A New Approach for Optimizing Traffic Signals in Networks Considering Rerouting

A New Approach for Optimizing Traffic Signals in Networks Considering Rerouting tài liệu, giáo án, bài giảng , luận văn,...

in Networks Considering Rerouting

Duc Quynh Tran1, Ba Thang Phan Nguyen2, and Quang Thuan Nguyen2

1 FITA, Vietnam National University of Agriculture, Hanoi, Vietnam

2 SAMI, Hanoi University of Science and Technology, Hanoi, Vietnam

tdquynh@vnua.edu.vn, phanbathang125692@gmail.com, thuan.nguyenquang@hust.vn

Abstract In traffic signal control, the determination of the green time

and the cycle time for optimizing the total delay time is an important problem We investigate the problem by considering the change of the as-sociated flows at User Equilibrium resulting from the given signal timings (rerouting) Existing models are solved by the heuristic-based solution methods that require commercial simulation softwares In this work, we build two new formulations for the problem above and propose two meth-ods to directly solve them These are based on genetic algorithms (GA) and difference of convex functions algorithms (DCA)

Keywords: DC algorithm, Genetic algorithm, Traffic signal control,

Bi-level optimization model

Traffic signal control plays an important role to reduce congestion, improve safety and protect environment [23] The determination of optimal signal timings have been continuously developed At the beginning, researchers studied isolated junc-tions [28] Thus, an urban network is signalized by considering all its juncjunc-tions independently Some work study the group of junctions such as the problem of green wave in which the traffic light at a junction depends on the others [21],[29] Normally, after finding an optimal signal timing, it is fixed Some systems, how-ever, use real time data to design signal timing that leads to a non-fixed time signal plan [8]

This work focuses on the fixed time plan process Signal timings are optimized

by using historical flows observed on links This bases on the assumption that the flow rates will not change after the new optimal timing is set Almond and Lott in 1968 showed that the assumption is not valid anymore for a wide area [1] The signal time makes a change on journey time on a certain route and thus the users may choose another route that is better It is theoretically explained by Wardrop user equilibrium condition [27] To reflect the dependency of flow rates

on signal timing change, when formulating optimization problem, an equilibrium model may be integrated as constraints to the problem

The problem of determining optimum signal timing is usually formulated as a bi-level optimization problem In the upper level, the objective function is often c

H.A Le Thi et al (eds.), Model Comput & Optim in Inf Syst & Manage Sci.,

Advances in Intelligent Systems and Computing 359, DOI: 10.1007/978-3-319-18161-5 _13

non-smooth and non-linear that optimizes some measures such as total delay, pollution, operating cost, This upper level problem is constrained by the lower level equilibrium problem in which transport users try to alter their travel choices

in order to minimize their travel costs Such an optimization problem may has multiple optima and finding an efficient method to even get local optima is diffi-cult [17] Many solution methods are studied to devise an efficient technique for solving the above problem: heuristic methods ([24],[5]), linearization methods ([10], [2]), sensitivity based methods ([7],[30]), Krash-Kuhn-Tucker based meth-ods ([26]), marginal function method ([18]), cutting plan method ([9]), stochastic search methods ([6], [4], [3])

One of the impressive researches is of Ceylan and Bell ([3],[5]) They use a signal timings optimization method in which rerouting is taken in to account Recall that the problem is formulated as a bi-level optimization problem in which the upper level objective is to minimize total travel time and the lower level problem is a traffic equilibrium problem The proposed solution method was heuristic, namely, a genetic algorithm (GA) for the upper level problem and the SATURN package for the lower level one SATURN is a simulation-assignment modeling software package [25] that gives an equilibrium solution by solving heuristically sub-routines Since SATURN is heuristic- based and a commercial software as well, it is necessary to find a more-efficient approach to solve the problem

In order to overcome the difficulty and to aim at getting a good equilibrium solution, we propose two new formulations that are directly solved by some effi-cient methods The first formulation is then solved by genetic algorithms (GA) while the second one is done by a combination of GA and DCA (Difference of Convex functions Algorithm) As known, DCA was first introduced by Pham Dinh Tao in 1985 and has been extensively developed since 1994 by Le Thi Hoai An and Pham Dinh Tao in their common works It has been successfully applied to many large-scale (smooth or nonsmooth) nonconvex programs in var-ious domains of applied science, and has now become classic and popular (see [11],[12],[15] and references therein) This motivates us using DCA to improve the solution quality in GA-DCA scheme

The paper is organized as follows After the introduction in Section 1, the mathematical problem is described in Section 2 Section 3 is devoted to the GA-based solution method A combined GA-DCA is presented in Section 4 Section

5 gives some conclusions

In this section, we present new mathematical models for optimizing traffic sig-nals in a network considering rerouting The problem is first formulated as an optimization problem with complementarity constraints The objective function

is the total travel time of all vehicles in the network

For the formulation, we use the following notations (see Table 1, Table 2) The parameters and variables are respectively defined in Table 1 and 2

Table 1 Parameters

p pathp = i p → i p → → i p

n(p),

w = (i, j) pair of origin i and destination j (OD pair),

P w set of paths fromi to j,

P = ∪P w set of all paths,

d w demand of origin destination pairw,

a = (u, v) link a,

δ a,p parameter equal to 1 if link a belongs to path p, 0 otherwise,

h junctionh,

S h total number of stages at junctionh,

I r,h inter-green between the end of green time for stager and the start of the next green,

 h,r,p parameter equal to 1 if the vehicles on pathp can cross junction h at stage r,

C min minimum of cycle time,

C max maximum of cycle time,

φ h,r,min minimum of duration green time of stager at junction h,

φ h,r,max maximum of duration green time of stager at junction h,

Table 2 Variables

q a flow on linka,

t a travel time on link a,

t p travel time on pathp,

f p flow on pathp,

t w travel time for OD pair w,

W T h,pwaiting time at junctionh associated to path p,

W T h,p0 initial waiting time at junctionh associated to path p,

z h,p integer variables, that is used to calculateW T h,p0 ,

ST h,r starting time of stager at junction h,

θ h offset of junctionh,

C common cycle time,

φ h,r duration of the green time for stage r at junction h

The total travel time is calculated by

p

t p f p=

w

d w t w

The cycle time, the green time and the offset must satisfy the following conditions:

The total of green time and inter-green time is equal to the cycle time

C =

S h



r=1

φ h,r+

S h



r=1

The flow on link (u, v) is the total of flows on all path p where (u, v) ∈ p.

q (u,v) =

p

The travel time on a path is the sum of the travel on links and the waiting time at junctions

t p=

n(p)−1

k=1

t (i p ,i p k+1)+

n(p)−1

k=2

The travel time on link (u, v), t (u,v) , linearly depends on flow q u,v

t (u,v) = t0(u,v) + α u,v q u,v ∀(u, v), (7)

where α u,v is a constant

For each OD pair, the demand is the total of the flows on used paths



p∈P w

For each OD pair, the travel time t wis equal to the one of all used paths and

the travel time on non-used path is greater than t w (user equilibrium)

Constraints (11)-(13) are introduced to determine the starting time of stages

ST h,r = ST h,r−1 + φ h,r−1 + I h,r−1 ∀h, ∀r ≥ 1 (13)

At junctions, vehicles must spend an initial waiting time that is the time from the arrival time to the beginning of the stage at which vehicle can cross the intersection to continue its journey Constraints (14)-(15) are used to estimate the initial waiting times for the junction after the second one of a path The

integer variables z i p ,p are used in order to assure that the initial waiting time is always smaller than the common cycle time

r

 i p ,r,p ST i p ,r −

r

 i p k−1 ,r,p ST i p

k−1 ,r − t (i p

k−1 ,i p)− z i k ,p C = W T i0p ,p , ∀p, k

(14)

0≤ W T0

Under the assumption that the arrival flow is under an uniform distribu-tion, the initial waiting time at the first junction on a path is estimated by constraints (16)

W T i0p

2,p=1

2[C −

r

 i p

2,r,p φ i p

The delay time at junctions depends on the initial waiting time and the number of vehicles crossing the junction This relation can be expressed by constraint (17)

W T i p ,p = W T i0p + β i p ,p 

p1

 i p ,r,p1.f p1, ∀p, (17)

where β i p ,pis a constant

The flows and the travel time are non-negatives, variables z i p ,pare integers

The aim of problem is to minimize the total travel time T T in the network.

Therefore, it is formulated as the following optimization problem

w

d w t w } s.t.(1) − (19)

This is a mixed integer non-linear program It is very difficult to solve due to

the complementarity constraint (10) and the integer variables z h,p In order to

overcome the difficulty above, Problem (P1) is transformed to Problem (P2) by using penalty techniques

Firstly, we define set D as below:

D = {ξ = (C, θ h , φ h,r , f p , t p , t w , t (u,v) , z h,p)|(1) − (9), (11) − (18)}

p

min{f p , t p − t w }, ν(ξ) =

h,p

sin2(z h,p π)

We see that constraint (10) and constraint (19) can be replaced by μ(ξ) ≤ 0

and ν(ξ) ≤ 0, respectively We consider the following problem

(P2)

min{T T (ξ) =

w

d w t w + λ.

p

min{f p , t p − t w } + λ.

h,p

sin2(z h,p π) } s.t. (1)− (9), (11) − (18)

where λ is a sufficiently large number.

It is clear that if an optimal solution ξ ∗ to (P2) satisfies μ(ξ ∗ ) = 0, ν(ξ ∗) =

0 then it is an optimal solution to the original problem On the other hand, according to the general result of the penalty method (see [16], pp 366-380),

for a given large number λ, the minimizer of (P2) should be found in a region

where μ(ξ), ν(ξ) are relatively small Thus, we will consider in the sequel the problem (P2) with a sufficiently large number λ Problem (P2) can be handled

by a genetic algorithm (in the next section)

Another way, to remove the difficulty in Problem (P1), is to transform it into

an equivalent problem as below

Since min

ξ∈D ν(ξ) = 0, Problem (P1) is equivalent to

(P3)

min{T T (ξ) =

w

d w t w } s.t (1)− (4)

ξ ∈ argmin{

h,p

sin2(z h,p π) } s.t. (5)− (9), (11) − (18) μ(ξ) ≤ 0.

In the lower level of Problem (P3), the constraint μ(ξ) ≤ 0 is still hard It is

tackled by using exact penalty techniques Theorem 1 is in order

Theorem 1 [13] Let Ω be a nonempty bounded polyhedral convex set, f be a

finite DC function on Ω and p be a finite nonnegative concave function on Ω Then there exists η0 ≥ 0 such that for η > η0 the following problems have the same optimal value and the same solution set

(P η) α(η) = min

f (x) + η.p(x) : x ∈ Ω,

f (x) : x ∈ Ω, p(x) ≤ 0.

For given (C, θ h , φ h,r ), denote Ω = {x = (f p , t p , t w , q u,v , t u,v , z h,p) |(5) −

(9), (11) − (18)} It is easy to see that μ(ξ) is concave and non negative on Ω.

Hence, the lower problem can be rewritten as a DC program

(P lower)

min{

h,p

sin2(z h,p π) + η.μ(ξ) } s.t. (5)− (9), (11) − (18)

where η > 0 is a sufficiently large number.

The original problem is equivalent to the following one

(P4)

min{T T (ξ) =

w

d w t w } s.t (1)− (4)

(f p , t p , t w , q u,v , t (u,v) , z h,p)∈ argmin{

h,p

sin2(z h,p π) + η.μ(ξ) } s.t. (5)− (9), (11) − (18)

The lower problem in (P4) is a DC program It can be solved by a deterministic method

3 A GA-Based Solution Method

3.1 Introduction to Genetic Algorithm

Genetic algorithm (GA) is a branch of evolutionary computation in which one imitates the biological processes of reproduction and natural selection to solve for the fittest solutions GA allows one to find solutions to problems that other optimization methods cannot handle due to a lack of continuity, derivatives, linearity, or other features Although GA may not provide a global solution, but the quality of solutions obtained by GA are acceptable in practice Moreover,

GA can be easily implemented and the executable time is reasonable Today genetic algorithms have become a classic in the field of computer science and applied successfully to solve a lot of problems in different areas The basic steps

to solve a problem using a genetic algorithm can be presented as follows:

Initialization

Coding each solution as an individual in the population One has different ways to do this One of the most popular way is using binary coding In the binary coding, each individual is encoded by a sequence of bits 0 or 1

Randomly generating an initial population

Repeat

Step 1: Decoding and Evaluating the quality of the population by a fitness

function In reality, we can choose the objective function as the fitness function

If stopping criteria are satisfied then STOP else goto Step 2

Step 2: Improving the quality of population through crossover and mutation

procedure (evolution) Goto Step 3

Step 3: Selecting a new population Go to Step 1.

In the next sub-session, we introduced a genetic algorithm for solving problem

(P2) The chromosome encoding and decoding are presented in Subsection 3.2 while the procedure of fitness function computation is described in Subsection 3.3 The crossover, mutation and selection are similar to the one in [5]

3.2 Chromosome Encoding and Decoding

Firstly, note that if the values of common cycle time C, duration of green time

φ h,r , offset θ h , flow f p are given then the others variables are computed In this

study, an individual is (C, θ h , φ h,r , f p) We use the binary coding for variables

C, θ h , φ h,r , f p Each variable is coded by a sequence of 8 bits Suppose that

C, θ h , φ h,r , f p are respectively the representations of variables C, θ h , φ h,r , f p

In the next paragraph, the decoding procedure is showed

Cycle time: is the proportion of the difference C max − C min plus C min:

C = C min+ (C)

28− 1 .(C max − C min ),

where(X) is 10 base equivalent of X.

Offset: for a junction h, it is the proportion of the cycle time

θ h= (θ h)

28− 1 .(C − 1)

Green times: for a stage at junction h, are defined as the sum of the minimum stage length and the proportion of the remaining green time, φ h,r,max −φ h,r,min,

as follows:

φ h,r = φ h,r,min+ (φ h,r)

S h



r=1

(φ h,r)

.(φ h,r,max − φ h,r,min ).

Here, φ h,r,min is a given constant and φ h,r,max is a parameter calculated by

φ h,r,max = C − S h

r=1

I h,r − Sh

y=1,y=r

φ h,r,min By this way, constraint (4) is always satisfied

Flow on path: for a path p ∈ P w flow on path p is defined as the proportion

of demand d w as follows:

f p=(f p)



p∈P w d w

By this way, constraint (8) always holds

3.3 Computing Other Variables and the Fitness Function

Variables q u,v are computed via f pby equation (5)

Variables t (u,v) are computed via q u,v by equation (7)

Variables S h,r are computed via θ h , φ h,r , I h,r by equations (11-13)

Variables W T i0p ,p and z i p ,p are calculated by equation (14) Specifically, W T i0p ,p and z i p ,p ∀k ≥ 3 are respectively the residual and integer part of number

1

C .[



r

 i p

,r,p ST i p

,r −

r

 i p k−1 ,r,p ST i p

k−1 ,r − t (i p

k−1 ,i p)].

Variables W T i0p

2,p , W T i p ,p and t p are calculated via (16),(17),(6)

Variables t w= min

p∈P w {t p }.

The fitness function F F is the objective function

F F = T T (ξ) =

w

d w t w + λ.

p

min{f p , t p − t w } + λ.

h,p

sin2(z h,p π) (20)

In order to improve the quality of individuals in GA, we use DCA for solving

the lower problem in Problem (P )

4.1 A Brief Presentation of DC Programming and DCA

To give the reader an easy understanding of the theory of DC programming

& DCA and our motivation to use them, we briefly outline these tools in this section

Let Γ0(IRn) denotes the convex cone of all lower semi-continuous proper con-vex functions on IRn Consider the following primal DC program:

(P dc) α = inf {f(x) := g(x) − h(x) : x ∈ IR n }, (21)

where g, h ∈ Γ0(IRn)

Let C be a nonempty closed convex set The indicator function on C, denoted

χ C , is defined by χ C (x) = 0 if x ∈ C, ∞ otherwise Then, the problem

can be transformed into an unconstrained DC program by using the indicator

function of C, i.e.,

where φ := g + χ C is in Γ0(IRn)

Recall that, for h ∈ Γ0(IRn ) and x0 ∈dom h := {x ∈ IR n |h(x0) < + ∞}, the

subdifferential of h at x0, denoted ∂h(x0), is defined as

∂h(x0) :={y ∈ IR n : h(x) ≥ h(x0) +x − x0, y , ∀x ∈ IR n }, (24) which is a closed convex set in IRn It generalizes the derivative in the sense that

h is differentiable at x0 if and only if ∂h(x0) is reduced to a singleton which is exactly{∇h(x0)}.

The idea of DCA is simple: each iteration of DCA approximates the concave part−h by its affine majorization (that corresponds to taking y k ∈ ∂h(x k)) and

minimizes the resulting convex problem (P k)

Generic DCA scheme

Initialization: Let x0∈ IR n be a best guess, 0← k.

Repeat

Calculate y k ∈ ∂h(x k)

Calculate x k+1 ∈ arg min{g(x) − h(x k)− x − x k , y k : x ∈ IR n } (P k)

k + 1 ← k

Until convergence of x k

Convergence properties of the DCA and its theoretical bases are described in [11,15,19,20]

4.2 DCA for Solving (P lower) and GA-DCA Algorithm

In Problem (P lower ), the objective function f (x) =

h,p

sin2(z h,p π) + η.μ(ξ) is a

DC function Consider function f h,p (x) = sin2(z h,p π), there exists a DC

decom-position f h,p (x) = τ.z h,p2 −(τ.z2

h,p −sin2

(z h,p π)), where τ > 2π2 Hence, we obtain

a DC decomposition of the objective function f (x) = g(x) − h(x) where g(x) =

τ.

h,p

z2

h,p and h(x) = τ.

h,p

z2

h,p −

h,p

sin2(z h,p π) + η.

h,p

max{−f p , −t p + t w } We

see that the subdifferential of h(x) can be easily computed.

DCA applied to (P lower) can be described as follows:

DCA

Initialization

Let  be a sufficiently small positive number Set  = 0 and x0is a starting point

Repeat

Calculate y  ∈ ∂h(x )

Calculate x +1) by solving a convex quadratic program min{g(x) s.t x ∈ Ω}

 ←−  + 1

Until x +1 − x  ≤  or f(x +1)− f(x ) ≤ .

In the combined GA-DCA, an individual is (C, θ h , φ h,r) The chromosome en-coding and deen-coding are similar to GA presented in Section 3 while the values of

the other variables (f p , t p , t w , q u,v , t (u,v) , z h,p ) are the optimal solution of P (lower)

by using DCA.

The combined GA-DCA scheme is described as follows:

GA-DCA

Initialization

Randomly generate an initial populationP

For an individual Id i = (C i , θ i

h , φ i h,r)∈ P, we solve problem (P lower) by DCA

to obtain (f i

p , t i

p , t i

w , q i u,v , t i

(u,v) , z i h,p ).

Compute the fitness of Id i by formula (20)

Repeat

Step 1: Check the stopping criteria If it is satisfied then STOP else go to

Step 2

Step 2: Launch crossover and mutation procedure (evolution) for improving

the quality of population

For a new individual Id l = (C l , θ l

h , φ l h,r) ∈ P, we solve problem (P lower) by

DCA to obtain (f p l , t l p , t l w , q l u,v , t l (u,v) , z h,p l )

Compute the fitness of Id l by formula (20)

Go to Step 3

Step 3: Select a new population Go to Step 1.

The work studied the problem of optimizing traffic signals considering rerouting The main contribution is to build two new formulations that are probably solved

by efficient methods We also proposed two algorithms to directly solve them

GA and a combination of GA-DCA are investigated and described in detail The effect of the parameters of the models and the algorithms on the numerical results are planned in the future work

Randomly generate an initial populationP

For an individual Id i = (C i , θ i

h , φ i h,r)∈... by DCA

to obtain (f i

p , t i

p , t i

w , q i... rerouting The main contribution is to build two new formulations that are probably solved

by efficient methods We also proposed two algorithms to directly solve them

GA and a combination