lpp_graphical1

8/9/2019 LPP_GRAPHICAL1

1/32

COPYRIGHT2006byLAVON

B.PAGE

Solvinglinear

programmin

g

problemsusingthe

graphicalm

ethod


2/32


B.PAGE

Example-designingadiet

Adietit

ianwantstodes

ignabreakfastmenufor

certain

hospitalpatients.

Themenuist

oinclude

twoitemsAa

ndB.

Su

pposethateach

ounceof

Ap

rovides2unitsofvitaminCand2unitsofiron

andeachounceofB

provides1unitofvitaminC

and2u

nitsofiron.

SupposethecostofAi

s

4/oun

ceandthecostofBi

s3/ounce.

Ifthe

breakfa

stmenumustp

rovideatleast8

unitsof

vitamin

Cand10units

ofiron,

howman

younces

ofeach

items

houldbe

providedinorde

rtomeet

theironandvitaminCrequirementsfor

theleast

cost?

Whatwillthisbre

akfastcost?


3/32


B.PAGE

x=#oz.ofA

y=#oz.ofB

vit.C:2x+y8

iron:

2x+2y1

0

x0,y0Cost=

C=4x+3y


4/32


B.PAGE

vit.

c

vit.

C:

2x+y8

iron:

2x+2y1

0

x

0,y0Cost=

C=4x+3y

x=#oz.ofA

y=#oz.ofB


5/32


B.PAGE

iron

vit.

C:

2x+y8

iron:

2x+2y1

0

x0,y0

vit.

c

Cost=

C=4x+3y

x=#oz.ofA

y=#oz.ofB


6/32


B.PAGE

iron

vit.

C:

2x+y8

iron:

2x+2y1

0

x0,y0

vit.

c

Cost=

C=4x+3y

x=#oz.ofA

y=#oz.ofB


7/32


B.PAGE

iron

vit.

C:

2x+y8

iron:

2x+2y1

0

x0,y0Cost=

C=4x+3y

The3b

luelinesare

48=4x+3y

36=4x+3y

24=4x+3y

vit.

cx=#oz.ofA

y=#oz.ofB


8/32


B.PAGE

iron

vit.

C:

2x+y8

iron:

2x+2y1

0

x0,y0Cost=

C=4x+3y

The3b

luelinesare

48=4x+3y

36=4x+3y

24=4x+3y

Thecostwillbeminimiz

edifthe

strategy

followedisthe

one

correspo

ndingtothisco

rner

point

vit.

cx=#oz.ofA

y=#oz.ofB


9/32


B.PAGE

iron

2x+y=8

2x+2y=10

vit.

C:

2x+y8

iron:

2x+2y1

0

x0,y0

vit.cx

=#oz.ofA

y=#oz.ofB


10/32


B.PAGE

vit.c

iron

Solution:x=3,y=2

C=4x+3y=

18

2x+y=8

2x+2y=10

vit.

C:

2x+y8

iron:

2x+2y1

0

x0,y0

x=#oz.ofA

y=#oz.ofB


11/32


B.PAGE

iron

co

rnerpt.

C=4

x+3y

(0,8

)

24ce

nts

(5,0

)

20ce

nts

(3,2

)

18ce

nts

vit.cx

=#oz.ofA

y=#oz.ofB


12/32


B.PAGE

x=#daysfac

toryAi

soperated

y=#daysfac

toryB

isoperat

ed

Example-bicyclefactories

Asma

llb

usinessmake

s3-speedand1

0-speed

bicyclesattwodifferentfactories.

Fac

toryA

produ

ces163-speedand2010-speed

bikesin

onedaywhilefactoryBp

roduces123

-speed

and2010-speedbikesdaily.

Itcosts

$1000

/daytooperate

factoryAa

nd$800/day

toope

ratefactoryB.

Anorderfor96

3-speed

bikes

and14010-spee

dbikeshasjust

arrived.

Howm

anydaysshouldeachfactorybe

operatedinordertofillt

hisorderata

minimum

cost?

Whatisthemin

imumc

ost?


13/32


B.PAGE

x=#daysfac

toryAi

soperated

y=#daysfac

toryB

isoperated


14/32


B.PAGE

3-

speedconstraint

:16x+12y9

6

x=#days

factoryAi

sop

erated

y=#days

factoryB

isoperated


15/32


B.PAGE

3-speedconstraint:16x+12

y9

6


0y1

40

x0,y0

x=#

daysfactoryA

isoperated

y=#

daysfactoryB

isoperated


16/32


B.PAGE

3-speedconstraint:16x+12y

9

6


y1

40

x0,y0

Minimize:C=

1000x+800y

x=#

daysfactoryAisoperated

y=#

daysfactoryB

isoperated


17/32


18/32


B.PAGE

3-speedconst

raint:16x+12y

9

6


y1

40

x0,y0

Minimize:C=

1000x+800y

10-speed

x=#daysfactoryAisoperated

y=#daysfactoryB

isoperated

3-speed


19/32


B.PAGE

3-speedconst

raint:16x+12y

9

6


y1

40

x0,y0

Minimize:C=

1000x+800y

10-speed

x=#daysfactoryAisoperated

y=#daysfactoryB

isoperated

3-speed


20/32


B.PAGE


9

6


y1

40

x0,y0

Minimize:C=

1000x+800y

cornerpts

C=1000x+800y

x=#


y=#

daysfactoryB

isoperated

10-speed

3-speed


21/32


B.PAGE


9

6


y1

40

x0,y0

Minimize:C=

1000x+800y

3-speedc

ornerpts

C=1000x+800y

(0

,8)

$6400

x=#


y=#

daysfactoryB

isoperated

10-speed


22/32


B.PAGE


9

6


y1

40

x0,y0

Minimize:C=

1000x+800y

cornerpts

C=1000x+800y

(0

,8)

$6400

(7

,0)

$7000

3-speedx

=#


y=#

daysfactoryB

isoperated

10-speed


23/32


B.PAGE

3-speedconst

raint:16x+12y

9

6


y1

40

x0,y0

Minimize:C=

1000x+800y

cornerpts

C=1000x+800y

(0,8

)

$6400

(7,0

)

$7000

(3,4

)

$6200

3-speedx

=#daysfactoryAisoperated

y=#daysfactoryB

isoperated

10-speed


24/32


B.PAGE

MichiganPolarProductsmakesdownhillandcross-

countryskis.

Apairof

downhillskisreq

uires2

man-hoursforcutting,

1man-hourforshapingand

3man-hoursforfinishingwhileapairo

fcross-

countryskisrequires2

man-hoursforc

utting,

2

man-hoursforshaping

and1man-hourfor

finishin

g.

Eachdaythe

companyhasavailable140

man-hoursforcutting,

120man-hours

forshaping

and15

0man-hoursfor

finishing.

How

manypairs

ofeach

typeofskishouldthecompany

manufactureeachday

inordertomaximizeprofitif

apairofdownhillskisy

ieldsaprofitof

$10anda

pairof

cross-countryskisyieldsaprofit

of$8?

Example-skimanufac

turing


25/32


B.PAGE

x=#

pairsofdownh

illskis

y=#

pairsofcrossc

ountryskis

cutting:

2x+2y1

40

shaping:

x+2y1

20

finishing:

3x+y1

50

x0,y

0

P=10x+8y


26/32


B.PAGE

cutting:

2x+2y

140

shaping:

x+2y1

20

finishing:

3x+y1

50

x0,y

0

P=

10x+8y

cutting

x=#

pairsofdownhillskis

y=#

pairsofcrossc

ountryskis


27/32


B.PAGE

cutting:

2x+2y

140

shaping:

x+2y1

20

finishing:

3x+y1

50

x0,y

0

P=

10x+8y

cuttings

hapin

g

x=#

pairsofdownhillskis

y=#

pairsofcrossc

ountryskis


28/32


B.PAGE

cutting:

2x+2y

140

shaping:

x+2y1

20

finishing:

3x+y1

50

x0,y

0

P=

10x+8y

cuttings

hapin

g

finishing

x=#

pairsofdownhillskis

y=#

pairsofcrossc

ountryskis


29/32


B.PAGE

cutting:

2x+2y

140

shaping:

x+2y1

20

finishing:

3x+y1

50

x0,y

0

P=

10x+8y

cuttings

hapin

g

finishing

x=#

pairsofdownhillskis

y=#

pairsofcrossc

ountryskis


30/32


B.PAGE

cutting:

2x+2y1

40

shaping:

x+2y1

20

finishing:

3x+y1

50

x0,y

0

P=10x+8y

cuttings

hapin

g

finishing

x=#

pairsofdownh

illskis

y=#

pairsofcrossc

ountryskis


31/32


B.PAGE

cutting:

2x+2y

140

shaping:

x+2y1

20

finishing:

3x+y1

50

x0,y

0

P=

10x+8y

cuttings

hapin

g

finishing

cornersP

=10x+8y

(0,0

)

(0,6

0)

(20,5

0)

(40,3

0)

(50,0

)

$0

$480

$600

$640

$500

x=#

pairsofdownhillskis

y=#

pairsofcrossc

ountryskis


32/32


B.PAGE

cutting:

2x+2y1

40

shaping:

x+2y1

20

finishing:

3x+y1

50

x0,y

0

P=10x+8y

cutting

finishing

cornersP

=10x+8y

(0,0

)

(0,6

0)

(20,5

0)

(40,3

0)

(50,0

)

$0

$480

$600

$640

$500

x=#

pairsofdownh

illskis

y=#

pairsofcrossc

ountryskis

Make40pairsof

downhillskis

and

30pairsofcross

countryskisf

ora

profitof$640

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