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Move /should/ to the optimizing part, and fill that part out
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committees.org
229
committees.org
@ -254,46 +254,29 @@ cry for help.
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return [0 <= comms[c] for c in comms]
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#+END_SRC
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** Committee constraints
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The $S$ relation is sort of odd. That one examiner /should/ form a committee
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with someone they relate to by $S$. Let's interpret it as $e_1$ should not
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form a committee to someone they are not related to by $S$.
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#+BEGIN_SRC python :tangle committees.py
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def should_correct_with_constraint(comms, S, C):
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examiners = set(range(len(C)))
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return [comms[frozenset([i, j])] == 0
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for i in S
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for j in examiners - S[i]
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if j != i]
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#+END_SRC
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The $A$ relation is similar, and easier.
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with someone they relate to by $S$. This is not an absolute requirement,
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which is not ideal for a satisfiability problem, so we will ignore this
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constraint for now. The $A$ relation is similar, but clearer. For any pair
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$(i,j) \in A$, we don't form a committee consisting of those examiners.
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#+BEGIN_SRC python :tangle committees.py
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def avoid_correct_with_constraint(comms, A):
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return [comms[frozenset([i, j])] == 0 for i, j in A]
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#+END_SRC
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Each committee will correct their exams to times, so if the sum of all the
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committees is $N$, then all exams have been corrected twice. Let's encode
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that as a constraint.
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** All exams are corrected constraint
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Each committee correct their exams two times (once by each examiner), so if
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the sum of all the committees is $N$, then all exams have been corrected
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twice (presumably by two different examiners). Let's encode that as a
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constraint.
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#+BEGIN_SRC python :tangle committees.py
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def all_corrected_constraint(comms, N):
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return [sum(comms.values()) == N]
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#+END_SRC
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Let's collect all the constraints in a single list.
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#+BEGIN_SRC python :tangle committees.py
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def constraints(instance):
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N, C, S, A = instance
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comms = committees(C)
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return (capacity_constraint(comms, C) +
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non_negative_constraint(comms) +
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all_corrected_constraint(comms, N) +
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should_correct_with_constraint(comms, S, C) +
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avoid_correct_with_constraint(comms, A))
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return sum(comms.values()) == N
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#+END_SRC
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** Invoking Z3
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@ -302,9 +285,15 @@ cry for help.
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#+BEGIN_SRC python :tangle committees.py
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def check_instance(instance):
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N, C, S, A = instance
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comms = committees(C)
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s = Solver()
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s.add(constraints(instance))
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s.add(capacity_constraint(comms, C))
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s.add(non_negative_constraint(comms))
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s.add(all_corrected_constraint(comms, N))
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s.add(avoid_correct_with_constraint(comms, A))
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s.check()
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return s.model()
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@ -333,12 +322,17 @@ cry for help.
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This is not especially readable, so let's write a quick prettyprinter.
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#+BEGIN_SRC python :tangle committees.py
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def prettyprint(m):
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for k in m:
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cname = k.name()[10:-1]
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cval = m[k].as_long()
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if cval > 0:
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print(cname + ':', cval)
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def prettyprint(instance, m):
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N, C, S, A = instance
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comms = committees(C)
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exams = [sum(m[comms[c]].as_long() for c in comms if i in c)
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for i in range(len(C))]
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examiners = '\n'.join(['%d: %d/%d' % (i, exams[i], C[i])
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for i in range(len(C))])
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cs = [(c, m[comms[c]].as_long()) for c in sorted(comms, key=sorted)]
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csstr = '\n'.join([', '.join(map(str, sorted(c))) + ': ' + str(cv)
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for c, cv in cs if cv > 0])
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print(examiners + '\n\n' + csstr)
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#+END_SRC
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This outputs the something like:
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@ -353,52 +347,145 @@ cry for help.
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#+END_EXAMPLE
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Note the /something like/. There are multiple ways to satisfy this set of
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constraints, and Z3 only guarantees to provide /some/ solution (if one
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exists).
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constraints, and Z3 only provide /some/ solution (if one exists).
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* Optimization
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So far, we have found a way to model the problem and satisfy all the
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constraint. However, it is preferable to have fewer committees, because all
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committees have to discuss the exams, causing administrative overhead. Z3
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also provides optimization, meaning that we can find a smallest or largest
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solution for numeric theories.
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So far, we have found a way to model the problem and satisfy the constraints.
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However, it is preferable to have fewer committees, because all committees
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have to discuss the exams, causing administrative overhead. Z3 also provides
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optimization, meaning that we can find a smallest or largest solution for
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numeric theories. The underlying theory for optimization is MaxSMT.
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** Minimize committees
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In our case, we want to minimize the number of committees.
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In our case, we want to minimize the number of committees. First we write a
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function to find the number of committees which we will soon minimize.
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#+BEGIN_SRC python :tangle committees.py
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def minimize_committees(comms):
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return sum(If(0 < comms[c], 1, 0) for c in comms)
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#+END_SRC
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#+BEGIN_SRC python :tangle committees.py
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def number_of_committees(comms):
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return sum(If(0 < comms[c], 1, 0) for c in comms)
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#+END_SRC
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Now we can invoke Z3, using an ~Optimize~ instance and adding our
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minimization constraint.
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Now we can invoke Z3, using an ~Optimize~ instance and adding our
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minimization constraint.
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#+BEGIN_SRC python :tangle committees.py
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def optimize_instance(instance):
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o = Optimize()
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#+BEGIN_SRC python :tangle committees.py
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def optimize_instance(instance):
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N, C, S, A = instance
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comms = committees(C)
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o.add(constraints(instance))
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o.minimize(minimize_committees(committees(instance)))
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o = Optimize()
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o.check()
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return o.model()
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#+END_SRC
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o.add(capacity_constraint(comms, C))
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o.add(non_negative_constraint(comms))
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o.add(all_corrected_constraint(comms, N))
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o.add(avoid_correct_with_constraint(comms, A))
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There is still more than one way to satisfy this model, but we are
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guaranteed to get a minimal number of committees (which is 6 in our
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example).
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o.minimize(number_of_committees(comms))
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#+BEGIN_EXAMPLE
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{2, 3}: 56
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{0, 1}: 137
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{2, 5}: 7
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{0, 5}: 23
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{2, 4}: 47
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{1, 4}: 13
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#+END_EXAMPLE
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o.check()
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return o.model()
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#+END_SRC
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There is still more than one way to satisfy this model, but we are
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guaranteed to get a minimal number of committees (which is 6 in our
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example).
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#+BEGIN_EXAMPLE
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{2, 3}: 56
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{0, 1}: 137
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{2, 5}: 7
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{0, 5}: 23
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{2, 4}: 47
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{1, 4}: 13
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#+END_EXAMPLE
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** Dealing with /should/
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Remember $S$, which maps examiners to other examiners they /should/ form a
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committee with. With optimization, we now have a way of expressing that some
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solution is more preferable than another. One way to model this is
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maximizing the number of exams given to committees that consists of an
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examiner $i$ that should be in a committee with examiner $j$. We want this
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for all such pairs $i,j$, and can achieve this by summing all such
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committees.
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#+BEGIN_SRC python :tangle committees.py
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def should_correct_with_weight(comms, S, C):
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return sum(comms[frozenset([i, j])] for i in S for j in S[i])
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#+END_SRC
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When adding multiple optimization objectives (or goals), Z3 defaults to
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order the objectives lexicographically, i.e. in the order they appear. If we
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place the minimization of committees before the
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~should_correct_with_weight~, then we still are guaranteed to get a minimal
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number of committees.
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#+BEGIN_SRC python :tangle committees.py
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def optimize_instance(instance):
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N, C, S, A = instance
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comms = committees(C)
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o = Optimize()
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o.add(capacity_constraint(comms, C))
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o.add(non_negative_constraint(comms))
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o.add(all_corrected_constraint(comms, N))
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o.add(avoid_correct_with_constraint(comms, A))
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o.minimize(number_of_committees(comms))
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o.maximize(should_correct_with_weight(comms, S, C))
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o.check()
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return o.model()
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#+END_SRC
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#+BEGIN_EXAMPLE
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{0, 2}: 50
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{1, 3}: 60
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{1, 5}: 7
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{2, 4}: 60
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{0, 5}: 23
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{0, 1}: 83
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#+END_EXAMPLE
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** Optimize for capacities
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Maybe we can try to satisfy (🙃) all the examiners by trying to close the
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gap between their capacity and the number of exams they end up correcting.
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Usually at the Department, there is quite a lot of flex in these capacities;
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if you are willing to correct $50$ exams, then you will most likely be okey
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with correcting $40$ and /actually/ willing to correct $52$. Therefore, we
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can try to add some slack to the capacity.
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#+BEGIN_SRC python :tangle committees.py
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def capacity_slack(comms, i, C):
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a = sum(comms[c] for c in comms if i in c)
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return If(a > C[i], a - C[i], C[i] - a)
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def capacity_weight(comms, C):
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return sum(capacity_slack(comms, i, C) for i in range(len(C)))
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#+END_SRC
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#+BEGIN_SRC python :tangle committees.py
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def optimize_instance(instance):
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N, C, S, A = instance
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comms = committees(C)
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o = Optimize()
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o.add(non_negative_constraint(comms))
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o.add(all_corrected_constraint(comms, N))
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o.add(avoid_correct_with_constraint(comms, A))
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o.minimize(capacity_weight(comms, C))
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o.minimize(number_of_committees(comms))
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o.maximize(should_correct_with_weight(comms, S, C))
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o.check()
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return o.model()
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#+END_SRC
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** COMMENT Implementation draft
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@ -412,7 +499,7 @@ cry for help.
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s.add(capacity_constraint(comms, C))
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s.add(non_negative_constraint(comms))
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s.add(all_corrected_constraint(comms, N))
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s.add(should_correct_with_constraint(comms, S, C))
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s.add(should_correct_with_weight(comms, S, C))
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s.add(avoid_correct_with_constraint(comms, A))
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s.check()
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