Skip to main content
SpringerLink
Log in

Stochastic multi-objective optimization: a survey on non-scalarizing methods

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

Currently, stochastic optimization on the one hand and multi-objective optimization on the other hand are rich and well-established special fields of Operations Research. Much less developed, however, is their intersection: the analysis of decision problems involving multiple objectives and stochastically represented uncertainty simultaneously. This is amazing, since in economic and managerial applications, the features of multiple decision criteria and uncertainty are very frequently co-occurring. Part of the existing quantitative approaches to deal with problems of this class apply scalarization techniques in order to reduce a given stochastic multi-objective problem to a stochastic single-objective one. The present article gives an overview over a second strand of the recent literature, namely methods that preserve the multi-objective nature of the problem during the computational analysis. We survey publications assuming a risk-neutral decision maker, but also articles addressing the situation where the decision maker is risk-averse. In the second case, modern risk measures play a prominent role, and generalizations of stochastic orders from the univariate to the multivariate case have recently turned out as a promising methodological tool. Modeling questions as well as issues of computational solution are discussed.

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

Notes

  1. Expressed in the terms used in Ben Abdelaziz (2012), this means that we focus on the “multi-objective method” instead of the “stochastic method”.

  2. The sample space Ω is the first component of the probability space \((\varOmega,\mathcal{A},P)\). Note that in stochastic optimization, the probability model given by \((\varOmega,\mathcal{A},P)\) is always assumed as known, and therefore in particular also the distributions of the considered random variables.

  3. The axioms proposed, in the convex setting, are essentially

    1. (i)

      Monotonicity: ρ(X)≤ρ(Y) provided that X≥Y almost everywhere,

    2. (ii)

      Convexity: ρ((1−λ)Y 0+λY 1)≤(1−λ)ρ(Y 0)+λρ(Y 1),

    3. (iii)

      Positive Homogeneity: ρ(λY)=λρ(Y) for all λ>0 and

    4. (iv)

      Translation Invariance: ρ(Y+c)=ρ(Y)−c.

  4. In Sects. 4.1.3 and 4.2.3, also problem formulations encompassing stochastic dominance constraints will be discussed. It is clear that these constraints cannot be considered anymore as “deterministic”. This case will nevertheless be included here, since stochastic dominance constraints are not so much externally defined conditions restricting the feasible set, but rather constitute an alternative way to express the preferences of a decision maker. We admit that the distinction is not sharp.

  5. An alternative way to judge risk-neutrality is to apply the Arrow-Pratt risk aversion measure defined by −u″(⋅)/u′(⋅) to each random variable f j . Evidently, the values vanish for the utility u given by (13). Using a more general observation (Theorem 1 in Gutjahr 2012a), it can be shown that on the additional constraint that u is nondecreasing and concave, (13) is the only form u can have for a decision maker who is risk-neutral toward all properly stochastic objective functions. This does not hold anymore if the restriction to concave utilities is dropped.

  6. Let us also mention the s-convex class given by , where u (s) is the s-th derivative of u. This class contains the function x↦x k as well as the function x↦−x k for every k<s. Notably, the relation thus ensures that the moments of order less than s coincide: \(\mathbb{E}[X^{k}] = \mathbb{E}[Y^{k}]\) for all k<s. The special case  of the 2-convex class is closely related to the class defined above, but note that it has an equality constraint for expected values instead of an inequality constraint.

  7. For example, one criterion for each , but also other representations of the criteria can be chosen, as it will be seen later. Under some special circumstances, some of these criteria may coincide in a way making the problem finite-dimensional even for infinite , but in general, this does not hold.

  8. The last part of the sentence says that an efficient solution x needs not necessarily to be stochastically nondominated itself, but it can only be stochastically dominated by a solution y with \(\mathbb{E}[f(x,\omega )] = \mathbb{E} [f(y,\omega)] \mbox{ and } \rho(f(x,\omega)) = \rho(f(y,\omega))\). Therefore, if the pre-image of a Pareto-optimal point of the mean-risk model (4) is finite, it contains at least one stochastically nondominated solution.

  9. Analogously as in the case mentioned in Sect. 2.2, it is often sufficient to have only one representative from the pre-image of each stochastically nondominated random vector in the solution set.

  10. Actually, the authors use the term CVaR instead of \(\mathsf {AV@R}\).

References

  • Abbas, M., & Bellahcene, F. (2006). Cutting plane method for multiple objective stochastic integer linear programming. European Journal of Operational Research, 168, 967–984.

    Article  Google Scholar 

  • Ahmed, S., Schütz, P., & Tomasgard, A. (2009). Supply chain design under uncertainty using sample average approximation and dual decomposition. European Journal of Operational Research, 199, 409–419.

    Article  Google Scholar 

  • Amodeo, L., Prins, C., & Sanchez, D. (2009). Comparison of metaheuristic approaches for multi-objective simulation-based optimization in supply chain inventory management. In Lecture notes in computer science. Applications of evolutionary computing. Berlin: Springer.

    Google Scholar 

  • Armbruster, B., & Luedtke, J. B. (2011). Models and formulations for multivariate dominance constrained stochastic programs (Technical Report).

  • Artzner, P., Delbaen, F., Eber, J., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9, 203–228.

    Article  Google Scholar 

  • Artzner, P., Delbaen, F., & Heath, D. (1997). Thinking coherently. Risk, 10, 68–71.

    Google Scholar 

  • Baesler, F. F., & Sepúlveda, J. A. (2001). Multi-objective simulation optimization for a cancer treatment center. In Proc. winter simulation conference 2001 (pp. 1405–1411).

    Google Scholar 

  • Basseur, M., & Zitzler, E. (2006). Handling uncertainty in indicator-based multiobjective optimization. International Journal of Computational Intelligence Research, 2, 255–272.

    Article  Google Scholar 

  • Bath, S. K., Dhillon, J. S., & Kothari, D. P. (2004). Stochastic multiobjective generation dispatch. Electric Power Components and Systems, 32, 1083–1103.

    Article  Google Scholar 

  • Ben Abdelaziz, F. (1992). L’efficacité en programmation multi-objectifs stochastique. PhD thesis, Université de Laval, Québec.

  • Ben Abdelaziz, F. (2012). Solution approaches for the multiobjective stochastic programming. European Journal of Operations Research, 216, 1–16.

    Article  Google Scholar 

  • Ben Abdelaziz, F., Aouni, B., & El Fayedh, R. (2007). Multiobjective stochastic programming for portfolio selection. European Journal of Operations Research, 177, 1811–1823.

    Article  Google Scholar 

  • Caballero, R., Cerda, E., Muños, M. M., & Rey, L. (2004). Stochastic approach versus multiobjective approach for obtaining efficient solutions in stochastic multiobjective programming problems. European Journal of Operational Research, 158, 633–648.

    Article  Google Scholar 

  • Cardona-Valdés, Y., Álvarez, A., & Ozdemir, D. (2011). A bi-objective supply chain design problem with uncertainty. Transportation Research Part C, 19, 821–832.

    Article  Google Scholar 

  • Chinchuluun, A., & Pardalos, P. M. (2007). A survey of recent developments in multiobjective optimization. Annals of Operations Research, 154, 29–50.

    Article  Google Scholar 

  • Dantzig, G. B. (1955). Linear programming under uncertainty. Management Science, 197–206.

  • Dentcheva, D., & Martinez, G. (2012). Two-stage stochastic optimization problems with stochastic ordering constraints on the recourse. European Journal of Operational Research, 219(1), 1–8.

    Article  Google Scholar 

  • Dentcheva, D., & Ruszczyński, A. (2003). Optimization with stochastic dominance constraints. SIAM Journal on Optimization, 14(2), 548–566.

    Article  Google Scholar 

  • Dentcheva, D., & Ruszczyński, A. (2009). Optimization with multivariate stochastic dominance constraints. Mathematical Programming Series B, 117, 111–127.

    Article  Google Scholar 

  • Ding, H., Benyucef, L., & Xie, X. (2006). A simulation-based multi-objective genetic algorithm approach for networked enterprises optimization. Engineering Applications of Artificial Intelligence, 19, 609–623.

    Article  Google Scholar 

  • Ehrgott, M. (2000). Multicriteria optimization. Berlin: Springer.

    Book  Google Scholar 

  • Ehrgott, M., & Gandibleux, X. (2000). A survey and annotated bibliography of multiobjective combinatorial optimization. OR Spektrum, 22, 425–460.

    Article  Google Scholar 

  • Eskandari, H., & Geiger, C. D. (2009). Evolutionary multiobjective optimization in noisy problem environments. Journal of Heuristics, 15, 559–595.

    Article  Google Scholar 

  • Eskandari, H., Rabelo, L., & Mollaghasemi, M. (2005). Multiobjective simulation optimization using an enhanced genetic algorithm. In Proceedings of the 37th conference on winter simulation, WSC’05, Orlando, Florida (pp. 833–841).

    Chapter  Google Scholar 

  • Fábián, C. I., Mitra, G., & Roman, D. (2011). Processing second order stochastic dominance models using cutting plane representations. Mathematical Programming, 130(1), 33–37.

    Article  Google Scholar 

  • Föllmer, H., & Schied, A. (2004). de Gruyter studies in mathematics: Vol. 27. Stochastic finance—an introduction in discrete time. Berlin: de Gruyter.

    Book  Google Scholar 

  • Fonseca, M., García-Sánchez, A., Ortega-Mier, M., & Saldanha da Gama, F. (2010). A stochastic bi-objective location model for strategic reverse logistics. TOP, 18, 158–184.

    Article  Google Scholar 

  • Gutjahr, W. J. (2005). Two metaheuristics for multiobjective stochastic combinatorial optimization. In Proc. SAGA 2005 (stochastic algorithms: foundations and applications) (pp. 116–125). Berlin: Springer.

    Google Scholar 

  • Gutjahr, W. J. (2009). A provably convergent heuristic for stochastic bicriteria integer programming. Journal of Heuristics, 15, 227–258.

    Article  Google Scholar 

  • Gutjahr, W. J. (2011). Recent trends in metaheuristics for stochastic combinatorial optimization. Central European Journal of Computer Science, 1, 58–66.

    Google Scholar 

  • Gutjahr, W. J. (2012a). Multi-objective stochastic optimization under partial risk neutrality (Technical Report). Univ. of Vienna.

  • Gutjahr, W. J. (2012b). A note on the utility-theoretic justification of the Pareto approach in stochastic multi-objective combinatorial optimization. In L. Zadnik Stirn, J. Zerovnik, J. Povh, S. Drobne, & A. Lisec (Eds.), Proc. 11th internat. symposium on operational research in Slovenia (SOR’11), Dolenjske Toplice, Slovenia, September 28–30, 2012.

    Google Scholar 

  • Gutjahr, W. J. (2012c). Runtime analysis of an evolutionary algorithm for stochastic multi-objective combinatorial optimization. Evolutionary Computation, 20, 395–421.

    Article  Google Scholar 

  • Gutjahr, W. J. (2012d, accepted for publication). A three-objective optimization approach to cost effectiveness analysis under uncertainty. In Proc. OR2012 (international annual conference of the German operations research society 2012).

  • Gutjahr, W. J., & Reiter, P. (2010). Bi-objective project portfolio selection and staff assignment under uncertainty. Optimization Methods & Software, 59, 417–445.

    Google Scholar 

  • Haskell, W. B., Shanthikumar, J. G., & Shen, Z. M. (2012a). Aspects of increasing concave stochastic order constrained optimization (Technical Report).

  • Haskell, W. B., Shanthikumar, J. G., & Shen, Z. M. (2012b). Optimization with stochastic objectives (Technical Report).

  • Hassan-Pour, H. A., Mosadegh-Khah, M., & Tavakkoli-Moghaddam, R. (2009). Solving a multi-objective multi-depot stochastic location-routing problem by a hybrid simulated annealing algorithm. Proceedings of the Institution of Mechanical Engineers. Part B, Journal of Engineering Manufacture, 1045–1054.

  • Heitsch, H., Römisch, W., & Strugarek, C. (2006). Stability of multistage stochastic programs. SIAM Journal on Optimization, 17, 511–525.

    Article  Google Scholar 

  • Hnaien, F., Delorme, X., & Dolgui, A. (2010). Multi-objective optimization for inventory control in two-level assembly systems under uncertainty of lead times. Computers & Operations Research, 20, 1835–1843.

    Article  Google Scholar 

  • Homem-de Mello, T. (2003). Variable-sample methods for stochastic optimization. ACM Transactions on Modeling and Computer Simulation, 13, 108–133.

    Article  Google Scholar 

  • Homem-de Mello, T., & Mehrotra, S. (2009). A cutting surface method for uncertain linear programs with polyhedral stochastic dominance constraints. SIAM Journal on Optimization, 20, 1250–1273.

    Article  Google Scholar 

  • Hu, J., Homem-de Mello, T., & Mehrotra, S. (2011). Risk-adjusted budget allocation models with application in homeland security. IIE Transactions, 43, 819–839.

    Article  Google Scholar 

  • Hu, J., Homem-de Mello, T., & Mehrotra, S. (2012). Sample average approximation of stochastic dominance constrained programs. Mathematical Programming, 133, 171–201.

    Article  Google Scholar 

  • Hughes, E. J. (2001). Evolutionary multi-objective ranking with uncertainty and noise. In Proc. EMO’01 (evolutionary multicriterion optimization) (pp. 329–343). Berlin: Springer.

    Google Scholar 

  • Köksalan, M., Wallenius, J., & Zionts, S. (2011). Multiple criteria decision making: from early history to the 21st century. Singapore: World Scientific.

    Book  Google Scholar 

  • Koopmans, T. C. (1951). An analysis of production as an efficient combination of activities. In Activity analysis of production and allocation. New York: Wiley.

    Google Scholar 

  • Laumanns, M., Thiele, L., & Zitzler, E. (2006). An efficient, adaptive parameter variation scheme for metaheuristics based on the epsilon-constraint method. European Journal of Operational Research, 169, 932–942.

    Article  Google Scholar 

  • Liefooghe, A., Basseur, M., Jourdan, L., & Talbi, E.-G. (2007a). Combinatorial optimization of stochastic multi-objective problems: an application to the flow-shop scheduling problem. In Lecture notes in computer science: Vol. 4403. Evolutionary multi-criterion optimization (pp. 457–471). Berlin: Springer.

    Chapter  Google Scholar 

  • Liefooghe, A., Basseur, M., Jourdan, L., & Talbi, E.-G. (2007b). ParadisEO-MOEO: a framework for evolutionary multi-objective optimization. In Lecture notes in computer science: Vol. 4403. Evolutionary multi-criterion optimization (pp. 386–400). Berlin: Springer.

    Chapter  Google Scholar 

  • Müller, A., & Stoyan, D. (2002). Wiley series in probability and statistics. Comparison methods for stochastic models and risks. Chichester: Wiley.

    Google Scholar 

  • Norkin, V. I., Ermoliev, Y. M., & Ruszczyński, A. (1998). On optimal allocation of indivisibles under uncertainty. Operations Research, 46, 381–395.

    Article  Google Scholar 

  • Noyan, N., & Rudolf, G. (2012). Optimization with multivariate conditional value-at-risk constraints (Technical Report).

  • Ogryczak, W., & Ruszczyński, A. (1999). From stochastic dominance to mean-risk models: semideviations as risk measures. European Journal of Operational Research, 116, 33–50.

    Article  Google Scholar 

  • Ogryczak, W., & Ruszczyński, A. (2001). On consistency of stochastic dominance and mean-semideviation models. Mathematical Programming Series B, 89, 217–232.

    Article  Google Scholar 

  • Ogryczak, W., & Ruszczyński, A. (2002). Dual stochastic dominance and related mean-risk models. SIAM Journal on Optimization, 13(1), 60–78.

    Article  Google Scholar 

  • Pflug, G. Ch. (2001). Scenario tree generation for multiperiod financial optimization by optimal discretization. Mathematical Programming, 89, 251–271.

    Article  Google Scholar 

  • Pflug, G. Ch. (2003). Stochastic optimization and statistical inference. In A. Ruszczyński & A. Shapiro (Eds.), Handbooks in operations research and management science: Vol. 10. Stochastic programming (Chap. 7, pp. 427–482). Amsterdam: Elsevier.

    Chapter  Google Scholar 

  • Pflug, G. Ch., & Pichler, A. (2011). In International series in operations research & management science: Vol. 163. Approximations for probability distributions and stochastic optimization problems (Chap. 15, pp. 343–387). New York: Springer.

    Google Scholar 

  • Pflug, G. Ch., & Pichler, A. (2012). A distance for multistage stochastic optimization models. SIAM Journal on Optimization, 22, 1–23.

    Article  Google Scholar 

  • Pflug, G. Ch., & Römisch, W. (2007). Modeling, measuring and managing risk. River Edge: World Scientific.

    Book  Google Scholar 

  • Pflug, G. Ch., & Wozabal, N. (2010). Asymptotic distribution of law-invariant risk functionals. Finance and Stochastics, 14, 397–418.

    Article  Google Scholar 

  • Rath, S., Gendreau, M., & Gutjahr, W. J. (2012). Bi-objective stochastic programming models for determining depot locations in disaster relief operations planning (Technical Report).

  • Roman, D., Darby-Dowman, K., & Mitra, G. (2007). Mean-risk models using two risk measures: a multi-objective approach. Quantitative Finance, 7, 443–458.

    Article  Google Scholar 

  • Saaty, Th., & Gass, S. (1954). Parametric objective function (part i). Operations Research, 316–319.

  • Schultz, R. (2011). Risk aversion in two-stage stochastic integer programming. In International series in operations research and management science: Vol. 150. Stochastic programming (pp. 165–187). New York: Springer.

    Chapter  Google Scholar 

  • Shaked, M., & Shanthikumar, J. G. (2007). Springer series in statistics. Stochastic order. Berlin: Springer.

    Book  Google Scholar 

  • Shapiro, A., & Xu, H. (2007). Uniform laws of large numbers for set-valued mappings and subdifferentials of random functions. Mathematical Analysis and Applications, 325, 1390–1399.

    Article  Google Scholar 

  • Shapiro, A., Ruszczyński, A., & Dentcheva, D. (2009). In MPS-SIAM series on optimization: Vol. 9. Lectures on stochastic programming.

    Chapter  Google Scholar 

  • Syberfeldt, A., Ng, A., John, R. I., & Moore, Ph. (2009). Multi-objective evolutionary simulation-optimisation of a real-world manufacturing problem. Robotics and Computer-Integrated Manufacturing, 25, 926–931.

    Article  Google Scholar 

  • Teich, J. (2001). Pareto-front exploration with uncertain objectives. In Proc. EMO’01 (evolutionary multicriterion optimization) (pp. 314–328). Berlin: Springer.

    Google Scholar 

  • Tricoire, F., Graf, A., & Gutjahr, W. J. (2012). The bi-objective stochastic covering tour problem. Computers & Operations Research, 39, 1582–1592.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics and Operations Research, University of Vienna, Universitaetsstrasse 5/9, 1010, Vienna, Austria

    Walter J. Gutjahr

  2. Norwegian University of Science and Technology, Trondheim, Norway

    Alois Pichler

Authors
  1. Walter J. Gutjahr
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alois Pichler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Walter J. Gutjahr.

Additional information

Invited Survey Article for Annals of Operations Research: Special Volume.

Second author gratefully acknowledges support of the Norwegian grant 207690/E20.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gutjahr, W.J., Pichler, A. Stochastic multi-objective optimization: a survey on non-scalarizing methods. Ann Oper Res 236, 475–499 (2016). https://doi.org/10.1007/s10479-013-1369-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-013-1369-5

Keywords

Search

Navigation