A new optimization algorithm based on mimicking the voting process for leader selection

Lecture review

Natural phenomena, the behavior of living things in nature, the biological sciences, genetic sciences, the laws of physics, the rules of the game, human behavior, and any evolutionary process that has an optimization process have been the source of inspiration in the design and development of metaheuristic algorithms. Accordingly, metaheuristic algorithms fall into nine groups: swarm-based, biology-based, physics-based, human-based, sport-based, math-based, chemistry-based, music-based, and the other hybrid approaches (Akyol & Alatas, 2017).

Behaviors of living organisms such as animals, birds, and insects have been the main source of ideas in the development of numerous swarm-based algorithms. The most common feature used in many swarm-based methods is the ability of living organisms to search for food sources. The most popular methods developed based on food search process modeling are the Particle Swarm Optimization (PSO) based on the search behavior of birds and fish (Kennedy & Eberhart, 1995), Ant Colony Optimization (ACO) based on ants search for the shortest path to food (Dorigo, Maniezzo & Colorni, 1996), Artificial Bee Colony (ABC) based on bee colony search behavior (Karaboga & Basturk, 2007), the Butterfly Optimization Algorithm (BOA) based on search and mating behavior of butterflies (Arora & Singh, 2019), and the Tunicate Search Algorithm (TSA) based on search behavior tonics (Kaur et al., 2020). The process of reproduction among bees and the scout bees search mechanism to find suitable new places for hives have been employed in the designing Fitness Dependent Optimizer (FDO) (Abdullah & Ahmed, 2019). Chimpanzee’s hunting strategy using operators such as emotional intelligence and sexual motivation has been the main source of inspiration in designing the Chimp Optimization Algorithm (ChOA) (Khishe & Mosavi, 2020).

Modeling of hunting strategies of living organisms in the wild has been a source of inspiration in designing various optimization approaches, including Grey Wolf Optimizer (GWO) based on gray wolf strategy (Mirjalili, Mirjalili & Lewis, 2014), Whale Optimization Algorithm (WOA) (Mirjalili & Lewis, 2016) based on humpback whales strategy, and Pelican Optimization Algorithm (POA) based on pelican behavior (Trojovský & Dehghani, 2022).

Applying the concepts of biology, genetics, and natural selection alongside random operators such as selection, crossover, and mutation has led to the development of biology-based algorithms. The process of reproduction, Darwin’s evolutionary theory, and natural selection are key concepts in the development of two widely used methods, the Genetic Algorithm (GA) (Goldberg & Holland, 1988) and the Differential Evolution (DE) algorithm (Storn & Price, 1997). The mechanism of the immune system in the face of diseases, viruses, and microbes has been the major inspiration in the development of the artificial immune system (AIS) method (Hofmeyr & Forrest, 2000).

Many phenomena, laws, and forces in physics science have been employed as inspiration sources for the development of physics-based metaheuristic algorithms. The phenomenon of melting and cooling of metals, which is known in physics as the refrigeration process, has been the main inspiration in the development of the Simulated Annealing (SA) approach (Van Laarhoven & Aarts, 1987). The phenomenon of the water cycle based on its physical changes in nature has inspired the Water Cycle Algorithm (WCA) (Eskandar et al., 2012). Gravitational force and Newton’s laws of motion have been the main concepts employed to introduce the method of Gravitational Search Algorithm (GSA) (Rashedi, Nezamabadi-Pour & Saryazdi, 2009). The application of Hook’s law and spring tensile force has been the main inspiration in Spring Search Algorithm (SSA) design (Dehghani et al., 2020a). Various physical theories and concepts have been the source of inspiration in the development of physics-based methods such as Multiverse Optimizer (MVO), inspired from cosmology concepts (Mirjalili, Mirjalili & Hatamlou, 2016), Big Bang-Big Crunch (BB-BC) inspired from Big Bang and Big Crunch theories (Erol & Eksin, 2006), Big Crunch Algorithm (BCA) inspired from Closed Universe theory (Kaveh & Talatahari, 2009), Integrated Radiation Algorithm (IRA) inspired from gravitational radiation concept in Einstein’s theory of general relativity (Chuang & Jiang, 2007), and Momentum Search Algorithm (MSA) inspired from momentum concept (Dehghani & Samet, 2020).

Behavior, thought, interactions, and collaborations in humans have been designed ideas in the development of human-based approaches. The most widely used human-based method is the Teaching-Learning-Based Optimization (TLBO) algorithm, which mimics the classroom learning environment and the interactions between students and teachers (Rao, Savsani & Vakharia, 2011). The competition between political parties and the efforts of parties to seize control of parliament is the source of inspiration in designing the Parliamentary Optimization Algorithm (POA) (Borji & Hamidi, 2009). The economic activities of the rich and the poor to gain wealth in society have been the main inspiration for the Poor and Rich Optimization (PRO) approach (Moosavi & Bardsiri, 2019). Influencing people in the community from the best successful people in the community has been the main idea of the Following Optimization Algorithm (FOA) (Dehghani, Mardaneh & Malik, 2020). The mechanism of admission of high school graduates to the university and the process of improving the educational level of students has been the main idea in designing the Learner Performance-based Behavior (LPB) algorithm (Rahman & Rashid, 2021). The cooperation of the members of a team to improve the performance of the team in performing their tasks and achieving the goal has been the main inspiration of the Teamwork Optimization Algorithm (TOA) (Dehghani & Trojovský, 2021). The efforts of human society to achieve felicity by changing and improving the thinking of individuals has been employed in the design of the Human Felicity Algorithm (HFA) (Veysari, 2022). The strategic movement of army troops during the war, using attack, defense, and troop relocation operations, has been a central idea in the design of War Strategy Optimization (WSO) (Ayyarao et al., 2022).

The rules governing various games, both individual and group, along with the activities of players, referees, coaches, and influential individuals, have been the main source of inspiration in the development of sport-based methods. The effort of the players in the tug-of-war competition have been the main idea in designing the Tug of War Optimization (TWO) technique (Kaveh & Zolghadr, 2016). The use of volleyball club interactions and the coaching process has been instrumental in designing the Volleyball Premier League (VPL) approach (Kaveh & Zolghadr, 2016). The players’ effort to find a hidden object was the main idea used in Hide Object Game Optimization (HOGO) (Dehghani et al., 2020b). The strategy that players and individuals use to solve the puzzle and arrange the puzzle pieces to complete it has been the source of inspiration in designing the Puzzle Optimization Algorithm (POA) (Zeidabadi & Dehghani, 2022).

Matheuristics (Boschetti et al., 2009) and the Base Optimization Algorithm (BOA) (Salem, 2012) are among math-based methods. Chemical Reaction Optimization (CRO) (Lam & Li, 2009) and the Artificial Chemical Reaction Optimization Algorithm (ACROA) (Alatas, 2011) are among chemistry-based methods. The Harmony Search Algorithm (HSA) (Geem, Kim & Loganathan, 2001) is music-based method. In addition, by combining metaheuristic algorithms with each other, researchers have developed hybrid metaheuristic approaches, including: the Sine-Cosine and Spotted Hyena-based Chimp Optimization Algorithm (SSC) (Dhiman, 2021) and the Hybrid Aquila Optimizer with Arithmetic Optimization Algorithm (AO-AOA) (Mahajan et al., 2022).

The literature review shows that numerous metaheuristic algorithms have been developed so far. However, according to the best knowledge of the literature, the voting process to determine the leader of the community has not yet been used in the design of any algorithm. This research gap motivated the authors of this article to develop a new human-based metaheuristic algorithm based on mathematical modeling of the electoral process and public movement.

Election-based optimization algorithm

This section is dedicated to introducing the proposed Election-Based Optimization Algorithm (EBOA) and then mathematical modeling of it.

Inspiration

An election is a process by which individuals in a community select a person from among the candidates. The person elected as the leader influences the situation of all members of that society, even those who did not vote for him. The more aware the community members are, the better they will be able to choose and vote for the better candidate. These expressed concepts of the election and voting process are employed in the design of the EBOA.

Algorithm initialization

EBOA is a population-based metaheuristic algorithm whose members are community individuals. In the EBOA, each member of the population represents a proposed solution to the problem. From a mathematical point of view, the EBOA population is represented by a matrix called the population matrix using Eq. (1). (1)${\displaystyle X={\left[\begin{array}{c}\hfill X_1\hfill \\ \hfill ⋮\hfill \\ \hfill X_i\hfill \\ \hfill ⋮\hfill \\ \hfill X_N\hfill \end{array}\right]}_{N\times m}={\left[\begin{array}{ccccc}\hfill x_{1,1}\hfill & \hfill \cdots \hfill & \hfill x_{1,j}\hfill & \hfill \cdots \hfill & \hfill x_{1,m}\hfill \\ \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill & \hfill ⋰\hfill & \hfill ⋮\hfill \\ \hfill x_{i,1}\hfill & \hfill \cdots \hfill & \hfill x_{i,j}\hfill & \hfill \cdots \hfill & \hfill x_{i,m}\hfill \\ \hfill ⋮\hfill & \hfill ⋰\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill x_{N,1}\hfill & \hfill \cdots \hfill & \hfill x_{N,j}\hfill & \hfill \cdots \hfill & \hfill x_{N,m}\hfill \end{array}\right]}_{N\times m},}$

where X refers to the EBOA population matrix, X_i refers to the ith EBOA member (i.e., the proposed solution), x_i,j refers to the value of the jth problem variable specified by the ith EBOA member, N refers to EBOA population size, and m refers to number of decision variables.

The initial position of individuals in the search space is determined randomly according to Eq. (2). (2)${\displaystyle x_{i,j}=l{b}_j+r\cdot \left(u{b}_j-l{b}_j\right),i=1,2,\dots ,N,j=1,2,\dots ,m,}$

where lb_j and ub_j refer to the lower bound and upper bound of the jth variable, respectively, and r is a random number in the interval $\left[0,1\right]$.

Based on the values proposed by each EBO member for the problem variables, a value can be evaluated for the objective function. These evaluated values for the objective function of the problem are specified using a vector according to Eq. (3). (3)${\displaystyle OF={\left[\begin{array}{c}\hfill O{F}_1\hfill \\ \hfill ⋮\hfill \\ \hfill O{F}_i\hfill \\ \hfill ⋮\hfill \\ \hfill O{F}_N\hfill \end{array}\right]}_{N\times 1}={\left[\begin{array}{c}\hfill OF\left(X_1\right)\hfill \\ \hfill ⋮\hfill \\ \hfill OF\left(X_i\right)\hfill \\ \hfill ⋮\hfill \\ \hfill OF\left(X_N\right)\hfill \end{array}\right]}_{N\times 1},}$

where OF refers to the vector of obtained objective function values of EBOA population and OF_i refers to the obtained objective function value for the ith EBOA member. The values of the objective function are the criterion for measuring the quality of the proposed solutions in such a way that the best value of the objective function specifies the best member while the worst value of the objective function specifies the worst member.

Mathematical model of EBOA

The main difference between metaheuristic algorithms is how members of the population are updated and the process that improves the proposed solutions in each iteration. The process of updating the algorithm population in EBOA has two phases of exploration and exploitation, which are discussed below.

Phase 1: Voting process and holding elections (exploration).

EBOA members, based on their awareness, participate in the election and vote for one of the candidates. People’s awareness can be considered as dependent on the quality and goodness of the value of the objective function. Accordingly, the awareness of individuals in the community is simulated using Eq. (4). In this awareness simulation process, individuals with better values of the objective function are more aware. (4)${\displaystyle A_i=\left\{\begin{array}{cc}\frac{O{F}_i-O{F}_{\mathrm{worst}}}{O{F}_{\mathrm{best}}-O{F}_{\mathrm{worst}}},\hfill & O{F}_{\mathrm{best}}\ne O{F}_{\mathrm{worst}};\hfill \\ 1,\hfill & \mathrm{else},\hfill \end{array}\right.}$

where A_i is the awareness of the ith EBOA member, OF_best and OF_worst are the best and worst values of the objective function, respectively. It should be noted that in minimization problems, OF_best is related to the minimum value of the objective function and OF_worst is related to the maximum value of the objective function, while in maximization problems, OF_best is related to the maximum value of the objective function and OF_worst is related to the minimum value of the objective function.

Among the members of the society, 10% of the most awareness individuals in the society are considered as election candidates. In the EBOA, it is assumed that the minimum number of candidates (N_C) is equal to 2 (i.e., N_C ≥ 2), meaning that at least two candidates will register for the election.

The implementation of the voting process in EBOA is such that the level of awareness of each person is compared to a random number, if the level of awareness of a person is higher than that random number, the person is able to vote for the best candidate (known as C₁). Otherwise, that person randomly votes for one of the other candidates. This voting process is mathematically modeled in Eq. (5). (5)${\displaystyle V_i=\left\{\begin{array}{cc}{C}_1,\hfill & A_i>r;\hfill \\ C_k,\hfill & \mathrm{else},\hfill \end{array}\right.}$

where V_i refers to the vote of the ith person in the community, C₁ refers to the best candidate, and C_k refers to the k th candidate, where k isa randomly selected number from the set $\left\{2,3,\dots ,N_C\right\}$.

At the end of the voting process, based on the counting of votes, the candidate who has received the highest number of votes is selected as elected (leader). This elected leader affects the situation of all members of the society and even those who did not vote for him. The position of individuals in the EBOA is updated under the influence and guidance of the elected leader. This leader directs the algorithm population to different areas in the search space and increases the EBOA’s exploration ability in the global search. The process of updating the EBOA population is led by the leader in such a way that firstly a new position is generated for each member. The newly generated position is acceptable for updating if it improves the value of the objective function. Otherwise, the corresponding member remains in the previous position. This update process in the EBOA is modeled using Eqs. (6) and (7).

(6)${\displaystyle x_{i,j}^{\mathrm{new},P1}=\left\{\begin{array}{cc}{x}_{i,j}+r\cdot \left(L_j-I\cdot x_{i,j}\right),\hfill & O{F}_L<O{F}_i;\hfill \\ x_{i,j}+r\cdot \left(x_{i,j}-L_j\right),\hfill & \mathrm{else},\hfill \end{array}\right.}$ (7)${\displaystyle X_i=\left\{\begin{array}{cc}{X}_i^{\mathrm{new}.P1},\hfill & O{F}_i^{\mathrm{new},P1}<O{F}_i;\hfill \\ X_i,\hfill & \mathrm{else},\hfill \end{array}\right.}$

where $X_i^{\mathrm{new}.P1}$ refers to a new generated position for the ith EBOA member, $x_{i,j}^{\mathrm{new},P1}$ is its jth dimension, $O{F}_i^{\mathrm{new},P1}$ is its value of the objective function, I is an integer selected randomly from the values 1 or 2, L refers to the elected leader, L_j is its jth dimension, and OF_L is its objective function value.

Phase 2: Public movement to raise awareness (exploitation).

The awareness of the people of the society has a great impact on their correct decisions in the election and voting process. In addition to the leader’s influence on people’s awareness, every person’s thoughts and activities can increase that person’s awareness. From a mathematical point of view, a better solution may be identified based on a local search adjacent to any proposed solution. Thus, the activities of community members to increase their awareness, lead to an increase in the EBOA’s exploitation ability in the local search and find better solutions to the problem. To simulate this local search process, a random position is considered in the neighborhood of each member in the search space. The objective function of the problem is then evaluated based on this new situation to determine if this new situation is better than the existing situation of that member. If the new position has a better value for the objective function, the local search is successful and the position of the corresponding member is updated. Improving the value of the objective function will increase that person’s awareness for better decision-making in the next election (in the next iteration). This update process to increase people’s awareness in the EBOA is modeled using Eqs. (8) and (9).

(8)${\displaystyle x_{i,j}^{\mathrm{new},P2}=x_{i,j}+\left(1-2r\right)\cdot R\cdot \left(1-\frac{t}{T}\right)\cdot x_{i,j},}$ (9)${\displaystyle X_i=\left\{\begin{array}{cc}{X}_i^{\mathrm{new},P2},\hfill & O{F}_i^{\mathrm{new},P2}<O{F}_i;\hfill \\ X_i,\hfill & \mathrm{else},\hfill \end{array}\right.}$

where $X_i^{\mathrm{new}.P2}$ refers to a new generated position for the ith EBOA member, $x_{i,j}^{\mathrm{new},P2}$ is its jth dimension, $O{F}_i^{new,P2}$ is its value of the objective function, R is the constant equals to 0.02, t refers to iteration contour, and T refers to maximum number of iterations.

Repetition process, pseudocode, and flowchart of EBOA

An EBOA iteration is completed after updating the status of all members of the population. The EBOA enters the next iteration with the newly updated values, and the population update process is repeated based on the first and second phases according to Eqs. (4) to (9) until the last iteration. Upon completion of the full implementation of the algorithm, EBOA introduces the best proposed solution found during the algorithm iterations as the solution to the problem. The EBOA steps are summarized as follows:

Start.

Step 1: Specify the given optimization problem information: objective function, constraints, and a number of decision variables.

Step 2: Adjust the number of iterations of the algorithm (T) and the population size (N).

Step 3: Initialize the EBOA population at random and evaluate the objective function.

Step 4: Update the best and worst members of the EBOA population.

Step 5: Calculate the awareness vector of the community.

Step 6: Determine the candidates from the EBOA population.

Step 7: Hold the voting process.

Step 8: Determine the elected leader based on the vote count.

Step 9: Update the position of EBOA members based on elected leader guidance in the search space.

Step 10: Update the position of EBOA members based on the concept of local search and public movement to raise awareness.

Step 11: Save the best EBOA member as the best candidate solution so far.

Step 12: If the iterations of the algorithm are over, go to the next step, otherwise go back to Step 4.

Step 13: Print the best-obtained candidate solution in the output.

End.

The flowchart of all steps of implementation of the EBOA is specified in Fig. 1 and its pseudocode is presented in Algorithm 1.

Computational complexity of EBOA

This subsection is devoted to examining the computational complexity of the EBOA. The computational complexity of EBOA initialization, including random population generation and initial evaluation of the objective function, is equal to O(Nm)where N is the size of the EBOA population and m is the number of problem variables. Holding the election and updating the EBOA population in the first phase has the computational complexity equal to O(NmT) where T is the number of iterations. Population update based on the second phase of EBOA to increase people’s awareness is equal to O(NmT). Accordingly, the total computational complexity of EBOA is equal to O(Nm(1 + 2T)).

Evaluation unimodal objective function

The results of applying EBOA and ten competitor metaheuristic algorithms to optimize F1 to F7 unimodal functions are reported in Table 6. The optimization outputs show the EBOA has provided the global optimal in solving the F1, F3, and F6 functions. EBOA is the first best optimizer in solving F2, F4, F5, and F7. The simulation results show that in handling the F1 to F7 functions, the EBOA performed better than the ten competitor metaheuristic algorithms and ranked first.

Evaluation of high-dimensional multimodal objective functions

The optimization results of F8 to F13 functions obtained from the implementation of EBOA and ten competing metaheuristic algorithms are released in Table 6. EBOA is able to converge to the global optimum in handling F9 and F11 functions. In optimizing the F10, F12, F13, and F14 functions, what is evident from the simulation results is that EBOA is the first best optimizer in these functions. In optimizing the F8 function, after GA and TLBO, the proposed EBOA is the third best optimizer of this function. What can be deduced from the results of Table 7 is that EBOA has a higher capability in optimizing high-dimensional multimodal functions compared to ten competitor algorithms and is ranked first as the best optimizer in functions 8 to F13.

Evaluation of fixed-dimensional multimodal objective functions

The results obtained from the implementation of EBOA and ten competitor metaheuristic algorithms on F14 to F23 functions are presented in Table 8. What emerges from the simulation output is that EBOA is the first best optimizer to handle F14 to F23 functions. Analysis and comparison of the obtained results indicate that the proposed EBOA approach has a superior performance over ten metaheuristic algorithms and among them, it has the first rank of the best optimizer.

The performance of the EBOA and the ten competitor metaheuristic algorithms implemented on the F1 to F23 objective functions are shown in Fig. 2 as the boxplot. For visual analysis of the ability to achieve the searched solution, Figs. 3 to 11 show the convergence curves of the EBOA and ten other competing algorithms in optimizing a number of objective functions.

Statistical analysis

Capability analysis of metaheuristic algorithms in terms of mean, best, std, median, and rank indices provides valuable information to compare their performance. However, a very small probability can be considered for the chance superiority of one method over another. In this study, the Wilcoxon rank sum test (Wilcoxon, 1992) and non-parametric t-test (Kim, 2015) are used to determine whether the superiority of the EBOA over any of the competing metaheuristic algorithms was statistically significant. The results of applying Wilcoxon rank sum test and non-parametric t-test on EBOA performance and competitor metaheuristic algorithms are released in Tables 9 and 10, respectively. In cases where the p-value is less than 0.05, it can be concluded that there is a significant difference between the two compared groups. What is clear from the results of the Wilcoxon rank sum test and non-parametric t-test is that the EBOA has a significant superiority in terms of statistical analysis over all ten competing algorithms in all objective function groups.

Sensitivity analysis

The proposed EBOA approach is a population-based metaheuristic algorithm that addresses optimization problems in a repetitive-based process. Thus, the two parameters of EBOA, population number (N) and the maximum number of iterations of the algorithm (T), affect EBOA performance. This subsection is dedicated to the sensitivity analysis of EBOA to changes in N and T parameters.

EBOA sensitivity analysis to parameter N has been studied by applying it to the handling of functions F1 to F23 for different values of parameter N equal to 20, 30, 50, and 80. The results of EBOA sensitivity analysis to parameter N are released in Table 11. The effect of the parameter N changes on EBOA convergence curves and how to achieve the solution is shown in Fig. 12. The simulation results reveal the fact that increasing the EBOA population size increases the search power of this algorithm, as it can be seen that by increasing the values of parameter N, the proposed EBOA achieves better solutions and as a result, the values of all objective functions decrease.

EBOA sensitivity analysis to the parameter T has been tested by implementing it on the handling of F1 to F23 functions for the parameter T equal to 100, 500, 800, and 1,000. The outputs of the EBOA sensitivity analysis for the T parameter value are shown in Table 12. In addition, the EBOA convergence curves, which show how to achieve the optimal solution under changes in the T parameter, are shown in Fig. 13. What can be understood from the simulation results is that increasing the number of iterations gives the EBOA more opportunity to be able to identify the main optimal area more accurately and to converge more towards the global optimal, which this reduced the values of the objective functions.

Evaluation of CEC 2019 suite objective functions

The implementation of EBOA on the functions F1 to F23 indicated the high ability of EBOA in optimization applications. In this subsection, the performance of the EBOA is evaluated in addressing the CEC 2019 objective functions, which consist of ten functions of CEC01 to CEC10. The optimization results of CEC 2019 functions using EBOA and competitor algorithms are presented in Table 13. EBOA is the first best optimizer in addressing the functions cec02, cec03, cec07, cec08, cec09, and cec10. The results of the Wilcoxon rank-sum test and t-test are reported in Table 14. In cases where the p-value in this table is less than 0.05, the proposed EBOA approach has a statistically significant superiority over the corresponding algorithm. Analysis of the simulation results shows that the proposed EBOA approach has a superior performance over competitor algorithms in handling most cases of CEC 2019 test functions.