In the example 4, the demands for each of the four months are month 1, 1 unit; month 2, 3 units; month 3, 2 units; month 4, 4 units. Setup cost of production is $3 and variable cost of $1 for every unit produced is incurred. Holding cost is $0.5 per unit. We solve the problem starting backwards.
Define as the minimum cost of meeting demands for months if units are on hand at the beginning of month t. We define to be cost of producing x units during a period. Then,
The production level during month t minimizes the total cost during months if units are on hand at the beginning of month t.
We use the same month 4 computations as in example 4. The month 4 cost is given as,
Now for each possible production level x during month 3, the total cost during months 3 and 4 is,
Thus we can use the table of computations for month 3 in the example 4 as given below,
Now for each possible production level x during month 2, the total cost during months 2, 3 and 4 is,
Thus we can use the table of computations for month 2 in the example 4 as given below,
Similarly, since the demand for month 1 is of 1 unit, using the above procedure, the table for month 1 computation could be formed as in example 4.
Now we can determine a production schedule that minimizes the total cost of meeting the demands for all four months on time. Since our initial inventory is 3 units, the minimum cost for the four months will be. To attain, we must produce units during month 1. Then the inventory at the beginning of the month 2 will be units. Thus, in the month 2, we should produce units.
Then at the beginning of month 3, our beginning inventory will be units. Hence during month 3, we need to produce units. Then month 4 will begin with units on hand. Thus, units should be produced during month 4.
So the optimal production schedule incurs a total cost of and produces 0 units during month 1, 3 units during month 2, 0 units during month 3 and 4 units during month 4.