\[O=6x_1+3x_2+x_3+2x_4\]
subject to the constraints\[x_1+x_2+x_3+x_4 \le 8\]
(1)\[2x_1+x_2+3x_3 \le 12\]
(2)\[5x_2+x_3+3x_4 \le 6\]
(3)\[0 \le x_1 \le 1\]
(4)\[0 \le x_2 \le 1\]
(5)\[0 \le x_3 \le 4\]
(6)\[0 \le x_4 \le 2\]
(7)When a problem is required to have integer solutions, there are a finite number of possible solutions, so it is possible to analyse every possible solution to find the maximum value of
\[O\]
.\[x_1\] |
\[x_2\] |
\[x_3\] |
\[x_4\] |
\[O\] |
(1) | (2) | (3) |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 6 | 1 | 2 | 0 |
| 0 | 1 | 0 | 0 | 3 | 1 | 1 | 5 |
| 1 | 1 | 0 | 0 | 9 | 2 | 3 | 5 |
| 0 | 0 | 1 | 0 | 1 | 1 | 3 | 1 |
| 1 | 0 | 1 | 0 | 7 | 5 | 2 | 5 |
| 0 | 1 | 1 | 0 | 4 | 2 | 4 | 6 |
| 1 | 1 | 1 | 0 | 10 | 3 | 6 | 6 |
| 0 | 0 | 2 | 0 | 2 | 2 | 6 | 2 |
| 1 | 0 | 2 | 0 | 8 | 3 | 8 | 2 |
| 0 | 1 | 2 | 0 | 5 | 3 | 7 | 7 |
| 1 | 1 | 2 | 0 | 11 | 4 | 9 | 7 |
| 0 | 0 | 3 | 0 | 3 | 3 | 9 | 3 |
| 1 | 0 | 3 | 0 | 9 | 4 | 11 | 3 |
| 0 | 1 | 3 | 0 | 6 | 4 | 10 | 8 |
| 1 | 1 | 3 | 0 | 12 | 5 | 12 | 8 |
| 0 | 0 | 4 | 0 | 4 | 4 | 12 | 4 |
| 1 | 0 | 4 | 0 | 10 | 5 | 14 | 4 |
| 0 | 1 | 4 | 0 | 7 | 5 | 13 | 9 |
| 1 | 1 | 4 | 0 | 13 | 6 | 15 | 9 |
| 0 | 0 | 0 | 1 | 2 | 1 | 0 | 3 |
| 1 | 0 | 0 | 1 | 8 | 2 | 2 | 3 |
| 0 | 1 | 0 | 1 | 5 | 2 | 1 | 8 |
| 1 | 1 | 0 | 1 | 11 | 3 | 3 | 6 |
| 0 | 0 | 1 | 1 | 3 | 2 | 3 | 4 |
| 1 | 0 | 1 | 1 | 9 | 3 | 5 | 4 |
| 0 | 1 | 1 | 1 | 6 | 3 | 4 | 9 |
| 1 | 1 | 1 | 1 | 12 | 4 | 6 | 9 |
| 0 | 0 | 2 | 1 | 4 | 3 | 6 | 5 |
| 1 | 0 | 2 | 1 | 10 | 4 | 8 | 5 |
| 0 | 1 | 2 | 1 | 7 | 4 | 7 | 10 |
| 1 | 1 | 2 | 1 | 13 | 5 | 9 | 10 |
| 0 | 0 | 3 | 1 | 5 | 4 | 9 | 6 |
| 1 | 0 | 3 | 1 | 11 | 5 | 11 | 6 |
| 0 | 1 | 3 | 1 | 8 | 5 | 10 | 11 |
| 1 | 1 | 3 | 1 | 14 | 6 | 12 | 11 |
| 0 | 0 | 4 | 1 | 6 | 5 | 12 | 7 |
| 1 | 0 | 4 | 1 | 12 | 6 | 14 | 7 |
| 0 | 1 | 4 | 1 | 9 | 6 | 13 | 12 |
| 1 | 1 | 4 | 1 | 15 | 7 | 15 | 12 |
| 0 | 0 | 0 | 2 | 4 | 2 | 0 | 6 |
| 1 | 0 | 0 | 2 | 10 | 3 | 2 | 6 |
| 0 | 1 | 0 | 2 | 7 | 3 | 1 | 11 |
| 1 | 1 | 0 | 2 | 13 | 4 | 3 | 11 |
| 0 | 0 | 1 | 2 | 5 | 3 | 3 | 7 |
| 1 | 0 | 1 | 2 | 11 | 4 | 5 | 7 |
| 0 | 1 | 1 | 2 | 8 | 4 | 4 | 12 |
| 1 | 1 | 1 | 2 | 14 | 5 | 6 | 12 |
| 0 | 0 | 2 | 2 | 6 | 4 | 6 | 8 |
| 1 | 0 | 2 | 2 | 12 | 5 | 8 | 8 |
| 0 | 1 | 2 | 2 | 9 | 5 | 7 | 13 |
| 1 | 1 | 2 | 2 | 15 | 6 | 9 | 13 |
| 0 | 0 | 3 | 2 | 7 | 5 | 9 | 9 |
| 1 | 0 | 3 | 2 | 13 | 6 | 11 | 9 |
| 0 | 1 | 3 | 2 | 10 | 6 | 10 | 14 |
| 1 | 1 | 3 | 2 | 16 | 7 | 12 | 14 |
| 0 | 0 | 4 | 2 | 8 | 6 | 12 | 10 |
| 1 | 0 | 4 | 2 | 14 | 7 | 14 | 10 |
| 0 | 1 | 4 | 2 | 11 | 7 | 13 | 15 |
| 1 | 1 | 4 | 2 | 17 | 8 | 15 | 15 |
\[O\]
and satisfies all the constraints is \[x_1=1, \: x_2=0, \: x_3=3, \: x_3=1\]
.