EUROPEAN JOURNAL OF OPERATIONAL RESEARCH ELSEVIER European Journal of Operational Research 82 (1995) 458-475 Theory and Methodology Preference programming through approximate ratio comparisons Ahti A. Salo, Raimo P. Hiirniiliiinen * Systems Analysis Laboratory, Helsinki University of Technology, Otakaari 1 M, 02150 Espoo, Finland Received July 1992;revised March 1993 Abstract In the context of hierarchical weighting, this paper operationalizes interval judgments which allow the decision maker to enter ambiguous preference statements by indicating the relative importance of factors as intervals of values on a ratio scale. Through such judgments the decision maker can capture the subjective uncertainty in his preferences and thus avoid the often cumbersome elicitation of exact ratio estimates. After each new statement the interval judgments are synthesized into dominance relations on the alternatives by solving a series of linear programming problems. This leads to an interactive process of preference programming which provides more detailed results as the decision maker gradually enters a more specific preference description. Moreover, the overall effort of preference elicitation is smaller than in the analytic hierarchy process because the most preferred alternative can usually be identified before all possible comparisons between pairs of factors have been completed. Keywords: Decision analysis; Analytic hierarchy process; Multiple criteria programming 1. Introduction Methods of hierarchical weighting typically de- compose the overall objectives of a problem into their lower level subobjectives until the resulting hierarchy provides a sufficiently detailed frame- work for the analysis. Within such a framework, the analytic hierarchy process (AHP) (Saaty, 1980) elicits preferences through pairwise comparisons in which the decision maker (DM) considers the relative importance of two factors at a time with respect to a common higher level criterion. For each comparison the DM indicates the intensity * Corresponding author. of preference of one factor over another as a point estimate on a ratio scale. Several researchers have acknowledged the difficulties in eliciting exact ratio estimates. Van Laarhoven and Pedrycz (1983), Buckley (1985) and Boender et al. (1989) address this problem by suggesting fuzzy sets for the assessment and anal- ysis of pairwise comparisons. Saaty and Vargas (1987), on the other hand, propose interval judg- ments which allow the DM to make approximate ratio statements as intervals of values on a ratio scale. Arbel (1989) interprets such judgments as linear constraints which at each criterion define a non-empty set of local priorities called the feasi- ble region. The present paper builds on Arbel's (1989, 0377-2217/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0377-2217(93)E0224-L