Advertisement

Dealing with Uncertainties in MCDA

  • Theodor J. StewartEmail author
  • Ian Durbach
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 233)

Abstract

This chapter presents various approaches to incorporating formal modelling of risks and uncertainties into multi-criteria decision analysis, in a theoretically valid but also operationally meaningful manner. We consider both internal uncertainties (in the formulation and modelling of the decision problem), and external uncertainties arising from exogenous factors, but with greater attention paid to the latter. After a broad discussion on the meaning of uncertainty, we first review approaches to sensitivity analysis, which is particularly, although not exclusively, relevant to internal uncertainties. We discuss the role, but also some limitations, of representing uncertainties in formal probabilistic structures, linked also to concepts of expected (multi-attribute) utility theory. Such probability/utility approaches may be used in explicitly identifying a most preferred solution, or simply to eliminate certain courses of action when stochastically dominated (in various senses) by others. In some contexts it may be useful to view minimization of various risk measures as additional criteria in more standard MCDA models, and we comment on advantages and disadvantages of such approaches. Finally we discuss the integration of MCDA with scenario planning, in order to deal with deeper uncertainties (not easily if at all representable by probability models), particularly in a strategic planning context. The emphasis throughout is on the practice of MCDA rather than on esoteric theoretical results.

Keywords

Multicriteria decision analysis Risk Uncertainty Sensitivity analysis Utility theory Scenario planning 

References

  1. 1.
    Abdellaoui, M.: Parameter-free elicitation of utility and probability weighting functions. Manag. Sci. 46(11), 1497–1512 (2000)CrossRefGoogle Scholar
  2. 2.
    Abdellaoui, M., Barrios, C., Wakker, P.: Reconciling introspective utility with revealed preference: experimental arguments based on prospect theory. J. Econ. 138(1), 356–378 (2007)CrossRefGoogle Scholar
  3. 3.
    Abdellaoui, M., Bleichrodt, H., Paraschiv, C.: Loss aversion under prospect theory: a parameter-free measurement. Manag. Sci. 53(10), 1659–1674 (2007)CrossRefGoogle Scholar
  4. 4.
    Angilella, S., Greco, S., Matarazzo, B.: Non-additive robust ordinal regression: a multiple criteria decision model based on the choquet integral. Eur. J. Oper. Res. 201(1), 277–288 (2010)CrossRefGoogle Scholar
  5. 5.
    Azondékon, S.H., Martel, J.M.: “Value” of additional information in multicriterion analysis under uncertainty. Eur. J. Oper. Res. 117, 45–62 (1999)CrossRefGoogle Scholar
  6. 6.
    Ballestero, E.: Stochastic goal programming: a mean-variance approach. Eur. J. Oper. Res. 131, 476–481 (2001)CrossRefGoogle Scholar
  7. 7.
    Banuelas, R., Antony, J.: Application of stochastic analytic hierarchy process within a domestic appliance manufacturer. Eur. J. Oper. Res. Soc. 58(1), 29 (2007)CrossRefGoogle Scholar
  8. 8.
    Bawa, V.S.: Optimal rules for ordering uncertain prospects. J. Financ. Econ. 2, 95–121 (1975)CrossRefGoogle Scholar
  9. 9.
    Bazerman, M.H.: Judgment in Managerial Decision Making, 5th edn. Wiley, New York (2002)Google Scholar
  10. 10.
    Bell, D.E.: One-switch utility functions and a measure of risk. Manag. Sci. 34, 1416–1424 (1988)CrossRefGoogle Scholar
  11. 11.
    Belton, V., Stewart, T.J.: Multiple Criteria Decision Analysis: An Integrated Approach. Kluwer Academic Publishers, Boston (2002)CrossRefGoogle Scholar
  12. 12.
    Ben, A.S., Jabeur, K., Martel, J.: Multiple criteria aggregation procedure for mixed evaluations. Eur. J. Oper. Res. 181(3), 1506–1515 (2007)CrossRefGoogle Scholar
  13. 13.
    Beynon, M., Curry, B., Morgan, P.: The Dempster-Shafer theory of evidence: an alternative approach to multicriteria decision modelling. OMEGA: Int. J. Manag. Sci. 28, 37–50 (2000)CrossRefGoogle Scholar
  14. 14.
    Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming, 2nd edn. Springer, New York (2011)CrossRefGoogle Scholar
  15. 15.
    Bleichrodt, H., Miyamoto, J.: A characterization of quality-adjusted life-years under cumulative prospect theory. Math. Oper. Res. 28(1), 181–193 (2003)CrossRefGoogle Scholar
  16. 16.
    Bleichrodt, H., Pinto, J.: A parameter-free elicitation of the probability weighting function in medical decision analysis. Manag. Sci. 46(11), 1485–1496 (2000)CrossRefGoogle Scholar
  17. 17.
    Bleichrodt, H., Pinto, J., Wakker, P.: Making descriptive use of prospect theory to improve the prescriptive use of expected utility. Manag. Sci. 47(11), 1498–1514 (2001)CrossRefGoogle Scholar
  18. 18.
    Bordley, R.F., Hazen, G.: Nonlinear utility models arising from unmodelled small world intercorrelations. Manag. Sci. 38, 1010–1017 (1992)CrossRefGoogle Scholar
  19. 19.
    Bordley, R., Kirkwood, C.: Multiattribute preference analysis with performance targets. Oper. Res. 52(6), 823 (2004)CrossRefGoogle Scholar
  20. 20.
    Boujelben, M., Smet, Y., Frikha, A., Chabchoub, H.: Building a binary outranking relation in uncertain, imprecise and multi-experts contexts: the application of evidence theory. Int. J. Approx. Reason. 50(8), 1259–1278 (2009)CrossRefGoogle Scholar
  21. 21.
    Castagnoli, E., Calzi, M.: Expected utility without utility. Theory Decis. 41(3), 281–301 (1996)CrossRefGoogle Scholar
  22. 22.
    Chang, N.B., Wang, S.: A fuzzy goal programming approach for the optimal planning of metropolitan solid waste management systems. Eur. J. Oper. Res. 99, 303–321 (1997)CrossRefGoogle Scholar
  23. 23.
    Chang, N.B., Wen, C., Chen, Y.: A fuzzy multi-objective programming approach for optimal management of the reservoir watershed. Eur. J. Oper. Res. 99, 289–302 (1997)CrossRefGoogle Scholar
  24. 24.
    D’Avignon, G.R., Vincke, P.: An outranking method under uncertainty. Eur. J. Oper. Res. 36, 311–321 (1988)CrossRefGoogle Scholar
  25. 25.
    Dembo, R., Rosen, D.: The practice of portfolio replication. a practical overview of forward and inverse problems. Ann. Oper. Res. 85, 267–284 (1999)Google Scholar
  26. 26.
    Dendrou, B., Dendrou, S., Houstis, E.: Multiobjective decision analysis for engineering systems. Comput. Oper. Res. 7, 301–312 (1980)CrossRefGoogle Scholar
  27. 27.
    Dentcheva, D., Ruszczynski, A.: Optimization with stochastic dominance constraints. SIAM J. Optim. 14(2), 548–566 (2003)CrossRefGoogle Scholar
  28. 28.
    Dimitras, A., Slowinski, R., Susmaga, R., Zopounidis, C.: Business failure prediction using rough sets. Eur. J. Oper. Res. 114, 263–280 (1999)CrossRefGoogle Scholar
  29. 29.
    Durbach, I.: A simulation-based test of stochastic multicriteria acceptability analysis using achievement functions. Eur. J. Oper. Res. 170, 923–934 (2006)CrossRefGoogle Scholar
  30. 30.
    Durbach, I., Davis, S.: Decision support for selecting a shortlist of electricity-saving options: a modified SMAA approach. ORiON 28(2), 99–116 (2012)CrossRefGoogle Scholar
  31. 31.
    Durbach, I., Stewart, T.: Using expected values to simplify decision making under uncertainty. Omega 37(2), 312–330 (2009)CrossRefGoogle Scholar
  32. 32.
    Durbach, I.N., Stewart, T.J.: Modelling uncertainty in multi-criteria decision analysis. Eur. J. Oper. Res. 223, 1–14 (2012)CrossRefGoogle Scholar
  33. 33.
    Fan, Z., Liu, Y., Feng, B.: A method for stochastic multiple criteria decision making based on pairwise comparisons of alternatives with random evaluations. Eur. J. Oper. Res. 207(2), 906–915 (2010)CrossRefGoogle Scholar
  34. 34.
    Fishburn, P.C.: Foundations of risk measurement. I. Risk as probable loss. Manag. Sci. 30, 396–406 (1984)Google Scholar
  35. 35.
    French, S.: Uncertainty and imprecision: modelling and analysis. J. Oper. Res. Soc. 46, 70–79 (1995)CrossRefGoogle Scholar
  36. 36.
    French, S., Maule, J., Papamichail, N.: Decision Behaviour, Analysis and Support. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  37. 37.
    Friend, J.: The strategic choice approach. In: Rosenhead, J., Mingers, J. (eds.) Rational Analysis for a Problematic World Revisited, 2nd edn. pp. 115–149. Wiley, Chichester (2001)Google Scholar
  38. 38.
    Gal, T., Stewart, T.J., Hanne, T. (eds.): Multicriteria Decision Making: Advances in MCDM Models, Algorithms, Theory, and Applications. Kluwer Academic Publishers, Boston (1999)Google Scholar
  39. 39.
    Goicoechea, A., Hansen, D.R., Duckstein, L.: Multiobjective Decision Analysis with Engineering and Business Applications. Wiley, New York (1982)Google Scholar
  40. 40.
    Goodwin, P., Wright, G.: Decision Analysis for Management Judgement, 4th edn. Wiley, Chichester (2009)Google Scholar
  41. 41.
    Greco, S., Matarazzo, B., Slowinski, R.: Rough approximation of a preference relation by dominance relations. Eur. J. Oper. Res. 117, 63–83 (1999)CrossRefGoogle Scholar
  42. 42.
    Greco, S., Matarazzo, B., Slowinski, R.: The use of rough sets and fuzzy sets in MCDM. In: Gal, T., Stewart, T.J., Hanne, T. (eds.) Multicriteria Decision Making: Advances in MCDM Models, Algorithms, Theory, and Applications, Chap. 14. Kluwer Academic Publishers, Boston (1999)Google Scholar
  43. 43.
    Greco, S., Matarazzo, B., Slowinski, R.: Rough sets theory for multicriteria decision analysis. Eur. J. Oper. Res. 129, 1–47 (2001)CrossRefGoogle Scholar
  44. 44.
    Greco, S., Matarazzo, B., Slowinski, R.: Rough sets methodology for sorting problems in presence of multiple attributes and criteria. Eur. J. Oper. Res. 138, 247–259 (2002)CrossRefGoogle Scholar
  45. 45.
    Greco, S., Mousseau, V., Słowiński, R.: Multiple criteria sorting with a set of additive value functions. Eur. J. Oper. Res. 207(3), 1455–1470 (2010)CrossRefGoogle Scholar
  46. 46.
    Greco, S., Słowiński, R., Figueira, J., Mousseau, V.: Robust ordinal regression. In: Greco, S., Ehrgott, M., Figuera, J. (eds.) Trends in Multiple Criteria Decision Analysis, Chap 9. Springer, New York (2010)Google Scholar
  47. 47.
    Greco, S., Kadziński, M., Mousseau, V., Słowiński, R.: ELECTRE-GKMS: robust ordinal regression for outranking methods. Eur. J. Oper. Res. 214(1), 118–135 (2011)CrossRefGoogle Scholar
  48. 48.
    Hadar, J., Russell, W.R.: Rules for ordering uncertain prospects. Am. Econ. Rev. 59, 25–34 (1969)Google Scholar
  49. 49.
    Harries, C.: Correspondence to what? Coherence to what? What is good scenario-based decision making. Technol. Forecast. Soc. Chang. 70, 797–817 (2003)CrossRefGoogle Scholar
  50. 50.
    Hokkanen, J., Lahdelma, R., Miettinen, K., Salminen, P.: Determining the implementation order of a general plan by using a multicriteria method. J. Multi-Criteria Decis. Anal. 7(5), 273–284 (1998)CrossRefGoogle Scholar
  51. 51.
    Hughes, N.: A historical overview of strategic scenario planning. Technical Report. UKERC and EON.UK/EPSRC Project on Transition Pathways to a Low Carbon Economy (2009)Google Scholar
  52. 52.
    Insua, D., French, S.: A framework for sensitivity analysis in discrete multi-objective decision-making. Eur. J. Oper. Res. 54(2), 176–190 (1991)CrossRefGoogle Scholar
  53. 53.
    Jacquet-Lagrèze, E., Siskos, Y.: Preference disaggregation: 20 years of MCDA experience. Eur. J. Oper. Res. 130, 233–245 (2001)CrossRefGoogle Scholar
  54. 54.
    Jia, J., Dyer, J.S.: A standard measure of risk and risk-value models. Manag. Sci. 42, 1691–1705 (1996)CrossRefGoogle Scholar
  55. 55.
    Jiménez, A., Mateos, A., Ríos-Insua, S.: Missing consequences in multiattribute utility theory. Omega Int. J. Manag. Sci. 37, 395–410 (2009)CrossRefGoogle Scholar
  56. 56.
    Kahneman, D., Tversky, A.: Prospect theory: an analysis of decision under risk. Econometrica 47, 263–291 (1979)CrossRefGoogle Scholar
  57. 57.
    Keeney, R.L., Raiffa, H.: Decisions with Multiple Objectives. Wiley, New York (1976)Google Scholar
  58. 58.
    Keown, A.J., Taylor III, B.W.: A chance-constrained integer goal programming model for capital budgeting in the production area. J. Oper. Res. Soc. 31, 579–589 (1980)CrossRefGoogle Scholar
  59. 59.
    Kirkwood, C.: Estimating the impact of uncertainty on deterministic multiattribute evaluation. Manag. Sci. 38(6), 819–826 (1992)CrossRefGoogle Scholar
  60. 60.
    Klein, G., Moskowitz, H., Ravindran, A.: Interactive multiobjective optimization under uncertainty. Manag. Sci. 36, 58–75 (1990)CrossRefGoogle Scholar
  61. 61.
    Klir, G.J., Folger, T.A.: Fuzzy Sets, Uncertainty and Information. Prentice Hall, Englewood Cliffs, NJ (1988)Google Scholar
  62. 62.
    Korhonen, A.: Strategic financial management in a multinational financial conglomerate: a multiple goal stochastic programming approach. Eur. J. Oper. Res. 128, 418–434 (2001)CrossRefGoogle Scholar
  63. 63.
    Krokhmal, P., Zabarankin, M., Uryasev, S.: Modeling and optimization of risk. Surv. Oper. Res. Manag. Sci. 16(2), 49–66 (2011)Google Scholar
  64. 64.
    Lahdelma, R., Salminen, P.: SMAA-2: stochastic multi-criteria acceptability analysis for group decision making. Oper. Res. 49(3), 444–454 (2001)CrossRefGoogle Scholar
  65. 65.
    Lahdelma, R., Salminen, P.: Prospect theory and stochastic multicriteria acceptability analysis (SMAA). Omega 37(5), 961–971 (2009)CrossRefGoogle Scholar
  66. 66.
    Lahdelma, R., Hokkanen, J., Salminen, P.: SMAA – Stochastic multiobjective acceptability analysis. Eur. J. Oper. Res. 106, 137–143 (1998)CrossRefGoogle Scholar
  67. 67.
    Lahdelma, R., Miettinen, K., Salminen, P.: Reference point approach for multiple decision makers. Eur. J. Oper. Res. 164(3), 785–791 (2005)CrossRefGoogle Scholar
  68. 68.
    Levary, R.R., Wan, K.: A simulation approach for handling uncertainty in the analytic hierarchy process. Eur. J. Oper. Res. 106, 116–122 (1998)CrossRefGoogle Scholar
  69. 69.
    Levary, R.R., Wan, K.: An analytic hierarchy process based simulation model for entry mode decision regarding foreign direct investment. Omega 27(6), 661–677 (1999)CrossRefGoogle Scholar
  70. 70.
    Linares, P.: Multiple criteria decision making and risk analysis as risk management tools for power systems planning. IEEE Trans. Power Syst. 17, 895–900 (2002)CrossRefGoogle Scholar
  71. 71.
    Liu, Y., Fan, Z., Zhang, Y.: A method for stochastic multiple criteria decision making based on dominance degrees. Inform. Sci. 181(19), 4139–4153 (2011)CrossRefGoogle Scholar
  72. 72.
    Mareschal, B.: Stochastic multicriteria decision making and uncertainty. Eur. J. Oper. Res. 26, 58–64 (1986)CrossRefGoogle Scholar
  73. 73.
    Martel, J.M., Zaras, K.: Stochastic dominance in multicriterion analysis under risk. Theory Decis. 39, 31–49 (1995)CrossRefGoogle Scholar
  74. 74.
    Martel, J., d’Avignon, G., Couillard, G.: A fuzzy outranking relation in multicriteria decision making. Eur. J. Oper. Res. 25, 258–271 (1986)CrossRefGoogle Scholar
  75. 75.
    Mateos, A., Ríos-Insua, S., Jiménez, A.: Dominance, potential optimality and alternative ranking in imprecise multi-attribute decision making. J. Oper. Res. Soc. 58(3), 326–336 (2006)CrossRefGoogle Scholar
  76. 76.
    Millet, I., Wedley, W.C.: Modelling risk and uncertainty with the analytic hierarchy process. J. Multi-Criteria Decis. Anal. 11, 97–107 (2002)CrossRefGoogle Scholar
  77. 77.
    Miranda, V., Proença, L.M.: Why risk analysis outperforms probabilistic choice as the effective decision support paradigm for power system planning. IEEE Trans. Power Syst. 13, 643–648 (1998)CrossRefGoogle Scholar
  78. 78.
    Miyamoto, J.M., Wakker, P.: Multiattribute utility theory without expected utility foundations. Oper. Res. 44, 313–326 (1996)CrossRefGoogle Scholar
  79. 79.
    Montibeller, G., Gummer, H., Tumidei, D.: Combining scenario planning and multi-criteria decision analysis in practice. J. Multi-Criteria Decis. Anal. 14, 5–20 (2006)CrossRefGoogle Scholar
  80. 80.
    Nowak, M.: Preference and veto thresholds in multicriteria analysis based on stochastic dominance. Eur. J. Oper. Res. 158(2), 339–350 (2004)CrossRefGoogle Scholar
  81. 81.
    Parnell, G.S., Jackson, J.A., Burk, R.C., Lehmkuhld, L.J., Engelbrecht, Jr. J.A.: R&D concept decision analysis: using alternate futures for sensitivity analysis. J. Multi-Criteria Decis. Anal. 8, 119–127 (1999)CrossRefGoogle Scholar
  82. 82.
    Pomerol, J.C.: Scenario development and practical decision making under uncertainty. Decis. Support Syst. 31, 197–204 (2001)CrossRefGoogle Scholar
  83. 83.
    Rios Insua, D.: Sensitivity Analysis in Multi-Objective Decision Making. Lecture Notes in Economics and Mathematical Systems, vol. 347. Springer, Berlin (1990)Google Scholar
  84. 84.
    Roman, D., Darby-Dowman, K., Mitra, G.: Portfolio construction based on stochastic dominance and target return distributions. Math. Program. 108(2), 541–569 (2006)CrossRefGoogle Scholar
  85. 85.
    Rosqvist, T.: Simulation and multi-attribute utility modelling of life cycle profit. J. Multi-Criteria Decis. Anal. 10, 205–218 (2001)CrossRefGoogle Scholar
  86. 86.
    Saltelli, A., Tarantola, A.S., Chan, K.: A role for sensitivity analysis in presenting the results from MCDA studies to decision makers. J. Multi-Criteria Decis. Anal. 8, 139–145 (1999)CrossRefGoogle Scholar
  87. 87.
    Sarin, R.K., Weber, M.: Risk-value models. Eur. J. Oper. Res. 70, 135–149 (1993)CrossRefGoogle Scholar
  88. 88.
    Shafer, G.:A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)Google Scholar
  89. 89.
    Stewart, T.J.: Simplified approaches for multi-criteria decision making under uncertainty. J. Multi-Criteria Decis. Anal. 4, 246–258 (1995)CrossRefGoogle Scholar
  90. 90.
    Stewart, T.J.: Robustness of additive value function methods in MCDM. J. Multi-Criteria Decis. Anal. 5, 301–309 (1996)CrossRefGoogle Scholar
  91. 91.
    Stewart, T.J.: Measurements of risk in fisheries management. ORiON 14, 1–15 (1998)Google Scholar
  92. 92.
    Stewart, T.J.: Concepts of interactive programming. In: Gal, T., Stewart, T.J., Hanne, T. (eds.) Multicriteria Decision Making: Advances in MCDM Models, Algorithms, Theory, and Applications, Chap. 10. Kluwer Academic Publishers, Boston (1999)Google Scholar
  93. 93.
    Stewart, T.J., French, S., Rios, J.: Integrating multicriteria decision analysis and scenario planning – review and extension. Omega Int. J. Manag. Sci. 41, 679–688 (2013). doi:10.1016/j.omega.2012.09.003CrossRefGoogle Scholar
  94. 94.
    Teghem, Jr. J., Dufrane, D., Thauvoye, M.: STRANGE: an interactive method for multi-objective linear programming under uncertainty. Eur. J. Oper. Res. 26, 65–82 (1986)CrossRefGoogle Scholar
  95. 95.
    Urli, B., Nadeau, R.: PROMISE/scenarios: an interactive method for multiobjective stochastic linear programming under partial uncertainty. Eur. J. Oper. Res. 155, 361–372 (2004)CrossRefGoogle Scholar
  96. 96.
    Van der Heijden, K.: Scenarios: The Art of Strategic Conversation. Wiley, Chichester (1996)Google Scholar
  97. 97.
    von Winterfeldt, D., Edwards, W.: Decision Analysis and Behavioral Research. Cambridge University Press, Cambridge (1986)Google Scholar
  98. 98.
    Wakker, P., Deneffe, D.: Eliciting von Neumann-Morgenstern utilities when probabilities are distorted or unknown. Manag. Sci. 42(8), 1131–1150 (1996)CrossRefGoogle Scholar
  99. 99.
    Walker, W.E., Harremoes, P., Rotmans, J., van der Sluijs, J.P., van Asselt, M.B.A., Janssen, P., Krayer von Krausswalker, M.P.: Defining uncertainty: a conceptual basis for uncertainty management in model based decision support. Integr. Asess. 4(1), 5–17 (2003)Google Scholar
  100. 100.
    Watkins, Jr. D.W., McKinney, D.C., Lasdon, L.S., Nielsen, S.S., Martin, Q.W.: A scenario-based stochastic programming model for water supplies from the highland lakes. Int. Trans. Oper. Res. 7, 211–230 (2000)CrossRefGoogle Scholar
  101. 101.
    Whitmore, G.A.: Third order stochastic dominance. Am. Econ. Rev. 60, 457–459 (1970)Google Scholar
  102. 102.
    Xu, D.: An introduction and survey of the evidential reasoning approach for multiple criteria decision analysis. Ann. Oper. Res. 195(1), 163–187 (2012)CrossRefGoogle Scholar
  103. 103.
    Yang, J.B.: Rule and utility based evidential reasoning approach for multiattribute decision analysis under uncertainties. Eur. J. Oper. Res. 131, 31–61 (2001)CrossRefGoogle Scholar
  104. 104.
    Yeh, C., Deng, H., Pan, H.: Multi-criteria analysis for dredger dispatching under uncertainty. J. Oper. Res. Soc. 50, 35–43 (1999)CrossRefGoogle Scholar
  105. 105.
    Yilmaz, M.R.: An information-expectation framework for decisions under uncertainty. J. Multi-Criteria Decis. Anal. 1, 65–80 (1992)CrossRefGoogle Scholar
  106. 106.
    Zank, H.: Cumulative prospect theory for parametric and multiattribute utilities. Math. Oper. Res. 26(1), 67–81 (2001)CrossRefGoogle Scholar
  107. 107.
    Zaras, K.: Rough approximation of a preference relation by a multi-attribute dominance for deterministic, stochastic and fuzzy decision problems. Eur. J. Oper. Res. 159(1), 196–206 (2004)CrossRefGoogle Scholar
  108. 108.
    Zhang, Y., Fan, Z., Liu,Y.: A method based on stochastic dominance degrees for stochastic multiple criteria decision making. Comput. Ind. Eng. 58(4), 544–552 (2010)CrossRefGoogle Scholar
  109. 109.
    Zimmermann, H.: An application-oriented view of modeling uncertainty. Eur. J. Oper. Res. 122, 190–198 (2000)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Statistical SciencesUniversity of Cape TownRondeboschSouth Africa
  2. 2.Research Center, African Institute for Mathematical SciencesMuizenbergSouth Africa

Personalised recommendations