Dominance, potential optimality, imprecise information, and hierarchical structure in multi-criteria analysis

Abstract

Incorporating the imprecision of both weights and values into multi-attribute value theory brings us some computational difficulties in evaluating alternatives to be considered. For instance, we involve treating a non-linear programming problem because of non-linear forms of objectives or constraints to represent inner product forms of imprecisely known weights and values. In this paper, we thus develop a technique to translate such non-linear programming problems into ordinary linear programming equivalents. Note that there exits an earlier method similar to those developed in the present paper. However, that method has a limitation where it is assumed that there exists at least one alternative that has the only value of unity to be maximal in the corresponding attribute. This limitation is removed in the present article and, further, an extension is made to handle hierarchical structures. Scope and purpose In multi-criteria (or attribute) decision analysis, linear programming techniques have been utilized to effect non-dominated and/or potentially optimal alternatives, when trade-off weights are not known exactly but marginal values are all known exactly. In some practice, however, the marginal values may also be imprecise like those in weights. For instance, some of the values may be known only within specified bounds, while other values may be known only in terms of ordinal relations. The purpose of this paper is hence to demonstrate how to deal with situations in which both weights and values are not known exactly. To cope with even more general situations, we further extend it to hierarchically structured value trees.