[Math] Converting linear programming problems into standard form

Question

I have the following linear programming problem:

1. 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!

Answer 1

@jjepsuomi $(1a)$: I think your third and fourth lines of constraints should be something like $x + z + c=5$ and $x+z-d=4$ instead.

Cheers!