risk-constrained cash-in-transit vehicle routing problem€¦ · rctvrp problem risk-constrained...
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Risk-constrained Cash-in-TransitVehicle Routing Problem
Luca Talarico, Kenneth Sorensen, Johan SpringaelVEROLOG 2012, Bologna, Italy
19 June 2012
University of Antwerp
Operations Research Group
ANT/OR
Summary
Introduction
Risk Constraint
Problem Formulation
Solution StrategiesStrategies DescriptionMetaheuristic Components
Experimental Results
Conclusions
Future works
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Introduction
Context =⇒ Cash and valuables in transportation sectorRisk =⇒ Crime is a real challenge
I In the U.K. alone, there is an estimated £ 500 billiontransported each year (£ 1.4 billion per day);
I In 2008, there were 1,000 documented attacks against CITcouriers in the UK;
I With the downturn in the global economy, CIT crime is on therise;
I Money stolen in CIT attacks is a major source of funding fororganized crime.
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Risk Constraint
RISK
Probability that the =⇒ Exposure of the vehicleunwanted event occurs outside of the depot
x x
Consequences of the =⇒ Loss of the moneyunwanted event or valuables transported
Risk threshold
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Risk Constraint
RISK
Probability that the =⇒ Exposure of the vehicleunwanted event occurs outside of the depot
x x
Consequences of the =⇒ Loss of the moneyunwanted event or valuables transported
Risk threshold
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Example Risk Constraint
0
A
1
B
2 C
5
D
4
E
5
F
10
10
Risk Threshold
I R=150 km·$
Current Values
I CC=1 $
I TT=10 km
I RE=0 · 10 =0 km·$
CC=cash carried TT=travel time RE=risk exposure
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Example Risk Constraint
0
A
1
B
2 C
5
D
4
E
5
F
10
10
8
Risk Threshold
I R=150 km·$
Current Values
I CC=3 $
I TT=18 km
I RE=1 · 8 =8 km·$
CC=cash carried TT=travel time RE=risk exposure
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Example Risk Constraint
0
A
1
B
2 C
5
D
4
E
5
F
10
10
8
10Risk Threshold
I R=150 km·$
Current Values
I CC=8 $
I TT=28 km
I RE=1 ·8+3 ·10 =38 km·$
CC=cash carried TT=travel time RE=risk exposure
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Example Risk Constraint
0
A
1
B
2 C
5
D
4
E
5
F
10
10
8
10
13
Risk Threshold
I R=150 km·$
Current Values
I CC=8 $
I TT=41 km
I RE=1 · 8+3 · 10+8 · 13 = 142 km·$
CC=cash carried TT=travel time RE=risk exposure
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Example Risk Constraint
0
A
1
B
2 C
5
D
4
E
5
F
10
10
8
10
13
12
Risk Threshold
I R=150 km·$
Current Values
I CC=4 $
I TT=12 km
I RE=0 · 12 =0 km·$
CC=cash carried TT=travel time RE=risk exposure
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Example Risk Constraint
0
A
1
B
2 C
5
D
4
E
5
F
10
10
8
10
13
127
Risk Threshold
I R=150 km·$
Current Values
I CC=9 $
I TT=19 km
I RE=4 · 7 =28 km·$
CC=cash carried TT=travel time RE=risk exposure
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Example Risk Constraint
0
A
1
B
2 C
5
D
4
E
5
F
10
10
8
10
13
127
13
Risk Threshold
I R=150 km·$
Current Values
I CC=9 $
I TT=32 km
I RE=4 ·7+9 ·13 =145 km·$
CC=cash carried TT=travel time RE=risk exposure
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Example Risk Constraint
0
A
1
B
2 C
5
D
4
E
5
F
10
10
8
10
1312
7
12
15
Risk Threshold
I R=150 km·$
Current Values
I CC=10 $
I TT=15 km
I RE=0 · 15 =0 km·$
CC=cash carried TT=travel time RE=risk exposure
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Example Risk Constraint
0
A
1
B
2 C
5
D
4
E
5
F
10
10
8
10
1312
7
12
15
15
Risk Threshold
I R=150 km·$
Current Values
I CC=10 $
I TT=30 km
I RE=10 · 15 =150 km·$
CC=cash carried TT=travel time RE=risk exposure
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RCTVRP Problem
Risk-constrained Cash-in-Transit Vehicle Routing Problem
Determine a set of routes, where a route is a tour that begins at thedepot, traverses a subset of the customers in a specified sequenceand returns to the depot. Each customer must be assigned toexactly one of the routes. The total risk for each route must notexceed the risk threshold. The routes should be chosen to minimizetotal travel cost.
[Talarico, Sorensen and Springael in 2012]
No single tour as a consequence of the risk threshold!
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RCTVRP Problem
Risk-constrained Cash-in-Transit Vehicle Routing Problem
Determine a set of routes, where a route is a tour that begins at thedepot, traverses a subset of the customers in a specified sequenceand returns to the depot. Each customer must be assigned toexactly one of the routes. The total risk for each route must notexceed the risk threshold. The routes should be chosen to minimizetotal travel cost.
[Talarico, Sorensen and Springael in 2012]
No single tour as a consequence of the risk threshold!
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Problem Formulation
A necessary and sufficient condition on the RCTVRP feasibility is:
T ≥ maxi∈N{pi · ti0}
The condition guarantees that at least the problem has a feasible solutioncontaining n routes.
01
p12
p2
3
p3
4
p4
i
pi
5
p5
n
pn
0
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Exact Approach
Small instances of the RCTVRP have been solved to optimality using CPLEX
I The time needed to solve small istances of the problem increasesexponentially;
I We were unable to solve problems with more than 10 nodes in a reasonabletime (Instances with 16 nodes solved in ' 15 min).
0
10
20
30
40
50
60
3 4 5 6 7 8 9
Tim
e (
s.)
Nodes
Time Expon. (Time)
We need efficient Metaheuristics! 16/39
Solving Strategies
Seven Metaheuristics have been developped and coded in JAVAcombining the following components:
I 4 Constructive Heuristics:I Modified Clarke & Wright with Grasp (Stochastic);I Nearest Neigborhood with Grasp (Stochastic);I TSP Nearest Neigborhood with Grasp plus Splitting
(Stochastic);I TSP Lin Kernighan plus Splitting (Deterministic).
I 1 Local Search Block (6 LS Operators);
I 2 Structures:I Multi-Start;I Perturbation.
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Solving Strategies
Seven Metaheuristics have been developped and coded in JAVAcombining the following components:
I 4 Constructive Heuristics:I Modified Clarke & Wright with Grasp (Stochastic);I Nearest Neigborhood with Grasp (Stochastic);I TSP Nearest Neigborhood with Grasp plus Splitting
(Stochastic);I TSP Lin Kernighan plus Splitting (Deterministic).
I 1 Local Search Block (6 LS Operators);
I 2 Structures:I Multi-Start;I Perturbation.
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Solving Strategies
Seven Metaheuristics have been developped and coded in JAVAcombining the following components:
I 4 Constructive Heuristics:I Modified Clarke & Wright with Grasp (Stochastic);I Nearest Neigborhood with Grasp (Stochastic);I TSP Nearest Neigborhood with Grasp plus Splitting
(Stochastic);I TSP Lin Kernighan plus Splitting (Deterministic).
I 1 Local Search Block (6 LS Operators);
I 2 Structures:I Multi-Start;I Perturbation.
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Metaheuristics Overview
MSMC&WG
Initialize Meta-Heuristic
start
Find an ISusing MC&WG
Improve theCS using LSO
Is CSbetter
than BS?
Update the BS
Is the #of restartreached?
Exit no
yes
yes
no
MC&WGP
Initialize Meta-Heuristic
start
Find a CSusing MC&WG
Improve theCS using LSO
Perturb the CS
Is CSbetter
than BS?
Update the BS
Is the #of restartreached?
Exit no
yes
yes
no
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Metaheuristics Overview
MSNNG
Initialize Meta-Heuristic
start
Find an ISusing NNG
Improve theCS using LSO
Is CSbetter
than BS?
Update the BS
Is the #of restartreached?
Exit no
yes
yes
no
NNGP
Initialize Meta-Heuristic
start
Find a CSusing NNG
Improve theCS using LSO
Perturb the CS
Is CSbetter
than BS?
Update the BS
Is the #of restartreached?
Exit no
yes
yes
no
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Metaheuristics Overview
MSTSPNNGS
Initialize Meta-Heuristic
start
Solve the beneathTSP using NNG
Find a CS split-ting the Big Tour
Improve theCS using LSO
Is CSbetter
than BS?
Update the BS
Is the #of restartreached?
Exit no
yes
yes
no
TSPNNGSP
Initialize Meta-Heuristic
start
Solve the beneathTSP using NNG
Find a CS split-ting the Big Tour
Improve theCS using LSO
Perturb the CS
Is CSbetter
than BS?
Update the BS
Is the #of restartreached?
Exit no
yes
yes
no
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Metaheuristics Overview
TSPLKSP
Initialize Meta-Heuristic
start
Solve the beneathTSP using LK
Find a CS split-ting the Big Tour
Improve theCS using LSO
Perturb the CS
Is CSbetter
than BS?
Update the BS
Is the #of restartreached?
Exit no
yes
yes
no
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Modified Clarke & Wright Heuristic
Route R1
Route R2
i
j
R1→
R2
R2 →
R1
i
j
i
j
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Local Search Block
Six different Local Search Operators have been used:
I 2 Intra-Route LSO;
I 4 Inter-Route LSO.
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Intra-Route Local Search Operators
Two-opt
0
A
pA
C
pC
B
pB
D
pD
0
A
pA
C
pC
B
pB
D
pD
Or-opt
0
A
pA
B
pB
C
pC
D
pD
E
pE
F
pF
0
A
pA
B
pB
C
pC
D
pD
E
pE
F
pF
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Inter-Route Search Operators
Two-opt
0
A
pA
D
pD
B
pB
C
pC
0 0
A
pA
D
pD
B
pB
C
pC
0
Relocate
0
A
pA
B
pB
C
pC
0
D
pE
E
pE
0
A
pA
B
pB
C
pC
0
D
pD
E
pE
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Inter-Route Local Search Operators
Exchange
0
A
pA
B
pB
C
pC
0
F
pF
E
pE
D
pD
0
A
pA
B
pB
C
pC
0
F
pF
E
pE
D
pD
Cross Exchange
0
A
pA
B
pB
C
pC
D
pD
0
E
pE
F
pF
G
pG
H
pH
0
A
pA
B
pB
C
pC
D
pD
0
E
pE
F
pF
G
pG
H
pH
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Instances Description
Since RCTVRP has not been studied before, no test instances are availablein the literature.
I 55 Instance have been used to test the 7 Metaheuristics:
I 11 Basic Instances taken from VRP lib;I 5 diffferent Risk Constraint Levels.
Name NodesE022-04g 22E026-08m 26E030-03g 30E036-11h 36E045-04f 45E051-05e 51E072-04f 72E101-08e 101E121-07c 121E135-07f 135E151-12b 151
Risk Level ValueRL1 T = maxi∈N {pi · ti0}RL2 1.5 · RL1RL3 2.0 · RL1RL4 2.5 · RL1RL5 3.0 · RL1
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Metaheuristic Parameters
Two different kind of parameters have been analyzed:
I Instance characteristics;
I Metaheuristics parameters;
Parameter Values Nr. of LevelsNodes 22, 26, 30, 36, 45, 51, 72, 101, 121, 135, 151 11Risk Level 1, 1.5, 2, 2.5, 3 5Restart 1, 2, 3, 4, 5 5TSP-Repetition 1, 2, 3, 4 4Perturbation 0%, 5%, 10%, 15%, 20%, 25% 6Or-Opt-Internal on, off 2Two-opt-Internal on, off 2Exchange on, off 2Relocate on, off 2Cross-Exchange on, off 2Two-opt-External on, off 2
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Computational Results
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3
3.25
3.5
3.75
4
22 26 30 36 45 51 72 101 121 135 151
Tim
e (
s)
Nodes
Computational Time Instances with RL1
MSNNG MSMC&WG MC&WGP NNGP MSTSPNNGS TSPNNGSP TSPLKSP
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Computational Results
30/39
Computational Results
31/39
Computational Results
32/39
Computational Results
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Conclusions
The main contributions of this works are:
I Introduction and mathematical formulation of a new Risk index;
I Introduction of a new unexplored variant of the vehicle routingproblem named RCTVRP;
I Mathematical formulation of the RCTVRP;
I Exact approach to solve small instances of the problem;
I 7 metaheuristic approaches to solve the RCTVRP in areasonable time.
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Future research
I Completing the statistical analysis of the results obtained
I Handling the risk as an additional objective to be minimized
I Finding some real applications of the problem involving CITcompanies
I Several extensions of the problem can be proposed for futureresearch taking in consideration some real-life constraints as:
I route length restrictions;I time windows;I precedence relations.
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Mathematical Formulation 1/3
minm∑r=0
n∑i=0
n∑j=0
tij xrij
s.t.
n∑j=0
x r0j =n∑
i=0
x ri0 r = 0, ...,m (1)
n∑j=0
x00j = 1 (2)
n∑i=0
x ri0 ≥n∑
j=0
x r+10j r = 0, · · · ,m − 1 (3)
n −
1 +m∑r=0
n∑i=0
n∑j=0
x rij
≤ n ·n∑
h=1
x r+10h r = 0, · · · ,m − 1 (4)
n∑h=0
x rhj −n∑
k=0
x rjk = 0j = 0, · · · , nr = 0, · · · ,m (5)
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Mathematical Formulation 2/3
Pr0 = 0 r = 0, · · · ,m (6a)
Pri + pj −
(1− x rij
)· HP ≤ Pr
j
i = 0, · · · , nj = 1, · · · , nr = 0, · · · ,m
(6b)
0 ≤ Pri ≤
n∑j=0
x rij ·n∑
i=0
pi i = 0, · · · , n (6c)
Rrj ≥ Pr
0 t0j −(
1− x r0j
)· HC j = 1, · · · , n
r = 0, · · · ,m (7a)
Rrj ≥ Rr
i + Pri tij −
(1− x rij
)· HC
i = 0, · · · , nj = 1, · · · , nr = 0, · · · ,m
(7b)
0 ≤ Rri ≤ T ·
n∑j=0
x riji = 0, · · · , nr = 0, · · · ,m (7c)
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Mathematical Formulation 3/3
uri − urj + n · x rij ≤ n −n∑
l=1
x r0l
i = 0, · · · , nj = 1, · · · , ni 6= jr = 0, · · · ,m
(8a)
ur0 = 0 r = 0, · · · ,m (8b)
0 ≤ uri ≤ (n − 1) ·n∑
j=0
x riji = 1, · · · , nr = 0, · · · ,m (8c)
urij ∈ {0, 1}i = 0, · · · , nj = 1, · · · , nr = 0, · · · ,m
(8d)
x rij ∈ {0, 1}i = 0, · · · , nj = 1, · · · , nr = 0, · · · ,m
(9)
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