0
$\begingroup$

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 :)

$\endgroup$
1
  • 1
    $\begingroup$ 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. $\endgroup$ Commented Jul 5, 2021 at 0:24

1 Answer 1

Reset to default
2
$\begingroup$

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

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

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).

enter image description here

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

$\endgroup$
6
  • $\begingroup$ could you simplify those actions a bit if possible, kinda new to LPP :) $\endgroup$ Commented Jul 5, 2021 at 11:46
  • $\begingroup$ Thank you, all the best mate :) $\endgroup$ Commented Jul 5, 2021 at 14:21
  • $\begingroup$ when we plug (3),(4) into F i think we get this $$F=2x_3+3x_4-4$$ not this $$F=2x_3+11x_4-4$$ or this $$F=2x_3-3x_4-4$$ $\endgroup$ Commented Jul 6, 2021 at 12:36
  • $\begingroup$ You are right. I have corrected it. $\endgroup$ Commented Jul 6, 2021 at 14:04
  • $\begingroup$ Even this sentence "Maximize $F=2x_3-3x_4-4$ under the constraints:" should it be "Maximize $F=2x_3+3x_4-4$ under the constraints:" (change sign) i think, sorry for bothering. All the best! $\endgroup$ Commented Jul 6, 2021 at 14:28

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.