Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Optimal quantity discount coordination for supply chain optimization with one manufacturer and multiple suppliers under demand uncertainty


In this paper, a coordination strategy is developed to integrate business decisions and manufacturing planning in supply chain management. We consider one manufacturer and multiple suppliers to determine production, prices, and inventory simultaneously with uncertain demands. This paper aims at providing an optimal discount policy derived from Stackelberg equilibrium to coordinate a manufacturer and multiple suppliers. The optimal discount coordination mechanism helps the manufacturer to select suppliers in order to maintain long-term relationship with the contracted suppliers under demand uncertainty. Noncooperative game is applied in order to resolve decision-making in order to determine quantities of components, price, production, and selection of suppliers simultaneously. Computational experiments are conducted to demonstrate the effectiveness and efficiency of the proposed approach.

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


  1. 1.

    Constantine EP (2006) The business of globalization and the globalization of business. J Comp Int Mgmt 9(1):3–18

  2. 2.

    Hussain AHA, Mohammad ON (2010) Supply chain integration: definition and challenges. Proc Int MultiConf Eng Comput Sci 978–988

  3. 3.

    Chopra S, Meindl P (2012) Supply chain management: strategy, planning, and operation. Education, Pearson

  4. 4.

    Yu Y, Huang GQ, Liang L (2009) Stackelberg game-theoretic model for optimizing advertising, pricing and inventory policies in vendor managed inventory (VMI) production supply chains. Comput Ind Eng 57:368–382

  5. 5.

    Alaei S, Alaei R, Salimi P (2014) A game theoretical study of cooperative advertising in a single-manufacturer-two-retailers supply chain. Int J Adv Manuf Technol 74:101–111

  6. 6.

    Esmaeili M, Aryanezhad M, Zeephongsekul P (2009) A game theory approach in seller-buyer supply chain. Eur J Oper Res 195:442–428

  7. 7.

    Xiao T, Shi K, Yang D (2010) Coordination of a supply chain with consumer return under demand uncertainty. Int J Prod Econ 124:171–180

  8. 8.

    Nishi T, Shinozaki R, Konishi M (2008) An augmented Lagrangian approach for distributed supply chain planning for multiple companies. IEEE Trans Autom Sci Eng 5:259–274

  9. 9.

    Tominaga H, Nishi T, Konishi M (2008) Effects of inventory control on bullwhip in supply chain planning for multiple companies. Int J Innov Comput I 4(3):513–529

  10. 10.

    Yang D, Choi T, Xiao T, Cheng T (2011) Coordinating a two- supplier and one-retailer supply chain with forecast updating. Automatica 47:1317–1329

  11. 11.

    Huang Y, Huang GQ, Newman ST (2010) Coordination pricing and inventory decisions in a multi-level supply chain: a game-theoretic approach. Transp Res Part E 47:115–129

  12. 12.

    Leng M, Parlar M (2010) Game-theoretic analyses of decentralized assembly supply chain: non-cooperative equilibria vs. coordination with cost-sharing contracts. Eur J Oper Res 204:96–104

  13. 13.

    Hennet JC, Arda Y (2008) Supply chain coordination: a game theory approach. Eng Appl Artif Intell 399–405

  14. 14.

    Xiao T, Jin J, Chen G, Shi J, Xie M (2010) Ordering, wholesale pricing and lead-time decisions in a three-stage supply chain under demand uncertainty. Comput Ind Eng 59(4):840–852

  15. 15.

    Pezeshki Y, Baboli A, Reza Akbari Jolar M (2013) Simultaneous coordination of capacity building and price decisions in a decentralized supply chain. Int J Adv Manuf Technol 64:961–976

  16. 16.

    Kokhlesian M, Hessameddin Zegordi S (2014) Application of multidivisional bi-level programming to coordinate pricing and inventory decisions in a multiproduct competitive supply chain. Int J Adv Manuf Technol 71:1975–1989

  17. 17.

    Sarmah SP, Acharya D, Goyal SK (2006) Buyer vendor coordination models in supply chain management. Eur J Oper Res 175:1–15

  18. 18.

    Li SX, Huang Z, Ashley A (1996) Improving buyer-seller system cooperation through inventory control. Int J Prod Econ 43:37–46

  19. 19.

    Viswanathan S, Wang Q (2003) Discount pricing decisions in distribution channels with price-sensitive demand. Eur J Oper Res 149:571–587

  20. 20.

    Qin Y, Tang H, Guo C (2007) Channel coordination and volume discounts with price-sensitive demand. Int J Prod Econ 105:43–53

  21. 21.

    Kim B, Leung JMY, Park KT, Zhang G, Lee S (2002) Configuring a manufacturing firm’s supply network with multiple suppliers. IIE Trans 34:663–677

  22. 22.

    Zhang G, Ma L (2009) Optimal acquisition policy with quantity discounts and uncertain demands. Int J Prod Res 47(9):2409–2425

  23. 23.

    Yin S, Nishi T (2012) Game theoretic approach for global manufacturing planning under risk and uncertainty. Procedia CIRP 3:251–256

  24. 24.

    Yin S, Nishi T (2014) A solution procedure for mixed integer nonlinear programming formulation for supply chain planning with quantity discounts under demand uncertainty. Int J Syst Sci 14(11): 2354–2365

  25. 25.

    Kuzdrall PJ, Britney RR (1982) Total setup lot sizing with quantity discounts. Dec Sci 13:101–112

  26. 26.

    Kalvelagen E (2004) New special functions in GAMS: http://amsterdamoptimization.com/pdf/dea.pdf

Download references

Author information

Correspondence to Tatsushi Nishi.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yin, S., Nishi, T. & Grossmann, I.E. Optimal quantity discount coordination for supply chain optimization with one manufacturer and multiple suppliers under demand uncertainty. Int J Adv Manuf Technol 76, 1173–1184 (2015). https://doi.org/10.1007/s00170-014-6298-1

Download citation


  • Supply chain coordination
  • Quantity discounts
  • Game theoretical model
  • Stackelberg game
  • Demand uncertainty