Transcript Document

An Introduction to
Stochastic Vehicle Routing
Michel Gendreau
CIRRELT and MAGI
École Polytechnique de Montréal
PhD course on Local Distribution Planning
Molde University College − March 12-16, 2012
Outline
1. Introduction
2. Basic Concepts in Stochastic Optimization
3. Modeling Paradigms
4. Problems with Stochastic Demands
5. Problems with Stochastic Customers
6. Problems with Stochastic Service or Travel Times
7. Conclusion and perspectives
Introduction to Stochastic Vehicle Routing
Acknowledgements

Walter Rei
CIRRELT and ESG UQÀM

Ola Jabali
CIRRELT and HEC Montréal

Tom van Woensel and Ton de Kok
School of Industrial Engineering
Eindhoven University of Technology
Introduction to Stochastic Vehicle Routing
Introduction
Vehicle Routing Problems

Introduced by Dantzig and Ramser in 1959

One of the most studied problem in the area of
logistics

The basic problem involves delivering given
quantities of some product to a given set of
customers using a fleet of vehicles with limited
capacities.

The objective is to determine a set of minimumcost routes to satisfy customer demands.
Introduction to Stochastic Vehicle Routing
Vehicle Routing Problems
Many variants involving different constraints or
parameters:

Introduction of travel and service times with route
duration or time window constraints

Multiple depots

Multiple types of vehicles

...
Introduction to Stochastic Vehicle Routing
What is Stochastic Vehicle Routing?
Basically, any vehicle routing problem in which one
or several of the parameters are not deterministic:
 Demands
 Travel or service times
 Presence of customers
 …
Introduction to Stochastic Vehicle Routing
Basic Concepts in Stochastic
Optimization
Dealing with uncertainty in optimization


Very early in the development of operations
research, some top contributors realized that :
 In many problems there is very significant
uncertainty in key parameters;
 This uncertainty must be dealt with explicitly.
This led to the development of :
 Chance-constrained programming (1951)
 Stochastic programming with recourse (1955)
 Dynamic programming (1958)
 Robust optimization (more recently)
Introduction to Stochastic Vehicle Routing
Information and decision-making
In any stochastic optimization problem, a key
issue is:

How do the revelation of information on the
uncertain parameters and decision-making
(optimization) interact?


When do the values taken by the uncertain
parameters become known?
What changes can I (must I) make in my plans on
the basis of new information that I obtain?
Introduction to Stochastic Vehicle Routing
Chance-constrained programming

Proposed by Charnes and Cooper in 1951.

The key idea is to allow some constraints to be
satisfied only with some probability.
E.g., in VRP with stochastic demands,
Pr{total demand assigned to route r ≤ capacity } ≥ 1-α
Introduction to Stochastic Vehicle Routing
Stochastic programming with recourse


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
Proposed separately by Dantzig and by Beale in
1955.
The key idea is to divide problems in different stages,
between which information is revealed.
The simplest case is with only two stages. The
second stage deals with recourse actions, which
are undertaken to adapt plans to the realization of
uncertainty.
Basic reference:
J.R. Birge and F. Louveaux, Introduction to
Stochastic Programming, 2nd edition, Springer, 2011.
Introduction to Stochastic Vehicle Routing
Dynamic programming





Proposed by Bellman in 1958.
A method developed to tackle effectively sequential
decision problems.
The solution method relies on a time decomposition
of the problem according to stages. It exploits the
so-called Principle of Optimality.
Good for problems with limited number of possible
states and actions.
Basic reference:
D.P. Bertsekas, Dynamic Programming and Optimal
Control, 3rd edition, Athena Scientific, 2005.
Introduction to Stochastic Vehicle Routing
Robust optimization

Here, uncertainty is represented by the fact that the
uncertain parameter vector must belong to a given
polyhedral set (without any probability defined)


E.g., in VRP with stochastic demands,
having set upper and lower bounds for each demand,
together with an upper bound on total demand.
Robust optimization looks in a minimax fashion for the
solution that provides the best “worst case”.
Introduction to Stochastic Vehicle Routing
Modelling paradigms
Real-time optimization
Also called re-optimization


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Based on the implicit assumption that information
is revealed over time as the vehicles perform their
assigned routes.
Relies on Dynamic programming and related
approaches (Secomandi et al.)
Routes are created piece by piece on the basis on
the information currently available.
Not always practical (e.g., recurrent situations)
Introduction to Stochastic Vehicle Routing
A priori optimization

A solution must be determined beforehand;
this solution is “confronted” to the realization of
the stochastic parameters in a second step.

Approaches:




Chance-constrained programming
(Two-stage) stochastic programming with recourse
Robust optimization
[“Ad hoc” approaches]
Introduction to Stochastic Vehicle Routing
Chance-constrained programming

Probabilistic constraints can sometimes be transformed into deterministic ones (e.g., in the case above
if customer demands are independent and Poisson).

This model completely ignores what happens when
things do not “turn out correctly”.
Introduction to Stochastic Vehicle Routing
Robust optimization

Not used very much in stochastic VRP up to now.

Model may be overly pessimistic.
Introduction to Stochastic Vehicle Routing
Stochastic programming with recourse

Recourse is a key concept in a priori optimization

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Solution methods:

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What must be done to “adjust” the a priori solution to the
values observed for the stochastic parameters!
Another key issue is deciding when information on the
uncertain parameters is provided to decision-makers.
Integer L-shaped (Laporte and Louveaux)
Heuristics (including metaheuristics)
Probably closer to actual industrial practices,
if recourse actions are correctly defined!
Introduction to Stochastic Vehicle Routing
VRP with stochastic demands
VRP with stochastic demands (VRPSD)



A probability distribution is specified for the
demand of each customer.
One usually assumes that demands are
independent
(this may not always be very realistic...).
Probably, the most extensively studied SVRP:


Under the reoptimization approach (Secomandi)
Under the a priori approach (several authors) using
both the chance-constrained and the recourse
models.
Introduction to Stochastic Vehicle Routing
VRP with stochastic demands

Probably, the most extensively studied SVRP:


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Classical recourse strategy:

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
Under the reoptimization approach (Secomandi et al.)
Under the a priori approach (several authors) using both
the chance-constrained and the recourse models.
Return to depot to restore vehicle capacity
Does not always seem very appropriate or “intelligent”
However, recently many authors have started
proposing more creative recourse schemes:


Pairing routes (Erera et al.)
Preventive restocking (Yang, Ballou, and Mathur)
Introduction to Stochastic Vehicle Routing
VRP with stochastic demands

Additional material:


M. Gendreau, W. Rei and P. Soriano, “A Hybrid
Monte Carlo Local Branching Algorithm for the
Single Vehicle Routing Problem with Stochastic
Demands”, presented at GOM 2008, August 2008.
W. Rei and M. Gendreau, “An Exact Algorithm for
the Multi-Vehicle Routing Problem with Stochastic
Demands”, presented at the TU Eindhoven,
November 2009.
Introduction to Stochastic Vehicle Routing
VRP with stochastic customers
VRP with stochastic customers (VPRSC)

Each customer has a given probability of
requiring a visit.

Problem grounded in the pioneering work of
Jaillet (1985) on the Probabilistic Traveling
Salesman Problem (PTSP).

At first sight, the VRPSC is of no interest under
the reoptimization approach.
Introduction to Stochastic Vehicle Routing
VRP with stochastic customers (VPRSC)

Recourse action:
 “Skip” absent customers

Has been extensively studied by Gendreau,
Laporte and Séguin in the 1990’s:
 Exact and heuristic solution approaches

Has also been used to model the Consistent VRP
(following slides).
Introduction to Stochastic Vehicle Routing
The Consistent VRP with Stochastic Customers
The consistent vehicle routing problem


First introduced by Groër, Golden, and Wasil (2009)
 Have the same driver visiting the same customers at roughly the
same time each day that these customers need service
 Focus is on the customer
 Planning is done for D periods, known demand, m vehicles
 Arrival time variation is no more than L
Minimize travel time over D periods
day 1
Depot
Introduction to Stochastic Vehicle Routing
day 2
Depot
Problem definition
The consistent vehicle routing problem with
stochastic customers


Each customer has a probability of occurring
 Same driver visits the same customers
 A delivery time window is quoted to the customer
→ (Self-imposed TW)
Cost structure
 Penalties for early and late arrivals
 Travel times
a priori approach
Stage 1
Plan routes and set targets
Stage 2
Compute travel times and penalties
Introduction to Stochastic Vehicle Routing
Depot
Problem data
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An undirected graph G=(V,A)
 V={v1,..,vn} is a set of vertices
 E={(vi, vj): vi, vj V, i<j} is a set of edges
 Vertex v1 corresponds to the depot
 Verticesv2,..,vn correspond to the potential clients
 cij is the travel time between i and j
m is the number of available vehicles
A vehicle can travel at most λ hours
pi is the probability that client i places an order
Ω is the set of possible scenarios associated with the
occurrences for all customers
Introduction to Stochastic Vehicle Routing
Model

First-stage decision variables :

xij = 1, if client j is visited immediately after client i for
2 ≤ i<j≤ n, and 0 otherwise

x1j can take the values 0,1 or 2

ti , target arrival time at customer i

ξ : a random vector containing all Bernoulli random variables
associated with the customers.
For each scenario ω Ω, let ξ(ω)T=[ξ2(ω), …, ξn(ω)]
 ξi(ω) = 1, if customer i is present and 0 otherwise.


Q(x) : second-stage cost (recourse)
Introduction to Stochastic Vehicle Routing
Model
min Q( x)  E Q( x,  ( ))
x
n
x
i 1
x
ik
ik
1i
 2m
  xik  2
j k
 l (S  ) 
xïj  S  


i, j
  
0  x0 j  2
vk V \ v1
S  V \ v1 , 2  S  n  2
v j  V \ v1
0  xij  1
1 i  j  n
xij integer
1 i  j  n
Introduction to Stochastic Vehicle Routing
Model
Reformulation the objective function:
min cT x  Q( x)
x ,t
c T x is a lower bound on the expected
travel time
Gendreau, Laporte and Séguin (1995)
And
Q( x)  Q( x)  cT x
Introduction to Stochastic Vehicle Routing
Model


Assumption: early arrivals do not wait for the time window
Evaluation of the second stage cost
Qr,δ: expected recourse cost corresponding to route r if orientation
δ is chosen
QPr,δ: total average penalties associated with time window deviations
for route r if orientation δ is chosen
QTr: total average travel time for route for route r
Q
r ,
r ,
P
 Q Q
r
T
m
Q( x)   min{Q r ,1 , Q r ,2 }
r 1
Introduction to Stochastic Vehicle Routing
Model
Given a route r, we relabel the vertices on the route according
to a given orientation δ as follows:
(v1  v1r , v2r ,..., vtr , vtr 1  v1 )
  (vi ) : the minimum expected penalty associated with
r

customer v ir
tr 1
QPr ,    (vir )
i 1
Introduction to Stochastic Vehicle Routing
Model

Setting of tir and evaluation of  (vir )

Parameters:

Ai
v0
v5
v7
v0
v5
v4
v0
v7
v0
– the collection of random events where customer vi
r
requires a visit
w
– half length of the time window
pω
– probability of   A 
ir

ai ( ) – arrival time at vir considering   A 
ir
β
– late arrival penalty
  (vi
v4
p5 p7
v4
p5 (1  p7 )
(1  p5 ) p7
(1  p5 )(1  p7 )
v4
r

)  min
r
r

 A 
p (eir ( )   lir ( ))
ir
Variables:

ti – target arrival time at customer vi
r

early arrival at customer vi by   Ai
ei ti–
r
r
li – late arrival at customer vi by   A 
r
r
r
r
r
s.t. [tir  w]  air  eir ( )
  Ai
r
air  [tir  w]  lir ( )
  Ai
eir ( ), lir ( )  0
  Ai
ir
tir  0
Introduction to Stochastic Vehicle Routing
r
r
Solution procedure
Based on the Integer 0-1 L-Shaped Method proposed by Laporte
and Louveaux (1993)

Variant of branch-and-cut



Assumption 1: Q(x) is computable
Assumption 2: There exists a finite value L = general lower bound for
the recourse function.
Operates on the current problem (CP) on each node of the search tree

In the VRP context, CP is relaxed:
I.
Integrality constraints
II.
Subtour elimination and route duration constraints
III.
ct x  Q( x)  ct x  
Introduction to Stochastic Vehicle Routing
0-1 Integer L-Shaped Algorithm
List contains initial relaxed
CP z : 
Choose next pendent node
z z
*
Apply branching
no
no
Introduce violated
feasibility constraints and
LBF
yes
Integer
Update
z
Introduce optimality cuts
Introduction to Stochastic Vehicle Routing
yes
Fathom node
General lower bound
l  min cij
i, j
p  min pi
i
q  min(1  pi )
i
We create an auxiliary graph with all distances equal
to l and all probabilities are set to p and q
→ a lower bound on average travel time is (n-1)pl
→ a lower bound on the penalties associated with time
window deviations can be determined also
Introduction to Stochastic Vehicle Routing
Lower bounding functionals
Introduced by Hjorring and Holt (1999) bound Q(x) using
partial routes
Partial route h consist of:
S  (v0 ,...., vS )
v0
v5
v7
T  (v0 ,...., vT )
v2
v4
v3
v8
v9
v6
U  S  {vS }
U  T  {vT }
Introduction to Stochastic Vehicle Routing
v1
v0
Lower bounding functionals
We look for a lower bound on the recourse associated with route h, Ph
•
Bounds on S → compute exactly
v0
v5
v7
v4
Bounds on U: assume each node, separately,
v5
v7
v4
v0
directly succeeds vS → U h  2 ps   (vs )
h
h
→ compute for each node in U
• Bounds on T: assume general sequence with U as in L
→ compute for a subset of scenarios where at most one customer is absent
•
lh 
v0
v5
v7
v4
vg
min
i , j{U vS vT }
vg
cij
vg
ph  min pi
iU
Introduction to Stochastic Vehicle Routing
v6
v8
v9
v1
v0
Lower bounding functionals
Let Rh  Sh  Th Uh
Let (vi , v j )  Sh or Th if vi and v j are consecutive in Sh and Th
Wh ( x) 

( vi , v j )Sh
xij 

( vi , v j )Th
xij 

vi , v j U h
xij  Rh  1
For r partial routes the following is a valid inequality:
r
P   Ph
h 1
 r

  L  ( P  L)  Wh ( x)  r  1
 h1

Introduction to Stochastic Vehicle Routing
Preliminary results
Experimental sets:

Vertices were generated similar to Laporte,
Louveaux and van Hamme (2002)

p values are randomly generated within 0.6 and 0.9

20 customers with 4 vehicles or 15 with 3 vehicles
Introduction to Stochastic Vehicle Routing
Preliminary results
Set
N
Initial
best
integer
Initial
best
node
1
2
3
4
5
6
7
9
10
11
12
13
15
16
17
8
14
15
15
15
15
15
20
20
20
20
20
20
20
20
20
20
20
20
356.7
352.5
397.3
363.3
407.7
616.2
486.4
476.5
520.0
449.4
526.9
474.0
571.9
444.3
442.5
494.4
522.8
303.1
264.3
360.3
303.0
357.0
554.1
486.4
405.4
414.0
368.5
475.3
436.5
472.1
397.7
390.9
401.0
449.5
Initial GAP Final solution Final GAP Run time
15.0%
25.0%
9.3%
16.6%
12.4%
10.1%
6.2%
14.9%
20.4%
18.0%
9.8%
7.9%
17.5%
10.5%
11.7%
18.9%
14.0%
332.8
285.2
388.1
312.8
393.3
597.2
461.0
451.3
455.1
397.8
478.6
448.0
486.7
416.2
414.2
433.1
503.5
Introduction to Stochastic Vehicle Routing
<1%
<1%
<1%
<1%
<1
<1%
<1%
<1%
<1
<1
<1
<1
9.35%
<1
3.13%
3.41%
1.53%
64
55
263
69
97
879
340
27564
64638
67501
2242
1244
25200
25200
25200
86400
86400
Future research
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
A subset of customers that occur with probability 1
Multiple partial routes
Improving the LBF
Improving the bound on the objective function
Sampling approach for larger sets
Introduction to Stochastic Vehicle Routing
VRP with stochastic service
or travel times
VRP with stochastic service or travel times


The travel times required to move between vertices
and/or service times are random variables.
The least studied, but possibly the most interesting of
all SVRP variants.

Reason: it is much more difficult than others, because
delays “propagate” along a route.

Usual recourse:


Pay penalties for soft time windows or overtime.
All solution approaches seem relevant, but present
significant implementation challenges.
Introduction to Stochastic Vehicle Routing
Conclusions and perspectives
Conclusion and perspectives





Stochastic vehicle routing is a rich and promising
research area.
Much work remains to be done in the area of recourse
definition.
SVRP models and solution techniques may also be useful
for tackling problems that are not really stochastic, but
which exhibit similar structures
Up to now, very little work on problems with stochastic
travel and service times, while one may argue that travel
or service times are uncertain in most routing problems!
Correlation between uncertain parameters is possibly a
major stumbling block in many application areas, but no
one seems to work on ways to deal with it.
Introduction to Stochastic Vehicle Routing