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,...

A New Approach for Optimizing Traffic Signals in Networks Considering Rerouting Duc Quynh Tran1 , Ba Thang Phan Nguyen2 , and Quang Thuan Nguyen2 FITA, Vietnam National University of Agriculture, Hanoi, Vietnam 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 associated 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 methods 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 Introduction 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 junctions [28] Thus, an urban network is signalized by considering all its junctions 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, however, 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 Springer International Publishing Switzerland 2015 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_13 143 144 D.Q Tran, B.T Phan Nguyen and Q.T Nguyen 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 difficult [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 methods ([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 efficient 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 various 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 Section is devoted to the GAbased solution method A combined GA-DCA is presented in Section Section gives some conclusions Mathematical Model In this section, we present new mathematical models for optimizing traffic signals 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 and Optimizing Traffic Signals in Networks Considering Rerouting 145 Table Parameters p w = (i, j) Pw P = ∪Pw dw a = (u, v) δa,p h Sh Ir,h h,r,p Cmin Cmax φh,r,min φh,r,max ip1 ip2 path p = → → → ipn(p) , pair of origin i and destination j (OD pair), set of paths from i to j, set of all paths, demand of origin destination pair w, link a, parameter equal to if link a belongs to path p, otherwise, junction h, total number of stages at junction h, inter-green between the end of green time for stage r and the start of the next green, parameter equal to if the vehicles on path p can cross junction h at stage r, minimum of cycle time, maximum of cycle time, minimum of duration green time of stage r at junction h, maximum of duration green time of stage r at junction h, Table Variables qa ta fp tw W Th,p W Th,p zh,p STh,r θh C φh,r flow on link a, travel time on link a, travel time on path p, flow on path p, travel time for OD pair w, waiting time at junction h associated to path p, initial waiting time at junction h associated to path p, integer variables, that is used to calculate W Th,p , starting time of stage r at junction h, offset of junction h, common cycle time, duration of the green time for stage r at junction h The total travel time is calculated by fp = TT = dw tw p w The cycle time, the green time and the offset must satisfy the following conditions: (1) Cmin ≤ C ≤ Cmax , ≤ θh ≤ C − 1, ∀h, (2) φh,r,min ≤ φh,r ≤ φh,r,max (3) The total of green time and inter-green time is equal to the cycle time Sh C= Sh φh,r + r=1 Ih,r , r=1 ∀h (4) 146 D.Q Tran, B.T Phan Nguyen and Q.T Nguyen The flow on link (u, v) is the total of flows on all path p where (u, v) ∈ p q(u,v) = δu,v,p fp (5) p The travel time on a path is the sum of the travel on links and the waiting time at junctions n(p)−1 = n(p)−1 t p (ip k ,ik+1 ) k=1 + W Tipk ,p ∀p (6) k=2 The travel time on link (u, v), t(u,v) , linearly depends on flow qu,v ∀(u, v), t(u,v) = t0(u,v) + αu,v qu,v (7) where αu,v is a constant For each OD pair, the demand is the total of the flows on used paths ∀w fp = dw (8) p∈Pw For each OD pair, the travel time tw is equal to the one of all used paths and the travel time on non-used path is greater than tw (user equilibrium) ≥ tw ∀p ∈ Pw fp (tp − tw ) = (9) ∀p ∈ Pw (10) Constraints (11)-(13) are introduced to determine the starting time of stages ST1,1 = STh,1 = STh−1,1 + θh (11) ∀h ≥ STh,r = STh,r−1 + φh,r−1 + Ih,r−1 ∀h, ∀r ≥ (12) (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 zipk ,p are used in order to assure that the initial waiting time is always smaller than the common cycle time Optimizing Traffic Signals in Networks Considering Rerouting STipk ,r ip k ,r,p − r STipk−1 ,r ip k−1 ,r,p − t(ipk−1 ,ipk ) − zikp ,p C = W Ti0p ,p , k r 147 ∀p, k (14) ≤ W Ti0p ,p ≤ C, ∀p, k = 3, , n(p) − k (15) Under the assumption that the arrival flow is under an uniform distribution, the initial waiting time at the first junction on a path is estimated by constraints (16) W Ti0p ,p = [C − φip2 ,r ] ip ,r,p ∀p (16) r 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 Tipk ,p = W Ti0p + βipk ,p k fp1 , ip k ,r,p1 ∀p, (17) p1 where βipk ,p is a constant The flows and the travel time are non-negatives, variables zipk ,p are integers f p , , tw ≥ zipk ,p ∈ Z ∀p, w (18) ∀p, k (19) 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 min{T T = (P1 ) w dw tw } 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 zh,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, θh , φh,r , fp , , tw , t(u,v) , zh,p )|(1) − (9), (11) − (18)} min{fp , − tw }, μ(ξ) = p sin2 (zh,p π) ν(ξ) = h,p We see that constraint (10) and constraint (19) can be replaced by μ(ξ) ≤ and ν(ξ) ≤ 0, respectively We consider the following problem min{T T (ξ) = (P2 ) w dw tw + λ p min{fp , − tw } + λ s.t (1) − (9), (11) − (18) where λ is a sufficiently large number sin2 (zh,p π)} h,p 148 D.Q Tran, B.T Phan Nguyen and Q.T Nguyen It is clear that if an optimal solution ξ ∗ to (P2 ) satisfies μ(ξ ∗ ) = 0, ν(ξ ∗ ) = 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 ν(ξ) = 0, Problem (P1 ) is equivalent to ξ∈D min{T T (ξ) = w dw tw } s.t (1) − (4) ξ ∈ argmin{ sin2 (zh,p π)} (P3 ) h,p s.t (5) − (9), (11) − (18) μ(ξ) ≤ In the lower level of Problem (P3 ), the constraint μ(ξ) ≤ is still hard It is tackled by using exact penalty techniques Theorem is in order Theorem [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 ≥ such that for η > η0 the following problems have the same optimal value and the same solution set α(η) = f (x) + η.p(x) : x ∈ Ω , (Pη ) α = f (x) : x ∈ Ω, p(x) ≤ (P ) For given (C, θh , φh,r ), denote Ω = {x = (fp , , tw , qu,v , tu,v , zh,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 sin2 (zh,p π) + η.μ(ξ)} min{ (Plower ) h,p s.t (5) − (9), (11) − (18) where η > is a sufficiently large number The original problem is equivalent to the following one min{T T (ξ) = w (P4 ) dw tw } s.t (1) − (4) (fp , , tw , qu,v , t(u,v) , zh,p ) ∈ argmin{ s.t (5) − (9), (11) − (18) sin2 (zh,p π) + η.μ(ξ)} h,p The lower problem in (P4 ) is a DC program It can be solved by a deterministic method Optimizing Traffic Signals in Networks Considering Rerouting 3.1 149 A GA-Based Solution Method 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 this One of the most popular way is using binary coding In the binary coding, each individual is encoded by a sequence of bits or 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 Step 2: Improving the quality of population through crossover and mutation procedure (evolution) Goto Step Step 3: Selecting a new population Go to Step 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 φh,r , offset θh , flow fp are given then the others variables are computed In this study, an individual is (C, θh , φh,r , fp ) We use the binary coding for variables C, θh , φh,r , fp Each variable is coded by a sequence of bits Suppose that C, θh , φh,r , fp are respectively the representations of variables C, θh , φh,r , fp In the next paragraph, the decoding procedure is showed Cycle time: is the proportion of the difference Cmax − Cmin plus Cmin : (C) (Cmax − Cmin ), 28 − where (X) is 10 base equivalent of X Offset: for a junction h, it is the proportion of the cycle time C = Cmin + θh = (θh ) (C − 1) 28 − 150 D.Q Tran, B.T Phan Nguyen and Q.T Nguyen 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, φh,r,max − φh,r,min , as follows: (φh,r ) φh,r = φh,r,min + Sh (φh,r,max − φh,r,min ) (φh,r ) r=1 Here, φh,r,min is a given constant and φh,r,max is a parameter calculated by φh,r,max = C − Sh r=1 Ih,r − Sh φh,r,min By this way, constraint (4) is always y=1,y=r satisfied Flow on path: for a path p ∈ Pw flow on path p is defined as the proportion of demand dw as follows: (fp ) fp = dw p∈Pw By this way, constraint (8) always holds 3.3 Computing Other Variables and the Fitness Function Variables Variables Variables Variables and zipk ,p qu,v are computed via fp by equation (5) t(u,v) are computed via qu,v by equation (7) Sh,r are computed via θh , φh,r , Ih,r by equations (11-13) W Ti0p ,p and zipk ,p are calculated by equation (14) Specifically, W Ti0p ,p k k ∀k ≥ are respectively the residual and integer part of number [ C STipk ,r ip k ,r,p − r STipk−1 ,r ip k−1 ,r,p − t(ipk−1 ,ipk ) ] r Variables W Ti0p ,p , W Tipk ,p and are calculated via (16),(17),(6) Variables tw = {tp } p∈Pw The fitness function F F is the objective function F F = T T (ξ) = w min{fp , − tw } + λ dw tw + λ p sin2 (zh,p π) (20) h,p Combination of GA and DCA In order to improve the quality of individuals in GA, we use DCA for solving the lower problem in Problem (P4 ) Optimizing Traffic Signals in Networks Considering Rerouting 4.1 151 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 convex functions on IRn Consider the following primal DC program: (Pdc ) α = inf{f (x) := g(x) − h(x) : x ∈ IRn }, (21) where g, h ∈ Γ0 (IRn ) Let C be a nonempty closed convex set The indicator function on C, denoted χC , is defined by χC (x) = if x ∈ C, ∞ otherwise Then, the problem inf{f (x) := g(x) − h(x) : x ∈ C}, (22) can be transformed into an unconstrained DC program by using the indicator function of C, i.e., inf{f (x) := φ(x) − h(x) : x ∈ IRn }, (23) n where φ := g + χC is in Γ0 (IR ) Recall that, for h ∈ Γ0 (IRn ) and x0 ∈dom h := {x ∈ IRn |h(x0 ) < +∞}, the subdifferential of h at x0 , denoted ∂h(x0 ), is defined as ∂h(x0 ) := {y ∈ IRn : h(x) ≥ h(x0 ) + x − x0 , y , ∀x ∈ IRn }, (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(xk )) and minimizes the resulting convex problem (Pk ) Generic DCA scheme Initialization: Let x0 ∈ IRn be a best guess, ← k Repeat Calculate y k ∈ ∂h(xk ) Calculate xk+1 ∈ arg min{g(x) − h(xk ) − x − xk , y k : x ∈ IRn } (Pk ) k+1←k Until convergence of xk Convergence properties of the DCA and its theoretical bases are described in [11,15,19,20] 4.2 DCA for Solving (Plower ) and GA-DCA Algorithm sin2 (zh,p π) + η.μ(ξ) is a In Problem (Plower ), the objective function f (x) = h,p DC function Consider function fh,p (x) = sin2 (zh,p π), there exists a DC decom2 position fh,p (x) = τ.zh,p −(τ.zh,p −sin2 (zh,p π)), where τ > 2π Hence, we obtain 152 D.Q Tran, B.T Phan Nguyen and Q.T Nguyen a DC decomposition of the objective function f (x) = g(x) − h(x) where g(x) = 2 τ zh,p and h(x) = τ zh,p − sin2 (zh,p π) + η max{−fp , −tp + tw } We h,p h,p h,p h,p see that the subdifferential of h(x) can be easily computed DCA applied to (Plower ) can be described as follows: DCA Initialization Let be a sufficiently small positive number Set = and x0 is a starting point Repeat Calculate y ∈ ∂h(x ) Calculate x +1 ) by solving a convex quadratic program min{g(x) s.t x ∈ Ω} ←− + Until x +1 − x ≤ or f (x +1 ) − f (x ) ≤ In the combined GA-DCA, an individual is (C, θh , φh,r ) The chromosome encoding and decoding are similar to GA presented in Section while the values of the other variables (fp , , tw , qu,v , t(u,v) , zh,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 population P For an individual Idi = (C i , θhi , φih,r ) ∈ P, we solve problem (Plower ) by DCA i i to obtain (fpi , tip , tiw , qu,v , ti(u,v) , zh,p ) i Compute the fitness of Id by formula (20) Repeat Step 1: Check the stopping criteria If it is satisfied then STOP else go to Step Step 2: Launch crossover and mutation procedure (evolution) for improving the quality of population For a new individual Idl = (C l , θhl , φlh,r ) ∈ P, we solve problem (Plower ) by l l , tl(u,v) , zh,p ) DCA to obtain (fpl , tlp , tlw , qu,v l Compute the fitness of Id by formula (20) Go to Step Step 3: Select a new population Go to Step Conclusions 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 Optimizing Traffic Signals in Networks Considering Rerouting 153 Acknowledgements This work is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.01-2013.10 References Almond, J., Lott, R.S.: The Glasgow experiment: Implementation and assessment Road Research Laboratory Report 142, Road Research Laboratory, Crowthorne (1968) Ben Ayed, O., Boyce, D.E., Blair, C.E.: A general bi-level linear programming formulation of the network design 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