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CHAPTER 1
LINEAR PROGRAMMING
Linear programming is a powerful quantitative technique (or operational
research technique) designs to solve allocation problem. The term linear
programming consists of the two words Linear and Programming.
The word 'Linear' is used to describe the relationship between decision
variables which are directl proportional. !or e"ample# if doubling (or tripling) the
production of a product will e"actl double (or triple) the profit and required
resources# then it is linear relationship.
The word 'programming' means planning of activities in a manner that
achieves some 'optimal' result with available resources. $ program is 'optimal' if it
ma"imi%es or minimi%es some measure or criterion of effectiveness such as
profit# contribution (i.e. sales&variable cost)# sales# and cost.
Thus# 'Linear Programming' indicates the planning of decision variables
which are directl proportional# to achieve the 'optimal' result considering the
limitations within which the problem is to be solved.
The minimization modelstarts with an obective function with the purpose
of minimi%ing a goal while maximization model starts with an obective function
with the purpose of ma"imi%ing a goal.
MAXIMIZATION
ou mae three inds of computers* +on# ,ell# and $pple. These sell for-/00# -1000# and -1200. The +on model requires 3 hours for circuit boardinstallation and hour to fit the peripheral equipment. The ,ell model requires
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hour for circuit boards and / hours for peripherals. The $pple model requires 3hours for circuit boards and 1 hours for peripherals. ou have 10 hoursavailable for circuit board wor and 40 hours for fitting peripherals. 5$675789P:;!7T.
GRAPHICAL METHOD
TALE 1!1DATA TALE
"in#$ o% Comp&er$
Con$rain$ " :? /00 3" = >? /00Dell / >? 1000 " = / >? 1000
Apple 3 1 >? 1200 3" = 1 >? 1200
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TALE 1!,GRAPHING RE*-LT
Corner Poin$X 8. 0 0.
/.. 0 40#000.. 200 12#000.
0,!2/31 31.21@4 44#21@./A
INTERPRETATION4 The Bompan should use 3C3 circuit boards# 31peripherals to meet the minimum cost of Php44# 21C. +ee graph result above.
*IMPLEX METHOD
TALE 1!0ITERATION*
"in#$ o% Comp&er *ol&ion
C5 DasicEariables 10" 40 0slac 0slac1 0slac3 FuantitIeraion1
c&% 10 40 0 0 0
. slac 3 0 0 #/00
. slac 1 / 0 0 1#000
. slac 3 3 1 0 0 1#200Ieraion
,c&% 0 10.0 &20 0 0
1,. 6 0.3333 0.3333 0 0 /00. slac 1 0 2.444A &
0.3333 0 #/00
. slac 3 0 & 0 C00Ieraion
0c&% 0 0 &3@./A &2.1@/ 0
1,. 6 0 0.3/A &0.0A 0 3C1.@/A6. 0 &
0.0A20.123 0 31.21@4
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. slac 3 0 0 &0.C1@4
&0.123 /A@./A2
TALE 1!7D-ALING
"in#$ o% Comp&er *ol&ion
Original Pro8lem
Ma(imi)e "
*on+ 3 >? /00Dell / >? 1000
Apple 3 1 >? 1200
D&al Pro8lem
+on ,ell $pple
Minimi)e /00 1000 1200
X 3 3 G? 10
9 / 1 G? 40
TALE 1!/RANGING
"in#$ o% Comp&er *ol&ion:aria8le Ealue :educed
Bost;riginal
EalLowerDound
HpperDound
X 3C1.@/A 0 10 1 @0
9 31.21@4 0 40 20 400Con$rain ,ual
Ealue+lacI+urplus ;riginal
EalLowerDound
HpperDound
*on+ 3@./A2 0 /00 200 113.0AADell 2.1@/A 0 1000 /00.000 2A00
Apple 0 /A@./A2 1200 @1.21C 7nfinit
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TALE 1!6*OL-TION LI*T
"in#$ o% Comp&er *ol&ion
:aria8le +tatus Ealue
X Dasic 3C1.@/A
9 Dasic 31.21@4$la;< 1 J;JDasic 0
$la;< , J;JDasic 0
$la;< 0 Dasic /A@./A2
Opimal :al&e =Z> 4421@./A
TALE 1!3O:ERALL LINEAR PROGRAMMING RE*-LT*
"in#$ o% Comp&er *ol&ion
6 :? /00 3@./A2Dell / >? 1000 2.1@/AApple 3 1 >? 1200 0*ol&ion?@ 3C1.@/A 31.21@4 4421@./A
INTERPRETATION4 The Bompan should use 3C3 circuit boards# 31peripherals to meet the ma"imum profit of Php44# 21C. +ee graph result above.
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MINIMIZATION
ou mae three inds of computers* +on# ,ell# and $pple. The totalmanufacturing cost for each computer is -/00# -1000# and -1200. The Bheapmodel requires 3 hours for circuit board installation and hour to fit theperipheral equipment. The Kood model requires hour for circuit boards and /hours for peripherals. The delu"e model requires 3 hours for circuit boards and 1
hours for peripherals. ou have 10 hours available for circuit board wor and 40hours for fitting peripherals ,etermine the best mi" that will 57J75789 ourB;+T.
TALE 1!1DATA TALE
"in#$ o% Comp&er
" :
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TALE 1!,GRAPHING RE*-LT
Corner Poin$X 8. /00 C0#000.
,... 0 20#000.
,.. C00 /@#000.61/!0276 1A4.C13 1@#C13.0@
INTERPRETATION4 The Bompan should use 4/ of circuit boards 1AA ofperipherals to meet the minimum cost of Php1C.
*IMPLEX METHOD
TALE 1!0ITERATION*
"in#$ o% Comp&er *ol&ion
C5 DasicEariables
10"
40
0 artfcl
0surplus
0 artfcl1
0surplus
1
0 artfcl3
0surplus
3
Fuantit
Ieraion1
c&% A @ 0 & 0 & 0 &
. artfcl 3 & 0 0 0 0 #/00
. artfcl 1 / 0 0 & 0 0 1#000
. artfcl 3 3 1 0 0 0 0 & 1#200
Ieraion,
c&% /.2 0 0 & &.4 0.4 0 &
. artfcl 1.@ 0 & &0.1 0.1 0 0 #006. 0.1 0 0 0.1 &0.1 0 0 200. artfcl 3 1.4 0 0 0 &0.2 0.2 & #400
Ieraion
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0c&% 0 0 &
.C1@4
0.C1@4 &.123 0.123 0 &
,. 6 0 0.3/A &0.3/A &0.0A2 0.0A2 0 0 3C1.@/A6. 0 &
0.0A20.0A2 0.123 &0.123 0 0 31.21@4
. artfcl 3 0 0 &0.C1@4
0.C1@4 &0.123 0.123 & /A@./A/
Ieraion7
c&% 0 0 & 0 &.0 0 &.0 0
,. 6 0 0 0 &0./3@ 0./3@ 0.3@24 &0.3@24 4/.3@246. 0 0 0 0.130@ &0.130@ &
0.0A4C0.0A4C 1A4.C13
. surplus 0 0 & &0.130@ 0.130@ .0A4C &.0A4C 413.0AA
Ieraion/
c&% 0 0 0 0 0.A4C1 &0.A4C1
3.0A4C &3.0A4C
,. 6 0 0 0 &0./3@ 0./3@ 0.3@24 &0.3@24 4/.3@246. 0 0 0 0.130@ &0.130@ &
0.0A4C0.0A4C 1A4.C13
. surplus 0 0 & &0.130@ 0.130@ .0A4C &.0A4C 413.0AA
TALE 1!7D-ALING
"in#$ o% Comp&er *ol&ion
Original Pro8lemMinimi)e 6
*on+ 3 G? /00Dell / G? 1000
Apple 3 1 G? 1200D&al Pro8lem
Bheap Kood ,elu"eMa(imi)e /00 1000 1200
X 3 3 >? 109 / 1 >? 40
TALE 1!/RANGING
"in#$ o% Comp&er *ol&ion:aria8le Ealue :educed
Bost;riginal
EalLowerDound
HpperDound
X 4/.3@24 0 10 1 C08
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9 1A4.C13 0 40 3.3333 00Con$rain
,ualEalue
+lacI+urplus ;riginalEal
LowerDound
HpperDound
*on+ 0 413.0A4C /00 &7nfinit 113.0AADell &0.A4C1 0 1000 @00 2A00
Apple &3.0A4C 0 1200 @1.21C 4000
TALE 1!6*OL-TION *ET
"in#$ o% Comp&er *ol&ion:aria8le +tatus Ealue
X Dasic 4/.3@249 Dasic 1A4.C13
$&rpl&$ 1 Dasic 413.0AA$&rpl&$ , J;JDasic 0$&rpl&$ 0 J;JDasic 0
Opimal :al&e =Z> 1@C13.0@
TALE 1!3O:ERALL LINEAR PROGRAMMING RE*-LT
"in#$ o% Comp&er *ol&ion
6 :
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CHAPTER ,PROECT E:AL-ATION RE:IEB TECHNI-E
CRITICAL PATH METHODPROECT CRA*HING
P9:T and BP5 techniques both involve dividing a large proect into a
series of smaller tass and activities. $lthough the two techniques differ in their
original forms# :ender and +tair cite si" steps common to both methods. These
steps are defining the proect and its significant activities developing a sequence
of these activities drawing a networ diagram that connects the activities
assigning time or cost estimates computing the longest time path through the
networ# nown as the critical path and using the networ to monitor and
manage the proect. 7f proect managers want to reduce total proect time# the
must reduce the length of some activit on the critical path identified through
P9:T or BP5# according to :ender and +tair. 5eanwhile# an dela of an
activit on that path will dela proect completion. P9:T is an e"cellent technique
for monitoring proect completion time but does not consider costs. $ modification
of this technique# P9:TIBost# allows proect managers to control costs and
proect time# according to :ender and +tair. BP5 approaches proect costs
through two estimates* normal and crash. Jormal time and cost are estimates of
proect time and cost under normal conditions. Proect crashing is the time andcost required to complete a proect on a deadline# sometimes through additional
e"penditures to reduce completion time.
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CRITICAL PATH METHOD =CPM>
+am is the 9vent 5anager of 5TE 5usic +ummit and she is tased toorgani%e the event and finish the preparation at the earliest possible time. +hewas able to determine the sequence of activities and the time in das needed tofinish each activit. The figures are summari%ed in Table..
TALE 1!1DATA TALE
CRITICAL PATH METHOD$ctivit time Prec Prec 1 Prec 3
A 1 1 $
C DD $E 2 D , 1 B 9G / !H 1 KI 1 K 1 K" < 7 ML N
M 1 L JN 4 NO 5P 3 J 1 PR 2 F
TALE 1!,CPM RE*-LT
CRITICAL PATH METHOD *OL-TION
$ctivittime
9arl+tart
9arl!inish
Late+tart
Late!inish
+lac
Pro5e; 23
A 1 0 1 0 1 0
1 1 2 1 2 0
C 2 / A @ 3
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D 1 3 3 2
E 2 2 @ 2 @ 0 1 @ 0 @ 0 0
G / 0 / 0 / 0
H 1 / A 1/ 1A 0
I 1 / 1A / 1A 0
1 / A 1/ 1A 0
" 1A 1@ 1A 1@ 0
L 1@ 1C 3C 20
M 1 32 34 20 21 4
N 4 1@ 32 1@ 32 0
O 34 3A 21 23 4
P 3 32 3A 32 3A 0
1 3A 3C 3A 3C 0
R 2 3C 23 3C 23 0
TALE 1!0CHART* O CRITICAL PATH METHOD
GANTT CHART?EARL9 TIME*
GANTT CHART?LATE TIME*
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GANTT CHART?EARL9 AND LATE TIME*
PRECEDENCE GRAPH
INTERPRETATION4 !rom above computations# the critical method is from$&D&9&!&K&7&N&;&P&F&:. +ee critical path method result and graphs above.PROGRAM E:AL-ATION AND RE:IEB TECHNI-E =PERT>
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5ann is the Proect 5anager of Darber +hop and he is tased to open anew branch in Datangas as part of the compans e"pansion program at theearliest possible time.
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G 22 @C 133 @C 133 0 .33H 1@.33 20 4@.33 102.4A 133 42.4A 2.33I / 133 12@ 133 12@ 0
TALE ,!0TA*" TIME COMP-TATION
PERT *OL-TION;ptimistic
time5ostLieltime
Pessimistictime
$ctivittime
+tandard,eviation
Eariance
A 32 3/ 21 34 .33 .A@
13 30 3 1C .33 .A@C C 10 1A 1 .33 .A@D 3A 20 23 20 E 2A // /A /2 .4A 1.A@ 23 /0 / 2C .33 .A@G 3@ 2/ 24 22 .33 .A@H 11 1/ 2@ 1@.33 2.33 @.A@I 1 / @ /
Pro5e;
Re$&l$Toal o%;rii;al
A;iiie$
1.4A
*F&areroo o%oal
3./4
TALE ,!7CHART* O PROGRAM E:AL-ATION AND RE:IEB TCHNI-E
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GANTT CHART?LATE TIME*
GANNT CHART?EARL9 AND LATE TIME*
PRECEDENCE GRAPH
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INTERPRETATION4 Hsing Proect evaluation and review technique critical path
method is $&D&B&9&!&K&7. +ee P9:T result# tas time computation and its graphabove.
PROECT CRA*HING
E# the 9vent 5anager of 5TE 5usic +ummit was informed that theproect must be finished in @4 das.
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TALE 0!,
CRA*HING RE*-LTCRA*HING *OL-TION
Jormaltime
Brashtime
JormalBost
BrashBost
BrashcostIpd
Brashb
Brashingcost
Pro5e; 24 1AA / 1 1/000 32000 3000 3 C000 2 1 30000 20000 /000 1 0000C 0 4 2/000 @000 C000 2 34000D / 3 30000 3@000 2000 1 @000E A 4 30000 3A000 A000 A000 / 3 10000 14000 3000 1 4000
G 2 1 3/000 22000 2/00 1 C000H 4 3 3/000 4/000 0000 3 30000
TOTAL* 1/0000 /000
TALE 0!0CRA*H *CHED-LE
CRA*HING *OL-TION
Pro5e; ime Periodcost
Bumulative cost $ D B , 9 ! K 100*e&pKOr#ering ;o$=*> 10Hol#ing ;o$=H> 30Q-ni ;o$ 1@
TALE 1!,IN:ENTOR9 RE*-LT*
Har#+?ran# aerie$ *ol&ion
Parameer Ealue Parameter EalueDeman# rae=D> 100 ;ptimal order quantit (FS) A/./C*e&pKOr#ering;o$=*>
10 5a"imum 7nventor Level (7ma") A/./C
Hol#ing;o$=H>0.
@.2 $verage inventor 3A.@
-ni ;o$ 1@ ;rders per period(ear) /.@A $nnual +etup cost 3A.2C
$nnual
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IINTERPRETATION4 7nventor results and cost curve answers are givenabove.
CHAPTER /
ORECA*TING
$ planningtoolthat helps management in its attemptsto cope with
the uncertaintof the future# reling mainl ondatafrom the past and present
and analsisof trends.!orecasting starts with certain assumptions based on the managements
e"perience andudgment. Theseestimatesare proected into the
coming monthsor ears usingone or more techniquessuch as Do"&Menins
models# ,elphi method# e"ponential smoothing# moving averages# regression
analsis# and trend proection. +ince an errorin the assumptions will resultin a
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similar or magnified error in forecasting# the technique of sensitivit analsisis
used which assignsa rangeof valuesto the uncertain factors(variables).
$ forecastshould not be confused with a budget.
NAI:E APPROACH
Kiven the following data below# prepare the forecast for the period 4 usingnaive approach.
NA:E ORECA*TING APPROACH
,emand()1 @0, @/0 A/7 A@/ @2
TALE 1!1
NAI:E ORECA*TING RE*-LTNA:E ORECA*TING APPROACH
Mea$&re EalueError Mea$&re$ia$ =Mean Error> MAD =Mean A8$ol&e Deiaion> 4
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M*E =Mean *F&are# Error> 21./*an#ar# Error =#enomn?,,> C.11MAPE =Mean A8$ol&e Per;en Error> .0@ore;a$ne( perio# @2
TALE 1!,NAI:E DETAIL* AND ERROR ANAL9*I*
NA:E ORECA*TING APPROACH
,emand()
!orecast 9rror 9rror 9rrorU1 Pct9rror
1 @0
, @/ @0 / / 1/ .04
0 A/ @/ &0 0 00 .3
7 A@ A/ 3 3 C .02
/ @2 A@ 4 4 34 .0A
TOTAL* 201 2 12 A0 .3
A:ERAGE @0.2 4 21./ .0@
Ne( perio#
%ore;a$
@2 (Dias) (5$,) (5+9) (5$P9)
+tderr
C.11
TALE 1!0NAI:E CONTROL =TRAC"ING *IGNAL>
NA:E ORECA*TING APPROACH
,emand()
!orecast 9rror :+!9 :+!9 Bum$bs
Bum5$,
Trac+ignal
1 @0
, @/ @0 / / / / /
0 A/ @/ &0 &/ 0 / A./ &.4A
7 A@ A/ 3 &1 3 @ 4 &.33
/ @2 A@ 4 2 4 12 4 .4A
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TALE 1!7NAI:E GRAPH
INTERPRETATION4Hsing naive approach the interpretation for period si" wouldbe @2. +ee naive details# error analsis# naive control (tracing signal) and naivegraph above.
MO:ING A:ERAGE APPROACH
Kiven the following data below# Prepare the forecast for the period 4 usingthree period moving average approach.
MO:ING A:ERAGE APPROACH
,emand()
1 @0
, @/
0 A/
7 A@
/ @2
TALE ,!1MO:ING A:ERAGE ORECA*TING RE*-LT
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MO:ING A:ERAGE APPROACH
Mea$&re EalueError Mea$&re$ia$ =Mean Error> .33
MAD =Mean A8$ol&e Deiaion> 3.33M*E =Mean *F&are# Error> 1.@C
*an#ar# Error =#enomn?,.> J$MAPE =Mean A8$ol&e Per;en Error> .02
ore;a$ne( perio# AC
TALE ,!,DETAIL* AND ERROR ANAL9*I*
MO:ING A:ERAGE APPROACH
,emand()
!orecast 9rror 9rror 9rrorU1 Pct9rror
1 @0
, @/
0 A/
7 A@ @0 &1 1 2 .03
/ @2 AC.33 2.4A 2.4A 1.A@ .04
TOTAL* 201 1.4A 4.4A 1/.A@ .0@
A:ERAGE @0.2 .33 3.33 1.@C .02
Ne( perio#%ore;a$
AC (Dias) (5$,) (5+9) (5$P9)
+td err J$
TALE ,!0ERROR* A* -NCTION O N
MO:ING A:ERAGE APPROACH
N Dias 5$, 5+9 +tandarderror
5$P9
1 4 21./ C.11 .0@
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, &.4A /.4A 3@.@3 0.AC .0A
0 .33 3.33 1.@C .027 2./ 2./ 10.1/ .0/
TALE ,!7CONTROL =TRAC"ING *IGNAL>
Moing Aerage approa; *ol&ion,emand(
)!orecast 9rror :+!9 :+!9 Bum
$bsBum5$,
Trac+ignal
1 @0, @/0 A/7 A@ @0 &1 &1 1 1 1 &/ @2 AC.33 2.4A 1.4A 2.4A 4.4A 3.33 .@
TALE ,!/MO:ING A:ERAGE GRAPH
INTERPRETATION4 Hsing moving average approach# the forecast for period si"would be AC. +ee moving average forecasting results# details and error analsis#errors as function of n# control tracing signal and moving average graph above.
EXPONENTIAL *MOOTHINGKiven the following data below# prepare the forecast for the period 4 using
e"ponential smoothing approach.
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EXPONENTIAL *MOOTHING APPRACH
,emand() !orecast1 13 @C, 31 00 1/@ 07 /1 0/ 13C 0
TALE 0!1
ORECA*TING RE*-LTEXPONENTIAL *MOOTHING APPROACHMea$&re EalueError Mea$&re$ ia$ =Mean Error> 12.MAD =Mean A8$ol&e Deiaion> /4.3@M*E =Mean *F&are# Error> 2/13.C*an#ar# Error =#enomn?,0> @4.@3MAPE =Mean A8$ol&e Per;en Error> .1/ore;a$ne( perio# 1C.2
TALE 0!,DETAIL* AND ERROR ANAL9*I*
EXPONENTIAL *OOTHING APPROACH,emand(
)!orecast 9rror 9rror 9rrorU1 Pct
9rror1 13 @C 12 12 /A4 ., 31 C/ A A 34@C .3@0 1/@ 112.1/ 33.A/ 33.A/ 3C.04 .37 /1 131.4C &@0.4C @0.4C 4/0.2A ./3/ 13C 11./1 14.2@ 14.2@ A0.21 .
TOTAL* A2 10.// 1@.C1 114/.C4 .14A:ERAGE 132.@ 12. /4.3@ 2/13.C .1/
Ne( perio#%ore;a$
1C.2 (Dias) (5$,) (5+9) (5$P9)
+td err @4.@3
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TALE 0!0ERROR* A* -NCTION O ALPHA
EXPONENTIAL *MOOTHING PPROACHAlpa Dias 5$, 5+9 +tandard
error5$P9
!.. 2/.@ 40.4 2@4A C0.04 .1/!.1 22.4@ 40.32 2@3.3/ @C./A .1/!., 23./@ 40.0@ 2A42.41 @C. .1/!.0 21./ /C.@2 2A10./A @@.A .1/!.7 2.24 /C.4 24@0.C1 @@.33 .1/
!./ 20.23 /C.3@ 242/.2/ @@ .1/!.6 3C.21 /C.4 243.C @A.4C .1/!.3 3@.23 /@.C/ 2/@4.0@ @A.23 .1/!.2 3A.2A /@.A/ 2/4.A4 @A.C .1/!. 34./1 /@.// 2/20.A/ @A .1/!1. 3/.4 /@.3A 2/11.@2 @4.@1 .1/!11 32.A /@.C 2/0A.@4 @4.4@ .1/!1, 33.@1 /@.01 22C/.42 @4./4 .1/!10 31.C4 /A.@/ 22@4.01 @4.2A .1/!17 31.1 /A.A 22A@.@1 @4.2 .1/!1/ 3.1C /A.// 22A3.C1 @4.3/ .1/
!16 30.2C /A.2 22A.4 @4.31 .1/!13 1C.A /A.1A 22A0.2 @4.31 .1/!12 1@.C2 /A.2 22A.// @4.33 .1/!1 1@.1 /A.0 22A2.24 @4.34 .1/!,. 1A.2A /4.@C 22AC.03 @4.2 .1/!,1 14.A4 /4.A@ 22@/./ @4.24 .1/!,, 14.0A /4.4A 22C1.A @4./3 .1/!,0 1/.2 /4./A 2/0.43 @4.41 .1/!,7 12.A/ /4.2@ 2/.@1 @4.A1 .1/!,/ 12. /4.3@ 2/13.C @4.@3 .1/
!,6 13.2C /4.3 2/3/.4A @4.C/ .1/!,3 11.@@ /4.11 2/2C.A @A.0A .1/!,2 11.3 /4.2 2/43.42 @A.1 .1/!, 1.A3 /4.04 2/AC @A.34 .1/!0. 1.A /4 2/C/.1 @A./ .1/!01 10.43 //.C3 241.1 @A.4@ .1/
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!31 C.4/ 42.04 /40.13 C4.41 .3!3, C./@ 42.2A /41C.A@ C4.@A .3!30 C./1 42.@@ /4/@.2 CA. .3!37 C.24 4/.1C /4@A. CA.34 .3!3/ C.21 4/.4C /A/.@A CA.4 .3!36 C.3A 44.0C /A22.4@ CA.@/ .3!33 C.32 44.2@ /AA3./2 C@.0C .3!32 C.3 44.@A /@01.23 C@.32 .3!3 C.1C 4A.1/ /@3.32 C@./@ .31!2. C.1@ 4A.43 /@40.1A C@.@3 .31!21 C.1A 4@ /@@C.C CC.0A .31!2, C.14 4@.3A /C@. CC.31 .31
!20 C.1A 4@.A2 /C2A CC./4 .31!27 C.1A 4C.0C /CA/.@/ CC.@ .31!2/ C.1C 4C.2/ 4002.4/ 00.02 .31!26 C.3 4C.AC 4033.3C 00.1@ .33!23 C.33 A0.3 4041.02 00./1 .33!22 C.3/ A0.2A 40C0.4 00.A/ .33!2 C.3C A0.@ 4C.0/ 00.CC .33!. C.21 A.1 42A.3A 0.11 .33!1 C.24 A.22 4A/.// 0.2/ .33!, C./ A.A/ 4103./4 0.4@ .33
!0 C./4 A1.0/ 413.3C 0.C .33!7 C.4 A1.3/ 41/C.03 01.2 .33!/ C.4A A1.42 41@4.2/ 01.34 .32!6 C.A3 A1.C3 433.43 01./@ .32!3 C.AC A3.1 4320./4 01.@ .32!2 C.@4 A3.2@ 434A.1 03.0 .32! C.C3 A3.A2 43C3./4 03.13 .32
1!.. 0 A2 42C.4 03.22 .32
TALE 0!7
CONTROL =TRAC"ING *IGNAL>E(ponenial *mooing *ol&ion
,emand()
!orecast 9rror :+!9 :+!9 Bum$bs
Bum5$,
Trac+ignal
1 13 @C 12 12 12 12 12 , 31 C/ A 2 A 2 A0./ 1
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0 1/@ 112.1/ 33.A/ A2.A/ 33.A/ A2.A/ /@.1/ 37 /1 232.69 &
@0.4CC2.04 @0.4C 1//.22 43.@4 .2A
/ 13C 212.52 14.2@ 10.// 14.2@ 1@.C1 /4.3@ 1.2
TALE 0!/EXPONENTIAL *MOOTHING GRAPH
CHAPTER 6
-EING K BAITING LINE*
Fueuing theor is the mathematical stud of waiting lines. There are
several related processes# arriving at the bac of the queue# waiting in the queue
(essentiall a storage process)# and being served b the server at the front of the
queue. 7t is applicable in transport and telecommunication and is occasionall
lined to ride theor. 7ncoming traffic to queuing theor sstems is modelled via
Poisson distribution# with the assumptions of pure chance traffic. Ball arrivals and
departures are random and independent events. +tatistical equilibrium
probabilities within the sstem do not change. $ll incoming traffic can be routed
to an other customer within the networ and congestion is cleared as soon as
servers are free.
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Fueuing theor is finding the best level of service b trade&off between the
cost of providing good service and the cost of customer waiting line. Oe will alsoassume a first in first out (!7!;) disciple where customers are serviced according
to order of their arrival.!our 5odels presented using Nendall notation*
. 5I5I1. 5I5Im3. 5I,I2. 5I5I with finite source
MKMK1Dran is the branch manager of 9ast&Oest Dan and he wants to improve
the service of the ban b reducing the average waiting time of the bans client.
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TALE 1!,BAITING LINE* RE*-LT*
EA*T?BE*T AN" *ol&ionParameer Ealue Parameter Ealue 5inutes +econds
MKMK1=e(ponenial
$eri;e ime$>
$verage serverutili%ation
.1
Arrialrae=lam8#a>
3 $verage number inthe queue(Lq)
.0/
*eri;e rae=m&> / $verage number inthe sstem(Ls)
.1/
N&m8er o% $erer$ $verage time in thequeue(Oq)
.01 40
*erer ;o$ Kime @ $verage time in thesstem(Os)
.0@ / 300
Baiing ;o$Kime
Bost (Labor = VwaitingSwait cost)
@.//
Bost (Labor = V insstemSwait cost)
0.A/
TALE 1!0TALE O PROAILITIE*
EA*T?BE*T AN" *ol&ion
< Prob (num in ss ?)
Prob (num in ss >?)
Prob (num in ssG)
. .@ .@ .1
1 .4 .C4 .02
, .03 0
0 0 0
7 0 0
/ 0 0
6 0 0
TALE 1!7GRAPH* O PROAILITIE*
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INTERPRETATION4 Oaiting lines and table of probabilities results using 55method are above.
MKMK$ MKMKmBlair# the Dan 5anager of 9$+T&O9+T Dan wants to improve the
service of the ban b hiring an additional teller. +he observed that the averagearrival and the average number of clients serviced per teller per hour remains thesame. Blair nows that with the additional teller# the number of clients waiting inline will decrease. +he observed that the tellers labour costs as well as theaverage opportunit cost of clients who are waiting remain the same. The figuresare summari%ed in Table1..
TALE ,!1DATA TALE
EA*T?BE*T AN"Parameer Ealue
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MKMK$Arrial rae=lam8#a> 3
*eri;e rae=m&> /N&m8er o% $erer$ 1*erer ;o$ Kime @Baiing ;o$ Kime
TALE ,!,BAITING LINE* RE*-LT
EA*T?BE*T AN" *ol&ionParameer Ealue Parameter Ealue 5inutes +econds
MKMK$ $verage serverutili%ation .
Arrialrae=lam8#a>
3 $verage number in thequeue(Lq)
0
*eri;erae=m&>
/ $verage number in thesstem(Ls)
.1
N&m8er o%$erer$
1 $verage time in thequeue(Oq)
0 .02 1.21
*erer ;o$Kime
@ $verage time in thesstem(Os)
.0A 2.02 121.21
Baiing ;o$
Kime
Bost (Labor = V
waitingSwait cost)
4.01
Bost (Labor = V insstemSwait cost)
@.11
TALE ,!0TALE O PROAILITIE*
EA*T?BE*T AN" *ol&ion< Prob (num in ss ?
)Prob (num in ss >?
)Prob (num in ss
G). .@1 .@1 .@
1 .4 .C@ .01, .01 00 0 07 0 0/ 0 0
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TALE ,!7
CO*T :*! *ER:ER*EA*T?BE*T AN" *ol&ion
N&m8er o% $erer$ Total cost based on waiting Total cost based on sstem
1 @.// 0.A/
, 4.01 @.11
0 12 14.1
7 31 32.1
/ 20 21.1
TALE ,!/GRAPH* O PROAILITIE*
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INTERPRETATION* Oaiting lines# table of probabilities and cost vs.server results using 55mI55s method are above.
MKDK1Krace# the Dranch 5anager of 9$+T&O9+T Dan wants to improve the
service of the ban b replacing the teller with an $T5. +he observed that theaverage arrival and the average number of clients serviced per hour remain thesame. Decause an $T5 processes according to a fi"ed ccle# she presumed thatthe service rate distribution is now constant. Lea nows that with the $T5 as theteller# the number of clients waiting in line will decrease. +he observed that the
$T5s service cost is equal to the labour cost of a teller and the averageopportunit cost of clients who are waiting remain the same. The figures aresummari%ed in Table3..
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TALE 0!,
BAITING LINE* RE*-LT*EA*T BE*T AN" *ol&ion
Parameer Ealue Parameter Ealue 5inutes +econdsMKDK1 =;on$an$eri;e ime$>
$verage serverutili%ation
.1
Arrialrae=lam8#a>
3 $verage number inthe queue(Lq)
.03
*eri;e rae=m&> / $verage number inthe sstem(Ls)
.13
N&m8er o%$erer$
$verage time in thequeue(Oq)
0 ./ 30
*erer ;o$Kime
@ $verage time in thesstem(Os)
.0@ 2./ 1A0
Baiing ;o$Kime
Bost (Labor = VwaitingSwait cost)
@.1@
Bost (Labor = V insstemSwait cost)
0.2@
INTERPRETATION4 Oaiting lines result using 55 method is above.
CHAPTER 3
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NETBOR"The term network low !rogram describes a tpe of model that is a special
case of the more general linear program. The class of networ flow programs
includes such problems as the shortest path problem# the ma"imum flow
problem# the pure minimum cost flow problem# and the generali%ed minimum
cost flow problem. 7t is an important class because man aspects of actual
situations are readil recogni%ed as networs and the representation of model is
much more compact than the general linear program. Ohen the situation can beentirel modelled as networ# ver efficient algorithm e"ists for the solution of the
optimi%ation problem# man times more efficient than linear programming in the
utili%ation of computer time and space resources.Jetwor models use nodes and arcs to mae decision.
Three techniques*. 5inimal&+panning Three Technique1. 5a"imal&!low Technique3. +hortest&:oute Technique
Minimal?*panning Tree Te;niF&eThe obective of "inimal#$!anning %ree %echni&'eis to connect each node to
at least one node while minimi%ing the total distance!
Ma(imal?loJ Te;niF&e 5a"imal !low Technique sees to determine the ma"imum
quantit that can go from one node to another in a networ at an given time.
*ore$?Ro&e Te;niF&eThe shortest&route technique determines the minimum distance
from one point of the networ to another.
MINIMAL *PANNING TREEDlair is the
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was able to determine the distance in ilometres from one town to another. Thedistance figures are summari%ed in Table ..
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E 1 / 1 1
3 2 3 3G 3 / @ @
H 3 4 1I 3 A C
3 @ 11
" 2 A 4 4
L / @ 2 2
M 4 A C C
N A C 10
O @ C
Toal C3
TALE 1!0*OL-TION *TEP
MINIMAL *PANNING TREE *OL-TION
M9?D*L +tartingnode
9ndingnode
Bost Bumulativecost
3 0 0
G 3 / @ @
E 1 / 1 30
3 2 3 23
L / @ 2 /A
O @ C 4@
" 2 A 4 @2
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M 4 A C C3
INTERPRETATION4 Hsing minimal spanning tree method / towns in Jasugbushould be connected from D(&3)# K(3&/)# 9(1&/)# !(3&2)# L(/&@)# ;(@&C)# N(2&A)#5(4&A) for a total of C3 ilometres. +ee minimal spanning tree networ result andsolution step above.
MAXIMAL LOB TECHNI-E
Dagito is the 7ndustrial 9ngineer of +abang Oater ,istrict and she is
tased to ensure that ma"imum amount of water flows from Oater Pump M. +hewas able to determine the capacit of the outflow and inflow of water from onepump to another in litres per second. The capacit figures are summari%ed inTable 1..
Ohat is the total capacit of the water flow
TALE ,!1
DATA TALEMAXIMAL LOB TECHNI-E
*AANG BATERDI*TRICT
+tartnode
9nd node Bapacit :eversecapacit
A 1 / C
2 3
C 1 3 0 4
D 1 / @ 1
E 3 2 4 A
3 4 2 @G 2 4 1 2
H / 4 A
I / A C /
4 A 3 0
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TALE ,!,NETBOR" RE*-LT
MAXIMAL LOB TECHNI-E*AANG BATER
DI*TRICT+tartnode
9ndnode
Bapacit :eversecapacit
!low
Ma(imal NeJor
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