LinearProgramming
Practice
Problems
1)A loggingcompanyhasonesawwithwhichit makesplywood.pnd
pressure-treated
wood.
Eachpieceof plywoodrequires20 minuteson the sawand useJ0 eight-footlogs.Eachpieceof
pressure-treated
woodrequires10 minuteson the sawand uses1 eight-footlog.The saw is
available
20ha day,andthe loggingcompanycanobtain160eight-foot
logsa day.The plywood
sellsfor $50 each,andthe pressure-treated
woodsellsfor $10each.Findthe maximum
solution.
tnls
3aw
Pluwood=
x
4
Pl^/ 2 0
fr'essurctruaY,eL=y
?T to
I
funhon
Obicctlr,e
-
60xr llq
J
(o,teo)= l2oo
@o,Bo)=,ooo
(,10,o) = 2ooo
The max.vylww
Logs.
4xtu -<lbo
r@
,
VrotiL,at * 2ooo
ls whenI0 st4eats
W
of plqwoo
d, ond no
lveat erL Woodis pnduu^ol.
Pteslsuro
requiretwo
2) A companymakestwo products,
televisions
and DVDplayers.The televisions
pertelevision.
is $300.The DVD
The profitpertelevision
hoursof laborand $100of materials
per DVDplayer.The profitper playeris
playerrequires
threehoursof laborand$50of materials
week
for the costof the material.lf an
The
each
company
can
afford
to
spend
$150.
$700
and/orDVDplayersshouldthe
employeeworksthirtyhoursa week,howmanytelevisions
employeemakein orderto maximizethe company'sprofit?
JV=x
DVD=y
obiecttve
fnnr#toh
-
7a0x
+l1Dy
(orto)= lsoo
(3,b):Ztoo
(+ro) = 2too
rc rnauiwrtkdr
?rofut,
bU proiluuYlA EWL
Xn'd 6 pvo,i _
ov by yrodvtctng
7 Tv's'.
Lnbour,
lx+3y(3o
\aabeYial'
l00X+ 1oqr(tuo
J
ys- $x*ro
ys-2xtf+
X2o
5 7'o
3) A companymakesdishwashers
anddryers.Thedishwashers
requirefivehoursof laborand
per
profit
per
wodh
of
materials
dishwasher.
The
dishwasher
is $400.The dryersrequire
$75
two hoursworthof laborand $75worthof materialsperdryer.The profitper dryeris $500.The
companycan affordto spend$2625eachweekfor the costof the materials.
The company
operates
for 100hoursperweek.Howmanyof eachproductshouldthe companyproducein
t+filileul
orderto maximizethe profit?
.,\,^,- r,
I 1-.4l!Upl tM*
5
75
15
7
,bi'4;twfunr,ttirw
tl00x+ 5Ny
(otgo)= l?5oo
-
=lb boo
Uo,aP)
(zoro)= fiooo
5o
Laboul'
5\ +2y.(too
lvlabnalt
X)ro
VK
7t O
V
Pro^f
iL B wavtmtutdl/ y,rEx+50
at''\lfrtt bg
- 'groduuy
bb dvtyvs.
.(-X+36
Long-term
calculator.
4) A calculator
calculator
anda graphing
companyproduces
a scientific
projections
and80 graphingcalculators
indicatean expected
demandof at least100scientific
and 170
production
no morethan200scientific
capacity,
eachday.Becauseof limitations
on
graphing
contract,
a totalof at least200
calculators
can be madedaily.To satisfya shipping
muchbe shippedeachday.
calculators
produces
a $5
calculator
lf eachscientific
calculator
soldresultsin a $2 loss,buteachgraphing
_' f'
X>,loo
'1
* "'tl !*--t
-
sc=x
3c=
!
X-<2"00
-
y<trc
Fmutvs*
Dbgrrhrve.
iia.
ll
-J ' -r - t.- ** *-*.-'iiti:)
ir
-2x +59
(foo)po): 3oa
( too,l?o)= b;o
=e
(zoo,0o)
(zoo,fto) = 4so
(lzo, bo) = 11oP
!>,to
X+y lzoo
y>_ x+200
?rofrtrcrnaxvntned"
dt ilbso by
Vrooluuvtg loo
9oownh' ft c- h, Ic'u.la hvs
on* t?.n"ry\\^*V
o*lutLadi*.
5) A goldprocessor
hastwo sourcesof goldoresourceA andsourceB. In orderto keep his
plantrunning,at leastthreetonsof oremustbe processed
eachday.OrefromsourceA costs
per
process,
ton
to
$20
and orefromsourceB costs$10 perton to process.Costsmustbe kept
to lessthan$80 per day.Moreover,FederalRegulations
requirethatthe amountof ore from
sourceB cannotexceedtwicethe amountof orefromsourceA. lf orefromsourceA yields 2 oz.
of goldper ton, and ore fromsourceB yields3 oz. of goldperton,howmanytons of ore from
bothsourcesmustbe processedeachdayto maximizethe amountof goldextractedsubjectto
the aboveconstraints?
=
Sourte
A x
Source
B=y
-
20x+lou( 8o
Fancfroin
oblacltve
2x+3y
y..<-2Xtoo
(l,z): B
(2,'+): lb
(3,0)= L
(4,o)= b
tswt(tLtmwctva*,lboz
?roiluchon
of qld"if souruA evtrads L lons
afiasourcebefirads l frons.
6) A minecompanyownstwo plantsthatproduceor.Aftercrushing,
the or is gradedintotwo
classes:high,and low-grade.
The companyhascontracted
to providea smeltingplantwith8
tonsof high-grade
and6 tonsof low-grade.
MineA produces
2 tonsof highgradeand 3 tonsof
lowgradeand hasa dailyoperating
costof $180.MineB produces
2 tonnesof highgradeand
1 tonneof lowgradeand hasa dailyoperating
costof $160. Howmanydaysperweekshould
eachmineoperateto fulfilthe plantcontractat minimalcost?
MtneA=x.
I
Filq H
2
row
3
z
I
'
HtEH
0biecfiw
fi,rnefiovr
T{f,2v}b
ffi'
"^ffi
Mtne B=Y
B
lgox+tbDy
(0.,u)=9bo
,
\
rzrr
= fll2b
(o,r)
,. -\
- / t ta
g) =bbg
(t,
k,o)=V2o
(+,o)
(",
l2bo
o;--l2bo
r"
I
L0W,
3X+Y >tb
I V>,-sxrt"
lx_.7 ,
G,i=LJbo,
p:l I
,,
Mtntmel
cxsf
of|ffiruachcd,
aryry@$a|aiiflanczt
'oPe,raVLfrrz
alau{J
'
when
pt;"; B-