Appointment Scheduling with Discrete Random Durations and Applications

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Appointment Schedulingwith Discrete Random
Durations and Applications

• Maurice Queyranne
• joint work with Mehmet A. Begen
• Sauder School of Business, University of British
  Columbia
• Part II is also joint work with Retsef Levi
• Sloan School of Management, MIT

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Scheduling Challenge

• Suppose you are a surgeon and surgery durations
  are uncertain
• Under-utilization cost is high
• for the surgeon the opportunity cost of being
  idle
• for the OR the opportunity cost of idle time
  (equipment and staffing costs, and long patient
  waiting list)
• Patient waiting and/or overtime cost are also
  high
• for the surgeon overtime cost
• for the patient waiting cost
• for the OR staffing and equipment overtime costs
• How would you schedule patients into your OR
  block?
• which patients? or how many patients? (determine
  total workload)
• in what order? (sequence the chosen surgeries)
• how much time to allocate to each surgery?
  (determine planned start times/appointment times
  of surgeries)
• appointment scheduling (for a given sequence of
  surgeries)

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Motivation

• Many real world applications (especially in
  healthcare)
• surgery scheduling
• physician or treatment (e.g., radiation)
  appointments
• healthcare diagnostic operations (such as CAT
  scan, MRI)
• project/contract scheduling
• container vessel and terminal operations
• airport gate or runway schedules
• Interesting and challenging problem
• New approach idea
• discrete time version
• Processing duration distributions may not known
  but only samples may be available in practice

Part 1
Part 2
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Outline

• 1. Problem Definition
• 2. Appointment Scheduling
• 3. Sampling Approach
• 4. An Overview of Current and Future Work
• 5. Conclusion

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1. Problem Definition

• n jobs on a single processor (e.g., n surgeries
  in an OR)
• Given processing order
• Random processing durations
• An important tradeoff between costs of early and
  late jobs
• overage (lateness wait overtime) costs
• underage (earliness idle time) costs
• Find an appointment schedule minimizing total
  (overage and underage) expected cost

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Illustrative Example
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Illustrative Example
A realization of durations
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Problem Definition

• When the distributions of the processing
  durations are
• known how can we find an optimum solution?
• Appointment Scheduling
• unknown but random samples are available how can
  we find a good solution, close to the true
  optimum?
• Sampling Approach

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2. Appointment Scheduling

• Appointment Scheduling with Discrete Random
  Durations

Joint work with Mehmet A. Begen (Sauder School of
Business, UBC)
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Previous Studies on Appointment Scheduling

• Several dozen papers,
• IE, OR/MS, OM literature
• medical lit., transportation lit.
• including notably
• Denton and Gupta (03), Robinson and Chen (03),
  Weiss (90),
• Elhafsi (02), Sabria and Daganzo (89),
• Wang (93 and 99), Vanden Bosch et al. (99),
  Kaandorp and Koole (07)
• Cayirli and Veral (03), Cardoen et. al. (08),
• Patrick et al. (08), Green et al. (06), Pinedo
  (83, 01)

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Previous Studies on Appointment Scheduling

• Existing work mostly
• situation specific and not portable
• uses identical and specific duration
  distributions (e.g., exponential, normal)
• uses identical costs
• uses continuous duration distributions (even the
  ones with discrete potential appointment times)
• experiences severe computational difficulties
• only very small instances solved to optimality
• heuristics and/or simulations
• We aim to develop an approach
• sufficiently generic, and
• computationally efficient (both in theory and
  practice)

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Results Overview Appointment Scheduling

• An algorithm that finds an optimal schedule using
  
• polynomial time and
• polynomial number of expected-cost evaluations
• When processing durations are independent,
  optimum schedule and its expected cost may be
  computed efficiently both in
• theory (in polynomial time) ? faster algorithm
• practice (fast computations in our preliminary
  experiments)
• Extensions to due dates, no-shows, emergencies

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Method Overview Appointment Scheduling

• A discrete time model
• Arbitrary (generic) discrete distributions of
  processing durations
• Existence of an integer optimum
• ? may optimize only over integer appointment
  schedules
• Discrete convexity (under a very mild assumption
  on cost coefficients) of the objective function
• ? an efficient algorithm
• Recursive computations for expected cost
  evaluations (if independent durations)

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Assumptions Appointment Scheduling

• There are n1 jobs that need to be sequentially
  processed on a single processor
• The processing sequence is given
• Processing durations are given by their discrete
  joint distribution
• bounded, non-negative and integer valued
• Non-negative cost coefficients

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Notation
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Objective Function
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Basic Properties

• Lemma (Continuity) F(.p) and F are continuous.
• Lemma (Existence of an Optimal Vector)
• continuity and compact feasible set (w.l.o.opt.)
• Lemma (Optimality of Non-increasing Appointment
  Dates)
• w.l.o.opt. A1 A2 .... An An1

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Appointment Vector Integrality

• Proof surprisingly non-trivial
• Given any non-integer appointment vector A
• generate two more vectors A and A from A
• move all jobs with same fractional parts as the
  first non-integer component
• maximum possible movement ?gt0 so there is no
  event change

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• expected cost changes
• linearly with ?
• between A and A

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Submodularity and L-convexity of F
Definition. A function f is submodular
if f (A?B)f (A?B) lt f (A) f (B) for all
A, B where (A?B)j minAj, Bj and (A?B)j
maxAj, Bj for all j
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Polynomial Time Algorithms
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Due Date, No-Shows, Emergencies

• A (given) due date for the end of processing
  (e.g., end of day for an operating room) instead
  of deciding one
• No-shows (e.g., in MRI)
• processing time becomes zero
• Emergencies (e.g., surgeries)

emergency job arrivals
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3. Sampling Approach

• A sampling-based approach to appointment
  scheduling

Joint work with Mehmet A. Begen (Sauder School of
Business, UBC) and
Retsef Levi (Sloan School of Management, MIT)
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Preliminaries for Sampling Approach

• What defines a good method?
• convergence
• convergence rate
• worst case performance
• input required (sample size)
• closeness to the true optimum
• How much is known about the true distributions?
• parametric or non-parametric?
• Our approach
• non-parametric
• determine number of independent samples required
  for a good solution with high probability

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Preliminaries for Sampling Approach

• Our approach is similar to Levi, Roundy Shmoys
  (2007) LRS, who presented
• a novel sampling-based DP framework for a
  multi-period newsvendor problem
• a provably near-optimal solution with high
  confidence
• in a sequential decision making context
• Unlike LRS,
• our modeling framework is not DP
• we determine all decision variables
  simultaneously at the beginning of the planning
  horizon (i.e., at time zero)
• although related, our problem is not a
  multi-period newsvendor
• For this, we fully characterize the
  subdifferential of the objective function

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Results Overview Sampling Approach

• Relaxation of the assumption of known
  distribution of the job durations
• non parametric
• determine sample size N(e,d,u,o) such that
• Fp(Â) (1e) Fp(A) with probability at least
  (1?d)
• Ability to handle objective function with a due
  date

exp. cost of a true optimal schedule (determined
from knowing the true distribution p)
exp. cost of the sample-based optimal solution
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Method Overview Sampling Approach

• Convexity of the objective function (as a
  function of continuous appointment vector)
• under assumption that cost rates (u,o) are
  ?-monotone
• Subdifferential characterization (in closed-form)
  
• useful for
• sampling approach
• non-smooth optimization
• Hoeffdings inequality
• upper bound on the probability of very large
  deviations

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Assumptions Sampling Approach

• There are n1 jobs that need to be sequentially
  processed on a single processor
• The processing sequence is given
• Only independent samples available for the
  discrete joint distribution of the processing
  durations
• bounded, non-negative and integer valued
• Non-negative cost coefficients

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Convexity of the Objective Function

• Lemma (Convexity) If the cost vectors (u,o) are
  a-monotone then F(.p) and F(.) are convex.

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Subdifferential Characterization

• Start from elementary components of the objective
  function
• Find the subdifferentials of Lj (Ap), Tj (Ap)
  and Mj (Ap).
• Use rules of subdifferential calculus to get
  ?Lj(A), ?Tj(A), ?Mj(A) where zj(A) Ep(zj(Ap))
  for z in L, T, M

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Subdifferential Characterization the gory
details
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Sampling Approach

• Find in polynomial time (in n, N, h, 1/e), a
  point  such that
• a small subgradient (of the true objective Fp )
  exists at Â, and
• Fp (Â) (1 e) Fp (A).
• Obtain N such that the absolute difference
  between the true (w.r.t. the true distribution)
  and estimated probabilities (w.r.t. samples) of
  certain events is small with high probability (1
  ? d ).

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4. An Overview of Current and Future Work
Current Work Appointment Scheduling and
Applications
Appointment Scheduling Polynomial Time Algorithm
Sampling Approach
Computational Study
Relax the assumption of known duration
distribution Sample-based approach Convexity
of the objective Subdifferential
characterization Obtain a provably near-optimal
solution with high probability
Obtain a subgradient in polynomial time Use
non-smooth convex optimization Hybrid
algorithms with integer rounding
Implementation and computational experiments
Expected cost computation Appointment vector
integrality Discrete convexity Polynomial time
algorithm
SODA09
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Future Work
Future Work Appointment Scheduling and
Applications
Sequencing Problem
Practical Application
Game Theoretical Approaches
Inventory Application
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5. Conclusion

• An important real world problem with many
  practical applications
• Sufficiently generic and portable approach
• Potential benefits
• reduced job/patient waiting times
• improved capacity utilization
• First polynomial time algorithm
• Relaxation of the known distribution assumption
• using only sample information on processing times

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Comments and Questions?