Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 541782, 8 pages

http://dx.doi.org/10.1155/2015/541782

## A Nondominated Genetic Algorithm Procedure for Multiobjective Discrete Network Design under Demand Uncertainty

^{1}Institute of Transportation Engineering, Tsinghua University, Beijing 100084, China^{2}China Academy of Urban Planning and Design, Beijing 100044, China

Received 9 December 2014; Accepted 12 January 2015

Academic Editor: Wei (David) Fan

Copyright © 2015 Bian Changzhi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper addresses the multiobjective discrete network design problem under demand uncertainty. The OD travel demands are supposed to be random variables with the given probability distribution. The problem is formulated as a bilevel stochastic optimization model where the decision maker’s objective is to minimize the construction cost, the expectation, and the standard deviation of total travel time simultaneously and the user’s route choice is described using user equilibrium model on the improved network under all scenarios of uncertain demand. The proposed model generates globally near-optimal Pareto solutions for network configurations based on the Monte Carlo simulation and nondominated sorting genetic algorithms II. Numerical experiments implemented on Nguyen-Dupuis test network show trade-offs among construction cost, the expectation, and standard deviation of total travel time under uncertainty are obvious. Investment on transportation facilities is an efficient method to improve the network performance and reduce risk under demand uncertainty, but it has an obvious marginal decreasing effect.

#### 1. Introduction

A collection of models dealing with decision problems related to transportation infrastructure investment is called a network design problem (NDP). The NDP is to improve transportation network by selecting facilities (e.g., entire-lane or new link) to add to a transportation network or to determine capacity enhancements of existing facilities of a transportation network, under certain investment constraints and considering route choice behaviors of the network users. Bilevel programming models are used for transportation investment decision problems because the bilevel structure enables us to describe both policy makers’ decisions and travelers’ behaviors. Yang and Bell [1] proposed a general form of a bilevel programming model for NDP as follows: where is implicitly defined by When the decision vector in the upper-level is discrete, the bilevel programming is a discrete network design problem (DNDP). When we assume the links with a certain capacity are to be either built or not built, the DNDP is a better fit for the planning problem.

There are a lot of literatures on developing formulations and solution algorithms for the deterministic DNDP. Leblanc [2] developed a branch-and-bound algorithm for the DNDP, but the bounding step was dependent on the assumption that additional link improvements would always reduce total user cost. Chen and Alfa [3] presented some computational experiences in solving DNDP where the route selection is based on a stochastic incremental traffic assignment approach. Yang and Bell [1] gave a thorough review on NDP model and various solution algorithms. Gao et al. [4] proposed a traditional bilevel programming model for DNDP and a new solution algorithm using the support function concept. Farvaresh and Sepehri [5] proposed a new branch-and-bound algorithm being able to find exact solution of the DNDP, and a lower bound for the upper-level objective and its computation method were developed. Wang et al. [6] developed a dynamic outer-approximation scheme to make use of the state-of-the-art mixed-integer linear programming solvers to solve the SO-relaxation formulation for DNDP.

Most of the work so far has primarily been concentrated in developing methodologies for the deterministic NDP. Practically, there are a number of uncertainties in the NDP. Uncertainty in travel demand is typical in long-term forecasting. Sun and Turnquist [7] proposed a model of investment planning for transportation networks to maximize expected system capacity subject to uncertainty that will occur in the future demand pattern. Ukkusuri et al. [8] addressed a robust network design model under demand uncertainty. Chung et al. [9] formulated a robust network design problem as a tractable linear programming model and demonstrated the model robustness by comparing its solution performance with the nominal solution from the corresponding deterministic model.

Moreover, many transportation planning problems involve multiple conflicting objectives that should be considered simultaneously. Chen et al. [10] developed a mean-variance model for determining the optimal toll and capacity in a BOT roadway project subject to traffic demand uncertainty. Lin and Xie [11] proposed a parameterization-based heuristic that resembles an iterative divide-and-conquer strategy to locate a Pareto optimal solution in each divided range of commensurate parameters to study equilibrium transportation network design problems with multiple objectives. Yang et al. [12] formulated a multiobjective discrete transportation network design model using the chance constrained model and the ideal point model.

In this paper, a new multiobjective discrete network design problem (MDNDP) under demand uncertainty is formulated using bilevel stochastic optimization approach and solved by nondominated sorting GA technique. This formulation takes into account not only expected total travel time but also the risk reflected by the variance of the total travel time and the construction cost. The results of the model provide a set of Pareto optimal solutions to be used by the decision makers to find the best configuration according to their preferences. This paper is organized as follows. In Section 2, the formulation of MDNDP under demand uncertainty is presented. In Section 3, we explain the nondominated sorting genetic algorithm II (NSGA II) technique to solve the multiobjective problem. Section 4 presents the computational experiments. Finally, the conclusion of the paper is drawn in Section 5.

#### 2. MDNDP under Uncertainty

##### 2.1. Multiobjective under Uncertainty

Uncertainty in long-term OD demand is considered here. It is important because the investment decisions made in present have a significant effect into the future. In urban transportation planning, there are several different goals faced by decision makers, such as total travel time, construction cost, consumer surplus, and accessibility. These goals can be considered simultaneously using the technique of multiobjective optimization.

The multiobjective optimization where the set of feasible solutions is not explicitly known in advance but is restricted by constraint functions can be formulated as follows:where is the decision vector, is the objective vector, is the domain of vector in dimension space, is the domain of vector in dimension space, and the constrains determine the feasible region of decision vector.

In multiobjective optimization problems, multiple objective functions need to be optimized simultaneously. Instead of aiming to find a single solution, the objective is to produce a set of good compromises from which the decision maker will select one. These solutions are known as nondominated, efficient, noninferior, or Pareto optimal solutions. Given a set of multiobjective solutions, some of this set will be dominated by others in this set. Those that are not dominated by any others in that set form what we call the Pareto set. In objective space, the set of objective vectors corresponding to the Pareto set is called the Pareto front.

In this study, the OD travel demands are assumed to be uncertain and can be described using a probability distribution. Expected total travel time (TTT) minimization is an important goal under the realization of all demand scenarios. However, if planners want to reduce the risk in investment decision on transportation facilities, minimization of standard deviation of TTT becomes another important goal. Moreover, the construction cost minimization of infrastructure is always one of the goals faced by decision makers and it is also considered in our model.

##### 2.2. Model Formulation

The main focus of the present paper is to present a formulation to MDNDP which is capable of handling investment decisions under multiobjective and demand uncertainty. Here, OD travel demand is supposed as a random variable submitting to the given probability distribution. In practical calculation, when Monte Carlo random sampling is used to form a demand scenario set , any demand scenario realization is . To analyze the trade-offs among construction cost, expectation, and the standard deviation of TTT, a new MDNDP model under OD demand uncertainty is formulated using bilevel programming. The upper-level model is to minimize three objectives simultaneously, the expectation of TTT, the standard deviation of TTT, and the construction cost under all realization scenarios of the random OD demand. The links to be built or expanded will be decided in the upper-level model. The lower-level model is the corresponding user equilibrium of each demand scenario under the improved network decided by the upper-level model. The resultant MDNDP becomes as follows: where is implicitly defined by the following lower-level model: The travel time of link is defined using BPR function: where we have the following: : links set of transportation network; : set of new built or expanded links; : trip generation nodes set; is one of generation nodes; : trip attraction nodes set; is one of attraction nodes; : OD between generation node and attraction node ; : paths set between origin node and destination node ; : traffic volume on link ; : travel time impedance function of link ; : traffic volume using path between OD pair ; : traffic time impedance on path between OD pair ; : indicator variable (1 if link belongs to path between OD , 0 otherwise); : decision variable (1 when link will be built or expanded, otherwise 0); : capacity of link ; : cost of new built or expanded link ; : all possible scenarios set of uncertain travel demand; : any realization of uncertain travel demand; : realization probability of uncertain travel demand scenario ; : travel time impedance on link under free flow in BRP function; : parameters in BPR function.

#### 3. Solution Methodology

Because the formulation is intractable with traditional optimization methods, the MDNDP is better suited for the application of metaheuristics. In this section, we make a brief overview of GA and its application in NDP. Then the solution procedure is presented for MDNDP formulated in previous section using nondominated sorting genetic algorithm II (NSGA II).

##### 3.1. Overview of GA and Its Application in NDP

Genetic algorithms (GA) developed by Holland [13] are one of the best known algorithms in evolutionary computation, which imitates living beings to develop powerful algorithms for difficult optimization problems.

Genetic algorithm is a search algorithm, which works starting from an initial collection of strings representing possible solutions of the problem. Each string of the populations is called a chromosome and has an associated value called a fitness function that contributes to the generation of new populations by means of genetic operators (denoted as reproduction, crossover, and mutation). The initial population is generated randomly, or it may consist of a number of known solutions, or a combination of both. The GA goes through a number of steps in which the population at the beginning of each step is replaced with another population, which hopefully will include better solutions to the problem. The chromosomes at each new generation are produced by a process called reproduction, in which the chromosomes of the old population are combined to create new ones. A detailed explanation of the working of GA can be found in Goldberg [14] and Deb [15].

In the past few years GA has been used in optimization problems in transportation such as NDP. Yin [16] proposed a genetic algorithm based approach to solve the bilevel optimization and compared the performance of the algorithm with the previous sensitivity-analysis based algorithms. Results show that the GA based approach is efficient and much simpler. Jeon et al. [17] proposed a new solution search procedure based on the selectorecombinative genetic algorithm for DNDP. Chen et al. [18] give a multiobjective genetic algorithm procedure for BOT network design problem.

##### 3.2. Nondominated Sorting Genetic Algorithm II

Nondominated sorting genetic algorithm II is an efficient algorithm to solve multiobjective optimization problems proposed by Deb et al. [19]. The procedure of NSGA II can be summarized as follows.

*Step 1 (initialization). *
Consider the following.*Step **1.1*. Determine the basic parameter of GA.*Step **1.2*. A random parent population is created. Selection, crossover, and mutation operators are used to create an offspring population . Both sizes of and are .

*Step 2 (for the generation). *Consider the following.*Step **2.1*. Combine the and to form population of size 2.*Step **2.2*. Sorting population according to nondomination and the best nondominated set is created.*Step **2.3*. Calculate the crowding distance for every individual in .*Step **2.4*. Choose to population , until the size of population exceeds ; the last nondominated set is .*Step **2.5*. Crowding distance sorting for every individual in , choose the best solutions of to fill all population slots.*Step **2.6*. Use the selection, crossover, and mutation operators for to create a new population .

*Step 3. **End,* the Pareto front is formed.

##### 3.3. Demand Simulation and UE Solution Algorithm

To handle travel demand uncertainty in the model, stochastic sampling technique is used to simulate uncertain OD demand based on a probability distribution with predefined expectation and variance. In this study, Monte Carlo simulation is used to generate random OD demand according to a truncation normal distribution. The parameters used are the expectation and variance coefficient in the distribution. It should be noted that other distributions could also be used. The potential OD demand is chosen as the only key exogenous input variables to reflect the uncertainty of travel demand. Under every scenario of OD demand realization, the UE model in lower-level is solved using Frank-Wolfe method presented in Sheffi [20].

##### 3.4. Algorithm Procedure for MDNDP

The algorithm procedure for MDNDP is proposed in Figure 1 and discussed in the following.