Stochastic Models Of Resource Allocation For Services
Stochastic Models of
Resource Allocation for Services
Ralph D. Badinelli
Virginia Tech
Stochastic Models Of Resource Allocation For Services
Motivation
Manufacturing
Service
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Product design
Process design
Capacity acquisition
Location/layout
Revenue management
Aggregate planning
P&IC
Shop floor control
Quality control
Service design
PSS design
Capacity acquisition
Revenue process design
Location/layout/IT design
Resource planning
Resource allocation
Resource dispatching
Quality control
Stochastic Models Of Resource Allocation For Services
INFORMS Service Science Section
 Formed in February 2007
 Meetings sponsored/co-sponsored
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National INFORMS 2007 (Seattle)
2008 Logic of Service Science (Hawaii)
Service, Operations, Logistics, Informatics SOLI 2008 (Beijing)
2008 Frontiers in Service (Washington)
National INFORMS 2008 (Washington, DC)
International Conference on Service Science (Hong Kong)
National INFORMS 2009 (San Diego)
 November, 2008 - New Quarterly Journal
 Service Science
 http://www.sersci.com/ServiceScience/
 2010 – First on-line INFORMS SIG conference
 Vice Chair/Chair-Elect = Ralph D. Badinelli
Stochastic Models Of Resource Allocation For Services
Purpose
 We develop a resource allocation model with general
forms of service technology functions
 We describe the relationship between inputs and
outputs of a process of co-creation of value by a
service provider and a service recipient.
 Model development is directed at providing useful
policy prescription for service providers
Stochastic Models Of Resource Allocation For Services
Contributions
 A useful optimization model for resource allocation
and dispatch
 Some basic guidelines for optimal resource
allocation/dispatching, for client involvement and
adaptation of resource management to process
learning
 A modeling framework for service processes that can
serve as a foundation for further model development
Stochastic Models Of Resource Allocation For Services
Service Process
Definition: A service process is a coordinated set of
activities which transforms a set of tangible and
intangible resources (inputs), which include the
contributions from the service recipient and the
service provider, into another set of tangible and
intangible resources (outputs).
 E.g., agile software development, IT consulting,
higher education
Stochastic Models Of Resource Allocation For Services
Technology functions
 A technology function for a service encounter is a
function that effectively maps inputs to outputs
according to the capabilities of the service
participants to transform inputs into outputs.
 We construct this functional relationship by
considering the inputs and outputs of a process to be
functions of the volume, or number of service
“cycles”, of the process which are simultaneously
executed. Athanossopoulus (1998)
Stochastic Models Of Resource Allocation For Services
Assumptions
 The set of inputs of a service process is comprised of
two sets of inputs
 provider inputs
 client inputs
 Resource constraints
 Awareness – the client/provider may not have full
knowledge of the technology function.
 Objective function - maximization of utility of the
service participants.
Stochastic Models Of Resource Allocation For Services
Efficiency & Returns to scale
DMU Technology Possibility Set - Scale Changes
8
DRS Region
7
CRS Region
6
5
Output
4
IRS Region
3
Current Location of DMU
2
1
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Input
Stochastic Models Of Resource Allocation For Services
Technology functions
 The general nonlinear (VRS) technology function:
 
 


T p  t , x pi
 y pj

i SIp 
j SOp

 The linear VRS technology function
  
Tp x  b  y

y j
x i
 T pji
Stochastic Models Of Resource Allocation For Services
The linear CRS technology function
 pi 
Benchmark technological coefficient of input i of
process p
 pj  Benchmark technological coefficient of input i of
process p
 pi 
 pj 

1
 pi
= benchmark usage rate of resource i per
cycle of process p
1
 benchmark generation of resource j per
 pj
cycle of process p
number of cycles of process that are executed
Stochastic Models Of Resource Allocation For Services
Basic I/O relationships (Benchmark PSS)
 pi  x i , i  1,..., m
 pj  y j , j  1,..., n
 pi
 pj
yj 
xi 
x i   pj  pi x i
 pj
 pi
T ji 
yj
xi

 pi
 pj

 pj
 pi
  pj  pi
Stochastic Models Of Resource Allocation For Services
Real PSS – performance and uncertainty
u pi   pi  pui
b pi   pi  bpi
g pj   pj  pgj
a pj   pj  paj
 p ..  random variable
b pi x pi   pi  pbi x pi  v p
a pj y pj   pj  paj y pj  v p
Problem P1
Stochastic Models Of Resource Allocation For Services
Resource allocation problem
ŷ pj
min   w pj 
x p 
p
j
ŷ pj
 y f y pj ( y ) dy  c T  x p
p
0
subject to:
r  xp  0
p
xp  0
for all p
ŷ p 
a vector of target outputs for process
f y pj 
the distribution of
r
y pj
, a function of the resource allocations
vector of capacities of available resources
Stochastic Models Of Resource Allocation For Services
Loss function
ŷ pj
  w pj  ŷ pj  y f y pj ( y )dy
p
j
0
Lemma 1: The loss function increases with inefficiency
Lemma 2: Loss is increasing in the targets, ŷ pj
Lemma 4: Loss is decreasing and convex in volume
Stochastic Models Of Resource Allocation For Services
Process uncertainty
Self adjusting assumption: after the process inputs are
allocated, the process usage rates are dispatched by
the service provider and the service recipient in such
a way that they mutually adjust to values that support
a certain volume and which are consistent with the
inefficiency of the bottleneck input.
b p1 x p1  b p 2 x p 2  ...  b pm x pm  v p

b p1
 p1
p 

bp2
 p2
x pi
 pi
 ... 
  pi x pi
b pm
 pm
 pb 
vp
p
Stochastic Models Of Resource Allocation For Services
Problem re-statement
Define,
z pj  g pj  p
ẑ pj
min   w pj  p 
 p 
p
j
0
ẑ pj
 z f z pj  z dz  c T   p p
p
subject to:
r    p p  0
p
p 0
for all p
Stochastic Models Of Resource Allocation For Services
Optimality conditions
First-order KKT conditions imply:


M p (  p )  c T  T  p
where,





M p (  p )    w pj  z pj  A z pj ( ẑ pj )  ẑ pj G z pj ( ẑ pj ) 


 j


G z pj ( z )  1  Fz pj ( z )   f z pj ( s )ds
z

A z pj ( z )   G z pj ( s )ds
z
ẑ pj 
ŷ pj
p
Stochastic Models Of Resource Allocation For Services
Optimal resource dispatch
,
Theorem 2: Processes that have lower usage
rates will be allocated higher proportions of
available input resources and achieve higher
volumes under an optimal policy.
x pi   pi  p
x pi
x p

 pi
  pi
i
Stochastic Models Of Resource Allocation For Services
Optimal effort vs. performance
Volume vs. Mean Imbalance
2.5
Volume
2
1.5
1
Process 1 effort
0.5
Process 2&3 effort
0
0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3
Mean Epsilon of Process 1 Outputs
Stochastic Models Of Resource Allocation For Services
The cost of poor performance
Objective vs. mean imbalance
Optimal Objective Function
120
100
80
60
40
20
0
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
Mean Epsilon of Process 1 Outputs
0.3
Stochastic Models Of Resource Allocation For Services
Optimal effort vs. uncertainty
Volume vs. Variance Imbalance
1.7
1.65
volume
1.6
1.55
1.5
1.45
Process 1 effort
1.4
Process 2&3 effort
1.35
1
2
3
4
5
6
7
8
9
10 11 12 13
Variance Imbalance
Stochastic Models Of Resource Allocation For Services
The cost of uncertainty
Objective vs. Variance Imbalance
Optimal Objective Function
50
48
46
44
42
40
38
36
34
32
30
1
2
3
4
5
6
7
8
9
Variance Imbalance
10
11
12
13
Stochastic Models Of Resource Allocation For Services
General outcomes
The need for model-based resource planning
Optimal allocation of input resources across
processes that are different in terms of their
efficiencies, uncertainties and/or output targets is
quite complex and, in some cases counter-intuitive
Conflict resolution
Service providers and service recipients should
make every attempt to educate themselves jointly
about the nature of a service process before they
engage in dispatching resources to it.
Stochastic Models Of Resource Allocation For Services
Outcome adaptive policies
The transition law is simple
ŷ t  ŷ t 1  y t 1  d t
Stochastic Models Of Resource Allocation For Services
Estimation adaptive policies
 Update estimates of the parameters of the
process pdf with each service period.
 Consider improvements in efficiency as well
as random variation.
 ARIMA (0,1,1) forecasting model?
 Non parametric updating
 Must use approximate DP due to
dimensionality of estimation state
Stochastic Models Of Resource Allocation For Services
Conclusions
 We began the modeling of stochastic, multiperiod resource allocation problems
 Service models can borrow much
mathematical structure from manufacturing
models
 The multi-dimensionality of service processes
introduces new mathematical features to
planning models
 In our lifetimes, a cure will be found!