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The problem is:

A production company requires different amounts of renewable resource on different days of the week. The units of renewable resource required on each day is given as follows: Monday: $17$ Tuesday: $13$ Wednesday: $15$ Thursday: $19$ Friday: $14$ Saturday: $16$ Sunday: $11$

For some production reason, the renewable resource must be used five consecutive days from the day which it begins to be used. And then it becomes unusable and needs two days to be renewed.

For example, $2.6$ units of resource which are used from Monday to Friday must be off for renewal on Saturday and Sunday. The company has to meet its daily requirements using the considered renewable resource. Formulate an LP for the company such that the total amount (in units) of renewable resource required can be minimized subject to the daily requirements.

I believe this should be the objective function

$x_k =$ renewable resource units used during day $k$

$MIN = 17x_1 + 13x_2 + 15x_3 + 19x_4 + 14x_5 + 16x_6 + 11x_7$

However I'm a bit lost when looking at how to interpret as a constraint, the renewable resource being used over $5$ days and not the next $2$ days.

Will it look something like this? $x_1 + x_2 + x_3 + x_4 + x_5 - x_6 - x_7 \leq 78$ etc?

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nelsun
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2 Answers2

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If the resource e.g. is a software engineer with five days shift and two days off :–), it makes sense:

Let $x_k$ be the amount which is put in use starting at day $k$.

The objective is to minimize $\sum_k x_k$.

At day $k$ we have $u_k$ of the resource in use where $$ u_k = \sum_{i=0}^4 x_{((k-1-i) \bmod 7)+1} $$ e.g $$ u_3 = x_3 + x_2 + x_1 + x_7 + x_6 $$ The requirements give the constraints $$ u_k \ge r_k $$ e.g. $$ x_3 + x_2 + x_1 + x_7 + x_6 \ge 15 $$

mvw
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  • Thank you for your answer, this seems correct. If i may ask, would there be a way to incorporate uk into the objective function? – nelsun May 21 '18 at 10:16
  • You could use $\sum u_k/5$ which should be the same as $\sum x_k$. – mvw May 21 '18 at 10:21
  • Note: This models a production cycle which is established. If one starts with production fresh on a monday then there are no prior resources in use. I assume one would need information on what day of week production starts and how long it would run to be precise. – mvw May 21 '18 at 10:31
  • So if i were to have uk/5 as the variable in the objective function, how would i interpret the constraints in terms of just uk, without using x? – nelsun May 21 '18 at 11:56
  • I think i've figured it out, using a similar version of yours, i posted as an answer – nelsun May 21 '18 at 12:04
  • The constraints are simply $u_k\ge r_k$, e.g, $u_1\ge 17$. You can formulate the problem in terms of the $u_k$, but you want to know the $x_k$. So I would expand everything in terms of the $x_k$. – mvw May 21 '18 at 12:07
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$MIN = x1 + x2 + x3 + x4 + x5 + x6 + x7$

$x1 + x7 + x6 + x5 + x4 >= 17 $

$x2 + x1 + x7 + x6 + x5 >= 13$

$x3 + x2 + x1 + x7 + x6 >= 15$

$x4 + x3 + x2 + x1 + x7 >= 19$

$x5 + x4 + x3 + x2 + x1 >= 14$

$x6 + x5 + x4 + x3 + x2 >= 16$

Jessica Tiberio
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nelsun
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