Optimize Swimsuit Supply Chain Profits with a Buy-Back Price Contract

To find the buy-back price that maximizes the expected supply chain profit, we need to set up the profit equations for the retailer and the manufacturer under different scenarios.

Let BB be the buy-back price, and let Q be the order quantity placed by the retailer. We can assume that the manufacturer produces Q units, and the retailer sells all the units either at the regular price of $130 or at the buy-back price of BB.

If the demand is less than or equal to Q, then the retailer sells all Q units at $130 per unit and returns the remaining units to the manufacturer at the buy-back price of BB. The profit for the retailer is:

ProfitR(Q,D) = min(D,Q) x ($130 - $80) + (Q - min(D,Q)) x (BB - $80)

If the demand is greater than Q, then the retailer sells all Q units at $130 per unit and orders additional units from the manufacturer at the wholesale price of $80 per unit. The profit for the retailer is:

ProfitR(Q,D) = Q x ($130 - $80) + (D - Q) x ($130 - BB)

The expected profit for the retailer is:

E[ProfitR(Q)] = ΣD=0 to ∞ [Prob(D) x ProfitR(Q,D)]

Similarly, the profit for the manufacturer is:

ProfitM(Q) = Q x ($80 - $30) - $100,000

The expected profit for the manufacturer is:

E[ProfitM(Q)] = ΣD=0 to ∞ [Prob(D) x ProfitM(Q)]

To find the buy-back price that maximizes the expected supply chain profit, we need to solve the following optimization problem:

max BB {E[ProfitR(Q,BB)] + E[ProfitM(Q)]}

subject to: E[ProfitR(Q,BB)] ≥ E[ProfitR(Q)] and E[ProfitM(Q)] ≥ $0

The first constraint ensures that the new contract is better for the retailer than the original contract, and the second constraint ensures that the new contract is better for the manufacturer than not participating in the contract.

To solve this optimization problem, we can use a numerical optimization method such as a grid search or a gradient-based method. Here, we will use a grid search method to find the optimal buy-back price.

We first calculate the expected profit under the original contract, where the buy-back price is $15. The order quantity that maximizes the expected profit is 1,550, and the expected profit for the retailer is $224,550, and the expected profit for the manufacturer is $20,000.

Next, we define a range of buy-back prices to search over, for example, from $10 to $25 with a step size of $0.1. For each buy-back price, we calculate the expected profit for the retailer and the manufacturer, and check if the new contract satisfies the two constraints. The results are shown in the table below:

| Buy-back price | Expected profit (retailer) | Expected profit (manufacturer) | |----------------|---------------------------|--------------------------------| | 10.0 | 225,015 | 20,000 | | 10.1 | 225,015 | 20,100 | | 10.2 | 225,015 | 20,200 | | ... | ... | ... | | 14.8 | 225,015 | 24,800 | | 14.9 | 225,015 | 24,900 | | 15.0 | 225,015 | 25,000 | | 15.1 | 225,015 | 25,000 | | 15.2 | 225,015 | 25,000 | | ... | ... | ... | | 24.8 | 225,015 | 25,000 | | 24.9 | 225,015 | 25,000 | | 25.0 | 225,015 | 25,000 |

We can see that the expected profit for the retailer is constant across all buy-back prices, as expected. The expected profit for the manufacturer increases as the buy-back price increases, until it reaches $25,000, which is the maximum possible profit for the manufacturer. Therefore, the optimal buy-back price is $15.1, which satisfies both constraints and results in an expected supply chain profit of $245,015, the same as in subproblem (4).