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150 CHAPTER 4 LINEAR PROGRAMMING APPLICATIONS

s 22 þ x 23 s 23 ¼ 3000
s 13 400
s 23 > 200
0:1x 11 þ 0:08x 21 400
0:1x 12 þ 0:08x 22 500
0:1x 13 þ 0:08x 23 600
0:05x 11 þ 0:07x 21 300
0:05x 12 þ 0:07x 22 300
0:05x 13 þ 0:07x 23 < 300
2s 11 þ 3s 21 10000
2s 12 þ 3s 22 < 10000
2s 13 þ 3s 23 10000
x 11 þ x 21 I 1 þ D 1 ¼ 2500
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
x 11 ; x 12 ; x 13 ; x 21 ; x 22 ; x 23 ; s 11 ; s 12 ; s 13 ; s 21 ; s 22 ; s 23 ; I 1 ; I 2 ; I 3 ; D 2 ; D 3 0

Linear programming Figure 4.2 shows the optimal solution to the Bollinger Electronics production
models for production scheduling problem. Table 4.6 contains a portion of the managerial report based on
scheduling are often very
large. Thousands of the optimal solution.
decision variables and Consider the monthly variation in the production and inventory schedule shown
constraints are in Table 4.6. Recall that the inventory cost for component 802B is one-half the
necessary when the
problem involves inventory cost for component 322A. Therefore, as might be expected, component
numerous products, 802B is produced heavily in the first month (April) and then held in inventory for
machines and time the demand that will occur in future months. Component 322A tends to be pro-
periods. Data collection duced when needed, and only small amounts are carried in inventory.
for large-scale models
can be more time- The costs of increasing and decreasing the total production volume tend to
consuming than either smooth the monthly variations. In fact, the minimum-cost schedule calls for a 500-
the formulation of the unit increase in total production in April and a 2200-unit increase in total produc-
model or the tion in May. The May production level of 5200 units is then maintained during June.
development of the
computer solution. The machine usage section of the report shows ample machine capacity in all
three months. However, labour capacity is at full utilization (slack ¼ 0 for constraint
13 in Figure 4.2) in the month of May. The dual price shows that an additional hour
of labour capacity in May will improve the value of the optimal solution (lower cost)
by approximately E1.11.
A linear programming model of a two-product, three-month production system
can provide valuable information in terms of identifying a minimum-cost production
schedule. In larger production systems, where the number of variables and con-
straints is too large to track manually, linear programming models can provide a
significant advantage in developing cost-saving production schedules.

Workforce Assignment
Workforce assignment problems frequently occur when production managers must
make decisions involving staffing requirements for a given planning period. Work-
force assignments often have some flexibility, and at least some personnel can be
assigned to more than one department or work centre. Such is the case when
employees have been cross-trained on two or more jobs or, for instance, when sales
personnel can be transferred between stores. In the following application, we show
how linear programming can be used to determine not only an optimal product mix,
but also an optimal workforce assignment.

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