A store-and-forward neural network to solve multicriteria optimal path problem in time-dependent networks

This paper introduces the constrained multi-objective optimal path problem in time-dependent networks. In the existing literatures, the constraints are all imposed on the objective function while the problem constraints are related to the non-objective function. It is the difference that makes the traditional algorithm unable to get a better solution quality. In this light, we propose a store-and-forward neural network (SFNN) that finds the better result. In the design of SFNN, the topology of neural network is the same as that of time-varying network, and each node is designed as store-and-forward neuron. Each neuron transmits information to other neurons by sending signals. The experimental results show that compared with the traditional methods, the accuracy is significantly improved when the calculation time is acceptable.


Introduction
The constrained multi-objective optimal path problem has a wide range of applications in different engineering fields such as the Internet of Things [1] and transportation network [2]. In this regard, traditional network algorithms approximate the actual network to a static model, and the solution obtained is far from the actual optimal solution. Therefore, attention to the dynamic characteristics of arc attributes in time-dependent networks has attracted more and more attention. The constraints are all imposed on the objective function in the existing literatures, while our constraints of the proposed problem is imposed on the non-objective function. This difference directly results in the worse solution quality for traditional algorithms in our experiments. In order to compare with other algorithms, the common indicators of multi-objective optimization solution set are used to analyze the results.
In time-dependent networks, Orda and Rom [3] have found that computing multicriteria optimal paths violates the subpath optimality property due to time-vaying arc weights. Kostreva and Wiecek [4] are the first to deal with the multicriteria optimal path problem in time-dependent networks. Their algorithm applies Bellman's optimality principle in backward direction. It effectively identifies whether or not a subpath can belong to one of the optimal paths. Most of the research works [5,6,7] have been carried out under their idea. Hamacher et al. [6] have proposed backward label setting algorithm for solving time-dependent bicriterion shortest path problem. This algorithm reverses the path from the target node to the starting node, pruning part of the path, but the disadvantage is that it is difficult to determine the route time range of the target node. Androutsopoulos and Zografos [5] have IOP Publishing doi:10.1088/1742-6596/2246/1/012071 2 solved multi-objective routing problem where multiple objectives are equally important. However, these methods always delete part of the Pareto optimal path in the calculation process under resource constraints, so that the complete set of optimal paths cannot be obtained.
Neural network approaches are gaining considerable attention in recent years. Different neural network approaches to solve route-related problems have been reported in the literature. Dynamic neural networks [8], time-delay neural networks [9] and convolutional neural networks [10,11] are examples of the reported methods. A time-delay neural networks [9] is proposed to solve timedependent shortest path problem. Based on the ideas, a new neural network algorithm is designed to solve our problem.
This paper designs the store-and-forward neural network to solve the multicriteria shortest path problem in time-dependent network. Structure of the paper is organized as follows: In Section 2 the problem definition is stated. The proposed SFNN algorithm is described in Section 3. In Section 4 the numerical experiments are studied using multicriteria evaluation index. In Section 5 we summarize the advantages of SFNN.

Problem definition
This section first describes the relevant definitions of time-dependent networks, and then defines the multi-objective shortest path problem in a time-varying environment.

Time-dependent Network
is a time-dependent network, where is a set of nodes, ⊆ , ∈ | ≠ is a set of arcs. is a set of cost functions which stands for a cost function for a tourist from node to node at the depart time . Each arc , ∈ is associated with ∈ weight variables , . . . , . The values of variables are determined by which is defined as : ℕ → ℝ , that is, = , . . . , . A route with n nodes can be represented as a sequence of tuples = ⟨ , ; , . . . , , ; , . . . , , ; ⟩, where ∈ with 1 ≤ ≤ , , ∈ is the ith edge on the path, and is the departure time from node for 1 ≤ ≤ − 1.

Multicriteria shortest path problem with resource constraints
Given a time-dependent network = , , , the constrained multicriteria shortest path problem(TCMPP) is to determine the optimal path from the source node ∈ to the destination node ∈ with limited battery power constraints to minimize the travel time and travel cost. The TCMPP problem can be formulate as (1)-(5).
Subject to: The objective function (2)  arc of a path. Constraint (4) describes the resource consumption of any path ∈ , to tourists shall not exceed the limit value. However, the access time of each node of the constraint (5) path is within a certain time range, and it can arrive in advance and then wait until the earliest access time.

Design of store-and-forward neural network
A SFNN is a store-and-forward neural network without data training. Each neuron serves as input and output ports from precursor neurons and successor neurons respectively to communicate information through the mechanism of forwarding and storing signals.

A general neuron's structure of SFNN
A neuron is composed of five parts: input, depart time selector, signal generator, signal encoder and output. Figure 1 depicts the overall structure of the general neuron of SFNN. (1) Input: The input receives the signals from all associated neurons and transmits them to the signal decoder.
(2)Signal decoder: The signal decoder consists of five parts: , , , , and . These variables can be regarded as parameters of a signal decoded by function .

(3)Depart time selector:
The depart time selector part uses the function ( ) for each received and decoded signal to calculate the time to send it out again. Among them, let be the set of all possible departure times along the arc ( , ), and represents the arrival time when the signal from the precursor neuron is received.
where c denotes the index of depart time, ij l represents the number of depart times on the arc ( ) From formula (7) and (8), we can see that the departure time of node along arc ( , ) to node is calculated by the function ( ). In particular, the departure of the signal should occur after neuron receives the corresponding precursor neuron signals. Otherwise, it is set to a given infinite number Q.
= ⟨ , ⟩ Neuron i will temporarily store the wave information in set . When certain conditions are met, they will send out waves to successor neuron and delete them from .
(5) Signal Encoder: A signal is encoded by function , which the expression can be stated as follows: (6) Output: The output will send signals to the corresponding subsequent nodes associated with the direction according to the wave information.