Solving linear programming with bounds by simplex algorithm

Question

I want to solve the following simplex algorithm :

Maximize $$z =x_1+x_2+2x_3-2x_4$$ Subject to $$ x_1+2x_3 \le 700 $$ $$ 2x_2-8x_3 \le 0 $$ $$ x_2-2x_3+2x_4 \ge 1 $$ $$ x_1+x_2+x_3+x_4 = 10 $$ where $$ 0 \le x_1 \le 10$$ $$ 0 \le x_2 \le 10$$ $$ 0 \le x_3 \le 10$$ $$ 0 \le x_4 \le 10$$

I know that the standard simplex problem has following form :

Maximize $$z=c_1x_1+c_2x_2+ \ldots + c_n+x_n$$ Subject to $$a_{11}x_1+a_{12}x_2+a_{13}x_3+\ldots+a_{1n}x_n \le b1$$ $$a_{11}x_1+a_{12}x_2+a_{13}x_3+\ldots+a_{1n}x_n \le b2$$ Subject to $$\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots $$ $$a_{m1}x_1+a_{m2}x_2+a_{m3}x_3+\ldots+a_{mn}x_n \le bm$$

where $$ x_j \ge 0 , j=1,2,3,\ldots,n$$

How can I convert my problem to standard form ? How can I solve my problems by simplex algorithm ?

My attempt:

I have replaced $0 \le x_1 \le 10 $ by two separate constraints $ 0 \le x_1 $ and $ x_1 \le 10 $. But after that, I can't reach to optimal solution although the problem has basic feasible solution.

Answer 1

Some software packages offer Linear Programming solvers. Each package has a typical formatting rules so for instance in MATHEMATICA we have the solver format

LinearProgramming[c, M, b]

in which the problem should be submitted as

$$ \max c x = c_1x_1+\cdots + c_n x_n\\ \mbox{subject to}\\ M x \ge b\\ x \ge 0 $$

with

$$ M = \left( \begin{array}{cccc} -1 & 0 & -2 & 0 \\ 0 & -2 & 8 & 0 \\ 0 & 1 & -2 & 2 \\ 1 & 1 & 1 & 1 \\ -1 & -1 & -1 & -1 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{array} \right), b = \left( \begin{array}{c} -700 \\ 0 \\ 1 \\ 10 \\ -10 \\ -10 \\ -10 \\ -10 \\ -10 \\ \end{array} \right), c = (1,1,2,-2) $$

and the result is

$$ x_0 = (0,0,0,10) $$

NOTE

The equality

$$ x_1+x_2+x_3+x_4 = 10 $$

is handled as

$$ x_1+x_2+x_3+x_4 \ge 10\\ x_1+x_2+x_3+x_4 \le 10 $$