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Westmead International School

School of Economics Business and Accountancy

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

1

Qualitative Technique


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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

2



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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


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


: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*


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*


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


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> [email protected]@

TALE 1!3O:ERALL LINEAR PROGRAMMING RE*-LT


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 [email protected] 20 [email protected] 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@ [email protected] 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

15



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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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http://www.businessdictionary.com/definition/planning.htmlhttp://www.businessdictionary.com/definition/tool.htmlhttp://www.businessdictionary.com/definition/attempt.htmlhttp://www.businessdictionary.com/definition/uncertainty.htmlhttp://www.businessdictionary.com/definition/data.htmlhttp://www.businessdictionary.com/definition/analysis.htmlhttp://www.businessdictionary.com/definition/trend.htmlhttp://www.businessdictionary.com/definition/judgment.htmlhttp://www.businessdictionary.com/definition/estimate.htmlhttp://www.businessdictionary.com/definition/month.htmlhttp://www.businessdictionary.com/definition/user.htmlhttp://www.businessdictionary.com/definition/technique.htmlhttp://www.businessdictionary.com/definition/Box-Jenkins-model.htmlhttp://www.businessdictionary.com/definition/Box-Jenkins-model.htmlhttp://www.businessdictionary.com/definition/delphi-method.htmlhttp://www.businessdictionary.com/definition/exponential-smoothing.htmlhttp://www.businessdictionary.com/definition/moving-average.htmlhttp://www.businessdictionary.com/definition/regression-analysis-RA.htmlhttp://www.businessdictionary.com/definition/regression-analysis-RA.htmlhttp://www.businessdictionary.com/definition/projection.htmlhttp://www.businessdictionary.com/definition/error.htmlhttp://www.businessdictionary.com/definition/result.htmlhttp://www.businessdictionary.com/definition/planning.htmlhttp://www.businessdictionary.com/definition/tool.htmlhttp://www.businessdictionary.com/definition/attempt.htmlhttp://www.businessdictionary.com/definition/uncertainty.htmlhttp://www.businessdictionary.com/definition/data.htmlhttp://www.businessdictionary.com/definition/analysis.htmlhttp://www.businessdictionary.com/definition/trend.htmlhttp://www.businessdictionary.com/definition/judgment.htmlhttp://www.businessdictionary.com/definition/estimate.htmlhttp://www.businessdictionary.com/definition/month.htmlhttp://www.businessdictionary.com/definition/user.htmlhttp://www.businessdictionary.com/definition/technique.htmlhttp://www.businessdictionary.com/definition/Box-Jenkins-model.htmlhttp://www.businessdictionary.com/definition/Box-Jenkins-model.htmlhttp://www.businessdictionary.com/definition/delphi-method.htmlhttp://www.businessdictionary.com/definition/exponential-smoothing.htmlhttp://www.businessdictionary.com/definition/moving-average.htmlhttp://www.businessdictionary.com/definition/regression-analysis-RA.htmlhttp://www.businessdictionary.com/definition/regression-analysis-RA.htmlhttp://www.businessdictionary.com/definition/projection.htmlhttp://www.businessdictionary.com/definition/error.htmlhttp://www.businessdictionary.com/definition/result.html


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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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http://www.businessdictionary.com/definition/sensitivity-analysis.htmlhttp://www.businessdictionary.com/definition/assign.htmlhttp://www.businessdictionary.com/definition/range.htmlhttp://www.businessdictionary.com/definition/values.htmlhttp://www.businessdictionary.com/definition/factor.htmlhttp://www.businessdictionary.com/definition/variable.htmlhttp://www.businessdictionary.com/definition/forecast.htmlhttp://www.businessdictionary.com/definition/sensitivity-analysis.htmlhttp://www.businessdictionary.com/definition/assign.htmlhttp://www.businessdictionary.com/definition/range.htmlhttp://www.businessdictionary.com/definition/values.htmlhttp://www.businessdictionary.com/definition/factor.htmlhttp://www.businessdictionary.com/definition/variable.htmlhttp://www.businessdictionary.com/definition/forecast.html


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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*


,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>


,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.


,emand()

1 @0

, @/

0 A/

7 A@

/ @2

TALE ,!1MO:ING A:ERAGE ORECA*TING RE*-LT

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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*


,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


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.// [email protected] 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 [email protected]/ @C./A .1/!., 23./@ 40.0@ 2A42.41 @C. .1/!.0 21./ /C.@2 2A10./A @@.A .1/!.7 2.24 /C.4 [email protected] @@.33 .1/

!./ 20.23 /C.3@ 242/.2/ @@ .1/!.6 3C.21 /C.4 243.C @A.4C .1/!.3 [email protected] /@.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.@/ [email protected] @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 [email protected] /A.2 22A.// @4.33 .1/!1 [email protected] /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 [email protected] .3!32 C.3 44.@A /@01.23 [email protected] .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 [email protected] /C@. CC.31 .31

!20 C.1A [email protected] /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 [email protected]/ 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@ [email protected] /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*

40



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41



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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


@.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@


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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