AN EXTENDED COPRAS MODEL FOR MULTIPLE ATTRIBUTE GROUP DECISION MAKING BASED ON SINGLE-VALUED NEUTROSOPHIC 2-TUPLE LINGUISTIC ENVIRONMENT

. In this article, we develop the COPRAS model to solve the multiple attribute group decision making (MAGDM) under single-valued neutrosophic 2-tuple linguistic sets (SVN2TLSs). Firstly, we introduce the relevant knowledge about SVN2TLSs in a nutshell, such as the definition, the operation laws, a few of fused operators and so on. Then, combine the traditional COPRAS model with SVN2TLNs, and structure as well as elucidate the computing steps of the SVN2TLN-COPRAS pattern. Furthermore, in this article, we propose a method for determining attribute weights in different situations relying on the maximizing deviation method with SVN2TLNs. Last but not least, a numerical example about assessing the safety of construction project has been designed. And for further demonstrating the advantage of the new designed method, we also select a number of existed methods to have comparisons.


Introduction
Social development has driven changes in the construction industry. In the increasingly fierce market competition, the construction progress is largely affect the construction cycle. Taking all kinds of factors in the process of construction has a crucial significance for managers of construction progress to effective control of the project from the beginning to the end. Thus, choosing the right construction scheme becomes a common multi-attribute decision making (MADM) issue. There is no doubt that the proposition of fuzzy set theory proposed by  (Wu et al., 2018). Let Then according to score and accuracy function of z 1 and z 2 , if ( ) ( )

The SVN2TLN-COPRAS model for MAGDM
In such section, the SVN2TLN-COPRAS model is designed for MAGDM. We define that the set of alternatives is { } Step 2. Based on the results of formula (11), we can obtain the normalized matrix Step 3. The Eq. (14) is useful in acquiring the corresponding weighted normalized matrix = fz g s C c ×     by making use of the information deriving from the normalized matrix Step 4. According to each alternative's type, we can respectively calculate the sum value of benefit attributes F f(b) and the sum value of cost attributes F f(c) , just as Eq. (15) and Eq. (16): where k is the number of the benefit attributes. Obviously, the best alternatives with bigger values of F f (b) and smaller values of F f(c) .
Step 5. Figure up the relative significance Q f of every alternative with the Eq. (17): 1 1 min( ) where the optimal alternatives with bigger values of Q f .
Step 6. Compute the values of utility degree U f based on each alternative's Q f by following equation: Step 7. The value of U f as a ranking criterion, the closer it is to 100%, the better the corresponding alternative is. It's clear that the utility degree of best alternative is 100%.

Determine the attribute's weights
It is almost impossible for decision makers to gather full information, which especially sticks out in the intricate decision making problems. To deal with such cases with incomplete attribute's weights, Wang (1998) proposed the maximizing deviation method which has been studied by a large amount of scholars (Wang et al., 2020;Yin et al., 2016)  , , , , 0 Based on the above conditions, we can calculate difference degree between any two alternatives in line with the maximizing deviation method just as the Eq. (19) shows: , 1,2, , , 1,2, , where w under attribute T z is calculated from different experts.

The maximizing deviation method with SVN2TLNs
In this section, the following nonlinear programming model is used to compute attribute's weight based on the maximizing deviation method.
Case 1. The following non-linear programming model is for cases where the partial attribute weights are accessible.
where k q depicts the weights of W q and By solving the above equations, we can get the first-rank attribute weighting vector Case 2. For the cases where the entire attribute weights are unreachable, the following nonlinear programming model comes into play.
The Lagrange function is necessary to build for solving this model, as follows: where p is the Lagrange multiplier.
Next, take the partial derivatives of w z and p in Eq. (24) respectively, and set the results equal to zero that we can get the following simultaneous equations.
Finally, the weighting of attributes is worked out shown in Eq. (25) according to the partial derivatives and Eq. (24). x k by normalizing the formula (25), the normalized results are listed as:

The decision-making model
To sum up, the SVN2TLN-COPRAS method for MAGDM issues for incompletely attribute's weights includes the following steps: Step 1. Establish the SVN2TLN evaluation matrix Step 2. Integrate all SVN2TLN decision matrices into the overall decision matrix with respect to Eq. (11).
Step 3. Take advantage of Eq. (21) or Eq. (26) respectively facing different cases to acquire the weighting of attributes.
Step 4. Compute the corresponding weighted normalized matrix = fz g s C c ×     by using Eq. (14).
Step 5. According to = fz g s C c ×     and the types of attributes, the sum of benefit attributes Step 6. Figure up the relative significance Q f of every alternative with the Eq. (17).
Step 7. Compute the utility degree U f based on every alternative's Q f by Eq. (18).
Step 8. The value of U f as a ranking criterion, the closer it is to 100%, the better the corresponding alternative is.

Numerical example
The ultimate goal of the construction project is to complete the construction task with high quality and low consumption within the time limit specified in the contract, and to put it into production or deliver it for use on schedule. Any construction project, objectively there are a variety of technically feasible construction plans. The construction plan is to determine the construction sequence, construction method and selection of construction machinery according to the established construction deployment, so that the project can achieve the best effect of short duration, good quality and low cost. Therefore, there must be the problem of optimal selection of construction scheme. In most construction schemes, the quality of construction depends mainly on the following factors: ① T 1 is the factor of management level of construction project construction unit; ② T 2 is the construction of machinery, materials and other resource factors; ③ T 3 is a natural and social environmental factor; ④ T 4 is the influencing factor of relevant construction units; ⑤ T 5 is a variety of accident risk factors in the construction process; ⑥ T 6 is the technical level factor of the construction unit. In order to consider both quantitative and qualitative evaluation information, in this section, a numerical example is provided to select best construction projects by COPRAS model with SVN2TLNs. The five possible construction projects ( ) , W W with senior construction experience and one university professor W 3 with certain academic attainments in the field of construction to form an expert group to assess these construction projects. Moreover, according to the expert's age and authority in their field, through internal discussion, the expert's weight is ( ) 0.35,0.25,0.40 .
Step 1. Based on the information of p decision makers, establish the SVN2TLN evaluation Tables 1-3).      Step 2. Take advantage of Eq. (21) or Eq. (26) respectively facing different cases to acquire the weighting of attributes.
Case 1. When the partial attribute weights are accessible: The following non-linear programming model could be obtained according to (M -1): ( ) ( ) Table 5. The weighted decision-making matrix C Step 5. According to the weighted matrix and the types of attributes, the sum of benefit attributes F f(b) and the sum of cost attributes F f(c) are computed by Eq. (15) and (16) which listed in Table 6. (Suppose T 1 , T 3 , T 4 and T 6 are benefit attributes and T 2 as well as T 5 are cost attributes.) Step 6. Figure up the Q f ( )

Compare SVN2TLN-COPRAS method with aggregation operators
In this section, we compare our proposed SVN2TLN-COPRAS method with SVN2TLWA operator, SVN2TLWG operator, SVN2TLHWA operator and SVN2TLHWG operator (Wu et al., 2018). Based on the attribute's weight and Table 4, the fused values by SVN2TLWA, SVN2TLWG, SVN2TLHWA and SVN2TLHWG operators are shown in Table 7. According to the score of SVN2TLNs, we could derive the score results (Table 8). The rankings of alternatives are collected in Table 9. From the above calculation process and the final result summary Table 9, it's easy for us to conclude that the optimal scheme is consistent, but in a slightly different order.
According to the operational formulas of these operators, we can find that the SVN2TLWA operator and SVN2TLWG operator are incapable of taking the relationship between being fused variables into account so that the integration results in practical decision problems are too rough and imprecise. Moreover, the SVN2TLHWA operator and SVN2TLHWG operator adopt parameter control to consider the relationship between integrated variables, but their operation formula is extremely complex, which will lead to excessive workload in the actual decision making, so it is difficult to be applied. Compared our proposed model with the above mentioned operators, although the decision results are slightly different, the SVN2TLN-COPRAS model has the precious merits in settling the inconsistent attributes by computing the utility degree shown as the percentage which depicts the alternative is better or worse than others, and the decision steps of this new model are very simple and more in line with the actual decision situation for consider the conflicting attributes.

Conclusions
In this paper, we develop the SVN2TLN-COPRAS model to solve the MAGDM and some basically concepts of SVN2TLSs. Firstly, we introduce the relevant knowledge about SVN2TLSs in a nutshell, such as the definition, the operation laws, a few of fused operators and so on. Then, combine the traditional COPRAS model with SVN2TLNs, and structure as well as elucidate the computing steps of the SVN2TLN-COPRAS pattern. Furthermore, in this article, we propose a method for determining attribute weights in different situations relying on the maximizing deviation method with SVN2TLNs. Last but not least, a numerical example about assessing the safety of construction project has been designed. And for further demonstrating the advantage of the new designed method, we also select a number of existed methods to have comparisons. In future research, we are committed to apply the SVN2TLN-COPRAS model to other fields so as to resolve more MAGDM problems.