# Decision-Making under Multiple and Conflicting Objectives

In compromise programming, the “satisficing” solution is defined as the solution that minimizes the distance from the set of Pareto optima to the so-called “ideal solution”. This ideal solution is defined as the solution that yields minimum (or maximum) values for all criteria. Such a solution does not exist, but is introduced in compromise programming as a target or a goal to get close to, although impossible to reach. The criterion used in compromise programming is the minimization of the deviation from the ideal solution f* measured by the family of Lp metrics (Koski 1984; Lounis and Cohn 1993). In this paper, a multi-criteria optimality or multi-objective index, “MOI”, is defined as the value of the weighted and normalized deviation from the ideal solution f* measured by the family of L metrics:

p

(9)

This family of Lp metrics is a measure of the closeness of the satisficing solution to the ideal solution. The value of the weighting factors wi of the optimization criteria f (i=1,…,m) depends primarily on the attitude of the decision-maker towards risk. The choice of p indicates the importance given to different deviations from the ideal solution. For example, if p=1, all deviations from the ideal solution are considered in direct proportion to their magnitudes, which corresponds to a group utility (Duckstein 1984). However, for p >2, a greater weight is associated with the larger deviations from the ideal solution, and L2 represents the Euclidian metric. For p=<», the largest deviation is the only

one taken into account and is referred to as the Chebyshev metric or mini-max criterion and Lx corresponds to a purely individual utility (Duckstein 1984; Koski 1984; Lounis and Cohn 1995; Lounis and Vanier 2000). In this paper, both the Euclidean and the Chebyshev metrics are used to determine the multi-objective optimality index and corresponding satisficing solution.