I need to solve the following problem: $$\max: x_1+12x_2+65x_3; $$ $$x_2+4x_3 \leq 200;$$ $$x_1+10x_2+60x_3 \leq 750; $$ $$x_1,x_2,x_3 \leq 50 $$ To solve it efficiently by hand, I have tried to make the following change of variable: $y_i=50 - x_i \ge 0$ for $i = 1,2,3$ So the problem I have to solve is: $$\max: y_1+12y_2+65y_3-3900;$$ $$y_2+4y_3 \geq 50;$$ $$y_1+10y_2+60y_3 \geq 2800;$$ And the solution is unbounded, which doesn't make sense. My question is, what's wrong with the change of variable?, and how do you properly do it?
Thanks in advance
If you replace $x_i$ with $50-y_i$ in the objective function, you will get $-1$ times the expression that you reported. You have also omitted the lower bounds $y_i \ge 0$ that result from the upper bounds $x_i \le 50$.