I have the following linear programming problem:
- Convert the following problems to standard form: $$\begin{align} \text{a)}&\text{minimize}&x+2y+3z\\ & \text{subject to}&2\le x+y\le 3\\ & &4\le x+z \le 5\\ & &x\ge 0, \,\,\,y\ge0\,\,\,z\ge0.\\ \\ \text{b)}&\text{minimize}&x+y+z\\ & \text{subject to}&2\le x+y\le 3\\ & &x\ge 1, \,\,\,y\ge2\,\,\,z\ge1.\\ \end{align}$$
Here is my attempted solution:
$(1a)$ First I note that: $x+y \ge 2, x+y \leq 3, x+y\ge 4, x+y \leq 5$ with $x,y,z \ge 0$. I transform the equation into standard form by selecting two surplus and two slack variables $a, b, c, d$. So I get:
$$\text{minimize} \;\;\; x+2y+3z$$ $$\text{subject to} \;\;\; x+y + a= 3$$ $$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; x+y - b= 2$$ $$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; x+y + c= 5$$ $$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; x+y - d= 4$$ $$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; x \ge 0, \;\; y \ge 0, \;\; z \ge 0, \;\; a \ge 0, \;\; b \ge 0, \;\; c \ge 0, \;\; d \ge 0 \;\;.$$
$(1b)$ I do a change of variables by setting $a = x-1, b=y-2, c=z-1$. Then I formulate the problem in terms of variables $a, b$ and $c$ :
$$\text{minimize} \;\;\; a+b+c+4$$ $$\text{subject to} \;\;\; a+2b + 3c= 2$$ $$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; a \ge 0, \;\; b \ge 0, \;\; c \ge 0, \;\;.$$
Now this problem is equivalent with the original problem and by solving the optimal values for $a, b$ and $c$ I can solve the values for $x, y$ and $z$.
I would appreciate if someone can correct my mistakes if any exist :) I'm new with linear and nonlinear programming and want verification whether I have understood the basic concepts or not :)
Thank you!
