During the selling season a swimsuit is sold to customers at $130 per unit The wholesale price paid by the retailer to the manufacturer is $80 per unit Any swimsuit not sold during the summer season h

(1) The optimal ordering quantity can be found by equating the retailer's marginal revenue to the marginal cost, which is the sum of the wholesale price and the variable production cost per unit. Thus, we have:

130 - Q = 80 + 30Q/1000 50Q/1000 = 50 Q* = 1000

Therefore, the optimal ordering quantity for the retailer is 1000 units. The expected profit of the retailer is:

(130 - 80) * 1000 = $50,000

The expected profit of the manufacturer is:

(80 - 30) * 1000 - 100,000 = -$30,000

The total supply chain profit is:

$50,000 - $30,000 = $20,000

(2) With the buy-back agreement, the salvage value of the swimsuits is now $60 instead of $15. The optimal ordering quantity can be found in the same way as in part (1):

130 - Q = 80 + 30Q/1000 + (60 - 80) * Q/1000 50Q/1000 = 10 Q* = 200

Therefore, the optimal ordering quantity for the retailer is 200 units. The expected profit of the retailer is:

(130 - 80) * 200 = $10,000

The expected profit of the manufacturer is:

(80 - 30) * 200 - 100,000 + (200 - 200) * 60 = -$8,000

The total supply chain profit is:

$10,000 - $8,000 = $2,000

(3) With the revenue-sharing agreement, the wholesale price is now $55 instead of $80, and the manufacturer receives 20% of the retailer's sales revenue. The optimal ordering quantity can be found in the same way as in part (1):

130 - Q = 55 + 30Q/1000 + 0.2 * 130 * Q 40Q/1000 = 13 Q* = 325

Therefore, the optimal ordering quantity for the retailer is 325 units. The expected profit of the retailer is:

(130 - 55 - 0.2 * 130) * 325 = $14,137.50

The expected profit of the manufacturer is:

(55 - 30) * 325 - 100,000 = $2,875

The total supply chain profit is:

$14,137.50 + $2,875 = $17,012.50

(4) With vertical integration, the supply chain profit is maximized when the marginal revenue equals the marginal cost, which is the sum of the fixed cost and the variable production cost per unit. Thus, we have:

130 - Q = 100 + 30Q/1000 40Q/1000 = 30 Q* = 750

Therefore, the retailer should order 750 units. The expected profit of the supply chain is:

(130 - 100) * 750 - 30 * 750 - 100,000 = $20,000

(5) Let P be the buy-back price that maximizes the supply chain profit. We know that the supply chain profit in part (4) is $20,000, so we want to find P such that the supply chain profit in part (2) is also $20,000. The expected profit of the manufacturer is:

(80 - 30) * 200 - 100,000 + (200 - Q) * P = -$8,000

Solving for P, we get:

P = ($8,000 + 100,000 - 50Q)/(200 - Q)

Plugging in Q = 750, we get:

P = $53.33

Therefore, the buy-back price that maximizes the supply chain profit is $53.33.

(6) Let W be the wholesale price and R be the revenue-sharing percentage that maximize the supply chain profit. We know that the supply chain profit in part (4) is $20,000, so we want to find W and R such that the supply chain profit in part (3) is also $20,000. The expected profit of the manufacturer is:

(55 - 30) * Q - 0.2 * 130 * Q - 100,000 + R * 130 * Q = $2,875

Solving for R, we get:

R = ($97,125 - 25Q)/(26,000Q)

The expected profit of the retailer is:

(130 - W - 0.8 * 130) * Q = (26 - 0.8W)Q

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

W = $67.50

Plugging in W = $67.50 and solving for Q, we get:

Q = 500

Plugging in Q = 500, we get:

R = 0.055

Therefore, the wholesale price and revenue-sharing percentage that maximize the supply chain profit are $67.50 and 5.5%, respectively.