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.

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD2 Contents Skills Based Routing (SBR) Model SBR Staffing Problem Stochastic Approximation (SA) Solution Numerical Experiments Future Work

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD3 Service System with SBR – Basic Model I – set of customer classes J – set of server pools Arrivals for class i : renewal (e.g. Poisson), rate λ i Servers in pool j : N j, iid Service of class i by pool j: (Im)patience of class i: SBR Model

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD4 Routing Arrival Control: upon customer arrival, which of the available servers, if any, should be assigned to serve the arriving customer Idleness Control: upon service completion, which of the waiting customers, if any, should be admitted to service SBR Model ??? ???

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD5 Cost-Optimization Formulation f k (N) – service level penalty functions Examples: f k (N) = c k λ k P N {ab k } – cost rate of abandonments f k (N) = λ k E N [c k (W k )] – waiting costs SBR Staffing Problem

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD6 Constraints-Satisfaction Formulation f k (N) – service level objective Examples f k (N) = P N {W k >T k } – probability of waiting more than T time units f k (N) = E N [W k ] – expected wait SBR Staffing Problem

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD7 Stochastic Approximation (SA) Uses Monte-Carlo sampling techniques to solve (approximate) - convex set ξ – random vector (probability distribution P) supported on set Ξ - almost surely convex SA Based Solution analytically intractable

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD8 SA Basic Assumptions 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) SA Based Solution

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD9 SBR Simulation 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.) Use random streams to enable common number generation SA Based Solution

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD10 SBR Simulation 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 SA Based Solution

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD11 Cost Optimization Algorithm ProblemSolution Use Robust SA For simulation, real- valued points are rounded to integers SA Based Solution

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD12 Constraints Satisfaction Algorithm ProblemSolution There exist a solution with cost C that satisfies the Service Level constraints if”f where Look for the minimal C via binary search SA Based Solution

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD13 Numerical Study Goal Examine algorithms performance Explore convexity and its affect on performance Method 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 Numerical Experiments

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD14 Simple Example: Penalizing Abandonments N-model ( I=2, J=2 ) Control: Static Priority Class 1: pool 1, pool 2 Pool 2: class 1, class 2 Optimization problem Numerical Experiments µ 11 =1 λ 1 =100 θ1= 1θ1= 1 λ 2 =100 µ 22 =2 µ 21 =1.5 θ2= 1θ2= 1

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD15 Simple Example: Objective Function Numerical Experiments

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD16 Simple Example: Solution Convergence Rate Solution: N=(98,58), 0.5% above optimal Numerical Experiments Convergence Point

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD17 Realistic Example 100’s-agents Call Center (US Bank: SEE Lab – open data source) 2 classes of calls Business Quick & Reilly (Brokerage) 2 pools of servers Pool 1- Dedicated to Business Pool 2 - Serves both Numerical Experiments

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD18 Realistic Example Arrival Process: Hourly Rates Numerical Experiments

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD19 Realistic Example Service Distribution (via SEE Stat) Numerical Experiments BusinessBrokerage LogN(3.7,3.4) LogN(3.9,4.3) Exp(mean=7.35min)Exp(mean=19.3min) Patience:

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD20 Realistic Example: Optimization Models Daily SLA Hourly SLA Numerical Experiments

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD21 Realistic Example: SLA Daily SLA Hourly SLA Numerical Experiments

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD22 Realistic Example: Staffing Levels Daily SLA Staffing cost: 510 Hourly SLA Staffing cost: 575 Numerical Experiments

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD23 Summary We developed simulation-based algorithms for optimizing staffing of systems with skills- based-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.December 2010Winter Simulation Conference, Baltimore, MD24 Future Work Incorporating scheduling mechanism Complex models Optimal Routing Enhance algorithms Relax convexity assumption More efficient Convexity Analysis

Backup

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD26 Cost Optimization Algorithm

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD27 Cost Optimization Algorithm Denote: Theorem: using, and we achieve

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD28 Constraints Satisfaction Algorithm

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD29 Constraints Satisfaction Algorithm Denote: Theorem: using, and we achieve

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD30 Constraints Satisfaction Algorithm

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD31 Summary Results

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD32 Summary Results

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD33 Constraint Satisfaction: Delay Threshold with FQR

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD34 Constraint Satisfaction: Delay Threshold with FQR Feasible region and optimal solution Algorithm solution: N=(91,60), cost=211

Feldman et. al.December 2010Winter Simulation Conference, Baltimore, MD35 Constraint Satisfaction: Delay Threshold with FQR Comparison of Control Schemes FQR controlSP control