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I came across some questions of this style and was not sure what the minimization function would be.

A hotel requires a known number of hand towels for guests to be given during the week and the goal is to meet the demand at minimum cost. Suppose the towels required are 100,80,120,150 and 130 for Monday through Friday respectively. New towels can be purchased for \$0.12 each and used towels can be washed. Towels that are washed for the next-day service have a cost of \$0.06 each, for use on the second day at \$0.03 each and for the third day at \$0.03 each. At the end of the week, all used towels can be sold for \$0.01 each.

I assumed the following: $$x_i = \text{the # of new towels purchased for day } i, \text{where } i = 1,2,3,4,5$$ $$x_{ij} = \text{the # of towels used on day } i, \text{and washed to be used again on day } j, = i+1,...,5$$

My constraints are:

$$x_{ij} \geq 0$$ $$x_1 \geq 100$$ $$x_2 + x_{12} \geq 80$$ $$x_3 + x_{13} + x_{23} \geq 120$$ $$x_4 + x_{14} + x_{24} + x_{34} \geq 150$$ $$x_5 + x_{15} + x_{25} + x_{35} + x_{45} \geq 130$$

I am not sure how to develop the cost function fully. I reached this far but not sure what to do with the selling of the towels:

$cost = 0.12 (x_1 + x_2 + x_3 + x_4 + x_5) + 0.06 (x_{12}+ x_{23}+ x_{34}+ x_{45}) + 0.04 (x_{13} + x_{24} + x_{35}) + 0.03 (x_{14} + x_{25})$

user2958395
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  • Ok deleted my old comments to avoid confusion. I think you need to subtract off $0.01(\sum_{i=1}^{4}x_{i5})$ from your cost function. – Grid Apr 05 '14 at 05:29
  • On second thought, maybe it's -$0.01(x_5 + \sum_{i=1}^{4} x_{i5})$. Hmm LP formulation is confusing, what do you think? Basically you want to treat that profit as a "negative-cost". – Grid Apr 05 '14 at 05:36

1 Answers1

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Your model looks correct, though it's also important to have the non-negativity constraint for $x_i \ge 0$ for $i = 1,2,...5$. The rest of your constraints are correct. Your objective function is almost correct, but you need to take account the 'profit' from the used towels on the final day. Note: what I suggested in the comments is incorrect, there is a simple fix in which you just replace the 0.12 with 0.12 - 0.01.

objective function:

min $0.12\sum_{i=1}^{5} x_i + 0.06(x_{12}+x_{23}+x_{34}+x_{45}) + 0.04(x_{13}+x_{24}+x_{35}) + 0.03(x_{14} + x_{25}) - 0.01(\sum_{i=1}^{5} x_i)$ =

min $0.11\sum_{i=1}^{5} x_i + 0.06(x_{12}+x_{23}+x_{34}+x_{45}) + 0.04(x_{13}+x_{24}+x_{35}) + 0.03(x_{14} + x_{25})$

Where the term $\sum_{i=1}^{5} x_i$ represents the number of towels bought in the week, and also the number of used towels remaining at the end of week (i.e., all the towels you bought are considered used at the end of week).

Grid
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  • Thank you, I understand what needs to be done now. I too initially thought it was 0.01 by the whole of the last row. This makes much more sense and yes LP formulation can become confusing. – user2958395 Apr 05 '14 at 15:57