Spring 200415.764 Theory of Operations

Management 1

A Method for Staffing Large Call Centers Based on

Stochastic Fluid Models

Written by:J. Michael Harrison

Assaf Zeevi

Presentation by:Stephen Shum

This summary presentation is based on: Harrison, J. Michael, and Assaf Zeevi. "A Method for Staffing Large Call Centers Based on Stochastic Fluid Models." Stanford University, 2003.

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Outline

Motivation and ProblemProposed ModelSupporting LogicNumerical ExamplesComments and Discussion

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MotivationLarge-Scale Call CentersExample: AT&T, VerizonLocal Phone, DSL, Cell Phone, etc.Billing, Technical Support, Termination, etc.Different Agents/Servers for different types of callsOther examples: Banks, Insurance Companies, Dell, IBM and many others

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Model

m customer classesr agent poolsbi agents in pool iService time depends on customer class and service pool

1

3

m

21

2

r

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Tasks

1. Staff Scheduling/ Capacity Planning*Number and types of agent pools# of agents in each pools

2. Dynamic RoutingCustomer arrival -> serve/bufferService Completion -> server idle/another customer

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AssumptionsScale is large enough to treat variables as continuous variablesThe capacity is determined in advance and cannot be revisedDemand is doubly stochastic Poisson All other uncertainty and variability are negligible compared to demand, i.e., assume everything else constant

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Doubly Stochastic Poisson Model

Ttttt m ≤≤ΛΛ=Λ 0)),(),...,(()( 1

Givenconditional distribution of arrivals in different

classes immediately after t are independent Poisson processes

),...,()( 1 mt λλ=Λ

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ObjectiveCost ck per agent in pool kPenalty pi per abandonment call per customer for class IWant to select bk to minimize

Problem:Difficult to find qi the total expected abandonment call per

class

∑ ∑= =

+m

k

r

iiikk qpbc

1 1

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Notations

(Please see the Harrison and Zeevi paper for notation explanations.)

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Proposed MethodGiven

*( , ) minimize ( )subject to: , , 0

m rb

b p RxRx Ax b x

λ

π λ π λλ

+ +∈ ∈

= = ⋅ −≤ ≤ ≥

Objective: minimize *

0

( ( ), )T

c b t b dtπ⎧ ⎫⎪ ⎪⋅ + Ε Λ⎨ ⎬⎪ ⎪⎩ ⎭∫

{ }0

1( ) : ( ) for T

mF t dtT

λ λ λ += Ρ Λ ≤ ∈∫Let

*( , ) ( ) : ( )mR

c b T b dF bπ λ λ φ+

⋅ + =∫minimize

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Proposed Method

)(bφ is a convex function on +rR

Proposition

Convex Optimization problemGradient-descent method combined with Monte Carlo simulation to find the numerical solution

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Supporting Logic

(See section 3 on page 9 of the Harrison and Zeevi paper)

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Numerical ExamplesHomogenous System

(See Figure 3, Figure 4, and Table 1 on pages 15-17 of the Harrison and Zeevi paper)

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Numerical ExamplesStylized Demand

(See Figures 5-6, and Tables 2-6 on pages 18-23 of the Harrison and Zeevi paper)

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Comments and CritiquesWell written paper with interesting under-research topicProposed a model for skill-based routingMajor Criticism:

Assume no variability in service timeNeglect the dependence between optimal policy and routing method -> most routing methods may only get a cost far above optimal given the theoretical policyWould be more interesting if they provide some interesting results using their methods