Linear Programming, with Perentage Restraints

[London]

Linear Programming, with Percentage Restraints

Good evening,

I am having some difficulty with my linear programming assignment and would appreciate any help to bring me in the right direction.

I have figured out the decision variables, but the question mentions costs and other constraints and I'm very confused.

The questions says:

The company has an order
from a customer to produce 120 tons of an alloy that contains four
metals (for simplicity, we will be calling them metals A, B, C, and D
instead of wolfram, palladium, iridium, and copper) according to the
following specifications:
• The alloy should contain at least 24% of metal A.
• The alloy should contain no more than 11% of metal B.
• The alloy should contain no more than 8% of metal C.
• The alloy should contain between 35% and 65% of metal D.
The proportion of the four metals in each of the six ores and the level of impurities in each ore
are provided in the following table.



I have come up with the following but I am not sure:

Objective Function: Minimize Cost 2750X1+ 2500X2+3250X3+2200X4+2100X5+2400X6

Constraints:

I thought that since they want to know the amount of ore they should use and only have specific amount of each metal, those would be the constraints.

Any help would be much appreciated!

Thanks

 

[DrPhil]

Good evening,

I am having some difficulty with my linear programming assignment and would appreciate any help to bring me in the right direction.

I have figured out the decision variables, but the question mentions costs and other constraints and I'm very confused.

The questions says:

The company has an order
from a customer to produce 120 tons of an alloy that contains four
metals (for simplicity, we will be calling them metals A, B, C, and D
instead of wolfram, palladium, iridium, and copper) according to the
following specifications:
• The alloy should contain at least 24% of metal A.
• The alloy should contain no more than 11% of metal B.
• The alloy should contain no more than 8% of metal C.
• The alloy should contain between 35% and 65% of metal D.
The proportion of the four metals in each of the six ores and the level of impurities in each ore
are provided in the following table.

View attachment 3403

I have come up with the following but I am not sure:

Objective Function: Minimize Cost 2750X1+ 2500X2+3250X3+2200X4+2100X5+2400X6

Constraints:
View attachment 3404
I thought that since they want to know the amount of ore they should use and only have specific amount of each metal, those would be the constraints.

Any help would be much appreciated!

Thanks

I would define all of the \(\displaystyle x_i\) to be ratios. Repeat the final constraint for the two limits.

\(\displaystyle \displaystyle \begin{array}{rrrrrrcl}x_1,& x_2,& x_3,& x_4,& x_5,& x_6& > & 0 \\
x_1 & + x_2 & + x_3 & + x_4 & + x_5 & + x_6 & = & 1\\
0.17x_1 & +0.43x_2 & +0.14x_3 & +0.22x_4 & & +0.12x_6 & \ge & 0.24 \ \ A\\
0.15x_1 & +0.09x_2 & & +0.12x_4 & +0.24x_5 & +0.18x_6 & \le & 0.11 \ \ B\\
0.12x_1 & +0.25x_2 & & & +0.10x_5 & +0.16x_6 & \le & 0.08 \ \ C\\
0.14x_1 & +0.07x_2 & 0.53x_3 & +0.18x_4 & +0.31x_5 & +0.25x_6 & \ge & 0.35 \ \ D\\
0.14x_1 & +0.07x_2 & 0.53x_3 & +0.18x_4 & +0.31x_5 & +0.25x_6 & \le & 0.65 \ \ D\\ \\
2750x_1 & +2500x_2 & +3250x_3 & +2200x_4 &+2100x_5 & +2400x_6 & = & Cost,\ $/ton
\end{array}\)

So I agree with your interpretation- good luck with the calculation!!