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.
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 [
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 [
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.
Let
Let
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 [
In extenics [
Suppose a finite interval
Here When When When When
A special case is when
Obviously, as shown in formula (
The extension simple dependent degree.
Suppose a finite interval
Suppose a finite interval
From Definition
Obviously, for a finite interval
Suppose a finite interval
The proof is very easy according to Theorem
According to the above content, we will deduce extension dependent degree for three-parameter interval and its mapping transformation method.
Suppose a finite interval
(1) When
(2) When
(3) When
(4) When
Obviously, when
Suppose a finite interval
(1) When
(2) When
(3) When
(4) When
Obviously, when
As above, when
Suppose a finite interval
Suppose a finite interval
Without loss of generality, suppose
(1) When
(2) When
The interval dependent degree mapping transformation.
Therefore, when
Obviously, for a finite interval
Suppose a finite interval
The proof is very easy according to Theorem
As we see, this theorem successfully transforms the three-parameter interval extension dependent degree calculation with the optimal point not at the endpoint to the calculation with the optimal point at the endpoint. That makes the three-parameter interval dependent degree calculation process very simple and uniform.
Different from the traditional interval theories such as gray correlation analysis, interval closeness, and the other measurement methods, the three-parameter interval extension dependent degree not only measures the relationship between two intervals, but also measures the relationship between the gravity points of them. Therefore, the measure is more comprehensive and reasonable. Compared to some existing methods in recent years such as three-parameter interval gray correlation degree, three-parameter interval closeness, three-parameter set pair connection number, and three-parameter interval projection sorting [
Suppose a benefit attribute interval
(1) when
(2) when
(3) when
Since
Suppose a benefit attribute interval
(1) when
(2) when
(3) when
The following part analyzes the practical significance of Corollary
As shown in Figure
An example of the interval dependent degree calculation for benefit attribute interval.
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Sorting result | | | |
The interval dependent degree of benefit attribute interval.
Suppose a benefit attribute interval
(1) when
(2) when
(3) when
The practical significance of Corollary
Suppose a cost attribute interval
(1) when
(2) when
(3) when
The proof is the same as that of Theorem
Suppose a cost attribute interval
(1) when
(2) when
(3) when
The practical significance of Corollary
Suppose a cost attribute interval
(1) when
(2) when
(3) when
The practical significance of Corollary
Suppose a fixed attribute interval
The proof is easy by Definition
Theorem
An example of the interval dependent degree calculation for fixed attribute interval.
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Sorting result | | | |
In the three-parameter interval extension dependent degree formula of Definition
Setting of preference attitude coefficient.
I | II | III | IV | V | VI | |
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| 0.333 | 0.25 | 0.4 | 0.1065 | 0.222 | 0.444 |
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| 0.333 | 0.25 | 0.4 | 0.1065 | 0.444 | 0.222 |
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| 0.333 | 0.5 | 0.2 | 0.787 | 0.333 | 0.333 |
Among them, type I represents decision makers having no preference. Type II represents that decision makers focusing more on the dependent degree of gravity values. Type III represents that decision makers preferring the dependent degree of the upper and the lower bounds. Type IV represents that decision makers preferring the dependent degree of gravity values and considering the evaluation values within the interval presenting a standard normal distribution. Type V and IV, respectively, represent decision makers tending to the dependent degree of the upper bound and the lower bound.
In type IV, assuming the evaluation value
Suppose, for multiattribute decision making, there are solution set
According to Theorems
Determine the coefficient setting of
Get the value range of each attribute
If the attribute weight set has been given
If the attribute weights have been given in the form of three-parameter interval number set
Perform uncertainty analysis for decision result through different settings of
Perform dynamics analysis and rule discovery on decision result through the interval extension transformation.
For convenience of comparison and illustration, the example uses the data from [
Evaluation value table.
candidate | | | | | | |
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(1) The evaluation matrix is obtained according to Table
(2) According to Step
The evaluation table after the interval mapping transformation.
candidate | | | | | | |
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(3) Determine the setting scheme of coefficient
(4) Determine the value range of each attribute according to Step
Dependent degree of the three-parameter interval attribute values.
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| 0.2500 | 0.6167 | 0.6667 | 0.8000 | 0.5500 | 0.8500 |
| 0.7500 | 0.5333 | 0.6167 | 0.6167 | 0.8167 | 0.6333 |
| 0.5667 | 0.3333 | 0.7000 | 0.6833 | 0.4500 | 0.6167 |
| 0.3667 | 0.6500 | 0.3833 | 0.4667 | 0.5167 | 0.6833 |
| 0.5000 | 0.6167 | 0.7000 | 0.6500 | 0.6333 | 0.3667 |
(5) In order to maintain comparison consistency and validity, the weights of the attributes are set the same as those in [
The three-parameter interval weight values.
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Dependent degree of the three-parameter interval weight values.
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The scheme sorting results and their stability test.
| | | | | sorting result | |
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| 0.3967 | 0.4417 | 0.3767 | 0.3150 | 0.3850 | |
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| 0.5833 | 0.6400 | 0.5442 | 0.4683 | 0.5583 | |
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| 0.7700 | 0.8383 | 0.7117 | 0.6217 | 0.7317 | |
(6) According to the different settings of
the sorting results under different attitude coefficient settings.
coefficient settings | | | | | | sorting result |
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I | 0.5833 | 0.6400 | 0.5442 | 0.4683 | 0.5583 | |
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II | 0.5850 | 0.6400 | 0.5425 | 0.4662 | 0.5569 | |
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III | 0.5820 | 0.6400 | 0.5455 | 0.4700 | 0.5595 | |
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IV | 0.5879 | 0.6400 | 0.5396 | 0.4627 | 0.5544 | |
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V | 0.5578 | 0.6111 | 0.5147 | 0.4400 | 0.5253 | |
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VI | 0.6089 | 0.6689 | 0.5736 | 0.4966 | 0.5913 | |
At above, the sorting method and uncertainty analysis of multiattribute decision making under static environment have been described precisely though building extension interval dependent degree function. In addition, according to Step
Suppose a three-parameter interval
Suppose a benefit attribute interval
Since
For multiattribute decision making, each evaluation value may be performed by the movement transformation. As a result, the combination result of all movement transformations tends to be quite complex. Here, we only discuss the impact from movement transformation of one attribute. For example, for a positive direction move transformation of one attribute of candidate
According to Theorem
Sorting reversion caused by the move transformation of evaluation of attribute
move transformation | | | sorting reversion |
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| 0.6433 | 0.6400 | |
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| 0.6458 | 0.6400 | |
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| 0.6433 | 0.6400 | |
Here is another example, by taking positive direction move transformation of one attribute of candidate
Sorting reversion caused by the move transformation of evaluation of attribute
move transformation | | | sorting reversion |
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| 0.5592 | 0.5583 | |
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| 0.5592 | 0.5583 | |
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| 0.5692 | 0.5583 | |
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| 0.5592 | 0.5583 | |
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| 0.5642 | 0.5583 | |
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| 0.5592 | 0.5583 | |
By adopting the extension dependent degree function, the paper researches the multiattribute decision making with attribute information being three-parameter interval number. It is a new thinking in the related research field. The main contributions are listed as follows: (1) Based on extension dependent function, a new decision making method and framework towards the multiattribute decision with attribute information being three-parameter interval number are put forward. (2) A formula of extension three-parameter interval dependent degree function is given, which reflects the different tendency from decision makers towards the lower bound, the upper bound, and the gravity point of the attribute evaluation by setting preference attitude coefficients. (3) Through defining the interval dependent degree mapping method, the calculation of the interval dependent degree with the optimal point not at the endpoint is transformed to the calculation with the optimal point at the endpoint, which has monotonic and simple process. (4) Six typical settings of attitude coefficients are given and the uncertainty analysis of decision results is made accordingly. (5) Based on the framework of the extension dependent degree calculation, dynamic analysis and rule discovery on decision results are performed through some extension element transformation.
The research work is quite abundant in the future. As a new thinking and framework, the decision method based on extension dependent function still needs further development and promotion. Next, the model will consider the combination of psychological behavior from decision makers such as prospect theory or regret theory, which can better reflect decision makers’ risk preferences. By improving the extension dependent function, the model can describe some more complex decision processes. The model may be revised to adapt to some more complex extension transformations. Furthermore, the model can also need to be expanded to many other decision environments including mixed type data, incomplete information, and fuzzy hesitant set.
The data used to support the findings of this study are included within the article.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This research was supported by Zhejiang Science Foundation Project of China (No. Y16G010010, LY18F020001) and Ningbo Innovative Team: the intelligent big data engineering application for life and health (Grant No. 2016C11024).