0.5) G 10 = (10, 1) 6 = (6, 0.5) H 6 = (6, 0.5) 2 = (2, 1.5) Table 2 Efficiency evaluation results Possibility DMU h A B C D E F G H It should be pointed out that, except in the cases of A, C and G (which are crisp-efficient), these efficiency scores are always less than those that would be obtained from a conventional crisp evaluation of the centers of the fuzzy triangular numbers (whose results are those in the h = 1 row in Table 2). Furthermore, we can use model (14) to analyze how the efficiency scores vary between these two extremal situations. In fact, Table 2 includes the values of the h-possibilistic efficiency scores of the eight DMUs for values of h from 0 to 1 by 0.1. We can see that the fuzzy set of efficient units is Ẽf ={(A, 1), (B, 0.3), (C, 1), (G, 1)}. We have not included D, E, F and H because their membership function values are null, that is these DMUs are inefficient regardless of the value of the possibility level. It is worth pointing out the case of DMU B, which is very inefficient in the crisp problem (for h = 1), φb = 1, but it becomes efficient for lower values of h (in fact, it can be shown that φb = 1 for h<0.4). In our opinion, the analysis of efficiency that we propose may be very useful to the decision maker. It allows us to incorporate uncertainty in the model formulation. Besides, Table 2 shows how the efficiency scores change from the case in which all the possible production scenarios are considered (h = 0) to the crisp case (h = 1) in which we recover the input relaxation model. Finally, we would like to comment the main advantage that, in our opinion, this method has versus other previous fuzzy approaches. This advantage is that it does not require any special software but the usual DEA or linear programming packages to be implemented. This can be very interesting, specially for DEA practitioners who are not too familiar with mathematical programming techniques. 4. Conclusion In this paper, we developed input relaxation model in fuzzy DEA. This approach has been based on ranking methods that compare α-cuts. The fuzzy version of the input relaxation model in DEA is transformed into equivalent crisp LP formulations. These auxiliary crisp problems can be solved by the usual DEA software. Using other ordering relations that provide different interpretations of the inequalities between fuzzy

22 402 M. Khodabakhshi and M. Hejrizadeh / An input relaxation measure of efficiency in fuzzy data envelopment analysis (FDEA) numbers might be explored (see for example [34, 35]). Besides, we also suggest to investigate the development of the additive input relaxation model into fuzzy DEA (see e.g., [5 7, 19, 29]). Furthermore, development of non-radial efficiency measure such as russell efficiency measure [26] to deal with imprecise data (stochastic and fuzzy) may be interesting. References [1] R.D. Banker, A. Charnes and W.W. Cooper, Some models for estimating technical and scale inefficiencies in data envelopment analysis, Management Science 30 (1984), [2] C. Carlsson and P. Korohnen, A parametric approach to fuzzy linear programming, Fuzzy Sets and Systems 20 (1986), [3] Chang and Lee, Fuzzy arithmetics and comparison of fuzzy numbers, in: Fuzzy Optimization: Recent Advances, Delgado, et al., eds, Springer, New York, [4] A. Charnes, W.W. Cooper and E. Rhodes, Measuring the Efficiencies of DMUs, European Journal of Operational Research 2(6) (1978), [5] W.W. Cooper, H. Deng, Z. Huang and Susan X. Li, Chance constrained programming approaches to technical efficiencies and inefficiencies in stochastic data envelopment analysis, Journal of the Operational Research Society 53 (2002), [6] W.W. Cooper, H. Deng, Z. Huang and S.X. Li, Chance constrained programming approaches to congestion in stochastic data envelopment analysis, European Journal of Operational Research 155(2) (2004), [7] W.W. Cooper, Z. Huang and S. Li, Satisfacing DEA models under chance constraints, Ann Oper Res 66 (1996), [8] D. Dubois, E. Kerre, R. Mesiar and H. Prade, Fuzzy interval analysis, in: Fundamentals of Fuzzy Sets, D. Dubois and H. Prade, eds, Kluwer Academic Publishers, Dordrecht, [9] D. Dubois and H. Prade, Systems of linear fuzzy constraints, Fuzzy Sets and Systems 3 (1980), [10] T. Entani, Y. Maeda and H. Tanaka, Dual models of interval DEA and its extension to interval date, European J Oper Res 136 (2002), [11] P. Guo and H. Tanaka, Fuzzy DEA: A perceptual evaluation method, Fuzzy Sets and Systems 119 (2001), [12] J.L. Hougaard, Fuzzy scores of technical efficiency, European J Oper Res 115 (1999), [13] M. Inuiguchi, H. Ichihasi and H Tanaka, Fuzzy programming: A survey of recent developments, in: Stochastic Versus Fuzzy Approaches to Multiobjective Mathematical Programming Under Uncertainity, R. Slowinski and J. Teghem, eds, Kluwer Academic Publishers, Dordrecht, [14] M. Inuiguchi and T. Tanino, Data envelopment analysis with fuzzy input-output data, in: Research and Practice in Multiple Criteria Decision Making, Y.Y. Haims and R.E. Steuer, eds, Springer-Verlag, Berlin, 1999, pp [15] G.R. Jahanshahloo and M. Khodabakhshi, Suitable combination of inputs for improving outputs in DEA with determining input congestion Considering textile industry of China, Applied Mathematics and Computation 151 (1) (2004), [16] G.R. Jahanshahloo and M. Khodabakhshi, Determining assurance interval for non-archimedean element in the improving outputs model in DEA, Applied Mathematics and Computation 151(2) (2004), [17] C. Kao and S.T. Liu, Fuzzy efficiency measures in data envelopment analysis, Fuzzy Sets and Systems 119 (2000), [18] M. Khodabakhshi, A super-efficiency model based on improved outputs in data envelopment analysis, Applied Mathematics and Computation 184 (2) (2007), [19] M. Khodabakhshi and M. Asgharian, An input relaxation measure of efficiency in stochastic data envelopment analysis, Applied Mathematical Modelling 33 (2009), [20] M. Khodabakhshi, A one-model approach based on relaxed combinations of inputs for evaluating input congestion in DEA, Journal of Computational and Applied Mathematics 230 (2009), [21] M. Khodabakhshi, Estimating most productive scale size with stochastic data in data envelopment analysis, Economic Modelling 26 (2009), [22] M. Khodabakhshi, M. Asgharian and G.N. Gregoriou, An input-oriented super efficiency measure in stochastic data envelopment analysis Expert System with Applications: Evaluating chief executive officers of US public banks and thrifts, Expert Systems with Applications 37 (2010), [23] M. Khodabakhshi, Y. Gholami and H. Khairollahi, An additive model approach for estimating returns to scale in imprecise data envelopment analysis, Applied Mathematical Modelling 34 (2010), [24] M. Khodabakhshi, Chance consetrained additive input relaxation model in stochastic data envelopment analysis, International Journal of Information and Systems Sciences 6(1) (2010), [25] M. Khodabakhshi, Super-efficiency in stochastic data envelopment analysis: An input relaxtion approach, Journal of Computational and Applied Mathematics, in press (2010). [26] Y.J. Lai and C.L. Hwang, Fuzzy Mathematical Programming: Theory and Applications, Springer, Berlin, [27] T. Leon, V. Liern, J.L. Ruiz and I. Sirvent, A fuzzy mathematical programming approach to the assessment of efficiency with DEA models, Fuzzy sets and Systems 139 (2003), [28] S. Lertworasirikul, S.-C. Fang, J.A. Joines and H.L.W. Nuttle, Fuzzy data envelopment analysis (DEA): A possibility approach, Fuzzy Sets and Systems 139(2) (2003), [29] Y. Leung, Spatial Analysis and Planning Under Imprecision, North-Holland, Amsterdam, [30] B. Liu, Uncertain Programming, A Wiley-Interscience Publication, New York, [31] M.K. Luhandjula, Fuzzy optimization: An appraisal, Fuzzy Sets and Systems 30 (1989), [32] J.T. Pastor, J.L. Ruiz and I. Sirvent, An enhanced DEA russell graph efficiency measure, European J Oper Res 115(3) (1999), [33] J. Ramik and J. Rimanek, Inequality relation between fuzzy numbers and its use in fuzzy optimization, Fuzzy sets and Systems 16 (1985), [34] H. Rommelfanger and R. Slovinski, Fuzzy linear programming with single or multiple objective functions, in: Fuzzy Sets in Desission Analysis, R. Slowinski, ed., Operations Research and Statistics, Kluwer Academic Publishers, Dordrecht, [35] J.K. Sengupta, Data envelopment analysis for efficiency measurment in stochastic case, Comput Oper Res 14 (1987), [36] J.K. Sengupta, A fuzzy systems approach in data envelopment analysis, Comput Math Appl 24 (1992),

23 M. Khodabakhshi and M. Hejrizadeh / An input relaxation measure of efficiency in fuzzy data envelopment analysis (FDEA) 403 [37] J.K. Slowinski, Fuzzy Sets in Decission Analysis, Operations Research and Statistics, Kluwer Academic Publishers, Dordrecht, [38] H. Tanaka, H. Ichihasi and K. Asai, A formulation of fuzzy linear programming problem based on comparison of fuzzy numbers, Control and Cybernetics 13 (1984), [39] K. Trianttis and O. Girod, A mathematical programming approach for measuring technical efficiency in a fuzzy environment, J Prod Analy 10 (1998), [40] X. Wang and E.E. Kerre, Reasonable properties for the ordering of fuzzy quantities (I), Fuzzy sets and Systems 118 (2001), [41] X. Wang and E.E. Kerre, Reasonable properties for the ordering of fuzzy quantities (II), Fuzzy sets and Systems 118 (2001), [42] L.A. Zadeh, Fuzzy Sets, Information and Control 8 (1965), [43] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978), [44] H.J. Zimmermann, Description and optimization of fuzzy system, Internat J General System 2 (1976), [45] H.J. Zimmermann, Fuzzy Set Theory and Its Applications, 3rd ed., Kluwer Academic Publishers, Boston, 1996.

24 Journal of Intelligent & Fuzzy Systems 21 (2010) DOI: /IFS IOS Press An adaptive fuzzy system for the control of the vergence angle on a robotic head Nikolaos Kyriakoulis a,*, Antonios Gasteratos a and Spyridon G. Mouroutsos b a Laboratory of Robotics and Automation, Department of Production and Management Engineering, Democritus University of Thrace, University Campus Kimmeria, Xanthi, Greece b Laboratory of Special Mechanics, Department of Electrical and Computer Engineering, Democritus University of Thrace, University Campus Kimmeria, Xanthi, Greece Abstract. An important issue in realizing robots with stereo vision is the efficient control of the vergence angle. In an active robotic vision system the vergence angle along with the pan and tilt ones determines uniquely the fixation point in the 3D space. The vergence control involves the adjustment of the angle between the two cameras axes towards the fixation point and, therefore, it enables the robot to perceive depth and to compute obstacle maps. Vergence movement is directly related to the binocular fusion. Additionally, the decision for convergence or divergence is extracted either by motion affine models or by mathematical ones. In this paper, a new method for extracting the cameras movement direction is presented. The movement decision is performed by an adaptive fuzzy control system, the inputs of which are the zero-mean normalized cross correlation (ZNCC) and the depth estimations at each time step. The proposed system is assessed on a 4 d.o.f. robotic head, yet it can be utilized in any active binocular system, since it is computationally inexpensive and it is independent to a priori camera calibration. Keywords: Vergence angle, computer vision, mechatronics, fuzzy logic 1. Introduction Among the contemporary mechatronics systems, robots are of the most complex in terms of structure and control. Also, they claim for an extensive sensorial system, which enables them with enhanced perception. In such a system, vision holds a dominant role and that s why it was utilized in robotic systems since their infancy [10]. Of special interest is the use of the binocular vision, that enables the biological systems to perceive depth and, thus, allows them a deeper knowledge of the scene. Vergence control supports the above capacity by turning the eyes, so that they are directed both towards the same point in the 3D space. Humans possess the me- Corresponding author. Nikolaos Kyriakoulis, Laboratory of Robotics and Automation, Department of Production and Management Engineering, Democritus University of Thrace, University Campus Kimmeria, GR Xanthi, Greece. Tel.: ; Fax: ; [email protected]. dial and lateral recti muscles to rotate their globes so that an image pair is projected onto their foveae [2]. In a robot system the cameras play the role of the eyes, the servo motors this of the muscles, and the optic sensors correspond to the foveae. This principle was realized in [7], where the same landmark is identified in both images to enable the fixation of the active cameras of a human-like set-up to that point. A stereoscopic vision system controls the vergence angle by initially placing the target of interest at the image center as shown in Fig. 1. A plethora of different techniques have been applied for calculating and controlling the vergence angle, which can be classified into two main categories: disparity based and correlation based ones. The disparity (stereo matching) techniques deal with the problem of stereo correspondence. The correlation based methods utilize a measure such as normalized cross correlation (NCC), zero-mean normalized cross correlation (ZNCC) or sum of absolute differences (SAD), to max /10/$ IOS Press and the authors. All rights reserved

25 386 N. Kyriakoulis et al. / An adaptive fuzzy system for the control of the vergence angle on a robotic head Fig. 1. The vergence angles in a stereoscopic vision system. imize the stereo fusion index [5, 13, 14]. Furthermore the techniques for the control of the vergence angle are distinguished according to the images that are utilized, i.e. disparity estimation with log-polar images [4, 6, 12, 18] provides better results than with Cartesian ones. In [1] the cortically magnified visual cortex is used to match the entire image. Vergence control is based on the human s ability to rotate their eyes, a neuron-based procedure, mimicked in robotics in [16], where responses of energy neurons are used for vergence control and for disparity estimation. Furthermore, energy neurons have also been applied for the simultaneous gaze and vergence control [17]. In similar applications, the correct vergence angle can be reached by saccading towards the target in the periphery map and subsequently by the cameras to be guided through the visual cortex [11]. Moreover, a dynamic vergence control for direct estimation of the disparity can be achieved with a robust binocular fusion computation [4]. As the visual data is the dominant element for the vergence control, the followed strategy that is interpreted and processed leads to different results. A vergence control strategy based on Hering s law has been adopted for the motor movements by interpreting the visual data [15], with remarkable results. In this paper, a new method is described for the control of the vergence angle. All the previous work has focused mostly on the optimal estimation of the 3D features between the stereo image pairs. We applied the proposed technique on a 4 d.o.f. robotic head [9]. The use of fuzzy logic on this scheme derives from the way the human eyes move towards a target, i.e. they converge when it approaches, and diverge when it draws away. The presented adaptive fuzzy system, is the optimization of the one presented in [11]. It has two inputs, the similarity and the depth, which determine whether the two cameras are aiming at the same target or not. If the two images are very similar it is believed that both cameras are focusing on the same target or scenery. The depth input, was selected due to the fact that it is easily manipulated, it is related with the cameras angles and, finally, it is related to the disparity of the current scene. The similarity method chosen for the vergence control is the zero-mean normalized cross correlation as it has exhibited smoother results and does not require camera calibration [8]. The current vergence angle is read directly on the encoders and the depth is calculated accordingly. The output of the adaptive fuzzy system is used as feedback for the vergence control.

26 N. Kyriakoulis et al. / An adaptive fuzzy system for the control of the vergence angle on a robotic head Mathematical formulation 2.1. Similarity measure The similarity criterion is essential in vergence control. It should be able to provide accurate measures in a fast manner. For that purpose, a fusion index technique has been adopted as it exploits the fact that if a target is correctly verged, the stereo images are very similar. The goal of the vergence control system is to maximize the similarity. The index of the binocular fusion is computed using the zero-mean normalized cross correlation ZNCC = 1 where Z is the depth of the scene, and b is the baseline of the stereo head. vergence is the vergence angle, while right and left are the angle of one of the right and the left camera, respectively. The above formulae are only valid for a parallel camera setup, i.e. zero cyclopean angle. In order to ensure the symmetric operation of the cameras, the two respective independent degrees of freedom are controlled by the same absolute motion commands. The only difference in the movement is the sign, which results into symmetrical verge or diverge. (u,v) W (I r(u, v) I r )(I l (u, v) I l ) (u,v) W (I r(u, v) I r ) 2 (u,v) W (I l(u, v) I l ) 2 (1) where I r and I l are the right and left images, respectively. I r and I l represent the mean values of the right and left images, respectively. The variables u and v are the image coordinates. The region in which the ZNCC is applied is the whole image: (u, v) W. ZNCC is invariant to the illumination changes and its range is normalized into [0 1]. ZNCC was selected as it has the smoother performance, even in the most difficult scenes. This measure is fast, accurate and robust in environmental alterations, as it compasses the correct vergence angle even with extremely low-resolution images, without making any topological rearrangement, such as log-polar mapping Depth estimation The depth is calculated via triangulation of the 3D feature correspondence on the image pair. An appropriate vergence angle leads to high ZNCC values, i.e. either the stereo cameras are directed to the infinity or to the same object. The capacity of knowing the cameras angles provides the better understanding of a captured image pair. Moreover, when the camera angles are known, the vergence angle is calculated simply geometrically (see Fig. 1). The mathematical equations for a symmetrical vergence system are: and Z = b 2tan (2) vergence = right + left = 2 (3) 3. Fuzzy vergence system We applied fuzzy logic to estimate the vergence angle in a simple and effective manner. The necessary data in our approach are the similarity of the two images and the depth of the scene. These two parameters are sufficient for computing the correct vergence angle, as the similarity determines whether the two cameras are aiming at the same target or not. The depth of the scene is easily extracted from the known angles, read from the encoders. From the formulae (2) and (3) it is obvious that the depth can be interpreted by means of the cameras angles, which are the dominant features for the vergence control. In Fig. 2 the block diagram of the proposed vergence control is depicted. The control of the camera angles is performed by a classic scheme. The proposed fuzzy system calculates the feedback control signal. The images from the left and right cameras are used for the computation of the ZNCC and the readings of the encoders of the actuators of the camera angles are used for the computation of the depth (Z). Both ZNCC and Z are then fed into the fuzzy system. The output of the system is the amplitude and the sign of the angle applied to the actuators. The disturbance in the block diagram corresponds to potential noise including light reflections resulting to a global brightness dissimilarity between the two images of the stereo pair. However, the system is still capable to provide correct vergence angles since the ZNCC input utilizes the mean value of the respective images. As a result, the difference between the pixel s intensity and the image s mean value

27 388 N. Kyriakoulis et al. / An adaptive fuzzy system for the control of the vergence angle on a robotic head Fig. 2. Block diagram of the proposed automatic vergence system. (I r (u, v) I r ) is the same under any brightness level. On the other hand, the system is sensitive to cases where the disturbance is caused by an occlusion to one of the cameras, since the main algorithm prerequisite is that the two cameras have to operate simultaneously and to observe the same environment. The design of a fuzzy system is highly depended on the application and the experience of the designer. Therefore, building such a system, relies on previous experience and exhaustive experimentation. This approach was followed also for the proposed system, where five Gaussian membership functions (MFs) are used for both inputs and five trapezoid MFs for the output, as they found to be more appropriate for the desired task. Apart from the type of the MFs, important role to the fuzzy system plays the possible adjustment methods, such as the defuzzification and the aggregation one. In the proposed system the aggregation method was set to sum. Other aggregation methods such as the maximum, the minimum, and the probabilistic OR (probor), were tested, but the sum provided a smoother output value in our tests. The defuzzification method was set to centroid, as it covers the output range more efficiently. The centroid defuzzification method provides the center value between the interacted rules, so in some cases the uttermost values of the output are not fired, reducing so the efficient range. We also assessed other defuzzification methods, namely: bisector, middle of maximum, largest of maximum, and smallest of maximum. The results acquired by the centroid exceeded the ones of the other methods and, thus, the centroid one was selected. Key features to the design of a fuzzy system are the MFs and the rule base. In the fuzzy vergence system, apart from the first input, all MFs are normally distributed to their range. The membership functions are displayed in Fig. 3 and the rules interaction is shown in Table 1. The first input (Fig. 3a) is the similarity measure (ZNCC). The odd distribution of the MFs of the first input aims at distinguishing between the cases with a very high and a very low ZNCC. As the similarity between the two images is the criterion for the correct vergence angle, this kind of distribution highlights any alterations. Furthermore, as the ZNCC values around the correct vergence angle have very small differences, it is very difficult even for humans to identify the best angle. Thus, the arrangement of the MFs with small deviations and almost with no overlaps, enables the system to identify more efficiently the highest similarity. In the cases of high ZNCC, the system remains to the current state or makes small movements around the highest similarity point. On the other hand, in the cases of low ZNCC the system covers big distances until it reaches a point with higher similarity. The reason that three MFs are close to each other at the middle point, is that the ZNCC in most cases varies dramatically in that range. By having a denser distribution of the MFs in that range, instead of distributing them normally, we have

28 N. Kyriakoulis et al. / An adaptive fuzzy system for the control of the vergence angle on a robotic head 389 ZNCC Input Depth Input Very Low Low Average High Very high Ver y Near Near Average Far Very Far 1 1 Degree of membership Degree of membership ZNCC (a) Depth (b) Fuzzy system output Diverge Much Diverge Zero Verge Verge Much 1 Degree of membership Vergence angle (c) Fig. 3. Membership functions of the fuzzy vergence system for (a) input 1, (b) input 2 and (c) output. Table 1 Rule base for the fuzzy kalman system Depth ZNCC Very low Low Average High Very high Very near DM DM D D Z Near DM D D D Z Average DM D D Z Z Far VM V V V Z Very far VM VM V V Z *DM = Diverge much, D = Diverge, Z = Zero, V = Verge, VM = Verge much. increased more the system s efficiency. The second input (Fig. 3b) is the depth and is set to be [0 4] meters, due to the nature of the application in hand. In cases where the environment is uniform, the range can easily be adapted to larger values. The output range was set to [ ] degrees. The minus sign implies the vergence while the plus sign the divergence movement. The fuzzy system is adaptive, as in every step the ranges of both inputs and the output are altered, depending on the values of the variables. The range is [x dx+ d] for all variables, where x is the value of the variable at the current time step, and d is the range coefficient. The d values are 0.1 for the ZNCC input, 1.2 m for the depth (Z) and 10 degrees for the output. The d values are derived from the nature of the application in hand. For the ZNCC input the main criterion is whether the two cameras are fixated to the same target or not. Extensive experiments showed that the 1/10 of the ZNCC range is sufficient for the system to decide efficiently whether to verge or to diverge. Moreover, a small d value of the ZNCC enables the system to respond accurately in cases where the ZNCC is high, i.e. close to the desired vergence angle. For the depth input d is set to 1.2 m in order for the system to seek the correct vergence angle in a wider space. In cases where

29 390 N. Kyriakoulis et al. / An adaptive fuzzy system for the control of the vergence angle on a robotic head the d receives smaller values than 1.2 m, the system is restricted to search in fewer positions for the correct vergence angle resulting to much more iterations than with a higher d value. Finally, the output s range is altered by d = 10 in order to keep the number of iterations as low as possible. It was found that when the output range is reduced below 10 degrees, the d values for the ZNCC and the Z have to be significantly reduced in order for the system to respond accurately, yet increasing the computation time and the iterations needed. For example concerning the ZNCC input, if its value is 0.82 the adaptive range will not be [0 1] but [ ] instead. In the cases where the ZNCC values exceed the maximum predefined range [0 1], there is a logical check, and the range limit is not violated, i.e. if the ZNCC is 0.99 the adaptive range will be [0.89 1]. The aforementioned restriction is applied to the inputs and the output. The adaptive range allows us to create a generic fuzzy system which is mostly dependent on the rule base and not on the variable values. The fuzzy rule base is shown in Table 1. The abbreviations DM, D, Z, V, and VM, are for Diverge Much, Diverge, Zero, Verge (Converge), and Verge Much, respectively. 4. Experimental results The experimental setup is a 4 d.o.f. robotic head shown in Figs 4 and 5. For its movements (pan, tilt and vergence), four respective harmonic drive actuation mechanisms are used controlled by four flexible and compact DSP based PID controllers. The actuators controllers are four respective high-performance intelligent drives, combining motion controller and PLC functionality in a single compact unit. A complete set of high level instructions permit to define and perform complex motion sequences from the host PC. Two distributed computers are connected with the head. One dedicated to its control and the other to run the image processing to extract the control signals [3]. This approach was chosen due to the high data volume for imaging, which unless a dedicated system is devoted, it might late the control of the head. Thus, the system functions are separated into information processing and motion control. Image processing possess a high computational burden and it is recourse demanding. On the other hand, motion control requires a real-time operation system. This demand for high mul A B C D E F Fig. 4. The robotic head used for vergence control.

30 N. Kyriakoulis et al. / An adaptive fuzzy system for the control of the vergence angle on a robotic head 391 Fig. 5. The experimental setup: On the left-hand side the 4-d.o.f. robotic head; in the middle the dedicated PC for the image processing; on the right-hand side the four controllers for each of the d.o.f.; finally in the background the control PC. timedia performance and real-time motion control led us to use a computer with Windows operating system for the image processing and one with RT-Linux for the control tasks. The computers are connected to each other with a high speed network in order to achieve real-time operation. In order to evaluate the performance of the control scenario presented in the previous Section, several tests were executed. The tests include different vergence experiments. In every time step the ZNCC along with the depth were estimated. Extracting the vergence angle and the depth from the encoders, a new angle was calculated by our adaptive fuzzy system, which was read by the PID controllers and finally it was reached by the servo motors. The new vergence angle was read from the motors encoders and the new ZNCC was calculated. The process continues perpetually as described above. The processing time of the fuzzy system in every time step is about 1.5 msec for a Pentium 4 PC, running at 1.8 GHz. In the first experiment a human was put in a fixed position during the whole process. The distance from the robotic head was set randomly and as close to the cyclopean axis as possible. In the second experiment, some objects were put in front of the robotic head to a randomly selected position near to the cyclopean axis. In the last experiment a more difficult scene was selected, i.e. there were not a specific target for the stereo vision system and the system should aim to the infinity without any obstacle intruding directly into the field of view. In all experiments the system started from a random symmetrical vergence configuration and operated until the correct vergence one to be achieved. There was a restriction to the servo motors for not diverging further than the zero angle point. In case of violating the aforementioned restriction the system was programmed to return to its initial position. In the presented experiments the images are non rectified ones. Notwithstanding, the correct vergence angle was approximated after 4 5 iterations of the system. The fused images after several consequent iterations of the fuzzy vergence system are illustrated in Fig. 6. The sequences represent the vergence angles reached before the correct ones (with the highest similarity), which are illustrated for each row at the last subfigure. For further evaluating and examining the adaptive fuzzy vergence system, every measurement was stored. Thus, all the experiments are evaluated not only by their visual results, but also by the record of the actuators encoders, concerning the maximization of the similarity and the time intervals needed. The acquired results of our system were compared with a known and used technique for controlling the vergence angle presented in [4]. The decision of verging or diverging is taken by examining the relation between the current and the previous correlation index values. This approach is based on minimizing the given correlation function; the movement direction is determined by the sign of the subtraction between the previous correlation values and the current ones. If this sign is positive the chosen direction is maintained otherwise it is reversed. The comparative results are shown in Fig. 7. The abscissa expresses the time in seconds, while the ordinate respective ZNCC values. The blue lines correspond to the fuzzy system s steady state output [11], while the black ones to the steady state output of the technique presented in [4]. The red lines represent the computed ZNCC values at each time step before any correction has taken place. Finally, the green lines correspond to the output of the proposed adaptive system. It is obvious that adaptive range has increased the efficiency of the fuzzy system presented in [11]. For demonstration reasons, the go-between ZNCC values are not illustrated, but only the final ones. Thus, for a vergence angle with an initial respective ZNCC, the diagrams in Fig. 7 represent the final state achieved.

31 392 N. Kyriakoulis et al. / An adaptive fuzzy system for the control of the vergence angle on a robotic head (a) (b) (c) Fig. 6. Fused image sequences during the convergence of the system for several cases: (a) a human standing at a fixed position; (b) some objects were put in front of the robotic head and (c) there is absent of a specific target for the stereo vision system. From left to right, is displayed the index of the consequent vergence angles. From Table 1, one can see that when a high ZNCC value is achieved, the system stands still. This is the reason of not reaching the highest possible similarity value. For example, in the Infinity experiment Fig. 7c and f, which had cluttered environment (Fig. 6c), and suffered from noise and occluded scenery, although the ZNCC values are quite high, the system responded accurately (the ZNCC is very high, about 0.92). The demonstrated results of all the experiments can be further improved by filtering the noise and by storing all history positions. In any new measurement a check is made, with the form of a third input, whether the current or the old angle has a higher similarity. Thus, the system does not reach a position with low similarity twice, but instead it moves to a higher similarity angle, reducing the iterations and the time intervals needed. 5. Conclusion A new vergence control, which uses an adaptive fuzzy system, is proposed. The adaptive fuzzy system decides whether to converge or diverge, depending on the given similarity and depth. The results show that the vergence direction was achieved during the whole process, while the amplitude of the exported vergence angle was the one which had provided the maximum similarity. The fuzzy system responded efficiently to all the experiments, considering the noise involved to the image processing. Its fast response renders it appropriate for real-time operation. To conclude fuzzy systems are a promising vergence control method due to their low response times and the high efficiency. Moreover, fuzzy logic incorporates the human way of thinking with simple if then rules and, therefore, it provides better understanding of the vergence control procedure.

32 N. Kyriakoulis et al. / An adaptive fuzzy system for the control of the vergence angle on a robotic head 393 (a) ZNCC Human standing experiment Time (sec) Fuzzy system [4] Old ZNCC ZNCC (b) Fuzzy system [4] Old ZNCC Bottle experiment Time (sec) ZNCC (c) Infinity experiment Fuzzy system [4] Old ZNCC Time (sec) (d) ZNCC Human standing experiment Time (sec) Fuzzy system [4] Old ZNCC Adaptive fuzzy ZNCC (e) Fuzzy system [4] Old ZNCC Adaptive fuzzy Bottle experiment Time (sec) ZNCC (f) Infinity experiment Fuzzy system [4] Old ZNCC Adaptive v fuzzy Time (sec) Fig. 7. The new ZNCC values acquired from the fuzzy system (gray thin lines) in relation with the old (black thin lines) and with the [4] ones (gray dashed lines) for the examined situations. The black bold lines represent the output of the adaptive fuzzy system. Acknowledgements This work was supported by the EC research projects: RESCUER, FP6-IST and AC- ROBOTER, FP6-IST References [1] E. Aho, J. Vanne, T.D. Hamalainen and K. Kuusilinna, Blocklevel parallel processing for scaling evenly divisible images, Circuits and Systems I: Regular Papers, IEEE Transactions on 52(12) (Dec. 2005), [2] T.L. Alvarez, J.L. Semmlow and C. Pedrono, Dynamic assessment of disparity vergence ramps, Computers in Biology and Medicine 37 (2007), [3] A. Amanatiadis, A. Gasteratos, C. Georgoulas, L. Kotoulas and I. Andreadis, Development of a stereo vision system for remotely operated robots: A control and video streaming architecture, in: Virtual Environments, Human-Computer Interfaces and Measurement Systems, VECIMS IEEE Conference on, July 2008, pp [4] A. Bernardino and J. Santos-Victor, Correlation based vergence control using log-polar images, International Symposium on Intelligent Robotic Systems, [5] M.Z. Brown, D. Burschka and G.D. Hager, Advances in computational stereo, IEEE Trans on Pattern Analysis and Machine Intelligence 25(8) (2003), [6] C. Capurro, F. Panerai and G. Sandini, Dynamic vergence using log-polar images, International Journal of Computer Vision 21 (1997), [7] S.-B. Choi, B.-S. Jung, S.-W. Ban, H. Niitsuma and M. Lee, Biologically motivated vergence control system using humanlike selective attention model, Neurocomputing 69(4 6) (2006), [8] D. Chrysostomou and A. Gasteratos, Comparison and evaluation of similarity measures for vergence angle estimation, International Workshop on Robotics in Alpe-Adria-Danube (RAAD2007), [9] A. Gasteratos, Tele-autonomous active stereo-vision head, International Journal of Optomechatronics 2 (2008), [10] B.K. Horn, Robot Vision, MIT press, Cambridge, Massachusetts, USA, [11] N. Kyriakoulis, A. Gasteratos and S.G. Mouroutsos, Fuzzy vergence control for an active binocular vision system, in: Cybernetic Intelligent Systems, CIS th IEEE International Conference on, Sept. 2008, pp [12] R. Manzotti, A. Gasteratos, G. Metta and G. Sandini, Disparity estimation on log-polar images and vergence control, Computer Vision and Image Understing 83(2) (2001),

33 394 N. Kyriakoulis et al. / An adaptive fuzzy system for the control of the vergence angle on a robotic head [13] L. Nalpantidis, G.Ch. Sirakoulis and A. Gasteratos, Review of stereo matching algorithms for 3D vision, International Symposium on Measurement and Control in Robotics, [14] H. Niitsuma and T. Maruyama, Real-time detection of moving objects, Lecture Notes in Computer Science 3203 (2004), [15] J.G. Samarawickrama and S.P. Sabatini, Version and vergence control of a stereo camera head by fitting the movement into the hering s law, in: Computer and Robot Vision, CRV 07. Fourth Canadian Conference on, May 2007, pp [16] W. Stürzl, U. Hoffmann and H.A. Mallot, Vergence control and disparity estimation with energy neurons: Theory and implementation, International Conference on Artificial Neural Networks (ICANN 02), Lecture Notes In Computer Science 2415 (2002), [17] E.K.C. Tsang, S.Y.M. Lam, Y. Meng and B.E. Shi, Robotic gaze and vergence control via disparity energy neurons, in: Cellular Neural Networks and their Applications, CNNA th International Workshop on, July 2008, p. 3. [18] A.X.J. Zhang, A.L.P. Tay and A. Saxena, Vergence control of 2 d.o.f. pan-tilt binocular cameras using a log-polar representation of the visual cortex, in: International Joint Conference on Neural Networks (2006),

34 Journal of Intelligent & Fuzzy Systems 21 (2010) DOI: /IFS IOS Press 379 Statistical summability (C, 1) for sequences of fuzzy real numbers and a Tauberian theorem 1 Y. Altin a,, M. Mursaleen b and H. Altinok a a Department of Mathematics, Firat University, Elazığ, Turkey b Department of Mathematics, Aligarh Muslim University, Aligarh, India Abstract. Statistical convergence for sequences of fuzzy real numbers has been studied by various authors. In this paper we study the concept of statistical summability (C, 1) for fuzzy real numbers which for real numbers was introduced by Moricz [18]. We also construct some interesting examples. AMS Mathematical Subject Classification: 40A30, 40E05, 94D05 Keywords: Fuzzy numbers, statistical convergence, Tauberian theorem 1. Introduction The concepts of fuzzy sets and fuzzy set operations were first introduced by Zadeh [28] and subsequently several authors have discussed various aspects of the theory and applications of fuzzy sets such as fuzzy topological spaces, similarity relations and fuzzy orderings, fuzzy measures of fuzzy events, fuzzy mathematical programming. Matloka [17] introduced bounded and convergent sequences of fuzzy numbers, studied some of their properties and showed that every convergent sequence of fuzzy numbers is bounded. In addition, sequences of fuzzy numbers have been discussed by Nuray and Savas [21], Kwon [16], Altin et al. [1, 2], Altinok et al. [3], Aytar [4], Et et al. [10] and many others. The notion of statistical convergence was introduced by Fast [11] and Schoenberg [24], independently. 1 This research was supported by FUBAP (The Management Union of the Scientific Research Projects of Firat University) when the second author visited Firat University under the Project Number Corresponding author. [email protected] (Y. Altin), [email protected] (M. Mursaleen), hifsialtinok@yahoo. Com (H. Altinok). Over the years and under different names statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory and number theory. Later on it was further investigated from the sequence space point of view and linked with summability theory by Fridy [12], Šalát [22], Aytar [5] and many others. In recent years, generalizations of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on locally compact spaces. Moreover, statistical convergence is closely related to the concept of convergence in probability. The existing literature on statistical convergence appears to have been restricted to real or complex sequences, but in Nanda [20], Kumar et al. [15], Tripathy and Dutta [27], and Et et al. [10] extended the idea to apply to sequences of fuzzy numbers. Fridy and Khan [13, 14], Edely and Mursaleen [9], Chen and Chang [6, 7], Moricz and Orhan [19] and Terán [26] have studied Tauberian theorems in relation to statistical convergence. Subrahmanyam [25] has studied Tauberian theorem for (C, 1) summability of fuzzy numbers /10/$ IOS Press and the authors. All rights reserved

35 380 Y. Altin et al. / Statistical summability (C, 1) for sequences of fuzzy real numbers and a Tauberian theorem The definitions of statistical convergence and strong p Cesàro convergence of a sequence of real numbers were introduced in the literature independently of one another and have followed different lines of development since their first appearance. It turns out, however, that the two definitions can be simply related to one another in general and are equivalent for bounded sequences. The idea of statistical convergence depends on the density of subsets of the set N of natural numbers. The density of a subset E of N is defined by 1 δ(e) = lim χ E (k) provided the limit exists, n n k=1 where χ E is the characteristic function of E. It is clear that any finite subset of N has zero natural density and δ (E c ) = 1 δ (E). A sequence (x k ) is said to be statistically convergent to L if for every ε>0,δ({k N: x k L ε}) = 0. In this case we write S lim x k = L. Let C(R n ) denote the family of all nonempty, compact, convex subsets of R n. If α, β R and A, B C(R n ), then α(a + B) = αa + αb, (αβ)a = α(βa), 1A = A and if α, β 0, then (α + β)a = αa + βa. The distance between A and B is defined by the Hausdorff metric δ (A, B) = max{ sup inf a b, sup inf a b }, a Ab B b Ba A where denotes the usual Euclidean norm in R n. It is well known that (C(R n ),δ ) is a complete metric space. Denote L(R n ) ={u :R n 0, 1], u satisfies (i) (iv) below}, where (i) u is normal, that is, there exists an x 0 R n such that u(x 0 ) = 1; (ii) u is fuzzy convex, that is, for x, y R n and 0 λ 1, u(λx + (1 λ)y) min[u(x),u(y)]; (iii) u is upper semicontinuous; (iv) the closure of {x R n : u(x) > 0}, denoted by [u] 0, is compact. If u L(R n ), then u is called a fuzzy number, and L(R n ) is said to be a fuzzy number space. For 0 <α 1, the α-level set [u] α of u is defined by [u] α ={x R n : u(x) α}. Then from (i) (iv), it follows that the α-level sets [u] α C(R n ). The arithmetic operations summation and substraction on the set of fuzzy numbers can be defined as follows: [ Let u, v L(R),k R and the α-level sets be [u] α = a α 1,b1 α ], [v] α = [ a2 α ],bα 2,α [0, 1]. Then [u + v] α = [ a α 1 + aα 2,bα 1 + bα 2], [u v] α = [ a α 1 bα 2,bα 1 aα 2], [ku] α = k[u] α The set of all real numbers can be embedded in L(R). For a R, â L(R) is defined by { 1, for x = a â (x) = 0, for x /= a. Define, for each 1 q<, 1 d q (u, v) = δ ([u] α, [v] α ) q dα 0 1/q and d (u, v) = sup 0 α 1 δ ([u] α, [v] α ), where δ is the Hausdorff metric. Clearly d (u, v) = lim q d q (u, v) with d q d s if q s [8]. d (cx, cy) = c d (X, Y) d (X + Z, Y + Z ) d (X, Y) + d (Z, Z ). A sequence X = (X k ) of fuzzy numbers is a function X from the set N of all positive integers into L(R n ). Thus, a sequence of fuzzy numbers X is a correspondence from the set of positive integers to a set of fuzzy numbers, i.e., to each positive integer k there corresponds a fuzzy number X(k). It is more common to write X k rather than X(k) and to denote the sequence by (X k ) rather than X. The fuzzy number X k is called the k-th term of the sequence. Let X = (X k ) be a sequence of fuzzy numbers. A sequence X = (X k ) of fuzzy numbers is said to be convergent to the fuzzy number X 0, written as lim k X k = X 0, if for every ε>0 there exists a positive integer k 0 such that d (X k,x 0 ) <εfor k>k 0. The natural density of a set K of positive integers is defined by δ (K) = lim n 1 n {k n : k K}, where {k n : k K} denotes the number of elements of K not exceeding n. We shall be concerned with integer sets having natural density zero [12, 22]. Definition 1.1. If X = (X k ) is a sequence that satisfies a property P for all k except a set of natural density zero,

36 Y. Altin et al. / Statistical summability (C, 1) for sequences of fuzzy real numbers and a Tauberian theorem 381 then we say that X k satisfies P for almost all k and we write by a.a.k. (see [12, 22]). Definition 1.2. Let X = (X k ) be a sequence of fuzzy numbers. Then the sequence X = (X k ) of fuzzy numbers is said to be statistically convergent to fuzzy number X 0 if for every ε>0, 1 lim n n {k n : d (X k,x 0 ) ε} = 0, (i.e. d (X k,x 0 ) < ε a.a.k.) where the vertical bars indicate the number of elements in the enclosed set. In this case we write X k X 0 (S F ) or S F lim X k = X 0. The set of all statistically convergent sequences of fuzzy numbers is denoted by S F (see [21]). Statistical convergence for double sequences of fuzzy numbers were studied in Savas and Mursaleen [23]. Note that lim X n = X 0 implies S F lim X k = X 0 but the converse does not hold (see Example 2.3). Definition 1.3. Let X = (X k ) be a sequence of fuzzy numbers. The sequence X is said to be a strongly Cesàro summable if there is a fuzzy number X 0 such that 1 lim nk=1 n n [(d (X k,x 0 ))] = 0. In this case we write (C, 1) lim X k = X 0. The set of all Cesàro summable sequences of fuzzy numbers is denoted by (C, 1). 2. Statistical summability (C, 1) for fuzzy real numbers The idea of statistical summability (C, 1) for real numbers was introduced by Moricz [18]. A sequence X k (x) = k k + 2 x + 2 2k Statistical convergence implies statistical summability (C, 1) but converse need not be true and Moricz [18] obtained Tauberian condition for the reverse implication. In this section we define statistical summability (C,1) for sequences of fuzzy real numbers. As in case of real numbers it is true for sequences of fuzzy real numbers that convergence statistical convergence statistical summability (C, 1) but converse implications fail which we demonstrate by consructing some interesting examples of fuzzy real numbers. Definition 2.1. A sequence of fuzzy real numbers (X k ) is said to be statistically (C, 1) summable to a fuzzy real number X 0 if the sequence (ˆσ ) j is statistically convergent to X 0 where ˆσ j = 1 j j k=1 X k (the sum is usual addition of fuzzy real numbers through α-level sets); that is (X k ) is statistically (C, 1) summable if for every ε>0, 1 lim { ) } j n : d (ˆσj,X 0 ε = 0. n n Remark 2.2. By the definition of metric d it is easy to see that for sequences of fuzzy numbers (C, 1) summability implies statistical (C, 1) summability (see Example 2.4). Furthermore, convergence implies statistical convergence and statistical convergence implies statistical (C, 1) summability. But converse need not be true (see Example 2.5). Example 2.3. Consider the fuzzy sequence X = (X k ) as follows: [ ] 2k 2 k + 2, for x, 3 k [ 3, 4k + 2 ] k k k + 2 x + 4k + 2 k + 2, for x 0, otherwise if k = n 2 (n = 1, 2, 3,...) X 0 (x), if k/= n 2 x = (x k ) of real numbers is said to be statistically summable (C, 1) to the number L if the sequence of its partial sums σ = (σ n ) is statistically convergent to L, where σ n := 1 n x k. k=1 where x 8, for x [8, 9] X 0 (x) = x + 10, for x [9, 10] 0, otherwise Thus, the sequence X = (X k ) is both bounded and statistical convergent to fuzzy number X 0, but this sequence is not convergent (Fig. 1).

37 382 Y. Altin et al. / Statistical summability (C, 1) for sequences of fuzzy real numbers and a Tauberian theorem 1 X1 X0 X9 X4 0 3/2 16/9 3 38/9 9/ Fig. 1. A sequence which is statistical convergent, but not convergent. Example 2.4. Consider the sequence (X k ) of fuzzy numbers as follows: kx + 1, 1 k x 0 X k (x) = kx + 1, 0 <x 1, for k = n 2 k 0, otherwise ˆ0, otherwise Then, for α (0, 1), the α-level set of this sequence is [ α 1 [X k ] α, 1 α ], for k = n 2 = k k. [0, 0], otherwise It is easy to see that (X k ) is (C, 1) summable to ˆ0 and hence statistically (C, 1) summable to ˆ0. Example 2.5. Consider the sequence (X k ) of fuzzy numbers as follows: 2kx + 1, 1 2k x 0 2kx + 1, 0 x 1 2k 0, otherwise X k (x) = kx 1 k + 1, 1 k x 0 k kx 1 k + 1, 0 x k 1 k 0, otherwise, for k = n 2, for k = n 2 1 ˆ0, otherwise We can construct the α-level sets of this sequence such that [ α 1 2k, 1 α ], for k = n 2 2k [X k ] α [ = 1 k (1 α), k 1 ] (1 α), for k = n 2 1 k k [0, 0] otherwise It is easy to see that the sequence (X k ) is statistically convergent to zero, but not convergent. Moreover, this sequence is (C, 1) summable to ˆ0 and hence statistically (C, 1) summable to ˆ0. 3. Tauberian theorem In this section we find a condition (Tauberian) on (X k ) so that statistical (C, 1) summability implies convergence. First we define the following: Definition 3.1. A sequence of fuzzy numbers (X n ) is said to be slowly oscillating if for every ε>0 there exists λ = λ (ε) > 1 (close to 1) such that limsup max d (X k,x n ) ε (1) n n<k λn or equivalently lim d ( ) Xmj,X nj = ˆ0 (2) j whenever n j, 1 < m j 1 as j. (3) n j Lemma 3.2. If d ( Xk, ˆ0 ) = O(1/k), then d ( ˆσn, ˆ0 ) = O(1/n), where ˆσ n = 1 nk=1 X k. ( Proof. d ˆσn, ˆ0 ) = d (ˆσn ˆσ n 1, ˆ0 ) ( = d 1n nk=1 X k n 1 1 ) n 1 k=1 X k, ˆ0. Therefore ( nd ˆσn, ˆ0 ) ( ) = d X k n n 1 X k, ˆ0 n 1 k=1 = ( 1 (n 1) d (n 1) k=1 ) n 1 X k n X k, ˆ0 k=1 k=1 1 = (n 1) d (j 1) X j, ˆ0 j=2

38 Y. Altin et al. / Statistical summability (C, 1) for sequences of fuzzy real numbers and a Tauberian theorem ( (j 1) d Xj, ˆ0 ) (n 1) j=2 1 = O (1) = O (1). (n 1) j=2 ( Hence d ˆσn, ˆ0 ) = O(1/n). The following Lemma is a generalization of the above lemma. Lemma 3.3. If (X k ) is slowly oscillating, then (ˆσ n ) is also slowly oscillating. Proof. We have following inequality by making use of Theorem 2 of Moricz [18] d (ˆσ k, ˆσ n ) k n ( ) d Xn,X j kn + 1 k k ( ) d Xj,X n. j=n+1 Hence it follows easily that (ˆσ n ) is slowly oscillating. ( Remark 3.4. Note that d Xk, ˆ0 ) = O(1/k) implies ( that (X k ) is slowly oscillating. Similarly d ˆσn, ˆ0 ) = O(1/n) implies that (ˆσ n ) is slowly oscillating. Hence we have ( Lemma 3.5. If d Xk, ˆ0 ) = O(1/k) then (ˆσ n ) is slowly oscillating. Theorem 3.6. A sequence (X k ) of fuzzy real numbers is statistically convergent to X 0 if and only if there exists a set K = {k i } N such that δ (K) = 1 and lim i X ki = X 0. Proof. Let (X k ) be statistically convergent to X 0. Then for ε>0 the set K ε := {k n : d (X k,x 0 ) ε} has natural density zero and hence its complement M ε := { ( ) } j n : d Xj,X 0 <ε has natural density 1, i.e. δ (M ε ) = δ (N) δ (K ε ) = 1 0 = 1 and K ε M ε =?. Now, we have to show that ( ) X j,j Mε is convergent to X 0. Suppose that ( ) X j is not convergent ( ) to X0. Then there exists ε 0 > 0 such that d Xj,X 0 ε0 for infinitely many terms. Let M ε :={j M ε : d (X j,x 0 ) ε}. Clearly? = M ε M ε. Then for all k and ε 0,wehave K ε :={k n : d (X k,x 0 ) ε 0 } {j n : d (X j,x 0 ) ε 0 }. Thus δ(m ε ) = 0, i.e. M ε K ε. Now for ε<ε 0, K ε K ε, which contradicts the fact that K ε M ε =?. Hence (X i ) is convergent to X 0. Conversely, suppose that there exists a subset K = {K i } N such that δ (K) = 1 and lim j X kj = X 0, i.e. there exists N N = X 0 such that for every ε>0, ( ) d Xkj,X 0 <ε, i N. Now, {k n : d (X k,x 0 ) ε} N {k N+1,k N+2,...}. Therefore δ ({k n : d (X k,x 0 ) ε}) = 1 1 = 0. Hence (X k ) is statistically convergent to X 0. Remark 3.7. If st lim k X k = X 0, then there exists a sequence (Y k ) such that lim k Y k = Y 0 and δ ({k n : Y k = X k }) = 1, i.e. 1 lim n n {k n : Y k /= X k } = 0. (4) Theorem 3.8. Let a sequence (X k ) of fuzzy real numbers be statistically (C, 1) summable to some X 0 and slowly oscillating, then (X k ) converges to X 0. Proof. By Lemma 3.3, we have that (ˆσ n ) is slowly oscillating. Now, let us start by setting n:=l m in (3.4), where 1 l 1 <l 2 <... is sequence of those indices k for which X k = Y k a.a.k. Therefore we have 1 {k l m : Y k = X k } = m 1 (m ), l m l m and consequently l m+1 lim = 1. (5) m l m By the definition of the subsequence (l m ), we have lim X l m m = lim Y l m m = X 0. (6) Since (ˆσ n ) is slowly oscillating, we have by (3.1) that for every ε>0 there exists λ = λ (ε) > 1 such that limsup max d (ˆσ k, ˆσ n ) ε, n n<k λn which together with (3.5) gives limsup max d (ˆσ k, ˆσ n ) ε. m l m <k<l m+1 Since ε is arbitrary, it follows that lim max ) d (ˆσk, ˆσ lm = ˆ0. (7) l m <k<l m+1 m

39 384 Y. Altin et al. / Statistical summability (C, 1) for sequences of fuzzy real numbers and a Tauberian theorem Combining (3.6) and (3.7) implies that the whole sequence (ˆσ k ) is convergent to X 0, i.e. (X k ) is (C, 1) summable to X 0, therefore the convergence of (X k ) follows from the Schmidt classical theorem [18]. Acknowledgement The present paper was completed when Professor Mursaleen visited Firat University (May June, 2007). The author is very much grateful to the Firat University for providing him hospitalities. References [1] Y. Altin, M. Et and R. Çolak, Lacunary statistical and lacunary strongly convergence of generalized difference sequences of fuzzy numbers, Comput Math Appl 52(6 7) (2006), [2] Y. Altin, M. Et and M. Basarir, On some generalized difference sequences of fuzzy numbers, Kuwait J Sci Engrg 34(1A) (2007), [3] H. Altinok, R. Çolak, and M. Et, -Difference sequence spaces of fuzzy numbers, Fuzzy Sets and Systems 160(21) (2009), [4] S. Aytar, Statistical limit points of sequences of fuzzy numbers, Inform Sci 165 (2004), [5] S. Aytar, Rough statistical convergence, Numerical Functional Analysis and Optimization 29(3 4) (2008), [6] C.P. Chen and C.T. Chang, Tauberian conditions under which the original convergence of double sequences follows from the statistical convergence of their weighted means, J Math Anal Appl 332(2) (2007), [7] C.P. Chen and C.T. Chang, Tauberian theorems in the statistical sense for the weighted means of double sequences, Taiwanese J Math 11(5) (2007), [8] P. Diamond and P. Kloeden, Metric spaces of fuzzy sets, Fuzzy Sets and Systems 35(2) (1990), [9] O.H.H. Edely and M. Mursaleen, Tauberian theorems for statistically convergent double sequences, Inform Sci 176(7) (2006), [10] M. Et, H. Altinok and R. Çolak, On -statistical convergence of difference sequences of fuzzy numbers, Inform Sci 176(15) (2006), [11] H. Fast, Sur la convergence statistique, Colloq Math 2 (1951), [12] J.A. Fridy, On statistical convergence, Analysis 5(4) (1985), [13] J.A. Fridy and M.K. Khan, Statistical extensions of some classical Tauberian theorems, Proc Amer Math Soc 128(8) (2000), [14] J.A. Fridy and M.K. Khan, Statistical gap Tauberian theorems in metric spaces, J Math Anal Appl 282(2) (2003), [15] V. Kumar, A. Sharma, K. Kumar and N. Singh, On I-limit points and I-cluster points of sequences of fuzzy numbers, Int Math Forum 2(57 60) (2007), [16] J.S. Kwon, On statistical and P Cesàro convergence of fuzzy numbers, Korean J Comput Appl Math 7(1) (2000), [17] M. Matloka, Sequences of fuzzy numbers, BUSEFAL 28 (1986), [18] F. Móricz, Ordinary convergence follows from statistical summability (C, 1) in the case of slowly decreasing or oscillating sequences, Colloq Math 99(2) (2004), [19] F. Móricz and C. Orhan, Tauberian conditions under which statistical convergence follows from statistical summability by weighted means, Studia Sci Math Hungarica 41 (2004), [20] S. Nanda, On sequences of fuzzy numbers, Fuzzy Sets and Systems 33 (1989), [21] F. Nuray and E. Savas, Statistical convergence of fuzzy numbers, Math Slovaca 45(3) (1995), [22] T. Šalát, On statistically convergent sequences of real numbers, Math Slovaca 30(2) (1980), [23] E. Savas and M. Mursaleen, On statistically convergent double sequences of fuzzy numbers, Inform Sci 162(3 4) (2004), [24] I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer Math Monthly 66 (1959), [25] P. V. Subrahmanyam, Cesàro summability for fuzzy real numbers, J Anal 7 (1999), [26] P. Terán, A reduction principle for obtaining Tauberian theorems for statistical convergence in metric spaces, Bull Belg Math Soc Simon Stevin 12(2) (2005), [27] B.C. Tripathy and A.J. Dutta, On fuzzy real-valued double sequence space 2 l p F, Mathematical and Computer Modelling 46 (2007) [28] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965),