# Simulation and Optimization for the Staircase Evacuation of a Cruise Ship Based on a Multigrid Model

Authors: Min Hu; Wei Cai
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

A cruise ship is a large public place, and it is very important to ensure the safety of passengers during the evacuation process in case of an emergency. This paper proposes a method to improve evacuation efficiency on cruise ships by controlling passengers’ density. According to the construction of the staircase, the space of the staircase is divided into the step and landing areas. On the basis of considering the influence of passengers’ view field and moving characteristics of passengers, the moving rules of passengers in two areas are established. Taking staircases of the cruise ship as the evacuation scenario, the evacuation process is simulated by using the established model. From simulation results, it is found that numbers of evacuated passengers between staircases are very unbalanced and too many passengers gather in one staircase, which lead to serious congestion. By controlling passengers’ density in stairs areas, the minimum evacuation time is the optimization objective and the optimization model is established by using the quantum-inspired evolutionary algorithm and genetic algorithm. The optimization results show that the evacuation time is significantly shortened when the passenger’s density on the staircase is kept within an appropriate range, which proves that the evacuation efficiency can be effectively improved by controlling the passengers’ density.


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## Body

## 1. Introduction

A cruise ship is a multistorey structure, and staircases are the main vertical evacuation channels connecting between decks. When an emergency occurs, staircases are tasked with evacuating large numbers of people. The staircase is a semi-closed space, and the walking speed of passengers will be affected by the structure and size of the staircase. Once passengers enter the staircase, their moving is almost fixed, and they will move downstairs to the destination floor.In order to obtain the characteristics of the staircase evacuation process, many researchers studied the staircase evacuation by experiments and simulations. Li et al. [1] proposed a stairs-unit mode for the evacuation of multistorey teaching buildings, and this model could describe the spatial relationship between stairwells and floors. Zeng et al. [2] modified the optimal steps model [3] and studied the process of pedestrian evacuation on the stair platform. They found that when the corridor was connected to the platform on the opposite side of the incoming stairs, this structure was conducive to the evacuation of pedestrians on the stairs. By extending the heuristic-based model [4], Guo et al. [5] indicated that pedestrians could adjust their walking speed according to the distance from the people around. They applied this model for simulation of pedestrian evacuation on the single-file staircase. Chen et al. [6] proposed a pedestrian footstep mode, which could describe pedestrian’s walking distance according to the depth and height of steps. Goyal et al. [7] collected the data of pedestrian flow on the stairs of three subway stations in Delhi and analyzed the influence of age, gender, carrying luggage, and different time periods on the speed of pedestrian movement on the stairs. Fu et al. [8–10] presented a fine discrete floor field cellular automata model; they considered influence factors that contained fatigue of pedestrians and visibility during the stairs evacuation process. Qu et al. [11] focused on the problem of the staircase evacuation of subway station and proposed an enhanced social force mode. In this model, pedestrians were described by a set of three circles, and rotation dynamics was added into the model to describe the congestion effect. Li et al. [12] proposed an extended social force model based on height map to simulate the moving state of pedestrians on the stairs. Through the simulation experiment, they found that the risk of clogging from transitions between flat ground and stair steps can be decreased by using ramps instead of staircases. Lei et al. [13] carried out an experiment of the stair evacuation process in the student dormitory, and FDS + Evac software [14] was used to simulate the same evacuation scenarios. They found that the simulation results were consistent with the experiment, and the stratification phenomenon of flow would appear as long as the width of the exit increased. Ronchi et al. [15] presented a conceptual model for stair evacuation, and they comprehensively considered the impact of pedestrian fatigue and psychological changes on the evacuation process. Larusdottir and Dederichs [16] studied the evacuation stair process of children in stairs from the flow rate, density, and walking speed.During the process of multistorey evacuation, pedestrians enter the staircase from different floors and leave the staircase from the exit on the same floor. The evacuation of the cruise ship is a typical multistorey evacuation problem. In the process of evacuation, merging occurs between passengers descending from upper decks and ones entering the staircase on lower decks. Some studies have shown the effect of merging on stairs evacuation. Considering the impact of merging ratio on pedestrian flows, Sano et al. [17] proposed a simplified mathematical model. This model could identify the effect of merging behavior on evacuation time of each floor. Chen et al. [18] studied the pedestrians’ movement speed of stairwell on the high-rise building. They found that the traveling distance in the stairwell had no obvious effect on the walking speed of pedestrians, while merging behavior would make the walking speed slower. Zeng et al. [19, 20] investigated the impact of initial pedestrians’ distribution between different floors and illumination in the stairwell area during pedestrian evacuation, and found that merging behavior would slow down the moving speed of pedestrians on upper floors. Huo et al. [21] studied pedestrian processes in two evacuation scenarios. In scenario one, all the pedestrians were evacuated from a selected floor. In scenario two, pedestrians were distributed on different floors. From results of scenario two, they found that the walking speed of the pedestrian from the upper floor would slow down as the lower floor pedestrian entered the stair area. Ding et al. [22] considered the impact of the merging behavior on stair evacuation, and carried out evacuation simulations of different initial distributions of pedestrians on five floors. They found that the evacuation time is almost linearly changed with the increase of floors.The layout of evacuation scenarios or the group psychology may cause pedestrians to move towards one exit, and then lead to congestion. It is necessary to guide pedestrians to move from the congested area to other exits. The evacuation guidance and evacuation optimization were studied by some researchers. Ding et al. [23] optimized the coordinated strategy for evacuation between stairs and elevators in high-rise buildings. Hou et al. [24] simulated pedestrian evacuation process by considering the effect of trained leaders and found that the evacuation efficiency had been significantly improved when there were just one or two leaders in the crowd. Jeon and Hong [25] studied the impact of phosphorescent guidance signs on the evacuation process and carried out an experiment in the subway station. The result showed that the walking speed of pedestrians increased only when the installation interval of signs was smaller than one step of pedestrians. Ma et al. [26, 27] applied the extended social force model to study the influence of guidance on pedestrian evacuation simulation, and the simulation result showed that guidance could improve evacuation efficiency when the pedestrian density of the neighborhood field was moderate. In order to optimize escape paths in the earthquake, Bernardini et al. [28] proposed a Seismic Pedestrians’ Evacuation Dynamic Guidance Expert System. Though the data were collected in the surrounding environment, the safest evacuation path could be obtained by using Dijkstra algorithm. Yang et al. [29, 30] simulated the evacuation process of large public places, and simulation results show that proper initial positions and numbers of guiders were more conducive to improve evacuation efficiency. Cao et al. [31] discussed the impact of types, numbers, distributions, and strategies of guiders on the evacuation process and found that guiders who were easily identified by pedestrians were more beneficial for evacuation. Based on hazard prediction data, Choi and Chi [32] obtained the fastest escape route by using improved A∗ algorithm. Zhou et al. [33] proposed a hybrid bi-level mode to optimize numbers, initial distributions, and paths of leaders during the evacuation process.From what has been discussed above, it can be seen that a lot of work has been done about stairs evacuation and evacuation optimization. However, there are few studies on pedestrian evacuation optimization by transferring pedestrians between staircases. This paper aims to establish an optimization method for the cruise staircase evacuation. Firstly, the passenger evacuation model is constructed based on a multigrid model and the staircase evacuation process of passengers is simulated by an established evacuation model. Secondly, the evacuation efficiency of passengers is analysed through the simulation results. Thirdly, an optimization method is proposed in order to improve the evacuation efficiency of passengers, and quantum-inspired evolutionary algorithm and genetic algorithm are used to optimize the evacuation process. Taking the passenger density in the staircase area as the variable and the evacuation time as the objective function, the influence of the passenger density on the evacuation efficiency is studied to verify the feasibility of improving the evacuation efficiency by controlling the passenger density in the staircase. The rest of this paper is organized as follows: the staircase evacuation model of a cruise ship is constructed in Section2. Section 3 carries on the evacuation simulation of the cruise ship scenario and the simulation result is discussed. In Section 4, the staircase evacuation process is optimized by quantum-inspired evolutionary algorithm and genetic algorithm. The last section presents the conclusions.


## 2. The Multigrid Staircase Evacuation Model

In evacuation studies, the area occupied by pedestrians is generally described with a fixed size. Typical models are widely applied by researchers, which include the cellular automata model [34–36], the lattice gas model [37, 38], the social force model [39, 40], and the multigrid model [41–43]. The multigrid model is a kind of refined grid model, and one pedestrian occupies multiple grids. Therefore, this model can be more accurate to simulate the behavior of pedestrians. According to characteristics of passenger movement in the staircase, the passenger model and moving rules could be defined, and the stair evacuation model of the cruise ship would be established based on the multigrid model.


### 2.1. The Passenger Model

In an ergonomic design, the size of doors or passageways is generally based on the value of the 95 percentile [44]. The 95th percentile means 95% of the population is equal to or less than this size, and 5% of the population is greater than this size. According to the 95th percentile, the horizontal size of the human body is about 0.5 m × 0.3 m [45]. Each passenger occupies fifteen grids if the grid size is 0.1 m × 0.1 m, as shown in Figure 1. The red grid represents the centroid of one passenger. The passenger can move forward, left, and right in one step.Figure 1
The passenger model.


### 2.2. The Moving Range

Passengers will judge states of surrounding grids and move towards empty grids. It is assumed that maximum moving distance of the passenger is three grids per step, and the moving range can be defined in Figure2. The orange grids are movable areas of the passenger centroid. The moving range of passengers is generally referred to as the neighborhood area. As shown in Figure 3, there are three areas in moving directions of the passenger, and each area contains three-layer grids. If grids are occupied by other passengers or obstacles, the moving range is limited and becomes smaller, as shown in Figure 4.Figure 2
The moving range.Figure 3
The neighborhood area.Figure 4
The limited neighborhood area.The moving range is represented by the distance that the passenger moves between grids. The moving range can be defined as follows:(1)Pm=∑n=1Nm∏i=1n∑j∈LimCj+∑j∈LimCj2∑j∈LimCj,where m=mf,ml,mr is the moving direction of the passenger; mf, ml, and mr represents move forward, left, and right, respectively; Pm is the maximum number of grids that the passenger can move towards the direction of m; Nm represents the number of layers of grids in the direction of m; Lim is the set of neighborhood grids that are i-layer in distance from the centroid of the passenger in the direction of m, and Cj represents the state of grid, j (Cj is equal to 1 when grid j is empty or Cj is equal to −1 if grid j is occupied).When front grids are occupied, the passenger will go around obstacles and move to either side. The passenger will take into full consideration distributions of other passengers and obstacles in the view field, and then decide the moving direction and target location in the next step [46]. Therefore, the passengers will select a location with fewer people in left front or right front, and ensure that follow-on areas are not to be occupied. Considering the passenger’s choice, the neighborhood area should be expanded, as shown in Figure 5.Figure 5
The expanded neighborhood area.The moving behavior of passengers is determined by their states and neighborhood grids. Each neighborhood grid has its own moving probability, and the passenger will move to the grid that has the maximum moving probability. In terms of the extended neighborhood, the moving probability can be defined as follows:(2)gk,m=∑i=1k∑j∈LimCj+∑i=1k∑j∈LimCj∑i=1k2∑j∈LimCj,where gk,m represents the moving probability of the grid that is k-layer in distance from the centroid of the passenger in the direction of m, and k≤Pm.


### 2.3. The Turning Rule

When passengers arrive at the landing area of the staircase, they need to enter the step area by making a turn. Figure6 shows processes of turning left and right. As occupied grids by passengers have been changed between before and after turning, grids of the turning neighborhood should be empty so that passengers can complete a turn. Therefore, shaded parts in Figure 6 should be empty before passengers make a turn.Figure 6
The turning rule of passengers. (a) Turning left. (b) Turning right.
(a)(b)Figure7 shows the construction of the staircase in this paper. Taking left half of the staircase as an example, this area is divided into six portions. Da, Db, Dd, and De are landing areas, and Dc and Df are step areas. Arrows in Figure 7 indicate the descending direction of each area. During the process of evacuation, passengers tend to move towards the descending direction of areas where they are located. Then, the relationship between moving directions of passenger and descending directions can be defined as follows:(3)sinAt=Ao⟶×Di⟶Ao⟶|×|Di⟶,where Ao⟶ and Di⟶ are two unit vectors. Ao⟶ represents the moving direction of the passenger, Di⟶ represents the descending direction of the area where the passenger is located and i=a,b,c,d,e,f, and sinAt represents the relationship between Ao⟶ and Di⟶.Figure 7
The construction of the staircase.If the passenger makes a turn, the turning neighborhood should be empty. Then, the turning rule can be defined as follows:(4)T=0,sinAt=0,sinAt∑j∈StmCj+∑j∈StmCj2∑j∈StmCj,sinAt≠0,where T is the turning rule. The passenger would keep the current direction if T=0, turn left if T>0, and turn right if T<0. Stm represents the set of turning neighborhood grids.


### 2.4. The Moving Rule

The average walking speed of passengers in the staircase is about 0.60 m/s [47]. The update frequency of the model in this paper is 0.5 seconds, and the maximum moving distance of passengers is three grids per step. According to the structure of the staircase, the staircase can be divided into the step and landing area. As these two areas have different structural characteristics, different moving rules are set for these two areas, respectively.The moving rule on the step area is as follows:Step 1: calculatingPm and gk,m according to equations (1) and (2).Step 2: ifPmf=Pml=Pmr=0, then go to Step 3. Otherwise, go to Step 4.Step 3: the passenger will stay still.Step 4: ifPmf≠0, then go to Step 5. Otherwise, go to Step 6.Step 5: the passenger will movePmf grids forward.Step 6: passengers will move to the grid which has the maximum value ofgk,m.The moving rule on the landing area is as follows:Step 1: calculatingPm, gk,m, At, and T according to equations (1)–(4).Step 2: ifPmf=Pml=Pmr=0, then go to Step 3. Otherwise, go to Step 4.Step 3: the passenger will stay still.Step 4: ifAt=0, then go to Step 5. Otherwise, go to Step 8.Step 5: ifPmf≠0, then go to Step 6. Otherwise, go to Step 7.Step 6: the passenger will movePmf grids forward.Step 7: passengers will move to the grid which has the maximum value ofgk,m.Step 8: ifT=0, then go to Step 7. Otherwise, go to Step 9.Step 9: ifT>0, then go to Step 10. Otherwise, go to Step 11.Step 10: the passenger would turn left.Step 11: the passenger would turn right.


## 2.1. The Passenger Model

In an ergonomic design, the size of doors or passageways is generally based on the value of the 95 percentile [44]. The 95th percentile means 95% of the population is equal to or less than this size, and 5% of the population is greater than this size. According to the 95th percentile, the horizontal size of the human body is about 0.5 m × 0.3 m [45]. Each passenger occupies fifteen grids if the grid size is 0.1 m × 0.1 m, as shown in Figure 1. The red grid represents the centroid of one passenger. The passenger can move forward, left, and right in one step.Figure 1
The passenger model.


## 2.2. The Moving Range

Passengers will judge states of surrounding grids and move towards empty grids. It is assumed that maximum moving distance of the passenger is three grids per step, and the moving range can be defined in Figure2. The orange grids are movable areas of the passenger centroid. The moving range of passengers is generally referred to as the neighborhood area. As shown in Figure 3, there are three areas in moving directions of the passenger, and each area contains three-layer grids. If grids are occupied by other passengers or obstacles, the moving range is limited and becomes smaller, as shown in Figure 4.Figure 2
The moving range.Figure 3
The neighborhood area.Figure 4
The limited neighborhood area.The moving range is represented by the distance that the passenger moves between grids. The moving range can be defined as follows:(1)Pm=∑n=1Nm∏i=1n∑j∈LimCj+∑j∈LimCj2∑j∈LimCj,where m=mf,ml,mr is the moving direction of the passenger; mf, ml, and mr represents move forward, left, and right, respectively; Pm is the maximum number of grids that the passenger can move towards the direction of m; Nm represents the number of layers of grids in the direction of m; Lim is the set of neighborhood grids that are i-layer in distance from the centroid of the passenger in the direction of m, and Cj represents the state of grid, j (Cj is equal to 1 when grid j is empty or Cj is equal to −1 if grid j is occupied).When front grids are occupied, the passenger will go around obstacles and move to either side. The passenger will take into full consideration distributions of other passengers and obstacles in the view field, and then decide the moving direction and target location in the next step [46]. Therefore, the passengers will select a location with fewer people in left front or right front, and ensure that follow-on areas are not to be occupied. Considering the passenger’s choice, the neighborhood area should be expanded, as shown in Figure 5.Figure 5
The expanded neighborhood area.The moving behavior of passengers is determined by their states and neighborhood grids. Each neighborhood grid has its own moving probability, and the passenger will move to the grid that has the maximum moving probability. In terms of the extended neighborhood, the moving probability can be defined as follows:(2)gk,m=∑i=1k∑j∈LimCj+∑i=1k∑j∈LimCj∑i=1k2∑j∈LimCj,where gk,m represents the moving probability of the grid that is k-layer in distance from the centroid of the passenger in the direction of m, and k≤Pm.


## 2.3. The Turning Rule

When passengers arrive at the landing area of the staircase, they need to enter the step area by making a turn. Figure6 shows processes of turning left and right. As occupied grids by passengers have been changed between before and after turning, grids of the turning neighborhood should be empty so that passengers can complete a turn. Therefore, shaded parts in Figure 6 should be empty before passengers make a turn.Figure 6
The turning rule of passengers. (a) Turning left. (b) Turning right.
(a)(b)Figure7 shows the construction of the staircase in this paper. Taking left half of the staircase as an example, this area is divided into six portions. Da, Db, Dd, and De are landing areas, and Dc and Df are step areas. Arrows in Figure 7 indicate the descending direction of each area. During the process of evacuation, passengers tend to move towards the descending direction of areas where they are located. Then, the relationship between moving directions of passenger and descending directions can be defined as follows:(3)sinAt=Ao⟶×Di⟶Ao⟶|×|Di⟶,where Ao⟶ and Di⟶ are two unit vectors. Ao⟶ represents the moving direction of the passenger, Di⟶ represents the descending direction of the area where the passenger is located and i=a,b,c,d,e,f, and sinAt represents the relationship between Ao⟶ and Di⟶.Figure 7
The construction of the staircase.If the passenger makes a turn, the turning neighborhood should be empty. Then, the turning rule can be defined as follows:(4)T=0,sinAt=0,sinAt∑j∈StmCj+∑j∈StmCj2∑j∈StmCj,sinAt≠0,where T is the turning rule. The passenger would keep the current direction if T=0, turn left if T>0, and turn right if T<0. Stm represents the set of turning neighborhood grids.


## 2.4. The Moving Rule

The average walking speed of passengers in the staircase is about 0.60 m/s [47]. The update frequency of the model in this paper is 0.5 seconds, and the maximum moving distance of passengers is three grids per step. According to the structure of the staircase, the staircase can be divided into the step and landing area. As these two areas have different structural characteristics, different moving rules are set for these two areas, respectively.The moving rule on the step area is as follows:Step 1: calculatingPm and gk,m according to equations (1) and (2).Step 2: ifPmf=Pml=Pmr=0, then go to Step 3. Otherwise, go to Step 4.Step 3: the passenger will stay still.Step 4: ifPmf≠0, then go to Step 5. Otherwise, go to Step 6.Step 5: the passenger will movePmf grids forward.Step 6: passengers will move to the grid which has the maximum value ofgk,m.The moving rule on the landing area is as follows:Step 1: calculatingPm, gk,m, At, and T according to equations (1)–(4).Step 2: ifPmf=Pml=Pmr=0, then go to Step 3. Otherwise, go to Step 4.Step 3: the passenger will stay still.Step 4: ifAt=0, then go to Step 5. Otherwise, go to Step 8.Step 5: ifPmf≠0, then go to Step 6. Otherwise, go to Step 7.Step 6: the passenger will movePmf grids forward.Step 7: passengers will move to the grid which has the maximum value ofgk,m.Step 8: ifT=0, then go to Step 7. Otherwise, go to Step 9.Step 9: ifT>0, then go to Step 10. Otherwise, go to Step 11.Step 10: the passenger would turn left.Step 11: the passenger would turn right.


## 3. Evacuation Simulation and Results

### 3.1. The Evacuation Scenario

As shown in Figure8, the cruise ship is divided into seven main vertical zones (MVZ) along the longitudinal direction of the ship. Red lines are locations of main fire barriers (MFB) on cruise ships. Cabins and passengers are distributed on decks 2, 3, 4, 5, 6, 8, and 9. Deck 1 is the emergency assembly area. Green dotted boxes are positions where stairwells are located in the cruise ship. Stairs 1, 2, 3, 4, 5, and 6 connect deck 1 to deck 6, and stairs 7 connect deck1 to deck9. The initial positions of passengers are set according to the night evacuation scene of the IMO guideline [47]. In the initial stage, passengers are distributed within cabins of MVZ 1–6. The number of passengers to be evacuated in each MVZ is shown in Table 1. In the event of an accident, passengers would move to deck 1 through the stairs.Figure 8
Locations of passengers and staircases.Table 1
Initial distribution of passengers.
ItemNumbers of passengers to be evacuatedMVZ 1MVZ 2MVZ 3MVZ 4MVZ 5MVZ 6Deck 2986468425692Deck 3986468549072Deck 49864685490120Deck 59864685490112Deck 6866468529058Deck 8————2684Deck 9—————50Passengers enter the landing area from entrances on either side of the staircase, as shown in Figure9(a). After entering the staircase, passengers will merge with the passengers from the upper deck in the merging area, and then form a new crowd. All of the passengers will leave from exits of staircases on Deck 1. When counting the accumulative number of evacuated passengers in the staircase of one floor, the starting and ending areas of evacuation are as shown in Figure 9(a). Figure 9(b) is an example that shows the counting area of the staircase on Deck 2. Table 2 presents the height of each deck.Figure 9
Areas of the staircase. (a) Evacuation areas of the staircase. (b) The counting area of the staircase.
(a)(b)Table 2
The height of decks.
ItemHeight (m)Deck 14Deck 22.8Deck 32.8Deck 42.8Deck 52.8Deck 62.8Deck 73.2Deck 82.8Deck 92.8


### 3.2. The Discussion of Results

Passengers start moving from the initial position at the beginning of evacuation, and the evacuation is completed when all passengers leave from exits of staircases on Deck 1. Figures10(a)–10(i) give a snapshot of the evacuation simulation process, and Figure 10(j) is a legend that uses different colors to distinguish original decks where the passengers are located. It is can be seen from Figure 10 that passengers entered from both sides of the landing area and merged with others on the lower deck, then formed a new passenger flow and continued to go downstairs.Figure 10
A snapshot of the evacuation simulation process. (a) Deck 9. (b) Deck 8. (c) Deck 7. (d) Deck 6. (e) Deck 5. (f) Deck 4. (g) Deck 3. (h) Deck 2. (i) Deck 1. (j) Legend.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)Figure11 shows the evacuation time of stairs on each deck. It can be found that the evacuation time of stairs 7 on each deck is longer than other stairs. After 250 seconds, only stairs 7 had not completed evacuation. Figure 12 is the accumulative number of evacuated passengers changed with the evacuation time. Before 200 seconds, the curve slope is relatively large, which indicates that the number of evacuees per unit time was relatively higher. After 200 seconds, it can be seen from Figure 11 that most of the passengers in staircases had completed evacuation. At this time, the number of remaining passengers was less. Therefore, the number of passengers evacuated in unit time became less and the curve slope also became smaller. All of the passengers were evacuated in 496 seconds.Figure 11
Evacuation time of stairs on each deck.Figure 12
The accumulative number of evacuated passengers changed with the evacuation time.Figures13 and 14 show that the accumulative number of passengers that entered and left stairs on each deck changed with the evacuation time, respectively. From these figures, it can be found that the accumulative number of passengers showed no difference in the same stairs of the same deck. However, the evacuation time of reaching the same accumulative number of passengers was different. The time point in Figure 14 was behind Figure 13 if accumulative number of passengers in two figures reached the same number, which indicated that passengers needed a certain time to leave stairs after they entered the stairs, and this phenomenon was completely consistent with the reality.Figure 13
The accumulative number of passengers that entered the stairs. (a) Stairs 1. (b) Stairs 2. (c) Stairs 3. (d) Stairs 4. (e) Stairs 5. (f) Stairs 6. (g) Stairs 7.
(a)(b)(c)(d)(e)(f)(g)Figure 14
The accumulative number of passengers that left the stairs. (a) Stairs 1. (b) Stairs 2. (c) Stairs 3. (d) Stairs 4. (e) Stairs 5. (f) Stairs 6. (g) Stairs 7.
(a)(b)(c)(d)(e)(f)(g)Slopes of curves in Figure13 were larger within 30 seconds. After 30 seconds, the curve slope began to decrease, which indicated passengers that entered the stairs per unit time became smaller. It means that in the early stage of evacuation, stairs were clear and passengers could enter the stairs quickly. With the progress of evacuation, the passenger density of staircases increased gradually. External passengers would merge with the existing ones in the stairs when they entered the stairs. Merging behavior would reduce the walking speed of passengers; then, the number of passengers that entered the stairs per unit time was also decreased.The slope of end curves of Decks 2 and 3 in Figures14(c) and 14(e) became smaller, which indicated that more passengers were evacuated in the early stage of evacuation in these areas. The number of remaining passengers was less, and fewer people were evacuated per unit time during the later stage of evacuation. The curve slopes of decks in Figures 14(a), 14(b), 14(d), and 14(f) had little change between the beginning and end of curves, showing that the passenger flow in these stairs were balanced and numbers of evacuated passengers were evenly distributed in the whole stage of evacuation.Figures13(g) and 14(g) show the evacuation process of stairs 7; the number of passengers that entered and left the staircase of Deck 7 was the same as that of Deck 8. As can be seen from Table 1, there was no passenger distribution on Deck 7 at the beginning of evacuation. Therefore, the accumulative number of passengers of stairs 7 on Deck 7 did not increase. There are several ranges where slopes of curves are zero, and these ranges contain among Deck 5 between 63 and 84 seconds in Figure 13(g), Deck 6 between 85 and 115 seconds in Figure 13(g), Deck 5 between 70 and 90 seconds in Figure 14(g), Deck 6 between 60 and 120 seconds in Figure 14(g) and Deck 7 between 80 and 116 seconds in Figure 14(g). The curve slope of these ranges is zero, indicating that the accumulative number of passengers that entered and left stairs was not changed during the corresponding time of these ranges.The reason of this phenomenon would be discussed from the time point passengers entering and leaving the stairs. In Figure13(g), passengers whose initial positions were located on Decks 8 and 9 entered stairs 7 within 100 seconds. In Figure 14(g), these passengers left stairs 7 on Decks 8 and 9 within 100 seconds, showing that these passengers entered stairs 7 on Deck 7 before 100 seconds. In addition, there is no range where curve slope is zero on Deck 7 in Figure 13(g). Therefore, the passenger flow was continuous when passengers entered stairs 7 on Deck 7 from Decks 8 and 9. However, the increase of accumulative number of passengers that left Stairs 7 on Deck 7 was stagnation, which indicated that stairs 7 on the lower deck were congested and made it impossible for passengers to descend.There was congestion time in both periods of passengers entering and leaving stairs 7 on Deck 6. After 60 seconds, passengers in stairs 7 on Deck 6 could not leave due to the congestion and external passengers also could not enter stairs 7 after 85 seconds. It means that there was still free space in stairs 7 on Deck 6 between 60 and 85 seconds, and passengers on Deck 6 and upper decks could enter stairs 7 during this time. However, stairs 7 on the lower deck was so congested that passengers already in stairs 7 could not move to the lower deck from the staircase.No passengers left stairs 7 on Deck 7 after 80 seconds, showing that the mid-landing area of Deck 6 was jammed after 80 seconds, which caused passengers from upper decks to be unable to enter stairs 7 on Deck 6. The landing area of Deck 6 was blocked after 85 seconds, which made it impossible for external passengers to enter stairs 7 through the entrance on Deck 6. From the above analysis, it can be found that stairs 7 was jammed due to too many passengers from external and upper stairs during a period of time. The consequences were external passengers could not enter the stairs, while passengers inside could not leave. Therefore, a serious congestion occurred in stairs 7 during the above several ranges where slopes of curves are zero.Figure15 is the statistics of the evacuated passengers from each staircase. It can be seen that the number of evacuated passengers from stairs 7 is about 2 or 3 times higher than other stairs. There were two main reasons for this phenomenon. Firstly, passengers on Decks 8 and 9 could only descend through stairs 7, so the vertical passenger flow was larger than other stairs. The second reason was related with the layout of the evacuation scenario. As can be seen from Table 1 and Figure 8, the number of passengers in MVZ 6 is the largest. However, there was only one staircase on the left side of this area, while staircases were distributed on both sides of other areas. Therefore, more passengers entered stairs 7 than other staircases, which exceeded the evacuation capacity of stairs 7 and caused a serious blockage. It can be found from the above analysis that stairs 7 was blocked due to an unbalanced number of evacuated passengers between staircases, which lead to the increase of the evacuation time.Figure 15
The evacuated number of passengers from each staircase.


## 3.1. The Evacuation Scenario

As shown in Figure8, the cruise ship is divided into seven main vertical zones (MVZ) along the longitudinal direction of the ship. Red lines are locations of main fire barriers (MFB) on cruise ships. Cabins and passengers are distributed on decks 2, 3, 4, 5, 6, 8, and 9. Deck 1 is the emergency assembly area. Green dotted boxes are positions where stairwells are located in the cruise ship. Stairs 1, 2, 3, 4, 5, and 6 connect deck 1 to deck 6, and stairs 7 connect deck1 to deck9. The initial positions of passengers are set according to the night evacuation scene of the IMO guideline [47]. In the initial stage, passengers are distributed within cabins of MVZ 1–6. The number of passengers to be evacuated in each MVZ is shown in Table 1. In the event of an accident, passengers would move to deck 1 through the stairs.Figure 8
Locations of passengers and staircases.Table 1
Initial distribution of passengers.
ItemNumbers of passengers to be evacuatedMVZ 1MVZ 2MVZ 3MVZ 4MVZ 5MVZ 6Deck 2986468425692Deck 3986468549072Deck 49864685490120Deck 59864685490112Deck 6866468529058Deck 8————2684Deck 9—————50Passengers enter the landing area from entrances on either side of the staircase, as shown in Figure9(a). After entering the staircase, passengers will merge with the passengers from the upper deck in the merging area, and then form a new crowd. All of the passengers will leave from exits of staircases on Deck 1. When counting the accumulative number of evacuated passengers in the staircase of one floor, the starting and ending areas of evacuation are as shown in Figure 9(a). Figure 9(b) is an example that shows the counting area of the staircase on Deck 2. Table 2 presents the height of each deck.Figure 9
Areas of the staircase. (a) Evacuation areas of the staircase. (b) The counting area of the staircase.
(a)(b)Table 2
The height of decks.
ItemHeight (m)Deck 14Deck 22.8Deck 32.8Deck 42.8Deck 52.8Deck 62.8Deck 73.2Deck 82.8Deck 92.8


## 3.2. The Discussion of Results

Passengers start moving from the initial position at the beginning of evacuation, and the evacuation is completed when all passengers leave from exits of staircases on Deck 1. Figures10(a)–10(i) give a snapshot of the evacuation simulation process, and Figure 10(j) is a legend that uses different colors to distinguish original decks where the passengers are located. It is can be seen from Figure 10 that passengers entered from both sides of the landing area and merged with others on the lower deck, then formed a new passenger flow and continued to go downstairs.Figure 10
A snapshot of the evacuation simulation process. (a) Deck 9. (b) Deck 8. (c) Deck 7. (d) Deck 6. (e) Deck 5. (f) Deck 4. (g) Deck 3. (h) Deck 2. (i) Deck 1. (j) Legend.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)Figure11 shows the evacuation time of stairs on each deck. It can be found that the evacuation time of stairs 7 on each deck is longer than other stairs. After 250 seconds, only stairs 7 had not completed evacuation. Figure 12 is the accumulative number of evacuated passengers changed with the evacuation time. Before 200 seconds, the curve slope is relatively large, which indicates that the number of evacuees per unit time was relatively higher. After 200 seconds, it can be seen from Figure 11 that most of the passengers in staircases had completed evacuation. At this time, the number of remaining passengers was less. Therefore, the number of passengers evacuated in unit time became less and the curve slope also became smaller. All of the passengers were evacuated in 496 seconds.Figure 11
Evacuation time of stairs on each deck.Figure 12
The accumulative number of evacuated passengers changed with the evacuation time.Figures13 and 14 show that the accumulative number of passengers that entered and left stairs on each deck changed with the evacuation time, respectively. From these figures, it can be found that the accumulative number of passengers showed no difference in the same stairs of the same deck. However, the evacuation time of reaching the same accumulative number of passengers was different. The time point in Figure 14 was behind Figure 13 if accumulative number of passengers in two figures reached the same number, which indicated that passengers needed a certain time to leave stairs after they entered the stairs, and this phenomenon was completely consistent with the reality.Figure 13
The accumulative number of passengers that entered the stairs. (a) Stairs 1. (b) Stairs 2. (c) Stairs 3. (d) Stairs 4. (e) Stairs 5. (f) Stairs 6. (g) Stairs 7.
(a)(b)(c)(d)(e)(f)(g)Figure 14
The accumulative number of passengers that left the stairs. (a) Stairs 1. (b) Stairs 2. (c) Stairs 3. (d) Stairs 4. (e) Stairs 5. (f) Stairs 6. (g) Stairs 7.
(a)(b)(c)(d)(e)(f)(g)Slopes of curves in Figure13 were larger within 30 seconds. After 30 seconds, the curve slope began to decrease, which indicated passengers that entered the stairs per unit time became smaller. It means that in the early stage of evacuation, stairs were clear and passengers could enter the stairs quickly. With the progress of evacuation, the passenger density of staircases increased gradually. External passengers would merge with the existing ones in the stairs when they entered the stairs. Merging behavior would reduce the walking speed of passengers; then, the number of passengers that entered the stairs per unit time was also decreased.The slope of end curves of Decks 2 and 3 in Figures14(c) and 14(e) became smaller, which indicated that more passengers were evacuated in the early stage of evacuation in these areas. The number of remaining passengers was less, and fewer people were evacuated per unit time during the later stage of evacuation. The curve slopes of decks in Figures 14(a), 14(b), 14(d), and 14(f) had little change between the beginning and end of curves, showing that the passenger flow in these stairs were balanced and numbers of evacuated passengers were evenly distributed in the whole stage of evacuation.Figures13(g) and 14(g) show the evacuation process of stairs 7; the number of passengers that entered and left the staircase of Deck 7 was the same as that of Deck 8. As can be seen from Table 1, there was no passenger distribution on Deck 7 at the beginning of evacuation. Therefore, the accumulative number of passengers of stairs 7 on Deck 7 did not increase. There are several ranges where slopes of curves are zero, and these ranges contain among Deck 5 between 63 and 84 seconds in Figure 13(g), Deck 6 between 85 and 115 seconds in Figure 13(g), Deck 5 between 70 and 90 seconds in Figure 14(g), Deck 6 between 60 and 120 seconds in Figure 14(g) and Deck 7 between 80 and 116 seconds in Figure 14(g). The curve slope of these ranges is zero, indicating that the accumulative number of passengers that entered and left stairs was not changed during the corresponding time of these ranges.The reason of this phenomenon would be discussed from the time point passengers entering and leaving the stairs. In Figure13(g), passengers whose initial positions were located on Decks 8 and 9 entered stairs 7 within 100 seconds. In Figure 14(g), these passengers left stairs 7 on Decks 8 and 9 within 100 seconds, showing that these passengers entered stairs 7 on Deck 7 before 100 seconds. In addition, there is no range where curve slope is zero on Deck 7 in Figure 13(g). Therefore, the passenger flow was continuous when passengers entered stairs 7 on Deck 7 from Decks 8 and 9. However, the increase of accumulative number of passengers that left Stairs 7 on Deck 7 was stagnation, which indicated that stairs 7 on the lower deck were congested and made it impossible for passengers to descend.There was congestion time in both periods of passengers entering and leaving stairs 7 on Deck 6. After 60 seconds, passengers in stairs 7 on Deck 6 could not leave due to the congestion and external passengers also could not enter stairs 7 after 85 seconds. It means that there was still free space in stairs 7 on Deck 6 between 60 and 85 seconds, and passengers on Deck 6 and upper decks could enter stairs 7 during this time. However, stairs 7 on the lower deck was so congested that passengers already in stairs 7 could not move to the lower deck from the staircase.No passengers left stairs 7 on Deck 7 after 80 seconds, showing that the mid-landing area of Deck 6 was jammed after 80 seconds, which caused passengers from upper decks to be unable to enter stairs 7 on Deck 6. The landing area of Deck 6 was blocked after 85 seconds, which made it impossible for external passengers to enter stairs 7 through the entrance on Deck 6. From the above analysis, it can be found that stairs 7 was jammed due to too many passengers from external and upper stairs during a period of time. The consequences were external passengers could not enter the stairs, while passengers inside could not leave. Therefore, a serious congestion occurred in stairs 7 during the above several ranges where slopes of curves are zero.Figure15 is the statistics of the evacuated passengers from each staircase. It can be seen that the number of evacuated passengers from stairs 7 is about 2 or 3 times higher than other stairs. There were two main reasons for this phenomenon. Firstly, passengers on Decks 8 and 9 could only descend through stairs 7, so the vertical passenger flow was larger than other stairs. The second reason was related with the layout of the evacuation scenario. As can be seen from Table 1 and Figure 8, the number of passengers in MVZ 6 is the largest. However, there was only one staircase on the left side of this area, while staircases were distributed on both sides of other areas. Therefore, more passengers entered stairs 7 than other staircases, which exceeded the evacuation capacity of stairs 7 and caused a serious blockage. It can be found from the above analysis that stairs 7 was blocked due to an unbalanced number of evacuated passengers between staircases, which lead to the increase of the evacuation time.Figure 15
The evacuated number of passengers from each staircase.


## 4. Evacuation Optimization

In order to shorten the evacuation time, it is necessary to optimize the process of evacuation and transfer passengers from the blocked stairs to other stairs through the passage. Then, the evacuated number of passengers between the stairs would be balanced and the evacuation efficiency would be also improved.


### 4.1. The Optimization Objective

Taking stairs 6 and 7 as an example to discuss the process of evacuation optimization, the distance between the two staircases is about 48 meters. Assuming that the walking speed of passengers in the passage is 1.2 m/s, it will take 40 seconds for them to completely transfer from one staircase to the other. Moreover, the transferring should also consider the proportion of passengers between the upper and lower staircases. If too many passengers are transferred from the staircase on the upper deck in a short time, they would merge with others on the lower deck, which would lead to congestion.In order to determine the appropriate opportunity for transferring, the passenger density in the staircase is used as a threshold. Outside threshold is defined as the critical density of passengers in the stair area when passengers should be transferred. Meanwhile, in order to avoid passengers that enter the staircase early, the inside threshold is defined as the critical density of a staircase that passengers are transferred into. When the passenger density in the one staircase is greater than the outside threshold and the passenger density in the other staircase is smaller than the inside threshold, the transferring of passengers between two staircases is started.The evacuation is a dynamic process, and the passenger density in the staircase changes at every moment. It is difficult to directly obtain thresholds; therefore, evolutionary algorithms are used to obtain optimal thresholds. The evolutionary algorithm is a global optimization method with wide applicability. Classical evolutionary algorithms include the genetic algorithm (GA) and some improved evolutionary algorithms [48, 49], which have been applied in many fields. The quantum-inspired evolutionary algorithm (QEA) is a polymorphic evolutionary algorithm, which has the characteristics of wide search space, and some researchers use it for engineering optimizations [50–52]. In this paper, GA and QEA are used to obtain inside and outside thresholds, and then the impact of thresholds on the evacuation process would be discussed.The concrete steps of optimization are as follows: Firstly, two algorithms are used to generate populations and chromosomes’ corresponding thresholds would be substituted into the established staircase evacuation model. Secondly, thresholds are applied to control the timing of pedestrians transferred between staircases in the evacuation simulation. Thirdly, evacuation times can be obtained by the evacuation simulation, and evacuation times would be returned into the algorithms as fitness values of chromosomes. Finally, the optimal evacuation time can be obtained through iterative calculations.


### 4.2. The Optimization of the Staircase Evacuation Process

As the size of the cruise ship is large, passengers’ transfer between staircases will take up time. Undoubtedly, the transferring time could be shortened if passengers transfer between adjacent staircases. Since Decks 2 and 3 are close to Deck 1 and the heights of these two decks are smaller, the descending time is short. If passengers transfer between staircases on these two decks, the evacuation time may be increased. Therefore, Decks 4, 5, and 6 are selected as the experimental object, and passengers in stairs 7 would be transferred out and then these passengers would be transferred into stairs 6.


#### 4.2.1. Operation Steps of QEA

Let the number of iterations betmax, and the operation steps of QEA be as follows:Step 1: initializing the generation countert=0.Step 2: the population size is set as 20. Initializing the populationQ0, let qubits of all chromosomes in Q0 be 1/2, which means each chromosome is in the same probability of superposition states.Step 3: observation of the qubit state. The solution of generationPQ0 can be obtained by observing Q0.Step 4: the threshold is obtained by converting the binary code of chromosomes inPQ0 into real numbers, and calculating the evacuation time by substituting the threshold into the evacuation model. The evacuation time is taken as the fitness value of the chromosome, and the chromosome with the minimum fitness value was assigned to the optimal chromosome Xbest.Step 5:t=t+1.Step 6: updating qubits. Comparing the solution ofPQt−1 with the current optimal solution, then the population Qt can be obtained by updating Qt−1 with the quantum rotation gate.Step 7: observation of the qubit state.PQt is generated by observing the Qt.Step 8: fitness values of generationt are obtained by PQt. The chromosome with the minimum fitness value of population in generation t is expressed as Xbestt, and Xbest=Xbestt if the fitness value of Xbestt is smaller than Xbest.Step 9: ifXbest have no improvement in the continuous nc generations, two probabilities of each qubit are exchanged to complete the mutation operation.Step 10: ift<tmax, then go to Step 5. Otherwise, output the current optimal chromosome and the minimum evacuation time.


#### 4.2.2. Operation Steps of GA

Let the number of iterations betmax′, and the operation steps of GA be as follows:Step 1: initializing the generation countert′=0.Step 2: randomly generate an initial populationPG0, the size of which is 20.Step 3: the threshold is obtained by converting the binary code of chromosomes inPG0 into real numbers, and calculating the evacuation time by substituting the threshold into the evacuation model. The evacuation time is taken as the fitness value of the chromosome, and the chromosome with the minimum fitness value was assigned to the optimal chromosome Ybest.Step 4:t′=t′+1.Step 5: the population is selected by using a roulette wheel.Step 6: the crossover probability is set as 0.4, and crossover operation is used to population and the fitness of chromosomes are calculated.Step 7: the mutation probability is set as 0.1, and mutation operation is used to population and the fitness of chromosomes are calculated.Step 8: the populationPGt of generation t′ can be obtained by crossover and mutation operation. The chromosome with minimum fitness value of population in generation t′ is expressed as Ybestt′, and Ybest=Ybestt′ If Ybestt′<Ybest.Step 9: go to Step5 Ift′<tmax′, else output the current optimal chromosome and the minimum evacuation time.


### 4.3. The Analysis of Optimization Results

The experiments are done on Inter i5-2320, 3.00 GHz. The machine uses Windows 10 as the operating system, and the program was developed by VC ++. The number of iterations is set as 100, and optimization time is about 87 hours for QEA and 73 hours for GA.Figure16 presents optimization results of the evacuation. The minimum evacuation time of the QEA result is 375 seconds, and that of the GA result is 387.5 seconds, which are 24.4% and 21.9% less than the origin result, respectively. As the search space of the QEA is larger [53], the final convergence value is slightly better than that of the GA, and the minimum evacuation time is 3.2% lesser than that of the GA.Figure 16
Optimization results of the evacuation.Figure17 shows changes of numbers of evacuated passengers in stairs 6 and 7. In the initial stage of iterations, the difference of numbers of evacuees between two staircases was relatively large, which indicated that the number of evacuees between the two staircases was unbalanced. With the increase of iterations, the difference between the two staircases gradually shrunk and the evacuation time also decreased. In results of QEA, the difference in the number of evacuees reached the minimum in the 45th generation. For results of GA, the difference in the number of evacuees reached the minimum in the 43rd generation. However, the evacuation time was not the minimum. It can be inferred that the evacuation time was not linearly related to the difference of the number of evacuees between two staircases, and it also might be related to the timing of passengers that transferred between the two staircases.Figure 17
Changes of numbers of evacuated passengers in stairs 6 and 7.Figure18 shows the changes of outside and inside thresholds with iterations. For both QEA and GA results, it can be found that the inside threshold on the same deck is greater than the outside threshold with the decreasing of the evacuation time. It indicates that when passengers from stairs 7 were moving through the passage, other passengers that were already in stairs 6 were going downstairs to the next deck. When passengers from stairs 7 arrived at stairs 6, stairs 6 has spare space for additional passengers from stairs 7. At the same time, stairs 7 would also be congested due to some passengers being transferred out.Figure 18
Changes of outside and inside thresholds. (a) The result of QEA. (b) The result of GA.
(a)(b)The QEA result is taken as an example to analyse the process of passengers that transferred between stairs. Figures19 and 20 show the accumulative number of passengers that entered and left stairs in the QEA result. There are two ranges of curve slopes of Deck 6 are zero, the one is in the range of 39 to 60 seconds of Deck 6 curve in Figure 19(a) and the other is in the range of 61 to 87 seconds of Deck 6 curve in Figure 20(a). The accumulative number of passengers that entered Deck 6 at 39 seconds was 84, which was the same as the total number of passengers that entered Deck 6 in Figure 13(f). It can be seen from Figure 20(a) that these 84 passengers had left stairs 6 on Deck 6 before 61 seconds, and stairs 6 on Deck 6 was completely empty at 61 seconds. Another 30 passengers entered after 60 seconds, so these additional passengers should be transferred from stairs 7. Therefore, the reason for curve slope being zero in these two ranges is that passengers were under the process of transferring and had not entered the staircase. Unlike Figure 13(g), this is not due to the congestion. Comparing with Figure 14(f), the accumulative number of passengers in stairs 6 on each deck has an increase in Figure 20(a). In contrast to stairs 6, the accumulative number of passengers in stairs 7 on each deck in Figure 20(b) is lower than that in Figure 14(g). This means that some passengers in stairs 7 had transferred to stairs 6. In addition, stairs 7 was only blocked for a short time on Deck 5, while the evacuation process on others decks was smooth. Evacuation efficiency was significantly improved compared to Figure 14(g).Figure 19
The accumulative number of passengers that entered the stairs in the QEA result. (a) Stairs 6. (b) Stairs 7.
(a)(b)Figure 20
The accumulative number of passengers that left the stairs in the QEA result. (a) Stairs 6. (b) Stairs 7.
(a)(b)As shown in Figure21, the curve slope of QEA and GA began to increase gradually compared to the origin curve after 100 seconds. It means that passengers had started to transfer between staircases before 100 seconds, and the number of passengers who achieved evacuation was more than the origin result. After 200 seconds, slopes of QEA and GA curves were significantly greater than that of the origin curve, indicating that passengers’ transfer behavior had a greater impact on the evacuation process and the number of evacuated passengers increased per unit time. Through the above analysis, it can be seen that the appropriate passenger transfer timing can be obtained by the threshold optimization. The evacuation efficiency of passengers changes with the inside and outside thresholds. When the inside threshold is greater than the outside threshold and both of them are within an appropriate range, the evacuation time of passengers can be significantly shortened and the evacuation efficiency also could be improved.Figure 21
Evacuation results before and after optimization.


## 4.1. The Optimization Objective

Taking stairs 6 and 7 as an example to discuss the process of evacuation optimization, the distance between the two staircases is about 48 meters. Assuming that the walking speed of passengers in the passage is 1.2 m/s, it will take 40 seconds for them to completely transfer from one staircase to the other. Moreover, the transferring should also consider the proportion of passengers between the upper and lower staircases. If too many passengers are transferred from the staircase on the upper deck in a short time, they would merge with others on the lower deck, which would lead to congestion.In order to determine the appropriate opportunity for transferring, the passenger density in the staircase is used as a threshold. Outside threshold is defined as the critical density of passengers in the stair area when passengers should be transferred. Meanwhile, in order to avoid passengers that enter the staircase early, the inside threshold is defined as the critical density of a staircase that passengers are transferred into. When the passenger density in the one staircase is greater than the outside threshold and the passenger density in the other staircase is smaller than the inside threshold, the transferring of passengers between two staircases is started.The evacuation is a dynamic process, and the passenger density in the staircase changes at every moment. It is difficult to directly obtain thresholds; therefore, evolutionary algorithms are used to obtain optimal thresholds. The evolutionary algorithm is a global optimization method with wide applicability. Classical evolutionary algorithms include the genetic algorithm (GA) and some improved evolutionary algorithms [48, 49], which have been applied in many fields. The quantum-inspired evolutionary algorithm (QEA) is a polymorphic evolutionary algorithm, which has the characteristics of wide search space, and some researchers use it for engineering optimizations [50–52]. In this paper, GA and QEA are used to obtain inside and outside thresholds, and then the impact of thresholds on the evacuation process would be discussed.The concrete steps of optimization are as follows: Firstly, two algorithms are used to generate populations and chromosomes’ corresponding thresholds would be substituted into the established staircase evacuation model. Secondly, thresholds are applied to control the timing of pedestrians transferred between staircases in the evacuation simulation. Thirdly, evacuation times can be obtained by the evacuation simulation, and evacuation times would be returned into the algorithms as fitness values of chromosomes. Finally, the optimal evacuation time can be obtained through iterative calculations.


## 4.2. The Optimization of the Staircase Evacuation Process

As the size of the cruise ship is large, passengers’ transfer between staircases will take up time. Undoubtedly, the transferring time could be shortened if passengers transfer between adjacent staircases. Since Decks 2 and 3 are close to Deck 1 and the heights of these two decks are smaller, the descending time is short. If passengers transfer between staircases on these two decks, the evacuation time may be increased. Therefore, Decks 4, 5, and 6 are selected as the experimental object, and passengers in stairs 7 would be transferred out and then these passengers would be transferred into stairs 6.


### 4.2.1. Operation Steps of QEA

Let the number of iterations betmax, and the operation steps of QEA be as follows:Step 1: initializing the generation countert=0.Step 2: the population size is set as 20. Initializing the populationQ0, let qubits of all chromosomes in Q0 be 1/2, which means each chromosome is in the same probability of superposition states.Step 3: observation of the qubit state. The solution of generationPQ0 can be obtained by observing Q0.Step 4: the threshold is obtained by converting the binary code of chromosomes inPQ0 into real numbers, and calculating the evacuation time by substituting the threshold into the evacuation model. The evacuation time is taken as the fitness value of the chromosome, and the chromosome with the minimum fitness value was assigned to the optimal chromosome Xbest.Step 5:t=t+1.Step 6: updating qubits. Comparing the solution ofPQt−1 with the current optimal solution, then the population Qt can be obtained by updating Qt−1 with the quantum rotation gate.Step 7: observation of the qubit state.PQt is generated by observing the Qt.Step 8: fitness values of generationt are obtained by PQt. The chromosome with the minimum fitness value of population in generation t is expressed as Xbestt, and Xbest=Xbestt if the fitness value of Xbestt is smaller than Xbest.Step 9: ifXbest have no improvement in the continuous nc generations, two probabilities of each qubit are exchanged to complete the mutation operation.Step 10: ift<tmax, then go to Step 5. Otherwise, output the current optimal chromosome and the minimum evacuation time.


### 4.2.2. Operation Steps of GA

Let the number of iterations betmax′, and the operation steps of GA be as follows:Step 1: initializing the generation countert′=0.Step 2: randomly generate an initial populationPG0, the size of which is 20.Step 3: the threshold is obtained by converting the binary code of chromosomes inPG0 into real numbers, and calculating the evacuation time by substituting the threshold into the evacuation model. The evacuation time is taken as the fitness value of the chromosome, and the chromosome with the minimum fitness value was assigned to the optimal chromosome Ybest.Step 4:t′=t′+1.Step 5: the population is selected by using a roulette wheel.Step 6: the crossover probability is set as 0.4, and crossover operation is used to population and the fitness of chromosomes are calculated.Step 7: the mutation probability is set as 0.1, and mutation operation is used to population and the fitness of chromosomes are calculated.Step 8: the populationPGt of generation t′ can be obtained by crossover and mutation operation. The chromosome with minimum fitness value of population in generation t′ is expressed as Ybestt′, and Ybest=Ybestt′ If Ybestt′<Ybest.Step 9: go to Step5 Ift′<tmax′, else output the current optimal chromosome and the minimum evacuation time.


## 4.2.1. Operation Steps of QEA

Let the number of iterations betmax, and the operation steps of QEA be as follows:Step 1: initializing the generation countert=0.Step 2: the population size is set as 20. Initializing the populationQ0, let qubits of all chromosomes in Q0 be 1/2, which means each chromosome is in the same probability of superposition states.Step 3: observation of the qubit state. The solution of generationPQ0 can be obtained by observing Q0.Step 4: the threshold is obtained by converting the binary code of chromosomes inPQ0 into real numbers, and calculating the evacuation time by substituting the threshold into the evacuation model. The evacuation time is taken as the fitness value of the chromosome, and the chromosome with the minimum fitness value was assigned to the optimal chromosome Xbest.Step 5:t=t+1.Step 6: updating qubits. Comparing the solution ofPQt−1 with the current optimal solution, then the population Qt can be obtained by updating Qt−1 with the quantum rotation gate.Step 7: observation of the qubit state.PQt is generated by observing the Qt.Step 8: fitness values of generationt are obtained by PQt. The chromosome with the minimum fitness value of population in generation t is expressed as Xbestt, and Xbest=Xbestt if the fitness value of Xbestt is smaller than Xbest.Step 9: ifXbest have no improvement in the continuous nc generations, two probabilities of each qubit are exchanged to complete the mutation operation.Step 10: ift<tmax, then go to Step 5. Otherwise, output the current optimal chromosome and the minimum evacuation time.


## 4.2.2. Operation Steps of GA

Let the number of iterations betmax′, and the operation steps of GA be as follows:Step 1: initializing the generation countert′=0.Step 2: randomly generate an initial populationPG0, the size of which is 20.Step 3: the threshold is obtained by converting the binary code of chromosomes inPG0 into real numbers, and calculating the evacuation time by substituting the threshold into the evacuation model. The evacuation time is taken as the fitness value of the chromosome, and the chromosome with the minimum fitness value was assigned to the optimal chromosome Ybest.Step 4:t′=t′+1.Step 5: the population is selected by using a roulette wheel.Step 6: the crossover probability is set as 0.4, and crossover operation is used to population and the fitness of chromosomes are calculated.Step 7: the mutation probability is set as 0.1, and mutation operation is used to population and the fitness of chromosomes are calculated.Step 8: the populationPGt of generation t′ can be obtained by crossover and mutation operation. The chromosome with minimum fitness value of population in generation t′ is expressed as Ybestt′, and Ybest=Ybestt′ If Ybestt′<Ybest.Step 9: go to Step5 Ift′<tmax′, else output the current optimal chromosome and the minimum evacuation time.


## 4.3. The Analysis of Optimization Results

The experiments are done on Inter i5-2320, 3.00 GHz. The machine uses Windows 10 as the operating system, and the program was developed by VC ++. The number of iterations is set as 100, and optimization time is about 87 hours for QEA and 73 hours for GA.Figure16 presents optimization results of the evacuation. The minimum evacuation time of the QEA result is 375 seconds, and that of the GA result is 387.5 seconds, which are 24.4% and 21.9% less than the origin result, respectively. As the search space of the QEA is larger [53], the final convergence value is slightly better than that of the GA, and the minimum evacuation time is 3.2% lesser than that of the GA.Figure 16
Optimization results of the evacuation.Figure17 shows changes of numbers of evacuated passengers in stairs 6 and 7. In the initial stage of iterations, the difference of numbers of evacuees between two staircases was relatively large, which indicated that the number of evacuees between the two staircases was unbalanced. With the increase of iterations, the difference between the two staircases gradually shrunk and the evacuation time also decreased. In results of QEA, the difference in the number of evacuees reached the minimum in the 45th generation. For results of GA, the difference in the number of evacuees reached the minimum in the 43rd generation. However, the evacuation time was not the minimum. It can be inferred that the evacuation time was not linearly related to the difference of the number of evacuees between two staircases, and it also might be related to the timing of passengers that transferred between the two staircases.Figure 17
Changes of numbers of evacuated passengers in stairs 6 and 7.Figure18 shows the changes of outside and inside thresholds with iterations. For both QEA and GA results, it can be found that the inside threshold on the same deck is greater than the outside threshold with the decreasing of the evacuation time. It indicates that when passengers from stairs 7 were moving through the passage, other passengers that were already in stairs 6 were going downstairs to the next deck. When passengers from stairs 7 arrived at stairs 6, stairs 6 has spare space for additional passengers from stairs 7. At the same time, stairs 7 would also be congested due to some passengers being transferred out.Figure 18
Changes of outside and inside thresholds. (a) The result of QEA. (b) The result of GA.
(a)(b)The QEA result is taken as an example to analyse the process of passengers that transferred between stairs. Figures19 and 20 show the accumulative number of passengers that entered and left stairs in the QEA result. There are two ranges of curve slopes of Deck 6 are zero, the one is in the range of 39 to 60 seconds of Deck 6 curve in Figure 19(a) and the other is in the range of 61 to 87 seconds of Deck 6 curve in Figure 20(a). The accumulative number of passengers that entered Deck 6 at 39 seconds was 84, which was the same as the total number of passengers that entered Deck 6 in Figure 13(f). It can be seen from Figure 20(a) that these 84 passengers had left stairs 6 on Deck 6 before 61 seconds, and stairs 6 on Deck 6 was completely empty at 61 seconds. Another 30 passengers entered after 60 seconds, so these additional passengers should be transferred from stairs 7. Therefore, the reason for curve slope being zero in these two ranges is that passengers were under the process of transferring and had not entered the staircase. Unlike Figure 13(g), this is not due to the congestion. Comparing with Figure 14(f), the accumulative number of passengers in stairs 6 on each deck has an increase in Figure 20(a). In contrast to stairs 6, the accumulative number of passengers in stairs 7 on each deck in Figure 20(b) is lower than that in Figure 14(g). This means that some passengers in stairs 7 had transferred to stairs 6. In addition, stairs 7 was only blocked for a short time on Deck 5, while the evacuation process on others decks was smooth. Evacuation efficiency was significantly improved compared to Figure 14(g).Figure 19
The accumulative number of passengers that entered the stairs in the QEA result. (a) Stairs 6. (b) Stairs 7.
(a)(b)Figure 20
The accumulative number of passengers that left the stairs in the QEA result. (a) Stairs 6. (b) Stairs 7.
(a)(b)As shown in Figure21, the curve slope of QEA and GA began to increase gradually compared to the origin curve after 100 seconds. It means that passengers had started to transfer between staircases before 100 seconds, and the number of passengers who achieved evacuation was more than the origin result. After 200 seconds, slopes of QEA and GA curves were significantly greater than that of the origin curve, indicating that passengers’ transfer behavior had a greater impact on the evacuation process and the number of evacuated passengers increased per unit time. Through the above analysis, it can be seen that the appropriate passenger transfer timing can be obtained by the threshold optimization. The evacuation efficiency of passengers changes with the inside and outside thresholds. When the inside threshold is greater than the outside threshold and both of them are within an appropriate range, the evacuation time of passengers can be significantly shortened and the evacuation efficiency also could be improved.Figure 21
Evacuation results before and after optimization.


## 5. Conclusions

The staircase is an important exit route in cruise ships in the event of an emergency. In this paper, a multigrid model is used to simulate the evacuation process in the staircase of the cruise ship. The imbalance between the numbers of passengers evacuated between staircases will cause congestion, resulting in an extension of the evacuation time.The QEA and the GA were used to optimize the evacuation process of staircases, and optimization results showed that there was a certain correlation between the evacuation time and the density of passengers in the staircase area. If the passenger density in the staircase area can be kept within an appropriate range, the congestion would decrease. Therefore, evacuation plans can be optimized based on considering the impact of the pedestrian density in the staircase area, which could provide a reference for the development of cruise ship evacuation strategies.



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*Source: 9961536-2021-05-17.xml*