Technical Note—Complementary Slackness Theorem in Multiple Objective Linear Programming

A complementary slackness theorem in multiple objective linear programming (MOLP) is proved. MOLP problems may have many efficient solutions, and the objective values of the solutions may not be equal to those of efficient solutions of their dual problems. We show that the necessary and sufficient condition for existence of a pair of primal and dual efficient solutions with equal objective values is as follows if a variable in the primal problem is positive, then the corresponding relation in the dual is equality, and if a relation in the dual problem is not equality, then the corresponding variable in the primal must be zero.