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?
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. $$ x_3 + x_2 + x_1 + x_7 + x_6 \ge 15 $$