Discrete Dynamics in Nature and Society

Volume 2018, Article ID 2916391, 12 pages

https://doi.org/10.1155/2018/2916391

## A Multiobjective Route Robust Optimization Model and Algorithm for Hazmat Transportation

^{1}School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, China^{2}Key Laboratory of Road Traffic Engineering of the Ministry of Education, Changsha University of Science and Technology, Changsha, Hunan 410114, China^{3}Hunan Provincial Key Laboratory of Intelligent Processing of Big Data on Transportation, Changsha University of Science and Technology, Changsha, Hunan 410114, China^{4}Department of Civil Engineering, City College of New York, NY 10031, USA

Correspondence should be addressed to Changxi Ma; nc.utjzl.liam@ixgnahcam

Received 12 June 2018; Accepted 13 September 2018; Published 9 October 2018

Guest Editor: Xiaobo Qu

Copyright © 2018 Changxi Ma 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

Aiming at route optimization problem of hazardous materials transportation in uncertain environment, this paper presents a multiobjective robust optimization model by taking robust control parameters into consideration. The objective of the model is to minimize not only transportation risk but also transportation time, and a robust counterpart of the model is introduced through applying the Bertsimas-Sim robust optimization theory. Moreover, a fuzzy C-means clustering-particle swarm optimization (FCMC-PSO) algorithm is designed, and the FCMC algorithm is used to cluster the demand points. In addition the PSO algorithm with the adaptive archives grid is used to calculate the robust optimization route of hazmat transportation. Finally, the computational results show the multiobjective route robust optimization model with 3 centers and 20 demand points’ sample studied and FCMC-PSO algorithm for hazmat transportation can obtain different robustness Pareto solution sets. As a result, this study will provide basic theory support for hazmat transportation safeguarding.

#### 1. Introduction

Hazardous materials (hazmat) refer to products with flammable, poisonous, and corrosive properties that can cause casualties, damage to properties, and environmental pollution and require special protection in the process of transportation, loading, unloading, and storage. In recent years, the demand for hazmat has increased, its freight volume has increased year by year, and the potential transportation risk is also expanding. Practice has proved that the optimization of the transportation route of hazmat can effectively reduce the transportation risk, and it has significant influence to ensure the safety of people along route and protect the surrounding ecological environment.

Many scholars have studied the transportation route optimization problem for hazmat. Rhyne (1994) conducted a statistical analysis of the hazmat transport accident using the diffusion formula [1]. Gordon et al. (2000) used the British transport accident to prove the different risk level in different places, which provided a realistic basis for the evacuation of the population and the balance of risks during the accident [2]. Power et al. (2000) established a risk-cost analysis model and conducted a systematic analysis to find out the transport plan [3]. Wu et al. (2002) studied the vehicle routing optimization problem of hazmat transportation with multidistribution center, and the clustering algorithm was used to solve the complex model [4]. Kara et al. (2003) presented a risk model based on the primitive roads to reduce the error of risk estimation, and then they proposed the minimum risk route selection algorithm [5]. Fabiano et al. (2005) used the experimental data to calculate the probability and consequence of the transport accident on the specific route of the experimental area and got the optimal transportation scheme [6]. Erkut and Ingolfsson (2005) studied the hazmat transport route by considering the shortest distance, the smallest population exposure, and the smallest accident risk [7]. Bubbico et al. (2006) analyzed the transportation risk of hazmat and obtained the safety route algorithm by using the experimental data from Italy [8]. Wei et al. (2006) analyzed the route selection model under the influence of time-varying conditions without considering the uncertain risk [9]. Verma (2009) presented a biobjective optimization model considering the cost and the risk and used the boundary algorithm to solve the model [10]_{.} Wang et al. (2009) established a hazmat transportation path optimization model based on a geographic information system [11]. Jassbi et al. (2010) developed a multiobjective optimization framework for the hazmat transportation by minimizing the transport mileage, the number of affected residents, social risks, the probability of accidents, and so on [12]. Pradhananga et al. (2014) created a dual-objective optimization model with time windows for the hazmat transportation and designed a heuristic algorithm to solve this model [13]. Suh-Wen Chiou (2016) proposed a dual-objective and dual-level signal control policy for the hazmat transportation [14]. Assadipour et al. (2016) proposed a toll-based dual-level programming method for the hazmat transportation and designed hybrid speed constraints for multiobjective particle swarm optimization algorithm [15]. Pamucar et al. (2016) proposed a multiobjective route planning method for hazmat transportation and designed a solution algorithm combining neurofuzzy and artificial bee colony approach [16]. Mohammadi et al. (2017) studied the hazmat transportation under uncertain conditions using a mixed integer nonlinear programming model and the metaheuristic algorithm was utilized to solve this model [17]. Kheirkhah et al. (2017) established a bilevel optimization model for the hazmat transportation, two heuristic algorithms were designed to solve the dual-level optimization model, and some randomly generated problems were used to verify the applicability and validity of the model [18]. Bula et al. (2017) focused on the heterogeneous fleet vehicle routing problem and designed a variant of the variable neighborhood search algorithm to solve the problem [19]. Ma et al. (2013) studied the route optimization models and algorithms for hazardous materials transportation under different environments [20]. Ma et al. (2018) proposed a road screening algorithm and created a distribution route multiobjective robust optimization for hazardous materials [21]. Obviously, although the route optimization of hazmat transportation has made many achievements, the robustness of solution is rarely considered in this field.

Ben-Tal and Neimirovski (1998) proposed robust optimization theory based on ellipsoid uncertainty set [22]. Bertsimas and Sim (2003) further put forward adjustable robustness robust optimal theory [23]. Based on the adjustable robust optimization theory, this paper will propose a route robust optimization model for hazmat transportation with multiple distribution centers, and the author also designed a kind of improved PSO algorithm.

The rest of this paper is structured as follows: Section 2 introduces the hazmat transportation route problem and establishes a multiobjective route robust optimization model of hazmat transportation. Section 3 presents the FCMC-PSO algorithm. Section 4 is the case study. At the end of the paper, the conclusion is proposed.

#### 2. Multiobjective Route Robust Optimization Model of Hazmat Transportation

##### 2.1. Problem Definition

Hazmat transportation route robust optimization for multidistribution center is defined as follows: there are several hazmat distribution centers, and each distribution center owns enough hazmat transport vehicles; meanwhile, multiple need points exist which should be assigned to the relevant hazmat distribution center. Vehicles from distribution center will service the corresponding demand points. Each vehicle can service several customer demand points while each customer demand point only can be serviced by one vehicle. After completing the transport mission, the vehicles must return to the distribution center [24].

Uncertainty of hazmat transport refers to the uncertainty of the transportation time and transportation risk, which may be caused by the traffic accident, weather, and traffic density of the road. Compared to ordinary goods transport, hazmat transport is more complicated, and it needs more security demands. Therefore, it is needed to set the goal of minimizing the total hazmat transportation risk. In the process of hazmat transportation, transport time reduction is also necessary. As a consequence, this paper will target minimizing the total transportation time. In conclusion, the scientific transportation routes should be found to guarantee the hazmat transported safely and quickly.

##### 2.2. Model

###### 2.2.1. Assumption

There are a few assumptions in this study:

(1) Multiple hazmat distribution centers are existent

(2) The supply of hazmat distribution centers is adequate

(3) Vehicle loading capacity is provided and the demand of each customer is specified

(4) Multiple vehicles of the distribution center can service the customer

(5) The transportation risk and transportation time are identified among the customer demand points, but they are uncertain number as interval number

###### 2.2.2. Symbol Definition

: set of the Hazmat distribution centers, where shows that the number distribution center is and the sequence number of nodes set is .

: set of customer demand points, where shows that the number of customer demand points is and the sequence number of nodes set is .

: all nodes set in the transportation network, where *.*

: available transportation vehicle set in the hazmat distribution center, where .

: road section set among nodes.

: demand of customer demand point *.*

: maximum load of transport vehicle* k *from hazmat distribution center* d. *

: variable transport risk from customer demand points* i* and* j*, where .

: transportation risk nominal value from customer demand points* i* and* j.*

: deviation of the variable transport risk to its nominal value from customer demand points* i* and* j*, where .

: travel time nominal value from customer demand points* i* and* j.*

: deviation of variable travel time to its nominal value from customer demand points and , where .

: variable transport risk from customer demand points* i* and* j*, where .

: the set of columns which all uncertain data belonging to the ith row of the variable risk matrix are in, here .

: parameter to adjust robust risk of robust discrete optimization method and control the risk degree of conservatism, where decimal is permitted.

: maximum integer less than .

*ψ*_{i}^{r}: the set of column subscripts* j *of uncertain data of line in the variable risk matrix .

: the set of columns with all uncertain data belonging to the* i*th row of the variable time matrix, where ≤*n.*

Γ_{i}^{t}: parameter to adjust robust time of robust discrete optimization method and control the time degree of conservatism, where decimal is permitted.

: the maximum integer less than .

*ψ*_{i}^{t}: the set of column subscript* j* of uncertain data of line* i* in variable time matrix .

###### 2.2.3. Multiobjective Route Robust Optimization Model

where the objective function (1) minimizes the total transportation risk of hazmat [25], and the objective function (2) minimizes the total transportation time of hazmat. Constraint (3) as the load constraint that means any vehicle of any distribution center should satisfy the corresponding load constraint, namely, cannot overload. Constraint (4) means that vehicle number of the distribution center is limited, and vehicles arranged to transport the hazmat should not exceed the distribution center owning. Constraint (5) expresses that the transportation risk of each section must be less than or equal to threshold* r*_{max} set by decision makers. Constraint (6) expresses that the transportation risk of each route must be less than or equal to threshold* R*_{max} set by decision makers. Constraint (7) indicates every hazmat vehicle departing from distribution center should return back to the original distribution center after finishing the transportation task. Constraint (8) and Constraint (9) guarantee that each demand point is served once and by one vehicle form a distribution center. Constraint (10) indicates hazmat transport vehicles cannot depart from one distribution center but back to another one. Constraint (11) defines the 0-1 integer variable , if the route of Hazmat transport vehicle* k* from distribution center containing the segment from node* i *to node* j*, =1; else =0.

###### 2.2.4. Robust Counterpart Model

Each objective function of the above multiobjective robust model corresponds to parameter Γ. The purpose is to control the degree of conservatism of the solution. Objective functions (1) and (2) of the robust optimization model contain “max” extreme value problem. Set feasible solution set* X*_{vrp} to satisfy all constraints, and robust discrete optimization criterion is used to transform the multiobjective route robust optimization model, and a new robust counterpart of the model is as follows [26, 27].

Objective function is

Constraint condition is

Then, the optimal objective function value can be obtained as and .

#### 3. Algorithm

In this section, we propose FCMC-PSO algorithm to solve the multiobjective route optimization problem of hazmat transportation in uncertain environment. The demand points is clustered by the fuzzy C means algorithm, and the transportation route for each demand points is determined based on the adaptive archives grid multiobjective particle swarm optimization [28–30].

##### 3.1. Fuzzy C-Means Clustering

Suppose that n data samples are ,* C*(*2≤C≤n*) is the number of types into which the data samples are to be divided, indicating the corresponding C categories,* U* is its similar classification matrix, the clustering centers of all categories are , and is the membership degree of sample to category* A*_{k} (abbreviated as ). Then the objective function can be expressed as follows:where , it is the synthetic weighted value of the transport risk and time after nondimensionalization between the* i*-th sample* X*_{i} and the* k*-th category center point;* m* is the characteristics number of the sample;* b* is the weighting parameter, and the value range is 1 ≤ b ≤ ∞. The fuzzy C-means clustering method is to find an optimal classification, so that the classification can produce the smallest function value . It requires a sample for the sum of the membership degree of each cluster is 1, which is satisfied:

Formulas (17) and (18) are used to calculate separately the membership degree of the sample* X*_{i} for the category* A*_{k} and C clustering centers :

Let , for all the i categories, , .

Using formulas (17) and (18) to repeatedly modify the cluster center, data membership, and classification, when the algorithm is convergent in theory, we can get the membership degree of the cluster center and each sample for each pattern class; thus the division of the fuzzy clustering is completed.

##### 3.2. Multiobject PSO Algorithm

Particle swarm algorithm is derived from the study of the predatory behavior of birds. It is used to solve the problem of path optimization. Each particle in the algorithm represents a potential solution, and the fitness value for each particle is determined by the fitness function, and the value of fitness determines the pros and cons of the particle. The particle moves in the N-dimensional solution space and updates the individual position by the individual extremum and the group extremum. In the algorithm, the velocity, position, and fitness value are used to represent the characteristics of the particle. The velocity of the particle determines the direction and distance of the particle movement, and the velocity is dynamically adjusted with the moving experience of its own and other particles. Once the position of the particle is updated, the fitness value will be calculated, and the individual extremum and the population extremum are updated by the fitness values of the new particles, the individual extremum, and the population extremum. Multiobjective particle swarm optimization algorithm is a method based on particle swarm optimization algorithm to solve multiobjective problem. At the same time, the best location of multiple populations exists in the population, and the optimal positions of multiple particles themselves are also found in the iterative process. Therefore,* gbest* and pbest also need to adopt certain strategies to choose. Aiming at the robust optimization model of hazmat transportation, the key elements of multiobjective particle swarm optimization are as follows.

*(**1) Individual Coding*. In this paper, the method of particle encoding adopts integer encoding, and each particle represents the experienced demand point. For example, when the number of required points is 9, the individual coding is [9 2 1 4 3 6 7 5 8], indicating that the requirement point traversal starting from the distribution center, followed by 9 2 1 4 3 6 7 5 8, and ultimately return to the distribution center.

*(**2) Fitness Value*. In the hybrid particle swarm algorithm, the fitness value is the criterion of judging the quality of the particle. And the fitness function is to facilitate the search and improve the performance of the algorithm. In the paper, the fitness value of the particle is expressed by the objective function of the built model.

*(**3) Crossover Operation*. Crossover operation is the process of replacing the partial structure of the parent individual and reorganizing the new individual. The design of the crossover operation is related to the representation of the coding, the cross-operation design based on the coarranged coding method of the demand point and the distribution center [31, 32]. The method of integer crossing is adopted. Set the two individuals of the parent as [9 2 1 4 3 6 7 5 8] and [8 3 2 4 1 5 9 7 6], Firstly, two crossover positions are selected, and then the individual is crossed; the operation process can be seen in Figure 1.