Swimsuit Supply Chain Optimization: Maximizing Profit with Vertical Integration

If the retailer and manufacturer are vertically integrated, then the supply chain profit is the same as the profit of the single firm. To maximize this profit, the single firm should order the quantity that maximizes the expected profit.

To find this quantity, we can use the newsvendor model. The optimal order quantity Q* is given by:

Q* = F^-1(p)

where F^-1 is the inverse of the cumulative distribution function of demand, evaluated at probability p. We want to choose p so that the expected profit is maximized.

Let's first calculate the expected profit for a given order quantity Q:

Expected revenue = min(Q, D) x $130 Expected cost = Q x $80 + max(0, Q-D) x $15 + $100,000 + (Q x $30) Expected profit = Expected revenue - Expected cost

where D is the random demand.

We can plug in the demand distribution and simplify to get:

Expected profit = $20,800Q - $1,300(Q)^2 - $100,000

To maximize this function, we can take the derivative with respect to Q, set it equal to zero, and solve for Q:

d/dQ(Expected profit) = $20,800 - $2,600Q = 0 Q* = 8, which is the optimal order quantity

To find the expected profit at this order quantity, we can plug Q* into the expected profit formula:

Expected profit = $20,800(8) - $1,300(8)^2 - $100,000 = $20,400

Therefore, the expected profit of the supply chain when vertically integrated is $20,400, and the retailer should order 8 swimsuits.