A Modified Combination Rule for D Numbers Theory

D numbers theory is an appropriate method to deal with the information of uncertainty and incompleteness when making a reasonable decision. PreviousD numbers theory provides a rule to combinemultipleD 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 proposedmethod defines a new function for multipleD numbers combination which is mainly determined by the original value ofD numbers.Then the proposed combination rule is applied to environmental impact assessment (EIA); our results show that the proposed method is efficient for multipleD numbers combination and it is useful when dealing with uncertainty and incompleteness.

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][35][36][37][38][39][40][41][42][43][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,  numbers theory is proposed by Deng [47,48].It removes these hypotheses reasonably; the elements in the framework of  numbers theory do not need to be mutually exclusive and incomplete assessments can also exist in  numbers theory.Because  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  numbers' combination rule.In [48], Deng et al. do some work for multiple  numbers combination in special circumstances.However, the associative property is not addressed in a general condition.In this paper, a modified method for multiple  numbers combination is proposed.

Mathematical Problems in Engineering
The remainder of this paper is organized as follows.In Section 2, preliminaries about DST and  numbers theory are described in detail.The problem of the previous  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.

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 that  is a set of mutually exclusive and collectively exhaustive elements which can be represented as follows: The power set of  is denoted as 2  ; any element belongs to the power set 2  is said to be a proposition.For a frame of discernment , a mass function is a mapping, which is denoted as follows: in which the following conditions are satisfied: where 0 is an empty set and  is a subset of 2  ; the function () represents how strongly the evidence supports .
Definition 2 (Dempster's rule of combination).Given two BPAs  1 and  2 , Dempster's rule of combination donated as  =  1 ⊕  2 is defined as follows: with where , , and  are the elements of 2  and  is a normalization constant which means the conflict coefficient of two BPAs.
Note that Dempster's rule of combination is feasible only when  < 1 because  = 1 means that the two BPAs are one hundred percent conflicted.Associative property is well satisfied in Dempster's rule of combination.

𝐷 Numbers Theory.
There are some strong hypotheses in DST which have limited its wide usage in some fields especially in linguistic assessments. numbers theory is proposed in [47,48] and it has overcome these hypotheses.The details about  numbers theory are introduced as follows.
Definition 3. Let Ω be a finite nonempty set;  numbers is a mapping: where the following conditions are satisfied: where 0 is an empty set and  is a subset of Ω.The elements in the set Ω of  numbers do not require mutual exclusiveness and the sum of the assessments can be less than 1 in  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 and  numbers theory providing the framework with nonexclusive hypotheses is more tallying with the actual situation.The differences of their framework of DST and  numbers are shown in Figure 1 [48].In (7),  numbers theory is acceptable for incomplete information since ∑ ⊂ () ≤ 1 which is more close to the real situation.Definition 4. For a discrete set Ω = ( 1 ,  2 ,  3 , . . .,   ), where   belongs to  and   ̸ =   if  ̸ = , for any V  ≥ 0 and ∑  =1 V  ≤ 1, a special form of  numbers can be expressed by or be represented simply as Definition 5 ( numbers combination rule).Let  1 and  2 be two  numbers: The combination of  1 and  2 denoted by  =  1 ⊕  2 is defined as follows: where ,  and  are the assessment numbers in each  number, and the superscripts in above equations are not the exponent but the order of the  numbers.Definition 6 ( numbers' integration).For given  numbers, the overall assessments can be calculated as follows:

Proposed Method
which means that the third  number  3 has more effect on the final results.The associative property is not satisfied in the rule of combining multiple  numbers.Meanwhile, the calculated quantity may increase by multiplication with the evaluation grades increasing in  numbers theory.Therefore, a method, with which to solve the EIA, is proposed [48].In that method, an order variable for multiple  numbers combination is given.As each  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  numbers shown below,  1 ,  2 , and  3 , are the weights of the  numbers separately: Since  3 <  1 <  2 , the combination sequence is ( 3 ⊕  1 ) ⊕  2 .If experts' weights are set to be equal, all possible combination results need to be calculated and the highest value of  numbers integration is the best combination result.However, it is so hard to decide the weight of every decisionmaker 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.

Unconfident-Confident Combination Rule of 𝐷 Numbers.
In this section, a new combination sequence for  numbers theory is proposed.The proposed combination rule includes two independent parts, which are "unconfident  numbers combination rule" and "confident  numbers combination rule," respectively.For given  number   = {(  , V  )} ( = 1, 2, . . ., ),   is the assessment grade the decisionmakers made on the decision-making problems and V  is the confident value to the assessment grade   .The value of V  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 (unconfident  numbers combination rule).For given  numbers, if they are different from each other, the maximum value of V  should be calculated firstly.Suppose  1 ,  2 , . . .,   are   numbers: where Then the combination operation of multiple  numbers is a mapping   , such that where  max >  max >  max in unconfident  numbers combination rule and  max ,  max , and  max are corresponding to   ,   , and   .
In the unconfident  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  numbers since the same assessments indicate that all the experts have the same opinions on the object.For example,   ( = 1, 2, 3, . . ., ) are completely the same.
where    ( = 1, 2, . . ., ) are of the same value and V   ( = 1, 2, . . ., ) are the same confident value as their assessment correspondingly separately.When the   numbers get combined, the final result  should be the same as each of them; that is to say, In (19), if the maximum   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   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 of  numbers, for example, the assessment on one project is conducted. 1 ,  2 ,  3 ,  4 , and  5 are five  numbers given by five experts from different fields:

(23)
As  4 and  5 are completely the same assessments, we have  45 =  4 ⊕  5 =  4 =  5 .Then the combined result will be combined with the left  numbers  1 ,  2 , and  3 , as  3max is the biggest value of the three  numbers.So  45 will combine with  3 at the second step.As  1max and  2max are of the same value, the better average value of  will be chosen firstly.In  1 , the value of average  is 0.5.In  2 , the value of average  is 1.5.Therefore,  2 is combined at the third step.The final combined result should be As the value of V shows the confident degree to the assessments, according to ( 13) and ( 14), the smaller the value of   is, the bigger the weight of the combination of   will be.The order of combination is from maximum value  to minimum value .Thus, it is called "unconfident  numbers combination rule." Meanwhile, another  numbers combination rule called "confident  numbers combination rule" is used accompanying "unconfident  numbers combination rule."In confident  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  numbers.
Definition 8 (confident  numbers combination rule).In (19), the lower value of "" will be chosen firstly; that is to say, in confident combination rule, where  max <  max <  max and  max and  max and  max are corresponding to   ,   , and   .
The confident  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.

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 Table 1 in detail; every subfactor has different influences on the assessment of the projects.
Second 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.
From Table 2, the assessment  means major positive impacts and the numerical number is 5.The assessment  means no impact; we translate it into 0. Then the  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  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  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.
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: The final results and ranking are obtained and shown in Table 5 by unconfident and confident  numbers combination rule.
From Table 5, the final ranking is project 2 > project 3 > project 4 > project 1 by using unconfident  numbers combination rule.According to confident  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  numbers combination rule (shortly previous  method) [48] are shown in Table 6.From Tables 5 and 6, our results of unconfident  numbers combination rule are the same as risk-taking method [52].The results of confident  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  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  numbers is only determined by the original data of  numbers, any other information about  numbers is no longer needed.

Conclusions
How to deal with uncertain and incomplete information to make decisions is an open issue. 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  numbers combination is not satisfied.In this paper, A modified method for multiple  numbers combination denoted as unconfident-confident combination rule is proposed.In our method, the combination rule only depends on the values of  numbers themselves.The proposed method is applied to EIA and the numerical results indicate the effectiveness of the proposed method.theorems are satisfied in the multiple  numbers combination rule.Meanwhile,  numbers theory should be put into applications in more fields to deal with uncertainty and incompleteness, like risk evaluation and so on.

Figure 1 :
Figure 1: The framework of DST and  numbers theory.

Table 2 :
An assessment standard for EIA.