About Linear programming

[EveQi]

I cannot understand linear programming. What is the maximize x for? How can I solve this problem? Is it using graphing or there are other ways to solve the question?

 

[HallsofIvy]

So that people won't have to keep opening that attachment, the problem is to
"maximize \(\displaystyle x_1\) subject to the constraints
\(\displaystyle x_1+ 2x_2+ x_3\le 10\)2
\(\displaystyle 2x_1+ 3x_2+ 3x_3\le 10\)
\(\displaystyle x_1, x_2, x_3\ge 0\)."

Those 5 planes form a "convex polyhedron". The basic theorem in "linear programming" is that the maximum (or minimum) of a linear function must occur at one of the vertices. The intersection of two planes is a line, the intersection of three planes is a point. It might help to graph the planes to be sure that the point of intersection of three planes does not lie outside the "feasible area" (the set of points, (x, y, z) that satisfy all inequalities) but it is not always necessary. In fact, there exist problems in linear programming with more than there variables which obviously cannot be "graphed" in three dimensions. And you could use the "simplex method" which does not require graphing.

Here, the three planes, given by \(\displaystyle x_1+ 2x_2+ x_3= 10\), \(\displaystyle 2x_1+ 3x_2+ 3x_3= 10\), and \(\displaystyle x_1= 0\) obviously intersect where \(\displaystyle 0+ 2x_2+ x_3= 10\) and \(\displaystyle 2(0)+ 3x_2+ 2x_3= 10\). Subtracting the second equation from twice the first equation gives \(\displaystyle x_2= 10\). Then \(\displaystyle 20+ x_3= 10\(\displaystyle so \(\displaystyle x_3= -10\). One vertex is at \(\displaystyle (0, 10, -10)\)

The three planes, given by \(\displaystyle x_1+ 2x_2+ x_3= 10\), \(\displaystyle 2x_1+ 3x_2+ 3x_3= 10\), and \(\displaystyle x_2= 0\) intersect where \(\displaystyle x_1+ 2(0)+ x_3= 10\) and \(\displaystyle 2x_1+ 3x_3= 10\(\displaystyle . Subtracting the second equation from twice the first equation, \(\displaystyle x_1= -10\). Then \(\displaystyle -10+ x_3= 10\(\displaystyle so \(\displaystyle x_3= 20\). Another vertex is at (-10, 0, 20).

The three planes, given by \(\displaystyle x_1+ 2x_2+ x_3= 10\), \(\displaystyle 2x_1+ 3x_2+ 3x_3= 10\), and \(\displaystyle x_3= 0\) intersect where \(\displaystyle x_1+ 2x_2+ 0= 10\) and \(\displaystyle 2x_1+ 3x_2+ 3(0)= 10\). Subtracting the second equation from twice the first equation, \(\displaystyle x_2 = 10\). Then \(\displaystyle x_1+ 20= 10\) so \(\displaystyle x_1= -10\). Another vertex is at (-10, 10, 0).

You will also need to look at points where \(\displaystyle x_1+ 2x_2+ x_3= 10\), \(\displaystyle x_1= 0\), and \(\displaystyle x_2= 0\), points where \(\displaystyle x_1+ 2x_2+ x_3= 10\), \(\displaystyle x_1= 0\), and \(\displaystyle x_3=0\), where \(\displaystyle x_1+ 2x_2+ x_3= 10\), \(\displaystyle x_2= 0\), and \(\displaystyle x_3= 0\), \(\displaystyle 2x_1+ 3x_2+ 3x_3= 10\), \(\displaystyle x_1= 0\), and \(\displaystyle x_2=0\), where \(\displaystyle 2x_1+ 3x_2+ 3x_3= 10\), \(\displaystyle x_1= 0\), and \(\displaystyle x_3= 0\), where \(\displaystyle 2x_1+ 3x_2+ 3x_3= 10\), \(\displaystyle x_2= 0\), and \(\displaystyle x_3= 0\).\

And, of course, (0, 0, 0) where the three coordinate planes intersect.

Evaluate the function \(\displaystyle f(x_1, x_2, x_3)= x_1\) at each of those vertices. Which give the largest value?\)\)\)\)\)\)

 

[EveQi]

To start: WHO gave you that problem?
Your math teacher?

Ya.