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Learn more: PMC Disclaimer | PMC Copyright Notice Sci Rep . 2026 Mar 6;16:12417. doi: 10.1038/s41598-026-40670-9 Search in PMC Search in PubMed View in NLM Catalog Add to search Police UAV path planning method based on improved PSO using AFS and HJS Dongxing Wang Dongxing Wang 1 Department of Computer and Information Security Management, Fujian Police College, Fuzhou, 350007 China 2 School of Intelligence Science and Technology, University of Science and Technology Beijing, Fuzhou, 100083 China Find articles by Dongxing Wang 1, 2, ✉ , Xiaoxiao Qian Xiaoxiao Qian 1 Department of Computer and Information Security Management, Fujian Police College, Fuzhou, 350007 China Find articles by Xiaoxiao Qian 1 Author information Article notes Copyright and License information 1 Department of Computer and Information Security Management, Fujian Police College, Fuzhou, 350007 China 2 School of Intelligence Science and Technology, University of Science and Technology Beijing, Fuzhou, 100083 China ✉ Corresponding author. Received 2025 Dec 10; Accepted 2026 Feb 16; Collection date 2026. © The Author(s) 2026 Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/ . PMC Copyright notice PMCID: PMC13083867 PMID: 41792264 Abstract Police UAV plays an increasingly important role in improving new quality combat capability of public security organs. To solve the path planning problem of police UAV (PU3P), this paper proposed a police UAV path planning method based on an improved Particle Swarm Optimization (PSO) using AFS and HJS. Firstly, a specific Adaptive Factor Strategy (AFS) was introduced into original PSO in order to strengthen overall exploration capability while Half Jumping Strategy (HJS) was adopted to enhance it local exploitation level, thus an improved PSO(AFS-HJS-PSO) using AFS and HJS was proposed. Secondly, a police UAV path planning method (AFS-HJS-PSO-PU3P) was proposed by utilizing AFS-HJS-PSO to optimize police UAV path planning fitness function (UPPF) abstracted from PU3P,devoted to get the global best solution for police UAV path planning. Finally, comparative experiments among PSO, Generic Algorithm (GA),Simulated Annealing Algorithm (SAA), a Nonlinear Programming Solver: FMINCON, AFS-PSO and AFS-HJS-PSO were independently conducted 30 times on 20 classical benchmark functions as well as the ones among PSO-PU3P,GA-PU3P,SAA-PU3P,AFS-PSO-PU3P and AFS-HJS-PSO-PU3P were conducted 30 times on UPPF, theoretical research and experimental results indicate that new proposed AFS-HJS-PSO is superior to PSO, GA, SAA, FMINCON in optimization accuracy, convergence speed and stability as well as AFS-HJS-PSO-PU3P can better solve PU3P than other compared methods. Keywords: Police UAV, Path planning, PSO, Adaptive factor, Half jumping Subject terms: Engineering, Mathematics and computing Introduction Police UAV is a newly emerging type of police tools while its importance in public security organs has become increasingly prominent, especially as an aerial operational machine 1 , police UAV has obvious operational advantages in complex urban environments while police UAV faced with numerous difficulties in path planning for searching multiple targets 2 . Many existing UAV path planning algorithms have been already applied in many fields,such as Wang Xiaodong et al. proposed a UAV data collection trajectory planning algorithm based on the nearest neighbor algorithm which greatly reducing data collection time, however, node distribution is extremely uneven and resulting in wasted energy and time 3 ;Li Yanru et al. adopted a DON-based deep reinforcement learning method for intelligent UAV flight route planning 4 ; Yang Yisen et al. used the ant colony algorithm to optimize UAV emergency rescue routes, but this algorithm heavily depends on the initial parameter settings and has low computational efficiency 5 ; Li Mingming et al. proposed an improved Tuna Optimization Algorithm 6 ; Zhou Cheng et al. used simulated annealing to plan multi-UAV network perimeter patrol paths 7 ; Gao Yaowen et al. proposed a bat algorithm integrating genetic factors, but experiments were conducted in a fixed steady-state environment, so its adaptability to dynamic conditions is questionable 8 ; Researchers adopted an improved A* algorithm to solve the UAV path planning problem under multiple types of no-fly zones 9 . Although these algorithms can partially address UAV path planning problems, each of them has its own limitations and cannot perfectly meet the path planning requirements of police drones in dealing with complex situations. For decades, due to the simple structure and high search efficiency, Particle Swarm Optimization (PSO) algorithm has been widely applied in various fields, such as Peng Yanli et al. constructed a PSO-GA model to predict leakage locations in water supply pipes while achieving synchronous detection of leakage location and leakage degree with high practicality 10 ; Zeng Chao et al. proposed a vehicle access and retrieval scheduling optimization model based on the PSO-OBL algorithm and opposition-based learning mechanism to precisely regulate vehicle management strategies and timing 11 . Gan Lintang et al. proposed a coal rock drillability prediction method based on PCA-PSO-SVM, using particlde swarm optimization and support vector machines to reasonably set prediction parameters 12 ; Zhao Quansheng et al. used the PSO-BP method to predict carbon emissions 13 ; Yuan Hongjun et al. proposed a VMD-PSO-LSTM multi-scale combined prediction model 14 ; Shi Xiufeng et al. used a sliding window to process raw data and converted it into an input matrix for the PSO-ELM model 15 ; Gao Ning et al. established a detection model based on the PSO-SVR algorithm 16 ; APSO 17 greatly enhances the performance by avoiding premature convergence and the tendency to fall into local optima. Unfortunately, various improved PSO lack in-depth research on discrete problems 18 – 21 , such as in 22 – 24 . Generally, both the existing problems in path planning problem of police UAV (PU3P) and the inherent advantages of PSO inspired us to consider employing a further improved PSO to solve the UAV path planning problem for police drones. This paper is organized as follows. Chapter 1 is about “Introduction”; Chapter 2 shows “Related works” about the new proposed method; Chapter 3 illustrates innovations about the new proposed method; Chapter 4 shows “Experiments and analysis”; Chapter 5 is about “Conclusion”. Related works Particle swarm optimization Inspired by the foraging behavior of birds in nature, PSO is designed as an optimization algorithm by starting from a group of random solutions and iteratively identifies the global optimal solution, which includes main the following steps, shown in Fig. 1 . Fig. 1. Open in a new tab Flowchart about original PSO. Step 1 Algorithm parameter initialization: The following key parameters need to be set before the PSO begins,such as Number of particles: N;Problem Dimension: D;Maximum number of iterations: maxIter;Inertia weight: w;Individual learning factor: c₁;Social learning factor: c₂;Speed range: [v_min, v_max];Position boundaries:[minBounds, maxBounds] determined based on the specific optimization function. Step 2 Particle swarm initialization: Parameters initialized for each particle and the detailed contents are shown following. The initialization of Position and Velocity about particle swarm can be got by related equations as well as individual optimal position and fitness. Step 3 Speed and position update: Accoring to the original PSO, the speed and position update for each particle follows related equations. Step 4 Fitness calculation and optimal update: In this progress, one main task is to update the personal best fitness and best position according to the relationship between the personal best fitness and the fitness of current particle in current iteration. Step 5 Iterative loop and termination conditions: The iteration condition will be judged at the end of each iteration, if j is not bigger than T (j means the order of current iteration) ,gBestFit has not yet converged to the theoretical optimal value or continuous iterations with significant improvement about gBestFit, then repeat Steps 3 and Step 4 until any of them is met. Finally, the final global optimal solution gBest and its corresponding optimal value gBestVal should be recorded as the result of the optimization process. Comparison algorithm Genetic Algorithms(GA) are population-based optimization algorithms inspired by the mechanisms of biological evolution, known for their excellent global search capability and high robustness, and in this paper GA will be implemented by the related tools in Matalab 2024a. Simulated Annealing Algorithms (SAA) is a population optimization algorithm inspired by the metal annealing process in physics, known for its excellent search capability and simplicity, which adoptes a ‘temperature’ parameter to control the randomness of the search process, and in this paper SAA will be implemented by a tool in Matalab 2024a. Fmincon function (FMINCON) is primarily utilized for solving constrained nonlinear multivariable function minimization problems and can handle optimization problems where the objective function or constraints involve nonlinear functions, which supports various algorithms including the interior-point method, trust-region-reflective algorithm, SQP algorithm and active-set algorithm. As same as GA and SAA,FMINCON will be implemented by calling related functions in tools from matalab2024a. Improvement Adaptive facotor strategy According to the original PSO, inertia weight ω is a fixed number upon the past experience, shown in Fig. 2 a and Eq. ( 1 ). Obviously, it’s insufficient to reflect the constantly changing situation to deal with complex optimization problems and bound to degrade the whole performance of optimization. 1 where, ω means inertia weight; fixed_number means the fixed numerical value in optimization process, which is usually greater than 0. Fig. 2. Open in a new tab Schematic diagram of inertia weight in different strategy ( a );( b );( c ). In recent years, the concept of adaptive size is widely used, such as: And in 25 – 28 , adaptive step-size is adopted to improve the performances of the related algorithms in different fields. Accordingly, the thought of adaptive size inspired authors to introduce a new adaptive factor into PSO to enhance its performance, thus, AFS is proposed to improve the performance in optimization process, which is shown in Eq. ( 2 ) following. 2 where, factor means the adaptive factor in optimization process; iter represents the current iteration count while maxIter expresses the max iteration count; ω means inertia weight. And Fig. 2 b shows the schematic diagram of adaptive factor while Fig. 2 c shows the one about inertia weight in AFS. Half jumping strategy During the iterative process, to find the global optimal around the global best position, particles are hoped to be closed to the global best one as soon as possible while the distance between them is neither too small to enable particle swarm fall into local optimum, nor too large so that the speed of convergence is too slow,the way to update the location is shown in Eqs. ( 3 ) and ( 4 ) above. 3 4 where, p(i) means the i-th particle and p(i).vel means the velocity of i-th particle;ω represents the inertia weight; rand() is a function that returns a random decimal between 0 and 1; p(i).pBest stands for the best position in its own history while p(i).pos means the current one of i-th particle; c1 is individual learning factor of i-th particle and c2 is the social one; gbest means the current best position of the particle swarm; max() is a function that returns the bigger one from the given parameters and min() returns the smaller one on the contrary, both of them are to limit speed range; vMin,vMax are respectively the lower limit of boundary and upper one about velocity as well as xMin and xMax are respectively the lower and upper limit about position; N means the number of particle swarm. 5 6 However, according to the existing original PSO, each particle seeks to find the global optimum based on its own velocity, current position, optimal position record, current global optimal position of the particle swarm as well as fixed inertia weights,individual and social learning factor;this searching method is somewhat rigid and cannot meet the needs of quickly approaching the vicinity of the current optimal position of the particle swarm and conducting a search. So Half Jumping Strategy(HJS) was proposed to make up for this deficiency,detailed in Eqs. ( 5 ) and ( 6 ) as well as Fig. 3 . Fig. 3. Open in a new tab Schematic Diagram of Searching Strategy. (1) Searching Strategy in Original PSO; (2) Searching Strategy about HJS. Flowchart about methods In this section,two flowcharts are designed about AFS-HJS-PSO and AFS-HJS-PSO-PU3P, which are separately to solve continuous issue and discrete issue. Flowchart of AFS-HJS-PSO Step 1 Algorithm parameter initialization: The following key parameters need to be set before the PSO begins,Number of particles: N = 50;Problem Dimension: D = 2;Maximum number of iterations: maxIter = 300;Inertia weight: ω = 0.72;Individual learning factor: c₁ = 1.49;Social learning factor: c₂ = 1.49;Speed range: [ v_min , v_max ] = [-0.1, 0.1];Position boundaries: [ minBounds, maxBounds ] determined based on the specific optimization function. Step 2 Particle swarm initialization: Parameters initialized for each particle and the detailed contents are shown following. The initialization of Position and Velocity about particle swarm can be got by Eq. ( 7 ) . 7 where, p(i).pos means the position about i-th particle while p(i).vel means its velocity; xMin and xMax are respectively the lower and upper limit about the position of particle swarm while vMin , vMax are respectively the lower limit and upper bourndary about velocity; rand() is a function that returns a random decimal between 0 and 1. p(i).fit is the fitness of the i-th particle, and get_fitness() is a function that returns the fitness upon the given parameters. Individual optimal position and fitness can be got by Eq. ( 8 ) as well as global optimal position and fitness are calculated by Eq. ( 9 ) 8 9 where, p(i).bestPos means the best position of i-th particle while p(i).bestFit means its best fitness accordingly on its own history record; p(i).pos(0) and p(i).fit(0) are respectively the position and corresponding fitness of the i-th particle in the first iteration; gBest means the global best position of the whole particle swarm while gBestFit means the corresponding global best fitness; min{} is a function that returns the minimum one according to the given parameters while argmin() returns the correponding position. Step 3 Speed and position update: Accoring to the original PSO, the speed and position update for each particle follows Eqs. ( 3 ) and ( 4 ) above,but in this paper the same process follows Eqs. ( 5 ) and ( 6 ), and the detailed was shown in chapter 3.2. Step 4 Fitness calculation and optimal update: 10 where, p(i).fit(j) and p(i).pos(j) means the fitness and position of i-th particle in j-th iteration respectively; p(i).bestFit means the best fitness of i-th particle on its own history record while p(i).bestPos means the corresponding position; i is the order of i-th particle and N is the amount of particle swarm; j is the order of current iteration and T is the maximum number about iterations. 11 where, gBestFit and gBest mean the global best fitness and position in j-th iteration respectively; other variables are same as described above. In this progress, one main task is to update the personal best fitness and best position according to the relationship between the personal best fitness and the fitness of current particle in current iteration, following Eq. ( 10 ); similarly, update the global best fitness of particle swarm is described in Eq. ( 11 ). Step 5 Iterative loop and termination conditions: The iteration condition will be judged at the end of each iteration, if j is not bigger than T ( j means the order of current iteration), gBestFit has not yet converged to the theoretical optimal value or continuous iterations with significant improvement about gBestFit , then repeat Steps 3 and Step 4 until any of them is met. Finally, the final global optimal solution gBest and its corresponding optimal value gBestVal should be recorded as the result of the optimization process. Figure 4 gives the flowchart about AFS-HJS-PSO. Fig. 4. Open in a new tab Flowchart of AFS-HJS-PSO. Flowchart of AFS-HJS-PSO-PU3P To adopt new methods to get the global best solution for police UAV path planning, Police UAV path planning fitness function (UPPF) was abstracted from the path planning problem of police UAV, and Eq. ( 12 ) was followed. 12 where, target_locations is the given target locations by specific tasks; And order is an arbitrary sequence regarding the order of target locations; Fitness is a function that returns the total distance calculated by target_locations and order;total_distance is a parameter that means the total distance returned from the function Fitness. Step 1 Algorithm parameter initialization: This process is completely same as the Step 1 in Chapter 3.3.1(Flowchart of AFS-HJS-PSO). Step 2 Particle swarm initialization: 13 where, len_locations means the length of target locations; length is a function that returns the length of the given parameters; particles(i).position shows the position of i- th particle and particles(i).velocity means the velocity of the same particle; randperm(len_locations) is a function that returns an unordered sequence from 1 to len_locations;rand() is a fucntion that return a random decimals between 0 and 1. Since UPPF is a discrete problem, for any particle, its position should be a sequence about target locations as well as its velocity should be some a degree of sequence variation, and the related parameters can be got by Eq. ( 13 ). Additionally, individual optimal position and fitness can be got by Eq. ( 8 ) as well as global optimal position and fitness are calculated by Eq. ( 9 ), just as the Step2 in Chapter 3.3.1(Flowchart of AFS-HJS-PSO). Step 3 Speed and position update: UPPF is a discrete problem, so any velocity should be an integer for any particle. According to Eq. ( 5 ), if current_order ∈ first_index, then Eq. ( 14 ) and ( 15 ) will be conducted, otherwise, if current_order ∈ second_index, Eq. ( 16 ) should be conducted. 14 15 16 where, ω means the inertia weight, c1 is the individual learning factor while c2 is the social learning one; particles(i).velocity means the velocity of i-th particle and ceil() is a function that returns the upward rounded value about the given number;rand() is a function that returns a random number between 0 and 1; v1 is the current velocity about the current particle, v2 stands for the velocity in the direction of individual learning while v3 stands for the velocity in the direction of social learning;pos1,pos2 and pos3 respectively mean the location based on v1,v2 and v3;datasample() is a function returns k observations sampled uniformly at random,with no replacement ;particles(i).position means the position of i-th particle; min_fitness() is a function that returns the minimum one according to the given parameters. It should be noted that in order to solve discrete problems (referred to as UPPF in this article), the classic steps in PSO are transformed into various formulas in this step while maintaining their basic meaning, which is also a highlight of this article. Step 4 Fitness calculation and optimal update: This process is completely same as the Step 4 in Chapter 3.3.1(Flowchart of AFS-HJS-PSO). Step 5 Iterative loop and termination conditions: This process is completely same as the Step 5 in Chapter 3.3.1(Flowchart of AFS-HJS-PSO). Figure 5 shows the flowchart about AFS-HJS-PSO-PU3P. Fig. 5. Open in a new tab Flowchart of AFS-HJS-PSO-PU3P. Experiments and analysis Experimental designment In this paper, all verified experiments are conducted on a computer equipped with Windows 11 operating system, 12th Gen Intel(R) Core (TM) i7-1260P (2.10 GHz) processor and 40G memory as well as Matlab 2024a, which is an integrated development environment with M programming language. PSO experiments were implemented according to the thought regarding original PSO; GA was implemented by the toolbox in Matlab 2024a as well as SAA and FMINCON regarded as a Nonlinear-Programming-Solver were implemented by the same platform,while imporved PSOs were implemented according to the theory illustrated above. To prove the good performance of the proposed algorithm, experiments were implemented for 30 times on any benchmark function for any optimization algorithm mentioned above, and Table 1 shows the parameters about comparative algorithms while Table 2 gives the 20 classical benchmark functions. Table 1. Configuration of algorithms. Order Algorithm name Configuration 1 PSO Number of particles is set at 50; the dimension of issue is set at 2; maximum number of iterations is set at 300; inertial weight is set at 0.72; individual learning factor is set at 1.49; social learning factor is set at 1.49; speed limit is set at 0.1; lower speed limit is set at -0.1; 2 GA The crossover probability is set at 0.7; the mutation probability is set at 0.01, while the generation gap is set at 0.95 3 SAA Default parameters about “SIMULANNEALBND” from the MATLAB R2024a toolbox 4 FMINCON Default parameters of FMINCON from the MATLAB R2024a toolbox 5 AFS-PSO Number of particles is set at 50; the dimension of issue is set at 2; maximum number of iterations is set at 300; inertial weight is set at 0.72; individual learning factor is set at 1.49; social learning factor is set at 1.49; speed limit is set at 0.1; lower speed limit is set at -0.1; 6 AFS-HJS-PSO Number of particles is set at 50; the dimension of issue is set at 2; maximum number of iterations is set at 300; inertial weight is set at 0.72; individual learning factor is set at 1.49; social learning factor is set at 1.49; speed limit is set at 0.1; lower speed limit is set at -0.1; Open in a new tab Table 2. Test functions. Order Function Expression Dimension Range Optimum 1 Bukin6 2 [− 15,3] Min f = 0 2 Drop-Wave 2 [− 5.12,5.12] Min f = -1 3 Eggholder 2 [− 512,512] Min f = -959.6407 4 Levy13 2 [− 10,10] Min f = 0 5 Schaffer2 2 [− 100,100] Min f = 0 6 Bohachevsky1 2 [− 100,100] Min f = 0 7 Booth 2 [− 10,10] Min f = 0 8 Matyas 2 [− 10,10] Min f = 0 9 Easom 2 [− 4,4] Min f = -1 10 Eggcrate 2 [− π, π] Min f = 0 11 Bohachevsky3 2 [− 100,100] Min f = 0 12 Perm0-d-β 2 [− 2,2] Min f = 0 13 Rotated Hyper-Ellipsoid 2 [− 65.536,65.536] Min f = 0 14 Sum Squares 2 [− 10,10] Min f = 0 15 Trid 2 [− 4,4] Min f = -2 16 Zakharov 2 [− 5,10] Min f = 0 17 Ackley 2 [− 32.768,32.768] Min f = 0 18 Rastrigin 2 [− 5.12,5.12] Min f = 0 19 Griewank 2 [− 600,600] Min f = 0 20 Levy 2 [− 10,10] Min f = 0 Open in a new tab Experimental results Table 3 shows the raw experimental results, which shows clearly the statistical data about “BestFitness”,”WorstFitness”,”MeanFitness”,”Standard Deviation” and ”Mean Time” of 30 independent experiments on 20 classical benchmark functions, it is the basis for subsequent analysis. Table 3. Experimental results. Function Order BestFitness WorstFitness MeanFitness Standard Deviation Mean Time F1 PSO 0.015883 0.12364 0.062946 0.00083942 0.12347 GA 0.036092 0.17391 0.09929 0.0015545 0.074698 SAA 0.036092 0.17391 0.09929 0.0015545 0.074698 FMINCON 0.12364 0.12364 0.12364 3.1877e-33 0.054972 AFS-PSO 0.0041034 0.1167 0.053462 0.0011091 0.15463 AFS-HJS-PSO 0.00090704 0.10455 0.044226 0.00092449 0.12026 F2 PSO − 1 − 0.93625 − 0.99787 0.00013549 0.21223 GA − 1 − 0.78575 − 0.93259 0.0019686 0.028688 SAA − 0.99996 − 0.85101 − 0.93017 0.00094969 0.092959 FMINCON − 0.93625 − 0.93625 − 0.93625 4.5904e-31 0.011201 AFS-PSO − 1 − 0.93625 − 0.9915 0.00048589 0.125 AFS-HJS-PSO − 1 − 0.93625 − 0.9915 0.00048587 0.11877 F3 PSO − 913.2351 − 482.0918 − 706.203 11,978.1499 0.12305 GA − 959.6407 − 716.6715 − 865.7628 7343.8133 0.041981 SAA − 559.7869 − 126.4075 − 294.9513 13,459.9124 0.10484 FMINCON − 66.8437 − 66.8437 − 66.8437 2.0891e-28 0.0084498 AFS-PSO − 931.0532 − 516.7351 − 722.5632 13,472.4453 0.12237 AFS-HJS-PSO − 913.6888 − 514.8785 − 740.1626 13,809.6619 0.11815 F4 PSO 1.3498e-31 7.1096e-28 4.6141e-29 1.7417e-56 0.12156 GA 2.1347e-11 0.10987 0.025637 0.0022341 0.032229 SAA 3.947e-07 0.10988 0.0057282 0.00045211 0.093961 FMINCON 6.0541e-15 6.0541e-15 6.0541e-15 1.03e-59 0.009462 AFS-PSO 1.3498e-31 1.3498e-31 1.3498e-31 4.4634e-93 0.12121 AFS-HJS-PSO 1.3498e-31 0.000252 8.4704e-06 2.1158e-09 0.1181 F5 PSO 6.4599e-08 0.1676 0.037837 0.0020296 0.12607 GA 1.8652e-14 0.29141 0.032502 0.0042457 0.032508 SAA 0.007905 0.007905 0.007905 0 0.045412 FMINCON 1.4655e-14 1.4655e-14 1.4655e-14 0 0.0066063 AFS-PSO 5.5052e-05 0.31513 0.043908 0.0043974 0.12164 AFS-HJS-PSO 0 0.0013995 8.4688e-05 8.955e-08 0.12189 F6 PSO 2.1372e-13 121.098 14.3148 861.5645 0.12796 GA 6.2674e-12 0.41293 0.013764 0.0056836 0.032524 SAA 3.7097e-09 0.41293 0.02046 0.0061946 0.08588 FMINCON 0.88281 0.88281 0.88281 1.1476e-31 0.010255 AFS-PSO 8.3028e-08 330.0092 62.9698 6571.0692 0.1242 AFS-HJS-PSO 0 2.2204e-16 1.4803e-17 3.1736e-33 0.11901 F7 PSO 0 1.0255e-29 7.0997e-31 3.9205e-60 0.12738 GA 2.4487e-11 4.8172e-09 1.3861e-09 1.508e-18 0.031931 SAA 7.1242e-09 0.00062 6.113e-05 2.2496e-08 0.082472 FMINCON 9.1639e-15 9.1639e-15 9.1639e-15 2.3175e-59 0.037216 AFS-PSO 0 0 0 0 0.12054 AFS-HJS-PSO 0 0 0 0 0.11789 F8 PSO 9.9309e-36 1.1351e-29 7.4405e-31 5.0515e-60 0.12427 GA 4.5895e-12 4.3782e-06 4.7961e-07 1.0181e-12 0.032459 SAA 4.7623e-07 0.027519 0.0024119 3.8888e-05 0.082087 FMINCON 4.8861e-19 4.8861e-19 4.8861e-19 1.5348e-67 0.076912 AFS-PSO 1.3734e-226 1.0032e-173 3.3441e-175 0 0.12371 AFS-HJS-PSO 1.786e-241 4.9065e-13 1.6355e-14 8.0245e-27 0.12214 F9 PSO − 1 − 1 − 1 0 0.12174 GA − 1 − 1 − 1 1.789e-19 0.030037 SAA − 0.99994 − 0.97977 − 0.99719 2.4755e-05 0.081593 FMINCON − 1 − 1 − 1 4.5904e-31 0.050807 AFS- PSO − 1 − 1 − 1 5.3299e-31 0.12119 AFS-HJS-PSO − 1 − 0.95857 − 0.99843 5.7498e-05 0.11909 F10 PSO 3.7019e-36 2.8529e-31 1.2118e-32 2.6811e-63 0.12267 GA 1.1253e-11 2.9037e-08 4.4956e-09 3.8244e-17 0.031575 SAA 4.8674e-10 0.20038 0.008734 0.0014353 0.085877 FMINCON 18.9764 18.9764 18.9764 0 0.03739 AFS-PSO 8.0646e-264 1.111e-233 3.7032e-235 0 0.12682 AFS-HJS-PSO 4.3286e-247 7.3263e-83 2.4421e-84 1.7892e-16 0.1243 F11 PSO 2.3169e-12 284.9789 14.0239 2831.0237 0.12547 GA 9.4974e-13 0.22626 0.015084 0.0032952 0.034419 SAA 4.1803e-07 0.0052834 0.00053261 1.2905e-06 0.082871 FMINCON 0.22626 0.22626 0.22626 7.1724e-33 0.038509 AFS-PSO 0.00074109 163.7141 19.1591 1215.8384 0.12534 AFS-HJS-PSO 0 0.00060962 2.0321e-05 1.2388e-08 0.11961 F12 PSO 0 0 0 0 0.12322 GA 1.1132e-11 5.113e-07 7.362e-08 1.2666e-14 0.037768 SAA 3.0393e-08 0.00048366 7.1862e-05 1.6968e-08 0.076877 FMINCON 7.1861e-14 7.1861e-14 7.1861e-14 1.648e-58 0.04748 AFS-PSO 0 7.8886e-31 1.0518e-31 7.439e-62 0.12254 AFS-HJS-PSO 0 2.2877e-29 8.1516e-31 1.7402e-59 0.12098 F13 PSO 1.3265e-22 1.0606 0.035352 0.037492 0.12444 GA 3.1469e-13 6.2168e-09 5.9576e-10 1.7174e-18 0.030699 SAA 7.314e-11 0.00023359 2.1423e-05 2.8631e-09 0.079632 FMINCON 9.6891e-14 9.6891e-14 9.6891e-14 4.1201e-57 0.0309 AFS-PSO 7.8915e-37 114.14 4.0125 433.6988 0.12532 AFS-HJS-PSO 8.7293e-236 4.3212e-116 1.4404e-117 6.2243e-23 0.12483 F14 PSO 8.9494e-36 1.229e-29 1.5576e-30 7.907e-60 0.12219 GA 2.0965e-13 2.051e-09 3.1987e-10 2.2882e-19 0.030852 SAA 4.3145e-09 0.000312 2.173e-05 3.9771e-09 0.090102 FMINCON 1.3211e-13 1.3211e-13 1.3211e-13 6.5921e-58 0.037891 AFS-PSO 1.9188e-227 2.7798e-211 1.0211e-212 0 0.12512 AFS-HJS-PSO 8.8448e-245 6.4543e-141 2.1514e-142 1.3886e-22 0.12543 F15 PSO − 2 − 2 − 2 0 0.12098 GA − 2 − 2 − 2 3.422e-19 0.028679 SAA − 2 − 2 − 2 0 0.0456 FMINCON − 2 − 2 − 2 0 0.039712 AFS-PSO − 2 − 2 − 2 0 0.12069 AFS-HJS-PSO − 2 − 2 − 2 0 0.11948 F16 PSO 1.5576e-36 1.4297e-30 1.3428e-31 8.2027e-62 0.12456 GA 6.2532e-14 3.0234e-09 3.5921e-10 4.1092e-19 0.030842 SAA 5.054e-08 0.00046064 3.6011e-05 9.991e-09 0.08062 FMINCON 1.4772e-16 1.4772e-16 1.4772e-16 6.2867e-64 0.014619 AFS-PSO 1.7977e-237 4.7443e-213 1.5814e-214 0 0.12804 AFS-HJS-PSO 1.3121e-247 3.008e-121 1.0027e-122 3.0161e-23 0.12846 F17 PSO 4.1642e-12 1.6996e-05 5.7615e-07 9.6184e-12 0.12409 GA 6.4427e-06 3.5745 0.20517 0.62655 0.039527 SAA 4.4203e-07 3.451 0.28712 0.78467 0.091572 FMINCON 1.3863e-10 1.3863e-10 1.3863e-10 0 0.020335 AFS-PSO 4.4409e-16 3.3044e-06 1.1073e-07 3.6385e-13 0.12463 AFS-HJS-PSO 4.4409e-16 3.9968e-15 7.9936e-16 1.1751e-30 0.12081 F18 PSO 0 0 0 0 0.12298 GA 7.4337e-11 1.9899 0.33165 0.43239 0.034428 SAA 1.9417e-08 1.9899 0.66332 0.43239 0.076417 FMINCON 0 0 0 0 0.0083913 AFS-PSO 0 0.0084349 0.0002934 2.3656e-06 0.12232 AFS-HJS-PSO 3.5018e-08 0.45754 0.057812 0.013846 0.1211 F19 PSO 0.082538 11.3609 2.4465 6.2492 0.12691 GA 7.6983e-13 0.1849 0.0096981 0.0012882 0.030379 SAA 0.0074837 0.50511 0.13038 0.014751 0.075685 FMINCON 0.007396 0.007396 0.007396 0 0.007551 AFS-PSO 0.15025 7.5927 2.3704 3.2055 0.12719 AFS-HJS-PSO 0.00025306 0.051526 0.011402 0.00010198 0.12418 F20 PSO 1.4998e-32 1.4363e-29 1.2037e-30 9.1077e-60 0.14432 GA 5.3499e-15 1.3128e-09 1.6048e-10 7.9638e-20 0.034806 SAA 2.4449e-06 0.011461 0.0012746 7.6196e-06 0.09 FMINCON 6.0495e-16 6.0495e-16 6.0495e-16 2.5147e-61 0.0076035 AFS-PSO 1.4998e-32 1.4998e-32 1.4998e-32 1.2398e-94 0.14272 AFS-HJS-PSO 1.4998e-32 1.4998e-32 1.4998e-32 1.2398e-94 0.14327 Open in a new tab Experimental comparative analysis To focus on “BestFitness”, Table 4 is formed by extracting “BestFitness” from Table 3 and Fig. 5 is the corresponding graphical representation about Table 4 . Table 4. Experimental results about BestFiness. Algorithm Function PSO GA SAA FMINCON AFS-PSO AFS-HJS-PSO F1 0.015883 0.036092 0.036092 0.12364 0.0041034 0.00090704 F2 − 1 − 1 − 0.99996 − 0.93625 − 1 − 1 F3 − 913.2351 − 959.6407 − 559.7869 − 66.8437 − 931.0532 − 913.6888 F4 1.35E-31 2.13E-11 3.95E-07 6.05E-15 1.35E-31 1.35E-31 F5 6.46E-08 1.87E-14 0.007905 1.47E-14 5.51E-05 0 F6 2.14E-13 6.27E-12 3.71E-09 0.88281 8.30E-08 0 F7 0 2.45E-11 7.12E-09 9.16E-15 0 0 F8 9.93E-36 4.59E-12 4.76E-07 4.89E-19 1.37E-226 1.79E-241 F9 − 1 − 1 − 0.99994 − 1 − 1 − 1 F10 3.70E-36 1.13E-11 4.87E-10 18.9764 8.06E-264 4.33E-247 F11 2.32E-12 9.50E-13 4.18E-07 0.22626 0.00074109 0 F12 0 1.11E-11 3.04E-08 7.19E-14 0 0 F13 1.33E-22 3.15E-13 7.31E-11 9.69E-14 7.89E-37 8.73E-236 F14 8.95E-36 2.10E-13 4.31E-09 1.32E-13 1.92E-227 8.84E-245 F15 − 2 − 2 − 2 − 2 − 2 − 2 F16 1.56E-36 6.25E-14 5.05E-08 1.48E-16 1.80E-237 1.31E-247 F17 4.16E-12 6.44E-06 4.42E-07 1.39E-10 4.44E-16 4.44E-16 F18 0 7.43E-11 1.94E-08 0 0 3.50E-08 F19 0.082538 7.70E-13 0.0074837 0.007396 0.15025 0.00025306 F20 1.50E-32 5.35E-15 2.44E-06 6.05E-16 1.50E-32 1.50E-32 Open in a new tab From Table 3 ,it can be seen that the global best fitnesses about PSO,AFS-PSO and AFS-HJS-PSO are better than the ones about GA,SAA and FMINCON respectively while the mean-time spent by PSO,AFS-PSO and AFS-HJS-PSO is more than the one spent by GA,SAA and FMINCON; In fact, GA, SAA, FMINCON and PSO are different types of algorithms, so it is difficulty to get best performance on all terms for some an algorithm. Accordingly, PSO,AFS-PSO and AFS-HJS-PSO should be listed together to evaluate the performance, so Table 5 is formed by focusing on “Mean Time” to compare the time-spent by PSO,AFS-PSO and AFS-HJS-PSO and Fig. 6 is a corresponding graphical analysis about Table 5 . Table 5. Time data comparison. Algorithm Function PSO AFS-PSO AFS-HJS-PSO AFS-PSO vs PSO AFS-HJS-PSO vs PSO AFS-HJS-PSO vs AFS-PSO F1 0.12347 0.15463 0.12026 − 25.24% 2.60% 22.23% F2 0.21223 0.125 0.11877 41.10% 44.04% 4.98% F3 0.12305 0.12237 0.11815 0.55% 3.98% 3.45% F4 0.12156 0.12121 0.1181 0.29% 2.85% 2.57% F5 0.12607 0.12164 0.12189 3.64% 3.32% − 0.21% F6 0.12796 0.1242 0.11901 2.94% 6.99% 4.18% F7 0.12738 0.12054 0.11789 5.37% 7.45% 2.20% F8 0.12427 0.12371 0.12214 0.45% 1.71% 1.27% F9 0.12174 0.12119 0.11909 0.45% 2.18% 1.73% F10 0.12267 0.12682 0.1243 − 3.38% − 1.33% 1.99% F11 0.12547 0.12534 0.11961 0.10% 4.67% 4.57% F12 0.12322 0.12254 0.12098 0.55% 1.82% 1.27% F13 0.12444 0.12532 0.12483 − 0.71% − 0.31% 0.39% F14 0.12219 0.12512 0.12543 − 2.40% − 2.65% − 0.25% F15 0.12098 0.12069 0.11948 0.24% 1.24% 1.00% F16 0.12456 0.12804 0.12846 − 2.79% − 3.13% − 0.33% F17 0.12409 0.12463 0.12081 − 0.44% 2.64% 3.07% F18 0.12298 0.12232 0.1211 0.54% 1.53% 1.00% F19 0.12691 0.12719 0.12418 − 0.22% 2.15% 2.37% F20 0.14432 0.14272 0.14327 1.11% 0.73% − 0.39% Open in a new tab Fig. 6. Open in a new tab Histograms of BestFiness on classical benchmark functions. ( a )-Bukin6(F1), ( b )-DropWave(F2), ( c )-Sumsquares(F3), ( d )-Levy13(F4), ( e )-Schaffer2(F5), ( f )-Bohachevsky1(F6), ( g )-Booth(F7), ( h )-Matyas(F8), ( i )-Easom(F9), ( j )-Eggcrate(F10), ( k )-Bohachevsky3(F11), ( l )-Perm0-d-β(F12), ( m )-RotatedHyper-Ellipsoid(F13), ( n )-Sum Squares(F14), ( o )-Trid(F15), ( p )-Zakharov(F16), ( q )-Ackley(F17), ( r )-Rastrigin(F18), ( s )-Griewank(F19), ( t )-Levy(F20);1:PSO,2:GA,3:SAA,4:FMINCON,5:AFS-PSO,6:AFS-HJS-PSO. Firstly, from the term of “Bestfitness” detailed in Table 4 and Fig. 6 , AFS-HJS-PSO can find the theoretical global optima on F2, F5, F6, F7, F9, F11,F12 and F15, and the amount reached 8, which is the highest among all algorithms,refer to the red part in Table 4 for details; On F1,F4,F8,F13,F14,F16,F17,F19,F20, AFS-HJS-PSO found best or smallest fitnesses than other algorithms although not theoretically optimal, refer to the blue part in Table 4 for details;And on F3, F10 and F18,AFS-HJS-PSO has not the best performances, but the results are basically on the same order of magnitude as that of other algorithms, epecially compared with AFS-PSO. In other words, AFS-HJS-PSO has the best performance on all classical benchmark functions except F3, F10 and F18, so it can be concluded that AFS-HJS-PSO has the best optimization accuracy. Moreover, from Table 5 and Fig. 7 ,focusing on the “Mean-Time” spent by PSO,AFS-PSO and AFS-HJS-PSO, the mean-time spent by AFS-HJS-PSO is smaller than the others’ on F1, F2, F3, F4, F6, F7, F8, F9, F11, F12, F15, F17, F18 and F19; And especially compared with AFS-PSO,AFS-HJS-PSO has better performance in term of “Mean-Time” on all classical benchmark functions except F5,F14,F16 and F20 while AFS-HJS-PSO has better performance in term of “BestFitness” on the F5,F14,F16 and F20, so it can be concluded that AFS-HJS-PSO has best convergence speed. Fig. 7. Open in a new tab Histograms of Mean-Time on classical benchmark functions. ( a )–F1, ( b )–F2, ( c )–F3, ( d )–F4, ( e )–F5, ( f )–F6, ( g )–F7, ( h )–F8, ( i )–F9, ( j )–F10, ( k )–F11, ( l )–F12, ( m )–F13, ( n )–F14, ( o )–F15, ( p )–F16, ( q )–F17, ( r )–F18, ( s )–F19, ( t )–F20; 1: AFS-PSO vs PSO, 2: AFS-HJS-PSO vs PSO, 3: AFS-HJS-PSO vs AFS-PSO. From Table 4 , it can be seen that the Standard Deviation(SD) about AFS-HJS-PSO is better than the one about AFS-PSO on F1,F2,F5,F6,F7,F11,F13,F15,F17,F19 and F20, more than half of the proportion. So it can be concluded that AFS-HJS-PSO has relatively better stability. In general, AFS-HJS-PSO has best performance in terms of optimization accuracy, convergence speed and relatively better stability. Experiments about solving PU3P To verify the performance of AFS-HJS-PSO solving PU3P, many groups of experiments have been conducted according to the steps Chapter3.3.2, and it should be noted that Fmincon is a solver specifically designed for continuous optimization problems and not suitable for solving discrete optimization problems while PU3P is a discrete optimization problem, so in this chapter only 5 algorithms are taken for further research. Table 6 shows the results of 30 independent comparative experiments on UPPF. Table 6. Results of Experiments on UPPF. Algorithm Best fitness Worst fitness Mean fitness Standard deviation Mean time PSO-UPPF 253.4865 302.0188 277.913 130.8768 1.2863 GA-UPPF 207.7258 254.2101 236.3762 135.5538 0.058628 SIMUL-UPPF 293.0595 353.8006 332.7101 186.4812 0.0058829 AFS-PSO-UPPF 236.7391 272.4409 252.0268 84.4029 1.0018 AFS-HJS-PSO-UPPF 203.0453 245.5122 224.7879 142.6733 0.69913 Open in a new tab From Table 6 , it can be seen that AFS-HJS-PSO-UPPF has the best performance in term of “BestFitness”, “WorstFitness” and “MeanFitness” as well as better performance in “Standard Deviation” and “Mean Time”, which indicates that AFS-HJS-PSO has the best performance although it may not be optimal in some terms. Conclusion and discussion To solve the PU3P, this paper proposed a police UAV path planning method based on an improved PSO using AFS and HJS, which has three main works: Firstly, AFSwas introduced into original PSO in order to strengthen overall exploration capability while HJS was adopted to enhance it local exploitation level, thus AFS-HJS-PSO using AFS and HJS was proposed. Secondly, AFS-HJS-PSO-PU3P based on AFS-HJS-PSO wasproposed by utilizing the new proposed PSO to optimize UPPF abstracted fromPU3P, devoted to get the global best solution for police UAV path planning. Finally, comparative experiments among PSO, Generic Algorithm(GA),Simulated Annealing Algorithm(SAA),a Nonlinear Programming Solver: FMINCON, AFS-PSO and AFS-HJS-PSO were independently conducted 30 times on 20 classical benchmark functions as well as compared experiments among PSO-PU3P,GA-PU3P,SAA-PU3P,AFS-PSO-PU3P and AFS-HJS-PSO-PU3P were independently conducted 30 times on UPPF. Theoretical research and experimental results indicate that new proposed AFS-HJS-PSO is superior to PSO, GA, SAA, FMINCON in terms of optimization accuracy, convergence speed and stability as well as AFS-HJS-PSO-PU3P can better solve PU3P,detailed in Chapter 4. However, both of AFS-HJS-PSO and AFS-HJS-PSO-PU3P still possess some shortcomings, such as AFS-HJS-PSO has poor performance in term of “Standard Deviation” and AFS-HJS-PSO-PU3P didn’t perform the best in terms of “Standard Deviation” and “Mean Time”.The main reason is that AFS’s global exploration capability is not stable, and HJS’s local mining consumes too much computing resources. In general, AFS-HJS-PSO is superior under tested conditions compared with other methods above as well as AFS-HJS-PSO-PU3P is the most excellent method among theall compared methods. In the future, authors will attempt to introduce innovations from other latest optimization algorithms such as SRA 29 and DA 30 to improve AFS-HJS-PSO, and promote its application in other engineering fields. Acknowledgements Authors are grateful for peer experts for full support of this paper and thank University of Science and Technology Beijing for supporting necessary scientific environment. Author contributions Dongxing Wang gave the conception and wrote the main manuscript text while Xiaoxiao Qian prepared Figs. 1, 2 and 3. All authors reviewed the manuscript. Funding This research was funded by Funds for the Innovation of Policing Science and Technology,Fujian province,grant number 2024Y0065. Data availability The data that supports the findings of this study are available in Table 1. And the data can be got from the website: https://www.sfu.ca/~ssurjano/optimization.html . (Accessed on 26 October 2025). Code availability AFS-HJS-PSO and AFS-HJS-PSO-PU3P are central to the conclusions of this study,and the custom code developed for them can be accessed through the Link: https://pan.baidu.com/s/1M-e4BKa7nWYDh3hDasxF6Q?pwd=SCRE ; Extraction code: SCRE. No restrictions apply to the access and non-commercial use of the code. Focus on pso_test.m,ga_test.m,simul_test.m,fmin_test.m,afs_pso.m and afs_hjs_pso.m, which are correspond to the function PSO,GA,SAA,FMINCON,AFS-PSO and AFS-HJS-PSO respectively. Similarly,Focus on pso_tsp.m,ga_tsp1.m,simul_tsp.m,afs_pso_tsp.m and afs_hjs_pso_tsp.m,which are correspond to function PSO-UPPF,GA-UPPF,SIMUL-UPPF,AFS-PSO-UPPF and AFS-HJS-PSO-UPPF respectively. Declarations Competing interests The authors declare no competing interests. Footnotes Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. References 1. Fu, C. 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AFS-HJS-PSO and AFS-HJS-PSO-PU3P are central to the conclusions of this study,and the custom code developed for them can be accessed through the Link: https://pan.baidu.com/s/1M-e4BKa7nWYDh3hDasxF6Q?pwd=SCRE ; Extraction code: SCRE. No restrictions apply to the access and non-commercial use of the code. Focus on pso_test.m,ga_test.m,simul_test.m,fmin_test.m,afs_pso.m and afs_hjs_pso.m, which are correspond to the function PSO,GA,SAA,FMINCON,AFS-PSO and AFS-HJS-PSO respectively. Similarly,Focus on pso_tsp.m,ga_tsp1.m,simul_tsp.m,afs_pso_tsp.m and afs_hjs_pso_tsp.m,which are correspond to function PSO-UPPF,GA-UPPF,SIMUL-UPPF,AFS-PSO-UPPF and AFS-HJS-PSO-UPPF respectively. 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