LESSON ON NETWORKS: Finding the Shortest Route and

1-Center Location

ELENA I. PASCUALEisenhower 9th Grade Center

Aldine ISDHouston, Texas

Mentors:

Dr. HALIT USTERDirector of logistics and Networked Systems

Research LaboratoryDepartment of Industrial Engineering

Texas A & M, College Station

BURCU B. KESKIN (Ph. D. expected 2006)

OOBJECTIVESBJECTIVES:: In this lesson, the students will:In this lesson, the students will:

1. become familiar with basic elements of a 1. become familiar with basic elements of a network;network;

2. integrate physics concepts and skills to 2. integrate physics concepts and skills to solve engineering problems;solve engineering problems;

3. find the shortest route and 1-center 3. find the shortest route and 1-center location using algorithms.location using algorithms.

4. design a network system to solve real life 4. design a network system to solve real life situation problems.situation problems.

BBACKGROUNDACKGROUND SSCIENCECIENCE

Before presenting this lesson, the Before presenting this lesson, the students should have mastered their students should have mastered their skills on physics concepts of skills on physics concepts of speed, speed, distance, displacement, velocity, and distance, displacement, velocity, and acceleration.acceleration.

The students have demonstrated The students have demonstrated mastery on problem solving and mastery on problem solving and critical thinking.critical thinking.

• Mayor White of Houston wants to improve the service efficiency of the Harris County

Fire Department. As such, he proposed that the location of the Harris County Fire

Department Headquarter must be relocated in such a way that the Fire Fighters must

be able to reach any of their assigned service areas, the easiest and the shortest time

possible.

• In order to solve the problem, and at the same time to please the mayor, Maj. Browne,

the newly assigned Head of the Fire Department , consulted Industrial Engr. Spencer to

decide where is the best location for them to move the Headquarter.

• Looking at the different service areas covered by Harris County, as shown in the

diagram, where should be the right location that Engr. Spencer would recommend to

Maj. Browne so as to solve the problem?

AACTIVITY No. 1CTIVITY No. 1

1-CENTER LOCATION ON TREE NETWORK

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1-center

The Physical Analogy ModelThe Physical Analogy Model1-Center Algorithm:1-Center Algorithm:

1. Pick a tip node, call it v.1. Pick a tip node, call it v.

2. Find the tip node farthest away from v, call 2. Find the tip node farthest away from v, call it v’.it v’.

3. Find the tip node farthest away from v’, call 3. Find the tip node farthest away from v’, call it v”.it v”.

4. Find the midpoint of the path v’-v”. This is 4. Find the midpoint of the path v’-v”. This is the optimum 1-center location.the optimum 1-center location.

Activity NO.2Activity NO.2Speedy Delivery:Speedy Delivery: Finding the Shortest Finding the Shortest

RouteRouteMr. Pete Zahat, the driver of a Pizza delivery in the Greater Houston area , wants to find the quickest route from the pizzeria (A) to the largest customer (E) before the pizza becomes cold. What route from A to E do you think requires the least time for him to take in order to satisfy his customers by delivering them really “Hot Pizzas” ?

Speedy Delivery:Speedy Delivery: Finding the Shortest Route Finding the Shortest Route

Activity No. 2Activity No. 2

Hint:• First, find the time it takes for him

to travel between each given distances denoted by the line segments, using the given speed for each specific location. Use the formula

v=d/t or t=d/v

• Label the time between each node in the graph, and then use Dijktra’s Algorithm to find the shortest route. Fill up Table B as you do the Algorithm.

DIAGRAM:DIAGRAM:

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D=3 km v=30km/h

D=4 km v=60 km/h

D=1.5km v=30km/h

D=3.5km v=30km/hD=3.75km

v=45km/h

D=2.75km v=55km/h

D=3.5km v=30km/hD=2km

v=30km/h

D=5km v=60km/h

D=4km v=60km/h

D= 5km v=60km/h

t= 6 min t= 4 min

t= 3 mint= 7 mint= 5 min

t=3 min

t= 7 mint= 4 mint= 5 min

t= 4 min

t= 5 min

Dijkstra’s Algorithm:Dijkstra’s Algorithm:

1.1. Circle the starting node(vertex). Examine all arcs (edges) Circle the starting node(vertex). Examine all arcs (edges) that have that node as an endpoint. that have that node as an endpoint.

2. Examine all uncircled nodes that are adjacent to the circled 2. Examine all uncircled nodes that are adjacent to the circled nodes in the graph.nodes in the graph.

3. Using only circled nodes, find lengths of each path from 3. Using only circled nodes, find lengths of each path from starting point to those nodes in starting point to those nodes in step 2. step 2. Choose the node Choose the node and arc that yield the shortest path. Circle this node. Ties and arc that yield the shortest path. Circle this node. Ties are broken arbitrarily ( if two or more paths have the same are broken arbitrarily ( if two or more paths have the same total length, then you can choose either of them).total length, then you can choose either of them).

4. Repeat steps 2 and 3 until all nodes are circled. Using the 4. Repeat steps 2 and 3 until all nodes are circled. Using the labels and distances next to each node, you can back trace labels and distances next to each node, you can back trace the shortest path from each node to your starting point.the shortest path from each node to your starting point.

STEPS:

STEPS:STEPS:

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6 4

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∞∞

∞∞

∞∞

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STEPS:STEPS:

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∞∞ 5A

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STEPS:STEPS:

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∞ 8F

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STEPS:STEPS:

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STEPS:STEPS:

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STEPS:STEPS:

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RESULT:RESULT:

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Dijkstra’s AlgorithmDijkstra’s AlgorithmCircled NodeCircled Node Adjacent Nodes Adjacent Nodes

(uncircled)(uncircled)Path (from A)Path (from A) Total TimeTotal Time

11stst A A 44

55

22ndnd B B FF

CCABFABF

ABCABC77

1010

33rdrd F F

GG

CCAFGAFG

AFCAFC

88

1212

1212

44thth D D ________________________

________________________

________________________

________________________

________________________

________________________

________________________

________________________

________________________

55thth C C NONENONE

66thth _____ _____ ________________ __________________ ________________________

77thth _____ _____ NONENONE

ABB

F AF

D AFD

RRESEARCH ESEARCH PPROJECTROJECTSituation:Situation:

It has been an observation that many residents in It has been an observation that many residents in different neighborhoods in the Houston area are different neighborhoods in the Houston area are victims of theft and burglary. Most of these victims of theft and burglary. Most of these victims have a burglar alarm system installed in victims have a burglar alarm system installed in their houses. However, most of the time when the their houses. However, most of the time when the alarm goes off, when intruders or robbers breaks alarm goes off, when intruders or robbers breaks in, before the police could even arrive, the in, before the police could even arrive, the robbers are long gone with all the most expensive robbers are long gone with all the most expensive and valuable belongings they could get. and valuable belongings they could get.

RRESEARCH ESEARCH PPROJECT cont…ROJECT cont…

Problem:Problem:

Design a network system in your area wherein Design a network system in your area wherein you could set the best possible location of a you could set the best possible location of a police station such that when their services are police station such that when their services are needed, they could reach the residents the needed, they could reach the residents the fastest and the earliest possible time.fastest and the earliest possible time.

Draw a model of your network system and Draw a model of your network system and explain your procedure on how to solve the explain your procedure on how to solve the problem.problem.

AACKNOWLEDGEMENTCKNOWLEDGEMENT EE3 3 Organizing Committee led by Organizing Committee led by Dr. Jan RinehartDr. Jan Rinehart at Texas A & M at Texas A & M

University, College Station, TxUniversity, College Station, Tx

Dr. Bruce HerbertDr. Bruce Herbert, Facilitator, Facilitator EE33 Summer Institute for Secondary Science and Math Teachers, 2006 Summer Institute for Secondary Science and Math Teachers, 2006

Dr. Halit UsterDr. Halit Uster, Director of the , Director of the Logistics and Networked Systems Research Logistics and Networked Systems Research Laboratory,Laboratory, Department of Industrial and Systems Engineering, Texas A & M Department of Industrial and Systems Engineering, Texas A & M University, College Station, TX.University, College Station, TX.

Burcu B. KeskinBurcu B. Keskin (Ph.D. candidate) (Ph.D. candidate)

Other Members of the Logistics and Networked Systems Research LaboratoryOther Members of the Logistics and Networked Systems Research Laboratory Gopal EaswaranGopal Easwaran (Ph.D. candidate) (Ph.D. candidate) Panitan KewcharoenwongPanitan Kewcharoenwong (Ph.D. student) (Ph.D. student) Hui LinHui Lin (Ph. D. student) (Ph. D. student) Richard A. IveyRichard A. Ivey (USRG, Summer 2006) (USRG, Summer 2006)

Annette CoronadoAnnette Coronado (Team member, E (Team member, E3 3 Summer Institute 2006) Summer Institute 2006)

RREFERENCESEFERENCES Krajewski, Lee., Ritzman, Larry P. “Operations Management Krajewski, Lee., Ritzman, Larry P. “Operations Management

Strategy and Analysis”,6Strategy and Analysis”,6thth edition, 2002 edition, 2002 Jay Heiser, Barry Render, “Operations Management”, 8Jay Heiser, Barry Render, “Operations Management”, 8THTH

edition, 2006edition, 2006 http://ie.tamu.edu/http://ie.tamu.edu/ http://hsor.org/modules.cfm?name=Speedy_Deliveryhttp://hsor.org/modules.cfm?name=Speedy_Delivery http://www.nasaexplores.com/show_912_teacher_st.php?idhttp://www.nasaexplores.com/show_912_teacher_st.php?id

=040402133024=040402133024 http://www.sciencejoywagon.com/physicszone/lessonch/o1http://www.sciencejoywagon.com/physicszone/lessonch/o1

motion/linear/velocity/avg.htmmotion/linear/velocity/avg.htm

http://www.teachengineering.comhttp://www.teachengineering.com