Multi objective optimization for genetic algorithm showing out of bound solutions.

Hello! I am comparatively new to Matlab.

I am working on a project. I need to optimize two objective functions which have two variables (x1,x2) in total.

x1 should be within 130 to 160, and x2 should be within 18 to 22.There are no other constraints. But the optimized solution is showing results x2 value of 160. I wrote the code with the help of some youtube videos and ChatGPT. What is the reason behind the value of x2 being more than 22?.

Here is the code:

% Define the first objective function
fun1 = @(x) -(1.10684 + 0.005818*x(1) + 0.498853*x(2));

% Define the second objective function
fun2 = @(x) -(82.015320468227 + 0.38563758777369*x(1) - 9.6030354792151*x(2) - 0.0095666666666666*x(1)*x(2) - 0.00055977777777776*x(1)^2 + 0.2517625*x(2)^2);

% Combine the objectives into a single function for multi-objective optimization
multiObjFun = @(x) [fun1(x), fun2(x)];

% Set lower and upper bounds for each parameter
lb = [130, 18];
ub = [160, 22];


% Use the genetic algorithm for multi-objective optimization
options = optimoptions('gamultiobj', 'Display', 'iter');
[x, fval, exitflag, output, population, scores] = gamultiobj(multiObjFun, 2, [], [], [], [], lb, ub, options);

Multi-objective optimization:
2 Variables

Options:
CreationFcn:       @gacreationuniform
CrossoverFcn:      @crossoverintermediate
SelectionFcn:      @selectiontournament
MutationFcn:       @mutationadaptfeasible

                              Average            Average
Generation   Func-count    Pareto distance    Pareto spread
    1            50                 1                 1
    2           100         0.0206928          0.411061
    3           150         0.0199406          0.221978
    4           200        0.00509167         0.0739517
    5           250         0.0103144          0.312047
    6           300        0.00535776         0.0785609
    7           350         0.0101285          0.145853
    8           400          0.010704          0.339981
    9           450        0.00562346          0.128856
   10           500        0.00644826         0.0969272
   11           550        0.00824844          0.123374
   12           600        0.00376789         0.0649277
   13           650         0.0102013           0.13323
   14           700        0.00809621          0.126988
   15           750         0.0141773          0.201989
   16           800        0.00746776         0.0974903
   17           850        0.00525844         0.0701876
   18           900        0.00572953          0.120595
   19           950         0.0150192          0.193395
   20          1000         0.0100848          0.126341
   21          1050        0.00803535          0.215489
   22          1100         0.0136081          0.200525
   23          1150        0.00541403         0.0886357
   24          1200         0.0040178         0.0620155
   25          1250        0.00699872          0.102564
   26          1300         0.0052385         0.0816008
   27          1350         0.0109806          0.153293
   28          1400        0.00582916          0.153478
   29          1450        0.00876156          0.122766
   30          1500         0.0124733           0.19677

                              Average            Average
Generation   Func-count    Pareto distance    Pareto spread
   31          1550        0.00679181          0.108716
   32          1600        0.00837598          0.124589
   33          1650        0.00662755         0.0902761
   34          1700         0.0059547          0.134663
   35          1750        0.00795628          0.120039
   36          1800         0.0081142          0.135508
   37          1850        0.00714794          0.134259
   38          1900         0.0118365          0.164218
   39          1950        0.00872715            0.1291
   40          2000         0.0147769          0.192475
   41          2050        0.00827088          0.129132
   42          2100        0.00694102           0.11361
   43          2150        0.00668206          0.105147
   44          2200         0.0114985          0.140688
   45          2250        0.00522078         0.0803578
   46          2300        0.00697735         0.0999024
   47          2350         0.0113815          0.153091
   48          2400         0.0115905          0.156521
   49          2450        0.00806875          0.116909
   50          2500        0.00796998           0.11933
   51          2550        0.00905485          0.133076
   52          2600        0.00549726         0.0878519
   53          2650        0.00664611          0.115028
   54          2700        0.00828181          0.148065
   55          2750        0.00677163           0.11915
   56          2800         0.0098354           0.12161
   57          2850         0.0120453          0.162291
   58          2900        0.00665377          0.106378
   59          2950        0.00637543         0.0988354
   60          3000         0.0104169           0.13678

                              Average            Average
Generation   Func-count    Pareto distance    Pareto spread
   61          3050        0.00627528         0.0792071
   62          3100         0.0068369         0.0978531
   63          3150        0.00464378         0.0684661
   64          3200        0.00591546         0.0897549
   65          3250        0.00738431          0.110535
   66          3300        0.00868434          0.192815
   67          3350        0.00518109         0.0645982
   68          3400        0.00560747         0.0883839
   69          3450        0.00824552          0.202331
   70          3500        0.00819077          0.109783
   71          3550        0.00531933         0.0782256
   72          3600        0.00702356          0.106576
   73          3650         0.0059714          0.122619
   74          3700        0.00886941          0.114288
   75          3750        0.00728946          0.101009
   76          3800        0.00425773         0.0627526
   77          3850        0.00736038          0.114394
   78          3900        0.00474414         0.0815989
   79          3950        0.00400237          0.067633
   80          4000        0.00997999          0.153975
   81          4050        0.00455801         0.0775172
   82          4100         0.0063894         0.0979791
   83          4150         0.0081187          0.129505
   84          4200          0.009662          0.139239
   85          4250        0.00575785         0.0768275
   86          4300        0.00579521         0.0873514
   87          4350        0.00843108           0.13242
   88          4400        0.00781309          0.122024
   89          4450        0.00748439          0.117754
   90          4500        0.00537968          0.076847

                              Average            Average
Generation   Func-count    Pareto distance    Pareto spread
   91          4550         0.0121643          0.146514
   92          4600        0.00746221         0.0992378
   93          4650         0.0068336         0.0979483
   94          4700         0.0041051         0.0666727
   95          4750        0.00604042          0.101174
   96          4800         0.0105444          0.157265
   97          4850        0.00633384          0.100006
   98          4900        0.00743112          0.126459
   99          4950         0.0111458          0.160848
  100          5000        0.00558786         0.0895414
  101          5050        0.00938994          0.131796
  102          5100        0.00876371          0.131211
  103          5150        0.00596723         0.0893314
  104          5200        0.00374933         0.0602251
  105          5250        0.00651905          0.108185
  106          5300        0.00611937          0.104669
  107          5350         0.0104159           0.14715
  108          5400        0.00645118         0.0997381
  109          5450        0.00710006          0.230089
  110          5500        0.00558316         0.0826734
  111          5550         0.0068663          0.106392
  112          5600        0.00650016          0.112754
  113          5650        0.00763674          0.104838
  114          5700        0.00527261           0.10487
  115          5750        0.00408311         0.0804505
gamultiobj stopped because the average change in the spread of Pareto solutions is less than options.FunctionTolerance.

% Display the Pareto front (trade-off solutions)
figure;
plot(scores(:, 1), scores(:, 2), 'o');
xlabel('Objective 1');
ylabel('Objective 2');
title('Pareto Front');

% Display information about the optimal solution(s)
fprintf('Optimal Solution:\n');

Optimal Solution:

fprintf('x1 = %.2f\nx2 = %.2f\n', x(1), x(2));

x1 = 159.98
x2 = 159.96

fprintf('Objective 1 Value: %.4f\n', -fval(1));

Objective 1 Value: 13.0075

fprintf('Objective 2 Value: %.4f\n', -fval(2));

Objective 2 Value: 11.0173

Answer 1

[Torsten]

If you plot x(:,2) against x(:,1), you will see that both respect the bounds you prescribed:

figure(2)
[x1,idx] = sort(x(:,1))
plot(x1,x(idx,2))

Printing x(1) and x(2) for your two-dimensional array x means to plot x(1,1) and x(2,1):

x = [3 4;5 6];
x(1)

ans = 3

x(2)

ans = 5

[Ahmed]

Okay, I have changed my display section by this way.Now it prints x1 and x2.

But I don't think it is actually optimizing both of the objective functions. Because x1 is always 159 or 160 (means always highest), and x2 is 18 or 22(means lowest or highest). And the objectve 2 keeps fluctuating.

% Display information about the optimal solution(s)
fprintf('Optimal Solution:\n');
fprintf('x1 = %.2f\nx2 = %.2f\n', x(1,1), x(1,2));
fprintf('Objective 1 Value: %.4f\n', -fval(1,1));
fprintf('Objective 2 Value: %.4f\n', -fval(1,2));

[Ahmed]

I got what's happening here.

The program is printing initial values of x1,x2, obj 1, obj 2. The pareto front graph of the objectives shows the optimal values, but the printing values are the initial values.

What is the solution for this?

[Ahmed]

here's the full code:

% Define the first objective function
fun1 = @(x) -(1.10684 + 0.005818*x(1) + 0.498853*x(2));

% Define the second objective function
fun2 = @(x) -(82.015320468227 + 0.38563758777369*x(1) - 9.6030354792151*x(2) - 0.0095666666666666*x(1)*x(2) - 0.00055977777777776*x(1)^2 + 0.2517625*x(2)^2);

% Combine the objectives into a single function for multi-objective optimization
multiObjFun = @(x) [fun1(x), fun2(x)];

% Set lower and upper bounds for each parameter
lb = [130, 18];
ub = [160, 22];


% Use the genetic algorithm for multi-objective optimization
options = optimoptions('gamultiobj', 'Display', 'iter');
[x, fval, exitflag, output, population, scores] = gamultiobj(multiObjFun, 2, [], [], [], [], lb, ub, options);

% Display the Pareto front (trade-off solutions)
figure;
plot(scores(:, 1), scores(:, 2), 'o');
xlabel('Objective 1');
ylabel('Objective 2');
title('Pareto Front');
figure(2);
[x1,idx] = sort(x(:,1));
plot(x1,x(idx,2));
% Display information about the optimal solution(s)
fprintf('Optimal Solution:\n');
fprintf('x1 = %.2f\nx2 = %.2f\n', x(1,1), x(1,2));
fprintf('Objective 1 Value: %.4f\n', -fval(1,1));
fprintf('Objective 2 Value: %.4f\n', -fval(1,2));

[Torsten]

You have two objective functions - so there is no single optimal solution as you think there is.

Did you read what the "Pareto Front" you get means ?

[Ahmed]

Yes, I think understand what it means. Pareto front gives all the possible solutions after optimization.

But I thought the code was printing the best out of it. I understand now that I was wrong.

So, how can I get the optimal values? should I point out every one of them from the graph? there must be some way to print all of them.

[Ahmed]

By the way. If there are multiple solutions, then what are the final printed values (x1,x2, obj1, obj2) ?

[Torsten]

There is no "best" solution. Each of the solution pairs you get in the arrays x(:,1) and x(:,2) are pareto-optimal. Read about the concept of what a "Pareto Front" means.

https://uk.mathworks.com/help/gads/what-is-multiobjective-optimization.html

Imagine you want to make profit and save nature at the same time and you have two objective functions - one for profit, the other for state of environment. There cannot be an optimum that optimizes both simultaneously because the two targets are in conflict with each other. But given an amount of profit, you can find a point that best saves nature for this profit. And given a level of safety for nature, you can maximize profit under this constraint.