When an LPP has an optimal value occurring at two corner points, both points must give the same value of the objective function.
The objective function is z=px+y with corner points (2,11) and (4,5).
At point (2,11):
z=p(2)+11
z=2p+11
At point (4,5):
z=p(4)+5
z=4p+5
Since both points give the optimal value:
2p+11=4p+5
11−5=4p−2p
6=2p
p=3
The optimal value is:
z=2p+11
z=2(3)+11
z=6+11
z=17
Therefore, the optimal value is 17.