Journal of Global Optimization

, Volume 59, Issue 2–3, pp 597–631

# Models and solution techniques for production planning problems with increasing byproducts

• Srikrishna Sridhar
• Jeffrey Linderoth
• James Luedtke
Article

## Abstract

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.

## Keywords

Mixed integer nonlinear programming Piecewise linear approximation Production planning

## Notes

### Acknowledgments

We thank Stephen J. Wright for several suggestions that helped improve this work, and we thank Ignacio Grossmann for bringing the reference [12] to our attention. We are grateful to the anonymous referees for suggestions that helped improve the paper.

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© Springer Science+Business Media New York 2014

## Authors and Affiliations

• Srikrishna Sridhar
• 1
• Jeffrey Linderoth
• 2
• James Luedtke
• 2
1. 1.School of Computer SciencesUniversity of Wisconsin-MadisonMadisonUSA
2. 2.Department of Industrial and Systems EngineeringUniversity of Wisconsin-MadisonMadisonUSA