Computational Economics 14: 197–218, 1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands.

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A Multicriteria Decision Aid Methodology for Sorting Decision Problems: The Case of Financial Distress CONSTANTIN ZOPOUNIDIS and MICHAEL DOUMPOS Technical University of Crete, Department of Production Engineering and Management, Financial Engineering Laboratory, University Campus, 73100 Chania, Greece (Accepted: 24 November 1998) Abstract. Sorting problems constitute a major part of real world decisions, where a set of alternative actions (solutions) must be classified into two or more predefined classes. Multicriteria decision aid (MCDA) provides several methodologies, which are well adapted in sorting problems. A well known approach in MCDA is based on preference disaggregation which has already been used in ranking problems, but it is also applicable in sorting problems. The UTADIS (UTilités Additives DIScriminantes) method, a variant of the UTA method, based on the preference disaggregation approach estimates a set of additive utility functions and utility profiles using linear programming techniques in order to minimize the misclassification error between the predefined classes in sorting problems. This paper presents the application of the UTADIS method in two real world classification problems concerning the field of financial distress. The applications are derived by the studies of Slowinski and Zopounidis (1995), and Dimitras et al. (1999). The obtained results depict the superiority of the UTADIS method over discriminant analysis, and they are also comparable with the results derived by other multicriteria methods. Key words: bankruptcy risk, discriminant analysis, financial distress, multicriteria decision aid, ordinal regression, sorting

1. Introduction and Review Generally, a decision problem involves the examination of a set of potential alternative actions (solutions) over a set of criteria in order to reach a decision. Decision problems can be categorized in the following four types (problematics) according to the objective of the decision: (i) selection of the most appropriate (best) alternative, (ii) sorting of the alternatives in predefined homogenous classes, (iii) ranking of the alternatives from the best one to the worst one, and (iv) description of the alternatives. Many practical decision problems such as fault diagnosis, medical diagnosis, pattern recognition, diagnosis of sales potential, etc., belong to the second of these problematics, i.e., the sorting. Multiple criteria decision aid (MCDA, Zeleny, 1982; Roy, 1985; Roy and Bouyssou, 1993; Roy, 1996) provides several powerful and effective tools for confronting sorting problems, such as the ELECTRE TRI

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method proposed by Yu (1992) and its variant proposed by Mousseau and Slowinski (1998), the N-TOMIC method presented by Massaglia and Ostanello (1991), the variant of the PROMETHEE method presented by Martel and Khoury (1994), and the trichotomic analysis proposed by Roy (1981) and Roy and Moscarola (1977). A significant approach in MCDA is based on preference disaggregation. The preference disaggregation approach aims at the estimation of an additive utility function through the analysis of the global judgments (ranking or grouping of the alternatives) of the decision maker. UTA (UTilités Additives, cf. Jacquet-Lagrèze and Siskos, 1982; Siskos and Yannacopoulos, 1985) is a well known preference disaggregation method based on ordinal regression analysis, which is mainly oriented towards ranking problems. Using a variant of the UTA method, it is possible to estimate a set of additive utility functions, which minimize the misclassification error between the classes. At the same time the profiles (utility thresholds), that distinguish the classes, can be calculated. Such a variant of the UTA method is the UTADIS method (UTilités Additives DIScriminantes, Devaud et al., 1980; JacquetLagrèze and Siskos, 1982; Jacquet-Lagrèze, 1995) for building additive utility functions in sorting problems. An application of the UTADIS method in the evaluation of research and development projects has been presented by Jacquet-Lagrèze (1995). A sorting problem of major practical and academic interest is that of financial distress. Financial distress involves a situation where the firm cannot fulfil its obligations to its creditors, suppliers, or because a bill is overdrawn, etc., resulting in a respite of the firm’s operation. Of course other definitions of financial distress could also be introduced according to a specific point of view (financial, legal, etc.). Financial distress interests the financial managers, credit and financial analysts, individual investors, and of course financial and operational researchers. Therefore, the need for the development of efficient tools for assessing the business failure risk is of vital importance. This was the motivation for several scientists and researchers to develop methods and techniques for predicting business failure as effectively and accurately as possible. Obviously, the main problem in the case of financial distress is to classify a set of firms in predefined classes distinguishing the firms according to their failure risk (healthy firms, bankrupt firms, uncertain firms). The first approach used in the prediction of financial distress was based on univariate and multivariate statistical techniques. The objective of univariate statistical techniques was to determine the most important financial ratio providing the higher predicting accuracy. According to the selected ratio and a corresponding cut-off value, the firms are classified in two groups, bankrupt and non-bankrupt. Using this approach, Beaver (1966) has concluded that the financial ratio ‘Total debt/Cash flow’, provides the higher discrimination ability. Then, multivariate statistical techniques were proposed, such as discriminant analysis (Altman, 1968, 1984; Libby, 1975), and cluster analysis (Jensen, 1971; Gupta and Huefner, 1972). Both univariate and multivariate statistical methods are based on restrictive stat-

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istical assumptions (distribution of the sample, independence of variables, etc.), which lead to several problems, such as the multicollinearity of the variables, the difficulty in the explanation of error rates, the selection of the a priori probabilities or costs of misclassification, etc. (Eisenbeis, 1977). Later multivariate conditional probability models were introduced, such as logit analysis (Martin, 1977; Peel, 1987; Keasey et al., 1990), and probit analysis (Casey et al., 1986; Skogsvik, 1990). A significant drawback of all these methods is the exclusion from the analysis of failure risk of substantial strategic variables such as quality of management, organization, market trend, market position, etc. Thus, the prediction and the analysis of business failure is based only on the examination of financial ratios, ignoring significant information which can not be assessed using quantitative criteria, such as financial ratios. In order to confront these problems new techniques were proposed, based on information and computer science (rough sets theory, expert systems, decision support systems). Slowinski and Zopounidis (1995), presented a rough set approach. The basic idea of the rough set theory (Pawlak, 1982) is to develop a set of decision rules describing a set of objects (firms) by a set of multi-valued attributes (financial ratios and qualitative criteria), in order to classify the objects in their original class (i.e., bankrupt and non-bankrupt firms). Expert systems (ESs) were also proposed as an effective tool for the assessment of business failure risk (Bouwman, 1983; Ben-David and Sterling, 1986; Elmer and Borowski, 1988; Messier and Hansen, 1988; Shaw and Gentry, 1988; Cronan et al., 1991; Michalopoulos and Zopounidis, 1993). ESs represent the knowledge that expert financial or credit analysts use in practical cases when assessing corporate failure risk. The symbolic reasoning and the explanation capabilities of ESs make them highly applicable in decision problems based on judgemental procedures such as the assessment of failure risk. Decision support systems (DSSs) were also used as a tool to provide support to decision makers in assessing business failure risk (Mareschal and Brans, 1991; Zopounidis et al., 1992; Siskos et al., 1994). The progress in computer science provided the necessary means for performing complex and time-consuming tasks very easily, as well as for accessing and handling large data bases. The implementation of DSSs in the assessment of failure risk was combined with the application of multicriteria decision aid methods, which are well adapted to decision problems, by analyzing the preferences of the decision makers. Recently a new type of system has been proposed, combining the symbolic reasoning and the explanation capabilities of the ESs’ technology with the powerful analytical tools and techniques already used in DSSs. This type of system, known as knowledge-based decision support system (KBDSS), has recently started to be applied in the assessment of failure risk (Duchessi and Belardo, 1987; Pinson, 1989, 1992; Srinivasan and Ruparel, 1990; Zopounidis et al., 1996a,b). Finally, another significant approach in the prediction of corporate failure is based on MCDA (Zopounidis, 1987; Dimitras, 1995; Dimitras et al., 1995). Sev-

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eral multicriteria methods applied in the evaluation of bankruptcy risk can be found in Zopounidis (1995), while a complete survey of all the methods that have been applied in this field of financial management can be found in Dimitras et al. (1996). The UTA multicriteria method has already been applied in several decision problems in the field of financial management, such as bankruptcy prediction (Zopounidis, 1987), venture capital investments (Siskos and Zopounidis, 1987), evaluation of country risk (Cosset et al., 1992), business financing (Siskos et al., 1994), portfolio management (Hurson and Zopounidis, 1995; Zopounidis et al., 1995), etc. This paper presents the application of the UTADIS method in the assessment of corporate failure risk. Initially, the method is described in Section 2. Then, in Section 3 two real case applications of the method are presented concerning financial distress. Some possible extensions of the method are described in Section 4, and finally the concluding remarks are discussed (Section 5). 2. The UTADIS Method Let g1 , g2 , . . . , gm be a consistent family of m evaluation criteria, and A = {a1 , a2 , . . . , an } a set of n alternatives to be classified in Q ordered classes C1 , C2 , . . . , CQ which are defined a priori: C1 P C2 . . . , CQ−1 P CQ , where, P denotes the strict preference relation, between the classes. For each evaluation criterion gi the interval Gi = [gi ∗ , gi∗ ] of its values is defined. gi ∗ and gi∗ are the less and the most preferred values, respectively, of criterion gi for all the alternatives belonging to A. The interval Gi is divided into j j +1 ai −1 equal intervals [gi , gi ], j = 1, 2, . . . , ai −1. ai , is defined by the decision maker as the number of estimated points for every marginal utility ui . Each point j gi can be calculated using linear interpolation: j

gi = gi ∗ +

j −1 ∗ (g − gi ∗ ). ai − 1 i

The aim is to estimate the marginal utilities at each of these points. Suppose that the j j +1 evaluation of an alternative a on criterion gi is gi (a) ∈ [gi , gi ]. The marginal utility of the alternative action a, ui [gi (a)], can be approximated through linear interpolation: j

j

ui [gi (α)] = ui (gi ) +

gi (α) − gi j +1 gi

−

j gi

j +1

[ui (gi

j

) − ui (gi )] .

(1)

To achieve the monotonicity of the criteria the following constraint must be satisfied: j +1

ui (gi

j

) − ui (gi ) ≥ 0, ∀i .

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Figure 1. Distribution of the classes on the assessed utility.

The monotonicity constraints are taken into account through the following transformations, as in the UTASTAR method (Siskos and Yannacopoulos, 1985):  j +1 j wij = ui (gi ) − ui (gi ) ≥ 0 ∀i, j       ui (gi ∗ ) = 0 (2) j −1  X  j   ui (gi ) = wik   k=1

Thus, the weights of each criterion can be computed as: ui (gi∗ ) = Using these transformations, (1) can be written as: ui [gi (α)] =

j −1 X

Pai −1 k=1

wik .

j

wik +

k=1

gi (α) − gi j +1

gi

j

− gi

wij .

The global utility U (a) of an alternative a ∈ A is of an additive form: U (a) =

m X

ui [gi (α)] .

(3)

i=1

There are two possible errors (misclassification errors) relative to the global utility U (a): the over-estimation error σ + (a) and the under-estimation error σ − (a). An over-estimation error exists in cases where an alternative according to its utility is classified to a lower class than the class that it belongs (e.g., an alternative is classified in class C2 while belonging in class C1 ). In such cases the amount σ + (a) should be added to the utility of this alternative in order to be correctly classified. An under-estimation error exists in cases where an alternative according to its utility is classified to a higher class than the class that it belongs (e.g., an alternative is classified in class C1 while belonging in class C2 ). In such cases the amount σ − (a) should be subtracted from the utility of this alternative so that it can be correctly classified. These two types of errors are better presented in Figure 1. The classification of the alternatives is achieved through the comparison of each utility with the corresponding utility thresholds ui (a1 > u2 > . . . > uQ−1 ), which

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distinguish the one class from the other: U (a) ≥ u1

⇒ a ∈ C1

u2 ≤ U (a) < u1 ⇒ a ∈ C2 ... ... ... ... ... ... ... uk ≤ U (a) < uk−1 ⇒ a ∈ Ck ... ... ... ... ... ... ... U (a) < uQ−1

⇒ a ∈ CQ

Taking into account (3) and the two types of errors that have been described, the above inequalities can be written as follows: m X

ui [gi (α)] − u1 + σ + (a) ≥ 0

i=1

   ui [gi (α)] − uk−1 − σ (α) ≤ −δ    

m X

(4)

∀a ∈ Ck

(5)

∀ ∈ CQ ,

(6)

−

i=1 m X

     

ui [gi (α)] − uk + σ + (α) ≥ 0

i=1 m X

∀a ∈ C1

ui [gi (α)] − uQ−1 − σ − (a) ≤ −δ

i=1

where δ is a small positive real number, used to ensure the strict inequality of U (a) to uk−1 (in the cases a ∈ Ck , k > 1) and uQ−1 (in the cases a ∈ CQ ). The aim is to estimate both the marginal utilities ui [gi (a)] and the utility thresholds uk that satisfy the above constraints (4), (5) and (6), minimizing the sum of all the errors (LP1). X X X Minimize F = σ + (α) + . . . + [σ + (α) + σ − (α)] + . . . + σ − (α) α∈C1

α∈Ck

α∈CQ

subject to: m X

ui [gi (α)] − u1 + σ + (a) ≥ 0

i=1 m X i=1 m X i=1

∀a ∈ C1

   ui [gi (α)] − uk−1 − σ − (α) ≤ −δ     ui [gi (α)] − uk + σ + (α) ≥ 0

     

∀a ∈ Ck

A MULTICRITERIA DECISION AID METHODOLOGY FOR SORTING DECISION PROBLEMS m X

ui [gi (α)] − uQ−1 − σ − (a) ≤ −δ

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∀a ∈ CQ

i=1 m aX i −1 X

wij = 1

i=1 j =1

uk−1 − uk ≥ s,

k = 2, 3, . . . , Q − 1

wij ≥ 0, σ + (a) ≥ 0, σ − (a) ≥ 0 The threshold s is used to denote the strict preference relation between the utility thresholds that distinguish the classes. In a second stage the sensitivity of the optimal solution F ∗ achieved by solving the above linear program, is examined through a post-optimality analysis. The aim is to find, if possible, multiple or generally near optimal solutions corresponding to error values lower than F ∗ + k(F ∗ ), where k(F ∗ ) is a small proportion of F ∗ . Therefore, the error objective is transformed into a new constraint of the type: X

σ + (α) + . . . +

α∈C1

X

[σ + (α) + σ − (α)] + . . . +

α∈Ck

X

σ − (α) ≤ F ∗ + k(F ∗ ) .

α∈CQ

(7)

The new objective is to maximize and minimize the weights for each criterion and the utility thresholds uk . In this way the sensitivity analysis of the weights of the criteria is achieved, and at the same time one can have an idea of the sensitivity of the utility thresholds:     aX Q−1 aX Q−1 i −1 i −1 X X max  wij + uk  and min  wij + uk  ∀i . i

j =1

k=1

i

j =1

k=1

Denoting as |C1 | the number of alternatives that belong in class C1 , as |Ck | the number of alternatives belonging in any intermediate class Ck and as |CQ | the numP ber of alternatives belonging in class CQ , LP1 has |C1 |+2 Q−1 k=2 |Ck |+|CQ |+Q−1 linear constraints (non-negativity constraints). Concerning the number of variables, P there are |C1 | + 2 Q−1 |C | + |C | variables involving the over-estimation and k Q k=2 under-estimation errors [σ +P (a) and σ − (a) respectively], Q − 1 variables involving m the utility thresholds and i=1 (ai − 1) variables wij . All these variables are continuous. Thus, the solution of LP1 is not computationally intensive even for large-scale problems. During the post-optimality stage, LP1 must be solved 2m times with the additional constraint (7), while the number of variables remains the same. Furthermore, it should be mentioned that the application of the UTADIS method to classify a large number of alternatives does not necessarily require the consideration of all the alternatives during the building of the additive utility model through the above procedure. Instead, a small reference set of alternatives may be selected to build the additive utility model, and if this model is considered satisfactory by the decision maker, then it can be extrapolated to classify the whole set

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of alternatives. This extrapolation ability of the models that are developed through the UTADIS method enables the decision maker to build and evaluate classification models easily through a computationally tractable procedure. Closing the discussion about the UTADIS method, it is important to note that apart from the classification of the alternatives, the decision maker can examine the competitive level between the alternatives of the same according to their global utilities (i.e., which are the best and the worst alternatives within a specific class). 3. Applications The UTADIS method has been applied in two real world classification problems concerning the evaluation of bankruptcy risk of firms financed by an industrial development bank in Greece and the prediction of business failure of Greek firms (cf. Slowinski and Zopounidis, 1995; Dimitras, 1995; Dimitras et al., 1999). 3.1.

THE EVALUATION OF BANKRUPTCY RISK

Data The first application of the UTADIS method in the evaluation of bankruptcy risk is originated by the study of Slowinski and Zopounidis (1995). The application involves 39 firms that were classified by the financial manager of a Greek industrial development bank called ETEVA in three predefined classes: • The acceptable firms, including firms that the financial manager would recommend for financing (class C1 ). • The uncertain firms, including firms for which further study was needed (class C2 ). • The unacceptable firms, including firms that the financial manager would not recommend to be financed by the bank (class C3 ). The sample of the 39 firms included 20 firms that were considered as acceptable firms (healthy firms, class C1 ), 10 firms for which a further study was needed (class C2 ), and finally, 9 firms that were considered as bankrupt (class C3 ). The firms were evaluated along 12 criteria (Table I). The evaluation criteria included six quantitative criteria (financial ratios) and six qualitative criteria (Siskos et al., 1994; Slowinski and Zopounidis, 1995). Presentation of Results The classification of the firms according to their global utilities and the utility thresholds u1 and u2 that are calculated by the UTADIS method are presented in Table II. Figure 2 presents the marginal utilities of the evaluation criteria.

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Figure 2. Marginal utilities of the evaluation criteria.

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Figure 2. (Continued).

Table I. Evaluation criteria (source: Slowinski and Zopounidis, 1995). Code

Evaluation criteria

G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12

Earnings before interest and taxes/Total assets Net income/Net worth Total liabilities/Total assets Total liabilities/Cash flow Interest expenses/Sales General and administrative expenses/Sales Managers’ work experience Firm’s market niche/position Technical structure-Facilities Organization-Personnel Special competitive advantages of firms Market flexibility

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Table II. Classification results by the UTADIS method. Firms

Original class

Utility

Estimated class

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20

C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1

0.6451 0.9796 0.8777 0.6527 0.6443 0.6467 0.6600 0.6604 0.6308 0.6227 0.6351 0.6452 0.6229 0.6314 0.6230 0.6436 0.6277 0.6435 0.6248 0.6321

C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1

Utility threshold u1 F21 F22 F23 F24 F25 F26 F27 F28 F29 F30

0.6226 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2

Utility threshold u2 F31 F32 F33 F34 F35 F36 F37 F38 F39

0.3836 0.3847 0.6102 0.3727 0.3859 0.3851 0.3862 0.3871 0.4001 0.3861

C2 C2 C2 C2 C2 C2 C2 C2 C2 C2

0.3726 C3 C3 C3 C3 C3 C3 C3 C3 C3

0.3096 0.3717 0.3717 0.3657 0.2004 0.3303 0.3382 0.2970 0.2286

C3 C3 C3 C3 C3 C3 C3 C3 C3

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Table III. Evaluation criteria. Code

Evaluation criteria

G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12

Net income/Gross profit Gross profit/Total assets Net income/Total assets Net income/Net worth Current assets/Current liabilities Quick assets/Current liabilities (Long term debt + current liabilities)/Total assets Net worth/(Net worth + long term debt) Net worth/Net fixed assets Inventories/Working capital Current liabilities/Total assets Working capital/Net worth

According to the achieved results there are no misclassifications (classification accuracy 100%). This fact implies that the solution of LP1 that was described in Section 2, along with the post-optimality analysis, resulted in an optimum solution corresponding to a classification error of zero. The obtained results are comparable with the results derived by the application of the rough set approach in the same problem (Slowinski and Zopounidis, 1995). Moreover, the criteria G1 (industrial profitability) and G7 (managers’ work experience) that have a weight of 12.46% and 28.81% respectively, were found to be the most important for the rough set method (they are included in the core, cf. Slowinski and Zopounidis, 1995). Another very important criterion is G2 (financial profitability) with a weight of 36.02%.

3.2.

THE BUSINESS FAILURE PREDICTION

Data The second application of the UTADIS method in the prediction of business failure is originated by the study of Dimitras (1995) and also Dimitras et al. (1999). A sample of 80 firms (40 bankrupt and 40 non-bankrupt) was used as the basic sample (for a five year period) and another sample of 38 firms (19 bankrupt and 19 non-bankrupt) was used as the control sample (for a three year period) to test the predictability of the method. The firms are evaluated along the 12 financial ratios presented in Table III.

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Figure 3. Marginal utilities of the evaluation criteria.

Presentation of Results The first year prior to the year of bankruptcy (year –1) for the basic sample was used to develop the additive utility model using the UTADIS method. The marginal utilities of the evaluation criteria are presented in Figure 3.

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Figure 3. (Continued).

Table IV. Error analysis for the application of the UTADIS method for the years –1, –2, –3, –4, –5 of the basic sample.

Type I error Type II error Total error

Year –1

Year –2

Year –3

Year –4

Year –5

0% 0% 0%

17.50% 15.00% 16.25%

20.00% 12.50% 16.25%

30.00% 20.00% 25.00%

40.00% 17.50% 28.75%

According to the marginal utilities, the most important criteria are G2 (Gross profit/Total assets), G1 (Net income/Gross profit), and G5 (Current assets/Current liabilities) with weights 23.409%, 15.863%, and 11.759% respectively. The predictability of the additive utility model developed using the data of year –1, was tested on the previous years (years –2, –3, –4, and –5). The obtained results (type I error, type II error, and overall error) for each year are presented in Table IV. The type I error means that a failed firm is classified as non-failed, while the type II error means that a non-failed firm is classified as failed. In that sense, type I error corresponds to an under-estimation error [σ − (a)], while type II error corresponds to an over-estimation error [σ + (a)].

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Table V. Error analysis for the application of the UTADIS method for the years –1, –2, –3 of the control sample.

Type I error Type II error Total error

Year –1

Year –2

Year –3

26.32% 47.37% 36.84%

47.37% 36.84% 42.11%

52.63% 21.05% 36.84%

According to the results of Table IV, in the first year prior to bankruptcy there are no misclassifications (similarly to the previous application, the solution of LP1 that was described in Section 2, along with the post-optimality analysis, resulted in an optimum classification error of zero). In years –2 and –3 the total error is 16.25%, while in years –4 and –5 the total error increases at 25%, and 28.75% respectively. The increase of the total error over the five years is mainly due to the significant increase of the type I error, while on the other hand the type II error is rather stable. The predictability of the model developed by the UTADIS method was also tested using the control sample of the 38 firms. The results concerning the type I error, type II error, and total error are presented in Table V. The total error in the first year prior to bankruptcy (year –1) is 36.84%, in year –2 the total error increases up to 42.11%, while in year –3 the total error decreases down to 36.84%. It is obvious that the obtained results are worse than the ones derived using the basic sample, but this is mainly caused by the differences between the two samples (Dimitras, 1995). Comparison Between UTADIS and Discriminant Analysis Dimitras et al. (1999) using the same sample of firms developed a discriminant analysis model to predict corporate failure. Discriminant analysis is a multivariate statistical technique that leads to the development of a linear discriminant function in order to maximize the ratio of among-group to within-group variability, assuming that the variables follow a multivariate normal distribution and that the dispersion matrices of the groups are equal. Clearly, both these assumptions create a significant problem regarding the application of discriminant analysis in real world situations, since they are difficult to meet. Nevertheless, discriminant analysis has found several applications in the field of finance as an approach for studying financial decision problems that require a grouping of a set of alternatives, which is the focal point of the issue in business failure prediction. In this case study, the discriminant analysis was applied following the same methodology that was used for the development of the business failure prediction model through the UTADIS method. The first year prior to failure for the basic

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Table VI. Discriminant function’s coefficients (source: Dimitras et al., 1999). Evaluation criteria

Coefficient

G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 Constant

0.0093 1.9154 2.4196 0.1245 1.2882 –0.9008 –0.7149 0.0004 0.0342 –0.0168 0.6294 0.0022 –1.1510

Table VII. Error analysis for the application of the discriminant analysis for the years –1, –2, –3, –4, –5 of the basic sample (source: Dimitras et al., 1999). Type I error Type II error Total error

12.50% 7.50% 10.00%

25.00% 12.50% 18.75%

32.50% 12.50% 22.50%

45.00% 15.00% 30.00%

45.00% 20.00% 32.50%

sample was used to develop a linear discriminant function through discriminant analysis. Since the aim of the application of discriminant analysis, was to compare it with the UTADIS method, it was decided not to use a stepwise procedure for selecting the financial ratios to be included in the discriminant function. Instead, all the 12 financial ratios are incorporated in the developed discriminant function so that the comparison between discriminant analysis and the UTADIS method will be performed on the same basis. Table VI presents the discriminant function’s coefficients of this model, Table VII presents the error analysis for years –1, –2, –3, –4, and –5 of the basic sample, while Table VIII presents the error analysis for years –1, –2, –3 of the control sample. According to the results of Table VII, discriminant analysis obtained a total error of 10% in the first year prior to bankruptcy of the basic sample. For the same sample and in the previous years, the total error increases considerably reaching 32.5% in year –5. As it has been observed in the results of the UTADIS method

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Table VIII. Error analysis for the application of the discriminant analysis for the years –1, –2, –3 of the control sample (source: Dimitras et al., 1999).

Type I error Type II error Total error

Year –1

Year –2

Year –3

36.84% 31.58% 34.21%

57.89% 26.32% 42.11%

63.16% 26.32% 44.74%

this increase is mainly due to the rapid increase of the type I error. As far as the control sample is concerned (Table VIII), there is a stable increase of the total error from 34.21% in year –1 to 44.74% in year –3. Moreover, it is interesting to point out that the results of DA seem to be biased, since the type I error is significantly higher than the type II error for all years of the analysis both in the basic and the holdout samples. Especially, in the case of the holdout sample in years –2 and –3 the type I error exceeds 57%, which is a rather disappointing result. Comparing the obtained results of the two different approaches, the superiority of the UTADIS method over discriminant analysis is clear, considering either the basic or the control sample. More specifically, as far as the basic sample is concerned the UTADIS method provides significantly lower error rates for all types of errors in the 5 years of the analysis, except for the type II error in year –2, where discriminant analysis provides a slightly lower error rate. As far as the control sample is concerned, discriminant analysis provides slightly lower rates in year –1. In year –2 the results are the same, but in the last year the UTADIS method provides considerably better results for all types of errors. It is also important to note that the UTADIS method provides substantially lower type I error rates (firms classified as non-bankrupt while they wend bankrupt) than the discriminant analysis in all years. Dimitras et al. (1999) also applied the rough set approach in the same problem. The obtained results by the rough set approach are comparable to the results of the UTADIS method. 4. Extensions of the Method In this section, some extensions of the UTADIS method are presented and discussed. 4.1.

MINIMIZING THE NUMBER OF MISCLASSIFICATIONS

In order to improve the performance of the UTADIS method it would be possible to minimize the total number of misclassification errors, instead of minimizing the

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amount of theseX errors. In this case the linear program can be formulated as follows: Minimize F = M + (α) + M − (α) α∈A

under the constraints: m X ui [gi (α)] − u1 + M + (a) ≥ 0 i=1 m X i=1 m X

   ui [gi (α)] − uk−1 − M (α) ≤ −δ     −

ui [gi (α)] − uk + M + (α) ≥ 0

i=1 m X

∀a ∈ C1

ui [gi (α)] − uQ−1 − M − (a) ≤ −δ

     

∀a ∈ Ck

∀a ∈ CQ

i=1 m aX i −1 X

wij = 1

i=1 j =1

uk−1 − uk ≥ s,

k = 2, 3, . . . , Q − 1

wij ≥ 0 M + (α) and M − (α) are boolean variables denoting the misclassification of an alternative a. If an alternative is correctly classified then M + (α) = 0 and M − (α) = 0. If an alternative is classified in a lower class than its original class then M + (α) = 1, otherwise if an alternative is classified in a higher class than its original class then M − (α) = 1 (Zopounidis and Doumpos, 1998). 4.2.

NON - MONOTONE PREFERENCES

The model could, also, be altered to handle non-monotone preferences. Often in many real world problems the preferences of decision makers concerning the evaluation of an action on a specific criterion are not monotone (increasing or decreasing) on its scale (Despotis and Zopounidis, 1995). For example, one would prefer a cup of coffee with 10 gr of sugar than a cup of coffee containing 5 gr of sugar, but he/she would not prefer a cup of coffee with 20 gr of sugar than a cup of coffee with 10 gr of sugar. In such cases the preferences of decision makers are monotonically increasing up to a desired level and monotonically decreasing for values exceeding this level. In such cases the range of the values of an attribute is divided into a number of intervals so that the preferences in each interval are monotone. Then, the marginal utility of each alternative is approximated by linear interpolation, taking into account the interval that it belongs (cf. Despotis and Zopounidis, 1995).

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4.3.

215

POST- OPTIMALITY ANALYSIS USING THE L∞ NORM

In the post-optimality stage the aim is to investigate the existence of sub-optimal solutions (in the sense of the total misclassification error) that may provide the same or better classification results (classification accuracy). Nevertheless, except for the total error F ∗ , it is also the dispersion of the individual errors that is determinant of the classification accuracy. Therefore, it would be possible in the post-optimality analysis to investigate sub-optimal solutions that minimize the differences between the maximum and the minimum error. This requirement can be satisfied by minimizing the maximum individual error (L∞ norm, cf. Despotis et al., 1990).

5. Concluding Remarks This paper presented the application of the UTADIS ordinal regression method for constructing additive utility functions in sorting problems to the case of financial distress. The obtained results on the two real world applications that have been presented are encouraging. They are superior to the results of discriminant analysis and comparable with the results of the rough set approach, indicating that the UTADIS method could be a useful and powerful tool for analyzing the decision makers’ preferences in sorting problems. The possible applications of this multicriteria method concern every financial classification decision problem, including the assessment of corporate failure risk, credit granting problems, venture capital investments, portfolio selection, financial planning, and other classification decision problems such as marketing of new products, sales strategy problems, environmental decisions, etc. Finally, the method could be incorporated in an integrated DSS, such as the FINEVA system for the assessment of corporate performance and viability (Zopounidis et al., 1996a,b), or in a specific DSS for the evaluation of bankruptcy risk, so that the decision makers could take advantage of the capabilities of the method, through powerful database management systems, graphical techniques, and friendly user interfaces. Such a DSS could save a significant amount of time for the decision makers, offering them the opportunity to examine further the possible solutions and plan their future actions.

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