Linear Programming Minimization Problem

P. A hotel rental service needs to have clean towels for each day of a three-day period. Some of the clean towels may be purchased new and some may be dirty towels from previous days that have been washed by a laundry service. The cost of new towels is $1 per towel, cost of a fast one day laundry service is 40 cents per towel and the cost of a slow two day laundry service is 25 cents per towel. If the rental service needs 300,200, and 400 clean towels for each of the next three days, respectively. how many towels should the rental service buy new and how many should rental service have washed by different laundry services so as to minimize total costs?

Now, clearly this is a minimization problem and I can solve for it using a minimization tableau. But I don't really know how to set up the problem. The back of the book says there are 4 variables to solve for, but I only count 3, so I have no idea how to proceed. Please shed some light on this problem. Thanks in advance.

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Originally Posted by bambamm

P. A hotel rental service needs to have clean towels for each day of a three-day period. Some of the clean towels may be purchased new and some may be dirty towels from previous days that have been washed by a laundry service. The cost of new towels is $1 per towel, cost of a fast one day laundry service is 40 cents per towel and the cost of a slow two day laundry service is 25 cents per towel. If the rental service needs 300,200, and 400 clean towels for each of the next three days, respectively. how many towels should the rental service buy new and how many should rental service have washed by different laundry services so as to minimize total costs?

Now, clearly this is a minimization problem and I can solve for it using a minimization tableau. But I don't really know how to set up the problem. The back of the book says there are 4 variables to solve for, but I only count 3, so I have no idea how to proceed. Please shed some light on this problem. Thanks in advance.

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There are five variables in the problem, the number bought in on days 2 and 3 (v,w), the number put to 1 day laundry from day 1 (x), the number put to 2 day laundry from day 1 (y) and the number put to 1 day laundry from day 2 (z). You could introduce more variables but it is obvious that they will end up as zero (for example the number put to 2 day laundry from day 2)

Now we can either leave the problem in this form or we can start using some of the constraints to eliminate variables. Since 400=w+y+z we can eliminate z. We could also use 200=v+x to eliminate x, which will leave us with three variables.

But of course we could have left one or other of these variables in the problem giving any of 3,4 or 5 variables.

CB

Thank you very much sir. You helped me immensely!