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A Noniterative Algorithm for the Linear-Quadratic Profit-Maximization Model for Smoothing Multiproduct Production


  • Jaya Singhal

    () (Merrick School of Business, University of Baltimore, Baltimore, Maryland 21201)

  • Kalyan Singhal

    () (Merrick School of Business, University of Baltimore, Baltimore, Maryland 21201)


In multiproduct-production and workforce-smoothing problems, the objective is to determine the levels of employment and production for each product in each period that will maximize the total profit over a planning horizon of N periods given quadratic revenue and cost functions for M products. The planned levels of production, in turn, determine the planned levels of inventory or backorders for each product in each period. A real-world problem may involve 20,000 products and 12 periods, leading to over 480,000 variables. This requires solving simultaneous equations with as many variables. We exploited the staircase structure of optimality conditions to develop an algorithm that requires solving, without iterations, simultaneous equations with only three variables. The computation time, regardless of the number of variables in the model, is of the order of one second on a Vax 11/780. This algorithm also facilitates sensitivity analysis and generation of alternate plans for production, the workforce, and sales.

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  • Jaya Singhal & Kalyan Singhal, 2008. "A Noniterative Algorithm for the Linear-Quadratic Profit-Maximization Model for Smoothing Multiproduct Production," INFORMS Journal on Computing, INFORMS, vol. 20(2), pages 169-178, May.
  • Handle: RePEc:inm:orijoc:v:20:y:2008:i:2:p:169-178
    DOI: 10.1287/ijoc.1070.0227

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    References listed on IDEAS

    1. Dobos, Imre, 2003. "Optimal production-inventory strategies for a HMMS-type reverse logistics system," International Journal of Production Economics, Elsevier, vol. 81(1), pages 351-360, January.
    2. Morton I. Kamien & Lode Li, 1990. "Subcontracting, Coordination, Flexibility, and Production Smoothing in Aggregate Planning," Management Science, INFORMS, vol. 36(11), pages 1352-1363, November.
    3. Warren H. Hausman & John O. McClain, 1971. "A Note on the Bergstrom-Smith Multi-Item Production Planning Model," Management Science, INFORMS, vol. 17(11), pages 783-785, July.
    4. Stephen C. Graves, 1982. "Using Lagrangean Techniques to Solve Hierarchical Production Planning Problems," Management Science, INFORMS, vol. 28(3), pages 260-275, March.
    5. Nam, Sang-jin & Logendran, Rasaratnam, 1992. "Aggregate production planning -- A survey of models and methodologies," European Journal of Operational Research, Elsevier, vol. 61(3), pages 255-272, September.
    6. P. R. Kleindorfer & C. H. Kriebel & G. L. Thompson & G. B. Kleindorfer, 1975. "Discrete Optimal Control of Production Plans," Management Science, INFORMS, vol. 22(3), pages 261-273, November.
    7. Rein Peterson, 1971. "Optimal Smoothing of Shipments in Response to Orders," Management Science, INFORMS, vol. 17(9), pages 597-607, May.
    8. Gabriel R. Bitran & Elizabeth A. Haas & Arnoldo C. Hax, 1981. "Hierarchical Production Planning: A Single Stage System," Operations Research, INFORMS, vol. 29(4), pages 717-743, August.
    9. Charles C. Holt, 2002. "Learning How to Plan Production, Inventories, and Work Force," Operations Research, INFORMS, vol. 50(1), pages 96-99, February.
    10. Sampath Rajagopalan & Jayashankar M. Swaminathan, 2001. "A Coordinated Production Planning Model with Capacity Expansion and Inventory Management," Management Science, INFORMS, vol. 47(11), pages 1562-1580, November.
    11. Simon, Herbert A, 1979. "Rational Decision Making in Business Organizations," American Economic Review, American Economic Association, vol. 69(4), pages 493-513, September.
    12. Gary L. Bergstrom & Barnard E. Smith, 1970. "Multi-Item Production Planning--An Extension of the HMMS Rules," Management Science, INFORMS, vol. 16(10), pages 614-629, June.
    13. Hamilton, James D., 2002. "On the interpretation of cointegration in the linear-quadratic inventory model," Journal of Economic Dynamics and Control, Elsevier, vol. 26(12), pages 2037-2049, October.
    14. C. Van De Panne & P. Bosje, 1962. "Sensitivity Analysis of Cost Coefficient Estimates: The Case of Linear Decision Rules for Employment and Production," Management Science, INFORMS, vol. 9(1), pages 82-107, October.
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