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.

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 N = { 1 , 2 , 3 , ⋯   } be the set of all natural numbers. Let I k = { 0 , 1 , 2 , ⋯   , k − 1 } and I k + = { 1 , 2 , 3 , ⋯   , k } for each k ∈ N . Let G = ( V , E , A ) be a connected simple mixed graph, where V is the set of vertices, E is the set of undirected edges and A is the set of arcs. Hereafter, the number of vertices in G is denoted as n and fixed to V = { 1,2 , … , n } . Here, we denote an undirected edge by { u , v } and an arc by ( u , v ) . The term “edge” refers to either an undirected edge or an arc, denoted by ⟨ u , v ⟩ . Thus, if ⟨ u , v ⟩ is an undirected edge, ⟨ u , v ⟩ = ⟨ v , u ⟩ ; if it is an arc, ⟨ v , u ⟩ ∉ A .

Let m ∞ be a sufficiently large positive integer. Then, let d be a function from V 2 to N satisfying the following conditions: for all u , v ∈ V

1. d ( u , v ) = d ( v , u ) ,

Here, d ( u , v ) denotes the distance between u and v if there exists an edge ⟨ u , v ⟩ (or ⟨ v , u ⟩ ). For convenience, d ( u , v ) = m ∞ when there is no edge between u and v .

Let H be a subset of E ∪ A . We consider that there exist important facilities on each edge in H that must be stopped at. Each edge in H is considered a high-security edge. A sequence s : v 0 , v 1 , v 2 , ⋯   , v k of vertices is considered a patrolling route on G if the following conditions hold:

1. The sequence s is a walk. That is, ⟨ v i , v i + 1 ⟩ ∈ E ∪ A for each i ∈ I k .

The length L ( s ) of a patrolling route s is the total sum of the distances of all edges on s and is calculated as follows: L s = ∑ i = 0 k − 1 d v i , v i + 1 .

For a patrolling route s , let V s = { v i ∈ V | i ∈ I k } and E s = { ⟨ v i , v i + 1 ⟩ ∈ E ∪ A | i ∈ I k } . Here, V s denotes the set of vertices on s , and E s denotes the set of edges traversed in s . Let ⟨ u , v ⟩ be an edge of G . If u ∈ V s or v ∈ V s , the edge is considered guarded. In particular, if ⟨ u , v ⟩ ∈ E s , said the edge is considered guarded by traversing; otherwise, if exactly one vertex of u and v is in V s , the edge is considered guarded by visual confirmation.

Let E s g = { ⟨ u , v ⟩ ∈ E ∪ A | u ∈ V s  or  v ∈ V s } be a set of edges guarded by a patrolling route s . Then, the total sum of the distances of all edges guarded by s is denoted as ∑ ⟨ u , v ⟩ ∈ E s g d ( u , v ) . The total sum of the distances of all edges of G is denoted as ∑ ⟨ u , v ⟩ ∈ E ∪ A d ( u , v ) ; thus, the covered ratio c o v ( s ) of G by s is defined as follows: c o v s = ∑ 〈 u , v 〉 ∈ E s g d u , v ∑ 〈 u , v 〉 ∈ E ∪ A d u , v − 1 .

Similarly, c o v ̄ ( s ) = 1 − c o v ( s ) is called the noncovered ratio of G by s . For any patrolling route s of G , 0 < c o v ( s ) ≤ 1 and 0 ≤ c o v ̄ ( s ) < 1 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.

Let v i ( i ∈ I k ) be a vertex on a patrolling route s : v 0 , v 1 , v 2 , ⋯   , v k of a mixed graph G = ( V , E , A ) . If ⟨ v i − 1 , v i ⟩ , ⟨ v i , v i + 1 ⟩ ∉ H and ⟨ v i − 1 , v i + 1 ⟩ ∈ E ∪ A , then the sequence s ′ : v 0 , v 1 , v 2 , ⋯   , v i − 1 , v i + 1 , ⋯   , v k − 1 , v k by removing v i from s is also a patrolling route of G (In the case i = 0 , s ′ : v 1 , v 2 , v 3 , ⋯   , v k − 1 , v 1 is a patrolling route if ⟨ v k − 1 , v k ⟩ , ⟨ v 0 , v 1 ⟩ ∉ H and ⟨ v k − 1 , v 1 ⟩ ∈ E ∪ A ). Some edges guarded by visual confirmation from v i on s may not be guarded on s ′ , although two edges removing from s are guarded by visual confirmation. That is, c o v ̄ ( s ) ≤ c o v ̄ ( s ′ ) holds. On the other hand, the increase or decrease in the length of the sequence v 0 , v 1 , v 2 , ⋯   , v k does not determine the increase or decrease in the length of the patrolling route. That is, L ( s ′ ) ≤ L ( s ) holds if the triangle inequality d ( v i − 1 , v i + 1 ) ≤ d ( v i − 1 , v i ) + d ( v i , v i + 1 ) ( d ( v k − 1 , v 1 ) ≤ d ( v k − 1 , v k ) + d ( v 0 , v 1 ) in the case i = 0 ) is satisfied, otherwise, L ( s ) ≤ L ( s ′ ) holds.

The GPOPP is a bi-objective optimization problem that obtains a patrolling route with the shortest length and the lowest noncovered ratio for a given connected simple mixed graph G = ( V , E , A ) , a set H ⊆ E ∪ A of high-security edges, and a distance function d : V 2 → Z + . The parameters n , c u , v , h u , v , and d u , v for the GPOPP are defined as follows:

(i) n : number of vertices.

If G has an undirected edge { u , v } , c u , v = c v , u = 1 ; if G has an arc ( u , v ) , c u , v = 1 and c v , u = 0 .

(iii) h u , v = 1 , if  ⟨ u , v ⟩ ∈ H  or  ⟨ v , u ⟩ ∈ H , 0 , otherwise .

Note that h u , v = h v , u = 1 even if ⟨ u , v ⟩ is an arc in H .

(iv) d u , v = d ( u , v ) for each ( u , v ) ∈ V 2 .

In other words, c u , v ′ = 1 when there is an edge that has endpoints u and v ; c u , v ′ = 0 when there is no edge between them in the mixed graph G . c u , v ′ is immediately obtained from c u , v .

The decision variables for the GPOPP are x u , v and y u , v . The decision variable x u , v represents the number of times the edge ⟨ u , v ⟩ is traversed from u to v . If there exists no edge between u and v , x u , v = 0 . The decision variable y u , v is expressed as follows:

(vi) y u , v = 1 2 , if  ∑ k = 1 n x u , k > 0  and  ∑ k = 1 n x v , k > 0 , 1 , if  ∑ k = 1 n x u , k > 0  and  ∑ k = 1 n x v , k = 0 , 0 , otherwise .

We remark that ∑ k = 1 n x u , k is strictly positive if and only if the vertex u is in V s . Thus, y u , v = y v , u = 1 2 when both u and v are on s ; y u , v = 1 and y v , u = 0 when u is on s even though v is not on s . Note that y u , v = y v , u = 1 2 does not mean that ⟨ u , v ⟩ or ⟨ v , u ⟩ are in E s . Then, ∑ u = 1 n ∑ v = 1 n c u , v ′ d u , v y u , v denotes the total length of edges guarded by the patrolling route s .

Let x = ( x 1,1 , x 1,2 , ⋯   , x n , n ) be an n 2 -tuple of nonnegative integers. The mathematical model for the GPOPP is formulated as follows: minimize f 1 x = ∑ u = 1 n ∑ v = 1 n d u , v x u , v , minimize f 2 x = 1 − ∑ u = 1 n ∑ v = 1 n c u , v ′ d u , v y u , v ∑ u = 1 n − 1 ∑ v = u + 1 n c u , v ′ d u , v − 1 .

The objective function f 1 is the function that minimizes the total length of the patrolling route, and f 2 is the function that minimizes the noncovered ratio.

The following constraints must be satisfied: x u , v ≥ 0 ∀ u , v ∈ V , (1) c u , v = 0 ⇒ x u , v = 0 ∀ u , v ∈ V , (2) h u , v = 1 ⇒ x u , v + x v , u > 0 ∀ u , v ∈ V , (3) ∑ v = 1 n x v , u = ∑ v = 1 n x u , v ∀ u ∈ V , (4) W ≠ ϕ ∧ W ≠ V x ⇒ ∑ u ∈ W ∑ v ∉ W x u , v > 0 ∀ W ⊆ V x , (5) where V x = u ∈ V ∑ v = 1 n x u , v > 0 is the same as V s defined in the previous section. Equation 1: The number of times to directly traverse from u to v is nonnegative. Equation 2: If there exists no edge between u and v or even if there exists an arc from v to u , it is not possible to directly traverse from u to v . Equation 3: The edge ⟨ u , v ⟩ (or ⟨ v , u ⟩ ) must be traversed if it is a high-security edge. Equation 4: For any vertex u , the number of times traversed from other vertices to u is the same as the number of times traversed from u to other vertices.

Suppose that x does not satisfy Equation 5. Then, there exists a nonempty proper subset W of V x that satisfies ∑ u ∈ W ∑ v ∉ W x u , v = 0 . This means that any edge ⟨ u , v ⟩ incident to u ∈ W and v ∈ V x − W is not traversed; therefore, x represents two or more separate walks. In other words, based on Equation 5, the patrolling route is a continuous closed walk.

The notation used in formulating the mathematical programming model for GPOPP is summarized as follows:

Parameters

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 N e x be the number of numerical experiments for each method. Let S  NSGA - II [ i ] , S  MoEA - HSS [ i ] and S  pMH [ i ] be the Pareto-optimal solution sets of NSGA-II, MoEA-HSS and pMH obtained by the i -th experiment, respectively. The reference solution set S * is defined as the set of nondominated solutions in the union of all Pareto-optimal solution sets ⋃ i ∈ I N e x + S  NSGA - II [ i ] ∪ S  MoEA - HSS [ i ] ∪ S  pMH [ i ] .

The three evaluation criteria we used are as follows:

1. Number of nondominated solutions. For each method α ∈ {NSGA-II, MoEA-HSS, pMH}, | S α [ i ] | is the number of nondominated solutions obtained by the i -th experiment for α . In general, the selection of the final solution is left to human judgment; thus, the number of Pareto-optimal solutions is an important evaluation criterion.