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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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2008 Prentice Hall, Inc. B 2
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
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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
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
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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
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LP -pplicationsLP -pplications
7575 0eeloping a production sc%edule t%at0eeloping a production sc%edule t%atwill satisfy future demands for a firm
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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
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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
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#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
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#ormulating LP Pro"lems#ormulating LP Pro"lems
@+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
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#ormulating LP Pro"lems#ormulating LP Pro"lems
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
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#ormulating LP Pro"lems#ormulating LP Pro"lems
$econd (onstraint3$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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)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"
erof2atc%1.
s
Hum"
erof2atc%1.
s
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
*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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)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"
erofBlueBerrys
Hum"
erofBlueBerrys
um"er of @+podsum"er of @+pods
@@11
@@22
Figure B.4Figure B.4
(0, 42)
(30, 0)(30, 0)
$210 = $7$210 = $7@@11+ $5+ $5@@22
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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"
erofBlueBeryys
Hum"
erofBlueBeryys
um"er of @+podsum"er of @+pods
@@11
@@22
Figure B.5Figure B.5
$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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LP -pplicationsLP -pplications
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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LP -pplicationsLP -pplications
= $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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LP -pplicationsLP -pplications
= $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