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PRODUCTION MANAGEMENT APPLICATIONS 147

After estimating the effects of employee layoffs, turnovers, reassignment training
costs and other costs associated with fluctuating production levels, Bollinger esti-
mates that the cost associated with increasing the production level for any month is
E0.50 per unit increase. A similar cost associated with decreasing the production
level for any month is E0.20 per unit. Thus, we write the third portion of the
objective function as:
Change-in-production-level costs ¼ 0:5I 1 þ 0:50I 2 þ 0:50I 3
þ 0:20D 1 þ 0:20D 2 þ 0:20D 3
Note that the cost associated with changes in production level is a function of the
change in the total number of units produced in month m compared to the total
number of units produced in month m 1. In other production scheduling appli-
cations, fluctuations in production level might be measured in terms of machine
hours or labour-hours required rather than in terms of the total number of units
produced.
Combining all three costs, the complete objective function becomes:

Min 20x 11 þ 20x 12 þ 20x 13 þ 10x 21 þ 10x 22 þ 10x 23 þ 0:30s 11
þ 0:30s 12 þ 0:30s 13 þ 0:15s 21 þ 0:50s 22 þ 0:15s 23 þ 0:50I 1
þ 0:50I 2 þ 0:50I 3 þ 0:20D 1 þ 0:20D 3 þ 0:20D 3

We now consider the constraints. First, we must guarantee that the schedule
meets customer demand. Because the units shipped can come from the current
month’s production or from inventory carried over from previous months, the
demand requirement takes the form:
0 1 0 1 0 1 0 1
Ending Ending
B inventory C B Current C B inventory C B This month’s C
B C þ B C B C ¼ B C
@ from previous A @ production A @ for this A @ demand A
month month

Suppose that the inventories at the beginning of the three-month scheduling period
were 500 units for component 322A and 200 units for component 802B. The demand
for both products in the first month (April) was 1000 units, so the constraints for
meeting demand in the first month become:

500 þ x 11 s 11 ¼ 1000
200 þ x 21 s 21 ¼ 1000

Moving the constants to the right-hand side, we have:
x 11 s 11 ¼ 500
x 21 s 21 ¼ 800
Similarly, we need demand constraints for both products in the second and third
months. We write them as follows.

Month 2
s 11 þ x 12 s 12 ¼ 3000
s 21 þ x 22 s 22 ¼ 500
Month 3

s 12 þ x 13 s 13 ¼ 5000
s 22 þ x 23 s 23 ¼ 3000

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