## Balanced Incomplete Block Design (clpe13)

```/***************************************************************/
/*                                                             */
/*          S A S   S A M P L E   L I B R A R Y                */
/*                                                             */
/*    NAME: clpe13                                             */
/*   TITLE: Balanced Incomplete Block Design (clpe13)          */
/* PRODUCT: OR                                                 */
/*  SYSTEM: ALL                                                */
/*    KEYS: OR                                                 */
/*   PROCS: CLP                                                */
/*    DATA:                                                    */
/*                                                             */
/* SUPPORT:                             UPDATE:                */
/*     REF:                                                    */
/*    MISC: Example 13 from the CLP Procedure chapter of the   */
/*          Constraint Programming book.                       */
/*                                                             */
/***************************************************************/

%macro bibd(v, b, r, k, lambda, out=bibdout);
/* Arrange v objects into b blocks such that:
(i) each object occurs in exactly r blocks,
(ii) each block contains exactly k objects,
(iii) every pair of objects occur together in exactly lambda blocks.

Equivalently, create a binary matrix with v rows and b columns,
with r 1s per row, k 1s per column,
and scalar product lambda between any pair of distinct rows.
*/

/* Check necessary conditions */
%if (%eval(&r * &v) ne %eval(&b * &k)) or
(%eval(&lambda * (&v - 1)) ne %eval(&r * (&k - 1))) or
(&v > &b) %then %do;
%put BIBD necessary conditions are not met.;
%goto EXIT;
%end;

proc clp out=&out(keep=x:) domain=[0,1] varselect=FIFO;
/* Decision variables: */
/* Decision variable X_i_c = 1 iff object i occurs in block c. */
var (
%do i=1 %to &v;
x&i._1-x&i._&b.
%end;
) = [0,1];

/* Mandatory constraints: */
/* (i) Each object occurs in exactly r blocks. */
%let q = %eval(&b.-&r.);  /* each row has &q 0s and &r 1s */
%do i=1 %to &v;
gcc( x&i._1-x&i._&b. ) = ((0,0,&q.) (1,0,&r.));
%end;

/* (ii) Each block contains exactly k objects. */
%let h = %eval(&v.-&k.);  /* each column has &h 0s and &k 1s */
%do c=1 %to &b;
gcc(
%do i=1 %to &v;
x&i._&c.
%end;
) = ((0,0,&h.) (1,0,&k.));
%end;

/* (iii) Every pair of objects occurs in exactly lambda blocks. */
%let t = %eval(&b.-&lambda.);
%do i=1 %to %eval(&v.-1);
%do j=%eval(&i.+1) %to &v;
/* auxiliary variable p_i_j_c =1 iff both i and j occur in c */
var ( p&i._&j._1-p&i._&j._&b. ) = [0,1];
%do c=1 %to &b;
reify p&i._&j._&c.: (x&i._&c. + x&j._&c. = 2);
%end;

gcc(p&i._&j._1-p&i._&j._&b.) = ((0,0,&t.) (1,0,&lambda.));
%end;
%end;

/* Symmetry breaking constraints: */
/* Break row symmetry via lexicographic ordering constraints. */
%do i = 2 %to &v.;
%let i1 = %eval(&i.-1);
lexico( (x&i._1-x&i._&b.) LEX_LT (x&i1._1-x&i1._&b.) );
%end;

/* Break column symmetry via lexicographic ordering constraints. */
%do c = 2 %to &b.;
%let c1 = %eval(&c.-1);
lexico( ( %do i = 1 %to &v.;
x&i._&c.
%end; )
LEX_LE
( %do i = 1 %to &v.;
x&i._&c1.
%end; ) );
%end;
run;
%put &_orclp_;
%EXIT:
%mend bibd;

%bibd(15,15,7,7,3);

%macro bibd_out(v, b, r, k, lambda, out=bibdout, transpose=0);
/* Create a binary matrix with v rows and b columns from the solution
of the bibd macro.  If transpose = 1 then the matrix will be transposed
for convenience of display.
*/
data bibdmat;
set &out;
array Block{&b.};
%do i = 1 %to &v.;
%do j = 1 %to &b.;
Block[&j.] = x&i._&j.;
%end;

output;
%end;

drop x:;
run;

%if &transpose %then %do;
/* Transposes the rows and columns of the binary matrix for
convenience of display.  */
proc transpose data=bibdmat
out=bibdmat2(rename=(_NAME_=Block))
prefix=Object;
run;

/* Print the solution */
proc print data=bibdmat2;
title "Balanced Incomplete Block Design Problem";
title2 "(&v, &b, &r, &k, &lambda)";
run;
%end;
%else %do;
/* Print the solution */
proc print data=bibdmat;
title "Balanced Incomplete Block Design Problem";
title2 "(&v, &b, &r, &k, &lambda)";
run;
%end;
%mend bibd_out;

%bibd_out(15,15,7,7,3);

```