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,

Possible Stream Sequences a b c.

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: f i + v i c –Pedestrian streams f i = start-up + crossing time v i = 0 –Vehicular streams f i = start-up and yellow lost time (3-4 s) v i = (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 Arc i-j: i and j are in conflict

Compatibility Cliques Maximal groups of mutually compatible streams “Stages” or “phases” abcd A sequence of stages a 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 u i = stream i’s start time arc i-j length = lower bound time, for every conflict Length of arc i-j depends on (yet undetermined) sequence; can be + or - Need to parameterize sequence

Representing Sequence u j – u i > a ij + v i c - c j ij'j' ujuj uiui If i follows j: c u j + c ij i'i' uiui ujuj If i doesn’t follow j: u j – u i > a ij + v i c Define: F ij = 1 if i follows j, 0 otherwise In general, u j – u i > a ij + v i c - F ij c

Formulation on Conflict Digraph Decision variables: c, u i, F ij Minimize c s.t. u j – u i > a ij + v i c - F ij c (1) time needs F ij = 0 or 1(2) sequence F ji = 1 – F ij (3) reciprocity c > 0 Non-linear, mixed integer

Solution Approach Branch and Bound 1.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 F ij ’s between the clique’s members cliq 1 cliq 2 cliq 3 perm 1 perm 2 perm n 1 ! perm 1 perm 2 perm n 2 ! perm 1 perm 2 perm n 2 ! cliq 3

Advantage of Branching on Conflict Cliques Ex: conflict clique with 4 streams 4! = 24 possible sequences (branches) Contrast binary branching on F ij ’s 4 * (4 - 1) / 2 = 6 conflict pairs 2 6 = 64 possible sets of F 40 sets of F violate transitivity, are contradictory If F ij = 1 and F jk = 1, it must be that F ik = 1

Branch & Bound Relaxation Position in B & B tree fixes some F ij ’s. For the rest: u j – u i > a ij + v i c - F ij c (1) …retain F ji = 1 – F ij (3) …RELAX F ij = 0 or 1(2) …reduces to F ij = F ji = 1(2') … so only constraint 1, with fixed F, remains. Result: relaxation is an LP in u i and c use Network Simplex very fast solution

Network Simplex for min c, given F Extreme point = Spanning tree solution –Longest Path tree determines u i ’s as function of c –Each non-tree arc has surplus ij = [u j (c) – u i (c) ] – [a ij + 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 a ij + v' ij c ij u i (c) u j (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 ij root tree arcs enter tree exit tree

Optimization Software

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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