Hindawi Journal of Optimization Volume 2018, Article ID 3605298, 14 pages https://doi.org/10.1155/2018/3605298
Research Article Extended GRASP-Capacitated �-Means Clustering Algorithm to Establish Humanitarian Support Centers in Large Regions at Risk in Mexico
Santiago-Omar Caballero-Morales , Erika Barojas-Payan, Diana Sanchez-Partida , and Jose-Luis Martinez-Flores Universidad Popular Autonoma del Estado de Puebla, A.C., Postgraduate Department of Logistics and Supply Chain Management, Puebla, Mexico Correspondence should be addressed to Santiago-Omar Caballero-Morales; [email protected]
Received 9 July 2018; Revised 25 November 2018; Accepted 4 December 2018; Published 20 December 2018
Guest Editor: Morteza Ghobakhloo
Copyright © 2018 Santiago-Omar Caballero-Morales et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 signifcant 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 Pacifc and Atlantic Oceans. Due to these situations, each year many communities are afected by diverse natural disasters in Mexico and efcient logistic systems are required to provide prompt support. Tis work is aimed at providing an efcient metaheuristic to determine the most appropriate location for support centers in the State of Veracruz, which is one of the most afected regions in Mexico. Te metaheuristic is based on the �-Means Clustering (KMC) algorithm which is extended to integrate (a) the associated capacity restrictions of the support centers, (b) a micro Genetic Algorithm �GA 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 fexibility to the assignation task, and (d) random-based decision model to further improve the fnal assignments. Tese extensions on the KMC algorithm led to the GRASP-Capacitated �-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. Te validation supported its suitability as alternative to standard metaheuristics such as Capacitated �-Means (CKM), Genetic Algorithms (GA), and Variable Neighborhood Search (VNS).
1. Introduction Mexico is located within the so-called Fire Belt of the Pacifc and within the tropical zone. Tis makes the country A phenomenon or disturbing agent is defned as an aggressive susceptible to a great variety of disturbing agents of natural andpotentiallyharmfulphysicalevent,naturalorderived origin [3]: from human activity, which can cause loss of life or injury, material damage, serious disruption of social and economic (a) Two-thirds of the country have signifcant seismic life, or environmental degradation. Tus, these agents can risks. have the following origins [1, 2]: (b) Coastal regions are frequently afected by hurricanes (a) Natural: geological, hydrometeorological, and astro- which are generated in the Pacifc and Atlantic nomical. Oceans. (b) Anthropogenic: chemical-technological, health-eco- Due to its geographical location, geological characteris- logical, and social-organizational. tics, and the complex morphology of its territory, the State of 2 Journal of Optimization
Table 1: Declarations of natural disasters: 2014-2017 [5].
Year Description FONDEN (Mexican Pesos) 2014 01. Severe rain and fuvial food 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 fuvial food in March 25-27 03. Severe rain and fuvial food in June 11-14 04. Hillside movement in July 9-13 05. Hillside movement in September 16-18 06. Severe rain and fuvial and rain food in October 16-21 07. Severe rain and fuvial and rain food in October 18-24 2016 01. Hillside movement in August 5-7 860,195,408.00 02. Severe rain and fuvial food 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 fuvial food in September 8-12 03. Hillside movement from September 27 to October 9 04. Severe rain and fuvial food from September 27 to October 9 05. Fluvial food in October 11-15 06. Severe rain in October 11 Total 3,161,486,971.20
Veracruz in Mexico is exposed to natural phenomena such which consist of inventory and location planning of as earthquakes, volcanic eruptions, foods, and landslides. resources, are performed. Te presence of hydrometeorological phenomena is very (2) Pre-impact phase: warning to the population based common in Veracruz, which leads to frequent afectations. In on prediction mechanisms and implementation of response to the presence of disturbing natural phenomena, in mitigating measures are performed. Mexico the Natural Disasters Fund (FONDEN) was created. Tisisafnancialinstrumentwhosepurposeistoprovide (3) Disaster impacts the community. relief supplies and assistance in emergency and disaster sit- (4) Emergency phase: isolation, rescue, and external uations. In Veracruz, the rules of the Fund for the Prevention assistance are performed. It is ofen the phase in which of Natural Disasters (FOPREDEN) are an instrument that local resources are overwhelmed and external aid is aims to revitalize initiatives aimed at preventing disasters and required to reduce the number of fatalities. seeks to optimize the use of available fnancial resources and (5) Reconstruction phase: activities focused on recov- magnify the results linked mainly to the preservation of the ering the normal duties of the community are per- life and physical integrity of people, as well as that of public formed. services and infrastructure and the environment [4]. As of 2017 FONDEN has authorized resources for more Before the disaster occurs, it is important to have facilities than three billion of Mexican pesos to support the road, with an optimal inventory of products of frst necessity to educational sectors, forestry, hydraulic, naval, housing, and support the survival of the people who will be afected. urban infrastructure in Veracruz due to the signifcant occur- Also, afer the disaster occurs, it is important to have the rence of natural disasters within the period 2014-2017.Table 1 infrastructure to resupply the facilities and transport afected presents an overview of the historical phenomena within this people to other facilities as needed. period and the resources provided. Hence, among the most critical decisions and resources Standard protocols to be performed before, during, and to provide relief to the afected communities in Veracruz, afer a disaster involve diferent logistic processes. Tese are prepositioning of warehouses must be performed. Tis allows performed in the diferent phases of a disaster [6, 7]: the protection of supplies and the efcient and timely supply of products to cover the basic needs of the people afected (1) Interdisaster phase: processes are performed in which by the disturbing phenomenon. Within the activities to the elaboration of the map of risks for the com- be performed in these warehouses or support centers, the munity is highlighted. Also the Plans of Emergency, following can be mentioned: Journal of Optimization 3
(1) Identifcation, labeling, and location of the necessary (ii) Te facility location problem was also addressed supplies to attend the emergency. by [17] that presented a multiobjective optimization model to solve a multidepot emergency facilities (2) Consolidation of load and change of means of trans- location-routing problem. Due to the inherent com- port. putational complexity of this model an approximate (3) Delivery scheduling for the supplies. method based on the metaheuristic of Genetic Algo- rithms (GAs) was developed. Likewise, the warehouse must have an information and (iii) Prepositioning or relief assets was studied in [11] to inventory control system which must be updated, through optimize the transportation of afected people to relief the control of inventories. Te activation of a prepositioned centers. Te proposed stochastic model considered warehouse is the responsibility of State Civil Protection optimization of resources such as personnel and with selection criteria for its allocation such as (a) being vehicles to minimize casualties. located outside the risk area, (b) having a solid and roofed construction in compliance with safety parameters, (c) being (iv) In [18] the aspect of considering containers as storage accessible through favorable conditions for transport loading facilities was studied, and a mathematical model was and unloading, (d) being ventilated, illuminated, and without proposed to determine the locations of supply points water seepage risk, (e) being located far away from food- and the quantity of containers and relief supplies prone areas, (f) being free of pollution or plague, and (g) assigned to each supply point under the minimum having space to facilitate the mobility, cleaning, and classi- distance criteria. fcation of products [8, 9]. Minimization of distance between (v) A stochastic inventory control strategy was proposed the afected regions and the prepositioned warehouses is an in [19] for the uncertain requirements of goods for important aspect of humanitarian relief planning because postdisaster conditions, in order to have the adequate communities must be able to reach these centers within short stock to serve the vital needs of afected communities. periods of time and distances due to the severity of the (vi) A conceptual model that integrated the aspect of disasters. agility in HL was presented in [20] to improve on In this regard, humanitarian logistics (HL) formally theresponseofHLtodisasterscenarios.While addresses the “process of planning, implementing, and con- no mathematical model was presented or discussed, trolling the efcient, cost-efective fow of and storage of the roles of people, processes, and technology were goods and materials as well as related information from identifed as agility enablers for the success of general point of consumption for the purpose of meeting the end models within HL. benefciary’s requirements” [10, 11]. Te need of HL for strategic planning has been recognized by important orga- In general for the determination and location of facilities nizations such as the U.S. Federal Emergency Management (i.e., support centers, warehouses, prepositioned inventory, Agency (FEMA) and the United Nations (UN) [11, 12]. In etc.) the following mathematical models have been consid- contrast to commercial logistics (CL), the main focus of HL ered: the Capacitated �-Median Problem (CPMP) [21, 22] is to save lives and provide benefciaries with aid instead of and the Capacitated Centered Clustering Problem (CCCP) maximizing profts. However, due to this characteristic, HL [23, 24]. Both models are focused on determining the location has disadvantages when compared to CL as it faces lower of � facilities in order to minimize the total weighted distance technology, challenging inventory control, unstable demand from the facilities to all demand points (customers). A patterns, zero lead time, and unpredictable supply resources demand point cannot be assigned to more than one facility, [13, 14]. and the points assigned to a facility cannot exceed its capacity. Hence, diferent strategies have been developed within Te main diference between both models is about the the feld of HL for the optimal operation of all the aspects features of the locations of the � facilities. In the CPMP the of the supply chain (SC) for the delivery of goods to afected location is determined at a median point while for the CCCP communities considering these disadvantages. In this con- the location is determined at a centroid. text, humanitarian relief organizations (HRO) have been Both models are difcult to be solved to optimality identifed as the best suited organizations for preparedness due to their NP-hard computational complexity. Particularly and recovery when compared to commercial and military for large problems, this has led to the development of organizations[13].Animportantaspectofpreparednessisthe metaheuristics to provide near-optimal solutions [25]. In prepositioning of inventories or warehouses for postdisaster the literature, metaheuristics based on Clustering Search relief. Among the most recent strategies, which are focused (CS) have been reported as the most competitive methods on transportation, planning, policies and procedures, and for the CCCP [24]. However, in this work we focus on inventory/warehousing [15], the following can be mentioned: providing an alternative to standard methods which are commonly implemented for practical situations. In the case of (i) In [16], a stochastic model was developed to deter- humanitarian relief actions, fast implementation is required, mine the location of Emergency Medical Service and we are considering the situation of Veracruz in Mexico, (EMS) systems. In order to solve this model, exact where 3837 communities with 526,954 people are at risk. and approximate (metaheuristics) methods were pro- Tus, in this work a metaheuristic based on the integra- posed. tion of the Greedy Randomized Adaptive Search Procedure 4 Journal of Optimization
(GRASP) and the �-Means Clustering (KMC) algorithm 2.1. Capacitated KMC. �-Means is one of the basic unsu- is presented to provide a suitable location planning for pervised learning algorithms that solve the well-known the support centers (prepositioned warehouses) for these clustering problem [28–30]. Tis model is similar to the also communities in Mexico. In order to provide more accurate well-known �-Nearest Neighbor (KNN) search algorithm solutions for large CCCP instances than those of standard [31]. Te KMC follows a simple procedure to classify a given methods, the proposed metaheuristic has the following fea- data set through a certain number of clusters � [28, 32]. tures: Within the context of the CCCP or CPMP the facility is located at the cluster’s centroid or median point, respectively. (a) capacity restrictions to the KMC algorithm for For multiple facilities, the frst problem to be solved is the the assignation of communities to support centers consolidation of clusters (i.e., groups of points) and the � (Capacitated -Means Clustering, CKMC); second is the determination of the median point or centroid. (b) micro Genetic Algorithm (�GA) that performs single Both problems can be addressed simultaneously by the �- executions of the CKMC to estimate a search interval Means Clustering (KMC) algorithm. Figure 2 presents the for the most suitable number of support centers; details of the standard KMC algorithm. (c) variable number of assigned communities to centers As presented in Figure 2 clustering involves the unique in order to add fexibility to the assignation task assignment of a point to the nearest cluster based on its through iterative executions of the CKMC; center (defned as the median point or the centroid). Te locations of the centers must be reestimated each time that (d) conditional decision process to perform insertion, new assignments are performed, and new assignments can deletion, and exchange of communities between cen- be generated each time that the reestimation process is ters for further improvement of the fnal assignments; performed as they afect the closeness of the centers to the (e) Earth’s arc length as distance metric to locate centers considered points. within measurable distance in kilometers. For the purpose of determining the locations of the support centers and their assigned communities, the standard Te details of this metaheuristic, termed as GRASP- KMC algorithm must integrate capacity restrictions. How- CKMC, are presented as follows: in Section 2 the technical ever this adds complexity to the assignment task because details of the GRASP-CKMC and its validation are presented. not all nearest points to a certain center can comply with Ten, in Section 3 the results on the instance of Veracruz its capacity restriction (thus, not all nearest points can be are presented and analyzed. Finally, our conclusions are assigned to this center). presented in Section 4. Approaches have been proposed to address the capac- itated task. In contrast to the circular regions shown in 2. GRASP-CKMC Figure 2, in [32] a rectangular region around the center was considered to determine the candidates for clustering. A Greedy Randomized Adaptive Search Procedure (GRASP) Tis reduces the number of points to be assigned to the is a metaheuristic which consists of two main phases: (a) the cluster and thus reduces the likelihood of not complying Construction Phase which consists in providing a feasible with the capacity restriction. Te points located outside the solution by combining a greedy function with a method rectangular region are omitted by this initial assignment of random selection and (b) the Local Search Phase which process. Afer this process is performed, a priority is assigned consists in iteratively improving the feasible solution [26, 27]. to the omitted points in order to be assigned to the clusters As presented in Figure 1 the proposed GRASP manages with available capacity in a fnal assignment process. Other three main algorithms for these phases: approaches involve an average distance for the reassignment � of points [33]. (i) Constructive Phase: a GA is performed to determine Te assignment of close points and reassignment of the lower and upper limits for the most suitable omitted points are procedures which can be performed number of clusters. Random selection is performed with some randomness to add fexibility to the local search for the creation of the initial population and the � process of KMC. Tus, the proposal to extend the KMC to number of nearest points to extend the KMC to perform the capacitated task consists in including a uniform comply with capacity restrictions (CKCM). random variable to control the ratio of acceptance for the (ii) Local Search Phase: the CKMC is iteratively per- KMC algorithm (and thus, of the � nearest locations). Tis formed with uniform random variation in � and the proposal is similar to the Variable Neighborhood Search number of clusters � restricted by the lower and (VNS) principle [34]. upper limits identifed in the previous phase. Figure 3 presents the general structure of the proposed (iii) Random decision process to exchange locations capacity-restricted KMC algorithm (CKMC). As presented between capacity-complying assignments for further in Figure 1, this CKMC algorithm is used in both phases of improvement of the fnal CCCP solution. the GRASP-CKMC metaheuristic. Tis is the reason of the adjustments stated in Figure 3: In the following sections the details of the main algo- rithms used for the phases of the GRASP are presented and (i) In the Constructive Phase, the CKMC is executed discussed. only once for two random � values which will be used Journal of Optimization 5
Construction Phase Local Search Phase
GA Extended KMC (Capacitated KMC) Extended KMC [Kmin,Kmax] with variable V and (Capacitated KMC) Best Solution multiple executions (most suitable assignments of with constant V and Minimum and maximum points to clusters) single executions limits for the most suitable number of clusters K Insertion, Deletion, Exchange (Decision Process)
Figure 1: General structure of the GRASP-CKMC algorithm.
(1)
2 Initialize the number of clusters (K) 5 1 3 Randomly generate K clusters and B 6 4 (1) determine their centroids, or generate K A random points as centroids. 11
8 9 10 (2) Assign each point to the nearest centroid 7
(3) Re-compute the new centroids 2 2 (2) 5 (3) 5 1 1 3 3 Stop No B B condition 6 4 6 4 A A met? 11 11
Yes 8 8 9 10 9 10
(4) Stop 7 7
2 (4) 5 1 3 B 6 4 A 11
8 9 10
7
Figure 2: Structure of the standard �-Means Clustering (KMC) algorithm.
by the �GA to determine the lower and upper limits the best found solution is improved by means of (����, ����)for�. Also, the number of nearest points insertion, deletion, and exchange operations which to each center (�)isconstantgivenby�. are controlled by a decision process.
(ii) In the Local Search Phase, the CKMC is iterated � 2.2. �GA. Te standard KMC algorithm considers that the times, and at each iteration, diferent values of � quantity of clusters is known apriori[30]. Within the context (within ���� and ����) and the ratio of acceptance � of the CCCP,the minimization of the objective function (total (which has an upper limit given by �) are considered. distance from each cluster to each assigned point) depends At each iteration, the best assignment of points on fnding the most suitable number of clusters. Hence, the (locations) to clusters (centroids) (as measured by proposed GRASP-CKMC includes an evolutionary mecha- its objective function value �)issaved.Aferthe nism to determine the suitable range of clusters which can � iterations of the CKMC algorithm are executed, minimize the total distance to the afected communities. 6 Journal of Optimization
K = round(random(0,200)) and K = round(random(200, 400)) (performed only in the Constructive Phase) or K = round(random(Kmin, Kmax)) (performed only in the Local Search Phase)
t = 1 Best_Distance = Infnite (performed only in the Local Search Phase)
V = X (Constant only in the Constructive Phase) or Re-compute the new centroids K= round(random(Kmin, Kmax)) V= round(random(0, X )) (Variable only in the Local Search Phase)
Assign the nearest V points to each centroid K. Compute the average k and standard deviation k of the distances between each centroid and its assigned points.
Te omitted points are sorted from nearest to farthest. Tese are assigned based on the adjusted ratio given by k + 1.2k
G≤ G Best_Distance Compute total distance from each centroid to its assigned points Yes Best_Distance Current Centroids + Assignments Objective Function Value G ? Best_Solution
No Yes Ｎ≤０ t = t+1 ?
No
Insertion, Deletion, Exchange (Decision Process)
(performed only in the Local Search Phase)
Figure 3: Structure of the proposed Capacitated �-Means Clustering (CKMC) algorithm.
Figure 4 presents the general structure of the �GA which Finally, the implementation of the metaheuristic was was developed to address this task. Te �GA is characterized performed with Octave and MATLAB in a HP Workstation by small populations which can lead to achieving faster with Intel Zeon CPU at 3.40 GHz with 8 GB RAM. convergence with less storing memory [35, 36]. In this case, � the individuals of the population of the GA consist only of 2.4. Assessment. Before proceeding to obtaining a solution � � pairs of values ( ���, ���) that can defne the lower and for our instance, we assessed the performance of the GRASP- � upper limits of a range that may contain the value that can CKMC metaheuristic with a selection of CCCP instances. lead to a total minimum distance on a single execution of the Due to the size of the instance (3837 communities), we CKMC algorithm. By using the random mutation and the considered the following SJC and DONI instances [37, 38]: linear crossover operators a diversifcation on these bounds SJC1 (100 points), SJC2 (200 points), SJC3a (300 points), � is obtained to estimate an interval for the local search of SJC4a (402 points), DONI1 (1000 points), DONI2 (2000 within the main GRASP-CKMC algorithm. An estimate for points), DONI3 (3000 points), DONI4 (4000 points), and � � � � ( ���, ���)isobtainedafer generations (in this case, DONI5 (5000 points) [39]. For comparison purposes, the � � =100)ofthe GA are executed, and within this range, is performance of the GRASP-CKMC metaheuristic was com- randomly selected. pared to standard and most recent methods, including the latest best known solutions as follows: 2.3. Insertion, Deletion, and Exchange. As presented in Fig- ures 1 and 3, the best solution found by the CKMC in the Local (i) Best known solutions as reported in [24]. Search Phase is improved by a decision algorithm which (ii) Best results obtained by CKM and GA as reported in performs insertion, deletion, and exchange of points between [38]. clusters. A conditional decision process was designed to avoid unnecessary tasks due to the random selection of points (iii) Best results obtained by VNS as reported in [23]. which can be inserted, deleted, or exchanged. Te description (iv) Best results obtained by TS (Tabu Search) and CS of this improvement process is presented in Figure 5. (Clustering Search) as reported in [24]. Journal of Optimization 7
Initial Population of the GA (t = 20 individuals) Start: ）Ｈ＞ＣＰＣ＞Ｏ；ＦＮ = gen=0 [round(random(0,200)) round(random(200, 400))]
at bt
For each individual Kt =round(random(at,bt)) and ftness assessment (objective function value) is computed by a single execution of the KMC with capacity restriction. Sort individuals in ascending order based on ftness (minimization).
Selection of Parents by Roulette a ,b a ,b Wheel ((i.e., [ 1 1] and [ 2 2])
Generation of Ofspring
Linear Crossover 1 a +a (1 ) |b+b (1 ) Ofspring = [ 1 2 - 1 2 - ] 2 a (1 )+a |b(1 )+b Ofspring = [ 1 - 2 1 - 2 ] Random Mutation 3 |b Ofspring = [round(random(0, 200 )) 1 ] 4 a | Ofspring = [ 2 round(random(200,400))]
Fitness assessment of each ofspring is computed by a single execution of the KMC with capacity restriction.
Population Update (20 individuals with the best ftness between the Initial gen ≤ m Yes Population and the Ofsprings). Sort fnal individuals in ? gen = gen+1 ascending order based on ftness (minimization). Stop: No top individual = [Kmin, Kmax]
Figure 4: Structure of the �GA for initialization of �.
(v) Best results obtained by the latest method known as Table 2: Parameters of the GRASP-CKMC. Adaptive Biased Random-Key Genetic Algorithm (A- Parameter Value BRKGA) as reported in [24]. � 10 Table 2 presents the parameters for the GRASP-CKMC � 50 algorithm. As mentioned in [24], metaheuristics have no � 10000 optimal values of parameters. Tus, recommended ranges are Random Number Generator Mersenne Twister 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. Tus, small values were considered for � (the a similar number of points to the considered instance of upper limit for the number of nearest points to each cluster), 3837 communities, the GRASP-CKMC metaheuristic is able the executions of the CKMC in the Local Search Phase (�) to obtain solutions with errors smaller than 5.0% (3.93% and and the number of pairs of points to be considered for 4.64%, respectively) within 10 runs. exchange, deletion, or insertion (�). Regarding the Mersenne Table 4 presents the comparison of the best results Twister random number generator, it was considered as obtained with the reviewed methods. When compared to recommended by the MATLAB documentation. CKM the proposed metaheuristic outperforms it in all Table 3 presents the results obtained for 10 runs of the instances. Tis is observed in the average error which is algorithm. It is observed that the average of the best results signifcantly higher for CKM in comparison to GRASP- is 3.03% while the average of the worst results is 5.59%. CKMC (10.58% > 3.03%). Te average performance of the Particularly for the instances DONI3 and DONI4, which have GA is similar to the performance of CKM (10.27% ≈ 10.00%). 8 Journal of Optimization .55% .53% 1.71% 7. 1 2 % 7. 3 7 % 2.61% Error(%) Error(%) Error(%) Average = 3.03% 5.59% 4.45% Best Worse Average Best Worst Average Runs Table 3: Results of 10 runs of the GRASP-CKMC metaheuristics on the SJC and DONI instances. Known 1 2 3 4 5 6 7 8 9 10 Instance Best 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 SJC2SJC3a 33181.65 45356.35SJC4a 33810.48 46993.46 34077.98 46165.69DONI1 61931.60 33568.98 46789.82 62774.78 3021.41 46503.93DONI2 33529.65 62710.35 46559.25 6080.70 33673.30 63064.16 46459.06 3088.71DONI3 62962.47 33342.50 46944.45 6487.85 8343.49 62844.49 46697.09DONI4 3165.04 33745.19 62899.00 6499.84 46250.59 10777.64 8706.45 63000.86 33330.23 46030.70DONI5 11540.02 6498.32 3104.18 63357.25 46030.70 34084.06 8971.18 11114.67 62926.63 11508.46 46993.46 6490.31 33731.71 3144.54 63369.95 11704.86 8953.30 12016.16 46539.40 62710.35 6459.86 33330.23 11568.63 3169.48 11991.16 8964.51 63369.95 34084.06 1.49% 6498.46 11541.06 12034.59 62990.99 8677.99 33689.41 3141.08 11846.86 11534.28 6487.18 3.61% 8691.23 1.26% 12022.82 3095.66 6484.72 11702.91 0.45% 11724.42 8894.92 11277.44 6483.75 3109.24 2.32% 11934.19 11392.09 2.72% 8782.56 6420.06 3157.25 11937.12 11685.20 9010.17 6420.06 11938.20 11277.44 1 3158.03 6499.84 8671.70 11893.37 11704.86 3088.71 11724.42 6481.04 8671.70 11545.49 12034.59 3169.48 9010.17 5.58% 4.64% 11933.89 3133.32 8832.40 8.60% 6.89% 5.49% 2.23% 3.93% 6.58% 8.28% 4.90% 7.99% 3.70% 5.86% Journal of Optimization 9
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%
Coding of Best_Solution into the single vector assignments of length N, where N = number of points to be assigned to K clusters, i = 1, …, N and assignments(i) stores the k-th cluster where the point i is assigned. u = 1 Randomly choose two points (i, j) where i ≠ j k and 1 assignments(i) ≠ assignments(j) 3891015 2 167131819 Decision Process 3 2 5 14 20 4 411121617
u ≤ Y Yes Coding ? u = u+1 No i=12345678910 11 12 13 14 15 16 17 18 19 20 Stop assignments 23143221114423144223 k i (4) ≠ j (14) assignments(4) = 4 = a ≠ assignments(14) = 3 = b
∙ Compute the distances between the points (i, j) and their centroids (a, b): [dia dib dja djb] Decision Process
∙ If dia
∙ If dia
Figure 5: Structure of the decision process of the GRASP algorithm.
However this is observed because the GA outperforms the the GRASP-CKMC metaheuristic outperforms the VNS, GA, CKM method for medium instances (SJC1-DONI1) while the and CKM methods. CKM signifcantly outperforms the GA for large instances When comparing the performance of the GRASP-CKMC (DONI2-DONI5). Better performance is observed with the with more updated metaheuristics such as TS, CS, and A- VNS method with an average error of 3.64%. Also, in BRKGA, these reported a better performance with average two instances the VNS method obtained the best known errors smaller than 1.0%. Tis is expected as the proposed solutions (error = 0.0%). Even though the GRASP-CKMC metaheuristic is based on the GRASP and KMC principles metaheuristic is not able to obtain the best known solution, and as such, it is proposed as an alternative to similar overall performance is better than VNS (3.03% < 3.64%). metaheuristics such as GA, KMC, and VNS. In general terms, Particularly for instances SJC3a, SJC4a, DONI3, and DONI4, the GRASP-CKMC performs in the middle between the 10 Journal of Optimization
20 20
19.8 19.8
19.6 19.6
19.4 19.4 ) Afected people per region: ∘ 260 Support Centers - Highlands: 301,987 (2279 communities) ) 19.2 - Capital: 224,960 (1558 communities) ∘ - Largest community: 19.2 Xalapa-Enríquez (42,476, Capital) 19 19 Latitude ( Latitude
18.8 ( Latitude 18.8 18.6 18.6 18.4 18.4 18.2 − − − − − − − 97.4 97.2 97 96.8 96.6 96.4 96.2 18.2 ∘ − − − − − − − Longitude ( ) 97.4 97.2 97 96.8 96.6 96.4 96.2 ∘ Highlands Longitude ( ) Capital Figure 7: Afected communities in Veracruz: assignment of support Figure 6: Afected communities in Veracruz: capital and highlands centers. regions. standard and the most recent methods for the CCCP with an 120 average best error of approximately 3.0%. 100 Due to these results, the proposed metaheuristic is 80 considered suitable to address the location of the support centers or prepositioned warehouses for the communities of 60 Veracruz. 40 20 0 3. Proposed Locations for Centers Support of Number Communities at Risk [ 2- 717) [ 717- 1432) Figure 6 presents general statistics regarding the people [3577- 4292 ) [1432- 2147) [2147- 2862 ) [2862- 3577 ) [4292- 5007 ) [5007- 5722 ) [5722- 6437 ) [6437- 7152 ) [ 7152 - 7867 ) [7867- 8582 ) [8582- 9297 ) [9297- 10000 ) afected by disasters in the considered regions of Veracruz, Intervals of Demand Mexico. In total, in the capital and highlands regions, there are 526,947 people at risk throughout 3837 communities Figure 8: Afected communities in Veracruz: number of centers vs. intervals of demand. where the community of Xalapa-Enr´ıquez has the largest amount with 42,476 people. Because support centers are considered to supply resources for a maximum of 10,000 in Table 5 while Figure 7 presents the visualization of the people, larger communities (such as Xalapa-Enr´ıquez) were assignments. segmented into equally-sized smaller communities. Tis led Based on these results, it was determined that the mean to a total of 3844 communities. distance that people at risk must travel from their community Due to the importance of minimizing the distance to its assigned center is approximately 2.08 Km with a between the afected communities and the location of the standard deviation of 0.60 Km. In this case, humanitarian centers, a reliable distance metric must be used. In this case, relief can be provided within a short period of time. the geographic arc length metric is considered because it Tese results also provide information to determine can provide accurate distances in kilometers based on the the most suitable capacities for each center. Although the spherical model of Earth’s surface which has a radius of �= GRASP-CKMC imply the establishment of 260 centers to 6,371 Km. With this metric, the arc length (distance) between provide supplies to a maximum of 10,000 afected people, the two locations (��,�) with geographic coordinates (��, ��)and people at risk within the communities assigned to each center (��, ��), where � is the latitude and � is the longitude in can be considered to determine its most suitable capacity. radians, is estimated as follows [40]: Figure 8 presents a histogram that represents the number of centers assigned for each interval or range of people at risk. ��,� =�×��,� =� (1) As observed, 103 centers serve communities with a minimum and maximum of 2 and 717 afected people, respectively. In × Arccos [cos �� cos �� cos (�� −��)+sin �� sin ��]. contrast, just 8 centers serve communities with a minimum With this data, the GRASP-CKMC metaheuristic deter- and maximum of 9297 and 10000 people, respectively. Tese mined a set of 260 centers to provide support to the 3837 results can be considered to make a better estimation of the communities (or extended 3844 communities) with mini- capabilities of the prepositioned warehouses and, thus, of the mum average total distance. Te general results are presented necessary inventory. Journal of Optimization 11 erage at Risk Distance Distance tance, and average distance from the at Risk Distance Distance at Risk Distance Distance at Risk Distance Distance Center Locations People Total1 Average2 Center Locations3 People4 Total 125 Average Center 56 Locations People 10 24767 Total 158 27.05 68 Average 4437 39 Center Locations 9 People 602 2.2510 18.75 8.38 23 Total11 167 44.27 14 66 Av 12 1.87 362 1.68 3203 513 5.16 2.95 4 18.19 21314 55.71 68 16 18 6715 584 69 1.72 2 25.19 2.02 194 2.4216 2 1927 1410 17 15.96 1217 19 70 1.80 13 5.09 71 7218 137 44.33 34.09 17 3.19 103419 88 656 10 4482 73 25 2.46 1.27 2.13 1163 1.1920 39.65 23 32 3 1561 42.91 74 29.4821 3.03 26.15 7 267 131 75 9411 7622 2.33 0.59 41.25 41 16 1269 2259 2.46 2.26 123323 23.04 28 26 1.52 63.99 2.01 54.62 78.58 584 133 2.4324 77 8874 9 132 17 33 2.46 6 79 488 2.3025 2.56 2371 134 78 4 24.53 911 82.90 2.37 2.46 8026 33.97 15 15 0.82 4714 8374 1074 14 71.08 11 921 135 2627 81 79.92 3.50 2.02 7 136 137 242 5 30.90 92.23 23.23 728 2.12 17 82 1194 2.54 3 2301 713 7.9129 3.07 9941 926 138 83 5 19 5.73 1105 2.79 1.82 5 2.58 17 34.9930 8 84 26.56 123 6 639 50.56 37.88 85 321 20.38 139 1931 1.32 16 11.96 14 1995 141 196 140 1.43 9 2.33 326 16 1.9032 13.16 24 9.42 312 2.53 29.46 1.94 70 175 1.85 16 779 45.6833 86 10000 1491 29 12 6 1.71 3.58 8.13 88 87 142 50.70 25 19 13 1.88 198 1.73 144 828 716 1.88 52.02 2130 18.37 20 8.30 14.21 197 32.65 2.40 279 23 1444 89 729 1.19 17 2.98 72.72 263 38.64 1.63 16.62 3.25 143 5207 3227 2962 2.30 199 23 90 14 19 2256 5 2.37 200 1.66 12 1.72 27.52 4 82.77 19 23.00 6639 55.78 53.64 29.88 201 145 3.03 2.42 6.15 36.76 1.85 434 202 91 23 93 2576 146 92 1.72 147 2312 10 1139 2.85 6 24 53.35 21 741 357 1.92 544 320 2.23 2.82 2.30 148 36.45 94 1.84 39.97 203 1.02 42.10 5 31.64 7 1035 149 4 28.42 95 2.32 3 1924 9 6.89 1860 18 46.62 204 5.23 2385 21 3 213 206 205 2.14 1.74 150 71.20 96 41 2.22 2.26 16 25.27 9 52.37 39.45 2.37 460 97 2.45 185 15 1.38 2996 9.57 22 1.31 804 12 81 258 4511 151 207 10 3 3.10 10 152 98 5836 2.53 2.18 16 4 1.88 46.03 9.81 691 6.71 46.10 153 9.60 15.56 25 209 1.59 790 4.22 4 111.30 304 154 215 5.38 991 3 3 2.56 155 42.14 198 1658 23 2 1.96 2.20 0.96 15.34 38.17 3.20 1.73 76 41.04 208 2.71 23 16.81 1.05 9 31.85 385 45.35 1.79 2.63 211 4.01 7 210 157 157 9011 62 158 4 2.54 1.70 156 3.58 1.87 213 58 1.40 3683 11 3.18 7.06 2.83 4179 212 42.20 1.34 3404 1.93 214 5.03 6 159 43.62 160 15 0.89 1235 9 8 10 1.38 19.71 161 1.77 5 911 19.11 1.83 12 14.53 1.90 0.64 1.68 215 19 972 7 2.19 920 0.69 7 1445 756 20.30 162 116 163 2.73 21 144 216 3.63 1711 11.23 32.07 15.22 20.69 218 28 217 380 13.90 1.85 379 639 219 27 11.74 1564 21.40 220 6 2.14 1.90 14 1.87 2.07 56.62 10.39 1.54 12.56 150 17 17 41.40 9448 2.35 1.78 3 223 12 2.98 221 518 50.60 222 585 1.48 84.59 1.79 1.97 1875 1270 224 27.98 585 9.81 627 1.81 3 40.34 3.13 225 31.63 6 226 30 26.54 2.00 4.06 16 2.37 1.63 227 1.86 1508 23 4 652 174 2.21 1.35 228 1250 1.91 70.32 4 11.57 9367 60 36.62 66.26 2.34 0.64 5 1.93 95 2.29 4.40 2.88 120 2.70 1.10 5.66 0.67 1.13 locations to the assigned center. Table 5: Afected communities in Veracruz: number of centers, number of locations (communities), and people at risk assigned to each center, total dis 12 Journal of Optimization erage at Risk Distance Distance at Risk Distance Distance Table 5: Continued. at Risk Distance Distance at Risk Distance Distance Center Locations People Total34 Average35 Center Locations36 People37 Total 1738 39 Average Center39 Locations 16 2277 People40 8353 8 Total 25.3141 19 9118 109.04 Average42 Center 12 Locations 1.49 1337 30.44 People43 2.80 14 945 Total44 21.39 12 2176 1.90 99 10045 Av 31.18 5 878 28.0346 2.67 10 101 81047 39.35 16 1.64 16 5 2.34 66148 102 26.20 8 1215 2.8149 103 4 4 2071 104 322 11.09 538 19.6850 20 2.18 38 941 41.36 10551 41.62 14 25 2.22 249 6.88 404 1.97 752 106 10 2326 3314 13.11 2.59 2.6053 24 13 107 7.08 5.87 588 49.74 1519 1.38 10854 99.33 1138 176 9 3 164 1.64 10955 53.60 26.33 1335 2.49 2766 8 1.77 1.47 165 22 2.61 21.0556 21.04 38 5 110 54.73 21.44 1479 2.14 1.8857 225 2 22 112 11 166 111 6305 167 243 2.10 3.0158 2 15.07 16 4231 2.28 1.65 168 39 1.51 11359 16 51.83 2053 5 14.90 1172 22 94.32 114 7 169 260 15 1.67 35 6 115 925 48.93 170 33 8.05 36.5761 2.36 0.50 1271 14 1.86 7 16 2.48 1034 38362 116 2.12 29 41.17 4994 1041 388 2.22 5 6 39.64 0.23 3.32 172 13 171 88 1.6163 64.23 9 22 117 6182 118 2714 71.98 4.55 9191 33.29 15.0064 2.57 1399 1.06 119 2.48 9 33 174 2319 2.92 15.03 292 27.62 0.12 10000 173 39.8765 1285 23 12 11.35 2.06 4 240 2.22 60 0.91 72.16 29 2.14 120 229 35.81 24 11.74 175 4990 9.44 29.71 1.97 177 230 51 8725 2.49 2.51 28 18.24 22 122 1.62 8 1191 176 2.49 561 7460 121 231 88.85 1599 1.63 137 16.34 1.96 7015 14 8 2.29 123 8 178 1.89 232 11 144.50 1966 57.54 28 2.03 7414 233 6 25.68 8947 234 67.88 2.69 11 37.22 12 154 4.48 125 1.82 19 179 78.85 6 180 124 2.41 52.33 696 54.23 2.50 235 853 15 1894 306 3846 13 2.34 5 2.14 25.07 13 181 1.55 19 137 1.12 126 1187 597 1.55 35.84 2926 21 182 81.85 1.87 16.74 2.46 19 127 12.83 18 362 9.31 15 183 237 26 1831 2334 10.67 3.13 128 26.93 226 26.20 1960 17.39 236 476 2.56 184 11 238 239 2.92 9613 1.52 15.15 6 2849 1.60 26.04 39.75 1298 1.16 5 35.09 5 3872 663 1.78 129 19 6.91 1.42 2.38 4 51.24 1.45 48.10 185 43.91 7 242 240 42.73 4 25 2.52 2.00 44 2.65 12 61.26 44 38.52 130 2.70 13 186 187 904 2.70 4613 241 1.38 2.29 47 7 2.31 2.37 34 19 28 4846 188 2.36 9940 19 441 17.58 2.57 162 13.07 52.11 69 8.60 12 190 243 18 12 79.86 244 8.31 15 63.23 245 13.20 2.76 7175 389 1.60 2218 189 4.63 2173 9 2.61 27.39 2.74 1.43 2991 3.19 24 1.44 83.06 2028 76.62 7 1.66 13.08 4723 787 43 1.89 39.79 0.69 246 12 2.28 247 24 1.16 29.61 44.00 191 448 44.96 249 2.44 6301 2.74 19.73 192 1.87 2.09 248 193 5216 244 1646 6 2.47 2859 2.44 13.68 40 55.37 3.00 250 1.64 24 87.43 14 194 27.97 16.24 43 72.49 34 20 195 251 1.52 2.31 6829 355 252 2.03 19 5724 2.33 2.32 1603 3.02 4528 11 105.91 7534 2260 255 253 16.88 15 54.87 23 30.47 100.74 86.96 47.66 2167 254 3 2.65 105 2.81 2.29 20 2.34 2.18 3379 29.69 6 780 2.56 2.38 13 28.17 579 46.16 257 30.59 3998 256 1.56 258 1300 5.51 2.56 50.98 884 2.01 2.04 18.06 27 17 9 2.55 27.78 259 1.84 260 3.01 3751 1281 2.14 68 13 63.21 9 29.53 12.89 2.34 1273 1.74 2005 24.20 1.43 18.30 1.86 2.03 Journal of Optimization 13
4. Conclusions and Future Work Data Availability Te present work addressed the location planning for prepo- Te databases used for the present work are publicly available sitioned warehouses or support centers for communities at in the referenced sources [39]. risk in Veracruz, Mexico. Tis was addressed by means of the Capacitated Centered Clustering Problem (CCCP) Conflicts of Interest [23] because minimization of distances between the afected regions and the prepositioned warehouses is an important Te authors declare that there are no conficts of interest aspect of humanitarian relief planning. regarding the publication of this paper. 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. Tis metaheuristic Acknowledgments integrated the principles of GRASP, GA, and KMC to pro- Article Processing Charge (APC) was funded by Universidad vide more suitable solutions than those obtained by similar Popular Autonoma del Estado de Puebla A.C. 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 References 3.03%. Although more complex algorithms such as CS and [1] E. V. Arteaga-Vega and I. P. L´opez-Delf´ın, Gaceta Ofcial: Ley A-BRKGA reported better results with errors smaller than Numero´ 856 de Proteccion´ Civil y la Reduccion´ del Riesgo de 1.00%, the performance of the GRASP-CKCM metaheuristic Desastres para el Estado de Veracruz, Gobierno del Estado de was more competitive when compared to standard methods Veracruz, 2013. such as GA, KMC, and VNS. Tus, the GRASP-CKCM can [2]V.E.R.Protecci´on, Instrumentos y Programas de Proteccion´ be considered as a more suitable strategy when compared to Civil, Gobierno del Estado de Veracruz: Secretar´ıa de Pro- these methods. tecci´on Civil, 2017. When the GRASP-CKMC was applied on the real [3] CENAPRED, Diagnostico´ de Peligros e Identifcacion´ de Riesgos instance with 3837 communities, the metaheuristic deter- de Desastres en Mexico:´ Atlas Nacional de Riesgos de la Republica´ mined a set of 260 centers to provide full coverage to all Mexicana,Secretar´ıa de Gobierno: Centro Nacional de Pre- communities. Tese results also provided insights regarding venci´on de Desastres (CENAPRED), 2014. the utilization of these centers considering the actual com- [4] Instituto Nacional de Ecolog´ıa y Cambio Clim´atico, munities assigned to them. Based on these insights, it was Vulnerabilidad al cambio climatico´ ,Secretar´ıa de Gobierno: determined that the facility location task could also support Instituto Nacional de Ecolog´ıa y Cambio Clim´atico, 2016, the decisions regarding the characteristics of the support https://www.gob.mx/inecc/acciones-y-programas/vulnerabilidad- centers by obtaining the estimation of the communities al-cambio-climatico-80125. assigned to each one of them. Tus, smaller centers or prepo- [5] Coordinaci´on Nacional de Protecci´on Civil, Declaratorias de sitioned warehouses can be considered for some regions. Desastre Natural: Fideicomiso No. 2003 FONDEN,Coordinaci´on Nacional de Protecci´on Civil: Direcci´onGeneralparalaGesti´on Tis can optimize the use of resources and improve relief de Riesgos, 2018, http://www.proteccioncivil.gob.mx/work/models/ eforts. ProteccionCivil/Resource/36/26/images/DGGR-RA2017-27MAR2018 Optimization of the supply chain for humanitarian .pdf. relief eforts is an extensive feld which requires contin- [6] P.I.ArcosGonz´alez,R.CastroDelgado,andF.Del-BustoPrado, uous advances in the logistics and production planning “Desastres y salud p´ublica: Un abordaje desde el marco te´orico processes. Tus, as future work, the following aspects are de la epidemiolog´ıa,” Revista Espanola˜ de Salud Publica´ ,vol.76, considered: no. 2, pp. 121–132, 2002. [7]A.Cozzolino,S.Rossi,andA.Conforti,“Agileandleanprinci- (i) Extending the CCCP model to consider heteroge- ples in the humanitarian supply chain: Te case of the United neous capacities for the centers. Nations World Food Programme,” Journal of Humanitarian Logistics and Supply Chain Management,vol.2,no.1,pp.16–33, (ii) Integrating route planning on the facility location 2012. problem to optimize the two-echelon supply chain. [8] Sistema Nacional de Protecci´on Civil, Centro de Acopio,Sistema Nacional de Protecci´on Civil (SINAPROC), 2012, http://sismos (iii) Multicriteria optimization to extend on the facility .gob.mx/en/sismos/Centro de acopio. location problem. [9] Sistema Nacional para el Desarrollo Integral de la Familia, Manual Operativo, Atencion´ a la Poblacion´ en Riesgo o (iv) Integrating the use of Artifcial Neural Networks Condicion´ de Emergencia APCE,SistemaNacionalparael (ANNs) to dynamically determine the number of Desarrollo Integral de la Familia (SNDIF): Unidad de Atenci´on clusters to improve speed and convergence of the alaPoblaci´on Vulnerable, Direcci´on General de Alimentaci´on y CKMC algorithm. Desarrollo Comunitario, 2011, http://www.dif.gob.mx/dgadc/ media/Manual%20de%20Operaci%C3%B3n%20APCE%202011 (v) Integrating the principles of the CS method to .pdf. enhance the performance of the GRASP metaheuris- [10] A. Tomas and M. Mizushima, “Logistics training: Necessity or tic. luxury?” Forced Migration Review,vol.22,pp.60-61,2005. 14 Journal of Optimization
[11] J. Salmer´on and A. Apte, “Stochastic optimization for natural of Information Science and Engineering, vol. 21, pp. 1157–1177, disaster asset prepositioning,” Production Engineering Research 2005. and Development,vol.19,no.5,pp.561–574,2010. [29]K.LiaoandD.Guo,“AClustering-basedapproachtothe [12] A. Cozzolino, “Humanitarian logistics and supply chain man- capacitated facility location problem,” Transactions in GIS,vol. agement,”in Springer Briefs in Business: Humanitarian Logistics, 12,no.3,pp.323–339,2008. pp. 5–16, Springer, Berlin, Heidelberg, 2012. [30] M. Javadi and J. Shahrabi, “New spatial clustering-based models [13] T. E. Fernandez and N. Suthikarnnarunai, “Control aspects for optimal urban facility location considering geographical in humanitarian logistics,” International Journal of Logistics obstacles,” Journal of Industrial Engineering International,vol. Systems and Management, vol. 28, no. 3, pp. 267–286, 2017. 10, no. 54, pp. 10–12, 2014. [14] G. Kov´acs and K. Spens, “Identifying challenges in humanitar- [31] C.-M. Pintea and P. C. Pop, “An improved hybrid algorithm for ian logistics,” International Journal of Physical Distribution and capacitated fxed-charge transportation problem,” Logic Journal Logistics Management,vol.39,no.6,pp.506–528,2009. of the IGPL. Interest Group in Pure and Applied Logics,vol.23, [15] R. E. Overstreet, D. Hall, J. B. Hanna, and R. Kelly Rainer, no. 3, pp. 369–378, 2015. “Research in humanitarian logistics,” Journal of Humanitarian [32] R. Sahraeian and P. Kaveh, “Solving capacitated p-median Logistics and Supply Chain Management,vol.1,no.2,pp.114– problembyhybridk-meansclusteringandfxedneighborhood 131, 2011. search algorithm,”in Proceedings of the International Conference [16] P. Beraldi and M. E. Bruni, “A probabilistic model applied on Industrial Engineering and Operations Management,pp.1–6, to emergency service vehicle location,” European Journal of 2010. Operational Research,vol.196,no.1,pp.323–331,2009. [33] G. R. Sutanto, S. Kim, D. Kim, and H. Sutanto, “A heuris- [17] B. Zhang, H. Li, S. Li, and J. Peng, “Sustainable multi-depot tic approach to handle capacitated facility location problem emergency facilities location-routing problem with uncertain evaluated using clustering internal evaluation,” IOP Conference information,” Applied Mathematics and Computation, vol. 333, Series: Materials Science and Engineering, vol. 332, no. 332, pp. 506–520, 2018. Article ID 012023, pp. 1–9, 2018. [18] A. S¸ahin, M. Alp Ertem, and E. Em¨ur, “Using containers as stor- [34] J. Brimberg, P. Hansen, N. Mladenovi˙c, and E. Taillard, age facilities in humanitarian logistics,” Journal of Humanitarian “Improvements and comparison of heuristics for solving the Logistics and Supply Chain Management,vol.4,no.2,pp.286– multisource weber problem,” Technical Report IDSIA-33-97, 307, 2014. 1997. [19] K. Ozbay and E. E. Ozguven, “Stochastic humanitarian inven- [35] R. Batres, “Generation of operating procedures for a mixing tory control model for disaster planning,” Transportation tank with a micro genetic algorithm,” Computers & Chemical Research Record,vol.2022,no.1,pp.63–75,2007. Engineering,vol.57,pp.112–121,2013. [20] C. L’Hermitte, M. Bowles, P. Tatham, and B. Brooks, “An [36]P.C.Ribas,L.Yamamoto,H.L.Polli,L.Arruda,andF.Neves- integrated approach to agility in humanitarian logistics,” Journal Jr, “A micro-genetic algorithm for multi-objective scheduling of Humanitarian Logistics and Supply Chain Management,vol. of a real world pipeline network,” Engineering Applications of 5,no.2,pp.209–233,2015. Artifcial Intelligence,vol.26,no.1,pp.302–313,2013. [21] B. Goldengorin, D. Krushinsky, and P. M. Pardalos, “Te p- [37]A.Palhano,M.Negreiros,andG.Laporte,“Aconstrained Median Problem,” in CellFormationinIndustrialEngineering: k-median procedure for the capacitated centered clustering Teory, Algorithms and Experiments,pp.25–73,Springer,2013. problem,” in Proceedings of the XIV Congreso Latino Ibero [22] F. Stefanello, O. de-Ara´ujo, and F. M¨uller, “Matheuristics for the Americano de Investigacion de Operaciones (CLAIO ’08),2008. capacitated p-median problem,” International Transactions in [38] A. Chaves, J. Goncalves, and L. Nogueira, “Adaptive biased Operational Research,2014. random-key genetic algorithm with local search for the capac- [23] M. Negreiros and A. Palhano, “Te capacitated centred cluster- itated centered clustering problem,” Computers & Industrial ing problem,” Computers & Operations Research,vol.33,no.6, Engineering,vol.124,pp.331–346,2018. pp. 1639–1663, 2006. [39] L. Nogueira, Problem Instances,INPE-InstitutoNacionalde [24] A. A. Chaves and L. A. Nogueira Lorena, “Hybrid evolutionary Pesquisas Espaciais, http://www.lac.inpe.br/lorena/instancias.html, algorithm for the Capacitated Centered Clustering Problem,” last accessed on July 9th 2018. Expert Systems with Applications,vol.38,no.5,pp.5013–5018, [40] L.Cazabal-Valencia,S.-O.Caballero-Morales,andJ.-L.Mart´ınez- 2011. Flores, “Logistic model for the facility location problem on el- [25] A. Mart´ınez-Gavara, D. Landa-Silva, V. Campos, and R. Mart´ı, lipsoids,” International Journal of Engineering Business Manage- “Randomized heuristics for the Capacitated Clustering Prob- ment,vol.8,pp.1–9,2016. lem,” Information Sciences, vol. 417, pp. 154–168, 2017. [26] R. Interian and C. C. Ribeiro, “A GRASP heuristic using path-relinking and restarts for the Steiner traveling salesman problem,” International Transactions in Operational Research, vol.24,no.6,pp.1307–1323,2017. [27] R. Dias, B. Guazzelli, and P. Henrique, “A GRASP heuristic in the choice of clusterheads for wireless sensor networks provided as a service,” in Proceedings of the International Conference on High Performance Computing & Simulation, pp. 160–167, IEEE, 2017. [28]M.C.Hung,J.Wu,J.H.Chang,andD.L.Yang,“Anefcientk- means clustering algorithm using simple partitioning,” Journal Advances in Advances in Journal of The Scientifc Journal of Operations Research Decision Sciences Applied Mathematics World Journal Probability and Statistics Hindawi Hindawi Hindawi Hindawi Publishing Corporation Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 http://www.hindawi.comwww.hindawi.com Volume 20182013 www.hindawi.com Volume 2018
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