Skip to main content
SpringerLink
Log in

Approximation algorithms for supply chain planning and logistics problems with market choice

  • Full Length Paper
  • Series A
  • Published:
Mathematical Programming Submit manuscript

Abstract

We propose generalizations of a broad class of traditional supply chain planning and logistics models that we call supply chain planning and logistics problems with market choice. Instead of the traditional setting, we are given a set of markets, each specified by a sequence of demands and associated with a revenue. Decisions are made in two stages. In the first stage, one chooses a subset of markets and rejects the others. Once that market choice is made, one needs to construct a minimum-cost production plan (set of facilities) to satisfy all of the demands of all the selected markets. The goal is to minimize the overall lost revenues of rejected markets and the production (facility opening and connection) costs. These models capture important aspects of demand shaping within supply chain planning and logistics models. We introduce a general algorithmic framework that leverages existing approximation results for the traditional models to obtain corresponding results for their counterpart models with market choice. More specifically, any LP-based α-approximation for the traditional model can be leveraged to a \({\frac{1}{1-e^{-1/\alpha}}}\) -approximation algorithm for the counterpart model with market choice. Our techniques are also potentially applicable to other covering problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aggarwal A., Park J.K.: Improved algorithms for economic lot-sizing problems. Oper. Res. 14, 549–571 (1993)

    Article  MathSciNet  Google Scholar 

  2. Arkin E., Joneja D., Roundy R.: Computational complexity of uncapacitated multi-echelon production planning problems. Oper. Res. Lett. 8, 61–66 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. Balas E.: The prize collection travelling salesman problem. Networks 19, 621–636 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bárány I., Van Roy T.J., Wolsey L.A.: Uncapacitated lot-sizing: the convex hull of solutions. Math. Program. Study 22, 32–43 (1984)

    MATH  Google Scholar 

  5. Bartal Y., Leonardi S., Spaccamela A.M., Sgall J., Stougie L.: Multiprocessor scheduling with rejection. SIAM J. Discret. Math. 13, 64–78 (2000)

    Article  MATH  Google Scholar 

  6. Bertsimas D., Teo C., Vohra R.: On dependent randomized rounding algorithms. Oper. Res. Lett. 25, 105–114 (1999)

    Article  MathSciNet  Google Scholar 

  7. Bienstock D., Goemans M.X., Simchi-Levi D., Williamson D.: A note on the prize collecting traveling salesman problem. Math. Program. 59, 413–420 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  8. Byrka, J.: An optimal bifactor approximation algorithm for the metric uncapacitated facility location problem. In: Proceedings of APPROX-RANDOM, pp. 29–43 (2007)

  9. Chan A., Muriel A., Shen Z.-J., Simchi-Levi D., Teo C.-P.: Effectiveness of zero inventory ordering policies for an one-warehouse multi-retailer problem with piecewise linear cost structures. Manage. Sci. 48, 1446–1460 (2000)

    Article  Google Scholar 

  10. Charikar, M., Khuller, S., Mount, D.M., Narasimhan, G.: Algorithms for facility location problems with outliers. In: Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 642–651 (2001)

  11. Chudak, F.A., Shmoys, D.B.: Improved approximation algorithms for the capacitated facility location problem. In: Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. S875–S876, (1999)

  12. Federgruen A., Tzur M.: A simple forward algorithm to solve general dynamic lot-sizing models with n periods in O(n log n) or O(n) time. Manage. Sci. 37, 909–925 (1991)

    Article  MATH  Google Scholar 

  13. Feige U.: A threshold of ln(n) for approximating set-cover. J. ACM 45, 634–652 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Geunes J., Romeijn H.E., Taaffe K.: Requirements planning with order selection flexibility. Oper. Res. 54, 394–401 (2006)

    Article  MATH  Google Scholar 

  15. Goemans, M.X.: Approximate solutions of hard combinatorial problems, Lecture notes in the school of ASHCOMP (1996)

  16. Goemans M.X., Williamson D.P.: A general approximation technique for constrained forest problems. SIAM J. Comput. 24, 296–317 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hoesel A.V., Wagelmans A., Kolen A.: A dual algorithm for the economic lot-sizing problem. Eur. J. Oper. Res. 52, 315–325 (1991)

    Article  MATH  Google Scholar 

  18. Krarup, J., Bilde, O.: Plant location, set covering and economic lot sizing: an O(mn) algorithm for structured problems. In: Numerische Methoden Bei Optimierungsaufgaben, vol. 3, pp. 155–180 (1977)

  19. Levi, R., Geunes, J., Romeijn, E.H., Shmoys, D.B.: Inventory and facility location models with market selection. In: Proceedings of the 12th IPCO, pp. 111–124, (2005)

  20. Levi R., Roundy R.O., Shmoys D.B.: Primal-dual algorithms for deterministic inventory problems. Math. Oper. Res. 31, 267–284 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Levi R., Roundy R.O., Shmoys D.B., Sviridenko M.: A constant approximation algorithm for the one-warehouse multi-retailer problem. Manage. Sci. 54, 763–776 (2008)

    Article  Google Scholar 

  22. Magnanti, T.L., Stratila, D.: Separable concave optimization approximately equals piecewise linear optimization. In: Proceedings of the 11th IPCO, pp. 234–243 (2004)

  23. Magnanti, T.L., Stratila, D.: Strongly polynomial algorithms for concave cost combinatorial optimization problems. (2007) (in preparation)

  24. Müller A., Stoyan D.: Comparison Methods for Stochastic Models and Risks. Wiley, London (2002)

    MATH  Google Scholar 

  25. Pochet Y., Wolsey L.A.: Lot-sizing with constant batches: formulation and valid inequalities. Math. Oper. Res. 18, 767–785 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  26. van den Heuvel, W., Kundakcioglu, O.E., Geunes, J., Romeijn, H.E., Sharkey, T.C., Wagelmans, A.P.M.: Integrated market selection and production planning: complexity and solution approaches. Unpublished manuscript (2009)

  27. Wagelmans A.P.M., van Hoesel C.P.M., Kolen A.: Economic lot-sizing: An O(n log n) algorithm that runs in linear time in the Wagner-Whitin case. Oper. Res. 40, 145–156 (1992)

    Article  MathSciNet  Google Scholar 

  28. Wagner H.M., Whitin T.M.: Dynamic version of the economic lot sizing model. Manage. Sci. 5, 89–96 (1958)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Industrial and Systems Engineering, University of Florida, P.O. Box 116595, Gainesville, FL, 32611-6595, USA

    Joseph Geunes

  2. Sloan School of Management, M.I.T., Cambridge, MA, 02139, USA

    Retsef Levi

  3. Department of Industrial and Operations Engineering, University of Michigan, 1205 Beal Avenue, Ann Arbor, MI, 48109-2117, USA

    H. Edwin Romeijn

  4. School of ORIE and Department of Computer Science, Cornell University, Ithaca, NY, 14853, USA

    David B. Shmoys

Authors
  1. Joseph Geunes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Retsef Levi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. Edwin Romeijn
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. David B. Shmoys
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Retsef Levi.

Additional information

The work of J. Geunes was supported by the National Science Foundation under Grants No. DMI-0322715 and DMI-0355533. Part of this work was done while R. Levi was a PhD student at the ORIE Department in Cornell University. The work of R. Levi was supported in part by NSF grants DMS-0732175 and CMMI-0846554 (CAREER Award), an AFOSR award FA9550-08-1-0369, an SMA grant and the Buschbaum Research Fund of MIT and by Motorola. The work of H. E. Romeijn was supported by the National Science Foundation under Grant No. DMI-0355533. The work of D. B. Shmoys was supported in part by the National Science Foundation under Grants No. CCR-0635121, DMI-0500263, DMS-0732196, and CCF-0832782.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Geunes, J., Levi, R., Romeijn, H.E. et al. Approximation algorithms for supply chain planning and logistics problems with market choice. Math. Program. 130, 85–106 (2011). https://doi.org/10.1007/s10107-009-0310-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-009-0310-9

Keywords

Mathematics Subject Classification (2000)

Search

Navigation