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Week8 Homework
Chapter4
1. BettyMalloy,owneroftheEagleTaverninPittsburgh,ispreparingforSuperBowlSunday,and she
mustdeterminehowmuchbeertostock.Betty
stocksthreebrandsofbeer-Yodel,Shotz,and Rainwater.The costpergallon (tothetavern owner)ofeach
brand isasfollows:
Brand
Cost/Gallon
Yodel
$1.50
Shotz
0.90
Rainwater
0.50
Thetavernhasabudgetof$2,000forbeerforSuperBowlSunday.BettysellsYodelatarateof
$3.00pergallon,Shotzat$2.50pergallon,andRainwaterat$1.75pergallon.Basedonpast footballgames, Bettyhasdeterminedthemaximumcustomerdemandtobe400gallonsofYodel,
500gallonsofshotz,and 300gallonsofRainwater. The tavernhas thecapacity
to
stock1,000 gallons ofbeer;Bettywants
to stockup
completely.
Bettywants to
determine thenumberofgallons
ofeach
brand
of
beerto ordersoasto maximize
profit.
a. Formulatealinearprogramming model for thisproblem. b.
Solve
the modelbyusingthe
computer.
2. Asresultofarecentlypassedbill,acongressman’sdistricthasbeenallocated$3millionfor
programsandprojects.Itisup tothecongressmantodecide how
to distributethemoney. The congressmanhasdecidetoallocate
the
money tofourongoing programsbecauseof their importancetohisdistrict-ajobtrainingprogram,aparksproject,asanitationproject,anda mobilelibrary.However,thecongressmanwantsto
distributethemoney
inamannerthatwill pleasethemostvoters,or, inotherwords,gainhim
the
mostvotesin theupcomingelection.His
staff’sestimatesofthenumberofvotesgainedperdollarspent forthevarious programsareas
follows.
Program
Votes/Dollar
Job training
0.03
Parks
0.08
Sanitation
0.05
Mobilelibrary
0.03
Inorderalsoto satisfyseverallocalinfluentialcitizenswhofinancedhiselection, heisobligated to observethefollowing guidelines:
· None
ofthe
programs can
receivemore than 30%of the
totalallocation
· Theamountallocatedtoparkscannotexceedthetotalallocatedtoboththesanitation
projectandthe
mobile library.
· Theamountallocatedtojobtrainingmustatleastequaltheamountspentonthesanitation project.
Anymoney notspentinthedistrictwillbereturnedtothegovernment;therefore,thecongressman
wantstospenditall.Theecongressmanwantstoknowtheamounttoallocatetoeachprogram
to maximize
his votes.
a. Formulatea linearprogramming
model for thisproblem.
b. Solve
the modelbyusingthe
computer.
3. AnnaBroderick is
the dietician
for the
StateUniversity
football team,andshe
isattempting to
determineanutritiouslunchmenufortheteam.Shehassetthefollowingnutritionalguidelines
for
eachlunch
serving:
· Between1,300 and
2,100
calories
· At least4mgof iron
· At least15
but
no more than 55gof fat
· At least30gofprotein
· At least60gofcarbohydrates
· No
more
than 35mgofcholesterol
Sheselectsthemenufromsevenbasicfooditems,asfollows,withthenutritionalcontributions
perpound and the
cost
as given:
Calories
(perlb.)
Iron
(mg/lb.)
Protein
(g/lb.)
Carbo-
hydrates
(g/lb.)
Fat
(g/lb.)
Cholesterol
(mg/lb)
Cost
($/lb.)
Chicken
500
4.2
17
0
30
180
0.85
Fish
480
3.1
85
0
5
90
3.35
Ground beef
840
0.25
82
0
75
350
2.45
Driedbeans
590
3.2
10
30
3
0
0.85
Lettuce
40
0.4
6
0
0
0
0.70
Potatoes
450
2.25
10
70
0
0
0.45
Milk(2%)
220
0.2
16
22
10
20
0.82
Thedieticianwantstoselectamenutomeetthenutritionalguidelineswhileminimizingthetotal
cost
perserving.
a. Formulatea
linearprogramming model for thisproblemand solve.
4. Dr. Maureen
Becker, the head administrator at Jefferson
County Regional Hospital, must determineaschedulefornursestomakesurethereareenoughofthemondutythroughoutthe
day.
Duringtheday,thedemandfornursesvaries. Maureenhasbrokenthedayintotwelve2- hourperiods.The slowesttimeoftheday encompassesthethreeperiods
from
12:00A.M. to6:00
A.M.,whichbeginningatmidnight;requireaminimumof30,20,and40nurses,respectively. Thedemandfornursessteadilyincreasesduringthenextfourdaytimeperiods.Beginningwith
the
6:00A.M.-8:00A.M.
period,aminimumof50,60,80,and80nursesarerequiredforthese fourperiods,respectively.After2:00P.M. thedemandfornursesdecreasesduring theafternoon
andeveninghours.Forthefive2-hourperiodsbeginningat2:00P.M.andendingmidnight,70,
70,60,50,and50nursesarerequired,respectively.Anursereportsforduty
atthebeginning
of oneof the2-hourperiods andworks8
consecutive hours
(which
is
required in thenurses’ contract).
Dr. Becker wants
to
determine
a
nursing schedule
that will meet
the
hospital’s
minimumrequirement throughout thedaywhile
usingthe
minimumnumberofnurses.
a. Formulatea
linearprogramming model for thisproblem.
b. Solve the
modelbyusingthe
computer.
5.
Theproduction managerofVideotechnics Companyis attemptingto
determinethe upcoming5-month
production schedule forvideo recorders. Past productionrecords indicate that
2,000 recorders
can beproduced permonth.
An additional 600 recorders can beproduced
monthlyonan
overtime basis. Unit cost is
$10 for recorders produced
duringregularworkinghours and
$15 forthose produced
on an overtime basis. Contracted
sales per month are as follows:
Month
Contracted Sales(units)
1
1200
2
2100
3
2400
4
3000
5
4000
Inventorycarryingcostsare$2
per recorder per month. Themanager
does
not want any inventorycarried
over past the fifth month. Themanagerwants
to know themonthly production
that willminimizetotal
production and inventorycosts.
a. Formulatealinear programmingmodel
for
this problem. b. Solve
the model byusingthe
computer.
