SCHEDULING WITH CANCELLATIONS VAN-ANH TRUONG (JOINT WORK WITH - PowerPoint PPT Presentation

MULTI-PRIORITY ONLINE SCHEDULING WITH CANCELLATIONS VAN-ANH TRUONG (JOINT WORK WITH XINSHANG WANG) COLUMBIA UNIVERSITY 29-MAY-15

THREE APPLICATIONS 1. Make-to-order photolithography mask-making facility (Rubino and Ata 2009)

THREE APPLICATIONS 1. Make-to-order photolithography mask-making facility (Rubino and Ata 2009) • Orders from multiple demand classes arrive over time. • Production only starts when there are outstanding orders. • No inventory is kept. • Manager can outsource an order upon its arrival or accept it • Enqueues the order and allocates the order for production as appropriate. • Each order incurs a waiting cost as it waits. • How to make outsourcing decisions to trade-off outsourcing and waiting costs?

THREE APPLICATIONS 2. Elective surgery facility ( Gerchak et al. 1996 )

THREE APPLICATIONS 2. Elective surgery facility ( Gerchak et al. 1996 ) • Hospital reserves a portion of each day for emergency surgeries • Remaining time used to performed elective surgeries • New elective surgery requests arrive each day • Cost for waiting • Scheduling too many cases per day exceeds day’s capacity • Overtime hours or surge capacity used • Aggregate-planning decision: How many cases to perform on each day in regular and over time?

THREE APPLICATIONS 3. Network routing for server applications

THREE APPLICATIONS 3. Network routing for server applications • Data requests arrive to server applications by local clients. • Servers make use of computing resources (CPU, disk I/O, etc) • Data requests not immediately processed are stored in the memory. • If the wait time is too long, some data requests might expire or lose the value of being processed • Priority of a data request determined by its expiration date or the probability of expiring. • Data packages can often be routed to remote (external) idle servers at a cost of propagation delay • Routing decision needs to be dynamically made to reduce the overall cost

MODEL • Adapted for surgical-scheduling terminology • There are T days in the planning horizon • There is some regular capacity on each day • Might be random and variable because of staffing • Unlimited amount of overtime capacity

MODEL: CAPACITY Regular capacity Overtime • Use of regular capacity Day 1 is free (sunk cost) • Use of overtime capacity Day 2 incurs a constant overtime cost per patient Day 3 …

MODEL: SCHEDULING cancel • Waiting patients Day 1 served might cancel randomly in each period Day 2 served Day 3 served …

MODEL: PRIORITIES Higher priority cancel Day 1 served Day 2 served Day 3 served …

MODEL: PRIORITIES • Each patient 𝑣 is associated with • A waiting cost 𝑥 𝑣 • A cancellation probability 𝑟 𝑣 • A cancellation cost 𝑠 𝑣 • Higher priority patients have higher values of 𝑥 𝑣 , 𝑟 𝑣 and 𝑠 𝑣 • It is optimal to serve some number of highest-priority patients on each day ( Ayvaz and Huh 2010, Min and Yih 2010 ) • We will study this class of policies

ONLINE FRAMEWORK • Distribution of demand/capacity unknown • Demand/capacity can be chosen by an adversary • Can be correlated over time • Cancellations are exogenous to all policies

ONLINE FRAMEWORK • OFF: Optimal offline policy. Knows the sequence of demand arrivals and capacities up front. • ON: My online policy. Observes demand arrivals and capacity one period at a time and makes adaptive decisions. • Goal is to characterize an upper bound on the worst-case ratio of expected costs 𝑊 ON / 𝑊 OFF called the competitive ratio.

WHY ONLINE ? • Is the approach overly pessimistic? 1. Past approaches have not been tractable for general models of demand • Stationary demands usually needed • In practice, demand usually non-stationary • In server applications (Tai et all 2011). • Call centers and hospitals (Kim and Whitt 2013)

WHY ONLINE ? • Is the approach overly pessimistic? 2. Even with stationary demand, there are few characterizations of theoretical performance

KNOWN RESULTS • Appointment scheduling literature • Stochastic dynamic programming framework. • Gerchak, Gupta and Henig (1996), Patrick, Putterman and Queyranne (2008), Min and Yih (2010), Ayvaz and Huh (2010), Gocgun and Ghate (2012), Huh, Liu, and Truong (2013). • Structural results known. • Few heuristics to compute policies. • No performance guarantee.

KNOWN RESULTS • Queuing literature • Blackburn (1972) • Rubino and Ata (2009) • Assume stationary Poisson arrivals • Fixed capacity • Solved problem in heavy-traffic regime • The only characterization of performance known

KNOWN RESULTS • Machine-scheduling literature • Online algorithms to minimize sum of energy usage and waiting costs • Yao et al. (1995), Chan et al. (2007), Bansal et al. (2007, 2009, 2011), Albers and Fujiwara (2007) • Continuous control of processing rate • Preemption • No cancellations

OUR RESULTS • First online algorithm with a competitive ratio of 2. • Proof that 2 is the best possible competitive ratio. • Algorithm easy to implement and has good empirical performance. • Analysis of performance provides new insights into problem.

DIFFICULTY IN THE ANALYSIS • Need to compare two different policies • Do not know OFF • System states high-dimensional • States take different paths over time • Cannot directly compare costs between different system states

A MOTIVATING EXAMPLE OFF ON Day 1 served served Day 2 served • Both policies serve a total of 3 patients over 2 days. • ON delays scheduling => ends with lower-priority patients

A MOTIVATING EXAMPLE OFF ON Day 1 served served Day 2 served Type 1: overtime cost under ON ≤ overtime cost under OFF

A MOTIVATING EXAMPLE OFF ON Day 1 served served Day 2 served Type 2: waiting cost under ON ≤ waiting cost under OFF

COMPARING 2 POLICIES Two policies differ only in number of overtime slots used in each period. A policy might be ``ahead’’ for part of the time, and behind for part of the time. How to tell when a policy is ahead generally? First, ignore cancellations.

COMPARING 2 POLICIES Cumulative number of overtime slots used 8 6 4 2 ON 0 1 2 3 4 5 6 7 8 9 OFF OFF - ON -2 -4

COMPARING 2 POLICIES Cumulative number of overtime slots used 8 6 4 2 ON 0 OFF 1 2 3 4 5 6 7 8 9 Recalculate OFF - ON -2 cumulative difference OFF is behind in period 1 starting in -4 period 2

COMPARING 2 POLICIES Cumulative number of overtime slots used 8 6 4 2 ON 0 OFF 1 2 3 4 5 6 7 8 9 Recalculate OFF - ON -2 cumulative difference OFF falls behind in periods 4 and 5 starting in -4 period 6

DISTANCE BETWEEN POLICIES Cumulative number of overtime slots used 8 6 4 2 ON 0 OFF 1 2 3 4 5 6 7 8 9 OFF - ON -2 OFF falls behind in periods 8 and 9 -4

COMPARING 2 POLICIES Cumulative number of overtime slots used 8 6 4 2 ON 0 OFF 1 2 3 4 5 6 7 8 9 OFF - ON -2 -4 OFF is behind

COMPARING 2 POLICIES Cumulative number of overtime slots used 8 6 4 2 ON 0 OFF 1 2 3 4 5 6 7 8 9 OFF - ON -2 -4 OFF is ahead

DISTANCE BETWEEN POLICIES Distance is 0 when OFF is behind 8 6 4 2 ON 0 OFF 1 2 3 4 5 6 7 8 9 OFF - ON -2 -4

DISTANCE BETWEEN POLICIES Distance equals adjusted cumulative difference when OFF is ahead 8 6 4 2 ON 0 OFF 1 2 3 4 5 6 7 8 9 OFF - ON -2 -4

DISTANCE BETWEEN POLICIES Distance function 8 6 4 2 ON 0 OFF 1 2 3 4 5 6 7 8 9 OFF - ON -2 -4

WHY DISTANCE FUNCTION? Physical interpretation: • In a period in which the distance is d=0 • OFF uses fewer overtime slots in that period • OFF accumulates more jobs in that period • In an interval in which the distance is d>0 • OFF uses d more overtime slots cumulatively in interval • But by serving d more jobs, ON can catch up to OFF

WHY DISTANCE FUNCTION? Physical interpretation: when the distance is d • If ON serves another d jobs in overtime, the remaining jobs in ON will have priorities less than or equal to those in OFF

WHY DISTANCE FUNCTION? Physical interpretation: when the distance is d=2 and ON serves 2 more jobs ON OFF

WHY DISTANCE FUNCTION? each remaining job in ON can be uniquely matched to a higher priority job in OFF ON OFF

WHY DISTANCE FUNCTION? ON becomes dominated by OFF ON OFF

PROPERTY OF DISTANCE FUNCTION Theorem: In a period in which distance is 0 • The jobs in ON are dominated by the jobs in OFF • OFF incurs higher waiting costs

PROPERTY OF DISTANCE FUNCTION Theorem: In an interval in which distance is positive • OFF uses more cumulative overtime slots in the interval • OFF incurs higher overtime cost in the interval

SIGNIFICANCE OF DISTANCE FUNCTION Provide an invariance between the costs of ON and OFF 1. No direct knowledge of OFF