Question

A retailer sells four items i = 1, 2, 3, 4. Their weekly demand characteristics are the same, with a mean demand of µi = 100 units, and a standard deviation of σi = 10 units. The four items are also independent of one another (wherever you need to, you can assume the covariance is 0). The retailer does not know exactly how the item demands are distributed, but they assume they follow a normal distribution. Answer the following questions. (a) Your supplier for item 1 takes one week to fulfill your replenishment orders. What is the probability a customer does not find item 1 if you place an order when you have 80 items left? (b) For the same supplier for item 1, what is the probability a customer does not find item 1 if you place an order when you have 125 items left? (c) What is the mean weekly demand for all four products? What is the standard deviation of the weekly demand for all four products? (d) Based on your calculations in (c), what is the probability there are more than or equal to 400 customers next week for all four items? What is the probability there are less than 350 customers next week for all four items?

Answer 1

Let x₁ be the weekly demand for item 1

Given : µ₁ = 100 , σ₁ = 10

a) P( x₁ > 80 ) =

= P( z > -2 ) = 1 - P( z < -2 ) = 1 - 0.0228

= 0.9772

b) P( x₁ > 125 ) =

= P( z > 2.5 ) = 1 - P( z < 2.5 ) = 1 - 0.9938

= 0.0062

c) mean weekly demand for all four products?

µ_∑x= n* µ = 4*100 = 400

Standard deviation of the weekly demand for all four products?

σ_∑x=n*σ = 4*10 = 40

d)

= P( z > 0 ) = 1 - P( z ≤ 0 )

= 1 - 0.5

= 0.5

The probability there are more than or equal to 400 customers next week for all four items is 0.5

= P( z < -1.25 )

= 0.1056

The probability there are less than 350 customers next week for all four items is 0.1056