The article explores the application of a newly developed Immune Plasma Algorithm (IPA) variant, called hospIPA, for optimizing the path planning of unmanned aerial vehicles (UAVs) and unmanned combat aerial vehicles (UCAVs). Traditional path planning techniques, while effective, face limitations such as local minima traps and detailed battlefield mapping requirements. The IPA, inspired by immune system responses, offers a promising alternative. The hospIPA variant introduces a hospitalization mechanism that quarantines poor solutions and a revised plasma treatment schema that enhances exploitation and exploration. The algorithm's performance is validated through extensive comparative studies, showcasing its superiority over standard meta-heuristics. This innovative approach not only improves path planning efficiency but also ensures safer and more fuel-efficient routes for UAVs and UCAVs.
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Abstract
Unmanned aerial vehicles (UAVs) and their specialized variants known as unmanned combat aerial vehicles (UCAVs) have triggered a profound change in the well-known military concepts and researchers from different disciplines tried to solve challenging problems of the mentioned vehicles. Path planning is one of these challenging problems about the UAV or UCAV systems and should be solved carefully by considering some optimization requirements defined for the enemy threats, fuel or battery usage, kinematic limitations on the turning and climbing angles in order to further improving the task success and safety of autonomous flight. Immune plasma algorithm (IP algorithm or IPA) modeling the details of a medical method gained popularity with the COVID-19 pandemic has been introduced recently and showed promising performance on solving a set of engineering problems. However, IPA requires setting the control parameters appropriately for maintaining a balance between the exploration and exploitation characteristics and does not design the particular treatment and hospitalization procedures by taking into account the implementation simplicity. In this study, IP algorithm was supported with a newly designed and realistic hospitalization mechanism that manages when an infected population member enters and discharges from the hospital. Moreover, the existing treatment schema of the algorithm was changed completely for improving the efficiency of the plasma transfer operations and removing the necessity of IPA specific control parameters and then a novel path planner called hospital IPA (hospIPA) was presented. For investigating the performance of hospIPA on solving path planning problem, a set of detailed experiments was carried out over twenty test cases belonging to both two and three-dimensional battlefield environments. The paths calculated by hospIPA were also compared with the calculated paths of other fourteen meta-heuristic based path planners. Comparative studies proved that the hospitalization mechanism making an exact discrimination between the poor and qualified solutions and modified treatment schema collecting the plasma being transferred by guiding the best solution give a tremendous contribution and allow hospIPA to obtain more safe and robust paths than other meta-heuristics for almost all test cases.
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1 Introduction
The recent advances on the microcomputers, remote sensing methods, communication technologies and smart munition production approaches started a revolution for the design and usage concepts of unmanned aerial vehicles (UAVs) and unmanned combat aerial vehicles (UCAVs). Even though the developed countries have strong air forces containing the trained pilots, fighter jets, bombers and helicopters, they also allocate defence budgets for producing UAV or UCAV systems or increasing their task performances and capabilities by trying to solve complex problems about these vehicles [1]. In order to increase the task performance of a UAV or UCAV system, the first and foremost problem that should be solved optimally is the path planning. For planning the optimal or near optimal paths before the flight of a UAV or UCAV being operated, some classical techniques including Artificial Potential Field (APF), Probabilistic Road Map (PRM), Rapid-Exploring Random Trees (RRT) and well-known graph based methods such as A*, D* and Voronoi diagram were successfully experimented [2]. However, all of these path planners have difficulties about trapping local minimum solutions and require detailed information for the battlefield in a map format [2]. Because of the mentioned limitations of the classical techniques, researchers tried to find alternative ways and discovered the potential of meta-heuristic algorithms to plan UAV paths or solve other complex engineering problems [3‐5].
The meta-heuristic algorithms introduced and used with the standard or modified implementations for solving path planning problem of UAV or UCAV systems inspire from the intelligent behaviors of different species such as birds, ants, bees, butterflies, moths and wolves or try to model evolutionary mechanisms including natural selection, mutation and crossover or guide physical or chemical phenomenons such as gravitation, explosion, burning and annealing. However, the new coronavirus first seen in Wuhan, China at the beginning of 2020 and caused a global health crisis altered the main focus of researchers from computer and information sciences and they investigated how a medical method or treatment approach used for the patients infected with the coronavirus can be referenced to design and develop modern meta-heuristic techniques [6]. Immune Plasma algorithm (IP algorithm or IPA) is the first meta-heuristic directly utilizing from the fundamental steps of a medical method called immune or convalescent plasma treatment as the given name implies [7]. The promising performance of the standard IPA has been validated recently on big data optimization [8], radio channel assignment [9], wireless sensor deployment [10], neural network training [11], in addition to the UAV or UCAV path planning [12]. Even though the standard implementation of the IPA shows promising performance on different optimization problems, it still requires configuring the control parameters responsible for determining the number of donors and receivers subtly or modeling and then integrating more detailed treatment procedures. In this study,
A new variant of the IPA called the hospital IPA (hospIPA) was proposed.
The hospIPA integrated a newly designed and realistic hospitalization mechanism into the workflow of the IPA for controlling when an infected population member will enter and discharge from the hospital.
The plasma collection and transfer schema was redesigned for hospIPA by aiming at increasing the treatment efficiency of the hospitalized individual or individuals.
Moreover, the proposed plasma collection and transfer schema removed the necessity of the IPA specific control parameters determining how many individuals will be receivers and how many individuals will be plasma donors.
The path planning performance of hospIPA was investigated by using twenty test cases in total belonging to both two and three-dimensional battlefield environments. The paths obtained by the hospIPA were compared with the calculated paths of a set of meta-heuristics including IPA, Genetic algorithm (GA) and Particle Swarm Optimization algorithm (PSO) based version of GA called GAPSO, Moth Flame Optimization (MFO), Salp Swarm algorithm (SSA), Pathfinder algorithm (PFA), Stain Bowerbird Optimization (SBO), Sine-Cosine algorithm (SCA), Grey Wolf Optimizer (GWO) and its hybridization with the Symbiotic Organism Search (SOS) known as HSGWO-MSOS, Artificial Ecosystem Optimizer (AEO) and adaptive neighborhood-based search enhanced AEO (NSEAEO), a strong implementation of the Teaching-Learning Based Optimization (TLBO) algorithm and finally the comprehensively improved PSO for short CIPSO. Comparative studies between hospIPA and other meta-heuristic based planners showed that hospIPA is capable of calculating more safe, fuel efficient and flyable paths for the vast majority of the test cases. While the newly designed hospitalization approach allows hospIPA to discriminate the poor solutions from the remaining part of the population and helps exploring the vicinity of qualified solutions more steadily, the proposed treatment schema significantly improves the exploitation performance and gives a chance to update a poor solution with a candidate better than the best solution found until the current cycle. The rest of the paper is organized as follows: Mathematical model of the path planning problem is explained in Sect. 2. Fundamental steps of the IP algorithm are given in Sect. 3. Details of the newly proposed hospitalization and treatment procedures are mentioned in Sect. 4. Section 5 is devoted to the experimental and comparative studies. Finally, in Sect. 6, some final remarks and future works about the IPA based path planners are presented.
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1.1 Related works
One of the first studies that illustrates how a meta-heuristic can plan optimal flight path of a UCAV was presented by Duan et al. over the Ant Colony Optimization (ACO) algorithm [13]. In another study, Duan et al. introduced a hybrid approach by combining ACO and Differential Evolution (DE) algorithms and experimented their path planner for a single UCAV being operated in a three-dimensional environment [14]. Ma and Lei related the values being assigned to the inertia weight of PSO algorithm with the second order oscillation links and second-order oscillating PSO (SOPSO) was presented [15]. Xu et al. designed a new Artificial Bee Colony (ABC) algorithm in which each employed forager searches food sources within the neighborhood of the current best solution by using chaotic random numbers and illustrated the effectiveness of new path planner against standard ABC [16]. For a more clear discrimination between the food sources of ABC algorithm, Zhang et al. scaled the raw fitness values in their path planning technique [17]. Zhang et al. also introduced Fitness-scaling Adaptive Chaotic PSO (FAC-PSO) algorithm [18]. Gravitational Search algorithm (GSA) was first taken as a base by P. Li and Duan and then combined with the memory and social information concepts of PSO algorithm [19]. Comparative studies showed that proposed GSA performs better than other UAV path planners depending on default implementation of GSA or PSO algorithm [19]. Fu served a PSO algorithm that calculates the velocity of each particle by guiding the best particle of a previously determined small solution group [20].
Wang et al. developed an efficient information sharing procedure between the qualified solutions of Firefly algorithm (FA) and presented modified FA for short MFA [21]. The UAV path planning performance of MFA was investigated and a detailed comparison with other meta-heuristics such as PSO, DE, ACO, GA, a modification of GA named stud GA (SGA), Biogeography-based Optimization (BBO), Evolutionary Strategies (ES) and Population-based Incremental Learning (PBIL) was given [21]. In another study, Wang et al. proposed a three-dimensional path planning technique by hybridizing DE and Cuckoo Search (CS) algorithms [22]. The contribution of Wang et al. to the literature of meta-heuristic based path planners continued with the Bat algorithm (BA) supported by the mutation operator of DE [23]. Wang et al. also developed a BA referenced three-dimensional path planner called improved BA (IBA) and compared IBA with basic BA over the visual representations of the calculated paths [24]. For further improving the path planning performance of FA, C. Liu et al. decided to adjust the parameter related with the attractiveness of fireflies adaptively [25]. Zhu and Duan assisted the BBO algorithm with the predator–prey concept and chaos theory [26]. The novel implementation of BBO, Chaotic Predator-Prey BBO (CPPBBO), was employed to plan UAV paths by considering the constraints about the yawing angle and total flight length [26]. Black Hole (BH) algorithm was guided by Heidari and Abbaspour for UCAV path planning [27]. Glowworm Swarm Optimization (GSO) was varied by Tang and Zhou with the equations coming from PSO and PGSO was introduced [28]. Detailed performance investigations about PGSO informed that PGSO obtains better paths than other methods when the number of segmentation points is kept relatively small [28].
Yu et al. tested TLBO for the planning a UAV being operated in a fixed altitude environment and proved its competitive performance against PSO, DE, ABC and GSO algorithms [29]. The mathematical model of the path planning problem was tuned with the difficult constraints about the various enemy threats such as anti-air guns, missiles, radars, terrain and non-flight zones, turning angle, climbing or gliding slope, flight altitude and total length by Zhang and Duan and they used a DE algorithm with \(\alpha\) level comparison based constraint-handling approach [30]. Zhou et al. designed a hybrid path planner by using Wolf Colony Search (WCS) and Complex method [31]. Duan and Qiao solved path planning problem with Pigeon-Inspired Optimization (PIO) algorithm [32]. B. Li et al. presented Balance-Evolution Strategy ABC algorithm for short BE-ABC in which trial counters are controlled when generating candidate solutions [33]. Zhang and Duan integrated the predator–prey concept into the PIO algorithm and tried to plan UCAV paths in a battlefield for which danger zones move dynamically [34]. The searching strategy of PSO algorithm using a kind of memory and mutation operator of GA that provides an extra support for avoiding solutions matched with the local optimums were referenced by Yongbo Chen et al. and a variant of the Central Force Optimization (CFO) was announced for planning of a rotary wing vertical take-off and landing (VTOL) system [35].
Zhou et al. utilized from the quantum gates for improving the performance of Wind Driven Optimization (WDO) algorithm and developed quantum WDO (QWDO) [36]. How the standard GWO performs on planning UAV paths was analyzed by Zhang et al. with two-dimensional battlefields [37]. The path planning capabilities of PSO algorithm was further improved by Liu et al. with adaptive sensitivity decision area method in which the high potential particles are determined and other candidates are removed to overcome the difficulties about the premature convergence [38]. In addition to this, Liu et al. addressed the defects of standard PSO algorithm to do with the trapping local optimums and slow convergence by Spatial Refined Voting Mechanism (SRVM) [39]. They further managed the possible collision with a newly introduced spatial-temporal collision avoidance technique when planning multiple UAVs by employing the mentioned PSO variant [39]. Luo et al. changed the representation of solutions in BA with quantum encoding and replaced the existing update and mutation models with quantum rotation gate and quantum not gate for a new path planner [40]. Q. Zhang et al. referenced Collection Decision Optimization algorithm (CDOA) and investigated its performance as a path planner [41].
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Alihodzic et al. focused to solve path planning problem with Elephant Herd Optimization (EHO) algorithm [42]. In another study, Alihodzic et al. determined the number of sparks and exploitation amplitude of Fireworks (FW) algorithm when designing a UCAV path planner [43]. Miao et al. combined advantageous sides of Simplex method and SOS and provided a rich set of experimental results about their technique for battlefields with static and random enemy threats [44]. Dolicanin et al. announced a Brain Storm Optimization (BSO) algorithm based path planner [45]. Pan et al. decided to adjust the fraction probability and scaling factor of CS algorithm with the sequences of Circle-type Chaotic Map and illustrated the better path planning capabilities of the mentioned CS implementation [46]. The valuable contribution of Pan et al. is not only limited with the CS based path planner. Pan et al. also utilized from Whale Optimization algorithm (WOA) after remodeling the encircling or searching procedures [47]. Another study successfully completed by Pan et al. was devoted to the development of CIJADE that brings together strong properties of two DE variants called CIPDE and JADE [48]. Position update procedure of BA was altered by Lin et al. with the guidance of APF [49]. They also used optimal success rate and chaos theory for further improving the local search performance of their path planner [49]. Qu et al. assumed that the members of GWO are the train agents of Reinforcement Learning (RL) and RLGWO was offered for calculating UAV paths [50]. The promising performance of GWO became source of inspiration to Qu et al. for another study in which GWO and SOS are coupled to develop a planner known as HSGWO-MSOS [51].
Yi et al. regenerated a set of low quality solutions in Monarchy Butterfly Optimization (MBO) with quantum operations and quantum inspired MBO (QMBO) was introduced [52]. Wu et al. reported an intelligent initialization schema by considering the physical limitations of the UAV being planning for ABC algorithm [53]. A Flower Pollination algorithm (FPA) guided path planner was declared by Yang Chen et al. and compared with the well-studied techniques including A*, APF and RRT [54]. After a detailed comparison between the path planning performances of BA, ABC, DE, FA, GWO, PSO, WOA, CS, a recent variant of MBO known as GSMBO, Harmony Search (HS) and Spider Monkey Optimization (SMO), Zhu et al. concluded that SMO plans more safe paths [55]. The performance of SMO tried to be further improved by Zhu et al. and Cooperation Co-evolution SMO (CESMO) was developed [56]. In order to plan a UAV path for a battlefield containing specifically designed enemy weapons and climate effects, Zhou et al. offered improved BA (IBA) [57]. Wu et al. serviced the Zaslavskii chaos map and a path planner called chaotic PSO was presented [58]. H. Xu et al. changed the critical stages of GSA by using adaptive alpha-adjusting strategy and Cauchy mutation for optimizing the interactions with enemy threats, reducing the total flight length and turning angles of a UAV [59].
Jiang et al. nearly fixed all details about the workflow of GWO by designing an efficient communication mechanism and \(\epsilon\)-level comparison for handing constraints [60]. The higher number of waypoints can increase the sensitivity of the calculated paths. However, increasing the number of segmentation points or waypoints bring extra difficulty and computational burden to the path planning problem. Jarray et al. tried to handle the mentioned complexity by integrating Cooperative Co-evolution mechanism that depends on splitting the decision variables or parameters of the problem into subgroups for solving them independently into parallel GWO algorithm [61]. Du et al. tried to address some troublesome stages of Chimp Optimization algorithm (ChOA) by inspiring mathematical models of Monkey algorithm and improved ChOA for short IChOA was designed [62]. The performance of IChOA was investigated over the numerical benchmark functions in addition to the three-dimensional UAV path planning problem [62]. Wang et al. concerned with the exploration capability and convergence speed of Mayfly algorithm (MA) and developed modified MA (modMA) in which exponent decreasing inertia weight strategy, adaptive Cauchy mutation and enhanced crossover operation are combined together [63]. Some experiments using two battlefields with eight and ten enemy threats showed that modMA is not only better than MA but also more stable than PSO, GWO and Butterfly Optimization algorithm (BOA) [63]. Niu et al. replaced commonly used neighborhood topologies such as star and ring with an approach called adaptive neighborhood search for their AEO based path planner [64]. Also, they improved the decomposition stage of AEO with a dynamic method selection technique and integrated quadratic interpolation for further enhancing the search capability [64]. The new AEO algorithm referenced path planner named NSEAEO was compared with the GA and PSO variants, an improved version of TLBO (ECTLBO), GA, HSGWO-MSOS, MFO, SSA, SBO, SCA, PFA in addition to the AEO over two and three-dimensional battlefield scenarios [64]. The investigations about the capabilities of AEO based path planners were continued by Niu et al. and they designed an adaptive mechanism that controls the distance between the current optimal and newly generated candidate solution and integrated it into the workflow of AEO algorithm [65]. Five complex three dimensional scenarios were generated to evaluate the performance of recent AEO on the path planning and experimental studies proved that the modifications significantly improve the search characteristics and allow to obtain better solutions than the solutions of GA, PSO, GWO, WOA, AEO, HSGWO-MSOS, IChOA and improved adaptive GWO (AGWO) based path planners [65].
The battlefield model was tried to be simplified with a creative idea of Jia et al. when solving path planning problem by executing their special PSO algorithm [66]. The specialized PSO algorithm or DLCRPSO utilized from a rotation strategy that improves the search efficiency for high-dimensional space [66]. Search and Rescue (SAR) optimization algorithm was combined with a heuristic crossover (HC) strategy that adjusts the range for improving the efficiency of candidate generation mechanism by C. Zhang et al. and HC-SAR was introduced [67]. Comparative studies between HC-SAR and other path planners based on SAR, DE, SSA, Ant Lion Optimizer (ALO) and Squirrel Search algorithm (SSA) showed that HC-SAR can serve as a consistent UAV path planner [67]. Ait-Saadi et al. used Simulated Annealing (SA) and Singer chaotic map with Aquila Optimization (AO) algorithm for developing Chaotic Aquila Optimization Simulated Annealing (CAOSA) and tested it on solving both two and three-dimensional UAV path planning problem [68]. Another GWO based path planner was announced by Yu et al. for which the characteristics of alpha, beta and delta wolves are changed in a manner that they search around the alpha wolf and the characteristics of the omega wolves are changed in a manner that they search around the best three wolves for enriching the exploitation [69]. Chowdhury and De introduced a GSO algorithm based path planner known as Reverse GSO (RGSO) in which the movements between the solutions represented with different glowworms are adjusted by checking the luciferin values of them [70].
Chen et al. first replaced the Cartesian coordinate system with a spherical coordinate system in which some constraints about the angle and velocity of a UAV are handled more clearly when planning a flight path [71]. They also introduced a mechanism called Truncated Mean Stabilization (TMS) for maintaining the population diversity by replacing some solutions with a newly determined one. The modernized BA that uses spherical coordinate system and TMS approach was named TMS-SBA by Chen et al. and tested for calculating paths in four different battlefield scenarios [71]. The potential of changing the Cartesian coordinate system to realize the characteristics of a UAV or UCAV was also utilized by Huang et al. for designing a new PSO algorithm based path planner, Adaptive Cylinder Vector PSO with DE (ACVDEPSO), and ACVDEPSO was experimented in three-dimensional environments generated with the Digital Elevation Model (DEM) maps [72]. Hu et al. improved the overall optimization performance of the standard Honey Badger algorithm (HBA) by invoking Bernoulli shift map for initialization of the population, piecewise optimal decreasing neighborhood for stabilizing the unbalanced convergence characteristics and finally horizontal crossing with strategy adaptation for the generation of new candidates [73]. They tested new HBA variant to plan UAV paths in different battlefield environments containing only circular or irregular obstacles [73].
2 Mathematical model of path planning problem
The calculation of a path for a UAV, UCAV or other similar aerial vehicle requires a strong mathematical description about the different enemy threats, their sensing, detecting or shooting capabilities, fuel consumption or battery usage and finally kinematic constraints on the turning and climbing maneuvers. In addition to the mathematical descriptions of the enemy threats, fuel or battery usage and kinematic constraints, a model that defines how a path can be generated and a score calculation schema for deciding which path is more better should also be supplied. Given that a UAV or UCAV starts flight from the point \(P_{s}=(x_{s},y_{s},z_{s})\) to find or destroy a target located at the point \(P_{t}=(x_{t},y_{t},z_{t})\) and a reference line between the \(P_{s}\) and \(P_{t}\) is drawn by considering the xy-plane.
When the operations to do with the drawing of a reference line are completed, it is divided equally into \(D+1\) segments by using D segmentation points [64]. Each segmentation point on the reference line is actually responsible for intersecting with a unique line that is perpendicular to the reference line. If the lines, each is perpendicular to the reference line and intersects only one segmentation point, are organized, a set of lines showed as \(L=\{L_{1},L_{2},\ldots ,L_{D-1},L_{D}\}\) can be obtained. The set L in which \(L_{1}\) corresponds to the vertical line passing through the first segmentation point, \(L_{2}\) corresponds to the vertical line passing through the second segmentation point, and so on opens a gate for the subsequent operations of the path planning. If only one point on each line in the set L is selected and then combined with the \(P_{s}\) and \(P_{t}\) by guiding that the \(P_{s}\) is the start point and \(P_{t}\) is the target point, a set of points or \(P=\{P_{s},P_{1},\ldots ,P_{D},P_{t}\}\) and an implicit path after connecting sequential pair of points in set P with a line segment are generated [64].
The method lying behind the definition of a UAV or UCAV path through the set L and set P depends on strong mathematical and geometrical backgrounds. All the lines in the set L require correct equations that satisfy the prerequisites about the segmentation points and reference line between \(P_{s}\) and \(P_{t}\). Also, it must be guaranteed that each point of the set P is selected in a manner that the point \(P_{i}\) is on the line \(L_{i}\) where i ranges from 1 to D and huge amount of computational burden arises. In order to reduce the computational effort about the sets of lines and points, an appropriate coordinate system transformation that converts the reference line into the horizontal axis of the new coordinate system by referencing Eq. (1) can be used [64]. In Eq. (1), \(x_{k}\), \(y_{k}\) and \(z_{k}\) represent the x-axis, y-axis and z-axis values of point \(P_{k}\) located at the original coordinate system, while \(\acute{x}_{k}\), \(\acute{y}_{k}\) and \(\acute{z}_{k}\) represent the \(\acute{x}\)-axis, \(\acute{y}\)-axis and \(\acute{z}\)-axis values of point \(\acute{P}_{k}\) and \(\acute{P}_{k}\) corresponds the transformed counterpart of point \(P_{k}\) for the new coordinate system. Finally, \(\theta\) is matched with the angle of rotation and calculated as \(arctan((y_{k}-y_{s}) / (x_{k}-x_{s}))\).
A three-dimensional battlefield (a), xy-plane (b), transformed counterpart (c), determined paths (d) and their provisions to the original battlefield (e)–(f)
One of the first advantages coming with the mentioned coordinate transformation is about the \(\acute{x}\)-axis values of the corresponding points in the set P. Because of each line in the set L is vertical to the reference line or horizontal axis of the new coordinate system and the distance between the subsequent pair of lines is equal, \(\acute{x}\)-axis value of any point on the line \(\acute{L}_{i}\) where \(\acute{L}_{i}\) shows the counterpart of \(L_{i}\) in the set L for new coordinate system can be calculated as \(i\vert P_{s}P_{t}\ \vert /(D+1)\). If the \(\acute{y}\)-axis values of the points on the lines in set L are selected and they are brought together with the vertical axis values of the transformed start and target points such as \(\{\acute{y}_{s},\acute{y}_{1},\acute{y}_{2},\ldots ,\acute{y}_{D-1},\acute{y}_{D},\acute{y}_{t}\}\), path planning can be turning into a D-dimensional optimization problem that requires minimization of difficult objectives about enemy threats, battery or fuel consumption measured over total flight length, turning and climbing angles. In Fig. 1, how the set of lines and set of points are utilized to represent a path and transitions between the initial and original coordinate systems are carried out is illustrated over a battlefield with four enemy threats.
A relatively small modification on one of the guessed points in the set P can cause a dramatic change for the corresponding path and a quality or score calculation schema taking into account the enemy threats and their properties, fuel or battery consumption, turning and climbing angles should be used to understand the appropriateness of a path and make a discrimination between more than one candidates as described in Eq. (2) [64]. In Eq. (2), \(C_{t}\) is used on behalf of the cost of all enemy threats and it is calculated by taking the integral of \(w_{t}\) from 0 to \(\ell\) where \(\ell\) shows the total length of the discovered path. Similarly, \(C_{f}\) is used on behalf of the cost of fuel or battery consumption of UAV or UCAV system and calculated by taking the integral of \(w_{f}\) from 0 to \(\ell\). While \(C_{s}\) represents the cost of kinematic limitations of considered aerial vehicle, it has two different parts one of which is related with the turning angles and the other is related with the climbing angles. Also, it should be noticed that \(C_{t}\), \(C_{f}\) and \(C_{s}\) are weighted with \(\lambda _{t}\), \(\lambda _{f}\) and \(\lambda _{s}\) whose sum is equal to 1 for adjusting their contributions on the total path cost showed as C.
The integral calculations to do with the \(C_{f}\) can be simplified by executing an accurate approximation. Because of the fuel consumption or battery usage of a UAV or UCAV is directly proportional to the length of path, \(w_{f}\) can be replaced with a constant such as 1 [64]. An accurate but more detailed approximation can also help the integral calculations about \(C_{t}\). Given that \(P_{i}\) and \(P_{j}\) are two adjacent points in the set P and the length of line segment between these points is found as \(L_{ij}\). Also, it is noted that the line segment of length \(L_{ij}\) is divided into ten equal subsegments with the help of nine subsegmentation points and then the first, third, fifth, seventh and ninth subsegmentation points are selected and called as 0.1, 0.3, 0.5, 0.7, 0.9 subsegmentation points. If the line segment of length \(L_{ij}\) is in the effect range of kth enemy threat with the grade \(t_{k}\), the cost of considered enemy threat for the line segment between \(P_{i}\) and \(P_{j}\) or \(C_{t,(ij),k}\) is found by using Eq. (3) [64]. While the Euclidean distance between 0.1 subsegmentation point and the center of the kth enemy threat is represented with \(d_{0.1,i,k}^{4}\) in Eq. (3), the Euclidean distances between the other selected segmentation points and the center of the kth enemy threat are showed with \(d_{0.3,i,k}^{4}\), \(d_{0.5,i,k}^{4}\), \(d_{0.7,i,k}^{4}\) and \(d_{0.9,i,k}^{4}\) for the same equation. After calculating the cost of each enemy threat for all of the line segments and then summing them, \(C_{t}\) is approximated successfully.
For tracking the calculated path, a UAV or UCAV should perform different maneuvers that necessitate aggressive turning and climbing with variable angles. However, a UAV or UCAV system has certain limitation about the turning and climbing angles and they should be considered when the overall path quality is calculated. Assume that three subsequent points such as \(P_{j}\), \(P_{j+1}\) and \(P_{j+2}\) are selected from the set P and two vectors such as \(\overrightarrow{p_{j}p_{j+1}}\) and \(\overrightarrow{p_{j+1}p_{j+2}}\) are generated by referencing these points. When \(P_{j}\), \(P_{j+1}\) and \(P_{j+2}\) are the first three points of set P, they correspond to \(P_{s}\), \(P_{1}\) and \(P_{2}\). In a similar manner, when \(P_{j}\), \(P_{j+1}\) and \(P_{j+2}\) are the last three points of set P, they correspond to \(P_{D-1}\), \(P_{D}\) and \(P_{t}\). The calculation of turning angle or \(\varnothing _{j}\) by considering the \(P_{j}\), \(P_{j+1}\) and \(P_{j+2}\) points and \(\overrightarrow{p_{j}p_{j+1}}\) and \(\overrightarrow{p_{j+1}p_{j+2}}\) vectors can be made with Eq. (4). If the absolute value of the \(\varnothing _{j}\) is less than or equal to the maximum angle or \(\varnothing _{max}\) of the UAV being operated, the effect of turning with the angle of \(\varnothing _{j}\) is simply ignored for the \(C_{s}\). Otherwise, absolute value of the calculated turning angle is summed with the absolute values of other turning angles violating the constraint about the maximum turning angle.
For calculating the climbing angle, subsequent points taken from set P and some vectors are needed. Assume that two subsequent points namely \(P_{j}\) and \(P_{j+1}\) are selected from set P and \(\overrightarrow{p_{j}p_{j+1}}\) is the vector generated by referencing these points. When \(P_{j}\) and \(P_{j+1}\) are the first two points of set P, they correspond to \(P_{s}\) and \(P_{1}\). In a similar manner, when \(P_{j}\) and \(P_{j+1}\) are the last two points of set P, they correspond to \(P_{D}\) and \(P_{t}\) respectively. The calculation of climbing angle or \(\Psi _{j}\) by considering \(P_{j}\), \(P_{j+1}\) points and \(\overrightarrow{p_{j}p_{j+1}}\) vector can be made with Eq. (5). If the absolute value of \(\Psi _{j} - \Psi _{j-1}\) operation is less than or equal to the maximum climbing angle or \(\Psi _{max}\) of the UAV, the effect of climbing is simply ignored for the \(C_{s}\). Otherwise, the absolute value of \(\Psi _{j} - \Psi _{j-1}\) operation is summed with the absolute values of other climbing angle calculations violating the constraint about the maximum climbing angle. The whole symbols used for the description and formulation of the path planning problem can be accessed in Table 1.
Used symbols and their descriptions for path planning
Symbols
Description
D
Number of segmentation points or parameters
\(P_{s},P_{t}\)
Start and target points
\(L, L_{i}\)
Line set and its ith member
\(P, P_{i}\)
Point set and its ith member
\(\theta\)
Rotation angle for coordinate transformation
\(\acute{L}_{i}\)
Counterpart of \(L_{i}\) for new coordinate system
C
Total cost of path
\(C_{f}\)
Cost of fuel consumption
\(C_{t}\)
Cost of enemy threats
\(C_{s}\)
Cost of turning and climbing maneuvers
\(\lambda _{f},\lambda _{t},\lambda _{s}\)
Weighting factors for \(C_{f}, C_{t}\) and \(C_{s}\)
\(\ell\)
Total length of path
\(L_{ij}\)
Length of line segment between \(P_{i}\) and \(P_{j}\) points
\(\varnothing _{j}\)
Cost of turning for \(P_{j}\), \(P_{j+1}\) and \(P_{j+2}\) points
\(\Psi _{j}\)
Cost of climbing for \(P_{j}\) and \(P_{j+1}\) points
\(\varnothing _\textrm{max}, \Psi _\textrm{max}\)
Maximum turning and climbing angles
3 Immune plasma algorithm
The immune system tries to protect a host by increasing the amount of plasma cells and their synthesis products also called antibodies. Antibodies are actually a type of proteins and can circulate in the blood as free-floating forms [74, 75]. When an antibody detects an antigen for which the antibodies are produced specially, it binds that antigen with the purpose of inactivating antigen functionalities. However, some persons who suffering from the immune system disorders have several difficulties for synthesizing remarkable amount of antibodies and when they are infected, hospitalization and intense care can be needed [74, 75]. In order to help the treatment operations of a critical person, the antibody rich part of the blood donated by an individual recovered previously can be utilized successfully.
The immune or convalescent plasma treatment is one of the strong medical methods guiding the fact that the antibodies can be transferred from the recovered individual or individuals to the critical patients or receivers and its efficiency was proven against the great influenza of 1918 pandemic more than a century ago and the recent global COVID-19 crisis [75, 76]. When the details of the immune or convalescent plasma treatment is controlled carefully, it is seen that there is an obvious analogy with the main operations known as exploration and exploitation of a meta-heuristic algorithm. By considering the mentioned analogy, Aslan introduced a new intelligent optimization technique called IP algorithm or for short IPA [7]. In IP algorithm, each person or individual of the population represents a possible solution of the optimization problem being solved. An infection can spread easily among the members of population and their immune responses are calculated according to the objective or cost function of the problem. While an individual with small objective function value corresponds to a qualified solution for a minimization problem, an individual with high objective function value corresponds to a qualified solution for a maximization problem [7]. Some individuals representing poor solutions are labeled as receivers and tried to be treated with the plasma taken from other individuals that are selected as donors because of their high quality immune responses. The mathematical models used by IP algorithm for distributing infection in the population, selecting receiver and donor individuals, applying plasma treatment and controlling the immune memories of donors were stated in the following subsections.
3.1 Initializing the members of population
Population based meta-heuristics such as IP algorithm starts the search operations by generating a set of solutions randomly. Given that IP algorithm with the population of size PS is employed for solving a D-dimensional optimization problem, kth individual also termed as \(x_{k}\) can be initialized by using Eq. (6) [7]. In Eq. (6), \(x_{kj}\) is matched with the \(j\textrm{th}\) parameter for which the lower and upper bounds are \(x_{j}^\textrm{min}\) and \(x_{j}^\textrm{max}\). Also, it should be noticed that rand(0, 1) is a random number taking its value between 0 and 1.
In an infection cycle of IP algorithm, there is a stage that is responsible for distributing infection from one individual to another with Eq. (7) where \(x_{k}\) is the individual being infected by the randomly selected \(x_{m}\) individual [7]. Moreover, it should be noted that \(x_{kj}\) and \(x_{mj}\) are the \(j\textrm{th}\) parameters of them and the j index is determined randomly from the set \(\{1,2,\ldots ,D\}\). For representing the infectious \(x_{k}\) individual, a temporary solution or \(x_{k}^{inf}\) is used in the same equation. All of the parameters belonging to \(x_{k}^{inf}\) are equal to the corresponding parameters of \(x_{k}\) except the jth one and the newly calculated jth parameter of \(x_{k}^{inf}\) is symbolized with \(x_{kj}^{inf}\).
The infection triggers the immune system of \(x_{k}\) individual and a special response in terms of antibodies is given. In order to evaluate the immune response of the infectious \(x_{k}\) individual or amount of synthesized antibodies, the value of the objective function f is utilized. If the immune response of the infectious \(x_{k}\) individual or \(f(x_{k}^{inf})\) is less than the antibody amount of the same individual before the infection or \(f(x_{k})\) by considering a minimization problem, it is decided that \(x_{k}\) individual is capable of handling infection and its immune memory is re-organized for a quick response to the similar infection as in Eq. (8) [7]. Otherwise, \(x_{k}\) individual and its jth parameter are left unchanged.
The second stage of an infection cycle in IPA is related with the selection of receivers and donors and then the application of plasma treatment. IP algorithm decides how many individuals will be receiver and how many individuals will be donors by introducing two control parameters called number of receivers or NoR and number of donors or NoD [7]. When IPA reaches the second stage of an infection cycle, it first sorts the individuals of the population by considering their objective function values in ascending order and then labels the last NoR individuals as critical patients or receivers and selects the first NoD individuals as plasma donors [7]. After determining the receiver and donor individuals, IPA starts plasma treatment. Given that \(x_{k}^{rcv}\) is the kth receiver from the receiver set of size NoR and \(x_{m}^{dnr}\) is the randomly selected donor from the donor set of size NoD. For the transfer of a dose of plasma from the \(x_{m}^{dnr}\) to the \(x_{k}^{rcv}\), a mathematical model as detailed in Eq. (9) where j is selected sequentially from the set \(\{1,2,\ldots ,D\}\) is used [7]. In Eq. (9), \(x_{k}^{rcv-p}\) is matched with the plasma transferred counterpart of \(x_{k}^{rcv}\) and jth parameters of them are \(x_{kj}^{rcv}\) and \(x_{kj}^{rcv-p}\). If the \(f(x_{k}^{rcv-p})\) is better than the \(f(x_{m}^{dnr})\) and proves the efficiency of treatment, \(x_{k}^{rcv}\) is updated with the parameters of \(x_{k}^{rcv-p}\) and second dose of plasma is prepared. Otherwise, \(x_{k}^{rcv}\) is updated with the parameters of \(x_{m}^{dnr}\) and treatment is completed for \(x_{k}^{rcv}\) [7].
The second or subsequent dose of plasma is transferred to \(x_{k}^{rcv}\) by using the mathematical model introduced for the transfer of first dose. However, in order to decide that whether the treatment will be continued with the third or subsequent dose of plasma or not, a comparison between the objective function values of \(x_{k}^{rcv-p}\) and \(x_{k}^{rcv}\) is carried out [7]. If the objective function value of \(x_{k}^{rcv}\) immediately after the second dose of plasma or \(f(x_{k}^{rcv-p})\) is better than the objective function value of \(x_{k}^{rcv}\) before the second dose of plasma or \(f(x_{k}^{rcv})\), \(x_{k}^{rcv}\) is updated with the parameters of \(x_{k}^{rcv-p}\) and third dose of plasma is prepared. Otherwise, the treatment of \(x_{k}^{rcv}\) is completed and the next receiver is selected if exists for starting the plasma transfer operations.
3.4 Updating immune memories of donors
The immune response or amount of synthesized antibodies by an individual who recovers shortly before and helps critical individuals for the treatment can change as time goes by or with the frequency of encountering to the same of similar infection. If the frequency of encountering to the infection increases with time, the immune memory recognizes the intruder quickly and a strong response in terms of synthesized antibodies is given. For integrating this type of mechanism into the workflow of the IPA, the ratio between \(t_{cr}\) and \(t_{max}\) and a random number generated between 0 and 1 were utilized [7]. While \(t_{cr}\) shows the current evaluation number and it is incremented by one for each request to the procedure calculating the objective function value, \(t_{max}\) demonstrates the maximum evaluation number and IPA terminates when \(t_{cr}\) becomes equal to \(t_{max}\). If the ratio between \(t_{cr}\) and \(t_{max}\) is less than the generated random number, it is decided that the immune memory of the mth donor individual or \(x_{m}^{dnr}\) still continues to learn details about the intruder causing infection and an entire re-initialization as in Eq. (6) is applied [7]. Otherwise, the immune memory of the \(x_{m}^{dnr}\) is changed slightly by using Eq. (10) where j index ranges from 1 to D [7]. As easily seen from the decision mechanism about how the donor individual is updated, the probability of execution Eq. (10) gets higher while the IPA reaches termination and allows a donor for protecting its memory partially.
4 Details of hospitalization mechanism for immune plasma algorithm
As stated previously, the standard implementation of the IP algorithm completes the treatment of an \(x_{k}^{rcv}\) individual if the first dose of plasma does not improve the antibody response of \(x_{k}^{rcv}\) as better as the antibody response of \(x_{m}^{dnr}\) donor. Moreover, when the IP algorithm decides that the treatment of \(x_{k}^{rcv}\) is ended immediately after the first dose of plasma from the \(x_{m}^{dnr}\) donor individual, the \(x_{k}^{rcv}\) is updated with the corresponding parameters of \(x_{m}^{dnr}\) for guaranteeing that at least one dose of plasma is transferred. Even though the idea lying behind the existing treatment schema of IPA ensures that the quality of the solution represented by a receiver individual becomes equal or better than the quality of the solution represented by the selected donor, it requires subtle configuration of the NoR and NoD parameters in order to maintain the population diversity while increasing the qualities of the existing solutions.
Algorithm 1
Distribution of infection by considering hospitalization
For further improving the performance of IPA and removing the necessity of both requirement and configuration of NoR and NoD parameters, a more efficient and realistic model by considering that a receiver or receivers can stay in a hospital rather than simply discharging them if the first dose of plasma does not generate the expected effect can be designed. At the end of the stage related with the distribution of infection in each cycle, the worst solution of the non-hospitalized individuals is first assumed as the patient and added to the set of hospitalized solutions or receivers. All of the hospitalized individuals are treated by transferring plasma. If the transferred dose of plasma gives a tremendous contribution to the antibody amount of a receiver individual, it is discharged from the hospital and becomes ready for the interaction of other healthy individuals in the next cycle. Otherwise, the mentioned receiver is isolated from other healthy individuals and stays at the hospital for the treatment operations of the subsequent cycle. When the number of hospitalized individuals or receivers is equal to \(PS-1\) for a population of size PS, it is easily understood that there is only one healthy individual and the operations to do with the distribution of infection are skipped. On the other hand, if the number of hospitalized individuals or receivers is not equal to \(PS-1\), non-hospitalized individuals still interact with each other and distribution of infection between these individuals can continue. In order to understand that how the discrimination between the hospitalized and non-hospitalized individuals is carried out when distributing the infection, Alg. (1) given can be examined.
Because of the newly introduced hospitalization mechanism adjusts the number of receivers dynamically for each cycle, an improved plasma treatment schema that is able to successfully handle the varying composition and number of receivers should be designed. Assume that \(x^{best}\) is the best solution found so far and \(x^{dnr}\) is the most qualified solution in the current population. For obtaining plasma being used for the treatment of hospitalized individuals, the mathematical representation given in Eq. (11) can be employed. While \(x^{pls-t}_{j}\) represents the newly determined \(j\textrm{th}\) parameter of the \(x^{pls}\) and \(x^{pls}\) corresponds to the plasma initialized with \(x^{best}\), \(x_{j}^{dnr}\) shows the \(j\textrm{th}\) parameter of the \(x^{dnr}\) individual. If the newly calculated \(j\textrm{th}\) parameter or \(x_{j}^{pls-t}\) of the \(x^{pls}\) improves the overall quality of the collected plasma, a greedy selection between \(x_{j}^{pls-t}\) and \(x_{j}^{pls}\) is executed. After controlling all of D different parameters sequentially and applying greedy selection between the new and existing ones, \(x^{pls}\) that is at least equal to or better than the \(x^{best}\) is obtained and ready to the usage for the treatment operations.
The default workflow of the IP algorithm uses the selected donor as the source of plasma and the treatment is modeled in a manner that each parameter of a receiver is changed with the information provided by its donor. However, in order to utilize from the information provided by the donor or collected plasma, some parameters of the receiver should be set to the corresponding parameters of the donor or collected plasma directly while the remaining ones are changed appropriately. A more efficient transfer approach that focuses on increasing the positive contribution of the treatment can be formulated as in Eq. (12). In Eq. (12), \(j_{rand}\) is used on behalf of a random number generated between 1 and D and it is compared with the j index chosen sequentially from the set \(\{1,2,\ldots ,D\}\). If the \(j_{rand}\) is found equal to the current value of j index, jth parameter of the \(x_k^{rcv-p}\) or \(x_{kj}^{rcv-p}\) is calculated again with the help of corresponding parameter of \(x^{pls}\). Otherwise, \(x_{kj}^{rcv-p}\) is set to the jth parameter of the \(x^{pls}\) or \(x_{j}^{pls}\) for a direct utilization from the valuable information provided by the \(x^{pls}\). When the transfer of plasma to the \(x_{k}^{rcv}\) is completed and the antibody level of \(x_{k}^{rcv}\) immediately after the treatment calculated as \(f(x_{k}^{rcv-p})\) is determined, a simple comparison between \(f(x_{k}^{rcv-p})\) and the antibody level of \(x^{pls}\) also calculated as \(f(x^{pls})\) is carried out. If \(f(x_{k}^{rcv-p})\) is better than \(f(x^{pls})\), \(x_{k}^{rcv}\) is updated with the \(x_{k}^{rcv-p}\) and \(x_{k}^{rcv}\) is discharged from the hospital. Otherwise, \(x_{k}^{rcv}\) continues to stay at hospital and waits the treatment operations of the next cycle. The details of the proposed treatment schema for the hospitalized individuals are presented in Alg. (2).
The IP algorithm that integrates the hospitalization mechanism into the workflow of it to dynamically determine the number of individuals who will be treated as receivers and the specialized plasma generation and transfer schema is named hospital IPA for short hospIPA. In the hospIPA, there is no need to the NoR and NoD parameters and their subtle adjustments. Because of the hospitalized individuals are not allowed to interact with the non-hospitalized individuals, the vicinity of the qualified solutions represented by the non-hospitalized individuals is explored more successfully. Also, it should be noticed that the treatment schema of a hospitalized or receiver individual is re-designed completely for handling the difficulty of dynamically determined set of receivers and increasing the effectiveness of plasma collection and transfer operations. When the operations related with the collection of plasma and its transfer to the receiver or receivers are carried out, hospIPA gets a chance of exploiting the solutions corresponding to the \(x^{best}\), \(x^{dnr}\) and \(x^{pls}\) implicitly. In Fig. 2, a hypothetical scenario with ten individuals was illustrated to describe the hospitalization mechanism and treatment schema of the hospIPA.
Fig. 2
A pictorial description of fundamental operations in hospIPA
The quality of a path being calculated by hospIPA changes according to the values of the algorithm specific control parameters, properties of the battlefields, enemy threats and finally number of segmentation points. Moreover, extra mechanisms executed by hospIPA effect the execution time and convergence performance of the same algorithm. For a more organized investigations about the path planning performance of hospIPA, the whole experimental studies were divided into four subsections. While the first and second subsections were devoted to the tests and comparative studies for the two and three-dimensional battlefield scenarios, some results about the execution times of hospIPA were shared in the third subsection. Finally, the convergence characteristics of hospIPA and statistical significance of its solutions were evaluated in the fourth subsection.
5.1 Planning paths for two-dimensional battlefields with hospIPA
The path planning performance of hospIPA after assuming that the altitude is fixed was investigated over three different battlefield scenarios each has four test cases generated by setting the number of segmentation points or D as 10, 15, 20 and 25. An enemy threat in a battlefield used for experiments is represented with a circle and the location of the circle center and radius are decided previously. Moreover, grades were assigned to the enemy threats for defining danger levels of them. The details about the battlefields and included enemy threats were given in Table 2 [64]. Because of only the population size or PS is adjustable for hospIPA, each test case was experimented by setting PS to 30, 40, 50, 75 and 100. A run of hospIPA that is terminated when the evaluation counter reaches to 6000 was repeated 30 times with random seeds and the best solution and its objective function value found at the end of a run were recorded [64]. By using the recorded objective function values, the best, worst, mean best objective function values and standard deviations were determined and then summarized in Table 3 for Scenario-1, Table 4 for Scenario-2 and finally Table 5 for Scenario-3.
Table 2
Details of battlefields used for fixed altitude path planning
Sc
Threat centers
Threat radius
Threat grade
Start-Target point
1
(12,48),(24,33),(27,58),(30,70),
(55,80),(59,52),(70,34),(70,65)
12,9,9,10,
9,10,12,7
1,12,3,2,
7,9,13,5
(10,15)
(80,75)
2
(20,70),(25,19),(25,39),(45,20),
(47,41),(50,61),(70,53),(75,74),(78,20)
20,9,9,9,
9,9,9,9,20
7,5,5,5,
5,5,5,5,7
(5,5)
(95,95)
3
(10,50),(20,20),(30,42),(30,80),(50,55),
(60,10),(60,80),(65,38),(75,65),(90,80)
10,9,8,10,10,
10,10,11,8,10
8,6,5,4,7,
6,7,6,8,10
(10,0)
(80,100)
Table 3
Results of hospIPA with varying PS values for Scenario-1
D
PS
30
40
50
75
100
10
Best
38.394
38.378
38.366
38.367
38.392
Worst
50.915
38.443
38.403
38.435
38.431
Mean
39.657
38.403
38.386
38.411
38.409
Std
3.817
0.023
0.016
0.025
0.017
15
Best
38.256
38.262
38.283
38.278
38.306
Worst
38.335
55.448
62.381
55.472
50.426
Mean
38.285
41.709
43.107
41.744
40.362
Std
0.028
6.987
9.802
6.981
4.578
20
Best
38.245
38.259
38.296
38.443
38.359
Worst
38.350
56.749
47.276
56.419
59.664
Mean
38.290
39.563
39.250
49.477
44.374
Std
0.038
4.672
2.722
7.153
6.921
25
Best
38.363
38.401
38.472
38.533
49.943
Worst
46.183
63.416
77.490
82.181
63.363
Mean
41.732
45.365
61.429
58.621
55.508
Std
3.852
8.956
12.009
16.920
5.172
Bold values show the better results
Table 4
Results of hospIPA with varying PS values for Scenario-2
D
PS
30
40
50
75
100
10
Best
57.584
57.785
57.559
57.644
57.698
Worst
63.757
61.398
63.962
60.526
60.298
Mean
60.632
59.123
60.316
58.356
59.276
Std
2.580
1.728
2.544
1.219
1.264
15
Best
54.412
54.215
54.195
54.211
56.364
Worst
58.646
58.367
58.573
56.448
60.634
Mean
56.493
56.200
56.204
55.365
58.323
Std
1.132
1.541
1.503
1.064
1.374
20
Best
53.813
53.756
53.677
53.998
53.767
Worst
64.771
57.127
64.403
64.728
57.217
Mean
55.374
54.944
56.904
56.253
54.304
Std
3.751
1.170
4.950
3.083
1.164
25
Best
53.648
53.744
53.595
53.721
53.686
Worst
63.181
63.156
60.563
64.823
62.755
Mean
55.984
56.352
55.405
55.721
57.713
Std
2.722
2.611
2.209
2.816
2.200
Bold values show the better results
Table 5
Results of hospIPA with varying PS values for Scenario-3
D
PS
30
40
50
75
100
10
Best
49.772
49.772
49.772
49.771
49.779
Worst
49.797
49.785
49.796
49.786
49.789
Mean
49.776
49.776
49.781
49.776
49.785
Std
0.009
0.004
0.010
0.004
0.004
15
Best
49.763
49.756
49.750
49.763
49.769
Worst
49.815
49.851
49.957
49.872
49.850
Mean
49.779
49.785
49.785
49.815
49.801
Std
0.016
0.034
0.062
0.042
0.029
20
Best
49.789
49.786
49.825
49.867
49.945
Worst
49.927
50.075
50.383
50.213
52.334
Mean
49.845
49.915
49.964
50.011
50.732
Std
0.055
0.113
0.176
0.088
0.900
25
Best
49.878
49.932
50.103
49.869
50.045
Worst
62.445
55.805
53.399
57.162
52.150
Mean
50.831
50.611
52.194
51.529
50.860
Std
3.158
1.443
1.078
2.151
0.689
Bold values show the better results
The results given in Tables 3, 4, 5 provide important information about the relatively stable and consistent performance of hospIPA. When the value being assigned to PS parameter is increased, a population based meta-heuristic can discover the search space more efficiently at the initial stage of a run and start subsequent operations with a set of solutions providing required diversities. However, the number of function evaluations spent per cycle, iteration or generation is directly proportional to the population size and a meta-heuristic terminates more quickly when its population size is set to higher values without executing algorithm specific search processes. On the other hand, when the population is configured with a small set of solutions, algorithm continues to search more longer and the probability of finding qualified solutions is boosted intrinsically. Even though the small set of solutions brings some advantages to the considered algorithm by allowing it for showing exclusive exploration and exploitation characteristics, the diversity of solutions can not be enough to represent the different regions of the space and convergence problems can arise from one run to another. As stated earlier, hospIPA hospitalizes the critical individuals corresponding to poor solutions of the problem and decreases the number of active individuals being used in the subsequent cycle. By executing this type of mechanism, hospIPA becomes capable of managing a population containing huge number of members. If hospIPA starts its optimization with a population containing a small number of members, the hospitalization mechanism can also decrease the solution diversity, but it should be noticed that hospIPA discharges some patients whose treatments conclude successfully and adjusts the number of active individuals dynamically. Also, hospIPA utilizes from a specialized treatment schema where the plasma being used for the patients is collected at the beginning of the second main stage in order to explore the neighborhood of the best solution discovered so far with the help of the the best solution of the current population and then transferred subtly to improve the exploitation characteristics of the algorithm.
The stable performance of hospIPA was validated over the results given in Tables 3, 4, 5 for varying PS parameter values. However, when the results of these tables are analyzed carefully, a subtle detail about the relationship between the PS, battlefield scenarios and their test cases getting intrinsically difficult with the higher values of D can also be detected. While hospIPA obtains slightly better paths for the test cases of Scenario-1 and Scenario-3 by setting the PS to 30 compared to the paths of the same algorithm by setting the PS to 40, 50, 75 or 100, it requires more than 30 individuals for planning more qualified paths related with the test cases of Scenario-2. The optimal paths being calculated for the test cases of Scenario-1 and Scenario-3 contain less maneuvers than the optimal paths being calculated for the test cases of Scenario-2 and setting PS to a small constant such as 30 allows hospIPA utilizing from specialized operations more and finding fine-tuned paths for a UAV or UCAV. If a test case similar to the test cases of Scenario-2 includes optimal path or paths with challenging maneuvers, assigning higher values to the PS increases the probability of obtaining initial solutions satisfying mentioned maneuvers partially or near fully. By combining the benefits of starting optimization with a huge number of initial solutions, newly designed hospitalization mechanism and treatment schema, it is seen that hospIPA calculates better paths for the test cases of Scenario-2 when its PS parameter is determined as 50, 75 or 100. For a visual representation of the battlefields and the paths found by hospIPA with 30 individuals, Figs. 3, 4, 5 can be controlled.
Fig. 3
The best and worst paths found by hospIPA for Scenario-1 with D equal to 10 (a), 15 (b), 20 (c) and 25 (d)
The quality evaluation of the discovered paths by hospIPA should be made over a comparison with other meta-heuristic based planners. For this purpose, a set of comparative studies between hospIPA and standard implementations of IPA, GA, MFO, SSA, PFA, SBO, SCA, GWO, AEO and improved variants of some of them such as GAPSO, ECTLBO, HSGWO-MSOS, CIPSO and NSEAEO was carried out. In order to guarantee that the comparative studies between hospIPA and other techniques are performed under the same conditions, each test case in Scenario-1, Scenario-2 and Scenario-3 was experimented 30 times by setting the population size to 30 and maximum evaluation number to 6000 and obtained results were presented in Tables 6, 7, 8. The first and foremost thing that can be extracted from the mentioned tables is the promising performance of hospIPA against its competitors. While hospIPA is determined as the best path planner among other algorithms with the average ranks equal to 1.750 for the Scenario-1 and Scenario-2, its superiority becomes more apparent for Scenario-3 and hospIPA is also determined as the best planner among other algorithms with the average rank equal to 1.000. Another important conclusion that can be extracted from Tables 6, 7, 8 is about that the performance of hospIPA increases generally compared to the other tested meta-heuristics when the number of segmentation points is chosen high enough. If the number of segmentation points is chosen high for the sensitivity, finding an optimum or near optimum path gets more difficult. However, the difficulty of path planning stemmed from the higher values of the number of segmentation points is handled successfully by hospIPA and its plasma generation and transfer schema. Because of the plasma generation depends on improving each parameter of the best solution with the help of selected donor, if the number of segmentation points is set to a relatively high value, collecting plasma that is qualifiable than the considered best solution and donor individual can be more probable. Nevertheless, it should be noticed that the cost of plasma generation in terms of consumed function evaluations rises and hospIPA can terminate without repeating its operations adequately. The mentioned drawback of hospIPA shows its effect on the path planning capabilities and hospIPA lags behind only NSEAEO and gets ranked as the second best technique for the test cases of Scenario-1 and Scenario-2 with D equal to 25.
Table 6
Comparison between hospIPA and other path planners for Scenario-1
D
hospIPA
IPA
GA
GAPSO
MFO
SSA
PFA
SBO
SCA
ECTLBO
HSGWO-MSOS
CIPSO
GWO
AEO
NSEAEO
10
Best
38.394
38.413
39.647
39.754
38.703
39.241
38.587
38.582
51.367
39.080
38.565
40.103
38.565
38.642
38.506
Worst
50.915
38.467
149.678
98.436
89.657
122.790
318.996
241.127
109.966
57.912
132.095
245.441
124.542
52.378
46.526
Mean
39.657
38.442
65.919
61.725
63.210
57.037
88.227
69.774
78.536
45.379
57.994
96.501
63.559
46.562
38.929
Std
3.817
0.022
34.300
24.078
15.100
22.882
68.817
52.248
15.700
5.016
20.323
52.917
25.320
2.902
1.526
Rank
3
1
11
8
9
6
14
12
13
4
7
15
10
5
2
15
Best
38.256
38.582
41.148
42.253
42.814
39.022
38.466
38.402
63.348
51.425
39.068
39.912
38.560
38.302
38.291
Worst
38.335
43.773
232.634
80.672
266.532
105.756
210.693
203.260
209.946
112.515
115.202
223.591
97.052
58.778
58.042
Mean
38.285
39.357
81.644
57.736
90.566
56.959
69.610
66.273
117.484
82.711
61.265
92.743
52.706
45.410
39.074
Std
0.028
1.764
50.303
13.416
44.985
19.697
45.982
38.427
30.720
17.811
13.564
49.986
15.555
9.140
3.584
Rank
1
3
11
7
13
6
10
9
15
12
8
14
5
4
2
20
Best
38.245
47.952
49.553
51.378
60.290
44.282
39.543
39.351
66.430
49.978
43.637
42.977
39.083
39.508
38.369
Worst
38.350
59.512
301.088
118.726
396.167
219.033
145.096
174.852
373.008
211.499
141.557
503.643
97.515
52.182
39.606
Mean
38.290
53.747
135.674
80.166
158.211
81.580
59.488
65.309
181.787
131.767
67.328
137.171
51.652
43.195
38.769
Std
0.038
3.362
59.459
14.844
84.234
40.491
25.292
31.510
71.888
44.952
22.644
93.903
11.872
3.994
0.293
Rank
1
5
12
9
14
10
6
7
15
11
8
13
4
3
2
25
Best
38.363
63.018
102.967
84.872
84.829
60.426
41.970
47.433
57.273
81.008
43.554
65.411
41.514
41.017
39.299
Worst
46.183
80.786
439.336
199.769
2246.004
276.890
166.076
137.645
580.501
454.877
167.021
647.234
90.889
51.240
40.108
Mean
41.732
76.723
173.069
122.905
282.449
122.920
77.936
68.971
252.098
187.488
81.527
205.902
50.307
42.697
39.615
Std
3.852
4.735
68.499
25.532
388.776
50.105
30.544
21.100
150.970
100.894
33.750
141.196
11.722
1.995
0.182
Rank
2
6
11
9
15
10
7
5
14
12
8
13
4
3
1
Average rank
1.750
3.750
11.250
8.250
12.750
8.000
9.250
8.250
14.250
9.750
7.750
13.750
5.750
3.750
1.750
Overall rank
1
3
12
8
13
7
10
8
15
11
6
14
5
3
1
Bold values show the better results
Table 7
Comparison between hospIPA and other path planners for Scenario-2
D
hospIPA
IPA
GA
GAPSO
MFO
SSA
PFA
SBO
SCA
ECTLBO
HSGWO -MSOS
CIPSO
GWO
AEO
NSEAEO
10
Best
57.584
57.802
61.668
61.721
59.158
59.726
57.896
58.135
88.336
60.541
57.869
59.643
61.239
58.590
56.204
Worst
63.757
61.205
312.343
130.981
270.787
221.180
598.749
525.925
221.805
99.426
109.680
1677.440
120.087
72.425
69.813
Mean
60.632
58.705
99.630
94.912
90.261
103.382
166.529
107.482
123.145
75.787
67.683
158.779
79.191
65.734
57.474
Std
2.580
1.299
47.060
20.382
40.877
48.416
159.441
104.792
26.882
10.033
13.555
292.587
14.446
6.742
2.659
Rank
3
2
10
9
8
11
15
12
13
6
5
14
7
4
1
15
Best
54.412
57.274
66.193
67.163
60.710
57.056
58.235
57.025
105.006
69.484
57.225
70.080
55.091
54.796
55.949
Worst
58.646
62.937
551.464
241.538
670.571
398.873
268.370
230.585
334.849
235.419
201.640
537.313
127.408
75.777
66.180
Mean
56.493
59.266
162.766
111.416
171.648
131.444
122.142
82.483
184.436
152.439
80.750
230.860
78.095
57.660
60.250
Std
1.132
1.993
106.642
47.446
154.733
79.183
68.468
45.101
56.189
36.838
32.343
134.219
17.432
4.269
2.654
Rank
1
3
12
8
13
10
9
7
14
11
6
15
5
2
4
20
Best
53.813
63.286
79.686
97.730
68.284
61.780
54.954
57.605
104.952
91.201
64.063
93.525
57.437
54.659
54.606
Worst
64.771
67.483
706.929
227.695
3271.735
369.486
353.765
218.618
842.337
505.444
216.044
2238.837
182.698
84.463
58.735
Mean
55.374
65.470
207.937
130.496
314.948
184.588
133.978
113.440
341.186
249.007
102.336
397.446
77.195
57.304
55.451
Std
3.751
1.589
148.025
27.769
576.712
92.907
82.512
51.390
193.248
87.854
42.546
471.670
23.585
5.344
0.861
Rank
1
4
11
8
13
10
9
7
14
12
6
15
5
3
2
25
Best
53.648
68.697
126.733
111.400
83.876
74.451
61.648
65.162
90.491
153.223
60.073
78.704
57.743
55.240
55.061
Worst
63.181
78.183
1609.607
272.376
2182.624
366.400
489.869
577.207
1503.886
905.050
432.937
1980.693
110.751
63.672
56.795
Mean
55.984
73.419
368.502
160.852
352.786
168.946
175.648
162.158
455.569
495.212
136.757
460.333
71.813
56.626
55.922
Std
2.722
3.364
285.403
33.317
370.204
67.626
107.691
107.937
283.937
211.767
85.818
401.653
14.139
1.532
0.433
Rank
2
5
12
7
11
9
10
8
13
15
6
14
4
3
1
Average rank
1.750
3.500
11.250
8.000
11.250
10.000
10.750
8.500
13.500
11.000
5.750
14.500
5.250
3.000
2.000
Overall rank
1
4
12
7
12
9
10
8
14
11
6
15
5
3
2
Bold values show the better results
Table 8
Comparison between hospIPA and other path planners for Scenario-3
D
hospIPA
IPA
GA
GAPSO
MFO
SSA
PFA
SBO
SCA
ECTLBO
HSGWO -MSOS
CIPSO
GWO
AEO
NSEAEO
10
Best
49.772
49.785
56.638
55.832
57.290
53.727
53.576
53.350
70.344
56.020
53.967
53.589
53.639
52.638
51.186
Worst
49.797
49.817
424.807
466.463
186.974
140.715
499.851
330.582
270.192
100.391
159.386
454.467
290.474
62.066
78.808
Mean
49.777
49.807
124.320
105.328
97.579
79.831
144.949
120.515
132.559
76.156
77.725
137.143
82.311
59.555
55.885
Std
0.009
0.012
100.904
89.177
28.811
28.453
121.329
82.407
55.161
12.884
21.098
103.686
43.614
4.187
5.391
Rank
1
2
12
10
9
7
15
11
13
5
6
14
8
4
3
15
Best
49.763
50.217
57.131
58.074
58.535
51.142
52.823
52.724
103.034
80.003
55.461
59.655
53.811
50.429
50.155
Worst
49.815
50.400
239.798
146.225
333.608
206.804
423.168
230.734
472.427
193.046
236.558
381.843
160.144
220.555
54.568
Mean
49.779
50.283
98.429
74.533
103.670
77.403
125.829
84.734
213.126
130.755
93.354
151.807
70.679
61.643
52.163
Std
0.016
0.064
42.726
17.151
58.943
42.210
90.463
50.770
90.269
21.543
44.605
83.722
25.917
31.234
1.434
Rank
1
2
10
6
11
7
12
8
15
13
9
14
5
4
3
20
Best
49.789
51.591
86.357
64.500
76.072
52.148
54.123
53.267
104.708
95.829
57.597
60.565
53.736
50.843
50.257
Worst
49.927
54.415
397.440
261.488
1048.996
303.329
435.366
306.776
573.087
380.192
203.090
779.707
365.019
64.468
54.184
Mean
49.845
52.841
180.662
96.442
236.951
91.752
150.474
113.194
258.060
215.535
108.225
224.292
82.149
54.906
52.492
Std
0.055
1.015
83.360
35.996
217.852
53.819
96.867
58.481
119.930
59.956
38.526
154.675
57.171
2.108
1.113
Rank
1
3
11
7
14
6
10
9
15
12
8
13
5
4
2
25
Best
49.878
60.958
118.945
95.917
96.659
57.896
55.473
51.546
83.408
113.601
59.155
100.647
54.723
53.19031
51.090
Worst
62.445
73.486
789.659
230.781
1150.201
262.789
461.069
296.169
926.916
688.007
239.553
888.148
166.014
57.591
54.327
Mean
50.831
65.097
227.699
138.983
395.133
105.547
129.069
112.060
356.789
347.054
111.295
367.574
74.745
55.191
53.648
Std
3.158
4.119
133.016
28.790
283.375
48.219
93.393
53.088
206.941
159.762
44.611
196.039
25.892
0.916
0.649
Rank
1
4
11
10
15
6
9
8
13
12
7
14
5
3
2
Average rank
1.000
2.750
11.000
8.250
12.250
6.500
11.500
9.000
14.000
10.500
7.500
13.750
5.750
3.750
2.500
Overall rank
1
3
11
8
13
6
12
9
15
10
7
14
5
4
2
Bold values show the better results
5.2 Planning paths for three-dimensional battlefields with hospIPA
The investigations about the path planning performance of hospIPA are continued with the experiments by using three-dimensional battlefield scenarios that are called Scenario-4 and Scenario-5. The Scenario-4 and Scenario-5 represent an enemy threat with a cylinder whose center, radius and height are known as detailed in Table 9 [64]. The test cases generated by assigning 10, 15, 20 and 25 constants to D for Scenario-4 and Scenario-5 were solved with hospIPA. The PS parameter of hospIPA was set to 30, 40, 50, 75 and 100 and 30 independent runs were carried out after determining maximum evaluation number as 6000. The best, worst, mean best objective function values and standard deviations of 30 independent runs were summarized in Table 10 for Scenario-4 and Table (11) for Scenario-5. From the results given in Table (10) and Table 11, it is seen that hospIPA is capable of protecting previously proven stable performance especially for the 10 and 15 dimensional cases of the considered scenarios. However, when the number of segmentation points is increased and determined as 20 and 25, hospIPA requires selection of PS parameter more carefully. The existence of z-coordinate and the higher number of segmentation points bring additional complexity to the path planning problem and hospIPA should iterate more by starting the search with relatively small PS values such as 30 or 40. If hospIPA iterates more by starting the search with relatively small PS values, the discrimination between the hospitalized and non-hospitalized individuals is carried out quickly. Moreover, repeating the plasma collection operations for each iteration allows hospIPA to explore the vicinity of the best solution and find more qualified plasma being used for the treatment of hospitalized individuals. The best and worst paths found by hospIPA with PS equal to 30 are depicted in Figs. 6, 7 for a pictorial investigations about the tested three-dimensional battlefields their and maneuver requirements.
Table 9
Details of battlefields used for three-dimensional path planning
Sc
Threat centers
Threat radius
Threat height
Threat grade
Start-Target point
4
(23,60),(30,15),(45,27),(50,75),
(60,10),(70,85),(78,62),(90,80)
11,9,10,11,
10,8,8,7
120,80,140,110,
130,100,144,160
8,10,8,11,
8,7,4,9
(0,0,10)
(100,90,75)
5
(25,15),(25,80),(39,40),(45,70),
(55,10),(70,80),(75,50),(85,25)
10,10,8,9,
12,7,11,11
80,60,100,120,
130,140,80,90
10,8,5,11,
3,6,13,4
(0,0,0)
(80,75,50)
Table 10
Results of hospIPA with varying PS values for Scenario-4
D
PS
30
40
50
75
100
10
Best
60.396
60.391
60.387
60.393
60.380
Worst
60.428
60.430
66.226
60.463
60.414
Mean
60.411
60.405
61.377
60.419
60.401
Std
0.012
0.016
2.205
0.027
0.016
15
Best
60.477
60.532
60.524
60.522
60.549
Worst
69.564
61.843
69.558
61.923
60.591
Mean
62.201
60.766
62.402
60.791
60.567
Std
3.381
0.490
3.304
0.517
0.015
20
Best
60.436
60.386
60.437
60.581
60.463
Worst
61.698
61.725
68.445
62.273
61.962
Mean
60.703
60.848
62.083
61.348
61.324
Std
0.505
0.597
2.935
0.661
0.665
25
Best
60.526
60.485
60.558
60.522
60.776
Worst
73.029
62.008
61.855
62.970
78.807
Mean
62.658
60.690
61.097
61.155
63.344
Std
3.549
0.451
0.503
0.685
4.385
Bold values show the better results
Table 11
Results of hospIPA with varying PS values for Scenario-5
D
PS
30
40
50
75
100
10
Best
49.203
49.241
49.236
49.185
49.185
Worst
52.350
49.296
49.417
49.256
49.882
Mean
50.168
49.263
49.320
49.216
49.416
Std
1.453
0.022
0.071
0.030
0.299
15
Best
49.129
49.141
49.149
49.160
49.186
Worst
53.674
49.234
53.508
49.266
53.486
Mean
49.905
49.171
50.069
49.187
49.824
Std
1.714
0.032
1.383
0.039
1.462
20
Best
49.121
49.152
49.146
49.147
49.342
Worst
52.523
52.604
49.526
74.710
56.957
Mean
49.616
49.465
49.248
52.620
50.686
Std
1.160
0.857
0.126
8.812
2.294
25
Best
49.092
49.098
49.342
49.283
49.582
Worst
58.451
73.209
90.681
52.457
63.061
Mean
50.067
53.968
57.025
50.163
52.808
Std
2.308
6.702
12.980
1.199
4.409
Bold values show the better results
Fig. 6
The best and worst paths found by hospIPA for Scenario-4 with D equal to 25
In order to decide that whether the promising performance of hospIPA for the fixed altitude battlefield scenarios against other meta-heuristic based path planners is also achieved on the three-dimensional battlefield scenarios or not, a comparison between hospIPA and IPA, GA, MFO, SSA, PFA, SBO, SCA, GWO, AEO, GAPSO, ECTLBO, HSGWO-MSOS, CIPSO and NSEAEO was made again. Each test case in Scenario-4 and Scenario-5 was solved 30 times with hospIPA and other mentioned meta-heuristic algorithms by assigning 30 and 6000 constants to the population size and maximum evaluation number and then their results were presented in Tables 12 and 13. When the results given in Tables 12 and 13 are investigated, it is seen that hospIPA obtains better paths than other competitors for seventy-five percent of all test cases about the three-dimensional battlefields and gets ranked as the best path planner. Moreover, for the remaining 25% of all test cases about the three-dimensional battlefields, hospIPA lags slightly behind only IP algorithm and gets ranked as the second best path planner. If the details of two test cases in which IPA performs better than hospIPA are controlled, it is understood that the number of segmentation points is equal to 10 or 15 and the difference between the mean best objective function values of the IPA based techniques is relatively small. However, if the details of the test cases in which hospIPA performs better than IPA are controlled, it is observed that the number of segmentation points for the majority of the cases is equal to 20 or 25 and the difference between the mean best objective function values of IPA based techniques is considerable high and demonstrates the effectiveness of hospIPA becoming more clearer with the growing difficulty of the path planning problem.
Table 12
Comparison between hospIPA and other path planners for Scenario-4
D
hospIPA
IPA
GA
GAPSO
MFO
SSA
PFA
SBO
SCA
ECTLBO
HSGWO -MSOS
CIPSO
GWO
AEO
NSEAEO
10
Best
60.396
60.428
76.756
67.111
75.029
67.312
70.768
71.199
91.852
68.907
71.534
70.308
64.749
62.555
60.845
Worst
60.428
60.625
158.217
102.925
132.448
129.544
198.560
119.071
190.916
124.651
154.507
241.003
158.006
95.048
68.715
Mean
60.411
60.454
110.254
77.969
97.308
88.945
117.956
86.433
127.029
92.787
97.594
133.476
82.311
69.734
63.080
Std
0.012
0.049
22.965
8.663
13.529
15.398
38.821
10.365
21.023
15.308
25.590
53.087
20.561
7.885
2.400
Rank
1
2
12
5
10
8
13
7
14
9
11
15
6
4
3
15
Best
60.477
60.704
113.068
94.349
112.402
90.111
75.349
66.831
169.526
80.845
77.291
73.293
75.104
69.351
63.000
Worst
69.564
62.177
400.388
225.290
229.757
249.239
321.773
166.029
387.570
228.600
233.960
470.726
188.100
116.736
71.903
Mean
62.201
61.545
236.271
130.595
158.716
171.551
169.957
103.489
256.540
158.629
127.876
166.228
97.445
85.650
66.879
Std
3.381
0.481
70.877
26.801
32.652
36.149
63.589
30.486
42.378
41.715
37.616
109.587
26.882
13.987
2.439
Rank
2
1
14
8
10
13
12
6
15
9
7
11
5
4
3
20
Best
60.436
69.583
212.118
166.114
147.388
163.500
153.119
84.385
260.958
115.410
78.344
104.200
82.119
75.350
69.317
Worst
61.698
83.571
645.485
242.612
513.245
370.773
605.612
277.449
528.958
383.258
287.395
494.316
208.406
177.231
82.313
Mean
60.703
76.906
352.004
201.938
241.840
245.420
242.891
164.267
381.926
262.510
153.699
221.071
125.478
101.527
73.513
Std
0.505
5.166
94.715
22.093
74.172
53.865
89.024
49.330
52.763
75.975
64.485
95.919
41.968
28.088
3.058
Rank
1
3
14
8
10
12
11
7
15
13
6
9
5
4
2
25
Best
60.526
72.321
360.871
229.277
173.405
193.243
152.942
128.840
375.237
209.783
80.544
140.185
79.462
82.095
73.671
Worst
73.029
124.608
815.003
386.461
653.905
483.707
459.100
411.034
1078.313
584.577
379.766
736.038
377.548
151.071
85.915
Mean
62.658
93.485
470.697
288.554
305.863
314.832
283.540
244.363
560.718
366.932
137.188
359.509
135.731
90.038
77.894
Std
3.549
19.641
109.875
35.624
101.403
62.828
64.289
60.948
145.646
100.359
75.610
133.501
60.112
12.873
2.698
Rank
1
4
14
9
10
11
8
7
15
13
6
12
5
3
2
Average rank
1.250
2.500
13.500
7.500
10.000
11.000
11.000
6.750
14.750
11.000
7.500
11.750
5.250
3.750
2.500
Overall rank
1
2
14
7
9
10
10
6
15
10
8
13
5
4
2
Bold values show the better results
Table 13
Comparison between hospIPA and other path planners for Scenario-5
D
hospIPA
IPA
GA
GAPSO
MFO
SSA
PFA
SBO
SCA
ECTLBO
HSGWO -MSOS
CIPSO
GWO
AEO
NSEAEO
10
Best
49.203
49.039
63.953
58.587
65.548
55.416
65.060
55.742
83.149
58.946
57.964
68.905
54.410
55.725
49.521
Worst
52.350
49.111
210.849
125.263
228.838
159.560
219.133
97.709
161.330
105.822
124.814
270.951
114.798
64.532
56.788
Mean
50.168
49.087
105.116
70.071
99.140
80.013
119.546
66.524
113.678
83.595
87.189
114.524
83.284
57.357
52.663
Std
1.453
0.032
33.480
13.037
31.422
22.023
38.797
11.571
20.770
16.218
13.503
44.396
15.829
1.826
2.634
Rank
2
1
12
6
11
7
15
5
13
9
10
14
8
4
3
15
Best
49.129
49.828
144.123
97.437
71.380
92.825
96.046
64.484
165.075
78.945
61.897
102.656
70.663
55.740
52.141
Worst
53.674
53.455
414.686
175.326
288.325
234.469
281.901
157.030
353.789
287.699
239.709
711.134
180.964
100.518
58.459
Mean
49.905
51.717
225.811
133.764
137.688
145.071
187.006
93.061
267.275
188.067
137.615
219.629
127.641
71.299
55.378
Std
1.714
1.311
58.365
15.323
55.691
30.044
41.949
22.135
44.212
56.586
48.832
144.745
30.012
11.9766
2.023
Rank
1
2
14
7
9
10
11
5
15
12
8
13
6
4
3
20
Best
49.121
57.629
278.666
153.305
77.246
163.633
139.168
83.314
231.104
160.508
61.037
107.833
62.021
60.427
55.016
Worst
52.523
78.618
590.340
269.286
570.129
298.376
712.942
207.097
661.466
466.870
386.527
1184.152
269.834
68.067
60.604
Mean
49.616
69.224
356.830
203.675
245.734
220.348
260.349
144.505
378.619
311.800
176.983
410.190
161.174
63.330
57.818
Std
1.160
8.138
77.825
33.018
109.796
37.509
103.924
34.728
98.131
92.835
87.104
302.314
54.730
1.865
1.277
Rank
1
4
13
8
10
9
11
5
14
12
7
15
6
3
2
25
Best
49.092
67.011
381.902
260.008
107.694
184.216
179.310
146.369
312.985
201.941
68.600
113.614
65.933
63.512
57.735
Worst
58.451
109.927
939.446
419.365
581.034
631.841
446.313
468.136
1431.116
697.660
455.734
1423.391
427.776
166.026
60.152
Mean
50.067
92.290
525.074
317.795
323.891
345.003
316.295
223.264
591.812
435.614
261.152
457.170
216.622
110.245
58.848
Std
2.308
16.163
122.076
35.055
121.414
117.991
59.176
62.073
252.107
136.389
117.664
322.862
60.112
31.083
0.652
Rank
1
3
14
9
10
11
8
6
15
12
7
13
5
4
2
Average rank
1.250
2.500
13.250
7.500
10.000
9.250
11.250
5.250
14.250
11.250
8.000
13.750
6.250
3.750
2.500
Overall rank
1
2
13
7
10
9
11
5
15
11
8
14
6
4
2
Bold values show the better results
5.3 Execution times of hospIPA
The hospitalization mechanism used by hospIPA completely changed the interactions between the members of population. Also, it should be noticed that the operations to do with the collection of plasma in hospIPA invoke D different calls to the procedure responsible for calculating the objective function value of a solution and a cycle consumes D more evaluations. If most of the individuals are in hospital, the phase related with the distribution of infection completes quickly and considerable amount of function evaluations are spent for the plasma collection and its transfer to the receivers. Moreover, if there are more than one hospitalized individual, hospIPA does not require compute intensive operations for selecting the most critical individual with the purpose of hospitalization or the best one as a donor. In order to analyze how the mentioned situations effect the execution time of hospIPA and generate a difference compared to the execution time of the standard IPA, 30 independent runs by taking PS and maximum evaluation number equal to 30 and 6000 were carried out. Both hospIPA and IPA were implemented in C programming language and experiments were conducted on a Fedora-34 computer with an Intel i5-10500 processor. For each run, the time elapsed until termination was recorded in terms of seconds and the best, worst, mean execution times and the calculated standard deviations were presented in Table 14. From the results in Table 14, it is seen that hospIPA requires less time than the standard IPA for 85 percent of all test cases. Executing a decision making approach and dynamically adjusting the number of receivers as in the mechanisms of hospIPA reduce the computational burden stemmed from the selection of the currently hospitalized individual and the donor being used for the collection of plasma.
Table 14
Elapsed times of hospIPA and IPA with PS equal to 30
D
Scenario-1
Scenario-2
Scenario-3
Scenario-4
Scenario-5
hospIPA
IPA
hospIPA
IPA
hospIPA
IPA
hospIPA
IPA
hospIPA
IPA
10
Best
0.057
0.073
0.067
0.068
0.061
0.076
0.063
0.068
0.063
0.066
Worst
0.083
0.103
0.116
0.128
0.103
0.101
0.085
0.100
0.097
0.092
Mean
0.062
0.069
0.076
0.084
0.069
0.084
0.071
0.078
0.071
0.075
Std
0.006
0.016
0.012
0.015
0.009
0.017
0.005
0.009
0.008
0.008
15
Best
0.079
0.090
0.090
0.099
0.092
0.104
0.100
0.099
0.096
0.098
Worst
0.107
0.126
0.141
0.125
0.173
0.138
0.134
0.135
0.129
0.138
Mean
0.085
0.097
0.101
0.109
0.116
0.115
0.106
0.110
0.104
0.109
Std
0.006
0.008
0.012
0.006
0.018
0.009
0.007
0.011
0.008
0.011
20
Best
0.107
0.117
0.116
0.131
0.123
0.136
0.118
0.132
0.125
0.133
Worst
0.144
0.144
0.160
0.161
0.151
0.174
0.159
0.180
0.169
0.153
Mean
0.117
0.125
0.129
0.141
0.131
0.150
0.136
0.144
0.139
0.139
Std
0.007
0.007
0.011
0.009
0.009
0.010
0.010
0.012
0.012
0.005
25
Best
0.136
0.145
0.144
0.161
0.147
0.171
0.162
0.160
0.158
0.162
Worst
0.170
0.186
0.228
0.189
0.226
0.207
0.201
0.208
0.203
0.189
Mean
0.149
0.155
0.161
0.168
0.160
0.183
0.173
0.176
0.170
0.170
Std
0.009
0.010
0.014
0.007
0.015
0.011
0.010
0.011
0.017
0.007
5.4 Convergence performance and statistical analysis of hospIPA
For a numerical analysis about the convergence characteristics of a meta-heuristic algorithm, Success rate (Sr) and Mean evaluations (Me) are the two common metrics [77]. If an algorithm finds a solution whose objective function value is better than the previously determined threshold until the end of a run, the considered algorithm is assumed as successful for this run and the ratio between the number of runs for which algorithm is successful and total number of runs corresponds to the Sr metric. When the minimum number of evaluations required to find a solution whose objective function is better than the threshold for each successful run is recorded and then averaged, the Me value is obtained. The threshold was determined as 60 and 70 for the two and three-dimensional battlefield scenarios respectively and the calculated Sr and Me values of hospIPA and IPA with 30 individuals were presented in Table 15. The Sr and Me values given in Table 15 prove the superior convergence performance of hospIPA compared to the convergence performance of IPA. The Sr value of hospIPA is found equal or higher than the Sr value of IPA for 19 of all 20 test cases. Also, it should be noted that the Me value of hospIPA is calculated less than the Me value of IPA for 16 of 19 test cases in which the Sr value of hospIPA is equal or higher than the Sr value of IPA. When the remaining 3 test cases in which IPA performs better than hospIPA by considering the Me values are analyzed, it should be emphasized that while IPA excesses threshold only for 10 percent of all runs, hospIPA excesses the threshold for 90 percent of all runs and shows its nine times stable and consistent performance against IPA. The better convergence performance of hospIPA even though IPA outperforms its competitor by evaluating the Me metric for some test cases can be further validated with the convergence curves of two and three-dimensional test cases containing 25 segmentation points over Fig. 8.
Table 15
Sr and Me values of hospIPA and IPA with PS equal to 30
D
Scenario-1
Scenario-2
Scenario-3
Scenario-4
Scenario-5
hospIPA
IPA
hospIPA
IPA
hospIPA
IPA
hospIPA
IPA
hospIPA
IPA
10
Sr
100.000
100.000
36.667
73.333
100.000
100.000
100.000
100.000
100.000
100.000
Me
604.700
875.200
810.000
3598.818
84.933
375.267
159.567
1489.933
173.033
771.333
15
Sr
100.000
83.333
100.000
90.000
100.000
100.000
100.000
100.000
100.000
100.000
Me
591.133
3849.560
901.533
4091.222
352.833
1548.967
483.233
2639.067
206.333
2083.500
20
Sr
100.000
100.000
86.667
3.333
100.000
100.000
100.000
33.333
100.000
70.000
Me
1401.433
3834.333
1429.500
1468.000
641.333
3590.567
758.033
3963.800
508.533
4289.238
25
Sr
100.000
56.667
90.000
10.000
93.333
10.000
90.000
10.000
100.000
26.667
Me
1708.600
2519.412
1848.889
752.000
1051.714
602.000
1756.630
641.333
1498.333
3685.000
Fig. 8
Convergence curves of hospIPA and IPA for Scenario-1 (a), Scenario-2 (b), Scenario-3 (c), Scenario-4 (d) and Scenario-5 (e)
Even though the positive contribution of the proposed hospitalization mechanism and treatment schema on the solving performance of hospIPA and its superiority against standard implementation of IPA can be demonstrated by checking the results of comparative studies, an appropriate statistical test should also be employed for proving the path planning capabilities of hospIPA. The Wilcoxon signed rank test is used commonly in order to decide that one of the compared techniques is statistically better [77]. If the significance level abbreviated as \(\rho\) is less than a constant that is usually chosen as 0.05, it is accepted that the difference between two techniques is enough to generate statistical significance in favor of one of them [77]. The Wilcoxon signed rank test results for the comparison of hospIPA and IPA with 30 individuals were given in Table 16. While the Z value corresponds to the test statistics, \(W+\) and \(W-\) show the sum of ranks for which IPA is better than hospIPA and the sum of ranks for which hospIPA is better than IPA by considering 30 independent runs respectively in Table 16. When the \(\rho\) values calculated for the comparison between hospIPA and IPA are evaluated, it is validated that hospIPA is able to calculate paths whose qualities statistically apparent for seventeen of twenty test cases. Only for the cases with 10 segmentation points belonging to Scenario-2 and Scenario-5, the Wilcoxon signed rank test indicates that the statistical significance is in favor of IPA. As seen from the properties and number of the test cases for which hospIPA is statistically better than IPA, the newly introduced variant manages the difficulties of the paths being planned in detail by calculating the convenient values for more than 15 segmentation points.
Table 16
Results of the Wilcoxon signed rank test for hospIPA and IPA
D
Scenario-1
Scenario-2
Scenario-3
Scenario-4
Scenario-5
hospIPA vs IPA
hospIPA vs IPA
hospIPA vs IPA
hospIPA vs IPA
hospIPA vs IPA
10
\(\rho\) Val
2.758e\(-\)03
6.010e\(-\)04
1.635e\(-\)06
1.501e\(-\)06
1.450e\(-\)06
Z Val
2.993
\(-\)3.431
4.793
4.811
\(-\)4.818
\(W+\)
87
399
0
0
465
\(W-\)
378
66
465
465
0
Sign
hospIPA
IPA
hospIPA
hospIPA
IPA
15
\(\rho\) Val
1.557e\(-\)06
1.727e\(-\)06
1.687e\(-\)06
7.243e\(-\)02
7.592e\(-\)06
Z Val
4.803
4.782
4.787
1.796
4.476
\(W+\)
0
0
0
145
15
\(W-\)
465
465
465
320
450
Sign
hospIPA
hospIPA
hospIPA
-
hospIPA
20
\(\rho\) Val
1.684e\(-\)06
1.766e\(-\)06
1.697e\(-\)06
1.702e\(-\)06
1.692e\(-\)06
Z Val
4.788
4.778
4.786
4.785
4.787
\(W+\)
0
0
0
0
0
\(W-\)
465
465
465
465
465
Sign
hospIPA
hospIPA
hospIPA
hospIPA
hospIPA
25
\(\rho\) Val
1.717e\(-\)06
1.729e\(-\)06
1.767e\(-\)06
1.781e\(-\)06
1.756e\(-\)06
Z Val
4.784
4.782
4.778
4.776
4.779
\(W+\)
0
0
0
0
0
\(W-\)
465
465
465
465
465
Sign
hospIPA
hospIPA
hospIPA
hospIPA
hospIPA
Bold values show the statistically significant algorithms
6 Conclusion
The advantages coming with the usage of UAVs and UCAVs caused strategical changes on the military projections of nations and immense budgets were released in order to improve the performance and task success of these modern vehicles. Because of the direct impact on the performance and task success of a UAV or UCAV system, solving a problem called path planning optimally by considering the enemy threats, fuel or battery consumption and some limitations about the maneuverability became more important. Immune Plasma algorithm (IP algorithm or IPA) has been introduced recently to the literature of intelligent optimization techniques. In this study, IP algorithm was powered with a hospitalization mechanism that generates a hospital, fills it with the critical patients corresponding to the poor solutions of the population and decides who will be discharged after the plasma transfer. Moreover, the existing treatment schema was remodeled in a manner that the plasma being transferred will be gathered over the best solution found so far and the donor chosen from the population. The new IPA variant supported with the mentioned hospitalization mechanism and treatment schema that together remove the requirement of NoR and NoD parameters was named the hospital IPA (hospIPA) and employed as a UAV or UCAV path planner.
The paths planned by hospIPA for two and three-dimensional battlefields were compared with the paths planned by a set of well-known meta-heuristics and some of their variants. The experimental studies allowed to conclude that hospIPA is more qualified path planner than the tested algorithms. While hospIPA outperforms the considered path planners for the fourteen of all twenty test cases, it is ranked as the second or third best solver for the six remaining cases and still proves its competitive performance. The hospitalization mechanism selecting the poor solutions and quarantining them in a hospital helps hospIPA to explore the vicinity of the qualified solutions steadily. Furthermore, if the newly designed treatment schema that models the collection of the plasma by exploiting the best solution found so far does not give a substantial contribution to a hospitalized individual or it is not enough to discharging, hospitalization is continued for the considered individual. As an expected result of the mentioned decision, the evaluations are consumed more effectively for the non-hospitalized individuals or qualified solutions in the population. The future works can be devoted to the researches about the IPA based path planners for which the hospitalization mechanism and treatment schema are selected adaptively by considering the properties of the population. Also, the performance of the hospIPA or similar variants can be investigated by planning paths for multiple UAV or UCAV systems in a battlefield with static and dynamic enemy threats, non-flight zones and its challenging cases containing relatively high number of segmentation points.
Acknowledgements
The author acknowledges to anonymous reviewer for their thoughtful suggestions and comments.
Declarations
Conflict of interest
The author has no competing interests to declare that are relevant to the content of this article.
Ethics approval and consent to participate
This article does not contain any studies with human participants or animals performed by any of the authors.
Consent for publication
This article does not contain any studies with human participants or animals performed by any of the authors.
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