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Received: February 13, 2020.Revised: March 17, 2020.246Shell Game Optimization: A Novel Game-Based AlgorithmMohammad Dehghani1*Zeinab Montazeri1Om Parkash Malik234Hadi GiviJosep M. Guerrero1Department of Electrical and Electronics Engineering, Shiraz University of Technology, Shiraz, Iran2Department of Electrical Engineering, University of Calgary, Calgary Alberta, Canada3Department of Electrical Engineering, Faculty of Engineering,University of Shahreza, Shahreza 86481-41143, Iran4Center for Research on Microgrids (CROM), Department of Energy Technology,Aalborg University, Aalborg, Denmark* Corresponding author’s Email: [email protected]: This article presents a new game-based optimization method entitled Shell Game Optimization (SGO). Thenovelty of this article is simulating the rules of a game known as shell game to design an algorithm for solvingoptimization problems in different fields of science. The key idea of the SGO is to find the ball hidden under one ofthe three shells, which should be guessed by players. The main feature and advantage of SGO is that it does not haveany control parameters and hence, there is no need to set parameters. SGO is mathematically modeled and implementedon 23 well-known benchmark test functions as well as on a real life-engineering problem entitled pressure vesseldesign problem. Moreover, SGO is compared with eight optimization algorithms: Genetic Algorithm (GA), ParticleSwarm Optimization (PSO), Gravitational Search Algorithm (GSA), Teaching Learning Based Optimization (TLBO),Grey Wolf Optimizer (GWO), Grasshopper Optimization Algorithm (GOA), Spotted Hyena Optimizer (SHO), andEmperor Penguin Optimizer (EPO). The results and data obtained from applying SGO and other mentioned algorithmson unimodal test functions, multimodal test functions, and pressure vessel design problem show that SGO is able toprovide better results in comparison with other well-known optimization algorithms. Moreover, results of Wilcoxonsigned rank test confirm that SGO achieves more accuracy in comparison with the mentioned algorithms.Keywords: Shell, Shell game, Shell game optimization, Optimization, Game-based algorithms.1. Introduction1.1 Physics-based algorithmsIn recent years, various algorithms have beenpresented in the literature in order to hms are applied by researchers in variousfields of science and technology such as energy [5, 7],power engineering [8-10], energy carriers [11, 12],and protection [13]. Population-based algorithms canbe generally classified into four categories includingPhysics-based, Evolutionary-based, Swarm-based,and Game-based algorithms.These algorithms have been developed using therules of physics. Simulated Annealing (SA) is basedon the gradual freezing technique. The gradualfreezing technique is a way to achieve a state, inwhich solid-state energy is minimized well anduniformly. This technique involves placing thesubstance at high temperature and then graduallylowering it [14]. Spring Search Algorithm (SSA) isinspired by Hooke's law. In SSA, search agents are agroup of weights, which are connected together withsprings [3]. Some of the other popular physics-basedalgorithms are Gravitation Search Algorithm (GSA)[15], Charged System Search (CSS) [16], Galaxy-International Journal of Intelligent Engineering and Systems, Vol.13, No.3, 2020DOI: 10.22266/ijies2020.0630.23

Received: February 13, 2020.Revised: March 17, 2020.based Search Algorithm (GbSA) [17], Curved SpaceOptimization (CSO) [18], Ray Optimization (RO)algorithm [19], Artificial Chemical ReactionOptimization Algorithm (ACROA) [20], SmallWorld Optimization Algorithm (SWOA) [21],Central Force Optimization (CFO) [22], Black Hole(BH) [23], and Big-Bang Big-Crunch (BBBC) [24].1.2 Evolutionary-based algorithmsThese algorithms combine aspects of naturalselection and continuity of coordination. Anevolutionary algorithm protects the population fromthe structures of the selection rules, recombination,change, and survival. These structures are based ongenetic operators. In this method, the environmentdetermines the coordination or performance of eachpopulation, and uses more consistent individuals toreproduce. Genetic Algorithm (GA) is one of themost popular evolutionary-based algorithms. GAsimulates the genetic evolution of living organisms[25]. Another evolutionary-based algorithm isDifferential Evolution (DE) that was presented toovercome the main flaw of the GA, the lack of localsearch. The main difference between GA and DE isin the selection operator [26]. Some of the otherEvolutionary-based algorithms are EvolutionStrategy (ES) [27], Genetic Programming (GP) [28],and Biogeography-based Optimizer (BBO) [29].1.3 Swarm-based algorithmsThese techniques are inspired by the naturalprocesses of plants, foraging behaviors of insects, andsocial behaviors of animals [30]. Particle SwarmOptimization (PSO) is in this category that simulatesthe bird’s behavior [31]. Ant Colony Optimization(ACO) is inspired by the ability of the ants to find theshortest route between the nest and a food source [32].Some of the other Swarm -based algorithms areArtificial Bee Colony (ABC) [33], Bat-inspiredAlgorithm (BA) [34], Spotted Hyena Optimizer(SHO) [35], Cuckoo Search (CS) [36], EmperorPenguin Optimizer (EPO) [37], Grey Wolf Optimizer(GWO) [38], Grasshopper Optimization Algorithm(GOA) [39], Group Optimization (GO) [40],‘Following’ Optimization Algorithm (FOA) [41],and Donkey Theorem Optimization (DTO) [42].1.4 Game-based algorithmsThese algorithms are developed based on therules of various games. Dehghani et al. suggestedOrientation Search Algorithm (OSA), which isinspired by the rules of the orientation game. In thisgame, players move in the orientation of the referee’s247hand [1, 43]. Dice Game Optimizer (DGO) is anothergame based algorithm that simulate an old gameentitled dice game [44].1.5 ContributionSo far, many algorithms have been proposed byresearchers in the first three categories (Physicsbased, Evolutionary-based, and Swarm-basedalgorithms), which are applied in various fields ofscience. The main idea of these algorithms is usingthe nature of different phenomena to achieve acommon goal. Since players strive to achieve a goal(called victory) in various individual and groupgames, the rules of these games are also very usefulto design optimization algorithms. In this regard, thecontribution of the authors is proposing a new gamebased optimization technique.This paper presents a novel game-basedalgorithm entitled Shell Game Optimization (SGO)for solving the optimization problems. SGO isinspired by the rules governing on a game called shellgame. Shell game is based on the precision andintelligence that each player should find the shell,under which the object is hidden. Many of thementioned optimization algorithms encounter withtwo challenges, setting of multiple control parametersand complexity of the equations. However, lack ofcontrol parameters and simplicity of the equations aswell as implementation are the important features ofSGO. Therefore, SGO can be easily applied to anyoptimization problem. The performance of SGO hasbeen compared to eight well-known optimizationtechniques considering twenty-three linear andnonlinear benchmark test functions. Moreover, SGOhas been tested on an engineering optimizationproblem to validate its effectiveness.1.6 Paper structureThe rest of this paper is organized as follows:Section 2 describes the shell game. SGO is explainedin section 3. The experimental results and discussionare presented in section 4. Finally, the conclusion isgiven in section 5.2. Shell gameShell game is an old game, in which the operatorprovides three shells and a small ball as shown in Fig.1. In this game, the curiosity of players is stimulated,which helps to increase the accuracy of the players.First, the operator invites several persons as players.Then the operator shows the ball to the players. Afterthat, puts the ball under one of the shells. Theoperator moves the shells on the table using handInternational Journal of Intelligent Engineering and Systems, Vol.13, No.3, 2020DOI: 10.22266/ijies2020.0630.23

Received: February 13, 2020.Revised: March 17, 2020.Figure. 1 Shell gamegestures. Now the operator asks the players to guessthe shell under which the ball is hidden. Each playermay choose the correct or wrong shell, depending onthe degree of accuracy and intelligence. More pointsare awarded to the player that recognizes the correctshell.In this paper, a new optimization method isintroduced inspired by this game.3. Shell Game Optimization (SGO)In this section, shell game is simulated to inventa new optimization algorithm called Shell GameOptimization (SGO). For this purpose, the followingassumptions are considered: In this game, a person is considered as thegame's operator. Three shells and one ball are available to theoperator. Each player has only two opportunities toguess the correct shell.3.1. Mathematical ModelNow, a set of N person is assumed as the game'splayers. In Eq. (1), the position ‘d’ of player ‘i’ isshown as 𝑥𝑖𝑑 .𝑋𝑖 (𝑥𝑖1 , , 𝑥𝑖𝑑 , , 𝑥𝑖𝑛 )(1)Here, 𝑋𝑖 is actually a random value for theproblem variables. Based on 𝑋𝑖 , the value of thefitness function is evaluated for each player.After calculating the fitness function value foreach player, the game's operator chooses three shellsthat one of the shells is related to the position of thebest player and two other shells is chosen randomlyby Eq. (2).𝑠ℎ𝑒𝑙𝑙1 𝑏𝑎𝑙𝑙 𝑋𝑏𝑒𝑠𝑡𝑠ℎ𝑒𝑙𝑙2 𝑋𝑘1𝑔𝑎𝑚𝑒 𝑠 𝑜𝑝𝑒𝑟𝑎𝑡𝑜𝑟: {𝑠ℎ𝑒𝑙𝑙3 𝑋𝑘2′(2)248Where, 𝑋𝑏𝑒𝑠𝑡 is the position of minimum (inminimization problems) or maximum (inmaximization problems) of fitness, 𝑋𝑘1 and 𝑋𝑘2 arepositions of two members of the population. 𝑘1 and𝑘2 are random numbers between 1 to N, which arechosen randomly.After calculating the fitness function andidentifying the shells for each player, intelligence andaccuracy of the players should be evaluated in thisstage. Each player guesses the shell based onaccuracy and intelligence. Accuracy and intelligenceof each player are simulated according to the fitnessnormalized value by Eq. (3).𝐴𝐼𝑖 𝑓𝑖𝑡𝑖 𝑓𝑖𝑡 (𝑋𝑤𝑜𝑟𝑠𝑡 )𝑁 𝑗 1[𝑓𝑖𝑡𝑗 𝑓𝑖𝑡 (𝑋𝑤𝑜𝑟𝑠𝑡 )](3)Where 𝐴𝐼𝑖 is the accuracy and intelligence ofplayer i and 𝑋𝑤𝑜𝑟𝑠𝑡 is the position of minimum (inmaximization problems) or maximum (inminimization problems) of fitness.Now, the player is ready to guess the ball. Giventhat the game is played with three shells and eachplayer has only two chances, there are three states ofguess for each player. In the first state, the first guessmay be correct and the location of the ball will berecognized. In the second state, the player after awrong guess in the first selection may guess the ball'slocation in the second time. Finally, in the third state,both guesses of player may be wrong and thus theplayer was unsuccessful to recognize the ball'slocation. The guess vector specified by 𝐺𝑣 issimulated by Eq. (4) for each player.𝑠𝑡𝑎𝑡𝑒 1: [1 0 0],𝑎𝑡 𝑓𝑖𝑟𝑠𝑡[0.5 0.5 0]𝐺𝑣 (𝑥) 𝑠𝑡𝑎𝑡𝑒 2: {, 𝑎𝑡 𝑠𝑒𝑐𝑜𝑛𝑑[0.5 0 0.5]{ 𝑠𝑡𝑎𝑡𝑒 3: [0 0.5 0.5],𝑒𝑙𝑠𝑒(4)The probability of choosing one of the states forshell selection is simulated by Eq. (5).𝑠𝑡𝑎𝑡𝑒 1: 𝑖𝑓 𝐴𝐼𝑖 𝑟𝑔1𝑠𝑡𝑎𝑡𝑒 { 𝑠𝑡𝑎𝑡𝑒 2: 𝑖𝑓 𝐴𝐼𝑖 𝑟𝑔2𝑠𝑡𝑎𝑡𝑒 3: 𝑒𝑙𝑠𝑒(5)Where 𝑟𝑔1 is the possibly of correct guess at thefirst selection and 𝑟𝑔2 denotes the possibly of correctguess at the second time.International Journal of Intelligent Engineering and Systems, Vol.13, No.3, 2020DOI: 10.22266/ijies2020.0630.23

Received: February 13, 2020.Revised: March 17, 2020.Finally, 𝑋𝑖 vector, which is assumed as thelocation of each member of population, is updatedaccording to Eqs. (6)-(9).𝑑𝑑𝑥𝑖,𝑏𝑎𝑙𝑙 𝑟1 (𝑏𝑎𝑙𝑙 𝑥𝑖𝑑 ) state (1,1)(6)𝑑𝑑𝑥𝑖,𝑠ℎ𝑒𝑙𝑙 𝑟2 (𝑠ℎ𝑒𝑙𝑙2𝑑 𝑥𝑖𝑑 )2 𝑠𝑖𝑔𝑛(𝑓𝑖𝑡𝑖 𝑓𝑖𝑡𝑠ℎ𝑒𝑙𝑙2 ) state (1,2)(7)𝑑𝑑𝑥𝑖,𝑠ℎ𝑒𝑙𝑙 𝑟3 (𝑠ℎ𝑒𝑙𝑙3𝑑 𝑥𝑖𝑑 )3 𝑠𝑖𝑔𝑛(𝑓𝑖𝑡𝑖 𝑓𝑖𝑡𝑠ℎ𝑒𝑙𝑙3 ) state (1,3)(8)𝑑𝑑𝑑𝑥𝑖𝑑 𝑥𝑖𝑑 𝑑𝑥𝑖,𝑏𝑎𝑙𝑙 𝑑𝑥𝑖,𝑠ℎ𝑒𝑙𝑙 𝑑𝑥𝑖,𝑠ℎ𝑒𝑙𝑙23(9)Where 𝑟𝑖 is a random value in the range of [0 1],𝑑𝑑𝑑𝑑𝑥𝑖,𝑏𝑎𝑙𝑙, 𝑑𝑥𝑖,𝑠ℎ𝑒𝑙𝑙, and 𝑑𝑥𝑖,𝑠ℎ𝑒𝑙𝑙are the23displacements of dimension ‘d’ of player ‘i’ based onshell1, shell2, and shell3.3.2. Steps of SGOThe steps of SGO are summarized as follows:Step 1: Random formation of initial population usingEq. (1)Step 2: Calculating the fitness value of agentsStep 3: Selection of i-th memberStep 4: Selecting three shells using Eq. (2)Step 5: Calculation of accuracy and intelligence (AI)using Eq. (3)Step 6: Simulating the state of guess using Eqs. (4)and (5)Step 7: Selection of d-th dimension of i-th member𝑑𝑑Step 8: Calculating 𝑑𝑥𝑖,𝑏𝑎𝑙𝑙, 𝑑𝑥𝑖,𝑠ℎ𝑒𝑙𝑙, 6)-(8)3Step 9: Updating location of d-th dimension of i-thmember using Eq. (9)Step 10: If all dimensions of i-th member are updated,going Step 11, else returning Step 7Step 11: If all members are updated, going Step 12,else returning Step 3Step 12: If the stop condition is established, goingStep 13, else returning Step 2Step 13: Printing the best optimal solution2494.1 Benchmark test functionsThe standard benchmark test functions utilized inthis section have been taken from [45].4.2 Algorithms used for comparisonPerformance of the SGO algorithm is comparedwith the following eight optimization algorithms. Genetic Algorithm (GA) [46]: GA is inspiredby genetic science and Darwinian evolutionbased on the survival of the highest or the naturalselection. A common use of GA is its utilizationas an optimization function. Particle Swarm Optimization (PSO) [47]: InPSO, the movement of the bird group issimulated as part of a sociological study thatstudies the concept of collective intelligence inthe biological community. Gravitational Search Algorithm (GSA) [15]:GSA is inspired by law of gravity in the nature.In this algorithm, search agents are a set ofobjects that can be thought as planets of a system. Teaching Learning Based Optimization(TLBO) [48]: TLBO is based on teaching andlearning, which is divided into two phases. Thefirst phase, which includes learning from theteacher, and the second phase, where studentslearn from each other's interaction. Grey Wolf Optimizer (GWO) [38]: GWO isa nature-inspired algorithm based on thehierarchical structure and wolf's social behaviorduring hunting. GrasshopperOptimizationAlgorithm(GOA) [39]: GOA is a nature-inspired algorithmthat imitates and simulates the behavior ofgrasshoppers in the nature and the swarmmovement of grasshoppers toward food sources. Spotted Hyena Optimizer (SHO) [35]: SHOis inspired by the behavior of spotted hyenas. Themain concept behind this algorithm is the socialrelationship between spotted hyenas and theircollaborative behavior. Emperor Penguin Optimizer (EPO) [37]:EPO simulates the behavior of the emperor'spenguins.4. Experimental results and discussion4.3 Performance comparisonThis section describes the experimentation ontwenty-three standard benchmark test functions toevaluate the performance of SGO. The detaileddescription of these benchmarks is presented in thefollowing. Moreover, the results of SGO arecompared with eight optimization algorithms.In order to demonstrate the effectiveness of SGO,it is compared with eight well-known optimizationalgorithms considering unimodal, multimodal, andfixed-dimension multimodal benchmark testfunctions [45].International Journal of Intelligent Engineering and Systems, Vol.13, No.3, 2020DOI: 10.22266/ijies2020.0630.23

Revised: March 17, 2020.SGO6.74 10-359.17 10-367.78 10-453.48 10-452.63 10-259.83 10-274.65 10-264.68 10-295.41 10-15.05 10-28.03 10-245.22 10-263.33 10-81.18 10-6International Journal of Intelligent Engineering and Systems, Vol.13, No.3, 2020F7F6In this section, SGO has been applied on anengineering design problem. Mathematical modelF54.3.4. Pressure vessel designF4Functions F14 to F23 in [45] have a low numberof dimensions and also low local answers. The resultsof 20 times implementation of SGO and otheralgorithms for these multimodal test functions arepresented in Table 3. These results show that SGOalso performs effectively for this type of functionsand is very competitive over other optimizationalgorithms. Convergence curves of SGO and otheroptimization algorithms for three models of thefunctions are illustrated in Fig. 2. For unimodalfunctions such as F5, multimodal test functions withhigh dimension such as F12, and multimodal testfunctions with low dimension such as F15, SGOconverges with more precision and speed in thesearch space due to its adaptive mechanism.F34.3.3. Evaluation of multimodal test functions with lowdimensionF2In multimodal test functions, the number of localresponses increases exponentially with increase ofthe function dimensions. Therefore, it is hardlypossible to achieve the minimum answer for this typeof functions. In this type of functions, reaching thenearest answer indicates the remarkable capability ofthe algorithm for passing the wrong local answers.The results of evaluating functions F8 to F13 [45] for20 independent runtimes are presented in Table 2. Forall of these functions, SGO has achieved a estdAvestd4.3.2. Evaluation of multimodal test functionsF1Functions F1 to F7 are Unimodal test functions.The average results obtained during 20 timesindependent implementation of the algorithms arepresented in Table 1. The results indicate that theSGO performance is better than other algorithms forall of the mentioned functions (F1 to F7) [45].EPO5.71 10-288.31 10-296.20 10-403.32 10-402.05 10-199.17 10-204.32 10-183.98 10-195.074.90 10-17.01 10-194.39 10-202.71 10-59.26 10-64.3.1. Evaluation of unimodal test functionsofthis problem has been taken from [49]. Tables 4 and5 show the performance of SGO and other algorithms.Table 1. Results for SGO and other algorithms considering Unimodal test functions.PSOGSATLBOGOAGWOSHO-9-16-2-1-104.98 101.16 103.55 102.81 107.86 104.61 10-231.40 10-86.10 10-171.06 10-11.11 10-18.11 10-97.37 10-23-4-1-5-1-207.29 101.70 103.23 103.96 105.99 101.20 10-341.84 10-39.29 10-18.57 10-51.41 10-11.11 10-171.30 10-341.40 10 14.16 10 24.91 10 34.31 10 19.19 10-51.00 10-14 2 3-47.131.56 103.89 108.976.16 104.10 10-146.00 10-11.121.87 10 18.80 10-18.73 10-12.02 10-141.72 10-19.89 10-18.212.50 10-11.19 10-12.43 10-14 1 1 2 2 24.93 103.85 107.37 101.18 108.91 102.79 10 13.89 10 13.47 10 11.98 10 31.43 10 22.97 10 21.84-9-16-1-179.23 101.08 104.883.15 108.18 106.58 10-1-8-17-1-2-181.78 104.00 109.75 109.98 101.70 103.38 10-16.92 10-27.68 10-13.88 10-22.02 10-25.37 10-17.80 10-42.87 10-22.775.79 10-27.43 10-31.89 10-13.85 10-4The experimentation has been done on MatlabR2014a (8.3.0.532) version in the environment ofMicrosoft Windows 7 using 64 bit Core i-7 processorwith 2.40 GHz and 16 GB main memory. Theaverage (Ave) and standard deviation (std) of the bestoptimal solution are mentioned in the tables. For eachbenchmark test function, SGO utilizes 30independent runs, in which each run employs 1000iterations.250GA1.95 10-122.01 10-116.53 10-185.10 10-177.70 10-107.36 10-99.17 10 15.67 10 15.57 10 24.16 10 13.15 10-19.98 10-26.79 10-43.29 10-3Received: February 13, 2020.DOI: 10.22266/ijies2020.0630.23

International Journal of Intelligent Engineering and Systems, Vol.13, No.3, 2020Table 3. Results for SGO and other algorithms considering Multimodal test functions with low 9.98 10 11.263.711.084.41 10-22.322.961.129.14 10-16.86 10-13.864.11 10-27.36 10-29.09 10-36.84 10-25.15 10-27.15 10-21.01 10-23.66 10-28.21 10-3-3-3-2-3-1-3-22.39 102.38 107.37 103.45 101.26 103.75 107.60 104.09 10-3-1.02-1.02-1.02-1.01-1.02-1.02-1.02-1.024.19 10-70.000.003.64 10-84.74 10-83.23 10-57.02 10-99.80 10-73.98 10-13.98 10-13.98 10-13.98 10-13.98 10-13.98 10-13.98 10-13.98 10-1-17-16-16-15-7-4-73.71 109.03 101.13 109.45 101.15 107.61 107.00 105.39 10-53.003.003.003.003.003.003.003.006.33 10-76.59 10-53.24 10-21.94 10-101.48 10 12.25 10-57.16 10-61.15 10-8-3.81-3.80-3.86-3.73-3.77-3.75-3.84-3.864.37 10-103.37 10-154.15 10-19.69 10-43.53 10-72.55 10-31.57 10-36.50 10-7-2.39-3.32-1.47-2.17-3.23-2.84-3.27-2.814.37 10-12.66 10-15.32 10-11.64 10-15.37 10-23.71 10-17.27 10-27.11 0-3.99-1.04-10.011.37 10-23.082.642.89 10-43.021.992.73 10-43.97 10-2-3.10-9.19-9.37-2.46-8.41-4.49-1.05 10 1-3.412.372.522.751.193.131.961.81 10-41.11 tdAvestdAvestdSGO9.98 10-17.64 10-123.3 10-41.25 10-5-1.035.12 10-103.98 10-14.56 10-213.001.15 10-18-3.865.61 10-10-3.314.29 10-5-10.151.25 10-2-10.403.65 10-7-10.535.26 10-6SGO-1.2 10 49.14 10-128.76 10-44.85 10-28.04 10-203.34 10-184.23 10-105.11 10-76.33 10-54.71 10-40.000.00Revised: March 17, 8EPO-8.76 10 25.92 10 16.90 10-14.81 10-18.03 10-162.74 10-144.20 10-54.73 10-45.09 10-33.75 10-31.25 10-82.61 10-7AvestdAvestdAvestdAvestdAvestdAvestdTable 2. Results for SGO and other algorithms considering Multimodal test functions.PSOGSATLBOGOAGWOSHO-5.01 10 2-2.75 10 2-3.81 10 2-6.92 10 2-4.69 10 1-6.14 10 24.28 10 15.72 10 12.83 10 19.19 10 10 13.94 10 19.32 10 1-1 1 1 2-21.20 103.35 102.23 101.01 104.85 104.34 10-1 1 1 1 1 14.01 101.19 103.25 101.89 103.91 101.665.20 10-118.25 10-91.55 10 11.152.83 10-81.63 10-141.08 10-101.90 10-98.117.87 10-14.34 10-73.14 10-15-6-1-1-53.24 108.193.01 105.74 102.49 102.29 10-3-5-1-1-44.11 103.702.89 101.12 101.34 105.24 10-3-8-1 1-58.93 102.65 105.21 101.271.34 103.93 10-2-7-1 2-44.77 103.14 102.47 101.026.23 102.42 10-2-2-32 2-2-86.26 105.73 102.81 106.60 109.94 104.75 10-1-2-32 2-2-74.39 108.95 108.63 104.33 102.61 102.38 10-1GA-5.11 10 24.37 10 11.23 10-14.11 10 15.31 10-111.11 10-103.31 10-64.23 10-59.16 10-84.88 10-76.39 10-24.49 10-2Received: February 13, 2020.251DOI: 10.22266/ijies2020.0630.23

Received: February 13, 2020.Revised: March 17, 2020.252Figure. 2 Convergence curves of SGO and other optimization algorithms on three benchmark test functionsTable 4. Comparison of results for pressure vessel design problemOptimum F4F5F6F7Optimum 137.372411550.29765890.32796550.0230Table 5. Statistical results for pressure vessel design problemBestMeanWorstStd. 05.4397657.523EPO-1-1-1-1-1-1-1Table 6. Wilcoxon signed rank tests for unimodal functions 1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1SGO provides optimal solution at onding fitness value equal to O-1-1-1-1-1-1-1GA-1-1-1-1-1-1-14.3.5. Wilcoxon Signed Rank TestWilcoxon signed rank test [50] is used to comparethe data in two groups dependent on each other.International Journal of Intelligent Engineering and Systems, Vol.13, No.3, 2020DOI: 10.22266/ijies2020.0630.23

Received: February 13, 2020.Revised: March 17, 2020.253F8F9F10F11F12F13EPO-1-1-1-1-1-1Table 7. Wilcoxon signed rank tests for multimodal functions -1-1-1-1-1-1-1Table 8. Wilcoxon signed rank tests for multimodal functions -1-1-1-1-1-1-1-1-1Table 9. Wilcoxon signed rank tests for pressure vessel design (PVD) DBased on the fitness function, the Wilcoxon test wasperformed at 95% confidence level (the zerohypothesis in this test indicates lack of difference andthe opposite hypothesis indicates the difference), andthe results show that SGO achieves more accuracy incomparison with the mentioned eight algorithms.Wilcoxon signed rank test results are presented inTables 6 to 9. In these tables, -1 means worse, 0means equal, and 1 means better.functions confirmed the superior exploitation andexploration capability of SGO.For future works, there are several ideas that issuggested by the authors for study. As an interestingfuture contribution, one can develop a binary versionof SGO. In addition, SGO can be applied to solvemany-objective real-life optimization as well asmulti-objective problems.Acknowledgments5. ConclusionIn this paper, a novel optimization methodentitled Shell Game Optimization was introduced.SGO is based on the rules of the Shell game. In thisgame, players try to find a ball that is hidden underone of the three Shells. SGO and eight otheroptimisation algorithms were tested on 23 benchmarktest functions. In addition, pressure vessel designproblem was considered to further evaluate theeffectiveness of the proposed algorithm. The resultsdemonstrate that SGO has good performancecompared to GA, PSO, GSA, TLBO, GWO, GOA,SHO, and EPO. Nevertheless, SGO was alsoanalyzed considering the Wilcoxon signed rank test.Based on the results obtained for SGO and otherlisted optimization algorithms; it was shown thatSGO is able to handle different types of constraintsvery efficiently and provides better solutions. Theresults obtained for unimodal and multimodal testJ. M. Guerrero was funded by a VillumInvestigator grant (no. 25920) from The VillumFonden.References[1] M. Dehghani, Z. Montazeri, O. P. Malik, A.Ehsanifar, and A. Dehghani, “OSA: OrientationSearch Algorithm”, International Journal ofIndustrial Electronics, Control and Optimization,Vol.2, pp.99-112, 2019.[2] M. Dehghani, Z. Montazeri, A. Dehghani, N.Nouri, and A. Seifi, “BSSA: Binary SpringSearch Algorithm”, In: Proc. of IEEE 4thInternational Conf. on Knowledge-BasedEngineering and Innovation (KBEI), pp. 220-224,2017.[3] M. Dehghani, Z. Montazeri, A. Dehghani, and A.Seifi, “Spring Search Algorithm: A New MetaHeuristic Optimization Algorithm Inspired byInternational Journal of Intelligent Engineering and Systems, Vol.13, No.3, 2020DOI: 10.22266/ijies2020.0630.23

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Abstract: This article presents a new game-based optimization method entitled Shell Game Optimization (SGO). The novelty of this article is simulating the rules of a game known as shell game to design an algorithm for solving optimization problems in different fields of science. The key idea of the SGO is to find the ball hidden under one of