Hub Location Problems (HLPs) which are known as "network optimization problems", identify hubs as collection and distribution centers [1]. In these problems, less and more direct connections are used to achieve the economic interests rather direct relationship between the two points and accordingly, demands with the same destination are aggregated and then distributed.

There are various other applications in the HLPs in addition to telecommunication and transportation systems, including production planning, retail management, wholesale management, and healthcare.

Teo and Shu [2-15] and many other works have been conducted in this field. However, along with the issues raised in all of these problems with a wide variety of applications, it should be noted that the disruption and revocation of a service flow will impose additional costs in addition to unfulfilled economic goal. The costs include increased direct and indirect costs, intensified dissatisfaction, and undermined system legitimacy [4]. Therefore, the reliability of the network communication routes and how to establish hub facilities based on reliability is important. However, Snyder et al. [16] have considered the effects of natural disasters, labor strikes, or terrorist threats among the factors affecting the reliability. The importance of this issue is increasing day by day by growing the industrial society and the increased volume of communications, and the growth of scientific research is possible in the field of reliable hub location. Kim and O'kelly [17] were the first ones who focused on promoting the reliability of communication routes in the hub location problem and An et al. [4] improved the hub facility reliability factor by defining the backup facilities. According to them, given the strategic level of HLPs, the solving method is of great importance. Therefore, they proposed two approaches of branch and bound and Lagrangian for their model.

One of these characteristics of the HLPs is the hierarchical structure proposed by Yaman [18], Alumur et al. [10] and Korani and Sahraeian [13]. However, multimodal transportation is another important characteristic that has been taken today beyond hub problems in the field of macro transportation. Competition in the field of accelerated services is one of the key indicators of global trade, led to inefficiencies in specialized and individual transportation [19]. Therefore, emphasis on multimodal transportation in hub has also been important as can be observed in recent research including Chen et al. [20], Alumur et al. [14], Onyemechi [21], SteadieSeifi et al. [22], Hanasusanto et al. [23], Ambrosino and Sciomachen [24], Fazayeli et al. [25] and Huang et al. [26]. However, simultaneous emphasis to these three characteristics causes new contingencies that should be considered in designing the model.

Given the complexity and time-consuming nature of the solving process of various problems in the real world, researchers have tried to overcome these complexities by creating new methods. One of these methods is the Artificial Neural Network (ANN), which is more useful than many models, and this is because of their higher level of speed and accuracy [27–29]. By the motivation of developing an alternative model that allows for a quick and reliable estimation of site overhead costs, Leśniak and Juszczyk [29] suggested an ANN-based regression model which was able to predict the site overhead cost index.

A new method for quick and accurate estimation and forecasting iron oxide (FeO) concentration in steel slag during discharge process in steelworks was also proposed utilizing two key tools of the ANN and infrared imaging [30]. This method has attempted to estimate and predict the value of material physical parameters based on the changes in the radiative parameters, which is most relevant to the amount of FeO in steel production.

An ANN model based on the input parameters of random data and strain values obtained from the static test was proposed to predict the static load applied on the wing rib [31]. The performance of their proposed ANN model has been also evaluated in predicting static load to a degree of accuracy.

Continuous and permanent monitoring of Particulate Matter (PM) at subway station is of particular importance for assessing the exposure level of PM to passengers. In this way, PM variations were predicted using ANN for 6 subway stations in Seoul. This ANN model has three inputs of outdoor PM10 concentrations, ventilation rates, and subway frequencies in the input layer, a hidden layer and an output layer [32].

Optimizing the design process of asphalt mix in the road construction industry is one of the interesting uses of ANN which used by Sebaaly et al. [33] to optimize the asphalt mix design process using past process database suggesting a mixed method of ANN and Genetic Algorithm, so that their results were consistent with the applicable specification requirements.

ANN was also used in a heuristic effort to estimate the operatinal parameters of the counter-rotating wind turbine in which, neural networks played an important role in enabling the optimization of the dual-rotor turbine parameters [34].

An ANN model was used to model air void content in an aggregate mixture, which was designed for feed-forward type using the error Back-Propagation Algorithm (BPA) [35].

ANN has also been used to predict gold price fluctuations, so that Kristjanpoller and Minutolo [36] presented a hybrid model of ANN and the generalized autoregressive conditional hetero-skedasticity (GARCH), which also analyzed the price fluctuations of the gold spot price and future price. This mixed model dramatically contributed to predict the process in order to determine the impact of financial variables on gold prices. ANN has also been considered in the field of agricultural industry, so that it has been used to predict the amount of sesame seed production. The prediction in this process was utilized two models of ANN and Multiple Regression Models (MLRs) that eventually, ANN prediction outcomes were far better than the MLR results based on the comparison results [37].

There are many instances of ANN application in literature; however the lack of the Location-Allocation Problems (LAPs) is so clear in literature. Hence, it is attempted in this paper, to implement ANN in the HLPs (as one of the LAPs subsets).

HLPs fall into the NP-Hard problems group [38], and increased reliability aspects of these problems causes doubling their complexity, and reliable HLPs can be surely placed in this group. So, the Lagrangian relaxation solution is used in the problem solving process. Accordingly, different algorithms and methods for these problems spend considerable time to solution to provide final result and problem solution. For instance, sometimes it is needed to compare the solution and select it from a set of solutions of the problem to select the final solution so that each solution is obtained by changing the problem parameters and re-executing the solving method, which requires considerable time. In this paper, we attempted to provide a linear modeling of Multimodal Hierarchical Hub Location Problem (MHHLP), utilizing ANN that provides a set of the problem different solutions for different values of parameters with a minimum error rate and in the least possible time.

Therefore, the most important contribution of this paper is designing and developing an ANN linear model to predict the objective function of problem, in addition to provide a multimodal hierarchical hub location model and developing the lagrangian relaxation algorithm for its solution process.

The reminder of the paper is structured as follows according to the well-known patterns in the literature: the problem statement and the mathematical model are presented in Section 2. The steps of solving the lagrangian relaxation are presented in Section 3. Section 4 describes the design stages of the ANN model. Section 5 describes the computational results and MHHLP route reliability prediction stages. Finally, conclusion and research suggestions are provided in Section 6.

As mentioned earlier, the MHHLP research problem is one of the specific problems and subsets of LAPs. Therefore, before designing the mathematical model of the problem, first, its assumptions will be described to determine how the model will be formed. Then various indices and sets are introduced, after which the parameters and variables of the problem are presented. Finally, the integer programming model is proposed.

There are various approaches based on mathematics to simplify the model and relax it from the variables and constraints that make the problem difficult to solve, including the Lagrangian relaxation method which is developed for the MHHLP model in the present paper. This methodology has been developed for several models in the HLPs literature including Aykin [43], Lee et al. [44], Marín [45], Contreras et al. [46], Ishfaq and Sox [47], Mohammadi et al. [48], Karimi and Setak [11], He et al. [49] and Neamatian Monemi et al. [5]. According to the literature, the problem constraints was identified, and it was observed that the relationship between the two decision variables and dramatically affect the time to solution which was observed in Constraint (13) and Constraint (15). Hence, two sets of Lagrange multipliers and were defined for the Constraint (13) and Constraint (15), respectively ( and ). With the help of Lagrange multipliers, the relaxed problem as LR was defined as Eq. (22).

(22)

Such that. (2)-(12),(14),(16) and (21)

The problem is a formulation based on the Lagrangian relaxation approach, resulting in a high bound for MHHLP. Therefore, this approach is an attempt to find the best bound for the problem to match or close the optimal solution. To create this improvement process, the proposed method of Neamatian Monemi et al. [5] was followed by the sub gradient optimization method. In this method, a six-step algorithm was developed in the form of Fig. 2, in which the signs and indexes were used, as follows:

LB: lower bound.

UB: upper bound.

iter: iteration index.

: a random response resulting from solving the problem relaxed from hard constraint clause.

: the output of lagrangian relaxation for the objective function in iter repeat.

: sub gradients of Constraint (13) in the iterth iteration.

: sub gradients of Constraint (15) in the iterth iteration.

: the movement step in the iterth iteration.

: the multiplicative which is used as a decreasing factor in each iteration.

In the next section, it is attempted to examine the inequity and proposed method performance, individually and in combination of all available modes with the help of the standard data in the research literature, so that the best combination can be achieved in order to reach the best possible response in the shortest possible time.

An information processing system is used in order to predict and estimate the route reliability, or the same value of the MHHLP objective function, which is based on a large number of super-integrated processing elements called neurons that work together to solve a problem and transmit information by synopses (electromagnetic communications); this system is called the ANN in literature and neural biology, which main idea is inspired by the structure of human brain activity [50]. In the last decade, ANN has been used as an effective tool to estimate problems, and given that ANN can be defined as a mathematical system and the neuron is interpreted as a processing unit, network architecture is created helping organizing neurons, many of which are designed in the literature; however, the most popular of them is the Multi-Layer Perceptron (MLP) networks, which are among the Feed Forward Neural Networks type [51].

These networks have input, hidden, and output layers in their topology [28], [51].

The neurons are placed on the layers with an activation function and each neuron is connected to the neurons in the next layer with the weighted connections.

The ANN weights are trained using a form of error feedback, which can be viewed as a generalization of the Learning Management System (LMS) [28].

An error BPA is used to train the network to correct these weights in the various layers of the MLP.

The complexity of the connection network is directly related to the number of layers and neurons of each layer, whose accurate selection has a significant impact on achieving the best possible responses at the best possible time. Therefore, the neural network sensitivity was analyzed based on different numbers of hidden layers, the number of neurons, activation functions, etc. in order to achieve the best topology for MHHLP design, The obtained result is the proposed MLP model structure with four inputs, a hidden layer and an output that is mapped to the CAB dataset in Fig. 3 and is exactly identical to the IAD dataset, however, 3 neurons are defined instead of the 4 neurons in the hidden layer.

In the context of a feed-forward network, a hidden layer neuron is a Pre-synaptic Neuron (PN) because it is connected to other neurons in the network which are post-synaptic in relation to its output. In feed-forward networks, neurons form layers, or slabs, which are connected to and from post- and pre- synaptic neurons. Such groupings of neurons that are themselves connected to other neurons are sometimes (traditionally) called hidden layers or hidden layer neurons1.

Input data is composed of problem parameters (i.e. the number of nodes, the number of hubs, hubbing factor of time and hubbing factor of reliability) and the results of the Lagrangian relaxtion method choosed as output data for training for the ANN model. The Levenberg–Marquardt (LM) algorithm used for the training rule of network. These data were divided into two sets: training set (eighty percent of data) and test set (twenty percent of data). The MATLAB R2013a 8.1.0.604 software used for the training of the ANN model. The implementing phases of the ANN model with BPA presents in the Fig. 4. The planned ANN model specifications have been presented in Table 2 for each data sets of IAD and CAB.

We use tangent sigmoid transfer function tansig function as activation function in MLP models. This function is wellknown in literature [28]. The formulation of Tansig defines in Eq. (23) as follow:

(23)

http://standoutpublishing.com/g/hidden-layer.html

The present paper uses two well-known dataset in the HLPs literature namely the Civil Aeronautics Board (CAB) and the Iranian Aviation Dataset (IAD). O'kelly [52] introduced the CAB data on the basis of the transport network of the 25 states of the United States.

Karimi and Bashiri [53] studied transportation network in 37 cities of Iran and provided the collected data in the form of IAD. These two data sets do not provide the reliability rate of the routes. Therefore, the reliability of data was adopted from Kim and O'kelly [17] and An et al. [4]. Also, the reciprocating success chance of each return route was assumed equal and, by were placed at a distance of (0,1) by normalizing the reliability data results. The reliability and travel time in the ground mode are reduced by 50% compared to air mode through calculations, considering the capability and ability of air mode to ground mode transportation. Tehran (the capital city of Iran) was selected as the central hub in the IAD dataset. Placing the only international airport in this city and possessing the largest number of population were the reasons for selecting this city (as the research Alumur et al. [10]). We selected the node 17 with the largest volume of communications (1447732) as the central hub for CAB data in a heuristic attempt regarding the dataset. The β value is obtained from the average of the problem feasible modes for its various values in the rigorous mode, which is similar to the decision-making process used by Yaman [18], Alumur et al. [14] and Korani Sahraeian [13] to determine β. Therefore, this amount was obtained 2640 and 2880, for the MHHLP structure for CAB and IAD data, respectively.

Three Objective Factor (ObFu), Response Improvement Rate (ReIm) and hub facilities (Hub) were considered to evaluate the performance of the Lagrangian approach. ReIm is obtained from the difference between the pure relaxed response and the lower bound of Lagrangian relaxation on the pure relaxation response.

The numerical initial value is considered constant considering the dimensions of the problem. For the subsets 5, 10, 15 and 20, this value was 9, and for the subsets of 25, 30, and 37 nodes, it was considered 90. The results of the Lagrangian relaxation method for IAD and CAB data are presented in Table 3 and Table 4, respectively.

To demonstrate the impressionability of the ObFu and step size changes process from , Fig. 5 is plotted for the 30-node set of IAD data, so that PH = 7, = 0.8 and = 0.8 α, by GAMS software outputs. Fig. 5 shows that the accuracy of the calculation is greater by increasing the discount factor in large data and the changes curve is more concave, and vice versa, the change slope of step size changes shifted from line shape to stairs, which guarantees achieving the better response.

Table 5 and Table 6 show the obtained results for the proposed ANN model for IAD and CAB datasets, respectively. So that, the Eqs. (24)-(26) are used to calculating the mean relative error percentage (MRE %), the Root Means Square Error (RMSE) and the mean absolute error percentage (MAE %) of the network, respectively.

(24)

(25)

(26)

All of the symbols and indexes are used in the Eqs. (24)-(26), define in Table 7.

Several numbers of neurons have been applied for the proposed ANN architectures, which best structure is selected. The results of the applied networks with different number of neurons in the hidden layer are illustrated in Fig. 6. This figure shows the MAE errors of the networks versus different number of neurons in the hidden layer. Each point in these figure represent the network with best MAE results during 200 times. These MAE errors are reported according to the normalized data.

According to Fig. 6, the numbers of 3 and 4 neurons, which have the best MAE error values, are considered for the hidden layers of IAD and CAB models, respectively. In the proposed networks, 80 percent of data set is used for training, while 20 percent of the data set is used for testing process of the presented model. So, we suggested ANN model for both datasets then train and test for everyone. Objective function actual and predicted values comparison for the IAD data, using the proposed MLP model for IAD is illustrated in Fig. 7 and Fig. 8. The objective function actual and predicted data for training and testing process for IAD data are listed in Table 8 and Table 9, respectively. Also, the conformity actual and predicted values for the IAD data using the proposed MLP model for IAD are illustrated in Fig. 8.

As can be seen from Fig. 7 and Fig. 8 and Table 8 and Table 9, the test and train data are predicted precisely and the predicted values of output MHHLP reliability by ANN model is the instances of near to the Objective function actual results, undoubtedly.

The outcomes endorse the accuracy and applicability of estimation of ANN as a reliable model to predict the proposed MLPs output MHHLP reliability from the number of nodes, the number of hubs, hubbing factor of time and hubbing factor of reliability validate by results.

Another model is propped for the CAB data. Actual and predicted values comparison for the CAB data, using the proposed MLP model for CAB is illustrated in Fig. 9 and Fig. 10. The objective function actual and predicted data for training and testing process for CAB data are listed in Table 10 and Table 11, respectively. As can be seen, the test and train data are predicted precisely.

Also, the conformity actual and predicted values for the CAB data using the proposed MLP model for CAB are illustrated in Fig. 10.

The most important benefits of the proposed ANN model are the ease of use, present output at the best time and the quality of the solution. Accordingly, the MHHLP reliability for the IAD with N=37 and CAB with N=25 was predicted that the different values of the three inputs PH, 1- ɤ and α. Hence, the following data are predicted by the proposed IAD and CAB MLP models which are listed in Table 12 and Table 13, respectively. These results have been calculated in best time vs solution time other methods.

The gap is determined in this paper through examining the literature on HLPs and a new model called MHHLP was introduced, which included three multimodal transport, hierarchical structure and reliability characteristics, simultaneously. The contribution of the proposed model is in terms of its constraints and problem variables design and that, the different routes of network are identified and formulated based on the demand flow in the network in defining its equations. Considering the problem placement in the range of NP-Hard problems, the Lagrangian Relaxation Algorithm was developed for it and the results were obtained, accordingly. Finally, the data gathered by the Multilayer Perceptron (MLP) neural network were transformed into a model.

Therefore, this paper has been presented a new feedforward method for estimating the reliability of MHHLP. For validation, the output reliability was predicted using proposed ANN model for the MHHLP. This proposed method is easily estimated the reliability of the MHHLP and has been validated by ANN model. Hence, the presented ANN model can be applied instead of the MHHLP model, for other implementations.

The author would like to express sincere gratitude to all those who contributed to this research, including colleagues and mentors for their valuable insights and constructive feedback.

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

The data used in this study, including the CAB and IAD datasets, are publicly available and have been obtained from previously published sources. Any additional data or computational models generated during this research can be made available upon reasonable request to the corresponding author.