1transportation modelslesson 4 lecture four transportation models
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1 Transportation Models Lesson 4
LECTURE FOURLECTURE FOUR
Transportation Transportation ModelsModels
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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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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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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– 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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• 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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• 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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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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• 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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• 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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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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• 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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• 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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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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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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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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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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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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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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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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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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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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• 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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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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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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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.
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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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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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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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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Solving Transportation Solving Transportation Networks ManuallyNetworks Manually
Worked example 5.10 (page 166 – 173) Worked example 5.10 (page 166 – 173)