Hybrid heuristic approach for generalized police officer patrolling problem

Abstract

In urban areas with many commercial facilities, patrolling by police officers or security guards is essential for crime prevention, in addition to the use of surveillance cameras. To address the challenge of planning effective patrol routes, Tohyama and Tomisawa introduced the Police Officer Patrolling Problem (POPP), an arc routing problem that allows for visual monitoring from intersections and is proven to be NP-complete. Building on this work, we propose the Generalized POPP (GPOPP), a more realistic bi-objective combinatorial optimization model. This model simultaneously minimizes the total patrol route length and maximizes the coverage of surveillance areas. The contributions of this paper are threefold: (1) we formulate the GPOPP by incorporating practical constraints, such as mandatory patrolling of high-security roads and visibility-based coverage from intersections; (2) we develop a novel hybrid heuristic method that combines a multi-objective evolutionary algorithm (MoEA-HSS) with an improved Jaya algorithm to solve the GPOPP effectively; and (3) we conduct comprehensive computational experiments using benchmark instances to evaluate the effectiveness and competitiveness of the proposed method. These contributions demonstrate the practicality and efficiency of our approach for addressing realistic urban patrolling problems.

1 Introduction

In the fields of information engineering and science, to solve various social and economic problems, these problems are generally structured as mathematical models, and solutions are found using algorithms that are suited to that structure. Many of these problems are modeled using discrete graphs, and there are many studies on them.

One of the problems modeled by discrete graphs is the routing problem. Routing problems are classified as node routing problems (NRPs), which traverse the nodes of a graph, and arc routing problems (ARPs), which traverse the edges (or arcs). A typical NRP is the traveling salesperson problem (TSP). The TSP is a problem that involves finding the minimum-cost route that visits every vertex exactly once. The vehicle routing problem (VRP) (Dantzig and Ramser, 1959) is a generalization of the TSP. This problem involves planning transportation from a distribution center to multiple customers using trucks or other transportation methods. Both the TSP and VRP are NP-hard; thus, evolutionary algorithms, such as genetic algorithms (GAs), have been studied (Elatar et al., 2023).

The most famous ARP is probably the Euler circuit problem. This problem determines whether there exists a circuit that traverses all edges exactly once for a given graph, and this problem is solvable in polynomial time. The Chinese postman problem (CPP) (Mei-Ko, 1962), which is a generalization of the Euler circuit problem, involves determining whether a tour exists for a post officer in a given area within a given amount of time that starts and ends at the post office. The post officer must traverse every street in the area at least once; however, they may traverse any street several times. The CPP on undirected or directed graphs can be solved in polynomial time (Edmonds and Johnson, 1973). Papadimitriou (1976) showed that the CPP on mixed graphs is NP-complete. Mixed graphs represent realistic situations in urban areas with both two- and one-way streets. The rural postman problem (RPP) is a generalization of the CPP with a given set of edges that must be traversed by a post officer. This problem considers the fact that, in rural areas, not every street has a delivery destination. Lenstra and Rinnooy-Kan (1976) and Lenstra and Rinnooy-Kan (1981) showed that the optimization version of the RPP on undirected or directed graphs is NP-hard. The capacitated ARP (CARP) is an ARP corresponding to the VRP, which belongs to the NRP. The CPP, RPP, and CARP correspond to mathematical models of real social problems such as postal delivery, delivery planning, snow shoveling, and garbage collection. Finding exact solutions for the CPP and RPP optimization problems is intractable, along with the TSP and VRP; thus, various heuristic methods have been proposed for these problems. Recent examples include methods using GAs (Gil-Gala et al., 2023), the Tabu search algorithm (Tang et al., 2024), and ant colony optimization (Sgarro and Grilli, 2024).

Police patrols play a crucial role in preventing crimes and accidents, thereby ensuring public safety within their jurisdictions. Recent studies such as (Kim et al., 2023; Dewinter et al., 2020; Samanta et al., 2022) have proposed methods for optimizing patrol routes. These approaches primarily employ heuristic algorithms to generate efficient patrol routes for multiple officers operating within the shared area.

Recently, Tohyama and Tomisawa (2022) proposed the police officer patrol problem (POPP) as a mathematical model of the patrolling route problem of police officers (or security guards), and showed that the decision problem is NP-complete (Tohyama and Tomisawa, 2022). Patrolling areas generally include one- and two-way streets; thus, the POPP is modeled using a mixed graph. At each intersection, police officers may conduct security checks visually even if they do not traverse the streets connecting to it. If the POPP is considered a CPP model, it is necessary to find a patrolling route that traverses all streets. The POPP model allows some streets to conduct visual security checks without traversing, making it possible to find more efficient patrolling routes. In addition, Tomisawa and Tohyama showed that the POPP on weighted digraphs is NP-complete (Tomisawa and Tohyama, 2024).

In this study, we introduce the generalized POPP (GPOPP) as a model to adapt the POPP to more realistic patrolling routes by police officers. The POPP model requires that all areas be guarded. However, in reality, some streets require security because important facilities are located there, and some roads do not necessarily require security (Chainey et al., 2021). In addition, there are cases where a patrolling route needs to be found that can be patrolled within a given time. Therefore, we define the GPOPP as an optimization problem with the following two objectives. The first objective is to find the shortest patrolling route among the routes that traverse all high-security streets. The second objective is to find a patrolling route that guards as large a given area as possible (maximizes coverage).

Many GAs have been proposed to solve multi-objective problems (Deb et al., 2002; Sardinas et al., 2006; Pizzuti, 2009; Ghoseiri and Ghannadpour, 2010; Aiello et al., 2012; Akyurt et al., 2015; Yu et al., 2015; Lu et al., 2019). The hybrid sampling strategy based multi-objective evolutionary algorithm (MoEA-HSS) (Zhang et al., 2014) is based on a hybrid sampling strategy that combines a vector-valued GA (VEGA) (Schaffer, 2014) and a sampling strategy according to the Pareto dominating and dominated relationship-based fitness function (PDDR-FF) (a goodness-of-fit function based on Pareto dominance–dominance relations). The MoEA-HSS has demonstrated effectiveness for several problems. The Jaya algorithm (Rao, 2016) is a meta-heuristic algorithm with a very simple structure based on the concept that solutions obtained for a particular problem progress toward the best solution and avoid the worst solution. We propose a hybrid heuristic approach that combines the MoEA-HSS with an improved Jaya algorithm, and demonstrate its effectiveness through numerical experiments.

The remainder of this paper is organized as follows. In Section 2, we formally define the Generalized Police Officer Patrolling Problem (GPOPP) and present the necessary graph-theoretical concepts. Section 3 provides a mathematical formulation of the GPOPP as a bi-objective optimization problem. In Section 4, we describe the proposed hybrid heuristic method that combines the MoEA-HSS and an improved Jaya algorithm. Section 5 presents the results of the numerical experiments conducted to evaluate the performance of the proposed method. Finally, Section 6 concludes the paper and discusses potential directions for future research.

2 Generalized police officer patrolling problem (GPOPP)

In this study, we introduce a bi-objective problem that can be applied to more realistic problems based on the POPP, which is NP-complete edge routing decision problem, and propose a heuristic algorithm to solve the problem. One police officer (or security guard or robot) is assigned to a security area, and each officer patrols his/her assigned area. Each street through which a police officer is traversed during a patrol is considered guarded. In addition, except for streets with important facilities, police officers are allowed to visually confirm each street adjacent to an intersection without traversing it. The GPOPP is a bi-objective optimization problem with the following two objectives: One is to find the patrolling route with the shortest length, and the other is to find the route with the largest guarded area. Here, we note that all high-security streets must be traversed. In this section, we define the notion in graph theory necessary to formulate the GPOPP.

Throughout this paper, let be the set of all natural numbers. Let and for each . Let be a connected simple mixed graph, where is the set of vertices, is the set of undirected edges and is the set of arcs. Hereafter, the number of vertices in is denoted as and fixed to . Here, we denote an undirected edge by and an arc by . The term “edge” refers to either an undirected edge or an arc, denoted by . Thus, if is an undirected edge, ; if it is an arc, .

Here, denotes the distance between and if there exists an edge (or ). For convenience, when there is no edge between and .

The length of a patrolling route is the total sum of the distances of all edges on and is calculated as follows:

For a patrolling route , let and . Here, denotes the set of vertices on , and denotes the set of edges traversed in . Let be an edge of . If or , the edge is considered guarded. In particular, if , said the edge is considered guarded by traversing; otherwise, if exactly one vertex of and is in , the edge is considered guarded by visual confirmation.

Let be a set of edges guarded by a patrolling route . Then, the total sum of the distances of all edges guarded by is denoted as . The total sum of the distances of all edges of is denoted as ; thus, the covered ratio of by is defined as follows:

Similarly, is called the noncovered ratio of by . For any patrolling route of , and holds.

An example of a patrolling route for a mixed graph is illustrated in Figure 1. The red edges represent high-security edges, and the blue line indicates a patrolling route. The green area is guarded by this patrolling route. In particular, the green area not on the patrolling route is guarded by visual confirmation. The graph shown in Figure 1 has 60 vertices and 104 edges. Let us assume that the distance between any two vertices is one. Then, the length of the patrolling route is 44. The total sum of the distances of the guarded edges is 84, and the covered and noncovered ratios are 0.808 and 0.192, respectively.

FIGURE 1

Let be a vertex on a patrolling route : , , , , of a mixed graph . If , and , then the sequence : , , , , , , , , by removing from is also a patrolling route of (In the case , : , , , , , is a patrolling route if , and ). Some edges guarded by visual confirmation from on may not be guarded on , although two edges removing from are guarded by visual confirmation. That is, holds. On the other hand, the increase or decrease in the length of the sequence , , , , does not determine the increase or decrease in the length of the patrolling route. That is, holds if the triangle inequality ( in the case ) is satisfied, otherwise, holds.

3 Formulation

In other words, when there is an edge that has endpoints and ; when there is no edge between them in the mixed graph . is immediately obtained from .

We remark that is strictly positive if and only if the vertex is in . Thus, when both and are on ; and when is on even though is not on . Note that does not mean that or are in . Then, denotes the total length of edges guarded by the patrolling route .

Let be an -tuple of nonnegative integers. The mathematical model for the GPOPP is formulated as follows:

The objective function is the function that minimizes the total length of the patrolling route, and is the function that minimizes the noncovered ratio.

The following constraints must be satisfied:where is the same as defined in the previous section. Equation 1: The number of times to directly traverse from to is nonnegative. Equation 2: If there exists no edge between and or even if there exists an arc from to , it is not possible to directly traverse from to . Equation 3: The edge (or ) must be traversed if it is a high-security edge. Equation 4: For any vertex , the number of times traversed from other vertices to is the same as the number of times traversed from to other vertices.

Suppose that does not satisfy Equation 5. Then, there exists a nonempty proper subset of that satisfies . This means that any edge incident to and is not traversed; therefore, represents two or more separate walks. In other words, based on Equation 5, the patrolling route is a continuous closed walk.

4 Proposed methods

4.1 Framework

The GPOPP requires two conflicting objectives to be considered simultaneously: minimizing the total length of the patrolling route and minimizing its noncoverage ratio. Many Pareto-optimal solutions with incomparable qualities must be generated for the decision-maker. In this section, we introduce a hybrid heuristic approach based on the MoEA-HSS and the improved Jaya algorithm for the GPOPP.

The MoEA-HSS is based on a hybrid sampling strategy that combines the VEGA and a sampling strategy according to the PDDR-FF. The sampling strategy of the VEGA is a natural extension of simple GAs in the sense that the individuals are divided and reproduced independently according to each objective function. It prefers the edge region of the Pareto front with less time complexity, and the qualities of the solution are not good because of the selection bias. Conversely, the PDDR-FF-based sampling strategy tends to converge toward the central area of the Pareto front. The combination of these two mechanisms is expected to maintain both the convergence rate and distribution performance. The Jaya algorithm modifies a given individual to move closer to the best solution and away from the worst solution based on the best and worst candidates in the population. The Jaya algorithm is expected to accelerate the convergence rate.

The main framework of the proposed method is shown in Figure 2. The assemblage of chromosomes in each generation is divided into two groups and . and are called Archive and Population, respectively. The Pareto solution set update shown in the figure follows the same procedure as in the MoEA-HSS (Zhang et al., 2014) framework.

FIGURE 2

Let be a positive constant. Initially, archive is set to empty, and the initial population consists of individuals. For each individual in population, the objective functions and and the evaluation function are calculated. We adopt the PDDR-FF as the evaluation function . Let be the number of individuals that can be dominated by the individual and be the number of individuals that can dominate the individual . Then,

Based on these values, the Pareto-optimal solution in the population is obtained and maintained.

Phases 1 – 3 of the proposed method are depicted in Figure 3.

FIGURE 3

4.2 Chromosome representation and notation

The most natural route expression is adopted as the chromosome expression in the proposed method. For example, if a sequence : , , , , of vertices is a legal patrolling route, its chromosome representation is a -tuple . The first component of is considered the starting point. By rotating the genes in , the chromosome at which the starting point is replaced with is denoted by . That is,

4.3 Initial population

The pseudocode for the procedure generating the initial population consisting of individuals is presented in Figure 4.

FIGURE 4

Let , , , , , be a patrolling route, and assume that and (, , ). Then, the edge connecting and is undirected. If , the edge is traversed three times on , as shown in the left panel of Figure 5a, and there exists a patrolling route that reduces the total length, as shown in the right panel of Figure 5a. Such a patrolling route traverses all edges on the original route; thus, the noncoverage rate remains unchanged. Similarly, even if , the edge is traversed three times, as shown in the left panel of Figure 5b, and there exists a patrolling route that reduces the total length without changing the noncoverage rate, as shown in the right panel of Figure 5b. In general, assume that undirected edge is traversed from to times and from to times. If , there exists a patrolling route that reduces both the number of times traversing from to and from to by times without changing the noncoverage rate. If , there exists a patrolling route that reduces both the number of times traversing from to and from to by times without changing the noncoverage rate. The pseudocode for the improvement procedure of the chromosome is shown in Figure 6.

FIGURE 5

FIGURE 6

4.4 Crossover

FIGURE 7

The pseudocode for the proposed crossover operation is presented in Figure 8.

FIGURE 8

4.5 Mutation

In the crossover operation, the walk of parent individuals tends to be inherited by their offspring. Therefore, it is seldom that generated offspring will traverse vertices or edges that their parents do not. To maintain the diversity of the population, we introduce a mutation operation (Figure 9).

FIGURE 9

FIGURE 10

4.6 Local search strategy based on the Jaya algorithm

The crossover operation introduced in Section 4.4 is an order-based operation in which the orders of traversing the high-security edges of parents and other edges are inherited by offspring. To compensate for the ability to converge the solutions of this crossover, we introduce an improvement strategy that moves dominated solutions closer to a Pareto-optimal solution by combining the strategy used in the mutation operation with the Jaya algorithm.

As mention in Section 2, if the triangle inequality holds, removing a vertex from the patrolling route tends to reduce the noncovered ratio since edges may be no longer guarded by visual confirmation from , although the total length of the patrolling route becomes shorter. Conversely, adding a vertex that is not on the patrolling route increases the total length; however, this is expected to improve the noncovered ratio because edges guarded by visual confirmation may increase.

FIGURE 11

Remark that is a hypothetical solution that does not always exist; thus, the operation is repeated until approaching the neighborhood of (inside a circle of radius centered on ), or the operation is repeated until the terminal condition is reached. The pseudocode for the improved Jaya algorithm is presented in Figure 12.

FIGURE 12

5 Numerical experiments

In this section, we report the results of numerical experiments conducted to verify the effectiveness of the proposed method (the hybrid strategy of MoEA-HSS and the improved Jaya algorithm) for the GPOPP. In these experiments, we compared the proposed method (called pMH) with the NSGA-II (called NSGA-II) proposed by Deb et al. (2002) and the original MoEA-HSS without the improved Jaya algorithm (called MoEA-HSS). All these methods incorporated the crossover and mutation operations proposed in this paper.

Evaluating the performance of multi-objective optimization methods is not easy since it is difficult to compare the Pareto-optimal solutions (also called the nondominated solutions) obtained by different methods. We compared pMH with NSGA-II and MoEA-HSS using three representative comparison methods. Let be the number of numerical experiments for each method. Let , and be the Pareto-optimal solution sets of NSGA-II, MoEA-HSS and pMH obtained by the -th experiment, respectively. The reference solution set is defined as the set of nondominated solutions in the union of all Pareto-optimal solution sets .

The benchmark problems used in the experiments were created based on the Mixed Rural Postman Problem (MRPP) benchmark problems RB422 (, , ), RB452 (, , ), RB472 (, , ), RB522 (, , ), RB552 (, , ) and RB572 (, , ) shown in Arc Routing Problems: Data Instances (https://www.uv.es/corberan/instancias.htm). The number of high-security edges (called required edges) defined in the MRPP benchmark problems is sufficiently large. Therefore, they are not suitable for use as benchmarks for methods against the GPOPP in which visual confirmation is allowed. To confirm the effect of visual confirmation, in these experiments, we changed the required edges of each benchmark problem to non-required edges, creating four patterns of benchmark problems with 10, 20, 30, and 40 high-security edges. We used these edges in the experiments. In addition, the ratio of the number of high-security undirected edges to high-security arcs was set to .

We ran 10 numerical experiments for each method (that is, ). Comparison results of pMH with NSGA-II and MoEA-HSS at 10, 50, 150 and 300 generations are shown in Table 1. The number of nondominated solutions, generational distance and CPU time are shown in (a) and these are the maximum, minimum, average and standard deviation obtained by 10 experiments. Here, the for each generation is calculated using the Pareto-optimal solution sets at the -th generation and the reference solution set at the 300th generation. The comparison results for the coverage are shown in (b) and these are calculated by 100 values of . The evolution of the Pareto-optimal solution set at the 1st experiment for each method is shown in Figure 13.

FIGURE 13

The number of nondominated solutions for NSGA-II and pMH are no significant difference, although the value for MoEA-HSS is slightly lower. If the Pareto-optimal solution set contains even a single solution that is far from , the generational distance will have a large value. Therefore, method cannot be considered inferior simply because the generational distance has a large value. On the other hand, when the minimum value of the generational distance is sufficiently small, method is considered superior because the nondominated solutions obtained by method are likely to be close to the solutions in the reference solution set . For these reasons, it is more appropriate to use the minimum value, rather than the average or maximum, when conducting evaluations with the generational distance. Therefore, from these experimental results, in evaluations based on the number of nondominated solutions and the generational distance, pMH can be considered superior to other methods. In particular, the comparison between MoEA-HSS and pMH based on these criteria implies that the ability of local search for the improved Jaya algorithm has achieved the expected results.

The coverage is the rate that solutions in are covered by solutions in . Hence, if the average of is higher than the average of , can be considered as better than under this criterion. Therefore, by comparing the average of two coverages and ({NSGA-II, MoEA-HSS}), the Pareto-optimal solution set is considered statistically superior to obtained by other methods at every generations.

The experiment results of comparing pMH with NSGA-II and MoEA-HSS when we ran each method until 300 generations for all benchmark problems are shown in Tables 2, 3.

Our method pMH achieved better results than other methods in all benchmark problems with respect to the generational distance and the coverage of two sets, although pMH and NSGA-II are evenly-matched at the criterion the number of nondominated solutions. Here, even if pMH is inferior to that of other methods at the criteria the number of nondominated solutions, since it is considered that the ratio of solutions in covered by solutions in is high, is not necessarily inferior to or . Based on these numerical experiments, we confirmed that the local search based on the improved Jaya algorithm is effective and that pMH can generate superior Pareto-optimal solutions.

6 Conclusion

In this study, we defined a new arc routing bi-objective optimization problem (GPOPP) that models the patrol security of police officers (or security guards) based on the POPP and proposed a hybrid heuristic approach for the GPOPP. The proposed method combines the hybrid sampling strategy MoEA-HSS, which combines sampling strategies based on the VEGA and PDDR-FF, with a solution improvement strategy based on the improved Jaya algorithm. The solutions of the MoEA-HSS approach to the true Pareto differ in various directions because the VEGA-based sampling strategy has a preference for the edge region of the Pareto front and the PDDR-FF-based sampling strategy tends to converge toward the center area of the Pareto front. The proposed method (pMH) improves convergence by combining the MoEA-HSS with the improved Jaya algorithm-based local search method. The numerical experimental results demonstrate that the proposed method can obtain better solutions than the NSGA-II and the MoEA-HSS. The remaining challenge for us is to improve efficiency by reducing CPU time while maintaining high solution quality. In addition, extending GPOPP model will enable us to more accurately replicate the complex challenges of real-world urban policing. One extension would be to model situations where multiple officers work together, and we consider to adapt our method to address such extended problems.

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