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To link to full-text access for this article, visit this link: http://dx.doi.org/10.1016/j.ejor.2008.06.035 Byline: Ying-Ming Wang Keywords: Dempster-Shafer theory of evidence; Combination and normalization of evidence; The evidential reasoning approach; Multiple attribute decision analysis; Interval belief degree Abstract: In a very recent note by Gao and Ni [B. Gao, M.F. Ni, A note on article 'The evidential reasoning approach for multiple attribute decision analysis using interval belief degrees', European Journal of Operational Research, in press, doi:10.1016/j.ejor.2007.10.0381], they argued that Yen's combination rule [J. Yen, Generalizing the Dempster-Shafer theory to fuzzy sets, IEEE Transactions on Systems, Man and Cybernetics 20 (1990) 559-570], which normalizes the combination of multiple pieces of evidence at the end of the combination process, was incorrect. If this were the case, the nonlinear programming models we proposed in [Y.M. Wang, J.B. Yang, D.L. Xu, K.S. Chin, The evidential reasoning approach for multiple attribute decision analysis using interval belief degrees, European Journal of Operational Research 175 (2006) 35-66] would also be incorrect. In this reply to Gao and Ni, we re-examine their numerical illustrations and reconsider their analysis of Yen's combination rule. We conclude that Yen's combination rule is correct and our nonlinear programming models are valid. Author Affiliation: School of Economics & Management, Tongji University, Shanghai 200092, PR China Article History: Received 10 November 2007; Accepted 30 June 2008

To link to full-text access for this article, visit this link: http://dx.doi.org/10.1016/j.ejor.2006.07.020 Byline: Ying-Ming Wang (a)(b), Zhi-Ping Fan (c), Zhongsheng Hua (d) Keywords: Fuzzy preference relation; Multiplicative preference relation; Group decision making; Chi-square method; Priority ranking Abstract: Decision makers (DMs)' preferences on decision alternatives are often characterized by multiplicative or fuzzy preference relations. This paper proposes a chi-square method (CSM) for obtaining a priority vector from multiplicative and fuzzy preference relations. The proposed CSM can be used to obtain a priority vector from either a multiplicative preference relation (i.e. a pairwise comparison matrix) or a fuzzy preference relation or a group of multiplicative preference relations or a group of fuzzy preference relations or their mixtures. Theorems and algorithm about the CSM are developed. Three numerical examples are examined to illustrate the applications of the CSM and its advantages. Author Affiliation: (a) School of Public Administration, Fuzhou University, Fuzhou 350002, PR China (b) Manchester Business School, The University of Manchester, Manchester M15 6PB, UK (c) School of Business Administration, Northeastern University, Shenyang 110004, PR China (d) School of Management, University of Science and Technology of China, Hefei 230026, PR China Article History: Received 3 October 2005; Accepted 18 July 2006 Article Note: (footnote) [star] This research was supported by the National Natural Science Foundation of China (NSFC) under the Grant Nos. 70171035 and 70371050, and also supported by National Science Fund for Distinguished Young Scholars of China under the Grant No. 70525002.

To link to full-text access for this article, visit this link: http://dx.doi.org/10.1016/j.ejor.2005.10.066 Byline: Ying-Ming Wang (a)(b), Taha M.S. Elhag (a) Keywords: Analytic hierarchy process; Multiple criteria decision making; Interval comparison matrix; Interval regression analysis; Normalization Abstract: Crisp comparison matrices lead to crisp weight vectors being generated. Accordingly, an interval comparison matrix should give an interval weight estimate. In this paper, a goal programming (GP) method is proposed to obtain interval weights from an interval comparison matrix, which can be either consistent or inconsistent. The interval weights are assumed to be normalized and can be derived from a GP model at a time. The proposed GP method is also applicable to crisp comparison matrices. Comparisons with an interval regression analysis method are also made. Three numerical examples including a multiple criteria decision-making (MCDM) problem with a hierarchical structure are examined to show the potential applications of the proposed GP method. Author Affiliation: (a) Project Management Division, School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, P.O. Box 88, Manchester M60 1QD, UK (b) School of Public Administration, Fuzhou University, Fuzhou 350002, PR China Article History: Received 2 March 2005; Accepted 10 October 2005 Article Note: (footnote) [star] This research was supported by the National Natural Science Foundation of China (NSFC) under the Grant No. 70171035 and also in part by the UK Engineering and Physical Sciences Research Council (EPSRC) under the Grant No. GR/S66770/01.

To link to full-text access for this article, visit this link: http://dx.doi.org/10.1016/j.ejor.2005.03.034 Byline: Ying-Ming Wang (a)(b)(c), Jian-Bo Yang (a), Dong-Ling Xu (a), Kwai-Sang Chin (d) Keywords: Multiple attribute decision analysis; The evidential reasoning approach; Uncertainty modeling; Interval degrees of belief; Interval data; Nonlinear optimization Abstract: Multiple attribute decision analysis (MADA) problems having both quantitative and qualitative attributes under uncertainty can be modeled using the evidential reasoning (ER) approach. Several types of uncertainties such as ignorance and fuzziness can be modeled in the ER framework. In this paper, the ER approach will be extended to model new types of uncertainties including interval belief degrees and interval data that could be incurred in decision situations such as group decision making. The Dempster-Shafer (D-S) theory of evidence is first extended, which is one of the bases of the ER approach. The analytical ER algorithm is used to combine all evidence simultaneously. Two pairs of nonlinear optimization models are constructed to estimate the upper and lower bounds of the combined belief degrees and to compute the maximum and the minimum expected utilities of each alternative, respectively. Interval data are equivalently transformed to interval belief degrees and are incorporated into the nonlinear optimization models. A cargo ship selection problem is examined to show the implementation process of the proposed approach. Author Affiliation: (a) Manchester Business School, The University of Manchester, P.O. Box 88, Manchester M60 1QD, UK (b) School of Public Administration, Fuzhou University, Fuzhou, Fujian 350002, PR China (c) Center for Accounting Studies of Xiamen University, Xiamen, Fujian 361005, PR China (d) Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Tat Chee, Kowloon, Hong Kong Article History: Received 8 September 2004; Accepted 8 March 2005

To link to full-text access for this article, visit this link: http://dx.doi.org/10.1016/j.ejor.2004.09.059 Byline: Ying-Ming Wang (a)(b), Jian-Bo Yang (a), Dong-Ling Xu (a) Keywords: Environmental impact assessment; Multiple criteria decision analysis; Uncertainty modelling; The evidential reasoning approach; Utility Abstract: Environmental impact assessment (EIA) problems are often characterised by a large number of identified environmental factors that are qualitative in nature and can only be assessed on the basis of human judgments, which inevitably involve various types of uncertainties such as ignorance and fuzziness. So, EIA problems need to be modelled and analysed using methods that can handle uncertainties. The evidential reasoning (ER) approach provides such a modelling framework and analysis method. In this paper the ER approach will be applied to conduct EIA analysis for the first time. The environmental impact consequences are characterized by a set of assessment grades that are assumed to be collectively exhaustive and mutually exclusive. All assessment information, quantitative or qualitative, complete or incomplete, and precise or imprecise, is modelled using a unified framework of a belief structure. The original ER approach with a recursive ER algorithm will be introduced and a new analytical ER algorithm will be investigated which provides a means for using the ER approach in decision situations where an explicit ER aggregation function is needed such as in optimisation problems. The ER approach will be used to aggregate multiple environmental factors, resulting in an aggregated distributed assessment for each alternative policy. A numerical example and its modified version are studied to illustrate the detailed implementation process of the ER approach and demonstrate its potential applications in EIA. Author Affiliation: (a) Manchester Business School, The University of Manchester, P.O. Box 88, Manchester, M60 1QD, UK (b) School of Public Administration, Fuzhou University, Fuzhou, Fujian, 350002, PR China Article History: Received 25 November 2003; Accepted 28 September 2004 Article Note: (footnote) [star] This research was supported by the UK Engineering and Physical Science Research Council under the Grant Nos: GR/N65615/01 and GR S85498 01, the European Commission under the Grant No: IPS-2000-00030, and also in part by the National Natural Science Foundation of China (NSFC) under the Grant No: 70171035 and Fok Ying Tung Education Foundation under the Grant No: 71080.

To link to full-text access for this article, visit this link: http://dx.doi.org/10.1016/j.ejor.2005.02.064 Byline: Dong-Ling Xu, Jian-Bo Yang, Ying-Ming Wang Keywords: Multiple attribute decision analysis; Evidential reasoning; Interval uncertainty modelling; Intelligent decision system; Decision support systems Abstract: The evidential reasoning (ER) approach is a method for multiple attribute decision analysis (MADA) under uncertainties. It improves the insightfulness and rationality of a decision making process by using a belief decision matrix (BDM) for problem modelling and the Dempster-Shafer (D-S) theory of evidence for attribute aggregation. The D-S theory provides scope and flexibility to deal with interval uncertainties or local ignorance in decision analysis, which is not explored in the original ER approach and will be investigated in this paper. Firstly, interval uncertainty will be defined and modelled in the ER framework. Then, an extended ER algorithm, IER, is derived, which enables the ER approach to deal with interval uncertainty in assessing alternatives on an attribute. It is proved that the original ER algorithm is a special case of the IER algorithm. The latter is demonstrated using numerical examples. Author Affiliation: Manchester Business School, The University of Manchester, PO Box 88, Manchester M60 1QD, UK Article History: Received 18 December 2003; Accepted 22 February 2005