transportation

Copyright 2006 John Wiley & Sons, Inc.Copyright 2006 John Wiley & Sons, Inc.Beni AsllaniBeni Asllani

University of Tennessee at ChattanoogaUniversity of Tennessee at Chattanooga

Operations Management - 6hh EditionOperations Management - 6hh Edition

Chapter 11 SupplementChapter 11 Supplement

Roberta Russell & Bernard W. Taylor, III

Operational Decision-Making Tools: Operational Decision-Making Tools: Transportation and Transshipment ModelsTransportation and Transshipment Models

Copyright 2006 John Wiley & Sons, Inc.Copyright 2006 John Wiley & Sons, Inc. Supplement 10-Supplement 10-22

Just how do you make Just how do you make decisions?decisions?

Emotional directionEmotional direction IntuitionIntuition Analytic thinkingAnalytic thinking Are you an intuit, an analytic, what???Are you an intuit, an analytic, what??? How many of you use models to make How many of you use models to make

decisions??decisions??

42

Arise whenever there is a perceived Arise whenever there is a perceived difference between difference between what is desiredwhat is desired and and what iswhat is in actuality. in actuality.

Problems serve as motivators for doing Problems serve as motivators for doing somethingsomething

Problems lead to decisionsProblems lead to decisions

ProblemsProblems


Model Classification CriteriaModel Classification Criteria

PurposePurpose PerspectivePerspective

Use the perspective of the targeted decision-makerUse the perspective of the targeted decision-maker Degree of AbstractionDegree of Abstraction Content and FormContent and Form Decision EnvironmentDecision Environment {This is what you should start any modeling {This is what you should start any modeling

facilitation meeting with}facilitation meeting with}


PurposePurpose

PlanningPlanning ForecastingForecasting TrainingTraining Behavioral researchBehavioral research


PerspectivePerspective

DescriptiveDescriptive ““Telling it like it is”Telling it like it is” Most simulation models are of this typeMost simulation models are of this type

PrescriptivePrescriptive ““Telling it like it should be”Telling it like it should be” Most optimization models are of this typeMost optimization models are of this type


Degree of AbstractionDegree of Abstraction

IsomorphicIsomorphic One-to-oneOne-to-one

HomomorphicHomomorphic One-to-manyOne-to-many


Content and FormContent and Form

verbal descriptionsverbal descriptions mathematical constructsmathematical constructs simulationssimulations mental modelsmental models physical prototypesphysical prototypes


Decision EnvironmentDecision Environment

Decision Making Under CertaintyDecision Making Under Certainty TOOL: all of mathematical programmingTOOL: all of mathematical programming

Decision Making under Risk and UncertaintyDecision Making under Risk and Uncertainty TOOL: Decision analysis--tables, trees, TOOL: Decision analysis--tables, trees,

Bayesian revisionBayesian revision

Decision Making Under Change and Decision Making Under Change and ComplexityComplexity TOOL: Structural models, simulation modelsTOOL: Structural models, simulation models


Mathematical ProgrammingMathematical Programming Linear programmingLinear programming Integer linear programmingInteger linear programming

some or all of the variables are integer variablessome or all of the variables are integer variables

Network programming (produces all integer Network programming (produces all integer solutions)solutions)

Nonlinear programmingNonlinear programming Dynamic programmingDynamic programming Goal programmingGoal programming The list goes on and onThe list goes on and on

Geometric ProgrammingGeometric Programming


A Model of this classA Model of this class

What would we include in it?What would we include in it?


Management Science ModelsManagement Science Models

A QUANTITATIVE REPRESENTATION A QUANTITATIVE REPRESENTATION OF A PROCESS THAT CONSISTS OF OF A PROCESS THAT CONSISTS OF THOSE COMPONENTS THAT ARE THOSE COMPONENTS THAT ARE

SIGNIFICANT FOR THE SIGNIFICANT FOR THE PURPOSEPURPOSE BEING CONSIDEREDBEING CONSIDERED


Mathematical programming models Mathematical programming models covered in Ch 11, Supplementcovered in Ch 11, Supplement

Transportation ModelTransportation Model Transshipment ModelTransshipment Model

Not included are:Shortest RouteMinimal Spanning TreeMaximal flowAssignment problemmany others


Transportation ModelTransportation Model

A transportation model is formulated for a class of A transportation model is formulated for a class of problems with the following characteristicsproblems with the following characteristics a product is transported from a number of sources to a a product is transported from a number of sources to a

number of destinations at the minimum possible costnumber of destinations at the minimum possible cost each source is able to supply a fixed number of units of each source is able to supply a fixed number of units of

productproduct each destination has a fixed demand for the producteach destination has a fixed demand for the product

Solution (optimization) AlgorithmsSolution (optimization) Algorithms stepping-stonestepping-stone modified distributionmodified distribution Excel’s SolverExcel’s Solver


Transportation Method: ExampleTransportation Method: Example


Problem Formulation Using Excel

Total Cost Formula


Using Solver from Tools

Menu


Solution


Modified Problem Solution


The Underlying NetworkThe Underlying Network


For problems in which there is an For problems in which there is an underlying network: underlying network:

There are easy (fast) solutionsThere are easy (fast) solutions An exception is the traveling salesman An exception is the traveling salesman

problemproblem

The solutions are always integer onesThe solutions are always integer ones {How about solving a 50,000 node {How about solving a 50,000 node

problem in less than a minute on a problem in less than a minute on a laptop??}laptop??}


CARLTON PHARMACEUTICALSCARLTON PHARMACEUTICALS

Carlton Pharmaceuticals supplies drugs and other Carlton Pharmaceuticals supplies drugs and other medical supplies.medical supplies.

It has three plants in: Cleveland, Detroit, It has three plants in: Cleveland, Detroit, Greensboro.Greensboro.

It has four distribution centers in: It has four distribution centers in: Boston, Richmond, Atlanta, St. Louis.Boston, Richmond, Atlanta, St. Louis.

Management at Carlton would like to ship cases of a Management at Carlton would like to ship cases of a certain vaccine as economically as possible.certain vaccine as economically as possible.


DataData Unit shipping cost, supply, and demandUnit shipping cost, supply, and demand

AssumptionsAssumptions Unit shipping cost is constant.Unit shipping cost is constant. All the shipping occurs simultaneously.All the shipping occurs simultaneously. The only transportation considered is between The only transportation considered is between

sources and destinations.sources and destinations. Total supply equals total demand.Total supply equals total demand.

To From Boston Richmond Atlanta St. Louis Supply Cleveland $35 30 40 32 1200 Detroit 37 40 42 25 1000 Greensboro 40 15 20 28 800 Demand 1100 400 750 750

To From Boston Richmond Atlanta St. Louis Supply Cleveland $35 30 40 32 1200 Detroit 37 40 42 25 1000 Greensboro 40 15 20 28 800 Demand 1100 400 750 750


NETWORK NETWORK

REPRESENTATIONREPRESENTATION Boston

Richmond

Atlanta

St.Louis

Destinations

Sources

Cleveland

Detroit

Greensboro

S1=1200

S2=1000

S3= 800

D1=1100

D2=400

D3=750

D4=750

37

40

42

32

35

40

30

25

3515

20

28


• The Associated Linear Programming Model The Associated Linear Programming Model

The structure of the model is:The structure of the model is:

Minimize <Total Shipping Cost>Minimize <Total Shipping Cost>

STST

[Amount shipped from a source] = [Supply at that source][Amount shipped from a source] = [Supply at that source]

[Amount received at a destination] = [Demand at that [Amount received at a destination] = [Demand at that destination]destination]

Decision variablesDecision variablesXXijij = amount shipped from source i to destination j. = amount shipped from source i to destination j.

where: i=1 (Cleveland), 2 (Detroit), 3 (Greensboro) where: i=1 (Cleveland), 2 (Detroit), 3 (Greensboro)

j=1 (Boston), 2 (Richmond), 3 (Atlanta), 4(St.Louis) j=1 (Boston), 2 (Richmond), 3 (Atlanta), 4(St.Louis)


Boston

Richmond

Atlanta

St.Louis

D1=1100

D2=400

D3=750

D4=750

The supply constraints

Cleveland S1=1200

X11

X12

X13

X14

Supply from Cleveland X11+X12+X13+X14 = 1200

DetroitS2=1000

X21

X22

X23

X24

Supply from Detroit X21+X22+X23+X24 = 1000

GreensboroS3= 800

X31

X32

X33

X34

Supply from Greensboro X31+X32+X33+X34 = 800


• The complete mathematical programming modelThe complete mathematical programming model

Minimize 35X11+30X12+40X13+ 32X14 +37X21+40X22+42X23+25X24+ 40X31+15X32+20X33+38X34

ST

Supply constrraints:X11+ X12+ X13+ X14 1200

X21+ X22+ X23+ X24 1000X31+ X32+ X33+ X34 800

Demand constraints: X11+ X21+ X31 1000

X12+ X22+ X32 400X13+ X23+ X33 750

X14+ X24+ X34 750

All Xij are nonnegative

===

====


Excel Optimal SolutionExcel Optimal Solution

CARLTON PHARMACEUTICALS

UNIT COSTSBOSTON RICHMOND ATLANTA ST.LOUIS SUPPLIES

CLEVELAND 35.00$ 30.00$ 40.00$ 32.00$ 1200DETROIT 37.00$ 40.00$ 42.00$ 25.00$ 1000GREENSBORO 40.00$ 15.00$ 20.00$ 28.00$ 800

DEMANDS 1100 400 750 750

SHIPMENTS (CASES)BOSTON RICHMOND ATLANTA ST.LOUIS TOTAL

CLEVELAND 850 350 0 0 1200DETROIT 250 0 0 750 1000GREENSBORO 0 50 750 0 800

TOTAL 1100 400 750 750 TOTAL COST = 84000


Range of optimality

WINQSB Sensitivity AnalysisWINQSB Sensitivity Analysis

If this path is used, the total cost will increase by $5 per unit shipped along it


Range of feasibility

Shadow prices for warehouses - the cost resulting from 1 extra case of vaccine demanded at the warehouse

Shadow prices for plants - the savings incurred for each extra case of vaccine available at the plant


Transshipment Transshipment Model Model


Transshipment Model: SolutionTransshipment Model: Solution


Copyright 2006 John Wiley & Sons, Inc.Copyright 2006 John Wiley & Sons, Inc.All rights reserved. Reproduction or translation All rights reserved. Reproduction or translation of this work beyond that permitted in section 117 of this work beyond that permitted in section 117 of the 1976 United States Copyright Act without of the 1976 United States Copyright Act without express permission of the copyright owner is express permission of the copyright owner is unlawful. Request for further information should unlawful. Request for further information should be addressed to the Permission Department, be addressed to the Permission Department, John Wiley & Sons, Inc. The purchaser may John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only make back-up copies for his/her own use only and not for distribution or resale. The Publisher and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, assumes no responsibility for errors, omissions, or damages caused by the use of these or damages caused by the use of these programs or from the use of the information programs or from the use of the information herein. herein.


DEPOT MAX DEPOT MAX

A General Network ProblemA General Network Problem

Depot Max has six stores.Depot Max has six stores. Stores 5 and 6 are running low on the Stores 5 and 6 are running low on the

model model 65A Arcadia workstation, and need a total 65A Arcadia workstation, and need a total of 25 additional units. of 25 additional units.

Stores 1 and 2 are ordered to ship a total Stores 1 and 2 are ordered to ship a total of 25 units to stores 5 and 6. of 25 units to stores 5 and 6.

Stores 3 and 4 are transshipment nodes Stores 3 and 4 are transshipment nodes with no demand or supply of their own.with no demand or supply of their own.


Other restrictionsOther restrictions There is a maximum limit for quantities There is a maximum limit for quantities

shipped on various routes.shipped on various routes. There are different unit transportation costs There are different unit transportation costs

for different routes.for different routes.

Depot Max wishes to transport the Depot Max wishes to transport the available workstations at minimum total available workstations at minimum total cost.cost.


1

2 4

3 5

6

5

10

20

6

15

12

7

15

117

Transportation unit cost

• DATA:

Network presentation–Supply nodes: Net flow out of the node] = [Supply at the node]X12 + X13 + X15 - X21 = 10 (Node 1)X21 + X24 - X12 = 15 (Node 2)

–Intermediate transshipment nodes: [Total flow out of the node] = [Total flow into the node]X34+X35 = X13 (Node 3)X46 = X24 + X34 (Node 4)

–Demand nodes:[Net flow into the node] = [Demand for the node]X15 + X35 +X65 - X56 = 12 (Node 5)X46 +X56 - X65 = 13 (Node 6)

Arcs: Upper bound and lower bound constraints:

0 X Uij ij


The Complete mathematical modelThe Complete mathematical modelMinimize X X X X X X X X X XST

5 12 10 13 20 15 6 21 15 24 12 34 7 35 15 46 11 56 7 65

X12 + X13 + X15 - X21 = 10

- X12 + X21 + X24 = 15

- X13 + X34 + X35 = 0

- X24 - X34 + X46 = 0

- X15 - X35 + X56 - X65 = -12

- X46 - X56 + X65 = -13

0 X12 3; 0 X13 12; 0 X15 6; 0 X21 7; 0 X24 10; 0 X34 8; 0 X35 8;

0 X46 17; 0 X56 7; 0 X65 5


WINQSB Input DataWINQSB Input Data


WINQSB Optimal SolutionWINQSB Optimal Solution


MONTPELIER SKI COMPANYMONTPELIER SKI COMPANY Using a Transportation model for production Using a Transportation model for production schedulingscheduling

Montpelier is planning its production of skis for the months Montpelier is planning its production of skis for the months of of July, August, and September.July, August, and September.

Production capacity and unit production cost will change Production capacity and unit production cost will change from from month to month.month to month.

The company can use both regular time and overtime to The company can use both regular time and overtime to produce skis.produce skis.

Production levels should meet both demand forecasts and Production levels should meet both demand forecasts and end-of-quarter inventory requirement.end-of-quarter inventory requirement.

Management would like to schedule production to minimize Management would like to schedule production to minimize its costs for the quarterits costs for the quarter..


Data:Data: Initial inventory = 200 pairsInitial inventory = 200 pairs Ending inventory required =1200 pairsEnding inventory required =1200 pairs Production capacity for the next quarter = 400 pairs in Production capacity for the next quarter = 400 pairs in

regular time.regular time.

= 200 pairs in = 200 pairs in overtime.overtime.

Holding cost rate is 3% per month per ski.Holding cost rate is 3% per month per ski.

Production capacity, and forecasted demand for this Production capacity, and forecasted demand for this quarter quarter (in pairs of skis), and production cost per unit (by (in pairs of skis), and production cost per unit (by months)months)

Forecasted Production Production Costs Month Demand Capacity Regular Time OvertimeJuly 400 1000 25 30August 600 800 26 32September 1000 400 29 37

Forecasted Production Production Costs Month Demand Capacity Regular Time OvertimeJuly 400 1000 25 30August 600 800 26 32September 1000 400 29 37


Analysis of demand:Analysis of demand: Net demand to satisfy in July = 400 - 200 = 200 pairsNet demand to satisfy in July = 400 - 200 = 200 pairs

Net demand in August = 600Net demand in August = 600 Net demand in September = 1000 + 1200 = 2200 pairsNet demand in September = 1000 + 1200 = 2200 pairs

Analysis of Supplies:Analysis of Supplies: Production capacities are thought of as supplies.Production capacities are thought of as supplies. There are two sets of “supplies”: There are two sets of “supplies”:

Set 1- Regular time supply (production capacity)Set 1- Regular time supply (production capacity) Set 2 - Overtime supply Set 2 - Overtime supply

Initial inventory

Forecasted demand In house inventory

• Analysis of Unit costs Unit cost = [Unit production cost] +

[Unit holding cost per month][the number of months stays in inventory] Example: A unit produced in July in Regular time and sold in

September costs 25+ (3%)(25)(2 months) = $26.50


Network representation

252525.7525.7526.5026.50 00 3030

30.9030.9031.8031.80

00+M+M

2626

26.7826.78

00

+M+M

3232

32.9632.96

00

+M+M

+M+M

2929

00

+M+M

+M+M

3737

00

ProductionMonth/period

Monthsold

JulyR/T

July O/T

Aug.R/T

Aug.O/T

Sept.R/T

Sept.O/T

July

Aug.

Sept.

Dummy

1000

500

800

400

400

200

200

600

300

2200

Demand

Prod

uctio

n Ca

pacit

y

July R/T


Source: July production in R/TDestination: July‘s demand.

Source: Aug. production in O/TDestination: Sept.’s demand

32+(.03)(32)=$32.96Unit cost= $25 (production)Unit cost =Production+one month holding cost


Summary of the optimal solutionSummary of the optimal solution In July produce at capacity (1000 pairs in R/T, and 500 In July produce at capacity (1000 pairs in R/T, and 500

pairs in O/T). Store 1500-200 = 1300 at the end of July.pairs in O/T). Store 1500-200 = 1300 at the end of July.

In August, produce 800 pairs in R/T, and 300 in O/T. In August, produce 800 pairs in R/T, and 300 in O/T.

Store additional 800 + 300 - 600 = 500 pairs.Store additional 800 + 300 - 600 = 500 pairs.

In September, produce 400 pairs (clearly in R/T). With In September, produce 400 pairs (clearly in R/T). With

1000 pairs 1000 pairs

retail demand, there will be retail demand, there will be

(1300 + 500) + 400 - 1000 = 1200 pairs available for (1300 + 500) + 400 - 1000 = 1200 pairs available for

shipment to shipment to

Ski Chalet Ski Chalet..Inventory + Production -

Demand


Problem 4-25Problem 4-25


6.3 The Assignment Problem6.3 The Assignment Problem

Problem definitionProblem definition m workers are to be assigned to m jobsm workers are to be assigned to m jobs

A unit cost (or profit) CA unit cost (or profit) Cijij is associated with worker i is associated with worker i

performing job j.performing job j.

Minimize the total cost (or maximize the total Minimize the total cost (or maximize the total profit) of assigning workers to job so that each profit) of assigning workers to job so that each worker is assigned a job, and each job is worker is assigned a job, and each job is performed.performed.


BALLSTON ELECTRONICSBALLSTON ELECTRONICS

Five different electrical devices produced on five Five different electrical devices produced on five production lines, are needed to be inspected.production lines, are needed to be inspected.

The travel time of finished goods to inspection The travel time of finished goods to inspection areas depends on both the production line and the areas depends on both the production line and the inspection area.inspection area.

Management wishes to designate a separate Management wishes to designate a separate inspection area to inspect the products such thatinspection area to inspect the products such that the total travel time is minimized.the total travel time is minimized.


Data: Travel time in minutes from Data: Travel time in minutes from assembly assembly lines to lines to inspection areas.inspection areas.

Inspection AreaA B C D E

1 10 4 6 10 12Assembly 2 11 7 7 9 14 Lines 3 13 8 12 14 15

4 14 16 13 17 175 19 17 11 20 19

Inspection AreaA B C D E

1 10 4 6 10 12Assembly 2 11 7 7 9 14 Lines 3 13 8 12 14 15

4 14 16 13 17 175 19 17 11 20 19


NETWORK REPRESENTATIONNETWORK REPRESENTATION

1

2

3

4

5

Assembly Line Inspection AreasA

B

C

D

E

S1=1

S2=1

S3=1

S4=1

S5=1

D1=1

D2=1

D3=1

D4=1

D5=1


Assumptions and restrictionsAssumptions and restrictions

The number of workers equals the number of jobs.The number of workers equals the number of jobs.

Given a balanced problem, each worker is Given a balanced problem, each worker is assigned exactly once, and each job is performed assigned exactly once, and each job is performed by exactly one worker.by exactly one worker.

For an unbalanced problem “dummy” workers (in For an unbalanced problem “dummy” workers (in case there are more jobs than workers), or case there are more jobs than workers), or “dummy” jobs (in case there are more workers than “dummy” jobs (in case there are more workers than jobs) are added to balance the problem.jobs) are added to balance the problem.


Computer solutionsComputer solutions A complete enumeration is not efficient even A complete enumeration is not efficient even

for moderately large problems (with m=8, m! > for moderately large problems (with m=8, m! > 40,000 is the number of assignments to 40,000 is the number of assignments to enumerate).enumerate).

The The Hungarian methodHungarian method provides an efficient provides an efficient solution procedure.solution procedure.

Special casesSpecial cases A worker is unable to perform a particular job.A worker is unable to perform a particular job. A worker can be assigned to more than one A worker can be assigned to more than one

job.job. A maximization assignment problem.A maximization assignment problem.


6.5 The Shortest Path Problem6.5 The Shortest Path Problem

For a given network find the path of minimum For a given network find the path of minimum distance, time, or cost from a starting point,distance, time, or cost from a starting point,the the start nodestart node, to a destination, the , to a destination, the terminal nodeterminal node..

Problem definitionProblem definition There are n nodes, beginning with start node 1 and There are n nodes, beginning with start node 1 and

ending with terminal node n.ending with terminal node n. Bi-directional arcs connect connected nodes i and jBi-directional arcs connect connected nodes i and j

with nonnegative distances, dwith nonnegative distances, d i j i j.. Find the path of minimum total distance that connectsFind the path of minimum total distance that connects

node 1 to node n. node 1 to node n.


Fairway Van LinesFairway Van Lines Determine the shortest route from Seattle to El Determine the shortest route from Seattle to El

Paso over the following network highways.Paso over the following network highways.


Salt Lake City

1 2

3 4

56

7 8

9

1011

1213 14

15

16

17 18 19

El Paso

Seattle

Boise

Portland

Butte

Cheyenne

Reno

Sac.

Bakersfield

Las VegasDenver

Albuque.

KingmanBarstow

Los Angeles

San Diego Tucson

Phoenix

599

691497180

432 345

440

102

452

621

420

526

138

291

280

432

108

469207

155114

386403

118

425 314


Solution - a linear programming approachSolution - a linear programming approach

Decision variablesDecision variables

X ij

10 if a truck travels on the highway from city i to city j otherwise

Objective = Minimize dijXij


7

2

Salt Lake City

1

3 4

Seattle

Boise

Portland

599

497180

432 345

Butte

[The number of highways traveled out of Seattle (the start node)] = 1X12 + X13 + X14 = 1

In a similar manner:[The number of highways traveled into El Paso (terminal node)] = 1X12,19 + X16,19 + X18,19 = 1

[The number of highways used to travel into a city] = [The number of highways traveled leaving the city]. For example, in Boise (City 4):X14 + X34 +X74 = X41 + X43 + X47.

Subject to the following constraints:

Nonnegativity constraints


WINQSB Optimal SolutionWINQSB Optimal Solution


Solution - a network approach Solution - a network approach

The Dijkstra’s algorithm:The Dijkstra’s algorithm: Find the shortest distance from the “START” node Find the shortest distance from the “START” node

to every other node in the network, in the order of to every other node in the network, in the order of the closet nodes to the “START”.the closet nodes to the “START”.

Once the shortest route to the m closest node is Once the shortest route to the m closest node is determined, the shortest route to the (m+1) closest determined, the shortest route to the (m+1) closest node can be easily determined.node can be easily determined.

This algorithm finds the shortest route from the start This algorithm finds the shortest route from the start to all the nodes in the network.to all the nodes in the network.


SEA.SEA.Salt Lake City

1 2

3 4

56

7 8

9

1011

1213 14

15

16

17 18 19

El Paso

Seattle

Boise

Portland

Butte

Cheyene

Reno

Sac.

Bakersfield

Las VegasDenver

Albuque.

KingmanBarstow

Los Angeles

San Diego Tucson

Pheonix

599

691497180

432 345

440

102

452

621

420

526

138

291

280

432

108

469207

155114

386403

118

425 314

BUT599

POR

180

497BOI

599

180

497POR.POR.

BOI432

SAC602

+

+

=

=

612

782

BOI

BOIBOI.BOI.

345SLC+ =

842

BUT.BUT.

SLC

420

CHY.691

+

+

=

=

1119

1290

SLC.

SLCSLC.

SAC.SAC.

An illustration of the Dijkstra’s algorithm

… and so on until the whole network is covered.


6.6 The Minimal Spanning Tree6.6 The Minimal Spanning Tree

This problem arises when all the nodes of a This problem arises when all the nodes of a given network must be connected to one given network must be connected to one another, without any loop.another, without any loop.

The minimal spanning tree approach is The minimal spanning tree approach is appropriate for problems for which appropriate for problems for which redundancy is expensive, or the flow along redundancy is expensive, or the flow along the arcs is considered instantaneous. the arcs is considered instantaneous.


THE METROPOLITAN TRANSIT DISTRICTTHE METROPOLITAN TRANSIT DISTRICT

The City of Vancouver is planning the development of The City of Vancouver is planning the development of

aa

new light rail transportation system. new light rail transportation system.

The system should link 8 residential and commercialThe system should link 8 residential and commercialcenters.centers.

The Metropolitan transit district needs to select the set The Metropolitan transit district needs to select the set of lines that will connect all the centers at a minimum of lines that will connect all the centers at a minimum total cost.total cost.

The network describes:The network describes: feasible lines that have been drafted,feasible lines that have been drafted, minimum possible cost for taxpayers per line.minimum possible cost for taxpayers per line.


5

2 6

4

7

81

3

West Side

North Side University

BusinessDistrict

East SideShoppingCenter

South Side

City Center

33

50

30

55

34

28

32

35

39

45

38

43

44

41

3736

40

SPANNING TREE NETWORK PRESENTATION


Solution - a network approachSolution - a network approach The algorithm that solves this problem is a very easy The algorithm that solves this problem is a very easy

(“trivial”) procedure.(“trivial”) procedure. It belongs to a class of “greedy” algorithms.It belongs to a class of “greedy” algorithms. The algorithm:The algorithm:

Start by selecting the arc with the smallest arc length.Start by selecting the arc with the smallest arc length. At each iteration, add the next smallest arc length to the set At each iteration, add the next smallest arc length to the set

of arcs already selected (provided no loop is constructed).of arcs already selected (provided no loop is constructed). Finish when all nodes are connected.Finish when all nodes are connected.

Computer solution Computer solution Input consists of the number of nodes, the arc length, Input consists of the number of nodes, the arc length,

and the network description.and the network description.


WINQSB Optimal Solution


ShoppingCenter

Loop

5

2 6

4

7

81

3

West Side

North Side

University

BusinessDistrict

East Side

South Side

City Center

33

50

30

55

34

28

32

35

39

45

38

43

44

41

3736

40

Total Cost = $236 million

OPTIMAL SOLUTIONNETWORKREPRESENTATION

transportation

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