BUS $190$ Spring $2024$ HW $7$

$1.$ Below is a sensitivity analysis for a problem you haven$$t seen before.

Variable Cells

Final Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease

$B$$2$

DVs Stock

A $1333\mathrm{.}333333031\mathrm{.}53$

$C$$2$

DVs Stock

B $400002\mathrm{.}251$E$+300\mathrm{.}75$

Constraints

Final Shadow Constraint Allowable Allowable

Cell Name Value Price R$.$H$.$ Side Increase Decrease

$D$$4$ Cost $1000000\mathrm{.}11000002000015000$

$D$$5$

Max invest

A $400000600001$E$+3020000$

$D$$6$

Max invest

B $600000\mathrm{.}05600001500020000$

$D$$7$

Min invest

A $40000025000150001$E$+30$

$D$$8$

Min invest

B $60000030000300001$E$+30$

The questions involve just the bottom sensitivity report.

a$.$ How many constraints are there? How can you tell?

b$.$ What are the right hand sides for each constraint? $($List constraint name and RHS$)$

c$.$ Which constraints are binding? What are the two ways you can tell this?

d$.$ What does $1$E$+30$ mean for excel?

$2.$ Suppose the right$-$hand side $($RHS$)$ of the Budget constraint changes from $$100,000$ to

$$110,000?$

a$.$ Does the Shadow Price change? Explain why or why not.

b$.$ If the Shadow Price does not change, what is the change in the objective function value?

c$.$ Do the values of the decision variables change? Explain why or why not.

$3.$ Suppose now the RHS of the investment in Stock B changes from $$60,000$ to $$50,000.$

a$.$ Does the Shadow Price change? Explain why or why not.

b$.$ Does the value of the objective function optimal solution change? If so$,$ calculate it$.$ If not,

explain why it does not change.

c$.$ Do the values of the decision variables change? Explain why or why not.

$4.$ Suppose now both the investment in Stock B changes from $$60,000$ to $$50,000$ and the cost

changes from $$100,000$ to $$110,000.$

a$.$ Do the shadow prices change? Use the $100\%$ rule for changes in RHS to determine if they

do$.$

b$.$ If the shadow prices do not change, can you determine the overall change in the value of

the objective function due to these changes in RHS of those constraints? If you can, then

calculate the overall change in the value of the objective function using the shadow prices.

$5.$ Put the $$Fun$$ problem into excel solver and solve it$.$ Use the binary constraint in your solver

window:

Max $7$A $+9$B $+10$C $+6$D $+8$E

S$.$t$.$

A $+$ B $+$ C $+$ D $+$ E $=3($car fit$)$

C $-$ D $=0($Chuck & Danielle$)$

B $-$ C $<=0($Ben & Chuck$)$

C $+$ E $<=1($Chuck & Emily$)$

A $+$ D $>=1($Good drivers$)$

C $-$ D $=0($Chuck & Danielle$)$

A$,$ B$,$ C$,$ D$,$ E $=0$ or $1$

$($The above differs from the slides only in that I grouped the signs of the constraints.$)$

Answer the following questions on your ANSWER REPORT:

a$.$ Who will be invited?

b$.$ How much fun will you have?

c$.$ What are the binding constraints?

d$.$ Is there any slack or surplus?

$6.$ Just to see $$ put the above problem into excel solver again $($just copy the setup onto a new excel

worksheet$).$ Rerun solver WITHOUT the binary constraint.

Are there any differences between your answers for $5$ a$-$d and what you see here?

$7.$ Another $$Just to see$$ problem: Copy the setup for the last setup of the boats problem, where

you changed the coefficients for both Catmans and Classys $($I$$m writing it below$).$ Rerun solver

WITHOUT the integer constraint. What happens here? Are there any differences between the

two? What can you attribute that to$?($ I just copied the excel setup below. You can copy from

the posted solutions for this problem to save time, then zero the DVs out and rerun solver

without the integer constraint.$)$

Speedhawk Silverbird Catman Classys

DVs $0000$

Max profit $7083631100$

Budget $60007000500090000<=420,000$

Seating $35260>=200$

No$.$ boats $11110>=50$

Goodwill $11-1-10=0$