Consider a swimsuit supply chain We assume that there are two companies involved in the supply chain a retailer that faces customer demand and a manufacturer who produces and sells swimsuits to the re

(1) The optimal ordering quantity of the retailer is 600.

To find this, we can use the formula for the economic order quantity (EOQ):

$Q^* = \sqrt{\frac{2DS}{H}}$

where $D$ is the demand, $S$ is the setup (ordering) cost, and $H$ is the holding (carrying) cost per unit. In this case, $D = 800$, $S = 80 \times 600 = 48000$, and $H = 0.5 \times (130-80) = 25$. Plugging these values into the formula, we get:

$Q^* = \sqrt{\frac{2 \times 800 \times 48000}{25}} \approx 600$

So the optimal ordering quantity is 600.

To compute the expected profits of the retailer, manufacturer, and total supply chain, we can use the following table:

| | Retailer | Manufacturer | Total Supply Chain | |---------------------|----------|--------------|---------------------| | Revenue | $130D$ | | $130D$ | | Cost of Goods Sold | | $80Q^$ | $80Q^$ | | Production Cost | | $100000 + 30Q^$ | $100000 + 30Q^$ | | Salvage Value | $15(600-D)$ | | $15(600-D)$ | | Holding Cost | $25 \times \frac{Q^}{2}$ | | $25 \times \frac{Q^}{2}$ | | Setup Cost | $48000$ | | $48000$ | | | | | | | Expected Profit | $130D - 80Q^* - 25 \times \frac{Q^}{2} - 48000 + 15(600-D)$ | $80Q^ - (100000 + 30Q^)$ | $50D - 20Q^ - 24000 + 15(600-D)$ |

Plugging in $D = 800$ and $Q^* = 600$, we get:

Expected profit of the retailer = $130 \times 800 - 80 \times 600 - 25 \times \frac{600}{2} - 48000 + 15(600-800) = $28,750$

Expected profit of the manufacturer = $80 \times 600 - (100000 + 30 \times 600) = $2,000$

Expected profit of the total supply chain = $50 \times 800 - 20 \times 600 - 24000 + 15(600-800) = $30,250$

(2) The optimal ordering quantity of the retailer is 668.

To find this, we need to adjust the salvage value for any unsold swimsuits, since the manufacturer agrees to buy them back for $60. The salvage value is now $60(600-D)$ instead of $15(600-D)$.

Using the same formula for EOQ and plugging in the new salvage value, we get:

$Q^* = \sqrt{\frac{2 \times 800 \times 48000}{55-30} \times \frac{130-60}{130}} \approx 668$

So the optimal ordering quantity is 668.

To compute the expected profits of the retailer, manufacturer, and total supply chain, we can use the same table as before, but with the salvage value changed to $60(600-D)$:

| | Retailer | Manufacturer | Total Supply Chain | |---------------------|----------|--------------|---------------------| | Revenue | $130D$ | | $130D$ | | Cost of Goods Sold | | $80Q^$ | $80Q^$ | | Production Cost | | $100000 + 30Q^$ | $100000 + 30Q^$ | | Salvage Value | $60(600-D)$ | | $60(600-D)$ | | Holding Cost | $25 \times \frac{Q^}{2}$ | | $25 \times \frac{Q^}{2}$ | | Setup Cost | $48000$ | | $48000$ | | | | | | | Expected Profit | $130D - 80Q^* - 25 \times \frac{Q^}{2} - 48000 + 60(600-D)$ | $80Q^ - (100000 + 30Q^)$ | $50D - 20Q^ - 24000 + 60(600-D)$ |

Plugging in $D = 800$ and $Q^* = 668$, we get:

Expected profit of the retailer = $130 \times 800 - 80 \times 668 - 25 \times \frac{668}{2} - 48000 + 60(600-800) = $29,750$

Expected profit of the manufacturer = $80 \times 668 - (100000 + 30 \times 668) = $4,304$

Expected profit of the total supply chain = $50 \times 800 - 20 \times 668 - 24000 + 60(600-800) = $33,054$

(3) The optimal ordering quantity of the retailer is 600.

With the new wholesale price of $55$ per unit and revenue-sharing percentage of 20%, the cost of goods sold for the retailer is $0.8\times 55 \times Q^$ and the revenue is $130D\times (1-0.2)$. The manufacturer's revenue is $0.2\times 130D$ and the production cost for the manufacturer is $100,000+30Q^$. Hence we can form the following table:

| | Retailer | Manufacturer | Total Supply Chain | |---------------------|----------|--------------|---------------------| | Revenue | $130D\times 0.8$ | $0.2\times 130D$ | $130D\times 0.8$ | | Cost of Goods Sold | | $55Q^$ | $55Q^$ | | Production Cost | | $100000 + 30Q^$ | $100000 + 30Q^$ | | Salvage Value | $15(600-D)$ | | $15(600-D)$ | | Holding Cost | $25 \times \frac{Q^}{2}$ | | $25 \times \frac{Q^}{2}$ | | Setup Cost | $48000$ | | $48000$ | | Revenue-Sharing | $130D\times 0.2$ | $130D\times 0.2$ | | | | | | | | Expected Profit | $130D\times 0.8 \times 0.8 - 55Q^* - 25 \times \frac{Q^}{2} - 48000 + 15(600-D) + 130D\times 0.2 \times 0.8$ | $0.2\times 130D - (55Q^ + 100000 + 30Q^)\times 0.2$ | $50D - 20Q^ - 24000 + 15(600-D) + 130D\times 0.2 \times 0.8$ |

Plugging in $D = 800$ and $Q^* = 600$, we get:

Expected profit of the retailer = $83,200-55\times 600-25 \times \frac{600}{2}-48000+15(600-800)+20,800 = $26,200$

Expected profit of the manufacturer = $20,800 - 0.2\times(55\times 600 + 100000 + 30\times 600) = $1,000$

Expected profit of the total supply chain = $40,800 - 20\times 600 - 24000 + 15(600-800) = $24,400$

(4) If the retailer and manufacturer are vertically integrated, then they should order enough swimsuits to maximize the total profit of the supply chain. This means setting the derivative of the total profit with respect to $Q$ equal to zero and solving for $Q$.

The total profit of the supply chain is:

$50D - 100000 - 30Q + 130D\min{Q,800}$

Taking the derivative with respect to $Q$ and setting it equal to zero, we get:

$-30 + 130D\begin{cases} 1 & Q < 800 \ 0 & Q \geq 800 \end{cases} = 0$

Solving for $Q$, we get $Q=800$.

Plugging in $Q=800$, we get:

Expected profit of the retailer = $130 \times 800 - 80 \times 800 - 25 \times \frac{800}{2} - 48000 + 15(600-800) = $23,750$

Expected profit of the manufacturer = $80 \times 800 - (100000 + 30 \times 800) = $8,000$

Expected profit of the total supply chain = $50 \times 800 - 20 \times 800 - 24000 + 15(600-800) = $26,250$

(5) To determine the buy-back price that makes the expected supply chain profit equal to the profit in part (4), we need to find a price $b$ such that:

$50D - 100000 - 30Q + 130D\min{Q,800} = 50D - 20 \times 800 - 24000 + b(600-D)$

Simplifying, we get:

$-30Q + 130D\min{Q,800} = -20 \times 800 - 24000 + b(600-D) + 100000$

We know that $Q=800$ maximizes the left-hand side, so plugging this in, we get:

$-20 \times 800 - 24000 + b(600-D) + 100000 = 130D$

Solving for $b$, we get:

$b = \frac{50D - 20 \times 800 - 24000 + 130D}{600-D} \approx 62.41$

So the buy-back price that makes the expected supply chain profit equal to the profit in part (4) is approximately $62.41$.

(6) To determine the wholesale price and revenue-sharing percentage that make the expected supply chain profit equal to the profit in part (4), we need to find values $w$ and $r$ such that:

$50D - 100000 - 30Q + 130D\min{Q,800} = 50D - 20 \times 800 - 24000 + (130D - 130Dr) - wQ$

Simplifying, we get:

$-30Q + 130D\min{Q,800} - wQ + 130Dr = -20 \times 800 - 24000$

We know that $Q=800$ maximizes the left-hand side, so plugging this in, we get:

$-20 \times 800 - 24000 + 130D - 800w + 130Dr = 130D$

Simplifying, we get:

$w = \frac{130 - 20 \times 800 - 24000}{800} \approx 26.25$

$r = \frac{130D - 100000 - 30Q + wQ + 15(600-D)}{130D} \approx 0.142$

So the wholesale price that makes the expected supply chain profit equal to the profit in part (4) is approximately $26.25$ per unit, and the revenue-sharing percentage is approximately 14.2%.