Weighted sum method with three objectives

GAMS
1

Hi everyone,
I can run my model with different weight sets for 2 objectives as follows;

set u /1*11/;

loop(u,
w1=1-(ord(u)-1)*0.1;
w2=1-w1;

However, when it comes to 3 objective, I was stuck. Could you help me about this issue? (For intervals of 0.1 similarly above statement)

2

Hi Kaidr
Why don’t you use the random number generator and generate a set of random weights? Something like this:

set u /1*11/;

Option Seed=12

parameter w1(u), w2(u), w3(u), delta(u);

w1(u) = uniformInt(0,10);
w2(u) = uniformInt(0,10);
w3(u) = uniformInt(0,10);
delta(u) = w1(u) + w2(u) + w3(u);
w1(u) = round(w1(u)/delta(u),1);
w2(u) = round(w2(u)/delta(u),1);
w3(u) = 1 - w1(u) - w2(u);

parameter check(u);

check(u) = w1(u) + w2(u) + w3(u);

display w1,w2, w3, check;

Cheers
Renger

3

Dear Renger,
Thank you for your answer. It absolutely works but, it may not generate all possible weight pairs, am I right? Or some weight pairs can repeat? This may cause missing some Pareto optimal solutions. Is there a way of producing all possible weights as increasing way such as:

w1 w2 w3
0.1 0 0.9
0.1 0.1 0.8
0.1 0.2 0.7
…
0.1 0.9 0
…
0.9 0.1 0.

Actually, such an approach will cause so many subproblems and will increase the solution effort and times. I think that I can use the epsilon constraint method but I am not sure how will I apply the epsilon constraint method to three objectives. Could you also help me with this issue?

4

Hi
According to me, you will have 55 combinations of weights. I used this code (there might be a more subtle way to do this):

set ln1 /1*10/; 
set cnt /1*110/  Counter;
parameter w,  w1, w2, w3, delta;


parameter counter;

* Set the first value (as not calculated in the loop)
w("1","1") = 1;
w("1","2") = 0;
w("1","3") = 0;    
counter = 0; 

loop(ln1,
    w1  = 1 - (ord(ln1) -1)*0.1 + EPS;
    delta = 1 - w1;
    w2 = 0;
    w3 = 0;
    counter = counter + 1;
    while(delta > 0.01,
        counter = counter + 1;
        w2 = w2 + 0.1;
        w(cnt,"1")$(ord(cnt) = counter) = w1;              
        w(cnt,"2")$(ord(cnt) = counter) = w2;
        w(cnt,"3")$(ord(cnt) = counter) = 1 - w1 - w2 + EPS;
        delta = 1 - w1 - w2;
    );
);
display w;

Cheers
Renger

5

Dear Renger,
Thank you very much. It is working perfectly.