ECV 408 Traffic Engineering II

ECV 408

TRAFFIC ENGINEERING II

KU-CIVIL ENG

ECV 408 TRAFFIC ENGINEERING II Prerequisites: ECV 400: Traffic Engineering I and ECU 201: Engineering Mathematics VI Course Outline 

Introduction of stochastic models, Poisson arrivals and diffusion approximations.



Planning, implementation, and operations of control technologies. t echnologies.



Design of traffic facilities; Freeways, Intersections; signalized & un-signalized, Interchanges and parking facilities. Stationary flows, flow conservation, traffic assignment, hierarchy structure of highway networks, time-dependent flows in bottlenecks bottlenecks.. Graphs, Graphs, theory, theory, shortag shortagee paths, paths, continu continuum um approx approximatio imations, ns, vehicle vehicle routing. routing.

Textbooks and References

1. Rogers P. Roess; Traffic Engineering. Prentice Hall, 3rd edt. 2004. ISBN:0130812935 ISBN:0130812935 2. Nicholas J. Garber & Lester A. Hoel; Traffic and Highway engineering. CengageEngineering,2001 3. Highway and traffic Vol.1 by C.A oflaherty 4. Highway and traffic Engineering in developing countries by Bant thageson 5. Principles of highway Eng and traffic analysis by Fred L. Mannering and Walter P. 6. Highway traffic analysis and design by R. J salter . 7. MOR. Design manual. 8. Mannering, F.L.; Kilareski, W.P. and Washburn, S.S. (2005). Principles of Highway Engineering and Traffic Analysis, Analysis, Third Edition. Chapter 5 9. Transportation Research Board. Board. (2000). Highway Capacity Manual Manual 2000. 2000 . National Research Council, Washington, D.C.

Tulatia Mungathia ©2016

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CHAPTER ONE THEORITICAL RELATIONSHIPS RELATIONSHIPS BETWEEN SPEED, FLOW AND VOLUME

The primary variables used to describe or charasterise char asterise the flow of vehicles on a path are volume, speed, concentration and headway - the term headway of two vehicles is defined as the time interval between the moment at a t which the front of one point to the next vehicle passes the same point. Another headway concept is the distance between the front of one vehicle and the front of the following vehicle and the front of the following vehicle at given moment in time. Speed and density (concentration) describe the quality of service ex perienced by the stream and the demand of highway facility. Speed is the space mean speed. Density is the number of vehicles per unit length of highway. Flow is the number of vehicles passing a given point on the highway highway per unit time.

The relationship

Consider a small section of highway of length L in which N vehicles p ass a point in the section during a time interval T. All the vehicles travelling in the same direction. directi on. Therefore volume flowing Q 

N T

Density D=average number of Veh. Travelling over L divide by L The average number of vehicle travelling over L is given b y N

 i  i ti T Where t is the time of travel of the i th veh over the length L.

D 

 N i  i ti TL

Densit Den sityy 

Flow Spacemeans peed pe ed


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FUNDAMENTAL FUNDAMENTAL DIAGRAMS OF ROAD TRAFFIC FLOW

Theoretical curve on vehicle spacing rule Speed (V)

Observed behaviour

Concentration K

a) Speed – Speed – mean mean space Vs Concentration

Speed V

Volume q

Concentration K b) Volume vs concentration

Volume q (c ) speed – speed – mean mean space vs volume

In fig a, based on observations of actual driver be haviour on freeways, there is a max speed at one extreme, point A under a concentration O, at which speed effecti vely approaches zero or a lock-up (jam concentration) at the other end point B. The ditched line is the relationship as it would be if drivers always maintained the minimum spacing dictated by the safety rules at each speed.. In fig b, at values of concentration near zero z ero the volume is necessarily necessaril y low; despite the high speed because there are so few vehicles (defined by the origin concept incurve b) and the high speed, zero volume volume intercept of the speed-volume curve of fig c. The three curves are referred to as the fundamental diagrams diagrams of road traffic flow. It however apples only only to flow on roads where the movement of traffic is not interrupted, as it would be by traffic lights or stop signs.


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BASIC CAPACITY

Is a marginal number of cars that can pass over a given section of a lane or a carriage way. Traffic condition at this level is unstable and minor disturbance in the traffic stream may cause stop and go- operations.

SPEED VEHICLE VS FLOW OF VEHICLES S Normal condition zone P

Speed B Unstable condition zone

C Forced condition zone 0 Flow

At concentration SP the curve is as flat as a reasonable as per circumstances under consideration so that is only relatively small drop in speed as flow increased within the design limit. OC – Speed are much lower and cars move under conditions of forced driving, the concentration is very high and hence the control exercise on each car by the one in OC has the most important influence on flow. Therefore OC is relatively independent on the standard geometric design of the road. PBC – flow can be very high but driving conditions are very unstable.

PATH CHARACTERISTICS

The characteristics of the path which influence vehicle motion and performance fall in three categories. (1) Vehicle exclusions or limitation of size and weight -

Prohibits certain vehicles from operating on certain path.

(2) Speed restrictions – max and min speed (3) Effects of environmental conditions i.e weather conditions maybe rainy or snow.


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In almost all transportation system, its movement of an y vehicle will be limited by the presence of other vehicles and thus its optimum performance may not actually be realized.

TRAFFIC CONGESTION & RESTRAINTS

In road traffic, peaking phenomenon is very pronounced giving rise to congestion where congestion is the impedance and delay imposed by the vehicle on the other. Costs are affected by congestion; delay to people freight is one component of congestion cost. Higher operating cost of vehicle is another one others are social and environmental costs. The necessity to control the use of the vehicle reduces in urban areas is brought about by: (i) The demand for the physical space (ii) The pollution effect of the vehicle exhaust (iii)The noise associated with road vehicles. The demand for the road space will always be greater than the supply. So even if the necessary financial resources were available, there would be conflicting demand for the available land. The fact that the demand for space is higher than the supply it results to congestion. There are three general ways in which restraints would be applied.

(i) Physical restraints – whereby the entry of the vehicles to certain area at certain times could be prohibited by administrative means. (ii) Regulatory measures i.e by use of parking regulations fiscal meas ures such as vehicle and fuel tax, parking charges, road pricing etc.

Benefits of parking control and restraints

-

Control in accidents

-

Increase in road capacity

-

Preservation of environment

-

Traffic restraints

Traffic characteristic refers to the composition of traffic streams at different times speed, journey times and turning moments.


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CONTROL OF VEHICLE FLOW

It is done to ensure that a collision doesn’t occur or at least that the probability is acceptably low. Means used to effect the desired control of vehicles varies considerably. The basic process is: identical. -

Actual or potential situation regulatory

-

Change in vehicle motion.

Deflection of that vehicle condition Time Change forces action on vehicles. Change vehicle speed directions

SEQUENCE OF EVENTS IN CONTROL OF VEHICLE MOTION

The goal is to enable the detection of any condition that might require a change in the movement of vehicle, change in speed, direction etc at a time and distance sufficiently in advance of the point of danger that the danger can be avoided. A common form of vehicle movement control is (i)

Channelization – this concept underlies separation of traffic on a street in 2 directions each placed at its own section of road using sample centre strips.

(ii)

Speed limits

(iii)

Warning signs – indicating the higher probability of dangerous condition along the roads water way and railway lines.

(iv)

Vehicle control on way links- these are prohibitation for ensuring that a vehicle doesn’t collide with another travelling on the same path; one way link ( btn intersections of links). This is a primary problem ensuring that vehicles follow one another in such a manner so as to avoid collisions. Second vehicle must follow the first at a distance and speed such that it can decelerate as necessary and to avoid hitting the 1st vehicle which may have to swarf or decelerate as to avoid an object on its path. The term following control is applied to cover its situation and the term following behaviour is used in those cases where human behaviour is an element in

the vehicle control and decision process.


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CHAPTER TWO

TRAFFIC ASSIGNMENT

It is a stage at which trip interchanges are allocated to different parts of network. In this stage:

The route to be travelled is determined.



The interzonal flows are assigned to the selected route.

APPLICATION OF TRAFFIC ASSIGNMENT

1. To determine the efficiency in the existing transportation system by assigning the future trips to the existing trips. 2. To evaluate the effects of the improvement and additions to the existing system by assigning estimated future trips to the improved network. 3. To develop construction priority by assigning estimate future trip to the proposed transportation system. 4. To test alternative transportation system proposal. 5. Provide design hourly traffic volume (Veh per hour) on highway and determine movement at the Junctions.

GENERAL PRINCIPLES

Assignment techniques are based on: a) Route selection which depends on criteria such as journey time, cost, convenience and safety which can be done by computer or manually. b) Highway network description – is described by a system of links and modes. A l ink is a section of highway network between two intersections which a node is either a centroid of a zone or the intersection of two or more links. c) Algorithm method of determining the shortest or least cost route ( path).


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ASSIGNMENT TECHNIQUES

They include:(i) All or nothing (free assignment or desire assignment)

The technique assumes that all the trips are assigned to the route having the shortest travel time, travel distance or lowest generalized cost. The technique assumes that the capacit y of each route is infinite so that the volume of traffic on it affects neither the time nor the cost of travel.

Example

The figure below shows the minimum path tree connecting zone centroids one with z one controids 2, 3 and 4. The traffic volumes from zone centroid 1 to zone controid 2 3 and 4 are given below:-

From Zone centroid

To Zone Centroid

Traffic volume Veh/Hr

1

2

2500

1

3

3000

1

4

4000 4

3

1

11

12

15

2

It is required to assign the flow from zone centroid (1) to zone centroid (2), (3) & (4).


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Solution

From the figure the traffic volumes assigned to th various links are:

Link

Traffic flow ( veh/hr)

1-11

9500

11-12

2500

12-2

2500

11-15

7000

15-18

7000

18-3

3000

18-4

4000

ADVANTAGES OF ALL OR NOTHING ASSIGNMENTS TECHNIQUE

-method is simple -The method can be used in the first stage of capacity

DIASDVANTAGES

1. Only one factor i.e time is used at any one time to determine the least cost ( minimum path) – other equally important factors such as reliability, cost, convenience and safety will be neglected. But a driver ma y attach more value to these neglected factors thus causing errors in the assigned flow. 2. Because of the very principle on which the technique is base too many vehicles tend to be assigned to many attractive routes. They may cause increasing congestion in these routes. 3. Where travel times or cost is used the methods tend to ignore the tendency of people to use superior facilities for longer journeys. 4. Small differences in journey times of different routes between the same origin and destination can result in unrealistic journey paths.


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MULTI – ROUTE ASSIGNMENT TECHNIQUE

It is based on assumptions i.

All road users may not be able to judge the minimum path for themselves.

ii. All road users may not have the same criteria for judging the shortest route. iii. Driver doesn’t know the least cost journey route, hence involving random selection criteria. In a multi-route assignment, interzonal flow is assigned to series of routes, the proportion of the total assigned to each route being a function of the length of that route in relation to the shortest route. Multiple rout models yield more accurate assignment more than all or nothing assignment.

MULTI- ROUTE ASSIGMENT ALGARITHMS (A)

Moore method

Aimed to assign a label to each note on the network (b)

Mchauglin

A drivers route selection criteria is used which is a function of (i)

Travel times

(ii)

Travel costs

(iii) Accident potential

The minimum resistance paths between origin and destination pair are calculated with all the links resistance set to values which correspond to a zero traffic volume. Resistance can be increased in terms of -Travel time

- Distance

-Travel cost

-A suitable combination of these parameters

III

CAPACITY RESTRAINT ASSIGNMENT TECHNIQUES

Is a process in which the travel resistance of a link is increased according to a relation between the practical capacity of the link and the volumes assigned to the link. This technique attempts to stimulate the real life situation and also takes account of congestion which builds up with increased traffic volumes. It has been developed to overcome the inherent weakness of all or nothing technique which takes no account of the system between a pair of zones. The capacity restraint system restrains the number of vehicles that can use any particular corridor if the volumes are beyond the capacity of the network and redistributes the traffic to realistic or alternative paths.


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Because of the interactive nature of the calculations involved, the capacity restraint technique is carried out entirely by computers. The first stage of this method involves building of minimum paths trees in the same way as they are build for all nothing techniques. Traffic is then assigned to these minimum path trees. As the assigned volume on each link increases, the computer automatically lowers the assumed speeds on the affected links through an interactive procedure in which loaded links information is used as a feedback to the tree building process as shown below:

Road Network (Link node, Q-V Condition) Interzonal movements for assignment

To set the speed of each link

Change the speed of each link

Interzonal movements stage (Divided)

To search the minimum travel time route

To assign the interzonal movements to each link

Are all trips assigned?

No

Yes

Interzonal movements stage (Divided)


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It is assumed that the relationship between the journey time (speed) and volume of each li nk in the road network is as shown below:-

Speed

Speed

(V)

(V)

Volume (Q)

Volume (Q)

An example of volume – speed condition

Some of the models based in capacity restrains technique used in transport planning study are:i) The Detroit model

In the Detroit traffic study an interactive procedure was used. Traffic was assigned to various links using all, or nothing approach. The speed assumed for initial assignment purpose was the free (unrstrained) speed. NB: Traffic on the network will not always operate under free flow conditions. The speed on each link is affected by flow. Therefore each link’s travel times was modified according to the function: v

1



T A



T 0 e

c

Where

T A = Adjusted travel time T 0 = Original assigned travel time ( a function of the desired operating speed) or the travel time on Link when v=c v = Assigned Volume c = Computed Capacity The second interaction was accomplished by using new travel times, to determine a new service of maximum paths. The volume so determined is then added to the results of the


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previously interaction and the average link load determined. Successive re-interactions recalculate the TA values based on the model using the average link volume for the v-value until a balanced network is obtained.

ii) The TRC trip assignment model

The model involves two travel times verses the volume relationships; used interactively to arrive at prediction of volume on separate routes b etween any two zones.

The equation, used for predicting the volume on a route, v, is given by

1 V r

t r



m

xV

1

 t r 1

r

Where Vr = volume of traffic on route r ( veh/hr/lane) tr = travel time on route r ( minutes) V = Total volume of traffic (trips) from zone I to j in all m routes

Example

The total trip volumes from zone 1 to 2 are 2000. Using data in table below, find the volumes on each route connecting the two zones, using the particulars given in the table below: Note: Use the TRC trip assignment model

Route No.

Length (km)

Speed (km/h)

1

2

30

2

1.7

20

3

1

5

4

2.5

15


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Solution 4

Zone 1

Zone 2

3

2

1 The above figure shows the four routes

Travel time on route 1 t r 1

Dis tan ce 

Speed

2 

30

x60



4 min

Travel time on route 2 t 21

Dis tan ce 

Speed

1.7 

20

x60



5.1 min


Dis tan ce 

Speed

1 

5

x60



12 min


Dis tan ce 

Speed

2.5 

15

x60



10 min

The TRC assignment model is

1

t r

V r 

m

1

xV --------------------------------------------------------------------------Eqn 1

 t r 1

r

Total volume between zone 1 and 2, V=2000 trips 4

But

1

 t r 1

 0.25  0.196  0.0833 0.1  0.63

r


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1

t 1

Volume on route V 1 

4

xV 

1

 t r 1

Volume on route V 2  Volume on route V 3  Volume on route V 4 

0.25 0.63

x 2000  794trips

r

0.19 6 0.63 0.23 3 0.63 0.1 0.63

x 2000 

623trips

x 2000 

26 5trips

x 2000  31 8trips

Hence Route No.

Length (km)

Speed (km/h)

Calculated travel

Calculated 1/tr

time tr Min 1

2

30

4

0.25

2

1.7

20

5.1

0.196

3

1

5

12

0.0833

4

2.5

15

10

0.100

Total

0.6293

The equation (i.e TRC) defines up the volumes of trips from zone i to zone j among various routes in accordance with the reciprocal of travel times. The decrease in t r thus leads into an increase of 1/tr trips assigned to route i. The value of tr for interactions is found by the following equipment.

t r  t rc 

d V r  V rc  V rc

xLr -----------------------------------------------------------------Eqn 2

Where trc = unit travel time at the critical volume (min/km) tr = Travel time on route r given in min Vr = critical volume for route r veh/hr/lane Lr = unit travel length at the critical volume (min/km) d= delay parameter (min/km) Where d is taken as follows

d= 0.5 for Vr < Vrc d= 10.0 for V r ≥ Vrc


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The values of tr found by equation 2 are put in equation 1 and the resulting V r is then put in equation 2. This cyclic procedure continues until the changes in volume of travel time become negligible.

EXAMPLE

Between the two zones (1) and (2) there are two routes whose characteristics are given in the table below. The total number of trips between the two zones is 1200 trips/hr

Route

No. of

Speed

Length

Critical

Critical

Ideal travel time

NO.

Tones

Limit

(km)

Vol. unit lane

travel

With no vol. min/km

1

One

30

3

600

3

2.5

2

One

50

4

1100

2

1.5

The assignment procedure starts using the ideal tra vel time for entire length of each route.

Ideal situations, no traffic t1=Total travel time on route 1=2.5x3=7.5 Min t2=Total travel time on route 2=1.5x4=6 Min

Using Equation 1

1

V r 

t r m

1

xV

 t r 1

r

Volume on route 1 1

V 1 

7.5 1 1



7.5

x1200 

53 2

6

Volume on route 2 1

V 2 

6 1 7.5



1

x1200 

66 8

6

Using the above values of V 1 and V2 in equation 2, we find the revised travel times.


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F irst I nteraction

t r  t rc 


xLr , here d=0.5 because Vr
Travel time on route 1, t 1  3 x3 



 x3  8.82

0.5 53 2  600 600

Travel time on route 2, d=0.5 because V r
t 2  2 x 4 



x

0.5 66 8  1100 1100

4  7.20

Going back to equation 1, using these, travel times, volumes after interaction can be calculated Volume on route 1 1

V 1 

8.82 1 8.82

1



x1200 

53 6vph

7.2

Volume on route 2 1

V 2 

7.2 1 8.82



1

x1200 

66 4vph

7.2

Using the results as inputs, use next set of interactions Second I nteraction

t r  t rc 


xLr , here d=0.5 because Vr
Travel time on route 1, t 1  3 x3 





0.5 536  600 60 0

x3  8.85 min

Travel time on route 2, d=0.5 because V r
t 2  2 x 4 





0.5 66 4  1100 1100

x4 

7.18 min

The last set of travel times do not differ significantly from the previous calculated 8.82 is almost equal to 8.85 min and 7.2 is almost equal to 7.18 min


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So the interaction procedure can be terminated at this point. so the final results are V1=536vph/lane and V 2=664vph/lane t1=8.85Min and t 2=7.18Min

NB This was a very simple case involving only two routes. With more routes and greater volume differences and varying travel times; this procedure requires considerable time and is done through computers.

IV

DIVERSION CURVE METHOD OF ASSIGNMENT

It is based on empirical data. They show the proportion of traffic start that is likely to be diverted on a new facility (bypass, new expressing, new street etc); once such a facility is constructed the data collected from the pattern of road usage in the first serve build up such curves – diversion curves. The proportion of diverted traffic is generally related to such parameters as distance, travel times, cost or speed. Diversion curves can be constructed using a variety of variables such as travel time saved, distance saved, travel time ratio, distance ratio, distance and speed ratio & travel cost ratio.


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CHAPTER THREE

HIERARCHY OF HIGHWAY NETWORKS

Individual urban streets can be located in an hierarchy according to the function thy perform. Common functions are – 

Primary distributor, district



Local distributor



Access roads



Access ways



Private drives



Cycle wages



Food ways

The standard to which ways should be constructed follows from its positi on in this hierarchy. Estate roads except private drives re maintained at the public expense. Therefore the local highway authority ensures that the construction standards adopted are appropriate. Industrial estate roads can be developed on hierarchical principal, although they are likely to be lower down the hierarchy than the residential local roads. Pavement structure need to be stronger and turning facilities less restricted than on a residential estate. Roads giving access to single industrial or commercial premises or to a modern light industrial estate or office park are unlikely to be adopted by the highway authority.

DISTRIBUTOR ROADS

In a purely theoretical road networks all roads would be identified as being distributors, access roads or residential. In such a system all trips would start from premises located in residential/access roads proceed up through the hierarchy of distributor roads for the main part of the journey and then go back on to access/residential roads to re ach their destination. Such a system cannot exist in practice. Where parking spaces are not assigned to a particular dwelling, it is important to ensure they are conveniently located; otherwise they would not be used. Where walking distances are excessive the result tends to be that vehicles are parked on verges leading to damage to kerbs manholes etc. Junctions on access roads should be carefully locate d with a minimum distance of between 45m and 90m i.e between the centre lines of side roads of the same side of main road. Kerb radius and junctions between access roads and distributor roads should generally be 10m unless a significant number of long vehicles are expected to use the function. Kerb radii


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should be restricted on purely residential roads as a means of reducing speed. Although a judgment will have to be made bearing in mind the damage caused by vehicle wheels striking the kerb.

RESIDENTIAL ROADS

Many considerations which apply to access roads will also apply to residential roads. Residential roads serve 1-15 dwellings. They will not normally provide through routes except in the case of a loop having both ends on the same acc ess roads. Carriage way will be between 2.8m and 4.5m. Shared surfaces without kerbs and separate footways will be likely although in such a situation there must be a verge or other facilities provided within the highway in order to accommodate services. Adequate facilities must be provided to ensure that any vehicle likely to be using the road can turn in order to exit in forward gear. The worst case is usually considered to be large removals of vehicles.

PEDESTRIAN AND CYCLE WAYS

In early designs, environmentally friendly estate concentrated on the desirably of separating vehicular traffic from pedestrian and cycle traffic. They developed these estates concentrated on segregated vehicular access giving direct acc ess to packing areas with an entirely network for pedestrians and cyclists. But these designs were seen to be primary designed for the benefit. Pedestrian routes tended to be tedious/torturous and to have problems with lighting, security and vandalism. At points where the earth way had to cross a road, it tended to be the path that disappeared underground into a dirty under path. This problem was overcome by dividing up the centre area by a grid formed by distributor roads. Each square within the grid is relatively self sufficient so that journey to local facilities can be undertaken by walking or cycling. Long journey to the city centre would be made by car or other mea ns.


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Primary road network

District Distributor

Local Distributor

Residential road network

Residential Collector

Traditional estate road or formal or informal transition

Accessory

Car ways

Private drive

Distributor road network

Major industrial road

Minor industrial road

Fig. 1 Hierarchy of highway networks


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CHAPTER FOUR QUEUING AND CONGESTION OF TRAFFIC Lighthill and Whitham’s theory


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Asst 1

The maximum capacity of a 2-lane carriage way of a 4-lane dual carriage way is 2400 veh/hr. Due to pipeline operations, the width of a 2-lane way is reduced restricting the vehicle to 1200vel/her. When the flow upstream beyond the influence of the bottleneck is reasonably steady and free flowing at 1700 veh/hr, find. (a) The mean speed of traffic in the bottleneck. (b) The rate at which the queue of congested conditions outside bottleneck grows. The mean space headway when the vehicles are stationary is 8m. The relationship between speed and concentration is linear.


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CHAPTER FIVE

NETWORK ANALYSIS

A set of points representing cities in network analysis is called nodes. The set of straight lines ar curves connecting the nodes is called link or arcs or branches. Links or branches are denoted by a pair of points e.g AB, CD, EF, BA, DC

C

5 Nodes 9 Links

A B

E

D

MINIMUM SPAN PROBLEMS

Involves any set of nodes (cities, destinations) and a set of branches in which AB is the same as BA. Each proposed branch (link) has a non negative cost or distance associated with it. The objective of the method is to construct a parti cular net containing all nodes such that the sum of the cost associated with these branches in the network is a minimum.

Example A,B, C,D,E,F G are seven cities, the distance between some of the cities where there are routes available are given below:From

To

Distance

A

B

20

A

E

100

A

D

20

A

C

40

B

D

10

C

D

40

C

G

30


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D

G

70

D

F

100

D

E

70

E

F

80

E

G

50

F

G

30

SOLUTION

A

B

C

D

E

F

G

A

-

20

40

10

100

-

-

B

20

-

-

10

-

-

-

C

-

-

-

40

-

-

30

D

10

10

40

-

70

100

70

E

100

-

-

70

-

80

50

F

-

-

-

100

80

-

30

G

-

-

30

70

50

30

-

Let us choose any one of the points supporting no restrictions. Lets us choose A as the starting point. Step I possible routes AB

20

AC

40

AD

10

AE

100

A

D

Step II . Two nodes are determined now. Get all the routes f rom A and from D (except AD and DA) Available routes from A AB

20

AC

40

AE

100


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Available routes from D DB

10

DC

40

DE

70

DF

100

DG

70

B

D

Select the route whose distance is minimum among the above routes: sel ect DB=10km

Step III . Three nodes A, B, D are determined now. Get all the routes from A, from B and from D (except AB, BA, BD, DB, AD and DA)

Available routes from A

Available routes from

Available routes from D

B AC

40

-

DC

40

AC

100

-

DE

70

-

-

DF

100

-

-

DG

70

C

D

C

A

Select the route whose distance is minimum among the above routes: sel ect either AC or DC=40km

Step IV . Four nodes are available, A, B, C, D. Get all the routes from A, from B, from C and from D (except those already written)

AE

100

CG

30

DG

C

70

DE

70

G

Select the route whose distance is minimum among the above routes: select CG=30km


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Step V. Five nodes are available, A, B, C, D, G. Get all the routes from the five nodes (except those already written)

AE

100

B-

C-

DE

70

GE

50

GF

30

F

G

Select the route whose distance is minimum among the above routes: sel ect GF=30km

Step VI. Six nodes are available, A, B, C, D, G, F. Get a ll the routes from the six nodes (except those already written)

AE

100

B-

C-

DE

70

FE

80

GE

50

E

G

Select the route whose distance is minimum among the above routes: sel ect GE=50km

B

F E

30 10

A

10

50 G

D 30

40

C

Total distance is 170km


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Assignment II The national park service plans to develop a wide area for tourism. Four locations in the area are very important. The distance between them ( in km) are given in the table below. The park service wants to minimize the km of the total distance to see all the four spaces. Determine how the road should be constructed to achieve this objective. P is the entrance to the park. P

W

M

S

L

P

-

7

20

19

26

W

7

-

8

16

13

M

20

8

-

18

5

S

19

16

18

-

17

L

26

13

5

17

-


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CHAPTER FIVE

SHORTEST ROUTE /DISTANCE (Lowest cost)

The shortest route problem involves a connected network having a non-negative number associated with cost or distance or time associated with each branch (between two nodes). Starting node is called a source and the final node is called destination (sink). The objective of the shortest route problem is to determine the path or the route joining the source and the sink; such that the sum of the cost or distance or time on a particular path is minimum. Example

An individual who lives in R and works in W seeks a route that will minimize the morning driving time. He has recorded the driving time in minutes between the intermediate nodes. There are no major roads directly linking some of the nodes as shown in the table, determine the best commuting route for the individual.

C

12

O 32

28

18

17 R

W

17 32 T 11 4 P

R

C

O

T

P

W

R

-

18

-

32

-

-

C

18

-

12

28

-

-

O

-

12

-

17

-

32

T

32

28

17

-

4

17

P

-

-

-

4

-

11

W

-

-

32

17

11

-


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SOLUTION Algorithm 1

Write down all possible routes from each node in increasing order under each node (Route back to R and the from W should be avoided) R

C

O

T

P

RC 18

CO 12

OC 12

TP 4

PT 4

RC 32

CT 28

OT 17

TO 17

PW 11

OW 32

TW 17

W

TC 28 Algorithm 2

Starting from R, select the route which has shortest time. Among the two routes, decide temporarily the route to C and cancel any other le ading to C. R

C

O

T

P

RC 18

CO 12

OC 12

TP 4

PT 4

RC 32

CT 28

OT 17

TO 17

PW 11

OW 32

TW 17

W

TC 28 Algorithm 3

Consider the cost for one more point, C, T and O nodes R to T = 32 C to O = 12 RCO=RC+CO=18+12=30 Select the route RCO which is equal to 30 rather than route RT=32. Circle route CO, cancel any other route leading to O


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