Lagrangian Relaxation Bounds for a Production-Inventory-Routing Problem

  • Agostinho AgraEmail author
  • Adelaide Cerveira
  • Cristina Requejo
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10122)


We consider a single item Production-Inventory-Routing problem with a single producer/supplier and multiple retailers. Inventory management constraints are considered both at the producer and at the retailers, following a vendor managed inventory approach, where the supplier monitors the inventory at retailers and decides on the replenishment policy for each retailer. We assume a constant production capacity. Based on the mathematical formulation we discuss a classical Lagrangian relaxation which allows to decompose the problem into four subproblems, and a new Lagrangian decomposition which decomposes the problem into just a production-inventory subproblem and a routing subproblem. The new decomposition is enhanced with valid inequalities. A computational study is reported to compare the bounds from the two approaches.


Inventory routing Lagrangian relaxation Lagrangian decomposition Lower bounds 



The research of the first and third authors was supported through CIDMA and FCT, the Portuguese Foundation for Science and Technology, within project UID/MAT/ 04106/2013. The research of the second author was financed by the ERDF - European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation - COMPETE 2020 Programme within project “POCI-01-0145-FEDER-006961”, and by FCT within project UID/EEA/50014/2013.


  1. 1.
    Adulyasak, Y., Cordeau, J., Jans, R.: The production routing problem: a review of formulations and solution algorithms. Comput. Oper. Res. 55, 141–152 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Adulyasak, Y., Cordeau, J., Jans, R.: Formulations and branch and cut algorithms for multi-vehicle production and inventory routing problems. Inf. J. Comput. 26(1), 103–120 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Agra, A., Andersson, H., Christiansen, M., Wolsey, L.: A maritime inventory routing problem: discrete time formulations and valid inequalities. Networks 62, 297–314 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Agra, A., Christiansen, M., Delgado, A.: Mixed integer formulations for a short sea fuel oil distribution problem. Transp. Sci. 47, 108–124 (2013)CrossRefGoogle Scholar
  5. 5.
    Agra, A., Christiansen, M., Delgado, A., Simonetti, L.: Hybrid heuristics for a short sea inventory routing problem. Eur. J. Oper. Res. 236, 924–935 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Agra, A., Christiansen, M., Ivarsoy, K., Solhaug, I., Tomasgard, A.: Combined ship routing and inventory management in the salmon farming industry. Ann. Oper. Res. (in press)Google Scholar
  7. 7.
    Andersson, H., Hoff, A., Christiansen, M., Hasle, G., Løkketangen, A.: Industrial aspects and literature survey: combined inventory management and routing. Comput. Oper. Res. 37(9), 1515–1536 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Archetti, C., Bertazzi, L., Laporte, G., Speranza, M.G.: A branch-and-cut algorithm for a vendor-managed inventory-routing problem. Transp. Sci. 41(3), 382–391 (2007)CrossRefGoogle Scholar
  9. 9.
    Bell, W.J., Dalberto, L.M., Fisher, M.L., Greenfield, A.J., Jaikumar, R., Kedia, P.: Improving the distribution of industrial gases with an on-line computerized routing and scheduling optimizer. Interfaces 13(6), 4–23 (1983)CrossRefGoogle Scholar
  10. 10.
    Christiansen, M.: Decomposition of a combined inventory and time constrained ship routing problem. Transp. Sci. 33(1), 3–16 (1999)CrossRefzbMATHGoogle Scholar
  11. 11.
    Christiansen, M., Fagerholt, K.: Maritime Inventory Routing Problems. In: Floudas, C., Pardalos, P. (eds.) Encyclopedia of Optimization, 2nd edn, pp. 1947–1955. Springer, New York (2009)Google Scholar
  12. 12.
    Eksioglu, S.D., Romeijn, H.E., Pardalos, P.M.: Cross-facility management of production and trasportation planning problem. Comput. Oper. Res. 33(11), 3231–3251 (2006)CrossRefzbMATHGoogle Scholar
  13. 13.
    Eppen, G.D., Martin, R.K.: Solving multi-item capacitated lot-sizing problems using variable redefinition. Oper. Res. 35, 832–848 (1997)CrossRefzbMATHGoogle Scholar
  14. 14.
    FICO Xpress Optimization SuiteGoogle Scholar
  15. 15.
    Fumero, F., Vercellis, C.: Synchronized development of production, inventory, and distribution schedules. Transp. Sci. 33(3), 330–340 (1999)CrossRefzbMATHGoogle Scholar
  16. 16.
    Geunes, J., Pardalos, P.M.: Network optimization in supply chain management and financial engineering: an annotated bibliography. Networks 42(2), 66–84 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Held, M., Wolfe, P., Crowder, H.P.: Validation of subgradient optimization. Math. Program. 6, 62–88 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Miller, C., Tucker, A., Zemlin, R.: Integer programming formulations and travelling salesman problems. J. Assoc. Comput. Mach. 7(4), 326–329 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Pei, J., Pardalos, P.M., Liu, X., Fan, W., Yang, S., Wang, L.: Coordination of production and transportation in supply chain scheduling. J. Ind. Manage. Optim. 11(2), 399–419 (2015)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Ruokokoski, M., Solyali, O., Cordeau, J.-F., Jans, R., Süral, H.: Efficient formulations and a branch-and-cut algorithm for a production-routing problem. GERAD Technical report G-2010-66. HEC Montréal, Canada (2010)Google Scholar
  21. 21.
    Shor, N.Z.: Minimization Methods for Non-Differentiable Functions. Springer, Heidelberg (1985)CrossRefzbMATHGoogle Scholar
  22. 22.
    Solyali, O., Süral, H.: A branch-and-cut algorithm using a strong formulation and an a priori tour-based heuristic for an inventory-routing problem. Transp. Sci. 45(3), 335–345 (2011)CrossRefGoogle Scholar
  23. 23.
    Solyali, O., Süral, H.: A relaxation based solution approach for the inventory control and vehicle routing problem in vendor managed systems. In: Neogy, S.K., Das, A.K., Bapat, R.B. (eds.) Modeling, computation and Optimization, pp. 171–189. World Scientific, Singapore (2009)Google Scholar
  24. 24.
    Solyali, O., Süral, H.: The one-warehouse multi-retailer problem: reformulation, classification and computational results. Ann. Oper. Res. 196(1), 517–541 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Agostinho Agra
    • 1
    Email author
  • Adelaide Cerveira
    • 2
    • 3
  • Cristina Requejo
    • 1
  1. 1.University of Aveiro and CIDMAAveiroPortugal
  2. 2.University of Trás-os-Montes e Alto-DouroVila RealPortugal
  3. 3.INESC TECPortoPortugal

Personalised recommendations