1transportation modelslesson 4 lecture four transportation models

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Page 1: 1Transportation ModelsLesson 4 LECTURE FOUR Transportation Models

1 Transportation Models Lesson 4

LECTURE FOURLECTURE FOUR

Transportation Transportation ModelsModels

Page 2: 1Transportation ModelsLesson 4 LECTURE FOUR Transportation Models

2 Transportation Models Lesson 4

IntroductionIntroduction

– Many business problems lend Many business problems lend themselves to a network formulation.themselves to a network formulation.

– Network problems can be efficiently Network problems can be efficiently solved by compact algorithms due to solved by compact algorithms due to there special mathematical structure, there special mathematical structure, even for large scale models. even for large scale models.

• The importance of network modelsThe importance of network models

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3 Transportation Models Lesson 4

A NETWORKNETWORK problem is one that can be represented by...

NodesNodes

ArcsArcs

8

9

10

10

7

6

Function on ArcsFunction on Arcs

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4 Transportation Models Lesson 4

Network TerminologyNetwork Terminology– Flow : the amount sent from node i to Flow : the amount sent from node i to

node j, over an arc that connects them. node j, over an arc that connects them.

– The following notation is used:The following notation is used:

• Xij = amount of flowXij = amount of flow

• Uij = upper bound of the flowUij = upper bound of the flow

• Lij = lower bound of the flowLij = lower bound of the flow

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5 Transportation Models Lesson 4

– Directed/undirected arcs : when flow is Directed/undirected arcs : when flow is allowed in one direction the arc is allowed in one direction the arc is directed (marked by an arrow). When directed (marked by an arrow). When flow is allowed in two directions, the arc flow is allowed in two directions, the arc is undirected (no arrows).is undirected (no arrows).

– Adjacent nodes : a node (j) is adjacent to Adjacent nodes : a node (j) is adjacent to another node (i) if an arc joins node i to another node (i) if an arc joins node i to node j.node j.

Network TerminologyNetwork Terminology

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6 Transportation Models Lesson 4

• Path / Connected nodesPath / Connected nodes

–Path is a collection of arcs formed by Path is a collection of arcs formed by a series of adjacent nodes. a series of adjacent nodes.

–The nodes are said to be connected if The nodes are said to be connected if there is a path between them.there is a path between them.

Network TerminologyNetwork Terminology

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7 Transportation Models Lesson 4

• Cycles / Trees / Spanning TreesCycles / Trees / Spanning Trees

– Cycle is a path starting at a certain node and Cycle is a path starting at a certain node and returning to the same node without using any returning to the same node without using any arc twice.arc twice.

– Tree is a series of nodes that contain no cycles.Tree is a series of nodes that contain no cycles.

– Spanning tree is a tree that connects all the Spanning tree is a tree that connects all the nodes in a network.nodes in a network.

Network TerminologyNetwork Terminology

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8 Transportation Models Lesson 4

The Transportation ProblemThe Transportation Problem

Transportation problems arise when a Transportation problems arise when a

cost-effective pattern is needed to ship cost-effective pattern is needed to ship

items from origins that have limited items from origins that have limited

supply to destinations that have supply to destinations that have

demand for the goods.demand for the goods.

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9 Transportation Models Lesson 4

• Problem definitionProblem definition

– There are m sources. Source i There are m sources. Source i has a supply capacity of Si.has a supply capacity of Si.

– There are n destinations. The There are n destinations. The demand at destination j is Dj.demand at destination j is Dj.

The Transportation ProblemThe Transportation Problem

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10 Transportation Models Lesson 4

• Objective:Objective:

Minimize the total shipping cost Minimize the total shipping cost of supplying the destinations of supplying the destinations with the required demand from with the required demand from the available supplies at the the available supplies at the sources.sources.

The Transportation ProblemThe Transportation Problem

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11 Transportation Models Lesson 4

CARLTON PHARMACEUTICALSCARLTON PHARMACEUTICALS

• Carlton Pharmaceuticals supplies drugs and other medical supplies.

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

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

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

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12 Transportation Models Lesson 4

• DataData– Unit shipping cost, supply, and demandUnit shipping cost, supply, and 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

CARLTON PHARMACEUTICALSCARLTON PHARMACEUTICALS

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13 Transportation Models Lesson 4

• AssumptionsAssumptions

– Unit shipping cost is constant.Unit shipping cost is constant.

– All the shipping occurs All the shipping occurs simultaneously.simultaneously.

– The only transportation considered is The only transportation considered is between sources and destinations.between sources and destinations.

– Total supply equals total demand.Total supply equals total demand.

CARLTON PHARMACEUTICALSCARLTON PHARMACEUTICALS

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14 Transportation Models Lesson 4

NETWORK REPRESENTATIONNETWORK REPRESENTATION

BostonBoston

RichmondRichmond

AtlantaAtlanta

St.LouisSt.Louis

DestinationsDestinations

SourcesSources

ClevelandCleveland

DetroitDetroit

GreensboroGreensboro

SS11=1200=1200

SS22=1000=1000

SS33= 800= 800

D1=1100

D2=400

D3=750

D4=750

3737

4040

4242

3232

3535

4040

3030

2525

40401515

2020

2828

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15 Transportation Models Lesson 4

The Mathematical Model The Mathematical Model

–Decision variablesDecision variablesXXijij = amount shipped from source i = amount shipped from source i

to destination j.to destination j.

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

3 (Greensboro) 3 (Greensboro)

j=1 (Boston), 2 (Richmond), j=1 (Boston), 2 (Richmond),

3 (Atlanta), 4(St.Louis)3 (Atlanta), 4(St.Louis)

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16 Transportation Models Lesson 4

BostonBoston

RichmondRichmond

AtlantaAtlanta

St.LouisSt.Louis

D1=1100

D2=400

D3=750

D4=750The Supply ConstraintsThe Supply Constraints

ClevelandCleveland SS11=1200=1200

X11X11

X12X12

X13X13

X14X14

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

DetroitDetroit

SS22=1000=1000

X21X21

X22X22

X23X23

X24X24

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

GreensboroGreensboroSS33= 800= 800

X31X31

X32X32

X33X33

X34X34

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

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17 Transportation Models Lesson 4

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 1100

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

X14+ X24+ X34 750

All Xij are nonnegative

===

====

The Complete Mathematical ModelThe Complete Mathematical Model

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18 Transportation Models Lesson 4

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

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19 Transportation Models Lesson 4

Range of feasibility

Shadow prices for plants (source)- the cost saving resulting from 1 case of vaccine that was NOT shipped out to the distribution centre (destination)

Shadow prices for distribution centre (destination) - the cost incurred for 1 extra case of vaccine shipped to the distribution centre (destination)

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20 Transportation Models Lesson 4

Sensitivity AnalysisSensitivity AnalysisReduced costsReduced costs

• The amount of transportation cost The amount of transportation cost reduction per unit that makes a given route reduction per unit that makes a given route economically attractive. economically attractive.

• If the route is forced to be used under the If the route is forced to be used under the current cost structure, for each item current cost structure, for each item shipped along it, the total cost increases by shipped along it, the total cost increases by an amount equal to the reduced cost.an amount equal to the reduced cost.

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21 Transportation Models Lesson 4

Shadow pricesShadow prices

• For the plants (source), shadow prices For the plants (source), shadow prices convey the cost savings realized for each convey the cost savings realized for each extra case of vaccine available at plant.extra case of vaccine available at plant.

• For the warehouses (destination), shadow For the warehouses (destination), shadow prices convey the cost incurred from prices convey the cost incurred from having an extra case demanded at the having an extra case demanded at the warehouse. warehouse.

Sensitivity AnalysisSensitivity Analysis

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22 Transportation Models Lesson 4

Special Cases of the Special Cases of the Transportation ProblemTransportation Problem

– Cases may arise that appear to violate the Cases may arise that appear to violate the assumptions necessary to solve the assumptions necessary to solve the transportation problem using standard transportation problem using standard methods.methods.

– Modifying the resulting models make it Modifying the resulting models make it possible to use standard solution methods.possible to use standard solution methods.

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23 Transportation Models Lesson 4

• Blocked routes - shipments along certain routes are Blocked routes - shipments along certain routes are prohibited. prohibited.

• Minimum shipment - the amount shipped along a Minimum shipment - the amount shipped along a certain route must not fall below a specified level.certain route must not fall below a specified level.

• Maximum shipment - an upper limit is placed on Maximum shipment - an upper limit is placed on the amount shipped along a certain route.the amount shipped along a certain route.

• Transshipment nodes - intermediate nodes that Transshipment nodes - intermediate nodes that may have demand , supply, or no demand and no may have demand , supply, or no demand and no supply of their own. supply of their own.

Special Cases of the Special Cases of the Transportation ProblemTransportation Problem

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24 Transportation Models Lesson 4

Solution of transportation problems

Two phases:

• 1st phase:– Find an initial feasible solution

• 2nd phase:– Check for optimality and improve the

solution

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25 Transportation Models Lesson 4

Vogel’s approximation methodOperational steps:Step 1: For each column and row, determine its penalty cost by subtracting their two of their least

cost

Step 2: Select row/column that has the highest penalty cost in step 1

Step 3: Assign as much as allocation to the selected row/column that has the least cost

Step 4: Block those cells that cannot be further allocatedStep 5: Repeat above steps until all allocations have been

assigned

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26 Transportation Models Lesson 4

Medical Supply Transportation Problem

• A Medical Supply company produces catheters in packs at three productions facilities.

• The company ships the packs from the production facilities to four warehouses.

• The packs are distributed directly to hospitals from the warehouses.

• The table on the next slide shows the costs per pack to ship to the four warehouses.

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Medical Supply

Seattle New York Phoenix Miami

FROMPLANT

Juarez $19 $ 7 $ 3 $21Seoul 15 21 18 6Tel Aviv 11 14 15 22

TO WAREHOUSE

CapacityJuarez 100Seoul 300Tel Aviv 200

DemandSeattle 150New York 100Phoenix 200Miami 150

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28 Transportation Models Lesson 4

Medical Supply Transportation Problem

J Xjs Xjn Xjp Xjm 100

S N P M

Xss Xsn Xsp Xsm

Xts Xtn Xtp Xtm

150 100 200 150

S 300

T 200

WarehouseDemand 600

TO WAREHOUSE PlantCapacity

From Plant

Number of constraints = number of rows + number of columnsTotal plant capacity must equal total warehouse demand.Although this may seem unrealistic in real world application, it is possible to construct any transportation problem using this model.

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29 Transportation Models Lesson 4

S N P M

150 100 200 150

J 100

S 300

T 200

Demand 600

CapacityFrom

To

19 7 3 21

6

22

18

15

21

14

15

11

Vogel’s approximation method

150

100 7-3=4

15-6=9

14-11=3

18-15=3

150

150

21-18=3

15-14=150 50

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30 Transportation Models Lesson 4

S N P M

150 100 200 150

J 100

S 300

T 200

Demand 600

CapacityFrom

To

19 7 3 21

6

22

18

15

21

14

15

11

Vogel’s approximation method

300

9002700

1650 700 750

300

3600

3100

C =7,000

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31 Transportation Models Lesson 4

Solving Transportation Solving Transportation Networks ManuallyNetworks Manually

Worked example 5.10 (page 166 – 173) Worked example 5.10 (page 166 – 173)