Dempster–Shafer belief structures for decision making under uncertainty Ronald R. Yager a,b,⇑ , Naif Alajlan c a Machine Intelligence Institute, Iona College, New Rochelle, NY 10801, United States b Visiting Distinguished Scientist, King Saud University, Riyadh, Saudi Arabia c Computer Engineering Dept, College of Computer and Information Sciences, King Saud University Riyadh, Saudi Arabia article info Article history: Received 9 October 2014 Received in revised form 15 December 2014 Accepted 30 December 2014 Available online xxxx Keywords: Uncertainty Decision-making Belief Structures Information modeling Fusion abstract We discuss the need for tools for representing various types of uncertain information in decision-making. We introduce the Dempster–Shafer belief structure and discuss how it provides a formal mathematical framework for representing various types of uncertain information. We provide some fundamental ideas and mechanisms related to these structures. We then investigate their role in the important task of decision-making under uncertainty. Ó 2015 Elsevier B.V. All rights reserved. 1. Introduction An important task in decision-making is the modeling and representation of uncertain information. While probability is the most common form of uncertainty representation the variety of types information available from modern technological sources makes it imperative that we have at our disposal mathematical formalisms to model various types of uncertain knowledge. The Dempster–Shafer belief structure provides a rather general formal framework in which we can represent various types of knowledge about uncertain variables [1–17]. In addition to probabilistic knowledge we can represent the type of possibilistic knowledge that arises from human sourced linguistic expression of informa- tion [18]. It also allows for the representation of imprecise probabilistic information, situations in which the probabilities are only available as values within ranges rather as precise values. This been recently referred to as type two probabilities [19–24]. Our objective here is to first introduce some ideas from the theory of belief structures and then to look at the task of decision making under Dempster–Shafer uncertain information. 2. Dempster–Shafer belief structures A Dempster–Shafer belief structure can be viewed as a general- ization of a probability distribution that can provide the ability to model sophisticated information about the uncertainty associated with a variable. Formally a Dempster–Shafer belief structure m on a space X is defined via a collection of non-empty subsets of X, F 1 ; ... ; F q , called focal elements, and a mapping mðF j Þ2½0; 1 such that mðF j Þ > 0 and P q j mðF j Þ¼ 1. Here mðF j Þ is known as the weight associated with F j . While there are various interpretations that have been associ- ated with the D–S belief structure, the interpretation that we shall find useful relates the D–S belief structure to a probability distribu- tion. In the case of a probability distribution on a finite space X, associated with each x i 2 X is a value p i indicating the probability of x i . We recall R i p i ¼ 1. In the D–S framework the value mðF j Þ indi- cates an amount of probability allocated to the elements in F j in some unknown matter. As shall subsequently become clear a Dempster–Shafer belief situation can be viewed as a probability distribution with imprecise probabilities. Thus instead of knowing that the probability of x i is p i we have a range, ½a i ; b i  in which the probability of x i is known to lie. Thus a Dempster–Shafer belief structure can be viewed as a piece of information that contains two types of uncertainty. One is probabilistic, randomness, and the other is impression in the parameters associated with the probability distribution. The imprecision in the information often requires additional subject information to perform some tasks especially those related to deci- sion-making. In the following we shall assume our interest is in a variable V whose domain X is finite, X ¼fx 1 ; ... ; x n g. A belief structure has the ability to represent in a unified way many different types of http://dx.doi.org/10.1016/j.knosys.2014.12.031 0950-7051/Ó 2015 Elsevier B.V. All rights reserved. ⇑ Corresponding author at: Machine Intelligence Institute, Iona College, New Rochelle, NY 10801, United States. E-mail addresses: yager@panix.com (R.R. Yager), najlan@ksu.edu.sa (N. Alajlan). Knowledge-Based Systems xxx (2015) xxx–xxx Contents lists available at ScienceDirect Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys Please cite this article in press as: R.R. Yager, N. Alajlan, Dempster–Shafer belief structures for decision making under uncertainty, Knowl. Based Syst. (2015), http://dx.doi.org/10.1016/j.knosys.2014.12.031