GENETIC ALGORIThM fOR SOLVING DYNAMIC SIMULTANEOUS ROUTE AND DEPARTURE TIME EqUILIBRIUM PROBLEM

. We present a genetic algorithm for solving dynamic simultaneous route and departure time equilibrium problem. Not only can a flow-swapping process in the algorithm guarantee the flow conservation constraints between OD pair, but also accelerate the convergence velocity of the algorithm. Finally, a simulation example shows feasibility and validity of genetic algorithm.


Introduction
The observations show the travelers do adjust routes and departure times to avoid peak period congestion (Hendrickson and Plank 1984). Furthermore, most existing studies (Arnott et al. 1990;Friesz et al. 1993;Wie et al. 1995;Huang and Lam 2002;Szeto and Lo 2004;Lin and Heydecker 2005) focus on dynamic simultaneous route and departure time problem. Many researchers used various models such as simulation and analytical models, and propose some algorithms for dynamic SRD problem. The algorithms are classified into two classes, class I can be described as the strict mathematical algorithm such that Szeto and Lo (2004) adopted Han and Lo (2003) decent direction method to solve the VIP; class II is heuristic algorithm based on flow-swapping rules (Wie et al. 1995;Huang and Lam 2002;Lin and Heydecker 2005). The basic idea of the algorithm is: for each OD pair, inflows on the non-cheapest time-dependent paths are moved to the cheapest paths. Due to non-monistic of the path travel cost function, the above algorithms only can converge to local optimal solution. In order to obtain global optimal solution, Sadek et al. (1997) used a genetic algorithm for dynamic traffic assignment problems; however, the algorithm is based on link. In this paper, we combine genetic algorithm with flow-swapping rules for solving a dynamic simultaneous route and departure time problem. This paper includes the following several sections, firstly we describe a dynamic simultaneous route and departure time model, then we present a variational in-equality formulation for SRD problem. In section 3, we propose a combined genetic algorithm for the proposed model. Section 4 provides a numerical example to demonstrate and verify the proposed solution algorithms.

Model formulation
Here, the dynamic network models of Freisz et al. (1993), Chabini (2001) Flow propagation constraints are used to describe the flow progression over time: Definitional constraint: Boundary conditions: Now we give the actual path travel time and actual path travel cost functions. The actual travel time to traverse path p={a 1 ,a 2 ,…,a n } for travelers entering into the network during interval k is calculated using the following nested function: where, let 1 a a a a a t t k t t k t k = = + for short. The schedule delay cost function can be expressed as follows: Denote [t s -Δ s , t s + Δ s ] as the desired time interval for arrival at the destination s in the network. Where t s -Δ s is the travelers' desired earliest arrival time, t s + Δ s is the desired latest arrival time at the destination s. β, γ is the unit cost of schedule delay (early, late) at the destination s, respectively. Therefore, the travel cost of a trip from origin r to destination s on path p for a traveler leaving origin at time interval k is: Where α is a convention factor to transform the path travel time into travel cost. In accordance with the empirical results, we assume that γ > α > β holds.
Travelers follow dynamic simultaneous path and departure time equilibrium (UE-SRD), expressed as: Where min ( ) rs c ⋅ is the minimum unit travel cost of travelers between origin r and destination s, (10) represents the flow conservation of travelers between origin r and destination s and equation (11) represents the non-negativity of all path inflow rates.
For travelers and for each origin-destination (OD) pair, the path travel costs experienced for travelers, regarding of departure times, is equal and minimum, and less than (or equal to) the path travel costs for travelers on any unused paths.
The above UE-SRD equilibrium condition of travelers can be expressed by a finite dimensional variational inequality formulation. Where, Ω is a closed convex:

Algorithm
GAs are search and optimization procedures motivated by natural principles and selection. Because of their simplicity, minimal problem restrictions, global perspective, and implicit parallelism, GAs have been applied to a wide variety of problem domains including engineering, sciences, and commerce (Yin 2000). In this section, we propose a genetic algorithm for solving VIP(12), which mainly include several parts: determination of the initial population, implementation of the constraints, convergence indicator, fitness function, and the crossover and mutation operation, etc.

The initial population
A population of chromosomes is initialized by the following equation: Rnd expresses the random count between [0,1]; K represents total time interval; rs P represents the path number between OD pair rs. The initial path inflow cannot follow the flow conservation constraint (10).

Implementation of the constraints
The implementation of constraints of VIP (12) is an important question to be considered both in initializing a population and designing an objective evaluation function. The implementation of constraints can be classified into two classed: one is by imposing moderate penalties on individuals that violate them, the other is by creating individuals directly satisfying them by means of a decoding procedure or decoder. In this paper, we propose flow equilibrium decoders that avoid the violation of the flow conversation constraint. A decoder is similar with the flow swapping processes given by Wie et al. 1995, Huang, Williams 2002 The population that violate the flow conservation constraint can be classified into two conditions, one is If the sum of the path inflow rate of an individual is more than actual OD demand for each OD pair rs. The inflows on the non-cheapest time-dependent path are subtracted according to a proportion in order to guarantee the satisfaction of the flow conservation constraint. The equations can be expressed as follows:

fitness function
We define the fitness function as: 1 where α is the accelerating parameter of the algorithm; The step-by-step procedure of our algorithm is given below: Step 1: Select at random the initial population, crossover and mutation probability.
Step 5: Reproduce a new population (n + 1) from population n according to the distribution of fitness function.
Step 6: Genetic Operators. Crossover: Cross two individuals chosen from population (n + 1) with a specified crossover probability, pc. Mutation: Select one individual from population (n + 1) with a specified probability, pm. Go to step 2.

Numerical example
The numeral example is a TF network as shown in Fig. 1. It consists of 13 nodes, 19 links and two OD pairs ((1.11),(3.13)). The link parameter is shown in Table 1 Table 2. Other input data are: α = 6.4; β = 3.9; t s = 9.0; Δ s = 0.25; ε = 0.00001, set T be from 6 to 10.A.M. and K = 100. The proposed solution algorithm was coded in Visual Basic 6.0, and run on a personal computer (P4 2.88G).
The value of the convergence indicator of the algorithm is decreasing as the iteration number increases as shown in Fig. 4. When the iteration number amounts to 1000, the value of the convergence indicator descends to 0,09. Figs. 4, 5 give the path inflow rates and the travel costs on path 1 of OD pairs (1, 11) and path 3 of OD pair (3, 13). We can find an approximate dynamic equilibrium pattern for all route inflow rates of travelers between two OD pairs. The other path flow and cost on network have similar dynamic equilibrium conditions. Due to limitation of page, we don't give all path flow patterns.