# Extended GRASP-CapacitatedK-Means Clustering Algorithm to Establish Humanitarian Support Centers in Large Regions at Risk in Mexico

Authors: Santiago-Omar Caballero-Morales; Erika Barojas-Payan; Diana Sanchez-Partida; Jose-Luis Martinez-Flores
Journal: Journal of Optimization
 (2018)

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## Abstract

Mexico is located within the so-called Fire Belt which makes it susceptible to earthquakes. In fact, two-thirds of the Mexican territory have a significant seismic risk. On the other hand, the country’s location in the tropical zone makes it susceptible to hurricanes which are generated in both the Pacific and Atlantic Oceans. Due to these situations, each year many communities are affected by diverse natural disasters in Mexico and efficient logistic systems are required to provide prompt support. This work is aimed at providing an efficient metaheuristic to determine the most appropriate location for support centers in the State of Veracruz, which is one of the most affected regions in Mexico. The metaheuristic is based on theK-Means Clustering (KMC) algorithm which is extended to integrate (a) the associated capacity restrictions of the support centers, (b) a micro Genetic Algorithm μGA to estimate a search interval for the most suitable number of support centers, (c) variable number of assigned elements to centers in order to add flexibility to the assignation task, and (d) random-based decision model to further improve the final assignments. These extensions on the KMC algorithm led to the GRASP-Capacitated K-Means Clustering (GRASP-CKMC) algorithm which was able to provide very suitable solutions for the establishment of 260 support centers for 3837 communities at risk in Veracruz, Mexico. Validation of the GRASP-CKMC algorithm was performed with well-known test instances and metaheuristics. The validation supported its suitability as alternative to standard metaheuristics such as Capacitated K-Means (CKM), Genetic Algorithms (GA), and Variable Neighborhood Search (VNS).


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## Body

## 1. Introduction

Aphenomenon ordisturbing agent is defined as an aggressive and potentially harmful physical event, natural or derived from human activity, which can cause loss of life or injury, material damage, serious disruption of social and economic life, or environmental degradation. Thus, these agents can have the following origins [1, 2]:(a)
Natural: geological, hydrometeorological, and astronomical.(b)
Anthropogenic: chemical-technological, health-ecological, and social-organizational.Mexico is located within the so-calledFire Belt of the Pacific and within the tropical zone. This makes the country susceptible to a great variety of disturbing agents of natural origin [3]:(a)
Two-thirds of the country have significant seismic risks.(b)
Coastal regions are frequently affected by hurricanes which are generated in the Pacific and Atlantic Oceans.Due to its geographical location, geological characteristics, and the complex morphology of its territory, the State of Veracruz in Mexico is exposed to natural phenomena such as earthquakes, volcanic eruptions, floods, and landslides. The presence of hydrometeorological phenomena is very common in Veracruz, which leads to frequent affectations. In response to the presence of disturbing natural phenomena, in Mexico the Natural Disasters Fund (FONDEN) was created. This is a financial instrument whose purpose is to provide relief supplies and assistance in emergency and disaster situations. In Veracruz, the rules of the Fund for the Prevention of Natural Disasters (FOPREDEN) are an instrument that aims to revitalize initiatives aimed at preventing disasters and seeks to optimize the use of available financial resources and magnify the results linked mainly to the preservation of the life and physical integrity of people, as well as that of public services and infrastructure and the environment [4].As of 2017 FONDEN has authorized resources for more than three billion of Mexican pesos to support the road, educational sectors, forestry, hydraulic, naval, housing, and urban infrastructure in Veracruz due to the significant occurrence of natural disasters within the period 2014-2017. Table1 presents an overview of the historical phenomena within this period and the resources provided.Table 1
Declarations of natural disasters: 2014-2017 [5].
Year Description FONDEN (Mexican Pesos) 2014 01. Severe rain and fluvial flood in October 13-16 193,636,015.00  2015 01. Severe rain in March 11-12 1,610,707,729.00 02. Severe rain in March 21-23 and severe rain and fluvial flood in March 25-27 03. Severe rain and fluvial flood in June 11-14 04. Hillside movement in July 9-13 05. Hillside movement in September 16-18 06. Severe rain and fluvial and rain flood in October 16-21 07. Severe rain and fluvial and rain flood in October 18-24  2016 01. Hillside movement in August 5-7 860,195,408.00 02. Severe rain and fluvial flood in August 5-7 03. Severe rain in September 27-28  2017 01. An earthquake with magnitude 8.2 on September 7 496,947,819.20 02. Hurricane “Katia” - severe rain and fluvial flood in September 8-12 03. Hillside movement from September 27 to October 9 04. Severe rain and fluvial flood from September 27 to October 9 05. Fluvial flood in October 11-15 06. Severe rain in October 11  Total 3,161,486,971.20Standard protocols to be performed before, during, and after a disaster involve different logistic processes. These are performed in the different phases of a disaster [6, 7]:(1)
Interdisaster phase: processes are performed in which the elaboration of the map of risks for the community is highlighted. Also the Plans of Emergency, which consist of inventory and location planning of resources, are performed.(2)
Pre-impact phase: warning to the population based on prediction mechanisms and implementation of mitigating measures are performed.(3)
Disaster impacts the community.(4)
Emergency phase: isolation, rescue, and external assistance are performed. It is often the phase in which local resources are overwhelmed and external aid is required to reduce the number of fatalities.(5)
Reconstruction phase: activities focused on recovering the normal duties of the community are performed.Before the disaster occurs, it is important to have facilities with an optimal inventory of products of first necessity to support the survival of the people who will be affected. Also, after the disaster occurs, it is important to have the infrastructure to resupply the facilities and transport affected people to other facilities as needed.Hence, among the most critical decisions and resources to provide relief to the affected communities in Veracruz, prepositioning of warehouses must be performed. This allows the protection of supplies and the efficient and timely supply of products to cover the basic needs of the people affected by the disturbing phenomenon. Within the activities to be performed in these warehouses or support centers, the following can be mentioned:(1)
Identification, labeling, and location of the necessary supplies to attend the emergency.(2)
Consolidation of load and change of means of transport.(3)
Delivery scheduling for the supplies.Likewise, the warehouse must have an information and inventory control system which must be updated, through the control of inventories. The activation of a prepositioned warehouse is the responsibility of State Civil Protection with selection criteria for its allocation such as (a) being located outside the risk area, (b) having a solid and roofed construction in compliance with safety parameters, (c) being accessible through favorable conditions for transport loading and unloading, (d) being ventilated, illuminated, and without water seepage risk, (e) being located far away from flood-prone areas, (f) being free of pollution or plague, and (g) having space to facilitate the mobility, cleaning, and classification of products [8, 9]. Minimization of distance between the affected regions and the prepositioned warehouses is an important aspect of humanitarian relief planning because communities must be able to reach these centers within short periods of time and distances due to the severity of the disasters.In this regard, humanitarian logistics (HL) formally addresses the “process of planning, implementing, and controlling the efficient, cost-effective flow of and storage of goods and materials as well as related information from point of consumption for the purpose of meeting the end beneficiary’s requirements” [10, 11]. The need of HL for strategic planning has been recognized by important organizations such as the U.S. Federal Emergency Management Agency (FEMA) and the United Nations (UN) [11, 12]. In contrast to commercial logistics (CL), the main focus of HL is to save lives and provide beneficiaries with aid instead of maximizing profits. However, due to this characteristic, HL has disadvantages when compared to CL as it faces lower technology, challenging inventory control, unstable demand patterns, zero lead time, and unpredictable supply resources [13, 14].Hence, different strategies have been developed within the field of HL for the optimal operation of all the aspects of the supply chain (SC) for the delivery of goods to affected communities considering these disadvantages. In this context, humanitarian relief organizations (HRO) have been identified as the best suited organizations for preparedness and recovery when compared to commercial and military organizations [13]. An important aspect of preparedness is the prepositioning of inventories or warehouses for postdisaster relief. Among the most recent strategies, which are focused on transportation, planning, policies and procedures, and inventory/warehousing [15], the following can be mentioned:(i)
In [16], a stochastic model was developed to determine the location of Emergency Medical Service (EMS) systems. In order to solve this model, exact and approximate (metaheuristics) methods were proposed.(ii)
The facility location problem was also addressed by [17] that presented a multiobjective optimization model to solve a multidepot emergency facilities location-routing problem. Due to the inherent computational complexity of this model an approximate method based on the metaheuristic of Genetic Algorithms (GAs) was developed.(iii)
Prepositioning or relief assets was studied in [11] to optimize the transportation of affected people to relief centers. The proposed stochastic model considered optimization of resources such as personnel and vehicles to minimize casualties.(iv)
In [18] the aspect of considering containers as storage facilities was studied, and a mathematical model was proposed to determine the locations of supply points and the quantity of containers and relief supplies assigned to each supply point under the minimum distance criteria.(v)
A stochastic inventory control strategy was proposed in [19] for the uncertain requirements of goods for postdisaster conditions, in order to have the adequate stock to serve the vital needs of affected communities.(vi)
A conceptual model that integrated the aspect of agility in HL was presented in [20] to improve on the response of HL to disaster scenarios. While no mathematical model was presented or discussed, the roles of people, processes, and technology were identified as agility enablers for the success of general models within HL.In general for the determination and location of facilities (i.e., support centers, warehouses, prepositioned inventory, etc.) the following mathematical models have been considered: the Capacitatedp-Median Problem (CPMP) [21, 22] and the Capacitated Centered Clustering Problem (CCCP) [23, 24]. Both models are focused on determining the location of p facilities in order to minimize the total weighted distance from the facilities to all demand points (customers). A demand point cannot be assigned to more than one facility, and the points assigned to a facility cannot exceed its capacity. The main difference between both models is about the features of the locations of the p facilities. In the CPMP the location is determined at a median point while for the CCCP the location is determined at a centroid.Both models are difficult to be solved to optimality due to their NP-hard computational complexity. Particularly for large problems, this has led to the development of metaheuristics to provide near-optimal solutions [25]. In the literature, metaheuristics based on Clustering Search (CS) have been reported as the most competitive methods for the CCCP [24]. However, in this work we focus on providing an alternative to standard methods which are commonly implemented for practical situations. In the case of humanitarian relief actions, fast implementation is required, and we are considering the situation of Veracruz in Mexico, where 3837 communities with 526,954 people are at risk.Thus, in this work a metaheuristic based on the integration of the Greedy Randomized Adaptive Search Procedure (GRASP) and theK-Means Clustering (KMC) algorithm is presented to provide a suitable location planning for the support centers (prepositioned warehouses) for these communities in Mexico. In order to provide more accurate solutions for large CCCP instances than those of standard methods, the proposed metaheuristic has the following features:(a)
capacity restrictions to the KMC algorithm for the assignation of communities to support centers (CapacitatedK-Means Clustering, CKMC);(b)
micro Genetic Algorithm (μGA) that performs single executions of the CKMC to estimate a search interval for the most suitable number of support centers;(c)
variable number of assigned communities to centers in order to add flexibility to the assignation task through iterative executions of the CKMC;(d)
conditional decision process to perform insertion, deletion, and exchange of communities between centers for further improvement of the final assignments;(e)
Earth’s arc length as distance metric to locate centers within measurable distance in kilometers.The details of this metaheuristic, termed as GRASP-CKMC, are presented as follows: in Section2 the technical details of the GRASP-CKMC and its validation are presented. Then, in Section 3 the results on the instance of Veracruz are presented and analyzed. Finally, our conclusions are presented in Section 4.


## 2. GRASP-CKMC

A Greedy Randomized Adaptive Search Procedure (GRASP) is a metaheuristic which consists of two main phases: (a) the Construction Phase which consists in providing a feasible solution by combining a greedy function with a method of random selection and (b) the Local Search Phase which consists in iteratively improving the feasible solution [26, 27].As presented in Figure1 the proposed GRASP manages three main algorithms for these phases:(i)
Constructive Phase: aμGA is performed to determine the lower and upper limits for the most suitable number of clusters. Random selection is performed for the creation of the initial population and the number of V nearest points to extend the KMC to comply with capacity restrictions (CKCM).(ii)
Local Search Phase: the CKMC is iteratively performed with uniform random variation inV and the number of clusters K restricted by the lower and upper limits identified in the previous phase.(iii)
Random decision process to exchange locations between capacity-complying assignments for further improvement of the final CCCP solution.Figure 1
General structure of the GRASP-CKMC algorithm.In the following sections the details of the main algorithms used for the phases of the GRASP are presented and discussed.


### 2.1. Capacitated KMC

K-Means is one of the basic unsupervised learning algorithms that solve the well-known clustering problem [28–30]. This model is similar to the also well-known K-Nearest Neighbor (KNN) search algorithm [31]. The KMC follows a simple procedure to classify a given data set through a certain number of clusters K [28, 32]. Within the context of the CCCP or CPMP the facility is located at the cluster’s centroid or median point, respectively. For multiple facilities, the first problem to be solved is the consolidation of clusters (i.e., groups of points) and the second is the determination of the median point or centroid. Both problems can be addressed simultaneously by the K-Means Clustering (KMC) algorithm. Figure 2 presents the details of the standard KMC algorithm.Figure 2
Structure of the standardK-Means Clustering (KMC) algorithm.As presented in Figure2 clustering involves the unique assignment of a point to the nearest cluster based on its center (defined as the median point or the centroid). The locations of the centers must be reestimated each time that new assignments are performed, and new assignments can be generated each time that the reestimation process is performed as they affect the closeness of the centers to the considered points.For the purpose of determining the locations of the support centers and their assigned communities, the standard KMC algorithm must integrate capacity restrictions. However this adds complexity to the assignment task because not all nearest points to a certain center can comply with its capacity restriction (thus, not all nearest points can be assigned to this center).Approaches have been proposed to address the capacitated task. In contrast to the circular regions shown in Figure2, in [32] a rectangular region around the center was considered to determine the candidates for clustering. This reduces the number of points to be assigned to the cluster and thus reduces the likelihood of not complying with the capacity restriction. The points located outside the rectangular region are omitted by this initial assignment process. After this process is performed, a priority is assigned to the omitted points in order to be assigned to the clusters with available capacity in a final assignment process. Other approaches involve an average distance for the reassignment of points [33].The assignment of close points and reassignment of omitted points are procedures which can be performed with some randomness to add flexibility to the local search process of KMC. Thus, the proposal to extend the KMC to perform the capacitated task consists in including a uniform random variable to control the ratio of acceptance for the KMC algorithm (and thus, of theV nearest locations). This proposal is similar to the Variable Neighborhood Search (VNS) principle [34].Figure3 presents the general structure of the proposed capacity-restricted KMC algorithm (CKMC). As presented in Figure 1, this CKMC algorithm is used in both phases of the GRASP-CKMC metaheuristic. This is the reason of the adjustments stated in Figure 3:(i)
In the Constructive Phase, the CKMC is executed only once for two randomK values which will be used by the μGA to determine the lower and upper limits (Kmin, Kmax) for K. Also, the number of nearest points to each center (V) is constant given by X.(ii)
In the Local Search Phase, the CKMC is iteratedP times, and at each iteration, different values of K (within Kmin and Kmax) and the ratio of acceptance V (which has an upper limit given by X) are considered. At each iteration, the best assignment of points (locations) to clusters (centroids) (as measured by its objective function value G) is saved. After the P iterations of the CKMC algorithm are executed, the best found solution is improved by means of insertion, deletion, and exchange operations which are controlled by a decision process.Figure 3
Structure of the proposed CapacitatedK-Means Clustering (CKMC) algorithm.


### 2.2.μGA

The standard KMC algorithm considers that the quantity of clusters is knowna priori [30]. Within the context of the CCCP, the minimization of the objective function (total distance from each cluster to each assigned point) depends on finding the most suitable number of clusters. Hence, the proposed GRASP-CKMC includes an evolutionary mechanism to determine the suitable range of clusters which can minimize the total distance to the affected communities.Figure4 presents the general structure of the μGA which was developed to address this task. The μGA is characterized by small populations which can lead to achieving faster convergence with less storing memory [35, 36]. In this case, the individuals of the population of the μGA consist only of pairs of values (Kmin, Kmax) that can define the lower and upper limits of a range that may contain the K value that can lead to a total minimum distance on a single execution of the CKMC algorithm. By using the random mutation and the linear crossover operators a diversification on these bounds is obtained to estimate an interval for the local search of K within the main GRASP-CKMC algorithm. An estimate for (Kmin, kmax) is obtained after m generations (in this case, m = 100) of the μGA are executed, and within this range, K is randomly selected.Figure 4
Structure of theμGA for initialization of K.


### 2.3. Insertion, Deletion, and Exchange

As presented in Figures1 and 3, the best solution found by the CKMC in the Local Search Phase is improved by a decision algorithm which performs insertion, deletion, and exchange of points between clusters. A conditional decision process was designed to avoid unnecessary tasks due to the random selection of points which can be inserted, deleted, or exchanged. The description of this improvement process is presented in Figure 5.Figure 5
Structure of the decision process of the GRASP algorithm.Finally, the implementation of the metaheuristic was performed with Octave and MATLAB in a HP Workstation with Intel Zeon CPU at 3.40 GHz with 8 GB RAM.


### 2.4. Assessment

Before proceeding to obtaining a solution for our instance, we assessed the performance of the GRASP-CKMC metaheuristic with a selection of CCCP instances. Due to the size of the instance (3837 communities), we considered the following SJC and DONI instances [37, 38]: SJC1 (100 points), SJC2 (200 points), SJC3a (300 points), SJC4a (402 points), DONI1 (1000 points), DONI2 (2000 points), DONI3 (3000 points), DONI4 (4000 points), and DONI5 (5000 points) [39]. For comparison purposes, the performance of the GRASP-CKMC metaheuristic was compared to standard and most recent methods, including the latest best known solutions as follows:(i)
Best known solutions as reported in [24].(ii)
Best results obtained by CKM and GA as reported in [38].(iii)
Best results obtained by VNS as reported in [23].(iv)
Best results obtained by TS (Tabu Search) and CS (Clustering Search) as reported in [24].(v)
Best results obtained by the latest method known as Adaptive Biased Random-Key Genetic Algorithm (A-BRKGA) as reported in [24].Table2 presents the parameters for the GRASP-CKMC algorithm. As mentioned in [24], metaheuristics have no optimal values of parameters. Thus, recommended ranges are usually considered for these cases. For the GRASP-CKMC algorithm it was considered to have a lean execution due to the size of the instance and the diverse algorithms which were developed. Thus, small values were considered for X (the upper limit for the number of nearest points to each cluster), the executions of the CKMC in the Local Search Phase (P) and the number of pairs of points to be considered for exchange, deletion, or insertion (Y). Regarding the Mersenne Twister random number generator, it was considered as recommended by the MATLAB documentation.Table 2
Parameters of the GRASP-CKMC.
Parameter Value  X 10 P 50 Y 10000 Random Number Generator Mersenne TwisterTable3 presents the results obtained for 10 runs of the algorithm. It is observed that the average of the best results is 3.03% while the average of the worst results is 5.59%. Particularly for the instances DONI3 and DONI4, which have a similar number of points to the considered instance of 3837 communities, the GRASP-CKMC metaheuristic is able to obtain solutions with errors smaller than 5.0% (3.93% and 4.64%, respectively) within 10 runs.Table 3
Results of 10 runs of the GRASP-CKMC metaheuristics on the SJC and DONI instances.
Instance Best      Runs     Best Worse Average Best Worst Average Known 1 2 3 4 5 6 7 8 9 10    Error(%) Error(%) Error(%) SJC1 17359.75 17807.73 18230.49 17742.19 18088.71 17791.75 18054.23 18169.28 17948.46 18132.05 17789.13 17742.19 18230.49 17975.40 2.20% 5.02% 3.55% SJC2 33181.65 33810.48 34077.98 33568.98 33529.65 33673.30 33342.50 33745.19 33330.23 34084.06 33731.71 33330.23 34084.06 33689.41 0.45% 2.72% 1.53% SJC3a 45356.35 46993.46 46165.69 46789.82 46503.93 46559.25 46459.06 46944.45 46697.09 46250.59 46030.70 46030.70 46993.46 46539.40 1.49% 3.61% 2.61% SJC4a 61931.60 62774.78 62710.35 63064.16 62962.47 62844.49 62899.00 63000.86 63357.25 62926.63 63369.95 62710.35 63369.95 62990.99 1.26% 2.32% 1.71% DONI1 3021.41 3088.71 3165.04 3104.18 3144.54 3169.48 3141.08 3095.66 3109.24 3157.25 3158.03 3088.71 3169.48 3133.32 2.23% 4.90% 3.70% DONI2 6080.70 6487.85 6499.84 6498.32 6490.31 6459.86 6498.46 6487.18 6484.72 6483.75 6420.06 6420.06 6499.84 6481.04 5.58% 6.89% 6.58% DONI3 8343.49 8706.45 8971.18 8953.30 8964.51 8677.99 8691.23 8894.92 8782.56 9010.17 8671.70 8671.70 9010.17 8832.40 3.93% 7.99% 5.86% DONI4 10777.64 11540.02 11508.46 11704.86 11568.63 11541.06 11534.28 11702.91 11277.44 11392.09 11685.20 11277.44 11704.86 11545.49 4.64% 8.60% 7.12% DONI5 11114.67 12016.16 11991.16 12034.59 11846.86 12022.82 11724.42 11934.19 11937.12 11938.20 11893.37 11724.42 12034.59 11933.89 5.49% 8.28% 7.37%  Average = 3.03% 5.59% 4.45%Table4 presents the comparison of the best results obtained with the reviewed methods. When compared to CKM the proposed metaheuristic outperforms it in all instances. This is observed in the average error which is significantly higher for CKM in comparison to GRASP-CKMC (10.58% > 3.03%). The average performance of the GA is similar to the performance of CKM (10.27% ≈ 10.00%). However this is observed because the GA outperforms the CKM method for medium instances (SJC1-DONI1) while the CKM significantly outperforms the GA for large instances (DONI2-DONI5). Better performance is observed with the VNS method with an average error of 3.64%. Also, in two instances the VNS method obtained the best known solutions (error = 0.0%). Even though the GRASP-CKMC metaheuristic is not able to obtain the best known solution, overall performance is better than VNS (3.03% < 3.64%). Particularly for instances SJC3a, SJC4a, DONI3, and DONI4, the GRASP-CKMC metaheuristic outperforms the VNS, GA, and CKM methods.Table 4
Performance of CKM, GA, VNS, GRASP-CKMC, A-BRKGA, TS, and CS on the SJC and DONI instances when compared to best-known solutions.
Instance Best-Known CKM GA VNS GRASP-CKMC A-BRKGA TS CS SJC1 17359.75 17.18% 0.02% 1.94% 2.20% 0.00% 0.00% 0.00% SJC2 33181.65 6.12% 0.83% 0.73% 0.45% 0.00% 0.00% 0.00% SJC3a 45356.35 11.54% 3.29% 5.80% 1.49% 0.00% 0.00% 0.00% SJC4a 61931.60 11.87% 4.92% 7.68% 1.26% 0.00% 0.10% 0.00% DONI1 3021.41 7.06% 3.88% 0.00% 2.23% -0.13% 0.12% 0.21% DONI2 6080.70 10.06% 14.88% 0.00% 5.58% 4.78% 5.00% 4.81% DONI3 8343.49 17.42% 15.70% 5.10% 3.93% 0.41% 0.00% 1.14% DONI4 10777.64 7.58% 23.66% 6.85% 4.64% 0.12% 0.00% 1.62% DONI5 11114.67 6.42% 25.24% 4.68% 5.49% 0.54% 0.00% 0.86%  Average = 10.58% 10.27% 3.64% 3.03% 0.64% 0.58% 0.96%When comparing the performance of the GRASP-CKMC with more updated metaheuristics such as TS, CS, and A-BRKGA, these reported a better performance with average errors smaller than 1.0%. This is expected as the proposed metaheuristic is based on the GRASP and KMC principles and as such, it is proposed as an alternative to similar metaheuristics such as GA, KMC, and VNS. In general terms, the GRASP-CKMC performs in the middle between the standard and the most recent methods for the CCCP with an average best error of approximately 3.0%.Due to these results, the proposed metaheuristic is considered suitable to address the location of the support centers or prepositioned warehouses for the communities of Veracruz.


## 2.1. Capacitated KMC

K-Means is one of the basic unsupervised learning algorithms that solve the well-known clustering problem [28–30]. This model is similar to the also well-known K-Nearest Neighbor (KNN) search algorithm [31]. The KMC follows a simple procedure to classify a given data set through a certain number of clusters K [28, 32]. Within the context of the CCCP or CPMP the facility is located at the cluster’s centroid or median point, respectively. For multiple facilities, the first problem to be solved is the consolidation of clusters (i.e., groups of points) and the second is the determination of the median point or centroid. Both problems can be addressed simultaneously by the K-Means Clustering (KMC) algorithm. Figure 2 presents the details of the standard KMC algorithm.Figure 2
Structure of the standardK-Means Clustering (KMC) algorithm.As presented in Figure2 clustering involves the unique assignment of a point to the nearest cluster based on its center (defined as the median point or the centroid). The locations of the centers must be reestimated each time that new assignments are performed, and new assignments can be generated each time that the reestimation process is performed as they affect the closeness of the centers to the considered points.For the purpose of determining the locations of the support centers and their assigned communities, the standard KMC algorithm must integrate capacity restrictions. However this adds complexity to the assignment task because not all nearest points to a certain center can comply with its capacity restriction (thus, not all nearest points can be assigned to this center).Approaches have been proposed to address the capacitated task. In contrast to the circular regions shown in Figure2, in [32] a rectangular region around the center was considered to determine the candidates for clustering. This reduces the number of points to be assigned to the cluster and thus reduces the likelihood of not complying with the capacity restriction. The points located outside the rectangular region are omitted by this initial assignment process. After this process is performed, a priority is assigned to the omitted points in order to be assigned to the clusters with available capacity in a final assignment process. Other approaches involve an average distance for the reassignment of points [33].The assignment of close points and reassignment of omitted points are procedures which can be performed with some randomness to add flexibility to the local search process of KMC. Thus, the proposal to extend the KMC to perform the capacitated task consists in including a uniform random variable to control the ratio of acceptance for the KMC algorithm (and thus, of theV nearest locations). This proposal is similar to the Variable Neighborhood Search (VNS) principle [34].Figure3 presents the general structure of the proposed capacity-restricted KMC algorithm (CKMC). As presented in Figure 1, this CKMC algorithm is used in both phases of the GRASP-CKMC metaheuristic. This is the reason of the adjustments stated in Figure 3:(i)
In the Constructive Phase, the CKMC is executed only once for two randomK values which will be used by the μGA to determine the lower and upper limits (Kmin, Kmax) for K. Also, the number of nearest points to each center (V) is constant given by X.(ii)
In the Local Search Phase, the CKMC is iteratedP times, and at each iteration, different values of K (within Kmin and Kmax) and the ratio of acceptance V (which has an upper limit given by X) are considered. At each iteration, the best assignment of points (locations) to clusters (centroids) (as measured by its objective function value G) is saved. After the P iterations of the CKMC algorithm are executed, the best found solution is improved by means of insertion, deletion, and exchange operations which are controlled by a decision process.Figure 3
Structure of the proposed CapacitatedK-Means Clustering (CKMC) algorithm.


## 2.2.μGA

The standard KMC algorithm considers that the quantity of clusters is knowna priori [30]. Within the context of the CCCP, the minimization of the objective function (total distance from each cluster to each assigned point) depends on finding the most suitable number of clusters. Hence, the proposed GRASP-CKMC includes an evolutionary mechanism to determine the suitable range of clusters which can minimize the total distance to the affected communities.Figure4 presents the general structure of the μGA which was developed to address this task. The μGA is characterized by small populations which can lead to achieving faster convergence with less storing memory [35, 36]. In this case, the individuals of the population of the μGA consist only of pairs of values (Kmin, Kmax) that can define the lower and upper limits of a range that may contain the K value that can lead to a total minimum distance on a single execution of the CKMC algorithm. By using the random mutation and the linear crossover operators a diversification on these bounds is obtained to estimate an interval for the local search of K within the main GRASP-CKMC algorithm. An estimate for (Kmin, kmax) is obtained after m generations (in this case, m = 100) of the μGA are executed, and within this range, K is randomly selected.Figure 4
Structure of theμGA for initialization of K.


## 2.3. Insertion, Deletion, and Exchange

As presented in Figures1 and 3, the best solution found by the CKMC in the Local Search Phase is improved by a decision algorithm which performs insertion, deletion, and exchange of points between clusters. A conditional decision process was designed to avoid unnecessary tasks due to the random selection of points which can be inserted, deleted, or exchanged. The description of this improvement process is presented in Figure 5.Figure 5
Structure of the decision process of the GRASP algorithm.Finally, the implementation of the metaheuristic was performed with Octave and MATLAB in a HP Workstation with Intel Zeon CPU at 3.40 GHz with 8 GB RAM.


## 2.4. Assessment

Before proceeding to obtaining a solution for our instance, we assessed the performance of the GRASP-CKMC metaheuristic with a selection of CCCP instances. Due to the size of the instance (3837 communities), we considered the following SJC and DONI instances [37, 38]: SJC1 (100 points), SJC2 (200 points), SJC3a (300 points), SJC4a (402 points), DONI1 (1000 points), DONI2 (2000 points), DONI3 (3000 points), DONI4 (4000 points), and DONI5 (5000 points) [39]. For comparison purposes, the performance of the GRASP-CKMC metaheuristic was compared to standard and most recent methods, including the latest best known solutions as follows:(i)
Best known solutions as reported in [24].(ii)
Best results obtained by CKM and GA as reported in [38].(iii)
Best results obtained by VNS as reported in [23].(iv)
Best results obtained by TS (Tabu Search) and CS (Clustering Search) as reported in [24].(v)
Best results obtained by the latest method known as Adaptive Biased Random-Key Genetic Algorithm (A-BRKGA) as reported in [24].Table2 presents the parameters for the GRASP-CKMC algorithm. As mentioned in [24], metaheuristics have no optimal values of parameters. Thus, recommended ranges are usually considered for these cases. For the GRASP-CKMC algorithm it was considered to have a lean execution due to the size of the instance and the diverse algorithms which were developed. Thus, small values were considered for X (the upper limit for the number of nearest points to each cluster), the executions of the CKMC in the Local Search Phase (P) and the number of pairs of points to be considered for exchange, deletion, or insertion (Y). Regarding the Mersenne Twister random number generator, it was considered as recommended by the MATLAB documentation.Table 2
Parameters of the GRASP-CKMC.
Parameter Value  X 10 P 50 Y 10000 Random Number Generator Mersenne TwisterTable3 presents the results obtained for 10 runs of the algorithm. It is observed that the average of the best results is 3.03% while the average of the worst results is 5.59%. Particularly for the instances DONI3 and DONI4, which have a similar number of points to the considered instance of 3837 communities, the GRASP-CKMC metaheuristic is able to obtain solutions with errors smaller than 5.0% (3.93% and 4.64%, respectively) within 10 runs.Table 3
Results of 10 runs of the GRASP-CKMC metaheuristics on the SJC and DONI instances.
Instance Best      Runs     Best Worse Average Best Worst Average Known 1 2 3 4 5 6 7 8 9 10    Error(%) Error(%) Error(%) SJC1 17359.75 17807.73 18230.49 17742.19 18088.71 17791.75 18054.23 18169.28 17948.46 18132.05 17789.13 17742.19 18230.49 17975.40 2.20% 5.02% 3.55% SJC2 33181.65 33810.48 34077.98 33568.98 33529.65 33673.30 33342.50 33745.19 33330.23 34084.06 33731.71 33330.23 34084.06 33689.41 0.45% 2.72% 1.53% SJC3a 45356.35 46993.46 46165.69 46789.82 46503.93 46559.25 46459.06 46944.45 46697.09 46250.59 46030.70 46030.70 46993.46 46539.40 1.49% 3.61% 2.61% SJC4a 61931.60 62774.78 62710.35 63064.16 62962.47 62844.49 62899.00 63000.86 63357.25 62926.63 63369.95 62710.35 63369.95 62990.99 1.26% 2.32% 1.71% DONI1 3021.41 3088.71 3165.04 3104.18 3144.54 3169.48 3141.08 3095.66 3109.24 3157.25 3158.03 3088.71 3169.48 3133.32 2.23% 4.90% 3.70% DONI2 6080.70 6487.85 6499.84 6498.32 6490.31 6459.86 6498.46 6487.18 6484.72 6483.75 6420.06 6420.06 6499.84 6481.04 5.58% 6.89% 6.58% DONI3 8343.49 8706.45 8971.18 8953.30 8964.51 8677.99 8691.23 8894.92 8782.56 9010.17 8671.70 8671.70 9010.17 8832.40 3.93% 7.99% 5.86% DONI4 10777.64 11540.02 11508.46 11704.86 11568.63 11541.06 11534.28 11702.91 11277.44 11392.09 11685.20 11277.44 11704.86 11545.49 4.64% 8.60% 7.12% DONI5 11114.67 12016.16 11991.16 12034.59 11846.86 12022.82 11724.42 11934.19 11937.12 11938.20 11893.37 11724.42 12034.59 11933.89 5.49% 8.28% 7.37%  Average = 3.03% 5.59% 4.45%Table4 presents the comparison of the best results obtained with the reviewed methods. When compared to CKM the proposed metaheuristic outperforms it in all instances. This is observed in the average error which is significantly higher for CKM in comparison to GRASP-CKMC (10.58% > 3.03%). The average performance of the GA is similar to the performance of CKM (10.27% ≈ 10.00%). However this is observed because the GA outperforms the CKM method for medium instances (SJC1-DONI1) while the CKM significantly outperforms the GA for large instances (DONI2-DONI5). Better performance is observed with the VNS method with an average error of 3.64%. Also, in two instances the VNS method obtained the best known solutions (error = 0.0%). Even though the GRASP-CKMC metaheuristic is not able to obtain the best known solution, overall performance is better than VNS (3.03% < 3.64%). Particularly for instances SJC3a, SJC4a, DONI3, and DONI4, the GRASP-CKMC metaheuristic outperforms the VNS, GA, and CKM methods.Table 4
Performance of CKM, GA, VNS, GRASP-CKMC, A-BRKGA, TS, and CS on the SJC and DONI instances when compared to best-known solutions.
Instance Best-Known CKM GA VNS GRASP-CKMC A-BRKGA TS CS SJC1 17359.75 17.18% 0.02% 1.94% 2.20% 0.00% 0.00% 0.00% SJC2 33181.65 6.12% 0.83% 0.73% 0.45% 0.00% 0.00% 0.00% SJC3a 45356.35 11.54% 3.29% 5.80% 1.49% 0.00% 0.00% 0.00% SJC4a 61931.60 11.87% 4.92% 7.68% 1.26% 0.00% 0.10% 0.00% DONI1 3021.41 7.06% 3.88% 0.00% 2.23% -0.13% 0.12% 0.21% DONI2 6080.70 10.06% 14.88% 0.00% 5.58% 4.78% 5.00% 4.81% DONI3 8343.49 17.42% 15.70% 5.10% 3.93% 0.41% 0.00% 1.14% DONI4 10777.64 7.58% 23.66% 6.85% 4.64% 0.12% 0.00% 1.62% DONI5 11114.67 6.42% 25.24% 4.68% 5.49% 0.54% 0.00% 0.86%  Average = 10.58% 10.27% 3.64% 3.03% 0.64% 0.58% 0.96%When comparing the performance of the GRASP-CKMC with more updated metaheuristics such as TS, CS, and A-BRKGA, these reported a better performance with average errors smaller than 1.0%. This is expected as the proposed metaheuristic is based on the GRASP and KMC principles and as such, it is proposed as an alternative to similar metaheuristics such as GA, KMC, and VNS. In general terms, the GRASP-CKMC performs in the middle between the standard and the most recent methods for the CCCP with an average best error of approximately 3.0%.Due to these results, the proposed metaheuristic is considered suitable to address the location of the support centers or prepositioned warehouses for the communities of Veracruz.


## 3. Proposed Locations for Communities at Risk

Figure6 presents general statistics regarding the people affected by disasters in the considered regions of Veracruz, Mexico. In total, in the capital and highlands regions, there are 526,947 people at risk throughout 3837 communities where the community of Xalapa-Enríquez has the largest amount with 42,476 people. Because support centers are considered to supply resources for a maximum of 10,000 people, larger communities (such as Xalapa-Enríquez) were segmented into equally-sized smaller communities. This led to a total of 3844 communities.Figure 6
Affected communities in Veracruz: capital and highlands regions.Due to the importance of minimizing the distance between the affected communities and the location of the centers, a reliable distance metric must be used. In this case, the geographic arc length metric is considered because it can provide accurate distances in kilometers based on the spherical model of Earth’s surface which has a radius ofR= 6,371 Km. With this metric, the arc length (distance) between two locations (di,j) with geographic coordinates (ϕi, θi) and (ϕj, θj), where ϕ is the latitude and θ is the longitude in radians, is estimated as follows [40]: (1)di,j=R×αi,j=R×Arccoscos⁡ϕicos⁡ϕjcos⁡θi-θj+sin⁡ϕisin⁡ϕj.With this data, the GRASP-CKMC metaheuristic determined a set of 260 centers to provide support to the 3837 communities (or extended 3844 communities) with minimum average total distance. The general results are presented in Table5 while Figure 7 presents the visualization of the assignments.Table 5
Affected communities in Veracruz: number of centers, number of locations (communities), and people at risk assigned to each center, total distance, and average distance from the locations to the assigned center.
Center Locations People Total Average Center Locations People Total Average Center Locations People Total Average Center Locations People Total Average at Risk Distance Distance   at Risk Distance Distance   at Risk Distance Distance   at Risk Distance Distance 1 12 2476 27.05 2.25 66 18 1927 44.33 2.46 131 9 1074 23.23 2.58 196 19 3227 53.64 2.82 2 5 68 8.38 1.68 67 12 656 29.48 2.46 132 11 926 20.38 1.85 197 19 320 46.62 2.45 3 10 4437 18.75 1.87 68 17 1034 39.65 2.33 133 15 713 37.88 2.53 198 12 741 28.42 2.37 4 15 602 44.27 2.95 69 13 1163 26.15 2.01 134 5 123 9.42 1.88 199 10 1924 25.27 2.53 5 3 167 5.16 1.72 70 25 9411 63.99 2.56 135 19 1995 45.68 2.40 200 21 2385 39.45 1.88 6 9 362 18.19 2.02 71 23 1269 54.62 2.37 136 17 312 50.70 2.98 201 5 460 9.81 1.96 7 23 3203 55.71 2.42 72 32 2259 78.58 2.46 137 8 175 18.37 2.30 202 4 81 4.22 1.05 8 14 213 25.19 1.80 73 41 8874 82.90 2.02 138 9 716 16.62 1.85 203 16 691 42.14 2.63 9 5 584 15.96 3.19 74 26 911 79.92 3.07 139 12 729 23.00 1.92 204 12 215 16.81 1.40 10 4 194 5.09 1.27 75 17 8374 30.90 1.82 140 13 2962 29.88 2.30 205 10 991 31.85 3.18 11 16 1410 34.09 2.13 76 33 4714 92.23 2.79 141 25 5207 55.78 2.23 206 3 198 4.01 1.34 12 2 137 1.19 0.59 77 14 2301 26.56 1.90 142 23 2576 39.97 1.74 207 3 62 5.03 1.68 13 2 88 3.03 1.52 78 7 639 13.16 1.88 143 6 213 9.57 1.59 208 11 911 20.30 1.85 14 19 4482 42.91 2.26 79 26 9941 50.56 1.94 144 4 544 5.23 1.31 209 23 3683 43.62 1.90 15 17 1561 41.25 2.43 80 3 14 3.58 1.19 145 7 185 6.71 0.96 210 15 920 32.07 2.14 16 10 267 23.04 2.30 81 5 70 8.30 1.66 146 18 2996 46.03 2.56 211 6 972 11.23 1.87 17 3 1233 2.46 0.82 82 19 1491 32.65 1.72 147 3 10 5.38 1.79 212 12 1711 21.40 1.78 18 7 584 24.53 3.50 83 16 828 38.64 2.42 148 41 5836 111.30 2.71 213 5 144 11.74 2.35 19 16 488 33.97 2.12 84 16 279 27.52 1.72 149 15 790 38.17 2.54 214 7 379 10.39 1.48 20 28 2371 71.08 2.54 85 6 263 6.15 1.02 150 4 76 3.58 0.89 215 28 150 50.60 1.81 21 6 921 7.91 1.32 86 17 434 36.45 2.14 151 3 157 1.93 0.64 216 14 585 27.98 2.00 22 4 242 5.73 1.43 87 14 1139 31.64 2.26 152 2 58 1.38 0.69 217 17 1270 31.63 1.86 23 15 1194 34.99 2.33 88 5 357 6.89 1.38 153 9 4179 19.71 2.19 218 17 1875 40.34 2.37 24 7 1105 11.96 1.71 89 23 1035 71.20 3.10 154 7 3404 19.11 2.73 219 3 585 4.06 1.35 25 17 321 29.46 1.73 90 24 1860 52.37 2.18 155 4 1235 14.53 3.63 220 12 627 26.54 2.21 26 5 326 8.13 1.63 91 9 258 15.56 1.73 156 10 756 20.69 2.07 221 6 174 11.57 1.93 27 6 779 14.21 2.37 92 21 4511 46.10 2.20 157 9 116 13.90 1.54 222 30 652 70.32 2.34 28 16 10000 52.02 3.25 93 3 804 9.60 3.20 158 8 1445 15.22 1.90 223 3 1508 1.91 0.64 29 24 2130 72.72 3.03 94 9 25 15.34 1.70 159 19 380 56.62 2.98 224 16 1250 36.62 2.29 30 29 1444 82.77 2.85 95 22 304 41.04 1.87 160 7 639 12.56 1.79 225 4 60 4.40 1.10 31 20 2256 36.76 1.84 96 16 1658 45.35 2.83 161 21 1564 41.40 1.97 226 23 9367 66.26 2.88 32 23 6639 53.35 2.32 97 4 385 7.06 1.77 162 27 9448 84.59 3.13 227 4 95 2.70 0.67 33 19 2312 42.10 2.22 98 23 9011 42.20 1.83 163 6 518 9.81 1.63 228 5 120 5.66 1.13 34 17 2277 25.31 1.49 99 16 2071 41.36 2.59 164 2 2 2.12 1.06 229 8 306 12.83 1.60 35 39 8353 109.04 2.80 100 5 538 6.88 1.38 165 2 33 0.23 0.12 230 6 137 10.67 1.78 36 16 9118 30.44 1.90 101 4 249 7.08 1.77 166 7 388 15.00 2.14 231 6 362 15.15 2.52 37 8 1337 21.39 2.67 102 38 3314 99.33 2.61 167 6 88 15.03 2.51 232 13 2334 26.04 2.00 38 19 945 31.18 1.64 103 25 1519 53.60 2.14 168 7 9191 11.35 1.62 233 13 1960 35.09 2.70 39 12 2176 28.03 2.34 104 7 176 21.04 3.01 169 29 1399 72.16 2.49 234 19 476 51.24 2.70 40 14 878 39.35 2.81 105 13 2766 21.44 1.65 170 9 240 18.24 2.03 235 26 3872 61.26 2.36 41 12 810 26.20 2.18 106 3 225 1.51 0.50 171 4 137 4.48 1.12 236 4 162 4.63 1.16 42 5 661 11.09 2.22 107 22 6305 51.83 2.36 172 12 561 25.68 2.14 237 19 4613 52.11 2.74 43 10 1215 19.68 1.97 108 5 39 8.05 1.61 173 22 8947 54.23 2.46 238 44 9940 63.23 1.44 44 16 322 41.62 2.60 109 11 1172 36.57 3.32 174 28 7414 52.33 1.87 239 12 69 27.39 2.28 45 8 941 13.11 1.64 110 16 1271 39.64 2.48 175 11 1894 16.74 1.52 240 19 2173 39.79 2.09 46 4 404 5.87 1.47 111 15 1041 33.29 2.22 176 12 1187 17.39 1.45 241 15 4723 44.96 3.00 47 20 2326 49.74 2.49 112 22 1034 64.23 2.92 177 28 3846 81.85 2.92 242 28 2218 76.62 2.74 48 14 588 26.33 1.88 113 16 2714 39.87 2.49 178 5 226 6.91 1.38 243 7 244 16.24 2.32 49 10 1138 21.05 2.10 114 6 2319 11.74 1.96 179 19 2849 43.91 2.31 244 43 5216 87.43 2.03 50 24 1335 54.73 2.28 115 13 1285 29.71 2.29 180 18 1298 42.73 2.37 245 12 1646 27.97 2.33 51 9 1479 15.07 1.67 116 33 4990 88.85 2.69 181 11 44 17.58 1.60 246 6 355 16.88 2.81 52 8 243 14.90 1.86 117 60 7460 144.50 2.41 182 5 904 13.07 2.61 247 40 6829 105.91 2.65 53 38 4231 94.32 2.48 118 29 1599 67.88 2.34 183 4 19 2.76 0.69 248 34 7534 86.96 2.56 54 22 2053 48.93 2.22 119 51 1966 78.85 1.55 184 25 4846 79.86 3.19 249 24 5724 54.87 2.29 55 16 925 41.17 2.57 120 14 696 35.84 2.56 185 34 7175 83.06 2.44 250 19 2167 29.69 1.56 56 5 383 4.55 0.91 121 19 2926 26.93 1.42 186 18 2028 44.00 2.44 251 23 3379 46.16 2.01 57 35 4994 71.98 2.06 122 11 597 26.20 2.38 187 12 787 19.73 1.64 252 3 579 5.51 1.84 58 14 6182 27.62 1.97 123 15 1831 39.75 2.65 188 9 448 13.68 1.52 253 6 1300 18.06 3.01 59 5 292 9.44 1.89 124 15 663 38.52 2.57 189 24 2859 72.49 3.02 254 13 884 27.78 2.14 60 22 10000 35.81 1.63 125 21 9613 48.10 2.29 190 24 6301 55.37 2.31 255 20 3998 50.98 2.55 61 9 8725 16.34 1.82 126 6 13 8.60 1.43 191 14 1603 30.47 2.18 256 17 1281 29.53 1.74 62 23 1191 57.54 2.50 127 5 47 8.31 1.66 192 43 4528 100.74 2.34 257 27 3751 63.21 2.34 63 24 7015 37.22 1.55 128 7 441 13.20 1.89 193 20 2260 47.66 2.38 258 9 68 12.89 1.43 64 8 154 25.07 3.13 129 7 389 13.08 1.87 194 11 105 28.17 2.56 259 13 1273 24.20 1.86 65 8 853 9.31 1.16 130 12 2991 29.61 2.47 195 15 780 30.59 2.04 260 9 2005 18.30 2.03Figure 7
Affected communities in Veracruz: assignment of support centers.Based on these results, it was determined that the mean distance that people at risk must travel from their community to its assigned center is approximately 2.08 Km with a standard deviation of 0.60 Km. In this case, humanitarian relief can be provided within a short period of time.These results also provide information to determine the most suitable capacities for each center. Although the GRASP-CKMC imply the establishment of 260 centers to provide supplies to a maximum of 10,000 affected people, the people at risk within the communities assigned to each center can be considered to determine its most suitable capacity. Figure8 presents a histogram that represents the number of centers assigned for each interval or range of people at risk. As observed, 103 centers serve communities with a minimum and maximum of 2 and 717 affected people, respectively. In contrast, just 8 centers serve communities with a minimum and maximum of 9297 and 10000 people, respectively. These results can be considered to make a better estimation of the capabilities of the prepositioned warehouses and, thus, of the necessary inventory.Figure 8
Affected communities in Veracruz: number of centers vs. intervals of demand.


## 4. Conclusions and Future Work

The present work addressed the location planning for prepositioned warehouses or support centers for communities at risk in Veracruz, Mexico. This was addressed by means of the Capacitated Centered Clustering Problem (CCCP) [23] because minimization of distances between the affected regions and the prepositioned warehouses is an important aspect of humanitarian relief planning.Due to the large set of communities (3837) and people at risk (526,947), a metaheuristic was developed to provide a suitable solution for this problem. This metaheuristic integrated the principles of GRASP, GA, and KMC to provide more suitable solutions than those obtained by similar local search metaheuristics. When tested with well-known large facility location instances, the metaheuristic termed as GRASP-CKMC was able to obtain a mean best error of 3.03%. Although more complex algorithms such as CS and A-BRKGA reported better results with errors smaller than 1.00%, the performance of the GRASP-CKCM metaheuristic was more competitive when compared to standard methods such as GA, KMC, and VNS. Thus, the GRASP-CKCM can be considered as a more suitable strategy when compared to these methods.When the GRASP-CKMC was applied on the real instance with 3837 communities, the metaheuristic determined a set of 260 centers to provide full coverage to all communities. These results also provided insights regarding the utilization of these centers considering the actual communities assigned to them. Based on these insights, it was determined that the facility location task could also support the decisions regarding the characteristics of the support centers by obtaining the estimation of the communities assigned to each one of them. Thus, smaller centers or prepositioned warehouses can be considered for some regions. This can optimize the use of resources and improve relief efforts.Optimization of the supply chain for humanitarian relief efforts is an extensive field which requires continuous advances in the logistics and production planning processes. Thus, as future work, the following aspects are considered:(i)
Extending the CCCP model to consider heterogeneous capacities for the centers.(ii)
Integrating route planning on the facility location problem to optimize the two-echelon supply chain.(iii)
Multicriteria optimization to extend on the facility location problem.(iv)
Integrating the use of Artificial Neural Networks (ANNs) to dynamically determine the number of clusters to improve speed and convergence of the CKMC algorithm.(v)
Integrating the principles of the CS method to enhance the performance of the GRASP metaheuristic.



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*Source: 3605298-2018-12-20.xml*