J Intell Manuf (2012) 23:2171–2178 DOI 10.1007/s10845-011-0563-z Decision-making for the best selection of suppliers by using minor ANP Toshimasa Ozaki · Mei-Chen Lo · Eizo Kinoshita · Gwo-Hshiung Tzeng Received: 31 July 2010 / Accepted: 20 June 2011 / Published online: 7 August 2011 © Springer Science+Business Media, LLC 2011 Abstract There are many studies on the application exam- ples and the definitions of the super-matrix in the Analytic Network Process (ANP). The ANP applies network struc- tures with the alternatives and the criteria to clarify a com- plex decision-making. It is important to obtain the consent from both the criteria and the alternatives because the eval- uation is relative and reciprocal. In addition, there are very few studies on the interaction between the criteria and the alternatives. There is no realistic interpretation of the max- imum eigenvalue in the ANP and the negative elements of the criteria matrix. Based on our study, the evaluation with only the alternatives consists of the missing values or the non-square matrix, it is not easy to make. The ANP that con- tains these irregular alternatives is defined as “Minor ANP”. Then, we describe these methods of making the priority of T. Ozaki (B ) Faculty of Commerce, Nagoyagakuin University, 1-25 Atsuta-Nishimachi, Atsuta, Nagoya, Aichi, Japan e-mail: ozaki@ngu.ac.jp M.-C. Lo Department of Business Management, National United University, 1, Lienda, Miaoli 36003, Taiwan e-mail: meichen_lo@yahoo.com E. Kinoshita Faculty of Urban Science, Meijo University, Kani, Gifu, Japan e-mail: kinoshit@urban.meijo-u.ac.jp G.-H. Tzeng Institute of Project Management, Kainan University, Taoyuan, Taiwan G.-H. Tzeng Institute of Management of Technology, Chiao Tung University, Hsinchu, Taiwan e-mail: ghtzeng@cc.nctu.edu.tw only the alternative’s matrix values by using the ANP. This study has proven that the missing values in the alternative matrix can be replaced by zeros such that the criteria matrix can be derived without using the inverse alternative matrix. Then, the priority of alternative selection can be obtained through the calculation of eigenvector. An empirical case of the best selection of supplier illustrates our proposed meth- ods for applying to the manufacturing industry. The result reveals the methods can be well adopted in the real world. Keywords Decision analysis · Analytic network process (ANP) · Multiple criteria decision making (MCDM) · Non-square matrix Introduction There are many studies on the application examples and the definitions of the super-matrix in the Analytic Network Pro- cess (ANP) (Saaty 1996). The ANP completely relies on human side to assist in solving social decision problems. The ANP applies network structures with the alternatives and the criteria to clarify a complex decision-making. It is important to obtain the consent from both the criteria and the alter- natives because the evaluation is relative and reciprocal. In addition, there are very few studies on the interaction between the criteria and the alternatives. Either, there is no realistic interpretation of the maximum eigenvalue in the ANP and the negative elements of the criteria matrix. Ozaki et al. (2009) proved the missing value in the alternative matrix which can be replaced as “zero” to be calculated without considering of criteria direct effect, instead of deriving the criteria matrix from inversed alternative matrix. Then, the priority of alter- native selection can be obtained through the calculation of eigenvector. 123