Integrated artificial intelligence-based resizing strategy and multiple criteria decision making technique to form a management decision in an imbalanced environment

Abstract

Classification in an imbalanced dataset is a current challenge in machine learning communities, as the class-imbalanced problem deteriorates the performance of numerous classifiers. This study introduces a two-stage intelligent data preprocessing approach to tackle the class-imbalanced problem. By modifying the penalty parameter of the support vector machine (SVM), the discriminating boundary will move toward the majority class and in turn misclassify the majority class examples as minority class examples. That is, more misclassifications for the majority class examples are equivalent to a greater number of minority class examples. Executing the SVM as a preprocessor can be used to overcome the class imbalanced problem. Sequentially, the modified dataset undergoes the random forest to defy the curse of dimensionality. Finally, the preprocessed data are fed into a rule-based classifier to generate comprehensive decision rules. According to the empirical results, the presented architecture is a promising alternative for the class-imbalanced problem.

Appendix A VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR)

VIKOR was proposed by Opricovic [37] and Opricovic and Tzeng [38] for multi-criteria optimization of complicated problems. Opricovic [37] indicated that the VIKOR ranked alternatives in the occurrence of conflicting criteria by generating the multi-criteria ranking index, which was ground on the specific evaluation of closeness to the ideal alternative. The VIKOR was expressed as follow [39].

Step 1 Calculate the best g ^* _i and the worst g ^- _i values of whole criterion functions, i = 1, … , n.

$$\begin{aligned} g_{i}^{*} = \left\{ \begin{aligned} &\mathop {Max}\limits_{j} \begin{array}{*{20}c} {} \\ \end{array} g_{ij} \begin{array}{*{20}c} {} \\ \end{array} for\begin{array}{*{20}c} {} \\ \end{array} benefit\begin{array}{*{20}c} {} \\ \end{array} criteria \hfill \\ &\mathop {Min}\limits_{j} \begin{array}{*{20}c} {} \\ \end{array} g_{ij} \begin{array}{*{20}c} {} \\ \end{array} for\begin{array}{*{20}c} {} \\ \end{array} \cos t\begin{array}{*{20}c} {} \\ \end{array} criteria \hfill \\ \end{aligned} \right\},\quad \begin{array}{*{20}c} {j = 1, \ldots ,J} \\ \end{array} \hfill \\ g_{i}^{ - } = \left\{ \begin{aligned} &\mathop {Max}\limits_{j} \begin{array}{*{20}c} {} \\ \end{array} g_{ij} \begin{array}{*{20}c} {} \\ \end{array} for\begin{array}{*{20}c} {} \\ \end{array} benefit\begin{array}{*{20}c} {} \\ \end{array} criteria \hfill \\ &\mathop {Min}\limits_{j} \begin{array}{*{20}c} {} \\ \end{array} g_{ij} \begin{array}{*{20}c} {} \\ \end{array} for\begin{array}{*{20}c} {} \\ \end{array} \cos t\begin{array}{*{20}c} {} \\ \end{array} criteria \hfill \\ \end{aligned} \right\},\quad \begin{array}{*{20}c} {j = 1, \ldots ,J} \\ \end{array} \hfill \\ \end{aligned}$$

(A1)

where the number of alternatives denotes as J, the number of criteria is expressed as n and the rating of i-th criterion function for alternative b _j .

Step 2 Calculate the values of X _j and Y _j , j = 1, …, J.

$$\begin{aligned} X_{j} = \sum\limits_{i = 1}^{n} {\left[ {w_{i} (g_{i}^{*} - g_{ij} )/(g_{i}^{*} - g^{ - } )} \right]} \hfill \\ Y_{j} = \mathop {Max}\limits_{i} \begin{array}{*{20}c} {\left[ {w_{i} (f_{i}^{*} - f_{ij} )/(f_{i}^{*} - f_{i}^{ - } )} \right]} \\ \end{array} \hfill \\ \end{aligned}$$

(A2)

where the weight of i-th criteria is expressed as w _i , the ranking evaluation are measured by X _j and Y _j .

Step 3 Calculate the value Z _j , j = 1, … , J.

$$\begin{aligned} Z_{j} &= \left[ {v(X_{j} - X^{*} )/(X^{ - } - X^{*} )} \right] + \left[ {(1 - v)(Y_{j} - Y^{*} )/(Y^{ - } - Y^{*} )} \right] \hfill \\ X^{*} &= \mathop {Min}\limits_{j} X_{j} ,\quad X^{ - } = \mathop {Max}\limits_{j} X_{j} , \hfill \\ Y^{*} &= \mathop {Min}\limits_{j} Y_{j} ,\quad Y^{ - } = \mathop {Max}\limits_{j} Y_{j} , \hfill \\ \end{aligned}$$

(A3)

where X ^*is the solution with the maximum group utility, Y ^*is the solution with a minimum single regret of the opponent, and the weight of the strategy of the majority of criteria is represented inv. This compromise solution is stable within a decision making process, which could be “voting by majority rule” (when v > 0.5 is need), or “by consensus” v ≈ 0.5 or “with veto” v < 0.5 [39]. Followed by the prior researches [39, 43], the value of v is set to 0.5.

Step 4 Ranking the alternatives in decreasing order. There are three ranking lists X, Y and Z.

Step 5 Generate the alternative b′, which was measured by Z and ranked the best, as a compromise solution if the following two conditions are satisfied [43]

• (a) Z(b″)–Z(b′) ≥ 1 – (J − 1)

• (b) Alternative b′ is ranked the best by X and/or Y.

If only the condition (b) is violated, the alternatives b′ and b″ are taken as compromise solutions, where b″ was measured by Z was ranked the second. If the condition (a) is violated, alternatives b′, … , b ^M were viewed as compromise solution, where b ^M was evaluated by Z was ranked the M-th and was according to the relation Z(b ^M) - Z(b′) < 1(J − 1)for maximum M.