Soft Computing

, Volume 22, Issue 17, pp 5791–5801 | Cite as

The information value and the uncertainties in two-stage uncertain programming with recourse

  • Mingfa Zheng
  • Yuan Yi
  • Xuhua Wang
  • Jian Wang
  • Sheng Mao


Based on uncertainty theory, this paper mainly studies the uncertainties and the information value in the two-stage uncertain programming with recourse. We first define three fundamental concepts and investigate their theoretical properties, based on which we present two optimal indices, i.e., EVPI and VUS. Then, we introduce a method to calculate the expected value of the second-stage objective function involving discrete uncertain variables. Due to the complexity of calculation, the upper bound and lower bound for the two indices are studied, respectively. Finally, two examples are given to illustrate these concepts clearly. The results obtained in this paper can provide theoretical basis for studying uncertainties and information value in decision-making process under uncertain systems.


Uncertainty theory Two-stage uncertain programming Expected value Expected value of perfect information 



The author gratefully acknowledges the financial support provided by State Key Laboratory Development Program of China (Grant No. 9140C890302) and National Natural Science Foundation of China (Grant Nos.61502523, 61502521).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors. This work was carried out in collaboration between all authors. All authors read and approved the final manuscript.


  1. Birge JR, Louveaux F (2011) Introduction to stochastic programming. Springer, BerlinCrossRefzbMATHGoogle Scholar
  2. Dantzig GB (1955) Linear programming under uncertainty. Manag Sci 1(3–4):197–206MathSciNetCrossRefzbMATHGoogle Scholar
  3. Eckermann S, Karnon J, Willan AR (2010) The value of value of information: best informing research design and prioritization using current methods. Pharmacoeconomics 28(9):699CrossRefGoogle Scholar
  4. Eckermann S, Willan AR (2007) Expected value of information and decision making in HTA. Health Econ 16(2):195CrossRefGoogle Scholar
  5. Fan Y et al (2015) Planning water resources allocation under multiple uncertainties through a generalized fuzzy two-stage stochastic programming method. IEEE Trans Fuzzy Syst 23(5):1488–1504CrossRefGoogle Scholar
  6. Hoomans T, Fenwick EA, Palmer S, Claxton K (2009) Value of information and value of implementation: application of an analytic framework to inform resource allocation decisions in metastatic hormone-refractory prostate cancer. Value Health 12(2):315–324CrossRefGoogle Scholar
  7. Leovey H, Romisch W (2015) Quasi-Monte Carlo methods for linear two-stage stochastic programming problems. Math Program 151(1):315–345MathSciNetCrossRefzbMATHGoogle Scholar
  8. Liu B (2007) Uncertainty theory, 2nd edn. Springer, BerlinzbMATHGoogle Scholar
  9. Liu B (2009a) Theory and practice of uncertain programming. Springer, BerlinCrossRefzbMATHGoogle Scholar
  10. Liu B (2009b) Some research problems in uncertainty theory. J Uncertain Syst 3:3–10Google Scholar
  11. Liu B (2010a) Uncertainty theory: a branch of mathematics for modeling human uncertainty. Springer, BerlinCrossRefGoogle Scholar
  12. Liu B (2010b) Uncertain risk analysis and uncertain reliability analysis. J Uncertain Syst 4(4):163–170Google Scholar
  13. Liu B (2013) Polyrectangular theorem and independence of uncertain vectors. J Uncertain Anal Appl, 1, Article 9Google Scholar
  14. Liu B, Chen XW (2015) Uncertain multiobjective programming and uncer- 340 tain goal programming. J Uncertain Anal Appl, 3, Article 10Google Scholar
  15. Liu B, Yao K (2015) Uncertain multilevel programming: algorithm and applications. Comput Ind Eng 89:235–240CrossRefGoogle Scholar
  16. Parisio A, Jones CN (2015) A two-stage stochastic programming approach to employee scheduling in retail outlets with uncertain demand. Omega 53:97–103CrossRefGoogle Scholar
  17. Romeijnders W, Stougie L, Vlerk MHVD (2014) Approximation in two-stage stochastic integer programming. Surv Oper Res Manag Sci 19(1):17–33MathSciNetGoogle Scholar
  18. Shapiro A, Dentcheva D (2014) Lectures on stochastic programming: mod- eling and theory, SiamGoogle Scholar
  19. Sheng Y, Yao K (2014) Some formulas of variance of uncertain random variable. J Uncertain Anal Appl 2, Article 12Google Scholar
  20. Wang Z, Guo J, Zheng M, Wang Y (2015) Uncertain multiobjective traveling salesman problem. Eur J Oper Res 241(2):478–489MathSciNetCrossRefzbMATHGoogle Scholar
  21. Wolf C, Fabian CI, Koberstein A, Suhl L (2014) Applying oracles of on-demand accuracy in two-stage stochastic programming-a computational study. Eur J Oper Res 239(2):437–448MathSciNetCrossRefzbMATHGoogle Scholar
  22. Zhang B, Peng J (2013) Uncertain programming model for uncertain optimal assignment problem. Appl Math Model 37(9):6458–6468MathSciNetCrossRefGoogle Scholar
  23. Zheng M, Yi Y, Wang Z, Liao T (2017a) Relations among efficient solutions in uncertain multiobjective programming. Fuzzy Optim Decis Mak, doi: 10.1007/s10700-016-9252-x, to be published
  24. Zheng M, Yi Y, Wang Z, Liao T (2016) Efficient solution concepts andtheir application in uncertain multiobjective programming. Appl SoftComputing, doi: 10.1016/j.asoc.2016.07.021, to be published
  25. Zheng M, Yi Y, Wang Z, Chen JF (2017b) Study on two-stage uncertain programming based on uncertainty theory. J Intell Manuf 28(3):633–642CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Mingfa Zheng
    • 1
    • 2
  • Yuan Yi
    • 1
  • Xuhua Wang
    • 3
  • Jian Wang
    • 4
  • Sheng Mao
    • 4
  1. 1.School of Mathematics and StatisticsXi’an Jiaotong UniversityXi’anP.R. China
  2. 2.College of ScienceAir Force Engineering UniversityXi’anP.R. China
  3. 3.China Xi’an Satellite Control CenterXi’anP.R. China
  4. 4.Equipment Management and Safety Engineering CollegeAir Force Engineering UniversityXi’anP.R. China

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