Thread: Probabilistic inventory models

The campus bookstore must decide how many textbooks to order for a course that will be offered only once. The number of students who will take the course is a random variable D whose distribution can be approximated by a (continuous) uniform distribution on the interval [35,75]. After the course starts, the value of D becomes known. If D exceeds the number of books available, the known shortfall is made up by placing a rush order at a cost of $150 plus $5 per book over the normal ordering cost. If D is less than the stock on hand, the extra books are return for their original ordering cost less $2 each. What is the order quantity that minimises the expected cost?

Originally Posted by veronica.white

The campus bookstore must decide how many textbooks to order for a course that will be offered only once. The number of students who will take the course is a random variable D whose distribution can be approximated by a (continuous) uniform distribution on the interval [35,75]. After the course starts, the value of D becomes known. If D exceeds the number of books available, the known shortfall is made up by placing a rush order at a cost of $150 plus $5 per book over the normal ordering cost. If D is less than the stock on hand, the extra books are return for their original ordering cost less $2 each. What is the order quantity that minimises the expected cost?

Let the initial order be $\displaystyle x$, and the number of students be $\displaystyle D$. Then the additional cost is:

$\displaystyle c(x,D)=\begin{cases}2(x-D)& x\ge D\\ 150+5(D-x)&x<D \end{cases}$

Expected cost:

$\displaystyle
E(c(x,.)) = \int_{D=35}^{75} c(x,D) (1/40) \ dD = $ $\displaystyle (1/40) \left[\int_{D=35}^x 2(x-D)\ dD + \int_{D=x}^{75}150+5(D-x) \ dD\right]
$

which you should be able to evaluate, and so find the $\displaystyle x$ that minimises this.

CB