Vollmer Manufacturing makes three components for sale to refrigeration companies. The components are processed on two machines: a shaper and a grinder. The times $($in minutes$)$ required on each machine are as follows:

Machine

Component Shaper Grinder

$164$

$245$

$342$

The shaper is available for $120$ hours, and the grinder is available for $110$ hours. No more than $200$ units of component $3$ can be sold, but up to $800$ units of each of the other components can be sold. In fact, the company already has orders for $600$ units of component $1$ that must be satisfied. The profit contributions for components $1,2,$ and $3$ are $$8,$ $$6,$ and $$9,$ respectively.

$($a$)$

Formulate and solve for the recommended production quantities.

component $1$

units

component $2$

units

component $3$

units

$($b$)$

What are the objective coefficient ranges for the three components? $($If there is no upper or lower limit$,$ enter NO LIMIT. Round your answers to the nearest integer.$)$

component $1$

to

units

component $2$

to

units

component $3$

to

units

Interpret these ranges for company management.

Changes in the profit coefficients within these ranges will not cause a change in the optimal number of components to produce.

Changes in the profit coefficients within these ranges will cause changes in the optimal number of components to produce.

Changes in the profit coefficients outside these ranges will not cause a change in the optimal number of components to produce.

$($c$)$

What are the right$-$hand$-$side ranges? $($Use minutes for time. If there is no upper or lower limit$,$ enter NO LIMIT. Round your answers to the nearest integer.$)$

shaper availability

to

grinder availability

to

max units of component $3$

to

max units of component $1$

to

max units of component $2$

to

existing orders for component $1$

to

Interpret these ranges for company management.

These are the ranges over which the shadow prices for the associated constraints are applicable.

These are the ranges over which the reduced costs for the associated constraints are applicable.

These are the ranges over which the shadow prices for the associated constraints are not applicable.

These are the ranges over which the reduced costs for the associated constraints are not applicable.

$($d$)$

If more time could be made available on the grinder, how much would it be worth?

Nothing, since there are $300$ minutes of slack time on the grinder at the optimal solution.

Every additional minute on the grinder decreases profits by $$1.$

Every additional minute on the grinder increases profits by $$1\mathrm{.}50.$

Every additional minute on the grinder increases profits by $$3.$

$($e$)$

If more units of component $3$ can be sold by reducing the sales price by $$4,$ should the company reduce the price?

Yes. At that price, it would be profitable to produce $700$ units of component $3.$

Yes. Every additional unit of component $3$ increases profits by $$3.$

No$.$ There is a slack of $300$ units so selling more would not improve profits.

No$.$ At that price, it would not be profitable to produce any of component $3.$