Skip to main content
Log in

The decision rule approach to optimization under uncertainty: methodology and applications

  • Original Paper
  • Published:
Computational Management Science Aims and scope Submit manuscript

Abstract

Dynamic decision-making under uncertainty has a long and distinguished history in operations research. Due to the curse of dimensionality, solution schemes that naïvely partition or discretize the support of the random problem parameters are limited to small and medium-sized problems, or they require restrictive modeling assumptions (e.g., absence of recourse actions). In the last few decades, several solution techniques have been proposed that aim to alleviate the curse of dimensionality. Amongst these is the decision rule approach, which faithfully models the random process and instead approximates the feasible region of the decision problem. In this paper, we survey the major theoretical findings relating to this approach, and we investigate its potential in two applications areas.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  • An Y, Zeng B, Zhang Y, Zhao L (2014) Reliable \(p\)-median facility location problem: two-stage robust models and algorithms. Transp Res B 64:54–72

    Google Scholar 

  • Ardestani-Jaafari A, Delage E (2016) Robust optimization of sums of piecewise linear functions with application to inventory problems. Oper Res 64(2):474–494

    Google Scholar 

  • Atamtürk A, Zhang M (2007) Two-stage robust network flow and design under demand uncertainty. Oper Res 55(4):662–673

    Google Scholar 

  • Ayoub J, Poss M (2016) Decomposition for adjustable robust linear optimization subject to uncertainty polytope. Comput Manag Sci 13(2):219–239

    Google Scholar 

  • Bandi C, Trichakis N, Vayanos P (2017) Robust multiclass queuing theory for wait time estimation in resource allocation systems. Manag Sci (forthcoming)

  • Ben-Tal A, Goryashko A, Guslitzer E, Nemirovski A (2004) Adjustable robust solutions of uncertain linear programs. Math Program 99(2):351–376

    Google Scholar 

  • Ben-Tal A, El Ghaoui L, Nemirovski A (2009) Robust optimization. Princeton University Press, Princeton

    Google Scholar 

  • Bertsekas DP (2001) Dynamic programming and optimal control. Athena Scientific, Belmont

    Google Scholar 

  • Bertsimas D, Caramanis C (2010) Finite adaptibility in multistage linear optimization. IEEE Trans Automat Contr 55(12):2751–2766

    Google Scholar 

  • Bertsimas D, Georghiou A (2015) Design of near optimal decision rules in multistage adaptive mixed-integer optimization. Oper Res 63(3):610–627

    Google Scholar 

  • Bertsimas D, Georghiou A (2018) Binary decision rules for multistage adaptive mixed-integer optimization. Math Program 167(2):395–433

    Google Scholar 

  • Bertsimas D, Goyal V (2010) On the power of robust solutions in two-stage stochastic and adaptive optimization problems. Math Oper Res 35(2):284–305

    Google Scholar 

  • Bertsimas D, Goyal V (2012) On the power and limitations of affine policies in two-stage adaptive optimization. Math Program 134(2):491–531

    Google Scholar 

  • Bertsimas D, de Ruiter FJCT (2016) Duality in two-stage adaptive linear optimization: faster computation and stronger bounds. INFORMS J Comput 28(3):500–511

    Google Scholar 

  • Bertsimas D, Iancu DA, Parrilo PA (2010) Optimality of affine policies in multi-stage robust optimization. Math Oper Res 35(2):363–394

    Google Scholar 

  • Bertsimas D, Brown DB, Caramanis C (2011) Theory and applications of robust optimization. SIAM Rev 53(3):464–501

    Google Scholar 

  • Bertsimas D, Goyal V, Sun XA (2011) A geometric characterization of the power of finite adaptability in multistage stochastic and adaptive optimization. Math Oper Res 36(1):24–54

    Google Scholar 

  • Bertsimas D, Iancu DA, Parrilo PA (2011) A hierarchy of near-optimal policies for multistage adaptive optimization. IEEE Trans Automat Contr 56(12):2809–2824

    Google Scholar 

  • Bertsimas D, Thiele A (2006) Robust and data-driven optimization: modern decision making under uncertainty. In: INFORMS TutoRials in operations research: models, methods, and applications for innovative decision making, pp 95–122

    Google Scholar 

  • Bertsimas D, Vayanos P (2017) Data-driven learning in dynamic pricing using adaptive optimization. Optimization Online

  • Birge JR, Louveaux F (1997) Introduction to stochastic programming. Springer, New York

    Google Scholar 

  • Campi MC, Garatti S (2011) A sampling-and-discarding approach to chance-constrained optimization: feasibility and optimality. J Optim Theory Appl 148(2):257–280

    Google Scholar 

  • Charnes A, Cooper WW (1959) Chance-constrained programming. Manag Sci 6(1):73–79

    Google Scholar 

  • Charnes A, Cooper WW, Symonds GH (1958) Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil. Manag Sci 4(3):235–263

    Google Scholar 

  • Chen X, Zhang Y (2009) Uncertain linear programs: extended affinely adjustable robust counterparts. Oper Res 57(6):1469–1482

    Google Scholar 

  • Chen P, Papageorgiou LG, Pinto JM (2008) Medium-term planning of single-stage single-unit multiproduct plants using a hybrid discrete/continuous-time MILP model. Ind Eng Chem Res 47(6):1925–1934

    Google Scholar 

  • Chen X, Sim M, Sun P, Zhang J (2008) A linear decision-based approximation approach to stochastic programming. Oper Res 56(2):344–357

    Google Scholar 

  • Chvátal V (1983) Linear programming. WH Freeman, New York

    Google Scholar 

  • Daskin MS, Snyder LV, Berger RT (2005) Logistics systems: design and optimization. In: Langevin A, Riopel D (eds) Facility location in supply chain design. Springer, New York, pp 39–65

    Google Scholar 

  • Delage E, Iancu DA (2015) Robust multistage decision making. In: The operations research revolution. INFORMS TutORials in operations research: models, methods, and applications for innovative decision making, pp 20–46

    Google Scholar 

  • Dupačová J, Consigli G, Wallace S (2000) Scenarios for multistage stochastic programs. Annal Oper Res 100(1):25–53

    Google Scholar 

  • Eisner M, Olsen P (1975) Duality for stochastic programming interpreted as L.P. in \({L}_p\)-space. SIAM J Appl Math 28(4):779–792

    Google Scholar 

  • Floudas CA, Lin X (2005) Mixed integer linear programming in process scheduling: modeling, algorithms, and applications. Annal Oper Res 139(1):131–162

    Google Scholar 

  • Gabrel V, Murat C, Thiele A (2014) Recent advances in robust optimization: an overview. Eur J Oper Res 235(3):471–483

    Google Scholar 

  • Garstka SJ, Wets RJB (1974) On decision rules in stochastic programming. Math Program 7(1):117–143

    Google Scholar 

  • Georghiou A, Wiesemann W, Kuhn D (2015) Generalized decision rule approximations for stochastic programming via liftings. Math Program 152(1):301–338

    Google Scholar 

  • Georghiou A, Tsoukalas A, Wiesemann W (2017) A primal-dual lifting scheme for two-stage robust optimization. Optimization Online

  • Georghiou A, Tsoukalas A, Wiesemann W (2018) Robust dual dynamic programming. Oper Res (forthcoming)

  • Gilboa I, Schmeidler D (1989) Maxmin expected utility with non-unique prior. J Math Econ 18(2):141–153

    Google Scholar 

  • Goh J, Sim M (2010) Distributionally robust optimization and its tractable approximations. Oper Res 58(4):902–917

    Google Scholar 

  • Gorissen BL, Yanğkoğlu İ, den Hertog D (2015) A practical guide to robust optimization. Omega 53:124–137

    Google Scholar 

  • Gounaris CE, Wiesemann W, Floudas CA (2013) The robust capacitated vehicle routing problem under demand uncertainty. Oper Res 61(3):677–683

    Google Scholar 

  • Guslitser E (2002) Uncertainty-immunized solutions in linear programming. Master’s thesis, Technion

  • Hadjiyiannis MJ, Goulart PJ, Kuhn DA (2011) scenario approach for estimating the suboptimality of linear decision rules in two-stage robust optimization. In: Proceedings of the 50th IEEE conference on decision and control and european control conference, pp 7386–7391

  • Hadjiyiannis MJ, Goulart PJ, Kuhn D (2011) An efficient method to estimate the suboptimality of affine controllers. IEEE Trans Automat Contr 56(12):2841–2853

    Google Scholar 

  • Hanasusanto GA, Kuhn D (2018) Conic programming reformulations of two-stage distributionally robust linear programs over Wasserstein balls. Oper Res 66(3):849–869

    Google Scholar 

  • Hanasusanto GA, Kuhn D, Wiesemann W (2015) \(K\)-adaptability in two-stage robust binary programming. Oper Res 63(4):877–891

    Google Scholar 

  • Hochreiter R, Pflug GC (2007) Financial scenario generation for stochastic multi-stage decision processes as facility location problems. Annal Oper Res 152(1):257–272

    Google Scholar 

  • IBM ILOG CPLEX Homepage (2015). https://www.ibm.com/analytics/cplex-optimizer

  • Kall P, Wallace S (1994) Stochastic programming. Wiley, New York

    Google Scholar 

  • Kong Q, Li S, Liu N, Teo C-P, Yan Z (2017) Robust multi-period vehicle routing under customer order uncertainty. http://www.columbia.edu/~nl2320/doc/Noshow-MS-1030c.pdf

  • Kuhn D, Wiesemann W, Georghiou A (2011) Primal and dual linear decision rules in stochastic and robust optimization. Math Program 130(1):177–209

    Google Scholar 

  • Kuhn D, Parpas P, Rustem B (2008) Stochastic optimization of investment planning problems in the electric power industry. In: Georgiadis M, Kikkinides E, Pistikopoulos E (eds) Process systems engineering: volume 5: energy systems engineering, Wiley-VCH, ch. 4, pp 215–230

  • Lappas NH, Gounaris CE (2016) Multi-stage adjustable robust optimization for process scheduling under uncertainty. AIChE J 62(5):1646–1667

    Google Scholar 

  • Liebchen C, Lübbecke M, Möhring RH, Stiller S (2009) The concept of recoverable robustness, linear programming recovery, and railway applications. In: Ahuja RK, Möhring RH, Zaroliagis CD (eds) Robust and online large-scale optimization: models and techniques for transportation systems. Springer, New York, pp 1–27

    Google Scholar 

  • Liu S, Pinto JM, Papageorgiou LG (2008) A TSP-based MILP model for medium-term planning of single-stage continuous multiproduct plants. Ind Eng Chem Res 47(20):7733–7743

    Google Scholar 

  • Lorca A, Sun XA (2015) Adaptive robust optimization with dynamic uncertainty sets for multi-period economic dispatch under significant wind. IEEE Trans Power Syst 30(4):1702–1713

    Google Scholar 

  • Lorca A, Sun XA (2017) Multistage robust unit commitment with dynamic uncertainty sets and energy storage. IEEE Trans Power Syst 32(3):1678–1688

    Google Scholar 

  • Meixell MJ, Gargeya VB (2005) Global supply chain design: a literature review and critique. Transp Res E 41(6):531–550

    Google Scholar 

  • Melo M, Nickel S, da Gama FS (2009) Facility location and supply chain management: a review. Eur J Oper Res 196(2):401–412

    Google Scholar 

  • Min H, Zhou G (2002) Supply chain modeling: past, present and future. Comput Ind Eng 43(1–2):231–249

    Google Scholar 

  • Nemirovski A, Shapiro A (2007) Convex approximations of chance constrained programs. SIAM J Optim 17(4):969–996

    Google Scholar 

  • Pereira MVF, Pinto LMVG (1991) Multi-stage stochastic optimization applied to energy planning. Math Program 52(1):359–375

    Google Scholar 

  • Pflug GC (2000) Some remarks on the value-at-risk and the conditional value-at-risk. Probabilistic constrained optimization. Springer, New York, pp 272–281

    Google Scholar 

  • Pflug GC (2001) Scenario tree generation for multiperiod financial optimization by optimal discretization. Math Program 89(2):251–271

    Google Scholar 

  • Pflug GC, Pichler A (2014) Multistage stochastic optimization. Springer, New York

    Google Scholar 

  • Pflug GC, Römisch W (2007) Modeling, measuring and managing risk. World Scientific, Singapore

    Google Scholar 

  • Rockafellar R, Uryasev S (2000) Optimization of conditional value-at-risk. J Risk 2:21–42

    Google Scholar 

  • Ruiz C, Conejo AJ (2015) Robust transmission expansion planning. Eur J Oper Res 242(2):390–401

    Google Scholar 

  • Santoso T, Ahmed S, Goetschalckx M, Shapiro A (2005) A stochastic programming approach for supply chain network design under uncertainty. Eur J Oper Res 167(1):96–115

    Google Scholar 

  • Shapiro A (2011) Analysis of stochastic dual dynamic programming method. Eur J Oper Res 209(1):63–72

    Google Scholar 

  • Shapiro A, Dentcheva D, Ruszczyński A (2009) Lectures on stochastic programming: modeling and theory. SIAM

  • Simchi-Levi D, Wang H, Wei Y (2018) Constraint generation for two-stage robust network flow problem

  • Subramanyam A, Gounaris C, Wiesemann W (2017a) \(K\)-adaptability in two-stage mixed-integer robust optimization. Optimization Online

  • Subramanyam A, Mufalli F, Pinto JM, Gounaris CE (2017b) Robust multi-period vehicle routing under customer order uncertainty. Optimization Online

  • Tsiakis P, Shah N, Pantelides CC (2001) Design of multi-echelon supply chain networks under demand uncertainty. Ind Eng Chem Res 40(16):3585–3604

    Google Scholar 

  • van der Vlerk MH (1996–2007) Stochastic programming bibliography. http://www.eco.rug.nl/mally/spbib.html

  • Vayanos P, Kuhn D, Rustem B (2012) A constraint sampling approach for multi-stage robust optimization. Automatica 48(3):459–471

    Google Scholar 

  • Vayanos P, Kuhn D, Rustem B (2011) Decision rules for information discovery in multi-stage stochastic programming. In: Proceedings of the 50th IEEE conference on decision and control and european control conference, pp 7368–7373

  • Verderame PM, Elia JA, Li J, Floudas CA (2010) Planning and scheduling under uncertainty: a review across multiple sectors. Ind Eng Chem Res 49(9):3993–4017

    Google Scholar 

  • Warrington J, Goulart P, Mariéthoz S, Morari M (2013) Policy-based reserves for power systems. IEEE Trans Power Syst 28(4):4427–4437

    Google Scholar 

  • Xu G, Burer S (2018) A copositive approach for two-stage adjustable robust optimization with uncertain right-hand sides. Comput Optim Appl 70(1):33–59

    Google Scholar 

  • Yanıkoğlu İ, Gorissen BL, den Hertog D (2017) Adjustable robust optimization–a survey and tutorial ResearchGate

  • Zeng B, Zhao L (2013) Solving two-stage robust optimization problems using a column-and-constraint generation method. Oper Res Lett 41(5):457–561

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Daniel Kuhn.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Georghiou, A., Kuhn, D. & Wiesemann, W. The decision rule approach to optimization under uncertainty: methodology and applications. Comput Manag Sci 16, 545–576 (2019). https://doi.org/10.1007/s10287-018-0338-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10287-018-0338-5

Keywords

Navigation