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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 _{0}

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 _{0}=[_{0},_{0}] and

_{0} and the interval

(1) When_{0} =_{0},

(2) When_{0} =_{0 }=

(3) When_{0} =_{0} =

(4) When

Obviously, when

Suppose a finite interval

(1) When_{0} =_{0} =_{0},

(2) When_{0} =_{0 }=_{0 }=

(3) When_{0} =_{0} =_{0 }=_{0} or_{0} =_{0},

(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 _{0}, and transforming 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 _{0} =_{1}, then,

(1) when

(2) when

(3) when

The following part analyzes the practical significance of Corollary

As shown in Figure _{0} (_{1}), which indicates that their overall evaluation is consistent. Although

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 _{0}=_{1},_{0}=_{1}, then,

(1) when_{0}=_{1}, there is

(2) when_{0}<_{1}, there is

(3) when_{0}>_{1}, there is

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 _{0 }=_{1}, then,

(1) when

(2) when

(3) when

The practical significance of Corollary

Suppose a cost attribute interval _{0}=_{1},_{0}=_{1}, then,

(1) when_{0}=_{1}, there is

(2) when_{0}<_{1}, there is

(3) when_{0}>_{1}, there is

The practical significance of Corollary

Suppose a fixed attribute interval _{0},_{1 }

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 [_{1},_{2},_{3},_{4},_{5}) according to 6 evaluation attributes (_{1},_{2},_{3},_{4},_{5},_{6}) which include moral quality, working attitude, working style, educational level, leadership, and development ability. Each attribute of each candidate was scored, respectively, from the group decisions and then the evaluation of each attribute of each one is obtained by some basic statistical processing. The score range was determined in advance. Obviously, the evaluation for the same candidate varies from person to person; as a result, the attribute value is given in the form of three-parameter interval number. The first five are benefit attributes which are the bigger the better, ranging from 0.80 to 1.00. The last one is a fixed attribute, ranging from 0.80 to 1.20. For the fixed attribute, 1.00 is the optimal score, and the score beyond it means too radical and vice versa (too conservative). The attribute values are shown in Table

Evaluation value table.

candidate | _{1} | _{2} | _{3} | _{4} | _{5} | _{6} |
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_{1} | | | | | | |

_{2} | | | | | | |

_{3} | | | | | | |

_{4} | | | | | | |

_{5} | | | | | | |

(1) The evaluation matrix is obtained according to Table

(2) According to Step

The evaluation table after the interval mapping transformation.

candidate | _{1} | _{2} | _{3} | _{4} | _{5} | _{6} |
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_{1} | | | | | | |

_{2} | | | | | | |

_{3} | | | | | | |

_{4} | | | | | | |

_{5} | | | | | | |

(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.

_{1} | _{2} | _{3} | _{4} | _{5} | _{6} | |
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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.

| _{1} | _{2} | _{3} | _{4} | _{5} | _{6} |

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Dependent degree of the three-parameter interval weight values.

| _{1} | _{2} | _{3} | _{4} | _{5} |

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The scheme sorting results and their stability test.

_{1} | _{2} | _{3} | _{4} | _{5} | 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 | _{1} | _{2} | _{3} | _{4} | _{5} | 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 _{1} of candidate_{1} changes from

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_{1}, the transformation is established as

According to Theorem _{1} wants to overtake that of_{2}, the evaluation of any of the attributes_{1},_{3},_{5} should be chosen to be promoted. Among them, attribute_{3} needs a minimum level of promotion. However, except for_{1},_{3},_{5}, promoting the evaluation of any of the other attributes will not change the results.

Sorting reversion caused by the move transformation of evaluation of attribute _{1}.

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_{3}, transformation analysis will be performed. By iteration calculation with step size of 0.01, the transformation rule and the threshold value that is able to bring some sorting reversion are obtained, as shown in Table _{3} is going to overtake that of_{5}, the evaluation of any of the attributes can be chosen to be promoted. Among them, attributes_{1},_{3},_{5},_{6} need relatively low levels of promotion. It must be noted that the interval of_{6} has been processed by mapping transformation and, as a result, and needs to map back after movement transformation.

Sorting reversion caused by the move transformation of evaluation of attribute _{3}.

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).