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% | ||
% This is the problem from the Choco v 2.1 example | ||
% sample.scheduling.Rehearsal, the one defined in main() . | ||
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num_pieces = 5; | ||
num_players = 3; | ||
duration = [4,6,3,5,7]; | ||
rehearsal = array2d(1..num_players, 1..num_pieces, | ||
[ | ||
1,1,0,1,0, | ||
0,1,1,0,1, | ||
1,1,0,1,1 | ||
]); |
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Type: MiniZinc data |
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% This is the problem from Barbara M. Smith's Rehearsal paper cited in the | ||
% model rehearsal.mzn | ||
num_pieces = 9; | ||
num_players = 5; | ||
duration = [2, 4, 1, 3, 3, 2, 5, 7, 6]; | ||
rehearsal = array2d(1..num_players, 1..num_pieces, | ||
[ | ||
1,1,0,1,0,1,1,0,1, | ||
1,1,0,1,1,1,0,1,0, | ||
1,1,0,0,0,0,1,1,0, | ||
1,0,0,0,1,1,0,0,1, | ||
0,0,1,0,1,1,1,1,0 | ||
]); | ||
|
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Type: MiniZinc data |
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% | ||
% Scheduling a Rehearsal in MiniZinc. | ||
% | ||
% From Barbara M. Smith: | ||
% "Constraint Programming in Practice: Scheduling a Rehearsal" | ||
% http://www.dcs.st-and.ac.uk/~apes/reports/apes-67-2003.pdf | ||
% """ | ||
% A concert is to consist of nine pieces of music of different durations | ||
% each involving a different combination of the five members of the orchestra. | ||
% Players can arrive at rehearsals immediately before the first piece in which | ||
% they are involved and depart immediately after the last piece in which | ||
% they are involved. The problem is to devise an order in which the pieces | ||
% can be rehearsed so as to minimize the total time that players are waiting | ||
% to play, i.e. the total time when players are present but not currently | ||
% playing. In the table below, 1 means that the player is required for | ||
% the corresponding piece, 0 otherwise. The duration (i.e. rehearsal time) | ||
% is in some unspecified time units. | ||
% | ||
% Piece 1 2 3 4 5 6 7 8 9 | ||
% Player 1 1 1 0 1 0 1 1 0 1 | ||
% Player 2 1 1 0 1 1 1 0 1 0 | ||
% Player 3 1 1 0 0 0 0 1 1 0 | ||
% Player 4 1 0 0 0 1 1 0 0 1 | ||
% Player 5 0 0 1 0 1 1 1 1 0 | ||
% Duration 2 4 1 3 3 2 5 7 6 | ||
% | ||
% For example, if the nine pieces were rehearsed in numerical order as | ||
% given above, then the total waiting time would be: | ||
% Player 1: 1+3+7=11 | ||
% Player 2: 1+5=6 | ||
% Player 3: 1+3+3+2=9 | ||
% Player 4: 4+1+3+5+7=20 | ||
% Player 5: 3 | ||
% giving a total of 49 units. The optimal sequence, as we shall see, | ||
% is much better than this. | ||
% | ||
% ... | ||
% | ||
% The minimum waiting time for the rehearsal problem is 17 time units, and | ||
% an optimal sequence is 3, 8, 2, 7, 1, 6, 5, 4, 9. | ||
% | ||
% """ | ||
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% | ||
% The data above is in | ||
% http://www.hakank.org/minizinc/rehearsal_smith.dzn | ||
% | ||
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% Here are all optimal sequences for Barbara M. Smith's problem | ||
% (total_waiting_time: 17) | ||
% | ||
% order: [9, 4, 6, 5, 1, 7, 2, 8, 3] | ||
% waiting_time: [3, 5, 0, 3, 6] | ||
% total_waiting_time: 17 | ||
% ---------- | ||
% order: [9, 4, 6, 5, 1, 2, 7, 8, 3] | ||
% waiting_time: [3, 5, 0, 3, 6] | ||
% total_waiting_time: 17 | ||
% ---------- | ||
% order: [9, 4, 5, 6, 1, 7, 2, 8, 3] | ||
% waiting_time: [3, 5, 0, 3, 6] | ||
% total_waiting_time: 17 | ||
% ---------- | ||
% order: [9, 4, 5, 6, 1, 2, 7, 8, 3] | ||
% waiting_time: [3, 5, 0, 3, 6] | ||
% total_waiting_time: 17 | ||
% ---------- | ||
% order: [3, 8, 7, 2, 1, 6, 5, 4, 9] | ||
% waiting_time: [3, 5, 0, 3, 6] | ||
% total_waiting_time: 17 | ||
% ---------- | ||
% order: [3, 8, 7, 2, 1, 5, 6, 4, 9] | ||
% waiting_time: [3, 5, 0, 3, 6] | ||
% total_waiting_time: 17 | ||
% ---------- | ||
% order: [3, 8, 2, 7, 1, 6, 5, 4, 9] | ||
% waiting_time: [3, 5, 0, 3, 6] | ||
% total_waiting_time: 17 | ||
% ---------- | ||
% order: [3, 8, 2, 7, 1, 5, 6, 4, 9] | ||
% waiting_time: [3, 5, 0, 3, 6] | ||
% total_waiting_time: 17 | ||
% ---------- | ||
% | ||
% Note that all waiting times are the same for | ||
% all sequences, i.e. player 1 always wait 3 units, etc. | ||
% | ||
% With symmetry breaking rule that order[1] < order[num_pieces] | ||
% there are 4 solutions where piece 2 and 7 can change place and | ||
% 5 and 6 can change place. | ||
% | ||
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% | ||
% This MiniZinc model was created by Hakan Kjellerstrand, hakank@gmail.com | ||
% See also my MiniZinc page: http://www.hakank.org/minizinc | ||
% | ||
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% Licenced under CC-BY-4.0 : http://creativecommons.org/licenses/by/4.0/ | ||
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include "globals.mzn"; | ||
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int: num_pieces; | ||
int: num_players; | ||
array[1..num_pieces] of int: duration; | ||
array[1..num_players, 1..num_pieces] of 0..1: rehearsal; | ||
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% | ||
% Decision variables | ||
% | ||
array[1..num_pieces] of var 1..num_pieces: rehearsal_order; | ||
array[1..num_players] of var 0..sum(duration): waiting_time; % waiting time for players | ||
array[1..num_players] of var 1..num_pieces: p_from; % first rehearsal | ||
array[1..num_players] of var 1..num_pieces: p_to; % last rehearsal | ||
var 0..sum(duration): total_waiting_time = sum(waiting_time); % objective | ||
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solve :: int_search( | ||
rehearsal_order % ++ waiting_time% ++ p_from ++ p_to ++ [total_waiting_time] | ||
, | ||
first_fail, % occurrence, % max_regret, % first_fail, | ||
indomain_max, % indomain_max, | ||
complete) | ||
minimize total_waiting_time; | ||
% satisfy; | ||
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% solve :: labelling_ff minimize total_waiting_time; | ||
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constraint | ||
all_different(rehearsal_order) :: domain | ||
/\ | ||
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% This solution is my own without glancing at Smith's models... | ||
forall(p in 1..num_players) ( | ||
% This versions is much faster than using exists (see below) | ||
% fix the range from..to, i.e. don't count all that start with 0 | ||
% or ends with 0. | ||
% This means that we collect the rehearsals with many 0 at the ends | ||
% | ||
p_from[p] < p_to[p] | ||
/\ | ||
% skipping rehearsal at start (don't come yet) | ||
forall(i in 1..num_pieces) ( | ||
i < p_from[p] -> (rehearsal[p, rehearsal_order[i]] = 0) | ||
) | ||
/\ | ||
% skipping rehearsal at end (go home after last rehearsal) | ||
forall(i in 1..num_pieces) ( | ||
i > p_to[p] -> (rehearsal[p, rehearsal_order[i]] = 0) | ||
) | ||
/\ % and now: count the waiting time for from..to | ||
waiting_time[p] = | ||
sum(i in 1..num_pieces) ( | ||
duration[rehearsal_order[i]] * bool2int( | ||
i >= p_from[p] /\ i <= p_to[p] | ||
/\ | ||
rehearsal[p,rehearsal_order[i]] = 0 | ||
) | ||
) | ||
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% % alternative solution with exists. | ||
% % More elegant (= declarative) in my book but slower. | ||
% exists(from, to in 1..num_pieces) ( | ||
% % skipping rehearsal at start (don't come yet) | ||
% forall(i in 1..from-1) ( | ||
% rehearsal[p, rehearsal_order[i]] = 0 | ||
% ) | ||
% /\ | ||
% % skipping rehearsal at end (go home after last rehearsal) | ||
% forall(i in to+1..num_pieces) ( | ||
% rehearsal[p, rehearsal_order[i]] = 0 | ||
% ) | ||
% /\ % and now: count the waiting time for from..to | ||
% waiting_time[p] = | ||
% sum(i in from..to) ( | ||
% duration[rehearsal_order[i]]* | ||
% bool2int( | ||
% rehearsal[p,rehearsal_order[i]] = 0 | ||
% ) | ||
% ) | ||
% ) | ||
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) | ||
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/\ % symmetry breaking | ||
rehearsal_order[1] < rehearsal_order[num_pieces] | ||
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% for all solutions | ||
% /\ total_waiting_time = 17 | ||
; | ||
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% | ||
% data | ||
% | ||
% | ||
% This is the problem from Barbara M. Smith's Rehearsal paper cited above: | ||
% (see rehearsal_smith.dta) | ||
% num_pieces = 9; | ||
% num_players = 5; | ||
% duration = [2, 4, 1, 3, 3, 2, 5, 7, 6]; | ||
% rehearsal = array2d(1..num_players, 1..num_pieces, | ||
% [ | ||
% 1,1,0,1,0,1,1,0,1, | ||
% 1,1,0,1,1,1,0,1,0, | ||
% 1,1,0,0,0,0,1,1,0, | ||
% 1,0,0,0,1,1,0,0,1, | ||
% 0,0,1,0,1,1,1,1,0 | ||
% ]); | ||
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% | ||
% This is the problem from the Choco v 2.1 example | ||
% sample.scheduling.Rehearsal, the one defined in main() . | ||
% (see rehearsal_choco.dta) | ||
% num_pieces = 5; | ||
% num_players = 3; | ||
% duration = [4,6,3,5,7]; | ||
% rehearsal = array2d(1..num_players, 1..num_pieces, | ||
% [ | ||
% 1,1,0,1,0, | ||
% 0,1,1,0,1, | ||
% 1,1,0,1,1 | ||
% ]); | ||
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output[ | ||
"order: " , show(rehearsal_order), "\n", | ||
"waiting_time: ", show(waiting_time), "\n", | ||
"total_waiting_time: " , show(total_waiting_time), "\n", | ||
] ++ | ||
[ | ||
if j = 1 then "\n" else " " endif ++ | ||
show(rehearsal[p, rehearsal_order[j]]) ++ " " | ||
| p in 1..num_players, j in 1..num_pieces, | ||
] ++ | ||
["\n"] | ||
; | ||
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Type: MiniZinc |