A multicriteria decision aid methodology for sorting decision problems: The case of financial distress

Computational Economics14: 197–218, 1999.© 1999Kluwer Academic Publishers. Printed in the Netherlands.

197

A Multicriteria Decision Aid Methodologyfor Sorting Decision Problems: The Caseof Financial Distress

CONSTANTIN ZOPOUNIDIS and MICHAEL DOUMPOSTechnical University of Crete, Department of Production Engineering and Management, FinancialEngineering 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 alternativeactions (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 knownapproach in MCDA is based on preference disaggregation which has already been used in rankingproblems, but it is also applicable in sorting problems. The UTADIS (UTilités Additives DIScrim-inantes) method, a variant of the UTA method, based on the preference disaggregation approachestimates a set of additive utility functions and utility profiles using linear programming techniquesin 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 problemsconcerning the field of financial distress. The applications are derived by the studies of Slowinskiand Zopounidis (1995), and Dimitras et al. (1999). The obtained results depict the superiority of theUTADIS method over discriminant analysis, and they are also comparable with the results derivedby 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 altern-ative actions (solutions) over a set of criteria in order to reach a decision. Decisionproblems can be categorized in the following four types (problematics) accordingto the objective of the decision: (i) selection of the most appropriate (best) alternat-ive, (ii) sorting of the alternatives in predefined homogenous classes, (iii) rankingof the alternatives from the best one to the worst one, and (iv) description of thealternatives.

Many practical decision problems such as fault diagnosis, medical diagnosis,pattern recognition, diagnosis of sales potential, etc., belong to the second ofthese problematics, i.e., the sorting. Multiple criteria decision aid (MCDA, Zeleny,1982; Roy, 1985; Roy and Bouyssou, 1993; Roy, 1996) provides several powerfuland effective tools for confronting sorting problems, such as the ELECTRE TRI

198 CONSTANTIN ZOPOUNIDIS AND MICHAEL DOUMPOS

method proposed by Yu (1992) and its variant proposed by Mousseau and Slow-inski (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. Thepreference disaggregation approach aims at the estimation of an additive utilityfunction through the analysis of the global judgments (ranking or grouping of thealternatives) of the decision maker. UTA (UTilités Additives, cf. Jacquet-Lagrèzeand Siskos, 1982; Siskos and Yannacopoulos, 1985) is a well known preferencedisaggregation method based on ordinal regression analysis, which is mainly ori-ented towards ranking problems. Using a variant of the UTA method, it is possibleto estimate a set of additive utility functions, which minimize the misclassificationerror between the classes. At the same time the profiles (utility thresholds), thatdistinguish the classes, can be calculated. Such a variant of the UTA method is theUTADIS method (UTilités Additives DIScriminantes, Devaud et al., 1980; Jacquet-Lagrèze and Siskos, 1982; Jacquet-Lagrèze, 1995) for building additive utilityfunctions in sorting problems. An application of the UTADIS method in the evalu-ation of research and development projects has been presented by Jacquet-Lagrèze(1995).

A sorting problem of major practical and academic interest is that of financialdistress. Financial distress involves a situation where the firm cannot fulfil its ob-ligations to its creditors, suppliers, or because a bill is overdrawn, etc., resultingin a respite of the firm’s operation. Of course other definitions of financial distresscould 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 riskis of vital importance. This was the motivation for several scientists and researchersto develop methods and techniques for predicting business failure as effectively andaccurately as possible. Obviously, the main problem in the case of financial distressis to classify a set of firms in predefined classes distinguishing the firms accordingto their failure risk (healthy firms, bankrupt firms, uncertain firms).

The first approach used in the prediction of financial distress was based onunivariate and multivariate statistical techniques. The objective of univariate stat-istical techniques was to determine the most important financial ratio providing thehigher predicting accuracy. According to the selected ratio and a correspondingcut-off value, the firms are classified in two groups, bankrupt and non-bankrupt.Using this approach, Beaver (1966) has concluded that the financial ratio ‘Totaldebt/Cash flow’, provides the higher discrimination ability. Then, multivariate stat-istical 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-

A MULTICRITERIA DECISION AID METHODOLOGY FOR SORTING DECISION PROBLEMS 199

istical assumptions (distribution of the sample, independence of variables, etc.),which lead to several problems, such as the multicollinearity of the variables, thedifficulty in the explanation of error rates, the selection of the a priori probabilitiesor costs of misclassification, etc. (Eisenbeis, 1977). Later multivariate conditionalprobability 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 offailure risk of substantial strategic variables such as quality of management, organ-ization, market trend, market position, etc. Thus, the prediction and the analysisof business failure is based only on the examination of financial ratios, ignoringsignificant information which can not be assessed using quantitative criteria, suchas financial ratios.

In order to confront these problems new techniques were proposed, based oninformation and computer science (rough sets theory, expert systems, decision sup-port 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 decisionrules describing a set of objects (firms) by a set of multi-valued attributes (financialratios 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 assessmentof business failure risk (Bouwman, 1983; Ben-David and Sterling, 1986; Elmer andBorowski, 1988; Messier and Hansen, 1988; Shaw and Gentry, 1988; Cronan etal., 1991; Michalopoulos and Zopounidis, 1993). ESs represent the knowledge thatexpert financial or credit analysts use in practical cases when assessing corporatefailure risk. The symbolic reasoning and the explanation capabilities of ESs makethem highly applicable in decision problems based on judgemental procedures suchas the assessment of failure risk.

Decision support systems (DSSs) were also used as a tool to provide supportto decision makers in assessing business failure risk (Mareschal and Brans, 1991;Zopounidis et al., 1992; Siskos et al., 1994). The progress in computer scienceprovided the necessary means for performing complex and time-consuming tasksvery easily, as well as for accessing and handling large data bases. The implement-ation of DSSs in the assessment of failure risk was combined with the applicationof 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 symbolicreasoning and the explanation capabilities of the ESs’ technology with the powerfulanalytical tools and techniques already used in DSSs. This type of system, knownas knowledge-based decision support system (KBDSS), has recently started to beapplied 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 isbased on MCDA (Zopounidis, 1987; Dimitras, 1995; Dimitras et al., 1995). Sev-


eral multicriteria methods applied in the evaluation of bankruptcy risk can befound in Zopounidis (1995), while a complete survey of all the methods that havebeen applied in this field of financial management can be found in Dimitras etal. (1996). The UTA multicriteria method has already been applied in several de-cision 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 ofcorporate failure risk. Initially, the method is described in Section 2. Then, in Sec-tion 3 two real case applications of the method are presented concerning financialdistress. Some possible extensions of the method are described in Section 4, andfinally the concluding remarks are discussed (Section 5).

2. The UTADIS Method

Let g1, g2, . . . , gm be a consistent family ofm evaluation criteria, andA ={a1, a2, . . . , an} a set ofn alternatives to be classified inQ ordered classesC1, C2, . . . , CQ which are defined a priori:

C1PC2 . . . , CQ−1PCQ ,

where,P denotes the strict preference relation, between the classes.For each evaluation criteriongi the intervalGi = [gi∗ , g∗i ] of its values is

defined.gi∗ and g∗i are the less and the most preferred values, respectively, ofcriteriongi for all the alternatives belonging toA. The intervalGi is divided intoai−1 equal intervals[gji , gj+1

i ], j = 1,2, . . . , ai−1. ai , is defined by the decisionmaker as the number of estimated points for every marginal utilityui . Each pointgj

i can be calculated using linear interpolation:

gj

i = gi∗ +j − 1

ai − 1(g∗i − gi∗).

The aim is to estimate the marginal utilities at each of these points. Suppose that theevaluation of an alternativea on criteriongi is gi(a) ∈ [gji , gj+1

i ]. The marginalutility of the alternative actiona, ui[gi(a)], can be approximated through linearinterpolation:

ui[gi(α)] = ui(gji )+gi(α)− gjigj+1i − gji

[ui(gj+1i )− ui(gji )] . (1)

To achieve the monotonicity of the criteria the following constraint must besatisfied:

ui(gj+1i )− ui(gji ) ≥ 0, ∀i .


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

The monotonicity constraints are taken into account through the followingtransformations, as in the UTASTAR method (Siskos and Yannacopoulos, 1985):

wij = ui(gj+1i )− ui(gji ) ≥ 0∀i, j

ui(gi∗) = 0

ui(gj

i ) =j−1∑k=1

wik

(2)

Thus, the weights of each criterion can be computed as:ui(g∗i ) =

∑ai−1k=1 wik .

Using these transformations, (1) can be written as:

ui[gi(α)] =j−1∑k=1

wik + gi(α)− gj

i

gj+1i − gji

wij .

The global utilityU(a) of an alternativea ∈ A is of an additive form:

U(a) =m∑i=1

ui[gi(α)] . (3)

There are two possible errors (misclassification errors) relative to the globalutility U(a): the over-estimation errorσ+(a) and the under-estimation errorσ−(a).An over-estimation error exists in cases where an alternative according to its utilityis classified to a lower class than the class that it belongs (e.g., an alternative isclassified in classC2 while belonging in classC1). In such cases the amountσ+(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 util-ity is classified to a higher class than the class that it belongs (e.g., an alternative isclassified in classC1 while belonging in classC2). In such cases the amountσ−(a)should be subtracted from the utility of this alternative so that it can be correctlyclassified. These two types of errors are better presented in Figure 1.

The classification of the alternatives is achieved through the comparison of eachutility with the corresponding utility thresholdsui(a1 > u2 > . . . > uQ−1), which


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 ∈ CQTaking into account (3) and the two types of errors that have been described,

the above inequalities can be written as follows:m∑i=1

ui[gi(α)] − u1+ σ+(a) ≥ 0 ∀a ∈ C1 (4)

m∑i=1

ui[gi(α)] − uk−1− σ−(α) ≤ −δm∑i=1

ui[gi(α)] − uk + σ+(α) ≥ 0

∀a ∈ Ck (5)

m∑i=1

ui[gi(α)] − uQ−1− σ−(a) ≤ −δ ∀ ∈ CQ , (6)

whereδ is a small positive real number, used to ensure the strict inequality ofU(a)

to uk−1 (in the casesa ∈ Ck, k > 1) anduQ−1 (in the casesa ∈ CQ). The aimis to estimate both the marginal utilitiesui[gi(a)] and the utility thresholdsuk thatsatisfy the above constraints (4), (5) and (6), minimizing the sum of all the errors(LP1).

Minimize F =∑α∈C1

σ+(α) + . . . +∑α∈Ck[σ+(α)+ σ−(α)] + . . . +

∑α∈CQ

σ−(α)

subject to:m∑i=1

ui[gi(α)] − u1+ σ+(a) ≥ 0 ∀a ∈ C1

m∑i=1

ui[gi(α)] − uk−1− σ−(α) ≤ −δm∑i=1

ui[gi(α)] − uk + σ+(α) ≥ 0

∀a ∈ Ck


m∑i=1

ui[gi(α)] − uQ−1− σ−(a) ≤ −δ ∀a ∈ CQ

m∑i=1

ai−1∑j=1

wij = 1

uk−1− uk ≥ s, k = 2,3, . . . ,Q− 1

wij ≥ 0, σ+(a) ≥ 0, σ−(a) ≥ 0

The thresholds is used to denote the strict preference relation between the utilitythresholds that distinguish the classes.

In a second stage the sensitivity of the optimal solutionF ∗ achieved by solvingthe above linear program, is examined through a post-optimality analysis. The aimis to find, if possible, multiple or generally near optimal solutions correspondingto error values lower thanF ∗ + k(F ∗), wherek(F ∗) is a small proportion ofF ∗.Therefore, the error objective is transformed into a new constraint of the type:∑

α∈C1

σ+(α)+ . . .+∑α∈Ck[σ+(α)+ σ−(α)] + . . .+

∑α∈CQ

σ−(α) ≤ F ∗ + k(F ∗) . (7)

The new objective is to maximize and minimize the weights for each criterionand the utility thresholdsuk. In this way the sensitivity analysis of the weights ofthe criteria is achieved, and at the same time one can have an idea of the sensitivityof the utility thresholds:

maxi

ai−1∑j=1

wij +Q−1∑k=1

uk

and mini

ai−1∑j=1

wij +Q−1∑k=1

uk

∀i .Denoting as|C1| the number of alternatives that belong in classC1, as|Ck| the

number of alternatives belonging in any intermediate classCk and as|CQ| the num-ber of alternatives belonging in classCQ, LP1 has|C1|+2

∑Q−1k=2 |Ck|+|CQ|+Q−1

linear constraints (non-negativity constraints). Concerning the number of variables,there are|C1| + 2

∑Q−1k=2 |Ck| + |CQ| variables involving the over-estimation and

under-estimation errors [σ+(a) andσ−(a) respectively],Q−1 variables involvingthe utility thresholds and

∑mi=1(ai − 1) variableswij . All these variables are

continuous. Thus, the solution of LP1 is not computationally intensive even forlarge-scale problems. During the post-optimality stage, LP1 must be solved 2m

times with the additional constraint (7), while the number of variables remains thesame. Furthermore, it should be mentioned that the application of the UTADISmethod to classify a large number of alternatives does not necessarily require theconsideration of all the alternatives during the building of the additive utility modelthrough the above procedure. Instead, a small reference set of alternatives may beselected to build the additive utility model, and if this model is considered satis-factory by the decision maker, then it can be extrapolated to classify the whole set


of alternatives. This extrapolation ability of the models that are developed throughthe UTADIS method enables the decision maker to build and evaluate classificationmodels easily through a computationally tractable procedure.

Closing the discussion about the UTADIS method, it is important to note thatapart from the classification of the alternatives, the decision maker can examinethe competitive level between the alternatives of the same according to their globalutilities (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 problemsconcerning the evaluation of bankruptcy risk of firms financed by an industrialdevelopment 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 riskis originated by the study of Slowinski and Zopounidis (1995). The applicationinvolves 39 firms that were classified by the financial manager of a Greek industrialdevelopment bank called ETEVA in three predefined classes:

• The acceptable firms, including firms that the financial manager wouldrecommend for financing (classC1).• The uncertain firms, including firms for which further study was needed(classC2).• The unacceptable firms, including firms that the financial manager would notrecommend to be financed by the bank (classC3).

The sample of the 39 firms included 20 firms that were considered as acceptablefirms (healthy firms, classC1), 10 firms for which a further study was needed (classC2), and finally, 9 firms that were considered as bankrupt (classC3).

The firms were evaluated along 12 criteria (Table I). The evaluation criteria in-cluded six quantitative criteria (financial ratios) and six qualitative criteria (Siskoset al., 1994; Slowinski and Zopounidis, 1995).

Presentation of Results

The classification of the firms according to their global utilities and the utilitythresholdsu1 andu2 that are calculated by the UTADIS method are presented inTable II. Figure 2 presents the marginal utilities of the evaluation criteria.


Figure 2. Marginal utilities of the evaluation criteria.


Figure 2. (Continued).

Table I. Evaluation criteria (source: Slowinski and Zo-pounidis, 1995).

Code Evaluation criteria

G1 Earnings before interest and taxes/Total assets

G2 Net income/Net worth

G3 Total liabilities/Total assets

G4 Total liabilities/Cash flow

G5 Interest expenses/Sales

G6 General and administrative expenses/Sales

G7 Managers’ work experience

G8 Firm’s market niche/position

G9 Technical structure-Facilities

G10 Organization-Personnel

G11 Special competitive advantages of firms

G12 Market flexibility


Table II. Classification results by the UTADIS method.

Firms Original class Utility Estimated class

F1 C1 0.6451 C1F2 C1 0.9796 C1F3 C1 0.8777 C1F4 C1 0.6527 C1F5 C1 0.6443 C1F6 C1 0.6467 C1F7 C1 0.6600 C1F8 C1 0.6604 C1F9 C1 0.6308 C1F10 C1 0.6227 C1F11 C1 0.6351 C1F12 C1 0.6452 C1F13 C1 0.6229 C1F14 C1 0.6314 C1F15 C1 0.6230 C1F16 C1 0.6436 C1F17 C1 0.6277 C1F18 C1 0.6435 C1F19 C1 0.6248 C1F20 C1 0.6321 C1

Utility thresholdu1 0.6226

F21 C2 0.3836 C2F22 C2 0.3847 C2F23 C2 0.6102 C2F24 C2 0.3727 C2F25 C2 0.3859 C2F26 C2 0.3851 C2F27 C2 0.3862 C2F28 C2 0.3871 C2F29 C2 0.4001 C2F30 C2 0.3861 C2

Utility thresholdu2 0.3726

F31 C3 0.3096 C3F32 C3 0.3717 C3F33 C3 0.3717 C3F34 C3 0.3657 C3F35 C3 0.2004 C3F36 C3 0.3303 C3F37 C3 0.3382 C3F38 C3 0.2970 C3F39 C3 0.2286 C3


Table III. Evaluation criteria.

Code Evaluation criteria

G1 Net income/Gross profit

G2 Gross profit/Total assets

G3 Net income/Total assets

G4 Net income/Net worth

G5 Current assets/Current liabilities

G6 Quick assets/Current liabilities

G7 (Long term debt+ current liabilities)/Total assets

G8 Net worth/(Net worth+ long term debt)

G9 Net worth/Net fixed assets

G10 Inventories/Working capital

G11 Current liabilities/Total assets

G12 Working capital/Net worth

According to the achieved results there are no misclassifications (classificationaccuracy 100%). This fact implies that the solution of LP1 that was described inSection 2, along with the post-optimality analysis, resulted in an optimum solutioncorresponding to a classification error of zero. The obtained results are comparablewith the results derived by the application of the rough set approach in the sameproblem (Slowinski and Zopounidis, 1995). Moreover, the criteriaG1 (industrialprofitability) andG7 (managers’ work experience) that have a weight of 12.46%and 28.81% respectively, were found to be the most important for the roughset method (they are included in the core, cf. Slowinski and Zopounidis, 1995).Another very important criterion isG2 (financial profitability) with a weight of36.02%.

3.2. THE BUSINESS FAILURE PREDICTION

Data

The second application of the UTADIS method in the prediction of business fail-ure 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 basicsample (for a five year period) and another sample of 38 firms (19 bankrupt and 19non-bankrupt) was used as the control sample (for a three year period) to test thepredictability of the method. The firms are evaluated along the 12 financial ratiospresented in Table III.


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 wasused to develop the additive utility model using the UTADIS method. The marginalutilities of the evaluation criteria are presented in Figure 3.


Figure 3. (Continued).

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

Year –1 Year –2 Year –3 Year –4 Year –5

Type I error 0% 17.50% 20.00% 30.00% 40.00%

Type II error 0% 15.00% 12.50% 20.00% 17.50%

Total error 0% 16.25% 16.25% 25.00% 28.75%

According to the marginal utilities, the most important criteria areG2 (Grossprofit/Total assets),G1 (Net income/Gross profit), andG5 (Current assets/Currentliabilities) 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 IIerror means that a non-failed firm is classified as failed. In that sense, type I errorcorresponds to an under-estimation error[σ−(a)], while type II error correspondsto an over-estimation error[σ+(a)].


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

Year –1 Year –2 Year –3

Type I error 26.32% 47.37% 52.63%

Type II error 47.37% 36.84% 21.05%

Total error 36.84% 42.11% 36.84%

According to the results of Table IV, in the first year prior to bankruptcy thereare no misclassifications (similarly to the previous application, the solution of LP1that was described in Section 2, along with the post-optimality analysis, resultedin an optimum classification error of zero). In years –2 and –3 the total error is16.25%, 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 thesignificant increase of the type I error, while on the other hand the type II error israther stable.

The predictability of the model developed by the UTADIS method was alsotested using the control sample of the 38 firms. The results concerning the type Ierror, 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 decreasesdown to 36.84%. It is obvious that the obtained results are worse than the onesderived using the basic sample, but this is mainly caused by the differences betweenthe two samples (Dimitras, 1995).

Comparison Between UTADIS and Discriminant Analysis

Dimitras et al. (1999) using the same sample of firms developed a discriminantanalysis model to predict corporate failure. Discriminant analysis is a multivari-ate statistical technique that leads to the development of a linear discriminantfunction in order to maximize the ratio of among-group to within-group variab-ility, assuming that the variables follow a multivariate normal distribution and thatthe dispersion matrices of the groups are equal. Clearly, both these assumptionscreate a significant problem regarding the application of discriminant analysis inreal world situations, since they are difficult to meet. Nevertheless, discriminantanalysis has found several applications in the field of finance as an approach forstudying 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 samemethodology that was used for the development of the business failure predictionmodel through the UTADIS method. The first year prior to failure for the basic


Table VI. Discriminant function’scoefficients (source: Dimitras et al.,1999).

Evaluation criteria Coefficient

G1 0.0093

G2 1.9154

G3 2.4196

G4 0.1245

G5 1.2882

G6 –0.9008

G7 –0.7149

G8 0.0004

G9 0.0342

G10 –0.0168

G11 0.6294

G12 0.0022

Constant –1.1510

Table VII. Error analysis for the application of the discriminant analysisfor the years –1, –2, –3, –4, –5 of the basic sample (source: Dimitras etal., 1999).

Type I error 12.50% 25.00% 32.50% 45.00% 45.00%

Type II error 7.50% 12.50% 12.50% 15.00% 20.00%

Total error 10.00% 18.75% 22.50% 30.00% 32.50%

sample was used to develop a linear discriminant function through discriminantanalysis. Since the aim of the application of discriminant analysis, was to compareit with the UTADIS method, it was decided not to use a stepwise procedure forselecting the financial ratios to be included in the discriminant function. Instead,all the 12 financial ratios are incorporated in the developed discriminant functionso that the comparison between discriminant analysis and the UTADIS method willbe performed on the same basis.

Table VI presents the discriminant function’s coefficients of this model, Table VIIpresents the error analysis for years –1, –2, –3, –4, and –5 of the basic sample, whileTable 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 totalerror of 10% in the first year prior to bankruptcy of the basic sample. For the samesample and in the previous years, the total error increases considerably reaching32.5% in year –5. As it has been observed in the results of the UTADIS method


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

Year –1 Year –2 Year –3

Type I error 36.84% 57.89% 63.16%

Type II error 31.58% 26.32% 26.32%

Total error 34.21% 42.11% 44.74%

this increase is mainly due to the rapid increase of the type I error. As far as thecontrol sample is concerned (Table VIII), there is a stable increase of the total errorfrom 34.21% in year –1 to 44.74% in year –3. Moreover, it is interesting to pointout that the results of DA seem to be biased, since the type I error is significantlyhigher than the type II error for all years of the analysis both in the basic and theholdout samples. Especially, in the case of the holdout sample in years –2 and –3the type I error exceeds 57%, which is a rather disappointing result.

Comparing the obtained results of the two different approaches, the superiorityof the UTADIS method over discriminant analysis is clear, considering either thebasic or the control sample. More specifically, as far as the basic sample is con-cerned the UTADIS method provides significantly lower error rates for all types oferrors in the 5 years of the analysis, except for the type II error in year –2, wherediscriminant analysis provides a slightly lower error rate. As far as the controlsample 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 providesconsiderably better results for all types of errors. It is also important to note that theUTADIS method provides substantially lower type I error rates (firms classified asnon-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 theUTADIS method.

4. Extensions of the Method

In this section, some extensions of the UTADIS method are presented anddiscussed.

4.1. MINIMIZING THE NUMBER OF MISCLASSIFICATIONS

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


amount of these errors. In this case the linear program can be formulated as follows:Minimize F =

∑α∈A

M+(α)+M−(α)

under the constraints:m∑i=1

ui[gi(α)] − u1+M+(a) ≥ 0 ∀a ∈ C1

m∑i=1

ui[gi(α)] − uk−1−M−(α) ≤ −δm∑i=1

ui[gi(α)] − uk +M+(α) ≥ 0

∀a ∈ Ck

m∑i=1

ui[gi(α)] − uQ−1−M−(a) ≤ −δ ∀a ∈ CQ

m∑i=1

ai−1∑j=1

wij = 1

uk−1− uk ≥ s, k = 2,3, . . . ,Q− 1

wij ≥ 0

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

4.2. NON-MONOTONE PREFERENCES

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


4.3. POST-OPTIMALITY ANALYSIS USING THE L∞ NORM

In the post-optimality stage the aim is to investigate the existence of sub-optimalsolutions (in the sense of the total misclassification error) that may provide thesame or better classification results (classification accuracy). Nevertheless, exceptfor the total errorF ∗, it is also the dispersion of the individual errors that isdeterminant of the classification accuracy. Therefore, it would be possible in thepost-optimality analysis to investigate sub-optimal solutions that minimize the dif-ferences between the maximum and the minimum error. This requirement can besatisfied by minimizing the maximum individual error (L∞ norm, cf. Despotis etal., 1990).

5. Concluding Remarks

This paper presented the application of the UTADIS ordinal regression methodfor constructing additive utility functions in sorting problems to the case of fin-ancial distress. The obtained results on the two real world applications that havebeen presented are encouraging. They are superior to the results of discriminantanalysis and comparable with the results of the rough set approach, indicatingthat the UTADIS method could be a useful and powerful tool for analyzing thedecision makers’ preferences in sorting problems. The possible applications ofthis multicriteria method concern every financial classification decision problem,including the assessment of corporate failure risk, credit granting problems, venturecapital investments, portfolio selection, financial planning, and other classificationdecision 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 theFINEVA system for the assessment of corporate performance and viability (Zo-pounidis et al., 1996a,b), or in a specific DSS for the evaluation of bankruptcyrisk, so that the decision makers could take advantage of the capabilities of themethod, through powerful database management systems, graphical techniques,and friendly user interfaces. Such a DSS could save a significant amount of time forthe decision makers, offering them the opportunity to examine further the possiblesolutions and plan their future actions.

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A multicriteria decision aid methodology for sorting decision problems: The case of financial distress