operations management module b

7/26/2019 Operations Management Module B

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2008 Prentice Hall, Inc. B 1

OperationsManagementModule B Module B Linear ProgrammingLinear Programming

PowerPoint presentation to accompanyPowerPoint presentation to accompany

Heizer/RenderHeizer/Render

Principles of Operations Management, 7ePrinciples of Operations Management, 7e

Operations Management, eOperations Management, e


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OutlineOutline

Re!uirements of a LinearRe!uirements of a LinearProgramming Pro"lemProgramming Pro"lem

#ormulating Linear Programming#ormulating Linear ProgrammingPro"lemsPro"lems

$%ader &lectronics &'ample$%ader &lectronics &'ample


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Outline (ontinuedOutline (ontinued

)rap%ical $olution to a Linear)rap%ical $olution to a LinearProgramming Pro"lemProgramming Pro"lem

)rap%ical Representation of)rap%ical Representation of(onstraints(onstraints

*so+Profit Line $olution Met%od*so+Profit Line $olution Met%od

(orner+Point $olution Met%od(orner+Point $olution Met%od


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Outline (ontinuedOutline (ontinued

Linear Programming -pplicationsLinear Programming -pplications

Production+Mi' &'ampleProduction+Mi' &'ample

0iet Pro"lem &'ample0iet Pro"lem &'ample

La"or $c%eduling &'ampleLa"or $c%eduling &'ample

1%e $imple' Met%od of LP1%e $imple' Met%od of LP


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Learning O"ectiesLearning O"ecties

2%en you complete t%is module you2%en you complete t%is module yous%ould "e a"le to3s%ould "e a"le to3

4545 #ormulate linear programming#ormulate linear programmingmodels, including an o"ectiemodels, including an o"ectiefunction and constraintsfunction and constraints

6565 )rap%ically sole an LP pro"lem wit%)rap%ically sole an LP pro"lem wit%

t%e iso+profit line met%odt%e iso+profit line met%od

55 )rap%ically sole an LP pro"lem wit%)rap%ically sole an LP pro"lem wit%t%e corner+point met%odt%e corner+point met%od


7/47 2008 Prentice Hall, Inc. B 7

Learning O"ectiesLearning O"ecties

2%en you complete t%is module you2%en you complete t%is module yous%ould "e a"le to3s%ould "e a"le to3

8585 *nterpret sensitiity analysis and*nterpret sensitiity analysis ands%adow pricess%adow prices

9595 (onstruct and sole a minimization(onstruct and sole a minimizationpro"lempro"lem

:5:5 #ormulate production+mi', diet, and#ormulate production+mi', diet, andla"or sc%eduling pro"lemsla"or sc%eduling pro"lems



Linear ProgrammingLinear Programming

- mat%ematical tec%ni!ue to- mat%ematical tec%ni!ue to

%elp plan and ma;e decisions%elp plan and ma;e decisionsrelatie to t%e trade+offsrelatie to t%e trade+offsnecessary to allocate resourcesnecessary to allocate resources

2ill find t%e minimum or2ill find t%e minimum orma'imum alue of t%e o"ectiema'imum alue of t%e o"ectie

)uarantees t%e optimal solution)uarantees t%e optimal solutionto t%e model formulatedto t%e model formulated


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LP -pplicationsLP -pplications

8585 $electing t%e product mi' in a factory$electing t%e product mi' in a factoryto ma;e "est use of mac%ine+ andto ma;e "est use of mac%ine+ andla"or+%ours aaila"le w%ile ma'imizingla"or+%ours aaila"le w%ile ma'imizing

t%e firm




7575 0eeloping a production sc%edule t%at0eeloping a production sc%edule t%atwill satisfy future demands for a firm



Re!uirements of anRe!uirements of an

LP Pro"lemLP Pro"lem4545 LP pro"lems see; to ma'imize orLP pro"lems see; to ma'imize or

minimize some !uantity >usuallyminimize some !uantity >usually

profit or cost? e'pressed as anprofit or cost? e'pressed as ano"ectie functiono"ectie function

6565 1%e presence of restrictions, or1%e presence of restrictions, or

constraints, limits t%e degree toconstraints, limits t%e degree tow%ic% we can pursue ourw%ic% we can pursue ouro"ectieo"ectie



Re!uirements of anRe!uirements of an

LP Pro"lemLP Pro"lem55 1%ere must "e alternatie courses1%ere must "e alternatie courses

of action to c%oose fromof action to c%oose from

8585 1%e o"ectie and constraints in1%e o"ectie and constraints inlinear programming pro"lemslinear programming pro"lemsmust "e e'pressed in terms ofmust "e e'pressed in terms of

linear e!uations or ine!ualitieslinear e!uations or ine!ualities



#ormulating LP Pro"lems#ormulating LP Pro"lems

1%e product+mi' pro"lem at $%ader &lectronics1%e product+mi' pro"lem at $%ader &lectronics

1wo products1wo products4545 $%ader @+pod, a porta"le music$%ader @+pod, a porta"le music

playerplayer

6565 $%ader BlueBerry, an internet+$%ader BlueBerry, an internet+

connected color telep%oneconnected color telep%one

0etermine t%e mi' of products t%at will0etermine t%e mi' of products t%at willproduce t%e ma'imum profitproduce t%e ma'imum profit




@+pods@+pods BlueBerrysBlueBerrys -aila"le Hours-aila"le Hours

0epartment0epartment ((@@11)) ((@@22)) 1%is 2ee;1%is 2ee;

Hours Re!uiredHours Re!uiredto Produce 4 Anitto Produce 4 Anit

&lectronic&lectronic 44 33 240240

-ssem"ly-ssem"ly 22 11 100100

Profit per unitProfit per unit

$7$7

$5$5

0ecision .aria"les30ecision .aria"les3

@@11 num"er of @+pods to "e produced num"er of @+pods to "e produced

@@22

num"er of BlueBerrys to "e produced num"er of BlueBerrys to "e produced

Table B.1Table B.1




O"ectie #unction3O"ectie #unction3

Ma'imize Profit Ma'imize Profit $7$7@@11CC $5$5@@22

1%ere are t%ree types of constraints

Apper limits w%ere t%e amount used is Dt%e amount of a resource

Lower limits w%ere t%e amount used is Et%e amount of t%e resource

&!ualities w%ere t%e amount used is t%e amount of t%e resource




$econd (onstraint3$econd (onstraint3

22@@11CC 11@@22 100 100 >%ours of assem"ly time?>%ours of assem"ly time?

-ssem"ly-ssem"lytime aaila"letime aaila"le

-ssem"ly-ssem"lytime usedtime used

is Dis D

#irst (onstraint3#irst (onstraint3

44@@11CC 33@@22 240 240 >%ours of electronic time?>%ours of electronic time?

&lectronic&lectronic

time aaila"letime aaila"le

&lectronic&lectronic

time usedtime used

is Dis D


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)rap%ical $olution)rap%ical $olution

4FF

=F=F

:F:F

8F8F

6F6F

G G G G G G G G G G G

FF 6F6F 8F8F :F:F =F=F 4FF4FF

Hum"

erofBlueBerry

s

Hum"

erofBlueBerrys

um"er of @+podsum"er of @+pods

@@11

@@22

-ssem"ly >constraint B?-ssem"ly >constraint B?

&lectronics >constraint -?&lectronics >constraint -?#easi"leregion

Figure B.3Figure B.3


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4FF

=F=F

:F:F

8F8F

6F6F



Hum"

erof2atc%1.

s

Hum"

erof2atc%1.

s


@@11

@@22

-ssem"ly >constraint B?-ssem"ly >constraint B?

&lectronics >constraint -?&lectronics >constraint -?#easi"leregion


*so+Profit Line $olution Met%od

(%oose a possi"le alue for t%eo"ectie function

$210 7@1C 5@2

$ole for t%e a'is intercepts of t%e functionand plot t%e line

@2= 42 @1= 30


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4FF

=F=F

:F:F

8F8F

6F6F



Hum"

erofBlueBerrys

Hum"

erofBlueBerrys


@@11

@@22


(0, 42)

(30, 0)(30, 0)

$210 = $7$210 = $7@@11+ $5+ $5@@22


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4FF

=F=F

:F:F

8F8F

6F6F



Hum"

erofBlueBeryys

Hum"

erofBlueBeryys


@@11

@@22


$210 = $7$210 = $7@@11+ $5+ $5@@22

$350 = $7$350 = $7@@11+ $5+ $5@@22

$420 = $7$420 = $7@@11+ $5+ $5@@22

$280 = $7$280 = $7@@11+ $5+ $5@@22


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(orner+Point Met%od(orner+Point Met%od 1%e optimal alue will always "e at a

corner point

#ind t%e o"ectie function alue at eac%

corner point and c%oose t%e one wit% t%e%ig%est profit

Point 1 (@1= 0,@2= 0) Profit $7(0) + $5(0) = $0

Point 2 (@1= 0,@2= 80) Profit $7(0) + $5(80) = $400

Point 4 (@1= 50,@2= 0) Profit $7(50) + $5(0) = $350


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(orner+Point Met%od(orner+Point Met%od 1%e optimal alue will always "e at a

corner point

#ind t%e o"ectie function alue at eac%

corner point and c%oose t%e one wit% t%e%ig%est profit

Point 1 (@1= 0,@2= 0) Profit $7(0) + $5(0) = $0

Point 2 (@1= 0,@2= 80) Profit $7(0) + $5(80) = $400

Point 4 (@1= 50,@2= 0) Profit $7(50) + $5(0) = $350

$ole for t%e intersection of two constraints

2@1C 1@

2 100 >assem"ly time?

4@1C 3@2 240 >electronics time?

4@1 C 3@2 = 240

! 4@1 +2@2 =!200

+ 1@2 = 40

4@1 C 3(40) = 240

4@1 C 120 = 240

@1 = 30


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$ensitiity Report$ensitiity Report

"r#gra B.1"r#gra B.1


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$oling Minimization$oling Minimization

Pro"lemsPro"lems #ormulated and soled in muc% t%e#ormulated and soled in muc% t%e

same way as ma'imizationsame way as ma'imization

pro"lemspro"lems *n t%e grap%ical approac% an iso+*n t%e grap%ical approac% an iso+

cost line is usedcost line is used

1%e o"ectie is to moe t%e iso+1%e o"ectie is to moe t%e iso+cost line inwards until it reac%es t%ecost line inwards until it reac%es t%elowest cost corner pointlowest cost corner point


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Minimization &'ampleMinimization &'ample

@@44 num"er of tons of "lac;+and+w%ite picturenum"er of tons of "lac;+and+w%ite picture

c%emical producedc%emical produced

@@66 num"er of tons of color picture c%emicalnum"er of tons of color picture c%emical

producedproduced

Minimize total costMinimize total cost == 2,5002,500@@11 ++ 3,0003,000@@22

%ub&e' #%ub&e' #

11 * 30* 30 tons of "lac;+and+w%ite ctons of "lac;+and+w%ite c

@@22 * 20* 20 tons of color c%emicaltons of color c%emical

@@11C @C @22 * 60* 60 tons totaltons total

@@

11,,@@

22 * $0* $0

nonnegatiity re!uiremennonnegatiity re!uiremen


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Minimization &'ampleMinimization &'ample

1otal cost at a1otal cost at a == 2,5002,500@@11 ++ 3,0003,000@@22== 2,500 (40)2,500 (40) ++ 3,000(20)3,000(20)

== $160,000$160,000

1otal cost at "1otal cost at " == 2,5002,500@@11 ++ 3,0003,000@@22== 2,500 (30)2,500 (30) ++ 3,000(30)3,000(30)

== $165,000$165,000

Lowest total cost is at point aLowest total cost is at point a


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La"or $c%eduling &'ampleLa"or $c%eduling &'ample

1ime1ime um"er ofum"er of 1ime1ime um"er ofum"er ofPeriodPeriod 1ellers Re!uired1ellers Re!uired PeriodPeriod 1ellers Re!uired1ellers Re!uired

-M-M+ 4F+ 4F-M-M 1010 44 PMPM+ 6+ 6 PMPM 1818

4F4F-M-M+ 44+ 44-M-M 1212 66 PMPM+ + PMPM 17174444-M-M+ oon+ oon 1414 PMPM+ 8+ 8 PMPM 1515

oon + 4oon + 4 PMPM 1616 88 PMPM+ 9+ 9 PMPM 1010

## #ull+time tellers #ull+time tellers

PP11 Part+time tellers starting at Part+time tellers starting at -M-M>leaing at 4>leaing at 4 PMPM??PP22 Part+time tellers starting at 4F Part+time tellers starting at 4F -M-M>leaing at 6>leaing at 6 PMPM??

PP33 Part+time tellers starting at 44 Part+time tellers starting at 44-M-M>leaing at >leaing at PMPM??

PP44 Part+time tellers starting at noon >leaing at 8 Part+time tellers starting at noon >leaing at 8 PMPM??

PP

55 Part+time tellers starting at 4 Part+time tellers starting at 4 PMPM

>leaing at 9>leaing at 9 PMPM

??


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= $75= $75## + $24(+ $24(PP11C PC P22C PC P33C PC P44C PC P55))Minimize total dailyMinimize total daily

manpower costmanpower cost

## C PC P11 * 10* 10 ((-M-M+ 4F+ 4F-M-Mneedsneeds))

## C PC P11 C PC P22 * 12* 12 ((4F4F-M-M+ 44+ 44-M-Mneedsneeds))

4/6 #4/6 # C PC P11 C PC P22 C PC P33 * 14* 14 ((4444-M-M+ 44+ 44-M-Mneedsneeds))4/6 #4/6 # C PC P11 C PC P22 C PC P33 C PC P44 * 16* 16 ((noon + 4noon + 4 PMPMneedsneeds))

## C PC P22 C PC P33 C PC P44 C PC P55 * 18* 18 ((44 PMPM+ 6+ 6 PMPMneedsneeds))

## C PC P33 C PC P44 C PC P55 * 1* 177 ((66 PMPM+ + PMPMneedsneeds))

## C PC P44 C PC P

55 * 15* 15 (( PMPM+ 7+ 7 PMPMneedsneeds))

## C PC P55 * 10* 10 ((88 PMPM+ 9+ 9 PMPMneedsneeds))

## 12 12

4(4(PP11C PC P22C PC P33C PC P44C PC P55) .50(10 + 12 + 14 + 16 + 18 + 17 + 15 + 10)) .50(10 + 12 + 14 + 16 + 18 + 17 + 15 + 10)


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= $75= $75## + $24(+ $24(PP11C PC P22C PC P33C PC P44C PC P55))Minimize total dailyMinimize total daily

manpower costmanpower cost

## C PC P11 * 10* 10 ((-M-M+ 4F+ 4F-M-Mneedsneeds))

## C PC P11 C PC P22 * 12* 12 ((4F4F-M-M+ 44+ 44-M-Mneedsneeds))

4/6 #4/6 # C PC P11 C PC P22 C PC P33 * 14* 14 ((4444-M-M+ 44+ 44-M-Mneedsneeds))4/6 #4/6 # C PC P11 C PC P22 C PC P33 C PC P44 * 16* 16 ((noon + 4noon + 4 PMPMneedsneeds))

## C PC P22 C PC P33 C PC P44 C PC P55 * 18* 18 ((44 PMPM+ 6+ 6 PMPMneedsneeds))

## C PC P33 C PC P44 C PC P55 * 1* 177 ((66 PMPM+ + PMPMneedsneeds))

## C PC P44 C PC P

55 * 15* 15 (( PMPM+ 7+ 7 PMPMneedsneeds))

## C PC P55 * 10* 10 ((88 PMPM+ 9+ 9 PMPMneedsneeds))

## 12 12

4(4(PP11 C PC P22 C PC P33 C PC P44 C PC P55)) .50(112) .50(112)

##,, PP11,,PP22,,PP33,,PP44,,PP55 * 0* 0


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1%e $imple' Met%od1%e $imple' Met%od

Real world pro"lems are tooReal world pro"lems are toocomple' to "e soled using t%ecomple' to "e soled using t%egrap%ical met%odgrap%ical met%od

1%e simple' met%od is an algorit%m1%e simple' met%od is an algorit%mfor soling more comple' pro"lemsfor soling more comple' pro"lems

0eeloped "y )eorge 0antzig in t%e0eeloped "y )eorge 0antzig in t%e

late 48Fslate 48Fs

Most computer+"ased LP pac;agesMost computer+"ased LP pac;agesuse t%e simple' met%oduse t%e simple' met%od

operations management module b

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