# A Modified Combination Rule forD Numbers Theory

Authors: Ningkui Wang; Fuyu Liu; Daijun Wei
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

D numbers theory is an appropriate method to deal with the information of uncertainty and incompleteness when making a reasonable decision. Previous D numbers theory provides a rule to combine multiple D numbers. However, the commutative law is not satisfied in the rule of combining multiple D numbers. In this paper, a modified method for multiple D numbers combination is proposed. The proposed method defines a new function for multiple D numbers combination which is mainly determined by the original value of D numbers. Then the proposed combination rule is applied to environmental impact assessment (EIA); our results show that the proposed method is efficient for multiple D numbers combination and it is useful when dealing with uncertainty and incompleteness.


---

## Body

## 1. Introduction

In the real world, it is difficult but necessary to make a comprehensive assessment to make a reasonable decision because much uncertainty and incompleteness are often involved in the assessments [1–4]. Several methods, such as probability theory [5, 6], fuzzy theory [7–13], rough set [14–17], uncertainty theory [18–20], and Dempster-Shafer theory of evidence (DST) [21–23], are widely used to deal with these problems. These methods have widely been used in kinds of fields, like supplier selection [24, 25], risk assessments [26–29], and so on [30, 31].The DST needs weaker conditions than the Bayesian theory of probability, it is often regarded as an extension of the Bayesian theory [32, 33]. For the frame of discernment, which consists of mutually exclusive and collective elements, the basic probability assignment (BPA) can distribute confident degree to the power set of the frame of discernment. Furthermore, an overall assessment can be obtained by combining pairs of BPAs in the DST. Therefore, the DST has been widely applied to multiple criteria decision-making [34–44]. However, some strong hypotheses obviously exist in the DST because of the definitions of the frame of discernment and the BPA. Firstly, the elements in the frame of discernment require being mutually exclusive, but it is hard to be satisfied in the real life especially in linguistic assessments, such as the evaluation on the subjects; “good” and “very good” are two common linguistic evaluations, but they are not completely mutually exclusive so that the DST is unable to handle them. At the same time, the sum of all the BPAs must be equal to 1. However, lacking of some professional knowledge and inadequacy judgements may lead to incompleteness everywhere in the real word. These shortcomings have limited its usage in some fields [45, 46].Regarded as the generalization of DST,D numbers theory is proposed by Deng [47, 48]. It removes these hypotheses reasonably; the elements in the framework of D numbers theory do not need to be mutually exclusive and incomplete assessments can also exist in D numbers theory. Because D numbers theory has the ability to deal with uncertainty and incompleteness, it has been used in EIA [48], failure mode and effects analysis [49], supplier selection [50], and curtain grouting efficiency assessment [51]. Nevertheless, associative property is not satisfied in the previous D numbers’ combination rule. In [48], Deng et al. do some work for multiple D numbers combination in special circumstances. However, the associative property is not addressed in a general condition. In this paper, a modified method for multiple D numbers combination is proposed.The remainder of this paper is organized as follows. In Section2, preliminaries about DST and D numbers theory are described in detail. The problem of the previous D numbers combination rule and the proposed method is shown in Section 3. An illustrative numerical example is presented in Section 4. Conclusions are given in Section 5.


## 2. Problem Statement and Preliminaries

### 2.1. Dempster-Shafer Theory

DST is proposed by Dempster and Shafer; some basic concepts are introduced as follows [21, 22].Definition 1.
Establish thatU is a set of mutually exclusive and collectively exhaustive elements which can be represented as follows:(1)
U
=
e
1
,
e
2
,
e
3
,
…
,
e
n
.The power set of U is denoted as 2
U; any element belongs to the power set 2
U is said to be a proposition. For a frame of discernment U, a mass function is a mapping, which is denoted as follows:(2)
m
:
2
U
⟶
0,1in which the following conditions are satisfied:(3)
m
∅
=
0
∑
A
⊂
U
m
A
=
1
,where ∅ is an empty set and A is a subset of 2
U; the function m
(
A
) represents how strongly the evidence supports A.Definition 2 (Dempster’s rule of combination).
Given two BPAsm
1 and m
2, Dempster’s rule of combination donated as m
=
m
1
⊕
m
2 is defined as follows:(4)
m
A
=
1
1
-
K
∑
B
∩
C
=
A
m
1
B
m
2
C
,
A
≠
∅
;
0
,
A
=
∅
;with(5)
K
=
∑
B
∩
C
=
∅
m
1
B
m
2
C
,where A, B, and C are the elements of 2
U and K is a normalization constant which means the conflict coefficient of two BPAs.Note that Dempster’s rule of combination is feasible only whenK
<
1 because K
=
1 means that the two BPAs are one hundred percent conflicted. Associative property is well satisfied in Dempster’s rule of combination.


### 2.2.D Numbers Theory

There are some strong hypotheses in DST which have limited its wide usage in some fields especially in linguistic assessments.D numbers theory is proposed in [47, 48] and it has overcome these hypotheses. The details about D numbers theory are introduced as follows.Definition 3.
LetΩ be a finite nonempty set; D numbers is a mapping:(6)
D
:
Ω
⟶
0,1
,where the following conditions are satisfied:(7)
D
∅
=
0
∑
B
⊂
Ω
D
B
≤
1
,where ∅ is an empty set and B is a subset of Ω. The elements in the set Ω of D numbers do not require mutual exclusiveness and the sum of the assessments can be less than 1 in D numbers theory.Suppose that five linguistic assessments “extremely poor (EP),” “poor (P),” “average (A),” “good (G),” and “very good (VG)” are used for the evaluation of a car. The framework of DST must be mutually exclusive andD numbers theory providing the framework with nonexclusive hypotheses is more tallying with the actual situation. The differences of their framework of DST and D numbers are shown in Figure 1 [48]. In (7), D numbers theory is acceptable for incomplete information since ∑
B
⊂
U
‍
D
(
B
)
≤
1 which is more close to the real situation.Figure 1
The framework of DST andD numbers theory.Definition 4.
For a discrete setΩ
=
(
b
1
,
b
2
,
b
3
,
…
,
b
n
), where b
i belongs to R and b
i
≠
b
j if i
≠
j, for any v
i
≥
0 and ∑
i
=
1
n
‍
v
i
≤
1, a special form of D numbers can be expressed by(8)
D
b
1
=
v
1
D
b
2
=
v
2
D
b
3
=
v
3
⋮
D
b
n
=
v
nor be represented simply as (9)
D
=
b
1
,
v
1
,
b
2
,
v
2
,
b
3
,
v
3
,
…
,
b
n
,
v
n
.Definition 5 (D numbers combination rule).
LetD
1 and D
2 be two D numbers:(10)
D
1
=
b
1
1
,
v
1
1
,
b
2
1
,
v
2
1
,
b
3
1
,
v
3
1
,
…
,
b
n
1
,
v
n
1
D
2
=
b
1
2
,
v
1
2
,
b
2
2
,
v
2
2
,
b
3
2
,
v
3
2
,
…
,
b
n
2
,
v
n
2
.The combination of D
1 and D
2 denoted by D
=
D
1
⊕
D
2 is defined as follows:(11)
D
b
=
v
(12)
b
=
b
i
1
+
b
j
2
2
(13)
v
=
v
i
1
+
v
j
2
2
×
c
(14)
C
=
∑
j
=
1
m
∑
i
=
1
n
v
i
1
+
v
j
2
2
∑
i
=
1
n
v
i
1
=
1
,
∑
i
=
1
m
v
j
1
=
1
;
∑
j
=
1
m
∑
i
=
1
n
v
i
1
+
v
j
2
2
+
∑
j
=
1
m
v
c
1
+
v
j
2
2
∑
i
=
1
n
v
i
1
<
1
,
∑
i
=
1
m
v
j
1
=
1
;
∑
j
=
1
m
∑
i
=
1
n
v
i
1
+
v
j
2
2
+
∑
i
=
1
n
v
i
1
+
v
c
2
2
∑
i
=
1
n
v
i
1
=
1
,
∑
i
=
1
m
v
j
1
<
1
;
∑
j
=
1
m
∑
i
=
1
n
v
i
1
+
v
j
2
2
+
∑
j
=
1
m
v
c
1
+
v
j
2
2
+
∑
i
=
1
n
v
i
1
+
v
c
2
2
∑
i
=
1
n
v
i
1
<
1
,
∑
i
=
1
m
v
j
1
<
1
,where v
c
1
=
1
-
∑
i
=
1
n
‍
v
i
1 and v
c
2
=
1
-
∑
j
=
1
m
‍
v
j
2, m and n are the assessment numbers in each D number, and the superscripts in above equations are not the exponent but the order of the D numbers.Definition 6 (D numbers’ integration).
For givenD numbers, the overall assessments can be calculated as follows:(15)
I
D
=
∑
i
=
1
n
b
i
v
i
.


## 2.1. Dempster-Shafer Theory

DST is proposed by Dempster and Shafer; some basic concepts are introduced as follows [21, 22].Definition 1.
Establish thatU is a set of mutually exclusive and collectively exhaustive elements which can be represented as follows:(1)
U
=
e
1
,
e
2
,
e
3
,
…
,
e
n
.The power set of U is denoted as 2
U; any element belongs to the power set 2
U is said to be a proposition. For a frame of discernment U, a mass function is a mapping, which is denoted as follows:(2)
m
:
2
U
⟶
0,1in which the following conditions are satisfied:(3)
m
∅
=
0
∑
A
⊂
U
m
A
=
1
,where ∅ is an empty set and A is a subset of 2
U; the function m
(
A
) represents how strongly the evidence supports A.Definition 2 (Dempster’s rule of combination).
Given two BPAsm
1 and m
2, Dempster’s rule of combination donated as m
=
m
1
⊕
m
2 is defined as follows:(4)
m
A
=
1
1
-
K
∑
B
∩
C
=
A
m
1
B
m
2
C
,
A
≠
∅
;
0
,
A
=
∅
;with(5)
K
=
∑
B
∩
C
=
∅
m
1
B
m
2
C
,where A, B, and C are the elements of 2
U and K is a normalization constant which means the conflict coefficient of two BPAs.Note that Dempster’s rule of combination is feasible only whenK
<
1 because K
=
1 means that the two BPAs are one hundred percent conflicted. Associative property is well satisfied in Dempster’s rule of combination.


## 2.2.D Numbers Theory

There are some strong hypotheses in DST which have limited its wide usage in some fields especially in linguistic assessments.D numbers theory is proposed in [47, 48] and it has overcome these hypotheses. The details about D numbers theory are introduced as follows.Definition 3.
LetΩ be a finite nonempty set; D numbers is a mapping:(6)
D
:
Ω
⟶
0,1
,where the following conditions are satisfied:(7)
D
∅
=
0
∑
B
⊂
Ω
D
B
≤
1
,where ∅ is an empty set and B is a subset of Ω. The elements in the set Ω of D numbers do not require mutual exclusiveness and the sum of the assessments can be less than 1 in D numbers theory.Suppose that five linguistic assessments “extremely poor (EP),” “poor (P),” “average (A),” “good (G),” and “very good (VG)” are used for the evaluation of a car. The framework of DST must be mutually exclusive andD numbers theory providing the framework with nonexclusive hypotheses is more tallying with the actual situation. The differences of their framework of DST and D numbers are shown in Figure 1 [48]. In (7), D numbers theory is acceptable for incomplete information since ∑
B
⊂
U
‍
D
(
B
)
≤
1 which is more close to the real situation.Figure 1
The framework of DST andD numbers theory.Definition 4.
For a discrete setΩ
=
(
b
1
,
b
2
,
b
3
,
…
,
b
n
), where b
i belongs to R and b
i
≠
b
j if i
≠
j, for any v
i
≥
0 and ∑
i
=
1
n
‍
v
i
≤
1, a special form of D numbers can be expressed by(8)
D
b
1
=
v
1
D
b
2
=
v
2
D
b
3
=
v
3
⋮
D
b
n
=
v
nor be represented simply as (9)
D
=
b
1
,
v
1
,
b
2
,
v
2
,
b
3
,
v
3
,
…
,
b
n
,
v
n
.Definition 5 (D numbers combination rule).
LetD
1 and D
2 be two D numbers:(10)
D
1
=
b
1
1
,
v
1
1
,
b
2
1
,
v
2
1
,
b
3
1
,
v
3
1
,
…
,
b
n
1
,
v
n
1
D
2
=
b
1
2
,
v
1
2
,
b
2
2
,
v
2
2
,
b
3
2
,
v
3
2
,
…
,
b
n
2
,
v
n
2
.The combination of D
1 and D
2 denoted by D
=
D
1
⊕
D
2 is defined as follows:(11)
D
b
=
v
(12)
b
=
b
i
1
+
b
j
2
2
(13)
v
=
v
i
1
+
v
j
2
2
×
c
(14)
C
=
∑
j
=
1
m
∑
i
=
1
n
v
i
1
+
v
j
2
2
∑
i
=
1
n
v
i
1
=
1
,
∑
i
=
1
m
v
j
1
=
1
;
∑
j
=
1
m
∑
i
=
1
n
v
i
1
+
v
j
2
2
+
∑
j
=
1
m
v
c
1
+
v
j
2
2
∑
i
=
1
n
v
i
1
<
1
,
∑
i
=
1
m
v
j
1
=
1
;
∑
j
=
1
m
∑
i
=
1
n
v
i
1
+
v
j
2
2
+
∑
i
=
1
n
v
i
1
+
v
c
2
2
∑
i
=
1
n
v
i
1
=
1
,
∑
i
=
1
m
v
j
1
<
1
;
∑
j
=
1
m
∑
i
=
1
n
v
i
1
+
v
j
2
2
+
∑
j
=
1
m
v
c
1
+
v
j
2
2
+
∑
i
=
1
n
v
i
1
+
v
c
2
2
∑
i
=
1
n
v
i
1
<
1
,
∑
i
=
1
m
v
j
1
<
1
,where v
c
1
=
1
-
∑
i
=
1
n
‍
v
i
1 and v
c
2
=
1
-
∑
j
=
1
m
‍
v
j
2, m and n are the assessment numbers in each D number, and the superscripts in above equations are not the exponent but the order of the D numbers.Definition 6 (D numbers’ integration).
For givenD numbers, the overall assessments can be calculated as follows:(15)
I
D
=
∑
i
=
1
n
b
i
v
i
.


## 3. Proposed Method

### 3.1. Problem of ExistingD Numbers Combination Rule

It has to be pointed out that the associative property is not satisfied in the previousD numbers combination rule; that is to say that the sequence of multiple D numbers has great effects on the final results when they get combined. As can be seen in the previous D numbers combination rule, b
=
(
b
i
1
+
b
j
2
)
/
2 and v
=
(
v
i
1
+
v
j
2
)
/
2
×
c; when three D numbers D
1, D
2, and D
3 get combined, the b and v of the combined results should be (16)
b
=
b
1
+
b
2
/
2
+
b
3
2
v
=
v
1
+
v
2
/
2
c
1
+
v
3
2
c
2which means that the third D number D
3 has more effect on the final results. The associative property is not satisfied in the rule of combining multiple D numbers. Meanwhile, the calculated quantity may increase by multiplication with the evaluation grades increasing in D numbers theory.Therefore, a method, with which to solve the EIA, is proposed [48]. In that method, an order variable for multiple D numbers combination is given. As each D number is given by a knowledgeable expert from different cultural or educational backgrounds, so all of them will be evaluated in different weights in the decision-making system. The higher the weight is, the more credible the expert should be. For example, three D numbers shown below, ω
1, ω
2, and ω
3, are the weights of the D numbers separately:(17)
D
1
=
0,0.6
,
1,0.4
,
ω
1
=
0.3
D
2
=
0,0.5
,
1,0.5
,
ω
2
=
0.5
D
3
=
0,0.2
,
1,0.8
,
ω
3
=
0.2
.Since ω
3
<
ω
1
<
ω
2, the combination sequence is (
D
3
⊕
D
1
)
⊕
D
2. If experts’ weights are set to be equal, all possible combination results need to be calculated and the highest value of D numbers integration is the best combination result. However, it is so hard to decide the weight of every decision-maker and deciding the weight will always involve human subjective judgements. What is more, when the weights are set to be equal, all possible combination results will have enormous computational complexity.


### 3.2. Unconfident-Confident Combination Rule ofD Numbers

In this section, a new combination sequence forD numbers theory is proposed. The proposed combination rule includes two independent parts, which are “unconfident D numbers combination rule” and “confident D numbers combination rule,” respectively. For given D number D
i
=
{
(
b
i
,
v
i
)
} (i
=
1,2
,
…
,
n), b
i is the assessment grade the decision-makers made on the decision-making problems and v
i is the confident value to the assessment grade b
i. The value of v
i being more close to 1 means that decision-maker is more confident about the assessment grade. Therefore, the proposed method is given as follows.Definition 7 (unconfidentD numbers combination rule).
For givenD numbers, if they are different from each other, the maximum value of v
i should be calculated firstly. Suppose D
1
,
D
2
,
…
,
D
n are n
D numbers:(18)
D
1
=
b
1
1
,
v
1
1
,
b
2
1
,
v
2
1
,
b
3
1
,
v
3
1
,
…
,
b
n
1
,
v
n
1
D
2
=
b
1
2
,
v
1
2
,
b
2
2
,
v
2
2
,
b
3
2
,
v
3
2
,
…
,
b
n
2
,
v
n
2
⋮
D
n
=
b
1
n
,
v
1
n
,
b
2
n
,
v
2
n
,
b
3
n
,
v
3
n
,
…
,
b
n
n
,
v
n
n
,where(19)
V
1
m
a
x
=
max
⁡
v
1
1
,
v
2
1
,
v
3
1
,
…
,
v
n
1
V
2
m
a
x
=
max
⁡
v
1
2
,
v
2
2
,
v
3
2
,
…
,
v
n
2
⋮
V
n
m
a
x
=
max
⁡
v
1
n
,
v
2
n
,
v
3
n
,
…
,
v
n
n
.Then the combination operation of multiple D numbers is a mapping f
D, such that(20)
f
D
D
1
,
D
2
,
D
3
,
…
,
D
n
=
⋯
D
i
⊕
D
j
⊕
D
k
⋯
,where V
i
m
a
x
>
V
j
m
a
x
>
V
k
m
a
x in unconfident D numbers combination rule and V
i
m
a
x, V
j
m
a
x, and V
k
m
a
x are corresponding to D
i, D
j, and D
k.In the unconfidentD numbers combination rule, if some assessments are completely the same, then these assessments should be combined at the first step. Meanwhile, the combinatorial results should be the same to each of the D numbers since the same assessments indicate that all the experts have the same opinions on the object. For example, D
i
(
i
=
1,2
,
3
,
…
,
n
) are completely the same.(21)
D
1
=
b
1
1
,
v
1
1
,
b
2
1
,
v
2
1
,
…
,
b
i
1
,
v
i
1
,
…
,
b
n
1
,
v
n
1
,
D
2
=
b
1
2
,
v
1
2
,
b
2
2
,
v
2
2
,
…
,
b
j
2
,
v
j
2
,
…
,
b
n
2
,
v
n
2
,
⋮
D
n
=
b
1
n
,
v
1
n
,
b
2
n
,
v
2
n
,
…
,
b
j
n
,
v
j
n
,
…
,
b
n
n
,
v
n
n
,where b
j
i
(
i
=
1,2
,
…
,
n
) are of the same value and v
j
i
(
i
=
1,2
,
…
,
n
) are the same confident value as their assessment correspondingly separately. When the n
D numbers get combined, the final result D should be the same as each of them; that is to say,(22)
D
=
D
1
⊕
D
2
⊕
⋯
⊕
D
n
=
D
1
=
D
2
=
⋯
=
D
n
.In (19), if the maximum V
i are of the same value, the better average assessment grades will be combined ahead of the lower average evaluated grades. That is to say, the order of combination is according to the value of average b
i from largest to smallest. The higher average assessment means evaluating it more positively and the lower average assessment means evaluating it more negatively.In order to illustrate the law of combination ofD numbers, for example, the assessment on one project is conducted. D
1, D
2, D
3, D
4, and D
5 are five D numbers given by five experts from different fields:(23)
D
1
=
0,0.3
,
1,0.5
,
V
1
m
a
x
=
0.5
D
2
=
1,0.5
,
2,0.4
,
V
2
m
a
x
=
0.5
D
3
=
0,0.6
,
1,0.4
,
V
3
m
a
x
=
0.6
D
4
=
1,0.8
,
2,0.2
,
V
4
m
a
x
=
0.8
D
5
=
1,0.8
,
2,0.2
,
V
5
m
a
x
=
0.8
.AsD
4 and D
5 are completely the same assessments, we have D
45
=
D
4
⊕
D
5
=
D
4
=
D
5. Then the combined result will be combined with the left D numbers D
1, D
2, and D
3, as V
3
m
a
x is the biggest value of the three D numbers. So D
45 will combine with D
3 at the second step. As V
1
m
a
x and V
2
m
a
x are of the same value, the better average value of b will be chosen firstly. In D
1, the value of average b is 0.5. In D
2, the value of average b is 1.5. Therefore, D
2 is combined at the third step. The final combined result should be (24)
D
=
D
4
⊕
D
5
⊕
D
3
⊕
D
2
⊕
D
1
.As the value ofv shows the confident degree to the assessments, according to (13) and (14), the smaller the value of V
i is, the bigger the weight of the combination of V
i will be. The order of combination is from maximum value V to minimum value V. Thus, it is called “unconfident D numbers combination rule.”Meanwhile, anotherD numbers combination rule called “confident D numbers combination rule” is used accompanying “unconfident D numbers combination rule.” In confident D numbers combination rule, the first step is the same as the unconfident method and all the same assessments should be combined with the same results as each of the D numbers.Definition 8 (confidentD numbers combination rule).
In (19), the lower value of “V” will be chosen firstly; that is to say, in confident combination rule,(25)
f
D
D
1
,
D
2
,
D
3
,
…
,
D
n
=
⋯
D
i
⊕
D
j
⊕
D
k
⋯
,where V
i
m
a
x
<
V
j
m
a
x
<
V
k
m
a
x and V
i
m
a
x and V
j
m
a
x and V
k
m
a
x are corresponding to D
i, D
j, and D
k.The confidentD numbers combination rule is contrary to the unconfident combination rule. Then when minimum values are of the same value, the lower average assessment grade will be combined ahead of the better average evaluated grade.


## 3.1. Problem of ExistingD Numbers Combination Rule

It has to be pointed out that the associative property is not satisfied in the previousD numbers combination rule; that is to say that the sequence of multiple D numbers has great effects on the final results when they get combined. As can be seen in the previous D numbers combination rule, b
=
(
b
i
1
+
b
j
2
)
/
2 and v
=
(
v
i
1
+
v
j
2
)
/
2
×
c; when three D numbers D
1, D
2, and D
3 get combined, the b and v of the combined results should be (16)
b
=
b
1
+
b
2
/
2
+
b
3
2
v
=
v
1
+
v
2
/
2
c
1
+
v
3
2
c
2which means that the third D number D
3 has more effect on the final results. The associative property is not satisfied in the rule of combining multiple D numbers. Meanwhile, the calculated quantity may increase by multiplication with the evaluation grades increasing in D numbers theory.Therefore, a method, with which to solve the EIA, is proposed [48]. In that method, an order variable for multiple D numbers combination is given. As each D number is given by a knowledgeable expert from different cultural or educational backgrounds, so all of them will be evaluated in different weights in the decision-making system. The higher the weight is, the more credible the expert should be. For example, three D numbers shown below, ω
1, ω
2, and ω
3, are the weights of the D numbers separately:(17)
D
1
=
0,0.6
,
1,0.4
,
ω
1
=
0.3
D
2
=
0,0.5
,
1,0.5
,
ω
2
=
0.5
D
3
=
0,0.2
,
1,0.8
,
ω
3
=
0.2
.Since ω
3
<
ω
1
<
ω
2, the combination sequence is (
D
3
⊕
D
1
)
⊕
D
2. If experts’ weights are set to be equal, all possible combination results need to be calculated and the highest value of D numbers integration is the best combination result. However, it is so hard to decide the weight of every decision-maker and deciding the weight will always involve human subjective judgements. What is more, when the weights are set to be equal, all possible combination results will have enormous computational complexity.


## 3.2. Unconfident-Confident Combination Rule ofD Numbers

In this section, a new combination sequence forD numbers theory is proposed. The proposed combination rule includes two independent parts, which are “unconfident D numbers combination rule” and “confident D numbers combination rule,” respectively. For given D number D
i
=
{
(
b
i
,
v
i
)
} (i
=
1,2
,
…
,
n), b
i is the assessment grade the decision-makers made on the decision-making problems and v
i is the confident value to the assessment grade b
i. The value of v
i being more close to 1 means that decision-maker is more confident about the assessment grade. Therefore, the proposed method is given as follows.Definition 7 (unconfidentD numbers combination rule).
For givenD numbers, if they are different from each other, the maximum value of v
i should be calculated firstly. Suppose D
1
,
D
2
,
…
,
D
n are n
D numbers:(18)
D
1
=
b
1
1
,
v
1
1
,
b
2
1
,
v
2
1
,
b
3
1
,
v
3
1
,
…
,
b
n
1
,
v
n
1
D
2
=
b
1
2
,
v
1
2
,
b
2
2
,
v
2
2
,
b
3
2
,
v
3
2
,
…
,
b
n
2
,
v
n
2
⋮
D
n
=
b
1
n
,
v
1
n
,
b
2
n
,
v
2
n
,
b
3
n
,
v
3
n
,
…
,
b
n
n
,
v
n
n
,where(19)
V
1
m
a
x
=
max
⁡
v
1
1
,
v
2
1
,
v
3
1
,
…
,
v
n
1
V
2
m
a
x
=
max
⁡
v
1
2
,
v
2
2
,
v
3
2
,
…
,
v
n
2
⋮
V
n
m
a
x
=
max
⁡
v
1
n
,
v
2
n
,
v
3
n
,
…
,
v
n
n
.Then the combination operation of multiple D numbers is a mapping f
D, such that(20)
f
D
D
1
,
D
2
,
D
3
,
…
,
D
n
=
⋯
D
i
⊕
D
j
⊕
D
k
⋯
,where V
i
m
a
x
>
V
j
m
a
x
>
V
k
m
a
x in unconfident D numbers combination rule and V
i
m
a
x, V
j
m
a
x, and V
k
m
a
x are corresponding to D
i, D
j, and D
k.In the unconfidentD numbers combination rule, if some assessments are completely the same, then these assessments should be combined at the first step. Meanwhile, the combinatorial results should be the same to each of the D numbers since the same assessments indicate that all the experts have the same opinions on the object. For example, D
i
(
i
=
1,2
,
3
,
…
,
n
) are completely the same.(21)
D
1
=
b
1
1
,
v
1
1
,
b
2
1
,
v
2
1
,
…
,
b
i
1
,
v
i
1
,
…
,
b
n
1
,
v
n
1
,
D
2
=
b
1
2
,
v
1
2
,
b
2
2
,
v
2
2
,
…
,
b
j
2
,
v
j
2
,
…
,
b
n
2
,
v
n
2
,
⋮
D
n
=
b
1
n
,
v
1
n
,
b
2
n
,
v
2
n
,
…
,
b
j
n
,
v
j
n
,
…
,
b
n
n
,
v
n
n
,where b
j
i
(
i
=
1,2
,
…
,
n
) are of the same value and v
j
i
(
i
=
1,2
,
…
,
n
) are the same confident value as their assessment correspondingly separately. When the n
D numbers get combined, the final result D should be the same as each of them; that is to say,(22)
D
=
D
1
⊕
D
2
⊕
⋯
⊕
D
n
=
D
1
=
D
2
=
⋯
=
D
n
.In (19), if the maximum V
i are of the same value, the better average assessment grades will be combined ahead of the lower average evaluated grades. That is to say, the order of combination is according to the value of average b
i from largest to smallest. The higher average assessment means evaluating it more positively and the lower average assessment means evaluating it more negatively.In order to illustrate the law of combination ofD numbers, for example, the assessment on one project is conducted. D
1, D
2, D
3, D
4, and D
5 are five D numbers given by five experts from different fields:(23)
D
1
=
0,0.3
,
1,0.5
,
V
1
m
a
x
=
0.5
D
2
=
1,0.5
,
2,0.4
,
V
2
m
a
x
=
0.5
D
3
=
0,0.6
,
1,0.4
,
V
3
m
a
x
=
0.6
D
4
=
1,0.8
,
2,0.2
,
V
4
m
a
x
=
0.8
D
5
=
1,0.8
,
2,0.2
,
V
5
m
a
x
=
0.8
.AsD
4 and D
5 are completely the same assessments, we have D
45
=
D
4
⊕
D
5
=
D
4
=
D
5. Then the combined result will be combined with the left D numbers D
1, D
2, and D
3, as V
3
m
a
x is the biggest value of the three D numbers. So D
45 will combine with D
3 at the second step. As V
1
m
a
x and V
2
m
a
x are of the same value, the better average value of b will be chosen firstly. In D
1, the value of average b is 0.5. In D
2, the value of average b is 1.5. Therefore, D
2 is combined at the third step. The final combined result should be (24)
D
=
D
4
⊕
D
5
⊕
D
3
⊕
D
2
⊕
D
1
.As the value ofv shows the confident degree to the assessments, according to (13) and (14), the smaller the value of V
i is, the bigger the weight of the combination of V
i will be. The order of combination is from maximum value V to minimum value V. Thus, it is called “unconfident D numbers combination rule.”Meanwhile, anotherD numbers combination rule called “confident D numbers combination rule” is used accompanying “unconfident D numbers combination rule.” In confident D numbers combination rule, the first step is the same as the unconfident method and all the same assessments should be combined with the same results as each of the D numbers.Definition 8 (confidentD numbers combination rule).
In (19), the lower value of “V” will be chosen firstly; that is to say, in confident combination rule,(25)
f
D
D
1
,
D
2
,
D
3
,
…
,
D
n
=
⋯
D
i
⊕
D
j
⊕
D
k
⋯
,where V
i
m
a
x
<
V
j
m
a
x
<
V
k
m
a
x and V
i
m
a
x and V
j
m
a
x and V
k
m
a
x are corresponding to D
i, D
j, and D
k.The confidentD numbers combination rule is contrary to the unconfident combination rule. Then when minimum values are of the same value, the lower average assessment grade will be combined ahead of the better average evaluated grade.


## 4. Examples and Applications

In this section, the proposed method is adopted to EIA. EIA usually contains four steps. Firstly the hierarchical structure model for assessment needs to be established, the second step is the assessment for each environmental impact factor, the third step is the calculation of all the evaluated factors, and the last step is to rank the entire projects. In an EIA example, the assessment on the impact of four projects for the conservation of the area of Rupa Tal is taken as follows [52, 53].Project  1. Keep it the way it is and do not make changes. The lake is disappearing and a small gorge is formed to control the streams because the present sedimentation is still continuing.Project  2. A high retaining dam is created to raise the overall water level along the southern edge and the in-lake areas created by sedimentation over the last few decades would be overflowed because of the build of retaining dam.Project  3. Between two precipices, a smaller high dam is built. This dam is smaller than that built in project 2 but has similar upstream effects.Project  4. A single large sedimentation reservoir is in the upstream area, or a series of smaller retaining walls which would be used to form a sedimentation cascade. The water area may remain intact by this project.In order to assess these four projects, each factor has some primary subfactors which is shown in Table1 in detail; every subfactor has different influences on the assessment of the projects.Table 1
The meanings of factors and subfactors in EIA in literature [52].
Factor
Subfactor
Physical/chemical (P/C)
P/C1
The impacts of lake water volume
P/C2
The impacts of the lake sedimentation
P/C3
The impacts of crop and grazing areas
Biological/ecological (B/E)
B/E1
The impacts of lake fisheries
B/E2
The impacts of biodiversity
B/E3
The impacts of primary production
B/E4
The impacts of aquatic macrophytes
B/E5
The impacts of disease vector populations
Sociological/cultural (S/C)
S/C1
The loss of housing
S/C2
The loss of shops/public buildings
S/C3
The impacts of accessing routes
S/C4
The impacts induced by changes of tourism patterns
S/C5
The impacts of water supplies
S/C6
The impacts of diet/nutrition
S/C7
The impacts of aesthetic landscapes
S/C8
The impacts of water/vector borne disease
S/C9
The impacts of upstream quality of life
S/C10
The impacts of downstream quality of life
Economic/operational (E/O)
E/O1
The impacts of crop-generated incomes
E/O2
The impacts of fishery generated incomes
E/O3
The convenience of operation and maintenance of option
E/O4
The cost of operation and maintenance of option
E/O5
The cost of resettlement/compensation for land loss
E/O6
The cost of rehabilitation and restoration of shops
E/O7
The cost of restoration of accessing routes
E/O8
The impacts of tourism-generated incomesSecond the calculation of the assessment should be done. Nevertheless most of the assessments are represented by linguistic grades like “good” and “poor” and “A,” “B,” and “C,” and so on. First of all, translating such a kind of assessment into numerical grade is necessary. In the existing world, a seven-point scale and five grades are presented [54]. In this method, 3, 2, 1, 0, −1, −2, and −3 represent “very good” to “moderate” to “very bad”. The original grades are represented by the letters “A,” “B,” “C,” and so on [52]. In [48], the grade is translated into numerical and shown in Table 2.Table 2
An assessment standard for EIA.
Assessment grade
Numerical rating
Description
E
5
Major positive impact
D
4
Signification positive impact
C
3
Moderately positive impact
B
2
Positive impact
A
1
Slight impact
N
0
No impact
-
A
-
1
Slightly negative impact
-
B
-
2
Negative impact
-
C
-
3
Moderately negative impact
-
D
-
4
Significant negative impact
-
E
-
5
Major negative impactFrom Table2, the assessment E means major positive impacts and the numerical number is 5. The assessment N means no impact; we translate it into 0. Then the D numbers are obtained from the assessment of experts. For example, when ten experts give the assessments for the conservation of Rupa Tal, six experts believe it is major positive impacts and other four evaluate it to be moderately positive impact; then D numbers should be {
5,0.6
,
(
3,0.4
)
}. If five experts assess it to be positive impact while four experts evaluate it to be no impact, the remaining expert does not give any evaluation because of lacking information; the D numbers can be {
(
3,0.5
)
,
(
0,0.4
)
}; this kind of information is incomplete. The assessment matrix for project 1 and project 2 and project 3 and project 4 are shown in Tables 3 and 4, respectively.Table 3
Assessment matrix of environment impact factors for projects 1 and 2 [52].
Environmental factors
Project 1
Project 2
Physical/chemical
P/C1
{
(
-
4,0.3
)
,
(
-
3,0.7
)
}
{
(
1,0.1
)
,
(
2,0.9
)
}
P/C2
{
(
-
2,0.8
)
,
(
-
1,0.2
)
}
{
(
1,0.1
)
,
(
2,0.85
)
}
P/C3
{
2,0.45
)
,
(
3,0.35
)
}
{
(
-
3,0.2
)
,
(
-
2,0.8
)
}
Biological/ecological
B/E1
{
(
-
3,0.5
)
,
(
-
2,0.4
)
}
B/E2
{
-
2,0.5
,
(
-
1,0.5
)
}
{
(
-
1,1.0
)
}
B/E3
{
(
-
2,1.0
)
}
{
(
-
2,1.0
)
}
B/E4
{
(
-
2,1.0
)
}
{
(
-
2,1.0
)
}
B/E5
{
(
1,1.0
)
}
Sociological/culture S/C
S/C1
{
(
0,1.0
)
}
{
(
-
1,1.0
)
}
S/C2
{
(
0,1.0
)
}
{
(
-
1,0.65
)
,
(
0,0.03
)
}
S/C3
{
(
1,0.5
)
,
(
2,0.5
)
}
{
(
-
1,1.0
)
}
S/C4
{
(
-
2,0.2
)
,
(
1,0.8
)
}
{
(
2,0.8
)
,
(
3,0.2
)
}
S/C5
{
(
-
2,0.3
)
,
(
1,0.7
)
}
{
(
3,1.0
)
}
S/C6
{
(
0,1.0
)
}
{
(
1,0.8
)
,
(
2,0.2
)
}
S/C7
{
(
-
2,1.0
)
}
{
(
2,1.0
)
}
S/C8
{
(
1,0.5
)
,
(
2,0.3
)
}
{
(
-
1,1.0
)
}
S/C9
{
(
0,1.0
)
}
{
(
1,1.0
)
}
S/C10
{
(
-
1,1.0
)
}
{
(
2,1.0
)
}
Economical/operational
E/O1
{
(
2,0.8
)
}
{
(
-
1,1.0
)
}
E/O2
{
(
-
2,1.0
)
}
{
(
2,1.0
)
}
E/O3
{
(
0,1.0
)
}
{
(
-
1,1.0
)
}
E/O4
{
(
0,1.0
)
}
{
(
-
1,1.0
)
}
E/O5
{
(
0,1.0
)
}
{
(
-
1,1.0
)
}
E/O6
{
(
0,1.0
)
}
{
(
-
1,1.0
)
}
E/O7
{
(
0,1.0
)
}
{
(
-
1,1.0
)
}
E/O8
{
(
-
1,1.0
)
}
{
(
3,0.7
)
}Table 4
Assessment matrix of environmental impact factors for projects 3 and 4 [52].
Environmental factors
Project 3
Project 4
Physical/chemical
P/C1
{
(
2,0.8
)
,
(
3,0.2
)
}
{
(
0,1
)
}
P/C2
{
(
-
1,0.85
)
,
(
3,0.15
)
}
{
1,0.3
,
(
2,0.7
)
}
P/C3
{
(
0,0.5
)
,
(
1,0.5
)
}
{
(
0,1
)
}
Biological/ecological
B/E1
{
(
0,0.2
)
,
(
1,0.8
)
}
{
(
0,1
)
}
B/E2
{
(
-
2,0.8
)
,
(
-
1,0.1
)
}
{
(
0,1
)
}
B/E3
{
(
-
1,1.0
)
}
{
(
0,1
)
}
B/E4
{
(
-
1,1.0
)
}
{
(
0,1
)
}
B/E5
{
(
1,1.0
)
}
{
(
-
1,0.4
)
,
(
0,0.5
)
}
Sociological/culture S/C
S/C1
{
(
-
1,1.0
)
}
{
(
0,1.0
)
}
S/C2
{
(
-
1,1.0
)
}
{
(
0,1.0
)
}
S/C3
{
(
0,1.0
)
}
S/C4
{
(
1,1.0
)
}
{
(
0,1.0
)
}
S/C5
{
(
1,1.0
)
}
{
(
1,0.8
)
}
S/C6
{
(
1,0.5
)
,
(
2,0.5
)
}
{
(
0,1.0
)
}
S/C7
{
(
-
1,0.85
)
,
(
3,0.15
)
}
{
(
0,1.0
)
}
S/C8
{
(
1,1.0
)
}
{
(
-
1,1.0
)
}
S/C9
{
(
0,0.2
)
,
(
1,0.7
)
}
{
(
-
1,1.0
)
}
S/C10
{
(
-
2,0.8
)
,
(
-
1,0.2
)
}
{
(
0,1.0
)
}
Economical/operational
E/O1
{
(
-
2,0.9
)
}
{
(
0,1.0
)
}
E/O2
{
(
0,1.0
)
}
{
(
0,1.0
)
}
E/O3
{
(
-
1,1.0
)
}
{
(
-
2,1.0
)
}
E/O4
{
(
-
1,1.0
)
}
{
(
-
2,1.0
)
}
E/O5
{
(
-
1,1.0
)
}
{
(
0,1.0
)
}
E/O6
{
(
-
1,1.0
)
}
{
(
0,1.0
)
}
E/O7
{
(
-
1,1.0
)
}
{
(
0,1.0
)
}
E/O8
{
(
1,1.0
)
}
{
(
0,1.0
)
}The overall assessment for different projects is calculated via unconfident and confidentD numbers combination rule, respectively. For example, in unconfident D numbers combination rule, for the evaluation of project 4, the environmental factors are biological and ecological. To the subfactors B
/
E
1, B
/
E
2, B
/
E
3, and B
/
E
4, all the assessments are {
(
0,1
)
}; to subfactor B
/
E
5, the assessment is {
(
-
1,0.4
)
,
(
0,0.5
)
}. Firstly, the same assessment should be combined: (26)
0,1
⊕
0,1
⊕
0,1
⊕
0,1
⊕
0,1
=
0,1
.Secondly, the combined result {
(
0,1
)
} needs to be fused with B
/
E
5: (27)
-
1,0.4
,
0,0.5
⊕
0,1
=
0.5
,
1.4
4
,
0
,
1.5
4
.Then, all assessments are combined by same process.Lastly, by (15), the last score can be calculated and the example above is taken into consideration: (28)
I
D
=
0.5
×
1.4
4
+
0
×
1.5
4
=
0.175
.The final results and ranking are obtained and shown in Table5 by unconfident and confident D numbers combination rule.Table 5
Overall environmental impacts and ranking of each project.
Assessment grade
Impact rating
Ranking
Unconfident method
Project 1
−0.18720
4
Project 2
0.09550
1
Project 3
0.04909
2
Project 4
−0.03760
3
Confident method
Project 1
−0.3453
4
Project 2
−0.0444
2
Project 3
0.2984
1
Project 4
−0.29727
3From Table5, the final ranking is project 2 > project 3 > project 4 > project 1 by using unconfident D numbers combination rule. According to confident D numbers combination rule, the ranking is project 3 > project 2 > project 4 > project 1. The results by the evidential reasoning approach (shortly ER approach) [52] and previous D numbers combination rule (shortly previous D method) [48] are shown in Table 6. From Tables 5 and 6, our results of unconfident D numbers combination rule are the same as risk-taking method [52]. The results of confident D numbers combination rule are the same as decision-optimistic method in [48]. Meanwhile, project 1 is always the worst choice for all methods. In ER approach, the best choice is project 2 or project 4. In previous D method, the best choice is project 3 or project 4. In our method, the best choice is project 2 and project 3; there is the same option for these researches. Furthermore, the unconfident-confident combination rule of D numbers is only determined by the original data of D numbers, any other information about D numbers is no longer needed.Table 6
Overall environmental impacts and ranking of each project.
Method
Ranking
ER approach
Risk-neutral
Project 2> project 4 > project 3 > project 1
Risk-taking
Project 2> project 3 > project 4 > project 1
Risk-average
Project 4> project 2 > project 3 > project 1
D method
Decision-optimistic
Project 3> project 2 > project 4 > project 1
Decision-pessimistic
Project 4> project 2 > project 3 > project 1


## 5. Conclusions

How to deal with uncertain and incomplete information to make decisions is an open issue.D numbers theory, which is an extension of DST, has the ability to combine multiple evidence and is wildly used to deal with uncertain and incomplete information problems. However, the associative property for multiple D numbers combination is not satisfied. In this paper, A modified method for multiple D numbers combination denoted as unconfident-confident combination rule is proposed. In our method, the combination rule only depends on the values of D numbers themselves. The proposed method is applied to EIA and the numerical results indicate the effectiveness of the proposed method.In the future, more work should be done for multipleD numbers combination. D numbers theory is regarded as the generation of DST; many mathematical theorems including the associative property are satisfied in DST. It is reasonable for us to believe that the associative property should be satisfied in D numbers theory. More attempts will be made to find out the solution in which many mathematical theorems are satisfied in the multiple D numbers combination rule. Meanwhile, D numbers theory should be put into applications in more fields to deal with uncertainty and incompleteness, like risk evaluation and so on.



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*Source: 3596517-2016-11-29.xml*