A new (λ,𝜖)-constraint method for min-max bi-objective programming problems

Abstract

Min-max bi-objective programming (MMBOP) has a wide range of applications in practical fields such as management and transportation; however, solving such problems is challenging. In this paper, we propose a new [Formula: see text]-constraint method for solving this class of problems. In our approach, the bound on the objective function is converted into an [Formula: see text]-constraint formed by a convex combination of the objective’s best and worst possible values. By adjusting the coefficient of the convex combination, the partitioning results of any grid points with respect to the current objective function can be obtained. To simultaneously optimize the objective function subject to the constraints, a penalty parameter [Formula: see text] is introduced. We augment the optimization objective function by adding the product of [Formula: see text] and the difference between the constrained objective function and its bound. A single-objective optimization model for MMBOP is then established, and a new [Formula: see text]-constraint method is proposed to solve it. Furthermore, we provide a theoretical proof that the proposed model yields better solutions than the traditional model. Finally, numerical results obtained from solving the test problems, which include knapsack problems of varying dimensions and a real-world portfolio selection problem, demonstrate the competitive and practical performance of the [Formula: see text]-constraint method.