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

2/39

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!

14/39

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

20/39

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

21/39

Modified Clarke & Wright Heuristic

Route R1

Route R2

i

j

R1→

R2

R2 →

R1

i

j

i

j

22/39

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

25/39

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

26/39

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

33/39

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.

34/39

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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Thank you !Questions?

luca.talarico@ua.ac.be

36/39

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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