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The Express heuristic for probabilistically constrained integer problems

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Abstract

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.

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References

  • Ahuja, R.K., Cunha, C.B.: Very large-scale neighborhood search for the K-constraint multiple knapsack. J. Heuristics 11, 465–481 (2005)

    Article  MATH  Google Scholar 

  • 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)

  • Balas, E., Jeroslow, R.: Canonical cuts on the unit hypercube. SIAM J. Appl. Math. 23(1), 61–69 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  • Beraldi, P., Bruni, M.E.: A probabilistic model applied to emergency service vehicle location. Eur. J. Oper. Res. 196, 323–331 (2009)

    Article  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • Beraldi, P., Ruszczyński, A.: A branch and bound method for stochastic integer problems under probabilistic constraints. Optim. Methods Soft. 17, 359–382 (2002)

    Article  MATH  Google Scholar 

  • Beraldi, P., Ruszczyński, A.: The probabilistic set covering problem. Oper. Res. 50, 956–967 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Beraldi, P., Ruszczyński, A.: Beam search heuristic to solve stochastic integer problems under probabilistic constraints. Eur. J. Oper. Res. 167(1), 35–47 (2005)

    Article  MATH  Google Scholar 

  • Beraldi, P., Bruni, M.E., Conforti, D.: Designing robust medical service via stochastic programming. Eur. J Oper. Res. 158(1), 183–193 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Beraldi, P., Bruni, M.E., Guerriero, F.: Network reliability design via joint probabilistic constraints. IMA J. Manag. Math. 21(2), 213–226 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Beraldi, P., Bruni, M.E., Violi, A.: Capital rationing problems under uncertainty and risk. Comput. Optim. Appl. 51(3), 1375–1396 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

  • 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)

    Article  Google Scholar 

  • 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)

    Article  MATH  Google Scholar 

  • 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 

  • Charnes, A., Cooper, W.W.: Deterministic equivalents for optimizing and satisficing under chance constraints. Oper. Res. 11(1), 18–39 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • Freville, A., Plateau, G.: An efficient preprocessing procedure for the multidimensional 0–1 knapsack problem. Discr. Appl. Math. 49(1–3), 189–212 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  • Gilmore, P.C., Gomory, R.E.: The theory and computation of knapsack functions. Oper. Res. 14, 1045–1075 (1966)

    Article  MathSciNet  Google Scholar 

  • Henrion, R., Möller, A.: Optimization of a continuous distillation process under random inflow rate. Comput. Math. Appl. 45(1–3), 247–262 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Herault, L., Privault, C.: Solving a real world assignment problem with a metaheuristic. J. Heuristics 4(4), 383–398 (1998)

    Article  MATH  Google Scholar 

  • Jagannathan, R., Rao, M.R.: A class of nonlinear chance-constrained programming models with joint constraints. Oper. Res. 21(1), 360–364 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • Klopfenstein, O., Nace, D.: Robust approach to the chance-constrained knapsack problem. Oper. Res. Lett. 36(5), 628–632 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Klopfenstein, O.: Tractable algorithms for chance-constrained combinatorial problems. RAIRO Oper. Res. 43(2), 157–187 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Küçükyavuz, S.: On mixing sets arising in chance-constrained programming. Math. Program. 132(1–2), 31–56 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Lejeune, M.A., Ruszczyński, A.: An efficient trajectory method for probabilistic inventory production distribution problems. Oper. Res. 55(2), 378–394 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

  • Lorie, J., Savage, L.: Three problems in capital rationing. J. Bus. 28(4), 229–239 (1995)

    Article  Google Scholar 

  • 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)

  • Luedtke, J., Ahmed, S.: A sample approximation approach for optimization with probabilistic constraints. SIAM J. Optim. 19(2), 674–699 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • Miller, B.L., Wagner, H.M.: Chance constrained programming with joint constraints. Oper. Res. 13(6), 930–945 (1965)

    Article  MATH  Google Scholar 

  • 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)

  • Nannicini, G., Belotti, P.: Rounding-based heuristics for nonconvex MINLPs. Math. Program. Comput. 4(1), 1–31 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • Prékopa, A.: Logarithmic concave measures with application to stochastic programming. Acta Sci. Math. 32, 301–315 (1971)

    MATH  Google Scholar 

  • Prékopa, A.: Programming under probabilistic constraints with a random technology matrix. Mathematische Operationsforschung und Statistik, Series Optimization 5, 109–116 (1974)

    Article  MATH  Google Scholar 

  • Prékopa, A.: Stochastic Programming. Kluwer, Dordrecht (1995)

    Book  Google Scholar 

  • Prékopa, A., Yoda, K., Subasi, M.M.: Uniform quasi-concavity in probabilistic constrained stochastic programming. Oper. Res. Lett. 39, 188–192 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Puchinger, J., Raidl, G.R., Pferschy, U.: The multidimensional knapsack problem: structure and algorithms. INFORMS J. Comput. 22(2), 250–265 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • Rossi, F., Van Beek, P., Walsh, T.: Handbook of Constraint Programming. Elsevier, Amsterdam (2006)

  • Ruszczyński, A.: Probabilistic programming with discrete distributions and precedence constrained knapsack polyhedra. Math. Program. A 93(2), 195–215 (2002)

    Google Scholar 

  • 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 

  • Sen, S.: Relaxations for probabilistically constrained programs with discrete random variables. Oper. Res. Lett. 11(2), 81–86 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Tanner, M., Beier, E.: A general heuristic method for joint chance-constrained stochastic programs with discretely distributed parameters. Optimization, online (2010)

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • Tsang, E.P.K.: Foundations of Constraint Satisfaction. Academic Press, London (1993)

    Google Scholar 

  • Walser, J.P.: Integer Optimization by Local Search: A Domain-Independent Approach. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Engineering for Mechanics, Energy and Management, University of Calabria, 87036 , Arcavacata di Rende, CS, Italy

    Maria Elena Bruni, Patrizia Beraldi & Demetrio Laganà

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  1. Maria Elena Bruni
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  2. Patrizia Beraldi
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  3. Demetrio Laganà
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Correspondence to Maria Elena Bruni.

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Bruni, M.E., Beraldi, P. & Laganà, D. The Express heuristic for probabilistically constrained integer problems. J Heuristics 19, 423–441 (2013). https://doi.org/10.1007/s10732-013-9218-x

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