# How to determine all the basic

How to determine all the basic solutions of an optimizationproblem, and classify them as feasible or infeasible?

As an example,

Maximize z= 2x1 + 3x2

Subject to:

x1 + 3x2 <= 6

3x1 + 2x2 <= 6

x1 + x2 >= 0

Some solutions are (2,0), (0,2), (0,0), (6/7, 12/7), but arethere more solutions? And how do you tell if they areinfeasile/feasible? Would you need to graph it?

The basic solutions are the solutions which solve any of theconstraints. However, feasible solutions are which satisfy allconstraints.So the basic solutions are all the points that lie on the lines

However out of this the feasible ones are that also satisfy theconstraints, so to do that find the intersecting points on thelines

X1 = 6/7

X2 = 22/7

So feasible solutions are the points that lie on either of theline but within the following range

So from our process above we find 4 points of interest. theseare actually the extreme points that satisfy our constraints. (Thiswill be a lot more clear in the graph)We have

(0,0), (0,2), (2,0), (6/7, 12/7)When we put all these points in the objective function, we findthat the maximum value occursat X2 = 12/7, X1 = 6/7 then z= 48/7we will just plot the graph and make things clear now

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