Stochastic Models Of Resource Allocation For Services Stochastic Models of Resource Allocation for...

Stochastic Models Of Resource Allocation For Services

Stochastic Models of Resource Allocation for Services

Ralph D. Badinelli

Virginia Tech


Motivation

Manufacturing Product design Process design Capacity acquisition Location/layout Revenue management Aggregate planning P&IC Shop floor control Quality control

Service Service design PSS design Capacity acquisition Revenue process design Location/layout/IT design Resource planning Resource allocation Resource dispatching Quality control


INFORMS Service Science Section Formed in February 2007 Meetings sponsored/co-sponsored

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


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


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


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


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)


Assumptions

The set of inputs of a service process is comprised of two sets of inputs provider inputsclient 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.


Efficiency & Returns to scale

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 29 300

1

2

3

4

5

6

7

8

DMU Technology Possibility Set - Scale Changes

Input

Output

Current Location of DMU

CRS Region

DRS Region

IRS Region


Technology functions

The general nonlinear (VRS) technology function:

The linear VRS technology function

OpIp SjpjSipip yx,tT

ybxTp

pjii

jT

x

y


The linear CRS technology function

pipi

1

= benchmark usage rate of resource i per

cycle of process p

pj

pj1

benchmark generation of resource j per

cycle of process p

pi Benchmark technological coefficient of input i of process p

pj Benchmark technological coefficient of input i of process p

number of cycles of process that are executed


Basic I/O relationships (Benchmark PSS)

m,...,1i,x ipi

n,...,1j,y jpj

ipipjipi

pji

pj

pij xxxy

pipjpi

pj

pj

pi

i

jji x

yT


Real PSS – performance and uncertainty

puipipiu

pgjpjpjg

bpipipib

pajpjpja

ppipbipipipi vxxb

ppjpajpjpjpj vyya

variable random ..p


Resource allocation problem

p

p

Typj

y

0pj

jpxxcdy)y(fyyw min

pj

pj

p

0xr pp

0x p

py

pjyf pjy

r

Problem P1

subject to:

for all p

a vector of target outputs for process

the distribution of , a function of the resource allocations

vector of capacities of available resources


Loss function

dy)y(fyywpj

pj

ypj

y

0pj

jp

Lemma 1: The loss function increases with inefficiency

Lemma 2: Loss is increasing in the targets, pjy

Lemma 4: Loss is decreasing and convex in volume


Process uncertaintySelf 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.

ppmpm2p2p1p1p vxb...xbxb

pm

pm

2p

2p

1p

1p b...

bb

pipipi

pip x

x

p

ppb

v


Problem re-statement

pp

p

Tzpj

z

0ppj

jpcdzzfzzw min

pj

pj

p

0r ppp

0p

subject to:

for all p

ppjpj gz Define,


Optimality conditions

pTT

pp c)(M

)z(Gz)z(Azw)(M pjzpjpjzpjpj

jpp pjpj

z

zzz ds)s(f)z(F1)z(Gpjpjpj

z

zz ds)s(G)z(Apjpj

First-order KKT conditions imply:

where,

p

pjpj

yz


Optimal resource dispatch

ppipix

ipi

pi

p

pi

x

x

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.

,


Optimal effort vs. performance

0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.30

0.5

1

1.5

2

2.5

Volume vs. Mean Imbalance

Process 1 effort

Process 2&3 effort

Mean Epsilon of Process 1 Outputs

Vol

ume


The cost of poor performance

0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.30

20

40

60

80

100

120

Objective vs. mean imbalance

Mean Epsilon of Process 1 Outputs

Op

tima

l Ob

ject

ive

Fu

nct

ion


Optimal effort vs. uncertainty

1 2 3 4 5 6 7 8 9 10 11 12 131.35

1.4

1.45

1.5

1.55

1.6

1.65

1.7Volume vs. Variance Imbalance

Process 1 effort

Variance Imbalance

volu

me


The cost of uncertainty

1 2 3 4 5 6 7 8 9 10 11 12 1330

32

34

36

38

40

42

44

46

48

50

Objective vs. Variance Imbalance

Variance Imbalance

Opt

imal

Obj

ectiv

e F

unct

ion


General outcomes

The need for model-based resource planningOptimal 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 resolutionService 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.


Outcome adaptive policies

t1t1tt dyyy

The transition law is simple


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


Conclusions

We began the modeling of stochastic, multi-period 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!

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