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Le design de réseaux logistiques
robustes : la prise en compte des
aléas et des périls
Alain Martel
Codirecteur, CIRRELT, et
Professeur titulaire,
Operations et systèmes de décision
Walid Klibi
Étudiant au doctorat, CIRRELT
Consortium de recherche FOR@C
Centre interuniversitaire de recherche sur les réseaux
d’entreprise, la logistique et le transport (CIRRELT)
Outline
1. Supply Chain Network (SCN) Design Problem
2.
Design Methodology
•
SCN risk analysis
•
Decision-making framework
•
Conceptual design model
3.
Solution Approach
•
Scenario building
•
Sample average design generation model
•
Design evaluation model
4. Ongoing Research
1-Supply Chain Network Design Problem
Raw material sources
...
Manufacturing
Process
Finished Products
Distribution
Channels
...
Markets
Deployed Supply
Chain Network
1
v V
Supply
Market
Potential Facilities
Inventory
sS
2
Bucking 3
Sawing 4
Drying 6
Chipping 5
Planing 7
Grading
sSd
s  S pd
Production-distribution site
Inventory
Distribution site
8
9
Sales
Market
d D
Optimal Supply Network
DOMTAR CASE
Design Objective
Design
Discounted
Cost
(Value)
Total Revenue
Value
added
(Profit)
Maximize
Economic
Value
Added
Total Cost
Robustness ?
Design Response Time
Large MIP model
2- Design Methodology
Second planning cycle
First planning cycle
Possible environments (w)
Design
Decisions
User
Decisions
• Location
• Capacity
• Technology
• Markets
• Mission
x1
Must be
2 anticipated
User
Decisions
• Demand management
• Supply
• Production
• Inventory
• Transportation…
y1
…
• Demand management
• Supply
• Production
• Inventory
• Transportation
•…
4
y2
Planning horizon
1
SCN Risk
Analysis
3
Deployment
Network design
decision point
Network available
for operations
Deployment
Structural
adaptation
decision point
x2
Adapted network
available
for operations
…
SCN’s Environment

Aleatory events (A)

Hazardous events (H)

Totally uncertain events (T)
Information
Perfect
probable
Imperfect
Deterministic Models
Stochastic
Programming
Models
Catastrophe Models
Very low
probability
Min Max Regret
Nil
None
Moderate
Normal
Catastrophic
Serious
Impact
SCN’s Environment Evolutionary Paths
Leads to the definition of a
set K of evolutionary paths
with probabilities
pk , k  K
Design Methodology: Concept definitions
• Environment: Compound events defined over a planning period t  Tˆ
• Scenario: A set of environments for a planning horizon T̂
•  = Set of all possible scenarios over horizon T̂
• p(w ) = Probability of occurrence of scenario w 
• K = Set of all possible evolutionary paths over horizon T̂
Ω is divided in 3 mutually exclusives and collectively exhaustive
subsets:
A
•  Scenarios including only aleatory events
H
•  Scenarios including aleatory and hazardous events
T
•  Scenarios including, in addition, totally uncertain events
Scenarios Tree for the Planning Horizon
(Fan of individual scenarios)
x 2 (w )
…
x N (w )
Aleatory
scenarios
w  A
Hazardous
scenarios
x1
w  H
Totally
uncertain
scenarios
w  T
1
t  T̂
Planning horizon

 Ak   Hk  Tk
k K
SCN Hazard Risk Analysis
• What can go wrong?
• Vulnerability sources identification and filtering
• Multihazard zones exposure index
• What are the consequences?
• Hazardous incidents damage on SCN resources
• What is the likelihood of that happening?
• Stochastic multihazard arrival processes
• Attenuation probabilities
Haimes (2004), Grossi & Kunreuther (2005), Banks (2006)
SCN Risk Analysis: What can go wrong?
Natural
X
Accidental
Hazards
Willful
1
2
3
4, 5
=>
Vulnerability
Sources Set
{1, 2, 3, 4, 5}
S
SCN Risk Modeling
What can go wrong?
Exposure level of
network node locations ?
Fund for Peace Failed State Index
Foreign Policy
Seismic Hazard Exposure Map
SCN Risk Modeling
SCN Risk Analysis:
What are the
consequences?
 Multihazard Incidents Severity Profile
SCN Vulnerability Sources (S = {1, 2, 3, 4, 5})
( )
Time to
recovery
( )
(2)
Plants
Unfilled
supply rate
Capacity
loss
rate
Time to
restoring
supplies
Time to
restarting
production
(3)
Warehouses
(4) Demand
source 1
(5) Demand
source 2
Capacity
loss
rate
Demand
inflation
rate
Demand
deflation
rate
Time to
restarting
distribution
Surge
duration
Drop
duration
Severity dimensions
metrics
Impact
intensity
(1)
Suppliers


,

~F
 l l  s(l ) ,g (l ) (.) l  Lsg , (s , g )  S  G
What are the
consequences?
SCN Risk Analysis:
Recovery Function Examples
Capacity
bs (  , ; t ) available
bs (  , ; t ) level
Demand
Capacity
when hit


t

Capacity loss
recovery function
Demand
when hit
t

Demand surge
recovery function
SCN Risk Analysis:
What is the likelihood
of that happening?
 Multihazard Likelihood Assessment
• Distinct multihazard non-stationary arrival process per
exposure level
• Poisson arrival process
• Exponential inter-arrival time Exp(mgkt) with an expected
time between multihazard mgkt
• Time pattern for an evolutionary path k superimposed on
process using a mean shaping function mgkt  k (mg1 ,t )
• Attenuation probability (azl) per network node based on
hazard zone granularity
Design methodology:
Decision-making framework
Design Level
• Investment
• Policy making
Anticipation
of expected
revenues and costs
Design
User Level
Synchronization of supply and demand
to minimize operations costs and
maximize revenues
Design methodology:
Decision time hierarchy for two planning cycles
Structural adaptation
decision
x2  X2 I ( 2 )
Design
decision
x1  X1 I (1 )
Design
level
1  
1 Deployment
lead time

2
Deployment
lead time
Usage period Tˆ1
ˆy1  Ŷ1x1
xˆ 2  X̂ x21
Usage period Tˆ2
ˆ xˆ 2
ˆy2  Y
2
T2u
T1u
User level

Tˆ  Tˆ1  Tˆ2
x*n( )
y  Y
I u ( )
Illustrative Case:
Multi-depot location-routing problem
• Daily stochastic orders from customers
• Depots vulnerable to extreme events
• How many warehouses and where ?
Compound
Stochastic
Hazard
Process
Plant
Potential DC
locations
l  DC
L
xl
xa
Ship-to
Ship
to
points
locations
r
yDemand zones
yr
(d 
D)
yr
p P
Compound Poisson Demand Process
Design methodology:
Decision-making framework (Rolling Horizon)


max R C ( x1 , .) I  (1 ) , C ( x1 , w )  C d ( x1 , w )  Cˆ du ( x1 , w ) , w  
Design
Model
x1X1
Cˆ
du
( x1 ,w ) , w 
Cˆ du ( x1 ,w ) 
Anticipated
AdaptationUsage Model
x1
opt
xˆ 2 ( .) ,...,xˆ N ( .)
yˆ 1 ( .) ,...,yˆ ( .)
Tˆ
 d

u
u
ˆ
ˆ
ˆ
C ( xˆ n (w ) )   C ( yˆ t (w ) ) 
ˆ C ( yˆ t (w ) )  
n 1 

tT1
tTˆn

ˆ xn1 (w ) n  1, yˆ (w )  Y
ˆ xn ( t ) (w ) t
s.t xˆ n (w )  X
n
t
t
w 
x1*
User Level
Decisions
I  (1 )

opt R C
x*n ( )
y Y
u
( y ) I ( )
u
I u ( )
 T u
Design Methodology: Design model
Using Stochastic Programming (Shapiro, 2007), Robust Optimization
(Kouvelis et al., 1997) and Risk Analysis (Haimes, 2004) concepts, the
design problem can be formulated as follows:


max R A A C ( x1 , .) , RH H C ( x1 , .) , RT T C ( x1 , .)
x1X1
Conditional
return measure
Conditional
dispersion measure
Conditional expected
value measure












R A A C ( x1 , .)  E A A C ( x1 , .)  AD A A C ( x1 , .) , A  0,1
RH H C ( x1 ,.)  EH H C ( x1 ,.)  H DH H C ( x1 ,.) , H  0 ,1
R
T
T
D C ( x , w )
C ( x , .)  Min
w
1
T
T
Robustness criterion
1
Multiparametric program
max wARA A C ( x1 ,.)  wH RH H C ( x1 ,.)  wT RT T C ( x1 ,.)
x1X1






3- Solution Approach
Design generation
( ) ,i…
SSA 
2
m
i
im , i  1,..., I
Small samples replications
Modeling approaches
( Anticipation; Resilience)
Status quo
x
0
1
x1j , j  1,...J
Design evaluation
SSA (  M , x1j ) , j
x*1
Monte Carlo
scenario
generation
…
3
Large sample
M
Solution methods
Worst Case Scenarios
m
T
1
Scenario building
A scenario w   is a compound event defined over all the environments of
the planning horizon
It is the juxtaposition of aleatory, hazardous and totally uncertain events
Scenarios w  mn
Aleatory
events (A)
(A)
Scenarios
A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1
Hazardous
events (H)
(H)
Scenarios
A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2
H1
Totally
uncertain (T)
events
H2
(T)
Scenarios
t  Tˆ
T1
1
ˆ
Environments t  T
Tˆ
Monte Carlo : Scenario (w, k ) generation
Monte-Carlo (w , k ) :
1)
For all, e  , t  Tˆ
a)
Generate the random number u et
b)
Compute Fekt (u et )
1
End For
2)
zZ
u  , ua , u  and u are random numbers)
a) Repeat : compute tez  Fg(z )kt -1 (u (z, t ))
Until t  Tˆ
ez
b) For all tez, l  Lz ua (l )  a zl
i. Compute l  Fs(l ) ,g (l )-1 (u (l ))
For all
(
ii.
End For
End For
(
Compute   f (  )  F  1 u ( l )
l
l

s( l )
)
Solution Approach:
Sample statistics design generation model
• Generate I samples i , i  1,..., I of m scenarios
partitioned into m A, m H and m T A, H and T-scenarios,
with associated weights w A , wH and wT
(Importance Sampling)
m
• Define X1 to take resilience opportunities into account
• For a given I, solve the design problem (for the case of a
dispersion neutral decision-maker)
max
x1X1
wA
m
A

w
mA
C ( x1 ,w ) 
wH
m
H
 C ( x ,w )  w
1
w
mH
T
MinT DT C ( x1 ,w )
wm
The complexity of the problem depends on the nature of C ( x1 ,w )
Solution Approach:
Design evaluation model
• Generate one large sample  of M scenarios
partitioned into M A , M H and M T scenarios
M
• For a given ( x1j ), solve the sample statistics design
evaluation model:
( )
Z x1j = max
  ( )
j
xˆ 2 ( .) ,...,xˆ N ( .)
yˆ 1 ( .) ,...,yˆ ( .)
 ( )
j
 ( )
j
R  A A C x1 , . , R H H C x1 , . , R T T C x1 , .
Tˆ
ˆ xn1 (w ) n  1, yˆ (w )  Y
ˆ xn ( t ) (w ) t
s.t xˆ n (w )  X
n
t
t
The most Effective and Robust SCN design is
*
x1
Z(x1* ) max{Z(x1j ), jJ }
w 
4- Ongoing Research
• Evaluate several anticipation types to explore the complexity
of models and the quality of related solutions
• Propose a modeling approach based on flexibility to improve
the SCN resilience under disruptions
• Redundancy
• Dual sources
• Operational flexibility
• Strategic emergency buffers (insurance inventories)
• Propose a general solution method for large scale problems
(heuristic approach)
Thank you for your attention
Questions ?