define subset in GAMS

Question

I have a problem about partitioning . with 3 set $i,j,k$ , $i=1,..,32 $ Represents the number of classes, $j=1,..,512 $ Represents the number of student, $k=1,..,40 $ Represents the type of student.

set $j=\{ 1,...,512\}$ , and subsets $t_k$ of $j$ ,Which indicates student $j$th belongs to which type. for example , if we consider type $k=1$, then all student with type $'1'$ are $t_1= \{ 2,10,78,69,114,500\}$.

These subsets are known. for example $t_1= \{ 2,10,78,69,114,500\}$ and $t_2=\{8,42,50\}$ ,...

I want to code in gams this sum in GAMS , $ \sum_{i,j} a_{i,j} x_{i,j} \ \ \ \ \forall i , \ \ j\in t_k$

I know when we want to use subset we can write

set j / 1*512/,
    t(j) 

but in this case, The subset has already been specified .

i use this way

set
     j/1*512/
     t(j)/2,10,78,69,114,500/
     ;

its right for one case "$t_1$ "

I don't know how can I define this type of subset such that Work for all sub set, and how can I define this summation?

Answer 1

In the first few iterations of this question the math was just wrong. It is no surprise that in that case the translation into GAMS is very difficult.

Now we have:

sets
    i 'classes' /class1*class32/
    j 'students' /student1*student512/
    k 'student type' /type1*type40/
; 

Of course for a real model the set elements would have proper meaning, like:

set i 'classes' /English,Biology,Math/;

Now, we can do:

set t(k,j) 'mapping between type and student' /
   type1.(student2,student10,student78,student69,student114,student500)
   type2.(student8,student42,student50)
/;     

We can use this in expressions as

equation 
   e(k).. v(k) =e= sum((i,j)$t(k,j), a(i,j)*x(i,j));

This would correspond to:

$$v_k = \sum_{i,j|t(k,j)} a_{i,j} x_{i,j}$$

I.e. $ is a 'such-that' operator, just like | in math.