Measuring Inconsistency of Pair-wise Comparison Matrix with Fuzzy Elements

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1 International Journal of Operations Research International Journal of Operations Research Vol., No., 8 (3) Measuring Inconsistency of Pair-wise Comparison Matri with Fuzzy Elements Jaroslav Ramík an Petr Korviny School of Business Aministration in Karviná, Silesian niversity in Opava niversity Square 934/3, Karviná, Czech Republic. Receive February 3; Revise April 3; Accepte May 3 Abstract This paper eals with measuring inconsistency an incompatibility of pair-wise comparison matri with fuzzy elements. Here we eal with some properties of such pair-wise comparisons, particularly, reciprocity consistency an compatibility. Moreover, we show how to measure it efining two new inices. The first ine of inconsistency is base on the classical concept of consistency, however, instea of principle eigenvector metho use in AHP we apply the logarithmic least squares metho. Defining the secon ine of incompatibility we look for a consistent matri in the form of the ratio matri with the maimal membership grae as close as possible to the original fuzzy matri. This leas to solving a nonlinear optimization problem which can be transforme to a sequence of P ones. We compare properties an application areas of both inices. The first ine FI is suitable for non-interactive elements of fuzzy matrices, particularly, when uncertainty of iniviual elements of the matri can be reflecte/measure. The secon ine GI is appropriate for interactive elements of fuzzy matrices. By GI a measure of compatibility of fuzzy matri with the closest consistent matri is epresse. Illustrating eamples an simulations are supplie to characterize the concepts an erive properties. Keywors ata analysis; ecision making; uncertainty; pair-wise comparison; inconsistency; incompatibility. INTRODCTION Pair-wise comparison is a popular metho for solving DM problems of fining the best alternative among more than ones. This metho is frequently use when ranking alternatives, evaluating the relative importance of the iniviual criteria in MCDM problem an/or when evaluating alternatives accoring to qualitative criteria, e.g. esign, taste, likeness, etc. The core of the metho is how to aggregate the results into the final prioritization or ranking. In this paper we eal with some properties of such pair-wise comparisons, particularly with consistency of pair-wise comparisons. Consistency means that if an element i is a-times better than element an element is b-times better than element k, then i is a.b-times better than k. If this property is vali for all compare elements i, an k, we say that the matri of pair-wise comparisons is consistent. If, at least for one triple of elements, the property is not satisfie, then the matri is inconsistent. It is important to measure intensity of inconsistency as in some cases the pair-wise comparison matri can be close to a consistent matri, in the other ones inconsistency can be strong, meaning that there eist numerous inconsistent triples eventually with large ifferences between corresponing values. For crisp pair-wise comparison matrices there eists numerous inconsistency measurement methos (inices), see e.g. Aguaron (3), Koczkoa (993), Gass (4), Stein (7), an others. However, in these papers it was proven that all the consistency inices are linear or nonlinear transformations of the original Saaty s consistency ratio (Saaty, 99). Moreover, these inices cannot be irectly use for measuring consistency of a matri with fuzzy elements. The earliest work in AHP using fuzzy sets as ata was publishe by van aarhoven an Perycz (983). They compare fuzzy ratios escribe by triangular membership functions. The metho of logarithmic least squares was use to erive local fuzzy priorities. ater on, using a geometric mean, Buckley et al. (985) etermine fuzzy priorities of comparison ratios whose membership functions were assume trapezoial. The issue of consistency in AHP using fuzzy sets as elements of the matri was first tackle by Salo (996). Departing from the fuzzy arithmetic approach, fuzzy weights using an auiliary mathematical programming formulation escribing relative fuzzy ratios as constraints on the membership values of local priorities were erive. ater on eung an Cao () propose a notion of tolerance eviation of fuzzy relative importance that is strongly relate to Saaty s consistency ratio. Instea of principal eigenvector metho use in classical AHP, here, we use the logarithmic least squares metho, see Van aarhoven, (983) an Ramik (). The first ine inconsistency ine is base on the logarithmic least squares Corresponing author s ramik@opf.slu.cz 83-73X Copyright ORSTW

2 Ramik an Korviny: Measuring inconsistency of pairwise comparison matri with fuzzy elements function. Defining the secon ine incompatibility ine - we look for a consistent crisp matri in the form of the ratio matri with the maimal membership grae as close as possible to the original fuzzy matri. This leas to solving a nonlinear optimization problem which can be transforme to a sequence of P ones. Recently, in the literature we can fin papers ealing with applications of pair-wise comparison metho where evaluations are performe by fuzzy values, particularly, evaluating regional proects, web pages, e-commerce proposals etc. This paper is organize as follows. In Section the basic concept of Triangular Fuzzy Positive Reciprocal (TFPR) matri is introuce. In Section 3, a general concept of inconsistency ine FI of TFPR matri base on the metric function is efine, in Section 4, the fuzzy logarithmic SQ metho is introuce an applie. Section 5, eals with the incompatibility ine GI of TFPR matri. Finally, in Section 6, illustrating eamples an simulation eperiments are supplie in orer to characterize the concepts an erive properties. Finally, in Section 7, concluing remarks of the paper are presente.. TRIANGAR FZZY POSITIVE RECIPROCA MATRIX Triangular fuzzy numbers are suitable for moeling uncertain values by DMs in practice. A triangular fuzzy number a can be equivalently epresse by a triple of real numbers, i.e. a = ( a ; a ; a ), where a is the ower number, a M is the Mile number, an a is the pper number, a a M a. If a = a M = a, then a is sai to be the crisp number (non-fuzzy number). Eviently, the set of all crisp numbers is isomorphic to the set of real numbers. If a a M a, then the membership function of is suppose to be continuous, strictly increasing in the interval [a, a M ] an strictly ecreasing in [a M, a ]. Moreover, the membership grae is equal to zero for [a, a ] an equal to one for = a M. As usual, the membership function is assume to be piece-wise linear, see Figure, where the evaluation moerately more important is epresse by the triangular fuzzy number on the scale S = [ / 9, 9]. If a = a M an/or a M = a, then the membership function is iscontinuous. It is well known that the arithmetic operations +, -, * an / can be etene to fuzzy numbers by the Etension principle, see e.g. Buckley (). The elements of Triangular Fuzzy Positive Reciprocal (TFPR) matri are positive triangular fuzzy numbers ~ M a a ( a ; a ; a ), where an é (; ; ) ( a ; a ; a ) ( a ; a ; a ) ù n n n ( ; ; ) (; ; ) ( a ; a ; a ) M n n n a a a A =. () ( ; ; ) ( ; ; ) (; ; ) M M êë a a a a a a ú n n n n n n û Arithmetic operations with fuzzy numbers are efine as follows, see Buckley (986) or Ramik (). et ~ ~ i a ( a ; a ; a ) an b ( b ; b ; b ), where a, b, be positive triangular fuzzy numbers. We efine the following arithmetic operations: Aition: a + b = ( a + b ; a M + b M ; a + b ); Subtraction: a - b = ( a -b ; a M -b M ; a -b ); Multiplication: M M a * b = ( a * b ; a * b ; a * b ); Division: M M a / b = ( a / b ; a / b ; a / b ). Particularly: æ ö = ; ; M a ç çèa a a ø

3 Ramik an Korviny: Measuring inconsistency of pairwise comparison matri with fuzzy elements It shoul be note, that for triangular fuzzy numbers with piece-wise linear membership functions the above formulae for multiplication (an also for ivision) are not obtaine by the Etension principle. They are only approimate ones, e.g. multiplication is a triangular fuzzy number with piece-wise linear membership function, whereas the membership function of the eact operation efine by the Etension principle woul be non-linear. For using matrices with triangular fuzzy elements there eist at least following reasons: The membership function of triangular fuzzy elements is piece-wise linear, i.e. it is easy to unerstan. Triangular fuzzy numbers can be easily manipulate, e.g. ae, multiplie. Crisp (non-fuzzy) numbers are special cases of triangular fuzzy numbers. The TFRP matri can be consiere by the DM as a moel for his/her fuzzy pair-wise preference representations concerning n elements (e.g. alternatives). In this moel, it is assume that only n(n-)/ ugments are neee, the rest is given by reciprocity conition. In practice, when interval-value matrices are employe, the DM often gives ranges narrower than his or her actual perception woul authorize, because he/she might be afrai of epressing information which is too imprecise. On the other han, triangular fuzzy numbers epress rich information because the DM provies both the support set of the fuzzy number as the range that the DM believes to surely contain the unknown ratio of relative importance, an the graes of possibility of occurrence (i.e. membership function) within this range. Triangular fuzzy numbers are appropriate in group ecision making where a can be interprete as the minimum possible value of DMs ugments, a is interprete as the maimum possible value of DMs ugments, an a M the geometric mean of the DMs ugments is interprete as the mean value, or, the most possible value of DMs ugments, see Buckley (985).. Figure. Elements of Triangular Fuzzy Positive Reciprocal (TFRP) matri 3. INCONSISTENCY INDEX FI OF TFPR MATRIX Construction of a new measure - inconsistency ine of the reciprocal matri with triangular fuzzy elements is base on the iea of istance of the TFPR matri to the ratio matri measure by a particular metric function, see Ramik (). et M be a set of n n matrices with triangular fuzzy elements, an let Φ be a real function efine on M M, i.e. Φ : M M R satisfying the 3 assumptions: (i) Φ( A, B ) for all A, B M. (ii) If Φ( A, B ) = then A = B. (iii) Φ( A, B ) + Φ( B,C ) Φ( A,C ) for all A, B,C M. Then Φ is calle the metric function on M. et X = { i }, i, =,,...,n, be a ratio matri with positive triangular fuzzy numbers = ( ; M ; ). The vector k k k k of positive triangular fuzzy numbers = (,..., ) n is calle the vector of fuzzy weights if

4 Ramik an Korviny: Measuring inconsistency of pairwise comparison matri with fuzzy elements 3 n M å =. () k k = Given A = { a }- n n reciprocal matri with triangular fuzzy elements (n>), where the support supp( a ) S, S = [ /, ], >, a = ( a ; a M ; a ), i, =,,...,n, an let Φ an Ψ be metric functions on M. The new inconsistency ine FI( A ) of A = { a } is esigne in two steps (Ramik, ): Step : Solve the following optimization problem: Φ( A, X ) min; (3) subect to n M å =, ³ M ³ ³, k =,,...,n. (4) k k k k k = Step : Set the inconsistency ine I n of A as I n ( A ) = inf{ Ψ( A,W ); W - optimal solution of (3), (4)}. (5) Remark. In Ramik (), Ramik an Korviny efine Φ an Ψ as follows: ì Φ( A, X æ ö ü i i M i ) = ma ï log log a, log log a, log loga ï å í - - M - ý, (6), ç è ø çè ø èç ø ïî Ψ( A,W ì w w w ü i i M i ) = g.maï íma{ -a, -a, -a } ï, M ý, (7) w w w ïî where is a normalizing constant. Here, in Step an Step, in contrast to Ramik (), we apply metric function Φ accoring to (6) an we set Ψ = Φ. Here, inconsistency ine of TFPR matri A is efine by (5) as follows: ì FI( A æ ö ü i i M i ) = ma ï log log a, log log a, log loga ï å í - - M - ý. (8), ç è ø çè ø ç è ø ïî This efinition was recommene also in Brunelli (). A TFRP matri is sai to be F-consistent if =. Clearly, this new ine (8) is ientical to the former inconsistency ine, for crisp matrices all three parts within the inner parentheses in (8) coincie. 4. FZZY OGARITHMIC SQ METHOD Instea of principle eigenvector metho use in classical AHP, here, we use the logarithmic least squares metho moifie for fuzzy values. The avantage of the logarithmic least squares metho over the classical least squares metho is the symmetry of values from the scale [, 9] an the reciprocal values from [/9, ]. et be a TFPR matri. Now, solve the optimization problem: ì æ ö ü i i M i ma ï log log a, log log a, log loga ï å í - - M - ý min; (9), ç è ø çè ø ç è ø ïî subect to M ³ ³ ³, k =,,...,n. () k k k / n æ n ö ç S a ç k = S Theorem. et w = ç k, S {,M,} an k=,,,n, Then w = ( w ; w M ; w ), k=,,,n, is an optimal k k k k çè ø solution of problem (9), ().

5 Ramik an Korviny: Measuring inconsistency of pairwise comparison matri with fuzzy elements 4 The proof is easy, see Ramik (). Notice, that optimal solution of problem (9), ():, k =,,...,n, can be use for ranking the alternatives an/or criteria in MCDM problem. The following properties of inconsistency ine FI( A ) of TFPR matri A follow irectly from Theorem : (i) FI( A ). (ii) If FI( A ) =, i.e. A is F-consistent, then A is crisp (i.e. nonfuzzy). Now, we will show that if fuzziness of the elements of a TFPR matri increases then the inconsistency ine FI of this matri increases, too. et A { } {( ; M ; M = a = a a a )} be a FPR matri an let an let A = {( a ; a ; a )} be calle -fuzzification of matri A = { a }. Now, we show that if fuzziness of the elements of the matri A increases then the inconsistency ine FI( A ) increases, too. In orer to show this property, we nee the following result. Proposition. et an w = w = w = w, i =,,...,n, be an optimal solution of (9), () with respect to crisp i i i i n æ w ö i matri A = {a }, FI( A) = å log loga - ç, = w çè. Then the inconsistency ine FI of -fuzzification of matri A = {a } ø satisfies the following formula: n æ w ö i FI( A ) = log log a log å - + ç, = w () çè ø The proof of formula () follows irectly from Theorem, as,for i, =,,...,n, i, an, for i=,,...,n. Now, it follows easily from () that inconsistency ine FI( A ) increases with increasing as n æ w ö i FI( A ) = log log a log å - + ç, = w log increases with increasing. çè ø 5. RESTS AND ANAYSIS Here we look for a (crisp) consistent matri in the form of the ratio matri with the maimal membership grae i.e. as close as possible to the original fuzzy matri. We efine an incompatibility ine GI base on Ohnishi (8). ater on, we will show that GI has ifferent behavior when comparing to FI. Again, let A = { a } = {( a ; a M ; a )} be a TFPR matri, where m is a membership function of wi ( ) a. et Gw (,..., w) = min { m, n}. () n w Then GI( A ), calle the incompatibility ine of, is efine as ì n ü GI( A ) = - ma ï íg ( w,..., w ) w =, w ³, i =,,..., nï n i ý å. (3) ï î = Theorem. et be a TFPR matri. * Then GI( A ) = -, where is an optimal solution of the following optimization problem: ma; (4) subect to a + ( a - a ) a - ( a - a ), (5) M i M

6 Ramik an Korviny: Measuring inconsistency of pairwise comparison matri with fuzzy elements 5, ³, i,, k n. (6) k The proof of Theorem is easy an it follows irectly from () an (3). Optimization problem (4) (6) is nonlinear in variables,,, n, however, it can be transforme into a sequence of P problems which can be solve by ichotomy metho, see Ohnishi (8). The following two properties of incompatibility ine GI( A ) of TFPR matri A follow irectly from () (6): (i) GI( A ). (ii) If GI( A ) =, then FI( A ) =, i.e. A is F-consistent. (iii) There eists a TFPR non-crisp matri A * with GI( A *) =. Now, we will prove the following property: Incompatibility ine GI( A ) coul ecrease when fuzziness of the elements of the matri woul increase. M et be a TFPR matri, an let A = {( a ; a ; a )} be -fuzzification of matri A = { a },. Notice that. et optimization problem (P ) be as follows: (P ) subect to ma; (7) a + ( a - a ) a - ( a - a ), (8) M i M n å =, (9) =, ³, i,, k n () k Proposition. et < < an let = (,,..., ) n be a vector of feasible solution of (P ), i.e. k satisfies (7)-() with =. Then is a feasible solution of ( P ), i.e. k satisfies (7) () with =. Proof: To prove the proposition it is sufficient to show that for all i, =,,...,n M M a + ( a - a ) a + ( a - a ), () an M M a + ( a - a ) a + ( a - a ). () Eviently, () is equivalent to the following inequality: æ ö a ( ) - -. (3) ç è ø Similarly, () is equivalent to a - ( - ). (4) ( ) As < <,, a, an a, inequations (3), (4) are satisfie for all i, =,,...,n, q.e.. From Proposition it is clear that if is the maimum value of the obective function of an is the maimum * value of the obective function of, an, then. Moreover, GI( A ) = - an * GI( A ) = -, then. Consequently, is non-increasing function of. Comparing this result to that of Proposition, inconsistency ine FI( A ) increases with increasing.

7 Ramik an Korviny: Measuring inconsistency of pairwise comparison matri with fuzzy elements 6 6. EXAMPES AND SIMATIONS In this section we present some illustrating eamples showing that the new inconsistency/incompatibility inices are convenient tools not only for measuring consistency/compatibility of pair-wise comparison matrices with fuzzy elements, but also for measuring consistency/compatibility of crisp pair-wise comparison matrices. The last eample is a simulation case stuy comparing the both presente inices. Eample. Consier the following TFPR (crisp) matri: é ù (;;) (;;) (6;6;6) é 6ù A = ( ; ; ) (;;) (3;3;3) 3 = ê 6 3 ú ( ; ; ) ( ; ; ) (;;) ë û êë úû et S = [ / 9, 9] be the scale. Then by Theorem we calculate: w = ( w ; w ; w ) =(.89;.89;.89), ~ ( ; ; ) =(.38;.38;.38). Consequently, FI ( A) =, then ( ; M ; w = w w w ) =(.45;.45;.45), w = w w w A ~ is F-consistent. Eample. Consier the following TFPR (fuzzy) matri: é ù (; ; ) (; 3; 4) (4; 5; 6) B = ( ; ;) (;;) (3; 4;5) 4 3. ( ; ; ) ( ; ; ) (;;) êë úû The scale is S = [ / 9, 9] an by Theorem we calculate: ( ; M ; w = w w w ) =(.587;.466;.884), w = ( w ; w ; w ) =(.99;.;.7), GI( B ) =.5. Eample 3. Consier the following TFPR (fuzzy) matri é a ù 3 (; ; ) ( ; ; ) ( ; a ; a ) A = ( ; ; ) (;;) ( ;3;3 ), ( ; ; ) ( ; ; ) (;;) a a a êë úû where δ [,9]. Figure. δ -fuzzification of element

8 Ramik an Korviny: Measuring inconsistency of pairwise comparison matri with fuzzy elements 7 Comparison of FI an GI ine for TFPR matri A : Figure 3. FI(δ ) Figure 4. GI(δ ) It is clear that the results epicte in Figure 3 an Figure 4 confirm the conclusions of Propositions an. 7. CONCSION When comparing FI an GI we can give the following recommenation: FI is suitable for non-interactive elements of fuzzy matrices; GI is appropriate also for interactive elements of fuzzy matrices; by FI uncertainty of iniviual elements of the matri can be reflecte/measure; by GI a measure of compatibility of fuzzy matri with the closest consistent matri is epresse. Consistency itself is a necessary conition for a better unerstaning of relations between elements in MCDM. In MCDM every criterion may have its own inconsistency/incompatibility ine. Then inconsistency/incompatibility of the whole problem is given as maimum of the iniviual inconsistencies/incompatibilities. ACKNOWEDGMENT This research has been supporte by GACR Proect No REFERENCES. Aguaron, J. an Moreno-Jimenez, J.M. (3). The geometric consistency ine: Approimate threshols. European Journal of Operational Research, 47: Bozoki, S. an Rapcsak, T. (7). On Saaty s an Koczkoa s inconsistencies of pairwise comparison matrices. WP 7-, June 7, 7 Bozoki Rapcsak.pf 3. Brunelli, M. (). A note on the article Inconsistency of pair-wise comparison matri with fuzzy elements base on geometric mean. Fuzzy Sets an Systems, 76():

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