INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL Online ISSN 1841-9844, ISSN-L 1841-9836, Volume: 17, Issue: 4, Month: August, Year: 2022 Article Number: 4749, https://doi.org/10.15837/ijccc.2022.4.4749 CCC Publications A Multi-objective Location Decision Making Model for Emergency Shelters Giving Priority to Subjective Evaluation of Residents Yiying Wang, Zeshui Xu Yiying Wang Business School Sichuan University Chengdu, Sichuan, 610064, China wanggyiying@163.com Zeshui Xu∗ Business School Sichuan University Chengdu, Sichuan, 610064, China xuzeshui@263.net ∗Corresponding author Abstract Earthquake is regarded as the most destructive and terrible disaster among all-natural disasters [1]. Experts agree that immediate emergency evacuation is the safest and most effective response to the earthquake disaster [2]. In the research of emergency evacuation planning, the influence of human subjectivity has gradually attracted researchers’ attention. In this paper, we take the human subjectivity as one of the most important factors for emergency evacuation planning. Based on the preferences of the residents at each demand point for the attributes of every candidate emergency shelter, the subjective score of each candidate emergency shelter is obtained. The preferences of residents will change with the refuge time, so do the weights of residents’ subjective scores of all attributes of candidate emergency shelters. Therefore, we use the subjective score function to describe the change of residents’ evaluations for the emergency shelter over time, and take the average value of subjective scores at all refuge times as the primary basis for location decision making. On these bases, we build a multi-objective location decision making model for emergency shelters giving priority to subjective evaluation of residents. In the model, we consider transfer distance, the efficiency of construction funds and the distribution of people among emergency shelters. Considering fairness, we minimize the standard deviation of the scores and the standard deviation of the transfer distances in the model. This model is applied to a case, which verifies its feasibility and shows that human subjectivity plays an important role in emergency evacuation planning. Keywords: Emergency shelter location; Multi-objective optimization; Multi-attribute; Sub- jective preference; Total refuge time. https://doi.org/10.15837/ijccc.2022.4.4749 2 1 Introduction The frequent occurrence of natural disasters such as earthquakes has caused a large number of casualties and property losses [3], [4]. Therefore, emergency evacuation planning has attracted more and more attention. Emergency planning includes evacuation of disaster victims, site selection and construction of emergency shelters, etc. Next, we review the related literature. The location problem originated from weber’s study, which selected a position for the warehouse by minimizing the distance between the warehouse and all customers [5]. From then on, scholars have proposed models such as p-median problem [6], [7], set covering location problem [8] and maximal covering location problem [9]. However, the location of emergency shelters and evacuation of disaster victims are complex issues, involving many factors, while the single-objective model ignores some im- portant objectives [10]. Therefore, the multi-objective model was put forward. Some scholars regarded minimizing the total evacuation distance and the total cost as the targets of emergency evacuation site selection [10], [11]. Coutinho-Rodrigues et al. presented a mixed integer linear programming model with such objectives as risks associated with paths and shelter locations, evacuation path lengths, and the final evacuation time [12]. Wang et al. proposed a model with three objectives to minimize construction cost, minimize traveling distance and maximize final number of survivors [13]. Wang et al. proposed a multi-objective optimization model, in which the first objective is to minimize the total service distance, the second objective is to maximize the network reliability level [14]. A central challenge in developing an evacuation plan is in determining the distribution of evacuees into the safe areas, that is, deciding where and from which road each evacuee should go [15]. There- fore, scholars have developed bilevel models with the multi-objective optimization in the upper-level problem or in the lower-level problem or both in the upper-level and lower-level problems. Xu et al. proposed scenario-based hybrid bilevel model that considers the dynamic number of evacuees and its implementation for earthquake emergency shelter location and allocation [10]. Kulshrestha et al. developed a bilevel model, in which the planning authority determines the shelter locations along with their capacities, whereas the evacuees choose the shelters and routes to evacuate [16]. From the previous literature review, we can see that scholars have begun to pay attention to residents’ willingness to transfer, such as the choice of transfer route. In fact, the decision-making theory based on subjective preference has been widely studied and applied in many aspects, such as commodity selection [17], investment decision [18]. However, subjectivity associated with factors for shelter location decisions are not given much research attention [19]. Recent research considered self-interest (non-cooperative behavior) of evacuees during large evacuations [10], [20]. In this paper, the subjective willingness of the affected residents to move in an emergency is not only reflected in the choice of the transfer route, but also in their satisfaction with the emergency shelters, including their satisfaction with the types of emergency shelters, supporting facilities, accessibility, environment and so on. Only when we take into account the affected residents’ willingness to transfer to emergency shelters, can they cooperate with the emergency management department to complete the implementation of various emergency management measures to the greatest extent. In addition to paying attention to the satisfaction of residents with the emergency shelters, equity is another concern in emergency planning. However, there is not any common definition for equity [21]. Ng and Park proposed a model that yielded shelter assignments, which accounts for fairness in the sense that the algorithm assigns evacuees to shelters that are as close as possible to their points of origin [22]. Tsai and Yeh thought that the regional allocation of disaster evacuation/refuge shelters must consider timeliness and fairness. The indicator for fairness is the number of victims each single shelter can accommodate; the more the victims, the higher the service [23]. Sabouhi et al. developed a model, in which equity in relief provided for various affected areas is achieved by defining objective functions that minimize the maximum time for the transportation of evacuees to shelters, the transportation of injured people to hospitals, and the distribution of relief commodities [21]. In this paper, the residents of each demand point evaluate the attributes of emergency shelters according to their own preferences, and get the score of each emergency shelter. For the fairness, we assume that the residents in a demand point should be transferred only to their higher-scored emergency shelters that are within the service distance of this demand point. In addition, we take the standard deviation of the subjective scores and the standard deviation of the transfer distance as the factors to https://doi.org/10.15837/ijccc.2022.4.4749 3 be considered, and minimize these two standard deviations as two goals in model, which is another embodiment of fairness. Therefore, in this paper the location planning and construction of emergency shelters proposed is based on the subjectivity of regional residents, and the construction of emergency shelters in our study is carried out in advance before the disaster, rather than the temporary planning and construction during and after the disaster [24]. In recent years, scholars have paid more and more attention to the standards of site selection and construction of emergency shelters. Nappi and Souza raised 10 possible criteria and their relevant aspects, including: location, optimal distribution, urban infrastructure, safety, physical adequacy, cultural adequacy, privacy, environmental comfort, universal accessibility and economic aspects [25]. Aman and Aytac defined 15 criteria to evaluate the suitability in terms of spatial, natural structure and accessibility of each potential open space for emergency gathering [26]. Wu et al. established an indicator system comprising five aspects: scale and location; risk of disaster; rescue facilities; feasibility; resident aspect [27]. Song et al. summarized the evaluation criteria for shelter-site selection, including location & logistic efficiency, costs, environmental conservation and social aspects [28]. In this paper, we combine the above-mentioned site selection and construction standards of emergency shelters, and refer to China Earthquake Administration’s standard which is called Site and Supporting Facilities for Earthquake Emergency Shelter [29], and put forward six site selection and construction indexes of emergency shelters, including distance, accessibility, scale, supporting facilities, environment and type. The contributions of this paper are as follows: we comprehensively use the methods of normal distribution, ordered weighted aggregation (OWA) operator [30], [31], [32] and analytic hierarchy process (AHP) [33] to form the subjective score and the initial weight of each attribute score; on the basis of reasonable assumptions, the dynamic weight of each attribute score is obtained; the concept of total refuge time is formed, and the subjective score function of each demand point for each candidate site is obtained with the total refuge time as the independent variable; and the average value of subjective scores at all refuge times is taken as the primary basis for location decision making. This paper expands the meaning of fairness in emergency shelter location, and brings the standard deviation of residents’ subjective scores and the standard deviation of residents’ transfer distances together into the scope of fairness in emergency shelter location. As human subjectivity is paid more and more attention, the model in this paper could be used as an important part of the government decision support system for emergency shelter location [34]. The rest of the paper is arranged as follows: Section 2 presents a methodology to determine the value and the weight of each attribute. Section 3 describes the formulation of the model. Then, in Section 4, the previous multi-objective model is applied to a case. Meanwhile, we conduct some analyses about the case. Finally, we make conclusions about our study in Section 5. 2 Methodology 2.1 Description of the problem and determining the value of each attribute The candidate emergency shelters in this paper are the sites built for other purposes, such as the park, the green space, the square, the stadium and the indoor public place. If a site needs to be used as an emergency shelter, it needs to be constructed to improve the function, so as to meet the requirements of being an emergency shelter. This paper focuses on people’s subjective preferences for the attributes of the emergency shelters. Emergency shelter has many attributes. We select six attributes to analyze people’s subjective pref- erences. These attributes include the distance from the demand point to the emergency shelter, the accessibility, scale, supporting facilities, environment and type of the emergency shelter. The first two are external attributes, and the last four are internal attributes. According to different evaluation methods, these attributes are divided into three categories: 1) the distance from the demand point to the emergency shelter; 2) the accessibility, the scale, the supporting facilities and the environment of the emergency shelters; 3) the types of the emergency shelters. Next, we discuss the subjective evaluation of these attributes by the affected residents in turn. https://doi.org/10.15837/ijccc.2022.4.4749 4 Figure 1: Graph of function y = 1/x. 1. Evaluation of the distance attribute: Some researchers studied people’s behaviors under the condition that the travel cost is pro- portional to distance [35]. Usually, people’s willingness to move to a place decreases with the increase of the distance, that is, the farther away the place is, the more reluctant people are to move, and people prefer to move to a place closer to themselves. This paper assumes that the score of the distance between the demand point i and the shelter j is inversely proportional to the distance between them, as shown in Fig 1. Let the distance between the demand point i and the shelter j be rij, and the distance between the demand point i and its nearest shelter be ri0, then the distance score of the demand point i to the shelter j could be expressed as xij,d = 100 · ri0rij . 2. Evaluation of the accessibility, the scale, the supporting facilities and the environ- ment: Emergency shelters are distributed in various areas of the city. Due to different actual construction conditions, their attributes are at different levels. According to the construction standards, we suppose that the levels of the accessibility, the scale, the supporting facilities and the environment are divided into five grades. The scores from grade 1 to grade 5 are set sequentially to be: 100, 90, 80, 70 and 60. Then, the scores of the accessibility, the scale, the supporting facilities and the environment of the shelter j could be obtained, which are expressed as xj,c, xj,sc, xj,su and xj,en respectively. When a candidate emergency site is selected as the emergency site, we can take measures to improve the level of the attributes of this emergency site to obtain a higher score. 3. Evaluation of the type: Different residents have different preferences for different types of sites. We use the combination of normal distribution and the OWA operator to evaluate different types of sites [30], [31], [32]. The evaluation process is as follows: Residents evaluate the type of each shelter according to their preferences, and get the subjective score of type ranging from 1 to 100. The scores obtained are arranged in descending order. We use k to indicate the type of the emergency shelter, i.e., k represents the park (PA), the green space (GR), the square (SQ), the stadium (ST) and the indoor public place (ID). Let u be the number of residents participating in the evaluation, and the following evaluation matrix is obtained: https://doi.org/10.15837/ijccc.2022.4.4749 5   hPA1 h GR 1 h SQ 1 h ST 1 h ID 1 hPA2 h GR 2 h SQ 2 h ST 2 h ID 2 ... ... ... ... ... hPAq h GR q h SQ q h ST q h ID q ... ... ... ... ... hPAu−1 h GR u−1 h SQ u−1 h ST u−1 h ID u−1 hPAu h GR u h SQ u h ST u h ID u   in which, hPAq , hGRq , hSQq , hSTq , hIDq respectively represent the score of the ith resident for the Park (PA), the green space (GR), the square (SQ), the stadium (ST), the indoor public place (ID), which could be uniformly expressed as hkq. The method of determining the OWA weight based on normal distribution refers to weighting those too large or too small data by small values to eliminate or reduce the impact of unfair evaluation on decision-making results [32]. The OWA weights of residents’ scores are solved by using the method of normal distribution, so as to obtain residents’ scores for different types of sites. Calculating the weight coefficient ωk,q, which is the evaluation of the resident q for the site type k (q = 1, 2, 3, . . . ,u): ωk,q = e − [ (hkq−µk) 2 /2σk 2 ] ∑u r=1 e − [ (hkr−µk) 2 /2σk 2 ] ,µk = ∑u q=1 h k q u ,σk = √√√√1 u u∑ q=1 ( hkq −µk )2 , in which, µk represents the average value of scores of all residents participating in the evaluation on the site type k; σk represents the standard deviation of scores of all residents participating in the evaluation on the site type k. We calculate the type score hk of the type k by the following formula: hk = ∑u q=1 ωk,qh k q. Therefore, the final evaluation matrix of each site type is: [ hPA hGR hSQ hST hID ] , in which, hPA, hGR, hSQ, hST , hID represent the final score of the park, the green space, the square, the stadium, the indoor public place respectively. Through the above analysis, we have obtained all the attribute scores of the shelter j. The distance score, the accessibility score, the scale score, the supporting facilities score, the envi- ronment score and the type score are expressed as xij,d, xj,c, xj,sc, xj,su, xj,en, xj,k (here, xj,k is the above mentioned hk) respectively. 2.2 Dynamic weights of the attributes AHP can quantify people’s experience judgment, and it is more practical when there are many and complicated factors [36]. Through AHP, the relative importance of pairwise attributes is judged, the weight judgment matrix is constructed, the consistency is checked, and the weights are normalized, so that the weight of each attribute can be obtained. Because this paper needs to deal with many attributes of the demand points, we use the AHP [33] to get the attribute weights of each demand point. In this paper, the attribute weight reflects the degree of resident’s concern on the attribute of the candidate emergency shelter: ωi(0) = (ωi,d(0),ωi,c(0),ωi,sc(0),ωi,su(0),ωi,en(0),ωi,k(0)), in which, ωi,d(0), ωi,c(0), ωi,sc(0), ωi,su(0), ωi,en(0), ωi,k(0) represent respectively the weight of the distance score, the accessibility score, the scale score, the supporting facilities score, the environment score and the type score that are evaluated by the demand point i, and meet: https://doi.org/10.15837/ijccc.2022.4.4749 6 Figure 2: The graph of function f1(t). Figure 3: The graph of function f2(t). ωi,d(0) + ωi,c(0) + ωi,sc(0) + ωi,su(0) + ωi,en(0) + ωi,k(0) = 1. In fact, for different total refuge time (i.e., the total time that the residents stayed in an emergency shelter at one time of refuge), the resident’s preference for each attribute of the emergency shelter is also different. That is, the weight of each attribute score of the emergency shelter will change with the total refuge time. Therefore, the weight obtained by the above AHP method is only taken as the initial weight. We use the dynamic weight to express the change of people’s preference for the attributes of the emergency shelters for different total refuge times. Furthermore, people’s preferences for external attributes and internal attributes change in the opposite direction with the change of the total refuge time. The longer the total refuge time, the more attention they pay to internal attributes, and the less attention they pay to external attributes. This paper assumes that the change of people’s preferences for external attributes and internal attributes meets f1(t) = 11+αt2 and f2(t) = 2 − 1 1+αt2 respectively, where t (t ≥ 0) represents the total refuge time, α (α > 0) is a constant. The following two figures are the graphs of f1(t) and f2(t) respectively. From Fig 2 and Fig 3, we could see that f1(t) is a decreasing function and f2(t) is an increasing function, which satisfy the change of resident’s external attribute preference and internal attribute preference respectively. The first and second derivatives of f1(t) = 11+αt2 are: f ′1(t) = − 2αt (1 + αt2)2 ,f ′′1 (t) = 2α(3αt2 − 1) (1 + αt2)3 . When f ′′1 (t) = 0, i.e., 2α(3αt2−1) (1+αt2)3 = 0, we have t = 1√ 3α . When t < 1√ 3α , we have f ′′1 (t) < 0; when t > 1√ 3α , we have f ′′1 (t) > 0. Therefore, when t = 1√ 3α , f ′1(t) get the minimum value, and when t = 1√ 3α , f1(t) decreases the fastest. In order to get the weight function of each attribute, it is necessary to determine the value of α. This paper assumes that f1(t) decreases the fastest when t = 3, then α = 127 . Therefore, the weight functions of the distance score, the accessibility score, the scale score, the https://doi.org/10.15837/ijccc.2022.4.4749 7 supporting facilities score, the environment score and the type score are as follows: ωi,d(t) = ωi,d(0)f1(t) = 27ωi,d(0) 27 + t2 , ωi,c(t) = ωi,c(0)f1(t) = 27ωi,c(0) 27 + t2 , ωi,sc(t) = ωi,sc(0)f2(t) = ωi,sc(0) ( 2 − 27 27 + t2 ) , ωi,su(t) = ωi,su(0)f2(t) = ωi,su(0) ( 2 − 27 27 + t2 ) , ωi,en(t) = ωi,en(0)f2(t) = ωi,en(0) ( 2 − 27 27 + t2 ) , ωi,k(t) = ωi,k(0)f2(t) = ωi,k(0) ( 2 − 27 27 + t2 ) . When t = 0, we have f1(0) = f2(0) = 1, the weight of each attribute score is the initial weight under this condition. When t > 0, the sum of the attribute weights of the demand point i is: ωi(t) = ωi,d(t) + ωi,c(t) + ωi,sc(t) + ωi,su(t) + ωi,en(t) + ωi,k(t) = f1(t)(ωi,d(0) + ωi,c(0)) + f2(t)(ωi,sc(0) + ωi,su(0) + ωi,en(0) + ωi,k(0)). Assume that f1(t) = f1(0) − ∆(t). Since f1(t) + f2(t) = 2, we have f2(t) = f2(0) + ∆(t). Then ωi(t) = (f1(0) − ∆(t))(ωi,d(0) + ωi,c(0)) + (f2(0) + ∆(t))(ωi,sc(0) + ωi,su(0) + ωi,en(0) + ωi,k(0)) = f1(0)(ωi,d(0) + ωi,c(0)) − ∆(t)(ωi,d(0) + ωi,c(0)) + f2(0)(ωi,sc(0) + ωi,su(0) + ωi,en(0) + ωi,k(0)) +∆(t)(ωi,sc(0) + ωi,su(0) + ωi,en(0) + ωi,k(0)) = 1 + ∆(t)[(ωi,sc(0) + ωi,su(0) + ωi,en(0) + ωi,k(0)) − (ωi,d(0) + ωi,c(0))]. When (ωi,sc(0) + ωi,su(0) + ωi,en(0) + ωi,k(0))−(ωi,d(0) + ωi,c(0)) 6= 0, we get ωi(t) 6= 1. Therefore, we need to normalize ωi(t): ωi,d(t) = ωi,d(t)/ωi(t),ωi,c(t) = ωi,c(t)/ωi(t),ωi,sc(t) = ωi,sc(t)/ωi(t), ωi,su(t) = ωi,su(t)/ωi(t),ωi,en(t) = ωi,en(t)/ωi(t),ωi,k(t) = ωi,k(t)/ωi(t), in which, ωi,d(t), ωi,c(t), ωi,sc(t), ωi,su(t), ωi,en(t), ωi,k(t) are the normalized weights of the distance score, the accessibility score, the scale score, the supporting facilities score, the environment score and the type score respectively. Here, ωi,d(t) + ωi,c(t) + ωi,sc(t) + ωi,su(t) + ωi,en(t) + ωi,k(t) = 1. 3 The formulation of the model 3.1 The formulation of the score function and the standard deviation function When the total refuge time is t, the distance score, the accessibility score, the scale score, the supporting facilities score, the environment score and the type score of the emergency shelter j that are evaluated by the demand point i could be expressed as follows: yij,d(t) = ωi,d(t) ·xij,d,yij,c(t) = ωi,c(t) ·xj,c, yij,sc(t) = ωi,sc(t) ·xj,sc,yij,su(t) = ωi,su(t) ·xj,su, https://doi.org/10.15837/ijccc.2022.4.4749 8 yij,en(t) = ωi,en(t) ·xj,en,yij,k(t) = ωi,k(t) ·xj,k. Therefore, when the total refuge time is t, the subjective score of the demand point i for the candidate shelter j is: yij(t) = yij,d(t) + yij,c(t) + yij,sc(t) + yij,su(t) + yij,en(t) + yij,k(t) = ωi,d(t) ·xij,d + ωi,c(t) ·xj,c + ωi,sc(t) ·xj,sc + ωi,su(t) ·xj,su + ωi,en(t) ·xj,en + ωi,k(t) ·xj,k. Assume that ωi(t) = (ωi,d(t),ωi,c(t),ωi,sc(t),ωi,su(t),ωi,en(t),ωi,k(t)), xj = (xij,d,xj,c,xj,sc,xj,su,xj,en,xj,k)T , then we get: yij(t) = ωi(t) · xj. Assume that there are m (i = 1, 2, . . . ,m) demand points, and n (j = 1, 2, . . . ,n) candidate shelters. Let x = (x1, x2, . . . , xn−1, xn), ω(t) =   ω1(t) ω2(t) ... ωm−1(t) ωm(t)  ,then the score matrix of all demand points for all candidate shelters is: y(t) = ω(t) · x =   ω1(t) ω2(t) ... ωm−1(t) ωm(t)   (x1, x2, . . . , xn−1, xn) Notice that, xij,d in xj should change with ωi(t). The total subjective score of the demand point i for all candidate emergency shelters is: y0i (t) = ∑n j=1 yij(t). It could be seen from the above analysis that when the total refuge time t changes, people’s sub- jective score for the emergency shelters also changes. Obviously, it is not feasible to select emergency shelters once for each t. Assume that the maximum value of t is T, it is a feasible method to derive the average value of subjective scores for all t within the interval [0,T], and use this average value to make location decision. We divide the interval [0,T] into N equal parts, and each equal part is ∆t. Then, we take a value of t for every interval T/N from 0, and let ts = Ts/N (s = 1, 2, . . . ,N). If the probability of each t in the interval [0,T] is the same, then, from 0 to T, the average value of the subjective scores for the emergency shelter j evaluated by the demand point i is: yij(T) = ∑N s=1 yij(ts) N = ∆t · ∑N s=1 yij(ts) ∆t ·N = ∑N s=1 yij(ts) · ∆t ∆t ·N . When ∆t → 0, the above formula becomes: yij(T) = ∫ T 0 yij (t)dt T . The total subjective score for all candidate emergency shelters evaluated by the demand point i is: y0i (T) = n∑ j=1 yij(T) = n∑ j=1 1 T ∫ T 0 yij(t)dt = 1 T ∫ T 0 ( n∑ j=1 yij(t))dt = 1 T ∫ T 0 y0i (t)dt. We define a binary variable Yij. When j provides service for the demand point i, Yij = 1; otherwise, Yij = 0. Then, the total subjective score of the demand point i for the selected shelters that provides services for i is: y1i (T) = n∑ j=1 yij(T)Yij = n∑ j=1 1 T ∫ T 0 yij(t)Yijdt = 1 T ∫ T 0 ( n∑ j=1 yij(t)Yij)dt. Let y1i (t) = ∑n j=1 yij(t)Yij, then y1i (T) = 1 T ∫T 0 y 1 i (t)dt. https://doi.org/10.15837/ijccc.2022.4.4749 9 Let pij = yij(T)Yij/y1i (t),y1ij(t) = yij(t)Yij, then pij = 1 T ∫T 0 yij(t)Yijdt 1 T ∫T 0 y 1 i (t)dt = ∫T 0 yij(t)Yijdt∫T 0 y 1 i (t)dt = ∫T 0 y 1 ij(t)dt∫T 0 y 1 i (t)dt . The larger the pij is, the more people transfer from the demand point i to the selected shelter j. We suppose that the number of people transferred from the demand point i to the selected shelter j is directly proportional to pij, so the number of people transferred from the demand point i to the selected shelter j is: aij = pij ·ai, where, ai represents the number of people at the demand point i. Here, ai = ∑n j=1 aij, which could be proven as follows: n∑ j=1 aij = n∑ j=1 pijai = ai n∑ j=1 pij = ai n∑ j=1 yij(T)Yij/y1i (T) = (ai/y 1 i (T)) n∑ j=1 yij(T)Yij = ai, then ai = ∑n j=1 aij holds. Let Z1i,s be the number of the emergency shelters that provide services for the demand point i, then: Z1i,s = n∑ j=1 Yij. The average subjective score of the emergency shelters that provide services for the demand point i is: y1i (T) = y1i (T) Z1i,s . Let Zs be the number of the selected sites, y(T)Zs be the total subjective score of all selected emergency shelters evaluated by all demand points, then y(T)Zs = m∑ i=1 aiy 1 i (T). Let a be the total number of residents at all demand points, then per capita score for all selected shelters is: y(T) Zs = ( m∑ i=1 aiy 1 i (T) ) /a = ( m∑ i=1 aiy 1 i (T) ) / m∑ i=1 ai. The variance of the subjective scores of residents at all demand points is: fy = 1 a m∑ i=1 ai ( y1i (T) −y(T) Zs )2 . The standard deviation of the subjective scores of residents at all demand points is: f ′y = √√√√1 a m∑ i=1 ai ( y1i (T) −y(T) Zs )2 . 3.2 The formulation of the transfer distance function and the standard deviation function Let dij be the shortest distance from the demand point i to the emergency shelter j, then the transfer distance of residents at the demand point i to the selected shelters that provide services for i is ∑n j=1 aijdij, and the total transfer distance of residents at all demand points is ∑m i=1 ∑n j=1 aijdij. So, the per capita transfer distance of residents at all demand points is: d = 1 a m∑ i=1 n∑ j=1 aijdij. https://doi.org/10.15837/ijccc.2022.4.4749 10 Therefore, the variance of the transfer distance of residents at all demand points is: fd = 1 a m∑ i=1 n∑ j=1 aij(dij −d)2. The standard deviation of the transfer distance of residents at all demand points is: f ′d = √√√√1 a m∑ i=1 n∑ j=1 aij(dij −d)2. 3.3 Cost and the formulation of the standard deviation function of the numbers of residents transferred to every emergency shelter Let bj,1 be the supporting construction cost of the emergency shelter j, bj,2 be the upgrading construction cost of the emergency shelter j, b1 be the total cost, then b1 = n∑ j=1 (bj,1 + bj,2)Yj, in which, Yj is a binary variable. When j is selected, Yj = 1; otherwise, Yj = 0. The variance of the numbers of residents that transfer to every selected shelter is: fp = 1 Zs n∑ j=1 ( m∑ i=1 aij − a Zs )2 . The standard deviation of the numbers of residents that transfer to every selected shelter is: f ′p = √√√√√ 1 Zs n∑ j=1 ( m∑ i=1 aij − a Zs )2 . 3.4 Goals and constraints The research shows that in the process of the large-scale emergency evacuation, the affected resi- dents do not transfer completely according to the government planning, and they follow self-interests [10], [20]. This paper comprehensively considers the subjective and objective factors affecting the location of emergency shelters. The subjective evaluation of residents at the demand point for the emergency shelter reflects their satisfaction with the emergency shelters. Therefore, the subjective evaluation of the residents will be reflected in their emergency transfer behavior. So, this paper takes the residents’ subjective evaluation for the emergency shelter as an important basis for location de- cision making, this is an aspect of humanistic consideration. On the other hand, in order to ensure fairness, we take the standard deviation of the scores of the residents and the standard deviation of the transfer distance as the factors for emergency shelter planning. To further ensure fairness, in the following model, we assume that a demand point can only be served by the emergency shelters that are within the service distance rd of this demand point. Among the selected emergency shelters, only those emergency shelters that meet this requirement can provide services for the corresponding demand points. In order to improve the efficiency of the emergency evacuation and save the transfer cost, we take the minimization of the transfer distance of the residents as a goal. In fact, minimizing the transfer distance is also the overall desire of the residents at all demand points in the region. In order to avoid residents’ excessive gathering in the emergency shelters, we also take the minimization of the standard deviation of the numbers of the residents that transfer to every selected emergency shelter as a goal in the model, which is a consideration of safety. In terms of the construction cost, previous studies mainly focused on planning new shelters [20]. This paper considers providing the supporting facilities and upgrading the site level on the basis of https://doi.org/10.15837/ijccc.2022.4.4749 11 the existing sites, so as to make full use of the existing resources and save the construction funds. Therefore, the cost in the model includes the cost of the supporting construction and the cost for upgrading the site level as analysis in Section 3.3. • Assumptions - in addition to some assumptions put forward in the previous analysis, the model also includes the following assumptions: 1. The candidate sites and the demand points are two different finite point sets in spatial network; 2. The spatial distribution and the number of the candidate sites are known; 3. Based on the road network, the shortest distance between any candidate site and demand point is known; 4. A candidate site can provide services for multiple demand points; 5. The candidate site has no capacity constraint. • Sets M: The set of the demand points in the network {i|i = 1, 2, 3, . . . ,m}. N: The set of the candidate sites {j|j = 1, 2, 3, . . . ,n}. • Parameters y(T) Zs: The per capita score of all residents for all selected shelters. y1i (T): The average subjective score of the sites that provide services for the demand point i. a: The total number of the residents at all demand points. ai: The number of residents at the demand point i, ∀i ∈ M. aij: The number of people transferred from the demand point i to the shelter j. dij: The shortest distance between the demand point i and the candidate site j, ∀i ∈ M, ∀j ∈ N. d: The per capita transfer distance of residents at all demand points. f ′d: The standard deviation of the transfer distance of residents at all demand points. f ′y: The standard deviation of the subjective scores of residents at all demand points. f ′p: The standard deviation of the numbers of residents that transfer to every selected shelter. rd: The service distance. b: The total cost. bj,1: The supporting construction cost of the site j. bj,2: The upgrading construction cost of the site j. Zs: The total number of the selected sites. Z0i,s: The maximal number of the selected sites that provide service for the demand point i. Z1i,s: The actual number of the selected sites that provide service for the demand point i. • Decision variables Yj = { 1, if the site is located at j,∀j ∈ N. 0, otherwise. Yij = { 1, if the site j provide service for the demand point i,∀i ∈ M,∀j ∈ N. 0, otherwise. https://doi.org/10.15837/ijccc.2022.4.4749 12 Objectives: MAX y(T) Zs = ∑m i=1 aiy 1 i (T) a = ∑m i=1 aiy 1 i (T)∑m i=1 ai (1) MIN f ′y = √√√√1 a m∑ i=1 ai ( y1i (T) −y(T) Zs )2 (2) MIN d = 1 a m∑ i=1 n∑ j=1 aijdij (3) MIN f ′d = √√√√1 a m∑ i=1 n∑ j=1 aij(dij −d)2 (4) MIN b1 = n∑ j=1 (bj,1 + bj,2)Yj (5) MIN f ′p = √√√√√ 1 Zs n∑ j=1 ( m∑ i=1 aij − a Zs )2 (6) Subject to: dij ≤ rdYij,∀i ∈ M,∀j ∈ N (7) Zs = n∑ j=1 Yj,∀j ∈ N (8) 1 ≤ n∑ j=1 Yij ≤ Z0i,s,∀i ∈ M (9) Yij ≤ Yj,∀i ∈ M,∀j ∈ M (10) Yi,Yij ∈{0, 1}, for ∀i ∈ M,∀j ∈ M (11) Objectives (1)-(6) have already been analyzed above. Constraint (7) means that a demand point can only be served by the emergency shelters that are within the service distance rd of this demand point. Constraint (8) requires that the total number of selected emergency sites is Zs. Constraint (9) requires that the number of the emergency shelters serving a demand point is at least 1 and at most Z0i,s. Constraint (10) means that a site could provide service for a demand point only when this site is selected. Constraint (11) means that both Yj and Yij are binary variables. 4 Case study 4.1 Basic data Some scholars have used the Sioux Falls Network (or a simplified Sioux Falls Network) to conduct research [37], [38]. The simplified Sioux Falls network is shown in Fig 4. In our case, we also use a simplified Sioux Falls Network (undirected) to test our model, and the number marked next to each side is the length of the side. Table 1 is the initial value of the attribute weight of each demand point for each attribute of the shelter. Table 2 shows the number of the residents at each demand point. Table 3 shows the supporting construction cost (SCC) and the upgrading construction cost (UCC) of each candidate site. Table 4 shows the attribute grade and score of each candidate site after upgrading. Table 5 shows the distance between each demand point and each candidate site and the corresponding distance score. https://doi.org/10.15837/ijccc.2022.4.4749 13 Figure 4: The network. Demand point External attributes Internal attributesDistance Accessibility Scale Supporting facilities Environment Type A 0.50 0.02 0.07 0.20 0.09 0.12 B 0.51 0.05 0.06 0.15 0.08 0.15 C 0.49 0.04 0.08 0.17 0.11 0.11 D 0.50 0.07 0.06 0.18 0.06 0.13 E 0.44 0.04 0.10 0.14 0.08 0.20 F 0.48 0.03 0.09 0.16 0.12 0.12 G 0.53 0.06 0.07 0.13 0.10 0.11 I 0.42 0.10 0.09 0.15 0.07 0.17 K 0.52 0.08 0.07 0.14 0.11 0.08 M 0.43 0.03 0.11 0.16 0.13 0.14 N 0.47 0.05 0.12 0.20 0.07 0.09 P 0.39 0.11 0.09 0.22 0.11 0.08 R 0.55 0.07 0.05 0.15 0.11 0.07 U 0.44 0.08 0.10 0.12 0.13 0.13 Table 1: The initial value of the attribute weight of each demand point for each attribute of the shelter. https://doi.org/10.15837/ijccc.2022.4.4749 14 Demand point The number of the residents Demand point The number of the residents A 1500 I 900 B 2000 K 1800 C 800 M 2000 D 900 N 1700 E 2500 P 2300 F 500 R 1800 G 1000 U 1100 Table 2: The number of the residents at each demand point. Candidate site SCC UCC Candidate site SCC UCC H 110 180 S 95 70 J 120 60 T 60 20 L 90 40 V 70 24 O 130 30 W 125 160 Q 105 100 X 85 30 Table 3: The supporting construction cost (SCC) and the upgrading construction cost (UCC) of each candidate site. Candidate Accessibility Scale Supporting facilities Environment Type site Grade Score Grade Score Grade Score Grade Score Name Score H 2 90 2 90 3 80 2 90 GR 68 J 1 100 2 90 3 80 2 90 SQ 72 L 2 90 4 70 2 90 2 90 ST 83 O 1 100 1 100 2 90 2 90 SQ 72 Q 2 90 3 80 3 80 2 90 GR 68 S 1 100 1 100 3 80 2 90 PA 78 T 2 90 2 90 1 100 1 100 ID 92 V 1 100 2 90 2 90 1 100 ID 92 W 4 70 3 80 3 80 2 90 GR 68 X 2 90 3 80 3 80 2 90 ST 83 Table 4: The attribute grade and score of each candidate site after upgrading. https://doi.org/10.15837/ijccc.2022.4.4749 15 i\ j H J L O Q S T V W A 24, 96 23, 100 78, 29 86, 27 143, 16 103, 22 151, 15 152, 15 130, 18 B 96, 27 49, 53 46, 57 26, 100 74, 35 99, 26 91, 29 92, 28 169, 15 C 150, 13 103, 19 54, 37 77, 26 20, 100 150, 13 100, 20 143, 14 172, 12 D 64, 83 60, 88 115, 46 53, 100 110, 48 70, 76 118, 45 119, 45 90, 59 E 155, 14 108, 19 117, 18 45, 47 102, 21 31, 68 28, 75 21, 100 99, 21 F 214, 9 167, 12 148, 14 104, 19 85, 24 90, 22 46, 43 38, 53 67, 30 G 176, 13 129, 18 184, 12 120, 19 128, 18 49, 47 89, 26 54, 43 24, 96 I 70, 43 85, 35 30, 100 102, 29 104, 29 165, 18 132, 23 168, 18 224, 13 K 76, 34 29, 90 26, 100 46, 57 94, 28 109, 24 111, 23 112, 23 179, 15 M 120, 20 73, 33 24, 100 50, 48 50, 48 123, 20 78, 31 116, 21 191, 13 N 82, 34 35, 80 90, 31 28, 100 85, 33 45, 62 93, 30 94, 30 115, 24 P 140, 19 93, 29 49, 55 30, 90 27, 100 103, 26 53, 51 96, 28 166, 16 R 150, 17 103, 24 77, 32 40, 63 55, 45 84, 30 25, 100 74, 34 138, 18 U 150, 15 103, 22 158, 15 96, 24 153, 15 23, 100 81, 28 32, 72 50, 46 Table 5: The distance between each demand point and each candidate site and the corresponding distance score. (Note: The first number in the table represents the distance and the last number represents the distance score.) 4.2 Analysis and discussions 4.2.1 Analysis In this case, we set rd = 120, Z0i,s = 2. We set min Zs = 2, and solve the model respectively under the condition of Zs = 2, 3, . . . , 10. Since the emergency evacuation time may be short-term or long-term [39], we set the maximum value of T be 20, and solve the model respectively under the condition of T = 1, 2, 3, . . . , 20. We set the priority of the objectives in the order of y(T) Zs, f ′y, d, f ′d, b and f ′ p to solve the model. Table 6 to Table 9 are the solutions of the model. Fig 5 is the three-dimensional figure of the y(T) Zs, in which T and Zs are both independent variables. Fig 6 is a figure which reflects the y(T) Zs changes with Zs, when T = 4, 8, 12, 16, 20. From Table 6, we could see that y(T) Zs shows an upward trend with the increase of T , when Zs remains unchanged; y(T) Zs shows an upward trend with the increase of Zs, when T remains unchanged. From Table 7, we could see that d shows an upward trend with the increase of T, when Zs remains unchanged; d shows a downward trend with the increase of Zs, when T remains unchanged. From Table 6, f ′y shows an overall downward trend with the increase of Zs, when T remains unchanged. From Table 7, f ′d shows an overall downward trend with the increase of Zs, when T remains unchanged. From Table 6, Fig 5 and Fig 6, we notice that when T is constant, the increment of y(T) Zs is fast first and then slow with the increase of Zs, on the whole. This indicate that, when T is constant, the utilization efficiency of costs is different with the change of Zs. Next, we analyze the utilization efficiency of the costs under the condition of constant T, based on the cost and the per capita score when Zs is 2. Let bZs1 (T) be the total cost, y(T) Zs be the per capita score, and βZs (T) be the increment of y(T) Zs generated by the unit increment of the cost when the biggest total refuge time is T and the number of selected sites is Zs. Then βZs (T) = ( y(T) Zs −y(T) 2) / ( bZs1 (T) − b21(T) ) , in which, Zs = 3, 4, . . . , 10. For example, when T = 4, from Table 8, we have b21(4) = 240, b31(4) = 370, b61(4) = 934, b91(4) = 1419. https://doi.org/10.15837/ijccc.2022.4.4749 16 T \ Zs 2 3 4 5 6 7 8 9 10 y f ′y y f ′ y y f ′ y y f ′ y y f ′ y y f ′ y y f ′ y y f ′ y y f ′ y 1 72.77 9.96 76.56 9.86 79.39 9.4 82.20 8.33 83.37 7.56 84.51 7.23 85.62 5.06 86.29 4.60 86.78 4.43 2 73.35 9.68 76.95 9.52 79.71 9.10 82.41 8.06 83.50 7.32 84.60 6.98 85.65 4.92 86.30 4.50 86.76 4.32 3 74.19 9.28 77.51 9.04 80.18 8.67 82.72 7.67 83.74 7.49 84.72 6.06 85.69 4.73 86.31 4.34 86.74 4.18 4 75.15 8.81 78.17 8.62 80.73 8.32 83.08 7.22 84.02 7.01 84.89 5.72 85.74 4.52 86.32 4.18 86.72 4.02 5 76.16 8.32 78.88 8.10 81.32 7.86 83.44 6.76 84.30 6.52 85.09 5.37 85.79 4.29 86.34 4.02 86.69 3.87 6 77.13 7.85 79.57 7.60 81.9 7.41 83.81 6.32 84.59 6.05 85.29 5.03 85.88 4.03 86.37 3.83 86.68 3.69 7 78.04 7.41 80.28 7.92 82.47 7.00 84.17 5.92 84.88 5.62 85.52 4.70 86.00 3.77 86.44 3.63 86.71 3.50 8 78.88 7.00 81.02 7.49 83.00 6.63 84.50 5.55 85.15 5.23 85.72 4.42 86.11 4.39 86.50 3.47 86.73 3.35 9 79.65 6.63 81.70 7.11 83.47 6.30 84.81 5.23 85.40 4.89 85.90 4.19 86.26 4.17 86.55 3.33 86.75 3.22 10 80.34 6.29 82.32 6.76 83.9 6.01 85.08 4.96 85.62 4.60 86.07 3.99 86.39 3.99 86.66 3.25 86.82 3.15 11 80.96 5.99 82.87 6.45 84.29 5.75 85.33 4.72 85.82 4.35 86.22 3.82 86.50 3.84 86.75 3.19 86.89 3.10 12 81.53 5.72 83.37 6.17 84.65 5.51 85.56 4.51 86.02 4.13 86.36 3.68 86.62 3.71 86.85 3.13 86.96 3.05 13 82.04 5.47 83.83 5.92 85.00 5.30 85.79 4.32 86.21 3.94 86.52 3.55 86.74 3.59 86.96 3.07 87.05 3.00 14 82.50 5.25 84.24 5.69 85.31 5.12 86.02 4.17 86.40 3.78 86.67 3.45 86.86 2.99 87.07 3.02 87.14 2.97 15 82.93 5.05 84.62 5.48 85.59 4.95 86.22 4.03 86.58 3.65 86.81 3.36 86.99 2.95 87.17 2.99 87.22 2.95 16 83.31 4.87 84.96 5.30 85.85 4.80 86.44 3.92 86.76 3.54 86.96 3.30 87.13 2.92 87.29 2.96 87.33 2.94 17 83.66 4.70 85.28 5.12 86.09 4.66 86.63 3.82 86.93 3.45 87.11 3.24 87.26 2.91 87.40 2.95 87.42 2.93 18 83.99 4.55 85.58 4.96 86.31 4.54 86.81 3.73 87.09 3.36 87.23 3.09 87.38 2.90 87.50 2.93 87.50 2.93 19 84.28 4.41 85.86 4.82 86.51 4.42 86.98 3.65 87.23 3.30 87.36 3.05 87.49 2.89 87.60 2.93 87.60 2.93 20 84.56 4.28 86.11 4.69 86.7 4.32 87.13 3.58 87.37 3.24 87.48 3.03 87.59 2.89 87.68 2.93 87.68 2.93 Table 6: The per capita score and the standard deviation of the subjective scores of the residents. (Note: here, y represents y(T) Zs.) Figure 5: The three-dimensional figure of the per capita score. Figure 6: The figure of the per capita score. https://doi.org/10.15837/ijccc.2022.4.4749 17 T \ Zs 2 3 4 5 6 7 8 9 10 d f ′d d f ′ d d f ′ d d f ′ d d f ′ d d f ′ d d f ′ d d f ′ d d f ′ d 1 62.42 22.77 51.45 22.62 45.96 18.65 39.88 13.62 36.89 11.86 35.13 11.83 32.42 7.51 31.77 7.39 31.16 7.53 2 62.51 22.76 51.50 22.61 46.01 18.65 40.47 14.10 37.48 12.54 35.17 11.85 32.45 7.53 31.80 7.41 31.19 7.55 3 62.65 22.74 51.57 22.60 46.09 18.66 40.56 14.16 38.21 14.03 35.49 11.35 32.49 7.54 31.84 7.43 31.22 7.57 4 62.80 22.73 52.27 22.33 46.80 18.56 40.66 14.23 38.27 14.07 35.54 11.38 32.54 7.57 31.89 7.46 31.27 7.59 5 62.96 22.71 52.39 22.35 46.93 18.61 40.77 14.29 38.33 14.11 36.37 12.12 33.36 8.90 31.94 7.48 31.31 7.61 6 63.11 22.70 53.74 21.76 48.29 18.31 42.11 14.43 40.22 15.02 38.27 13.44 36.04 11.75 34.60 11.05 33.96 11.29 7 63.24 22.69 57.80 21.91 48.42 18.36 42.22 14.50 40.31 15.08 38.35 13.51 36.14 11.87 34.70 11.17 34.05 11.41 8 63.37 22.68 57.94 21.93 48.54 18.41 42.32 14.56 40.38 15.14 38.43 13.57 36.99 13.21 34.78 11.28 34.12 11.52 9 63.48 22.67 58.06 21.95 48.65 18.45 42.41 14.62 40.45 15.19 38.51 13.63 37.05 13.26 36.26 11.30 35.60 11.62 10 63.58 22.67 58.16 21.97 48.75 18.49 42.50 14.67 40.51 15.23 38.57 13.68 37.11 13.31 36.33 11.38 35.66 11.70 11 63.67 22.66 58.26 21.98 48.84 18.53 42.57 14.72 40.56 15.27 38.63 13.72 37.16 13.35 36.40 11.45 35.72 11.77 12 63.75 22.66 58.34 22.00 50.93 18.10 44.65 15.04 42.62 15.84 40.69 14.62 39.22 14.48 38.47 12.89 37.79 13.28 13 63.82 22.65 58.42 22.01 51.02 18.13 45.93 16.97 43.89 17.82 42.06 16.82 40.58 16.81 39.84 15.54 39.15 15.93 14 63.88 22.65 58.49 22.02 51.10 18.16 46.01 17.02 43.95 17.87 42.12 16.88 41.39 15.69 39.91 15.61 39.22 16.00 15 63.94 22.65 58.55 22.04 51.17 18.19 48.32 16.78 46.25 17.90 44.43 17.16 43.70 16.10 42.22 16.23 41.53 16.70 16 63.99 22.65 58.60 22.05 51.24 18.21 48.40 16.82 46.31 17.95 44.49 17.21 43.78 16.16 42.29 16.30 41.60 16.77 17 64.04 22.64 61.75 24.74 51.72 18.96 48.89 17.71 46.80 18.84 44.98 18.18 43.85 16.21 42.36 16.35 41.66 16.83 18 64.09 22.64 61.80 24.74 51.78 18.98 48.95 17.74 46.85 18.87 45.72 17.12 43.91 16.26 42.42 16.41 41.71 16.88 19 64.13 22.64 61.84 24.74 51.82 18.99 49.01 17.77 46.90 18.90 45.78 17.16 43.96 16.31 42.47 16.45 42.47 16.45 20 64.16 22.64 61.88 24.74 51.87 19.01 49.06 17.80 46.94 18.93 45.83 17.20 44.23 16.27 42.74 16.43 42.74 16.43 Table 7: The per capita transfer distance and the standard deviation of the transfer distances of the residents. T \ Zs 2 3 4 5 6 7 8 9 10 b1 f ′ p b1 f ′ p b1 f ′ p b1 f ′ p b1 f ′ p b1 f ′ p b1 f ′ p b1 f ′ p b1 f ′ p 1 240 1483 420 2392 514 1872 644 977 849 1173 1139 1296 1254 1313 1419 1438 1704 1471 2 240 1455 420 2395 514 1877 644 1332 849 1434 1139 1297 1254 1314 1419 1440 1704 1472 3 240 1416 420 2399 514 1883 644 1342 934 1434 1049 1384 1254 1317 1419 1442 1704 1474 4 240 1371 370 2133 464 1752 644 1353 934 1436 1049 1383 1254 1319 1419 1445 1704 1476 5 240 1326 370 2136 464 1761 644 1364 934 1438 1049 1523 1254 1385 1419 1448 1704 1478 6 240 1283 370 1756 464 1715 644 1300 934 1645 1049 1636 1254 1383 1419 1397 1704 1432 7 240 1244 334 3031 464 1729 644 1313 934 1651 1049 1640 1254 1388 1419 1400 1704 1434 8 240 1208 334 3037 464 1743 644 1324 934 1657 1049 1644 1214 1646 1419 1403 1704 1436 9 240 1176 334 3042 464 1755 644 1334 934 1662 1049 1647 1214 1647 1419 1658 1704 1662 10 240 1147 334 3047 464 1766 644 1343 934 1666 1049 1650 1214 1649 1419 1661 1704 1664 11 240 1122 334 3051 464 1775 644 1352 934 1670 1049 1653 1214 1651 1419 1663 1704 1666 12 240 1099 334 3056 464 2204 644 1738 934 1944 1049 1839 1214 1781 1419 1800 1704 1789 13 240 1078 334 3060 464 2216 644 1955 934 2110 1049 2041 1214 1940 1419 1941 1704 1923 14 240 1060 334 3064 464 2227 644 1964 934 2117 1049 2046 1254 2087 1419 1946 1704 1927 15 240 1044 334 3067 464 2237 644 2395 934 2460 1049 2328 1254 2331 1419 2156 1704 2118 16 240 1029 334 3070 464 2246 644 2403 934 2467 1049 2334 1254 2337 1419 2160 1704 2122 17 240 1015 334 2427 464 2389 644 2507 934 2534 1049 2394 1254 2342 1419 2164 1704 2126 18 240 1003 334 2431 464 2396 644 2514 934 2539 1139 2536 1254 2347 1419 2168 1704 2130 19 240 992 334 2435 464 2402 644 2520 934 2543 1139 2540 1254 2351 1419 2172 1704 2174 20 240 981 334 2439 464 2408 644 2526 934 2547 1139 2545 1254 2379 1419 2198 1704 2197 Table 8: The total cost and the standard deviation of the numbers of residents that transfer to each selected shelter. https://doi.org/10.15837/ijccc.2022.4.4749 18 T \ Zs 2 3 4 5 6 7 8 9 10 1 OT JOT JOTV JLOTV JLOQTV HJLOQTV HJLOQTVX HJLOQSTVX HJLOQSTVWX 2 OT JOT JOTV JLOTV JLOQTV HJLOQTV HJLOQTVX HJLOQSTVX HJLOQSTVWX 3 OT JOT JOTV JLOTV HJLOTV HJLOTVX HJLOQTVX HJLOQSTVX HJLOQSTVWX 4 OT LOT LOTV JLOTV HJLOTV HJLOTVX HJLOQTVX HJLOQSTVX HJLOQSTVWX 5 OT LOT LOTV JLOTV HJLOTV HJLOTVX HJLOQTVX HJLOQSTVX HJLOQSTVWX 6 OT LOT LOTV JLOTV HJLOTV HJLOTVX HJLOQTVX HJLOQSTVX HJLOQSTVWX 7 OT OTV LOTV JLOTV HJLOTV HJLOTVX HJLOQTVX HJLOQSTVX HJLOQSTVWX 8 OT OTV LOTV JLOTV HJLOTV HJLOTVX HJLOSTVX HJLOQSTVX HJLOQSTVWX 9 OT OTV LOTV JLOTV HJLOTV HJLOTVX HJLOSTVX HJLOQSTVX HJLOQSTVWX 10 OT OTV LOTV JLOTV HJLOTV HJLOTVX HJLOSTVX HJLOQSTVX HJLOQSTVWX 11 OT OTV LOTV JLOTV HJLOTV HJLOTVX HJLOSTVX HJLOQSTVX HJLOQSTVWX 12 OT OTV LOTV JLOTV HJLOTV HJLOTVX HJLOSTVX HJLOQSTVX HJLOQSTVWX 13 OT OTV LOTV JLOTV HJLOTV HJLOTVX HJLOSTVX HJLOQSTVX HJLOQSTVWX 14 OT OTV LOTV JLOTV HJLOTV HJLOTVX HJLOQTVX HJLOQSTVX HJLOQSTVWX 15 OT OTV LOTV JLOTV HJLOTV HJLOTVX HJLOQTVX HJLOQSTVX HJLOQSTVWX 16 OT OTV LOTV JLOTV HJLOTV HJLOTVX HJLOQTVX HJLOQSTVX HJLOQSTVWX 17 OT OTV LOTV JLOTV HJLOTV HJLOTVX HJLOQTVX HJLOQSTVX HJLOQSTVWX 18 OT OTV LOTV JLOTV HJLOTV HJLOQTV HJLOQTVX HJLOQSTVX HJLOQSTVWX 19 OT OTV LOTV JLOTV HJLOTV HJLOQTV HJLOQTVX HJLOQSTVX HJLOQSTVWX 20 OT OTV LOTV JLOTV HJLOTV HJLOQTV HJLOQTVX HJLOQSTVX HJLOQSTVWX Table 9: The combination of the selected sites. From Table 6, we have y(4) 2 = 75.15, y(4) 3 = 78.17, y(4) 6 = 84.02, y(4) 9 = 86.32. So β3(4) = (78.17 − 75.15)/(370 − 240) = 0.023231, β6(4) = (84.02 − 75.15)/(934 − 240) = 0.012781, β9(4) = (86.32 − 75.15)/(1419 − 240) = 0.009474. The value of βZs (T) is shown in Table 10. We draw the figure of βZs (T) under T = 4, 8, 12, 16, 20, as shown in Fig 7. From Table 10 and Fig 7, we could see that the value of βZs (T) is relatively large when Zs is small; and the value of βZs (T) is relatively small when Zs is large. This indicates that, the increment of b has relatively large influence on y(T) Zs when Zs is small. However, the increment of b has relatively small influence on y(T) Zs when Zs is large. So, we could see that, the effect of b on y(T) Zs is obvious when Zs is small, and the effect of b on y(T) Zs is not obvious when Zs is large. We also notice that, when T is constant, the reduction of d is fast first and then slow with the increase of Zs, on the whole. Similar to the previous analysis about y(T) Zs, let dZs (T) be the per capita transfer distance, γZs (T) be the reduction of d generated by the unit increment of the cost when the biggest total refuge time is T and the number of selected sites is Zs. Then γZs (T) = (d2(T) −dZs (T))/(bZs1 (T) − b21(T)), in which, Zs = 3, 4, . . . , 10. For example, when T = 4, from Table 8, we have b21(4) = 240, b31(4) = 370, b61(4) = 934, b91(4) = 1419. From Table 7, we have d2(4) = 62.8, d3(4) = 52.27, d6(4) = 38.27, d9(4) = 31.89. Then γ3(4) = (62.8 − 52.27)/(370 − 240) = 0.081, γ6(4) = (62.8 − 38.27)/(934 − 240) = 0.035346, γ9(4) = (62.8 − 31.89)/(1419 − 240) = 0.026217. The value of γZs (T) is shown in Table 11. We draw the figure of γZs (T) under T = 4, 8, 12, 16, 20, as shown in Fig 8. From Table 11 and Fig 8, we could see that, on the whole, the smaller the Zs, the larger the γZs (T), and the larger the Zs, the smaller the γZs (T). Then, we draw the conclusion that the increment of b has relatively large influence on d when Zs is small. While the increment of b has relatively small influence on d when Zs is large. Therefore, the effect of b on d is obvious when Zs is small, and the effect of b on d is not obvious when Zs is large. https://doi.org/10.15837/ijccc.2022.4.4749 19 Figure 7: The graph of βZs (T). Figure 8: The graph of γZs (T). T \Zs 3 4 5 6 7 8 9 10 1 0.021056 0.024161 0.023342 0.017406 0.013059 0.012673 0.011467 0.009570 2 0.020000 0.023212 0.022426 0.016667 0.012514 0.012130 0.010984 0.009160 3 0.018444 0.021861 0.021114 0.013761 0.013016 0.011341 0.010280 0.008572 4 0.023231 0.024911 0.019629 0.012781 0.012040 0.010444 0.009474 0.007903 5 0.020923 0.023036 0.018020 0.011729 0.011038 0.009497 0.008634 0.007193 6 0.018769 0.021295 0.016535 0.010749 0.010087 0.008629 0.007837 0.006523 7 0.023830 0.019777 0.015173 0.009856 0.009246 0.007850 0.007125 0.005922 8 0.022766 0.018393 0.013911 0.009035 0.008455 0.007423 0.006463 0.005362 9 0.021809 0.017054 0.012772 0.008285 0.007726 0.006786 0.005852 0.004850 10 0.021064 0.015893 0.011733 0.007608 0.007083 0.006211 0.005360 0.004426 11 0.020319 0.014866 0.010817 0.007003 0.006502 0.005688 0.004911 0.004051 12 0.019574 0.013929 0.009975 0.006470 0.005970 0.005226 0.004512 0.003709 13 0.019043 0.013214 0.009282 0.006009 0.005538 0.004825 0.004173 0.003422 14 0.018511 0.012545 0.008713 0.005620 0.005155 0.004300 0.003876 0.003169 15 0.017979 0.011875 0.008144 0.005259 0.004796 0.004004 0.003596 0.002930 16 0.017553 0.011339 0.007748 0.004971 0.004512 0.003767 0.003376 0.002746 17 0.017234 0.010848 0.007351 0.004712 0.004265 0.003550 0.003172 0.002568 18 0.016915 0.010357 0.006980 0.004467 0.003604 0.003343 0.002977 0.002398 19 0.016809 0.009955 0.006683 0.004251 0.003426 0.003166 0.002816 0.002268 20 0.016489 0.009554 0.006361 0.004049 0.003248 0.002988 0.002646 0.002131 Table 10: The increment of per capita score generated by the unit increment of cost. https://doi.org/10.15837/ijccc.2022.4.4749 20 T \Zs 3 4 5 6 7 8 9 10 1 0.060944 0.060073 0.055792 0.041921 0.030356 0.029586 0.025997 0.021352 2 0.061167 0.060219 0.054554 0.041100 0.030412 0.029645 0.026047 0.021393 3 0.061556 0.060438 0.054678 0.035216 0.033572 0.029744 0.026132 0.021469 4 0.081000 0.071429 0.054802 0.035346 0.033696 0.029842 0.026217 0.021537 5 0.081308 0.071563 0.054926 0.035490 0.032868 0.029191 0.026310 0.021619 6 0.072077 0.066161 0.051980 0.032983 0.030705 0.026696 0.024182 0.019911 7 0.057872 0.066161 0.052030 0.033040 0.030766 0.026726 0.024207 0.019939 8 0.057766 0.066205 0.052104 0.033127 0.030828 0.027084 0.024249 0.019980 9 0.057660 0.066205 0.052153 0.033184 0.030865 0.027136 0.023087 0.019044 10 0.057660 0.066205 0.052178 0.033242 0.030915 0.027177 0.023113 0.019071 11 0.057553 0.066205 0.052228 0.033300 0.030952 0.027218 0.023130 0.019092 12 0.057553 0.057232 0.047277 0.030447 0.028504 0.025185 0.021442 0.017732 13 0.057447 0.057143 0.044282 0.028718 0.026897 0.023860 0.020339 0.016851 14 0.057340 0.057054 0.044233 0.028718 0.026897 0.022179 0.020331 0.016844 15 0.057340 0.057009 0.038663 0.025490 0.024116 0.019961 0.018422 0.015307 16 0.057340 0.056920 0.038589 0.025476 0.024104 0.019931 0.018405 0.015294 17 0.024362 0.055000 0.037500 0.024841 0.023560 0.019911 0.018388 0.015287 18 0.024362 0.054955 0.037475 0.024841 0.020434 0.019901 0.018380 0.015287 19 0.024362 0.054955 0.037426 0.024827 0.020412 0.019892 0.018372 0.014795 20 0.024255 0.054866 0.037376 0.024813 0.020389 0.019655 0.018168 0.014631 Table 11: The reduction of per capita distance generated by the unit increment of cost. 4.2.2 Comparison We have set y(T) Zs prior to d and solved the model above. Next, we set the priority of the objectives in the order of d, f ′d, y(T) Zs, f ′y, b and f ′p (which is noted as the second order) to solve the model, i.e., d has priority over y(T) Zs. The combinations of the selected sites under the second order are shown in Table 12. We could derive the differences of y(T) Zs (noted as ∆y) and the differences of d (noted as ∆d) respectively under the two orders for each T and each Zs. The results are shown in Table 13. We could see from Table 13 that, the solutions of y(T) Zs and d under the first order are both not lower than that under the second order, for the same T and Zs. From Table 9 and Table 12, we know that the combinations of the selected sites are the same under the two orders, when the difference is 0 in Table 13. Otherwise, the combinations of the selected sites are not the same under the two orders, when the difference is not 0 in Table 13. 4.2.3 Discussions In this case, we set the different orders of priority for the six objectives to solve the model. In the first order, we take the y(T) Zs as the highest priority, from the solutions, we know that, when T remains constant, the y(T) Zs and the d shows upward and downward trend respectively with the increase of Zs. It could be seen that the satisfaction of the residents and the transfer efficiency are both constantly rising with the increase of Zs. So, in order to improve the satisfaction of the residents and the transfer efficiency, we could consider selecting more emergency shelters. Meanwhile, we could see that, on the whole, f ′y and f ′d both show downward trend with the increase of Zs when T remains constant. This reveals that more emergency sites mean greater fairness. So, in order to improve the fairness in location decision making, we could consider selecting more emergency sites. From the previous analysis, we also notice that, when T remains constant, the same increase of fund could produce larger increment of βZs (T) and larger reduction of γZs (T) when Zs is relatively small, then the efficiency of funds is obvious. Otherwise, the efficiency of funds is not obvious when Zs is relatively big. So, in order to make full use of funds, we should avoid selecting excessive emergency shelters. In addition, from the analysis in 4.2.2, we know that, if we take y(T) Zs as the highest priority of the objectives in the model, the satisfaction of the residents for the emergency sites will be improved. Thus, https://doi.org/10.15837/ijccc.2022.4.4749 21 T \Zs 2 3 4 5 6 7 8 9 10 1 OW LOS JLOV JLOVX JLOQVX JLOQTVX HJLOQTVX HJLOQSTVX HJLOQSTVWX 2 OW LOS LOSV JLOVX JLOQVX HJLOQVX HJLOQTVX HJLOQSTVX HJLOQSTVWX 3 OW LOS LOSV JLOVX JLOQTV JLOQTVX HJLOQTVX HJLOQSTVX HJLOQSTVWX 4 OW LOS LOSV JLOVX JLOQTV JLOQTVX HJLOQTVX HJLOQSTVX HJLOQSTVWX 5 OW LOS LOSV JLOVX JLOQTV HJLOQTV HJLOQSTV HJLOQSTVX HJLOQSTVWX 6 OW LOS LOSV JLOVX JLOQVX HJLOQVX HJLOQSVX HJLOQSVWX HJLOQSTVWX 7 OW LOS LOSV JLOVX JLOQVX HJLOQVX HJLOQSVX HJLOQSVWX HJLOQSTVWX 8 OW LOS LOSV JLOVX JLOQVX HJLOQVX HJLOQSVX HJLOQSVWX HJLOQSTVWX 9 OW LOS LOSV JLOVX JLOQVX HJLOQVX HJLOQSVX HJLOQSVWX HJLOQSTVWX 10 OW LOS LOSV JLOVX JLOQVX HJLOQVX HJLOQSVX HJLOQSVWX HJLOQSTVWX 11 OW LOS LOSX JLOSX JLOQVX HJLOQVX HJLOQSVX HJLOQSVWX HJLOQSTVWX 12 OW LOS LOSX JLOSX JLOQVX HJLOQVX HJLOQSVX HJLOQSVWX HJLOQSTVWX 13 OW LOS LOSX JLOSX JLOQVX HJLOQVX HJLOQSVX HJLOQSVWX HJLOQSTVWX 14 OW LOS LOSX JLOSX JLOQVX HJLOQVX HJLOQSVX HJLOQSVWX HJLOQSTVWX 15 OW LOS LOSX JLOSX JLOQVX HJLOQVX HJLOQSVX HJLOQSVWX HJLOQSTVWX 16 OW LOS LOSX JLOSX JLOQVX HJLOQVX HJLOQSVX HJLOQSVWX HJLOQSTVWX 17 OW LOS LOSX JLOSX HJLOSX HJLOQSX HJLOQSWX HJLOQSVWX HJLOQSTVWX 18 OW LOS LOSX JLOSX HJLOSX HJLOQSX HJLOQSWX HJLOQSVWX HJLOQSTVWX 19 OW LOS LOSX JLOSX HJLOSX HJLOQSX HJLOQSWX HJLOQSTVX HJLOQSTVWX 20 OW LOS LOSX JLOSX HJLOSX HJLOQSX HJLOQSWX HJLOQSTVX HJLOQSTVWX Table 12: The combinations of the selected sites under the second order. T \Zs 2 3 4 5 6 7 8 9 10 ∆y ∆d ∆y ∆d ∆y ∆d ∆y ∆d ∆y ∆d ∆y ∆d ∆y ∆d ∆y ∆d ∆y ∆d 1 0.19 6.00 1.23 2.62 0.55 2.42 1.76 0.51 1.48 0.58 0.03 0.95 0.00 0.00 0.00 0.00 0.00 0.00 2 0.35 6.02 1.28 2.60 1.07 2.20 1.78 0.51 1.50 0.58 1.51 0.58 0.00 0.00 0.00 0.00 0.00 0.00 3 0.59 6.06 1.35 2.58 1.15 2.22 1.82 0.51 0.04 0.64 0.05 0.65 0.00 0.00 0.00 0.00 0.00 0.00 4 0.85 6.09 1.45 3.17 1.24 2.86 1.86 0.50 0.10 0.60 0.09 0.60 0.00 0.00 0.00 0.00 0.00 0.00 5 1.14 6.13 1.57 3.19 1.35 2.92 1.89 0.50 0.15 0.56 0.08 1.05 0.14 0.38 0.00 0.00 0.00 0.00 6 1.42 6.17 1.70 4.43 1.47 4.06 1.94 1.73 1.75 2.14 1.67 2.07 1.34 1.20 1.52 0.40 0.00 0.00 7 1.68 6.19 1.87 8.40 1.60 4.12 2.00 1.74 1.83 2.13 1.76 2.08 1.38 1.24 1.55 0.45 0.00 0.00 8 1.92 6.22 2.12 8.45 1.73 4.18 2.05 1.75 1.91 2.11 1.83 2.10 1.41 2.03 1.57 0.48 0.00 0.00 9 2.15 6.25 2.35 8.49 1.83 4.24 2.11 1.76 1.98 2.10 1.89 2.12 1.49 2.04 1.58 1.92 0.00 0.00 10 2.35 6.27 2.56 8.52 1.92 4.29 2.15 1.78 2.04 2.08 1.95 2.13 1.56 2.06 1.66 1.95 0.00 0.00 11 2.53 6.29 2.75 8.56 3.36 3.49 3.43 0.70 2.10 2.06 2.00 2.14 1.61 2.06 1.72 1.98 0.00 0.00 12 2.7 6.31 2.91 8.58 3.45 5.53 3.47 2.72 2.17 4.07 2.05 4.16 1.68 4.09 1.79 4.02 0.00 0.00 13 2.84 6.33 3.07 8.61 3.55 5.57 3.53 3.94 2.24 5.28 2.13 5.50 1.75 5.41 1.88 5.36 0.00 0.00 14 2.98 6.34 3.21 8.64 3.63 5.61 3.60 3.97 2.32 5.29 2.21 5.52 1.82 6.19 1.96 5.40 0.00 0.00 15 3.11 6.35 3.34 8.66 3.70 5.64 3.66 6.23 2.41 7.55 2.28 7.80 1.91 8.48 2.04 7.69 0.00 0.00 16 3.22 6.36 3.45 8.67 3.77 5.68 3.75 6.27 2.50 7.57 2.37 7.83 2.01 8.53 2.14 7.74 0.00 0.00 17 3.32 6.38 3.57 11.79 3.84 6.13 3.82 6.72 3.82 6.73 3.69 6.77 3.71 6.21 2.22 0.98 0.00 0.00 18 3.42 6.39 3.67 11.81 3.90 6.16 3.89 6.54 3.89 6.54 3.75 7.28 3.78 6.03 2.26 0.99 0.00 0.00 19 3.50 6.40 3.78 11.82 3.96 6.17 3.96 6.57 3.95 6.57 3.83 7.31 3.85 6.06 0.00 0.00 0.00 0.00 20 3.59 6.41 3.87 11.83 4.01 6.20 4.01 6.20 4.02 6.20 3.90 6.95 3.91 5.92 0.00 0.00 0.00 0.00 Table 13: The ∆y and the ∆d under the two orders. https://doi.org/10.15837/ijccc.2022.4.4749 22 the behavior of residents will better follow the planning of the government in emergency planning. In the case, we only take f ′p as the lowest priority among the six objectives of the model. If we upgrade the priority level of f ′p, the numbers of the residents among all emergency shelters will be more balanced. Thus, the residents will not be excessively concentrated in some emergency shelters. Therefore, emergency planner should consider the priority order of the six objectives, according to the actual situation. 5 Conclusion This paper study the emergency planning problem considering the subjective preference. We select six attributes to analyze the subjective preference of the residents. According to different evaluation methods, the six attributes are divided into three categories, then initial subjective score of every demand point for every candidate emergency site is obtained. Base on this, we assume that the subjective preferences of the residents for the attributes of the emergency shelters change with the total refuge time, i.e., the preference weights of the residents for the attributes of the emergency shelters are dynamic. Therefore, the subjective scores of the residents for the emergency shelters also change with the total refuge time. In the emergency planning, it is impossible for us to make a location decision for each total refuge time. Under the condition that the probability of each total refuge time is the same, we take the average value of the scores at all total refuge times as the primary basis for location decision making. For improving the transfer efficiency, we minimize the per capita transfer distance. Considering fairness, we minimize respectively the standard deviations of the scores and the distances of the residents at all demand points. Considering safety, the standard deviation of the numbers of the residents among the selected shelters is taken as a factor in emergency planning, so as to avoid the excessive gathering of the residents. In addition, we propose the objective of minimizing the total cost. The model is applied to a case, we find that, the more the emergency shelters, the higher the satisfaction degree of the residents, and the shorter the per capita distance, the more fairness is satisfied. However, the utilization effect of funds decreases with the increase of the number of the emergency shelters. Therefore, in order to improve residents’ satisfaction and transfer efficiency, and avoid the inefficient utilization of funds, we should consider selecting an appropriate number of emergency shelters. In this paper, we assume the emergency shelter has no capacity constraint. 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Capacity Building Lessons from a Decade of Transitional Settlement and Shelter. International Journal of Strategic Property Management, 2009, 13(3): 247–265. https://doi.org/10.15837/ijccc.2022.4.4749 25 Copyright ©2022 by the authors. Licensee Agora University, Oradea, Romania. This is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial 4.0 International License. Journal’s webpage: http://univagora.ro/jour/index.php/ijccc/ This journal is a member of, and subscribes to the principles of, the Committee on Publication Ethics (COPE). https://publicationethics.org/members/international-journal-computers-communications-and-control Cite this paper as: Yiying Wang, Zeshui Xu; (2022). A Multi-objective Location Decision Making Model for Emer- gency Shelters Giving Priority to Subjective Evaluation of Residents, International Journal of Com- puters Communications & Control, 17(4), 4749, 2022. https://doi.org/10.15837/ijccc.2022.4.4749 Introduction Methodology Description of the problem and determining the value of each attribute Dynamic weights of the attributes The formulation of the model The formulation of the score function and the standard deviation function The formulation of the transfer distance function and the standard deviation function Cost and the formulation of the standard deviation function of the numbers of residents transferred to every emergency shelter Goals and constraints Case study Basic data Analysis and discussions Analysis Comparison Discussions Conclusion