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meatballs merged 1 commit into from
Aug 20, 2017
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Implement equity objective function #103

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56 changes: 47 additions & 9 deletions docs/background/mathematical_model.rst
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Original file line number Diff line number Diff line change
Expand Up @@ -130,8 +130,8 @@ However, it might be desirable to also optimise a given objective function.
Objective functions
+++++++++++++++++++

Optimising to avoid room overflow
---------------------------------
Efficiency: Optimising to avoid total room overflow
---------------------------------------------------

Demand for events might be known: this will be captured using a vector
:math:`d\in\mathbb{R}_{\geq 0}^{M}`. Similarly capacity for rooms might be
Expand All @@ -141,19 +141,57 @@ with parallel sessions delegates might not go where they originally intended) we
can aim to minimise the expected overflow given by the following expression:

.. math::
:label: overflow_objective_function
:label: total_overflow_objective_function

\sum_{i=1}^{M}\sum_{j=1}^{N}X_{ij}(c_j - d_i)
\sum_{i=1}^{M}\sum_{j=1}^{N}X_{ij}(d_i - c_j)

Using this, our optimisation problem to give a desirable schedule is obtained by
solving the following problem:

Minimise :eq:`overflow_objective_function` subject to :eq:`all_events_scheduled_constraint`,
:eq:`all_slots_at_most_1_event_constraint`,
:eq:`slot_constraint` and :eq:`event_constraint`.
Minimise :eq:`total_overflow_objective_function` subject to
:eq:`all_events_scheduled_constraint`,
:eq:`all_slots_at_most_1_event_constraint`, :eq:`slot_constraint` and
:eq:`event_constraint`.

Equity: Optimising to avoid worse room overflow
-----------------------------------------------

Minimising :eq:`total_overflow_objective_function` might still leave a given
slot with a very large overflow relative to all over slots.
We
can aim to minimise the maximum overflow in a given slot given by the following
expression:

.. math::

\max_{i,j}X_{ij}(d_i - c_j)

Note that it is not possible to use :math:`\max` in the objective function for a
linear program (it is none linear). However, instead we can define another
variable: :math:`\beta` as the upper bound for the overflow in each slot:

.. math::
:label: overflow_constraints

X_{ij}(d_i - c_j) \leq \beta \text{ for all }0\leq i\leq N\text{ for all }1\leq j\leq M

The objective function then becomes to minimize:

.. math::
:label: worse_overflow_objective_function

\beta

Using this, our optimisation problem to give a desirable schedule is obtained by
solving the following problem:

Minimise :eq:`worse_overflow_objective_function` subject to
:eq:`all_events_scheduled_constraint`,
:eq:`all_slots_at_most_1_event_constraint`, :eq:`slot_constraint`,
:eq:`event_constraint` and :eq:`overflow_constraints`.

Minimise change from a previous schedule
----------------------------------------
Consistency: Minimise change from a previous schedule
-----------------------------------------------------

Once a schedule has been obtained and publicised to all delegates, a new
constraint might arise (modifying :eq:`all_events_scheduled_constraint`,
Expand Down
163 changes: 96 additions & 67 deletions docs/tutorial/index.rst
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Expand Up @@ -65,7 +65,7 @@ duration/location of the slots we know some of them are unavailable for a given

>>> events = [Event(name='Talk 1', duration=30, tags=['beginner'], unavailability=outside_slots[:], demand=50),
... Event(name='Talk 2', duration=30, tags=['beginner'], unavailability=outside_slots[:], demand=130),
... Event(name='Talk 3', duration=30, tags=['beginner'], unavailability=outside_slots[:], demand=3000),
... Event(name='Talk 3', duration=30, tags=['beginner'], unavailability=outside_slots[:], demand=200),
... Event(name='Talk 4', duration=30, tags=['beginner'], unavailability=outside_slots[:], demand=30),
... Event(name='Talk 5', duration=30, tags=['intermediate'], unavailability=outside_slots[:], demand=60),
... Event(name='Talk 6', duration=30, tags=['intermediate'], unavailability=outside_slots[:], demand=30),
Expand All @@ -75,7 +75,7 @@ duration/location of the slots we know some of them are unavailable for a given
... Event(name='Talk 10', duration=30, tags=['advanced'], unavailability=outside_slots[:], demand=30),
... Event(name='Talk 11', duration=30, tags=['advanced'], unavailability=outside_slots[:], demand=30),
... Event(name='Talk 12', duration=30, tags=['advanced'], unavailability=outside_slots[:], demand=30),
... Event(name='Workshop 1', duration=60, tags=['testing'], unavailability=outside_slots[:], demand=300),
... Event(name='Workshop 1', duration=60, tags=['testing'], unavailability=outside_slots[:], demand=40),
... Event(name='Workshop 2', duration=60, tags=['testing'], unavailability=outside_slots[:], demand=40),
... Event(name='City tour', duration=90, tags=[], unavailability=talk_slots[:] + workshop_slots[:], demand=100),
... Event(name='Boardgames', duration=90, tags=[], unavailability=talk_slots[:] + workshop_slots[:], demand=20)]
Expand Down Expand Up @@ -137,49 +137,79 @@ Avoiding room overcrowding

The data we input in to the model included information about demand for a talk;
this could be approximated from previous popularity for a talk. However, the
scheduler has put :code:`Talk 2` and :code:`Talk 3` (which have high demand) in
scheduler has put :code:`Talk 3` (which have high demand) in
the small room (which has capacity 50). We can include an objective function in
our scheduler to minimise the difference between room capacity and demand::

>>> from conference_scheduler.lp_problem import objective_functions
>>> func = objective_functions.capacity_demand_difference
>>> func = objective_functions.efficiency_capacity_demand_difference
>>> schedule = scheduler.schedule(events, slots, objective_function=func)

>>> schedule.sort(key=lambda item: item.slot.starts_at)
>>> for item in schedule:
... print(f"{item.event.name} at {item.slot.starts_at} in {item.slot.venue}")
Talk 1 at 15-Sep-2016 09:30 in Big
Talk 12 at 15-Sep-2016 09:30 in Small
Talk 2 at 15-Sep-2016 10:00 in Big
Talk 6 at 15-Sep-2016 10:00 in Small
Talk 4 at 15-Sep-2016 12:30 in Small
Talk 11 at 15-Sep-2016 12:30 in Big
Talk 5 at 15-Sep-2016 13:00 in Small
Talk 10 at 15-Sep-2016 13:00 in Big
Talk 7 at 16-Sep-2016 09:30 in Big
Talk 4 at 15-Sep-2016 09:30 in Big
Talk 5 at 15-Sep-2016 09:30 in Small
Talk 3 at 15-Sep-2016 10:00 in Big
Talk 9 at 15-Sep-2016 10:00 in Small
Talk 6 at 15-Sep-2016 12:30 in Big
Talk 11 at 15-Sep-2016 12:30 in Small
Talk 2 at 15-Sep-2016 13:00 in Small
Talk 7 at 15-Sep-2016 13:00 in Big
Talk 8 at 16-Sep-2016 09:30 in Big
Workshop 2 at 16-Sep-2016 09:30 in Small
Talk 8 at 16-Sep-2016 10:00 in Big
Talk 3 at 16-Sep-2016 12:30 in Big
Talk 12 at 16-Sep-2016 10:00 in Big
Talk 1 at 16-Sep-2016 12:30 in Big
Boardgames at 16-Sep-2016 12:30 in Outside
Talk 9 at 16-Sep-2016 13:00 in Big
Talk 10 at 16-Sep-2016 13:00 in Big
Workshop 1 at 16-Sep-2016 13:00 in Small
City tour at 16-Sep-2016 13:00 in Outside

We see that :code:`Talk 3` has moved to the bigger room but that all other
constraints still hold. Note however that this has also moved :code:`Talk 2`
(which has relatively high demand) to a small room. This is because we have
minimised the overall overcrowding. This can have the negative effect of leaving
one slot with a high overcrowding for the benefit of overall efficiency. We can
however include a different objective function to minimise the maximum
overcrowding in any given slot::

We see that those talks have moved to the bigger room but that all other
constraints still hold.
>>> from conference_scheduler.lp_problem import objective_functions
>>> func = objective_functions.equity_capacity_demand_difference
>>> schedule = scheduler.schedule(events, slots, objective_function=func)

>>> schedule.sort(key=lambda item: item.slot.starts_at)
>>> for item in schedule:
... print(f"{item.event.name} at {item.slot.starts_at} in {item.slot.venue}")
Talk 1 at 15-Sep-2016 09:30 in Small
Talk 9 at 15-Sep-2016 09:30 in Big
Talk 3 at 15-Sep-2016 10:00 in Big
Talk 10 at 15-Sep-2016 10:00 in Small
Talk 4 at 15-Sep-2016 12:30 in Small
Talk 7 at 15-Sep-2016 12:30 in Big
Talk 2 at 15-Sep-2016 13:00 in Big
Talk 8 at 15-Sep-2016 13:00 in Small
Talk 6 at 16-Sep-2016 09:30 in Big
Workshop 2 at 16-Sep-2016 09:30 in Small
Talk 12 at 16-Sep-2016 10:00 in Big
Talk 11 at 16-Sep-2016 12:30 in Big
Boardgames at 16-Sep-2016 12:30 in Outside
Talk 5 at 16-Sep-2016 13:00 in Big
Workshop 1 at 16-Sep-2016 13:00 in Small
City tour at 16-Sep-2016 13:00 in Outside

Now, both :code:`Talk 2` and :code:`Talk 3` are in the bigger rooms.

Coping with new information
---------------------------

This is fantastic! Our schedule has now been published and everyone is excited
about the conference. However, as can often happen, one of the speakers now
informs us of a particular new constraints. For example, the speaker for
:code:`Talk 11` is unable to speak on the first day.
:code:`Talk 11` is unable to speak on the second day.

We can enter this new constraint::

>>> events[10].add_unavailability(*slots[:9])
>>> events[10].add_unavailability(*slots[9:])

We can now solve the problem one more time from scratch just as before::

Expand All @@ -188,20 +218,20 @@ We can now solve the problem one more time from scratch just as before::
>>> alt_schedule.sort(key=lambda item: item.slot.starts_at)
>>> for item in alt_schedule:
... print(f"{item.event.name} at {item.slot.starts_at} in {item.slot.venue}")
Talk 1 at 15-Sep-2016 09:30 in Small
Talk 9 at 15-Sep-2016 09:30 in Big
Talk 5 at 15-Sep-2016 10:00 in Big
Talk 3 at 15-Sep-2016 09:30 in Big
Talk 12 at 15-Sep-2016 09:30 in Small
Talk 2 at 15-Sep-2016 10:00 in Big
Talk 10 at 15-Sep-2016 10:00 in Small
Talk 4 at 15-Sep-2016 12:30 in Big
Talk 7 at 15-Sep-2016 12:30 in Small
Talk 6 at 15-Sep-2016 13:00 in Big
Talk 12 at 15-Sep-2016 13:00 in Small
Talk 2 at 16-Sep-2016 09:30 in Big
Talk 1 at 15-Sep-2016 12:30 in Big
Talk 8 at 15-Sep-2016 12:30 in Small
Talk 5 at 15-Sep-2016 13:00 in Big
Talk 9 at 15-Sep-2016 13:00 in Small
Talk 11 at 16-Sep-2016 09:30 in Big
Workshop 2 at 16-Sep-2016 09:30 in Small
Talk 8 at 16-Sep-2016 10:00 in Big
Talk 3 at 16-Sep-2016 12:30 in Big
Talk 4 at 16-Sep-2016 10:00 in Big
Talk 7 at 16-Sep-2016 12:30 in Big
Boardgames at 16-Sep-2016 12:30 in Outside
Talk 11 at 16-Sep-2016 13:00 in Big
Talk 6 at 16-Sep-2016 13:00 in Big
Workshop 1 at 16-Sep-2016 13:00 in Small
City tour at 16-Sep-2016 13:00 in Outside

Expand All @@ -210,27 +240,26 @@ completely different schedule with a number of changes. We can however solve the
problem with a new objective function which is to minimise the changes from the
old schedule::


>>> func = objective_functions.number_of_changes
>>> similar_schedule = scheduler.schedule(events, slots, objective_function=func, original_schedule=schedule)

>>> similar_schedule.sort(key=lambda item: item.slot.starts_at)
>>> for item in similar_schedule:
... print(f"{item.event.name} at {item.slot.starts_at} in {item.slot.venue}")
Talk 1 at 15-Sep-2016 09:30 in Big
Talk 12 at 15-Sep-2016 09:30 in Small
Talk 2 at 15-Sep-2016 10:00 in Big
Talk 6 at 15-Sep-2016 10:00 in Small
Talk 1 at 15-Sep-2016 09:30 in Small
Talk 9 at 15-Sep-2016 09:30 in Big
Talk 3 at 15-Sep-2016 10:00 in Big
Talk 10 at 15-Sep-2016 10:00 in Small
Talk 4 at 15-Sep-2016 12:30 in Small
Talk 9 at 15-Sep-2016 12:30 in Big
Talk 5 at 15-Sep-2016 13:00 in Small
Talk 10 at 15-Sep-2016 13:00 in Big
Talk 7 at 16-Sep-2016 09:30 in Big
Talk 7 at 15-Sep-2016 12:30 in Big
Talk 2 at 15-Sep-2016 13:00 in Big
Talk 8 at 15-Sep-2016 13:00 in Small
Talk 11 at 16-Sep-2016 09:30 in Big
Workshop 2 at 16-Sep-2016 09:30 in Small
Talk 8 at 16-Sep-2016 10:00 in Big
Talk 3 at 16-Sep-2016 12:30 in Big
Talk 12 at 16-Sep-2016 10:00 in Big
Talk 6 at 16-Sep-2016 12:30 in Big
Boardgames at 16-Sep-2016 12:30 in Outside
Talk 11 at 16-Sep-2016 13:00 in Big
Talk 5 at 16-Sep-2016 13:00 in Big
Workshop 1 at 16-Sep-2016 13:00 in Small
City tour at 16-Sep-2016 13:00 in Outside

Expand All @@ -246,34 +275,35 @@ with the original. Firstly, we can see which events moved to different slots::
>>> event_diff = scheduler.event_schedule_difference(schedule, alt_schedule)
>>> for item in event_diff:
... print(f"{item.event.name} has moved from {item.old_slot.venue} at {item.old_slot.starts_at} to {item.new_slot.venue} at {item.new_slot.starts_at}")
Talk 1 has moved from Big at 15-Sep-2016 09:30 to Small at 15-Sep-2016 09:30
Talk 10 has moved from Big at 15-Sep-2016 13:00 to Small at 15-Sep-2016 10:00
Talk 11 has moved from Big at 15-Sep-2016 12:30 to Big at 16-Sep-2016 13:00
Talk 12 has moved from Small at 15-Sep-2016 09:30 to Small at 15-Sep-2016 13:00
Talk 2 has moved from Big at 15-Sep-2016 10:00 to Big at 16-Sep-2016 09:30
Talk 4 has moved from Small at 15-Sep-2016 12:30 to Big at 15-Sep-2016 12:30
Talk 5 has moved from Small at 15-Sep-2016 13:00 to Big at 15-Sep-2016 10:00
Talk 6 has moved from Small at 15-Sep-2016 10:00 to Big at 15-Sep-2016 13:00
Talk 7 has moved from Big at 16-Sep-2016 09:30 to Small at 15-Sep-2016 12:30
Talk 9 has moved from Big at 16-Sep-2016 13:00 to Big at 15-Sep-2016 09:30
Talk 1 has moved from Small at 15-Sep-2016 09:30 to Big at 15-Sep-2016 12:30
Talk 11 has moved from Big at 16-Sep-2016 12:30 to Big at 16-Sep-2016 09:30
Talk 12 has moved from Big at 16-Sep-2016 10:00 to Small at 15-Sep-2016 09:30
Talk 2 has moved from Big at 15-Sep-2016 13:00 to Big at 15-Sep-2016 10:00
Talk 3 has moved from Big at 15-Sep-2016 10:00 to Big at 15-Sep-2016 09:30
Talk 4 has moved from Small at 15-Sep-2016 12:30 to Big at 16-Sep-2016 10:00
Talk 5 has moved from Big at 16-Sep-2016 13:00 to Big at 15-Sep-2016 13:00
Talk 6 has moved from Big at 16-Sep-2016 09:30 to Big at 16-Sep-2016 13:00
Talk 7 has moved from Big at 15-Sep-2016 12:30 to Big at 16-Sep-2016 12:30
Talk 8 has moved from Small at 15-Sep-2016 13:00 to Small at 15-Sep-2016 12:30
Talk 9 has moved from Big at 15-Sep-2016 09:30 to Small at 15-Sep-2016 13:00


We can also look at slots to see which now have a different event scheduled::

>>> slot_diff = scheduler.slot_schedule_difference(schedule, alt_schedule)
>>> for item in slot_diff:
... print(f"{item.slot.venue} at {item.slot.starts_at} will now host {item.new_event.name} rather than {item.old_event.name}" )
Big at 15-Sep-2016 09:30 will now host Talk 9 rather than Talk 1
Big at 15-Sep-2016 10:00 will now host Talk 5 rather than Talk 2
Big at 15-Sep-2016 12:30 will now host Talk 4 rather than Talk 11
Big at 15-Sep-2016 13:00 will now host Talk 6 rather than Talk 10
Big at 16-Sep-2016 09:30 will now host Talk 2 rather than Talk 7
Big at 16-Sep-2016 13:00 will now host Talk 11 rather than Talk 9
Small at 15-Sep-2016 09:30 will now host Talk 1 rather than Talk 12
Small at 15-Sep-2016 10:00 will now host Talk 10 rather than Talk 6
Small at 15-Sep-2016 12:30 will now host Talk 7 rather than Talk 4
Small at 15-Sep-2016 13:00 will now host Talk 12 rather than Talk 5

Big at 15-Sep-2016 09:30 will now host Talk 3 rather than Talk 9
Big at 15-Sep-2016 10:00 will now host Talk 2 rather than Talk 3
Big at 15-Sep-2016 12:30 will now host Talk 1 rather than Talk 7
Big at 15-Sep-2016 13:00 will now host Talk 5 rather than Talk 2
Big at 16-Sep-2016 09:30 will now host Talk 11 rather than Talk 6
Big at 16-Sep-2016 10:00 will now host Talk 4 rather than Talk 12
Big at 16-Sep-2016 12:30 will now host Talk 7 rather than Talk 11
Big at 16-Sep-2016 13:00 will now host Talk 6 rather than Talk 5
Small at 15-Sep-2016 09:30 will now host Talk 12 rather than Talk 1
Small at 15-Sep-2016 12:30 will now host Talk 8 rather than Talk 4
Small at 15-Sep-2016 13:00 will now host Talk 9 rather than Talk 8


We can use this facility to show how using :code:`number_of_changes` as our objective function
Expand All @@ -282,9 +312,8 @@ resulted in far fewer changes::
>>> event_diff = scheduler.event_schedule_difference(schedule, similar_schedule)
>>> for item in event_diff:
... print(f"{item.event.name} has moved from {item.old_slot.venue} at {item.old_slot.starts_at} to {item.new_slot.venue} at {item.new_slot.starts_at}")
Talk 11 has moved from Big at 15-Sep-2016 12:30 to Big at 16-Sep-2016 13:00
Talk 9 has moved from Big at 16-Sep-2016 13:00 to Big at 15-Sep-2016 12:30

Talk 11 has moved from Big at 16-Sep-2016 12:30 to Big at 16-Sep-2016 09:30
Talk 6 has moved from Big at 16-Sep-2016 09:30 to Big at 16-Sep-2016 12:30


Scheduling chairs
Expand Down
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