Skip to main content
SpringerLink
Log in

A multiobjective metaheuristic for a mean-risk static stochastic knapsack problem

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

In this paper we address two major challenges presented by stochastic discrete optimisation problems: the multiobjective nature of the problems, once risk aversion is incorporated, and the frequent difficulties in computing exactly, or even approximately, the objective function. The latter has often been handled with methods involving sample average approximation, where a random sample is generated so that population parameters may be estimated from sample statistics—usually the expected value is estimated from the sample average. We propose the use of multiobjective metaheuristics to deal with these difficulties, and apply a multiobjective local search metaheuristic to both exact and sample approximation versions of a mean-risk static stochastic knapsack problem. Variance and conditional value-at-risk are considered as risk measures. Results of a computational study are presented, that indicate the approach is capable of producing high-quality approximations to the efficient sets, with a modest computational effort.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ahmed, M.A., Alkhamis, T.M.: Simulation-based optimization using simulated annealing with ranking and selection. Comput. Oper. Res. 29(4), 387–402 (2002)

    Article  MATH  Google Scholar 

  2. Ahmed, S.: Convexity and decomposition of mean-risk stochastic programs. Math. Program. 106(3), 433–446 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Alrefaei, M.H., Andradottir, S.: A simulated annealing algorithm with constant temperature for discrete stochastic optimization. Manag. Sci. 45(5), 748–764 (1999)

    Article  Google Scholar 

  4. Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Financ. 9(3), 203–228 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Battiti, R., Tecchiolli, G.: Parallel biased search for combinatorial optimization—genetic algorithms and TABU. Microprocess. Microsyst. 16(7), 351–367 (1992)

    Article  Google Scholar 

  6. Carraway, R.L., Schmidt, R.L., Weatherford, L.R.: An algorithm for maximizing target achievement in the stochastic knapsack-problem with normal returns. Nav. Res. Logist. 40(2), 161–173 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chang, T.J., Meade, N., Beasley, J.E., Sharaiha, Y.M.: Heuristics for cardinality constrained portfolio optimisation. Comput. Oper. Res. 27(13), 1271–1302 (2000)

    Article  MATH  Google Scholar 

  8. Cheng, L.F., Subrahmanian, E., Westerberg, A.W.: Multi-objective decisions on capacity planning and production—inventory control under uncertainty. Ind. Eng. Chem. Res. 43(9), 2192–2208 (2004)

    Article  Google Scholar 

  9. Claro, J., Sousa, J.P.: An object-oriented framework for multiobjective local search. In: Sousa, J.P. (ed.) MIC’2001 4th Metaheuristics Internat. Conf., Porto, Portugal, pp. 231–236 (2001)

  10. Cohn, A.M., Barnhart, C.: The stochastic knapsack problem with random weights: a heuristic approach to robust transportation planning. In: Triennial Sympos. Transportation Anal. (TRISTAN III), San Juan, Puerto Rico (1998)

  11. Costa, D., Silver, E.A.: Tabu search when noise is present: An illustration in the context of cause and effect analysis. J. Heuristics 4(1), 5–23 (1998)

    Article  MATH  Google Scholar 

  12. Crama, Y., Schyns, M.: Simulated annealing for complex portfolio selection problems. Eur. J. Oper. Res. 150(3), 546–571 (2003)

    Article  MATH  Google Scholar 

  13. Czyzak, P., Jaszkiewicz, A.: Pareto simulated annealing—a metaheuristic technique for multiple-objective combinatorial optimization. J. Multi-Criteria Decis. Anal. 7(1), 34–47 (1998)

    Article  MATH  Google Scholar 

  14. Das, I.: Robustness optimization for constrained nonlinear programming problems. Eng. Optim. 32(5), 585–618 (2000)

    Article  Google Scholar 

  15. Das, S., Ghosh, D.: Binary knapsack problems with random budgets. J. Oper. Res. Soc. 54(9), 970–983 (2003)

    Article  MATH  Google Scholar 

  16. Deb, K., Gupta, H.: Introducing robustness in multiobjective optimization. Evol. Comput. 14(4), 463–494 (2006)

    Article  Google Scholar 

  17. Ehrgott, M., Gandibleux, X.: A survey and annotated bibliography of multiobjective combinatorial optimization. OR Spectrum 22(4), 425–460 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  18. Ehrgott, M., Klamroth, K., Schwehm, C.: An MCDM approach to portfolio optimization. Eur. J. Oper. Res. 155(3), 752–770 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  19. Eichhorn, A., Romisch, W.: Polyhedral risk measures in stochastic programming. SIAM J. Optim. 16(1), 69–95 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Freville, A.: The multidimensional 0–1 knapsack problem: An overview. Eur. J. Oper. Res. 155(1), 1–21 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  22. Gandibleux, X., Freville, A.: Tabu search based procedure for solving the 0–1 multiobjective knapsack problem: The two objectives case. J. Heuristics 6(3), 361–383 (2000)

    Article  MATH  Google Scholar 

  23. Gutjahr, W.J.: A converging ACO algorithm for stochastic combinatorial optimization. In: Stochastic Algorithms: Foundations and Applications, vol. 2827, pp. 10–25 (2003)

  24. Hanafi, S., Freville, A.: An efficient tabu search approach for the 0–1 multidimensional knapsack problem. Eur. J. Oper. Res. 106(2–3), 659–675 (1998)

    Article  MATH  Google Scholar 

  25. Hansen, M.P.: Tabu search for multiobjective combinatorial optimization: TAMOCO. Control Cybern. 29(3), 799–818 (2000)

    MATH  Google Scholar 

  26. Haugen, K.K., Lokketangen, A., Woodruff, D.L.: Progressive hedging as a meta-heuristic applied to stochastic lot-sizing. Eur. J. Oper. Res. 132(1), 116–122 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  27. Henig, M.I.: Risk criteria in a stochastic knapsack-problem. Oper. Res. 38(5), 820–825 (1990)

    Article  MathSciNet  Google Scholar 

  28. Hill, I.D.: Algorithm AS66: The normal integral. Appl. Stat. 22(3), 424–427 (1973)

    Article  Google Scholar 

  29. Jaszkiewicz, A.: On the performance of multiple-objective genetic local search on the 0/1 knapsack problem—a comparative experiment. IEEE Trans. Evol. Comput. 6(4), 402–412 (2002)

    Article  Google Scholar 

  30. Jin, Y., Branke, J.: Evolutionary optimization in uncertain environments—a survey. IEEE Trans. Evol. Comput. 9(3), 303–317 (2005)

    Article  Google Scholar 

  31. Jin, Y.C., Sendhoff, B.: Trade-off between performance and robustness: An evolutionary multiobjective approach. In: Evolutionary Multi-Criterion Optimiz., Proc., vol. 2632, pp. 237–251 (2003)

  32. Jones, D.F., Mirrazavi, S.K., Tamiz, M.: Multi-objective meta-heuristics: An overview of the current state-of-the-art. Eur. J. Oper. Res. 137(1), 1–9 (2002)

    Article  MATH  Google Scholar 

  33. Jorion, P.: Risk2: Measuring the risk in value-at-risk. Financ. Anal. J. 52(6), 47–56 (1996)

    Article  Google Scholar 

  34. Kellerer, H., Mansini, R., Speranza, M.G.: Selecting portfolios with fixed costs and minimum transaction lots. Ann. Oper. Res. 99(1), 287–304 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  35. Kleywegt, A.J., Shapiro, A., Homem-De-Mello, T.: The sample average approximation method for stochastic discrete optimization. SIAM J. Optim. 12(2), 479–502 (2002)

    Article  MathSciNet  Google Scholar 

  36. Kristoffersen, T.K.: Deviation measures in linear two-stage stochastic programming. Math. Methods Oper. Res. 62(2), 255–274 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  37. Markert, A., Schultz, R.: On deviation measures in stochastic integer programming. Oper. Res. Lett. 33(5), 441–449 (2004)

    Article  MathSciNet  Google Scholar 

  38. Markowitz, H.M.: Portfolio Selection: Efficient Diversification of Investments. Wiley, New York (1959)

    Google Scholar 

  39. Martello, S., Toth, P.: Knapsack Problems: Algorithms and Computer Implementations. Wiley, New York (1990)

    MATH  Google Scholar 

  40. Morton, D.P., Wood, R.K.: On a stochastic knapsack problem and generalizations. In: Woodruff, D.L. (ed.) Adv. in Comput. and Stochastic Optim., Logic Programming, and Heuristic Search: Interfaces in Comput. Sci. and Oper. Res., pp. 149–168. Kluwer Academic, Dordrecht (1998)

    Google Scholar 

  41. Ogryczak, W., Ruszczynski, A.: From stochastic dominance to mean-risk models: Semideviations as risk measures. Eur. J. Oper. Res. 116(1), 33–50 (1999)

    Article  MATH  Google Scholar 

  42. Ogryczak, W., Ruszczynski, A.: Dual stochastic dominance and related mean-risk models. SIAM J. Optim. 13(1), 60–78 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  43. Phelps, S., Koksalan, M.: An interactive evolutionary metaheuristic for multiobjective combinatorial optimization. Manag. Sci. 49(12), 1726–1738 (2003)

    Article  Google Scholar 

  44. Ray, T.: Constrained robust optimal design using a multiobjective evolutionary algorithm. In: Proc. 2002 Congress on Evolutionary Comput., 2002, CEC ’02, vol. 1, pp. 419–424 (2002)

  45. Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2(3), 21–41 (2000)

    Google Scholar 

  46. Rockafellar, R.T., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank. Financ. 26(7), 1443–1471 (2002)

    Article  Google Scholar 

  47. Rockafellar, R.T., Wets, R.J.B.: Scenarios and policy aggregation in optimization under uncertainty. Math. Oper. Res. 16(1), 119–147 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  48. Rosen, S.L., Harmonosky, C.M.: An improved simulated annealing simulation optimization method for discrete parameter stochastic systems. Comput. Oper. Res. 32(2), 343–358 (2005)

    MATH  MathSciNet  Google Scholar 

  49. Ruszczynski, A., Shapiro, A.: Optimization of convex risk functions. Math. Oper. Res. 31(3), 433–452 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  50. Sahinidis, N.V.: Optimization under uncertainty: state-of-the-art and opportunities. Comput. Chem. Eng. 28(6–7), 971–983 (2004)

    Article  Google Scholar 

  51. Schlottmann, F., Seese, D.: A hybrid heuristic approach to discrete multi-objective optimization of credit portfolios. Comput. Stat. Data Anal. 47(2), 373–399 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  52. Schultz, R.: Stochastic programming with integer variables. Math. Program. 97(1–2), 285–309 (2003)

    MATH  MathSciNet  Google Scholar 

  53. Schultz, R., Tiedemann, S.: Conditional value-at-risk in stochastic programs with mixed-integer recourse. Math. Program. 105(2–3), 365–386 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  54. Shapiro, A., Ahmed, S.: On a class of minimax stochastic programs. SIAM J. Optim. 14(4), 1237–1249 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  55. Silva, C.G., Climaco, J., Figueira, J.: A scatter search method for bi-criteria {0,1}-knapsack problems. Eur. J. Oper. Res. 169(2), 373–391 (2006)

    Article  MATH  Google Scholar 

  56. Sniedovich, M.: Preference order stochastic knapsack-problems: Methodological issues. J. Oper. Res. Soc. 31(11), 1025–1032 (1980)

    MATH  MathSciNet  Google Scholar 

  57. Sniedovich, M.: Some comments on preference order dynamic-programming models. J. Math. Anal. Appl. 79(2), 489–501 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  58. Steinberg, E., Parks, M.S.: Preference order dynamic program for a knapsack problem with stochastic rewards. J. Oper. Res. Soc. 30(2), 141–147 (1979)

    MATH  Google Scholar 

  59. Steuer, R.E., Na, P.: Multiple criteria decision making combined with finance: A categorized bibliographic study. Eur. J. Oper. Res. 150(3), 496–515 (2003)

    Article  MATH  Google Scholar 

  60. Takriti, S., Ahmed, S.: On robust optimization of two-stage systems. Math. Program. 99(1), 109–126 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  61. Teghem, J., Tuyttens, D., Ulungu, E.L.: An interactive heuristic method for multi-objective combinatorial optimization. Comput. Oper. Res. 27(7–8), 621–634 (2000)

    Article  MATH  Google Scholar 

  62. Wichura, M.J.: Algorithm AS241: The percentage points of the normal distribution. Appl. Stat. 37(3), 477–484 (1988)

    Article  Google Scholar 

  63. Zitzler, E., Thiele, L.: Multiobjective optimization using evolutionary algorithms—a comparative case study. In: Parallel Problem Solving from Nature—PPSN V, vol. 1498, pp. 292–301 (1998)

Download references

Author information

Authors and Affiliations

  1. Faculdade de Engenharia da Universidade do Porto, INESC Porto–Instituto de Engenharia de Sistemas e Computadores do Porto, Rua Dr. Roberto Frias, 4200-465, Porto, Portugal

    João Claro & Jorge Pinho de Sousa

Authors
  1. João Claro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jorge Pinho de Sousa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to João Claro.

Additional information

The work reported in this paper has been supported by FCT project POCI/EGE/61362/2004.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Claro, J., de Sousa, J.P. A multiobjective metaheuristic for a mean-risk static stochastic knapsack problem. Comput Optim Appl 46, 427–450 (2010). https://doi.org/10.1007/s10589-008-9197-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-008-9197-2

Keywords

Search

Navigation