capacities are $5,0006-$cylinder connecting rods and $8,0004-$cylinder connecting rods. Let

$\backslash ($ x$\_\{4\}=\backslash )$ the number of $4-$cylinder connecting rods produced next week

$\backslash ($ x$\_\{6\}=\backslash )$ the number of $6-$cylinder connecting rods produced next week

$\backslash ($ s$\_\{4\}=1\backslash )$ if the production line is set up to produce the $4-$cylinder connecting rods; $0$ if otherwise

$\backslash ($ s$\_\{6\}=1\backslash )$ if the production line is set up to produce the $6-$cylinder connecting rods; $0$ if otherwise

$($a$)$ Using the decision variables $\backslash ($ x$\_\{4\}\backslash )$ and $\backslash ($ s$\_\{4\}\backslash ),$ write a constraint that limits next week's production of the $4-$cylinder connecting rods to either $0$ or $8,000$ units.

$($b$)$ Using the decision variables $\backslash ($ x$\_\{6\}\backslash )$ and $\backslash ($ s$\_\{6\}\backslash ),$ write a constraint that limits next week's production of the $6-$cylinder connecting rods to either $0$ or $5,000$ units.

$($c$)$ Write a third constraint that, taken with the constraints from parts $($a$)$ and $($b$),$ limits the production of connecting rods for next week.

$($d$)$ Write an objective function for minimizing the cost of production for next week.