1 Introduction
2 Brief review of the literature
3 MURAME for creditworthiness evaluation
3.1 Step I: reference profiles
3.2 Step II: concordance, discordance and outranking indexes
3.3 Step III: net flow and firms ranking
4 The optimization problem for the endogenous parameter specification
4.1 The optimization problem
-
It is quite hard to write an exact analytical expression in closed form for \({\mathcal {I}}(w_1,\dots ,w_n)\), in terms of the decision variables;
-
The function \({\mathcal {I}}(w_1,\dots ,w_n)\) in general might be neither smooth nor Lipschitz/Holder continuous;
-
The use of gradient-based methods for the solution of (8) is discouraged, since the derivatives of \({\mathcal {I}}(w_1,\dots ,w_n)\) are unavailable;
-
An evolutionary optimizer to fast approximately solve (8) might be more appropriate, since expensive accurate solutions are possibly unnecessary.
4.2 Balance sheet data on SMEs
\(I_{1}\) | Cost of debt: Financial costs/Bank debts |
\(I_{2}\) | Return on equity (ROE): Net profit before tax/Total equity |
\(I_{3}\) | Total assets turnover: Sales/Total assets |
\(I_{4}\) | R&D costs/Total asset |
\(I_{5}\) | Income tax/Profit before taxes |
\(I_{6}\) | Equity − Equipment |
\(I_{7}\) | Rate of increase of revenues from sales and services |
\(I_{8}\) | Liabilities/Total assets |
\(I_{9}\) | Cash/Total assets |
\(I_{10}\) | Working capital/Total assets |
\(I_{11}\) | Intangible/Total assets |
\(I_{12}\) | EBITDA/Total assets |
\(I_{13}\) | Retained earnings/Total assets |
\(I_{14}\) | Net income/Sales |
\(I_{15}\) | Short term debt/Equity |
\(I_{16}\) | EBITDA/Interest expenses |
\(I_{17}\) | Account payable/Sales |
\(I_{18}\) | Account receivable/Liabilities |
\(I_{19}\) | Sales/Personnel costs |
Years | Active | Bankrupt | Total |
---|---|---|---|
firms | firms | ||
2006 | 6625 | 1089 | 7714 |
2007 | 6766 | 925 | 7691 |
2008 | 6933 | 696 | 7629 |
Accounting ratio category | Variable |
---|---|
Leverage | Short term debt/Equity book value |
Liquidity | Cash/Total assets |
Profitability | EBITDA/Total assets |
Coverage | Retained earnings/Total assets |
Activity | EBITDA/Interest expenses |
5 Using PSO evolutionary method to solve the optimization problem (8)
5.1 A nonlinear reformulation of the optimization problem (8)
5.2 The penalty function approach, for comparison purposes
-
The proposal in Subsect. 5.1 always ensures that the found (sub)-optimal solution of problem (8) is always feasible; on the contrary, the last property does not necessarily hold for the penalty function-based approach;
-
The approach in Subsect. 5.1 does not require the burdensome assessment of the penalty parameter \(\epsilon \);
-
As a well known drawback from the literature (see, e.g., Corazza et al. 2013), in the penalty function-based approach the use of small values of the penalty parameter \(\epsilon \) may likely yield ill-conditioning; conversely, large values of \(\epsilon \) might not be satisfactory in order to guarantee the feasibility for the found solution.
5.3 The initialization procedures
Orthoinit
and Orthoinit+
, recently proposed in Corazza et al. (2015a) and Diez et al. (2016), respectively. The idea behind these two novel initializations is to scatter particle trajectories in the search space in the early iterations, in order to better initially explore the search space, and to obtain approximate solutions that are not grouped in a reduced sub–region of the feasible set. In the Appendix, we detail a brief summary of the theoretical results supporting these initializations (see Diez et al. 2016 for a more complete report).Orthoinit
) for PSO particlesOrthoinit+
) has been proposed in Diez et al. (2016). Here, the vectors \(z_i(k)\), \(i=1,\dots , 2n\) in (21) and (22) are replaced by the following ones6 Numerical results
Orthoinit
and Orthoinit+
allows a better minimization of \({\mathcal {I}}(\cdot , \ldots , \cdot )\) within the early iterations, compared to the one obtained with the usual random initialization of the PSO particles. This fact, if verified, could be useful to quickly finding approximate solutions when the size of \(A'\) is particularly large, and consequently, the time needed for each iteration of the PSO-based solver is quite long.Random | Orthoinit | Orthoinit+ | |
---|---|---|---|
\(|A'|=250\) | |||
\(\mu (A')\) | 0.534783 | 0.495652 | 0.547826 |
\(\sigma (A')\) | 0.055168 | 0.070644 | 0.078261 |
\(\mu (A)\) | 0.681466 | 0.643247 | 0.626437 |
\(\sigma (A)\) | 0.048448 | 0.047240 | 0.022899 |
\(|A'|=2500\) | |||
\(\mu (A')\) | 0.580263 | 0.522807 | 0.587719 |
\(\sigma (A')\) | 0.015451 | 0.042920 | 0.015445 |
\(\mu (A)\) | 0.608046 | 0.594109 | 0.603879 |
\(\sigma (A)\) | 0.011598 | 0.019901 | 0.004454 |
6.1 Results: the creditworthiness classifications with decision variables \({\mathbf {w}}\)
Orthoinit
” and “Orthoinit+
”). In particular, Tables 5 and 6 provide the results considering all the indicators of Table 1, while Tables 7 and 8 consider only the Altman’s variables of Table 3. We remind that the values of the weights \(\{w_j({\mathbf {t}})\}\) have been detected applying the solution algorithm to a reference set of firms \(A'\) of size 2500, and that the classification results reported here refer to the application of MURAME once set with the obtained weights to the entire population of firms. Lastly, for comparison purposes, in the first row of each table (labeled “Standard”) we give the classification results got using the standard exogenous specification of weights adopted in Corazza et al. (2016), that is \(w_j=1/n\) for the weights and \(q_j=s_j/6\) for the indifference thresholds, with \(j=1,\dots ,n\).Initialization | Class 1 (%) | Class 2 (%) | Class 3 (%) | Class 4 (%) | Class 5 (%) | |
---|---|---|---|---|---|---|
Standard | Bankrupt | 20.40 | 21.70 | 20.98 | 18.39 | 18.53 |
All | 23.59 | 20.85 | 20.71 | 20.23 | 14.62 | |
PSO-Random | Bankrupt | 5.60 | 12.36 | 19.40 | 31.03 | 31.61 |
All | 20.40 | 18.17 | 18.18 | 23.38 | 19.87 | |
PSO- Orthoinit | Bankrupt | 1.29 | 15.23 | 24.57 | 32.04 | 26.87 |
All | 13.07 | 24.76 | 25.32 | 21.37 | 15.48 | |
PSO- Orthoinit+ | Bankrupt | 9.63 | 15.09 | 18.53 | 27.59 | 29.17 |
All | 21.09 | 19.71 | 19.29 | 19.94 | 19.96 |
Initialization | Class 1 (%) | Class 2 (%) | Class 3 (%) | Class 4 (%) | Class 5 (%) | |
---|---|---|---|---|---|---|
Standard | Bankrupt | 20.40 | 21.70 | 20.98 | 18.39 | 18.53 |
All | 23.59 | 20.85 | 20.71 | 20.23 | 14.62 | |
PSO-Random | Bankrupt | 9.48 | 11.78 | 15.80 | 23.28 | 39.66 |
All | 20.87 | 20.79 | 17.63 | 18.67 | 22.05 | |
PSO- Orthoinit | Bankrupt | 2.16 | 2.16 | 6.18 | 65.09 | 24.43 |
All | 14.17 | 10.46 | 9.04 | 48.46 | 17.87 | |
PSO- Orthoinit+ | Bankrupt | 9.34 | 11.78 | 16.24 | 23.13 | 39.51 |
All | 20.89 | 20.02 | 18.29 | 18.52 | 22.28 |
Initialization | Class 1 (%) | Class 2 (%) | Class 3 (%) | Class 4 (%) | Class 5 (%) | |
---|---|---|---|---|---|---|
Standard | Bankrupt | 6.03 | 13.07 | 20.26 | 27.30 | 33.33 |
All | 20.20 | 20.04 | 19.98 | 19.90 | 19.88 | |
PSO-Random | Bankrupt | 5.60 | 13.51 | 19.97 | 27.59 | 33.33 |
All | 20.09 | 20.02 | 20.02 | 19.94 | 19.94 | |
PSO- Orthoinit | Bankrupt | 5.60 | 13.36 | 19.68 | 27.87 | 33.48 |
All | 20.02 | 19.99 | 19.99 | 20.00 | 20.00 | |
PSO- Orthoinit+ | Bankrupt | 5.60 | 13.51 | 19.68 | 27.73 | 33.48 |
All | 20.07 | 20.02 | 19.99 | 19.98 | 19.95 |
Initialization | Class 1 (%) | Class 2 (%) | Class 3 (%) | Class 4 (%) | Class 5 (%) | |
---|---|---|---|---|---|---|
Standard | Bankrupt | 6.03 | 13.07 | 20.26 | 27.30 | 33.33 |
All | 20.20 | 20.04 | 19.98 | 19.90 | 19.88 | |
PSO-Random | Bankrupt | 5.75 | 13.36 | 19.97 | 27.59 | 33.33 |
All | 20.13 | 20.02 | 19.99 | 19.94 | 19.92 | |
PSO- Orthoinit | Bankrupt | 5.60 | 13.36 | 19.68 | 27.87 | 33.48 |
All | 20.02 | 19.99 | 19.99 | 20.00 | 20.00 | |
PSO- Orthoinit+ | Bankrupt | 5.75 | 13.36 | 19.68 | 27.87 | 33.33 |
All | 20.12 | 20.00 | 19.96 | 19.99 | 19.92 |
-
When considering as criteria either all the indicators of Table 1 or only the Altman’s variables and using either the inconsistency measure \({\mathcal {I}}_1\) or the inconsistency measure \({\mathcal {I}}_2\), the (sub-)optimal values of the weights \(\{w_j({\mathbf {t}})\}\) determined by our PSO-based solver allow improvements in terms of the creditworthiness distribution of bankrupt firms with respect to the exogenous specification of the same weights (see Tables 5 and 6);
-
The overall quality of the creditworthiness distribution of bankrupt firms depends on the PSO initialization procedure applied. Indeed, in the case of the
Orthoinit
one, when all the nineteen indicators of Table 1 are considered and either using \({\mathcal {I}}_1\) or \({\mathcal {I}}_2\), the obtained classification (slightly) dissatisfies the aforementioned general principle, since it shows an higher concentration of bankrupt firms in the fourth rating class (see Tables 5 and 6); -
Conversely, when considering only the five Altman’s variables, some little improvements in the bankrupt firms’ creditworthiness classification are observed, generally regardless of the initialization procedure or the inconsistency measure adopted, with respect to the results obtained when using all the 19 indicators of Table 1 (see Tables 7 and 8).
6.2 Results: the creditworthiness classifications with decision variables \({\mathbf {w}}\) and \({\mathbf {q}}\)
Initialization | Class 1 (%) | Class 2 (%) | Class 3 (%) | Class 4 (%) | Class 5 (%) | |
---|---|---|---|---|---|---|
Standard | Bankrupt | 6.03 | 13.07 | 20.26 | 27.30 | 33.33 |
All | 20.20 | 20.04 | 19.98 | 19.90 | 19.88 | |
PSO-Random | Bankrupt | 0.43 | 18.25 | 58.48 | 19.83 | 3.02 |
All | 1.31 | 19.67 | 60.13 | 16.53 | 2.36 | |
PSO- Orthoinit | Bankrupt | 0.57 | 18.10 | 10.34 | 22.84 | 48.13 |
All | 7.20 | 39.70 | 14.84 | 14.10 | 24.16 | |
PSO- Orthoinit+ | Bankrupt | 0.43 | 16.67 | 63.22 | 16.95 | 2.73 |
All | 1.27 | 19.31 | 62.76 | 14.44 | 2.22 |
Initialization | Class 1 (%) | Class 2 (%) | Class 3 (%) | Class 4 (%) | Class 5 (%) | |
---|---|---|---|---|---|---|
Standard | Bankrupt | 6.03 | 13.07 | 20.26 | 27.30 | 33.33 |
All | 20.20 | 20.04 | 19.98 | 19.90 | 19.88 | |
PSO-Random | Bankrupt | 2.01 | 5.89 | 15.23 | 32.90 | 43.97 |
All | 20.02 | 20.09 | 20.06 | 19.90 | 19.94 | |
PSO- Orthoinit | Bankrupt | 0.72 | 20.11 | 10.06 | 21.98 | 47.13 |
All | 8.01 | 39.93 | 14.65 | 13.41 | 24.00 | |
PSO- Orthoinit+ | Bankrupt | 2.01 | 5.89 | 15.52 | 32.76 | 43.82 |
All | 20.08 | 20.08 | 20.11 | 19.83 | 19.90 |
-
In general, good results are obtained in terms of the percentages of bankrupt firms classified in the best creditworthiness class, either using the inconsistency measure \({\mathcal {I}}_1\) or the inconsistency measure \({\mathcal {I}}_2\) and regardless of the initialization procedure (see Tables 9 and 10, respectively). Moreover, these percentages are all better than the corresponding ones reported in Tables 7 and 8, respectively;
-
Nevertheless, when using \({\mathcal {I}}_1\) and independently of the initialization procedure employed, the creditworthiness classifications obtained always fail to satisfy the general principle considered in Sect. 4.1;
-
Conversely, when using \({\mathcal {I}}_2\), the same general principle as above is dissatisfied only once, in correspondence of the initialization procedure
Orthoinit
(see Table 10). Furthermore, again when using \({\mathcal {I}}_2\) and regardless of the initialization procedure, remarkable improvements are achieved in terms of the percentages of bankrupt firms classified in the worst creditworthiness class, with respect to the corresponding ones presented in Table 8.
6.3 Results: the endogenously determined MURAME parameters
Initialization | \(I_1\) (%) | \(I_2\) (%) | \(I_3\) (%) | \(I_4\) (%) | \(I_5\) (%) |
---|---|---|---|---|---|
PSO-Random | 0.01 | 34.22 | 14.76 | 0.01 | 11.56 |
PSO- Orthoinit | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
PSO- Orthoinit+ | 0.06 | 57.17 | 38.41 | 0.06 | 3.45 |
\(I_6\) (%) | \(I_7\) (%) | \(I_8\) (%) | \(I_9\) (%) | \(I_{10}\) (%) | |
---|---|---|---|---|---|
PSO-Random | 0.00 | 4.84 | 2.50 | 10.92 | 4.37 |
PSO- Orthoinit | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
PSO- Orthoinit+ | 0.06 | 0.06 | 0.06 | 0.06 | 0.06 |
\(I_{11}\) (%) | \(I_{12}\) (%) | \(I_{13}\) (%) | \(I_{14}\) (%) | \(I_{15}\) (%) | |
---|---|---|---|---|---|
PSO-Random | 0.02 | 0.03 | 3.27 | 0.59 | 3.34 |
PSO- Orthoinit | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
PSO- Orthoinit+ | 0.06 | 0.06 | 0.06 | 0.06 | 0.06 |
\(I_{16}\) (%) | \(I_{17}\) (%) | \(I_{18}\) (%) | \(I_{19}\) (%) | ||
---|---|---|---|---|---|
PSO-Random | 2.75 | 4.99 | 0.41 | 1.40 | |
PSO- Orthoinit | 100.00 | 0.00 | 0.00 | 0.00 | |
PSO- Orthoinit+ | 0.06 | 0.06 | 0.06 | 0.06 |
Initialization | \(I_{9}\) | \(I_{12}\) | \(I_{13}\) | \(I_{15}\) | \(I_{16}\) | |
---|---|---|---|---|---|---|
PSO-Random | \({\mathbf {w}}\) | 0.92% | 56.54% | 15.33% | 1.28% | 25.93% |
\({\mathbf {q}}\) | 6.994 | 5.468 | 2.886 | 20.714 | 0.021 | |
PSO- Orthoinit | \({\mathbf {w}}\) | 0.00% | 0.00% | 50.53% | 0.00% | 49.47% |
\({\mathbf {q}}\) | 0.000 | 0.000 | 0.000 | 0.000 | 1.549 | |
PSO- Orthoinit+ | \({\mathbf {w}}\) | 13.97% | 21.51% | 21.51% | 21.51% | 21.51% |
\({\mathbf {q}}\) | 0.161 | 0.161 | 0.161 | 22.829 | 0.161 |
-
A drawback related to the use of the initialization procedure
Orthoinit
emerges, as already highlighted in Corazza et al. (2015a): namely, the sparsity of the achieved solutions8. Indeed, while both the other two initialization procedures give dense solutions9, those provided byOrthoinit
lead to a creditworthiness classification model in which just one or at most two criteria play some role; -
Then, from Table 11 one can infer that a careless initialization of PSO, like that performed by the initialization procedure Random, may be unable to lead to clear indications about the most relevant criteria. Indeed, using Random a too large number of criteria is suggested, which does not help to rank them by their importance. This motivates the introduction of the initialization procedure
Orthoinit+
, whose performance generally constitutes a balance between the extreme ones provided by Random andOrthoinit
. -
Finally, again with reference to Table 11, one can note that the PSO-based solver initialized through
Orthoinit+
identifies \(I_2\), \(I_3\) and \(I_5\) as relevant criteria, while assigns the value 0.06% to the other weights, suggesting the scarce relevance of the latter. We highlight that, even if the obtained values for the endogenously determined MURAME parameters may appear unusual in a creditworthiness evaluation model, two aspects have to be remarked. First aspect: such values represent a (sub-)optimal solution determined by a metaheuristic. In general, ceteris paribus, the solution performances of a metaheuristic worsen when the dimension of the search space increases, as in the considered case. Therefore, as per operational practice, in order to improve the quality of the solution it might be necessary to increase the number of PSO iterations. Of course, this would have negative consequences in terms of the computational time required. Second aspect: we highlight that the determined values of the MURAME parameters are however significantly informative. Indeed, recalling that the main aim of a MCDA-based model is to support the DM in the decision process, the obtained values of the weights can represent a tentative assessment for a better understanding of the relevance of the various criteria. Similarly, the achieved values of the indifference thresholds can represent a basis for a better understanding of the preference structure implicit in the used data.
Orthoinit+
initialization seems preferable. This can be explained by noting that Orthoinit+
adopts an initial population which proves to be more scattered on the search space, with respect to both Orthoinit
and Random initializations. Thus, better approximate solutions can be obtained by PSO scheme, inasmuch as a better exploration of the search space is expected.6.4 Some in-depth-analyses about the deterministic initialization procedures
-
Especially when using the initialization procedure
Orthoinit+
, a fast decrease of \({\mathcal {I}}_2\) occurs after the very first iterations of PSO, that is when the effects of the uniformly linearly independent choice of the initialization of the PSO particles are still relevant; -
Then, further improvements of the solution quality generally require an additional number of PSO iterations. This could mean that the evolutionary process might be entangled in a plateau of \({\mathcal {I}}_2\). Notice that it happens regardless of the chosen initialization procedure;
-
Anyway, the quality of the results achieved within a moderate number of PSO iterations (see the previous tables) may be considered satisfactory for practical applications. This is a positive feature of our MCDA-based method, since the search of a definitely acceptable solution might require a really heavy computational burden.
7 Comparisons and some proposals
7.1 The comparison with the penalty function approach
Orthoinit
, Orthoinit+
)—Inconsistency measure (\({\mathcal {I}}_1\), \({\mathcal {I}}_2\)) – Criteria (all indicators of Table 1, Altman’s variables of Table 3)—Decision variables (\({\mathbf {w}}\), \({\mathbf {w}}\) and \({\mathbf {q}}\)).” The importance for the reader of the last note will become clearer in a while.Initialization | Class 1 (%) | Class 2 (%) | Class 3 (%) | Class 4 (%) | Class 5 (%) | |
---|---|---|---|---|---|---|
Standard | Bankrupt | 20.40 | 21.70 | 20.98 | 18.39 | 18.53 |
All | 23.59 | 20.85 | 20.71 | 20.23 | 14.62 | |
PSO-Random | Bankrupt | 4.89 | 18.25 | 24.71 | 23.56 | 28.59 |
All | 17.43 | 23.40 | 22.53 | 20.33 | 16.31 | |
PSO- Orthoinit | Bankrupt | 2.30 | 14.66 | 23.99 | 30.60 | 28.45 |
All | 14.86 | 24.30 | 23.73 | 20.63 | 16.48 | |
PSO- Orthoinit+ | Bankrupt | 18.82 | 18.97 | 19.40 | 20.26 | 22.56 |
All | 20.33 | 18.42 | 19.77 | 22.58 | 18.90 |
Initialization | Class 1 (%) | Class 2 (%) | Class 3 (%) | Class 4 (%) | Class 5 (%) | |
---|---|---|---|---|---|---|
Standard | Bankrupt | 20.40 | 21.70 | 20.98 | 18.39 | 18.53 |
All | 23.59 | 20.85 | 20.71 | 20.23 | 14.62 | |
PSO-Random | Bankrupt | 32.76 | 0.00 | 0.00 | 0.00 | 67.24 |
All | 55.22 | 0.00 | 0.00 | 0.00 | 44.78 | |
PSO- Orthoinit | Bankrupt | 31.47 | 0.00 | 0.00 | 0.00 | 68.53 |
All | 54.80 | 0.00 | 0.00 | 0.00 | 45.20 | |
PSO- Orthoinit+ | Bankrupt | 38.94 | 17.82 | 6.47 | 1.87 | 34.91 |
All | 30.99 | 16.31 | 6.13 | 2.04 | 44.53 |
Initialization | Class 1 (%) | Class 2 (%) | Class 3 (%) | Class 4 (%) | Class 5 (%) | |
---|---|---|---|---|---|---|
Standard | Bankrupt | 6.03 | 13.07 | 20.26 | 27.30 | 33.33 |
All | 20.20 | 20.04 | 19.98 | 19.90 | 19.88 | |
PSO-Random | Bankrupt | 6.03 | 13.07 | 20.26 | 27.30 | 33.33 |
All | 20.17 | 20.04 | 20.02 | 19.88 | 19.88 | |
PSO- Orthoinit | Bankrupt | 3.16 | 14.08 | 27.59 | 24.14 | 31.03 |
All | 20.28 | 23.76 | 22.39 | 16.88 | 16.69 | |
PSO- Orthoinit+ | Bankrupt | 6.03 | 13.07 | 20.26 | 27.30 | 33.33 |
All | 20.17 | 20.02 | 20.03 | 19.88 | 19.90 |
Initialization | Class 1 (%) | Class 2 (%) | Class 3 (%) | Class 4 (%) | Class 5 (%) | |
---|---|---|---|---|---|---|
Standard | Bankrupt | 6.03 | 13.07 | 20.26 | 27.30 | 33.33 |
All | 20.20 | 20.04 | 19.98 | 19.90 | 19.88 | |
PSO-Random | Bankrupt | 6.03 | 13.22 | 20.11 | 27.30 | 33.33 |
All | 20.09 | 20.06 | 20.06 | 19.91 | 19.88 | |
PSO- Orthoinit | Bankrupt | 70.26 | 0.00 | 0.00 | 0.00 | 29.74 |
All | 78.73 | 0.00 | 0.00 | 0.00 | 21.27 | |
PSO- Orthoinit+ | Bankrupt | 5.75 | 13.36 | 19.68 | 27.73 | 33.48 |
All | 20.09 | 20.02 | 19.99 | 19.95 | 19.95 |
Initialization | Class 1 (%) | Class 2 (%) | Class 3 (%) | Class 4 (%) | Class 5 (%) | |
---|---|---|---|---|---|---|
Standard | Bankrupt | 6.03 | 13.07 | 20.26 | 27.30 | 33.33 |
All | 20.20 | 20.04 | 19.98 | 19.90 | 19.88 | |
PSO-Random | Bankrupt | 0.43 | 36.35 | 29.02 | 17.53 | 16.67 |
All | 4.82 | 47.21 | 24.39 | 12.81 | 10.76 | |
PSO- Orthoinit | Bankrupt | 0.14 | 30.32 | 35.92 | 19.25 | 14.37 |
All | 0.72 | 35.52 | 37.70 | 15.77 | 10.29 | |
PSO- Orthoinit+ | Bankrupt | 2.01 | 19.83 | 48.28 | 23.28 | 6.61 |
All | 4.34 | 26.86 | 44.38 | 19.10 | 5.32 |
Initialization | Class 1 (%) | Class 2 (%) | Class 3 (%) | Class 4 (%) | Class 5 (%) | |
---|---|---|---|---|---|---|
Standard | Bankrupt | 6.03 | 13.07 | 20.26 | 27.30 | 33.33 |
All | 20.20 | 20.04 | 19.98 | 19.90 | 19.88 | |
PSO-Random | Bankrupt | 7.76 | 3.02 | 0.00 | 0.00 | 89.22 |
All | 43.90 | 3.55 | 0.00 | 0.00 | 52.55 | |
PSO- Orthoinit | Bankrupt | 9.77 | 0.29 | 0.14 | 0.43 | 89.37 |
All | 47.16 | 0.66 | 0.63 | 0.20 | 51.36 | |
PSO- Orthoinit+ | Bankrupt | 1.72 | 6.61 | 20.55 | 36.06 | 35.06 |
All | 19.33 | 20.79 | 22.91 | 20.59 | 16.37 |
-
The endogenous determination of the MURAME parameters obtained applying the penalty function approach allows improvements in terms of the percentages of bankrupt firms classified in the best creditworthiness class, with respect to the exogenous specification of the same parameters, only in 9 over 18 of the considered configurations. On the contrary, when using the nonlinear reformulation approach, this percentage reaches its maximum value, i.e. in 18 cases over 18;
-
The same endogenous determination of the MURAME parameters permits improvements in terms of the percentages of bankrupt firms classified in the worst creditworthiness class, with respect to the exogenous specification of the same parameters, in 10 over 18 configurations. Differently, when using the nonlinear reformulation approach, this performance increases, yielding a satisfactory result in 13 cases over 18;
-
The obtained creditworthiness classifications fail in satisfying the general principle stated above in 14 over 18 of the considered configurations, even if in some cases the displacement is minimal. Conversely, when using the nonlinear reformulation approach, the performance decreases up to 6 over 18.
-
Some of the parameters may get negative values (which are not negligibly small), and one weight assumes a value which is even considerably greater than 100%;
-
In correspondence with some configurations the summation of the weights may significantly differ from 100%.
7.2 A tentative use of the nonbankrupt firms’ informative content
Orthoinit+
as initialization procedures; Altman’s variables as criteria; \({\mathbf {w}}\) and \({\mathbf {q}}\) as decision variables. Lastly, having no argumentations to prefer bankrupt firms’ informative content to the nonbankrupt firms’ one and vice versa, we set \(\lambda = 0.5\).Initialization | Class 1 (%) | Class 2 (%) | Class 3 (%) | Class 4 (%) | Class5 (%) | |
---|---|---|---|---|---|---|
Standard | Bankrupt | 6.03 | 13.07 | 20.26 | 27.30 | 33.33 |
All | 20.20 | 20.04 | 19.98 | 19.90 | 19.88 | |
PSO-Random | Bankrupt | 0.72 | 21.26 | 60.78 | 15.23 | 2.01 |
All | 1.68 | 26.82 | 58.26 | 11.78 | 1.45 | |
PSO- Orthoinit+ | Bankrupt | 0.57 | 23.99 | 48.56 | 20.83 | 6.03 |
All | 2.66 | 31.37 | 45.29 | 15.76 | 4.93 |
Initialization | Class 1 (%) | Class 2 (%) | Class 3 (%) | Class 4 (%) | Class 5 (%) | |
---|---|---|---|---|---|---|
Standard | Bankrupt | 6.03 | 13.07 | 20.26 | 27.30 | 33.33 |
All | 20.20 | 20.04 | 19.98 | 19.90 | 19.88 | |
PSO-Random | Bankrupt | 19.54 | 5.03 | 9.77 | 21.26 | 44.40 |
All | 34.89 | 15.02 | 13.95 | 13.16 | 22.98 | |
PSO- Orthoinit+ | Bankrupt | 3.59 | 14.08 | 13.65 | 23.13 | 45.55 |
All | 25.21 | 21.01 | 17.30 | 15.55 | 20.93 |
-
The overall quality of the creditworthiness distribution of bankrupt firms depends on the inconsistency measure applied. Indeed, in the case of \(\widetilde{{\mathcal {I}}}_1\), the obtained classifications heavily dissatisfy the general principle concerning the bankrupt firms themselves, regardless of the initialization procedures (see Table 19). Conversely, this principle dissatisfaction is less evident when using \(\widetilde{{\mathcal {I}}}_2\) and almost disappears when applying the initialization procedure
Orthoinit+
(see Table 19). These findings confirm some of the general conclusions already presented. Indeed, in terms of creditworthiness classification: \(\widetilde{{\mathcal {I}}}_2\), which derives from \({\mathcal {I}}_2\), seems preferable to \(\widetilde{{\mathcal {I}}}_1\), which derives from \({\mathcal {I}}_1\), as \({\mathcal {I}}_2\) resulted preferable to \({\mathcal {I}}_1\);Orthoinit+
generally performs better than Random; -
The endogenous determination of the MURAME parameters based on \(\widetilde{{\mathcal {I}}}_1\) and on \(\widetilde{{\mathcal {I}}}_2\), respectively, leads to a quality of the bankrupt firms’ creditworthiness distributions which is not much worse than the one of the corresponding distributions achieved through the application of \({\mathcal {I}}_1\) and of \({\mathcal {I}}_2\), respectively;
-
The (sub-)optimal values of the weights and of the indifference thresholds (not present here) do not show noteworthy features and, in any case, not particularly different from those detectable by the corresponding results achieved when using \({\mathcal {I}}_1\) and \({\mathcal {I}}_2\), respectively.