Parametric Forecasting and Stochastic Programming Models

Parametric Forecasting and StochasticProgramming Models for Call-Center Workforce

SchedulingWhen OM meets Statistics ...

Haipeng Shen

Statistics & Operations Research Innovation & Information ManagementUniversity of North Carolina School of Business

University of Hong Kong

Partially supported by NSF, NIH, The Xerox Foundation, Stanley Ho Alumni Challenge Fund

June 12, 2015Haipeng Shen (UNC/HKU) Arrival Rate Uncertainty Mostly OM, Beijing 1 / 39

Research ProgramsStatistical Methodology - high dimensional inference, dimensionreduction

I US NSF Grant DMS-0606577, DMS-1106912, DMS-1407655

Business Analytics - call centers, mobile marketing

I US NSF Grant CMMI-0800575, The Xerox Foundation UAC Award, StanleyHo Challenge Fund

Healthcare Analytics - neurology, cardiology

I NIH Challenge Grant 1 RC1 DA029425-01, NSF of ChinaI In collaboration with Beijing Tiantan Hospital

F China National Stroke Registry (CNSR I), Golden Bridge (CNSR II),CNSR III (recruiting)

F Big data - network of hospitals, digital view of patient visit/healthcareenvironment

Haipeng Shen (UNC/HKU) Arrival Rate Uncertainty Mostly OM, Beijing 2 / 39

Call center industry

Economy - dominated by service sector

Call center: major communication channelCall-Center Environment: Service Network

10

Other service systems: healthcare delivery systems, ...


Queueing model for a single call center

Tutorial, background...

Queueing model associated with asingle location:

retrials

arrivals

abandon

queue

busy

lost calls

retrials

lost calls returns

N = 3 CSR-servers

5 = (k – N) places in queue

w = 5 work stations

k = 8 trunk lines (not visible)

Call-center hardware Queueing model parameters

4

Gans, Koole and Mandelbaum (2003)


The Erlang-C Model: M/M/N +∞Review, basic model...

Performance estimate uses M/M/N/∞ model:

x xx

xxx

x• no blocking, abandonment, or retrials

• fixed arrival and service rates

• exponential interarrival and service times

• measures of stationary performance

10

No blocking, No abandonment, No retrials

Fixed arrival rate λi and service rate µi for time period i

Exponential inter-arrival and service times


Call-by-call data enable investigating ...

Are the Erlang-C assumptions valid in real call centers?

Poisson arrivals?

Exponential service durations?

No abandonment?

Brown, Gans, Mandelbaum, Sakov, Shen, Zeltyn and Zhao (2005)

A small Israeli banking call center

580 citations (Google Scholar)


Poisson arrivals?Consider short time intervals such as quarter hours

Yes, but rates are inhomogeneous and random

Hence, a doubly stochastic Poisson process, or a Cox process

Kim and Whitt (13, 14a, 14b, 15, 15+) - new tests, callcenters/hospitals

Dependence among the rates:

Interday (day-to-day) dependence: today/tomorrow, weekly, . . .

Intraday (within-day) dependence: morning/afternoon/night, . . .

Inter call type dependence: feed off, cross training, . . .

Such dependence is essential for interday rate forecasting andintraday updating. (more later)


Exponential service times?Beyond Averages (+ The Human Factor)

Histogram of Service Times in an Israeli Call Center

January-October November-December

Beyond Data Averages Short Service Times

AVG: 200 STD: 249

AVG: 185 STD: 238

7.2 % ? Jan – Oct:

Log-Normal AVG: 200 STD: 249

Nov – Dec:

27

Beyond Data Averages Short Service Times

AVG: 200 STD: 249

AVG: 185 STD: 238

7.2 % ? Jan – Oct:

Log-Normal AVG: 200 STD: 249

Nov – Dec:

27

I 7.2% Short-Services: Agents’ “Abandon" (improve bonus, rest)I Distributions, not only Averages, must be measured (seconds).I Lognormal service times prevalent in call centers (Why?)

8

Quick Hang: Agents “Abandon" (incentive, rest)Distributions, not just Averages, must be measured

I Service times are Lognormal


How about abandonment? Patience?

0 100 200 300 400

0.00

10.

002

0.00

30.

004

0.00

50.

006

Regular CustomersPriority Customers

Hazard Rate: Empirical (Im)Patience

x-axis: waiting time in queue

y-axis: instantaneous probability of abandonment, (or hazard rate)

Probability of abandonment peaks at times of announcement

VIP are more patient ???


Time-varying perspective ...

3 30 60 90 120150180210240270300

79

1113

1517

190

2

4

6

8x 10

−3

Waiting Time (Seconds)

ADMM Hazard Surface for Customer Patience

Time−of−Day (Hours)

Haz

ard

Rat

e

Li, Huang, and Shen (2015)Haipeng Shen (UNC/HKU) Arrival Rate Uncertainty Mostly OM, Beijing 10 / 39

“Standard” model for call center capacity planning

1 Forecast offered load (e.g., by 1/2-hour)

{Rt = λt/µt : t = 1, . . . ,T}

where λt : arrival rate, µt : service rate.

2 Find minimum numbers of agents to satisfy QoS constraint

st = min{s | P{Abandonment} ≤ α}

3 Find minimum cost assignment of agents to schedules

min{cy |Ay ≥ s; y ≥ 0; y integer}

where A: 0-1 schedule matrix, y : # of agents for each schedule, c: schedule cost.


Outline

1 Arrival Rate Uncertainty

2 Forecasting Model and Stochastic Scheduling

3 Forecast Updating and Scheduling Recourse

4 Real Data Application


Outline






The arrival rate is not known with certainty'

&

$

%

Figure 1: Plot of # of Arrivals during 2 12 -min intervals in 2002

Time

# A

rriv

als

10 15 20

010

020

030

040

0

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

MondayTuesdayWednesdayThursdayFridaySaturdaySunday

3

Bank with a network of 4 call centers in Northeast US

300K calls/day, 60K/day seeking agents, 1K agents in peak hours


Two arrival streams

Introduction Single Queue Multiple Queues

Multiple Arrival Streams

Israel telecom company – Majority arrival streams: Privatecustomers(30%), Business customers(18%)

Two queues are strongly correlated.

10 15 20

510

1520

Private

Time of a day

Volum

e

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

SunMonTueWedThu

10 15 20

05

1015

Business

Time of a day

Volum

e

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

SunMonTueWedThu

850 900 950 1000 1050

560

580

600

620

640

660

680

Scatter Plot of Daily Totals

Private

Bus

ines

s

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

Han Ye Call Center: Forecast and Management 9 / 16Two major arrival streams: Private (30%), Business (18%)

215 days (Mon-Thu) between 06/19/2004 and 04/14/2005


Single-stream arrival rate uncertainty

OM, operational performanceI Whitt (99), Chen and Henderson (01), Jongbloed and Koole (01),

Harrison and Zeevi (05), Whitt (06), Bassamboo, Harrison andZeevi (05, 06, 09), Steckley et al. (09), Robbins et al. (10), Robbinsand Harrison (10), Bertsimas and Doan (10), Mehrotra et al. (10),Zan et al. (2013), Ding and Koole (2014), etc.

Statistics, forecasting performanceI Avramidis et al. (04), Brown et al. (05), Weinberg et al. (07), Taylor

(07), Shen and Huang (08a, 08b), Aldor-Noiman et al. (09),Matteson et al. (11), Taylor (12), Oreshkin et al. (2014), etc.

OM + Statistics, both operational and forecastingI Gans, Shen, Zhou, Genesys (15): forecasting/updating +

stochastic programming with recourse


Our goal

Develop distributional forecasts for arrival ratesI update given additional information

Perform stochastic scheduling using the distributional forecastsI recourse actions after forecast updating

I adjust staffing assignmentsF send agents home early ... → reduce costF call in part-time agents ... → better achieve QoS measure

Test the approach in large-scale real systemsI computational speed


Outline






Forecasting Model

Arrival counts: Nd ,t ∼ Poisson(Λd ,t )

I Xd,t ≡√

Nd,t + 1/4 ∼ N(√

Λd,t , σ20)

I days: d = 1,2, . . . ,D

I intervals: t = 1,2, . . . ,T

I day-of-week index: wd

The Model:Xd ,t =

√Λd ,t + εd ,t , εd ,t

i.i.d.∼ N(0, σ20),√

Λd ,t ≡ Θd ,t = ud fwd ,t , fwd ,t ≥ 0,T∑

t=1fwd ,t = 1,

ud − αwd = b (ud−1 − αwd−1) + zd , zdi.i.d.∼ N(0, σ2)


Distributional arrival-rate forecasts

500

1000

1500

Time

Rat

e

7 9 11 13 15 17 19 21 23

actual volumeforecast


Stochastic program for scheduling agentsDistribution of the Λt ’s determined from the forecast, t = 1, . . . ,T

α∗: upper bound on expected abandonment rate

y : # of agents on each of the possible schedules, with cost c

aty : # of agents working in period t

With distributions for Λt ’s, solve the stochastic program

min {cy}s.t.

T∑t=1

EΛt [g(Λt ,aty)] ≤ α∗T∑

t=1

E [Λt ]

y ≥ 0; y integer,

where g(Λt ,aty) is abandonment count in period t


Gaussian quadrature vs. sampling: daily staffing costEuropean banking center, 176 weekdays in 2007, last 76 days forout-of-sample testingT = 26 half-hour intervalsConsider 1, 4, 9, 16, 25, 49, 100, 225, 400 scenarios


Gaussian quadrature vs. sampling: abandonment rate

Target abandonment rate: α = 3%


Outline






Night-before forecasts can sometimes be off

500

1000

1500

Time

Rat

e

7 9 11 13 15 17 19 21 23


Setup for intra-day forecast updates

History: Day 1 to Day D

Forecasting window: h day ahead

Forecasting distribution of uD,D+h:

N(µD,D+h, σ2D,D+h)

Perform update after interval T0 on Day D + h

Additional data observed:

I whole day data - xD+1, . . . ,xD+h−1

I early part of day D + h, t = 1, . . . ,T0, - xD+he

Derive an updated distribution for uD,D+h


Intra-day forecast updates

A sequence of h one-day-ahead forecast updates + AR(1) innovation

Start with uD,D+1

With xD+1, obtain uD+1,D+1

Given AR(1) innovation, obtain uD+1,D+2

With xD+2, obtain uD+2,D+2

. . .

Given AR(1) innovation, obtain uD+h−1,D+h

With xD+he , obtain uD+he,D+h


A key result

For day d , ud−1,d ∼ N(µd−1,d , σ2d−1,d )

We observe {Xd ,t |t = 1, . . . ,T0} on day d

Define

Ad ,T0 =

T0∑t=1

fwd ,tXd ,t and νd ,T0 =

T0∑t=1

f 2wd ,t

Then, the updated distribution has mean and variance

µdT0 ,d =σ2

d−1,d Ad ,T0 + σ20 µd−1,d

σ2d−1,dνd ,T0 + σ2

0,

σ2dT0 ,d =

σ20 σ

2d−1,d

σ2d−1,dνd ,T0 + σ2

0


Forecast updates significantly reduce error anduncertainty

500

1000

1500

Time

Rat

e

7 9 11 13 15 17 19 21 23

actual volumeoriginal forecast12:00PM update


Stochastic programming with recourse - Stage 1

Solve the same stochastic program

min {cy}s.t.

T∑t=1

EΛt [g(Λt ,aty)] ≤ α∗T∑

t=1

E [Λt ] t = 1, . . . ,T

y ≥ 0; y integer

Calculate

αl =

∑Tt=T0+1 EΛt [g(Λt ,aty)]∑T

t=T0+1 E [Λt ],

expected abandonment rate over late part of the planning horizon


Stochastic programming with recourse - Stage 2

Updated forecast Λ∗t

Adjust staffing assignmentsI update original staffing, y , to updated vector zI cost of change, d(y − z)

Solve stochastic program with recourse

min {cy + d(y − z)}s.t.

T∑t=T0+1

EΛ∗t

[g(Λ∗t ,atz)] ≤ αl

T∑t=T0+1

E [Λ∗t ]

z ≥ 0; z integer


Recourse program that uses 2-stage forecast

Idea: account for recourse actions in initial schedule

Example:I it costs little to send agents home after T0

I it costs a lot to increase staffing after T0

I then, should initially staff high and send people home, if necessary

In two-stage program:I 1st-stage periods as before: initial staffing y fixed across scenariosI 2nd-stage periods more complex: for each initial scenario,

2nd-stage action z varies


Recourse program that uses 2-stage forecast

Initial staffing vector y , and random update vector z

Stochastic program with recourse

min{

cy + EΛ1,...,ΛT0[d(y − z)]

}s.t.T0∑

t=1

EΛt [g(Λt ,aty)]

+T∑

t=T0+1

EΛ1,...,ΛT0

[EΛt |Λ1,...,ΛT0

[g(Λt ,atz)]]≤ α∗

T∑t=1

E [Λt ]

y , z ≥ 0; y , z integer


Outline






We test nine scheduling schemes

Three schemes with no updatingI 1 scenario = SP1I 4 scenarios = SP4I 100 scenarios = SP100

Three schemes with an afternoon update of the original scheduleI 1 scenario = UP1I 4 scenarios = UP4I 100 scenarios = UP100

Three schemes that update an original schedule with recourseI 1 scenario = RP1I 4 scenarios = RP4I 100 scenarios = RP100


Testing the value of the scheduling schemes

1 Preliminary forecast using previous n days of data

2 Solve 6 scheduling problems based on initial forecastI SP1, SP4, and SP100I 1st phase of RP1, RP4, and RP100

3 Update forecast based on 1st part of day

4 Update solutions based on revised forecastI SP1⇒ UP1, SP4⇒ UP4, and SP100⇒ UP100I 2nd phase of RP1, RP4, and RP100

5 Simulate using schedules and actual arrival counts


1st set of empirical testsA network of four large retail-banking call centers in US

I service rate µ = 14.6 calls/30-min, abandonment rate θ = 3.93calls/30-min

I 8am-9pm (13 hours) a day, 5 days a week, schedule updates at11am

Shift structure and costsI 262 feasible daily schedules (7 and 9-hour shifts, with breaks)

F cost of 1 per agent per 1/2-hour intervalI 4,973 potential recourse actions (with 1/2-hour costs)

F send home (-0.75), overtime (1.5), call in (2.0)

Arrival data, forecasts, and QoS targetI last 110 days as testing setI forecasts based on previous (rolling) 100 days of dataI target expected abandonment rate of 3% across scenarios


Updating systematically lowers cost per call

For abandonment rate: point forecast leads to upward biasFor cost: pair-wise comparison suggests significant reduction

I UP100→ SP100: average cost 2.9% reductionI RP100→ SP100: average cost 3.7% reduction


2nd set of empirical tests

A European retail bank’s call centerI call volume about 15% of the US bankI other parameters remain the same

F schedules, costsF service rate, abandonment rate

Arrival data, forecasts, and QoS targetI last 76 days as testing setI forecasts based on previous (rolling) 100 days of dataI target expected abandonment rate of 3% across scenarios


Major improvement on abandonment rate

For abandonment rate: point forecast leads to large upward biasFor cost: pair-wise comparison suggests significant reduction

I UP100→ SP100: average cost 2.1% reductionI RP100→ SP100: average cost 4.1% reduction


Different messages learned ...

US center:I arrival rate uncertainty has less impactI updating/recourse makes more difference

European center: opposite holds

Explanation: scale of uncertainty

Minimum 25% Median Mean 75% MaximumEur 0.1141 0.1235 0.1313 0.1338 0.1407 0.1663US 0.0284 0.0321 0.0333 0.0333 0.0346 0.0391

Table : Distribution of Daily Forecast-Dist CVs over Out-of-Sample Tests


Take Home Messages

Fruitful marriage: OM + Statistics

Distributional forecasts necessary

Model the right dependence

I Inter-day, intra-day, inter-stream, additional covariates, ...

Future workI Theoretical properties of the data-driven math programsI Scheduling (with recourse) for multiple arrival streamsI Poisson-Gamma formulation on the original scaleI Healthcare settings


Key References

L. D. Brown, N. Gans, A. Mandelbaum, A. Sakov, H. Shen, S. Zeltyn, L.H. Zhao (2005), Statistical Analysis of a Telephone Call Center: AQueueing-Science Perspective, JASA, 100, 36-50.

H. Shen, J. Z. Huang (2008), Interday Forecasting and IntradayUpdating of Call Center Arrivals, MSOM, 10, 391-410.

R. Ibrahim, P. L’Ecuyer (2013), Forecasting Call Center Arrivals:Fixed-Effects, Mixed-Effects, and Bivariate Models, MSOM, 15, 72-85.

N. Gans, H. Shen, Y. Zhou, Genesys (2015), Parametric Forecasting

and Stochastic Programming Models for Call-Center Workforce

Scheduling, MSOM, accepted.


Documents

Parametric Forecasting and Stochastic Programming Models