Using the sum function in GAMS to sum over a subset of variables

Question

I am working with maximazation problems in GAMS where I will choose X=(x_1,x2,...,x_n) such that f(X)=c_1*x_1+...c_n*x_n is maximized. The c's are known scalars and I know n (10 in my case). I want my constraints to be such that the first (n-1)=9 x's should sum up to one and the last one should be less than 10. How do I use the sum to do so?

This is what I have tried:

SET C / c1 .... c2 /;
ALIAS(Assets,i)

Parameter Valuesforc(i) 'C values'/
*( here are my values typed in for all the C1)

POSITIVE VARIABLES
    x(i);

EQUATIONS
    Const1 First constraint
    Const1 Second constraint
    Obj    The Object;
* here comes the trouble:
Const1 ..   x(10) =l= 10
Const2 ..   sum((i-1),x(i)) =e= 1

The code is not done all the way but I believe the essential setup is typed in. How do you make the summation to find x_1+x_1 + .... x_(n-1) and how do you refer to x_10?

Answer 1

Try this:

Const1 ..   x('10') =l= 10;
Const2 ..   sum(i$(ord(i)<card(i)),x(i)) =e= 1;

Edit: Here are some notes to explain what happens in Const2, especially in the "$(ord(i) < card(i))" part.

• The "$" starts a condition, so it excludes certain elements of i from the sum (see: https://www.gams.com/latest/docs/UG_CondExpr.html#UG_CondExpr_TheDollarCondition)
• The operator ord returns the relative position of a member in a set (see: https://www.gams.com/latest/docs/UG_OrderedSets.html#UG_OrderedSets_TheOrdOperator)
• The operator card returns the number of elements in a set (see: https://www.gams.com/latest/docs/UG_OrderedSets.html#UG_OrderedSets_TheCardOperator)

So, all in all, there is a condition saying that all elements of i should be included in the sum except for the last one.