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BLENDING, DIET AND FEED-MIX PROBLEMS 161

and thus:

0:40x 1r þ 0:60x 2r 0:40x 3r 0

Similarly, we write the four remaining blending specifications listed in Table 4.10 as:
0:20x 1r 0:20x 2r þ 0:80x 3r 0
þ0:75x 1p 0:25x 2p 0:25x 3p 0
0:40x 1p þ 0:60x 2p 0:40x 3p 0
0:30x 1p 0:30x 2p þ 0:70x 3p 0
The constraint for at least 10 000 litres of regular fuel is:

x 1r þ x 2r þ x 3r 10 000

The complete linear programming model with six decision variables and ten con-
straints is then:

Max 0:50x 1r þ 0:40x 2r þ 0:16x 3r þ 0:58x 1p þ 0:48x 2p þ 0:24x 3p
s:t:
x 1r þ x 1p 5 000
þ 10 000
x 2r x 2p
x 3r þ x 3p 10 000
0
0:70x 1r 0:30x 2r 0:30x 3r
0:40x 1r þ 0:60x 2r 0:40x 3r 0
0
0:20x 1r 0:20x 2r þ 0:80x 3r
0:75x 1p 0:25x 2p 0:25x 3p 0
0:40x 1p þ 0:60x 2p 0:40x 3p 0
0:30x 1p 0:30x 2p þ 0:70x 3p 0
x 1r þ x 2r þ x 3r 10 000
x 1r ; x 2r ; x 3r ; x 1p ; x 2p ; x 3p 0

Try Problem 11 as The optimal solution to the DOC blending problem is shown in Figure 4.5. The
another example of a optimal solution, which provides a daily profit of $9300, is summarized in Table 4.11.
blending model.
The optimal blending strategy shows that 10 000 litres of regular fuel should be
produced. The regular fuel will be manufactured as a blend of 8000 litres of com-
ponent 2 and 2000 litres of component 3. The 15 000 litres of premium fuel will be
manufactured as a blend of 5000 litres of component 1, 2000 litres of component 2
and 8000 litres of component 3.
The interpretation of the slack and surplus variables associated with the product
specification constraints (constraints 4–9) in Figure 4.5 needs some clarification.
If the constraint is a constraint, the value of the slack variable can be inter-
preted as the litres of component use below the maximum amount of the
component use specified by the constraint. For example, the slack of 3000.000
for constraint 4 shows that component 1 use is 3000 litres below the maximum
amount of component 1 that could have been used in the production of 10 000
litres of regular fuel. If the product specification constraint is a constraint, a
surplus variable shows the litres of component use above the minimum amount
of component use specified by the blending constraint. For example, the surplus
of 4000.000 for constraint 5 shows that component 2 use is 4000 litres above the
minimum amount of component 2 that must be used in the production of 10 000
litres of regular fuel.

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