Linear Programming minimization problem

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TeachSteve

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Hello,

I am having trouble with this linear programming minimization problem and was hoping someone could help me.

A politician wants to create an advertising campaign. They would like to buy television and radio ads of at least 80 minutes and less than 120 minutes. It is estimated that television ads reach 50,000 people per minute and radio ads reach 10,000 people per minute. They would like to reach at least 1,500,000 people. The cost per minute is $800.00 for television ads and $200.00 for radio ads. If the variable t is the amount of television ad minutes and the variable r is the radio ad minutes, find the combination that would minimize the cost the cost of the campaign while reaching their goals.

I determined the cost function to be $800t + $200r and am having trouble determining the constraints. The obvious ones are t and r are greater than zero. I also came with t+r is greater than or equal to 80 and t+r is less than or equal to 120. Lastly, I came up with 50,000t + 10,000r is greater than or equal to 1,500,000. These don't seem correct to me and I don't see how to solve for the vertex points. Can someone please help me?

Thanks,

Steve
 
Everything you have looks right to me. You now have a bounded feasible region which guarantees you the existence of a minimum. Just sketch the region, find the corner points, and substitute into the cost function.
 
Thank you Dr. Steve,

I have graphed them and do see a feasible region, but I am confused about how to find the vertex points. I can see that (0,120) is one of them and (0,80) is another, but there are two more that I am having trouble with. Can you give me some advice on how to find those. They look like (20,60) and maybe (10,110). I would normally find one of the variables by working through the system of equations and plugging it into the other, but that doesn't seem to work here. I'm not sure why this one seems different to me?

Steve
 
TeachSteve said:
They look like (20, 60) and maybe (10, 110).
Click to expand...

It looks like you've chosen to graph t horizontally and r vertically.

For example, that choice gives actual values of (17.5, 62.5) for your estimated coordinates (20, 60).

Here's how I found the actual values.

You posted the following relationships between t and r:

t + r = 80

50000t + 10000r = 1500000

We can solve both of these equations for r.

r = -t + 80

r = -5t + 150

Plotting these two lines shows an intersection point. (I think this point is the lower-right vertex of your feasible region.)

To find the t-coordinate of this point, we equate the two expressions above for r and solve for t.

-t + 80 = -5t + 150

t = 17.5

Substitution gives the vertical coordinate.

r = -17.5 + 80

r = 62.5

Please check my arithmetic. 8-)
 
Yes. I worked it out and come up with the same vertex coordinates. (7.5,112.5) and (17.5,62.5). Since they want to reach greater than 1,500,000 people, I believe the other two coordinates of the feasible region are (0,80) and (0,120). I plug all four coordinates into the cost function and come up with $28,500 for (7.5, 112.5), $26,500 for (17.5, 62.5), $64,000 for (0,80) and $96,000 for (0,120) so I guess the minimum cost to reach 1,500,000 would be $26,500 with 17.5 minutes of television and 62.5 minutes of radio advertising. One thing I don't understand though is that if 50,000 people are reached per minute on television the point (0,120) seems like it would reach 6,000,000 people for $96,000. If I only want to reach 1,500,000 I could do it for $24,000 through 30 minutes of television advertising, but then I guess then it would not be in the feasible region of at least 80 minutes. Is this correct?

Thank you,

Steve
 
I don't believe that (0,80) and (0,120) are in the feasible region (they don't satisfy your last inequality).

I think the correct corner points are (80,0) and (120,0).
 
Thank you for your help Dr. Steve. I finally got it. I actually did mean (80,0) and (120,0), but wrote them incorrectly.

Much appreciated,

Steve
 
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