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Third Party Logistics Planning with Routing and Inventory Costs

  • Alexandra M. Newman
  • Candace A. Yano
  • Philip M. Kaminsky
Part of the Applied Optimization book series (APOP, volume 98)

Abstract

We address a scheduling and routing problem faced by a third-party logistics provider in planning its day-of-week delivery schedule and routes for a set of existing and/or prospective customers who need to make shipments to their customers (whom we call “end-customers”). The goal is to minimize the total cost of transportation and inventory while satisfying a customer service requirement that stipulates a minimum number of visits to each customer each week and satisfaction of time-varying demand at the end-customers. Explicit constraints on the minimum number of visits to each customer each week give rise to interdependencies that result in a dimension of problem difficulty not commonly found in models in the literature. Our model includes two other realistic factors that the third-party logistics provider needs to consider: the cost of holding inventory borne by end-customers if deliveries are not made “just-in-time” and the possibility of multiple vehicle visits to an end-customer in the same period (day).

We develop a solution procedure based on Lagrangian relaxation in which the particular form of the relaxation provides strong bounds. One of the subproblems that arises from the relaxation serves to integrate the impact of the timing of deliveries to the various end-customers with inventory decisions, which not only contributes to the strong lower bound that the relaxation provides, but also yields a mathematical structure with some unusual characteristics; we develop an optimal polynomial-time solution procedure for this subproblem. We also consider two variants of the original problem with more restrictive assumptions that are usually imposed implicitly in many vehicle routing problems. Computational results indicate that the Lagrangian procedure performs well for both the original problem and the variants. In many realistic cases, the imposition of the additional restrictive assumptions does not significantly affect the quality of the solutions but substantially reduces computational effort.

Keywords

Third-party logistics vehicle scheduling period vehicle routing problem inventory routing problem delivery scheduling 

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References

  1. Aggarwal, A. and J.K. Park, 1990. Improved Algorithms for Economic Lot-Size Problems. Working Paper, Laboratory for Computer Science, MIT, Cambridge, MA.Google Scholar
  2. Anily, S. and M. Tzur, 2002. Shipping Multiple Items by Capacitated Vehicles-An Optimal Dynamic Programming Approach. Working paper, Faculty of Management, Tel Aviv University, Tel Aviv, Israel. To appear in Transportation Science.Google Scholar
  3. 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(16), 1710–1720.zbMATHGoogle Scholar
  4. Bard, J.F., L. Huang, P. Jaillet and M.A. Dror, 1998. A Decomposition Approach to the Inventory Routing Problem with Satellite Facilities. Transportation Science 32(2), 189–203.zbMATHGoogle Scholar
  5. Bell, W.J., L.M. Dalberto, M.L. Fisher, A.J. Greenfield, R. Jaikumar, P. Kedia, R.G. Mack and P.J. Prutzman, 1983. Improving the Distribution of Industrial Gases with an On-Line Computerized Routing and Scheduling Optimizer. Interfaces 13(6), 4–23.Google Scholar
  6. Camerini, P., L. Fratta and F. Maffioli, 1975. On Improving Relaxation Methods by Modified Gradient Techniques. Mathematical Programming Study 3, 26–34.MathSciNetGoogle Scholar
  7. Carter, M.W., J.M. Farvolden, G. Laporte and J. Xu, 1996. Solving an Integrated Logistics Problem Arising in Grocery Distribution. INFOR 34(4), 290–306.zbMATHGoogle Scholar
  8. Chan, L.M., A. Federgruen and D. Simchi-Levi, 1998. Probabilistic Analyses and Practical Algorithms for Inventory-Routing Models. Operations Research 46(1), 96–106.zbMATHGoogle Scholar
  9. Chandra, P., 1993. A Dynamic Distribution Model with Warehouse and Customer Replenishment Requirements. Journal of the Operational Research Society 44(7), 681–692.zbMATHGoogle Scholar
  10. Chandra, P. and M.L. Fisher, 1994. Coordination of Production and Distribution Planning. European Journal of Operational Research 72(3), 503–517.CrossRefzbMATHGoogle Scholar
  11. Chao, I.M., B.L. Golden and E.A. Wasil, 1995. A New Heuristic for the Period Traveling Salesman Problem. Computers and Operations Research 22(5), 553–565.CrossRefzbMATHGoogle Scholar
  12. Chien, T.W., A. Balakrishnan and R.T. Wong, 1989. An Integrated Inventory Allocation and Vehicle Routing Problem. Transportation Science 23(2), 67–76.MathSciNetzbMATHGoogle Scholar
  13. Dror, M., M. Ball and B. Golden, 1985. A Computational Comparison of Algorithms for the Inventory Routing Problem. Annals of Operations Research 4, 3–23.MathSciNetCrossRefGoogle Scholar
  14. Dror, M. and L. Levy, 1986. A Vehicle Routing Improvement Algorithm Comparison of a “Greedy” and a Matching Implementation for Inventory Routing. Computers and Operations Research 13(1), 33–45.CrossRefzbMATHGoogle Scholar
  15. Dror, M. and P. Trudeau, 1996. Cash Flow Optimization in Delivery Scheduling. European Journal of Operational Research 88, 504–515.CrossRefzbMATHGoogle Scholar
  16. Federgruen, A. and G. van Ryzin, 1997. Probabilistic Analysis of a Combined Aggregation and Math Programming Heuristic for a General Class of Vehicle Routing and Scheduling Problems. Management Science 43(8), 1060–1078.zbMATHGoogle Scholar
  17. Federgruen, A. and P. Zipkin, 1984. A Combined Vehicle Routing and Inventory Allocation Problem. Operations Research 32(5), 1019–1037.MathSciNetzbMATHGoogle Scholar
  18. Federgruen, A., G. Prastacos and P.H. Zipkin, 1986. An Allocation and Distribution Model for Perishable Products. Operations Research 34(1), 75–82.zbMATHGoogle Scholar
  19. Federgruen, A. and M. Tzur, 1991. A Simple Forward Algorithm to Solve General Dynamic Lot Sizing Models with n periods in O(n log n) or O(n) Time. Management Science 37(8), 909–925.zbMATHGoogle Scholar
  20. Florian, M. and M. Klein, 1971. Deterministic Production Planning with Concave Costs and Capacity Constraints. Management Science 18(1), 12–20.MathSciNetGoogle Scholar
  21. Garfinkel, R.S. and G.L. Nemhauser, 1969. The Set Partitioning Problem: Set Covering with Equality Constraints. Operations Research 17, 848–856.zbMATHGoogle Scholar
  22. Gaudioso, M. and G. Paletta, 1992. A Heuristic for the Periodic Vehicle-Routing Problem. Transportation Science 26(2), 86–92.zbMATHGoogle Scholar
  23. Held, M., P. Wolfe and H. Crowder, 1974. Validation of Subgradient Optimization. Mathematical Programming 6, 62–88.MathSciNetCrossRefzbMATHGoogle Scholar
  24. Herer, Y.T. and R. Levy, 1997. The Metered Inventory Routing Problem, an Integrative Heuristic Algorithm. International Journal of Production Economics 51(1, 2), 69–81.CrossRefGoogle Scholar
  25. Herer, Y. and R. Roundy, 1997. Heuristics for a One-Warehouse Multiretailer Distribution Problem with Performance Bounds. Operations Research 45(1), 102–115.zbMATHGoogle Scholar
  26. Larson, R.C., 1988. Transporting Sludge to the 106-mile Site: An Inventory/Routing Model for Fleet Sizing and Logistics System Design. Transportation Science 22(3), 186–198.Google Scholar
  27. Lee, C.-G., Y.A. Bozer and C.C. White, III, 2003. A Heuristic Approach and Properties of Optimal Solutions to the Dynamic Inventory Routing Problem. Working paper, Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario, Canada.Google Scholar
  28. Lee, C.-Y., 1989. A Solution to the Multiple Set-Up Problem with Dynamic Demand. HE Transactions 21(3), 266–270.Google Scholar
  29. Lippman, S., 1969. Optimal Inventory Policy with Multiple Set-up Costs. Management Science 16(1), 118–138.zbMATHMathSciNetGoogle Scholar
  30. Love, S.F., 1973. Bounded Production and Inventory Models with Piecewise Concave Costs. Management Science 20(3), 313–318.zbMATHMathSciNetGoogle Scholar
  31. Metters, R.D., 1996. Interdependent Transportation and Production Activity at the United States Postal Service. Journal of the Operational Research Society 47(1), 27–37.Google Scholar
  32. Pochet, Y. and L.A. Wolsey, 1993. Lot-Sizing with Constant Batches: Formulation and Valid Inequalities. Mathematics of Operations Research 18(4), 767–785.MathSciNetzbMATHGoogle Scholar
  33. Qu, W.W., J.H. Bookbinder, and P. Iyogun, 1999. An Integrated Inventory-Transportation System with Modified Periodic Policy for Multiple Products. European Journal of Operational Research 115, 254–269.CrossRefzbMATHGoogle Scholar
  34. Russell, R.A. and D. Gribbin, 1991. A Multiphase Approach to the Period Vehicle Routing Problem. Networks 21(7), 747–765.zbMATHGoogle Scholar
  35. Viswanathan, S. and K. Mathur, 1997. Integrating Routing and Inventory Decisions in One-Warehouse Multiretailer Multiproduct Distribution Systems. Management Science 43(3), 294–312.zbMATHGoogle Scholar
  36. Wagelmans, A., S. Van Hoesel and A. Kolen, 1992. Economic Lot Sizing: An O(n log n) Algorithm that Runs in Linear Time in the Wagner Whitin Case. Operations Research 40, Suppl. No. 1, S145–S156.Google Scholar
  37. Wagner, H.M. and T.M. Whitin, 1958. Dynamic Version of the Economic Lot Size Model. Management Science 5(1), 89–96.MathSciNetCrossRefGoogle Scholar
  38. Webb, I.R. and R.C. Larson, 1995. Period and Phase of Customer Replenishment-A New Approach to the Strategic Inventory/Routing Problem. European Journal of Operational Research 85(1), 132–148.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Alexandra M. Newman
    • 1
  • Candace A. Yano
    • 2
  • Philip M. Kaminsky
    • 3
  1. 1.Division of Economics and BusinessColorado School of MinesGolden
  2. 2.Department of Industrial Engineering and Operations Research and The Haas School of BusinessUniversity of CaliforniaBerkeley
  3. 3.Department of Industrial Engineering and Operations ResearchUniversity of CaliforniaBerkeley

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