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 ContentMulti-Objective Evolutionary Algorithms(MOEA)MOEA solving KnapsackMOEA Application: Automated Antenna DesignMulti-Objective Evolutionary Algorithms(MOEA)Multi-Objective ProblemsMOEA’s Difference from Single-Objective EASelection: Non-Dominated Sorting <非劣排序>MOEA solving KnapsackProblem DescriptionA Test CaseMOEA Application: Automated Antenna Design

3. Multi-Objective ProblemsMulti-Objective Evolutionary Algorithms(MOEA)MOEA solving KnapsackMOEA Application: Automated Antenna DesignMulti-Objective ProblemsMOEA‘s Difference from Single-Objective EASelection: Non-Dominated Sorting <非劣排序>With multiple objectivesExamples:- Minimize y,z 凤姐征婚：Maximize[学历，相貌，国际视野，……]

4. Knapsack with multiple values

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

6. 选修课：Minimize [所花精力] Maximize [最终得分]Solution methods for Multi-Objective problemsMulti-Objective Evolutionary Algorithms(MOEA)MOEA solving KnapsackMOEA Application: Automated Antenna DesignMulti-Objective ProblemsMOEA‘s Difference from Single-Objective EASelection: Non-Dominated Sorting <非劣排序>Constructing a single aggregate objective function (AOF) <聚合为单目标>Normal Boundary Intersection (NBI) methodNormal Constraint (NC) methodSuccessive Pareto Optimization (SPO) methodMultiobjective Optimization using Evolutionary Algorithms (MOEA).PGEN (Pareto surface generation for convex multiobjective instances)IOSO (Indirect Optimization on the basis of Self-Organization)Reference: Wikipedia[Multiobjective optimization] - http://en.wikipedia.org/wiki/Multiobjective_optimization

7. General Framework of Evolutionary AlgorithmsMulti-Objective Evolutionary Algorithms(MOEA)MOEA solving KnapsackMOEA Application: Automated Antenna DesignMulti-Objective ProblemsMOEA‘s Difference from Single-Objective EASelection: Non-Dominated Sorting <非劣排序>Initialize PopulationEvaluationRandomly create a certain number of solutions(individuals)Breeding OperationApply breeding operators on individuals in the population, producing a group of new individuals.EvaluationCrossover, 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.SelectionTermination Conditions?Termination(Output)

8. MOEA’s Difference from Single-Objective EAMulti-Objective Evolutionary Algorithms(MOEA)MOEA solving KnapsackMOEA Application: Automated Antenna DesignMulti-Objective ProblemsMOEA‘s Difference from Single-Objective EASelection: Non-Dominated Sorting <非劣排序>Initialize PopulationEvaluationBreeding OperationSingle Objective  Multiple Objectivesdouble 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>SelectionTermination Conditions?内涵在这里！Termination(Output)Optimization Result: one optimal solution a group of non-dominated solutions

9. Selection: Non-Dominated SortingMulti-Objective Evolutionary Algorithms(MOEA)MOEA solving KnapsackMOEA Application: Automated Antenna DesignMulti-Objective ProblemsMOEA‘s Difference from Single-Objective EASelection: 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 ValuesMulti-Objective Evolutionary Algorithms(MOEA)MOEA solving KnapsackMOEA Application: Automated Antenna DesignProblem DescriptionA Test CaseDefinitionGiven a set of itemseach with a set of costs [e.g. Weight, Volume, etc. ]and a set of valuesdetermine 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 DataMulti-Objective Evolutionary Algorithms(MOEA)MOEA solving KnapsackMOEA Application: Automated Antenna DesignProblem DescriptionA Test CaseTotal Evaluation Used: 240300PropertiesAlgorithmObjectivesProblemConstraints

12. Knapsack - ResultMulti-Objective Evolutionary Algorithms(MOEA)MOEA solving KnapsackMOEA Application: Automated Antenna DesignProblem DescriptionA Test CaseDistribution of ObjectivesGeneration = 0Generation = 100Generation = 200Generation = 400Generation = 600Generation = 800

13. Knapsack - ResultEvolutionary progress of average value of each objectiveMulti-Objective Evolutionary Algorithms(MOEA)MOEA solving KnapsackMOEA Application: Automated Antenna DesignProblem DescriptionA Test CaseObjectives’ evolutionary progress in same scale

14. Demand of Micro AntennasMulti-Objective Evolutionary Algorithms(MOEA)MOEA solving KnapsackMOEA 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

15. Objectives and Contraints of X-Band 5Multi-Objective Evolutionary Algorithms(MOEA)MOEA solving KnapsackMOEA Application: Automated Antenna DesignVSWR(Objectives)

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