GAMS — The Solver Manuals - Available Software

124 CONOPT
of 4.1**2, which means that Jacobian values outside the range EXP(-4.1)=0.017 to EXP(+4.1)=60.4 are about
as common at values inside. This range is for most models acceptable, while a variance of 5, corresponding to
about half the derivatives outside the range EXP(-5)=0.0067 to EXP(+5)=148, can be dangerous.
6.5.1 Scaling of Intermediate Variables
Many models have a set of variables with a real economic or physical interpretation plus a set of intermediate
or helping variables that are used to simplify the model. We have seen some of these in section 6.3 on Simple
Expressions. It is usually rather easy to select good scaling units for the real variables since we know their order
of magnitude from economic or physical considerations. However, the intermediate variables and their defining
equations should preferably also be well scaled, even if they do not have an immediate interpretation. Consider
the following model fragment where X, Y, and Z are variables and Y is the intermediate variable:
SET P / P0*P4 /
PARAMETER A(P) / P0 211, P1 103, P2 42, P3 31, P4 6 /
YDEF .. Y =E= SUM(P, A(P)*POWER(X,ORD(P)-1));
ZDEF .. Z =E= LOG(Y);
X lies in the interval 1 to 10 which means that Y will be between 211 and 96441 and Z will be between 5.35 and
11.47. Both X and Z are reasonably scaled while Y and the terms and derivatives in YDEF are about a factor 1.e4
too large. Scaling Y by 1.e4 and renaming it YS gives the following scaled version of the model fragment:
YDEFS1 .. YS =E= SUM(P, A(P)*POWER(X,ORD(P)-1))*1.E-4;
ZDEFS1 .. Z =E= LOG(YS*1.E4);
The Z equation can also be written as
ZDEFS2 .. Z
=E= LOG(YS) + LOG(1.E4);
Note that the scale factor 1.e-4 in the YDEFS1 equation has been placed on the right hand side. The mathematically
equivalent equation
YDEFS2 .. YS*1.E4 =E= SUM(P, A(P)*POWER(X,ORD(P)-1));
will give a well scaled YS, but the right hand side terms of the equation and their derivatives have not changed
from the original equation YDEF and they are still far too large.
6.5.2 Using the Scale Option in GAMS
The rules for good scaling mentioned above are exclusively based on algorithmic needs. GAMS has been developed
to improve the effectiveness of modelers, and one of the best ways seems to be to encourage modelers to write
their models using a notation that is as ”natural” as possible. The units of measurement is one part of this
natural notation, and there is unfortunately often a conflict between what the modeler thinks is a good unit and
what constitutes a well scaled model.
To facilitate the translation between a natural model and a well scaled model GAMS has introduced the concept
of a scale factor, both for variables and equations. The notation and the definitions are quite simple. First
of all, scaling is by default turned off. To turn it on, enter the statement ”.SCALEOPT = 1;” in your
GAMS program somewhere after the MODEL statement and before the SOLVE statement. ”” is the name
of the model to be solved. If you want to turn scaling off again, enter the statement ”.SCALEOPT = 0;”
somewhere before the next SOLVE.
The scale factor of a variable or an equation is referenced with the suffix ”.SCALE”, i.e. the scale factor of
variable X(I) is referenced as X.SCALE(I). Note that there is one scale value for each individual component of a
multidimensional variable or equation. Scale factors can be defined in assignment statements with X.SCALE(I)