14.11
Solution of Multiobjective Problems Using MATLAB
767
with the weights satisfying the normalization condition
k
,
wi = 1
i=1
14.11
SOLUTION OF MULTIOBJECTIVE PROBLEMS USING
MATLAB
The MATLAB function fgoalattain can be used to solve a multiobjective optimization problem using the goal attainment method. The following example illustrates the
procedure.
Example 14.3 Find the solution of the following three-objective optimization problem
using goal attainment method using the MATLAB function fgoalattain.
Minimize
Š2
2)
(x
+
3
13
f2 = 1751 (x 1 + x 2 Š 3
17 2
f3 = 8 ( x 1 Š 2x 2 + 4
31 + 27
f1 =
21
(x
+
1
1
+1
2
2)
(+ x 2 Š x 1) 2
Š 13
2) 1 (x
1Š x 2+1
+ 15
2) 1
2)
subject to
Š4_x
4x
1
Šx
x
1
i
+x
1
_ 4; i = 1, 2
2
Š4_0
Š1_0
Šx
2
Š2_0
Assume the initial design variables to be x 1 = x
w2 = 0 .5, and w 3 = 0.3, and the goals to be b
= 0.1, the weights to be w 1 = 0.2,
=
5, b 2 = Š8, and b 3 = 20.
1
2
SOLUTION
Step 1: Create an m-file for the objective functions and save it as fgoalattain_obj.m
function f = fgoalattainobj(x)
f(1) = (x(1)-2)^2/2+(x(2)+1)^2/13+3
f(2) = (x(1)+x(2)-3)^2/175+(2*x(2)-x(1))^2/17-13
f(3) = (3*x(1)-2*x(2)+4)^2/8+(x(1)-x(2)+1)^2/27+15
Step 2: Create an m-file for the constraints and save it as fgoalattain_con.m
function [c ceq] = fgoalattaincon(x)
c= [- 4- x(1); ...
x(1)- 4; ...
768
Practical Aspects of Optimization
- 4- x(2); ...
x(2)- 4; ...
x(2)+4*x(1)- 4; ...
- 1- x(1); ...
x(1)- 2- x(2)]
ceq = [];
Step 3:
Ctreate an m-file for the main program and save it as fgoalat-
tain_main.m
clc; clear all;
x0 = [0.1 0.1]
weight = [0.2 0.5 0.3]
goal = [5 -8 20]
x,fval,attainfactor,exitflag] = fgoalattain (@fgoalattainobj,
x0,goal,weight,[],[],[],[],[],[],@fgoalattaincon)
Step 4: Run the program fgoalattain_main.m to obtain the following result:
Initial design vector:
Initial objective values:
Constraints at initial design:
0.1,0.1
4.8981 -12.9546 17.1383
-4.1000
-3.9000
-4.1000
-3.9000
-3.5000
-1.1000
-2.0000
Optimum design vector:
Optimum objective values:
Constraints at optimum design:
0.8308 0.6769
3.8999 -12.9712 18.3498
-4.8308
-3.1692
-4.6769
-3.3231
-0.0000
-1.8308
-1.8462
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