Problem $3-23$

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

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 $1000$ 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 $\backslash (\backslash$$ $8\backslash ),\backslash (\backslash$$ $6\backslash ),$ and $\backslash (\backslash$$ $9\backslash ),$ respectively.

a$.$ Formulate and solve for the recommended production quantities. If the constant is $"1"$ it must be entered in the box.

Let $\backslash ($ C$\_\{1\}=\backslash )$ units of component $1$ manufactured

$\backslash ($ C$\_\{2\}=\backslash )$ units of component $2$ manufactured

$\backslash ($ C$\_\{3\}=\backslash )$ units of component $3$ manufactured The optimal solution is

Profit $\backslash (=\backslash$$ $\backslash )$

b$.$ What are the objective coefficient ranges for the three components? If there is no lower or upper limit$,$ then enter the text $"$NA$"$ as your answer. Enter $"0"$ if your answer is zero.

Interpret the above ranges for company management.

As long as the profit contribution of component $1$ is less than $\backslash (\backslash$$ $\backslash )$ then the optimal solution will not change. As long as the profit contribution of component $2$ is between $\backslash (\backslash$$ $\backslash )$ and $\backslash$$ then the optimal solution will not change. As long as the profit contribution of component $3$ is more than $\backslash$$ then the optimal solution will not change. The assumption is that only variable$($s$)$ will change at a time.

c$.$ What are the right$-$hand$-$side ranges? If there is no lower or upper limit$,$ then enter the text $"$NA$"$ as your answer. If required, round your answers to two decimal places. Interpret the above ranges for company management.

The right$-$hand$-$side ranges are the ranges over which the $?$ for the associated constraints are applicable.

d$.$ If more time could be made available on the grinder, how much would it be worth? Enter $"0"$ if your answer is zero.

$\backslash$$ $,$ since the dual value is on the grinder at the optimal solution.

e$.$ If more units of component $3$ can be sold by reducing the sales price by $\backslash (\backslash$$ $4\backslash ),$ should the company reduce the price?

$?,$ since at that price it$.$ be profitable to produce any of component $3.$