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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 Staffing of Systems with Skills-Based-Routing

  • 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 Staffing of Systems with Skills-Based-Routing – 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

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Routing

Arrival Control Staffing of Systems with Skills-Based-Routing: 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

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Cost optimization formulation

SBR Staffing Problem Staffing of Systems with Skills-Based-Routing

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 Staffing of Systems with Skills-Based-Routing

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

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Stochastic approximation sa

SA Based Solution Staffing of Systems with Skills-Based-Routing

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

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Sa basic assumptions

SA Based Solution Staffing of Systems with Skills-Based-Routing

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 Staffing of Systems with Skills-Based-Routing

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

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Sbr simulation1

SA Based Solution Staffing of Systems with Skills-Based-Routing

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

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Cost optimization algorithm

Problem Staffing of Systems with Skills-Based-Routing

Solution

Use Robust SA

For simulation, real-valued points are rounded to integers

SA Based Solution

Cost Optimization Algorithm

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Constraints satisfaction algorithm

Problem Staffing of Systems with Skills-Based-Routing

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

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Numerical study

Numerical Experiments Staffing of Systems with Skills-Based-Routing

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

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Simple example penalizing abandonments

λ Staffing of Systems with Skills-Based-Routing2 =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

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Simple example objective function

Numerical Experiments Staffing of Systems with Skills-Based-Routing

Simple Example: Objective Function

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Simple example solution

Numerical Experiments Staffing of Systems with Skills-Based-Routing

Simple Example: Solution

  • Convergence Rate

  • Solution: N=(98,58), 0.5% above optimal

Convergence Point

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Realistic example

Numerical Experiments Staffing of Systems with Skills-Based-Routing

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

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Realistic example1

Numerical Experiments Staffing of Systems with Skills-Based-Routing

Realistic Example

  • Arrival Process: Hourly Rates

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Realistic example2

Numerical Experiments Staffing of Systems with Skills-Based-Routing

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)

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Realistic example optimization models

Daily SLA Staffing of Systems with Skills-Based-Routing

Hourly SLA

Numerical Experiments

Realistic Example: Optimization Models

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Realistic example sla

Daily SLA Staffing of Systems with Skills-Based-Routing

Hourly SLA

Numerical Experiments

Realistic Example: SLA

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Realistic example staffing levels

Daily SLA Staffing of Systems with Skills-Based-Routing

Staffing cost: 510

Hourly SLA

Staffing cost: 575

Numerical Experiments

Realistic Example: Staffing Levels

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Summary
Summary Staffing of Systems with Skills-Based-Routing

  • 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 Staffing of Systems with Skills-Based-Routing

  • Incorporating scheduling mechanism

  • Complex models

  • Optimal Routing

  • Enhance algorithms

    • Relax convexity assumption

    • More efficient

  • Convexity Analysis

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Backup

Backup Staffing of Systems with Skills-Based-Routing


Cost optimization algorithm1
Cost Optimization Algorithm Staffing of Systems with Skills-Based-Routing

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Cost optimization algorithm2
Cost Optimization Algorithm Staffing of Systems with Skills-Based-Routing

  • Denote:

  • Theorem: using , and we achieve

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Constraints satisfaction algorithm1
Constraints Satisfaction Algorithm Staffing of Systems with Skills-Based-Routing

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Constraints satisfaction algorithm2
Constraints Satisfaction Algorithm Staffing of Systems with Skills-Based-Routing

  • Denote:

  • Theorem: using , andwe achieve

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Constraints satisfaction algorithm3
Constraints Satisfaction Algorithm Staffing of Systems with Skills-Based-Routing

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Summary results
Summary Results Staffing of Systems with Skills-Based-Routing

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Summary results1
Summary Results Staffing of Systems with Skills-Based-Routing

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Constraint satisfaction delay threshold with fqr
Constraint Satisfaction: Delay Threshold with FQR Staffing of Systems with Skills-Based-Routing

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Constraint satisfaction delay threshold with fqr1
Constraint Satisfaction: Delay Threshold with FQR Staffing of Systems with Skills-Based-Routing

  • Feasible region and optimal solution

  • Algorithm solution: N=(91,60), cost=211

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Constraint satisfaction delay threshold with fqr2
Constraint Satisfaction: Delay Threshold with FQR Staffing of Systems with Skills-Based-Routing

  • Comparison of Control Schemes

FQR control

SP control

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