The corner points of the bounded feasible region are (0, 0), (5, 0), (3, 4) and (0, 5). For the objective function Z=px+qy where p,q>0, the maximum occurs at two corner points when both points give equal Z values.
For maximum at both (5, 0) and (3, 4), these points must yield the same value of Z.
At point (5, 0):
Z=p(5)+q(0)
Z=5p
At point (3, 4):
Z=p(3)+q(4)
Z=3p+4q
For maximum to occur at both points:
5p=3p+4q
5p−3p=4q
2p=4q
p=2q
This can also be written as 2q=p.
In linear programming, when the maximum occurs at two corner points, the objective function line is parallel to the edge connecting those points. The condition 2q=p ensures this property.
Therefore, the condition is 2q=p.