Multicriteria decision making under uncertainty: a visual approach

In their basic form, multicriteria decision making problems have a simple structure: given a finite number of decisions and a finite number of criteria, each decision is evaluated on each criterion, resulting in some outcome such as profit, environmental impact, sales, or expected costs. Most often, these evaluations are assumed to be quantitative. As dominances rarely occur and multidimensional comparisons are difficult, weights are typically used to aggregate the evaluations of the criteria to reduce the problem to a simple single-dimensional comparison between decisions. A large number of (mostly strategic) decision-making problems can be and has been reduced to this format. Popular examples include the location of landfills, hazmat facilities and power plants, expansions of companies into new regions, the choice of technologies, and similar decisions. A variety of methods has been devised to solve such problems. Among others, they include outranking methods by the " Belgian school " with proponents Roy and his ELECTRE method and Brans with his PROMETHEE technique, multiattribute value theory developed and popularized by Keeney and Raiffa, preference cones, and Saaty's analytic hierarchy method for problems with inconsistent preference estimates. All of these approaches share the assumption that the evaluations of the decisions with respect to the criteria are deterministic. Our work introduces stochasticity into the problem. More specifically, we assume that the outcome of the decision on each of the criteria is a random variable with an arbitrary underlying probability distribution. The main idea is now to use two components for each decision – criterion pair: rather than resorting to the usual descriptors " expected outcome " and " risk " as, for instance, used in the original portfolio selection models, this presentation utilizes the expected outcome and the probability that the outcome satisfies at least one benchmark value, a target value specified by the decision maker as a lowest acceptable bound. Regardless of the dimensionality of the problem, i.e., the number of decisions and the number of criteria, each decision can then be plotted as a polygon in the two-dimensional space of expected outcome and the probability that the outcome exceeds the prespecified target value, a number that can easily be computed for any arbitrary distribution. It is worth noting that limitations that the decision maker may specify on the weights or relative importance of the criteria can be incorporated as well. The polygons of the decisions that are computed in this manner …