Game Theory Based Evolutionary Algorithms: A Review with Nash Applications in Structural Engineering Optimization Problems

A general review of game-theory based evolutionary algorithms (EAs) is presented in this study. Nash equilibrium, Stackelberg game and Pareto optimality are considered, as game-theoretical basis of the evolutionary algorithm design, and also, as problems solved by evolutionary computation. Applications of game-theory based EAs in computational engineering are listed, with special emphasis in structural optimization and, particularly, in skeletal structures. Additionally, a set of three problems are solved: reconstruction inverse problem, fully stressed design problem and minimum constrained weight, for discrete sizing of frame skeletal structures. We compare panmictic EAs, Nash EAs using 4 different static domain decompositions, including also a new dynamic domain decomposition. Two frame structural test cases of 55 member size and 105 member size are evaluated with the linear stiffness matrix method. Numerical experiments show the efficiency of the Nash EAs approach, confirmed with statistical significance analysis of results, and enhanced with the dynamic domain decomposition.

[1]  Qian Jixin,et al.  A new hybrid genetic algorithm for global minimax optimization , 2001, 2001 International Conferences on Info-Tech and Info-Net. Proceedings (Cat. No.01EX479).

[2]  Yan Zhang,et al.  Nash-optimization enhanced distributed model predictive control applied to the Shell benchmark problem , 2005, Inf. Sci..

[3]  Elia Daniele,et al.  Equilibrium strategies via GA to stackelberg games under multiple follower's best reply , 2012, Int. J. Intell. Syst..

[4]  Michael M. Kostreva,et al.  Linear optimization with multiple equitable criteria , 1999, RAIRO Oper. Res..

[5]  R. Aumann.  Subjectivity and Correlation in Randomized Strategies , 1974 .

[6]  A. Michell.  LVIII. The limits of economy of material in frame-structures , 1904 .

[7]  Kalyanmoy Deb,et al.  A fast and elitist multiobjective genetic algorithm: NSGA-II , 2002, IEEE Trans. Evol. Comput..

[8]  T. Vincent,et al.  Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics , 2005 .

[9]  Miguel Galante.  UN ALGORITMO GENETICO SIMPLE PARA LA OPTIMIZACION DE ESTRUCTURAS PLANAS ARTICULADAS , 1993 .

[10]  Nachol Chaiyaratana,et al.  Multi-objective Co-operative Co-evolutionary Genetic Algorithm , 2002, PPSN.

[11]  Richard K. Belew,et al.  New Methods for Competitive Coevolution , 1997, Evolutionary Computation.

[12]  S. Rajeev,et al.  Discrete Optimization of Structures Using Genetic Algorithms , 1992 .

[13]  Kenneth A. De Jong,et al.  Cooperative Coevolution: An Architecture for Evolving Coadapted Subcomponents , 2000, Evolutionary Computation.

[14]  Manoj Kumar.  OPTIMIZATION USING GENETIC ALGORITHMS , 1998 .

[15]  Carlos A. Coello Coello,et al.  A non-cooperative game for faster convergence in cooperative coevolution for multi-objective optimization , 2015, 2015 IEEE Congress on Evolutionary Computation (CEC).

[16]  John J. Grefenstette,et al.  Optimization of Control Parameters for Genetic Algorithms , 1986, IEEE Transactions on Systems, Man, and Cybernetics.

[17]  J. H. Argyris,et al.  Energy theorems and structural analysis , 1960 .

[18]  Subramaniam Rajan,et al.  Sizing, Shape, and Topology Design Optimization of Trusses Using Genetic Algorithm , 1995 .

[19]  El-Ghazali Talbi,et al.  Metaheuristics for Bi-level Optimization , 2013 .

[20]  David Greiner,et al.  Advances in Evolutionary and Deterministic Methods for Design, Optimization and Control in Engineering and Sciences , 2016 .

[21]  Patrick Murren,et al.  Design-driven harmony search (DDHS) in steel frame optimization , 2014 .

[22]  Rudolf Paul Wiegand,et al.  An analysis of cooperative coevolutionary algorithms , 2004 .

[23]  Zong Woo Geem,et al.  Mathematical and Metaheuristic Applications in Design Optimization of Steel Frame Structures: An Extensive Review , 2013 .

[24]  J. Periaux,et al.  Genetic Algorithms and Game Theory for High Lift Multi-Airfoil Design Problems in Aerodynamics , 2001 .

[25]  Phil Husbands,et al.  Simulated Co-Evolution as the Mechanism for Emergent Planning and Scheduling , 1991, ICGA.

[26]  G. Hommel,et al.  Improvements of General Multiple Test Procedures for Redundant Systems of Hypotheses , 1988 .

[27]  Andy J. Keane,et al.  Coevolutionary architecture for distributed optimization of complex coupled systems , 2002 .

[28]  W. Daniel Hillis,et al.  Co-evolving parasites improve simulated evolution as an optimization procedure , 1990 .

[29]  Dumitru Dumitrescu,et al.  Lorenz equilibrium: equitability in non-cooperative games , 2012, GECCO '12.

[30]  Siamak Talatahari,et al.  Particle swarm optimizer, ant colony strategy and harmony search scheme hybridized for optimization of truss structures , 2009 .

[31]  Enrique Alba,et al.  Parallelism and evolutionary algorithms , 2002, IEEE Trans. Evol. Comput..

[32]  Patrick Siarry,et al.  A survey on optimization metaheuristics , 2013, Inf. Sci..

[33]  David Greiner,et al.  Truss topology optimization for mass and reliability considerations—co-evolutionary multiobjective formulations , 2012 .

[34]  T. Başar,et al.  Off-Line Computation of Stackelberg Solutions with the Genetic Algorithm , 1999 .

[35]  Xiu Zhang,et al.  Nested genetic algorithm for resolving overlapping spectra , 2001, Fresenius' journal of analytical chemistry.

[36]  Scott A. Burns,et al.  Fully stressed frame structures unobtainable by conventional design methodology , 2001 .

[37]  Carlos A. Coello Coello,et al.  Evolutionary multi-objective optimization: a historical view of the field , 2006, IEEE Comput. Intell. Mag..

[38]  M. Turner.  Stiffness and Deflection Analysis of Complex Structures , 1956 .

[39]  Kalyanmoy Deb,et al.  Optimal design of a welded beam via genetic algorithms , 1991 .

[40]  Kwee-Bo Sim,et al.  Game Theory Based Coevolutionary Algorithm: A New Computational Coevolutionary Approach , 2004 .

[41]  Kalyanmoy Deb,et al.  Multiobjective optimization , 1997 .

[42]  Carlos Alberto Conceição António,et al.  Optimization of laminated composite structures using a bilevel strategy , 1995 .

[43]  Jordan B. Pollack,et al.  A Game-Theoretic Memory Mechanism for Coevolution , 2003, GECCO.

[44]  Hong-Shuang Li,et al.  Discrete Optimum Design for Truss Structures by Subset Simulation Algorithm , 2015 .

[45]  Eckart Zitzler,et al.  HypE: An Algorithm for Fast Hypervolume-Based Many-Objective Optimization , 2011, Evolutionary Computation.

[46]  James Clerk Maxwell,et al.  I.—On Reciprocal Figures, Frames, and Diagrams of Forces , 2022, Transactions of the Royal Society of Edinburgh.

[47]  David Greiner,et al.  Gray Coding in Evolutionary Multicriteria Optimization: Application in Frame Structural Optimum Design , 2005, EMO.

[48]  Singiresu S. Rao.  Game theory approach for multiobjective structural optimization , 1987 .

[49]  Kalyanmoy Deb,et al.  Efficient Evolutionary Algorithm for Single-Objective Bilevel Optimization , 2013, ArXiv.

[50]  H. Adeli,et al.  Concurrent genetic algorithms for optimization of large structures , 1994 .

[51]  Lionel Amodeo,et al.  New multi-objective method to solve reentrant hybrid flow shop scheduling problem , 2010, Eur. J. Oper. Res..

[52]  Gary B. Lamont,et al.  Evolutionary Algorithms for Solving Multi-Objective Problems , 2002, Genetic Algorithms and Evolutionary Computation.

[53]  H. Adeli,et al.  Augmented Lagrangian genetic algorithm for structural optimization , 1994 .

[54]  Kapil Khandelwal,et al.  Comparison of robustness of metaheuristic algorithms for steel frame optimization , 2015 .

[55]  G. Winter,et al.  Single and multiobjective frame optimization by evolutionary algorithms and the auto-adaptive rebirth operator , 2004 .

[56]  R. Lewontin.  Evolution and the theory of games. , 1961, Journal of theoretical biology.

[57]  Rodica Ioana Lung,et al.  An evolutionary approach for detecting Aumann equilibrium in Congestion games , 2010, 2010 11th International Symposium on Computational Intelligence and Informatics (CINTI).

[58]  Wei Zhong,et al.  Topology and sizing optimization of discrete structures using a cooperative coevolutionary genetic algorithm with independent ground structures , 2016 .

[59]  Jean-Antoine Désidéri,et al.  Concurrent Aerodynamic Optimization of Rotor Blades Using a Nash Game Method , 2016 .

[60]  Sibel Sirakaya,et al.  On-line computation of Stackelberg equilibria with synchronous parallel genetic algorithms , 2003 .

[61]  Vladimir Kobelev,et al.  On a game approach to optimal structural design , 1993 .

[62]  Baoding Liu,et al.  Stackelberg-Nash equilibrium for multilevel programming with multiple followers using genetic algorithms , 1998 .

[63]  Mourad Sefrioui,et al.  Algorithmes evolutionnaires pour le calcul scientifique : application a l'electromagnetisme et a la mecanique des fluides numeriques , 1998 .

[64]  Ali Kaveh,et al.  Genetic algorithm for discrete‐sizing optimal design of trusses using the force method , 2002 .

[65]  J. Périaux,et al.  Increasing parallelism of evolutionary algorithms by Nash games in design inverse flow problems , 2013 .

[66]  Zhongping Wan,et al.  Genetic algorithm based on simplex method for solving linear-quadratic bilevel programming problem , 2008, Comput. Math. Appl..

[67]  David Greiner,et al.  A Study of Nash-Evolutionary Algorithms for Reconstruction Inverse Problems in Structural Engineering , 2015 .

[68]  Jacques Periaux,et al.  MOO Methods for Multidisciplinary Design Using Parallel Evolutionary Algorithms, Game Theory and Hierarchical Topology: Theoretical, numerical and practical aspects , 2006 .

[69]  S Rajeev,et al.  GENETIC ALGORITHMS - BASED METHODOLOGY FOR DESIGN OPTIMIZATION OF TRUSSES , 1997 .

[70]  Jean-Antoine Désidéri,et al.  Distributed Optimization using Virtual and Real Game Strategies for Aerodynamic Design , 2002 .

[71]  Min Liu,et al.  Fully Stressed Design of Frame Structures and Multiple Load Paths , 2002 .

[72]  Kenneth A. De Jong,et al.  A Cooperative Coevolutionary Approach to Function Optimization , 1994, PPSN.

[73]  Jean-Antoine Désidéri.  Cooperation and competition in multidisciplinary optimization , 2012, Comput. Optim. Appl..

[74]  Chih-Chin Lai,et al.  An Optimal Material Distribution System Based on Nested Genetic Algorithm , 2004, IEICE Trans. Inf. Syst..

[75]  Rubén Saborido,et al.  Global WASF-GA: An Evolutionary Algorithm in Multiobjective Optimization to Approximate the Whole Pareto Optimal Front , 2017, Evolutionary Computation.

[76]  T. Narendran,et al.  A genetic algorithm approach to the machine-component grouping problem with multiple objectives , 1992 .

[77]  Qingfu Zhang,et al.  MOEA/D: A Multiobjective Evolutionary Algorithm Based on Decomposition , 2007, IEEE Transactions on Evolutionary Computation.

[78]  G. Anandalingam,et al.  Genetic algorithm based approach to bi-level linear programming , 1994 .

[79]  M. Galante,et al.  GENETIC ALGORITHMS AS AN APPROACH TO OPTIMIZE REAL‐WORLD TRUSSES , 1996 .

[80]  J. M. Smith.  The theory of games and the evolution of animal conflicts. , 1974, Journal of theoretical biology.

[81]  Hojjat Adeli,et al.  Concurrent Structural Optimization on Massively Parallel Supercomputer , 1995 .

[82]  David Greiner,et al.  Comparing the Fully Stressed Design and the Minimum Constrained Weight Solutions in Truss Structures , 2015 .

[83]  Carlos A. Coello Coello,et al.  A coevolutionary multi-objective evolutionary algorithm , 2003, The 2003 Congress on Evolutionary Computation, 2003. CEC '03..

[84]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

[85]  Nicola Beume,et al.  SMS-EMOA: Multiobjective selection based on dominated hypervolume , 2007, Eur. J. Oper. Res..

[86]  Adam Wierzbicki,et al.  Equitable aggregations and multiple criteria analysis , 2004, Eur. J. Oper. Res..

[87]  Abderrahmane Habbal,et al.  Multidisciplinary topology optimization solved as a Nash game , 2004 .

[88]  Mitchell A. Potter,et al.  The design and analysis of a computational model of cooperative coevolution , 1997 .

[89]  Tadeusz Burczynski,et al.  Biologically-inspired Methods and Game Theory in Multi-criterion Decision Processes , 2011, Intelligent Decision Systems in Large-Scale Distributed Environments.

[90]  R. Paul Wiegand,et al.  A Visual Demonstration of Convergence Properties of Cooperative Coevolution , 2004, PPSN.

[91]  A. Koh.  Differential Evolution Based Bi-Level Programming Algorithm for Computing Normalized Nash Equilibrium , 2011 .

[92]  Ivo F. Sbalzariniy,et al.  Multiobjective optimization using evolutionary algorithms , 2000 .

[93]  Yafeng Yin,et al.  Genetic-Algorithms-Based Approach for Bilevel Programming Models , 2000 .

[94]  P. Hajela,et al.  Genetic search strategies in multicriterion optimal design , 1991 .

[95]  Anthony Chen,et al.  A simulation-based multi-objective genetic algorithm (SMOGA) procedure for BOT network design problem , 2006 .

[96]  G. Winter,et al.  Optimising frame structures by different strategies of genetic algorithms , 2001 .

[97]  J M Smith,et al.  Evolution and the theory of games , 1976 .

[98]  Manolis Papadrakakis,et al.  Large scale structural optimization: Computational methods and optimization algorithms , 2001 .

[99]  J. Redmond,et al.  Actuator placement based on reachable set optimization for expected disturbance , 1996 .

[100]  Ali Kaveh,et al.  Advances in Metaheuristic Algorithms for Optimal Design of Structures , 2014 .

[101]  Shengxiang Yang,et al.  Pareto or Non-Pareto: Bi-Criterion Evolution in Multiobjective Optimization , 2016, IEEE Transactions on Evolutionary Computation.

[102]  P. Hajela.  Genetic search - An approach to the nonconvex optimization problem , 1990 .

[103]  Marco P. Schoen,et al.  System Identification and Robust Controller Design Using Genetic Algorithms for Flexible Space Structures , 2009 .

[104]  Hyeonsoo Yeo,et al.  Investigation of Rotor Vibratory Loads of a UH-60A Individual Blade Control System , 2016 .

[105]  T. Riechmann.  Genetic algorithm learning and evolutionary games , 2001 .

[106]  T. L. Vincent,et al.  Game Theory as a Design Tool , 1983 .

[107]  K. Wong,et al.  Analyzing oligopolistic electricity market using coevolutionary computation , 2006, IEEE Transactions on Power Systems.

[108]  Gara Miranda,et al.  Using multi-objective evolutionary algorithms for single-objective optimization , 2013, 4OR.

[109]  M. Sefrioui,et al.  Nash genetic algorithms: examples and applications , 2000, Proceedings of the 2000 Congress on Evolutionary Computation. CEC00 (Cat. No.00TH8512).

[110]  Jacques Periaux,et al.  Parallel Evolutionary Computation for Solving Complex CFD Optimization Problems : A Review and Some Nozzle Applications , 2003 .

[111]  Lionel Amodeo,et al.  Lorenz versus Pareto Dominance in a Single Machine Scheduling Problem with Rejection , 2011, EMO.

[112]  C. Darwin.  On the Origin of Species by Means of Natural Selection: Or, The Preservation of Favoured Races in the Struggle for Life , 2019 .

[113]  Fang Yan,et al.  Nested DE based parameter estimation for multiple vortex ring microburst model , 2013 .

[114]  Keith Michael Mueller.  Sizing of Members in the Fully Stressed Design of Frame Structures , 2000 .

[115]  J. Morgan,et al.  A theoretical approximation scheme for Stackelberg problems , 1989 .

[116]  Kalyanmoy Deb,et al.  An integrated approach to automated innovization for discovering useful design principles: Case studies from engineering , 2014, Appl. Soft Comput..

[117]  Jordan B. Pollack,et al.  A Game-Theoretic Approach to the Simple Coevolutionary Algorithm , 2000, PPSN.

[118]  Xiaodong Li,et al.  A Cooperative Coevolutionary Multiobjective Algorithm Using Non-dominated Sorting , 2004, GECCO.

[119]  Carlos A. Coello Coello,et al.  Multiobjective Evolutionary Algorithms in Aeronautical and Aerospace Engineering , 2012, IEEE Transactions on Evolutionary Computation.

[120]  J. Nash.  NON-COOPERATIVE GAMES , 1951, Classics in Game Theory.

[121]  Daisuke Sasaki,et al.  Visualization and Data Mining of Pareto Solutions Using Self-Organizing Map , 2003, EMO.

[122]  Jacques Periaux,et al.  Combining game theory and genetic algorithms with application to DDM-nozzle optimization problems , 2001 .

[123]  Kalyanmoy Deb,et al.  Finding optimal strategies in a multi-period multi-leader-follower Stackelberg game using an evolutionary algorithm , 2013, Comput. Oper. Res..

[124]  J. Herskovits,et al.  Contact shape optimization: a bilevel programming approach , 2000 .

[125]  Rajan Filomeno Coelho,et al.  Co-Evolutionary Optimization for Multi-Objective Design Under Uncertainty , 2013 .

[126]  Patrice Perny,et al.  A decision-theoretic approach to robust optimization in multivalued graphs , 2006, Ann. Oper. Res..

[127]  Jean-Antoine Désidéri,et al.  Optimisation aéro-structurale de la voilure d’un avion d’affaires par un jeu de Nash et un partage adapté des variables , 2010 .

[128]  M. Friedman.  The Use of Ranks to Avoid the Assumption of Normality Implicit in the Analysis of Variance , 1937 .

[129]  Kwee-Bo Sim,et al.  Game Theory Based Co-evolutionary Algorithm (GCEA) for Solving Multiobjective Optimization Problems , 2004, IEICE Trans. Inf. Syst..

[130]  Andrew Lewis,et al.  Novel performance metrics for robust multi-objective optimization algorithms , 2015, Swarm Evol. Comput..

[131]  Marc Gravel,et al.  Efficient solutions to the cell-formation problem with multiple routings via a double-loop genetic algorithm , 1998, Eur. J. Oper. Res..

[132]  Min Liu,et al.  Multiple fully stressed designs of steel frame structures with semi‐rigid connections , 2003 .

[133]  Jun Dong,et al.  Distributed optimization using virtual and real game strategies for multi-criterion aerodynamic design , 2008 .

[134]  Scott A. Burns,et al.  Recent advances in optimal structural design , 2002 .

[135]  Eric Goodman,et al.  Investigations in meta-GAs: panaceas or pipe dreams? , 2005, GECCO '05.

[136]  Patrice Marcotte,et al.  An overview of bilevel optimization , 2007, Ann. Oper. Res..

[137]  Singiresu S. Rao,et al.  Game theory approach for the integrated design of structures and controls , 1988 .

[138]  J. Périaux,et al.  Multicriterion Aerodynamic Shape Design Optimization and Inverse Problems Using Control Theory and Nash Games , 2007 .

[139]  Phil Husbands,et al.  Distributed Coevolutionary Genetic Algorithms for Multi-Criteria and Multi-Constraint Optimisation , 1994, Evolutionary Computing, AISB Workshop.

[140]  Helio J. C. Barbosa.  A Coevolutionary Genetic Algorithm for a Game Approach to Structural Optimization , 1997, ICGA.

[141]  Urszula Boryczka,et al.  The Differential Evolution with the Entropy Based Population Size Adjustment for the Nash Equilibria Problem , 2013, ICCCI.

[142]  J. Nash.  Equilibrium Points in N-Person Games. , 1950, Proceedings of the National Academy of Sciences of the United States of America.

[143]  Michael G.H. Bell,et al.  Genetic algorithm solution for the stochastic equilibrium transportation networks under congestion , 2005 .

[144]  Krishna R. Pattipati,et al.  A novel congruent organizational design methodology using group technology and a nested genetic algorithm , 2006, IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans.

[145]  Antonio J. Nebro,et al.  A survey of multi-objective metaheuristics applied to structural optimization , 2014 .

[146]  H. Stackelberg,et al.  Marktform und Gleichgewicht , 1935 .

[147]  J. Périaux,et al.  Efficient Parallel Nash Genetic Algorithm for Solving Inverse Problems in Structural Engineering , 2016 .

[148]  H. Barbosa.  A coevolutionary genetic algorithm for constrained optimization , 1999, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406).

[149]  Hojjat Adeli,et al.  Distributed Genetic Algorithm for Structural Optimization , 1995 .

[150]  Kalyanmoy Deb,et al.  Solving Bilevel Multicriterion Optimization Problems With Lower Level Decision Uncertainty , 2016, IEEE Transactions on Evolutionary Computation.

[151]  David Greiner,et al.  Improving Computational Mechanics Optimum Design Using Helper Objectives: An Application in Frame Bar Structures , 2007, EMO.

[152]  Jacques Periaux,et al.  Evolutionary Optimization and Game Strategies for Advanced Multi-Disciplinary Design , 2015 .

[153]  Manolis Papadrakakis,et al.  Engineering and Applied Sciences Optimization , 2015 .

[154]  Jacques Periaux,et al.  Constraints handling in Nash/Adjoint optimization methods for multi-objective aerodynamic design , 2014 .

[155]  Mingjun Zhang,et al.  Evolutionary game based control for biological systems with applications in drug delivery. , 2013, Journal of theoretical biology.

[156]  Mathias Stolpe,et al.  Truss optimization with discrete design variables: a critical review , 2016 .

[157]  Kemper Lewis,et al.  The other side of multidisciplinary design optimization: Accomodating a multiobjective, uncertain and non-deterministic world , 1998 .

[158]  H. Adeli,et al.  Integrated Genetic Algorithm for Optimization of Space Structures , 1993 .

[159]  J. Aubin.  Mathematical methods of game and economic theory , 1979 .

[160]  R. Baldick,et al.  Hybrid coevolutionary programming for Nash equilibrium search in games with local optima , 2004, IEEE Transactions on Evolutionary Computation.

[161]  A. Kaveh,et al.  Charged system search for optimal design of frame structures , 2012, Appl. Soft Comput..

[162]  S. García,et al.  An Extension on "Statistical Comparisons of Classifiers over Multiple Data Sets" for all Pairwise Comparisons , 2008 .

[163]  Luis Felipe Gonzalez,et al.  Efficient Hybrid-Game Strategies Coupled to Evolutionary Algorithms for Robust Multidisciplinary Design Optimization in Aerospace Engineering , 2011, IEEE Transactions on Evolutionary Computation.

[164]  Osvaldo M. Querin,et al.  Bilevel Optimization of Blended Composite Wing Panels , 2011 .

[165]  Dario Bauso,et al.  Game Theory with Engineering Applications , 2016 .

[166]  M. N. Vrahatis,et al.  Computing Nash equilibria through computational intelligence methods , 2005 .

[167]  Jan Paredis,et al.  Coevolutionary computation , 1995 .

[168]  Tomasz Arciszewski,et al.  Evolutionary computation and structural design: A survey of the state-of-the-art , 2005 .

[169]  Jiang Feng Wang,et al.  Optimisation distribuee multicritere par algorithmes genetiques et theorie des jeux et application a la simulation numerique de problemes d'hypersustentation en aerodynamique , 2001 .

[170]  Jacques Periaux,et al.  Distributed evolutionary optimization using Nash games and GPUs – Applications to CFD design problems , 2013 .

[171]  M. Ohsaki.  Local and global searches of approximate optimal designs of regular frames , 2006 .

[172]  J. Clerk Maxwell.  1. On Reciprocal Figures, Frames, and Diagrams of Forces. , 1872 .

[173]  David Greiner,et al.  A comparison of minimum constrained weight and fully stressed design problems in discrete cross-section type bar structures , 2014 .

[174]  Dumitru Dumitrescu,et al.  Evolutionary detection of aumann equilibrium , 2010, GECCO '10.

[175]  Siamak Talatahari,et al.  A particle swarm ant colony optimization for truss structures with discrete variables , 2009 .

[176]  David E. Goldberg,et al.  ENGINEERING OPTIMIZATION VIA GENETIC ALGORITHM, IN WILL , 1986 .

[177]  Nachol Chaiyaratana,et al.  Using a co-operative co-evolutionary genetic algorithm to solve a three-dimensional container loading problem , 2001, Proceedings of the 2001 Congress on Evolutionary Computation (IEEE Cat. No.01TH8546).