Advances in Systems Science and Applications (2014) Vol.14 No.4 388-395 Optimal Location Model and Algorithm of the Emergency Exits Hong-jie Wang and Yan Gao School of Management, University of Shanghai for Science and Technology, 334 Jun Gong Road Shanghai 200093, China Abstract In evacuation, the emergency exits are important. The exit’s location is a key which decides the success of evacuation. So how to build the emergency exits is worth to researching. In this paper, the effect of the emergency exit and main door are considered. The emergency exit’s ordinate is defined as the independent variable. Then a model of exits is proposed and the golden section algorithm is used to find the most suitable location. At last, it is verified that the method is effective and feasible. Keywords Evacuation Emergency exits Golden section algorithm Simulation. 1 Introduction Emergency evacuation is a mass movement of people from areas affected by dis- asters, such as earthquake, hurricane, and fire, to safer in a timely manner to mitigate disastrous consequences. It involves the choice of routes. If people can quickly find the exit and through the door, then hope of escape is big in the danger. So the location of emergency exit is important. And the right location of the exit can increase the rate of evacuation. In recently, the researchers have found many factors which affect the evacuation [1-5] and provide many evacua- tion models [6-9]. Although some researches show that the building’s structure can affect the evacuation [10-11], the most of the research presupposes the build- ing’s layout. They didn’t consider the effect of the exit’s position. It means the emergency exit, main door and access are designed and don’t change in the re- searches. Obviously, the effect of layout is ignored. If we don’t consider the main door in evacuation process, obviously the middle sides is the most suitable exit’s location in some symmetrical architect. But if main door can be used well in the evacuation, the new evacuation route is added. So the exits and main door is in some suitable locations, maybe it can shorten the evacuation time. It’s useful to evacuation. So it’s important to finding the suitable locations. In this paper, we try to find the suitable location of exit’s location. So we only consider the effect of exit’s location. And in the research, we also consider the main door in the evacuation. In order to finding the suitable location of exit, we use the golden section algorithm. At last, we discuss and conclude the numerical results by numerical examples. Advances in Systems Science and Applications (2014) Vol.14 No.4 389 2 Description The authors consider a population of N individuals and assumed that each in- dividual nk(k = 1, 2, · · · , N) moving by the speed Vkin the emergency condition and randomly select the evacuation route. Generally, the evacuation time is that how many times is used between the beginning of escape and all people arriving at the safe place. Because we hope that we can save more people in shortest time, so the evacuation problem can be described by the object function as follows: min f = max{Ti(ni) | i = 1, 2, · · · , N } (1) where Ti(ni) is the evacuation time of the ith individual. In the articles, we consider the situation as follow. All individuals were ran- domly distributed in space according to the population density, and their initial velocities equal to zero. Everyone has two choose, main door or emergency exits, to escape. We assume that each individual is a particle without quality and each individuals speed is same. We set up a coordinate system as Fig.1 and use the array (xi, yi) to show the location of the ith particle. (a) the rectilinearly shaped floor (b) the vertical shaped floor Fig.1 The initial distribution of the people on the floor We assume the number of main door is l,and the one of exits is w.Then the kth particle’s evacuation time by the each door can be as:(k = 1, 2, · · · , N) ti(nk) = Ski Vk i = 1, 2 · · · , l (2) tj(nk) = Skj Vk j = l + 1, l + 2, · · · , l + w (3) which the Ski is the shortest distance between theith main door and the kth par- ticle,the Skj is the shortest distance between the jth exit and the kth particle.So the kth particle′s evacuation time Tk is satisfied the equation: Tk(nk) ≤ max{ti(nk) |i = 1, 2 · · · l, l + 1, l + 2, · · · , l + w} (4) 390 Hong-jie Wang: Optimal Location Model And Algorithm Of The Emergency Exits Summary, the evacuation problem can be as: min f = max{Tk(nk) | k = 1, 2, · · · , N } ≤min max 1≤k≤N {ti(nk) |i = 1, 2, · · · l, l + 1, l + 2, · · · , l + w} (5) Obviously, when the each people choose the shortest path, the equation is estab- lished. The people’s location is fixed. Perhaps, the each evacuation time maybe change, if the emergency exit’s location is changed. Because the distances be- tween the exits and the kth particle are changed.So the location of emergency exit effects the evacuation. As follow, we consider the exit’s location. We assume(x, y) is the coordinate of emergency exit and f(x) is the max evacuation time. In ar- ticles, we only consider the situation that there are two emergency exits which are distributed on the right and left sides of building respectively. In the situ- ation, the coordinate y is 0 and the x is the independent variable. So we only consider the coordinate of emergency exit x.Absolutely, the f(x) is decided by x. Because we hope to find the most suitable location of emergency exit, which the evacuation time is the shortest, so we can setup the function as follows: min f(x) = max{Tk(nk, x) | k = 1, 2, · · · , N } s.t. a ≤ x ≤ b (6) where a is the index of wall-width,b is the superscript of wall-width, and x is the coordinate of emergency exit’s center point. 3 Algorithm According to the shortest path theory, when everyone selects the shortest path, the each individual’s evacuation time and total time are the shortest. We select the shortest path as the each individual’s escaping path. Let each individuals distance function as: (k = 1, 2, · · · , N) Sk(x) = min{Ski(nk, x) |i = 1, 2, · · · , l, l + 1, l + 2, · · · , l + w} (7) Absolutely, the function Sk(x) is the linear and the f(x) = max{Tk(nk, x) |k = 1, 2, · · · , N} is the linear, too. We can solve the problem by the optimization al- gorithm. We hope that we can exclude the unsuitable position in the process of searching. So we select the golden section algorithm to solving the problem. The golden section algorithm is a kind of optimization algorithms for solving one-dimensional problem as follow: min f(x) s.t. a ≤ x ≤ b (8) Advances in Systems Science and Applications (2014) Vol.14 No.4 391 where f(x) is a uni-modal descent function. The method is a iteration algorithm and find the best solution by shorting the independent variables interval in each iteration, which reduced probability is the 0.618. The algorithms step is as follows: Step1:define the a,b,ε and (xi, yi), i = 1, 2, · · · , N ; Step2:define: x2 = a+ 0.618 ∗ (b− a) (9) Sk(x2) = min{Ski(nk, x2) |i = 1, 2, · · · , l, l + 1, l+2, · · · , l + w} (10) f2 = max{Tk(x2) = Sk(x2) Vk | k = 1, 2, · · · , N } (11) go to step3; step3:define x1 = a+ 0.382 ∗ (b− a) (12) Sk(x1) = min{Ski(nk, x1) |i = 1, 2, · · · , l, l + 1, l + 2, · · · , l + w} (13) f1 = max{Tk(x1) = Sk(x1) Vk | k = 1, 2, · · · , N } (14) go to step4; Step4: if | b− a | ≤ ε,define x∗ = a+ b 2 (15) and stop; else go to step5; Step5: if f1 < f2, define b = x2, x2 = x1, f2 = f1 then go to step3; if f1 = f2, define b = x2, x2 = x1,f2 = f1, then go to step3; if f1 > f2, define a = x1, x1 = x2, f1 = f2,then go to step6; Step6: define x2 = a+ 0.618 ∗ (b− a) (16) Sk(x2) = min{Ski(nk, x2) |i = 1, 2, · · · , l, l + 1, l + 2, · · · , l + w} (17) f2 = max{Tk(x2) = Sk(x2) Vk | k = 1, 2, · · · , N } (18) then go to step4. 392 Hong-jie Wang: Optimal Location Model And Algorithm Of The Emergency Exits 4 Simulation results 4.1 Architectural attributes To design the suitable position of exit, the authors have designed the building layout to be two shaped. One is the rectilinearly shaped building, which each floor is a 10×30 units orthogonal area. The another is the vertical shaped building, which major semi-axis of floor is 14units, and the minor semi-axis is the 7units. In the two layouts, we consider the one kind of scenario that main entrance is placed on the bottom middle-south side of the floor. Because the exits are usually placed on the right and left sides in most buildings, we consider the exits on the right and left sides, and they are opposite. The space occupancy levels vary in square meter per person ranging within the limits of space and occupancy density standards. In the two scenarios, the authors used occupancy densities of 0.15, 0.25, 0.5, 0.7, and 1.2 person/units 4.2 Simulation and numerical tests To verify the feasible and effective of the way, it’s run each scenario for six hundred times, take the average of the evacuation time, and analyze the all results. In the all tables, where x, is the center point’s coordinate of emergency exit,P is occupancy densities, and p is the percentage of people selecting the emergency exit. In order to clearly verify our conclusion, we compare the number results to the evacuation time of building with emergency exits in center sides. Simulation numerical results reveal the following: Table 1.1 The results of rectilinearly shaped with one main entrance situation of emergency exit P Number of Number of Number of Number of Number of Number of( person/m2 ) 0 < x < 1 1 ≤ x < 2 2 ≤ x < 3 3 ≤ x < 4 4 ≤ x < 4.9 4.9 ≤ x < 5.1 0.15 1 7 4 176 44 3 0.25 0 1 5 98 55 14 0.5 0 0 0 128 39 8 0.7 0 0 0 100 49 12 1.2 0 0 0 87 34 12 Table 1.2 The results of rectilinearly shaped with one main entrance. situation of emergency exit P Number of Number of Number of Number of Number of( person/m2 ) 5.1 ≤ x < 6 6 ≤ x < 7 7 ≤ x < 8 8 ≤ x < 9 9 ≤ x < 10 0.15 40 171 50 81 53 0.25 0.45 191 63 70 32 0.5 82 208 50 50 35 0.7 103 201 53 44 38 1.2 128 200 66 40 33 Advances in Systems Science and Applications (2014) Vol.14 No.4 393 Table 2 The results of vertical shaped with one main entrance. occupancy Building with emergency exits densities in center sides Results of simulations the percentage of the percentage of P Average-time people selecting the Average-time people selecting the( person/m2 ) (unit) emergency exit P (unit) emergency exit P 0.15 6.51 0.51966 6.43 0.51049 0.25 6.66 0.51896 6.56 0.51319 0.5 6.78 0.52090 6.67 0.51368 0.7 6.82 0.51860 6.71 0.51167 1.2 6.89 0.51890 6.77 0.51450 Table 3.1 The results of vertical shaped with one main entrance. situation of emergency exit P Number of Number of Number of Number of Number of Number of Number of( person/m2 ) 0 < x ≤ 1 1 < x ≤ 2 2 < x ≤ 3 3 < x ≤ 4 4 < x ≤ 5 5 < x ≤ 6 6 < x ≤ 6.9 0.15 13 9 20 8 10 15 2 0.25 11 5 10 13 4 7 1 0.5 2 0 7 6 2 3 0 0.7 0 2 8 3 9 4 1 1.2 0 0 1 2 0 2 0 Table 3.2 The results of vertical shaped with one main entrance. situation of emergency exit P Number of Number of Number of Number of Number of Number of Number of( person/m2 ) 6.9 < x ≤ 7.1 7.1 ≤ x < 9 9 ≤ x < 10 10 ≤ x < 11 11 ≤ x < 12 12 ≤ x < 13 13 ≤ x < 14 0.15 7 19 22 115 99 209 52 0.25 6 11 11 103 115 256 47 0.5 1 2 8 101 65 341 62 0.7 0 2 3 87 52 378 60 1.2 0 2 4 79 22 411 77 Table 4 The compare of vertical shaped with one main entrance. occupancy Building with emergency exits densities in center sides Results of simulations the percentage of the percentage of P Average-time people selecting the Average-time people selecting the( person/m2 ) (unit) emergency exit P (unit) emergency exit P 0.15 7.78 0.64214 5.62 0.68798 0.25 8.22 0.64292 5.94 0.67642 0.5 8.62 0.65034 6.18 0.68156 0.7 8.81 0.64968 6.26 0.68131 1.2 9.01 0.64573 6.37 0.68303 394 Hong-jie Wang: Optimal Location Model And Algorithm Of The Emergency Exits 5 Discussions and Conclusion From the table1.1, table 1.2, table 3.1, and table 3.2, it can be known that the x arent distributed randomly, but most in some areas. The occupancy density is more, the number of x’s appearance is larger in this areas. In the rectilinearly shaped building, there are about 50% x in the area from 5.1 to 7. In the vertical shaped building, the x is the most in the areas from 12 to 13. And both of numbers are rising with the occupancy density’s raising. In the centre area, the percentage of x’s appearance is less than 2.5% in each simulation, both of two shaped. Obviously, the more the occupancy density is, the smaller the individual’s distributive space is. So the regularity is clearer as the density raising. These show that the effect of emergency exit is existent and the center point is not the suitable place for buildings of this size and shape. Because that if the effect isn’t existent, the distribution of x is random and no regular. From the table 2 and table 4, the evacuation times of simulations are shorter than the one of situations which emergency exit is in the center of right and left sides, although the p are close in each comparison. It shows the location of emergency exits is effect the evacuation, and the center point is not necessarily the most suitable position. The suitable location of emergency exit can shorten the evacuation time. So the authors conclude the conclusions as follow: 1. The location of emergency exit effects the evacuation in some way. 2. The suitable place of emergency exit is related to the shape of building, occupancy density, and the layout and people distribution. The center point of left and right sides is not necessarily the most suitable position in different situation. 3.Golden section algorithm about searching the emergency exit is effective. Because the distribution of people is random and irregular in each simulation, so the results are not clustering some point. 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Corresponding author Author can be contacted at: gaoyan@usst.edu.cn