Abstract: The paper presents a concept and the outline of the implementation of a hybrid approach to modelling and solving constrained problems. Two environments of mathematical programming (in particular, integer programming) and declarative programming (in particular, constraint logic programming) were integrated. The strengths of integer programming and constraint logic programming, in which constraints are treated in a different way and different methods are implemented, were combined to use the strengths of both. The hybrid method is not worse than either of its components used independently. The proposed approach is particularly important for the decision models with an objective function and many discrete decision variables added up in multiple constraints. To validate the proposed approach, two illustrative examples are presented and solved. The first example is the authors' original model of cost optimisation in the supply chain with multimodal transportation. The second one is the two-echelon variant of the well-known capacitated vehicle routing problem.
B I B L I O G R A F I A[1] Kanyalkar A.P., Adii G.K. - An integrated aggregate and detailed planning in a multi-site production environment using linear programming. International Journal of Production Research 43, 2005, pp. 4431-4454.[Crossref]
[2] Perea-Lopez E., Ydstie B.E., Grossmann I.E. - A model predictive control strategy for supply chain optimization. Computers and Chemical Engineering 27, 2003, pp. 1201-1218.
[3] Christian Lang J. - Production and Operations Management: Models and Algorithms. Production and Inventory Management with Substitutions, Lecture Notes in Economics and Mathematical Systems, Volume 636, 2010, pp 9-79.
[4] Dang Q., Nielsen I.A, Steger-Jensen K., Madsen O. - Scheduling a Single Mobile Robot for PartFeeding Tasks of Production Lines. Journal of Intelligent Manufacturing, 2013, DOI: 10.1007/s10845-013-0729-y.[Crossref]
[5] Apt K., Wallace M. - Constraint Logic Programming using Eclipse. Cambridge University Press, 2006.
[6] Sitek P., Wikarek J. - A Declarative Framework for Constrained Search Problems. New Frontiers in Applied Artificial Intelligence. Lecture Notes in Artificial Intelligence, Nguyen, NT., et al. (Eds.), Vol. 5027, Springer-Verlag, Berlin-Heidelberg 2008, pp. 728-737.
[7] Bocewicz G., Wójcik R., Banaszak Z. - AGVs distributed control subject to imprecise operation times, In: Agent and Multi-Agent Systems: Technologies and Applications, Lecture Notes in Artificial Intelligence, LNAI, Springer-Verlag, Vol. 4953, 2008, pp. 421-430.
[8] Sitek P., Zaborowski M. - Grouping products in a follow-up production control system for parallel partitioned flow production lines. Intelligent manu-factoring systems IMS 2001: 6th IFAC Workshop,: Pergamon, 2001, New York, pp.122-126.
[9] Sitek P. Wikarek J. - The concept of decision support system structures for the distribution center. MPER (Management and Production Engineering Review), Vol. 1, No. 3, 2010, pp.63-69.
[10] Sitek P., Wikarek J. - Cost optimization of supply chain with multimodal transport. Federated Conference on Computer Science and Information Systems (FedCSIS), 2012, pp. 1111-1118.
[11] Sitek P., Wikarek J. - Supply chain optimization based on a MILP model from the perspective of a logistics provider. Management and Production Engineering Review, 2012, pp. 49-61.
[12] Sitek P., Wikarek J. - The Declarative Framework Approach to Decision Support for Constrained Search Problems. INTECH, 2011, pp 163-182.
[13] Jain V., Grossmann I.E. - Algorithms for hybrid MILP/CP models for a class of optimization problems. INFORMS Journal on Computing 13(4), 2001, pp. 258-276.[Crossref]
[14] Milano M., Wallace M. - Integrating Operations Research in Constraint Programming. Annals of Operations Research, Vol. 175, Issue 1, 2010, pp. 37 - 76.[WoS]
[15] Achterberg T., Berthold T., Koch T., Wolter K. -Constraint Integer Programming: A New Approach to Integrate CP and MIP. Lecture Notes in Computer Science, Volume 5015, 2008, pp. 6-20.
[16] Bockmayr A., Kasper T. - Branch-and-Infer: A Framework for Combining CP and IP. Constraint and Integer Programming Operations Research/Computer Science Interfaces Series, Volume 27, 2004, pp. 59-87.
[17] Sitek P., Wikarek J. - A hybrid approach to supply chain modeling and optimization. Federated Conference on Computer Science and Information Systems (FedCSIS), 2013, pp. 1223-1230.
[18] Perboli G., Tadei R., Vigo D. - The Two-Echelon Capacitated Vehicle Routing Problem: Models and Math-Based Heuristics. Transportation Science, Vol. 45, 2011, pp. 364-380.[WoS]
[19] Crainic T., Ricciardi, N., Storchi, G. - Advanced freight transportation systems for congested urban areas. Transportation Research part C 12, 2004, pp. 119-137.
[20] Schrijver A. - Theory of Linear and Integer Programming. ISBN 0-471-98232-6, John Wiley & Sons, 1998.
[21] www.eclipse.org
[22] www.lindo.com[PubMed]
[23] Ricciardi N., Tadei R., Grosso A. - Optimal facility location with random throughput costs. Computers and Operations Research 29 (6), 2002, pp. 593607.
[24] Christofides N., Elion S. - An algorithm for the vehicle dispatching problem. Operational Research Quarterly 20, 1969, pp. 309-318.[Crossref]
[25] http://www.orgroup.polito.it/
[26] Rossi F., Van Beek P., Walsh T. - Handbook of Constraint Programming (Foundations of Artificial Intelligence), Elsevier Science Inc. New York, NY, USA 2006.
[27] Relich M. - Project prototyping with application of CP-based approach. Management, 2011, Vol. 15, No. 2, pp. 364-377. 
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