$3.$ Graphical Solution and Range of Optimality for Objective Function Coefficients.

Consider the linear program below and answer the following.

Min $8$X $+12$Y

s$.$t$.$

$1$X$+3$Y $>=9$

$2$X$+2$Y $>=10$

$6$X $+2$Y $>=18$

A$,$ B $>=0$

$4.$ Graphical Solution and Range of Feasibility for Constraints. Consider the linear

program in Problem $3.$ The value of the optimal solution is $48.$ Suppose that the righthand

side for constraint $1$ is increased from $9$ to $10.$

a$.$ Use the graphical solution procedure to find the new optimal solution.

b$.$ Use the solution to part $($a$)$ to determine the dual value for constraint $1.$

c$.$ The computer solution for the linear program in Problem $3$ provides the following

right$-$hand$-$side range information:

Constraint RHS Value Allowable Increase. Allowable Decrease

$19\mathrm{.}000002\mathrm{.}000004\mathrm{.}00000$

$210\mathrm{.}000008\mathrm{.}000001\mathrm{.}00000$

$318\mathrm{.}000004\mathrm{.}00000$ Infinite

What does the right$-$hand$-$side range information for constraint $1$ tell you about the

dual value for constraint $1?$

d$.$ The dual value for constraint $2$ is $3.$ Using this dual value and the right$-$hand$-$side

range information in part $($c$),$ what conclusion can be drawn about the effect of

changes to the right$-$hand side of constraint $2?$

$6.$ Number of Baseball Gloves to Produce $($revisited$).$ Refer to the computer solution

of the Kelson Sporting Equipment problem in Figure $3\mathrm{.}13($see Problem $5).$

a$.$ Determine the objective coefficient ranges.

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

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

Variable Value Reduced Cost

$----------------------------------------------$

R $500\mathrm{.}000000\mathrm{.}00000$

C $150\mathrm{.}000000\mathrm{.}00000$

Constraint Slack$/$Surplus Dual Value

$----------------------------------------------$

$1175\mathrm{.}000000\mathrm{.}00000$

$20\mathrm{.}000003\mathrm{.}00000$

$30\mathrm{.}0000028\mathrm{.}00000$

Objective Allowable Allowable

Variable Coefficient Increase Decrease

$------------------------------------------$

R $5\mathrm{.}000007\mathrm{.}000001\mathrm{.}00000$

C $8\mathrm{.}000002\mathrm{.}000004\mathrm{.}66667$

RHS Allowable Allowable

Constraint Value Increase Decrease

$------------------------------------------$

$1900\mathrm{.}00000$ Infinite $175\mathrm{.}00000$

$2300\mathrm{.}00000100\mathrm{.}00000166\mathrm{.}66667$

$3100\mathrm{.}0000035\mathrm{.}0000025\mathrm{.}00000$

c$.$ Interpret the right$-$hand$-$side ranges.

d$.$ How much will the value of the optimal solution improve if $20$ extra hours of packaging

and shipping time are made available?

$15.$ Producing Injection$-$Molded Parts $($revisited$).$ Refer to the computer solution to

Problem $14$ in Figure $3\mathrm{.}18$

a$.$ Interpret the ranges of optimality for the objective function coefficients.

b$.$ Suppose that the manufacturing cost increases to $$11\mathrm{.}20$ per case for model A$.$

What is the new optimal solution?

c$.$ Suppose that the manufacturing cost increases to $$11\mathrm{.}20$ per case for model A and

the manufacturing cost for model B decreases to $$5$ per unit. Would the optimal

solution change?