Journal of Global Optimization

, Volume 59, Issue 2–3, pp 597–631 | Cite as

Models and solution techniques for production planning problems with increasing byproducts

  • Srikrishna Sridhar
  • Jeffrey Linderoth
  • James Luedtke


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.


Mixed 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 [12] to our attention. We are grateful to the anonymous referees for suggestions that helped improve the paper.


  1. 1.
    Balakrishnan, A., Graves, S.C.: A composite algorithm for a concave-cost network flow problem. Networks 19(2), 175–202 (1989)CrossRefGoogle Scholar
  2. 2.
    Beale, E.M.L., Forrest, J.J.H.: Global optimization using special ordered sets. Math. Program. 10(1), 52–69 (1976)CrossRefGoogle Scholar
  3. 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
  4. 4.
    Belotti, P., Lee, J., Liberti, L., Margot, F., Wächter, A.: Branching and bounds tighteningtechniques for non-convex MINLP. Optim. Methods Softw. 24(4–5), 597–634 (2009)CrossRefGoogle Scholar
  5. 5.
    Borghetti, A., D’Ambrosio, C., Lodi, A., Martello, S.: An MILP approach for short-term hydro scheduling and unit commitment with head-dependent reservoir. IEEE Trans. Power Syst. 23(3), 1115–1124 (2008)CrossRefGoogle Scholar
  6. 6.
    Burkard, R.E., Hamacher, H.W., Rote, G.: Sandwich approximation of univariate convex functions with an application to separable convex programming. Naval Res, Logist, 38, 911–924 (1991)CrossRefGoogle Scholar
  7. 7.
    Camponogara, E., de Castro, M.P., Plucenio, A., Pagano, D.J.: Compressor scheduling in oil fields. Optim. Eng. 12(1–2), 153–174 (2011)CrossRefGoogle Scholar
  8. 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
  9. 9.
    Geissler, B., Martin, A., Morsi, A., Schewe, L.: Using piecewise linear functions for solving MINLPs. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming, vol. 154 of the IMA Volumes in Mathematics and its Applications, pp. 287–314. Springer, New York (2012)CrossRefGoogle Scholar
  10. 10.
    Geoffrion, A.M.: Objective function approximations in mathematical programming. Math. Program. 13(1), 23–37 (1977)CrossRefGoogle Scholar
  11. 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
  12. 12.
    Gupta, V., Grossmann, I.E.: An efficient multiperiod MINLP model for optimal planning of offshore oil and gas field infrastructure. Ind. Eng. Chem. Res. 51(19), 6823–6840 (2012)CrossRefGoogle Scholar
  13. 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
  14. 14.
    Iyer, R.R.: Optimal planning and scheduling of offshore oil field infrastructure investment and operations. Ind. Eng. Chem. Res. 37(4), 1380–1397 (1998)CrossRefGoogle Scholar
  15. 15.
    Lee, J., Wilson, D.: Polyhedral methods for piecewise-linear functions I: the lambda method. Discrete Appl. Math. 108(3), 269–285 (2001)CrossRefGoogle Scholar
  16. 16.
    Markowitz, H.M., Manne, A.S.: On the solution of discrete programming problems. Econ. J. Econ. Soc., 25, 84–110 (1957)Google Scholar
  17. 17.
    Martín, M., Grossmann, I.E.: Energy optimization of hydrogen production from lignocellulosic biomass. Comput. Chem. Eng. 35(9), 1798–1806 (2011)CrossRefGoogle Scholar
  18. 18.
    Orero, S.O., Irving, M.R.: A genetic algorithm modelling framework and solution technique for short term optimal hydrothermal scheduling. IEEE Trans. Power Syst. 13(2), 501–518 (1998)CrossRefGoogle Scholar
  19. 19.
    Padberg, M.W., Rijal, M.P.: Location, Scheduling, Design and Integer Programming, vol. 19. Kluwer Academic Norwell, Dordrecht (1996)CrossRefGoogle Scholar
  20. 20.
    Rote, G.: The convergence rate of the sandwich algorithm for approximating convex functions. Computing 48, 337–361 (1992)Google Scholar
  21. 21.
    Sridhar, S., Linderoth, J., Luedtke, J.: Locally ideal formulations for piecewise linear functions with indicator variables. Oper. Res. Lett. 41(6), 627–632 (2013)CrossRefGoogle Scholar
  22. 22.
    Tarhan, B., Grossmann, I.E., Goel, V.: Stochastic programming approach for the planning of offshore oil or gas field infrastructure under decision-dependent uncertainty. Ind. Eng. Chem. Res. 48(6), 3078–3097 (2009)CrossRefGoogle Scholar
  23. 23.
    Tawarmalani, M., Sahinidis, N.V.: Global optimization of mixed-integer nonlinear programs: a theoretical and computational study. Math. Program. 99(3), 563–591 (2004)CrossRefGoogle Scholar
  24. 24.
    Van Den Heever, S.A., Grossmann, I.E.: An iterative aggregation/disaggregation approach for the solution of a mixed-integer nonlinear oilfield infrastructure planning model. Ind. Eng. Chem. Res. 39(6), 1955–1971 (2000)CrossRefGoogle Scholar
  25. 25.
    Van Den Heever, S.A., Grossmann, I.E., Vasantharajan, S., Edwards, K.: Integrating complex economic objectives with the design and planning of offshore oilfield infrastructures. Comput. Chem. Eng. 24(2), 1049–1055 (2000)CrossRefGoogle Scholar
  26. 26.
    Vielma, J.P., Ahmed, S., Nemhauser, G.: Mixed-integer models for nonseparable piecewise-linear optimization: unifying framework and extensions. Oper. Res. 58(2), 303–315 (2010)CrossRefGoogle Scholar
  27. 27.
    Vielma, J.P., Keha, A.B., Nemhauser, G.L.: Nonconvex, lower semicontinuous piecewise linear optimization. Discrete Optim. 5(2), 467–488 (2008)CrossRefGoogle Scholar

Copyright information

© 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

Personalised recommendations