Newsboy problem

A newspaper is concerned with controlling the number of papers to be distributed to newstands. The cost of a paper varies (i.e., Sunday vs. daily), and the demand is a random variable, $LaTeX: q,$ with probability function $LaTeX: P(q).$ Unsold papers are returned, with no salvage value the next day, losing millions of dollars in the production cost. The demand for newspapers is a random variable, with probability function $LaTeX: P(q)$ = probability that demand equals $LaTeX: q.$ It is possible, however, for a newstand to order more papers the same day. There are holding and shortage (penalty) costs. The problem is to determine a reorder policy so as to minimize total expected cost. This problem was used to consider a reorder policy with a 2-parameter decision rule:

1. $LaTeX: s$ = inventory level at which an order is placed;
2. $LaTeX: S$ = inventory level to which to order.

Then, the decision rule to be employed is the following policy:

Order nothing if the inventory of papers is $LaTeX: \ge s;$
Order $LaTeX: S-s$ if there are s papers on hand and $LaTeX: s < S.$

The significance of this problem is that it was used to introduce the notion (and optimality) of an $LaTeX: (s, S)$ policy in inventory theory.