Best Price in Town (BPT) is an electronics store in Ann Arbor. They are currently deciding how many units of a new digital camera, called Model Z, to order. Let Q denote this stock level choice. BPT has a single ordering opportunity for Model Z. Once Model Z arrives at BPT, it will be sold for three months, and then will be replaced by some future model. At the end of the three months, the leftover inventory for Model Z will have no value, and will be discarded.
The total customer demand for Model Z over the next three months in Ann Arbor is a random variable, denoted by D. Let F denote the cumulative distribution function of D, and f the probability density function. Of all the customers in Ann Arbor who want to buy Model Z, only a fraction k will demand the product from BPT. The rest will go to other electronics stores. (For example, if D turns out to be 100 and k = 0.6. then the demand for Model Z at BPT will be 60.)
The unit retail price of the camera is p, and the unit purchase price that BPT pays to the supplier of the camera is c.
(a) Write an expression for the expected sales of Model Z at BPT for a given stock level choice Q, i.e., the expected number of Model Z's that BPT will sell in the next three months. (This expression should be detailed enough that if we knew BPT's stock level choice (Q). the fraction of total demand that BPT gets (k), the functions F and f, and had a tool at our disposal to evaluate integrals, then we could evaluate this expression.)
(b) Using your answer to part (a), write an expression for the expected prot BPT will make from Model Z.
(c) Using your answer to part (b), derive a relationship that needs to be satised by BPT's optimal stock level.