Joe\'s Coffee Shoppe orders fresh doughnuts each morning. Joe pays $3.20 per doz

Question

Joe's Coffee Shoppe orders fresh doughnuts each morning. Joe pays $3.20 per dozen doughnuts and sells doughnuts for $0.4 each. The leftover doughnuts are salvaged at half the selling price. a) Suppose that the demand for doughnuts is uniformly distributed from 16 dozens to 30 dozens. How many doughnuts (in dozens) should Joe stock each day? b) A detailed analysis of past data shows that the number of doughnuts sold per day is better described by a normal distribution, with a mean 23 dozens and standard deviation 10 dozens. Now, how many doughnuts should Joe stock each day? c) The expected demand was the same in parts (a) and (b), but the optimal order quantities should have been different. What accounted for this difference?

Explanation / Answer

Cost per dozen, c = $3.2
Selling Price per dozen, p = $0.4 * 12 = $4.8
Salvage value per dozen, s = $4.8/2 = $2.4

Unit cost of shortage (cost of underage), Cu = p - c = $4.8 - $3.2 = $1.6
Unit cost of excess inventory (cost of overage), Co = c - s = $3.2 - $2.4 = $0.8

Cumulative density function of expected demand,
F(Q) = Cu / (Cu + Co) = 1.6 / (1.6 + 0.8) = 0.667

(a)
If Demand follows a uniform distribution from 16 dozens to 30 dozens
F(Q) = (Q - 16) / (30 - 16)
0.667 = (Q - 16) / (30 - 16)
Q = 25.338
So, he should stock 25.34 dozens of doughnuts each day.

(b)
If Demand follows a normal distribution with mean of 23 dozens and standard deviation 10 dozens,
F(Q) = 0.667 => z = 0.4316

So, Q = SD * z + mean
= 10 * 0.4316 + 23 = 27.316
So, he should stock 27.32 dozens of doughnuts each day.

(c)
Standard deviation of normal distribution = 10
Variance of of uniform distribution = (30 - 16)^2 / 12 = 16.33
Standard deviation of uniform distribution = sqrt(16.33) = 4.04

Even though both the Normal and the Uniform distributions have the same mean of 23, we get different optimal order quantities because of the variance (equivalently, standard deviation) and the shape of the distribution. The shape of normal and uniform distribution is different. The standard deviation of normal distribution is 10 but that of uniform distribution is 4.04

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