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Electronic Notes in Discrete Mathematics 64 (2018) 175–184 www.elsevier.com/locate/endm

Methods for solving road network problems with disruptions Yipeng Huang a,1,2 , Andr´ea Cynthia Santos a,1,3 , Christophe Duhamel b,1,4 a b

ICD-LOSI, Universit´e de Technologie de Troyes, Troyes, France.

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

Abstract In this study, we provide two approximate methods for solving both Unidirectional and Multidirectional Road Network problems with Disruptions and connecting requirements. These problems aim at defining alternative paths to deviate traffic flows, allowing the arc orientation to be reversed, such that the network remains strongly connected. The objectives are set to minimize the total travel cost and the number of reversed arcs. Two methods, a Biased Random Key Genetic Algorithm and an Iterated Local Search, are proposed. Numerical experiments with single objective and bi-objective versions are performed and results are hence compared to the exact method studied in previous works. Keywords: Road network, Heuristics, Biased Random Key Genetic Algorithm, Iterated Local Search.

1 2 3 4

This project is funded by the Beijing YuanZhi Tiancheng Technology Co. Ltd, China. Email: [email protected] Email: [email protected] Email: [email protected]

https://doi.org/10.1016/j.endm.2018.01.019 1571-0653/© 2018 Elsevier B.V. All rights reserved.

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Introduction

In a real urban network, maintaining a fluid traffic is a key point due to the density of population in urban areas, and the population needs concerning the multiple daily travels. This issue will become relevant and difficult to manage since the concentration of population living in urban areas by 2030 will be about 5 billions of people over 8,3 billions, according to the United Nation Organization. Moreover, the urban sprawl and its road networks cannot be extended indefinitely. In this context, we investigate methods to address disruptions on urban networks by defining alternative paths to deviate the traffic flow. Disruption of routes can break the network strong connectivity, i.e. existence of a path between every pair of nodes. Thus, alternative paths may require the direction of vehicles’ flow to be reversed for some routes, named arc reversals. Another practical solution consists in alternating the traffic. We investigate the Road Network Problem with Disruptions (RND) that is a generic term used to address the Unidirectional Road Network Problem with Disruptions and connecting requirements (URND) and the Multidirectional Road Network Problem with Disruptions and connecting requirements (MRND). A unidirectional road network relies on a simple, directed and connected graph, while a multidirectional road network is modeled by means of a directed, connected, and loopless multigraph. In both problems URND and MRND, two goals are considered: (i) minimizing the number of arc reversals and (ii) minimizing the total travel distance, such that arc reversals are allowed and the road network remains strongly connected. This situation can disturb the users’ habits, which justifies the minimization of arc reversals. The major difference between URND and MRND is that the input graph for URND is a representation of one-way road networks, a situation usually found in downtown; while for MRND, the graph is a multigraph, that can model lanes in any urban network. If the objectives (i) and (ii) are optimized simultaneously, the URND and MRND are referred as bi-URND and bi-MRND. The URND complexity, a mathematical model and Pareto fronts using the -constraint method were studied in [9]. In [10], the authors show that any instance of the MRND can be represented by a 2-directed multigraph, and present an extended mathematical model. Here, the contributions are the following. We propose heuristic methods able to handle both bi-URND and bi-MRND. For this purpose, a Biased Random Key Genetic Algorithm (BRKGA) [7] and an Iterated Local Search (ILS) [11] were developed. In order to allow the ILS to manage a bi-objective function, the objectives are

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normalized and addressed by an aggregation of the two objectives. In addition, the -constraint method based on the mathematical model proposed in [9,10] is used, resp. for the bi-URND and bi-MRND. This allows assessing the quality of the proposed heuristics. In addition, a dynamic shaking has been integrated to ILS in order to improve diversity and guide the search in the feasible biobjective space. The remainder of manuscript is organized as follows, biURND and bi-MRND are formally defined in Section 2, together with a brief description of the existing literature review. Then, the proposed methods are presented in Section 3, followed by numerical experiments and results in Section 4. Concluding remarks are provided in Section 5.

2

Problem definition and related works

Let G = (N, A) be a connected simple digraph and G = (N  , A ) be a connected loopless multigraph. Graph G models URND problems, where N is the set of nodes, corresponding typically to crossroads or roundabouts; A is the set of arcs defining one-way streets. Moreover, MRND problems are modeled by graph G , where the set A of arcs can contain multiple arcs between a pair of nodes belonging to the set N  of nodes, i.e. (n1 , n2 ) ∈ N  × N  stands for street lanes of a two-way street. Disruptions can block several streets on a given network. Hence these blocked arcs are removed from the graph, potentially breaking the strong connectivity. In addition, travel paths on the network may also be impacted which implicitly increases travel costs. Thus, reversal of road directions may be necessary in order to restore the strong connectivity and to optimize the total travel cost of the network. In this model, solving bi-URND and biMRND problems relies on searching for strong orientations 5 minimizing the total travel cost (C1 objective) and the number of arc reversals (C2 objective). In terms of complexity, both bi-URND and bi-MRND are NP-hard, since defining a strong orientation with total distance minimization is NP-hard as shown in [3,5]. It is worth mentioning that the minimization of the number of arc reversals is proved to be polynomial by [1]. Relevant theoretical works were done on strong orientation in simple digraphs and multigraphs by [1,2,13]. Besides, a Depth First Search (DFS)based algorithm for orientation problems on simple graphs was proposed by [14]. Theoretical results, mathematical formulations and simple algorithms for bi-URND and bi-MRND were focused by [9,10]. A review on network design 5

i.e. an orientation such that the resulting graph is strongly connected

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problems for urban network applications is found in [6].

3

Heuristics for bi-URND and bi-MRND

Two heuristic methods, a BRKGA and an ILS, are proposed here to solve both bi-URND and bi-MRND in mono- and bi-objective versions. The main components shared by these heuristics are the way of building feasible solutions and the evaluation of the two objectives. A feasible solution is a strongly connected graph. Thus, an adaptation of the DFS-based algorithm [14] is used to set orientations in a given graph. The objective value C1 is computed using an All Pair Shortest Path algorithm to evaluate the total distance, while for the objective value C2 , a basic comparison with the input graph can quickly evaluate the number of arc reversals. The mono-objective version evaluates the value C1 or C2 independently, while the multi-objective version of the algorithms addresses the 2-dimensional search space of C1 and C2 . 3.1

Biased Random Key Genetic Algorithm

We apply initially a standard Genetic Algorithm (GA) BRKGA [7]. This algorithm is elitist and manages a population (solutions) that evolves by means of selection and application of operators such as crossover. The main difference with traditional GAs is that chromosomes are encoded as vectors of random weights (keys) that have to be decoded to provide a solution for a specific problem. This makes the core algorithm almost independent from modeling, data structure and implementation, thus reusable across different problems. For both bi-URND and bi-MRND, a key in (0, 1] is associated with each node of the graph. Then, a DFS-based algorithm is used to set an orientation for each edge, visiting the node with the highest weight in priority. The DFS search can pass through a node already visited if it allows orientating a missing edge. This strategy guarantees the strong connectivity by definition [14]. An example is given in Figure 1, where an input graph is depicted in Figure 1-(a). Suppose the following random weights {0.9, 0.6, 0.8, 0.4, 0.5} are associated respectively with nodes {0, 1, 2, 3, 4} as depicted in Figure 1-(b). The DFS starts from the node with the highest weight (node 0). Then, it visits node 1 and orientates the corresponding arc, instead of visiting node 3 due to its weight. The search continues from node 1 to node 2, ending with the orientation in Figure 1-(b). Figure 1-(c) is another orientation obtained by applying this method with another set of weights. For the multi-objective version, non-dominated solutions found by BRKGA

Y. Huang et al. / Electronic Notes in Discrete Mathematics 64 (2018) 175–184

(a) A road network

(b) One orientation

179

(c) Another orientation

{}

{}

{0.6}

{0.8}

{0.4}

{0.8}

1

2

1

2

1

2

4

4

4

{}

{0.5}

{0.2}

0

3

0

3

0

3

{}

{}

{0.9}

{0.4}

{0.9}

{0.6}

Fig. 1. Orientation using a DFS-based algorithm and random keys

are stored, following a typical multi-objective GA, in order to build an approximate Pareto-optimal front. 3.2

Iterated Local Search

We also propose an ILS method [11,12] due to its simplicity and performance. Given an initial feasible solution, the ILS performs a perturbation followed by a local search, until a stopping condition is reached. Then, the incumbent solution is returned. For computing feasible initial solutions, we adapt the DFS-based algorithm for decoding random keys as previously defined. A Variable Neighborhood Descent (VND) [8] is used as local search. Given a set {Ni , i = 1, 2, · · · , l} of neighborhoods, the VND starts with an initial solution s0 and N0 . Each time an improving solution s is found on Nk (s), the method moves to s and sets k = 1. Otherwise, s is kept and k ← k + 1. The method stops when k = l + 1. The DFS-based algorithm manages to find feasible solutions for both biURND and bi-MRND. For bi-URND, the following three neighborhoods adapted from [15] are used in the local search. A move in N1 changes arcs direction of one cycle. A move in N2 reverses all arcs incident to a node as shown in Figure 2, and a move in N3 reverses an arc. If the incumbent solution is improved, the method updates the solution, following a first improvement search strategy. Moves in N1 ensure the strong connectivity, while they do not in N2 and N3 . Hence a feasibility check is done using Tarjan’s Algorithm [4], to accept only strongly connected neighbors. No repair procedure is applied. For bi-MRND, the first move slightly differs from its bi-URND version. Since cycles with smallest number of arcs contains 2 arcs in opposite directions between two nodes in a multigraph, this may not change the solution, according to C1 and C2 . Thus only cycles with at least 3 arcs are considered.

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Y. Huang et al. / Electronic Notes in Discrete Mathematics 64 (2018) 175–184

(a) Original orientation 1

(b) Modified orientation

2

1

4

0

2

4

3

0

3

Fig. 2. N2 move at node 4

Three perturbations are used and randomly managed by ILS. The first perturbation generates a different orientation from a new set of random keys, which provides diversity to the algorithm. The second one reverses a cycle which contains one or several arcs belonging to other cycles as well. The third perturbation completely changes the incumbent solution and helps escaping local optima, by reversing all directions of the currently best orientation. The ILS is not a multi-objective method. To this extend, we propose a simple and dedicated way to guide the search in a bi-dimensional space using this method. The two objectives are aggregated into a single objective function as shown in Equation 1. The parameter α is varied between [0.05, 0.95] with steps of 0.05. This allows to explore different regions in the bi-dimensional search space and produce a set of supported solutions. As a consequence, an approximate Pareto-optimal front can be obtained. In addition, a memory mechanism is applied to store non-supported solutions. min C = αfC1 (s) + (1 − α)fC2 (s) ∀α ∈ [0, 1] s∈S

4

(1)

Numerical experiments

Experiments have been performed on existing instances from [9,10], using a 4 Intel Xeon CPU at 2.00 GHz, with 32 cores (64 threads), 128 GB RAM under a CentOS Linux. A time limit of 2h is set for CPLEX. 4.1

Results for the mono-objective methods

Tables 1 and 2 present respectively the results for the URND and the MRND minimizing C1 . In both tables, grid size, number of arcs and number of disruptions are referred in columns ‘Size’, ‘|A|’ and ‘|B|’, respectively. In addition, Table 2 uses column ‘%M’ to indicate the percentage of multiple arcs. For each method, CPLEX solver, BRKGA and ILS, columns ‘C1 *’, ‘C2 ’ and ‘t(s)’

Y. Huang et al. / Electronic Notes in Discrete Mathematics 64 (2018) 175–184

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provide respectively the optimum total travel distance, the number of reversals and the processing time in seconds. Results indicate that ILS performs better than BRKGA as shown in Table 1. On small instances (i.e. 4x4 and 5x5), ILS found 7 out of 8 global optima, while BRKGA found 5 out of 8 global optima. On medium sized instances (i.e. 6x6), CPLEX solved only 1 out of 4 instances in the time limit. In terms of computational time, results show that ILS is less time-consuming than BRKGA on almost all instances. This is due to the fact that BRKGA addresses one population per iteration instead of one solution per iteration as in ILS. Instance Size |A|

4x4 24

5x5 40

6x6 60

CPLEX

BRKGA

ILS

|B|

C1 *

C2

t(s)

C1 *

C2

t(s)

C1 *

C2

t(s)

1

912

1

38.61

912

3

5.93

912

1

2.36

2

968

5

14.73

968

5

4.86

968

5

2.70

4

1392

8

6.62

1392

8

4.02

1392

8

1.42

6

1304

7

3.45

1388

6

3.89

1304

7

0.57

1

2638

14

333.11

2668

23

21.87

2638

14

30.03

2

2748

16

268.27

2748

16

21.17

2758

11

14.06

4

3032

23

197.70

3032

23

23.55

3032

13

15.96

6

3116

4

210.95

3140

28

16.31

3116

4

9.74

1

≤ 6250

-

7200

6244

22

81.26

6274

20

75.82

2

≤ 6554

-

7200

6356

32

73.33

6374

30

64.17

4

≤ 6820

-

7200

6780

35

68.32

6764

16

85.93

6

7006

35

3656.6

7006

19

67.62

7006

35

46.76

Table 1 C1 minimization on small and medium instances (URND)

Both two approximate methods are able to find optimal solutions for all 4 out of 4 small instances (i.e. 4x4 and 5x5), as shown in Table 2. With less time used than BRKGA, ILS can even find better solutions in terms of the objective C2 on several instances.

Y. Huang et al. / Electronic Notes in Discrete Mathematics 64 (2018) 175–184

182

Instance Size |A| %M 4x4 24 25

5x5 40 25

6x6 60 25

CPLEX

BRKGA

ILS

|B|

C1 *

C2

t(s)

C1 *

C2

t(s)

C1 *

C2

t(s)

1

832

5

4.47

832

5

8.40

832

5

10.01

2

850

12

3.12

850

6

7.52

850

6

5.51

1

2406

14

72.20

2406

10

59.17

2406

10

35.51

2

2450

14

56.50

2450

18

69.16

2450

12

31.32

1

≤ 5834

-

7200

5948

40

243.25

5854

39

299.09

2

≤ 5954

-

7200

5978

32

255.90

5954

31

144.91

Table 2 C1 minimization on small multigraph instances (MRND)

4.2

Bi-objective optimization

Pareto-optimal fronts have been generated to evaluate the BRKGA and the ILS. Table 3 reports the results obtained by the -constraint method, the BRKGA and the ILS. The columns ‘Q’ and ‘HV’ present respectively the number of solutions and the hyper volume. All other parameters have been previously defined. Results indicate that the fronts found by ILS are optimal or very close to the optimal fronts, confirmed by the small gap for HV (less than 1%). On the other hand, the BRKGA does not approximate well Pareto-optimal fronts. This is clearly illustrated in Figures 3-(a) and 3-(b) for instances 5x5. -constraint

Instance Size |A| 4x4 24

5x5 40

6x6 40

BRKGA

ILS

|B|

Q

HV

t(s)

Q

HV

t(s)

Q

HV

t(s)

2

4

0.50

25.78

4

0.50

14.97

4

0.50

60.83

4

5

0.72

17.15

5

0.72

10.14

5

0.72

41.15

2

13

0.87

2785.17

12

0.81

79.06

11

0.86

552.95

4

11

0.82

1317.45

10

0.70

82.55

9

0.81

450.91

22

0.66

8310.29

11

0.85

9519.04

15

0.77

3717.46

13

0.80

2325.32

2 4

Out of memory

Table 3 Performance comparison of Pareto fronts

Y. Huang et al. / Electronic Notes in Discrete Mathematics 64 (2018) 175–184

(a) instance with 2 disruptions


(b) instance with 4 disruptions



    




183


   !"#


 
 







 














































 





































Fig. 3. Comparison of Pareto-optimal fronts on instances with 25 nodes and 40 arcs

5

Concluding remarks

In this study, we investigate mono- and bi-objective methods for bi-URND and bi-MRND, where arc reversals are allowed in order to restore the strong connectivity due to disruptions and to reduce travel costs. Two objectives are optimized: the number of reversals and the total travel cost. We propose a standard BRKGA and an ILS to address both bi-URND and bi-MRND. Multi-objective optimization is also taken into account, with a dedicated mechanism to manage the search on bi-dimensional space integrated on the ILS. Results show the ability of ILS to converge to high quality solutions in mono and bi-objective versions. Further investigations can address theoretical aspects of the bi-URND and bi-MRND. One can mention the lack of a simple polynomial algorithm to minimize the number of arc reversals. In addition, there is room to improve the proposed heuristics, especially to reduce the computational time and to add other multi-objective operators to BRKGA. We are investigating other realistic constraints found in the general RND problem.

References [1] Bang-Jensen, J. and G. Z. Gutin, “Digraphs: Theory, Algorithms and Applications,” Springer Publishing Company, Incorporated, 2008, 2nd edition. [2] Boesch, F. and R. Tindell, Robbins’s theorem for mixed multigraphs, The American Mathematical Monthly 87 (1980), pp. 716–719. [3] Burkard, R., K. Feldbacher, B. Klinz and G. Woeginger, Minimum-cost strong network orientation problems: Classification, complexity, and algorithms,

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Networks 33 (1999), pp. 57–70. [4] Chung, F., M. Garey and R. Tarjan, Strongly connected orientations of mixed multigraphs, Networks 15 (1985), pp. 477–484. [5] Chv´atal, V. and C. Thomassen, Distances in orientations of graphs, Journal of Combinatorial Theory, Series B 24 (1978), pp. 61–75. [6] Farahani, R., E. Miandoabchi, W. Szeto and H. Rashidi, A review of urban transportation network design problems, European Journal of Operational Research 229 (2013), pp. 281 – 302. [7] Gon¸calves, J. F. and M. G. C. Resende, Biased random-key genetic algorithms for combinatorial optimization, Journal of Heuristics 17 (2011), pp. 487–525. [8] Hertz, A. and M. Mittaz, A variable neighborhood descent algorithm for the undirected capacitated arc routing problem, Transportation science 35 (2001), pp. 425–434. [9] Huang, Y., A. C. Santos and C. Duhamel, A bi-objective model to address disruptions on unidirectional road networks, in: 8th IFAC Conference on Manufacturing Modelling, Management and Control MIM 2016, 12 49, Troyes, France, 2016, pp. 1620–1625. [10] Huang, Y., A. C. Santos and C. Duhamel, Disruptions management in multidirectional road networks, in: Proceedings of the 9th Triennial Symposium on Transportation Analysis (TRISTAN), Aruba, Netherlands, 2016, p. 4. [11] Louren¸co, H. R., O. C. Martin and T. St¨ utzle, “Iterated Local Search,” Springer US, Boston, MA, 2003 pp. 320–353. [12] Louren¸co, H. R., O. C. Martin and T. St¨ utzle, “Iterated Local Search: Framework and Applications,” Springer US, Boston, MA, 2010 pp. 363–397. [13] Robbins, H., A theorem on graphs, with an application to a problem on traffic control, The American Mathematical Monthly 46 (1939), pp. 281–283. [14] Roberts, F. S., “Graph theory and its applications to problems of society,” CBMS-NSF Regional Conference Series in Applied Mathematics 29, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1978. [15] Santos, A. C., C. Duhamel and C. Prins, Heuristics for setting directions in urban networks, in: The X Metaheuristics International Conference (MIC), Singapore, 2013, pp. 1–4.