# A Clustering-based Method for Business Hall Efficiency Analysis

Authors: Tianlin Huang; Ning Wang
Journal: Scientific Programming
 (2021)

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## Abstract

Excessive or insufficient business hall resources may result in unreasonable resource allocation, adversely affecting the value of an entity business hall. Therefore, proper characteristic parameters are the key factors for analyzing the business hall, which strongly affect the final analysis results. In this study, a characteristic analysis method for the economic operation of a business hall is developed and the feature engineering is established. Because of its simplicity and versatility, thek-means algorithm has been widely used since it was first proposed around 50 years ago. However, the classical k-means algorithm has poor stability and accuracy. In particular, it is difficult to achieve a suitable balance between of the centroid initialization and the clustering number k. We propose a new initialization (LSH-k-means) algorithm for k-means clustering. This algorithms is mainly based on locality-sensitive hashing (LSH) as an index for computing the initial cluster centroids, and it reduces the range of the clustering number. Furthermore, an empirical study is conducted. According to the load intensity and time change of the business hall, an index system reflecting the optimization analysis of the business hall is established, and the LSH-k-means algorithm is used to analyze the economic operation of the business hall. The results of the empirical study show that the LSH-k-means that the clustering method outperforms the direct prediction method, provides expected analysis results as well as decision optimization recommendations for the business hall, and serves as a basis for the optimal layout of the business hall.


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## Body

## 1. Introduction

An entity business hall is where a company directly conducts specific business activities, such as commodity trade, business handling, and service. However, owing to rapid urbanization and economic development, unreasonable resource allocation is becoming increasingly prevalent. For example, the number of entity business halls is excessive in some places and insufficient in others. Hence, the deployment of new commercial outlets (halls) or resource allocation optimization for existing retail outlets often needs to be performed manually. Therefore, how to evaluate the efficiency of business halls has emerged as a major concern for many enterprises.To this end, many researchers have attempted to overcome the disadvantages of human judgment, which is highly subjective. Brandeau and Chiu [1] considered the transportation cost and the distance between the warehouse and the customer and used a gradient-like algorithm to study the location issue. Wang et al. [2] used nearest-neighbor clustering and the function of Ripley [3] to analyze the layout of commercial outlets and suggested that business type, land price, and traffic accessibility are the critical factors. Gerard [4] analyzed the service needs and waiting demand of customers for bank halls and attempted to shorten the perceived waiting time of customers on the basis of the customers’ business types. Thus, customer satisfaction was improved. Anderson et al. [5] used the queuing model to optimize the queuing service system of banks. They determined the optimal number of service windows by acquiring and presenting a large amount of data. Lin et al. [6] studied the relationship between retail stores and street centrality and pointed out that besides the transport network, which has a strong impact on the retailer’s location, the street centrality influences the type of retail store. Kang [7] analyzed the changes in warehouses from central urban areas to the urban periphery over time and studied the main factors affecting the warehouse location. Hui [8] used data mining to establish the channel analysis model for an electricity business hall and optimized the resource allocation. Based on the statistics of customer queuing time, business processing time, customer satisfaction, and so on, Yan et al. [9] established an intelligent access platform for the business data and improved the service efficiency. However, there is no unified standard for the business hall index system.Clustering is a key technique in data mining, and its applications include pattern recognition [10, 11], image processing [12], and recommendation [13]. Clustering aims to partition data into different categories based on a measure of similarity. The k-means algorithm is widely used owing to its simplicity and effectiveness. However, the different settings of the parameters and random selection of the initial clustering centers make the classical k-means algorithm unstable.The classical clustering algorithm involves two problems: the first problem is to classify a given dataset on the basis of the prespecified cluster numberk; hence, the problem of determining the “correct cluster number” has attracted considerable interest. Although several methods have been developed for estimating the number of data clusters [14–17], it is difficult to use them in practical applications. Therefore, determining the correct number of clusters has long been an important research topic in cluster analysis. The second problem is to determine the initial clustering center, which has a significant impact on the clustering effect. Studies conducted thus far have explored several initialization methods for the thek-means algorithm. For example, the k-means++ algorithm [18] has been proposed to avoid this issue. This algorithm randomly selects the first centroid, and the other centroids are selected as far away as possible from the first centroid. However, random selection is still widely used in practice [19]. Erisoglu et al. [20] proposed an incremental approach for computing the initial clustering centers. In this approach, the reduced dataset is partitioned until the number of clusters equals the predefined number of clusters. However, the number of clusters must be known in advance. The compressed k-means (CKM) algorithm [21] is initialized by locality-sensitive hashing (LSH) [22], and the distance is calculated using the Hamming distance between binary codes. The LSH link [23] can rapidly find a nearby cluster to be connected through the LSH algorithm. David et al. [24] proposed a new LSH scheme adapted to the x2 distance for approximate nearest neighbors (ANN) search in high-dimensional spaces.In summary, there is no unified standard for the index system of business halls at present. Therefore, we establish an index system for analyzing the efficiency of a business hall. To address the problem ofk-means initialization sensitivity as well as the difficulty in determining the number of clusters, we initialize the k-means centroid on the basis of LSH. Accordingly, we implement the relevant algorithms and present the optimal allocation scheme for the business hall.The main contributions of this study are as follows:(1)
According to the average waiting time, ticketing time, and business type of a business hall, we analyze the average load rate of the business hall and use the relevant characteristic variables to describe the load of the business hall. Finally, we propose a general business hall index system.(2)
By combining the characteristics ofk-means and LSH, We propose a new initialization (LSH-k-means) algorithm for k-means clustering. The model can get the load classification of each business hall by inputting the relevant index variables for the optimization of business hall distribution.(3)
The results of our empirical analysis verify the validity of the proposed LSH-k-means approach. Thus, LSH-k-means can be efficiently used for the operational analysis of a business hall.The remainder of this paper is organized as follows: Section2 introduces the required preliminaries, definitions, and models. Section 3 describes the proposed initialization methodology. Section 4 presents, compares, and discusses the experimental results. Finally, Section 5 concludes the paper.


## 2. Preliminaries

### 2.1.k-means Algorithm

The notations used in this paper are defined in Table1. The k-means [25] method is the most well-known clustering method because of its simplicity. It has been identified as one of the top 10 algorithms in data mining [26]. Given a dataset D=x1,x2,…,xn, k-means aims to partition it into k different clusters C=C1,C2,…,Ck, where k≤n is a predefined number. The objective of the k-means clustering algorithm is to minimize the sum of squared errors (SSE) [27] over all k clusters. The SSE is defined as follows:(1)SSE=∑j∑x∈Cjx−Ωj22,where Ωj denotes the j-th cluster centroid, which is computed as the mean of points in Cj, and x∈Cj is the data object in the j-th cluster.(2)Ωj=1Cj∑xi∈Cjxi,where Cj denotes the number of data points in the j-th cluster.Table 1
The notations used throughout this paper.
NotationRepresentationDThe training set withN points and d dimensionsxiThe data pointCjThej-th clustersSSEThe sum of squared errorsΩjDenotes thej-th cluster centroidCjThe number of data points in thej-th clusterOtkndDenotes the complexity ofk-meansHA family of hash functionsPrThe probabilitysimx,yThe similarity betweenx and yMThe maximum load of business hallpiThe proportion of a specific businesswiThe average time to process the businessnThe number of business typesKThe ratio of actual daily load to maximum loadSwThe working timeSThe number of staffs in the business hallAiRepresents the actual load during peak period of the business hallMSThe ratio of average load to the maximum loadWThe number of working daysATThe actual load trendflDenotes the actual load curve fitting functionb0A constantb1The regression coefficientBHThe proportion of high-value businessHBViThe high-value business volumeTBViThe total business volumeFHThe high-frequency loadhThe high thresholdlThe low thresholdTnowRepresent the current timeTlatestRepresent the latest timeFLThe low-frequency loadRHThe latest high-load intervalRLThe latest low-load intervalUThe copy training setNThe size of training datasetTThe minimum number data point in one clusterLThe maximum distance in one clusterBnA two-dimensional arrayQThe nearest neighbor dataqueryxiCalculate the similarity or distance betweenxikThe number of clustersdxi,xjDistance betweenxi and xjKMConductk-means algorithmdiThei-th datasetR+Represents the sum of rank, which better than the otherR−Opposite toR+SU,kThe tight and separative indicatorXBThe XB indexObjminObjective functionFThe quantitative value of factorshiThe weightTo solve equation (1), an expectation–maximization (EM)-like optimization method is adopted by updating Ix or Ω and simultaneously fixing the other [28]. In general, the clustering procedure involves three steps: (1) initialize k cluster centroids; (2) assign each sample to its closest centroid; and (3) recompute the cluster centroids with the assignments produced in Step 2 and go back to Step 2 until convergence. This is known as the Lloyd iteration procedure [29]. Such an iterative optimization approach has several drawbacks. First, it is sensitive to the initialization, which may lead to an inferior result for a given poorly initialized Ω. Many methods have been proposed to obtain a stable solution, including the k-means++ algorithm [18]. Second, finding the optimal solution to k-means is an NP-hard problem. Some variants of k-means have been proposed, such as various parametric k-means, including fuzzy c-means [30, 31]. Third, k-means cannot handle new data, which requires the entire dataset to be observed. The complexity is Ot⋅k⋅n⋅d , where t, n, k, and d denote the number of iterations, size of the dataset, number of clusters, and dimensionality, respectively. This complexity is considerably higher than that of other well-known clustering algorithms such as DBSCAN [32] and mean shift [33].


### 2.2. LSH

LSH is a well-known solution for the approximate nearest neighbor problem in high-dimensional spaces. LSH was first introduced for the Hamming metric by Indyk and Motwani [34]. Data points are assigned to individual hash buckets in each hash function. The idea of LSH is that closer data points are mapped to the same hash bucket with high probability. LSH has been shown to be effective even for high-dimensional data, both theoretically and experimentally [35]. H are a family of hash functions. Each hash function H must satisfy the LSH property: PrHx=Hy=simx,y, where simx,y is the similarity between x and y. These hash functions must meet the following two conditions:(1)
Ifdx,y≤r, then PrHx=Hy≥p1(2)
Ifdx,y≥cr, then PrHx=Hy≤p2wheredx,y represents the distance measure between x and y, r>1, and c<1. The definition implies that x and y are hashed into the same bucket in the projection with a very high probability ≥p1. Regardless of whether they are close to each other, they will be hashed into the same bucket ≤p2 with a low probability. A r,cr,p1,p2-sensitive family of hash functions is useful when the collision probabilities p1, p1 satisfy p1>p2. Figure 1 shows an example of hashing key space.Figure 1
Illustration of hashing key space.


## 2.1.k-means Algorithm

The notations used in this paper are defined in Table1. The k-means [25] method is the most well-known clustering method because of its simplicity. It has been identified as one of the top 10 algorithms in data mining [26]. Given a dataset D=x1,x2,…,xn, k-means aims to partition it into k different clusters C=C1,C2,…,Ck, where k≤n is a predefined number. The objective of the k-means clustering algorithm is to minimize the sum of squared errors (SSE) [27] over all k clusters. The SSE is defined as follows:(1)SSE=∑j∑x∈Cjx−Ωj22,where Ωj denotes the j-th cluster centroid, which is computed as the mean of points in Cj, and x∈Cj is the data object in the j-th cluster.(2)Ωj=1Cj∑xi∈Cjxi,where Cj denotes the number of data points in the j-th cluster.Table 1
The notations used throughout this paper.
NotationRepresentationDThe training set withN points and d dimensionsxiThe data pointCjThej-th clustersSSEThe sum of squared errorsΩjDenotes thej-th cluster centroidCjThe number of data points in thej-th clusterOtkndDenotes the complexity ofk-meansHA family of hash functionsPrThe probabilitysimx,yThe similarity betweenx and yMThe maximum load of business hallpiThe proportion of a specific businesswiThe average time to process the businessnThe number of business typesKThe ratio of actual daily load to maximum loadSwThe working timeSThe number of staffs in the business hallAiRepresents the actual load during peak period of the business hallMSThe ratio of average load to the maximum loadWThe number of working daysATThe actual load trendflDenotes the actual load curve fitting functionb0A constantb1The regression coefficientBHThe proportion of high-value businessHBViThe high-value business volumeTBViThe total business volumeFHThe high-frequency loadhThe high thresholdlThe low thresholdTnowRepresent the current timeTlatestRepresent the latest timeFLThe low-frequency loadRHThe latest high-load intervalRLThe latest low-load intervalUThe copy training setNThe size of training datasetTThe minimum number data point in one clusterLThe maximum distance in one clusterBnA two-dimensional arrayQThe nearest neighbor dataqueryxiCalculate the similarity or distance betweenxikThe number of clustersdxi,xjDistance betweenxi and xjKMConductk-means algorithmdiThei-th datasetR+Represents the sum of rank, which better than the otherR−Opposite toR+SU,kThe tight and separative indicatorXBThe XB indexObjminObjective functionFThe quantitative value of factorshiThe weightTo solve equation (1), an expectation–maximization (EM)-like optimization method is adopted by updating Ix or Ω and simultaneously fixing the other [28]. In general, the clustering procedure involves three steps: (1) initialize k cluster centroids; (2) assign each sample to its closest centroid; and (3) recompute the cluster centroids with the assignments produced in Step 2 and go back to Step 2 until convergence. This is known as the Lloyd iteration procedure [29]. Such an iterative optimization approach has several drawbacks. First, it is sensitive to the initialization, which may lead to an inferior result for a given poorly initialized Ω. Many methods have been proposed to obtain a stable solution, including the k-means++ algorithm [18]. Second, finding the optimal solution to k-means is an NP-hard problem. Some variants of k-means have been proposed, such as various parametric k-means, including fuzzy c-means [30, 31]. Third, k-means cannot handle new data, which requires the entire dataset to be observed. The complexity is Ot⋅k⋅n⋅d , where t, n, k, and d denote the number of iterations, size of the dataset, number of clusters, and dimensionality, respectively. This complexity is considerably higher than that of other well-known clustering algorithms such as DBSCAN [32] and mean shift [33].


## 2.2. LSH

LSH is a well-known solution for the approximate nearest neighbor problem in high-dimensional spaces. LSH was first introduced for the Hamming metric by Indyk and Motwani [34]. Data points are assigned to individual hash buckets in each hash function. The idea of LSH is that closer data points are mapped to the same hash bucket with high probability. LSH has been shown to be effective even for high-dimensional data, both theoretically and experimentally [35]. H are a family of hash functions. Each hash function H must satisfy the LSH property: PrHx=Hy=simx,y, where simx,y is the similarity between x and y. These hash functions must meet the following two conditions:(1)
Ifdx,y≤r, then PrHx=Hy≥p1(2)
Ifdx,y≥cr, then PrHx=Hy≤p2wheredx,y represents the distance measure between x and y, r>1, and c<1. The definition implies that x and y are hashed into the same bucket in the projection with a very high probability ≥p1. Regardless of whether they are close to each other, they will be hashed into the same bucket ≤p2 with a low probability. A r,cr,p1,p2-sensitive family of hash functions is useful when the collision probabilities p1, p1 satisfy p1>p2. Figure 1 shows an example of hashing key space.Figure 1
Illustration of hashing key space.


## 3. Proposed LSH-Based Initialization Algorithm

The proposed framework involves three steps: (1) an index system for the efficiency analysis of a business hall is established in Section3.1. (2) To overcome the problems of poor stability and low accuracy of the classical algorithm, a boost k-means algorithm based on LSH initialization is proposed in Section 3.2. (3) The k-means algorithm is implemented to obtain the clustering results. The details of these three steps are illustrated in Figure 2.Figure 2
Architecture of the proposed LSHk-means model.


### 3.1. Establishment of the Index System

Through the load analysis of the business hall, we can determine the high and low loads and optimize the business hall. The average utilization rate of each business hall is analyzed according to multiple indicators (including average waiting time, ticketing time for business, and business type). Thus, we can use the relevant characteristic variables to describe the load of the business hall. By applying the clustering algorithms, we can obtain the load categories of different business halls, which provides a basis for planning the locations of the business halls. First, the following two essential features are extracted: the maximum load of the business hall (M) and the ratio of the actual daily load to the maximum load K.(1)
The maximum load of the business hallM is given by(3)M=S×Sw∑i=1npiwi,wherepi is the proportion of a specific business, wi is the average time for the clerk to handle the business, n is the number of business types, Sw is the working time of the clerk, and S is the number of clerks in the business hall. The variables are taken from the peak period. This value represents the maximum business volume that a business hall can withstand during the peak period. The peak period can be obtained by measuring the historical data of each business hall.(2)
The ratio of the actual daily load to the maximum loadK is given by(4)K=AM,whereA represents the actual daily load of the business hall and M is the maximum load in one day.By combining the essential characteristics of the business hall and based on the analysis of historical data, we can obtain the calculation indicators of the business hall to prepare for the subsequent model input. Therefore, the feature engineering for the business hall efficiency analysis is established, and the critical indexes extracted are as follows:(1)
The ratio of the average load to the maximum loadMS is given by(5)MS=∑i=1WAiW×M,whereW is the number of days, and A and M are the same as above. This index denotes the ratio of the average actual load to the maximum load over some time.(2)
The actual load trendAT is given by(6)fl=b0+b1t,AT=b1+∑fl×t−n×fl×t∑ti2−n×t2,wherefl denotes the actual load curve fitting, b0 is a constant, b1 is the regression coefficient, t is a time-independent variable, and n is the number of statistical data. This index indicates whether the load trend of the business hall will be rising, flat, or declining for some time. A fitting curve fl can be used to characterize the load trend of the business hall over some time, and the slope b1 represents the trend state. Our method includes a commercial center, residential center, new urban area, and other factors.(3)
The proportion of high-value businessBH is given by(7)BH=∑i=1WHBVi∑i=1WTBVi,whereHBVi is the high-value business volume and TBVi is the total business volume. Thus, this index denotes the proportion of high-value business to total business in the peak period.(4)
The high-frequency loadFH is given by(8)FH=countKi>hW,i=1,2,…,W,whereh is a high threshold and FH represents the load of K exceeding h within a period. Furthermore, h can be obtained by statistical analysis of the historical data of the business hall.(5)
The low-frequency loadFL is given by(9)FL=countKi<lW,i=1,2,…,W,wherel is a low threshold and FL denotes the frequency that K is less than l for some time. Furthermore, l can be obtained by statistical analysis the of historical data of the business hall.(6)
The latest high-load intervalRH is given by(10)RH=Tnow−TlatestKi>h,i=1,2,…,W,whereTnow represents the current time, h denotes a high threshold, and TlatestKi>h refers to the time when the latest K is greater than h. Furthermore, RH denotes the interval from Tnow to TlatestKi>h.(7)
The latest low-load intervalRL is given by(11)RL=Tnow−TlatestKi<l,i=1,2,…,W,whereTnow represents the current time, l denotes a low threshold, and TlatestKi>h refers to the time when the latest K is greater than h. Furthermore, RL denotes the interval from Tlatest to TlatestKi>h.


### 3.2. LSH-k-Means

The main purpose of clustering is to divide data into clusters in which objects in the same cluster are close to one another, whereas objects in different clusters are far from one another. Two factors affect the quality ofk-means clustering. Before applying the algorithm, we need to specify the number of clusters k and select the initial cluster centroid. Selecting an appropriate initial cluster centroid can improve the quality of clustering. To this end, a critical study was conducted by Vassilvitskii et al. [18, 36]. If the initial cluster centroid is selected carefully, the k-means algorithm converges to a better local optimal solution. Furthermore, careful selection of the initial cluster centroid makes the k-means iteration converge faster [18]. However, to make the initial centroid adapt to the data distribution, it is necessary to scan k rounds. Therefore, although the number of scanning wheels in [36] has been reduced to a small value, the additional computing cost is still inevitable. Our algorithm exploits LSH. The algorithm minimizes the path by adding the nearest neighbor, and LSH can effectively search for the nearest group features in the path. The average time complexity of the hash-based search is O1. LSH scans the data records and finds the nearest points; the average values are computed after the nearest points are classified as a category. Algorithm 1 describes the process of obtaining the initialization centroids in our proposed LSH-k-means scheme. The main steps are as follows:(1)
Suppose that we have a set of pointsD∈ℝd via the index system in Section 3.1. We use LSH to index the feature vectors extracted from the dataset D to reduce the search time for the nearest neighbor of each query. This is based on the hash mapping function, hash functions, and hash table L [37]. Constructing an effective LSH index structure for approximate nearest neighbor search depends on the number of hash tables L and the number of bits V of the hash codes.(2)
To facilitate the statistics of nonclustered data points, in Algorithm1, we copy a dataset U from D. Randomly select one data point xi from U as the centroid. Then, xi is merged into the set An and removed from the dataset U, where n is the n-th cluster. After obtaining xi points, query the corresponding bucket number according to the hash table L in Step 1 and take out the data in bucket number V. Calculate the similarity or distance between xi and the data points xi in the bucket and return the nearest neighbor data Q=queryxi.(3)
Take data pointxj from Q, whose distance to xi does not exceed L. Put xj merged into Bn, that is, Bn=Bn∪xj, and remove it from the dataset U.(4)
Repeat Step 3 until the other data pointxj in Q reaches a certain threshold; the threshold can be computed as follows:(12)L≥dxi,xj(5)
Repeat Steps 2-3 until the length of theU dataset is less than the threshold L. As shown in Algorithm 1, countU≤T.(6)
The arithmetic mean values for the finalk sets of samples are computed; then, we can obtain the clustering centers for all the categories in this way:(13)Cn=SumBnBn,n=1,2,…,k.Therefore, based on the aforementioned steps, we will have two algorithms to choose from: “best” movement and “fast” movement [38]. For the “best” movement, we can use equation (13) and the k value in Algorithm 1 as the initial clustering center of the classical k-means input KMk,ci, and run the algorithm; the result is the final result. For “fast” movement, the divided categories can be regarded as approximate clustering results and directly used as the classification results. Because the initial clustering center is determined and the initial category is obtained, the result of the algorithm is more stable and accurate, and it requires a relatively short running time.Algorithm 1: LSH-based k-means.
Required: Training dataset D∈ℝd; the size of dataset N; the minimum number T of data points in one cluster; the maximum distance in one cluster L; the closest set Bn, where n∈1,2,…,k and A is a two-dimensional array; k is the number of clusters.Output: The final clustering result.(1)
Initialize LSH.(2)
Index datasetD via LSH.(3)
Let0←k.(4)
LetU=D.//copy D(5)
whilecountU≤Tdo(6)
k←k+1.(7)
Randomly select one pointxi from dataset U.(8)
Bn=Bn∪xi.(9)
xi is removed from U.(10)
Q = query xi.(11)
Let0←q(12)
forq≤countQdo(13)
q←q+1.(14)
ifL≥dQqthen(15)
break;(16)
end if(17)
ifQq∈Uthen(18)
Qq removed from U.(19)
Bk=Bk∪Qq.(20)
end if(21)
end for(22)
end while(23)
Compute the centeroidsci via Bn.(24)
FunctionKMk,ci(25)
The final clustering result.C=C1,C2,…,Ck.


## 3.1. Establishment of the Index System

Through the load analysis of the business hall, we can determine the high and low loads and optimize the business hall. The average utilization rate of each business hall is analyzed according to multiple indicators (including average waiting time, ticketing time for business, and business type). Thus, we can use the relevant characteristic variables to describe the load of the business hall. By applying the clustering algorithms, we can obtain the load categories of different business halls, which provides a basis for planning the locations of the business halls. First, the following two essential features are extracted: the maximum load of the business hall (M) and the ratio of the actual daily load to the maximum load K.(1)
The maximum load of the business hallM is given by(3)M=S×Sw∑i=1npiwi,wherepi is the proportion of a specific business, wi is the average time for the clerk to handle the business, n is the number of business types, Sw is the working time of the clerk, and S is the number of clerks in the business hall. The variables are taken from the peak period. This value represents the maximum business volume that a business hall can withstand during the peak period. The peak period can be obtained by measuring the historical data of each business hall.(2)
The ratio of the actual daily load to the maximum loadK is given by(4)K=AM,whereA represents the actual daily load of the business hall and M is the maximum load in one day.By combining the essential characteristics of the business hall and based on the analysis of historical data, we can obtain the calculation indicators of the business hall to prepare for the subsequent model input. Therefore, the feature engineering for the business hall efficiency analysis is established, and the critical indexes extracted are as follows:(1)
The ratio of the average load to the maximum loadMS is given by(5)MS=∑i=1WAiW×M,whereW is the number of days, and A and M are the same as above. This index denotes the ratio of the average actual load to the maximum load over some time.(2)
The actual load trendAT is given by(6)fl=b0+b1t,AT=b1+∑fl×t−n×fl×t∑ti2−n×t2,wherefl denotes the actual load curve fitting, b0 is a constant, b1 is the regression coefficient, t is a time-independent variable, and n is the number of statistical data. This index indicates whether the load trend of the business hall will be rising, flat, or declining for some time. A fitting curve fl can be used to characterize the load trend of the business hall over some time, and the slope b1 represents the trend state. Our method includes a commercial center, residential center, new urban area, and other factors.(3)
The proportion of high-value businessBH is given by(7)BH=∑i=1WHBVi∑i=1WTBVi,whereHBVi is the high-value business volume and TBVi is the total business volume. Thus, this index denotes the proportion of high-value business to total business in the peak period.(4)
The high-frequency loadFH is given by(8)FH=countKi>hW,i=1,2,…,W,whereh is a high threshold and FH represents the load of K exceeding h within a period. Furthermore, h can be obtained by statistical analysis of the historical data of the business hall.(5)
The low-frequency loadFL is given by(9)FL=countKi<lW,i=1,2,…,W,wherel is a low threshold and FL denotes the frequency that K is less than l for some time. Furthermore, l can be obtained by statistical analysis the of historical data of the business hall.(6)
The latest high-load intervalRH is given by(10)RH=Tnow−TlatestKi>h,i=1,2,…,W,whereTnow represents the current time, h denotes a high threshold, and TlatestKi>h refers to the time when the latest K is greater than h. Furthermore, RH denotes the interval from Tnow to TlatestKi>h.(7)
The latest low-load intervalRL is given by(11)RL=Tnow−TlatestKi<l,i=1,2,…,W,whereTnow represents the current time, l denotes a low threshold, and TlatestKi>h refers to the time when the latest K is greater than h. Furthermore, RL denotes the interval from Tlatest to TlatestKi>h.


## 3.2. LSH-k-Means

The main purpose of clustering is to divide data into clusters in which objects in the same cluster are close to one another, whereas objects in different clusters are far from one another. Two factors affect the quality ofk-means clustering. Before applying the algorithm, we need to specify the number of clusters k and select the initial cluster centroid. Selecting an appropriate initial cluster centroid can improve the quality of clustering. To this end, a critical study was conducted by Vassilvitskii et al. [18, 36]. If the initial cluster centroid is selected carefully, the k-means algorithm converges to a better local optimal solution. Furthermore, careful selection of the initial cluster centroid makes the k-means iteration converge faster [18]. However, to make the initial centroid adapt to the data distribution, it is necessary to scan k rounds. Therefore, although the number of scanning wheels in [36] has been reduced to a small value, the additional computing cost is still inevitable. Our algorithm exploits LSH. The algorithm minimizes the path by adding the nearest neighbor, and LSH can effectively search for the nearest group features in the path. The average time complexity of the hash-based search is O1. LSH scans the data records and finds the nearest points; the average values are computed after the nearest points are classified as a category. Algorithm 1 describes the process of obtaining the initialization centroids in our proposed LSH-k-means scheme. The main steps are as follows:(1)
Suppose that we have a set of pointsD∈ℝd via the index system in Section 3.1. We use LSH to index the feature vectors extracted from the dataset D to reduce the search time for the nearest neighbor of each query. This is based on the hash mapping function, hash functions, and hash table L [37]. Constructing an effective LSH index structure for approximate nearest neighbor search depends on the number of hash tables L and the number of bits V of the hash codes.(2)
To facilitate the statistics of nonclustered data points, in Algorithm1, we copy a dataset U from D. Randomly select one data point xi from U as the centroid. Then, xi is merged into the set An and removed from the dataset U, where n is the n-th cluster. After obtaining xi points, query the corresponding bucket number according to the hash table L in Step 1 and take out the data in bucket number V. Calculate the similarity or distance between xi and the data points xi in the bucket and return the nearest neighbor data Q=queryxi.(3)
Take data pointxj from Q, whose distance to xi does not exceed L. Put xj merged into Bn, that is, Bn=Bn∪xj, and remove it from the dataset U.(4)
Repeat Step 3 until the other data pointxj in Q reaches a certain threshold; the threshold can be computed as follows:(12)L≥dxi,xj(5)
Repeat Steps 2-3 until the length of theU dataset is less than the threshold L. As shown in Algorithm 1, countU≤T.(6)
The arithmetic mean values for the finalk sets of samples are computed; then, we can obtain the clustering centers for all the categories in this way:(13)Cn=SumBnBn,n=1,2,…,k.Therefore, based on the aforementioned steps, we will have two algorithms to choose from: “best” movement and “fast” movement [38]. For the “best” movement, we can use equation (13) and the k value in Algorithm 1 as the initial clustering center of the classical k-means input KMk,ci, and run the algorithm; the result is the final result. For “fast” movement, the divided categories can be regarded as approximate clustering results and directly used as the classification results. Because the initial clustering center is determined and the initial category is obtained, the result of the algorithm is more stable and accurate, and it requires a relatively short running time.Algorithm 1: LSH-based k-means.
Required: Training dataset D∈ℝd; the size of dataset N; the minimum number T of data points in one cluster; the maximum distance in one cluster L; the closest set Bn, where n∈1,2,…,k and A is a two-dimensional array; k is the number of clusters.Output: The final clustering result.(1)
Initialize LSH.(2)
Index datasetD via LSH.(3)
Let0←k.(4)
LetU=D.//copy D(5)
whilecountU≤Tdo(6)
k←k+1.(7)
Randomly select one pointxi from dataset U.(8)
Bn=Bn∪xi.(9)
xi is removed from U.(10)
Q = query xi.(11)
Let0←q(12)
forq≤countQdo(13)
q←q+1.(14)
ifL≥dQqthen(15)
break;(16)
end if(17)
ifQq∈Uthen(18)
Qq removed from U.(19)
Bk=Bk∪Qq.(20)
end if(21)
end for(22)
end while(23)
Compute the centeroidsci via Bn.(24)
FunctionKMk,ci(25)
The final clustering result.C=C1,C2,…,Ck.


## 4. Experimental Results

First, we use the UCIhttps://archive-beta.ics.uci.edu/ml/datasets datasets [39] to verify the performance of the proposed algorithm, and we state the verification criteria. In addition, we use the Mall-Customers dataset https://www.kaggle.com/shwetabh123/mall-customers for the value range of the number of clusters k of the proposed LSH-k-means model. Our experimental results demonstrate the effectiveness and superiority of the proposed LSH-k-means. Then, we compare it with the actual business hall dataset and present an example to optimize the business hall operation.


### 4.1. Experimental Design

To verify the aforementioned points and evaluate the effectiveness of the proposed LSH-k-means model, numerous experiments were conducted on the UCI datasets, which consist of Balance, Wine, Breast, Diabet, Iris, Hayes-roth, Tic-tac-toe, and Bupa. We followed the experiments conducted in a previous study [40]. We briefly review the existing baselines as follows:(1)
k-means [25] is derived from the classical k-means.(2)
Enhancedk-means [38] enhances the classical k-means algorithm. The initial cluster centers are determined in advance instead of random selection.(3)
The AC algorithm [41] for clustering can assume each sample as a pattern; by computing the similarity between patterns, the more similar patterns are grouped into one class, and the less similar patterns are classified into different classes. The difference between two patterns in AC clustering is usually measured by the distance function, including the Euclidean distance or Hamming distance. In the experiment, the AC algorithm is implemented by the KnowledgeMiner Software [41].There are 200 samples in the Mall-Customers dataset. It includes gender, customer ID, age, annual income, and expenditure scores. In addition, it collects insights from the data and groups them according to their behaviors. The elbow method [42] is a well-known method for determining the optimal value of k. As shown in Figure 3(a), the optimum number of clusters of the Mall-Customers dataset is 5. According to Algorithm 1, we set the minimum number T=20 and the maximum distance L=50.0. Owing to the small amount of data, we set the number of buckets to 1. After 10 LSH-based initializations, we get the value of k between 4−6. Figure 3(b) shows the results of LSH k-means clustering. The black dots represent the centroids.Figure 3
Results on the Mall-Customers dataset. (a) Elbow point graph, (b) Results of LSHk-means clustering.
(a)(b)There are 525 samples in the Balance dataset. For the classicalk-means algorithm, the number of clustering categories that match the real categories is 271, and the matching rate is 51.62%. The corresponding values of the LSH k-means algorithm are 288 and 54.87%, respectively. Similarly, the results of the other UCI datasets are listed in Table 2. To determine whether there are significant differences between algorithms, we use the Wilcoxon signed-rank test [43]. It is a nonparametric statistical test. The Wilcoxon test has been widely used in many fields, especially in algorithm comparison and analysis [40]. It is expressed as follows:(14)R+=∑dj>0rankdj+12∑dj=0rankdj,R−=∑dj<0rankdj+12∑dj=0rankdj,where dj is the difference in clustering performance between the two algorithms on the j-th dataset, and the absolute values of their difference are arranged in the ascending order. If the rank is the same, we take the average value. R+ implies that the sum of ranks for the algorithm is better than the other, and R− implies the opposite.Table 2
Experimental results on UCI datasets.
Datasetsk-means [25] (%)Enhancedk-means [38] (%)LSHk-means (this work) (%)Difference (%)RankBalance51.6253.1654.871.717Wine57.3065.1870.225.048Breast93.8593.9994.430.442Diabet66.2867.7666.72−1.044Iris89.3390.6789.77−0.903Hayes-roth34.0935.1935.190.001Tic-tac-toe54.0256.0157.171.165Bupa44.6444.9346.431.506The calculations for the eight aforementioned datasets are presented below.(15)R+=7+8+2+5+6=28,R−=4+1+3=8.LetT=minR+,R−=8; we get T=8. According to the critical value table of the Wilcoxon test, we can judge that the difference between algorithms is significant under the condition a=0.05. Furthermore, as shown in Table 2, there are five datasets for the LSH-based k-means, which is hence better than the enhanced k-means; thus, in terms of quantity, the LSH-based k-means algorithm outperforms the enhanced k-means algorithm. Therefore, we can judge that the efficiency of LSH-based k-means is significant.In addition to comparison with actual categories, we further distinguish the clustering effects ofk-means clustering and the AC algorithm. A tight and separative indicator is used to evaluate the clustering results [44], which is defined as follows:(16)SU,k=1/n∑i=1n∑j=1kuijxi−cj2minp,q=1,2,…,kcp−cq,where cp, cq, and cj denote the cluster centers, xi is any sample in the dataset, k is the number of clusters, and U is the sample set. The Xie–Beni (XB) index [45, 46] is based on intracluster and intercluster distances; it is formulated in terms of the cluster compactness and separation between the clusters. We use the XB index for the evaluation of the cluster effects, and it is defined as follows:(17)XB=maxSU,k,k=2,3,…,n−1,where SU,k is the ratio of the average distance between data objects and their corresponding clustering centers to the minimum distance of the cluster centers. The smaller the value of SU,k, the higher is the clustering quality. The results are summarized in Table 3.Table 3
XB evaluation value.
AlgorithmsIterations similarityCategoryXBC1C2C2k-means [25]827102490.7067Enhancedk-means [38]44769620.7045AC algorithm [41]95%27126252.1694LSHk-means (this work)34868620.7033From the XB value calculated in Table3, we can conclude that the difference between the algorithms is significant. The XB value of the AC algorithm is the largest, while that of the LSH-based k-means algorithm is the smallest, which implies that the LSH k-means algorithm outperforms the other algorithms in the experiment. Thus, the experimental results verify the effectiveness and superiority of the proposed method. Therefore, it can finally be applied to the empirical analysis. In the next section, we describe the application of LSH k-means to business hall analysis.


### 4.2. Business Hall Analysis

In reality, business hall resource allocation may be unreasonable. For example, some business halls may be busy, while others may be idle. This may be caused by overlapping user coverage in different business halls, unreasonable location of the business halls, and a large proportion of low-value businesses. In this section, we experimentally verify the effectiveness of our index system and analyze the results of the proposed LSH-k-means model.


#### 4.2.1. Business Hall Clustering

When the index system is established as described in Section3.1, we get the characteristic information of the business hall. After data preprocessing, the number of clusters is determined subjectively. Consider the load intensity and time change information for the business hall. The load intensity can be categorized into High, Medium, and Low, and the load trend can be categorized into No change, Slow grouth, Fast grouth. Thus, a nine-square grid (Figure 4(b)) map can be obtained. At the same time, by referring to the knowledge of field experts, the number of clusters can be defined as 9 for the subjective clustering methods. After the clusters are determined, the extracted feature indicators can be taken as the input, and the clustering model is implemented. For the LSH k-means algorithm, the distance parameter was selected as the Euclidean distance, the maximum number of iterations was set to 500, the number of seeds was set to 10, and the number of the clusters was set to 9. Then, the outcomes were obtained, as shown in Table 4 and Figure 4(a).Figure 4
Results of LSHk-means clustering on business hall dataset. (a) Diagram of clustering results, (b) Nine-square grid map.
(a)(b)Table 4
Clustering result of LSH-k-means.
No.Full data123456789AttributesCluster centroidsS0.550.250.600.200.100.960.500.950.500.90Q0.000.51−0.510.01−0.500.500.53−0.530.010.01V0.620.750.500.450.400.800.600.800.650.70FH0.320.050.080.030.010.950.250.600.080.09FL0.350.850.100.800.950.020.250.100.060.02RH0.370.100.700.800.010.020.500.600.650.03RL.0.510.300.100.040.980.900.500.080.800.90Meanwhile, in the case of different predetermined cluster numbers for the subjective clustering methods, the AC algorithm determined the clustering number automatically, which was computed on the basis of the similarity between the samples. Here, the similarity was set at 95%, and the algorithm was implemented at the same time. Thus, the result was exactly consistent with that of the LSHk-means algorithm. The details are presented in Table 5. For example, the first and sixth samples, the second sample, and the third sample were clustered into the same category.Table 5
Clustering result of the AC algorithm.
ResultSQVFHFLRHRLC10.950.50.80.950.010.020.9C20.890.010.690.890.020.020.89C20.900.70.90.010.030.9C30.94−0.560.790.590.090.590.9C40.50.50.60.250.250.50.5C10.960.490.790.940.020.010.89……………………C90.09−0.490.390.010.940.010.97C90.1−0.50.400.9500.98C80.59−0.510.490.070.090.690.09The final classification results obtained from the model can provide the load grades and decision-making suggestions, which can serve as a basis for site planning optimization of the business halls. In addition, the results of the two algorithms were consistent, which indicate that the LSHk-means algorithm is effective and a stable result was obtained. Accordingly, further optimization action can be implemented.


#### 4.2.2. Optimization Analysis

As shown in Figure5(a), the 16th and 17th business halls are both in Class I. This category indicates that the current load is Low, and the load trend remains unchanged, implying that the business hall resources are redundant in this area. The business halls in this class are idle, and the site may be unreasonable. In addition, the merger of business halls, relocation, and reduction of resource input in this area should be considered.Figure 5
Illustration of business hall location optimization. (a) Location of business halls in classA; (b) Suggested location for supplementary business hall.
(a)(b)By contrast, for ClassA (the red part of Figure 5(b)), it can be seen that the current load and load trend are both High, which implies that the business volumes of the business halls are large, and the load trend change is still on the rise. Currently, the first and sixth business halls belong to this category. The future trend is still likely to be growing, and the business volumes will keep increasing. Therefore, this area is where more business hall resources need to be input, and the optimal site planning of business halls should be considered accordingly. We can define the objective function of the optimal site for the input business hall resources as follows:(18)Objmin=∑k=1nFkn+∑i=lhidvi,x,where Fk is the quantitative value of factors that affect the rationality of business hall location, n is the number of factors, dvi,x is the distance function, hi is the weight, and x is the target point to be solved. Thus, according to the objective function and relevant coordinate information of key units in the area, we can compute the optimal planning location of the business hall using the optimization algorithm. Here, the optimal location of the business hall was computed as 100.0644,128.8199, and the optimal solution was 527.0368. The details are shown in Figure 5.The 9th and 12th business halls belong to the Class E, which shows that the current load and load trends are both normal, and the status is stable. Therefore, the business halls in this class are not the current focus of optimization. In addition, the other classes are similar to this category, which is also not the current focus. The main objects are Class I and Class A, that is, excessive or insufficient business hall resources are mainly concentrated in these two classes, which are the focus of our optimization analysis.


## 4.1. Experimental Design

To verify the aforementioned points and evaluate the effectiveness of the proposed LSH-k-means model, numerous experiments were conducted on the UCI datasets, which consist of Balance, Wine, Breast, Diabet, Iris, Hayes-roth, Tic-tac-toe, and Bupa. We followed the experiments conducted in a previous study [40]. We briefly review the existing baselines as follows:(1)
k-means [25] is derived from the classical k-means.(2)
Enhancedk-means [38] enhances the classical k-means algorithm. The initial cluster centers are determined in advance instead of random selection.(3)
The AC algorithm [41] for clustering can assume each sample as a pattern; by computing the similarity between patterns, the more similar patterns are grouped into one class, and the less similar patterns are classified into different classes. The difference between two patterns in AC clustering is usually measured by the distance function, including the Euclidean distance or Hamming distance. In the experiment, the AC algorithm is implemented by the KnowledgeMiner Software [41].There are 200 samples in the Mall-Customers dataset. It includes gender, customer ID, age, annual income, and expenditure scores. In addition, it collects insights from the data and groups them according to their behaviors. The elbow method [42] is a well-known method for determining the optimal value of k. As shown in Figure 3(a), the optimum number of clusters of the Mall-Customers dataset is 5. According to Algorithm 1, we set the minimum number T=20 and the maximum distance L=50.0. Owing to the small amount of data, we set the number of buckets to 1. After 10 LSH-based initializations, we get the value of k between 4−6. Figure 3(b) shows the results of LSH k-means clustering. The black dots represent the centroids.Figure 3
Results on the Mall-Customers dataset. (a) Elbow point graph, (b) Results of LSHk-means clustering.
(a)(b)There are 525 samples in the Balance dataset. For the classicalk-means algorithm, the number of clustering categories that match the real categories is 271, and the matching rate is 51.62%. The corresponding values of the LSH k-means algorithm are 288 and 54.87%, respectively. Similarly, the results of the other UCI datasets are listed in Table 2. To determine whether there are significant differences between algorithms, we use the Wilcoxon signed-rank test [43]. It is a nonparametric statistical test. The Wilcoxon test has been widely used in many fields, especially in algorithm comparison and analysis [40]. It is expressed as follows:(14)R+=∑dj>0rankdj+12∑dj=0rankdj,R−=∑dj<0rankdj+12∑dj=0rankdj,where dj is the difference in clustering performance between the two algorithms on the j-th dataset, and the absolute values of their difference are arranged in the ascending order. If the rank is the same, we take the average value. R+ implies that the sum of ranks for the algorithm is better than the other, and R− implies the opposite.Table 2
Experimental results on UCI datasets.
Datasetsk-means [25] (%)Enhancedk-means [38] (%)LSHk-means (this work) (%)Difference (%)RankBalance51.6253.1654.871.717Wine57.3065.1870.225.048Breast93.8593.9994.430.442Diabet66.2867.7666.72−1.044Iris89.3390.6789.77−0.903Hayes-roth34.0935.1935.190.001Tic-tac-toe54.0256.0157.171.165Bupa44.6444.9346.431.506The calculations for the eight aforementioned datasets are presented below.(15)R+=7+8+2+5+6=28,R−=4+1+3=8.LetT=minR+,R−=8; we get T=8. According to the critical value table of the Wilcoxon test, we can judge that the difference between algorithms is significant under the condition a=0.05. Furthermore, as shown in Table 2, there are five datasets for the LSH-based k-means, which is hence better than the enhanced k-means; thus, in terms of quantity, the LSH-based k-means algorithm outperforms the enhanced k-means algorithm. Therefore, we can judge that the efficiency of LSH-based k-means is significant.In addition to comparison with actual categories, we further distinguish the clustering effects ofk-means clustering and the AC algorithm. A tight and separative indicator is used to evaluate the clustering results [44], which is defined as follows:(16)SU,k=1/n∑i=1n∑j=1kuijxi−cj2minp,q=1,2,…,kcp−cq,where cp, cq, and cj denote the cluster centers, xi is any sample in the dataset, k is the number of clusters, and U is the sample set. The Xie–Beni (XB) index [45, 46] is based on intracluster and intercluster distances; it is formulated in terms of the cluster compactness and separation between the clusters. We use the XB index for the evaluation of the cluster effects, and it is defined as follows:(17)XB=maxSU,k,k=2,3,…,n−1,where SU,k is the ratio of the average distance between data objects and their corresponding clustering centers to the minimum distance of the cluster centers. The smaller the value of SU,k, the higher is the clustering quality. The results are summarized in Table 3.Table 3
XB evaluation value.
AlgorithmsIterations similarityCategoryXBC1C2C2k-means [25]827102490.7067Enhancedk-means [38]44769620.7045AC algorithm [41]95%27126252.1694LSHk-means (this work)34868620.7033From the XB value calculated in Table3, we can conclude that the difference between the algorithms is significant. The XB value of the AC algorithm is the largest, while that of the LSH-based k-means algorithm is the smallest, which implies that the LSH k-means algorithm outperforms the other algorithms in the experiment. Thus, the experimental results verify the effectiveness and superiority of the proposed method. Therefore, it can finally be applied to the empirical analysis. In the next section, we describe the application of LSH k-means to business hall analysis.


## 4.2. Business Hall Analysis

In reality, business hall resource allocation may be unreasonable. For example, some business halls may be busy, while others may be idle. This may be caused by overlapping user coverage in different business halls, unreasonable location of the business halls, and a large proportion of low-value businesses. In this section, we experimentally verify the effectiveness of our index system and analyze the results of the proposed LSH-k-means model.


### 4.2.1. Business Hall Clustering

When the index system is established as described in Section3.1, we get the characteristic information of the business hall. After data preprocessing, the number of clusters is determined subjectively. Consider the load intensity and time change information for the business hall. The load intensity can be categorized into High, Medium, and Low, and the load trend can be categorized into No change, Slow grouth, Fast grouth. Thus, a nine-square grid (Figure 4(b)) map can be obtained. At the same time, by referring to the knowledge of field experts, the number of clusters can be defined as 9 for the subjective clustering methods. After the clusters are determined, the extracted feature indicators can be taken as the input, and the clustering model is implemented. For the LSH k-means algorithm, the distance parameter was selected as the Euclidean distance, the maximum number of iterations was set to 500, the number of seeds was set to 10, and the number of the clusters was set to 9. Then, the outcomes were obtained, as shown in Table 4 and Figure 4(a).Figure 4
Results of LSHk-means clustering on business hall dataset. (a) Diagram of clustering results, (b) Nine-square grid map.
(a)(b)Table 4
Clustering result of LSH-k-means.
No.Full data123456789AttributesCluster centroidsS0.550.250.600.200.100.960.500.950.500.90Q0.000.51−0.510.01−0.500.500.53−0.530.010.01V0.620.750.500.450.400.800.600.800.650.70FH0.320.050.080.030.010.950.250.600.080.09FL0.350.850.100.800.950.020.250.100.060.02RH0.370.100.700.800.010.020.500.600.650.03RL.0.510.300.100.040.980.900.500.080.800.90Meanwhile, in the case of different predetermined cluster numbers for the subjective clustering methods, the AC algorithm determined the clustering number automatically, which was computed on the basis of the similarity between the samples. Here, the similarity was set at 95%, and the algorithm was implemented at the same time. Thus, the result was exactly consistent with that of the LSHk-means algorithm. The details are presented in Table 5. For example, the first and sixth samples, the second sample, and the third sample were clustered into the same category.Table 5
Clustering result of the AC algorithm.
ResultSQVFHFLRHRLC10.950.50.80.950.010.020.9C20.890.010.690.890.020.020.89C20.900.70.90.010.030.9C30.94−0.560.790.590.090.590.9C40.50.50.60.250.250.50.5C10.960.490.790.940.020.010.89……………………C90.09−0.490.390.010.940.010.97C90.1−0.50.400.9500.98C80.59−0.510.490.070.090.690.09The final classification results obtained from the model can provide the load grades and decision-making suggestions, which can serve as a basis for site planning optimization of the business halls. In addition, the results of the two algorithms were consistent, which indicate that the LSHk-means algorithm is effective and a stable result was obtained. Accordingly, further optimization action can be implemented.


### 4.2.2. Optimization Analysis

As shown in Figure5(a), the 16th and 17th business halls are both in Class I. This category indicates that the current load is Low, and the load trend remains unchanged, implying that the business hall resources are redundant in this area. The business halls in this class are idle, and the site may be unreasonable. In addition, the merger of business halls, relocation, and reduction of resource input in this area should be considered.Figure 5
Illustration of business hall location optimization. (a) Location of business halls in classA; (b) Suggested location for supplementary business hall.
(a)(b)By contrast, for ClassA (the red part of Figure 5(b)), it can be seen that the current load and load trend are both High, which implies that the business volumes of the business halls are large, and the load trend change is still on the rise. Currently, the first and sixth business halls belong to this category. The future trend is still likely to be growing, and the business volumes will keep increasing. Therefore, this area is where more business hall resources need to be input, and the optimal site planning of business halls should be considered accordingly. We can define the objective function of the optimal site for the input business hall resources as follows:(18)Objmin=∑k=1nFkn+∑i=lhidvi,x,where Fk is the quantitative value of factors that affect the rationality of business hall location, n is the number of factors, dvi,x is the distance function, hi is the weight, and x is the target point to be solved. Thus, according to the objective function and relevant coordinate information of key units in the area, we can compute the optimal planning location of the business hall using the optimization algorithm. Here, the optimal location of the business hall was computed as 100.0644,128.8199, and the optimal solution was 527.0368. The details are shown in Figure 5.The 9th and 12th business halls belong to the Class E, which shows that the current load and load trends are both normal, and the status is stable. Therefore, the business halls in this class are not the current focus of optimization. In addition, the other classes are similar to this category, which is also not the current focus. The main objects are Class I and Class A, that is, excessive or insufficient business hall resources are mainly concentrated in these two classes, which are the focus of our optimization analysis.


## 4.2.1. Business Hall Clustering

When the index system is established as described in Section3.1, we get the characteristic information of the business hall. After data preprocessing, the number of clusters is determined subjectively. Consider the load intensity and time change information for the business hall. The load intensity can be categorized into High, Medium, and Low, and the load trend can be categorized into No change, Slow grouth, Fast grouth. Thus, a nine-square grid (Figure 4(b)) map can be obtained. At the same time, by referring to the knowledge of field experts, the number of clusters can be defined as 9 for the subjective clustering methods. After the clusters are determined, the extracted feature indicators can be taken as the input, and the clustering model is implemented. For the LSH k-means algorithm, the distance parameter was selected as the Euclidean distance, the maximum number of iterations was set to 500, the number of seeds was set to 10, and the number of the clusters was set to 9. Then, the outcomes were obtained, as shown in Table 4 and Figure 4(a).Figure 4
Results of LSHk-means clustering on business hall dataset. (a) Diagram of clustering results, (b) Nine-square grid map.
(a)(b)Table 4
Clustering result of LSH-k-means.
No.Full data123456789AttributesCluster centroidsS0.550.250.600.200.100.960.500.950.500.90Q0.000.51−0.510.01−0.500.500.53−0.530.010.01V0.620.750.500.450.400.800.600.800.650.70FH0.320.050.080.030.010.950.250.600.080.09FL0.350.850.100.800.950.020.250.100.060.02RH0.370.100.700.800.010.020.500.600.650.03RL.0.510.300.100.040.980.900.500.080.800.90Meanwhile, in the case of different predetermined cluster numbers for the subjective clustering methods, the AC algorithm determined the clustering number automatically, which was computed on the basis of the similarity between the samples. Here, the similarity was set at 95%, and the algorithm was implemented at the same time. Thus, the result was exactly consistent with that of the LSHk-means algorithm. The details are presented in Table 5. For example, the first and sixth samples, the second sample, and the third sample were clustered into the same category.Table 5
Clustering result of the AC algorithm.
ResultSQVFHFLRHRLC10.950.50.80.950.010.020.9C20.890.010.690.890.020.020.89C20.900.70.90.010.030.9C30.94−0.560.790.590.090.590.9C40.50.50.60.250.250.50.5C10.960.490.790.940.020.010.89……………………C90.09−0.490.390.010.940.010.97C90.1−0.50.400.9500.98C80.59−0.510.490.070.090.690.09The final classification results obtained from the model can provide the load grades and decision-making suggestions, which can serve as a basis for site planning optimization of the business halls. In addition, the results of the two algorithms were consistent, which indicate that the LSHk-means algorithm is effective and a stable result was obtained. Accordingly, further optimization action can be implemented.


## 4.2.2. Optimization Analysis

As shown in Figure5(a), the 16th and 17th business halls are both in Class I. This category indicates that the current load is Low, and the load trend remains unchanged, implying that the business hall resources are redundant in this area. The business halls in this class are idle, and the site may be unreasonable. In addition, the merger of business halls, relocation, and reduction of resource input in this area should be considered.Figure 5
Illustration of business hall location optimization. (a) Location of business halls in classA; (b) Suggested location for supplementary business hall.
(a)(b)By contrast, for ClassA (the red part of Figure 5(b)), it can be seen that the current load and load trend are both High, which implies that the business volumes of the business halls are large, and the load trend change is still on the rise. Currently, the first and sixth business halls belong to this category. The future trend is still likely to be growing, and the business volumes will keep increasing. Therefore, this area is where more business hall resources need to be input, and the optimal site planning of business halls should be considered accordingly. We can define the objective function of the optimal site for the input business hall resources as follows:(18)Objmin=∑k=1nFkn+∑i=lhidvi,x,where Fk is the quantitative value of factors that affect the rationality of business hall location, n is the number of factors, dvi,x is the distance function, hi is the weight, and x is the target point to be solved. Thus, according to the objective function and relevant coordinate information of key units in the area, we can compute the optimal planning location of the business hall using the optimization algorithm. Here, the optimal location of the business hall was computed as 100.0644,128.8199, and the optimal solution was 527.0368. The details are shown in Figure 5.The 9th and 12th business halls belong to the Class E, which shows that the current load and load trends are both normal, and the status is stable. Therefore, the business halls in this class are not the current focus of optimization. In addition, the other classes are similar to this category, which is also not the current focus. The main objects are Class I and Class A, that is, excessive or insufficient business hall resources are mainly concentrated in these two classes, which are the focus of our optimization analysis.


## 5. Conclusion

Excessive or insufficient business hall resources may result in unreasonable resource allocation, which adversely affects the value of an entity business hall. Therefore, proper characteristic parameters are the key factors for analyzing the business hall, which strongly affect the final analysis results. According to the time change and load trend, multiple variables such as average load rate, actual load trend, and high-frequency load are extracted as the characteristic indexes of the business hall. In this study, a characteristic analysis method for the economic operation of a business hall was developed, and the specific calculation process was presented; accordingly, the feature engineering was established. Moreover, based on the load intensity and time change information of business halls, we built an index system and performed further optimization analysis. The key characteristic indicators extracted were the average waiting time, ticket handling time, and business type, and a model for evaluating business hall efficiency was established. The model obtained the load grading of each business hall by the relevant variable input, which provided a basis for optimal site planning of the business halls.An empirical study showed that the LSH-k-means clustering method outperforms the direct prediction method, provides expected analysis results and decision optimization suggestions for business halls, and serves as a basis for the optimal layout of business halls. In addition, by considering the load intensity and time change information, the cluster number was determined according to the characteristic analysis results, with a certain theoretical and practical significance. In the future, we will explore and develop a general method to automatically determine the parameters and use it in practical applications.



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*Source: 7622576-2021-10-01.xml*