002: Template Design

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% Template design
% Problem 002 in CSPLib
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% Based on "ILP and Constraint Programming Approaches to a Template
% Design Problem", Les Proll and Barbara Smith, School of Computing
% Research Report 97.16, University of Leeds, May 1997.
%-----------------------------------------------------------------------------%

include "globals.mzn";

int: S;   		% Number of slots per template.
int: t;   		% Number of templates.
int: n;   		% Number of variations.
array[1..n] of int: d; 	% How much of each variation we must print?

% Lower and upper bounds for the total production.
%
int: llower = ceil(sum(i in 1..n)(int2float(d[i]))/int2float(S));
int: lupper = 2*llower; % If t>1, this should be the optimal Production_{t-1}-1.

% # Slots allocated to variation i in template j
array[1..n,1..t] of var 0..S: p;

% # Pressings of template j.
array[1..t] of var 1..lupper: R;

% Sum of all Rj.
var llower..lupper: Production;

% Production x S - sum(d[i])
var 0..lupper-llower: Surplus;

% First, set up Production to be the sum of the Rj
constraint
	Production = sum(i in 1..t)(R[i]);

% the limits on production
constraint
	Production >= llower /\ Production <= lupper;

% The number of slots occupied in each template is S.
constraint
	forall(j in 1..t)
		 (sum(i in 1..n)(p[i,j]) = S);

% Enough of each variation is printed.
constraint
	forall(i in 1..n)
		 (sum(j in 1..t)(p[i,j]*R[j]) >= d[i]);

% Symmetry constraints.
% Variations with the same demand are symmetric.
constraint
	forall(i in 1..n-1) (
		if d[i] == d[i+1] then
			lex_lesseq([p[i,  j] | j in 1..t],
				[p[i+1,j] | j in 1..t])
		else
			true
		endif
	);

% pseudo symmetry
constraint
	forall(i in 1..n-1) (
		if d[i] < d[i+1] then
		       sum (j in 1..t) (p[i,j]*R[j])
		     < sum (j in 1..t) (p[i+1,j]*R[j])
		else
			true
		endif
	);

% implied constraints on the surplus

% These are presented in the paper as necessary to get good
% performance for this model, but I think bounds consistency on the
% sum(R[i]) constraint would produce the same amount of propagation

% Set up surplus, which is bounded as production is bounded.
constraint
	Surplus = Production*S - sum(i in 1..n)(d[i]);

% The surplus of each variation is also limited by the surplus.
constraint
	forall(k in 1..n)
		 (sum(j in 1..t)(p[k,j]*R[j]-d[k]) <= Surplus);

% The surplus of the first k variations is limited by the surplus.
constraint
	forall(k in 2..n-1)
		 (sum(j in 1..t, m in 1..k)( p[m,j]*R[j]-d[m] ) <= Surplus);

% Implied constraints on the run length.
constraint
	if t=2 then (
		R[1] <= Production div 2
	/\	R[2] >= Production div 2
	) else true endif;

constraint
	if t=3 then (
		R[1] <= Production div 3
	/\	R[2] <= Production div 2
	/\	R[3] >= Production div 3
	) else true endif;

% Minimize the production.
solve :: int_search(array1d(1..n*t,p) ++ R, input_order, indomain_min, complete)
    minimize Production;

output [
    if v = 1 then "template #" ++ show(i) ++ ": [" else "" endif ++
    show(p[v, i]) ++
    if v = n then "], pressings: " ++ show(R[i]) ++ "\n" else ", " endif
	| i in 1..t, v in 1..n]
    ++ ["Total pressings: ", show(Production), "\n%\n"];

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