Natural disasters are threats to human life and the ecosystem in general. Climate changes as well as environmental changes—e.g., deforestation—increase the frequency and intensity of natural disasters such as hurricanes, floods, and landslides [1
]. Such extreme catastrophes cause many losses in lives, affect the economy, and leave many damages to the affected area. However, disaster effects can be reduced if the society is prepared and plans—e.g., for evacuation—are in place.
Evacuation is a means to save lives and is incorporated in both preparedness and response phases of disaster operations management [2
]. However, evacuation planning is a complex process and more crucial in urban areas due to the high population density and complex urban settlement. A critical challenge in evacuation planning is to find optimum evacuation time and a proper shelter allocation such that they have enough space for evacuees and other basic living requirements. This means that evacuation planning is considered as a complex multi-criteria decision problem with conflicting objectives, constraints, and spatial aspects. Usually, such problems are modeled by multi-objective optimization techniques, which often provide decision-makers with quick responses and reliable solutions to the problem.
Various studies reported the complexity of evacuation planning in disaster operations management (DOM) and proposed techniques to solve them [3
]. Thus, depending on the type of disaster and according to the aim of emergency planners, evacuation problems have been modeled as network/routing problems [5
], transportation problems [7
], or location-allocation problems [9
]. This study considers evacuation planning as a location-allocation problem.
There are two approaches for solving multi-objective optimization evacuation problems: exact methods and metaheuristic methods. The exact methods—such as linear programming, goal programming, mixed-integer programming, and weighted summation—have been widely used for many decades in disaster operations [12
]. These methods combine the criteria/objectives of a multi-objective optimization problem with a set of weights provided by decision-makers. Doing so, a single-objective optimization model is created and then a conventional mathematical programming algorithm can be used to solve the problem. Cova and Johnson [14
] presented a network flow model for lane-based evacuation routing. Mixed-Integer programming was used as a solution method to identify an optimal lane-based evacuation routing plan in a complex road network.
Although these old and traditional methods have been widely used to solve multi-objective optimization problems, they have limitations when applied to real-world problems. For example, these techniques are often extremely time-consuming to solve real-world problems with large dimensions, hardly constrained problems, and multimodal problems. In addition, the final solution is highly influenced by, and biased towards, the initial weights provided by experts at the early stage of the algorithm. To overcome these limitations, researchers have used multi-objective optimization instead of single-objective optimization in order to design and solve evacuation problems. Metaheuristic algorithms have the ability to produce good quality solutions in reasonable computation time, good enough for a practical purpose [15
]. They are also not biased with the preferences of experts since no initial weighting of criteria is needed. However, not all of these algorithms are efficient, a few algorithms have proved their capacities for solving real-world problems [17
]. Moreover, each algorithm has its own limitations. Therefore, it is very important to conduct a comparative study of metaheuristic algorithms on a specific real-world problem.
Saeidian et al. [19
] compared two metaheuristic algorithms for location-allocation of earthquake relief centers—genetic algorithm (GA) and bees algorithm (BA). Their results show that BA converges faster than GA, while GA is more favorable in terms of repeatability of the algorithm. Also, Saeidian et al. [20
] compared particle swarm optimization (PSO) and ant colony optimization (ACO) using different criteria. The study found that PSO outperformed ACO in terms of quality of solutions, better convergence, and consistency. Xu et al. [21
] applied a modified particle swarm optimization (PSO) algorithm combined with a simulated annealing (SA) algorithm to derive solutions using the hybrid bi-level model and conventional multi-objective model for shelters location-allocation problems. The hybrid bi-level model was proven to be useful for optimal shelter allocation. As mentioned in the study by Caunhye et al. [22
], the multi-objective approaches are less used and more advanced algorithms are needed to solve many problems in DOM including evacuation.
This paper aims to compare the performance of four metaheuristic algorithms extended from the standard algorithms of simulated annealing (archive multi-objective simulated annealing—AMOSA), artificial bee colony (multi-objective artificial bee colony—MOABC), genetic algorithm (non-dominated sorted genetic algorithm-II—NSGA-II), and particle swarm optimization (multi-objective version of standard particle swarm optimization—MSPSO) for evacuation planning. The four algorithms along with geographic information systems (GIS) are used to solve an urban evacuation problem on a study area in the city of Kigali, Rwanda. Their performance is evaluated based on effectiveness, efficiency, consistency, and computational time for each algorithm.
The remainder of this paper is organized as follows: In Section 2
we review the metaheuristic algorithms and give an overview of the tested four algorithms; Section 3
describes the study area and data preparation; Section 4
explains the methodology used in this study; Section 5
presents the results and analysis, and Section 6
concludes the paper and provides future research directions.
2. An Overview of Metaheuristic Algorithms
Multi-objective optimization problems (MOOP) involve more than one objective function that is to be minimized or maximized. An answer to these types of problems is to find a set of solutions that define the best tradeoff between conflicting objectives. In recent decades, there has been a trend in the scientific community to solve MOOPs by using metaheuristic methods over exact methods. A metaheuristic is defined as a procedure or technique designed for finding the approximate solution in a short time (low computation time) [23
]. Metaheuristic approaches categorized as population-based metaheuristics are emerged to find optimal solutions through the iterative process of generating a new population through natural selection. According to Fister Jr. et al. [24
], evolutionary algorithms or bio-inspired-based and swarm-intelligence-based algorithms are the most interesting and widely used approaches in population-based metaheuristics. GA and its variants represent a group of evolutionary algorithms, while ABC, ACO, and PSO are three approaches grouped in swarm-intelligence-based algorithms. Those four algorithms are commonly used to solve real-world problems [15
]. Another category of metaheuristics is physics/chemistry-based algorithms, which mimic certain physical and or chemical phenomena, including for instance electrical charges, temperature changes, and gravity or river systems. Such algorithms solve a problem based on the process of improving a single solution. SA is the commonly used algorithm in this category [25
]. These five metaheuristic algorithms are all global optimization methods and can solve higher-dimensional problems; they are robust with respect to the complexity of the evaluation of functions. They can easily be adjusted to the problem at hand. On the other hand, although a lot of research has used these algorithms, the question of finding which one is the best suited for a specific problem has not been answered satisfactorily. Furthermore, maintaining the diversity of optimal solutions and premature convergence of solutions to local optima are still crucial to population-based algorithms.
In order to evaluate all categories, this study used the multi-objective version of four approaches, that is NSGA-II to represent evolutionary algorithms, MOABC and MSPSO to represent swarm-intelligence-based, and AMOSA to represent physics/chemistry-based algorithms. A brief review of each approach is discussed in the following.
2.1. Archive Multi-Objective Simulated Annealing Algorithm
Archive multi-objective simulated annealing (AMOSA) is a global optimization algorithm adapted from the process of annealing in metallurgy. Bandyopadhyay et al. [27
] proposed the AMOSA algorithm based on the principle of the original Simulated Annealing (SA) algorithm [28
]. In AMOSA, the Pareto dominance approach is adopted and uses the concept of an archive to store all non-dominated solutions. The archive size is limited with two parameters known as hard limit (HL) and soft limit (SL). The HL is the maximum size of the archive on termination, and it is equal to the number of non-dominated solutions required by the user; while SL is the maximum size to which the archive may be filled before clustering is used. The algorithm starts with the set of solutions randomly initialized and refined in the archive by using a hill-climbing technique. A solution is added in the archive if it dominates the previous one and exceeds the HL. If the archive reaches the SL size, then the well-known single-linkage clustering is used to reduce the size of the archive to HL in order to keep a diversity of non-dominated solutions [29
]. In the main loop of AMOSA, three cases can occur in dominance:
The current solution dominates the new solution and k points from the archive dominate the new solution. In this situation, a new solution can be accepted as the current solution with a given probability.
The current solution and the new solution are non-dominating with respect to each other. Here, the domination status of a new solution and members of the archive are checked through three situations: when a new solution is dominated by k points in the archive, the new solution is non-dominating with respect to the points in the archive, and when new solution dominates k points of the archive.
The new solution dominates k points of the archive. Here the new solution is selected as the current solution and also added to the archive, while all the k dominated points in the archive are removed. The process in the main loop is repeated through the number of iterations for each temperature, which is reduced to at each iteration using the cooling rate alpha until the minimum temperature is reached. Thereafter, the process stops and the resulting archive contains the final non-dominated solutions.
AMOSA algorithm is capable of solving problems with many objective functions. It has been used to solve medical and engineering-related problems [30
], but so far there is no literature on AMOSA applied to solve evacuation problems.
2.2. Multi-Objective Artificial Bee Colony Algorithm
Akbari et al. [32
] proposed a multi-objective artificial bee colony algorithm (MOABC) based on the standard ABC algorithm developed by Karaboga [33
]. Recently, a variant version of MOABC developed based on ABC has been used to solve evacuation problems [34
]. In this study, the MOABC colony consists of three groups of artificial bees: employed, onlookers, and scout bees. This algorithm generates a number of solutions and works through optimizing them. First, a number of scout bees explore the search space of the problem randomly and generate solutions as the initial population. The quality of the solutions is evaluated (fitness value) and the best solutions are stored in the external memory (archive). The scout bees that have high fitness are selected to act as employed bees. Each employed bee explores the neighborhood to update its position. Onlooker bees select a solution with a high amount of fitness from the neighborhood of employed bees. A new scout bee makes a new generation of the solution if the onlooker failed to update the quality of the solution. Then, the fitness values of all bees are compared to select the best solution and store it to the archive. The Pareto-based approach proposed by Deb et al. [36
] has been used to rank the non-dominated solutions into Pareto fronts. The archive is updated by non-dominated solutions, at each iteration. The MOABC algorithm terminates when the termination conditions are met, and the archive returns the final best solutions as output.
2.3. Multi-Objective Standard Particle Swarm Optimization Algorithm
Particle swarm optimization (PSO) is a population-based metaheuristic algorithm introduced by Kennedy and Eberhart [37
]. PSO is a swarm intelligence algorithm inspired by the social behavior of bird flocks, fish school. The algorithm has many variants due to its flexibility and robustness in terms of updating the way the velocity of the particle is updated [38
]. This velocity is the speed of a particle which is used to find its next position in search space. A particle updates its position through topological relationships in the neighborhood. The links between particles facilitate to share information about the previous best position of particles from one to another. PSO has been adapted in many studies related to evacuation planning [40
In this study, we used the recent standard PSO (SPSO) that was proposed to provide common procedures and guidance to improve the original PSO [44
]. However, the proposed SPSO is not for solving complex problems with many objectives. Therefore, we applied a Pareto-based method to evaluate the two objectives simultaneously, and the algorithm is named multi-objective SPSO (MSPSO). MSPSO starts by initializing a random swarm of particles. Each particle is stored in memory with its position, its fitness, and its initial velocity. Then, at each iteration, the velocity of each particle is re-calculated using an equation that contains: (i) the current position of the particle (pbest); (ii) the current velocity; and (iii) the previous best position in the neighborhood (gbest). The fitness is calculated based on new positions of particles found at each iteration. The algorithm can be stopped if a given maximum number of iterations is met.
2.4. Non-Dominated Sorting Genetic Algorithm-II
The NSGA-II algorithm proposed by Deb et al. [36
] is the best known multi-objective optimization genetic algorithm and widely used to solve evacuation planning [11
]. This algorithm belongs to the class of evolutionary algorithms (EA), in the subclass of genetic algorithms (GA), solving the optimization problem through an evolutional process of the population of individuals.
Initially, a random population of size N is initialized, evaluated, and sorted on the basis of non-domination. The fitness of each solution is set to a level number; where level 1 is the best, level 2 is the second-best, and so on. The binary tournament selection, crossover, and mutation operators are applied over to generate an offspring population of with size N. A solution of wins a tournament with another solution xj if solution xi has a better rank or if it has the same rank but solution xi has better crowding distance than the solution . After generating offspring , the main loop of NSGA-II starts by combining the two populations and sort with the size of 2 N on the basis of non-domination. Then, the elitist selection is applied to select the new population with size N from the highest fronts of . This main loop is repeated as many times as needed until the satisfaction of an end criterion (i.e., the number of iterations) is reached. NSGA-II has advantages including its low overall complexity of .
The objective of this study was to compare the performance of four multi-objective optimization algorithms (AMOSA, MOABC, MSPSO, NSGA-II respectively) for a given spatial problem, namely evacuation planning. In our study, the evacuation problem was aiming to minimize the accumulated distance from high-risk zones to shelters and to minimize the total capacity overload cost of shelters. The higher the minimum fitness values of both capacity and distance are, the better are the obtained alternatives for assigning people to appropriate shelters.
In terms of algorithm performance, all algorithms generated the optimization in a consistent way, and no results were obtained that could suggest that some of them were trapped in a local minimum. By evaluating the convergence speed of the fitness variation of the four algorithms (see Figure 8
), we found that AMOSA and NSGA-II followed by MOABC converge faster and smoother towards the final optimal solutions. This justifies not only the competence of NSGA-II, which has been used in the literature to a larger extent than the other algorithms [60
]. However, the competence of AMOSA and MOABC shows the capacity of solving multi-objective optimization problems including evacuation problems.
The presented metaheuristic methods and others of its type are not meant to find a ‘single perfect solution’ but a set of ‘good enough’ solutions in an efficient way, and therefore, it is possible that a more optimal solution can be achieved by using alternative methods. Decision-makers must be aware of this aspect, in order to properly assess the benefits and limitations of these techniques.
A suggestion for future work, as an alternative approach dealing with this type of spatial multi-objective optimization problems, is to modify the classical algorithms to better fit the problem in hand. For example, based on the results obtained by MOABC and the comparison made to other algorithms, MOABC could be an interesting algorithm to modify in order to solve complex problems such as evacuation planning. It is also important to consider the use of other methods, such as recoverable robustness, to solve evacuation planning. Iris and Lam [61
] proposed a recoverable robust optimization approach for the weekly berth and quay crane planning problem. The results proved the strength of the proposed model for solving a spatial problem.