Mixed Integer Linear Programming Model for Vehicle Routing Problem for Hazardous Materials Transportation**Universidad Nacional de Colombia. Universite de Technologie de Troyes.

Abstract

This paper presents a mathematical model to solve the Heterogeneous Vehicle Routing Problem (HVRP) in the context of hazardous materials (HazMat) transportation. To evaluate the model a linear approximation of the total routing risk is used as objective function. In the first stage a routing risk measure is proposed as a nonlinear function of the truck load. This function is approximated by means of two different piecewise linear functions (PLF). A genetic algorithm is employed to estimate the interval limits of PLF. These two functions are utilized to approximate the total routing risk for the best known solution for the benchmark instances of HVRP with fixed costs and unlimited fleet, both approaches are compared with the nonlinear risk function value. In the second stage the best piecewise linear approximation of the routing risk is integrated to a mixed integer linear programming (MILP) model for solving the risk optimization problem. The final model is tested on HVRP instances with 20 nodes. Results show that total cost minimization and total risk minimization appear to be conflicting objectives.

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IFAC-PapersOnLine 49-12 (2016) 538–543
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10.1016/j.ifacol.2016.07.691
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2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Mixed Integer Linear Programming Model
for Vehicle Routing Problem for Hazardous
Materials Transportation 
Gustavo Alfredo Bula∗,∗∗ Fabio Augusto Gonzalez ∗
Caroline Prodhon∗∗ H. Murat Afsar ∗∗
Nubia Milena Velasco ∗∗∗
∗System and Industrial Engineering Deparment, Universidad Nacional
de Colombia, Bogot´a, Colombia (e-mail: {gabula,fagonzalezo}@
unal.edu.co)
∗∗ ICD-LOSI, Universit´e de Technologie de Troyes, Troyes Cedex
10004, France (e-mail: {caroline.prodhon, murat.afsar}@ utt.fr)
∗∗∗ School of Management, Universidad de los Andes, Bogot´a,
Colombia (e-mail: nvelasco@ uniandes.edu.co)
Abstract: This paper presents a mathematical model to solve the Heterogeneous Vehicle
Routing Problem (HVRP) in the context of hazardous materials (HazMat) transportation. To
evaluate the model a linear approximation of the total routing risk is used as objective function.
In the first stage a routing risk measure is proposed as a nonlinear function of the truck load.
This function is approximated by means of two different piecewise linear functions (PLF). A
genetic algorithm is employed to estimate the interval limits of PLF. These two functions are
utilized to approximate the total routing risk for the best known solution for the benchmark
instances of HVRP with fixed costs and unlimited fleet, both approaches are compared with
the nonlinear risk function value. In the second stage the best piecewise linear approximation of
the routing risk is integrated to a mixed integer linear programming (MILP) model for solving
the risk optimization problem. The final model is tested on HVRP instances with 20 nodes.
Results show that total cost minimization and total risk minimization appear to be conflicting
objectives.
Keywords: Transportation science, Operations Research, Risk minimization
1. INTRODUCTION
Estimation and analysis of risks is an important aspect
of hazardous materials (HazMat) transportation manage-
ment. Selection of the safest routes is a main focus research
in this area, Erkut et al. (2007). Routing in HazMat
transportation are grouped in two categories: shortest path
and vehicle routing, and scheduling problems. While the
first type of routing problems has been substantially stud-
ied, scarce work has been found on the second type, see
Androutsopoulos and Zografos (2012) and Pradhananga
et al. (2014). The selection of a routing solution in HazMat
transportation depends critically on the model adopted
for the quantification of risk, Bell (2006). Therefore, a
combined model considering both HazMat transportation
risk and routing problem should be developed.
Tarantilis and Kiranoudis (2001) made one of the first
studies that has explicitly considered HazMat transporta-
tion risk in vehicle routing problems. In this work the risk
is defined as the number of people exposed to a HazMat
transportation accident. The authors utilize an aggregated
measure for computing the risk on an arc of the routing
network. The traditional risk model is used in most of pos-
Universidad Nacional de Colombia. Universit´e de Technologie de
Troyes.
terior studies, see Zografos and Androutsopoulos (2004),
Zografos and Androutsopoulos (2008) and Pradhananga
et al. (2010). The risk is computed as the product of
the hazardous materials accident probability on an arc by
the population exposed to the impacts of an accident on
that arc. In the work of Androutsopoulos and Zografos
(2010) both, the probability of accident and expected
consequences arising in case of an accident occurring on
an arc, are considered time dependent. In a later work,
Androutsopoulos and Zografos (2012) considers that con-
sequences expected radius of a hazardous materials acci-
dents depends on the truck load, but the assessment of the
computational performance of the solving algorithm was
based on load-invariant risk values.
The volume of transported HazMat plays an important
role in determining the likelihood of occurrence of an
incident. In HazMat material distribution, the vehicle load
is reduced by a quantity equals to the customer demand
each time a client is visited, affecting the optimal path in
risk minimization problems. Other important aspects to
consider in HazMat transportation risk estimation are the
probability of a truck tank accident, and the probability
of a material release. Truck tank accident probability
depends on the type of truck and the road conditions;
meanwhile release probability depends on the nature of
IFAC Conference on Manufacturing Modelling,
Management and Control
June 28-30, 2016. Troyes, France
Copyright © 2016 IFAC 538
Mixed Integer Linear Programming Model
for Vehicle Routing Problem for Hazardous
Materials Transportation 
Gustavo Alfredo Bula∗,∗∗ Fabio Augusto Gonzalez ∗
Caroline Prodhon∗∗ H. Murat Afsar ∗∗
Nubia Milena Velasco ∗∗∗
∗System and Industrial Engineering Deparment, Universidad Nacional
de Colombia, Bogot´a, Colombia (e-mail: {gabula,fagonzalezo}@
unal.edu.co)
∗∗ ICD-LOSI, Universit´e de Technologie de Troyes, Troyes Cedex
10004, France (e-mail: {caroline.prodhon, murat.afsar}@ utt.fr)
∗∗∗ School of Management, Universidad de los Andes, Bogot´a,
Colombia (e-mail: nvelasco@ uniandes.edu.co)
Abstract: This paper presents a mathematical model to solve the Heterogeneous Vehicle
Routing Problem (HVRP) in the context of hazardous materials (HazMat) transportation. To
evaluate the model a linear approximation of the total routing risk is used as objective function.
In the first stage a routing risk measure is proposed as a nonlinear function of the truck load.
This function is approximated by means of two different piecewise linear functions (PLF). A
genetic algorithm is employed to estimate the interval limits of PLF. These two functions are
utilized to approximate the total routing risk for the best known solution for the benchmark
instances of HVRP with fixed costs and unlimited fleet, both approaches are compared with
the nonlinear risk function value. In the second stage the best piecewise linear approximation of
the routing risk is integrated to a mixed integer linear programming (MILP) model for solving
the risk optimization problem. The final model is tested on HVRP instances with 20 nodes.
Results show that total cost minimization and total risk minimization appear to be conflicting
objectives.
Keywords: Transportation science, Operations Research, Risk minimization
1. INTRODUCTION
Estimation and analysis of risks is an important aspect
of hazardous materials (HazMat) transportation manage-
ment. Selection of the safest routes is a main focus research
in this area, Erkut et al. (2007). Routing in HazMat
transportation are grouped in two categories: shortest path
and vehicle routing, and scheduling problems. While the
first type of routing problems has been substantially stud-
ied, scarce work has been found on the second type, see
Androutsopoulos and Zografos (2012) and Pradhananga
et al. (2014). The selection of a routing solution in HazMat
transportation depends critically on the model adopted
for the quantification of risk, Bell (2006). Therefore, a
combined model considering both HazMat transportation
risk and routing problem should be developed.
Tarantilis and Kiranoudis (2001) made one of the first
studies that has explicitly considered HazMat transporta-
tion risk in vehicle routing problems. In this work the risk
is defined as the number of people exposed to a HazMat
transportation accident. The authors utilize an aggregated
measure for computing the risk on an arc of the routing
network. The traditional risk model is used in most of pos-
Universidad Nacional de Colombia. Universit´e de Technologie de
Troyes.
terior studies, see Zografos and Androutsopoulos (2004),
Zografos and Androutsopoulos (2008) and Pradhananga
et al. (2010). The risk is computed as the product of
the hazardous materials accident probability on an arc by
the population exposed to the impacts of an accident on
that arc. In the work of Androutsopoulos and Zografos
(2010) both, the probability of accident and expected
consequences arising in case of an accident occurring on
an arc, are considered time dependent. In a later work,
Androutsopoulos and Zografos (2012) considers that con-
sequences expected radius of a hazardous materials acci-
dents depends on the truck load, but the assessment of the
computational performance of the solving algorithm was
based on load-invariant risk values.
The volume of transported HazMat plays an important
role in determining the likelihood of occurrence of an
incident. In HazMat material distribution, the vehicle load
is reduced by a quantity equals to the customer demand
each time a client is visited, affecting the optimal path in
risk minimization problems. Other important aspects to
consider in HazMat transportation risk estimation are the
probability of a truck tank accident, and the probability
of a material release. Truck tank accident probability
depends on the type of truck and the road conditions;
meanwhile release probability depends on the nature of
IFAC Conference on Manufacturing Modelling,
Management and Control
June 28-30, 2016. Troyes, France
Copyright © 2016 IFAC 538
Mixed Integer Linear Programming Model
for Vehicle Routing Problem for Hazardous
Materials Transportation 
Gustavo Alfredo Bula∗,∗∗ Fabio Augusto Gonzalez ∗
Caroline Prodhon∗∗ H. Murat Afsar ∗∗
Nubia Milena Velasco ∗∗∗
∗
System and Industrial Engineering Deparment, Universidad Nacional
de Colombia, Bogot´a, Colombia (e-mail: {gabula,fagonzalezo}@
unal.edu.co)
∗∗ ICD-LOSI, Universit´e de Technologie de Troyes, Troyes Cedex
10004, France (e-mail: {caroline.prodhon, murat.afsar}@ utt.fr)
∗∗∗ School of Management, Universidad de los Andes, Bogot´a,
Colombia (e-mail: nvelasco@ uniandes.edu.co)
Abstract: This paper presents a mathematical model to solve the Heterogeneous Vehicle
Routing Problem (HVRP) in the context of hazardous materials (HazMat) transportation. To
evaluate the model a linear approximation of the total routing risk is used as objective function.
In the first stage a routing risk measure is proposed as a nonlinear function of the truck load.
This function is approximated by means of two different piecewise linear functions (PLF). A
genetic algorithm is employed to estimate the interval limits of PLF. These two functions are
utilized to approximate the total routing risk for the best known solution for the benchmark
instances of HVRP with fixed costs and unlimited fleet, both approaches are compared with
the nonlinear risk function value. In the second stage the best piecewise linear approximation of
the routing risk is integrated to a mixed integer linear programming (MILP) model for solving
the risk optimization problem. The final model is tested on HVRP instances with 20 nodes.
Results show that total cost minimization and total risk minimization appear to be conflicting
objectives.
Keywords: Transportation science, Operations Research, Risk minimization
1. INTRODUCTION
Estimation and analysis of risks is an important aspect
of hazardous materials (HazMat) transportation manage-
ment. Selection of the safest routes is a main focus research
in this area, Erkut et al. (2007). Routing in HazMat
transportation are grouped in two categories: shortest path
and vehicle routing, and scheduling problems. While the
first type of routing problems has been substantially stud-
ied, scarce work has been found on the second type, see
Androutsopoulos and Zografos (2012) and Pradhananga
et al. (2014). The selection of a routing solution in HazMat
transportation depends critically on the model adopted
for the quantification of risk, Bell (2006). Therefore, a
combined model considering both HazMat transportation
risk and routing problem should be developed.
Tarantilis and Kiranoudis (2001) made one of the first
studies that has explicitly considered HazMat transporta-
tion risk in vehicle routing problems. In this work the risk
is defined as the number of people exposed to a HazMat
transportation accident. The authors utilize an aggregated
measure for computing the risk on an arc of the routing
network. The traditional risk model is used in most of pos-
Universidad Nacional de Colombia. Universit´e de Technologie de
Troyes.
terior studies, see Zografos and Androutsopoulos (2004),
Zografos and Androutsopoulos (2008) and Pradhananga
et al. (2010). The risk is computed as the product of
the hazardous materials accident probability on an arc by
the population exposed to the impacts of an accident on
that arc. In the work of Androutsopoulos and Zografos
(2010) both, the probability of accident and expected
consequences arising in case of an accident occurring on
an arc, are considered time dependent. In a later work,
Androutsopoulos and Zografos (2012) considers that con-
sequences expected radius of a hazardous materials acci-
dents depends on the truck load, but the assessment of the
computational performance of the solving algorithm was
based on load-invariant risk values.
The volume of transported HazMat plays an important
role in determining the likelihood of occurrence of an
incident. In HazMat material distribution, the vehicle load
is reduced by a quantity equals to the customer demand
each time a client is visited, affecting the optimal path in
risk minimization problems. Other important aspects to
consider in HazMat transportation risk estimation are the
probability of a truck tank accident, and the probability
of a material release. Truck tank accident probability
depends on the type of truck and the road conditions;
meanwhile release probability depends on the nature of
IFAC Conference on Manufacturing Modelling,
Management and Control
June 28-30, 2016. Troyes, France
Copyright © 2016 IFAC 538
Mixed Integer Linear Programming Model
for Vehicle Routing Problem for Hazardous
Materials Transportation

Gustavo Alfredo Bula∗,∗∗ Fabio Augusto Gonzalez ∗
Caroline Prodhon∗∗ H. Murat Afsar ∗∗
Nubia Milena Velasco ∗∗∗
∗System and Industrial Engineering Deparment, Universidad Nacional
de Colombia, Bogot´a, Colombia (e-mail: {gabula,fagonzalezo}@
unal.edu.co)
∗∗ ICD-LOSI, Universit´e de Technologie de Troyes, Troyes Cedex
10004, France (e-mail: {caroline.prodhon, murat.afsar}@ utt.fr)
∗∗∗ School of Management, Universidad de los Andes, Bogot´a,
Colombia (e-mail: nvelasco@ uniandes.edu.co)
Abstract: This paper presents a mathematical model to solve the Heterogeneous Vehicle
Routing Problem (HVRP) in the context of hazardous materials (HazMat) transportation. To
evaluate the model a linear approximation of the total routing risk is used as objective function.
In the first stage a routing risk measure is proposed as a nonlinear function of the truck load.
This function is approximated by means of two different piecewise linear functions (PLF). A
genetic algorithm is employed to estimate the interval limits of PLF. These two functions are
utilized to approximate the total routing risk for the best known solution for the benchmark
instances of HVRP with fixed costs and unlimited fleet, both approaches are compared with
the nonlinear risk function value. In the second stage the best piecewise linear approximation of
the routing risk is integrated to a mixed integer linear programming (MILP) model for solving
the risk optimization problem. The final model is tested on HVRP instances with 20 nodes.
Results show that total cost minimization and total risk minimization appear to be conflicting
objectives.
Keywords: Transportation science, Operations Research, Risk minimization
1. INTRODUCTION
Estimation and analysis of risks is an important aspect
of hazardous materials (HazMat) transportation manage-
ment. Selection of the safest routes is a main focus research
in this area, Erkut et al. (2007). Routing in HazMat
transportation are grouped in two categories: shortest path
and vehicle routing, and scheduling problems. While the
first type of routing problems has been substantially stud-
ied, scarce work has been found on the second type, see
Androutsopoulos and Zografos (2012) and Pradhananga
et al. (2014). The selection of a routing solution in HazMat
transportation depends critically on the model adopted
for the quantification of risk, Bell (2006). Therefore, a
combined model considering both HazMat transportation
risk and routing problem should be developed.
Tarantilis and Kiranoudis (2001) made one of the first
studies that has explicitly considered HazMat transporta-
tion risk in vehicle routing problems. In this work the risk
is defined as the number of people exposed to a HazMat
transportation accident. The authors utilize an aggregated
measure for computing the risk on an arc of the routing
network. The traditional risk model is used in most of pos-
Universidad Nacional de Colombia. Universit´e de Technologie de
Troyes.
terior studies, see Zografos and Androutsopoulos (2004),
Zografos and Androutsopoulos (2008) and Pradhananga
et al. (2010). The risk is computed as the product of
the hazardous materials accident probability on an arc by
the population exposed to the impacts of an accident on
that arc. In the work of Androutsopoulos and Zografos
(2010) both, the probability of accident and expected
consequences arising in case of an accident occurring on
an arc, are considered time dependent. In a later work,
Androutsopoulos and Zografos (2012) considers that con-
sequences expected radius of a hazardous materials acci-
dents depends on the truck load, but the assessment of the
computational performance of the solving algorithm was
based on load-invariant risk values.
The volume of transported HazMat plays an important
role in determining the likelihood of occurrence of an
incident. In HazMat material distribution, the vehicle load
is reduced by a quantity equals to the customer demand
each time a client is visited, affecting the optimal path in
risk minimization problems. Other important aspects to
consider in HazMat transportation risk estimation are the
probability of a truck tank accident, and the probability
of a material release. Truck tank accident probability
depends on the type of truck and the road conditions;
meanwhile release probability depends on the nature of
IFAC Conference on Manufacturing Modelling,
Management and Control
June 28-30, 2016. Troyes, France
Copyright © 2016 IFAC 538
Mixed Integer Linear Programming Model
for Vehicle Routing Problem for Hazardous
Materials Transportation 
Gustavo Alfredo Bula∗,∗∗ Fabio Augusto Gonzalez ∗
Caroline Prodhon∗∗ H. Murat Afsar ∗∗
Nubia Milena Velasco ∗∗∗
∗System and Industrial Engineering Deparment, Universidad Nacional
de Colombia, Bogot´a, Colombia (e-mail: {gabula,fagonzalezo}@
unal.edu.co)
∗∗ ICD-LOSI, Universit´e de Technologie de Troyes, Troyes Cedex
10004, France (e-mail: {caroline.prodhon, murat.afsar}@ utt.fr)
∗∗∗ School of Management, Universidad de los Andes, Bogot´a,
Colombia (e-mail: nvelasco@ uniandes.edu.co)
Abstract: This paper presents a mathematical model to solve the Heterogeneous Vehicle
Routing Problem (HVRP) in the context of hazardous materials (HazMat) transportation. To
evaluate the model a linear approximation of the total routing risk is used as objective function.
In the first stage a routing risk measure is proposed as a nonlinear function of the truck load.
This function is approximated by means of two different piecewise linear functions (PLF). A
genetic algorithm is employed to estimate the interval limits of PLF. These two functions are
utilized to approximate the total routing risk for the best known solution for the benchmark
instances of HVRP with fixed costs and unlimited fleet, both approaches are compared with
the nonlinear risk function value. In the second stage the best piecewise linear approximation of
the routing risk is integrated to a mixed integer linear programming (MILP) model for solving
the risk optimization problem. The final model is tested on HVRP instances with 20 nodes.
Results show that total cost minimization and total risk minimization appear to be conflicting
objectives.
Keywords: Transportation science, Operations Research, Risk minimization
1. INTRODUCTION
Estimation and analysis of risks is an important aspect
of hazardous materials (HazMat) transportation manage-
ment. Selection of the safest routes is a main focus research
in this area, Erkut et al. (2007). Routing in HazMat
transportation are grouped in two categories: shortest path
and vehicle routing, and scheduling problems. While the
first type of routing problems has been substantially stud-
ied, scarce work has been found on the second type, see
Androutsopoulos and Zografos (2012) and Pradhananga
et al. (2014). The selection of a routing solution in HazMat
transportation depends critically on the model adopted
for the quantification of risk, Bell (2006). Therefore, a
combined model considering both HazMat transportation
risk and routing problem should be developed.
Tarantilis and Kiranoudis (2001) made one of the first
studies that has explicitly considered HazMat transporta-
tion risk in vehicle routing problems. In this work the risk
is defined as the number of people exposed to a HazMat
transportation accident. The authors utilize an aggregated
measure for computing the risk on an arc of the routing
network. The traditional risk model is used in most of pos-
Universidad Nacional de Colombia. Universit´e de Technologie de
Troyes.
terior studies, see Zografos and Androutsopoulos (2004),
Zografos and Androutsopoulos (2008) and Pradhananga
et al. (2010). The risk is computed as the product of
the hazardous materials accident probability on an arc by
the population exposed to the impacts of an accident on
that arc. In the work of Androutsopoulos and Zografos
(2010) both, the probability of accident and expected
consequences arising in case of an accident occurring on
an arc, are considered time dependent. In a later work,
Androutsopoulos and Zografos (2012) considers that con-
sequences expected radius of a hazardous materials acci-
dents depends on the truck load, but the assessment of the
computational performance of the solving algorithm was
based on load-invariant risk values.
The volume of transported HazMat plays an important
role in determining the likelihood of occurrence of an
incident. In HazMat material distribution, the vehicle load
is reduced by a quantity equals to the customer demand
each time a client is visited, affecting the optimal path in
risk minimization problems. Other important aspects to
consider in HazMat transportation risk estimation are the
probability of a truck tank accident, and the probability
of a material release. Truck tank accident probability
depends on the type of truck and the road conditions;
meanwhile release probability depends on the nature of
IFAC Conference on Manufacturing Modelling,
Management and Control
June 28-30, 2016. Troyes, France
Copyright © 2016 IFAC 538

Gustavo Alfredo Bula et al. / IFAC-PapersOnLine 49-12 (2016) 538–543 539
Mixed Integer Linear Programming Model
for Vehicle Routing Problem for Hazardous
Materials Transportation 
Gustavo Alfredo Bula∗,∗∗ Fabio Augusto Gonzalez ∗
Caroline Prodhon∗∗ H. Murat Afsar ∗∗
Nubia Milena Velasco ∗∗∗
∗System and Industrial Engineering Deparment, Universidad Nacional
de Colombia, Bogot´a, Colombia (e-mail: {gabula,fagonzalezo}@
unal.edu.co)
∗∗ ICD-LOSI, Universit´e de Technologie de Troyes, Troyes Cedex
10004, France (e-mail: {caroline.prodhon, murat.afsar}@ utt.fr)
∗∗∗ School of Management, Universidad de los Andes, Bogot´a,
Colombia (e-mail: nvelasco@ uniandes.edu.co)
Abstract: This paper presents a mathematical model to solve the Heterogeneous Vehicle
Routing Problem (HVRP) in the context of hazardous materials (HazMat) transportation. To
evaluate the model a linear approximation of the total routing risk is used as objective function.
In the first stage a routing risk measure is proposed as a nonlinear function of the truck load.
This function is approximated by means of two different piecewise linear functions (PLF). A
genetic algorithm is employed to estimate the interval limits of PLF. These two functions are
utilized to approximate the total routing risk for the best known solution for the benchmark
instances of HVRP with fixed costs and unlimited fleet, both approaches are compared with
the nonlinear risk function value. In the second stage the best piecewise linear approximation of
the routing risk is integrated to a mixed integer linear programming (MILP) model for solving
the risk optimization problem. The final model is tested on HVRP instances with 20 nodes.
Results show that total cost minimization and total risk minimization appear to be conflicting
objectives.
Keywords: Transportation science, Operations Research, Risk minimization
1. INTRODUCTION
Estimation and analysis of risks is an important aspect
of hazardous materials (HazMat) transportation manage-
ment. Selection of the safest routes is a main focus research
in this area, Erkut et al. (2007). Routing in HazMat
transportation are grouped in two categories: shortest path
and vehicle routing, and scheduling problems. While the
first type of routing problems has been substantially stud-
ied, scarce work has been found on the second type, see
Androutsopoulos and Zografos (2012) and Pradhananga
et al. (2014). The selection of a routing solution in HazMat
transportation depends critically on the model adopted
for the quantification of risk, Bell (2006). Therefore, a
combined model considering both HazMat transportation
risk and routing problem should be developed.
Tarantilis and Kiranoudis (2001) made one of the first
studies that has explicitly considered HazMat transporta-
tion risk in vehicle routing problems. In this work the risk
is defined as the number of people exposed to a HazMat
transportation accident. The authors utilize an aggregated
measure for computing the risk on an arc of the routing
network. The traditional risk model is used in most of pos-
Universidad Nacional de Colombia. Universit´e de Technologie de
Troyes.
terior studies, see Zografos and Androutsopoulos (2004),
Zografos and Androutsopoulos (2008) and Pradhananga
et al. (2010). The risk is computed as the product of
the hazardous materials accident probability on an arc by
the population exposed to the impacts of an accident on
that arc. In the work of Androutsopoulos and Zografos
(2010) both, the probability of accident and expected
consequences arising in case of an accident occurring on
an arc, are considered time dependent. In a later work,
Androutsopoulos and Zografos (2012) considers that con-
sequences expected radius of a hazardous materials acci-
dents depends on the truck load, but the assessment of the
computational performance of the solving algorithm was
based on load-invariant risk values.
The volume of transported HazMat plays an important
role in determining the likelihood of occurrence of an
incident. In HazMat material distribution, the vehicle load
is reduced by a quantity equals to the customer demand
each time a client is visited, affecting the optimal path in
risk minimization problems. Other important aspects to
consider in HazMat transportation risk estimation are the
probability of a truck tank accident, and the probability
of a material release. Truck tank accident probability
depends on the type of truck and the road conditions;
meanwhile release probability depends on the nature of
IFAC Conference on Manufacturing Modelling,
Management and Control
June 28-30, 2016. Troyes, France
Copyright © 2016 IFAC 538
Mixed Integer Linear Programming Model
for Vehicle Routing Problem for Hazardous
Materials Transportation 
Gustavo Alfredo Bula∗,∗∗ Fabio Augusto Gonzalez ∗
Caroline Prodhon∗∗ H. Murat Afsar ∗∗
Nubia Milena Velasco ∗∗∗
∗System and Industrial Engineering Deparment, Universidad Nacional
de Colombia, Bogot´a, Colombia (e-mail: {gabula,fagonzalezo}@
unal.edu.co)
∗∗ ICD-LOSI, Universit´e de Technologie de Troyes, Troyes Cedex
10004, France (e-mail: {caroline.prodhon, murat.afsar}@ utt.fr)
∗∗∗ School of Management, Universidad de los Andes, Bogot´a,
Colombia (e-mail: nvelasco@ uniandes.edu.co)
Abstract: This paper presents a mathematical model to solve the Heterogeneous Vehicle
Routing Problem (HVRP) in the context of hazardous materials (HazMat) transportation. To
evaluate the model a linear approximation of the total routing risk is used as objective function.
In the first stage a routing risk measure is proposed as a nonlinear function of the truck load.
This function is approximated by means of two different piecewise linear functions (PLF). A
genetic algorithm is employed to estimate the interval limits of PLF. These two functions are
utilized to approximate the total routing risk for the best known solution for the benchmark
instances of HVRP with fixed costs and unlimited fleet, both approaches are compared with
the nonlinear risk function value. In the second stage the best piecewise linear approximation of
the routing risk is integrated to a mixed integer linear programming (MILP) model for solving
the risk optimization problem. The final model is tested on HVRP instances with 20 nodes.
Results show that total cost minimization and total risk minimization appear to be conflicting
objectives.
Keywords: Transportation science, Operations Research, Risk minimization
1. INTRODUCTION
Estimation and analysis of risks is an important aspect
of hazardous materials (HazMat) transportation manage-
ment. Selection of the safest routes is a main focus research
in this area, Erkut et al. (2007). Routing in HazMat
transportation are grouped in two categories: shortest path
and vehicle routing, and scheduling problems. While the
first type of routing problems has been substantially stud-
ied, scarce work has been found on the second type, see
Androutsopoulos and Zografos (2012) and Pradhananga
et al. (2014). The selection of a routing solution in HazMat
transportation depends critically on the model adopted
for the quantification of risk, Bell (2006). Therefore, a
combined model considering both HazMat transportation
risk and routing problem should be developed.
Tarantilis and Kiranoudis (2001) made one of the first
studies that has explicitly considered HazMat transporta-
tion risk in vehicle routing problems. In this work the risk
is defined as the number of people exposed to a HazMat
transportation accident. The authors utilize an aggregated
measure for computing the risk on an arc of the routing
network. The traditional risk model is used in most of pos-
Universidad Nacional de Colombia. Universit´e de Technologie de
Troyes.
terior studies, see Zografos and Androutsopoulos (2004),
Zografos and Androutsopoulos (2008) and Pradhananga
et al. (2010). The risk is computed as the product of
the hazardous materials accident probability on an arc by
the population exposed to the impacts of an accident on
that arc. In the work of Androutsopoulos and Zografos
(2010) both, the probability of accident and expected
consequences arising in case of an accident occurring on
an arc, are considered time dependent. In a later work,
Androutsopoulos and Zografos (2012) considers that con-
sequences expected radius of a hazardous materials acci-
dents depends on the truck load, but the assessment of the
computational performance of the solving algorithm was
based on load-invariant risk values.
The volume of transported HazMat plays an important
role in determining the likelihood of occurrence of an
incident. In HazMat material distribution, the vehicle load
is reduced by a quantity equals to the customer demand
each time a client is visited, affecting the optimal path in
risk minimization problems. Other important aspects to
consider in HazMat transportation risk estimation are the
probability of a truck tank accident, and the probability
of a material release. Truck tank accident probability
depends on the type of truck and the road conditions;
meanwhile release probability depends on the nature of
IFAC Conference on Manufacturing Modelling,
Management and Control
June 28-30, 2016. Troyes, France
Copyright © 2016 IFAC 538
Mixed Integer Linear Programming Model
for Vehicle Routing Problem for Hazardous
Materials Transportation 
Gustavo Alfredo Bula∗,∗∗ Fabio Augusto Gonzalez ∗
Caroline Prodhon∗∗ H. Murat Afsar ∗∗
Nubia Milena Velasco ∗∗∗
∗System and Industrial Engineering Deparment, Universidad Nacional
de Colombia, Bogot´a, Colombia (e-mail: {gabula,fagonzalezo}@
unal.edu.co)
∗∗ ICD-LOSI, Universit´e de Technologie de Troyes, Troyes Cedex
10004, France (e-mail: {caroline.prodhon, murat.afsar}@ utt.fr)
∗∗∗ School of Management, Universidad de los Andes, Bogot´a,
Colombia (e-mail: nvelasco@ uniandes.edu.co)
Abstract: This paper presents a mathematical model to solve the Heterogeneous Vehicle
Routing Problem (HVRP) in the context of hazardous materials (HazMat) transportation. To
evaluate the model a linear approximation of the total routing risk is used as objective function.
In the first stage a routing risk measure is proposed as a nonlinear function of the truck load.
This function is approximated by means of two different piecewise linear functions (PLF). A
genetic algorithm is employed to estimate the interval limits of PLF. These two functions are
utilized to approximate the total routing risk for the best known solution for the benchmark
instances of HVRP with fixed costs and unlimited fleet, both approaches are compared with
the nonlinear risk function value. In the second stage the best piecewise linear approximation of
the routing risk is integrated to a mixed integer linear programming (MILP) model for solving
the risk optimization problem. The final model is tested on HVRP instances with 20 nodes.
Results show that total cost minimization and total risk minimization appear to be conflicting
objectives.
Keywords: Transportation science, Operations Research, Risk minimization
1. INTRODUCTION
Estimation and analysis of risks is an important aspect
of hazardous materials (HazMat) transportation manage-
ment. Selection of the safest routes is a main focus research
in this area, Erkut et al. (2007). Routing in HazMat
transportation are grouped in two categories: shortest path
and vehicle routing, and scheduling problems. While the
first type of routing problems has been substantially stud-
ied, scarce work has been found on the second type, see
Androutsopoulos and Zografos (2012) and Pradhananga
et al. (2014). The selection of a routing solution in HazMat
transportation depends critically on the model adopted
for the quantification of risk, Bell (2006). Therefore, a
combined model considering both HazMat transportation
risk and routing problem should be developed.
Tarantilis and Kiranoudis (2001) made one of the first
studies that has explicitly considered HazMat transporta-
tion risk in vehicle routing problems. In this work the risk
is defined as the number of people exposed to a HazMat
transportation accident. The authors utilize an aggregated
measure for computing the risk on an arc of the routing
network. The traditional risk model is used in most of pos-
Universidad Nacional de Colombia. Universit´e de Technologie de
Troyes.
terior studies, see Zografos and Androutsopoulos (2004),
Zografos and Androutsopoulos (2008) and Pradhananga
et al. (2010). The risk is computed as the product of
the hazardous materials accident probability on an arc by
the population exposed to the impacts of an accident on
that arc. In the work of Androutsopoulos and Zografos
(2010) both, the probability of accident and expected
consequences arising in case of an accident occurring on
an arc, are considered time dependent. In a later work,
Androutsopoulos and Zografos (2012) considers that con-
sequences expected radius of a hazardous materials acci-
dents depends on the truck load, but the assessment of the
computational performance of the solving algorithm was
based on load-invariant risk values.
The volume of transported HazMat plays an important
role in determining the likelihood of occurrence of an
incident. In HazMat material distribution, the vehicle load
is reduced by a quantity equals to the customer demand
each time a client is visited, affecting the optimal path in
risk minimization problems. Other important aspects to
consider in HazMat transportation risk estimation are the
probability of a truck tank accident, and the probability
of a material release. Truck tank accident probability
depends on the type of truck and the road conditions;
meanwhile release probability depends on the nature of
IFAC Conference on Manufacturing Modelling,
Management and Control
June 28-30, 2016. Troyes, France
Copyright © 2016 IFAC 538
Mixed Integer Linear Programming Model
for Vehicle Routing Problem for Hazardous
Materials Transportation 
Gustavo Alfredo Bula∗,∗∗ Fabio Augusto Gonzalez ∗
Caroline Prodhon∗∗ H. Murat Afsar ∗∗
Nubia Milena Velasco ∗∗∗
∗System and Industrial Engineering Deparment, Universidad Nacional
de Colombia, Bogot´a, Colombia (e-mail: {gabula,fagonzalezo}@
unal.edu.co)
∗∗ ICD-LOSI, Universit´e de Technologie de Troyes, Troyes Cedex
10004, France (e-mail: {caroline.prodhon, murat.afsar}@ utt.fr)
∗∗∗ School of Management, Universidad de los Andes, Bogot´a,
Colombia (e-mail: nvelasco@ uniandes.edu.co)
Abstract: This paper presents a mathematical model to solve the Heterogeneous Vehicle
Routing Problem (HVRP) in the context of hazardous materials (HazMat) transportation. To
evaluate the model a linear approximation of the total routing risk is used as objective function.
In the first stage a routing risk measure is proposed as a nonlinear function of the truck load.
This function is approximated by means of two different piecewise linear functions (PLF). A
genetic algorithm is employed to estimate the interval limits of PLF. These two functions are
utilized to approximate the total routing risk for the best known solution for the benchmark
instances of HVRP with fixed costs and unlimited fleet, both approaches are compared with
the nonlinear risk function value. In the second stage the best piecewise linear approximation of
the routing risk is integrated to a mixed integer linear programming (MILP) model for solving
the risk optimization problem. The final model is tested on HVRP instances with 20 nodes.
Results show that total cost minimization and total risk minimization appear to be conflicting
objectives.
Keywords: Transportation science, Operations Research, Risk minimization
1. INTRODUCTION
Estimation and analysis of risks is an important aspect
of hazardous materials (HazMat) transportation manage-
ment. Selection of the safest routes is a main focus research
in this area, Erkut et al. (2007). Routing in HazMat
transportation are grouped in two categories: shortest path
and vehicle routing, and scheduling problems. While the
first type of routing problems has been substantially stud-
ied, scarce work has been found on the second type, see
Androutsopoulos and Zografos (2012) and Pradhananga
et al. (2014). The selection of a routing solution in HazMat
transportation depends critically on the model adopted
for the quantification of risk, Bell (2006). Therefore, a
combined model considering both HazMat transportation
risk and routing problem should be developed.
Tarantilis and Kiranoudis (2001) made one of the first
studies that has explicitly considered HazMat transporta-
tion risk in vehicle routing problems. In this work the risk
is defined as the number of people exposed to a HazMat
transportation accident. The authors utilize an aggregated
measure for computing the risk on an arc of the routing
network. The traditional risk model is used in most of pos-
Universidad Nacional de Colombia. Universit´e de Technologie de
Troyes.
terior studies, see Zografos and Androutsopoulos (2004),
Zografos and Androutsopoulos (2008) and Pradhananga
et al. (2010). The risk is computed as the product of
the hazardous materials accident probability on an arc by
the population exposed to the impacts of an accident on
that arc. In the work of Androutsopoulos and Zografos
(2010) both, the probability of accident and expected
consequences arising in case of an accident occurring on
an arc, are considered time dependent. In a later work,
Androutsopoulos and Zografos (2012) considers that con-
sequences expected radius of a hazardous materials acci-
dents depends on the truck load, but the assessment of the
computational performance of the solving algorithm was
based on load-invariant risk values.
The volume of transported HazMat plays an important
role in determining the likelihood of occurrence of an
incident. In HazMat material distribution, the vehicle load
is reduced by a quantity equals to the customer demand
each time a client is visited, affecting the optimal path in
risk minimization problems. Other important aspects to
consider in HazMat transportation risk estimation are the
probability of a truck tank accident, and the probability
of a material release. Truck tank accident probability
depends on the type of truck and the road conditions;
meanwhile release probability depends on the nature of
IFAC Conference on Manufacturing Modelling,
Management and Control
June 28-30, 2016. Troyes, France
Copyright © 2016 IFAC 538
Mixed Integer Linear Programming Model
for Vehicle Routing Problem for Hazardous
Materials Transportation 
Gustavo Alfredo Bula∗,∗∗ Fabio Augusto Gonzalez ∗
Caroline Prodhon∗∗ H. Murat Afsar ∗∗
Nubia Milena Velasco ∗∗∗
∗System and Industrial Engineering Deparment, Universidad Nacional
de Colombia, Bogot´a, Colombia (e-mail: {gabula,fagonzalezo}@
unal.edu.co)
∗∗ ICD-LOSI, Universit´e de Technologie de Troyes, Troyes Cedex
10004, France (e-mail: {caroline.prodhon, murat.afsar}@ utt.fr)
∗∗∗ School of Management, Universidad de los Andes, Bogot´a,
Colombia (e-mail: nvelasco@ uniandes.edu.co)
Abstract: This paper presents a mathematical model to solve the Heterogeneous Vehicle
Routing Problem (HVRP) in the context of hazardous materials (HazMat) transportation. To
evaluate the model a linear approximation of the total routing risk is used as objective function.
In the first stage a routing risk measure is proposed as a nonlinear function of the truck load.
This function is approximated by means of two different piecewise linear functions (PLF). A
genetic algorithm is employed to estimate the interval limits of PLF. These two functions are
utilized to approximate the total routing risk for the best known solution for the benchmark
instances of HVRP with fixed costs and unlimited fleet, both approaches are compared with
the nonlinear risk function value. In the second stage the best piecewise linear approximation of
the routing risk is integrated to a mixed integer linear programming (MILP) model for solving
the risk optimization problem. The final model is tested on HVRP instances with 20 nodes.
Results show that total cost minimization and total risk minimization appear to be conflicting
objectives.
Keywords: Transportation science, Operations Research, Risk minimization
1. INTRODUCTION
Estimation and analysis of risks is an important aspect
of hazardous materials (HazMat) transportation manage-
ment. Selection of the safest routes is a main focus research
in this area, Erkut et al. (2007). Routing in HazMat
transportation are grouped in two categories: shortest path
and vehicle routing, and scheduling problems. While the
first type of routing problems has been substantially stud-
ied, scarce work has been found on the second type, see
Androutsopoulos and Zografos (2012) and Pradhananga
et al. (2014). The selection of a routing solution in HazMat
transportation depends critically on the model adopted
for the quantification of risk, Bell (2006). Therefore, a
combined model considering both HazMat transportation
risk and routing problem should be developed.
Tarantilis and Kiranoudis (2001) made one of the first
studies that has explicitly considered HazMat transporta-
tion risk in vehicle routing problems. In this work the risk
is defined as the number of people exposed to a HazMat
transportation accident. The authors utilize an aggregated
measure for computing the risk on an arc of the routing
network. The traditional risk model is used in most of pos-
Universidad Nacional de Colombia. Universit´e de Technologie de
Troyes.
terior studies, see Zografos and Androutsopoulos (2004),
Zografos and Androutsopoulos (2008) and Pradhananga
et al. (2010). The risk is computed as the product of
the hazardous materials accident probability on an arc by
the population exposed to the impacts of an accident on
that arc. In the work of Androutsopoulos and Zografos
(2010) both, the probability of accident and expected
consequences arising in case of an accident occurring on
an arc, are considered time dependent. In a later work,
Androutsopoulos and Zografos (2012) considers that con-
sequences expected radius of a hazardous materials acci-
dents depends on the truck load, but the assessment of the
computational performance of the solving algorithm was
based on load-invariant risk values.
The volume of transported HazMat plays an important
role in determining the likelihood of occurrence of an
incident. In HazMat material distribution, the vehicle load
is reduced by a quantity equals to the customer demand
each time a client is visited, affecting the optimal path in
risk minimization problems. Other important aspects to
consider in HazMat transportation risk estimation are the
probability of a truck tank accident, and the probability
of a material release. Truck tank accident probability
depends on the type of truck and the road conditions;
meanwhile release probability depends on the nature of
IFAC Conference on Manufacturing Modelling,
Management and Control
June 28-30, 2016. Troyes, France
Copyright © 2016 IFAC 538
the accident, the type of HazMat, and the container
conditions. In this context the probability of a HazMat
incident depends on the load and type of the truck.
In this paper we focus on expected population exposure
risk mitigation via selection of routes by solving a vari-
ant of heterogeneous vehicle routing problem (HVRP).
The remainder of this paper is organized as follows. The
next section provides a problem definition and presents
the mixed integer linear programming (MILP) formula-
tion of the HVRP in HazMat transportation. Section 3
presents the routing risk measure and it is formulated as
a piecewise linear approximation. Section 4 presents the
computational results of risk approximation and MIPL
model application to the well-known benchmark instances
of a HVRP variant. The final section concludes the paper
and discusses future research issues.
2. PROBLEM DEFINITION
The vehicle routing problem for HazMat transportation
using a heterogeneous fleet can be defined as the determi-
nation of the safest routes assigned to a fleet of different
vehicles transporting a specific HazMat from a depot to a
set of clients. This problem is characterized as the HVRP
proposed by Golden et al. (1984) but with the introduc-
tion of the transportation risk objective function, that is
expressed as the expected consequences of a hazardous
materials accident.
To model the problem, a Mixed Integer Linear Program-
ming (MILP) based on the proposal of Gheysens et al.
(1984) and Baldacci et al. (2008) is proposed.
In this model the HVRP is defined on a complete directed
graph G(N,L). The node set N={0,1,2,...,n}includes
the depot node 0, and a set of customer nodes (service
stations), C. Each client i∈Chas a demand diand it is
connected with other node j∈N by an arc (i, j)∈L.
Each arc is characterized by a length alij , a cost cij , and
a number of persons exposed to the consequences of a
HazMat release PD
ij . To satisfy the demands there is a
set of Kdifferent types of trucks. A truck type k∈Kis
characterized by a maximal capacity Qk, a fixed cost fk,
and an accident rate TTAR
k.
A solution is composed of a set of routes SR satisfying all
customers demands once. Each route r∈ SR starts and
ends at the depot, and respects the vehicle capacity Qk.
Split deliveries are not allowed.
Two types of decision variables are defined:
yk
ij : flow of goods from node ito node j
in a vehicle of type k
xk
ij :1 if a vehicle of type ktravels the arc (i, j)
0 otherwise
The HVRP for HazMat transportation is formulated as
follows:
z=
r∈SR
R(r) (1)
subject to:

k∈K 
i∈N
xk
ij =1,∀j∈N \{0}(2)

i∈N
xk
ij −
i∈N
xk
ji =0,∀k∈K,∀j∈C (3)

k∈K 
i∈N
yk
ij −
k∈K 
i∈N
yk
ji =dj,∀j∈C (4)
dj
k∈K
xk
ij ≤
k∈K
yk
ij ∀i, j ∈N,i=j(5)
yk
ij ≤xk
ij (Qk−di)∀i, j ∈N i=j, ∀k∈K (6)
yk
ij ≥0,∀k∈K,∀(i, j)∈L (7)
xk
ij ∈{0,1},∀k∈K,∀(i, j)∈L (8)
Equation (1) expresses objective function for minimizing
the total risk, being R(r) the risk associated to route r.
The set of constraints (2) ensures that each customer is
visited exactly once, and the set (3) and (4) represents the
conservation flux constraints. Additionally, (4) guarantees
demands satisfaction. Constraints (5) and (6) state that no
goods are transported from ito jif no vehicle is serving
the arc (i, j), and (6) define the load of the vehicle kwhen
traversing arc (i, j).
3. TRANSPORTATION RISK ASSESSMENT
In order to include the risk in the MILP proposed on
section 2, a model for the route risk must be defined and
after it must be linearized.
3.1 Route Risk Model
The proposed route risk model is based on the traditional
one that uses the expected consequence as a measure of
risk, Erkut et al. (2007). In this model population exposure
is the consequence measure but other exposed receptors
can be considered as the environment, or the properties in
vicinity of the HazMat transportation incident.
Let rbe a route composed of (r1,r
2),(r2,r
3),··· ,(rn−1,r
n)
arcs, where rirepresents the client visited on i−th position
on the route. To evaluate the risk associated to a route r
we use the total expected consequences expressed as:
R(r)=
n−1

u=1 u−1

v=1
(1 −PIk
rvrv+1 )PIk
ruru+1 PD
ruru+1 (9)
Where PIk
ruru+1 is the probability of occurrence of an
incident using a type of vehicle k, and PDruru+1 is the
exposed population within a buffer distance surrounding
the arc (ruru+1), see Erkut et al. (2007). To estimate
R(r) we considered that PIk
ruru+1 PIk
rvrv+1
∼
=0 for all
pair of consecutive arcs (ruru+1),(rvrv+1), given the fact
that HazMat incident probability takes small values, Erkut
et al. (2007). In consequence, (9) is reduced to:
IFAC MIM 2016
June 28-30, 2016. Troyes, France
539

540 Gustavo Alfredo Bula et al. / IFAC-PapersOnLine 49-12 (2016) 538–543
R(r)=
n−1

u=1
PIk
ruru+1 PD
ruru+1 =
(i,j)∈r
PIk
ij PD
ij (10)
To evaluate the incident probability PIk
ij and consequence
PDk
ij it is necessary to consider that an accident occurs,
generating a material release that could have several
outcomes (jet-fire, pool fire, toxic cloud, explosion, etc.)
affecting a population, as is presented on Fig. 1. All the
events involved in a HazMat incident have an associated
probability that is used to estimate the risk.
Fig. 1. Risk assessment in HazMat Transportation
PIk
ij is computed by using (11) where, δk
ij is the incident
rate for a truck type k, and alij is the length of arc (i, j).
PIk
ij =δk
ij ×alij (11)
The incident rate per kilometer for a truck type k(δk
ij ) is
function of the incident probability given a truck accident
on an arc (i, j)(IPk
ij ), and the truck tank accident rate
(TTAR
k), that depends on the vehicle type, Button and
Reilly (2000):
δk
ij =TTAR
k×IPk
ij (12)
Equation (13) defines IPk
ij as the product of the release
probability of HazMat in a truck accident (Prelease ) and
the probability of a certain outcome arising as a conse-
quence of the initiating event, Ronza et al. (2007).
IPij =Pr elease ×(β×(yk
ij )α) (13)
As mentioned above yk
ij is the load of a truck type
ktraversing the arc (ij), while αand βare constant
values that depend on the type of material. The release
probability in a truck accident, Pr elease, is obtained from
the combination of probabilities of accident outcomes with
the probabilities of different types of accidents for a truck
carrying a HazMat, Saccomanno et al. (1993), Button and
Reilly (2000) and Kazantzi et al. (2011b). The probability
of a certain consequence of the initiating event as fire or
explosion is estimated using the model proposed by Ronza
et al. (2007). This model is based on empirical approaches
to predict ignition and explosion probabilities for land
transportation spills as a function of the substance, the
load, and the transportation mode.
The population exposure in a radius (buffer size) is defined
depending on the type of HazMat transported and the
maximum volume that can be carried on a truck.
Combining (10)-(13), the route risk for a given route ris
estimated as:
R(r)=TTAR
k×Prelease ×β×

(i,j)∈r
(yk
ij )α×(alij ×PD
ij )(14)
The objective risk function for the mathematical model
becomes:
z=Prelease ×β×

(i,j)∈L 
k∈K
TTAR
k×(yk
ij )α×alij ×PD
ij (15)
As this is a nonlinear function on, yk
ij , a piecewise linear
approximation is used.
3.2 Linear Approximation
Let [q0,q
M] be a bounded interval for yk
ij , this interval
is divided into an increasing sequence of Mbreakpoints
{l0,...,l
M}. The value of (yk
ij )αis then approximated by
using linear interpolations over the Msegments according
to (16).
(yk
ij )α:= am+bmyk
ij ,
yk
ij ∈[lm−1,l
m]∀m∈{1,...,M}(16)
where am∈R,bm∈Rare the intercepts and the slopes of
the linear functions, respectively, and l0<l
1<··· <l
M.
Two different kinds of piecewise linear approximations are
performed: first looks for define an straight line joining
two points on the function ([lm−1,lm]) and the second one
looks for the tangent line to one point (tpm) on the curve,
see Fig. 2.
3.3 Genetic Algorithm for Piecewise Linear Approximation
Parameters
Using a genetic algorithm (GA) the values of the limits
of each interval are established. In both piecewise linear
approaches, a GA is used for finding the solution that
minimizes the total sum of squared errors.
A solution Sis represented by an array composed of rp
arrays of binary numbers of the same size lb. Each rp array
encodes the position of a required point on the array of
integer values, qi∈[q0,q
M], that represents the different
possible values of the truck load. On the first approach,
these points correspond to the intervals limits and on the
second one, they correspond to the tangent points, see
Fig. 3. The chromosome for each individual of the initial
population Pt=0 is randomly generated using a Bernoulli
distribution with parameter p=0.5.
The evaluation function is the sum of the squared errors
of function approximation, see (17). For this, each linear
function equation and the interval limits are computed.
IFAC MIM 2016
June 28-30, 2016. Troyes, France
540

Gustavo Alfredo Bula et al. / IFAC-PapersOnLine 49-12 (2016) 538–543 541
R(r)=
n−1

u=1
PIk
ruru+1 PD
ruru+1 =
(i,j)∈r
PIk
ij PD
ij (10)
To evaluate the incident probability PIk
ij and consequence
PDk
ij it is necessary to consider that an accident occurs,
generating a material release that could have several
outcomes (jet-fire, pool fire, toxic cloud, explosion, etc.)
affecting a population, as is presented on Fig. 1. All the
events involved in a HazMat incident have an associated
probability that is used to estimate the risk.
Fig. 1. Risk assessment in HazMat Transportation
PIk
ij is computed by using (11) where, δk
ij is the incident
rate for a truck type k, and alij is the length of arc (i, j).
PIk
ij =δk
ij ×alij (11)
The incident rate per kilometer for a truck type k(δk
ij ) is
function of the incident probability given a truck accident
on an arc (i, j)(IPk
ij ), and the truck tank accident rate
(TTAR
k), that depends on the vehicle type, Button and
Reilly (2000):
δk
ij =TTAR
k×IPk
ij (12)
Equation (13) defines IPk
ij as the product of the release
probability of HazMat in a truck accident (Prelease ) and
the probability of a certain outcome arising as a conse-
quence of the initiating event, Ronza et al. (2007).
IPij =Pr elease ×(β×(yk
ij )α) (13)
As mentioned above yk
ij is the load of a truck type
ktraversing the arc (ij), while αand βare constant
values that depend on the type of material. The release
probability in a truck accident, Pr elease, is obtained from
the combination of probabilities of accident outcomes with
the probabilities of different types of accidents for a truck
carrying a HazMat, Saccomanno et al. (1993), Button and
Reilly (2000) and Kazantzi et al. (2011b). The probability
of a certain consequence of the initiating event as fire or
explosion is estimated using the model proposed by Ronza
et al. (2007). This model is based on empirical approaches
to predict ignition and explosion probabilities for land
transportation spills as a function of the substance, the
load, and the transportation mode.
The population exposure in a radius (buffer size) is defined
depending on the type of HazMat transported and the
maximum volume that can be carried on a truck.
Combining (10)-(13), the route risk for a given route ris
estimated as:
R(r)=TTAR
k×Prelease ×β×

(i,j)∈r
(yk
ij )α×(alij ×PD
ij )(14)
The objective risk function for the mathematical model
becomes:
z=Prelease ×β×

(i,j)∈L 
k∈K
TTAR
k×(yk
ij )α×alij ×PD
ij (15)
As this is a nonlinear function on, yk
ij , a piecewise linear
approximation is used.
3.2 Linear Approximation
Let [q0,q
M] be a bounded interval for yk
ij , this interval
is divided into an increasing sequence of Mbreakpoints
{l0,...,l
M}. The value of (yk
ij )αis then approximated by
using linear interpolations over the Msegments according
to (16).
(yk
ij )α:= am+bmyk
ij ,
yk
ij ∈[lm−1,l
m]∀m∈{1,...,M}(16)
where am∈R,bm∈Rare the intercepts and the slopes of
the linear functions, respectively, and l0<l
1<··· <l
M.
Two different kinds of piecewise linear approximations are
performed: first looks for define an straight line joining
two points on the function ([lm−1,lm]) and the second one
looks for the tangent line to one point (tpm) on the curve,
see Fig. 2.
3.3 Genetic Algorithm for Piecewise Linear Approximation
Parameters
Using a genetic algorithm (GA) the values of the limits
of each interval are established. In both piecewise linear
approaches, a GA is used for finding the solution that
minimizes the total sum of squared errors.
A solution Sis represented by an array composed of rp
arrays of binary numbers of the same size lb. Each rp array
encodes the position of a required point on the array of
integer values, qi∈[q0,q
M], that represents the different
possible values of the truck load. On the first approach,
these points correspond to the intervals limits and on the
second one, they correspond to the tangent points, see
Fig. 3. The chromosome for each individual of the initial
population Pt=0 is randomly generated using a Bernoulli
distribution with parameter p=0.5.
The evaluation function is the sum of the squared errors
of function approximation, see (17). For this, each linear
function equation and the interval limits are computed.
IFAC MIM 2016
June 28-30, 2016. Troyes, France
540
Fig. 2. Piecewise linear approximation functions
Fig. 3. Genetic Algorithm Solution Encoding
The fitness of a individual is defined as the inverse of the
evaluation function value.

∀qi∈[q0,qM]
[qα
i−(am+bm×qi)]2,
qi∈[lm−1,l
m]∀m∈{1,...,M}
(17)
Next generations Pt+1 are produced thanks to recombina-
tion and mutation processes.
3.4 Picewise Linear Approximation Modelling
Let tk
ij be the piecewise linear approximation value of
(yk
ij )αand t0=(l0)α. Then, the piecewise-linear functions
of the road segment risk can be transformed into MILP
model by introducing binary variables hm
ijk and continuous
variables λm
ijk ,m=1,··· ,M. The hm
ijk indicates the
comparison between yk
ij and lm−1, and the λm
ijk variable
evaluates the distance between yk
ij and lm−1, following
Padberg (2000). The model for each (yk
ij )αis given as
follows:
tk
ij =t0+
M

m=1
bmλm
ijk ∀i, j ∈N i=j, ∀k∈K (18)
yk
ij =l0+
M

m=1
λm
ijk ∀i, j ∈N i=j, ∀k∈K (19)
λ1
ijk ≤l1−l0∀i, j ∈N i=j, ∀k∈K (20)
λm
ijk ≥(lm−lm−1)hm
ijk ∀i, j ∈N i=j, ∀k∈K
m=1,··· ,M −1(21)
λm+1
ijk ≤(lm+1 −lm)hm
ijk ∀i, j ∈N i=j, ∀k∈K
m=1,··· ,M −1
(22)
λm
ijk ≥0,∀k∈K,∀(i, j )∈L,m=1,··· ,M (23)
hm
ijk ∈{0,1},∀k∈K,∀(i, j )∈L,m=1,··· ,M (24)
tk
ij ≥0,∀k∈K,∀(i, j)∈L (25)
total risk function for a set of routes is now defined as:
z=Prelease ×β×

(i,j)∈L 
k∈K
TTAR
k×alij ×PD
ij ×tk
ij (26)
Equations (18)-(25) are included as constraints into the
model defined in subsection 2.1.
4. NUMERICAL TESTS
4.1 Problem Instances
To assess the quality of the approximation risk function,
we computed the total risk of best known solution routes
for HVRP instances proposed by Golden et al. (1984). 1
In this work we assumed that the nodes demand is ex-
pressed in hundreds of gallons of gasoline, gasoline density
is 2.805Kg/Gallon, and one unit of distance is equal of
100m. A release probability equals to 0.02487845, from
Kazantzi et al. (2011a) and Button and Reilly (2000), and
values of 0.72 and 0.00027 for αand βare used, see Ronza
et al. (2007). The risk objective (26) becomes:
z=0.02487845 ×0.0027×

(i,j)∈L 
k∈K
TTAR
k×alij ×PD
ij
10 ×tk
ij (27)
The exposed population surrounding an arc (i, j ), PD
ij ,
is represented by a quantity in the square grid (dimension
1The cost-optimal solution for instances 3, 4, 5, 14 and 15 see
Gendreau et al. (1999); instances 13 and 17 see Brand˜ao (2009);
instance 16 see Penna et al. (2013); instance 18 see Prins (2009);
and the best know solution for instance 20 see Subramanian et al.
(2012).
IFAC MIM 2016
June 28-30, 2016. Troyes, France
541

542 Gustavo Alfredo Bula et al. / IFAC-PapersOnLine 49-12 (2016) 538–543
4×4 distance units) that contains the arc. A decay function
from the center to the exterior is used in order to represent
an urban area where the population density is decreasing
towards peripheral zones. PD
ij value generator is shown
in (28).
PD
ij = 9500u(1 −maxDist −popDist
maxDist ) + 500 (28)
where uis generated using a uniform distribution, u∼
U(0.4,0.6), maxDist is the maximum between the length
and the width of the rectangle that contains the squared
grid, and popDist is the maximum value between the
difference of the abscise and ordinate coordinates of the
left superior corner of the current square grid and the
coordinates of left superior corner of the central square
grid.
The HazMat incident rate of a vehicle on an arc is about
1×10−6per (vehicle −Km), see Button and Reilly
(2000) and Kazantzi et al. (2011b). In this case we use
TTAR
k∼U[0.6,1.0]10−6per (vehicle −Km)∀k∈K.
The GA was code in Java SE 8 and executed in an Intel
Core i7 Processor 2.4 GHz with 16 GB of RAM running
Windows 10. The MILP formulation was implemented
using a comercial solver (Gurobi) for Python 2.7.
For the piece-wise linear approximation of the incident
probability function, the number of linear functions (M),
is fixed at four corresponding to small load, a medium-
size load, a large load and a very-large load. After a
parameter tuning for a truck load that goes from 1
(minimum demand) to 400 (maximum vehicle capacity),
the parameters of GA were: a population size of 100
individuals, a mutation rate of 0.02, and 500 generations.
The value assigned to lb is 10.
After running the algorithm, the results for the first
piecewise linear approximation function are:
b1=0.07349097; 280.5≤(280.5yk
ij )≤9256.5
b2=0.04590467; 9256.5≤(280.5yk
ij )≤30574.5
b3=0.03555368; 30574.5≤(280.5yk
ij )≤64795.5
b4=0.02979203; 64795.5≤(280.5yk
ij )≤112200.0
and for the second function are:
b1=0.06835302; 280.5≤(280.5yk
ij )≤9995.2
b2=0.04468697; 9995.2≤(280.5yk
ij )≤31855.6
b3=0.03515308; 31855.6≤(280.5yk
ij )≤65838.6
b4=0.02964824; 65838.6≤(280.5yk
ij )≤112200.0
Given that the truck load varies according to the demand
of the clients to visit, and the range of these values is
the same for all the studied instances, the same piecewise
linear approximation was used in all of them.
4.2 Results
Tables 1 and 2 show the expected consequences estimated
using (15) and both PLF. In these tables, Inst. denotes
the name of the test-problem, ndenotes the number
of customers, BKS represents the best known solution
(optimal solutions in some cases). The approximation and
Table 1. Total routing risk for optimal-cost
solution using the first piecewise linear approx-
imation
Risk
BKS Exact Approximation (27)
Inst. n cost value (15) Val ue Gap %
×10−6×10−6
3 20 961.026 307.65 296.526 3.62
4 20 6437.331 248.85 233.875 6.02
5 20 1007.051 431.687 418.693 3.01
6 20 6516.468 261.625 248.483 5.02
13 50 2406.361 768.897 754.072 1.93
14 50 9119.28 571.197 554.861 2.86
15 50 2586.37 700.91 673.843 3.86
16 50 2720.433 790.004 762.671 3.46
17 75 1734.531 1479.142 1454.417 1.67
18 75 2369.646 1229.726 1198.711 2.52
19 100 8661.808 1171.318 1131.362 3.41
20 100 4032.81 1189.733 1148.8 3.44
Table 2. Total routing risk for optimal-cost
solution using the second piecewise linear ap-
proximation
Risk
BKS Exact Approximation (27)
Inst. n cost value (15) Val ue Gap %
×10−6×10−6
3 20 961.026 307.65 313.025 1.75
4 20 6437.331 248.85 252.622 1.52
5 20 1007.051 431.687 437.342 1.31
6 20 6516.468 261.625 266.198 1.75
13 50 2406.361 768.897 775.776 0.89
14 50 9119.28 571.197 577.295 1.07
15 50 2586.37 700.91 710.516 1.37
16 50 2720.433 790.004 801.02 1.39
17 75 1734.531 1479.142 1491.461 0.83
18 75 2369.646 1229.726 1242.815 1.06
19 100 8661.808 1171.318 1187.278 1.36
20 100 4032.81 1189.733 1205.337 1.31
the exact value of the expected consequences for each cost-
optimal solutions is presented, and the gap between both
approaches and the exact value. All the possible route
sequence combinations were considered in order to find
the minimum risk for a solution.
Given that the average difference in percentage between
the exact value (by using (15)) and the piecewise linear
approximation using the first function (3.4%) is greater
than the average difference (1.3%) for the second function,
this last approach is used on optimization problem.
The optimal risk for the first four instances is obtained
using the above presented MILP model of the risk problem
optimization shown in table 3. For other instances the
CPU time is more than 15 hours. From table 3, it is
remarkable that for instance 5, the total cost for the
optimal-risk solution is more than twice the optimal-cost
solution. Also, the optimal risk value obtained for each
instance is greater than the total risk value obtained when
the optimal-risk solution is evaluated by using (15).
5. CONCLUSIONS
The present study addresses the risk minimization prob-
lem for vehicle routing in hazardous materials (HazMat)
transportation using heterogeneous fleet of vehicles. We
IFAC MIM 2016
June 28-30, 2016. Troyes, France
542

Gustavo Alfredo Bula et al. / IFAC-PapersOnLine 49-12 (2016) 538–543 543
4×4 distance units) that contains the arc. A decay function
from the center to the exterior is used in order to represent
an urban area where the population density is decreasing
towards peripheral zones. PD
ij value generator is shown
in (28).
PD
ij = 9500u(1 −maxDist −popDist
maxDist ) + 500 (28)
where uis generated using a uniform distribution, u∼
U(0.4,0.6), maxDist is the maximum between the length
and the width of the rectangle that contains the squared
grid, and popDist is the maximum value between the
difference of the abscise and ordinate coordinates of the
left superior corner of the current square grid and the
coordinates of left superior corner of the central square
grid.
The HazMat incident rate of a vehicle on an arc is about
1×10−6per (vehicle −Km), see Button and Reilly
(2000) and Kazantzi et al. (2011b). In this case we use
TTAR
k∼U[0.6,1.0]10−6per (vehicle −Km)∀k∈K.
The GA was code in Java SE 8 and executed in an Intel
Core i7 Processor 2.4 GHz with 16 GB of RAM running
Windows 10. The MILP formulation was implemented
using a comercial solver (Gurobi) for Python 2.7.
For the piece-wise linear approximation of the incident
probability function, the number of linear functions (M),
is fixed at four corresponding to small load, a medium-
size load, a large load and a very-large load. After a
parameter tuning for a truck load that goes from 1
(minimum demand) to 400 (maximum vehicle capacity),
the parameters of GA were: a population size of 100
individuals, a mutation rate of 0.02, and 500 generations.
The value assigned to lb is 10.
After running the algorithm, the results for the first
piecewise linear approximation function are:
b1=0.07349097; 280.5≤(280.5yk
ij )≤9256.5
b2=0.04590467; 9256.5≤(280.5yk
ij )≤30574.5
b3=0.03555368; 30574.5≤(280.5yk
ij )≤64795.5
b4=0.02979203; 64795.5≤(280.5yk
ij )≤112200.0
and for the second function are:
b1=0.06835302; 280.5≤(280.5yk
ij )≤9995.2
b2=0.04468697; 9995.2≤(280.5yk
ij )≤31855.6
b3=0.03515308; 31855.6≤(280.5yk
ij )≤65838.6
b4=0.02964824; 65838.6≤(280.5yk
ij )≤112200.0
Given that the truck load varies according to the demand
of the clients to visit, and the range of these values is
the same for all the studied instances, the same piecewise
linear approximation was used in all of them.
4.2 Results
Tables 1 and 2 show the expected consequences estimated
using (15) and both PLF. In these tables, Inst. denotes
the name of the test-problem, ndenotes the number
of customers, BKS represents the best known solution
(optimal solutions in some cases). The approximation and
Table 1. Total routing risk for optimal-cost
solution using the first piecewise linear approx-
imation
Risk
BKS Exact Approximation (27)
Inst. n cost value (15) Val ue Gap %
×10−6×10−6
3 20 961.026 307.65 296.526 3.62
4 20 6437.331 248.85 233.875 6.02
5 20 1007.051 431.687 418.693 3.01
6 20 6516.468 261.625 248.483 5.02
13 50 2406.361 768.897 754.072 1.93
14 50 9119.28 571.197 554.861 2.86
15 50 2586.37 700.91 673.843 3.86
16 50 2720.433 790.004 762.671 3.46
17 75 1734.531 1479.142 1454.417 1.67
18 75 2369.646 1229.726 1198.711 2.52
19 100 8661.808 1171.318 1131.362 3.41
20 100 4032.81 1189.733 1148.8 3.44
Table 2. Total routing risk for optimal-cost
solution using the second piecewise linear ap-
proximation
Risk
BKS Exact Approximation (27)
Inst. n cost value (15) Val ue Gap %
×10−6×10−6
3 20 961.026 307.65 313.025 1.75
420 6437.331 248.85 252.622 1.52
5 20 1007.051 431.687 437.342 1.31
6 20 6516.468 261.625 266.198 1.75
13 50 2406.361 768.897 775.776 0.89
14 50 9119.28 571.197 577.295 1.07
15 50 2586.37 700.91 710.516 1.37
16 50 2720.433 790.004 801.02 1.39
17 75 1734.531 1479.142 1491.461 0.83
18 75 2369.646 1229.726 1242.815 1.06
19 100 8661.808 1171.318 1187.278 1.36
20 100 4032.81 1189.733 1205.337 1.31
the exact value of the expected consequences for each cost-
optimal solutions is presented, and the gap between both
approaches and the exact value. All the possible route
sequence combinations were considered in order to find
the minimum risk for a solution.
Given that the average difference in percentage between
the exact value (by using (15)) and the piecewise linear
approximation using the first function (3.4%) is greater
than the average difference (1.3%) for the second function,
this last approach is used on optimization problem.
The optimal risk for the first four instances is obtained
using the above presented MILP model of the risk problem
optimization shown in table 3. For other instances the
CPU time is more than 15 hours. From table 3, it is
remarkable that for instance 5, the total cost for the
optimal-risk solution is more than twice the optimal-cost
solution. Also, the optimal risk value obtained for each
instance is greater than the total risk value obtained when
the optimal-risk solution is evaluated by using (15).
5. CONCLUSIONS
The present study addresses the risk minimization prob-
lem for vehicle routing in hazardous materials (HazMat)
transportation using heterogeneous fleet of vehicles. We
IFAC MIM 2016
June 28-30, 2016. Troyes, France
542
Table 3. Optimal risk values for some instances
of HVRP with unlimited fleet
Risk Optimization
Inst. n MILP Exact value (15) Cost Time
×10−6×10−6value Sec.
3 20 156.817 154.08 2047.102 17
4 20 168.238 165.597 33556.056 68
5 20 251.837 248.793 1305.316 133
6 20 196.801 193.361 14541.962 1839
propose a mixed integer linear programming model that
incorporates a piecewise linear approximation of the trans-
portation risk objective. The objective function captures
some variations in risk which are not considered in pre-
vious models in HazMat transportation. This function
includes an estimation of the HazMat transportation inci-
dent probability as a function of the truck load and type.
The proposed model is a more realistic approach and it
can be used for reducing the risk in HazMat distribution.
Comparing the risk of the cost-optimal solution to the risk-
optimal solution for the small instances of Heterogeneous
Vehicle Routing Problem with Fixed Costs and unlimited
fleet, the minimization of total cost and the minimization
of total risk appear as to be conflicting ob jectives.
Given the NP-hard nature of VRP problems, the develop-
ment of heuristics and meta-heuristics techniques that can
deal with problems with a greater number of clients is a
promising aspect of future research in using heterogeneous
fleet of truck for HazMat transportation.
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