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PRODUCTION MANAGEMENT APPLICATIONS 155

Table 4.8 Cross-Training Ability and Capacity Information
Cross-Training Transfers
Permitted to Department

Maximum Hours
From Department 1 2 3 4 Transferable

1 — yes yes — 400
2 — — yes yes 800
3 — — — yes 100
4 yes yes — — 200

0
0:65P 1 þ 0:95P 2 b 1
0:45P 1 þ 0:86P 2 b 2 0
1:00P 1 þ 0:70P 2 b 3 0
0:15P 1 þ 0:30P 2 b 4 0
The labour-hours ultimately allocated to each department must be determined by a
series of labour balance equations, or constraints, that include the number of hours
initially assigned to each department plus the number of hours transferred into the
department minus the number of hours transferred out of the department. Using
department 1 as an example, we determine the workforce allocation as follows:
0 1 0 1 0 1
Hours Hours Hours
b 1 ¼ @ initially in A þ @ transferred into A @ transferred out of A
department 1 department 1 department 1

Table 4.7 shows 6500 hours initially assigned to department 1. We use the transfer
decision variables t i1 to denote transfers into department 1 and t 1j to denote transfers
from department 1. Table 4.8 shows that the cross-training capabilities involving
department 1 are restricted to transfers from department 4 (variable t 41 ) and transfers
to either department 2 or department 3 (variables t 12 and t 13 ). Thus, we can express the
total workforce allocation for department 1 as:

b 1 ¼ 6500 þ t 41 t 12 t 13
Moving the decision variables for the workforce transfers to the left-hand side, we
have the labour balance equation or constraint:

b 1 t 41 þ t 12 þ t 13 ¼ 6500

This form of constraint will be needed for each of the four departments. Thus, the
following labour balance constraints for departments 2, 3 and 4 would be added to the
model.
b 2 t 12 t 42 þ t 23 þ t 24 ¼ 6000
¼ 7000
b 3 t 13 t 23 þ t 34
b 4 t 24 t 34 þ t 41 þ t 42 ¼ 1400
Finally, Table 4.8 shows the number of hours that may be transferred from each
department is limited, indicating that a transfer capacity constraint must be added
for each of the four departments. The additional constraints are:

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