A new approach for fire and non-fire aerosols discrimination based on multilayer perceptron trained by modified bonobo optimizer

Abstract

Fire smoke detection plays a pivotal role in our life. For the optical fire smoke detector, it is easy to produce false alarms due to the misidentification of non-fire aerosols, such as dust and water mist. In this research, four multilayer perceptron (MLP) models are established and applied to discriminate the fire smokes and non-fire aerosols. A new hybrid algorithm (HBOBES) is proposed to optimize the parameters of MLP, which incorporates three search phases, including space selecting, promiscuous and restrictive mating, and consortship and extra-group mating, which are derived from the basic bonobo optimizer (BO) and bald eagle search algorithm (BES). Moreover, a adaptive tent chaos mapping technique is introduced into the first two phases to increase the population diversity. In the experiments, eight standard classification datasets and sixteen self-established aerosol classification datasets are applied to evaluate HBOBES’s optimization performance on training the MLP models. The results show that the proposed HBOBES ranks first overall, showing the merits in training MLP models compared to eight other algorithms. For the aerosol classification problem, it is found that both the optimized 3-7-2 and 3-7-3 MLP models have achieved the highest classification accuracy of approximately 95%. Black smokes can be identified with 100% classification accuracy. Therefore, this paper provides an effective and feasible approach to distinguish fire and non-fire aerosols by using the MLP classifier optimized by optimization algorithm, which has practical significance for the development of intelligent optical smoke detector.

1 Introduction

Early fire detection is vital to protect people’s lives and property. Up to now, various sensors have been developed to provide effective fire warning, including heat sensors [1], gas sensors [2], flame sensors [3], smoke sensors [4], graphene oxide based sensors [5] and so on. Among them, the smoke sensors (or called smoke detectors) are the most common one, which are effective for early fire alarm. According to the working principles of the smoke detectors, it can be mainly divided into three types: photoelectric type [6], ionization type [7] and video type [8, 9]. Among these different types of smoke detectors, the photoelectric type has the merits of environmental protection, simple structure and low cost. Thus it is widely used in the household, industries, electronic equipment rooms and other places [10]. With the widespread use of photoelectric smoke detectors, the industry’s requirements are becoming more and more stringent. In the 2013 Edition of NFPA 72, household smoke detectors are required to resist common nuisance sources, especially the residential cooking nuisances [10]. The Society of Automotive Engineers (SAE) published the Aerospace Standard (AS) 8036 REV [11], which requires that the smoke detectors used in aircraft cargo compartments, galleys, electronic equipment bays and other similar installations should have adequate immunity ability to false alarms.

For the smoke detectors, high detection sensitivity and low false alarm rate are the most wanted. However, these two goals seem to be contradictory in a certain degree. Early and highly sensitive smoke detection requirements will also make smoke detectors more sensitive to non-fire aerosols and thus easier to produce false alarms. For instance, in application of the aircraft cargo compartment, fire smoke detection with high accurate and sensitivity are crucial to the flight safety. The visual instructions to the flight crew are required within 60 s for the fire detection system [12]. As a result, the fire smoke detection system in the aircraft cargo compartment suffer from the problem of high false alarm rate due to various nuisance sources (such as, dust, water mist, fiber, villus, etc.), which come from the cargoes or are generated under the variable-pressure variable-oxygen environment [13].

Since the invention of scattering type smoke detectors, there are two major problems that need to be addressed, i.e., nuisance alarms triggered by non-fire aerosols and unbalanced response between black smokes and white smokes [14]. Traditional photoelectric type smoke detectors are not able to discriminate fire smokes and other non-fire aerosols due to the principle of particle light scattering [15]. Moreover, the black smokes and white smokes have significantly different refractive indices, resulting in a large response difference between them. With regard to the optical smoke detectors, low sensitivity to black smokes may bring missing alarm or delayed alarm problem. According to the statistical results of Duisburg Fire Brigade [16], dusts and water steam lead to 9.3% and 9.8% of false alarms, respectively. The cost caused by false alarms cannot be ignored and has aroused great attention in various industries and fields. The unwanted false alarms resulted by non-fire aerosols and unbalanced response have become the urgent problems to be solve.

In this works, the multilayer perceptron (MLP), which is one type of the feed-forward ANNs [17], is applied for the discrimination of fire and non-fire aerosols. The datasets of various aerosols are created according to our previous works [14]. An improved hybrid algorithm (HBOBES) base on the bonobo optimizer (BO) [18] and bald eagle search algorithm (BES) [19] is proposed to optimize the parameters of MLP models.

There are two motivations for this work. First, there is an urgent need of accurate identification method of fire smoke and non-fire aerosol for the optical fire smoke detectors. The false alarm rate of the smoke detector will be effectively reduced and the balanced response of black and white smoke in the fire will also be achieved if the fire smoke types and nuisance aerosols (such as dusts, water mist) can be accurately identified. The second is that both the BO and BES algorithms have shown good performance on training MLP models and can obtain good solutions on the aerosol classification problems. In BO, the fission-fusion pattern and reproductive strategies shows outstanding exploitation capability. For the BES, the selecting stage can help to prevent local optima. According to the No Free Lunch (NFL) theorem [20], although many optimization algorithms have been proposed to solve various optimization problems, new optimization techniques are always required to be developed for solving emerging optimization tasks. Thus it is necessary to develop advanced and effective optimization algorithm for the aerosol classification problem in this paper. By proposing the HBOBES, we can synthesize the advantages of BO and BES to obtain better optimization algorithm, which is conducive to train the MLP more efficiently and provide a new insight to develop low false alarm rate and balanced fire smoke detection method. To sum up, the main contributions of this paper are summarized as follows:

• 1. A new hybrid algorithm HBOBES is proposed to optimized the weights and biases of MLP models, which combines the space selecting stage of BES and promiscuous and restrictive mating strategy and consortship and extra-group mating strategy of BO.

• 2. A adaptive tent chaos mapping technique is employed to improve the population diversity of HBOBES.

• 3. Sixteen aerosol classification datasets including four fire smokes and four non-fire aerosols in each dataset are established.

• 4. Four MLP models with different structures are investigated for solving the aerosol classification problem.

• 5. Compared to eight other well-known algorithms, HBOBES shows the superior performance on training the MLP models for solving the standard and aerosol classification datasets.

The rest of this paper is organized as follows: the related work of MLP is introduced in Sect. 2. The basic principles of BO and BES algorithms are introduced in Sect. 3. Then the proposed hybrid algorithm HBOBES is described in Sect. 4. In Sect. 5, the optimization performance of HBOBES on standard classification datasets is analyzed. In Sect. 6, the aerosol classification problem is presented and the results are discussed in detail. Finally, Sect. 7 gives the conclusions and provides future directions of this paper.

2 Related work

2.1 Traditional method

Philipp et al. [21] pointed out that the ratio between two different wavelengths reveals the information on the aerosol particle size, and the ratio between two different scattering angles indicates the information on the aerosol particle shape and color. A three-channel detection method was proposed for identifying fire and non-fire aerosols. To identify the types of fire smokes, Meyer et al. [22] proposed a moment method based on particle size distribution in the microgravity environment of the international space station. Wang et al. [6] proposed a multi-parameter aerosol identification method based on Baron’s ‘Three regions’ law. The Sauter mean diameter (SMD) of aerosol particles was used to distinguish large size particles, such as dust particles and water droplets. A four-channel scattered light detection method was further designed to address the unbalanced response of smoke detectors to black smokes and white smokes [23]. Schultze and Marcius et al. [24, 25] proposed a discrimination method of non-fire aerosols using the fog rainbow effect of water droplets and the depolarization effect of dust particles. In our previous works, a discrimination and identification method was also developed based on the asymmetry ratio of particle scattered light [14, 26]. By using multiple polarized or unpolarized scattered light asymmetry ratios, white smokes, black smokes, dusts and water droplet aerosols can be discriminated to a certain extent. However, the false alarm problem of smoke detectors has not been effectively solved when applied in some complex and harsh environments (such as the aircraft cargo compartment) [27], and advanced technical methods are urgently needed.

2.2 Artificial neural network technology

The accurate identification of fire smokes and non-fire aerosols is a feasible method for solving the current problems (i.e., false alarms and unbalanced response) of optical smoke detectors. In recent years, artificial intelligence techniques have achieved remarkable achievements in various fields, such as the disease diagnosis [28], language identification [29], and images recognition [30]. The artificial neural networks (ANNs) provides a promising technique in solving the early, anti-false alarm and balanced fire smoke detection. In the ANN-based smoke detection, Okayama [31] pointed out that neural nets can be used to realize the precise fire discrimination, especially distinguish the non-real fire phenomena. Liu et al. [32] designed a 1-Dimension CNN classifier to avoid false alarms caused by water steam, which can achieve more than 50% chance of successfully to identify the water steam. Qu et al. [33] proposed a multi-parameter method to identify the fire using the combination of RNN, CNN, etc. Ai et al. [34] proposed an improved transformer model to capture fire information. The fire detection accuracy can reach 0.995.

2.3 Optimization method of ANNs

The MLP has unidirectional connections between neurons that can be used to solve the classification problems. The weights and biases of MLP can be optimized by using the metaheuristic algorithms (MAs) and their improved versions [35], such as GWO-MLP [36], SHO-MLP [37], IBOA-MLP [38], and IAMO-MLP [39]. Moreover, MAs also can be used to optimize the structures of MLP [39]. In recent years, many MAs have been developed to solve challenging optimization problems, which can obtain the acceptable optimal estimation while avoiding the local minimum. Among them, population-based algorithms are one of the most popular branches. Hashim et al. proposed a novel nature-inspired metaheuristic algorithm named as snake optimizer (SO) [40], which is inspired from the mating behavior of snakes. Daliri et al. [41] designed a novel metaheuristic called water optimization algorithm (WAO) to solve the continuous optimization problems, which is based on the chemical and physical characteristics of water molecules. Lian et al. [42] developed a parrot optimizer (PO) to tackle the optimization problems in the medical field. Bouaouda et al. [43] proposed a novel optimizer called pied kingfisher optimizer (PKO), which shows superior exploration and exploitation ability in handling the intricate optimization problems. Abdollahzadeh et al. [44] developed the puma optimizer (PO) according to the puma’s natural hunting behaviors and apply it to solve the machine learning problems. Moreover, El-kenawy et al. proposed greylag goose optimization (GGO) algorithm [45], iHow optimization algorithm (iHowOA) [46] and football optimization algorithm (FbOA) [47], which have shown superior performance for solving various optimization problems. For the improved algorithms, Hashim et al. [48] introduced a modified sea horse optimizer (mSHO) by using an innovative local search strategy. The numerical and statistical results confirmed its superior performance on intricate optimization challenges. Abed-alguni et al. [49] proposed an improved salp swarm algorithm (ISSA) that includes four modifications, the Gaussian perturbation, the mixed opposition-based learning, the highly disruptive polynomial mutation method, and the Laplace crossover operator.

3 Background

This section introduces the basic principles of multilayer perceptron (MLP), bonobo optimizer (BO) and bald eagle search (BES) algorithm.

3.1 Multilayer perceptron

Multilayer perceptron (MLP) is one type of the feedforward network (FNN) [50], in which the transmission of information is one-way, i.e., from the input layer to the output layer. A MLP model consists of one input layer, one or more hidden layers and one output layer. In each layer, there are multiple nodes for the computation of transmitted signals. The weights between any two nodes of different layers and biases for nodes except for those in input layer are two key parameters for a given MLP model, which are required to be optimized [38]. In addition, the structures of MLP model can also be adjusted for better performance on practical issues, such as modifying the number of nodes in each layer and the number of layers [39]. The concrete computation procedures of MLP are provided as follows [51].

The first step is to calculate the weighted sums of input signals using the following equation:

$$\begin{aligned} {s_{j}=\sum _{i=1}^{n}\left( W_{ij} \times X_{i}\right) -\theta _{j}, j=1,2, \ldots , h} \end{aligned}$$

(1)

where n and h denotes the number of the input signals and nodes in the hidden layer. $W_{ij}$ is the weight between the i-th node and the j-th node. $X_{i}$ represents the value of input signal. And $\theta _{j}$ is the bias of the j-th node. The next is the calculation of activation function using sigmoid function for the j-th node, which is as follows:

$$\begin{aligned} S_{j}=\operatorname {sigmoid}\left( s_{j}\right) =\frac{1}{\left( 1+\exp \left( -s_{j}\right) \right) }, j=1,2, \ldots , h \end{aligned}$$

(2)

And then the same calculation is performed for the nodes in the next layer, which can be the next hidden layer or output layer. The equations are as follows:

$$\begin{aligned} O_{k}= & \sum _{j=1}^{h}\left( w_{j k} \cdot S_{j}\right) -\theta _{k}{\prime }, k=1,2, \ldots , m \end{aligned}$$

(3)

$$\begin{aligned} O_{k}= & \operatorname {sigmoid}\left( o_{k}\right) =\frac{1}{\left( 1+\exp \left( -o_{k}\right) \right) }, k=1,2, \ldots , m \end{aligned}$$

(4)

where m denotes the number of nodes in the next layer. $\theta _{j}{\prime }$ is the bias of the j-th node.

By using the Eqs. (1)-(4), the correspondence between inputs and outputs can be established. It also can be seen that the performance of MLP depends on the parameter values of weights and biases, which can be optimized by the metaheuristic algorithms. In this case, the variables that need to be optimized are weights and biases. And the objective function can be the average of the mean square error (MSE), which is defined as follows:

$$\begin{aligned} {\overline{MSE}}=\sum _{k=1}^{s} \frac{\sum _{i=1}^{m}\left( o_{i}^{k}-d_{i}^{k}\right) }{s} \end{aligned}$$

(5)

where s is the number of training samples. m is the number of nodes in output layer. $o_{m}^{k}$ is the actual output in the i-th output node for the k-th training sample. And $d_{k}^{k}$ is the correct output in the i-th output node for the k-th training sample. Therefore, the training of MLP model using the metaheuristic methods is namely to find the optimal weights and biases to minimize the average of MSE.

3.2 Bonobo optimizer

Bonobo optimizer (BO) is a powerful optimization technique inspired by the social behaviors and mating strategies of bonobos in nature [18]. The mathematical models of BO are introduced in the following sections.

3.2.1 Fission-fusion social strategy

The first step of BO is the utilization of the fission-fusion social strategy, which is used to select the bonobo for mating with current individual. This selected bonobo (p-th bonobo) is chosen from a relatively small temporary sub-group. At first, the maximum size $tsgs_{max}$ of the sub-group is calculated as follows:

$$\begin{aligned} tsgs_{max}=\operatorname {maximum}\left( 2,\left( tsgs_{factor} \times N\right) \right) \end{aligned}$$

(6)

where $tsgs_{factor}$ denotes the factor of the temporary sub-group, N represents the size of population. The $tsgs_{factor}$ is a dynamic parameter which will be introduced in Subsection 3.2.4, and its initial value is initialized as follows:

$$\begin{aligned} tsgs_{factor\_initial}=0.5\times tsgs_{factor\_max} \end{aligned}$$

(7)

where $tsgs_{factor\_max}$ denotes the maximum value of $tsgs_{factor}$, which is set to 0.07 in the basic BO.

3.2.2 Promiscuous and restrictive mating strategy

A new bonobo will be created by using the promiscuous and restrictive mating strategy when the population of bonobos is in the positive phase (PP). The mathematical model is constructed as follows:

$$\begin{aligned} \begin{aligned} new\_bonobo_{j}&= bonobo_{j}^{i}+r_{1} \times \operatorname {scab} \times \left( \alpha _{j}^{ bonobo } - bonobo _{j}^{i}\right) \\&+\left( 1-r_{1}\right) \times \operatorname {scsb} \times flag \times \left( bonoboo_{j}^{i}-bonoboo _{j}^{p}\right) \end{aligned} \end{aligned}$$

(8)

where $\alpha _{j}^{ bonobo }$, bonobo $_{j}^{i}$, and bonobo $_{j}^{p}$ are the positions in the j-th dimension of the best bonobo, current bonobo, and selected bonobo, respectively. $r_{1}$ is a random number uniformly distributed between 0 and 1. scab and scsb are the sharing coefficients for the $\alpha _{bonobo}$ and the p-th bonobo, respectively. flag is set to -1 or 1. To be specific, when the current bonobo is better than the p-th bonobo, flag will be set as 1 and then the p-th bonobo is re-selected from the temporary sub-group. Otherwise, flag is set as -1.

3.2.3 Consortship and extra-group mating strategy

In the other case, i.e., negative phase (NP), the consortship or extra-group mating strategies will be adopted for the generation of new bonobo. When a random value $r_{2}$ between 0 and 1 is greater than the parameter $p_{xgm}$, the consortship mating strategy will be applied, which is shown below:

$$\begin{aligned} \begin{aligned} new\_bonobo_{j}={\left\{ \begin{array}{ll} bonobo_{j}^{i}+flag \times e^{-r_{3}} \times \left( bonobo_{j}^{i}-bonobo_{j}^{p}\right) , \\ \text{ if }\left( flag=1 \Vert r_{4} \le p_{d}\right) \\ bonobo_{j}^{p}, otherwise \end{array}\right. } \end{aligned} \end{aligned}$$

(9)

where $r_{3}$ and $r_{4}$ are the random numbers in [0,1]. If $r_{2}$ is found to be smaller than or equal to the $p_{xgm}$, the extra-group mating strategy is going to be performed, which is provided below:

$$\begin{aligned} & \left\{ \begin{array}{l} \beta _{1}=e^{\left( r_{5}^{2}+r_{5}-2 / r_{5}\right) } \\ \beta _{2}=e^{\left( -r_{5}^{2}+2 \times r_{5}-2 / r_{5}\right) } \end{array}\right. \end{aligned}$$

(10)

$$\begin{aligned} & new\_bonobo_{j}=\left\{ \begin{array}{c} bonobo_{j}^{i}+\beta _{1} \times \left( b_{j}-bonobo_{j}^{i}\right) , \\ if \left( \alpha _{j}^{ bonobo } \ge bonobo_{j}^{i} and r_{6} \le p_{d}\right) \\ bonobo_{j}^{i}-\beta _{2} \times \left( bonobo_{j}^{i}-lb_{j}\right) , \\ if \left( \alpha _{j}^{ bonobo } \ge bonobo_{j}^{i} and r_{6}>p_{d}\right) \\ bonobo_{j}^{i}-\beta _{1} \times \left( bonobo_{j}^{i}-lb_{j}\right) , \\ if \left( \alpha _{j}^{ bonobo }< bonobo _{j}^{i} and r_{6} \le p_{d}\right) \\ bonobo_{j}^{i}+\beta _{2} \times \left( ub_{j}-bonobo_{j}^{i}\right) , \\ if \left( \alpha _{j}^{ bonobo } < bonobo _{j}^{i} and r_{6}>p_{d}\right) \end{array}\right. \end{aligned}$$

(11)

where $r_{5}$ and $r_{6}$ are the random numbers between 0 and 1. $\beta _{1}$ and $\beta _{2}$ are two intermediate parameters which can indicate the strength of the extra-group mating. $ub_{j}$ and $lb_{j}$ are the upper and lower boundaries in the j-th dimension. $p_{d}$ is the directional probability.

3.2.4 Parameters’ updating

The update of parameters is very important for the BO algorithm, which makes the performance of BO flexible, stable and efficient. According to whether the current iteration finds a new optimal, BO provides two kinds of parameters’ updating forms. If a better $\alpha _{bonobo}$ is found, the parameters will be modified as follows:

$$\begin{aligned} \left\{ \begin{array}{l} npc=0, ppc=ppc+1, cp=minimum(0.5, ppc \times rcpp)\\ p_{xgm}=p_{xgm\_initial}, p_{p}=0.5+cp, p_{d}=p_{p}\\ tsgs_{factor}=minimum \left( tsgs_{factor\_max},\left( tsgs_{factor\_initial}+ppc \times rcpp^{2}\right) \right) \end{array}\right. \end{aligned}$$

(12)

where rcpp denotes the rate of change in positive phase and negative phase. It can be seen that in the positive phase, $p_{p}$ will increase slowly to enhance the exploitation ability of BO, which helps the bonobos find better solution in the next iteration. In another case (i.e., negative phase), better $\alpha _{bonobo}$ is not discovered, these parameters are updated as follows:

$$\begin{aligned} \left\{ \begin{array}{l} ppc=0, npc=npc+1, cp=-( mimimum (0.5, npc \times rcpp))\\ p_{xgm}=minimum \left( 0.5,\left( p_{xgm\_initial}+npc \times rcpp^{2}\right) \right) , p_{p}=0.5+cp\\ p_{d}=p_{p}, tsgs_{factor}=\operatorname {maximum}\left( 0,\left( tsgs_{factor-initial}-npc \times rcpp^{2}\right) \right) \end{array}\right. \end{aligned}$$

(13)

By using the Eq. (13), the exploration ability of BO can be improved due to the small $p_{p}$ (less than 0.5).

3.3 Bald eagle search algorithm

Bald eagle search (BES) algorithm is another optimization method which is developed based on the hunting behaviors of bald eagles [19]. In the basic BES, bald eagles seek the food through three stages: select stage, search stage, and swooping stage. The mathematical models are described below.

3.3.1 Selecting stage

In the selecting stage, each bald eagle selects the area based on the best position (i.e., position of food) and average position, which is formulated as follows:

$$\begin{aligned} P_{i, new}=P_{best}+\alpha \times r_{6} \times \left( P_{mean}-P_{i}\right) \end{aligned}$$

(14)

where $P_{best}$, $P_{mean}$, and $P_{i}$ indicate the location of food, average position, and position of current individual, respectively. $\alpha$ is a constant value between 1.5 and 2, and $r_{6}$ is random value between 0 and 1.

3.3.2 Searching stage

The next step is the searching stage, which is used to search the food around the current individual location, i.e., the previously selected position. The expression is as follows:

$$\begin{aligned} & P_{i, new}=P_{i}+y(i) \times \left( P_{i}-P_{i+1}\right) +x(i) \times \left( P_{i}-P_{mean}\right) \end{aligned}$$

(15)

$$\begin{aligned} & \left\{ \begin{array}{l} x(i)=\frac{xr(i)}{\max (|xr|)}, y(i)=\frac{y r(i)}{\max (|yr|)}\\ xr(i)=r(i) \times \sin (\theta (i)), yr(i)=r(i) \times \cos (\theta (i))\\ \theta (i)=a \times \pi \times r_{7}, r(i)=\theta (i)+R \times r_{8} \end{array}\right. \end{aligned}$$

(16)

where a and R are the parameters lying in [5,10] and [0.5,2]. $r_{7}$ and $r_{8}$ are the random values in [0, 1].

3.3.3 Swooping stage

The final step is the swooping stage. A polar equation is embedded in this process to simulate the swing behavior during hunting. This stage is formulated as follows:

$$\begin{aligned} & P_{i, new}=r_{9} \times P_{best}+x_{1}(i) \times \left( P_{i}-c_{1} \times P_{mean}\right) +y_{1}(i) \nonumber \\ & \quad \times \left( P_{i}-c_{2} \times P_{best}\right) \end{aligned}$$

(17)

$$\begin{aligned} & \left\{ \begin{array}{l} x_{1}(i)=\frac{x r(i)}{\max (|x r|)}, y_{1}(i)=\frac{y r(i)}{\max (|y r|)}\\ x r(i)=r(i) \times \sinh (\theta (i)), y r(i)=r(i) \times \cosh (\theta (i))\\ \theta (i)=a \times \pi \times r_{10}, r(i)=\theta (i) \end{array}\right. \end{aligned}$$

(18)

where $r_{9}$ and $r_{10}$ are random values in [0,1]. $c_{1}$ and $c_{2}$ indicate the movement intensity of swooping, which take between 1 and 2.

4 The proposed HBOBES

4.1 Motivation for proposing HBOBES

In the face of actual complex optimization problems, existing optimization techniques are easy to fall into local optima. For the aerosol classification task of fire smoke detection in this paper, many optimization techniques show poor capability during exploring or exploiting and fail to find the satisfactory optimal solution. According to the preliminary tests, the BO and BES methods have displayed some optimization performance on this problem. However, the exploration and exploitation abilities, convergence speed and accuracy of BO and BES are still need to be further improved to meet the requirements of practical application. Therefore, in this works, a new hybrid approach called HBOBES is developed, which is applied to optimized the parameters of MLP models for solving the classification tasks.

4.2 Architecture of HBOBES

To tackle the shortcomings basic BO and BES, the proposed HBOBES is based on the classical BO and select stage of BES. Like the basic BES, the proposed HBOBES can be divided into three stages, followed by the space selecting, promiscuous and restrictive mating, and consortship and extra-group mating. And the greedy selection is adopted in each stage for the improvement of optimal solution. Moreover, to improve population randomness and distribution uniformity, a adaptive tent chaos mapping technique is embedded into the promiscuous and restrictive mating strategy of BO and the select stage of BES [52]. The specific definition is as follows:

$$\begin{aligned} Atent\_m_{j+1}=\left\{ \begin{array}{ll} 1-r_{11}, & if \left( m_{j}=0\right) \\ 1-2 \times m_{j}, & if \left( 0<m_{j}<0.5\right) \\ 2 \times \left( 1-m_{j}\right) , & if \left( 0.5<m_{j}<1\right) \end{array}\right. \end{aligned}$$

(19)

where $r_{11}$ is a random value between 0 and 1. In the HBOBES, random values $r_{1}$ in Eq. (8) and $r_{6}$ in Eq. (14) are generated by using the Eq. (19), which makes the these two random values more evenly distributed. Therefore, the population diversity of HBOBES is improved.

As shown in Fig. 1, the flowchart of the HBOBES is represented. In the next, the execution process of the proposed HBOBES is further detailed.

Fig. 1

The flowchart of proposed HBOBES

• 1. Step 1: The positions of population are initialized randomly within the search interval. And the parameters $\alpha , scab, scsb, p_{xgm\_initial}, tsgs_{factor\_max}$, and $p_{d}$ are set for the running of algorithm.

• 2. Step 2: Calculate the fitness of all individuals and identify the best one, i.e., $\alpha ^{bonoobo}$.

• 3. Step 3: When the number of iterations t is less than or equal to the maximum number of iterations T, fission-fusion social strategy is applied to determine the p-th individual based on Eq. (6).

• 4. Step 4: all individuals enter the selection phase firstly and update their positions using Eq. (14) and Eq. (19).

• 5. Step 5: The modified promiscuous and restrictive mating strategy based on Eq. (8) and Eq. (19) is performed for each individual.

• 6. Step 6: The consortship and extra-group mating strategy based on Eqs. (9)-(11) is executed for each individual.

• 7. Step 7: Updates related parameters based on whether a better global optimal is found using Eqs. (12)-(13).

• 8. Step 8: Determine whether the termination condition is met. If t is greater than T, then stop the iteration and output the optimal solution. Otherwise, return to the Step 3.

Algorithm 1 gives the pseudo-code of the proposed HBOBES. Lines 6-7 are the fission-fusion social strategy. Lines 9-14 belong to the process of space selecting phase. Lines 16-21 give the promiscuous and restrictive mating phase. And lines 23-33 represent the execute procedures of consortship and extra-group mating strategy.

4.3 The computational complexity of HBOBES

The computational complexity of the optimization techniques depends on the processes of initialization, evaluation of fitness function, and updating of the positions. In the initialization phase of the proposed HBOBES, the computational complexity is O(N), where N is the size of the population. The computational complexity of the fitness evaluation relies on the specific problems, which is not considered here. And then the computational complexity in each phase (space selecting phase, promiscuous and restrictive mating phase, and consortship and extra-group mating phase) should be $O(T \cdot N)$. Therefore, the overall computational complexity of three phases is $O(3 \cdot T \cdot N)$. Finally, the computational complexity of HBOBES is $O(N \cdot (1+3 \cdot T))$.

Algorithm 1

The pseudo-code of proposed HBOBES

5 Application of the HBOBES on standard datasets

In this section, the application of proposed HBOBES on practical problems is performed to verify its optimizing performance. In the experiment, all methods are developed and implemented in Matlab-R2021b on Windows 11 with an Intel(R) Core(TM) i7-10700 CPU @ 2.90 GHz and 32.00 GB RAM.

5.1 Experimental settings

To better illustrate the advantages of HBOBES, eight well-known algorithms are selected for the performance comparison, including BO [18], BES [52], GTO [53], DO [54], MPA [55], GWO [56], SSA [57], and PSO [55]. Table 1 shows the details of parameter settings for these algorithms as well as the HBOBES. For a fair comparison, the population size and the maximum number of iterations during optimization process are set to 30 and 500, respectively. All algorithms run 15 times independently for each test case to eliminate the experimental randomness.

Table 1 Parameter settings for the optimization algorithms

5.2 Dataset description

The proposed HBOBES is applied to train the MLP models for solving eight standard datasets, including five classification problems (XOR, Balloon, Iris, Breast cancer, Heart classification datasets) and three function-approximation problems (Sigmoid, Cosine, and Sine), which are obtained from the University of California at Irvine (UCI) Machine Learning Repository [36]. The details are listed in Table 2 [37, 58].

Table 2 The details of standard datasets and corresponding structures of MLP models

5.3 Performance measures

In this study, the experimental results are analyzed based on the criteria of the Best, Mean, Worst, and Stand deviation (Std). The performance indicators of Mean and Std are given in the following equations.

$$\begin{aligned} Mean= & \frac{1}{R}\sum _{i=1}^RF(i) \end{aligned}$$

(20)

$$\begin{aligned} Std= & \sqrt{ \frac{\sum _{i=1}^R(F(i)-Mean)}{R}} \end{aligned}$$

(21)

where R is the number of run times, and F(i) is the obtained best fitness value or classification accuracy at i-th run.

The results of classification accuracy for the classification problems are also treated as another metric for comparison, and is formulated as follows:

$$\begin{aligned} Accuracy=\frac{TP+TN}{TP+TN+FP+FN} \end{aligned}$$

(22)

where TP denotes the true positive, TN denotes the true negative, FP is the false positive, and FN means the false negative [28].

5.4 Results of standard classification problems

5.4.1 Statistical analysis

The statistical training results and Friedman ranking test of proposed HBOBES and other algorithms are shown in Table 3. The bold data are the minimum values of mean MSE, indicating the best results. In the aspect of the mean values, the HBOBES outperforms other algorithms on Iris dataset and Breast cancer dataset. It is worth noting that the MLP models corresponding to these two datasets have relatively complex structures. HBOBES also obtains the third rank on XOR and Ballon datasets, as well as the second rank on Heart dataset. Therefore, the results of mean MSE show the overall superiority for solving these classification problems. Compared with the basic BO and BES, the HBOBES gives the best optimization results on the XOR, Iris, and Breast cancer datasets and comparable performance on the Balloon and Heart datasets, which demonstrates the effectiveness of strategies applied in HBOBES.

Table 3 Training results for the standard classification datasets

Table 4 gives the results of Wilcoxon signed-rank test. According to the ranking results of mean values, the Wilcoxon signed-rank test results of BO, BES, GTO, DO, MPA, GWO, SSA and PSO are 2/2/1, 3/2/0, 5/0/0, 5/0/0, 4/0/1, 4/1/0, 5/0/0, and 3/0/2. Overall, HBOBES show competitive performance on training the MLP models for solving the problems of standard classification datasets.

Table 4 The Wilcoxon signed-rank test results for the standard classification datasets

Table 5 gives the best, mean, worst, std and rank results of classification accuracy on the five standard classification datasets, and the bold data mean the maximum values of mean value. According to Table 5, the HBOBES obtains the first rank with 100% classification rates on XOR and Ballon datasets, the second rank on Iris dataset, and the best results on Breast cancer and Heart datasets, which also show the merits of the HBOBES for solving classification problems.

Table 5 Classification accuracy (%) for the standard classification datasets

5.4.2 Convergence analysis

Figure 2 shows the convergence curves of and other compared algorithms. It is observed that HBOBES converges faster and obtains the lowest MSE values on Iris, Breast cancer, and Heart classification datasets. For the XOR and Balloon classification datasets, HBOBES outperforms other algorithms except the PSO. Overall, HBOBES shows advantages in finding the optimal solutions for standard classification datasets.

5.4.3 Boxplot analysis

The boxplots of all algorithms are shown in Fig. 3, which are used to reveal the distribution characteristics of the 15 independent optimal solutions. The first and third quartile points of optimal solutions correspond to the bottom and top edges of the box. And the red symbol “+” means the rest values. PSO obtains the theoretical value zero on Balloon classification datasets, thus the boxplots of PSO are not displayed in Fig. 3. HBOBES has narrower box, lower position or smaller median on these classification datasets compared to other algorithms except the PSO. In addition, HBOBES has fewer outliers on these datasets, which indicates that HBOBES has better robustness.

Fig. 2

Convergence trends of HBOBES and compared algorithms for the standard classification datasets

Fig. 3

Box plots of HBOBES and compared algorithms for the standard classification datasets

5.5 Results of functional approximation problems

Moreover, the performance of proposed HBOBES has been evaluated using three functions, including Sigmoid function, Cosine function, and Sine function. The training results are reported in Table 6. The best outcomes of mean values are marked in bold. It is observed that HBOBES ranks first on Sigmoid and Cosine functions. However, the results of training on the sine function are not very good.

Fig. 4 presents the results of test errors for three functional approximation problems. It is shown that HBOBES achieves lower test errors on the first two functions and comparable results in the sine function. In summary, HBOBES also shows very competitive results for the functional approximation problems.

Table 6 Training results for the functional approximation datasets

Fig. 4

Box plots of test errors for the functional approximation datasets

6 Application of the HBOBES on aerosol classification datasets

In this section, the proposed HBOBES is applied to solve the aerosol classification problem, which is a critical step for the accurate and reliable detection of optical smoke detectors. The scheme of identifying fire and non-fire aerosols is shown in Fig. 5. Various aerosol samples are collected as the inputs of MLP models. Then the HBOBES is applied as the MLP trainer, optimizing the weights and bias parameters. As shown in Fig. 5, the classification performance of four MLP models was investigated, including two cases of divided into two categories (i.e., fire and non-fire aerosols) and three categories (i.e., black smokes, white smokes and non-fire aerosols). It is worth mentioning that the case of divided into three categories are more desirable in practice, because it helps to achieve a balanced response of the smoke detectors to black and white smokes. The details for solving the aerosol classification problem are described in the following.

Fig. 5

The illustration for the classification of fire and non-fire aerosols using MLP models trained by optimization algorithms

6.1 Establishment of aerosol classification datasets

The unpolarized light scattering signal data of fire and non-fire aerosols are drawn from our previous experiments [14], including those of fire smokes generated by cotton lamp wick, beech wood, polyurethane foam and n-Heptane, and nuisance aerosols, including DMT dolomite 90, ISO A1 test dust, water mist and water steam. By using the different combinations of two scattering angles ($45^{\circ }$, $135^{\circ }$) and two wavelengths (405 nm, 850 nm), four detection light path signals can be obtained, i.e., [405 nm, $45^{\circ }$], [405 nm, $135^{\circ }$], [850 nm, $45^{\circ }$], and [850 nm, $135^{\circ }$]. As shown in Fig. 6, some distribution differences between different aerosols can be observed under different scattered light intensity ratios. In particularly, it is observed that there is no intersection between the black smokes and other aerosols. Thus the scattered light intensity ratio can be used as the identification index of aerosol types.

Moreover, the utilization of multiple ratios is a feasible way to distinguish different types of aerosols. In this study, we mainly consider two cases in which two and three scattered light intensity ratios are used for the discrimination of different aerosols. As shown in Table 7, eight different combinations of the scattered light intensity ratios are considered as the input parameters of MLP models. For the case of applying two scattered light intensity ratios, the first scattered light intensity ratio indicates the ratio of scattered light intensity at different angles of the same wavelength, and the second scattered light intensity ratio indicates the ratio of scattered light intensity at different wavelengths of the same angle. According to the Philipp’s study [21], the first feature can reveal the information of particle’s shape and color, while the second feature can show the information of particle size. For the case of applying three scattered light intensity ratios, one of the four scattered light intensity signals is taken as the reference signal, then the obtained three scattered light intensity ratios are used as the selected features of aerosols. In theory, it is understandable that using three ratios as the inputs can reflect more physical and chemical properties information of aerosols than the method using two ratios. To make the test results more reliable, 200 training samples and 50 test samples in each aerosol dataset are collected from previous study [14] for the experiments. The preprocessing flow of aerosol classification datasets is shown in Fig. 7.

Table 7 The feature details of aerosol classification datasets

Fig. 6

Scatter diagram of typical fire and non-fire aerosols

Fig. 7

The flow chart of aerosol sample data collection and preprocessing

6.2 Establishment of MLP classification models

In the construction of MLP models, the details are given in Table 8. Four types of MLP structure with only one hidden layer are designed to solve the aerosol classification problem, which are 2-5-2, 3-7-2, 2-5-3 and 3-7-3. The number of neurons in hidden layer is determined by empirical value $2\times n$+1 [59], where n is the number of neurons in input layer. The sigmoid function is selected as the activation function of each neuron node, which is a common and effective activation function that enables neural networks to have the ability of nonlinear mapping.

As shown in Table 8, two classification cases of aerosols are investigated in this paper. The first case is the discrimination of fire and non-fire aerosols, i.e., dividing into two categories, corresponding to the structures of 2-5-2 and 3-7-2. And the other one is the distinguish between white smokes, black smokes and non-fire aerosols, i.e., dividing into three categories, corresponding to the structures of 2-5-3 and 3-7-3. It is worth noting that the second case is conducive to realize the balanced response between white and black smokes for fire smoke detectors due to the ability of identifying the fire smoke types.

The accuracy convergence curve of training MLP model is shown in Fig. 8. It is noted that the curve for accuracy is not monotonically increasing with the number of iterations. This is because the average of MSE is used as the objective function during the training process of MLP model. And a lower MSE does not guarantee higher accuracy. However, accuracy will generally increases as the average of MSE decreases.

Table 8 The details of MLP models for aerosol classification problem

Fig. 8

Convergence curve of accuracy during the training process of MLP model

6.3 Results for dividing into two categories

The performance of MLP models with structures 2-5-2 and 3-7-2 are investigated in this section. As previous mentioned, there are four cases with different combinations of the scattered light intensity ratios for each MLP model.

6.3.1 Statistical analysis of training results

Table 9 gives the best, mean, worst, std, and ranking results whening training the MLP models. The bold data are the minimum mean values. It is clearly observed that HBOBES achieves the best mean value on cases 1, 2, 3, 4 on MLP model with 2-5-2 and case 3 on MLP model with 3-7-2. For the rest cases, HBOBES also gets good results and ranks second. According to the ranking results, HBOBES, BO and BES obtain the first, third and second results in the final ranking, which indicates HBOBES has been enhanced by hybridizing the BO and BES.

Table 10 gives the results of Wilcoxon signed-rank test. It can be seen that HBOBES has better performance than basic BO and BES with the Wilcoxon signed-rank test results 6/2/0 and 4/3/1, respectively. Moreover, HBOBES outperforms algorithms GTO, DO, MPA, GWO, SSA, and PSO on all cases with the Wilcoxon signed-rank test result 8/0/0. This means that the HBOBES has the advantages for training the MLP models compared to other algorithms.

Table 9 Training results for the aerosol classification datasets (two categories)

Table 10 The Wilcoxon signed-rank test results between HBOBES and other algorithms for the aerosol classification datasets (two categories)

6.3.2 Convergence analysis

Figure 9 shows the convergence curves of MSE during the training process of MLP models with structures 2-5-2 and 3-7-2. It can be observed that the proposed HBOBES algorithm obtained the lowest fitness values on most of the cases except for the case 1 (2-5-2), which indicates that the HBOBES has better optimizing ability for this task. The results of box plots are shown in Fig. 10. The proposed HBOBES has concentrated and lower box distribution compared to other algorithms except for case 2 (3-7-2), which indicates its stable optimizing performance.

Fig. 9

Convergence trends of HBOBES and other algorithms when training the MLP models (two categories)

Fig. 10

Box plots of of HBOBES and other algorithms for the aerosol classification datasets (two categories)

6.3.3 Statistical analysis of classification accuracy

The statistical results of classification accuracy under various cases are given in Table 11, where the data in bold indicate the best result for the mean value. As shown in Table 11, MPA obtains the highest classification accuracy of 94.75% at case 2 with 3-7-2 MLP structure, where the backward scattered light intensities of both wavelengths are selected as the reference signal. It can be figured out that the backward scattered signal of aerosol is more stable than the forward scattered signal and thus helps to improve the classification accuracy. BES gets the highest average classification accuracy of 83.36% at case 2 and 85.35% at case 4 with 3-7-2 MLP structure. Moreover, the HBOBES ranks the first on mean value for the rest cases. The average value shows that the HBOBES has a clear advantage on optimizing the MLP structures and obtains better classification results compared to the other eight algorithms. According to the mean rank and final ranking, HBOBES ranks the first with mean rank of 1.25. It should be noted that the BO and BES also obtain good performance with mean rank of 2.5 and 2.25, respectively, which indicates that the search mechanisms in BO and BES are suitable for solving the aerosol classification problem. The search capability of HBOBES is strengthened by combining the effective strategies in BO and BES.

Table 11 Overall classification accuracy (%) for testing samples (two categories)

6.3.4 Analysis of confusion matrix and ROC results

Figure 11 shows the confusion matrix for the best outcomes of 15 runs. Categories 1 and 2 indicate the fire smokes and non-fire aerosols, respectively. It is found that the proposed HBOBES has the ability to classify 180 out of 200 fire smoke samples and 188 out of 200 non-fire aerosol samples. The results of HBOBES are very competitive with those of other algorithms.

In addition, the proposed HBOBES has been evaluated in terms of ROC curves. Figure 12 shows the ROC curves for the best results of 15 runs obtained by HBOBES and compared algorithms. The proposed approach has attained 0.92 of area under curve (AUC) value, which is higher than those of BO (0.91), GTO (0.895), DO (0.9125), GWO (0.8475), SSA (0.895), and PSO (0.885) and only lower than those of BES (0.945) and MPA (0.9475).

Fig. 11

Confusion matrix for the best outcomes of HBOBES and other algorithms (two categories)

Fig. 12

ROC for the the best outcomes of HBOBES and other algorithms

6.4 Results for dividing into three categories

In this section, aerosols are classified into three categories using the MLP models with structures 2-5-3 and 3-7-3. To evaluate the performance of the proposed HBOBES, a series of MLP training and testing experiments were also carried out.

6.4.1 Statistical analysis of training results

Table 12 gives the best, mean, worst, std, and ranking results. The lowest mean values are marked in bold. It is noted that HBOBES achieves the best mean value on all cases except the case 3 on MLP model with 3-7-3, in which HBOBES ranks the second. And then HBOBES ranks first in the final ranking, which shows the remarkable training ability on MLP models.

Table 13 lists the results of Wilcoxon signed-rank test. It is observed that HBOBES outperforms algorithms GTO, DO, MPA, GWO, SSA, and PSO on all cases. HBOBES also shows better performance than basic BO and BES with the Wilcoxon signed-rank test results 7/1/0 and 4/4/0, respectively.

Table 12 Training results for the aerosol classification datasets (three categories)

Table 13 The Wilcoxon signed-rank test results between HBOBES and other algorithms for the aerosol classification datasets (three categories)

6.4.2 Convergence analysis

Figure 13 presents the convergence curves of MSE during the training process of MLP models with structures 2-5-3 and 3-7-3. It can be observed that the proposed HBOBES algorithm obtained the lowest fitness values on most of the cases except for case 4 (3-7-3). The outstanding exploration capability can be observed on cases 1 (2-5-3), cases 2 (2-5-3), cases 3 (2-5-3), cases 1 (3-7-3), and cases 3 (3-7-3). And the continuous exploitation capability is also shown throughout the training process. Therefore, the HBOBES exhibits better performance for the training of MLP models.

6.4.3 Boxplot analysis

Furthermore, the box plot results are shown in Fig. 14. The proposed HBOBES shows concentrated and lower box distribution compared to other algorithms except for the case 1 (3-7-3), where the BES algorithm shows better results. This demonstrates the effectiveness and stability of the HBOBES when solving the aerosol classification problem when divided into three categories.

Fig. 13

Convergence trends of HBOBES and other algorithms when training the MLP models (three categories)

Fig. 14

Box plots of HBOBES and other algorithms for the aerosol classification datasets (three categories)

6.4.4 Statistical analysis of classification accuracy

The statistical results of overall classification accuracy for all aerosol testing samples are given in Table 14. In Table 14, it can be seen that HBOBES obtains the highest classification accuracy of 94.25% at case 2 with 3-7-3 MLP structure and highest average classification accuracy of 85.15% at case 4 with 3-7-3 MLP structure. As listed in the Table 7, the backward scattered light intensity is selected as the reference signal, which helps in more accurate classification. In addition, the HBOBES ranks the first on mean value on all cases except the case 2 with 3-7-3 MLP structure, on which the HBOBES ranks the second. And in the aspect of mean rank, HBOBES ranks the first with the value of 1.125. Therefore, compared with other methods, HBOBES also represents the superior performance in solving the classification problem that the aerosols are divided into three categories.

Table 14 Overall classification accuracy (%) for testing samples (three categories)

6.4.5 Analysis of confusion-matrix

Fig. 15 presents the confusion matrix for the best outcomes of all algorithms. Categories 1, 2, and 3 indicate white smokes, black smokes and non-fire aerosols, respectively. It is clear that all samples of black smokes can be accurately identified, which is consistent with the aerosol distribution shown in Fig. 6. For the proposed HBOBES, it is able to classify 92 out of 100 white smoke samples and 185 out of 200 non-fire aerosol samples. Compared to other algorithms, HBOBES obtains the best results.

Fig. 15

Confusion matrix for the best outcomes of HBOBES and other algorithms (three categories)

6.5 Discussions

According to the previous experimental results, the proposed HBOBES shows excellent performance on the training of MLP models for solving classification problems, including the standard classification datasets as well as the aerosol classification datasets established in this paper. Eight well-know optimization algorithms are selected for the performance comparison with proposed HBOBES, including BO, BES, GTO, DO, MPA, GWO, SSA, and PSO.

6.5.1 Analysis of overall ranking

For a more comprehensively comparison, the relationships between the obtained best results of mean classification accuracy and selected features (i.e., scattered light intensity ratios) for all cases are shown in Fig. 16. It is observed that the overall accuracy results of 3-7-2 and 3-7-3 MLP models are better those of 2-5-2 and 2-5-3 MLP models. In particular, the case 4 of 3-7-2 and 3-7-3 MLP models, which uses the scattered light intensity of wavelength at 850 nm and scattered angle at $135^{\circ }$, shows better outcomes.

In addition, Fig. 17 shows the average ranking of all algorithms on standard classification datasets and aerosol classification problem, including cases of dividing into two categories and three categories. In Fig. 17, the bottom number 1 and 2 mean the training and testing mean ranking of standard classification datasets. The bottom number 3 and 4 indicate the training and testing mean ranking of MLP models with structures 2-5-2 and 3-7-2. And bottom number 5 and 6 indicate the training and testing mean ranking of MLP models with structures 2-5-3 and 3-7-3. It is clear that the HBOBES ranks first in all cases, indicating that the HBOBES has the merits on training the MLP models and finding the optimal weights and biases for solving the classification problems.

Fig. 16

The best mean classification accuracy and corresponding features for all cases

Fig. 17

Comprehensive ranking of proposed HBOBES and compared algorithms

6.5.2 Analysis of computation time

Further, the computation time for training the MLP models are assessed in this section. Table 15 gives the results of HBOBES and compared algorithms for solving the aerosol classification datasets. It is shown that the proposed HBOBES requires more time than other methods when training these four MLP models. However, compared to the BES algorithm, the increased time is relatively small. On the other hand, HBOBES achieves significantly better optimization performance according to previous experiments, which can be acceptable to a certain extent.

Table 15 Comparison of average execution-time of training MLP models for 15 times

6.5.3 Analysis of iteration times and population size

Moreover, the effects of iteration times T and population size N on classification accuracy were further analyzed, which are shown in Fig. 18. MLP models with structures 3-7-2 and 3-7-3 trained by HBOBES are considered due to their better performance. With the increasing of T and N, the classification accuracy rates of both models reach more than 90%. It is also noted that 3-7-2 MLP model has a slightly higher accuracy (close to 95%), presumably because it is easier to classify aerosols into two categories (fire and non-fire aerosols), rather than three categories (white smokes, black smokes and non-fire aerosols). Moreover, it is also noted that the classification accuracy can not reach or close to 100%, possibly because the the characteristics of white smokes and nuisance aerosols are too similar, and there is a certain crossover in distribution (as shown in Fig. 6).

To sum up, in the application scenarios of optical smoke detectors, such as aviation or industrial settings, the accurate identification of fire smokes and nuisance aerosols are the most needed and can significantly reduce operating costs and improve operational safety. According to the above training and test results, the HBOBES-MLP can be a promising method for the effective recognition of different fire smokes and non-fire aerosols, which makes it possible to reduce false alarms and respond equally to black and white smokes for fire smoke detectors. Therefore, the proposed method in this paper has remarkable practical significance and broad application prospect. At last, the nomenclatures used in this paper are given in Table 16.

Fig. 18

Trends of aerosol classification accuracy with iteration times (T) and population size (N): (a) MLP model with structure 3-7-2; (b) MLP model with structure 3-7-3

Table 16 Nomenclatures used in this paper

7 Conclusions and future remarks

In this study, a new approach for solving the aerosol identification problem in the field of fire smoke detection is provided. A hybrid algorithm HBOBES is developed to optimize the weights and biases of MLP model, which is utilized to solve the aerosol classification problem. The proposed HBOBES incorporates three phases: space selecting phase, promiscuous and restrictive mating phase, and consortship and extra-group mating phase. Moreover, a adaptive tent chaos mapping technique is applied to the phases of space selecting and promiscuous and restrictive mating to extend the diversity of population and improve the exploration ability of algorithm. The effectiveness of HBOBES is demonstrated through its application to train MLP models for solving standard classification datasets as well as aerosol classification datasets, which are established based on our previous works. By comparing to eight other advanced algorithms, the results show that HBOBES has superior performance for optimizing the parameters of MLP models. Moreover, it is found that the short wavelength and backward scattered light intensity are helpful for the classification of fire smokes and non-fire aerosols. MLP models with structures 3-7-2 and 3-7-3 show better aerosol classification performance, with an average classification accuracy of about 85% and the highest classification accuracy close to 95%, which has practical significance for the anti-false alarm and balanced fire smoke detection.

In the future, the structural parameters of MLP models for the aerosol classification can be further optimized using advanced optimization algorithms, including the number of hidden layers, the number of neurons in the hidden layer, and the selection of transfer function. The proposed HBOBES can be applied to solve more classification datasets of different aerosol types or real-world classification datasets. Further, the optimized MLP model can be deployed to the smoke detector prototype to verify its actual performance on the identification of fire smokes and nuisance aerosols. In addition, it is important to accurately identify black smoke, white smoke and interference aerosols respectively, and there is need to consider more characteristics of aerosol optical properties to improve more accurate and reliable identification results, such as the polarization features of aerosols. Therefore, the aerosol classification problem also can be treated as feature selection and multi-objective optimization problem [60,61,62,63], which can be another direction of this work.