Models and solution techniques for production planning problems with increasing byproducts
- 168 Downloads
We consider a production planning problem where the production process creates a mixture of desirable products and undesirable byproducts. In this production process, at any point in time the fraction of the mixture that is an undesirable byproduct increases monotonically as a function of the cumulative mixture production up to that time. The mathematical formulation of this continuous-time problem is nonconvex. We present a discrete-time mixed-integer nonlinear programming (MINLP) formulation that exploits the increasing nature of the byproduct ratio function. We demonstrate that this new formulation is more accurate than a previously proposed MINLP formulation. We describe three different mixed-integer linear programming (MILP) approximation and relaxation models of this nonconvex MINLP, and we derive modifications that strengthen the linear programming relaxations of these models. We also introduce nonlinear programming formulations to choose piecewise-linear approximations and relaxations of multiple functions that share the same domain and use the same set of break points in the domain. We conclude with computational experiments that demonstrate that the proposed formulation is more accurate than the previous formulation, and that the strengthened MILP approximation and relaxation models can be used to obtain provably near-optimal solutions for large instances of this nonconvex MINLP. Experiments also illustrate the quality of the piecewise-linear approximations produced by our nonlinear programming formulations.
KeywordsMixed integer nonlinear programming Piecewise linear approximation Production planning
We thank Stephen J. Wright for several suggestions that helped improve this work, and we thank Ignacio Grossmann for bringing the reference  to our attention. We are grateful to the anonymous referees for suggestions that helped improve the paper.
- 3.Beale, E.M.L., Tomlin, J.A.: Special facilities in a general mathematical programming system for non-convex problems using ordered sets of variables. Oper. Res. 69, 447–454 (1970)Google Scholar
- 8.Dantzig, G.B.: On the significance of solving linear programming problems with some integer variables. Econ. J. Econ. Soc., 28, 30–44 (1960)Google Scholar
- 11.Goldberg, N., Kim, Y., Leyffer, S., Veselka, T.D.: Adaptively refined dynamic program for linear spline regression. Technical Report Preprint ANL/MCS-P3040-0912, Argonne National Laboratory (2012)Google Scholar
- 13.Imamoto, A., Tang, B.: Optimal piecewise linear approximation of convex functions. In: Proceedings of the World Congress on Engineering and Computer Science, pp. 1191–1194 (2008)Google Scholar
- 16.Markowitz, H.M., Manne, A.S.: On the solution of discrete programming problems. Econ. J. Econ. Soc., 25, 84–110 (1957)Google Scholar
- 20.Rote, G.: The convergence rate of the sandwich algorithm for approximating convex functions. Computing 48, 337–361 (1992)Google Scholar