2 A company faces the following demands during the next three periods: period 1, 20 units; period 2, 10 units; period 3, 15 units. The unit production cost during each period is as follows: period 1—$13; period 2—$14; period 3—$15.

A holding cost of $2 per unit is assessed against each period’s ending inventory. At the beginning of period 1, the company has 5 units on hand.

In reality, not all goods produced during a month can be used to meet the current month’s demand. To model this fact, we assume that only one half of the goods produced during a period can be used to meet the current period’s demands.

Formulate an LP to minimize the cost of meeting the demand for the next three periods. (Hint: Constraints such as i₁ = x₁ + 5 - 20 are certainly needed. Unlike our example, however, the constraint i1  0 will not ensure that period 1’s demand is met. For example, if x1  20, then i1

 0 will hold, but because only 

1/2

(20) = 10 units of period 1 production can be used to meet period 1’s demand, x1 

20 would not be feasible. Try to think of a type of constraint that will ensure that what is available to meet each period’s demand is at least as large as that period’s demand.)