Journal of Heuristics

, Volume 19, Issue 3, pp 423–441 | Cite as

The Express heuristic for probabilistically constrained integer problems

  • Maria Elena Bruni
  • Patrizia Beraldi
  • Demetrio Laganà


Integer problems under joint probabilistic constraints with random coefficients in both sides of the constraints are extremely hard from a computational standpoint since two different sources of complexity are merged. The first one is related to the challenging presence of probabilistic constraints which assure the satisfaction of the stochastic constraints with a given probability, whereas the second one is due to the integer nature of the decision variables. In this paper we present a tailored heuristic approach based on alternating phases of exploration and feasibility repairing which we call Express (Explore and Repair Stochastic Solution) heuristic. The exploration is carried out by the iterative solution of simplified reduced integer problems in which probabilistic constraints are discarded and deterministic additional constraints are adjoined. Feasibility is restored through a penalty approach. Computational results, collected on a probabilistically constrained version of the classical 0–1 multiknapsack problem, show that the proposed heuristic is able to determine good quality solutions in a limited amount of time.


Joint probabilistic constraints Discrete distribution  Integer programming Random technology matrix 


  1. Ahuja, R.K., Cunha, C.B.: Very large-scale neighborhood search for the K-constraint multiple knapsack. J. Heuristics 11, 465–481 (2005)MATHCrossRefGoogle Scholar
  2. Alastair, A., Levine, J., Long, D.: Constraint directed variable neighbourhood search In: Proceedings of the 4th International Workshop on Local Search Techniques in Constraint Satisfaction, pp. 348–371 (2007)Google Scholar
  3. Balas, E., Jeroslow, R.: Canonical cuts on the unit hypercube. SIAM J. Appl. Math. 23(1), 61–69 (1972)MathSciNetMATHCrossRefGoogle Scholar
  4. Beraldi, P., Bruni, M.E.: A probabilistic model applied to emergency service vehicle location. Eur. J. Oper. Res. 196, 323–331 (2009)MATHCrossRefGoogle Scholar
  5. Beraldi, P., Bruni, M.E.: An exact approach for solving integer problems under probabilistic constraints with random technology matrix. Ann. Oper. Res. 177(1), 127–137 (2010)MathSciNetMATHCrossRefGoogle Scholar
  6. Beraldi, P., Ruszczyński, A.: A branch and bound method for stochastic integer problems under probabilistic constraints. Optim. Methods Soft. 17, 359–382 (2002)MATHCrossRefGoogle Scholar
  7. Beraldi, P., Ruszczyński, A.: The probabilistic set covering problem. Oper. Res. 50, 956–967 (2002)MathSciNetMATHCrossRefGoogle Scholar
  8. Beraldi, P., Ruszczyński, A.: Beam search heuristic to solve stochastic integer problems under probabilistic constraints. Eur. J. Oper. Res. 167(1), 35–47 (2005)MATHCrossRefGoogle Scholar
  9. Beraldi, P., Bruni, M.E., Conforti, D.: Designing robust medical service via stochastic programming. Eur. J Oper. Res. 158(1), 183–193 (2004)MathSciNetMATHCrossRefGoogle Scholar
  10. Beraldi, P., Bruni, M.E., Guerriero, F.: Network reliability design via joint probabilistic constraints. IMA J. Manag. Math. 21(2), 213–226 (2010)MathSciNetMATHCrossRefGoogle Scholar
  11. Beraldi, P., Bruni, M.E., Violi, A.: Capital rationing problems under uncertainty and risk. Comput. Optim. Appl. 51(3), 1375–1396 (2012)MathSciNetMATHCrossRefGoogle Scholar
  12. Branda, M.: On relations between chance constrained and penalty function problems under discrete distributions. Math. Methods Oper. Res. doi: 10.1007/s00186-013-0428-7 (2012)
  13. Bruni, M.E., Conforti, P., Beraldi, P., Tundis, E.: Probabilistically constrained models for efficiency and dominance in DEA. Int. J. Prod. Econ. 117(1), 219–228 (2009a)CrossRefGoogle Scholar
  14. Bruni, M.E., Guerriero, F., Pinto, E.: Evaluating project completion time in project networks with discrete random activity durations. Comput. Oper. Res. 36, 2716–2722 (2009b)MATHCrossRefGoogle Scholar
  15. Bruni, M.E., Beraldi, P., Guerriero, F., Pinto, E.: A Heuristic approach for resource constrained projects with uncertain activity durations. Comput. Oper. Res. 38(9), 1305–1318 (2011)Google Scholar
  16. Charnes, A., Cooper, W.W.: Deterministic equivalents for optimizing and satisficing under chance constraints. Oper. Res. 11(1), 18–39 (1963)MathSciNetMATHCrossRefGoogle Scholar
  17. Cheon, M.S., Ahmed, S., Al-Khayyal, F.: A branch-reduce-cut algorithm for the global optimization of probabilistically constrained linear programs. Math. Program. B 108(2–3), 617–634 (2006)Google Scholar
  18. Freville, A., Plateau, G.: An efficient preprocessing procedure for the multidimensional 0–1 knapsack problem. Discr. Appl. Math. 49(1–3), 189–212 (1994)MathSciNetMATHCrossRefGoogle Scholar
  19. Gilmore, P.C., Gomory, R.E.: The theory and computation of knapsack functions. Oper. Res. 14, 1045–1075 (1966)MathSciNetCrossRefGoogle Scholar
  20. Henrion, R., Möller, A.: Optimization of a continuous distillation process under random inflow rate. Comput. Math. Appl. 45(1–3), 247–262 (2003)MathSciNetMATHCrossRefGoogle Scholar
  21. Herault, L., Privault, C.: Solving a real world assignment problem with a metaheuristic. J. Heuristics 4(4), 383–398 (1998)MATHCrossRefGoogle Scholar
  22. Jagannathan, R., Rao, M.R.: A class of nonlinear chance-constrained programming models with joint constraints. Oper. Res. 21(1), 360–364 (1973)MathSciNetMATHCrossRefGoogle Scholar
  23. Kaparis, K., Letchford, A.N.: Local and global lifted cover inequalities for the 0–1 multidimensional knapsack problem. Eur. J. Oper. Res. 186(1), 91–103 (2008)MathSciNetMATHCrossRefGoogle Scholar
  24. Klopfenstein, O., Nace, D.: Robust approach to the chance-constrained knapsack problem. Oper. Res. Lett. 36(5), 628–632 (2008)MathSciNetMATHCrossRefGoogle Scholar
  25. Klopfenstein, O.: Tractable algorithms for chance-constrained combinatorial problems. RAIRO Oper. Res. 43(2), 157–187 (2009)MathSciNetMATHCrossRefGoogle Scholar
  26. Küçükyavuz, S.: On mixing sets arising in chance-constrained programming. Math. Program. 132(1–2), 31–56 (2012)MathSciNetMATHCrossRefGoogle Scholar
  27. Lejeune, M.A., Ruszczyński, A.: An efficient trajectory method for probabilistic inventory production distribution problems. Oper. Res. 55(2), 378–394 (2007)MathSciNetMATHCrossRefGoogle Scholar
  28. Liberti, L., Cafieri, S., Tarissan, F.: Reformulations in mathematical programming: A computational approach. In Abraham, A., Hassanien, A.-E., Siarry, P., Engelbrecht, A. (eds.) Foundations of Computational Intelligence: Vol. 3(203) of Studies in Computational Intelligence, pp. 153–234. Springer, Berlin (2009)Google Scholar
  29. Lorie, J., Savage, L.: Three problems in capital rationing. J. Bus. 28(4), 229–239 (1995)CrossRefGoogle Scholar
  30. Luedtke, J.: An integer programming and decomposition approach to general chance-constrained mathematical programs. Lecture Notes in Computer Science, vol. 6080, pp. 271–284 (2010)Google Scholar
  31. Luedtke, J., Ahmed, S.: A sample approximation approach for optimization with probabilistic constraints. SIAM J. Optim. 19(2), 674–699 (2008)MathSciNetMATHCrossRefGoogle Scholar
  32. Luedtke, J., Ahmed, S., Nemhauser, G.L.: An integer programming approach for linear programs with probabilistic constraints. Mathe. Program. A 122(2), 247–272 (2010)Google Scholar
  33. Miller, B.L., Wagner, H.M.: Chance constrained programming with joint constraints. Oper. Res. 13(6), 930–945 (1965)MATHCrossRefGoogle Scholar
  34. Murr, M.R., Prékopa, A.: Solution of a product substitution problem using stochastic programming. In: Uryasev, S. P. (eds.) Probabilistic Constrained Optimization: Methodology and Applications, pp. 252–271. Kluwer, Dordrecht (2000)Google Scholar
  35. Nannicini, G., Belotti, P.: Rounding-based heuristics for nonconvex MINLPs. Math. Program. Comput. 4(1), 1–31 (2012)MathSciNetMATHCrossRefGoogle Scholar
  36. Patel, J., Chinneck, J.W.: Active-constraint variable ordering for faster feasibility of mixed integer linear programs. Math. Programm. A 110(3), 445–474 (2007)Google Scholar
  37. Prékopa, A.: Logarithmic concave measures with application to stochastic programming. Acta Sci. Math. 32, 301–315 (1971)MATHGoogle Scholar
  38. Prékopa, A.: Programming under probabilistic constraints with a random technology matrix. Mathematische Operationsforschung und Statistik, Series Optimization 5, 109–116 (1974)MATHCrossRefGoogle Scholar
  39. Prékopa, A.: Stochastic Programming. Kluwer, Dordrecht (1995)CrossRefGoogle Scholar
  40. Prékopa, A., Yoda, K., Subasi, M.M.: Uniform quasi-concavity in probabilistic constrained stochastic programming. Oper. Res. Lett. 39, 188–192 (2011)MathSciNetMATHCrossRefGoogle Scholar
  41. Puchinger, J., Raidl, G.R., Pferschy, U.: The multidimensional knapsack problem: structure and algorithms. INFORMS J. Comput. 22(2), 250–265 (2010)MathSciNetMATHCrossRefGoogle Scholar
  42. Quadri, D., Soutif, E., Tolla, P.: Exact solution method to solve large scale integer quadratic multidimensional knapsack problems. J. Combin. Optimiz. 17(2), 157–167 (2009)MathSciNetMATHCrossRefGoogle Scholar
  43. Rossi, F., Van Beek, P., Walsh, T.: Handbook of Constraint Programming. Elsevier, Amsterdam (2006)Google Scholar
  44. Ruszczyński, A.: Probabilistic programming with discrete distributions and precedence constrained knapsack polyhedra. Math. Program. A 93(2), 195–215 (2002)Google Scholar
  45. Saxena, A., Goyal, V., Lejeune, M.A.: MIP reformulations of the probabilistic set covering problem. Math. Program. A 121(1), 1–31 (2009)Google Scholar
  46. Sen, S.: Relaxations for probabilistically constrained programs with discrete random variables. Oper. Res. Lett. 11(2), 81–86 (1992)MathSciNetMATHCrossRefGoogle Scholar
  47. Tanner, M., Beier, E.: A general heuristic method for joint chance-constrained stochastic programs with discretely distributed parameters. Optimization, online (2010)Google Scholar
  48. Tanner, M.W., Ntaimo, L.: IIS Branch-and-cut for joint chance-constrained programs with random technology matrices. Eur. J. Oper. Res. 207(1), 290–296 (2010)MathSciNetMATHCrossRefGoogle Scholar
  49. Tsang, E.P.K.: Foundations of Constraint Satisfaction. Academic Press, London (1993)Google Scholar
  50. Walser, J.P.: Integer Optimization by Local Search: A Domain-Independent Approach. Springer, Berlin (1999)MATHCrossRefGoogle Scholar
  51. Watson, J.P., Wets, R.J.-B., Woodruff, D.L.: Scalable heuristics for a class of chance-constrained stochastic programs. INFORMS J. Comput. 22(4), 543–554 (2010)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Maria Elena Bruni
    • 1
  • Patrizia Beraldi
    • 1
  • Demetrio Laganà
    • 1
  1. 1.Department of Engineering for Mechanics, Energy and ManagementUniversity of CalabriaArcavacata di RendeItaly

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