Supply chain planning for hurricane response
with information updates
Except where reference is made to the work of others, the work described in this
dissertation is my own or was done in collaboration with my advisory committee. This
dissertation does not include proprietary or classified information.
Selda Taskin
Certificate of Approval:
Robert L. Bulfin
Professor
Industrial and Systems Engineering
Emmett J. Lodree, Chair
Assistant Professor
Industrial and Systems Engineering
Chan S. Park
Ginn Distinguished Professor
Industrial and Systems Engineering
David M. Carpenter
Associate Professor
Mathematics and Statistics
George T. Flowers
Interim Dean
Graduate School
Supply chain planning for hurricane response
with information updates
Selda Taskin
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Doctor of Philosophy
Auburn, Alabama
August 9, 2008
Supply chain planning for hurricane response
with information updates
Selda Taskin
Permission is granted to Auburn University to make copies of this dissertation at its
discretion, upon the request of individuals or institutions and at
their expense. The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
Vita
Selda Taskin, daughter of Sevim Taskin and Osman Nuri Taskin, was born in December
1, 1979, in Istanbul, Turkey. She graduated from Kadik?oy Anatolian High School in 1997.
She received her Bachelor of Science degree in Industrial and Systems Engineering from
Yildiz Technical University, Turkey in 2001. She earned her M. Sc. degree in 2003 from the
same university. During her graduate study at Yildiz Technical University, she worked as
a Teaching Assistant. After graduation, she started her Ph.D degree at Mississippi State
University, MS in 2003 and worked as a Research Assistant for one semester. She continued
her Ph.D study at Auburn University. She worked as Teaching/Research Assistant at
Auburn University between 2003 and 2008.
iv
Dissertation Abstract
Supply chain planning for hurricane response
with information updates
Selda Taskin
Doctor of Philosophy, August 9, 2008
(M.A., Yildiz Technical University?Istanbul, Turkey, 2003)
(B.S., Yildiz Technical University?Istanbul, Turkey, 2001)
116 Typed Pages
Directed by Emmett J. Lodree
Planning inventories of supplies for the hurricane season can be challenging. For in
stance, in 2004, manufacturing and retail firms experienced stock outs because they were not
prepared for responding to the demand caused by several hurricanes that swept through the
state of Florida in the southeastern United States. In 2005, these firms again experienced
shortages due to the extreme demand surge caused by Hurricane Katrina. These experi
ences motivated firms to be proactive and more aggressive in their approach to stocking
hurricane supplies in 2006, resulting in large amounts of excess inventory because of an
inactive hurricane season.
While there are many issues, such as evacuation decisions and cooperation among
government agencies that need to be addressed in terms of developing effective plans for re
sponding to disastrous hurricanes, this research investigates stochastic production/inventory
control problems that are relevant to planning for potential disaster relief activities associ
ated with hurricane events. In particular, this study considers supply chain organizations
v
who experience demand surge for items such as flashlights, batteries, and gaspowered gen
erators, where the magnitude of the demand surge is influenced by various characteristics of
a hurricane season and/or a specific hurricane. These organizations are faced with challeng
ing procurement and production decisions since the hurricane logistics planning process is
complicated by the uncertainties associated with the number of hurricanes that will occur
during a hurricane season, hurricane intensities, and locations affected during the season.
This study aims to assist major corporations to quickly and cost effectively respond to
demand surges caused by hurricanes. In this dissertation, two different types of stochastic
inventory models are introduced to determine the appropriate hurricane stocking levels for
these organizations. The first two models address a hurricane stocking problem that is
relevant to disaster recovery planning. In this context, the disaster recovery plan requires
to determine optimal ordering/production policies for supply chain organizations for whom,
the magnitude of the demand surge is influenced by various characteristics of an observed
storm during the hurricane season. The third model introduces a multi period hurricane
inventory control problem that allows the managers to adjust inventory decisions during
the preseason periods as demand realizes to reserve for the hurricane season demand. The
model enables decision makers (DMs) to determine the optimal level of reserved hurricane
stock while satisfying the demand associated with the preseason periods. Finally, the work
accomplished for each chapter of this dissertation is summarized with their relevance and
usefulness, and possible extensions of this research and future study are proposed.
vi
Acknowledgments
First and foremost the author would like to thank Dr. Emmett J. Lodree for his
guidance, support, corrective comments and suggestions on her dissertation. She would
like to extend her thanks to Dr. Robert L. Bulfin, Dr. Chan S. Park, and Dr. David
M. Carpenter for their valuable comments on her dissertation. She also thanks Dr. Alice
E. Smith for her support throughout her doctoral study. She thanks her parents Sevim
and Osman Nuri Taskin, and her sister Beyza Taskin Akg?ul for their years of support, and
encouragement. She thanks Mert Serkan for giving her the motivation to accomplish her
degree.
vii
Style manual or journal used Computers and Operations Research (together with the
style known as ?aums?). Bibliograpy follows van Leunen?s A Handbook for Scholars.
Computer software used The document preparation package TEX (specifically LATEX)
together with the departmental stylefile aums.sty.
viii
Table of Contents
List of Figures xi
List of Tables xii
1 Introduction 1
2 Emergency Inventory Planning During the Hurricane Season 7
2.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Background and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.2 Single Period Loss Function . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.3 Bayes Risk Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.4 Recursive Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Algorithm Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.1 Optimal Order/Production Quantity . . . . . . . . . . . . . . . . . . 23
2.4.2 Optimal Stopping Time . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Empirical Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5.1 Empirical Likelihood Densities . . . . . . . . . . . . . . . . . . . . . 26
2.5.2 Simulating Wind Speeds . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6 Extension to Ordering Disruption . . . . . . . . . . . . . . . . . . . . . . . . 31
2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Multi Location Inventory Model with Wind Speed Forecast Updates 36
3.1 Hurricane Prediction Literature . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Hurricane prediction model . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 Sequential Statistical Decision Model . . . . . . . . . . . . . . . . . . . . . . 45
3.4 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.2 Single Period Loss Function . . . . . . . . . . . . . . . . . . . . . . . 50
3.4.3 Bayes Risk Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.5 Solution Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.5.1 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
ix
4 Advanced Inventory Planning for the Hurricane Season 61
4.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Stochastic Programming Model . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.1 Demand Scenario Probabilities . . . . . . . . . . . . . . . . . . . . . 69
4.2.2 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3 Scenario Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3.1 Heuristic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.4 Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5 Conclusions and Proposed Future Study 93
Bibliography 98
x
List of Figures
3.1 NHC windspeed probability map. . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Windspeed probability map at t = 0h. . . . . . . . . . . . . . . . . . . . . . 55
3.3 Windspeed probability map at t = 30h. . . . . . . . . . . . . . . . . . . . . 56
3.4 Windspeed probability map at t = 60h. . . . . . . . . . . . . . . . . . . . . 56
3.5 Windspeed probability map at t = 90h. . . . . . . . . . . . . . . . . . . . . 57
3.6 Windspeed probability map at t = 120h. . . . . . . . . . . . . . . . . . . . 57
4.1 WinBugs posterior regression coefficients. . . . . . . . . . . . . . . . . . . . 75
4.2 WinBugs predictive hurricane count rates. . . . . . . . . . . . . . . . . . . . 76
4.3 WinBugs predictive hurricane counts. . . . . . . . . . . . . . . . . . . . . . 76
xi
List of Tables
2.1 Example results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1 Scenario probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 CLIPER Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3 Predicted windspeed probabilities . . . . . . . . . . . . . . . . . . . . . . . 54
3.4 Demand information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.5 Example results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.1 Hurricane landfall count and AprilMay NAO and AMO index derived by
the 1950?1979 data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Hurricane landfall count and AprilMay NAO and AMO index derived by
the 1980?2007 data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3 Demand distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4 Results of the original model . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.5 Probability distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.6 Euclidean distances (c metric) . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.7 Results of reduced models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.8 Optimal values (solutions) of reduced models . . . . . . . . . . . . . . . . . 86
4.9 Euclidean distance matrix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.10 Euclidean distance matrix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.11 Euclidean distance matrix 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.12 Euclidean distance matrix 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
xii
Chapter 1
Introduction
This dissertation is motivated by the impact of increased hurricane activity in the
United States, particularly in the Gulf Coast region. Many government agencies, notfor
profit organizations, and private corporations assume leading roles in positioning supplies,
equipment, and personnel both during and after a major hurricane. These organizations are
faced with challenging supply chain and logistics decisions to ensure that supplies, equip
ment, and personnel are readily available at the right places, at the right times, and in the
right quantities. In addition to the complexities associated with supply chain and logis
tics planning in general, inventory planning decisions made before the hurricane season are
complicated by forecasts related to the number of hurricanes that are expected to develop
during the ensuing season. Similarly, supply chain decisions made during the hurricane
season (after a tropical disturbance or depression is initially observed) are especially com
plicated by the dynamics and uncertainties associated with various hurricane characteristics
such as its diameter, its projected path, and its intensity along the path. Therefore, the
objective of this research can be stated more generally as determining the optimal level of
supply chain readiness with respect to hurricane preparedness.
Chapter 2 investigates a disaster recovery planning problem encountered by manufac
turing and retail organizations whose demand for products such as batteries, flashlights,
and gaspowered generators is significantly influenced by the characteristics of an observed
storm during the hurricane season. The planning horizon begins during the initial stages of
storm development, when a particular tropical depression or disturbance is first observed,
1
and ends when the storm dissipates. If an observed tropical disturbance or depression
materializes into a major storm, then manufacturing and retail firms will often experience
demand surge for hurricane supplies caused by increased consumer activity and additional
requests from service organizations who use these and other supplies to carry out initial re
sponse operations. Without an effective disaster recovery plan in place, these manufacturing
and retail firms are not likely to satify a hurricane induced demand surge. Consequently,
the results of an ineffective or nonexistent disaster recovery plan include (i) stock outs of
hurricane supplies such as those mentioned above, (ii) low service levels with respect to
fulfilling hurricane related demand, and (iii) an extended recovery period associated with
rebuilding inventory levels capable of supporting regular demand and order fulfillment not
directly related to the storm. From this perspective, the disaster recovery plan for a man
ufacturing or retail firm with respect to preparing for a potential demand surge caused by
an observed storm entails determining appropriate inventory levels for hurricane supplies
that will enable the firm to (i) fulfill hurricane related demands during and immediately
following the storm and (ii) minimize the time and resources invested in rebuilding target
inventory levels to support normal operations after the storm.
In order to address these issues, this chapter proposes a framework that explicitly and
dynamically incorporates hurricane predictions associated with an observed storm into the
decision process, which also accounts for the inherent tradeoff between hurricane forecast
accuracy and logistics cost efficiency as a function of time. More specifically, predictions
associated with a storm during its initial stages of development are often uninformative with
respect to disaster recovery planning, at least relative to predictions during the later stages.
From this perspective, it is more beneficial to postpone decisions until an accurate forecast
2
is observed some time during the later stages of the planning horizon. On the other hand,
it is also beneficial to implement supply chain decisions during the earlier stages of storm
development to avoid potential inefficiencies and complications that may arise during the
later stages. For instance, manufacturing firms constrained by production capacities may be
forced into expensive overtime labour or outsourcing in order to respond to demand surge
caused by a major hurricane. Manufacturing and retail firms may also incur additional
expenses for faster modes of transportation in order to ensure that hurricane supplies are
available when and where they are needed. Under more extreme circumstances, it might
be impossible to deliver hurricane supplies during the later stages of the planning horizon
because primary components of the transportation network could be inaccessible due to
damages caused by the storm. Given the abovementioned problems encountered in practice
and tradeoff between forecast accuracy and logistics cost, the objective of this research is
to determine the optimal inventory level for hurricane supplies and how long after a tropical
depression or disturbance is observed this inventory decision should be postponed such that
the tradeoff between logistics cost efficiency and hurricane forecast accuracy is optimized.
The forecast updating approach to hurricane supply chain planning described in this
chapter emphasizes preparing for potential extreme hurricanes such as Hurricane Katrina.
For the purposes of this chapter, an extreme hurricane is characterized by two specific
features: (i) abnormal surge in demand for hurricane supplies and (ii) statistically low
probability of occurrence. In order to minimize the expected costs associated with supply
chain planning for a potential extreme event, a statistical decision model based on the
traditional expected value approach that incorporates the cost and probability associated
with the extreme event is not likely to lead to an appropriate and practical decision, just
3
as the mean is often not an accurate indicator of central tendency for skewed data that
contain outliers. However, the proposed Bayesian updating framework allows the decision
maker to postpone his decision until there is enough information available to accurately
predict whether or not a hurricane will become extreme, which is more likely to lead to an
appropriate decision than the traditional expected value approach.
Chapter 3 addresses a disaster recovery planning (DRP) problem for a one supplier
multi retailer supply chain as a result of a hurricane event. This chapter investigates the
suppliers? hurricane related inventory decisions to ensure that the right amount of hurri
cane supplies are readily available at the right retailers, at the right times, and in the right
quantities. More specifically, the supplier is considered as the decisionmaker (DM), for
whom the DRP horizon begins when a particular tropical depression is first observed, and
ends with the average storm lifecycle time (120h). The order/production decisions imple
mented earlier on the planning horizon will be less expensive for the supplier relative to
those implemented in the later stages. On the other hand, the tropical cyclone informa
tion acquired during its initial development stage is not accurate enough to make stocking
decisions. From this perspective, it is necessary for the supplier to balance the tradeoff
between cost efficiency and hurricane forecast accuracy. More specifically, the inventory
decision should be postponed until this kind of optimization is achieved to minimize the
financial risks associated with over/under preparation.
This chapter is an extension of previous work [58], which only focuses on the wind
speed data to characterize the hurricane event. More specifically, this chapter accounts for
the tropical cyclone path information in addition to the wind speed data to enhance the
accuracy of predictions. In order to achieve this, the National Hurricane Center?s (NHC)
4
new wind speed probability model, which is based on the official forecasts of the cyclone?s
center position and intensity (maximum 1min surface wind speed) is used. The official
forecasts are issued every 6 hours, and each contains projections at cumulative 12 hour
forecast periods up to 5 days. These forecasts are compared with the best track data, which
are obtained after the NHC?s post storm analysis of all available storm data. The NHC
predicts hurricane?s tracks and intensity by implementing various statistical, dynamical or
combined models. Since 2005, the NHC has been making storm forecasts for locations
based on wind speed probability maps. The wind speed probability model allows DMs to
predict the current chance that a location will be hit by damaging tropical cyclone winds.
Therefore, businesses and industry can better evaluate the risks associated with a tropical
cyclone at their locations. For instance, insurers may obtain realtime information on the
likelihood of a potential loss for their portfolios ([70]). This quantitative information is also
important for many organizations that provide hurricane supplies to the retailers whose
demands are affected by hurricanes. The NHC?s wind speed probability predictions assist
these organizations in their inventory related decisions, since they provide information as to
how the demand will fluctuate at their locations. Therefore, they can have more accurate
estimates of the characteristics associated with the hurricane season.
Chapter 4 presents a stochastic programming inventory model, which is based on gen
eral predictions regarding the ensuing hurricane season such as those issued by the NHC.
This research considers allocating hurricane stock to meet the hurricane season demand
while meeting the period?s demand. In order to achieve this objective in a cost efficient
way, this chapter introduces a stochastic programming model that is converted to a deter
ministic linear program where the potential realizations of the discrete demand distribution
5
are introduced as the scenarios of the demand process. Additionally, the hurricane season
demand distribution is described by implementing a Markov chain approach. The states
of the Markov chain are defined by a finite number of hurricane landfall count rates. The
hurricane count rate predictive probabilities are introduced as the stationary transition
probabilities of the Markov chain. The underlying demand distribution is then described
with respect to the preseason and hurricane season demand distributions. The stochastic
programming model is used to determine the optimal quantity and timing of the inventory
decisions. The scenario reduction approach introduced by [38] is also implemented to de
termine the optimal ordering policy. The optimal values obtained from the solution of the
stochastic programming model are compared with the results of the reduced models. The
stochastic model presented in this chapter enables DMs to determine an appropriate stock
level that should be available at the beginning of the hurricane season and allows multiple
production/procurement decisions.
6
Chapter 2
Emergency Inventory Planning During the Hurricane Season
This chapter introduces a stochastic inventory control problem that is relevant to proac
tive disaster recovery planning as it relates to preparing for potential hurricane activity. The
disaster recovery planning (DRP) problem encountered by manufacturing and retail organi
zations is characterized by (i) the financial risks associated with overpreparing if inventory
levels exceed the demand caused by a hurricane, (ii) the financial risk of expensive emer
gency procurement / production if inventory levels are not sufficient to meet demand, (iii)
the social risks characterized by long customer waiting times and prolonged human suffering
also due to insufficient inventory levels, and (iv) the financial risks associated with restor
ing business continuity after a hurricane. As an example, the author has interacted with
production and logistics managers for large scale manufacturing and retail organizations
(who wish to remain anonymous) that experienced demand surges due to hurricane events.
These firms felt that they could have been better prepared for responding to demand surge
for hurricane related supplies caused by several hurricanes that affected the southeastern
United States during the 2004 hurricane season, and again in 2005 after Hurricane Katrina.
On the other hand, these organizations were stuck with large amounts of excess inventory
at the end of the inactive 2006 hurricane season. In order to assist these organizations
in their inventory related decisions, an optimal stopping model with Bayesian hurricane
information updates is introduced. Additionally, a dynamic programming algorithm is im
plemented to solve the optimal stopping problem. The planning horizon for the Bayesian
inventory model initiates when a tropical depression or disturbance is observed, and ends
7
with the hurricane?s lifecycle time. The objective is to determine the optimal ordering
policy by leveraging the hurricane related information updates.
The remainder of this chapter is organized as follows: Section 2.1 reviews relevant lit
erature, which includes disaster recovery planning, disaster relief planning, and inventory
control with Bayesian updates. In section 2.2, the fundamental concepts related to optimal
stopping problems are introduced, along with the notational conventions used in this chap
ter. In section 2.3, the mathematical formulation of the problem is presented, followed by
the solution approach. Then, a numerical example is given to illustrate how the method
ology can be implemented in practice. In section 2.6, an extension of the base model that
accounts for the inability to carry out an ordering decision as a result of damages caused
by the hurricane is presented. Finally in section 2.7, the conclusions and future research
directions are presented.
2.1 Literature Review
Disruptions in business continuity caused by natural and manmade disasters demon
strate the need for organizations to develop effective Disaster Recovery Plans (DRPs). For
example, immediately following the World Trade Center attacks of September 11, 2001,
the United States government?s protective measures inhibited the daily operations of many
corporations. One such corporation was Ford Motor Company who eventually closed five
U.S. plants and reported a 13% decline in vehicle production ([60]). Another example of
business continuity disruption is Telefon AB L.M. Ericsson, a mobile phone manufacturer
who estimated $400 million in lost sales during the year 2000 when a lightening induced
8
fire shut down its sole supplier (Royal Phillips Electronics) of one of the chips needed for
production ([23]).
The disaster recovery planning research literature with respect to developing an orga
nization?s business continuity plan seems to have originated with computer network security
applications, although there is some early evidence of DRP in manufacturing environments
(e.g., [43]). A noticeable increase of DRP applications in supply chain environments, which
is often referred to as disruption management or disruption planning, occurred shortly after
the World trade Center attacks mentioned above (e.g., [23], [49], and [73]). Quantitative
approaches to DRP in supply chain management based on Operations Research and Man
agement Science (ORMS) methods are discussed in [57] and [89]. ORMS approaches to
DRP in airline operations, project management, machine scheduling, and other environ
ments are presented in [89]. Finally, general frameworks for integrating DRP and ORMS
are presented in [6], [16] and [79].
This research is also applicable to disaster relief planning problems encountered by mil
itary, government, and service organizations. While the focus of DRP is business continuity
planning, disaster relief planning falls under the more general topic of emergency manage
ment. Therefore, the author also reviews relevant research from the emergency management
/ disaster relief logistics literature.
In terms of hurricane logistics planning, the majority of the literature entails the lo
gistical aspects of the emergency evacuation process. Examples include a simulation based
optimization approach to hurricane evacuation planning for Ocean City, Maryland ([92]),
a methodology for establishing evacuation zones in the New Orleans, Louisiana area ([85]),
and a Markov decision model that uses hurricane predictions to determine when and if an
9
evacuation should be ordered ([67]). For a comprehensive review of hurricane evacuation
planning and management protocols, the reader is referred to [86] and [87].
Hurricane logistics issues outside the realm of evacuation has received considerably
less attention. Sheppard [74] discusses disaster relief logistics associated with the storage
and distribution of public water utilities in the U.S. Virgin Islands, and [4] reports the
experiences of firms such as Home Depot, Walmart, and CVS Pharmacy who were able to
successfully support relief operations after Hurricane Katrina. To the best of the author?s
knowledge, there are no other published materials that address disaster relief logistics spe
cific to hurricanes (except for other hurricane evacuation research). Additionally, note that
neither [4] nor [74] involves quantitative approaches to hurricane logistics planning.
General frameworks for disaster relief logistics planning have also been presented in the
research literature. For example, [81] discusses the role of private sector supply chains and
their interactions with humanitarian organizations in providing logistics support to victims
of disaster. Thomas [78] proposed a reliability and decision analysis framework for assessing
the readiness of a contingency logistics network, and [39] proposed interaction network opti
mization as a general framework for disaster management. Disaster relief logistics planning
associated with hazards other than hurricanes has also been addressed in the literature.
Examples include helicopter logistics operations ([9]), inventory control for longterm hu
manitarian response ([13]), responding to pandemic outbreaks ([39]), and assigning electric
power repair crews and depots to various areas in need after a natural disaster ([69]).
Finally, relevant research from the inventory control literature is reviewed. More specif
ically, the author discusses inventory models characterized by Bayesian forecast updates
10
(Sethi et al. [72] summarize nonBayesian approaches to forecast updating that have ap
peared in the inventory literature). Bayesian updating has consistently been an active area
of research in the inventory control literature since the pioneering work of Dvoretzky et
al. [29]. A comprehensive review of this literature would then prove to be quite extensive.
Hence, the related presentation is limited only to the more recent advances in Bayesian in
ventory research (some earlier papers that are often cited include [7], [8], [59], [62], [33], [42],
and [71]). The motivation for applying Bayesian methodologies to problems of inventory
control with stochastic demand is to make informed stocking decisions based on accurate
demand forecasts. The general approach to doing so is to represent one or more of the
demand distribution parameters as a random variable, but these parameter distributions
are estimates. Bayesian inventory modeling then involves using early sales or demand in
formation as it becomes available to improve the estimated parameter(s) distribution(s),
thereby resulting in more information rich stocking decisions. This approach has recently
been applied to spare parts inventory management [5], quick response inventory control
[22], partially observed demand [27], supply chain contract design [88], quantity and pricing
decisions [90], and products with short life cycles ([91]). The approach adopted in this
chapter, which involves updating hurricane intensity predictions based on wind speed in
formation updates, can be classified as a Bayesian inventory model with multiple delivery
modes ([20], [21], [72]). Of these, this chapter is most closely related to [21], who model
an inventory problem as an optimal stopping problem with Bayesian updates and normal
demand distributions. This research is more general than [21] in the sense that neither
the demand distribution nor its parameters are limited to normal distributions. Also, this
11
research contributes to the inventory control literature in that hurricane predictions are
explicitly incorporated into the Bayesian updating framework and inventory decision.
2.2 Background and Notations
The framework of sequential statistical decision problems is ideally suited to model
the hurricane supply chain planning problem described in Section 1. Therefore, relevant
concepts and terminology related to sequential decision problems based on [14] and [24] are
first introduced and a model related to hurricane planning is presented later as a special
case.
Consider a decision problem in which a decisionmaker (DM) must specify a onetime
decision ? that minimizes some loss function L. The loss function depends on a random
variableE that has a known probability density function (pdf)g(e,?), where ? is a random
variable from an unknown distribution characterized by pdf h(?). Before deciding upon a ?
that minimizes L, DM has the opportunity to obtain more information about the unknown
distribution parameter ? by observing a sequential random sample Wt = (W1,W2,...,Wt)
from ?, where the cost of observing Wj is Cj. Note that Wt = (W1,W2,...,Wt) is called
a sequential random sample if the Wj?s are independently and identically distributed.
The definition of the sequential statistical decision problem is based on two fundamental
ideas: (i) the stopping time, T, and (ii) the decision rule, ? . The stopping time T is derived
from the concept of a stopping rule, ? , which is a series of functions ?0,?1(w1),?2(w2),...
such that?t(wt) is the probability of terminating the sampling processes aftertobservations.
Note that ?0 is interpreted as the probability of giving the decision without sampling. In
12
[14] (page 442), the stopping time T is defined as
TparenleftbigWtparenrightbig = mint?0 braceleftbig?tparenleftbigWtparenrightbig = 1bracerightbig (2.1)
In words, T ? {0,1,...,t} is the number of observations such that sampling is stopped and
a decision ?t is given as opposed to observing Wt+1.
Now the decision rule ? is a set of functions ?0,?1parenleftbigw1parenrightbig,?2parenleftbigw2parenrightbig,... that specify which
action is to be taken once sampling is stopped and the observed values from the sequential
random sample is given by wt = (w1,w2,...,wt). The sequential statistical decision prob
lem then involves determining a stopping rule ? and decision rule ? that minimizes the loss
function Lparenleftbig?,?tparenleftbigwtparenrightbig,tparenrightbig based on a sequential random sample W1,...,Wt.
More formally, define the risk function as the expected value of the loss function, de
notedR(?,d), where d = (?,?) . Also define ? = braceleftbigt? 1 : ?tparenleftbigwtparenrightbig = 1 and?parenleftbigwjparenrightbig = 0bracerightbig for
allj 0.5. The DMs risk behavior can also be taken into account using this approach by
choosing threshold values other than 0.5. However, an alternative loss function L(Q) would
then apply.
2.4.2 Optimal Stopping Time
In order to determine the optimal stopping time, the approach presented in [21] is
adopted by using the concept of cutting points p?t. These cutting points are based on the
23
fact that according to Theorem 2.1, it is optimal to stop only if r0(ht,t) ? rT?t(ht,t)
for t = 0,1,...,T, where t = 0 is interpreted as giving an immediate decision without any
sampling. A cutting point is then the pointp?t that satisfiesr0(ht,t) = rT?t(ht,t), which can
be considered a decision threshold that specifies whether or not sampling should continue
after pt is observed. Note that the parameter pt in rj(ht,t) is expressed as an expected
value in terms of W using the posterior distribution h(ptwt). Thus, to determine cutting
point values, the loss function, which is expressed explicitly in terms of pt, is used. Once
the sample point pt is observed, r0(ht,t) ? rT?t(ht,t) becomes Lt(Qt) ? Lt+1(Qt+1) and
the cutting point p?t satisfies Lt(Qt) = Lt+1(Qt+1). That is, Theorem 2.1 suggests that it is
optimal to stop and place an order ifLt(Qt) ?Lt+1(Qt+1) and the cutting point is obtained
by solving
NBt0 +ptparenleftbigNBt1 ?NBt0parenrightbig = NBt+10 +ptparenleftbigNBt+11 ?NBt+10 parenrightbig (2.20)
Note that NBtk = NBtk (Qt). Solving Eq. (2.20) for pt yields the following cutting point p?t.
p?t = NB
t+1
0 ?NBt0parenleftbig
NBt+10 ?NBt0parenrightbig?parenleftbigNBt+11 ?NBt1parenrightbig (2.21)
Solving the inequality Lt(Qt) ? Lt+1(Qt+1) yields pt ? p?t. Thus the optimal procedure in
Theorem 2.1 as applied to our hurricane stocking problem can be restated as follows.
Theorem 2.3 If a sample update pt is observed in period t and satisfies pt ?p?t, where pt
is given by Eq. (2.8) and p?t is given by Eq. (2.21), it is optimal to terminate sampling and
give an immediate decision by stocking Qt. Otherwise if tp?t, do not order/produce and continue sampling. Increment t and
repeat steps 3 and 4.
Note that Theorem 2.3 requires the sequential random sample Wt = (W1,...,Wt) to
be expressed as Pt = (P1,...,Pt), which is the reason for Step 3 in the above procedure.
Also note the condition p?T = 1 ensures that an order is always placed. That is, if no order
has been placed by period T, the condition pT ? 1 will always hold and force an order QT.
Although the cutting point formula (2.21) is structurally the same as in [21], the details are
more complicated because this research considers a more general case in which Bayesian
updates are not conjugate. Furthermore, additional complexities arise because the optimal
stocking policy in Theorem 2.2 cannot be expressed in closed form.
2.5 Empirical Study
In this section, the solution methodology presented in Section 2.4 is demonstrated
using real hurricane data from the HURDAT database. The objective is to use historical
wind speed data to simulate the evolution of the wind speeds associated with an observed
tropical depression, and then apply the solution approach to determine a onetime stocking
25
decision, as well as which period this stocking decision should be given. A sample ofN = 143
hurricanes comprises our data set spanning the 10year period 1995  2004. The sample is
represented as a matrix WT?N with T rows and N columns, where each row corresponds to
a period t and each column corresponds to a particular hurricane n in the sample. Thus an
entry wtn represents the wind speed of hurricane n during period t. This data is used for
two purposes: (i) to determine appropriate parameters for the Normal likelihood densities
(Assumption 4) and (ii) to simulate wind speeds at 6hour intervals for creating problem
instances.
2.5.1 Empirical Likelihood Densities
Calculation of posterior densities requires that the distribution parameters associated
with the likelihood densities htparenleftbigwtpparenrightbig be specified. To ensure the condition p in htparenleftbigwtpparenrightbig
is satisfied, initially all hurricanes in the sample WTN with W ? 131 (recall that W is
the maximum recorded wind speed associated with a specific hurricane over all periods t)
are identified. The resulting sample is denoted WT?NE, where NE = 23 is the number of
extreme hurricanes from the sample WTN. To empirically determine parameters (using the
maximum likelihood method) for the likelihood densityhtparenleftbigwtpparenrightbigthat corresponds to period
t, the wind speedswtn forn = 1,...,nE are used resulting inT likelihood density functions.
These likelihoods can then be used to transform an observation wt into its corresponding
pt, which is needed for Step 3 of the algorithm presented in Section 2.4.2.
2.5.2 Simulating Wind Speeds
For illustrative purposes, the following elementary approach is applied to simulate
hurricane wind speeds at 6hour intervals for creating problem instances as follows (note
26
that there is no need to simulate wind speeds in practice; wind speeds are simulated here
only to create example problem instances):
Step 1: Initialize: Set t = 1.
Step 2: Simulate a random number rt between 1 and N.
Step 3: Let wt be the simulated wind speed period t and wt,rt be the wind speed
associated with period t for the rtht hurricane in WTN. Then wt = wt,rt.
Step 4: Repeat steps 2 and 3 until t = T.
It is observed that the first several wind speed observations for many of the sample
hurricanes (both extreme and otherwise) are either 30 mph or 35 mph. Since there is little
to no distinction between storms in the initial stages, the analysis is limited to periods in
which there are observable differences in wind speeds by defining
t = min{t : wt ? 131} (2.22)
In words, t is the earliest period in which enough information is available to classify a
hurricane as extreme. From a practitioners perspective, t represents the period in which
managers should begin to take notice of wind speed updates and predictions.
Recall from Assumption 3 that hurricane demand is X1 if W ? 131 and X0 otherwise.
Therefore, if some observationwt is at least 131, there is no need to continue observing wind
speed updates. As a result, the number of wind speed simulations required to implement
the analysis is limited by defining
t = max{t : wt ? 131} (2.23)
27
By replacing t = 1 with t = t in Step 1 of the above wind speed simulation procedure, and
also replacing t = T with t = t in Step 5, the number of unnecessary simulated wind speed
updates are reduced.
Now recall that the parameters associated with the likelihood densities are empirically
derived based on 6hour intervals between hurricane forecast updates because HURDAT
data is recorded every six hours (Assumption 4). Although the available hurricane data
is limited to 6hour intervals, the Bayesian method for updating distribution parameters
in general has no restrictions related to the times between successive information updates
nor is there any restrictions on the number of updates. Furthermore, the amount of time
between observed hurricane forecast updates in practice may be more than six hours during
the earlier stages of the planning horizon, but the frequency of updates during the later
stages is likely to increase resulting in times between observed updates that are less than
six hours. The above procedure for simulating wind speed data can be modified as follows
in order to account for hurricane predictions that are observed every k hours (or decision
periods of length k), where k can also vary during the planning horizon.
1. If k > 6 is an integer, simulate wind speed data using the above procedure (which is
based on forecast updates given every six hours) and solve the problem as usual. Then
the modified decision period ?t, which is based on updates given every k > 6 hours,
is given by ?t = max{q,1}, where q is the integer quotient resulting from the division
6t/k. For example, suppose wind speed updates are observed every k = 10 hours, and
the optimal decision period for the 6hour interval problem is t = 7. First of all, the
period t = 7 corresponds to 42 hours after the hurricane?s initial observation. For the
28
case k = 10, this corresponds to period ?t = max{4,1} = 4 since the integer quotient
when 42 is divided by 10 is 4.
2. If k < 6, then more wind speed data points must be simulated during period t as
follows.
(a) If k = 1 between the 6hour periods t and t+ 1, then simulate s = 6 random
numbers rt1,...,rts in Step 2 of the above wind speed simulation procedure,
where each rtj ? {1,...,N} and j = 1,...,s.
(b) Then in Step 3, there will be s = 6 simulated wind speeds wt1,...,wts, where
wtj = wt,rtj, wt,rtj is the wind speed in period t associated with the rthtj hurricane
in WTN, and j = 1,...,s.
(c) For k = 2 and k = 3, steps (a) and (b) can be repeated with s = 3 and s = 2
respectively. Also, it can be argued that s = 2 applies to both k = 4 and k = 5.
(d) The optimal decision period is then determined by solving the problem with
these additional wind speed updates, and will be based on the jth hour of period
t.
2.5.3 Numerical Example
The above wind speed generating procedure was used to produce 14 wind speed sam
ples, WTN. As shown in Table 2.1, seven of the examples were extreme and seven were
not. The following data was used to solve each of the example problems, and the results
are shown in Table 2.1: c1 = 20,ct+1 = ct + 1,s = 100,h = 15,X0 ? N(98,152), and
X1 ? Gumbel(?,?) = (153,16). Note that p0 = Pr{W ? 131} = 0.16,t = 11, and t = 51
for our sample of N = 143 hurricanes. Note that the Gumbel distribution, which is an
29
extreme value distribution used for low probability events, was assumed for the demand as
sociated with an extreme hurricane. Also note that Matlab?was used to perform numerical
integration calculations associated with the posterior densities given by Eq. (2.14) for the
example problems. Matlab?was also used to solve Eq. (2.19) to obtain decision rules for
the example problems.
Table 2.1: Example results
Example Hurricane Type Update Interval Decision Period t? Stocking Quantity Qt
1 Not Extreme 6hour 4 104.51
2 Not Extreme 6hour 17 39.13
3 Not Extreme 6hour 9 103.15
4 Not Extreme 6hour 18 105.05
5 Not Extreme 6hour 4 104.88
6 Not Extreme 3hour 3 135.82
7 Not Extreme 10hour 8 111.31
8 Extreme 6hour 3 111.95
9 Extreme 6hour 2 109.83
10 Extreme 6hour 7 180.53
11 Extreme 6hour 4 135.37
12 Extreme 6hour 3 135.82
13 Extreme 3hour 3 153.57
14 Extreme 10hour 2 185.86
Table 2.1 suggests that the DMs are inclined to order/produce early for the extreme
cases relative to the nonextreme cases (note that the decision periodt? in Table 2.1 actually
corresponds to period t? + 11 since t = 11). More specifically, the HURDAT data shows
that wind speeds tend to increase more quickly for extreme hurricanes than for nonextreme
hurricanes during the hurricane?s evolution. The results are also consistent with intuition in
that the optimal order quantities associated with extreme hurricanes are larger, on average,
than the optimal order quantities associated with nonextreme hurricanes.
30
2.6 Extension to Ordering Disruption
In this section, the base model presented in Section 2.4 is extended such that damages
from an observed storm could prevent an ordering / producing decision from being carried
out. That is, if the solution to the base model suggests ordering / producing a quantity Qt
in period t, then the extended model accounts for possible disruptions, such as damages to
the transportation network or inaccessible overtime labour, that would prevent the decision
from being implemented. In order to extend the base model, it is assumed that ordering
disruptions are caused by the characteristics of an observed storm. That is, the disruptions
caused by events other than the observed storm are not considered, which is consistent with
Assumption 1. More specifically, it is assumed that an ordering disruption is a function of
an observed storm?s maximum sustained wind speed during period t, which is consistent
with Assumption 2. Additionally, it is assumed that if an ordering disruption occurs during
period t, then no order / production can occur during periods t,t+ 1,...,T, which means
that the ordering disruption cannot be resolved until after the storm dissipates.
Now let Zt be a Bernoulli random variable, with parameter zt, that assumes the value
1 if an ordering disruption occurs in period t and 0 otherwise. Then zt = Pr(Zt = 1) and
1?zt = Pr(Zt = 0). Furthermore, the parameterzt is also assumed to be a random variable
with prior probability zt0, and the following likelihood and posterior densities
gtparenleftbigwtzparenrightbig = PrparenleftbigWt = wtZt = 1parenrightbig (2.24)
gtparenleftbigzwtparenrightbig = PrparenleftbigZt = 1Wt = wtparenrightbig (2.25)
31
Note that unlike the Bernoulli random variable E introduced in Assumption 4, there are T
random variables Zt, each of which is unknown at the beginning of period t and known with
certainty at the end of period t. Also note that the random sample wt is observed at the
beginning of period t, which is then used to calculate the posterior probability associated
with zt. Additionally, note that if an order / production decision is given in period t, it is
given at the beginning of the period before the realization of Zt. If Zt = 0, then the order /
production quantity satisfies Theorem 2.2. However, if Zt = 1 for some period t = s, then
Qt = 0 for each t = s,s+ 1,...,T since Zt = 1 implies that no ordering or producing can
take place in period t or after period t. Under these conditions, the loss function Lt(Q) is:
Lt(Q) = (1?zt)?Mt(Q)+zt ?Mt(0) (2.26)
where
Mt(Q) = (1?pt)?NBt0(Q)+pt ?NBt1(Q)
Mt(0) = (1?pt)?NBt0(0)+pt ?NBt1(0)
Similar to the base model, the cutting point for period t (see Section 2.4.2) can be
derived by solving the equation Lt(Qt) = Lt+1(Qt+1), where Lt(Qt) is given by Eq. (2.26).
The resulting cutting point is:
p?t = Nt(Q)N
t(Q)?Rt(Q)
(2.27)
32
where
Nt(Q) = NBt+10 (Q)?NBt0(Q)+ztbracketleftbigNBt0(Q)?NBt0(0)bracketrightbig?zt+1bracketleftbigNBt+10 (Q)?NBt+10 (0)bracketrightbig
Rt(Q) = NBt+11 (Q)?NBt1(Q)+ztbracketleftbigNBt1(Q)?NBt1(0)bracketrightbig?zt+1bracketleftbigNBt+11 (Q)?NBt+11 (0)bracketrightbig
The cutting point given by Eq. (2.27) can be used in a variation of the algorithm
following Theorem 2.3 to compute the optimal stopping time. The modified algorithm
requires an additional step that converts sample wind speeds wt into their corresponding
posterior probabilities zt for each t. Consequently, the resulting algorithm would involve a
more complex Bayesian updating environment that accounts forpand eachzt. Furthermore,
additional data would need to be collected and compared to HURDAT in order to construct
meaningful prior and likelihood distributions for each zt.
2.7 Conclusion
In response to increased hurricane activity in the United States, particularly the devas
tating impact of Hurricane Katrina during the year 2005, this chapter addresses a disaster
recovery planning problem encountered by manufacturing and retail organizations who ex
perience demand surge for various products if an observed storm evolves into a catastrophic
hurricane. The proposed model and solution method are also applicable to a closely related
disaster relief planning problem relevant to the military, electric power companies, and other
service organizations. Instead of formulating a general model that is applicable to preparing
for any kind of disaster, the proposed approach leverages hurricane predictions to develop
a disaster recovery plan that is most appropriate for managing the risks that are specific to
33
hurricane events. This approach is consistent with fundamental concepts from emergency
management in that risk identification is an integral part of the planning process (see [36]).
Relative to hazards such as earthquakes, terrorist attacks, and tornadoes, it is rea
sonable to expect more reliable disaster recovery plans for hurricanes because of (i) an
abundance of historical data, (ii) the availability of sophisticated prediction models, (iii)
the increasing accuracy of hurricane predictions as a storm evolves after its initial develop
ment, and (iv) the length of the planning horizon after a potential threat is first identified.
This research leverages these characteristics by formulating the inventory problem as an
optimal stopping problem with dynamic hurricane prediction updates. In particular, his
torical hurricane data is used to develop a statistical model for predicting whether or not an
observed tropical depression or disturbance will evolve into an catastrophic hurricane. The
prediction model entails Bayesian updates of hurricane wind speeds and is integrated with a
decision model that specifies the optimal quantity and timing of the inventory decision such
that the tradeoff between forecast accuracy (better with time) and cost efficiency (worse
with time) is optimized.
The framework presented in this chapter represents the initial stages of research needed
to develop disaster recovery plans that would be effective in practice with respect to prepar
ing for hurricane events. One possible extension is to integrate a more sophisticated hur
ricane prediction model into our decision framework, particularly one that includes both
track and intensity predictions. This extension is more realistic because the magnitude
of demand surge obviously depends on more than one hurricane characteristic. However,
advanced statistical techniques would be required to facilitate a more complex Bayesian
34
updating process. This chapter also introduces several other opportunities for further re
search. For example, the current model emphasizes planning for extreme hurricane events
by considering two demand classes. A natural extension is to consider five demand classes
(one for each hurricane category) to plan for any type of hurricane. Another possibility is to
explore a multiple product version of the model. Finally, this chapter can also be extended
based on other decision rules or loss functions, such as those described in Section 2.6 and
in [57].
35
Chapter 3
Multi Location Inventory Model with Wind Speed Forecast Updates
This chapter investigates an inventory stocking problem encountered by suppliers whose
demand for hurricane supplies is influenced by the hurricane season. More specifically, these
suppliers? inventory related decisions made after a tropical depression is initially observed
are affected by the uncertainties associated with the tropical cyclone?s projected path, and
its intensity along the path. Therefore, it is necessary for them to develop effective and
efficient disaster recovery plans before the hurricane season initiates. In this chapter, the
National Hurricane Center?s (NHC) windspeed probability model is used along with the
Climatology and Persistence (CLIPER) tropical cyclone track prediction model to forecast
various hurricane characteristics. These hurricane related forecasts and forecast updates
are introduced as a Bayesian model. Then, this information is incorporated into an optimal
stopping model to assist these suppliers in their inventory decisions for hurricane supplies.
This chapter discusses the DRP problem introduced in the previous chapter for a one
supplier multi retailer supply chain. In this study, more accurate hurricane predictions
are able to be obtained compared to the ones from ([58]). Lodree and Taskin [58] use
historical windspeed HURDAT data to simulate the evolution of the windspeeds. In other
words, an empirical methodology is implemented to obtain hurricane windspeed forecast
updates. However, in this chapter a widely accepted statistical prediction model known
as the wind speed probability model is used. Starting in 2006, the windspeed probabilities
are issued by the Tropical Prediction Center/National Hurricane Center (TPC/NHC) for
each hurricane season in the Atlantic and Eastern North Pacific basins. It is assumed
36
that the supplier?s DRP horizon corresponds to the average lifecycle time of hurricanes
(120h). In the absence of having an effective disaster recovery plan in place, the supplier
under consideration is not likely to withstand a hurricane induced demand surge. Lodree
and Taskin [58] discuss the results of an ineffective or nonexistent disaster recovery plan.
They mention that it might be impossible to deliver hurricane supplies to some locations
within the supply chain network because of the inaccessibility of the transportation system.
Additionally, the supplier may incur expenses for overtime, outsourcing or faster modes of
transportation to accommodate the demand surges caused by hurricane events. Therefore,
the supplier should determine the optimum inventory levels for the hurricane supplies such
that the financial risks associated with over/under preparation are minimized as a result of
improved hurricane forecast accuracy.
The remainder of this chapter is organized as follows: Section 3.1 reviews the hurricane
prediction literature. In section 3.2, the hurricane prediction model is introduced. In section
3.3, the sequential statistical decision model associated with the optimal stopping problem
is presented. In section 3.4, the mathematical formulation of the problem is presented,
followed by the solution approach. Then, numerical example problems are given to illustrate
how the methodology can be implemented in practice. Finally, conclusions are presented in
section 3.6.
3.1 Hurricane Prediction Literature
There exists a substantial amount of research on hurricane prediction. Vickery and
Twisdale [82] develop a simulation methodology using windfield and filling models to obtain
hurricane windspeeds associated with various return periods. Simulation results reveal
37
that the subregion identification is a critical factor for windspeed prediction. Lehmiller,
Kimberlain, and Elsner [56] use a multivariate discriminant analysis for making forecasts
of hurricane activity both in the Gulf of Mexico and Caribbean Sea. The results of their
statistical model identify different subsets of predictors within different prediction locations.
Elsner, Niu, and Tsonis [44] develop an empirical Bayesian prediction algorithm to assess
the potential usage of multiseason forecasts for the North Atlantic hurricane activity. Their
analysis of the correlation values of fitted univariate time series reveals that the hurricane
attributes can be well fitted by an univariate autoregressive moving average. Jagger, Niu,
and Elsner [45] apply a spacetime count process model to annual North Atlantic hurricane
activity. They use the besttrack data set of historical hurricane positions and intensities
together with climate variables to determine the local spacetime coefficients of a right
truncated Poisson process. The results show that on average, model forecast probabilities
are larger in regions, in which hurricanes occur. Additionally, it is determined that there
exists a hurricane path persistence among seasons. The results also indicate that this
modeling procedure can be useful as a climate prediction tool since forecast skill above
climatology is observed.
Klotzbach and Gray [51] develop an updated statistical scheme for forecasting tropical
cyclone activity in the Atlantic basin. Their statistical findings reveal that the hurricane
landfall probability shows considerable forecast skill from 1951?2000 based on the net trop
ical cyclone activity prediction and the weighted North Atlantic See Surface Temperatures
(SST). Elsner and Jagger [30] develop a hierarchical Bayesian strategy for modeling annual
U.S. hurricane counts from 1851?2000. The Bayesian analysis reveals that hurricane counts
only from the twentieth century together with noninformative priors compares favourably
38
to a traditional approach. Through the implementation of the Bayesian approach, climate
relationships to U.S. hurricanes are also examined. The results of the Bayesian model also
confirms a statistical relationship between climate patterns and coastal hurricane activity.
Weber [83] develops a method known as The Probabilistic Ensemble System for the
Prediction of Tropical Cyclones (PEST) to develop geographical strike probability maps.
The results of the model indicate that the mean annual errors of the deterministic position
forecasts are comparable in quality to that of the current consensus approaches. In another
paper, [84] presents a method for the maximum windspeed prediction of tropical cyclones
using PEST. In this study, he makes deterministic intensity predictions for all global tropical
cyclone events during subsequent forecast periods of years 2001 and 2002, respectively.
Postanalysis results reveal that the sizes of all intensity probability intervals give reliable
estimates of future storm intensities. The model?s deterministic forecasts are determined to
have the same quality of the majority of all dynamical models. Nevertheless, the intensity
predictions with PEST is observed to have lower overall quality than the position predictions
with PEST discussed in [83].
Regnier[66]surveyexistingresearchonweatherforecaststoassistDMsintheirweather
sensitive decisions using Operations Research and Management Science (OR&MS) tools.
DeMaria et al. [26] develop an experimental version of Statistical Hurricane Intensity Pre
diction Scheme (SHIPS). This new version includes the satellite observations for the 2002
and 2003 hurricane season. Predictors selected based on this version include the brightness
temperature information from Geostationary Operational Environmental Science (GOES).
The storm decay information is also incorporated to SHIPS to increase the accuracy of
the hurricane forecast. The results of the analysis demonstrate that the inclusion of the
39
effects of the decay over land beginning in year 2000 reduce the intensity errors up to 15%.
Additionally, the combination of GOES and satellite altimetry improve the Atlantic fore
casts by up to 3.5%. Regnier and Harr [68] develop a decision model to prepare for an
oncoming hurricane where the DM monitors an evolving hurricane. The results indicate
that the DM who has the flexibility to wait for an updated hurricane forecast can gain
substantial value by adopting a dynamic approach to anticipate the improving forecast ac
curacy. Elsner and Jagger [31] develop a modeling strategy that uses MayJune averaged
values representing the North Oscillation Index (NOI), Southern Oscillation Index (SOI),
and the Atlantic Multidecadal Oscillation (AMO) to predict the probabilities of observing
U.S. hurricanes in the months ahead (JulyNovember). A Bayesian approach is used to
examine three different models that take the advantage of historical records extending back
to 1851. These models are (i) a full model that includes all three predictors (NOI, SOI and
AMO) (ii) a reduced model that includes NOI and SOI and (iii) a singlepredictor model
that includes only NOI. The statistical findings show that the NOI and SOI combination
model and the NOI single predictor model performs best. The results of the model also
show forecast skill above climatology for the years in which there are no hurricanes or more
than two hurricanes. Additionally, it is determined that all three models capture annual
variation in hurricane counts better than climatology does. Other recent research includes
([80, 64, 50, 52, 55, 51, 65, 25, 17]).
3.2 Hurricane prediction model
In this chapter, the National Hurricane Center (NHC)?s wellrecognized hurricane pre
diction model, which is based on the windspeed probability predictions, is used. The
40
hurricane wind speed probability graphs show the probabilities of sustained surface wind
speeds of 74 mph (hurricane force) at different locations. More specifically, each graphic
provides cumulative probabilities that windspeeds of 74 mph will occur at any location
officially during cumulative 12 hour intervals (i.e., 0?12h, 0?24h, 0?36h,...,0?120h)
and extends through a 5 day forecast. Figure 3.1 demonstrates an example windspeed
probability map made by the NHC.
Figure 3.1: NHC windspeed probability map.
As can be seen in Figure 3.1, the windspeed probability graphs are organized such that the
cumulative probabilities are given in percent from 1% to 100% in colorcoded 10% bands,
which indicate the probabilities of sustained surface winds that corresponds to 74 mph. Al
though, the focus of this research is on hurricane force winds only, windspeed probability
predictions also exist for tropical storm force windspeeds.
41
The windspeed probabilities are based on the official track, intensity, and wind radii
forecasts, and on their corresponding forecast errors issued by the NHC. These forecast
errors are determined by comparing the official forecasts with the best track database 1.
The track forecast error is determined as the circular distance between a cyclone?s forecast
position and the best track position. These errors are obtained by the distribution of along
track (AT) and cross track (CT) forecast errors ([40]). While AT errors give an indication of
the forecast of the tropical cyclone movement, CT errors are used to determine whether the
model changes the path of the hurricane so frequently or not ([40]). On the other hand, the
intensity error is forecasted as the absolute difference between the forecast and the best track
intensity at each forecast verifying time. These forecasts and forecast errors can be found
from the NHC?s public resources, the Tropical Cyclone Forecast/Advisory and NHC Official
Forecast Error Database. In addition to forecasting the center position and intensity of a
tropical cyclone, the variability in tropical cyclone size (wind radii) is incorporated into the
track forecasts via a climatological wind radii forecast model developed by ([35]). Based on
their model, Monte Carlo (MC) simulation of the windspeed probabilities are accomplished
by creating a large sample of storm tracks and intensities relative to a given forecast track
along with the climatological variations of tropical cyclone size. The corresponding wind
radii forecast model formulated by [35] is:
V(r,?) = (Vm ??)?(rm/r)x +??[cos(???o)] (3.1)
where V(r,?) is the windspeed threshold, r is the radius from the storm center ?wind
radius? (nmi), ? is the angle measured counterclockwise starting from a direction 90o to the
1obtained by the NHC?s poststorm analysis of all observed storm data
42
right of the storm motion, Vm is the maximum wind (mph), rm is the radius of maximum
wind (nmi), xis a size parameter (nondimensional), and? is an asymmetry parameter (kt)
that is a function of the storm speed of motion.
Gross, DeMaria and Knaff [35] analyze the climatological model by fitting x,rm,?,?o
to the NHC wind radii forecasts for the 1988?2002 Atlantic storms. They determine that
rm and x can be estimated in terms of Vm and latitude ?, and also ? can be expressed as a
function of the storm speed of motion c as given below.
rm = 35.37?0.111?Vm +0.570?(??25) (3.2)
x = 0.285+0.0028?Vm (3.3)
? = 0.337?c?0.003?c2 (3.4)
The wind radii for each MC track sample is simulated based on the error distribution of the
xparameter. The initial value of the error inX is chosen as the difference between the value
from the above climatological model and the best fit values to the observed radii at each
storm quadrant. Various error values at forecasting times are then determined as a linear
combination of the initial value and a random component to develop the error distribution.
Then, the windspeed probabilities are determined by counting the fraction of grid points
that fall within the radius of a given windspeed threshold (39,74) mph.
Initially, the Climatology and Persistence (CLIPER) is implemented to predict the
storm center coordinates at forecasting periods. This model uses the current path of a
tropical cyclone and an average of historical paths of similar cyclones to come up with
a track. Additionally, CLIPER takes into account the size of the cyclone at the start of
the forecast period as well as changes in size as the cyclone evolves in strength, motion
43
and other factors. Two sets of regression equations are implemented to predict the storm
track, where each predictand is either the zonal or meridional displacement observed at
time t. Aberson [1] determine that the initial latitude, initial longitude, initial intensity,
initial day number, initial zonal motion and the initial meridional motion impact the topical
cyclone track. Among these predictors, [1] chose the significant predictors for the meridional
displacement M, and zonal displacement Z in the Atlantic basin. Based on these selected
predictors, regression analysis is implemented for various hurricane scenarios to forecast the
storm centers. Eq. (3.5) gives the regression equations:
Z = ?0 +?1 ?U +?2 ?LAT +?3 ?(LON ?V)+?4 ?(LAT ?V) (3.5)
+?5 ?(LON ?DAY)?6 ?(LAT ?U)+?7 ?(LAT ?LAT)+?8 ?(DAY ?U)
+?9 ?(U ?V)+?10 ?(INT ?V)
M = ?0 +?1 ?V +?2 ?U +?3 ?(INT ?U)+?4 ?(LAT ?INT)+?5 ?(INT ?V)
+?6 ?(U ?V)+?7 ?(DAY ?U)+?8 ?(LON ?DAY)+?9 ?(LAT ?V)
where LAT is the initial latitude, LON is the initial longitude, INT is the initial intensity,
DAY is the initial day number, U is the initial zonal motion, and finally V corresponds to
the initial meridional motion.
In order to develop hurricane scenarios, mean values of these predictors and predictands
are used for the chosen dependent data (1931 ? 1995). Then, the windspeed probability
maps are simulated at each storm center to predict the probability of having hurricanes at
different locations within the selected region.
44
3.3 Sequential Statistical Decision Model
In this chapter, a sequential statistical decision problem is considered to determine a
stopping rule ? and a decision rule ? that minimizes the loss function Lparenleftbigp,?tparenleftbigwtparenrightbig,tparenrightbig based
on a sequential random samplewt. The decision loss depends on random variable vectorWvectorH
that has a known probability density function with a parameter vector vectorP. Before deciding
upon a ? that minimizes the loss function, the DM has the opportunity to obtain more
information about the unknown distribution parameter vector vectorP by observing a sequential
random sample Wt:
Wt =
?
??
??
??
??
??
W11 W12 ... W1t
W21 W22 ... W2t
.....................
Wn1 Wn2 ... Wnt
?
??
??
??
??
??
(3.6)
where the cost of observing vectorWj = (Wj1,...,Wjn) is Cj. Note that vectorWj?s are independently
and identically distributed.
In this study, the stopping time T is defined as the cumulative 30h interval such that
sampling is stopped and a decision ?t is given whether to observe vectorWt+1. Then, the sampling
cost of the statistical sequential model is determined as
Tsummationdisplay
j=30
Cj. Eq. (3.7) expresses the
stopping time T as
TparenleftbigWtparenrightbig = min
t?{0,30,60,90,120}
braceleftbig?
t
parenleftbigWtparenrightbig = 1bracerightbig (3.7)
with ?t being the probability of stopping the sampling process after Wt is observed. Then
the risk function associated with the sequential decision procedure d = (t?,Q) is the
45
expected loss:
R(vectorP,d) = E
bracketleftBig
L
parenleftBigvector
P,?T parenleftbigWtparenrightbig,T
parenrightBigbracketrightBig
(3.8)
= P(T = 0)?L(vectorP,?0,0)+
120summationdisplay
t=30
integraldisplay
?
L(vectorP,?t(wt),t)dHt(wtvectorP)
+
?summationdisplay
t=1
tsummationdisplay
j=1
CjP(T = t)
where the sequential sample wt is
wt =
?
??
??
??
??
??
w11 w12 ... w1t
w21 w22 ... w2t
...................
wn1 wn2 ... wnt
?
??
??
??
??
??
(3.9)
The Bayesian updating of the prior estimates ?vectorp of the random variable vectorP is achieved at
each forecasting period t to obtain the corresponding posterior estimates. The Bayes risk
function of the problem is developed using these predictions as given in Eq. (3.10).
r(?vectorpt,d,t) = E[R(vectorP,d)] (3.10)
Then the sequential decision problem can be formulated as follows.
r(?vectorpt,t) = inf
d
r(?vectorpt,d,t) (3.11)
46
3.4 Model Formulation
In this section, an optimal stopping framework is introduced to solve the hurricane
supply stocking problem such that both the track and the intensity of the tropical cyclone
are taken into account to determine an optimal inventory decision. Below, the assumptions
and the details of the model are presented.
3.4.1 Assumptions
The derivations of the loss and risk functions are based on the following assumptions.
Assumption 6 The supplier?s demand is a random variable X, where X is a function of
the observed storm?s intensity along its path.
Let Xi,i = 1,...,n, be a random variable that represents the demand of retailer i, where
n is the number of retailer locations whose demand can potentially be affected by the
observed storm. Then X is a convolution of the random variables Xi. Each Xi depends
on the intensity of the storm surrounding location i, which includes the possibility that the
storm does not threaten the location at all. LetWi be a random variable that represents the
maximum sustained windspeed at locationi. Then Assumption 6 implies thatXi = X(Wi)
and X = summationtextXi(Wi).
Assumption 7 Two classes of demand are considered at each retailer location: demand
associated with hurricane force winds and demand that corresponds to no hurricane force
winds.
Assumption 7 implies that the demand distribution at location i is one of two categories.
Let Yi denote demand at location i if no hurricane force winds are experienced and Zi
47
be demand at location i if hurricane force winds are experienced, where both Yi and Zi
are random variables for each i = 1,...,n. Then Xi ? {Yi,Zi}. This represents the initial
approach to modeling the impact of an observed storm on each retailer?s demand, and hence
the supplier?s demand.
Assumption 8 Let vectorH = (H1,...,Hn) be a multivariate Bernoulli random variable such
that Hi = 1 indicates hurricane force winds at location i, and Hi = 0 otherwise, where
i = 1,...,n. Let vectorP = (P1,...,Pn) be the parameter vector associated with vectorH, where
Pi = Pr{Hi = 1}, and Pit = Pr{Hit = 1vectorAit = vector?it} is the probability of having hurricane
at location i after observing the tth windspeed probability update, vector?it. Then
1. vectorP is a random vector with prior densities hi(pi), likelihood densities hti(vector?tipi), and
posterior densities hti(pivector?ti). Here vector?it is a vector that represents an observation of
storm attributes (namely location, intensity, and radius) that are used in windspeed
probability calculations, t = 1,...,T and T is the number of hurricane prediction up
dates (T is also interpreted as the number of periods during the planning horizon...see
Assumption 10).
2. The likelihood and posterior densities, hti(vector?tipi) and hti(pivector?ti) respectively, are based
on hurricane prediction updates that are given every 30 hours after a tropical depres
sion or disturbance is initially observed. Thus t progresses in 30h intervals.
NHC publishes windspeed probability maps (see Figure 3.1) typically in 12 hour intervals
as an observed storm evolves. In this study, the windspeed probability maps are updated
in 30 hour intervals. Each map is generated based on an observation of vector?ti, given that t?1
48
maps have been published before. Thus each windspeed map is a posterior probability
calculation, which is represented mathematically based on the notation as hti(pivector?ti).
Assumption 9 Let ct be the production cost associated with giving a decision after t hours.
Then ct ?ct+30 for all t = 0,30,...,120.
Assumption 9 relates to the difficulty of implementing a production decision during the
latter stages of the planning horizon. In reality, the problem with waiting for very accurate
storm information is that there may not be enough time to meet demands because of limited
production capacity. If the decision about the target inventory level is determined earlier,
then the supplier can schedule production over a few days without paying a premium for
additional capacity. Thus Cj = cj+30?cj forj = 0,30,...,90 andc0 = 0 can be interpreted
as the premium that the supplier has to pay for additional capacity in order to reach target
inventory levels such that finished goods can be shipped to the disaster area immediately
after the disaster strikes.
Assumption 10 Demand realization happens at each location i exactly 5 days (120 hours)
after the storm is initially observed.
Assumption 10 allows us to specify that the length of the supplier?s decision horizon is
known with certainty. That is, T = 120. In actuality, the length of the supplier?s decision
horizon is also uncertain. However, useful results from sequential Bayesian decision theory
are leveraged to model and solve the inventory control problem by assuming that T is
known with certainty. Additionally, this assumption is used to introduce convoluted demand
distributions.
49
3.4.2 Single Period Loss Function
Assumption 8 allows to introduce a multivariate Bernoulli random variable vectorH =
(H1,...,Hn) with density of g(vectorh,vectorP) = producttextni=1Phii ? (1 ? Pi)(1?hi) and hi ? {0,1}, that
specifies whether a hurricane is observed (Hi = 1) or not (Hi = 0) at location i. Then, a
hurricane demand Zi with probability Pi is considered at location i. Similarly, a regular
demand Yi with probability 1?Pi is applicable at location i. Since the demand at different
locations are independent and identically distributed, the loss function L(Q) is formed using
a convolution of the random variables that represent demand at each respective location.
The probability of observing a specific convoluted demand k at time t is introduced as the
scenario probability qkt. Once the hurricane demand probabilities vectorpt = {p1t,p2t,...,pnt} at
each location i = 1,2,...,n are observed, the corresponding scenario probabilities qkt are
calculated using the multiplication rule. Then, the expected loss at time t is expressed as:
Lt(Qt) =
k=2nsummationdisplay
k=1
qkt ?NBtk (3.12)
where n is the number of locations, and NBtk is the expected cost function for the newsboy
problem with convoluted demand random variable Xk for k = 1,...,2n. Note that exactly
one scenario can occur out of k = 2n number of scenarios at time t. If ct is the unit
order/production cost, ht the unit holding (or overstocking) cost, st the unit shortage cost
at time t, and fk(xk) the density of Xk, then NBtk is
NBtk = Qt ?ct +ht ?
integraldisplay Qt
0
(Qt ?xk)fk(xk)dxk +st ?
integraldisplay ?
Qt
(xk ?Qt)fk(xk)dxk (3.13)
50
Table 3.1 demonstrates the scenario probabilities for a two location problem at timet. Note
that for a n location problem, there will be n+1 different convoluted demand distributions
given that the retailers have the same demand distribution parameters for the two classes
of demand.
Table 3.1: Scenario probabilities
Scenario k Scenario probability qkt Retailer1 Retailer2 Convoluted Demand Xk
1 q1t = p1t ? p2t Z1 Z2 Z1 +Z2
2 q2t = (1 ? p1t) ? p2t Y1 Z2 Y1 +Z2
3 q3t = (1 ? p1t) ? (1 ? p2t) Y1 Y2 Y1 +Y2
4 q4t = p1t ? (1 ? p2t) Z1 Y2 Z1 +Y2
3.4.3 Bayes Risk Function
In order to determine the ordering quantity Q, the Bayes risk function of the problem
r(?vectorpt,d,t) is formalized by revising the loss function given in Eq. (3.12) such that the pos
terior predictions of vectorP = ?vectorp, and the cost of observing a sequential sample Wt associated
with vectorP are incorporated into the function. As a result, r(?vectorpt,d,t) is expressed as
r(?vectorpt,d,t) =
k=2nsummationdisplay
k=1
qkt ?NBtk +
tsummationdisplay
j=30
Cj t = 30,...,T (3.14)
Then, the sequential problem is bounded such that
rT(?vectorpt,t) = inf
d
r(?vectorpt,d,t) (3.15)
with T being the maximum number of observations taken at locations. Additionally, define
r0(?vectorpt,t) as the minimum Bayes risk associated with giving an immediate decision in period
51
t, and rT?t(?vectorpt,t) as the minimum Bayes risk associated with (T ?t) more observations at
locations. Theorem 3.1 is introduced to state the sequential decision problem recursively.
Theorem 3.1 (Degroot [24]): Among all sequential decision procedures in which not more
than T observations can be taken, the following procedure is optimal: If r0(?vectorpt,0) ?rT(?vectorpt,0),
a decision ?0 is chosen immediately without any observations. Otherwise, vectorW1 is observed.
Furthermore, for t = 30,60,...,T ? 30, suppose the sequential random sample Wt has
been observed. If r0(?vectorpt,t) ?rT?t(?vectorpt,t), a decision ?t is chosen immediately without further
observations. Otherwise, vectorWt+1 is observed. If sampling has not been terminated earlier, it
must be terminated after vectorWT is observed.
Then, the recursive formulation of the sequential decision problem is
rj(?vectorpt,t) = min
d
braceleftBig
r0(?vectorpt,t),Ebracketleftbigrj?30(vectorptWt+1,t+1)bracketrightbig
bracerightBig
j = 60,...,T (3.16)
3.5 Solution Methodology
In this section, the methodology used to determine the order/production quantity Qt
and single order/production period t? that minimizes the expected loss associated with
ordering/producing, overstocking, and understocking is described. Initially, posterior wind
speed probabilities ?vectorpt at locations are predicted based on the CLIPER?s updated storm
center forecast and the generated windspeed probability map at this forecasted storm
center. Then, the optimal order/production quantity is determined. More specifically, once
vectorpt is observed, the optimal decision is Qt that minimizes the loss function Lt(Qt) with
order/production cost ct. Theorem 3.2 describes the resulting decision rule.
52
Theorem 3.2 (Lodree and Taskin [57]): Let Fk(xk), where k = 1,...,2n be the cumulative
distribution function of random demand Xk ? {Y,Z}. Then the optimal decision ?t = Qt
that minimizes the loss function Lt(Qt) with order/production cost ct satisfies
k=2nsummationdisplay
k=1
qkt ?Fk(Q?t) = s?cs+h (3.17)
Theorem 3.1 suggests that the optimal decision rule ?t = (Q1,...,Qt) should be de
termined first to search for the optimal stopping time t?. Then, it is optimal to stop and
place an order if Lt(Qt) ?Lt+1(Qt+1) is satisfied.
3.5.1 Numerical Example
In this section, the previously mentioned tropical cyclone forecasting methods are used
to determine the windspeed probabilities. More specifically, various windspeed proba
bility scenarios are considered to demonstrate the solution methodology. The CLIPER
track prediction model is used to forecast the fictitious storm?s track. The hypothetical
tropical cyclone is assumed to start at a predetermined location for simplicity. The initial
zonal (longitude) coordinate is x = 75Wo and the initial meridional (latitude) coordinate is
y = 25No. Then, the CLIPER regression equations are run for a large sample size at each
forecasting time to come up with a track scenario. Table 3.2 demonstrates the zonal and
meridional displacements over 30h forecasting intervals.
Table 3.2: CLIPER Output
Forecast Time Zonal Displacement Meridional Displacement Storm Center Coordinate (Wo,No)
30 3.76 0.33 (78.76,25.33)
60 6.30 1.82 (85.06,23.51)
90 3.27 0.69 (88.33,22.82)
120 4.50 1.73 (92.83,21.09)
53
Based on the CLIPER storm track forecast, the windspeed probability model is implemented
to predict vectorP = (P1,...,Pn) at each forecasting period. Then, the predicted windspeed
probabilities, which are determined from the simulated maps, are considered for the follow
ing example problems. These values are presented as a matrix PT?N with T rows and N
columns, where each row corresponds to a period t = 0,30,60,90,120h and each column
corresponds to a particular location. Thus an entry ptn represents the windspeed probabil
ity at locationn during periodt. The predicted wind speed probabilities at each forecasting
period are presented in Table 3.3.
Table 3.3: Predicted windspeed probabilities
Example PTxN Example PTxN
1
?
??
??
?
0.6 0.62 0.64 0.66 0.68
0.5 0.52 0.54 0.56 0.58
0.4 0.42 0.44 0.46 0.48
0.3 0.32 0.34 0.36 0.38
0.2 0.22 0.24 0.26 0.28
?
??
??
?
2
?
??
??
?
0.2 0.22 0.24 0.26 0.28
0.3 0.32 0.34 0.36 0.38
0.4 0.42 0.44 0.46 0.48
0.5 0.52 0.54 0.56 0.58
0.6 0.62 0.64 0.66 0.68
?
??
??
?
3
?
??
??
?
0.5 0.45 0.6 0.55 0.7
0.5 0.45 0.6 0.55 0.7
0.5 0.45 0.6 0.55 0.7
0.2 0.25 0.35 0.4 0.55
0.3 0.35 0.4 0.45 0.5
?
??
??
?
4
?
??
??
?
0.9 0.6 0.45 0.7 0.85
0.95 0.7 0.65 0.6 0.7
0.9 0.75 0.6 0.65 0.75
0.85 0.8 0.65 0.75 0.8
0.9 0.85 0.7 0.7 0.75
?
??
??
?
5
?
??
??
?
0.7 0.65 0.55 0.48 0.1
0.6 0.65 0.15 0.2 0.9
0.2 0.25 0.25 0.15 0.55
0.2 0.25 0.25 0.15 0.55
0.3 0.35 0.4 0.35 0.5
?
??
??
?
6
?
??
??
?
0 0 0 0 0.9
0.3 0 0 0 0.8
0 0.25 0 0 0.55
0.2 0.25 0 0 0.55
0.3 0.35 0.4 0 0.5
?
??
??
?
7
?
??
??
?
0.45 0 0.25 0.3 0.4
0.3 0 0.25 0.15 0.2
0.1 0 0 0.17 0.3
0.06 0 0 0.1 0.9
0.01 0 0 0 0.8
?
??
??
?
8
?
??
??
?
0.2 0.22 0.24 0.26 0.28
0 0 0.24 0.26 0.28
0.2 0.1 0 0 0.3
0.1 0 0 0.46 0.48
0.2 0 0 0.56 0.58
?
??
??
?
9
?
??
??
?
0.2 0.5 0.6 0.4 0.3
0.04 0.4 0.9 0 0
0.45 0.6 0.25 0 0
0.03 0.2 0.8 0 0
0.1 0.4 0.9 0 0
?
??
??
?
10
?
??
??
?
0.1 0.4 0.6 0.5 0.3
0 0 0.6 0.55 0.45
0 0 0.34 0.36 0.38
0 0 0.2 0.36 0.38
0 0 0.54 0.56 0.58
?
??
??
?
54
As can be seen in Table 3.3, the first five examples consider having hurricane force wind
speeds at all locations during forecasting periods. The other five also include locations that
are expected to have no hurricane force winds speeds at a specific forecasting period.
Figures 3.2, 3.3, 3.4, 3.5 and 3.6 display the Matlab?windspeed probability maps for
Example 1 created at 0h, 30h, 60h, 90h, and 120h, respectively. Note that the green line
corresponds to the forecasted storm track, and the red line represents the realized storm
path up to that forecasting period. Figure 3.2 shows only the forecasted track since it is
formed at the start of the forecasting period t = 0h. More specifically, it gives the prior
windspeed probability map, which is updated in 30h intervals to predict the posterior wind
speed probability maps as more windspeed information becomes available. In fact, each
wind speed probability map corresponds to a prior of its subsequent one. Similar maps can
be created for the other example problems.
West
North
102030405060708090100
10
20
30
40
50
60
70
80
90
100
Figure 3.2: Windspeed probability map at t = 0h.
55
West
North
102030405060708090100
10
20
30
40
50
60
70
80
90
100
Figure 3.3: Windspeed probability map at t = 30h.
West
North
102030405060708090100
10
20
30
40
50
60
70
80
90
100
Figure 3.4: Windspeed probability map at t = 60h.
56
West
North
102030405060708090100
10
20
30
40
50
60
70
80
90
100
Figure 3.5: Windspeed probability map at t = 90h.
West
North
102030405060708090100
10
20
30
40
50
60
70
80
90
100
Figure 3.6: Windspeed probability map at t = 120h.
57
The following data is used to solve each of the example problems: c1 = 20,ct+1 =
ct + 1,s = 100,h = 15,n = 5 location. The demand information at locations are given in
Table 3.4.
Table 3.4: Demand information
Location i Regular Demand Yi Hurricane Demand Zi
1 N(500,25) N(1000,100)
2 N(550,30) N(1500,125)
3 N(600,35) N(2000,150)
4 N(650,40) N(2500,175)
5 N(700,45) N(3000,200)
Note that the DMs have the option of giving an order each time an updated windspeed
probability map is obtained. Therefore, only the cumulative 30 hour interval windspeed
probability updates are considered for the example problems. Mathematica 5.2?is also used
to obtain decision rules for these problems. The results are shown in Table 3.5.
Table 3.5: Example results
Example Decision Period t? Stocking Quantity Qt Expected Cost EC
1 1 8567.79 $198,995
2 1 6299.83 $140,188
3 1 7685.85 $203,198
4 1 8619.32 $174,569
5 2 6751.26 $196,656
6 3 5295.66 $155,180
7 3 7156.1 $132,302
8 3 5829.81 $122,755
9 4 4884.36 $112,849
10 4 6250.24 $145,707
The results suggest that the DMs are inclined to wait to give their order/production
decisions related to the hurricane supplies as no hurricane force windspeeds are predicted
at some locations during a forecasting period. This finding is consistent with intuition in
that the DMs would like to reduce the risk associated with demand uncertainty by obtaining
more accurate windspeed probability predictions as the storm evolves over time.
58
Table 3.5 also suggests that the optimal order quantities associated with the examples,
inwhichalllocations are expectedtohavehurricaneforcewindspeeds are larger, onaverage,
than the other examples where cyclone windspeeds that are less than hurricane force wind
speeds are predicted for some locations. Finally, the results are encouraging in the sense
that they are consistent with the previous findings ([58]).
3.6 Summary
This chapter discusses a disaster recovery planning problem encountered by suppliers
who experience a demand surge for various products such as flashlights, batteries, and gas
powered generators in case of a hurricane event. The decision framework is also applicable
to a disaster relief problem faced by various service organizations. This chapter presents a
more sophisticated hurricane prediction model than [58] by including both tropical cyclone
track and intensity predictions into the decision framework. More specifically, the hurricane
stocking problem is formulated as an optimal stopping problem with Bayesian prediction
updates based on these hurricane characteristics. The proposed model and its solution
methodology is based on windspeed probabilities as opposed to maximum sustained wind
speeds used in [58]. The proposed approach utilizes windspeed probability predictions to
form a disaster recovery plan to manage the hurricane related risks. The Bayesian wind
speed probability updates are integrated into an optimal stopping framework to determine
the optimal ordering policy such that a balance is maintained between forecast accuracy and
cost efficiency. This approach provides suppliers a better forecast of the hurricane season
so that they can accommodate demand fluctuations on time at their retailer locations at a
minimum cost.
59
The solution methodology presented in this chapter will lead suppliers to establish
disaster recovery plans that are most appropriate for managing the risks associated with
hurricane events. An area for future research is to search for new methods, other than the
windspeed probabilities predicted within certain regions, to enhance the efficiency of the
hurricane prediction model. Another possibility is to explore a multiple product version of
the model with more sophisticated decision rules or loss functions.
60
Chapter 4
Advanced Inventory Planning for the Hurricane Season
This chapter addresses a stochastic inventory control problem for manufacturing and
retail firms who expose to challenging procurement and production decisions caused by
the hurricane events. The hurricane stocking decisions made in advance of the season are
affected by the general predictions regarding the ensuing hurricane season, such as the ex
pected number of hurricane landfalls. These kinds of predictions are issued by the NHC up
to six months in advance of the season. The inventory planning problem is characterized by
multiple periods before the hurricane season in which the inventory manager has the option
of adjusting the inventory level during each of these planning periods. More specifically, it
is assumed that the production / inventory planning horizon spans several months before
the beginning of the hurricane season. During these prehurricane season months, manu
facturing and retail organizations plan emergency supply inventory levels for the ensuing
season in addition to satisfying demands for these products that occur before the season.
In this chapter, it is assumed that hurricane season demand predictions are revised at
the beginning of each preseason planning period, and that these demand predictions are
correlated to landfall hurricane count rate predictions. A stochastic programming model is
introduced to determine production / inventory decisions that account for prehurricane sea
son demands as well as anticipated demands during the hurricane season. Hurricane season
demand predictions, which are updated and observed at the beginning of each preseason
period, are represented as a Markov chain based on a hurricane landfall count prediction
model. The historical hurricane landfall counts along with hurricane prediction related index
61
values are used as for the historical data matrix. The states of the Markov chain are de
fined by a finite number of hurricane count rate predictions. The predicted hurricane count
rate probabilities are empirically analyzed to calculate the stationary transition probabili
ties. The longrun properties of Markov chains are used to come up with the steadystate
probabilities, over which the hurricane season demand distribution is described. Then, the
underlying demand distribution is described over the weighted probabilities that are deter
mined based on the hurricane season demand and preseason demand distributions. The
model and solution methodology described in this chapter optimize the tradeoff between
hurricane forecast accuracy and cost efficiency as a function of time. More specifically,
the DMs can make more accurate decisions as the season draws near. On the other hand,
procurement, production and logistics costs are more efficient during the earlier stages of
planning.
In section 4.1 a review of related literature is presented. In section 4.2, the stochastic
programming inventory model is explained. In section 4.2.1, the hurricane count prediction
model and the selected predictors are introduced. In addition to this, the general idea behind
the Markov Chain approach in generating demand scenarios is discussed. A numerical
example is also implemented in section 4.2.2. In section 4.3, the scenario reduction approach
in stochastic programming is introduced followed by a hurricane stocking example problem.
In section 4.3.1, the numerical example is solved with a heuristic algorithm. Comparisons
with optimal and heuristic reduction methodologies are also made. Finally, conclusions and
managerial implications are presented in section 4.4.
62
4.1 Literature Review
In this section, relevant research from the supply chain inventory literature that is most
applicable to this problem and to our solution approach is summarized. Relevant research
includes inventory control for humanitarian relief and supply chain management, stochastic
inventory models with Markovian demand predictions, and inventory models with more
than one period to prepare for the selling season.
Beamon [10] compares and contrasts the commercial supply chain and the humanitar
ian relief chain to identify the challenges of relief logistics planning. Beamon [13] addresses
this issue by developing a multiple supplier inventory model that determines optimal order
quantities and reorder points for longterm emergency relief response. The expected num
ber of units held per cycle and corresponding expected cycle lengths are determined both
with and without emergency orders. Initially, the optimal order quantity is determined
using a prespecified stock out risk. Then the reorder quantity and level are optimized
based on reordering, holding and backorder costs. Beamon and Balcik [12] evaluate in
ventory management strategies applied to emergency cases in South Sudan. They develop
quantitative inventory management strategies for humanitarian relief. Kov?acs and Spens
[54] describe the unique characteristics of humanitarian logistics, and emphasize that the
humanitarian logistics should benefit from business logistics. Oloruntoba and Gray [63]
identify the characteristics of business supply chains, and apply them to the humanitarian
aid supply chain. They develop an agile supply chain model for humanitarian aid. Kapucu
[46] examines the role of nonprofit organizations with respect to responding to a catas
trophic disaster via a case study. Lodree and Taskin [58] introduce newsvendor variants to
assess the risks and benefits associated with inventory decisions with respect to preparing
63
for supply chain disruptions or disaster relief efforts. Alexander Smirnov et al. [76] develop
a decisionmaking approach for disaster response operations, and present the similarities
of industrial environment and disaster relief operations in decisionmaking. Beamon and
Balcik [11] develop an effective performance measurement system for the relief sector. They
compare performance measurement in the humanitarian relief chain with in the commercial
supply chain to develop new performance metrics for the humanitarian relief chain.
The inventory models where the demand distribution is defined via a Markovian process
is also relevant to this research. Karlin and Fabens [47] introduce a Markovian demand
model. They claim that if each demand state is defined by different numbers, a basestock
type inventory policy can be obtained. Iglehart and Karlin [41] prove that a basestock
policy is optimal for a demand process modeled by a discretetime Markov chain. Song and
Zipkin [77] examine an inventory model, in which the fluctuations in the demand rate is
represented by a continuoustime Markov chain. They determine the optimal ordering policy
for a linear cost model through a modified valueiteration algorithm. They also show that
their algorithm yield slightly better solutions than a standard value iteration approach. A
fixed ordering cost model is also explored and it is shown that a statedependent basestock
policy is optimal. Beyer et al. [15] show the existence of an optimal Markov policy for the
discounted and averagecost problems where the demand is introduced as unbounded, and
costs have polynomial growth. They also prove that the basestock policy is optimal even
when the ordering cost consists of additional components to that of the fixed cost given
a convex surplus cost function. Cheng and Sethi [19] examine an inventorypromotion
decision problem, in which the demand state is represented both by the environmental
factors and the promotion decisions. In this study, they determine a threshold inventory
64
level such that if this level is exceeded, it is favorable to give a promotion decision. They
use dynamic programming to come up with the optimal inventory and promotion decisions
for the finite horizon problem. Abhyankar and Graves [2] consider a Markovmodulated
Poisson demand process, and determine closedform approximations both for the inventory
and the service level. In order to hedge against cyclic demand variability, they suggest using
an intermediatedecoupling inventory. Additionally, they develop an optimization model to
determine the tradeoffs between inventory investments and customer service. Finally, [18]
examine a serial multistage inventory problem with Markovmodulated demand. They prove
that statedependent basestock policy is optimal. Additionally, they develop an algorithm
to determine the optimal basestock levels.
This study can also be described as an inventory model with more than one period
to prepare for the selling season. The reader is referred to [75] for an extensive list of
references related to this problem. In this section, representative papers for this type of
inventory model are presented. For instance, [62] examine the sales potential of a product,
which is treated as a subjective random variable whose distribution is updated adaptively
using Bayes?s rule as the sales data becomes available. They formulate this problem as
a dynamic program and introduce computationally efficient procedures for special cases.
Hausman and Peterson [37] extend the work in [62] to the case of multiple products with
limited production capacity in each period. They develop and compare three heuristics to
solve the multiproduct production planning problem. Bitran et al. [34] investigate a system
that produces several families of style goods. The problem is formulated as a deterministic
mixed integer programming problem that provides an approximate solution. Matuso [61]
65
uses a continuous treatment of time and formulated the problem examined by [34] as a two
stage stochastic sequencing model. A heuristic procedure is developed to solve the problem.
Kodama [53] derives the optimality condition for a singleperiod problem in which partial
returns and purchases are allowed in case of a surplus and shortage, respectively. The
author shows that this problem is a special case of the singleperiod problem when surplus
inventory cannot be carried over between periods.
4.2 Stochastic Programming Model
The objective of this study is to determine an optimal ordering policy such that (i)
demand at each prehurricane season period is met and (ii) reserved supplies are stored
for the ensuing hurricane season in a cost effective way. The preseason planning problem
is introduced as a stochastic programming model, in which the procurement/production
decisions are given to minimize the expected total cost. The assumptions of the stochastic
inventory model are given as follows.
Assumption 11 The annual hurricane landfall count ?nh is assumed to follow a Poisson
distribution with rate ?.
The Poisson distribution is used to express the probability of hurricane counts occurring in
a fixed period of time. From a statistical standpoint, it is appropriate to describe the distri
bution of the hurricane counts as a Poisson distribution because they occur independently
of the time since the last event, and with a known average rate.
Assumption 12 Hurricane season demand is a linear function of predicted hurricane land
fall count rates ?t during month t.
66
Assumption 12 enables us to define the hurricane season demand distribution in terms of
the hurricane count rate probabilities. In real applications, the demand is influenced by
various attributes such as hurricane windspeeds, radius of the storm, and the population
of the locations hit by hurricanes. For illustrative purposes, the hurricane season demand
is introduced as a linear function of ?t.
Assumption 13 Preseason demand and hurricane season demand are introduced as in
dependent random variables.
As consistent with intuition, demand tends to be higher during the hurricane season com
pared to the demand observed during the preseason months. More specifically, they are
not correlated, and should be described as independent variables.
The nature of the stochastic problem requires making multiperiod ordering decisions
by considering the uncertainty associated with demand realizations. The inventory control
theory introduces dynamic programming, optimal control and stochastic programming as
the main approaches to solve multiperiod inventory problems. In this study, the inventory
control problem is formulated as a multistage stochastic programming with recourse that
can be reduced to a discreteequivalent linear program. Dupa?cov?a et al. [28] formulate the
twostage and multistage stochastic programs with recourse as follows:
min
d?D
EPf(x,d) =
integraldisplay
X
f(x,d)P(dx) (4.1)
whereD is the set of feasible firststage decisions, andX ?D. f(?,d),d ? D is the objective
function of the stochastic model. P is the probability measure on the Borel ?field and the
subset X.
67
The stochastic programs determine the optimal decisions d given a realization of the
stochastic process x. The basic idea behind this approach is the concept of recourse. Re
course gives DMs the flexibility to make further decisions after the realization of the stochas
tic elements of the problem. Recourse decisions preserve the feasibility of the constraints of
the problem. For instance, a twostage recourse problem requires to (i) choose one decision
variable for each decision that must be made in stageone (ii) determine the possible states
of the world that might be realized next period (iii) take some recourse action after the real
izations of the stochastic elements. This stochastic programming problem can be extended
to a multistage case, in which the realizations of the stochastic elements are represented
as scenarios. In this case, the stochastic elements are often considered to have discrete
distributions.
Now the stochastic programming inventory problem is described as follows: A demand
realization occurs at the end of each period t. Let Qkt denote the order quantity at the
beginning of period t under scenario k, and let ct be the associated unit cost. Let Xkt
be a discrete random variable representing the total demand for the item during period t
under scenario k, and qkt the corresponding scenario probability. Let vkt denote the excess
inventory observed at the end of period t under scenario k, and let ht be the associated unit
holding cost. Similarly, ukt is the observed number of shortages at the end of period t under
scenario k with corresponding unit cost st. Then the multistage stochastic programming
problem can be expressed as the following linear program. The details about the model can
68
be found in [3].
min
Tsummationdisplay
t=1
Ksummationdisplay
k=1
qkt ?(ct ?Qkt +ht ?vkt +st ?ukt) (4.2)
Qkt +vk(t?1) +ukt ?vkt = xkt
vk0 = 0, k = 1,...,K
Qkt,vkt,ukt ? 0, t = 1,...,T, k = 1,...,K
This problem can be shortened by adding the nonanticipativity constraints:
Qkt = Qkprimet, ukt = ukprimet, and vkt = vkprimet for all k,kprime for which xk,[1,t] = xkprime,[1,t],t = 1,...,T
These constraints ensure that the decisions taken at period t do not depend on the future
observations of the stochastic process, but on the information available up to periodt, x[1,t].
4.2.1 Demand Scenario Probabilities
In order to generate scenarios for the demand process, the Markov chain associated with
hurricane count rates is used. The stationary transition probability pij of the Markov chain
corresponds to the probability of predicting hurricane landfall count rate of j = 0,...,5
during the current preseason month given that the previous month?s prediction is i =
0,...,5. These probabilities are predicted based on the hurricane count prediction model
developed by [30]. Elsner et al. [32] further extend [30] to provide a sixmonth forecast
horizon for annual hurricane counts along the U.S. coastline. Elsner et al. [32] determine
that the North Atlantic Oscillation (NAO) and the Atlantic SeaSurface Temperature (SST)
variations are the most significant predictors of the annual hurricane landfall. The NAO
is measured based on the fluctuations in the difference of sealevel pressure between the
69
subtropical high and the polar low. The corresponding NAO index data is used to predict
the general path hurricanes will take when they form. On the other hand, SST variations
referred to as the Atlantic Multidecadal Oscillation (AMO) indicate how much temperatures
depart from what is normal for that time of year. Therefore, AMO index is used to predict
how active the basin will be in terms of the number of hurricanes. These AMO data
are determined as the difference between the current observation and the corresponding
climatological value. A Bayesian approach to regression analysis is conducted to generate
samples of posterior parameters ?t+1 given the prior estimates of these parameters ?t at
period t. Eq. (4.3) shows the ensuing Poisson regression equation:
log(?) = ?0t +?1t ?AMO+?2t ?NAO+?3t ?(AMO?NAO) (4.3)
where the response is the observed annual hurricane landfall count nh = 0,...,51, and
the predictors are the NAO and AMO index. Note that the regression coefficients ?t =
(?0t,?1t,?2t,?3t) are introduced as random variables as opposed to constants.
Thepriordistributionfor?t isspecifiedbyamultivariatenormaldistribution,MVN(?t,
??1t ). In order to determine bootstrap prior estimates for?t and?t, the regression equation
is fitted to the set of hurricane counts using only the April index for NAO and AMO, and
a large number of bootstrap samples are generated with replacement. Then, landfall count
rate ?t is predicted during preseason month t for the forecasted hurricane season using
these bootstrap samples and the observed NAOt and AMOt data during month t as given
1Note that response is nh when observed data is used
70
in Eq. (4.4).
log(?t) = ?0t +?1t ?AMOt +?2t ?NAOt +?3t ?(AMOt ?NAOt) (4.4)
The regression equation given in Eq. (4.3) is run with the corresponding May index data
to form the likelihood function gt(?t?t). The posterior density of ?t is then determined
conditioned on these observed hurricane landfall counts f(?t+1?t). In order to analyze the
posterior distribution f(?t+1?t), the Gibbs Sampler algorithm is implemented. The Gibbs
Sampler is a MCMC method used to generate samples from the joint distribution of two or
more variables for highdimensional situations. The sampler works by iteratively generating
values from each distribution in turn, using parameter values from the previous iteration
to generate successive values. The Gibbs Sampler is run to generate samples of regression
coefficients. The posterior coefficient values ?t+1 = [?0t+1,?1t+1,?2t+1,?3t+1] are obtained
after convergence of the Markov chain is achieved. These stationary coefficient values are
used to predict landfall count rate ?t+1 for the ensuing hurricane season. Eq. (4.5) is used
to generate predictive samples given the observed NAO and AMO index values associated
with month t+1.
log(?t+1) = ?0t+1 +?1t+1 ?AMOt+1 +?2t+1 ?NAOt+1 +?3t+1 ?(AMOt+1 ?NAOt+1) (4.5)
Eq. (4.6) gives the predictive density of hurricane landfall count rate ht+1(?t+1?t):
ht+1(?t+1?t) =
integraldisplay
gt+1(?t+1?t+1)ft+1(?t+1?t)d? (4.6)
71
where ?t corresponds to an observed prediction, and ?t+1 is the predicted prediction. Ad
ditionally, gt+1(?t+1?t+1) is the density function of ?t+1 given the posterior parameter
estimates ?t+1, and ft+1(?t+1?t) is the posterior distribution of ?t+1.
Predictive samples of ?t+1 are then generated using Gibbs sampler. The predictive
probabilities associated with landfall count rate are used to determine the stationary tran
sition probabilities. More specifically, the stationary transition probabilities, pij, i,j =
0,...,5, are obtained by evaluating P{?t+1 = j?t = i}. Then, the steadystate equations
are formed based on these stationary transition probabilities. Eq. (4.7) gives the steady
state equations:
pij =
5summationdisplay
i=0
piipij for j = 0,1,...,5 (4.7)
5summationdisplay
j=0
pij = 1
where pij is referred to as the steady state probability of the Markov chain.
The steadystate probabilities give the predictive probabilities of observing hurricane
count states. For instance, pi1 gives the predictive probability of observing exactly one
hurricane during the hurricane season. These probabilities are used to calculate the scenario
probabilities for the demand process. The inventory control problem is then solved using
the stochastic programming model described in the previous section.
4.2.2 Numerical Example
For the numerical example, the preseason months of April and May are considered as
the inventory planning periods. The hurricane season (June 1November 30) as a whole is
72
considered to be one period. Table 4.1 gives the historical NAO and AMO index values
associated with the preseason months of April and May along with the observed landfall
counts for each hurricane season.
Table 4.1: Hurricane landfall count and AprilMay NAO and AMO index derived by the
1950?1979 data
April May
Year nh NAO AMO NAO AMO
1950 3 1.61 0.16 1.73 0.34
1951 3 0.45 0.43 2.11 0.29
1952 0 2.79 0.06 0.94 0.28
1953 1 1.6 0.33 0.75 0.14
1954 3 0.26 0.15 0.91 0.14
1955 3 2.4 0.18 0.33 0.11
1956 3 1.9 0.11 4.54 0.14
1957 1 0.62 0.46 0.84 0.23
1958 1 1.79 1.02 1.1 0.78
1959 0 1.51 0.08 2.22 0.36
1960 3 1.93 0.13 0.07 0.1
1961 2 0.71 0.08 0.94 0.11
1962 1 0.74 0.33 0.1 0.43
1963 0 0.46 0.44 1.91 0.23
1964 1 0.95 0.05 2.51 0
1965 4 2.14 0.16 0.08 0.27
1966 1 1.18 0.47 1.51 0.2
1967 2 0.76 0.18 0.46 0.12
1968 1 0.71 0.18 1.5 0.14
1969 1 1.11 0.91 0.23 0.78
1970 2 2.52 0.39 1.87 0.47
1971 1 3.15 0.23 0.62 0.36
1972 3 0.22 0.08 1.24 0.34
1973 1 2.61 0.02 0.37 0.11
1974 0 2.3 0.9 0.01 0.99
1975 1 0.84 0.47 2.42 0.72
1976 1 1.53 0.44 1.2 0.68
1977 1 1.07 0.19 1.62 0.14
1978 1 3.12 0.53 0.37 0.14
1979 0 0.79 0.4 1 0.46
This data is used to predict the hurricane landfall count rate for the forecasted hurricane
season. The NAO values are obtained from the Climatic Research Unit and the AMO val
ues are obtained from the Climatic Diagnostics Center. These data are used as inputs for
the WinBUGS (Windows version of Bayesian inference using Gibbs Sampling) to analyze
73
the predictive distribution of hurricane count rates via a Bayesian analysis. The hurricane
landfall count rate is bounded such that ? = 0,1,...,5. Initially, Bootstrap priors are
specified for the model coefficients. These bootstrap priors and the NAO and AMO index
data associated with period t are used to estimate ?t, which corresponds to the predicted
hurricane landfall count rate at period t. Then, the Gibbs sampler is run to update the pri
ors until the convergence of the posterior coefficients is achieved. Table 4.2 is continuation
of Table 4.1, and demonstrates the 1980?2007 hurricane related data.
Table 4.2: Hurricane landfall count and AprilMay NAO and AMO index derived by the
1980?2007 data
April May
Year nh NAO AMO NAO AMO
1980 3 0.03 0.42 2.26 0.73
1981 1 3.04 0.48 0.05 0.56
1982 0 0.99 0.12 1.1 0.1
1983 0 1.01 0.74 0.57 0.57
1984 1 0.33 0.33 2.34 0.33
1985 5 0.34 0.55 2.13 0.57
1986 2 0.93 0.45 2.16 0.45
1987 1 2.59 0.39 0.81 0.36
1988 1 2.39 0.33 1.24 0.15
1989 3 0.48 0.7 1.16 0.69
1990 0 1.77 0.2 1.19 0.34
1991 1 1.48 0.29 0.04 0.26
1992 1 1.32 0.16 0.8 0
1993 1 0.83 0.08 2.59 0.05
1994 0 1.38 0.45 1.43 0.53
1995 2 1.81 0.41 0.36 0.55
1996 2 0.31 0.46 1.5 0.38
1997 1 0.97 0.2 1.35 0.23
1998 3 0.39 0.69 1.26 0.92
1999 3 0.43 0.01 1.03 0.05
2000 0 3.34 0.19 0.31 0.17
2001 0 1.24 0.14 0.09 0.09
2002 1 0.91 0.17 0.05 0.08
2003 2 1.74 0.03 1.17 0.09
2004 5 1.08 0.53 0.67 0.27
2005 5 0.71 1 0.13 1.18
2006 0 0.57 0.44 0.22 0.5
2007 2 0.1 0.47 0.62 0.19
74
iteration
8000 10000 15000 20000
]
1
[
a
t
e
b
5
.
0

5
.
0
0
.
1
5
.
1
Prior Density Posterior Density
beta[1] sample: 15000
beta[1]
0.5 0.0 0.5 1.0 1.5
)
]
1
[
a
t
e
b
(
P
0
.
0
0
.
2
beta[1] sample: 15000
beta[1]
0.5 0.0 0.5 1.0 1.5
)
]
1
[
a
t
e
b
(
P
0
.
0
0
.
2
Figure 4.1: WinBugs posterior regression coefficients.
Figure 4.1 shows the obtained prior and posterior density function of the regression coeffi
cient ?1. Similar density functions are obtained for the other coefficients.
Visual inspections reveal that the convergence of the chain is observed after 15,000 simu
lations. The predictive inference for ?t+1 is made by setting the hurricane landfall count
rate to NA (not available) for the forecasted hurricane season. More specifically, the Gibbs
sampler is run once again to generate ?t+1 conditional on the posterior coefficients ?t+1
and the observed NAO and AMO index data at period t+ 1. Figure 4.2 demonstrates the
WinBugs output associated with the landfall count rate predictions, and Figure 4.3 shows
the corresponding hurricane landfall count predictions.
75
iteration
8000 10000 15000 20000
a
d
b
m
a
l
0
.
0
0
.
1
0
.
2
0
.
3
Prior Density Posterior Density
lambda_t sample: 15000
lambda_t
0.0 5.0
)
t
_
a
d
b
m
a
l
(
P 0
.
0
4
.
0
lambda_t+1 sample: 15000
lambda_t+1
0.0 1.0 2.0 3.0 4.0
)
1
+
t
_
a
d
b
m
a
l
(
P
0
.
0
0
.
1
0
.
2
Figure 4.2: WinBugs predictive hurricane count rates.
iteration
8000 10000 15000 20000
)
1
+
t
(
h
_
n
0
.
0
0
.
2
0
.
4
0
.
6
0
.
8
Prior Density Posterior Density
n_ht sample: 15000
n_ht
0 5 10
)
t
h
_
n
(
P
0
.
0
2
.
0
n_h(t+1) sample: 15000
n_h(t+1)
0 3 6 9
)
)
1
+
t
(
h
_
n
(
P 0
.
0
2
.
0
4
.
0
Figure 4.3: WinBugs predictive hurricane counts.
76
The predicted hurricane count rate probabilities are used to determine the stationary
transition probabilities of the sixstate Markov chain. For instance, p11 is obtained by
evaluating P{?t+1 = 1?t = 1} = 0.29 empirically. Remaining entries are obtained in a
similar manner. Eq. (4.8) shows the resulting transition matrix. Recall that the states of
the Markov chain correspond to predicting exactly ? = 0,...,5, respectively.
P =
?
??
??
??
??
??
??
??
??
?
0.23 0.29 0.22 0.15 0.09 0.02
0.22 0.29 0.22 0.15 0.09 0.03
0.23 0.29 0.22 0.13 0.1 0.03
0.21 0.28 0.22 0.16 0.1 0.03
0.2 0.3 0.25 0.18 0.05 0.02
0.21 0.31 0.18 0.18 0.1 0.02
?
??
??
??
??
??
??
??
??
?
(4.8)
By substituting pij values into the steadysate equations, the following set of equations
are obtained.
pi0 = 0.23pi0 +0.22pi1 +0.23pi2 +0.21pi3 +0.2pi4 +0.21pi5 (4.9)
pi1 = 0.29pi0 +0.29pi1 +0.29pi2 +0.28pi3 +0.3pi4 +0.31pi5
pi2 = 0.22pi0 +0.22pi1 +0.22pi2 +0.22pi3 +0.25pi4 +0.18pi5
pi3 = 0.15pi0 +0.15pi1 +0.13pi2 +0.16pi3 +0.18pi4 +0.18pi5
pi4 = 0.09pi0 +0.09pi1 +0.1pi2 +0.1pi3 +0.05pi4 +0.1pi5
pi5 = 0.02pi0 +0.03pi1 +0.03pi2 +0.03pi3 +0.02pi4 +0.02pi5
1 = pi0 +pi1 +pi2 +pi3 +pi4 +pi5
77
The simultaneous solutions to the last six equations provide a unique solution as
pi0 = 0.22, pi1 = 0.29, pi2 = 0.22 (4.10)
pi3 = 0.15, pi4 = 0.09, pi5 = 0.03
The steadystate probabilities given in Eq. (4.10) are used to define the hurricane season
demand distribution.
Suppose the likely outcomes of the hurricane season demand are 200,250,300,350,400,
and 450 corresponding to ? = 0,1,...,5, respectively. Assuming that each period?s pre
season demand is equally likely to be 100 or 150, the underlying stochastic demand distri
bution can be described as shown in Table 4.3.
Table 4.3: Demand distribution
Preseasonal demand Probability Hurricane season demand Probability Demand Weighted probability
100 0.5 200 0.22 650 0.17
150 0.5 250 0.29 750 0.27
300 0.22 850 0.23
350 0.15 950 0.17
400 0.09 1050 0.12
450 0.03 1150 0.04
Using the data in Table 4.3, the stochastic programming model (4.2) becomes
min
2summationdisplay
t=1
36summationdisplay
k=1
qkt ?(ct ?Qkt +ht ?vkt +st ?ukt) (4.11)
Qkt +vk(t?1) +ukt ?vkt = xkt, t = 1,...,2, k = 1,...,36
vk0 = 0, k = 1,...,36
Qkt,vkt,ukt ? 0, t = 1,...,2, k = 1,...,36
78
plus the nonanticipativity constraints:
Qk1 = Q1, k = 1,...,36 (4.12)
Qk2 = Q21, uk2 = u21, vk2 = v21, k = 1,...,6
Qk2 = Q22, uk2 = u22, vk2 = v22, k = 7,...,12
Qk2 = Q23, uk2 = u23, vk2 = v23, k = 13,...,18
Qk2 = Q24, uk2 = u24, vk2 = v24, k = 19,...,24
Qk2 = Q25, uk2 = u25, vk2 = v25, k = 25,...,30
Qk2 = Q26, uk2 = u26, vk2 = v26, k = 31,...,36
where t = 1,2 corresponds to the preseason months of April and May, respectively. The
weighted probabilities are determined using the preseason and hurricane season demand
probabilities. For instance, 0.17 is calculated as follows. Similar calculations are made to
develop the demand distribution.
0.17 = 300?(0.22?0.5)+350?(0.22?0.5)300?0.11+350?0.255+400?0.255+450?0.185+500?0.12+550?0.06+600?0.015
(4.13)
Excel Solver?is used to obtain the optimal ordering policy for the numerical example.
The following data are used to solve the linear program, and the results are shown in Table
4.4: c1 = 20,ct+1 = ct ?4,s = 300,ht = ct/2.
79
Table 4.4: Results of the original model
Q?1 Scenarios Q?2 Expected Cost
1700 1 0 $87,243.6
2 0
3 0
4 100
5 200
6 300
Note that the nonanticipativity constraints are added to the problem. In other words,
there is only one first period decision, namely Q1, and there are 6 second period ordering
and recourse decisions, one for each scenario. There are 91 variables and 42 constraints in
the linear program. The solution yields a total expected cost of $87,243.6. The optimal
solution values in Table 4.4 can be interpreted as follows: Order/produce 1700 units at
the beginning of the current period (April). If the observed demand associated with the
month of April is 950, then order/produce 100 units at the beginning of May. Similarly,
order/produce 200 and 300 units at the beginning of May if April?s demand is 1050 and
1150, respectively.
4.3 Scenario Reduction
In reallife problems the true probability distribution can have many realizations. In
order to numerically solve such problems, it is necessary to find an approximation of the
stochastic process that is defined by a finite number of realizations. This discretization
process is referred to as a scenario tree. The scenario tree serves to model the uncertainty
associated with the stochastic process. A scenario is then defined as a possible realization
of the underlying stochastic process. In the literature, there exists a wide range of scenario
generation methods such as moment matching, conditional sampling, bootstrap, Monte
80
Carlo sampling, and Markov chain. Note that the performance of the scenario generation
method can also be improved by increasing the initial number of scenarios, or simply by
improving the sampling method as discussed in ([48]).
In most practical cases, the original tree has a large scale branching structure. There
fore, the size of the tree should be decreased to eliminate the computational burden. For
these situations, [38] and [28] introduce an optimal scenario reduction methodology based
on the probability metric minimization. They define the optimal scenario reduction of a
given discrete approximation as the determination of a scenario subset of prescribed car
dinality that is closest to the original distribution. Heitsch and R?omisch [38] show that
the stochastic programs are stable in terms of a FortetMourier probability metric. Let
P = summationtextNi=1pi ??xi be the original discrete probability distribution, and Q = summationtextjnegationslash?J qj ??xj be
its optimal reduced distribution where ?x denotes the Dirac measure assigning unit mass to
x. Then the probability metric with the FortetMourier structure is defined in [38] as
?c(P,Q) := sup
f?Fc

integraldisplay
X
f(x)P(dx)?
integraldisplay
X
f(x)Q(dx) (4.14)
with Fc being the class of continuous functions having the form
Fc := {f : X ? R : f(x)?f(?x) ?c(x,?x) for all x,?x?X} (4.15)
Here, c is a continuous symmetric function that is selected such that the following Lipschitz
condition described in [38] is satisfied given a nondecreasing function g : R+ ? R+\{0}
f(x,d)?f(?x,d) ?g(d)?c(x,?x) (4.16)
81
In this model, c is defined as a metric such that c(x,?x) = x??x, and ? is the Euclidean
norm.
Dupa?cov?a et al. [28], and Heitsch and R?omisch [38] prove that the Kantorovich function
??c(P,Q) is an estimate of the upper bound value of ?c(P,Q), i.e. ?c(P,Q) ? ??c(P,Q). They
use the Kantorovich function ??c, which represents the optimal value of a finitedimensional
linear program, to develop an optimal scenario reduction approach for a given discrete
approximation Q of P. The optimal reduction approach described in their papers suggests
considering the following Kantorovich probability distance:
??c(P,Q) = min
??
?
??
Nsummationdisplay
i,j=1
jnegationslash?J
c(xi,xj)??ij : ?ij ? 0,
Nsummationdisplay
i=1
?ij = qj,
Nsummationdisplay
j=1
jnegationslash?J
?ij = pi
??
?
?? (4.17)
D(J;q) := ??c
?
?
Nsummationdisplay
i=1
pi ??xi,
summationdisplay
jnegationslash?J
qj ??xj
?
?
where J ? {1,...,N} is the index of withdrawn scenarios with fixed cardinality. Based
on the optimal reduction concept, the optimal index set J?, and the optimal weights q?
are determined such that D(J;q) is minimized. Then, the new probabilistic weights qj,j ?
{1,...,N}\J are assigned to each remaining scenario xj,j negationslash?J using the following optimal
redistribution rule.
Theorem 4.1 (Heitsch and R?omisch [38]): Given J ? {1,...,N}, the probability distance
is defined as follows.
DJ = min
?
?D(J;q) : qj ? 0,summationdisplay
jnegationslash?J
qj = 1
?
? = summationdisplay
i?J
pi ?min
jnegationslash?J
c(xi,xj) (4.18)
82
and the minimum value is attained at:
q?j = pj +
summationdisplay
i?Jj(i)
pi for each j negationslash?J (4.19)
where j(i) ? argminjnegationslash?J c(xi,xj) for each i?J.
Theorem 4.1 implies that the new probability of a kept scenario is equal to the sum of its
original probability and of all probabilities of the closest withdrawn scenarios determined
based on the c metric. Then, the optimal index set J? for scenario reduction with given
cardinality is determined by solving the following problem formulated by ([38]).
min
braceleftBigg
DJ :=
summationdisplay
i?J
pi ?min
jnegationslash?J
c(xi,xj) : J ? {1,...,N},n(J) = N ?n
bracerightBigg
(4.20)
As discussed by [38] when n(J) = N ?1, Eq. (4.20) reduces to
min
j?{1,...,N}
Nsummationdisplay
i=1
pi ?c(xi,xj) (4.21)
Eq. (4.21) yields the best possible deterministic approximation of the initial distribution
such that the redistribution rule assigns q?j = 1 to the preserved scenario.
Now, the numerical example presented in the previous section is resolved using the
optimal scenario reduction concept. The number of deleted scenarios is fixed as n(J) = 4.
Table 4.5 gives the selected index sets and their corresponding probability distances, and it
reveals that J? = {1,3,5,6} gives the minimum distance with D?J = 60. Therefore, these
scenarios should be removed from the original set of scenarios.
83
Table 4.5: Probability distances
J DJ J DJ J DJ
{1,2,3,4} 212 {1,2,5,6} 81 {2,3,4,5} 119
{1,2,3,5} 140 {1,3,4,5} 86 {2,3,4,6} 94
{1,2,3,6} 132 {1,3,4,6} 61 {2,3,5,6} 70
{1,2,4,5} 90 {1,3,5,6} 60 {2,4,5,6} 80
{1,2,4,6} 82 {1,4,5,6} 70 {3,4,5,6} 109
TheDJ values shown in Table 4.5 are calculated using the Euclidean distances. For instance,
D?J is determined as follows.
D?J = 100?0.17+100?0.23+100?0.12+200?0.04 = 60 (4.22)
Table 4.6 gives the Euclidean distances c(xi,xj) used to determine the optimal weights for
the remaining scenarios.
Table 4.6: Euclidean distances (c metric)
(i,j) 2 4
1 100 300
3 100 100
5 300 100
6 400 200
The optimal weights for scenarios 2 and 4 are calculated using Eq. (4.19).
q?2 =
3summationdisplay
i=1
= 0.17+0.27+0.23 = 0.67 (4.23)
q?4 =
6summationdisplay
i=4
= 0.17+0.12+0.04 = 0.33
Stochastic programs can have more than one candidate scenario that has the same prob
ability distance to another scenario. For instance, in this example different values can be
assigned to optimal weights by incorporating the probability of deleted scenario i = 3 to
84
the scenario j = 4 since i = 3 has the same proximity both to j = 2,4. Eq. (4.24) shows
the resulting optimal weight values.
q?2 =
2summationdisplay
i=1
= 0.17+0.27 = 0.44 (4.24)
q?4 =
6summationdisplay
i=3
= 0.23+0.17+0.12+0.04 = 0.56
Then, the reduced versions of the stochastic programming models are solved using the pre
viously defined unit costs. The reduced models defined by n = 2 consist of 15 variables, and
6 constraints obtained by considering the nonanticipativity property of stochastic programs.
Table 4.7 demonstrates the results of these reduced models.
Table 4.7: Results of reduced models
Reduced model Q?1 Scenario i q?i Q?2 Expected Cost
1 1700 2 0.67 0 $76,180.8
4 0.33 0
2 1700 2 0.44 0 $82,290
4 0.56 200
The reduced model 1 recommends ordering 1700 units at the beginning of April, and no
order should be given in May. Similar to this model, the reduced model 2 indicates that
initially 1700 units should be ordered. However, if the demand realization at the end of
April is 950, then this model suggests ordering an additional 200 units. Otherwise no or
der should be given at the beginning of May. Table 4.8 presents the solutions for all the
potential reduced models given that n(J) = N ?n.
85
Table 4.8: Optimal values (solutions) of reduced models
n J? D?J Q?1 Expected Cost Relative Error
1 J = {1,2,4,5,6} 114 1700 $63,750 114/114 = 100%
2 J = {1,3,5,6} 60 1700 $76,180.8 60/114 = 52.63%
3 J = {1,5,6} 37 750 $127,695 37/114 = 32.46%
4 J = {5,6} 20 750 $153,660 20/114 = 17.54%
5 J = {6} 4 750 $163,260 4/114 = 3.50%
Table 4.8 also indicates that with increasing J, i.e., with increasing number of deleted
scenarios, the accuracy of the approximation decreases. The relative error is defined based
on the probability distances. More specifically, the error from the approximation of the
initial distribution is defined relative to the minimum distance associated with the deter
ministic approximation given by the reduced model with n = 1. For detailed information
about the relative error concept, the reader is referred to ([38]).
Another important issue that is worth exploring in stochastic programming is the
evaluation of the quality of the reduced scenario trees. In this context, one does not search
for the best approximation of the initial distribution but for the quality of the optimal
solutions (values). In this study, the accuracy is defined as the ratio of the optimal first
stage decisions obtained from the solutions of the reduced model and the original model.
Recall that only the (deterministic) first stage is the appropriate outcome of the stochastic
program. The tree serves to model the demand. uncertainty. Table 4.8 reveals that the
models carried by the number of scenarios n = 1,2 are reduced in an optimal way since
the value of firststage optimal decisions obtained from both of the reduced models are
exactly the same as that of the original model. On the other hand, reduced models having
n = 3,4,5 scenarios have an accuracy of 7501700 ?100 ? 44%. These results suggest that the
accuracy of the reduced trees tends to increase as they are supported with a small number
of scenarios. In other words, while the stochastic process is approximated with less number
86
of scenarios by implementing the scenario reduction approach, the accuracy of the values
(solutions) obtained from reduced models increases. This finding is consistent with intuition
such that one would expect to obtain more accurate results as the uncertainty associated
with the stochastic process reduces. Similar interpretations can be made for the expected
costs associated with the reduced models.
In order to evaluate the performance of the reduced models, the stochastic inventory
problem is initially solved on the reduced tree. Then, the values of all the firststage(root)
variables are fixed, and resolved on the original tree. As a result, the outofsample perfor
mance of the reducedtree solution is obtained assuming that the original tree is a good
enough approximation of the true distribution. The reduced models carried by n = 1,2 give
the same optimal expected cost value as the initial optimum value ($87,243.6). The reduced
models having 44% solution accuracy result in an expected cost of ($174,975). These find
ings indicate that as the accuracy of the optimal first stage solutions of the reduced model
decreases so too does the cost efficiency.
4.3.1 Heuristic Algorithm
In most of the stochastic problems where the stochastic process is represented by many
scenarios, Eq. (4.20) can not be solved optimally. Therefore, [28] and [38] develop heuristic
algorithms to approximate solutions of Eq. (4.20). In this study, the simultaneous backward
reduction algorithm, which includes all the previously deleted scenarios in each backward
step, is used. The algorithm determines an index set J to be removed from the original set
87
of scenarios based on the solution of the following equation given in ([38]):
lk ? arg min
lnegationslash?J[k?1]
summationdisplay
i?J[k?1]?{l}
pi ? min
jnegationslash?J[k?1]?{l}
c(xi,xj) (4.25)
where J[k?1] = {l1,...,lk?1} is defined as the index set of deleted scenarios up to and
including step k?1.
The stochastic programming inventory model is solved by implementing the simulta
neous backward reduction algorithm to illustrate the application of the heuristic algorithm.
The first step requires the deletion of only one scenario. Through the following steps, the
index lk is determined given that the previous index set {l1,...,lk?1} is optimal. Finally,
the optimal distribution rule given by Theorem 4.1 is implemented. Then, the number of
withdrawn scenarios is set asn(J) = 4. In step 1 of the algorithm, initially the scenarios are
defined as l = 1,2,3,4,5,6. Then, the optimal scenario to be removed is selected based on
the Euclidean probability distances. Table 4.9 shows the resulting distance matrix. Table
4.9 indicates that the minimum distance is achieved at l1 = 6 with D?J1 = 0.04?100 = 4.
Table 4.9: Euclidean distance matrix 1
(i,j) 1 2 3 4 5 6 DJ1 = pi ? c(xi,xj)
1  100 200 300 400 500 17
2 100  100 200 300 400 27
3 200 100  100 200 300 23
4 300 200 100  100 200 17
5 400 300 200 100  100 12
6 500 400 300 200 100  4
In step 2, the Euclidean probability distances are calculated for the kept scenarios
l = 1,2,3,4,5 as DJ2 = 21,31,27,21,20, respectively. The minimum distance is obtained
as D?J2 = 20 at l2 = 5. Table 4.10 shows the corresponding Euclidean distance matrix.
88
Table 4.10: Euclidean distance matrix 2
(i,j) 1 2 3 4 D?J2 =summationtexti?{6,5} pi ? minj?{1,2,3,4} c(xi,xj)
6 500 400 300 200 D?J2 = 0.04 ? 200 + 0.12 ? 100 = 20
5 400 300 200 100
In step 3, the remaining scenarios are revised as l = 1,2,3,4. Then, the corresponding
probability distances are calculated as DJ3 = 37,47,43,53. The minimum distance is
D?J3 = 37 that corresponds to l3 = 1. Table 4.11 gives the Euclidean distances associated
with D?J3.
Table 4.11: Euclidean distance matrix 3
(i,j) 2 3 4 summationtexti?{6,5,4} pi ? minj?{2,3,4} c(xi,xj)
6 400 300 200 D?J3 = 0.04 ? 200 + 0.12 ? 100 + 0.17 ? 100 = 37
5 300 200 100
1 100 200 300
In step 4, the scenarios l = 2,3,4 are considered for reduction. The probability dis
tances are calculated as DJ4 = 81,60,70, respectively. It can be inferred that the minimum
distance is D?J4 = 60 with l4 = 3. Table 4.12 gives the Euclidean distances used to deter
mine this minimum distance.
Table 4.12: Euclidean distance matrix 4
(i,j) 2 4 summationtexti?{6,5,1,3} pi ? minj?{2,4} c(xi,xj)
6 400 200 D?J4 = 0.04 ? 200 + 0.12 ? 100 + 0.17 ? 100 + 0.23 ? 100 = 60
5 300 100
1 100 300
3 100 100
Since n(J) = 4 is achieved, the algorithm is terminated. The reduced stochastic model is
solved for the remaining scenarios j = 2,4. As can be seen in Table 4.12, the optimal redis
tribution rule will result in the same optimal weights. Therefore, the obtained stochastic
89
programming model will have the same arguments as that of the optimally reduced one. Ad
ditionally, the simultaneous backward reduction algorithm yields optimal solution (value)
except for the reduced model supported by n = 1. This arises from the fact that the best
possible scenario i = 3 has already been deleted in the previous backward step. In other
words, while the optimal reduction method directly solves Eq. (4.20), the heuristic algo
rithm implements the scenario reduction process recursively in a stepwise fashion. For this
deterministic problem, the algorithm yields an optimal order quantity ofQ?1 = 1900 with an
expected cost of $71,250. The index set i = 6,5,1,3,2 is deleted and so the reduced model
is defined only by the scenarioj = 4. These results indicate that the simultaneous backward
reduction algorithm works reasonably well, and can be used in lieu of optimal reduction
approach to reduce the number of scenarios where the initial distribution is represented by
many scenarios.
4.4 Summary and Future Work
This chapter presents a stochastic inventory model that will assist organizations in
determining their optimal ordering policies as related to an upcoming hurricane season. In
this study, the preseason demand distribution is assumed to be known to the inventory
manager at the beginning of the inventory planning horizon. However, the hurricane season
demand distribution is based on monthly information updating. The hurricane landfall
count rate predictive probabilities, which are used to define the hurricane season demand
distribution as a Markov chain, are estimated via a widelyaccepted hurricane prediction
model developed by ([30]).
90
This chapter considers making preparations for the hurricane season demand two
months ahead of the season. The objective is to allocate some reserved stock to meet
the hurricane season demand while satisfying each period?s demand in a cost efficient way.
The uncertainty in the stochastic model is represented by a finite number of discrete demand
realizations. The cost minimization function, together with the constraints, constitutes the
structure of the model. Depending on the context of the inventory decision model, a dif
ferent type of solution approach can be implemented. For instance, dynamic programming
has been widely used to solve inventory problems with the application of the principle of
optimality due to its computational efficiency. However, a dynamic programming represen
tation of this problem could not be found since the randomness required to be defined in
the model increases with the addition of reserved stock and the Markovian hurricane sea
son demand random variables. Therefore, a stochastic programming model that is written
as a deterministic linear program is developed to determine the optimal ordering policy.
Different scenarios of the stochastic process are defined based on the underlying demand
distribution.
In reallife applications, the stochastic process is represented by many scenarios and/or
stages. For these situations, the stochastic programming approach becomes less efficient,
and requires the implementation of other algorithms. In this study, the scenario reduction
approach introduced by [38] is implemented to find the optimal set of scenarios to represent
the underlying distribution. It is determined that the optimum scenario reduction method
yield approximately 44% accuracy when at least half of the scenarios are removed. Addi
tionally, It is shown that both the accuracy and the performance of the reduced models
91
increase as more scenarios are withdrawn from the original distribution. This finding sug
gests that as the uncertainty associated with the demand process decreases, the accuracy
of the solution tends to increase.
In this study, it is also determined that the simultaneous backward reduction algorithm
yields optimum results for the considered reduced models except for the reduced model cor
responding to the deterministic approximation of the initial distribution n = 1. Therefore,
it can be inferred that the heuristic algorithm developed by [38] can be used to find the
reduced approximations of the stochastic processes introduced by many scenarios. However,
as the problem gets larger, the running time of the algorithm substantially increases.
This chapter introduces a stochastic inventory model that will enable quickresponse
logistics decisions as related to hurricane disaster relief. For future study, one might want
to develop approximations of a demand process described by many scenarios through the
implementation of the scenario generators. It is also worth exploring the quality of the
reduced scenario model where the discrete demand process is represented by many scenarios.
Additionally, a case can be developed in which a different demand distribution is introduced
for each state. Finally, the existence of an optimal statedependent basestock policy can
be investigated.
92
Chapter 5
Conclusions and Proposed Future Study
This dissertation discusses supply chain organizations disaster recovery problem whose
demand for hurricane supplies is influenced by various attributes of hurricane events. In this
context, the objective of disaster recovery planning is to minimize interruption to business
continuity during and after a hurricane. The proposed approach is applicable to predictable
disasters and leverages hurricane predictions to develop disaster recovery plans. In a hur
ricane, it is reasonable to expect various logistics issues that affect the supplier?s ability
to adequately address the order at a given time. For instance, transportation may not be
possible and overtime may be unavailable or available but inaccessible due to transporta
tion issues. Therefore, the objective of this dissertation can be stated more generally as
determining the optimal level of supply chain readiness with respect to hurricane prepared
ness(i.e. levels of supplies, equipment or personnel) and how long this preparation decision
should be postponed such that the tradeoff between logistics cost efficiency and hurricane
forecast accuracy is optimized.
Although this dissertation is presented as a stochastic production / inventory control
problem from the perspective of disaster recovery plan associated with a manufacturing
facility or retail organization, the framework is also relevant to disaster relief planning
problems encountered by service organizations. For example, military organizations and
electric power companies often preposition manpower and equipment in anticipation of a
potential disasterreliefoperation. This prepositioning decisionalso inherits the risk of over
preparation if prepositioning efforts exceed the demands of the disasterrelief operation,
93
and the risk of underpreparation if demands exceed prepositioning efforts. Notforprofit
service organizations such as the American Red Cross face similar risks and decisions related
to stocking and staffing evacuation shelters. Therefore, the primary intention in this study
is to develop an information updating framework as is relates to disaster recovery planning
for managing the hurricane related risks faced by different types of organizations.
In this dissertation, three models are introduced to assist these organizations in their
hurricane related inventory decisions. The first two model focus on an emergency inventory
planing problem in which the ordering decision is given sometime during the season when a
storm is first observed until it dissipates. The first model considers a one location problem.
The second model is introduced as an extension of the first model where the multi location
problem is taken into account. For both of these models, the unit ordering/production cost
is introduced as an increasing function of time, which is influenced by the expectation of
a demand surge during the evolution of an observed storm. Since hurricane characteristics
can be predicted with more accuracy during the later stages of the planning horizon rela
tive to the earlier stages, the inventory control problem is formulated as optimal stopping
problem with Bayesian updates, where the updates are based on hurricane predictions. The
information updating framework is introduced by applying a sequential statistical decision
approach with fixed sample size. The samples consist of observed maximum windspeeds
at a specific location and sustained windspeeds at different locations, respectively. Two
different classes of demand are defined over the observed sequential samples. In the first
model, empirical methodologies are implemented to illustrate the proposed approach. On
the other hand, the second model is developed based on a widelyrecognized statistical pre
diction model to investigate the managerial applications inferred from the decision process.
94
Additionally, Matlab?and Mathematica?software programs are used to solve the hurricane
stocking problems and the corresponding optimal ordering policies are determined. The
results of the models are encouraging since they are consistent. The obtained ordering
policies reveal that the DMs tend to wait to give their inventory related decisions as no
extreme hurricanes or hurricane force windspeeds are observed at a specific location or
at different locations, respectively. For these situations, the ordering quantities are also
relatively smaller as consistent with intuition.
The extended model that accounts for the ordering disruptions is planned to be inves
tigated based on other decision rules or loss functions. In this study, the stocking quantity
represents either supplies, equipment or personnel. For future research, the author might
explore an aggregated model that account for all the resources. Additionally, the demand
rate or multiple demand classes will be defined over multiple hurricane attributes to make
the model more realistic. The author also plans to examine the case with random demand
realizations at different locations. In reallife applications the magnitude of the demand
surge is influenced by the fluctuations in the market as a result of hurricane events. There
fore, the author plans to examine the relationship between the hurricane demand rate and
the market value. In this context, the value of information may be investigated under a
?Real Options? framework. In the first two models, Bayesian sequential decision models
are developed based on a fixed sample size. As an extension of this base model, the author
might also randomize the duration of an observed storm?s evolution to account for various
sample sizes.
The third model investigates preseason inventory planning problem with respect to
preparing for a potential hurricane activity. The objective is to determine an optimal
95
ordering policy such that an appropriate amount of hurricane supplies are reserved for
the ensuing season while meeting the period?s demand in a cost efficient way. In order to
determine the optimal ordering policies for the preseason months, a stochastic programming
model is introduced. The proposed model and solution approach gives DMs the flexibility
to adjust their ordering decisions considering the hurricane season demand predictions as
demand realizations occur.
In this model, the underlying demand distribution is approximated with a small number
of scenarios corresponding to the demand realizations. As an extension, the author plans to
investigate the case where the stochastic demand process is continuous or approximated with
many scenarios. Therefore, the author will implement the scenario reduction approaches
presented in this study to determine appropriate ordering policies for those organizations
who make stocking decisions in advance of the hurricane season to prepare themselves
for a potential demand surge. Additionally, the author plans to make sensitivity analysis
to compare the accuracy (efficiency) of the optimal solutions (values) among the reduced
models. She would also like to search for optimal statedependent basestock policies defined
over the Markovian hurricane count distribution. The author might also explore optimal
ordering policies using a Value at Risk (VaR) approach.
The hurricane inventory planning models and the information updating framework pre
sented in this dissertation are also applicable for other predictable disasters such as floods
and droughts. The disaster recovery plans associated with these kinds of predictable disas
ters are more reliable compared to the ones proposed for the hazards such as earthquakes,
terrorist attacks, and tornadoes. For future research, the author plans to develop new de
cision models to account for earthquake events. Earthquakes are a common problem for
96
all mankind and it is of vital importance to support research activities concerning them.
In recent years, academic and scientific institutions develop various forecasting models as
sociated with earthquakes. For instance, Istanbul Technical University conducts a project
to develop an earthquake prediction system that is based on the electrical stress measure
ments of rocks. The objective of this project is to forecast the approximate place and
time of earthquakes based on the analysis of the collected data. Afterwards, early warning
studies will be initiated. The author plans to look for potential research collaborations
with this project team. The author? s aim is to examine the application of this earthquake
prediction model to develop decision models for supply chain organizations that are under
the threat of earthquakes. The author may also explore an earthquake risk model to assess
the economical impacts of earthquakes on profitdriven organizations.
Finally, the author plans to do research on ?Humanitarian Relief Logistics?. This is a
new emerging topic in the disaster management and relief planning area. Therefore, there
exist a scant amount of relevant research. In the long term, the author aims to develop
statistical decision models such that the human factor is incorporated into the decision
process.
97
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