Jones Furniture Company Produces Beds and Desks for College Students $$\begin{array} { l } X _ { 1 } = \text { Number of Beds to produce } \\X _ { \mathbf { 2 } } = \text { Number of Desks to produce }\end{array}$$

Essay

Jones Furniture Company produces beds and desks for college students. The production process requires carpentry and varnishing. Each bed requires 6 hours of carpentry and 4 hour of varnishing. Each desk requires 4 hours of carpentry and 8 hours of varnishing. There are 36 hours of carpentry time and 40 hours of varnishing time available. Beds generate $30 of profit and desks generate $40 of profit. Demand for desks is limited so at most 8 will be produced.
Suppose the company can purchase more varnishing time for $3.00, should it be purchased and how much can be bought before the value of the additional time is uncertain? Base your response on the following Risk Solver Platform (RSP) sensitivity output.
Laet
$$\begin{array} { l } X _ { 1 } = \text { Number of Beds to produce } \\X _ { \mathbf { 2 } } = \text { Number of Desks to produce }\end{array}$$ The LP model for the problem is
$\text { MAX: } \quad 30 \mathrm{X}_{1}+40 \mathrm{X}_{2}$
$\begin{array}{l}\text { Subject to: }\\\begin{array} { l } 6 X _ { 1 } + 4 X _ { 2 } \leq 36 \text { (carpentry) } \\4 X _ { 1 } + \mathbf { 8 } X _ { 2 } \leq 40 \text { (varnishing } \\X _ { 2 } \leq \mathbf { 8 } \left( \text { demand for } X _ { 2 } \right) \\X _ { 1 } , X _ { 2 } \geq 0\end{array}\end{array}$ $\text { Changing Cells}$
$\begin{array}{llrrrrr}&&\text { Final } & \text { Reduced } & \text { Objective } & \text { Allowable } & \text { Allowable }\\\text { Cell } & \text { Name } & \text { Value } & \text { Cost } & \text { Coefficient } & \text { Increase } & \text { Decrease } \\\hline \$ \mathrm{~B} \$ 4 & \text { Number to make: Beds } & 4 & 0 & 30 & 30 & 10 \\\$ \mathrm{C} \$ 4 & \text { Number to make: Desks } & 3 & 0 & 40 & 20 & 20\end{array}$

$\text { Constraints }$
$\begin{array}{llrrrrr}&&\text { Final } & \text {Shadow } & \text {Constraint } & \text {Allowable } & \text {Allowable }\\\text { Cell } & \text { Name } & \text { Value } & \text { Price } & \text { R.H. Side } & \text { Increase } & \text { Decrease } \\ \hline\$ \mathrm{D} \$ 8 & \text { Carpentry Used } & 36 & 2.5 & 36 & 24 & 16 \\\$ \mathrm{D} \$ 9 & \text { Varnishing Used } & 40 & 3.75 & 40 & 26.67 & 16 \\\$ \mathrm{D} \$ 10 & \text { Desk demand Used } & 3 & 0 & 8 & 1 \mathrm{E}+30 & 5\end{array}$