Mathematical Problems in Engineering

Volume 2018, Article ID 1831086, 12 pages

https://doi.org/10.1155/2018/1831086

## Extension Dependent Degree Method with Mapping Transformation for Three-Parameter Interval Number Decision Making

^{1}Zhejiang University Ningbo Institute of Technology, Ningbo, Zhejiang 315000, China^{2}Guangdong University of Technology, Guangzhou, Guangdong 510006, China

Correspondence should be addressed to Li-Bo Xu; moc.361@obil_ux

Received 22 July 2018; Accepted 28 August 2018; Published 12 September 2018

Academic Editor: Carlos Llopis-Albert

Copyright © 2018 Li-Bo Xu 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

In view of the multiattribute decision making problem that the attribute values and weights are both three-parameter interval numbers, a new decision making approach and framework based on extension simple dependent degree are proposed. According to traditional extension simple dependent function, the new approach proposes a new extension dependent function for three-parameter interval number. Then through an interval mapping transformation method, the process for obtaining dependent degree for the interval with its optimal value not being the endpoint is transformed to the monotonous process for the interval with its optimal value being the endpoint. The method can not only perform uncertain analysis of decision results by different settings of attitude coefficients, but also take dynamic analysis and rule finding by some extension transformation. At last, an example is presented to examine the effectiveness and stability of our method.

#### 1. Introduction

In multiattribute decision making (MADM), attributes information often shows some ambiguity and uncertainty due to the complexity from objective things and the finiteness from decision makers, so it is difficult to describe by some accurate numerical values. Therefore, several methods that can describe this uncertainty information, such as interval number [1], fuzzy number [2], gray number [3], and connection number [4, 5], have been widely studied and applied. Among them, interval number theory, as one of the most possible solutions, has produced many valuable research results. However, the traditional interval number is too rough to describe the uncertain information. It only focuses on the upper and lower bounds of the interval and possibly ignores its intrinsic preference information from users, which greatly limits its practicability in many actual application scenarios. In comparison, the three-parameter interval number [6] not only retains the upper and lower bounds of the interval, but also emphasizes the gravity value which has the maximum hit possibility in all values. Therefore, it is superior to the traditional interval number method in describing the uncertainty information. In recent years, the three-parameter interval number theory has become the research focus in uncertainty decision making domain. Literature [7–9] establishes the concept of three-parameter interval gray number, correlation degree, and closeness calculation method according to the gray system theory. In literature [10], the ratio of the sums of three parameters is directly used as the interval comparison results, and then the pairwise comparison matrix is built to sort the scheme. In literature [11], the traditional interval likelihood ranking method is extended to the three-parameter interval ranking, and the TOPSIS model is established for three-parameter interval number. Literature [12] also adopts TOPSIS method to sort three-parameter interval numbers by redefining an Euclidean distance. In literature [13], combining the center value, the gravity value, and the interval’s length of the three-parameter interval gray number, an exact score function is defined as the basis for comparison. Literature [14] proposes to convert the three-parameter interval into traditional two-parameter interval, so that the likelihood sorting method for the traditional interval can be used directly. In some earlier literature [15, 16], some triangle fuzzy numbers are used to describe the three-parameter interval, and the scheme order was determined based on the fuzzy number operation process and the distance measure of ideal point. Literature [17] defines the concept of three-parameter interval fuzzy set, its operation rules, and distance measure formula. Literature [18] proposes a three-parameter interval gray linguistic variable decision making method based on projection model and prospect theory.

In general, although the existing research has made some great progress, there are still some things to be improved. First, most of the research is based on the fuzzy number correlation theory including score function, fuzzy distance, fuzzy similarity, and likelihood method. These fuzzy number concepts and measures are only extended to the three-parameter interval number field, so there is no new method and framework. Second, many decision making models are too deterministic and lose their uncertainty in the process. Hence, it is difficult to carry out stability checking and uncertainty analysis for the decision results in the later stage. In this regard, the set pair analysis method is proposed [19], which converts the three-parameter interval number into the connection number expression and maintains the uncertainty of the result and the calculation simplicity. Therefore, it is another idea worthy of further study. Third, the existing decision making model is basically static. Researches on dynamic decision judgment and rule discovery are very inadequate.

In order to solve the above problems, the paper attempts to propose a new multiattribute decision making method and framework based on extension dependent degree. This thinking and method are rarely seen in the existing research literature. In this method, firstly, according to the extension simple dependent degree calculation method and its mapping transformation rule, the dependent degree calculation expression for the three-parameter interval and its interval map transformation method are given. This will transform the process of calculating dependent degree of the interval with its optimal value not being the endpoint to the monotonous process of the interval with its optimal value being the endpoint. It not only makes the calculation process simple and unified, but also expresses a new three-parameter interval sorting method. Secondly, six typical coefficient setting schemes are given for the attitude coefficient in the dependent degree calculation expression, which can reflect the different preference attitudes from the decision makers for the upper, lower, and average evaluation scores. It can make the model perform some uncertainty analysis for the decision results. After that, the comparison between our method and the existing other research results is shown by numerical examples, which illustrates the effectiveness and stability of the proposed method and its ability to perform uncertainty analysis. Finally, based on the extension dependent degree calculation, the dynamic analysis and rule discovery of the decision process through extension transformation are proposed, which shows the dynamic applicability of our method.

#### 2. Extension Simple Dependent Degree for Three-Parameter Interval

##### 2.1. Three-Parameter Interval Number

*Definition 1. *Let* R* be the real number set, and for any , , is noted as interval number. Here* a* is the lower bound of the interval, and* b* is the upper bound. When* a* =* b*,* X* degenerates into a real number.

*Definition 2 (see [10–12]). *Let* R* be a real number set, and for any , , is noted as a three-parameter interval number. Here* a* and* b*, respectively, represent the lower and upper bound of the interval number.* m* represents a special point with some statistical meaning in this interval such as mean value or maximum possibility value, which is called gravity value or ideal value.

In fact, the three-parameter interval number is the expansion of the traditional interval number but may describe uncertainty and fuzzy numbers more abundantly and accurately. For example, there is an evaluation score for the performance of a product noted as a three-parameter interval number [4, 6, 8]. The number may mean that the lower and upper bounds from user group evaluations are 4 and 8, respectively. It can also mean that the score from one user covers between 4 and 8, and 6 is the most preferred score. It can also indicate that the lowest score and highest score for single attribute are 4 and 8, and the average score is 6. By some means, the three-parameter interval number is similar to the triangular fuzzy number [20, 21] expression. Both of them consider that the left and right interval endpoints are the critical points, and the middle points are the most representative ones. However, the triangular fuzzy number focuses more on the degree of the elements belonging to some fuzzy set. The three-parameter interval number pays more attention to the probability or the statistical meaning for the interval, and so its meaning is more universal and rich. It is worthy of further research and application.

##### 2.2. Extension Simple Dependent Degree and Mapping Transformation

In extenics [22], extension simple dependent function describes the relationship between a point* x* and interval covering* X*. Its calculation is based on the given range of values and does not need to rely on subjective judgment or empirical value from decision makers, so it is convenient to quantitatively describe the nature of things. It has been used in some evaluation and forecasting applications [23, 24].

*Definition 3 (see [22]). *Suppose a finite interval , and its optimal value is* m ∈X*; thenHere

*k*(

*x*,

*X*) is called extension simple dependent degree of the point

*x*and interval

*X*, as shown in Figure 1, and it satisfies the following properties:(1)When

*x*=

*m*,

*k*(

*x*,

*X*) reaches the maximum value and

*k*(

*x*,

*X*)=

*k*(

*m*,

*X*) =1.(2)When

*x∈X*and

*x*≠

*a*,

*b*, then

*k*(

*x*,

*X*) > 0.(3)When

*x∉X*and

*x*≠

*a*,

*b*, then

*k*(

*x*,

*X*) < 0.(4)When

*x*=

*a*or

*x*=

*b*, then

*k*(

*x*,

*X*) = 0. A special case is when

*m*=

*a*or

*b*, and

*x*; thenObviously, as shown in formula (2), when the optimal point

*∈*X*m*is just at the endpoint of the interval, extension simple dependent degree formula becomes a monotonic increasing or monotonic decreasing function, and its calculation process is very concise and intuitive. However, as shown in formula (1), when the optimal point

*m*is not at the endpoint, the dependent degree formula must be changed according to the location of

*m*, which will bring some complexity and inconveniences. Therefore, here we propose a mapping transformation method which will transform the dependent degree calculation with the optimal point not being the endpoint into the calculation with the optimal point being the endpoint, so as to keep the monotonicity and simplicity of the process.