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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. Contents. Skills Based Routing (SBR) Model

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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 .

contents
Contents
  • Skills Based Routing (SBR) Model
  • SBR Staffing Problem
  • Stochastic Approximation (SA) Solution
  • Numerical Experiments
  • Future Work

Winter Simulation Conference, Baltimore, MD

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 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

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

Routing

Winter Simulation Conference, Baltimore, MD

cost optimization formulation

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

constraints satisfaction formulation

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

stochastic approximation sa

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

sa basic assumptions

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

sbr simulation

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

sbr simulation1

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

cost optimization algorithm
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

constraints satisfaction algorithm
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

numerical study

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

simple example penalizing abandonments

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

simple example objective function

Numerical Experiments

Simple Example: Objective Function

Winter Simulation Conference, Baltimore, MD

simple example solution

Numerical Experiments

Simple Example: Solution
  • Convergence Rate
  • Solution: N=(98,58), 0.5% above optimal

Convergence Point

Winter Simulation Conference, Baltimore, MD

realistic example

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

realistic example1

Numerical Experiments

Realistic Example
  • Arrival Process: Hourly Rates

Winter Simulation Conference, Baltimore, MD

realistic example2

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

realistic example optimization models
Daily SLA

Hourly SLA

Numerical Experiments

Realistic Example: Optimization Models

Winter Simulation Conference, Baltimore, MD

realistic example sla
Daily SLA

Hourly SLA

Numerical Experiments

Realistic Example: SLA

Winter Simulation Conference, Baltimore, MD

realistic example staffing levels
Daily SLA

Staffing cost: 510

Hourly SLA

Staffing cost: 575

Numerical Experiments

Realistic Example: Staffing Levels

Winter Simulation Conference, Baltimore, MD

summary
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

future work
Future Work
  • Incorporating scheduling mechanism
  • Complex models
  • Optimal Routing
  • Enhance algorithms
    • Relax convexity assumption
    • More efficient
  • Convexity Analysis

Winter Simulation Conference, Baltimore, MD

cost optimization algorithm1
Cost Optimization Algorithm

Winter Simulation Conference, Baltimore, MD

cost optimization algorithm2
Cost Optimization Algorithm
  • Denote:
  • Theorem: using , and we achieve

Winter Simulation Conference, Baltimore, MD

constraints satisfaction algorithm1
Constraints Satisfaction Algorithm

Winter Simulation Conference, Baltimore, MD

constraints satisfaction algorithm2
Constraints Satisfaction Algorithm
  • Denote:
  • Theorem: using , andwe achieve

Winter Simulation Conference, Baltimore, MD

constraints satisfaction algorithm3
Constraints Satisfaction Algorithm

Winter Simulation Conference, Baltimore, MD

summary results
Summary Results

Winter Simulation Conference, Baltimore, MD

summary results1
Summary Results

Winter Simulation Conference, Baltimore, MD

constraint satisfaction delay threshold with fqr
Constraint Satisfaction: Delay Threshold with FQR

Winter Simulation Conference, Baltimore, MD

constraint satisfaction delay threshold with fqr1
Constraint Satisfaction: Delay Threshold with FQR
  • Feasible region and optimal solution
  • Algorithm solution: N=(91,60), cost=211

Winter Simulation Conference, Baltimore, MD

constraint satisfaction delay threshold with fqr2
Constraint Satisfaction: Delay Threshold with FQR
  • Comparison of Control Schemes

FQR control

SP control

Winter Simulation Conference, Baltimore, MD

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