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PRODUCTION MANAGEMENT APPLICATIONS 149

We can find the amount of the change in production for April from the
relationship:
April production March production ¼ Change

Using the April production variables, x 11 and x 21 , and the March production of 2500
units, we have:
ðx 11 þ x 21 Þ 2500 ¼ Change

Note that the change can be positive or negative. A positive change reflects an
increase in the total production level, and a negative change reflects a decrease in
the total production level. We can use the increase in production for April, I 1 , and
the decrease in production for April, D 1 , to specify the constraint for the change in
total production for the month of April:
ðx 11 þ x 21 Þ 2500 ¼ I 1 D 1
Of course, we cannot have an increase in production and a decrease in produc-
tion during the same one-month period; thus, either, I 1 or D 1 will be zero. If April
requires 3000 units of production, I 1 ¼ 500 and D 1 ¼ 0. If April requires 2200 units
of production, I 1 ¼ 0 and D 1 ¼ 300. This approach of denoting the change in
production level as the difference between two nonnegative variables, I 1 and D 1 ,
permits both positive and negative changes in the total production level. If a single
variable (say, c m ) had been used to represent the change in production level, only
positive changes would be possible because of the nonnegativity requirement.
Using the same approach in May and June (always subtracting the previous
month’s total production from the current month’s total production), we obtain
the constraints for the second and third months of the production scheduling period:

ðx 12 þ x 22 Þ ðx 11 þ x 21 Þ¼ I 2 D 2
ðx 13 þ x 23 Þ ðx 12 þ x 22 Þ¼ I 3 D 3
Placing the variables on the left-hand side and the constants on the right-hand side yields
the complete set of what are commonly referred to as production-smoothing constraints:
I 1 þ D 1 ¼ 2500
x 11 þ x 21
x 11 x 21 þ x 12 þ x 22 I 2 þ D 2 ¼ 0
x 12 x 22 þ x 13 þ x 23 I 3 þ D 3 ¼ 0
The initially rather small, two-product, three-month scheduling problem has now
Problem 14 involves a developed into an 18-variable, 20-constraint linear programming problem. Note that in
production scheduling this problem we were concerned only with one type of machine process, one type of
application with labour-
smoothing constraints. labour and one type of storage area. Actual production scheduling problems usually
involve several machine types, several labour grades and/or several storage areas,
requiring large-scale linear programmes. For instance, a problem involving 100 prod-
ucts over a 12-month period could have more than 1000 variables and constraints.
The full model formulation is then:
Min 20x 11 þ 20x 12 þ 20x 13 þ 10x 21 þ 10x 22 þ 10x 23 þ 0:3s 11 þ 0:3s 12 þ 0:3s 13 þ 0:15s 21 þ
0:5s 22 þ 0:15s 23 þ 0:5I 1 þ 0:5I 2 þ 0:5I 3 þ 0:2D 1 þ 0:2D 2 þ 0:2D 3
s:t:
x 11 s 11 ¼ 500
x 21 s 21 ¼ 800
s 11 þ x 12 s 12 ¼ 3000
s 21 þ x 22 s 22 ¼ 500
s 12 þ x 23 s 13 ¼ 5000

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