$1($a$)$ A restaurant is open seven days per week. Based on experience, the number of staff needed on a particular day is as follows:

Day: Mon Tue Wed Thu Fri Sat Sun

Number: $14131516191811$

Symbol: n$1$ n$2$ n$3$ n$4$ n$5$ n$6$ n$7$

Every worker works five consecutive days and then takes two days off, repeating

this pattern indefinitely. Formulate a Linear Program to minimize the number of

workers that staff the restaurant.

Hint: A worker who starts a shift on $($say$)$ Tuesday $($Day $2)$ is only available for

work on Days $26.$

$($i$)$ Make a choice of $$intermediate$$ variables, i$.$e$.$ variables that are useful when

tracking the day$-$to$-$day variation of the problem.

$($ii$)$ Make a choice of decision variables for the problem.

$($iii$)$ Express the intermediate variables in terms of the decision variables $$ a simple $7-$day table may be helpful.

$($iv$)$ What are the constraints $$ expressed in terms of the intermediate variables?

$($Use the symbols ni$,$ i $=1,...,7.)$

$($v$)$ Express the constraints in terms of the decision variables, just write out the first

two constraints.

$($vi$)$ Express your first constraint as an equality constraint. What is the coefficient

of the slack variable? Explain the reason for this choice.

$($vii$)$ Write the objective function z in terms of the decision variables only.

$($viii$)$ Explain whether z is to be maximised or minimised.

$($ix$)$ What other constraints must be applied to the decision variables for the problem

to be a Standard Form L$.$P$.?$