1 Optimal Staffing of Systems with Skills- Based-Routing Master Defense, February 2 nd, 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 Introduction to SBR Systems I – set of customer classes J – set of server pools Arrivals for class i : renewal (e.g. Poisson) processes, rate λ i Servers in pool j : N j, statistical identical Service of class i by pool j: D S i,j (Im)patience of class i: D P i Schematic Representation

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’ k λ k P N {ab k } – cost of abandonments per time unit f k (N) = λ k E N [c’ k (W k )] – waiting costs

7 The Optimization Problem Constraints Satisfaction Problem 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

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 Cost Optimization Algorithm ProblemSolution Use Robust SA Simulation is used with rounded points

13 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 in a binary search fashion

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 Cost Optimization: Penalizing Abandonments N model ( I=2,J=2 ) λ 1 = λ 2 =100 µ 11 =1, µ 21 =1.5, µ 22 =2 θ 1 = θ 2 =1 Static Priority: class1 customers prefer pool 1 over pool 2. Pool 2 servers prefer class 1 customers over class

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) BusinessQuick & 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 Realistic Example: Optimization Models Hourly SLA Daily SLA

26 Realistic Example: SLA Hourly SLA Daily SLA

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

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 x 0 from X – Step 1 Generate F k (x i,ξ i ) and G k (x i,ξ i ) using simulation – Step 2 x i+1 ←Π X (x i - γG k (x i,ξ i )) – 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 ←C max, x←x max /2, x* ←x max Step 1 If dC<δ return the solution x*, 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, and we achieve

33 Constraint Satisfaction: Delay Threshold with FQR N model ( I=2,J=2 ) λ 1 = λ 2 =100 µ 11 =1, µ 21 =1.5, µ 22 =2 T 1 =0.1, α 1 =0.2; T 2 =0.2, α 2 =0.2 FQR: pool 2 admits to service customer from class i which maximize Q i - p i ∑Q j, p=(1/3,2/3); Class 1 will go to pool j which maximize I j - q j ∑ I k 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 controlSP control