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PRODUCTION MANAGEMENT APPLICATIONS 153

Table 4.7 Departmental Labour-Hours Per Unit and Total Hours Available for the
McCormick Manufacturing Company
Labour-Hours per Unit

Department Product 1 Product 2 Total Hours Available
1 0.65 0.95 6 500
2 0.45 0.85 6 000
3 1.00 0.70 7 000
4 0.15 0.30 1 400

The linear programme is:

Max 10P 1 þ 9P 2
s:t:
0:65P 1 þ 0:95P 2 6500
0:45P 1 þ 0:85P 2 6000
1:00P 1 þ 0:70P 2 7000
0:15P 1 þ 0:30P 2 1400
P 1 ; P 2 0

The optimal solution to the linear programming model is shown in Figure 4.3.
After rounding, it calls for 5744 units of product 1, 1795 units of product 2 and a
total profit of E73 590. With this optimal solution, departments 3 and 4 are operat-
ing at capacity, and departments 1 and 2 have a slack of approximately 1062 and
1890 hours, respectively. We would anticipate that the product mix would change
and that the total profit would increase if the workforce assignment could be revised
so that the slack, or unused hours, in departments 1 and 2 could be transferred to
the departments currently working at capacity. However, the production manager
may be uncertain as to how the workforce should be reallocated among the four
departments. Let us expand the linear programming model to include decision
variables that will help determine the optimal workforce assignment in addition to
the profit-maximizing product mix.
However, the company are keen to improve productivity and are launching a
training programme for their workforce. The cross-training programme will train a
number of workers in each department to undertake the task of workers in other
departments. This will provide the company with some labour flexibility. By taking
advantage of the cross-training skills, a limited number of employees and labour-
hours may be transferred from one department to another. For example, the cross-
training permits transfers as shown in Table 4.8. Row 1 of this table shows that some
employees assigned to department 1 have cross-training skills that permit them to be
transferred to department 2 or 3. The right-hand column shows that, for the current
production planning period, a maximum of 400 hours can be transferred from
department 1. Similar cross-training transfer capabilities and capacities are shown
for departments 2, 3 and 4.
When workforce assignments are flexible, we do not automatically know how
many hours of labour should be assigned to or be transferred from each department.
We need to add decision variables to the linear programming model to account for
such changes.

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