Soft Computing

, Volume 22, Issue 17, pp 5879–5887 | Cite as

Belief degree of optimal models for uncertain single-period supply chain problem

  • Sibo Ding
Methodologies and Application


Mathematical model formulations for single-period supply chain problem depend on how to describe the demand. This paper applies uncertainty theory, which is a branch of axiomatic mathematics for dealing with human uncertainty, to model demand distribution. Uncertain decentralized management model and uncertain centralized management model are developed. Unique closed-form solutions for the two models are derived. The belief degree of “order quantity being less than the supply chain optimal order quantity” is proposed and the lower bound of the belief degree is obtained and carefully analyzed. Finally, some examples are presented to illustrate our method.


Uncertainty theory Uncertain demand Uncertain variable Newsboy problem 



This work was supported by the National Natural Science Foundation of China (Grant No. U1404701), the Scholarship Programm of China Scholarship Council (Grant No. 201509895007), the Soft Science Research Program of Henan Province (Grant No. 152400410447), the Science Foundation of Henan University of Technology (Grant No. 2017RCJH11) and the Key Research Base of Humanities and Social Sciences for Universities in Henan Province.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Human and animal rights

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. Araneda-Fuentes C, Lustosa LJ, Minner S (2015) A contract for coordinating capacity decisions in a business-to-business (B2B) supply chain. Int J Prod Econ 165:158–171CrossRefGoogle Scholar
  2. Awudu I, Zhang J (2013) Stochastic production planning for a biofuel supply chain under demand and price uncertainties. Appl Energy 103:189–196CrossRefGoogle Scholar
  3. Chen TH (2011) Coordinating the ordering and advertising policies for a single-period commodity in a two-level supply chain. Comput Ind Eng 61(4):1268–1274CrossRefGoogle Scholar
  4. Chen XW (2011) American option pricing formula for uncertain financial market. Int J Oper Res 8(2):32–37MathSciNetGoogle Scholar
  5. Ding SB (2013) Uncertain multi-product newsboy problem with chance constraint. Appl Math Comput 223:139–146MathSciNetzbMATHGoogle Scholar
  6. Ding SB (2014) Uncertain random newsboy problem. J Intell Fuzzy Syst 26(1):483–490MathSciNetzbMATHGoogle Scholar
  7. Dominey MJG, Hill RM (2004) Performance of approximations for compound Poisson distributed demand in the newsboy problem. Int J Prod Econ 92(2):145–155CrossRefGoogle Scholar
  8. Edgeworth FY (1888) The mathematical theory of banking. J Roy Stat Soc 51(1):113–127Google Scholar
  9. Ehrhardt R, Taube L (1987) An inventory model with random replenishment quantity. Int J Prod Res 25(12):1795–1803zbMATHGoogle Scholar
  10. Gallego G, Moon I (1993) The distribution free newsboy problem: review and extensions. J Oper Res Soc 44(8):825–834CrossRefzbMATHGoogle Scholar
  11. Gao XL, Gao Y (2013) Connectedness index of uncertain graphs. Int J Uncertain Fuzziness Knowl Based Syst 21(1):127–137MathSciNetCrossRefzbMATHGoogle Scholar
  12. Gao Y, Wen ML, Ding SB (2013) (\(s\), \(S\)) policy for uncertain single period inventory problem. Int J Uncertain Fuzziness Knowl Based Syst 21(6):945–953MathSciNetCrossRefzbMATHGoogle Scholar
  13. Gao Y, Yang LX, Li SK, Kar S (2015) On distribution function of the diameter in uncertain graph. Inf Sci 296:61–74MathSciNetCrossRefzbMATHGoogle Scholar
  14. Hadley G, Whitin TM (1963) Analysis of inventory systems. Prentice-Hall, New JerseyzbMATHGoogle Scholar
  15. Hill RM (1997) Applying Bayesian methodology with a uniform prior to the single period inventory model. Eur J Oper Res 98(3):555–562CrossRefzbMATHGoogle Scholar
  16. Hua ZS, Li SJ (2008) Impacts of demand uncertainty on retailers dominance and manufacturer-retailer supply chain cooperation. Omega Int J Manag Sci 36(5):697–714CrossRefGoogle Scholar
  17. Hua ZS, Li SJ, Liang L (2006) Impact of demand uncertainty on supply chain cooperation of single-period products. Int J Prod Econ 100(2):268–284CrossRefGoogle Scholar
  18. Huang KL, Kuo CW, Lu ML (2014) Wholesale price rebate vs. capacity expansion: the optimal strategy for seasonal products in a supply chain. Eur J Oper Res 234(1):77–85MathSciNetCrossRefzbMATHGoogle Scholar
  19. Kabak IW, Schiff AI (1978) Inventory models and management objectives. Sloan Manag Rev 19(2):53–59Google Scholar
  20. Kalpana P, Kaur A (2011) Optimal ordering decisions and revenue sharing in a single period split order supply chain. Technol Prod Oper 2(2):61–79CrossRefGoogle Scholar
  21. Lariviere MA, Porteus EL (2001) Selling to the newsvendor: an analysis of price-only contracts. Manuf Serv Oper Manag 3(4):293–305CrossRefGoogle Scholar
  22. Lau HS (1980) Some extensions of Ismail-Louderback’s stochastic CVP model under optimizing and satisfying criteria. Decis Sci 11(3):557–561CrossRefGoogle Scholar
  23. Lin J, Ng TS (2011) Robust multi-market newsvendor models with interval demand data. Eur J Oper Res 212(2):361–373MathSciNetCrossRefzbMATHGoogle Scholar
  24. Liu B (2007) Uncertainty theory, 2nd edn. Springer, BerlinzbMATHGoogle Scholar
  25. Liu B (2009) Some research problems in uncertainty theory. J Uncertain Syst 3(1):3–10Google Scholar
  26. Liu B (2010) Uncertainty theory: a branch of mathematics for modeling human uncertainty. Springer, BerlinCrossRefGoogle Scholar
  27. Mahoney JF, Sivazlian BD (1980) Probability of shortage for Erlang distributed demand in a \((\sigma, S)\) inventory problem. SIAM J Appl Math 38(1):156–162MathSciNetCrossRefzbMATHGoogle Scholar
  28. Pasternack BA (1985) Optimal pricing and return policies for perishable commodities. Mark Sci 4(2):166–176CrossRefGoogle Scholar
  29. Petruzzi NC, Dada M (1999) Pricing and the newsvendor problem: a review with extensions. Oper Res 47(2):183–194CrossRefzbMATHGoogle Scholar
  30. Qin ZF (2015) Mean-variance model for portfolio optimization problem in the simultaneous presence of random and uncertain returns. Eur J Oper Res 245(2):480–488MathSciNetCrossRefzbMATHGoogle Scholar
  31. Qin ZF, Kar S (2013) Single-period inventory problem under uncertain environment. Appl Math Comput 219(18):9630–9638MathSciNetzbMATHGoogle Scholar
  32. Qin Y, Wang R, Vakharia AJ, Chen Y, Seref MMH (2011) The newsvendor problem: review and directions for future research. Eur J Oper Res 213(2):361–374MathSciNetCrossRefzbMATHGoogle Scholar
  33. Scarf HE (1958) A min-max solution of an inventory problem. In: Arrow KJ, Karlin S, Scarf HE (eds) Studies in the mathematical theory of inventory and production. Stanford University Press, Stanford, pp 201–209Google Scholar
  34. Tadikamalla PR (1978) Applications of the Weibull distributions in inventory control. J Oper Res Soc 29(1):77–83MathSciNetCrossRefzbMATHGoogle Scholar
  35. Tadikamalla PR (1979) The lognormal approximation to the lead time demand in inventory control. Omega Int J Manag Sci 7(6):553–556CrossRefGoogle Scholar
  36. Vairaktarakis GL (2000) Robust multi-item newsboy models with a budget constraint. Int J Prod Econ 66(3):213–226CrossRefGoogle Scholar
  37. Wang ZF, Qin S Kar (2015) A novel single-period inventory problem with uncertain random demand and its application. Appl Math Comput 269:133–145MathSciNetGoogle Scholar
  38. Wang D, Qin ZF (2016) Multi-product newsvendor problem with hybrid demand and its applications to ordering pharmaceutical reference standard materials. Int J Gen Syst 45(3):271–285MathSciNetCrossRefzbMATHGoogle Scholar
  39. Yang XF, Gao JW (2013) Uncertain differential games with application to capitalism. J Uncertain Anal Appl 1:17CrossRefGoogle Scholar
  40. Yao K, Ralescu DA (2013) Age replacement policy in uncertain environment. Iran J Fuzzy Syst 10(2):29–39MathSciNetzbMATHGoogle Scholar
  41. Yu J, Sarker BR, Duan Q, Wu B (2012) Single-manufacturer, multi-retailer consignment policy for retailers generalized demand distributions. J Oper Res Soc 63(12):1708–1719CrossRefGoogle Scholar
  42. Zhu YG (2010) Uncertain optimal control with application to a portfolio selection model. Cybern Syst 41(7):535–547CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Henan University of TechnologyZhengzhouChina

Personalised recommendations