1

15053 Thursday March 14

bull Introduction to Network Flows

bull Handouts Lecture Notes

2

Network Models

bull Linear Programming models that exhibit a very special structure

bull Can use this structure to dramatically reduce computational complexity

bull First widespread application of LP to problems of industrial logistics1048698bull Addresses huge number of diverse applications

3

Notation and Terminology

Note Network terminology is not (and never will be)standardized The same concept may be denoted inmany different ways

Called bull NETWORK bull directed graph bull digraph bull graphClass Handouts (AhujaMagnanti Orlin)Network G = (NA)Node set N = 1234Arc Set (12)(13)(32)(34)(24)

Also SeenGraph G = (VE)Vertex set V = 1234Edge set A=1-21-33-23-42-4

4

Directed and Undirected Networks

An Undirected Graph A Directed Graph

bull Networks are used to transport commodities bull physical goods (products liquids) bull communication bull electricity etc

bull The field of Network Optimization concerns optimization problems on networks

5

An Overview of Some Applications ofNetwork Optimization

Pipelines

Applications Physical analogof nodes

Physical analogof arcs Flow

Communicationsystems

phone exchangescomputers

transmissionfacilities satellites

Cables fiber opticlinks microwave

relay links

Voice messages

Data

Video transmissions

Hydraulic systemsPumping stationsReservoirs Lakes

Water Gas OilHydraulic fluids

Integratedcomputer circuits

Gates registersprocessors Wires Electrical current

Mechanical systems JointsRods Beams

Springs Heat Energy

Transportationsystems

IntersectionsAirports

Rail yards

HighwaysAirline routes

Railbeds

Passengersfreight

vehiclesoperators

6

Examples ofterms

Path Example 5 2 3 4(or 5 c 2 b 3 e 4)Note that directions are ignored

Directed Path Example 1 2 3 4 (or 1 a 2 b 3 e) Directions are important

Two paths a-b-e (or 1-2-3-4)and a-c-d-e (or 1-2-5-3-4)

a-b-c-d (or 1-2-3-4-1)b-a-d-c (or 3-2-1-4-3)e-b-a (or 1-3-2-1)c-d-e (or 3-4-1-3)

Cycles (loops)

Cycle or circuit (or loop) 1 2 3 1 (or 1 a 2 b 3 e)Note that directions are ignored

Directed Cycle (1 2 3 4 1) or 1 a 2 b 3 c 4 d 1Directions are important

7

More Definitions

A network is connected if every nodecan be reached from every othernode by following a sequence ofarcs in which direction is ignored

A spanning tree is a connected subset of a networkincluding all nodes but containing no loops

8

The Minimum Cost Flow Problem

Network G = (N A) ndash Node set N arc set A ndash Capacities uij on arc (ij) ndash Cost cij on arc (ij) ndash Supplydemand bi for node i (Positive indicates supply)

A network with costscapacities supplies demands

Minimize the cost of sending flow st Flow out of i - Flow into i = bi Flow on arc (ij) le uij

9

The Minimum Cost Flow Problem

Let xij be the flow on arc (ij)

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 le xij le uij

Minimizest for all i

for all i-j

10

Example Formulation

Min -3 x12 + 8 x13 + 7 x23 + 3 x24 + 2 x34

st x12 + x13 = 4 x23+ x24 - x12 = 3 x34 - x13 - x23 = -5 - x24 - x34 = -2 0 le x12 le 6 0 le x13 le 5 0 le x23 le 2 0 le x24 le 4 0 le x34 le 7

11

An Application of the Minimum CostFlow Problem

Ship from suppliers to customers possibly throughwarehouses at minimum cost to meet demands

Warehouses

Customers

Demands

PlantsSupplies

12

Useful Facts About The MinimumCost Flow Problem

bull Suppose the following properties of the constraint matrix A (ignoring simple upper and lower variable bounds such as x le 7) hold (1) all entries of A are 0 or 1 or -1 (2) there is at most one 1 in any column and at most one -1bull Then this is a minimum cost flow problem

13

Useful Facts (contrsquod)

Theorem If one carries out the simplex algorithm on the minimum cost flow problem with integer valued capacities and RHS then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1

Corollary The optimal LP solution is integer valued

14

The Minimum Cost Flow Problem

Network G = (N A) ndash Node set N arc set A ndash Capacities uij on arc (ij) ndash Cost cij on arc (ij) ndash Supplydemand bi for node i (Positive indicates supply)

A network with costscapacities supplies demands

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

Flow on arc (ij) le uij

15

The Transportation Problem

Suppose that one wants to ship from warehouses toretailers

In this example 3 warehouses 4 retailersai is the supply at warehouse ibj is the demand at retailer jcij is the cost of shipping from i to jThere are no capacities on the arcs

Let xij be the amount of flow shippedfrom warehouse i to retailer jHow do we formulate an LP

16

The Transportation Problem is a MinCost Flow Problem

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 lexij leuij

Flow out occurs at the supply nodesFlow in occurs at demand nodesCapacities are infinite uij= infin

17

The Transportation Problem

In general the LP formulation is given as

Minimize

All arcs arefrom a node inS to a node inD anduncapacitated

S Supply nodes

D Demand nodes

18

Useful Facts About TransportationProblem

Suppose that (1) the constraint matrix can be partitioned into A1x = b1 and A2x = b2 (2) all entries of A1 and A2 are 0 or 1 (3) there is at most one 1 in any column of A1 or A2Then this is a transportation problemTheorem If one carries out the simplex algorithm on the transportation problem then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1 The costs and RHS are both integer valuedCorollary The optimal solution to the LP is integer valued

19

The Assignment Problem

Suppose that one wants to assign tasks to persons

Tasks Persons

In this example 4 tasks 3 personsNo two tasks to the same personEach person gets a taskcij is the ldquocostrdquo of assigning task ito person j

Let xij = 1 if task i is assigned to jLet xij = 0 otherwiseHow do we formulate an LP

20

The Assignment Problem

In general the LP formulation is given as

Minimize

Each supply is 1

Each demand is 1

21

More on the Assignment Problem

Tasks PersonsThe assignment problem isa special case of thetransportation problem

The simplex algorithm cansolve the LP relaxation andit will give integer answersthat is it will solve theassignment problem

22

An Application of the AssignmentProblem

Suppose that there are moving targets in space You can identifyeach target as a pixel on a radar screen Given two successivepictures identify how the targets have moved

23

The Maximum Flow Problem

Network G = (N A)

ndash Source s and sink t

ndash Capacities uij on arc (ij)

ndash Variable Flow xij on arc (ij) Graph with capacities

Maximize the flow leaving sst Flow out of i - Flow into i = 0 for i ne s t

0 le xij le uij

24

The Max Flow Problem

In general the LP formulation is given as

Minimize v

otherwise

This is not formulated as a special case of aminimum cost flow formulationCan we reformulate it in this way

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

2

Network Models

bull Linear Programming models that exhibit a very special structure

bull Can use this structure to dramatically reduce computational complexity

bull First widespread application of LP to problems of industrial logistics1048698bull Addresses huge number of diverse applications

3

Notation and Terminology

Note Network terminology is not (and never will be)standardized The same concept may be denoted inmany different ways

Called bull NETWORK bull directed graph bull digraph bull graphClass Handouts (AhujaMagnanti Orlin)Network G = (NA)Node set N = 1234Arc Set (12)(13)(32)(34)(24)

Also SeenGraph G = (VE)Vertex set V = 1234Edge set A=1-21-33-23-42-4

4

Directed and Undirected Networks

An Undirected Graph A Directed Graph

bull Networks are used to transport commodities bull physical goods (products liquids) bull communication bull electricity etc

bull The field of Network Optimization concerns optimization problems on networks

5

An Overview of Some Applications ofNetwork Optimization

Pipelines

Applications Physical analogof nodes

Physical analogof arcs Flow

Communicationsystems

phone exchangescomputers

transmissionfacilities satellites

Cables fiber opticlinks microwave

relay links

Voice messages

Data

Video transmissions

Hydraulic systemsPumping stationsReservoirs Lakes

Water Gas OilHydraulic fluids

Integratedcomputer circuits

Gates registersprocessors Wires Electrical current

Mechanical systems JointsRods Beams

Springs Heat Energy

Transportationsystems

IntersectionsAirports

Rail yards

HighwaysAirline routes

Railbeds

Passengersfreight

vehiclesoperators

6

Examples ofterms

Path Example 5 2 3 4(or 5 c 2 b 3 e 4)Note that directions are ignored

Directed Path Example 1 2 3 4 (or 1 a 2 b 3 e) Directions are important

Two paths a-b-e (or 1-2-3-4)and a-c-d-e (or 1-2-5-3-4)

a-b-c-d (or 1-2-3-4-1)b-a-d-c (or 3-2-1-4-3)e-b-a (or 1-3-2-1)c-d-e (or 3-4-1-3)

Cycles (loops)

Cycle or circuit (or loop) 1 2 3 1 (or 1 a 2 b 3 e)Note that directions are ignored

Directed Cycle (1 2 3 4 1) or 1 a 2 b 3 c 4 d 1Directions are important

7

More Definitions

A network is connected if every nodecan be reached from every othernode by following a sequence ofarcs in which direction is ignored

A spanning tree is a connected subset of a networkincluding all nodes but containing no loops

8

The Minimum Cost Flow Problem

Network G = (N A) ndash Node set N arc set A ndash Capacities uij on arc (ij) ndash Cost cij on arc (ij) ndash Supplydemand bi for node i (Positive indicates supply)

A network with costscapacities supplies demands

Minimize the cost of sending flow st Flow out of i - Flow into i = bi Flow on arc (ij) le uij

9

The Minimum Cost Flow Problem

Let xij be the flow on arc (ij)

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 le xij le uij

Minimizest for all i

for all i-j

10

Example Formulation

Min -3 x12 + 8 x13 + 7 x23 + 3 x24 + 2 x34

st x12 + x13 = 4 x23+ x24 - x12 = 3 x34 - x13 - x23 = -5 - x24 - x34 = -2 0 le x12 le 6 0 le x13 le 5 0 le x23 le 2 0 le x24 le 4 0 le x34 le 7

11

An Application of the Minimum CostFlow Problem

Ship from suppliers to customers possibly throughwarehouses at minimum cost to meet demands

Warehouses

Customers

Demands

PlantsSupplies

12

Useful Facts About The MinimumCost Flow Problem

bull Suppose the following properties of the constraint matrix A (ignoring simple upper and lower variable bounds such as x le 7) hold (1) all entries of A are 0 or 1 or -1 (2) there is at most one 1 in any column and at most one -1bull Then this is a minimum cost flow problem

13

Useful Facts (contrsquod)

Theorem If one carries out the simplex algorithm on the minimum cost flow problem with integer valued capacities and RHS then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1

Corollary The optimal LP solution is integer valued

14

The Minimum Cost Flow Problem

Network G = (N A) ndash Node set N arc set A ndash Capacities uij on arc (ij) ndash Cost cij on arc (ij) ndash Supplydemand bi for node i (Positive indicates supply)

A network with costscapacities supplies demands

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

Flow on arc (ij) le uij

15

The Transportation Problem

Suppose that one wants to ship from warehouses toretailers

In this example 3 warehouses 4 retailersai is the supply at warehouse ibj is the demand at retailer jcij is the cost of shipping from i to jThere are no capacities on the arcs

Let xij be the amount of flow shippedfrom warehouse i to retailer jHow do we formulate an LP

16

The Transportation Problem is a MinCost Flow Problem

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 lexij leuij

Flow out occurs at the supply nodesFlow in occurs at demand nodesCapacities are infinite uij= infin

17

The Transportation Problem

In general the LP formulation is given as

Minimize

All arcs arefrom a node inS to a node inD anduncapacitated

S Supply nodes

D Demand nodes

18

Useful Facts About TransportationProblem

Suppose that (1) the constraint matrix can be partitioned into A1x = b1 and A2x = b2 (2) all entries of A1 and A2 are 0 or 1 (3) there is at most one 1 in any column of A1 or A2Then this is a transportation problemTheorem If one carries out the simplex algorithm on the transportation problem then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1 The costs and RHS are both integer valuedCorollary The optimal solution to the LP is integer valued

19

The Assignment Problem

Suppose that one wants to assign tasks to persons

Tasks Persons

In this example 4 tasks 3 personsNo two tasks to the same personEach person gets a taskcij is the ldquocostrdquo of assigning task ito person j

Let xij = 1 if task i is assigned to jLet xij = 0 otherwiseHow do we formulate an LP

20

The Assignment Problem

In general the LP formulation is given as

Minimize

Each supply is 1

Each demand is 1

21

More on the Assignment Problem

Tasks PersonsThe assignment problem isa special case of thetransportation problem

The simplex algorithm cansolve the LP relaxation andit will give integer answersthat is it will solve theassignment problem

22

An Application of the AssignmentProblem

Suppose that there are moving targets in space You can identifyeach target as a pixel on a radar screen Given two successivepictures identify how the targets have moved

23

The Maximum Flow Problem

Network G = (N A)

ndash Source s and sink t

ndash Capacities uij on arc (ij)

ndash Variable Flow xij on arc (ij) Graph with capacities

Maximize the flow leaving sst Flow out of i - Flow into i = 0 for i ne s t

0 le xij le uij

24

The Max Flow Problem

In general the LP formulation is given as

Minimize v

otherwise

This is not formulated as a special case of aminimum cost flow formulationCan we reformulate it in this way

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

3

Notation and Terminology

Note Network terminology is not (and never will be)standardized The same concept may be denoted inmany different ways

Called bull NETWORK bull directed graph bull digraph bull graphClass Handouts (AhujaMagnanti Orlin)Network G = (NA)Node set N = 1234Arc Set (12)(13)(32)(34)(24)

Also SeenGraph G = (VE)Vertex set V = 1234Edge set A=1-21-33-23-42-4

4

Directed and Undirected Networks

An Undirected Graph A Directed Graph

bull Networks are used to transport commodities bull physical goods (products liquids) bull communication bull electricity etc

bull The field of Network Optimization concerns optimization problems on networks

5

An Overview of Some Applications ofNetwork Optimization

Pipelines

Applications Physical analogof nodes

Physical analogof arcs Flow

Communicationsystems

phone exchangescomputers

transmissionfacilities satellites

Cables fiber opticlinks microwave

relay links

Voice messages

Data

Video transmissions

Hydraulic systemsPumping stationsReservoirs Lakes

Water Gas OilHydraulic fluids

Integratedcomputer circuits

Gates registersprocessors Wires Electrical current

Mechanical systems JointsRods Beams

Springs Heat Energy

Transportationsystems

IntersectionsAirports

Rail yards

HighwaysAirline routes

Railbeds

Passengersfreight

vehiclesoperators

6

Examples ofterms

Path Example 5 2 3 4(or 5 c 2 b 3 e 4)Note that directions are ignored

Directed Path Example 1 2 3 4 (or 1 a 2 b 3 e) Directions are important

Two paths a-b-e (or 1-2-3-4)and a-c-d-e (or 1-2-5-3-4)

a-b-c-d (or 1-2-3-4-1)b-a-d-c (or 3-2-1-4-3)e-b-a (or 1-3-2-1)c-d-e (or 3-4-1-3)

Cycles (loops)

Cycle or circuit (or loop) 1 2 3 1 (or 1 a 2 b 3 e)Note that directions are ignored

Directed Cycle (1 2 3 4 1) or 1 a 2 b 3 c 4 d 1Directions are important

7

More Definitions

A network is connected if every nodecan be reached from every othernode by following a sequence ofarcs in which direction is ignored

A spanning tree is a connected subset of a networkincluding all nodes but containing no loops

8

The Minimum Cost Flow Problem

Network G = (N A) ndash Node set N arc set A ndash Capacities uij on arc (ij) ndash Cost cij on arc (ij) ndash Supplydemand bi for node i (Positive indicates supply)

A network with costscapacities supplies demands

Minimize the cost of sending flow st Flow out of i - Flow into i = bi Flow on arc (ij) le uij

9

The Minimum Cost Flow Problem

Let xij be the flow on arc (ij)

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 le xij le uij

Minimizest for all i

for all i-j

10

Example Formulation

Min -3 x12 + 8 x13 + 7 x23 + 3 x24 + 2 x34

st x12 + x13 = 4 x23+ x24 - x12 = 3 x34 - x13 - x23 = -5 - x24 - x34 = -2 0 le x12 le 6 0 le x13 le 5 0 le x23 le 2 0 le x24 le 4 0 le x34 le 7

11

An Application of the Minimum CostFlow Problem

Ship from suppliers to customers possibly throughwarehouses at minimum cost to meet demands

Warehouses

Customers

Demands

PlantsSupplies

12

Useful Facts About The MinimumCost Flow Problem

bull Suppose the following properties of the constraint matrix A (ignoring simple upper and lower variable bounds such as x le 7) hold (1) all entries of A are 0 or 1 or -1 (2) there is at most one 1 in any column and at most one -1bull Then this is a minimum cost flow problem

13

Useful Facts (contrsquod)

Theorem If one carries out the simplex algorithm on the minimum cost flow problem with integer valued capacities and RHS then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1

Corollary The optimal LP solution is integer valued

14

The Minimum Cost Flow Problem

Network G = (N A) ndash Node set N arc set A ndash Capacities uij on arc (ij) ndash Cost cij on arc (ij) ndash Supplydemand bi for node i (Positive indicates supply)

A network with costscapacities supplies demands

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

Flow on arc (ij) le uij

15

The Transportation Problem

Suppose that one wants to ship from warehouses toretailers

In this example 3 warehouses 4 retailersai is the supply at warehouse ibj is the demand at retailer jcij is the cost of shipping from i to jThere are no capacities on the arcs

Let xij be the amount of flow shippedfrom warehouse i to retailer jHow do we formulate an LP

16

The Transportation Problem is a MinCost Flow Problem

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 lexij leuij

Flow out occurs at the supply nodesFlow in occurs at demand nodesCapacities are infinite uij= infin

17

The Transportation Problem

In general the LP formulation is given as

Minimize

All arcs arefrom a node inS to a node inD anduncapacitated

S Supply nodes

D Demand nodes

18

Useful Facts About TransportationProblem

Suppose that (1) the constraint matrix can be partitioned into A1x = b1 and A2x = b2 (2) all entries of A1 and A2 are 0 or 1 (3) there is at most one 1 in any column of A1 or A2Then this is a transportation problemTheorem If one carries out the simplex algorithm on the transportation problem then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1 The costs and RHS are both integer valuedCorollary The optimal solution to the LP is integer valued

19

The Assignment Problem

Suppose that one wants to assign tasks to persons

Tasks Persons

In this example 4 tasks 3 personsNo two tasks to the same personEach person gets a taskcij is the ldquocostrdquo of assigning task ito person j

Let xij = 1 if task i is assigned to jLet xij = 0 otherwiseHow do we formulate an LP

20

The Assignment Problem

In general the LP formulation is given as

Minimize

Each supply is 1

Each demand is 1

21

More on the Assignment Problem

Tasks PersonsThe assignment problem isa special case of thetransportation problem

The simplex algorithm cansolve the LP relaxation andit will give integer answersthat is it will solve theassignment problem

22

An Application of the AssignmentProblem

Suppose that there are moving targets in space You can identifyeach target as a pixel on a radar screen Given two successivepictures identify how the targets have moved

23

The Maximum Flow Problem

Network G = (N A)

ndash Source s and sink t

ndash Capacities uij on arc (ij)

ndash Variable Flow xij on arc (ij) Graph with capacities

Maximize the flow leaving sst Flow out of i - Flow into i = 0 for i ne s t

0 le xij le uij

24

The Max Flow Problem

In general the LP formulation is given as

Minimize v

otherwise

This is not formulated as a special case of aminimum cost flow formulationCan we reformulate it in this way

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

4

Directed and Undirected Networks

An Undirected Graph A Directed Graph

bull Networks are used to transport commodities bull physical goods (products liquids) bull communication bull electricity etc

bull The field of Network Optimization concerns optimization problems on networks

5

An Overview of Some Applications ofNetwork Optimization

Pipelines

Applications Physical analogof nodes

Physical analogof arcs Flow

Communicationsystems

phone exchangescomputers

transmissionfacilities satellites

Cables fiber opticlinks microwave

relay links

Voice messages

Data

Video transmissions

Hydraulic systemsPumping stationsReservoirs Lakes

Water Gas OilHydraulic fluids

Integratedcomputer circuits

Gates registersprocessors Wires Electrical current

Mechanical systems JointsRods Beams

Springs Heat Energy

Transportationsystems

IntersectionsAirports

Rail yards

HighwaysAirline routes

Railbeds

Passengersfreight

vehiclesoperators

6

Examples ofterms

Path Example 5 2 3 4(or 5 c 2 b 3 e 4)Note that directions are ignored

Directed Path Example 1 2 3 4 (or 1 a 2 b 3 e) Directions are important

Two paths a-b-e (or 1-2-3-4)and a-c-d-e (or 1-2-5-3-4)

a-b-c-d (or 1-2-3-4-1)b-a-d-c (or 3-2-1-4-3)e-b-a (or 1-3-2-1)c-d-e (or 3-4-1-3)

Cycles (loops)

Cycle or circuit (or loop) 1 2 3 1 (or 1 a 2 b 3 e)Note that directions are ignored

Directed Cycle (1 2 3 4 1) or 1 a 2 b 3 c 4 d 1Directions are important

7

More Definitions

A network is connected if every nodecan be reached from every othernode by following a sequence ofarcs in which direction is ignored

A spanning tree is a connected subset of a networkincluding all nodes but containing no loops

8

The Minimum Cost Flow Problem

Network G = (N A) ndash Node set N arc set A ndash Capacities uij on arc (ij) ndash Cost cij on arc (ij) ndash Supplydemand bi for node i (Positive indicates supply)

A network with costscapacities supplies demands

Minimize the cost of sending flow st Flow out of i - Flow into i = bi Flow on arc (ij) le uij

9

The Minimum Cost Flow Problem

Let xij be the flow on arc (ij)

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 le xij le uij

Minimizest for all i

for all i-j

10

Example Formulation

Min -3 x12 + 8 x13 + 7 x23 + 3 x24 + 2 x34

st x12 + x13 = 4 x23+ x24 - x12 = 3 x34 - x13 - x23 = -5 - x24 - x34 = -2 0 le x12 le 6 0 le x13 le 5 0 le x23 le 2 0 le x24 le 4 0 le x34 le 7

11

An Application of the Minimum CostFlow Problem

Ship from suppliers to customers possibly throughwarehouses at minimum cost to meet demands

Warehouses

Customers

Demands

PlantsSupplies

12

Useful Facts About The MinimumCost Flow Problem

bull Suppose the following properties of the constraint matrix A (ignoring simple upper and lower variable bounds such as x le 7) hold (1) all entries of A are 0 or 1 or -1 (2) there is at most one 1 in any column and at most one -1bull Then this is a minimum cost flow problem

13

Useful Facts (contrsquod)

Theorem If one carries out the simplex algorithm on the minimum cost flow problem with integer valued capacities and RHS then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1

Corollary The optimal LP solution is integer valued

14

The Minimum Cost Flow Problem

Network G = (N A) ndash Node set N arc set A ndash Capacities uij on arc (ij) ndash Cost cij on arc (ij) ndash Supplydemand bi for node i (Positive indicates supply)

A network with costscapacities supplies demands

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

Flow on arc (ij) le uij

15

The Transportation Problem

Suppose that one wants to ship from warehouses toretailers

In this example 3 warehouses 4 retailersai is the supply at warehouse ibj is the demand at retailer jcij is the cost of shipping from i to jThere are no capacities on the arcs

Let xij be the amount of flow shippedfrom warehouse i to retailer jHow do we formulate an LP

16

The Transportation Problem is a MinCost Flow Problem

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 lexij leuij

Flow out occurs at the supply nodesFlow in occurs at demand nodesCapacities are infinite uij= infin

17

The Transportation Problem

In general the LP formulation is given as

Minimize

All arcs arefrom a node inS to a node inD anduncapacitated

S Supply nodes

D Demand nodes

18

Useful Facts About TransportationProblem

Suppose that (1) the constraint matrix can be partitioned into A1x = b1 and A2x = b2 (2) all entries of A1 and A2 are 0 or 1 (3) there is at most one 1 in any column of A1 or A2Then this is a transportation problemTheorem If one carries out the simplex algorithm on the transportation problem then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1 The costs and RHS are both integer valuedCorollary The optimal solution to the LP is integer valued

19

The Assignment Problem

Suppose that one wants to assign tasks to persons

Tasks Persons

In this example 4 tasks 3 personsNo two tasks to the same personEach person gets a taskcij is the ldquocostrdquo of assigning task ito person j

Let xij = 1 if task i is assigned to jLet xij = 0 otherwiseHow do we formulate an LP

20

The Assignment Problem

In general the LP formulation is given as

Minimize

Each supply is 1

Each demand is 1

21

More on the Assignment Problem

Tasks PersonsThe assignment problem isa special case of thetransportation problem

The simplex algorithm cansolve the LP relaxation andit will give integer answersthat is it will solve theassignment problem

22

An Application of the AssignmentProblem

Suppose that there are moving targets in space You can identifyeach target as a pixel on a radar screen Given two successivepictures identify how the targets have moved

23

The Maximum Flow Problem

Network G = (N A)

ndash Source s and sink t

ndash Capacities uij on arc (ij)

ndash Variable Flow xij on arc (ij) Graph with capacities

Maximize the flow leaving sst Flow out of i - Flow into i = 0 for i ne s t

0 le xij le uij

24

The Max Flow Problem

In general the LP formulation is given as

Minimize v

otherwise

This is not formulated as a special case of aminimum cost flow formulationCan we reformulate it in this way

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

5

An Overview of Some Applications ofNetwork Optimization

Pipelines

Applications Physical analogof nodes

Physical analogof arcs Flow

Communicationsystems

phone exchangescomputers

transmissionfacilities satellites

Cables fiber opticlinks microwave

relay links

Voice messages

Data

Video transmissions

Hydraulic systemsPumping stationsReservoirs Lakes

Water Gas OilHydraulic fluids

Integratedcomputer circuits

Gates registersprocessors Wires Electrical current

Mechanical systems JointsRods Beams

Springs Heat Energy

Transportationsystems

IntersectionsAirports

Rail yards

HighwaysAirline routes

Railbeds

Passengersfreight

vehiclesoperators

6

Examples ofterms

Path Example 5 2 3 4(or 5 c 2 b 3 e 4)Note that directions are ignored

Directed Path Example 1 2 3 4 (or 1 a 2 b 3 e) Directions are important

Two paths a-b-e (or 1-2-3-4)and a-c-d-e (or 1-2-5-3-4)

a-b-c-d (or 1-2-3-4-1)b-a-d-c (or 3-2-1-4-3)e-b-a (or 1-3-2-1)c-d-e (or 3-4-1-3)

Cycles (loops)

Cycle or circuit (or loop) 1 2 3 1 (or 1 a 2 b 3 e)Note that directions are ignored

Directed Cycle (1 2 3 4 1) or 1 a 2 b 3 c 4 d 1Directions are important

7

More Definitions

A network is connected if every nodecan be reached from every othernode by following a sequence ofarcs in which direction is ignored

A spanning tree is a connected subset of a networkincluding all nodes but containing no loops

8

The Minimum Cost Flow Problem

Network G = (N A) ndash Node set N arc set A ndash Capacities uij on arc (ij) ndash Cost cij on arc (ij) ndash Supplydemand bi for node i (Positive indicates supply)

A network with costscapacities supplies demands

Minimize the cost of sending flow st Flow out of i - Flow into i = bi Flow on arc (ij) le uij

9

The Minimum Cost Flow Problem

Let xij be the flow on arc (ij)

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 le xij le uij

Minimizest for all i

for all i-j

10

Example Formulation

Min -3 x12 + 8 x13 + 7 x23 + 3 x24 + 2 x34

st x12 + x13 = 4 x23+ x24 - x12 = 3 x34 - x13 - x23 = -5 - x24 - x34 = -2 0 le x12 le 6 0 le x13 le 5 0 le x23 le 2 0 le x24 le 4 0 le x34 le 7

11

An Application of the Minimum CostFlow Problem

Ship from suppliers to customers possibly throughwarehouses at minimum cost to meet demands

Warehouses

Customers

Demands

PlantsSupplies

12

Useful Facts About The MinimumCost Flow Problem

bull Suppose the following properties of the constraint matrix A (ignoring simple upper and lower variable bounds such as x le 7) hold (1) all entries of A are 0 or 1 or -1 (2) there is at most one 1 in any column and at most one -1bull Then this is a minimum cost flow problem

13

Useful Facts (contrsquod)

Theorem If one carries out the simplex algorithm on the minimum cost flow problem with integer valued capacities and RHS then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1

Corollary The optimal LP solution is integer valued

14

The Minimum Cost Flow Problem

Network G = (N A) ndash Node set N arc set A ndash Capacities uij on arc (ij) ndash Cost cij on arc (ij) ndash Supplydemand bi for node i (Positive indicates supply)

A network with costscapacities supplies demands

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

Flow on arc (ij) le uij

15

The Transportation Problem

Suppose that one wants to ship from warehouses toretailers

In this example 3 warehouses 4 retailersai is the supply at warehouse ibj is the demand at retailer jcij is the cost of shipping from i to jThere are no capacities on the arcs

Let xij be the amount of flow shippedfrom warehouse i to retailer jHow do we formulate an LP

16

The Transportation Problem is a MinCost Flow Problem

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 lexij leuij

Flow out occurs at the supply nodesFlow in occurs at demand nodesCapacities are infinite uij= infin

17

The Transportation Problem

In general the LP formulation is given as

Minimize

All arcs arefrom a node inS to a node inD anduncapacitated

S Supply nodes

D Demand nodes

18

Useful Facts About TransportationProblem

Suppose that (1) the constraint matrix can be partitioned into A1x = b1 and A2x = b2 (2) all entries of A1 and A2 are 0 or 1 (3) there is at most one 1 in any column of A1 or A2Then this is a transportation problemTheorem If one carries out the simplex algorithm on the transportation problem then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1 The costs and RHS are both integer valuedCorollary The optimal solution to the LP is integer valued

19

The Assignment Problem

Suppose that one wants to assign tasks to persons

Tasks Persons

In this example 4 tasks 3 personsNo two tasks to the same personEach person gets a taskcij is the ldquocostrdquo of assigning task ito person j

Let xij = 1 if task i is assigned to jLet xij = 0 otherwiseHow do we formulate an LP

20

The Assignment Problem

In general the LP formulation is given as

Minimize

Each supply is 1

Each demand is 1

21

More on the Assignment Problem

Tasks PersonsThe assignment problem isa special case of thetransportation problem

The simplex algorithm cansolve the LP relaxation andit will give integer answersthat is it will solve theassignment problem

22

An Application of the AssignmentProblem

Suppose that there are moving targets in space You can identifyeach target as a pixel on a radar screen Given two successivepictures identify how the targets have moved

23

The Maximum Flow Problem

Network G = (N A)

ndash Source s and sink t

ndash Capacities uij on arc (ij)

ndash Variable Flow xij on arc (ij) Graph with capacities

Maximize the flow leaving sst Flow out of i - Flow into i = 0 for i ne s t

0 le xij le uij

24

The Max Flow Problem

In general the LP formulation is given as

Minimize v

otherwise

This is not formulated as a special case of aminimum cost flow formulationCan we reformulate it in this way

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

6

Examples ofterms

Path Example 5 2 3 4(or 5 c 2 b 3 e 4)Note that directions are ignored

Directed Path Example 1 2 3 4 (or 1 a 2 b 3 e) Directions are important

Two paths a-b-e (or 1-2-3-4)and a-c-d-e (or 1-2-5-3-4)

a-b-c-d (or 1-2-3-4-1)b-a-d-c (or 3-2-1-4-3)e-b-a (or 1-3-2-1)c-d-e (or 3-4-1-3)

Cycles (loops)

Cycle or circuit (or loop) 1 2 3 1 (or 1 a 2 b 3 e)Note that directions are ignored

Directed Cycle (1 2 3 4 1) or 1 a 2 b 3 c 4 d 1Directions are important

7

More Definitions

A network is connected if every nodecan be reached from every othernode by following a sequence ofarcs in which direction is ignored

A spanning tree is a connected subset of a networkincluding all nodes but containing no loops

8

The Minimum Cost Flow Problem

Network G = (N A) ndash Node set N arc set A ndash Capacities uij on arc (ij) ndash Cost cij on arc (ij) ndash Supplydemand bi for node i (Positive indicates supply)

A network with costscapacities supplies demands

Minimize the cost of sending flow st Flow out of i - Flow into i = bi Flow on arc (ij) le uij

9

The Minimum Cost Flow Problem

Let xij be the flow on arc (ij)

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 le xij le uij

Minimizest for all i

for all i-j

10

Example Formulation

Min -3 x12 + 8 x13 + 7 x23 + 3 x24 + 2 x34

st x12 + x13 = 4 x23+ x24 - x12 = 3 x34 - x13 - x23 = -5 - x24 - x34 = -2 0 le x12 le 6 0 le x13 le 5 0 le x23 le 2 0 le x24 le 4 0 le x34 le 7

11

An Application of the Minimum CostFlow Problem

Ship from suppliers to customers possibly throughwarehouses at minimum cost to meet demands

Warehouses

Customers

Demands

PlantsSupplies

12

Useful Facts About The MinimumCost Flow Problem

bull Suppose the following properties of the constraint matrix A (ignoring simple upper and lower variable bounds such as x le 7) hold (1) all entries of A are 0 or 1 or -1 (2) there is at most one 1 in any column and at most one -1bull Then this is a minimum cost flow problem

13

Useful Facts (contrsquod)

Theorem If one carries out the simplex algorithm on the minimum cost flow problem with integer valued capacities and RHS then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1

Corollary The optimal LP solution is integer valued

14

The Minimum Cost Flow Problem

Network G = (N A) ndash Node set N arc set A ndash Capacities uij on arc (ij) ndash Cost cij on arc (ij) ndash Supplydemand bi for node i (Positive indicates supply)

A network with costscapacities supplies demands

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

Flow on arc (ij) le uij

15

The Transportation Problem

Suppose that one wants to ship from warehouses toretailers

In this example 3 warehouses 4 retailersai is the supply at warehouse ibj is the demand at retailer jcij is the cost of shipping from i to jThere are no capacities on the arcs

Let xij be the amount of flow shippedfrom warehouse i to retailer jHow do we formulate an LP

16

The Transportation Problem is a MinCost Flow Problem

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 lexij leuij

Flow out occurs at the supply nodesFlow in occurs at demand nodesCapacities are infinite uij= infin

17

The Transportation Problem

In general the LP formulation is given as

Minimize

All arcs arefrom a node inS to a node inD anduncapacitated

S Supply nodes

D Demand nodes

18

Useful Facts About TransportationProblem

Suppose that (1) the constraint matrix can be partitioned into A1x = b1 and A2x = b2 (2) all entries of A1 and A2 are 0 or 1 (3) there is at most one 1 in any column of A1 or A2Then this is a transportation problemTheorem If one carries out the simplex algorithm on the transportation problem then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1 The costs and RHS are both integer valuedCorollary The optimal solution to the LP is integer valued

19

The Assignment Problem

Suppose that one wants to assign tasks to persons

Tasks Persons

In this example 4 tasks 3 personsNo two tasks to the same personEach person gets a taskcij is the ldquocostrdquo of assigning task ito person j

Let xij = 1 if task i is assigned to jLet xij = 0 otherwiseHow do we formulate an LP

20

The Assignment Problem

In general the LP formulation is given as

Minimize

Each supply is 1

Each demand is 1

21

More on the Assignment Problem

Tasks PersonsThe assignment problem isa special case of thetransportation problem

The simplex algorithm cansolve the LP relaxation andit will give integer answersthat is it will solve theassignment problem

22

An Application of the AssignmentProblem

Suppose that there are moving targets in space You can identifyeach target as a pixel on a radar screen Given two successivepictures identify how the targets have moved

23

The Maximum Flow Problem

Network G = (N A)

ndash Source s and sink t

ndash Capacities uij on arc (ij)

ndash Variable Flow xij on arc (ij) Graph with capacities

Maximize the flow leaving sst Flow out of i - Flow into i = 0 for i ne s t

0 le xij le uij

24

The Max Flow Problem

In general the LP formulation is given as

Minimize v

otherwise

This is not formulated as a special case of aminimum cost flow formulationCan we reformulate it in this way

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

7

More Definitions

A network is connected if every nodecan be reached from every othernode by following a sequence ofarcs in which direction is ignored

A spanning tree is a connected subset of a networkincluding all nodes but containing no loops

8

The Minimum Cost Flow Problem

Network G = (N A) ndash Node set N arc set A ndash Capacities uij on arc (ij) ndash Cost cij on arc (ij) ndash Supplydemand bi for node i (Positive indicates supply)

A network with costscapacities supplies demands

Minimize the cost of sending flow st Flow out of i - Flow into i = bi Flow on arc (ij) le uij

9

The Minimum Cost Flow Problem

Let xij be the flow on arc (ij)

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 le xij le uij

Minimizest for all i

for all i-j

10

Example Formulation

Min -3 x12 + 8 x13 + 7 x23 + 3 x24 + 2 x34

st x12 + x13 = 4 x23+ x24 - x12 = 3 x34 - x13 - x23 = -5 - x24 - x34 = -2 0 le x12 le 6 0 le x13 le 5 0 le x23 le 2 0 le x24 le 4 0 le x34 le 7

11

An Application of the Minimum CostFlow Problem

Ship from suppliers to customers possibly throughwarehouses at minimum cost to meet demands

Warehouses

Customers

Demands

PlantsSupplies

12

Useful Facts About The MinimumCost Flow Problem

bull Suppose the following properties of the constraint matrix A (ignoring simple upper and lower variable bounds such as x le 7) hold (1) all entries of A are 0 or 1 or -1 (2) there is at most one 1 in any column and at most one -1bull Then this is a minimum cost flow problem

13

Useful Facts (contrsquod)

Theorem If one carries out the simplex algorithm on the minimum cost flow problem with integer valued capacities and RHS then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1

Corollary The optimal LP solution is integer valued

14

The Minimum Cost Flow Problem

Network G = (N A) ndash Node set N arc set A ndash Capacities uij on arc (ij) ndash Cost cij on arc (ij) ndash Supplydemand bi for node i (Positive indicates supply)

A network with costscapacities supplies demands

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

Flow on arc (ij) le uij

15

The Transportation Problem

Suppose that one wants to ship from warehouses toretailers

In this example 3 warehouses 4 retailersai is the supply at warehouse ibj is the demand at retailer jcij is the cost of shipping from i to jThere are no capacities on the arcs

Let xij be the amount of flow shippedfrom warehouse i to retailer jHow do we formulate an LP

16

The Transportation Problem is a MinCost Flow Problem

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 lexij leuij

Flow out occurs at the supply nodesFlow in occurs at demand nodesCapacities are infinite uij= infin

17

The Transportation Problem

In general the LP formulation is given as

Minimize

All arcs arefrom a node inS to a node inD anduncapacitated

S Supply nodes

D Demand nodes

18

Useful Facts About TransportationProblem

Suppose that (1) the constraint matrix can be partitioned into A1x = b1 and A2x = b2 (2) all entries of A1 and A2 are 0 or 1 (3) there is at most one 1 in any column of A1 or A2Then this is a transportation problemTheorem If one carries out the simplex algorithm on the transportation problem then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1 The costs and RHS are both integer valuedCorollary The optimal solution to the LP is integer valued

19

The Assignment Problem

Suppose that one wants to assign tasks to persons

Tasks Persons

In this example 4 tasks 3 personsNo two tasks to the same personEach person gets a taskcij is the ldquocostrdquo of assigning task ito person j

Let xij = 1 if task i is assigned to jLet xij = 0 otherwiseHow do we formulate an LP

20

The Assignment Problem

In general the LP formulation is given as

Minimize

Each supply is 1

Each demand is 1

21

More on the Assignment Problem

Tasks PersonsThe assignment problem isa special case of thetransportation problem

The simplex algorithm cansolve the LP relaxation andit will give integer answersthat is it will solve theassignment problem

22

An Application of the AssignmentProblem

Suppose that there are moving targets in space You can identifyeach target as a pixel on a radar screen Given two successivepictures identify how the targets have moved

23

The Maximum Flow Problem

Network G = (N A)

ndash Source s and sink t

ndash Capacities uij on arc (ij)

ndash Variable Flow xij on arc (ij) Graph with capacities

Maximize the flow leaving sst Flow out of i - Flow into i = 0 for i ne s t

0 le xij le uij

24

The Max Flow Problem

In general the LP formulation is given as

Minimize v

otherwise

This is not formulated as a special case of aminimum cost flow formulationCan we reformulate it in this way

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
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• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

8

The Minimum Cost Flow Problem

Network G = (N A) ndash Node set N arc set A ndash Capacities uij on arc (ij) ndash Cost cij on arc (ij) ndash Supplydemand bi for node i (Positive indicates supply)

A network with costscapacities supplies demands

Minimize the cost of sending flow st Flow out of i - Flow into i = bi Flow on arc (ij) le uij

9

The Minimum Cost Flow Problem

Let xij be the flow on arc (ij)

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 le xij le uij

Minimizest for all i

for all i-j

10

Example Formulation

Min -3 x12 + 8 x13 + 7 x23 + 3 x24 + 2 x34

st x12 + x13 = 4 x23+ x24 - x12 = 3 x34 - x13 - x23 = -5 - x24 - x34 = -2 0 le x12 le 6 0 le x13 le 5 0 le x23 le 2 0 le x24 le 4 0 le x34 le 7

11

An Application of the Minimum CostFlow Problem

Ship from suppliers to customers possibly throughwarehouses at minimum cost to meet demands

Warehouses

Customers

Demands

PlantsSupplies

12

Useful Facts About The MinimumCost Flow Problem

bull Suppose the following properties of the constraint matrix A (ignoring simple upper and lower variable bounds such as x le 7) hold (1) all entries of A are 0 or 1 or -1 (2) there is at most one 1 in any column and at most one -1bull Then this is a minimum cost flow problem

13

Useful Facts (contrsquod)

Theorem If one carries out the simplex algorithm on the minimum cost flow problem with integer valued capacities and RHS then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1

Corollary The optimal LP solution is integer valued

14

The Minimum Cost Flow Problem

Network G = (N A) ndash Node set N arc set A ndash Capacities uij on arc (ij) ndash Cost cij on arc (ij) ndash Supplydemand bi for node i (Positive indicates supply)

A network with costscapacities supplies demands

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

Flow on arc (ij) le uij

15

The Transportation Problem

Suppose that one wants to ship from warehouses toretailers

In this example 3 warehouses 4 retailersai is the supply at warehouse ibj is the demand at retailer jcij is the cost of shipping from i to jThere are no capacities on the arcs

Let xij be the amount of flow shippedfrom warehouse i to retailer jHow do we formulate an LP

16

The Transportation Problem is a MinCost Flow Problem

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 lexij leuij

Flow out occurs at the supply nodesFlow in occurs at demand nodesCapacities are infinite uij= infin

17

The Transportation Problem

In general the LP formulation is given as

Minimize

All arcs arefrom a node inS to a node inD anduncapacitated

S Supply nodes

D Demand nodes

18

Useful Facts About TransportationProblem

Suppose that (1) the constraint matrix can be partitioned into A1x = b1 and A2x = b2 (2) all entries of A1 and A2 are 0 or 1 (3) there is at most one 1 in any column of A1 or A2Then this is a transportation problemTheorem If one carries out the simplex algorithm on the transportation problem then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1 The costs and RHS are both integer valuedCorollary The optimal solution to the LP is integer valued

19

The Assignment Problem

Suppose that one wants to assign tasks to persons

Tasks Persons

In this example 4 tasks 3 personsNo two tasks to the same personEach person gets a taskcij is the ldquocostrdquo of assigning task ito person j

Let xij = 1 if task i is assigned to jLet xij = 0 otherwiseHow do we formulate an LP

20

The Assignment Problem

In general the LP formulation is given as

Minimize

Each supply is 1

Each demand is 1

21

More on the Assignment Problem

Tasks PersonsThe assignment problem isa special case of thetransportation problem

The simplex algorithm cansolve the LP relaxation andit will give integer answersthat is it will solve theassignment problem

22

An Application of the AssignmentProblem

Suppose that there are moving targets in space You can identifyeach target as a pixel on a radar screen Given two successivepictures identify how the targets have moved

23

The Maximum Flow Problem

Network G = (N A)

ndash Source s and sink t

ndash Capacities uij on arc (ij)

ndash Variable Flow xij on arc (ij) Graph with capacities

Maximize the flow leaving sst Flow out of i - Flow into i = 0 for i ne s t

0 le xij le uij

24

The Max Flow Problem

In general the LP formulation is given as

Minimize v

otherwise

This is not formulated as a special case of aminimum cost flow formulationCan we reformulate it in this way

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
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• Slide 14
• Slide 15
• Slide 16
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• Slide 18
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• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

9

The Minimum Cost Flow Problem

Let xij be the flow on arc (ij)

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 le xij le uij

Minimizest for all i

for all i-j

10

Example Formulation

Min -3 x12 + 8 x13 + 7 x23 + 3 x24 + 2 x34

st x12 + x13 = 4 x23+ x24 - x12 = 3 x34 - x13 - x23 = -5 - x24 - x34 = -2 0 le x12 le 6 0 le x13 le 5 0 le x23 le 2 0 le x24 le 4 0 le x34 le 7

11

An Application of the Minimum CostFlow Problem

Ship from suppliers to customers possibly throughwarehouses at minimum cost to meet demands

Warehouses

Customers

Demands

PlantsSupplies

12

Useful Facts About The MinimumCost Flow Problem

bull Suppose the following properties of the constraint matrix A (ignoring simple upper and lower variable bounds such as x le 7) hold (1) all entries of A are 0 or 1 or -1 (2) there is at most one 1 in any column and at most one -1bull Then this is a minimum cost flow problem

13

Useful Facts (contrsquod)

Theorem If one carries out the simplex algorithm on the minimum cost flow problem with integer valued capacities and RHS then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1

Corollary The optimal LP solution is integer valued

14

The Minimum Cost Flow Problem

Network G = (N A) ndash Node set N arc set A ndash Capacities uij on arc (ij) ndash Cost cij on arc (ij) ndash Supplydemand bi for node i (Positive indicates supply)

A network with costscapacities supplies demands

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

Flow on arc (ij) le uij

15

The Transportation Problem

Suppose that one wants to ship from warehouses toretailers

In this example 3 warehouses 4 retailersai is the supply at warehouse ibj is the demand at retailer jcij is the cost of shipping from i to jThere are no capacities on the arcs

Let xij be the amount of flow shippedfrom warehouse i to retailer jHow do we formulate an LP

16

The Transportation Problem is a MinCost Flow Problem

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 lexij leuij

Flow out occurs at the supply nodesFlow in occurs at demand nodesCapacities are infinite uij= infin

17

The Transportation Problem

In general the LP formulation is given as

Minimize

All arcs arefrom a node inS to a node inD anduncapacitated

S Supply nodes

D Demand nodes

18

Useful Facts About TransportationProblem

Suppose that (1) the constraint matrix can be partitioned into A1x = b1 and A2x = b2 (2) all entries of A1 and A2 are 0 or 1 (3) there is at most one 1 in any column of A1 or A2Then this is a transportation problemTheorem If one carries out the simplex algorithm on the transportation problem then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1 The costs and RHS are both integer valuedCorollary The optimal solution to the LP is integer valued

19

The Assignment Problem

Suppose that one wants to assign tasks to persons

Tasks Persons

In this example 4 tasks 3 personsNo two tasks to the same personEach person gets a taskcij is the ldquocostrdquo of assigning task ito person j

Let xij = 1 if task i is assigned to jLet xij = 0 otherwiseHow do we formulate an LP

20

The Assignment Problem

In general the LP formulation is given as

Minimize

Each supply is 1

Each demand is 1

21

More on the Assignment Problem

Tasks PersonsThe assignment problem isa special case of thetransportation problem

The simplex algorithm cansolve the LP relaxation andit will give integer answersthat is it will solve theassignment problem

22

An Application of the AssignmentProblem

Suppose that there are moving targets in space You can identifyeach target as a pixel on a radar screen Given two successivepictures identify how the targets have moved

23

The Maximum Flow Problem

Network G = (N A)

ndash Source s and sink t

ndash Capacities uij on arc (ij)

ndash Variable Flow xij on arc (ij) Graph with capacities

Maximize the flow leaving sst Flow out of i - Flow into i = 0 for i ne s t

0 le xij le uij

24

The Max Flow Problem

In general the LP formulation is given as

Minimize v

otherwise

This is not formulated as a special case of aminimum cost flow formulationCan we reformulate it in this way

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
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• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

10

Example Formulation

Min -3 x12 + 8 x13 + 7 x23 + 3 x24 + 2 x34

st x12 + x13 = 4 x23+ x24 - x12 = 3 x34 - x13 - x23 = -5 - x24 - x34 = -2 0 le x12 le 6 0 le x13 le 5 0 le x23 le 2 0 le x24 le 4 0 le x34 le 7

11

An Application of the Minimum CostFlow Problem

Ship from suppliers to customers possibly throughwarehouses at minimum cost to meet demands

Warehouses

Customers

Demands

PlantsSupplies

12

Useful Facts About The MinimumCost Flow Problem

bull Suppose the following properties of the constraint matrix A (ignoring simple upper and lower variable bounds such as x le 7) hold (1) all entries of A are 0 or 1 or -1 (2) there is at most one 1 in any column and at most one -1bull Then this is a minimum cost flow problem

13

Useful Facts (contrsquod)

Theorem If one carries out the simplex algorithm on the minimum cost flow problem with integer valued capacities and RHS then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1

Corollary The optimal LP solution is integer valued

14

The Minimum Cost Flow Problem

Network G = (N A) ndash Node set N arc set A ndash Capacities uij on arc (ij) ndash Cost cij on arc (ij) ndash Supplydemand bi for node i (Positive indicates supply)

A network with costscapacities supplies demands

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

Flow on arc (ij) le uij

15

The Transportation Problem

Suppose that one wants to ship from warehouses toretailers

In this example 3 warehouses 4 retailersai is the supply at warehouse ibj is the demand at retailer jcij is the cost of shipping from i to jThere are no capacities on the arcs

Let xij be the amount of flow shippedfrom warehouse i to retailer jHow do we formulate an LP

16

The Transportation Problem is a MinCost Flow Problem

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 lexij leuij

Flow out occurs at the supply nodesFlow in occurs at demand nodesCapacities are infinite uij= infin

17

The Transportation Problem

In general the LP formulation is given as

Minimize

All arcs arefrom a node inS to a node inD anduncapacitated

S Supply nodes

D Demand nodes

18

Useful Facts About TransportationProblem

Suppose that (1) the constraint matrix can be partitioned into A1x = b1 and A2x = b2 (2) all entries of A1 and A2 are 0 or 1 (3) there is at most one 1 in any column of A1 or A2Then this is a transportation problemTheorem If one carries out the simplex algorithm on the transportation problem then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1 The costs and RHS are both integer valuedCorollary The optimal solution to the LP is integer valued

19

The Assignment Problem

Suppose that one wants to assign tasks to persons

Tasks Persons

In this example 4 tasks 3 personsNo two tasks to the same personEach person gets a taskcij is the ldquocostrdquo of assigning task ito person j

Let xij = 1 if task i is assigned to jLet xij = 0 otherwiseHow do we formulate an LP

20

The Assignment Problem

In general the LP formulation is given as

Minimize

Each supply is 1

Each demand is 1

21

More on the Assignment Problem

Tasks PersonsThe assignment problem isa special case of thetransportation problem

The simplex algorithm cansolve the LP relaxation andit will give integer answersthat is it will solve theassignment problem

22

An Application of the AssignmentProblem

Suppose that there are moving targets in space You can identifyeach target as a pixel on a radar screen Given two successivepictures identify how the targets have moved

23

The Maximum Flow Problem

Network G = (N A)

ndash Source s and sink t

ndash Capacities uij on arc (ij)

ndash Variable Flow xij on arc (ij) Graph with capacities

Maximize the flow leaving sst Flow out of i - Flow into i = 0 for i ne s t

0 le xij le uij

24

The Max Flow Problem

In general the LP formulation is given as

Minimize v

otherwise

This is not formulated as a special case of aminimum cost flow formulationCan we reformulate it in this way

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

11

An Application of the Minimum CostFlow Problem

Ship from suppliers to customers possibly throughwarehouses at minimum cost to meet demands

Warehouses

Customers

Demands

PlantsSupplies

12

Useful Facts About The MinimumCost Flow Problem

bull Suppose the following properties of the constraint matrix A (ignoring simple upper and lower variable bounds such as x le 7) hold (1) all entries of A are 0 or 1 or -1 (2) there is at most one 1 in any column and at most one -1bull Then this is a minimum cost flow problem

13

Useful Facts (contrsquod)

Theorem If one carries out the simplex algorithm on the minimum cost flow problem with integer valued capacities and RHS then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1

Corollary The optimal LP solution is integer valued

14

The Minimum Cost Flow Problem

Network G = (N A) ndash Node set N arc set A ndash Capacities uij on arc (ij) ndash Cost cij on arc (ij) ndash Supplydemand bi for node i (Positive indicates supply)

A network with costscapacities supplies demands

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

Flow on arc (ij) le uij

15

The Transportation Problem

Suppose that one wants to ship from warehouses toretailers

In this example 3 warehouses 4 retailersai is the supply at warehouse ibj is the demand at retailer jcij is the cost of shipping from i to jThere are no capacities on the arcs

Let xij be the amount of flow shippedfrom warehouse i to retailer jHow do we formulate an LP

16

The Transportation Problem is a MinCost Flow Problem

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 lexij leuij

Flow out occurs at the supply nodesFlow in occurs at demand nodesCapacities are infinite uij= infin

17

The Transportation Problem

In general the LP formulation is given as

Minimize

All arcs arefrom a node inS to a node inD anduncapacitated

S Supply nodes

D Demand nodes

18

Useful Facts About TransportationProblem

Suppose that (1) the constraint matrix can be partitioned into A1x = b1 and A2x = b2 (2) all entries of A1 and A2 are 0 or 1 (3) there is at most one 1 in any column of A1 or A2Then this is a transportation problemTheorem If one carries out the simplex algorithm on the transportation problem then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1 The costs and RHS are both integer valuedCorollary The optimal solution to the LP is integer valued

19

The Assignment Problem

Suppose that one wants to assign tasks to persons

Tasks Persons

In this example 4 tasks 3 personsNo two tasks to the same personEach person gets a taskcij is the ldquocostrdquo of assigning task ito person j

Let xij = 1 if task i is assigned to jLet xij = 0 otherwiseHow do we formulate an LP

20

The Assignment Problem

In general the LP formulation is given as

Minimize

Each supply is 1

Each demand is 1

21

More on the Assignment Problem

Tasks PersonsThe assignment problem isa special case of thetransportation problem

The simplex algorithm cansolve the LP relaxation andit will give integer answersthat is it will solve theassignment problem

22

An Application of the AssignmentProblem

Suppose that there are moving targets in space You can identifyeach target as a pixel on a radar screen Given two successivepictures identify how the targets have moved

23

The Maximum Flow Problem

Network G = (N A)

ndash Source s and sink t

ndash Capacities uij on arc (ij)

ndash Variable Flow xij on arc (ij) Graph with capacities

Maximize the flow leaving sst Flow out of i - Flow into i = 0 for i ne s t

0 le xij le uij

24

The Max Flow Problem

In general the LP formulation is given as

Minimize v

otherwise

This is not formulated as a special case of aminimum cost flow formulationCan we reformulate it in this way

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

12

Useful Facts About The MinimumCost Flow Problem

bull Suppose the following properties of the constraint matrix A (ignoring simple upper and lower variable bounds such as x le 7) hold (1) all entries of A are 0 or 1 or -1 (2) there is at most one 1 in any column and at most one -1bull Then this is a minimum cost flow problem

13

Useful Facts (contrsquod)

Theorem If one carries out the simplex algorithm on the minimum cost flow problem with integer valued capacities and RHS then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1

Corollary The optimal LP solution is integer valued

14

The Minimum Cost Flow Problem

Network G = (N A) ndash Node set N arc set A ndash Capacities uij on arc (ij) ndash Cost cij on arc (ij) ndash Supplydemand bi for node i (Positive indicates supply)

A network with costscapacities supplies demands

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

Flow on arc (ij) le uij

15

The Transportation Problem

Suppose that one wants to ship from warehouses toretailers

In this example 3 warehouses 4 retailersai is the supply at warehouse ibj is the demand at retailer jcij is the cost of shipping from i to jThere are no capacities on the arcs

Let xij be the amount of flow shippedfrom warehouse i to retailer jHow do we formulate an LP

16

The Transportation Problem is a MinCost Flow Problem

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 lexij leuij

Flow out occurs at the supply nodesFlow in occurs at demand nodesCapacities are infinite uij= infin

17

The Transportation Problem

In general the LP formulation is given as

Minimize

All arcs arefrom a node inS to a node inD anduncapacitated

S Supply nodes

D Demand nodes

18

Useful Facts About TransportationProblem

Suppose that (1) the constraint matrix can be partitioned into A1x = b1 and A2x = b2 (2) all entries of A1 and A2 are 0 or 1 (3) there is at most one 1 in any column of A1 or A2Then this is a transportation problemTheorem If one carries out the simplex algorithm on the transportation problem then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1 The costs and RHS are both integer valuedCorollary The optimal solution to the LP is integer valued

19

The Assignment Problem

Suppose that one wants to assign tasks to persons

Tasks Persons

In this example 4 tasks 3 personsNo two tasks to the same personEach person gets a taskcij is the ldquocostrdquo of assigning task ito person j

Let xij = 1 if task i is assigned to jLet xij = 0 otherwiseHow do we formulate an LP

20

The Assignment Problem

In general the LP formulation is given as

Minimize

Each supply is 1

Each demand is 1

21

More on the Assignment Problem

Tasks PersonsThe assignment problem isa special case of thetransportation problem

The simplex algorithm cansolve the LP relaxation andit will give integer answersthat is it will solve theassignment problem

22

An Application of the AssignmentProblem

Suppose that there are moving targets in space You can identifyeach target as a pixel on a radar screen Given two successivepictures identify how the targets have moved

23

The Maximum Flow Problem

Network G = (N A)

ndash Source s and sink t

ndash Capacities uij on arc (ij)

ndash Variable Flow xij on arc (ij) Graph with capacities

Maximize the flow leaving sst Flow out of i - Flow into i = 0 for i ne s t

0 le xij le uij

24

The Max Flow Problem

In general the LP formulation is given as

Minimize v

otherwise

This is not formulated as a special case of aminimum cost flow formulationCan we reformulate it in this way

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

13

Useful Facts (contrsquod)

Theorem If one carries out the simplex algorithm on the minimum cost flow problem with integer valued capacities and RHS then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1

Corollary The optimal LP solution is integer valued

14

The Minimum Cost Flow Problem

Network G = (N A) ndash Node set N arc set A ndash Capacities uij on arc (ij) ndash Cost cij on arc (ij) ndash Supplydemand bi for node i (Positive indicates supply)

A network with costscapacities supplies demands

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

Flow on arc (ij) le uij

15

The Transportation Problem

Suppose that one wants to ship from warehouses toretailers

In this example 3 warehouses 4 retailersai is the supply at warehouse ibj is the demand at retailer jcij is the cost of shipping from i to jThere are no capacities on the arcs

Let xij be the amount of flow shippedfrom warehouse i to retailer jHow do we formulate an LP

16

The Transportation Problem is a MinCost Flow Problem

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 lexij leuij

Flow out occurs at the supply nodesFlow in occurs at demand nodesCapacities are infinite uij= infin

17

The Transportation Problem

In general the LP formulation is given as

Minimize

All arcs arefrom a node inS to a node inD anduncapacitated

S Supply nodes

D Demand nodes

18

Useful Facts About TransportationProblem

Suppose that (1) the constraint matrix can be partitioned into A1x = b1 and A2x = b2 (2) all entries of A1 and A2 are 0 or 1 (3) there is at most one 1 in any column of A1 or A2Then this is a transportation problemTheorem If one carries out the simplex algorithm on the transportation problem then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1 The costs and RHS are both integer valuedCorollary The optimal solution to the LP is integer valued

19

The Assignment Problem

Suppose that one wants to assign tasks to persons

Tasks Persons

In this example 4 tasks 3 personsNo two tasks to the same personEach person gets a taskcij is the ldquocostrdquo of assigning task ito person j

Let xij = 1 if task i is assigned to jLet xij = 0 otherwiseHow do we formulate an LP

20

The Assignment Problem

In general the LP formulation is given as

Minimize

Each supply is 1

Each demand is 1

21

More on the Assignment Problem

Tasks PersonsThe assignment problem isa special case of thetransportation problem

The simplex algorithm cansolve the LP relaxation andit will give integer answersthat is it will solve theassignment problem

22

An Application of the AssignmentProblem

Suppose that there are moving targets in space You can identifyeach target as a pixel on a radar screen Given two successivepictures identify how the targets have moved

23

The Maximum Flow Problem

Network G = (N A)

ndash Source s and sink t

ndash Capacities uij on arc (ij)

ndash Variable Flow xij on arc (ij) Graph with capacities

Maximize the flow leaving sst Flow out of i - Flow into i = 0 for i ne s t

0 le xij le uij

24

The Max Flow Problem

In general the LP formulation is given as

Minimize v

otherwise

This is not formulated as a special case of aminimum cost flow formulationCan we reformulate it in this way

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

14

The Minimum Cost Flow Problem

Network G = (N A) ndash Node set N arc set A ndash Capacities uij on arc (ij) ndash Cost cij on arc (ij) ndash Supplydemand bi for node i (Positive indicates supply)

A network with costscapacities supplies demands

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

Flow on arc (ij) le uij

15

The Transportation Problem

Suppose that one wants to ship from warehouses toretailers

In this example 3 warehouses 4 retailersai is the supply at warehouse ibj is the demand at retailer jcij is the cost of shipping from i to jThere are no capacities on the arcs

Let xij be the amount of flow shippedfrom warehouse i to retailer jHow do we formulate an LP

16

The Transportation Problem is a MinCost Flow Problem

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 lexij leuij

Flow out occurs at the supply nodesFlow in occurs at demand nodesCapacities are infinite uij= infin

17

The Transportation Problem

In general the LP formulation is given as

Minimize

All arcs arefrom a node inS to a node inD anduncapacitated

S Supply nodes

D Demand nodes

18

Useful Facts About TransportationProblem

Suppose that (1) the constraint matrix can be partitioned into A1x = b1 and A2x = b2 (2) all entries of A1 and A2 are 0 or 1 (3) there is at most one 1 in any column of A1 or A2Then this is a transportation problemTheorem If one carries out the simplex algorithm on the transportation problem then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1 The costs and RHS are both integer valuedCorollary The optimal solution to the LP is integer valued

19

The Assignment Problem

Suppose that one wants to assign tasks to persons

Tasks Persons

In this example 4 tasks 3 personsNo two tasks to the same personEach person gets a taskcij is the ldquocostrdquo of assigning task ito person j

Let xij = 1 if task i is assigned to jLet xij = 0 otherwiseHow do we formulate an LP

20

The Assignment Problem

In general the LP formulation is given as

Minimize

Each supply is 1

Each demand is 1

21

More on the Assignment Problem

Tasks PersonsThe assignment problem isa special case of thetransportation problem

The simplex algorithm cansolve the LP relaxation andit will give integer answersthat is it will solve theassignment problem

22

An Application of the AssignmentProblem

Suppose that there are moving targets in space You can identifyeach target as a pixel on a radar screen Given two successivepictures identify how the targets have moved

23

The Maximum Flow Problem

Network G = (N A)

ndash Source s and sink t

ndash Capacities uij on arc (ij)

ndash Variable Flow xij on arc (ij) Graph with capacities

Maximize the flow leaving sst Flow out of i - Flow into i = 0 for i ne s t

0 le xij le uij

24

The Max Flow Problem

In general the LP formulation is given as

Minimize v

otherwise

This is not formulated as a special case of aminimum cost flow formulationCan we reformulate it in this way

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

15

The Transportation Problem

Suppose that one wants to ship from warehouses toretailers

In this example 3 warehouses 4 retailersai is the supply at warehouse ibj is the demand at retailer jcij is the cost of shipping from i to jThere are no capacities on the arcs

Let xij be the amount of flow shippedfrom warehouse i to retailer jHow do we formulate an LP

16

The Transportation Problem is a MinCost Flow Problem

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 lexij leuij

Flow out occurs at the supply nodesFlow in occurs at demand nodesCapacities are infinite uij= infin

17

The Transportation Problem

In general the LP formulation is given as

Minimize

All arcs arefrom a node inS to a node inD anduncapacitated

S Supply nodes

D Demand nodes

18

Useful Facts About TransportationProblem

Suppose that (1) the constraint matrix can be partitioned into A1x = b1 and A2x = b2 (2) all entries of A1 and A2 are 0 or 1 (3) there is at most one 1 in any column of A1 or A2Then this is a transportation problemTheorem If one carries out the simplex algorithm on the transportation problem then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1 The costs and RHS are both integer valuedCorollary The optimal solution to the LP is integer valued

19

The Assignment Problem

Suppose that one wants to assign tasks to persons

Tasks Persons

In this example 4 tasks 3 personsNo two tasks to the same personEach person gets a taskcij is the ldquocostrdquo of assigning task ito person j

Let xij = 1 if task i is assigned to jLet xij = 0 otherwiseHow do we formulate an LP

20

The Assignment Problem

In general the LP formulation is given as

Minimize

Each supply is 1

Each demand is 1

21

More on the Assignment Problem

Tasks PersonsThe assignment problem isa special case of thetransportation problem

The simplex algorithm cansolve the LP relaxation andit will give integer answersthat is it will solve theassignment problem

22

An Application of the AssignmentProblem

Suppose that there are moving targets in space You can identifyeach target as a pixel on a radar screen Given two successivepictures identify how the targets have moved

23

The Maximum Flow Problem

Network G = (N A)

ndash Source s and sink t

ndash Capacities uij on arc (ij)

ndash Variable Flow xij on arc (ij) Graph with capacities

Maximize the flow leaving sst Flow out of i - Flow into i = 0 for i ne s t

0 le xij le uij

24

The Max Flow Problem

In general the LP formulation is given as

Minimize v

otherwise

This is not formulated as a special case of aminimum cost flow formulationCan we reformulate it in this way

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

16

The Transportation Problem is a MinCost Flow Problem

Minimize the cost of sending flowst Flow out of i - Flow into i = bi

0 lexij leuij

Flow out occurs at the supply nodesFlow in occurs at demand nodesCapacities are infinite uij= infin

17

The Transportation Problem

In general the LP formulation is given as

Minimize

All arcs arefrom a node inS to a node inD anduncapacitated

S Supply nodes

D Demand nodes

18

Useful Facts About TransportationProblem

Suppose that (1) the constraint matrix can be partitioned into A1x = b1 and A2x = b2 (2) all entries of A1 and A2 are 0 or 1 (3) there is at most one 1 in any column of A1 or A2Then this is a transportation problemTheorem If one carries out the simplex algorithm on the transportation problem then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1 The costs and RHS are both integer valuedCorollary The optimal solution to the LP is integer valued

19

The Assignment Problem

Suppose that one wants to assign tasks to persons

Tasks Persons

In this example 4 tasks 3 personsNo two tasks to the same personEach person gets a taskcij is the ldquocostrdquo of assigning task ito person j

Let xij = 1 if task i is assigned to jLet xij = 0 otherwiseHow do we formulate an LP

20

The Assignment Problem

In general the LP formulation is given as

Minimize

Each supply is 1

Each demand is 1

21

More on the Assignment Problem

Tasks PersonsThe assignment problem isa special case of thetransportation problem

The simplex algorithm cansolve the LP relaxation andit will give integer answersthat is it will solve theassignment problem

22

An Application of the AssignmentProblem

Suppose that there are moving targets in space You can identifyeach target as a pixel on a radar screen Given two successivepictures identify how the targets have moved

23

The Maximum Flow Problem

Network G = (N A)

ndash Source s and sink t

ndash Capacities uij on arc (ij)

ndash Variable Flow xij on arc (ij) Graph with capacities

Maximize the flow leaving sst Flow out of i - Flow into i = 0 for i ne s t

0 le xij le uij

24

The Max Flow Problem

In general the LP formulation is given as

Minimize v

otherwise

This is not formulated as a special case of aminimum cost flow formulationCan we reformulate it in this way

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

17

The Transportation Problem

In general the LP formulation is given as

Minimize

All arcs arefrom a node inS to a node inD anduncapacitated

S Supply nodes

D Demand nodes

18

Useful Facts About TransportationProblem

Suppose that (1) the constraint matrix can be partitioned into A1x = b1 and A2x = b2 (2) all entries of A1 and A2 are 0 or 1 (3) there is at most one 1 in any column of A1 or A2Then this is a transportation problemTheorem If one carries out the simplex algorithm on the transportation problem then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1 The costs and RHS are both integer valuedCorollary The optimal solution to the LP is integer valued

19

The Assignment Problem

Suppose that one wants to assign tasks to persons

Tasks Persons

In this example 4 tasks 3 personsNo two tasks to the same personEach person gets a taskcij is the ldquocostrdquo of assigning task ito person j

Let xij = 1 if task i is assigned to jLet xij = 0 otherwiseHow do we formulate an LP

20

The Assignment Problem

In general the LP formulation is given as

Minimize

Each supply is 1

Each demand is 1

21

More on the Assignment Problem

Tasks PersonsThe assignment problem isa special case of thetransportation problem

The simplex algorithm cansolve the LP relaxation andit will give integer answersthat is it will solve theassignment problem

22

An Application of the AssignmentProblem

Suppose that there are moving targets in space You can identifyeach target as a pixel on a radar screen Given two successivepictures identify how the targets have moved

23

The Maximum Flow Problem

Network G = (N A)

ndash Source s and sink t

ndash Capacities uij on arc (ij)

ndash Variable Flow xij on arc (ij) Graph with capacities

Maximize the flow leaving sst Flow out of i - Flow into i = 0 for i ne s t

0 le xij le uij

24

The Max Flow Problem

In general the LP formulation is given as

Minimize v

otherwise

This is not formulated as a special case of aminimum cost flow formulationCan we reformulate it in this way

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

18

Useful Facts About TransportationProblem

Suppose that (1) the constraint matrix can be partitioned into A1x = b1 and A2x = b2 (2) all entries of A1 and A2 are 0 or 1 (3) there is at most one 1 in any column of A1 or A2Then this is a transportation problemTheorem If one carries out the simplex algorithm on the transportation problem then at every iteration of the simplex algorithm each coefficient in the tableau (except for costs and RHS) is either 0 or -1 or 1 The costs and RHS are both integer valuedCorollary The optimal solution to the LP is integer valued

19

The Assignment Problem

Suppose that one wants to assign tasks to persons

Tasks Persons

In this example 4 tasks 3 personsNo two tasks to the same personEach person gets a taskcij is the ldquocostrdquo of assigning task ito person j

Let xij = 1 if task i is assigned to jLet xij = 0 otherwiseHow do we formulate an LP

20

The Assignment Problem

In general the LP formulation is given as

Minimize

Each supply is 1

Each demand is 1

21

More on the Assignment Problem

Tasks PersonsThe assignment problem isa special case of thetransportation problem

The simplex algorithm cansolve the LP relaxation andit will give integer answersthat is it will solve theassignment problem

22

An Application of the AssignmentProblem

Suppose that there are moving targets in space You can identifyeach target as a pixel on a radar screen Given two successivepictures identify how the targets have moved

23

The Maximum Flow Problem

Network G = (N A)

ndash Source s and sink t

ndash Capacities uij on arc (ij)

ndash Variable Flow xij on arc (ij) Graph with capacities

Maximize the flow leaving sst Flow out of i - Flow into i = 0 for i ne s t

0 le xij le uij

24

The Max Flow Problem

In general the LP formulation is given as

Minimize v

otherwise

This is not formulated as a special case of aminimum cost flow formulationCan we reformulate it in this way

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
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• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

19

The Assignment Problem

Suppose that one wants to assign tasks to persons

Tasks Persons

In this example 4 tasks 3 personsNo two tasks to the same personEach person gets a taskcij is the ldquocostrdquo of assigning task ito person j

Let xij = 1 if task i is assigned to jLet xij = 0 otherwiseHow do we formulate an LP

20

The Assignment Problem

In general the LP formulation is given as

Minimize

Each supply is 1

Each demand is 1

21

More on the Assignment Problem

Tasks PersonsThe assignment problem isa special case of thetransportation problem

The simplex algorithm cansolve the LP relaxation andit will give integer answersthat is it will solve theassignment problem

22

An Application of the AssignmentProblem

Suppose that there are moving targets in space You can identifyeach target as a pixel on a radar screen Given two successivepictures identify how the targets have moved

23

The Maximum Flow Problem

Network G = (N A)

ndash Source s and sink t

ndash Capacities uij on arc (ij)

ndash Variable Flow xij on arc (ij) Graph with capacities

Maximize the flow leaving sst Flow out of i - Flow into i = 0 for i ne s t

0 le xij le uij

24

The Max Flow Problem

In general the LP formulation is given as

Minimize v

otherwise

This is not formulated as a special case of aminimum cost flow formulationCan we reformulate it in this way

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

20

The Assignment Problem

In general the LP formulation is given as

Minimize

Each supply is 1

Each demand is 1

21

More on the Assignment Problem

Tasks PersonsThe assignment problem isa special case of thetransportation problem

The simplex algorithm cansolve the LP relaxation andit will give integer answersthat is it will solve theassignment problem

22

An Application of the AssignmentProblem

Suppose that there are moving targets in space You can identifyeach target as a pixel on a radar screen Given two successivepictures identify how the targets have moved

23

The Maximum Flow Problem

Network G = (N A)

ndash Source s and sink t

ndash Capacities uij on arc (ij)

ndash Variable Flow xij on arc (ij) Graph with capacities

Maximize the flow leaving sst Flow out of i - Flow into i = 0 for i ne s t

0 le xij le uij

24

The Max Flow Problem

In general the LP formulation is given as

Minimize v

otherwise

This is not formulated as a special case of aminimum cost flow formulationCan we reformulate it in this way

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

21

More on the Assignment Problem

Tasks PersonsThe assignment problem isa special case of thetransportation problem

The simplex algorithm cansolve the LP relaxation andit will give integer answersthat is it will solve theassignment problem

22

An Application of the AssignmentProblem

Suppose that there are moving targets in space You can identifyeach target as a pixel on a radar screen Given two successivepictures identify how the targets have moved

23

The Maximum Flow Problem

Network G = (N A)

ndash Source s and sink t

ndash Capacities uij on arc (ij)

ndash Variable Flow xij on arc (ij) Graph with capacities

Maximize the flow leaving sst Flow out of i - Flow into i = 0 for i ne s t

0 le xij le uij

24

The Max Flow Problem

In general the LP formulation is given as

Minimize v

otherwise

This is not formulated as a special case of aminimum cost flow formulationCan we reformulate it in this way

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

22

An Application of the AssignmentProblem

Suppose that there are moving targets in space You can identifyeach target as a pixel on a radar screen Given two successivepictures identify how the targets have moved

23

The Maximum Flow Problem

Network G = (N A)

ndash Source s and sink t

ndash Capacities uij on arc (ij)

ndash Variable Flow xij on arc (ij) Graph with capacities

Maximize the flow leaving sst Flow out of i - Flow into i = 0 for i ne s t

0 le xij le uij

24

The Max Flow Problem

In general the LP formulation is given as

Minimize v

otherwise

This is not formulated as a special case of aminimum cost flow formulationCan we reformulate it in this way

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

23

The Maximum Flow Problem

Network G = (N A)

ndash Source s and sink t

ndash Capacities uij on arc (ij)

ndash Variable Flow xij on arc (ij) Graph with capacities

Maximize the flow leaving sst Flow out of i - Flow into i = 0 for i ne s t

0 le xij le uij

24

The Max Flow Problem

In general the LP formulation is given as

Minimize v

otherwise

This is not formulated as a special case of aminimum cost flow formulationCan we reformulate it in this way

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

24

The Max Flow Problem

In general the LP formulation is given as

Minimize v

otherwise

This is not formulated as a special case of aminimum cost flow formulationCan we reformulate it in this way

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

25

More on the maximum flow problem

Is the current flow optimal

An s-t cut is a separation ofthe nodes into two parts Sand T with s in S and t in T

The capacity of the cut isthe sum of the capacitiesfrom S to T

The max flow from s to t isat most the capacity of anys-t cut

Graph with capacitiesand flows (underlined)

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

26

The Shortest Path Problem

What is the shortest path from an origin or sourcenode (often denoted as s) to a destination or sink node(often denoted as t) What is the shortest path fromnode 1 to node 6Assumptions for now 1 There is a path from node s to all other nodes 2 All arc lengths are non-negative

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

27

Direct Applications

bull What is the path with the shortest driving time from 77 Massachusetts Avenue to Boston City Hall

bull What is the path from Building 7 to Building E40 that minimizes the time spent outside

bull1048698What is the communication path from i to j that is the fastest (taking into account congestion at nodes)

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

28

Formulation as a linear program

In general the LP formulation is given as

Minimize

otherwise

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

29

The Shortest Path Problem1048698bull Fact The Shortest path problem is a special case of the minimum cost flow problem

bull Lots of interesting applications (coming up)

bull Very fast algorithm (coming up)

bull Connection to dynamic programming (several lectures from now)

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Slide 23
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

30

Conclusions

bull Advantages of the transportation problem and the minimum cost flow problem ndash Integer solutions ndash Very fast solution methods ndash Extremely common in modeling

bull Today we saw the following ndash The minimum cost flow problem ndash The transportation problem ndash The assignment problem ndash The maximum flow problem ndash The shortest path problem

• Slide 1
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
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• Slide 29
• Slide 30