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The requirement of the road services and transportation network development planning came into existence with the development of civilization. In the modern urban transport scenario with the forever mounting amount of vehicles, it is very much essential to tackle network congestion and to minimize the travel time. This work is based on determining the optimal wait time at traffic signals for the microscopic discrete model. The problem is formulated as a bilevel model. The upper layer optimizes the travel time by reducing the wait time at traffic signal and the lower layer solves the stochastic user equilibrium. Soft computing techniques like Genetic Algorithms, Ant Colony Optimization, and many other biologically inspired techniques prove to give good results for bilevel problems. Here this work uses Bat Intelligence to solve the transport network design problem. The results are compared with the existing techniques.

Nowadays the ever more increasing number of vehicles creates a challenge in the modern urban transportation scenario. For a road network with n number of junctions, there are

The need for the transport and road network planning came on track with the expansion of civilization. Abdullaal [

Allsop [

It is not comprehensible a priori which path through the network has the shortest travel time. We can conclude that the responses of the vehicle user can be predicted not dictated. Biologically inspired techniques have proven to give good results in such scenarios. The nature provides a wide range of inspiration in many unusual forms, sizes, and attributes.

Ceylan and Bell [

Xin-She Yang [

This paper is organized into 5 sections. The first section introduces the paper and discusses some related work. Section

The road network can be taken as a directed graph G= (N, a), where ‘N’ is the set of nodes; i.e., the road junctions ‘a’ is the links connecting the junctions as shown in Figure

An intersection of the network showing an O-D pair connected by a 3-way and a 4-way junction. The green arrows denote the signal values.

We assume that the links connecting nodes have a travel time function

Bilevel model for traffic network determination problem.

This model can be formulated mathematically as shown below for both the layers.

Bat algorithm is an innovative technique proving to give better solution than many popular traditional and heuristic algorithms [

Echolocation behavior of the microbats.

The bats emit loud ultrasonic sound waves and listen to the echo that reflects back from the surrounding objects. The bat algorithm uses some idolized rules for simplicity.

Bats use echolocation to sense prey, predator, or any barriers in the path and distance.

Bats fly with a velocity

As they get close to the prey, pulse increases and loudness decreases.

Figure

Bat algorithm.

The projected method for solving the TNDP is based on bilevel model. Figure

Flowchart for the traffic signal optimization problem using bat algorithm.

The frequency of bats

This paper considers 3 sets of test cases.

Five test cases were adapted from [

A 12-node test network with 4 intersections showing the signal value of 5 different phases.

16-, 20-, 24-, and 28-node test network.

16-node network

20-node network

24-node network

28-node network

The termination condition for BA is taken as 100 iterations. This termination condition was set up by experiment on several run time results. Intersections were implemented using Origin-Destination nodes with extra features like wait time at signal, signal values, travel time on the link, positional information on the links attached, and other delays.

There were certain assumptions made to simplify the implementation, maintaining the integrity of the problem. Every Origin-Destination pair in the chosen networks is connected through at least one intersection. Lower layer statistics were generated randomly to simulate its function as an input to the upper layer. Table

A comparative analysis of best, average, and worst solution for all the 5 networks.

12 Nodes | 16 Nodes | 20 Nodes | 24 Nodes | 28 Nodes | |
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| 62.25 | 256.82 | 443.10 | 480.03 | 596.50 |

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| 70.15 | 222.25 | 338.03 | 459.54 | 541.21 |

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ACO- | 75.40 | 218.29 | 332.10 | 421.29 | 542.15 |

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| 64.81 | 212.21 | 320.21 | 430.31 | 538.10 |

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| 90.14 | 265.02 | 443.00 | 480.27 | 569.17 |

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| 110.21 | 259.17 | 450.41 | 485.17 | 560.11 |

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ACO- | 85.01 | 242.33 | 421.36 | 479.33 | 542.65 |

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| 122.02 | 273.36 | 465.22 | 511.25 | 588.17 |

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| 92.14 | 215.65 | 425.39 | 465.14 | 560.34 |

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| 103.21 | 245.24 | 425.37 | 438.54 | 540.74 |

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ACO- | 87.27 | 220.14 | 414.26 | 450.94 | 528.28 |

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| 93.12 | 242.94 | 399.58 | 465.79 | 561.16 |

The results for BA are weighed against ACO, GA, and hybrid ACO-GA [

Best solution.

Figure

Worst solution.

Figure

Average solution.

The ranges of all the four algorithms, ACO, GA, hybrid ACO-GA, and BA, are put side by side in Figure

Range of solutions of ACO, GA, hybrid ACO-GA, and BA.

Several researchers have tested the performance of continuous network design problem on multiple networks. A widely used 16-link network with 6 nodes is adapted from Suwansirikul et al. [

16-link, 6-node network.

The continuous network design problem is executed for 3 test cases with different demand scenarios for the given network. The travel demands are shown in Table

Travel demand scenario.

Demand from node 1 to 6 | Demand from node 6 to 1 | Total demand | |
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Case 1 | 5 | 10 | 15 |

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Case 2 | 10 | 20 | 30 |

The 16-link network problem is estimated using several techniques like traditional H-J, EDO, SA, and CS by different researchers as mentioned in Table

Techniques and references of the test case 2.

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Modular In-core Nonlinear Optimization System | MINOS | Suwansirikul et al. [ |

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Hooke-Jeeves algorithm | H-J | Abdulaal and LeBlanc [ |

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Equilibrium Decomposed Optimization | EDO | Suwansirikul et al. [ |

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Simulated Annealing algorithm | SA | Friesz et al. [ |

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Cuckoo Search Algorithm with Lévy Flights | CS | Ozgur Baskan [ |

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Particle Swarm Optimization | PSO | Hu Hui [ |

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Bat Algorithm | BA | This paper |

Solution of demand scenario 1 for TNDP.

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Solution of demand scenario 2 for TNDP.

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A more realistic data for road network is adapted from the city Sioux Falls, South Dakota, situated in the USA. The network is much more complex and appealing to the researchers working on the transport network problem [

Techniques and references of test case 3.

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Hooke-Jeeves algorithm | H-J | Abdulaal and LeBlanc [ |

Simulated Annealing algorithm | SA | Friesz et al. [ |

Gradient Projection method | GP | Chiou [ |

Genetic Algorithm | GA | Mathew and Sarma [ |

Cuckoo Search Algorithm with Lévy Flights | CS | Ozgur Baskan [ |

Harmony Search | HS | Ozgur Baskan[ |

Artificial Bee Colony | ABC | Ozgur Baskan[ |

Differential Evolution | DE | Ozgur Baskan[ |

Bat Algorithm | BA | This paper |

Sioux Falls network.

The results for Sioux Falls network is shown in Table

Comparative analysis of TNDP for Sioux Falls network.

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Convergence of BA.

The simulation work was carried out for various sizes of multiple networks for variant test cases and a number of times. As per the results of test case 1, it can be concluded that BA explores a wide range of solution set and gives better results than GA, ACO, and hybrid ACO-GA. Although the hybrid ACO-GA outperforms ACO, GA, and BA for average solution. BA was compared on a 16-link problem in test case 2. In the

In the third test case, BA is compared with H-J, SA, GP, GA, CS, HS, ABC, and DE. For the best solution of the objective function value, BA outperforms all the mentioned techniques. For the average value among HS, ABC, and DE, BA gives the higher objective function value. The range of solution for BA seems to be on the higher side. A value of worst solution for HS is given which is better than the worst solution of BA. In future more improvements can be carried out on the proposed algorithm and can be implemented to give a much better solution

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.