WSC 2010, Baltimore, Maryland Avishai Mandelbaum (Technion)

1. 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 .

2. Contents • Skills Based Routing (SBR) Model • SBR Staffing Problem • Stochastic Approximation (SA) Solution • Numerical Experiments • Future Work Winter Simulation Conference, Baltimore, MD

3. 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: SBR Model Service System with SBR – Basic Model Winter Simulation Conference, Baltimore, MD

4. 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 Routing Winter Simulation Conference, Baltimore, MD

5. SBR Staffing Problem Cost-Optimization Formulation • f k(N) – service level penalty functions • Examples: • f k(N) = ckλkPN{abk}–cost rate of abandonments • f k(N) = λkEN[ck(Wk)]– waiting costs Winter Simulation Conference, Baltimore, MD

6. SBR Staffing Problem Constraints-Satisfaction Formulation • 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 Winter Simulation Conference, Baltimore, MD

7. SA Based Solution 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 analytically intractable Winter Simulation Conference, Baltimore, MD

8. SA Based Solution 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) Winter Simulation Conference, Baltimore, MD

9. SA Based Solution 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 Winter Simulation Conference, Baltimore, MD

10. SA Based Solution 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 byFinite Differences Winter Simulation Conference, Baltimore, MD

11. Problem Solution Use Robust SA For simulation, real-valued points are rounded to integers SA Based Solution Cost Optimization Algorithm Winter Simulation Conference, Baltimore, MD

12. Problem Solution 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 Constraints Satisfaction Algorithm Winter Simulation Conference, Baltimore, MD

13. Numerical Experiments 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 Winter Simulation Conference, Baltimore, MD

14. λ2 =100 λ1 =100 θ2=1 θ1=1 µ21=1.5 µ11=1 µ22=2 Numerical Experiments 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 Winter Simulation Conference, Baltimore, MD

15. Numerical Experiments Simple Example: Objective Function Winter Simulation Conference, Baltimore, MD

16. Numerical Experiments Simple Example: Solution • Convergence Rate • Solution: N=(98,58), 0.5% above optimal Convergence Point Winter Simulation Conference, Baltimore, MD

17. Numerical Experiments 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 Winter Simulation Conference, Baltimore, MD

18. Numerical Experiments Realistic Example • Arrival Process: Hourly Rates Winter Simulation Conference, Baltimore, MD

19. Numerical Experiments Realistic Example • Service Distribution (via SEE Stat) Business Brokerage LogN(3.9,4.3) LogN(3.7,3.4) Patience: Exp(mean=7.35min) Exp(mean=19.3min) Winter Simulation Conference, Baltimore, MD

20. Daily SLA Hourly SLA Numerical Experiments Realistic Example: Optimization Models Winter Simulation Conference, Baltimore, MD

21. Daily SLA Hourly SLA Numerical Experiments Realistic Example: SLA Winter Simulation Conference, Baltimore, MD

22. Daily SLA Staffing cost: 510 Hourly SLA Staffing cost: 575 Numerical Experiments Realistic Example: Staffing Levels Winter Simulation Conference, Baltimore, MD

23. 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 Winter Simulation Conference, Baltimore, MD

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

25. Backup

26. Cost Optimization Algorithm Winter Simulation Conference, Baltimore, MD

27. Cost Optimization Algorithm • Denote: • Theorem: using , and we achieve Winter Simulation Conference, Baltimore, MD

28. Constraints Satisfaction Algorithm Winter Simulation Conference, Baltimore, MD

29. Constraints Satisfaction Algorithm • Denote: • Theorem: using , andwe achieve Winter Simulation Conference, Baltimore, MD

30. Constraints Satisfaction Algorithm Winter Simulation Conference, Baltimore, MD

31. Summary Results Winter Simulation Conference, Baltimore, MD

32. Summary Results Winter Simulation Conference, Baltimore, MD

33. Constraint Satisfaction: Delay Threshold with FQR Winter Simulation Conference, Baltimore, MD

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

35. Constraint Satisfaction: Delay Threshold with FQR • Comparison of Control Schemes FQR control SP control Winter Simulation Conference, Baltimore, MD