Optimal Sequencing of Traffic Streams

at a Signalized Intersection

Peter G. Furth

Dept. of Civil & Environmental Engineering

Northeastern University, Boston, MA

Visiting Scholar at ULB, 2004-2005

23

5

9

8

11

2

3 8 11

9 5

Possible Stream Sequences

23

5

9

8

11

2

3 8 11

9 5a.

2

38 11

9 5b.

2

3 8

95

11c.

What Drives the Problem?

• Some stream pairs conflict, some don’t

• Select optimal sequence for a pre-timed cycle

• Objective: minimize cycle length c – Tends to minimize delay to vehicles,

pedestrians– Tends to maximize intersection capacity

What Drives the Problem?

• Each stream appears exactly once per cycle

• Stream i minimum duration: fi + vi c – Pedestrian streams

fi = start-up + crossing time

vi = 0

– Vehicular streamsfi = start-up and yellow lost time (3-4 s)

vi = (traffic volume) / (discharge rate)

• Asymmetric clearance time for conflict pairs (typically 0 to 3 s)

Researchers have tackled this problem since 1962 …

Incompatibility Cliques

• Maximal groups of mutually conflicting streams

• Place lower bound on cycle length

• Usually determine minimum cycle length – but not always

2

5

11

3 8

9

Arc i-j: i and j are in conflict

Compatibility Cliques

• Maximal groups of mutually compatible streams

• “Stages” or “phases”

2

3 8 11

9 5

a b c d

A sequence of stages

2

8 9

3

11

5a

b

c

d

arc i-j: i and j are not in conflict

• Optimal sequence is usually a sequence of compatibility cliques - but not always

Directed Graph of Conflicts(Activity Graph)

• node = stream’s green start

• ui = stream i’s start time

• arc i-j length = lower bound time, for every conflict

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5

11

3 8

9

• Length of arc i-j depends on (yet undetermined) sequence; can be + or -

• Need to parameterize sequence

Representing Sequence

uj – ui > aij + vi c - cj i j'uj ui

If i follows j:

c

uj + c

i j i'

ui uj

If i doesn’t follow j:uj – ui > aij + vi c

Define:

Fij = 1 if i follows j, 0 otherwise

In general,

uj – ui > aij + vi c - Fij c

Formulation on Conflict DigraphDecision variables: c, ui, Fij

Minimize c

s.t.

uj – ui > aij + vi c - Fij c (1) time needs

Fij = 0 or 1 (2) sequence

Fji = 1 – Fij (3) reciprocity

c > 0Non-linear, mixed integer

Solution Approach

Branch and Bound1. Build B&B tree using conflict cliques

2. Relaxation that yields a LP

3. Specialized network simplex to solve relaxations

Building the B & B Tree

• Node = Conflict clique

• Branch = permutation (sequence) for the clique

• Permutation specifies Fij’s between the clique’s members

cliq 1

cliq 2

cliq 2

cliq 3

cliq 3

perm

1

perm 2

perm n1 !

perm

1

perm 2

perm n2 !

perm

1

perm 2

perm n2 !

cliq 3

cliq 3

Advantage of Branching on Conflict Cliques

• Ex: conflict clique with 4 streams4! = 24 possible sequences (branches)

• Contrast binary branching on Fij’s

4 * (4 - 1) / 2 = 6 conflict pairs

26 = 64 possible sets of F

• 40 sets of F violate transitivity, are contradictory

• If Fij = 1 and Fjk = 1, it must be that Fik = 1

Branch & Bound Relaxation Position in B & B tree fixes some Fij’s.

For the rest: uj – ui > aij + vi c - Fij c (1) … retain

Fji = 1 – Fij (3) … RELAX

Fij = 0 or 1 (2) … reduces to

Fij = Fji = 1 (2') …

so only constraint 1, with fixed F, remains.

Result: relaxation is an LP in ui and c • use Network Simplex• very fast solution

Network Simplex for min c, given F

• Extreme point = Spanning tree solution– Longest Path tree determines ui’s as function of c– Each non-tree arc has

surplusij = [uj(c) – ui (c) ] – [aij + v'ij c] > 0

– c is lowered until, for one arc i-j, surplus = 0• “Pivot arc” is the one with smallest critical ratio• Creates critical (zero-length) circuit

aij + v'ij ci j

ui(c) uj(c)root

tree arcs

Network Simplex for min c, given F

• Improvement step– Pivot arc enters tree, creating circuit

• If it’s a directed circuit, STOP: optimum reached

– Circuit arc co-incident to end of pivot arc leaves tree– Lower c to find next extreme point

i jroot

tree arcs

enter tree

exit tree

Optimization Software

Feedback:

Results So Far …

• Virtually immediate solution for– “typical” test problem– Test problem in the literature– Awaiting test on “large” problem

• Practical implications– Better fixed-cycle timing plans for complex

intersections– Possible guidance into optimal sequence for

actuated signals