A MIN-MAX GOAL PROGRAMMING APPROACH TO PRIORITY

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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, 18534, Greece DIMITRIS DERPANIS Department of Informatics, University of Piraeus, 80 Karaoli & Dimitriou Str. Piraeus, 18534, 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 Saaty1 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 judgements2 , the integration of AHP with data envelopment analysis (DEA) in a MCDM framework3 , the derivation of group 1

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welfare functions4 and the design of a web-based multicriteria electoral system that incorporates intensity of preferences5 . In the AHP context, a comparison matrix is an nxn positive reciprocal matrix R = (rij ) 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 rij 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 ≤ rij ≤ 9) proposed by Saaty. The basic method proposed by Saaty for deriving the priorities w = (w1 , w2 , . . . , wn ) 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 Vargas6 for a comparative study) and the minmax goal programming method7 . An important issue addressed in the literature8,9 is the approximate articulation of preferences in the AHP context. In such a situation, the decision maker provides a range of values (interval) [lij , uij ] instead of a single point rij 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 rij in R as fuzzy numbers with triangular10,11 or trapezoidal12 membership functions. Saaty and Vargas9 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 Mikhailov13 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

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collected on a matrix R having the general form 

1

i [l12 , u12 ] [l13 , u13 ]  1 1 1 [l23 , u23 ]  u12 , l12 h i h i  1 1 1 1 R =  u13 , l13 1 u23 , l23   h ··· i h ··· i h ··· i 1 1 1 1 1 1 u1n , l1n u2n , l2n u3n , l3n h

· · · [l1n , u1n ]



 · · · [l2n , u2n ]    · · · [l3n , u3n ]   ··· ···   ··· 1

where lij and uij 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 lji = 1/uij and uji = 1/lij . The preference programming method of Arbel8 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 Hamalainen14,15 . According to the preference programming method the priority vector w = (w1 , w2 , . . . , wn ) is obtained as a solution to the following set of linear inequalities S = {w = (w1 , w2 , . . . , wn )/lij ≤ wi /wj ≤ uij , i, j = 1, . . . , n, w1 +w2 +· · ·+wn = 1}. Arbel8 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 Hamalainen14,15 use a linear programming technique to compute the minimum wLi and the maximum wU i values that each priority (wi , i = 1, . . . , n) can attain. The resultant priority intervals are used to express the decision makers preferences. Saaty and Vargas9 propose a simulation technique to compute these priority intervals. They assume that the interval judgments are uniformly distributed. Sampling randomly values from the intervals [lij , uij ], 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 = (w1 , w2 , . . . , wn ) such that the ratios wi /wj lie in the corresponding intervals [lij , uij ] 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 approach18 and the lexicographic goal programming (LGP) approach19 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.

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2.1. The lexicographic goal programming approach to weight estimation As mentioned in the previous section, the interval judgments [lij , uij ] in the matrix R give rise to the following system of inequalities: lij ≤ wi /wj ≤ uij , i = 1, . . . , n − 1, j = i + 1, . . . , n

(2.1)

Separating the two-sided inequalities and introducing the non-negative devia0 tional variables qij , nij , qij and n0ij , the above inequalities are transformed to the following set of linear equalities: −wi + lij wj + nij − qij = 0 wi − uij wj + n0ij − qij = 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): ! n−1 n P P 0 min α = qnn + nnn , (qij + qij ) i=1 j=i+1

s.t. −wi + lij wj + nij − qij = 0 0 wi − uij wj + n0ij − qij =0 n P wi + nnn − qnn = 1

(2.3)

i=1

0 wi , nij , qij , n0ij , qij ≥0 i = 1, . . . , n − 1, j = i + 1, . . . , n

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 (qnn + nnn = 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: wi = wj lij + wj sij (uij − lij ), sij ∈ [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 sij . This new variable is used to express the ratio wi /wj in terms of the left and the right extreme of the interval [lij , uij ]. With the constraint that the values of the variable

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sij are in [0, 1], we adopt the assumption that the ratio wi /wj lies in the interval [lij , uij ]. To linearize the equations (3.4), we replace the terms wj sij with the variables pij (pij = wj sij ). For the new variables pij holds that 0 ≤ pij ≤ wj . This is true in the case of consistent intervals, as it is sij = pij /wj , wj > 0 and 0 ≤ sij ≤ 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): wi = wj lij + pij (uij − lij ) pij − wj ≤ 0 pij ≥ 0, i = 1, . . . , n − 1, j = i + 1, . . . , n

(3.5)

Notice that for pij = 0 it is wi /wj = lij and for pij = wj it is wi /wj = uij . 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 wi /wj lie in the intervals [lij , uij ] 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 pij should be allowed to take values greater than wj . Similarly, the variables pij 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, wj ]. With the introduction of the variable z the following linear program is solvable in any case. minz wi − wj lij − pij (uij − lij ) = 0 pij − wj − z ≤ 0 −pij − z ≤ 0 i = 1, . . . , n − 1, j = i + 1, . . . , n n P wj = 1

(3.6)

j=1

wj ≥ 0, z ≥ 0 pij free After calculating the priorities w = (w1 , w2 , . . . , wn ) by the model (3.6), the priority ratios wi /wj may or may not lie in the intervals [lij , uij ]. 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 wi /wj that violates the concerned interval, it comes as close as possible to the upper bound uij from the right or to the lower bound lij 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

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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 w1 , w2 , . . . , wm of model (3.6), such that lij ≤ wik /wjk ≤ uij , i = 1, . . . , n − 1, j = i + 1, . . . , n, k = 1, . . . , m, then the average weight vector wa , 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 lij ≤ wia /wja ≤ uij , i = 1, . . . , n − 1, j = i + 1, . . . , 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 pairs19 : A B C D

A B C D 1 [1, 2] [1, 2] [2, 5] 1 [2, 5] [4, 5] 1 [2, 3] 1

Solving model (3.6) for the preference data in the above matrix, we get the priorities w(1) = (w1 = 0.3636, w2 = 0.3636, w3 = 0.1818, w4 = 0.0909). 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 w1 , that is w(2) = (w1 = 0.3704, w2 = 0.3704, w3 = 0.1852, w4 = 0.0741). 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 w1 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) = (w1 = 0.3478, w2 = 0.3913, w3 = 0.1594, w4 = 0.1014). The optimal value of the variable z = 0.0435. This verifies that the comparison matrix is inconsistent. The corresponding solution obtained by the LGP method is w1 = 0.3000, w2 = 0.4500, w3 = 0.1500, w4 = 0.1000. Both solutions provide the same priority ranking to the decision elements A, B, C and D.

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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) = (w1 = 0.3103, w2 = 0.4138, w3 = 0.2069, w4 = 0.0690) with z = 0.1034. The solution obtained by the LGP method on the same data matrix is w1 = 0.3030, w2 = 0.4545, w3 = 0.1515, w4 = 0.0910. 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 = (rij ) with exact preference estimates and then assume intervals generated by estimates distributed around rij . 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) 245–261. 3. 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) 263–276. 4. T. L. Saaty and L. G. Vargas, The possibility of group welfare functions, International Journal of Information Technology and Decision Making 4(2005) 167–176. 5. A. Zahir, Making public policy decisions using a web-based multi-criteria electoral system, International Journal of Information Technology and Decision Making 1(2002) 293–309. 6. T. L. Saaty and L. G. Vargas, Comparisons of eigenvalue, logarithmic least squares and least squares methods in estimating ratios, Mathematical Modelling 5(1984) 309–324. 7. 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) 989–999.

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