Question

Colony Shop is stocking heavy coats for next winter. Colony pays $50 for a coat and sells it for $110. At the end of the winter season Colony offers the coats at $55 each. The demand for the coats during winter is a discrete uniform random variable 20 and 40 (including both).

a) What is the expected demand?

b) What is the optimal number of coats should the Colony stock to maximize its expected profit?

c) What is the maximum expected profit?

Can someone answer these questions step by step?

Answer 1

a)

Demand ~ Uniform(20,40)

By Uniform distribution,

Expected demand = (20 + 40)/2 = 30

b)

Let Co be Unit cost of excess inventory (cost of overage) and Cu be Unit cost of shortage (cost of underage)

Given, Unit cost, c = $50

Selling price, p = $110

Salvage value, s = $55

Cu = p - c = $110 - $50 = $60

Co = c - s = $50 - $55 = -$5

As, Co is negative, we conclude that shopkeepr still earns profit of $5 per coat after the end of the winter season.

So, the shopkeeper should order the maximum quantity to earn maximum profit. The maximum value of demand is 40.

So, the optimal number of coats should the Colony stock to maximize its expected profit is 40.

c)

Maximum expected profit = Expected Demand * Profit + (Optimal order - Expected Demand) * Salvage value

= 30 * ($110 - $50) + (40 - 30) * ($55 - 50)

= 30 * $60 + 10 * $5

= $1850