Multi-strategy Jaya algorithm for industrial optimization tasks

Abstract

Jaya, a simple heuristic algorithm, has shown attractive features, especially parameter-free. However, the simple structure of Jaya algorithm may result in poor performances, to boost the performance, a multi-strategy Jaya (MJaya) algorithm based on multi-population has been proposed in this paper. Three strategies correspond to three groups of solutions. The first strategy based on the first population is to introduce an adaptive weight parameter to the position-updating equation to improve the local search. The second strategy is based on rank-based mutation to enhance the global search. The third strategy is to exploit around the best solution to reinforce the local search. Three strategies cooperate well during the evolution process. The experimental results based on CEC 2014 have proven that the proposed MJaya is superior compared with Jaya and its latest variants. Then, the proposed MJaya algorithm is used to solve three industrial problems and the results have shown that the proposed MJaya algorithm can also solve complex industrial applications effectively.

Keywords

Jaya algorithm multi-strategy optimization adaptive

1 Introduction

In the past several years, various meta-heuristic algorithms have been developed. Nowadays, most of them have been successfully used in industrial applications. Compared with the conventional mathematical methods, these algorithms do not demand to calculate the gradient information. They are simple, flexible, and derivation-free, which make them attractive when solving these optimization problems.

It is necessary to determine the algorithm-specific parameter values when using these meta-heuristic algorithms. For example, Particle Swarm Optimization (PSO) utilizes inertia weight, cognitive and social parameters, Genetic algorithm (GA) uses mutation and crossover probabilities, Differential Evolution (DE) employs scale factor and crossover rate, and so on. Parameter tuning is a very tedious and time-consuming task. Compared with these algorithms, Jaya is a novel algorithm, which doesn’t have any algorithm-specific parameters. Jaya is a novel algorithm, proposed by Rao in 2016 [1]. The algorithm optimizes problems by approaching the best solution and keeping away from the worst solution. As the algorithm is parameter-free and has a very simple structure, it has been utilized to solve plenty of industrial problems, such as photovoltaic models [2], job shop scheduling problems [3], text document clustering [4], traveling salesman problem [5], reservoir operation [6], and so on.

Like most meta-heuristic algorithms, Jaya may suffer from premature and be trapped into the local optimum [7]. The industrial optimization problems are often complex and hence more improvement algorithms based on Jaya have been put forward [8]. The first improvement is based on the chaotic technique, which is used as a random mechanism to improve the exploration of Jaya. Farah and Belazi [9] integrated the technique into Jaya algorithm to develop chaotic Jaya, with the aim of alleviating the drawbacks of Jaya. Yu et al., [10] developed a multi-population based on the chaotic approach and Jaya to guarantee the optimal solution of the problem. Jiana and Weng [11] developed a logistic chaotic Jaya algorithm on the basis of the logistic chaotic map method, which can boost the diversity of the population. Lévy flight was used [12] as a chaotic method to push the Jaya algorithm to have a large jump, which can start a new search in the different search regions and avoid local optimum. Migallón et al., [13] pointed out that it was necessary to speed up the convergence of Jaya and then, a chaotic method was used. To balance the global and local search capabilities of the Jaya algorithm, Leghari [14] introduced a dynamic weight parameter to update the expression of the entire solutions.

The second improvement is the adaptive approach. The method can automatically tune the parameters of meta-heuristic algorithms. In the Jaya algorithm, the approach is mainly used to tune the population size. Rao and More [15] developed a self-adaptive Jaya algorithm to dynamically adjust the population size and use the adaptive Jaya to optimize the mechanical draft cooling tower system. A similar technique is used in the Enhanced Jaya algorithm(EJaya) [16] which was applied to estimate the parameters of photovoltaic solar cells and modules. In another version of EJaya, the local exploitation is according to the defined lower and upper local attractors while global exploration is based on the historical population [17]. Premkumar et al., [18] used an adaptive weight to adjust the tendency of the current population to be close to the best solution and avoid the worst solution, logistic, tent map, and sine chaotic methods to optimize the best solution. Pervez et al., [19] realized an adaptive Jaya, in which two adaptive parameters were used to solve the power point tracking of the photovoltaic system. Surrogate-assisted Jaya based on historical learning mechanism and optimal directional guidance is developed to solve optimization problems [20]. Comprehensive learning Jaya is designed for solving engineering problems, in which the information population is fully used [21].

The third improvement is the elite-based Jaya algorithm. It uses the survival of the fittest to select the powerful solution toward optimality. Raut and Mishra [22] proposed an elitist–Jaya algorithm, in which elitism strategy was used to adjust solutions on the basis of the neighborhood search and an adaptive linear inertia weight was used to balance local and global search. Rao and Saroj [23] implemented an elite-based Jaya algorithm with a single evolution phase and simple structure. In another work, Rao and Saroj [24] developed an elitist-based multi-objective Jaya algorithm to optimize the effectiveness and total cost of the heat exchangers.

These improvements have boosted the performance of the conventional Jaya algorithm. It is also noticed that most of these improvements are based on a single updating strategy, which keeps away from the worst solution and approaches the best solution. Multi-strategies and multi-population are rarely involved in these improvements. On the one hand, the single updating strategy of Jaya algorithm may result in poor diversity of the population, poor exploration, and exploitation. The algorithm may be trapped into the local optimum although the convergence may be fast. The search capability of the single strategy is limited. On the other hand, plenty of research in the community of meta-heuristic algorithms has indicated that multi-population based on multi-strategy is a useful approach, which can restrain disadvantages of a single strategy and utilize advantages of each strategy [25 –27]. What’s more, different optimization problems demand different evolution strategies. Different strategies may be better during evolution than a single strategy [28]. Based on these observations, a Multi-strategy Jaya (MJaya) algorithm is developed in this paper. In the proposed MJaya algorithm, three updating strategies are involved. The first one is to extend the conventional updating strategy of Jaya algorithm by introducing an adaptive weight parameter to enhance the exploration at the early stage and boost the exploitation at the later stage. The second strategy is to enhance the global search capacity by introducing the ranking-based mutation strategy. The third strategy is to mend the best solution, which is the most important solution to guide the search direction. A chaotic method is adopted to update the best solution to boost the exploitation. Through these three updating strategies, the local and global search capacity of MJaya is greatly boosted. We use the proposed MJaya algorithm to solve the CEC 2014 and these three typical industrial problems related to frequency-modulated sound waves, environmental economic dispatch, and parameters identification of photovoltaic models. These experimental results demonstrate that the proposed MJaya algorithm is superior and can solve complex industrial problems.

Therefore, the main the novelty of the paper can be summarized as follows:

An adaptive weight parameter is introduced into Jaya to adjust the global and local search.

A rank-based mutation mechanism is integrated into Jaya to boost the exploration.

Multi-strategy Jaya (MJaya) algorithm based on multi-population has been proposed.

Three industrial applications have demonstrated the effectiveness of MJaya.

The paper is organized as follows: In Section 2, the Jaya algorithm is briefly presented. In Section 3, the proposed MJaya algorithm is elaborately designed. In Section 4, the experiments are performed and the conclusions are made in Section 5.

2 Jaya algorithm

Jaya is one of the novel population-based optimization algorithms, which was developed by Rao in 2016 [1]. Its main feature is that it has no additional parameters, which makes it different from the conventional evolutionary algorithms. It is known that tuning parameters is a tedious task for evolutionary algorithms. During the evolution, Jaya optimizes solutions by approaching the best solution and keeping away from the worst solution.

Let f (x) be the objective function with D dimension (j = 1, 2, 3, …, D), x_ij be the value of the ith solution on jth dimension (i = 1, 2, …, NP) and NP be the number of solutions, in which x_i = (x_i,1, x_i,2, …, x_i,D) is the position of the solution i. Let x_best = (x_best,1, x_best,2, …, x_best,D) be the best solution found so far, it has the best objective function. For the minimization optimization problem, the f (x_best) is smaller than the remaining candidate solutions. On the contrary, let x_worst = (x_worst,1, x_worst,2, …, x_worst,D) be the worst solution, it has the worst objective function. Thereforethe is more than the remaining candidate solutions for the minimization problems. Like other populatn-based meta-heuristic algorithms, the first step is to initialize the population as follows:

$x_{i, j} = {LB}_{j} + rand \times ({UB}_{j} - {LB}_{j})$ (1) where rand is a random number and the range is bwn 0 and 1, UB_j and LB_j are the upper and lower boundaries of the jth dimension, respectively.

Then, the population is evolved by the Jaya algorithm.h solution x_i updates the position based on the following equation:

$\begin{matrix} x_{i, j}^{'} = x_{i, j} + {rand}_{1} \times (x_{best, j} - | x_{i, j} |) - {rand}_{2} \\ \times (x_{worst, j} - | x_{i, j} |) \end{matrix}$ (2) where rand₁andrand₂ are two random numbers and both ranges are between 0 and 1, x_best,j and x_worst,j represent the value of the jth dimension on the best, worst solutions found so far, |x_i,j| is the absolute value of x_i,j. The eression rand₁ × (x_best,j - |x_i,j|) is used to reveal the tendency of the solution to approach to the best solution while -rand₂ × (x_worst,j - |x_i,j|) is to represent the tendency of the solution to keep away from the worst solution.

If the objective function of $x_{i}^{'} = {x_{i, 1}^{'}, x_{i, 2}^{'}, \dots, x_{i, j}^{'}, \dots, x_{i, D}^{'}}$ is superior to that of the x_i, the $x_{i}^{'}$ will replace x_i. Otherwise, x_i will maintain eame.

$x_{i} = {\begin{matrix} x_{i}^{'}, iff (x_{i}^{'}) & < f (x_{i}) \\ x_{i}, & otherwise \end{matrix}$ (3)

3 Multi-strategy Jaya (MJaya)

The updating strategy used in Jaya is Eq. (2). It is a se strategy, which may result in poor population diversity and may be trapped into the local optimu For meta-heuristic algorithms, it is necessary to boost and balance the exploration vs exploitation, simultaneously. The exploration is to push candidate solutions search as broadly as possible while the exploitation is to use the existing knowledge. As the search direction is controlled by the best and worst solutions, the Jaya algorithm has relatively poor exploration and exploitation. Various improvements, such as introducing chaotic approaches, adaptive methods, and elite-based methods re proposed [8 –21]. Most of the improvements are based on a single updating strategy, which ls the performance of the algorithm. Many research have demonstrated that multi-strategy is a useful method to boost the performance of meta-heuristic algorithms [25 –27]. Based on the observations, we develop Multi-strategy Jaya (MJaya) algorithm, in which three strategies are ensembled. The first strategy is to introduce an adaptive weight parameter to update Eq.(2) with the aim of enhancing the exploitation. The second strategy is based on ranking mutation strategy, which can improve exploration while keeping the exploitation. The third strategy is to improve the best solution, which can further boost the exploitation.

3.1 The adaptive weight parameter

In population-based meta-heuristic algorithms, the desirable approach to converge to the optimal solution can be classified into two stages. The first stage is the beginning of evolution, the candidate solutions are encouraged to move through the whole search space. The second stage pushes the solutions search in the promising region. Based on the motivation, we introduce an adaptive weight parameter as follows:

$w = {\begin{matrix} {(\frac{f (x_{best})}{mean (f)})}^{2} mean (f) \neq 0 \\ 1 otherwise \end{matrix}$ (4)

$\begin{matrix} x_{i, j}^{'} = & x_{i, j} + {rand}_{1} \times (x_{best, j} - | x_{i, j} |) \\ - w \times {rand}_{2} \times (x_{worst, j} - | x_{i, j} |) \end{matrix}$ (5) where f (x_best) is the best objective function, mean (f) is the mean value of the objective functions from all current solutions. At first, the adaptive weight w is small as the difference between the objective function of the best solution and the mean objective function is large. The algorithm can search around the potential promising area and the range of the area is often large. Later, the difference between the objective function of the best solution and the mean objective function becomes small and the weight w approaches to 1. The difference between the best and worst solution also becomes small so that the local search can be reinforced. The adaptive weight parameter also adjusts the global and local search.

3.2 The ranking-based mutation

This strategy is to improve the exploration. Most of the exploration can be implemented by the mutation operator. However, too much exploration has a high risk. Better solutions should be assigned more priority to better individuals to guide the search direction. The ranking-based mutation can control the search direction by assigning more hts to better solutions. It is necessary to rank solutions depending on their fitness values from the best to the worst [29]. The selection probability can be calculated as follows:

$p_{i} = \exp^{- i / NP}$ (6) where i is the rank index of the ith solution, NP is the population size, p_i is the selection probability of the ith solution. The better the solution, the higher the probability. The better solutions have higher probabilities to be selected, which can produce better results after mutation. The mutation operator is as follows:

$u_{i} = x_{l} + rand \times (x_{m} - x_{n})$ (7) where l, m, n are randomly generated in the range of 1 and population size NP, i = 1, 2, … , NP andl ≠ m ≠ n ≠ i, u_i = { u_i1, u_i2, …, u_ij, …, u_i,D } is the trial vector, D is the dimension of the problem. The two solutions x_l and x_m are chosen depending on their probability calculated in Eq. (6). Let’s take x_l as an exampl Womly generate an integer l in the range of 1 and NP. Then, a random number in the range of [0, 1] is generated. We compare the p (l) and the random number. If the p (l) is mo than the random number, we accept the integer l and the corresponding individual x (l, :). Oerwise, we have to re-generate the integer numberl again. The operator can give better solutions more chances to be selected so that the search direction towards the promising regions cane controlled. The generation method of the parameter n is different from that of l and m in tt it is randomly generated without comparing with the random number. The purpose is to oid the local optimum. Then, the information from u_i and x_i is combined as follows:

$x_{i, j}^{″} = t \times u_{i, j} + (1 - t) \times x_{i, j}$ (8)

$t = {\begin{matrix} 0 rand < 0.5 \\ 1 otherwise \end{matrix}$ (9) where x_i,j is the value of ith solution on jth mension, u_i,j is the value of u_i on jth dimension, t is used to mix the value from u_i,j and x_i,j, rand is a random number in the range of 0 and 1.

3.3 Chaotic perturbation

Table 1

Algorithm The proposed MJaya algorithm

Input: The population size NP, the maximal function evaluaons FES_max

Output: The optimal solution x_best

1: Initialize the population

2: Calculate the objective function f_i of x_i (i = 1, 2, …, NP)

3: FES = FES + NP

4: While FES ⩽ FES_max

5: Calculate the mean value of fitness values mean (f)

6: Identity the best and the worst solution (x_best and x_worst)

7: Generate the first trial vector $x_{i}^{'} (i = 1, 2, \dots, NP)$ by Eq. (4-5)

8: Calculate the selection probability p_i (i = 1, 2, …, NP) according to Eq.(6)

9: Generate the second trial vector $x_{i}^{″} (i = 1, 2, \dots, NP)$ according to Eqs. (7-9)

10: Check the boundary of the two trial vectors $x_{i}^{'} and x_{i}^{″} (i = 1, 2, \dots, NP)$

11: Evaluate the first and second trial vectors $x_{i}^{'} and x_{i}^{″} (i = 1, 2, \dots, NP)$ , obtain the corresponding fitness values $f_{i}^{'} and f_{i}^{″} (i = 1, 2, \dots, NP)$

12: FES = FES+ 2 × NP ;

13: for i = 1 : NP

14: if ( $f_{i}^{'}$ < f_i)

15: x_i= $x_{i}^{'}$ ;f_i= $f_{i}^{'}$ ;

16: end

17: if ( $f_{i}^{″}$ < f_i)

18:x_i= $x_{i}^{″}$ ;f_i= $f_{i}^{″}$ ;

19: end

20: end

21: Find out the best and the worst solution (x_best and x_worst)

22: Use the chaotic perturbation to generate the vector x^* by Eqs. (10–12)

23: Iff (x^) < f (x_worst)

24: x_worst = x^ ;

25: f (x_worst) = f (x^) ;

26: end

27: FES* = FES+ 1 ;

28: end

Algorithm The proposed MJaya algorithm
Input: The population size NP, the maximal function evaluaons FES_max
Output: The optimal solution x_best
1: Initialize the population
2: Calculate the objective function f_i of x_i (i = 1, 2, …, NP)
3: FES = FES + NP
4: While FES ⩽ FES_max
5: Calculate the mean value of fitness values mean (f)
6: Identity the best and the worst solution (x_best and x_worst)
7: Generate the first trial vector $x_{i}^{'} (i = 1, 2, \dots, NP)$ by Eq. (4-5)
8: Calculate the selection probability p_i (i = 1, 2, …, NP) according to Eq.(6)
9: Generate the second trial vector $x_{i}^{″} (i = 1, 2, \dots, NP)$ according to Eqs. (7-9)
10: Check the boundary of the two trial vectors $x_{i}^{'} and x_{i}^{″} (i = 1, 2, \dots, NP)$
11: Evaluate the first and second trial vectors $x_{i}^{'} and x_{i}^{″} (i = 1, 2, \dots, NP)$ , obtain the corresponding fitness values $f_{i}^{'} and f_{i}^{″} (i = 1, 2, \dots, NP)$
12: FES = FES+ 2 × NP ;
13: for i = 1 : NP
14: if ( $f_{i}^{'}$ < f_i)
15: x_i= $x_{i}^{'}$ ;f_i= $f_{i}^{'}$ ;
16: end
17: if ( $f_{i}^{″}$ < f_i)
18:x_i= $x_{i}^{″}$ ;f_i= $f_{i}^{″}$ ;
19: end
20: end
21: Find out the best and the worst solution (x_best and x_worst)
22: Use the chaotic perturbation to generate the vector x^* by Eqs. (10–12)
23: Iff (x^*) < f (x_worst)
24: x_worst = x^* ;
25: f (x_worst) = f (x^*) ;
26: end
27: FES = FES+ 1 ;
28: end

The quality of the population is very important, which can determine the performance of the algorithm. It is further necessary to boost the exploitation of Jaya algorithm by searching around the best solution. The chaotic perturbation approach is used toind the potential solution around the current optimal solution [7 , 30]. The logistic map method as the chaotic sequence is proved to be useful in previous research [7]. It is also adopted into theroposed MJaya algorithm. The self-adaptive chaotic mechanism is implemented in the proposed MJaya algorithm in order to boost the exploitation. At the beginning of the algorithm, more chaotic perturbation is performed on the best solution in order to find a better solution. At the latter stageevolution,etter solutions are found. Less chaotic perturbation is used since the current best solution is very close to the optimal solution. The process can be mathematically presented as follows:

${\begin{matrix} z_{k} = rand, k = 1 \\ z_{k} = 4 \times z_{k - 1} \times (1 - z_{k - 1}), k > 1 \end{matrix}$ (10)

$x_{j}^{*} = {\begin{matrix} x_{best, j} + rand \times (2 \times z_{k} - 1), & rand < 1 - (FES / {FES}_{\max}) \\ x_{best, j}, & otherwise \end{matrix}$ (11) where the FES is the current function evaluations, FES_max is the maximal function evaluations, z₁ is randomly generated in the range of 0 and 1. At the beginning of the algorithm, FES is small, the chaotic perturbation is performed more frequently. As the FES becomes more and more, the chaotic perturbation is performed less. If the objective function of the x^* is superior to the worse solution in the current population, the x^* will take the place of the worse solution.

$x_{worst} = {\begin{matrix} x^{*}, iff (x^{*}) & < f (x_{worst}) \\ x_{worst}, & otherwise \end{matrix}$ (12) where x^* is the generated solution from the chaotic meod,. is the worst solution among the current solutions. The self-adaptive mechanism can improve the quality of the population by pushing them to move towards the promising area.

3.4 The proposed MJaya algorithm

The proposed MJaya algorithm mainly consists of the above three strategies: introducing adaptive weight parameter, ranking-based mutation, and adtive chaotic perturbation. In the optimization process, multi-population is adopted. The first population is evolved by the conventional Jaya while the second population is evolved by the ranking-based mutation. Two populations will construct a new population by the greedy selection.

The pseudo code of the proposed MJaya algorithm is prested as follows:

The flowchart of the proposed MJaya algorithm is plott in Fig. 1.

3.5 The computational complexity

Fig. 1

The flowchart of the proposed MJaya algorithm.

The computational complexity is analyzed in this sub-Section. Let the number of solutions be NP and the dimension of the problem is D. For each iteration, the calculation the mean fitness value demands O (NP); generating the first trial vector needs O (NP × D); ranking solutions requires O (NPlog (NP)) ; generating the second trial vector demands O (NP + NP × D); the chaotic perturbation requires O (D). The total complexity of each iteration is O (NP + NP × D + NPlog (NP) + D). As the log (NP) is often smaller than D, the total complexity is O (NP × D), which is the same as that of the traditional Jaya algorithm.

4 Experiments and analysis

4.1 The experimental results on the CEC 2014 testing suite

The first experiment is performed on CEC 2014 testinguite. In the test suite, there are 30 testing functions, which can be used to test the exploitation and exploration ability of the algorithm. The testing functions consist of unimodal, simple multi-modal, hybrid, and composition functions. Four new meta-heuristic algorithms, including Jaya and its two variants, Grey Wolf Optimizer (GWO) are used. The GWO algorithm, proposed by Mirjalili et al [31], is on the basis of the hierarchy and social status of the grey wolf. It also uses fewer parameters and a simple structure, which makes it attractive. The two Jaya variants are Improved Jaya (IJaya) [2] and Performance-Guided Jaya (PGJaya) [7]. The experience-based learning technique is adopted in IJaya while the performance-based evolution process is used in the PGJaya algorithm. Both algorithms are used previous information to guide the search direction. This is the feedback mechanism, which is often used in evolutionary algorithms.

Each algorithm runs 50 times independently to avoid randomness. The population size is 50, the dimension is 30, and the maximal function evaluations are 300 000. The best results x_best can be obtained. We can calculate the error Δf = f (x_best) - f (x^*), in which x^* is the optimal solution to the problem and x_best is the best solution obtained by the algorithm. We can get Δf during each run. We compute the mean value of Δf for 50 runs. The mean results of five algorithms are listed in Table 1, in which the best results are highlighted.

Table 2
The results of GWO, Jaya, IJaya, PGJaya, and MJaya algorithms

Fction GWO Jaya IJaya ya MJaya

f ₁ 6.8216E+7 3.0213E+8 2.5297E+7 1.5641E+5 5.9923E+5

f ₂ 1.6922E+9 4.0705e+10 3.4035E+6 E+3 0

f ₃ 2.8819E+4 1.9817E+5 2.6146E+4 560 1.1369E-13

f ₄ 201.8449 9.7781E+3 202.3763 60 23.9648

f ₅ 20.9546 20.8927 20.4384 20.0000 20.0000

f ₆ 11.4942 39.7443 30.6969 23.2154 11.1882

f ₇ 11.7569 164.1617 0.8627 0.0058 9.0311E-4

f ₈ 75.7928 299.0069 107.2064 6822 25.5041

f ₉ 106.6213 354.6439 120.3753 8.3350 72.8640

f ₁₀ 2.1729E+3 4.7151E+3 4.0677E+3 2.6390E+3 577.4275

f ₁₁ 3.1167E+3 6.9613E+3 4.2835E+3 26E+3 3.4769E+3

f ₁₂ 1.3735 2.5338 1.2299 0876 0.0709

f ₁₃ 0.3882 5.4897 0.5595 0.4248 0.3004

f ₁₄ 1.6750 65.6622 0.2173 59 0.2797

f ₁₅ 108.1569 1.1888E+3 20.3454 .0394 5.5582

f ₁₆ 11.8620 13.2130 12.5778 12.3267 12.0438

f ₁₇ 1.9694E+6 3.1142E+7 6.2510E+5 11E+4 1.0030E+5

f ₁₈ 5.6049E+6 8.6837E+8 5.5686E+3 3107E+3 2.4807E+3

f ₁₉ 39.3488 124.5587 12.4097 8.1090 4.7929

f ₂₀ 1.4666E+4 9.2053E+4 1.7815E+3 7.8118 105.5451

f ₂₁ 8.6130E+5 7.1501E+6 1.0086E+5 5698E+4 2.4544E+4

f ₂₂ 382.8517 1.2499E+3 437.2950 524.1617 538.3361

f ₂₃ 331.3008 677.0857 318.8727 2441 315.2441

f ₂₄ 200.0016 247.5569 235.9675 0.0288 232.6098

f ₂₅ 209.8384 254.5091 211.7366 1653 210.3537

f ₂₆ 143.5371 105.1939 100.4796 2073 100.3134

f ₂₇ 619.0657 1.3307E+3 443.5866 600.7439 467.3119

f ₂₈ 1.1843E+3 2.0188E+3 1.4713E+3 1.7695E+3 1.0158E+3

f ₂₉ 1.1358E+6 3.1118E+7 6.9679E+3 3342E+3 1.3036E+3

f ₃₀ 4.5960E+4 3.8187E+5 1.0510E+4 1.9794E+3 2.4386E+3

+/-/= 25/5/0 30/0/0 26/4/0 22/6/2

Fction	GWO	Jaya	IJaya	ya	MJaya
f ₁	6.8216E+7	3.0213E+8	2.5297E+7	1.5641E+5	5.9923E+5
f ₂	1.6922E+9	4.0705e+10	3.4035E+6	E+3	0
f ₃	2.8819E+4	1.9817E+5	2.6146E+4	560	1.1369E-13
f ₄	201.8449	9.7781E+3	202.3763	60	23.9648
f ₅	20.9546	20.8927	20.4384	20.0000	20.0000
f ₆	11.4942	39.7443	30.6969	23.2154	11.1882
f ₇	11.7569	164.1617	0.8627	0.0058	9.0311E-4
f ₈	75.7928	299.0069	107.2064	6822	25.5041
f ₉	106.6213	354.6439	120.3753	8.3350	72.8640
f ₁₀	2.1729E+3	4.7151E+3	4.0677E+3	2.6390E+3	577.4275
f ₁₁	3.1167E+3	6.9613E+3	4.2835E+3	26E+3	3.4769E+3
f ₁₂	1.3735	2.5338	1.2299	0876	0.0709
f ₁₃	0.3882	5.4897	0.5595	0.4248	0.3004
f ₁₄	1.6750	65.6622	0.2173	59	0.2797
f ₁₅	108.1569	1.1888E+3	20.3454	.0394	5.5582
f ₁₆	11.8620	13.2130	12.5778	12.3267	12.0438
f ₁₇	1.9694E+6	3.1142E+7	6.2510E+5	11E+4	1.0030E+5
f ₁₈	5.6049E+6	8.6837E+8	5.5686E+3	3107E+3	2.4807E+3
f ₁₉	39.3488	124.5587	12.4097	8.1090	4.7929
f ₂₀	1.4666E+4	9.2053E+4	1.7815E+3	7.8118	105.5451
f ₂₁	8.6130E+5	7.1501E+6	1.0086E+5	5698E+4	2.4544E+4
f ₂₂	382.8517	1.2499E+3	437.2950	524.1617	538.3361
f ₂₃	331.3008	677.0857	318.8727	2441	315.2441
f ₂₄	200.0016	247.5569	235.9675	0.0288	232.6098
f ₂₅	209.8384	254.5091	211.7366	1653	210.3537
f ₂₆	143.5371	105.1939	100.4796	2073	100.3134
f ₂₇	619.0657	1.3307E+3	443.5866	600.7439	467.3119
f ₂₈	1.1843E+3	2.0188E+3	1.4713E+3	1.7695E+3	1.0158E+3
f ₂₉	1.1358E+6	3.1118E+7	6.9679E+3	3342E+3	1.3036E+3
f ₃₀	4.5960E+4	3.8187E+5	1.0510E+4	1.9794E+3	2.4386E+3
+/-/=	25/5/0	30/0/0	26/4/0	22/6/2

From Table 1, it can be found that GWO has gotten the best performance on four functions, i.e., f₁₁, f₁₆, f₂₂ and f₂₄. GWO searches for the optimal solutions mainly depending on the top three wolves. The exploitation ability is good compared with its exploration. Unfortunately, the Jaya algorithm cannot achieve any better results on thirty testing functions. The position-updating mechanism of the Jaya algorithm is simple, which results in pr local and global search capacity. The performances of IJaya and PGJaya are better than Jaya. IJaya adopts the experience-based learning method, which can enhance the diversity of the poputn. However, it does not control the search direction in this process, which deteriorates the convergence. The performance is also not very good. Compared with IJaya, PGJaya is better as it has obtained superior results on eight functions, i.e., f₁, f₄, f₅, f₁₇, f₂₁, f₂₃, f₂₅, and f₃₀. The reason is that PGJaya adopts two strategies randomly to boost the search capacity. The proposed MJaya algorithm is the best one among these five algorithms in that it has obtained better results on eighteen functions, i.e., f₂, f₃, f₅ - f₁₀, f₁₂, f₁₃, f₁₅, f₁₈, f₁₉, f₂₀, f₂₃, f₂₆, f₂₈, and f₂₉. The proposed MJaya algorithm even finds the optimal solutions of function 2, which is significantly better than the four algorithms. What’s more, the results from the proposed MJaya algorithm on f₁, f₄, f₁₁, f₁₄, f₁₆, f₁₇, f₂₁, f₂₄, f₂₇ and f₃₀. rank the second compared with the remaining four algorithms. The multi-strategy based on multi-population can attribute to superior results. During the optimization process, the first strategy based on the first population introduces the adaptive weight parameter to the tendency of the worst solution, which can boost the exploitation. The second strategy based on the second population uses the ranking-based mutation technique to control the search direction so that the exploration is enhanced. By the dy selection, two populations can reconstruct a new population. During the process, the advantages of both strategies are fully made use of. At last, the self-adaptive chaotic perturbation method is performed on the best solution to further reinforce the exploitation. Therefore, the proposed MJaya algorithm can achieve better performances compared with the four algorithms.

Two statistical approaches, Wilcoxon signed-rank test and t Friedman test, are used to test the significant differences between the proposed MJaya algor and its opponents. The Wilcoxon test is based on 0.05 level, and the results of the comparison are listed in Table 2, in which three flags ‘+’, ‘–’, and ‘=’ denote that the proposed MJaya algorithm is superior to, inferior to, and similar to its opponents. Based on the last row of Table 1, it can be found that the proposed MJaya algorithm is significantly superior to its opponents. Then, the ranking of the Friedman test is presented in Table 2. The average ranking of the proposed MJaya algorithm is the lowest, which indicates the performance of the algorithm is the best. The results of the two statistical methods are consistent and prove

Table 3

The average ranking of five algorithms

Algorithm	Ranking
GWO	3.1667
Jaya	4.9000
IJaya	3.0667
PGJaya	2.3333
MJaya	1.5333

Fig. 2

The convergence curves of five algorithms on f₁ - f₁₂.

the superior performances of the proposed MJaya algorithm compared with its four opponents.

The convergence rate of the proposed MJaya algorithm is evaluated by the convergence curves shown in Figs. 1 –3. The convergence curves are presented by t mean of log(Δf) during the 50 runs. We use the log function to make the differences among the five algorithms more significant.

Fig. 3

The convergence curves of five algorithms on f₁₃ - f₃₀.

It can be seen that the proposed MJaya algorithm has the best convergence performances while Jaya is the worst one during evolution. The trend of the convergence curve fluctuates fast, which demonstrates that a multi-strategy based on multi-population is effective to update positions of solutions and balance the local and global search.

4.2 The industrial application

In this section, three typical industrial applications: frequency-modulated sound waves, environmental economic dispatch, and parameters identification of photovoltaic models are solved by the above five algorithms. These applications can further prove the usefulness of meta-heuristic algorithms and the effectiveness of the proposed MJaya algorithm.

Table 4
The results of five algorithms on paramete determination issue

Algorithm Maximum Minimum an

GWO 25.0980 2.0290 16.3643(+)

JAYA 31.0140 16.9546 27.1680(+)

IJAYA 26.8423 13.1882 17.4844(+)

PGJAYA 27.5148 10.5134 18.1001(+)

MJAYA 15.9785 1.5672E-7 3.1656

Algorithm	Maximum	Minimum	an
GWO	25.0980	2.0290	16.3643(+)
JAYA	31.0140	16.9546	27.1680(+)
IJAYA	26.8423	13.1882	17.4844(+)
PGJAYA	27.5148	10.5134	18.1001(+)
MJAYA	15.9785	1.5672E-7	3.1656

4.2.1 Frequency-modulated sound waves

Wave synthesis is very important in the modern music system. In this problem, six parameters y₁, y₂, y₃, y₄, y₅, y₆ are required to be optimized with theurpose of minimizing the square errors between the target wave Y₀ (t) and the estimated wave Y (t) [32]. The square errors can be mathematically presented as follows:

$\begin{matrix} Minimize F (Y) = \sum_{t = 1}^{100} {(Y (t) - Y_{0} (t))}^{2} \\ Y = (y_{1}, y_{2}, y_{3}, y_{4}, y_{5}, y_{6}) \\ - 6.40 ⩽ y_{1}, y_{2}, y_{3}, y_{4}, y_{5}, y_{6} ⩽ 6.35 \\ Y_{0} (t) = 1.0 \sin (5 t φ + 1.5 \sin (4.8 t φ + 2 sin (4.9 t φ))) \\ Y (t) = y_{1} \sin (y_{2} t φ + y_{3} \sin (y_{4} t φ + y_{5} sin (y_{6} t φ))) \end{matrix}$ (13) where the parameter φ = 2π/100. Above five algorithm i., GWO, Jaya, IJaya, PGJaya, and MJaya are employed to solve the problem. The maximal iterations are 400. Thirty runs are performed for each algorithm. The statistical results maxima minimal, and mean results are presented in Table 3 and Fig. 3. There is no doubt that three statistical indicators: the maximum, minimum, and mean of the proposed MJaya are the least, which indicates the superiority of the algorithm. In addition, the Wilcoxon test is also presented in Table 3. It can be found that the proposed MJaya algorithm ranks first compared with the rest four algorithms. The convergence curve of five algorithms is plotted in Fig. 4. The convergence ra of the proposed MJaya algorithm is the fastest in the five algorithms.

Fig. 4

The statistical results of five algorithms.

4.2.2 Environmental economic dispatch

The global economy develops very quickly, which demands lots of energy. During the consumption of these energies, it is indispensable to generate some pollutants. Nowadays, the public and government are more concerned about environmental issues and pollution control. We want to achieve harmony between the environment and the development of the economy. In order to realize the goal, we have to control pollutants emission from thermal power plants while satisfying the requirement. It is the environmental economic dispatch problem. The problem is to minimize the operating and emission cost while meeting a number of constraints, such as the system load, the minimal and maximal output of each generator, and so on.

The problem can be mathematically demonstrated as follows:

$MinF = \sum_{i = 1}^{N_{g}} F_{i} (P_{G_{i}}) = \sum_{i = 1}^{N_{g}} a_{i} P_{G_{i}}^{2} + b_{i} P_{G_{i}} + c_{i}$ (14)

Subject to:

The energy balance between generated power and demand power:

$\sum_{i = 1}^{N_{g}} P_{i} = P_{D}$ (15)

The boundary constraints of each generator unit:

$P_{i}^{\min} ⩽ P_{i} ⩽ P_{i}^{\max}$ (16) where F is the objective function, N_g is the numr of the generators, P_i is the generated power of ith generator, $P_{i}^{\min}$ and $P_{i}^{\max}$ are the lower and upper boundaries of the ith generator, a_i, b_i, c_i are the coefficients of the ith generator. B₀₀, B_i0, B_ij are the B-coefficients.

Table 5

The range of each generated power

P _i	Lower bound	Uer bound
P ₁	10
P ₂	10
P ₃	35	225
P ₄	35	210
P ₅	130	3
P ₆	125	3

Fig. 5

The convergence curve of five algorithms.

Fig. 6

The maximum, minimum and mean results of five algorithms when fuel load is 500 MW.

The problem is a typical non-linear, non-convex and non-smooth optimization problem, which cannot be effectively solved by the conventional

Table 6

The solutions obtained from five algorithms when fuel load is 500 MW

Algorithm	P ₁	P ₂	P ₃	P ₄	P ₅	P ₆
GWO	12.8984	11.0471	65.6530	66.7825	183.6865	159.8398
Jaya	30.0221	10.0000	71.7560	35.0000	195.0054	158.1594
IJaya	17.9839	10.5119	57.9378	86.5017	147.8382	179.1358
PGJaya	16.3369	11.8159	56.7485	76.7873	175.5568	162.6550
MJaya	17.4646	10.8994	47.4080	83.7412	181.4542	158.9689

Table 7

The solutions obtained from five algorithms when fuel load is 1100 MW

Algorithm	P ₁	P ₂	P ₃	P ₄	P ₅	P ₆
GWO	26.0177	52.7803	195.2975	186.7239	325.0000	314.1001
Jaya	25.9806	10.0000	225.0000	210.0000	325.0000	303.9491
IJaya	47.6510	60.9999	184.7496	200.3785	322.0930	284.1305
PGJaya	53.9863	31.7014	218.1481	161.7012	319.4541	314.9741
MJaya	40.9188	24.4491	215.7747	187.3324	316.9542	314.4727

mathematical methods, such as the derivative-based optimization method. At this moment, we have to turn to computational intelligence and meta-heuristic algorithms. Therefore, we use the above five algorithm i.e., GWO, Jaya, IJaya, PGJaya, and MJaya to optimize the problem. The maximal function evaluations are 300 000. The range of each generated power is listed in Table 4.

We set the fuel load to 500 and 1100, respectively. The stistical results of the five algorithms are plotted in Figs. 5 6. When the demand is 500 MW, thPGJaya algorithm has obtained the best results in terms of minimum and mean values while MJaya has obtained the superior performance in terms of maximum and minimum. When the dand is 1100 MW, PGJaya, GWO, and MJaya have obtained the best maximum, minimum, anmean values. The corresponding solutions obtained by five algorithms are listed in Tables 5, 6. In all, the proposed MJaya and PGJaya have ranked top two in terms of statistical results.

4.2.3 Parameters identification of photovoltaic models

Nowadays, the globe has become warmer and warmer, which is one of the critical challenges to the sustainable development of the world. New energy resources have become alternatives to take the place of traditional energy, in which solar energy is a promising one as it is clean, cheap, and inexhaustible. Solar energy is transformed into electrical energy by photovoltaic systems. It is necessary to measure the efficiency of photovoltaic systems. We need some mathematical models to evaluate the behavior of the system, in which the single anudde models are commonly used [7]. The single diode model is as follows:

${\begin{matrix} f = I_{ph} - I_{sd} \cdot [exp (\frac{q \cdot (V_{L} + R_{S} \cdot I_{L})}{n \cdot k \cdot T}) - 1] - \frac{V_{L} + R_{S} \cdot I_{L}}{R_{sh}} - I_{L} \\ x = {I_{ph}, I_{sd}, R_{S}, R_{Sh}, n} \end{matrix}$ (17) where I_ph is the total current from the solar cell, I_sd is the reverse current of diode, q and k are constants, V_L is the output voltage of the cell, R_S is the series resistance, R_sh is the shunt resistance, I_L is the output current of the solar cell, T is the temperature of the cell. There are five parameters x ={ I_ph, I_sd, R_S, R_Sh, n }, which are needed to be optimized. The double diode model is as follows:

${\begin{matrix} f = I_{ph} - I_{sd 1} \cdot [exp (\frac{q \cdot (V_{L} + R_{S} \cdot I_{L})}{n_{1} \cdot k \cdot T}) - 1] - I_{sd 2} \cdot [exp (\frac{q \cdot (V_{L} + R_{S} \cdot I_{L})}{n_{2} \cdot k \cdot T}) - 1] - \frac{V_{L} + R_{S} \cdot I_{L}}{R_{sh}} - I_{L} \\ x = {I_{ph}, I_{sd 1}, I_{sd 2}, R_{S}, R_{Sh}, n_{1}, n_{2}} \end{matrix}$ (18) where I_sd1 is the diffusion current, I_sd2 is the sation current, n₁ is the diffusion factor, and n₂ is the recombination factor. ere are seven parameters x ={ I_ph, I_sd1, I_sd2, R_S, R_Sh, n₁, n₂ }, which need to be optized.

In order to identify these parameters, two problems can be converted into an optimization problem by using the root mean square error (RMSE) as the jective function [7]. The function is to quantify the

Table 8

The search range of decision variables

Parameter	Lower bound	Upper bound
I_ph (A)	0	1
I_sd, I_sd1, I_sd2 (μA)	0	1
R_S (Ω)	0	0.5
R_sh (Ω)	0	100
n, n₁, n₂	1	2

Table 9

The mean and std values of five algorithms for two models

Algorithm	Single diode model		Double diode model
Mean	Std	Mean	Std
GWO	5.1E-3	4.8585E-5	8.0 E-3	1.2920E-4
Jaya	1.6E-3	2.8550E-8	2.3E-3	2.8586E-7
IJaya	1.0E-3	5.7390E-10	1.10E-3	8.1421E-9
PGJaya	9.8609E-4	1.3646E-14	9.8634E-4	4.0200E-12
MJaya	9.8608E-4	1.4382E-14	9.9511E-4	1.9508E-10

difference between the simulated and experimental data as follows:

$RMSE (x) = \sqrt{\frac{1}{N} \sum_{k = 1}^{N} f_{k} {(V_{L}, I_{L}, x)}^{2}}$ (19) where x is the decision variable, N is the amount of data.

The proposed MJaya algorithm, GWO, Jaya, IJaya, and PGJaya are used to optimize this problem. The data needed in the experiment are from ref [33]. The boundaries of decision variables are listed in Table 7, q = 1.60217646 × 10^-19C and k = 1.3806503 × 10^-23J/K. The maximal function evaluation is 50 000 and the populatiosize is 50. After running 50 times for each algorithm on the single and double diode models, theean and standard deviation of five algorithms on double and single diodes are listed in Table 8. It can be observed that GWO and Jaya have achieved the worst performance as the structure of e twgorithms is simple and the evolution strategy is sole. On the contrary, the proposed MJaya algorithm is much better and PGJaya ranks second. The proposed MJaya algorithm adopts multi-strategy and multi-population to reinforce the search capacity, which significantly enhances the results. The convergence curve of the above five algorithms is also given in Figs. 7 8,. The value of the y-axis is log (RMSE) function, which is used to different the algorithms used. It can be found that the convergence speed of the proposed MJaya algorithm is comparatively faster. The finding further indicates that the algorithm is effective.

Fig. 7

The maximum, minimum and mean results of five algorithms when fuel load is 1100 MW.

Fig. 8

The convergence curves of five algorithms on a single, double diode model.

For the three industrial applications, the proposed MGJaya algorithm has achieved superior results. The main reasons behind the results are the multi-strategy and multi-population. Three strategies are performed during the evolution. The first strategy introduces an adaptive weight parameter for the positing-updating equation for the first population, which can boost the exploitation. The second strategy uses rank-based mutation, which can reinforce the exploration. The third strategy adopts chaotic techniques based on the best solution, which can enhance the quantity of population. The three strategies play different roles during the evolution and cooperate well. Hence, the very good results can be achieved by the proposed MJaya algorithm.

5 Conclusions

Jaya is a very simple and promising meta-heuristic algorithm as it is parameter-free. The simple structure of the algorithm may result in poor performance when solving complex problems. In this paper, a novel algorithm, MJaya is developed to cope with the problem by using multi-strategies and multi-population. The first strategy is used to boost the exploitation by introducing an adaptive weight parameter for the original position-updating equation. The rank-based mutation is the second strategy, which can boost the exploration. The adaptive chaotic technique is the third one, which pushes the population search around the best solution and enhances the exploitation.

The proposed MJaya algorithm is tested on 30 CEC 2014 functions. Its opponents are GWO, Jaya, IJaya, and PGJaya algorithms. The experiments based on benchmark testing functions have revealed that the proposed MJaya algorithm is competitive compared with its opponents. Then, the five algorithms are used to solve three typical industrial problems. The experimental results have proven that the proposed MJaya algorithm is superior. In the next research, we will employ the proposed MJaya algorithm to solve more complex industrial problems, such as supply chain models [34].

Footnotes

Acknowledgments

The authors would like to thank professor R. Venkata Rao, who proposed Jaya algorithm, for his very constructive comments and suggestions. This research was funded by the China Natural Science Foundation (No. 71974100), Major Project of Philosophy and Social Science Research in Colleges and Universities in Jiangsu province (2019SJZDA039), and Natural Science Foundation in Jiangsu Province (No. BK20191402) and Higher school in jiangsu province college students' practice innovation training programs (XJDC202110300130).

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Multi-strategy Jaya algorithm for industrial optimization tasks

Abstract

Keywords

1 Introduction

2 Jaya algorithm

3.1 The adaptive weight parameter

3.5 The computational complexity

4.1 The experimental results on the CEC 2014 testing suite

Table 4 The results of five algorithms on paramete determination issue Algorithm Maximum Minimum an GWO 25.0980 2.0290 16.3643(+) JAYA 31.0140 16.9546 27.1680(+) IJAYA 26.8423 13.1882 17.4844(+) PGJAYA 27.5148 10.5134 18.1001(+) MJAYA 15.9785 1.5672E-7 3.1656

Footnotes

Acknowledgments

References

Table 4
The results of five algorithms on paramete determination issue

Algorithm Maximum Minimum an

GWO 25.0980 2.0290 16.3643(+)

JAYA 31.0140 16.9546 27.1680(+)

IJAYA 26.8423 13.1882 17.4844(+)

PGJAYA 27.5148 10.5134 18.1001(+)

MJAYA 15.9785 1.5672E-7 3.1656