Multiobjective Site Selection Model for Emergency Shelter Facilities in Urban Areas


 Industrial and economic development is primarily applied to densely populated urban areas. If a sudden disaster occurs in such areas, the consequences can be severe. Shelter facility location affects the implementation of postdisaster relief work. This study explored residents’ perceived utility of evacuation time, their risk utility for road blocking, and the cost factors associated with constructing shelter facilities in the context of governance. A location model for emergency shelter facilities in cities was established on the basis of the aforementioned factors. Because the resolution of the random-weighted genetic algorithm (RWGA) is susceptible to influence from random weights, a robustness random-weighted method (RRWM) was developed. The validity and feasibility of the location model were examined through numerical analysis. Finally, the convergence of the RRWM was analyzed and compared with that of the RWGA and a single-objective genetic algorithm. The results revealed that the proposed algorithm exhibited satisfactory performance and can assist in evaluation and decision-making related to the selection of urban shelter facility locations.


Introduction 25
Climate change and social disruption may lead to natural or artificial disasters 26 such as typhoons, earthquakes, and nuclear accidents, all of which can substantially 27 affect urban economies [1]. After a sudden large-scale disaster or major accident, the 28 affected population must immediately seek refuge in safe and shelter facilities. 29 Therefore, the selection of shelter facility locations is crucial to urban disaster 30 prevention and urban development, and the locations of such facilities should be 31 based on pedestrian patterns. 32 Existing places of refuge should be selected as locations for facilities with short-33 and medium-term sheltering functions. Resources can then be allocated to improve 34 and reconstruct these facilities and to stockpile supplies in order to meet safety 35 standards. Therefore, numerous factors must be considered during the site selection 36 process. Accordingly, the "Easy-to-Do" program of the National Science and 37 Technology Center for Disaster Reduction (Taiwan) considers the physical 38 environment, shelter-related facilities, and factors such as distance to a sheltered 39 facility (https://easy2do.ncdr.nat.gov.tw/easy2do/). Several studies have suggested 40 that the time required to seek refuge is a critical factor for site selection [2,3]. 41 However, unsuspecting city residents may be influenced by psychological factors 42 such as panic and fear during a disaster because of the chaos caused by the 43 interruption of urban traffic networks and the breakdown of communication. Lin et al. 44 [4] have thus suggested that panic following a disaster hinders individual judgment. 45 The public may not act entirely rationally when confronted by an unexpected disaster 46 Because road networks may be blocked or destroyed following a disaster, the 55 urban space undergoes drastic changes. When residents with bounded rationality are 56 confronted by these changes, they may experience difficulty in making accurate risk 57 assessments while on the road. The risk of road network blockage affects the safe 58 travel of residents to shelter facilities; therefore, this phenomenon should be 59 considered during site selection [8]. Moreover, medium-and short-term shelter 60 facilities should be constructed even if no urban disasters are expected. In the 61 context of budgeting [1], resources should be allocated to reconstruction and the 62 establishment of shelter environments that conform to disaster resistance standards. 63 The aforementioned factors directly affect the efficiency of shelter selection 64 influencing shelter site selection, Zhu and Wang [21] have improved the conventional 144 spatial location selection problem by establishing a road emergency evacuation index 145 and a road risk coefficient to quantify road network risks. In addition, Hsu and Lu 146 [22] determined the risk of earthquake-induced road blockages. They created a joint 147 utility function by combining this risk with the influence of traffic load on traffic 148 congestion to obtain the path of least risk for earthquake relief. By combining a 149 After an appropriate urban shelter site is selected from various alternatives 156 (excluding potentially dangerous sites), resources are invested to construct facilities 157 according to the required medium-or short-term functions of the site. Therefore, the 158 cost of construction and the distance from the affected area are factors that warrant 159 consideration in shelter site selection processes for construction. Karatas  is not exceeded and that the distance between residences and shelters is minimized. 173 Their results indicated that the model with capacity constraints was superior to that 174 without such constraints because it could more accurately reflect the real-life situation 175 of the problem. programs and compared the advantages and disadvantages of these algorithms. 220 Among these methods, the aggregation function was the first to be developed and is 221 the most direct approach for solving multiobjective optimization problems. In this 222 method, a single-objective solution is obtained for the multiobjective problem by 223 adjusting the weight coefficient through combination or aggregation. The RWGA 224 proposed by Murata and Ishibuchi [9] are based on weight summation. Murata and 225 Ishibuchi [9] compared the RWGA with the vector-evaluated GA (VEGA). Their 226 results revealed that the RWGA yielded more optimal solutions than did the VEGA. 227

Comprehensive evaluation and analysis
In the present study, the public perception of evacuation time, road availability,

Research model construction 253
In the conventional maximum coverage problem, the aim is to minimize the 254 average travel distance between the demand and service nodes. Therefore, demand 255 nodes that are located outside the maximum service distance from a given service 256 node must be covered by other service nodes. However, in the context of traveling to 257 shelter facilities after disasters, the service level of shelter facility j is not limited to its 258 spatial distance from disaster site i. The evacuation time tij between disaster site i and 259 shelter facility j should be the main basis for assessment. Therefore, ij i tL  indicates 260 that victims at disaster node i feel safe traveling to shelter node j. The term Li denotes 261 the longest time that people at disaster node i are willing to accept when evacuating to shelter facility j. 263 Disaster victims decide to leave the affected area to seek shelter according to their 264 expectations of the facility and environmental factors. In this case, distance is the 265 main consideration in shelter site selection processes, and the travel time between the 266 disaster site and the shelter facility is ignored. However, in practice, urban spaces 267 change substantially after disasters, and the members of the public are in a state of 268 bounded rationality. Therefore, an accurate perception of the distance to shelter 269 facilities may be difficult for the public to obtain.
We can assume that the utility function of perceived evacuation time is nonlinear. The 292 value of ki can be considered the sensitivity coefficient for evacuation time. This 293 parameter represents the sensitivity of people in different regions (e.g., cities and rural 294 areas) to the evacuation time. The higher the value of ki is, the higher the gradient of the utility function of perceived evacuation time is, which indicates greater time 296 sensitivity. Ma et al. [20] suggested that ki should be between 0.5 and 1.5. The effect 297 of sensitivity coefficient ki is illustrated in Figure 1. Individual perceptions of 298 evacuation time can vary even in the same area. However, the aim of the present 299 study was not to estimate individual heterogeneity. Therefore, the utility function of 300 known roadblock risk, a utility function is used to convert the risk value into a utility 307 value. Herein, the utility function for roadblock risk is a decreasing exponential utility 308 function. If the roadblock probability is 0, the road section is unaffected, and the 309 utility value is 1. If the roadblock probability is 1, the road section has been severely 310 damaged, and its safety and reliability are extremely low; thus, the utility value is 0. 311 The utility function of roadblock risk for road section a is defined as follows: where Ra is the roadblock probability for road section a. The utility value of the 314 roadblock risk for a section reflects the safety and reliability of the road. It can also be 315 considered to be the probability of being able to pass through a road section. The 316 higher the utility of the roadblock risk is for a road section, the higher is the 317 probability of disaster victims being able to use this section to reach shelter facilities. 318 Therefore, on the basis of the discussion on road passability by Shen et al. [7], the 319 utility of the roadblock risk of road section a is defined as the passability of road 320 section a, as presented in Eq. (4). Moreover, ij k u is defined as the utility value of 321 roadblock risk for path k from disaster node i to shelter node j. Similarly, ij k p is 322 defined as the probability that path k can be used to travel from disaster node i to 323 shelter node j. Therefore, the risk of a roadblock in a road section and the utility value 324 of the risk of a roadblock are defined as follows: 325 Eqs. (6) and (7) can then be combined as follows: 334

Algorithm steps 379
In the RWGA, random weights are initialized, and an optimal solution is searched for through the evolution of each weight [8]. However, because this method is 381 susceptible to random values, the quality and efficiency of its solutions can be 382 inconsistent. Therefore, we developed the RRWM, which is based on the RWGA. The 383 RRWM has two components. EA is designed to tend toward the global solution. Therefore, the fitness function for a 393 multiobjective problem can be redefined as follows to determine the closest ideal 394 solution based on the CPM and AWA: 395 where SOL is the solution set for the multiobjective problem, q is the number of 397 objectives, and k i z is the value of the ith objective function of the kth solution in 398 SOL. If objective i is fixed (e.g., i = 1), k i z can be considered the result of the 399 standardization of the kth solution in the solution set for objective i. Therefore, we can 400 standardize each objective function as follows: 401 Eqs. the multiobjective problem is presented in Eq. (27). 413 In the second component of the RRWM, the set of Pareto optimal solutions 414 produced in each generation is adjusted according to the weights randomly generated 415 in the current generation. This effect is reflected in the quality of the current 416 generation's solution and that of the overall multiobjective solution. Therefore, the 417 elitist strategy is adopted to select the superior solution from the set of Pareto optimal 418 solutions in each generation. Finally, the elite Pareto optimal solution set is obtained 419 to normalize the quality of the Pareto optimal solutions. The steps of the RRWM are 420 described as follows: 421 Step 1: Initiate the algorithm. 422 Step 2: Calculate the network values. 423 Based on given postdisaster information, the optimistic evacuation time (tij,optimistic) , 424 the actual evacuation time (tij) , and the longest evacuation time (tij,longest) between 425 disaster node i and shelter node j is obtained using the shortest path algorithm, and the 426 utility function of perceived evacuation time is derived according to Eq. (4). 427 roadblock risk value Ra for each road section a. 429 Step 3: Encode the network nodes. 430 Binary gene encoding [0,1] is applied with the decision variable yij under the 431 assumption that chromosome length is equal to the total number of shelter and 432 disaster nodes, where 0 represents a disaster node and 1 represents a shelter node. 433 Each chromosome represents a feasible solution-a configuration of shelter nodes. 434 Step 4: Randomly generate an initial population of chromosomes and place the initial 435 population in Npop, and then set the total number of generations T. 436 Step 5: Evolve the chromosomes. Step 6: Calculate the fitness value. 442 Eq. (28) is used to obtain the random weights, which are then substituted into Eq. (27)  Step 7: Select the elite chromosomes (Nelite) from the Pareto optimal solution set. 449 The chromosomes with the highest fitness values in the Pareto optimal solution set are 450 selected as the elite chromosomes (Nelite). 451 Step 8: Perform mating. 452 The single-point mating method is applied to the selected chromosomes with a mating 453 rate RC of 0.8 and randomly selected mating sites. Two new chromosomes are 454 produced, with the mating site serving as the baseline. This mating mechanism yields 455 new chromosomes for the population Npop. 456 Step 9: Perform mutation. 457 A certain number of genes in the chromosome are mutated at a mutation rate Rm of 458 0.06. The selected genes are mutated from 0 to 1 or from 1 to 0. 459 Step 10: Apply the elitist strategy. 460 Npop. Next, Nelite additional chromosomes are randomly selected from the current 462 Pareto optimal solution set and added to Npop to replace the chromosomes that were 463 randomly removed. 464 Step 11: Terminate the algorithm according to the condition test. 465 The condition for termination in this model is reaching the maximum number of 466 generations T. If this condition is satisfied, the algorithm is terminated. If the 467 condition is not satisfied, set t = t +1 and return to Step 4. 468 This algorithm yields a set of elite Pareto optimal solutions, and the most suitable 469 compromise solution can be selected from this set. 470

Evaluation of solution sets for the multiobjective problem 471
In the MOGA, solutions are obtained by approaching the Pareto optimal front 472 through continuous evolution. The present study adopted the assessment of solution

Test network data 496
This study used Zhongzheng District, Taipei City, as a test network. Figure 2  497 shows that this network contains 31 villages, 153 nodes, and 481 road links. The

4.2.Testing and analysis 506
In this study, the RRWM was used to solve the multiobjective problem of 507 selecting urban shelter facility sites. The number of selected shelter facilities should 508 not exceed the total number of available sites (i.e., 32). The total number of 509 chromosomes in Npop was set to 500, the number of generations was set to 500, the 510 mating rate Rc was set to 0.8, and the mutation rate Rm was set to 0.06. Key 511 information from the test results is presented subsequently. 512 With 500 generations, the RRWM was able to search the entire solution space. 513 The total computation time was 249 s, and 500 optimal solution sets were obtained. 514 The minimum adaptive value was 0.58 and was obtained after solution set 340; thus, 515 this set was the superior compromise solution set. The node numbers of the refuge 516 facilities corresponding to the optimal compromise solution included 122, 123, 126, 517 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 143, 144, 147, 148, 149, 518 150, 152, and 153. According to Eqs. (9)-(11), the total utility of perceived 519 evacuation time was derived as 19257.55 (Objective 1), the total utility of roadblock 520 risk was 67.92 (Objective 2), and the total construction cost was NT$77 million 521 (Objective 3). 522 Table 3 presents the assignment of victims from disaster to shelter facilities. For 523 example, the equivalent number of victims at node 8 was 235. Because of capacity 524 constraints, 144 equivalent number of victims were assigned to node 132 for shelter. 525 On the basis of the developed allocation mechanism, the remaining 91 equivalent 526 number of victims were assigned to node 140, which exhibited the second-greatest 527 utility of perceived evacuation time. Moreover, node 140 was located near node 132. 528 Overall, the sum of the equivalent number of victims assigned to the different shelter 33 nodes was equal to the equivalent number of victims at the disaster nodes. Thus, the 530 solution satisfied the constraint for the multiobjective model presented in Eq. (18). 531 Table 4 shows the sum of the equivalent number of victims from each 532 disaster-affected node to the shelter node, which meets the shelter facilities'  Pareto optimal solution set were input into Eq. (31), and a spatial distribution of 545 solutions are uniformly distributed in the three-dimensional space defined by the 547 normalized objective values. We used STATISTICA 6.1 software to plot the spatial 548 distribution of the solution set ( Figure 3). The distribution of the solution set near the 549 origin was similar to that of the Pareto optimal front, demonstrating the suitability of 550 the RRWM. 551 Table 5   and 0.04 (as shown in Figure 5). However, this solution set was not forming a Pareto 566 frontier either. By contrast, the spatial distribution value of the RRWM was 567 suboptimal; however, the solution set was close to a Pareto optimal front when plotted 568 in the space defined by the normalized objective function values. Moreover, the 569 solution set of the RWGA was oriented in the direction of the 1 Z  -axis with a 570 suboptimal spatial distribution. For other weighting strategies, the solution sets were 571 oriented in the direction of the axis with the highest enactment value (i.e., aligned 572 according to the specific weight ratio). 573

Conclusions and recommendations for future work 574
This study presents a model for site selection for shelter facilities after large-scale 575 urban disasters. A trade-off exists among the utility of perceived evacuation time, the 576 utility of roadblock risk, and the cost of shelter construction. Accordingly, the 577 relationships between these parameters were modeled using a multiobjective model. 578 The locations selected for shelter facilities should be the optimal compromise between 579 the aforementioned parameters. The number of people required to move from disaster 580 nodes to shelter nodes and the capacities of the shelter facilities are also included in the developed model. 582 In contrast to the RWGA, the proposed RRWM includes an elitist mechanism and 583 is designed to evolve an evenly distributed trade-off frontier defined by nonconvex 584 functions. The RRWM yields a nondominated solution set with satisfactory 585 distribution; hence, it may provide valuable assistance to decision-makers. This 586 finding demonstrates the flexibility of the proposed method for practical planning 587 problems and its effectiveness for evaluating decision schemes. 588 This study was limited to a single category of displaced people. In the future, 589 different identities can be included to solve problems cause by multiple identities. 590 Information on the utility of perceived evacuation time and the utility of roadblock 591 risk can be collected during regular household surveys and urban environmental 592 audits. The main focus of the present study was on modeling and algorithm design. In 593 future studies, the proposed model can be applied and evaluated by adding parameters 594