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156 CHAPTER 4 LINEAR PROGRAMMING APPLICATIONS

t 12 þ t 13 400
t 23 þ t 24 800
t 34 100
t 41 þ t 42 200

The complete linear programming model has two product decision variables (P 1
and P 2 ), four department workforce assignment variables (b 1 , b 2 , b 3 and b 4 ), seven
transfer variables (t 12 , t 13 , t 23 , t 24 , t 34 , t 41 and t 42 ) and 12 constraints. The full
formulation is then:
Max 10P 1 þ 9P 2 þ 0b 1 þ 0b 2 þ 0b 3 þ 0b 4 þ 0t 12 þ 0t 13 þ 0t 23 þ 0t 24 þ 0t 34 þ 0t 41 þ 0t 42

s:t:
0:65P 1 þ 0:95P 2 b 1 0
0:45P 1 þ 0:85P 2 b 2 0
P 1 þ 0:7P 2 b 3 0
0:15P 1 þ 0:3P 2 b 4 0
b 1 þ t 12 þ t 13 t 41 ¼ 6500
b 2 t 12 þ t 23 þ t 24 t 42 ¼ 6000
b 3 t 13 t 23 þ t 34 ¼ 7000
b 4 t 24 t 34 þ t 41 þ t 42 ¼ 1400
t 12 þ t 13 400
t 23 þ t 24 800
t 34 100
t 41 þ t 42 200
P 1 ; P 2 ; b 1 ; b 2 ; b 3 ; b 4 ; t 12 ; t 13 ; t 23 ; t 24 ; t 34 ; t 41 ; t 42 0

Notice that the objective function shows all the decision variables even though only
two of them have non-zero coefficients, since it is only production of P 1 and P 2 that
contribute financially to profit. Figure 4.4 shows the optimal solution to the problem.
McCormick’s profit can be increased by E84 011 E73 590 ¼ E10 421 by taking
Variations in the advantage of cross-training and workforce transfers. The optimal product mix of
workforce assignment 6825 units of product 1 and 1751 units of product 2 can be achieved if t 13 ¼ 400
model could be used in hours are transferred from department 1 to department 3; t 23 ¼ 651 hours are
situations such as
allocating raw material transferred from department 2 to department 3; and t 24 ¼ 149 hours are transferred
resources to products, from department 2 to department 4. The resulting workforce assignments for
allocating machine time departments 1–4 would provide 6100, 5200, 8051 and 1549 hours, respectively.
to products and
allocating salesforce time If a manager has the flexibility to assign personnel to different departments,
to stores or sales reduced workforce idle time, improved workforce utilization and improved profit
territories. should result. The linear programming model in this section automatically assigns
employees and labour-hours to the departments in the most profitable manner.

4.3 Blending, Diet and Feed-Mix Problems

A large category of LP applications fall into the area generally known as blending, or
diet or feed-mix problems. Typically such problems revolve around the requirement
to mix together a variety of ingredients in order to produce some final product in
such a way that the final product meets certain specified criteria. Applications of
blending problems range from the oil industry where crude oils have to be blended
with other ingredients to make a variety of petrols and related fuels, through the

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