Metaheuristic algorithms are one of the most widely used stochastic approaches in solving optimization problems. In this paper, a new metaheuristic algorithm entitled Billiards Optimization Algorithm (BOA) is proposed and designed to be used in optimization applications. The fundamental inspiration in BOA design is the behavior of the players and the rules of the billiards game. Various steps of BOA are described and then its mathematical model is thoroughly explained. The efficiency of BOA in dealing with optimization problems is evaluated through optimizing twenty-three standard benchmark functions of different types including unimodal, high-dimensional multimodal, and fixed-dimensional multimodal functions. In order to analyze the quality of the results obtained by BOA, the performance of the proposed approach is compared with ten well-known algorithms. The simulation results show that BOA, with its high exploration and exploitation abilities, achieves an impressive performance in providing solutions to objective functions and is superior and far more competitive compared to the ten competitor algorithms.
Optimization is the process of finding the best solution among all feasible solutions for a problem. In fact, a problem that has more than one “feasible solution” is known as an optimization problem. With advances in science and technology, scientists encounter newer optimization applications that must be addressed using effective optimization methods. Deterministic and stochastic approaches are considered as effective tools for optimization problems. Deterministic approaches perform well in solving linear, convex, continuous, and derivative optimization problems. However, most real-world optimization applications have features such as nonlinear, non-convex, non-derivative, discrete search space, and high dimensions. The weakness of deterministic approaches to handle optimization problems with such features has led to the emergence of stochastic approaches as well as metaheuristic algorithms [1].
Metaheuristic algorithms are stochastic-based approaches that are able to provide optimal solutions to optimization problems through utilizing random search, random operators, and trial and error processes [2]. Due to the random search process of optimization algorithms, there is no guarantee that the solution achieved by these methods is exactly the global optimum [3]. However, these are acceptable candidate solutions known as quasi-optimal solutions [4]. The optimization process in metaheuristic algorithms begins with the production of a number of random candidate solutions. Then, during the iterations of the algorithm, the candidate solutions are improved so that at the end of the algorithm implementation, the best candidate solution is identified as the solution to the problem [5]. Exploration with the concept of global search and exploitation with the concept of local search are two key factors in the optimization process accomplished by metaheuristic algorithms. Balancing these two capabilities during the iterations of the algorithm plays an important role in the algorithm success to achieve the optimal solution.
Metaheuristic algorithms have been used in many scientific applications, including prediction of linear dynamical systems [6], solving systems of singular boundary value problems [7], solutions of Troesch’s and Bratu’s problems [8], deep learning algorithms [9–11], energy carriers [12,13], electrical engineering [14–19], protection [20], and energy management [21–24]. The design of metaheuristic algorithms is inspired by various natural phenomena, animal behavior in nature, physical laws, biological sciences, rules of games, and so on [25]. Particle Swarm Optimization (PSO) is a nature-inspired approach that is modeled based on the search behavior and strategy of fishes and birds for food [26]. Genetic Algorithm (GA) is a well-known metaheuristic algorithm based on mathematical modeling of biological sciences [27]. Gravitational Search Algorithm (GSA) is a metaheuristic algorithm designed based on modeling the laws of physics and the force of gravity [28]. Teaching-Learning Based Optimization (TLBO) is an optimization approach introduced based on the simulation of human activities [29]. Volleyball Premier League (VPL) is an optimization approach developed on the basis of modeling the rules and conditions governing volleyball league competitions [30].
The main research question is that considering various metaheuristic algorithms that have been investigated so far, is there still a need to introduce newer metaheuristic algorithms? In response to this question, No Free Lunch (NFL) theorem [31] explains that there is no guarantee for the same performance of a metaheuristic algorithm in all optimization problems. Based on the NFL theorem, a metaheuristic algorithm may provide very favorable results for a set of objective functions, but it may fail to address some other optimization problems. Hence, based on the NFL theorem, there is no assumption about the success or failure of implementing a metaheuristic algorithm on an optimization problem. The NFL theorem motivates researchers to provide suitable solutions for optimization problems through designing new metaheuristic algorithms.
The innovation and novelty of this paper is the design of a new game-based metaheuristic algorithm called Billiards Optimization Algorithm (BOA) to deal with optimization applications in different fields of science and technology. The contribution of this article is as follows. The basic idea of BOA is the rules of the billiards game and the strategy of the players during this game. Different steps of BOA implementation are described and mathematically modeled. The effectiveness of BOA in solving optimization problems is tested for twenty-three standard benchmark functions of unimodal and multimodal types. The results obtained by BOA are compared with the solutions achieved by ten well-known metaheuristic algorithms.
The rest of the article is as follows: In Section 2, literature review is provided. The proposed BOA is introduced in Section 3. Simulation studies and results are presented in Section 4. Finally, conclusions and some suggestions for future investigations are provided in Section 5.
Lecture Review
Metaheuristic algorithms can be divided into five groups based on the source of inspiration in their design: swarm-based, evolutionary-based, physics-based, game-based, and human-based algorithms.
Swarm-based algorithms are inspired by nature, and the swarming behaviors of animals, birds, and other living things. PSO is one of the most popular and one of the first optimization methods, which was developed based on modeling the movement of birds and fishes in search of food. The ability of ants to discover the shortest path between food sources and nests has been a central idea in Ant Colony Optimization (ACO) design [32]. The strategy of grey wolves in hunting and attacking prey is employed in the Grey Wolf Optimizer (GWO) design [33]. The humpback whale net-bubble hunting strategy has been a major source of inspiration in Whale Optimization Algorithm (WOA) [34]. The strategy of pelicans in hunting and trapping prey has been the main inspiration of Pelican Optimization Algorithm (POA) [35]. Some other swarm-based algorithms proposed so far are: Fennec Fox Optimization (FFO) [36], Reptile Search Algorithm (RSA) [37], Cat and Mouse based Optimizer (CMBO) [38], Good Bad Ugly Optimizer (GBUO) [39], Marine Predator Algorithm (MPA) [40], Tasmanian Devil Optimization (TDO) [41], Mutated Leader Algorithm (MLA) [42], Tunicate Search Algorithm (TSA) [43], Northern Goshawk Optimization (NGO) [44], Donkey Theorem Optimizer (DTO) [45], Rat Swarm Optimization (RSO) [46], All Members Based Optimizer (AMBO) [47], Red Fox Optimization (RFO) [48], Mixed Best Members Based Optimizer (MBMBO) [49], and Mixed Leader Based Optimizer (MLBO) [50].
Evolutionary-based algorithms are inspired by the concepts of natural selection, biology, and genetics. GA and Differential Evolution (DE) [51] are among the most widely used and well-known evolutionary algorithms, inspired by the process of reproduction, Darwin’s theory of evolution, and random operators such as selection, crossover, and mutation. Some other evolutionary-based algorithms can be listed as: Average and Subtraction-Based Optimizer (ASBO) [52], Genetic Programming (GP) [53], Search Step Adjustment Based Algorithm (SSABA) [54], Evolutionary Programming (EP) [55], Selecting Some Variables to Update-Based Algorithm (SSVUBA) [56], and Artificial Immune System (AIS) technique [57].
Physics-based algorithms are arisen from simulating processes, laws, and concepts in physics. Simulated Annealing (SA) is one of the most famous optimization algorithms that is inspired by the physical process of melting and cooling metals known as annealing [58]. Simulation of the law of curvature and spring force was the main idea for designing Spring Search Algorithm (SSA) [59], simulation of momentum and collision of objects has been the main inspiration of Momentum Search Algorithm (MSA) [60], simulation of gravitational force and laws of motion between objects at different distances has been the main inspiration of GSA. Some other physics-based algorithms are: Multi-Verse Optimizer (MVO) [61], Binary Spring Search Algorithm (BSSA) [62], Equilibrium Optimizer (EO) [63], and Henry Gas Solubility Optimization (HGSO) [64].
Game-based algorithms are inspired by modeling the rules of various individual and team games, the behavior of players, coaches, and referees, as well as holding competitions. Simulating the organization of the football league as well as the behavior of the players, and the interactions of the clubs with each other has been the main idea in the design of Football Game Based Optimization (FGBO) [65]. The players’ effort to put the puzzle pieces together has been the major inspiration in Puzzle Optimization Algorithm (POA) design [66]. Hide Object Game Optimizer (HOGO) is an optimization approach developed based on modeling players' behavior to find a hidden object in the hide object game [67]. Some other game-based algorithms are: VPL, Darts Game Optimizer (DGO) [68], Binary Orientation Search algorithm (BOSA) [69], Shell Game Optimization (SGO) [70], Orientation Search algorithm (OSA) [71], and Ring Toss Game Based Optimizer (RTGBO) [72].
Human-based algorithms are inspired by human behaviors, the interactions of people in a community with each other, and the relationships between humans. TLBO is one of the most widely used optimization techniques that is designed based on modeling the space of a classroom with the presence of a teacher and a number of students in two phases of teaching and learning. Interactions between doctor and patients for examination, prevention, and treatment have been the main inspiration of Doctor and Patient Optimization (DPO) [73]. Modeling and following the people of a society by the leader in order to develop that society has been the source of inspiration in designing Following Optimization Algorithm (FOA) [74]. Some other human-based algorithms are: Teamwork Optimization Algorithm (TOA) [75], Group Optimization (GO) [76], Archery Algorithm (AA) [77], Poor and Rich Optimization (PRO) technique [78], and Skill Optimization Algorithm (SOA) [79].
Based on the best knowledge gained from the above literature review, the behavior of players in the game of billiards has not been used in the design of any metaheuristic algorithm so far. While billiards players try to place balls in pockets is an intelligence process that has a good potential to design an optimizer. In order to address this research gap, in this study, based on the mathematical modeling of this process, a new metaheuristic algorithm has been designed, which is discussed in the next section.
Billiards Optimization Algorithm
In this section, theory of Billiards Optimization Algorithm (BOA) is described and then its mathematical modeling for use in optimization applications is presented.
Inspiration of BOA
Billiards is a fascinating sport that has become a favorite of many people. However, different games are based on the rules and conditions of billiards. In general, in this game, players try to place the balls inside the pockets of the billiards table. The players’ strategy to perform this activity is an intelligence behavior that has a potential to be designed as an optimizer. BOA is based on the simulation of players’ behavior in this game, which is discussed below.
Initialization of BOA
The BOA approach is a population-based metaheuristic algorithm whose members are billiards balls. Each BOA member is a candidate solution that assigns values to problem variables based on its position in the search space. BOA members can be mathematically modeled using a matrix according to Eq. (1). BOA members are randomly initialized at the beginning of the algorithm implementation using Eq. (2).
Here, X is the population matrix of BOA, Xi indicates ith candidate solution, xi,d represents the value of dth variable proposed by ith population member, N is the number of BOA’s members, m denotes the number of variables, r is a random number in interval [0−1], lbd and ubd are the lower bound and upper bound of dth variable, respectively.
Given that each member of the population is a candidate solution to the problem, a value for the objective function is calculated for each member of the population. The values obtained for the objective function can be represented mathematically using a vector according to Eq. (3).
F=[F1⋮Fi⋮FN]N×1=[F(X1)⋮F(Xi)⋮F(XN)]N×1,
Here, F is the vector of objective function values and Fi represents the obtained value for the objective function based on ith population member.
Considering the values obtained for the objective function, the best value identifies the best member of the population. Given that in each iteration, the positions of the population members are updated and new values are evaluated for the objective function, the best member of the population must be updated in each iteration.
Mathematical Model of BOA
In BOA, members of the population are updated based on the players’ behavior in placing balls in the pockets of the game table. Since the billiards table has 8 pockets, in each repetition, among the members of the population, 8 better members are considered as the pockets. The rest of the population is considered as balls that have to be moved towards these pockets. The player selects one of these pockets to hit each ball. The BOA design assumes that the pocket is chosen at random. In order to simulate this process, a proposed random position is first calculated according to Eq. (4) for each member of the population. Then, if the value of the objective function is improved in the new position, it replaces the previous position of that member according to Eq. (5).
Xinew:xi,dnew=xi,d+r×(Pi,d−I×xi,d)
Xi={Xinew,Finew<FiXi,else,
Here, Xinew is the new calculated status of ith population member, xi,dnew indicates its dth dimension, Finew represents its corresponding objective function value, Pi is the selected pocket for ith population member, Pi,d denotes its dth dimension, r is a random number in interval [0−1], and I is a random number, which is selected randomly from set of {1,2}.
Repetition Process, Pseudocode, and Flowchart of BOA
After updating all members of the population, the first iteration of the BOA is completed. The algorithm then enters the next iteration based on the new positions obtained for the population members. The process of updating members of the BOA population is repeated until the end of the algorithm implementation according to Eqs. (4) and (5). After complete implementation of the BOA, the best candidate solution obtained during the algorithm iterations is the output, which is the solution to the problem. The BOA implementation steps are shown through a flowchart in Fig. 1 and the corresponding pseudo-code is presented in Algorithm 1.
Flowchart of Billiards Optimization Algorithm (BOA)
Pseudo-code of Billiards Optimization Algorithm (BOA)
Start BOA.
1. Input the optimization problem information.
2. Set T (number of iterations) and N (number of population members).
3. Generate the initial position of the BOA members using Eq. (2). xi,d←lbd+r×(ubd−lbd).
4. Evaluate the objective function.
5. fort = 1 to Tdo
7. Determine the position of the pockets based on eight better members.
8. for i = 1 to Ndo
9. Select the target pocket at random among eight pockets.
10. Calculate new status of ith candidate solution using Eq. (4). xi,dnew←xi,d+r×(Pi,d−I×xi,d);
11. Update ith candidate solution using Eq. (5). Xi←{Xinew,Finew<FiXi,else;
12. end for
13. Save the best candidate solution so far.
14. end for
15. Output the best obtained solution.
End BOA.
Computational Complexity
In this subsection, the computational complexity of the BOA is analyzed. BOA initialization has a computational complexity of O(Nm), where N is the BOA population size and m is the number of problem variables. The process of updating the algorithm population has a computational complexity equal to O(NmT), where T is the number of algorithm iterations. Accordingly, the total computational complexity of BOA is equal to O(Nm(1 + T)).
Simulation Studies and Results
In this section, the performance of BOA in solving optimization problems is evaluated. To this end, BOA has been implemented on twenty-three standard objective functions of unimodal, high-dimensional multimodal, and fixed-dimensional multimodal types. Comprehensive information of these test functions including the mathematical model, dimensions, constraints, optimal value of the objective function, and other details has been presented in [80]. The capability of BOA in providing solutions to these functions has been compared with the performance of ten well-known metaheuristic algorithms including GA, WOA, RSA, PSO, GWO, MPA, MVO, GSA, TLBO, and TSA. The proposed BOA approach and ten competitor algorithms are each implemented on F1 to F23 functions in twenty independent implementations, while each implementation contains 1000 iterations. The optimization results are presented using six statistical indicators of mean, best, worst, standard deviation (std), median, and rank.
Evaluation of Unimodal Functions
Unimodal functions F1 to F7 are suitable for testing local search capability or the exploitation of metaheuristic algorithms since they have only one optimal solution. The results of optimization of F1 to F7 functions using BOA and the mentioned competitor algorithms are reported in Table 1. The results show that BOA has provided the global optimum for F1 to F6 functions. Also, in solving F7, the BOA approach is the first best optimizer for this function. Based on the presented simulation results, it is clear that BOA has a high capability to solve unimodal problems and its performance is superior to competitor algorithms.
Evaluation results for unimodal functions
GA
PSO
GSA
TLBO
GWO
MVO
WOA
TSA
MPA
RSA
BOA
F1
mean
34.5437
0.011351
9.92E-17
4.15E-75
8.45E-59
0.141392
6.1E-150
1.26E-47
1.88E-49
6.46E-84
0
best
21.8428
2.18E-05
5.36E-17
1.46E-76
5.78E-61
0.085887
1.1E-172
1.9E-50
2.3E-52
9.43E-93
0
worst
48.27454
0.097758
2.23E-16
2.18E-74
6.53E-58
0.229047
1.2E-148
9.02E-47
1.75E-48
1.19E-82
0
std
8.103596
0.021904
4.27E-17
5.75E-75
1.47E-58
0.035688
2.7E-149
2.43E-47
4.01E-49
2.64E-83
0
median
33.39374
0.004052
8.44E-17
1.9E-75
4.14E-59
0.134685
2.2E-160
7.09E-49
1.92E-50
3.69E-88
0
rank
11
9
8
4
5
10
2
7
6
3
1
F2
mean
2.893674
1.469507
5.21E-08
3.96E-39
9.56E-35
0.256823
4.8E-104
1.05E-28
5.4E-28
6.78E-46
0
best
1.771522
0.129615
3.14E-08
8.03E-40
1.9E-36
0.162297
3.4E-117
1.05E-30
1.58E-29
4.79E-49
0
worst
4.187515
10.91586
7.28E-08
1.2E-38
2.83E-34
0.393994
9.4E-103
6.38E-28
2.66E-27
5.43E-45
0
std
0.688131
2.427296
1.18E-08
3.01E-39
8.16E-35
0.061766
2.1E-103
1.6E-28
7.3E-28
1.51E-45
0
median
2.928265
0.819186
4.9E-08
2.99E-39
7.25E-35
0.25831
8.9E-108
3.99E-29
2.14E-28
3.56E-47
0
rank
11
10
8
4
5
9
2
6
7
3
1
F3
mean
2151.287
874.6891
474.5464
1.19E-24
6.36E-15
13.69546
19397.34
1.96E-12
2.7E-12
4.76E-58
0
best
1306.053
38.39523
191.6011
8.36E-29
7.31E-19
6.427129
1155.268
2.74E-17
1.58E-21
1.19E-69
0
worst
3690.226
5365.03
1028.324
1.56E-23
5.42E-14
23.71885
46521.37
2.32E-11
2.68E-11
5.35E-57
0
std
651.9863
1532.811
210.8921
3.52E-24
1.38E-14
5.422219
11275.71
5.22E-12
7.22E-12
1.3E-57
0
median
2013.684
279.0787
413.2552
3.35E-26
1.58E-16
12.17659
22075.51
9.99E-14
1.21E-13
1.49E-61
0
rank
10
9
8
3
4
7
11
5
6
2
1
F4
mean
3.182379
6.409232
1.347784
4.73E-30
1.34E-14
0.575497
45.73347
0.006311
3.38E-19
1.34E-35
0
best
2.460207
2.625176
1.93E-08
8.45E-32
7.64E-16
0.203873
0.047089
6.36E-06
3.7E-20
3.83E-40
0
worst
4.320177
9.826729
3.852453
2.31E-29
1.1E-13
0.98408
88.30133
0.074488
8.65E-19
1.66E-34
0
std
0.440358
2.122281
1.08719
5.62E-30
2.5E-14
0.180899
32.22163
0.016314
2.17E-19
3.82E-35
0
median
3.158793
6.156931
1.060589
2.21E-30
4.31E-15
0.59779
42.59176
0.001452
2.94E-19
2.7E-37
0
rank
9
10
8
3
5
7
11
6
4
2
1
F5
mean
512.3849
4685.587
26.4173
26.85637
26.80292
237.6288
27.19238
28.27683
23.63643
27.45887
0
best
227.6302
6.358709
25.84867
25.97881
25.29612
26.95008
26.45993
26.48034
22.44977
26.21217
0
worst
1904.539
90133.63
27.6268
28.55723
27.93213
1709.385
28.5018
28.86228
24.27026
28.59278
0
std
355.6761
20115.89
0.455677
0.70839
0.739022
398.9057
0.541389
0.706209
0.4456
0.72896
0
median
440.5026
81.59334
26.32633
26.69676
27.11089
63.20481
27.04987
28.63414
23.65578
27.18532
0
rank
10
11
3
5
4
9
6
8
2
7
1
F6
mean
34.86707
0.060897
1.25E-16
1.078098
0.705754
0.136757
0.077372
3.848483
1.86E-09
1.54416
0
best
16.97404
9.5E-06
4.71E-17
0.54459
1.77E-05
0.067418
0.009203
2.589849
1.03E-09
0.862897
0
worst
73.13031
0.835192
4.43E-16
1.595351
1.724524
0.237401
0.713129
4.796125
4.73E-09
2.393213
0
std
17.35489
0.184214
9.08E-17
0.298311
0.459725
0.03697
0.153834
0.592668
8.37E-10
0.399298
0
median
31.4086
0.009074
8.57E-17
1.088354
0.739962
0.14036
0.036663
4.050477
1.71E-09
1.639428
0
rank
11
4
2
8
7
6
5
10
3
9
1
F7
mean
0.010645
0.164599
0.061373
0.001623
0.000817
0.009972
0.001116
0.005855
0.000615
0.000401
3.56E-05
best
0.003995
0.095988
0.019791
0.000455
0.00012
0.0056
7.13E-05
0.00218
0.000215
2.99E-05
5.88E-07
worst
0.017568
0.285254
0.112113
0.003976
0.001565
0.018206
0.003835
0.013816
0.00151
0.000953
0.000105
std
0.004044
0.052236
0.023729
0.001106
0.000407
0.003234
0.001128
0.003043
0.000314
0.000307
3.34E-05
median
0.00971
0.153973
0.056558
0.001292
0.000774
0.009512
0.000817
0.004586
0.000571
0.000317
2.71E-05
rank
9
11
10
6
4
8
5
7
3
2
1
sum rank
71
64
47
33
34
56
42
49
31
28
7
mean rank
10.14286
9.142857
6.714286
4.714286
4.857143
8
6
7
4.428571
4
1
total rank
11
10
7
4
5
9
6
8
3
2
1
Evaluation of High-Dimensional Multimodal Functions
High-dimensional multimodal functions F8 to F13 are appropriate for testing global search capability or exploration of metaheuristic algorithms due to having multiple local optimal solutions. The results of BOA implementation and competitor algorithms on F8 to F13 functions are presented in Table 2. The optimization results show that BOA has provided the global optimum for F9 and F11 functions. In solving the functions F8, F10, F12, and F13, the proposed approach is the first best optimizer. Based on the simulation results, it is concluded that BOA has a high ability to solve high-dimensional multimodal problems and its global search power and exploration capability is superior than the considered competitor algorithms.
Evaluation results for high-dimensional multimodal functions
GA
PSO
GSA
TLBO
GWO
MVO
WOA
TSA
MPA
RSA
BOA
F8
mean
−8348.04
−7174.66
−2512.77
−5400.22
−6359.7
−7991.99
−10779.6
−5925.69
−9619.16
−7548.39
−12569.5
best
−9571.05
−8778.83
−2969.06
−6984.4
−7834.43
−9028.56
−12569.5
−7583.05
−10355.8
−9259.4
−12569.5
worst
−6569.32
−5206.21
−2015.67
−4465
−4357.11
−7097.08
−8026.53
−4728.78
−9025.09
−5383.42
−12569.5
std
757.8547
895.3058
260.3632
685.2514
877.8481
632.0867
1645.717
716.9043
417.5433
1154.307
1.87E-12
median
−8603.98
−7160.36
−2498.05
−5345.39
−6523.61
−7972.62
−10889.5
−5954.46
−9629.9
−7805.26
−12569.5
rank
4
7
11
10
8
5
2
9
3
6
1
F9
mean
58.86846
65.44741
29.15228
0
0.980934
103.7514
0
160.3345
0
0
0
best
16.8933
27.85958
16.9143
0
0
80.65951
0
115.0893
0
0
0
worst
95.10693
126.4306
46.76302
0
9.129372
141.3836
0
231.9054
0
0
0
std
19.16566
23.37939
7.881373
0
2.70233
15.81913
0
31.28116
0
0
0
median
58.96616
62.19288
26.86388
0
0
102.5599
0
165.7714
0
0
0
rank
4
5
3
1
2
6
1
7
1
1
1
F10
mean
3.546679
2.925435
8.19E-09
4.44E-15
1.58E-14
0.44077
3.2E-15
1.468278
4.09E-15
4.54E-13
8.88E-16
best
2.698017
1.897756
6.07E-09
4.44E-15
1.15E-14
0.087453
8.88E-16
7.99E-15
8.88E-16
8.88E-16
8.88E-16
worst
4.12547
4.878091
1.12E-08
4.44E-15
2.22E-14
2.141281
7.99E-15
3.546227
4.44E-15
9.03E-12
8.88E-16
std
0.430288
0.938045
1.35E-09
0
2.96E-15
0.611269
2.09E-15
1.672925
1.09E-15
2.02E-12
0
median
3.502708
2.817299
8.05E-09
4.44E-15
1.51E-14
0.133476
4.44E-15
2.22E-14
4.44E-15
8.88E-16
8.88E-16
rank
11
10
7
4
5
8
2
9
3
6
1
F11
mean
1.582293
0.320849
8.822246
0
0.005007
0.421495
0.009403
0.006064
0
0
0
best
1.214307
0.006832
3.255758
0
0
0.253372
0
0
0
0
0
worst
2.154375
2.005863
18.13903
0
0.047681
0.592456
0.081925
0.017241
0
0
0
std
0.243938
0.466655
4.109461
0
0.011945
0.079737
0.023873
0.006718
0
0
0
median
1.535265
0.099446
9.042558
0
0
0.423356
0
0.004493
0
0
0
rank
7
5
8
1
2
6
4
3
1
1
1
F12
mean
0.157037
1.423618
0.127676
0.072387
0.037371
0.960236
0.011252
6.958465
2.31E-10
0.069238
1.62E-32
best
0.035385
4.47E-05
3.7E-19
0.03343
0.006546
0.000691
0.001059
0.404251
7.31E-11
0.012096
1.57E-32
worst
0.359563
3.854956
0.635088
0.178688
0.073536
3.859424
0.083424
15.28632
5.8E-10
0.179779
2.54E-32
std
0.091936
1.221873
0.213301
0.030545
0.019632
0.990186
0.01782
4.73628
1.16E-10
0.039794
2.16E-33
median
0.147131
1.2003
1.07E-18
0.064612
0.036212
0.641166
0.00666
7.798794
1.93E-10
0.061529
1.57E-32
rank
8
10
7
6
4
9
3
11
2
5
1
F13
mean
2.614757
3.869959
0.100869
0.983304
0.633228
0.030971
0.261863
2.988147
0.001652
1.803955
7.65E-32
best
1.149958
0.263545
4.86E-18
0.58094
0.113218
0.016912
0.049589
2.280729
1.28E-09
1.051985
1.35E-32
worst
4.837813
17.42028
1.222664
1.450667
1.136582
0.066663
0.519796
5.169063
0.010987
2.793816
6.07E-31
std
1.063186
4.538492
0.28389
0.239798
0.282332
0.012548
0.141363
0.695106
0.004024
0.41072
1.61E-31
median
2.458018
1.703829
1.46E-17
0.929117
0.638573
0.028099
0.253052
2.728009
3.35E-09
1.694537
1.35E-32
rank
9
11
4
7
6
3
5
10
2
8
1
sum rank
43
48
40
29
27
37
17
49
12
27
6
mean rank
7.166667
8
6.666667
4.833333
4.5
6.166667
2.833333
8.166667
2
4.5
1
total rank
8
9
7
5
4
6
3
10
2
4
1
Evaluation of Fixed-Dimensional Multimodal Functions
Fixed-dimensional multimodal functions F14 to F23 are suitable for evaluating the ability of metaheuristic algorithms to strike a balance between exploration and exploitation. The results obtained for F14 to F23 functions using BOA and the mentioned competitor algorithms are released in Table 3. BOA approach is the first best optimizer for F15 and F20 functions. Considering other functions of this group, where BOA achieves values similar to some competitor algorithms for the mean index, it provides better values for the std index. Analysis of the simulation results shows that BOA has a superior ability to balance exploration and exploitation characteristics compared to competitor algorithms.
Evaluation results for fixed-dimensional multimodal functions
GA
PSO
GSA
TLBO
GWO
MVO
WOA
TSA
MPA
RSA
BOA
F14
mean
1.012829
3.645876
4.086478
1.196416
4.866265
0.998004
2.816492
8.893169
0.998004
4.823742
0.998004
best
0.998004
0.998004
1.019228
0.998004
0.998004
0.998004
0.998004
0.998004
0.998004
0.998004
0.998004
worst
1.194013
11.7187
9.831309
2.982105
12.67051
0.998004
10.76318
12.67051
0.998004
12.67051
0.998004
std
0.045865
3.732496
2.561839
0.610693
4.26894
3.16E-12
2.996068
4.787012
5.09E-17
3.851995
0
median
0.998006
1.992031
3.970242
0.998004
2.982105
0.998004
1.495018
12.67051
0.998004
3.96825
0.998004
rank
3
6
7
4
9
2
5
10
1
8
1
F15
mean
0.00603
0.001432
0.002131
0.000453
0.005459
0.004558
0.00063
0.008535
0.000311
0.005053
0.000307
best
0.000644
0.000307
0.001143
0.00031
0.000307
0.000308
0.000312
0.000308
0.000308
0.000307
0.000307
worst
0.023479
0.019276
0.004428
0.001241
0.020363
0.020363
0.002178
0.020942
0.000316
0.022553
0.000307
std
0.007244
0.00422
0.000696
0.0003
0.008834
0.008113
0.000457
0.009974
2.25E-06
0.008991
1.90E-19
median
0.004113
0.000307
0.00203
0.000315
0.000308
0.000627
0.000481
0.001072
0.000311
0.000653
0.000307
rank
10
5
6
3
9
7
4
11
2
8
1
F16
mean
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03005
−1.03163
−0.99082
−1.03163
best
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
worst
−1.03161
−1.03163
−1.03163
−1.03162
−1.03163
−1.03163
−1.03163
−1
−1.03163
−0.21546
−1.03163
std
4.59E-06
1.35E-16
1.35E-16
1.59E-06
2.98E-09
3.18E-08
8.38E-11
0.007072
2.28E-16
0.1825
8.31E-17
median
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
−1.03163
rank
7
1
1
6
4
5
3
8
2
9
1
F17
mean
0.422023
0.65439
0.397887
0.403109
0.397898
0.397887
0.397888
0.397909
0.397887
0.397887
0.397887
best
0.397887
0.397887
0.397887
0.397893
0.397887
0.397887
0.397887
0.397888
0.397887
0.397887
0.397887
worst
0.832817
1.937365
0.397887
0.500697
0.3981
0.397888
0.39789
0.397946
0.397887
0.397887
0.397887
std
0.09697
0.526896
0
0.02297
4.74E-05
7.33E-08
7.31E-07
1.7E-05
0
8.97E-16
0
median
0.398304
0.397887
0.397887
0.397965
0.397888
0.397887
0.397887
0.397907
0.397887
0.397887
0.397887
rank
8
9
1
7
5
3
4
6
1
2
1
F18
mean
8.421245
3
3
3
7.050008
3
3.000018
14.20181
3
13.8
3
best
3
3
3
3
3.000001
3
3
3.000001
3
3
3
worst
30.31682
3
3
3.000002
84.00001
3.000001
3.000147
92.03579
3
84
3
std
11.11524
2.82E-15
2.79E-15
4.58E-07
18.11215
3.24E-07
3.46E-05
26.59322
1E-15
20.3563
2.88E-16
median
3.000586
3
3
3
3.000005
3
3.000005
3.000009
3
3
3
rank
8
2
3
5
7
4
6
10
1
9
1
F19
mean
−3.86265
−3.82413
−3.86278
−3.86051
−3.8621
−3.86278
−3.86068
−3.86225
−3.86278
−3.74604
−3.86278
best
−3.86278
−3.86278
−3.86278
−3.8627
−3.86278
−3.86278
−3.86276
−3.86278
−3.86278
−3.86278
−3.86278
worst
−3.86161
−3.08976
−3.86278
−3.85474
−3.8549
−3.86278
−3.8549
−3.85501
−3.86278
−3.08976
−3.86274
std
0.000338
0.172852
1.92E-15
0.003385
0.002056
2.03E-07
0.002361
0.001753
2.28E-15
0.282864
9.02E-16
median
−3.86278
−3.86278
−3.86278
−3.86238
−3.86277
−3.86278
−3.86162
−3.86273
−3.86278
−3.86278
−3.86278
rank
4
9
1
8
6
3
7
5
2
10
1
F20
mean
−3.19843
−3.30089
−3.322
−3.27123
−3.25578
−3.26246
−3.25729
−3.25227
−3.322
−3.19517
−3.322
best
−3.31774
−3.322
−3.322
−3.31452
−3.32199
−3.32199
−3.32181
−3.32165
−3.322
−3.322
−3.322
worst
−3.02507
−3.13764
−3.322
−3.15712
−3.13762
−3.20273
−3.08687
−3.08336
−3.322
−1.9217
−3.322
std
0.081657
0.052988
3.67E-16
0.056572
0.06962
0.06108
0.091694
0.075807
3.81E-16
0.311345
1.41E-17
median
−3.19362
−3.322
−3.322
−3.30495
−3.26241
−3.26254
−3.32111
−3.26131
−3.322
−3.322
−3.322
rank
9
3
1
4
7
5
6
8
2
10
1
F21
mean
−4.05229
−6.6523
−6.21031
−6.63717
−9.6474
−8.12535
−9.89304
−5.79327
−10.1532
−8.78928
−10.1532
best
−7.88766
−10.1532
−10.1532
−9.30299
−10.153
−10.1532
−10.1529
−10.1034
−10.1532
−10.1532
−10.1532
worst
−2.294
−2.63047
−2.68286
−4.07333
−5.09985
−5.05516
−5.05519
−2.62401
−10.1532
−0.88199
−10.1532
std
2.072922
3.659234
3.702041
2.127264
1.555137
2.54809
1.138781
3.016364
2.41E-15
3.181731
2.07E-17
median
−2.62469
−7.62699
−4.18158
−6.69275
−10.1527
−10.1531
−10.1516
−4.95475
−10.1532
−10.1524
−10.1532
rank
11
7
9
8
4
6
3
10
2
5
1
F22
mean
−6.76101
−8.1775
−9.6989
−8.06962
−10.4024
−8.04873
−8.69025
−7.01929
−10.4029
−8.05397
−10.4029
best
−10.2388
−10.4029
−10.4029
−9.9566
−10.4029
−10.4029
−10.4029
−10.3661
−10.4029
−10.4029
−10.4029
worst
−2.5174
−2.75193
−4.67391
−3.94552
−10.402
−2.76589
−2.76539
−2.68875
−10.4029
−0.90808
−10.4029
std
3.154309
3.169789
1.757215
1.582401
0.000256
3.030069
2.723616
3.582683
3.65E-15
3.599306
1.61E-16
median
−7.94704
−10.4029
−10.4029
−8.43317
−10.4025
−10.4029
−10.4006
−9.45916
−10.4029
−10.3962
−10.4029
rank
11
6
4
7
3
9
5
10
2
8
1
F23
mean
−8.18721
−5.6555
−10.5364
−8.03277
−10.536
−9.99793
−9.6541
−7.04655
−10.5364
−7.32853
−10.5364
best
−10.3471
−10.5364
−10.5364
−9.58103
−10.5363
−10.5364
−10.5362
−10.5028
−10.5364
−10.5364
−10.5364
worst
−2.66877
−2.42173
−10.5364
−3.95463
−10.5356
−5.12847
−3.83473
−2.41642
−10.5364
−1.85948
−10.5363
std
2.385851
3.382098
1.58E-15
1.775707
0.000194
1.657273
2.150189
3.807051
2.31E-15
4.034066
1.95E-16
median
−8.74676
−3.83543
−10.5364
−8.62434
−10.536
−10.5363
−10.5346
−10.1711
−10.5364
−10.508
−10.5364
rank
6
10
1
7
3
4
5
9
2
8
1
sum rank
77
58
34
59
57
48
48
87
17
77
10
mean rank
7.7
5.8
3.4
5.9
5.7
4.8
4.8
8.7
1.7
7.7
1
total rank
8
6
3
7
5
4
4
9
2
8
1
The boxplot diagrams for the performance of BOA and competitor algorithms in optimizing functions F1 to F23 are shown in Fig. 2.
Boxplots of performances for Billiards Optimization Algorithm (BOA) and competitor algorithms
Statistical Analysis
In this subsection, a statistical analysis is presented to evaluate the BOA performance against the considered competitor algorithms. The Wilcoxon rank sum test [81], which is a non-parametric test, is used to accomplish this analysis. In this test, using an index called p-value, it is shown whether there is a significant difference between the mean of the two data samples.
The results of Wilcoxon rank sum test statistical analysis on the outputs of BOA and competitor algorithms are reported in Table 4. It can be seen from the simulation results that in cases where the p-value is less than 0.05, the proposed BOA has a significant statistical superiority over the corresponding competitor algorithm. Considering the obtained p-values and since the value of this index is less than 0.05 in all cases, it is concluded that BOA has a significant statistical superiority over all ten competitor algorithms in optimizing all three types of objective functions including unimodal, high-dimensional multimodal, and fixed-dimensional multimodal functions.
<italic>p</italic>-values obtained by Wilcoxon sum rank test
Compared Algorithms
Unimodal
High-Multimodal
Fixed-Multimodal
BOA vs. GA
1.00E-24
1.95E-21
0.000691
BOA vs. PSO
1.00E-24
1.95E-21
1.93E-05
BOA vs. GSA
1.00E-24
1.95E-21
5.43E-11
BOA vs. TLBO
1.00E-24
6.91E-15
2.43E-05
BOA vs. GWO
1.00E-24
1.16E-16
2.11E-12
BOA vs. MPO
1.00E-24
1.95E-21
2.07E-18
BOA vs. WOA
1.00E-24
5.11E-14
6.38E-12
BOA vs. MPA
1.00E-24
8.64E-20
0.222474
BOA vs. TSA
1.00E-24
1.52E-14
1.43E-34
BOA vs. RSA
1.00E-24
5.12E-12
0.002713
Conclusions and Future Works
In this paper, a new game-based metaheuristic algorithm entitled Billiards Optimization Algorithm (BOA) was introduced. The main inspiration in BOA design is the behavior of the players in the billiards game to place the balls in the pockets of the game table. Different steps of BOA were described and mathematically modeled. The performance of the proposed BOA approach was evaluated in solving twenty-three standard benchmark functions of unimodal, high-dimensional multimodal, and fixed-dimensional multimodal types. The optimization results showed that BOA has a high ability in exploration, exploitation, and balancing these features during the optimization process. The performance of BOA was compared with ten well-known metaheuristic algorithms. Analysis of the simulation results showed that the proposed BOA approach has superior performance over the competitor algorithms by providing better results.
The authors also offer some proposals for future research. Design and development of binary and multi-objective versions of BOA is an attractive research topic. Application of BOA for optimization problems in various fields of science and real-world issues is also another research proposal for future investigations.
Funding Statement: The research and article are supported by Specific Research project 2022 Faculty of Education, University of Hradec Králové, Czech Republic.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
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