A MODIFIED D NUMBERS METHODOLOGY FOR ENVIRONMENTAL IMPACT ASSESSMENT

Environmental impact assessment (EIA) is usually evaluated by many factors influenced by various kinds of uncertainty or fuzziness. As a result, the key issues of EIA problem are to represent and deal with the uncertain or fuzzy information. D numbers theory, as the extension of Dempster-Shafer theory of evidence, is a desirable tool that can express uncertainty and fuzziness, both complete and incomplete, quantitative or qualitative. However, some shortcomings do exist in D numbers combination process, the commutative property is not well considered when multiple D numbers are combined. Though some attempts have made to solve this problem, the previous method is not appropriate and convenience as more information about the given evaluations represented by D numbers are needed. In this paper, a data-driven D numbers combination rule is proposed, commutative property is well considered in the proposed method. In the combination process, there does not require any new information except the original D numbers. An illustrative example is provided to demonstrate the effectiveness of the method.


Introduction
In the 21th century, environment problems are the hot issues that draw many countries' attention. EIA problem is an identification of many factors involved in both harmful and beneficial about the project should be put into effect or which plan should be chosen in the decision making system. Some previous studies show the EIA framework has been developed in many fields (De Boer 2003;Lenzen et al. 2003;Dreyer et al. 2006;Wang et al. 2006;Kiliç et al. 2011;Zapp et al. 2012;Sueyoshi, Goto 2012;Bigum et al. 2012;Xu et al. 2014;Jordan et al. 2014;Zavadskas et al. 2015) and many methods are applied in EIA such as Life cycle assessment (Tukker 2000;Riga et al. 2015), group decision-making methods (Rikhtegar et al. 2014) and so on (Miao et al. 2014;Ma et al. 2014;Ni et al. 2014). In the real world, many potential environment assessment factors cannot be quantified accurately, they are qualitative or linguistic forms which lead to uncertainty, fuzziness and incompleteness. So one key problem in the EIA is to handle various kinds of forms of uncertainty. Up to now, many methods have been used to deal with uncertainty, such as fuzzy set theory (Zadeh 1965;Wood et al. 2007;Zavadskas et al. 2014;Ju, Yoo 2014;Jiang et al. 2015a), rough set (Pawlak, Skowron 2007;Hu, Lu 2009;Morón et al. 2009), uncertain theory (Liu 2014;Ahmadi et al. 2015;Deng 2015b), Dempster-Shafer theory of evidence (Dempster 1967;Shafer 1976;Deng 2015a;Jiang et al. 2015b), D numbers theory (Deng 2012;Deng et al. 2014c) and so on (Zolfani et al. 2013;Rabbani et al. 2014;Kaplinski et al. 2014;Baleženti, T., Baleženti, A. 2014;Akhavan et al. 2015;Su et al. 2015;Deng et al. 2015a;Mardani et al. 2015).
Literature (Wang et al. 2006) applied Dempster-Shafer theory of evidence to EIA to handle various kinds of uncertainty caused by the subjective judgments, the lacking of information and the incapability of experts to give the accurate assessments. Dempster-Shafer theory of evidence is also called the theory of evidence or evidence theory, it can represent uncertainty directly and the Dempster's rule of combination can combine multiple pieces of evidence into one. Because of such advantages, evidence theory has been used in many fields especially in data fusion (Tian et al. 2005;Deng et al. 2011aDeng et al. , 2011b, water assessment (Sadiq et al. 2006;Sueyoshi, Goto 2012), decision making problems (Beynon et al. 2001;Yang, Xu 2002;Taroun, Yang 2011;Fu, Yang 2012;Fu, Chin 2014) and so on (Wei et al. 2013;Liu et al. 2014b;Yager, Alajlan 2015;Deng et al. 2015b). However, in the classical evidence theory, there are some limitations which may hinder its further applications. For example, the concept basic probability assignment (BPA) which is used to represent the uncertainty must be independent, the sum of all the BPA must be equal to 1 and the frame of discernment must be mutually exclusive and collectively exhaustive. These conditions are usually hard to be satisfied in our real lives and these shortcomings have limited the application of evidence theory to a certain degree (Deng 2012;Deng et al. 2014b).
Recently, a new methodology called D numbers theory, which can deal with both the exclusiveness hypothesis and completeness constraint, has been proposed (Deng 2012;Deng et al. 2014c). This theory extends the classical Dempster-Shafer theory. D numbers theory can represent uncertain information effectively and the exclusive property does not need to be satisfied in the frame of discernment, at the same time, completeness constraint is released which means that the overall assessments does not need to be equal to 1. In the real word, most of the assessment data appears to be incomplete and hardly to be mutually exclusive, these two improvements in the methodology can be greatly beneficial. With these two advantages, D numbers theory has been used in bridge assessment (Deng et al. 2014a), EIA (Deng et al. 2014b), Supplier selection (Deng et al. 2014c) and so on (Deng et al. 2014d;Liu et al. 2014a;Fan et al. 2016).
D numbers theory has some desirable properties. However, commutative property is not well addressed when multiple D numbers get combined. In (Deng et al. 2014b), order variables have been defined for multiple D numbers combination, but this method is not property and convenient for some new information accompany with D numbers should be given before the combination process. In this paper, a new data-driven method to deal with D numbers fusion problem is proposed. This method is completely based on the evaluation grades and any other new information about the assessments is not required. Meanwhile, the proposed method has made full use of the information contained in the given D numbers in the combination process. In order to overcome that deficiency, in this paper, a new method based on the overall assessment grades to deal with D numbers fusion problem is proposed.
The rest of the paper is organized as follows. In Section 1, some preliminaries are described in detail. The proposed method based on assessment grade to combine the uncertain data is presented in Section 2. In Section 3, the proposed method for EIA based on D numbers theory is developed to show the effectiveness of the method. Some conclusions are given in the last Section.

Dempster-Shafer theory of evidence (Dempster 1967; Shafer 1976)
Many theories have been developed to handle with various kinds of uncertainty with desirable properties. However, there are some drawbacks that cannot be ignored. For example, with the inherent advantage to represent the uncertainty and the ability to combine pieces of information into one final assessment, the Dempster-Shafer theory of evidence has been applied to many fields. In the mathematical framework of evidence theory, the BPA is defined to represent the uncertain information, the problem domain is defined by a frame of discernment which is a finite and mutually exclusive non-empty set, let 2 Q denote the power set of Q and each element in the power set 2 Q is called a proposition which can be used to represent the uncertainty, the BPA is a mapping from 2 Q to [0,1] and the follow conditions are satisfied: Each element in the power set 2 Q is a proposition that can represent uncertainty directly because BPA has the ability to represent the belief degree to the composed subset of the element in Q rather than the individual subset in Q. But any element in Q must be exclusive that can hardly be satisfied in real life, this shortcoming has limited its application widely. For example, some linguistic assessment grades are given as "very good", "good", "medium", "bad" and "very bad" by human subjectivity judgment. These assessment grades are not mutually exclusive as the intersection is not empty. Dempster-Shafer theory of evidence is limited in this situation which means that we cannot give the BPA as m(very good, good) = 0.1, m(medium) = 0.1. Another shortcoming is that the sum of BPAs must be equal to 1, but lots of assessments are not given because of uncertainty or lacking of information due to different background. D numbers theory is an extension of Dempster-Shafer theory of evidence. It has overcome the limitations in classical Dempster-Shafer theory of evidence and appears to handle uncertain information effectively.
With two belief structures m 1 and m 2 , the Dempster's rule of combination denoted as = ⊕ 1 2 m m m , is defined as follows: K is a normalization constant, it shows the conflict coefficient of two BPAs. Dempster-Shafer theory of evidence are only available when K < 1. Dempster's rule of combination is the core of Dempster-Shafer theory of evidence, multiple pieces of BPAs can be combined into one by this rule and commutative and associative properties are satisfied in it.

D numbers theory
Let U be a finite nonempty set, D numbers is a mapping D: →[0, 1] U , satisfying (Deng 2012;Deng et al. 2014c): where ∅ is an empty set and B is a non-empty subset of U. As can be found that the definition of D number is so similar to the definition of BPA in evidence theory. In fact they are different, in D numbers theory, set U is not required to be mutually exclusive, at the same time, the sum of the assessment can be less than 1 in D numbers theory. An example is given to show the differences, supposing a assessment is conducted and the assessment score is in the interval [0, 100], an expert gives his evaluation in the frame of Dempster-Shafer theory as follows: ] the intersection between a 1 , a 2 , and a 3 is the empty set and (a 1 , a 2 , a 3 ) is the frame of discernment in evidence theory, the sum of m i equals to 1 means it is complete. Meanwhile, another expert gives his assessment by D numbers as follows: and b 3 are not mutually exclusive and they are not a frame of discernment. The sum of D i equals to 0.9, such kind of information is called incomplete information.
Definition 1. For a discrete set = 1 2 3 ( , , ) , a special form of D numbers can be expressed by: . . .

= ({ })
n n D b v or be represented simply as: Definition 2. For a given D number, the overall assessment can be calculated as: Definition 3. Let D 1 and D 2 be two D numbers, the combination of D 1 and D 2 denoted by = ⊕ 1 2 D D D , defined as:  Note that the associative property is not satisfied in the D numbers combine rule. Let D 1 , D 2 and D 3 be three D numbers, where: D D D , these three combination results are different from each other, the sequence has great influence on the final result. In literature (Deng et al. 2014b), an order variable is defined to deal with this problem and put that method in EIA, the expert's weight is regard as the order variable. In the real circumstance, deciding the weight of the knowledge experts will involves in various kinds of subjectivity and it is so hard to decide the weight of every decision maker. So finding out a subjective way for the order variable of the combination is necessary.

Proposed method
In order to fuse multiple D numbers correctly and efficiently, a subjective method based on evaluation grades which is called "positive-negative method" is proposed. In the decision making system, any human assessments represented by D numbers, b i reflects the evaluation grade. The higher the value of b i is, the more positive the experts have assessed on the object. On the contrary, the experts are not confident about the attribute of the object.
When multiple D numbers get combined, half of the first two D numbers contribute to the second combined results by Eq. (7). Then when the combined results are fused with the third D numbers, half of both the combined results and the third D numbers contribute to the second combined results by Eq. (7). It is to say, the final combined results consist of a quarter of the first two D numbers and half of the third one. Namely, the D numbers combined ahead contribute less and the latter ones are more influential for the final com-bined assessments. So in the positive method, the higher assessments are more contributive to the final assessments and those D numbers with higher average assessments should be fused latter than those of lower average assessments. Similarly, the lower assessments are more influential to the final combined results and those D numbers need to be fused after the better assessments are combined in the negative methods. Here the positive-negative method is detail described step by step.
Step 1: For two given D numbers, if they are completely same, which means that two experts gave the same evaluation on the same problem, they need to be fused firstly and the combination result should be the same as the original D numbers. Let D 1 and D 2 be two D numbers: D 1 and D 2 are completely the same, they should be combined firstly, D is the combined result of the two D numbers, Step 2: When two D numbers are different from each other, the first step is to find the maximum average value i b , the average value of D number shows the average assessment about the problem. Let 1 2 , ... n D D D be n D numbers: then, the average value i b can be calculated as follows: where i b is the average assessment. Then the combination operation of multiple D numbers is a mapping f D , such that: where < < Step 3: When the average assessments are the same, we need to find out the maximum value of confidence to the assessment grade. The higher value of the average assessment is combined later in the positive method and combined firstly in the negative method. We find that = = The process of combining multiple D numbers is described in Figure 1.

Examples and applications
In this section, the proposed method will be adopted to the EIA, meanwhile, the results obtained by the proposed method are compared with the other methods to show the effectiveness of the proposed method. Generally, four phases are necessary in EIA. Firstly the hierarchical structure model for assessment needs to be established. In this phase, some affect factors that influence the EIA problem need to be identified carefully. Usually these factors can be classified into two parts, the natural factors and the man-made factors (Canter 1996), some necessary work should be done to insure that all the influence factors are included in the primary factors. Then the assessments for each environmental impact factors need to be given by the knowledgeable expert. The third step is the calculation of all the evaluated factors and the last step is to rank the entire project. Here the initial environmental assessment to Rupa Tal Lake is re-investigated as an example to demonstrate the efficiently of the proposed D numbers combination method (Pastakia, Jensen 1998). Some description and evaluation are mainly based on the published work by Refs. (Pastakia, Jensen 1998).
Rupa Tal is a hot tourism of Nepal and provides substantial incomes. In recent years, lake is undergoing sedimentation at a rapid rate. Four projects are considered for conservation of the lake area: -project 1: No action. The present sedimentation is allowed to continue, so that the lake is disappearing completely and a small gorge is created to take the inflow/outflow streams. -project 2: Along the southern margin, a high retaining dam is built to raise the overall water level. Due to the build of retaining dam, the in-lake areas created by sedimentation over the last few decades would be inundated. -project 3: A smaller, high dam is built between two bluffs, it would be about one third of the way up from the southern shore. This partial dam is smaller than that built in project 2 but has similar upstream effects. -project 4: A single large sedimentation reservoir in the upstream area, or a series of smaller retaining walls, would be used to form a sedimentation cascade. By carrying on this option, the water area may be remaining intact. In order to assess these four projects, a hierarchical structure model for EIA was established in literature (Wang et al. 2006). From four aspects, it is shown in Figure 2. Each factor has some sub-factors which is detailed in Table 1

Environmental factors Sub-factors
Physical/Chemical (P/C) 1 (P/C) 2 (P/C) 3 The impacts of lake water volume The impacts of the lake sedimentation The impacts of crop and grazing areas The impacts of lake fisheries The impacts of biodiversity The impacts of primary production The impacts of aquatic macrophytes The impacts of disease vector populations Sociological/Culture (S/C) 1 (S/C) 2 (S/C) 3 (S/C) 4 (S/C) 5 (S/C) 6 (S/C) 7 (S/C) 8 (S/C) 9 (S/C) 10 The loss of housing The loss of shops/public buildings The impacts of accessing routes The impacts induced by changes of tourism patterns The impacts of water supplies The impacts of diet/nutrition The impacts of aesthetic landscapes The impacts of water/vector borne disease The impacts of upstream quality of life The impacts of downstream quality of life The impacts of crop-generated incomes The impacts of fishery generated incomes The convenience of operation and Maintenance of option The cost of operation and Maintenance of option The cost of resettlement and compensation for land loss The cost of rehabilitation The cost of restoration of accessing routes The impacts of tourism-generated incomes Every sub-factor has different influence on the assessment of the projects. So at second the calculation of the assessment should be done. Nevertheless most of the assessments are represented by linguistic grades such as "good" and "poor", "A", "B", "C" and so forth. First of all, translating such assessment into numerical grade is so necessary. In the existing literature (Khan, Fitzcharles 1998), a seven points scale from "-3" to "+3" was used to represent the impacts from "High influence" to "low influence", in Ref. (Pun et al. 2003), the numerical ratings of "10" to "6", "5" to "1", "0", "-1" to "-5" and "-6" to "-10" represented five grades from "very high impact" to "very low impact" respectively. In general terms, the translating level is different from specific problems. Literature (Wang et al. 2006), translated the original grades "A", "B", "C" and so on into numerical grades which are shown in Table 2 so that D numbers theory can deal with such kinds of uncertainty. For example, when ten experts give the assessments for the conservation of the Rupa Tal, four experts believe it is major positive impacts and the other six evaluate it to be Moderately positive impact, then D numbers should be {(5,0.4) (3,0.6)}, this information is complete because all experts have given their opinions. Another five experts assess it to be positive impact while two experts evaluate it to be no impact, the left experts don't give any evaluation because of lacking of information, the D numbers can be {(2, 0.5),(0, 0.2)}. This kind of information is said to be incomplete. Table 3 shows the assessment matrixes. Table 2. An assessment standard for EIA (Pastakia, Jensen 1998;Wang et al. 2006) Assessment grade Numerical rating Description  Physical/Chemical (P/C) 1 (P/C) 2 (P/ The third step is to calculate the overall assessment for different projects. It is based on the Eqs (5)-(14). For example, for the evaluation of project 3, the environmental factors are physical and chemical. With respect to primacy (P/C) 1 , the assessment is {(2,0.8),(3,0.2)}, for sub-factor (P/C) 2 , the assessed is {(-1,0.85)(0,0.15)}. the assessment {(0,0.5),(1,0.5)} to the sub-factor (P/C) 3 . Firstly, the average assessment grades should be obtained: (P/C) 1 : 2.5; (P/C) 2 : -0.5; (P/C) 3 : 0.5 then with the positive method, the fusion should be ⊕ ⊕ By using the same method, the overall assessment results are calculated. And the last step is to rank all projects according to the values of integration representation of projects' overall value. By Eq. (5), the last score can be calculated. Table 4 shows the final results and ranking by both positive and negative method.
In order to show the effective of the proposed method, the results obtained by the proposed method is combined with the previous methods which are shown in Table 5. As is shown, In literature (Wang et al. 2006), the ranking is project 2 > project 4 > project 3 > project 1 for a risk-neutral decision maker. project 2 > project 3 > project 4 > project 1 for a risk-taking decision maker and project 4 > project 2 > project 3 > project 1 for a riskaverse decision maker. Anyhow, project 1 is always the worst choice, project 2 and project 4 are tend to be chosen.  In literature (Deng et al. 2014b), the ranking is project 3 > project 2 > project 4 > project 1 for a decision optimistic maker and project 4 > project 2 > project 3 > project 1 for a decision pessimistic maker.
In this paper, both the positive and negative method shows project 3 > project 2 > project 4 > project 1. From the comparison of results of different methods, both of them shows project 1 is the worst choice, there is only little differences between the final options. In reference (Wang et al. 2006), the best choice is project 2 or project 4. In reference (Deng et al. 2014b), the best choice is project 3 or project 4. In this paper, the best choice is project 3 for both positive and negative method. In the real word, this kind of problem needs further consideration because it is still an open issue and our methods is also reasonable because of the uncertainty of the assessments.

Conclusions
Most EIA problems contain a lot of human subjective judgments and different kinds of uncertainty, complete or incomplete, quantitative or qualitative. These kinds of information increases the complexity and difficulty of EIA process. So powerful in the methodology and the capacities of dealing with uncertainty are so necessary for EIA problems. D numbers theory provides an ideal, reliable and flexible way for this problem.
D numbers theory, as the extension of Dempster-Shafer theory of evidence, has removed the necessary hypothesis in evidence theory reasonable and it is more powerful when handling the information of both uncertainty and incomplete. However, shortcomings do exist in the D numbers combination rule because commutative property is not well satisfied. In this paper, the proposed method is effective for multiple D numbers combination. In combination process, the same D numbers should be combined firstly. For the left D numbers, the worse average evaluations are combined ahead the better average assessments in the positive method while the better average assessments come first in the negative method. While the average assessments are the same, the confidence degree about the evaluations is taken into consideration. The modified data-driven D numbers combination method proposed in this paper can perfect the D numbers theory itself and can be used in many decision situations such as optimization and so on forth.
Meanwhile, the proposed D numbers combination rule has many advantages. It is a data-driven method and makes full use of the information contained in the given D numbers in the combination process, no more information about the evaluations denoted by D numbers are needed. At the same time, the proposed combination rule can provide two results for the decision makers from both positive and negative aspects, it is of great use for the decision makers to make the final decision.
In the next step, the proposed method to handle multiple D numbers combination will be extend to handle more uncertain and incomplete problems, such as the risk assessments, water assessments, data fusion and so on. We all believe that the proposed method is powerful and appropriate in these fields.