Annals of Operations Research

, Volume 110, Issue 1–4, pp 83–106 | Cite as

An Inventory-Location Model: Formulation, Solution Algorithm and Computational Results

  • Mark S. Daskin
  • Collette R. Coullard
  • Zuo-Jun Max Shen


We introduce a distribution center (DC) location model that incorporates working inventory and safety stock inventory costs at the distribution centers. In addition, the model incorporates transport costs from the suppliers to the DCs that explicitly reflect economies of scale through the use of a fixed cost term. The model is formulated as a non-linear integer-programming problem. Model properties are outlined. A Lagrangian relaxation solution algorithm is proposed. By exploiting the structure of the problem we can find a low-order polynomial algorithm for the non-linear integer programming problem that must be solved in solving the Lagrangian relaxation subproblems. A number of heuristics are outlined for finding good feasible solutions. In addition, we describe two variable forcing rules that prove to be very effective at forcing candidate sites into and out of the solution. The algorithms are tested on problems with 88 and 150 retailers. Computation times are consistently below one minute and compare favorably with those of an earlier proposed set partitioning approach for this model (Shen, 2000; Shen, Coullard and Daskin, 2000). Finally, we discuss the sensitivity of the results to changes in key parameters including the fixed cost of placing orders. Significant reductions in these costs might be expected from e-commerce technologies. The model suggests that as these costs decrease it is optimal to locate additional facilities.


Distribution Center Fixed Cost Lagrangian Relaxation Integer Programming Problem Safety Stock 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Axsater, Using the deterministic EOQ formula in stochastic inventory control, Management Science 42 (1996) 830-834.Google Scholar
  2. [2]
    M. Balinski, Integer programming: Methods, uses, computation, Management Science 13 (1965) 253-313.Google Scholar
  3. [3]
    R.T. Berger, C.R. Coullard and M.S. Daskin, Modeling and solving location-routing problems with route-length constraints, Working paper, Northwestern University (1998).Google Scholar
  4. [4]
    O. Berman, P. Jaillet and D. Simchi-Levi, Location-routing problems with uncertainty, in: Facilities Location, ed. Z. Drezner (Springer, 1995) pp. 427-452.Google Scholar
  5. [5]
    L.M.A. Chan, A. Federgruen and D. Simchi-Levi, Probabilistic analysis and practical algorithms for inventory routing models, Operations Research 46 (1998) 96-106.Google Scholar
  6. [6]
    C. Daganzo, Logistics Systems Analysis, Lecture Notes in Economics and Mathematical Systems, eds. M. Beckmann and W. Krelle (Springer, Berlin, 1991).Google Scholar
  7. [7]
    M.S. Daskin, Network and Discrete Location: Models, Algorithms, and Applications (Wiley, New York, 1995).Google Scholar
  8. [8]
    M.S. Daskin and S.H. Owen, Location models in transportation, in: Handbook of Transportation Science, ed. R. Hall (Kluwer Academic, Norwell, MA, 1999) pp. 311-360.Google Scholar
  9. [9]
    Z. Drezner, ed., Facility Location: A Survey of Applications and Methods (Springer, New York, 1995).Google Scholar
  10. [10]
    G. Eppen, Effects of centralization on expected costs in a multi-location newsboy problem, Management Science 25(5) (1979) 498-501.Google Scholar
  11. [11]
    S.J. Erlebacher and R.D. Meller, The interaction of location and inventory in designing distribution systems, IIE Transactions 32 (2000) 155-166.Google Scholar
  12. [12]
    D. Erlenkotter, A dual-based procedure for uncapacitated facility location, Operations Research 14 (1978) 361-368.Google Scholar
  13. [13]
    A. Federgruen and D. Simchi-Levi, Analytical analysis of vehicle routing and inventory routing problems, in: Network Routing, eds. M. Ball, T. Magnanti, C. Monma and G. Nemhauser, Handbooks in Operations Research and Management Science, Vol. 8 (North-Holland, Amsterdam, 1995) pp. 297-373.Google Scholar
  14. [14]
    A. Federgruen and P. Zipkin, A combined vehicle routing and inventory allocation problem, Management Science 32 (1984) 1019-1036.Google Scholar
  15. [15]
    M.L. Fisher, The Lagrangian relaxation method for solving integer programming problems, Management Science 27 (1981) 1-18.Google Scholar
  16. [16]
    M.L. Fisher, An applications oriented guide to Lagrangian relaxation, Interfaces 15(2) (1985) 2-21.Google Scholar
  17. [17]
    S.C. Graves, A.H.G. Rinnooy Kan and P.H. Zipkin, Logistics of Production and Inventory (Elsevier, Amsterdam, 1993).Google Scholar
  18. [18]
    S. Hakimi, Optimum location of switching centers and the absolute centers and medians of a graph, Operations Research 12 (1964) 450-459.Google Scholar
  19. [19]
    S. Hakimi, Optimum location of switching centers in a communications network and some related graph theoretic problems, Operations Research 13 (1965) 462-475.Google Scholar
  20. [20]
    W. Hopp and M.L. Spearman, Factory Physics: Foundations of Manufacturing Management (Irwin, Chicago, 1996).Google Scholar
  21. [21]
    A.J. Kleywegt, V.S. Nori and M.L.P. Savelsbergh, The stochastic inventory routing problem, Working paper, Georgia Institute of Technology, 2000.Google Scholar
  22. [22]
    M. Korkel, On the exact solution of large-scale simple plant location problems, European Journal of Operations Research 39 (1989) 157-173.Google Scholar
  23. [23]
    G. Laporte, Location-routing problems, in: Vehicle Routing: Methods and Studies, eds. B.L. Golden and A.A. Assad (North-Holland, Amsterdam, 1988) pp. 163-197.Google Scholar
  24. [24]
    G. Laporte and P.J. Dejax, Dynamic location-routing problems, Journal of the Operations Research Society 40(5) (1989) 471-482.Google Scholar
  25. [25]
    H. Lee, Information distortion in a supply chain: The bullwhip effect, Management Science 43 (1996) 546-558.Google Scholar
  26. [26]
    H. Lee, V. Padmanabhan and S. Whang, The bullwhip effect in supply chains, Sloan Management Review, Spring (1997) 93-102.Google Scholar
  27. [27]
    H. Min, V. Jayaraman and R. Srivastava, Combined location-routing problems: A synthesis and future research directions, European Journal of Operational Research 108 (1998) 1-15.Google Scholar
  28. [28]
    D.C. Montgomery, G.C. Runger and N.F. Hubele, Engineering Statistics (Wiley, New York, 1998).Google Scholar
  29. [29]
    S. Nahmias, Production and Operations Management, 3rd edn. (Irwin, Chicago, 1997).Google Scholar
  30. [30]
    Y. Sheffi, Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods (Prentice-Hall, Englewood Cliffs, NJ, 1985).Google Scholar
  31. [31]
    Z.J. Shen, Efficient algorithms for various supply chain problems, Ph.D. dissertation, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL (2000).Google Scholar
  32. [32]
    Z.J. Shen, C.R. Coullard and M.S. Daskin, A joint location-inventory model, to appear in Transportation Science.Google Scholar
  33. [33]
    D. Simchi-Levi, P. Kaminsky and E. Simchi-Levi, Designing and Managing the Supply Chain: Concepts, Strategies and Case Studies (Irwin McGraw Hill, Boston, MA, 2000).Google Scholar
  34. [34]
    M.B. Teitz and P. Bart, Heuristic methods for estimating generalized vertex median of a weighted graph, Operations Research 16 (1968) 955-961.Google Scholar
  35. [35]
    C.P. Teo, J. Ou and M. Goh, Impact on inventory costs with consolidation of distribution centers, IIE Transactions 33 (2001) 99-110.Google Scholar
  36. [36]
    S. Viswanathan and K. Mathur, Integrating routing and inventory decisions in one-warehouse multiretailer multiproduct distribution system, Management Science 43 (1997) 294-312.Google Scholar
  37. [37]
    Y.-S. Zheng, On properties of stochastic inventory systems, Management Science 38(1) (1992) 87-103.Google Scholar
  38. [38]
    P.H. Zipkin, Foundations of Inventory Management (Irwin, Burr Ridge, IL, 1997).Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Mark S. Daskin
    • 1
  • Collette R. Coullard
    • 1
  • Zuo-Jun Max Shen
    • 1
  1. 1.Department of Industrial Engineering and Management SciencesNorthwestern UniversityEvanstonUSA

Personalised recommendations