Inventory Planning for Hurricane Events Emmett Lodree, Jr. and Selda - - PowerPoint PPT Presentation

Inventory Planning for Hurricane Events

Emmett Lodree, Jr. and Selda Taskin

Department of Industrial & Systems Engineering Auburn University

Humanitarian Logistics: Networks for Africa May 5-9, 2008

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion

Outline

1

Introduction Motivation Problem Description and Research Questions Hurricanes and their Characteristics

2

Inventory Planning with Hurricane Forecast Updates Model Development Solution Methodology Extension

3

Conclusion

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics

Outline

1

Introduction Motivation Problem Description and Research Questions Hurricanes and their Characteristics

2

Inventory Planning with Hurricane Forecast Updates Model Development Solution Methodology Extension

3

Conclusion

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics

First Responders

NEW ORLEANS: Members of a Texas Task Force One and FEMA search and resuce team call for a man they were told had been calling for help from his attic all

• night. When they arrived they found that he was dead.

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics

Supplies

STAFF PHOTO BY ALEX BRANDON

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics

Hurricane Katrina: Personal Account of Humanitarian Logistics

My family did not evacuate prior to Katrina

No provisions for senior citizen bound to wheelchair Recent unnecessary evacuations to downtown

Trapped with no electricity, running water, police protection, communication, etc. for three full days Rescued by helicopter on day 4 (Thursday), thanks to US Coast Guard

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics

Public and Private Sector Logistics

Supply chain and logistics management:

Right products, in the right places, at the right times and in the right quantities for meeting consumer demands Includes procurement, production, transportation, storage, sales

Public Sector Logistics

Responsiveness is primary Cost is secondary

Private Sector Logistics

Profit driven Meet and create customer demands at minimum cost

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics

Humanitarian Relief Logistics

The FEMA Logistics Directorate

Established as a directorate in 2007; inspired by slow logistics response to Hurricane Katrina Works with public sector organizations such as American Red Cross and US Army Core of Engineers Working towards partnering with private sector organizations

Public and private sector organizations must work together to facilitate effective humanitarian logistics response

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics

Research Questions

What is the appropriate inventory level of hurricane supplies that should be available before the hurricane season? What is the appropriate inventory level of hurricane supplies that should be available before an observed storm makes landfall? How should inventories be prepositioned before the hurricane season? How should inventories be prepositioned before an observed storm makes landfall?

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics

Research Questions

What is the appropriate inventory level of hurricane supplies that should be available before an observed storm makes landfall?

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics

Research Questions

What is the appropriate inventory level of hurricane supplies that should be available before an observed storm makes landfall? This research adopts the perspective of the private sector firm

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics

Humanitarian and Economic Risks

The risk of being under-prepared The risk of being over-prepared

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics

Humanitarian and Economic Risks

The risk of being under-prepared

(Humanitarian performance metrics) Slow response, human suffering (Economic performance metrics) Opportunity cost, lost sales, lost market share

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics

Humanitarian and Economic Risks

The risk of being over-prepared

Excess inventories, excess costs, lower profits $12 million ice melted two years after Katrina

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics

One-Size-Fits-All Disaster Relief Planning

Emergency Management: hazard identification, risk assessment Hazards: Hurricanes, earthquakes, act of war or terrorism Exploit unique characteristics for better planning and response

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics

Abundance of historical data (e.g., HURDAT) Abundance of hurricane prediction models Sufficient time to react after potential threat is initially

• bserved

Hurricanes can be predicted with greater accuracy as time passes

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics

Hurricane Prediction

Model types

Statistical Dynamical Statistical-dynamical Ensemble

Prediction types

Path Intensity Other (e.g., storm surge, damage) How many hurricanes

Time horizons

After tropical depression is observed Up to 6 months before the hurricane season

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics

Time to React: 3- and 5-Day Forecasts

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics

Time to React: Wind Speed Probabilities

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Motivation Problem Description and Research Questions Hurricanes and their Characteristics

Forecast Accuracy vs. Time

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Model Development Solution Methodology Extension

Outline

1

Introduction Motivation Problem Description and Research Questions Hurricanes and their Characteristics

2

Inventory Planning with Hurricane Forecast Updates Model Development Solution Methodology Extension

3

Conclusion

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Model Development Solution Methodology Extension

Problem Description

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Model Development Solution Methodology Extension

Problem Description

A tropical storm (i.e., a potential threat for demand surge) has been observed . . . Now what??? We monitor the storm’s evolution and try to make profit maximizing production/inventory decisions. Assuming production/inventory decision can be given at most

• nce, the questions are

What is the profit maximizing (cost minimizing) inventory decision? How long should we wait during the storm’s evolution to give this inventory decision (forecast accuracy vs. cost efficiency)?

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Model Development Solution Methodology Extension

Assumptions for Bayesian Modeling Framework

1

Demand, ˜ ξ, is a function of an observed storm’s attributes,

• α = (˜

α1, . . . , ˜ αm).

2

For illustrative purposes, α = ˜ w, where ˜ w is an observed storm’s maximum wind speed during the life of the storm. Thus ˜ ξ = ˜ ξ(˜ w).

3

˜ ξ(˜ w) ∈ {˜ ξ0(˜ w), ˜ ξ1(˜ w)}

4

Let ˜ x be a Bernoulli random variable with Ω = {0, 1} and density f (x; ˜ θ) such that ˜ x = i ⇒ ˜ ξ = ˜ ξi(˜ w), where i ∈ {0, 1}. Then π(θ) is the prior density of ˜ θ. θn = π(θ| xn) is the posterior density after observing xn.

5

Let cn be the cost for giving an order or production decision in stage

• n. Then cn ≤ cn+1 for all n = 0, 1, . . ..

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Model Development Solution Methodology Extension

Loss Function

Newsboy Problem NB(a) = ca + L(a; ξ) = ca + h a (a − ξ)dΦ(ξ) + s ∞

a

(ξ − a)dΦ(ξ) where ˜ ξ ∼ φ(ξ) and Φ(ξ) = P(˜ ξ ≤ ξ).

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Model Development Solution Methodology Extension

Loss Function

Newsboy Problem NB(a) = ca + L(a; ξ) = ca + h a (a − ξ)dΦ(ξ) + s ∞

a

(ξ − a)dΦ(ξ) where ˜ ξ ∼ φ(ξ) and Φ(ξ) = P(˜ ξ ≤ ξ). Newsboy Problem with Demand Distribution ˜ ξk, k ∈ {0, 1} NBk(a) = ca + h a (a − ξ)dΦ

˜ ξk(ξ) + s

∞

a

(ξ − a)dΦ

˜ ξk(ξ)

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Model Development Solution Methodology Extension

Development of Bayes’ Risk from Loss Function

For illustration, consider the fixed sample size approach. L(a; θ) = (1 − θ) · NB0(a) + θ · NB1(a) (1) L(a, n; θ) = L(a; θ) +

n

• i=1

(cn − cn−1) (2) R(δ; θ) = E˜

x[L(δ(˜

x); θ)] = 1 NB0f (x)dx + 1 (NB1 − NB0)dF(x) (3) ρ(a, π(θ|x)) = Eπ(θ|x)[L(δ(˜ x); θ)] = 1 NB0π(θ|x)dθ + 1 (NB1 − NB0)dF π(θ|x)(θ) (4)

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Model Development Solution Methodology Extension

Bayes’ Decision Rule: General Results

LF(a, n; θ) := loss if action a is taken after observing realizations of ˜ xn = (˜ x1, . . . , ˜ xn) if ˜ θ = θ. n is determined beforehand (fixed), not one at a time. RF(δn, n; θ) := E˜

xn[LF(δn(

xn), n; θ)] =

• Ωn

LF(δn( xn), n; θ)dF ˜

xn(

xn; θ) Bayes decision rule, δπ

n , satisfies RF(δπ n , n; θ) = inf δn RF(δn, n; θ)

r(δ, n, π) = E

˜ θ[RF(δn, n, θ)] = E ˜ θE˜ xnLF(δn(

xn), n; θ) r(n, π) = E

˜ θ[RF(δπ n , n, θ)] = E ˜ θE˜ xnLF(δπ n (

xn), n; θ) r(n∗, π) = inf

n∈Z;n≥0 r(n, π).

Our model considers case in which ˜ xn is observed in sequence.

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Model Development Solution Methodology Extension

Decision Rule

Theorem

Let Φk(ξ), k ∈ {0, 1}, be the distribution function of hurricane demand ˜ ξk. Then the Bayes decision rule δπ

n (

xn) for the fixed sample size problem with observations ˜ x1, . . . , ˜ xn and loss functions given by Eq. (2) (and Eq. (1)) can be obtained by selecting the action an for each x such that an satisfies (1 − θn) · Φ0(an) + θn · Φ1(an) = s − c s + h The Bayes sequential decision procedure is then δπ = {δπ

0 , δπ 1 , . . .}.

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Model Development Solution Methodology Extension

Stopping Rule

Based on Theorem 2, it can be shown that the stopping rule is Ln(an, n, π) ≤ Ln+1(an+1; n + 1; π) and the cutting point (Choi et al., 2004) is obtained by solving NBn

0 + θn(NBn 1 − NBn 0 ) = NBn+1

+ θn(NBn+1

1

− NBn+1 ) The cutting point is θ∗

n =

NBn+1 − NBn (NBn+1 − NBn

0 ) − (NBn+1 1

− NBn

1 )

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Model Development Solution Methodology Extension

Algorithm

Theorem

If a sample update θn is observed in period n and satisfies θn ≤ θ∗

n,

it is optimal to terminate sampling and give an immediate decision by stocking an. Otherwise if n < N, it is optimal to observe the next update θn+1. Note that the observations are wind speeds, ˜ wn, which are in turn converted into probabilities ˜ θn.

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Model Development Solution Methodology Extension

Empirical Hurricane Wind-Speed Prediction Model

Storm NOT NAMED is number 1 of the year 1851 ***** *************** ******** ********** ********* * Month Day Hour Lat. Long. Dir.

• Speed ----
• Wind
• Pres

sure

• -----------Type----

June 25 0 UTC 28.0N 94.8W degree

• mph
• kph

90 mph 150 kph

• mb

Hurricane - Category June 25 6 UTC 28.0N 95.4W 270 deg 5 mph 9 kph 90 mph 150 kph

• mb

Hurricane - Category June 25 12 UTC 28.0N 96.0W 270 deg 5 mph 9 kph 90 mph 150 kph

• mb

Hurricane - Category June 25 18 UTC 28.1N 96.5W 285 deg 4 mph 7 kph 90 mph 150 kph

• mb

Hurricane - Category June 26 0 UTC 28.2N 97.0W 285 deg 4 mph 7 kph 80 mph 130 kph

• mb

Hurricane - Category June 26 6 UTC 28.3N 97.6W 280 deg 5 mph 9 kph 70 mph 110 kph

• mb

Tropical Storm June 26 12 UTC 28.4N 98.3W 280 deg 6 mph 11 kph 70 mph 110 kph

• mb

Tropical Storm June 26 18 UTC 28.6N 98.9W 290 deg 5 mph 9 kph 60 mph 90 kph

• mb

Tropical Storm June 27 0 UTC 29.0N 99.4W 310 deg 5 mph 9 kph 60 mph 90 kph

• mb

Tropical Storm June 27 6 UTC 29.5N 99.8W 325 deg 6 mph 11 kph 50 mph 70 kph

• mb

Tropical Storm June 27 12 UTC 30.0N 100.0W 340 deg 5 mph 9 kph 50 mph 70 kph

• mb

Tropical Storm June 27 18 UTC 30.5N 100.1W 350 deg 5 mph 9 kph 50 mph 70 kph

• mb

Tropical Storm June 28 0 UTC 31.0N 100.2W 350 deg 5 mph 9 kph 50 mph 70 kph

• mb

Tropical Storm

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Model Development Solution Methodology Extension

Numerical Example

HURDAT Sample

η = 143 hurricanes ηE = 23 Covers the years 1995 - 2004 Matrix ˜ wN×η

Empirical likelihood densities hn (wn|θ) Probability hn (θ|wn) used to compute probability πn (θ|xn)

N different likelihoods Simulate hurricane wind speeds

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Model Development Solution Methodology Extension

Results

Table: Example results.

Example Hurricane Type Decision Period t Stocking Quantity Qt 1 Not Extreme 4 104.5148 2 Not Extreme 17 39.1287 3 Not Extreme 9 103.1482 4 Not Extreme 18 105.0467 5 Not Extreme 4 104.8796 6 Extreme 3 111.9450 7 Extreme 2 109.8291 8 Extreme 7 180.5285 9 Extreme 4 135.3721 10 Extreme 3 135.8230

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Model Development Solution Methodology Extension

Multi-Location Model with Real Hurricane Predictions

Storm location and intensity data are used to enhance the accuracy of the demand forecast Demand forecast is based on the wind-speed probability map used by the National Hurricane Center Wind speed probability model is used to predict hurricane tracks at cumulative 12-h forecast period up to 5 days Well-known The North Atlantic climatology and persistence (CLIPER) model is used to predict hurricane tracks in the wind-speed probability model

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Model Development Solution Methodology Extension

NHC Wind Speed Probabilities

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Model Development Solution Methodology Extension

Bayesian Decision Framework

1

Demand, ˜ ξ, is a function of an observed storm’s attributes,

• α = (˜

α1, . . . , ˜ αm).

2

• α = (˜

w, l). Thus ˜ ξ = ˜ ξ(˜ w, l).

3

˜ ξ(˜ w, l) ∈ {˜ ξ0(˜ w, l), ˜ ξ1(˜ w, l)}

4

Let X be a multivariate Bernoulli random vector. A realization xi = 1 of component ˜ xi indicates hurricane force winds at location i. Denote the density of X as f ( x; θ). Then π( θ) is the prior density of Θ.

• θn = π(

θ | xn

i ) is the posterior density after observing

xn

i for

each location i = 1, . . . , n.

5

Let cn be the cost for giving an order or production decision in stage

• n. Then cn ≤ cn+1 for all n = 0, 1, . . ..

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Model Development Solution Methodology Extension

Demand Scenarios

Scenario Location 1 Location 2 Total Probability 1 ξ11 ξ12 ξ11 + ξ12 p1 = θ1θ2 2 ξ11 ξ02 ξ11 + ξ02 p2 = θ1(1 − θ2) 3 ξ01 ξ12 ξ01 + ξ12 p3 = (1 − θ1)θ2 4 ξ01 ξ02 ξ01 + ξ02 p4 = (1 − θ1)(1 − θ2)

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion Model Development Solution Methodology Extension

Model and Solution

Loss Function is L(a; θ) =

2m

• s=1

psNBs The functions R(δn, n; θ), r(δ, n, π), . . . , and r(n∗, π) are of the same form as in the previous model, but now based on the above loss function. The decision rule is

2m

• s=1

psΦsn(a∗

n) = ψ − ct

ψ + h . The stopping rule is Ln(an) ≤ Ln+1(an+1).

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion

Outline

1

Introduction Motivation Problem Description and Research Questions Hurricanes and their Characteristics

2

Inventory Planning with Hurricane Forecast Updates Model Development Solution Methodology Extension

3

Conclusion

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion

Summary

Lodree and Taskin. Supply chain planning for hurricane response with wind-speed information updates. Computers and Operations Research: Special Issue on Disaster Recovery Planning (in press). Specifies inventory level for a hurricane supply product based on dynamic evolution of a storm Bayesian sequential decision process Extensions include Multiple hurricane attributes Multiple demand classes Multiple order/production opportunities Randomizing the duration of an observed storm’s evolution

Lodree and Taskin Inventory Planning for Hurricane Events

Introduction Inventory Planning with Hurricane Forecast Updates Conclusion

Questions

Lodree and Taskin Inventory Planning for Hurricane Events