An approximation algorithm for multi-objective optimization problems using a box-coverage

For a continuous multi-objective optimization problem, it is usually not a practical approach to compute all its nondominated points because there are infinitely many of them. For this reason, a typical approach is to compute an approximation of the nondominated set. A common technique for this approach is to generate a polyhedron which contains the nondominated set. However, often these approximations are used for further evaluations. For those applications a polyhedron is a structure that is not easy to handle. In this paper, we introduce an approximation with a simpler structure respecting the natural ordering. In particular, we compute a box-coverage of the nondominated set. To do so, we use an approach that, in general, allows us to update not only one but several boxes whenever a new nondominated point is found. The algorithm is guaranteed to stop with a finite number of boxes, each being sufficiently thin.

A Pseudo-Polynomial Time Algorithm for a Special Multiobjective Order Picking Problem

In this study, we work on the order picking problem (OPP) in a specially designed warehouse with a single picker. Ratliff and Rosenthal [Operations Research31(3) (1983) 507–521] show that the special design of the warehouse and use of one picker lead to a polynomially solvable case. We address the multiobjective version of this special case and investigate the properties of the nondominated points. We develop an exact algorithm that finds any nondominated point and present an illustrative example. Finally we conduct a computational test and report the results.