Many labor-intensive production operations experience a learning curve effect. The learning curve specifies that the cost to produce a unit is a function of the unit number; that is, as production volume increases, the cost to produce each unit drops. One form of the learning curve is as follows: C_i = a(i)^b where Ci is the cost of unit i, a is called the first unit cost, and b is the learning “slope” parameter. The total cost of producing a batch of size x can then be approximated by (ax^1+b) / (1 + b). Now consider a production setting where there is learning. We have the following single-product production-planning data: demands for the next five periods are 100, 150, 300, 200, 400. Holding cost per unit per period is $0.30 and production cost follows a learning curve with a = 15 and b = −0.2.

a. Assume that there is no transfer of learning between time periods and that, at most, one batch is produced per time period. Solve the production-planning problem of minimizing the sum of production and inventory costs, while satisfying demand. What are the optimal batch sizes?
b. Solve the same production-planning problem, ignoring the learning curve, that is, assume that every unit costs $15 dollars.
c. Assume we must have an ending inventory in period 5 of at least 50. Re-solve the problem in part (a). What are the optimal batch sizes? How much of a required ending inventory in period 5 induces a change in the optimal batch sizes?