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Alloy mixture optimization
(Minimize Expenses)
There are four metals A, B, C and D that have the following properties:
| Metal |
Density (den) |
Carbon (crb), % |
Phosphor (psh), % |
Price, $ / kg |
| A |
6500 |
0.2 |
0.05 |
2.0 |
| B |
5800 |
0.35 |
0.015 |
2.5 |
| C |
6200 |
0.15 |
0.065 |
1.5 |
| D |
5900 |
0.11 |
0.1 |
2.0 |
The is a requirement to produce an alloy at minimal cost that has the following
property ranges:
| Range |
Density (den) |
Carbon (crb), % |
Phosphor (psh), % |
| Min |
5950 |
0.1 |
0.045 |
| Max |
6050 |
0.3 |
0.055 |
The decision variables are part of each metals in the alloy: X_A, X_B,
X_C and X_D.
The objective function that to be minimized is:
F = X_A * A_Price + X_B * B_Price + X_C * C_Price + X_D * D_Price.
The density constraints are:
X_A * A_den + X_B * B_den + X_C * C_den + X_D * D_den >= 5950
X_A * A_den + X_B * B_den + X_C * C_den + X_D * D_den <= 6050
The carbon concentration constraints are:
X_A * A_crn + X_B * B_crn + X_C * C_crn + X_D * D_crn >= 0.1
X_A * A_crn + X_B * B_crn + X_C * C_crn + X_D * D_crn <= 0.3
The phosphor concentration constraints are:
X_A * A_psh + X_B * B_psh + X_C * C_psh + X_D * D_psh >= 0.045
X_A * A_psh + X_B * B_psh + X_C * C_psh + X_D * D_psh <= 0.055
We have to define the part of each metals in the alloy, therefore the last constraints is:
X_A + X_B + X_C + X_D = 1
The problem has been solved successfully with the following plan:
| Metal |
Metal Part |
| A |
- |
| B |
0.285 |
| C |
0.595 |
| D |
0.12 |
The Total cost of the alloy is $1.84
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the example in GIPALS format
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