(page 2 of 9)

 

We will first look at what happens as we change the righthand side (RHS) of a constraint.   Lets look at our third example from the section on graphing 2-dimensional LPs.

Minimize

x + 1/3 y

Subject to:

x + y ³ 20

 

-2 x + 5 y £ 150

 

x ³ 5

 

x ³ 0 y ³ 0

 

To the right you have the graphical solution that we developed previously.   The optimal solution to this problem was x = 5, y = 15 which has an objective value of 10.

Now lets look at what happens if we decrease the right hand side of the second constraint from 20 to10.

 

To the right we have re-graphed the constraint.   The dotted line represents its old location, while the solid line is its new one.   Notice that decreasing the right hand side translates into moving the constraint out, while keeping the same slope.

Any time we move a constraint, our feasible region changes.   In this case, our feasible region enlarges to include the darker green area.

With this new feasible region, is our old solution (the blue dot) still optimal?

 

Notice that now that our feasible region is larger, we can push our objective function (the blue line) even further than we could before.

In fact, we can move it from the dotted line, to the solid line.   This gives us a new optimal solution with a new objective value.

Notice that when we decrease the right hand side of this constraint, we find a solution with a lower objective value.   This is becausea decrease in the right hand side makes this constraint less restrictive, therefore the size of our feasible region increases and we can expect that we might be able to find a "better" solution.