Adapted cross entropy method to investigate costly price-changes in pricing and production planning

Abstract

In this paper, we propose an adaptation of the cross entropy (CE) method, called ACE, to solve an integrated production planning and pricing problem in which price change is not costless. More specifically, we consider a firm with a common production capacity shared amongst multiple products. We show that by using a chance constrained approach we can convert the case of uncertain price dependent demand to a model similar to the deterministic case. Both systems of fixed and variable price change costs are studied. This problem arises in the management of manufacturing systems where it is necessary to find a policy that is both economical and operational from the production perspective. The above problem is mathematically formulated as a mixed integer nonlinear program. Solving such problems is algorithmically very challenging. In fact, commercial codes fail to solve or even find a feasible solution to realistic size problems. The challenge originates from the fact that an optimum should be found despite the difficulty of finding even a feasible solution. The ACE method shows promise in solving optimisation problems regardless of continuity or other assumptions. In our approach, we sample the integer variables using the CE mechanism, and solve simplified nonlinear programming problems (NLP) to obtain corresponding continuous variables. Numerical results, on a range of test problems with sufficient complexity to reflect the difficulty of practical size problems demonstrate the effectiveness of our methodology.