introduction to gams

7/31/2019 Introduction to GAMS

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Introduction to GAMS:

Conditionals, Subsets and Tuples in

GAMS

Bruce A. McCarl

&Levan ElbakidzeDepartment of Agricultural

Economics

Texas A&M University


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Conditionals allow GAMS modelers to write

expressions that operate over less than full seor incorporate various model features

conditionally depending on data.

Fundamentally, they are used to:

Control whether an item is calculatedOn a set element by element basis

Control the inclusion ofterms in equations

In = calculations

In a .. model equation specificationsControl inclusion ofset 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 fi

Here we cover ways to do this first with

conditionals and later with a set or subset

Conditionals, Subsets and Tuples in GAMS


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Basics of a $ Conditional

A conditional is entered into GAMS by adding a

$ into a GAMS statement. That causes an actionto occur if the conditional is true. The basic $

conditional form isterm$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 followsX $ (Y gt 0) = 10;

which says set X = 10only ifthe scalar y is

greater than zero.Many different logical condition forms are

possible as I review later in these notes.


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Conditionals, Subsets and Tuples in GAMSControlling whether an item is calculated

$conditionals can appear in a model to restrict

whether calculations are done. General format isnamedparameter$logicalcondition=term;

and specifies that the named parameter is set equal t

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 istrue. Above if X started at 7 and qq0 then X would = 3. The last 2

statements compute some but not all QThis is known in GAMS terminology as a conditional

on the left hand side.


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Conditionals, Subsets and Tuples in GAMSControlling inclusion of terms in equations

$ conditionals can restrict whether terms are included in

eplacements or model equations. General formatterm$logicalcondition

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 pinkwill equal zero when

he conditional is not true but in the maroon case will

etain their prior value. This is important left versus right

hand side conditional characteristic.


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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 elementdependent terms. The general format in a SUM context

is

sum(namedset$set dependentlogical condition,term)

where for example

0/ iqi

iaXis written as

X=sum(I$(q(i) gt 0),a(i));

Similarly one can writeLoop((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.


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Conditionals, Subsets and Tuples in GAMSControlling whether equations are included

conditionals can be used to restrict whether modelquations are active in a model. The general format isquationname$logicalcondition.. equationspecification;

nd specifies that the equation is active in the model only i

he 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);n these cases, the whole equation called equationname

see the Eq1, Eq2 and Eq5 cases above) will be entered

nto the model only if the logical condition is true. When

he conditional involves set dependent terms then only the

pecific cases associated with true conditionals will occur

see the Eq3 case above)

When the $ is on the left hand side, the equation is only

ncluded in the model and calculated if the logical

ondition is true. This can make the program faster


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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$conditionlistofitems;

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 abovedisplay$(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;);
http://c/classsource/basic/report_writing.dochttp://c/classsource/basic/report_writing.doc


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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 > termbLess than LT or < Is terma < termb

Less or = LE or >= Is terma < termb


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Numerical $ condition forms

Relation GAMS operator ExplanationEquality 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 < termbLess 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 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)


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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 valuexx$yx = 3; equivilant to xx$(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 itemxx$sum((i,j)$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)


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Conditionals, Subsets and Tuples in GAMSSet element position

Logical conditions can be formed which depend on setsin 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 likeFIRSTBAL(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;);


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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.


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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 rueNested $ - statements can involve multiple $

conditions

Examplesu(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) ;


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Conditionals, Subsets and Tuples in GAMSNesting logical condition

Nested operators are acted on in a fixed precedence

order and are also operated on in an order relative tomathematical operators. The default precedence order

in the absence of parentheses is shown below in

decreasing order.

Operation Operator Precedence

Exponentiation ** 1Multiplication, Division * / 2

Addition, Subtraction + - 3

Numerical Relationships < ,=, >

lt , le ,eq,ne , ge, gt

Not not 5

And and 6

Or, Xor or, xor 7

When operators with the same precedence appear inan 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);


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The conditional alternativethe Subset

Mathematically one often sees algebraic

statements where the index positions to address aresubsets 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

1i

iZX

we can define a subset of I containing elements 1-20 viaset subseti(i) /i1*i20/;

then we can writeX=sum(I$subseti(i),z(i));

orX=sum(subseti,z(subseti));

orX=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 subsetSubseti(i)$(ord(i) le 20)=yes;

and then use as above


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The conditional alternativethe tuple

Often the conditionals required in a model are complex and

re used in a repetitive manner. It can be simpler tomploy a tuple (a set that is defined over other sets see the

ets chapter of the documentation) that encapsulates the

onditionals and only use it. In the model below the

ogical condition in red that appears repetitively can be

eplaced with the tuple making the model visually simpler

COSTEQ.. TCOST =E= SUM((PLANT,MARKET)

$(supply(plant) and demand(market)

and distance(plant,market))

, SHIPMENTS(PLANT,MARKET)* COST(PLANT,MARKET));

UPPLYEQ(PLANT).. SUM(MARKET

$(supply(plant) and demand(market) anddistance(plant,market)),SHIPMENTS(PLANT,

MARKET)) =L= SUPPLY(PLANT);

EMANDEQ(MARKET).. SUM(PLANT

$(supply(plant)and demand(market)

and Distance(plant,market))

,SHIPMENTS(PLANT, MARKET)) =G= DEMAND(MARKET);

where the alternative with the tuple iset thistuple(plant,market)tuple expressing conditional;histuple(plant,market)$(supply(plant)and demand(market)and

istance(plant,market)) =yes;

COSTEQ2..TCOST =E= SUM(thistuple(plant,market)

, SHIPMENTS(PLANT,MARKET)*COST(PLANT,MARKET));

UPPLYEQ(PLANT).. SUM(thistuple(plant,market)

,SHIPMENTS(PLANT, MARKET)) =L= SUPPLY(PLANT);

EMANDEQ(MARKET).. SUM(thistuple(plant,market)



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The conditional alternativethe tupleA tuple refers to a set defined over other sets. The set

may either be a one dimensional subset or amultidimensional 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)


The statementSUM(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.
http://c/WINDOWS/advanced/calculating.dochttp://c/WINDOWS/advanced/conditional.dochttp://c/WINDOWS/advanced/conditional.dochttp://c/WINDOWS/advanced/calculating.doc


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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 repeatedlyexecute a block of statements until a logical condition is

satisfied.

while(x=10) ;

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