The item dependent stockingcost constraint


In a previous work we introduced a global StockingCost constraint to compute the total number of periods between the production periods and the due dates in a multi-order capacitated lot-sizing problem. Here we consider a more general case in which each order can have a different per period stocking cost and the goal is to minimise the total stocking cost. In addition the production capacity, limiting the number of orders produced in a given period, is allowed to vary over time. We propose an efficient filtering algorithm in O(n log n) where n is the number of orders to produce. On a variant of the capacitated lot-sizing problem, we demonstrate experimentally that our new filtering algorithm scales well and is competitive wrt the StockingCost constraint when the stocking cost is the same for all orders.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12


  1. 1.

    in the same way as minimumAssignment [6, 7] generalizes allDifferent [18], cost-gcc [24] generalizes gcc [22, 23], or cost-regular [4] generalizes regular [19], etc.

  2. 2.

    In typical applications of this constraint, assuming that ct is O(1), the number of orders n is on the order of the horizon T: nO(T).

  3. 3.

    A constraint is bound consistent if, for each minimum and maximum values, there exists a solution wrt the constraint by considering the domains of other variables without holes.

  4. 4.

    The monotonicity ensures that if we prune the upper bound of a variable to a given value, all other values greater than this value in the domain of the variable are inconsistent.

  5. 5.

    A reversible variable is a variable that can restore its domain when backtracks occur during the search.

  6. 6.

    item: order type.

  7. 7.

    idle period: period in which there is no production.


  1. 1.

    Armentano, V.A., Franca, P.M., de Toledo, F.M.B. (1999). A network flow model for the capacitated lot-sizing problem. Omega, 27, 275–284.

    Article  Google Scholar 

  2. 2.

    Barany, I., Roy, T.J.V., Wolsey, L.A. (1984). Strong formulations for multi-item capacitated lot sizing. Management Science, 30, 1255–1261.

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Belvaux, G., & Wolsey, L.A. (2001). Modelling practical lot-sizing problems as mixed integer programs. Management Science, 47, 724–738.

    Article  MATH  Google Scholar 

  4. 4.

    Demassey, S., Pesant, G., Rousseau, L.M. (2006). A cost-regular based hybrid column generation approach. Constraints, 4(11), 315–333.

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Drexl, A., & Kimms, A. (1997). Lot sizing and scheduling - survey and extensions. European Journal of Operational Research, 99, 221–235.

    Article  MATH  Google Scholar 

  6. 6.

    Ducomman, S., Cambazard, H., Penz, B. (2016). Alternative filtering for the weighted circuit constraint: Comparing lower bounds for the tsp and solving tsptw. In 13th AAAI conference on artificial intelligence.

  7. 7.

    Focacci, F., Lodi, A., Milano, M. (1999). Cost-based domain filtering. In Principles and practice of constraint programming–CP’99 (pp. 189–203). Springer.

  8. 8.

    Gay, S., Hartert, R., Lecoutre, C., Schaus, P. (2015). Conflict ordering search for scheduling problems. In Principles and practice of constraint programming - CP 2015 (pp. 144–148). Springer.

  9. 9.

    German, G., Cambazard, H., Gayon, J.P., Penz, B. (2015). Une contrainte globale pour le lot sizing. In Journée francophone de la programation par contraintes - JFPC 2015 (pp. 118–127).

  10. 10.

    Ghomi, S.M.T.F., & Hashemin, S.S. (2001). An analytical method for single level-constrained resources production problem with constant set-up cost. Iranian Journal of Science and Technology, 26(B1), 69–82.

    Google Scholar 

  11. 11.

    Gicquel, C. (2008). Mip models and exact methods for the discrete lot-sizing and scheduling problem with sequence-dependent changeover costs and times. Paris: Ph.D. thesis, Ecole centrale.

    Google Scholar 

  12. 12.

    Harris, F.W. (1913). How many parts to make at once. Factory, The magazine of management, 10(2), 135–136.

    Google Scholar 

  13. 13.

    Houndji, V.R., Schaus, P., Wolsey, L. Cp4pp: Constraint programming for production planning.

  14. 14.

    Houndji, V.R., Schaus, P., Wolsey, L., Deville, Y. (2014). The stockingcost constraint. In Principles and practice of constraint programming–CP 2014 (pp. 382–397). Springer.

  15. 15.

    Jans, R., & Degraeve, Z. (2006). Modeling industrial lot sizing problems: A review. International Journal of Production Research.

  16. 16.

    Karimi, B., Ghomi, S.M.T.F., Wilson, J. (2003). The capacitated lot sizing problem: a review of models. Omega, The international Journal of Management Science, 31, 365–378.

    Article  Google Scholar 

  17. 17.

    Leung, J.M.Y., Magnanti, T.L., Vachani, R. (1989). Facets and algorithms for capacitated lot sizing. Mathematical Programming, 45, 331–359.

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    López-Ortiz, A., Quimper, C.G., Tromp, J., van Beek, P. (2003). A fast and simple algorithm for bounds consistency of the alldifferent constraint. In International joint conference on artificial intelligence – IJCAI03.

  19. 19.

    Pesant, G. (2004). A regular language membership constraint for finite sequences of variables. In International conference on principles and practice of constraint programming (pp. 482–495). Springer.

  20. 20.

    Pesant, G., Gendreau, M., Potvin, J.Y., Rousseau, J.M. (1998). An exact constraint logic programming algorithm for the traveling salesman problem with time windows. Transportation Science, 32(1), 12–29.

    Article  MATH  Google Scholar 

  21. 21.

    Pochet, Y., & Wolsey, L. (2005). Production planning by mixed integer programming. Springer.

  22. 22.

    Quimper, C.G., Van Beek, P., López-Ortiz, A., Golynski, A., Sadjad, S.B. (2003). An efficient bounds consistency algorithm for the global cardinality constraint. In Principles and practice of constraint programming–CP 2003 (pp. 600–614). Springer.

  23. 23.

    Régin, J.C. (1996). Generalized arc consistency for global cardinality constraint. In Proceedings of the 13th national conference on artificial intelligence-Volume 1 (pp. 209–215). AAAI Press.

  24. 24.

    Régin, J.C. (2002). Cost-based arc consistency for global cardinality constraints. Constraints, 7(3–4), 387–405.

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Shaw, P. (1998). Using constraint programming and local search methods to solve vehicle routing problems. In International conference on principles and practice of constraint programming (pp. 417–431). Springer.

  26. 26.

    Oscar Team (2012). Oscar: Scala in or

  27. 27.

    Ullah, H., & Parveen, S. (2010). A literature review on inventory lot sizing problems. Global Journal of Researches in Engineering, 10, 21–36.

    Google Scholar 

  28. 28.

    Van Cauwelaert, S., Lombardi, M., Schaus, P. (2015). Understanding the potential of propagators. In Integration of AI and OR techniques in constraint programming for combinatorial optimization problems - CPAIOR 2015 (pp. 427–436). Springer.

Download references

Author information



Corresponding author

Correspondence to Vinasetan Ratheil Houndji.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Houndji, V.R., Schaus, P. & Wolsey, L. The item dependent stockingcost constraint. Constraints 24, 183–209 (2019).

Download citation


  • StockingCost constraint
  • Production planning
  • Lot-sizing
  • Scheduling
  • Constraint programming
  • Global constraint
  • Optimization contraint
  • Cost-based filtering