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

Combination of the Data Envelopment Analysis and the Discriminant Analysis for Evaluating... FUZZY INFORMATION AND ENGINEERING 2022, VOL. 14, NO. 2, 212–227 https://doi.org/10.1080/16168658.2022.2117514 RESEARCH ARTICLE Combination of the Data Envelopment Analysis and the Discriminant Analysis for Evaluating Bankrupt Business in a Fuzzy Environment a a,b b Navid Torabi , Reza Tavakkoli-Moghaddam and Ali Siadat a b School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran; Arts et Métiers Institute of Technology, Université de Lorraine, LCFC, HESAM Université, Metz, France ABSTRACT ARTICLE HISTORY Received 1 April 2019 This paper presents a combination of the data envelopment analysis Revised 21 December 2021 (DEA) and discriminant analysis (DA) to evaluate the bankrupt busi- Accepted 23 August 2022 ness in a fuzzy environment. The DEA is a non-parametric method that can be used for various assessments. The DA is a statistical KEYWORDS method that can predict an appropriate group for new observations. Data envelopment analysis; The combination of DEA and DA methods creates a powerful method discriminant analysis; that includes the advantages of both methods. According to the spe- bankrupt business; fuzzy condition cial features of this method (e.g. high resolution and assessment accuracy), it can be used for a bankruptcy assessment of organisa- tions. In normal conditions, accurate measurement of data is very difficult, which is why considering the uncertainty conditions in mod- els can make them more applied. Using a fuzzy condition in models can help this issue. Finally, the results are illustrated and discussed. 1. Introduction The need to study and assess the status of an operating company and organisation is essen- tial 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 com- pany 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 organ- isation 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, CONTACT Reza Tavakkoli-Moghaddam tavakoli@ut.ac.ir © 2022 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society, Guangdong Province Operations Research Society of China. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. FUZZY INFORMATION AND ENGINEERING 213 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 signif- icant 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 impor- tant 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. 2. 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 effi- ciency 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 exis- tence 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 214 N. TORABIETAL. 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 efficiency- based 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 per- formed 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. 3. 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. 3.1. 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 rela- tionship 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 observa- tions of x to mono-variable observations of y,sothat 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 inte- ger programming method by defining integer variables classifies and assigns observations. The advantages of this method are capable of using negative data [19]. 3.2. 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 decision- making unit that can produce Y output with the minimum input. Assume a set of observed DMUs, {DMU j; j = 1, ... ,n} is associated with m inputs, {x ; i = 1, ... , m} and s outputs, ij FUZZY INFORMATION AND ENGINEERING 215 {y ; r = 1, ... , s}. Model 1 shows the CCR model in an input nature [8]. rj Min θ (1) s.t. : λ x ≤ θx , i = 1, ... , s j ij io j=1 λ y ≥ y , r = 1, ... , s j rj ro j=1 λ ≥ 0, j = 1, ... , n 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. Min θ (2) s.t. : λ x ≤ θx , i = 1, ... , s (1) j ij io j=1 λ y ≥ y , r = 1, ... , s (2) j rj ro j=1 λ =1(3) j=1 λ ≥ 0, j = 1, ... , n (4) 3.3. 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 , G are 2 1 considered, which have n observations in sum, where (j = 1, ... , n) that n + n , where G 1 2 1 is observations with financial failure (j = 1, ... , n ) and G is observations without finan- 1 2 cial failure (j = 1, ... , n ). Each observation is determined by h independent factor (f = 1, ... , h) by z and so λ indicates weight of the f-th factor that in general will operate as in fj f Model 3: Min s (3) + − s.t. : (λ − λ )z − d + s ≥ 0, jD fj f f f =1 + − (λ − λ )z − d − s ≤ 0, jD fj f f f =1 216 N. TORABIETAL. + − (λ + λ ) = 1 f f f =1 + + + ζ ≥ λ ≥ εζ , f = 1, ... , h f f f − − − ζ ≥ λ ≥ εζ , f = 1, ... , h f f f + − ζ + ζ ≤ 1, f = 1, ... , h f f + − λ + λ ≥ ε, f = 1, ... , h f f + − d&s : free in sign, ζ , ζ : 0 or 1, other variables ≥ 0 f f The objective function in this model minimises s, which represents the size of the overlap between G , G .Overlapby d-s has been limited as the lower limit of G and as the upper 2 1 1 limit of G . 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 of the j-th company are connected fj + − by the discriminant function of (λ − λ )z and the weights are so limited that the total fj f f f =1 ∗ ∗ amount λ is 1. After solving the model, if s ≥ 0 indicates that there are overlap and s < 0 + + + indicates a lack of overlap. ζ and εζ indicate the upper and lower limits for λ variable, f f f respectively. And the same conditions are established for λ . Due to the numerical value of + − ζ variable that is zero or one it is not possible to measure ζ , ζ simultaneously. Because of f f + − + this condition, it is not possible to measure λ , λ as well. Similarly, the condition of λ + f f f − + − λ ≥ ε is to avoid the simultaneous zero value of λ , λ . After solving the aforementioned f f f ∗ ∗ 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 and G surely. 1 2 ⎧ ⎫ ⎨ ⎬ ∗ ∗ ∗ C = j ∈ G | λ z > d + s (4) 1 1 fj ⎩ ⎭ f =1 ⎧ ⎫ ⎨ ⎬ ∗ ∗ ∗ C = j ∈ G | λ z < d − s (5) 2 2 fj ⎩ ⎭ f =1 Members of C series are fully owned by G and also members of C are fully owned by 1 1 2 G . After this step is finished D = G − C and D = G − C must be determined. D ∪ D 2 1 1 1 2 2 2 1 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. Second step: the classification of data that overlap: This step is performed using Model 6. Min y + w y (6) j j j∈D j∈D 1 2 + − s.t. : (λ − λ )z − C + My ≥ o, jD fj j 1 f f f =1 + − (λ − λ )z − C − My ≤−ε, jD j 2 fj f f f =1 FUZZY INFORMATION AND ENGINEERING 217 + − (λ + λ ) = 1 f f f =1 + + + ζ ≥ λ ≥ εζ , f = 1, ... , h f f f − − − ζ ≥ λ ≥ εζ , f = 1, ... , h f f f + − ζ + ζ ≤ 1, f = 1, ... , h f f + − λ + λ ≥ ε, f = 1, ... , h f f + − C : free in sign, ζ , ζ , y : 0 or 1, other variables ≥ 0 f f Variable y 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 , λ should not be zero at the same time, for this case, f f + − the control condition of λ + λ ≥ ε must be used. Also, w is the weight to validate two f f ∗ ∗ 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. ∗ ∗ If λ z ≥ c then j ∈ G (7) fj 1 f =1 ∗ ∗ If λ z ≤ c − ε then j ∈ G (8) fj 2 f =1 So, at the end of this step, all observations are classified in G and G . 1 2 3.4. 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 Sen- gupta [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: Inputs x ˜ and the outputs y ˜ of the DMUj are fuzzy variables, j = 1, ... , n. Since the fuzzy j j T T constraints v y ˜ ≤ u x ˜ do not define a deterministic feasible set, a natural idea is to provide j j 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: T T Cr{v y ˜ ≤ u x ˜ }≥ 1 − α, j = 1, 2, ... , n j j 218 N. TORABIETAL. Considering the chance constraints (4), the fuzzy DEA model can be written as follows: v y max θ = Cr ≥ 1 u,v u x ˜ T T ˜ ˜ s.t : Cr{v y ≤ u x }≥ 1 − α, j = 1, 2, ... , n j j u ≥ 0 v ≥ 0 where α ∈ (0, 0.5] The greater the optimal objective is, the more efficient DMU0 is ranked. 3.5. 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. a ˜ as limitation coefficients and b as the right number), which are fuzzy. This method uses Relation (9) for this operation. a b ⎪0if E − E < 0 ⎪ 2 1 a b E −E 2 1 a b a b μ (a ˜, b) = if 0 ∈ [E − E , E − E ] (9) a b a b 1 2 2 1 E −E −(E −E ) 2 1 1 2 a b 1if E − E > 0 1 2 ˜ ˜ When μ (a ˜, b)>α, it can be said that a ˜ ≥ b and regarding the degree of α is written as a ˇ ≥ b. Based on the definition of fuzzy equation studied by Para et al. [27], for every pair of ˜ ˜ fuzzy numbers a ˜ and b itcanbesaidthat a ˇ is equal to b in α degree if the above equations ˜ ˜ ∝ ∝ simultaneously exist as a ˇ ≤ b, a ˇ ≥ b. These equations can be written by: 2 2 α α ≤ μ (a ˜, b) ≤ 1 − (10) 2 2 If the sample fuzzy model is in the form of Model 11: Min z = c ˜ x (11) s.t. : a x ≥ b , i = 1, ... , l i i a ˜ x = b , i = l + 1, ... , m i i x ≥ 0 According to research carried out by Heilpern [26], the decision vector of xR is justified in α degree, if the condition Min {μ (a ˜ x, b )}= α is satisfied. Based on Relations (9) i=1,...,m M i i ˜ ˜ and (11), the equations of a ˜ x ≥ b and a ˜ x = b are equal to Equations (12) and (13). i i i i a x b i i E − E 2 1 ≥ α; i = 1, ... , l (12) a x a x b b i i i i E − E + E − E 2 1 2 1 FUZZY INFORMATION AND ENGINEERING 219 a x b i i E − E α α 2 1 ≤ ≤ 1 − ; i = l + 1, ... , m (13) a x a x b b i i i i 2 2 E − E + E − E 2 1 2 1 These equations can be written by: a x a b b i i i i [(1 − α)E + αE ]x ≥ αE + (1 − α)E ; i = 1, ... , l (14) 2 1 2 1 α α α α a x a b b i i i i 1 − E + E x ≥ E + 1 − E ; i = l + 1, ... , m (15) 2 1 2 1 2 2 2 2 α α α α a x a b b i i i i E + 1 − E x ≥ 1 − E + E ; i = l + 1, ... , m (16) 2 1 2 1 2 2 2 2 Using the rating and using Jimenez [22] and Jimenez et al. [23], a justifiable solution of x 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 x≥ b , i = 1, ... , l i α i and a x≈ b , i = l + 1, ... , m and x ≥ 0 that in this case, Equation (17) is achieved: i α i t t c x > c x (17) So, x is the best option that is opposed to other justifiable vectors at minimum degree of . Accordingly, Relation (15) can be rewritten by: t t t 0 t 0 c x c x c x c x E + E E + E 2 1 2 1 ≥ (18) 2 2 The outcome is that using the above explanations, Model 11 can be written as Model 19: Min EV(c ˜) x (19) a x a b b i i i i s.t. : [(1 − α)E + αE ]x ≥ αE + (1 − α)E , i = 1, ... , l 2 1 2 1 α α α α a x a b b i i i i 1 − E + E x ≥ E + 1 − E , i = l + 1, ... , m 2 1 2 1 2 2 2 2 α α α α a x a b b i i i i E + 1 − E x ≥ 1 − E + E , i = l + 1, ... , m 2 1 2 1 2 2 2 2 x ≥ 0 3.6. Proposed Model The characteristics of the proposed fuzzy model are as the definite model and it is com- posed 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 is created. fj By applying fuzzification changes on this factor, the definite model changes and is written as model 20. Min s (20) + − s.t. : (λ − λ )z ˜ − d + s 0, jD ∼ 1 fj f f f =1 220 N. TORABIETAL. + − (λ − λ )z ˜ − d − s 0, jD fj ∼ 2 f f f =1 + − (λ + λ ) = 1 f f f =1 + + + ζ ≥ λ ≥ εζ , f = 1, ... , h f f f − − − ζ ≥ λ ≥ εζ , f = 1, ... , h f f f + − ζ + ζ ≤ 1, f = 1, ... , h f f + − λ + λ ≥ ε, f = 1, ... , h f f + − d&s : free in sign, ζ , ζ : 0 or 1, other variables ≥ 0 f f 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) z z + − fj fj s.t. : (λ − λ )[(1 − α)E + αE ] − d + s 0, jD ∼ 1 f f 2 1 f =1 z z fj fj + − (λ − λ )[(1 − α)E + αE ] − d − s 0, jD ∼ 2 2 1 f f f =1 + − (λ + λ ) = 1 f f f =1 + + + ζ ≥ λ ≥ εζ , f = 1, ... , h f f f − − − ζ ≥ λ ≥ εζ , f = 1, ... , h f f f + − ζ + ζ ≤ 1, f = 1, ... , h f f + − λ + λ ≥ ε, f = 1, ... , h f f + − d&s : free in sign, ζ , ζ : 0 or 1, other variables ≥ 0 f f ∗ ∗ ∗ 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 and G . 2 1 ⎧  ⎫ ⎪  ⎪ z z ⎨  fj fj ⎬ λ [(1 − α)E + αE ] 2 1 C = j = G (22) 1 1 f =1 ⎪  ⎪ ⎩ ∗ ∗ ⎭ > d + s ⎧  ⎫ ⎪  ⎪ z z ⎨  fj fj ⎬ λ [(1 − α)E + αE ] 2 1 C = j = G (23) 2 2 f =1 ⎪  ⎪ ⎩ ∗ ∗ ⎭ < d − s FUZZY INFORMATION AND ENGINEERING 221 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: Min y + w y (24) j j j∈D j∈D 1 2 z z fj fj + − s.t.: (λ − λ )[(1 − α)E + αE ] − C + My ≥ 0, jD j 1 2 1 f f f =1 z z + − fj fj (λ − λ )[(1 − α)E + αE ] − C − My ≤−ε, jD j 2 f f 2 1 f =1 + − (λ + λ ) = 1 f f f =1 + + + ζ ≥ λ ≥ εζ , f = 1, ... , h f f f − − − ζ ≥ λ ≥ εζ , f = 1, ... , h f f f + − ζ + ζ ≤ 1, f = 1, ... , h f f + − λ + λ ≥ ε, f = 1, ... , h f f + − C : freeinsign, ζ , ζ , y : 0 or 1, other variables ≥ 0 f f ∗ ∗ After solving the model, the values of λ , c are obtained by solving Equations (25) and (26), the final classification can be achieved. z z fj fj ∗ ∗ if λ [(1 − α)E + αE ] ≥ c then j ∈ G (25) f 2 1 f =1 z z fj fj ∗ ∗ if λ [(1 − α)E + αE ] ≤ c − ε then j ∈ G (26) f 2 1 f =1 4. 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 and 10-second companies are classified in group G .In 1 2 other words, we have: f = 1, 2, j = 1, ... , 10, j = 11, ... , 20. Numbers of factors are given 1 2 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 and then two random numbers between 0.2 and 0.8 are considered as r andr respectively. The left triangular fuzzy num- 2 1 0 m ber is obtained by using c = (1 + r )c and the right triangular fuzzy number is obtained p m using c = (1 − r )c . 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 222 N. TORABIETAL. Table 1. Number of the financial factors. Number of The first The second Number of The first The second company factor factor company factor factor 15 9 11 3.5 5 2 5.6 8.5124.5 4.5 35 8 13 4.5 5 4 5.5 7.5143 4.5 56 6 15 2.5 3.5 6 5.8 8.3163.5 2.5 7 7.5 12 17 1.5 2.5 86.5 7 18 4 3 97 8 19 3.5 3 10 6 7.5 20 2.5 5 Table 2. Number of a fuzzy financial factor with α = 0.2. The first factor The second factor The first factor The second factor pmo p m o p m o p M o 1 0.8 1 1.2 7.2 9 10.8 11 2.8 3.5 4.2 4 5 6 2 0.8 1 1.2 6.8 8.5 10.2 12 3.6 4.5 5.4 3.6 4.5 5.4 34 5 6 6.48 9.613 3.6 4.5 5.4 4 5 6 4 4.4 5.5 6.6 6 7.5 9 14 2.4 3 3.6 3.6 4.5 5.4 54.8 6 7.24.86 7.215 2 2.5 3 2.8 3.5 4.2 6 4.64 5.8 6.96 6.64 8.3 9.96 16 2.8 3.5 4.2 2 2.5 3 7 6 7.59 9.612 14.4 171.21.51.82 2.53 8 5.2 6.5 7.8 5.6 7 8.4 18 3.2 4 4.8 2.4 3 3.6 9 5.6 7 8.4 6.4 8 9.6 19 2.8 3.5 4.2 6.4 8 9.6 10 4.8 6 7.2 6 7.5 9 20 2 2.5 3 6.4 8 9.6 Table 3. Number of a fuzzy financial factor with α = 0.5. The first factor The second factor The first factor The second factor pm O p m O pmo p M o 1 0.5 1 1.5 4.5 9 13.5 11 1.75 3.5 5.25 2.5 5 7.5 2 0.5 1 1.5 4.25 8.5 12.75 12 2.25 4.5 6.75 2.25 4.5 6.75 3 2.5 5 7.5 4 8 12 13 2.25 4.5 6.75 2.5 5 7.5 4 2.75 5.5 8.25 3.75 7.5 11.25 14 1.5 3 4.5 2.25 4.5 6.75 5 3 6 9 3 6 9 15 1.25 2.5 3.75 1.75 3.5 5.25 6 2.9 5.8 8.7 4.15 8.3 12.45 16 1.75 3.5 5.25 1.25 2.5 3.75 7 3.75 7.5 11.25 6 12 18 17 0.75 1.5 2.25 1.25 2.5 3.75 8 3.25 6.5 9.75 3.5 7 10.5 18 2 4 6 1.5 3 4.5 9 3.5 7 10.5 4 8 12 19 1.75 3.5 5.25 4 8 12 10 3 6 9 3.75 7.5 11.25 20 1.25 2.5 3.75 4 8 12 values of Tables 5, 6,and 7 the optimal responses are obtained, the optimal responses are shown in Table 8. Also, we have: ∗ ∗ ∗ ∗ d + s = 6.25, d − s = 6 Crisp Model ∗ ∗ ∗ ∗ d + s = 6.85, d − s = 6.36 Fuzzy Model α=0.2 ∗ ∗ ∗ ∗ d + s = 6. 5, d − s = 6 Fuzzy Model α=0.5 ∗ ∗ ∗ ∗ d + s = 0.00738, d − s = 0.00682 Fuzzy Model α=0.8 FUZZY INFORMATION AND ENGINEERING 223 Table 4. Number of a fuzzy financial factor with α = 0.8. The first factor The second factor The first factor The second factor pm O p m O pmo p M o 1 0.2 1 1.8 1.8 9 16.2 11 0.7 3.5 6.3 1 5 9 2 0.2 1 1.8 1.7 8.5 15.3 12 0.9 4.5 8.1 0.9 4.5 8.1 3 1 5 9 1.6 8 14.4 13 0.9 4.5 8.1 1 5 9 4 1.1 5.5 9.9 1.5 7.5 13.5 14 0.6 3 5.4 0.9 4.5 8.1 5 1.2 6 10.8 1.2 6 10.8 15 0.5 2.5 4.5 0.7 3.5 6.3 6 1.16 5.8 10.44 1.66 8.3 14.94 16 0.7 3.5 6.3 0.5 2.5 4.5 7 1.5 7.5 13.5 2.4 12 21.6 17 0.3 1.5 2.7 0.5 2.5 4.5 8 1.3 6.5 11.7 1.4 7 12.6 18 0.8 4 7.2 0.6 3 5.4 9 1.4 7 12.6 1.6 8 14.4 19 0.7 3.5 6.3 1.6 8 14.4 10 1.2 6 10.8 1.5 7.5 13.5 20 0.5 2.5 4.5 1.6 8 14.4 Table 5. Defuzzified coefficients of EI1 and EI2 based on α = 0.2. Unit 123456789 11 E1 Factor1 0.9 0.9 4.5 4.95 5.4 5.22 6.75 5.85 6.3 5.4 Factor2 8.1 7.65 7.2 6.75 5.4 7.47 10.8 6.3 7.2 6.75 E2 Factor1 1.1 1.1 5.5 6.05 6.6 6.38 8.25 7.15 7.7 6.6 Factor2 9.9 9.35 8.8 8.25 6.6 9.13 13.2 7.7 8.8 8.25 Unit 11 12 13 14 15 16 17 18 19 20 E1 Factor1 3.15 4.05 4.05 2.7 2.25 3.15 1.35 3.6 3.15 2.25 Factor2 4.5 4.05 4.5 4.05 3.15 2.25 2.25 2.7 7.2 7.2 E2 Factor1 3.85 4.95 4.95 3.3 2.75 3.85 1.65 4.4 3.85 2.75 Factor2 5.5 4.95 5.5 4.95 3.85 2.75 2.75 3.3 8.8 8.8 Table 6. Defuzzified coefficients of EI1 and EI2 based on α = 0.5. Unit 123456789 11 E1 Factor1 0.75 0.75 3.75 4.125 4.5 4.35 5.625 4.875 5.25 4.5 Factor2 6.75 6.375 6 5.625 4.5 6.225 9 5.25 6 5.625 E2 Factor1 1.25 1.25 6.25 6.875 7.5 7.25 9.375 8.125 8.75 7.5 Factor2 11.25 10.625 10 9.375 7.5 10.375 15 8.75 10 9.375 Unit 11 12 13 14 15 16 17 18 19 20 E1 Factor1 2.625 3.375 3.375 2.25 1.875 2.625 1.125 3 2.625 1.875 Factor2 3.75 3.375 3.75 3.375 2.625 1.875 1.875 2.25 6 6 E2 Factor1 4.375 5.625 5.625 3.75 3.125 4.375 1.875 5 4.375 3.125 Factor2 6.25 5.625 6.25 5.625 4.375 3.125 3.125 3.75 10 10 Table 7. Defuzzified coefficients of EI1 and EI2 based on α = 0.8. Unit 123456789 11 E1 Factor1 0.6 0.6 3 3.3 3.6 3.48 4.5 3.9 4.2 3.6 Factor2 5.4 5.1 4.8 4.5 3.6 4.98 7.2 4.2 4.8 4.5 E2 Factor1 1.4 1.4 7 7.7 8.4 8.12 10.5 9.1 9.8 8.4 Factor2 12.6 11.9 11.2 10.5 8.4 11.62 16.8 9.8 11.2 10.5 Unit 11 12 13 14 15 16 17 18 19 20 E1 Factor1 2.1 2.7 2.7 1.8 1.5 2.1 0.9 2.4 2.1 1.5 Factor2 3 2.7 3 2.7 2.1 1.5 1.5 1.8 4.8 4.8 E2 Factor1 4.9 6.3 6.3 4.2 3.5 4.9 2.1 5.6 4.9 3.5 Factor2 7 6.3 7 6.3 4.9 3.5 3.5 4.2 11.2 11.2 224 N. TORABIETAL. Table 8. Optimal values of the fuzzy and definite models. + − + − + − + − sd λ λ λ λ ζ ζ ζ ζ 1 1 2 2 1 1 2 2 Crisp model 0.25 6.25 0.333 0 0.666 0 1 0 1 0 Fuzzy model α = 0.2 0.265 6.625 0.333 0 0.666 0 1 0 1 0 Fuzzy model α = 0.5 0.25 6.25 0.333 0 0.666 0 1 0 1 0 Fuzzy model α = 0.8 0.00028 0.0071 0.49975 0.49925 0.001 0 1 0 1 0 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. Table 9. Classification results of the crisp observations in groups. 1 2 3 4 5 678 9 10 Group Overlap Overlap G G Overlap G G G G G 1 1 1 1 1 1 1 M 6.33 5.99 6.99 6.8 5.99 7.46 10.49 6.83 7.66 6.99 11 12 13 14 15 16 17 18 19 20 Group G G G G G G G G Overlap Overlap 2 2 2 2 2 2 2 2 N 4.49 4.49 4.83 3.99 3.16 2.83 2.16 3.33 6.49 6.16 Table 10. Classification results of the fuzzy observations in groups α = 0.2. 1 2 3 4 5 678 9 10 Group Overlap Overlap G G Overlap G G G G G 1 1 1 1 1 1 1 M 6.71 6.35 7.41 7.34 6.35 7.91 11.12 7.24 8.12 7.41 11 12 13 14 15 16 17 18 19 20 Group G G G G G G G G Overlap Overlap 2 2 2 2 2 2 2 2 N 4.76 4.6 5.12 4.23 3.35 3 2.29 3.53 6.88 6.53 Table 11. Classification results of the fuzzy observations in groups α = 0.5. 1 2 3 4 5 678 9 10 Group Overlap Overlap G G Overlap G G G G G 1 1 1 1 1 1 1 M 6.33 5.99 6.99 6.8 5.99 7.46 10.49 6.83 7.66 6.99 11 12 13 14 15 16 17 18 19 20 Group G G G G G G G G Overlap Overlap 2 2 2 2 2 2 2 2 N 4.49 4.49 4.83 3.99 3.16 2.83 2.16 3.33 6.49 6.16 Table 12. Classification results of the fuzzy observations in groups α = 0.8. 1 23 456 7 89 10 Group Overlap Overlap G G Overlap G G G G G 1 1 1 1 1 1 1 M 0.0072 0.0068 0.0079 0.0077 0.0068 0.0085 0.0119 0.0077 0.0087 0.0079 11 12 13 14 15 16 17 18 19 20 Group G G G G G G G G Overlap Overlap 2 2 2 2 2 2 2 2 N 0.0051 0.0051 0.0055 0.0045 0.0036 0.0032 0.0024 0.0038 0.0074 0.007 Table 13. Optimum results obtained by implementing the second phase. + − + − + − + − C λ λ λ λ ζ ζ ζ ζ 1 1 2 2 1 1 2 2 Fuzzy model α = 0.2 −6.36 0 0.363 0 0.636 0 1 0 1 Fuzzy model α = 0.5 −6 0 0.363 0 0.636 0 1 0 1 Fuzzy model α = 0.8 −4.56 0 0.363 0 0.636 0 1 0 1 Crisp model −6 0 0.363 0 0.636 0 1 0 1 FUZZY INFORMATION AND ENGINEERING 225 Table 14. Results of the final classification. Observation1 Observation2 Observation5 Observation19 Observation20 Group G G G G G 1 2 1 1 2 Fuzzy model α = 0.2 −6.45 −6.11 −6.36 −6.74 6.35 Group G G G G G 1 2 1 1 2 Fuzzy model α = 0.5 −6.087 −5.769 −6 −0.635 −5.99 Group G G G G G 1 2 2 1 2 Fuzzy model α = 0.8 −4.62 −4.38 −4.55 −4.83 −4.55 Group G G G G G 1 2 1 1 2 Crisp model −6.087 −5.76 −6 −6.35 5.99 Table 15. Percentage of correct allocation. Crisp model Fuzzy model α = 0.2 Fuzzy model α = 0.5 Fuzzy model α = 0.8 Percentage 0.9 0.9 0.9 0.85 5. 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 impor- tant 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 per- centage 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 partici- pants 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). 226 N. TORABIETAL. Notes on Contributors Reza Tavakkoli-Moghaddam is a Professor of Industrial Engineering at the College of Engineering, University of Tehran, Iran. He obtained his Ph.D., M.Sc. and B.Sc. degrees in Industrial Engineering from the Swinburne University of Technology in Melbourne (1998), the University of Melbourne in Melbourne (1994), and the Iran University of Science and Technology in Tehran (1989), respectively. He serves as the Editor-in-Chief of the Journal of Industrial Engineering published by the University of Tehran and as the Editorial Board member of nine reputable academic journals. He is the recipient of the 2009 and 2011 Distinguished Researcher Awards and the 2010 and 2014 Distinguished Applied Research Awards at the University of Tehran, Iran. He has been selected as the National Iranian Dis- tinguished Researcher in 2008 and 2010 by the MSRT (Ministry of Science, Research, and Technology) in Iran. He has obtained an outstanding rank as the top 1% scientist and researcher in the world elite group since 2014. He also received the Order of Academic Palms Award as a distinguished educa- tor and scholar for the insignia of Chevalier dans l’Ordre des Palmes Academiques by the Ministry of National Education of France in 2019. He has published 5 books, 37 book chapters and more than 1000 journal and conference papers. ORCID Reza Tavakkoli-Moghaddam http://orcid.org/0000-0002-6757-926X References [1] Sueyoshi T, Goto M. DEA-DA for bankruptcy-based performance assessment: misclassification analysis of the Japanese construction industry. Eur J Oper Res. 2009;199:561–575. [2] Charnes A, Cooper WW, Wei QL, et al. Cone ratio data envelopment analysis and multi-objective programming. Int J Syst Sci. 1989;7:1099–1118. [3] Charles V, Kumar M, Kavitha SI. Measuring the efficiency of assembled printed circuit boards with undesirable outputs using data envelopment analysis. Int J Prod Econ. 2012;136(1):194–206. [4] Sueyoshi T. DEA-discriminant analysis in the view of goal programming. Eur J Oper Res. 1999;115:564–582. [5] Altman EI. Financial ratios, discriminant analysis and the prediction of corporate bankruptcy. J Finance. 1968;23:589–609. [6] Lo A. Logit versus discriminant analysis: a specification test and application to corporate bankruptcies. J Econom. 1986;31:151–178. [7] Keasey K, Watson R. Financial distress prediction models: a review of their usefulness. Br J Manage. 1991;2:89–102. [8] Charnes A, Cooper WW, Rhodes E. Measuring the efficiency of decision-making units. Eur J Oper Res. 1978;2:429–444. [9] Agarwal S. Efficiency measure by fuzzy data envelopment analysis model. Fuzzy Inf Eng. 2014;6(1):59–70. [10] Babazadeh R, Razmi J, Pishvaee MS. Sustainable cultivation location optimization of the Jatropha curcas L. under uncertainty: a unified fuzzy data envelopment analysis approach. Measurment. 2016;89:252–260. [11] Kordrostami S, Jahani Sayyad Noveiri M. Evaluating the multi-period systems efficiency in the presence of fuzzy data. Fuzzy Inf Eng. 2017;9(3):281–298. [12] Ashrafi A, Mansouri Kaleibar M. Cost, revenue and profit efficiency models in generalized fuzzy data envelopment analysis. Fuzzy Inf Eng. 2017;9(2):237–246. [13] Khanjarpanah H, Jabbarzadeh A. Sustainable wind plant location optimization using fuzzy cross- efficiency data envelopment analysis. Energy. 2019;170:1004–1018. [14] Tavassoli M, Farzipoor Saen R. Predicting group membership of sustainable suppliers via data envelopment analysis and discriminant analysis. Sustain Prod Consum. 2019;18:41–52. [15] Sueyoshi T, Qu J, Li A, et al. Understanding the efficiency evolution for the Chinese provincial power industry: a new approach for combining data envelopment analysis-discriminant analysis with an efficiency shift across periods. J Cleaner Prod. 2020;277:122371. FUZZY INFORMATION AND ENGINEERING 227 [16] Atris A. Assessment of oil refinery performance: application of data envelopment analysis- discriminant analysis. Resour Policy. 2020;65:101543. [17] Fallah M, Hosseinzadeh Lotfi F, Hosseinzadeh M. Discriminant analysis and data envelopment analysis with specific data and Its application for companies in the Iranian stock exchange. Iran J Oper Res. 2020;11(1):144–156. [18] Fisher RA. The use of multiple measurements in taxonomy problems. Ann Eugen. 1936;7: 179–188. [19] Sueyoshi T. DEA-discriminant analysis: methodological comparison among eight discriminant analysis approaches. Eur J Oper Res. 2006;169:247–272. [20] Sengupta JK. A fuzzy systems approach in data envelopment analysis. Comput Math Appl. 1992;24:259–266. [21] Zadeh LA. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst. 1978;1:3–28. [22] Jimenez M. Ranking fuzzy numbers through the comparison of its expected intervals. Int J Uncertain, Fuzziness Knowl Based Syst. 1996;4(4):379–388. [23] Jimenez M, Arenas A, Bilbao A, et al. Linear programming with fuzzy parameters: an interactive method resolution. Eur J Oper Res. 2007;177:1599–1609. [24] Yager R. A procedure for ordering fuzzy subsets of the unit interval. Inf Sci (Ny). 1981;24:143–161. [25] Dubois D, Prade H. The mean value of a fuzzy number. Fuzzy Sets Syst. 1987;24:279–300. [26] Heilpern S. The expected value of a fuzzy number. Fuzzy Sets Syst. 1992;47:81–86. [27] Parra MA, Terol AB, Gladish BP, et al. Solving a multi-objective possibilistic problem through compromise programming. Eur J Oper Res. 2005;164:748–759. [28] Lai YJ, Hwang CL. A new approach to some possibilistic linear programming problems. Fuzzy Sets Syst. 1992;49:121–133. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

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

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FUZZY INFORMATION AND ENGINEERING 2022, VOL. 14, NO. 2, 212–227 https://doi.org/10.1080/16168658.2022.2117514 RESEARCH ARTICLE Combination of the Data Envelopment Analysis and the Discriminant Analysis for Evaluating Bankrupt Business in a Fuzzy Environment a a,b b Navid Torabi , Reza Tavakkoli-Moghaddam and Ali Siadat a b School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran; Arts et Métiers Institute of Technology, Université de Lorraine, LCFC, HESAM Université, Metz, France ABSTRACT ARTICLE HISTORY Received 1 April 2019 This paper presents a combination of the data envelopment analysis Revised 21 December 2021 (DEA) and discriminant analysis (DA) to evaluate the bankrupt busi- Accepted 23 August 2022 ness in a fuzzy environment. The DEA is a non-parametric method that can be used for various assessments. The DA is a statistical KEYWORDS method that can predict an appropriate group for new observations. Data envelopment analysis; The combination of DEA and DA methods creates a powerful method discriminant analysis; that includes the advantages of both methods. According to the spe- bankrupt business; fuzzy condition cial features of this method (e.g. high resolution and assessment accuracy), it can be used for a bankruptcy assessment of organisa- tions. In normal conditions, accurate measurement of data is very difficult, which is why considering the uncertainty conditions in mod- els can make them more applied. Using a fuzzy condition in models can help this issue. Finally, the results are illustrated and discussed. 1. Introduction The need to study and assess the status of an operating company and organisation is essen- tial 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 com- pany 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 organ- isation 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, CONTACT Reza Tavakkoli-Moghaddam tavakoli@ut.ac.ir © 2022 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society, Guangdong Province Operations Research Society of China. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. FUZZY INFORMATION AND ENGINEERING 213 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 signif- icant 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 impor- tant 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. 2. 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 effi- ciency 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 exis- tence 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 214 N. TORABIETAL. 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 efficiency- based 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 per- formed 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. 3. 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. 3.1. 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 rela- tionship 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 observa- tions of x to mono-variable observations of y,sothat 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 inte- ger programming method by defining integer variables classifies and assigns observations. The advantages of this method are capable of using negative data [19]. 3.2. 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 decision- making unit that can produce Y output with the minimum input. Assume a set of observed DMUs, {DMU j; j = 1, ... ,n} is associated with m inputs, {x ; i = 1, ... , m} and s outputs, ij FUZZY INFORMATION AND ENGINEERING 215 {y ; r = 1, ... , s}. Model 1 shows the CCR model in an input nature [8]. rj Min θ (1) s.t. : λ x ≤ θx , i = 1, ... , s j ij io j=1 λ y ≥ y , r = 1, ... , s j rj ro j=1 λ ≥ 0, j = 1, ... , n 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. Min θ (2) s.t. : λ x ≤ θx , i = 1, ... , s (1) j ij io j=1 λ y ≥ y , r = 1, ... , s (2) j rj ro j=1 λ =1(3) j=1 λ ≥ 0, j = 1, ... , n (4) 3.3. 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 , G are 2 1 considered, which have n observations in sum, where (j = 1, ... , n) that n + n , where G 1 2 1 is observations with financial failure (j = 1, ... , n ) and G is observations without finan- 1 2 cial failure (j = 1, ... , n ). Each observation is determined by h independent factor (f = 1, ... , h) by z and so λ indicates weight of the f-th factor that in general will operate as in fj f Model 3: Min s (3) + − s.t. : (λ − λ )z − d + s ≥ 0, jD fj f f f =1 + − (λ − λ )z − d − s ≤ 0, jD fj f f f =1 216 N. TORABIETAL. + − (λ + λ ) = 1 f f f =1 + + + ζ ≥ λ ≥ εζ , f = 1, ... , h f f f − − − ζ ≥ λ ≥ εζ , f = 1, ... , h f f f + − ζ + ζ ≤ 1, f = 1, ... , h f f + − λ + λ ≥ ε, f = 1, ... , h f f + − d&s : free in sign, ζ , ζ : 0 or 1, other variables ≥ 0 f f The objective function in this model minimises s, which represents the size of the overlap between G , G .Overlapby d-s has been limited as the lower limit of G and as the upper 2 1 1 limit of G . 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 of the j-th company are connected fj + − by the discriminant function of (λ − λ )z and the weights are so limited that the total fj f f f =1 ∗ ∗ amount λ is 1. After solving the model, if s ≥ 0 indicates that there are overlap and s < 0 + + + indicates a lack of overlap. ζ and εζ indicate the upper and lower limits for λ variable, f f f respectively. And the same conditions are established for λ . Due to the numerical value of + − ζ variable that is zero or one it is not possible to measure ζ , ζ simultaneously. Because of f f + − + this condition, it is not possible to measure λ , λ as well. Similarly, the condition of λ + f f f − + − λ ≥ ε is to avoid the simultaneous zero value of λ , λ . After solving the aforementioned f f f ∗ ∗ 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 and G surely. 1 2 ⎧ ⎫ ⎨ ⎬ ∗ ∗ ∗ C = j ∈ G | λ z > d + s (4) 1 1 fj ⎩ ⎭ f =1 ⎧ ⎫ ⎨ ⎬ ∗ ∗ ∗ C = j ∈ G | λ z < d − s (5) 2 2 fj ⎩ ⎭ f =1 Members of C series are fully owned by G and also members of C are fully owned by 1 1 2 G . After this step is finished D = G − C and D = G − C must be determined. D ∪ D 2 1 1 1 2 2 2 1 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. Second step: the classification of data that overlap: This step is performed using Model 6. Min y + w y (6) j j j∈D j∈D 1 2 + − s.t. : (λ − λ )z − C + My ≥ o, jD fj j 1 f f f =1 + − (λ − λ )z − C − My ≤−ε, jD j 2 fj f f f =1 FUZZY INFORMATION AND ENGINEERING 217 + − (λ + λ ) = 1 f f f =1 + + + ζ ≥ λ ≥ εζ , f = 1, ... , h f f f − − − ζ ≥ λ ≥ εζ , f = 1, ... , h f f f + − ζ + ζ ≤ 1, f = 1, ... , h f f + − λ + λ ≥ ε, f = 1, ... , h f f + − C : free in sign, ζ , ζ , y : 0 or 1, other variables ≥ 0 f f Variable y 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 , λ should not be zero at the same time, for this case, f f + − the control condition of λ + λ ≥ ε must be used. Also, w is the weight to validate two f f ∗ ∗ 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. ∗ ∗ If λ z ≥ c then j ∈ G (7) fj 1 f =1 ∗ ∗ If λ z ≤ c − ε then j ∈ G (8) fj 2 f =1 So, at the end of this step, all observations are classified in G and G . 1 2 3.4. 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 Sen- gupta [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: Inputs x ˜ and the outputs y ˜ of the DMUj are fuzzy variables, j = 1, ... , n. Since the fuzzy j j T T constraints v y ˜ ≤ u x ˜ do not define a deterministic feasible set, a natural idea is to provide j j 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: T T Cr{v y ˜ ≤ u x ˜ }≥ 1 − α, j = 1, 2, ... , n j j 218 N. TORABIETAL. Considering the chance constraints (4), the fuzzy DEA model can be written as follows: v y max θ = Cr ≥ 1 u,v u x ˜ T T ˜ ˜ s.t : Cr{v y ≤ u x }≥ 1 − α, j = 1, 2, ... , n j j u ≥ 0 v ≥ 0 where α ∈ (0, 0.5] The greater the optimal objective is, the more efficient DMU0 is ranked. 3.5. 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. a ˜ as limitation coefficients and b as the right number), which are fuzzy. This method uses Relation (9) for this operation. a b ⎪0if E − E < 0 ⎪ 2 1 a b E −E 2 1 a b a b μ (a ˜, b) = if 0 ∈ [E − E , E − E ] (9) a b a b 1 2 2 1 E −E −(E −E ) 2 1 1 2 a b 1if E − E > 0 1 2 ˜ ˜ When μ (a ˜, b)>α, it can be said that a ˜ ≥ b and regarding the degree of α is written as a ˇ ≥ b. Based on the definition of fuzzy equation studied by Para et al. [27], for every pair of ˜ ˜ fuzzy numbers a ˜ and b itcanbesaidthat a ˇ is equal to b in α degree if the above equations ˜ ˜ ∝ ∝ simultaneously exist as a ˇ ≤ b, a ˇ ≥ b. These equations can be written by: 2 2 α α ≤ μ (a ˜, b) ≤ 1 − (10) 2 2 If the sample fuzzy model is in the form of Model 11: Min z = c ˜ x (11) s.t. : a x ≥ b , i = 1, ... , l i i a ˜ x = b , i = l + 1, ... , m i i x ≥ 0 According to research carried out by Heilpern [26], the decision vector of xR is justified in α degree, if the condition Min {μ (a ˜ x, b )}= α is satisfied. Based on Relations (9) i=1,...,m M i i ˜ ˜ and (11), the equations of a ˜ x ≥ b and a ˜ x = b are equal to Equations (12) and (13). i i i i a x b i i E − E 2 1 ≥ α; i = 1, ... , l (12) a x a x b b i i i i E − E + E − E 2 1 2 1 FUZZY INFORMATION AND ENGINEERING 219 a x b i i E − E α α 2 1 ≤ ≤ 1 − ; i = l + 1, ... , m (13) a x a x b b i i i i 2 2 E − E + E − E 2 1 2 1 These equations can be written by: a x a b b i i i i [(1 − α)E + αE ]x ≥ αE + (1 − α)E ; i = 1, ... , l (14) 2 1 2 1 α α α α a x a b b i i i i 1 − E + E x ≥ E + 1 − E ; i = l + 1, ... , m (15) 2 1 2 1 2 2 2 2 α α α α a x a b b i i i i E + 1 − E x ≥ 1 − E + E ; i = l + 1, ... , m (16) 2 1 2 1 2 2 2 2 Using the rating and using Jimenez [22] and Jimenez et al. [23], a justifiable solution of x 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 x≥ b , i = 1, ... , l i α i and a x≈ b , i = l + 1, ... , m and x ≥ 0 that in this case, Equation (17) is achieved: i α i t t c x > c x (17) So, x is the best option that is opposed to other justifiable vectors at minimum degree of . Accordingly, Relation (15) can be rewritten by: t t t 0 t 0 c x c x c x c x E + E E + E 2 1 2 1 ≥ (18) 2 2 The outcome is that using the above explanations, Model 11 can be written as Model 19: Min EV(c ˜) x (19) a x a b b i i i i s.t. : [(1 − α)E + αE ]x ≥ αE + (1 − α)E , i = 1, ... , l 2 1 2 1 α α α α a x a b b i i i i 1 − E + E x ≥ E + 1 − E , i = l + 1, ... , m 2 1 2 1 2 2 2 2 α α α α a x a b b i i i i E + 1 − E x ≥ 1 − E + E , i = l + 1, ... , m 2 1 2 1 2 2 2 2 x ≥ 0 3.6. Proposed Model The characteristics of the proposed fuzzy model are as the definite model and it is com- posed 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 is created. fj By applying fuzzification changes on this factor, the definite model changes and is written as model 20. Min s (20) + − s.t. : (λ − λ )z ˜ − d + s 0, jD ∼ 1 fj f f f =1 220 N. TORABIETAL. + − (λ − λ )z ˜ − d − s 0, jD fj ∼ 2 f f f =1 + − (λ + λ ) = 1 f f f =1 + + + ζ ≥ λ ≥ εζ , f = 1, ... , h f f f − − − ζ ≥ λ ≥ εζ , f = 1, ... , h f f f + − ζ + ζ ≤ 1, f = 1, ... , h f f + − λ + λ ≥ ε, f = 1, ... , h f f + − d&s : free in sign, ζ , ζ : 0 or 1, other variables ≥ 0 f f 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) z z + − fj fj s.t. : (λ − λ )[(1 − α)E + αE ] − d + s 0, jD ∼ 1 f f 2 1 f =1 z z fj fj + − (λ − λ )[(1 − α)E + αE ] − d − s 0, jD ∼ 2 2 1 f f f =1 + − (λ + λ ) = 1 f f f =1 + + + ζ ≥ λ ≥ εζ , f = 1, ... , h f f f − − − ζ ≥ λ ≥ εζ , f = 1, ... , h f f f + − ζ + ζ ≤ 1, f = 1, ... , h f f + − λ + λ ≥ ε, f = 1, ... , h f f + − d&s : free in sign, ζ , ζ : 0 or 1, other variables ≥ 0 f f ∗ ∗ ∗ 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 and G . 2 1 ⎧  ⎫ ⎪  ⎪ z z ⎨  fj fj ⎬ λ [(1 − α)E + αE ] 2 1 C = j = G (22) 1 1 f =1 ⎪  ⎪ ⎩ ∗ ∗ ⎭ > d + s ⎧  ⎫ ⎪  ⎪ z z ⎨  fj fj ⎬ λ [(1 − α)E + αE ] 2 1 C = j = G (23) 2 2 f =1 ⎪  ⎪ ⎩ ∗ ∗ ⎭ < d − s FUZZY INFORMATION AND ENGINEERING 221 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: Min y + w y (24) j j j∈D j∈D 1 2 z z fj fj + − s.t.: (λ − λ )[(1 − α)E + αE ] − C + My ≥ 0, jD j 1 2 1 f f f =1 z z + − fj fj (λ − λ )[(1 − α)E + αE ] − C − My ≤−ε, jD j 2 f f 2 1 f =1 + − (λ + λ ) = 1 f f f =1 + + + ζ ≥ λ ≥ εζ , f = 1, ... , h f f f − − − ζ ≥ λ ≥ εζ , f = 1, ... , h f f f + − ζ + ζ ≤ 1, f = 1, ... , h f f + − λ + λ ≥ ε, f = 1, ... , h f f + − C : freeinsign, ζ , ζ , y : 0 or 1, other variables ≥ 0 f f ∗ ∗ After solving the model, the values of λ , c are obtained by solving Equations (25) and (26), the final classification can be achieved. z z fj fj ∗ ∗ if λ [(1 − α)E + αE ] ≥ c then j ∈ G (25) f 2 1 f =1 z z fj fj ∗ ∗ if λ [(1 − α)E + αE ] ≤ c − ε then j ∈ G (26) f 2 1 f =1 4. 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 and 10-second companies are classified in group G .In 1 2 other words, we have: f = 1, 2, j = 1, ... , 10, j = 11, ... , 20. Numbers of factors are given 1 2 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 and then two random numbers between 0.2 and 0.8 are considered as r andr respectively. The left triangular fuzzy num- 2 1 0 m ber is obtained by using c = (1 + r )c and the right triangular fuzzy number is obtained p m using c = (1 − r )c . 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 222 N. TORABIETAL. Table 1. Number of the financial factors. Number of The first The second Number of The first The second company factor factor company factor factor 15 9 11 3.5 5 2 5.6 8.5124.5 4.5 35 8 13 4.5 5 4 5.5 7.5143 4.5 56 6 15 2.5 3.5 6 5.8 8.3163.5 2.5 7 7.5 12 17 1.5 2.5 86.5 7 18 4 3 97 8 19 3.5 3 10 6 7.5 20 2.5 5 Table 2. Number of a fuzzy financial factor with α = 0.2. The first factor The second factor The first factor The second factor pmo p m o p m o p M o 1 0.8 1 1.2 7.2 9 10.8 11 2.8 3.5 4.2 4 5 6 2 0.8 1 1.2 6.8 8.5 10.2 12 3.6 4.5 5.4 3.6 4.5 5.4 34 5 6 6.48 9.613 3.6 4.5 5.4 4 5 6 4 4.4 5.5 6.6 6 7.5 9 14 2.4 3 3.6 3.6 4.5 5.4 54.8 6 7.24.86 7.215 2 2.5 3 2.8 3.5 4.2 6 4.64 5.8 6.96 6.64 8.3 9.96 16 2.8 3.5 4.2 2 2.5 3 7 6 7.59 9.612 14.4 171.21.51.82 2.53 8 5.2 6.5 7.8 5.6 7 8.4 18 3.2 4 4.8 2.4 3 3.6 9 5.6 7 8.4 6.4 8 9.6 19 2.8 3.5 4.2 6.4 8 9.6 10 4.8 6 7.2 6 7.5 9 20 2 2.5 3 6.4 8 9.6 Table 3. Number of a fuzzy financial factor with α = 0.5. The first factor The second factor The first factor The second factor pm O p m O pmo p M o 1 0.5 1 1.5 4.5 9 13.5 11 1.75 3.5 5.25 2.5 5 7.5 2 0.5 1 1.5 4.25 8.5 12.75 12 2.25 4.5 6.75 2.25 4.5 6.75 3 2.5 5 7.5 4 8 12 13 2.25 4.5 6.75 2.5 5 7.5 4 2.75 5.5 8.25 3.75 7.5 11.25 14 1.5 3 4.5 2.25 4.5 6.75 5 3 6 9 3 6 9 15 1.25 2.5 3.75 1.75 3.5 5.25 6 2.9 5.8 8.7 4.15 8.3 12.45 16 1.75 3.5 5.25 1.25 2.5 3.75 7 3.75 7.5 11.25 6 12 18 17 0.75 1.5 2.25 1.25 2.5 3.75 8 3.25 6.5 9.75 3.5 7 10.5 18 2 4 6 1.5 3 4.5 9 3.5 7 10.5 4 8 12 19 1.75 3.5 5.25 4 8 12 10 3 6 9 3.75 7.5 11.25 20 1.25 2.5 3.75 4 8 12 values of Tables 5, 6,and 7 the optimal responses are obtained, the optimal responses are shown in Table 8. Also, we have: ∗ ∗ ∗ ∗ d + s = 6.25, d − s = 6 Crisp Model ∗ ∗ ∗ ∗ d + s = 6.85, d − s = 6.36 Fuzzy Model α=0.2 ∗ ∗ ∗ ∗ d + s = 6. 5, d − s = 6 Fuzzy Model α=0.5 ∗ ∗ ∗ ∗ d + s = 0.00738, d − s = 0.00682 Fuzzy Model α=0.8 FUZZY INFORMATION AND ENGINEERING 223 Table 4. Number of a fuzzy financial factor with α = 0.8. The first factor The second factor The first factor The second factor pm O p m O pmo p M o 1 0.2 1 1.8 1.8 9 16.2 11 0.7 3.5 6.3 1 5 9 2 0.2 1 1.8 1.7 8.5 15.3 12 0.9 4.5 8.1 0.9 4.5 8.1 3 1 5 9 1.6 8 14.4 13 0.9 4.5 8.1 1 5 9 4 1.1 5.5 9.9 1.5 7.5 13.5 14 0.6 3 5.4 0.9 4.5 8.1 5 1.2 6 10.8 1.2 6 10.8 15 0.5 2.5 4.5 0.7 3.5 6.3 6 1.16 5.8 10.44 1.66 8.3 14.94 16 0.7 3.5 6.3 0.5 2.5 4.5 7 1.5 7.5 13.5 2.4 12 21.6 17 0.3 1.5 2.7 0.5 2.5 4.5 8 1.3 6.5 11.7 1.4 7 12.6 18 0.8 4 7.2 0.6 3 5.4 9 1.4 7 12.6 1.6 8 14.4 19 0.7 3.5 6.3 1.6 8 14.4 10 1.2 6 10.8 1.5 7.5 13.5 20 0.5 2.5 4.5 1.6 8 14.4 Table 5. Defuzzified coefficients of EI1 and EI2 based on α = 0.2. Unit 123456789 11 E1 Factor1 0.9 0.9 4.5 4.95 5.4 5.22 6.75 5.85 6.3 5.4 Factor2 8.1 7.65 7.2 6.75 5.4 7.47 10.8 6.3 7.2 6.75 E2 Factor1 1.1 1.1 5.5 6.05 6.6 6.38 8.25 7.15 7.7 6.6 Factor2 9.9 9.35 8.8 8.25 6.6 9.13 13.2 7.7 8.8 8.25 Unit 11 12 13 14 15 16 17 18 19 20 E1 Factor1 3.15 4.05 4.05 2.7 2.25 3.15 1.35 3.6 3.15 2.25 Factor2 4.5 4.05 4.5 4.05 3.15 2.25 2.25 2.7 7.2 7.2 E2 Factor1 3.85 4.95 4.95 3.3 2.75 3.85 1.65 4.4 3.85 2.75 Factor2 5.5 4.95 5.5 4.95 3.85 2.75 2.75 3.3 8.8 8.8 Table 6. Defuzzified coefficients of EI1 and EI2 based on α = 0.5. Unit 123456789 11 E1 Factor1 0.75 0.75 3.75 4.125 4.5 4.35 5.625 4.875 5.25 4.5 Factor2 6.75 6.375 6 5.625 4.5 6.225 9 5.25 6 5.625 E2 Factor1 1.25 1.25 6.25 6.875 7.5 7.25 9.375 8.125 8.75 7.5 Factor2 11.25 10.625 10 9.375 7.5 10.375 15 8.75 10 9.375 Unit 11 12 13 14 15 16 17 18 19 20 E1 Factor1 2.625 3.375 3.375 2.25 1.875 2.625 1.125 3 2.625 1.875 Factor2 3.75 3.375 3.75 3.375 2.625 1.875 1.875 2.25 6 6 E2 Factor1 4.375 5.625 5.625 3.75 3.125 4.375 1.875 5 4.375 3.125 Factor2 6.25 5.625 6.25 5.625 4.375 3.125 3.125 3.75 10 10 Table 7. Defuzzified coefficients of EI1 and EI2 based on α = 0.8. Unit 123456789 11 E1 Factor1 0.6 0.6 3 3.3 3.6 3.48 4.5 3.9 4.2 3.6 Factor2 5.4 5.1 4.8 4.5 3.6 4.98 7.2 4.2 4.8 4.5 E2 Factor1 1.4 1.4 7 7.7 8.4 8.12 10.5 9.1 9.8 8.4 Factor2 12.6 11.9 11.2 10.5 8.4 11.62 16.8 9.8 11.2 10.5 Unit 11 12 13 14 15 16 17 18 19 20 E1 Factor1 2.1 2.7 2.7 1.8 1.5 2.1 0.9 2.4 2.1 1.5 Factor2 3 2.7 3 2.7 2.1 1.5 1.5 1.8 4.8 4.8 E2 Factor1 4.9 6.3 6.3 4.2 3.5 4.9 2.1 5.6 4.9 3.5 Factor2 7 6.3 7 6.3 4.9 3.5 3.5 4.2 11.2 11.2 224 N. TORABIETAL. Table 8. Optimal values of the fuzzy and definite models. + − + − + − + − sd λ λ λ λ ζ ζ ζ ζ 1 1 2 2 1 1 2 2 Crisp model 0.25 6.25 0.333 0 0.666 0 1 0 1 0 Fuzzy model α = 0.2 0.265 6.625 0.333 0 0.666 0 1 0 1 0 Fuzzy model α = 0.5 0.25 6.25 0.333 0 0.666 0 1 0 1 0 Fuzzy model α = 0.8 0.00028 0.0071 0.49975 0.49925 0.001 0 1 0 1 0 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. Table 9. Classification results of the crisp observations in groups. 1 2 3 4 5 678 9 10 Group Overlap Overlap G G Overlap G G G G G 1 1 1 1 1 1 1 M 6.33 5.99 6.99 6.8 5.99 7.46 10.49 6.83 7.66 6.99 11 12 13 14 15 16 17 18 19 20 Group G G G G G G G G Overlap Overlap 2 2 2 2 2 2 2 2 N 4.49 4.49 4.83 3.99 3.16 2.83 2.16 3.33 6.49 6.16 Table 10. Classification results of the fuzzy observations in groups α = 0.2. 1 2 3 4 5 678 9 10 Group Overlap Overlap G G Overlap G G G G G 1 1 1 1 1 1 1 M 6.71 6.35 7.41 7.34 6.35 7.91 11.12 7.24 8.12 7.41 11 12 13 14 15 16 17 18 19 20 Group G G G G G G G G Overlap Overlap 2 2 2 2 2 2 2 2 N 4.76 4.6 5.12 4.23 3.35 3 2.29 3.53 6.88 6.53 Table 11. Classification results of the fuzzy observations in groups α = 0.5. 1 2 3 4 5 678 9 10 Group Overlap Overlap G G Overlap G G G G G 1 1 1 1 1 1 1 M 6.33 5.99 6.99 6.8 5.99 7.46 10.49 6.83 7.66 6.99 11 12 13 14 15 16 17 18 19 20 Group G G G G G G G G Overlap Overlap 2 2 2 2 2 2 2 2 N 4.49 4.49 4.83 3.99 3.16 2.83 2.16 3.33 6.49 6.16 Table 12. Classification results of the fuzzy observations in groups α = 0.8. 1 23 456 7 89 10 Group Overlap Overlap G G Overlap G G G G G 1 1 1 1 1 1 1 M 0.0072 0.0068 0.0079 0.0077 0.0068 0.0085 0.0119 0.0077 0.0087 0.0079 11 12 13 14 15 16 17 18 19 20 Group G G G G G G G G Overlap Overlap 2 2 2 2 2 2 2 2 N 0.0051 0.0051 0.0055 0.0045 0.0036 0.0032 0.0024 0.0038 0.0074 0.007 Table 13. Optimum results obtained by implementing the second phase. + − + − + − + − C λ λ λ λ ζ ζ ζ ζ 1 1 2 2 1 1 2 2 Fuzzy model α = 0.2 −6.36 0 0.363 0 0.636 0 1 0 1 Fuzzy model α = 0.5 −6 0 0.363 0 0.636 0 1 0 1 Fuzzy model α = 0.8 −4.56 0 0.363 0 0.636 0 1 0 1 Crisp model −6 0 0.363 0 0.636 0 1 0 1 FUZZY INFORMATION AND ENGINEERING 225 Table 14. Results of the final classification. Observation1 Observation2 Observation5 Observation19 Observation20 Group G G G G G 1 2 1 1 2 Fuzzy model α = 0.2 −6.45 −6.11 −6.36 −6.74 6.35 Group G G G G G 1 2 1 1 2 Fuzzy model α = 0.5 −6.087 −5.769 −6 −0.635 −5.99 Group G G G G G 1 2 2 1 2 Fuzzy model α = 0.8 −4.62 −4.38 −4.55 −4.83 −4.55 Group G G G G G 1 2 1 1 2 Crisp model −6.087 −5.76 −6 −6.35 5.99 Table 15. Percentage of correct allocation. Crisp model Fuzzy model α = 0.2 Fuzzy model α = 0.5 Fuzzy model α = 0.8 Percentage 0.9 0.9 0.9 0.85 5. 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 impor- tant 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 per- centage 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 partici- pants 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). 226 N. TORABIETAL. Notes on Contributors Reza Tavakkoli-Moghaddam is a Professor of Industrial Engineering at the College of Engineering, University of Tehran, Iran. He obtained his Ph.D., M.Sc. and B.Sc. degrees in Industrial Engineering from the Swinburne University of Technology in Melbourne (1998), the University of Melbourne in Melbourne (1994), and the Iran University of Science and Technology in Tehran (1989), respectively. He serves as the Editor-in-Chief of the Journal of Industrial Engineering published by the University of Tehran and as the Editorial Board member of nine reputable academic journals. 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Journal

Fuzzy Information and Engineering – Taylor & Francis

Published: Apr 3, 2022

Keywords: Data envelopment analysis; discriminant analysis; bankrupt business; fuzzy condition

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