A stochastic production planning problem with nonlinear cost

Most production planning models are deterministic and often assume a linear relation between production volume and production cost. In this paper, we investigate a production planning problem in a steel production process considering the energy consumption cost which is a nonlinear function of the production quantity. Due to the uncertain environment, the production demands are stochastic. Taking a scenario-based approach to express the stochastic demands according to the knowledge of planners on the demand distributions, we formulate the stochastic production planning problem as a mixed integer nonlinear programming (MINLP) model. Approximated with the piecewise linear functions, the MINLP model is transformed into a mixed integer linear programming model. The approximation error can be improved by adjusting the linearization ranges repeatedly. Based on the piecewise linearization, a stepwise Lagrangian relaxation (SLR) heuristic for the problem is proposed where variable splitting is introduced during Lagrangian relaxation (LR). After decomposition, one subproblem is solved by linear programming and the other is solved by an effective polynomial time algorithm. The SLR heuristic is tested on a large set of problem instances and the results show that the algorithm generates solutions very close to optimums in an acceptable time. The impact of demand uncertainty on the solution is studied by a computational discussion on scenario generation. (C) 2011 Published by Elsevier Ltd.