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160 CHAPTER 4 LINEAR PROGRAMMING APPLICATIONS

Table 4.9 Petroleum Cost and Supply for the DOC Blending Problem
Petroleum Component Cost/litre Maximum Available
1 $0.50 5 000 litres
2 $0.60 10 000 litres
3 $0.84 10 000 litres

Table 4.10 Product Specifications for the DOC Blending Problem

Product Specifications
Regular fuel At most 30% component 1
At least 40% component 2
At most 20% component 3
Premium fuel At least 25% component 1
At most 40% component 2
At least 30% component 3

We develop the objective function of maximizing the profit contribution by
identifying the difference between the total revenue from both fuels and the total
cost of the three petroleum components. By multiplying the $1.00 per litre price by
the total litres of regular fuel, the $1.08 per litre price by the total litres of premium
fuel, and the component cost per litre figures in Table 4.9 by the total litres of each
component used, we obtain the objective function:

Max 1:00ðx 1r þ x 2r þ x 3r Þþ 1:08ðx 1p þ x 2p þ x 3p Þ
0:50ðx 1r þ x 1p Þ 0:60ðx 2r þ x 2p Þ 0:84ðx 3r þ x 3p Þ

When we combine terms, the objective function becomes:
Max 0:50x 1r þ 0:40x 2r þ 0:16x 3r þ 0:58x 1p þ 0:48x 2p þ 0:24x 3p
The limitations on the availability of the three petroleum components are:

x 1r þ x 1p 5 000 Component 1
x 2r þ x 2p 10 000 Component 2
x 3r þ x 3p 10 000 Component 3
Six constraints are now required to meet the product specifications stated in
Table 4.10. The first specification states that component 1 can account for no more
than 30 per cent of the total litres of regular fuel produced. That is,

x 1r 0:30ðx 1r þ x 2r þ x 3r Þ
Rewriting this constraint with the variables on the left-hand side and a constant on
the right-hand side yields:

0:70x 1r 0:30x 2r 0:30x 32 0
The second product specification listed in Table 4.10 becomes:
x 2r 0:40ðx 1r þ x 2r þ x 3r Þ

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