Solve this Linear Programming problem(LPP) using Graphical method :
$F=2x_1-x_2-2x_3+6x_4$
$x_1+x_2+x_3+3x_4=16$
$-x_1+x_2+3x_3-x_4=8$
$x_1\geq0 ;x_2\geq0;x_3\geq0;x_4\geq0$
Max $F=?$
I've searched and learned that graphical method can be use when we have two variables, otherwise if we have 4 variables like my example is it preferable to use Simplex method. But Problem asks to solve it by Graphical method.
P.s I know how to solve if we would have 2 variables. Need a bit help with 4.
Thank you in advance :)
$\endgroup$ 11 Answer
$\begingroup$$x_1+x_2+x_3+3x_4=16 \tag{1}$
$-x_1+x_2+3x_3-x_4=8 \tag{2}$
Adding and subtracting (1) and (2) gives:
$$x_1=x_3-2x_4+4\tag{3}$$
$$x_2=-2x_3-x_4+12\tag{4}$$
Now plug (3)+(4) into function $F$ which becomes:
$$F=2x_1-x_2-2x_3+6x_4=2(x_3-2x_4+4)-(-2x_3-x_4+12)-2x_3+6x_4$$
$$F=2x_3+3x_4-4$$
You have now to solve (graphically) the linear programming problem with 2 variables $x_3,x_4$ :
Maximize $F=2x_3+3x_4-4$ under the constraints:
$$\left\{\begin{array}{llll}using \ (3): \ & \ \ x_3-2x_4+4& \ge& 0\\using \ (4): \ &-2x_3-x_4+12&\ge&0\\& x_3& \ge &0\\&x_4 &\ge&0\end{array}\right.\tag{5}$$
You should find $x_3=x_4=4$ and then deduce values of $x_1,x_2$ from (3) and (4).
The "feasibility polygon" corresponding to inequalities (5).
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