单源多汇疏散问题的数学模型及安全出口优化

doi:10.16265/j.cnki.issn1003-3033.2025.03.1994

摘要/Abstract

摘要：

为提高公共建筑物疏散效率,减少因突发事故造成人员伤亡,针对建筑物安全出口及疏散方案的优化问题,提出单源多汇疏散问题的数学模型及其算法。首先,运用通路法,得到所有疏散通路,并按照疏散时间排序;其次,根据用户均衡原理(UE)、k最短路思想,得到计算疏散时间的递推公式,再分析各疏散通路在交叉点处容量分配问题;然后,得到建筑物安全出口的优化策略;最后,以某建筑物为例,建立疏散网络并进行计算,验证算法和优化策略的有效性。结果表明:随着疏散人数增加,建筑网络疏散效率逐渐增大;且当疏散人数达到某个特定值时,其变化率近似为恒值。疏散效率与特定的安全出口总的通行能力有关,优化特定安全出口,能够提高疏散网络的疏散效率,减少疏散时间,且疏散人数越多,优化效果越明显。

关键词: 单源多汇, 安全出口, 疏散模型, 用户均衡原理, 通路法

Abstract:

In order to improve the evacuation efficiency of public buildings and reduce casualties caused by emergencies, in view of the optimization problems of emergency exits and evacuation plans, a mathematical model and algorithm for the single-source multi-sink evacuation problem were proposed. Firstly, all evacuation paths were identified using path algorithm and sorted based on evacuation time. Then, the recursive formula for calculating evacuation time was derived based on the User Equilibrium (UE) principle and k-shortest path. The capacity allocation issues of each evacuation path at intersections were analyzed. After that then the optimization strategy for emergency exits is obtained. Finally, taking a building as an example, the evacuation network was established and calculated to validate the effectiveness of the algorithm and optimization strategy. The results show that the evacuation efficiency of the building network increases gradually with the increase in the number of evacuees, and the rate of change stabilizes after reaching a certain threshold value, Evacuation efficiency is closely associated with the total traffic capacity of specific emergency exits. Through the optimization of specific emergency exits, the evacuation efficiency of the network is enhanced, evacuation time can be reduced, and the optimization effect becomes more pronounced as the number of evacuees increases.

Key words: single-source multi-destinations, emergency exit, evacuation model, User Equilibrium principle, path algorithm

中图分类号:

X913.4安全系统工程

周令昌, 梁茵, 赖艳萍. 单源多汇疏散问题的数学模型及安全出口优化[J]. 中国安全科学学报, 2025, 35(3): 187-193.

ZHOU Lingchang, LIANG Yin, LAI Yanping. Mathematical model of single-source multi-destinations evacuation and optimization of emergency exits[J]. China Safety Science Journal, 2025, 35(3): 187-193.

图/表 9

图1

图2

图3

表1

表2

优化前的疏散通路及运动时间

序号	通路编号	包含的疏散弧	运动时间
1	①	e₇、e₁₁、e₂₃、e₂₈	4
2	②	e₁₇、e₁₈、e₂₂、e₃₁	6
3	③	$e 14 、 e 15 、 e 26 、$ e₃₀	7
4	⑥	e₁₆、e₁₉、e₂₁、e₂₂、e₃₁	8
5	⑦	e₁、e₄、e₁₁、e₂₄、e₂₇	9
6	⑧	e₃、e₄、e₁₁、e₂₃、e₂₈	9
7	⑨	e₁₁、e₁₂、e₁₄、e₂₆、e₃₀	9
8	⑬	e₈、e₁₁、e₂₅、e₂₉	11

表2

表3

表4

图4

图5

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序号	人数	疏散时间 T/s		疏散效率/ (人·s^-1)		ΔN∶ΔT/ (人·s^-1)
序号	人数	优化前	优化后	优化前	优化后	优化前	优化后
1	0	4	4	—	—	—	—
2	5	6.2	6.2	0.81	0.81	2.27	2.27
3	10	7.14	7.14	1.4	1.4	5.32	5.32
4	15	7.86	7.86	1.9	1.9	6.94	6.94
5	20	8.5	8.44	2.35	2.37	7.81	8.62
6	25	9.07	9.0	2.76	2.78	8.77	8.93
7	30	9.46	9.31	3.17	3.22	12.82	16.00
8	35	9.85	9.63	3.55	3.64	12.82	16.00
9	40	10.23	9.94	3.91	4.03	13.16	16.00
10	45	10.61	10.24	4.24	4.4	13.16	16.79
11	50	11	10.53	4.54	4.75	12.82	17.00
12	55	11.33	10.82	4.8	5.08	15.02	16.97
13	60	11.67	11.12	5.14	5.4	14.97	17.01
14	65	12	11.41	5.4	5.7	15.02	17.01
15	70	12.33	11.7	5.68	5.98	15.02	17.01
16	75	12.67	12.0	5.92	6.25	14.97	17.01
17	80	13	12.29	6.15	6.51	15.02	17.01
18	85	13.33	12.59	6.38	6.75	15.02	17.01
19	90	13.67	12.88	6.58	6.99	14.98	16.99
20	95	14	13.18	6.79	7.21	15.00	17.00
21	100	14.33	13.47	6.98	7.42	15.02	17.00
22	105	14.67	13.77	7.16	7.63	14.97	16.98
23	110	15	14.06	7.33	7.82	15.02	17.02
24	115	15.33	14.35	7.5	8.01	15.02	17.00
25	120	15.67	14.65	7.66	8.196	14.97	17.01