Abstract
We propose a new algorithm for dynamic lot size models (LSM) in which production and inventory cost functions are only assumed to be piecewise linear. In particular, there are no assumptions of convexity, concavity or monotonicity. Arbitrary capacities on both production and inventory may occur, and backlogging is allowed. Thus the algorithm addresses most variants of the LSM appearing in the literature. Computational experience shows it to be very effective on NP-hard versions of the problem. For example, 48 period capacitated problems with production costs defined by eight linear segments are solvable in less than 2.5 minutes of Vax 8600 cpu time.
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References
Aggarwal, A. and J. K. Park (1990), Improved Algorithms for Economic Lot-Size Problems, Working paper, IBM Thomas J. Watson Research Center, Yorktown Heights, New York.
Baker, K. R., P. Dixon, M. J. Magazine, and E. A. Silver (1978). An Algorithm for the Dynamic Lot-Size Problem with Time-Varying Production Capacity Constraints,Management Science 24, 1710–1720.
Bitran, G. R. and H. H. Yanasse (1982), Computational Complexity of the Capacitated Lot Size Problem,Management Science 28, 1174–1186.
Chand, S. and S. Sethi (1983), Finite Production Rate Inventory Models with First and Second Shift Setups,Naval Research Logistics Quarterly 30, 401–414.
Chen, H.-D. and C.-Y. Lee (1991), A Simple Algorithm for the Error Bound of the Dynamic Lot Size Model Allowing Speculative Motive, to appear inIIE Transactions.
Chen, H.-D., D. W. Hearn, and C.-Y. Lee (1994), A New Dynamic Programming Algorithm for Single Item Capacitated Dynamic Lot Size Model,Journal of Global Optimization 4, 285–300.
Chung, C. S. and C. H. M. Lin (1988), AnO(T2) Algorithm for the NI/G/NI/ND capacitated Lot Size Problem,Management Science 34, 420–426.
Chung, C.-S., J. Flynn, and C.-H. M. Lin (1990), An Efficient Algorithm for the Capacitated Lot Size Problem, Working paper, College of Business Administration, Cleveland State University, Cleveland, Ohio.
Denardo, E. V. (1982),Dynamic Programming: Models and Applications. Prentice-Hall, Englewood Cliffs, New Jersey.
Dongarra, J. J. (1989), Performance of Various Computers Using Standard Linear Equations Softwares, Technique Report CS-89-85, Mathematical Science Section, Oak Ridge National Laboratory, Oak Ridge, TN.
Erenguc, S. S. and S. Tufekci (1987), A Branch and Bound Algorithm for a Single-Item Multi-Source Dynamic Lot Sizing Problem with Capacity Constraints,IIE Transactions 19, 73–80.
Erenguc, S. S. and Y. Aksoy (1990), A Branch and Bound Algorithm for a Single Item Nonconvex Dynamic Lot Sizing Problem with Capacity Constraints,Computers and Operations Research 17, 199–210.
Federguren, A. and M. Tzur (1991), A Simple Forward Algorithm to Solve General Dynamic Lot Sizing Models withn Periods inO(n log n) orO(n) Time,Management Science 37, 909–925.
Florian, M. and M. Klein (1971), Deterministic Production Planning with Concave Costs and Capacity Constraints,Management Science 18, 12–20.
Florian, M., J. K. Lenstra, and A. H. G. Rinnooy Kan (1980), Deterministic Production Planning: Algorithms and Complexity,Management Science 26, 669–679.
Jagannathan, R. and M. R. Rao (1973), A Class of Deterministic Production Planning Problems,Management Science 19, 1295–1300.
Kirca, Ö. (1990), An Efficient Algorithm for the Capacitated Single Item Dynamic Lot Size Problem,European Journal of Operational Research 45, 15–24.
Lambert, A. and H. Luss (1982), Production Planning with Time-Dependent Capacity Bounds,European Journal of Operational Research 9, 275–280.
Lambrecht, M. and J. Vander Eecken (1978), A Capacity Constrained Single-Facility Dynamic Lot-Size Model,European Journal of Operational Research 2, 132–136.
Lee, S.-B. and P. H. Zipkin (1989), A Dynamic Lot-Size Model with Make-or-Buy Decisions,Management Science 35, 447–458.
Lippman, S. A. (1969), Optimal Invenstory Policy with Multiple Set-Up Costs,Management Science 16, 118–138.
Love, S. F. (1973), Bounded Production and Inventory Models with Piecewise Concave Costs,Management Science 20, 313–318.
Sethi, S. and S. Chand (1981), Multiple Finite Production Rate Dynamic Lot Size Inventory Models,Operations Research 29, 931–944.
Swoveland, C. (1975), A Deterministic Multi-Period Production Planning Model with Piecewise Concave Production and Holding-Backorderer Costs,Management Science 21, 1007–1013.
Wagelmans, A., S. Van Hoesel, and A. Kolen (1992), Economic Lot-Sizing: anO(n log n)) Algorithm that Runs in Linear Time in the Wagner-Whitin Case,Operations Research 40, S145-S156.
Wagner, H. M. and T. M. Whitin (1958), Dynamic Version of the Economic Lot Size Model,Management Science 5, 89–96.
Zangwill, W. (1966), A Deterministic Multi-Period Production Scheduling Model with Backlogging,Management Science 13, 105–119.
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Chen, HD., Hearn, D.W. & Lee, CY. A dynamic programming algorithm for dynamic lot size models with piecewise linear costs. J Glob Optim 4, 397–413 (1994). https://doi.org/10.1007/BF01099265
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DOI: https://doi.org/10.1007/BF01099265