Preference-Based Methods of MOO Problems

Preference-based methods are classical optimization techniques, that integrate decision maker's preferences at some stage of the resolution process. These preferences are required before the resolution process begins. In this class of methods, we find the value function method, the -constraint method, goal programming, and the generalized center methods. The foundation of the value function method is the individual choices theory. DM’s preferences are based on rationality assumptions. The DM compares pairs of alternatives (i.e., pairs of objective functions), and ranks them according to preference relations. The different types of preferences are Leontief, Cobb-Douglas or CES preferences. The -constraint method requires the selection of one objective while all other objectives are constrained to some value. A payoff table is constructed by solving for each chosen objective a SOO problem with additional constraints. Goal programming is another preference-based method for which DM decides a particular goal for each objective. The programming problem is to minimize the total deviation of solutions from the targets using a specified distance. In the generalized center method, level constraints on the objective function value restrict the feasible region by successive steps. Less performant half-spaces level sets are discarded in the process. The method can be described as a sequence of unconstrained minimization problems using a distance function. Numerical examples from literature illustrate the different classical methods.

Keywords: Basic feasible region, Chebyshev’s problem,  -constrained method, Generalized center method, Goal programming, Neighboring nonbasic solution, Neighboring point, Non-inferior solution, Payoff table, Pivot, Scalarization, Simplex tableau, Simplex-based Algorithm, Slack variable, Value function method, Weakly Pareto-optimal, Weighted exponential method, Weighted metric method, Weighted sum method, Zeleny’s simplex algorithm.