Staffing Service Systems via Simulation Julius Atlason, Marina Epelman University of Michigan Shane...
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Transcript of Staffing Service Systems via Simulation Julius Atlason, Marina Epelman University of Michigan Shane...
Staffing Service Systemsvia
Simulation
Julius Atlason, Marina Epelman
University of Michigan
Shane Henderson
Cornell University
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
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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
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WEN
WyES
WNWN
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g(y, j)
Sample Average Approximation
• Can’t compute ESj(y, W1)
• Replace it with a sample average
• Fix W1, W2, …, Wn
• Solve
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yAx
xc
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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?
Cutting Planes
g(y) 0g(y*)+ G(y*)T(y – y*)
cuts (linear)
s/t
min
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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