Optimal Staffing of Systems with Skills-Based-Routing

1. Optimal Staffing of Systems with Skills-Based-Routing Master Defense, February 2nd, 2009 Zohar Feldman Advisor: Prof. Avishai Mandelbaum

2. Contents • Skills-Based-Routing (SBR) Model • The Optimization Problem • Related Work • Optimization Algorithm (Stochastic Approximation) • Experimental Results • Future Work

3. I – set of customer classes J – set of server pools Arrivals for class i: renewal (e.g. Poisson) processes, rate λi Servers in pool j: Nj, statistical identical Service of class i by pool j: DSi,j (Im)patience of class i: DPi Schematic Representation Introduction to SBR Systems

4. Introduction to SBR Systems • 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

5. The Optimization Problem • We consider two optimization problems: • Cost Optimization • Constraints Satisfaction

6. The Optimization Problem • Cost Optimization Problem • f k(N) – service level penalty functions • Examples: • f k(N) = c’kλkPN{abk} – cost of abandonments per time unit • f k(N) = λkEN[c’k(Wk)] – waiting costs

7. The Optimization Problem • Constraints Satisfaction Problem • 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

8. Related Work • Call Centers Review (Gans, Koole, Mandelbaum) • V model (Gurvich, Armony, Mandelbaum) • Inverted-V model (Armony, Mandelbaum) • FQR (Gurvich, Whitt) • Gcµ (Mandelbaum, Stolyar) • Simulation & Cutting Planes (Henderson, Epelman) • Staffing Algorithm (Whitt, Wallace) • ISA (Feldman, Mandelbaum, Massey, Whitt) • Stochastic Approximation (Juditsky, Lan, Nemirovski, Shapiro)

9. Stochastic Approximation (SA) • Uses Monte-Carlo sampling techniques to solve (approximate) • - convex set • ξ – random vector, probability distribution P supported on set Ξ • - convex almost surely

10. Stochastic Approximation – Basic Assumptions • f(x) is analytically intractable • 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)

11. Optimization Algorithms • Let Ω be 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) • Subgradient is approximated by Independent of N

12. Problem Solution Use Robust SA Simulation is used with rounded points Cost Optimization Algorithm

13. Problem Solution There exist a solution with cost C that satisfies the Service Level constraints if”f where Look for the minimal C in a binary search fashion Constraints Satisfaction Algorithm

14. Experimental Results • Goals • Examine algorithm performance • Explore the geometry of the service level functions, validate convexity • Method • Construct SL functions by simulation • Compare algorithm solution to optimal

15. 100 100 1 2 1.5 Cost Optimization: Penalizing Abandonments • N model (I=2,J=2) • λ1= λ2=100 • µ11=1, µ21=1.5, µ22=2 • θ1= θ2=1 • Static Priority: class1 customers prefer pool 1 over pool 2. Pool 2 servers prefer class 1 customers over class2. 1 1

16. Cost Optimization: Penalizing Abandonments • Problem Formulation

17. Cost Optimization: The Objective Function

18. Cost Optimization: Algorithm Evolution

19. Cost Optimization: Convergence Rate

20. Cost Optimization: Penalizing Abandonments • Algorithm Solution: N=(98,57) cost=219 • Optimal Solution: N*=(102,56) cost*=218

21. Realistic Example • Medium-size Call Center (US Bank: SEE lab) • 2 classes of calls • Business • Quick & Reilly • 2 pools of servers • Pool 1- Dedicated to Business • Pool 2 - Serves both

22. Realistic Example • Arrival Rates

23. Realistic Example • Service Distribution (via SEE Stat) Business Quick & Reilly LogN(3.7,3.4) LogN(3.9,4.3)

24. Realistic Example • Patience – survival analysis shows that Exponential distribution fits both classes • Business: Exp(mean=7.35min) • Quick: Exp(mean=19.3min)

25. Hourly SLA Daily SLA Realistic Example: Optimization Models

26. Hourly SLA Daily SLA Realistic Example: SLA

27. Hourly SLA Solution – total cost 575 Daily SLA Solution – total cost 510 Realistic Example: Staffing

28. Future Work • Incorporating scheduling mechanism • Complex models • Optimal Routing • Enhance algorithms • Relax convexity assumption • More efficient

29. Cost Optimization Algorithm • Initialization i ← 0; Choose x0from X • Step 1 Generate Fk(xi,ξi) and Gk(xi,ξi) using simulation • Step 2 xi+1←ΠX(xi- γGk(xi,ξi)) • Step 3 i ← i+1 • Step 4 If i < J go to Step 1. • Step 5

30. Cost Optimization Algorithm • Denote: • Theorem: using , and we achieve

31. CS Algorithm – Formal Procedure • Initialization dC ←Cmax , x←xmax/2, x* ←xmax • Step 1 IfdC<δreturn the solutionx*, dC←dC/2 • Step 2 If Feasible(x)=true, C ← C-dC, x* ← x, , go to Step 1 • Step 3 x←MirrorSaddleSA(C) • Step 4 If Feasible(x)=true, C ← C-dC, x* ← x, , go to Step 1 • Step 5 C ← C+dC, go to Step 1

32. CS Algorithm • Denote: • Theorem: using , andwe achieve

33. 100 100 1 2 1.5 Constraint Satisfaction: Delay Threshold with FQR • N model (I=2,J=2) • λ1= λ2=100 • µ11=1, µ21=1.5, µ22=2 • T1=0.1, α1=0.2; T2=0.2, α2=0.2 • FQR: pool 2 admits to service customer from class i which maximize Qi - pi∑Qj, p=(1/3,2/3); Class 1 will go to pool j which maximize Ij - qj ∑Ik q=(1/2,1/2)

34. Constraint Satisfaction: Delay Threshold with FQR • Problem Formulation

35. Constraint Satisfaction: Delay Threshold with FQR

36. Constraint Satisfaction: Delay Threshold with FQR • Feasible region and optimal solution • Algorithm solution: N=(91,60), cost=211

37. Constraint Satisfaction: Delay Threshold with FQR • Comparison of Control Schemes FQR control SP control