Integration of Z-numbers and Bayesian decision theory: A hybrid approach to decision making under uncertainty and imprecision

For over four decades decision support systems have been emerging to facilitate decision making under uncertain conditions and in a rapidly changing unstructured environment. Most decision support approaches, such as Bayesian decision theory and computing with words (CW), compare and analyze the consequences of different decision alternatives. Bayesian decision methods use probability theory to handle uncertainty and have been widely used in different areas for estimation and prediction as well as offering decision support. On the other hand, computing with words and approximate reasoning apply fuzzy set theory to deal with imprecise measurements and inexact information and are most concerned with propositions stated in natural language. The concept of a Z-number [1] has been recently introduced to represent propositions and their reliability in natural language. This work proposes a methodology that integrates Z-numbers and Bayesian decision theory to provide decision support when precise measurements and exact values of parameters and probabilities are not available. The relationships and computing methods required for such integration are derived and mathematically proved. The proposed hybrid methodology benefits from both approaches and provides a decision support system based on imprecise and uncertain information drawn from natural language. In order to demonstrate the proof of concept, the proposed methodology has been applied to a realistic case study on breast cancer diagnosis.