10.4 Exercise 4

Optimal stocking of merchandise. A merchant is planning for the Christmas season. He intends to stock m units of a certain item at a cost of c per unit. Experience indicates demand can be represented by a random variable $D \sim$ Poisson $\left(\mu\right)$. If units remain in stock at the end of the season, they may be returned with recovery of r per unit. If demand exceeds the number originally ordered, extra units may be ordered at a cost of s each. Units are sold at a price p per unit. If $Z = g \left(D\right)$ is the gain from the sales, then

1. For $t \leq m , g \left(t\right) = \left(p - c\right) t - \left(c - r\right) \left(m - t\right) = \left(p - r\right) t + \left(r - c\right) m$
2. For $t > m , g \left(t\right) = \left(p - c\right) m + \left(t - m\right) \left(p - s\right) = \left(p - s\right) t + \left(s - c\right) m$

Let $M = \left(- \infty , m\right]$. Then

Suppose $\mu = 50 m = 50 c = 30 p = 50 r = 20 s = 40.$.
Approximate the Poisson random variable D by truncating at 100. Determine $P \left(500 \leq Z \leq 1100\right)$.