9.3 THE SIMPLEX METHOD: MAXIMIZATION
9.3 THE SIMPLEX METHOD: MAXIMIZATION
9.3 THE SIMPLEX METHOD: MAXIMIZATION
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502 CHAPTER 9 LINEAR PROGRAMMING<br />
subject to the constraints<br />
4x 1 3x 2 3x 3 30<br />
2x 1 3x 2 3x 3 60<br />
2x 1 2x 2 3x 3 40<br />
where x1 0, x2 0, and x3 0.<br />
Solution Using the basic feasible solution<br />
x 1 , x 2 , x 3 , s 1 , s 2 , s 3 0, 0, 0, 30, 60, 40<br />
the initial simplex tableau for this problem is as follows. (Note that s1 is an artificial variable,<br />
rather than a slack variable.)<br />
Basic<br />
x1 x2 x3 s1 s2 s3 b Variables<br />
3<br />
4 1 1 1 0 0 30 s 1 ← Departing<br />
2 3 1 0 1 0 60 s 2<br />
1 2 3 0 0 1 40 s 3<br />
↑<br />
Entering<br />
2<br />
1<br />
0 0 0 0<br />
x1 x2 x3 s1 s2 s3 b<br />
Basic<br />
Variables<br />
1 0 0 x1 0 1 0 45 s2 ← Departing<br />
0 0 1 s3 0<br />
↑<br />
Entering<br />
<br />
3<br />
4 0 0<br />
45<br />
2<br />
1<br />
4<br />
5<br />
<br />
65<br />
2<br />
4<br />
1<br />
<br />
7<br />
4<br />
11<br />
4 4<br />
1<br />
1<br />
4<br />
1<br />
4<br />
1<br />
4<br />
15<br />
2<br />
5<br />
2<br />
1<br />
2 2<br />
Basic<br />
x 1 x 2 x 3 s 1 s 2 s 3 b Variables<br />
1<br />
1 0 0 3 x1 0 1 0 18 x2 0 0 1 1 s3 7<br />
<br />
2<br />
5<br />
12 1<br />
1<br />
5 10 10<br />
1<br />
5 5<br />
5<br />
0 0 0 2 2 0 45<br />
This implies that the optimal solution is<br />
10<br />
x 1 , x 2 , x 3 , s 1 , s 2 , s 3 3, 18, 0, 0, 0, 1<br />
3<br />
1<br />
1<br />
10<br />
1<br />
and the maximum value of z is 45. (This solution satisfies the equation given in the constraints<br />
because 43 118 10 30.
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