Computational Study of state of the art path based assignment algorithms

1. Computational study of state-of-
the-art path-based
traffic assignment algorithms
Submitted by
Pankaj Kumar
120040112
4th year undergraduate
IIT Bombay

2. Content:
• Introduction
• Assignment Problem
• Assignment Algorithms
• Results
• Conclusion

3. Introduction:
Path-based algorithms, new area in solving traffic assignment problem, which has been made possible due to the
dramatic advances in computing power. It has been found that path-based algorithms is a viable approach for traffic
assignment problems with reasonably large network sizes.
Path-formulated traffic assignment problem:
Consider an urban traffic network represented as a graph G = (N,A), where N and A are the sets of nodes and links,
respectively. Let R and S be subsets of N and qrs be the steady-state demand generated from origin r ∈ R to
destination s ∈ S. The independent variables are a set of path flows 𝑓𝑘
𝑟𝑠
that satisfy the travel demand:
• 𝑓𝑘
𝑟𝑠
= qrs, ∀ r ∈ R, s ∈ S , k∈Krs (1)
where Krs is a set of cycle-free paths connecting r and s. Let xa be the traffic flow on link a ∈ A, then the total traffic
flow on each link can be written as:
xa = 𝑟∈𝑅 𝑠∈𝑆 𝑓𝑘
𝑟𝑠
for all k∈Krs (2) where
• δrs= 1 if link a is on path k connecting r and s
0 otherwise
To ensure meaningful solutions, all path flows must be non-negative: 𝑓𝑘
𝑟𝑠
≥ 0, ∀ k ∈ Krs, r ∈ R, s ∈ S (3)
The functional form is given by ta(xa) = tf(1+0.15(
𝑥𝑎
𝐶𝑎
)^4) (4)
• ZUE = 0
𝑥𝑎
𝑡𝑎 (𝑤)𝑑𝑤 for all a∈A (5)
• ZSO = 𝑡𝑎 (𝑥𝑎)𝑥𝑎 for all a∈A (6)

4. We are focusing on two particular algorithms:
• Disaggregate simplicial decomposition (DSD) algorithm
• Gradient projection (GP) algorithm
Factors taken into consideration for sensitivity analysis are:
• Network sizes
• Congestion levels
• Number of origin-destination (OD) pairs
• Solution accuracy
Disaggregate simplicial decomposition (DSD) algorithm: It belongs to the general class of
simplicial decomposition algorithm, which states that any point in a bounded polyhedral set can
be described by a convex combination of the extreme points. Its generalized solution iterates
between a linear subproblem and a master problem to determine the next solution point. The
subproblem solves a linear approximation of the objective function to obtain extreme points.
• minZ(λ) = 𝑎∈𝐴 0
𝑡𝑎(𝑤) d𝑤 where upper limit is 𝑟=𝑅 𝑠∈𝑆 𝑞𝑟𝑠 𝑘€ˆ𝐾𝑟𝑠 𝜆𝑟𝑠𝛿𝑟𝑠 (7)
• subject to
• 𝑘∈𝐾𝑟𝑠 𝜆 𝑘
𝑟𝑠
= 1, ∀ r ∈ R, s ∈ S (8)
• 𝜆 𝑘
𝑟𝑠
≥ 0, ∀ k ∈ ˆKrs, r ∈ R, s ∈ S (9)
• Path flows are readily available once 𝜆 𝑘
𝑟𝑠
are solved.

5. Gradient projection (GP) algorithm: GP algorithm is based on the Goldstein–Levitin–Polyak GP
method formulated by Bertsekas for general nonlinear multi-commodity problems. The algorithm
operates directly in the space of path flows with moves are in the direction of the negative gradient,
scaled by the second derivative Hessian. The main idea is to eliminate the travel demand constraints
which is done by reformulating the path-flow variables in terms of the non-shortest path flows which
make the projection operation simpler.
• 𝑓𝐾𝑟𝑠
𝑟𝑠
= 𝑞 𝑟𝑠 − 𝑘 ∈ 𝐾𝑟𝑠,𝑘 =.𝑘𝑟𝑠
𝑓𝑘
𝑟𝑠
, ∀ r ∈ R, s ∈ S (10)
• Embedding constraints into the objective function, we obtain a new formulation with just the
nonnegativity constraints on the non-shortest path flows as the decision variables.
• min ˜ Z( ˜ f ) (11)
• subject to
• 𝑓𝑘
𝑟𝑠
≥ 0, ∀ k ∈ Krs, k _= .krs, ∀ r ∈ R, s ∈ S (12)
• where ˜Z is the transformed objective function expressed in terms of the non-shortest path flows ˜
f for all OD pairs. In each iteration, the scaled GP algorithm updates the path flows according to the
following iteration equations:
• 𝑓𝑘
𝑟𝑠
𝑛 + 1 = [𝑓𝑘
𝑟𝑠
𝑛 −
𝛼 𝑛
𝑠 𝑘
𝑟𝑠 𝑛
(𝑑 𝑘
𝑟𝑠
𝑛 − 𝑑 𝑘 𝑟𝑠(𝑛)
𝑟𝑠
)] ∀ k ∈ Krs, k _= .krs, ∀ r ∈ R, s ∈ S (13)
• 𝑓𝐾 𝑟𝑠(𝑛)
𝑟𝑠
𝑛 + 1 = 𝑞 𝑟𝑠 −, ∀ 𝑘 ∈ 𝐾𝑟𝑠,𝑘 =.𝑘𝑟𝑠
𝑓𝑘
𝑟𝑠
(𝑛 + 1), ∀ r ∈ R, s ∈ S (14)

6. Experiment and Result:
A real and large-scale network, known as the ADVANCE network in the Chicago area, is used for
generating computational results from DSD and GP algorithms for empirically convincing that these
algorithms are viable in real world. Other than this, 5 randomly generated networks are used to do
sensitivity analysis of these algorithms.
Table 1: Provides the statistics of these networks. Objective value is used as a common criterion to
measure the performance of the algorithms.
Table 2 reports the computational times and number of iterations for both algorithms based on the above
termination criterion. DSD did not switch in any of the cases to the second-order operations. In Table 2,
two versions of DSD, denoted by DSD1 for the first-order method and DSD2 for the second-order method,
are provided and compared with GP.

7. Conclusion:
Researchers have implemented two path-based algorithms (GP and DSD) for the traffic assignment
problem and compared their efficiency using sensitivity analysis and Advance network. Results
confirmed that GP performed better than DSD of both versions (DSD2 & DSD1) in all the cases.
GP DSD
Number of paths used in each iteration is less Number of path used in each iteration is high
It uses one-at a time flow update strategy to avoid
expensive line searches by keeping second
derivative information
DSD quickly obtain a near optimal solution by using
first order reduced gradient method
Less time taken to find a solution More time taken to find the ultimate solution
GP avoids performing expensive line searches and
remove unused paths by employing column
dropping and maintain a much smaller path
Main time spent on optimizing the switching
strategy

8. Thank you
Any questions?