Optimal policy with a total order quantity commitment contract in the presence of a spot market

  • Wei Zhang
  • Youhua (Frank) Chen
  • Zhongsheng Hua
  • Weili Xue


In this paper, the optimal policy is considered when the buyer faces two supply sources: one is the contract supplier from which the buyer orders over a specific contract period (say, a year) at a pre-agreed price, and the other is the spot market. However, when ordering from the contract supplier, the buyer must fulfill a pre-determined total order quantity, or the so-called definite total order quantity commitment, over the whole contract period. In other words, the commitment secures the buyer a fixed price but obliges him/her a total order quantity over the contract period. Although the spot market gives the buyer more flexibility in terms of order quantities, its prices are volatile. Such a combination of contract and spot procurements is often observed in practice. Within the contract period, there are multiple sub-periods, during each of which the buyer reviews the inventory, issues an individual order, and uses the on-hand inventory to meet the random demand. Thus, in each (ordering) period, the buyer will weigh between the current known spot price (by procuring from the spot market) and a lower future price (by waiting while consuming the remaining commitment). An optimal dual ordering policy is characterized for each period, depending on the on-hand inventory level, the spot price, and the remaining commitment quantity. The optimal policy in each period is also shown to be independent of the contract price. Through a numerical study, the inventory cost is demonstrated to be (1) insensitive to the contract price when the total commitment quantity is lower than the total expected demand over the contract period and (2) non-increasing in the variability of spot prices.


Total ordering quantity commitment (TOQC) spot market procurement policy 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Ann, T.H. & Maguire, F. (2005). BHP to decide on $2.3 billion ore expansion plans. Bloomberg News, June 2nd, 2005Google Scholar
  2. [2]
    Anupindi, R. & Bassok, Y. (1998). Supply contracts with quantity commitments and stochastic demands. In: Tayur, S., Ganeshan, R., Magazine, M. (eds.), Quantitative Models for Supply Chain Management. Kluwer Academic Publishers, DordrechtGoogle Scholar
  3. [3]
    Arnold, J., Minner, S. & Eidam, B. (2006). Raw material procurement with fluctuating prices. Technical ReportGoogle Scholar
  4. [4]
    Bassok, Y. & Schuster, D.B. (1995). Supply contract with early termination. In: INFORMS New Orleans Conference, October 29–November 1, 1995Google Scholar
  5. [5]
    Bassok, Y. & Anupindi, R. (1997). Analysis of supply contracts with total minimum commitment. IIE Transactions, 29: 373–381Google Scholar
  6. [6]
    Bonser, J.S. & Wu, S.D. (2001). Procurement planning to maintain both short-term adaptiveness and long-term perspective. Management Science, 47: 769–786CrossRefGoogle Scholar
  7. [7]
    Chen, F.Y. & Krass, D. (2001). Analysis of supply contracts with minimum total order quantity commitments and non-stationary demands. European Journal of Operational Research, 131: 309–323CrossRefMATHMathSciNetGoogle Scholar
  8. [8]
    Fabian, T., Fisher, J.L., Sasieni, M.W. & Yardeni, A. (1959). Purchasing raw materials on a fluctuating market. Operations Research, 7: 107–122CrossRefGoogle Scholar
  9. [9]
    Feng, Q. & Sethi, S. (2008). Procurement flexibility under price uncertainty. McCombs Research Paper Series No. IROM-04-09Google Scholar
  10. [10]
    Golabi, K. (1995). Optimal inventory policies when ordering prices are random. Operations Research, 33: 575–588CrossRefMathSciNetGoogle Scholar
  11. [11]
    Guzel, A. (2004). Optimal commodity procurement under stochastic prices. Technology, 7: 29–39Google Scholar
  12. [12]
    Hannon, D. (2005). Why natrual gas prices remain so volatile. Purchasing, January 13th, 2005. Available via DIALOG. http://www.purchasing.com
  13. [13]
    Haksoz, C. & Seshadri, S. (2007). Supply chain operations in the presence of a spot market: a review with discussion. Journal of the Operations Research Society, 58: 1412–1429CrossRefGoogle Scholar
  14. [14]
    Hill, L. (2010). Canada’s Canpotex sells potash to India at $370/t. Miningweekly.com, February 19th, 2010
  15. [15]
    HKTDC. (2007). Cost escalation and trends for export price increase — a look at the rising production costs in the PRD. Available via DAILOG. http://info.hktdc.com/econforum/tdc/tdc070901.htm.
  16. [16]
    Inderfurth, K. & Kelle, P. (2008). Capacity reservation under spot market price uncertainty. Working paperGoogle Scholar
  17. [17]
    Kalymon, B.A. (1971). Stochastic prices in a single-item inventory purchasing model. Operations Research, 19: 1434–1458CrossRefMATHMathSciNetGoogle Scholar
  18. [18]
    Kingsman, G.G. (1969). Commodity purchasing. Operations Research Quarterly, 20: 59–79CrossRefGoogle Scholar
  19. [19]
    Magirou, V.F. (1982). Stockpiling under price uncertainty and storage capacity constraints. European Journal of Operational Research, 11: 233–246CrossRefMATHMathSciNetGoogle Scholar
  20. [20]
    Minner, S. (2003). Multiple-supplier inventory models in supply chain management: a review. International Journal of Production Economics, 81: 265–279CrossRefGoogle Scholar
  21. [21]
    Porteus, E.L. (2002). Foundations of Stochastic Inventory Theory. Stanford University Press. CaliforniaGoogle Scholar
  22. [22]
    Seifert, R.W., Thonemann, U.W. & Hausman, W.H. (2004). Optimal procurement strategies for online spot markets. European Journal of Operations Research, 152: 781–799CrossRefMATHMathSciNetGoogle Scholar
  23. [23]
    Tally Metal Sales Inc. (2010). Available via DIALOG. http://www.tally-metal.com/. Cited on June 2nd, 2010
  24. [24]
    Yi, J. & Scheller-Wolf, A. (2003). Dual sourcing from a regular supplier and a spot market. Working paperGoogle Scholar

Copyright information

© Systems Engineering Society of China and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Wei Zhang
    • 1
  • Youhua (Frank) Chen
    • 2
  • Zhongsheng Hua
    • 1
  • Weili Xue
    • 3
  1. 1.School of ManagementUniversity of Science and Technology of ChinaHefeiChina
  2. 2.Department of Systems Engineering and Engineering ManagementThe Chinese University of Hong Kong, Shatin, N. T.Hong KongChina
  3. 3.Yurun Corp.NanjingChina

Personalised recommendations