Archimedean Goal Programming Using Objective Functions
In this, the final case, we optimize a weighted sum of objective functions. In other words we solve a multi–objective problem. For consider the following:
| Weights | Sense | ||||
|---|---|---|---|---|---|
| goal 1 (OBJ1): | 5x + 2y | – | 20 | 100 | max |
| goal 2 (OBJ2): | –3x + 15y | – | 48 | 1 | min |
| goal 3 (OBJ3): | 1.5x + 21y | – | 3.8 | 0.01 | max |
| LIMIT: | 42x + 13y | ≤ | 100 |
In this case we have three different objective functions that will be combined into a single objective function by weighting them by the values given in the weights column. The solution of this model is one that minimizes:
| 1(–3x + 15y – 48) – 100(5x + 2y – 20) – 0.01(1.5x + 21y – 3.8) |
The resulting values that each of the objective functions will have are as follows:
| OBJ1: | 5x + 2y – 20 | = | –4.615389 |
| OBJ2: | –3x + 15y – 48 | = | 67.384613 |
| OBJ3: | 1.5x + 21y – 3.8 | = | 157.738464 |
The solution is x=0.0 and y=7.692308.