Computational Intelligence and Neuroscience

Volume 2015, Article ID 364512, 11 pages

http://dx.doi.org/10.1155/2015/364512

## Expected Utility Based Decision Making under -Information and Its Application

^{1}Department of Mathematics, Eastern Mediterranean University, Famagusta, Northern Cyprus, Mersin 10, Turkey^{2}Department of Computer-Aided Control Systems, Azerbaijan State Oil Academy, 20 Azadlig Avenue, 1010 Baku, Azerbaijan

Received 19 September 2014; Accepted 25 December 2014

Academic Editor: Rahib H. Abiyev

Copyright © 2015 Rashad R. Aliev et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Real-world decision relevant information is often partially reliable. The reasons are partial reliability of the source of information, misperceptions, psychological biases, incompetence, and so forth. -numbers based formalization of information (-information) represents a natural language (NL) based value of a variable of interest in line with the related NL based reliability. What is important is that -information not only is the most general representation of real-world imperfect information but also has the highest descriptive power from human perception point of view as compared to fuzzy number. In this study, we present an approach to decision making under -information based on direct computation over -numbers. This approach utilizes expected utility paradigm and is applied to a benchmark decision problem in the field of economics.

#### 1. Introduction

Decision making is one of the attractive research areas in the last decades. The complexity and uncertainty are persistent phenomenon in the real world, and the fuzzy set [1–3] is widely used in decision making process [3, 4]. Much of decision based information is uncertain. Human has a high capability of making logical decisions based on uncertain, incomplete, and/or inaccurate information [5].

-number is a sufficient formalization of real-world information that should roughly be considered in light of its reliability. The critical issue is that the reliability of information is not considered properly. Zadeh has proposed a new notion -number which is more appropriate to describe the uncertainty. -number takes both restraint and reliability. In comparison with the classical fuzzy number, -number has more ability to describe the real information of human [6].

-numbers were firstly presented by Zadeh in 2011 [5], and afterwards the researchers started to discuss -numbers in decision making under uncertainty and in many other fields. One of the main goals of -number is to produce fuzzy numbers with degree of self-confidence in order to know the real information. By using the -number the knowledge of human can be represented in a better way [7].

The computations with -numbers can be viewed as a generalization of computations with numbers, intervals, fuzzy numbers, and random numbers. As specified, the levels of generality can be separated as follows: computation with numbers (ground level zero); computation with intervals (level one); computation with fuzzy numbers (level two) [2]; computation with random numbers (level two); and computation with -numbers (level three). The capability of building realistic models of real-world systems is increased by the increase of the generality level, especially in the realms of economics, risk assessment, decision analysis, planning, and analysis of causality [5].

In [4] the authors suggest an approach to use -numbers for solving multicriteria decision making problem. For computation over -numbers some operations are suggested that are based on Zadeh’s extension principle [5]. -numbers are also used for the purpose of reasoning [8]. In [6] proposed approach is intended to use -numbers for the expected utility application to solve decision making problems. An approach to use -numbers for answering questions and decisions making is considered in [9]. -numbers converted into classical fuzzy numbers are suggested in [4, 9]. In [7], -numbers are converted into classical fuzzy numbers and the fuzzy numbers are converted into crisp numbers. In [10] the theoretical approach for computing arithmetic operations over discrete -numbers is proposed.

In [11] authors suggest general and computationally effective theoretic approach to computations with discrete -numbers. The authors provide strong motivation of the use of discrete -numbers as an alternative to the continuous counterparts. In particular, the motivation is based on the fact that NL based information has a discrete framework. The suggested arithmetic of -numbers includes basic arithmetic operations and important algebraic operations over -numbers. The proposed approach allows dealing with -information directly.

This paper focuses on investigating an approach for decision making which generalizes the expected utility approach of -information. This approach is based on direct computation over -numbers without converting them to fuzzy numbers and differed from the existing works used for decision making problems. The direct computation of -numbers without conversion eliminates the loss of information. In this research we recommend an approach based on expected utility to solve the decision making problems with -information. This approach is based on computation over -numbers according to operations suggested in [5, 10]. At the end, we provide a numerical example of the proposed approach to solve a benchmark problem.

This paper is organized as follows. The preliminaries for -numbers are reviewed in Section 2. Section 3 describes the numerical computations with discrete -numbers. Section 4 is devoted to statement and solution of a considered decision problem with -information. Section 5 consists of application, and the conclusions are revealed in Section 6.

#### 2. Preliminaries

*Definition 1 (a discrete fuzzy number [12–14]). *A fuzzy subset of the real line with membership function is a discrete fuzzy number if its support is finite; that is, there exist with , such that and there exist natural numbers with satisfying the following conditions:(1) for any natural number with ;(2) for any natural numbers with ;(3) for any natural numbers with .

*Definition 2 (probability measure of a discrete fuzzy number [15]). *Let be a discrete fuzzy number. A probability measure of denoted by is defined as
Below we present the definition of addition of discrete fuzzy numbers suggested in [12–14, 16], where noninteractive fuzzy numbers are considered.

*Definition 3 (addition of discrete fuzzy numbers [12–14, 16]). *The addition of discrete fuzzy numbers is a discrete fuzzy number whose -cut is given as [12–14, 16]
where

*Definition 4 (multiplication of discrete fuzzy numbers [10, 11]). *The multiplication of discrete fuzzy numbers is a discrete fuzzy number whose -cut is given as [10]
where

*Definition 5 (discrete probability distribution). *The discrete probability distribution is defined as a function where if we suppose a discrete random variable taking different values with probability that defined to be , the probability must satisfy for each and [17].

*Definition 6 (convolution of discrete probability distributions). *Suppose and are two discrete random variables with distribution functions and . The distribution function is given as [17]

*Definition 7 (a discrete -number [11]). *A discrete -number is defined as an ordered pair , where and are discrete fuzzy numbers, is a fuzzy constraint on values that a random variable may take, and which has a membership function is a fuzzy constraint on the probability measure of :
The concept of a restriction has more generality than the concept of a constraint [18]. A restriction may be observed as a generalized constraint. A probability distribution is a restriction but is not a constraint [19].

-number concept is related to discrete -number; that is, -number is a pair of fuzzy number and random number to be defined as
where plays the same role as in discrete -number , and plays the role of the probability distribution such that [10]

#### 3. Computation with Discrete -Numbers

##### 3.1. General Review

Zadeh has suggested a general approach for computations with -numbers according to Zadeh’s extension principle [5]. This study is very complex in comparison with the previous one. The researchers look into using -numbers, but the lack of a direct and easy way to compute -numbers forced them to start thinking about a way to convert them into fuzzy numbers.

In [9] authors suggest an approach to convert -numbers into classical fuzzy numbers. They convert the second part to crisp number, but this leads to loss of original information.

The studies [4, 7, 20] are used according to what has been put forward in the study [9], but in fact this method does not give the results of high reliability. Therefore, the researchers looked for a new and simple way to calculate -numbers directly without conversion, based on what has been suggested in the study [5].

##### 3.2. Addition and Multiplication of Discrete -Numbers

Assume and be discrete -numbers describing values of uncertain real valued variables and . The addition and multiplication of -numbers are determined as follows [11]. Let and be given. Then where and are represented by discrete probability distributions (Definition 5):

is a sum (or multiplication) of fuzzy numbers defined on the basis of Definition 3 (Definition 4) and is a convolution of probability distribution defined on the basis of Definition 6.

Next, we should construct by solving the following problem: subject to

Thus, is obtained as [10, 11].

##### 3.3. Ranking of Discrete -Numbers [11]

Ranking of discrete -numbers is a necessary operation in arithmetic of -numbers and is a challenging practical issue. Zadeh addresses the problem of ranking -numbers as a very important problem [5]. In contrast to real numbers, -numbers are ordered pairs, for ranking of which there can be no unique approach. We suggest considering comparison of -numbers on the basis of fuzzy optimality (FO) principle. Let -numbers and be given. First, it is needed to calculate the functions , , which evaluate how much one of the -numbers is better, equivalent, and worse than the other one with respect to the first and the second components [11]: where , : where , . As always holds, one always has , where is the number of components of a -number; that is, . The membership functions of , , are shown in Figure 1 [11].