chp12networksmodel-121128084640-phpapp01

7/25/2019 chp12networksmodel-121128084640-phpapp01

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2008 Prentice-Hall, Inc.

Chapter 12

To accompanyQuantitative Analysis for Management, Tenth Edition,

by Render, Stair, and HannaPower Point slides created by Je Heyl

Network Models

200! Prentice-Hall, Inc.


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200! Prentice-Hall, Inc. "2 # 2

Learning !"e#tives

". $onnect all points o a networ% w&ileminimi'in( total distance )sin( t&e minimal-

spannin( tree tec&ni*)e2. +etermine t&e maim)m low t&ro)(& a

networ% )sin( t&e maimal-low tec&ni*)e

. ind t&e s&ortest pat& t&ro)(& a networ%

)sin( t&e s&ortest-ro)te tec&ni*)e/. nderstand t&e important role o sotware in

sol1in( networ% problems

ter completin( t&is c&apter, st)dents will be able to3ter completin( t&is c&apter, st)dents will be able to3


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200! Prentice-Hall, Inc. "2 #

Chapter utline

"2.""2." Introd)ction

"2.2"2.2 4inimal-Spannin( Tree Tec&ni*)e

"2."2. 4aimal-low Tec&ni*)e

"2./"2./ S&ortest-Ro)te Tec&ni*)e


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200! Prentice-Hall, Inc. "2 # /

T&is c&apter co1ers t&ree networ% models t&atcan be )sed to sol1e a 1ariety o problems

T&e minimal$spanning tree te#hni%ueminimal$spanning tree te#hni%uedeterminesa pat& t&ro)(& a networ% t&at connects all t&epoints w&ile minimi'in( t&e total distance

T&e ma&imal$flow te#hni%uema&imal$flow te#hni%ueinds t&e maim)mlow o any *)antity or s)bstance t&ro)(& anetwor%

T&e shortest$route te#hni%ueshortest$route te#hni%uecan ind t&es&ortest pat& t&ro)(& a networ%

'ntrodu#tion


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6ar(e scale problems may re*)ire &)ndreds ort&o)sands o iterations ma%in( eicientcomp)ter pro(rams a necessity

ll types o networ%s )se a common terminolo(y

T&e points on a networ% are called nodesnodesandmay be represented as circles o s*)ares

T&e lines connectin( t&e nodes are called ar#sar#s

'ntrodu#tion


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Minimal$(panning Tree Te#hni%ue

T&e minimal-spannin( tree tec&ni*)e in1ol1esconnectin( all t&e points o a networ% to(et&erw&ile minimi'in( t&e distance between t&em

T&e 6a)derdale $onstr)ction $ompany isde1elopin( a &o)sin( proect

T&ey want to determine t&e least epensi1e wayto pro1ide water and power to eac& &o)se

T&ere are ei(&t &o)ses in t&e proect and t&e

distance between t&em is s&own in i()re "2."


7/29 200! Prentice-Hall, Inc. "2 # 9


Steps or t&e minimal-spannin( treetec&ni*)e

". Select any node in t&e networ%

2. $onnect t&is node to t&e nearest node t&atminimi'es t&e total distance

. $onsiderin( all t&e nodes t&at are nowconnected, ind and connect t&e nearest nodet&at is not connected. I t&ere is a tie, select

one arbitrarily. tie s)((ests t&ere may bemore t&an one optimal sol)tion.

/. Repeat t&e t&ird step )ntil all nodes areconnected


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:etwor% or 6a)derdale $onstr)ction

2

2

/

2

5

7

9

"

5

"

2

/

5

7

9

8

;)li()re "2."


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Start by arbitrarily selectin( node "

T&e nearest node is node at a distance o 2 e pic% node 2 and connect it to node


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ollowin( t&is same process we connect romnode 2 to node 5

>e t&en connect node to node 7

:ode 7 will connect to node 8 T&e last connection to be made is node 8 to node

9

T&e total distance is o)nd by addin( )p t&edistances in t&e arcs )sed in t&e spannin( tree

2 ? 2 ? ? ? ? " ? 2 @ "7


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ll iterations or 6a)derdale $onstr)ction

i()res "2.2 # "2.5

2

2

/

2

5

7

9

"

5

"

2

/

5

7

9

8

;)l


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Ma&imal$)low Te#hni%ue

T&e maimal-low tec&ni*)e allows )s todetermine t&e maim)m amo)nt o a material t&atcan low t&ro)(& a networ%

>a)%es&a >isconsin is in t&e process ode1elopin( a road system or t&e downtown area

T&ey want to determine t&e maim)m n)mber ocars t&at can low t&ro)(& t&e town rom west toeast

T&e road networ% is s&own in i()re "2.9 T&e n)mbers by t&e nodes indicate t&e n)mber o

cars t&at can low fromfromt&e node


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o)r steps o t&e 4aimal-low Tec&ni*)e". Pic% any pat& rom t&e start


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o)r steps o t&e 4aimal-low Tec&ni*)e. or eac& node on t&is pat&, decrease t&e low

capacity in t&e direction o low by t&eamo)nt $. or eac& node on t&e pat&,increase t&e low capacity in t&e re1ersedirection by t&e amo)nt $.

/. Repeat t&ese steps )ntil an increase in low isno lon(er possible


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200! Prentice-Hall, Inc. "2 # "5


Road networ% or >a)%es&a

$apacity in H)ndredso $ars per Ho)r

>estPoint

AastPoint

i()re "2.7

"0

0 2

"

"

"

"

22"

7

02

0 "

"

"

2

/

5

7


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>e start by arbitrarily pic%in( t&e pat& "#2#7w&ic& is at t&e top o t&e networ%

T&e maim)m low is 2 )nits rom node 2 tonode 7

T&e pat& capacity is ad)sted by addin( 2 to t&ewestbo)nd lows and s)btractin( 2 rom t&eeastbo)nd lows

T&e res)lt is t&e new pat& in i()re "2.9 w&ic&

s&ows t&e new relati1e capacity o t&e pat& att&is sta(e


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$apacity ad)stment or pat& "#2#7 iteration "

i()re "2.9

22

"

"

2

7

/0

"

"

2

7

Bld Pat&

:ew Pat&

dd 2

S)btract 2


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>e repeat t&is process by pic%in( t&e pat& "#2#/#7

T&e maim)m capacity alon( t&is pat& is "

T&e pat& capacity is ad)sted by addin( " to t&ewestbo)nd lows and s)btractin( " rom t&eeastbo)nd lows

T&e res)lt is t&e new pat& in i()re "2.8

>e repeat t&is process by pic%in( t&e pat& "##

5#7 T&e maim)m capacity alon( t&is pat& is 2

i()re "2.! s&ows t&is ad)sted pat&


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200! Prentice-Hall, Inc. "2 # "!


Second iteration or >a)%es&a road system

i()re "2.8

"0

0 2

"

"

2

0

/0

/

0

7

02

0 2

0

"

2

/

5

7

"

"

"

"

"

"

2

/

7

Bld Pat&

:ew :etwor%

dd "

S)btract "


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T&ird and inal iteration or >a)%es&a roadsystem

i()re "2.!

8

2 0

"

2

0

/0

/

0

/

22

0 2

0

"

2

/

5

7


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200! Prentice-Hall, Inc. "2 # 2"


T&ere are no more pat&s rom nodes " to 7 wit&)n)sed capacity so t&is represents a inaliteration

T&e maim)m low t&ro)(& t&is networ% is 500

cars

PTH 6B>


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(hortest$*oute Te#hni%ue

T&e shortest$route te#hni%ueshortest$route te#hni%ueinds &ow a personor item can tra1el rom one location to anot&erw&ile minimi'in( t&e total distance tra1eled

It inds t&e s&ortest ro)te to a series odestinations

Ray +esi(n, Inc. transports beds, c&airs, andot&er )rnit)re rom t&e actory to t&e ware&o)se

T&ey wo)ld li%e to ind t&e ro)te wit& t&e s&ortest

distance T&e road networ% is s&own in i()re "2."0


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Roads rom RayCs plant to ware&o)se

Plant

>are&o)se

"00

200

50

/0

"00

200

"00

"00

"50"

2

/

5

7

i()re "2."0


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200! Prentice-Hall, Inc. "2 # 2/


Steps o t&e s&ortest-ro)te tec&ni*)e". ind t&e nearest node to t&e ori(in


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>e can see t&at t&e nearest node to t&e plant isnode 2

>e connect t&ese two nodes

ter in1esti(ation, we ind node is t&e net

nearest node b)t t&ere are two possible pat&s T&e s&ortest pat& is "#2# wit& a distance o "50

>e repeat t&e process and ind t&e net node isnode 5 by (oin( t&ro)(& node

T&e net nearest node is eit&er / or 7 and 7 t)rnso)t to be closer

T&e s&ortest pat& is "#2##5#7 wit& a distance o2!0 miles


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irst iteration or Ray +esi(n

Plant

>are&o)se

"00

200

50

/0

"00

200

"00

"00

"50"

2

/

5

7

i()re "2.""

"00


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Second iteration or Ray +esi(n

i()re "2."2

Plant

>are&o)se

"00

200

50

/0

"00

200

"00

"00

"50"

2

/

5

7

"00

"50


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T&ird iteration or Ray +esi(n

i()re "2."

Plant

>are&o)se

"00

200

50

/0

"00

200

"00

"00

"50"

2

/

5

7

"00

"50 "!0


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o)rt& and inal iteration or Ray +esi(n

i()re "2."/

Plant

>are&o)se

"00

200

50

/0

"00

200

"00

"00

"50"

2

/

5

7

"00

"50 "!0

2!0

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chp12networksmodel-121128084640-phpapp01