I have been given the following problem.
Maximize $C = S+T$ using duality
Subject to the $S + 3T \geq 8$ and $3S + T \geq 8 , S,T \geq 0.$
The duality of this problem is following.
Minimize $C' = 8S' + 8T'$ using duality
Subject to the $S' + 3T' \leq 1$ and $3S' + T' \leq 1 , S,T \geq 0.$
I have solved this by using Graphical method. I got Min $C' = 4$ when $(S' , T') = (\frac{1}{4} , \frac{1}{4})$.
So I can say Max $C=4$ . But what will be $(S , T)$ for which Max $C=4$ ? Can anyone please help me?
By complementary slackness condition, we have
$$S'(S+3T-8)=0$$
$$T'(3S+T-8)=0$$
Hence knowing that $S'$ and $T'$ are non-zero, we have
$$S'>0 \implies S+3T=8$$ $$T' >0 \implies3S+T=8$$
Solving the simultaneous equation, we get $S=T=2$.