Expert: Steve Holleran - 4/9/2008

Question
Steve,

I am not sure if you can help me with this but I hope you can. It is a Linear Programming Problem.

A car repair shop blends oil from two suppliers. Supplier I can supply at most 46 gal with 3.8% detergent. Supplier II can supply at most 66 gal with 3.1% detergent. How much can be ordered from each to get at most 100gal of oil with maximum detergent?

I have come up with different objective functions and constraints, but nothing is working.  I think I am missing something.

Thanks

Answer
Hi Melissa,

It certainly has been awhile since I've worked this, but I think I came up with a solution here:

Let x = # gallons from Supplier I

   y = # gallons from Supplier II

Then we have :

  x <= 46, x >=0      y <= 66, y >=0

  x + y <= 100   or y <= -x + 100


  The function we want to maximize is

                 D = .038x + .031y

If you plot the lines x = 0, y = 0 , x = 46, y = 66 and y = -x + 100,

you'll get a 5-sided region with lattice points

A(46,0)    B(46,54)   C(34,66)  D(0,66) and E(0,0)

I got B by substituting x = 46 in y = -x + 100
and   C by substituting y = 66 in y = -x + 100.

When you test each point in the Objective function, you get:

 A:  D = 1.748

 B:  D = 3.422

 C:  D = 3.338

 D:  D = 2.046

 E:  D = 0

So it looks like the max is at B(46, 54).

I hope this is what you needed.

Steve