The Envelope Approach for Multiobjective Optimization Problems

Multiobjective optimization problems are generally solved by finding the set of all noninferior solutions. A new methodology--termed the envelope approach--is presented. An investigation of the use of the envelope approach in the multiobjective dynamic programming method and the parametric decomposition method shows that this approach is most suitable for the analysis of multiobjective optimization problems. Under the assumption of separability and monotonicity, the theorem of the principle of optimality for multiobjective dynamic systems is proved, and this gives a theoretical basis for a new multiobjective dynamic programming method which uses envelope analysis. A parametric decomposition theorem is proved and an algorithm is given. The original problem is split into a family of subproblems by temporarily fixing the values of certain variables according to a strategy that uses parametric decomposition and the envelope approach. This provides a powerful means of finding all noninferior solutions by identifying the envelope of the family of objective curves.