Question

A refinery blends three petroleum components into three grades of gasoline -regular, premium, and diesel. The maximum quantities available of each component and the cost per barrel are as follows: Component Cost/Barrel Maximum Barrels Available/Day 6,000 3,000 4,500 10 To ensure that each gasoline grade retains certain essential characteristics, the refinery has put limits on the percentages of the components in each blend. The limits as well as the selling prices for the various grades are as follows:

Grade Selling Price/Barrel Comen R (regular) 18 Component Specifications Not less than 30% of A Not more than 30% of B Not less than 30% of C Not less than 60% of C Not more than 50% of B less than 10% of A P (premium) 25 D (diesel) 15 The refinery wants to produce at least 5,000 barrels of each grade of gasoline. Management wishes to determine the optimal mix of the three components that will maximize profit.
1. Define the decision variables.
2. Build the objective function.
3. Build all the constraints.

Answer 1

Step-by-step explanation:

1)

xij = barrels of component i used in grade j per day for i = A, B, C and j = R, P, D

2)

Max Z = 18 (x AR + x BR + x CR) + 25 (x AP + x BP + x CP) + 15 (x AD + x BD + x CD) – 9 (x AR + x AP + x AD) – 7 (x BR + x BP + x BD) – 10 (x CR + x CP + x CD)

or,

Max Z = 9 x AR + 11 x BR + 8 x CR + 16 x AP + 18 x BP + 15 x CP + 6 x AD + 8 x BD + 5 x CD

3)

Subject to,

x AR + x AP + x AD <= 6000

x BR + x BP + x BD <= 3000

x CR + x CP + x CD <= 4500

x Aj + x Bj + x Cj >= 5000 for j = R, P, D

x AR >= 30%*(x AR + x BR + x CR) or, 0.7 x AR - 0.3 x BR - 0.3 x CR >= 0

x BR <= 30%*(x AR + x BR + x CR) or, -0.3 x AR + 0.7 x BR - 0.3 x CR <= 0

x CR >= 30%*(x AR + x BR + x CR) or, -0.3 x AR - 0.3 x BR + 0.7 x CR >= 0

x CP >= 60%*(x AP + x BP + x CP) or, -0.6 x AP - 0.6 x BP + 0.4 x CP >= 0

x BD <= 50%*(x AD + x BD + x CD) or, -0.5 x AD + 0.5 x BD - 0.5 x CD <= 0

xAD >= 10%*(xAD + xBD + xCD) or, 0.9 xAD - 0.1 xBD - 0.1 xCD >= 0

xij >= 0 for i = A, B, C and j = R, P, D