Set addressing and references

GAMS employs a sparse matrix data storage scheme.  A parameter like X(A,B,C) is stored in the order

a1 b1 c1

a1 b1 c2

...

a1 b1 cm

a1 b2 c1

...

a1 b2 cm

...

a1 bn cm

a2 b1 c1

...

ak bn cm

Note the entries are stored internally in systematic order with the last entry varied the fastest then the second then the first.  GAMS withdraws entries from memory fastest if they are referenced in the order most consistent with the storage order.  As a consequence the calculation specification

Y(a,b,c)=X (a,b,c);

is faster than

Y(a,b,c)=X(b,c,a).

Referencing in a manner inconsistent with the storage order slows things down.  One should arrange set reference order so that across calculations, equation definitions etc, the sets are always referenced in the same order.  We then arrive at speed tip number one.  To increase speed modelers should endeavor to arrange definitions, calculations, sums, and equation references to sets in a consistent order.  Consider the example  (gamsslow.gms versus gamsfast.gms)

parameter x(e,d,c,b,a);

X(e,d,c,b,a)=10;

parameter z(a,b,c,d,e);

z(a,b,c,d,e)=x(e,d,c,b,a);

parameter y;

                 Y=sum ((a,b,c,d,e),z(a,b,c,d,e)*x(e,d,c,b,a));

variables obj

Positive variables var(e,b,a);

equations objeq

          R( b,c,d)

          q(a,b,c);

objeq.. obj=e=sum((a,b,c,d,e),z(a,b,c,d,e)*x(e,d,c,b,a)*var(e,b,a));

r(b,c,d).. sum((a,e),Var(e,b,a))=l=sum((a,e),x(e,d,c,b,a)*z(a,b,c,d,e));

q(a,b,c).. sum((d,e),var(e,b,a)/x(e,d,c,b,a)*z(a,b,c,d,e))=l=20;

model slow /all/;

*option lp=bdmlp;

slow.workspace=10;

solve slow maximizing obj using lp;

parameter sumofvar;

sumofvar=sum((a,b,c,d,e),z(a,b,c,d,e)*x(e,d,c,b,a)*var.l(e,b,a));

which yields the Profile lines where the coloring corresponds to source file lines.

----     13 ASSIGNMENT x             0.090     0.090 SECS    5.7 Mb  172800

----     15 ASSIGNMENT z             0.520     0.610 SECS    9.9 Mb  172800

----     17 ASSIGNMENT y             0.561     1.171 SECS    9.9 Mb

----     29 ASSIGNMENT slow          0.000     1.171 SECS    9.9 Mb

----     30 SOLVE INIT slow          0.000     1.171 SECS    9.9 Mb

----     24 EQUATION   objeq         1.142     2.313 SECS   10.5 Mb      1

----     25 EQUATION   R             0.861     3.174 SECS   17.8 Mb   1200

----     26 EQUATION   q             1.142     4.316 SECS   18.3 Mb   1440

----     30 SOLVE FINI slow          0.120     4.436 SECS   18.3 Mb

----      1           EXEC-INIT      0.000     0.000 SECS    9.2 Mb

solve appeared here

----     30 SOLVE READ slow          0.010     0.010 SECS    9.7 Mb

----     32 ASSIGNMENT sumofvar      0.701     0.711 SECS   10.5 Mb

A rearrangement of set indexes to a consistent order in gamsfast.gms yields the profile information

----     13 ASSIGNMENT x             0.050     0.050 SECS    3.7 Mb  100000

----     15 ASSIGNMENT z             0.090     0.140 SECS    6.3 Mb  100000

----     17 ASSIGNMENT y             0.110     0.250 SECS    6.3 Mb

----     29 ASSIGNMENT slow          0.000     0.250 SECS    6.3 Mb

----     30 SOLVE INIT slow          0.000     0.250 SECS    6.3 Mb

----     24 EQUATION   objeq         0.410     0.660 SECS    6.3 Mb      1

----     25 EQUATION   R             0.401     1.061 SECS   11.0 Mb   1000

----     26 EQUATION   q             0.411     1.472 SECS   11.0 Mb   1000

----     30 SOLVE FINI slow          0.070     1.542 SECS   11.0 Mb

----     30 GAMS FINI                0.030     1.572 SECS   11.0 Mb

----     30 SOLVE READ slow          0.010     0.010 SECS    6.1 Mb

----     32 ASSIGNMENT sumofvar      0.150     0.160 SECS    6.8 Mb

Where note the redefinition of the line numbered 15

z(a,b,c,d,e)=x(e,d,c,b,a);

into

z(a,b,c,d,e)=x(a,b,c,d,e);

drops execution time from 0.52 to 0.09 seconds.  Substantial percentage reductions were achieved in all the time consuming cases by the reordering and improved consistency of set addressing.