An improved Jaya optimization algorithm with ring topology and population size reduction

1 Introduction

Computational intelligence optimization is an active field of research that, over the years, has proposed a plethora of successful techniques variously inspired to biological or physical phenomena such as swarm intelligence (SI) and evolutionary computation (EC) [1]. Some recent noteworthy examples of metaphor-inspired techniques include, but are not limited to:

1. novel bio-inspired algorithms, e.g., the Harris hawks optimization algorithm [2], the slime mould algorithm [3], the Hunger Games Search [4], monarch butterfly optimization [5], the moth search algorithm [6], or the iterative topographical global optimization algorithm [7];

2. variations of existing techniques, such as variants of the flower pollination algorithm [8], differential evolution [9], fruit fly optimization algorithm [10], or backtracking search algorithm [11,12];

3. hybrid algorithms obtained by combining, for instance ant colony optimization with chaotic sequences [13], particle swarm optimization (PSO) with local solvers [14], or the bat algorithm with extremal optimization [15].

While metaphor-based algorithms have shown successful results on a broad range of problems and massively increased the popularity of this field, there is a growing consensus in the research community on the fact that nature-inspired metaphors should not be considered anymore as the only way to explain new algorithms [16]. In fact, even though such metaphors may provide very intuitive analogies to describe the concepts behind metaheuristics, some researchers now believe that using metaphor-free descriptions should be preferred, especially when addressing practitioners or researchers from other fields.

A current trend is therefore to design metaphor-less algorithms that, rather than being based on (sometimes complex) nature-inspired analogies, are built on the principle of simplicity [17], also referred to as the Ockham’s razor (“entities are not to be multiplied without necessity”^[1]) applied to algorithmic design [18]. It is important to note however that, while the simplicity of design can be seen as a kind of “universal goal” that can applied to various algorithmic aspects, this property has also an inherently subjective nature. Therefore, here, we will focus on two quantifiable aspects of simplicity: (1) the ease of implementation (e.g., measured in terms of number of mathematical operations), and (2) the use of as few algorithmic-specific parameters as possible, ideally none. These two features are intuitively useful per se, but also from the perspective of both algorithmic designers and practitioners.

From an algorithmic designers’ point of view, the design of purposely simple (and, possibly metaphor-less) algorithms can be indeed an easier task. Moreover, studying the behavior of these algorithms, hybridizing them with other techniques, or modifying some of their inner principles to improve the performance, can also be facilitated.

From the practitioners’ perspective, these algorithms are attractive since they do not require a strong background in optimization, and thus, they can be applied easily to different problems in domains that are far away from computer science and applied mathematics. Moreover, practitioners may find easier to adapt, if needed, these algorithms, e.g., by incorporating domain knowledge. Or, they may find easier to port the code of the optimizer to a specific hardware platform, operating system, programming language, etc. Another practical consequence is that these algorithms may use less data structures, thus less memory, and simpler mathematical operations, hence being suitable also for hardware-constrained devices such as embedded systems [19,20] and wireless sensor networks [21]. As for the use of few or no parameters, this is desirable in that it relieves the user from a possibly time-demanding parameter tuning.

The two aforementioned features are the reasons for the popularity of a population-based algorithm such as DE [22], which in the original form uses only two algorithm-specific parameters (the scale factor and crossover rate) and has an extremely simple implementation. Another well-known example of this kind of algorithms can be considered simulated annealing [23], along with some of its variants such as that proposed in ref. [24]. Further instances are single particle algorithms [25,26,27], although these tend to have limited performances, especially on large-scale problems. The second issue that quite often affects these algorithms is premature convergence, i.e., they tend to quickly lose diversity [28] and obtain stuck in local optima. This is, probably, the main drawback that comes with simplicity, such that one may say that, to some extent, there is a trade-off between simplicity and performance.

One algorithm that follows this design philosophy aiming at simplicity, and that has recently attracted a growing interest, is Jaya [29]. Jaya is a population-based optimization algorithm whose main design principle is to use as few algorithm-specific parameters as possible. In particular, it only requires two parameters, i.e., the population size and the maximum number of generations, and it is extremely easy to implement. Despite these merits, Jaya is known to suffer from premature convergence, as shown, e.g., in refs [30,31]. Several modifications have been proposed to overcome this issue, but, as we will discuss later, most of these introduce various algorithmic components, more or less sophisticated, that somehow contradict the original simplicity of Jaya.

2 Background: Jaya and its variants

Similar to other metaheuristics such as PSO [33] and DE, Jaya operates on a population of candidate solutions, each one assimilated to a particle with a given position in the search space, determined by the solution’s variables. The algorithm proceeds iteratively by updating these positions to obtain them to the most promising regions of the search space, i.e., those characterized by better objective function values. In essence, the rationale of the Jaya search process is to move at each iteration (generation) each candidate solution far away from the worst solution in the current population, and closer to the best one.^[2] This logic is controlled by the following equation:

where x i , j is the value of the j th variable of the i th particle, x best , j and x worst , j are, respectively, the value of the j th variable of the best and worst solutions in the current population, r 1 and r 2 are two random numbers sampled uniformly in [ 0 , 1 ] , and x i , j ′ is the j th variable of the new solution to be evaluated. Of note, after the application of equation (2), possibly out-of-bounds variables are corrected by saturating them at their respective lower and upper bounds, l j and u j . While various works [34,35] have shown that this correction mechanism can introduce a bias in an evolutionary search process, here (and in our proposal, described in the next section) we stick to this mechanism, as it is used in the original Jaya implementation [29].

After the solution update operation, in case of improvement, the new solution, x i ′ , replaces the old one, x i . This process is applied to each solution in the current population and is continued for a maximum number of generations.

For completeness, the pseudo-code of Jaya is given in Algorithm 1, where it can be seen that the algorithm does not require any other parameter except the population size ( P ) and the maximum number of generations ( G ). Furthermore, its implementation in terms of SLOC (source lines of code) is rather straightforward.

As discussed earlier, these two advantages have recently made Jaya quite popular in the field of numerical optimization, as also shown in various studies on engineering applications [36,37], urban traffic light scheduling [38], image classification [39,40], and binary problems [41]. Nevertheless, several works [30,31] have collected evidence on the fact that this algorithm has an inherent tendency to premature convergence. Therefore, various improved Jaya variants have been proposed in the recent literature, in the attempt to alleviate this issue. In ref. [42], a self-adaptive multi-population Jaya variant (SAMP-Jaya) was proposed, which uses a relatively complex mechanism for dividing the population into multiple sub-populations based on the quality of the solutions. This algorithm was later extended in ref. [43] to include elitism. An even more complex multi-population Jaya was proposed lately in ref. [44], where the number of sub-populations is adapted at runtime based on a convergence criterion, and each sub-population uses a different perturbation mechanism (in addition to equation (2)), the worst of which is eventually updated. A slightly simpler approach was instead proposed in ref. [30], where a Jaya variant uses three modified versions of equation (2) where the two uniform random numbers r 1 and r 2 are replaced by numbers generated by a chaotic map. In ref. [45], Cellular Automata (CA) [46] was used to organize the solutions as a tripodal mesh. Each solution has a number of neighbors determined by the topological structure used; x best and x worst are chosen from the solution’s neighbors to avoid premature convergence. The approach generally outperforms Jaya on a set of relatively simple functions. Finally, a Lévy Flight Jaya algorithm (LJA) was recently introduced in ref. [31], which replaces r 1 and r 2 with random numbers sampled from the Lévy distribution to overcome premature convergence by allowing the algorithm to perform occasional “jumps”: notably, this is one of the few recent Jaya variants that do not really add complexity to the original algorithm. In fact, several other and much more complex variants exist; however, reviewing all of them is out of the scope of this article. We refer the interested reader to the recent survey [47].

3 The proposed approach: Jaya2

As seen earlier, Jaya is easy to implement and requires no algorithm-specific parameters, besides the population size P and the maximum number of generations G . However, the performance of Jaya suffers from premature convergence. The main reason for this behavior is the use of equation (2), which guides each solution to move toward the best solution in the population and move away from the worst one. This behavior is similar to that of the gbest PSO [48], where all solutions are directly connected to the best solution in the population (i.e., a star topology). The star topology (Figure 1) allows each solution in the population to share information globally. In a global neighborhood, information is instantaneously disseminated to all solutions, quickly attracting the entire population to the same search region with the high risk of premature convergence. To tackle this problem, the ring topology (Figure 2) makes use of local neighborhoods like the one used in the lbest PSO [48], i.e., each particle communicates only with its immediate neighbors. Therefore, the population needs more iterations to converge, but better solutions can be found. For example, Figure 2 shows that solution x 1 has two neighbors, x 2 and x 6 , while solution x 4 has x 3 and x 5 as its neighbors. Thus, each solution interacts with its two neighbors (rather than with the whole population). Hence, the ring lattice, known as lbest, is an indirect communication pattern where the best solution and worst solution, respectively, x best and x worst , are chosen from the adjacent neighbors of each solution (i.e., for x i , x best , and x worst are chosen from { x i − 1 , x i , x i + 1 } ). This way we purposely “slow down” the convergence speed of Jaya, but reduce the possibility of premature convergence. In fact, the ring splits the population into many small-sized neighborhoods (of size 3), such that the exchange of information occurs only within each neighborhood, rather than at the global level. So, while in the star topology (i.e., a fully connected topology with one single neighborhood encompassing the entire population), the effect of biasing the search toward the global best would lead to a greedy/exploitative approach, in the ring topology, exploration is favored over exploitation, and hence, in this sense, this topology “slows down” the convergence, the advantage being avoiding premature convergence. This advantage has made the use of the ring topology quite popular in the metaheuristics research. Recently, Lynn et al. [49] identified the use of population topologies, including the ring one, as one of the main aspects to focus research on. Some early attempts at using the ring topology beyond PSO have been made in refs [50,51], where ring neighborhoods were used in self-adaptive DE schemes, and in ref. [52], where the ring topology was used in a hybrid PSO-DE algorithm called “bare bones” DE. In all cases, the ring topology was proved to produce performance improvements in terms of optimization results. Yet, to the best of our knowledge, the ring topology has never been applied to Jaya.

Figure 1

A star topology with six solutions.

Figure 2

A ring topology with six solutions.

Another modification we introduce here is the use of a linearly decreasing population size. This mechanism has been used to improve the performance of many metaheuristics, see, e.g., [53,54], since it allows a stronger exploration pressure at the early stages of the optimization process, and a stronger exploitation behavior as long as the process advances. Hence, to improve the performance of Jaya, we reduce the population size linearly as a function of the number of objective function evaluations, as in L-SHADE [54]. More specifically, after each generation g = 1 , 2 , … , G − 1 , the new population size P g + 1 is defined as follows:

where P max indicates the initial population size and P min indicates the population size at the last generation. The latter is set to 3, which is the smallest possible value for Jaya (given that equation (2) requires three solutions). n f e is the current number of objective function evaluations, and n f e max is the maximum number of objective function evaluations. Whenever it occurs that P g + 1 < P g , the population is sorted and the worst ( P g − P g + 1 ) are discarded.

One last modification is related to the absolute value operator in equation (2). In fact, the use of the absolute value operator makes Jaya not translation independent, i.e., the moves and the final solution are dependent on the coordinate system. To show this, we use the approach proposed by ref. [55], where the optimization algorithm is tested on the two identical landscapes of the following functions:

1. Function 1: f ( x ) = x 2 , x ∈ [ − 100 , 100 ] .

2. Function 2: f ( x ) = ( x + 100 ) 2 , x ∈ [ − 200 , 0 ] .

We run Jaya under the following conditions [55]:

1. population size = 25;

2. number of runs = 15; and

3. number of iterations/run = 5.

On Function 1, the average best solution found by Jaya is 0.1300263, while on Function 2, it is only 1.686978.

To fix this problem, we propose to replace equation (2) with the following equation:

Now when testing Jaya using equation (4) (instead of equation (2)) on Function 1 and Function 2, we obtain the same result (namely, 0.2853698). Thus, the translation dependency problem of Jaya has been removed.

The new proposed variant is called Jaya2, and its detailed pseudo-code is listed in Algorithm 2. A flowchart describing the Jaya2 algorithm is shown in Figure 3.

Figure 3

Flowchart of the Jaya2 algorithm.

4 Experimental results

Two different sets of problems, namely, the 10 benchmark functions used at the CEC 2020 competition on single-objective bound-constrained numerical optimization [56] and the 22 real-world optimization problems from the CEC 2011 competition on testing evolutionary algorithms on real-world optimization problems [57], have been used in the experiments.

Table 1 summarizes the 10 CEC 2020 benchmark functions. The first four functions are shifted and rotated well-known functions. The next three functions are hybrid, which means they are linear combinations of several basic functions. Finally, the last three functions are compositions, i.e., couplings between the previous functions considered in different parts of their optimization domains. All functions are scalable, which means that they can be computed for different numbers of problem dimensions D . Accordingly, the search range for all functions is [ − 100 , 100 ] D . In our experiments, we set D to 10. The mathematical description of each function can be found in ref. [56].

Table 2 summarizes the 22 CEC 2011 real-world problems. These include various engineering problems, such as optimization of chemical systems, optimal design of TLC devices, optimal control, and spacecraft trajectory optimization. In this case, the number of problem dimensions depends on the specific problem, ranging from 1 ( T 3 , T 4 ) to 215 ( T 11.2 ), and apart from a few cases, the function type and its optimum are in general unknown. We should note that we did not use T 3 in our experiments due to its large computing time. However, this exclusion does not affect the overall results of our comparative analysis reported next. Furthermore, as detailed in Table 2, some problems include equality and/or inequality constraints other than the bound constraints. These were handled with a static penalty function proportionate to the sum of the constraint violations, as in the original Matlab formulation provided by the CEC 2011 competition organizers. For more details, the interested reader is referred to ref. [57].

We have chosen these two benchmark sets for the following reasons:

1. The CEC 2020 benchmark offers a broad set of hard-to-solve optimization functions with different characteristics (e.g., uni-modal, multi-modal), which helps examining the performance of the proposed method on different types of problems.

2. The CEC 2011 benchmark, on the other hand, allows to assess the applicability of the proposed method in real-world problems from diverse fields.

The numerical experiments were performed on an HP Desktop with 3.6 GHz Intel Core i 7, 32 GB RAM, running Matlab R2017b on MS Windows 10 Pro.

The detailed statistics of Jaya2, in terms of best, median, mean, standard deviation (std.), and worst objective function values (across the best function values obtained at the end of the allotted budget in all the available runs for each problem) on the CEC 2020 benchmark functions and the CEC 2011 real-world problems are reported in Tables 3 and 4, respectively.

For the comparative analysis presented in the rest of this section, we used the pairwise Wilcoxon signed rank test [58] with a 5% significance level, to verify the existence of statistical differences between the competing approaches. Given that the Wilcoxon test confronts the median values of two samples, in Tables 5–21, reported next, we show the comparative results in terms of median function values (also in this case across the best function values obtained at the end of the allotted budget in all the available runs of each algorithm for each problem), along with the corresponding p -values obtained with the Wilcoxon test, which represent the probability of two selected samples of algorithm results being statistically equivalent ( α = 0.05 ). We should remark that, for each comparison, Jaya2 was separately rerun: this explains the small fluctuations on the median function values shown in Tables 3–21, which anyway do not affect the conclusions drawn on the comparative analysis.

In what follows, first, we present the comparison on both testbeds between the proposed Jaya2 algorithm and both the original Jaya and its recent variant LJA. Then, we present the comparison between Jaya2 and three alternative few-parameter approaches from the literature. This is followed by a comparison with three complex, state-of-the-art algorithms. Finally, we conclude with an analysis of the contribution to the performance of each algorithmic element introduced in Jaya2.

4.1 Jaya2 vs the original Jaya and LJA

In this section, Jaya2 is compared against the original Jaya and the recently proposed Jaya variant LJA [31]. We have chosen LJA because it is one of the most recently proposed Jaya variant and because, contrary to the other Jaya variants discussed in Section 2, it kept the simplicity of Jaya almost intact. It only introduces one parameter for the Lévy distribution, i.e., β . Moreover, ref. [31] shows that LJA generally outperforms Jaya on the 30 CEC 2014 functions. We have used a population size of 30 for Jaya and LJA, while the initial population size of Jaya2 ( P max ) was set to 100. For LJA, β was set to 1.8 as recommended in ref. [31].

The median function values of the objective functions of Jaya2, Jaya, and LJA on the 10 functions of CEC 2020 are reported in Table 5. Table 6 shows the results of the statistical analysis based on the Wilcoxon test. The results of the two tables show that Jaya2 outperformed Jaya and LJA on almost all the functions.

Table 5

The CEC 2020 ( D = 10 ) median function values achieved by Jaya2, Jaya, and LJA over 30 runs and 100,000 function evaluations

Function	Jaya2	Jaya	LJA
f 1	4.63 × 10 2	1.32 × 1 0 8	9.86 × 1 0 3
f 2	1.12 × 10 3	2.20 × 1 0 3	2.09 × 1 0 3
f 3	7.14 × 10 2	7.52 × 1 0 2	7.36 × 1 0 2
f 4	1.90 × 10 3	1.90 × 10 3	1.90 × 10 3
f 5	2.15 × 10 3	1.19 × 1 0 4	4.23 × 1 0 3
f 6	1.60 × 10 3	1.60 × 10 3	1.60 × 10 3
f 7	2.11 × 10 3	3.05 × 1 0 3	2.43 × 1 0 3
f 8	2.30 × 10 3	2.32 × 1 0 3	2.30 × 10 3
f 9	2.73 × 10 3	2.77 × 1 0 3	2.76 × 1 0 3
f 10	2.90 × 10 3	2.95 × 1 0 3	2.90 × 10 3

The boldface indicates the lowest median function value per function.

Table 6

The CEC 2020 ( D = 10 ) statistical analysis showing the Wilcoxon test p -value

Function	Jaya	LJA
f 1	1.73 × 1 0 − 6 (+)	1.92 × 1 0 − 6 (+)
f 2	1.73 × 1 0 − 6 (+)	1.73 × 1 0 − 6 (+)
f 3	1.73 × 1 0 − 6 (+)	1.73 × 1 0 − 6 (+)
f 4	1.73 × 1 0 − 6 (+)	1.73 × 1 0 − 6 (+)
f 5	1.73 × 1 0 − 6 (+)	2.35 × 1 0 − 6 (+)
f 6	5.75 × 1 0 − 6 (+)	8.47 × 1 0 − 6 (+)
f 7	1.73 × 1 0 − 6 (+)	1.73 × 1 0 − 6 (+)
f 8	3.11 × 1 0 − 5 (+)	1.73 × 1 0 − 6 (+)
f 9	1.73 × 1 0 − 6 (+)	1.73 × 1 0 − 6 (+)
f 10	1.02 × 1 0 − 5 (+)	1.63 × 1 0 − 1 (=)
Wins	10	9
Draws	0	1
Loses	0	0

Each p -value is followed by a “ + ” if Jaya2 is significantly better than the competing algorithm, “ − ” if significantly worse, or “ = ” if statistically equivalent.

Figure 4 shows the convergence behavior on six selected CEC 2020 benchmark functions (i.e., the best function value found at each step of the algorithm, averaged across 30 runs) for the three Jaya-based approaches. The figure shows that Jaya2 generally reaches better solutions faster than the two other approaches. This suggests that Jaya2 has a good balance between exploration and exploitation, focusing on exploration at the beginning of the run and then switching to exploitation. Figure 5 shows the average diversity (across 30 runs) of the population during the progress of the three Jaya algorithms on the same six functions shown in Figure 4. The diversity of the population is measured as in ref. [59], namely:

(7) DI = 1 P ∑ i = 1 P ∑ j = 1 D ( x i , j − x ¯ j ) 2 ,

where x ¯ j is the j th component of the mean vector of the solutions in the population. The figure shows that Jaya2 tends to start with a high diversity, while toward the end of the run, the diversity drops rapidly, allowing for exploitation of good solutions. This behavior is due to a combination of the effects of the ring topology and the linearly decreasing population size. On the contrary, both Jaya and LJA do not make use of these mechanisms; thus, they keep large distances between solutions over all generations, which prevents them from exploiting good regions in the search space.

Figure 4

Best function value curves (averaged across 30 runs) of Jaya2, Jaya, and LJA for selected CEC 2020 benchmark functions ( D = 10 ).

Figure 5

Diversity curves (averaged across 30 runs) of Jaya2, Jaya, and LJA for selected CEC 2020 benchmark functions ( D = 10 ).

The three Jaya-based approaches were then tested on the CEC 2011 real-world problems. The results, summarized in Tables 7 and 8, confirm the superiority of Jaya2 compared to Jaya and LJA. The only two problems where Jaya managed to statistically outperform Jaya2 were T 8 and T 11.4 . LJA performed better than Jaya2 on only T 11.4 . Figures 6 and 7 depict the convergence behavior and the diversity of the three approaches. The figures are generally similar to Figures 4 and 5; thus, similar conclusions can be drawn.

Table 7

The CEC 2011 median function values achieved by Jaya2, Jaya, and LJA over 25 runs and 150,000 function evaluations

Problem	Jaya2	Jaya	LJA
T 1	0.00 × 10 0	1.79 × 1 0 1	1.72 × 1 0 1
T 2	-2.65 × 10 1	− 8.07 × 1 0 0	− 6.39 × 1 0 0
T 4	1.38 × 10 1	1.39 × 1 0 1	1.39 × 1 0 1
T 5	− 3.15 × 10 1	− 2.01 × 1 0 1	− 2.16 × 1 0 1
T 6	− 2.30 × 10 1	− 1.44 × 1 0 1	− 1.53 × 1 0 1
T 7	1.70 × 1 0 0	1.72 × 1 0 0	1.67 × 10 0
T 8	2.20 × 10 2	2.20 × 10 2	2.20 × 10 2
T 9	1.56 × 10 3	1.79 × 1 0 4	6.98 × 1 0 3
T 10	− 2.14 × 10 1	− 1.07 × 1 0 1	− 1.14 × 1 0 1
T 11.1	5.25 × 1 0 4	5.23 × 10 4	6.28 × 1 0 4
T 11.2	1.76 × 1 0 7	1.76 × 1 0 7	1.75 × 10 7
T 11.3	1.55 × 10 4	1.55 × 10 4	1.55 × 10 4
T 11.4	1.93 × 1 0 4	1.91 × 10 4	1.91 × 10 4
T 11.5	3.28 × 10 4	3.28 × 10 4	3.28 × 10 4
T 11.6	1.35 × 1 0 5	1.33 × 10 5	1.33 × 10 5
T 11.7	1.99 × 10 6	1.99 × 10 6	2.05 × 1 0 6
T 11.8	1.01 × 10 6	1.38 × 1 0 6	1.53 × 1 0 6
T 11.9	1.44 × 10 6	1.49 × 1 0 6	1.85 × 1 0 6
T 11.10	1.01 × 10 6	1.38 × 1 0 6	1.53 × 1 0 6
T 12	1.54 × 10 1	2.93 × 1 0 1	3.07 × 1 0 1
T 13	1.93 × 10 1	2.88 × 1 0 1	2.96 × 1 0 1

The boldface indicates the lowest median function value per problem.

Table 8

The CEC 2011 statistical analysis showing the Wilcoxon test p -value

Problem	Jaya	LJA
T 1	3.22 × 1 0 − 5 (+)	2.26 × 1 0 − 5 (+)
T 2	1.23 × 1 0 − 5 (+)	1.23 × 1 0 − 5 (+)
T 4	1.91 × 1 0 − 2 (+)	3.97 × 1 0 − 1 (=)
T 5	1.23 × 1 0 − 5 (+)	1.23 × 1 0 − 5 (+)
T 6	1.23 × 1 0 − 5 (+)	1.23 × 1 0 − 5 (+)
T 7	3.26 × 1 0 − 1 (=)	7.78 × 1 0 − 1 (=)
T 8	3.13 × 1 0 − 2 ( − )	1.00 × 1 0 0 (=)
T 9	2.00 × 1 0 − 5 (+)	1.23 × 1 0 − 5 (+)
T 10	1.23 × 1 0 − 5 (+)	1.23 × 1 0 − 5 (+)
T 11.1	6.77 × 1 0 − 1 (=)	1.23 × 1 0 − 5 (+)
T 11.2	7.36 × 1 0 − 2 (=)	3.00 × 1 0 − 1 (=)
T 11.3	3.22 × 1 0 − 5 (+)	3.28 × 1 0 − 4 (+)
T 11.4	8.08 × 1 0 − 4 ( − )	1.08 × 1 0 − 3 ( − )
T 11.5	8.27 × 1 0 − 2 (=)	1.83 × 1 0 − 1 (=)
T 11.6	4.43 × 1 0 − 1 (=)	2.42 × 1 0 − 1 (=)
T 11.7	3.82 × 1 0 − 1 (=)	3.96 × 1 0 − 2 (+)
T 11.8	5.13 × 1 0 − 5 (+)	1.23 × 1 0 − 5 (+)
T 11.9	1.58 × 1 0 − 1 (=)	1.26 × 1 0 − 4 (+)
T 11.10	5.13 × 1 0 − 5 (+)	1.23 × 1 0 − 5 (+)
T 12	1.39 × 1 0 − 5 (+)	1.23 × 1 0 − 5 (+)
T 13	1.57 × 1 0 − 5 (+)	1.57 × 1 0 − 5 (+)
Wins	12	14
Draws	7	6
Loses	2	1

Each p -value is followed by a “ + ” if Jaya2 is significantly better than the competing algorithm, “ − ” if significantly worse, or “ = ” if statistically equivalent.

Figure 6

Best function value curves (averaged across 25 runs) of Jaya2, Jaya, and LJA for selected CEC 2011 real-world problems.

Figure 7

Diversity curves (averaged across 25 runs) of Jaya2, Jaya, and LJA for selected CEC 2011 real-world problems.