Optimal b k -stable interval in VPRS-based group decision-making: A further application Gang Xie a,⇑ , Shouyang Wang a , K.K. Lai b a Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China b Department of Management Sciences, City University of Hong Kong, Hong Kong article info Keywords: b k -Stable interval Group consensus Variable precision rough set Petroleum investment Risk management abstract This study extends the extant research on use of variable precision rough set (VPRS) for group decision- making (GDM), where the optimal b k -stable interval is derived for the best group consensus. Firstly, we introduce the basic concepts a VPRS model encompasses, and the approach of VPRS-based GDM. Next, using a mathematical programming approach, we derive the optimal b k -stable interval for DM k . Then, an application in petroleum project investment risk management, including risk-based project selection and risk ranking, is investigated. The results suggest that b k -stable intervals have significant impacts on risk management. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction Reaching consensus is necessary in all group decision-making (GDM) processes because achieving general consensus on selected options is a desirable goal (Desanctis & Gallupe, 1987; Herrera, Herrera-Viedma, & Verdegay, 1997; Khorshid, 2010). Many differ- ent approaches to measure the degree of consensus have been pre- sented, including linguistic consistency measures (Herrera et al., 1997), dynamic consensus model (Eklund, Rusinowski, & De Swart, 2007), and absolute order-of-magnitude qualitative model (Ro- selló, Prats, Agell, & Sánchez, 2010). Generally, it is preferable that the set of experts reach a high degree of consensus before applying the selection process in GDM (Cabrerizo, Moreno, Pérez, & Herrera- Viedma, 2010). Therefore, how to achieve the best consensus in GDM is an interesting topic. Normally, there is a moderator (the director of a GDM) who guides the consensus process in GDM. To achieve the maximum possible agreement, Herrera-Viedma, Alonso, Chiclana, and Herrera (2007) proposed an advanced consensus approach allow- ing generation of recommendations to help DMs change their opin- ions. Based on multiplicative preference relations and fuzzy preference relations, Fan, Ma, Jiang, Sun, and Ma (2006) presented a new approach to achieve the best group consensus in GDM, where goal programming was used to achieve a collective opinion as close to each individual decision-maker’s opinion as possible. In the Rough Set Theory (RST), attributes are reduced on the ba- sis of quality of classification (QoC). In this case, QoC does not need any priori information and can acquire significance of attributes from decision tables automatically (Pawlak, 1982). On the other hand, RST can not treat inconsistent information well, which is al- most inevitable in a complex decision-making process (Pawlak, 1991). In order to solve this problem, Ziarko (1993) presented a variable precision rough set (VPRS) model, which is an extension of RST. If a threshold value b (0.5 < b 6 1) is set, then the misclassi- fication rate tolerated in decision tables will be 1  b. Due to this advantage, VPRS model has been introduced into GDM. In extant research on the VPRS model for GDM, experts are invited to evaluate the risk exposure (RE) of risk indices (condition attributes) and the projects (decision attributes). Then decision tables consisting of RE are established (Xie, Zhang, & Lai, 2005, 2006b). Assuming the DMs have the same, as well as different weights, Xie, Zhang, and Lai (2006a, 2006c) used VPRS to process data in decision tables and obtain the significance of each risk in- dex. In GDM, integrated risk exposure (IRE) of projects and risk indices is calculated on the basis of integration of RE, significance of risk indices and weights of DMs. Furthermore, Xie, Zhang, Lai, and Yu (2008) presented the concept of the b k -stable interval, which is the range of b k within which classifications of DM k ’s deci- sion results do not vary. In addition, they examined a new applica- tion in credit risk management. However, they did not analyze the problem with optimal b k -Stable intervals for the best group consensus. In this paper, for the best group consensus, we explore optimal b k -stable interval for DM k . Firstly, we introduce VPRS-based GDM which is used to evaluate condition and decision attributes. Group decision tables are established, and weight vector of condition attribute sets is derived in each b k -stable interval. Next, a mathe- matical programming approach is used to derive the optimal b k -Stable intervals, where the objective is to minimize the degrees 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.04.177 ⇑ Corresponding author. Tel.: +86 10 62545830; fax: +86 10 62541823. E-mail address: gxie@amss.ac.cn (G. Xie). Expert Systems with Applications 38 (2011) 13757–13763 Contents lists available at ScienceDirect Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa