Incorporating a non-additive decision making method into multi-layer neural networks and its application to financial distress analysis Yi-Chung Hu * Department of Business Administration, Chung Yuan Christian University, 200, Chung Pei Road, Chung-Li 32023, Taiwan Received 10 October 2006; accepted 20 February 2008 Available online 4 March 2008 Abstract This paper presents a novel multi-layer perceptron using a non-additive decision making method and applies this model to the finan- cial distress analysis, which is an important classification problem for a business, and the multi-layer perceptron has played a significant role in financial distress analysis. Traditionally, an activation function of an output neuron performs an additive method, namely the weighted sum method. Since the assumption of additivity among individual variables may not be reasonable, this paper uses a non-addi- tive method, Choquet fuzzy integral, the fuzzy integral, to replace the weighted sum. In order to determine appropriate parameter spec- ifications in the proposed model, a genetic algorithm is designed by considering the maximization of the number of correctly classified training patterns and the minimization of the training errors. The sample data obtained from Moody’s Industrial Manuals are employed to examine the classification ability of the proposed model. The results demonstrate that the proposed model performs well in compar- ison with the traditional multi-layer perceptron and some multivariate techniques. Ó 2008 Elsevier B.V. All rights reserved. Keywords: Neural networks; Fuzzy integral; Non-additive measure; Genetic algorithm; Multi-layer perceptron 1. Introduction It is known that financial distress analysis or bankruptcy prediction with two classes (i.e., bankruptcy and non-bank- ruptcy) has long been one of the major business classifica- tion problems. The empirical literature of financial distress analysis has recently gained more attention from financial institutions. Academics and practitioners have realized that the problem of asymmetric information between banks and firms lies at the heart of important market failures such as credit rationing, and that improvement in monitoring tech- niques represents a valuable alternative to any incomplete contractual arrangement aimed at reducing the borrowers’ moral hazard [1,2]. Since a problem of financial distress analysis may be nonlinearly separable, a well-known tool of approximating functions like regression [3], namely the multi-layer percep- tron (MLP) with single output node, has been the fre- quently used neural network for financial distress analysis [4,5]. The sigmoid function, whose output ranges from zero to one, is commonly used as each neuron’s activation func- tion. Furthermore, the back-propagation algorithm (BP) using gradient descent [3,6] can be utilized to train the MLP by measuring the differences between the actual and desired outputs of individual training patterns. It is known that the main problem of using gradient descent is that the training is likely to get stuck in a local minimum [7]. In the sigmoid function, a weighted sum method (WSM) is performed. WSM is a widely used method of multi-crite- ria decision making (MCDM), whereas the additivity prop- erty [8] of the interaction among variables is assumed in WSM. Nevertheless, because variables are not always inde- 0950-7051/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.knosys.2008.02.002 * Tel.: +886 3 2655130. E-mail address: ychu@cycu.edu.tw www.elsevier.com/locate/knosys Available online at www.sciencedirect.com Knowledge-Based Systems 21 (2008) 383–390