Keywords

1 Introduction

Computer science is a transversal discipline for many areas of study and human activities. One of them is the construction area. Many constructive designs are based on models that predict how the behavior of the design will be, in order to avoid risks or unexpected results. The more data representative of reality is able to capture the model, it will be possible to generate information to make better decisions. However, the quality of the information depends on the quality of the data and the accuracy in them is a fundamental requirement. This is where computing comes into play, by processing data more accurately and quickly, delivering timely, useful and error-free information.

A bridge is a structure built with the purpose of saving a geographical accident, road, water course or other obstacle that comes before it. Its design depends on its function and the nature of the terrain on which it rests. They are usually built on the basis of metallic structures, reinforced concrete, wood or a combination of these. Basically, the forms that the bridges adopt are three, being directly related to the efforts that support their constructive elements. These are: beam bridges, arches and pendants. The present paper will address the reinforcement of a beam bridge through an arch with tension hangers.

Every structure created by the human being has a useful life. In the case of bridges vary depending on various causes. In the case of beam bridges, the undermining of the piers is one of the most frequent causes and damage to these structures can be very serious, such as collapse. Repairing them has a high impact on cost, time and use, so a suitable solution alternative is one that has the least impact. The design of a cable-stayed bridge with a lower deck is a viable solution. This consists of an arc that covers the longitudinal extension of the bridge from which hangings hang anchored to the board, supporting the weight of the structure, in this way it is possible to do without the piers where it rested.

This paper is organized in the following way, in Sect. 2 the problem to be solved is presented, in Sect. 3 Moth Search algorithm is explained, in Sect. 4 the integration of metaheuristics to the bridges problem, and in Sect. 5 the results obtained with their respective statistical tests to evaluate performance, ending with the conclusion and future work.

2 Problem

The problem to solve was proposed by Valenzuela [10,11,12] using Algorithm Genetic [4] and then by Black Hole [3] in order to determine which of the algorithms achieved better results. The best of both turned out to be Black Hole. Now we will use Moth Search (MS) [13] to determine if it achieves better results than Black Hole [7].

The structure of the bridge is modeled in a CAD application called SAP2000 [1], which provides an API that allows it to be operated externally by our algorithm. Thanks to this, the application will perform many complex structural calculations, so that our algorithm will provide the necessary data to perform these calculations and, thus, obtain a solution.

2.1 Objective Function

The objective function is defined as the sum of the tense difference between the original and the modified bridge for each of K cuts in each of the beams, minimizing this difference.

$$\begin{aligned} min\sum _{i=1}^{2}\sum _{k=1}^{K}\left| \sigma o_{i,k}-\sigma m_{i,k}\right| ;\ i\in \left\{ 1,2\right\} ,\ k\in \left\{ 1,2,\ldots ,k\right\} \end{aligned}$$
(1)

Minimize the difference between the stresses of the original bridge and the modified arch bridge of the hangers. As long as the optimum result tends to zero, it means that the stress calculations will tend to equal the stresses of the models of both bridges (original and modified), preserving the original properties of the bridge. The tensions are obtained through the following formulas.

$$\begin{aligned} \sigma {o}_{i,k}=\frac{Mo_{i,k\ \cdot {\ v}_i}}{Io_i};\ \ \ \ \ \ i\in \left\{ 1,2\right\} ,\ k\in \left\{ 1,2,\ldots ,k\right\} \ \end{aligned}$$
(2)
$$\begin{aligned} {\sigma m}_{i,k}=\frac{P}{A}+\frac{P\cdot e\cdot v_i}{Im_{TOTAL}}+\frac{Mm_{i,k\ }\cdot v_i}{Im_i}\ ;\ \ \ \ \ \ \ i\in \left\{ 1,2\right\} ,\ k\in \left\{ 1,2,\ldots ,k\right\} \end{aligned}$$
(3)

2.2 Decision Variable

For a bridge of three hanging will be 6 decision variables, three to indicate the order of tension and the other three to determine the magnitude of tension of each hanger.

$$\begin{aligned} {ord}_1,{ord}_2,\ldots ,{ord}_n\ \ \in \{1,2,\ldots ,n\} \ \ \ (Orders) \end{aligned}$$
(4)
$$\begin{aligned} {mag}_1,{mag}_2,\ldots ,{mag}_n\ \in \ [T_{min},\ T_{max}\ ] \ \ \ (Magnitudes) \end{aligned}$$
(5)

2.3 Constraints

The problem have two constraints that must be met to satisfy the objective function.

  • The hangers cannot be jacking simultaneously.

    $$\begin{aligned} {ord}_w\ne ord_j\ ;\ \forall w,j\ \ \ \ con\ w\ne j\ \ \ \ \ \ w,j\in \left\{ 1,2,\ldots ,n\right\} \ \end{aligned}$$
    (6)

  • The effort of the modified bridge deck should not pass the limits of the BAM:

    $$\begin{aligned} \sigma m_i,_k \ge \sigma o \end{aligned}$$
    (7)
    $$\begin{aligned} \sigma m_i,_k \ge f_{ct} \end{aligned}$$
    (8)
    $$\begin{aligned} \sigma m_i,_k \le f_{cmax2} \ \ \ (in\ internmediate\ stages) \end{aligned}$$
    (9)
    $$\begin{aligned} \sigma m_i,_k \le f_{cmax} \ \ \ (in\ final\ stages) \end{aligned}$$
    (10)

  • \( \sigma m_i,_k\): Tension (top or bottom) in the beams of the cable-stayed bridge.

  • \( f_{ct}\): Maximum tension of admissible traction by the concrete.

  • \( f_{cmax}\): Maximum compressive stress admissible by the concrete.

  • \( f_{cmax2}\): Maximum compressive stress admissible for expanded concrete.

3 Moth Search

Moth Search is a bio-inspired metaheuristic algorithm to respond to global optimization problems. It was created by Wang [13].

Moths are a type of insects that belong to the order Lepidoptera. Among the various characteristics of moths, phototaxis and Levy flights are the most representative characteristics described below.

Phototaxis is the orientation reaction of free cellular organisms in response to a luminous stimulus. In general, moths tend to fly around the light source, in the form of Lévy flights.

3.1 Lévy Flights

Lévy flights are one of the most important flight patterns in nature. Many species move following a flight pattern of Lévy and describe a type of random walks, whose length steps are taken from the distribution of Lévy. The distribution of Lévy can be expressed mathematically in the form of a power law formula.

$$\begin{aligned} L(s) \sim \left| {s}\right| ^{- \beta } \end{aligned}$$
(11)

where \(1< \beta <= 3\)

Lévy flights can maximize the efficiency of finding resources in uncertain environments. Therefore, \(\beta \) = 1.5 is used to optimize benchmarks and engineering cases.

The moths, which have a smaller distance than the best, will fly around it in the form of Lévy flights. In other words, their positions are updated by Lévy flights.

For moth i in each variable, it can be updated as:

$$\begin{aligned} x^{t+1}_{i} = x^{t}_{i} + \alpha L(S) \end{aligned}$$
(12)
$$\begin{aligned} x^{t+1}_{i} = x^{t}_{i} + \alpha (\frac{(\beta - 1) \varGamma (\beta - 1) sin(\frac{\pi (\beta - 1)}{2})}{\pi S^\beta }) \end{aligned}$$
(13)

Where \(x_i ^ {t + 1}\) and \(x_i ^ t\) are respectively the updated original position in generation t, and t is the current generation. L(s) is the passage of Lévy flights. The parameter is the scale factor related to the problem of interest. In our current work, it can be given as:

$$\begin{aligned} \alpha = \frac{S_{max}}{t^2} \end{aligned}$$
(14)

Where \(S_{max}\) is the maximum walking step and its value is established according to the given problem. Lévy distribution L(s) in equation can be formulated as follows:

$$\begin{aligned} L(s) = \frac{(\beta - 1) \varGamma (\beta - 1) sin(\frac{\pi (\beta - 1)}{2})}{\pi S^\beta } \end{aligned}$$
(15)

Where s is greater than 0, \(\varGamma (s)\) it is the gamma function.

3.2 Straight Flight

Certain moths that are far from the source of light will fly towards that source of light in line. This process can be described below.

For moth i in each variable, its flights can be formulated as:

$$\begin{aligned} x_i^{t+1} = \lambda (x_i^t+ \phi (x_{best}^t- x_i^t )) \end{aligned}$$
(16)

where \(x_{best} ^ t\) is the best moth in generation t, and \(\phi \) is an acceleration factor established in golden relation in our current work. \(\lambda \) is a scale factor. On the other hand, the moth can fly towards the final position that is beyond the light source. For this case, the final position for moth i can be formulated as:

$$\begin{aligned} x_i^{t+1} = \lambda (x_i^t+ \frac{1}{\phi } (x_{best}^t- x_i^t )) \end{aligned}$$
(17)

For simplicity, for moth i, its position will be updated by Eqs. 16 or 17 with the possibility of 50%. In addition, these two update processes mentioned above can be represented in (Fig. 1a, b) respectively. In Fig. 1, \(x_{best}\), \(x_i\) and \(x_{i,new}\) are respectively the best, original and updated position for moth i, and are considered as a light source, start point and end point. \(\lambda \) is a scale factor that can control the speed of convergence of the algorithm and improve the diversity of the population. In our current work, the scale factor is set to a random number drawn by the standard uniform distribution.

4 Integration

In the MS method, for simplicity, the entire population of moths is divided into two equal subpopulations (Subpopulation 1 and Subpopulation 2) according to their suitability, and they are updated according to the Lévy flights or in a straight line, respectively. This is equivalent to saying that the moths in Subpopulation 1 are much closer to the better than Subpopulation 2. In addition, like many other metaheuristic algorithms, an elitism strategy is incorporated in order to accelerate the convergence of the MS method. MaxIterations is the initial maximum generation that can be considered as the term condition.Algorithm 1 describes this process.

In synthesis, two models will be used: one of the original bridge and another with the modifications to compare it with the first one. Therefore, the metaheuristic will provide the voltage magnitudes for SAP2000 to perform and thus obtain a solution. The communication between metaheuristics and SAP2000 will be made through the API provided by the latter.

figure a

The configuration was parameters used for the execution, it was obtained through parametric sweep are shown in the Table 1 [5].

Table 1. Parameters for execution.
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5 Experimental Results

The Tables 2, 3 and 4 summarize the best solutions achieved in each of the 15 executions for instance.

Table 2. Fitness comparation (a)
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Table 3. Fitness comparation (b)
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Table 4. Fitness comparation (c)
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5.1 Comparing Results

To determine which algorithm achieves the best results, the best solutions of the 15 executions made to each instance have been collected.

This allows obtaining 2 sets of data to be compared for each instance: those obtained with BH and those obtained with MS. These two samples are subjected to two statistical tests. The first allows to determine if both samples are independent, which would clear the doubt if the samples are reliable to compare, and the second to verify the veracity of the hypothesis formulated with respect to the best result obtained.

The Kolmogorov-Smirnov test, with Lilliefors correction [6], is used to test if a data set fits a normal distribution or not, in our the test concluded that the samples are independent.

Then, the Mann-Whitney-Wilcoxon test [2] will be applied to the same group of data to determine the veracity of the hypothesis with respect to which algorithm of the comparators presents the best results, based on the following hypotheses:

  • \(H_0:\) MS is better than BH

  • \(H_1:\) BH is better tha MS

If the p-value of a hypothesis of one sample with respect to the other is less than 0.05, it can not be assumed to be true.

When applying this test, the results presented in Table 5 were obtained.

Table 5. p-value Mann-Whitney-Wilcoxon test
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5.2 Distribution Comparison

The best solutions obtained in each execution are presented in the following graphs, where you can see a comparison between Black Hole and Moth Search. The boxes of blue color represent the solutions given by BH, the red ones correspond to those of MS.

Fig. 1.
figure 1

Best fitness instances

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In all situations, the results obtained by MS do not surpass BH. This is due to a number of insufficient iterations, a small population and parameter settings that allow greater precision (progress at a smaller step).

6 Conclusions

The use of optimization techniques, present great advantages when solving problems of great complexity [8, 9]. The solutions given by Moth Search in the vast majority of cases do not exceed those achieved by Black Hole. However, in some situations the proximity has been close, so it is possible to infer that by making the necessary adjustments to the algorithm, results such as Black Hole can be achieved.

In two of the eleven instances it is not concluded that Moth Search is unable to achieve results like those of Black Hole, which would lead to promising results if improvements are applied to the algorithm and execution parameters.

In conclusion, it is proposed to make improvements in the parameters such as Population Size, Number of Iterations, Acceleration Factors, Disturbance Operators and in the algorithm as Elitist Strategies, Convergence Acceleration Strategies, Selective Population, Population Grouping, others.