Question

Given the constraints
A+B + C ≤ 24, B +C ≥ 8 and A ≥ 0, B ≥ 0, C ≥ 0.
Maximize 24-A-B - C
A: amount of time spent on school work
B: amount of time spent on fun
C: amount of time spent on pay work

Answer 1

$max\quad 24-A-B-C$

subject to,

$A+B + C \le24\\ B +C\ge8\\ A ≥ 0, B ≥ 0, C ≥ 0$

$max\quad 24-A-B-C+0S_1+0S_2-Ma_1$

subject to,

$A+B + C+S_1\hspace{2.1 em} =24\\ \hspace{2 em}B +C-S_2+a_1=8\\ A , B, C,S_1,S_2,a_1 ≥ 0$

Iteration-1

$a_1$ is goin out from the basis and B is come into the basis.

Iteration-2

since all the $Z-C_j\ge0$ optimum solution is obtain.

optimum solution is,

$\bold{A=0\quad B=8\quad C=0}\\ Z_{max}=24-8\\ \bold{Z_{max}=16}$