Meta-Heuristic Algorithms for a Multi-Product Dynamic Lot-Sizing Problem with a Freight Container Cost

  • Received : 2012.05.20
  • Accepted : 2012.08.03
  • Published : 2012.09.30


Lot sizing and shipment scheduling are two interrelated decisions made by a manufacturing plant and a third-party logistics distribution center. This paper analyzes a dynamic inbound ordering problem and shipment problem with a freight container cost, in which the order size of multiple products and single container type are simultaneously considered. In the problem, each ordered product placed in a period is immediately shipped by some freight containers in the period, and the total freight cost is proportional to the number of containers employed. It is assumed that the load size of each product is equal and backlogging is not allowed. The objective of this study is to simultaneously determine the lot-sizes and the shipment schedule that minimize the total costs, which consist of production cost, inventory holding cost, and freight cost. Because the problem is NP-hard, we propose three meta-heuristic algorithms: a simulated annealing algorithm, a genetic algorithm, and a new population-based evolutionary meta-heuristic called self-evolution algorithm. The performance of the meta-heuristic algorithms is compared with a local search heuristic proposed by the previous paper in terms of the average deviation from the optimal solution in small size problems and the average deviation from the best one among the replications of the meta-heuristic algorithms in large size problems.


Dynamic Lot-Sizing;Shipment Scheduling;Multi-Products;Meta-Heuristics


  1. Anily, S. and Tzur, M. (2005), Shipping multiple items by capacitated vehicles: an optimal dynamic programming approach, Transportation Science, 39(2), 233-248.
  2. Barbarosoglu, G. and Ozdamar, L. (2000), Analysis of solution space-dependent performance of simulated annealing: the case of the multi-level capacitated lot sizing problem, Computers and Operations Research, 27(9), 895-903.
  3. Bitran, G. R. and Yanasse, H. H. (1982), Computational complexity of the capacitated lot size problem, Management Science, 28(10), 1174-1186.
  4. Florian, M., Lenstra, J. K., and Rinnooy Kan, A. H. G. (1980), Deterministic production planning: algorithms and complexity, Management Science, 26(7), 669-679.
  5. Fumero, F. and Vercellis, C. (1999), Synchronized development of production, inventory, and distribution schedules, Transportation Science, 33(3), 330- 340.
  6. Goldberg, D. E. (1989), Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley Pub., Reading, MA
  7. Hwang, H. and Sohn, K. I. (1985), An optimal policy for the dynamic transportation-inventory model with deteriorating items, IIE Transactions, 17(3), 233-241.
  8. Jaruphongsa, W., Cetinkaya, S., and Lee, C. Y. (2005), A dynamic lot-sizing model with multi-mode replenishments: polynomial algorithms for special cases with dual and multiple modes, IIE Transactions, 37(5), 453-467.
  9. Joo, C. M. and Kim, B. S. (2012), Non-identical parallel machine scheduling with sequence and machine dependent setup times using meta-heuristic algorithms, Industrial Engineering and Management Systems, 11(1), 114-122.
  10. Kim, B. S. and Lee, W. S. (2012), A multi-product dynamic inbound ordering and shipment scheduling problem at a third-party warehouse, International Journal of Industrial Engineering, Forthcoming.
  11. Kirkpatrick, S., Gelatt, C. D. Jr., and Vecchi, M. P. (1983), Optimization by simulated annealing, Science, 220 (4598), 671-680.
  12. Lee, C. Y. (1989), A solution to the multiple set-up problem with dynamic demand, IIE Transactions, 21(3), 266-270.
  13. Lee, C. Y., Cetinkaya, S., and Jaruphongsa, W. (2003), A dynamic model for inventory lot sizing and outbound shipment scheduling at a third-party warehouse, Operations Research, 51(5), 735-747.
  14. Lee, W. S., Han, J. H., and Cho, S. J. (2005), A heuristic algorithm for a multi-product dynamic lot-sizing and shipping problem, International Journal of Production Economics, 98(2), 204-214.
  15. Lippman, S. A. (1969), Optimal inventory policy with multiple set-up costs, Management Science, 16(1), 118-138.
  16. Van Norden, L. and van de Velde, S. (2005), Multi-product lot-sizing with a transportation capacity reservation contract, European Journal of Operational Research, 165(1), 127-138.
  17. Wagner, H. M. and Whitin, T. M. (1958), Dynamic version of the economic lot size model, Management Science, 5(1), 89-96.
  18. Wolsey, L. A. (1998), Integer Programming, Wiley, New York, NY.
  19. Zangwill, W. I. (1969), A backlogging model and a multiechelon model of a dynamic economic lot size production system: a network approach, Management Science, 15(9), 506-527.