Problem $3-06$

Kelson Sporting Equipment, Inc., makes two different types of baseball gloves: a regular model and a catcher's model. The firm has $900$

hours of production time available in its cutting and sewing department, $300$ hours available in its finishing department, and $100$ hours

available in its packaging and shipping department. The production time requirements and the profit contribution per glove are given in the

following table:

Letting

$R=$ number $of$ regular gloves

$C=$ number $of$ catcher's mitts

leads to the following formulation:

Max $5R+8C$

$s.t.$

$R+\frac{3}{2}C900$ Cutting and sewing

$\frac{1}{2^{R+\frac{1}{3}}}C300$ Finishing

$\frac{1}{8}R+\frac{1}{4}C100$ Packaging and shipping

$R,C0,$

The sensitivity report is shown in figure below.

Optimal Objective Value $=,3700\mathrm{.}00000$

Variable Value Reduced Cost

R $5000$

C $1500$

Constraint Slack$/$surplus Dual Value

$11750$

$203$

$3028$

Variable Objective coefficient increase decrease

R $571$

C $824\mathrm{.}66666$

Constraint RHS value increase decrease

$1900$ Infinite $175$

$2300100166\mathrm{.}6666$

$310035$

a$.$ Determine the objective coefficients ranges.

Round your answers to two decimal places.

b$.$ Interpret the ranges in part $($a$).$

As long as the profit per regular glove is between $$4$ and $$12,$ the optimal solution of

$500$ regular and $150$ catcher's gloves will not change.

Or as long as the profit per catcher's mitt is between $$3\mathrm{.}33$ and $$10,$ the optimal solution of

$500$ regular and $150$ catcher's $$ gloves will not change.

The assumption is that only one variable is $$ changed at the same time.

c$.$ Interpret the right$-$hand$-$side ranges. If there is no lower or upper limit$,$ then enter the text $"$NA$"$ as your answer. Round your answers to

two decimal places. Enter $"0"$ if your answer is zero.

As long as the number of hours available for cutting and sewing $($Constraint $1)$ are $7725$ or above $$ the change in the optimal

value of the solution per unit increase in the right$-$hand side of the constraint is

As long as the number of hours available for finishing $($Constraint $2)$ are Between $133\mathrm{.}33$ and $400,$ the change in the optimal

value of the solution per unit increase in the right$-$hand side of the constraint is

As long as the number of hours available for packaging and shipping $($Constraint $3)$ are Between $75$ and $135,$ the change in the

optimal value of the solution per unit increase in the right$-$hand side of the constraint is

d$.$ How much will the value of the optimal solution improve if $20$ extra hours of packaging and shipping time are made available?

Amount: $