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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 :)

Zenit
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    Pick two variables to keep, and use the equations to eliminate the other two. The bounds on the eliminated variables yield inequalities for the kept variables. – RobPratt Jul 05 '21 at 00:24

1 Answers1

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$x_1+x_2+x_3+3x_4=16 \tag{1}$

$-x_1+x_2+3x_3-x_4=8 \tag{2}$

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).

enter image description here

The "feasibility polygon" corresponding to inequalities (5).

Jean Marie
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