Probabilistic inventory models

**veronica.white** · May 12th 2009, 08:58 PM

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?

**CaptainBlack** · May 16th 2009, 12:45 AM

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 $x$ , and the number of students be $D$ . Then the additional cost is:

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

Expected cost:

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

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

CB

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