Distributed Timetabling

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Distributed Timetabling

• Eliezer Kaplansky and Amnon Meisels
• Dept. of Computer Science
• Ben-Gurion University

2
Overview

• The need for Distributed Timetabling
• Two real world examples
• University timetabling
• Employee timetabling
• Current status of DisTTP at BGU
• Distributed search for DisETP
• Iterative improvement of transportation cost
• Cooperative schedules improvement by SAs
• What next ?

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The need

• Many real world Timetabling Problems (TTPs) are
  composed of organizational parts that need to
  timetable their activities in an independent way,
  while adhering to some global constraints
• The timetables of all the parts must be combined
  to yield a coherent, consistent solution

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Why not solve as a single problem ?

• Organization The work-flow of the organization
  is distributed
• Privacy There are private constraints that
  employees prefer to keep inside the department
• Local control Department head wants the final
  say on the weight of local constraints and
  quality of solution
• Too big Central management refuses to take into
  account local details

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Two Case studies

• Over the last two years we have investigated two
  large scale problems
• BGU Course timetabling at Ben-Gurion University
• Departmental timetables (per semester)
• Conflicts among shared courses
• Shared rooms over all campus
• Soroka Nurse timetabling transportation in
  Soroka hospital, Beer-sheva
• Weekly timetables for each ward
• Transportation sharing among multiple wards

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Departmental Schedules

• University Departments prefer to do their Course
  Scheduling independently (dynamic private...)
• Departmental Schedules are constrained by other
  departments
• Students groups take courses in several
  departments ?
• conflicting classes between two departments
• Conflicts can arise from teachers that teach
  courses to different departments

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CS Dept. - a typical (large) example

• Schedule done every semester
• 120 classes are scheduled PLUS 400 TA sessions
• About 100 weekly time-slots
• Four study programs x 3 years 12 different
  student groups (some mixing occurs)
• TA sessions conflict with graduate courses (TAs
  are graduate students)
• Teachers (and TAs) have preferences and personal
  constraints that rule out multiple time-slots
  from the domains of the classes
• Typical Objectives lunch break, less than 8
  hours per day finish early on Thursday time to
  prepare

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CS Math. (1st year) Engineering

• Many courses are taken by students of all 3
  departments, at multiple levels
• Any change of schedule generates a need for
  changes in other departments, that share a common
  course
• Conflicts among schedules are negotiated by the
  Programs of CS., Math., SE, BioInformatics,
  Engineering
• Schedulers spend a few weeks of negotiations

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Rooms a global constraint

• Scheduler submits a list of required
  lecture-halls, based on a complete schedule
• The central room authority collects all
  departmental requirements in the university
• BGU has 80 departments and 180 lecture halls in
  the university pool
• Using a simple data base, the central authority
  grants most requests and rejects some
• Typically, the CS scheduler will receive a few
  rejections (10)

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Rooms the negotiation process

• Departmental schedulers try to move rejected
  sessions to less popular time-slots
• Moving scheduled classes clashes with personal
  constraints, with conflicting classes, with TA
  sessions
• schedulers must negotiate the proposed changes
  with constraining schedulers
• no central information on options to exchange
  scarce lecture-halls among different schedules is
  available i.e. a waiting list
• Many departments are forced to break some
  constraints in order to get rooms for all classes

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Multi-Agent Systems (MAS)

• A new paradigm for modeling, designing, and
  implementing software solutions
• Agents are sophisticated computer programs that
  act autonomously on behalf of their users, across
  distributed environments
• A MAS is a loosely coupled network of software
  agents that interact to solve a global problem
  that is beyond the individual capacities or
  knowledge of each agent

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Scheduling Agents for BGU

• A separate agent monitors the schedule of each
  department first generating it
• Cooperates with constraining departments to
  arrive at the best feasible schedule
• Every departmental agent negotiates with the
  rooms authority limited room capacity
• Currently, 13 departments are test-running the
  departmental course timetabling construction and
  solving program
• The departmental system is complex, including all
  assignments, preferences, courses, etc. on a
  distributed database server

13
Nurses Timetabling

• Multiple hospital wards
• Nurses have a weekly timetable
• Each ward schedules its nurses independently
• Hospital regulations require certain (combined)
  number of nurses of specific seniority and
  expertise for each shift
• Timetables of wards must be coordinated, to
  reduce transportation cost

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The Nurses ETP

• wards timetable 30 nurses to shifts
• Three shifts per day plus weekend shifts
• Nurses have different qualifications and are
  assigned to certain tasks in shifts
• Each nurse is assigned 3-5 shifts per week
• Each nurse is assigned 3-5 shifts per week
• Nurses have preferred shifts for each week and
  forbidden shifts, due to personal constraints
• Problem size 100 weekly assignments

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Nurses transportation

• Nurses are scheduled independently by each ward
• The hospital works 3 shifts a day, 7 days a week
• Transportation for nurses, for each shift
• Transportation lines grouped into destinations
• Cab/bus capacities are fixed
• To achieve cost effectiveness, the Scheduling
  Agents (SAs) must coordinate their timetables
• SAs are not aware of inter-ward transportation
  constraints

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Scheduling Agents for ETPs

• Each SA solves the weekly timetable of one ward
• to achieve coordination with other SAs many
  alternative solutions are needed
• Standard distributed search algorithms assume
  very simple agents that generate solutions in
  negligible time
• Distributed search algorithms for complex SAs
  must be investigated

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Inter-agent Cooperation

• Important Need improvement steps in the quality
  of the global solution
• Search mixes the global improvement with local
  schedules quality
• To enable an efficient search, appoint one
  responsible agent for each resource
• In Soroka we use a line manager for each
  transportation line

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Choosing A Resource Manager

• Among all resource-users select the Agent with
  the more complex problem to solve
• It makes sense to let a ward that has 7 nurses
  that live in a nearby town (compared to an
  average of 1 per ward) to manage the
  transportation line to that town
• The line manager sets the days the line is
  available and waits for other wards to request a
  change

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Inter-agent Cooperation

• First stage Find a local solution for the
  timetabling problem of each agent
• Second stage Improve the cost of transportation
  for shared lines
• Run the improvement process concurrently for all
  resources (seats on transportation lines)
• Third stage Enable SAs to improve their
  schedules, controlling transportation costs

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Stage 2 Distributed iterative improvement

• Each transportation line has a number of
  requested seats, per each shift, from every ward
• To improve the cost of a given line, some of its
  consumer wards have to change their schedules
• Changed schedules (satisfying imposed seats
  constraints) have different qualities
• An improvement step must have an overall quality
  that is above a given threshold

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Stage 3 Requests for Change

• Each agent tries to improve its local timetable
  by re-solving its local problem
• When it finds a better solution, it sends a
  Request for Change (RfC) to the Resource Manager
  
• The RfC message is accompanied by an Expected
  Gain (EG) and a list of changes in needed
  seats, for all shifts
• line manager approves RfCs with a lower
  transportation cost and a non-lower total quality

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Toy problem experiment one line

• 4 wards, each with 30 nurses
• 21 shifts per week
• 5 lines of transportation
• Cabs have a capacity of 1-7 passengers
• Different costs for lines (and shifts)
• Transportation cost is the sum of the number of
  needed cabs in each line
• filling-up cabs leads to optimality of cost
• The number of passengers depends on all SAs
  Timetables

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Search for a better cost (Toy problem)
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Managing one line

• To avoid cabs with one passenger, add a
  constraint to the SAs, limit number of nurses for
  the line
• Add a constraint that prevents the SA from
  adding a nurse to a line with full cabs
• SAs solve dynamic CSPs
• A structural change to SAs
• Add/remove constraints
• Improve quality of timetable
• SAs can be either local searchers or IRAs

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Dynamically adding constraints
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Transportation Cost vs. Quality of Schedules
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Distributed nature of management

• Line manager can request participating SAs
  sequentially or concurrently
• Sequential improvement can be complete
• Changing schedules is too complex to enforce
  completeness
• Requesting changes concurrently generates a need
  for coordination, but, can save time

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Synchronized protocol ? 10 SA utilization
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Concurrent protocol ? 50 SA utilization
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Concurrency generates improvement faster
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Controlling concurrency

• In stage 3 SAs request changes, trying to
  improve the quality of their schedules
• Requests are naturally concurrent
• Must be treated carefully, as requests of
  different agents are conflicting
• Currently, requests are processed one at a time
• Agents are committed to their proposed changes,
  as part of their requests

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Some degree of freedom to SAs

• Enable SAs to express special quality
  differences about parts of their schedules
• Such personal quality differences must have an
  impact on the coordinated schedules
• Different SAs assign a different value to the
  use of the shared resource, for special parts of
  their schedule
• Can be expressed by bidding for the resource
• Make it fair by distributing a fixed amount of
  tokens for each SA

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Bidding for resources

• Enables SAs to fine-tune their schedules
• High quality local improvements are achieved by
  paying high for the resource
• Explored for university room sharing (among
  departments) by Andrea Schaerf et al.
• In Udine bidding is for rooms of other
  departments, schedules are not involved
• A mechanism for bidding is still unexplored
• binary vs. resource-wide
• fixed of tokens vs. Dynamic
  (difficulty-based)

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Conclusions

• Two real world distributed timetabling problems
  have been identified
• University DisTTPs occur twice a year
• Room sharing is managed globally and can benefit
  from interacting departmental schedules
• Medical DisETPs are common shared
  transportation shared employees
• Weekly schedules generate frequent use
• Concurrent resource managers can create a large
  improvement in shared costs

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Distributed resource management