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Mathematical Programming Models and Formulations for Deterministic Production Planning Problems

  • Yves Pochet
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2241)

Abstract

We study in this lecture the literature on mixed integer programming models and formulations for a specific problem class, namely deterministic production planning problems. The objective is to present the classical optimization approaches used, and the known models, for dealing with such management problems.

We describe first production planning models in the general context of manufacturing planning and control systems, and explain in which sense most optimization solution approaches are based on the decomposition of the problem into single-item subproblems.

Then we study in detail the reformulations for the core or simplest subproblem in production planning, the single-item uncapacitated lot-sizing problem, and some of its variants. Such reformulations are either obtained by adding variables - to obtain so called extended reformulations - or by adding constraints to the initial formulation. This typically allows one to obtain a linear description of the convex hull of the feasible solutions of the subproblem. Such tight reformulations for the subproblems play an important role in solving the original planning problem to optimality.

We then review two important classes of extensions for the production planning models, capacitated models and multi-stage or multi-level models. For each, we describe the classical modeling approaches used. Finally, we conclude by giving our personal view on some new directions to be investigated in modeling production planning problems. These include better models for capacity utilization and setup times, new models to represent the product structure - or recipes - in process industries, and the study of continuous time planning and scheduling models as opposed to the discrete time models studied in this review.

Keywords

Setup Time Production Planning Valid Inequality Mathematical Programming Model Production Planning Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    P. Afentakis, B. Gavish and U. Karmarkar, “Computationally efficient optimal solutions to the lot-sizing problem in multistage assembly systems”, Management Science 30(2), 222–239, 1984.zbMATHGoogle Scholar
  2. 2.
    P. Afentakis and B. Gavish, “Optimal lot-sizing algorithms for complexpro duct structures”, Operations Research 34, 237–249, 1986.MathSciNetzbMATHGoogle Scholar
  3. 3.
    A. Aggarwal and J. Park, “Improved algorithms for economic lot-size problems”, Operations Research 41, 549–571, 1993.MathSciNetzbMATHGoogle Scholar
  4. 4.
    A. Agra and M. Constantino, “Lotsizing with backlogging and start-ups: the case ofWagner-Whitin costs”, Operations Research Letters 25 (2), 81–88, 1999.MathSciNetzbMATHGoogle Scholar
  5. 5.
    R.N. Anthony, “Planning and control systems: a framework for analysis”, Harvard University Press, Cambridge, Mass., 1965.Google Scholar
  6. 6.
    I. Barany, T.J. Van Roy and L.A. Wolsey, “Uncapacitated lot sizing: the convex hull of solutions”, Mathematical Programming Study 22, 32–43, 1984.MathSciNetzbMATHGoogle Scholar
  7. 7.
    I. Barany, T.J. Van Roy and L.A. Wolsey, “Strong formulations for multi-item capacitated lot-sizing”, Management Science 30, 1255–1261, 1984.MathSciNetzbMATHGoogle Scholar
  8. 8.
    C. Batta and J. Teghem, “Optimization of production scheduling in plastics processing industry”, Jorbel (Belgian Journal of operations research, statistics and computer science) 34 (2), 55–78, 1994.zbMATHGoogle Scholar
  9. 9.
    P.J. Billington, J.O. McClain, L.J. Thomas, “áMathematical Programming approaches to capacity constrained MRP systemsá: review, formulation and problem reductioná”, Management Science 29 (10), 1126–1141, 1983.zbMATHGoogle Scholar
  10. 10.
    W.-H. Chen and J.-M. Thizy, “Analysis of relaxations for the multi-item capacitated lot-sizing problem”, Annals of Operations Research 26, 29–72, 1990.MathSciNetzbMATHGoogle Scholar
  11. 11.
    A.J. Clark and H. Scarf, “Optimal policies for multi echelon inventory problems” Management Science 6, 475–490, 1960.Google Scholar
  12. 12.
    M. Constantino, “A cutting plane approach to capacitated lot-sizing with startup costs”, Mathematical Programming 75 (3), 353–376, 1996.MathSciNetzbMATHGoogle Scholar
  13. 13.
    M. Constantino, “Lower bounds in lot-sizing models: A polyhedral study”, Mathematics of Operations Research 23 (1), 101–118, 1998.MathSciNetzbMATHGoogle Scholar
  14. 14.
    W.B. Crowston, M.H. Wagner, “Dynamic lot size models for multi stage assembly systems”, Management Science 20(1), 14–21, 1973.zbMATHGoogle Scholar
  15. 15.
    W.B. Crowston, M.H. Wagner, J.F. Williams, “Economic lot size determination in multi stage assembly systems”, Management Science 19(5), 517–527, 1973.zbMATHGoogle Scholar
  16. 16.
    Z. Degraeve, F. Roodhooft, “Improving the efficiency of the purchasing process using total cost of ownership informationá: The case of heating electrodes at Cockerill-Sambre S.A.á”, European Journal of Operational Research 112, 42–53, 1999.zbMATHGoogle Scholar
  17. 17.
    M. Diaby, H.C. Bahl, M.H. Karwan and S. Zionts, “A Lagrangean relaxation approach to very large scale capacitated lot-sizing”, Management Science 38(9), 1329–1340, 1992.zbMATHGoogle Scholar
  18. 18.
    S.E. Elmaghraby, “The economic lot-scheduling problem (ELSP): reviews and extensions”, Management Science 24, 587–598, 1978.zbMATHGoogle Scholar
  19. 19.
    G.D. Eppen and R.K. Martin, Solving multi-item lot-sizing problems using variable definition, Operations Research 35, 832–848, 1987.zbMATHGoogle Scholar
  20. 20.
    A. Federgrun and M. Tsur, “A simple forward algorithm to solve general dynamic lot-size models with n periods in O(nlogn) or O(n) time”, Management Science 37, 909–925, 1991.Google Scholar
  21. 21.
    B. Fleischmann, “The discrete lotsizing and scheduling problem”, European Journal of Operational Research 44(3), 337–348, 1990.MathSciNetzbMATHGoogle Scholar
  22. 22.
    B. Fleischmann, “The discrete lotsizing and scheduling problem with sequencedependent setup costs”, European Journal of Operational Research, 1994.Google Scholar
  23. 23.
    M. Florian and M. Klein, “Deterministic production planning with concave costs and capacity constraints”, Management Science 18, 12–20, 1971.MathSciNetzbMATHGoogle Scholar
  24. 24.
    M.X. Goemans, “Valid inequalities and separation for mixed 0-1 constraints with variable upper bounds”, Operations Research Letters 8, 315–322, 1989.MathSciNetzbMATHGoogle Scholar
  25. 25.
    M. Grotschel, L. Lovasz and A. Schrijver, “The ellipsoid method and its consequences in combinatorial optimization”, Combinatorica 1, 169–197, 1981.MathSciNetzbMATHGoogle Scholar
  26. 26.
    O. Gunluk and Y. Pochet, “Mixing mixed integer rounding inequalities”, CORE discussion paper 9811, Université catholique de Louvain, Belgium, 1998. (to appear in Mathematical Programming)zbMATHGoogle Scholar
  27. 27.
    Haase, “Lotsizing and scheduling for production planning”, Lecture notes in economics and mathematical systems 408, Springer, Berlin, 1994.Google Scholar
  28. 28.
    F.W. Harris, “How many parts to make at once”, Factory, the Magazine of Management 10(2), 1913.Google Scholar
  29. 29.
    A.C. Haxand H.C. Meal, “Hierarchical integration of production planning and scheduling”, in M. Geisler editor, TIMS studies in Management Science, chapter 1, North Holland/American Elsevier, New York, 1975.Google Scholar
  30. 30.
    S. Kang, K. Malik, L.J. Thomas, “Lotsizing and scheduling in parallel machines with sequence dependent setup costs”,Working paper 97–07, Johnson Graduate school of Management, Cornell university, 1997.Google Scholar
  31. 31.
    U.S. Karmarkar and L. Schrage, “The deterministic dynamic product cycling problem”, Operations Research 33, 326–345, 1985.zbMATHGoogle Scholar
  32. 32.
    A. Kimms, “Multi-level lot sizing and scheduling: methods for capacitated, dynamic and deterministic models”, Physica-Verlag (production and logistics series), Heidelberg, 1997.Google Scholar
  33. 33.
    E. Kondili, C.C. Pantelides, R.W.H. Sargent, “A general algorithm for shortterm scheduling of batch operations-1. MILP formulation”, Computers Chemical Engineering 17, 211–227, 1993.Google Scholar
  34. 34.
    J. Krarup and O. Bilde, “Plant location, set covering and economic lot sizes: an O(mn) algorithm for structured problems”, in “Optimierung bei Graphentheoretischen und Ganzzahligen Probleme”, L. Collatz et al. eds, Birkhauser Verlag, Basel, 155–180, 1977.Google Scholar
  35. 35.
    R. Kuik, M. Salomon and L.N. van Wassenhove, “Batching decisions: structure and models”, European Journal of Operational Research 75, 243–263, 1994.Google Scholar
  36. 36.
    L.S. Lasdon and R.C. Terjung, “An efficient algorithm for multi-item scheduling”, Operations Research 19, 946–969, 1971.MathSciNetzbMATHGoogle Scholar
  37. 37.
    J. Leung, T.M. Magnanti and R. Vachani, “Facets and algorithms for capacitated lot-sizing”, Mathematical Programming 45, 331–359, 1989.MathSciNetzbMATHGoogle Scholar
  38. 38.
    M. Loparic, Y. Pochet and L.A. Wolsey, “Uncapacitated lot-sizing with sales and safety stocks”, Mathematical Programming 89 (3), 487–504, 2001.MathSciNetzbMATHGoogle Scholar
  39. 39.
    M. Loparic, H. Marchand and L.A. Wolsey, “Dynamic knapsack sets and capacitated lot-sizing”, CORE discussion paper 2000/47, Université catholique de Louvain, Louvain-la-Neuve, Belgium, 2000.Google Scholar
  40. 40.
    L. Lovasz, “Graph theory and integer programming”, Annals of Discrete Mathematics 4, 141–158, 1979.MathSciNetzbMATHGoogle Scholar
  41. 41.
    S.F. Love, “A facilities in series inventory model with nested schedules”, Management Science 18(5), 327–338, 1972.zbMATHGoogle Scholar
  42. 42.
    T.M. Magnanti and R. Vachani, “A strong cutting plane algorithm for production scheduling with changeover costs”, Operations Research 38, 456–473, 1990.MathSciNetzbMATHGoogle Scholar
  43. 43.
    H. Marchand and L.A. Wolsey, “The 0-1 knapsack problem with a single continuous variable”, Mathematical Programming 85, 15–33, 1999.MathSciNetzbMATHGoogle Scholar
  44. 44.
    R.K. Martin, “Generating alternative mixed-integer programming models using variable redefinition”, Operations Research 35, 331–359, 1987.MathSciNetzbMATHGoogle Scholar
  45. 45.
    R.K. Martin, “Using separation algorithms to generate mixed integer model reformulations”, Operations Research Letters 10, 119–128, 1991.MathSciNetzbMATHGoogle Scholar
  46. 46.
    A.J. Miller, G.L. Nemhauser and M.W.P. Savelsbergh, “On the polyhedral structure of a multi-item production planning model with setup times”, CORE discussion paper 2000/52, Université catholique de Louvain, Louvain-la-Neuve, Belgium, 2000.Google Scholar
  47. 47.
    G.L. Nemhauser and L.A. Wolsey, “Integer and combinatorial optimization”, Wiley, New York, 1988.zbMATHGoogle Scholar
  48. 48.
    G.L. Nemhauser and L.A. Wolsey, “A recursive procedure for generating all cuts for 0-1 mixed integer programs”, Mathematical Programming 46, 379–390, 1990.MathSciNetzbMATHGoogle Scholar
  49. 49.
    J. Orlicky, “Material Requirements planning”, McGraw-Hill, New York, 1975.Google Scholar
  50. 50.
    M.W. Padberg, T.J. Van Roy and L.A. Wolsey, “Valid inequalities for fixed charge problems”, Operations Research 33, 842–861, 1985.MathSciNetzbMATHGoogle Scholar
  51. 51.
    Pinto and Grossmann, “A logic-based approach to scheduling problems with resource constraints”, Computers and Chemical Engineering 21 (8), 801–818, 1997.Google Scholar
  52. 52.
    Y. Pochet, “Lot-sizing problems: reformulations and cutting plane algorithms”, PhD Thesis, Université Catholique de Louvain, Belgium, 1987.Google Scholar
  53. 53.
    Y. Pochet, “Valid inequalities and separation for capacitated economic lotsizing”, Operations Research Letters 7, 109–116, 1988.MathSciNetzbMATHGoogle Scholar
  54. 54.
    Y. Pochet and L.A. Wolsey, “Lot-size models with backlogging: Strong formulations and cutting planes”, Mathematical Programming 40, 317–335, 1988.MathSciNetzbMATHGoogle Scholar
  55. 55.
    Y. Pochet and L.A. Wolsey, “Solving multi-item lot sizing problems using strong cutting planes”, Management Science 37, 53–67, 1991.zbMATHGoogle Scholar
  56. 56.
    Y. Pochet and L.A. Wolsey, “Lot-sizing with constant batches: Formulation and valid inequalities”, Mathematics of Operations Research 18, 767–785, 1993.MathSciNetzbMATHGoogle Scholar
  57. 57.
    Y. Pochet and L.A. Wolsey, “Polyhedra for lot-sizing with Wagner-Whitin costs”, Mathematical Programming 67, 297–323, 1994.MathSciNetzbMATHGoogle Scholar
  58. 58.
    Y. Pochet and L.A. Wolsey, “Algorithms and reformulations for lot-sizing problems”, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 20, 245–293, 1995.MathSciNetzbMATHGoogle Scholar
  59. 59.
    Y. Pochet, T. Tahmassebi and L.A. Wolsey, “Reformulation of a single stage packing model”, Report, Memips: Esprit project 20118, June 1996.Google Scholar
  60. 60.
    R.L. Rardin and U. Choe, “Tighter relaxations of fixed charge network flow problems”, report J-79-18, Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia, 1979.Google Scholar
  61. 61.
    R. Rardin and L.A. Wolsey, “Valid inequalities and projecting the multicommodity extended formulation for uncapacitated fixed charge network flow problems”, European Journal of Operations Research, Nov 1993.Google Scholar
  62. 62.
    M. Salomon, “Deterministic lotsizing models for production planning”, PhD. Thesis, Erasmus Universiteit Rotterdam, The Netherlands, 1990.zbMATHGoogle Scholar
  63. 63.
    M. Salomon, L.G. Kroon, R. Kuik, L.N. Van Wassenhove, “Some extensions of the discrete lotsizing and scheduling problem”, Management Science 37(7), 801–812, 1991.zbMATHGoogle Scholar
  64. 64.
    N.C. Simpson, S.S. Erenguc, “Production planning in multiple stage manufacturing environments with joint costs, limited resources nad set-up times”, Technical report, Department of Management Science and Systems, University of Buffalo, 1998.Google Scholar
  65. 65.
    H. Stadtler, “Mixed integer programming model formulations for dynamic multi-item multi-level capacitated lotsizing”, European Journal of Operational Research 94 (3), 561–581, 1996.MathSciNetzbMATHGoogle Scholar
  66. 66.
    H. Tempelmeier and M. Derstro., “A Lagrangean-based heuristic for dynamic multilevel multiitem constrained lotsizing with setup times”, Management Science 42 (5), 738–757, 1996.zbMATHGoogle Scholar
  67. 67.
    J.M. Thizy and L.N. Van Wassenhove “Lagrangean relaxation for the multiitem capacitated lot-sizing problem: a heuristic implementation”, IIE Transactions 17 (4), 308–313, 1985.Google Scholar
  68. 68.
    W. Trigeiro, L.J. Thomas and J.O. McClain, “Capacitated lot sizing with setup times”, Management Science 35(3), 353–366, 1989.Google Scholar
  69. 69.
    F. Vanderbeck, “Lot-sizing with start up times”, Management Science 44 (10), 1409–1425, 1998.zbMATHGoogle Scholar
  70. 70.
    W. Van de Velde, Private communication, 1997.Google Scholar
  71. 71.
    C.P.M. van Hoesel, “Models and algorithms for single-item lot sizing problems”, Ph.D. Thesis, Erasmus Universiteit, Rotterdam, 1991.Google Scholar
  72. 72.
    S. van Hoesel, A. Wagelmans and L.A. Wolsey, “Economic lot-sizing with startup costs: the convex hull”, SIAM Journal of Discrete Mathematics, 1994.Google Scholar
  73. 73.
    S. van Hoesel and A. Wagelmans, “An O(T-3) algorithm for the economic lotsizing problem with constant capacities”, Management Science 42 (1), 142–150, 1996.zbMATHGoogle Scholar
  74. 74.
    T.J. Van Roy and L.A. Wolsey, “Valid inequalities and separation for uncapacitated fixed charge networks”, Operations Research Letters 4, 105–112, 1985.zbMATHGoogle Scholar
  75. 75.
    A.F. Veinott, “Minimum concave cost solution of Leontief substitution models of multi-facility inventory systems”, Operations Research 17(2), 262–291, 1969.MathSciNetzbMATHGoogle Scholar
  76. 76.
    T.E. Vollman, W.L. Berry, and D.C. Whybark, “Manufacturing Planning and Control Systems”, Third Edition, Richard D. Irwin., 1997.Google Scholar
  77. 77.
    A.P.M. Wagelmans, C.P.M. van Hoesel and A.W.J. Kolen, “Economic lotsizing: an O(nlogn) algorithm that runs in linear time in the Wagner-Whitin case”, Operations Research 40, Supplement 1, 145–156, 1992.MathSciNetzbMATHGoogle Scholar
  78. 78.
    H.M. Wagner and T.M. Whitin, “Dynamic version of the economic lot size model”, Management Science 5, 89–96, 1958.MathSciNetzbMATHGoogle Scholar
  79. 79.
    H. Westenberger and J. Kallrath, “Formulation of a jobshop problem in process industry”, Preprint, Bayer, Leverkusen, January 1995.Google Scholar
  80. 80.
    R.H. Wilson, “A scientific routine for stock control”, Harvard Business Review 13, 1934.Google Scholar
  81. 81.
    L.A. Wolsey, “Uncapacitated lot-sizing problems with start-up costs”, Operations Research 37, 741–747, 1989.MathSciNetzbMATHGoogle Scholar
  82. 82.
    L.A. Wolsey, “MIP modelling of changeovers in production planning and scheduling problems”, European Journal of Operational Research 99, 154–165, 1997.zbMATHGoogle Scholar
  83. 83.
    L.A. Wolsey, “Integer programming”, Wiley, New York, 1999.zbMATHGoogle Scholar
  84. 84.
    W.I. Zangwill, “Minimum concave cost flows in certain networks”, Management Science 14, 429–450, 1968.MathSciNetzbMATHGoogle Scholar
  85. 85.
    W.I. Zangwill, “A backlogging model and a multi-echelon model of a dynamic economic lot size production system-a network approach”, Management Science 15(9), 506–527, 1969.MathSciNetzbMATHGoogle Scholar
  86. 86.
    X. Zhang and R.W.H. Sargent, “A new unified formulation for process scheduling”, AIChE annual meeting, Paper 144c, St Louis, Missouri, 1993.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Yves Pochet
    • 1
  1. 1.CORE and IAGUniversité Catholique de LouvainBelgium

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