![20
algorithm, Gravitational Search Algorithm (GSA), Biogeography-Based Optimization (BBO), Grenade
Explosion Method (GEM), etc. (Rao and Patel, 2012, 2013).
All the evolutionary and swarm intelligence based algorithms are probabilistic algorithms and require
common controlling parameters like population size, number of generations, elite size, etc. Besides the
common control parameters, different algorithms require their own algorithm-specific control
parameters. For example, GA uses mutation probability, crossover probability, selection operator; PSO
uses inertia weight, social and cognitive parameters; ABC uses number of onlooker bees, employed bees,
scout bees and limit; HS algorithm uses harmony memory consideration rate, pitch adjusting rate, and
the number of improvisations. Similarly, the other algorithms such as ES, EP, DE, BFO, AIA, SFL, ACO,
etc. need the tuning of respective algorithm-specific parameters. The proper tuning of the algorithm-
specific parameters is a very crucial factor which affects the performance of the above mentioned
algorithms. The improper tuning of algorithm-specific parameters either increases the computational
effort or yields the local optimal solution. Considering this fact, Rao et al. (2011) introduced the teaching-
learning-based optimization (TLBO) algorithm which does not require any algorithm-specific
parameters. The TLBO algorithm requires only common controlling parameters like population size and
number of generations for its working. The TLBO algorithm has gained wide acceptance among the
optimization researchers (Rao, 2015).
Keeping in view of the success of the TLBO algorithm, another algorithm-specific parameter-less
algorithm is proposed in this paper. However, unlike two phases (i.e. teacher phase and the learner phase)
of the TLBO algorithm, the proposed algorithm has only one phase and it is comparatively simpler to
apply. The working of the proposed algorithm is much different from that of the TLBO algorithm. The
next section describes the proposed algorithm.
2. Proposed algorithm
Let f(x) is the objective function to be minimized (or maximized). At any iteration i, assume that there
are ‘m’ number of design variables (i.e. j=1,2,…,m), ‘n’ number of candidate solutions (i.e. population
size, k=1,2,…,n). Let the best candidate best obtains the best value of f(x) (i.e. f(x)best) in the entire
candidate solutions and the worst candidate worst obtains the worst value of f(x) (i.e. f(x)worst) in the entire
candidate solutions. If Xj,k,iis the value of the jth
variable for the kth
candidate during the ith
iteration, then
this value is modified as per the following Eq. (1).
X'j,k,i= Xj,k,i+ r1,j,i (Xj,best,i- │Xj,k,i│) - r2,j,i (Xj,worst,i- │Xj,k,i│) , (1)
where, Xj,best,iis the value of the variable j for the best candidate and Xj,worst,iis the value of the variable j
for the worst candidate. X'j,k,iis the updated value of Xj,k,i and r1,j,i and r2,j,i are the two random numbers
for the jth
variable during the ith
iteration in the range [0, 1]. The term “r1,j,i ( (Xj,best,i- │Xj,k,i│)” indicates
the tendency of the solution to move closer to the best solution and the term “-r2,j,i (Xj,worst,i- │Xj,k,i│)”
indicates the tendency of the solution to avoid the worst solution. X'j,k,iis accepted if it gives better
function value. All the accepted function values at the end of iteration are maintained and these values
become the input to the next iteration.
Fig.1 shows the flowchart of the proposed algorithm. The algorithm always tries to get closer to success
(i.e. reaching the best solution) and tries to avoid failure (i.e. moving away from the worst solution). The
algorithm strives to become victorious by reaching the best solution and hence it is named as Jaya (a
Sanskrit word meaning victory). The proposed method is illustrated by means of an unconstrained
benchmark function known as Sphere function in the next section.](https://image.slidesharecdn.com/ijiec201532-190422150057/85/imple-and-new-optimization-algorithm-for-solving-constrained-and-unconstrained-optimization-problems-2-320.jpg)
![R. Venkata Rao / International Journal of Industrial Engineering Computations 7 (2016) 29
Table 13
Holm-Sidak test for the ‘Best’ and the ‘Mean’ solutions obtained for G01-G13 functions
Test for best solution Test for mean solution
Algorithma p-value Algorithma p-value
1-5 0.02019 1-5 0.03481
1-2 0.36309 1-9 0.03692
1-3 0.37918 1-2 0.4158
1-9 0.41125 1-3 0.6321
1-7 0.97138 1-4 0.8273
1-4 0.98204 1-6 0.9108
1-6 0.98327 1-7 0.9522
1-8 0.98502 1-11 0.9613
1-11 0.98575 1-8 0.9782
1-10 0.98689 1-10 0.9881
a 1-Jaya, 2-HM, 3-ASCHEA, 4-SMES, 5-GA, 6-PSO, 7-DE, 8-ABC, 9-BBO, 10-TLBO, 11-HTS
Table 14
Holm-Sidak test for the ‘Best’ and the ‘Mean’ solutions obtained for G14-G24 functions
Test for best solution Test for mean solution
Algorithma p-value Algorithma p-value
1-3 0.01319 1-3 0.00071
1-5 0.15423 1-5 0.30019
1-4 0.21998 1-4 0.43427
1-2 0.22105 1-2 0.44549
1-7 0.96502 1-7 0.82612
1-6 0.97806 1-6 0.86512
a 1-Jaya, 2-PSO, 3-BBO, 4-DE, 5-ABC, 6-TLBO, 7-HTS
Table 15
Unconstrained benchmark functions considered
No. Function Formulation D Search range C
F 1 Sphere
2
min
1
D
i
i
F x
30 [-100, 100] US
F 2 SumSquares
2
min
1
D
i
i
F ix
30 [-10, 10] US
F 3 Beale
2 22 2 3
min 1 1 2 1 1 2 1 1 2
1
1.5 2.25 2.625
D
i
F x x x x x x x x x
5 [-4.5, 4.5] UN
F 4 Easom 2 2
min 1 2 1 2cos cos expF x x x x 2 [-100, 100] UN
F 5 Matyas 2 2
min 1 2 1 20.26 0.48F x x x x 2 [-10, 10] UN
F 6 Colville
2 222 2
min 1 2 1 3 3 4
2 2
2 4 1 2 2 4
100 1 1 90
10.1 1 1 0.48 19.8 1 1
F x x x x x x
x x x x x x
4 [-10, 10] UN
F 7 Trid 6 2
min 1
1 2
1
D D
i i i
i i
F x x x
6 [-D2
, D2
] UN
F 8 Trid 10 2
min 1
1 2
1
D D
i i i
i i
F x x x
10 [-D2
, D2
] UN
F 9 Zakharov
2 4
2
min
1 1 1
0.5 0.5
D D D
i i i
i i i
F x ix ix
10 [-5, 10] UN
F 10 Schwefel 1.2
2
2
min
1 1
D i
j
i j
F x
30 [-100, 100] UN
F 11 Rosenbrock 2 2 2
min 1
1
[100( ) (1 ) ]
D
i i i
i
F x x x
30 [-30, 30] UN
F 12 Dixon-Price
22 2
min 1 1
2
1 2
D
i i
i
F x i x x
30 [-10, 10] UN
D: Dimension, C: Characteristic, U: Unimodal, M: Multimodal, S: Separable, N: Non-separable](https://image.slidesharecdn.com/ijiec201532-190422150057/85/imple-and-new-optimization-algorithm-for-solving-constrained-and-unconstrained-optimization-problems-11-320.jpg)
![30
Table 15
Unconstrained benchmark functions considered (Continued)
No. Function Formulation D Search range C
F 13 Foxholes
1
25
min 2
6
1
1
1 1
500 j
i ij
i
F
j x a
2 [-65.536, 65.536]MS
F 14 Branin
2
2
min 2 1 1 12
5.1 5 1
6 10 1 cos 10
84
F x x x x
2 [-5, 10] [0, 15] MS
F 15 Bohachevsky 1 2 2
min 1 2 1 22 0.3cos 3 0.4cos 4 0.7F x x x x 2 [-100, 100] MS
F 16 Booth
2 2
min 1 2 1 22 7 2 5F x x x x 2 [-10, 10] MS
F 17 Michalewicz 2
20
2
min 1
1
sin sin
D
i
i
ix
F x
2 [0, π] MS
F 18 Michalewicz 5
20
2
min 1
1
sin sin
D
i
i
ix
F x
5 [0, π] MS
F 19 Bohachevsky 2 2 2
min 1 2 1 22 0.3cos 3 4 0.3F x x x x 2 [-100, 100] MN
F 20 Bohachevsky 3 2 2
min 1 2 1 22 0.3cos 3 4 0.3F x x x x 2 [-100, 100] MN
F 21 GoldStein-Price
2 2 2
min 1 2 1 1 2 1 2 2
2 2 2
1 2 1 1 2 1 2 2
1 1 19 14 3 14 6 3
30 2 3 18 32 12 48 36 27
F x x x x x x x x
x x x x x x x x
2 [-2, 2] MN
F 22 Perm
2
min
1 1
1
kD D
k i
k i
xF i
i
4 [-D, D] MN
F 23 Hartman 3
4 3
2
min
1 1
expi ij j ij
i j
F c a x p
3 [0, 1] MN
F 24 Ackley
2
min
1 1
1 1
20exp 0.2 exp cos2 20
D D
i i
i i
F x x e
D D
30 [-32, 32] MN
F 25 Penalized 2
2 2 2
1
1
2
min 1
2
1
1
( 1) 1 sin (3 ) ( 1)
0.1 sin ( )
1 sin (2 )
( ) ,
( ,5,100,4), ( , , , ) 0, ,
( ) ,
D
i i D
i D
m
i iD
i i i
i m
i i
x x x
F x
x
k x a x a
u x u x a k m a x a
k x a x a
30 [-50, 50] MN
F 26 Langerman 2
2 2
min
1 1 1
1
exp cos
D D D
i j ij j ij
i j j
F c x a x a
2 [0, 10] MN
F 27 Langerman 5
2 2
min
1 1 1
1
exp cos
D D D
i j ij j ij
i j j
F c x a x a
5 [0, 10] MN
F 28 Langerman 10
2 2
min
1 1 1
1
exp cos
D D D
i j ij j ij
i j j
F c x a x a
10 [0, 10] MN
F 29 FletcherPowell 5
2
min
1
1 1
sin cos , sin cos
D
i i
i
D D
i ij j ij j i ij j ij j
j j
F A B
A a b B a x b x
5 [-π, π] MN
F 30 FletcherPowell 10
2
min
1
1 1
sin cos , sin cos
D
i i
i
D D
i ij j ij j i ij j ij j
j j
F A B
A a b B a x b x
10 [-π, π] MN
D: Dimension, C: Characteristic, U: Unimodal, M: Multimodal, S: Separable, N: Non-separable
Friedman rank test is used to rank the algorithms based on the results obtained by the algorithms.
However, this test does not specify any statistical difference in the results and hence Holm-Sidak test is
used to determine the statistical difference between the algorithms. Table 13 shows the Holm-Sidak test
for the ‘Best’ and the ‘Mean’ solutions obtained for G01-G13 functions and Table 14 shows the](https://image.slidesharecdn.com/ijiec201532-190422150057/85/imple-and-new-optimization-algorithm-for-solving-constrained-and-unconstrained-optimization-problems-12-320.jpg)
imple and new optimization algorithm for solving constrained and unconstrained optimization problems
- 1. * Corresponding author. 91-261-2201661, Fax: 91-261-2201571 E-mail:[email protected](R. Venkata Rao) © 2016 Growing Science Ltd. All rights reserved. doi: 10.5267/j.ijiec.2015.8.004 International Journal of Industrial Engineering Computations 7 (2016) 19–34 Contents lists available at GrowingScience International Journal of Industrial Engineering Computations homepage: www.GrowingScience.com/ijiec Jaya: A simple and new optimization algorithm for solving constrained and unconstrained optimization problems R. Venkata Rao* Department of Mechanical Engineering, S.V. National Institute of Technology, Ichchanath, Surat, Gujarat – 395 007, India C H R O N I C L E A B S T R A C T Article history: Received July 21 2015 Received in Revised Format August 16 2015 Accepted August17 2015 Available online August18 2015 A simple yet powerful optimization algorithm is proposed in this paper for solving the constrained and unconstrained optimization problems. This algorithm is based on the concept that the solution obtained for a given problem should move towards the best solution and should avoid the worst solution. This algorithm requires only the common control parameters and does not require any algorithm-specific control parameters. The performance of the proposed algorithm is investigated by implementing it on 24 constrained benchmark functions having different characteristics given in Congress on Evolutionary Computation (CEC 2006) and the performance is compared with that of other well-known optimization algorithms. The results have proved the better effectiveness of the proposed algorithm. Furthermore, the statistical analysis of the experimental work has been carried out by conducting the Friedman’s rank test and Holm-Sidak test. The proposed algorithm is found to secure first rank for the ‘best’ and ‘mean’ solutions in the Friedman’s rank test for all the 24 constrained benchmark problems. In addition to solving the constrained benchmark problems, the algorithm is also investigated on 30 unconstrained benchmark problems taken from the literature and the performance of the algorithm is found better. © 2016 Growing Science Ltd. All rights reserved Keywords: Jaya algorithm Optimization CEC 2006 Constrained benchmark problems Unconstrained benchmark problems Statistical tests 1. Introduction The population based heuristic algorithms have two important groups: evolutionary algorithms (EA) and swarm intelligence (SI) based algorithms.Some of the recognized evolutionary algorithms are: Genetic Algorithm (GA), Evolution Strategy (ES), Evolution Programming (EP), Differential Evolution (DE), Bacteria Foraging Optimization (BFO), Artificial Immune Algorithm (AIA), etc. Some of the well- known swarm intelligence based algorithms are:Particle Swarm Optimization (PSO), Shuffled Frog Leaping (SFL), Ant Colony Optimization (ACO), Artificial Bee Colony (ABC), Fire Fly (FF) algorithm, etc. Besides the evolutionary and swarm intelligence based algorithms, there are some other algorithms which work on the principles of different natural phenomena. Some of them are: Harmony Search (HS)
- 2. 20 algorithm, Gravitational Search Algorithm (GSA), Biogeography-Based Optimization (BBO), Grenade Explosion Method (GEM), etc. (Rao and Patel, 2012, 2013). All the evolutionary and swarm intelligence based algorithms are probabilistic algorithms and require common controlling parameters like population size, number of generations, elite size, etc. Besides the common control parameters, different algorithms require their own algorithm-specific control parameters. For example, GA uses mutation probability, crossover probability, selection operator; PSO uses inertia weight, social and cognitive parameters; ABC uses number of onlooker bees, employed bees, scout bees and limit; HS algorithm uses harmony memory consideration rate, pitch adjusting rate, and the number of improvisations. Similarly, the other algorithms such as ES, EP, DE, BFO, AIA, SFL, ACO, etc. need the tuning of respective algorithm-specific parameters. The proper tuning of the algorithm- specific parameters is a very crucial factor which affects the performance of the above mentioned algorithms. The improper tuning of algorithm-specific parameters either increases the computational effort or yields the local optimal solution. Considering this fact, Rao et al. (2011) introduced the teaching- learning-based optimization (TLBO) algorithm which does not require any algorithm-specific parameters. The TLBO algorithm requires only common controlling parameters like population size and number of generations for its working. The TLBO algorithm has gained wide acceptance among the optimization researchers (Rao, 2015). Keeping in view of the success of the TLBO algorithm, another algorithm-specific parameter-less algorithm is proposed in this paper. However, unlike two phases (i.e. teacher phase and the learner phase) of the TLBO algorithm, the proposed algorithm has only one phase and it is comparatively simpler to apply. The working of the proposed algorithm is much different from that of the TLBO algorithm. The next section describes the proposed algorithm. 2. Proposed algorithm Let f(x) is the objective function to be minimized (or maximized). At any iteration i, assume that there are ‘m’ number of design variables (i.e. j=1,2,…,m), ‘n’ number of candidate solutions (i.e. population size, k=1,2,…,n). Let the best candidate best obtains the best value of f(x) (i.e. f(x)best) in the entire candidate solutions and the worst candidate worst obtains the worst value of f(x) (i.e. f(x)worst) in the entire candidate solutions. If Xj,k,iis the value of the jth variable for the kth candidate during the ith iteration, then this value is modified as per the following Eq. (1). X'j,k,i= Xj,k,i+ r1,j,i (Xj,best,i- │Xj,k,i│) - r2,j,i (Xj,worst,i- │Xj,k,i│) , (1) where, Xj,best,iis the value of the variable j for the best candidate and Xj,worst,iis the value of the variable j for the worst candidate. X'j,k,iis the updated value of Xj,k,i and r1,j,i and r2,j,i are the two random numbers for the jth variable during the ith iteration in the range [0, 1]. The term “r1,j,i ( (Xj,best,i- │Xj,k,i│)” indicates the tendency of the solution to move closer to the best solution and the term “-r2,j,i (Xj,worst,i- │Xj,k,i│)” indicates the tendency of the solution to avoid the worst solution. X'j,k,iis accepted if it gives better function value. All the accepted function values at the end of iteration are maintained and these values become the input to the next iteration. Fig.1 shows the flowchart of the proposed algorithm. The algorithm always tries to get closer to success (i.e. reaching the best solution) and tries to avoid failure (i.e. moving away from the worst solution). The algorithm strives to become victorious by reaching the best solution and hence it is named as Jaya (a Sanskrit word meaning victory). The proposed method is illustrated by means of an unconstrained benchmark function known as Sphere function in the next section.
- 3. R. Venkata Rao / International Journal of Industrial Engineering Computations 7 (2016) 21 Fig. 1. Flowchart of the Jaya algorithm 2.1 Demonstration of the working of Jaya algorithm To demonstrate the working of Jaya algorithm, an unconstrained benchmark function of Sphere is considered. The objective function is to find out the values of xi that minimize the value of the Sphere function. 2 1 min ( ) n i i i f x x subject to (2) -100≤ xi≤ 100 The known solution to this benchmark function is 0 for all xi values of 0. Now to demonstrate the Jaya algorithm, let us assume a population size of 5 (i.e. candidate solutions), two design variables x1 and x2 and two iterations as the termination criterion. The initial population is randomly generated within the ranges of the variables and the corresponding values of the objective function are shown in Table 1. As it is a minimization function, the lowest value of f(x) is considered as the best solution and the highest value of f(x) is considered as the worst solution. Table 1 Initial population Candidate x1 x2 f(x) Status 1 -5 18 349 2 14 63 4165 3 70 -6 4936 worst 4 -8 7 113 best 5 -12 -18 468 From Table 1 it can be seen that the best solution is corresponding the 4th candidate and the worst solution is corresponding to the 3rd candidate. Now assuming random numbers r1 = 0.58 and r2 = 0.81 for x1 and r1 = 0.92 and r2 = 0.49 for x2, the new values of the variables for x1 and x2 are calculated using Eq.(1) and Initialize population size, number of design variables and termination criterion Identify best and worst solutions in the population Modify the solutions based on best and worst solutions Is the termination criterion satisfied? Report the optimum solution Is the solution corresponding to ?better than that corresponding to Accept and replace the previous solution Keep the previous solution Yes No YesNo
- 4. 22 are placed in Table 2. For example, for the 1st candidate, the new values of x1 and x2 during the first iteration are calculated as shown below. X'1,1,1= X1,1,1+ r1,1,1 (X1,4,1 - │X1,1,1│) - r2,1,1 (X1,3,1 - │X1,1,1│) = -5 + 0.58 (-8-│-5│) -0.81 (70-│-5│) = -65.19 X'2,1,1= X2,1,1+ r1,2,1 (X2,4,1 - │X2,1,1│) – r2,2,1 (X2,3,1 - │X2,1,1│) = 18 + 0.92 (7-│18│) – 0.49 (-6-│18│) = 19.64 Similarly, the new values of x1 and x2 for the other candidates are calculated. Table 2 shows the new values of x1 and x2 and the corresponding values of the objective function. Table 2 New values of the variables and the objective function during first iteration Candidate x1 x2 f(x) 1 -65.19 19.64 4635.466 2 -44.12 45.29 3997.76 3 24.76 0.8 613.697 4 -67.5 13.37 4735 5 -70.58 -16.36 5249.186 Now, the values of f(x) of Tables 1 and 2 are compared and the best values of f(x) are considered and placed in Table 3. This completes the first iteration of the Jaya algorithm. Table 3 Updated values of the variables and the objective function based on fitness comparison at the end of first iteration Candidate x1 x2 f(x) Status 1 -5 18 349 2 -44.12 45.29 3997.76 worst 3 24.76 0.8 613.697 4 -8 7 113 best 5 -12 -18 468 From Table 3 it can be seen that the best solution is corresponding the 4th candidate and the worst solution is corresponding to the 2nd candidate. Now, during the second iteration, assuming random numbers r1 = 0.27 and r2 = 0.23 for x1 and r1 = 0.38 and r2 = 0.51 for x2, the new values of the variables for x1 and x2 are calculated using Eq.(1). Table 4 shows the new values of x1 and x2 and the corresponding values of the objective function during the second iteration. Table 4 New values of the variables and the objective function during second iteration Candidate x1 x2 f(x) 1 2.7876 -0.0979 7.7803 2 -37.897 30.74 2381.13 3 31.757 -19.534 1390.08 4 -0.3324 -12.528 157.06 5 -4.4924 -36.098 1323.247 Now, the values of f(x) of Tables 3 and 4 are compared and the best values of f(x) are considered and placed in Table 5. This completes the second iteration of the Jaya algorithm. Table 5 Updated values of the variables and the objective function based on fitness comparison at the end of second iteration Candidate x1 x2 f(x) Status 1 2.7876 -0.0979 7.7803 best 2 -37.897 30.74 2381.13 worst 3 24.76 0.8 613.697 4 -8 7 113 5 -12 -18 468
- 5. R. Venkata Rao / International Journal of Industrial Engineering Computations 7 (2016) 23 From Table 5 it can be seen that the best solution is corresponding the 1st candidate and the worst solution is corresponding to the 2nd candidate. It can also be observed that the value of the objective function is reduced from 113 to 7.7803 in just two iterations. If we increase the number of iterations then the known value of the objective function (i.e. 0) can be obtained within next few iterations. Also, it is to be noted that in the case of maximization function problems, the best value means the maximum value of the objective function and the calculations are to be proceeded accordingly. Thus, the proposed method can deal with both minimization and maximization problems. The above demonstration is for an unconstrained optimization problem. However, the similar steps can be followed in the case of constrained optimization problem. The main difference is that a penalty function is used in the constrained optimization problem to take care of the violation of each constraint and the penalty is operated upon the objective function. The next section deals with the experimentation of the proposed algorithm on 24 constrained benchmark problems given in CEC 2006 (Liang et al., 2006; Karaboga&Basturk, 2007; Karaboga&Akay, 2011). 3. Experiments on constrained benchmark problems The objectives and constraints of the considered 24 benchmark functions of CEC 2006 have different characteristics such as linear, nonlinear, quadratic, cubic and polynomial. The number of design variables and their ranges are different for each problem (Liang et al., 2006; Karaboga and Basturk, 2007; Karaboga and Akay, 2011). To evaluate the performance of the proposed Jaya algorithm, the results obtained by using the Jaya algorithm are compared with the results obtained by the other optimization algorithms such as homomorphous mapping (HM), adaptive segregational constraint handling evolutionary algorithm (ASCHEA), simple multi-membered evolution strategy (SMES), genetic algorithm (GA), particle swarm optimization (PSO), differential evolution (DE), artificial bee colony (ABC), biogeography based optimization (BBO) available in the literature (Karaboga&Basturk, 2007; Karaboga&Akay, 2011). However, these algorithms were experimented only on 13 functions of CEC 2006. Patel and Savsani (2015) extended the application of PSO, BBO, DE and ABC to the remaining 11 functions also. In addition, they had applied TLBO and heat transfer search (HTS) algorithms for all the 24 constrained benchmark functions. Now, the computational experiments are conducted to identify the performance of the proposed Jaya algorithm and the performance of the algorithm is compared with the above-mentioned algorithms. A common platform is provided by maintaining the identical function evolutions for different algorithms considered for the comparison. Thus, the consistency in the comparison is maintained while comparing the performance of Jaya algorithm with other optimization algorithms. However, in general, the algorithm which requires less number of function evaluations to get the same best solution can be considered as better as compared to the other algorithms. However, in this paper, to maintain the consistency in the comparison of competitive algorithms, a common experimental platform described by Patel and Savsani (2015) is provided by setting the maximum number of function evaluations as 240000 for each benchmark function and the static penalty method is used as a constraint handling technique. Just like other algorithms, the proposed Jaya algorithm is executed 100 times for each benchmark function and the mean results obtained are compared with the other algorithms for the same number of runs. 3.1. Results and discussion on constrained benchmark functions Table 6 presents the comparative results of G01-G13 test functions obtained by HM, ASCHEA, SMES, GA, PSO, DE, ABC, BBO, TLBO, HTS and Jaya algorithms for 240000 function evaluations averaged over 100 runs.
- 6. 24 Table 6 Comparative results of G01-G13 benchmark functions obtained by different algorithms Function HMa ASCHEAb SMESc GAc PSOc DEc ABCc BBOb HTSb TLBOb TLBOb (Corrected results) Jaya G01 (-15.00) B -14.7864 -15.000 -15 14.44 -15 -15 -15 -14.977 −15 -15 -15 -15 W --- --- --- --- -13 -11.828 -15 -14.5882 - 15 -6 -15 -15 M -14.7082 -14.84 -15 -14.236 -14.71 -14.555 -15 -14.7698 -15 -10.782 -15 -15 G02 (-0.803619) B -0.79953 -0.785 -0.803601 -0.796321 -0.669158 -0.472 -0.803598 -0.7821 - 0 .7517 -0.7835 -0.7899 -0.803605 W --- --- --- --- -0.299426 -0.472 -0.749797 -0.7389 -0.5482 -0.5518 -0.5985 -0.7542 M -0.79671 -0.59 -0.785238 -0.78858 -0.41996 -0.665 -0.792412 -0.7642 -0.6437 -0.6705 -0.6882 -0.7968 G03 (-1.0005) B 0.9997 1 1 0.99 1 -0.99393 -1 -1.0005 -1.0005 -1.0005 -1.0005 -1.005 W --- --- --- --- -0.464 -1 -1 -0.0455 0 0 -1 -1 M 0.9989 0.99989 1 0.976 0.764813 -1 -1 -0.3957 -0.9004 -0.8 -1 -1 G04 (-30665.539) B -30664.5 -30665.5 - -30626.053 -30665.539 -30665.539 -30665.539 -30665.539 -30665.5387 -30665.5387 -30665.5387 -30665.5387 W --- --- --- --- -30665.539 -30665.539 -30665.539 -29942.3 -30665.5387 -30665.5387 -30665.5387 -30665.5387 M -30655.3 -30665.5 - -30590.455 -30665.539 -30665.539 -30665.539 -30411.865 -30665.5387 -30665.5387 -30665.5387 -30665.5387 G05 (5126.486) B NF 5126.5 5126.599 NF 5126.484 5126.484 5126.484 5134.2749 5126.486 5126.486 5126.486 5126.486 W NF --- --- NF 5249.825 5534.61 5438.387 7899.2756 5126.6831 5127.714 5127.4197 5126.635 M NF 5141.65 5147.492 NF 5135.973 5264.27 5185.714 6130.5289 5126.5152 5126.6184 5126.5146 5126.5061 G06 (-6961.814) B -6952.1 -6961.81 -6961.814 -6952.472 -6961.814 -6954.434 -6961.814 -6961.8139 -6961.814 -6961.814 -6961.814 -6961.814 W --- --- --- --- -6961.814 -6954.434 -6961.805 -5404.4941 -6961.814 -6961.814 -6961.814 -6961.814 M -6342.6 -6961.81 -6961.284 -6872.204 -6961.814 -6954.434 -6961.813 -6181.7461 -6961.814 -6961.814 -6961.814 -6961.814 G07 (24.3062) B 24.62 24.3323 24.327 31.097 24.37 24.306 24.33 25.6645 24.3104 24.3101 24.3100 24.3062 W --- --- --- --- 56.055 24.33 25.19 37.6912 25.0083 27.6106 26.7483 24.8932 M 24.826 24.6636 24.475 34.98 32.407 24.31 24.473 29.829 24.4945 24.837 24.6482 24.3092 G08 (-0.095825) B 0.095825 0.095825 0.095825 0.095825 -0.095825 -0.095825 -0.095825 -0.095825 -0.095825 -0.095825 -0.095825 -0.095825 W --- --- --- --- -0.095825 -0.095825 -0.095825 -0.095817 -0.095825 -0.095825 -0.095825 -0.095825 M 0.0891568 0.095825 0.095825 0.095799 -0.095825 -0.095825 -0.095825 -0.95824 -0.095825 -0.095825 -0.095825 -0.095825 G09 (680.6301) B 680.91 680.63 680.632 685.994 680.63 680.63 680.634 680.6301 680.6301 680.6301 680.6301 680.6301 W --- --- --- --- 680.631 680.63 680.634 721.0795 680.644 680.6456 680.6334 680.6301 M 681.16 680.641 680.643 692.064 680.63 680.63 680.634 692.7162 680.6329 680.6336 680.6313 680.6301 G10 (7049.28) B 7147.9 7061.13 7051.903 9079.77 7049.481 7049.548 7053.904 7679.0681 7049.4836 7250.9704 7052.329 7049.312 W --- --- ---- --- 7894.812 9264.886 7604.132 9570.5714 7252.0546 7291.3779 7152.0813 7119.632 M 8163.6 7497.434 7253.047 10003.225 7205.5 7147.334 7224.407 8764.9864 7119.7015 7257.0927 7114.4893 7104.6201 G11 (0.7499) B 0.75 0.75 0.75 0.75 0.749 0.752 0.75 0.7499 0.7499 0.7499 0.7499 0.7499 W --- --- --- --- 0.749 1 0.75 0.92895 0.7499 0.7499 0.7499 0.7499 M 0.75 0.75 0.75 0.75 0.749 0.901 0.75 0.83057 0.7499 0.7499 0.7499 0.7499 G12 (-1) B --- --- -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 W --- --- --- --- -0.994 -1 -1 -1 -1 -1 -1 -1 M --- --- -1 -1 -0.998875 -1 -1 -1 -1 -1 -1 -1 G13 (-0.05394) B --- --- 0.054986 0.134057 0.085655 0.385 0.76 0.62825 0.37319 0.44015 0.003631 0.003625 W --- --- --- --- 1.793361 0.99 1 1.45492 0.79751 0.95605 0.003632 0.003631 M --- --- 0.166385 --- 0.569358 0.872 0.968 1.09289 0.66948 0.69055 0.003631 0.003627 Result in boldface indicates better performing algorithm. (-) indicates that the results are not available. NF: means that no feasible solutions were found. The bold values indicate best results. a: used decoder-based penalty method for constraint handling; b: used static penalty method for constraint handling.; c: used Deb’s method for constraint handling.
- 7. R. Venkata Rao / International Journal of Industrial Engineering Computations 7 (2016) 25 For the TLBO algorithm, Patel and Savsani(2015) used a population size of 50 for all the 24 benchmark functions and thus reported inferior values for the TLBO algorithm for many benchmark functions even though the function evaluations were kept as 240000 in each case. It may be noted that population size and the number of generations are common control parameters (i.e. not algorithm-specific parameters) and variation in the common control parameters may also affect the results. Patel and Savsani (2015) had not considered different combinations of population sizes and the number of generations in the case of TLBO algorithm while maintaining the number of function evaluations as 240000. Hence, the results of TLBO algorithm are corrected now and included in Table 6. The corrected results for TLBO algorithm presented in this paper correspond to the population sizes of 100, 70, 100, 40, 60, 100, 50, 30, 100, 70, 50, 60, 80, 80, 50, 100, 70, 100, 50, 100, 100, 50, 70 and 80 respectively for G01-G24 functions and the number of generations are decided accordingly keeping the number of function evaluations same as 240000 in each case. It can be seen that the corrected results of TLBO algorithm are better than those reported by Patel and Savsani (2015). Employing a population size of 50 for all the algorithms for solving all benchmark functions by Patel and Savsani (2015) might not have revealed the true potential of the algorithms. It is important to keep the same number of function evaluations for all the algorithms for a benchmark function (instead of same population size for all the algorithms) and this procedure can be applied to other benchmark functions also. The results of the proposed Jaya algorithm for each benchmark function by employing appropriate population sizes while maintaining the function evaluations of 240000 are included in the last column of Table 6. The Table 6 shows the comparative results of G01-G13 benchmark functions obtained by different algorithms for 240000 function evaluations averaged over 100 runs. The ‘best (B)’, ‘worst (W)’ and ‘mean (M)’ values of the 13 constrained benchmark functions (i.e. G01 – G13) attempted by HM, ASCHEA, SMES, GA, PSO, DE, ABC, BBO, HTS, TLBO and Jaya algorithm are shown. The global optimum values expected are given within brackets under each function. It can be observed that the proposed Jaya algorithm has obtained global optimum values for all the benchmark functions except for G10. However, in this case also, the global optimum value obtained by Jaya algorithm is superior to the remaining algorithms. Furthermore, it can be observed that the ‘mean (M)’ values of all the 13 benchmark functions obtained by Jaya algorithm are better than all other algorithms. Table 7 shows the comparative results of G14-G24 benchmark functions obtained by different algorithms for 240000 function evaluations averaged over 100 runs. Here also it can be observed that the proposed Jaya algorithm has obtained global optimum values for all the benchmark functions except for G14, G19, G20, GG21, G22 and G23. However, in these cases also, the global optimum value obtained by Jaya algorithm for each of these functions is superior to those given bythe remaining algorithms. Furthermore, it can also be observed that the ‘mean (M)’ values of all the 11 benchmark functions obtained by Jaya algorithm are better than all other algorithms. It may be mentioned here that the HTS algorithm has employed elitism while all other algorithms have not employed the concept of elitism. In fact, a fair comparison means that only the non-elitist algorithms should be compared. However, as only the results of elitist HTS are available in Patel and Savsani (2015), the comparison is made with the same. Even then it can be observed that Jaya algorithm has performed better than the other algorithms including the elitist HTS algorithm in the case of G14-G24 benchmark functions. It may be mentioned here that the elitist HTS algorithm is not an algorithm-specific parameter-less algorithm (as it contains conduction, convection and radiation factors and the results vary with the variation in the values of these factors). Table 8 shows the success rates of various algorithms for G01-G24 functions over 100 runs. The comparison is made between PSO, BBO, DE, ABC, HTS and TLBO algorithms as these algorithms are applied for all the 24 benchmark functions. The success rate obtained by all the algorithms is 0 in the case of 8 benchmark functions (i.e. G02, G10, G13, G14, G19, G20, G22 and G23). In the case of all remaining 16 benchmark functions, the success rate obtained by the proposed Jaya algorithm is either equal or superior to the other algorithms of PSO, BBO, DE, ABC, HTS and TLBO algorithms.
- 8. 26 Table 7 Comparative results of G14-G24benchmark functions obtained by different algorithms for 240000 function evaluations averaged over 100 runs Function PSO BBO DE ABC HTS TLBO TLBO (Corrected Jaya G14 (–47.764) Best -44.9343 54.6679 -45.7372 -44.6431 -47.7278 -46.5903 - 46.7339 -47.7322 Worst -37.5000 257.7061 -12.7618 -23.3210 -45.0648 -17.4780 -18.1581 -46.2908 Mean -40.8710 175.9832 -29.2187 -40.1071 -46.4076 -39.9725 -40.2854 -46.6912 SD 2.29E+00 7.90E+01 1.36E+01 7.14E+00 8.53E-01 1.15E+01 6.38E-01 2.98E-01 G15 (961.715) Best 961.7150 962.6640 961.7150 961.7568 961.7150 961.7150 961.7150 961.7150 Worst 972.3170 1087.3557 962.1022 970.3170 962.0653 964.8922 961.7150 961.7150 Mean 965.5154 1001.4367 961.7537 966.2868 961.7500 962.8641 961.7150 961.7150 SD 3.72E+00 4.74E+01 1.22E-01 3.12E+00 1.11E-01 1.49E+00 1.03E-01 0.9E-01 G16 (–1.9052) Best -1.9052 -1.9052 -1.9052 -1.9052 -1.9052 -1.9052 -1.9052 -1.9052 Worst -1.9052 -1.1586 -1.9052 -1.9052 -1.9052 -1.9052 -1.9052 -1.9052 Mean -1.9052 -1.6121 -1.9052 -1.9052 -1.9052 -1.9052 -1.9052 -1.9052 SD 2.3E-16 2.58E-01 2.34E-16 2.34E-16 2.34E-16 2.134E-16 2.08E-16 1.62E-16 G17 (8853.5396) Best 8857.5140 9008.5594 8854.6501 8859.7130 8853.5396 8853.5396 8853.5396 8853.5396 Worst 8965.4010 9916.7742 8996.3215 8997.1450 8932.0712 8919.6595 8919.6595 8902.203 Mean 8899.4721 9384.2680 8932.0444 8941.9245 8877.9175 8876.5071 8876.5071 8872.5402 SD 3.79E+01 3.06E+02 4.68E+01 4.26E+01 3.09E+01 3.02E+01 2.87E+01 1.87E+01 G18 (–0.86603) Best -0.86603 -0.65734 -0.86531 -0.86603 -0.86603 -0.86603 -0.86603 -0.86603 Worst -0.51085 -0.38872 -0.85510 -0.86521 -0.67468 -0.86294 -0.86304 -0.86601 Mean -0.82760 -0.56817 -0.86165 -0.86587 -0.77036 -0.86569 -0.86601 -0.86602 SD 1.11E-01 8.55E-02 3.67E-03 3.37E-04 1.01E-01 9.67E-04 8.45E-05 3.56E-06 G19 (32.6555) Best 33.5358 39.1471 32.6851 33.3325 32.7132 32.7916 32.7194 32.6803 Worst 39.8443 71.3106 32.9078 38.5614 33.2140 36.1935 35.9893 32.8776 Mean 36.6172 51.8769 32.7680 36.0078 32.7903 34.0792 33.9912 32.7512 SD 2.04E+00 1.12E+01 6.28E-02 1.83E+00 1.53E-01 9.33E-01 7.72E-01 1.87E-01 G20 (0.204979) Best 0.24743 1.26181 0.24743 0.24743 0.24743 0.24743 0.24385 0.24139 Worst 1.8720 1.98625 0.28766 1.52017 0.27331 1.84773 1.54249 0.25614 Mean 0.97234 1.43488 0.26165 0.80536 0.25519 1.22037 1.03492 0.24385 SD 6.34E-01 2.20E-01 1.91E-02 5.93E-01 1.25E-02 5.89E-01 1.12E-02 0.98E-02 G21 (193.274) Best 193.7311 198.8151 193.7346 193.7343 193.7264 193.7246 193.6539 193.5841 Worst 409.1320 581.2178 418.4616 330.1638 320.2342 393.8295 368.3786 202.3262 Mean 345.6595 367.2513 366.9193 275.5436 256.6091 264.6092 260.4783 193.7219 SD 6.36E+01 1.34E+02 9.13E+01 6.05E+01 6.63E+01 9.23E+01 4.82E+01 1.63E+01 G22 (236.4309) Best 1.68E+22 1.02E+15 1.25E+18 2.82E+08 2.16E+03 4.50E+17 3.39E+16 5.66E+02 Worst 3.25E+23 6.70E+16 2.67E+19 1.25E+18 1.33E+07 4.06E+19 3.97E+18 3.21E+06 Mean 1.63E+23 1.41E+16 1.78E+19 4.10E+17 1.36E+06 1.61E+19 2.93E+18 2.19E+03 SD 9.17E+22 1.96E+16 1.17E+19 4.72E+17 4.20E+06 1.51E+19 3.91E+09 1.91E+05 G23 (–400.055) Best -105.9826 2.3163 -72.6420 -43.2541 -390.6472 -385.0043 -390.8342 -391.5192 Worst 0 74.6089 0 0 0 0 0 -132.49 Mean -25.9179 22.1401 -7.2642 -4.3254 -131.2522 -83.7728 -142.5837 -381.2312 SD 4.30E+01 2.51E+01 2.30E+01 1.37E+01 1.67E+02 1.59E+02 2.37E+01 1.29E+01 G24 (–5.5080) Best -5.5080 -5.5080 -5.5080 -5.5080 -5.5080 -5.5080 -5.5080 -5.5080 Worst -5.5080 -5.4857 -5.5080 -5.5080 -5.5080 -5.5080 -5.5080 -5.5080 Mean -5.5080 -5.4982 -5.5080 -5.5080 -5.5080 -5.5080 -5.5080 -5.5080 SD 9.3E-16 6.75E-03 9.36E-16 9.36E-16 9.36E-16 9.36E-16 9.36E-16 8.15E-17 The bold values indicate best results. Table 9 shows the ‘Mean number of function evaluations’ required to reach global optimum value by the competitive algorithms for G01-G24 benchmark functions over 100 independent runs (except G02, G10, G13, G14, G19, G20, G22 and G23 functions for which the data is not available in the literature). Here also it can be observed that the proposed Jaya algorithm has obtained better results (i.e. minimum number of mean function evaluations required to reach global optimum value) for all the benchmark functions except for G01 and G12 as compared to PSO, BBO, DE, ABC, HTS and TLBO algorithms. The values of standard deviation of function evaluations for the Jaya algorithm are also comparatively better in the case of many functions.
- 9. R. Venkata Rao / International Journal of Industrial Engineering Computations 7 (2016) 27 Table 8 Success rate of various algorithms for G01-G24 functions over 100 runs Function PSO BBO DE ABC HTS TLBO TLBO (Corrected results) Jaya G01 38 0 94 100 100 26 100 100 G02 0 0 0 0 0 0 0 0 G03 59 23 41 67 86 74 81 96 G04 100 16 100 100 100 100 100 100 G05 61 0 93 28 95 92 96 98 G06 100 21 100 100 100 100 100 100 G07 21 0 26 28 37 23 34 41 G08 100 94 100 100 100 100 100 100 G09 84 26 95 89 96 91 92 100 G10 0 0 0 0 0 0 0 0 G11 100 57 19 100 100 100 100 100 G12 100 100 100 100 100 100 100 100 G13 0 0 0 0 0 0 0 0 G14 0 0 0 0 0 0 0 0 G15 53 0 73 42 83 81 86 87 G16 100 18 100 100 100 100 100 100 G17 0 0 0 0 26 58 61 74 G18 56 0 61 73 47 64 67 84 G19 0 0 0 0 0 0 0 0 G20 0 0 0 0 0 0 0 0 G21 12 0 24 36 48 35 51 64 G22 0 0 0 0 0 0 0 0 G23 0 0 0 0 0 0 0 0 G24 100 27 100 100 100 100 100 100 Table 9 Mean number of function evaluations required to reach global optimum value by comparative algorithms for G01-G24 over 100 independent runs Function PSO BBO DE ABC HTS TLBO TLBO (Corrected results) Jaya G01 Mean_FE 33750 --- 6988.89 13200 12570 9750 9678 8972 Std_FE 3872.01 --- 157.674 708.676 6312.43 4171.93 4067.43 4038.56 G02 Mean_FE --- --- --- --- --- --- --- --- Std_FE --- --- --- --- --- --- --- --- G03 Mean_FE 84610 157950 136350 121950 67400 178083 66985 61747 Std_FE 35670 5939.7 78988.3 64296.1 29776.2 43819.3 28564.5 27612.1 G04 Mean_FE 14432 189475 14090 29460 10135 5470 4958 4210 Std_FE 309.23 35390.7 1499.22 2619.25 1413.63 804.225 787.388 697.248 G05 Mean_FE 57921 --- 108572 197749 43356 46888 42583 41734 Std_FE 14277.4 --- 41757.1 20576.8 3416.3 19623.2 3356.6 2849.2 G06 Mean_FE 14923 140150 17540 69310 15395 11600 10953 9372 Std_FE 1789.32 22273.9 1214.91 3753.65 2566.61 2056.43 2011.84 1992.32 G07 Mean_FE 97742 --- 147650 114351 92916.7 147550 90382.57 87237.65 Std_FE 2984.2 --- 4737.62 11384.4 17237.3 5020.46 4839.58 4695.01 G08 Mean_FE 622 4290 725 670 635 680 604 567 Std_FE 189.78 4418.32 259.54 249.666 171.675 181.353 167.327 161.864 G09 Mean_FE 34877 194700 57205 149642 23235 37690 21949 20754 Std_FE 12280.1 29557.1 10779.1 73436.8 10806.2 26350.6 9943.6 9746.3 G10 Mean_FE --- --- --- --- --- --- --- --- Std_FE --- --- --- --- --- --- --- --- G11 Mean_FE 23312 35490 205250 29140 53270 3000 2957 2864 Std_FE 1231.41 30627.4 8273.15 12982.5 18215.2 1354.83 1294 1163 G12 Mean_FE 1204 1865 1150 1190 2190 2480 2167 2086 Std_FE 341.3 2240.54 263.523 747.514 824.554 917.484 811.135 789.326 G13 Mean_FE --- --- --- --- --- --- --- --- Std_FE --- --- --- --- --- --- --- --- G14 Mean_FE --- --- --- --- --- --- --- --- Std_FE --- --- ---- --- --- --- --- --- G15 Mean_FE 41972 --- 36391.7 157800 36756.3 52287.5 32593.6 30388.2 Std_FE 4073.9 --- 5509.21 57558.5 28670.6 47937.1 26399.1 24053.8 G16 Mean_FE 7114 85200 12565 19670 13045 7840 7612 6990 Std_FE 643.3 16122 1155.19 714.998 1358.6 2709.74 1328.5 1287.4 G17 Mean_FE --- --- --- --- 65600 126980 62943 60483 Std_FE --- --- --- --- 65053.8 46591.8 44936.3 42521.8 G18 Mean_FE 23769 --- 170140 114120 35360 19226 18395 17943 Std_FE 1009.78 --- 20227.7 58105.8 7731.14 5762.16 5693.91 5395.23 G19 Mean_FE --- --- --- --- --- --- --- --- Std_FE --- --- --- --- --- --- --- --- G20 Mean_FE --- --- --- --- --- --- --- --- Std_FE --- --- --- --- --- --- --- --- G21 Mean_FE 39937 --- 89500 99150 28037.5 108533 27664.8 26739.4 Std_FE 4302.2 --- 14283.6 3647.94 7032.35 8677.17 7011.76 6894.21 G22 Mean_FE --- --- --- --- --- --- --- --- Std_FE --- --- --- --- --- --- --- --- G23 Mean_FE --- --- --- --- --- --- --- --- Std_FE --- --- --- --- ---- --- --- --- G24 Mean_FE 2469 84625 4855 5400 3715 2710 2704 2689 Std_FE 245.5 2015.25 429.761 618.241 575.929 864.677 543.589 512.943 --- indicate that algorithm is failed to obtain a global optimum value for that function Mean_FE= Mean number of function evaluations. Std_FE=Standard deviation of function evaluations.
- 10. 28 4. Statistical tests It is observed from the results presented in Tables 6-9 that the performance of the proposed Jaya algorithm is better than the other competitive algorithms. However, it is necessary to conduct the statistical tests like Friedman rank test (Joaquin et al., 2011) and Holm-Sidak test (Holm, 1979) to prove the significance of the proposed algorithm. Table 10 shows Friedman rank test for the ‘Best’ and ‘Mean’ solutions obtained for G01-G13 functions. Table 11 shows the Friedman rank test for the ‘Best’ and ‘Mean’ solutions obtained for G14-G24 functions. The G05, G12, G13 functions were omitted by the previous researchers as the results of these functions are not available for some of the competitive algorithms and G22 function was omitted as none of the competitive algorithms had produced feasible solution for this function. Hence the same approach is used in the present work and it can be easily observed from Tables 10 and 11 that the proposed Jaya algorithm has got the 1st rank in the case of “Best” and “Mean’ solutions of all benchmark functions considered. The corrected TLBO algorithm has obtained the 2nd rank. Table 12 shows the results of Friedman rank test for the ‘Success Rate’ solutions obtained. The proposed Jaya algorithm has obtained the 1st rank again followed by TLBO. Table 10 Friedman rank test for the ‘Best’ and ‘Mean’ solutions obtained for G01-G13 functions Test for best solution Test for mean solution Algorithms Friedman value Normalized value Rank Algorithms Friedman value Normalized value Rank HM 85 2.61 10 HM 78 2.94 9 ASCHEA 70 2.15 9 ASCHEA 71 2.67 8 SMES 58.5 1.8 6 SMES 65 2.45 7 GA 92 2.83 11 GA 88 3.32 10 PSO 57 1.75 5 PSO 59 2.22 6 DE 60 1.84 7 DE 55.5 2.09 5 ABC 53 1.63 4 ABC 42.5 1.60 3 BBO 62 1.90 8 BBO 88 3.32 10 HTS 45.5 1.4 3 HTS 46 1.73 4 TLBO (corrected) 44.5 1.36 2 TLBO (corrected) 40.5 1.52 2 Jaya 32.5 1 1 Jaya 26.5 1 1 Table 11 Friedman rank test for the ‘Best’ and ‘Mean’ solutions obtained for G14-G24 functions Test for best solution Test for mean solution Algorithms Friedman value Normalized value Rank Algorithms Friedman value Normalized value Rank PSO 42.5 2.02 4 PSO 42 2.70 4 BBO 64 3.04 6 BBO 70 4.51 7 DE 42.5 2.02 4 DE 43 2.77 5 ABC 49.5 2.35 5 ABC 46 2.96 6 HTS 31.5 1.5 3 HTS 32 2.06 3 TLBO (corrected) 29 1.38 2 TLBO (corrected) 31.5 2.03 2 Jaya 21 1 1 Jaya 15.5 1 1 Table 12 Friedman rank test for the ‘Success Rate’ of the solutions obtained Algorithms Friedman value Normalized value Rank PSO 106 1.58 6 BBO 136.5 2.03 7 DE 100 1.49 5 ABC 95.5 1.42 4 HTS 83 1.23 3 TLBO (corrected) 80 1.19 2 Jaya 67 1 1
- 11. R. Venkata Rao / International Journal of Industrial Engineering Computations 7 (2016) 29 Table 13 Holm-Sidak test for the ‘Best’ and the ‘Mean’ solutions obtained for G01-G13 functions Test for best solution Test for mean solution Algorithma p-value Algorithma p-value 1-5 0.02019 1-5 0.03481 1-2 0.36309 1-9 0.03692 1-3 0.37918 1-2 0.4158 1-9 0.41125 1-3 0.6321 1-7 0.97138 1-4 0.8273 1-4 0.98204 1-6 0.9108 1-6 0.98327 1-7 0.9522 1-8 0.98502 1-11 0.9613 1-11 0.98575 1-8 0.9782 1-10 0.98689 1-10 0.9881 a 1-Jaya, 2-HM, 3-ASCHEA, 4-SMES, 5-GA, 6-PSO, 7-DE, 8-ABC, 9-BBO, 10-TLBO, 11-HTS Table 14 Holm-Sidak test for the ‘Best’ and the ‘Mean’ solutions obtained for G14-G24 functions Test for best solution Test for mean solution Algorithma p-value Algorithma p-value 1-3 0.01319 1-3 0.00071 1-5 0.15423 1-5 0.30019 1-4 0.21998 1-4 0.43427 1-2 0.22105 1-2 0.44549 1-7 0.96502 1-7 0.82612 1-6 0.97806 1-6 0.86512 a 1-Jaya, 2-PSO, 3-BBO, 4-DE, 5-ABC, 6-TLBO, 7-HTS Table 15 Unconstrained benchmark functions considered No. Function Formulation D Search range C F 1 Sphere 2 min 1 D i i F x 30 [-100, 100] US F 2 SumSquares 2 min 1 D i i F ix 30 [-10, 10] US F 3 Beale 2 22 2 3 min 1 1 2 1 1 2 1 1 2 1 1.5 2.25 2.625 D i F x x x x x x x x x 5 [-4.5, 4.5] UN F 4 Easom 2 2 min 1 2 1 2cos cos expF x x x x 2 [-100, 100] UN F 5 Matyas 2 2 min 1 2 1 20.26 0.48F x x x x 2 [-10, 10] UN F 6 Colville 2 222 2 min 1 2 1 3 3 4 2 2 2 4 1 2 2 4 100 1 1 90 10.1 1 1 0.48 19.8 1 1 F x x x x x x x x x x x x 4 [-10, 10] UN F 7 Trid 6 2 min 1 1 2 1 D D i i i i i F x x x 6 [-D2 , D2 ] UN F 8 Trid 10 2 min 1 1 2 1 D D i i i i i F x x x 10 [-D2 , D2 ] UN F 9 Zakharov 2 4 2 min 1 1 1 0.5 0.5 D D D i i i i i i F x ix ix 10 [-5, 10] UN F 10 Schwefel 1.2 2 2 min 1 1 D i j i j F x 30 [-100, 100] UN F 11 Rosenbrock 2 2 2 min 1 1 [100( ) (1 ) ] D i i i i F x x x 30 [-30, 30] UN F 12 Dixon-Price 22 2 min 1 1 2 1 2 D i i i F x i x x 30 [-10, 10] UN D: Dimension, C: Characteristic, U: Unimodal, M: Multimodal, S: Separable, N: Non-separable
- 12. 30 Table 15 Unconstrained benchmark functions considered (Continued) No. Function Formulation D Search range C F 13 Foxholes 1 25 min 2 6 1 1 1 1 500 j i ij i F j x a 2 [-65.536, 65.536]MS F 14 Branin 2 2 min 2 1 1 12 5.1 5 1 6 10 1 cos 10 84 F x x x x 2 [-5, 10] [0, 15] MS F 15 Bohachevsky 1 2 2 min 1 2 1 22 0.3cos 3 0.4cos 4 0.7F x x x x 2 [-100, 100] MS F 16 Booth 2 2 min 1 2 1 22 7 2 5F x x x x 2 [-10, 10] MS F 17 Michalewicz 2 20 2 min 1 1 sin sin D i i ix F x 2 [0, π] MS F 18 Michalewicz 5 20 2 min 1 1 sin sin D i i ix F x 5 [0, π] MS F 19 Bohachevsky 2 2 2 min 1 2 1 22 0.3cos 3 4 0.3F x x x x 2 [-100, 100] MN F 20 Bohachevsky 3 2 2 min 1 2 1 22 0.3cos 3 4 0.3F x x x x 2 [-100, 100] MN F 21 GoldStein-Price 2 2 2 min 1 2 1 1 2 1 2 2 2 2 2 1 2 1 1 2 1 2 2 1 1 19 14 3 14 6 3 30 2 3 18 32 12 48 36 27 F x x x x x x x x x x x x x x x x 2 [-2, 2] MN F 22 Perm 2 min 1 1 1 kD D k i k i xF i i 4 [-D, D] MN F 23 Hartman 3 4 3 2 min 1 1 expi ij j ij i j F c a x p 3 [0, 1] MN F 24 Ackley 2 min 1 1 1 1 20exp 0.2 exp cos2 20 D D i i i i F x x e D D 30 [-32, 32] MN F 25 Penalized 2 2 2 2 1 1 2 min 1 2 1 1 ( 1) 1 sin (3 ) ( 1) 0.1 sin ( ) 1 sin (2 ) ( ) , ( ,5,100,4), ( , , , ) 0, , ( ) , D i i D i D m i iD i i i i m i i x x x F x x k x a x a u x u x a k m a x a k x a x a 30 [-50, 50] MN F 26 Langerman 2 2 2 min 1 1 1 1 exp cos D D D i j ij j ij i j j F c x a x a 2 [0, 10] MN F 27 Langerman 5 2 2 min 1 1 1 1 exp cos D D D i j ij j ij i j j F c x a x a 5 [0, 10] MN F 28 Langerman 10 2 2 min 1 1 1 1 exp cos D D D i j ij j ij i j j F c x a x a 10 [0, 10] MN F 29 FletcherPowell 5 2 min 1 1 1 sin cos , sin cos D i i i D D i ij j ij j i ij j ij j j j F A B A a b B a x b x 5 [-π, π] MN F 30 FletcherPowell 10 2 min 1 1 1 sin cos , sin cos D i i i D D i ij j ij j i ij j ij j j j F A B A a b B a x b x 10 [-π, π] MN D: Dimension, C: Characteristic, U: Unimodal, M: Multimodal, S: Separable, N: Non-separable Friedman rank test is used to rank the algorithms based on the results obtained by the algorithms. However, this test does not specify any statistical difference in the results and hence Holm-Sidak test is used to determine the statistical difference between the algorithms. Table 13 shows the Holm-Sidak test for the ‘Best’ and the ‘Mean’ solutions obtained for G01-G13 functions and Table 14 shows the
- 13. R. Venkata Rao / International Journal of Industrial Engineering Computations 7 (2016) 31 corresponding values for G14-G24 functions. The pairwise p-vaues obtained from the Holm-Sidak test for all the algorithms show the statistical difference between the proposed Jaya algorithm and the other algorithms. The statistical difference between the proposed Jaya algorithm and the TLBO algorithm is smaller. Table 16 Results obtained by the Jaya algorithm for 30 bench mark functions over 30 independent runs with 500000 function evaluations No. Function Optimum Best Worst Mean SD F 1 Sphere 0 0 0 0 0.00E+00 F 2 SumSquares 0 0 0 0 0.00E+00 F 3 Beale 0 0 0 0 0.00E+00 F 4 Easom -1 -1 -1 -1 0.00E+00 F 5 Matyas 0 0 0 0 0.00E+00 F 6 Colville 0 0 0 0 0.00E+00 F 7 Trid 6 -50 -50 -50 -50 0.00E+00 F 8 Trid 10 -210 -210 -210 -210 0.00E+00 F 9 Zakharov 0 0 0 0 0.00E+00 F 10 Schwefel 1.2 0 0 0 0 0.00E+00 F 11 Rosenbrock 0 0 0 0 0.00E+00 F 12 Dixon-Price 0 0 0 0 0.00E+00 F 13 Foxholes 0.998 0.998004 0.998004 0.998004 0.00E+00 F 14 Branin 0.398 0.397887 0.397887 0.397887 0.00E+00 F 15 Bohachevsky 1 0 0 0 0 0.00E+00 F 16 Booth 0 0 0 0 0.00E+00 F 17 Michalewicz 2 -1.8013 -1.801303 -1.801303 -1.801303 0.00E+00 F 18 Michalewicz 5 -4.6877 -4.687658 -4.645895 -4.680138 1.58E-02 F 19 Bohachevsky 2 0 0 0 0 0.00E+00 F 20 Bohachevsky 3 0 0 0 0 0.00E+00 F 21 GoldStein-Price 3 3 3 3 0.00E+00 F 22 Perm 0 0 0 0 0.00E+00 F 23 Hartman 3 -3.86 -3.862780 -3.862780 -3.862780 0.00E+00 F 24 Ackley 0 0 0 0 0.00E+00 F 25 Penalized 2 0 0 0 0 0.00E+00 F 26 Langerman 2 -1.08 -1.080938 -1.080938 -1.080938 0.00E+00 F 27 Langerman 5 -1.5 -1.5 -0.482871 -1.240500 0.312691 F 28 Langerman 10 NA -1.5 -0.278661 -0.620503 0.319588 F 29 FletcherPowell 5 0 0 0.002794 0.000159 5.2E-04 F 30 FletcherPowell 10 0 0 5.33E-03 5.43E-04 9.87E-04 SD = Standard deviation 5. Experiments on unconstrained benchmark problems The performance of the proposed Jaya algorithm is tested further on 30 unconstrained benchmark functions well documented in the optimization literature. These unconstrained functions have different characteristics like unimodality/multimodality, separability/non-separability, regularity/non-regularity, etc. The number of design variables and their ranges are different for each problem. Table 15 shows the 30 unconstrained benchmark functions.To evaluate the performance of the proposed Jaya algorithm, the results obtained by using the Jaya algorithm are compared with the results obtained by the other optimization algorithms such as GA, PSO, DE, ABC and TLBO. A common platform is provided by maintaining the identical function evaluations for different algorithms considered for the comparison. Thus, the consistency in the comparison is maintained while comparing the performance of Jaya algorithm with other optimization algorithms. However, in general, the algorithm which requires less number of function evaluations to get the same best solution can be considered as better as compared to the other algorithms. However, in this paper, to maintain the consistency in the comparison of competitive algorithms, a common experimental platform is provided by setting the maximum number of function evaluations as 500000 for each benchmark function. Just like other algorithms, the proposed Jaya algorithm is executed 30 times for each benchmark function by choosing suitable population sizes and the mean results obtained are compared with the other algorithms for the same number of runs.
- 14. 32 Table 17 Comparative results of Jaya algorithm with other algorithms over 30 independent runs GA PSO DE ABC TLBO Jaya GA PSO DE ABC TLBO Jaya F1 M 1.11E+03 0 0 0 0 0 F13 M 0.998004 0.9980039 0.9980039 0.9980039 0.9980039 0.998004 SD 74.214474 0 0 0 0 0.00E+00 SD 0 0 0 0 0 0.00E+00 F2 M 1.48E+02 0 0 0 0 0 F14 M 0.397887 0.3978874 0.3978874 0.3978874 0.3978874 0.397887 SD 12.409289 0 0 0 0 0.00E+00 SD 0 0 0 0 0 0.00E+00 F3 M 0 0 0 0 0 0 F15 M 0 0 0 0 0 0 SD 0 0 0 0 0 0.00E+00 SD 0 0 0 0 0 0.00E+00 F4 M -1 -1 -1 -1 -1 -1 F16 M 0 0 0 0 0 0 SD 0 0 0 0 0 0.00E+00 SD 0 0 0 0 0 0.00E+00 F5 M 0 0 0 0 0 0 F17 M -1.8013 -1.5728692 -1.801303 -1.8013034 -1.801303 -1.801303 SD 0 0 0 0 0 0.00E+00 SD 0.00E+00 0.11986 0 0 0 0.00E+00 F6 M 0.014938 0 0.0409122 0.0929674 0 0 F18 M -4.64483 -2.4908728 -4.683482 -4.6876582 -4.6726578 -4.680138 SD 0.007364 0 0.081979 0.066277 0 0.00E+00 SD 0.09785 0.256952 0.012529 0.00E+00 4.74E-02 1.58E-02 F7 M -49.9999 -50 -50 -50 -50 -50 F19 M 0.06829 0.00 0.00 0.00 0.00 0 SD 2.25E–5 0 0 0 0 0.00E+00 SD 0.078216 0.00 0.00 0.00 0.00 0.00E+00 F8 M 0.193417 0 0 0 0 -210 F20 M 0.00 0.00 0.00 0.00 0.00 0 SD 0.035313 0 0 0 0 0.00E+00 SD 0.00 0.00 0.00 0.00 0.00 0.00E+00 F9 M 0.013355 0 0 0.0002476 0 0 F21 M 5.870093 3 3 3 3 3 SD 0.004532 0 0 0.000183 0 0.00E+00 SD 1.071727 0 0 0 0 0.00E+00 F10 M 7.40E+03 0 0 0 0 0 F22 M 0.302671 0.0360516 0.0240069 0.0411052 0.0006766 0 SD 1.14E+03 0 0 0 0 0.00E+00 SD 0.193254 0.048927 0.046032 0.023056 0.0007452 0.00E+00 F11 M 1.96E+05 15.088617 18.203938 0.0887707 1.62E-05 0 F23 M -3.86278 -3.6333523 -3.862782 -3.8627821 -3.862782 -3.862780 SD 3.85E+04 24.170196 5.036187 0.07739 3.64E-05 0.00E+00 SD 0.00E+00 0.116937 0 0 0 0.00E+00 F12 M 1.22E+03 0.6666667 0.6666667 0 0.6666667 0 F24 M 14.67178 0.1646224 0 0 0 0 SD 2.66E+02 E–8 E–9 0 0 0.00E+00 SD 0.178141 0.493867 0 0 0 0.00E+00 F25 M 125.0613 0.0076754 0.0021975 0 2.34E-08 0 F28 M -0.63644 -0.0025656 -1.0528 -0.4460925 -0.64906 -0.620503 SD 12.0012 0.016288 0.004395 0 0 0.00E+00 SD 0.374682 0.003523 0.302257 0.133958 0.1728623 0.319588 F26 M -1.08094 -0.679268 -1.080938 -1.0809384 -1.080938 -1.080938 F29 M 0.004303 1457.8834 5.988783 0.1735495 2.2038134 0.000159 SD 0 0.274621 0 0 0 0.00E+00 SD 0.009469 1269.3624 7.334731 0.068175 4.3863209 5.2E-04 F27 M 0.287548 0.213626 0 0.000208 1.55E-05 -1.240500 F30 M 29.57348 1364.4556 781.55028 8.2334401 35.971004 5.43E-04 SD 0.052499 0.039003 0 3.80E–05 2.83E-06 0.312691 SD 16.02108 1325.3797 1048.8135 8.092742 71.284369 9.87E-04
- 15. R. Venkata Rao / International Journal of Industrial Engineering Computations 7 (2016) 33 5.1. Results and discussion on unconstrained benchmark functions The results of Jaya algorithm corresponding to each benchmark function are presented in Table 16 in the form of best solution, worst solution, mean solution and standard deviation obtained in 30 independent runs on each benchmark function. The performance of Jaya algorithm is compared with the other well- known optimization algorithms such as GA, PSO, DE, ABC and TLBO and the results are given in Table 17. The results of GA, PSO, DE and ABC are taken from Karaboga and Akay (2009) and the results of TLBO are taken from Rao and Patel (2013) where the authors had experimented benchmark functions each with 500000 function evaluations with best setting of algorithm-specific parameters (except for TLBO for which, like Jaya algorithm, there are no algorithm-specific parameters). It can be observed from Tables 16 and 17 that the proposed Jaya algorithm has obtained better results in terms of “best’, “mean’ and ‘worst’ values of each objective function and ‘standard deviation’. Furthermore, it can be observed that the performance of the Jaya algorithm is found either equal or better than all other algorithms in all the 30 unconstrained benchmark functions. 6. Conclusions All the evolutionary and swarm intelligence based algorithms require proper tuning of algorithm-specific parameters in addition to tuning of common controlling parameters. A change in the tuning of the algorithm-specific parameters influences the effectiveness of the algorithm. The recently proposed TLBO algorithm does not require any algorithm-specific parameters and it only requires the tuning of the common controlling parameters of the algorithm for its working. Keeping in view of the success of the TLBO algorithm in the field of optimization, another algorithm-specific parameter-less algorithm is proposed in this paper and is named as ‘Jaya algorithm’. However, unlike two phases (i.e. teacher phase and the learner phase) of the TLBO algorithm, the proposed algorithm has only one phase and it is comparatively simpler to apply. The proposed algorithm is implemented on 24 well defined constrained optimization problems having different characteristics given in CEC 2006 competition. The results obtained by the proposed Jaya algorithm are compared with the results of well-known optimization algorithms such as HM, ASCHEA, SMES, GA, PSO, DE, ABC, BBO, HTS and TLBO algorithms for the considered constrained benchmark problems. Results have shown the satisfactory performance of Jaya algorithm for the constrained optimization problems. The statistical tests have also supported the performance supremacy of the proposed method. The proposed algorithm is also implemented on 30 well defined unconstrained optimization problems documented in the optimization literature. These unconstrained optimization problems have different characteristics and the results obtained by the proposed Jaya algorithm are compared with the results of well-known optimization algorithms such as GA, PSO, DE, ABC and TLBO algorithms. Results have shown the satisfactory performance of Jaya algorithm for the considered unconstrained optimization problems. It may be concluded that the proposed Jaya algorithm can be used for solving the constrained as well as unconstrained optimization algorithms. It is emphasized here that the proposed Jaya algorithm is not claimed as the ‘best’ algorithm among all the optimization algorithms available in the literature. In fact, there may not be any such ‘best’ algorithm existing for all types and varieties of problems! However, the Jaya algorithm is a newly proposed algorithm and is believed to have strong potential to solve the constrained and unconstrained optimization problems. If the algorithm is found having certain limitations then the efforts of the researchers should be to find out the ways to overcome the limitations and to further strengthen the algorithm. The efforts should not be in the form of destructive criticism. What can be said with more confidence at present about the Jaya algorithm is that it is simple to apply, it has no algorithm-specific parameters and it provides the optimum results in comparatively less number of function evaluations. Researchers are
- 16. 34 encouraged to make improvements to the Jaya algorithm so that the algorithm can become much more powerful with much improved performance. It is hoped that the researchers belonging to different disciplines of engineering and sciences (physical, life and social) will find the Jaya algorithm as a powerful tool to optimize the systems and processes. Readers interested to get the MATLAB code of the Jaya algorithm may contact the author. Readers interested to get the MATLAB code of the Jaya algorithm may refer to https://sites.google.com/site/jaya-algorithm/ Acknowledgement The author gratefully acknowledges the support of his present Ph.D. students (particularly, Mr. Dhiraj P. Rai for helping him in executing the code and Mr. G. G. Waghmare for providing the corrected TLBO results). References Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scandanavian Journal of Statistics, 6(2), 65-70. Joaquin, D., Salvador, G., Daniel, M., & Francisco, H. (2011). A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm and Evolutionary Computation, 1(1), 3–18. Karaboga, D., &Basturk, B. (2007). Artificial Bee Colony (ABC) Optimization Algorithm for Solving Constrained Optimization Problems, LNAI, SpringerVerlag Berlin, 4529, 789-798. Karaboga, D., &Akay, B. (2009). A comparative study of Artificial Bee Colony algorithm. Applied Mathematics and Computation, 214(1) 108-132. Karaboga, D., &Akay, B. (2011). A modified Artificial Bee Colony (ABC) algorithm for constrained optimization problems. Applied Soft Computing, 11, 3021-3031. Liang, J.J., Runarsson, T.P., Mezura-Montes, E., Clerc, M., Suganthan, P.N., Coello, C.A.C., & Deb, K. (2006). Problem Definitions and Evaluation Criteria for the CEC 2006 Special Session On Constrained Real-Parameter Optimization, Technical Report, Nanyang Technological University, Singapore, http://www.ntu.edu.sg/home/EPNSugan. Patel, V.K., &Savsani, V.J. (2015). Heat transfer search (HTS): a novel optimization algorithm. Information Sciences, 324, 217-246. Rao, R.V. (2015). Teaching Learning Based Optimization And Its Engineering Applications. Springer Verlag, London. Rao, R.V., & Patel, V. (2012). An elitist teaching-learning-based optimization algorithm for solving complex constrained optimization problems. International Journal of Industrial Engineering Computations, 3(4), 535-560. Rao, R.V., & Patel, V. (2013). Comparative performance of an elitist teaching-learning-based optimization algorithm for solving unconstrained optimization problems. International Journal of Industrial Engineering Computations, 4(1), 29-50. Rao, R.V., Savsani, V.J. &Vakharia, D.P. (2011). Teaching-learning-based optimization: A novel method for constrained mechanical design optimization problems. Computer-Aided Design, 43(3), 303-315.