Abstract
Uncertain data due to imprecise measurements is commonly specified as bounded intervals in a constraint decision or optimization problem. Dependencies do exist among such data, e.g. upper bound on the sum of uncertain production rates per machine, sum of traffic distribution ratios from a router over several links. For tractability reasons existing approaches in constraint programming or robust optimization frameworks assume independence of the data. This assumption is safe, but can lead to large solution spaces, and a loss of problem structure. Thus it cannot be overlooked. In this paper we identify the context of matrix models and show how data dependency constraints over the columns of such matrices can be modeled and handled efficiently in relationship with the decision variables. Matrix models are linear models whereby the matrix cells specify for instance, the duration of production per item, the production rates, or the wage costs, in applications such as production planning, economics, inventory management. Data imprecision applies to the cells of the matrix and the output vector. Our approach contributes the following results: 1) the identification of the context of matrix models with data constraints, 2) an efficient modeling approach of such constraints that suits solvers from multiple paradigms. An illustration of the approach and its benefits are shown on a production planning problem.
This research was partly support by the Marie Curie CIG grant, FP7-332683.
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References
Benhamou, F.: Inteval constraint logic programming. In: Podelski, A. (ed.) Constraint Programming: Basics and Trends. LNCS, vol. 910, pp. 1–21. Springer, Heidelberg (1995)
Benhamou, F., Goualard, F.: Universally quantified interval constraints. In: Dechter, R. (ed.) CP 2000. LNCS, vol. 1894, pp. 67–82. Springer, Heidelberg (2000)
Ben-Tal, A., Nemirovski, A.: Robust solutions of uncertain liner programs. Operations Research Letters 25, 1–13 (1999)
Bertsimas, D., Brown, D.: Constructing uncertainty sets for robust linear optimization. Operations Research (2009)
Bordeaux, L., Monfroy, E.: Beyond NP: Arc-consistency for quantified constraints. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, pp. 371–386. Springer, Heidelberg (2002)
Brown, K., Miguel, I.: Chapter 21: uncertainty and change. In: Handbook of Constraint Programming. Elsevier (2006)
Cheadle, A.M., Harvey, W., Sadler, A.J., Schimpf, J., Shen, K., Wallace, M.G.: ECLiPSe: An Introduction. Tech. Rep. IC-Parc-03-1, Imperial College London, London, UK
Chinneck, J.W., Ramadan, K.: Linear programming with interval coefficients. J. Operational Research Society 51(2), 209–220 (2000)
Fargier, H., Lang, J., Schiex, T.: Mixed constraint satisfaction: A framework for decision problems under incomplete knowledge. In: Proc. of AAAI-1996 (1996)
Gent, I., Nightingale, P., Stergiou, K.: QCSP-solve: A solver for quantified constraint satisfaction problems. In: Proc. of IJCAI (2005)
Gervet, C., Galichet, S.: On combining regression analysis and constraint programming. In: Laurent, A., Strauss, O., Bouchon-Meunier, B., Yager, R.R. (eds.) IPMU 2014, Part II. CCIS, vol. 443, pp. 284–293. Springer, Heidelberg (2014)
Inuiguchi, M., Kume, Y.: Goal programming problems with interval coefficients and target intervals. European Journl of Oper. Res. 52 (1991)
Medina, A., Taft, N., Salamatian, K., Bhattacharyya, S. Diot, C.: Traffic matrix estimation: existing techniques and new directions. In: Proceedings of ACM SIGCOMM02 (2002)
Oettli, W.: On the solution set of a linear system with inaccurate coefficients. J. SIAM: Series B, Numerical Analysis 2(1), 115–118 (1965)
Ratschan, S.: Efficient solving of quantified inequality constraints over the real numbers. ACM Trans. Computat. Logic 7(4), 723–748 (2006)
Rossi, F., van Beek, P., Walsh, T.: Handbook of Constraint Programming. Elsevier (2006)
Saad, A., Gervet, C., Abdennadher, S.: Constraint reasoning with uncertain data using CDF-intervals. In: Lodi, A., Milano, M., Toth, P. (eds.) CPAIOR 2010. LNCS, vol. 6140, pp. 292–306. Springer, Heidelberg (2010)
Tarim, S., Kingsman, B.: The stochastic dynamic production/inventory lot-sizing problem with service-level constraints. International Journal of Production Economics 88, 105–119 (2004)
Van Hentenryck, P., Michel, L., Deville, Y.: Numerica: A Modeling Language for Global Optimization. The MIT Press, Cambridge Mass (1997)
Yorke-Smith, N., Gervet, C.: Certainty closure: Reliable constraint reasoning with uncertain data. In: ACM Transactions on Computational Logic 10(1) (2009)
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Gervet, C., Galichet, S. (2015). Uncertain Data Dependency Constraints in Matrix Models. In: Michel, L. (eds) Integration of AI and OR Techniques in Constraint Programming. CPAIOR 2015. Lecture Notes in Computer Science(), vol 9075. Springer, Cham. https://doi.org/10.1007/978-3-319-18008-3_12
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