Suppose a certain manufacturing company produces connecting rods for $4-$ and $6-$cylinder automobile engines using the same production line. The cost required to set up the production line to produce the $4-$cylinder connecting rods is $$2,600,$ and the cost required to set up the production line for the $6-$cylinder connecting rods is $$3,900.$ Manufacturing costs are $$16$ for each $4-$cylinder connecting rod and $$19$ for each $6-$cylinder connecting rod. Hawkins makes a decision at the end of each week as to which product will be manufactured the following week. If a production changeover is necessary from one week to the next, the weekend is used to reconfigure the production line. Once the line has been set up$,$ the weekly production capacities are $5,0006-$cylinder connecting rods and $7,0004-$cylinder connecting rods. Let

x$4=$ the number of $4-$cylinder connecting rods produced next week

x$6=$ the number of $6-$cylinder connecting rods produced next week

s$4=1$ if the production line is set up to produce the $4-$cylinder connecting rods; $0$ if otherwise

s$6=1$ if the production line is set up to produce the $6-$cylinder connecting rods; $0$ if otherwise

$($a$)$

Using the decision variables x$4$ and s$4,$ write a constraint that limits next week's production of the $4-$cylinder connecting rods to either $0$ or $7,000$ units.

$($b$)$

Using the decision variables x$6$ and s$6,$ 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.