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Fig. 3. Computation of pairwise dominance. A.A. Salo, R.P. Himdldinen / European Journal of Operational Research 82 (1995) 458-475 regions, provided that these regions are convex and closed sets. As a result even other types of constraints, such as bounds on the components of the local priorities (e.g. 0.25 < w, < 0.80), could be employed to characterize the DM’s prefer- ences and the dominance relations could still be computed as before. In the sequel, however, it is assumed that the DM describes his preferences through interval judgments. Other types of con- straints are excluded in favor of pairwise compar- isons which have been found an efficient ap- proach to preference assessment both in the AHP and ratio-based techniques of multiattribute value measurement.

Figure 3 Computation of pairwise dominance. A.A. Salo, R.P. Himdldinen / European Journal of Operational Research 82 (1995) 458-475 regions, provided that these regions are convex and closed sets. As a result even other types of constraints, such as bounds on the components of the local priorities (e.g. 0.25 < w, < 0.80), could be employed to characterize the DM’s prefer- ences and the dominance relations could still be computed as before. In the sequel, however, it is assumed that the DM describes his preferences through interval judgments. Other types of con- straints are excluded in favor of pairwise compar- isons which have been found an efficient ap- proach to preference assessment both in the AHP and ratio-based techniques of multiattribute value measurement.

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In terms of interval judgments, the comparison matrix can be written as where the lower and upper bounds satisfy a re- ciprocity condition analogous to that of usual comparison matrices (Saaty and Vargas, 1987). That is, if the DM considers the i-th subelement to be more important than the j-th subelement (;,;= )), then the j-th subelement has to be less important than the i-th subelement (u,;= 1). Ex- tending this argument to the general case shows that /,,u,,= 1 for i #j.
In terms of interval judgments, the comparison matrix can be written as where the lower and upper bounds satisfy a re- ciprocity condition analogous to that of usual comparison matrices (Saaty and Vargas, 1987). That is, if the DM considers the i-th subelement to be more important than the j-th subelement (;,;= )), then the j-th subelement has to be less important than the i-th subelement (u,;= 1). Ex- tending this argument to the general case shows that /,,u,,= 1 for i #j.
A.A. Salo, R.P. Hamdldinen / European Journal of Operational Research 82 (1995) 458-475 intervals as solutions to non-linear programs in which all the local priorities in the hierarchy are included as decision variables. As a result, the time needed to solve these optimization problems increases with the size of the hierarchy, and therefore this approach, which in the consistent case would produce weight intervals similar to ours, is unappealing for the development of inter- active decision support.
A.A. Salo, R.P. Hamdldinen / European Journal of Operational Research 82 (1995) 458-475 intervals as solutions to non-linear programs in which all the local priorities in the hierarchy are included as decision variables. As a result, the time needed to solve these optimization problems increases with the size of the hierarchy, and therefore this approach, which in the consistent case would produce weight intervals similar to ours, is unappealing for the development of inter- active decision support.
Fig. 2. Computation of weight intervals.
Fig. 2. Computation of weight intervals.
Fig. 4. An algorithm for revising the results. The algorithm in Fig. 4 has been implemented into a mouse-driven program called INPRE (Salo The DM can perform the pairwise compar- isons either a top-down or bottom-up fashion, or in some other order that he finds convenient. In
Fig. 4. An algorithm for revising the results. The algorithm in Fig. 4 has been implemented into a mouse-driven program called INPRE (Salo The DM can perform the pairwise compar- isons either a top-down or bottom-up fashion, or in some other order that he finds convenient. In
Interval judgments at third level criteria
Interval judgments at third level criteria
A.A. Salo, R.P. Hamiildinen / European Journal of Operational Research 82 (1995) 458-475 correspond to three linear equality constraints which do not have a common solution due to the inconsistency of the matrix (17). Yet any two constraints do define a unique local priority vec- tor: the entries a,,, 4,3, for example, correspond to the constraints w, = 7w, and w, = 4w, which together with the normalization requirement give the vector At the lowest level criteria Mr. Vennamo’s judgments were converted:into feasible regions by forming the convex. hulls of the priority vectors defined by pairs of ratio comparisons (see Salo, 1993). At criterion capital resources, for instance, the pairwise comparisons of the matrix
A.A. Salo, R.P. Hamiildinen / European Journal of Operational Research 82 (1995) 458-475 correspond to three linear equality constraints which do not have a common solution due to the inconsistency of the matrix (17). Yet any two constraints do define a unique local priority vec- tor: the entries a,,, 4,3, for example, correspond to the constraints w, = 7w, and w, = 4w, which together with the normalization requirement give the vector At the lowest level criteria Mr. Vennamo’s judgments were converted:into feasible regions by forming the convex. hulls of the priority vectors defined by pairs of ratio comparisons (see Salo, 1993). At criterion capital resources, for instance, the pairwise comparisons of the matrix
Fig. 6. The feasible region at criterion capital resources. A.A. Salo, R.P. Hamdldinen / European Journal of Operational Research 82 (1995) 458-47.
Fig. 6. The feasible region at criterion capital resources. A.A. Salo, R.P. Hamdldinen / European Journal of Operational Research 82 (1995) 458-47.
Fig. 7. Interval judgments and consistency bounds. For each pairwise comparison the interval judgments J and the consistency intervals J con- tain up to four different bounds some of which may not be integers in the one to nine range. Instead of the two separate matrix representa- tions these bounds can be displayed together as
Fig. 7. Interval judgments and consistency bounds. For each pairwise comparison the interval judgments J and the consistency intervals J con- tain up to four different bounds some of which may not be integers in the one to nine range. Instead of the two separate matrix representa- tions these bounds can be displayed together as
In view of Section 4, these bounds show that the feasible region would not change if the DM were to state that the first alternative is better than the third (J, =1<2=1,,), and that the DM would be inconsistent if he preferred the second alter- native to the third (@,, = +<1=TJ},). From (15) and (19) the ambiguity index of the feasible re- gion in Fig. 6 is found as tain up to four different bounds some of which
In view of Section 4, these bounds show that the feasible region would not change if the DM were to state that the first alternative is better than the third (J, =1<2=1,,), and that the DM would be inconsistent if he preferred the second alter- native to the third (@,, = +<1=TJ},). From (15) and (19) the ambiguity index of the feasible re- gion in Fig. 6 is found as tain up to four different bounds some of which
Absolute bounds ¥ and vy at third level criteria Table 2 A.A. Salo, R.P. Hiimiildinen / European Journal of Operational Research 82 (1995) 458-475 of the first and second level criteria contain the local priority vectors (1, 0, 0), (0, 1, 0), (0, 0, 1). As a result, the bounds for the alternatives’ weight intervals
Absolute bounds ¥ and vy at third level criteria Table 2 A.A. Salo, R.P. Hiimiildinen / European Journal of Operational Research 82 (1995) 458-475 of the first and second level criteria contain the local priority vectors (1, 0, 0), (0, 1, 0), (0, 0, 1). As a result, the bounds for the alternatives’ weight intervals
Pairwise dominance 7 at third level criteria Table 3
Pairwise dominance 7 at third level criteria Table 3
Absolute bounds » and vy at second level criteria Table 4 In the computation of revised weight intervals, the absolute bounds are updated first at the second level criteria (see Table 4) and then at the topmost criterion. For example, at criterion eco- nomic factors the above interval judgments I,, = [3, 5], I<¢ =[1, 3] impose the constraints 3w, < w,
Absolute bounds » and vy at second level criteria Table 4 In the computation of revised weight intervals, the absolute bounds are updated first at the second level criteria (see Table 4) and then at the topmost criterion. For example, at criterion eco- nomic factors the above interval judgments I,, = [3, 5], I<¢ =[1, 3] impose the constraints 3w, < w,
Pairwise bounds 7 at second level criteria Table 5 indicates that the second alternative can receive
Pairwise bounds 7 at second level criteria Table 5 indicates that the second alternative can receive
Fig. 8. The feasible region at criterion economic factors. A.A. Salo, R.P. Hiimdldinen / European Journal of Operational Research 82 (1995) 458-475
Fig. 8. The feasible region at criterion economic factors. A.A. Salo, R.P. Hiimdldinen / European Journal of Operational Research 82 (1995) 458-475
Fig. 9. The feasible region at the topmost criterion. Finally, assume that the DM considers envi- ronmental factors to be more important than economic factors, and up to four times more important than political factors. These statements define the feasible region in Fig. 9 which is char- acterized by the linear constraints w,<w, and Ww, < Ww, <4w, and the extreme points (0.00, 0.50, 0.50), (0.00, 0.80, 0.20), (0.33, 0.33, 0.33), (0.44, 0.44, 0.11).
Fig. 9. The feasible region at the topmost criterion. Finally, assume that the DM considers envi- ronmental factors to be more important than economic factors, and up to four times more important than political factors. These statements define the feasible region in Fig. 9 which is char- acterized by the linear constraints w,<w, and Ww, < Ww, <4w, and the extreme points (0.00, 0.50, 0.50), (0.00, 0.80, 0.20), (0.33, 0.33, 0.33), (0.44, 0.44, 0.11).
A.A. Salo, R.P. Himdldinen / European Journal of Operational Research 82 (1995) 458-475
A.A. Salo, R.P. Himdldinen / European Journal of Operational Research 82 (1995) 458-475
For those local priorities which minimize (10)- (11) the above inequalities become equalities. This completes the proof. O For those local priorities which minimize (10)- (11) the above inequalities become equalities. This completes the proof. O
For those local priorities which minimize (10)- (11) the above inequalities become equalities. This completes the proof. O For those local priorities which minimize (10)- (11) the above inequalities become equalities. This completes the proof. O
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Analytic Hierarchy ProcessLINEAR PROGRAMPreference Elicitation
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