Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 832814, 11 pages

http://dx.doi.org/10.1155/2014/832814

## Model and Method for Multiobjective Time-Dependent Hazardous Material Transportation

^{1}School of Management, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China^{2}Laboratoire d’Informatique, Biologie Intégrative et Systèmes Complexes, Université d’Evry Val d’Essonne, 91020 Evry Cedex, France^{3}Laboratoire Génie Industriel, Ecole Centrale Paris, 92295 Châtenay-Malabry Cedex, France

Received 10 July 2014; Accepted 27 October 2014; Published 14 December 2014

Academic Editor: Yi-Kuei Lin

Copyright © 2014 Zhen Zhou et al. 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

In most of the hazardous material transportation problems, risk factors are assumed to be constant, which ignores the fact that they can vary with time throughout the day. In this paper, we deal with a novel time-dependent hazardous material transportation problem via lane reservation, in which the dynamic nature of transportation risk in the real-life traffic environment is taken into account. We first develop a multiobjective mixed integer programming (MIP) model with two conflicting objectives: minimizing the impact on the normal traffic resulting from lane reservation and minimizing the total transportation risk. We then present a cut-and-solve based -constraint method to solve this model. Computational results indicate that our method outperforms the -constraint method based on optimization software package CPLEX.

#### 1. Introduction

Hazardous materials are a kind of goods with physical, chemical, and biological properties, which could cause lots of accidents, such as flame, explosion, and leak, in the course of production, storage, and transportation. With the development of industry, more and more hazardous substances, including raw materials, intermediate, final products, and wastes, are produced and moved daily through the transportation network in different transportation modes, such as road, rail, water, air, and pipeline. Their transportation risk imposed on environment and human is widening and deepening year by year. Hazardous material transportation has become a serious problem worldwide and attracts many researchers’ attentions in the related field.

It has been pointed out in much relevant literature that the essential objective of hazardous material transportation is to minimize the transportation risk due to the nature of this problem. As we know, the selection of routes in a network for hazardous material transportation can affect its risk factors, such as the probability of hazardous material accidents and the risk exposure to the surrounding population and environment. Therefore, appropriate routing decisions are very important for hazardous material transportation management. In the last couple of decades, various applications of operations research models to hazardous material transportation have focused on risk reduction and fruitful achievements in this area have been published; please see [1] for details. However, in almost all of the hazardous material transportation problems, the transportation risk is considered to be time-invariant. That is to say, the risk of a road segment in a transportation network is assumed to be constant, which fails to capture the dynamic nature of the real-life traffic environment. In real life, risk on road segments is time-dependent on population density subject to time-of-day variation, peak and off-peak periods, various weather conditions, and so on. The time-dependent risk is one of the important features of hazardous material transportation. The time-dependent transportation problem is to decide the path for each shipment and its starting time so as to minimize the transportation risk.

An important branch of the time-dependent transportation concerns hazardous materials. Time-dependent transportation problems can be distinguished into deterministic and stochastic settings. The dynamic characteristic of transportation networks usually depends on one or more traffic quantities [2], such as travel time, link volume, and queue length. For deterministic time-dependent transportation problems, part or all of the traffic quantities are assumed to be variant but all of the traffic quantities are known for a road segment. For stochastic time-dependent transportation problems, the traffic quantities are considered as random variables with time-dependent distribution functions, such as in [2–8]. For time-dependent hazardous material transportation, traffic quantities are usually travel time and transportation risk. According to optimization criteria, time-dependent transportation problems can be divided into single-objective ones and multiobjective ones.

For the hazardous material transportation routing problem, a lot of works have concentrated on time-invariant risk and travel time. However, time-dependent hazardous material transportation problems have not been widely studied and only a few related publications can be found in the literature. Jia et al. [9] investigated a hazardous material transportation problem with deterministic time-dependent risk for minimizing the transportation risk. The proposed model guaranteed the minimum distance between hazardous material shipments at any time. They transformed the considered problem into a number of time-dependent shortest path problems for each truck and proposed an iterative heuristic. Erkut and Alp [10] proposed an integrated routing and scheduling problem for hazardous material transportation in a network with stochastic time-dependent accident probability, population exposure, and travel time. The model aimed to minimize transportation risk while imposing a constraint on the total travel time of the shipment. Meng et al. [11] examined a similar problem with multiple objectives. They transformed it into a time-dependent multiobjective shortest path problem subject to three types of time constraints. A dynamic programming approach was constructed to solve the problem. Both of the methods of references [10, 11] were pseudopolynomial. Nozick et al. [12] developed an approach for a hazardous material routing and scheduling problem with deterministic time-dependent risk. But their approach could not guarantee generating all Pareto-optimal paths. Chang et al. [13] proposed an effective algorithm for finding a path in stochastic time-dependent networks that could address multiple optimization criteria. In their work, travel time, transportation risk, and other traffic quantities along paths were random variables. However, the performance of the algorithm was sensitive to some parameters and the computational burden increased with the number of the dominated paths. Miller-Hooks and Mahmassani [14] proposed an optimal routing algorithm for a single hazardous material shipment in a stochastic time-dependent network, where the travel time and risk followed time-dependent normal distribution functions. They presented several procedures for determining the best compromise path to minimize the total travel time and risk (population exposure).

Lane reservation is considered as a flexible and economic strategy for special events or situations, like sport games and emergencies. As stated in [15], it has been successfully applied to the Olympic Games in 2000 in Sydney [16] and in 2004 in Athens [17]. The principle of lane reservation is to reserve lanes on some road segments and/or in some specific time periods in a transportation network. Only special transportation tasks are allowed to pass through them. A probable consequence of this strategy is that it may worsen traffic congestions on general lanes, which is also an important feature of the lane reservation problem. Research works about lane reservation involving simulation tools and statistical methods can be found in the literature [18–20]. Some recent works focused on minimizing the traffic impact by linear programming models and optimal methods for various applications, such as automated truck, large sportive events, and hazardous material transportation [15, 21–24]. Our previous work [15] investigated a hazardous material transportation problem via lane reservation, in which the risk along a road segment was assumed to be constant. The goal was to minimize the transportation risk and the impact on normal traffic due to lane reservation. Remarkably, it has been shown in [15] that the lane reservation strategy can greatly reduce the transportation risk at a reasonable cost of its traffic impact. The problem proposed in this paper, in which the transportation risk is considered to be deterministic and time-dependent, is more realistic. To the best of our knowledge, this is the first work for time-dependent hazardous material transportation via lane reservation.

In this paper, we investigate a novel multiobjective hazardous material transportation problem via lane reservation with a deterministic time-dependent transportation risk. As we know, the factors of the transportation risk generally include the hazardous material accident probability and the population exposure to the accidents. The accident probability estimation is influenced by the nature of roads, characteristics of the trucks, transportation environment, driver conditions [25], and so forth. Estimating the accident probability is a complicated and difficult work. For simplification, the probability of an accident is regarded to be time-invariant in this paper. Population exposure is determined by population density and area. In real life, the population density along a road segment strongly depends on time and space. As we know, the population density in hospitals, schools, factories, and so on in day time is greater than that at night, and the opposite happens in residential areas. In this work, the accident probability on reserved lanes is assumed to be constant and the population exposure is assumed to be time-dependent. Therefore, the transportation risk varies with time and space. This work is motivated by the dynamic characteristic of risk and it is a natural extension of our previous work.

The contributions of this work can be summarized as follows. Firstly, we propose a novel multiobjective MIP model for the time-dependent hazardous material transportation problem via lane reservation. Secondly, we develop a cut-and-solve based -constraint method for the considered problem. Computational results show the effectiveness of the method.

The remainder of this paper is organized as follows. Section 2 describes the time-dependent hazardous material transportation problem via lane reservation and a new multiobjective model is formulated with the objectives of minimizing the total impact on the normal traffic and the total transportation risk. In Section 3, some properties are analyzed to reduce the search space of solutions and a cut-and-solve based -constraint method is developed. Section 4 reports the computational results of experiments. Section 5 concludes the paper and discusses future research directions.

#### 2. Problem Formulation

##### 2.1. Problem Description

Consider a bidirectional graph , where is the set of nodes and is the set of arcs. The graph represents the transportation network in which the vehicles carrying hazardous materials are allowed to move. Arc denotes a road segment from node to node . kinds of hazardous materials will be transported from origin nodes to destination nodes .

Before formulation of the problem, we make the following nonrestrictive assumptions. There are at least two lanes on a road segment such that one lane is allowed to be reserved. From the point of view of transport safety, any path for hazardous material shipments consists of only reserved lanes. Vehicles with hazardous materials travel on the reserved lanes without congestion. Consequently, travel time on reserved lanes is time-invariant. Any two hazardous material shipments on the same road segment must maintain a minimum time interval, called* safety time interval*. The accident probability on a reserved lane is constant and hazardous material accidents happen independently.

In the network, nonreserved lanes are called general lanes, as shown in Figure 1. From assumption , the travel time on a reserved lane is time-invariant throughout the day. Nevertheless, the risk on a road segment should be time-dependent because the population exposure varies with time in nature. The population exposure on each arc at* time period *, denoted as , depends on the departure time from node . Without loss of generality, set as the beginning time of the first period. In a day, there are usually only several time periods [26]. So travel time on the reserved lane is far less than the length of a time period; that is, *.* The problem is to choose lanes on the existing network to be reserved, select the path for each hazardous material shipment, and decide the travel time period for each shipment on its selected path. The objective of the problem is to seek the best trade-off for minimizing the total traffic impact on the normal traffic and the total transportation risk.