Entropy method of determining the attribute weights of interval numbers based on relative superiority

Abstract

In view of the multi-attribute decision making problems which the attribute values are in the forms of interval numbers, the paper presents an entropy method to obtain the attribute weights using the relative superiority concept. Firstly, the concept of this kind of problem is explained; Then in the light of the basic principle of the traditional entropy value method and train of thought, it given the calculation steps of weights using the relative superiority about the attribute value is interval number multiple attribute decision making problems. Its core is that relative superiority judgment matrix is obtained by comparing with two sets of interval numbers under the same indicator, which the group of interval numbers is equivalently mapped to the exact value form with the merits of relationship, then the weights of each indicator are calculated. Finally, the method is illustrated by giving an example.

Keywords

Relative superiority interval numbers entropy method weights

1. Introduction

Decision making is to choose from a number of feasible alternatives. The correctness of decision making is deeply influenced by the objectivity of the scheme itself, the subjectivity of the decision maker or the group of decision makers and the scientific nature of the decision making procedure. The decision making state of single attribute and single goal appears less and less in modern management. Due to the complexity of decision making environment and the uncertainty of decision makers, the multi-objective and multi-attribute behavior of decision making has become common. This kind of multiple attribute decision making in economic management, operations research, systems engineering, decision theory occupies the important position in the research of disciplines and is widely applied in the optimization scheme and investment decisions, the location of the factory, project evaluation, and many other fields, with extensive theoretical value and application prospects. In multi-attribute decision making, the importance of each attribute is determined by the weight of the attribute. The larger the weight is, the more important the attribute is; on the contrary, the smaller the weight is, the less important the attribute is. Therefore, attribute weight determination plays a key role in multi-attribute decision making.

According to different data, the method of determining attribute weight can be segmented into three categories. The first is the subjective weighting method. The data based on this method is determined by the decision maker’s subjective will, but has nothing to do with the decision indicator of attributes. The second category is the objective weighting method, which is based on data obtained by attribute decision indicators and has nothing to do with the subjective preference of decision makers. The third method is the combination method, it is a combination of the previous two methods, both based on the subjective preference and considered decision objective indicators data.

The objective weighting method has no subjective arbitrariness, and the index weight entirely depends on the value of decision attributes and the application of mathematical theoretical methods. For the multi-attribute decision making whose attribute value is determine values, experts have suggested many applicable methods, just like dispersion weighting method, entropy weighting method, coefficient of variation method, mean square error method, principal component analysis method, multi-objective programming weighting method, scheme satisfaction method, scheme proximity degree method, etc. Attribute value is an uncertain value, which has many forms at present. Interval number is a common form of attribute value uncertainty. The decision of such number is called interval number multi-attribute decision. On the one hand, this kind of uncertainty is due to the uncertainty, randomness and fuzziness of the multi-attribute decision making problem itself. On the other hand, due to the diversity and uncertainty of the decision-making environment, the complexity of the social system, and the fuzziness of human thinking, the information given by decision makers is mostly expressed in the form of interval Numbers. The research of multi – attribute decision – making problem with interval number has attracted wide attention [1, 2, 3, 4, 5, 6, 7, 8, 9]. For example, literature [10] proposed a comprehensive decision-making model of interval number. Literature [11] considered an interval number multi-attribute decision-making method with incomplete attribute weight information. Literature [12] established a dynamic multi-attribute decision making method based on interval number.

For interval number multi-attribute decision making, the general research approach is to extend deterministic multi-attribute decision making to uncertain multi-attribute decision making, that is, to improve on the original method. Literature [13] by using similarity theory to deal with attribute is interval number, for literature [14], the weight of the given interval number reciprocal judgment matrix, literature [15, 16, 17] interval number is obtained by using the possible degrees, literature [18] adopt include degree and the possible degree for the operation of interval number, based on literature [19] probability credibility to deal with interval Numbers, the literature [20] calculated, using deviation degree of interval number from literature [21] is used to calculate the interval number.

In short, there are many methods to process the problem of determine the problem of determine the weight with interval numbers, but the objective weighting method is not complete.This paper extended the traditional entropy method to interval numbers, by using the concept of relative superiority to present an entropy method with the property is interval numbers. The entropy obtained by this method can provide useful information to decision makers and decision analyst.

2. Algorithms of interval number

Interval number is the number expressed by interval, which is actually a set composed of all real Numbers on a closed interval, and its algorithm is generally similar to the algorithm of set. Interval number reflects a kind of uncertainty and has great application potential in various fields at present. For example, the interval number is applied to uncertain multiple attribute decision making. Add Interval number to form the uncertainty of mathematical programming optimization model.

Definition 1: Real axis interval $a=[a^{L},a^{U}]$ , which $a^{L}$ , $a^{U}$ are real numbers, so we define $a$ as interval number. If $a=[{a^{L},a^{U}}]=\{{x|0<a^{L}\leqslant x\leqslant a^{U}}\}$ , so we define $a$ as positive interval number. Specifically, if $a^{L}=a^{U}$ , then $a$ degraded to a real number. Under normal circumstances, the interval number is not a real number, so the algorithms of real number can not be directly used as arithmetic definition of interval numbers, must definition of the interval number algorithm must be redefined [22, 23, 24, 25, 26, 27].

Definition 2: Let $a=[{a^{L},a^{U}}]$ , $b=[{b^{L},b^{U}}]$ be any two interval numbers, then interval number satisfies the following laws:

(1)
Addition: $a+b=[{a^{L}+b^{L},a^{U}+b^{U}}]$
(2)
Subtraction: $a-b=[{a^{L}-b^{L},a^{U}-b^{U}}]$
(3)
Multiplication:

$\displaystyle a\cdot b=[{a^{L},a^{U}}]\cdot[{b^{L},b^{U}}]=[\min({a^{L}b^{L},a% ^{L}b^{U},a^{U}b^{L},a^{U}b^{U}}),\max(a^{L}b^{L},a^{L}b^{U},a^{U}b^{L},a^{U}b% ^{U})]$

Of particular note, when $a$ and $b$ are positive numbers, this formula is changed into:

$\displaystyle a\cdot b=[{a^{L}b^{L},a^{U}b^{U}}]$
(4)
Division operation:

$\displaystyle\frac{a}{b}=\frac{\left[{a^{L},a^{U}}\right]}{\left[{b^{L},b^{U}}% \right]}=\left[{a^{L},a^{U}}\right]\cdot\left[{\frac{1}{b^{L}},\frac{1}{b^{U}}% }\right]$

Of particular note, when $a$ and $b$ are positive numbers, this formula is degraded to:

$\displaystyle\frac{a}{b}=\frac{\left[{a^{L},a^{U}}\right]}{\left[{b^{L},b^{U}}% \right]}=\left[{\frac{a^{L}}{b^{L}},\frac{a^{U}}{b^{U}}}\right]$

3. The definition and properties of superiority

Definition 3: Let $a=[{a^{L},a^{U}}]$ , $b=[{b^{L},b^{U}}]$ be any two interval numbers, so we define:

$\displaystyle P\left({a>b}\right)=\left\{\begin{array}[]{lll}1-\frac{1}{2e^{x-% 0.5}}&\textit{when}&x=\displaystyle\frac{a^{U}-b^{L}}{a^{U}-a^{L}+b^{U}-b^{L}}% \geqslant 0.5\\ \frac{1}{2e^{0.5-x}}&\textit{when}&x=\displaystyle\frac{a^{U}-b^{L}}{a^{U}-a^{% L}+b^{U}-b^{L}}<0.5\\ \end{array}\right.$ (1)

as the relative superiority of interval number $a$ and interval number $b$ when $a$ is larger then $b$ .

Relative superiority has the following properties:

Theorem 1: Let $a=\left[{a^{L},a^{U}}\right]$ , $b=\left[{b^{L},b^{U}}\right]$ be any two interval numbers, so

(1)

$0<P\left({a>b}\right)<1,0<P\left({b>a}\right)<1$ l;

(2)

$P\left({a>a}\right)=\frac{1}{2}$ ;

(3)

$P\left({a>b}\right)+P\left({b>a}\right)=1$ ;

(4)

$P\left({a>b}\right)\geqslant\frac{1}{2}$ , only when $a^{L}+a^{U}\geqslant b^{L}+b^{U}$ .

Theorem 2: Let $a=\left[{a^{L},a^{U}}\right]$ , $b=\left[{b^{L},b^{U}}\right],\ldots,a_{n}=\left[{a_{n}^{L},a_{n}^{U}}\right]$ be any n interval number, comparing with two sets of interval numbers according to Eq. (1), and denoted as $a_{ij}=P\left({a_{i}>a_{j}}\right),i,j=1,2,\cdots,n$ . The matrix $A=\left({a_{ij}}\right)_{n\times n}$ with $a_{ij}$ as the element is the relative dominance judgment matrix. The relative superiority judgment matrix $A=\left({a_{ij}}\right)_{n\times n}$ must be a fuzzy complementary matrix.

Theorem 3: Let $A=\left({a_{ij}}\right)_{n\times n}$ be the relative dominance judgment matrix obtained by comparing the interval number $a_{1}=\left[{a_{1}^{L},a_{1}^{U}}\right]$ , $a_{2}=\left[{a_{2}^{L},a_{2}^{U}}\right],\ldots,a_{n}=\left[{a_{n}^{L},a_{n}^{% U}}\right]$ according to the Eq. (1). Of which $a_{ij}$ represent the relative superiority when $a_{i}>a_{j}$ . The sequence of $w_{1},w_{2},\cdots,w_{n}$ calculated by Eq. (2) reflects the merits of the order of n interval numbers.

$\displaystyle w_{i}=\frac{1}{n}-\frac{n}{4\alpha\left({n-1}\right)}+\frac{1}{2% \alpha\left({n-1}\right)}\sum\limits_{k=1}^{n}{a_{ik}},i=1,2,\cdots,n$ (2)

Of which, $\alpha$ is the parameter [6, 7, 8, 9] at satisfied $\alpha\geqslant\frac{n-1}{2}$ [28, 29, 30, 31].

4. The steps of entropy method

Entropy, first proposed and applied by the German physicist Clausius in 1850, describes the uniformity of the distribution of energy in space. Entropy belongs to the discipline of thermodynamics and is used to measure the disorder of the system.

Since its birth, the concept of entropy has been widely used in different research fields.

Entropy is used in information theory to measure uncertainty. The entropy value increases with the decrease of information quantity and the increase of uncertainty, and the higher the entropy value, the more chaotic the system, and vice versa. Usually, the entropy value is calculated to judge the random probability and chaos degree of an event according to its size.

When entropy is applied to decision making, it is used to indicate the degree of dispersion of an index. The index has a great influence on the comprehensive evaluation. With the increase of the dispersion degree, the influence of the index on the comprehensive evaluation also increases, and vice versa. Therefore, on the premise that the variation degree of each indicator is known, the basic principle of information entropy can be used to obtain the weight value of each indicator, providing a basis for the comprehensive evaluation of multiple indicators.

The weight value obtained by entropy method is objective, and compared with other subjective weighting methods, it avoids subjective arbitrariness to some extent. According to the magnitude of information contained in each index value, the relationship between data, and the impact of the relative change degree of indicators on the system as a whole, the weight of indicators is determined. The index with relatively large change has a larger weight, and this method is now widely used in statistics and other fields, which has a strong research value.

With respect to multiple attribute decision making problems, for program $x^{i}$ , measure it according to attribute $u^{j}$ , get the attribute values $x^{i}$ about $u^{j}$ (Where $x^{ij}$ is the interval number, $x^{ij}=({x_{ij}^{L},x_{ij}^{U}})$ ), so as to constitute a decision matrix $X=(X_{ij})_{m\times n}$ .

$\displaystyle X=\left[{{\begin{array}[]{cccc}{\left[{x_{11}^{L},x_{11}^{U}}% \right]}&{\left[{x_{12}^{L},x_{12}^{U}}\right]}&\cdots&{\left[{x_{1n}^{L},x_{1% n}^{U}}\right]}\\ {\left[{x_{21}^{L},x_{21}^{U}}\right]}&{\left[{x_{22}^{L},x_{22}^{U}}\right]}&% \cdots&{\left[{x_{2n}^{L},x_{2n}^{U}}\right]}\\ \cdots&\cdots&\cdots&\cdots\\ {\left[{x_{m1}^{L},x_{m1}^{U}}\right]}&{\left[{x_{m2}^{L},x_{m2}^{U}}\right]}&% \cdots&{\left[{x_{mn}^{L},x_{mn}^{U}}\right]}\\ \end{array}}}\right]$

On the above matrix, $x_{ij}$ represents the indicator value of the j-th indicator of the i-th alternative, (1 $\leqslant i\leqslant m$ , 1 $\leqslant j\leqslant n$ ), its calculation procedure is as follows [32, 33, 34, 35, 36].

(1)
Normalized the decision matrix. The most common attribute type is benefit-type and cost-type. Let $I_{j}\left({j=1,2}\right)$ represent benefit-type and cost-type respectively. First, the decision results should be avoided to be constrained by different physical dimensions, we can turn decision matrix $X$ into a matrix $Y=\left({y_{ij}}\right)_{m\times n}$ by vector standardized method, in which $y_{ij}=\left({y_{ij}^{L},y_{ij}^{U}}\right)$ .

$\displaystyle\left\{{\begin{array}[]{l}y_{ij}^{L}={x_{ij}^{L}}\mathord{\left/{% \vphantom{{x_{ij}^{L}}{\sum\limits_{i=1}^{m}{x_{ij}^{U}}}}}\right.\kern-1.2pt}% {\sum\limits_{i=1}^{m}{x_{ij}^{U}}}\\ y_{ij}^{U}={x_{ij}^{U}}\mathord{\left/{\vphantom{{x_{ij}^{U}}{\sum\limits_{i=1% }^{m}{x_{ij}^{L}}}}}\right.\kern-1.2pt}{\sum\limits_{i=1}^{m}{x_{ij}^{L}}}\\ \end{array}}\right.\quad i\in m,j\in I_{1}$

$\displaystyle\left\{{\begin{array}[]{l}y_{ij}^{L}=\left({1\mathord{\left/{% \vphantom{1{x_{ij}^{U}}}}\right.\kern-1.2pt}{x_{ij}^{U}}}\right)\mathord{\left% /{\vphantom{{\sum\limits_{i=1}^{m}{\left({1\mathord{\left/{\vphantom{1{x_{ij}^% {L}}}}\right.\kern-1.2pt}{x_{ij}^{L}}}\right)}}}}\right.\kern-1.2pt}{\sum% \limits_{i=1}^{m}{\left({1\mathord{\left/{\vphantom{1{x_{ij}^{L}}}}\right.% \kern-1.2pt}{x_{ij}^{L}}}\right)}}\\ y_{ij}^{U}=\left({1\mathord{\left/{\vphantom{1{x_{ij}^{L}}}}\right.\kern-1.2pt% }{x_{ij}^{L}}}\right)\mathord{\left/{\vphantom{{\sum\limits_{i=1}^{m}{\left({1% \mathord{\left/{\vphantom{1{x_{ij}^{U}}}}\right.\kern-1.2pt}{x_{ij}^{U}}}% \right)}}}}\right.\kern-1.2pt}{\sum\limits_{i=1}^{m}{\left({1\mathord{\left/{% \vphantom{1{x_{ij}^{U}}}}\right.\kern-1.2pt}{x_{ij}^{U}}}\right)}}\\ \end{array}}\right.\quad i\in m,j\in I_{2}$

After such a transformationï¼Œturn cost-type indicators into benefit-type, achieve consistency in the nature of the indicator.
(2)
Calculate the corresponding relative dominance judgment matrix according to the Eq. (1) for each attribute $u_{j}$ of the decision matrix $Y$ . According to Eq. (2) the mapping matrix $P=(P_{ij})_{m\times n}$ of the interval number to the exact value under the attribute is obtained.
(3)
Calculate the entropy of the j-th index value.

$\displaystyle e_{j}=-\frac{1}{\ln m}\sum\limits_{i=1}^{m}{p_{ij}\ln p_{ij}}(1% \leqslant j\leqslant n)$ (3)
(4)
Calculate the difference coefficient of the j-th index value. As can be seen from the above, the greater the difference of index values, the greater the evaluation effect on the scheme results, and the smaller the corresponding entropy value, and vice versa. Therefore, the definition of the difference coefficient is:

$\displaystyle g_{j}=1-e_{j}(1\leqslant j\leqslant n)$
(5)
Determine the weight of each index. The weight of the j-th index is:

$\displaystyle w_{j}=\frac{g_{j}}{\sum\limits_{j=1}^{n}{g_{j}}}(1\leqslant j% \leqslant n)$

5. Example analysis

Considering the power company’s preference for electric construction contractors, there are three candidate contractors X1 (Electric Construction Company 1), X2 (Electric Construction Company 2), and X3 (Electric Construction Company 3). For each scenario, there are 4 metrics, namely engineering quality (u1), project progress (u2), company reputation (u3), and engineering risk (u4). Of which, u4 is cost index, the rest is benefit index. If 1 represents the best satisfaction of the power company for the engineering contractor, the evaluation value of each index of the electric construction contractor is given as the interval number, as shown in Table 1.

Table 1
The assessed value of power construction contractor indicators

Contractor $\backslash$ index	Quality of the project u1	Progress of the project u2	Company reputation u3	Risks of the project u4
ContractorX1	[0.70,0.74]	[0.85,0.89]	[0.63,0.83]	[0.10,0.15]
ContractorX2	[0.85,0.90]	[0.88,0.95]	[0.73,0.93]	[0.07,0.17]
ContractorX3	[0.75,0.79]	[0.80,0.83]	[0.77,0.97]	[0.10,0.13]

Standardized the decision matrix.

$\displaystyle Y=\left[\begin{array}[]{cccc}(0.28807,0.32174)&(0.31835,0.35178)% &(0.23077,0.38957)&(0.19444,0.49404)\\ (0.34979,0.39130)&(0.32959,0.37549)&(0.26740,0.43662)&(0.17157.0.70577)\\ (0.30864,0.34348)&(0.29963,0.32806)&(0.28205,0.45540)&(0.22436,0.49404)\end{% array}\right]$

(2) Calculate the corresponding relative dominance judgment matrix of each attribute of the decision matrix $Y$ , let $\text{P}_{i}\left({1\leqslant i\leqslant 4}\right)$ represent the judgment matrix of the i-th attribute,

$\displaystyle\text{P}_{1}=\left({{\begin{array}[]{ccc}{0.50000}&{0.20883}&{0.3% 6717}\\ {0.79117}&{0.50000}&{0.72079}\\ {0.63283}&{0.27931}&{0.50000}\\ \end{array}}}\right)$ $\displaystyle\text{P}_{2}=\left({{\begin{array}[]{ccc}{0.50000}&{0.40115}&{0.6% 4519}\\ {0.59885}&{0.50000}&{0.70291}\\ {0.35481}&{0.29709}&{0.50000}\\ \end{array}}}\right)$ $\displaystyle\text{P}_{3}=\left({{\begin{array}[]{ccc}{0.50000}&{0.44021}&{0.4% 1927}\\ {0.55979}&{0.50000}&{0.47619}\\ {0.58073}&{0.52391}&{0.50000}\\ \end{array}}}\right)$ $\displaystyle\text{P}_{4}=\left({{\begin{array}[]{ccc}{0.50000}&{0.44646}&{0.4% 8703}\\ {0.55354}&{0.50000}&{0.54706}\\ {0.51297}&{0.45294}&{0.50000}\\ \end{array}}}\right)$

Use Eq. (2) to get the interval number under the attribute to the exact value mapping matrix, and complete the normalization process.

$\displaystyle\text{D}=\left({{\begin{array}[]{cccc}{0.22733}&{0.34492}&{0.2982% 0}&{0.31671}\\ {0.46132}&{0.40877}&{0.34233}&{0.35848}\\ {0.31137}&{0.24631}&{0.35947}&{0.32481}\\ \end{array}}}\right)$

Calculate the entropy of the indicator.

$\displaystyle\text{e}_{1}=0.96209,\text{e}_{2}=0.98120,\text{e}_{3}=0.99723,% \text{e}_{4}=0.99867$

Calculate the difference coefficient of the index.

$\displaystyle\text{g}_{1}=0.03791,\text{g}_{2}=0.01880,\text{g}_{3}=0.00277,% \text{g}_{4}=0.00133$

Determine indicator weights.

$\displaystyle w_{1}=0.62348,w_{2}=0.30921,w_{3}=0.04550,w_{4}=0.02181$

6. Comparative analysis

In order to verify the effectiveness and feasibility of the entropy method based on the weight of the interval number attribute proposed in this paper, the interval number entropy method based on the degree of likelihood and the mathematical programming method based on the degree of relative advantage proposed in literature [15, 23] are respectively used to calculate the index weight of the above mentioned power construction contractors. According to the entropy method of interval number attribute weight based on probability degree proposed in literature [15], the weight calculation results of the above four indexes are 0.5237, 0.4169, 0.1026, 0.0983, respectively. According to the mathematical programming method based on the degree of relative superiority proposed in the paper [23], the weight calculation results of the above four indexes are 0.4412, 0.4117, 0.3976 and 0.3241 respectively.

As you can see, in this paper, and the method of literature [15] weight to calculate the entropy value method, only the means of interval number processing with different, the resulting weight slightly different, is due to the relative superiority degree method to consider the size of the gap between interval Numbers, with gap value set associated with the threshold value of 0.5, and the literature [15] no set standard; Compared using the method of literature [23] by weight is also slightly different, because the literature [23] processing interval number though is relative superiority degree method is used, but it USES mathematical programming for the calculation of weight, though it is a method of weight, but can’t do it on information mining distinguish the difference between the data. It is worth noting that although the weights of the four indexes calculated in literature [15] and literature [23] are different from each other, the trend is consistent, which indicates that the entropy method of interval number attribute weight based on relative dominance is effective and feasible.

7. Conclusion

In this article, a new analysis method is constructed for the attribute value is interval number multiple attribute decision making problems, following the basic idea of the traditional entropy weight method. This method has the following two characteristics:

(1)

Relative dominance when comparing the size of two intervalsï¼Œit avoids the deficiency of the possibility method of interval number comparison in the size of measurement interval number, and the numerical expression is more precise, which reduces the loss of evaluation informationï¼ŒReduce the likelihood of inaccurate results.

(2)

This method extends the analysis method of the multi-attribute decision making problems which the attribute values are in the forms of interval numbers. It has clear concept and the calculation is simple.

Objective weight is based on objective data, it reflects the characteristics of data distribution and the relationship between the objective weight simply reflects the difference of attribute index, failed to reflect the decision maker’s subjective will, and decision making is a complex system problem. The decision making process is affected by the decision making object, subject and many factors, and the results of the decision making have far-reaching effects. Therefore, the objective weights obtained from the data alone should not be used alone. Generally, the decision-maker’s preference should be reflected in the use to make the decision making results more reasonable and scientific. So research in this area should be the focus of future research.

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Entropy method of determining the attribute weights of interval numbers based on relative superiority

Abstract

Keywords

1. Introduction

2. Algorithms of interval number

Table 1 The assessed value of power construction contractor indicators

7. Conclusion

References

Table 1
The assessed value of power construction contractor indicators