Nomenclature

UAV

unmanned aerial vehicle

UCAV

unmanned combat aerial vehicle

IPA

immune plasma algorithm

gintIPA

greedy initialiser immune plasma algorithm

2.0 Formulation of the Geometrical Path Planning Problem

Critical details of an enemy threat about how it detects or senses a UAV or UCAV, how the detection or sensing range and danger level are appointed, and how a path from the start point to the target point is represented are all related to the mathematical model of the path planning problem. A common mathematical model that can also be called the geometrical model for the UAV or UCAV path planning problem represents an enemy threat by a circle for which the centre coordinate and radius showing the sensing range are known in advance. Moreover, for representing a UAV or UCAV path, a set of points whose members are chosen subtly after considering the minimisation requirements to do with the enemy threats, fuel or battery consumption, flight length and turning manoeuvres is employed.

The path planning operations for the geometrical description of the problem are started by drawing a reference line from the start point or ${p_s} = \{ {{x_s},{y_s}} \}$ to the target point or ${p_t} = \left\{ {{x_t},{y_t}} \right\}$ and dividing it into $D + 1$ line segments with the help of $D$ segmentation points. When the segmentation of the reference line is concluded, a line named as the ${L_1}$ passing through the first segmentation point and intersecting the reference line vertically, a line named as the ${L_2}$ passing through the second segmentation point and intersecting the reference line vertically and so on are drawn and then a line set $L = \left\{ {{L_1},{L_2}, \ldots ,{L_{D - 1}},{L_D}} \right\}$ is obtained. The set $L$ becomes a beacon for the subsequent operations of the path planning problem. If each line in the set $L$ is used in a manner that the point ${p_1}$ from the line ${L_1}$ , the point ${p_2}$ from the line ${L_2}$ and so on are chosen and then combined with the ${p_s}$ and ${p_t}$ appropriately, a point set $P = \left\{ {{p_s},{p_1},{p_2}, \ldots ,{p_{D - 1}},{p_D},{p_t}} \right\}$ can be organised. The set $P$ is capable of intrinsically describing a UAV or UCAV path. When the neighbour point pairs of the set $P$ are connected with a line segment, a path from the point ${p_s}$ to the point ${p_t}$ is generated [Reference Niu, Yan, Wang and Niu47].

Some calculations required to find the members of the set $L$ and set $P$ should be supported with a strong geometrical background. Each line in the set $L$ must vertically intersect the reference line by only passing through the matched segmentation point. Furthermore, the points of the set $P$ ranging from the ${p_1}$ to ${p_D}$ must satisfy the equation of the corresponding line in the set $L$ . For reducing the calculation difficulties about the members of set $L$ and set $P$ , a new coordinate system that accepts the reference line between the ${p_s}$ and ${p_t}$ as the horizontal axis can be utilised. To transform the point ${p_k}$ in the original coordinate system into the new coordinate system, Equation (1) can be referenced. While the ${x_k}$ and ${y_k}$ show the horizontal or $x$ -axis value and vertical or $y$ -axis value of the point ${p_k}$ , the ${\dot x_k}$ and ${\dot x_y}$ represent the $\dot x$ -axis and $\dot y$ -axis values of the transformed point ${p_k}$ or ${\dot p_k}$ in Equation (1) [Reference Niu, Yan, Wang and Niu47]. Also, the angle between the $x$ -axis and reference line is symbolised with the $\theta $ term and $arctan(({y_t} - {y_s})/( {{x_t} - {x_s}} ))$ trigonometric operation yields its value.

(1) \begin{align}\left[ {\begin{array}{l}{{{\dot x}_k}}\\{{{\dot y}_k}}\end{array}} \right] = \left[ {\begin{array}{l@{\quad}l}{cos( \theta )} & {}{sin( \theta )}\\{ - sin( \theta )} & {}{cos( \theta )}\end{array}} \right] \times \left[ {\begin{array}{l}{{x_k} - {x_s}}\\{{y_k} - {y_s}}\end{array}} \right]\end{align}

The major effect of the coordinate system exchange is seen on the calculation of point ${\dot p_i}$ that is the transformed counterpart of the point ${p_i}$ found on the line ${L_i}$ of the set $L$ . Because of the lines in the set $L$ are vertical to the reference line, and they pass through the segmentation points equally dividing the reference line, the $\dot x$ -axis value of the point ${\dot p_i}$ becomes also the same as the $\dot x$ -axis value of the transformed counterpart of the $ith$ segmentation point, and it is simply found as the result of $i| {{P_s}{P_t}} |/( {D + 1} )$ operation. If only $\dot y$ -axis values of the points are predicted and stored in a set such as $\left\{ {{{\dot y}_s},{{\dot y}_1},{{\dot y}_2}, \ldots ,{{\dot y}_{D - 1}},{{\dot y}_D},{{\dot y}_t}} \right\}$ , a path can be defined by using the previously calculated $\dot x$ -axis values and the guessed $\dot y$ -axis values of the points. A visual inspection of how the coordinate system transformation changes the obtainment procedure of the set $P$ can be carried out with Fig. 1.

Figure 1.

Original coordinate system for a battlefield with four enemy threats (a), its conversion to the new coordinate system (b), different paths in the new coordinate system (c) and their equivalences in the original coordinate system (d).

The points being selected from the lines of the set $L$ are in the continuous search space, and ignorable changes on the decisions can trigger tremendous modifications on the corresponding path. For making a discrimination between the paths and deciding which one is more appropriate, a cost or score requiring minimisation should be assigned to each path. When calculating the cost or quality score to a path, Equation (2) can be guided. In Equation (2), the cost stemming from the fuel consumption is represented with the ${C_f}$ , the ${C_t}$ and ${C_s}$ are used on behalf of the costs stemming from being sensed by the enemy threats and violations of the turning manoeuvres. While the weight of the ${C_f}$ on the overall path cost or $C$ is adjusted with the ${\lambda _f}$ coefficient, the weight of the ${C_t}$ and the weight of the ${C_s}$ are calibrated with the ${\lambda _f}$ and ${\lambda _s}$ coefficients. Finally, the sum of all three coefficients must be equal to $1$ .

(2) \begin{align}C = {\lambda _f}{C_f} + {\lambda _t}{C_t} + {\lambda _s}{C_s} = {\lambda _f}\mathop \smallint \nolimits_0^\ell \,{w_f}d\ell + {\lambda _t}\mathop \smallint \nolimits_0^\ell {w_t}d\ell + {\lambda _s}\mathop \sum \limits_{j = 1}^D {\emptyset _j}\end{align}

The integration of the ${w_f}$ term from $0$ to the $\ell $ showing the length of the path can be purified while keeping the calculation accuracy at the desired level. If the ${w_f}$ term is replaced with a positive integer constant such as $1$ , the value of the ${C_f}$ becomes equal to the $\ell $ and increases proportionally to the length of the path. Another purification can also be made for the integration of the ${w_t}$ term. Given that ${p_i}$ and ${p_j}$ are two consecutive points in the set $P$ and ${L_{ij}}$ is the length of line segment between them. Furthermore, the line segment of length ${L_{ij}}$ is divided equally into ten partitions and the first, third, fifth, seventh and ninth partition points are marked for a subsequent usage. If the line segment between the ${p_i}$ and ${p_j}$ is in the effect or sensing range of the $kth$ enemy threat, the cost of being sensed by the $kth$ enemy threat or ${C_{t,\left( {ij} \right),k}}$ is determined with Equation (3) [Reference Niu, Yan, Wang and Niu47]. In Equation (3), ${t_k}$ shows the threat grade or danger level. The Euclidean distances between the marked partition points and the centre of the $kth$ enemy threat are represented as the $d_{0.1,i,k}^4$ , $d_{0.3,i,k}^4$ , $d_{0.5,i,k}^4$ , $d_{0.7,i,k}^4$ and $d_{0.9,i,k}^4$ terms. After checking each line segment for all the enemy threats and summing their partial costs, the value of the ${C_t}$ is estimated.

(3) \begin{align}{C_{t,\left( {ij} \right),k}} = \frac{{{L_{ij}}{t_k}}}{5}\left( {\frac{1}{{d_{0.1,i,k}^4}} + \frac{1}{{d_{0.3,i,k}^4}} + \frac{1}{{d_{0.5,i,k}^4}} + \frac{1}{{d_{0.7,i,k}^4}} + \frac{1}{{d_{0.9,i,k}^4}}} \right)\end{align}

A path being tracked by a UAV or UCAV system necessitates a set of turning manoeuvres successfully [Reference Oktay, Arik, Turkmen, Uzun and Celik70, Reference Oktay and Coban71]. However, a UAV or UCAV has obvious kinematic limitations and some of the turning manoeuvres to track the path thoroughly can not be handled [Reference Arik, Turkmen and Oktay72]. Assume that ${p_j}$ , ${p_{j + 1}}$ and ${p_{j + 2}}$ are three successive points in the set $P$ and $\overleftarrow {{p_j}{p_{j + 1}}} $ and $\overleftarrow {{p_{j + 1}}{p_{j + 2}}} $ are two vectors drawn by referencing these points. If the first three points are considered, ${p_j}$ , ${p_{j + 1}}$ and ${p_{j + 2}}$ correspond to the ${p_s}$ , ${p_1}$ and ${p_2}$ elements. Similarly, if the last three points are considered, ${p_j}$ , ${p_{j + 1}}$ and ${p_{j + 2}}$ correspond to the ${p_{D - 1}}$ , ${p_D}$ and ${p_t}$ , respectively. When the absolute value of the turning angle ${A_j}$ or $\left| {{A_j}} \right|$ found as in Equation (4) is less than the maximum turning angle or ${\emptyset _{max}}$ , it is accepted that the turning manoeuvre with the ${A_j}$ degrees is within the limitations of the UAV system. Otherwise, the cost stemming from the turning with the ${A_j}$ degrees is added to the existing value of the ${C_s}$ [Reference Niu, Yan, Wang and Niu47].

(4) \begin{align}{A_j} = arctan\left( {\frac{{\parallel \overleftarrow {{p_j}{p_{j + 1}}} \times \overleftarrow {{p_{j + 1}}{p_{j + 2}}} \parallel }}{{\overleftarrow {{p_j}{p_{j + 1}}} \cdot \overleftarrow {{p_{j + 1}}{p_{j + 2}}} }}} \right)and{\rm{\;\;}}{\emptyset _j} = \left| {{A_j}} \right|,\left| {{A_j}} \right| \gt {\emptyset _{max}}0,otherwise\end{align}

3.0 Immune Plasma Algorithm

The COVID-19 pandemic taught the whole world that a medical method called immune or convalescent plasma treatment is still influential for helping critical patients. Even though the first practical usage of the immune or convalescent plasma treatment dates back more than a century ago and is associated with the 1918 Great Influenza, patients infected with MERS, Ebola, SARS and H1N1 have also tried to be cured by applying the plasma treatment. The idea biologically strong and ahead of its time lying behind the immune plasma treatment is transferring antibodies from a person recovered shortly before to a patient. Antibodies that are actually protein structures specifically synthesised by the immune system cells can float in the blood for binding to the invaders triggering the infection and healing a critical patient can be supported with the transfer of the donated blood or antibody-rich part of the blood, also known as plasma.

The fundamental steps of the plasma treatment perfectly conform to two main stages namely exploration and exploitation of a meta-heuristic. Aslan and Demirci analysed the mentioned conformation and IP algorithm or IPA that brought intelligent optimisation techniques to the literature [Reference Aslan and Demirci7, Reference Aslan and Demirci8]. IP algorithm sets an analogy between a solution of the problem and an individual or member of its population. The objective function of the problem is used to determine the immune response or antibody level of an individual. When the IPA decides to treat an individual labelled as a receiver because of poor quality or weak immune response, it first selects one of the top-qualified individuals from the population as a donor and then the plasma transfer from the donor to the receiver individual is conducted by guiding a mathematical model. Detailed explanations of IPA about the distribution of infection between the population members, selection of receivers and donors, implementation of plasma treatment and modification of donor individuals are given in the following subsections.

3.1 Generating individuals of population

An individual of population in IP algorithm is represented with a $D$ -dimensional vector. Each element of the $D$ -dimensional vector corresponds to a decision parameter of the optimisation problem with $D$ different decision parameters. When the first value of the $jth$ parameter for the $ith$ individual in the population of size $PS$ or ${x_{ij}}$ is determined, a simple calculation process described with Equation (5) is driven [Reference Aslan and Demirci7]. In Equation (5), $x_j^{min}$ and $x_j^{max}$ are the lower and upper bounds of the $jth$ parameter and $rand\left( {0,1} \right)$ is replaced with a random fractional number generated between $0$ and $1$ .

(5) \begin{align}{x_{kj}} = x_j^{min} + rand( {0,1} )\left( {x_j^{max} - x_j^{min}} \right)\end{align}

3.2 Infecting individuals of population

A pathogen causing an infection can spread from one individual to another. For modelling the spreading or distribution of infection, IP algorithm selects two individuals. While one of them is infected, the other is accepted as susceptible [Reference Aslan and Demirci7]. Given that ${x_k}$ is the $kth$ individual of the population and ${x_m}$ is another individual chosen randomly from the same population. For infecting the ${x_k}$ individual due to the sudden encounter with the ${x_m}$ , IPA offers the usage of Equation (6) [Reference Aslan and Demirci7]. While $x_{kj}^{inf}$ represents the randomly determined $jth$ parameter of $x_k^{inf}$ and $x_k^{inf}$ is matched with the infectious counterpart of ${x_k}$ , ${x_{kj}}$ and ${x_{mj}}$ represent the $jth$ parameters of ${x_k}$ and ${x_m}$ individuals, respectively. Finally, $rand( { - 1,1} )$ is a random fractional number between $ - 1$ and $1$ . As easily understood from the mentioned equation, IPA models the distribution of infection by changing only one parameter of the ${x_k}$ individual and $x_k^{inf}$ actually same as ${x_k}$ except the newly calculated $jth$ parameter.

(6) \begin{align}x_{kj}^{inf} = {x_{kj}} + rand( { - 1,1} )( {{x_{kj}} - {x_{mj}}} )\end{align}

When the ${x_k}$ individual meets with the infection, its immune system gives a response by synthesising antibodies. For measuring how strong the immune response of the ${x_k}$ individual is, IPA utilises from the objective function $f$ of the optimisation problem and the immune response of the ${x_k}$ individual to the infection or synthesised antibody amount is measured as $f\big( {x_k^{inf}} \big)$ [Reference Aslan and Demirci7]. If the immune response of the ${x_k}$ individual against to the infection is better than the immune response of the same individual before the infection calculated as $f\left( {{x_k}} \right)$ , the immune memory of ${x_k}$ individual is powered by changing ${x_{kj}}$ with $x_{kj}^{inf}$ . Otherwise, ${x_k}$ individual protects the value of $jth$ parameter as easily seen from Equation (7) for a minimisation problem [Reference Aslan and Demirci7].

(7) \begin{align}{x_{kj}} = x_{kj}^{inf},if{\rm{\;\;}}f( {x_k^{inf}} ) \lt f( {{x_k}} ){x_{kj}},otherwise\end{align}

3.3 Treating critical individuals

IP algorithm continues to distribute infection until all of $PS$ individuals are affected. While some of the individuals are capable of handling the infection because of their strong immune systems, some of them require intense care or even the help of other individuals who recovered shortly before. For deciding who are capable of handling the infection and who requires a treatment, IPA introduces two control parameters called $NoD$ and $NoR$ [Reference Aslan and Demirci7]. The $NoD$ parameter gets its name from the word group number of donors, and the best $NoD$ individuals are assumed to have already recovered and are ready to become donors [Reference Aslan and Demirci7]. On the other hand, the $NoR$ parameter gets its name from the word group number of receivers and the worst $NoR$ individuals are labelled as receivers and tried to be treated with the plasma donations [Reference Aslan and Demirci7].

Given that $x_k^{rcv}$ is the $kth$ receiver from the set of receivers containing $NoR$ individuals and $x_m^{dnr}$ is the randomly selected $mth$ donor from the set of donors containing $NoD$ individuals. In order to represent the transfer of the first plasma dose from the $x_m^{dnr}$ to the $x_k^{rcv}$ , IPA executes Equation (8) where $j$ is chosen sequentially from the set $\{ {1,2, \ldots ,D} \}$ [Reference Aslan and Demirci7]. In Equation (8), $x_{kj}^{rcv - p}$ shows the $jth$ parameter of $x_k^{rcv - p}$ and $x_k^{rcv - p}$ is used on behalf of the plasma transferred counterpart of the $x_k^{rcv}$ individual. Also, $x_{kj}^{rcv}$ and $x_{mj}^{dnr}$ are matched with the $jth$ parameters of the $x_k^{rcv}$ and $x_m^{dnr}$ individuals. If the immune response of the $x_k^{rcv}$ immediately after the first dose of plasma also measured as $f\left( {x_k^{rcv - p}} \right)$ is better than the immune response of donor individual measured as $f\left( {x_m^{dnr}} \right)$ , $x_k^{rcv}$ is updated with the parameters of $x_k^{rcv - p}$ appropriately. Moreover, the preparation for the transfer of second dose begins. Otherwise, $x_k^{rcv}$ is updated with the parameters of $x_m^{dnr}$ and treatment is simply concluded [Reference Aslan and Demirci7].

(8) \begin{align}x_{kj}^{rcv - p} = x_{kj}^{rcv} + rand( { - 1,1} )( {x_{kj}^{rcv} - x_{mj}^{dnr}} )\end{align}

The second dose of plasma from the $x_m^{dnr}$ is transferred to the $x_k^{rcv}$ as described in Equation (8). However, the decision about whether the treatment will be continued with the third dose of plasma or not is made in a different manner when compared to the previous case. If the immune response of the $x_k^{rcv}$ after the second dose of plasma is better than the immune response of the current $x_k^{rcv}$ , a procedure modifying the parameters of $x_k^{rcv}$ with the parameters of $x_k^{rcv - p}$ is executed and then the treatment is maintained with the third dose of plasma [Reference Aslan and Demirci7]. Otherwise, the treatment is completed for the $x_k^{rcv}$ , and next receiver if it exists is taken to start similar plasma transfer operations.

3.4 Modification of donor individuals

If an individual who recovered previously encounters the same infection frequently as time goes by, their immune memory can be specialised more to recognise the pathogen causing infection. As an expected result of this type of specialisation, the required amount of antibodies is synthesised immediately. In contrast, recognising the pathogen and synthesising the required amount of antibodies by the immune system of the considered individual takes longer. In the IP algorithm, only the individuals chosen as donors are assumed as recovered and their immune memories are specialised fully or partially according to a probabilistic mechanism. The probabilistic mechanism responsible for deciding which type of specialisation procedure is applied compares the ratio of ${t_{cr}}$ and ${t_{max}}$ with a random number between $0$ and $1$ [Reference Aslan and Demirci7]. While the ${t_{cr}}$ shows the current evaluation number, the ${t_{max}}$ represents the maximum evaluation number. If the ratio of ${t_{cr}}$ and ${t_{max}}$ is less than the random number, the donor individual is regenerated by using Equation (5). Otherwise, each parameter of the donor individual is changed somewhat as formulated in Equation (9) [Reference Aslan and Demirci7].

(9) \begin{align}x_{mj}^{dnr} = x_{mj}^{dnr} + rand( { - 1,1} )x_{mj}^{dnr}\end{align}

When the details about the model for specialising the immune memories of the donors are investigated, it is seen that the probability of updating immune memories partially with Equation (9) increases, while the run of the IPA reaches termination [Reference Aslan and Demirci7]. Because the ratio of ${t_{cr}}$ and ${t_{max}}$ gets close and then becomes equal to $1$ at the end of a run, the probability of generating a random number less than ${t_{cr}}/{t_{max}}$ and applying a partial update with Equation (9) also increases and immune memories of the donors are preserved more.

4.0 Greedy Initialiser Immune Plasma Algorithm

The geometric description of the path planning problem requires intense calculations to do with the lines of the set $L$ and their counterparts for the transformed coordinate system. However, all of these lines provide critical information about the battlefield and can be further referenced for generating an initial path or paths being used by a population-based meta-heuristic algorithm. Given that ${\dot L_i}$ is the $ith$ line from the set $\dot L$ and it is divided equally into $D + 1$ line segments with $D$ different points similar to the segmentation of the reference line [Reference Aslan73]. The points dividing the line ${\dot L_i}$ are also combined into a set such as ${\dot P^i} = \left\{ {\dot p_1^i,\dot p_2^i, \ldots ,\dot p_{D - 1}^i,\dot p_D^i} \right\}$ . If the first line is taken into account and a point from the point set ${\dot P^1}$ will be selected for a UAV or UCAV path, a greedy heuristic in which a point that is the nearest one to the ${\dot p_s}$ and outside the effect range of all enemy threats will be selected is employed [Reference Aslan73]. When the line ${\dot L_i}$ where $i$ is greater than $1$ is considered and a point from the point set ${\dot P^i}$ will be selected for a path, the mentioned greedy approach in which a point that is nearest to the taken point in the set ${\dot P^{i - 1}}$ of line ${\dot L_{i - 1}}$ and outside the effect range of all enemy threats is chosen again [Reference Aslan73]. The formulations for the selection of the first and other points by the heuristic method are given in Equations (10) and (11), respectively.

(10) \begin{align}{\dot p_1} = \dot p_j^1,\mathop {d( {t,\dot p_j^1} )}\limits_{\forall t \in T} \gt {r_t} \wedge d\left( {{{\dot p}_s},\dot p_j^1} \right) \lt \mathop {d( {{{\dot p}_s},\dot p_m^1} )}\limits_{\forall \dot p_m^1 \ne \dot p_j^1 \in {{\dot P}^1}} \dot p_1^1,otherwise\end{align}

(11) \begin{align}{\dot p_i} = \dot p_j^i,\mathop {d( {t,\dot p_j^i} )}\limits_{\forall t \in T} \gt {r_t}\dot p_k^i,\mathop {d\left( {t,\dot p_k^i} \right)}\limits_{\forall t \in T} \gt {r_t} \wedge d\left( {{{\dot p}_{i - 1}},\dot p_k^i} \right) \lt \mathop {d\left( {{{\dot p}_{i - 1}},\dot p_m^i} \right)}\limits_{\forall \dot p_m^i \ne \dot p_k^i \in {{\dot P}^i} - \left\{ {\dot p_j^i} \right\}} first{\rm{\;}}of{\dot P^i} - \left\{ {\dot p_j^i} \right\},otherwise\\[10pt]\nonumber \end{align}

The fundamental idea lying behind the introduced greedy approach is keeping the distance between the subsequent points and manoeuvre costs related to the turnings at minimum. Also, while the path safety is guaranteed partially when the line segment between two selected points intersects with the circles representing the enemy threats even though the selected points are outside the circles or fully when the line segment between two selected points does not intersect with any circles [Reference Aslan73]. Because of this reason, the first point being controlled from the set ${\dot P^i}$ should be $\dot p_j^i$ , if the point $\dot p_j^{i - 1}$ in the set ${\dot P^{i - 1}}$ where $i$ is greater than $1$ is chosen. If the point $\dot p_j^i$ is outside the effect range of all enemy threats, it is taken quickly without controlling other segmentation points in the set ${\dot P^i}$ of line ${\dot L_i}$ . In Fig. 2, a graphical example of how the points are selected by the heuristic method is given. When the selection of the point from the line ${\dot L_i}$ concludes with the point $\dot p_j^i$ , it should be noted that the length of line segment between $\dot p_j^{i - 1}$ and $\dot p_j^i$ is minimised and safety of UAV or UCAV for this line segment is ensured. In addition to this, a turning manoeuvre is not carried out by considering that the point $\dot p_j^{i - 2}$ from the line ${\dot L_{i - 2}}$ is chosen. Otherwise, the remaining points in the set ${\dot P^i}$ are investigated for finding a point that is closest to the $\dot p_j^{i - 1}$ and outside the effect range of all enemy threats.

Figure 2.

Selecting the points from the lines ${\dot L_1}$ (a), ${\dot L_2}$ (b), ${\dot L_3}$ (c) and ${\dot L_{D - 1}}$ (d) according to the greedy heuristic.

A path generated by the greedy heuristic can be referenced to populate individuals of the IP algorithm. For maintaining the information provided by the greedy approach and satisfying the required population diversity, the individuals of IPA is created in a manner that a randomly determined one is equalised to the path of greedy heuristic and each remaining member of the population is created by adding randomly produced number between $ - 1$ and $1$ to each parameter of the path inherited from the greedy heuristic. By slightly modifying the inherited path when generating the $PS - 1$ members of the population, the new path candidates without executing compute intensive operations are discovered. Subsequently, an initial population that contains the promising solutions is passed to the IP algorithm for the following phases or stages.

The population generated as detailed above and passed to the IP algorithm should be exploited more carefully for converging to the optimum solution or solutions. When the properties of the IP algorithm are investigated, it is seen that the fundamental phase related to the plasma treatment maintains the exploitation dominant operations. However, IPA uses a generalised model for providing an algorithmic simplicity by skipping some subtle parts about the acquisition of plasma. If the IP algorithm is supported with an extra mechanism that models the acquisition of the plasma and its transfer to a receiver, the vicinity of the promising solutions in the population is searched particularly. Given that ${x^b}$ is the best solution found by the algorithm until the current evaluation and $x_1^{dnr}$ and $x_2^{dnr}$ are two donors chosen from the existing population. For modeling the acquisition or collection of the plasma that is represented as ${x^p}$ and initialised with the ${x^b}$ , Equation (12) given below can be served. In Equation (12), $x_j^{p - t}$ is the newly calculated $jth$ parameter of ${x^p}$ with the help of $jth$ parameters of $x_1^{dnr}$ and $x_2^{dnr}$ . When the newly calculated $jth$ parameter of ${x^p}$ improves the solution quality, a replacement between $x_j^p$ and $x_j^{p - t}$ is approved. After controlling all the $D$ parameters and updating them if required, ${x^p}$ becomes ready for the transfer.

(12) \begin{align}x_{j}^{p - t} = x_1^{dnr} + rand( { - 1,1} )\left( {x_1^{dnr} - x_2^{dnr}} \right)\end{align}

The quality of the solution represented with the ${x^p}$ is equal to or most likely better than the quality of the solution represented with the ${x^b}$ and its transfer to a receiver should be carried out by considering the conservation of ${x^p}$ or a better candidate for utilising it in future cycles. Assume that ${x^{rcv}}$ is the most critical individual of the population. The transfer of plasma or ${x^p}$ to the ${x^{rcv}}$ individual can be described as in Equation (13). In Equation (13), $x_j^{rcv - p}$ is used on behalf of the $jth$ parameter of ${x^{rcv - p}}$ and ${x^{rcv - p}}$ shows the plasma transferred counterpart of the ${x^{rcv}}$ individual. While $x_j^{rcv}$ and $x_j^p$ are the $jth$ parameters of ${x^{rcv}}$ and ${x^p}$ , ${j_r}$ is a random integer between $1$ and $D$ . Since only one parameter index will be equal to the ${j_r}$ , ${x^{rcv - p}}$ directly configures its $D - 1$ different parameters with the corresponding parameter values of ${x^p}$ . If the objective function value of ${x^{rcv}}$ immediately after the treatment or $f\left( {{x^{rcv - p}}} \right)$ is better than the objective function value of the same individual before the treatment or $f\left( {{x^{rcv}}} \right)$ , ${x^{rcv}}$ is updated with the ${x^{rcv - p}}$ and treatment is ended. Otherwise, ${x^{rcv}}$ is updated with the ${x^p}$ and then treatment is concluded.

(13) \begin{align}x_j^{rcv - p} = {\rm{\;}}x_j^p,if{\rm{\;\;}}j \ne {j_r}\;x_j^{rcv} + rand( { - 1,1} )\left( {x_j^{rcv} - x_j^p} \right),otherwise\end{align}

The details of the proposed treatment schema that models how the plasma is collected and then transferred also give some clues about the changed requirement trend of the $NoR$ and $NoD$ control parameters of the IPA. As stated previously, the proposed treatment schema selects two donors from the population and transfers the collected plasma to the most critical individual. By considering this preset for the donors and receivers, there is no further need to the $NoR$ and $NoD$ control parameters and their sensitive configurations according to the population size, number of parameters and specific properties of the problem being solved optimally. The operational steps about how the plasma is collected first and transferred in the renovated treatment schema can be examined by guiding Algorithm 1. The IP algorithm, where the greedy heuristic is used to initially generate the population and the proposed treatment scheme is responsible for the collection and transfer of plasma, is called greedy initialisation IPA for short gintIPAFootnote ¹.

Algorithm 1.

Details of the treatment schema in gintIPA

5.0 Experimental Studies

The performance of the gintIPA can vary with the properties of the battlefield in terms of number of enemy threats, their centre locations, sensing or detecting ranges determined by radius, danger levels or grades. For evaluating how the performance of gintIPA varies, three battlefield scenarios detailed in Table 1 were used [Reference Niu, Yan, Wang and Niu47]. Each battlefield contains four test cases generated by configuring the number of segmentation points or $D$ as $10$ , $15$ , $20$ and $25$ . The gintIPA was experimented with $30$ times for each test case with the different $PS$ values including $30$ , $40$ , $50$ , $75$ and $100$ . When completing a run or test after consuming $6000$ evaluations, the best solution was found and its objective function value was recorded. The best, worst, mean best objective function values and standard deviations calculated over $30$ records were presented in Table 2 for the cases of Scenario- $1$ , Table 3 for the cases of Scenario- $2$ and finally Table 4 for the cases of Scenario- $3$ .

Table 1.

Information about the battlefields used in the experiments

Table 2.

Results of gintIPA with varying $PS$ values for Scenario- $1$

Table 3.

Results of gintIPA with varying $PS$ values for Scenario- $2$

Table 4.

Results of gintIPA with varying $PS$ values for Scenario- $3$

Figure 3.

The best and worst paths found by gintIPA for $10$ (a), $15$ (b), $20$ (c) and $25$ -dimensional (d) cases of Scenario- $1$ .

Figure 4.

The best and worst paths found by gintIPA for $10$ (a), $15$ (b), $20$ (c) and $25$ -dimensional (d) cases of Scenario- $2$ .

Figure 5.

The best and worst paths found by gintIPA for $10$ (a), $15$ (b), $20$ (c) and $25$ -dimensional (d) cases of Scenario- $3$ .

The results given in Tables 2, 3 and 4 provide critical information about the relatively consistent solving capabilities of the gintIPA. The population or colony size of a meta-heuristic algorithm can impact the quality of the final solutions. If the population or colony size is chosen high, the solution space is sampled more and the algorithm gets a chance of starting the subsequent operations with a set of qualified candidates. However, it should be noticed that the number of evaluations per cycle or iteration increases and a run of the algorithm terminates without executing specially designed operations enough. On the other hand, if the population or colony size is chosen small, the probability of starting a run of a meta-heuristic with a set of qualified initial candidates decreases. Even though the probability of initialising the population with qualified candidates decreases, the considered algorithm iterates more compared to the previously mentioned configuration about population or colony size. From the mean best objective function values in Tables 2, 3 and 4, it is seen that gintIPA protects the qualities of final solutions while the different values are assigned to the $PS$ parameter. One of the main reasons behind the stable performance of gintIPA is related to the employed greedy initialisation method that works by guiding only number of segmentation points. When the greedy initialiser determines its path, one of the individuals in the gintIPA is configured with the determined path, and the remaining members of the population are also produced randomly in a narrow neighbourhood of the path of the greedy initialiser. By generating the population of gintIPA with the help of a greedy initialiser, algorithm starts its phases about the distribution of infection, plasma treatment and modification of the donor or donors with the promising population members. Moreover, it is important to declare that if the $PS$ parameter of gintIPA is set to a higher value, an intrinsic advantage about the searching the neighbourhood of the qualified path determined by greedy initialiser is gained and gintIPA can protect the qualities of its solutions for the running configurations causing the huge amount of evaluation consumption per cycle.

The stable performance of gintIPA for different values assigned to the $PS$ parameter is also related to the modernised treatment schema that combines both newly introduced plasma collection mechanism and transfer approach and removes the necessity of the inherited control parameters from IPA called $NoR$ and $NoD$ . When the gintIPA decides to treat the most critical individual of the population, it first collects the plasma. The plasma is actually a new solution generated by guaranteeing its objective function value is at least equal to or less than the objective function value of the best solution found so far. Also, the transfer approach used for the plasma performs an extra search and tries to find a solution better than the solution represented by the plasma. As an expected result of the detailed exploitation of the eligible solutions in the population for the collection of plasma and subtle search when transferring it, gintIPA handles some problems originating from the quick termination without executing treatment operations adequately. The small differences between the best and worst paths found by gintIPA with $30$ individuals can be visually investigated over Fig. 3 for Scenario- $1$ , Fig. 4 for Scenario- $2$ and Fig. 5 for Scenario- $3$ .

In order to understand how successful the gintIPA is as a path planner and how eligible the calculated paths by gintIPA are, a set of comparative studies should be carried out. For this purpose, the paths found by the gintIPA were compared with the paths of fourteen different meta-heuristic techniques including IPA, GA, GAPSO, MFO, SSA, PFA, SBO, SCA, ECTLBO, HSGWO-MSOS, CIPSO, GWO, AEO and NSEAEO whose control parameters are set as in Table 5. Each test case belonging to the Scenario- $1$ , Scenario- $2$ and Scenario- $3$ was solved $30$ times after equalising the running conditions by setting the population or colony size to $30$ and maximum evaluation number to $6000$ for the gintIPA and other mentioned algorithms [Reference Niu, Yan, Wang and Niu47, Reference Aslan63]. When the results given in Tables 6, 7 and 8 are investigated, the competitive performance of gintIPA against other path planners can be seen clearly. While the gintIPA outperforms all of other techniques for 11 of 12 test cases and gets ranked as the best path planner, it lags behind IPA and NSEAEO only for one test case belonging to the Scenario- $2$ and gets ranked as the third best path planner and allows to state that gintIPA is still capable of obtaining better solutions than remaining 12 algorithms. The visually apparent contribution of the greedy initialiser as validated from Figs 3, 4 and 5 and improved exploitation performance with the newly designed treatment schema allow gintIPA to perform better than the considered algorithms in $91.6{\rm{\% }}$ of all test cases. The comparative studies between gintIPA and other path planners also provide important information about the advantageous sides of gintIPA becoming more evident with the increasing number of segmentation points. When the number of segmentation points and difficulty of the path planning problem are increased, the greedy initialiser of gintIPA can determine more points on the path reducing the cost of enemy threats, fuel or battery consumption or turning manoeuvres. In addition to this, when the number of segmentation points is increased, the chance of collecting more qualified plasma is also increased intrinsically due to the sequential control of all parameters with the help of the selected donors. As an expected result of the self-configurable searching characteristics according to the number of segmentation points, gintIPA handles the difficulties of the path planning problem more robustly and calculates better paths than its competitors.

Table 5.

Compared algorithms with gintIPA and their parameter adjustments

The greedy initialiser used with the gintIPA brings an extra computational burden. Also, the modernised treatment schema changes the number of evaluations spent per infection cycle and gintIPA can iterate less than the standard implementation of the IPA even though the termination criteria and other running configurations are the same. In order to analyse that how the modifications change the execution time, gintIPA and IPA were experimented with $30$ times by setting the population size to $30$ and maximum evaluation number to $6000$ for each test case of Scenario- $1$ , Scenario- $2$ and Scenario- $3$ and the time passed until the end of a run was recorded. Both gintIPA and IPA were coded in C programming language and a computer equipped with an Intel i5- $10500$ processor and managed by Fedora- $34$ operating system was used. The best, worst, mean elapsed times in terms of seconds and standard deviations were determined by using the records and then summarised in Table 9. When the mean execution times in Table 9 are investigated, it is understood that gintIPA requires more time compared to the IPA. While the differences between the mean execution times of the gintIPA and IPA are $0.002$ and $0.006$ seconds at most for $10$ and $15$ -dimensional cases, the differences between the mean execution times of the gintIPA and IPA are $0.008$ and $0.010$ seconds at most for $20$ and $25$ -dimensional cases and indicate the direct relationship with the computational complexity of the greedy initialiser and number of segmentation points.

Table 6.

Comparison between gintIPA and other path planners for Scenario- $1$

Table 7.

Comparison between gintIPA and other path planners for Scenario- $2$

Table 8.

Comparison between gintIPA and other path planners for Scenario- $3$

The proposed greedy initialiser and treatment schema can also affect the convergence performance of the algorithm. For evaluating the convergence performance of a meta-heuristic algorithm, the success rate abbreviated as $Sr$ and mean evaluations abbreviated as $Me$ are referenced frequently [Reference Aslan63]. If a run of the considered meta-heuristic concludes with a solution whose objective function value is better than the previously determined threshold, this run is accepted as successful and the ratio between the number of successful runs and the total number of runs corresponds to the $Sr$ metric. If a run is determined as successful, the minimum number of evaluations required to find a solution whose objective function value is less than the threshold is stored. The average of the stored values represents the value of the $Me$ metric. The $Sr$ and $Me$ metrics belonging to the gintIPA and IPA by setting the threshold to $50$ for the Scenario- $1$ and Scenario- $3$ and $60$ for the Scenario- $2$ were calculated and presented in Table 10. The $Sr$ and $Me$ metric values given in Table 10 prove the superior convergence performance of gintIPA against the IP algorithm. While the $Sr$ metric of gintIPA is better than the $Sr$ metric of IPA for 9 of 12 test cases, the $Me$ metric of gintIPA is better than the metric of IPA for all test cases. However, it should be noted that even though the $Me$ metric value of gintIPA is better than the $Me$ metric value of IPA for $10$ -dimensional test case of Scenario- $2$ , gintIPA obtains final solutions whose objective function values are less than the threshold for only 10 of 30 runs.

Table 9.

Comparison between the execution times of gintIPA and IPA

Table 10.

$Sr$ and $Me$ metrics of gintIPA and IPA

Table 11.

Results of the Wilcoxon signed rank test for gintIPA and IPA

The comparative studies between the gintIPA and IPA indicate that gintIPA is capable of planning more qualified paths than IPA by bringing negligible amount of computational burden. However, demonstrating the effectiveness of the gintIPA against IPA with an appropriate statistical test is important to prove the critical assistance of the modifications. Wilcoxon signed rank test is a well-known non-parametric statistical test and has a widespread usage for the comparison of meta-heuristics [Reference Aslan63]. If the significance level ( $\rho $ ) of Wilcoxon signed rank test is found less than a constant usually fixed to $0.05$ , the difference between the compared algorithms is enough to state that one of the competitors is statistically better [Reference Aslan63]. The test results of the comparison between the gintIPA and IPA were given in Table 11 where $Z$ represents the test statistics and $W + $ and $W - $ show the sum of ranks for which gintIPA lags behind IPA and the sum of ranks for which gintIPA outperforms IPA. As easily seen from the $\rho $ values in Table 11, gintIPA is statistically better than IPA for all cases of Scenario- $1$ and Scenario- $3$ . Moreover, the Wilcoxon signed rank test validates that the difference between the qualities of the paths found by gintIPA and IPA is enough to generate a statistical significance in favour of the gintIPA for $15$ , $20$ and $25$ -dimensional cases of Scenario- $2$ . Utilising from the properties of the geometrical model as in the introduced greedy initialiser and redesigning the exploitation dominant operations by considering the gains related to the initialisation boosts the planning capabilities of the gintIPA and assist it to obtain solutions whose superiorities are statistically prominent compared to the solutions found by the standard IPA.