This document introduces multi-objective evolutionary algorithms (MOEAs). It discusses how MOEAs can be applied to solve multi-objective optimization problems like the knapsack problem and automated antenna design. The key differences between single and multi-objective evolutionary algorithms are selection methods like non-dominated sorting to find a Pareto front of solutions instead of a single optimal solution. Examples show using a MOEA to optimize a knapsack problem with multiple costs and values, and designing an X-band antenna with over 1,000 objectives and constraints.

1. Introduction to Multi-Objective Evolutionary Algorithms

2. Table of Content Multi-Objective Evolutionary Algorithms(MOEA) MOEA solving Knapsack MOEA Application: Automated Antenna Design Multi-Objective Evolutionary Algorithms(MOEA) Multi-Objective Problems MOEA’s Difference from Single-Objective EA Selection: Non-Dominated Sorting <非劣排序> MOEA solving Knapsack Problem Description A Test Case MOEA Application: Automated Antenna Design

4. Knapsack with multiple values

5. TSP<旅行商问题> Minimize [Time, Distance, Cost, Risk]

7. General Framework of Evolutionary Algorithms Multi-Objective Evolutionary Algorithms(MOEA) MOEA solving Knapsack MOEA Application: Automated Antenna Design Multi-Objective Problems MOEA‘s Difference from Single-Objective EA Selection: Non-Dominated Sorting <非劣排序> Initialize Population Evaluation Randomly create a certain number of solutions(individuals) Breeding Operation Apply breeding operators on individuals in the population, producing a group of new individuals. Evaluation Crossover, Mutation, etc. Calculate the objective value based on decision variables of a individual. Apply selection operator on individuals in the population, shrink the population size to the initial size. Selection Termination Conditions? Termination(Output)

8. MOEA’s Difference from Single-Objective EA Multi-Objective Evolutionary Algorithms(MOEA) MOEA solving Knapsack MOEA Application: Automated Antenna Design Multi-Objective Problems MOEA‘s Difference from Single-Objective EA Selection: Non-Dominated Sorting <非劣排序> Initialize Population Evaluation Breeding Operation Single Objective  Multiple Objectives double Evaluate(vector<double> dna) vector<double> Evaluate(vector<double> dna) Crossover, Mutation, etc. Normal Sorting Non-Dominated Sorting <非劣排序> Producing a group of Pareto Fronts <Pareto 前沿> Selection Termination Conditions? 内涵在这里！ Termination(Output) Optimization Result: one optimal solution a group of non-dominated solutions

9. Selection: Non-Dominated Sorting Multi-Objective Evolutionary Algorithms(MOEA) MOEA solving Knapsack MOEA Application: Automated Antenna Design Multi-Objective Problems MOEA‘s Difference from Single-Objective EA Selection: Non-Dominated Sorting <非劣排序> Non-dominated comparison 对于给定的两个非全等向量:A(x1, x2, …, xn), B(y1, y2, …, yn) 当且仅当∀i: xi ≥ yi，有A 大于B 当且仅当∀i: xi ≤ yi， 有B大于A 其余情况，即 ∃i, ∃j: i≠j AND xi > yi AND xj < yj 则A与B不可比 学历 (2, 3) (3, 2) 国际视野

10. Knapsack with Multiple Values Multi-Objective Evolutionary Algorithms(MOEA) MOEA solving Knapsack MOEA Application: Automated Antenna Design Problem Description A Test Case Definition Given a set of items each with a set of costs [e.g. Weight, Volume, etc. ] and a set of values determine the number of each item to include in a collection, so that the total costs are less than given limits and the total values are as large as possible. 给定一组物品 每种物品有确定的 { 数量 成本[重量，体积，神秘成本] 价值[价值1，价值2] } 确定每种物品的选取数量，在满足总成本在给定的限定范围之内的前提下，极大化总价值[价值1，价值2]

11. Parameters & Knapsack Data Multi-Objective Evolutionary Algorithms(MOEA) MOEA solving Knapsack MOEA Application: Automated Antenna Design Problem Description A Test Case Total Evaluation Used: 240300 Properties Algorithm Objectives Problem Constraints

12. Knapsack - Result Multi-Objective Evolutionary Algorithms(MOEA) MOEA solving Knapsack MOEA Application: Automated Antenna Design Problem Description A Test Case Distribution of Objectives Generation = 0 Generation = 100 Generation = 200 Generation = 400 Generation = 600 Generation = 800

13. Knapsack - Result Evolutionary progress of average value of each objective Multi-Objective Evolutionary Algorithms(MOEA) MOEA solving Knapsack MOEA Application: Automated Antenna Design Problem Description A Test Case Objectives’ evolutionary progress in same scale

14. Demand of Micro Antennas Multi-Objective Evolutionary Algorithms(MOEA) MOEA solving Knapsack MOEA Application: Automated Antenna Design 用于微型卫星 空间和能量都十分宝贵 对天线各种性能要求极高

16. Minimize: T_VSWR(r0, r1, …, r4, x1, y1, z1, …, x4, y4, z4)

17. Minimize: R_VSWR(r0, r1, …, r4, x1, y1, z1, …, x4, y4, z4)

18. Gain(Objectives)

19. Maximize: T_Gainθ,φ(r0, r1, …, r4, x1, y1, z1, …, x4, y4, z4)

20. Maximize: R_Gainθ,φ(r0, r1, …, r4, x1, y1, z1, …, x4, y4, z4)

21. Constraints

22. T_Gainθ,φ≥ 0

23. R_Gainθ,φ ≥ 0

24. T_VSWR < 1.5

25. R_VSWR < 1.5

26. Diameter < 15.24cm

27. Height < 15.24cm

28. Antenna Mass < 165g

29. Θ= 5.0i, i = 8,9,…,16

30. φ = 5.0j, j = 0,1,…,71变量维数:17 目标个数:1298 约束条件个数:1301 问题规模庞大

31. Designing with MOEA Multi-Objective Evolutionary Algorithms(MOEA) MOEA solving Knapsack MOEA Application: Automated Antenna Design MOEA chromosome : vector<double> Evaluator Encoding Electromagnetic Simulator {NEC, HFSS, etc.}

32. Comparison between traditional and evolvable Antenna Multi-Objective Evolutionary Algorithms(MOEA) MOEA solving Knapsack MOEA Application: Automated Antenna Design QHAHandmade Antenna NASAX-Band 5 Antenna designed by EA

33. Resources Multi-Objective Evolutionary Algorithms(MOEA) MOEA solving Knapsack MOEA Application: Automated Antenna Design MOEA Competition @ CEC 2009 http://dces.essex.ac.uk/staff/qzhang/moeacompetition09.htm Knapsack problem http://en.wikipedia.org/wiki/Knapsack_problem NASA – Evolvable Systems http://www.nasa.gov/centers/ames/research/exploringtheuniverse/exploringtheuniverse-evolvablesystems.html