A combinatorial optimization model for post-disaster emergency resource allocation using meta-heuristics

Disasters are large-scale disruptions for the society that cause damage to human lives, infrastructure, and the ecosystem. Multiple emergencies emerge, which need to be handled and resolved efficiently to minimize the impact of the disaster. Therefore, the need for an effective and efficient emergency response system that optimally allocates resources and emergency services is essential. This paper proposes a methodology to model this situation as an optimization problem that can be solved using meta-heuristic algorithms. In the proposed model, four meta-heuristics, namely Particle Swarm Optimization, Cuckoo Search, Grey Wolf Optimizer, and iLSHADE-RSP algorithm, have been used to allocate resources and services. A benchmark dataset consisting of 16 situations is prepared to analyze the proposed model. The conducted empirical analysis demonstrates the applicability of the meta-heuristic algorithms in locating near-optimum solutions for the considered situations. The convergence analysis and statistical tests have been performed to test the validity and significance of the conducted experiments.


Introduction
Natural disasters are large-scale harm-causing events that occur due to the earth's processes. Every year, disasters cause harm to both human life and nature itself. They cause a massive number of casualties along with economic losses measured in billions of dollars annually. In 1998-2017, the financial loss caused by natural disasters was US$ 2,908 billion (Pascaline Wallemacq 2018). Substantial loss of human life and infrastructure raises the urgent need to address such issues caused by natural disasters. Activities and processes done to mitigate this damage and preparing for future disasters come under disaster management. It can be considered as a four-phase process: mitigation, preparedness, response, and recovery (Altay and Green III 2006).
Researchers (Ajami and Fattahi 2009;Altay and Green III 2006;Ransikarbum and Mason 2016) have classified activities and challenges in natural disaster management into three phases. These phases are as follows: 1. Preparation Phase before the disaster 2. Response Phase During and immediately after a disaster 3. Recovery Phase Long time after a disaster The preparation stage involves reinforcing the existing infrastructure, setting up emergency warning systems, protocols, services, and other similar activities (Gasparini et al. 2007;Sutton and Tierney 2006;Erbeyoglu and Bilge 2020;Das 2018;Acar and Kaya 2019).
The response stage includes evacuation, relief operations, emergency shelter, and health operations. Research in this stage aims to handle the situation to minimize loss of life and damage to infrastructure (Abdel-Basset et al. 2020;Pi et al. 2020;Chowdhury et al. 2017;Nayeri et al. 2019).
During the recovery stage, various personnel carry out activities like repair work, finding missing people, and providing emergency services. Research in this stage attempts to aid this recovery phase and get the disaster-struck area back to normalcy as quickly as possible (Eid and El-Adaway 2018;Miles et al. 2019;Kanno et al. 2019;Saleem et al. 2008).
After these three phases, the cycle comes back to the mitigation phase. In this phase, the data collected from a disaster are used to modify existing plans and policies and create new ones. The reasoning behind this is to ensure that the system is ready for any impending disaster, and its effect on human life, infrastructure, and the ecosystem is minimal.
This paper focuses on the second phase of disaster management, i.e., the response phase. The emergencies that arise after a disaster could be medical emergencies, fire emergencies, crowd control, infrastructure collapse, urgent resource requirements, and many others. Primarily, this phase attempts to allocate resources to emergency sites to reduce the losses as much as possible caused due to a disaster. It is a postdisaster activity that involves allocating resources just after a disaster. Dealing with emergencies as they come up and optimally using the resources available on hand reduces the damage caused by the disaster and is a critical step in disaster management.
Different response units are assigned to the various emergency sites to deal with the crisis. As there is no unit capable of dealing with all kinds of emergencies, the problem of allocating these units becomes a tedious task. Therefore, emergency managers must assign units to all emergencies with the condition that the unit must be capable of dealing with the emergency allotted to it. Traditional disaster management models use a simple severity factor, and units are allocated based on these severity values. This process was done manually without any decision support model (DSM). (Nayeri et al. 2019) Multiple disaster support systems have been proposed to manage the challenges and activities. These systems model the problem in various ways and use applied statistical methods, probability theory, and mathematical programming approaches to solve them. In this paper, the proposed model uses multiple factors to determine which response unit is suitable for a particular situation. For this, a fitness score is allotted to a possible allocation of units. The objective of the problem then becomes to minimize this fitness value. This can be considered an NP-hard problem; hence, metaheuristic algorithms are suited for this problem.

Literature review
The past few years have seen the development of decision support systems (DSS) to help make better decisions in times of difficulty before, during, and after a disaster. Kondaveti et al. (2009) proposed a decision support system for resource allocation in disaster management built on rapid information collection and resource tracking function-alities. This was intended to be used by emergency managers. Fikar et al. (2016) proposed a simulation and optimizationbased DSS for coordinated disaster relief distribution. The goal of this paper was to have a DSS that facilitates disaster relief coordination between private and relief organizations. Hadigunaa et al. (2014) created a web-based DSS for disaster logistics based on a case study in Indonesia. The DSS aimed to assess the extent to which public facilities can be used as evacuation centers for the victims of an earthquake and tsunami. Jung et al. (2020) proposed an intelligent DSS for smart city disaster management. They created a new conceptual framework of an intelligent DSS for disaster management, with particular attention paid to wildfires and cold and heatwaves. Turǧut et al. (2011) proposed a fuzzy AHP-based DSS for disaster center location selection. Rakes et al. (2014) proposed a DSS for post-disaster housing. This DSS assigned families to interim housing units. Mete et al. (2010) proposed a methodology for optimizing medical supply location and distribution in disaster management. Nayeri et al. (2019) proposed a fatigue effectbased model for scheduling rescue units in case of disasters. The authors compared the rescue operations immediately after a disaster to a job scheduling optimization problem. They used severity and time taken to reach and attend to the emergency as factors affecting the optimum solution. Researchers have also developed multicriteria approaches, i.e., competitive and cooperative mechanisms. Seraji et al. (2021) proposed a location-allocation model for humanitarian logistics. Fiedrich et al. introduced optimization models for the NDM problem (Fiedrich et al. 2000). Another method that has been explored is the application of artificial intelligence in decision support systems (Leifler 2008;Van de Walle and Turoff 2008).
Afkhamiaghda (Afkhamiaghda 2021) proposed a DSS for creating post-disaster temporary housing. Meng et al. proposed a DSS using lightweight blockchain which enhances the computation efficiency of peer-to-peer computing. Gharib et al. (2021) proposed a programming model for post-disaster management which tried to maximize a patient's survival while minimizing completion time and total costs.
In the literature, different optimization algorithms have been suggested to solve combinatorial optimization problems such as Genome Sequencing (Indumathy et al. 2015), Knapsack Problem (Luo and Zhao 2019;Ke et al. 2010), Traveling Salesman Problem (Ouaarab et al. 2014), and the Minimum Spanning Tree problem (Hu et al. 2015) among others. The disaster management problem also comes under this category. In past literature, metaheuristic algorithms have been suggested to solve various problems in disaster management (Rolland et al. 2010;Yi and Kumar 2007;Dávila de León et al. 2019).

Contributions
This work proposes a novel methodology to model the problem of allocating emergency services and resources in a post-disaster crisis. Meta-heuristic algorithms can be used to solve the proposed model to obtain near optimum solutions rapidly, thus saving the time it takes to find the best allotment of service units. The model is flexible with different types of emergencies and allocates units based on them.

Paper organization
The remainder of this paper is organized as follows. Section 2 represents the formulation of the problem statement. Section 3 describes the formulation of the proposed mathematical model. Section 4 represents the results and analysis of the proposed model. Section 5 represents the conclusion and future work.

Problem statement
As dealing with emergencies that come up immediately after a disaster is crucially important, the designed model should be as close as possible to a real-life situation. Therefore, a few entities must be translated into objects to model the situation. The entities are the emergency itself, the emergency service location, the unit that deals with the emergency, and the optimum allocation of these units. A detailed description of each object is given below.

Emergency
Each emergency object represents one emergency that the model must allocate units to. This object consists of the coordinates of the site of emergency, the severity of the emergency, and the type of aid required.

Emergency service location
The emergency service location object represents the home of the units deployed to deal with an emergency. This could be a hospital, a fire station, a police station, among others. Each emergency service location object records the service provider's location, a specific number of unit objects, and the type of services available.

Unit
A unit object represents the entity that deals with the emergency. This could be an ambulance, a paramedic vehicle, a police department vehicle, or a fire engine. Each object records the type of emergency unit and the maximum severity value that the unit can deal with.

Allocation
Each possible solution is considered as an allocation object. It consists of the mapping of the selected units and emergencies along with the details of which unit is associated with which emergency. The goal of the optimization problem then becomes to achieve an allocation object with the best fitness score (Refer, Sect. 3.1).
Moreover, a small number of units must be kept reserved for future emergencies. Hence, each of the objects and criteria discussed is formulated in such a way that we can represent the relationships as closely as possible to real life.
The proposed representation fits the skeleton of a combinatorial optimization problem that can be subjected to metaheuristic algorithms. Using metaheuristic algorithms, near-optimum solutions can be obtained for this model. Figure 1 represents the objects that are given to the model. Figure 2 represents the structure of the allocation map of the allocation object.

Smallest position value (SPV) rule
The proposed model is a permutation-based problem and thus is discrete. By converting this problem to a continuous one, any optimization algorithm can be used to solve it. Here, the SPV rule is used to do this. The smallest position value (SPV) rule is used to represent one permutation of the given position vector through an array of numbers.
For example, let us assume that the position vector of a solution is X id = {2.34, 4.63, −8.45, 0.64, −2.54}. This is a five-dimensional position vector. If the original order of the objects is i.e., the sequence vector, then using the SPV rule, we assign the first fragment to the shortest value in X id and so on. Thus, the sequence vector for this solution becomes In this problem, X id = x 1 , x 2 · · · x n represents a potential candidate solution for the considered problem. Here, x 1 , x 2 · · · x n are entities and not numbers, and hence cannot be used directly. The SPV rule converts it into a position vector. Figure 3 shows an example of the SPV rule.

Confluence
After representation, the next phase of the model is the optimum allocation of the resource unit to the emergency locations. Broadly, the resource allocation can be done in two ways: units to emergencies and emergencies to units.

Units to emergencies
In this method, emergencies are considered constant, and the units are allotted to each emergency. Generally, the number of available units for deployment will be greater than the Thus, each allocation object contains an array of units whose length equals the number of emergencies. Each unit corresponds to one emergency site.

Emergencies to units
In this method, units are taken as constant, and the emergency sites are allocated to the units. Generally, the number of units is greater than the number of emergencies. Hence, this difference in number must be addressed while creating the objects. Here, the proposed allocation object is an array. The length of an array is equal to the number of units available. Each element in the array represents the allocation of the corresponding unit.

Mathematical model
The mathematical models for both cases are mentioned in Sec. 2.3. They have the same inspiration and working principle; however, they differ in terms of the structure of the objects.

Fitness function
In both the above-discussed cases, the fitness function is governed by some common factors. The factors are as follows: • Severity of the emergency • Maximum severity that a particular unit can handle.
• Euclidean distance between the emergency service location and emergency site.
The fitness function has been derived as the summation of the score of each allocated emergency and unit pair. The mathematical formulation for the score of each pair is as follows: where D e,u represents the Euclidean distance between emergency site e and emergency service location of unit u. S e represents the severity of the emergency e, and S u represents the maximum severity that unit u can handle. The fitness is calculated as the sum of the scores of all emergencies and their allocated units.
where e 1 , e 2 · · · e n are the n emergencies in the input situation and u 1 , u 2 · · · u n are the n selected units out of all the units to be allocated to these n emergencies.
Here, the objective of this proposed model is to minimize the formulated objective function.

Units to emergency
In this case, the allocation object consists of: • The Input situation This object represents the given situation in the form of a map which contains the locations of the emergencies, service locations, severity of each emergency, and specifications regarding each unit. • Allocated Units It is a fraction of units from all available units chosen for the particular allocation object. • Solution Vector It is a vector of the form X i = x 1 , x 2 , x 3 · · · x n , where n is the number of emergencies for which units need to be allocated. It represents the solution in its raw form. • Allocation Map The actual solution is obtained by applying the SPV rule on the solution vector. It contains all the emergency-unit pairs corresponding to the selected unit and their destinations.

Constraints
The constraints considered for the proposed model are summarized below: 1. The number of units selected should be equal to the number of emergencies. 2. Each unit can be assigned to only one emergency point.
This constraint has been added for simulation purposes.
In a real-life implementation, it can be replaced with a function of severity. Moreover, the kind of units available and each unit's capabilities can also be considered. 3. The ratio of the number of units not allocated to the total number of units should be greater than the reserve ratio (this condition is flexible and must be changed as per the location, emergency situation, and the history of emergency situations in the past).

Emergencies to units
In this case, the allocation object is defined differently from the previous case.
• Input-situation This object represents the given situation in the form of a map which contains the locations of the emergencies, service locations, severity of each emergency, and specifications regarding each unit. This is the same as in the first case. • Solution vector This is a vector of the form where n is the total number of units. This vector represents the solution in its raw form. The length of each vector is n, consisting of m emergencies and n − m 0 values. Each element in this vector represents either the allocation or the absence of a unit. Here, 0 represents the absence of the unit, whereas 1 represents an allocated unit. The units assigned 0 are not considered for fitness function calculation. • Allocation map This is a map corresponding to the units and their allocated emergencies. This contains all the unit-emergency pairs corresponding to the selected emergency and its assigned unit. It represents the object that is evaluated when calculating the fitness.

Constraints
The constraints considered for the proposed model are summarized below: 1. The number of units selected should be equal to the number of emergencies.

Confluence
The mathematical model of the research problem is as follows: In this paper, only one unit is assigned to one emergency, therefore a = n. Table 8 in the appendix shows the notations used in the above model. Figure 4 depicts a visual representation of the allocation process of both the proposed methodologies.

Meta-heuristics algorithms
This paper uses four popular meta-heuristics: particle swarm optimization (PSO), grey wolf optimizer (GWO), cuckoo search and the improved L-SHADE algorithm to solve the defined problem.

Particle swarm optimization
Particle swarm optimization is a well-known swarm intelligence algorithm used to address combinatorial optimization problems. It optimizes a problem by iteratively optimizing the candidate solutions using a given fitness function. Kennedy and Eberhart proposed the PSO algorithm (Eberhart and Kennedy 1995) by simulating the social behavior of bird flocks or fish schools. In PSO, each particle is a potential candidate solution for the given problem. The set of solutions or the particles is called the swarm. Here, X represents the particle's position, which is updated in each iteration for all particles. The best particle in the swarm is represented by gbest, and each particle's personal best is represented by pbest. The position of each particle is updated by computing the velocity v from pbest and gbest. The mathematical formulation of the velocity and position update is as follows: where v i (k) represents the velocity of ith particle in kth iteration and X i (k) represents the position of particle i in iteration k. gbest represents the best particle in the current iteration and pbest represents the best position of the ith particle. c 1 and c 2 are constants that impact the social and cognitive behavior of the swarm. r 1 and r 2 are random numbers in the range of [0, 1]. PSO is a widely used and recognized algorithm to solve many real-life problems. Researchers have proposed multiple variants to improve various aspects of this algorithm (Zhang et al. 2015). Hence, in this paper, the PSO algorithm has been used to solve the resource allocation problem in disaster management.

Cuckoo search
Cuckoo search (CS) (Yang and Deb 2010) is a well-known metaheuristic algorithm that Yang and Deb developed in 2009 inspired by the breeding behavior of cuckoo birds. Cuckoo birds do not lay their eggs in their nests. They find other birds and lay their eggs in their nests. It leads to two possibilities: the host bird discovers the egg or does not. If Fig. 4 Example of the proposed methodologies the host bird identifies the foreign egg, it will either throw the egg or abandon its nest.
The three ideal rules for CS are as follows: 1. Each cuckoo lays one egg at a time and deposits it in a randomly chosen nest. 2. The best nests with high-quality eggs will carry over to the next generations. 3. The number of available host nests is fixed, and there is a probability that a host can discover an alien egg. In this case, the host bird can either throw the egg away or abandon the nest to build an entirely new nest in a new location.
The iterative improvement in the cuckoo search algorithm occurs using a combination of local random walk and a global random walk algorithm. The parameter p a controls the value of the random walk. p a represents the probability that the host bird discovers the egg. Higher the value of p a , higher is the exploitation in that specific iteration. Low values of p a show that the algorithm will favor exploration more than exploitation. The mathematical formulation of the local random walk is mentioned below.
where x t j and x t k are the two different solutions selected randomly from all nests in the algorithm. H (u) is the Heaviside Function, and is a random number drawn between 0 and 1. ⊗ represents entry-wise multiplication operation, and β is a scaling factor.
The global random walk is represented by where α is a random number between 0 and 1, and L(s, λ) represents the Levy function.
3.6 Grey Wolf optimizer Mirjalili et al. proposed grey wolf optimizer (Mirjalili et al. 2014) inspired by the hunting and leadership strategies of grey wolf wolves. The leaders of the pack are the alphas.
The betas are next in hierarchy as subordinates to the alphas.
Omegas are the lowest ranking wolves that play the role of scapegoat. Delta wolves submit to alphas and betas but not to omegas. The mathematical model for this algorithm consists of social hierarchy, encircling prey, and attacking prey.

Social hierarchy
The fittest solution is deemed as the alpha (α). Consequently, the second and third best are deemed as beta (β) and delta (δ), while the rest are omegas (ω).

Encircling prey
The following equations govern the encircling behavior of the algorithm.
where t indicates the iteration, A and C are coefficient vectors, X p is the position vector of the prey, and X indicates the position vector.
where all the components of a decrease linearly from 2 to 0. r 1 and r 2 are random vectors.

Attacking prey
The hunting behavior of the agents is modeled using the following equations.

iLSHADE-RSP
iLSHADE-RSP (Choi and Ahn 2021) is one of the most powerful variants of differential evolution. This technique perturbs a target vector with the Cauchy distribution based on a jumping rate. iLSHADE-RSP has four main aspects: mutation strategy, Linear Population Size Reduction, Adaptive Parameter Control and Cauchy Perturbation.

Mutation strategy
The algorithm uses a mutation strategy called DE/current-topbest/r. where The probability of the i th individual being selected can be calculated as follows: where

Linear population size reduction
At end of each iteration, the population size for the next iteration is calculated as follows: where NP init and NP fin denote the initial and final population sizes, respectively.

Adaptive parameter control
iLSHADE-RSP uses a learning process-based adaptive parameter control to adjust the control parameters F and C R automatically.

Cauchy perturbation
iLSHADE-RSP uses a modified recombination operator. The mathematical formulation of this operator is given by where rndc j i denotes the Cauchy distribution.

Experimental results
In this section, the performance of the proposed model is tested with particle swarm optimization, cuckoo search, grey wolf optimizer, and iL-SHADE optimizer. Since this is a novel model for this situation, there is no benchmark to evaluate its performance. Therefore, particle swarm optimization, cuckoo search, grey wolf optimizer and the iLSHADE-RSP algorithm have been used to assess the proficiency of this model. For an unbiased analysis, each algorithm was run 20 times on each dataset with different random seeds and the results were averaged for the final analysis. All algorithms were implemented using Python 3.7.9 and performed on a 3.1 GHz Dual-Core Intel Core i5 with 8GB of memory running Mac OS 11.2.

Data generation
The authors have created sixteen scenarios with varying numbers, coordinates, and severity's of emergencies, service locations, and units for experimental analysis. The parameters of these scenarios are shown in Table 1. This dataset is available on the Harvard Dataverse repository (Jain and Kumari 2022). Table 2 shows the relevant parameters for cuckoo search, particle swarm optimization, grey wolf optimizer, and the iL-SHADE optimization algorithm. The number of iterations  has been chosen as the termination criterion. The swarm population is constant for both algorithms.

Analysis of results and discussion
Tables 3, 4 and 5 show the results of particle swarm optimization and cuckoo search on the 16 different considered scenarios (Refer Sect. 4.1). Bold values in these tables represent the best solutions in each case. Since there is no benchmark fitness to work with, four popular meta-heuristics have been used to solve this problem. The fitness calculated in each situation is independent of the fitness of other situations; hence, they cannot be compared with each other. Thus, one can only compare the results of a particular optimization algorithm on a given situation with the results of other algorithms on the same situation. This does not impact the effectiveness of the proposed model because, in real-life applications, an input situation is fixed and unchangeable. Table 6 shows the average amount of time taken by the algorithms to run on the situations in the benchmark dataset twenty times over different random seeds.
In each iteration, the optimization algorithms try to improve their fitness score. Figure 6 shows how the score is improved on a mock situation over the iterations.
It is observed that iLSHADE-RSP outperforms the other 3 algorithms in most cases. It consistently provides a better average over different random seed with low variance. In Fig. 6a, line connecting a service location and an emergency represents the allocation of a unit from that specific service location to that emergency. This follows the same legend as in Fig. 5. It can be observed in Fig. 6 that a unit from a service location farther from the closest service location is allocated in some cases. This is done because the score of a farther service location might be higher if the maximum severity that can be handled by the units available is insufficient to handle a given emergency. This is also the reason the fitness function contains the expression exp(S e − S u ). Consequently, the preference is given to a unit that can handle the emergency compared to a unit with a lower maximum severity than the severity of the given emergency. Each of the two methodologies discussed in this paper has its advantages and disadvantages and would perform better in some cases.   The units-to-emergencies methodology is characterized by having a small memory requirement. This is because the number of units is greater than the number of emergencies and the array of objects in the model is equal to the number of emergencies. However, this methodology lacks in its exploration ability when compared to the other. Since in each allocation object, the units selected are fixed when the object is initialized, during any meta-heuristic algorithm, the solution represented by that particular object will be restricted to having the selected units only. This can be offset by increasing the population parameter of the metaheuristic used.
On the other hand, the emergency-to-units methodology requires more memory, but it offers a better exploration behavior than the previous methodology. This is because each solution object represents an array that contains relations to all emergency and unit objects. It explores the entire sample space by each agent in the population. Since this methodology requires having all emergency and unit objects related, it requires more memory. This becomes significant if there are a large number of units and emergencies or if there are various types of units and emergencies available on hand. Tables 3, 4 and 5 show the comparison between both these methods. In the conducted experiments, both these methodologies offer similar results in all ten situations that were solved.

Convergence analysis
Convergence analysis is a popular qualitative metric for analyzing the performance of a particular algorithm on a given problem. It represents the best fitness achieved by the algorithm at each iteration in a line graph. This helps conduct exploration-exploitation analysis and find out if an algorithm converges prematurely. Figure 7 represents the convergence graph of the considered algorithms. It is observed that all the algorithms improve themselves in the first few iterations and later become stagnant for the rest of the iterations. The iL-SHADE-RSP algorithm demonstrates better convergence than the other considered algorithms.

Statistical analysis
Student's T -test has been utilized to check the statistical significance of the results of the experiments conducted. Each of the algorithms considered is paired with another to check the pairwise significance. A significance value α = 0.05 is set. Table 7 shows the results of the T -test performed on the results obtained from the experiments done. T-Test results show that the algorithms, when compared with each other, gave significant results in 42 cases. The reason for this might be the complexity of the problem. From the convergence analysis, for all four algorithms, the optimum is reached around 700th for most cases, after which there is seldom any improvement. This means that after this point no better solution was found by the algorithms which is why the algorithms all converged to solutions that are close to each other.

Conclusion
This research targets the allocation of emergency services and resources in a post-disaster crisis. It is crucial to optimally allocate resources since every delay can significantly affect human lives and infrastructure. The model proposed in this manuscript can be used with meta-heuristic algorithms to rapidly obtain a near-optimum allocation strategy for a given situation. The model can be implemented using two strategies. By choosing one of the two proposed methodologies, the exploration ability of the algorithm used can be enhanced at the expense of taking more computing resources. In this research, four meta-heuristics, particle swarm optimization (PSO), grey wolf optimizer (GWO), cuckoo search (CS), and iLSHADE-RSP optimization algorithm, have been used to test this model. The algorithms were evaluated using statistical measures of their results over 20 random seeds. The experiments show that iLSHADE-RSP returns the best results on average, while PSO takes the least CPU time on average. Convergence analysis and statistical testing were performed to analyze the progress and significance of the conducted experiments.
Future work in this direction includes the elaboration of the considered factors. The proposed model can also be modified and tested using real-world data to set up benchmarks for future results. Researchers can also modify optimization algorithms to solve this combinatorial optimization problem more successfully.