Mathematical Problems in Engineering

Volume 2017, Article ID 4063810, 7 pages

https://doi.org/10.1155/2017/4063810

## A Modified Floor Field Model and Pareto Optimum of Pedestrian Evacuation Efficiency

^{1}School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot 010070, China^{2}School of Economics and Management, Fuzhou University, Fuzhou 350108, China

Correspondence should be addressed to Gui Yong; moc.361@iuggnoymn

Received 3 December 2016; Accepted 30 May 2017; Published 27 June 2017

Academic Editor: Alessandro Gasparetto

Copyright © 2017 Yan Xu 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

A modified floor field model is proposed to simulate the pedestrian evacuation behavior in a room with multiple exits. The modification relies upon introduction of a so-called semidynamical floor field which additionally embodies two cognition coefficients related to exit width and pedestrian distribution around exits. The energy consumption and time requirement of evacuees are numerically investigated and the Pareto optimum of evacuation efficiency is obtained by selecting the combinations of the two cognition coefficients.

#### 1. Introduction

Study on pedestrian dynamics has attracted many physicists over the past few years. Various collective behaviors and self-organization phenomena have been observed from the view point of complexity of the pedestrian flow system. For understanding these phenomena, many models have been developed, for example, the social force model [1–3], the fluid dynamic model [4–9], and the cellular automata model [10–21].

The social force model is a deterministic continuum model in which the interactions between pedestrians are governed by the social force or social field [1, 2]. Using this model, such phenomena as the arching around exit, the lane formation in counter flow, and the oscillation of flow direction at narrow door can be reproduced.

The fluid dynamic model describes how density and velocity change over time with the use of partial differential equation. This model can be used to investigate exactly the dynamics at the exits during an evacuation [7] and study some typical pedestrian behaviors, for example, the “thinking fluid” behavior and the overcompression effect [8, 9].

The rule-based cellular automata model is discrete in space and time. All pedestrians are represented by particles. Such typical phenomena as arching, jamming, kin behavior, faster-is-slower, lane formation, and sidle effect can be simulated by this kind of models [10–21]. Floor field model is one kind of discrete simulation models. At each time step, every pedestrian moves from one lattice to the neighboring lattice in terms of a transition probability. This model can be used to investigate various collective effects and the self-organization encountered in pedestrian dynamics [10, 11].

Modeling the evacuation process of pedestrians in a room with multiple exits has to consider the strategy of selecting exits. In the conventional floor field models, the exit selection strategy is mainly based on the herding behavior and the use of knowledge about the shortest path to each exit [10, 11]. In some works, the occupant density around each exit is considered as an important factor affecting the exit selection strategy [15, 16]. In this paper, we improve the floor field model through introducing two cognition coefficients associated with the exit width and the pedestrian distributions around exits, respectively.

The energy consumed in evacuation process is seldom studied by previous researches. However, if a space for evacuation is large enough to make some pedestrians expend more energy than their capabilities, he or she will fail to evacuate successfully and, on the contrary, they may become obstacles to other pedestrians. In this study, we will investigate the energy consumption of evacuees besides the time requirement and find the Pareto optimum of evacuation efficiency.

#### 2. Model

Let a space for evacuation be represented by two-dimensional square lattices. The size of each lattice is approximately cm^{2}. Each lattice can be either empty or occupied by one pedestrian. Each time step is real time of evacuation based on different movement velocity of pedestrians. The length of one time step is 0*.*3 seconds in this study. This implies a walking speed of approximately 1.33 m/s.

In each time step, pedestrians move only one lattice site in the forward, backward, left, or right direction or remain unmoving. They probabilistically select a neighboring lattice site in these directions to move using the transition probability. The transition probability is computed using the static and dynamic floor fields of those lattices.

In the simulation of evacuation process, the static floor field represents the degree of attractiveness of each lattice for pedestrians. The static floor field does not evolve with time and is not changed by the presence of the pedestrians. It is often given by the value depending on the distance from the exit. The dynamic floor field represents the characteristic that pedestrians tend to follow their predecessors. This is implemented using bosons dropped by pedestrians, having their own dynamics through diffusion and decay controlled by probabilities *σ* and *δ*. The dynamic floor field is the number of bosons remaining on the cell at each time step. These two types of floor fields have been historically used as fundamental and essential components of the floor field model.

The static floor field defined above reflects the attractiveness degree of each lattice for pedestrians, but the exit width and the pedestrian distribution around exits have not been considered. These two factors should have impacts on the attractiveness of some lattices and eventually influence the movement behavior of pedestrians. For this reason, the floor field model has to be improved so as to simulate the evacuation process more precisely.

##### 2.1. Occupant Density

The occupant density around exit is introduced to characterize the pedestrian distribution around exits in the evacuation process. The occupant density of exit* m* can be defined as the number of occupied cells in the effect area of exit* m* with radius* r*. The effect area is a special region around the exit. Figure 1 provides an illustration for calculating the occupant density [16]. The width of the exit is 2 cells. The effect area contains sixteen intact cells which are located within a half-round with radius of 4 cells. Five black colored cells are occupied. Thus, the occupant density of this exit is 5.