Manual

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User Manual:

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I. Introduction to PlatEMO
II. How to Use PlatEMO
- A. Use PlatEMO without GUI
- B. Use PlatEMO with GUI
III. How to Extend PlatEMO

Evolutionary Multi-Objective

Optimization Platform

User Manual

Ye Tian

April 10, 2017

Thank you very much for selecting PlatEMO. The copyright of PlatEMO belongs to the BIMK Group,

and all publications using the platform should acknowledge the use of “PlatEMO” and reference the

following literature:

Ye Tian, Ran Cheng, Xingyi Zhang, and Yaochu Jin, “PlatEMO: A MATLAB

Platform for Evolutionary Multi-Objective Optimization,” 2016.

If you have any comment or suggestion to PlatEMO or the MOEAs in PlatEMO, please send it to

field910921@gmail.com (Ye Tian). If you want to add your MOEA or MOP to PlatEMO, please send

the MATLAB code and the relevant literature to field910921@gmail.com as well.

We sincerely hope this platform is helpful to your research in evolutionary multi-objective

optimization. Your support is the greatest impetus for us!

Contents

I. Introduction to PlatEMO ..................................................................................... 1

A. Evolutionary Multi-Objective Optimization ............................................. 1

B. PlatEMO ................................................................................................... 2

C. File structure of PlatEMO ......................................................................... 3

II. How to Use PlatEMO .......................................................................................... 4

A. Use PlatEMO without GUI ....................................................................... 4

B. Use PlatEMO with GUI ............................................................................ 6

III. How to Extend PlatEMO ................................................................................... 10

A. Architecture of PlatEMO ........................................................................ 10

B. Add Algorithms ....................................................................................... 11

C. Add Problems.......................................................................................... 13

D. Add Operators ......................................................................................... 15

E. Add Performance Indicators ................................................................... 17

Section I. Introduction to PlatEMO

I. Introduction to PlatEMO

A. Evolutionary Multi-Objective Optimization

Multi-objective optimization problems (MOPs), which involve two or more

conflicting objectives to be optimized, can be formulated as follows:







Whereis the decision vector (i.e. solution), and is

the known decision space. is the objective vector, and is the

unknown objective space.includes inequality constraints. Each element

indenotes a decision variable, andis the number of decision variables. Each

element indenotes a single-objective optimization problem, andis the number

of objectives.

Solutionis said to dominate solution (denoted by) if and only if

 

 

And solutionis said to be Pareto optimal if and only if



All the Pareto optimal solutions inconstitute the Pareto optimal set, and the objective

values of all the solutions in Pareto optimal set constitute the Pareto front.

Many metaheuristics have been verified in solving MOPs in the last two decades,

including genetic algorithm, differential evolution, particle swarm optimization,

memetic algorithm, estimation of distribution algorithm and so on, which are

collectively known as multi-objective evolutionary algorithms (MOEAs). An MOEA

usually maintains a population consisting of a set of individuals, where an individual

represents a solution together with its objective values and constraint values. The

PlatEMO User Manual

population is updated in each generation during the evolution, where new individuals

are generated by operators (e.g. crossover and mutation), and the population together

with new individuals is truncated by environmental selection. The goal of MOEAs is to

make the population approximate the Pareto optimal set with good convergence and

diversity.

B. PlatEMO

PlatEMO is an open source and free MATLAB-based platform for evolutionary

multi-objective optimization, which is available at http://bimk.ahu.edu.cn/index.php?

s=/Index/Software/index.html. It can be run on any operating system able to run

MATLAB®. PlatEMO provides two running modes to users: command mode and GUI

mode. In command mode, no GUI is displayed, users should set the parameters and

execute the algorithms by commands. In GUI mode, a GUI is displayed, users can set

the parameters and execute the algorithms on the GUI. In order to successfully run the

GUI mode, the version of the MATLAB® software should not be lower than R2014b.

The main features of PlatEMO are:

· It includes more than 60 existing MOEAs, most of which are representative

algorithms published in top journals after 2010.

· It includes more than 100 popular MOPs, which provide a variety of difficulties

for testing the MOEAs.

· It includes a lot of operators for different encodings, which can be combined

with all the MOEAs.

· It includes many performance indicators for numerically assessing the

performance of MOEAs.

· It is easy to be used that uses need not establish any project or write any code to

run PlatEMO, but just invoke the interface function main().

· It provides a powerful experimental module in the GUI, which can help users

perform experiments on multiple MOEAs and MOPs, and obtain the statistical

results in the format of Excel or LaTeX directly.

Section I. Introduction to PlatEMO

C. File structure of PlatEMO

Fig. 1 The file structure of PlatEMO

As shown in Fig. 1, there are six folders and one .m file in the root directory of

PlatEMO, the functions of which are given in the following.

· main.m. The only interface of PlatEMO, invoke this function to run the platform.

· Algorithms. For storing the source codes of all the MOEAs.

· GUI. For storing the codes to establish the GUI of PlatEMO, which need not be

read or modified by users.

· Metrics. For storing the source codes of all the performance indicators.

· Operators. For storing the source codes of all the operators.

· Problems. For storing the source codes of all the MOPs.

· Public. For storing some public classes and functions.

All the files in these folders are .m files, each of which represents one MATLAB

function or MATLAB class with detailed comments. All the files are open source except

for the ones in the folder GUI.

PlatEMO User Manual

II. How to Use PlatEMO

A. Use PlatEMO without GUI

Users can run the command mode of PlatEMO by invoking the interface function

main() with input parameters. While if main() is invoked without any input parameter,

the GUI mode will be run. All the acceptable parameters for main() are listed in Table

1. Note that users need not assign all the parameters since each of them has a default

value.

Table 1 The acceptable parameters for main()

Parameter Name

Data Type

Default Value

Description

-algorithm

function handle

@NSGAII

MOEA function

-problem

function handle

@DTLZ2

MOP function

-operator

function handle

@EAreal

Operator function

-N

positive integer

100

Population size

-M

positive integer

Number of objectives

-D

positive integer

Number of variables

-evaluation

positive integer

10000

Number of evaluations

-mode

1, 2 or 3

Run mode

-run

positive integer

Run No.

-outputFcn

function handle

@GLOBAL.show

Function invoked after each

generation

-X_parameter

cell

N/A

Parameters for function X

· -algorithm. The function handle of the MOEA to be executed.

· -problem. The function handle of the MOP to be solved by the MOEA.

· -operator. The function handle of the operator to be used in the MOEA. Note

that some MOEAs specify the operator by themselves (e.g. MOPSO.m), hence

this parameter is invalid for these MOEAs.

· -N. The size of the population generated by the MOEA after termination. Note

that the population size is fixed to some special values in some MOEAs (e.g.

MOEAD.m), hence the actual size of the population generated by these MOEAs

may not exactly equal to this parameter.

Section II. How to Use PlatEMO

· -M. The number of objectives of the MOP. Note that the number of objectives

is constant in unscalable MOPs (e.g. ZDT1.m), hence this parameter is invalid

for these MOPs.

· -D. The number of decision variables of the MOP. Note that the number of

decision variables is constant or fixed to some integers in some MOPs (e.g.

ZDT5.m), hence the actual number of decision variables may not exactly equal

to this parameter.

· -evaluation. The maximum number of fitness evaluations.

· -mode. If this parameter is set to 1, a figure showing the result will be displayed

after termination. And if this parameter is set to 2, the result will be saved to a

file after termination and no figure will be displayed, where the filename is

Data\algorithm\algorithm_problem_M_run.mat with algorithm, problem, M,

and run denoting the MOEA name, the MOP name, the number of objectives,

and the run number, respectively. If this parameter is set to 3, the result will be

neither displayed nor saved, instead, the operation acting on the result is

determined by the function outputFcn.

· -run. The run number. If users want to record multiple results for the same

parameters of algorithm, problem and M, modify this parameter in each run so

that the filenames of the results are different.

· -outputFcn. The function invoked after each generation. In particular, when

mode is set to 1, this parameter will be forced to @GLOBAL.show, which can

display the result after termination; when mode is set to 2, this parameter will

be forced to @GLOBAL.save, which can save the result after termination.

· -X_parameter. For setting the specific parameters in each MOEA, MOP or

operator. For example, the algorithm LMEA.m has three parameters nSel, nPer

and nCor, using main(…, ’-LMEA_parameter’, {5,50,5}, …) to set the values

of the three parameters to 5, 50 and 5, respectively. If the command is

main(…, ’-LMEA_parameter’, {[],50}, …), the first parameter nSel and the

third parameter nCor will equal to their default values. Note that the parameter

values should be put in a cell array like {5,50,5}, but not an array like [5,50,5].

The specific parameters for each MOEA, MOP and operator can be found in the

comments in the head of the corresponding function.

For example, use the following command to run KnEA on DTLZ2 with a population

size of 200 and 10 objectives, and the final result will be displayed:

1. main(- @KnEA, - @DTLZ2, - 200, - 10);

PlatEMO User Manual

Use the following command to run RVEA on UF8 with DE operator for 10 times, the

parameters in DE.m, i.e. CR, F, proM and disM, are set to 1, 0.6, 1 and 10, respectively,

and the final results will be saved:

1. for r = 1 : 10

2. -algorithm, @RVEA, - @UF8, - 2, - r, 

3. - {1,0.6,1,10});

4. end

B. Use PlatEMO with GUI

Users can run the GUI mode of PlatEMO by the following command:

1. main();

Then two modules can be seen on the GUI, i.e. the test module and the experimental

module. The test module is used to execute one MOEA on an MOP each time, and the

result will be displayed in a figure. The experimental module is used to execute multiple

MOEAs on several MOPs at the same time, and the statistical results will be listed in a

table and saved to files.

Fig. 2 The interface of the test module

The interface of the test module of PlatEMO is shown in Fig. 2. The functions of the

Section II. How to Use PlatEMO

controls in each region are:

· Region A. Select the MOEA, MOP and operator to be executed. Note that some

MOEAs specify the operator by themselves (e.g. MOPSO.m), hence the

operator cannot be selected by users when selecting these MOEAs.

· Region B. Set the parameters of the selected MOEA, MOP and operator. The

value of each parameter should be a scalar. Note that here the common

parameters N, M, D and evaluation are regarded as the parameters of MOPs. A

parameter will equal to its default value if the value assigned by users is empty.

· Region C. Execute the algorithm according to the current configuration.

· Region D. Show the introduction of the parameter in Region B which the cursor

is moving over.

· Region E. Show the current population during the optimization.

· Region F. Zoom in, zoom out, pan or rotate the axis in Region E.

· Region G. Open the axis in Region E in a new standard MATLAB figure, thus

more operations can be acted on the axis, e.g. saving the axis.

· Region H. Select the data to be displayed in the axis in Region E, including the

Pareto front of the population, the Pareto set of the population, the true Pareto

front of the MOP, and the evolutionary trajectory of any performance indicator.

· Region I. Control the optimization procedure, e.g. start, pause, stop, backward

and forward.

· Region J. Select one of the historical results to be displayed.

· Region K. Show the value of any performance indicator of the final population

of the result.

· Region L. Show the detailed information of the result.

After open the text module, users should first select the MOEA, MOP and operator

to be executed in Region A, and set their parameters in Region B, then press the button

in Region C to execute the algorithm. The real-time population will be displayed in the

axis in Region E, and users can use the buttons in Region I to control the optimization

procedure. After the algorithm has been terminated, all the historical results can be

redisplayed by selecting the popup menu in Region J.

The interface of the experimental module of PlatEMO is shown in Fig. 3. The

functions of the controls in each region are:

PlatEMO User Manual

Fig. 3 The interface of the experimental module

· Region A. Select the MOEAs and MOPs to be executed and specify the run

times. Note that multiple MOEAs and MOPs can be selected here, but the

operators cannot be selected since only the default operators can be used.

· Region B. Set the folder for saving the results. Uses can also open an existing

configuration file so that all the MOEAs and MOPs to be executed and their

parameter values can be automatically set. The configuration file Setting.mat

will be automatically created in the specified folder when starting to execute the

experiment.

· Region C. Set the parameters of the selected MOEAs, MOPs and operators.

Note that the value of each parameter in the MOPs can be a vector, thus the

MOEAs can be executed on the same MOP with different settings at the same

time. Note that the length of the vector should be 0, 1 or the same to the number

of distinct values of parameter M.

· Region D. Execute the experiment according to the current configuration.

· Region E. Show the introduction of the parameter in Region B which the cursor

is moving over.

· Region F. Show the statistical results of the experiment.

· Region G. Specify the type of data to be shown in the table in Region F.

· Region H. Save the table in Region F in the format of Excel or LaTeX.

· Region I. Select the data to be shown in the table in Region F, including the

values of any performance indicator of the final populations of the results.

Section II. How to Use PlatEMO

· Region J. Control the optimization procedure.

After opening the experimental module, users should first select the MOEAs and

MOPs to be executed in Region A and set their parameters in Region C, or load existing

configuration by pressing the button in Region B, then press the button in Region D to

execute the experiment. The statistical results will be displayed in the table in Region

F, and users can use the buttons in Region J to control the optimization procedure. After

the experiment has been terminated, the data shown in the table in Region F can be

saved in the format of Excel or LaTeX by pressing the button in Region H.

PlatEMO User Manual

III. How to Extend PlatEMO

A. Architecture of PlatEMO

GLOBAL

-algorithm

-problem

-operator

... ...

+Initialization()

+Variation()

... ...

Algorithm

Problem

Operator

NSGA-II

DTLZ1

INDIVIDUAL

-dec

-obj

-con

-add

+Individual()

... ...

1 1

Fig. 4 Architecture of PlatEMO

The architecture of PlatEMO is shown in Fig. 4, where there are only two classes in

the implementation of PlatEMO, i.e. GLOBAL and INDIVIDUAL. Note that each

algorithm, problem and operator is just a function, but not an object.

GLOBAL represents the configuration of the current run, the source code and the

detailed comments of which can be found in Public\GLOBAL.m. During each execution,

one GLOBAL object is maintained to store all the parameter setting and the result data,

including the function handles of the executed MOEA, MOP and operator, the

population size, the number of objectives, the number of decision variables, the

maximum number of fitness evaluations, the number of evaluated fitness, and so on.

GLOBAL also provides some methods which can be invoked by the MOEAs, for

instance, GLOBAL.Initialization() can generate a randomly initial population, and

GLOBAL.Variation() can generate a set of offsprings according to specified parents.

INDIVIDUAL represents one individual, the source code and the detailed comments

of which can be found in Public\INDIVIDUAL.m. An INDIVIDUAL object stores the

decision variables dec, the objective values obj, the constraint values con, and the

additional property values add of one individual. The values of dec and add are assigned

Section III. How to Extend PlatEMO

when invoking the constructor, then the values of obj and con are calculated

automatically. Each of the above properties is a row vector, and use P.decs, P.objs,

P.cons or P.adds can obtain a matrix of the decision variables, objective values,

constraint values or additional property values of an array of INDIVIDUAL objects P,

respectively, where each row of the matrix denotes one individual and each column

denotes one dimension of the values. When the number of instantiated INDIVIDUAL

objects exceeds the maximum number of fitness evaluations, the algorithm will be

forced to be terminated.

B. Add Algorithms

An MOEA function is represented by an .m file in PlatEMO, which should be put in

the folder Algorithms. For example, the source code of NSGAII.m is

1. function NSGAII(Global)

2. Population = Global.Initialization();

3. FrontNo = NDSort(Population.objs, inf);

4. CrowdDis = CrowdingDistance(Population.objs, FrontNo);

5. while Global.NotTermination(Population)

6. MatingPool = TournamentSelection(2, Global.N, FrontNo, -CrowdDis);

7. Offspring = Global.Variation(Population(MatingPool));

8. 

9. EnvironmentalSelection([Population, Offspring], Global.N);

10. end

11. end

Note that the common codes for all the MOEAs are underlined. To begin with, an

MOEA function has one input parameter denoting the GLOBAL object and none output

parameter. Then an initial population Population is generated (line 2), and the non-

dominated front number and the crowding distance of each individual are calculated

(line 3-4). In each generation, Global.NotTermination() is invoked to check whether the

number of evaluated fitness exceeds the maximum number of fitness evaluation, and

Population is passed to the function to be the final output (line 5). Afterwards, the

mating pool selection, generating offsprings, and environmental selection are

performed in sequence (line 6-9).

As shown in the code of NSGA-II, one MOEA should perform three operations at

least: obtaining an initial population via Global.Initialization(), checking the

PlatEMO User Manual

optimization state and passing Population via Global.NotTermination(), and generating

offsprings via Global.Variation(). Besides, the function CrowdingDistance() and

EnvironmentalSelection() are specific to NSGA-II, and NDSort() and

TournamentSelection() are public functions stored in the folder Public.

For decomposition based MOEAs, a set of reference points should be generated

before the optimization. For example in MOEAD.m, it uses the following command to

generate the reference points:

1. [W, Global.N] = UniformPoint(Global.N, Global.M);

Where UniformPoint() is a public function in the folder Public for generating about

Global.N uniformly distributed points with Global.M objectives on the unit hyperplane.

W is the set of reference points, and the population size Global.N is reset to the same to

the number of reference points in W.

The comments in the head of the MOEA functions (as well as MOP, operator and

performance indicator functions) should be written in a specified form such that it can

be identified by PlatEMO. For example in MOEADDE.m, the comments in the head of

the function are

1. function MOEADDE(Global)

2. % <algorithm> <H-N>

3. % Multiobjective Optimization Problems With Complicated Pareto Sets, MOEA/D

4. % and NSGA-II

5. % delta --- 0.9 --- The probability of choosing parents locally

6. % nr --- 2 --- Maximum number of solutions replaced by each offspring

7. % operator --- DE

Line 2 gives two labels of the function, the first label <algorithm> indicates that this is

an MOEA function (it is <problem>, <operator> and <metric> for MOP, operator and

performance indicator function, respectively), and the second label <H-N> can be an

arbitrary string. Line 3-4 give the title of the relevant literature. Line 5-6 define the

parameters for MOEA/D-DE, where the name of the parameter is located on the first

column of each line, the default value of the parameter is located on the second column,

and the introduction to the parameter is located on the third column; the columns are

divided by --- . Line 7 specifies the operator for MOEA/D-DE, where this line is

optional.

MOEA/D-DE then receives the parameter setting from users by the following

Section III. How to Extend PlatEMO

command:

1. [delta, nr] = Global.ParameterSet(0.9, 2);

The detailed introduction to Global.ParameterSet() can be found in Public\GLOBAL.m.

An MOEA may use its specified operator to generate offsprings, for example in

MOEADDE.m, the following command is used to generate offsprings by DE.m:

1. Offspring = Global.Variation(Population(MatingPool), 1, @DE);

The detailed introduction to Global.Variation() can be found in Public\GLOBAL.m.

Note that the number of evaluated fitness will be increased once this function is invoked.

For surrogate-assisted MOEAs, the following command can be used to generate new

decision variables according to the decision variables of parents, while no

INDIVIDUAL object will be instantiated:

1. OffDec = Global.VariationDec(Population(MatingPool).decs);

The detailed introduction to Global.VariationDec() can be found in Public\GLOBAL.m.

C. Add Problems

An MOP function is represented by an .m file in PlatEMO, which should be put in

the folder Problems. For example, the source code of DTLZ2.m is

1. function varargout = DTLZ2(Operation, Global, input)

2. switch Operation

3. 

4. Global.M = 3;

5. Global.D = Global.M + 9;

6. Global.lower = zeros(1, Global.D);

7. Global.upper = ones(1, Global.D);

8. Global.operator = @EAreal;

9. PopDec = rand(input, Global.D);

10. varargout = {PopDec};

11. 

12. PopDec = input;

13. M = Global.M;

14. PopObj = zeros(size(PopDec, 1), M);

PlatEMO User Manual

15. g = sum((PopDec(:, M:end)-0.5).^2, 2);

16. for m = 1 : M

17. PopObj(:, m) = (1+g).*prod(cos(PopDec(:, 1:M-m)*pi/2), 2);

18. if m > 1

19. PopObj(:, m) = PopObj(:, m).*sin(PopDec(:, M-m+1)*pi/2);

20. end

21. end

22. PopCon = [];

23. varargout = {PopDec, PopObj, PopCon};

24. c

25. f = UniformPoint(input, Global.M);

26. f = f./repmat(sqrt(sum(f.^2, 2)), 1, Global.M);

27. varargout = {f};

28. end

29. end

Note that the common codes for all the MOPs are underlined. To begin with, an MOP

function has three input parameters and one output parameter, where Operation denotes

the operation to be done, Global denotes the GLOBAL object, and input and varargout

may have one of the three meanings when Operation is set to different values. To be

specific, if Operation is set to ‘init’, the MOP function will assign the default values of

Global.M, Global.D, Global.lower, Global.upper and Global.operator (line 4-8), and

return the decision variable matrix of a random population with size input (line 9-10).

If Operation is set to ‘value’, the parameter input will denote the decision variable

matrix of a population, and the objective values PopObj and constraint values PopCon

of the population are calculated (line 14-22), note that here PopCon is set to empty

since DTLZ2 does not have any constraint. Then the matrices of decision variables,

objective values and constraint values of the population are returned (line 23). If

Operation is set to ‘PF’, input uniformly distributed reference points will be sampled

on the true Pareto front of the MOP and returned (line 25-27). Note that if the true

Pareto front of the MOP is unknown, the MOP should also return a nadir point for the

calculation of hypervolume (e.g. MOKP.m).

The number of objectives and decision variables may be fixed to some special values

in some MOPs. For example in ZDT5.m, these two values are assigned by

1. Global.M = 2;

2. Global.M = 2;

3. Global.D = 80;

4. Global.D = ceil(max(Global.D-30,1)/5)*5 + 30;

Section III. How to Extend PlatEMO

In line 1, the function sets the default value of Global.M to 2; if Global.M has already

been assigned by users, this line of code will be invalid. In line 2, the function sets the

value of Global.M to 2 again, then Global.M will equal to 2 no matter whether users

have assigned it. Analogously, line 3 sets the default value of Global.D to 80, and line

4 forces the value of Global.D to be a multiple of 5 after being subtracted by 30.

It is worth to note that a decision variable is illegal in three cases. First, for continuous

MOPs, it may be greater than the upper bound Global.upper or less than the lower

bound Global.lower, in this case it will be set to the boundary value by INDIVIDUAL

class, so MOP functions need not handle this case. Second, for combinational MOPs, it

may be an illegal character, in this case it should be modified and returned by the MOP

function when calculating the objective values and constraint values (e.g. MOKP.m).

Thirdly, it may not fulfill the constraints (a positive constraint value indicates that this

constraint is not fulfilled), while in this case it will not be modified.

An MOP function can receive parameter setting similar to MOEA functions, for

example in WFG1.m, the following command is used to receive parameter setting from

user:

1. K = Global.ParameterSet(Global.M-1);

While in some combinatorial MOP functions, the parameter is a random matrix and

should be identical in different runs, therefore such parameters should be randomly

generated and then saved to files. For example in MOKP.m, the following command is

used to set such parameters:

1. [P, W] = Global.ParameterFile(springf(-M%d-, M, D)

2. randi([10, 100], M, D), randi([10, 100], M, D));

To be specific, if the specified data file does not exist, the two parameters will be set to

the random matrices, and the matrices will be saved to the specified data file; while if

the file exists, the two parameters will be set to the matrices saved in the file. The

detailed introduction to Global.ParameterFile() can be found in Public\GLOBAL.m.

D. Add Operators

An operator function is represented by an .m file in PlatEMO, which should be put

in the folder Operators. For example, the source code of EAbinary.m is

PlatEMO User Manual

1. function Offspring = EAbinary(Global, Parent)

2. Parent = Parent(min(1:ceil(end/2)*2, end));

3. parentDec = Parent.decs;

4. [N,D] = size(parentDec);

5. Parent1Dec = parentDec(1:N/2, :);

6. Parent2Dec = parentDec(N/2+1:end, :);

7. k = repmata(1:D, N/2, 1) > repmat(randi(D, N/2, 1), 1, D);

8. Offspring1Dec = Parent1Dec;

9. Offspring2Dec = Parent2Dec;

10. Offspring1Dec(k) = Parent2Dec(k);

11. Offspring2Dec(k) = Parent1Dec(k);

12. OffspringDec = [Offspring1Dec; Offspring2Dec];

13. site = rand(N, D) < 1/D;

14. OffspringDec(site) = ~OffspringDec(site);

15. Offspring = INDIVIDUAL(OffspringDec);

16. end

Note that the common codes for all the operators are underlined. To begin with, an

operator function has two input parameters and one output parameter, where Global

denotes the GLOBAL object, Parent denotes the parent population, and Offspring

denotes the new population generated based on Parent. Since the number of parents for

crossover should be a multiple of two, the operator function first makes the number of

parents be an even (line 2). Then, it calculates the decision variables of the offsprings

according to the decision variables of the parents, where the single-point crossover and

bitwise mutation are performed in line 7-12 and line 13-14, respectively. Finally, the

objects of the offsprings are generated by INDIVIDUAL class and returned (line 15).

In particular, an individual may have additional property in some operators. For

example in PSO.m, it uses the following command to obtain the values of the speed

property of all the parents:

1. Speed = Parent.adds(zeros(N, D));

Where the input parameter zeros(N, D) is the matrix of default values for the speed

property of all the parents, that is, set the speed property of a parent to the default value

if it does not include this property. Then, the function uses the following command to

assign the speed property of offsprings:

1. Offspring = INDIVIDUAL(NewDec, NewSpeed);

Section III. How to Extend PlatEMO

Where NewDec is the matrix of decision variables of all the offsprings, and NewSpeed

is the matrix of speed property of all the offsprings. Note that an individual can have

more than one additional properties, while the operator functions need not specify the

name of each additional property, because the name of an additional property is

automatically set to the same to the name of the operator function.

E. Add Performance Indicators

A performance indicator function is represented by an .m file in PlatEMO, which

should be put in the folder Metrics. For example, the source code of IGD.m is

1. function score = IGD(PopObj, PF)

2. score = mean(min(pdist2(PF, PopObj), [], 2);

3. end

Note that the common codes for all the performance indicators are underlined. To begin

with, a performance indicator function has two input parameters and one output

parameter, where PopObj denotes the matrix of objective values of the population, PF

denotes the set of reference points sampled on the true Pareto front of the MOP, and

score denotes the scalar performance indicator value. Then, the performance indicator

value of the population is calculated according to the reference points and returned (line

2).

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