8 Consider the following probabilistic inventory model:

a At the beginning of each period, a firm observes its inventory position.

b Then the firm decides how many units to produce during the current period. It costs c(x) dollars to produce x units during a period.
c With probability q(d), d units are demanded during the period. From units on hand (including the current period’s production), the firm satisfies as much of the demand as possible. The firm receives r dollars for each unit sold. For each unit of demand that is unsatisfied, a penalty cost p is incurred. All unsatisfied demand is assumed to be lost. For example, if the firm has 20 units available and current demand is 30, a revenue of 20r would be received, and a penalty of 10p would be incurred.
d If ending inventory is positive, a holding cost of $1 per unit is incurred.
e The next period now begins.
The firm’s inital inventory is zero, and its goal is to minimize the expected cost over a 100-period horizon. Formulate a dynamic programming recursion that will help the firm accomplish its goal.