International Journal of Information Technology & Decision Making c World Scientific Publishing Company

Transcription

1 International Journal of Information Technology & Decision Making c World Scientific Publishing Company A MIN-MAX GOAL PROGRAMMING APPROACH TO PRIORITY DERIVATION IN AHP WITH INTERVAL JUDGEMENTS DIMITRIS K. DESPOTIS Department of Informatics University of Piraeus 80 Karaoli & Dimitriou Str. Piraeus Greece DIMITRIS DERPANIS Department of Informatics University of Piraeus 80 Karaoli & Dimitriou Str. Piraeus Greece Received (Day Month Year) Revised (Day Month Year) We deal with the problem of priority elicitation in the analytic hierarchy process (AHP) on the basis of imprecise pair-wise comparison judgements on decision elements. We propose a minmax goal programming formulation to derive the AHP priorities in the case that the decision maker provides preference judgements in the form of interval numbers. By applying variable transformations we formulate a linear programming model that is capable of estimating the priorities from both consistent and inconsistent interval judgements. The proposed method is illustrated by numerical examples. Keywords: Analytic hierarchy process; Priority setting; Interval judgements; Goal programming. 1. Introduction A key issue addressed in multicriteria decision analysis (MCDA) is the assignment of priorities (weights) to decision elements. The analytic hierarchy process (AHP) introduced by Saaty 1 is one of the most widely used approaches for deriving such priorities through pair-wise comparisons of decision elements. The AHP proceeds in four steps: (a) break down the decision problem into a hierarchy of decision elements (general goal criteria sub-criteria alternatives); (b) construct the pairwise comparison matrices for the decision elements in each level of the hierarchy with respect to one decision element at a time in a level immediately above it; (c) derive local priorities for the decision elements from the pair-wise matrices and (d) synthesize the local priorities to derive global priorities of the alternatives with respect to the general goal of the problem. Recent developments on AHP include among others the design of a visual interface for the elicitation of preference judgements 2 the integration of AHP with data envelopment analysis (DEA) in a MCDM framework 3 the derivation of group 1

2 2 Dimitris K. Despotis Dimitris Derpanis welfare functions 4 and the design of a web-based multicriteria electoral system that incorporates intensity of preferences 5. In the AHP context a comparison matrix is an nxn positive reciprocal matrix R = (r ij ) of paired comparisons of n decision elements in a certain level of the hierarchy with respect to a decision element in a level immediately above it. In the original AHP each entry r ij represents a judgement concerning the perceived dominance (relative importance or preference) of decision element i over j and is provided as a crisp number in the bounded discrete scale (1/9 r ij 9) proposed by Saaty. The basic method proposed by Saaty for deriving the priorities w = (w 1 w 2... w n ) of the n decision elements from the matrix R is the eigenvector method but there are several other scaling methods to assess these priorities. Among them are the geometric mean the least squares and the logarithmic least squares methods (c.f. Saaty and Vargas 6 for a comparative study) and the minmax goal programming method 7. An important issue addressed in the literature 89 is the approximate articulation of preferences in the AHP context. In such a situation the decision maker provides a range of values (interval) [l ij u ij ] instead of a single point r ij on the scale to express her/his preference of a decision element i over an element j. In this paper we focus on the steps (b) and (c) of the AHP when preferences are stated by means of interval pair-wise judgments. The choice of the scale is not restrictive in our case. In the second section we provide a brief review of the relative literature. We present in some details the lexicographic goal programming approach as it has common methodological roots with our approach. In the third section we develop and illustrate our approach to deriving priorities from interval pair-wise judgments. The paper ends with some concluding remarks. 2. Dealing with approximate preferences in AHP Approximate preferences in AHP have been dealt first by considering the entries r ij in R as fuzzy numbers with triangular 1011 or trapezoidal 12 membership functions. Saaty and Vargas 9 introduced the interval numbers to handle approximate preferences and used a simulation approach to derive priority intervals from paired comparisons matrices with interval numbers. Recently Mikhailov 13 introduced linear or non-linear membership functions to derive crisp priorities from interval pair-wise comparisons. With interval judgments and if the Saatys 1 9 scale scale is assumed the decision maker can make statements such as the element i is at least 3 but no more than 7 times as preferable as the element j. Such pair-wise comparisons are

3 A min-max goal programming approach to priority derivation in AHP with interval judgements 3 collected on a matrix R having the general form [ [ R = [ 1 ] [l 12 u 12 ] [l 13 u 13 ] [l 1n u 1n ] l 12 1 [l 23 u 23 ] [l 2n u 2n ] ] [ ] 1 1 l 13 u 23 l 23 1 [l 3n u 3n ] ] [ ] [ ] l 1n u 2n l 2n u 3n l 3n u u u 1n where l ij and u ij are the lower and upper bounds defined on the scale (the Saaty s scale for example) that the decision maker uses to express the relative importance of the element i over the element j. The matrix R is reciprocal in the sense that l ji = 1/u ij and u ji = 1/l ij. The preference programming method of Arbel 8 is a linear programming approach to derive priorities from such a matrix R of interval numbers. An extension of Arbels method is given by Salo and Hamalainen According to the preference programming method the priority vector w = (w 1 w 2... w n ) is obtained as a solution to the following set of linear inequalities S = {w = (w 1 w 2... w n )/l ij w i /w j u ij i j = 1... n w 1 +w 2 + +w n = 1}. Arbel 8 suggested that the feasible region S itself can be viewed as a representation of the decision makers preferences on the decision elements that he compares. Salo and Hamalainen 1415 use a linear programming technique to compute the minimum w Li and the maximum w Ui values that each priority (w i i = 1... n) can attain. The resultant priority intervals are used to express the decision makers preferences. Saaty and Vargas 9 propose a simulation technique to compute these priority intervals. They assume that the interval judgments are uniformly distributed. Sampling randomly values from the intervals [l ij u ij ] they compute the priority vector of the resulting matrices and then they construct a confidence interval for each component of the priority vector. In Ref. 16 a link between the Arbel s and Vargas and Saaty s approaches is provided. The feasible region S is non-empty if the above system of inequalities is solvable i.e. if there exists at least one priority vector w = (w 1 w 2... w n ) such that the ratios w i /w j lie in the corresponding intervals [l ij u ij ] for all i and j. This is the case of consistent intervals or in other words the case of a consistent comparison matrix R. However in case of inconsistent comparisons in R the feasible region S is empty and Arbel s method is not applicable. Lee et al. 17 introduce uncertainty in the comparisons and propose a stochastic model and an iterative process to determine the priorities. The extended region approach 18 and the lexicographic goal programming (LGP) approach 19 are two alternative techniques to derive priorities from inconsistent matrices of paired comparisons. In the following we present the LGP method as it has common methodological roots with our approach and can be directly compared with it.

4 4 Dimitris K. Despotis Dimitris Derpanis 2.1. The lexicographic goal programming approach to weight estimation As mentioned in the previous section the interval judgments [l ij u ij ] in the matrix R give rise to the following system of inequalities: l ij w i /w j u ij i = 1... n 1 j = i n (2.1) Separating the two-sided inequalities and introducing the non-negative deviational variables q ij n ij q ij and n ij the above inequalities are transformed to the following set of linear equalities: w i + l ij w j + n ij q ij = 0 w i u ij w j + n ij q ij = 0 (2.2) Then the lexicographic goal-programming model for estimating the weights from the interval matrix R is as follows (see Ref. 19 for further details and properties of the model): ( ) min α = q nn + n nn n 1 n (q ij + q ij ) i=1 j=i+1 s.t. w i + l ij w j + n ij q ij = 0 w i u ij w j + n ij q ij n = 0 w i + n nn q nn = 1 i=1 w i n ij q ij n ij q ij 0 i = 1... n 1 j = i n (2.3) As stated in Ref. 19 although the first priority goal i.e. the satisfaction of the normalization constraint is always achieved at a zero deviation (q nn + n nn = 0) the lexicographic nature of the model is kept just to show the ability of introducing additional priority levels if the DM is willing to keep some specific deviational variables at a higher priority than the others. In the next section we develop an alternative approach for priority setting in the presence of interval judgments regardless of their consistency. 3. A min-max goal-programming approach to priority setting Initially let as assume that the system (2.1) in the previous section is solvable i.e. the matrix R of interval judgments is consistent. The system (2.1) can then be expressed as follows: w i = w j l ij + w j s ij (u ij l ij ) s ij [0 1] i = 1... n 1 j = 1... n (3.4) The system (3.4) is non-linear due to the introduction of the variable s ij. This new variable is used to express the ratio w i /w j in terms of the left and the right extreme of the interval [l ij u ij ]. With the constraint that the values of the variable

5 A min-max goal programming approach to priority derivation in AHP with interval judgements 5 s ij are in [0 1] we adopt the assumption that the ratio w i /w j lies in the interval [l ij u ij ]. To linearize the equations (3.4) we replace the terms w j s ij with the variables p ij (p ij = w j s ij ). For the new variables p ij holds that 0 p ij w j. This is true in the case of consistent intervals as it is s ij = p ij /w j w j > 0 and 0 s ij 1 for all i and j. With these variable transformations the non-linear system (3.4) takes the following linear form (see Ref. 20 for further details and properties although in a different context): w i = w j l ij + p ij (u ij l ij ) p ij w j 0 p ij 0 i = 1... n 1 j = i n (3.5) Notice that for p ij = 0 it is w i /w j = l ij and for p ij = w j it is w i /w j = u ij. As mentioned above the solution space of the system (3.4) is non-empty only in the consistent case. However extending the formulation introduced above we can handle also the case of inconsistent interval judgements that is the case where the system (3.5) has no feasible solution. We relax the assumption that all the ratios w i /w j lie in the intervals [l ij u ij ] by allowing these weight ratios to lie outside the intervals. Particularly to model the situation that a weigh ratio exceeds the upper bound of the interval the corresponding auxiliary variable p ij should be allowed to take values greater than w j. Similarly the variables p ij should be allowed to take negative values in order to model the situation that the weight ratio exceeds the lower bound of the interval. Let z be a non-negative variable expressing the maximum deviation on either side of the range of values [0 w j ]. With the introduction of the variable z the following linear program is solvable in any case. minz w i w j l ij p ij (u ij l ij ) = 0 p ij w j z 0 p ij z 0 i = 1... n 1 j = i n n w j = 1 j=1 w j 0 z 0 p ij free (3.6) After calculating the priorities w = (w 1 w 2... w n ) by the model (3.6) the priority ratios w i /w j may or may not lie in the intervals [l ij u ij ]. In case of inconsistencies some of the priority ratios may exceed the upper bound of the interval; others may lie below the lower bound. In model (3.6) however the priorities are estimated in a manner that the maximal of the deviations is minimized. That is in case of inconsistent interval judgments the ratio w i /w j that violates the concerned interval it comes as close as possible to the upper bound u ij from the right or to the lower bound l ij from the left. This is the min-max goal programming approach to deriving the priorities. The value of z is an indication of inconsistency. It gets a zero value in case of consistent interval judgments and a strictly positive value

6 6 Dimitris K. Despotis Dimitris Derpanis in case of inconsistencies in the matrix of paired interval comparisons. Moreover the higher is the value of the variable z in the optimal solution the higher is the inconsistency in the matrix. Thus z provides an ordinal measure of inconsistency. In case of a consistent pair-wise interval matrix (z = 0) model (3.6) has multiple optimal solutions. Indeed any possible solution to the system (2.1) is a feasible solution to LP model (3.6) and also an optimal solution of it (z = 0). As stated in Ref. 19 if there are multiple (m say) alternative weight vectors w 1 w 2... w m of model (3.6) such that l ij w k i /wk j u ij i = 1... n 1 j = i n k = 1... m then the average weight vector w a i.e. the vector whose components are the average of the components of the m alternative vectors is a feasible solution of (3.6) in the consistent case. That is l ij w a i /wa j u ij i = 1... n 1 j = i n k = 1... m. According to this property when there are multiple optimal solutions in (3.6) one can seek for a finite number of characteristic optimal solutions such as for example those that maximize one weight at a time and then calculate the average solution to derive the final priorities. To illustrate the proposed approach consider the following interval judgment matrix (only the upper triangular part of the matrix is presented) where four decision elements A B C D are compared in pairs 19 : A B C D A 1 [1 2] [1 2] [2 5] B 1 [2 5] [4 5] C 1 [2 3] D 1 Solving model (3.6) for the preference data in the above matrix we get the priorities w (1) = (w 1 = w 2 = w 3 = w 4 = ). The optimal value of the variable z = 0. This is an indication that the interval judgments in the above matrix are consistent that is all the ratios of the estimated priorities are in the corresponding intervals. Since there are multiple optimal solutions (consistent case) we suggest exploring a number of characteristic optimal solutions. One such solution is for example the solution that maximizes the weight w 1 that is w (2) = (w 1 = w 2 = w 3 = w 4 = ). The very same solution is exhibited as well in Ref. 19 as a solution to the LGP model. One can easily verify that the above solution vector w (2) is the solution that maximizes the weight w 1 in the Arbel s system of linear inequalities S. Modifying the intervals in the above matrix for example setting for the paired comparison (A D) the interval [2 3] and for the pair (B C) the interval [3 5] we get an inconsistent comparison matrix. Solving the model (3.6) for the new matrix we get the priorities w (3) = (w 1 = w 2 = w 3 = w 4 = ). The optimal value of the variable z = This verifies that the comparison matrix is inconsistent. The corresponding solution obtained by the LGP method is w 1 = w 2 = w 3 = w 4 = Both solutions provide the same priority ranking to the decision elements A B C and D.

7 A min-max goal programming approach to priority derivation in AHP with interval judgements 7 If we perturb further the matrix by assuming for example for the comparison of (C D) the interval [6 8] we get the priorities w (4) = (w 1 = w 2 = w 3 = w 4 = ) with z = The solution obtained by the LGP method on the same data matrix is w 1 = w 2 = w 3 = w 4 = Both priority vectors provide again the same ranking to the decision elements. 4. Conclusions Approximate articulation of preferences is the means to overcome the decision maker s inability due to fuzziness or uncertainty to provide point estimates on the scale when she/he compares criteria or decision alternatives in the AHP framework. In such a setting crisp priorities must be estimated from interval pair-wise comparisons. The approach proposed in this paper for solving this problem is based on a min-max goal programming formulation that enables the estimation of local priorities for the decision elements regardless the consistency of the matrix of pairwise comparisons. Moreover the underlying priorities derive through an optimality criterion. A measure of inconsistency is also obtained that enables the analyst to locate the inconsistencies in the process of preference elicitation and probably provide advice to the decision maker in order to eliminate these inconsistencies. The degree of inconsistency is highly depended on the size of the intervals. Let us start for example from an inconsistent matrix R = (r ij ) with exact preference estimates and then assume intervals generated by estimates distributed around r ij. The highest is the length of the intervals the more likely is to be consistent. On the other hand the highest is the length of the intervals the more imprecise are the preference judgments. So it is clear that in approximate articulation of preferences one has to balance between consistency of the interval judgements and preference accuracy. References 1. T. L. Saaty The analytic hierarch process (McGraw-Hill New York 1980). 2. S. Zahir Eliciting ratio preferences for the analytic hierarchy process with visual interfaces: a new mode of preference measurement International Journal of Information Technology and Decision Making 5(2006) N. Ahmad D. Berg and G. R. Simons The integration of analytical hierarchy process and data envelopment analysis in a multi-criteria decision-making problem International Journal of Information Technology and Decision Making 5(2006) T. L. Saaty and L. G. Vargas The possibility of group welfare functions International Journal of Information Technology and Decision Making 4(2005) A. Zahir Making public policy decisions using a web-based multi-criteria electoral system International Journal of Information Technology and Decision Making 1(2002) T. L. Saaty and L. G. Vargas Comparisons of eigenvalue logarithmic least squares and least squares methods in estimating ratios Mathematical Modelling 5(1984) D. K. Despotis Fractional goal programming: A unified approach to priority estimation and preference analysis in MCDM Journal of the Operational Research Society 47(1996)

8 8 Dimitris K. Despotis Dimitris Derpanis 8. A. Arbel Approximate articulation of preference and priority derivation European Journal of Operational Research 43(1989) T. L. Saaty and L. G. Vargas Uncertainty and rank order in the analytic hierarchy process European Journal of Operational Research 32(1987) P. G. Van Laarhoven and W. Pedrycz A fuzzy extension of Saaty s priority theory Fuzzy Sets and Systems 11 (1983) M-F Chen G-H Tzeng and T-I Tang Fuzzy MCDM approach for evaluation of Expatriate Assignments International Journal of Information Technology and Decision Making 4(2005) J. J. Buckley Fuzzy hierarchical analysis. Fuzzy Sets and Systems 17(1987) L. Mikhailov A fuzzy approach to deriving priorities from interval pairwise comparison judgements European Journal of Operational Research 159(2004) A. A. Salo and R.P. Hamalainen Processing interval judgments in the analytic hierarchy process in Multicriteria Decision Making Proceedings of the IX International Conference on MCDM eds. Goicoechea Duckstein and Zionts (Springer Berlin 1991) pp A. A. Salo and R. P. Hamalainen Preference programming through approximate ratio comparisons European Journal of Operational Research 82(1995) A. Arbel and L.G. Vargas Preference simulation and preference programming: robustness issues in priority derivation European Journal of Operational Research 69(1993) M. Lee H. Pham and X. Zhang A methodology for priority setting with application to software development process European Journal of Operational Research 118(1999) A. A. Salo Inconsistency analysis by approximately specified priorities Mathematical and Computer Modelling 17(1993) R. Islam and M. P Biswal S. S. Alam Preference programming and inconsistent interval judgments European Journal of Operational Research 97(1997) D. K Despotis and Y. G. Smirlis Data envelopment analysis with imprecise data European Journal of Operational Research 140(2002)