Added comments for easier reading (erroneously added value symmetry breaking, which is actually found in the table_seating_imp.mzn file)

predicates/tableseating/table_seating.mzn

@@ -2,47 +2,60 @@ enum SCHOLAR;
 array[SCHOLAR] of int: reputation;
 int: maxreputation = max(reputation);
 
-int: T;
+int: T; % number of tables
 set of int: TABLE = 1..T;
-int: S; % tables size
+int: S; % common table size
 array[int,1..2] of SCHOLAR: enemies;
 array[int,1..2] of SCHOLAR: friends;
 
 array[TABLE] of var set of SCHOLAR: table;
 
-constraint forall(t in TABLE)(card(table[t]) != 1);
-
 predicate not_same_table(SCHOLAR: p1, SCHOLAR: p2) =
           forall(t in TABLE)(not ({p1,p2} subset table[t]));
 
+% No table occupied by a a scholar all by his lonesome
+% and no table occupied beyond its capacity
+constraint forall(t in TABLE)(card(table[t]) != 1);
 constraint forall(t in TABLE)(card(table[t]) <= S);
 
+% Every scholar is sitting on at least 1 table
 constraint forall(p in SCHOLAR)(exists(t in TABLE)(p in table[t]));
+
+% A scholar can only sit at at most 1 table ("two different
+% tables cannot seat the same scholar")
 constraint forall(t1, t2 in TABLE where t1 < t2)
    (table[t1] intersect table[t2] = {});
 
-% enemies not on same table
+% Enemies not on same table
 constraint forall(c in index_set_1of2(enemies))
                  (not_same_table(enemies[c,1],enemies[c,2]));
 
-% friends on same table
+% Friends on same table
 constraint forall(c in index_set_1of2(friends))
                  (not(not_same_table(friends[c,1],friends[c,2])));
 
-% important persons (reputation 10) not on same table
+% Important persons (reputation 10) not on same table
 constraint forall(p1,p2 in SCHOLAR where p1 < p2 /\ reputation[p1] = 10 /\ reputation[p2] = 10)
    (not_same_table(p1,p2));
 
-% Use as few tables as possible
+% Primary objective: use as few tables as possible
 var int: obj1;
 constraint obj1 = sum(t in TABLE)(card(table[t]) != 0);
 
+% Secondary objective: minimize the difference in reputation over all tables
 var int: obj2;
 constraint obj2 = sum(t in TABLE)
    (let {var int: minRep = min([reputation[p]*(p in table[t]) +  maxreputation*(1-(p in table[t])) | p in SCHOLAR]); 
          var int: maxRep = max([reputation[p]*(p in table[t]) | p in SCHOLAR]); } in
     if minRep = maxreputation then 0 else maxRep - minRep endif);
 
+% Hierarchy of objectives: 
+% obj1(sol1) < obj1(sol2) -> obj(sol1) < obj(sol2)
+% obj1(sol1) = obj1(sol2) -> (obj2(sol1) < obj2(sol2) -> obj(sol1) < obj(sol2))
+
+% var obj = (obj1*(obj2_range+1) + obj2);
+% int: obj2_range = S * (maxreputation - minreputation) = 5 * 9 = 45; % choose 100
+
 solve minimize (obj1*100 + obj2);
 
 output ["Table \(t): \(table[t])\n" | t in TABLE] ++ ["Obj1: \(obj1) and Obj2: \(obj2)\n"];

predicates/tableseating/table_seating_imp.mzn

@@ -6,49 +6,64 @@ int: maxreputation = max(reputation);
 
 int: T; % number of tables
 set of int: TABLE = 1..T;
-int: S; % tables size
+int: S; % common table size
 array[int,1..2] of SCHOLAR: enemies;
 array[int,1..2] of SCHOLAR: friends;
 
 array[TABLE] of var set of SCHOLAR: table;
 array[SCHOLAR] of var TABLE: seat;
 
-% No for lonely table
-constraint forall(t in TABLE)(card(table[t]) != 1);
-
-% value symmetry breaking
-constraint value_precede_chain([t | t in 1..T], seat);
-
 predicate not_same_table(SCHOLAR: p1, SCHOLAR: p2) =
                         seat[p1] != seat[p2];
 
+% The scholar->table "seat" mapping is dual to the table->{scholar} "table" subsetting
+constraint forall(t in TABLE, p in SCHOLAR)(p in table[t] <-> seat[p] = t);
+
+% No table occupied by a a scholar all by his lonesome
+constraint forall(t in TABLE)(card(table[t]) != 1);
+
+% Each table seats between 0 and 5 scholars
+% (p.292 of the handbook)
 constraint global_cardinality_low_up(seat, [t|t in TABLE], [0|t in TABLE], [S|t in TABLE]);
 
-% enemies not on same table
+% Enemies not on same table
 constraint forall(c in index_set_1of2(enemies))
                  (not_same_table(enemies[c,1],enemies[c,2]));
 
-% friends on same table
+% Friends on same table
 constraint forall(c in index_set_1of2(friends))
                  (not(not_same_table(friends[c,1],friends[c,2])));
 
-% important persons (reputation 10) not on same table
+% Important persons (reputation 10) not on same table
 constraint forall(p1,p2 in SCHOLAR where p1 < p2 /\ reputation[p1] = 10 /\ reputation[p2] = 10)
    (not_same_table(p1,p2));
 
-% Use as few tables as possible
+% Value symmetry breaking.
+% (p.298 of the handbook)
+% If seat[scholar_high] = t[i+1] then
+% there is at least one scholar_low < scholar_high such that
+% seat[scholar_low] = t[i]
+constraint value_precede_chain([t | t in 1..T], seat);
+
+% Primary objective: use as few tables as possible
 var int: obj1;
 constraint obj1 = sum(t in TABLE)(card(table[t]) != 0);
 
-% minimize the difference in reputation in each table
+% Secondary objective: minimize the difference in reputation over all tables
 var int: obj2;
 constraint obj2 = sum(t in TABLE)
    (let {var int: minRep = min([reputation[p]*(seat[p] = t) +  maxreputation*(seat[p] != t) | p in SCHOLAR]); 
          var int: maxRep = max([reputation[p]*(seat[p] = t) | p in SCHOLAR]); } in
     if minRep = maxreputation then 0 else maxRep - minRep endif);
 
-constraint forall(t in TABLE, p in SCHOLAR)(p in table[t] <-> seat[p] = t);    
+% Hierarchy of objectives: 
+% obj1(sol1) < obj1(sol2) -> obj(sol1) < obj(sol2)
+% obj1(sol1) = obj1(sol2) -> obj2(sol1) < obj2(sol2) -> obj(sol1) < obj(sol2)
+
+% var obj = (obj1*(obj2_range+1) + obj2);
+% int: obj2_range = S * (maxreputation - minreputation) = 5 * 9 = 45; % choose 100
 
 solve minimize (obj1*100 + obj2);
 
 output ["Table \(t): \(table[t])\n" | t in TABLE] ++ ["Obj1: \(obj1) and Obj2: \(obj2)\n"];
+