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Parametric Mixed Integer Programming: An Application to Solid Waste Management

Author

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  • Larry Jenkins

    (Royal Military College of Canada, Kingston, Ontario)

Abstract

A method is developed for carrying out parametric analysis on a mixed integer linear program (MILP) as either objective function coefficients or right-hand-side values of the constraints are varied continuously. The method involves solving MILPs at point values of the parameters of variation and joining the results by LP parametric analysis. The procedure for parametric analysis on the objective function can be continued until a theoretical result proves that the analysis is complete. However, a heuristic rule that is presented may greatly reduce the number of individual MILPs that have to be solved and thus reduce the total computational effort. The rule assumes that if the same values of the integer variables are optimal at two different values of the change parameter, these same integer variable values will also be optimal at all intermediate parameter values. If the rule is applied, a complete parametric analysis requires solving 2n different MILPs, where n is the number of different sets of optimal integer variable values found. We determine the savings in number of MILPs solved using the heuristic rule compared to continuing the procedure to obtain theoretical proof of completeness. The paper also proposes various possible "advanced start" aids for solving the individual MILPs. Previously there has existed no method that could be used to perform continuous right-hand-side parametric analysis on an MILP of realistic size. A heuristic rule presented here makes possible such an analysis. The procedure again requires solving 2n different MILPs. The procedures for both objective function and right-hand-side analysis can be implemented using any software that can solve MILPs and perform LP parametric analysis. Computational results are reported using commercially available packages. The procedures are applied to a large fixed-charge model that was developed to analyze where to locate resource recovery plants in Southeastern Ontario. In that model it was essential to consider very wide variation in values of the parameters related to resource recovery. It was found that the results of the parametric analyses could be distilled into clear recommendations regarding which plants to build if a resource recovery policy is implemented.

Suggested Citation

  • Larry Jenkins, 1982. "Parametric Mixed Integer Programming: An Application to Solid Waste Management," Management Science, INFORMS, vol. 28(11), pages 1270-1284, November.
    • Handle: RePEc:inm:ormnsc:v:28:y:1982:i:11:p:1270-1284
    DOI: 10.1287/mnsc.28.11.1270
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    Citations

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    Cited by:

    1. Crema, Alejandro, 1998. "A procedure to verify the completeness of the right-hand-side parametric analysis for a mixed integer linear programming problem," European Journal of Operational Research, Elsevier, vol. 108(3), pages 684-695, August.
    2. Huang, G. H. & Baetz, B. W. & Patry, G. G., 1997. "A response to "A comment on 'Grey integer programming: An application to waste management planning under uncertainty"' by Larry Jenkins," European Journal of Operational Research, Elsevier, vol. 100(3), pages 638-641, August.
    3. Tcha, Dong-wan & Myung, Young-soo & Chung, Ki-ho, 1995. "Parametric uncapacitated facility location," European Journal of Operational Research, Elsevier, vol. 86(3), pages 469-479, November.
    4. Jie Xiong & Shuming Wang & Tsan Sheng Ng, 2021. "Robust Bilevel Resource Recovery Planning," Production and Operations Management, Production and Operations Management Society, vol. 30(9), pages 2962-2992, September.
    5. Su, Jun-Pin & Chiueh, Pei-Te & Hung, Ming-Lung & Ma, Hwong-Wen, 2007. "Analyzing policy impact potential for municipal solid waste management decision-making: A case study of Taiwan," Resources, Conservation & Recycling, Elsevier, vol. 51(2), pages 418-434.
    6. Mitsos, Alexander & Barton, Paul I., 2009. "Parametric mixed-integer 0-1 linear programming: The general case for a single parameter," European Journal of Operational Research, Elsevier, vol. 194(3), pages 663-686, May.
    7. Crema, Alejandro, 2002. "The multiparametric 0-1-integer linear programming problem: A unified approach," European Journal of Operational Research, Elsevier, vol. 139(3), pages 511-520, June.
    8. Crema, Alejandro, 1995. "Average shadow price in a mixed integer linear programming problem," European Journal of Operational Research, Elsevier, vol. 85(3), pages 625-635, September.
    9. Mukherjee, Saral & Chatterjee, A.K., 2006. "The average shadow price for MILPs with integral resource availability and its relationship to the marginal unit shadow price," European Journal of Operational Research, Elsevier, vol. 169(1), pages 53-64, February.
    10. Crema, Alejandro, 2000. "An algorithm for the multiparametric 0-1-integer linear programming problem relative to the objective function," European Journal of Operational Research, Elsevier, vol. 125(1), pages 18-24, August.
    11. Pati, Rupesh Kumar & Vrat, Prem & Kumar, Pradeep, 2008. "A goal programming model for paper recycling system," Omega, Elsevier, vol. 36(3), pages 405-417, June.
    12. Dai, C. & Li, Y.P. & Huang, G.H., 2012. "An interval-parameter chance-constrained dynamic programming approach for capacity planning under uncertainty," Resources, Conservation & Recycling, Elsevier, vol. 62(C), pages 37-50.
    13. Jenkins, L., 2000. "Selecting scenarios for environmental disaster planning," European Journal of Operational Research, Elsevier, vol. 121(2), pages 275-286, March.
    14. Jenkins, Larry, 1996. "A comment on "Grey integer programming: an application to waste management planning under uncertainty" by Guo H. Huang, Brian W. Baetz, Gilles G. Patry : European Journal of Operational Rese," European Journal of Operational Research, Elsevier, vol. 89(3), pages 671-670, March.
    15. Tsan Sheng Ng & Shuming Wang, 2017. "Recycling systems design using reservation incentive data," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 68(10), pages 1236-1258, October.
    16. Crema, Alejandro, 2002. "An algorithm to perform a complete parametric analysis relative to the constraint matrix for a 0-1-integer linear program," European Journal of Operational Research, Elsevier, vol. 138(3), pages 484-494, May.
    17. Shuming Wang & Tsan Sheng Ng & Manyu Wong, 2016. "Expansion planning for waste‐to‐energy systems using waste forecast prediction sets," Naval Research Logistics (NRL), John Wiley & Sons, vol. 63(1), pages 47-70, February.
    18. Crema, Alejandro, 1999. "An algorithm to perform a complete right-hand-side parametrical analysis for a 0-1-integer linear programming problem," European Journal of Operational Research, Elsevier, vol. 114(3), pages 569-579, May.
    19. Myung, Young-Soo & Kim, Hu-gon & Tcha, Dong-wan, 1997. "A bi-objective uncapacitated facility location problem," European Journal of Operational Research, Elsevier, vol. 100(3), pages 608-616, August.
    20. Wang, Hsiao-Fan & Horng, Jyh-Shing, 1996. "Structural approach to parametric analysis of an IP on the case of the right-hand side," European Journal of Operational Research, Elsevier, vol. 92(1), pages 148-156, July.

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