The palm tree optimization: Algorithm and applications

Abstract

A novel metaheuristic algorithm has been presented based on the physical significance of palm tree leaves and petioles, which can themselves water and fertilize with their unique architecture. Palm tree leaves collect almost all the raindrops that fall on the tree, which drags the nutrient-rich dropping of crawlers and birds that inhabit it and funnel them back to the palm tree’s roots. The proposed Palm Tree Optimization (PTO) algorithm is based on two main stages of rainwater before it reaches the trunk. Stage one is that the rainwater drops search for petioles in the local search space of a particular leaf, and stage two involves that the rainwater drops after reaching the petioles search for trunk to funnel back to the root along with nutrients. The performance of PTO in searching for global optima is tested on 33 Standard Benchmark Functions (SBF), 29 constrained optimization problems from IEEE-CEC2017 and real-world optimization problems from IEEE-CEC2011 competition especially for testing the evolutionary algorithms. Mathematical benchmark functions are classified into six groups as unimodal, multimodal, plate & valley-shaped, steep ridges, hybrid functions and composition functions which are used to check the exploration and exploitation capabilities of the algorithm. The experimental results prove the effectiveness of the proposed algorithm with better search ability over different classes of benchmark functions and real-world applications.

Keywords

PTO-palm tree optimization exploration exploitation petioles crankshaft

1 Introduction

Solving real-world optimization problems is complex using classical methods [1]. Classical techniques are numerical techniques that rely on straightforward mathematical equivalent models. Although the classical methods are capable of locating the best analytical solutions, they depend on gradient information and a suitable starting point to function properly [2]. Because these methods are extremely sensitive to initial points, choosing the wrong initial points for problems with multiple and sharp peaks often results in the search becoming unstable. Because of this, traditional approaches to solving them fall short, yet nature provides a wealth of ideas that can be used to develop artificial intelligence techniques. All nature-inspired algorithms find the best overall solutions to challenging real-world optimization problems, regardless of the initial points because they are fixed at random in all of them. Furthermore, the engineering design optimization problems, which are constrained and nonlinear with both continuous and discrete variables, are difficult to solve using classical methods [3]. Nature-inspired metaheuristic algorithms are excellent in dealing with such nonlinear and complex optimization problems. Optimization problems have been solved by a wide range of computation methods inspired by nature for decades [4]. However, a chief drawback of most metaheuristics is that they frequently exhibit delicate sensitivity to user-defined parameter settings. Another disadvantage of metaheuristics is that they do not always converge to global optimization [5]. When compared to other cutting-edge algorithms, the proposed PTO algorithm converges more quickly in many of the most challenging composition functions of IEEE-CEC2017, such as the combination of hybrid functions as essential functions and continuity around local as well as global as a trap. However, the proposed PTO algorithm is also sensitive to user-defined parameter settings.

The metaheuristic algorithms operate on rules, and their random behavior [6] can be categorized under four metaphors such as swarm intelligence-based, evolution-based, laws of physics-based and human related techniques. Evolution-based algorithms are inspired by the law of biological evolution by nature, like Genetic Algorithms (GA), which have been proposed first by Holland [7]. Many other popular algorithms have been developed since then like Differential Evolution (DE) [8], Backtracking Search Optimization (BSO) [9], Biogeography-Based Optimization (BBO) [10], Evolution Strategies (ES) [11], Fruit-fly optimization (FFO) [12], Selfish Gene Theory [13], Shuffled Frog Leap Algorithm (SFLA) [14].

Many algorithms have been developed based on swarm intelligence, inspired by the movement behavior of birds and animals with the highest level of complex communications. Particle Swarm Optimization (PSO) [15] is the first algorithm for swarm intelligence. Some of the other successful optimization algorithms inspiring the flocking behaviours are Grey Wolf Optimizer (GWO) [16], Artificial Bee Colony Optimization algorithm (ABC) [17], Whale optimization algorithm (WOA) [18], Social Spider Optimization Algorithm (SSO) [19], Ant Colony Optimization (ACO) [20], Dolphin Echolocation Algorithm (DEA) [21], Grasshopper Optimization Algorithm (GOA) [22], Social Engineering Optimizer (SEO) [23] and Cuckoo search algorithm (CSA) [24].

Simulated Annealing (SA) [25] is the first optimization algorithm based on physical laws. Following SA, many more algorithms like Atomic Orbital Search Algorithm (AOS) [26], Wind-Driven Optimization technique (WDO) [27], Electromagnetic Field Optimization (EFO) [28], Galaxy-Based Search algorithm (GBS) [29], Gravitational Search Algorithm (GSA) [30], Integrated Radiation Optimization (IRO) [31], Water Evaporation Optimization (WEO) [32], Central Force Optimization (CFO) [33] General framework of Artificial Physics Optimization Algorithm (APO) [34] have been developed. Furthermore, some of the existing algorithms belonging to each category are listed in Table 1.

Table 1
Popular optimization algorithms from recent literature

Swarm Intelligence based Evolution based Laws of Physics based

Algorithm Source of Inspiration Algorithm Source of Inspiration Algorithm Source of Inspiration

Monarch Butterfly Optimization [48] Migration of Monarch Butterflies Plant intelligence-based Algorithms [59] Review of all Plant based algorithms Spring search algorithm [65] Based on Hooke’s law

Slime Mould algorithm [49] Bio-Oscillator of Slime Mould Rooted Tree Optimization [60] Water searching behavior of Tree roots Momentum search algorithm [66] Law of Conservation of Momentum

RUN Beyond the Metaphor [51] Algorithm Based on Runge Kutta Method The Arithmetic Optimization Algorithm. [61] Distribution behavior of Arithmetic Operators Galactic Swarm Optimization [67] Motion of stars in Galaxies

Harris hawk’s optimization [52] Chasing strategies of Harris Hawks Intrusive tumor growth Algorithm. [62] Principle of tumor growth Henry gas solubility optimization [69] Based on Henry’s Law on gas solubility

Reptile Search Algorithm [54] Hunting behavior of Crocodile Synergistic fibroblast optimization [63] Behavior of fibroblast cell in healing wounds Thermal Exchange Optimization Algorithm [70] Based on Newton’s Law of Cooling

Pity Beetle Algorithm [55] Ability of Beetle to locate and harvest trees. Stochastic Fractal Search Algorithm [64] Diffusion property of random fractals Archimedes optimization algorithm [71] Based on Archimedes’ Principle

Mouth Brooding Fish algorithm [56] Dispersion and protection behavior Big Bang-Big Crunch optimization algorithm [68] Evolution of Universe

Squirrel search algorithm [57] Dynamic jumping and gliding strategies Backtracking optimization algorithm [72] Based on Brute force approach

Golden Eagle Optimizer [58] Prey catching strategies of Golden Eagle.

Swarm Intelligence based	Evolution based	Laws of Physics based
Monarch Butterfly Optimization [48]	Migration of Monarch Butterflies	Plant intelligence-based Algorithms [59]	Review of all Plant based algorithms	Spring search algorithm [65]	Based on Hooke’s law
Slime Mould algorithm [49]	Bio-Oscillator of Slime Mould	Rooted Tree Optimization [60]	Water searching behavior of Tree roots	Momentum search algorithm [66]	Law of Conservation of Momentum
RUN Beyond the Metaphor [51]	Algorithm Based on Runge Kutta Method	The Arithmetic Optimization Algorithm. [61]	Distribution behavior of Arithmetic Operators	Galactic Swarm Optimization [67]	Motion of stars in Galaxies
Harris hawk’s optimization [52]	Chasing strategies of Harris Hawks	Intrusive tumor growth Algorithm. [62]	Principle of tumor growth	Henry gas solubility optimization [69]	Based on Henry’s Law on gas solubility
Reptile Search Algorithm [54]	Hunting behavior of Crocodile	Synergistic fibroblast optimization [63]	Behavior of fibroblast cell in healing wounds	Thermal Exchange Optimization Algorithm [70]	Based on Newton’s Law of Cooling
Pity Beetle Algorithm [55]	Ability of Beetle to locate and harvest trees.	Stochastic Fractal Search Algorithm [64]	Diffusion property of random fractals	Archimedes optimization algorithm [71]	Based on Archimedes’ Principle
Mouth Brooding Fish algorithm [56]	Dispersion and protection behavior	Big Bang-Big Crunch optimization algorithm [68]	Evolution of Universe
Squirrel search algorithm [57]	Dynamic jumping and gliding strategies	Backtracking optimization algorithm [72]	Based on Brute force approach
Golden Eagle Optimizer [58]	Prey catching strategies of Golden Eagle.

Many algorithms are developed based on human related techniques. Social Network Search (SNS) [40], Hunger Games Search (HGS) [50] and Gaining-sharing knowledge-based algorithm (GSK) [53] are some of the recent algorithms based on Human related techniques.

Though there are many widely recognized metaheuristic algorithms for constrained and unconstrained optimization problems available, new and improved metaheuristic algorithms are needed, since the complexity of modern real world optimization problems has been increasing every day. Consequently, new algorithms to meet out the need of such problems deserve exploration and enhancement. The majority of well-known algorithms, including PSO, GA, CSA, and ABC, begin with a random solution and then, attempt to improve it to an optimum value while improving the answer. The two primary characteristics of the metaheuristic algorithms are Exploitation and Exploration. The exploitation phase revolves around the most effective search agents’ present optimal solutions. The exploration phase covers the full search space to ensure a comprehensive investigation of the entire search region. Considering the absolute opposite nature of both traits, every optimization technique must achieve the ideal balance between them in order to be effective. According to the no-free-lunch theorem of optimization [76], no optimization technique can provide the optimum results for every optimization problem, which motivates this research, as developing new and improved swarm-inspired optimization algorithm is necessary to tackle complex real-world problems, which are growing exponentially in number.

As in no-free-lunch theorem, though PTO is not able to perform best in all optimization algorithms, it performs better in most of the complex constrained multimodal optimization problems, where other state-of the art algorithms struggle to attain global optimum solutions. The effectiveness of the suggested PTO algorithm is based on its ability to assess and enhance each swarm’s capacity for exploration and exploitation. Each swarm is examined by the proposed PTO algorithm to classify the population into two groups: (1) Search agents in need of high exploitation capability, and (2) Search agents in need of high exploration capability.

A novel PTO optimization algorithm has been proposed in the present research and it is compared with other standard and well-known state-of-the-art optimization algorithms such as PSO [15], GA [7], CSA [24], ABC [17], TLBO [73], ES [11], GSK [53], GWO [16], BBO [10], ACO [20]. These algorithms are used to analyze the performance of the PTO algorithm. Experiments have been conducted on 33 SBFs [47], 29 constrained optimization problems from IEEE-CEC2017 [75] and real-world optimization problems from IEEE-CEC2011 [74] competition for testing the evolutionary algorithms.

The rest of research is depicted as follows: Section 2 provides inspiration and details about the PTO’s communal life. Section 3 explains the PTO algorithm’s mathematical model and the calculation procedure. Section 4 displays the results of PTO for resolving various SBFs. Section 5 discusses the PTO results of real-world optimization problems, and Section 6 concludes with some valuable perspectives.

2 Palm tree

The palmyra palm is one of the most economically significant species distributed mainly across Asian countries and dry African tropical regions [35]. Almost all parts of the palm tree, like roots, trunk, petioles, leaves, palm fruit, palm rafters, fibrous palm web, palm spathes, palm flower, and palm leaf ribs, are helpful to people [37]. Most palm trees have fibrous root systems, however palmyra has a tap root system that shoots straight downward vertically [36]. It could store a massive volume of water in its tubular roots and increase the locality’s water table level. Thus, it has a greater capacity to convert arid land into highly fertile land with rich groundwater resources [38]. Raindrops that fall over palm tree leaves are channeled to their scoop-shaped petioles, like rain gutters. The rainwater runs down in each gutter-like petiole to the crown shaft or junction point of petioles. The crownshaft is where the petioles are attached to the trunk, as in Fig. 1(b). After reaching the crownshaft, water pours through these gaps in the trunks directly to the base of the palm tree. [39] Christopher K Bunbury says that the raindrops falling over the palm tree crown flow straight to the tree’s base. The petiole funnels the rainwater through a split, and the water reaching the tree’s root is not just plain water. From the crown of the palm, rainwater carries all the nutrient-rich droppings of crawlers and birds to the roots. Kaiser-Bunbury and his coauthors have gone to Seychelles for in-depth research at a stand of coco de Mer palms to study how water recycling and the nutrient system of palm trees work. They have studied the nutrient concentration allocation of the palm tree across every tissue to see how it allocates its resources by taking the samples of leaves, flowers, fruits and trunk. In the soil under the palm tree, moisture and nutrients are measured. The research by the crew lasts for four years to track palm tree growth. In conclusion, Kaiser-Bunbury’s experimental results show that, even in nutrient-poor soils, palm trees can maintain high levels of nutrient supply by funneling particulate material along with rainwater to the base of the tree because of its distinctive morphological features. This particular feature, which makes the palm tree a high fertile host, aids the growth of many trees whose seeds will be transferred through bird droppings from palm tree leaves, as shown in Fig. 1(a).

Fig. 1

a) Palm tree being host to other tree species. b) Parts of Palm tree crown.

3 Proposed algorithm: palm tree optimization algorithm

This section discusses the introduction of PTO and its mathematical model. The proposed PTO algorithm comes under the swarm intelligence algorithm inspired by the nature of the flow of water observed over the distinctive morphological feature of the palm tree crown. This nature-inspired stochastic optimization technique generates a random population and then, moves towards an optimal solution by updating its position. The proposed algorithm considers every physical significance of the palm tree to maximize the water flow to the root. Raindrops falling over the palm tree crown are search agents and with the initial population generated all over, the crown becomes the search space for any continuous and discrete optimization problem. These random raindrops with the initial solution can update their position to a better solution. The position of the palm tree crownshaft is considered the globally optimal solution.

The palm tree crown comprises young leaves, petioles, leaf sheath and old leaves. As shown in Fig. 2, like water on gutters, raindrops falling over the young leaves and petioles flowing towards the crownshaft are perfect search agents with high fitness value. However, since the older fronds and petioles hang over the ground, the raindrops falling over them will not reach the crownshaft, but they fall on the ground and are considered poor search agents with low fitness value. Hence to improve the exploration phase of the algorithm, these search agents are discarded, and fresh search agents will be generated. Search agents are classified as agents over lamina of leaf, agents over petioles, and agents over dry fronds. Midribs of palm leaves in lamina are naturally arranged, as numerous concurrent lines meet at the costa of a leaf with steep slopes path for search agents towards the costa, the junction of midribs. After reaching the petioles, agents flow further towards the crown shaft, as the petioles are arranged as a second layer of concurrent lines with steep slope water gutters meeting at the crown shaft and it is the globally optimum solution. The general flow of PTO is shown in Fig. 3 and Algorithm 1.

Fig. 2

Concurrent Search behavior of Raindrops over Palm tree crown.

Fig. 3

PTO algorithm flowchart.

3.1 Mathematical model

The Mathematical model of PTO is derived in six steps.

3.1.1 Initialization of algorithm parameters

The PTO algorithm requires eight parameters to be initialized.

Number of raindrops (Search agents). (N)

Dimension of decision variables. (D)

Decision variable range (lb, ub)

Dry fronds fitness (P)

Palm leaf slope (M)

Crown shaft accelerator (m₂)

Number of searches. (R)

Termination criteria. (T)

3.1.2 Raining process

This step involves the generation of search agent’s matrix with N×D dimension of feasible solutions constrained by upper and lower limits.

Search agent matrix, $Population of Raindrops = [\begin{matrix} R_{1}^{1} & R_{1}^{2} & \dots & R_{1}^{D} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ R_{i}^{1} & R_{i}^{2} & \dots & R_{i}^{D} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ R_{N}^{1} & R_{N}^{2} & \dots & R_{N}^{D} \end{matrix}] .$ (1)

Algorithm 1 Pseudocode of PTO
Initialization of parameters
Random initialization of Population of Raindrops
Random Initialization of costa F = f(R_i)
R_ipbest=Population of Raindrops
R_Gbest=Raindrop with min F
for t = 1: Termination criteria
for i = 1: Population size
R_i(t+1) = C * A₁ + m₁ * rand * (R_ipbest - R_i(t))
F = f(R_i)
${fitness}_{i} = \frac{1}{1 + F_{i}}$
$P = \frac{{fitness}_{i}}{\max (fitness)}$
end
for i = 1: Population size
if(P > Dry fond fitness)
R_i(t+1) = C * A₁ + m₂ * rand * (R_Gbest - R_i)
R_i(t)=max (R_i(t), ub)
R_i(t)=min (R_i(t), lb)
F = f (R_i)
If R_i(t) better than R_ipbest
R_ipbest=R_i(t)
If R_ipbest better than R_Gbest
R_Gbest=R_ipbest
end
end
else
R_i(t)=lb+rand * (ub –lb)
F = f (R)
end
end
end
return (R_Gbest)

D = Problem dimension, N = Number of raindrops Each of the search agents is an array of size 1×D for problems with D dimension, Process of causing rain on palm tree crown is done using the below equation $R = lb + rand * (ub - lb)$ (2)

During the raining process, raindrops are randomly distributed all over the search space using equation 2, Where lb and ub are decision variable ranges, which are decided by the lower bound and upper bound of the objective function solution, respectively and rand represents uniformly generated random values between 0 and 1 by MATLAB.

3.1.3 Random positioning of costa

Costa is randomly generated during initialization and it acts as the meeting point of all concurrent lines or junction points for all possible locus of raindrops. Raindrops falling on leaves are guided over the midribs to costa. $C = lb + rand * (ub - lb)$ (3)

Where, C is a matrix with N×D dimension of feasible costa positions constrained by upper and lower limits and rand is an impartially distributed random number between 0 and 1 by MATLAB.

As the basic idea of this algorithm is to find the optimum position of a particular search space by randomly selecting the positions all over the search space, testing the fitness of e positions, and updating the positions towards the optimum position based on the natural phenomenon of rainwater flow towards the crownshaft over the palm crown. Hence in the mathematical modelling of the algorithm, the speed ofainfall is not considered, as it is uncertain and also, it does not play a significant role in searching crownshaft.

Equations (3) are general form of equations to randomly initiate variables within a particular range, which are mentioned as upper bound and lower bound. Significance of equation (2) is that they are the raindrops or search agents positioned randomly within the range depending on individual optimization problems. Significance of equation (3) is that they are costa of palm tree crown, positioned randomly within the range depending on individual optimization problems.

3.1.4 Fitness evaluation of raindrops

The cost of each raindrop is calculated using the objective function [f(x)] of any optimization problem. $F_{i} = f (R_{i}^{1} R_{i}^{2} \dots \dots R_{i}^{D})$ (4)

3.1.5 Raindrops flow towards costa

Consider the entire search space of any optimization problem as the crown of a palm tree. Raindrops initialized all over the leaves will reach the nearby midribs and then, start flowing towards the costa, by updating the local best position of a specific region covered by that particular leaf to which the costa belongs. At every iteration, the position of raindrop is updated over the midribs. As the formation of midribs grown from the costa is like concurrent lines meeting at costa, the locus of search agent positions initialized over that particular leaf will also be concurrent. Hence, the mathematical model of the flow of raindrops over midribs is $R_{i (t + 1)} = C * A_{1} + m_{1} * rand * (R_{ipbest} - R_{i (t)})$ (5)

Where $m_{1} = M * e^{\frac{- \propto t}{T}}$ , t = Current iteration and T = Maximum iteration

The slope of the palm leaves is governed by M, and as the iterations move closer to the termination criteria, the slope begins to exponentially decrease, indicating the matured palm lves’ behaviour due to ageing. Values of m₁ and A₁ (Costa accelerator) vary from 0.5 to 1. ∝ (Palm leaf age factor) varies from 2 to 4. M vary from 0 to 1. R_ipbest is an array of size 1×D for problems with D dimension. R_ipbest is the individual best position of any raindrop that has been attained until the current iteration, with the highest fitness function compared with all of its fitness values of previous iterations. In the above equation (5), raindrops move towards the local best solution at every iteration from their initial positions and its exploitation around local best is improved by A₁.

D to two physical properties of palm leaves, almost all raindrops, which fall over palm leaves, reach costa. Firstly, it moves with the steep slope nature towards costa. Secondly, the midribs are grown like many water gutters joined at one point, where the raindrops falling over the leaves reach. The paths of raindrops over the palm leaves are considered as concurrent lines meeting at costa. Hence, equation (is inspired by the general equation of concurrent lines.

The area of exploitation inside the search space of any individual raindrop is $\frac{R_{ipbest}}{m_{1}} - C * A_{1} ⩽ R_{i} ⩽ \frac{R_{ipbest}}{m_{1}} + C * A_{1}$ (6)

The general equation of the straight line $(y = \frac{- x}{m} + k)$ serves as an inspiration for equation 6 because the route of raindrops over the palm tree’s crown and towards the crownshaft is always a straight line. C * A₁being constant throughout the process acts as an intersection point of the straight-line path of raindrops at the x-axis. The possible locus of R_i and R_pbest is a straight-line intersecting x-axis at C * A₁. As m₁ is exponentially decaying in nature, the slope of the raindrops locus decreases. On reducing the slope of line for every iteration, the line sweeps around the search space between ub and lb with a fixed mid-point C * A₁ over the x-axis. As shown in Fig. 4, the current position of R_i is considered the best at the origin. Every straight-line path that a raindrop could take is concurrent to the origin, and its length is constrained by its upper and lower bounds while its slope is determined by m₁. The area covered by two horizontal triangles with an apex at origin is the only unexplored area of search agents, if the global optimum is at origin, but it will be covered by other possible crown shafts being updated while converging towards optimum from lower limit or upper limit.

Fig. 4

Search space of raindrops flowing towards crownshaft at origin.

3.1.6 Re-initialization of raining process for raindrops falling over old leaves

Due to the ageing process, older palm leaves hang onto the crownshaft and ready to fall off. Raindrops falling on the dead or older leaves will not reach the crownshaft, as the slope of matured palm tree leaves is usually less concerning the crownshaft. Subsequently, the search agents with lower fitness values are considered as raindrops on dead or matured leaves and the process of raining is reinitialized using equation (2). Raindrops on older leaves are identified by calculating dry fond fitness using an objective function. The probability value (Dry frond fitness) can be between 0 and 1. The formula for calculating the probability is given in equation (8). ${fitness}_{i} = \frac{1}{1 + F_{i}}$ (7) $Probability = \frac{{fitness}_{i}}{\max (fitness)}$ (8)

As dry frond fitness is one of the parameters of algorithm to be initialized at the beginning, if the probability of an individual raindrop is less than dry frond fitness, that raindrop is considered as a poor fitness search agent, and the raining process will be initiated for it, till it falls on young leaves. This re-initialization process helps PTO to explore the search space by discarding poor fitness search agents.

3.1.7 Raindrops flow towards crownshaft

The position of the crownshaft will be at the global best and at every iteration, the position of the crownshaft is updated with the global best solution. $R_{i (t + 1)} = C * A_{1} + m_{2} * rand * (R_{Gbest} - R_{i (t)})$ (9) The exploration phase of PTO is improved by m₂, whose value can be varied between 0 to 2.5. Raindrops, at crownshaft after a concurrent search through steeply sloped palm tree leaves and petioles, will exploit the global best solution with the search space of, $\frac{R_{Gbest}}{m_{1}} - C * A_{1} ⩽ R_{i} ⩽ \frac{R_{Gbest}}{m_{1}} + C * A_{1}$ (10)

The critical research contributions are summarized as follows.

PTO is inspired by the physical significance of palm tree leaves and petioles, which can themselves water and fertilize with their unique architecture, which includes two primary processes i) Flow of raindrops towards costa and ii) Flow of raindrops towards crownshaft and it makes the algorithm effective in solving optimization problems.

The flow of raindrops over each palm tree leaves towards its corresponding costa drifts the population towards the best solution around their corresponding position. This process makes the PTO algorithm effective with high exploitation capability.

The flow of raindrops over each petiole towards the crownshaft drifts the raindrops with high fitness values towards a globally optimum solution. This process further increases the exploitation capability of search agents with high fitness values.

Raindrops that fall over matured leaves are considered spilt to the ground and are reinitialized with raining process which gives another chance to the candidates with poor fitness solutions. This process makes the PTO algorithm more effective with high exploration capability.

PTO is straightforward to implement, and as it gives another chance to the candidates with poor fitness solutions, the proposed PTO algorithm shows outstanding performance in solving composition functions of IEEE-CEC2017, which needs high exploration and exploitation capabilities for any optimization algorithm.

The process of improving the exploitation behavior of high fitness agents and exploration behavior of low fitness agents, which may be trapped in local optima, makes the algorithm unique and hence, it performs better than other algorithms

4 Experimental results and discussions

A detailed experimental evaluation as well as comparison with ten cutting-edge algorithms such as PSO, GA, CSA, ABC, TLBO, ES, GSK, GWO, BBO, and ACO has been carried out to investigate the performance of the proposed PTO algorithm. All algorithms are implemented in MATLAB R2018a on Ryzen 3 3200 G 3.60 GHz processor with 8GB of RAM. Problems used for testing the performance of algorithms are classified into two, SBF and complex constrained optimization problems in real-world. Mathematical benchmark functions used to test the proposed algorithm are grouped based on their physical properties and shapes like unimodal, multimodal, plate and valley shaped, steep ridges, hybrid functions and composition functions.

The performance of proposed PTO algorithm has been compared in two stages. Firstly, with most popular optimization algorithms, which are extensively used for comparison in solving SBFs in many recent optimization algorithms. Though many recent optimization algorithms are proved better than ABC, CUCKOO, GA and PSO in solving SBFs, solving those unconstrained benchmark functions are considered as initial stage of testing for any optimization methods. Secondly the proposed algorithm is compared with constrained IEEE-CEC2017 and IEEE-CEC2011, which are very difficult to solve and are the latest testing methods to evaluate the performance. So, the proposed algorithm is compared with recent optimization algorithms in solving IEEE-CEC2017 and IEEE-CEC2011.

Extensive investigation of the foraging behavior of the PTO algorithm is done through the optimization task using 33 SBF, each with D = 10, 30, 50, and 100 dimensions, to increase the complexity of the optimization task and to compare the results with ABC, CSA, GA and PSO. The details of SBF used are given in Table 3 along with their range and optimum solution. In order to maintain the competition fair, and equal complexity, all the algorithms are executed with a population size of 30 and each of the SBF is executed for 30 runs of 100 iterations, each corresponding to 3000 function evaluations (FEs). The best solution from each run is selected and the averages of the best solutions and the standard deviation for 3000 FEs are calculated.

Table 3
Standard Benchmark functions

S. No. Benchmark Formula Range f_min

functions

f ₁ Ackley -a exp(-b $\sqrt{\frac{1}{d} \sum_{i = 1}^{d} x_{i}^{2}}$ ) - exp ( $\frac{1}{d} \sum_{i = 1}^{d} \cos (c x_{i}$ )+a+exp (1) x_i ∈ [-35, 35] 0

f ₂ Cross-in-tray -0.0001 (|sin(x) sin(y) exp (|100 - $\frac{\sqrt{x^{2} + y^{2}}}{π}$ |) |+1)^0.1 x_i ∈ [-10, 10] –2.0626

f ₃ Griewank 1+ $\sum_{i = 1}^{d} \frac{x_{i}^{2}}{4000} - \prod_{i = 1}^{d} \cos (\frac{x_{i}}{\sqrt{i}})$ x_i ∈ [-100, 100] 0

f ₄ Holder table - |sin(x) cos(y) exp (|1 - $\frac{\sqrt{x^{2} + y^{2}}}{π}$ |) | x_i ∈ [-10, 10] –19.2085

f ₅ Rastrigin 10d+ $\sum_{i = 1}^{d} [x_{i}^{2} - 10 \cos \cos (2 π x_{i})]$ x_i ∈ [-5.12, 5.12] 0

f ₆ ModSchaffer1 $0.5 + \frac{si n^{2} (y_{1}^{2} + y_{2}^{2}) - 0.5}{{[1 + 0.001 (y_{1}^{2} + y)]}^{2}}$ y_i ∈ [-100, 100] 0

f ₇ ModSchaffer2 $0.5 + \frac{si n^{2} (y_{1}^{2} - y_{2}^{2}) - 0.5}{{[1 + 0.001 (y_{1}^{2} + y_{2}^{2})]}^{2}}$ y_i ∈ [-100, 100] 0

f ₈ ModSchaffer3 $0.5 + \frac{\sin^{2} (cos [| y_{1}^{2} - y_{2}^{2} |]) - 0.5}{{[1 + 0.001 (y_{1}^{2} + y)]}^{2}}$ y_i ∈ [-100, 100] 0.0015

f ₉ ModSchaffer4 $0.5 + \frac{co s^{2} (\sin [| y_{1}^{2} - y_{2}^{2} |]) - 0.5}{{[1 + 0.001 (y_{1}^{2} + y_{2}^{2})]}^{2}}$ y_i ∈ [-100, 100] 0.2925

f ₁₀ Schwefel – $\sum_{i = 1}^{d} x_{i} \sin (\sqrt{| x_{i} |})$ x_i ∈ [-500, 500] –837.9658

f ₁₁ Carrom table $- \frac{1}{30} e^{2 | 1 - \frac{\sqrt{x_{1}^{2} + x_{2}^{2}}}{π} |}$ cos²(x₁) cos²(x₂) x_i ∈ [-10, 10] –24.1568

f ₁₂ Chichinaze x_i ∈ [-30, 30] –42.9443

f ₁₃ Himmelblau ( $y_{1}^{2}$ +y₂ -11)²+(y₁+ $y_{2}^{2}$ - 7)² y_i ∈ [–5, 5] 0

f ₁₄ levi13 sin²(3Πx₂)+(x₁ –1) [1 + sin²(3Πx₂)] x_i ∈ [-10, 10] 0

+(x₂ -1)² [1 + sin²((2Πx₂)]

f ₁₅ Penholder $- e^{| coscos (x_{1}) coscos (x_{2}) e^{| 1 - \frac{\sqrt{x_{1}^{2} + x_{2}^{2}}}{π} |}}$ x_i ∈ [-11, 11] –0.9635

f ₁₆ Test-tube holder (x₁) coscos (x₂) $e^{| \cos (\frac{x_{1}^{2} + x_{2}^{2}}{200}) |}$ x_i ∈ [-10, 10] –10.8723

f ₁₇ Sphere $Σ_{i = 1}^{d} x_{i}^{2}$ x_i ∈ [-100, 100] 0

f ₁₈ Rotated Hyper-Ellipsoid $Σ_{i = 1}^{d} Σ_{j = 1}^{i} x_{j}^{2}$ x_i ∈ [-100, 100] 0

f ₁₉ Booth (y₁+2y₂ –7)²+(2y₁+y₂ –5)² y_i ∈ [-10, 10] 0

f ₂₀ Matyas 0.26(y₁²+y₂²) –0.48y₁y₂ y_i ∈ [-10, 10] 0

f ₂₁ McCormick Sin (y₁ + y₂) + (y₁ + y₂) ² - 1.5y₁ + 2.5y₂ + 1 y_i ∈ [-1.5, 4] –1.9133

f ₂₂ Three-hump Camel 2x₁² –1.05x₁⁴+ $\frac{x_{1}^{6}}{6}$ +x₁x₂+x₂² x_i ∈ [-5, 5] 0

f ₂₃ Six-hump camel $(4 - 2.1 x_{1}^{2} + \frac{x_{1}^{4}}{3}) x_{1}^{2} + x_{1} x_{2} + (- 4 + 4 x_{2}^{2}) x_{2}^{2}$ x_i ∈ [-5, 5] –1.0316

f ₂₄ Rosenbrock $\sum_{i = 1}^{d - 1}$ y_i ∈ [-100, 100] 0

f ₂₅ Cube 100(x₂ –x₁³)²+(1 –x₁)² x_i ∈ [-100, 100] 0

f ₂₆ Zettl $\frac{1}{4} x_{1} + {(x_{1}^{2} - 2 x_{1} + x_{2}^{2})}^{2}$ x_i ∈ [-5, 5] –0.003791

f ₂₇ Easom -cos(x) cos(y) exp (-(x - π)² - (y-π)²) x_i ∈ [-100, 100] –1

f ₂₈ Beale (1.5-y₁+y₂)²+(2.25 –y₁+y₁y₂²)²+(2.625-y₁ + y₁y₂³)² x_i ∈ [-4.5, 4.5] 0

f ₂₉ Golstein-price [1+(x + y+1)² (19 –14x+3x² –14y+6xy+3y²)] * x_i ∈ [-2, 2] 3

[30+(2x+3y)² (18 –32x+12x² ⁺ 4y –36xy+27y²)]

f ₃₀ Powell $Σ_{i = 1}^{d / 4} [{(x_{4 i - 3} + 10 x_{4 i - 2})}^{2} + 5 {(x_{4 i - 1} - x_{4 i})}^{2} + {(x_{4 i - 2} - 2 x_{4 i - 1})}^{4}$ x_i ∈ [-4, 5] 0

+10 (x_4i-3 - x_4i) ⁴]

f ₃₁ Bird sin(x) exp((1-cos(y))²)+cos(y) exp((1-sin(x))²)+(x –y)² x ∈ [–2π,2π], –106.7645

y∈ [-2π,2π]

f ₃₂ Cross-Leg Table $\frac{1}{{(| sin (x_{1}) sin (x_{2}) e^{| 100 - \frac{\sqrt{x_{1}^{2} + x_{2}^{2}}}{π} |} | + 1)}^{0.1}}$ x_i ∈ [-10, 10] –1

f ₃₃ Leon 100 (x₂ –x₁³)² + (1 - x₁)² x_i ∈ [-1.2, 1.2] 0

S. No.	Benchmark	Formula	Range	f_min
f ₁	Ackley	-a exp(-b $\sqrt{\frac{1}{d} \sum_{i = 1}^{d} x_{i}^{2}}$ ) - exp ( $\frac{1}{d} \sum_{i = 1}^{d} \cos (c x_{i}$ )+a+exp (1)	x_i ∈ [-35, 35]	0
f ₂	Cross-in-tray	-0.0001 (\|sin(x) sin(y) exp (\|100 - $\frac{\sqrt{x^{2} + y^{2}}}{π}$ \|) \|+1)^0.1	x_i ∈ [-10, 10]	–2.0626
f ₃	Griewank	1+ $\sum_{i = 1}^{d} \frac{x_{i}^{2}}{4000} - \prod_{i = 1}^{d} \cos (\frac{x_{i}}{\sqrt{i}})$	x_i ∈ [-100, 100]	0
f ₄	Holder table	- \|sin(x) cos(y) exp (\|1 - $\frac{\sqrt{x^{2} + y^{2}}}{π}$ \|) \|	x_i ∈ [-10, 10]	–19.2085
f ₅	Rastrigin	10d+ $\sum_{i = 1}^{d} [x_{i}^{2} - 10 \cos \cos (2 π x_{i})]$	x_i ∈ [-5.12, 5.12]	0
f ₆	ModSchaffer1	$0.5 + \frac{si n^{2} (y_{1}^{2} + y_{2}^{2}) - 0.5}{{[1 + 0.001 (y_{1}^{2} + y)]}^{2}}$	y_i ∈ [-100, 100]	0
f ₇	ModSchaffer2	$0.5 + \frac{si n^{2} (y_{1}^{2} - y_{2}^{2}) - 0.5}{{[1 + 0.001 (y_{1}^{2} + y_{2}^{2})]}^{2}}$	y_i ∈ [-100, 100]	0
f ₈	ModSchaffer3	$0.5 + \frac{\sin^{2} (cos [\| y_{1}^{2} - y_{2}^{2} \|]) - 0.5}{{[1 + 0.001 (y_{1}^{2} + y)]}^{2}}$	y_i ∈ [-100, 100]	0.0015
f ₉	ModSchaffer4	$0.5 + \frac{co s^{2} (\sin [\| y_{1}^{2} - y_{2}^{2} \|]) - 0.5}{{[1 + 0.001 (y_{1}^{2} + y_{2}^{2})]}^{2}}$	y_i ∈ [-100, 100]	0.2925
f ₁₀	Schwefel	– $\sum_{i = 1}^{d} x_{i} \sin (\sqrt{\| x_{i} \|})$	x_i ∈ [-500, 500]	–837.9658
f ₁₁	Carrom table	$- \frac{1}{30} e^{2 \| 1 - \frac{\sqrt{x_{1}^{2} + x_{2}^{2}}}{π} \|}$ cos²(x₁) cos²(x₂)	x_i ∈ [-10, 10]	–24.1568
f ₁₂	Chichinaze	x_i ∈ [-30, 30]	–42.9443
f ₁₃	Himmelblau	( $y_{1}^{2}$ +y₂ -11)²+(y₁+ $y_{2}^{2}$ - 7)²	y_i ∈ [–5, 5]	0
f ₁₄	levi13	sin²(3Πx₂)+(x₁ –1) [1 + sin²(3Πx₂)]	x_i ∈ [-10, 10]	0
		+(x₂ -1)² [1 + sin²((2Πx₂)]
f ₁₅	Penholder	$- e^{\| coscos (x_{1}) coscos (x_{2}) e^{\| 1 - \frac{\sqrt{x_{1}^{2} + x_{2}^{2}}}{π} \|}}$	x_i ∈ [-11, 11]	–0.9635
f ₁₆	Test-tube holder	(x₁) coscos (x₂) $e^{\| \cos (\frac{x_{1}^{2} + x_{2}^{2}}{200}) \|}$	x_i ∈ [-10, 10]	–10.8723
f ₁₇	Sphere	$Σ_{i = 1}^{d} x_{i}^{2}$	x_i ∈ [-100, 100]	0
f ₁₈	Rotated Hyper-Ellipsoid	$Σ_{i = 1}^{d} Σ_{j = 1}^{i} x_{j}^{2}$	x_i ∈ [-100, 100]	0
f ₁₉	Booth	(y₁+2y₂ –7)²+(2y₁+y₂ –5)²	y_i ∈ [-10, 10]	0
f ₂₀	Matyas	0.26(y₁²+y₂²) –0.48y₁y₂	y_i ∈ [-10, 10]	0
f ₂₁	McCormick	Sin (y₁ + y₂) + (y₁ + y₂) ² - 1.5y₁ + 2.5y₂ + 1	y_i ∈ [-1.5, 4]	–1.9133
f ₂₂	Three-hump Camel	2x₁² –1.05x₁⁴+ $\frac{x_{1}^{6}}{6}$ +x₁x₂+x₂²	x_i ∈ [-5, 5]	0
f ₂₃	Six-hump camel	$(4 - 2.1 x_{1}^{2} + \frac{x_{1}^{4}}{3}) x_{1}^{2} + x_{1} x_{2} + (- 4 + 4 x_{2}^{2}) x_{2}^{2}$	x_i ∈ [-5, 5]	–1.0316
f ₂₄	Rosenbrock	$\sum_{i = 1}^{d - 1}$	y_i ∈ [-100, 100]	0
f ₂₅	Cube	100(x₂ –x₁³)²+(1 –x₁)²	x_i ∈ [-100, 100]	0
f ₂₆	Zettl	$\frac{1}{4} x_{1} + {(x_{1}^{2} - 2 x_{1} + x_{2}^{2})}^{2}$	x_i ∈ [-5, 5]	–0.003791
f ₂₇	Easom	-cos(x) cos(y) exp (-(x - π)² - (y-π)²)	x_i ∈ [-100, 100]	–1
f ₂₈	Beale	(1.5-y₁+y₂)²+(2.25 –y₁+y₁y₂²)²+(2.625-y₁ + y₁y₂³)²	x_i ∈ [-4.5, 4.5]	0
f ₂₉	Golstein-price	[1+(x + y+1)² (19 –14x+3x² –14y+6xy+3y²)] *	x_i ∈ [-2, 2]	3
		[30+(2x+3y)² (18 –32x+12x² ⁺ 4y –36xy+27y²)]
f ₃₀	Powell	$Σ_{i = 1}^{d / 4} [{(x_{4 i - 3} + 10 x_{4 i - 2})}^{2} + 5 {(x_{4 i - 1} - x_{4 i})}^{2} + {(x_{4 i - 2} - 2 x_{4 i - 1})}^{4}$	x_i ∈ [-4, 5]	0
		+10 (x_4i-3 - x_4i) ⁴]
f ₃₁	Bird	sin(x) exp((1-cos(y))²)+cos(y) exp((1-sin(x))²)+(x –y)²	x ∈ [–2π,2π],	–106.7645
			y∈ [-2π,2π]
f ₃₂	Cross-Leg Table	$\frac{1}{{(\| sin (x_{1}) sin (x_{2}) e^{\| 100 - \frac{\sqrt{x_{1}^{2} + x_{2}^{2}}}{π} \|} \| + 1)}^{0.1}}$	x_i ∈ [-10, 10]	–1
f ₃₃	Leon	100 (x₂ –x₁³)² + (1 - x₁)²	x_i ∈ [-1.2, 1.2]	0

The ranking of algorithms is based on the average value, whichever is near the optimum solution from the lowest to the highest. If there is a tie, the standard deviation (SD) is considered. Statistical results, i.e., the average cost function and its corresponding SD for SBF problems are depicted in Tables 4 5. The Ribbon chart of Ranking in Fig. 5 shows that the PTO has performed well in most SBF.

Table 4

Statistical result comparison of different algorithms for SBFs (10 & 30 Dimensions)

S.No	Parameters	10D					30D
		ABC	CUCKOO	GA	PSO	PTO	ABC	CUCKOO	GA	PSO	PTO
1	MEAN	0.00121	0.08852	2.6520416	0.00012378	0.001549	0.0521	0.043946	2.6478054	7.90E-05	0.0002264
	SD	0.006592	0.338109	0	0.00067338	0.000271	0.2211	0.239631	0.0196574	0.0004301	0.0002656
2	MEAN	–2.06261	–2.06208	–2.060421	–2.06261	–2.06261	–2.062484	–2.062294	–2.062337	–2.062612	–2.06261
	SD	5.91E-07	0.002124	9.22E-16	1.6739E-05	0	3.78E-05	0.001739	1.44E-05	4.62E-08	9.38E-09
3	MEAN	6.8E-05	0.229287	1.6863133	8.9391E-07	4.68E-08	0068	171441	0.1462488	3.80E-05	1.34E-09
	SD	0.000372	0.659752	1.02E-12	4.8961E-06	1.41E-06	0.0294	0.436122	0.0006771	0.0002079	0
4	MEAN	–19.2083	–19.178	–17.36836	–19.208503	–19.2058	–19.1974	–19.20634	–19.16471	–19.2085	–19.15465
	SD	0.001193	0.165638	5.27E-7	8.808E-08	0.00044	0.0507	0.011608	0	4.73E-09	0.0040458
5	MEAN	2.12E-06	0.000119	8.9668736	3.8657E-06	2.62E-06	0.0413	3.37E-05	0.0744924	1.04E-06	3.49E-07
	SD	1.16E-05	0.000594	0	2.1173E-05	3.38E-05	0.1954	0.000179	0	5.70E-05	1.24E-07
6	MEAN	0.000189	0.10286	0.1921372	8.6742E-07	0	0.0021	0.084893	0.0062242	1.60E-09	1.20E-11
	SD	0.000753	0.095052	1.56E-12	4.751E-06	3.01E-09	0.0044	0.097684	1.84E-10	8.78E-09	0
7	MEAN	9.97E-05	0.000191	0.2855263	1.2148E-08	3.02E-09	0.0018	0.030785	0.0022128	4.67E-09	6.72E-11
	SD	0.000406	0	1.13E-16	6.6535E-08	2.98E-07	0.0037	0.035217	0.0011982	2.56E-08	1.66E-10
8	MEAN	0.001762	0.0743	0.0112747	0.00157632	0.001567	0.0062	0.055729	0.0182494	0.0015672	0.0015673
	SD	0.000655	0.063357	5.57E-16	4.4418E-05	4.44E-07	0.0159	0.061247	0	2.07E-06	1.98E-06
9	MEAN	0.293062	0.319773	0.3021793	0.29258538	0.292579	05	214011	0.2952589	0.2925803	0.2925788
	SD	0.001775	0.022643	6.35E-17	3.6953E-05	3.14E-06	0.0038	0.182625	0	9.19E-06	0
10	MEAN	–837.946	–710.51	–836.9588	–837.96577	–837.96	–837.9235	–789.3075	–601.0766	–837.9657	–837.8789
	SD	0.109653	27.47522	4.63E-13	8.2021E-06	3.11E-05	0.2245	0	0	1.50E-04	2.24E-03
11	MEAN	–24.1568	–24.135	–20.82397	–24.156816	–24.155	–24.11039	–24.15622	–24.10096	–24.15683	–24.15682
	SD	6.14E-06	0.116866	6.58E-12	7.27E-09	0.000256	0	0.002227	0	0	1.08E-14
12	MEAN	–42.9436	–42.11	–28.51204	–42.497173	–42.9443	–42.70239	–41.50848	–41.36829	–42.49717	–42.9436
	SD	0.004415	1.640084	0.00737335	2.2654E-08	1.42E-08	0.2987992	4.053657	0	1.05E-09	0.0008389
13	MEAN	0.000247	4.78E-05	0.0112834	9.6052E-09	2.95E-05	0.0821998	0.000102	0.814202	0	0.0013656
	SD	0.001352	0.00026	0.00626940	5.261E-08	0.005519	0.0938729	0.00055	0	3.92E-10	0.0013225
14	MEAN	4.2E-06	0.000372	0.1482836	5.0394E-09	1.2E-05	8.00E-04	0.000291	1.0923882	1.32E-10	4.25E-05
	SD	2.3E-05	0.002034	7.30E-14	2.7602E-08	0.000272	0.0038	0.001524	0	6.89E-10	9.85E-05
15	MEAN	–0.96353	–0.96263	–0.957812	–0.9635348	–0.96353	–0.963276	–0.963358	–0.963533	–0.963535	–0.963468
	SD	3.66E-06	0.004919	6.78E-16	2.3101E-11	3.95E-05	9.91E-05	0.000969	0	0	0.0001376
16	MEAN	–10.8721	–10.87	–10.8689	–10.871639	–10.8722	–10.8484	–10.8529	–10.86791	–10.8716	–10.87221
	SD	0.000899	0.011576	0.04263716	0.00361795	0.000383	0.0169	4.34E-04	0.0007858	0.0036	0.0001677
17	MEAN	4.81E-05	231.7759	8.88E-08	2.105E-08	2.3E-11	19.76924	0.00023	534.66795	1.9227094	0.0334889
	SD	0.000176	3.32584	2.84E-14	1.153E-07	3.83E-08	4.0409282	0.001206	4.4835535	8.9165381	0.0096079
18	MEAN	0.000528	3.65897	11.402937	6.338E-07	7.42E-07	0.2177475	0.02214	4.755612	3.19E-07	0.0129856
	SD	0.002825	3.65E-13	2.15E-12	3.4715E-06	1.79E-06	0.8292721	0.115628	5.65E-13	1.75E-06	0.0134486
19	MEAN	0.001187	2.87E-05	0.3456661	8.246E-09	1.71E-05	0.0205737	0.000119	0.0067832	1.22E-07	0.0002281
	SD	0.00576	0.000146	0.00098974	4.5165E-08	5.83E-05	0.0060825	0.000625	0	6.70E-07	0.0008301
20	MEAN	4.23E-05	2.96E-05	0.0004925	1.3441E-12	1.54E-13	0.003301	0.000552	0.0010882	3.09E-10	1.01E-10
	SD	6.59E-05	9.48E-05	0	7.3597E-12	6.15E-08	0.0094336	0.003018	0	1.69E-09	1.22E-09
21	MEAN	–1.91051	–1.85688	–1.910507	–1.9105075	–1.91351	–1.9077	–1.9007	–1.7863	–1.9132	–1.913224
	SD	3.53E-06	0.116691	0	8.2465E-17	5.98E-11	0.0031	0.048	1.36E-15	5.44E-10	0.0001644
22	MEAN	5.01E-07	3.58E-05	0.2993628	2.4408E-10	1.68E-07	0.0104212	2.06E-06	0.0122073	6.56E-10	2.39E-10
	SD	2.55E-06	0.000176	1.69E-16	1.3369E-09	3.57E-08	0.0312289	7.24E-06	0	3.59E-09	3.30E-09
23	MEAN	–1.03163	–1.03015	–0.942382	–1.0316285	–1.03162	–1.029028	–1.031564	–1.018981	–1.031628	–1.031628
	SD	7.12E-06	0.007971	3.23E-14	5.6686E-11	2.05E-05	0.0058979	0.00033	0	2.70E-08	4.97E-06
24	MEAN	0.18939	0.021236	17570.967	14.6022199	0.000119	2.6208	0.020664	1.4760072	0.0008216	0.0004981
	SD	0.092582	0.123568	1.40E-08	30.791228	0.137602	4.6981	0.106598	0	0.0044888	0.0007662
25	MEAN	0.083503	0.037897	6079246.1	0.00199879	0.001291	2.6325	0.042212	9.0993668	0.0069879	0.0034071
	SD	2.82E-17	0.214537	598260.041	0.01086224	0.1244	7.0369	0.222152	0	0.0375635	0.0132056
26	MEAN	–0.00357	–0.00373	0.1648699	–0.0037912	–0.00379	–0.001208	–0.003723	–0.002305	–0.003791	–0.003791
	SD	0.001197	0.000237	0.00032974	7.28E-07	6.58E-08	0.0060075	0.000242	0.0003713	2.99E-09	2.98E-09
27	MEAN	–0.84313	–0.44254	–6.79E-06	–0.9995369	–0.99994	–0.690102	–0.063598	–0.667594	–0.999682	–1
	SD	0.347748	0.48151	1.5500E-06	3.7748E-06	0.000408	0.1875905	0.042342	0	1.72E-06	0.000726
28	MEAN	0.000225	0.000231	1.29E-06	1.9313E-09	2.79E-06	0.0150122	7.43E-06	0.1194171	1.03E-08	1.84E-05
	SD	0.000253	0.001256	4.66E-07	1.0578E-08	4.42E-05	0.0316923	3.28E-05	9.24E-14	5.63E-08	7.56E-05
29	MEAN	3.000056	5.82E-05	4.765017	3.00000006	3.000117	3.1184733	1.30E-05	31.566722	3.0000012	3
	SD	0.000213	0.000317	0	3.441E-07	0.002296	0.3738272	6.18E-05	0	2.17E-07	0.0004964
30	MEAN	0.000721	1.71E-05	9.2300671	6.2152E-06	3.81E-06	0.0015	2.58E-05	37.609237	1.74E-05	2.64E-06
	SD	0.002976	8.91E-05	0.04041511	2.8985E-05	2.93E-05	0.0026	0.000183	0.0734116	8.70E-05	0
31	MEAN	–106.764	–106.763	–96.49331	–106.76453	–106.7645	–106.6136	–106.7612	–106.7391	–106.7645	–106.7645
	SD	0.000241	0.010712	2.67E-13	4.1752E-08	0	0.214	0.0176	2.89E-14	2.96E-08	0.0377137
32	MEAN	–0.03451	–0.00068	–0.000705	–0.072481	–0.00026	–0.000207	–0.000804	–0.000232	–0.0723	–0.000328
	SD	0.025874	0.000204	0.00027559	0.02877148	4.14E-05	2.76E-05	0.000221	2.98E-05	0.028	6.43E-06
33	MEAN	0.006607	7.29E-07	0.1446569	2.1465E-07	0.000107	0.002171	5.17E-06	0.8271014	5.99E-08	3.41E-05
	SD	0.007185	3.35E-06	4.2223E-05	1.1646E-06	0.000861	0.0015455	2.41E-05	0	2.92E-07	5.25E-05

Table 5

Statistical result comparison of different algorithms for SBFs (50 & 100 Dimensions)

S.No	Parameters	50D					100D
		ABC	CUCKOO	GA	PSO	PTO	ABC	CUCKOO	GA	PSO	PTO
1	MEAN	0.203836	0.091797	0.418882	7.9887E-05	0.001256	0.72298951	0.119637	1.084321	2.05E-05	0.0700121
	SD	0.983626	0.440784	2.82E-16	0.00043747	0.001321	1.24327274	0.649965	0.215873	0.000109	0.5050599
2	MEAN	–2.06256	–2.06136	–2.05413	–2.0626119	–2.062611	–2.062339	–2.061605	–2.06228	–2.06261	–2.062611
	SD	0.000198	0.006873	9.03E-16	1.0528E-08	4.85E-07	0.00096169	0.004645	0.000352	1.31E-09	0.0002242
3	MEAN	0.030861	0.15197	1.090112	1.1567E-06	8.84E-07	0.12155436	0.029791	0.721688	1.88E-07	0.0008769
	SD	0.132396	0.390703	4.84E-13	6.3357E-06	9.04E-06	0.22781158	0.093709	4.1E-13	1.03E-06	0.0591036
4	MEAN	–19.1851	–19.1832	–19.191	–18.020717	–19.20835	–19.046197	–19.20832	–19.1947	–19.2085	–19.20852
	SD	0.118934	0.10504	0.001506	4.3661E-14	0.003607	0.36959693	0.000867	0.001548	7.72E-08	0.0540174
5	MEAN	0.04408	0.000499	1.38798	1.4799E-07	4.86E-08	0.24479064	0.000445	2.361826	2.49E-06	0.0013201
	SD	0.213217	0.002722	2.92E-14	8.1059E-07	2.58E-05	0.62539907	0.002436	3.76E-14	1.36E-05	0.0716581
6	MEAN	0.025764	0.051447	0.003129	5.1821E-09	4.39E-10	0.02885372	0.046044	0.07761	2.03E-09	2.70E-10
	SD	0.039304	0.084793	0	2.8384E-08	9.28E-09	0.02537355	0.057776	3.04E-05	1.11E-08	0.0051055
7	MEAN	0.005418	0.123377	0.063813	3.3094E-10	5.51E-11	0.04034702	0.026303	0.404065	9.79E-6	9.128E-11
	SD	0.00893	0.123637	4.23E-17	1.8126E-09	8.88E-09	0.06658279	0.079602	1.21E-13	5.36E-6	3.24E-8
8	MEAN	0.013623	0.043176	0.048285	0.00158422	0.001567	0.04333538	0.032687	0.009237	0.001568	0.0018125
	SD	0.034091	0.034655	0	9.5117E-05	7.86E-06	0.10217329	0.034958	0	5.34E-06	0.0031941
9	MEAN	0.298594	0.229251	0.386301	0.29257875	0.292579	0.30309421	0.181814	0.292607	0.292584	0.2925406
	SD	0.013291	0.142531	0.000163	6.3711E-07	1.01E-06	0.01507569	0.156538	5.15E-05	3.13E-05	0.0015037
10	MEAN	–835.368	–791.285	–818.067	–837.96577	–837.9515	–820.80234	–829.57	–617.882	–837.961	–837.9658
	SD	9.119449	57.77796	2.31E-13	1.1091E-07	0.00031	33.5452383	26.53536	7.19E-13	48.0292	4.302E-09
11	MEAN	–24.0339	–24.1564	–6.75316	–20.984789	–24.15547	–24.071443	–23.72224	–11.1008	–24.1568	–24.15169
	SD	0.606456	0.001574	0.000801	1.68E-13	0.003345	0.29013815	2.23601	1.45E-11	1.31E-09	0.0731402
12	MEAN	–42.8312	–42.6892	–42.2461	–42.944387	–42.94405	–42.50825	–42.7952	–42.3443	–42.4971	–42.94017
	SD	0.489862	0.814896	2.17E-14	2.5244E-06	0.13639	1.97422018	0.220283	1.62E-14	8.06E-08	0.1527678
13	MEAN	0.016465	0.000323	0.079301	9.3567E-09	0.000169	0.05134669	8.62E-05	0.53085	2.84E-07	5.13E-08
	SD	0.085235	0.001763	8.19E-13	5.1248E-08	0.00548	0.21689679	0.000469	1.18E-12	1.56E-06	0.0192251
14	MEAN	0.000206	7.37E-05	0.053389	4.0718E-07	7.01E-06	0.1056678	0.000131	1.776167	3.72E-08	0.0001702
	SD	0.000752	0.000399	2.46E-13	2.23E-06	2.17E-05	0.28517305	0.0007	0.017602	2.04E-07	0.0167194
15	MEAN	–0.96349	–0.96353	–0.96352	–0.9635348	–0.963533	–0.9634007	–0.963523	–0.96325	–0.96353	–0.963531
	SD	0.000176	3.62E-05	5.65E-16	1.135E-11	3.02E-05	0.00039487	6.45E-05	0	1.45E-12	8.891E-05
16	MEAN	–10.8631	–10.8713	–10.8027	–10.8723	–10.8722	–10.811751	–10.87089	–10.6039	–10.8721	–10.8723
	SD	0.021819	0.002098	5.42E-15	1.2292E-08	0.00095	0.10447952	0.002852	0.000579	0.013017	5.518E-07
17	MEAN	0.00189	0.1247	7.81E-11	5.3251E-05	4.48E-10	0.03249981	0.00179	0.000455	3.27E-08	0.0491904
	SD	0.006448	0.03258	3.87E-19	0.00028879	0.003771	0.03100522	0.23589	0.00173	1.79E-07	0.0022013
18	MEAN	1.567259	1.74528	255.5396	4.7515E-07	1.37E-07	2.45884845	6.9854	141.4281	2.84E-09	10.811337
	SD	5.47795	3.2658	6.45E-05	2.6025E-06	4.18E-05	6.50960183	1.3589	8.93E-12	1.56E-08	2.7346
19	MEAN	0.018516	0.000106	2.992086	5.0485E-09	3.01E-05	0.1005725	0.000366	0.096077	1.61E-09	2.454E-05
	SD	0.035509	0.000573	4.52E-16	2.7649E-08	0.000353	0.1608478	0.001977	0.004552	8.83E-09	0.0039
20	MEAN	0.001415	0.000248	0.020894	1.7676E-09	1.64E-09	0.00845949	0.000437	0.080237	1.791E07	1.98E-10
	SD	0.005715	0.001001	0	9.6813E-09	1.67E-08	0.02202692	0.002349	7.78E-05	0.0002282	1.08E-09
21	MEAN	–1.91042	–1.86696	–1.91051	–1.9105075	–1.911508	–1.9104634	–1.871229	–1.91049	–1.91050	–1.911441
	SD	0.000229	0.096999	1.13E-15	4.0061E-12	1.78E-07	0.00023853	0.102037	9.03E-16	1.02E-12	0.000794
22	MEAN	0.000534	0.000166	0.252974	5.6635E-09	0	0.0073658	2.24E-05	0.014275	1.47E-06	9.482E-09
	SD	0.0027	0.000911	4.15E-14	3.102E-08	2.99E-07	0.01852335	8.73E-05	2.17E-15	0.000411	5.193E-08
23	MEAN	–1.031	–1.03014	–0.97449	–1.0316283	–1.031628	–1.0291381	–1.031607	–0.45109	–1.03162	–1.031615
	SD	0.002484	0.007121	7.03E-14	7.0852E-07	8.36E-06	0.00627678	0.000116	6.37E-14	7.54E-09	0.0017513
24	MEAN	34.97964	24.568	0.984684	0.0006	7.79E-05	13.4384543	48.6398	307019.8	2.566781	8.3161
	SD	135.621	12.36589	0.107908	0.00325736	0.096449	29.273647	19.32456	5.46E-08	13.56761	2.15483
25	MEAN	24.10293	31.9857	0.083514	0.00086657	0.001134	1307.25374	17.0235	26.28554	0.000394	11.640093
	SD	119.5913	78.32498	0.188468	0.00474523	0.038767	6981.02272	12.0236	2.23E-12	0.002139	2.7742538
26	MEAN	–0.00356	–0.00366	0.082071	–0.0037912	–0.003791	–0.0029258	–0.003657	2.866731	–0.00379	–0.003791
	SD	0.000474	0.00057	0.000218	1.7308E-09	2.04E-07	0.00133073	0.000486	5.14E-13	5.32E-10	0.0003874
27	MEAN	–0.38659	–0.49662	–4.6E-07	–0.999997	–0.999997	–3.701E-05	–0.038957	–8.54E-08	–0.99255	–1
	SD	0.21692	0.469842	2.84E-19	1.6603E-05	0	1.6565E-05	0.042359	8.24E-09	0.200487	4.386E-07
28	MEAN	0.001741	0.00016	0.768218	6.1891E-11	4.09E-05	0.01196555	8.1E-05	0.882987	5.09E-06	7.041E-09
	SD	0.003811	0.000849	0.000393	3.3899E-10	5.84E-05	0.00831909	0.00044	1.49E-13	0.233568	3.857E-08
29	MEAN	3.288125	7.28E-06	3.391648	3.00000003	3.000009	5.38977295	3.74E-05	36.56553	3.000011	3
	SD	1.255786	3.99E-05	0.002278	1.6981E-07	0.001307	6.65782136	0.000204	3.68E-12	0.010642	6.843E-08
30	MEAN	0.097618	0.000185	4.868413	1.8344E-05	3.55E-06	1.25843688	5.64E-05	1.030654	9.34E-06	0.0254193
	SD	0.475797	0.001001	0.117057	9.1247E-05	3.66E-05	5.3209965	0.000303	0.064298	4.06E-05	0.1659129
31	MEAN	–106.701	–106.764	–100.865	–106.76454	–106.7632	–106.26458	–106.7643	–105.362	–106.764	–106.762
	SD	0.323819	0.00015	2.89E-14	2.9667E-08	0.088386	1.32341059	0.000688	5.78E-14	3.4E-09	3.5573412
32	MEAN	–0.00025	–0.00044	–0.00017	–0.0724088	–0.000256	–0.0002059	–0.000751	–0.00024	–0.06818	–0.000236
	SD	5.62E-05	4.27E-05	3.07E-06	0.02788931	2.51E-05	3.7164E-05	0.000169	7.38E-12	0.030295	2.679E-05
33	MEAN	0.008546	8.7E-06	3.86E-06	1.0605E-06	2.56E-07	0.05500199	3.23E-06	0.013998	1.18E-06	1.12E-06
	SD	0.001673	4.51E-05	8.52E-15	5.8076E-06	0.000615	0.07889816	1.63E-05	0.000106	6.49E-06	0.0006089

Fig. 5

Spider chart for ranks among all compared algorithms in solving SBFs.

The proposed algorithm is tested with 29 constrained optimization test functions from IEEE-CEC2017 and seven CEC2011 benchmarks and two real-world engineering design optimization problems are also employed. To evaluate the performance, the solution error measure (f(x)-f(x*)) is considered, where f(x) is the best solution obtained after one run of algorithms and f(x*) is the global optimum of the corresponding SBF. If the value of errors and its standard deviations are less than 10e-8, they are considered as zero [75]. Each benchmark functions undergoes a 10,000XD number of FEs with a population size of 100 for 51 independent runs. The performance of the proposed algorithm in solving CEC2017 and CEC2011 is statistically compared with the results of TLBO, ES, GSK, GWO, BBO, and ACO. For analyzing the numerical efficiency of the PTO algorithm, SBF and CEC2017 benchmark functions are divided into six types: unimodal, multimodal, plate & valley shaped, steep ridges, hybrid and composition functions. The control parameters of all the algorithms are given in Table 2 based on the recommendation of their original references. Besides comparing different algorithms qualitatively, in order to analyze the behavior of algorithms, their results are tested using a non-parametric statistical method called the Wilcoxon signed rank test at a significance level of 0.05, as shown in Table 9.

Table 2

Parameters setting for ABC, CSA, GA, PSO, TLBO, ES, GSK, GWO, BBO, ACO

S.No	Algorithms	Parameters
1.	ABC	Number of food sources (NFS)=100, Modification rate = 0.8
		$Limit = \frac{(NFS X D)}{2}$
2.	CSA	Transition probability coefficient = 0.1, Transition clusters = 5, Step length = 0.01, Levy distribution = 1.5
3.	GA	Roulette wheel selection, Single point crossover probability = 1
		Mutation probability = 0.01
4.	PSO	Population=100
		C₁ = 2; C₂ = 2; W = 0.6.
5.	TLBO	No
6.	ES	σ = 1, λ = 10
7.	GSK	P=0.1, Knowledge factor = 0.5, Knowledge ratio = 0.9, Knowledge rate = 10
8.	GWO	a –Linear decrement from 2 to 0
9.	BBO	Mutation probability = 0.05; Mutation probability = 0.05, Habitat modification probability = 1; Step size = 1
		Maximum immigration = 1; Migration rate = 1
10.	ACO	Initial pheromone = 1e^-6, Exploration constant = 1, Visibility sensitivity = 5, Pheromone update constant = 20, Global pheromone decay rate = 0.9, Local pheromone decay rate = 0.5, Pheromone sensitivity = 1
11.	PTO	A₁ = M = 0.8, α = 4, P = 0.2, m₂ = 1.5

Table 9

Non-parametric Statistical test result by Wilcoxon signed-rank test (Significant level = 0.05)

Types of Functions	Algorithms	R⁺	R^–	p value	r	+	–	=
Unimodal	PTO vs ABC	3	0	3.20E-06	6.12E-01	2	0	0
Functions	PTO vs CSA	1	2	4.78E-01	4.21E-01	1	1	0
SBF	PTO vs GA	3	0	4.60E-06	6.32E-01	2	0	0
(f₁₇ & f₁₈	PTO vs PSO	2	1	3.45E-01	2.68E-01	1	1	0
CEC2017	PTO vs TLBO	0	3	6.80E-02	1.24E-01	0	2	0
(f₁ & f₃)	PTO vs ES	3	0	3.90E-06	7.12E-01	2	0	0
	PTO vs GSK	0	3	6.71E-01	1.36E-01	0	2	0
	PTO vs GWO	2	1	2.54E-02	4.87E-01	1	1	0
	PTO vs BBO	2	1	3.20E-06	5.52E-01	1	1	0
	PTO vs ACO	2	1	4.63E-05	5.08E-01	2	0	0
Multimodal	PTO vs ABC	125	11	3.60E-06	7.02E-01	14	2	0
Functions SBF	PTO vs CSA	117	19	1.30E-07	6.74E-01	14	2	0
(f₁ - f₁₆	PTO vs GA	122	14	1.54E-07	6.82E-01	14	2	0
CEC2017	PTO vs PSO	81	55	1.56E-01	5.07E-01	8	8	0
(f₄ - f₁₀)	PTO vs TLBO	27	1	5.80E-07	6.12E-01	6	1	0
	PTO vs ES	28	0	1.56E-06	6.50E-01	7	0	0
	PTO vs GSK	10.5	4.5	7.30E-06	4.10E-01	4	1	2
	PTO vs GWO	28	0	3.20E-08	6.41E-01	7	0	0
	PTO vs BBO	26	2	5.60E-08	5.24E-01	6	1	0
	PTO vs ACO	28	0	6.50E-06	7.10E-01	7	0	0
Plate &Valley	PTO vs ABC	36	0	4.25E-07	6.40E-01	8	0	0
Shaped	PTO vs CSA	33.5	2.5	4.80E-06	5.58E-01	7	1	0
Functions SBF	PTO vs GA	36	0	2.30E-07	5.15E-01	8	0	0
(f₁₉ - f₂₆)	PTO vs PSO	29	7	2.30E-02	4.18E-01	7	1	0
Steep Ridges	PTO vs ABC	27	1	2.50E-06	4.72E-01	6	1	0
and other	PTO vs CSA	23	5	3.60E-02	5.17E-01	6	1	0
Functions SBF	PTO vs GA	27	1	5.40E-03	5.04E-01	6	1	0
(f₂₇ - f₃₃)	PTO vs PSO	19.5	8.5	1.56E-02	3.68E-01	4	3	0
Hybrid	PTO vs TLBO	51	4	2.60E-08	5.18E-01	8	2	0
Functions	PTO vs ES	55	0	4.80E-07	6.14E-01	10	0	0
CEC2017	PTO vs GSK	19	36	3.56E-01	2.14E-01	5	5	0
(f₁₁ - f₂₀)	PTO vs GWO	55	0	3.60E-07	5.50E-01	10	0	0
	PTO vs BBO	48	7	2.30E-07	4.78E-01	9	1	0
	PTO vs ACO	55	0	3.20E-06	7.10E-01	10	0	0
Composition	PTO vs TLBO	55	0	2.80E-04	6.32E-01	10	0	0
Functions	PTO vs ES	55	0	2.60E-07	6.12E-01	0	0
CEC2017	PTO vs GSK	52	3	6.30E-07	6.28E-01	9	1	0
(f₂₁ - f₃₀)	PTO vs GWO	55	0	3.60E-07	7.00E-01	10	0	0
	PTO vs BBO	55	0	2.50E-06	6.74E-01	10	0	0
	PTO vs ACO	55	0	2.50E-07	6.35E-01	10	0	0

4.1 Comparison with other algorithms using SBF

The statistical result comparisons with four state-of-the-art algorithms on SBFs with 10D,30D,50D and 100D, are summarized in Tables 4 5, respectively. and they include the mean of best solutions and the SD of 3000FEs. The best solutions of all the functions are highlighted in bold.

Multimodal SBFs have many local optima. Algorithms with poor exploration capability will be trapped in one of many local optima of multimodal functions. If the problem’s dimension is increased, it will exponentially increase the local optima number. Therefore, increased dimension multimodal function is a better performance indicator for analyzing the exploration capability of any optimization algorithm. As reported in Tables 4 5, it is evident that the PTO has outperformed most of the functions of many local optima and it is the second-best in most of the remaining functions in all dimensions. The PTO algorithm has outperformed other algorithms in 10, 7, 9, and 9 functions in 10D, 30D, 50D, and 100D multimodal SBFs, respectively and it is almost 55% on average of all multimodal SBF problems. The SD for multimodal SBFs ranges from 0.00 to 5.00E-03 for 10D, 30D and 50D problems whereas for 100D problems, SD varies from 9.12E-11 to 5.05E-01.

Plate valley-shaped (f₁₉ - f₂₆) and steep ridges (f₂₇ - f₃₃) SBFs are used to test the precision, rate of convergence and exploitation capability of optimization algorithms. They have a very minimum slope to the vast area around their global minima or an extensive area with near best fitness value and it is proved from the contour plot of valley-shaped SBF are reported in Table 7. In specific, algorithms need high exploitation capability to find the global optima of this kind of SBF. As reported in Tables 4 5, considering the 10D, 30D, 50D, and 100D of steep ridges SBFs, the proposed algorithm outperforms other algorithms in 4, 5, 4 & 4 functions, respectively and it is on average 61% of all steep ridges’ problems. In the case of plate & valley-shaped SBFs, the performance of PTO algorithm for 10D, 30D, 50D, and 100D problems is 6, 7, 6, and 5, respectively, for 75% of all plate & valley shaped SBFs. This proves the incredible exploitation capability of the proposed algorithm. SD varies from 0.00 to 0.12 for 10D, 30D and 50D, whereas for 100D problems, SD varies from 1.98E-10 to 3.55.

Tables 6 7 show the search history and convergence curve of all SBFs with 30D along with positions of all search agents during iterations are provided, and the sparse distribution of the population shows the ability to explore around search space, and the dense population shows the ability to exploit the global and the local optima. As depicted in Tables 6 7, in view of raindrops positions column, the blue dots indicate the final position of raindrops over contour plots of different SBFs after the termination criteria are met. Contour plots of SBFs (f₄, f₁₀, f₁₆) in Table 6 are scattered with a large number of dots all over the search sp It indicates that only a few raindrops have attained the global optimum position and most of the raindrops explored all over the search space to attain a global optimum position with regularly distributed populations are trapped in local optima and it is the proof of high exploration capability.

Table 6
Surface Plot, Position of Population, convergence curve for 30 independent runs of PTO on Many local minima Benchmark functions

Table 7

Surface Plot, Position of Population, convergence curve for 30 independent runs of PTO on bowl-shaped, plate-shaped, valley shaped, steep drops and others

Functions with steep ridges and plate-shaped with long narrow valley-like structure are complicated to search the global optimum. Contour plots of such functions are illustrated in Table 7, which shows excellent exploitation behaviour of PTO, with a dense population position around the global optimum contour plots of SBFs (f₁₇, f₁₈, f₂₀, f₂₂, f₂₆). in Table 7 with less dots compared to other SBFs, which means most of the raindrops have reached the global optimum position or crownshaft, showing higher exploitation behaviour and only a few raindrops, which are scattered with low fitness values, are trapped in local optima. Contours plots of SBFs (f₂, f₅, f₁₁, f₁₅, f₁₉, f_31,f₃₂.) in Tables 6 7 show that most of the raindrops have gathered around the global optimum, as the space around is a flat or low slope, showing high exploration and exploitation behaviours of the proposed algorithm.

The ranking of algorithms through performance over SBFs is based on the percentage of problems that they outperform with other algorithms in each category with good mean solutions and standard deviations. PTO is ranked in the first good position with better performance in the maximum percentage of problems. PSO is ranked second, as it outperforms other algorithms with a maximum percentage next to PTO. Figure 5 clearly shows the ranking of all the five state-of-the-art algorithms based on the average of best solutions of 30 independent runs. The spider chart is split into four sections corresponding to number of dimensions of each SBFs. Each section signifies the rank of the algorithms in solving corresponding SBFs with 10, 30, 50 and 100 dimensions in clockwise.

Moving outside of the inner-most web denotes a decline in algorithm performance, with the worst performing algorithm receiving the maximum rank of five, which is denoted in the outer-most web. The inner-most web represents rank one for a good solution. Figure 5 shows that the PTO algorithms ranked 1 in most of the SBF with better performance than other algorithms, with PSO being at rank two next to PTO. CSA, GA, and ABC are given third, fourth and fifth ranks, respectively.

4.2 Convergence behaviour of PTO

Analysis of PTO’s convergence behaviour is done based on comparing convergence plots of various algorithms considered in this paper with the PTO algorithm, as shown in Figs. 6 7. SBFs considered for convergence comparison are selected mainly from many local optima, plate-shaped and valley-shaped SBF. In all category SBF, the proposed PTO algorithm converges faster than other algorithms. PTO converges below 50 Iterations in most problems, while others need more iterations.

Fig. 6

Convergence plots comparison of SBFs (f₂, f₃, f₆, f₇, f₈, f₁₂). with other algorithms for 1000 Iterations.

Fig. 7

Convergence plots comparison of SBFs (f₁₆, f₂₁, f₂₃, f₂₄, f₂₆, f₃₁). with other algorithms for 1000 Iterations.

For plate and valley-shaped SBF, the rate of convergence towards global optima is more critical than the final solution, as it decides the exploitation capability of the algorithm. From Figs. 6 7, the convergence comparison plots of f₂, f₃, f₆, f₇, f₈, f₁₂, f₂₁, f₂₃, f₂₄, f₂₆ prove that the proposed algorithm has outperformed other algorithms in the rate of convergence.

4.3 Comparison with other algorithms using CEC2017

The statistical result comparison of CEC2017 benchmark funcons is given in Table 8, which includes the best error value and standard deviations of objective function values of PTO and other six state-of-the-art algorithms, each with 51 independent runs. Best solutions are highlighted in bold for all 29 benchmark functions. The ranking of all the algorithms is based on giving the best solution for the maximum number of benchmark functions in each category. Under the category of Multimodal benchmark functions from IEEE-CEC2017 (f04_cec17 - f10_cec17) as in Table 8, the proposed PTO algorithm can find the best solution for six functions except (f06_cec17) of multimodal category of CEC2017. It can be proven that the PTO algorithm performs better in 85% of benchmark functions under the multimodal category than other algorithms.

Table 8
Statistical result comparison of different algorithms for constrained optimization test functions from IEEE-CEC2017 (fXX_cec17)

fXX _cec17 TLBO ES GSK GWO O ACO PTO

f01_cec17 2.01E+03±2.44E+03 8.16E+09±3.21E+09 0.00E+00 ± 0.00E+00 1.35E+09±7.12E+07 1.56E+06±2.35E+05 1.52E+10±9.95E+09 2.73E+07±3.28E+07

f03_cec17 0.00E+00±0.00E+00 3.31E+04±6.35E+03 0.00E+00 ± 0.00E+00 5.24E+02±8.15E+02 7.89E+02±7.63E+02 6.56E+03±1.54E+03 1.46E+02±1.02E+02

f04_cec17 2.19E-01±2.25E-01 6.32E+02±2.44E+02 0.00E+00 ± 0.00E+00 2.12E+01±2.35E+01 6.35E+00±2.15E+00 1.80E+02±4.30E+01 0.00E+00 ± 0.00E+00

f05_cec17 7.94E+00±4.18E+00 7.59E+01±1.55E+01 2.12E+01±2.54E+00 2.96E+01±5.12E+00 8.01E+00±2.79E+00 6.86E+01±8.53E+00 6.21E+00 ± 5.28E+00

f06_cec17 3.21E-02±2.56E-01 7.80E+01±8.96E+00 0.00E+00 ± 0.00E+00 7.71E+00±2.25E+00 9.28E-01±1.08E+00 3.59E+01±3.58E+00 5.13E+00±1.41E+00

f07_cec17 1.95E+01±6.15E+00 3.11E+02±3.68E+01 3.81E+01±2.63E+00 4.15E+01±5.52E+00 3.37E+01±3.48E+00 1.23E+02±2.21E+01 1.30E+01 ± 6.19E+00

f08_cec17 7.32E+00±4.52E+00 8.36E+01±8.92E+00 3.04E+01±3.20E+00 3.15E+01±4.69E+00 18E+00±3.82E+00 4.63E+01±6.98E+00 5.52E+00 ± 3.89E+00

f09_cec17 1.95E-01±3.63E-01 1.58E+03±2.58E+02 0.00E+00 ± 0.00E+00 1.36E+01±4.25E+00 15E-01±9.12E-02 5.56E+02±3.35E+02 0.00E+00 ± 0.00E+00

f10_cec17 5.10E+02±2.53E+02 9.56E+02±2.56E+03 9.92E+02±5.56E+02 2.35E+03±2.54E+02 2.27E+02 ± 1.76E+02

f11_cec17 7.15E+00±6.15E+00 9.86E+02±3.52E+03 0.00E+00 ± 0.00E+00 4.33E+01±1.12E+01 69E+00±5.89E+00 1.66E+02±8.24E+01 2.05E+01±6.83E+00

f12_cec17 1.56E+05±9.38E+04 3.25E+08±4.25E+08 9.21E+01 ± 6.35E+01 2.61E+06±4.25E+05 12E+05±2.14E+05 1.15E+09±9.51E+08 1.99E+05±6.78E+05

f13_cec17 2.54E+03±3.10E+03 4.25E+06±2.35E+06 6.74E+00±1.58E+00 2.14E+04±7.96E+03 02E+05±8.32E+04 6.35E+06±2.58E+07 3.56E+00 ± 2.86E+00

f14_cec17 4.02E+01±2.14E+01 5.24E+03±4.23E+03 6.25E+00±3.12E+00 6.12E+02±1.58E+03 48E+04±6.23E+04 5.24E+02±2.35E+02 3.24E+00 ± 9.02E+00

f15_cec17 4.96E+01±3.52E+01 2.21E+04±3.10E+04 2.32E-01 ± 1.96E-01 8.15E+02±2.14E+03 25E+04±2.35E+04 44E+03±1.15E+03 4.63E+01±3.10E+01

f16_cec17 1.74E+01±3.36E+01 5.14E+02±1.63E+02 4.56E+00±4.97E+00 6.58E+01±1.14E+02 1.05E+02±6.38E+01 1.57E+02±4.25E+01 3.57E+00 ± 5.38E+01

f17_cec17 2.96E+01±7.74E+00 3.54E+02±4.26E+01 1.58E+01±6.23E+00 5.58E+01±7.69E+00 3.34E+01±2.39E+01 2.16E+02±1.13E+01 7.97E+00 ± 7.77E+00

f18_cec17 2.45E+04±2.54E+03 6.61E+07±7.25E+05 2.56E-01 ± 2.01E-01 4.58E+04±3.21E+04 16E+05±1.18E+05 78E+08±2.19E+08 7.72E+02±6.09E+03

f19_cec17 3.21E+01±1.54E+01 1.14E+04±2.24E+05 1.62E-01±3.31E-01 1.18E+04±3.67E+03 42E+04±4.58E+04 2.24E+04±2.14E+04 1.23E-01 ± 1.59E+01

f20_cec17 1.91E+01±2.13E+01 2.47E+02±5.24E+01 1.19E+00±3.92E+00 8.14E+01±2.35E+01 1.30E+01±5.90E+00 1.15E+02±3.11E+01 1.18E+00 ± 6.55E+00

f21_cec17 1.34E+02±4.16E+01 1.23E+02±5.32E+00 1.86E+02±4.96E+01 1.96E+02±5.01E+01 2.14E+02±3.06E+01 1.52E+02±1.56E+01 1.51E+01 ± 4.24E+01

f22_cec17 9.18E+01±2.21E+01 8.96E+02±2.74E+02 1.21E+02±4.87E+00 3.14E+02±5.98E+01 4.86E+01 ± 1.63E+01

f23_cec17 3.55E+02±5.13E+00 4.68E+02±1.14E+01 3.48E+02±3.95E+00 3.53E+02±4.12E+00 3.23E+02±3.85E+00 3.96E+02±1.02E+01 3.02E+02 ± 7.97E+00

f24_cec17 3.23E+02±3.11E+01 4.21E+02±8.35E+01 3.35E+02±1.55E+01 19E+02±3.14E+01 2.94E+02±3.15E+01 1.16E+02 ± 1.15E+01

f25_cec17 4.31E+02±2.45E+01 8.24E+02±3.36E+02 5.27E+02±1.05E+01 4.42E+02±2.15E+01 4.39E+02±2.16E+01 6.76E+02±4.30E+01 4.03E+02 ± 1.99E+01

f26_cec17 4.12E+02±4.92E+01 2.14E+03±2.55E+02 3.23E+02±1.35E+00 3.96E+02±2.32E+02 4.01E+02±1.41E+02 7.45E+02±9.19E+01 1.13E+02 ± 3.98E+01

f27_cec17 3.54E+02±1.52E+00 5.23E+02±8.01E+00 3.86E+02±2.05E-01 3.96E+02±1.15E+00 3.77E+02±2.58E+00 4.51E+02±9.25E+00 2.79E+02 ± 1.11E+01

f28_cec17 4.22E+02±1.15E+02 8.32E+02±9.34E+01 3.12E+02±3.20E+01 6.47E+02±1.02E+02 5.51E+02±1.12E+02 4.96E+02±4.85E+01 2.24E+02 ± 2.17E+01

f29_cec17 2.95E+02±1.35E+01 5.84E+02±5.36E+01 2.51E+02±4.72E+00 3.84E+02±2.98E+01 2.63E+02±1.45E+01 3.85E+02±2.41E+01 2.01E+02 ± 1.39E+01

f30_cec17 1.03E+05±2.63E+05 8.84E+05±1.12E+00 3.21E+02 ± 4.15E+01 5.18E+05±8.14E+05 5.12E+05±4.28E+05 3.21E+08±6.68E+07 1.58E+04±7.91E+04

fXX _cec17	TLBO	ES	GSK	GWO	O	ACO	PTO
f01_cec17	2.01E+03±2.44E+03	8.16E+09±3.21E+09	0.00E+00 ± 0.00E+00	1.35E+09±7.12E+07	1.56E+06±2.35E+05	1.52E+10±9.95E+09	2.73E+07±3.28E+07
f03_cec17	0.00E+00±0.00E+00	3.31E+04±6.35E+03	0.00E+00 ± 0.00E+00	5.24E+02±8.15E+02	7.89E+02±7.63E+02	6.56E+03±1.54E+03	1.46E+02±1.02E+02
f04_cec17	2.19E-01±2.25E-01	6.32E+02±2.44E+02	0.00E+00 ± 0.00E+00	2.12E+01±2.35E+01	6.35E+00±2.15E+00	1.80E+02±4.30E+01	0.00E+00 ± 0.00E+00
f05_cec17	7.94E+00±4.18E+00	7.59E+01±1.55E+01	2.12E+01±2.54E+00	2.96E+01±5.12E+00	8.01E+00±2.79E+00	6.86E+01±8.53E+00	6.21E+00 ± 5.28E+00
f06_cec17	3.21E-02±2.56E-01	7.80E+01±8.96E+00	0.00E+00 ± 0.00E+00	7.71E+00±2.25E+00	9.28E-01±1.08E+00	3.59E+01±3.58E+00	5.13E+00±1.41E+00
f07_cec17	1.95E+01±6.15E+00	3.11E+02±3.68E+01	3.81E+01±2.63E+00	4.15E+01±5.52E+00	3.37E+01±3.48E+00	1.23E+02±2.21E+01	1.30E+01 ± 6.19E+00
f08_cec17	7.32E+00±4.52E+00	8.36E+01±8.92E+00	3.04E+01±3.20E+00	3.15E+01±4.69E+00	18E+00±3.82E+00	4.63E+01±6.98E+00	5.52E+00 ± 3.89E+00
f09_cec17	1.95E-01±3.63E-01	1.58E+03±2.58E+02	0.00E+00 ± 0.00E+00	1.36E+01±4.25E+00	15E-01±9.12E-02	5.56E+02±3.35E+02	0.00E+00 ± 0.00E+00
f10_cec17	5.10E+02±2.53E+02	9.56E+02±2.56E+03	9.92E+02±5.56E+02			2.35E+03±2.54E+02	2.27E+02 ± 1.76E+02
f11_cec17	7.15E+00±6.15E+00	9.86E+02±3.52E+03	0.00E+00 ± 0.00E+00	4.33E+01±1.12E+01	69E+00±5.89E+00	1.66E+02±8.24E+01	2.05E+01±6.83E+00
f12_cec17	1.56E+05±9.38E+04	3.25E+08±4.25E+08	9.21E+01 ± 6.35E+01	2.61E+06±4.25E+05	12E+05±2.14E+05	1.15E+09±9.51E+08	1.99E+05±6.78E+05
f13_cec17	2.54E+03±3.10E+03	4.25E+06±2.35E+06	6.74E+00±1.58E+00	2.14E+04±7.96E+03	02E+05±8.32E+04	6.35E+06±2.58E+07	3.56E+00 ± 2.86E+00
f14_cec17	4.02E+01±2.14E+01	5.24E+03±4.23E+03	6.25E+00±3.12E+00	6.12E+02±1.58E+03	48E+04±6.23E+04	5.24E+02±2.35E+02	3.24E+00 ± 9.02E+00
f15_cec17	4.96E+01±3.52E+01	2.21E+04±3.10E+04	2.32E-01 ± 1.96E-01	8.15E+02±2.14E+03	25E+04±2.35E+04	44E+03±1.15E+03	4.63E+01±3.10E+01
f16_cec17	1.74E+01±3.36E+01	5.14E+02±1.63E+02	4.56E+00±4.97E+00	6.58E+01±1.14E+02	1.05E+02±6.38E+01	1.57E+02±4.25E+01	3.57E+00 ± 5.38E+01
f17_cec17	2.96E+01±7.74E+00	3.54E+02±4.26E+01	1.58E+01±6.23E+00	5.58E+01±7.69E+00	3.34E+01±2.39E+01	2.16E+02±1.13E+01	7.97E+00 ± 7.77E+00
f18_cec17	2.45E+04±2.54E+03	6.61E+07±7.25E+05	2.56E-01 ± 2.01E-01	4.58E+04±3.21E+04	16E+05±1.18E+05	78E+08±2.19E+08	7.72E+02±6.09E+03
f19_cec17	3.21E+01±1.54E+01	1.14E+04±2.24E+05	1.62E-01±3.31E-01	1.18E+04±3.67E+03	42E+04±4.58E+04	2.24E+04±2.14E+04	1.23E-01 ± 1.59E+01
f20_cec17	1.91E+01±2.13E+01	2.47E+02±5.24E+01	1.19E+00±3.92E+00	8.14E+01±2.35E+01	1.30E+01±5.90E+00	1.15E+02±3.11E+01	1.18E+00 ± 6.55E+00
f21_cec17	1.34E+02±4.16E+01	1.23E+02±5.32E+00	1.86E+02±4.96E+01	1.96E+02±5.01E+01	2.14E+02±3.06E+01	1.52E+02±1.56E+01	1.51E+01 ± 4.24E+01
f22_cec17	9.18E+01±2.21E+01	8.96E+02±2.74E+02	1.21E+02±4.87E+00			3.14E+02±5.98E+01	4.86E+01 ± 1.63E+01
f23_cec17	3.55E+02±5.13E+00	4.68E+02±1.14E+01	3.48E+02±3.95E+00	3.53E+02±4.12E+00	3.23E+02±3.85E+00	3.96E+02±1.02E+01	3.02E+02 ± 7.97E+00
f24_cec17	3.23E+02±3.11E+01	4.21E+02±8.35E+01	3.35E+02±1.55E+01		19E+02±3.14E+01	2.94E+02±3.15E+01	1.16E+02 ± 1.15E+01
f25_cec17	4.31E+02±2.45E+01	8.24E+02±3.36E+02	5.27E+02±1.05E+01	4.42E+02±2.15E+01	4.39E+02±2.16E+01	6.76E+02±4.30E+01	4.03E+02 ± 1.99E+01
f26_cec17	4.12E+02±4.92E+01	2.14E+03±2.55E+02	3.23E+02±1.35E+00	3.96E+02±2.32E+02	4.01E+02±1.41E+02	7.45E+02±9.19E+01	1.13E+02 ± 3.98E+01
f27_cec17	3.54E+02±1.52E+00	5.23E+02±8.01E+00	3.86E+02±2.05E-01	3.96E+02±1.15E+00	3.77E+02±2.58E+00	4.51E+02±9.25E+00	2.79E+02 ± 1.11E+01
f28_cec17	4.22E+02±1.15E+02	8.32E+02±9.34E+01	3.12E+02±3.20E+01	6.47E+02±1.02E+02	5.51E+02±1.12E+02	4.96E+02±4.85E+01	2.24E+02 ± 2.17E+01
f29_cec17	2.95E+02±1.35E+01	5.84E+02±5.36E+01	2.51E+02±4.72E+00	3.84E+02±2.98E+01	2.63E+02±1.45E+01	3.85E+02±2.41E+01	2.01E+02 ± 1.39E+01
f30_cec17	1.03E+05±2.63E+05	8.84E+05±1.12E+00	3.21E+02 ± 4.15E+01	5.18E+05±8.14E+05	5.12E+05±4.28E+05	3.21E+08±6.68E+07	1.58E+04±7.91E+04

Hybrid functions of CEC2017 (f11_cec17 - f20_cec17) are nctns of variables which are made into specific subcomponents randomly with specific essential functions from multi-modal and unimodal categories as different subcomponents. So, it is challenging to find the best solution r optimization algorithms, as it needs higher capability in both exploration and exploitation processes. However, Table 8 shows that the proposed algorithm outperforms all other algorithms in 60% of all functions under the hybrid function category.

Composition functions of CEC2017 (f_cec21 - f_cec30) are complicated to solve, as they combine the properties of hybrid functions as essential functions and maintain continuity around global as well as local optima, and as a trap, one of the local optima is set to the origin. However, the proposed PTO algorithm is very efficient and robust compared to other algorithms. As per Table 8, PTO outperforms all algorithms by giving better solutions to 90% of composition functions from CEC2017. Standard deviation of PTO algorithm in composition functions varies from 7.97E+00 to 3.98E+01, which shows the robust behavior of PTO algorithm.

Ranking algorithms’ performance over CEC2017 benchmark functions are based on the percentage of problems and it outperforms other algorithms. For example, the PTO algorithm is ranked first in a good position, as it can give better solutions in 21 functions of CEC2017, which is 72% of all problems. On the other hand, GSK is ranked second, as it outperforms other algorithms with a maximum percentage next to PTO.

Table 9 reports the r and p-values determined by a non-parametric statistical hypothesis test and Wilcoxon signed-rank test to analyse the pairwise performance comparison of all algorithms with the proposed PTO with the mean of best solutions of corresponding independent runs. The null hypothesis test has been conducted at a significant level of 0.05. If the p-value is less than 0.05, the null hypothesis can be rejected, and the PTO is considered as the winner. While considering the Wilcoxon signed-rank test for PTO Vs ABC, it is revealed that the mean of the best solutions is significantly lower for ABC with a considerably larger effect size (r = 0.612). The lower value of r (<0.5) and higher value of p (>0.05) indicate the mean of the best solutions of PTO being significantly lower in performance with its pair. From Table 9, it is evident that p and r of the maximum percentage of problems under all categories favour the better performance of PTO except for unimodal functions, where the null hypothesis cannot be rejected, and the PTO algorithm is significantly lower compared to CSA, PSO and GSK.

The last two rows of Table 9 make it clear that PTO is ranked first, outperforming other methods. The number of functions PTO has won across six categories-unimodal, multimodal, plate-and-valley shaped, steep ridges, hybrid functions, and composition functions is displayed in the “+” column. The number of functions PTO lost to its rival method is indicated in the “-” column. The number of functions where PTO and its rival method perform identically is shown in the “=” column and is regarded as a tie. For instance, the first row of the table displays the numerical proof that PTO performs better than ABC in unimodal functions, demonstrating tt PTO wins both unimodal functions and that none of the functions are deemed to be draws.

5 Real-world optimization problems

This section evaluates the performance of PTO by testing it with two complex constrained optimization problems in classic engineering design and seven real-world optimization problems from IEEE-CEC2011. The statistical results are compared with the data published in the literature. Since this problem has many constraints, a constraint handling method is needed to employ. There are different constraint handling methods like the Death penalty method, static penalty method, Dynamic penalty method, Barrier penalty method, and Deb feasibility method. The Death penalty method is used, which assigns a very high value, if a solution violates at least one constraint, and it will be called as an infeasible solution. This method is simple and most applicable with low computational cost compared to other penalty methods.

5.1 Piston Lever problem

The objective function of this problem is to minimize the oi, when the piston lever is lifted up with the constraint of locating the piston components as shown in Fig. 8 where, H, B, D, and X, which are expressed as x₁, x₂, x₃ and x₄ in below equations, respectively. From Tables 10 11, it is evident that the PTO optimizer outperforms all algorithms in finding minimum oil volume to lift the piston lever.

Fig. 8

Schematic representation of piston lever problem.

Table 10

Optimal solutions of Piston lever problem

Algorithm	DE [40]	GA [40]	PSO [40]	HPSO [40]	CSA [41]	PTO
x1	129.4	250	133.3	135.5	0.05	0.05
x2	2.43	3.96	2.44	2.48	2.043	2.042
x3	119.8	60.03	117.14	116.62	120	120
x4	4.75	5.91	4.75	4.75	4.085	4.0974
g₁ (X)	NA	NA	NA	NA	–1744.912	–1.20E+04
g₂ (X)	NA	NA	NA	NA	–600,000	–600,000
g₃ (X)	NA	NA	NA	NA	–117.185	–117.1868
g₄ (X)	NA	NA	NA	NA	–4.50E –04	0.0067
f (X)	159	161	122	162	8.427	8.410

Table 11

Statistical result comparison of PTO with other algorithms (Piston lever problem)

Algorithm	Worst	Mean	Best	SD	NFE
DE [40]	199	187	159	14.2	50,000
GA [45]	216	185	161	18.2	50,000
PSO [45]	294	166	122	51.7	50,000
HPSO [40]	197	187	162	13.4	50,000
HPSOQ [40]	168	151	129	13.4	50,000
CSA [41]	168.592	40.2319	8.4271	59.0552	50,000
ISA [42]	610.6	226.5	8.4	111.2	12,500
CGO [40]	167.47281	45.04866	8.4128138	67.24763	1,00,000
AOS [43]	167.66499	33.741276	8.4191427	93.466747	1,00,000
MGA [44]	167.47321	32.468893	8.4134067	29.963704	1,00,000
SNS [40]	167.47277	24.318974	8.4126983	47.717926	5000
PTO	13.5598	8.7594	8.4105	1.017	3000

The problem can be written mathematically as below,

Minimize: $f (X) = \frac{1}{4} π x_{3}^{2} (L_{2} - L_{1})$ ,

subject to:

g ₁ ( X ) = Q L cos θ - n R × F ⩽ 0, g ₂ ( X ) = Q (L - x₄) - M_max ⩽ 0,

g ₃ ( X ) = 1.2 (L₂ - L₁) - L₁ ⩽ 0, $g_{4} (X) = \frac{x_{3}}{2} - x_{2} ⩽ 0,$

Were,

$R = \frac{| - x_{4} (x_{4} \sin θ + x_{1}) + x_{1} (x_{2} - x_{4} \cos θ) |}{\sqrt{{(x_{4} - x_{2})}^{2} + x_{1}^{2}}}$ , $F = \frac{π {Px}_{3}^{2}}{4}$ , $L_{1} = \sqrt{{(x_{4} - x_{2})}^{2} + x_{1}^{2}}$ ,

$L_{2} = \sqrt{{(x_{4} \sin θ + x_{1})}^{2} + {(x_{2} - x_{4} \cos θ)}^{2}}$ , θ = 45°, Q = 10, 000 lbs, L = 240 in,

M _max = 1.8 × 10⁶ lbs in, P = 1500 psi,

variable range: 0.05≤x₁, x₂, x₄ ≤500, 0.05≤x₃≤120.

Tables 11 show that the PTO optimizer outperforms all algorithms in finding the minimum oil volume to lift the piston lever.

5.2 Design of tubular column

The design of Tubular column includes six inequality constraints. The main objective of this problem is to design a uniform tubular column to carry a compressive load with minimum material and construction costs. As shown in Fig. 9, this structural optimization problem includes two parameters such as mean diameter of column section (x₁) average thickness of column section (x₂). This constrained optimization problem is formulated as,

Fig. 9

Schematic representation of tubular column design problem.

Minimize:

f (X) = 9.8x₁x₂ + 2x₁,

Subject to:

$g_{1} (X) = \frac{P}{π x_{1} x_{2} σ_{y}} - 1 ⩽ 0,$ $g_{2} (X) = \frac{8 P L^{2}}{π E x_{1} x_{2} (x_{1}^{2} + x_{2}^{2})} - 1 ⩽ 0,$ $g_{3} (X) = \frac{2.0}{x_{1}} - 1 ⩽ 0,$

$g_{4} (X) = \frac{x_{1}}{14} - 1 ⩽ 0$ , $g_{5} (X) = \frac{0.2}{x_{2}} - 1 ⩽ 0$ , $g_{6} (X) = \frac{x_{2}}{8} - 1 ⩽ 0$ ,

Variable range:

2≤x₁≤14, 0.2≤x₂≤0.8.

From Tables 13, it is proved that PTO outperforms all other algorithms in finding minimum material and construction costs to design a uniform tubular column.

Table 12

Best results for the optimal design of tubular column problem

Algorithms	x ₁	x ₂	g₁ (X)	g₂ (X)	g₃ (X)	g₄ (X)	g₅ (X)	g₆ (X)	f (X)
CSA [41]	5.45139	0.29196	–0.0241	–0.1095	–0.633	.610	–0.315	–0.635	26.53217
ISA [40]	5.4511562	0.2919655	–2.5E –10	–1.8E –10	–0.6331	106	–0.3149	–0.635	26.5313
SNS [40]	5.4511562	0.2919655	–2.6E –10	–1.8E –10	–0.6331	–0.6106	–0.3149	–0.635	26.499497
PTO	5.4527	0.2918	-0.0011	-0.0419	-0.6332	-0.6105	-0.3184	-0.6332	26.49592

Table 13

Statistical result comparison of PTO with other algorithms for the tubular column example

Algorithm	worst	Mean	Best	SD	NFE
ISA [40]	26.532	26.531	26.531	1.70E –04	3000
CSA [41]	26.53972	26.53504	26.53217	1.93E –03	15,000
FA [46]	NA	28.74	26.52	2.08	3000
AOS [43]	26.6083136	26.531614	26.5313783	1.0300E –03	1,00,000
PTO	26.74849	26.5375	26.49592	2.5300E –05	2000

5.3 Designing of gear train problem

The objective of this design optimization problem is to optimize the ratio of the output shaft angular velocity to the input shaft angular velocity. The number of teeth of gears shown in the Fig. 10 are N_A, N_B, N_C, N_D of gears A, B, C and D, respectively. The optimization problem is formulated as follows,

Fig. 10

Schematic representation of Design train problem.

$f (X) = {(\frac{1}{6.931} - \frac{x_{3} x_{2}}{x_{1} x_{4}})}^{2},$ Variable range: $x_{1}, x_{2}, x_{3}, x_{4} \in {12, 13, 14, . . ., 60} .$

From Tables 15, it is proved that PTO can solve discrete optimization problems and its performance is compared with all other algorithms in finding optimum solution.

Table 14

Optimal Solutions for Design of gear train

Algorithm	x ₁	x ₂	x ₃	x ₄	f (X)
ABC [45]	49	16	19	43	2.70e-12
GA [40]	49	19	16	43	2.70e-12
PSO [40]	34	13	20	53	2.31e-12
CSA [41]	43	16	19	49	2.70e-12
ICA [48]	43	16	19	49	2.70e-12
MSFWA [46]	49	19	16	43	2.70e-12
MBA [47]	43	16	19	49	2.70e-12
BBO [48]	53	26	15	51	2.31e-11
NNA [48]	49	16	19	43	2.70e-12
GWO [48]	49	19	16	43	2.70e-12
ISA [42]	43	19	16	49	2.70e-12
APSO [49]	43	16	19	49	2.70e-12
IAPSO [49]	43	16	19	49	2.70e-12
MVO [50]	43	16	19	49	2.70e-12
MFO [51]	43	19	16	49	2.70e-12
ALO [52]	49	19	16	43	2.70e-12
PSOSCALF [53]	49	19	16	43	2.70e-12
PTO	43	19	16	49	2.70e-12

Table 15

Statistical result comparison of PTO with other algorithms for Design of gear train

Algorithm	worst	Mean	Best	SD	NFE
GA [40]	1.5247E –08	1.6212E –09	2.7009E –12	3.2174E –09	50,000
PSO [40]	1.0222E –06	7.9383E –08	2.3078E –11	1.8147E –07	50,000
ICA [48]	2.3576E –09	8.0417E –10	2.7009E –12	7.7862E –10	50,000
UPSO [40]	NA	3.80562E –08	2.700857E –12	1.09E –07	100,000
MFO [40]	2.7265E –08	7.5337E –09	2.3078E –11	9.3539E –09	50,000
SCA [40]	2.3576E –09	8.8113E –10	2.7009E –12	6.4529E –10	50,000
SSA [40]	2.7265E –08	1.9822E –09	2.7009E –12	4.5748E –09	50,000
BBO [48]	4.2018E –07	4.5418E –08	2.3078E –11	7.2953E –08	50,000
CBO [48]	1.1173E –08	2.1032E –09	2.3078E –11	2.4025E –09	50,000
NNA [48]	2.3576E –09	6.1128E –10	2.7009E –12	6.1167E –10	50,000
GWO [48]	9.9216E –10	3.3777E –10	2.7009E –12	4.0956E –10	50,000
WOA [48]	6.5123E –09	9.6633E –10	2.7009E –12	1.1296E –09	50,000
PTO	1.36549E-09	1.69543E-10	2.7009E-12	1.4754E-09	3000

5.4 IEEE-CEC 2011 real-world optimization problems

CEC2011 real-world optimization problems are very competitive, and the optimization algorithms must be very efficient, as the problems are highly composed and complex. CEC2011 problems are highly complex multimodal functions. Furthermore, CEC2011 problems are continuous and non-differentiable and these diverse characteristics make CEC2011 real-world problems more complex. Table 16 provides the best solutions obtained through PTO and the standard deviations of fitness values of objective function values of PTO as well as other five state-of-the-art algorithms with over 25 independent runs for all the seven benchmark functions. From Table 16, it is clearly shown that the proposed algorithm outperforms all other algorithms in four real-world problems with better solutions and standard deviations.

Table 16
Statistical result comparison of different algorithms for constrained optimization test functions from IEEE-CEC2011 with 1,50,000 FES

fn TLBO ES GSK GWO ACO PTO

1 6.63E+00±6.56E+00 3.99E+00±5.11E+00 3.14E+00±6.17E+00 2.07E+01±6.12E+00 4.12E+01±9.95E-01 3.12E+00 ± 3.30E+00

2 -2.09E+01 ± 3.26E+00 -2.44E+00±6.35E+03 -1.15E+01±2.06E+00 -1.54E+01±2.14E+00 -6.56E+00±1.54E-01 -1.32E+01±1.02E+00

3 1.15E-05 ± 2.35E-08 1.15E-05 ± 4.16E-07 1.15E-05 ± 1.36E-13 1.15E-05 ± 5.36E-11 1.15E-05 ± 0.00E+00 1.15E-05 ± 2.62E-14

4 0.00E+00 ± 0.00E+00 0.00E+00 ± 0.00E+00 0.00E+00 ± 0.00E+00 0.00E+00 ± 0.00E+00 0.00E+00 ± 0.00E+00 0.00E+00 ± 0.00E+00

5 -2.79E+01±3.56E+00 -2.85E+01±7.35E+00 -2.15E+01±1.14E+00 -2.34E+01±1.58E+00 -1.51E+01±3.58E+00 -2.10E+01±0.00E+00

6 -1.95E+01±6.24E+00 -18.9E+01±3.68E+00 -6.86E+00±2.63E+00 -1.94E+01±1.36E+00 -1.13E+01±2.21E+00 -1.53E+01±0.00E+00

7 1.20E+00±3.15E-01 2.34E+00±2.54E-02 1.75E+00±1.35E-01 1.76E+00±1.02E-01 1.89E+00±9.98E-02 1.15E+00 ± 0.00E+00

fn	TLBO	ES	GSK	GWO	ACO	PTO
1	6.63E+00±6.56E+00	3.99E+00±5.11E+00	3.14E+00±6.17E+00	2.07E+01±6.12E+00	4.12E+01±9.95E-01	3.12E+00 ± 3.30E+00
2	-2.09E+01 ± 3.26E+00	-2.44E+00±6.35E+03	-1.15E+01±2.06E+00	-1.54E+01±2.14E+00	-6.56E+00±1.54E-01	-1.32E+01±1.02E+00
3	1.15E-05 ± 2.35E-08	1.15E-05 ± 4.16E-07	1.15E-05 ± 1.36E-13	1.15E-05 ± 5.36E-11	1.15E-05 ± 0.00E+00	1.15E-05 ± 2.62E-14
4	0.00E+00 ± 0.00E+00	0.00E+00 ± 0.00E+00	0.00E+00 ± 0.00E+00	0.00E+00 ± 0.00E+00	0.00E+00 ± 0.00E+00	0.00E+00 ± 0.00E+00
5	-2.79E+01±3.56E+00	-2.85E+01±7.35E+00	-2.15E+01±1.14E+00	-2.34E+01±1.58E+00	-1.51E+01±3.58E+00	-2.10E+01±0.00E+00
6	-1.95E+01±6.24E+00	-18.9E+01±3.68E+00	-6.86E+00±2.63E+00	-1.94E+01±1.36E+00	-1.13E+01±2.21E+00	-1.53E+01±0.00E+00
7	1.20E+00±3.15E-01	2.34E+00±2.54E-02	1.75E+00±1.35E-01	1.76E+00±1.02E-01	1.89E+00±9.98E-02	1.15E+00 ± 0.00E+00

6 Conclusion and potential future works

This paper has utilized a novel metaheuristic swarm intelligent algorithm based on the unique feature of an effective funneling mechanism of palm trees to improve the nutrient and water supply to the base of the plant. The proposed palm tree optimization algorithm (PTO) includes three behaviors to mimic the palm tree’s structural significance and they are Flow of water over palm leaves towards costa, Flow of water over petioles towards crownshaft and reduction in the slope of palm tree branches due to ageing.

The results and insightful implications are summarized as follows:

The proposed algorithm has been tested with 33 classical SBF, 29 constrained benchmark functions from IEEE-CEC2017, two classical real-world engineering design problems, seven real-world problems from CEC2011 and finally compared with the performances of ten well-known state-of-the-art algorithms.

In order to increase the complexity of mathematical SBF, the algorithms are tested with 10, 30, 50 and 100 dimensions.

The PTO algorithm outperforms other metaheuristics algorithms in converging to the global optimum solutions for solving mathematical test functions.

The PTO algorithm can provide better solutions with very low standard deviations, specifically in multimodal and plate valley-shaped SBFs, even in higher dimensions and it is evident for three things. First is high exploration capability to solve multimodal SBFs. Second is high exploitation capability to solve plate valley-shaped SBFs and third is highly robust and reliable, with low standard deviations in multimodal and plate valley-shaped SBFs.

The PTO algorithm can provide better solutions in almost all composition functions from the IEEE-CEC2017 competition. As the composition functions use hybrid functions as their base function, they have unique properties for every variable subcomponent and they make them very difficult for optimization algorithms to perform better. However, the proposed PTO algorithm performs better in almost all composition functions than other state-of-the-art algorithms.

The PTO algorithm is also tested with two classic real-world problems and seven CEC2011 real-world problems. It shows efficient performance in solving real-world problems compared to other algorithms

While comparing all the statistical results, it is convincing that the PTO is statistically superior, very competitive and performs better in most optimization problems than the well-known and recent state-of-the-art algorithms. The significant drawback of the proposed algorithm is that it performs equally or sometimes underperforms in unimodal functions of CEC2017 with only a third position next to TLBO and GSK. Future research works can be carried out to improve the performance of PTO without adding additional parameters by focusing on better performance in unimodal functions of CEC2017 and solving high dimension benchmark functions with less execution time. Also, it can be extended to solve multi-objective, discrete and more complex real-life problems. Hybridizing the PTO algorithm with other well-known and recent state-of-the-art algorithms will also make the algorithm effective.

Conflicts of interest

The authors declare that they have no conflicts of interest to this work.

Data availability

The Data used to support the findings of this study are included in the article.

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The palm tree optimization: Algorithm and applications

Abstract

Keywords

1 Introduction

3.1.1 Initialization of algorithm parameters

3.1.2 Raining process

Table 6 Surface Plot, Position of Population, convergence curve for 30 independent runs of PTO on Many local minima Benchmark functions

5.1 Piston Lever problem

Conflicts of interest

Data availability

References

Table 6
Surface Plot, Position of Population, convergence curve for 30 independent runs of PTO on Many local minima Benchmark functions