Preference Structure Representation Using Convex Cones in Multicriteria Integer Programming
提出一种用凸锥表示决策者偏好结构的新方法，用于求解多准则整数规划问题，能精确表示拟凹效用函数，避免近似误差，并给出含80变量40约束的双准则问题的计算经验。
A new efficient system of representing the decision-maker's preference structure in solving multicriteria integer programming problems is developed. The problem is solved by an interactive branch-and-bound method that employs the procedure of Zionts and Wallenius (Zionts, S., J. Wallenius. 1983. An interactive multiple objective linear programming method for a class of underlying nonlinear utility functions. Management Sci. 29(5).) for multicriteria linear programming. The decision-maker's underlying utility function is assumed to be pseudoconcave, and his pairwise comparisons of decision alternatives are used to determine his preference structure in terms of certain convex cones in the objective function space and constraints on the weights on the objectives in the weight space. The two forms of preference structure representation are interrelated, and their underlying theory is developed. The primary objective of a representation scheme is exactness, and, in this respect, it is shown that the constraints on the weights are not adequate for representing nonlinear utility functions. On the other hand, the convex cones exactly represent any quasiconcave utility function and clearly avoid the approximations and inaccuracies in other utility assessment systems. Accordingly, an efficient ordered representation using convex cones is developed. An algorithmic framework for multicriteria integer programming that integrates the representation using convex cones with the branch-and-bound solution procedure is developed. Computational experience with bicriteria problems having up to 80 variables and 40 constraints is presented.