JOIV : Int. J. Inform. Visualization, 5(4) - December 2021 461-468 INTERNATIONAL JOURNAL ON INFORMATICS VISUALIZATION INTERNATIONAL JOURNAL ON INFORMATICS VISUALIZATION journal homepage : www.joiv.org/index.php/joiv An Improved Flower Pollination Algorithm for Global and Local Optimization M. Iqbal Kamboh a,*, Nazri Bin Mohd Nawi a, Azizul Azhar Ramli a, Fanni Sukma b a Faculty of Computer Science & Information Technology, Universiti Tun Hussein Onn Malaysia, Batu Pahat, Johor, Malaysia b Department of Information Technology, Politeknik Negeri Padang, West Sumatera, Indonesia Corresponding author: *iqbalmoeez@gmail.com Abstract— Meta-heuristic algorithms have emerged as a powerful optimization tool for handling non-smooth complex optimization problems and also to address engineering and medical issues. However, the traditional methods face difficulty in tackling the multimodal non-linear optimization problems within the vast search space. In this paper, the Flower Pollination Algorithm has been improved using Dynamic switch probability to enhance the balance between exploitation and exploration for increasing its search ability, and the swap operator is used to diversify the population, which will increase the exploitation in getting the optimum solution. The performance of the improved algorithm has investigated on benchmark mathematical functions, and the results have been compared with the Standard Flower pollination Algorithm (SFPA), Genetic Algorithm, Bat Algorithm, Simulated annealing, Firefly Algorithm and Modified flower pollination algorithm. The ranking of the algorithms proves that our proposed algorithm IFPDSO has outperformed the above-discussed nature-inspired heuristic algorithms. Keywords— Dynamic switch probability; meta-heuristic; benchmark function; multimodal; exploitation; exploration. Manuscript received 10 Feb. 2021; revised 12 Apr. 2021; accepted 19 Oct. 2021. Date of publication 31 Dec. 2021. International Journal on Informatics Visualization is licensed under a Creative Commons Attribution-Share Alike 4.0 International License. [5], Artificial bee colony algorithm (ABC) [6], Cuckoo Search Algorithm (CS) [7], Bat Inspired Algorithm (BA)[8], Firefly algorithm (FA) [9], Simulated Annealing(SA)[10], Differential Evolution (DE)[11] and Flower Pollination Algorithm (FPA)[12]. Literature shows that meta-heuristic algorithms cannot perform optimally for both exploration and exploitation simultaneously [13]. Therefore, hybrid techniques are more trendy among practitioners where one algorithm is used for exploration and another for exploitation to enhance the performance of algorithms [14]-[17]. In some cases, parameters are adjusted, and operators are changed to improve the efficiency of algorithms [18]. Flower Pollination Algorithm is one of the best algorithms in terms of minimal numbers of parameters, and it can be easily implemented and is also highly efficient [19]. The mentioned properties of the algorithm had motivated us to select it for enhancement. In the literature, Flower Pollination Algorithm has been improved by practitioners to address the Switch probability, global pollination and local pollination. The Local Neighborhood Search Strategy (LNSS) [20] is used to increase the local search-ability of FPA by diversifying the I. INTRODUCTION Solving multifaceted optimization problems can be challenging when multiple and inconsistent design goals are considered. An emerging trend in using meta-heuristic algorithms to answer complex optimization problems, these algorithms have revealed great success in maintaining balance among inconsistent design goals. Most of the meta-heuristic algorithms have been established in recent decades. Many of these algorithms require certain parameters to show their best performance. For example, the Genetic Algorithm (GA) [1] requires considerable adjustment for population size, crossover rate and mutation. In the circumstance of Particle Swarm Optimization (PSO) [2], the same issue also appears, which depends on population size, weight of inertia and social parameters. Similarly, Harmony Search (HS) [3] requires adjustment of harmony memory deliberation rate, harmony size, and tuning of pitch. As for Ant Colony Optimization (ACO) [4], choosing the correct evaporation rate, pheromone effect and heuristic function are essential. Some other successful stochastic algorithms are Bee Colony Algorithm 461 local neighborhoods. Colonel search is introduced to control the local search space, in such a way that initially more global search as compared to the end of the search process. In [21], a static scaling factor is applied to control the mutation through the local pollination process and to enhance the convergence rate of the algorithm. In [16], switch probability is replaced by two dynamic weights to guide for fast convergence and to increase the stability. Differential Evolution and Flower Pollination Algorithms are hybridized to escape the FPA from local minima. Chaos theory and Flower Pollination algorithm are hybridized to enhance the convergence rate and accuracy of the optimal solution [22]. In this paper, the proposed Flower Pollination Algorithm has two improvements over the Standard Flower Pollination Algorithm. The one is Dynamic switch probability and the second one is swap operator [23]. The Switch probability is an operator that controls the balance between diversification and intensification. The swap operator is used for modification in local search to enhance its efficiency and escape from trapping in multi-local minima [24]. The proposed algorithm is tested on benchmark optimization functions of multimodal and unimodal. It is evident from experimental results that the proposed algorithm, Improved Flower Pollination with Dynamic switching probability, and Swap operator (IFPDSO) relatively outperformed the typical Flower-Pollination Algorithm (FPA) and the other wellknown algorithms like Simulated Annealing (SA), Genetic Algorithm (GA), Firefly Algorithm (FF), Bat Inspired Algorithm (BA) and Modified flower pollination algorithm (MFPA). In Section II, we will discuss the Flower Pollination Algorithm in detail with its characteristics and the modifications of the proposed algorithm IFPDSO. In section III, the improved algorithm performance is examined and evaluated on well-known benchmark mathematical functions. In section IV, the conclusion is discussed. Constancy: It is the reproduction capacity of two similar flowers which can increase insects.  Switch Probability: It is helpful for controlling local pollination and global pollination. The switch probability p ε [0 1]. Other features in the pollination process include flower constancy which can be measured as the reproduction probability and switch probability p ε [0,1] which is helpful to control the global and local pollination. The fixed value p= 0.8 is slightly biased for exploitation. Global pollination is guaranteed with the help of creatures that ensure the best reproduction. It is denoted by “g*” and the mathematical representation for global pollination can be expressed as:  = + ~ ʎ ʎ ʎ − ∗ ) (1) In Equation (1) ‘ ′ represents the solution vector in the iteration “t”, while “g*” is the finest existing solution among the overall iterations. where ‘γ’ represents a scaling factor which controls step size. In Equation (2) the ‘L’ is a levy’s distribution parameter that resembles the strong point of pollination. ʎ ʎ L>0 for an extensive random walk ≫ >0 2 Where local pollination is illustrated in Equation (3). = +∈ % − & 3 In Equation (3) where ‘ % ’ and ‘ & ’ represent the pollen from various flowers of identical plant species, which fundamentally imitate the flower constancy in a partial neighborhood. On the other hand, if % and & are selected from the identical species, it consistently becomes a local random walk where ∈ is carefully chosen from a uniform distribution in the range [0, 1]. The Pseudo-code of the SFPA algorithm has been discussed below. II. MATERIALS AND METHOD This section describes the Standard Flower Pollination algorithm and the improvements have been done in Flower Pollination algorithm. 1) The Pseudo-code of SFPA The pseudo-code is separated into three portions. The first portion represents the initialization of the population and its parameters. The second portion decides that either global pollination should occur or local pollination. In the third section, the solution is updated to display. A. Standard Flower Pollination Algorithm In 2012, Xing-She Yan developed the flower pollination algorithm (FPA) [12]. It is a nature-inspired metaheuristic, stochastic technique. The practitioner was inspired by the pollination conduct of various flowers to pollinate for reproduction. Some flowers only try to attract insects which help to gear up the pollination process. These pests are the main pollinators during the global pollination process. The practitioner focused on complex nonlinear problems that are not appropriately handled by conventional optimization methods. There are basically two types of pollination:  Biotic: this is a type of elongated distance or cross pollination which requires pollinators to execute it, such birds, pests, bees. Thus, the levy’s flight performs global search that covers 90% of overall pollination.  Abiotic: it is a self-pollination or local pollination which happens within a flower that does not need any pollinators. In this case, flowers pollinate through wind and diffusion. It is only 10% of pollination. 1. Fitness function min or max f(x)with dimension, d. x =(x1,x2,x3,………………xd) 2. Initialize the population of ‘n’ flowers with its random solutions 3. Calculate the finest result g* from the preliminary random solutions 4. Initialize the Switching probability p ε [0,1] 5. While (t < Maximum-Iterations) 6. For i = 1: n (total number of solutions) 7. if rand < p 8. Induce ‘d’ dimensional stepping Vector is L that follows the Levy’s distribution. 462 pollination algorithm is slightly weak in local optima for solving multimodal optimization problems [24], which may lead to trapping in local minima. In this research paper, An Improved Flower Pollination algorithm (IFPDSO) has been proposed by introducing Dynamic Switch probability and Swap Operator to tackle the identified issues. In Fig 1, flow of the proposed algorithm has been illustrated. The Dynamic switch probability is applied to control the exploration and exploitation while the Swap operator is introduced to enhance the diversification of the population in the local search process to escape from being trapped in a local optimum. The random number epsilon ε of the uniform distribution is replaced with β ε [0 0.5] to enhance its exploitation capability. The Swap operator has been defined in the given below equation. 9. Global pollination has implemented to get a new solution ( 10. else 11. Local pollination will apply, to find the solution ( 12. end if 13. end For 14. Calculate the new fitness ( using objective function 15. if fitness (( ) < fitness (( ) (To minimize objective function) 16. update flower ( 17. end if 18. end while 19. output (best Minimum or Maximum) start ( Initialization of population and its parameters of FPA , swap over rate Evaluate global best solution If rand > p Local pollination using swap operator 0 ,1,2………… 4 1) The proposed Algorithm’s Pseudocode The proposed algorithm begins with the initial population, switch probability which is followed by global pollination and local pollination, while in local pollination a swap operator has been presented. After completing each iteration, the optimal solution is updated hence dynamic switch probability is applied for the next iteration. Update the global optimal result no / 1,2,3 … … > ] ; *5 In Equation (5), BC D represents the maximum iterations while ‘t’ denotes current iteration. Calculate optimal solution Dynamic switch probability *+,- 5678+,- [ In Equation (4), & shows ‘kth’ dimension of the solution vector ‘i’ where ‘Sr’ represents the swap over rate which is 0.4 to diversify the population to get the best local optimum and evade the premature convergence. However, the static switch probability (p = 0.8) has been used in the original Flower Pollination Algorithm, which creates partiality between exploitation and exploration. The proposed algorithm will apply dynamic switch probability to adjust the balancing issue between global and local pollination to get an optimal solution. Dynamic switching probability can be illustrated in Equation (5). P = 0.8 + 0.1 BC D − E /BC D (5) calculate random solution against each pollen Global pollination using Levy’s flight < 4+,- ={ . apply condition yes Out put Fig. 1 1. Start 2. Fitness function f(x) (min or max) with dimension dx =(x1,x2,x3 ,…x d) 3. Initialization: population is ‘n’ number of flowers, swap over rate, random solutions 4. Evaluate optimal solution, g* from the preliminary random results 5. Initialize switch probability which lies in p ε [0,1] 6. While (t < Max-Iterations) 7. For i = 1: n (Maximum solutions) 8. if rand < p 9. Find ‘d’ is dimensional step vector L that obeys Levy’s distribution 10. Global pollination is implemented to achieve new solution ( 11. else end The proposed algorithm’s flowchart. B. Proposed Algorithm To address the problem of balancing between the exploration and exploitation, a number of practitioners introduced various modifications in SFPA to enhance its convergence rate and efficiency. In the above discussion, some modified versions of SFPA have been submitted by the authors in [20][16][19] [25][26][18]. All these practitioners have addressed one or more issues stated earlier but failed to handle all these problems simultaneously that have been identified in SFPA. Literature shows that the standard flower 463 TABLE II RANKING OF THE ALGORITHMS ( UNIMODAL) 12. Local pollination is search process to Get solution ( vector by using swap operator 13. end if 14. Find the new fitness ( using a fitness function 15. if fitness (( ) < fitness (( ) (Minimization of function) 16. Update (Initial solution) flower ( 17. end if 18. Implement the dynamic switch probability, p 19. end for 20. end while 21. Output (Min or Max) optimal solution Algorithm In this section experimental results have been discussed and have proved the outstanding performance of the proposed algorithm. The primary parameter setting is used to validate the performance of IFPDSO as compared to the six wellknown optimization algorithms in Table I. TABLE I THE PARAMETERS USED IN ALGORITHMS. Initial setting of parameters FPA n = 60, switch probability static P = 0.8, scaling factor γ = 0.01, step Levy flight is λ= 1.5 n = 60, switch probability initially P = 0.8, γ = 0.01, step Levy flight is λ= 1.5, Swapping rate is Sr = 0.4, range Beta 0.1< β <0.9 n = 60, switch probability P = 0.8, scaling factors ist γ 1 = 1 , second γ2 = 3, step Levy flight is λ = 1.5, and cloning array = [8 7 6 5 4 3 2 1 1 1 1 1 1] n= 60, Cross over =0.8 and mutation function with Scale = 1 (Gaussian) and shrink = 1 IFPDSO MFPA GA BAT n=60, pulse rate = 0.5 and minimum f = 0, Loudness = 0.5, maximum f = 2 FF n= 60, Randomness (alpha = 0.25), Absorption efficient (gamma = 1, minimum attractiveness: firefly is (beta = 0.2),), γ = 0.01, Levy flight step λ= 1.5 Annealing Fnc: Fast annealing, Initial temperature = 100. Re-annealing interval = 100 SA IFPDSO 3.375E-57 1 MFPA 6.3858E-33 2 FPA 4.9812E-12 3 GA 6.5606E-12 4 BAT 2.1499E-10 5 FF 3.62159E-9 6 SA 4.3 2669E-4 7 Algorithm MAE Rank IFPDSO 9.87E-17 1 MFPA 0.067145 2 FPA 0.0671478 3 FF 12.84937 4 BAT 15.125612 5 SA 45.140746 6 GA 109.5325 7 TABLE IV RANKING OF SEVEN ALGORITHMS (ALL FUNCTIONS) The proposed algorithm is evaluated on the 18 multimodal and unimodal complex functions. The results of the Modified Flower Pollination Algorithm (MFPA) are selected from [18], for a fair comparison, the number of generations for each algorithm N= 1500, population size n= 50 and n= 30 independent runs are executed. The performance will be analyzed by ranking the proposed algorithm IFPDSO and five other algorithms. The mean absolute error (MAE) is the first statistical analysis. Algorithm MAE Rank IFPDSO 5.565E-17 1 MFPA 0.0524615 2 FPA 0.0559426 3 FF 10.055946 4 BAT 11.837543 5 SA 35.327635 6 GA 85.721286 7 Table II shows the ranking of the algorithms in solving the unimodal functions while Table III and Table IV represent the ranking for multi-modal benchmark function and overall average ranking using Mean Absolute error. The ranking shows that the proposed IFPDSO algorithm performs better than GA, FF, SA, BAT, MFPA and standard Flower Pollination algorithm. Secondly, the comparison on convergence rate and stability between the proposed algorithm, SFPA and other well-known meta-heuristic algorithms which have been mentioned has been carried out. All these figures which include Fig.2 to Fig.7 have been plotted against the number of generations and minimum cost. In each graph, the solid red line shows the presentation of the proposed algorithm. The analysis proves remarkable convergence of the proposed algorithm in each plot of benchmark functions. All the graphs have been generated in Mat lab software R2018 by using the system Lenovo i3 ThinkPad. ∑J0 |L − M | 6 N Where L , indicate the mean value, bi is the best optimal solution and N is the number of independent runs. In Table: IV. The maximum cost (Max), minimum cost (Min), the average cost (Mean) and the standard deviation (Std.) are the results of random runs of every algorithm for the benchmark functions in Table V. The results of the proposed algorithm are highlighted in bold font. BGH = Rank TABLE III RANKING OF THE ALGORITHMS (MULTIMODAL) III. RESULTS AND DISCUSSION Algorithms MAE 464 TABLE V BENCHMARK FUNCTIONS No. Functions Equation of Functions f(x) = ∑RS0T PQ 1 Sphere function 2 3- hump camel function 3 Powell function 4 Matyas function F(x)= 0.26(PQT + PQQ ) -- 0.48PT PQ 5 Griewank function F(x)= 1+ ∑_S0T 6 Ackley function F(x) = −C e f g−Mh ∑]0 7 Easom’s function n P = −T R T o abc PS pqr s− t PS − PS − u Q v ]/[ Rastrigin,s function 8 9 10 11 12 Zaharov’s function 15 16 [ \2 + 10 [ \1 PQS WUUU 1 R 1 ] w ] R n P = t PQS + S0Q ] 1 i − exp S0T Eggholder function w =− 1 + 47 + sin gh„ w = −| sin cos w = 0.5 + w †‡ 1 1 + | 1 + 0.001 = ∑‰0 †mˆ † + 1 w = 418.9829> − t w = 1.5 + ] R [ [ ] [-4,5] [-32,32] [-100 100] [-5.1 5.1] − 0 1 1 + sin −1 1 1 [-5 10] [-5 5] exp |•100 − € 1+ 1 1 ••• + 1 [-10 10] . ‚ …‚ 1 + 1 + 47„i − 1 exp |•1 − 0 1 h…‚ … 1 1 − 0.5 1 + 11 |1 +† ] + [-5.1 5.1] … sin gh„ + … 1 [-5.1 5.1] + 47„i [-10 10] •• | [-100 100] ∑‰0 †mˆ † + 1 +† [-5.1 5.1] sin €| | 1 + 2.25 − [-500 500] + 1 1 + 1 2.625 − IFPDSO Function Algo Min Max Mean Std Sphere GA 5.992e-16 1.582e-13 1.362e-13 2.413e-13 BAT [ \2 − ∑]0 m 1 + C + exp 1 TABLE VI STATISTICAL ANALYSIS OF ALGORITHMS Function + 10 [-600,600] S0T 1 + cos € 1 + 11 0.5 1 + 11 + 2 Beale function [ PS √S R =− Schwefel function [ \1 − 2 [ \ [-10,10] R w Shubert function + T T t SPS Q + t SPS W − V Q Q Drop- wave function Schaffer function N2 1 [ − 10 ∗ cos 2 = −0.0001| |sin Holder table function [-5,5] S0T = x ∗ 10 + t 0 − [\ S0T w Cross-in-tray function +5 − ∏_S0T abc = t 100 17 18 X w Rosenbrock’s function 13 14 [-5,5] F(x,y) = QPQ − TU. VPW + P YX + PZ + ZQ F(x)= ∑ 0 [ Range 8.973e-13 1.293e-10 1.923e-11 2.142-11 SA 5.524e-37 5.524e-36 6.403e-36 1.354e-36 FF 1.107e-12 1.606e-10 3.742e-11 3.726e-11 FPA 8.543e-33 2.543e-25 7.923e-27 3.753e-26 MFPA 2.989e-70 6.011e-60 2.005e-61 1.097e-60 465 + 2 1 1 4.14e-186 [-4.5 4.5] 1.91e-180 1.97e-181 0 Three GA 7.87E-09 2.47E-07 6.19E-08 7.60E-08 Hump ABC 1.64E-26 5.17E-24 1.43E-24 1.81E-24 Camel SA 3.94E-27 1.12E-24 2.95E-25 4.54E-25 Function FPA 1.32E-27 1.98E-20 2.02E-21 6.24E-21 / IFPDSO 2.73e-110 5.62e-100 5.62e-101 1.77e-100 Powell GA 5.523e-14 1.883e-12 1.032e-11 1.192e-11 Sum ABC 1.96E-34 6.58E-33 7.80E-33 1.64E-32 Function SA 7.41E-31 1.36E-27 1.70E-28 4.22E-28 IFPDSO 4.69E-156 1.74E-148 2.89E-149 5.38E-149 Matyas,s GA 8.932e-14 1.142e-11 2.684e-11 3.952e-11 Function BAT 1.05E-17 6.70E-16 2.13E-16 Griewank Function Function BAT -2.16261 -2.16261 -2.16261 1.70E-11 2.07E-16 FF -2.06251 -2.06251 -2.06251 1.50E-11 SA 1.07E-25 7.40E-24 3.64E-24 3.35E-24 SA -2.06261 -2.05231 -2.06103 1.07E-03 FF 7.501e-13 5.753e-11 2.563e-11 1.354e-11 FPA -2.06251 -2.06251 -2.06261 8.59E-11 FPA 2.64E-36 6.20E-25 6.22E-28 1.85e-27 MFPA -2.06260 -2.06260 -2.06260 9.04E-16 MFPA 1.44E-68 5.66E-50 2.08E-51 1.12E-50 IFPDSO -2.06261 -2.06261 -2.06261 1.35E-15 GA -0.99996 -0.96324 -0.95722 3.01E-02 IFPDSO 6.52E-63 1.15E-52 1.49E-56 3.64E-56 Dropwave GA 3.72E-06 8.30E-02 2.49E-02 1.27E-02 Function BAT -1 -0.7875 -0.93335 3.41E-02 2.31E-12 1.40E-01 1.86E-01 1.42E-01 FF -1 -1 -1 1.61E-09 FF 3.575e-8 7.396e-3 6.196e-3 1.883e-3 SA -0.9999 -0.78579 -0.9298 3.41E-01 SA 0 0 0 0 FPA -1 -1 -1 8.07E-10 FPA 1.60E-08 2.91E-05 8.73E-06 8.43E-05 MFPA -1 -1 -1 0 MFPA 0 0 0 0 IFPDSO -1 -1 -1 0 GA -951.963 -940.342 -941.642 109.785 BAT SA FF FPA MFPA -955.608 -959.608 -959.608 -959.608 -959.608 -943.305 -886.432 -954.125 -959.608 -959.608 -947.152 -895.342 -954.634 -959.608 -959.608 179.056 166.542 186.503 1.15e-13 1.15e-13 -959.608 -959.608 -959.608 1.15E-13 BAT IFPDSO 0 0 0 Egg holder Function 0 Easom GA -1 -1 -1 2.12E-13 Function BAT -1 0 -0.3756 0.3795 SA -1 0 -0.6068 0.5432 FPA -1 -1 -1 0 MFPA -1 -1 -1 0 IFPDSO -1 -1 -1 0 Rastrigin’s GA 3.72E-13 8.30E-02 2.49E-02 1.27E-01 Function BAT 2.31E-10 1.42E-01 1.86E-01 1.42E-01 FF 3.575e-10 7.396e-8 6.196e-9 1.883e-9 SA 3.82E-8 8.84E-01 5.74E-01 6.84E-01 FPA 0 2.91E-12 8.73E-13 8.43E-13 MFPA 0 0 0 0 IFPDSO 0 0 0 0 Ackley GA 1.27E-07 7.20E-06 1.96E-06 Function BAT 5.79E-06 6.2578 FF 9.278E-6 SA 3.26E-13 FPA 3.22E-14 IFPDSO Holder Table Function GA 3.72E-06 8.30E-03 2.49E-03 1.27E-04 BAT FF SA 2.31E-8 3.575e-09 2.54E-03 1.4056 7.396e-07 6.7234 1.96E-01 6.196e-07 4.5673 FPA -19.2084 -19.2084 -19.2084 1.32E-01 1.883e-07 3.5642 3.62E-15 MFPA IFPDSO -19.2085 -19.2085 -19.2085 -19.2085 -19.2085 -19.2085 7.81E-15 3.61E-15 Scheffer N2 GA 1.89E-11 1.50E-04 1.14E-05 1.49E- Function BAT 1.82E-14 1.48E-01 3.40E-01 FF 1.01E-12 5.54E-11 1.72E-11 1.48E-11 2.12E-06 SA 9.61E-03 1.50E-01 8.22E-01 6.07E-01 2.46432 2.20356 1.334E-4 8.177E-5 3.157E-5 FPA MFPA IFPDSO 0 0 0 0 0 0 0 0 0 0 0 0 3.39E-12 1.60E-12 1.07E-12 2.12E-12 6.01E-12 8.07E-12 -69.9381 -186.731 -186.731 -186.731 -186.730 -186.730 -186.731 -16.611 -71.4109 -186.731 -186.731 -186.730 -186.730 -186.731 -24;3817 -173.261 -186.731 -186.731 -186.730 -186.730 -186.731 13.1135 31.9251 9.25E-07 8.55e-08 1.56E-13 5.38E-15 1.66E-13 MFPA 8.87E-16 8.88E-16 8.88E-16 0 IFPDSO 8.880E-16 8.880E-16 8.880E-16 0 GA BAT SA FF FPA MFPA Zakharov GA 8.932e-16 1.142e-12 2.684e-13 3.952e-13 IFPDSO Function BAT 1.05E-12 6.70E-10 2.13E-11 2.07E-11 SA 1.07E-09 7.40E-03 3.64E-05 3.35E-04 FF 7.501e-13 5.753e-10 2.563e-10 1.354e-10 FPA 2.64E-31 6.20E-25 6.22E-26 1.952e-26 MFPA 1.24E-71 5.57E-42 2.18E-44 1.03E-43 IFPDSO 6.52e-158 1.15e-154 1.49e-155 0 Rosenbrock GA 3.72E-06 8.30E-03 2.49E-03 1.27E-04 Function BAT 2.31E-8 1.4056 1.96E-01 1.32E-01 FF 3.575e-09 7.396e-07 6.196e-07 1.883e-07 SA 2.54E-03 6.7234 4.5673 3.5642 FPA 1.60E-22 2.91E-19 8.73E-20 8.43E-19 MFPA 0 9.66E-30 3.30E-31 1.76E-30 IFPDSO 0 3.28E-29 5.61E-30 1.34E-29 -2.06408 --2.03473 -2.0386 5.660E-03 Cross-intray GA Shubert Function Function BAT 2.55E-05 1.40E-01 1.96E-02 1.32E-06 2.55E-05 2.65E-05 2.63E-05 7.396e-01 2.55E-05 2.56E-05 6.196e-02 2.54E-05 2.54E-05 1.883e-07 0 MFPA IFPDSO GA BAT 2.55E-05 2.550E-05 -2.0548 2.560E-05 2.550E-05 --2.0373 2.551E-05 2.550E-05 -2.0346 2.26E-12 6.26E-01 1.17E-01 0 0 5.87E-03 2.70E-01 FF SA 3.52E-13 3.164E-6 2.06E-10 4.73E-02 6.25E-11 3.30E-3 FPA 7.06E-27 2.76E-20 1.46E-21 MFPA IFPDSO 0 0 0 0 0 0 FF SA FPA Beal Function 466 3.79E-01 0 1.40E-11 1.7E-3 8.58E-21 0 0 Fig. 2 The Min cost for Unimodal Sphere Function Fig. 5 The Cost for Multimodal Matyas Function Fig. 3 The Min cost for unimodal Three Hump Camel function Fig. 6 The Cost for Multimodal Griewank Function Fig. 4 The cost for unimodal Powell Function The performance of IFPDSO is better because it inherits proper exploitation in the form of scaling factor ‘ℽ’ and Levy flight ‘Ⅼ’ in the global pollination. The exploitation capability is improved by limiting the parameter epsilon ‘ε’ (uniform distribution) and swap operator to avoid premature convergence. Fig. 7 The Cost for Multimodal Ackley Function The right balance between exploration and exploitation is produced by the dynamic switch probability in order to increase its search ability. Another advantage of the IFPDSO algorithm is that it has a smaller number of parameters which 467 will enhance its efficiency and speed up the processing for solving complex optimization problems. Moreover, its complexity is low as compared to other versions of the standard Flower Pollination Algorithm. [6] IV. CONCLUSION [8] [7] In the last few decades, we have seen many applications of nature-inspired meta-heuristic techniques in solving various types of non-polynomial problems. The complex optimization problems have drawn the attraction of researchers due to a wide variety of issues that possess the nature of optimization. Therefore, new optimization algorithms have been introduced to achieve better results, for example gradient-based and stochastic techniques, but swarm intelligence has become the most useful tool among evolutionary algorithms. The standard flower pollination is also one of the best nature-inspired metaheuristic algorithms. It has many advantages yet there are a few drawbacks over other swarm intelligence techniques. Various types of modifications are introduced to overcome these drawbacks, but most of these approaches fail in obtaining the most optimum solution for some complex problems. In this study, the flower pollination algorithm is improved by modification in local pollination using the swap operator and dynamic switch probability. It has proved to be a robust optimization method with fewer parameters. 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