Introduction to GAMS: Conditionals, Subsets and Tuples in GAMS Bruce A. McCarl & Levan Elbakidze Department of Agri

1. Introduction to GAMS: Conditionals, Subsets and Tuples in GAMS Bruce A. McCarl & Levan Elbakidze Department of Agricultural EconomicsTexas A&M University

2. 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 we would write or we might want to compute 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));

3. 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

4. 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 = 10only if the scalar y is greater than zero. Many different logical condition forms are possible as I review later in these notes.

5. Conditionals, Subsets and Tuples in GAMS Controlling whether an item is calculated $conditionals can appear in a model to restrict whether calculations are done. General format is namedparameter$logicalcondition=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.

6. 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$logicalcondition causing terms to be included only if the condition holds. Eq4.. xvar + yvar$(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 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.

7. 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 dependentlogical condition,term) where for example is written as 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.

8. 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

9. 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 $condition is true. display$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 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;);

10. 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

11. 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);

12. 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)

13. 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;);

14. 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”.

15. 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) ;

16. 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);

17. 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 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

18. 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);

19. 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.

20. 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) ;