Staffing Service Systems via Simulation Julius Atlason, Marina Epelman University of Michigan Shane Henderson Cornell University.

Staffing Service Systemsvia

Simulation

Julius Atlason, Marina Epelman

University of Michigan

Shane Henderson

Cornell University

The Problem

Response Time

Staff Cost

Outline

• The staffing process

• Service level constraints

• Sample average approximation

• Concavity of service levels

• The algorithm

• Gradient Estimation

Staffing Process

1. Work requirements (# agents per period)– Educated guess– Queueing models– Simulation

2. Scheduling(selecting lines of work)

3. Rosters (who works which lines)

yAx

xcT

s/t

min

y

Why Aggregate?• Over-coverage

05

101520253035404550

9:00-9:15 9:15-9:30 9:30-9:45 9:30-9:45Time Period

012345678910

Why Simulate?• Complexity

– Several agent classes

– Several call classes

– Call routing

– Complexity of arrival processes

– Absenteeism

• Linkage between periods– If service times are moderate to large

– Lag-max method doesn’t handle all cases well

A Simple Model

• But you said…

• M(t)/G/s(t)

• No reneging

• Infinite number of trunk lines

• One class of server

• Service level constraints…

Service Level Constraints

• y = vector of staffing levels in each period

• W = random “stuff” for one day’s operation

• Sj(y, W) = # customers arriving in period j that reach agent in less than 10 seconds

• Nj(W) = number of calls in period j

Service Level Constraints

• Over n days

• We want

)(

),(

)()(

),(),(

1

1

1

1

WEN

WyES

WNWN

WySWyS

j

j

njj

njj

0)(8.0),(8.0)(

),(11

1

1 WENWyESorWEN

WyESjj

j

j

g(y, j)

Sample Average Approximation

• Can’t compute ESj(y, W1)

• Replace it with a sample average

• Fix W1, W2, …, Wn

• Solve

0)(

s/t

min

yg

yAx

xc

n

T

Sample Average Approximation

• n simulated days– Solution to sampled problem xn*

– Set of optimal solutions to true problem S*

• Under very mild conditions– xn* S* for n > N (N is random)

– P(xn* S* ) to 1 exponentially fast in n

• But how do we solve the sampled problem?

Concavity of the Service Levels

• Want g(y) 0

• g is nondecreasing

• g is concave in each component?

• g is jointly concave?

• Checked numerically in our algorithm

• But does it hold?

The “S” Curve (Empirical)

yj

gj(y)

Cutting Planes

g(y) 0g(y*)+ G(y*)T(y – y*)

cuts (linear)

s/t

min

yAx

xcT

g(y*)+ G(y*)T(y – y*) 0

“Solve” IP Simulate

y

cuts

Converges in finite # iterations

Phew Point

“Solve” IP Simulate

y

cuts

•Assume that g is concave in y (check via LP)

•How do we get the subgradients?

Subgradients via Differences

• Treat the simulation as a black box

• Compute g(y+ej) – g(y) for each j

• Subgradient?

Estimating Subgradients: IPA

• IPA (smoothed) differentiates sample path

• But servers are discrete

• Use service rate as a proxy for # servers

• Need #servers fixed over entire horizon to ensure interchange is ok!

• Vary service rate with period to match true service capacity

Estimating Subgradients: LR

• Lots of heuristics with smoothed IPA

• Likelihood ratio (score function) method?

• Seems to apply more easily, but still some less-than-ideal modeling assumptions

• Overall: subgradient estimation unresolved

Some Computational Results

• Only (very) small problems thus far

• Requires very few iterations

• Differencing seems to work!

• Smoothed IPA, LR: Jury still out

Summary To Date

• Very few iterations are needed to “zero in” on good staffing levels

• Have convergence theory both for fixed n, and as n increases

• Subgradient estimation remains a challenge

• Working with Ann Arbor Police on patrol car staffing

Future Research

• Concavity, S curves

• Why does differencing work?

• Can we relax concavity assumption to quasiconcavity (modify algorithm)?

• Integer programming takes a while…