Journal of Advanced Transportation

Volume 2018, Article ID 4295419, 13 pages

https://doi.org/10.1155/2018/4295419

## Dynamic Route Network Planning Problem for Emergency Evacuation in Restricted-Space Scenarios

^{1}Information Engineering College, Beijing Institute of Petrochemical Technology, Beijing 102617, China^{2}School of Information, Renmin University of China, Beijing 100872, China^{3}National Engineering Research Center of Coal Mine Water Hazard Controlling, China University of Mining and Technology, Beijing, Beijing 100083, China

Correspondence should be addressed to Yi Hong; nc.ude.tpib@iygnoh

Received 22 March 2018; Revised 14 May 2018; Accepted 17 May 2018; Published 27 June 2018

Academic Editor: Zhi-Chun Li

Copyright © 2018 Yi Hong 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

We investigate a dynamic route planning problem in restricted-space evacuation, namely, the* Multiobjective Dynamic Route Network Planning (MODRNP) problem*. It models the multisource to multidestination evacuation in restricted-space scenarios, with the objectives of minimizing the whole evacuation delay and maximizing the evacuation efficiency. We study the problem in 3D scenarios, which can provide intuition vision for the geographic space and contribute to the evacuation plan and implementation. Based on the auxiliary graph transformation, we propose a heuristic algorithm referred to the classical problem, Minimum Weighted Set Cover. We finally conduct extensive experiments to evaluate the performance of the proposed algorithm and give an application instance on a typical kind of restricted-space scenarios. The results indicate that the proposed algorithm outperforms the existing alternatives in terms of the utilization as well as timeliness.

#### 1. Introduction

As a significant research problem, disaster prevention and relief became one of the new requirements for social development in recent years, and emergency evacuation planning is an important part of rescue and relief work. Emergency evacuation generally contains the evacuee evacuation [1] and the relief resource deployment [2], and the evacuee evacuation is the most urgent work of disaster relief. An effective evacuation path planning strategy can not only effectively reduce the evacuation time consumption, but also prevent unnecessary congestion in the evacuation process, even secondary damage. Thus, the optimization design and theoretical analysis for evacuation path planning are of great practical significance [3].

According to the disaster scenarios, evacuation path planning can be classified into free-space (e.g., urban streets and squares) planning and restricted-space (e.g., underground pipeline, tunnel, and mine laneway) planning [4, 5]. The former one has most applications in city-security and antiterrorist, which has been well studied based on one-source to one-destination model or one-source to multidestination model.

The later one involves in a large range of industrial applications, i.e., the evacuation planning in fire disaster, gas explosion, or water inrush in underground mine. The restricted-space has several complex characteristics which are different from the free space:** (a) the space and time consuming limitations on entrance and evacuation. The density of personnel is sparser; (b) the unsuitability of long operating time. The reliability of temporary shelters is weaker; (c) the trapped-vulnerability of internal structure in disasters.** When a disaster occurs, the number of available evacuation paths is much less. Thus, the restricted-space has higher accident rate and brings more difficulties and challenges to rescue and relief work, which has been urgent to be studied. To improve the quality of emergency response in restricted-space planning, we focus on dynamic path planning based on the multisource to multidestination model.

In practical, restricted-space emergency evacuations have different evacuation requirements according to the accident type, nature, and distribution. On the one hand, considering the limitations and unsuitability of restricted spaces, it may cause local or global congestions when evacuation traffic load is too large, e.g., the evacuees intensively rushing to a certain exit. Thus, evacuation requirements do not only include the shortest evacuation time consumption, but also the unobstructed connectivity of route network. The goals of the restricted-space emergency evacuation are minimizing the global evacuation time consumption and maximizing the utilization rate of each exit (balancing among the exits’ traffic loads).** To minimize the global evacuation time consumption**, each evacuee tends to choose the shortest path and nearest exit, which may cause the imbalance of exit utilization, i.e., most evacuees intensively rushing to a certain exit via their shortest paths.** To maximize the utilization rate of each exit**, the evacuees should dispersedly choose the exits, which may cause that some evacuee chooses the suboptimal path rather than the shortest one and its evacuation time consumption will be lengthened. Thus, the two objectives are conflicting and they need a tradeoff to balance their realization, i.e., each evacuee should choose the global optimum rather than local optimum. On the other hand, due to the trapped-vulnerability of restricted spaces, the static planning strategies cannot adapt to the dynamic spreading of disasters or accidents and other environment changes. Thus, it is necessary to design dynamic evacuation planning based on environment changes. Based on the above considerations, we propose new evacuation routes planning problem, the Multiobjective Dynamic Route Network Planning (MODRNP) problem, and aim to design dynamic route network planning strategies and give theoretical analysis.

In this paper, firstly we introduce the restricted-space model and formulate the Multiobjective Dynamic Route Network Planning (MODRNP) problem in restricted-space evacuation. Secondly, we propose dynamic and global-optimized strategies to construct the evacuation route network, whose performance will be evaluated by the simulations and instance experiments at last. The rest of the paper is organized as follows. We summarize the related work in Section 2. Section 3 introduces the restricted-space model and problem definitions. Section 4 introduces the strategy framework for MODRNP Problem and describes the algorithms. Simulation results and corresponding discussions are given in Section 5. Section 6 concludes this paper.

#### 2. Related Works

As a crucial kind of combinational optimization problem, the path planning problem has been widely studied such like recent research works on traveling salesman problem (TSP)[6], traffic assignment/vehicle routing problem [7–9], and evolutionary optimization problems, e.g., general transportation planning problems, facility location problem, and roadway repair problem [10]. Based on the existing theoretical achievements, the research on evacuation path planning can be classified as static planning and dynamic planning.

Research on the** static path planning problem provides theoretical guideline for the emergency analysis and escaping exercise preparation.** Most algorithms for it are based on Dijkstra algorithm, improved Dijkstra algorithm, Floyd algorithm, the first shortest path algorithm, algorithm, and dynamic programming. Under the static problem model, the solutions generally simplified the disaster type, evacuee psychological pattern, and other factors by introducing the path equivalent length, with the assumption that the disaster would be spreading steadily.

The studies on the** dynamic path planning problem** are the basis of real-time decision making in emergency response and management in disaster/accident. The solutions can be classified into two kinds, the approximate algorithms with high theoretical precision and the heuristic ones with high efficiency.

To design approximate algorithms for the dynamic problem model, most researchers firstly transformed the problem into the classical network flow problem model [11–14] and then utilized the classical polynomial time algorithms to design the path selection strategy.** By applying the maximum flow algorithm**, Dunn and Newton [15] constructed a path network with the goal of guiding the most survivors in accordance with the constructed path network under the condition of permission capacity.** By applying the minimum cost flow algorithm**, Yamada [16] assigned evacuation traffic aimed at minimizing the overall length of the chosen evacuation paths, and Cova and Johnson[17] focused on complex path network with the multiple objectives of minimizing the intersections between the evacuation paths and minimizing the overall evacuation length.** By applying the fast flow algorithm**, the authors in [18, 19] aimed at the application scenarios modeled as directed graph. But the above algorithms could be only applied to the case of grid structure path network and unified path capacity.** By applying the contraflow algorithm**, the authors in [20] considered the setup time of the contraflow operation and proposed a two-layer algorithm to minimize the evacuation time consumption and the setup time for flow patterns. Since the algorithms for the flow problems have high time or space complexity, most of these strategies lack high efficiency in practice.

To design heuristic algorithms for the dynamic problem model, researchers in [21, 22] designed simulated evolutionary algorithms. They took full advantages of the positive feedback information mechanism of ant colony algorithm and the fast convergence of the genetic algorithm [23]. For the airport emergency evacuation scenarios, the authors in [24] performed a series of simulations based on an agent-based model (ABM) to determine the collective behaviors and the overall evacuation time, which are affected by the environment and complex structures. And [25] was motivated by the improvement of the marine evacuation system plan in a practical case, Dalian offshore airport of China, and developed a modeling framework to evaluate and optimize the emergency evacuation capability. For the crowded building emergency evacuation scenarios, the authors in [26] studied the human behavior of crowd evacuation to guide the design of buildings and the reinforcement of security management. The proposed algorithms can solve the path planning problem in densely populated scenarios. Due to the sparsity of personnel in restricted spaces, these strategies cannot be applied directly in the evacuation planning.

The above related work had limitations at three aspects. Firstly, most research works considered application scenarios for two-dimensional (2D) planes, in which the proposed algorithms cannot adapt to three-dimensional (3D) application scenarios. Secondly, most existing solutions for the path network construction problem were with high time or space complexity, which may bring low efficiency in restrict space evacuation. Thirdly, the existing researches on dynamic planning of emergency evacuation paths were still limited; e.g., the authors in [27] conducted research on the vehicle routing problem for postdisaster scenarios. And their application effect in practical is not ideal in terms of dynamic adjustment and timely feedback. Therefore, we intend to introduce a new dynamic planning problem model and utilized new technology in solution for restricted-space scenarios.

#### 3. Problem Formulation

##### 3.1. Restricted-Space Model

We focus on the emergency evacuation in confined space works and model such a densely populated restricted-space, as a 3D connected graph , where is the set of predetermined observation vertices in the space, is the set of bidirectional edges, and for each , and for each , (). For any vertex subset , is the subgraph of induced by . Similarly, is the subgraph of induced by an edge subset . The source set and the destination set are important subsets of the vertex set in , which are composed of the evacuees’ initial positions and the feasible exits, respectively. Here we assume and are predefined.

When a disaster or an accident happens, the evacuees need to escape via the edge/segment in the restricted-space in the shortest possible time. And the traffic capacity of each edge is affected by several influence factors, which need to be considered in the decision of the most reliable evacuation paths. We list the main influence factors of the restricted-space as follows.(a)* Edge types: *.The properties like the shape and slope of the edge/section decide the different types of the edge. Different edge types have different level effects on evacuee evacuation. Here we consider the scope of the edge, and for any pair of edges and , ().(b) *Passibility: **.* The passibility of the edge is greatly influenced by the obstacles and the vehicle of the evacuees. We assume that the evacuees adopt a uniform activity manner and have the same traveling speed. For any pair of edges and , ().(c) *Exit escaping priority: *. Due to the exits’ security levels, structural characteristics, or functional characteristics, different exits have different escaping priorities which have influence on the edge’s traffic.(d)* Evacuation priority: **.* The factor is generated from the incident relationship with the destinations in , i.e., if edge is an incident edge with more than one in , . For any pair of edges and , ().(e)* Travel length: **.* The geometric length of the edge. For any pair of edges and , ().

Based on these influence factors, the traffic capacity on an edge can be expressed in terms of the equivalent length/edge weight: , . Thus, the constrained space can be further modeled as an edge-weighted graph . Here we adopt travel length instead of traveling time to be the basis of traffic capacity for the reason that the traveling speed of each evacuee is regarded as a uniform value.

##### 3.2. Disaster Spreading Model

When the disaster or accident happens, the spreading speed and mode have great influence on the decision of evacuation route. Generally, the disaster spreading pattern is determined by several influence elements, e.g., the fire spreading pattern is effected by carbon monoxide in the volume flow of smoke and fire gases , oxygen in the fire place , heat flow, and so on. And the spreading pattern is directly reflected in the speed of disaster on each edge. Generally the spreading pattern can be modeled as a statistical model depending on the disaster itself.

To this end, we value the maximum spreading time duration of the disaster as ; i.e., it takes T from the disasters happening to its whole coverage. Note that we assume that all the evacuees can find the feasible evacuation path and their evacuation time consumption is not exceeding . We also assume that can be splitted as several timeslots according to the spreading rule and the evacuation time can be measured as the same timeslot. Thus, the disaster spreading pattern can be modeled as a function of , where is the temporal space subgraph of determined by disaster spreading speed in the -th timeslot. And the influence factor of the disaster is as follows.(f)* Disaster spreading status: *. In each timeslot, we value according to . For any pair of edges and , ().

Based on the influence factor, the edge weight in can be further expressed as , . Note that the edge weight is a time-related measure only in depending on disaster spreading status.

##### 3.3. Problem Definitions

We consider a Multiobjective Dynamic Route Network Planning problem in a restricted-space evacuation, which is to design a staged and adjusted construction scheme of evacuation path network from all the evacuees in to the emergency exit set in a 3D space graph .

Firstly, the problem is motivated from two points of view: the one is for each evacuee and the goal is to decide its globally ideal escape exit based on evacuation priority and find the escaping path to the ideal exit; the other one is for each escape exit. In most of practical applications, the feasible escape exits have different evacuation priorities; e.g., in underground coal mine evacuation, the auxiliary and main shafts have higher priority than chambers or mobile capsules. Thus, to enhance the utilization rate of each exit, it is needed to assign relatively balanced count of evacuees to the exit such that the case of congesting in some an overdrawn exit can be avoided.

Secondly, the problem has two objectives; one is in time dimension, i.e., minimizing the whole evacuation travel length. Here* the whole evacuation travel length* is defined as the travel length consumed by the latest evacuee, which can be referred to as the length of the longest escaping path, . And the other one is in space dimension, i.e., maximizing the priority-oriented evacuation efficiency.* The priority-oriented evacuation efficiency* is a global measure representing the balance performance among each exit’s utilization efficiency; i.e., a smaller efficiency stands for a higher balance performance. And the priority-oriented evacuation efficiency is defined as the maximum value among the utilization efficiency differences between any pairs of exits, , where represents two meanings, the exit itself and the set of the sources in escaping from finally.

Thirdly, the problem has dynamic constrains, i.e., disaster spreading status, which is the most influential one among the factors for traffic capacity and is varying along with the evacuee evacuation. In each timeslot, the disaster spreading pattern is regarded as an input item of the path planning process and it can be modeled as a function of , where is the temporal version of .

Our problem in restricted-space evacuation is defined as follows, which is a staged problem model.

*Definition 1 (Multiobjective Dynamic Route Network Planning (MODRNP) problem). ***Input:** Given a 3D restricted-space model, graph , source set , and destination set with its priority set, , the disaster spreading pattern .**Find:** Find the route network in each timeslot from all the sources to their optimal destinations, and the whole route network is constructed with two goals, minimizing the whole evacuation travel length and maximizing the priority-oriented evacuation efficiency .

Theorem 2. *MODRNP Problem in restricted-space evacuation is NP-hard.*

*Proof. *In order to proof the hardness of MODRNP Problem, we firstly decompose the problem in each timeslot as a subproblem and rewrite the subproblem into a mathematical formulation as below: In a 3D connected and edge-weighted graph , given two vertex subsets of , the source set and the destination set , and ’s priority set of , the problem is to find a path network from to such that(i)the maximum difference among ’s utilization efficiencies, , can be minimized, where is not only a destination in , but also a set of nodes in and connected to by the constructed paths,(ii)the maximum path length among all sources and destinations, , can be minimized.Then we consider a special case of it: Case (a). , s are a uniform value; i.e., the maximum value in (ii) is unified as a constant. Thus, the minimization of the maximum value can be regarded a default implement. Based on Case (a), we transform these isometric paths into the corresponding one-hop edges; i.e., is transformed into a bipartite graph with the two vertex sets and , as shown in Figure 1.

Case (b). , , i.e., the objective measure in (i) . For each node , can be used to represent its accessible nodes in , i.e., . When each is assigned the weight of , the problem becomes to find a conditional set cover in the bipartite graph .

Based on the above problem variant, the subproblem in Cases (a) and (b) is to find a subcollection (here ) such that and the total weight of in is minimized. Here we review an important and classical NP-hard problem,** Minimum Weighted Set Cover (MWSC) Problem**: Given a set composed of elements, a collection of subsets of (), and each () has a weight , the problem is to find a subcollection such that and is minimized. It is easy to discover that this special version of each subproblem is equivalent to MWSC problem, which was proved to be NP-hard [28]. Therefore, MODRNP Problem is NP-hard in general.