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Journal of Intelligent Computing Volume 9 Number 2 June 2018 54

Minimum Deviation Method for Single-valued Neutrosophic Multiple Attribute Deci-sion Making with Preference Information on Alternatives

Dong-Sheng Xu, Cun WeiSouthwest Petroleum [email protected]@qq.comGui-Wu WeiSichuan Normal [email protected]

ABSTRACT: In this paper, we investigate the single-valued neutrosophic multiple attribute decision making (MADM)problems with preference information on alternatives, some operational laws of single-valued neutrosophic numbers, scorefunction where accuracy function of single-valued neutrosophic numbers are introduced. An optimization model based onthe minimum deviation method, by which the attribute weights can be determined, is established. For the special situationswhere the information about attribute weights is completely unknown, we establish another optimization model. By solvingthis model, we get a simple and exact formula, which can be used to determine the attribute weights. We utilize the single-valued neutrosophic weighted averaging (SVNWA) operator to aggregate the single-valued neutrosophic informationcorresponding to each alternative, and then rank the alternatives and select the most desirable one(s) according to the scorefunction and accuracy function. The method can sufficiently utilize the objective information, and meet decision makers’subjective preference and can be easily performed on computer. Furthermore, we have extended the above results to intervalneutrosophic environment. Finally, an illustrative example for potential evaluation of emerging technology commercializationis given to verify the developed approach and to demonstrate its practicality and effectiveness.

Keywords: Multiple Attribute Decision Making (MADM), Single-valued Neutrosophic Numbers, Interval Neutrosophic Numbers,Weight Information, Preference

Received: 19 October 2017, Revised 29 November 2017, Accepted 9 December 2017

DOI: 10.6025/jic/2018/9/1/54-75

© 2018 DLINE. All Rights Reserved


1. Introduction

In many cases, it is difficult for decision-makers to precisely express a preference when solving MADM problems with inaccurate,uncertain or incomplete information. Under these circumstances, Zadeh’s fuzzy sets (FSs) [1], where the membership degree isrepresented by a real number between zero and one, are regarded as an important tool for solving MADM problems [2,3], fuzzylogic and approximatereasoning [4], and pattern recognition [5]. However, FSs cannot handle certain cases where it is hard todefine the membership degree using one specific value. In order to overcome the lack of knowledge of non-membership degrees,Atanassov [6] introduced intuitionistic fuzzy sets (IFSs), an extension of Zadeh’s FSs. To date, IFSs have been widely appliedin solving MCDM problems [9-24]. IFSs simultaneously take into account the membership degree, non-membership degree andhesitation degree. Therefore, they are more flexible and practical when addressing fuzziness and uncertainty than traditionalFSs. Moreover, in some actual cases, the membership degree, non-membership degree and hesitation degree of an element inIFSs may not be a specific number; hence, they were extended to interval-valued intuitionistic fuzzy sets (IVIFSs) [25]. BecauseIFSs and IVIFSs cannot represent indeterminate and inconsistent information, Smarandache [26-27] introduced a neutrosophicset (NS) from a philosophical point of view to express indeterminate and inconsistent information. In a NS A, its truth-membershipfunction TA(x), indeterminacy-membership IA(x) and falsity-membership function FA(x) are represented independently, which liein real standard or nonstandard subsets of ]−0, 1+ [ , that’s, TA(x): X → ] −0, 1+ [ IA(x): X → ] −0, 1+ [ and FA(x): X → ] −0, 1+ [ . Thus,the nonstandard interval ] −0, 1+ [ may result in the difficulty of actual applications. Based on the real standard interval [0,1],therefore, the concepts of the concepts of a single-valued neutrosophic set (SVNS) [28] and an interval neutrosophic set (INS)[29] was presented as subclasses of NS to be easily used for actual applications, and then Ye [30] introduced a simplifiedneutrosophic set (SNS), including the concepts of SVNS and INS, which are the extension of IFS and IVIFS. Obviously, SNS isa subclass of NS, while SVNS and INS are subclasses of SNS. As mentioned in the literature [26-30], NS is the generalization ofFS, IFS and IVIFS. However, SNS (SVNS and INS) is very suitable for the expression of incomplete, indeterminate, and inconsistentinformation in actual applications. Recently, SNSs (INSs, and SVNSs) have been widely applied in many areas. Majumdar et al.[31] introduced a measure of entropy of SVNSs. Furthermore, the correlation coefficients of SVNSs as well as a decision-makingmethod using SVNSs were introduced [32]. Broumi and Smarandache [33] discussed the correlation coefficient of INSs andZhang et al. [34] developed a MCDM method based on aggregation operators within an interval neutrosophic environment.Furthermore, Ye [35-39] proposed the similarity measures between SVNSs and INSs, which were based on the relationshipbetween similarity measures and distances. Ye[40] investigated the interval neutrosophic multiple attribute decision-makingmethod based on the possibility degree ranking method and ordered weighted aggregation operators. Ye[41] proposed theprojection and bidirectional projection measures of single valued neutrosophic sets for MADM. Ye[42] proposed intervalneutrosophic multiple attribute decision-making method with credibility information. Zhang et al.[43] studied an improvedweighted correlation coefficient based on integrated weight for interval neutrosophic sets in MADM problems. Peng et al.[44]gave an outranking approach for multi-criteria decision-making problems with simplified neutrosophic sets. Zhang et al.[45]proposed an outranking approach for multi-criteria decision-making problems with interval-valued neutrosophic sets. Tian etal.[46] investigated the multi-criteria decision-making method based on a cross-entropy with interval neutrosophic sets. Biswaset al.[47] proposed the TOPSIS method for multi-attribute group decision-making under single-valued neutrosophic environment.Zhang et al.[48]developed the interval neutrosophic number weighted averaging (INNWA)operator and interval neutrosophicnumber weighted geometric(INNWG) operator. Liu and Wang[49] proposed a single-valued neutrosophic normalized weightedBonferroni mean (SVNNWBM) operator on the basis of Bonferroni mean, the weighted Bonferroni mean (WBM), and thenormalized WBM and developed the models solve the multiple attribute decision-making problems with SVNNs based on theSVNNWBM operator. Liu et al.[50] combined Hamacher operations and generalized aggregation operators to NS, and proposedsome new aggregation operators: the generalized neutrosophic number Hamacher weighted averaging (GNNHWA) operator,generalized neutrosophic number Hamacher ordered weighted averaging (GNNHOWA) operator, and generalized neutrosophicnumber Hamacher hybrid averaging (GNNHHA) operator, and explored some properties of these operators and analyzed somespecial cases of them. Zhao et al.[51] proposed improved aggregation operation rules for IVNSs, and extended the generalizedweighted aggregation (GWA) operator to work congruently with IVNS data which the aggregated results are also expressed asIVNSs, which are characterized by truth-membership degree, indeterminacy-membership degree, and falsity-membership degreeand the proposed method is proved to be the maximum approximation to the original data, and maintains most of the informationduring data processing. Sun et al.[52] proposed a ranking index according to its geometrical structure and established theinterval neutrosophic number Choquet integral (INNCI) operator. On the basis of prioritized aggregated operator and prioritizedordered weighted average (POWA) operator, Liu and Wang[53] further presented interval neutrosophic prioritized orderedweighted aggregation (INPOWA) operator with respect to interval neutrosophic numbers (INNs). Wu et al.[54] defined theprioritized weighted average operator and prioritized weighted geometric operator for simplified neutrosophic numbers (SNNs)and then proposed two novel effective cross-entropy measures for SNSs and proposed the ranking methods for SNSs to solve


MADM problems based on the proposed prioritized aggregation operators and cross-entropy measures. Ye[55] proposed aninterval neutrosophic weighted exponential aggregation (INWEA) operator and a dual interval neutrosophic weighted exponentialaggregation (DINWEA) operator based on the exponential operational laws, introduced comparative methods based on cosinemeasure functions for INNs and dual INNs and developed decision-making methods based on the INWEA and DINWEAoperators. Li et al.[56] proposed the improved generalized weighted Heronian mean (IGWHM) operator and improved generalizedweighted geometric Heronian mean (IGWGHM) operator based on crisp numbers, and prove that they can satisfy somedesirable properties, such as reducibility, idempotency, monotonicity and boundedness. They proposed the single valuedneutrosophic number improved generalized weighted Heronian mean (NNIGWHM) operator and single valued the neutrosophicnumber improved generalized weighted geometric Heronian mean (NNIGWGHM) operator for multiple attribute group decisionmaking (MAGDM) problems in which attribute values take the form of SVNNs.

In the process of single-valued neutrosophic MADM with preference information on alternatives, sometimes, the attributevalues and preference values on alternatives take the form of single-valued neutrosophic numbers, and the information aboutattribute weights is incompletely known or completely unknown because of time pressure, lack of knowledge or data, and theexpert’s limited expertise about the problem domain. All of the above methods, however, will be unsuitable for dealing with suchsituations. Therefore, it is necessary to pay attention to this issue. The aim of this paper is to develop a method, based on theminimum deviation method, to overcome this limitation. The remainder of this paper is set out as follows. In the next section, weintroduce some basic concepts related to single-valued neutrosophic sets. In Section 3 we introduce single-valued neutrosophicMADM problems with preference information on alternatives, in which the information about attribute weights is incompletelyknown, and the attribute values and preference values on alternatives take the form of single-valued neutrosophic numbers. Todetermine the attribute weights, an optimization model based on the minimum deviation method, by which the attribute weightscan be determined, is established. For the special situations where the information about attribute weights is completelyunknown, we establish another optimization model. By solving this model, we get a simple and exact formula, which can be usedto determine the attribute weights. We utilize the single-valued neutrosophic weighted averaging (SVNWA) operator to aggregatethe single-valued neutrosophic information corresponding to each alternative, and then rank the alternatives and select themost desirable one(s) according to the score function and accuracy function. In Section 4, we introduce interval neutrosophicMADM problems with preference information on alternatives, in which the information about attribute weights is incompletelyknown, and the attribute values and preference values on alternatives take the form of interval neutrosophic numbers. In Section5, an illustrative example for potential evaluation of emerging technology commercialization is pointed out. In Section 6 weconclude the paper and give some remarks.

2. Preliminaries

In this section, some basic concepts and definitions, including definitions of neutrosophic sets (NSs) and single-valuedneutrosophic sets (SVNSs), are introduced. The operations of single-valued neutrosophic numbers (SVNNs) are also providedas they will be utilized in the rest of the paper.

Definition 1[26-27]: Let X be a space of points (objects) with a generic element in fix set X, denoted by x. A neutrosophic sets(NSs) A in X is characterized by a truth-membership function TA(x), an indeterminacy-membership IA(x) and a falsity-membershipfunction FA(x). The functions TA(x), IA(x) and FA(x) are real standard or nonstandard subsets of ]−0, 1+ [ , that’s, TA(x): X → ] −0, 1+

[ IA(x): X → ] −0, 1+ [ and FA(x): X → ] −0, 1+ [ . There is no restriction on the sum of TA(x), IA(x) and FA(x), so 0− ≤ sup TA(x) + sup TA(x)+ sup FA(x) ≤ 3+.

The neutrosophic sets(NSs) was presented from philosophical point of view. Obviously, it was difficult to apply in the realapplications. Wang [28] further proposed the single-valued neutrosophic sets (SNSs) from scientific or engineering point ofview, which is a generalization of classical set, fuzzy set, intuitionistic fuzzy set and paraconsistent sets etc., and it was definedas follows.

Definition 2: Let X be a space of points (objects) with a generic element in fix set X, denoted by x. A single-valued neutrosophicsets (SVNSs) A in X is characterized as following:

(1)


where the truth-membership function TA(x), indeterminacy-membership IA(x) and falsity-membership function FA(x), are singlesubintervals/subsets in the real standard [0, 1], that is, TA(x): X →[0, 1], IA(x): X → [0, 1] and FA(x): X → [0, 1]. And the sum of TA(x),IA(x) and FA(x) satisfies the condition 0 ≤ TA(x) + IA(x) + FA(x) ≤ 3. Then a simplification of A is denoted by

, which is a SVNS.

For a SVNS , the ordered triple components , are described as a single-valued neutrosophic number (SVNN), and each SVNN can be expressed as , where ,and .

Definition 3[34]: Let , be a SVNN a score function S of a SVNN can be represented as follows:

(2)

Definition 4[34]: Let be a SVNN, an accuracy function H of a SVNN can be represented as follows:

(3)

to evaluate the degree of accuracy of the SVNN , where . The larger the value of H(A) is, the morethe degree of accuracy of the SVNN A.

Based on the score function S and the accuracy function H, in the following, Zhang et al.[34] gave an order relation between twoSVNNs, which is defined as follows:

Definition 5[34]: Let and be two SVNNs, and be the

scores of A and B, respectively, and let and be the accuracy degrees of A and B, respectively, thenif , then A is smaller than B, denoted by A < B; if S(A) = S(B), then

(1) If H(A) = H(B), then A and B represent the same information, denoted by A = B;

(2) If H(A) < H(B), A is smaller than B, denoted by A < B.

Definition 6[28]: Let and be two SVNNs, and some basic operations on them are defined asfollows:

Based on the Definition 6, we can derive the following properties easily.


Theorem 1[28]: Let and be two SVNNs, then

Definition 7[57]: Let be a collection of SVNNs, and let SVNWA: if

where be the weight vector of , and , , then SVNWA is called the single-

valued neutrosophic weighted averaging (SVNWA) operator.

Definition 8[31]: Let and be two SVNNs, then the normalized Hamming distance between

and is defined as follows:

(5)

3. Minimum Deviation Method for Single-valued Neutrosophic Madm Problems with Preference Information onAlternatives

The following assumptions or notations are used to represent the single valued neutrosophic MADM problems with incompleteweight information.

(1) The alternatives are known. Let be a discrete set of alternatives;

(2) The attributes are known. Let be a set of attributes;

(4)


(3) The subjective preference information on alternatives is known, and let be subjective preference value

are SVSNs, which is subjective preference values on alternative .

(4) The information about attribute weights is incompletely known. Let be the weight vector of

attributes, where , H is a set of the known weight information, which can be constructed by the

following forms [58-65], for : Form 1. A weak ranking: Form 2. A strict ranking: , ; Form 3. Aranking of differences: for ; Form 4. A ranking with multiples: , ; Form 5. Aninterval form:

Suppose that is the single-valued neutrosophic decision matrix, where indicates thetruth-membership function, indeterminacy-membership function and falsity-membership function given by the decision maker,

.

Definition 9: Let is the single-valued neutrosophic decision matrix, be the vectorof attribute values corresponding to the alternative, then we call

(6)

(7)

the overall value of the alternative Ai, where is the weight vector of attributes.

In the situation where the information about attribute weights is completely known, i.e., each attribute weight can be providedby the expert with crisp numerical value, we can weight each attribute value and aggregate all the weighted attribute valuescorresponding to each alternative into an overall one by using Eq. (6). Based on the overall attribute values ri of the alternatives

, we can rank all these alternatives and then select the most desirable one(s). The greater , the better thealternative Ai will be.

Because of the complexity of objects, the fuzziness of thought, and the finiteness of knowledge, it is difficult for decision makersto derive the attribute weights, and sometimes, attribute weight information is completely unknown. In this situation, in order toreflect the decision maker’s subjective preference and objective information, an optimization model is developed to get theattribute weight. However, there are some differences to the some extent between decision maker’s subjective preference andobjective information. For the more reasonable decision-making, to select attribute weight vector is to minimize total deviationbetween objective information and decision maker’s subjective preference.

The minimum deviation method is selected here to compute the differences between decision maker’s subjective preference andobjective information. For the attribute , the deviation of alternative Ai to decision maker’s subjective preference can bedefined as follows:


Let

Then Di(w) represent the deviation value of the alternatives Ai to decision maker’s subjective preference value .

Based on the above analysis, we have to choose the weight vector w to minimize all deviation values for all the alternatives. Todo so, we can construct a linear programming model as follows:

where .

By solving the model (M-1), we get the optimal solution , which can be used as the weight vector ofattributes.

If the information about attribute weights is completely unknown, we can establish another programming model:

To solve this model, we construct the Lagrange function:

(8)

where is the Lagrange multiplier.

Differentiating Eq. (8) with respect to and , and setting these partial derivatives equal to zero, the followingset of equations is obtained:

By solving Eq. (9), we get a simple and exact formula for determining the attribute weights as follows:

(9)


(10)

By normalizing be a unit, we have

(11)

Based on the above models, we develop a practical method for solving the MADM problems with preference information onalternatives, in which the information about attribute weights is incompletely known or completely unknown, and the attributevalues and preference values on alternatives take the form of SVNNs. The method involves the following steps:

(Procedure one)

Step 1: Let be a single-valued neutrosophic decision matrix, where , which is an attribute

value, given by an expert, for the alternative with respect to the attribute be the weight vectorof attributes, where H is a set of the known weight information, which can be constructed by the forms1-5. Let be subjective preference value, are SVNNs, which are subjective preference values onalternatives .

Step 2: If the information about the attribute weights is partly known, then we solve the model (M-1) to obtain the attributeweights. If the information about the attribute weights is completely unknown, then we can obtain the attribute weights by usingEq. (11).

Step 3: Utilize the weight vector and by Eq. (7), we obtain the overall values of the alternative.

Step 4: Calculate the scores of the overall SNNs to rank all the alternatives and then toselect the best one(s) (if there is no difference between two scores and , then we need to calculate the accuracydegrees and of the overall SNNs ri and rj, respectively, and then rank the alternatives Ai and Aj in accordance with theaccuracy degrees and .

Step 5: Rank all the alternatives and select the best one(s) in accordance with and .

Step 6: End.

4. Minimum Deviation Method for Interval Neutrosophic Madm Problems with Preference Information on Alternatives

In the real applications, sometimes, it is not easy to express the truth-membership, indeterminacy-membership and falsity-membership by crisp numbers, and they can be expressed by interval numbers. Wang et al. [29] further defined intervalneutrosophic sets (INSs) shown as follows.

Definition 10[29]: Let X be a space of points (objects) with a generic element in fix set X, denoted by x . An interval neutrosophic


sets (INSs) in X is characterized as follows:

(12)

where the truth-membership function , indeterminacy-membership and falsity-membership function , whichare interval values in the real standard [0, 1], that is, and . And the sum of and satisfies the condition .

For a INSs , the ordered triple components , are described as an interval

neutrosophic number (INNs), and each INN can be expressed as , where

, and .

Definition 11[66]: Let be an INN, a score function S of an INN can be represented asfollows:

(13)

Definition 12[66]: Let be an INN, an accuracy function H of an INN can be represented asfollows:

(14)

to evaluate the degree of accuracy of the INN , where . The larger the value

of is, the more the degree of accuracy of the INN .

Based on the score function S and the accuracy function H , in the following, Tang[66] gave an order relation between two INNs,which is defined as follows:

Definition 13[66]: Let and be two INNs,

and be the scores of and ,

respectively, and let and be the accuracy degrees of and ,

respectively, then if , then is smaller than , denoted by ; if , then

(2) If , then and represent the same information, denoted by ; (2) if , is smaller than ,

denoted by .

Definition 14[29,36]: Let and be two INNs, and somebasic operations on them are defined as follows:


Based on the Definition 14, we can derive the following properties easily.

Theorem 2[29,36]: Let and be two INNs, ,then

Definition 15[34]. Let be a collection of INNs, and let INWA:, if

(15)

where be the weight vector of , and , , then INWA is called the interval

neutrosophic weighted averaging (INWA) operator.


Definition 16[66]: Let and be two INNs, then the

normalized Hamming distance between and is defined asfollows:

(16)

Let A, G, w and H be presented as in section 3. Suppose that is the interval

neutrosophic decision matrix, where indicates the truth-membership function, indeterminacy-

membership function and falsity-membership function given by the decision maker, ,

, . The subjective preference information on alternatives is known, and let

be subjective preference value are INNs, which is subjective preference

values on alternative .

Definition 17: Let is the interval neutrosophic decision matrix,

be the vector of attribute values corresponding to the alternative then we call

(17)

the overall value of the alternative Ai, where is the weight vector of attributes.

In the situation where the information about attribute weights is completely known, i.e., each attribute weight can be providedby the expert with crisp numerical value, we can weight each attribute value and aggregate all the weighted attribute valuescorresponding to each alternative into an overall one by using Eq. (17). Based on the overall attribute values of the alternatives

, we can rank all these alternatives and then select the most desirable one(s). The greater , the better thealternative Ai will be.

Because of the complexity of objects, the fuzziness of thought, and the finiteness of knowledge, it is difficult for decision makersto derive the attribute weights, and sometimes, attribute weight information is completely unknown. In this situation, in order toreflect the decision maker’s subjective preference and objective information, an optimization model is developed to get theattribute weight. However, there are some differences to the some extent between decision maker’s subjective preference andobjective information. For the more reasonable decision-making, to select attribute weight vector is to minimize total deviationbetween objective information and decision maker’s subjective preference.

The minimum deviation method is selected here to compute the differences between decision maker’s subjective preference andobjective information. For the attribute , the deviation of alternative Ai to decision maker’s subjective preference can bedefined as follows:


(18)

Let

Then Di(w) represent the deviation value of the alternatives Ai to decision maker’s subjective preference value .

Based on the above analysis, we have to choose the weight vector w to minimize all deviation values for all the alternatives. Todo so, we can construct a linear programming model as follows:

where .

By solving the model (M -3), we get the optimal solution , which can be used as the weight vector ofattributes.

If the information about attribute weights is completely unknown, we can establish another programming model:

To solve this model, we construct the Lagrange function:

(19)

where is the Lagrange multiplier.

Differentiating Eq. (19) with respect to , and , setting these partial derivatives equal to zero, the following setof equations is obtained:

(20)


By solving Eq. (20), we get a simple and exact formula for determining the attribute weights as follows:

(21)

By normalizing be a unit, we have

(22)

Based on the above models, we develop a practical method for solving the MADM problems with preference information onalternatives, in which the information about attribute weights is incompletely known or completely unknown, and the attributevalues and preference values on alternatives take the form of INNs. The method involves the following steps:

(Procedure two)

Step 1: Let be an interval neutrosophic decision matrix, where

, which is an attribute value, given by an expert, for the alternative with respect to

the attribute , be the weight vector of attributes, where H is a set of the

known weight information, which can be constructed by the forms 1-5. Let be subjective preference value,

are INNs, which are subjective preference values on alternatives .

Step 2: If the information about the attribute weights is partly known, then we solve the model (M-3) to obtain the attributeweights. If the information about the attribute weights is completely unknown, then we can obtain the attribute weights by usingEq. (22).

Step 3: Utilize the weight vector and by Eq. (17), we obtain the overall values of the alternative.

Step 4: Calculate the scores of the overall INNs to rank all the alternatives and then to

select the best one(s) (if there is no difference between two scores and , then we need to calculate the accuracy

degrees and of the overall INNs and , respectively, and then rank the alternatives and in accordance with

the accuracy degrees and .

Step 5: Rank all the alternatives and select the best one(s) in accordance with and .

Step 6: End.


5. Numerical Example

Thus, in this section we shall present a numerical example to show potential evaluation of emerging technology commercializationwith single-valued neutrosophic information in order to illustrate the method proposed in this paper. There is a panel with fivepossible emerging technology enterprises to select. The experts selects four attribute to evaluate the fivepossible emerging technology enterprises: is the technical advancement; is the potential market and market risk; is the industrialization infrastructure, human resources and financial conditions; is the employment creation and thedevelopment of science and technology. The five possible emerging technology enterprises are to be evaluatedusing the SVNNs by the decision maker under the above four attributes, as listed in the following matrix.

Decision maker’s subjective preference value on alternative as follows:

In the following, we utilize the approach developed to show potential evaluation of emerging technology commercialization offive possible emerging technology enterprises.

Case 1: The information about the attribute weights is partly known and the known weight information is given as follows:

Step 1: Utilize the model (M-1) to establish the following single-objective programming model:

Solving this model, we get the weight vector of attributes:

Step 2. Utilize the weight vector and by Eq. (6), we obtain the overall values of the emerging technologyenterprises .


Step 3: Calculate the scores of the overall SVNs .

Step 4: Rank all the emerging technology enterprises in accordance with the scores of theoverall SVNNs and thus the most desirable emerging technology enterprise is A5.

Case 2: If the information about the attribute weights is completely unknown, we utilize another approach developed to get themost desirable emerging technology enterprise(s).

Step 1: Utilize the Eq. (11) to get the weight vector of attributes:

Step 2: Utilize the weight vector and by Eq. (6), we obtain the overall values of the emerging technologyenterprises .

Step 3: Calculate the scores of the overall SVNNs .

Step 4: Rank all the emerging technology enterprises in accordance with the scores ,and thus the most desirable emerging technology enterprise is A2.

If the five possible emerging technology enterprises are to be evaluated using the interval neutrosophicinformation by the decision maker under the above four attributes, as listed in the following matrix.


Decision maker’s subjective preference value on alternative as follows:

In the following, we utilize the approach developed to show potential evaluation of emerging technology commercialization offive possible emerging technology enterprises.

Case 1: The information about the attribute weights is partly known and the known weight information is given as follows:

Step 1: Utilize the model (M-3) to establish the following single-objective programming model:

Solving this model, we get the weight vector of attributes:

Step 2: Utilize the weight vector and by Eq. (17), we obtain the overall values of the emerging technologyenterprises .

Step 3. Calculate the scores of the overall INNs

Step 4: Rank all the emerging technology enterprise in accordance with the scores of the

overall INNs , and thus the most desirable alternative is A2..


Case 2: If the information about the attribute weights is completely unknown, we utilize another approach developed to get themost desirable emerging technology enterprise (s).

Step 1: Utilize the Eq. (22) to get the weight vector of attributes:

Step 2. Utilize the weight vector and by Eq. (17), we obtain the overall values of the emerging technologyenterprise .

Step 3: Calculate the scores of the overall INNs .

Step 4: Rank all the emerging technology enterprise in accordance with the scores ,and thus the most desirable emerging technology enterprise is A2.

6. Conclusion

In this paper, we have investigated single-valued neutrosophic MADM problems with preference information on alternatives,in which the information about attribute weights is incompletely known, and the attribute values and preference values onalternatives take the form of SVNNs. To determine the attribute weights, an optimization model based on the minimum deviationmethod, by which the attribute weights can be determined, is established. For the special situations where the information aboutattribute weights is completely unknown, we establish another optimization model. By solving this model, we get a simple andexact formula, which can be used to determine the attribute weights. We utilize the single-valued neutrosophic weightedaveraging (SVNWA) operator to aggregate the single-valued neutrosophic information corresponding to each alternative, andthen rank the alternatives and select the most desirable one(s) according to the score function and accuracy function. Themethod can sufficiently utilize the objective information, and meet decision makers’ subjective preference, can be easily performedon computer. Furthermore, we have extended the above results to interval neutrosophic environment. Finally, a practical examplefor potential evaluation of emerging technology commercialization is given to verify the developed approach and to demonstrateits practicality and effectiveness.

In the future, the application of the proposed models and methods of SVNSs and INSs needs to be explored in the decisionmaking[67-75], risk analysis[76-84] and many other uncertain and fuzzy environment [85-96].

Acknowledgment

The work was supported by the National Natural Science Foundation of China under Grant No. 61174149 and 71571128 and theHumanities and Social Sciences Foundation of Ministry of Education of the People’s Republic of China (17XJA630003) and theConstruction Plan of Scientific Research Innovation Team for Colleges and Universities in Sichuan Province (15TD0004).


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