Using Simulationbased Stochastic Approximation to Optimize Staffing of

Using Simulation-based Stochastic Approximation to Optimize Staffing of Systems with Skills-Based-Routing WSC 2010, Baltimore, Maryland Avishai Mandelbaum (Technion) Zohar Feldman (Technion, IBM Research Labs) Technion SEE Laboratory December 2010 Winter Simulation Conference, Baltimore, MD .

Contents l l l Skills Based Routing (SBR) Model SBR Staffing Problem Stochastic Approximation (SA) Solution Numerical Experiments Future Work Feldman et. al. Winter Simulation Conference, Baltimore, MD December 2010 2

SBR Model Service System with SBR – Basic Model l l l Feldman et. al. I – set of customer classes J – set of server pools Arrivals for class i: renewal (e. g. Poisson), rate λi Servers in pool j: Nj, iid Service of class i by pool j: (Im)patience of class i: Winter Simulation Conference, Baltimore, MD December 2010 3

SBR Model Routing l Arrival Control: upon customer arrival, which of the available servers, if any, should be assigned to serve the arriving customer l Idleness Control: upon service completion, which of the waiting customers, if any, should be admitted to service ? ? Feldman et. al. ? ? Winter Simulation Conference, Baltimore, MD December 2010 4

SBR Staffing Problem Cost-Optimization Formulation l l f k(N) – service level penalty functions Examples: • f k(N) = ckλk. PN{abk} – cost rate of abandonments • f k(N) = λk. EN[ck(Wk)] – waiting costs Feldman et. al. Winter Simulation Conference, Baltimore, MD December 2010 5

SBR Staffing Problem Constraints-Satisfaction Formulation l l f k(N) – service level objective Examples • • f k(N) = PN{Wk>Tk} – probability of waiting more than T time units f k(N) = EN[Wk] – expected wait Feldman et. al. Winter Simulation Conference, Baltimore, MD December 2010 6

SA Based Solution Stochastic Approximation (SA( l Uses Monte-Carlo sampling techniques to solve (approximate) analytically intractable l l l - convex set ξ – random vector (probability distribution P) supported on set Ξ - almost surely convex Feldman et. al. Winter Simulation Conference, Baltimore, MD December 2010 7

SA Based Solution SA Basic Assumptions l l There is a sampling mechanism that can be used to generate iid samples from Ξ There is an Oracle at our disposal that returns for any x and ξ • The value F(x, ξ) • A stochastic subgradient G(x, ξ) Feldman et. al. Winter Simulation Conference, Baltimore, MD December 2010 8

SA Based Solution SBR Simulation l Simulation Artifacts • Service Consumer: arrival process, patience distribution • Resource: availability function • Resource Skills: service distribution depending on resource type and requestor type • Router: arrival control, idleness control • Event Engine: sorts and executes events (arrivals, service completions, abandonment, shift change…) • Statistics: data series gathered by intervals (e. g. number of arrivals, number of abandonment, waiting times etc. ) l Use random streams to enable common number generation Feldman et. al. Winter Simulation Conference, Baltimore, MD December 2010 9

SA Based Solution SBR Simulation l l Ω - the probability space formed by arrival, service and patience times. f(N) can be represented in the form of expectation. For instance, D(N, ω) is the number of Delayed customers A(ω) is the number of Arrivals Use simulation to generate samples ω and calculate F(N, ω) Sub-gradients are approximated by Finite Differences Feldman et. al. Winter Simulation Conference, Baltimore, MD December 2010 10

SA Based Solution Cost Optimization Algorithm Problem Solution l l Feldman et. al. Use Robust SA For simulation, realvalued points are rounded to integers Winter Simulation Conference, Baltimore, MD December 2010 11

SA Based Solution Constraints Satisfaction Algorithm Problem Solution l There exist a solution with cost C that satisfies the Service Level constraints if”f l Feldman et. al. where Look for the minimal C via binary search Winter Simulation Conference, Baltimore, MD December 2010 12

Numerical Experiments Numerical Study l Goal l Method • Examine algorithms performance • Explore convexity and its affect on performance • Run the algorithms by several examples • For each example run simulation • To identify the best solution by calculating confidence intervals of all possible solutions • To evaluate solutions and approximate gradients to test for convexity Feldman et. al. Winter Simulation Conference, Baltimore, MD December 2010 13

Numerical Experiments Simple Example: Penalizing Abandonments l l l N-model (I=2, J=2) Control: Static Priority λ 1 =100 θ 1= 1 • Class 1: pool 1, pool 2 • Pool 2: class 1, class 2 µ 11=1 λ 2 =100 θ 2= 1 µ 21=1. 5 µ 22=2 Optimization problem Feldman et. al. Winter Simulation Conference, Baltimore, MD December 2010 14

Numerical Experiments Simple Example: Objective Function Feldman et. al. Winter Simulation Conference, Baltimore, MD December 2010 15

Numerical Experiments Simple Example: Solution l Convergence Rate Convergence Point l Solution: N=(98, 58), 0. 5% above optimal Feldman et. al. Winter Simulation Conference, Baltimore, MD December 2010 16

Numerical Experiments Realistic Example l 100’s-agents Call Center (US Bank: SEE Lab – open data source) 2 classes of calls l 2 pools of servers l • Business • Quick & Reilly (Brokerage) • Pool 1 - Dedicated to Business • Pool 2 - Serves both Feldman et. al. Winter Simulation Conference, Baltimore, MD December 2010 17

Numerical Experiments Realistic Example l Arrival Process: Hourly Rates Feldman et. al. Winter Simulation Conference, Baltimore, MD December 2010 18

Numerical Experiments Realistic Example l Service Distribution (via SEE Stat( Business Brokerage Log. N(3. 9, 4. 3) Log. N(3. 7, 3. 4) Patience: Exp(mean=7. 35 min) Feldman et. al. Exp(mean=19. 3 min) Winter Simulation Conference, Baltimore, MD December 2010 19

Numerical Experiments Realistic Example: Optimization Models l Daily SLA Feldman et. al. l Hourly SLA Winter Simulation Conference, Baltimore, MD December 2010 20

Numerical Experiments Realistic Example: SLA l Daily SLA Feldman et. al. l Hourly SLA Winter Simulation Conference, Baltimore, MD December 2010 21

Numerical Experiments Realistic Example: Staffing Levels l Daily SLA l Hourly SLA l Staffing cost: 510 l Staffing cost: 575 Feldman et. al. Winter Simulation Conference, Baltimore, MD December 2010 22

Summary l l l We developed simulation-based algorithms for optimizing staffing of systems with skillsbased-routing These algorithms apply to very general settings, including time-varying models and general distributions In most cases, the algorithms attained the optimal solutions even when the service levels were not convex Feldman et. al. Winter Simulation Conference, Baltimore, MD December 2010 23

Future Work l Incorporating scheduling mechanism Complex models Optimal Routing Enhance algorithms l Convexity Analysis l l l • Relax convexity assumption • More efficient Feldman et. al. Winter Simulation Conference, Baltimore, MD December 2010 24

Backup December 2010 Winter Simulation Conference, Baltimore, MD

Cost Optimization Algorithm Feldman et. al. Winter Simulation Conference, Baltimore, MD December 2010 26

Cost Optimization Algorithm l Denote: l Theorem: using we achieve Feldman et. al. , and Winter Simulation Conference, Baltimore, MD December 2010 27

Constraints Satisfaction Algorithm Feldman et. al. Winter Simulation Conference, Baltimore, MD December 2010 28

Constraints Satisfaction Algorithm l Denote: l Theorem: using we achieve , and • • Feldman et. al. Winter Simulation Conference, Baltimore, MD December 2010 29

Constraints Satisfaction Algorithm Feldman et. al. Winter Simulation Conference, Baltimore, MD December 2010 30

Summary Results Feldman et. al. Winter Simulation Conference, Baltimore, MD December 2010 31

Summary Results Feldman et. al. Winter Simulation Conference, Baltimore, MD December 2010 32

Constraint Satisfaction: Delay Threshold with FQR Feldman et. al. Winter Simulation Conference, Baltimore, MD December 2010 33

Constraint Satisfaction: Delay Threshold with FQR l Feasible region and optimal solution l Algorithm solution: N=(91, 60), cost=211 Feldman et. al. Winter Simulation Conference, Baltimore, MD December 2010 34

Constraint Satisfaction: Delay Threshold with FQR l Comparison of Control Schemes FQR control Feldman et. al. Winter Simulation Conference, Baltimore, MD SP control December 2010 35