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