The arithmetics of two dimensional belief functions

Real-world information is imperfect and usually characterized by uncertainty and partial reliability. Given these limitations, Z-numbers is introduced as a more suitable concept for describing real-world information. In recent years, Z-numbers has received immense attention. However, the general approach for computations over Z-number is too complex. Thus a simpler mathematical model has been proposed—Two Dimensional Belief Function (TDBF). A TDBF is an ordered pair of belief functions; the first belief function is on the frame of discernment of the values that a variable can take, and the second belief function is a measure of reliability of the first belief function. However, the processing of TDBF-based information requires a new theory to be developed, together with new approaches and procedures for computation with TDBF. In this paper, we propose a general framework for computations over TDBF, comprising addition, subtraction, multiplication, division, square and square root of TDBFs. In particular, a general approach to implementing combination over two TDBFs is also given. Our proposed method is validated by a variety of numerical examples, with an application in decision making problem.