Hazardous Materials Transportation using
Bi-level Linear Programming
Case-study of Liquid Fuel Distribution
Madalena S. Rodrigues
1
, Marta C. Gomes
1
, Alexandre B. Gonçalves
2
and Sílvia Shrubsall
1
1
CESUR,
Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, Lisboa, Portugal
2
ICIST, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, Lisboa, Portugal
Keywords: Hazardous Materials Transportation, Bi-level Linear Programming, Road Safety in Urban Areas,
Geographical Information Systems (GIS).
Abstract: Hazardous materials (hazmats) are essential for the competitiveness of contemporary societies, however
their transportation is potentially dangerous and expensive. In Portugal, despite the interest of both
academia and industry, studies enabling the identification of preferable road routes for hazmats distribution
were not identified. Hence, this research aims at contributing to advancing knowledge in identifying these
routes in the national current context by balancing two frequently intrinsically conflicting aspects of
hazmats transportation: the safety and the economic viability of the available routes. For that, a bi-level
linear programming model was implemented in the GAMS modelling system and applied to a real-world
case study using petrol and diesel fuels delivery data from a prominent energy group acting in the country.
The company shared data of distribution loads over one calendar year to both petrol stations and direct
clients in Lisbon. A geographical information system (GIS) was used to map Lisbon road network, which
was found to be significantly larger than other networks used in similar studies described in the literature.
The model was solved to optimality in a short computation time leading to the clear identification of the
preferable road routes for liquid fuel distribution in the Lisbon district of Olivais. The success of the
methodology applied in this study, including the generic implementation of the bi-level linear programming
model, offers an optimistic prospect for a gradual increase of the geographical coverage, assessed risks and
general complexity of the initial model.
1 INTRODUCTION
Hazardous materials (hazmats) are substances
dangerous to handle because of their flammable,
explosive or toxic nature, comprising serious threats
to human safety and health, to property or to the
environment (HMCRP, 2009). However, the
economic success of societies requires the
transportation of considerable amounts of hazmats:
indeed, in Portugal, 10% of the total materials
carried by road are hazmats (ANPC, n/d). Although
the transportation of hazmats is associated to
relatively few accidents, their consequences can be
serious due to the nature of the cargo (PHMSA,
2011). The transportation of hazmats is, therefore,
subject to special safety requirements to protect
populations and the environment, and it has been
recognized as important to find a balance between
these and the economic viability of the operation.
The hazmats industry generally places safety at the
centre of business and the analysis of safe route
definitely requires further studies beyond those
presented in the literature.
A political tool commonly used to reduce the risk
of hazmats transportation is the interdiction by the
regulator of certain road sections identified as more
vulnerable; the carrier can then choose the best
routes in the available network. Erkut et al. (2007),
in a survey on hazardous materials transportation,
name this the hazmat transportation network design
problem and state that it started receiving the
attention of researchers with the work of Kara and
Verter (2004). These authors developed a bi-level
linear programming model where hazmats are
grouped into categories according to the risk impact.
The model, which assigns an available road network
to each hazmat category, was applied to the region
376
S. Rodrigues M., C. Gomes M., B. Gonçalves A. and Shrubsall S..
Hazardous Materials Transportation using Bi-level Linear Programming - Case-study of Liquid Fuel Distribution.
DOI: 10.5220/0005223703760382
In Proceedings of the International Conference on Operations Research and Enterprise Systems (ICORES-2015), pages 376-382
ISBN: 978-989-758-075-8
Copyright
c
2015 SCITEPRESS (Science and Technology Publications, Lda.)
of Western Ontario, Canada.
Later on, the same authors proposed a linear
programming model based on the shortest path
formulation (Verter and Kara, 2008). In this
formulation, routes considered economically
infeasible were left out of the model, assuring that
the carriers would not be forced to use routes that
were their least preferable choices. Other approaches
to this problem are the ones of Erkut and Alp (2007)
and Erkut and Gzara (2008). The first authors
developed an algorithm in two phases: in the first
stage a minimum risk network is found, while in the
second stage the network is expanded in an iterative
procedure. This enables the regulator to control the
density of the hazmats network and the freedom
given to the carriers. Erkut and Gzara (2008) used a
flow problem formulation in a bi-level network, and
compare four networks scenarios related to different
decision levels: non-regulated model, over-regulated
model, two-step model and bi-level model.
In the present research the model of Kara and
Verter (2004) was implemented so as to identify safe
routes for the transportation of hazmats using real
data provided by a major company of the energy
sector in Portugal. Particularly noteworthy is the
case study dimension, which significantly surpasses
the ones of similar studies described in the reviewed
literature and posed a significant challenge to model
implementation.
2 A BI-LEVEL LINEAR
PROGRAMMING MODEL FOR
HAZARDOUS MATERIALS
TRANSPORTATION
This section defines the problem under study and
presents the bi-level linear programming formulation
of Kara and Verter (2004), followed by an
equivalent mixed-integer linear programming
(MILP) model, which was the one implemented and
solved with available computational tools.
2.1 Problem Definition
A bi-level model consists of two optimization
problems that are hierarchically related and belong
to two distinct decision makers, in which the optimal
decision for one decision maker is constrained by
the choices of the other one (Bianco et al., 2009). In
the present problem the regulator assumes the
leading role, as its decisions are taken at the first
level and the choice of routes by the carriers will
depend on them. Thus, the outer-level problem is the
regulator concern and determines the links to include
in the network to be made available, according to the
criteria of total risk minimization, while the inner-
level problem, pertaining to the carriers, consists of
the route choice in this network. Kara and Verter
(2004) present a model that determines the network
of minimum total risk and assumes the cost
minimization of the carriers, achieving significant
risk reductions in the transportation of hazmats in
Western Ontario, Canada.
An accident resulting in a release of the hazmat
is called an incident (Erkut et al., 2007). The model
of Kara and Verter (2004) assumes that the
undesired consequences of an incident involving
hazmats occur within a given distance from the
place where it happened, which varies according to
the type of hazmat. Thus, hazmats are grouped into
categories according to the impact of incidents
associated with each of them.
To assess the risk associated to hazmats
transportation, additivity of impacts is assumed. The
risk for each link is considered to be known and
independent of the direction of each shipment. It is
also assumed that every point in the same link has
the same incident probability and level of
consequences. The sum of risk of the transportation
activity in each link then results in the linearity of
the objective function (Erkut et al., 2007). In Kara
and Verter (2004) model the risk is measured by
population exposure and the travelled distance is the
criteria for the choice of routes by the carriers, but
the methodology can be easily used with other risk
and cost measures.
One of the strategies to solve bi-level linear
programming problems consists in the application of
the Karush-Kuhn-Tucker conditions (KKT) that
transform the bi-level model into a single level one.
This model, which is equivalent to the initial
formulation, can then be solved with a commercial
solver (Bianco et al., 2009). Kara and Verter (2004)
used the KKT transformation to solve the proposed
bi-level model.
2.2 Mathematical Formulation of the
Model
Based on the problem description, the following
sets, parameters and variables are defined:
Indices
c – shipment
p – population centre
i,j,k – node
m – type of hazmat
HazardousMaterialsTransportationusingBi-levelLinearProgramming-Case-studyofLiquidFuelDistribution
377
Sets
C – all shipments across the network, c ∈C.
Each shipment is characterized by an origin
node, a destination node and a type of hazmat
transported
P – population centre, p ∈ P . Set of population
centres affected by the activity of transport of
hazmats
N – nodes, i,j ∈ N
A – links, (i,j) ∈ A, where (i,j) designates the link
connecting nodes i and j, with direction i → j
M – hazmat types, m ∈ M
Parameters
,
– number of people in p exposed to a truck
carrying hazmat m through link (i, j)
– length of link (i,j)
– number of trucks used for shipment c
R – a large positive number
Auxiliary variables
These variables appear only in the transformed
model with KKT conditions.
and
are positive real variables, while
is a real variable (positive or negative).
Decision variables
The model decision variables are binary:
= 1 if link (i,j) is available for transportation
of hazmat type m,
= 0 otherwise.
= 1 if link (i,j) is used for shipment c,
= 0
otherwise.
2.2.1 Bi-level Linear Programming Model
Using the above definitions, the bi-level model is
formulated as follows:
Objective function
,
∈,∈∈
(1)
Subject to:
∈
0,1
∀,
∈
,∈
(2)
Where
solves:
,∈
∈
(3)
Subject to:
1
1
0
,
∈
,
∈
∀∈,∈
(4)
Where:
o(c) – origin node of shipment c
d(c) – destination node of shipment c
∀
,
∈
,∈
(5)
m(c) – hazmat carried in shipment c
∈
0,1
∀
,
∈
,∈
(6)
The outer-level problem (with objective function
(1)) regards the decisions of which links should be
made available for hazmats transportation, while the
inner-level problem, represented by objective
function (3) and constraints (4) - (6), deals with the
choice of the transportation routes by the carriers.
The binary decision variables (
) of the outer-level
problem are parameters for the inner-level problem,
wherefore given the values of
the inner problem
consists of determining the minimum cost flow in
the network, by minimizing the total distance
covered by the trucks (objective function (3)).
Constraints (2) establish the binary nature of
decision variables
.
The requirements of flow balance are expressed
in constraints (4). For an intermediate node (which is
neither the origin nor the destination node of a
shipment) equation (4) ensures the hazmat enters
and leaves the node (right hand side equal to zero).
For an origin node, it ensures the hazmat leaves the
node (right hand side equal to +1) while for a
destination node it guarantees the hazmat enters the
node (right hand side equal to −1).
Constraints (5) assure that only the links that the
regulator makes available for a given hazmat
(
= 1) can be used by the carriers in a shipment
of that hazmat (
= 1).
Constraint (6) sets decision variables
to be
binary.
2.2.2 Transformed Model with KKT
Conditions (MILP Model)
Objective function
,
∈,∈∈
(1)
Subject to:
1
1
0
,∈,∈
∀∈,∈
(2)
∀
,
∈
,∈
(3)
ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems
378
0
∀∈,,
∈
(4)
1
∀∈,,
∈
(5)
1
∀∈,,
∈
(6)
0,
0∀∈,,
∈
(7)
∈
∀∈,∈
(8)
∈
0,1
∀
,
∈
,∈
(9)
∈
0,1
∀
,
∈
,∈
(10)
The model comprising expressions (1) to (10) is
a mixed-integer linear programming model (MILP),
with binary and continuous variables, and consists in
the problem of identifying routes for hazmat
transportation that minimize the risk of accident
without compromising economic viability.
3 MODEL APPLICATION AND
RESULTS
This section describes the MILP model application
to obtain optimal routes for white oils (petrol and
diesel) distribution to the company clients in the
Olivais district of Lisbon.
3.1 Data Collection and Processing
The Lisbon area road network available for this
study (mapped in the ArcGIS software) comprised
35,981 links. This was foreseen as much larger than
what could be supported in a successful
computational implementation of the MILP model,
and hence the network dimension was reduced. To
this end, network parts (links and nodes) that neither
belonged to Lisbon city nor were one of the main
accesses to it were excluded. Route alternatives in
zones without petrol stations or direct clients of the
company were also eliminated and crossings
reconfigurated (by doing link junctions). As a result,
a 7,276 link network was obtained, depicted in
figure 1. Since this number was still very large, to
achieve a successful model implementation in the
first stage of this on-going research project, only the
Olivais district and its main access routes (even if
outside of the district) were considered (figure 2).
The road network of this case study comprises 682
links and 461 nodes.
To characterise the white oils distribution, data
concerning the location of clients in Lisbon
municipality and the amount of fuel delivered daily
during one calendar year was collected at the
company, and then aggregated to obtain annual
values. Clients comprised petrol stations and direct
clients supplied by the company in Lisbon.
To implement the objective function (1), where
the risk is proportional to the amount of fuel
delivered, the number of equivalent trucks was
considered. These are the number of trucks needed
to deliver the annual amount of fuel to each client
and is obtained by the division of the total fuel
distributed per year by the capacity of one truck (30
m
3
).
In order to incorporate risk in the network,
census 2011 data was used to quantify the
population living in each census ward (BGRI in
Portuguese, meaning "geographical information
referencing basis"). Population density was obtained
by dividing the population by the area of each
census ward polygon. Population exposure was
computed with the following expression:
.
5050,
where Pop exp. is the exposed population, l
link
is the
arc or link length, Density is the population density
and 50+50 corresponds to a buffer of 50m for each
side of the arc called evacuation distance.
Several strategies were used to deal with the
complexity of integrating the company data with
GIS software data. Thereby, the following
simplifications were assumed:
Transported materials
• All white oils belong to the same model
category;
• Each truck containing the same type of hazmat
imposes the same risk.
Population density
• The population density around a road segment
is constant;
• For links that cross two census wards, the
population density considered corresponds to
the average of both values;
• Accident probability is constant in each link.
Road network
• The shipments origin was considered to be the
A1 highway entrance in the city of Lisbon (all
the company shipments enter the city through
this highway);
• Trucks were assumed to circulate at the
maximum speed allowed for heavy trucks in
each network link, lowered by a degradation
coefficient of 5% to take traffic congestion
into account;
• Speed in curves was assumed to be equal to
speed in straight links;
HazardousMaterialsTransportationusingBi-levelLinearProgramming-Case-studyofLiquidFuelDistribution
379
• Additivity of impacts was assumed.
Destination points
• The destination points (petrol stations and
direct clients) were represented by the
projection of their real location in the nearest
node of the road network;
• Only the 6 shipments to fuel stations and
direct clients of the company in the Olivais
district were considered.
Figure 1 displays the graphic representation of
the road network and the liquid fuel destinations.
Although dozens of destinations appear in figure 1,
only those in the Olivais district (figure 2) were
considered when solving the model.
Figure 1: Lisbon road network and location of the
company liquid fuel destinations. The area depicted in
Figure 2 is highlighted.
Figure 2: Detail of the Lisbon road network and location
of the Olivais district destinations (highlighted area).
Finally, table 1 depicts the number of equivalent
trucks per year of white oils delivered to Olivais
district clients.
Table 1: Number of equivalent trucks per year in Olivais
district destinations/clients.
Destinations No. of trucks
1 264
2 216
3 68
4 97
5 22
6 186
3.2 Results and Discussion
The MILP model (transformed model with KKT
conditions) was implemented in GAMS modelling
system and solved with CPLEX version 12.4, on an
Intel Core computer with an i3-2350M processor
(2.3 GHz), 6 GB of RAM and running Windows 7
Home Premium. It should be highlighted the
extremely useful xls2gms and gdx GAMS
functionalities for data input and output in
spreadsheet format (Excel files).
Table 2 presents the numeric characteristics of
the model (number of variables and constraints), the
CPU time and number of iterations of the branch-
and-bound search, the optimality gap and the
objective function value.
Table 2: Summary of numeric characteristics and results
of the MILP model.
Characteristic/result Value
No. of variables 15,763
No. of binary variables 4,788
No. of constraints 33,007
No. of iterations 3,222
CPU time (s) 1.92
Optimality gap (%) 0
Value of objective function 838,335
Table 3: Detailed characteristics of the MILP model
solution.
Destination No. of links Travel time
(min)
Population
exposed
1 58 21.1 966
2 38 18.5 706
3 43 19.5 710
4 73 27.9 1,117
5 71 27.1 1,117
6 72 27.8 1,342
Total/year
19,366 838,335
Total/truck/year
22.7 983
Table 3 presents the solution characteristics. For
each destination, the number of links of the chosen
route, the corresponding travel time and measure of
ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems
380
exposed population are shown. By multiplying these
values by the number of trucks, the total travelling
time spent and exposed population were obtained
(line before the last in Table 3). Note that the latter
value (838,335) is equal to the objective function
value when solving the model (Table 2). By dividing
these values by the total number of trucks, average
travel time and exposed population per truck for the
Olivais district were finally obtained (last line in
Table 3).
As shown in table 2, the model was solved to
optimality (0% gap) in less than 2s of CPU time, for
a network dimension significantly larger than the
ones used in similar studies. In fact, the Olivais
district network displays 461 nodes and 682 links,
while the network in the Kara and Verter (2004)
study features 48 nodes and 57 links. This
computational performance underlines the powerful
tools that are currently available to solve
optimization models.
As a final step in addressing this case study, an
analysis was made to parameter R (often termed
Big-M in the literature), varying it by powers of ten
between 10 and 10
12
. For R values below 10
4
the
model is infeasible. Feasible models were solved
three times (for each R value) and the average of
CPU time computed. This varied between 1.6s and
9.7s, with no observable increasing or decreasing
trend regarding the R value variation.
4 CONCLUSIONS
It is the authors’ understanding that there is a
consensual belief, amongst the academia and the
industry, that both further knowledge and practical
tools to identify safe and economic viable routes for
hazmats distribution, including in complex urban
systems, requires further developments. This study
applied a bi-level linear programming model in a
consolidated area of Lisbon using data from one
energy group operating in Portugal, aiming to
contribute to find a balance between population risk
and the economic viability of white oils (petrol and
diesel fuels) transportation. The dimension of the
road network (number of nodes and links), which
was mapped in a GIS software, significantly
surpasses those of similar studies described in the
literature. The optimal solution (computed in
negligible CPU time) displays the road links to be
used for hazmats transportation in the Olivais district
of Lisbon and identifies the routes for each company
shipment in this area. Model results also quantify
population exposure to risk and the routes travel
time.
As future developments, the authors intend to
address the problem of liquid fuel distribution by
this company in the city of Lisbon, which totals 26
destinations. A simplified road network may be then
considered, presenting a smaller number of arcs than
the aforementioned 7,276. Additionally, risk may be
broken down in relation to population vulnerability
in different stretches of roads. Indeed, the current
study considered the resident population, when
greater accuracy would be provided by considering
the population effectively present in any area during
the day, by identifying generator poles of each zone,
such as services and jobs/schools. It is also
recommended for future analysis the comparison
between the routes currently used by the carriers
trucks with those resulting from the optimization
model.
This is a pioneer study in Portugal, which
benefited from a successful collaboration between
the academia and the industry, and is expected to be
a first step in a hopefully gradual expansion of
applied knowledge.
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