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MULTI-CRITERIA GROUP DECISION MAKING MODEL USING SINGLE-VALUED NEUTROSOPHIC SET

Background: In this article, we introduce some approaches for decision making in the neutrosophic set. The purpose of this study is to develop a neutrosophic multi-criteria group decision-making (MCGDM) model based on hybrid score-accuracy functions for approving a tender for construction under a simplified neutrosophic environment. Five criteria have been selected from experts’ opinions to be considered for the distribution of tender. In this paper, we use the score functions, the accuracy functions, and the hybrid score-accuracy functions of single-valued neutrosophic numbers (SVNNs) and ranking method for SVNNs, those will help for making a decision. Methods: Decision making under uncertain situation is an important aspect of those days. For this, we have developed the multi-criteria decision-making model using a single-valued neutrosophic set. The main aim is to select an appropriate tender for assigning the work to be done, so that the output will be the best one, under the available resources. Results: We have developed an algorithm for taking proper decisions for the selection of a contractor for the construction of a public/government work. Conclusions: We have verified our algorithm with the help of an example. We have considered five criteria. However, this algorithm can be applied for multi-criteria decision making. Also, it can be applied to other case studies too.


INTRODUCTION
By using multiple criteria decision making (MCDM) methods, group decision-makers can choose the best alternative given multiple criteria. For that, a strategic method needs to be implemented to this decision made in uncertainty. In MCDM difficulties, a group decision matrix is built by aggregating the individual evaluation of each decision-maker to find a group adequate solution that is most preferred by the decision-makers.
Logistics management is a component of supply chain management. It plans, implements, and manages the efficient, effective forward and reverse flow and storage of goods, services, and related information between the point of creation and the point of consumption in order to meet customers' requirements. In this connection our model is expected to be useful for the logistic practices for decision making.
The main fields within logistics are Procurement logistics, production logistics, distribution logistics, disposal logistics. Our work can help in different fields of logistics [Swierczek 2019].
Horizontal logistics collaboration allows a great opportunity for companies to diminish their distribution charges. By forming a combination, companies have the potential to become more productive. However, the selection of a coalition structure is a difficult job for decision-makers. The decision-maker needs to distinguish and choose the best workable partner(s) to carry out a joint plan concerning many patterns. This paper aims to propose a unique combination of group decision making [Sun, 2019, Harish, 2020.
For the choice of any object to a bye, we have decided from the available object and other opportunities like home delivery and time management quality. Since logistics management is a component of supply chain management, decision-makers have to ensure the flow of work management quality.
The fuzzy sets theory proposed by Zadeh in 1965. It has been very successful in dealing with difficulties involving uncertainty. Fuzzy set theory can be used to model imprecision in MCDM problems. [Pramanik, Mukhopadhyay 2011] performed an intuitionistic fuzzy MCDM strategy for teachers selection based on the grey relational analysis.
Till now fuzzy and intuitionistic fuzzy MCDM difficulties are investigated by many researchers. Uncertainty performs a vital role in group decision-making problems. Presently multiple researchers use uncertainty in the model formulation of different MCDM problems. So neutrosophic sets should be utilized in the decision-making method. The idea of the neutrosophic set was acquainted by [Smarandache 2005].
Distribution of tender for some construction can be considered as multi-criteria group decision-making (MCGDM) problem that generally consists of selecting the most desirable alternative from all the given alternatives. Classical MCGDM approaches deal with crisp numbers i.e. the weights of criteria are measured by crisp numbers. However, it is not always possible to present the information by crisp numbers. In order to deal with such a situation, the notion of the fuzzy set was introduced by Zadeh and Atanassov extended the concept of fuzzy sets (FSs) to intuitionistic fuzzy sets (IFSs) in 1986. The distribution of tender generally involves subjective judgment of experts, which makes the accuracy of the results highly questionable. In order to tackle the problem, a new methodology is needed. Liang and Wang studied the fuzzy multi-criteria decision making (MCDM) algorithm for personnel selection. Gunor and others developed an analytical hierarchy process (AHP) for personnel selection.
Since fuzzy and intuitionistic fuzzy MCDM problems are widely studied, but indeterminacy should be incorporated in the model formulation of the problems. Indeterminacy plays an important role in the decision-making process. So neutrosophic set, the generalization of intuitionistic fuzzy sets should be incorporated in the decision making process. Neutrosophic set [Smarandache 2005] was introduced to represent the mathematical model of uncertainty, imprecision, and decision making.  presented the entropy-based grey relational analysis method for multi-criteria decision making under single-valued neutrosophic assessment. ] also studied a new methodology to deal with neutrosophic multi-criteria decision-making problems. [Ye 2013] proposed the correlation coefficient of SVNSs for single-valued neutrosophic multicriteria decision-making problems [Cyplik, 2011, Karaaslan, 2020.
The ranking order of alternatives plays an important role in the decision-making process. In this study, we present a multi-criteria group decision-making approach for giving a tender for the construction with known weights based on the score functions, the accuracy functions, and the hybrid score-accuracy functions proposed by [Ye 2013] under a simplified neutrosophic environment.
The rest of the paper has been divided into different sections. Section 2, is the preliminaries and the definitions. In this section, we procure the definitions and the preliminary results used in this article. Section 3 is on multi-criteria group decision-making methods based on hybrid score-accuracy function. In this section, we discuss the multicriteria decision-making method in the singlevalued neutrosophic environment. Also, we have formulated the algorithm for this. Section 4 deals with the validation of the developed model. In this section we consider an example to verify our model. In section 5 we have talked about the difference and advantage of our model. In section 6 is the conclusion on the work done in this article.

PRELIMINARIES AND DEFINITIONS
Definition 2.1: Let X be a non-empty set. A neutrosophic set A in X is characterized by truth-membership function TA, indeterminacymembership function IA and falsitymembership function FA. TA(x), IA(x), FA(x) are real standard or non-standard subsets of ] -0,1 + [. That is

2) Two SVNSs A and B are equal (A=B) if
and only if A ⊆B and B ⊇A.
Definition 2.5: Let = (T( ), I( ), F( )) be a SVNN. Then the score function and the accuracy function of the SVNN can be represented respectively, as follows: For the score function of a SVNN , if the truth-membership T( ) is bigger and the indeterminacy-membership I( ) are smaller, then the score value of a SVNN is greater. For the accuracy function of a SVNN , if the sum of T( ),1-F( ),1-I( ) is bigger, compared to the other SVNN, then the statement is more affirmative, i.e. the accuracy of the SVNN is higher.
Proposition 2.1: Let a1, a2 be two SVNNs. Then the ranking method can be defined as follows: Operational definitions of the terms stated in the problem: 1. Technical approach: Technical Approach is the method of energy-saving tools that have large significance for a tender worker as well as industrial work. It gives the maximum advantage of the facility of the energy savings potential. 2. Management approach: Management or managerial system of any program is highly important because the quality of the supply and performance of work is handle by the technique of management system which is also beneficial for criteria of quick performance, technical and costeffectiveness. neutrosophic set. LogForum 16 (3), 421-429. http://doi.org/10.17270/J.LOG.2020.446 424 3. Quick performance: Quick performance is the main key of any work and the development of the society as well as personal. The good performance of a tender shows the degree of sincerity of the tenderer. 4. Price selection: The Price selection is the most important issue to select tender. The theory of evolution and natural selection of Price equation in such a way that the desire of the tender committee and host will have the optimum solution. 5. Credentials: This is a qualification, achievement, personal quality, or aspect of a person's background, typically when used to indicate that they are suitable for the tender or not.

MULTI-CRITERIA GROUP DECISION-MAKING METHODS BASED ON HYBRID SCORE-ACCURACY FUNCTION
In a multi-criteria group decision-making problem, let {A1,A2,A3,…….Am} be the set of alternatives and let {C1,C2,……Cn} be the set of all criteria. Let the committee of decisionmakers assigned the weights of all criteria previously. In such a case, we develop two methods based on the hybrid score-accuracy functions for multiple-criteria group decisionmaking problems with known weights under single-valued neutrosophic environment and interval neutrosophic environment.

Multi-criteria group decision-making method in single valued neutrosophic environment:
In the group decision process under single valued neutrosophic environment, if a group of t decision makers is required in the evaluation process, then the k th decision maker can provide the evaluation information of the alternatives Ai (i=1,2,….,m) on the criteria Cj (j=1,2,……,n), which is represented by the form of a SVNS for k =1,2,…,t, j=1,2,…..,n, i =1,2,….,m.

Step3: Decision maker's weights
In decision making problems, the decision makers may have personal biases and some individuals may give unduly high or low preference values with respect to their preferred or repugnant objects. In this case we will assign very low weights to these false or biased opinions. Since the "mean value" is the "distributing centre" of all elements in a set, the average matrix Y A is the maximum compromise among all individual decisions of the group. It mean sense, a hybrid scoreaccuracy matrix Y k is closer to the average matrix Y A . Then, the preference value (hybrid score-accuracy value) of the k th decision maker is closer to the average value and evaluation is more reasonable and more important, thus the weight of the k th decision maker is bigger. Hence a weight model for decision makers can be defined as: where 0≤ 1 ≤1, ∑ 1 % = 1 for k=1,2,…,t.

Step5: Weights for criteria
The Weight of criteria for this model has to be decided by the experts or committee of decision makers according to their requirements which is denoted by wj (j=number of criteria). The weight of criteria is based on the significance of the importance of the criteria and the total weight is always one.

Step7: Ranking the alternatives
To rank the alternatives, we can sum all values in each row of weighted hybrid scoreaccuracy matrix and find the overall weighted score-accuracy value of each alternative Ai (i=1,2,….,m): According to the overall weighted scoreaccuracy values M(Ai) of each alternatives Ai (i=1,2,…,m) we can rank the alternatives Ai (i=1,2,…,m) in descending order and choose the best alternative.

VALIDATION OF THE DEVELOPED MODEL
In this section we present an example to validate our developed model.

Example of tender distribution for construction
Suppose that the central government or any state government of our country is going to construct a national highway or any building, then they need to choose the best construction company to build that highway or building for the use of the public. Then the government gives an advertisement in some well-circulated newspaper or the particular website of the government. Some interested construction companies may submit for the tender. After initial screening, four construction companies A1, A2, A3, A4 remain for further evaluation. A committee of four decision-makers D1, D2, D3, D4 has been formed to conduct the interview and choose the better construction company. Decision-makers consider five criteria to evaluate the better alternative. The five criteria are namely, technical approach (C1), management approach (C2), credentials (C3), past performance (C4), price (C5). If four decision-makers Dq (q=1,2,3,4) are required in the evaluation process, then the five possible alternatives Ai (i=1,2,3,4) are evaluated by the form of SVNNs under the five criteria on fuzzy concept "excellence". Thus the four singlevalued neutrosophic decision matrix can be obtained from the four experts and expressed respectively as follows.
Single valued neutrosophic decision matrix for D1: Now we determine the weights of the four decision makers, by using the eq (5) and eq (6) as follows: 1 = 0.2498 , 1 6 = 0.2508 , 1 = 0.2497 , 1 C = 0.2497 Therefore, by using the eq (7) the hybrid score-accuracy values of the four decision makers' evaluations are aggregated and the following collective hybrid score-accuracy matrix can be obtained as follows: Now for this multi-criteria decision making problem we assume the weights for the given criteria. We take the weight vector of the attributes as: Das S., Das R., Tripathy B.Ch., 2020. Multi-criteria group decision making model using Single-valued neutrosophic set. LogForum 16 (3), 421-429. http://doi.org/10.17270/J.LOG.2020 Now we can calculate the overall weighted hybrid score-accuracy values M(Ai) for each alternatives Ai, i=1, 2, 3, 4 by using the equation (10) According to the values of M(Ai), i=1, 2, 3, 4, the ranking order of the alternatives Ai , i=1, 2, 3, 4, is A2 > A1 > A4 > A3. Hence the alternative A2 is the best choice to give the tender for construction.

DIFFERENCE AND ADVANTAGES OF THE DEVELOPED MODEL
In 2014, Mondal and Pramanik introduced a multi-criteria group decision-making algorithm for teacher recruitment in higher education under a simplified neutrosophic environment. In that algorithm, they used hybrid score-accuracy function, where the score function depends on the truthmembership and falsity-membership values. Again they used completely unknown weights for each criterion in the algorithm. However, for our model, we use a new score function which depends on truth and indeterminacy membership values. By using the proposed score function we get a more accurate score value for a single-valued neutrosophic number. Again we have used the completely known weights for each criterion, which is important for any decision-making problem.
In our paper, we have taken neutrosophic indeterminacy function which is very unique and interesting for the decision if the value of indeterminacy membership in score function is very less then the truth membership value then the decision will more accurate.

CONCLUSIONS
In this paper, we used some suitable criteria for decision making to a better choice of a tender among the available tenders and we use function namely score function, accuracy function and hybrid score-accuracy function of SVNNs to select the better construction company to give the tender for construction under the neutrosophic environment, where the weights of the decision-makers are completely unknown and the weights of the criteria are completely known. This method can be used for group decision making with single-valued neutrosophic information is provide simple calculations and good flexibility but also handled with the group decision-making problems with known weights by comparisons with other relative decision-making methods under single-valued neutrosophic environments.
We have established a formula for making a decision. The data used in this paper has not taken from any source. We have considered these numbers for the verification of our algorithm. However, this algorithm can apply for any real source data.