Formulate the problem as a linear program define the decisio

Formulate the problem as a linear program (define the decisions variables clearly, write the objective function and constraints in terms of those variables.)

Two gasoline types A and B have octane ratings of 80 and 92, respectively. Type A costs $0.83 per liter and type B costs $0.98 per liter. Determine the blend of minimum cost with an octane rating of at least 89.

Hint: The octane rating of the blend equals the sum of the octane rating of each type of gasoline multiplied by its percentage in the blend. For example, a blend of 30% A and 70% B will have an octane level of 0.3(80)+0.7(92) = 88.4 and will cost 0.3(0.83)+0.7(0.98) = 0.935 per liter.

Solution

Let a be the quantity of gasoline type A and b be the quantity if gasoline type B

So the objective function for the combined price is

0.83a + 0.98b = P

The constraints are

80a + 92b = 89 …… equation i

Since the blends are in ratio

a + b = 1 . . . . . . equation ii

b = 1 – a

Equating b in equation i

80 a + 92(1 – a) = 89

80 a + 92 – 92a = 89

12 a = 3

a = 0.25

Equating a in equation ii we get

b = 0.75

Minimum cost for a blend of octane rating 89

0.83a + 0.98b = P

0.83 * 0.25 + 0.98 * 0.75

0.207 + 0.735

= 0.942