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

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.