Download - Effective Route Guidance in Traffic Networks · C(f ) = 104 but fair! 51 51 1 1 1 2 100 c a =1 c =1 c 7 selfish users central planner the goal own travel time welfare fair, not

Transcript
Page 1: Effective Route Guidance in Traffic Networks · C(f ) = 104 but fair! 51 51 1 1 1 2 100 c a =1 c =1 c 7 selfish users central planner the goal own travel time welfare fair, not

Outline

Route Guidance; User

Optimum;

Capacities.

Constrained

c© 1

• G = ( )k demands (oi, di) ri

• fP .

• ta(·) → and → L

C(f) := �

a∈A

ta(fa)fa

2x x + 1

2

2

2

2

1

0 0

00

2 2

2 · 4 + 2 ·

c© 2

UE unique UE

UE/SO ≥ α(L) UE/SO

UE/SO ≤ α(L) BUE/SO ≤ α(L)

c© 3

Effective Route Guidance in Traffic Networks

Lectures developed by Andreas S. Schulz and Nicolas Stier

May 13, 2004

2004 Massachusetts Institute of Technology

Lecture 1

Equilibrium; System

User Equilibria in Networks with

Lecture 2

System Optimum; Dantzig-Wolfe

Decomposition; Constrained Shortest Paths;

Computational Results.

2004 Massachusetts Institute of Technology

Review of Traffic Model

Directed graph V,A with capacities, with rate

Flows on paths Can be non-integral.

Traversal times: latency functions continuous nondecreasing belong to a given set (e.g. linear)

The total travel time of a flow is:

1 = 10

2004 Massachusetts Institute of Technology

Review of First Lecture

No capacities With capacities

Set of

may be non-convex

unbounded

2004 Massachusetts Institute of Technology

Page 2: Effective Route Guidance in Traffic Networks · C(f ) = 104 but fair! 51 51 1 1 1 2 100 c a =1 c =1 c 7 selfish users central planner the goal own travel time welfare fair, not

optimize optimize system

c© 4

SO

51

51

1 1

1 22

100

ca = 1

ca = 1

c© 5

SO

SO C(SO) = . Unfair!

51

51

1 1

1 22

100

ca = 1

ca = 1

c© 6

SO

C(f ) = 104 but fair!

51

51

1 1

1 22

100

ca = 1

ca = 1

c© 7

selfish users central planner the goal own travel time welfare

fair, not efficient efficient, not fair fair, efficient

2004 Massachusetts Institute of Technology

Long Detours in

Instance with constant latencies:

2004 Massachusetts Institute of Technology

Long Detours in

routes 1 unit along each path: 100 + 3

2004 Massachusetts Institute of Technology

Long Detours in

Compare to routing 1 unit along the other paths:

2004 Massachusetts Institute of Technology

Page 3: Effective Route Guidance in Traffic Networks · C(f ) = 104 but fair! 51 51 1 1 1 2 100 c a =1 c =1 c 7 selfish users central planner the goal own travel time welfare fair, not

Constrained

c© Constrained 8

• SO cannot due to unfairness

• UE

Use instead!

• CSO = min total

c© Constrained 9

c© 10

• UE

Notation:

• �a

• �P = �

a∈P

�a

c© Constrained 11

System Optimum

2004 Massachusetts Institute of Technology System Optimum

Route Guidance

be implemented in practice

does not take into account the global welfare

constrained SO

travel time s.t. demand satisfied

users are assigned to “fair” routes capacity constraints

2004 Massachusetts Institute of Technology System Optimum

Technological Requirements

exact knowledge of the current position

2-way communication to a main server

2004 Massachusetts Institute of Technology Constrained System Optimum

Constrained SO: Normal Lengths

Normal lengths: a-priori belief of network

Geographic distances

Free-flow travel times (times in empty network)

Travel times under

normal length of arc:

normal length of path:

2004 Massachusetts Institute of Technology System Optimum

Page 4: Effective Route Guidance in Traffic Networks · C(f ) = 104 but fair! 51 51 1 1 1 2 100 c a =1 c =1 c 7 selfish users central planner the goal own travel time welfare fair, not

Constrained SO: Definition

• Fix a tolerance ε ≥ 0

• A path P ∈ Pi is valid if �P ≤ (1 + ε) × min Q∈Pi�Q

• Definition:

CSOε = min total travel time

s.t.�

P∈Pi:P valid

fP =ri for all i

P�a

fP ≤ca for a ∈ A

fP ≥0

c©2004 Massachusetts Institute of Technology Constrained System Optimum 12

CSO Example

SO CSO

c©2004 Massachusetts Institute of Technology Constrained System Optimum 13

Remarks about CSO

• It is a non-linear, convex, minimization problem over a polytope

(constrained min-cost multi-commodity flow problem)

• We solve it using the Frank-Wolfe algorithm:

we solve a sequence of linear programs

• No need to consider all path variables simultaneously:

we use column generation

c©2004 Massachusetts Institute of Technology Constrained System Optimum 14

Computing CSO

• Each algorithm uses the next as a subroutine:

1. Frank-Wolfe algorithm: linearize using current gradient

2. Simplex algorithm to solve resulting LP

3. Column generation to handle exponentially many paths

4. Constrained Shortest Path Problem (CSPP) algorithm for pricing

5. Dijkstra’s algorithm as a routine for CSPP

c©2004 Massachusetts Institute of Technology Constrained System Optimum 15

Page 5: Effective Route Guidance in Traffic Networks · C(f ) = 104 but fair! 51 51 1 1 1 2 100 c a =1 c =1 c 7 selfish users central planner the goal own travel time welfare fair, not

0. x0 k = 0 and LB = −∞.

1. UB = C(xk) and x = xk .

2. z ∗ = min{C(x) + ∇C(x)T (x − x) : x feasible }. Let x ∗

3. xk+1 = x+α(x ∗ −x).

4. LB { ∗}.

5. |UB −LB| ≤ tolerance, STOP! k = k + 1

c© 16

• Let ta = ∂C(f i) ∂fa

fa

• subset P ′⊆ P of :

min�

a∈A

ta fa

s.t.�

P �a

fP = fa a ∈ A

valid P ∈P ′ i

fP = ri i = 1, . . . , k (1)

fa ≤ ca a ∈ A (2)

fP ≥ 0 P ∈ P ′

c© Constrained 17

• i σi

• a πa ≥ 0

• ⇔ �

a∈P

(ta + πa) ≥ σi ∀ valid P ∈ Pi

• :

i Pi

σi

” !

c© 18

σi and πa the optimal solution

i: Pi in Pi ta + πa �

a∈Pi(ta + πa) ≥ σi i

5.

7. P ′

8. Pi with �

a∈Pi(ta + πa) < σi to P ′

9.

c© Constrained 19

Frank-Wolfe Algorithm

Initialization: start with flow . Set

Update upper bound: set

Compute next iterate:

be the optimal flow.

Solve the line-search problem and set

Update lower bound: set = max LB, z

Check stopping criteria: if Otherwise, set and go to step 1.

2004 Massachusetts Institute of Technology Constrained System Optimum

Linear Problem and Column Generation

be the objective coefficient of in the LP

As there are exponentially many paths in the LP, we form LP’ with a valid paths

for all

for all

for all

for all

2004 Massachusetts Institute of Technology System Optimum

Linear Problem and Column Generation II

For each demand , let be the dual variable corresponding to (1)

For each arc , let be the dual variable corresponding to (2)

Solution optimal in LP

The Pricing Problem

For every commodity , either find a valid path in with modified cost less than or assert that no such path exists.

Can be solved as a “Constrained Shortest Path Problem

2004 Massachusetts Institute of Technology Constrained System Optimum

Algorithm for Solving the LP

1. Solve the linear program LP’

2. Let be simplex multipliers of the current

3. for all find shortest valid path w.r.t. arc costs

4. if for all

Solution is optimal for LP. STOP

6. else

Remove one or more non-basic variables from

Add at least one path

goto 1

2004 Massachusetts Institute of Technology System Optimum

Page 6: Effective Route Guidance in Traffic Networks · C(f ) = 104 but fair! 51 51 1 1 1 2 100 c a =1 c =1 c 7 selfish users central planner the goal own travel time welfare fair, not

• as many new to

⇒ Add all

one

c© 20

4.

5. Dijkstra

c© Constrained 21

c© 22

• : let d(j) j ∈ A

• For j ∈ A d∗(j) from 1 to j

– – Let T = { } ⇒ d(j) ≥ d∗(j) ∀j ∈ T

– is – Let S = { } ⇒ d(j) = d∗(j) ∀j ∈ S

c© 23

Observations for Solving the LP

Empirically, very advantageous to add columns restricted master problem as possible

paths that price favorably until we run out of space

Non-basic variables removed when their slots are needed for new candidate paths

We observed a reduction in computation time by factors of about 50, compared to always adding a single column and removing another

2004 Massachusetts Institute of Technology Constrained System Optimum

The Pricing Problem

1. Frank-Wolfe algorithm: linearize using current gradient

2. Simplex algorithm to solve resulting LP

3. Column generation to handle exponentially many paths

Constrained Shortest Path Problem (CSPP) algorithm for pricing

’s algorithm as a routine for CSPP

2004 Massachusetts Institute of Technology System Optimum

Shortest Path Problem

2004 Massachusetts Institute of Technology Shortest Path Problem

Distance Labels

We want the shortest paths from node 1 to all other nodes

Throughout the run, nodes will have distance labels

denote the label of

, let denote the shortest distance

Labels can be :

Temporary: shortest distance found so far temporarily labeled nodes

Permanent: when the label the shortest distance permanently labeled nodes

2004 Massachusetts Institute of Technology Shortest Path Problem

Page 7: Effective Route Guidance in Traffic Networks · C(f ) = 104 but fair! 51 51 1 1 1 2 100 c a =1 c =1 c 7 selfish users central planner the goal own travel time welfare fair, not

d

d(j) ≤ d(i) + tij ∀ ( ) ∈ A

0

1

2

3

4

5

6

2

4

2 1

3

4

2

3

2

2

3 4

6

6

c© 24

d(i) i, (i) i

(i)

( ) ∈ A do if d(j) > d(i) + tij then

d(j) := d(i) + tij ; (j) := i;

10 i

j

2

4

j

15 12

13

c© 25

1

S := {1}; T := V \ {1}; d ; d(j) := ∞ j = 2, 3, . . . , n;

1); while S �= V do

i := { d(j) : j ∈ T }; S := S ∪ {i}; T := T \ {i};

i);

c© 26 c© 27

Optimality Conditions

The distance labels are shortest path distances iff

i, j

2004 Massachusetts Institute of Technology Shortest Path Problem

Dijkstra’s Algorithm: Update

Given a label for node Update

improves the labels of ’s neighbors:

Procedure Update

for each i, j

pred

2004 Massachusetts Institute of Technology Shortest Path Problem

Dijkstra’s Algorithm: Main

This routine computes shortest paths from node to all other nodes:

(1) := 0 for update(

// find minimum temporary labeled node and update it argmin

update(

2004 Massachusetts Institute of Technology Shortest Path Problem

Shortest Path Example

2004 Massachusetts Institute of Technology Shortest Path Example

Page 8: Effective Route Guidance in Traffic Networks · C(f ) = 104 but fair! 51 51 1 1 1 2 100 c a =1 c =1 c 7 selfish users central planner the goal own travel time welfare fair, not

Constrained

c© 28

1 to 6 (tij, �ij)

(1,10)

(1,1)

(1,7)(2,3)

(10,3)

(12,3)

(2,2)

(1,2) (10,1) (5,7)

1

2 4

53

6

i j

c© 29

• d(j) d(j) = (dt(j), d�(j))

– dt(j)– d�(j) 1 to j

• d(j) dominates d′(j) iff dt(j) ≤ d′ t(j) and d�(j) ≤ d′

�(j).

• j T (j) of S(j) of

• Let T and S and

c© 30

Given a d(i) i, the i’s

i d(i))

if dt(i) ≥ 1 to n return;

( ) ∈ A do dnew(j) := d(i) + (tij , �ij ); j if dnew

� (j) ≤ L and dnew j add dnew to T (j);

T (j);

c© 31

Shortest Path

2004 Massachusetts Institute of Technology Constrained Shortest Path

Constrained Shortest Path Example

Find the fastest path from node node with a length of at most 10

2004 Massachusetts Institute of Technology Constrained Shortest Path

Labels

A label is now a tuple

is the travel time the length of a path from node

A node may have several labels at the same time

A label

In the algorithm, every node has a set temporary labels and a set permanent labels

be the sets of all temporary permanent labels, resp.

2004 Massachusetts Institute of Technology Constrained Shortest Path

Update

label for node Update improves labels of neighbors:

Procedure Update(node , label

min. time of a feasible path from node so far

for each i, j

// new label for is not dominated by other labels in

delete dominated labels from

2004 Massachusetts Institute of Technology Constrained Shortest Path

Page 9: Effective Route Guidance in Traffic Networks · C(f ) = 104 but fair! 51 51 1 1 1 2 100 c a =1 c =1 c 7 selfish users central planner the goal own travel time welfare fair, not

1 n �(path) ≤ L:

S {(0, 0)}; 1, (0, 0));

while S(n) l l

d := { dt : d ∈ T }; i :=

d from T (i) to S(i); i, d);

enumeration

c© 32

Constrained

c© 33

Idea:

c© 34

CSOε = min total

s.t. �

P ∈Pi:P valid

fP =ri i

P �a

fP ≤ca a ∈ A

fP ≥0

• ⇒ C(f ) = ∞

• → !

CSO

c© 35

Labeling Algorithm

This routine computes a fastest path from node to node such that

(1) := update(

is empty do // find minimum temporary abe ed node

argmin corresponding node;

move the label update(

This can degenerate into a huge

2004 Massachusetts Institute of Technology Constrained Shortest Path

Shortest Path Example

2004 Massachusetts Institute of Technology

Alternative Algorithm for CMCFP

Forget about Capacity Constraints

1. Frank-Wolfe algorithm: linearize using current gradient

2. Simplex algorithm to solve resulting LP

3. Column generation to handle exponentially many paths

4. Constrained Shortest Path Problem (CSPP) algorithm for pricing

5. Dijkstra’s algorithm as a routine for CSPP

2004 Massachusetts Institute of Technology

Relaxing Capacity Constraints

travel time

for all

for

Capacity constraint violated because latency is infinity

Minimization takes care of making the solution feasible

No capacity constraints the problem is separable

can be found with a sequence of CSPP

2004 Massachusetts Institute of Technology

Page 10: Effective Route Guidance in Traffic Networks · C(f ) = 104 but fair! 51 51 1 1 1 2 100 c a =1 c =1 c 7 selfish users central planner the goal own travel time welfare fair, not

Computational Experience

c©2004 Massachusetts Institute of Technology Computational Experience 36

Computational Experiments

We used real-world instances obtained from DaimlerChrysler (Berlin)

and from the Transportation Network Test Problems website:http://www.bgu.ac.il/~bargera/tntp/

Instance Name |V | |A| |K| |A| · |K|Sioux Falls 24 76 528 40K

Friedrichshain 224 523 506 265K

Winnipeg 1,067 2,975 4,344 13M

Neukolln 1,890 4,040 3,166 13M

Mitte, Prenzlauerberg

& Friedrichshain 975 2,184 9,801 21M

Chicago Sketch 933 2,950 83,113 245M

Berlin 12,100 19,570 49,689 972M

c©2004 Massachusetts Institute of Technology Computational Experience 37

Part of an Instance

Demand Solution

c©2004 Massachusetts Institute of Technology Computational Experience 38

Unfairness

• Normal unfairness of path P for OD-pair i = �Pmin Q∈Pi

�Q

→ 1 ≤ normal unfairness ≤ 1 + ε

• Loaded unfairness of path P for OD-pair i = tP (f)min Q∈Pi

tQ(f)

→ 1 ≤ loaded unfairness

• UE unfairness of path P for OD-pair i = tP (f)min Q∈Pi

tQ(BUE)

→ 0 ≤ UE unfairness

c©2004 Massachusetts Institute of Technology Computational Experience 39

Page 11: Effective Route Guidance in Traffic Networks · C(f ) = 104 but fair! 51 51 1 1 1 2 100 c a =1 c =1 c 7 selfish users central planner the goal own travel time welfare fair, not

influenced

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

1.2

UE 1 1.1 1.2 1.3SO

percentile 99th

97.5th

95th

1 + ε)

1

1.1

1.2

1.3

1.4

1.5

1.6

UE 1 1.1 1.2 1.3 SO

percentile 99th

97.5th

95th

1 + ε)

c© Computational 40

C(CSOε) tP (f )

min Q∈Pi tQ(f )

2500

3000

3500

4000

UE 1.05 1.1 1.2 1.3 1.4 1.5 2 SO 1

1.2

1.4

1.6

Cost

Loaded Unfairness

1 + ε)

c© Computational 41

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1 1.2 1.4 1.6 1.8 2

prob

abili

ty

loaded unfairness

Tolerance UE

1.05 1.1 1.2 1.3 1.4 1.5

2 SO

0

0.2

0.4

0.6

0.8

1

0.6 0.7 0.8 0.9 1 1.1 1.2

prob

abili

ty

UE unfairness

Tolerance UE 1.05 1.1 1.2 1.3 1.4 1.5 2 SO

tP (f ) min Q∈Pi

tQ(f ) tP (f )

min Q∈Pi tQ(BUE)

c© Computational 42

C(CSOε) tP (f )

min Q∈Pi tQ(f )

2500

3000

3500

UE 1.01 1.02 1.03 1.05 1.1 1.2 1.3 1.4 1.5 2 SO 1

1.2

1.4

1.6

Cost UE

Loaded Unfairness

Cost Free-flow

1 + ε)

c© Computational 43

Unfairness Percentiles

normal unfairness: controlled directly

loaded unfairness:

tolerance ( tolerance (

2004 Massachusetts Institute of Technology Experience

Free-flow Normal Lengths: High Cost

tolerance (

2004 Massachusetts Institute of Technology Experience

Unfairness Distributions: High Travel Times

2004 Massachusetts Institute of Technology Experience

UE Travel Times: Good Normal Lengths

tolerance (

2004 Massachusetts Institute of Technology Experience

Page 12: Effective Route Guidance in Traffic Networks · C(f ) = 104 but fair! 51 51 1 1 1 2 100 c a =1 c =1 c 7 selfish users central planner the goal own travel time welfare fair, not

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1 1.2 1.4 1.6 1.8 2

prob

abili

ty

loaded unfairness

Tolerance UE

1.01 1.02 1.03 1.05

1.1 SO

0

0.2

0.4

0.6

0.8

1

0.6 0.7 0.8 0.9 1 1.1 1.2

prob

abili

ty

UE unfairness

Tolerance UE 1.01 1.02 1.03 1.05 1.1 SO

tP (f ) min Q∈Pi

tQ(f ) tP (f )

min Q∈Pi tQ(BUE)

c© Computational 44

Convergence

0

2000

4000

6000

8000

10000

12000

14000

UE 1 1.1 1.2 1.3 SO

seco

nds

gap 0.5

1.0

2.0

4.0

1 + ε)

0

2000

4000

6000

8000

10000

12000

14000

0.5 1 1.5 2 2.5 3 3.5 4

seco

nds

gap

tolerance UE

1.01

1.02

1.03

1.05

1.1

1.2

1.3

SO

c© Computational 45

CSO

1

1.2

1.4

1.6

1.8

2

2.2

1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16

Instance

SO

UE

CSO

SF F

W N

MPF CS B

◦ ◦

C(x)/C(SO )

Solutions

‘◦’ denote CSO1.02

c© Computational 46

Review

• Results:

2500

3000

3500

4000

UE 1.05 1.1 1.2 1.3 1.4 1.5 2 SO

tolerance

1

1.2

1.4

1.6

Cost

Loaded Unfairness

2500

3000

3500

4000

UE 1.01 1.02 1.03 1.05 1.1 1.2 1.3 SO

tolerance

1

1.2

1.4

1.6

Cost

Loaded Unfairness

c© 47

Unfairness Distributions: Fair Enough

2004 Massachusetts Institute of Technology Experience

More difficult with bigger tolerance

Runtime grows exponentially

tolerance (

2004 Massachusetts Institute of Technology Experience

allows us to control the tradeoff between efficiency and unfairness

load

ed u

nfa

irnes

s

marked with

2004 Massachusetts Institute of Technology Experience

Free-flow normal length UE normal length

2004 Massachusetts Institute of Technology

Page 13: Effective Route Guidance in Traffic Networks · C(f ) = 104 but fair! 51 51 1 1 1 2 100 c a =1 c =1 c 7 selfish users central planner the goal own travel time welfare fair, not

Conclusion

– SO

– UE

– CSO

– UE

– UE

c© 48

• can optimized

c© 49

c© 50

1982 2000

34% 58%

c© 51

Optimization Approach to Route Guidance

Conventional route guidance methods focus on the individual

not implementable

not efficient

is a better alternative: efficient and fair

Demand-dependent normal lengths are a better choice

Considered Networks with Capacities

Multiple equilibria

Worst is unbounded

Guarantee for best is as good as without capacities

2004 Massachusetts Institute of Technology

Summary

In principle, the system performance be while obeying individual needs and systems response.

In fact, many different tools, from non-linear optimization, from linear programming, and from discrete optimization, nicely complement each other to lead to a fairly efficient algorithm for huge (static) instances.

Yet, more (dynamic) ideas needed before technology ready for field test.

2004 Massachusetts Institute of Technology

THE END

2004 Massachusetts Institute of Technology

2002 Urban Mobility Study shows we could be better (http://mobility.tamu.edu/ums)

time penalty for peak period travelers 16 hours 62 hours

period of time with congestion 4.5 hours 7 hours

volume of roadways with congestion

2004 Massachusetts Institute of Technology

Page 14: Effective Route Guidance in Traffic Networks · C(f ) = 104 but fair! 51 51 1 1 1 2 100 c a =1 c =1 c 7 selfish users central planner the goal own travel time welfare fair, not

tolerance cost

UE 2915 1.056 1.01 2800 1.348 1.02 2738 1.375 1.03 2726 1.424 1.05 2694 1.456 1.10 2676 1.517 1.20 2657 1.545 1.30 2657 1.538 SO 2657 1.546

c© 52

2650

2700

2750

2800

2850

2900

2950

3000

UE 1 1.1 1.2 1.3 SO

cost

tolerance

gap 0.5

1.0

2.0

4.0

c© 53

UE Travel Times: Good Normal Lengths

99th percentile loaded unfairness

2004 Massachusetts Institute of Technology

Solution Quality: UE normal lengths are good

2004 Massachusetts Institute of Technology