An Estimation of an Acceptable Efficiency Frontier Having an Optimum Resource Management Approach, with a Combination of the DEA-ANN-GA Technique (A Case Study of Branches of an Insurance Company)

Abstract

In this paper, a novel artificial intelligence technique for the estimation of near-optimal resource management is proposed. The model utilizes a two-stage data envelopment analysis to find the best-practice frontier of the decision-making units. By employing this data, a supervised multi-layer Artificial Neural Network is exercised. This network is capable of predicting the frontier for the near future by receiving input and mediator variables. In the next step, a genetic algorithm is formed to find an optimal input value for the artificial neural network, such that the overall performance of decision-making units in the near future is maximized. The proposed algorithm allows the managers to set some restrictions on the whole system, including the minimum efficiency and the maximum change on resources. The performance of the presented technique is reviewed on 31 branches of an insurance company, during the years 2015 to 2018. The results show that the developed algorithm can efficiently maximize the overall performance of decision-making units.

1. Introduction

Insurance companies are known as one of the most crucial fiscal institutions, having a special standing in the economic growth and development of countries; such that, the efficient performance of this sector also has an impact on the other economic sectors as well. Thereby, one of the elements of growth and development depends or rests on the expansion and the advancement of that country’s insurance; and highly developed countries usually have an innovative insurance industry. Given its due importance, it can be easily comprehended that inefficiency in the insurance industry will not only have an influence on the quality of life but can also obstruct the improvement of efficiency in economic sectors and this trait can significantly contribute to a lack of accessibility to sustainable development targets.

The lack of resources and amenities from the distant past has always been discussed. To set itself as a strong example in socioeconomic conditions in the future and impose it, optimal use of available facilities and resources, as well as increasing efficiency in order to achieve the predicted goals and meet the growing needs of insurance has changed the problem.

It is essential that accurate supervision is conducted for the earning or income and expenditures (costs) of insurance companies and a specification of their weak aspects, which is an absolute necessity at this point of time and even much more than before. Insurance managers are always seeking to increase the efficiency of branches, the recognition of inefficient units, and their growth potential.

To date, numerous studies have been performed in concern with measuring the relative efficiency of insurance companies [1,2].

In past research, to analyze productivity and efficiency of the insurance market, two methods were utilized, the parametric study, such as [3,4,5,6,7,8,9]. In data envelopment analysis (DEA) non-parametric approaches, studies such as [1,2,5,10,11,12,13,14,15,16,17,18,19,20,21].

However, in recent years, DEA techniques have been one of the most widely used tools in modeling spheres and for computing the relative performance of decision-making units (DMUs). It is also employed in assessing the performance of various insurance, banking, medical, and other fields; if the organization has several different outputs of varied scales, this method does not have a problem in assessing efficiency. In respect to parametric models, the major advantage of this model is that there are no indexes to convert them into each other in conditions where several applicable outputs are present, and synchronously, general conformity, regarding weight or the importance does not exist [22].

In yet another study, a two-stage DEA method was utilized for analyzing productivity and the efficiency of insurance companies (studies such as [23,24,25,26,27,28,29,30]. In some cases, managers define various indexes and their measuring to take action for appraising the efficiency of the sub-set sectors, to elevate the efficiency, and as a result, productivity. Though these assessments offer or transfer an image of the organization’s efficiency to the managers, considering these indexes individually and in being inattentive to the affiliation between them could cause the creation of an imprecise image with regard to the subject under survey. By using DEA techniques, this problem is eliminated [31].

Nevertheless, despite the intensity and accuracy of the methodology, DEA has a defect in its actual implementation. Even upon accommodating the hypothetical legitimacy of this approach, the solutions arising from DEA are not always practical or realistic. Similarly, a dearth of predictive capacity has been indicated as an imperative shortcoming for the model, which averts further development of the real world, in terms of problem-solving methods [31,32,33].

DEA measures the efficiency of the precedent DMUs. Hence, the efficiency scores and ranking are correlated to the prior efficiency of the DMUs. Though, there is no assurance that the efficiency in the past is repeated in the succeeding planning periods; and to estimate the inputs and outputs of the ensuing time period, an accurate approach is required. Thereby, in recent years, artificial neural networks (ANN) have been considered as a favorable and powerful managerial tool in predicting the behavior of the system and likewise, in combination with DEA, in assessing the efficiency of units.

The initiative of combining ANN and DEA was initially discussed in [34]. After which, innumerable works of research are available proposing a hybrid of DEA and ANN models to assess efficiency [35,36,37].

Although in previous research, the DEA-ANN hybrid technique was used to predict and estimate a state, so to evaluate the efficiency of branches, there is a possibility of inefficiency of some DMU’s due to previous limitations. Factors such as managerial allocations, resource availability, etc. cannot prove that they have the ability to improve their performance instantly based on the frontier of best first-level performance.

Thence, the innovation which has taken place in this research supports the proposed hybrid DEA model, ANN, and the genetic algorithm (GA), which seeks and determines the optimum efficiency frontier by defining an acceptable limit and controlling a condition for minimum modifications in input resources and the near-optimal efficiency frontier.

In Section 2 of this case study, we will present a literature review and in Section 3, a combined algorithm and methodology will be rendered. Finally, in Section 4, the conclusion of the case study under discussion will be surveyed.

2. Literature Review

Hwang and Kao used the two-stage DEA method to assess the managerial efficiency for 24 non-life insurance companies in Taiwan. In this research, the efficiency of the first stage, that is, the marketing capacity, and the efficiency of the second stage, signifying the profitability capacity, were independently and mutually measured by the DEA method. The results gained by them showed that instead of measuring the efficiency of an insurance company on an overall basis only and merely once, it would be better to gauge the efficiency of an insurance company in two stages. This task would cause an enhanced managerial efficiency to be demonstrated and will assist in enlightening the benefits and weaknesses of their processes for perception [27].

Abdul Kader et al. investigated the financial returns of the Takaful non-life insurance companies which are active in 10 Islamic countries. The author, likewise, has engaged a two-stage approach which utilizes DEA in the first stage and fixed regression in the second stage, to survey the impact of the company’s special characteristics as to these efficiencies [38].

Barros et al. has assessed the efficiency of insurance companies in Greece by utilizing the DEA technique. They have considered labor costs and costs not involving labor, as well as net capital as input, reinsurance reserves, personal deposits, assets donated to investors, and actual damage as an output of the model. They surveyed the performance or efficiency of 710 insurance units in the years 1994 to 2003. Results illustrated that the productivity of Greek insurance companies has increased by a rate of 16.1 percent, whilst in the non-life insurance companies’ productivity had faced a 6.5 percent growth [1].

Mahlberg and Url analyzed the German insurance industry within the years 1991 to 2006. They illustrated that the average productivity had increased by 18 percent [21].

Bertoni and Croce utilized DEA, in the years between 1997 to 2004 in an active group of life insurance companies in five European countries, namely, Germany, France, Italy, Spain, and the UK. By using the Malmquist technique, the authors conclude that the annual productivity has increased by 6.71 percent and this increase is mostly due to innovation (technology change); and this is in a condition where the optimum modes of management only have a 0.04 percent impact on productivity growth [39].

Ansah-Adu et al. has examined the effectiveness of the insurance companies in Ghana, by utilizing a two-stage method to study the factors affecting insurance companies [40].

Bai-King et al. (2012) has studied the efficiency of 56 insurance units in China during the years 2006 to 2010. In this research, the inputs of the DEA model, total assets, number of employees, costs that have been taken into consideration; and also, the model outputs, final reserves, income of investments made, and the insurance profit ratio are under consideration [23].

Shahroudi et al. analyzed the efficiency of 14 Iranian private insurance companies with the two-stage DEA method and utilized the Kao-Hwang approach [41].

Eling and Huang investigated the insurance returns in the BRIC countries and utilized the two-stage DEA approach for their survey and analysis, as well as specified the mutual productivity stimulus in the second stage [42].

Barros and Wanke have assessed insurance companies in Mozambique in the years 2002 to 2011, using the DEA approach. The inputs of the model comprise operational costs, wages, investment, and the number of employees; whereas, the outputs of the model consist of paid claims, insurance profit ratio, premiums received, and reinsurance ceded [43].

Chen et al. studied the modifications of productivity in the communal insurance companies in Malaysia from the years 2008 to 2011, so as to survey the impacts of the intellectual capital concerning the changes in productivity, by using the OLS and Tobit regression [44].

Jaloudi employed the DEA technique to assess the technical efficiency of 22 insurance companies during the period of the years 2000 to 2016, in the Jordanian insurance market and surveyed the internal and external influential factors. The results showed that the owners’ equities are amongst the most important internal determinants for the efficiency of companies and there is a significant correlation between the type, size, and return of assets of the insurer as well as its efficiency [45].

Allahviranloo et al. presented a decision support system with the help of DEA for ranking research centers in Iran. Considering the actual implementation and checking the accuracy of the results, they showed that the ability of the model is high, and it can determine the efficiency and ranking of research centers well [46].

Similarly, in recent years, a combination of DEA-ANN has also been taken into consideration by researchers. In an article, Wu et.al. combined DEA and ANN to assess the relative efficiency of branches of a large Canadian bank. The results which came to hand showed that the efficiency attained with the neural network model and DEA is greatly correlated with that of the results of the DEA efficiency [33].

In his article, Mustafa surveyed the efficiency of 85 high-ranking Arab banks with two quantitative DEA models and neural networks, similar to the studies of Wu et al., in [33]. In this study, inputs such as assets and investments and outputs such as net profit, the relative ratio of return assets, and the relative return of investment or capital have been utilized. A comparison of the results of the neural networks and DEA demonstrates that there is a correlation of 94 percent between the rankings of the two methods [32].

In an article by Ozdemir and Temur design an ANN, which is supported by DEA, that provided a professional system to assist suppliers. As a result of this research and by creating an appropriate neural network, lengthy DEA computations are avoided, and the efficiencies of the suppliers have been predicted and assessed easily [47].

Wu et al. simulated the efficiency of production and investment of the Canadian life and health insurance companies using the DEA approach. In this research, a new DEA model was presented, so as to simultaneously appraise the efficiency of production and investment; and this has been proposed to appraise the performance or efficiency of the Canadian life and health insurance companies. The results of this study indicated that the Canadian life and health insurance industry has operated relatively efficiently during the duration of the survey [48].

Wanke and Barros have investigated the efficiency of the Brazilian insurance industry within the years 1995 to 2013. In this research, the authors have concentrated on the two-stage method, which consists of determining a DEA approach in the first stage and utilizing several data mining techniques in the second stage. Results illustrate that the overall efficiency in the insurance sector of Brazil is apparently high and shows that the minimum is 0.806, and emphasizes that the Brazilian insurance companies must have a criterion approach in order to assess their efficiency, in terms of the regions or areas offered and the type of insurance obtainable [49].

Kwon et al. utilized the DEA-ANN hybrid model to assess the efficiency of the Japanese electronics industry. Their findings illustrated that BPNN together with DEA has promising estimation capacities in predicting efficiency scores and the optimum efficiency benchmarks for the DMUs under assessment [50].

Kwon et al. have manipulated a three-stage model by utilizing DEA and BPNN (back-propagation neural networks) to assess the efficiency of 148 general hospitals for acute care in Pennsylvania in 2012. The results of the analysis illustrate that, although great endeavors have been made for improving the efficiency of health care in acute care hospitals, there is an immense scope available for alleviation and improvement and an enhanced performance benchmarking could be an advantageous option [51].

Similarly, studies have also been conducted in a combined model of DEA and GA. Chuang et al. rendered a hybrid efficiency assessment model using the DEA technique and multidimensional genetic algorithm (MOGA) to design the optimal combination of partner chains in order to achieve the target of minimizing the cost and time period for the development of the new product, as well as maximizing the reliability of production [52].

In another study, Udhayakumar et al. used DEA application in the banking sector and assumed the randomness of inputs and outputs. By applying the developed GA, they used one-point crossovers and perturbation mutation operators to solve the P model of the restricted technique and confirmed the feasibility of chance constraints by simulation techniques [53].

In a study, Azadeh et al. evaluated and optimized inputs in power transmission units using a DEA-GA combined model. By considering two different aspects of efficiency and cost, and by using ultra-efficient DEA models of cost allocation, they analyzed sensitivity and determined necessary inputs [54].

Madhanagopal and Chandrasekaran investigated the Indian banking sector and discussed a new approach to select an appropriate set of DEA input and output variables using a GA. In total, they selected the best model by surveying 35 models and their impact on the efficiency score of banks [55].

Zhang et al. also proposed a performance prediction model based on DEA (SBM) along with eleven machine learning models. To confirm the rationality of the model approach, they predicted the efficiency of carbon emission in 30 provinces of China in 2019 [56].

In other research, Zhang et al. proposed a multi-dimensional performance evaluation model by combining DEA and SVM techniques based on five dimensions of operation, innovation, finance, financing, and public opinion and by studying 156 Chinese companies. They showed that with an early warning model it is possible to identify and predict the financial crisis for listed companies [57].

2.1. Data Envelopment Analysis (DEA)

DEA is a method to determine the performance of DMUs evaluations. Each unit produces multiple outputs by consuming multiple inputs. Basic models in DEA do not only determine the efficiency and inefficiency of the model, but also provide a model for inefficient units that can be transformed into an efficient unit by reducing inputs or increasing outputs [46].

In general, DEA models are divided into two groups, “input-oriented” and “output-oriented”. Input-based models are models that use fewer inputs to get the same amount of output without modifying the outputs, and output-driven models are those that produce more output without changing the input rate.

The alternative of whether the model is an input-oriented or output-oriented one depends on the market distinctiveness of insurance companies. In competitive markets, as a general rule, the output-oriented model is utilized to assess the DMUs. This is due to the fact that the inputs are assumed to be under the control of the DMUs and the target is to maximize the outputs that are not under the control of the DMUs.

Figure 1 shows a basic DEA model that uses inputs X_ij (i = 1,…,m) to generate outputs Y_rj (r = 1,…,s).

In cases where there are negative values in the data, special DEA models should be used, one of the most common of which is the SBM (slack-based measure) model, which is shown in the objective function of model (1).

$$Min\frac{1-(\frac{1}{m}{{\displaystyle \sum }}_{i=1}^m\frac{s_i^{-}}{R_i^{-}})}{1+(\frac{1}{S}{\displaystyle {\sum }_{r=1}^s}\frac{s_r^{+}}{R_r^{+}})}$$

(1)

$$\begin{array}{l}s.t.{\displaystyle \sum _{j=1}^n}{\lambda }_j{x}_{ij}+s_i^{-}=x_{ip},i=1,\dots ,m\\ {\displaystyle \sum _{j=1}^n}{\lambda }_j{y}_{rj}-s_r^{+}=y_{rp},r=1,\dots ,s\\ {\displaystyle \sum _{j=1}^n}{\lambda }_j=1, j=1,\dots ,n\\ \lambda \ge 0,j=1,\dots ,n\\ \lambda ,s_i^{-},s_r^{+}\ge 0;\forall j=1,\dots ,n \& i=1,\dots ,m \& r=1,\dots ,s\end{array}$$

2.2. Artificial Neural Networks (ANN)

ANNs are mathematical models, which mimic the manner of performance of the human brain, and their capacity for the extraction of benchmarks or patterns in observed data is achieved without having to make presumptions about the correlations between variables. They function comprehensively and are flexible, as well as being a powerful tool for data analysis and modeling high-accuracy, non-linear correlations that process information like the brain. A key component of this idea is the novel structure of the information processing system. The system comprises a large number of highly interconnected processing fundamentals, which work together in harmony so as to resolve a problem [58].

ANNs are formed by a set of interconnected computational units (neurons) [59]. Each neuron conducts two successive calculations. One is a linear combination of its inputs and consecutively, a non-linear computation of the result, in order to achieve the output values, which is hereafter sent to the other neurons in the network [60]. Similar to the partial least squares method, modeling is done with the result by an intermediate set of unusual variables known as ‘concealed layers’ [61]. These concealed layers are situated between the first layer, which comprises input neurons and the final layer contains the neural network predictions for each case presented in its input neurons [60,62]. Figure 2 demonstrates an example of a multi-layer neural network.

2.3. Genetic Algorithm (GA)

The GA was introduced by Professor John Holland and his students in 1975 [63,64,65]. In simple words, GAs are probabilistic modes of searching and are designed for work in large spaces, and as such are designed to be demonstrated by strings [66].

The GA method starts from a random population of models, with good local solutions with imitation, leading to the selection of a natural process and the use of mechanisms such as higher rates for the proliferation of chromosomes, mutations for different crops and leads to the creation between species for reduction, and improves compounds [66].

The GA begins from a set of responses, which are illustrated by chromosomes. This set is known as the responses of the initial population. In this algorithm, the responses are from one population, which is utilized to produce the subsequent population. In this process there is hope that there is a better selection concerning a new population relative to the prior one. This process will continue until the conditions are determined, such as the sum of populations or the number of improved responses. In this algorithm, with due attention to the problem, the variables which have to be determined are specified. Next, these variables are coded in an appropriate manner and are demonstrated in the form of chromosomes. Based on the objective function, a fitness function is defined for the chromosome and a desirable initial population is also selected randomly. This is followed by computation in relevance with the amount of fitness function for each chromosome of the initial population. Then, the phase illustrated in Figure 3 is conducted [67].

3. Modeling (DEA-ANN-GA)

The production process of insurance companies, in general, comprises of two activities; one is marketing, which is relevant to marketing, the sale of insurance policies, and receiving premiums. The other activity involves the profitability of branches, which encompasses the payment for damages (losses) and profits from the insurance process. However, in this research, the two-stage DEA technique has been utilized to measure the efficient branch of one of the insurance companies active in Iran; and to identify the efficient and inefficient branches, which can offer adequate information, in order to gain access to the advantages and weaknesses of competitive strategies, and to be at the disposal of managers. But the studies performed show that despite the fact that DEA, which has immense capacities in practice, does also have weaknesses, but its combinations with other techniques could address the elimination of these flaws.

One of these failings of DEA is that it forms an efficient frontier on the basis of previous performance. If the inputs and outputs are estimated prior to their occurrence, the efficiency frontier is formed and based on more pragmatic data. Thus, ANN can be utilized as a precise method for the estimation of inputs and outputs ahead of occurrence. In actual fact, given that a DEA model is a linear one and neural networks are highly capacitated to approximate nonlinear functions, ANN’s are good tools to be used for such problems. ANNs are one of the prominent data mining models for prediction [68].

Thereby the use of an ANN for measuring the efficiency of DMUs is appropriate. One of the other challenges in the DEA technique in actual fact is that, if statistical perturbations are conducted to the data, the boundaries computed by the DEA may diverge [69]. However, its merging with ANN eliminates such challenges. This is because neural networks are resilient to statistical perturbations and outdated data. Likewise, neural networks have certain advantages which are on the grounds of a non-parametric approach. As forethought, it does not call for any theory as to the possible distribution or structures of the production function. Similarly, in DEA an inefficient unit can plan for an efficient target that is much further off. This implies that to reach the target, it may necessitate that the inputs are decreased drastically and there is an immense increase in the outputs. When there are vast changes in planning the inputs and outputs, it could be possible that performing this all at once would prove to be problematic [70].

In other words, a possibility does exist that, some of the DMUs are inefficient due to prior limitations, such as management proficiency, scarcity of resources etc., and will be incapable of improving their performance immediately, on the basis of the first level best-practice frontier [71].

This challenge is also noticed in the combined DEA-ANN technique, as neural networks have a high degree of predictive aspects and the estimation, which is carried out, is based on the inputs according to which they have been tutored. Thereby, there is a possibility that achieving the targets which are set in the hybrid technique will necessitate immense modifications in inputs or outputs, which shall not be feasible in practice.

The innovation made in this research to eliminate the expressed failings is that, whilst taking benefit of all the advantages of DEA and ANN and its power in estimating data and the efficiency frontier, the GA is also used in the hybrid model, which is shown in Figure 4; this algorithm acts as a “function optimizer”. A fitness function is first formulated using the objective function and is employed in consecutive genetic operations in generations of genetic algorithms. In utilizing this model, managers have the ability to handle the limited resources in access; and impose the essential controls, to minimize the modifications of resources, as well as the efficiency of all the branches, based on the efficiency frontier and elevating to the desired significance under consideration. In this paper, a set of different efficiencies for the branches is used during the entire period of study. Thus, the improvement of the efficiency of all branches is synchronously achieved and is above the set efficiency determined.

In a nutshell, the proposed algorithm uses ANN to predict the target variables Y using the input variables X and Z. Using this model, a GA is utilized to generate a good forecast of variables X and Z for the near future. In the following sections the details of the proposed method are described.

3.1. Calculation of Efficiency

According to the purpose of evaluation that each DMU is a branch of one of Iran’s insurance companies, 31 branches were selected, and each branch has a two-stage structure, the first stage is marketing, and the second stage is profitability. The first stage has three inputs and two outputs (intermediate production), which are the outputs of the first stage, the inputs of the second stage, and finally three outputs are considered for the second stage, which are the inputs, intermediate products, and the final output as shown in Figure 5. In the objective function of model (2), due to the existence of negative data among the outputs, the modified SBM model (SORM) is used. This model is stable to unit modifications in both input and output. In other words, the SORM model has more stability in transmissions. This signifies that, if the inputs and outputs are transferred in this model, the objective function and the efficiency attained will not change.

In this objective function, the maximum improvement is calculated for the input and output of the DMU, and the value of the relative efficiency objective function, which includes technical and combined efficiency, is calculated at the same time. The first set of constraints (2a) shows how an input less than the input of the first stage of the unit under evaluation (DMUP) can be found in the production possibility set. The second set of constraints (2b) indicates how to find more output in the second stage for the unit under evaluation (DMUP) in the production possibility set.

The third category of constraints (2c) also shows how to increase the possibility of production considering that the intermediate production (DMU_P), which is the output of the first stage, and the fourth category of constraints (2d) also shows how to increase the possibility of production. Considering that intermediate production is the input of the second stage, it is possible to reduce production in the production possibility set. The fifth and sixth category of constraints (2e and 2f) also shows the convexity condition on the combination coefficients, which means that the type of efficiency is assumed to be variable on the scale of the first stage, second stage, and the entire of the production possibility set.

$$Min\frac{1-(\frac{1}{3}{{\displaystyle \sum }}_{i=1}^3\frac{s_i^{-}}{R_i^{-}})}{1+(\frac{1}{3}{{\displaystyle \sum }}_{r=1}^3\frac{s_r^{+}}{R_r^{+}})}$$

(2)

$$s.t.{\displaystyle {\sum }_{j=1}^{31}}{\lambda }^1{}_j{x}_{ij}+s_i^{-}=x_{ip},i=1,2,3$$

(2a)

$${\displaystyle {\sum }_{j=1}^{31}}{\lambda }^2{}_j{y}_{rj}-s_r^{+}=y_{rp},r=1,2,3$$

(2b)

$${\displaystyle {\sum }_{j=1}^{31}}{\lambda }^1{}_j{z}_{dj}-{\widehat{s}}_d=z_{dj},d=1,2$$

(2c)

$${\displaystyle {\sum }_{j=1}^{31}}{\lambda }^2{}_j{z}_{dj}+{\widehat{s}}_d=z_{dj},d=1,2,$$

(2d)

$${\displaystyle {\sum }_{j=1}^{31}}{\lambda }^1{}_j=1,$$

(2e)

$${\displaystyle {\sum }_{j=1}^{31}}{\lambda }^2{}_j=1,$$

(2f)

$$\begin{array}{c}{\lambda }^1{}_j,{\lambda }^2{}_j\ge 0,j=1,\dots ,31,\\ s_i^{-},s_r^{+},{\widehat{s}}_d,{\tilde{s}}_d\ge 0,i,r=1,2,3 \& d=1,2.\\ R_i^{-}=Max\{x_{ij}:{\forall }_j\}-Min\{x_{ij}:{\forall }_j\}\\ R_r^{+}=Max\{y_{rj}:{\forall }_j\}-Min\{y_{rj}:{\forall }_j\}\end{array}$$

If $\left({\lambda }^1{}_j,{\lambda }^2{}_j,s_i^{-},s_r^{+},{\widehat{s}}_d,{\tilde{s}}_d\right),j=1,\dots ,31 \& i,r=1,2,3 \& d=1,2$ is the optimal solution of model (2), then the coordinates of the B.M (Benchmarking) point will be in the form of model (3) and the efficiency of the first, second stage, and the total efficiency will be calculated in the form of models (4,5,6).

$$\begin{array}{c}{x}_{Bi}={\displaystyle {\sum }_{j=1}^{31}}{\lambda }^{1*}{}_j{x}_{ij}=x_{ip}-s_i^{-*},i=1,2,3\\ z_{Bd}={\displaystyle {\sum }_{j=1}^{31}}{\lambda }^{1*}{}_j{z}_{dj}=z_{dp}+{\widehat{s}}_d^{*},d=1,2,\\ y_{Br}={\displaystyle {\sum }_{j=1}^{31}}{\lambda }^{2*}{}_j{y}_{rj}=y_{rp}+s_r^{+*},r=1,2,3\end{array}$$

(3)

The efficiency of Stage 1:

$$E_1=\frac{1-(\frac{1}{3}{\displaystyle {\sum }_{i=1}^3}\frac{s_i^{-*}}{x_{ip}})}{1+(\frac{1}{2}{\displaystyle {\sum }_{d=1}^2}\frac{{\widehat{s}}_d^{*}}{z_{dp}})}$$

(4)

The efficiency of Stage 2:

$$E_2=\frac{1-(\frac{1}{2}{\displaystyle {\sum }_{d=1}^2}\frac{{\widehat{s}}_d^{*}}{z_{dp}})}{1+(\frac{1}{3}{\displaystyle {\sum }_{r=1}^3}\frac{s_r^{+}}{y_{rp}})}$$

(5)

Total efficiency:

$$E_{overall}=\frac{1-(\frac{1}{3}{\displaystyle {\sum }_{i=1}^3}\frac{s_i^{-*}}{x_{ip}})}{1+(\frac{1}{3}{\displaystyle {\sum }_{r=1}^3}\frac{s_r^{+*}}{y_{rp}})}$$

(6)

We have indicated the efficiency of each DMU for the first stage with the symbol E₁, for the second stage as E₂, and the overall efficiency has been denoted with the symbol E_overall (ET). The data of the research is relevant to the indexes of 31 branches of an insurance company within the duration of the years 2014 to 2018. These are the input, intermediate, and output indexes of the model according to Figure 5.

Table 1. The average efficiency of the branches of the insurance company during the period from 2014 to 2018.

	2014–2015	2015–2016	2016–2017	2017–2018	2014–2018
DMU 1	0.869	0.718	0.766	0.759	0.778
DMU 2	0.767	0.722	0.713	0.612	0.704
DMU 3	0.654	0.604	0.570	0.621	0.612
DMU 4	0.522	0.447	0.420	0.435	0.456
DMU 5	0.679	0.634	0.600	0.574	0.622
DMU 6	0.777	0.727	0.774	0.746	0.756
DMU 7	0.872	0.825	0.897	0.922	0.879
DMU 8	0.753	0.601	0.619	0.563	0.634
DMU 9	0.695	0.716	0.681	0.682	0.694
DMU 10	0.877	0.918	0.899	0.902	0.899
DMU 11	0.610	0.546	0.539	0.549	0.561
DMU 12	0.682	0.680	0.698	0.604	0.666
DMU 13	0.592	0.530	0.505	0.521	0.537
DMU 14	0.780	0.702	0.672	0.652	0.701
DMU 15	0.863	0.856	0.919	0.903	0.885
DMU 16	0.531	0.574	0.475	0.509	0.522
DMU 17	0.847	0.780	0.840	0.841	0.827
DMU 18	0.781	0.670	0.702	0.637	0.697
DMU 19	0.979	0.943	0.967	0.904	0.948
DMU 20	0.738	0.679	0.766	0.719	0.726
DMU 21	0.495	0.544	0.498	0.512	0.512
DMU 22	0.823	0.795	0.791	0.748	0.789
DMU 23	0.810	0.757	0.781	0.808	0.789
DMU 24	0.721	0.703	0.705	0.687	0.704
DMU 25	0.686	0.645	0.596	0.599	0.632
DMU 26	0.640	0.612	0.611	0.619	0.620
DMU 27	0.809	0.728	0.709	0.647	0.723
DMU 28	0.645	0.649	0.630	0.683	0.651
DMU 29	0.741	0.715	0.697	0.630	0.696
DMU 30	0.926	0.774	0.889	0.774	0.841
DMU 31	0.701	0.651	0.608	0.643	0.651
The average efficiency of the branches					0.70038

illustrates the (E_overall) average total efficiency of each branch per annum. Likewise, the average efficiency of the entire branches throughout the period from 2014 to 2018, has also been calculated, which is equivalent to 0.70038. This value is utilized as an example of optimum efficiency frontier in the proposed model.

3.2. Data Estimation

In order to estimate data, the ANN, which is a powerful tool, is used for predictive purposes. The aim of this process is to learn a complex function of X and Z that is able to predict target variable Y. As stated above, by utilizing the GA, the X and Z parameters are determined. But since, in the insurance process, indexes such as damage coefficients (Y1), amount of damages (Y2), and as a result, the profit gained (Y3) are not under the control of the insurance companies, and are subject to losses and damages for those insured, it is crucial to predict these values. To do such prediction, we design three neural network architectures, named ANN1, ANN2, and ANN3 that are responsible for forecasting Y1, Y2, and Y3, respectively. Although the architecture of these networks is similar, the weights of the networks are different. The input and outputs of the network along with its hyper parameters are set according to

Table 2. Architecture of the network.

Name	Value
Input Dimension	5
Output Dimension	1
Activation Function	Tanh
Optimizer	Levenberg Marquardt
Iterations	1000
Architecture	[5-6-1]
Loss Function	Mean Squared Error

. It’s worth noting that the hyper parameters are found using cross-validation.

Despite the many solutions that have been published and used for a long time, normality testing remains an important issue for researchers. Despite the many results obtained, new approaches are being developed and improved [72,73,74]. In a study, Song et al. presented a method called JBsum (modified Jarque-Bera method) to test the normality of the data. They showed that the JBsum method is more stable than many other compared methods, and both in low and high dimensions, the proposed method performs well in testing normal and non-normal data [74]. Therefore, we also used this method in this research and the results indicated that the data were normal.

The index values for 31 branches have been employed, in the past four years, for the network training. This has been conducted in such a manner that from amongst the entire educative data, which was previously normalized, 70 percent has been used for network training and 30 percent for validation appraisal.

The way in which neural function works is that network inputs simultaneously form the first layer of the network. The outputs that are weighed in the first layer create the middle layer, and the output layer inputs are composed of middle layer outputs.

After being multiplied by the relevant weights and the transfer function imposed, the final output of the network is attained. It is essential to note that the number of neurons of the intermediate layer is determined on a trial-and-error basis, such that, it has the minimum average of assessment error, and is being utilized as the optimal neuron to determine the model. The number of optimal neurons, in accordance with the assessment error results, equates to 6, which has the lowest amount of error, being 0.0324. Hence, the trained network predicts the output value Y, by receiving the input values of X and Z.

The structure of ANN2 and ANN3 neural network is also the same as ANN1 (Table 2). But the number of optimal neurons of ANN2 network with the lowest amount of error being 0.0359 is equal to 5 and for ANN3 with the lowest amount of error being 0.0318 is also equal to 5.

3.3. Optimization of Efficiency

The next step is optimizing the efficient units, or in other words, optimizing the efficiency of all the branches is pursued. In this research, optimization has been conducted with the GA, which basically differs from the traditional optimization methods. The method of work followed in this algorithm is that, initially, the parameters that the search space created are in the form of strings called chromosomes. Each chromosome represents a response to the problem under consideration. The representation of data as a chromosome is shown in Figure 6.

Chromosomes together, as an aggregate, form a set called population and at the beginning of the operation, usually the initial population elements are selected randomly. It operates in an iterative manner on the population element and imposes two performances of crossover and mutation, and from one population a new one generates. Eventually, after a finite repetition, the responses under consideration are produced in the final generation. It is obvious that all the answers are surely not the optimum responses. In order to determine the degree of optimality of each response, a criterion known as the objective function is utilized. In practice, the objective function assigns a value to each population of chromosomes of a generation, giving it an amount of relativity and this amount determines the fitness of this response in relevance to the remaining responses in concern with the same generation.

The optimization of targets is to gain access to the most advantageous efficient units or in other words, maximize the total values of efficiency of the branches. However, as mentioned, if the distance between inefficient branches and the optimum efficiency frontier is significant, in practice, the possibility of extensive changes is not possible due to limited resources, and achieving efficiency with the least change in resources is always under contemplation by managers. Thereby, another case in defining the cost function of the GA is to reach the efficiency frontier with minimal changes in the input indexes X and Z.

Since the initial population genetic algorithms are usually created randomly with due attention to the limitations of the problem, it is a prerequisite for accepting the generated response as an initial answer, or that, this could be acceptable in the subsequent steps. For this purpose, every unacceptable response was taken as to be the conventional answer for the next generations. Though, in calculating the cost function, a penalty proportional to the amount of its encroachment or violation from the permissible limits is supplemented to the related cost function. In this way, the possibility of selecting an unwarranted response as one of the parents for the production of future generations is intensely reduced. However, this possibility for it remains that, if it is grafted with other responses present in the population for better production of children, this child shall take place as a new response in the following generation. Thus, the penalty function is selected in a manner, so that the variables exceeding the permissible limits prevent the algorithm, by way of its magnification, as well as decrease the reproduction of unsuitable chromosomes. Thus, we consider the acceptable range for efficiency values, and similarly, the maximum modifications in resources into view. The cost function algorithm is explained in Algorithm 1.

Algorithm 1. Cost function of the GA.

Input: Predicted input (X) and middle (Z) variables of 31 DMUs Output: Cost of this prediction based on predicted efficiencies and constraints violations.

1) Input: $X_{in}$ 2) Extract $X1, X2, X3$, and $Z1, Z2$ from $X_{in}$ based on Figure 6 3) Predict $Y1, Y2, Y3$ using $ANN1, ANN2$, and $ANN3$ 4) Solve the corresponding DEA 5) Compute efficiencies $E1, E2, ET$ 6) $TotalEfficiencies = {\displaystyle \sum }{E}_T$ 7) $MinEfficiency = MEAN\left(ET\right)$ for last month frontier (e.g., 0.7) 8) $EfficiencyConstraint = {\displaystyle {\sum }_{i=1}^{31}}\left|E{T}_i-\mathrm{MinEfficiency}\right|$ 9) $DeltaX =$ difference between the last values of $X$ and the current predicted $X$ 10)   $DeltaZ =$ difference between the last values of $Z$ and the current predicted $Z$ 11)   $CostLast = DeltaX + DeltaZ$ 12)   Violations = Sum of input restriction violation of $X$ and $Z$ (e.g., more than 50% of changes) 13)   $Constraints = EfficiencyConstraint + Violations$ 14)   $Cost = -TotalEfficiencies + CostLast + {10}^3 * Constraints$

This function receives a 155 vector in the form of Figure 6 and the variables X1, X2, X3, Z1, and Z2 are extracted from it. In this model, the total efficiency value is considered equal to ${\displaystyle \sum }{E}_T$ and the minimum acceptable efficiency is equal to the average efficiency of all branches during the study period (0.7), and finally we have set Efficiency Constraint value in line 8 as the sum of the difference between the efficiency value calculated by DEA and the minimum acceptable efficiency. Next, we calculated ΔZ and ΔX based on the difference between the model prediction and the latest available data (real data). This helped us to make the variables proposed by the genetic algorithm as similar to the real data as possible. According to line 12, we saved the amount of violation of the maximum allowed changes for X and Z values (50%) in Violations. It is worth mentioning that the value of CostLast = ΔZ + ΔX and finally the output of the algorithm will be Cost = −TotalEfficiencies + CostLast + 10³ × Constraints.

Since it is practically infeasible to achieve 100 percent efficiency for the entire number of branches, various runs of GA with different parameters have been applied to the cost function. We will discuss these parameters in the next section.

4. Results

At the first stage, we should train the neural networks. The training process uses the scaled data as the input and output. These data are obtained from the frontier computed by a DEA model. Using 70% of the data for the train phase and 30% for test case, the mean squared error loss of the model is reported in

Table 3. Results of the three neural networks.

Model	Train MSE Loss	Test MSE Loss
ANN1	1.85 × 10−2	1.97 × 10−2
ANN2	1.0628 × 10−4	1.1302 × 10−4
ANN3	9.6 × 10−3	2.9950 × 10−4

. The performance of the models is also shown in Figure 7. It can be seen that the proposed ANN model can accurately approximate the frontier target variables. This is because we have used the frontier data in the training phase. In a nutshell, the proposed architecture can accurately predict the target variables using the input and mediator variables. But this is not sufficient to maximize the efficiency of all DMUs. To handle this issue, as proposed in the previous section, a GA is designed in order to find the best X and Z variables for the near future.

Using these models, the GA can be employed to generate the approximate input variables X and Z such that the efficiency of branches is maximized for the near future. To show the stability of the GA, various sets of parameters have been used to optimize the cost function. The parameters are explained in

Table 4. GA Parameters.

Parameter-Set Number	Population Size	Crossover Fraction	Max Generations
#1	10	0.1	50
#2	25	0.1	50
#3	50	0.1	50
#4	10	0.8	50
#5	25	0.8	50
#6	50	0.8	50
#7	50	0.3	100
#8	100	0.3	85

.

Table 5. The best efficiencies given by optimizers. Best values are highlighted in bold.

Algorithm	Parameters	Cost	Cost Last	Constraints	E1		E2		ET
					Mean	Std.	Mean	Std.	Mean	Std.
fmincon	interior-point	−4.393	23.231	0	0.927	0.074	0.852	0.099	0.891	0.073
fmincon	sqp	2.918	31.367	0	0.972	0.044	0.856	0.090	0.918	0.052
fmincon	sqp	2.968	31.367	0	0.970	0.045	0.855	0.089	0.916	0.052
Ga	#1	207.797	42.4700	0.191	0.824	0.153	0.868	0.130	0.847	0.118
Ga	#2	10.6838	38.3486	0	0.856	0.146	0.926	0.094	0.892	0.098
Ga	#3	11.5586	38.4782	0	0.805	0.165	0.924	0.097	0.868	0.090
Ga	#4	17.1953	40.0725	0.0035	0.852	0.143	0.852	0.141	0.852	0.094
Ga	#5	4.5829	33.4537	0	0.888	0.133	0.971	0.049	0.931	0.074
Ga	#6	2.3208	31.0665	0	0.896	0.126	0.957	0.070	0.927	0.067
Ga	#7	4.5853	32.9614	0	0.856	0.130	0.968	0.049	0.915	0.075
Ga	#8	2.0215	30.4371	0	0.878	0.130	0.952	0.061	0.917	0.072

reports the results of these parameters of the GA using maximum difference on X and Z set to 50% and minimum efficiency to 0.7. For comparison purpose, two different mathematical optimizers have been employed to minimize the cost function. This table reveals that the numerical optimizers usually focus on the efficiency of the first stage, while the GA tries to find the most efficient inputs such that the E_T is maximized.

Figure 8 shows the box plot of the predicted efficiencies for the near future. In order to reach this overall efficiency, the input and mediator variables should be updated. These changes are reported in Figure 9, in terms of percent change to the last timestamp. Applying these changes, the predicted output variables, using the neural network models is also included in this figure. In fact, Figure 9 shows the percentage of changes that must be applied to the index of each branch so that all branches reach the efficiency frontier. For example, indexes X3 and X1 in most branches should increase and the value of X2 should decrease. Also, the value of Z2 for most branches should be accompanied by an increase.

Figure 10 illustrates the convergence process of GA. In observing the graph, the convergence speed in the initial iterations is exceptionally great, but gradually, this velocity reduces and finally, the intensity of convergence stabilizes.

It is distinctive that the optimal convergence fitness function demonstrates a diverse convergence for itself at each point in time. The results attained show that the GA converges to the correct response. As observed in this figure, the algorithm converges to the correct response after approximately twenty generations of convergence.

In the next experiment, a different set of parameters on the cost function is applied. It is seen that the model cannot achieve 0.8 total efficiencies with a maximum of 25% or even 50% changes on X and Z.

Table 6. Results of different parameters of cost function.

Max-Difference	Min-Efficiency	Cost	Cost Last	Constraint	Mean (E1)	Mean (E2)	Mean (ET)
25%	0.6	−9.27	17.945	0	0.839	0.913	0.878
25%	0.7	−10.02	17.517	0	0.844	0.928	0.888
25%	0.8	44.16	20.704	0.052	0.877	0.980	0.932
50%	0.6	−9.21	17.850	0	0.879	0.866	0.873
50%	0.7	4.58	33.454	0	0.888	0.971	0.931
50%	0.8	9.831	35.357	0.003	0.865	0.966	0.919
75%	0.6	4.98	33.996	0	0.916	0.955	0.936
75%	0.7	24.84	52.934	0	0.889	0.923	0.906
75%	0.8	22.84	51.925	0	0.905	0.969	0.938

shows the detailed results of this experiment using fifth parameter-set introduced in Table 4.

The analysis of best solutions given by GA and the numerical optimizers, show that the numerical methods usually apply higher changes of X and Z variables, while the GA tries to maintain the original variables with minimum changes. See

Table 7. Total changes, given by different optimizers.

	GA			Numerical Optimizer
Variable	Min	Mean	Max	Min	Mean	Max
X	1%	50%	105%	27%	68%	133%
Z	0%	24%	73%	7%	40%	65%
Y	23%	264%	949%	52%	242%	467%

for more details.

Finally, the predicted efficiency is compared with the average efficiency of the branches using the proposed model. The results in

Table 8. A comparison of the efficiency results predicted for 31 branches of an insurance company, with the average efficiency of the branches throughout the period.

DMU	Emean	ET	DMU	Emean	ET
DMU1	0.778	1.000	DMU17	0.827	0.922
DMU2	0.704	0.858	DMU18	0.697	0.984
DMU3	0.612	0.921	DMU19	0.948	1.000
DMU4	0.456	0.760	DMU20	0.726	0.917
DMU5	0.622	0.985	DMU21	0.512	0.854
DMU6	0.756	0.966	DMU22	0.789	1.000
DMU7	0.879	1.000	DMU23	0.789	0.987
DMU8	0.634	0.817	DMU24	0.704	1.000
DMU9	0.694	0.941	DMU25	0.632	0.840
DMU10	0.899	0.965	DMU26	0.620	0.919
DMU11	0.561	0.869	DMU27	0.723	0.999
DMU12	0.666	0.924	DMU28	0.651	1.000
DMU13	0.537	0.952	DMU29	0.696	0.861
DMU14	0.701	1.000	DMU30	0.841	1.000
DMU15	0.885	0.735	DMU31	0.651	1.000
DMU16	0.522	0.893

show that the efficiency in all branches exceeds the acceptable efficiency of 0.7 determined by the optimum efficiency frontier.

5. Conclusions

In this article, the combined DEA-ANN-GA model was used to estimate the efficiency frontier of 31 branches of an insurance company in Iran. In the first step, the efficiency of the branches was calculated using the technique of DEA, and then in the second step, the future data was estimated using the predictive power of the ANN, and finally, the efficiency frontier was estimated using the optimization capacity of the GA.

As a result, it became possible to build the efficiency frontier of the next period using new input and output parameters after estimating the outputs, and to find the best performance model and achieve the near-optimal efficiency frontier (average efficiency 0.7) with the limitation of resource changes of a maximum 50%.

The results showed that achieving the efficiency frontier with the lowest cost will be possible for all branches simultaneously in the future. Therefore, the managers of the company can provide the possibility of achieving an acceptable level of efficiency in the future by planning correctly and formulating the appropriate vision and goals for each branch.

According to the presented methods and models, the following suggestions are provided for future research.

• Development of mathematical relations, in order to analyze the sensitivity in DEA evaluation models, in such a way that it measures the sensitivity in the parameters relatively. In this way, before the linearization of DEA fractional models, whose objective function is the result of dividing the composition of outputs by the composition of inputs, it is possible to analyze the sensitivity of the output parameters to the changes in the input parameters and, as a result, analyze the sensitivity of the efficiency to the changes made in the inputs and outputs.
• Using deep learning models such as LSTM (Long Short-Term Memory) and GRU (Gated Recurrent Unit) as well as recurrent networks such as RNN (Recurrent Neural Networks) to estimate the efficiency frontier in such a way that nonlinear and hidden features of the data are modeled. Learn non-linearity and facilitate mapping and decision-making.
• Using other meta-heuristic algorithms such as “cuckoo optimization algorithm”, “ant colony algorithm”, and “artificial bee colony algorithm” in order to increase the accuracy of the model and also reduce its execution time.