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 $$2,100,$ and the cost required to set up the production line for the $6-$cylinder connecting rods is $$3,500.$

Manufacturing costs are $$15$ for each $4-$cylinder connecting rod and $$18$ 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 $8,0004-$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 $8,000$ 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.

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