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Appointment Systems - a Stochastic and Fluid Approach

Appointment Systems - a Stochastic and Fluid Approach. Michal Penn The William Davidson Faculty of Industrial Engineering and Management Technion - Israel Institute of Technology Joint work with Yossi Luzon and Avishai Mandelbaum January 23, 2008. Appointment Systems.

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Appointment Systems - a Stochastic and Fluid Approach

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  1. Appointment Systems - a Stochastic and Fluid Approach Michal Penn The William Davidson Faculty of Industrial Engineering and Management Technion - Israel Institute of Technology Joint work with Yossi Luzon and Avishai Mandelbaum January 23, 2008

  2. Appointment Systems Airline Services Health Care Why do Appointment Systems exist?

  3. Why do appointment systems exist? Economically efficient Quality of care Long service times Uncertainty

  4. Macro View Waiting time at home stochastic customersrequests for service Customers departure scheduling appointments service tool: appointment book Waiting time at server

  5. Stochastic arrivals

  6. Novelty: using fluid approximation in the context of appointment systems. Difficulties • Appointment system scheduling problems are: • Large • Dynamic • Combinatorial • Stochastic • To overcome the problem’s complexity we suggest using fluid approximation.

  7. General framework Deterministic Fluid approximation Appointment book Scheduling customers based on a.b. Heterogeneous stochastic arrivals Service; Stochastic service time Combinatorial scheduling problem Imitation of the fluid solution

  8. General framework Based on Expectation (ignore variance) Slots based on service times Objective function Heterogeneous stochastic arrivals Deterministic Fluid approximation Appointment book Scheduling customers based on a.b. service Combinatorial scheduling problem Imitation of the fluid solution Aim: asymptotically optimal

  9. Finite time horizon Periodicity of customers behavior • History repeats itself • Days • Weeks • Months… 0 T Finite time horizon Cyclic nature of history + solution for finite time horizon solution to the problem

  10. Single server – minimum waiting time Discrete problem NP hard Fluid model solved – rule

  11. Two servers – minimum makespan Appointment books

  12. Two servers – minimum makespan - Proportion devoted By Fluid model solved to customer 2 - work conserving

  13. Single Server – Minimum Waiting Time We solved this system and found optimal -s

  14. Single Server – Minimum Waiting Time

  15. 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.

  16. Constructing Optimal Control Optimal control - What do we have so far? • Single Server – Minimum Waiting Time • Tandem Network of Two Servers – Minimum Makespan These are special cases of…

  17. The General Network

  18. The Fluid Control Optimization Problem (minimum makespan)

  19. Aims: • Develop appointment books that are near optimal. • Prove theoretically the quality of our procedure. • Demonstrate by simulation the quality of our procedure.

  20. Literature Review • Appointment Systems – RelatedWork. • Time Dependent Stochastic Networks and Fluid Control • Scheduling via Fluid Approximations

  21. Appointment Systems – RelatedWork. Performance 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.

  22. Time Dependent Stochastic Networks and Fluid Control • Performance Analysis • Approximations: Newell, Keller, Massey, Dai, ... • strong approximations: Mandelbaum-Massey, ... • alternating load: Harchol-Balter, ... • Control • multi-class, static overload: Avram-Bertsimas-Ricard, Kelly, Weiss, … • multi-class, transient overload: Chang-Ayhan-Dai-Xia • Scheduling via Fluid Approximations • Job Shop • Makespan: Bertsimas-Gammarnik, Bertsimas-Sethurman, Boudoukh-Penn-Weiss, … • Holding cost: Bertsimas-Gammarnik-Sethurman,…

  23. Single Server – Minimum Waiting Time

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