A new VIKORbased insampleoutofsample classifier with application in bankruptcy prediction
Abstract
Nowadays, business analytics has become a common buzzword in a range of industries, as companies are increasingly aware of the importance of high quality predictions to guide their proactive planning exercises. The financial industry is amongst those industries where predictive analytics techniques are widely used to predict both continuous and discrete variables. Conceptually, the prediction of discrete variables comes down to addressing sorting problems, classification problems, or clustering problems. The focus of this paper is on classification problems as they are the most relevant in riskclass prediction in the financial industry. The contribution of this paper lies in proposing a new classifier that performs both insample and outofsample predictions, where insample predictions are devised with a new VIKORbased classifier and outofsample predictions are devised with a CBRbased classifier trained on the risk class predictions provided by the proposed VIKORbased classifier. The performance of this new nonparametric classification framework is tested on a dataset of firms in predicting bankruptcy. Our findings conclude that the proposed new classifier can deliver a very high predictive performance, which makes it a real contender in industry applications in finance and investment.
Keywords
Insample prediction Outofsample prediction VIKOR classifier CBR kNearest neighbour classifier Bankruptcy Risk class prediction1 Introduction
VIKOR is a multicriteria method originally designed for ranking a number of alternatives, say \( m \), under multiple noncommensurable (i.e., measured on different scales or in different units) and often conflicting criteria, say \( n \), where criteria are conflicting in the sense that improving a criterion is only achievable at the expense of at least another criterion; therefore, tradeoffs between conflicting criteria is the way to reach an acceptable solution. VIKOR is grounded into compromise programming, as it is designed to devise a solution that is the closest to an ideal one. In sum, VIKOR benchmarks all alternatives against an ideal solution and makes use of the relative closeness, as measured by the \( L_{p} \) distance, from the ideal solution—typically virtual and infeasible—to construct an index for each alternative or entity \( i \), say \( Q_{i} \), which is a convex combination of the standardized distance between entity \( i \) and the alternative with the best (observed) average performance and the standardized distance between entity \( i \) and the entity with the least (observed) regret. Since the publication of seminal paper by Duckstein and Opricovic (1980), several hundreds of papers were published on VIKOR and its variants. Application papers apart, papers on methodological contributions on the crisp version of VIKOR could be divided into two main categories; namely, VIKOR and variants for multicriteria decision problems where all alternatives are assessed based on a common set of criteria, and VIKOR and variants for multicriteria decision problems where different alternatives are assessed based on different sets of criteria. Examples of contributions in the first category include the original VIKOR (Duckstein and Opricovic 1980); VIKOR enhanced with Weight Stability Analysis and Tradeoffs Analysis (Opricovic and Tzeng 2007); VIKOR with Choiceless and Discontent Utilities (Huang et al. 2009); VIKOR with Logic Judgment (Chang 2010); and VIKOR for criteria with a Normal Reference Range (Zeng et al. 2013). On the other hand, examples of contributions in the second category include the modified VIKOR by Liou et al. (2011) and the modified VIKOR by Anvari et al. (2014). For reviews on VIKOR application areas, we refer the reader to Mardani et al. (2016), Gul et al. (2016), and Yazdani and Graeml (2014).
The remainder of this paper unfolds as follows. In Sect. 2, we provide a detailed description of the proposed integrated insample and outofsample framework for VIKORbased classifiers and discuss implementation decisions. In Sect. 3, we empirically test the performance of the proposed framework in bankruptcy prediction of companies listed on the London Stock Exchange (LSE) and report on our findings. Finally, Sect. 4 concludes the paper.
2 An integrated framework for designing and implementing VIKORbased classifiers

Input: A set of \( n \) entities (e.g., LSE listed firmyear observations) to be assessed on \( m \) prespecified criteria (e.g., financial criteria) along with their measures (e.g., financial ratios), where the measure of each criterion could either be minimized or maximized. Thus, each entity, say \( i \) (\( i = 1, \ldots ,n \)), is represented by an \( m \)dimensional vector of (observed) measures of the criteria under consideration, say \( x_{i} = \left( {x_{ij} } \right) \), where \( x_{ij} \) denote the observed measure of criterion \( j \) for entity \( i \) and the set of \( x_{i} \) s shall be denoted by \( X \). An observed riskclass membership, say \( Y \), is also available for all entities. The historical sample \( X \) is divided into a training sample, say \( X^{E} \), and a test sample, say \( X^{T} \).
2.1 Phase 1: VIKORbased insample classifier
2.1.1 Step 1: Compute the best and worst virtual alternatives or benchmarks
2.1.2 Step 2: Compute measures of the average performance behavior of alternatives
Also, compute \( S^{ + } = \mathop {\hbox{min} }\limits_{i} S_{i} \) and \( S^{  } = \mathop {\hbox{max} }\limits_{i} S_{i} \).
Note that \( S_{i} \) is an \( L_{p} \)metric based aggregation function for \( p = 1 \) that quantifies how close entity \( i \) is from the positive ideal alternative and could be interpreted as a utility function of entity \( i \). Note also that an aggregating function is used here instead of a utility function, because in many multicriteria decision problems it is not possible to obtain a mathematical representation of the decision maker’s utility function. Finally, notice that \( \left( {r_{j}^{ + }  x_{i,j}^{E} } \right)\bigg{/}\left( {r_{j}^{ + }  r_{j}^{  } } \right) \) is the deviation from the best virtual alternative on criterion \( j \), \( \left( {r_{j}^{ + }  x_{i,j}^{E} } \right) \), standardized by the distance between the best and the worst virtual alternatives on criterion \( j \), \( \left( {r_{j}^{ + }  r_{j}^{  } } \right) \); therefore, \( S_{i} \) is the weighted sum over all criteria of standardized deviations from the best virtual alternative, where \( w_{j} \) denote the weight assigned to criterion \( j \). In sum, \( S_{i} \) reflects the average performance behavior of entity \( i \), which allows for full compensation between criteria. Since \( S_{i} \) reflects the average performance behavior of entity \( i \), \( S^{ + } = \mathop {\hbox{min} }\limits_{i} S_{i} \) and \( S^{  } = \mathop {\hbox{max} }\limits_{i} S_{i} \) are the best and worst observed average performance behavior across all entities insample, respectively. Note that \( S^{ + } \) is often interpreted as the maximum group utility of the “majority”. Let \( i^{ + } = argmin_{i} S_{i} \) and \( i^{  } = argmax_{i} S_{i} \). Thus, \( \left( {S_{i}  S^{ + } } \right)/\left( {S^{  }  S^{ + } } \right) \) is the distance between entity \( i \) and the best (on average) observed performer; i.e., entity \( i^{ + } \), standardized by the distance between the best and worst (on average) observed performers; i.e., entities \( i^{ + } \) and \( i^{  } \), respectively.
2.1.3 Step 3: Compute measures of the worst performance behavior of alternatives
Also, compute \( R^{ + } = \mathop {\hbox{min} }\limits_{i} R_{i} \) and \( R^{  } = \mathop {\hbox{max} }\limits_{i} R_{i} \).
Note that \( R_{i} \) is also an \( L_{p} \)metric based aggregation function with \( p = \infty \) that quantifies how far, in the extreme case, entity \( i \) is from the positive ideal alternative and could be interpreted as a regret function of entity \( i \). In fact, unlike \( S_{i} \), \( R_{i} \) is the maximum over all criteria of the weighted standardized deviations from the best virtual alternative and thus reflects the worst performance behavior of entity \( i \). Note also that \( R_{i} \) does not allow for any compensation between criteria. Since \( R_{i} \) reflects the worst performance behavior of entity \( i \), \( R^{ + } = \mathop {\hbox{min} }\limits_{i} R_{i} \) and \( R^{  } = \mathop {\hbox{max} }\limits_{i} R_{i} \) represent the least and most observed individual regrets amongst all entities insample, respectively. Let \( i^{ + + } = argmin_{i} R_{i} \) and \( i^{   } = argmax_{i} R_{i} \). Thus, \( \left( {R_{i}  R^{ + } } \right)/\left( {R^{  }  R^{ + } } \right) \) is the distance between entity \( i \) and the observed entity with the least regret; i.e., entity \( i^{ + + } \), standardized by the distance between the observed entities with the least and the most regrets; i.e., entities \( i^{ + + } \) and \( i^{   } \), respectively.
2.1.4 Step 4: Compute a VIKOR score for each alternative
Sample of commonly used weighting schema in VIKOR
Type of weighting process  Description  Sample of references 

Subjective assignment of weights  Direct assignment of weights  Vinodh et al. (2014), Devi (2011), Peng et al. (2015), Anvari et al. (2014), Vučijak et al. (2015), Mela et al. (2012), Tošić et al. (2015), Vahdani et al. (2013), Bashiri et al. (2013), Chatterjee et al. (2009), Jahan et al. (2011), Bahraminasab and Jahan (2011), Yazdani and Payam (2015) and Chang and Hsu (2011) 
Analytical hierarchy process (AHP) based methods  Chatterjee et al. (2010), Zhu et al. (2015), Parameshwaran et al. (2015), Liu et al. (2015), Bairagi et al. (2014), Tzeng and Huang (2012), Mousavi et al. (2013), Büyüközkan and Görener (2015), Mohammadi et al. (2014), Ebrahimnejad et al. (2012), Hsu et al. (2012), Jahan et al. (2011), Çalışkan et al. (2013), Cavallini et al. (2013), Çalışkan (2013), Liu et al. (2014), Ray (2014), Rezaie et al. (2014), Wu et al. (2011a), Wu et al. (2009), Chen and Chen (2010), Zolfani et al. (2013), Dincer and Hacioglu (2013), Tsai and Chang (2013), Liu et al. (2012), Ren et al. (2015) and San Cristobal (2011)  
PROMETHEE II  Feng et al. (2013)  
SWARA (stepwise weight assessment ratio analysis)  Zolfani et al. (2013)  
Modified digital logic approach (MDL)  Bahraminasab and Jahan (2011)  
Objective/datadriven assignment of weights  Equal weights  Zeng et al. (2013) 
Entropy weight method  Liu et al. (2015), Chatterjee et al. (2009), Jahan et al. (2011), Chauhan and Vaish (2012), Çalışkan et al. (2013), Çalışkan (2013), Hsu (2014, 2015), Chou et al. (2014), Ranjan et al. (2015), Shemshadi et al. (2011) and Geng and Liu (2014)  
Coefficient of variation weight method  Zavadskas and Antuchevičiene (2004)  
Data envelopment analysis 
2.1.5 Step 5: Compute insample classification of alternatives
Use the performance scores, \( Q_{i} \) s, computed in the previous step to classify alternatives \( i \) in the training sample \( X^{E} \) according to a userspecified classification rule into, for example, risk (e.g., bankruptcy) classes, say \( \hat{Y}^{E} \). Then, compare the VIKOR based classification of alternatives in \( X^{E} \) into risk classes; that is, the predicted risk classes, \( \hat{Y}^{E} \), with the observed risk classes of alternatives in the training sample, \( Y^{E} \), and compute the relevant insample performance statistics. The choice of a decision rule for classification depends on the nature of the classification problem; that is, a twoclass problem or a multiclass problem. In this paper, we are concerned with a twoclass problem; therefore, we shall provide a solution that is suitable for these problems. In fact, we propose a VIKOR scorebased cutoff point procedure to classify entities in \( X_{E} \). The proposed procedure involves solving an optimization problem whereby the VIKOR scorebased cutoff point, say \( \kappa \), is determined so as to optimize a given classification performance measure, say \( \pi \) (e.g., Type I error, Type II error, Sensitivity, Specificity), over an interval with a lower bound, say \( \kappa_{LB} \), equal to the smallest VIKOR score of entities in \( X_{E} \) and an upper bound, say \( \kappa_{UB} \), equal to the largest VIKOR score of entities in \( X_{E} \). Any derivativefree unidimensional search procedure could be used to compute the optimal cutoff score, say \( \kappa^{*} \)—for details on derivativefree unidimensional search procedures, the reader is referred to Bazaraa et al. (2006). The optimal cutoff score \( \kappa^{*} \) is used to classify observations in \( X_{E} \) into two classes; namely, bankrupt and nonbankrupt firms. To be more specific, the predicted risk classes \( \hat{Y}^{E} \) are determined so that firms with VIKOR scores greater than \( \kappa^{*} \) are assigned to a bankruptcy class and those with VIKOR scores less than or equal to \( \kappa^{*} \) are assigned to a nonbankruptcy class. Note that an important feature of the design of our VIKOR scorebased cutoff point procedure for classification lies in the determination of a cutoff score to optimise a specific performance measure of the classifier.
2.2 Phase 2: CBRbased outofsample classifier
2.2.1 Step 6: Compute outofsample classification of alternatives
Use an instance of casebased reasoning (CBR); namely, the knearest neighbour (kNN) algorithm, to classify alternatives in \( X^{T} \) into risk classes (i.e., bankruptcy class, nonbankruptcy class), say \( \hat{Y}^{T} \). Then, compare the predicted risk classes \( \hat{Y}^{T} \) with the observed ones \( Y^{T} \) and compute the relevant outofsample performance statistics. A detailed description of kNN is hereafter outlined:
We would like to stress out that, when the decision maker is not confident enough to provide a value for \( \alpha \) in step 5, one could automate the choice of \( \alpha \). In fact, an optimal value of \( \alpha \) with respect to a specific performance measure (e.g., Type 1 error, Type 2 error, Sensitivity, or specificity) to be optimized either insample only or both insample and outofsample could be obtained by using a derivativefree unidimensional search procedure, which calls either a procedure that consists of steps 4 and 5 to optimize insample performance, or a procedure that consists of steps 4 to 6 to optimize both insample and outofsample performances simultaneously.
Finally, note that VIKOR outcome depends on the choice of the ideal solution, whose calculation depends on the given set of alternatives \( X^{E} \). Therefore, inclusion or exclusion of one or several alternative; e.g., \( X^{T} \), would affect the VIKOR outcome unless the ideal solution is chosen or fixed at the outset by the decision maker independently from \( X^{E} \). This is the main reason for choosing a CBR framework for the outofsample classification instead of VIKOR.
In the next section, we shall report on our empirical evaluation of the proposed VIKORCBR integrated prediction framework.
3 Empirical results
Dataset composition
Sample period 2010–2014  Bankrupt firmyear observations  NonBankrupt firmyear observations  

Industry  Nb.  %  Nb.  % 
Basic materials  100  1.51  907  13.73 
Consumer goods  29  0.44  515  7.80 
Consumer services  52  0.79  1101  16.67 
Health care  40  0.61  457  6.92 
Industrials  88  1.33  1648  24.95 
Oil and gas  62  0.94  691  10.46 
Technology  35  0.53  790  11.96 
Telecommunications  1  0.02  89  1.35 
Total  407  6.16  6198  93.84 
In our experiment, we reworked a standard and well known parametric model within the proposed VIKORCBR framework; namely, the multivariate discriminant analysis (MDA) model of Taffler (1984), to provide some empirical evidence on the merit of the proposed framework. Recall that Taffler’s model makes use of four explanatory variables or bankruptcy drivers which belong to the same category; namely, liquidity. These drivers are current liabilities to total assets, number of credit intervals, profit before tax to current liabilities, and current assets to total liabilities. Note that lower values are better than higher ones for Current Liabilities to Total Assets and Number of Credit Intervals, whereas higher values of Current Assets to Total Liabilities and Profit Before Tax to Current Liabilities are better than lower ones. We report on the performance of the proposed framework using four commonly used metrics; namely, Type I error (T1), Type II error (T2), Sensitivity (Sen) and Specificity (Spe), where T1 is the proportion of bankrupt firms predicted as nonbankrupt, T2 is the proportion of nonbankrupt firms predicted as bankrupt, Sen is the proportion of bankrupt firms predicted as bankrupt, and Spe is the proportion of nonbankrupt firms predicted as nonbankrupt.
Implementation decisions for VIKOR and kNN
Decision  Options considered and justification, if relevant 

VIKOR  
Value for \( \alpha \)  We performed tests for \( \alpha = 0, 0.25, 0.5, 0.75, 1 \) 
Weighting scheme  Equal weights \( w_{j} \) s 
Classification rule  VIKOR scorebased cutoff point procedure, where the choice of the cutoff point optimises a specific performance measure (i.e., T1, T2, Sen, Spe) 
kNN  
Metric \( d_{k  NN} \)  Euclidean, Cityblock, Mahalanobis 
Classification criterion  Majority vote. Several criteria could have been used such as a Weighted Vote, but once again our choice is made so as to avoid any personal (subjective) preferences 
Size of the neighbourhood \( k \)  \( k = \) 3; 5; 7. The results reported are for \( k = 3 \) since higher values delivered very close performances but required more computations 
Summary statistics of the performance of the proposed framework for \( \alpha = 0 \) (noncompensating scheme)
Insample performance  

Statistics  T1 (%)  T2 (%)  Sen. (%)  Spe. (%) 
Min  0  0.0484  100  99.9032 
Max  0  0.0968  100  99.9516 
Average  0  0.0750  100  99.9250 
SD  0  0.0147  0  0.0147 
Distance metric  Outofsample performance  

Statistics  T1 (%)  T2 (%)  Sen. (%)  Spe. (%)  
Euclidean  Min  0  0  100  100 
Max  0  0  100  100  
Average  0  0  100  100  
SD  0  0  0  0  
Cityblock  Min  0  0  100  100 
Max  0  0  100  100  
Average  0  0  100  100  
SD  0  0  0  0  
Mahalanobis  Min  0  0  100  100 
Max  0  0  100  100  
Average  0  0  100  100  
SD  0  0  0  0 
Summary statistics of the performance of the proposed framework for \( \alpha = 0.25 \) (mixed scheme)
Insample performance  

Statistics  T1 (%)  T2 (%)  Sen. (%)  Spe. (%) 
Min  0  0.0242  100  99.8549 
Max  0  0.1451  100  99.9758 
Average  0  0.0395  100  99.9605 
SD  0  0.0356  0  0.0356 
Distance metric  Outofsample performance  

Statistics  T1 (%)  T2 (%)  Sen. (%)  Spe. (%)  
Euclidean  Min  0  0  100  100 
Max  0  0  100  100  
Average  0  0  100  100  
SD  0  0  0  0  
Cityblock  Min  0  0  100  100 
Max  0  0  100  100  
Average  0  0  100  100  
SD  0  0  0  0  
Mahalanobis  Min  0  0  100  99.8547 
Max  0  0.1453  100  100  
Average  0  0.0194  100  99.9806  
SD  0  0.0469  0  0.0469 
Summary statistics of the performance of the proposed framework for \( \alpha = 0.5 \) (mixed scheme)
Insample performance  

Statistics  T1 (%)  T2 (%)  Sen. (%)  Spe. (%) 
Min  0  0.0242  100  99.9516 
Max  0  0.0484  100  99.9758 
Average  0  0.0274  100  99.9726 
SD  0  0.0084  0  0.0084 
Distance metric  Outofsample performance  

Statistics  T1 (%)  T2 (%)  Sen. (%)  Spe. (%)  
Euclidean  Min  0  0  100  100 
Max  0  0  100  100  
Average  0  0  100  100  
SD  0  0  0  0  
Cityblock  Min  0  0  100  100 
Max  0  0  100  100  
Average  0  0  100  100  
SD  0  0  0  0  
Mahalanobis  Min  0  0  100  100 
Max  0  0  100  100  
Average  0  0  100  100  
SD  0  0  0  0 
Summary statistics of the performance of the proposed framework for \( \alpha = 0.75 \)(mixed scheme)
Insample performance  

Statistics  T1 (%)  T2 (%)  Sen. (%)  Spe. (%) 
Min  0  0.0242  100  99.9516 
Max  0  0.0484  100  99.9758 
Average  0  0.0274  100  99.9726 
SD  0  0.0084  0  0.0084 
Distance metric  Outofsample performance  

Statistics  T1 (%)  T2 (%)  Sen. (%)  Spe. (%)  
Euclidean  Min  0  0  100  100 
Max  0  0  100  100  
Average  0  0  100  100  
SD  0  0  0  0  
Cityblock  Min  0  0  100  100 
Max  0  0  100  100  
Average  0  0  100  100  
SD  0  0  0  0  
Mahalanobis  Min  0  0  100  100 
Max  0  0  100  100  
Average  0  0  100  100  
SD  0  0  0  0 
Summary Statistics of the Performance of the Proposed Framework for \( \alpha = 1 \)(compensating scheme)
Insample performance  

Statistics  T1 (%)  T2 (%)  Sen. (%)  Spe. (%) 
Min  0  0.0242  100  99.9758 
Max  0  0.0242  100  99.9758 
Average  0  0.0242  100  99.9758 
SD  0  0  0  0 
Distance metric  Outofsample performance  

Statistics  T1 (%)  T2 (%)  Sen. (%)  Spe. (%)  
Euclidean  Min  0  0  100  100 
Max  0  0  100  100  
Average  0  0  100  100  
SD  0  0  0  0  
Cityblock  Min  0  0  100  100 
Max  0  0  100  100  
Average  0  0  100  100  
SD  0  0  0  0  
Mahalanobis  Min  0  0  100  100 
Max  0  0  100  100  
Average  0  0  100  100  
SD  0  0  0  0 
Summary statistics of the performance of MDA
Statistics  T1 (%)  T2 (%)  Sen. (%)  Spe. (%) 

Insample performance  
Min  97.0500  0.1900  0  99.3700 
Max  100  0.6300  2.9500  99.8100 
Average  98.8200  0.2600  1.1800  99.7400 
SD  0.6700  0.0900  0.6700  0.0900 
Outofsample performance  
Min  0  0  0  0.1500 
Max  100  99.8500  100  100 
Average  82.2000  17.0100  17.8000  82.9900 
SD  37.4300  37.6600  37.4300  37.6600 
On the other hand, the performance of the classifier outofsample is also outstanding—see Tables 4, 5, 6, 7 and 8. In fact, for all values of \( \alpha \) or compensation schema, all bankrupt and nonbankrupt firms are correctly classified. Note however that the performance of the outofsample classifier CBR trained on the insample classification provided by VIKOR seems to be marginally affected by the choice of the distance metric; to be more specific, the Mahalanobis distance seems to have slightly affected the performance – see Table 5, where the average type II error increased from 0 to 0.02% and the average specificity decreased from 100 to 99.98%. These differences in performance are however marginal to recommend that the Mahalanobis distance be avoided in implementing CBR. In sum, the performance of CBR trained on VIKOR classifier’s output is robust to the choice of the distance metric.
To conclude, our results suggest that the predictive performance of the proposed classification framework is by far superior to the predictive performance of multivariate discriminant analysis—see Table 8.
4 Conclusions
The analytics toolbox of risk management is crucial for the financial industry amongst others. In this paper, we extended such toolbox with a new nonparametric classifier for predicting risk class belonging. The proposed new integrated classifier has several appealing characteristics. First, it performs both insample and outofsample predictions, where insample predictions are devised with a first VIKORbased classifier and outofsample predictions are devised with a CBRbased classifier. Both the newly proposed VIKORbased classifier and CBRbased classifier are nonparametric and thus do not have the limitations of the usual statistical assumptions underlying the parametric classifiers. Second, the proposed VIKORbased classifier delivers an outstanding empirical performance suggesting that VIKOR scores are highly informative, on one hand, and the computation of the thresholds for classification using a nonlinear programming algorithm are optimised, on the other hand. Third, the empirical performance of the CBRbased classifier is enhanced by training it on the highquality risk class predictions provided by the VIKORbased classifier. Fourth, the proposed VIKOR classifier is based on a benchmarking framework, which contributes to its design’s strength. In fact, VIKOR benchmarks alternatives against the positive ideal solution by measuring the average and the maximum deviations from it, respectively, standardized by the distance between the positive and negative ideals. These deviations or distances from the positive ideal are then used to compute the distance between the performance behavior of each alternative and the behavior of the best observed performer, standardized by the distance between the behaviors of the best and worst observed performers, and the distance between the regret behavior of each alternative and the behavior of the observed entity with the least regret, standardized by the distance between the observed entities with the least and the most regrets. A convex combination of these behavioral measures is then used as the VIKOR score. Last, but not least, the basic concepts behind both VIKOR and CBR are easy to explain to managers.
We assessed the performance of the proposed VIKORCBR framework using a UK dataset of bankrupt and nonbankrupt firms. Our results support its outstanding predictive performance. In addition, the outcome of the proposed framework is robust to a variety of implementation decisions; namely, the choice of the value of \( \alpha \), the choice of the weighting Scheme, and the choice of the classification rule for VIKOR, and the choice of the distance metric \( d_{k  NN} \), the choice of the classification criterion, and the choice of the size \( k \) of the neighbourhood for kNN instance of CBR. Last, but not least, the proposed classification framework delivers a high performance similar to the DEAbased classifier proposed by Ouenniche and Tone (2017) and the MCDM classifiers proposed by Ouenniche et al. (2018a, b, c).
In sum, this research relates to both the field of MCDM and the field of AI. In fact, this paper proposes a hybrid design that integrates MCDM and artificial intelligence (AI) techniques, where a VIKORbased classifier is proposed for the first time and the output of VIKOR is used to train a CBR outofsample classifier. Empirical evidence supports our claim that the hybridisation of MCDM and AI fields is promising.
Notes
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