Maximal confidence intervals of the interval-valued belief structure and applications

An unsolved problem in the interval-valued belief structure is the existence of maximal confidence intervals, referring to the credible and maximal interval-valued beliefs assigned to focal elements, in which every point can be reached by a combination rule. In this study, the existence of maximal confidence intervals of the interval-valued belief structure was investigated and validated. In addition, some applications of the maximal confidence interval in constructing valid and normalized interval-valued belief structures are discussed. A numerical experiment was conducted to illustrate the existence of maximal confidence intervals. Several propositions and a theorem were further proposed to demonstrate the validity of the maximal confidence interval. By using the concept of the maximal confidence interval, a procedure was proposed to construct valid and normalized interval-valued belief structures, and some meaningful guidance for assigning interval-valued beliefs was also suggested to the experts. Finally, a series of well-designed examples was presented as applications of the maximal confidence interval, the proposed procedure and the suggestions. They indicated that valid and normalized interval-valued belief structures can be achieved accurately and efficiently in practical applications with the proposed procedure and guidance on the basis of the concept of the maximal confidence interval.