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Bilevel programming approaches to production planning for multiple products with short life cycles

Abstract

We provide decision-making models for a manufacturer which plans to produce multiple short life cycle products with the one-shot decision theory. The obtained optimal production quantities are based on the most appropriate scenarios for the manufacturer. Since the models are the bilevel programming problems with the max–min or min–max operator in the lower levels, we propose two approaches to translate them into general single-level optimization problems such that they can be solved via the commonly used optimization methods. The effectiveness of our approaches is examined from the theoretical and computational aspects.

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References

  • Alem DJ, Morabito R (2012) Production planning in furniture settings via robust optimization. Comput Oper Res 39(2):139–150

    Article  Google Scholar 

  • Allais M Le (1953) Comportement de l’homme rationnel devant le risque: critique des postulats et axiomes de l’ecole americaine. Econometrica 21(4):503–546

    Article  Google Scholar 

  • Allende GB, Still G (2013) Solving bilevel programs with the KKT-approach. Math Program 138(1–2):309–332

    Article  Google Scholar 

  • Alvarez PP, Vera JR (2014) Application of robust optimization to the sawmill planning problem. Ann Oper Res 219(1):457–475

    Article  Google Scholar 

  • Aouam T, Brahimi N (2013) Integrated production planning and order acceptance under uncertainty: a robust optimization approach. Eur J Oper Res 228(3):504–515

    Article  Google Scholar 

  • Ardjmand E, Weckman GR, Young WA II, Sanei Bajgiran O, Aminipour B (2016) A robust optimisation model for production planning and pricing under demand uncertainty. Int J Prod Res 54(13):3885–3905

    Article  Google Scholar 

  • Bagnoli M, Bergstrom T (2005) Log-concave probability and its applications. Econ Theory 26(2):445–469

    Article  Google Scholar 

  • Bard JF (1991) Some properties of the bilevel programming problem. J Optim Theory Appl 68(2):371–378

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Book  Google Scholar 

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

    Google Scholar 

  • Bordalo P, Gennaioli N, Shleifer A (2012) Salience theory of choice under risk. Q J Econ 127(3):1243–1285

    Article  Google Scholar 

  • Bornapour M, Hooshmand RA (2015) An efficient scenario-based stochastic programming for optimal planning of combined heat, power, and hydrogen production of molten carbonate fuel cell power plants. Energy 83:734–748

    Article  Google Scholar 

  • Busse MR, Lacetera N, Pope DG, Silva-Risso J, Sydnor JR (2013) Estimating the effect of salience in wholesale and retail car markets. Am Econ Rev 103(3):575–579

    Article  Google Scholar 

  • Carvalho AN, Oliveira F, Scavarda LF (2016) Tactical capacity planning in a real-world ETO industry case: a robust optimization approach. Int J Prod Econ 180:158–171

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Chen X, Sim M, Sun P (2007) A robust optimization perspective on stochastic programming. Oper Res 55(6):1058–1071

    Article  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

    Article  Google Scholar 

  • Colson B, Marcotte P, Savard G (2005) Bilevel programming: a survey. 4OR 3(2):87–107

    Article  Google Scholar 

  • Dantzig GB (1955) Linear programming under uncertainty. Manag Sci 1(3–4):197–206

    Article  Google Scholar 

  • Dempe S (2002) Foundations of bilevel programming. Springer, Berlin

    Google Scholar 

  • Dempe S, Zemkoho AB (2012) On the Karush–Kuhn–Tucker reformulation of the bilevel optimization problem. Nonlinear Anal Theory Methods Appl 75(3):1202–1218

    Article  Google Scholar 

  • Dempe S, Zemkoho AB (2013) The bilevel programming problem: reformulations, constraint qualifications and optimality conditions. Math Program 138(1–2):447–473

    Article  Google Scholar 

  • Ellsberg D (1961) Risk, ambiguity and savage axioms. Q J Econ 75(4):643–669

    Article  Google Scholar 

  • Fisher ML (1997) What is the right supply chain for your product? Harv Bus Rev 75:105–116

    Google Scholar 

  • Gorissen BL, Den Hertog D (2013) Robust counterparts of inequalities containing sums of maxima of linear functions. Eur J Oper Res 227(1):30–43

    Article  Google Scholar 

  • Graves SC (2011) Uncertainty and production planning. In: Kempf KG, Keskinocak P, Uzsoy R (eds) Planning production and inventories in the extended enterprise, vol 1. Springer, US, pp 83–101

    Chapter  Google Scholar 

  • Guo P (2010) One-shot decision approach and its application to duopoly market. Int J Inf Decis Sci 2(3):213–232

    Google Scholar 

  • Guo P (2011) One-shot decision theory. IEEE Trans Syst Man Cybern Part A Syst Hum 41(5):917–926

    Article  Google Scholar 

  • Guo P, Li Y (2014) Approaches to multistage one-shot decision making. Eur J Oper Res 236(2):612–623

    Article  Google Scholar 

  • Guo P, Ma X (2014) Newsvendor models for innovative products with one-shot decision theory. Eur J Oper Res 239(2):523–536

    Article  Google Scholar 

  • Guo P, Yan R, Wang J (2010) Duopoly market analysis within one-shot decision framework with asymmetric possibilistic information. Int J Comput Intell Syst 3(6):786–796

    Article  Google Scholar 

  • Guo L, Lin GH, Ye JJ (2015) Solving mathematical programs with equilibrium constraints. J Optim Theory Appl 166(1):234–256

    Article  Google Scholar 

  • Gyulai D, Pfeiffer A, Monostori L (2016) Robust production planning and control for multi-stage systems with flexible final assembly lines. Int J Prod Res 55(13):3657–3673

    Article  Google Scholar 

  • Hanasusanto GA, Roitch V, Kuhn D, Wiesemann W (2015) A distributionally robust perspective on uncertainty quantification and chance constrained programming. Math Program 151(1):35–62

    Article  Google Scholar 

  • Higle JL, Kempf KG (2011) Production planning under supply and demand uncertainty: a stochastic programming approach. In: Infanger G (ed) Stochastic programming. Springer, New York, pp 297–315

  • Ho CJ (1989) Evaluating the impact of operating environments on MRP system nervousness. Int J Prod Res 27(7):1115–1135

    Article  Google Scholar 

  • Hoheisel T, Kanzow C, Schwartz A (2013) Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints. Math Program 137(1–2):257–288

    Article  Google Scholar 

  • Kahneman D, Tversky A (1979) Prospect theory: an analysis of decision under risk. Econometrica 47(2):263–292

    Article  Google Scholar 

  • Kazemi Zanjani M, Nourelfath M, Ait-Kadi D (2010) A multi-stage stochastic programming approach for production planning with uncertainty in the quality of raw materials and demand. Int J Prod Res 48(16):4701–4723

    Article  Google Scholar 

  • Koca E, Yaman H, Akturk MS (2015) Stochastic lot-sizing with controllable processing times. Omega 53:1–10

    Article  Google Scholar 

  • Lin GH, Xu M, Ye JJ (2014) On solving simple bilevel programs with a nonconvex lower level program. Math Program 144(1–2):277–305

    Article  Google Scholar 

  • Luo ZQ, Pang JS, Ralph D (1996) Mathematical programs with equilibrium constraints. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Miller BL, Wagner HM (1965) Chance constrained programming with joint constraints. Oper Res 13(6):930–945

    Article  Google Scholar 

  • Mula J, Poler R, Garcia-Sabater JP, Lario FC (2006) Models for production planning under uncertainty: a review. Int J Prod Econ 103(1):271–285

    Article  Google Scholar 

  • Nasiri GR, Zolfaghari R, Davoudpour H (2014) An integrated supply chain production distribution planning with stochastic demands. Comput Ind Eng 77:35–45

    Article  Google Scholar 

  • Nemirovski A, Shapiro A (2006) Convex approximations of chance constrained programs. SIAM J Optim 17(4):969–996

    Article  Google Scholar 

  • Orquin JL, Loose SM (2013) Attention and choice: a review on eye movements in decision making. Acta Psychol 144(1):190–206

    Article  Google Scholar 

  • Outrata JV (1990) On the numerical solution of a class of Stackelberg problems. Math Methods Oper Res 34(4):255–277

    Article  Google Scholar 

  • Pagnoncelli BK, Ahmed S, Shapiro A (2009) Sample average approximation method for chance constrained programming: theory and applications. J Optim Theory Appl 142(2):399–416

    Article  Google Scholar 

  • Savage LJ (1954) The foundations of statistics. Wiley, New York

    Google Scholar 

  • Scheel H, Scholtes S (2000) Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. Math Oper Res 25(1):1–22

    Article  Google Scholar 

  • Sen A (2008) The US fashion industry: a supply chain review. Int J Prod Econ 114(2):571–593

    Article  Google Scholar 

  • Shapiro A, Dentcheva D, Ruszczyński A (2009) Lectures on stochastic programming: modeling and theory. Society for Industrial and Applied Mathematics, Philadelphia

    Book  Google Scholar 

  • Shi J, Zhang G, Sha J (2011) Optimal production planning for a multi-product closed loop system with uncertain demand and return. Comput Oper Res 38(3):641–650

    Article  Google Scholar 

  • Sodhi MS, Tang CS (2009) Modeling supply-chain planning under demand uncertainty using stochastic programming: a survey motivated by asset-liability management. Int J Prod Econ 121(2):728–738

    Article  Google Scholar 

  • Starmer C (2000) Developments in non-expected utility theory: the hunt for a descriptive theory of choice under risk. J Econ Lit 38(2):332–382

    Article  Google Scholar 

  • Stephen B, Vandenberghe L (2004) Convex optimization. Cambridge University Press, Cambridge

    Google Scholar 

  • Stewart N, Hermens F, Matthews WJ (2016) Eye movements in risky choice. J Behav Decis Mak 29(2–3):116–136

    Article  Google Scholar 

  • Tang L, Che P, Liu J (2012) A stochastic production planning problem with nonlinear cost. Comput Oper Res 39(9):1977–1987

    Article  Google Scholar 

  • Thomassey S (2010) Sales forecasts in clothing industry: the key success factor of the supply chain management. Int J Prod Econ 128(2):470–483

    Article  Google Scholar 

  • von Neumann J, Morgenstern O (1944) Theory of games and economic behavior. Princeton University Press, Princeton

    Google Scholar 

  • Wang C, Guo P (2017) Behavioral models for first-price sealed-bid auctions with the one-shot decision theory. Eur J Oper Res 261(3):994–1000

    Article  Google Scholar 

  • Wazed M, Ahmed S, Nukman Y (2010) A review of manufacture resources planning models under different uncertainties: state-of the-art and future directions. S Afr J Ind Eng 21(1):17–33

    Google Scholar 

  • Wiesemann W, Kuhn D, Sim M (2014) Distributionally robust convex optimization. Oper Res 62(6):1358–1376

    Article  Google Scholar 

  • Yanikoǧlu I, Kuhn D (2018) Decision rule bounds for two-stage stochastic bilevel programs. SIAM J Optim 28(1):198–222

    Article  Google Scholar 

  • Ye JJ, Zhu DL (1995) Optimality conditions for bilevel programming problems. Optimization 33(1):9–27

    Article  Google Scholar 

  • Zhu X, Guo P (2016) The one-shot decision theory based production planning models. In: Proceedings of IEEE international conference on industrial engineering and engineering management, December 4–7. pp 789-792

  • Zhu X, Guo P (2017) Approaches to four types of bilevel programming problems with nonconvex nonsmooth lower level programs and their applications to newsvendor problems. Math Methods Oper Res 86(2):255–275

    Article  Google Scholar 

  • Zymler S, Kuhn D, Rustem B (2013) Distributionally robust joint chance constraints with second-order moment information. Math Program 137(1–2):167–198

    Article  Google Scholar 

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Acknowledgements

This research was supported by JSPS KAKENHI Grant Number 15K03599.

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Correspondence to Peijun Guo.

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Zhu, X., Guo, P. Bilevel programming approaches to production planning for multiple products with short life cycles. 4OR-Q J Oper Res 18, 151–175 (2020). https://doi.org/10.1007/s10288-019-00407-z

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  • DOI: https://doi.org/10.1007/s10288-019-00407-z

Keywords

  • Bilevel programming
  • One-shot decision theory
  • Production planning
  • Uncertainty

Mathematics Subject Classification

  • 90B50
  • 90C26
  • 90C30
  • 91B06