A self-adaptive population Rao algorithm for optimization of selected bio-energy systems

This work proposes a metaphor-less and algorithm-specific parameter-less algorithm, named as self-adaptive population Rao algorithm, for solving the single, multi-, and many-objective optimization problems. The proposed algorithm adapts the population size based on the improvement in the fitness value during the search process. The population is randomly divided into four sub-population groups. For each sub-population, a unique perturbation equation is randomly allocated. Each perturbation equation guides the solutions toward different regions of the search space. The performance of the proposed algorithm is examined using standard optimization benchmark problems having different characteristics in the single-and multi-objective optimization scenarios. The results of the application of the proposed algorithm are compared with those obtained by the latest advanced optimization algorithms. It is observed that the results obtained by the proposed method are superior. Furthermore, the proposed algorithm is used to identify optimum design parameters through multi-objective optimization of a fertilizer-assisted microalgae cultivation process and many-objective optimization of a compression ignition biodiesel engine system. From the results of the computational tests, it is observed that the performance of the self-adaptive population Rao algorithm is superior or competitive to the other advanced optimization algorithms. The performances of the considered bio-energy systems are improved by the application of the proposed optimization algorithm. The proposed optimization algorithm is more robust and may be easily extended to solve single, multi-, and many-objective optimization problems of different science and engineering disciplines.


Introduction
Solving complex optimization problems to optimality using conventional optimization techniques takes much time and tiresome efforts. Achieving global optimum solutions may not be guaranteed by conventional techniques. Achieving a global optimum or a nearglobal optimum solution can be done by using metaheuristic techniques in relatively less time and effort. Metaheuristic techniques are the procedures developed to find satisfactorily superior solutions to the optimization problems. During the past few decades, various metaheuristic techniques have been developed by researchers and experts. Some of the prominent metaheuristics are genetic algorithm (GA), differential evolution (DE), ant colony optimization (ACO), artificial bee colony algorithm (ABC), cuckoo search, particle swarm optimization (PSO), firefly algorithm (FF), harmony search (HS), memetic algorithm (MA), biogeography-based optimization (BBO) algorithm, shuffled frog leaping (SFL) algorithm, and gravitational search algorithm (GSA). In addition, various hybrid optimization techniques, such as opposition-based DE, generalized generation gap model GA, hybrid DE algorithms with the estimation of (2019). To avoid premature convergence in the whale optimization algorithm (WOA), Mostafa and Yazdani (2019) hybridized the WOA with DE. Also, an adaptive parameter called the search mode was added for better balancing of the exploration and exploitation of the hybrid WOA. Misaghi and Yaghoobi (2019) proposed an improved invasive weed optimization algorithm based on the chaotic mapping schemes. Shukla, Singh, and Vardhan (2020) introduced adaptive inertia weight parameter to enhance the performance of the teachinglearning-based optimization algorithm. Liang and Cuevas Juarez (2020) introduced adaptation methods for the number of potent viruses, growth rate, and perturbation values in the virus optimization algorithm and enhanced the performance of the virus optimization algorithm. Doerr, Witt, and Yang (2020) proposed a self-adaptive evolutionary algorithm in which the mutation rate was adapted by encoding it within the individual solution. Raj, Kalpana, and Singh (2020) proposed a non-oscillating self-adaptive PSO algorithm by introducing an adaptive mechanism for inertia weight and by employing the best-achieved solution to guide the population. Wang, Morsidi, Ng, Budiman, and Neoh (2020) investigated the relationship between the parameter configurations and performance measures of the DE. Furthermore, the authors had proposed a self-adaptive variant of the DE, which reduces the dependency on the user for preparing the appropriate configuration for mutation strategy and control parameters. Similarly,  proposed an adaptive scheme that adjusts the mutation strategies used in the DE to bring equilibrium between population diversity and convergence.
From the literature, it can be observed that different adaptation techniques had been proposed to eliminate the process of selecting appropriate control parameters. However, much work is focused on the adaptive population size and multi-populations. Hence, in this work, to eliminate the adjustment of population size, a new algorithm named as "self-adaptive population Rao algorithm" is proposed. This algorithm is inspired by Rao algorithms. Rao (2020) proposed three algorithm-specific parameterless algorithms called the Rao algorithms (Rao-1, . Rao algorithms are metaphor-less algorithms. These algorithms move the population in the search space on the basis of interactions made by the population with best, worst, and randomly selected solutions. Rao (2020) opined that it would be better if the researchers focus on developing simple optimization techniques that can provide effective solutions to complex problems instead of looking for developing metaphor-based algorithms.
The performances of the Rao algorithms in solving the benchmark problems and engineering design problems were presented in the articles (Rao, 2020;Rao & Pawar, 2020a,b). From these articles, it is observed that these algorithms have different characteristics in exploration and exploitation. Inspired by the performances of these three Rao algorithms, another Rao algorithm named as  algorithm is proposed in this work. Furthermore, in this work, to bring the characteristics of these four algorithms together in one algorithm and to eliminate the adjustment of population size, a new algorithm named as "self-adaptive population Rao algorithm" is proposed. The performance of the proposed algorithm is tested using single-, multi-, and many-objective optimization scenarios and compared with that of the Rao algorithms, GSA, GA and its variant real coded genetic algorithm (RGA), PSO algorithm and its variants, and Jaya algorithm and its variants. The contributions of this work are: r A self-adaptive population (SAP)-Rao algorithm for single-, multi-, and many-objective optimization problems is proposed. r The proposed algorithm adapts the population size based on the improvement in the fitness value during the search process. r The efficiency of the proposed algorithm is demonstrated by solving standard single-and multi-optimization benchmark problems.
r The proposed algorithm is employed to identify optimum design parameters through many-objective optimization of a compression ignition biodiesel engine system, and multi-objective optimization of a fertilizer-assisted microalgae cultivation process.
r Furthermore, the SAP-Rao algorithm's effectiveness in solving multi-and many-objective optimization problems is presented in terms of the performance metrics such as inverted generational distance, spacing, hypervolume, and coverage.
The subsequent section presents the working principle of Rao optimization algorithms. Rao (2020) introduced the three Rao algorithms: Rao-1,  In this work, one more Rao algorithm named as  algorithm is proposed by the motivation of the Rao algorithms. Similar to the first three Rao algorithms, this algorithm also has no metaphorical explanation and algorithm-specific parameters. The flow of these four algorithms is similar, but the perturbation equation used is different for each algorithm. Figure 1 shows the Rao algorithms flowchart. The essential steps for optimization of an objective function Z are as follows:

Rao Optimization Algorithms
Step-I: Define the quantity of population (P); Define the quantity design variables (N) and their boundaries: Lower (LB), Upper (UB); Termination criterion: it can be the number of function evaluations or iterations.
Step-II: Randomly initialize the population of size P, and evaluate the objective function Z for all the population.
Step-III: Select the best and worst solutions from the current population on the basis of their Z value. If the Z is a minimization function, then the solution with the smallest Z value is the best solution, and the solution with the largest Z value is the worst solution and vice versa if the Z is a maximization function.
Step-IV: Locate new solutions for all the population (m = 1, 2. . . P): during the i th iteration, let s n, m, i be the value of the n th variable of m th solution, s n, b, i be the value of the n th variable of the best solution, s n, w, i be the value of the n th variable of the worst solution, and s n, m, i be the newly located value of s n, m, i . As per the Rao-1 algorithm, the new solutions are found using the following equation.
As per the Rao-2 algorithm, the new solutions are located using the following equation. s n,m,i = s n,m,i + r 1,n,i (s n,b,i − s n,w,i ) + r 2,n,i s n,m,i or s n,l,i − s n,l,i or s n,m,i As per the Rao-3 algorithm, the new solutions are located using the following equation. s n,m,i = s n,m,i + r 1,n,i s n,b,i − s n,w,i + r 2,n,i s n,m,i or s n,l,i − (s n,l,i or s n,m,i ) As per the Rao-4 algorithm, the new solutions are located using the following equation: s n,m,i = s n,m,i + r 1,n,i (s n,b,i − s n,w,i ) + 0.5 r 2,n,i (s n,w,i − s n,m,i ) + r 3,n,i (s n,b,i − s n,m,i ) − r 4,n,i (s n,w,i − s n,m,i ) where, r 1, n, i , r 2, n, i , r 3, n, i , and r 4, n, i are random numbers in the range [0, 1] for the n th variable during the i th iteration. In Equations (2) and (3), the third term on the right-hand side represents the interaction between the current solution (m th ) and a random solution (l th ) selected from the current population. These two terms are dependent on the Z values of the current (m th ) and randomly selected (l th ) solutions. If the current solution Z value is superior to the randomly selected solution Z value, then the third term in Equation (2) becomes r 2, n, i (|s n, m, i | − |s n, l, i |) and the third term in Equation (3) becomes r 2, n, i (|s n, m, i | − (s n, l, i )). Similarly, if the randomly selected solution Z value is superior to the current solution Z value, then the third term in Equation (2) becomes r 2, n, i (|s n, l, i | − |s n, m, i |) and the third term in Equation (3) becomes r 2, n, i (|s n, l, i | − (s n, m, i )).
Step-V: Evaluate Z values for the new population and apply a greedy selection process. If the Z value corresponding to the new solution (s n, m, i ) is superior to that of the old solution (s n, m, i ), then replace the old solution with the new solution, if not discard the new solution.
Step-VI: Verify the stopping criterion. If the termination criterion is satisfied, report the optimum solution from the final population, else go to Step-III.
Here, it can be noted that the Rao algorithms move the population in the search space based on interactions made by the current solution with best, worst, and randomly selected solutions. In the Rao-1 algorithm, the current solution will interact with only the best and worst solutions in the population. In the Rao-2, Rao-3, and Rao-4 algorithms, the current solution will interact with the best and worst solutions, including a randomly selected solution in the population. In addition, based on the fitness values of the current and randomly selected solutions, the movement equations for current solution transformation will be different. For more details about the Rao algorithms, the readers may refer to https://sites.google.com/view/raoalgorithms/.
In this work, for solving multi-and many-objective optimization problems, the posteriori version of the Rao algorithms is implemented. In this version of the Rao algorithms, the new solutions are located in the same manner as in Rao algorithms. However, the supremacy among the solutions is identified on the basis of non-dominance rank and crowding distance evaluation ap-proach (Deb, Pratap, Agarwal, & Meyarivan, 2002;Rao, Rai, & Balic, 2017). In this version of Rao algorithms, the set of P solutions is ranked using dominance principles, and proximity of the solutions with each other is calculated using crowding distance measurement. The solution that has the best rank (rank = 1) and the highest amount of crowding distance is considered as the best solution. In contrast, the solution that has the worst rank and the smallest amount of crowding distance is considered as the worst solution.
After identifying the best and worst solutions, a new set of P solutions is located using movement equations of the respective Rao algorithm. Now, the set of new solutions is combined with the set of earlier solutions forming solutions set of size 2P. Then, these combined solutions are again ranked using the dominance principles, and the crowding distances are calculated for all solutions. Based on the new ranks and crowding distance values, a set of P solutions will be selected for the next iteration. For more details about non-dominance ranking and crowding distance calculation, readers may refer to Deb et al. (2002) and Rao, Rai, and Balic (2017). The following section presents the working of the proposed SAP-Rao algorithm in single-and multi-objective scenarios.

Self-Adaptive Population Rao Algorithm
The Rao algorithms are population-based algorithms and have no algorithm-specific parameters. Besides, adjusting the population size for individual case studies is a tiresome task. The SAP-Rao algorithm has no control parameters. It does not require adjustment of the population size. The key features of the proposed algorithm are as follows: r During the iterative search process, the population is randomly divided into four sub-population groups. For each sub-population, a unique perturbation equation from Equations (1) to (4) is allocated. As these equations have different performance characteristics, each sub-population is moved toward a different region of search space. Furthermore, in all iterations, different solutions will enter into different sub-population sets, and perturbation equations will also change. This will ensure diversity in the exploration and exploitation of the search space.
r During the search process, the proposed algorithm adapts the population size based on the improvement in the fitness value. Let, Z be a minimization objective function, Z best be the minimum value of the objective function in the previous iteration, and Z best be the minimum value of the objective function in the current iteration. If |modulo(Z best , Z best )| < 0.1, then the current population is reduced by 10%, else the current population is increased by 10%. Figure 2 shows the SAP-Rao algorithm flowchart. The steps of SAP-Rao algorithm for single-objective optimization (Z) are as follows: Step-I: Define the quantity of population (P = P old ); Define the quantity design variables (N) and their boundaries: Lower (LB), Upper (UB); Termination criterion: it can be the number of function evaluations or iterations.
Step-II: Randomly initialize the population of size P, and evaluate the objective function Z for all the population. If Z is a minimization function, then take the minimum value of the objective function as Z best . If Z is a maximization function, then take the maximum value of the objective function as Z best .
Step-III: Randomly divide the population into four groups. Assign a unique equation for each group from Equations (1) to (4).
Step-IV: For each group, select the best and worst solutions from the group population on the basis of their Z value.
Step-VI: Combine the population of all groups together and evaluate Z values for the new population and apply a greedy selection process. If the Z value corresponding to the new solution s n, m, i is superior to that of the respective old solution, then replace the old solution with the new solution, if not discard the new solution.
Step-VII: Similar to the Z best value presented in Step-II, identify the Z best value from the new population. Then, compare the Z best with Z best . If |modulo(Z best , Z best )| < 0.1, then reduce the current population by 10% (P new = P old − round(0.1 × P old )), else increase the current population by 10% (P new = P old + round(0.1 × P old )).
Step-VIII: If P new < P old , then select P new amount of best solutions for the next iteration from the current population P old . If P new > P old , then the extra solutions (P extra = P new − P old ) needed are selected from the current population itself. The P extra amounts of elite solutions are duplicated in the current population, and the total population will be considered for the next iteration. Else, the current population will be considered in the next iteration.
Step-IX: Verify the stopping criterion. If the stopping criterion is satisfied, report the best solution from the final population, else go to Step-III.
The SAP-Rao algorithm simultaneously uses Equations (1) to (4) as the movement equations to guide the current population in the search space. However, Equations (1) to (4) use the best, worst, and random solutions to move the population in search space; a minimum of 20 population size is maintained in all iterations. This is because when the population gets divided into four sub-population groups, each sub-population can at least have a population of size of 5. Then, the best, worst, and random solutions can be selected from these five solutions. If the population size falls below 20, at least one sub-population will have less than five solutions. In such situations, there is more probability of selecting the best solution or the worst solution as the random solution, which will reduce the effectiveness of the algorithm. Hence, it is required to maintain a minimum population size of 20. In solving the single-objective optimization problems using the SAP-Rao algorithm, the term |modulo(Z best , Z best )| indicates the significance of the improvement in the fitness value, and it is defined as follows.
Here, it can be observed that the outcome of Equation (5) is dependent on the fitness values. In addition, depending on the problem type, fitness values will also vary. Hence, the threshold value for Equation (5) can be varied based on the problem type. However, in this work, the threshold value is arbitrarily taken as 0.1, and achieved better performances.
It may be mentioned here that the Rao-1 algorithm uses only the difference in the best and worst values of the variable, multiplied by a random number, for updating the value of the variable and there is no other mechanism present in Equation (1). However, sometimes, due to the absence of any other terms in Equation (1), the algorithm may get stuck up at local optimum in some cases. The self-adaptive population concept of SAP-Rao algorithm will avoid this drawback to a large extent.
The SAP-Rao algorithm adapts the population size based on the current population size. If there is no significant improvement in the fitness value for successive iterations, then the population size is increased by 10% for every iteration, and if there is a significant improvement in the fitness value for successive iterations, then the population size is reduced by 10% for every iteration. Here, it can be noted that the change in the population size is not constant, and it is varying based on the current population size. If the current population size is a considerably huge number, then the population size will also be changed by huge quantity. For example, if during an iteration, the population size is 200, and there is no significant improvement in the fitness value, then the population size in the next iteration will become 220 (200 + 10% of 200). Even with the increased population, if there is no significant improvement in the fitness value of the next iteration, then the population size in the next iteration will become 242. In this scenario, if 50% of increment is taken instead of 10%, then the population sizes for the above two iterations will become 300 and 450, respectively. This will consume more function evaluations. Similarly, if the significant improvement is observed in all the iterations of the above scenario, then the population sizes (with 50% decrement) will become 200, 100, and 50, respectively. This may reduce the consumption of the function evaluations, but there is a possibility of removing the solutions at distant and unique locations. Hence, the percentage of change in population size is kept to a minimum value (10%). As it is desired to maintain a minimum population size of 20 for the SAP-Rao algorithm, the minimum change in the population size with 10% will be 2. The general framework of the SAP-Rao algorithm is given below.

BEGIN
Initialize 'P = P old '-number of solutions, 'N'-number of design variables, 'LB'-lower boundaries of design variables, 'UB'-upper boundaries of design variables, and 'FE max'-termination criterion; Generate the initial candidate solutions and find fitness values (Z), find best fitness value-Z best , FE = 0, i = 0; While FE < FE Max i = i + 1; FE = FE + P; Divide the population into 4 groups randomly: P s1 , P s2 , P s3 , and P s4 Assign a unique equation to each subpopulation from Equations (1) Select P new number of best solutions from the current population to the next iteration Else If P new > P old P extra = P new − P old P extrapop = Select P extra number of best solutions from the current population P newpop = P currentpop + P extrapop Else Solutions in the current population will be considered for the next iteration End If End While Report the final solutions END Similar to the Rao algorithms for solving multi-and many-objective optimization problems, a posteriori version of the SAP-Rao algorithm is used. In this version, the new solutions are located in the same manner as in the SAP-Rao algorithm. Also, the best and worst solutions are identified using the same manner used in multi-objective Rao algorithms. However, the population size is updated differently.
After identifying the best solution and worst solution for each group, a new set of P/4 solutions is located for each group. Then, these new solutions are combined and evaluated for objective function values. The non-dominance ranks are found and the crowding distances are calculated for each solution. Now, the population size is updated based on the number of solutions ranked one. Let S R1 be the number of solutions ranked one in the combined population before updating the solutions, and S R1 be the number of solutions ranked one in the combined population after updating the solutions. If S R1 > S R1 , then increase the current population by 10% (P new = P old + round(0.1 × P old )), else reduce the current population by 10% (P new = P old − round(0.1 × P old )). Now, the set of new solutions is combined with the set of earlier solutions forming solutions set of size 2P. Then, these combined solutions are again ranked using dominance principles, and crowding distance values are calculated for all solutions. Based on the new ranks and crowding distance values, a set of P new solutions will be selected for the next iteration. Now, the proposed SAP-Rao algorithm is tested using standard optimization benchmark problems in a single-and multi-objective optimization scenario. Also, the proposed algorithm is tested in multi-and many-objective optimization scenarios using a case study of a compression ignition biodiesel engine system and a case study of a fertilizer-assisted microalgae cultivation process. The following section presents the computational results of the proposed SAP-Rao algorithm in single-and multi-objective optimization.

Computational Results in Single-and Multi-Objective Optimization
The proposed algorithms are coded in MATLAB-R2016b and the computational tests are performed by using a CPU with 3.40 GHz Intel R Core i5-7500 processor and 8 GB RAM. The performance of the proposed algorithm is tested using single-and multi-objective unconstrained benchmark problems first and then applied to multi-objective optimization case studies of a compression ignition biodiesel engine system and a microalgae cultivation process. In the computational experiments of the Rao-4 algorithm for solving single-objective optimization benchmark problems, the population size is varied from 10 to 100 for different problems. In the case of the SAP-Rao algorithm, the initial population size is taken as 50 for solving all the test problems and considered case studies.
The performance of the SAP-Rao algorithm in solving the single-objective optimization problems is compared with the performances of the Rao algorithms as well as the GSA, GA, RGA, PSO, extraordinariness particle swarm optimizer (EPSO), cooperative PSO (CPSO), comprehensive learning PSO (CLPSO), fully informed particle swarm (FIPS), Frankenstein's PSO (F-PSO), adaptive inertia weight PSO (AIWPSO), animal migration optimization (AMO), orthogonal learning-based cuckoo search (OLCS), Jaya, and self-adaptive multi-population Jaya (SAMP-Jaya) algorithms. Similarly, the performance of the SAP-Rao algorithm in solving the multi-objective optimization problems is compared with the performances of the Rao algorithms as well as the multi-objective dragonfly algorithm (MODA), multi-objective Grasshopper optimization algorithm (MOGOA), multi-objective ant lion optimizer (MOALO), multi-objective particle swarm optimization (MOPSO), non-dominated genetic algorithm (NSGA-II), Jaya, and adaptive multi-team perturbation guiding Jaya (AMTPG-Jaya). Also, the performance of the SAP-Rao algorithm in multi-objective optimization is evaluated on the basis of four performance indicators, namely inverted generational distance (IGD), hypervolume, spacing, and coverage. For more details about these performance metrics, readers may refer to .
The coverage (Cov) metric compares the two Pareto fronts on the basis of the quantity of dominated solutions. The spacing metric enumerates the consistency in distributing the solutions along a Pareto front. For a specified reference point, the hypervolume metric of a Pareto front indicates the volume of the search domain dominated by the Pareto front. When two Pareto fronts are compared on the basis of these three indicators, the Pareto front that has the higher hypervolume and coverage value can be considered as the best Pareto front. The Pareto front that has the best spacing value may not be the superior Pareto front, but the Pareto front that has the higher hypervolume and coverage values can be considered as the dominant Pareto front. The IGD indicator measures the convergence and diversity performance of the algorithms in objective space, and the algorithm with the least IGD value can be considered as the best. The subsequent section presents the analysis of the computational results of the SAP-Rao algorithm in solving the unconstrained benchmark problems.

Computational experiments and results on unconstrained benchmark problems
The performance of the SAP-Rao algorithm is tested using 25 unconstrained benchmark problems. These problems have characteristics such as separability, unimodality, non-separability, and multi-modality. A detailed description, including the number of independent variables and their ranges, was presented in Rao (2020). The Rao algorithms were tested on these problems by taking 500 000 function evaluations as the termination criterion. The statistical results were presented for 30 consecutive runs (Rao, 2020). Hence, for a fair comparison of the performances of these algorithms, the SAP-Rao algorithm is tested for the same number of function evaluations and runs.
Tables 1 and 2 present a comparison of the results achieved by the SAP-Rao algorithm with Rao algorithms. In all the tables, the term "B" refers to the best solution, the term "W" refers to the worst solution, the term "M" refers to the mean value of the final objective function values in 30 runs, the term "SD" refers to the standard deviation of the final objective function values in 30 runs, the term "MFE" refers to the mean of the function evaluations taken by the algorithm to reach the final reported solution in each run, and the term "R (%)" refers to the percentage of reduction in MFEs by the SAP-Rao algorithm. However, all the algorithms have attained the global optimum solutions in 13 out of 25 unconstrained benchmark problems. Hence, the results for these problems are presented in terms of MFE and R (%) in Table 1.
The statistical results for the remaining 12 problems are presented in Table 2. It can be observed that the performance of the Rao-4 algorithm is similar to the performances of the Rao-1, 4,8,9,10,11,12,17,19,and 22, the Rao-4 algorithm achieved better results using the least number of mean function evaluations. In solving the Rosenbrock test problem, the Rao-4 algorithm has achieved much better performance compared with the other Rao algorithms.
In addition, from Tables 1 and 2, it can be observed that the SAP-Rao algorithm is converging faster than the Rao algorithms. In 24 out of 25 test problems, the SAP-Rao algorithm has achieved global optimum solutions in fewer MFEs when compared with those of the Rao-1 algorithm. Similarly, the SAP-Rao algorithm has consumed less MFEs in 23 out of 25 test problems when compared with those of the Rao-2 and Rao-4 algorithms, and in 19 out of 25 problems when compared with those of the Rao-3 algorithm. Furthermore, the SAP-Rao algorithm has attained better or the same results in terms of the best, worst, mean, and standard deviation values in 23 problems when compared with those of the Rao algorithms. For the test problems 17, 24, and 25, the SAP-Rao algorithm has achieved better performance compared with that of Rao algorithms. For the test problems 18 and 21, the performance of the SAP-Rao algorithm is better or competitive when compared with that of the Rao algorithms.
In addition, to draw attention to the convergence and variation of the population size, the convergence and population size variation plots of the SAP-Rao algorithm are presented in Fig. 3. The unconstrained sphere function is used to present these plots. The initial population size for the SAP-Rao algorithm is taken as 50. Also, the population size and objective function values are recorded at the end of each iteration for the plot. The algorithm is executed for 10 independent runs taking 50 000 function evaluations as the termination criterion. The mean values of the objective function and population sizes are plotted against function evaluations in Fig. 3.
Here, an observation can be made that the variation in the population size is more frequent during the initial search process, and once it gets converged to the optimum solution, then there is no change in the population size. The peaks and valleys in the plot of the population size variation indicate that the SAP-Rao algorithm can avoid trapping in the local minima by changing the population size to the appropriate quantity. Once it is converged, the proposed algorithm is exploiting the search space by keeping a minimum number of population. Also, the proposed algorithm can converge toward the optimum solution in less number of function evaluations. The following subsection presents the analysis of the computational results of the SAP-Rao algorithm in solving some additional unimodal, multi-modal, and fixed-dimension multimodal optimization problems.

Computational experiments and results on unimodal and multimodal optimization problems
In addition to the unconstrained benchmark test problems, the performance of the SAP-Rao algorithm is tested using 23 unimodal and multimodal optimization test problems. The performance of the algorithms in the unimodal test problems reveals the exploitation capability of the algorithm, and the performance in the multimodal problems reveals the exploration capability of the algorithm. A detailed description, including the number of independent variables and their ranges, is presented in Rao (2020). Here, the performance of the proposed algorithm is compared with those of the AMO (Li, Zhang, & Yin, 2014b) and OLCS (Li, Wang, & Yin, 2014a) algorithms. In the computational experiments of the AMO and OLCS algorithms, the maximum function evaluations were taken differently for each problem. Hence, in this work, computations are performed by taking the same number of function evaluations as taken by the previous researchers for AMO and OLCS. The computational results are presented in Table 3. In Table 3, the performance of the SAP-Rao algorithm is compared with those of the AMO and OLCS algorithms in terms of the mean and standard deviation of the final objective function values in 50 runs. Here, an observation can be made that the SAP-Rao algorithm has better exploitation capability when compared with that of the AMO and OLCS algorithms. The SAP-Rao algorithm has achieved better mean and standard deviation values in unimodal problems (except P4) when compared with those of the AMO and OLCS algorithms. Furthermore, the exploration capability of the SAP-Rao algorithm is better or competitive to the AMO and OLCS algorithms. For the test problems P1, P2, P3, P5, P7, P17, P21, P22, and P23, the SAP-Rao algorithm has achieved better mean and standard deviation values when compared with those of the AMO and OLCS algorithms. For the problems P6, P11, P14, P16, P18, and P19, the SAP-Rao algorithm has achieved better or the same performance in terms of the mean and standard deviation when compared with those of the AMO and OLCS algorithms.
In addition, the performance of the SAP-Rao algorithm is compared with PSO (Eberhart & Kennedy, 1995), EPSO (Ngo, Sadollah, & Kim, 2016), CPSO (van den Bergh & Engelbrecht, 2004), CLPSO (Liang et al., 2006), FIPS (Mendes, Kennedy, & Neves, 2004), F-PSO (Montes de Oca, Stützle, Birattari, & Dorigo, 2009), AIWPSO (Nickabadi, Ebadzadeh, & Safabakhsh, 2011), GSA (Rashedi, Nezamabadi-pour, & Saryazdi, 2009), RGA (Haupt & Haupt, 2004), Jaya, and SAMP-Jaya (Rao & Saroj, 2017) algorithms using 15 unimodal and multimodal optimization test functions. These test functions are minimization problems and taken from Rao and Saroj (2017). Tables 4 and 5 present the comparison of the results for 50 000 and 200 000 function evaluations, respectively. The results are presented in terms Table 2: Comparison of the results of the SAP-Rao algorithm with the results of Rao algorithms achieved in the optimization of the unconstrained problems 14-25.   Rao-2, and Rao-3: Rao (2020). of the mean values, the standard deviation of the final objective function values in 30 runs. The test functions TF1 to TF7 are unimodal test functions, and TF8 to TF15 are multimodal test functions. All these test functions are minimization problems. From Table 4, it can be observed that in all the unimodal test cases (TF1 to TF7), the SAP-Rao algorithm has achieved better or same mean and standard deviation values when compared with those of the RGA, GSA, EPSO, Jaya, and SAMP-Jaya algorithms. This indicates that the exploitation capability of the SAP-Rao algorithm is better than that of the other algorithms compared. For the multimodal problem TF10, the proposed algorithm has achieved better mean and standard deviation values when compared with those of the other algorithms. For the remaining multimodal test functions, the proposed algorithm has attained better or competitive results in terms of mean and standard deviation values.
From Table 5, it can be observed that in all the unimodal test cases (TF1 to TF7), the SAP-Rao algorithm has achieved better or same results in terms of mean and standard deviation values when compared with those of the PSO and its variants, Jaya algorithm, and its variant SAMP-Jaya algorithm. This indicates that the exploitation capability of the SAP-Rao algorithm is better than that of the other algorithms compared. For test functions TF1, TF2, TF4, TF5, and TF7, the performance of the SAP-Rao algorithm is superior to those of the other compared algorithms. For functions TF3, TF6, and TF11, the proposed algorithm is able to find the global optimum solutions. Furthermore, for multimodal functions TF10, TF11, and TF15, the performance of the proposed algorithm is better or the same when compared with that of the other algorithms. In addition, for the problems TF9, TF12, and TF14, the proposed algorithm performance is better or competitive when compared with that of the other algorithms.
In these computational experiments, for all the unimodal problems considered here, the SAP-Rao algorithm is able to achieve better results when compared with those of the other algorithms from the literature. The performance of the SAP-Rao algorithm in solving multi-modal problems is better or competitive when compared with those of the other algorithms from the literature. The following section presents the analysis of the computational results of the SAP-Rao algorithm in solving the multi-objective optimization benchmark problems.

Computational experiments and results on multi-objective benchmark problems
The performance of the SAP-Rao algorithm is tested using five unconstrained multi-objective benchmark problems. The performances of the proposed algorithms are tested using five challenging multi-objective optimization test problems known as ZDT (Zitzler, Deb,   and Thiele) test problems (Zitzler, Deb, & Thiele, 2000). These problems are ZDT-1, ZDT-2, ZDT-3, ZDT-1 with linear Pareto front (ZDT-1L), and ZDT-2 with three objectives (ZDT2-3O). The ZDT-1 problem has a convex shaped Pareto front. The ZDT-2 problem has a concave-shaped Pareto front, and the ZDT-3 problem has a discrete convex Pareto front. A detailed description, including the number of independent variables and their ranges of the considered multi-objective optimization problems, is presented by Mirjalili (2016). The computational results of the proposed algorithms are presented in terms of IGD values and are compared with those of the basic algorithms as well as the MODA, MOGOA, MOALO, MOPSO, and NSGA-II. The IGD indicator measures the convergence and diversity performance of the algorithms in objective space, and the algorithm with the least IGD value can be considered as the best. The results of the MOGOA, MOPSO, and NSGA-II are taken from Mirjalili, Mirjalili, Saremi, Faris, and Aljarah (2018). The results of the MOALO and MODA are taken from Mirjalili, Jangir, and Saremi (2017) and Mirjalili (2016), respectively.
In the computational experiments of these problems, the function evaluations taken by the MODA and MOALO are 15 000 and 10 000, respectively. The function evaluations taken by the MOGOA were not specified in the literature. In this work, for a fair comparison of the performances of these algorithms, the proposed algorithm, along with the Rao algorithms, is tested on these problems by taking 10 000 function evaluations as the termination criterion. The population sizes of all the algorithms are taken as 100. In the case of multi-objective optimization using the SAP-Rao algorithm, 100 best-ranked solutions at termination are selected as the final population. The statistical results for the MODA, MOGOA, MOALO, MOPSO, and NSGA-II algorithms are presented for 10 consecutive runs. Hence, in this work also, the statistical results of the IGD values of the Pareto fronts achieved by the proposed algorithms are presented for 10 consecutive runs. The computational results of various algorithms in solving the multi-objective ZDT test problems are presented in Table 6.
The statistical results are presented in terms of best (B), mean values (M), and standard deviation (SD). From Table 6, it can be observed that in all the ZDT test problems, the SAP-Rao algorithm has achieved better or competitive results in terms of best, mean, and standard deviation values when compared with those of the MOGOA, MODA, MOALO, NSGA-II, MOPSO, and Rao algorithms. For the ZDT-3 problem, the SAP-Rao algorithm has outperformed all the algorithms compared in terms of best, mean, and SD. For ZDT-1 and ZDT-1L problems. The SAP-Rao algorithm has achieved better mean IGD values and SD than those of the other algorithms compared. For the ZDT2-3O problem, the SAP-Rao algorithm has achieved better values in terms of best and mean than those of the  other algorithms compared. For the ZDT-2 test problem, the SAP-Rao algorithm has achieved better mean values than the MOGOA, MODA, MOALO, NSGA-II, and Rao algorithms. Furthermore, the proposed SAP-Rao algorithm outperformed the other algorithms in solving the ZDT (except ZDT-2) problems. Also, the performances of the Rao algorithms are better than those of the MOGOA, MOPSO, NSGA-II, MODA, and MOALO in solving the ZDT (except ZDT-2) problems in terms of mean IGD value. The SAP-Rao algorithm achieved better mean IGD values than the other algorithms for the problems ZDT-1, ZDT-3, ZDT-1L, and ZDT2-3O. The best IGD values for the proposed algorithms are superior or competitive to those achieved by the other algorithms from the literature. However, the proposed improved algorithms have achieved better mean values than the other algorithms. Furthermore, mean IGD values are relatively closer to the best values. It indicates that the proposed algorithms are consistent in finding the optimal Pareto front. In addition, for qualitative assessment of the performances of the proposed algorithms, the Pareto fronts achieved by the proposed algorithms are presented in Figs 4-8. Here, it can be observed that the proposed algorithm's Pareto fronts are consistent and along with the true Pareto fronts.
The following section presents the computational complexity of the SAP-Rao algorithm, along with the Rao algorithms.

Computational complexity of the algorithms
The computational complexities of the SAP-Rao algorithm and Rao algorithms are evaluated as per the guidelines presented in the CEC-2017 technical report (Wu, Mallipeddi, & Suganthan, 2016). As per the guidelines to evaluate the computational complexities, the 28 test problems (with 10, 30, 50, and 100 dimensions) presented in the CEC-2017 technical report are solved using the proposed algorithms. The proposed algorithms and the complexity test function provided by the CEC-2017 technical report are executed using the same computer for 10 000 function evaluations. In executing the Rao algorithms and the complexity function, the population size is taken as 50. For the SAP-Rao algorithm, the initial population size is taken as 50. Now, the complexity of the algorithms is assessed using the values of T 1 , T 2 , and T 2 , which are expressed by the following equations.
Here, t 1i is the computing time of the complexity function for the problem i and t 2i is the complete computing time for the algorithm for the problem i. The computational complexity values achieved by the SAP-Rao and Rao algorithms are presented in Table 7. It is evident that the computational time will increase if the dimension of the problem increases. It can be observed from the computational times (T 2 ) of the proposed algorithms. However, the change in the computational time (T 3 ) as the dimension of the problem changes will not be similar for different algorithms. Here, it can be observed that the computational time for the Rao-1 algorithm is increased by 35% when the dimension is increased from 10 to 30. Also, for the Rao-1 algorithm, the T 3 value is increased by 22% and 47% when the dimension is increased from 30 to 50 and 50 to 100, respectively. Similarly, for the Rao-2 algorithm, the computational time is increased by 10%, 5%, and 19% when the dimension is increased to 30, 50, and 100, respectively. For the Rao-3 algorithm, the computational time is increased by 20%, 2%, and 22% when the dimension is increased to 30, 50, and 100, respectively. For the Rao-4 algorithm, the computational time is increased by 10%, 5%, and 26% when the dimension is increased to 30, 50, and 100, respectively. For the SAP-Rao algorithm, the computational time is increased by 2%, 3%, and 13% when the dimension is increased to 30, 50, and 100, respectively. The SAP-Rao algorithm has additional phases when compared with the Rao algorithms. Thus, the computational time of the SAP-Rao algorithm is more when compared with that of the Rao algorithms. However, if the dimension of the problems changes, the change in the computational time is relatively lesser than that of the Rao algorithms. The following section presents the analysis of the computational results of the SAP-Rao algorithm in multi-objective optimization of selected bio-energy applications.

Application of SAP-Rao algorithm for the multi-objective optimization case study of a microalgae-based biomass cultivation process
Biofuels can be considered as sustainable and environmentally friendly alternative to petroleum-based fuels and can be produced using biomass. Nowadays, producing biofuels from microalgae is gaining popularity (Agarwal, Agarwal, & Gupta, 2017). In this work, a multi-objective optimization case study of fertilizer-assisted cultivation of the Nannochloropsis species biomass production for biofuel feedstock is considered. This case study is presented by Banerjee, Guria, and Maiti (2016). The objectives of this case study include the maximization of biomass, eicosapentaenoic acid (EPA), and lipid productions. Banerjee et al. (2016) presented the mathematical models of these objectives considering the cultivation light intensity (X 1 in μmol/m 2 /s), temperature (X 2 in • C), and concentrations of NaCl (X 3 in M), NaHCO 3 (X 4 in g/L), and NPK-10:26:26 fertilizer (X 5 in g/L) as the design variables. The regression models for the biomass production, lipid, and EPA generations in terms of the considered design variables are as follows.
The spacing and hypervolume values achieved by various algorithms in multi-objective optimization of the biomass productivity, lipid, and EPA generations are presented in Table 9. The SAP-Rao algorithm achieved better spacing values in terms of best, the Rao-1 algorithm achieved better spacing values in terms of the worst and mean values, and the Rao-4 algorithm has the least standard deviation. The mean of the spacing values achieved by the SAP-Rao algorithm is 0.021043, which is 4.77%, 6.93%, 0.03%, and 5.37% lesser when compared with that of the Rao-2, Rao-3, Jaya, and Rao-4 algorithms, respectively. Furthermore, the SAP-Rao algorithm has achieved higher hypervolume values in terms of best, worst, mean, and standard deviation values.
The coverage values achieved by various algorithms in multi-objective optimization are presented in Table 10. In this optimization scenario, the coverage values achieved by the SAP-Rao algorithm are better than those of the other compared algorithms. Furthermore, from the coverage values of the SAP-Rao algorithm, it can be observed that 30% of the AMTPG-Jaya algorithm's solutions, 23% of the  Hence, the performance of the SAP-Rao algorithm can be considered as better.
The following subsection presents the many-objective optimization case study of the compression ignition biodiesel engine and associated computational results.

Application of SAP-Rao algorithm for the many-objective optimization case study of a compression ignition biodiesel engine with an exhaust gas recirculation system
Enhancing engine performance and reducing exhaust emissions of diesel engines is another crucial area in biodiesel research. The major biodiesel engine performance characteristics are engine torque, brake specific fuel consumption (BSFC), exhaust gas temperature, brake thermal efficiency, power output, and brake power. The exhaust gas emissions are carbon monoxide (CO), carbon dioxide  (CO 2 ), unburned hydrocarbons (HC), nitrogen oxides (NO x ), smoke opacity (S m ), and particulate matter (PM). Enhancing the performance and reducing the emissions can be achieved by modifying combustion chamber geometry, the timing of fuel injection, compression ratio, installing exhaust gas recirculation systems, and formulating biodiesel blends that have appropriate physicochemical properties (Damanik, Ong, Tong, Mahlia, & Silitonga, 2018).
In this work, a case study of a compression ignition biodiesel engine with an exhaust gas recirculation system is considered for multi-objective optimization to see if there can be any improvement in the considered system performance. This case study is presented by Jaliliantabar, Ghobadian, Najafi, Mamat, and Carlucci (2019). The exhaust gas recirculation rate (ER), engine load percentage (EL), engine speed (ES) in rpm, and biodiesel percentage (BP) are considered as the design variables in this case study. These design variables ranges are as follows: 0 ≤ E R ≤ 30, 25 ≤ E L ≤ 75, 1800 ≤ E L ≤ 2400, and 0 ≤ B P ≤ 15. The objective functions of this case study are maximization of power output (P), and minimization of BSFC, CO, NO x , HC, and S m , which are given in the following equations.
Jaliliantabar et al. (2019) reported a single Pareto optimal solution for this case study by implementing the NSGA-II algorithm. In this work, the proposed SAP-Rao algorithm, along with Jaya, AMTPG-Jaya, and Rao algorithms, is employed to identify the optimum design parameters through many-objective (six-objective) optimization to see if there can be any improvement in the considered objectives. Similar to the previous case study, the termination criterion in this case study is taken as 10 000 function evaluations. The population sizes of all the algorithms are taken as 30. In the case of the SAP-Rao algorithm, 30 best-ranked solutions at termination are selected as the final population. The spacing and hypervolume performance metrics are calculated for 10 independent runs, and the statistical results are presented in terms of best (B), worst (W), mean (M), and standard deviation (SD). Now, six-objective simultaneous optimization results obtained by the SAP-Rao and NSGA-II algorithms are reported in Table 11. In this scenario, the power is varied between 0.86 and 3.53 kW, the BSFC is varied from 218 to 577 g/kWh, the CO emission is varied from 0.00044% to 2.16%, the NO x emission is varied from 86 to 387 ppm, HC emission is varied from 16.01 to 250 ppm, and smoke emission is varied from 2.3 to 17.4 m −1 .
The solutions reported in Table 11 are non-dominated. To compare the solutions achieved by both the algorithms, Pareto optimal solutions ranking method suggested by Rao and Keesari (2019) is used. As per this method, each objective function is considered as a performance criterion, and each solution is considered as an alternative. Now, the solutions are ranked by employing various multi- attribute decision-making methods. Finally, the average rank is calculated based on the ranks given by different decision-making methods. The solution with the best (least) average rank is considered as the best alternative out of the available solutions. In this manner, the solutions of SAP-Rao and NSGA-II algorithms are ranked, and the ranks achieved by each method are presented in Table 12. For ranking of these solutions, simple additive weighting (SAW), grey relational analysis (GRA), technique for order preference by similarity to ideal solution (TOPSIS), preference ranking organization method for enrichment evaluations (PROMETHEE), complex proportional assessment (COPRAS), modified TOPSIS technique (MTOPSIS), compromise ranking method (VIKOR), and weighted product method (WPM) methods are employed. The readers may refer to Rao (2013) for more details of these methods. From Table 12, it can be observed that the NSGA-II algorithm solution achieved 16th position with an average rank value of 15.75. Solution-4 of the SAP-Rao algorithm has achieved the first position with an average rank of 1.625. Furthermore, the solution-4 of the SAP-Rao algorithm achieved better results in five out of six objectives when compared with those of the NSGA-II solution. With the solution-4 of the SAP-Rao algorithm, the power is increased by 75%, CO emission is reduced by 34.21%, NO x emission is reduced by 2.1%, HC emission is reduced by 53.31%, and smoke is reduced by 37.22%.
The spacing values achieved by various algorithms in six-objective optimization are presented in Table 13. The SAP-Rao algorithm achieved better spacing values in terms of mean value, the Rao-3 algorithm achieved better spacing values in terms of best, and the Rao-4 algorithm has the least standard deviation. The mean of the spacing values achieved by the SAP-Rao algorithm is 11.16%, 5.32%, 2.08%, 3.01%, 12.30%, and 6.99% lesser when compared with that of the Jaya, AMTPG-Jaya, Rao-1, Rao-2, Rao-3, and Rao-4 algorithms, respectively. The coverage values achieved by various algorithms in six-objective optimization are presented in

Discussion on the Computational Results
From the computational results of the SAP-Rao algorithm in single-, multi-, and many-objective optimization scenarios, it can be inferred that the SAP-Rao algorithm has accumulated the benefits of the four Rao algorithms in a single algorithm. The Rao algorithms have different movement (transformation) equations that have different characteristics in moving the population in the search space. These four movement equations can explore more search space compared with the individual movement equations. The SAP-Rao algorithm simultaneously uses these four movement equations to guide the current population in the search space. This will ensure the exploration of the search space. Furthermore, in all the iterations, the SAP-Rao algorithm divides the population into four sub-population groups and assigns a unique movement equation. Here, it can be noted that the movement equations of the Rao algorithms use the best, worst, and randomly selected solutions to move the population. In the SAP-Rao algorithm, during an iteration, each equation is assigned to a different subpopulation group and each movement equation will have different best and worst solutions. This indicates that the SAP-Rao algorithm is exploring and exploiting multiple regions of the search space simultaneously. Furthermore, in all iterations, different solutions will enter into different sub-population groups, and movement equations will also change. This will ensure diversity in the exploration and exploitation of the search space.
In addition, the SAP-Rao algorithm adapts the population size. If there is a significant improvement in the fitness value of the current iteration than in that of the previous iteration, then the population size is reduced. Also, if there is no significant improvement in the fitness value, then the population size will be increased. This implies that the algorithm increases the population to diversify the exploration of the search space when needed, and the algorithm decreases the population to preserve the function evaluations in exploiting the search space. Therefore, the SAP-Rao algorithm preserves the function evaluations when there is significant improvement in the solution and utilizes the preserved function evaluations when there is no significant improvement in the solution. This way the SAP-Rao algorithm achieves better solutions more efficiently and effectively. The SAP-Rao algorithm has the additional phases when compared with the Rao algorithms. Thus, the computational time of the SAP-Rao algorithm is more when compared with that of the Rao algorithms. However, if the dimension of the problems changes, the change in the computational time is relatively lesser than that of the Rao algorithms.
In solving the unconstrained benchmark problems, the SAP-Rao algorithm has achieved global optimum solutions in fewer MFEs when compared with those of the Rao algorithms. This indicates that the SAP-Rao algorithm is converging faster than the Rao algorithms. Similarly, in solving the unconstrained unimodal and multimodal test problems, it can be observed that the SAP-Rao algorithm has achieved better mean values in 9 out of 23 test problems, achieved the same mean values in 6 out of the 23 test problems, and for the remaining test problems competitive values are achieved when compared with those of the AMO and OLCS algorithms. Furthermore, for all the unimodal problems (TF1 to TF7) considered, the SAP-Rao algorithm is able to achieve better results when compared with those of the PSO, RGA, GSA, EPSO, CPSO, CLPSO, FIPSO, F-PSO, AIWPSO, Jaya, and SAMP-Jaya. Similarly, the performance of the SAP-Rao algorithm in solving multi-modal problems (TF8 to TF15) is better or competitive when compared with those of the PSO, RGA, GSA, EPSO, CPSO, CLPSO, FIPSO, F-PSO, AIWPSO, Jaya, and SAMP-Jaya. In addition, the SAP-Rao algorithm achieved better mean IGD values in solving the multi-objective benchmark (ZDT) problems when compared with those of the MOGOA, MOPSO, NSGA-II, MODA, MOALO, and Rao algorithms. Furthermore, mean IGD values achieved by the SAP-Rao and Rao algorithms are relatively closer to the best values, indicating that the proposed algorithms are consistent in finding the optimal Pareto front. Also, the proposed algorithm's Pareto fronts are consistent and along with the true Pareto fronts.
In the case study of the compression ignition biodiesel engine system, the proposed algorithm has achieved a better Pareto optimal solution. The power is increased by 75%, CO emission is reduced by 34.21%, NO x emission is reduced by 2.1%, HC emission is reduced by 53.31%, and smoke is reduced by 37.22% when compared with those reported by the NSGA-II algorithm. Furthermore, in the manyobjective optimization scenario, the proposed algorithm attained better or competitive results in terms of spacing and coverage values. Similarly, in the case study of fertilizer-assisted microalgae cultivation process, the proposed algorithm has attained better or competitive results in terms of spacing, hypervolume, and coverage values. The following section presents the conclusions of this work.

Conclusions
This work proposes an algorithm-specific parameter-less algorithm named "self-adaptive population Rao algorithm" for single-, multi-, and many-objective optimization problems. The key features of the proposed algorithm are as follows: r The algorithm has no algorithm-specific control parameters except the common control parameter of number of iterations. The proposed algorithm adapts the population size based on the improvement in the fitness value during the search process.
r The population is divided into four sub-population groups randomly. For each sub-population, a unique perturbation equation is allocated randomly.
r Each perturbation equation guides the solutions toward a different region of the search space.
The performance of the proposed algorithm is examined using the standard optimization benchmark problems in the singleobjective optimization scenario. The performance is compared with the recent state-of-art algorithms such as Jaya, SAMP Jaya, AMO, OLCS, PSO, EPSO, CPSO, CLPSO, FIPS, F-PSO, AIWPSO, GSA, and RGA. Also, the performance of the proposed algorithm in multiobjective optimization is evaluated on the well-known ZDT test problems and compared with those of the MOGOA, MOPSO, NSGA-II, MODA, MOALO, and Rao algorithms. The results of application of the proposed optimization algorithm are found superior in majority of the benchmark problems. Furthermore, the proposed algorithm is used to identify optimum design parameters through multi-objective optimization of a fertilizer-assisted microalgae cultivation process and many-objective optimization of a compression ignition biodiesel engine system. From the results of the computational tests, it is observed that the performance of the proposed algorithm is superior or competitive to the other optimization algorithms such as NSGA-II. The SPA-Rao algorithm solution has the better compromise among the various objectives of the many-objective optimization problem. In addition, the SAP-Rao algorithm has achieved better results in terms of coverage, hypervolume, and spacing values in multi-and many-objective optimization case studies.
The proposed algorithm is comparatively simpler and takes less time to give optimum reliable results. The idea of the SAP-Rao algorithm is simple, straightforward, and effective, and this may be attempted for solving the optimization problems of various science and engineering disciplines.