Linear Programming Lecture Notes - Penn State Personal Web Server
Linear Programming Lecture Notes - Penn State Personal Web Server
Linear Programming Lecture Notes - Penn State Personal Web Server
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The cost information becomes:<br />
c T BB −1 b = 720 c T BB −1 N = 6 0 c T BB −1 N − cN = 6 0 <br />
This yields the tableau:<br />
⎡<br />
z x1 x2<br />
z ⎢ 1 0 0<br />
(5.41) s1 ⎢ 0 1 0<br />
s2 ⎣ 0 0 1<br />
s1<br />
6<br />
0<br />
1<br />
s2<br />
0<br />
0<br />
0<br />
s3<br />
0<br />
1<br />
−3<br />
⎤<br />
RHS<br />
720 ⎥<br />
35 ⎥<br />
15 ⎦<br />
0 0 0 −2 1 5 95<br />
s3<br />
Unlike example 5.9, the reduced cost for s3 is 0. This means that if we allow s3 to enter<br />
the basis, the objective function value will not change. Performing the minimum ratio test<br />
however, we see that s2 will still leave the basis:<br />
⎡<br />
⎤<br />
z x1 x2 s1 s2 s3 RHS<br />
z ⎢ 1 0 0 6 0 0 720 ⎥<br />
(5.42) x1 ⎢ 0 1 0 0 0 1 35 ⎥<br />
x2 ⎣ 0 0 1 1 0 −3 15 ⎦<br />
s2 0 0 0 −2 1 5 95<br />
Therefore any solution of the form:<br />
s3 ∈ [0, 19]<br />
⎡<br />
(5.43)<br />
⎣ x1<br />
⎤ ⎡<br />
x2⎦<br />
= ⎣ 35<br />
⎤ ⎡<br />
15⎦<br />
− ⎣<br />
95<br />
1<br />
⎤<br />
−3⎦<br />
s3<br />
5<br />
s2<br />
MRT (s3)<br />
is an optimal solution to the linear programming problem. This precisely describes the edge<br />
shown in Figure 5.3.<br />
Figure 5.3. Infinite alternative optimal solutions: In the simplex algorithm, when<br />
zj − cj ≥ 0 in a maximization problem with at least one j for which zj − cj = 0,<br />
indicates an infinite set of alternative optimal solutions.<br />
85<br />
35<br />
−<br />
19