**5. Case Study**

To verify the validity and efficiency of the PSEP algorithm, we conducted two tests. One was used to verify the correctness and efficiency of the algorithm for the single-exit evacuation; the other to discuss the rationality of the partition method of the PSEP algorithm and to compare its performance with an existing algorithm. Since the PSEP algorithm is based on the single-exit evacuation algorithm, both tests are valuable to illustrate the advantages of the PSEP algorithm. Test data is the three-dimensional path network of the teaching building J6 of Shandong University of Science and Technology (SDUST), as shown in Figure 7, where each vertex (i.e., node) in the network represents an escape group and each edge (i.e., arc) represents a segmen<sup>t</sup> of indoor path. The network model consists of five layers, 818 nodes and 853 edges. On the first floor, there are three safety exits: *E1*, *E2* and *E3*). During the

tests, nodes in the route network are randomly selected as the starting nodes in order to simulate the evacuation environment in reality.

**Figure 7.** The indoor network model of the teaching building J6 of SDUST.

All involved algorithms were developed in C-Sharp and run on the portable notebook whose configuration were as follows: CPU i7-6500u, main frequency 2.5 GHz, running memory 12 G and the solid-state disk with capacity of 256 G.

#### *5.1. Tests Based on Single-Exit Network*

The PSEP algorithm is mainly inspired by the algorithm of [25]. We firstly compare and discuss their efficiency and results. Additionally, the algorithm of [25] is only suitable for the single-exit network, so we choose a part of the teaching building J6 (i.e., an evacuation zone with one safety exit) as the test zone to test the influence of the number of escape groups, the size of groups, the escape speed and other factors on the total evacuation time, as well as the efficiency of the two algorithms. The test network includes only one exit *E1*, 210 edges and 210 vertices. When the PSEP algorithm is applied to the single-exit network, the first procedure of the algorithm is omitted for the network has only one exit.

#### 5.1.1. Influence of the Number of Groups on Total Evacuation Time

The size of all groups is 15 m, the escape speed is fixed as 3 m/s, and the number of evacuation groups is set to 90, 130, 170, 210 respectively. The test results are shown in Figure 8.

**Figure 8.** Comparison of their total evacuation time varying with the number of evacuation groups.

Figure 8 shows that when the group size and the escape speed of each evacuation group are fixed, their total evacuation times increase linearly with the number of evacuation groups for both algorithms. At the same time, the total evacuation times of the two algorithms are equal.

5.1.2. Influence of Evacuation Speed on Total Evacuation Time

The number of evacuation groups is 210, the group size is 15 m, and the escape speed is set to 2, 3, 4, and 5 m/s respectively. The test results are shown in Figure 9.

**Figure 9.** Comparison of their total evacuation time varying with the escape speed.

Figure 9 illustrates that when the number of evacuation groups and the group size are fixed, the total evacuation times of the two algorithms decrease with the increase of the escape speed.

5.1.3. Influence of Group Size on Total Evacuation Time

The number of evacuation groups is 210, the escape speed is 3 m/s, and the group size is set to 6, 15, 24, and 33 m. The test results are shown in Figure 10.

**Figure 10.** Comparison of their total escape time varying with the group size.

Figure 10 illustrates that when the number of evacuation groups and the escape speed are fixed, their total evacuation times increase linearly with the increase of the group size for both algorithms.

#### 5.1.4. Comparison of Operating E fficiency

The group size is 15 m, the escape speed is 3 m/s, and the number of evacuation groups is set to 120, 280, 440, 600, 760, 920 and 1080 respectively. The test statistics are shown in Table 1. *Td* represents the time consumed by the algorithm of [25] and *Tp* represents that by the PSEP algorithm. Figure 11a shows the curves of the time consumed by the two algorithms, from which we can see that their time consumption is increasing with the increase of the number of evacuation groups, but the time consumed by the PSEP algorithm is significantly less than that by the algorithm of [25]. Figure 11b indicates the ratio curve of the time consumed by the two algorithms, from which we can see that the more evacuation groups there are, the more obvious the e fficiency advantage of the PSEP algorithm over the algorithm of [25] is. When the number of groups is 1080, *Td*/*Tp* reaches 41465.

**Table 1.** Statistical results of time consumed by the two algorithms.

**Figure 11.** Comparison of the e fficiency of the two algorithms. (**a**) The curves of the consumed time by the two algorithms; (**b**) The e fficiency ratio of the two algorithms

It can be seen from the first three tests that the total evacuation time of the two algorithms is the same regardless of the evacuation condition, which proves the correctness of the proposed algorithm. Test 4 illustrates that the PSEP algorithm is much more e fficient than the algorithm of [25]. The reason is that there are a lot of repeated calculations in the algorithm of [25]. Every time the departure time of an evacuation group is determined, it is necessary for each evacuation group whose departure time has not been determined to recalculate its time windows at all nodes on its evacuation path, while the PSEP algorithm only needs to calculate the time window of each group occupying the exit to determine the departure time of all escape groups. In addition, the tests above also prove the correctness of Theorems 1 and 2.

#### *5.2. Tests Based on Multi-Exit Network*

According to the principle of the PSEP algorithm, the partition method and the density of indoor evacuees are two important factors that a ffect the overall evacuation time. Therefore, we tested their influence on evacuation e fficiency. Furthermore, the relation between the evacuation path length and delayed time of each evacuation group is tested. Its performance is also compared with that of an existing algorithm.

#### 5.2.1. Influence of Partitioning Methods on Evacuation E fficiency

In theory, the partition method based on the principle of "nearest evacuation" will increase the overall evacuation time when the density of indoor evacuees is larger. In order to verify the theory, we apply the principles of "nearest evacuation" and "balanced evacuation" to the PSEP algorithm respectively to test their influence on evacuation. Figure 12a shows the partitioned indoor network only based on "nearest evacuation" that will find the nearest exit for each group, and Figure 12b shows that based on "balanced evacuation" that makes every exit have the approximately equal number of evacuees by considering both the number of evacuees and their path length. It can be seen from Figure 12 that the partition of some nodes in the route network has changed. Many nodes originally belonging to the zone E3 are assigned to E1, while the zone E1 also occupies part of the nodes in the original zone E2, and the zone E2 regains part of the nodes from the original zone E3. Furthermore, these adjusted nodes are mainly distributed in the adjacent area of the original zones.

**Figure 12.** Comparison of results partitioned by two different partitioning methods. (**a**) The partitioned network based on nearest evacuation; (**b**) The partitioned network based on balanced evacuation

Table 2 shows the number of evacuation groups, the number of evacuees of each evacuation zone and their total evacuation time when the PSEP algorithm adopts the two partitioning methods respectively. It can be seen from the table that the number of evacuation groups of each exit when using the principle of "nearest evacuation" is not balanced while that are balanced when using the principle of "balanced evacuation". The nearest evacuation makes exits E1 and E2 are not fully utilized, which increases the overall evacuation time by 265 seconds compared with that of the balanced evacuation. Therefore, we can conclude that the partition strategy of evacuation zones will greatly a ffect the evacuation e fficiency of the evacuation scheme.

**Table 2.** The number of evacuees, the number of evacuation groups in di fferent zones and their total evacuation time based on balanced evacuation and nearest evacuation


5.2.2. Influence of Evacuation Density on Total Evacuation Time

Let the evacuation density *ER* = *EL*/*NL*, where *EL* is the total length of all evacuation groups and *NL* is the total length of all edges of the evacuation network. To test the influence of the evacuation density on the total evacuation time of different evacuation plans, we implemented the nearest evacuation and the balanced evacuation respectively with different evacuation densities. The total length of the evacuation route network is 5443.3 m, and the number of evacuation groups is 818. The length of all evacuation groups is set as 0.5, 1, 2, 3, 4 and5mrespectively. The test results are shown in Table 3, where *Tn* represents the total evacuation time of nearest evacuation and *Tb* represents that of balanced evacuation. We can see that balanced evacuation has obvious advantages over nearest evacuation when the density of evacuees is large but nearest evacuation has a shorter overall evacuation time when the density of evacuees is small. Figure 13 shows that the greater the density of evacuees, the more advantageous balanced evacuation will be.


**Table 3.** Comparison of the total evacuation time of two strategies under different evacuation density

**Figure 13.** Comparison of the total evacuation time of two strategies under different evacuation density. (**a**) The change of their total evacuation time with the increase of evacuation density; (**b**) The change of the total evacuation time difference with the increase of evacuation density

#### 5.2.3. Evacuation Process Simulation

In order to visually verify the effectiveness of our algorithm, we took the whole teaching building J6 of SDUST as the test scene to simulate the emergency evacuation process using our evacuation simulation software. In the simulation scene, color is used to identify different evacuation groups, and the length of line segmen<sup>t</sup> represents group size. Assuming that the start time of evacuation is t0, escape speed is 3 m/s, group size is random and the sum of evacuation groups is 818. In case of emergency, the visual simulation results of evacuation process of all groups starting to escape at the same time are shown in Figure 14, where (a)–(d) are the distribution of all escape groups at t0 + 8 s, t0 + 16 s, t0 + 24 s, and t0 + 32 s, respectively. From Figure 14, we can see that many colorful line segments are mixed together, which indicates that there is a large area of congestion. Obviously, serious congestion will reduce the escape speed of evacuees and ultimately lead to the extension of the total evacuation time.

*ISPRS Int. J. Geo-Inf.* **2020**, *9*, 46

**Figure 14.** The simulation of simultaneous evacuation for the multi-exit network. Four screenshots are given at different times as follows: (**a**) t0 + 8 s; (**b**) t0 + 16 s; (**c**) t0 + 24 s; (**d**) t0 + 32 s.

Figure 15 shows the visual simulation results of the PSEP algorithm, where (a)–(f) are the distribution of all escape groups at t0 + 32 s, t0 + 64 s, t0 + 112 s, t0 + 144 s, t0 + 232 s, t0 + 272 s, respectively. The simulation process shows that all groups escape orderly according to the assigned departure time without any congestion on the way and all groups pass through the emergency exit successively. All of these ensure the efficient operation of the whole evacuation process and reduce the overall evacuation time.

**Figure 15.** *Cont.*

*ISPRS Int. J. Geo-Inf.* **2020**, *9*, 46

**Figure 15.** The simulation of partitioned and staged evacuation for the multi-exit network. Six screenshots are given at different times as follows: (**a**) t0 + 32 s; (**b**) t0 + 64 s; (**c**) t0 + 112 s; (**d**) t0 + 144 s; (**e**) t0 + 232 s; (**f**) t0 + 272 s.

The PSEP algorithm adopts the partitioned and staged evacuation strategy. Once evacuation partition is completed, the evacuation process in each zone is independent of each other. The total evacuation time of all escape groups is the maximum evacuation time of each zone. Table 4 shows the relationship between the evacuation path length and the delayed departure time of some evacuation groups in the zone E1, and Figure 16 shows that of all evacuation groups in zones E1, E2, and E3 respectively. We can see that the delayed departure time of all groups in each zone increases with the increase of the evacuation path length.


**Table 4.** The relationship of the evacuation path length and the delayed departure time of some groups planned by the PSEP algorithm when evacuation density is large.

**Figure 16.** The relationship of the evacuation path length and the delayed departure time of all groups in each zone when evacuation density is large. (**a**) Zone E1; (**b**) Zone E2; (**c**) Zone E3.

5.2.4. Relation between Evacuation Path Length and Delayed Time

Table 4 shows the relationship between the length of the evacuation path and the delayed time in the case of a large density of evacuees. To ge<sup>t</sup> a more comprehensive picture of their relationship, we conducted the other test in the case of a low density of evacuees. Let all evacuation groups be 2 m in size and the other test conditions will remain the same. Table 5 shows the evacuation path length and the delayed departure time of the partial evacuation groups in zone E1. Figure 17 shows the relationship between the evacuation path length and the delayed departure time for all evacuation groups in zones E1, E2, and E3, respectively.



**Figure 17.** The relationship of the evacuation path length and the delayed departure time of all groups in each zone when evacuation density is low. (**a**) Zone E1; (**b**) Zone E2; (**c**) Zone E3.

Table 5 shows that the delayed time of the evacuation groups will not increase completely with the increase of their evacuation path length, and some of them will be exceptional. This exception occurs because of the time interval between some of the adjacent evacuation groups (Figure 3), which will reduce the delayed departure time of the group that arrives later. For example, if the time windows of two groups A and B occupying the same exit are [23.0, 25.0] and [29.0, 32.0] respectively when they start to escape at the same time, they will not conges<sup>t</sup> with each other. But, if the groups ahead of A makes it have to delay 5 s to depart in order to avoid congestion at the exit, the time window of the group A occupying the exit becomes [28.0, 30.0]. To avoid congestion with the group A, and the group B should be delayed by 1 second that is less than the delayed time of the group A.
