Computing the weights of criteria with interval-valued fuzzy sets for MCDM problems Chen-Tung Chen 1, Kuan-Hung Lin 2, Hui-Ling Cheng 3

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1 Compting the weights of criteria with interval-valed fzzy sets for MCDM problems Chen-ng Chen Kan-Hng in 2 Hi-ing Cheng 3 Department of Information Management National nited niversity aiwan 2 Specialist 2 Qanta Compter Inc 2 en Hwa 2nd Road Keishan aoyan aiwan 3 Coege of General Edcation Hng Kang niversity aiwan Abstract In the real world many inflenced factors or criteria shold be considered in the mltiple criteria decisionmaking (MCDM) process Face to the ncertain environment the decision makers or experts always cannot express their opinions exactly in the decision-making process nder this sitation interval-valed fzzy sets are sitable sed to represent the sbective dgments of decision makers In this paper the fzzy AHP method is applied to compte the fzzy weights of each criterion based on the interval-valed fzzy sets he proposed method can provide a more flexible way for decision makers to express their sbective opinions A systematic method is presented here to compte the fzzy weights of criteria for dealing with a MCDM problem An example is presented to istrate the comptational procedre of the proposed method Conclsions and ftre research direction wi be discssed at the end of this paper Keywords MCDM Interval-valed fzzy set Fzzy AHP I INRODCION Decision making is one of the most complex administrative processes in management area (Ashtiani et al 2009) In some sitations becase of time pressre lack of knowledge and the decision maker s limited attention and information processing capabilities (X 2006) the decision maker cannot make the optimal decision easily In fact both qalitative and qantitative criteria shold be considered simltaneosly (Erol and Ferre Jr 2003) in the decision making process It cas mltiple criteria decision making (MCDM) process Saaty (980) proposed analytic hierarchy process (AHP) method for determining the weights of criteria in a MCDM problem by a hierarchical strctre and systematization According to the pairwise comparison reslt the weight of each criterion can be compted for a hierarchical problem strctre De to the sbective dgments of the decision makers are often ncertain in determining the pairwise comparisons the fzzy AHP are proposed to overcome the drawback (Bckley 985; Bckley et al 200) he fzzy AHP aowed the experts to se fzzy ratios in place of exact ratios to make the pairwise comparisons Based on the concepts there are varios fzzy AHP methods have been developed sch as the geometric mean method (Bckley 985) the ambda-max method (Cstora and Bckley 200) fzzy logarithmic least sqares method (ang et al 2006) the linear goal programming method (Jng 20) and the logarithmic fzzy preference programming method (Rezaei et al 203) However it is often difficlt for experts to exactly qantify their opinions as a crisp nmber in decision making process herefore the reasonable way is to represent the degree of certainty by an interval (Ashtiani et al 2009; Vahdani et al 200) he membership vale express as an interval vale can more fit the real world sitations Interval-valed fzzy nmbers have been applied in many fields sch as R&D leader selection (Ashtiani et al 2009) risk analysis (ei and Chen 2009) enterprise partner selection (Ye 200) traditional spplier selection (Chen 20) bs enterprise performance evalation (Ko and iang 202) etc herefore interval-valed fzzy sets are sitable sed to represent the fzziness of membership vale of sbective opinions In this paper a novel method is proposed by combining interval-valed fzzy set with fzzy AHP to determine the fzzy weights of each criterion he organization of this paper is described as foows In section 2 we wi present some notations of basic definitions and operators sch as trianglar fzzy nmber α-ct of fzzy nmber defzzified method and interval valed fzzy sets he proposed method wi be discssed at the section 3 For istrating the proposed method clearly an example wi be implemented Finay the conclsion and ftre research are smmarized at the end of this paper II 2 rianglar fzzy nmber FZZY NMBERS AND INGISIC VARIABES rianglar fzzy nmber (FN) can be defined as l m ) where l m hen l > 0 then is a positive FN Page 5

2 (PFN) (Zimmerman 99; Chen 2000) he membership fnction of positive FN can be defined as foow x l l x m m l x ( x) m x () m 0 otherwis he α-ct is sed to transform a fzzy set into crisp interval (Zimmerman 99) he α-ct can define as foow [( m l) l ( m) ] 0 (2) In a short it can be represented as [ l ] If is a trianglar fzzy nmber the area measrement method can be defined as (Yager 98) I( ) ( I ( ) I ( ))/ 2 (3) R where the I ( ) represents the area bonded by the left-shape fnction of the area inclde the x-axis the y-axis the horizontal line α= and the line ( x) he I R ( ) represents the area bonded by the right-shape fnction the area inclde the x-axis the y-axis the horizontal line by α= and the line R ( x) 22 Interval-valed fzzy set he definition of interval-valed fzzy set can be described as foow (Ashtiani et al 2009; Chen 202) {( x[ ( x) ( x)])} : X [0] x X (4) Yao and in (2002) presented the interval-valed trianglar fzzy nmber [ ] [( m ; )( l m ; )] (shown as Fig ) where the membership and the denotes the lower limit degree of membership and the denotes the pper limit degree of ( x) 0 l m FIG INERVA-VAED RIANGAR FZZY NMBER In order to express the fzziness more exactly Ashtiani et al (2009) and Vahdani et al (200) sed trianglar intervalvaled fzzy nmbers to represent the opinions of decision-makers In many cases an expert cannot exactly express the opinion by a crisp membership vale nder this sitation the interval-valed fzzy nmber can express the ncertain opinion as an interval vale he membership vale of x can be extended as [ ( x) ( x)] In this paper we sppose that the lower limit degree of membership and pper limit degree of membership at m are eqal to It represents a normal interval-valed trianglar fzzy nmber (IFVNs) Page 52

3 Assme that there are two interval-valed trianglar fzzy nmbers and 2 where ([ ] [ ] and 2 ([ 2 l2] m2[ 2 2] the arithmetic operations can be calclated as foows (Ko and iang 202) 2 ([ 2 l2] m2[ 2 2]) (5) 2 ([ 2 2] m2[ l2 2]) (6) 2 ([ 2 l2] m2[ 2 2]) (7) 2 ([ 2 2] m2[ l2 2]) (8) III PROPOSED MEHOD Based on interval-valed fzzy sets the ambda-max method (Cstora and Bckley 200) is applied to compte the fzzy weight of each criterion he comptational procedre is shown as foows Step Establish hierarchical strctre Establish a hierarchical framework of decision-making problem ithin the hierarchical framework there may contain several criterion and sb-criterion Step 2 he expert ses the interval-valed lingistic variable to make the pairwise comparison in the criteria and sb-criteria levels Step 3 sing interval-valed fzzy AHP the weights of criteria and sb-criteria can be compted and expressed as intervalvaled fzzy nmbers he comptational procedre of calclation steps as foow ) Constrct the fzzy positive reciprocal matrix he experts are aowed to se interval-valed fzzy ratios (see able ) in place of exact ratios he i i can now be interval-valed fzzy nmbers in any positive reciprocal matrix he fzzy positive reciprocal matrix is defined as [ i ] nn where i i i i 2 n If i ([ l] m[ ]) then ([ ] m [ l i ]) ABE INGISIC VARIABE FOR IMPORANCE Symbol ingistic variables Interval-vale fzzy nmber Extremely Eqal Importance ([] []) Eqal Importance ([] [23]) Intermediate vales ([] 2 [34]) eak Importance ([2] 3 [45]) Intermediate vales ([23] 4 [56]) Essential Importance ([34] 5 [67]) Intermediate vales ([45] 6 [78]) Very Strong Importance ([56] 7 [89]) Intermediate vales ([67] 8 [99]) Absolte Importance ([78] 9 [99]) i Page 53

4 2) Compte the interval-valed fzzy weights of criteria According to the ambda-max method (Cstora and Bckley 200) a seqence of fzzy positive reciprocal matrix is applied to determine the relative weights of each criterion he procedre of a calclation steps as foow a et α = sing α-ct to obtain m t im nn matrix and determine the fzzy weight b et α = 0 sing α-ct to obtain the n n it represent the decision-makers give the crisp positive reciprocal for i 2 n m 0 t i m im 0 t i nn 0 t i nn and 0 l t il n n accordance with the fzzy positive reciprocal matrix Next compting the fzzy weights l and where i l il i and i i 2 n c In order to make sre the fzzy weights are sti normal fzzy sets the foowing formla is sed to adst these im weights First we can determine lower bonds trianglar fzzy weights as Ql min{ i n} and Q im max{ i n} i il in he constants Q l and Q are sed to calclate the new weight as il Q l il and i Q i Second we se the new fzzy weight of lower limit and pper limit to determine the pper bonds trianglar fzzy weights il i as Q min{ i n} and Q max{ i n} i hen the constants and i Q i Q and i Q are sed to determine the new pper bonds trianglar fzzy weights as i Q i i l il i i Finay we can get the new fzzy weights [ ] [ ] [ ] and [ ] for i=2 n d o integrate l m and i i il im i i we can obtain the positive trianglar fzzy weight matrix for decisionmakers he fzzy weights of i-th criterion can be represented as ([ ] [ ]) e Defzzy the interval-valed fzzy weights Applying the area measrement method (Yager 98) to transfer the interval-valed fzzy nmber into interval vale as min max [ ] where represents the interval weight of criterion the lower limit of the interval and max min 2 m represents the pper limit of the interval 4 2 m l represents the 4 3) Repeating the process of step (B) the weights of sb-criteria can be compted with respect to each criterion IV NMERICA EXAMPE Sppose that a company desires to select an information system to serve their cstomer A committee of three decisionmakers (or experts) (p p 2 p 3 ) has been formed to evalate the importance of each criterion After the reviewing by the committee three criteria and nine sb-criteria (C C 2 C 33 ) are considered to select the sitable vendor he comptation process of proposed method is shown as foows Step According to the description the hierarchical strctre of problem is shown as Fig2 Step 2 Each expert ses the interval-vale lingistic variables to make the pairwise comparisons among criteria as able 2 Page 54

5 Sitable vendor Cost C Secrity C 2 Software C Hareware C 2 abor C 3 Control C 2 Privacy C 22 Reliability C 23 Operatio n C 3 Scalability C 3 Interoperability C 32 Performance C 33 P P 2 P 3 FIG 2 HE EVAAION SRCRE ABE 2 PAIRISE COMPARISONS OF HREE EXPERS C C 2 C 3 C C 2 C 3 C 2 C 22 C 23 C 3 C 32 C 33 C C - - C 2 - C 3 - C C 2 - C C 32 C 3 - C 3 C 23 C C C 2 C 3 C C 2 C 3 C 2 C 22 C 23 C 3 C 32 C 33 C C - - C 2 - C 3 C 2 - C 2 - C 22 C 32 - C C 3 C C C C 2 C 3 C C 2 C 3 C 2 C 22 C 23 C 3 C 32 C 33 C - - C C 2 C C 2 - C 2 - C 22 - C 32 - C 3 C C C 33 Step 3 Compte the interval-valed fzzy weights of criteria i he fzzy positive reciprocal matrix is shown as able 2 ii According to the fzzy reciprocal matrix for three criteria the criteria weights of the expert P can be compted as foows 2 4 () et α = sing α-ct to obtain m / 2 and determine the fzzy weight as m [ ] / 4 (2) et α = 0 sing α-ct to obtain the positive reciprocal matrix as / / / 4 / 3 / l / 3 / 2 / 5 6 Next we can compte the fzzy weights of each matrix as 0 [ ] l 0 [ ] 0 [ ] 0 [ ] (3) In order to make sre the weights are sti normal fzzy sets first we adst lower bonds trianglar fzzy weights and find the constants Q l =0849 and Q =056 And then obtain the new weights l [ ] and Page 55

6 [ ] Next we adst pper bonds trianglar fzzy weights and find the constants Q =0763 and Q =236 hen obtain the new weights [ ] and [ ] l (4) o combine the weight matrix [ ] [ ] we can obtain the interval-valed fzzy weight of expert P for three criteria as ([ ]0584[ ]) m 2 ([084084]084[ ]) 3 ([044073]0232[ ]) (5) Integrate the interval-valed fzzy weights of three experts; the area measrement method is applied to transform the interval-valed fzzy set into interval vale he weights of criteria can be compted as =[ ] 2=[ ] and 3 =[ ] iii he comptational procedres of sb-criteria weights are istrated as step B and the reslt can be shown as able 3 ABE 3 HE EIGHS OF SB-CRIERIA Sb-criteria eights Sb-criteria eights Sb-criteria eights C [ ] C 2 [ ] C 3 [ ] C 2 [ ] C 22 [ ] C 32 [ ] C 3 [ ] C 23 [ ] C 33 [ ] V CONCSIONS AND FRE RESEARCH In general many qantitative and qalitative factors wi inflence the expert to make a decision in the decision-making process nder this sitation decision makers always difficlt to express their opinions by the membership vale exactly herefore the interval-valed fzzy sets are sitable to express the sbective opinion of each decision maker in a decision environment In this paper the fzzy AHP method is extended to compte the fzzy weights of a criteria based on the interval-vale fzzy sets Applying the proposed method a systematic way is presented here to compte the fzzy weights of criteria In the ftre a decision analysis system wi be designed and developed based on this proposed method to redce the comptational time and improve the decision-making qality ACKNOEDGEMEN his work was spported partiay by Ministry of Science and echnology aiwan he proect No is MOS H MY2 REFERENCES [] B Ashtiani F Haghighirad A Maki and G A Montazer Extension of fzzy OPSIS method based on interval-valed fzzy sets Applied Soft Compting Vol pp [2] J J Bckley Fzzy hierarchical analysis Fzzy Sets and Systems Vo7 985 pp [3] J JBckley Fering and Y Hayashi Fzzy hierarchical analysis revisited Fzzy Sets and Systems Vo pp [4] C Chen Extensions of the OPSIS for grop decision-making nder fzzy environment Fzzy sets and systems Vo4 No 2000 pp-9 [5] Y Chen Optimistic and pessimistic decision making with dissonance redction sing interval-valed fzzy sets Information Sciences Vo8 No3 20 pp [6] Y Chen Comparative analysis of SA and OPSIS based on interval-valed fzzy sets: Discssions on score fnctions and weight constraints Expert Systems with Applications Vol39 No2 202 pp [7] R Cstora and J J Bckley Fzzy hierarchical analysis: the ambda-max method Fzzy Sets and Systems Vo pp8-95 [8] I Erol and Jr G Ferre A methodology for selection problems with mltiple conflicting obectives and both qalitative and qantitative criteria International Jornal of Prodction Economics Vol86 No pp87-99 [9] H Jng A fzzy AHP GP approach for integrated prodction-planning considering manfactring partners Expert systems with Applications Vol38 No5 20 pp Page 56

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