Backup Contracts under Supply Risks and Stochastic Demand Jing Hou

Outline Literature Review Model Sensitivity Analysis Comparison Study

Literature Review Supply disruption management Decision-making under supply and demand uncertainty Backup Agreement

Contributions First, we study the contract-based coordination mechanisms between a buyer and a backup supplier and address the buyer’s decisions on the optimal order quantity under two different contract types. Second, unlike most existing research efforts that study recurrent supply uncertainties and disruptions risks separately or treat the two risks indifferently, we take both risk types into account and compare their impacts. Finally, we provide an in-depth analysis of the impact of the supply uncertainties and compare the effects of two contracts on mitigating supply and demand risks: a stand- by contract and a make-to-order contract, which helps the buyer to choose a better way of using a backup supplier.

Models Consider a single period problem where the buyer faces stochastic demand and two kinds of supply risks from the main supplier: supply disruptions and recurrent supply uncertainties. During every cycle, the buyer faces an actual demand of x units. Denote F(x) is the distribution function of the demand at the market and its average is u D. The buyer (the retailer) has two supply options: one cheaper but unreliable (this supplier is referred to as the main supplier), and the other is perfectly reliable and responsive, but is more expensive (this one is referred to as the backup supplier). The main supplier is prone to disruption with a probability of p, in which case the main supplier becomes totally unavailable. If there is no disruption, random y units are yielded with distribution function of G(y) having a mean of S and standard deviation σ. The buyer sells the goods to the market at the price of s and the wholesale price from the main supplier is c m.

1 The 1 st type of Backup Supply Contract: a Stand- by Contract In a stand-by contract, in the normal situations, the buyer only orders from its main supplier, and the backup supplier holds a certain inventory level of M units (referred to as the reservation quantity) which is paid at e per unit by the buyer. When supply disruption occurs, the buyer has to order from the backup supplier to satisfy the demand. If the actual demand x is smaller than M, the buyer only orders x units at the wholesale price of c b ; else if x is larger than M, he has to order the extra units (x - M) at the price of c p, the value of which is higher than c b.

1- 期望利润 The first term is the revenue without supply disruption but with recurrent supply uncertainty, which includes the premium of the reservation at the backup supplier and the profit of selling the goods available from the main supplier. The second term is the revenue when there is supply disruption at the main supplier.

1- 最优决策 Proposition 1: For the buyer, the optimal reservation quantity M * at the backup supplier is given by Proposition 2: The optimal reservation quantity M * increases with the disruption probability p, the wholesale price difference (c p -c b ), but decreases with the premium e.

2 The 2 nd type of Backup Supply Contract: a Make-to-order Contract In a make-to-order contract, the buyer uses the backup supplier as a regular producer during normal situations, that is, has a proportion (  ) of the demand to be produced in a make-to-order fashion by the backup supplier at regular times; and the main supplier has an order of (1 -  ) of the demand. When supply disruption happens, the buyer will simply order more to satisfy the total demand. During the whole cycle, the wholesale price from the backup supplier is c p.

2- 期望利润 The first term is the revenue of selling the goods bought from the backup supplier. The second term is the revenue of selling the goods bought from the main supplier when there is no disruption.

2- 最优决策 Proposition 3: The optimal proportion  * for the buyer does not change with the disruption probability p. Proposition 4: For the buyer, the optimal proportion  * is given by

Sensitivity Analysis In this section we particularly consider the different impacts of σ and p on the decision- making of the buyer under the two contracts. the demand is assumed to follow a uniform distribution U[300,700] the main supplier’s delivery quantity is assumed to follow a normal distribution with S = 500. c m = 80, c b = 100, s = 200, c p = 120, e = 11.

1) The 1 st type of Backup Supply Contract: a Stand-by Contract The buyer ’ s optimal reservation quantity M * vs. the supply disruption p The buyer ’ s optimal expected value vs. the supply uncertainties

Impact In the following analysis, we define the impact of σ (or p) on the buyer’s expected profit as: So the impact of σ changes from 6.1% (p = 0) to % (p = 0.9); the impact of p changes from 28.4% (σ = 15) to 21.9% (σ = 150) in this case.

2) The 2 nd type of Backup Supply Contract: a Make-to-order Contract The optimal proportion, vs. the recurrent supply uncertainty, σ The buyer ’ s optimal expected value vs. the supply uncertainties

The Comparison of the Two Contracts First, about the buyer’s decision-making under two contracts: M * and, the supply uncertainties from the main supplier have different impacts. (1) The optimal reservation quantity M * increases with the disruption probability but has no relationship with the recurrent uncertainty. That is because the buyer does not order from the backup supplier during normal situations to deal with the recurrent supply risk. The reservation is only used for mitigating supply disruption risk. (2) The optimal proportion  * does not change with the disruption probability p, and it does not strictly increase with the recurrent supply uncertainty. This is because the buyer can use the backup supplier to mitigate the recurrent supply uncertainty, and during disruption period, the backup supplier’s wholesale price does not change with the order quantity.

Second, the supply uncertainties have different impacts on the expected profits of the buyer under two contracts: (1) the impacts of σ are similar. (2) the disruption probability p has more impacts on the buyer’s expected profit under the second contract, because when the buyer makes the decision of  *, it does not change with p; the buyer’s expected profit is directly impacted by p. But the buyer changes the value of M * with p, which in some way mitigates the impact of p on the buyer’s expected profit.

Third, on the choice of the two contacts, we can see from the numerical analysis that the buyer’s expected profits are higher under the 2 nd contract except when p = 0.9, which is almost impossible to happen, which means the make-to-order contract is better than the stand-by contract for the buyer under such pricing decisions of the backup supplier. One of the reasons, we may say, is that under the make-to-order contract, the backup supplier is used to mitigate not only the supply disruptions, but also the recurrent supply uncertainty. But under the stand-by contract, the buyer has no way to deal with the recurrent supply uncertainty.