Model and methods to address urban road network problems with disruptions

Intl. Trans. in Op. Res. 27 (2020) 2715–2739 DOI: 10.1111/itor.12641

TRANSACTIONS IN OPERATIONAL

RESEARCH

Model and methods to address urban road network problems

with disruptions

Yipeng Huang

, Andr´ea Cynthia Santos

and Christophe Duhamel

b_{Universit´e Clermont Auvergne, CNRS, LIMOS, F-63000 Clermont-Ferrand, France}

E-mail: yipeng.huang@utt.fr [Huang]; andrea.duhamel@utt.fr [Santos]; christophe.duhamel@isima.fr [Duhamel]

Received 11 June 2018; received in revised form 29 January 2019; accepted 29 January 2019

Abstract

Disruptions in urban road networks can quickly and significantly reduce the quality of the whole trans-portation network, and impact urban mobility for light vehicles, public transtrans-portation, etc. In this study, we consider both unidirectional and multidirectional road network problems with disruptions and connecting requirements. These problems aim at reconfiguring the urban network in terms of road direction in order to maintain a path among all points of the network (strong connectivity). The former is defined on simple graphs, mainly modeling part of a city such as historical centers, while the latter relies on multigraphs, mod-eling more general networks. Restoring the network (strong connectivity) after some disruptions may require the modification of the orientation of some streets, that is, arc reversals. Such actions can disturb users’ driving habits. Thus, two objectives are considered separately: minimizing the total travel distance and minimizing the number of arc reversals. We define formally both problems and propose two metaheuristics, a biased random key genetic algorithm and an iterated local search. Numerical experiments have been performed on a set of generated instances and on the urban network of Troyes (France).

Keywords: network design; metaheuristics; biased random key genetic algorithm; iterated local search; smart mobility

1. Introduction

Due to the growing concentration of population in urban areas, the management of urban road networks has become a major concern around the world. According to the World Urbanization Prospects of the United Nations in 2014, the urban population will increase to about 6.3 billion over 9.5 billion of people by 2050 (nearly 66%). Unless major and radical changes in transportation habits happen, this implicitly leads to a sharp growth of traffic flows in urban areas, which may increase the traffic congestion on the road networks in metropolitan areas from medium to large cities. Extending road networks infrastructures (adding new lanes or even brand new streets) can be a partial solution

∗_{Corresponding author.} C

2019 The Authors.

Fig. 1. Example of strong connectivity restoration through arc reversal.

to this issue, even if it is expensive and time-consuming. However, it cannot always be performed because urban sprawl and its urban network cannot be extended indefinitely. The situation becomes more complicated when a disruption, either predictable (planned road work, social event, etc.) or not (accident, bad weather condition, etc.), happens on road networks. It strongly impacts urban mobility, particularly for light vehicles, public transportation, etc. since it reduces the traffic capacity in the area, causes congestion, and possibly breaks travel paths between some locations (loss of strong connectivity). Therefore, adapted strategies should be studied to manage urban road networks properly, especially under such disruptions, as an essential requirement of smart mobility. An urban network has a physical layer (infrastructure) as well as a logical layer over the infra-structure that is made up of lane directions to carry the flow of vehicles. This work focuses on the logical layer, especially on the road network problems with disruptions (RND) and connecting requirements. According to the target zone in the logical urban network, the problem can be defined as unidirectional RND (URND) or multidirectional RND (MRND). The former often corresponds to road networks in historical city centers, where streets are almost one way and the network can hence be modeled as a simple digraph. The latter addresses more general road networks using a loopless directed multigraph model, where streets can be bidirectional and have multiple lanes in each direction.

For the sake of clarity, it is important to mention the relation between the situation in practice and modeling the logical layer. For instance, a road network can be modeled as a graph and must have a path allowing the access to any point of the networks. In graph theory, this characteristic relates to the so-called connectedness property for undirected graphs and the strong connectivity property for directed graphs. Moreover, changing the orientation of a lane corresponds to modifying the orientation of the corresponding arc, referred here as arc reversal. One may note that arc reversals are likely to disturb users’ driving habits. As a consequence, few modifications should be done on the network such that the strong connectivity is guaranteed. Last but not least, a disruption on the road network is defined as an edge that becomes unavailable in undirected graphs, respectively, an arc that becomes unavailable on digraphs.

An example of a disruption impact is given in Fig. 1. An initial road network with 6 nodes and 11 arcs is presented in Fig. 1(a). As seen in Fig. 1(b), a disruption happens on arc (5,2) and it is now unavailable for carrying traffic (dashed line). The resulting digraph is not strongly connected anymore since no path is possible between each pair of nodes. For instance, node 2 cannot be reached from any other node. Reversing the orientation of arc (2,5) restores the strong connectivity, as shown

in Fig. 1(c). Note that other arc reversals could have been set as well, for instance simultaneously on arcs (1,4) and (2,1). This would have resulted in a higher perturbation in terms of modifications on the road network.

Let G= (N, A) be a simple connected bridgeless digraph, where N is the set of nodes and A is

the set of arcs. G can model the URND with nodes being road intersections and arcs standing for

road segments. Let G= (N, A) be a connected loopless bridgeless multigraph, where Nis the set

of nodes and Ais the set of multiple arcs such that there can be more than one arc between a pair

of nodes. The MRND is modeled by graph G, where the nodes model crossroads or roundabouts

and the arcs correspond to either a direction lane on a multilane street or a direction on a two-way street. When disruptions happen, several road segments can be blocked or left unavailable for the traffic. Thus, they are first removed from the given network. As a consequence, the network may not be strongly connected anymore. Moreover, the impact on travel paths may increase the travel distance. To address these two issues, one can change the direction of a road. This may be necessary to restore the strong connectivity and to optimize the total travel distances of every travel paths on the network. However, reversing the orientation of an arc can disturb the user’s habits, as mentioned above, leading to poor local rerouting decisions. This can lead to other issues, for instance when onboard navigation systems data are not updated. Thus, arc reversals should be avoided as much as possible. In this work, arc’s reversals are allowed and two separate optimization criteria are considered in our models for solving both URND and MRND, minimizing the total travel distance

(c1criterion) or minimizing the number of arc reversals (c2criterion). Each criterion is optimized

separately. They correspond to very different evaluations from the traffic manager point of view:

minimizing c1 aims at globally reducing the travel distances, while minimizing c2 provides as few

network modifications as possible.

The following contributions are made in this study: a mathematical model with respect to the

two criteria, c1 and c2, are proposed, as well as two metaheuristics, a biased random key genetic

algorithm (BRKGA) and an iterated local search (ILS). A random restart and a dynamic shaking have also been integrated into the ILS in order to improve diversification and the search on the solution space. Several analyses are done to measure the quality and performance of the proposed method. In addition, an application to an urban road network with real data is performed.

The remainder of this paper is structured as follows. Related works are reviewed in Section 2. In the sequel, a mathematical formulation is presented in Section 3, before describing the proposed metaheuristics applied to URND and MRND in Section 4. Numerical results are shown in Section 5, followed by concluding remarks in Section 6.

2. Related works

In the literature, problems related to URND and MRND are found in the fields of graph theory, network design, and urban applications, which are detailed below. In graph theory, several topics are relevant for RND, such as the strong connectivity, orientation problems, and arc reversal. In the seminal work of Robbins (1939), the theorem stating the condition for a simple graph to admit a strong orientation is first presented. Motivated by the traffic control problem for modern cities, the graph orientability is defined as the fact that there is an orientation that makes it strongly connected. Besides, a bridge is an edge whose removal makes the resulting graph no longer connected. Then,

only bridgeless graphs admit strong orientations. This theorem is commonly considered as the first keystone for the strong orientation problems that have been studied in the last decades.

Furthermore, since an orientable graph can be oriented in several different ways, there should be metrics by which these orientations can be compared. To this extent, Chv´atal and Thomassen (1978) study a distance metric for strong orientations, which can be used when converting a two-way network into a one-way network that may rely on longer detours. A good lower bound is proposed for the radius of such orientations and a classification of orientation problems with different distance constraints is presented. The authors also prove that finding a strong orientation whose radius (or

diameter) is at most k isN P-hard. Also following Robbins (1939), Boesch and Tindell (1980) extend

the strong orientation problem to the case of mixed multigraphs. Thus, a network containing both one-way and two-way streets can be addressed. The authors prove that a strong orientation of a mixed multigraph exists iff there are no bridges on its corresponding undirected multigraph. In a real context, such a bridge corresponds to a road segment whose orientation cannot be changed since it would leave some locations unreachable and isolated on the network.

Chung et al. (1985) is another extension of Robbins (1939) combining the study of Chv´atal and Thomassen (1978) on distance with the study of Boesch and Tindell (1980) on mixed multigraphs. The major contributions are adapting the orientation radius lower bound of Boesch and Tindell (1980) on mixed multigraphs and proposing a linear-time algorithm to check the orientability of a given graph. Authors prove two lemmas following the conditions for the orientability proposed by Boesch and Tindell (1980). The first states that orienting an undirected edge according to the direction of the cycle it belongs to does not violate the orientability of the given graph. The second shows that there should be at least one outgoing directed arc and one incoming directed arc between any two connected components. Based on these lemmas, the orientability check algorithm is designed by extending a depth-first search (DFS) (Tarjan, 1972) to orient the given graph and reachability checks on all vertices. This algorithm is an adaptation of Tarjan’s strongly connected components (SCC) algorithm developed in Tarjan (1972). Besides, using a DFS algorithm to orient simple graphs is also proposed by Roberts (1978).

A recent work by Conte et al. (2016) proposes an algorithm for enumerating strong orientations

such that finding one strong orientation is done inO(m) amortized time for a given mixed

multi-graph with, respectively, m edges. The authors define a specific multi-graph transformation technique by merging a directed cycle into an auxiliary node such that undirected self-loops are formed. A strong orientation for the rest of the graph can be extended by assigning a direction to each self-loop, thus producing a strong orientation of the whole graph. Next, following the theorems of Robbins (1939) and Boesch and Tindell (1980), the one-way cut and forcing cut are defined depending on the status of an possible bridge. Using the linear-time algorithm of Italiano et al. (2012) for finding strong bridges, the algorithm proposed in this work orients all the bridges in such a way that the rest of the graph is orientable. Then, the algorithm recurs with the removal of an arbitrary edge and outputs every strong orientation found during the procedure.

Reversing the orientation of an arc has been investigated in graph theory, mainly to assess the impact on some graph invariants. For instance, Ad´am (1964) introduces a conjecture on the existence of an arc whose reversal decreases the number of elementary cycles in a digraph by at least one cycle. The conjecture has been validated on some class of graphs, for example, digraphs containing a 2-arc cycle. However, counterexamples are also found for some other classes of graphs, by Jir´asek (1987), Thomassen (1987), and Jir´asek (2005). According to these works, the conjecture remains

open on simple digraphs. More recently, Bogdanowicz (2011) shows that the reversal of any arc in a balanced directed loopless multigraph decreases the number of non-necessarily elementary cycles.

Concerning network design problems, the strong network orientation problem (SNOP) in Santos et al. (2013) is closely related to URND. The objective of the SNOP is to set an orientation for a given graph, such that the resulting digraph is strongly connected and the distance between all pairs of nodes is minimized (or maximized). A number of constructive heuristics and a set of metaheuristics (ILS; evolutionary local search, ELS; and relaxed ELS) were proposed in Santos et al. (2013) for SNOP. The main differences between SNOP and URND are the following: first, URND graph already has an initial orientation that corresponds to the existing road directions; second, the URND looks for reconfiguring the network such that it remains strongly connected, arc reversals

are allowed and criteria c1 and c2 are optimized. In Martins et al. (2017), an improved compact

mathematical model based on aggregated flow and a column-generation algorithm are proposed for SNOP. The column-generation subproblem consists in defining an arborescence rooted at each node. Interestingly, a large number of works are found in the context of automated guided vehicles (AGV) for production systems, addressing the so-called flow path design problem, introduced by Gaskins and Tanchoco (1987). The classical flow path design consists in designing a unique path between pickup and delivery points. The authors propose a mixed integer linear programming (MILP) formulation to define orientation in a given graph between all pairs of pickup and delivery points such that the traveling distance is minimized. In the following, a number of works dedicated to AGV systems appear motivated by several applications in manufacturing, distribution, trans-shipment, and transportation systems. Among others, aspects such as designing routes for AGV, balancing the charge, defining positioning policies, and scheduling were studied. Two good entry points surveying these aspects are found in Qiu et al. (2002) and Vis (2006). Moreover, a few studies, such as Gaskins and Tanchoco (1987) and Venkataramanan and Wilson (1991), use the concept of connectivity between pickup and delivery points.

The impact of network disruptions on the shortest path problem in terms of travel time is addressed by Sever et al. (2013). Each arc is associated with a disruption probability. Thus, the disruption state of each vulnerable arc is modeled by a random variable. A generic framework is proposed using a dynamic programming approach to evaluate both online and offline routing policies. Online information on disruption states may or may not be available in the Markov decision process.

Miandoabchi and Farahani (2011) study the lane allocation problem, considering the reserve ca-pacity concept (i.e., the ability to route additional flow). They propose a bilevel model following the widely used framework of Yang and Bell (1998) in order to solve the lane addition and orientation problem with reserve capacity. In particular, lane allocation was considered in two variations ac-cording to the layout symmetry constraint on both sides of a two-way road. Two metaheuristics have been proposed: a hybrid genetic algorithm and an evolutionary simulated annealing. Experiments using theoretical networks indicate that the first metaheuristic has a better performance.

Urban network applications are found for organizing traffic by optimizing signal settings in Cantarella et al. (2006), and for improving the network infrastructure considering the cost invest-ment in Gallo et al. (2012). To the best of our knowledge, either in graph theory or in network design, URND and MRND problems have not received much attention, especially when considering the possibility of arc reversals. Preliminary works have investigated URND and MRND complexity, as well as a first compact MILP formulation (see Huang et al., 2016a, 2016b). In addition, it is

shown in Huang et al. (2016b) that any instance of the MRND can be reduced to an instance of a

2-directed multigraph when considering c₁and c₂criteria. Another work, focusing on URND, has

proposed some results for MRND on a small set of instances Huang et al. (2018). Here, we propose an improved compact formulation for URND and MRND, some methodological improvements on

BRKGA and ILS, for both URND and MRND considering c₁or c₂at a time. The numerical

exper-iments have also been improved by including realistic instances, newer results, and metaheuristics convergence analysis using time-to-target plots (tttplots).

3. Problem definition and formulation

Given the graph G= (N, A), as previously defined in Section 1, used to model an urban road

network as follows: arcs represent streets with their orientation and nodes are the crossroads. Thus,

a one-way street is associated with a single arc (i, j) and a two-way street is associated with two

arcs in opposite direction, that is, (i, j) and ( j, i). The cost θi j of an arc (i, j) ∈ A corresponds

to its distance, for instance. Let P= {{o, d} | o, d ∈ N, o = d} be the set of all (o)rigin-(d)estination

pairs in G and let B be the list of the p blocked arcs (disruptions). In a preprocessing step, all arcs

(i, j) ∈ B are removed from G, leading to the modified graph H = (N, A_{), with A}_{= A \ B.}

We present In Equations (1)–(10) a mathematical formulation for URND. Strong connectivity in

H is ensured by setting a unit flow for each pair{o, d} ∈ P. Any arc in Acan be reversed in order to

restore the strong connectivity. Furthermore, the proposed model uses boolean variables x_{i j} ∈ {0, 1}

stating whether arc(i, j) ∈ A is reversed (x_{i j} = 1) or not (x_{i j} = 0). Variables fod

i j ∈ IR define the

flow from o to d on the arc(i, j), taking into account its final orientation. Assuming (i, j) is used

to send flow from o to d, f_{i j}od > 0 if the arc is not reversed and f_{i j}od < 0 otherwise. In this way, flow

conservation constraints still hold on variables f . Variables yod

i j ≥ 0 keep the amount of flow from o

to d going through arc(i, j), thus are independent from the orientation of the arc (i, j). The model

is as follows: min c1=  {o,d}∈P  (i, j)∈A θi jyodi j (1) min c2=  (i, j)∈A x_{i j} s.t. (2)  l:(l,o)∈A f_lood−  l:(o,l )∈A f_olod = −1 ∀{o, d} ∈ P (3)  l:(l,i)∈A f_liod−  l:(i,l )∈A f_ilod = 0 ∀{o, d} ∈ P, ∀i = o, d (4) −yod i j ≤ fi jod ≤ yodi j ∀{o, d} ∈ P, ∀(i, j) ∈ A (5) yod_{i j} − 2x_{i j} ≤ f_{i j}od ≤ −2x_{i j}− yod_{i j} + 2 ∀{o, d} ∈ P, ∀(i, j) ∈ A (6) 2019 The Authors.

x_{i j} ≤  {o,d}∈P yod_{i j} ∀(i, j) ∈ A (7) x_{i j} ∈ {0, 1} ∀(i, j) ∈ A (8) f_{i j}od ∈ IR ∀{o, d} ∈ P, ∀(i, j) ∈ A (9) yod_{i j} ≥ 0 ∀{o, d} ∈ P, ∀(i, j) ∈ A. (10)

The total traveling distance is minimized in (1) by computing the cost of routing one unit of flow for each pair of origin and destination. Minimizing the arc reversals is addressed in (2). Constraints

(3) and (4) are the flow conservation constraints on f_{i j}od variables. Inequalities (5) and (6) link

variables fod

i j with variables yodi j and xi j. Redundant inequalities (7) forbid an arc with no flow to be

reversed, in order to avoid large objective values of c2 when optimizing the c1 criterion. Variables

are defined from (8) to (10). This model contains O(|P| × |A|) constraints and variables.

URND and MRND complexity depends on the criterion. Finding a graph orientation minimizing

the total distance c₁isN P-hard (see Chv´atal and Thomassen, 1978). On the other hand, finding

a graph orientation minimizing the number of arc reversals c₂ isP (see Bang-Jensen and Gutin,

2008). Indeed, the problem is shown to be equivalent to the minimization of a submodular function,

which is polynomial. Yet, and to the best of our knowledge, even if minimizing c₂for URND and

MRND isP, no corresponding LP model and no dedicated algorithm has been found.

Note that this mathematical model can be conveniently adapted to MRND by adding lane indices in variables, to accurately and independently represent the flow and direction of each arc.

For example, x_{i j}becomes x_{i jk}where(i, j) ∈ N × N and k ∈ IN is the lane number (index). As shown

in Huang et al. (2016b), any multigraph can be reduced to a 2-directed multigraph in the context of

c₁or c2optimization. Thus, using the reduction as a preprocessing step, the size of the multigraph

considered in the mathematical formulation is reduced.

4. Methods for URND and MRND

Heuristics are a classical and efficient way to handle real-scale instances for combinatorial prob-lems, for instance in network design. In a limited CPU time, good quality solutions or even near-optimal/optimal solutions can be found. In this section, the methods for solving URND and MRND are proposed. A DFS-based orientation algorithm is first presented in Subsection 4.1, since it is a common feature in our two metaheuristics, followed by a BRKGA and an ILS.

4.1. DFS-based strong orientation algorithm

Given an existing road network in which there exists at least one path between each two nodes, that is, the digraph modeling this network is strongly connected. Let us consider that an arc can no longer be used to carry traffic flows. Thus, it is removed from the graph and the resulting graph may not be strongly connected anymore. In this context, a first strategy to handle this issue is to recompute a strong orientation for the initial network such that all locations are still reachable even

if some links are not available, keeping into account that modifying the number of arcs orientation and travel distances are minimized.

The strong orientation algorithm proposed in Roberts (1978) extends the classical DFS algorithms for graphs (Cormen et al., 2009) and it is determinist. Given an undirected graph and a root vertex, it generates one strong orientation. However, DFS has potential for more diversity. In fact, the strong connectivity of the resulting orientation is guaranteed irrespective of the order adjacent vertices are visited from the current vertex. At the same time, different orders result in different graph traversals. Thus, varying this order increases the possibility to build different strong orientations.

In this work, the variation on the order is set by assigning a weight to every vertex. When scanning neighbors from the current vertex, its available adjacent vertices (through a nonoriented edge) are sorted in decreasing order, according to their weight. In this way, a different vector of weights might result in a different strong orientation. This also allows the construction of a randomized heuristic,

if every weight is randomly generated with respect to a given distribution for instanceU[0,1]. Thus, the

final method, detailed in Algorithm 4.1, guarantees that its output is a strongly connected digraph since it extends the initial algorithm proposed for simple graphs in Roberts (1978).

The input arguments of Algorithm 4.1 are the following: an undirected graph G (either simple for URND or multiple for MRND), a vector of vertex weights W , and a root vertex u. Note that any vertex can be used as root. Here, the vertex with the highest weight is considered as the initial root vertex. As described in the previous paragraphs, the selecting order is defined in lines 1 and 2 by sorting the adjacent vertices in decreasing order of their weight. Then, following this order, lines

3–8 examine every neighbor v of u and propagate the visit to v if the edge(u, v) has not been already

oriented (see line 4). In line 5, an orientation for the edge(u, v) is set and in line 6 a recursive call

to the strong orientation algorithm is done with v as the new root vertex.

Algorithm 4.1. StrongOrient(G,W, u) Input: G= (V, E ): undirected graph

W : vertices weight u∈ V : root vertex

Output: G with a strong orientation 1. N(u) ← list of adjacent vertices of u

2. sort N(u) by decreasing order of weights in W 3. for each v∈ N(u) do

4. if(u, v) ∈ E not oriented then 5. set orientation from u to v 6. StrongOrient(G,W, v) 7. endif

8. endforeach

An example is given in Fig. 2, an initial undirected network with 9 vertices and 15 edges is given

in Fig. 2(a). Suppose the following weights {0.6, 0.9, 0.7, 0.3,0.5, 0.2, 0.8, 0.6, 0.1} are assigned,

respectively, to vertices {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} as depicted in Fig. 2(b). Figure 2(c) illustrates some steps to obtain the corresponding orientation. The algorithm starts at vertex 1, since its weight is the largest. Its neighbors list is{0, 2, 4}. Vertex 2 is first visited because of its higher weight (0.7) and the edge (1,2) is oriented from 1 to 2. The algorithm continues from vertex 2 to vertex 5 and from vertex 5 to vertex 4. The heuristic ends up with the strong orientation illustrated in Fig. 2(d).

Fig. 2. Example of DFS-based orientation algorithm.

4.2. Metaheuristics

The two proposed metaheuristics, a BRKGA and an ILS for URND and MRND, are detailed below. These methods were chosen based on their successful results for a significant number of applications, such as Resende (2012), Stefanello et al. (2017), Reis et al. (2011), and Mor´an-Mirabal et al. (2014) for BRKGA, and Lourenc¸o et al. (2002), Afsar et al. (2014), and Alvarez et al. (2018) for ILS.

4.2.1. BRKGA

BRKGA has been proposed by Gonc¸alves and Resende (2011) and Resende (2012). It extends the random key genetic algorithm (RKGA) from Bean (1994), which is a variant of the genetic

algorithm (GA) (see Goldberg, 1989). It relies on a population of individuals represented by vectors of random keys in [0,1]. Depending on the specification of the given problem, a vector of random keys should be decoded into a feasible solution by calling a decoder. In this way, the problem-specific part is separated from the main algorithmic scheme by means of the decoder. In the general structure of BRKGA, a population individuals evolves at each iteration using classic genetic operators (mutation, crossover). The specific point in BRKGA is the use of random keys to encode a chromosome. Both BRKGA and RKGA can hence be reused in different problems, requiring only to develop problem-specific decoders. In addition, elitism is used in BRKGA and the selection is biased in such a way that an offspring is always produced from an elite individual and a nonelite individual in the crossover operator. Moreover, mutants are introduced into the population in each generation by randomly creating a given number of new individuals. This mechanism maintains diversity in the population and allows diversification.

For the problems in this study, the randomized constructive algorithm (Algorithm 4.1) can be used as a decoder, since the weight vector can be interpreted as a vector of random keys. Thus, a BRKGA version for URND and MRND is developed, using this algorithm as a basis for the decoding heuristic. In our proposal, a random key is associated with each vertex of the

undirected graph H, built from a given directed graph H by removing orientation of every arc.

The parameterized uniform crossover of Spears and Jong (1991) is used, where one individual is always selected from the elite set. Moreover, the probability of a solution inherits the allele from the elite individual is equal to 0.7 as suggested by Gonc¸alves and Resende (2011). Both the number of mutants and the elite set size follow the work of Gonc¸alves and Resende (2011). For the former, the recommendation is to introduce a small number of mutants into the population at each generation. Thus, the number of mutants is set to 10% of the overall population. For the latter, Gonc¸alves and Resende (2011) show that elite sets rating from 15% to 25% of the overall population tend to produce a better performance of BRKGA, and it is set to 20% in this study.

Using the Algorithm 4.1 and setting the vertex with the largest weight as root vertex u, a vector of random keys can thus be decoded into a feasible solution of the problem. The decoding procedure is detailed in Algorithm 4.2. Depending on the optimization objective, the solution’s fitness is

evaluated in the function fitness(g) (value for c1 or c2) in line 4. The c1 criterion is evaluated by

computing the shortest distance between all pairs of nodes. Given a graph of n nodes and m arcs, this

can be done inO(n3_{) using Floyd–Warshall algorithm or in O(n × log n + n × m) using Johnson’s}

algorithm (see Cormen et al., 2009). The c2criterion is evaluated by counting the number of arcs

whose direction differs from the original orientation. This is done inO(m).

Algorithm 4.2. Decode(G, c)

Input: G= (N, A): graph (simple or multiple) with initial orientation c: vector of random keys

Output: fitness of the orientation decoded from c Variables: g: temporary graph

1: g← removeOrientation(G)

2: u← index of the highest random key in c 3: StrongOrient(g, c, u)

4: f ← fitness(g) 5: return f

Since the chromosomes depend on random keys, the decoded solutions can show a large dif-ference in the configuration. It has been observed that the graphs obtained by the decoding

pro-cedure often contain a lot of arc reversals, which is not suitable when minimizing the c₂criterion

in neither URND nor MRND. Thus, a repairing procedure is added to the decoding

heuris-tic, especially for c₂. It aims at reducing the arc reversals as much as possible. According to the

guideline of BRKGA, a solution should always be feasible. Therefore, this procedure also ensures that the resulting graph is strongly connected using a check based on Tarjan’s SCC algorithm

in Tarjan (1972). Modifications on Algorithm 4.2 lead to Algorithm 4.3 when optimizing the c2

criterion. Since the complexity of Tarjan’s algorithm is O(n + m), the procedure in lines 4–10

has a complexity of O(n × m + m2_{) in the worst case when all arcs have to be reversed back}

to the original orientation. Note that the order of reversed arcs considered in line 4 has an im-pact on the reduction. Thus, this procedure is heuristic as it does not guarantee the optimality on c2.

Algorithm 4.3. Decode W ith Repair c₂(G, c)

Input: G= (N, A): graph (simple or multiple) with initial orientation c: vector of random keys

Output: fitness of the orientation decoded from c Variables: g: temporary graph

1: g← removeOrientation(G)

2: u← index of the highest random key in c 3: StrongOrient(g, c, u)

4: for each reversed arc e in g do 5: reverse e back to its initial direction 6: nb scc← computeSCC(g) 7: if nb scc> 1 then 8: reverse e 9: end if 10: end for 11: return fitness(g) 4.2.2. ILS

ILS was proposed in Lourenc¸o et al. (2002) with major applicative contributions in Lourenc¸o et al. (2010). Given an initial feasible solution, ILS first improves it using a local search. Then the method performs each iteration a perturbation on the current solution, followed by a call to the local search. The incumbent solution is updated whenever the resulting solution is better. These two steps stop under a predefined stopping condition and the incumbent solution is returned. This method can thus be separated into three independent components: a procedure to generate a feasible initial solution, a local search procedure and a perturbation operator.

The DFS-based orientation algorithm (Algorithm 4.1) is used to generate the initial solution for ILS. First, blocked arcs are removed from the initial digraph. To comply to the algorithms’ input requirements, a weight is randomly generated for each vertex of the given graph and the graph is set unoriented. Thus, the resulting digraph is strongly connected and this initial solution is feasible by construction.

Fig. 3. N1move in cycle{(2, 3), (3, 4), (4, 2)}.

Fig. 4. N₂move at node 4.

Fig. 5. N3move at arc(4, 1).

The local search used is a variable neighborhood descent (VND), see Hertz and Mittaz (2001),

which works as follows. Given a predefined set of l neighborhoods{Ni, i = 1, 2, . . . , l}, the VND

starts with an initial solution s₀ and N₁. When an improving solution s is found in the active

neighborhoodN_k(s) of the solution s in the current iteration, the method moves to sand resumes

to the first neighborhood, that is, k← 1. Otherwise, the search proceeds with the next neighborhood

(k← k + 1) of s, until all the l neighborhoods have been scanned. Here, a first improvement search

strategy is applied in the VND.

Adapted from Santos et al. (2013), three neighborhoods are developed for URND and MRND.

A move in N1 reverses the orientation of all arcs in a cycle found in the given network. An

example is depicted in Fig. 3, where the cycle {(2, 3), (3, 4), (4, 2)} in Fig. 3(a) is reversed into

{(3, 2), (2, 4), (4, 3)} in Fig. 3(b). A move in N2reverses all arcs incident to a node. In Fig. 4(a),

all arcs incident to node 4 have their orientation modified as shown in Fig. 4(b). The last moveN₃

reverses one arc, as illustrated in Fig. 5(a), where arc (4,1) is reversed by (1,4) in Fig. 5(b). Feasibility

is guaranteed for moves in N1since the solution remains strongly connected, while it is not inN2

and N₃. Hence, Tarjan’s SCC algorithm is used for the feasibility check to accept only strongly

connected neighbors. Infeasible solutions are discarded.

A random restart mechanism is integrated in our ILS in order to provide diversity. This approach allows to randomly restart ILS from the initial solution generation step.

In the perturbation component of our ILS, two shaking procedures are developed. The first procedure tries to reverse a cycle having several arcs that also belong to other cycles, such that the

perturbation cannot be easily invalidated by aN1move in local search. It is applied in the first 50%

iterations and allows a slight perturbation on the current solution without moving too far in the solution area. The second shaking changes the current orientation of the graph by reversing all arcs direction, which helps to escape local optima. This function is used in the last 50% iterations. The strong connectivity and some other structural properties are still ensured by these perturbations.

5. Numerical results

The main goals of our numerical experiments are to assess the computational performance and limits of the proposed model, as well as for the metaheuristics. In this work, four experiments have been done: first, a parameter tuning for the proposed metaheuristics, BRKGA and ILS, is performed; second, the model is tested using the CPLEX commercial solver and the metaheuristics are tested on a set of generated small- and medium-sized instances; third, theoretical instances with a larger number of nodes and arcs are generated instances, then tested only with the metaheuristics; fourth, we built some instances from the Troyes road network (a medium-sized city in France with about 61,000 inhabitants in 2015) and we performed tests on them. All the experiments are done on a machine with an Intel Core i7-4710MQ CPU at 2.50 GHz with four cores, 16 GB RAM under Windows 7. A time limit of two hours is set for CPLEX and defaults parameters are used. All the procedures are coded in C++ and compiled with Visual C++ version 12.6.

The generated instances are grid graphs with arcs cost set to 1. They include simple graph instances for URND and multigraph instances for MRND. For the latter, multiple arcs are randomly

generated. For URND instance, their name is set toαxα_bβ, where α refers to the width and height

of the grid andβ corresponds to the number of blocked arcs. For MRND, the name is extended to

αxα_bβ_γ , where γ is the percentage of multiple arcs. In every instance, disruptions are selected

on a random basis and the number of disruptions varies in{1, 2, 4, 6}. In addition, a check using

Tarjan’s SCC algorithm is done to ensure the resulting instance is feasible.

In the tables, columns “Instance,” “c₁,” “c₂,” “t (seconds),” “LR,” and “Gap (%)” stand for

instance names, c₁value, c₂value, processing time in seconds, linear relaxation value, and relative

gap toward the optimum, respectively. Furthermore, columns “c∗₁” and “c∗₂” correspond to optimal

values for c₁ and c₂ criteria, respectively. Results highlighted in bold indicate the corresponding

method found the best result compared to the others.

5.1. Parameter tuning for the metaheuristics

A tuning phase is essential before getting into the experiments in order to figure out good settings for the proposed metaheuristics. We selected several generated instances and ran the algorithms with

Table 1

Results of BRKGA parameter tuning

R_B J_B 1.25 1.45 1.75 6 12 24 Perf. indicators c₁ t (seconds) c₁ t (seconds) c₁ t (seconds) c₁ t (seconds) c₁ t (seconds) c₁ t (seconds) Best 246 0.13 246 0.19 246 0.17 246 0.08 246 0.19 246 0.30 Worst 3032 2.18 3032 2.72 3058 3.70 3032 1.48 3032 2.72 3032 4.41 Average 1251.33 0.94 1254.67 1.24 1259.00 1.45 1254.67 0.62 1251.33 1.24 1254.67 1.96 Table 2

Results of ILS parameters tuning

JI

6 12 24

Perf. indicators c₁ t (seconds) c₁ t (seconds) c₁ t (seconds)

Best 246 0.14 246 0.23 246 0.44

Worst 3032 4.10 3032 6.97 3032 14.76

Average 1251.33 1.46 1251.33 2.52 1251.33 4.59

different parameter settings. In every calibration experiment, the worst, the best, and the average

solution values when optimizing c1and the average running time in seconds are used as performance

indicators, referred in Tables 1 and 2 as “Perf. indicators.”

For BRKGA, the tuning is done on the population size and on the stopping condition: the former

is set as a multiplier of the number of nodes(RB); the latter is defined by the number of generations

without improvement(J_B). For ILS, the tuning is performed on the stopping criterion represented

by the number of iterations without improvement(JI).

Tables 1 and 2 present the calibration results obtained by tuning the parameters for, respectively,

BRKGA and ILS for c1criterion. Results in Table 1 indicate that the performance with RB = 1.25

dominates the other settings on the parameter in terms of both objective values and computational

time. Besides, the settings for J_B do not seem to have an impact on the solution quality for the

tested instances, while a relatively low value (JB= 6) allows to save the processing time. For ILS,

the performance in terms of the processing time is generally better when setting J_I = 6. Similarly,

J_I does not seem to have an impact on the solution quality. This set of parameters is also kept for

minimizing c₂criterion, since the corresponding optimization problem is easier to solve.

5.2. Results for URND and MRND (I)

Tables 3 and 4 present, respectively, the results for the URND minimizing c1and c2on simple graph

instances with less than 50 nodes. Then, results for the MRND minimizing c₁ and c₂ are listed,

respectively, in Tables 5 and 6 on multigraphs with at most 36 nodes. In Table 3, the value for c2

could not be obtained when CPLEX fails to find the optimal solution for c₁, which is indicated by

“–.”

Table 3

c1Minimization for URND (I)

CPLEX BRKGA ILS

Instance c∗1 c2 t (seconds) LR Gap (%) c1 c2 t (seconds) c1 c2 t (seconds)

4× 4-b1 912 22 22.40 786 13.82 912 3 0.82 912 1 0.47 4× 4-b2 968 5 11.34 836 13.64 968 5 1.17 968 17 0.56 4× 4-b4 1392 8 3.99 1216 12.64 1392 8 0.60 1392 8 0.42 4× 4-b6 1304 7 1.72 1088 16.56 1388 6 0.58 1304 7 0.28 5× 5-b1 2638 14 414.62 2292 13.12 2668 23 2.42 2638 25 4.16 5× 5-b2 2748 16 373.96 2404 12.52 2748 16 2.64 2758 11 2.28 5× 5-b4 3032 23 182.74 2636 13.06 3032 23 2.68 3032 13 2.82 5× 5-b6 3116 4 253.86 2672 14.25 3140 28 3.35 3116 4 1.64 6× 6-b1 ≤6250 – 7200.00 5506 11.90 6244 22 8.60 6394 36 8.57 6× 6-b2 ≤6554 – 7200.00 5632 14.07 6356 32 7.70 6470 24 7.71 6× 6-b4 ≤6820 – 7200.00 5956 12.67 6780 35 10.94 6854 9 8.34 6× 6-b6 7006 35 3894.85 6394 8.74 7006 19 7.29 7006 35 6.14 7× 7-b1 Out of memory 13,282 30 37.32 13,344 43 49.90 7× 7-b2 13,592 35 33.10 13,688 29 23.68 7× 7-b4 14,206 49 31.64 14,214 30 24.80 7× 7-b6 14,824 45 24.54 14,494 50 34.29 Table 4

c2Minimization for URND (I)

CPLEX BRKGA ILS

Instance c₁ c∗₂ t (seconds) LR Gap (%) c₁ c₂ t (seconds) c₁ c₂ t (seconds)

4× 4-b1 960 0 0.44 0 0 960 0 0.48 960 0 0.77 4× 4-b2 1060 0 0.42 0 0 1060 2 0.52 1060 0 0.33 4× 4-b4 1442 2 0.51 2 0 1538 3 0.33 1442 2 0.38 4× 4-b6 1388 6 0.39 6 0 1388 6 0.30 1388 6 0.21 5× 5-b1 3388 0 3.84 0 0 3388 0 1.72 3388 0 2.50 5× 5-b2 3960 0 3.51 0 0 3980 2 1.20 3960 0 2.39 5× 5-b4 3844 1 3.70 1 0 3844 1 1.01 3844 1 2.03 5× 5-b6 3322 2 3.28 2 0 3732 2 0.86 3322 2 1.85 6× 6-b1 7770 2 20.39 2 0 8180 2 2.75 7770 2 10.77 6× 6-b2 7636 1 21.25 1 0 7636 1 3.09 7636 1 9.69 6× 6-b4 7606 1 20.90 1 0 7606 1 2.31 7606 1 8.78 6× 6-b6 7814 3 18.78 3 0 8522 3 3.40 7814 3 11.54 7× 7-b1 Out of memory 15,312 2 8.10 15,684 2 34.83 7× 7-b2 16,700 1 12.28 16,650 1 18.92 7× 7-b4 17,240 2 6.93 17,638 2 15.46 7× 7-b6 18,034 4 10.64 18, 044 2 15.94

5.2.1. Results for URND (I)

Results in Tables 3 and 4 indicate that ILS performs generally better than BRKGA. On 4 × 4

and 5 × 5 instances, ILS found seven of eight global optima when minimizing c₁ (Table 3) and

Table 5

c1minimization for MRND (I)

CPLEX BRKGA ILS

Instance c1* c2 t (seconds) LR Gap (%) c1 c2 t (seconds) c1 c2 t (seconds) 5× 5-b1-25 2406 27 56.69 2270 5.65 2406 10 2.11 2406 10 2.23 5× 5-b1-50 2296 12 12.67 2226 3.05 2296 22 2.36 2296 15 3.35 5× 5-b1-75 2158 19 7.75 2142 0.74 2158 19 2.64 2158 16 4.05 5× 5-b1-100 2024 20 3.65 2024 0.00 2024 20 2.92 2024 20 1.73 5× 5-b2-25 2450 12 41.09 2318 5.39 2450 24 5.15 2498 13 2.19 5× 5-b2-50 2260 26 23.90 2194 2.92 2264 19 2.90 2264 21 2.34 5× 5-b2-75 2126 16 8.10 2114 0.56 2126 16 2.62 2126 16 4.48 5× 5-b2-100 2036 20 3.81 2036 0.00 2036 20 2.92 2036 20 2.41 6× 6-b1-25 Out of memory 5466 – 5954 34 6.29 5952 37 6.29 6× 6-b1-50 5396 – 5536 46 7.46 5526 23 6.01 6× 6-b1-75 5268 41 41.48 5268 0.00 5268 40 10.34 5280 31 13.43 6× 6-b1-100 5060 35 20.22 5060 0.00 5060 35 7.43 5060 35 4.78 6× 6-b2-25 Out of memory 5592 – 5986 22 8.81 5962 23 5.92 6× 6-b2-50 5478 – 5436 26 11.84 5434 27 17.17 6× 6-b2-75 5322 34 45.12 5322 0.00 5322 33 6.76 5338 40 9.65 6× 6-b2-100 5092 38 21.61 5092 0.00 5092 38 7.36 5092 38 4.56 Table 6

c2Minimization for MRND (I)

CPLEX BRKGA ILS

Instance c₁ c∗₂ t (seconds) LR Gap (%) c₁ c₂ t (seconds) c₁ c₂ t (seconds)

5× 5-b1-25 2936 0 1.45 0 0 2936 0 1.61 2936 0 3.34 5× 5-b1-50 2814 0 1.36 0 0 2814 0 1.25 2814 0 5.01 5× 5-b1-75 2458 0 1.31 0 0 2458 0 1.78 2458 0 6.82 5× 5-b1-100 2408 0 1.19 0 0 2408 0 1.70 2408 0 9.00 5× 5-b2-25 2868 0 1.33 0 0 2868 0 1.37 2868 0 3.35 5× 5-b2-50 2676 0 1.37 0 0 2676 0 1.84 2676 0 5.43 5× 5-b2-75 2468 0 1.33 0 0 2468 0 2.40 2468 0 6.66 5× 5-b2-100 2430 0 1.19 0 0 2430 0 1.69 2430 0 8.56 6× 6-b1-25 7636 1 26.40 1 0 7636 1 2.29 7636 1 12.62 6× 6-b1-50 6550 2 22.04 2 0 6550 2 6.47 6550 2 16.90 6× 6-b1-75 6344 1 20.48 1 0 6344 1 6.24 6344 1 24.57 6× 6-b1-100 5942 1 16.01 1 0 5942 1 3.78 5982 1 30.06 6× 6-b2-25 7308 1 20.28 1 0 7308 1 2.81 7308 1 9.66 6× 6-b2-50 6986 0 5.76 0 0 6986 0 5.02 6986 0 15.66 6× 6-b2-75 6678 0 17.89 0 0 6678 0 4.77 6678 0 39.05 6× 6-b2-100 6116 0 5.07 0 0 6116 0 3.71 6116 0 32.28

five of eight global optima when minimizing c₁ (Table 3) and five of eight global optima when

minimizing c₂ (Table 4). For 6 × 6 instances, CPLEX solved only one of four instances in the

time limit when minimizing c₁ and both BRKGA and ILS were able to find this optimal value.

When minimizing c₂, CPLEX could easily solve all four instances of 6 × 6 to optimality and the

two heuristics were also able to find all these global optima. For such instances, ILS and BRKGA

found one and two upper bounds better than CPLEX, respectively. As for the instances of 7× 7,

CPLEX runs out of memory for both criteria, while the two metaheuristics could handle the

computation on all the instances for both c1and c2criteria. For 7× 7 instances, in c1optimization,

BRKGA found better solutions than ILS on three of four instances and ILS performs better on

the other instance. When minimizing c2, the two methods resulted in similar objective values for

three of four instances, while ILS found a solution with a better c₂value than BRKGA in the other

instance.

Not surprisingly, for time consumption, BRKGA and ILS are better than CPLEX in the biggest

problems considering c₁criterion. Comparing the two heuristics, the running time is generally in

the same scale with a slight difference for some instances. ILS is less time-consuming than BRKGA

in c₁ minimization on a number of instances and its computational time increases slightly faster

than BRKGA when the instance size grows. However, BRKGA solved the problems for c2globally

faster than ILS (see Table 4) because the local search operators in ILS lead to a lot of arc reversals, which is hard to control and does not comply with the arc reversals minimization.

Tests were also carried out to compute tttplots, as proposed by Aiex et al. (2002, 2007), in order to compare the robustness of the proposed metaheuristics. The idea of tttplots is to compare the ability of the methods to reach a given target value considering a number of runs (here 200) with different random seeds. The selected target value should neither be too easy nor too difficult to reach in order to get meaningful results. Thus, the selection of target values is done to allow conclusions in these experiments. Here, for each instance, we fixed a small deviation from the optimum value in which both methods were able to reach in order to get the target values. The abscissas and ordinates are, respectively, the running time and the probability to reach the target value. For each method, the more straight the curve is, the higher the probability for the method to reach the target value is.

Illustrative plots, representative for the instances tested, and obtained by minimizing c1objective for

two URND instances (5× 5-b6 and 6 × 6-b2) are given in Fig. 6. For these URND instances, the

target is 1.6% of the optimal value, while it is 2.8% from the best solution value found by BRKGA

for 6× 6-b2. Results indicate that ILS seems to perform globally better than BRKGA for URND

as it reaches target values faster. Moreover, one may note that BRKGA curve is not as smooth as ILS.

5.2.2. Results for MRND (I)

For the experiments on multigraph instances shown in Tables 5 and 6, ILS found 8 of the 12 global

optima for the 5× 5 and 6 × 6 instances that can be solved by CPLEX in the two-hour limit when

minimizing c₁. It solves to optimality all the 16 instances solvable by CPLEX when minimizing

c₂. On the same instances, BRKGA performs better for c₁, finding 11 of the 12 global optima.

On the other hand, it can also solve all the 16 instances to optimality for c2. Interestingly, ILS is

slightly better than BRKGA for the instances CPLEX could not solve. Given a number of nodes, it seems the problem is getting easier to solve when the amount of multiple arcs increases. This can be explained by the fact that setting arcs in opposite direction between two adjacent nodes is optimal

when optimizing c₁, irrespective of the remaining part of the graph. However, the more multiple

arcs an instance has, the higher the time consumed by the two heuristics is. They seem to be sensitive to the increase of the arc number. The computational time for BRKGA and ILS is fairly similar for

Fig. 6. Time-to-target plots of BRKGA and ILS for URND c1objective.

c₁ minimization. But ILS is slightly more time-consuming than BRKGA when minimizing c2 on

most of the instances.

The tttplots presented in Fig. 7 illustrate the behavior of the two metaheuristics for MRND.

Generated multigraph instances 5× 5-b1-25 and 5 × 5-b2-25 are used in these experiments. Results

indicate that BRKGA is more robust on the 5× 5 instance with one disruption, presented in the

Fig. 7. Time-to-target plots of BRKGA and ILS for MRND c₁objective.

upper tttplot, while its performance becomes worse than ILS on an instance with the same numbers of nodes and arcs but with two disrupted arcs, as shown in the lower tttplot. Note in this last figure, BRKGA has 5% of probability of spending more than two seconds, while this probability is nearly 0% for ILS. Overall, it seems that BRKGA is more sensitive to the number of disruptions than ILS. A behavior on the results similar to URND is the noncontinuous curve found by BRKGA.

Table 7

c1Minimization for URND (II)

BRKGA ILS

Instance c1 c2 t (seconds) c1 c2 t (seconds)

11× 11-b1 122,672 106 1187.38 121,660 89 864.46 11× 11-b2 121,830 123 1421.04 122,278 89 912.95 11× 11-b4 123,798 98 1001.62 122,578 88 1058.84 11× 11-b6 126,970 104 811.03 124,494 108 718.79 12× 12-b1 188,368 159 1680.56 187,300 134 1460.52 12× 12-b2 190,872 122 677.76 187,384 119 1105.15 12× 12-b4 188,164 122 2989.76 187,690 111 1527.31 12× 12-b6 195,228 126 791.19 188,124 116 1613.10 Table 8

c₂Minimization for URND (II)

BRKGA ILS

Instance c1 c2 t (seconds) c1 c2 t (seconds)

11× 11-b1 184,828 3 88.86 184,736 1 276.94 11× 11-b2 187,024 1 93.37 185,134 1 275.28 11× 11-b4 186,314 1 26.01 186,314 1 279.04 11× 11-b6 186,312 5 34.80 186,692 1 272.38 12× 12-b1 245,874 1 186.83 245,946 1 464.90 12× 12-b2 245,720 3 233.83 246,060 1 441.26 12× 12-b4 251,400 1 246.42 251,400 1 429.66 12× 12-b6 252,998 4 176.00 253,366 2 419.81 Table 9

c1Minimization for MRND (II)

BRKGA ILS

Instance c1 c2 t (seconds) c1 c2 t (seconds)

9× 9-b1-25 43,310 61 213.25 42,370 95 990.45 9× 9-b1-50 41,106 101 319.88 41,180 101 798.19 9× 9-b1-75 39,876 87 213.14 39,870 85 440.53 9× 9-b1-100 38,952 97 64.58 38,952 97 93.30 9× 9-b2-25 43,520 66 205.17 43,324 71 344.37 9× 9-b2-50 41,770 81 68.08 41,254 88 958.42 9× 9-b2-75 39,946 81 198.28 40,030 87 514.05 9× 9-b2-100 38,992 92 64.40 38,992 92 90.76

5.3. Results for URND and MRND (II)

Tables 7 and 8 report the results for larger URND instances with more than 100 nodes (11× 11 and

12× 12 instances). Moreover, Tables 9 and 10 refer to the MRND instances with 81 nodes and 288

Table 10

c2Minimization for MRND (II)

BRKGA ILS

Instance c1 c2 t (seconds) c1 c2 t (seconds)

9× 9-b1-25 53,256 0 45.74 53,256 0 124.25 9× 9-b1-50 53,350 0 56.05 53,350 0 198.09 9× 9-b1-75 48,388 0 90.68 48,388 0 270.24 9× 9-b1-100 49,930 0 21.23 49,930 0 382.88 9× 9-b2-25 49,496 1 39.41 49,496 1 116.06 9× 9-b2-50 49,922 1 63.98 49,756 1 180.80 9× 9-b2-75 48,952 0 76.29 48,952 0 264.37 9× 9-b2-100 46,962 0 20.98 46,962 0 359.64

arcs 9× 9 instances). These instances cannot be solved by CPLEX due to memory issues. Hence,

experiments were done using BRKGA and ILS.

Results of c₁optimization for URND reported in Table 7 indicate that ILS performs better than

BRKGA in terms of solution quality, since BRKGA found better values than ILS on only one of

eight instances, while the processing time is very similar for both methods. When minimizing c2,

more time is consumed by ILS, which find better or equal objective values than BRKGA on all the eight instances, as presented in Table 8. In addition, ILS also produces solutions with a better quality than BRKGA on six of eight multigraph instances, but its time consumption is higher for

optimizing either c₁ or c₂. In general, ILS performs better than BRKGA for most of the larger

URND and MRND instances. However, BRKGA is faster.

5.4. An application to a real urban network

In order to provide a glance on the real context of urban networks, we also extracted real geographical data (from OpenStreetMap) for the urban network of Troyes. We performed a data preprocessing that removes unnecessary information and merges intermediate nodes, such that the resulting graph has a relatively reasonable size in terms of node and arc numbers without altering its pertinence. The resulting graph with 58 nodes and 155 arcs is illustrated in Fig. 8. Note that the nodes may not correspond to exact real locations since the previously mentioned preprocessing may change the coordinates for merged nodes. Yet, real distances are kept. Troyes instance names contain the

number of nodes (here 58) and the number of blocked arcs in{1, 2, 4, 6}.

A set of instances is then created from this graph that can hence be tested by the proposed metaheuristics for both URND and MRND. Disruptions are randomly selected in a way to ensure that the resulting graph is still orientable, that is, does not contain bridges. We consider there is a travel demand for every origin–destination pair. Thus, there should be a path between every pair of nodes.

According to the results reported in Tables 11 and 12, the behavior of the two metaheuristics is similar to what is described in the previous subsections. Results in Table 11 indicate that, even if it

Fig. 8. Graph of historical Troyes downtown road network with 58 nodes and 155 arcs.

Table 11

c1Minimization on troyes-v58 instances

BRKGA ILS

Instance c₁ c₂ t (seconds) c₁ c₂ t (seconds)

troyes-v58-b1 2,082,930 24 50.31 2,061,660 43 127.20

troyes-v58-b2 2,079,430 24 53.52 2,064,860 43 119.33

troyes-v58-b4 2,094,690 13 72.12 2,072,890 41 112.18

troyes-v58-b6 2,141,170 20 77.10 2,108,600 49 215.11

Table 12

c2Minimization on troyes-v58 instances

BRKGA ILS

Instance c1 c2 t (seconds) c1 c2 t (seconds)

troyes-v58-b1 2,172,160 0 23.90 2,172,160 0 37.88

troyes-v58-b2 2,184,720 0 24.87 2,184,720 0 37.17

troyes-v58-b4 2,185,640 2 13.99 2,199,210 0 39.76

troyes-v58-b6 2,259,330 0 16.61 2,259,330 0 42.96

the four instances. BRKGA provides solutions around 1% above the solution values computed by

ILS. When minimizing c₂, BRKGA finds the same objective values as ILS on three of four instances,

presented in Table 12.

6. Concluding remarks

In this work, URND and MRND problems are addressed. They correspond to the particular situation in downtown city or historical city center. The reversal of arc direction is introduced to handle the issue of strong connectivity due to disruptions and to potentially reduce the travel cost. As a way to handle disruptions in urban context, this approach has not been widely studied. However, a reversal often involves changes of the traffic structure. Hence, it disturbs the users’ driving habits, which may slow down the traffic and even increase the risk of accident. In addition, it implies more signaling procedures for the technical services. Therefore, the number of reversals should stay as limited as possible.

A mathematical model is proposed to minimize the total travel cost in terms of distance and number of arc reversals. Two metaheuristics, a BRKGA and an ILS, are developed to solve both URND and MRND problems. Numerical experiments are performed on a set of generated instances up to 144 nodes and 264 arcs for URND and up to 81 nodes and 288 arcs for MRND. Methods are also applied on instances extracted from the real network of the French Troyes city, with 58 nodes and 155 arcs. Results show that the limit for CPLEX on URND instances is around 25 nodes and 40 arcs. For MRND, its limit is around 36 nodes and 120 arcs. On the other hand, the metaheuristics are able to solve instances with more nodes and arcs. Compared to BRKGA, ILS seems more robust for both URND and MRND, even when the number of disruptions increases.

A number of possibilities are opened for future works. First, a simple polynomial time algorithm to minimize the number of arc reversals is still an open issue. We intend to address these problems in a multiobjective context. In addition, the mathematical formulations can also be strengthened by means of valid inequalities.

Acknowledgments

The authors would like to thank the anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions. This project is funded by the Beijing YuanZhi Tiancheng Technology Co. Ltd., China, and the Innov’Action project from Champagne-Ardenne region in France.

References

Ad´am, ´A., 1964. Problem 2. In Fiedler, M. (ed.) Theory of Graphs and its Applications: Proceedings of the Symposium

held in Smolenice in June 1963. Czech Academy of Sciences, Prague, p. 157.

Afsar, H.M., Prins, C., Santos, A.C., 2014. Exact and heuristic algorithms for solving the generalized vehicle routing problem with flexible fleet size. International Transactions in Operational Research 21, 1, 153–175.

Aiex, R.M., Resende, M.G., Ribeiro, C.C., 2002. Probability distribution of solution time in GRASP: an experimental investigation. Journal of Heuristics 8, 3, 343–373.

Aiex, R.M., Resende, M.G.C., Ribeiro, C.C., 2007. TTT plots: a perl program to create time-to-target plots. Optimization

Letters 1, 4, 355–366.

Alvarez, A., Munari, P., Morabito, R., 2018. Iterated local search and simulated annealing algorithms for the inventory routing problem. International Transactions in Operational Research 25, 6, 1785–1809.

Bang-Jensen, J., Gutin, G.Z., 2008. Digraphs: Theory, Algorithms and Applications (2nd edn). Springer, Berlin.

Bean, J.C., 1994. Genetic algorithms and random keys for sequencing and optimization. ORSA Journal on Computing 6, 2, 154–160.

Boesch, F., Tindell, R., 1980. Robbins’s theorem for mixed multigraphs. The American Mathematical Monthly 87, 9, 716–719.

Bogdanowicz, Z.R., 2011. On arc reversal in balanced digraphs. Discrete Mathematics 311, 6, 435–436.

Cantarella, G., Pavone, G., Vitetta, A., 2006. Heuristics for urban road network design: lane layout and signal settings.

European Journal of Operational Research 175, 3, 1682–1695.

Chung, F., Garey, M., Tarjan, R., 1985. Strongly connected orientations of mixed multigraphs. Networks 15, 4, 477–484. Chv´atal, V., Thomassen, C., 1978. Distances in orientations of graphs. Journal of Combinatorial Theory, Series B 24, 1,

61–75.

Conte, A., Grossi, R., Marino, A., Rizzi, R., Versari, L., 2016. Directing road networks by listing strong orientations. In M¨akinen, V., Puglisi, S.J., Salmela, L. (eds) Combinatorial Algorithms, Proceedings of 27th International Workshop, IWOCA 2016, Helsinki, Finland, August 17–19, Springer, Berlin, pp. 83–95.

Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C., 2009. Introduction to Algorithms (3rd edn). MIT Press, Cambridge, MA.

Gallo, M., D’Acierno, L., Montella, B., 2012. A meta-heuristic algorithm for solving the road network design problem in regional contexts. Procedia —Social and Behavioral Sciences 54, 84–95.

Gaskins, R.J., Tanchoco, J.M.A., 1987. Flow path design for automated guided vehicle systems. International Journal of

Production Research 25, 667–676.

Goldberg, D.E., 1989. Genetic Algorithms in Search, Optimization and Machine Learning (1st edn). Addison-Wesley Longman, Boston, MA.

Gonc¸alves, J.F., Resende, M.G.C., 2011. Biased random-key genetic algorithms for combinatorial optimization. Journal

of Heuristics 17, 5, 487–525.

Hertz, A., Mittaz, M., 2001. A variable neighborhood descent algorithm for the undirected capacitated arc routing problem. Transportation Science 35, 4, 425–434.

Huang, Y., Santos, A.C., Duhamel, C., 2016a. A bi-objective model to address disruptions on unidirectional road networks. 8th IFAC Conference on Manufacturing Modelling, Management and Control MIM 2016, Vol. 49, Troyes, France, pp. 1620–1625.

Huang, Y., Santos, A.C., Duhamel, C., 2016b. Disruptions management in multidirectional road networks. Proceedings

of the 9th Triennial Symposium on Transportation Analysis (TRISTAN), Aruba, Netherlands, p. 4.

Huang, Y., Santos, A.C., Duhamel, C., 2018. Methods for solving road network problems with disruptions. Electronic

Notes in Discrete Mathematics 64, 175–184.

Italiano, G.F., Laura, L., Santaroni, F., 2012. Finding strong bridges and strong articulation points in linear time.

Theoretical Computer Science 447, 74–84.

Jir´asek, J., 1987. On a certain class of multidigraphs, for which reversal of no arc decreases the number of their cycles.

Commentationes Mathematicae Universitatis Carolinae 28, 1, 185–189.

Jir´asek, J., 2005. Arc reversal in nonhamiltonian circulant oriented graphs. Journal of Graph Theory 49, 1, 59–68. Lourenc¸o, H.R., Martin, O.C., St ¨utzle, T., 2002. Iterated local search. In Glover, F., Kochenberger, G. (eds) Handbook

of Metaheuristics, International Series in Operations Research and Management Science, Vol. 57. Kluwer Academic,

Dordrecht, pp. 321–353.

Lourenc¸o, H.R., Martin, O.C., St ¨utzle, T., 2010. Iterated Local Search: Framework and Applications. Springer, Boston, MA.

Martins, A.X., Duhamel, C., Santos, A.C., 2017. A column generation approach for the strong network orientation problem. Electronic Notes in Discrete Mathematics 62, 75–80.

Miandoabchi, E., Farahani, R.Z., 2011. Optimizing reserve capacity of urban road networks in a discrete network design problem. Advances in Engineering Software 42, 12, 1041–1050.

Mor´an-Mirabal, L., Gonz´alez-Velarde, J., Resende, M., 2014. Randomized heuristics for the family traveling salesperson problem. International Transactions in Operational Research 21, 1, 41–57.

Qiu, L., Hsu, W.J., Huang, S.Y., Wang, H., 2002. Scheduling and routing algorithms for AGVs: a survey. International

Journal of Production Research 40, 3, 745–760.

Reis, R., Ritt, M., Buriol, L.S., Resende, M.C., 2011. A biased random-key genetic algorithm for OSPF and DEFT routing to minimize network congestion. International Transactions in Operational Research 18, 3, 401–423. Resende, M.G.C., 2012. Biased random-key genetic algorithms with applications in telecommunications. TOP 20, 1,

130–153.

Robbins, H., 1939. A theorem on graphs, with an application to a problem on traffic control. The American Mathematical

Monthly 46, 281–283.

Roberts, F., 1978. Graph theory and its applications to problems of society. CBMS-NSF Regional Conference Series in

Applied Mathematics, Vol. 29. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

Santos, A.C., Duhamel, C., Prins, C., 2013. Heuristics for setting directions in urban networks. The X Metaheuristics

International Conference (MIC), Singapore, pp. 1–4.

Sever, D., Dellaert, N., van Woensel, T., de Kok, T., 2013. Dynamic shortest path problems: hybrid routing policies considering network disruptions. Computers & Operations Research 40, 12, 2852–2863.

Spears, V.M., Jong, K.A.D., 1991. On the virtues of parameterized uniform crossover. Proceedings of the Fourth

Interna-tional Conference on Genetic Algorithms, San Diego, CA, pp. 230–236.

Stefanello, F., Buriol, L.S., Hirsch, M.J., Pardalos, P.M., Querido, T., Resende, M.G.C., Ritt, M., 2017. On the minimiza-tion of traffic congesminimiza-tion in road networks with tolls. Annals of Operaminimiza-tions Research 249, 1, 119–139.

Tarjan, R., 1972. Depth-first search and linear graph algorithms. SIAM Journal on Computing 1, 2, 146–160.

Thomassen, C., 1987. Counterexamples to Ad´am’s conjecture on arc reversals in directed graphs. Journal of Combinatorial

Theory, Series B 42, 1, 128–130.

Venkataramanan, M.A., Wilson, K.A., 1991. A branch-and-bound algorithm for flow-path design of automated guided vehicle systems. Naval Research Logistics (NRL) 38, 3, 431–445.

Vis, I.F., 2006. Survey of research in the design and control of automated guided vehicle systems. European Journal of

Operational Research 170, 3, 677–709.

Yang, H., Bell, M.G.H., 1998. Models and algorithms for road network design: a review and some new developments.