Combination of the Data Envelopment Analysis and the Discriminant Analysis for Evaluating Bankrupt Business in a Fuzzy Environment

ABSTRACT This paper presents a combination of the data envelopment analysis (DEA) and discriminant analysis (DA) to evaluate the bankrupt business in a fuzzy environment. The DEA is a non-parametric method that can be used for various assessments. The DA is a statistical method that can predict an appropriate group for new observations. The combination of DEA and DA methods creates a powerful method that includes the advantages of both methods. According to the special features of this method (e.g. high resolution and assessment accuracy), it can be used for a bankruptcy assessment of organisations. In normal conditions, accurate measurement of data is very difficult, which is why considering the uncertainty conditions in models can make them more applied. Using a fuzzy condition in models can help this issue. Finally, the results are illustrated and discussed.


Introduction
The need to study and assess the status of an operating company and organisation is essential in terms of financial issues, particularly from the point of view of bankruptcy. If this issue is not predicted prior to the occurrence, it will lead to high costs for the owners of the company and its shareholders. Now, the question is that given the urgent need to study the bankruptcy of organisations, which method will be useful for this type of assessment. Much discussion is raised to answer to this question that one of the most up-to-date issues is the use of mathematical methods of the data envelopment analysis (DEA) and a combination of this method with other methods that this combination can be used in order to assess the bankruptcy of organisations [1].
The DEA is a non-parametric method that can be used for various assessments [2]. This method can be used in various types of assessments in different branches [3]. Using the DEA is a useful tool for determining the relative effectiveness and weaknesses of the organisation in various indicators. After a while of using this method, it was determined that this approach has had some weaknesses; therefore, proposals were put forward among, which it could be noted that by combining the DEA with other science topics, including statistics, we can reach a powerful method for the evaluations, including the evaluation of bankruptcy of organisation [1]. Using the DEA method by combining it with the discriminant analysis (DA) method can be a useful tool in financial studies, particularly in assessing bankruptcy of organisations. This method was originally proposed by Sueyoshi [4], and completed over time until finally this method was used by Sueyoshi and Goto [1] to assess bankruptcy.
Due to the high costs of the bankruptcy of organisations, a detailed assessment, prior to the occurrence of bankruptcy, is very important. In general, researches on bankruptcy assessment are classified into three groups. The first group that includes the most significant studies investigates the current processes in the organisation [5]. The second group focuses on a specific model, in which the related studies investigate the performance of a model in predicting the failure compared to another model [6]. Finally, the third group focuses on selecting the appropriate variable to be used in a specific model. It is important to select an appropriate sample and variable to assess bankruptcy [7]. In Section 2 of this paper, the literature review is discussed. In Section 3, the basic issues are discussed, and also the proposed model is discussed that has a new approach based on the existing models and fuzzy conditions. Also, in Section 4, a numerical example is presented with its solution and compared with other methods to elaborate further investigation. In Section 5, the conclusions and recommendations for future research are presented.

Literature Review
In this section, the research background and background of the fundamental issues used in this research are discussed. The DEA model proposed by Charnes [8] has been used by a group of researchers. Agarwal [9] proposed the traditional DEA model to a fuzzy framework using a fuzzy DEA model based on the α-cut approach to deal with the efficiency measuring and ranking the problem with the given fuzzy input and output data. Babazadeh et al. [10] proposed a unified fuzzy DEA (UFDEA) for sustainable cultivation location optimisation under uncertainty. Kordrostami et al. [11] proposed a method for measuring the overall and period efficiencies of DMUs under uncertainty. The proposed approach is illustrated and clarified by two numerical examples. Ashrafi and Mandouri Kaleibar [12] proposed the generalised cost, revenue and profit efficiency models in a fuzzy DEA. Khanjarpanah and Jabbarzadeh [13] developed a novel approach entailing DEA with cross-efficiency and fuzzy-cross-efficiency models to find the most suitable locations for wind plants establishment.
Predicting group membership of sustainable suppliers via the DEA and DA is another research. This article proposed a novel super-efficiency stochastic DEA model for measuring the relative efficiency of suppliers in presence of zero data. By proposing the model, all suppliers are classified into two efficient and inefficient groups based on their efficiency score. Then, to predict a group membership of a new supplier, a novel stochastic MIP model is presented [14].
Another article proposed an approach that combined the Data Envelopment Analysis-Discriminant Analysis (DEA-DA), DEA environmental assessment and a rank sum test. The proposed approach is designed to overcome the following difficulties: (a) how to classify various decision-making units (DMUs) into different groups, (b) how to identify the existence of group heterogeneity across DMUs, (c) how to measure unified efficiencies of a power industry in different regions of China, (d) how to separate among various unified efficiency measures and (e) how to unify these measures into a single measure which expresses total efficiency [15].
In another study, an oil refinery performance was assessed by the DEA-DA. This study examined the operational efficiency of refineries and conducted an efficiency-based rank assessment by using an unbalanced panel dataset comprised of oil and gas refineries in four global regions (i.e. U.S. and Canada; Europe; Asia-Pacific; Africa and the Middle East). This study applied a combination of the DEA and DEA-DA to examine the efficiencybased rank for oil and gas refineries [16]. Fallah et al. [17] designed a new modelling to find hyper planes for separating two sets by using the DEA and DA. Modelling was performed based on the different criteria that have existed, and each one applies in certain circumstances.
In the following, the properties of the designed model are expressed and proved. The specific conditions of the criteria have become limitations that have been added to the multiplicative form of the designed model.

Research Methods
In this section, the basic principles of the issue and all of the principles used in this research are mentioned, such as the theoretical background and models required for the study. Also, the proposed model is presented.

Discriminant Analysis
The DA is a statistical method for a classification that is used to assign observations to adequate groups. This method has different classification models that a few of them are mentioned here. The linear diagnostic analysis or Fisher's method tries to find a linear relationship of discriminant features of observation so that be able to assign a new observation to an appropriate group. This method works in a way that converts multivariable observations of x to mono-variable observations of y, so that y obtained from the communities is separated as much as possible [18]. Using the DA features of a goal programming approach in the diagnostic analysis is one of the issues raised in classification issues. This method was presented for the DA and researchers comparison with the model group. This method has the disadvantage that it can include the inability to classify observations noted that the data are negative [19]. This method was presented for the DA and researchers began to compare it with the additive model in the DEA. This method has some disadvantages, such as the inability to classify observations with negative data [19]. The DA based on the mixed integer programming method by defining integer variables classifies and assigns observations. The advantages of this method are capable of using negative data [19].

Data Envelopment Analysis
The DEA is a nonparametric method that can assess the decision-making unit, which has several inputs and outputs. Two basic models of DEA are described here. The purpose of the CCR (Charnes, Cooper & Rhodes) model in an input nature is to find virtual decisionmaking unit that can produce Y 0 output with the minimum input. Assume a set of observed DMUs, {DMU j; j = 1, ... ,n} is associated with m inputs, {x ij ; i = 1, . . . , m} and s outputs, {y rj ; r = 1, . . . , s}. Model 1 shows the CCR model in an input nature [8].
The BCC (Banker, Charnes & Cooper) model in an input nature was developed with the CCR model. The possibility of using this model is obtained through the elimination of the infinite ray principles from the series of principles of the DEA. Model 2 shows the BCC model is the input nature.

Combination of the DEA and DA
The method of the combination of the DEA and DA consists of two steps [1].
First step: the classification of data that do not overlap: the first two groups of G 2 , G 1 are considered, which have n observations in sum, where (j = 1, . . . , n) that n 1 + n 2 , where G 1 is observations with financial failure (j = 1, . . . , n 1 ) and G 2 is observations without financial failure (j = 1, . . . , n 2 ). Each observation is determined by h independent factor (f = 1, . . . , h) by z fj and so λ f indicates weight of the f-th factor that in general will operate as in Model 3: The objective function in this model minimises s, which represents the size of the overlap between G 2 , G 1 . Overlap by d-s has been limited as the lower limit of G 1 and as the upper limit of G 2 . The diagnostic rate is considered by the numerical value of d for the classification of groups. Also, d and s are infinite variables. The very small number of is used to avoid problems in segmentation. All financial variables of z fj of the j-th company are connected by the discriminant function of After solving the aforementioned model, the values of λ * , s * , d * are obtained that Equations (4) and (5) must be checked by using them to determine observations related to G 1 and G 2 surely.
Members of C 1 series are fully owned by G 1 and also members of C 2 are fully owned by G 2 . After this step is finished D 1 = G 1 − C 1 and D 2 = G 2 − C 2 must be determined. D 1 ∪ D 2 indicate observations that are not included in the classification due to overlapping that in order to solve this problem it is needed to enter the second phase of the model.
s.t. : Variable y j determines the number of variables that have been classified wrongly and minimises them. For accurate calculation, a large number of M and a small number of ε must be considered in the model. Similarly, a diagnostic rate of c should be used in the model, which is a free sign symptomatically. This model is an alternative to d in the first model. It should be noted that λ + f , λ − f should not be zero at the same time, for this case, the control condition of λ + f + λ − f ≥ ε must be used. Also, w is the weight to validate two groups of observations. After solving the model, the values of λ * , c * are obtained, which by solving Equations (7) and (8), the final classification can be achieved.
So, at the end of this step, all observations are classified in G 1 and G 2 .

Fuzzy Data Envelopment Analysis
Research on the fuzzy DEA began from a study by Sengupta [20]. This study proposes two approaches for solving the DEA model, in which indefinite data are used. The first is a probable approach to solve problems and the second is an approach based on fuzzy systems. The second approach is based on researches by Zadeh [21]. In the study by Sengupta [20], two membership functions are discussed for fuzzy numbers. The advantage of a linear membership function is the ability to take advantage of the linear programme in solving DEA problems in an indefinite condition. In this paper, an approach based on linear programming in the model discussed is also used. One of the basic fuzzy models in DEA can be considered as follows: Inputsx j and the outputsỹ j of the DMUj are fuzzy variables, j = 1, . . . , n. Since the fuzzy constraints v Tỹ j ≤ u Tx j do not define a deterministic feasible set, a natural idea is to provide a confidence level 1-α at which it is desired that the fuzzy constraints hold. In other words, the constraints will be violated at most α. Thus we have some chance constraints as follows: Considering the chance constraints (4), the fuzzy DEA model can be written as follows: The greater the optimal objective is, the more efficient DMU0 is ranked.

Defuzzification Method
One of the defuzzification methods of fuzzy models is a method offered by Jimenez [22] and Jimenez et al. [23]. This method operates based on defining the expected value and expected distance in fuzzy numbers, which was developed by Yager [24] and Dubois and Pradein [25] and was followed by Jimenez [22] and Heilpernin [26]. Based on this method it is possible to defuzzify the two sides of the equation (i.e.ã as limitation coefficients andb as the right number), which are fuzzy. This method uses Relation (9) for this operation.
When μ M (ã,b) > α, it can be said thatã ≥b and regarding the degree of α is written aš a≥ ∝b . Based on the definition of fuzzy equation studied by Para et al. [27], for every pair of fuzzy numbersã andb it can be said thatǎ is equal tob in α degree if the above equations simultaneously exist asǎ≤ ∝ 2b ,ǎ≥ ∝ 2b . These equations can be written by: If the sample fuzzy model is in the form of Model 11: According to research carried out by Heilpern [26], the decision vector of x R n is justified in α degree, if the condition Min i=1,...,m {μ M (ã i x,b i )} = α is satisfied. Based on Relations (9) and (11), the equations ofã i x ≥b i andã i x =b i are equal to Equations (12) and (13).
These equations can be written by: Using the rating and using Jimenez [22] and Jimenez et al. [23], a justifiable solution of x 0 as the optimal solution of α is acceptable. It is possible to prove the justifiable solution for this model, if and only if for every justifiable decision vector it is said x as a i x≥ α b i , i = 1, . . . , l and a i x≈ α b i , i = l + 1, . . . , m and x ≥ 0 that in this case, Equation (17) is achieved: So, x 0 is the best option that is opposed to other justifiable vectors at minimum degree of 1 2 . Accordingly, Relation (15) can be rewritten by: The outcome is that using the above explanations, Model 11 can be written as Model 19:

Proposed Model
The characteristics of the proposed fuzzy model are as the definite model and it is composed of two steps that the first step does the classification. If there is an overlap, it is needed to enter the second step of the model. The model proposed in this paper acts using fuzzy data. Changes of the model defuzzification on the financial factor specified by z fj is created. By applying fuzzification changes on this factor, the definite model changes and is written as model 20.
Min s (20) s.t. : To convert the above fuzzy model to the definite model, the previously described defuzzification method is used. After using this method, Model 20 is written as Model 21: Min s (21) s.t. : d&s : free in sign, ζ + f , ζ − f : 0 or 1, other variables ≥ 0 After solving the above model, the values of λ * , s * , d * are obtained, in which Equations (22) and (23) must be checked using them to surely determine the observations related to G 2 and G 1 .
For the second step of the model, it is possible to convert its fuzzy mode to non-fuzzy model using the method described, whose non-fuzzy model is as follows:

Numerical Example
An example discussed in this section is taken from Sueyoshi [4]. Items included in this example include 20 companies and two financial factors for each of the companies. 10 first companies are classified in group G 1 and 10-second companies are classified in group G 2 . In other words, we have: f = 1, 2, j 1 = 1, . . . , 10, j 2 = 11, . . . , 20. Numbers of factors are given in Table 1.
Considering the necessity to use fuzzy numbers in the model, the above numbers must be converted to fuzzy numbers. To do this, the research carried out by Lai and Huang in [28] is used. This method considers the definite number as C m and then two random numbers between 0.2 and 0.8 are considered as r 2 andr 1 respectively. The left triangular fuzzy number is obtained by using c 0 = (1 + r 1 )c m and the right triangular fuzzy number is obtained using c p = (1 − r 2 )c m . The obtained numbers for α = 0.2, 0.5 and 0.8 are given in Tables  2-4, respectively. Also, the numbers used as a coefficient and according to defuzzification of the model are given in Tables 5, 6, 7 with names EI2 and EI1. After solving the model according to the  values of Tables 5, 6, and 7 the optimal responses are obtained, the optimal responses are shown in Table 8. Also, we have:      Classification is done using formulas (20) and (21) and according to the optimal values obtained and the results of this classification are presented in Tables 9-12

Conclusions
In this study, the combination of the DEA and the DA was discussed in a fuzzy environment for the evaluation of bankrupt business. The need to investigate the status of companies in terms of finance and commerce, especially in terms of bankruptcy, could have an important role in determining the financial future of the organisation. In this study, a model was created based on the existing models in order to be able to assess companies in terms of bankruptcy, in a case where definite data were not available. The present model that was in a definite mode was performed in a classification in two phases. Then, by a new approach, a fuzzy model was first created and then using a special approach, previously discussed, the existing fuzzy model was defuzzified. In many cases, the lack of definitive data lead to the deficiency of models in a definite mode; therefore, the existence of a model that can assess using indefinite data was necessary. In this study, using a DEA-DA model in fuzzy conditions, a numerical example was examined to evaluate the adequacy of the proposed model. To prove the efficiency of the model using financial data taken from Sueyoshi [4], a numerical example was solved and the results of it were compared with the results of solving the crisp model. Due to the special relationship to determine the proper allocation percent, the percentage of correct assignment of crisp and fuzzy based on α = 0.2, 0.5 models is obtained 90% and fuzzy based on α = 0.8 obtained 0.85%. For future research and development of this model. It is recommended to study models that deal with two groups in classification and operate with indefinite data that can create a more complete model and more applied to classify the observations into several groups and to use it in indefinite conditions. This model can also be used in a type-2 fuzzy condition or interval data and by creating a new model extend the usage range of this model.

Ethical Approval
The authors certify that they have no affiliation with or involvement with human participants or animals performed by any of the authors in any organisation or entity with any financial or non-financial interest in the subject matter or materials discussed in this paper.

Disclosure statement
No potential conflict of interest was reported by the author(s).