Post on 04-Jan-2016
Appointment Systems - a Stochastic Appointment Systems - a Stochastic and Fluid Approachand Fluid Approach
Michal Penn
The William Davidson Facultyof Industrial Engineering and ManagementTechnion - Israel Institute of Technology
Joint work with Yossi Luzon and Avishai Mandelbaum
January 23, 2008
Appointment SystemsAppointment Systems
Airline ServicesAirline Services
Why do Why do Appointment Appointment
Systems Systems exist?exist?
Health CareHealth Care
Long service timesLong service times
Why do appointment systems exist?
Economically efficientEconomically efficient
UncertaintyUncertaintyQuality of careQuality of care
Macro View
scheduling appointment
sservice stochastic
customers requests for
service
Customers departure
tool: appointment book Waiting
time at server
Stochastic Stochastic arrivalsarrivals
DifficultiesDifficultiesAppointment system scheduling problems are:
• Large
• Dynamic
• Combinatorial
• Stochastic
To overcome the problem’s complexity we suggest using fluid approximation.
Novelty: using fluid approximation in the Novelty: using fluid approximation in the context of appointment systems.context of appointment systems.
General framework
Appointment book
Combinatorial scheduling problem
Scheduling customers
based on a.b.
Service;Stochastic
service time
Imitation of the fluid solution
Heterogeneous stochastic arrivals
Deterministic Fluid
approximation
General framework
Appointment book
Combinatorial scheduling problem
Scheduling customers
based on a.b.
service
Imitation of the fluid solution
Heterogeneous stochastic arrivals
Deterministic Fluid
approximation
Based on Expectation
(ignore variance)
Objectivefunction
Slots basedon service
times
Aim: asymptoticallyoptimal
Finite time horizonFinite time horizon
History repeats itself
• Days
• Weeks
• Months… 0 T
Cyclic nature of history + solution for finite Cyclic nature of history + solution for finite time horizon solution to the time horizon solution to the problemproblem
Finite time horizon
Periodicity of customers behavior
Single server – minimum waiting timeSingle server – minimum waiting time
Fluid model solved – Fluid model solved – rulerule
Discrete problem NP hard
c
Two servers – minimum makespanTwo servers – minimum makespan
Appointment books
Two servers – minimum makespanTwo servers – minimum makespan
:1m
Fluid model solvedFluid model solved
- work conserving
2,1u - Proportion devoted By to customer 21m
2m}max{:2,
1,1
i
im
Single Server – Minimum Waiting TimeSingle Server – Minimum Waiting Time
We solved this system and found optimal -We solved this system and found optimal -s s
iT t
Single Server – Minimum Waiting TimeSingle Server – Minimum Waiting Time
Algorithm: General idea
Fluid Based Dispatching Rule
Assume F is a feasible solution for a given fluid appointment system with its given time dependent expected arrival rates.
In the discrete appointment system, if server i is idle at time t and there is a customer type available, then assign the next slot to the customer type with the largest deviation from its fluid solution at time t.
Constructing Optimal ControlConstructing Optimal Control
Optimal control - What do we have so far?
• Single Server – Minimum Waiting TimeSingle Server – Minimum Waiting Time
• Tandem Network of Two Servers – Tandem Network of Two Servers – Minimum MakespanMinimum Makespan
These are special cases of…These are special cases of…
The General NetworkThe General Network
The Fluid Control Optimization Problem The Fluid Control Optimization Problem (minimum makespan)(minimum makespan)
Aims:
1. Develop appointment books that are near optimal.
2. Prove theoretically the quality of our procedure.
3. Demonstrate by simulation the quality of our procedure.
Literature ReviewLiterature Review
• Appointment Systems – Related Work.
• Time Dependent Stochastic Networks and Fluid Control
• Scheduling via Fluid Approximations
1. Appointment Systems – Related Work.Performance Analysis and OptimizationPerformance Analysis and Optimization Bailey (1952), Jackson (1964): Appointment intervals, worked on
balancing a trade-off between server idle times and patient waiting times. Used simulation.
Peterson-Bertsimas-Odoni (1995): Aircraft landings, used a Markov/semi-Markov model for the changes in weather. Computed moments of queues.
Bosch-Van den-Dietz-Simeoni (2000): Outpatient systems, worked on minimizing operating costs of wait and overtime. Offered a scheduling algorithm, used submodularity.
Wang (1993): AS of a single server, computed the expected customers delay time recursively, used stochastic decreasing convexity.
Patrick-Puterman-Queyranne (2007 under review): Public health care, worked on dynamically scheduling multi-priority patients. Used MDPs to allocate available capacity to incoming demand so that waiting time targets are achieved.
2. Time Dependent Stochastic Networks and Fluid Control• Performance AnalysisPerformance Analysis
Approximations: Newell, Keller, Massey, Dai, ... strong approximations: Mandelbaum-Massey, ... alternating load: Harchol-Balter, ...
• ControlControl multi-class, static overload: Avram-Bertsimas-Ricard, Kelly, Weiss, … multi-class, transient overload: Chang-Ayhan-Dai-Xia
3. Scheduling via Fluid Approximations• Job ShopJob Shop
Makespan: Bertsimas-Gammarnik, Bertsimas-Sethurman, Boudoukh-Penn-Weiss, …
Holding cost: Bertsimas-Gammarnik-Sethurman,…
Single Server – Minimum Waiting TimeSingle Server – Minimum Waiting Time