344 A. Davoodi / IJIM Vol. 1, No. 4 (2009)

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1 Available online at Int. J. Industrial Mathematics Vol., No. (009) -0 On Inconsistency of a Pairise Comparison Matrix A. Davoodi Department of Mathematics, Islamic Azad University, Neyshabur Branch, Neyshabur, Iran. Received September 009; revised Desember 009; accepted Desember Abstract Analytical Hierarchy Process is a method of solving a Multiple Attribute Decision Making. This method is based on the pairise comparisons beteen the alternatives. These comparisons form a matrix named pairise comparison matrix or judgment matrix. The eights of mentioned alternatives ill be determined from this mentioned above matrix. In the case of consistency of the judgments, the cited eights are reliable and trusty. On the other hand, if the consistency of the judgment matrix is not acceptable, the result sucient level of trust. In this case it is requested that the decision maker revise the judgment. In this paper e propose a method to approximate a consistent judgment matrix from an inconsistent one ithout the revision of the judgments. This method is based on decreasing the eects of the mistakes made by the decision maker. Keyords : Decision analysis; Analytical hierarchy process; Consistency. { Introduction Analytical Hierarchy Process (AHP) is a method for solving a multiple criteria decision making problem and is idely used no (See Poh and Ang [], Chang et al [], Wong and Li [9] and Vidal et al []). In this method the Decision Maker expresses his/her concept(opinion) as a matrix named Pairise Comparison Matrix (Saaty []). A typical form of a hierarchy ith k criteria and n alternatives is presented in Fig.. address: alirzd@yahoo.com, Tell:(+989)000

2 A. Davoodi / IJIM Vol., No. (009) -0 Fig.. A typical hierarchy ith levels. In this method every pair of alternative (criterion) must be compared ith each other ith respect to the common alternative at their higher level. In each comparison this question must be ansered: hich one is more important and ho much? Saaty [] introduced the numbers f; ; :::; 9g to express the rating preferences beteen each pair of stimulus. The pairise comparison matrices have some interesting properties: )The elements on the main diagonal are equal to to indicate the preference of each alternative over itself. ) If the preference of A i (alternative i) over A j (alternative j) is equal to h, then the preference of A j over A i is equal to. This property is called reciprocally. h These properties sho the elements of the judgment matrix are positive. Also suppose a ij ; i; j = ; :::; n are the elements of a pairise comparison matrix for n alternatives. This matrix is called consistent if the equation a ij = a ik a kj is held for all i; j; k f; :::; ng. But in reality the comparison of each three alternatives may lead to creating inconsistency. Saaty [] also introduced a Consistency Ratio(CR) for a pairise comparison matrix. By this denition if CR exceeds 0%, he recommended that the decision maker revise his/her elicited preferences (Saaty and Mariano [], Wind and Saaty [8]). Some of the limits on AHP ere presented by Murphy [] including the errors created by the scale of pairise comparison. Inconsistency causes errors and lack of certainty to get the logical and true results. There are some approaches trying to convert an inconsistent matrix into a consistent one( or by CR less than 0%) as the modied matrix has the least dierence compared to the primeval matrix (Koczkodaj and Orloski [8]). Kiesieleicz and Uden [9] have presented the relationship beteen inconsistent and contradictory of matrices of data and it shos that even if a matrix can pass a consistency test successfully, it may be contradictory. A ne denition of inconsistency described in (Koczkodaj [], Duszak and Koczkodaj []) allos us to locate the roots of inconsistency and it is claimed that this denition is easier to interpret than the current one. There are to sources for inconsistency of a judgment matrix have been introduced by Dadkhah and Zahedi [] including the decision maker's preference and the ay of eliciting the decision maker's preference. Furthermore, this paper provides an algorithm for moving toards consistency and shos mathematically hy the algorithm orks. Several open mathematical questions about the structure of the set of positive reciprocal matrices ere considered by Deturck []. Moreover, Aczel and Saaty [] found the geometric mean to be the ay to synthesize the judgments of several individuals to conform ith the reciprocal property of paired comparisons. Also a ne consistency measure, the harmonic consistency index that has been introduced by Stein and Mizzi [], as obtained for every positive reciprocal matrix

3 A. Davoodi / IJIM Vol., No. (009) -0 in the AHP and it as shon that this index varies ith changes in any matrix element. In this paper e propose a simple approach to get a consistent approximation of an inconsistent matrix ithout the revision of decision maker's preferences. Section presents some preliminaries in AHP. In section our method and its properties are stated. In section this method is applied to an inconsistent matrix. Finally section concludes. Preliminaries AHP is a method for solving an MCDM problem in discrete space. This method includes steps: the rst step creats a hierarchy that shos the problem graphically. The goal is at the top level, the criteria are in the intermediate levels and the alternatives are in the loest level. The elements of each level ill be compared to the elements in that level ith respect to the common elements of their higher level. These comparisons lead to create pairise comparison matrices shoing the ratio preferences and value of alternatives. These ratio preferences are stated as numbers of the set f; :::; 9g. AHP is based on calculating local eights and nal eights. The local eights of each comparison matrix are determined by the eigenvector corresponding to the greatest eigenvalue of the cited matrix. In other ords if A is a consistent pairise comparison matrix, then the vector W determined by AW = max W is the vector of local eights. P The nal eight of each alternative is calculated as follos: Final eight of A j = i (local eight of criterion i)(local eight of A j ith respect of criterion i). Alternatives could be sorted by these nal eights. The more the nal eight, the more the priority. Suppose P nn is a typical pairise comparison matrix of n alternatives and ; ; ::: n are their corresponding eights. The element at the ro i and the column j is i j, so the element at the ro j and the column i ill be j i (reciprocally property). Also the elements on the main diagonal are equal to. Moreover this matrix is consistent if the equation i j = i k k j is held for all i; j; k, otherise it is inconsistent. The Consistency Ratio (CR) has been introduced to sho the value of the consistency of a matrix. It has been shon that for a consistent judgment matrix, its greatest eigenvalue is equal to the dimension of the matrix. So the distance beteen max and n can be used as a measure for inconsistency. The consistency ratio is calculated as CR= max n, here RI is a random inconsistency index, hose value varies ith the order of pairise comparison matrix. (n )RI Table. shos this index for some orders. RI is the basis for dening the consistency ratio. It has been discussed by Lane and Verdini(989) and Golden and Wang(990). Also Tummala and Wan (99) presented a closed-form expression for the largest eigenvalue of a three-dimensional random pairise comparison matrix and the exact value of the mean and variance of the cited matrix. If CR 0., the pairise comparison matrix is thought to have an acceptable consistency otherise, it needs to be revised. For a consistent matrix all the methods of deriving eights have the same results. In this case the elements of each column of normalized corresponding matrix are the eights.

4 A. Davoodi / IJIM Vol., No. (009) -0 Table Random inconsistency index for pairise comparison matrices ith the order from to 0 n RI An Approach to get consistency First e present an important discussion that is useful for the rest of this paper.. Inconsistency Suppose the folloing judgment matrix beteen three alternatives: R = = x = =x and ; ; are the corresponding eights respectively. Suppose the Decision Maker rst presents his/her opinion about the preference of alternative over others by lling ro. Since the matrix must be reciprocal, it is sucient to ll the elements above the main diagonal. If this matrix is consistent the folloing equation must be held: = = = (=) = = So if x = = the judgment ill be consistent. Here the rst reason of inconsistency appears. The general scale of AHP to compare to stimulus has the form of f=9; =8; :::; =; ; ; :::; 9g, and / does not belong to this scale. This example shos one of the reasons of inconsistency is the lack of an appropriate scale. To overcome this problem, it is more appropriate to use the numbers of the form fa=b; a; b f; :::; 9gg. This modied scale allos the clever decision makers to present a consistent judgment. No suppose the matrix R again. Decision maker judged three times: Comparison beteen alternative and alternative, and, and nally beteen and. In spite of using the ne scale numbers, if the matrix is still inconsistent, it can be concluded that the decision maker judged rong at least in one of his/her comparisons. No consider the folloing three situations: (a) The comparisons beteen alternative and alternative and beteen and are logically correct, so the mistake(error) has happened in comparison beteen alternative and alternative. (b)the comparison beteen and, and are correct, so the mistake happened in comparison beteen and. (c)the mistake occurred hile alternatives and is compared. Situation (a) states that the rst ro of the pairise comparison matrix involves the reliable data(its elements are consistent ith the subjective judgment of decision maker). No consider situation (b), since by comparison of alternatives and the preference of alternative over alternative is recognized(by the reciprocally), as a result the elements of the second ro of the pairise comparison matrix are knon. We should keep in mind that elements on the main diagonal are equal to. Considering the preceding discussion, the elements of the third ro of the matrix are knon by situation (c). Here e sho that in general if the preferences of just one alternative over the others are apparent, then a consistent pairise comparison matrix ill be created. In other ords, each ro of a matrix conduces to a consistent matrix. Consider an n dimension pairise comparison matrix and suppose the elements of the ro ;

5 A. Davoodi / IJIM Vol., No. (009) -0 o; o f; :::; ng are knon. These elements are o ; o ; :::; o n, in hich i ; i = ; :::; n is the eight of alternative i. With the consistency condition, e can determine the elements of the ro k as follos: k i = k o o i = o o i k this procedure for i = ; :::; n, in hich o i ; 8i and the o k are knon by ro k. We can repeat i = o. Here the reciprocally property of matrix helps us a lot. Example: Consider a pairise comparison matrix ith ros and columns. The second ro is as = e no try to create three other ros as: =, = = =, = = (=) = =, = = (=)(=) = = Third ro: = = = (=) = =, = = =, =, = = (=) (=) = =8 In the same manner the elements of ro ill be,, 8 and. So the created consistent matrix is as: = = = = T = = = =8 8 This example shos an interesting note. If a typical pairise comparison matrix is consistent, the elements of each ro are in fact conrmation for other ros. For instance if e ant to make a consistent matrix by the forth ro as 8 ; the produced matrix is the same as matrix T. If that matrix is consistent, the decision maker certies his/her opinion iteratively by lling each ro of the judgment matrix,. No consider an inconsistent pairise comparison matrix again. The decision maker made at least one mistake hile lling this matrix, but here? The goal is to decrease the eects of this mistake during the decision process. It is not logically true to request the decision maker to revise his/her rst opinion. Since if he/she as able to recognize the mistake, probably he/she could be able to remove that in the rst place. Here e propose our method to decrease the eects of carelessness of decision maker. Consider an n-dimensional inconsistent pairise comparison matrix(p0). Suppose the decision maker judged right at comparing the rst alternative ith the others. In other ords the elements of the rst ro are logically true. We may set this ro as criterion ro. By the method described in the previous discussion, e can construct a consistent pairise comparison matrix named P. No matrix P is created by setting the second ro as the criterion ro. The matrices P; :::; P n ill be constructed in the same manner. Finally these matrices must be combined ith each other. We propose the geometric mean as follos: consider matrix P = [p ij ]; i; j = ; :::; n. No dene p ij = (p ij p ij ::: pn ij ) n, in hich P k = [p k ij ]; for i; j; k = ; :::; n. Theorem... The matrix P created by the geometric mean satisfy the folloing properties: a) elements on the main diagonal equal to

6 8 A. Davoodi / IJIM Vol., No. (009) -0 b) reciprocally c) consistency proof: Suppose i k is the eight of the alternative i determined by the matrix P k. Since matrices P; :::; P n are consistent pairise comparison matrices, they are satised in conditions a) and b), so p ii = ( ::: ) n =, for i = ; :::; n thus the rst condition is held. Also ) n = = p ji, p ij = (p ij p ij :::pn ij ) n = ( i j i j ::: n i j n this is the reciprocally condition. p ij = p ik p kj is held for i; j; k = ; :::; n. p ij = ( i j i j ::: n i j n ) n = (( i k k j ) n = ( j ( j j j n i i i i ::: i n ) n For the third condition e must sho equation ) ( i k k j j ::: n j n i ) ::: ( n i n k n k j n )) n = ( i k i k ::: n i k n ) n ( k j k j ::: n k j n ) n = p ik p kj for i; j; k = ; :::; n. In hich the second equality is held by the consistency property of each P k for k = ; :::; n. Here e summarize these discussions as the folloing algorithm: Algorithm: Input: An n dimensional inconsistent pairise comparison matrix P0. output: Matrix P that is a consistent approximation of the matrix P0. For i = to n Set ro i as the criterion ro and construct matrix P i. End for By the geometric mean construct matrix P by matrices P; P; :::; P n. This method in fact decreases the eects of errors that ere created by the decision maker's mistakes. Moreover the elements of the nal matrix do not necessarily need to be numbers ; :::; 9. At the example in the next section e ill study this situation. Numerical Example Matrix P0 is inconsistent and its ratio inconsistency is equal to 0.8 (Also the value of the greatest eigenvalue is equal to 8.). Although the matrix is inconsistent, e calculate its corresponding eight vector for comparison ith the one calculated by the proposed algorithm. The (inconsistent) vector of eights is equal to (0., 0.099, 0., 0.0, 0.). We no deal ith our algorithm to get a consistent matrix. First matrices P; :::; P are created and then the nal matrix P ill be determined by the geometric mean of these matrices. P = P 0 = = = = = = = = 9 = = = =9 = = = = = = = 8 = = = =8 ; P = = =8 =0 = = 8 8 = = =8 =0 0 = 0 8

7 P = A. Davoodi / IJIM Vol., No. (009) -0 9 P = =8 = = = 8 = = = = =8 = = 8 P = = = =9 8= = = = = 8 9 =8 = =8 = ; P = =8 8 = =0 = = 8= =8 = =8 = = =8 0:90 0:0 0:0 0: :099 0: 0: 0:9 :00 : :9 : :9 = 0:9 :8 : 0: :0 It is easy to sho that the ratio inconsistency for intermediate matrices P; P; P; P; P is equal to zero. The value of the greatest eigenvalue for all intermediate and nal matrices is equal to. The numbers in the matrix P have been rounded. Moreover this matrix is satised in the conditions of theorem... Finally the revised vector of eights is equal to (0., 0., 0.8, 0.8, 0.9). This vector must be replaced by the rst inconsistent vector in the decision process. Conclusion The existence of the right judgments of the decision maker has an important role in creating the pairise comparison matrix in Analytical Hierarchy Process. If the judgment matrix does not have an acceptable level of consistency, the results are not reliable. There are some approaches producing consistent approximation of an inconsistent matrix. In this paper e proposed a simple objective to determine a consistent judgment matrix from an inconsistent one. This method is based on decreasing the eects of the mistakes made by the decision maker. In fact decision maker ith a correct and logical judgment certies his/her opinion by lling each ro of the judgment matrix. So the elements of the matrix can be determined directly from just one ro. But in the case of inconsistency of the judgment matrix, elements of some ros can not be determined from others, moreover the elements of some ros have conicts ith some others. In this case it is not clearly knon here the decision maker has some mistakes. The method in this paper can decrease the eects of these mistakes in a sequence of simple computations. References [] J. Aczel,T. L. Saaty, Procedures for synthesizing ratio judgements, Journal of Mathematical Psychology (98) 9-0. [] C. W. Chang, Cheng-Ru Wu, Chin-Tsai Lin, Huang-Chu Chen, An application of AHP and sensitivity analysis for selecting the best slicing machine, Computers and Industrial Engineering () (00) 9-0. [] K. M. Dadkhah, F. Zahedi, A Mathematical treatment of inconsistency in the Analytic Hierarchy Process, Mathematicl and Computer Modelling, (99) -. 9

8 0 A. Davoodi / IJIM Vol., No. (009) -0 [] D. M. Deturck, The approach to consistency in the Analytic Hierarchy Process, Mathematical Modelling 9 (98) -. [] Z. Duszak, W. W. Koczkodaj, Generalization of a ne denition of consistency for pairise comparisons, Information Processing Letters (99) -. [] Bruce L. Golden, Q. Wang, An Alternative Measure of Consistency. In The Analytic Hierarchy Process: Application and Studies, 8-8. Ne York: Springer- Verlag, 990. [] W. W. Koczkodaj, A Ne Denition of Consistency of Pairise Comparisons, Mathematical and Computer Modelling 8 (99) 9-8. [8] W. W. Koczkodaj, M. Orloski, Computing a Consistent Approximation to a Generalized Pairise Comparisons Matrix, Computers and Mathematics ith Applications (999) 9-8. [9] M. Kiesieleicz, E. V. Uden, Inconsistent and contradictory judgements in pairise comparison method in the AHP, Computers and Operations Research (00) - 9. [0] E. F. Lane, W.A. Verdini, A Consistency Test for AHP Decision Makers, Decision Sciences 0 (989) -90. [] C. K. Murphy, Limits on the analytic hierarchy process from its consistency index, European Journal of Operational Research (99) 8-9. [] K. L. Poh, B. W. Ang, Transportation fuels and policy for Singapore: an AHP planning approach, Computers and Industrial Engineering () (999) 0-. [] T. L. Saaty, R. S. Mariano, Bationing energy to industries: Priorities and inputoutput dependence Energy Syafems and Policy, (99) 8-. [] T. L. Saaty, The Analytic Hierarchy Process, Ne York: McGra-Hill, 980. [] W. E. Stein, P. J. Mizzi, The harmonic consistency index for the analytic hierarchy process, European Journal of Operational Research (00) [] V. M. Rao. Tummala, Yat-ah Wan, On The Mean Random Inconsistency Index Of Analytic Hierarchy Process (AHP), Computers and Industrial Engineering (99) 0-0. [] L. A. Vidal, E. Sahin, N. Martelli, M. Berhoune, B. Bonan, Applying AHP to select drugs to be produced by anticipation in a chemotherapy compounding unit, Expert Systems ith Applications doi:0.0/j.esa (009). [8] Y. Wind, T. L. Saaty, Marketing applications of the analytic hierarchy process, Management Sciences (980) -8. [9] J. K. W. Wong, Heng Li, Application of the analytic hierarchy process (AHP) in multi-criteria analysis of the selection of intelligent building systems, Building and Environment () (008)