Optimal Staffing of Systems with Skills-Based-Routing

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Optimal Staffing of Systems with
Skills-Based-Routing

• Temporary Copy
• Do not circulate

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Optimal Staffing of Systems with
Skills-Based-Routing

• OR seminar, July 21, 2008
• Zohar Feldman
• Advisor Prof. Avishai Mandelbaum

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Contents

• Skills-Based-Routing (SBR) Models
• The Optimization Problem
• Related Work
• Optimization Algorithm (Stochastic Approximation)
• Experimental Results
• Future Work

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Introduction to SBR Systems

• I set of customer classes
• J set of server pools
• Arrivals for class i renewal (e.g. Poisson)
  processes, rate ?i
• Servers in pool j Nj, statistical identical
• Service of class i by pool j DSi,j
• (Im)patience of class i DPi

• Schematic Representation

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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

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The Optimization Problem

• We consider two optimization problems
• Cost Optimization
• Constraints Satisfaction

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The Optimization Problem

• Cost Optimization Problem
• f k(N) service level penalty functions
• Examples
• f k(N) ck?kPNabk cost of abandonments per
  time unit
• f k(N) ?kENck(Wk) waiting costs

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The Optimization Problem

• Constraints Satisfaction Problem
• f k(N) service level objective
• Examples
• f k(N) PNWkgtTk probability of waiting more
  than T time units
• f k(N) ENWk expected wait

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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)

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Simulation Approach

• Need to evaluate
• Generate samples ?1, ?2,
• Estimate

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Stochastic Approximation (SA)

• Uses Monte-Carlo sampling techniques to solve
  (approximate)
• - convex set
• ? random vector, probability distribution P
  supported on set ?
• - convex almost surely

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Stochastic Approximation Basic Assumptions

• f(x) is analytically intractable
• There is a sampling mechanism that can be used to
  generate iid samples from ?
• There is an Oracle at our disposal that returns
  for any x and ?
• The value F(x,?)
• A stochastic subgradient G(x,?)

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SA for SBR

• f(x) analytically intractable
• Sampling mechanism for generating samples ?1,
  ?2,
• Oracle returns F(x, ?i)
• Oracle returns G(x, ?i)
• F convex a.s. !

• f(N) analytically intractable
• SBR simulation generates sample paths ?1, ?2,
• Simulation calculates performance measures
  F(N,?i)
• G(N,?i) - finite differences
• F convex a.s. ?

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Stochastic Approximation - Algorithms

• Basic SA
• ? depends on strong convexity constant
• Objective convergence rate - O(j-1). (error of
  solution O(j-0.5))

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Stochastic Approximation - Algorithms

• Robust SA
• ? does not depend on any parameter (robust)
• Objective convergence rate - O(J-0.5)

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Stochastic Approximation - Algorithms

• Mirror Descent SA
• In order to guarantee accuracy e with confidence
  level d, one should use J of order O(e-2)
  dependent on d logarithmically

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Stochastic Approximation - Algorithms

• Minimax Problemswhich is the same as solving
  the Saddle Point problem

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Stochastic Approximation - Algorithms

• Mirror SA for Saddle Point Problem
• In order to guarantee accuracy e with confidence
  level d, one should use J of order O(e-2)
  dependent on d logarithmically

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Optimization Algorithms

• Let ? 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,?) is the
  number of Delayed customers A(?) is the number
  of Arrivals
• Use simulation to generate samples ? and
  calculate F(N,?)
• Subgradient is approximated by

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Cost Optimization Algorithm

• Initialization i ? 0 Choose x0 from X
• Step 1 Generate Fk(xi,?i) and Gk(xi,?i) using
  simulation
• Step 2 xi1??X(xi- ?Gk(xi,?i))
• Step 3 i ? i1
• Step 4 If i lt J go to Step 1.
• Step 5

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Cost Optimization Algorithm

• Denote
• Theorem using , and we
  achieve

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Constraints Satisfaction Algorithm

• Basic concept
• There exist a solution with cost C that satisfies
  the Service Level constraints iff where
• Look for the minimal C in a binary search fashion

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CS Algorithm Formal Procedure

• Initialization dC ?Cmax , x?xmax/2, x ?xmax
• Step 1 If dCltd 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 ? CdC, go to Step 1

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CS Algorithm MirrorSaddle Subprocedure

• Sub-procedure MirrorSaddleSA

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CS Algorithm MirrorSaddle Subprocedure

• Sub-procedure MirrorSaddleSA
• Mapping Function

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CS Algorithm Feasible Subprocedure

• Sub-procedure Feasible
• lbk(n), ubk(n) confidence interval for fk
  based on n samples
• If ubk(n)akd for all k1,,K return true
• If lbk(n)gtakd for some k1,,K return false
• Generate sample, n ?nbatch, calculate confidence
  interval lbk(n), ubk(n). Go to 1.

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CS Algorithm

• Denote
• Theorem using , andwe achieve

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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

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Cost Optimization 1 Penalizing Wait

• N model (I2,J2)
• ?1 ?2100
• µ111, µ211.5, µ222
• Static Priority class1 customers prefer pool 1
  over pool 2. Pool 2 servers prefer class 1
  customers over class2.

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Cost Optimization 1 Penalizing Wait

• Problem Formulation

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Cost Optimization 1 The Objective Function
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Cost Optimization 1 Penalizing Wait

• Algorithm solution N(88,47), cost190
• Optimal solution N(80,54), cost188

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Cost Optimization 2 Penalizing Abandonments

• N model (I2,J2)
• ?1 ?2100
• µ111, µ211.5, µ222
• ?1 ?21
• Static Priority class1 customers prefer pool 1
  over pool 2. Pool 2 servers prefer class 1
  customers over class2.

1
1
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Cost Optimization 2 Penalizing Abandonments

• Problem Formulation

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Cost Optimization 2 The Objective Function
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Cost Optimization 2 Algorithm Evolution
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Cost Optimization 2 Convergence Rate
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Cost Optimization 2 Penalizing Abandonments

• Algorithm Solution N(98,57) cost219
• Optimal Solution N(102,56) cost218

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Constraint Satisfaction 1 Delay Threshold with
FQR

• N model (I2,J2)
• ?1 ?2100
• µ111, µ211.5, µ222
• T10.1, a10.2 T20.2, a20.2
• 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)

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Constraint Satisfaction 1 Delay Threshold with
FQR

• Problem Formulation

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Constraint Satisfaction 1 Delay Threshold with
FQR
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Constraint Satisfaction 1 Delay Threshold with
FQR

• Feasible region and optimal solution
• Algorithm solution N(91,60), cost211

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Constraint Satisfaction 1 Delay Threshold with
FQR

• Comparison of Control Schemes

FQR control
SP control
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Constraints Satisfaction 2 Time-Varying Model

• N model
• ?1(t) ?2(t)1000200sin(t)
• µ1110, µ2115, µ2220
• Static Priority class1 customers prefer pool 1
  over pool 2. Pool 2 servers prefer class 1
  customers over class2.

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Constraints Satisfaction 2 Time-Varying Model

• Problem Formulation

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Constraints Satisfaction 2 Time-Varying Model

• Arrivals and Staffing

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Constraints Satisfaction 2 Time-Varying Model

• Performance Measures

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Constraints Satisfaction 2 Time-Varying Model

• System Statistics

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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

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Realistic Example

• Arrival Rates

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Realistic Example

• Service Distribution (via SEE Stat)

Business
Quick Reilly
LogN(3.7,3.4)
LogN(3.9,4.3)
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Realistic Example

• Patience survival analysis shows that
  Exponential distribution fits both classes
• Business Exp(mean7.35min)
• Quick Exp(mean19.3min)

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Realistic Example Hourly SLA

• Problem Formulation

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Realistic Example Hourly SLA

• Solution total cost 575

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Realistic Example Hourly SLA

• SLA

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Realistic Example Daily SLA

• Problem Formulation

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Realistic Example Daily SLA

• Solution total cost 510 (11 reduction)

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Realistic Example Daily SLA

• SLA

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Future Work

• Incorporating scheduling mechanism
• Complex models
• Enhance algorithms
• Independent of convexity assumptions
• Work faster