1. ## Linear Programming

I understand the problems that give various constraints and ask to maximize or minimize something, but not this type.

A chem manufacturer wants to rent a fleet of 24 tank cars with a combined carrying capacity of 504,000 gallons. Tank cars with three capacities are available: 7000 gallons, 14,000 gallons, 28,000 gallons. How many of each should be leased?

Set x, y and z as the three different capacity cars.

x+y+z=24
7000x+14000y+28000z=504000

There are various answers. For example it can be 18 z's, or 17 z's and 2 y's, or 17z's and 4 x's.

x=_t+_ y=_t+_ and z=t where _<=t<=_

that's something times t plus something, etc. where something <=t<=something

2. ## Re: Linear Programming

7000 gallon car = x1
14000 gallon car = x2
28000 gallon car = x3

Two equations from the given information:

x1+x2+x3=24 The number of the combined tank cars must be 24

7000x1+14000x2+28000x3 = 504000 Their combined capacity must be 504000

There are two equations with three unknowns. That can't be solved unless we select a value for one variable.

Call x3=t. Now solve for the other two variables in terms of t.

x1+x2+t=24
x1=24-x2-t

7000x1+14000x2+28000t=504000
7000(24-x2-t)+14000x2+28000t=504000
168000-7000x2-7000t+14000x2+28000t=504000
7000x2+21000t=336000
7000x2=336000-21000t

x2=48-3t= -3t + 48

x1+48-3t+t=24
x1-2t=-24
x1=2t-24

To find the constraints on t, set each equation (x1 and x2 in terms of t) >= 0. Then solve for t.

x1 = 2t-24 >= 0
2t>=24
t>=12

x2=-3t+48>=0
-3t>=48
t<=16

So t is in the range 12<=t<=16

t>=0, but that's redundant.

x1 = 2t-24
x2 = -3t+48
x3=t
12<=t<=16

3. ## Re: Linear Programming

Only 3 solutions if at least one of each:
2,9,13
4,6,14
6,3,15