Introduction to GAMS:
Conditionals, Subsets and Tuples in
GAMS
Bruce A. McCarl
&
Levan Elbakidze
Department of Agricultural
Economics
Texas A&M University
Conditionals, Subsets and Tuples in GAMS Basics
Conditionals allow GAMS modelers to write expressions
that operate over less than full sets or incorporate various
model features conditionally depending on data.
In ordinary algebra we have ways of controlling the
terms considered in indexed expressions. In particular, if
we had 40 elements and we wished to sum elements 1 to
20
20 we would write
X   Zi
i 1
or we might want to compute
b
pivot  min  i  for all i such that a i  0
 ai 
GAMS introduces that qualifications on indexes like i via
the use of conditionals ($ and if) or special sets we will
discuss both here.
Conditionals for the above follow (conditional.gms)
X=sum(I$(ord(i) le 20),z(i));
Pivot=smin(i$(a(i)>0),b(i)/a(i));
Conditionals, Subsets and Tuples in GAMS
Conditionals allow GAMS modelers to write
expressions that operate over less than full sets
or incorporate various model features
conditionally depending on data.
Fundamentally, they are used to:
Control whether an item is calculated
On a set element by element basis
Control the inclusion of terms in equations
In = calculations
In a .. model equation specifications
Control inclusion of set dependent terms in sums
Control inclusion of equations in a model
On a set element by element basis
Control whether a display is output to the LST file
Here we cover ways to do this first with
conditionals and later with a set or subset
Conditionals, Subsets and Tuples in GAMS
Basics of a $ Conditional
A conditional is entered into GAMS by adding a
$ into a GAMS statement. That causes an action
to occur if the conditional is true. The basic $
conditional form is
term$logical condition
which says include the term only if the logical
condition is true.
Suppose in a model we wish to set the parameter
X equal to 10 only if the parameter y is greater
than zero. A very basic conditional
(conditional.gms) that imposes this is as follows
X
$
(Y gt 0)
= 10;
which says set X = 10 only if the scalar y is
greater than zero.
Many different logical condition forms are
possible as I review later in these notes.
Conditionals, Subsets and Tuples in GAMS
Controlling whether an item is calculated
$conditionals can appear in a model to restrict
whether calculations are done. General format is
namedparameter$logical condition=term;
and specifies that the named parameter is set equal to
the term only if the logical condition is true.
Examples of whether an item is computed
X $(qq gt 0)
=3;
qq $(sum(I,q(i)) gt 0) =4;
a(i)$(qq gt 0)
=q(i)/qq;
qq $(sum(i,abs(a(i))) gt 0) =1/sum(i,a(i));
Examples of whether elements of an item are
computed
Q(i)$(a(i) gt 0) = -a(i);
a(i) $(abs(a(i)) gt 0) = q(i)/a(i);
The replacement is active only if the condition is
true. Above if X started at 7 and qq<0 then X would
finish = 7. But if qq>0 then X would = 3. The last 2
statements compute some but not all Q
This is known in GAMS terminology as a conditional
on the left hand side.
Conditionals, Subsets and Tuples in GAMS
Controlling inclusion of terms in equations
$ conditionals can restrict whether terms are included in
replacements or model equations. General format
term$logical condition
causing terms to be included only if the condition holds.
Eq4.. xvar + yvar$(qq gt 0)=e=3;
X = sum(I,q(i))$(qq gt 0) + 4;
Q(i)= a(i) + 1$(a(i) gt 0);
The right versus left hand side conditional case
merits mention here. Consider
Q(i)= (a(i) + 1)$(a(i) gt 0);
X = [sum(I,q(i))+4]$(qq gt 0);
Which are treated as
Q(i)= 0+ (a(i) + 1)$(a(i) gt 0);
X = 0+ [sum(I,q(i))+4]$(qq gt 0);
These are somewhat like .
Q(i) $(a(i) gt 0)= (a(i) + 1);
X $(qq gt 0) = [sum(I,q(i))+4];
But in above cases the items in pink will equal zero when
the conditional is not true but in the maroon case will
retain their prior value. This is important left versus right
hand side conditional characteristic.
Conditionals, Subsets and Tuples in GAMS
Controlling inclusion of set dependent terms
$conditionals are frequently used to restrict the elements
of a set that will be explicitly entered into a sum, loop,
or other set operation (e.g. prod, smax,smin). In
particular there may be a restriction one wishes to place
on the set elements that can validly enter the calculation.
Inclusion is controlled by a logical condition. Generally
this logical condition will incorporate set element
dependent terms. The general format in a SUM context
is
sum(namedset$set dependent logical condition,term)
where for example
X
is written as

i / qi  0
ai
X=sum(I$(q(i) gt 0),a(i));
Similarly one can write
Loop((I,j)$(cost(I,j) lt 0),X=x+cost(I,j));
X=smin(I$(sum(j,cost(I,j)) gt 0),q(i));
X=sum(I$(q(i) gt 5),1));
All include the term associated with particular set
elements in the sum or loop or smin only if the
logical condition is true.
Conditionals, Subsets and Tuples in GAMS
Controlling whether equations are included
$conditionals can be used to restrict whether model
equations are active in a model. The general format is
equationname$logicalcondition.. equationspecification;
and specifies that the equation is active in the model only if
the logical condition is true.
Eq1 $(qq gt 0)..
xvar=e=3;
Eq2 $(sum(I,q(i)) gt 0).. yvar=l=4;
Eq3(i)$(a(i) gt 0)..
ivar(i)=g=a(i);
Eq5(i)$(qq gt 0)..
ivar(i)=g=a(i);
In these cases, the whole equation called equationname
(see the Eq1, Eq2 and Eq5 cases above) will be entered
into the model only if the logical condition is true. When
the conditional involves set dependent terms then only the
specific cases associated with true conditionals will occur
(see the Eq3 case above)
When the $ is on the left hand side, the equation is only
included in the model and calculated if the logical
condition is true. This can make the program faster
Conditionals, Subsets and Tuples in GAMS
Conditionally displaying information
Displays may be conditioned to appear only if a
logical condition is met. (Note the display command
is discussed in the Report Writing chapter of the
documentation).
One can have the keyword display followed by a
$condition. Therein the display will only occur if the
$condition is true.
display$condition listofitems;
One can also use the if syntax as
if(condition, display listofitems);
Both are illustrated below
scalar x /0/,y /10/;
if(x gt 0,
display "X and y if x is positive",x,y;);
*note next command is redundant to above
display$(x+y le 0)
"display when sum of x and y le 0",x,y;
x=2;
display$(x > -1)
"display$ at third place",x;
*note will not get to next line
if((x+y > 2),
display "X and y if x+y is greater than 2",x,y;);
Conditionals, Subsets and Tuples in GAMS
Forms of $ logical conditions
Logical conditions can involve numerical comparisons,
existence of numbers or set elements, and index
characteristics in sets. Each case is slightly different
and will be discussed separately.
Numerical comparisons
Logical conditions can compare two numerical
expressions . The general form is
Term$(terma operator termb)
The operators are as follows:
Relation
GAMS operator
Explanation
Equality
EQ or =
Is terma = termb
Not Equal
NE or <>
Is terma ≠ termb
Greater than GT or >
Is terma > termb
Greater or = GE or >=
Is terma > termb
Less than
LT or <
Is terma < termb
Less or =
LE or >=
Is terma < termb
Conditionals, Subsets and Tuples in GAMS
Numerical $ condition forms
Relation
GAMS operator
Explanation
Equality
EQ or =
Is terma = termb
Not Equal
NE or <>
Is terma ≠ termb
Greater than GT or >
Is terma > termb
Greater or = GE or >=
Is terma > termb
Less than
LT or <
Is terma < termb
Less or =
LE or >=
Is terma < termb
Examples
Eq3$(x=2).. zz=e=3;
Xx=Sum(I$(x eq sum(j,y(I,j))+2),1);
Xx=Sum(I$(x <> sum(j,y(I,j))+2),1);
Xx=Sum(I$(x ne 22),1);
Eq4$(x<2).. zz=e=3;
Xx=Sum(I$( sum(j,y(I,j)) lt yx+2), z+1);
Loop(I$(x > yx+2),z=z+1);
Loop(I$(x gt yx+2),z=z+1);
Eq5$(x>=2).. zz=e=3;
Xx=Sum(I$(x ge yx+2),z=z+1);
Eq6(i,j)$(y(I,j)<=2).. y(I,j)*zz=e=3;
Loop(I$(x le yx+2),z=z+1);
Conditionals, Subsets and Tuples in GAMS
Element or item presence condition forms
General form
Action$Number or numeric expression
Action$Subsetelement
Such conditions can be formed which are true if
1. A data item exists with a nonzero value
xx$yx = 3;
equivilant to xx$(yx ne 0)=3;
2.A numerical expression exists (again with existence
being defined as the expression result being nonzero)
xx$sum((i,j),y(i,j)) = 3;
Important subcase is are there some data for an item
xx$sum((i,j)$y(i,j),1) = 3;
or
xx$sum((i,j),1$y(i,j)) = 3;
or
xx$sum((i,j),abs(y(i,j))) = 3;
3.A subset element exists
rx(i)$subsetofi(i) = 3;
4. Some of a group of set elements exist
xx$sum(i$subsetofi(i),1)=3; (note $ nesting)
Conditionals, Subsets and Tuples in GAMS
Set element position
Logical conditions can be formed which depend on sets
in terms of the relative placement of the element being
worked on and element text comparisons.
Element position
Ord returns number of an element from 1 to N
Card returns total number of elements in a set.
Use with numerical conditions from above
Example
If one wishes to do special things with respect to the
first, last and intermediate elements in a set one could
use code like
FIRSTBAL(time)$(ord(time) eq 1)..
SELL(time) + STORE(time) =L= INVENTORY;
INTERBAL(TIME)
$(ORD(time) GT 1 and ORD(time) lt
CARD(time))..
SELL(TIME)=L=STORE(TIME-1)STORE(TIME);
LASTBAL(TIME)$(ORD(time) eq CARD(time))..
SELL(TIME)=L= STORE(TIME-1);
or
loop(time$(ORD(time) GT 1 and
ORD(time) lt CARD(time)),Z=z+1;);
Conditionals, Subsets and Tuples in GAMS
Set element text contents
Logical conditions can test the text content of set
element names matching a particular text string or
matching up with names for a set element in another set.
These involve the sameas or diag commands.
SAMEAS(setelement,othersetelement)
is true if setelement name is the same as name of
othersetelement and is false otherwise.
sameas(asetelement,”texttotest”) is true if name of
asetelement equals texttotest and false otherwise.
DIAG works identically but returns numerical values of
one if the text matches and zero otherwise.
Set
cityI / “new york”, Chicago, boston/;
Set
cityj /boston/;
Scalar ciz,cir,cirr;
ciZ=sum(sameas(cityI,cityj),1);
ciR=sum((cityI,cityj)$ sameas(cityI,cityj),1);
ciRR=sum(sameas(cityI,”new york”),1);
ciZ=sum((cityi,cityj)$diag(cityI,cityj) ,1);
ciRR=sum(cityi$diag(cityI,"new york"),1);
The red uses of sameas and diag will only permit I and j to be
part of the sum when elements are both Boston. The blue
cases will only operate for the element of I associated with the
name “new york”.
Conditionals, Subsets and Tuples in GAMS
Nesting logical condition
Sometimes complex logical conditions are needed
to insure several things are true or at least one of
several things. Such complex conditionals are
called nested. Common nesting operators are
And, Or, Not.
and - perform an action if two or more
conditionals to apply simultaneously
or - perform an action if at least one of two or
more conditionals apply
not - do something when a conditional is not rue
Nested $ - statements can involve multiple $
conditions
Examples
u(k)$(s(k) and t(k)) = a(k);
u(k)$(s(k) and u(k) and t(k)) = a(k);
loop(k,if(s(k) lt 0 and t(k), u(k) = a(k)));
u(k)$(s(k) or t(k)) = a(k);
u(k)$(s(k) or u(k) or t(k)) = a(k);
loop(k,if(s(k) lt 0 or t(k), u(k) = a(k)) );
u(k)$(not s(k)) = a(k);
loop(k,if(not (s(k) lt 0), u(k) = a(k)) );
u(k)$(s(k)$t(k)) = a(k) ;
Conditionals, Subsets and Tuples in GAMS
Nesting logical condition
Nested operators are acted on in a fixed precedence
order and are also operated on in an order relative to
mathematical operators. The default precedence order
in the absence of parentheses is shown below in
decreasing order.
Operation
Operator
Precedence
Exponentiation
**
1
Multiplication, Division * /
2
Addition, Subtraction
+3
Numerical Relationships < ,<=, = ,
4
<>, >=, >
lt , le ,eq,
ne , ge, gt
Not
not
5
And
and
6
Or, Xor
or, xor
7
When operators with the same precedence appear in
an expression or logical condition they are performed
from left to right. It is always advisable to use
parentheses rather than relying on the precedence
order of operators.
u(k)$([not s(k)] and {u(k) or t(k)}) = a(k);
u(k)$(s(k) or {u(k) and t(k)}) = a(k);
u(k)$(s(k) or [not {u(k) and t(k)}]) = a(k);
Conditionals, Subsets and Tuples in GAMS
The conditional alternative – the Subset
Mathematically one often sees algebraic
statements where the index positions to address are
subsets of larger sets of index positions. This can
be done in GAMS using subsets and tuples.
Namely suppose in a case where we have 40
elements we wish to sum elements 1 to 20
20
X   Zi
i 1
we can define a subset of I containing elements 1-20 via
set subseti(i) /i1*i20/;
then we can write
X=sum(I$subseti(i),z(i));
or
X=sum(subseti,z(subseti));
or
X=sum(subseti(i),z(i));
The first and last of which vary I over values that
appear in the subset and the middle which addresses
only the subset.
One can also compute the subset
Subseti(i)$(ord(i) le 20)=yes;
and then use as above
Conditionals, Subsets and Tuples in GAMS
The conditional alternative –the tuple
Often the conditionals required in a model are complex and
are used in a repetitive manner. It can be simpler to
employ a tuple (a set that is defined over other sets see the
sets chapter of the documentation) that encapsulates the
conditionals and only use it. In the model below the
logical condition in red that appears repetitively can be
replaced with the tuple making the model visually simpler
TCOSTEQ..
TCOST =E= SUM((PLANT,MARKET)
$(supply(plant) and demand(market)
and distance(plant,market))
, SHIPMENTS(PLANT,MARKET)* COST(PLANT,MARKET));
SUPPLYEQ(PLANT).. SUM(MARKET
$(supply(plant) and demand(market) and
distance(plant,market)),SHIPMENTS(PLANT,
MARKET)) =L= SUPPLY(PLANT);
DEMANDEQ(MARKET).. SUM(PLANT
$(supply(plant)and demand(market)
and Distance(plant,market))
,SHIPMENTS(PLANT, MARKET)) =G= DEMAND(MARKET);
where the alternative with the tuple is
set thistuple(plant,market)tuple expressing conditional;
thistuple(plant,market)$(supply(plant)and demand(market)and
distance(plant,market))
=yes;
TCOSTEQ2..TCOST =E= SUM(thistuple(plant,market)
, SHIPMENTS(PLANT,MARKET)*COST(PLANT,MARKET));
SUPPLYEQ(PLANT).. SUM(thistuple(plant,market)
,SHIPMENTS(PLANT, MARKET)) =L= SUPPLY(PLANT);
DEMANDEQ(MARKET).. SUM(thistuple(plant,market)
,SHIPMENTS(PLANT, MARKET)) =G= DEMAND(MARKET);
Conditionals, Subsets and Tuples in GAMS
The conditional alternative –the tuple
A tuple refers to a set defined over other sets. The set
may either be a one dimensional subset or a
multidimensional set. Tuples are useful in calculations
and in imposing conditionals.
set thistuple(plant,market) tuple expressing conditional;
thistuple(plant,market)$( supply(plant) and demand(market)
and distance(plant,market))
=yes;
TCOSTEQ2..
TCOST =E= SUM(thistuple(plant,market)
,
SHIPMENTS(PLANT,MARKET)*COST(PLANT,MARKET));
SUPPLYEQ(PLANT).. SUM(thistuple(plant,market)
,SHIPMENTS(PLANT, MARKET)) =L= SUPPLY(PLANT);
DEMANDEQ(MARKET).. SUM(thistuple(plant,market)
,SHIPMENTS(PLANT, MARKET)) =G= DEMAND(MARKET);
The statement
SUM(thistuple(plant,market)
,SHIPMENTS(PLANT, MARKET))
Adds the terms by summing the PLANT and MARKET
set cases that appear in the tuple thistuple(plant,market)
and given the computation above correspond to a plant
with nonzero supply a market with nonzero demand
linked by a nonzero distance.
Conditionals, Subsets and Tuples in GAMS
Other conditional implementation possibilities
Conditionals appear in a few other forms
If -- One can use the if statement syntax that also
involves the else and elseif statements. In general, if
statements can be written as $ conditions, but the use of
if can make GAMS code more readable.
if (x ne 0, DATA(I)=12 ; );
While -- One can use the while statement to repeatedly
execute a block of statements until a logical condition is
satisfied.
while(x<10,
x=x+0.01; );
Repeat -- One can use the repeat to execute a block of
statements over and over until a logical condition is
satisfied.
repeat( x=x+0.01; until x>=10) ;