**1. Introduction**

With the rapid development of urban construction and building technology, more and more large buildings have been built in cities, and their internal structures are increasingly complex. When an emergency or disaster occurs inside the buildings, their complex internal structure makes it di fficult for indoor occupants to evacuate as quickly as a disaster occurs outside, which leads to frequent tragedies. It is critical for emergency rescuers and evacuees to plan an e ffective emergency evacuation plan [1]. The reason for this is that the plan can not only provide a reasonable escape path for evacuees in the event of a disaster, but also provide a basis for rescuers to make a rescue plan. Additionally, it can also provide reasonable suggestions for the layout of fire control facilities and the design of escape routes inside the buildings [2,3].

Due to the rapid occurrence and spread of disasters, it is necessary to make the best emergency evacuation plan in the shortest time. Therefore, two key objectives of a practical evacuation plan are to ensure the shortest overall escape time and to design the plan as quickly as possible. So far evacuation plans can be roughly classified as optimization-oriented and simulation-oriented [2]. Our research belongs to the former category and aims at developing an optimal method to design evacuation plans. In this paper, we deeply analyze the staged-evacuation process in crowded indoor environments and present a simple and e fficient algorithm for staged-evacuation path planning that is able to cope with multi-exit networks. Generally, indoor evacuation is a multi-exit evacuation problem. The algorithm first transforms the multi-exit evacuation problem into a single-exit problem by balancing the loads of evacuees at all emergency exits, then performs the single-exit evacuation. Our contribution includes: 1) for multi-exit indoor evacuation, a partitioned and staged evacuation planning approach is proposed, which e ffectively realizes the transform above and simplifies the planning of multi-exit evacuation; 2) for single-exit indoor evacuation, a new idea of determining the escape sequence of evacuees according to their shortest path length is proposed and verified, which improves the e fficiency of developing evacuation plan. Furthermore, the e fficient single-exit evacuation will e ffectively improve the e fficiency of the multi-exit evacuation, because the multi-exit evacuation is composed of multiple single-exit evacuations in this paper.

The remainder of this paper is organized as follows. Section 2 reviews related work. Section 3 describes the problem. Section 4 gives related definitions and theorems and presents our method. Section 5 illustrates the results of the algorithm, evaluates its performance and e ffectiveness by a series of tests, and gives a testing simulation. Section 6 concludes the paper.

## **2. Related Work**

From the perspective of implementation, Li et al. classify evacuation plans into two major types: spontaneous evacuation plans and organized evacuation plans [2]. The former is carried out by controlling the evacuation infrastructure (e.g., fire emergency lighting and dispersal indicator) while evacuees move spontaneously but are guided by the infrastructure. The latter is realized by controlling evacuees including their departure time, routes to safe exits, and so on. Each type of evacuation plans is applicable to a particular scenario.

Regarding spontaneous evacuation plans, many simulation models have been put forward to analyze the significant factors or parameters that influence the evacuation process or can be used in evaluating the evacuation performance under di fferent scenarios and strategies. Existing typical models include network flow based models [4], cellular-automata (CA) models [5,6], agent-based models [7–9], social-force models [10,11], lattice gas (LG) models [12,13], and so on. These models have been successfully applied to study crowd evacuation under various situations because of their grea<sup>t</sup> ability in representing some key elements influencing human behaviors during evacuation process, such as the impact of the occupant density around exits [14,15] and spatial distance on human behaviors. Flow based models are easy to construct while they lack social interaction between evacuees, human behavior in emergency conditions and hazards representation [4]. CA models are very flexible and e ffective in simulating evacuation process under complex environment while in contrast with multi-agent system, they have more primitive agents that are arranged on a rigid grid and interacting with each other by very simple rules. LG models present a special case of cellular automata modes that utilize biased random rules to simulate counter flow in channels, or to evaluate the impact of building parameters to the evacuation e fficiency. Agent-based or multi-agent-based models can represent various types of agents with di fferent attributes and their interactions are more complex [8], and the disadvantages of them are generally more computationally expensive than cellular automata. Social force models are a kind of continuous model applying Newton's second law to simulate pedestrian evacuation and are good at modeling interactions among pedestrians, but have low computational efficiency in simulating evacuation in complex buildings [6].

This paper focuses on organized evacuation planning requiring a comprehensive and e ffective escape plan for particular evacuation objectives according to di fferent escape environments. Generally, these objectives include reducing tra ffic conflicts and minimizing the whole clearance time of all evacuees or the evacuation time of each evacuee. Network flow models, such as maximum-flow models and minimum-cost flow models [3,16–18], are the most widely used in optimizing the flow of evacuees, but they target the whole network and attempt to organize the origin, destination, and routes of evacuation flow at a mesoscopic level. The integer programming or linear programming method [16,19], as an exact algorithm, is applicable to small-scale problems and usually requires additional parameters (e.g., lower or upper bounds, etc.) that are generally di fficult to estimate in advance. For large-scale evacuations, heuristic methods and scheduling algorithms are often adopted. The former, such as evolutionary algorithms [20] and ant colony optimization [21–23], are limited in terms of the quality of solutions and computing time. The latter are generally exact methods and are used to integrate the objectives and the constraints into the design of algorithms [2,24–26].

In the process of evacuation, if there is congestion, it will inevitably lead to the decrease of escape e fficiency and even trample accident. In order to avoid congestion, waiting is necessary [27]. There are two ways of waiting in the strategies of scheduling algorithms. One is waiting at the starting point and the other is waiting on the way. Li et al. proposed an innovative method to make a staged-evacuation plan for emergency situations, but it is only applied to the network with a single safe exit and it is assumed that the speed of evacuees is constant and equal [25]. Later, they extended the staged-evacuation plan method from two aspects. One is to make it apply to multi-exit evacuation based on the time-extended network model by balancing tra ffic loads to di fferent exits [26]. The other is to make it fit multi-speed evacuations with a single exit [2]. Although these algorithms ge<sup>t</sup> excellent results, iterative computation of the time-extended network results in their low e fficiency.

It should be noted that another research direction closely related to indoor emergency evacuation is to dynamically plan an indoor evacuation path based on the real-time perceived situation information about the spread of a disaster [28–31]. However, so far, the research results in this direction are more applicable to the situation without indoor congestion. Additionally, the acquisition technology of real-time disaster environmental information in the case of fire has made grea<sup>t</sup> progress, but remains a challenging work. Encouragingly, the arrival of smart city o ffers real-time access to indoor evacuation information such as the distribution of evacuees and the development of an indoor disaster, which provides a data base for the real-time design of an evacuation scheme. This is one of the reasons why we pay attention to the e fficiency of the algorithm.

## **3. Problem Description**

Once an emergency occurs in a building with multiple exits, evacuees would choose the nearest exit to escape if there are a few occupants in the building. However, if there are more occupants, due to the limitation of the capacity of the escape path, it is prone for evacuees to conges<sup>t</sup> at the corners or intersections of the path or safety exits, which reduces their escape speed, prolongs the overall evacuation time, and increases the probability of risk for them [25]. Therefore, the problem of indoor emergency evacuation studied in this paper is how to let all evacuees escape from the dangerous buildings with multiple safe exits in the shortest time when the capacity of indoor route is limited and congestion may occur during evacuation. Figure 1 shows the abstract representation of the studied evacuation problem. There are three safety exits namely E1, E2 and E3 in the indoor route network whose edges contain both path cost and capacity. *Ai* represents the room node. In this paper, it is assumed that the capacities of all locations in the route network are equal.

**Figure 1.** The indoor network with multiple exits.
