Multi-modal multi-target emergency material dispatching optimization method

Abstract

The invention discloses a multi-modal multi-target emergency material scheduling optimization method. The method comprises the following steps: step 1, establishing an emergency transit point site selection optimization model; when emergency materials need to be transported from a transit point to a demand point, the materials at the transit point can be sent to the demand point as soon as possible in a coverage range of scheduling resources, and the covered demand point has a relatively high emergency priority level; 2, performing preliminary calculation on the model established in the step 1 to obtain the overall coverage rate of the transfer points to the demand points and the average material delivery time from the alternative transfer points to all the demand points; and step 3, solving the model by adopting a multi-modal multi-objective optimization algorithm based on local center declustering. According to the method, basic problem modeling is carried out on the site selection problem of the transit point, the problem is solved by using a multi-modal multi-objective optimization algorithm based on local center declustering, and a plurality of local optimal site selection positions can be provided while the global optimal site selection position is found.

Description

Technical Field

[0001] The present invention belongs to the technical field of emergency material scheduling, and in particular relates to an optimization method for multi-modal and multi-objective emergency material scheduling. Background Art

[0002] In real life, disasters usually occur without warning, but they can have a huge impact on society and the economy. When such disastrous emergencies occur, minimizing the loss of property and human lives as much as possible has always been the most important goal in building an emergency management system. Once a large-scale emergency occurs, the country needs to immediately coordinate various local departments to formulate appropriate emergency scheduling plans, reasonably handle rescue materials from multiple directions and in large quantities within a short period of time, and provide timely rescue to the affected objects during the golden rescue time. An excellent emergency management system can have the fastest scheduling speed and the largest rescue scope in a short period of time, flexibly handle various problems that occur during the scheduling process, bring great help to subsequent emergency rescue work, and significantly reduce the possible casualties and property losses caused by disasters. Therefore, the research on the problem of emergency material scheduling is of great strategic significance for people's life safety and social and economic development.

[0003] The problem of emergency material scheduling itself is an integrated problem, involving multiple fields such as path planning, location optimization, material allocation, etc. It is a complex system composed of multiple subjects, multiple scheduling nodes, multiple scheduling resources, and multiple demand objects. Among them, the location selection of material scheduling nodes is a key link in the scheduling plan. After receiving the rescue information, the emergency scheduling system should be able to quickly dispatch materials and vehicles to the rescue site. The time interval from receiving the information from the scheduling system to the arrival of the rescue materials at the site is called the response time of the emergency material scheduling system. And the response time of the emergency material scheduling system largely depends on the number and location distribution of material scheduling nodes. If there are not enough material scheduling nodes or the location selection of the scheduling nodes is inappropriate, it may delay the golden rescue time and lead to an increase in the mortality rate of diseases and accidents.

[0004] Emergency material scheduling targets sudden emergencies and may encounter various complex situations. For example, the information of material demand points is insufficient, the demand points are relatively scattered, the geographical location of the demand points is difficult to reach, and the demand situations in different disaster-stricken areas are different. At this time, if it is possible to provide as much location selection information of appropriate scheduling nodes as possible based on the known distribution information of demand points and material demand situations, it can provide a more flexible scheduling plan for subsequent material scheduling work. Summary of the Invention

[0005] The present invention studies the location selection of dispatching nodes in emergency material dispatching, models the basic problem of the location selection of transfer points, and uses a multi-modal multi-objective optimization algorithm based on local center solution clustering to solve the problem. The algorithm can provide multiple locally optimal location selection positions while finding the globally optimal location selection position.

[0006] The technical solution adopted by the present invention is as follows:

[0007] A multi-modal multi-objective emergency material dispatching optimization method, comprising the following steps:

[0008] Step 1, establish an optimization model for the location selection of emergency transfer points; when emergency materials need to be transported from transfer points to demand points, the materials at the transfer points can be delivered to the demand points as quickly as possible within the range covered by dispatching resources, and the covered demand points have a relatively high emergency priority level;

[0009] Step 2, perform preliminary calculations on the model established in Step 1 to obtain the overall coverage rate of transfer points to demand points and the average material delivery time from candidate transfer points to all demand points;

[0010] Step 3, use a multi-modal multi-objective optimization algorithm based on local center solution clustering to solve the model.

[0011] Further, the specific content of Step 1 is as follows:

[0012] Within the coverage range of the emergency transfer point, the number of demand points that can be covered is

[0013]

[0014] Within the coverage range of the emergency transfer point, the sum of the emergency priorities of all covered demand points is

[0015]

[0016] The constraint conditions include:

[0017] (1) The distance between the location selection position and the demand points that can be covered is less than the maximum coverage distance

[0018] (2) When there are demand points that can be covered at the location selection position, the location selection position is valuable; otherwise, the values of objective function F1 and objective function F2 are both infinite, that is

[0019] If Then F1 = ∞, F2 = ∞;

[0020] (3) The maximum coverage range r of the transfer point = v a t max , where va is the driving speed of the truck, and t max is the maximum waiting time for the demand point to receive supplies;

[0021] (4) The decision variables x and y are the coordinates of the transfer point, and their value ranges should be determined according to the geographical coordinates of the actual scheduling area and the distribution of demand points to ensure that the selected location of the transfer point can cover all key demand points;

[0022] Among them, D is the set of demand points, D = {D i | i = 1, 2,..., n}; is the coordinate of the i-th demand point; C i is the emergency priority of the demand point D i ; D x is the set of all demand points that the transfer point X can cover.

[0023] Furthermore, the evaluation indicators in calculating the model in step 2 include:

[0024] (1) Demand point coverage rate:

[0025]

[0026] Among them, n is the total number of demand points for material scheduling;

[0027] The number of demand points that the transfer point can cover:

[0028]

[0029] Among them, X P = (x, y), X p represents the selected location of the transfer point, and D i represents the location coordinates of the i-th demand point. When there is an alternative transfer point within the maximum range where the demand point can be covered, it is considered that the demand point can be covered by the transfer point, and the number of demand points that can be covered is calculated;

[0030] (2) Average material delivery time:

[0031]

[0032] Among them, n is the total number of demand points for material scheduling; T XD refers to the time required for the material to reach the demand point when the nearest transfer point is selected from the alternative transfer points for material transportation;

[0033] T XD The specific calculation formula of is:

[0034]

[0035] where \(p\in(1,2,\ldots,popsize)\) and \(i\in(1,2,\ldots,n)\). Here, \(p\) represents the candidate transfer points found by the algorithm, and the number of solutions in the final solution set obtained by the algorithm represents the number of candidate transfer points found by the algorithm; \(|X p D i | represents the Euclidean distance between the \(p\)-th candidate transfer point and the \(i\)-th demand point, and \(V a represents the driving speed of the truck during transportation.

[0036] Furthermore, step 3 includes the following steps:

[0037] Step 3.1: Randomly generate an initial population of NP individuals in the search space, and calculate the fitness values of the population individuals;

[0038] Step 3.2: Divide the local niche according to the neighborhood radius, and use the dominance information among all individuals in the local niche to screen out the local central solutions; when the number of local central solutions is insufficient, the algorithm calculates the convergence quality of each individual in its local neighborhood through a formula, that is, the local convergence quality is equal to the ratio of the number of dominant individuals in the local neighborhood to the total number of individuals in the local neighborhood; after this screening step, if the number of local central solutions is lower than the required number, the algorithm will select individuals with higher local convergence quality from the remaining population to supplement the local central solutions;

[0039] Step 3.3: When the number of local central solutions is greater than the size of the sub-population, perform environmental selection on the population;

[0040] Step 3.4: The selected population has basically covered all global and local optimal potential regions, and perform NCM clustering on the individuals in each potential region;

[0041] Step 3.5: After NCM clustering, individuals will be divided into multiple local populations, and an adaptive local search strategy triggered by the quantitative relationship between the number of local populations and the sub-population threshold includes:

[0042] When the number of local populations in the region is less than the sub-population threshold, select the neighbor with the largest crowding distance of the current individual in the decision space as the base vector to generate offspring;

[0043] When the number of local populations in the region is greater than the sub-population threshold, select the neighbor with the largest crowding distance of the current individual in the objective space as the base vector to generate offspring.

[0044] Furthermore, in step 3.2, the screening method for local central solutions includes the following steps:

[0045] Step 3.2.1: Divide the neighborhood of each individual according to the niche partitioning method;

[0046] Step 3.2.2: Perform non-dominated sorting in the neighborhood and mark all the dominated solutions in the neighborhood;

[0047] Step 3.2.3: Delete all the marked dominated solutions in the population.

[0048] Furthermore, in Step 3.2, the method for supplementing the local central solutions includes the following steps:

[0049] Step 3.2.4: After partitioning the local niche, calculate the local convergence quality of each individual in the local niche through the following formula:

[0050]

[0051] where, Nbest represents the number of individuals in the neighborhood population whose dominance relationship is better than this individual, and Nlp represents the number of individuals in the neighborhood population;

[0052] When the number of local central solutions screened out in the first step is too small, supplement the solutions with better local convergence quality in the remaining population.

[0053] Furthermore, in Step 33, the method for environmental selection of the population is: use the SCD calculation method to rank the crowding situation of the population individuals, and the calculation method of SCD is

[0054]

[0055] where, CD i,x represents the crowding distance of an individual in the decision space, CD i,f represents the crowding distance of an individual in the objective space, CD aug,x represents the average crowding distance in the decision space, CD aug,f represents the average crowding distance in the objective space.

[0056] Furthermore, in Step 34, at the beginning of NCM clustering, first select an unmarked individual as the starting core point; set a neighborhood radius, and all individuals within the neighborhood of the core point are regarded as density-direct individuals and grouped into the same cluster, and mark these individuals to avoid repeated selection; then, the newly added individuals in the cluster are also regarded as new core points, thereby expanding the cluster; in this process, the unmarked individuals that are density-direct or density-connected to the new core point continue to be added to the current cluster and are marked accordingly; repeat this process until all individuals are marked and grouped into the corresponding clusters.

[0057] The beneficial effects of the present invention are:

[0058] 1. The present invention designs a screening mechanism for local central solutions by using the method of marking neighborhood-dominated solutions, divides local niches, so that the solutions in the global optimal region do not directly participate in the competition with the solutions in other regions, and maximally ensures the dominance of local Pareto solutions in the local optimal region, thereby greatly increasing the probability that local Pareto solutions are retained; meanwhile, the local convergence quality of individuals is calculated, and potential solutions with better local convergence quality are supplemented in the population of local central solutions.

[0059] 2. The present invention clusters the screened local central solutions, protecting the diversity of the population distribution while restricting the spread of population information; meanwhile, in order to better explore the optimal region, an adaptive local search strategy considering the population distributions in both the decision space and the objective space is designed; among them, this strategy can adaptively select different search strategies according to the exploration of the decision space and the objective space of the current local population in this region.

[0060] Compared with the multi-modal multi-objective algorithm that can only find the global optimal location, the present invention can provide more transfer point location information at suitable positions, providing more flexible choices for decision-makers in complex and changeable emergency situations. Brief Description of the Drawings

[0061] Figure 1 It is a geographical location map of emergency material demand points;

[0062] Figure 2a It is a fitness topographic map of the emergency transfer point location problem;

[0063] Figure 2b It is a top view of the fitness topographic map of the emergency transfer point location problem;

[0064] Figure 2c It is the global and local Pareto optimal regions of the emergency transfer point location problem;

[0065] Figure 3 It is an explanatory diagram of the local central solution screening strategy;

[0066] Figure 4 It is a density relationship diagram;

[0067] Figure 5 It is a convergence diagram of MO_Ring_PSO_SCD during the optimization process;

[0068] Figure 6 It is a convergence diagram of MMOGA during the optimization process;

[0069] Figure 7 It is a convergence diagram of the present invention during the optimization process;

[0070] Figure 8 It is a distribution map of emergency transfer point locations. Detailed implementation manners

[0071] The present invention will be further described below with reference to the accompanying drawings.

[0072] The following is an embodiment of the present invention. In this embodiment, 16 cities under the jurisdiction of a certain province are used as emergency material demand points, and a location optimization model is established for the location selection problem of emergency transfer points in the demand area based on the geographical location information and emergency priority information of the demand points.

[0073] Table 1 shows the index data of 16 demand points C1 - C16 in a certain province.

[0074] The actual geographical location map of the demand points is as shown in Figure 1 Shown. Considering the index data of the demand points, the emergency priority levels of each demand point obtained by using the emergency priority evaluation index system and the position coordinates of the demand points are shown in Table 2.

[0075] Table 1 Demand point index data

[0076]

[0077] Table 2 Position coordinates and emergency priorities of demand points

[0078]

[0079] Based on the geographical location coordinates of the demand points, in this embodiment, the x-axis coordinates 109 - 116 and the y-axis coordinates 29 - 33 are selected as the location selection area for the transfer points, and the location of the emergency transfer points in this area is optimized. The position distribution map of the demand points in the location selection area is as shown in Figure 1 Shown.

[0080] The purpose of model construction is to find as many suitable locations for emergency transfer points as possible. When emergency materials need to be transported from the transfer points to the demand points, it is first hoped that the materials at the transfer points can be delivered to the demand points as quickly as possible within the coverage range of the dispatching resources. At the same time, when conducting emergency dispatching of medical materials, the material shortage situations in different regions at different levels should be considered. Due to the differences in medical levels between demand points, there are different levels of urgency in material dispatching, and it is necessary to meet the material shortage situations of different demand points when setting up emergency transfer points. Therefore, when optimizing the location selection of emergency transfer points, two basic problem requirements should be met, that is, as many demand points as possible should be covered around the emergency transfer points, and the covered demand points have relatively high emergency priority levels.

[0081] According to the above requirements, the two optimization objectives for the location selection problem of emergency transfer points are as follows: First, within the coverage range of the emergency transfer point, the more demand points that can be covered, the better; Second, within the coverage range of the emergency transfer point, the greater the sum of the emergency priorities of all covered demand points, the better. Assume that when transporting materials from the transfer point to the demand point, the transportation tool is an ordinary truck, and the transportation from the transfer point to the demand point is a direct transportation without considering the time consumption involved in complex road conditions during the transportation process and the loading and unloading time of materials at the transfer point and the demand point.

[0082] Based on the above assumptions and optimization objectives, an optimization model for the location selection of emergency transfer points is constructed.

[0083] Decision variable: X(x, y), the coordinates of the transfer point;

[0084] Objective function:

[0085]

[0086] Constraint conditions:

[0087] 1. This constraint indicates that the distance between the selected location and the demand points that can be covered is less than the maximum coverage distance;

[0088] 2. If Then F1 = ∞, F2 = ∞; This constraint indicates that only when there are demand points that can be covered at the selected location, the selected location is valuable, otherwise the values of the objective function F1 and the objective function F2 are both infinite.

[0089] 3. r = v a t max , where v a = 72 km / h, t max = 1h; This constraint limits the maximum coverage range of the selected location. In the actual geographical location, one degree of longitude is approximately equal to 111 km, and one degree of latitude is approximately equal to 111.11 km. In the current emergency material scheduling problem, assume that the transportation tool for transporting materials from the transfer point to the demand point is a truck, and the driving speed of the truck is set to 72 km / h. Because when the demand point needs material transportation, it is required that the demand point can receive the medical materials transported from the transfer point as soon as possible within a short time, so the maximum waiting time for the demand point to receive materials in the problem is set to one hour. Therefore, the transportation distance of the truck within one hour at this speed is the maximum coverage range of the transfer point set in the problem.

[0090] 4. 109 < x < 116, x ∈ R and 29 < y < 33, y ∈ R; This constraint represents the value range of the decision variable, restricting the selection of the transfer point location between the x-axis coordinate of 109 - 116 and the y-axis coordinate of 29 - 33.

[0091] Among them, D is the set of demand points, D = {D i | i = 1, 2, …, n}; is the coordinate of the i-th demand point; C i is the emergency priority of the demand point D i ; r is the maximum coverage distance; D x is the set of all demand points that the transfer point X can cover.

[0092] The objective function F1 represents the number of demand points that the selected location can cover.

[0093] The objective function F2 represents the sum of the emergency priorities of all demand points covered by the selected location.

[0094] Since the multi-objective optimization problem generally hopes to obtain the minimum value of each objective, a negative sign is added in front of both objective functions when constructing the objective functions. When F1 and F2 obtain the minimum values, the transfer point selected location parsed by this solution can cover the most demand points and the sum of the emergency priorities of the covered demand points is the largest.

[0095] The schematic diagram of the constructed problem model is shown in Figure 2. The data in the picture are 100 points evenly sampled on the horizontal and vertical coordinates of the decision space, and finally 10,000 sampling data of 100×100 are obtained. The objective function is used to calculate the objective value corresponding to each sampling point, and the non-dominated rank sorting is performed on all objective values, and then the corresponding fitness terrain map and global and local optimal regions are drawn according to the non-dominated rank results. Figure 2a is the fitness terrain map of the problem, Figure 2b is the top view of the fitness terrain map of the problem. x1 and x2 represent the horizontal and vertical coordinates of the selected location, and the vertical axis represents the non-dominated sorting rank of the sampling point data. The darker the color in the figure, the more forward the non-dominated rank of the sampling points in that area. The more forward the non-dominated rank of the sampling point, the better the objective value corresponding to the sampling point. Therefore, the bottom basin area in the fitness terrain map is the global and local optimal area of this problem; Figure 2c shows the global and local Pareto optimal regions of the problem. The abscissa is the x-axis coordinate of the selected location, and the ordinate is the y-axis coordinate of the selected location. The purple area in the figure is the global optimal region for the selection of emergency transfer points, the blue area is the local optimal region for the selection of emergency transfer points, and the dark blue points show the positions of the demand points in the emergency scheduling problem in the decision space.

[0096] When using the algorithm to optimize the selected location of the emergency transfer point, it is usually hoped that the algorithm can find as many suitable transfer points as possible within the controllable range of resources, that is, it is hoped that the solution found by the algorithm can cover as many Figure 2bThe global and local optimal regions. From the problem schematic diagram, it can be seen that this problem has both a global Pareto region and multiple local Pareto regions, and the non-dominated sorting levels of some local Pareto regions are relatively low compared to the global Pareto region. Therefore, this problem is a relatively complex multi-modal multi-objective optimization problem with a local Pareto solution set, which poses higher requirements for the algorithm when solving the problem.

[0097] In this invention, all the selected point coordinates finally obtained by the algorithm are used as candidate transfer points to evaluate the location selection. In order to better evaluate the performance of different algorithms in this problem, this invention proposes new evaluation indicators for the characteristics of the emergency transfer point location selection problem, namely the overall coverage rate rcv of the transfer point to the demand points, and the average material delivery time tAD from the candidate transfer points to all demand points. The new evaluation indicators take into account the overall coverage of the demand points by the distribution of the transfer points, and the impact of the location distribution of the transfer points on the material scheduling time in the emergency material scheduling. Therefore, it can comprehensively evaluate all the transfer point location selection positions finally found by the algorithm.

[0098] The specific information of the evaluation indicators is as follows:

[0099] 1. Demand point coverage rate

[0100]

[0101] Among them, n is the total number of demand points for material scheduling.

[0102] is the number of demand points that can be covered by the transfer points found by the algorithm. The specific calculation method is as follows:

[0103]

[0104] Among them, Xp = (x, y), Xp is the location selection position of the transfer point, D i is the location coordinate of the i-th demand point. When there is a candidate transfer point within the maximum coverage range that can be covered by the demand point, it is considered that the demand point can be covered by the transfer point, and the number of demand points that can be covered is calculated.

[0105] The larger the rcv of the algorithm, the greater the coverage rate of the candidate selected points to the demand points, that is, the better the performance of the algorithm in this indicator.

[0106] 2. Average material delivery time

[0107]

[0108] Among them, n is the total number of demand points for material scheduling; T XDIt refers to the time required for the materials to reach the demand point when the transfer point closest to the demand point is selected from the candidate transfer points for material transportation.

[0109] T XD The specific calculation formula of

[0110]

[0111] is as follows: where p ∈ (1, 2,..., popsize), i ∈ (1, 2,..., n), p is the candidate transfer point found by the algorithm, and the number of solutions in the final solution set obtained by the algorithm is the number of candidate transfer points found by the algorithm; |X p D i | is the Euclidean distance between the p-th candidate transfer point and the i-th demand point; V a is the driving speed of the truck during transportation.

[0112] By calculating the time when all demand points can receive materials fastest, the average material delivery time that can be achieved by the candidate transfer points is obtained. The smaller the average material delivery time, the more uniform the distribution of the candidate transfer points, that is, the more reasonable the location of the emergency transfer point selected by the algorithm.

[0113] There are two key points in solving the multi-modal multi-objective optimization problem with local Pareto solution sets: 1. How to retain as many local Pareto solutions as possible on the premise of finding the global Pareto solution. 2. How to prevent the algorithm from falling into the local optimal region during the algorithm convergence process. To solve the multi-modal multi-objective optimization problem with local Pareto solution sets, the present invention designs a new multi-modal multi-objective optimization algorithm MMO_local based on local center solution clustering.

[0114] For the first key point of the problem, the algorithm designs a screening mechanism for local center solutions by using the method of marking neighborhood dominant solutions, divides local niches, so that the solutions in the global optimal region do not directly participate in the competition with the solutions in other regions, and maximally ensures the dominance of local Pareto solutions in the local optimal region, thereby greatly increasing the probability of retaining local Pareto solutions. At the same time, the local convergence quality of individuals is calculated, and potential solutions with better local convergence quality are supplemented in the population of local center solutions.

[0115] For the second key point of the problem, the algorithm clusters the selected local center solutions, protecting the diversity of the population distribution while restricting the spread of population information. At the same time, in order to better explore the optimal region, an adaptive local search strategy considering the population distribution in both the decision space and the objective space is designed. Among them, this strategy can adaptively select different search strategies according to the exploration of the decision space and the objective space of the current local population in this region.

[0116] Combined with the above strategies, the present invention can not only ensure the diversity in the decision space but also well balance the convergence in the objective space, thereby searching for more global and local Pareto solution sets.

[0117] The following table describes the overall algorithm framework of the present invention. First, an initial population of NP quantity is randomly generated in the search space, and the fitness values of the population are evaluated. Then, according to the neighborhood radius, local niches are divided, and the domination information among all individuals in the local niches is used to screen out the local central solutions. In particular, when the number of local central solutions is insufficient, the local convergence quality information of the population individuals is used to supplement the local central solutions. When the number of local central solutions is greater than the size of the sub-population, environmental selection is performed on the population. At this time, the selected population has basically covered all potential regions of the global and local optima. The individuals in each potential region are clustered, and in the subsequent population exploration, the information of the population in the local potential regions is used to better explore the global and local optimal regions.

[0118]

[0119] First, the method of dividing local niches and marking dominated solutions is used to screen out the high-quality local central solutions in the population. Then, solutions with good local convergence quality are selected to supplement the population of the found local central solutions, enabling the algorithm to achieve the purpose of simultaneously retaining the solutions in the global and local optimal regions. The following describes the specific methods of the local central solution screening strategy and the supplement strategy.

[0120] The local central solution screening strategy adopted by the present invention first divides the neighborhood of each individual according to the niche division method. Then, non-dominated sorting is performed in the neighborhood to mark all the dominated solutions in the neighborhood. Finally, all the marked dominated solutions are deleted from the population. This strategy ensures that in the further selection process, once an individual is dominated by any individual in the neighborhood, it will be deleted. Eventually, only the global and local Pareto optimal solutions are retained because they are not dominated by the individuals in their neighborhoods.

[0121] Such as Figure 4As shown, the circles drawn with dashed lines in the figure represent different neighborhoods divided in the decision space. Neighborhoods with global Pareto optimal solutions and local Pareto optimal solutions are represented by solid circles. There is no overlap between them, so the solutions in these two circles will neither mate nor compete. They evolve generation after generation and finally find the global and local Pareto optimal solutions in the circles respectively. The disadvantage of such screening is that in circles without global or local Pareto optimal solutions, non-dominated solutions will still be retained. And these solutions are only the climbing solutions in the global and local Pareto optimal regions and are not needed by decision-makers. To address this problem, this strategy further uses a selection method to delete these unnecessary solutions. In further selection, the dominated solutions in each neighborhood are marked. All marked solutions will be deleted from the selection of the final solutions. Since there is overlap between adjacent circles in the figure, the domination relationship can spread from one to another. For example, the top part of circle A is dominated by some solutions at the bottom, so the top part is marked. A similar situation also occurs in circle B. Therefore, all solutions in the circles that are not in the local and global Pareto optimal regions are marked. Finally, only the global and local Pareto optimal solutions will be retained.

[0122] By dividing local niches and marking dominated solutions, it can largely avoid the competition between local Pareto solutions and global Pareto solutions, and at the same time delete the hillside solutions in the middle of the global and local optimal regions in the fitness landscape, so as to retain high-quality global and local Pareto optimal solutions. However, in the fitness landscapes of different problems, there may be a situation where the local Pareto solution is too dominant, resulting in too few local Pareto solutions directly found by this method. For example, in some problems, the bottom of the optimal region in the fitness landscape is very narrow, and with a slight change, the performance of individuals will deteriorate sharply. At this time, if only the method of marking dominated solutions is used for selection, a large number of potential solutions will be deleted, wasting population information. Therefore, it is necessary to supplement the population of local central solutions initially screened out.

[0123] After dividing local niches, calculate the local convergence quality of each individual in the local niche through the following formula. When the number of local central solutions screened out in the first step is too small, supplement the solutions with better local convergence quality in the remaining population.

[0124]

[0125] Among them, Nbest represents the number of individuals in the neighborhood population whose domination relationship is better than that of this individual, and Nlp represents the number of individuals in the neighborhood population. Through the formula, the domination situation of this individual in the local niche can be calculated. local c The larger it is, the smaller the proportion of individuals dominated by this individual in the local niche; local cThe smaller it is, the higher the selection priority of the individual in the local niche, local c The smaller, the better.

[0126] When selecting local solutions at the initial stage of the algorithm, due to the randomness of the population distribution and the different difficulties of exploring local regions in the fitness landscape, during the evolution process, there will be different exploration degrees of population individuals for each local region. When the distances between two local optimal regions are relatively close, two individuals located in different local regions will be grouped into the same local niche for screening local solutions. At this time, when there are different exploration difficulties between these two local optimal regions, it may cause the individual with worse evolution in its own local region to be dominated by another individual, and thus be marked as a dominated solution and deleted during the subsequent screening process, resulting in the population losing the exploration information of this local region. Although this individual is dominated by individuals in adjacent local regions, in its own local region, this individual can dominate more other solutions, so it will have better local convergence quality. When supplementing the individuals of the next-generation population through the supplementary strategy of local central solutions, the potential individuals in these local regions have better local convergence quality and will be preferentially retained.

[0127] Combining the above screening strategy and supplementary strategy, the algorithm has strong local solution retention ability and can successfully retain the potential solutions in all global and local Pareto-optimal regions, thus providing a prerequisite for the subsequent exploration of global and local Pareto solution sets.

[0128] When the number of selected local central solutions is greater than the population size, perform environmental selection operations on the selected local central solutions. After the screening operation of local central solutions, the solutions with poor convergence quality in the population have been deleted. At this time, the remaining central solutions are all potential solutions in their respective local regions. Therefore, when the algorithm performs environmental selection, it no longer considers the convergence quality and non-dominated relationship of the solutions in the current population, and only deletes the solutions in overly crowded regions according to the crowding situation of the current population in the decision space and the objective space. The environmental selection mechanism is specifically described in Algorithm 3. The algorithm uses the widely used SCD calculation method to rank the crowding situation of population individuals, and the calculation method of SCD is as follows.

[0129]

[0130]

[0131]

[0132] Among them, CD i,x represents the crowding distance of an individual in the decision space, CD i,f represents the crowding distance of an individual in the objective space, CDaug,x represents the average crowding distance in the decision space, CDaug,f Represents the average crowding distance of the target space.

[0133] SCD can select the crowding distance according to the distribution in the individual decision space and the target space. When the crowding distance of an individual in a space is greater than the average level, the crowding distance of the larger space will be selected as the SCD. Therefore, it can balance the distribution of the population in the decision space and the target space, and improve the survival probability of multimodal solutions in environmental selection.

[0134] The following algorithm pseudocode describes the adaptive local search mechanism of the MMO_local algorithm. First, local population clustering is performed on the potential solutions in the global and local optimal regions. Specifically, when the number of solutions in the local population is less than the size of the subpopulation, all individuals in the neighborhood populations of the current individual in the decision space are selected as parents, and the DE / potential / 1 strategy is used to generate offspring individuals. When the number of solutions in the local population is greater than or equal to the size of the subpopulation, the individuals in the neighborhood of the target space in the current local population are selected as parents, and the DE / local / 1 strategy is used to generate offspring individuals. The following specifically introduces the clustering method, search mechanism, and mutation strategy involved in the adaptive local search mechanism.

[0135]

[0136]

[0137] Before performing adaptive local search, a clustering strategy needs to be used to analyze the neighborhood information of individuals in the population, so as to identify different local regions and determine in which regions the next local search step should be carried out. NCM clustering is a density-based clustering method that can divide them into multiple clusters through the connection relationship between population individuals. Compared with other clustering methods, NCM clustering has two advantages: 1. It does not require specifying the number of clusters in advance. It can automatically divide the population according to the density distribution of individuals in the population. 2. The shape of the clusters after division is not fixed. It can divide the subpopulation into any shape according to the distribution. Even if the individuals have a large gap, they may be divided into the same cluster. After the algorithm finds the local optimal solution set in the population, NCM clustering is used in the decision space. At this time, the population individuals are basically gathered in the potential regions of their respective optimal regions. Using NCM clustering can cluster the individuals in each local region and distinguish the global or local Pareto solution sets. The following combines Figure 4 , and gives a simple introduction to some basic concepts in the NCM clustering method.

[0138] Density direct reach: Given a population set P, for any x ∈ P, all individuals within the neighborhood range of x are density direct reach individuals of x.

[0139] Density reachability: For [x1, x2, x3] ∈ P, if individual x1 is a density-reachable individual of x2 and individual x2 is a density-reachable individual of x3, then individuals x1 and x3 are called density-reachable individuals.

[0140] Density connection: For x ∈ P, if both individual x1 and individual x2 are density-reachable individuals of x, then x1 and x2 are called density-connected individuals.

[0141] As Figure 4 shown, both individuals B and D are within the neighborhood of individual C. Therefore, both individuals B and D are density-reachable individuals of individual C; individual B is within the neighborhood of individual C and is density-reachable with individual C. At the same time, individual A is density-reachable with individual B, then it is said that individual A and individual C are density-reachable; individual A and individual C are density-reachable, and individual C and individual E are density-reachable, then it is said that individual A and individual E are density-connected through individual C.

[0142] The specific clustering process is as follows:

[0143] 1. First, select any unmarked individual as the core point and expand it starting from the core point. Given the neighborhood radius, all individuals within the neighborhood range are density-reachable individuals of this individual.

[0144] 2. Start clustering. All individuals that can be density-reachable with the core point are selected and put into the first cluster. At the same time, the core point and the points selected during the clustering process are marked.

[0145] 3. After the clustering starts, all individuals that are density-reachable and density-connected with the core point will be divided into the same cluster and marked in the population. Then, take the new individuals put into the cluster as the core point to expand this cluster, continue to select individuals from the unmarked population and put them into this cluster, and finally divide all qualified individuals into this cluster.

[0146] 4. Repeat the second and third steps until all individuals in the population are marked. When the population clustering is completed, all individuals will be divided into their respective clusters.

[0147] In order to enable the algorithm to better develop the global and local potential regions after finding them, the MMO_local algorithm proposed in this invention designs an adaptive local search mechanism, which can select different parent generations and mutation strategies to generate offspring individuals according to the exploration situation of the current local population in the region.

[0148] In the mutation step of the DE algorithm, there are several different mutation strategies, such as DE / rand / 1, DE / best / 1, DE / best / 2, and DE / rand / 2. Since the multi-modal multi-objective problem with local Pareto solution sets requires obtaining multiple optimal solution sets, the mutation strategy must have strong exploration ability. Therefore, this invention improves DE / rand / 2 with better diversity, and some mechanisms are embedded in DE / rand / 2. To improve the diversity of the decision space and the objective space, the following two mutation strategies are specifically used to adaptively select and generate differential vector individuals.

[0149] DE / potential / 1:

[0150] v i = x potemtial + F[(x r2 - x r3 )+(x r4 - x r5 )]

[0151] where x potemtial represents the individual with the largest crowding distance in the neighborhood of the decision space of the individual, guiding the individual to explore the potential area of the decision space.

[0152] DE / local / 1:

[0153] u i = x local + F[(x r2 - x r3 )+(x r4 - x r5 )]

[0154] where x local represents the individual with the largest crowding distance in the objective space among the individuals in the current local population, guiding the individual to deeply develop the local optimal area.

[0155] Specifically: when the number of individuals in the local population is less than the population threshold, the algorithm selects DE / potential / 1 to generate offspring individuals. According to the Euclidean distance between the current individual and all individuals in the entire population in the decision space, individuals satisfying the population threshold number are selected as neighborhood individuals. Then, five individuals are randomly selected from the neighborhood individuals as parent individuals, and the individual with the largest crowding distance in the decision space is selected as x potemtial , and the other four individuals are respectively used as x2, x3, x4, and x5.

[0156] When the number of individuals in the local population is greater than the population threshold, the algorithm selects DE / local / 1 to generate offspring individuals. According to the Euclidean distance between the current individual and all individuals in the local population to which it belongs in the objective space, individuals satisfying the population threshold number are selected as neighborhood individuals. Then, five individuals are randomly selected from the neighborhood individuals as parent individuals, and the individual with the largest crowding distance in the objective space is selected as x local . The other four individuals are respectively used as x2, x3, x4, and x5.

[0157] After individuals are clustered by NCM, they are divided into multiple local populations. When the individuals in the local area are sparsely distributed and the number of local populations is less than the sub-population threshold, it means that the current population has less exploration of this area. At the same time, the exploration resources allocated to this area in the population are also relatively few, and it is necessary to further explore this area with the help of the exploration information of the surrounding populations. At this time, the neighbors of the current individual in the decision space are selected, and the neighbor with the largest crowding distance in the decision space is selected as the base vector, which can improve the diversity of the decision space and strengthen the exploration of the population in this local area. When the number of local populations in the area is greater than the sub-population threshold, it means that the algorithm has explored this area relatively sufficiently. At this time, the neighborhood of the individual in the objective space of the local population is selected, and the neighbor with the largest crowding distance in the objective space is selected as the base vector, which can improve the diversity of the objective space. Selecting parent individuals in the clustered local population can restrict the evolution of individuals within the niche of the local area, better explore each local area, enhance the convergence of the local population, and at the same time avoid the competition between local solutions and global solutions during the evolution process, protecting the diversity of the population. Selecting mating individuals and mutation strategies in the above method is to balance the exploitation and exploration capabilities of the algorithm for local areas, and improve the diversity of both the decision space and the objective space. At the same time, the algorithm clusters the population, enabling the individuals in the local population to evolve in the local areas where they are distributed respectively, restricting the information propagation speed between populations, protecting the population diversity, and at the same time enabling the individuals in each local population to be assigned appropriate exploration strategies, so that each global and local optimal area can be effectively exploited.

[0158] In the embodiment, six multi-modal multi-objective optimization algorithms are used to solve this problem, namely the multi-modal multi-objective optimization algorithm MMO_local based on local center solution clustering, the multi-modal multi-objective optimization algorithm DN_NSGAII_Uc based on uniform crossover operator, the multi-modal multi-objective optimization algorithm MO_Ring_PSO_SCD adopting a ring topology structure, the multi-modal multi-objective evolutionary algorithm Tri-MOEATAR based on double archives, the evolutionary algorithm HREA with a hierarchical ranking method, and the multi-modal multi-objective genetic algorithm MMOGA. The population size of each algorithm is set to 200, and the maximum number of iterations is 50. To avoid the contingency of algorithm results, each algorithm is independently run 31 times on this problem. Two indicators, the demand point coverage rate rcv and the average material delivery time tAD, are used to evaluate the results obtained by different algorithms in solving this problem. The average values of the six algorithms after 31 experiments are shown in the following table.

[0159] Table 3 Average values of indicators of different algorithms in 31 experiments

[0160]

[0161] Combined with the data results in Table 3, it can be seen that the set of candidate transfer points obtained by MMO_local has a higher coverage rate for the overall demand points. At the same time, the set of candidate transfer points obtained can also have a shorter average material transportation time when transporting materials to the demand points, and its performance in the two model indicators is significantly better than other algorithms. In the statistics of 31 experimental results, the average coverage rate of the set of candidate transfer points obtained by MMO_local for the overall demand points reached 83.67%. During the process of material transportation, almost all demand points can receive emergency materials from the transfer points in the first time. When transporting materials to all demand points, the average material delivery time can be controlled within about half an hour. However, since MO_Ring_PSO_SCD, TriMOEATAR, and DN_NSGAII_Uc do not have a mechanism to retain the local optimal region, they can only retain the transfer points in a global optimal region. Therefore, for the set of candidate transfer points obtained by these algorithms, about half of the demand points cannot be covered. When emergency materials are needed, material scheduling can only be carried out from transfer points farther away, resulting in the demand points not being able to receive emergency materials in the first time, and the average material waiting time for the demand points exceeding one hour. Especially for TriMOEATAR, the average material waiting time for the demand points exceeds two hours, far exceeding the maximum waiting time set by the problem. Although HREA has a hierarchical mechanism that can retain the local Pareto solution set, it cannot accurately identify the local Pareto front. Therefore, in this problem, it does not retain the real local optimal region. And MMOGA only retains a small number of solutions in the local optimal region, and the vast majority of the solutions are still limited within the global optimal region. Therefore, it also fails to find a location area that can better cover the demand points, and the average material waiting time also exceeds one hour.

[0162] To visually show the process of the algorithm in solving this problem, the present invention respectively selects the distribution diagrams of the first generation, the fifth generation, and the last generation populations of three algorithms, namely MO_Ring_PSO_SCD, MMOGA, and MMO_local, during the optimization process as Figures 5 to 7 shown. The abscissa in the figure is the x-axis coordinate of the geographical location selected by the individual, and the ordinate is the y-axis coordinate of the geographical location selected by the individual. The blue and purple regions respectively correspond to the local optimal and global optimal regions in the location area, and the red hollow points are the solutions found by the algorithm.

[0163] As Figure 5As shown, it demonstrates the distribution of the population in the decision space during the convergence process of MO_Ring_PSO_SCD. It can be seen from the figure that part of the local optimal region was lost when the population started to explore at the beginning; at the 5th generation, the population began to converge to a region, where most individuals gathered in the global optimal region, and only a small number of individuals were not in the global optimal region but were distributed around it; at the 50th generation, the population could basically cover the global optimal region completely, but all local optimal regions were lost. This is because MO_Ring_PSO_SCD does not have the ability to retain local Pareto solutions. Even if the algorithm searches for local optimal regions during the search process, these local Pareto solutions will be deleted in the environmental selection because they are dominated by global Pareto solutions. MMOGA has a certain ability to retain local solutions. The convergence graph of its population is as Figure 6 shown. It can be seen that MMOGA has better diversity than MO_Ring_PSO_SCD. At the 1st generation, the population individuals spread throughout the decision space, and there were individuals distributed in all global and local optimal regions; at the 5th generation, the population distribution was still relatively dispersed, and some individuals began to gather around some global and local optima. Although the global and local optimal regions were not lost, the number of individuals distributed in these potential regions was still relatively small, and most individuals still wandered outside the global and local optimal regions; at the 50th generation, most individuals gathered in the global optimal region, and a small number of individuals in the local optimal regions were retained, but a large number of local optimal regions were still lost. The convergence graph of MMO_local during the optimization process is as Figure 7 shown. It can be seen that compared with MO_Ring_PSO_SCD and MMOGA, the algorithm has better diversity and convergence. At the 1st generation, the population covered all global and local optimal regions; at the 5th generation, individuals had basically gathered in all global and local optimal regions, and only a small number of individuals were still exploring potential regions outside the global and local optima; at the 50th generation, the population had a more uniform distribution in the global and local optimal regions, and at the same time, no local region was lost. This is because the algorithm balances local Pareto solutions and global Pareto solutions. While retaining local Pareto solutions, it can cluster individuals in local optimal regions and guide individuals around this region to better explore this optimal region. Therefore, MMO_local can find more global and local optimal regions and has better performance when solving the problem of the location selection of emergency transfer points.

[0164] Furthermore, the population obtained from MMO_local is processed. When making a decision on the location selection plan of the transfer points, in order to retain as much information about suitable locations as possible, an individual closest to the surrounding demand points is selected from each individual in the optimal region as the location of the transfer point. Finally, 8 emergency transfer points at suitable locations are obtained. The specific geographical distribution of the emergency transfer points is as shown in Figure 8 shown, and the specific location selection information of the transfer points is shown in Table 4.

[0165] Table 4 Transfer Point Location Selection Information

[0166]

[0167] Table 4 shows the geographical location information of each emergency transfer point and the situation of covering demand points. The location selection plan obtained from MMO_local can cover almost all demand points. Among them, P1, P2, P3, and P4 are located in areas where demand points are relatively concentrated, and each transfer point can cover multiple demand points at the same time. Therefore, when conducting material dispatching, more emergency materials can be equipped at the transfer points of P1, P2, P3, and P4, and the emergency materials stored at the transfer point can be flexibly dispatched according to the material demand quantities of the cities covered by different transfer points. At the same time, for demand cities that are relatively remote but have a high emergency priority, such as C6, C3, C5, and C12, the location selection plan also provides corresponding emergency transfer points, which can ensure that these demand areas can receive emergency materials in the first time when there is a shortage of materials. Among the 16 demand cities, C1 has the largest material demand quantity and the first emergency priority. Therefore, when selecting the location of the emergency transfer point, the location selection plan also provides more emergency transfer points around C1. When the transportation pressure of the P1 transfer point is relatively large, the P2 transfer point can also deliver materials to the demand area of C1 in the first time.

[0168] Table 5 Material Transportation Time

[0169]

[0170] Table 5 shows the shortest transportation time that can be achieved when using this location selection plan to select suitable transfer points to transport materials to demand points. It can be seen from the table that the vast majority of demand cities can receive emergency materials from the transfer point within one hour. The situation of C4 is relatively lighter than that of other cities, and it has a lower emergency level when conducting material distribution, so the material transportation time is longer, but it can also be delivered within one and a half hours. Generally speaking, the first-time material coverage rate of this location selection plan reaches 93.75%, and it enables the uncovered demand areas to also receive sufficient emergency materials within an acceptable time range.

[0171] In summary, MMO_local can retain the location information of transfer points in both the global and local optimal regions. During the process of emergency material dispatching, it provides the dispatching system with more transfer point location options at suitable positions, improves the coverage rate of transfer points for demand regions, and significantly reduces the material transportation time of emergency transfer points, providing a more flexible and efficient solution for the dispatching system.

[0172] In summary, this invention conducts an applied research on the local optimization of the location problem of emergency dispatching nodes. Taking the severe shortage of emergency materials in various cities of a certain province as the application background, an optimization model for the location of emergency transfer points is established. The designed algorithm of this invention is used to solve the problem, and the performance of different algorithms in solving this problem is analyzed by calculating two model evaluation indicators, namely the material coverage rate and the average material delivery time. At the same time, the specific location scheme obtained by MMO_local is given. It can be seen from the experimental results that the transfer point location scheme obtained by MMO_local can cover both the global and local optimal location regions, providing a more efficient solution for the dispatching system.

Claims

1. A multi-modal and multi-objective emergency material dispatch optimization method, characterized in that: The following steps are involved: Step 1: Establish an optimization model for the location of emergency transfer points; When emergency supplies need to be transported from a transit point to a demand point, the transit point supplies can be delivered to the demand point as quickly as possible within the scope of the dispatching resources, and the demand point covered has a relatively high emergency priority level; Step 2: Perform preliminary calculations on the model established in step 1 to obtain the overall coverage rate of transfer points to demand points and the average material delivery time from the alternative transfer points to all demand points; Step 3: Use a multi-modal multi-objective optimization algorithm based on local center de-clustering to solve the model.

2. The multi-modal multi-objective emergency material dispatch optimization method according to claim 1, characterized in that: The step 1 is specifically as follows: Within the coverage of the emergency transfer point, the number of demand points that can be covered is Within the coverage of the emergency transfer point, the sum of the emergency priorities of all covered demand points is Constraints include: (1) The distance between the selected location and the covered demand point is less than the maximum coverage distance (2) The selected location is valuable only when there are demand points that can be covered. Otherwise, the values of the objective function F1 and the objective function F2 are both infinite, that is, if Then F1=∞,F2=∞; (3) Maximum coverage of the transfer point r = v a t max , where v a is the speed of the truck, t max The maximum waiting time for receiving supplies at the demand point; (4) Decision variables x and y are the coordinates of the transfer points. Their value range should be determined according to the geographical coordinates of the actual dispatching area and the distribution of demand points to ensure that the location of the transfer point can cover all key demand points; Where D is the set of demand points, D={D i |i=1,2,…,n}; is the coordinate of the i-th demand point; C i Demand point D i Emergency priority; D x is the set of all demand points that can be covered by transit point X.

3. The multi-modal multi-objective emergency material dispatch optimization method according to claim 1, characterized in that: The evaluation metrics used when calculating the model in step 2 include: (1) Demand point coverage rate: Among them, n is the number of overall demand points for material scheduling; The number of demand points that a transit point can cover: Among them, X P =(x,y), X p Represents the location of the transfer point, D i Represents the location coordinates of the i-th demand point. When there is an alternative transit point within the maximum range that the demand point can be covered, the demand point is considered to be covered by the transit point, and the number of all demand points that can be covered is calculated; (2) Average delivery time of materials: Among them, n is the number of overall demand points for material scheduling; T XD It refers to the time required for materials to arrive at the demand point when the transfer point closest to the demand point is selected from the alternative transfer points for material transportation; T XD The specific calculation formula is: Among them, p∈(1,2,…,popsize),i∈(1,2,…,n), p represents the alternative turning point found by the algorithm, and the number of solutions in the solution set finally obtained by the algorithm represents the number of alternative turning points found by the algorithm; |X p D i | represents the Euclidean distance between the pth alternative transit point and the i-th demand point, V a Represents the speed of the truck during delivery.

4. The multi-modal multi-objective emergency material dispatch optimization method according to claim 1, characterized in that: The step 3 comprises the following steps: Step 3.1, randomly generate an initial population of NPs in the search space and calculate the fitness values of the individuals in the population; In step 3.2, the local microhabitat is divided according to the neighborhood radius. The dominance information among all individuals in the local microhabitat is used to screen out the local central solution. When the number of local central solutions is insufficient, the algorithm calculates the convergence quality of each individual in its local neighborhood using a formula: the local convergence quality is equal to the ratio of the number of dominant individuals in the local neighborhood to the total number of individuals in the local neighborhood. If the number of local central solutions after this screening step is lower than the required number, the algorithm selects individuals with higher local convergence quality from the remaining population to supplement the local central solution. Step 3.3, when the number of local center solutions is greater than the subpopulation size, perform environmental selection on the population; In step 3.4, the selected population has basically covered all the global and local optimal potential areas, and the individuals in each potential area are clustered using NCM. In step 3.5, individuals are divided into multiple local populations after NCM clustering. The adaptive local search strategy is triggered by the quantitative relationship between the number of local populations and the subpopulation threshold, including: When the number of local populations in the region is less than the subpopulation threshold, the neighbor with the largest crowding distance in the decision space is selected as the basis vector to generate offspring; When the number of local populations in the region is greater than the subpopulation threshold, the neighbor with the largest crowding distance of the current individual in the target space is selected as the basis vector to generate offspring.

5. The multi-modal multi-objective emergency material dispatch optimization method according to claim 4, characterized in that: In step 3.2, the method for screening the local central solution includes the following steps: Step 3.2.1, divide the neighborhood of each individual according to the niche partitioning method; Step 3.2.2, perform non-dominated sorting in the neighborhood and mark all dominated solutions in the neighborhood; In step 3.2.3, delete all marked dominated solutions from the population.

6. The multi-modal multi-objective emergency material dispatch optimization method according to claim 4, characterized in that: In step 3.2, the method for supplementing the local central solution includes the following steps: In step 3.2.4, after dividing the local microhabitat, the local convergence quality of each individual in the local microhabitat is calculated using the following formula: Among them, Nbest represents the number of individuals in the neighborhood population whose dominance relationship is better than that of the individual, and Nlp represents the number of individuals in the neighborhood population; When the number of local central solutions screened out in the first step is too small, solutions with better local convergence quality are added from the remaining population.

7. The multi-modal multi-objective emergency material dispatch optimization method according to claim 4, characterized in that: In step 3.3, the method for selecting the environment of the population is to use the SCD calculation method to sort the crowding of the individuals in the population. The SCD calculation method is: Among them, CD i,x represents the crowding distance of an individual in the decision space, CD i,f represents the crowding distance of an individual in the target space, CD aug,x represents the average crowding distance of the decision space, CD aug,f Represents the average crowding distance of the target space.

8. The multi-modal multi-objective emergency material dispatch optimization method according to claim 4, characterized in that: In step 3.4, at the beginning of NCM clustering, an unmarked individual is first selected as the starting core point; a neighborhood radius is set, and all individuals within the neighborhood of the core point are regarded as density-direct individuals and are classified into the same cluster. At the same time, these individuals are marked to avoid repeated selection; then, the newly added individuals in the cluster are also regarded as new core points, thereby expanding the cluster; in this process, unmarked individuals that are density-direct or density-connected to the new core point continue to be added to the current cluster and marked accordingly; this process is repeated until all individuals are marked and classified into the corresponding cluster.

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