# Problem Set 4

**Topics:**Normal distribution, Probability density function, Probability theory

**Pages:**4 (793 words)

**Published:**March 27, 2013

1. Cu = 24-11 = $13

Co = 11-7 = $4

Critical ratio = 13/(13+4) = 0.7647

μ = 30,000

σ = 10,000

Using normal distribution function (=norminv(0.7647,30000,10000)), the optimum order quantity is 37,216 jerseys to maximize profit.

2. Quantity = 32,000

First, we normalize the order quantity to find the z-statistic z=Q-μσ=32,000-25,00010,000=0.7

We then look up the standard normal loss function. The expected lost sale is given by. Lz=0.1429

Therefore, the expected lost sales = 10,000 * 0.1429 ≈ 1,429 Expected sales = 30,000 – 1,429 = 30,571 jerseys

3. First, we normalize the order quantity to find its z-statistic z=Q-μσ=28,000-20,00010,000=0.8

In-stock probability = normdist(0.8, 0, 1, 1) = 0.788145

Therefore, the probability of filling all demand is 0.788145

4. Quantity = 8,000

First, we normalize the order quantity to find the z-statistic z=Q-μσ=8,000-15,00010,000=-0.7

We then look up the standard normal loss function. The expected lost sale is given by. Lz=0.8429

Therefore, the expected lost sales = 10,000 * 0.8429 = 8,429 Expected sales = 15,000 – 8,429 = 6,571 jerseys

Expected leftover = 8000 – 6,571 = 1429 jerseys

Therefore, Nike has to sell 1,429 on discount

5. Cu = 16-11 = $5

Co = 9-7 = $2

Critical ratio = 5/(5+2) = 0.7143

μ = 40,000

σ = 10,000

Using normal distribution (=norminv(0.7143,40000,10000)), the optimum order quantity is 45660 jerseys in the first order to maximize profit.

6. Cu = 0.02-0.01 = $0.01

Co = 0.01-0 = $0.01

Critical ratio = 0.01/(0.01+0.01) = 0.5

μ = 300

σ = 75

Using normal distribution (=norminv(0.5,300,75)), the optimum order quantity is 300 million minutes from Vmail to minimize the expected capacity expense.

7. Q = 375

First, we normalize the order quantity to find the z-statistic z=Q-μσ=375-30075=1

We then look up the standard normal loss function. The expected lost sale is given by. Lz=0.0833

Therefore, the expected lost sales = 75 * 0.0833 = 6.2498...

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