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Journal of Intelligent & Fuzzy Systems 26 (2014) 2459–2466 DOI:10.3233/IFS-130916 IOS Press

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A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets Jun Ye∗

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Department of Electrical and Information Engineering, Shaoxing University, Shaoxing, Zhejiang Province, P.R. China

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Abstract. The paper introduces the concept of simplified neutrosophic sets (SNSs), which are a subclass of neutrosophic sets, and defines the operational laws of SNSs. Then, we propose some aggregation operators, including a simplified neutrosophic weighted arithmetic average operator and a simplified neutrosophic weighted geometric average operator. Based on the two aggregation operators and cosine similarity measure for SNSs, a multicriteria decision-making method is established in which the evaluation values of alternatives with respective to criteria are represented by the form of SNSs. The ranking order of alternatives is performed through the cosine similarity measure between an alternative and the ideal alternative and the best one(s) can be determined as well. Finally, a numerical example shows the application of the proposed method.

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1. Introduction

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Keywords: Neutrosophic set, simplified neutrosophic set, operational laws, aggregation operator, cosine similarity measure, multicriteria decision-making

Intuitionistic fuzzy sets [9] and interval-valued intuitionistic fuzzy sets [10] can only handle incomplete information but not the indeterminate information and inconsistent information which exist commonly in real situations. Then, the neutrosophic set proposed by Smarandache is a powerful general formal framework which generalizes the concept of the classic set, fuzzy set [11], interval valued fuzzy set [6], intuitionistic fuzzy set [9], interval-valued intuitionistic fuzzy set [10], paraconsistent set [1], dialetheist set [1], paradoxist set [1], and tautological set [1]. So the notion of neutrosophic sets is more general and overcomes the ∗ Corresponding author. Jun Ye, Department of Electrical and Information Engineering, Shaoxing University, 508 Huancheng West Road, Shaoxing, Zhejiang Province 312000, P.R. China. Tel.: +86 575 88327323; E-mail: [email protected].

aforementioned issues. In the neutrosophic set, indeterminacy is quantified explicitly and truth-membership, indeterminacy-membership, and false-membership are independent. This assumption is very important in many applications such as information fusion in which the data are combined from different sensors. Recently, neutrosophic sets have been applied to image thresholding, image denoise applications, and image segmentation. Cheng and Guo [3] proposed a thresholding algorithm based on neutrosophy, which could select the thresholds automatically and effectively. Guo et al. [18] defined some concepts and operators based on neutrosophic sets and applied them for image denoising, which can process not only noisy images with different levels of noise, but also images with different kinds of noise well. Guo and Cheng [19] applied neutrosophic sets to process the images with noise and proposed a novel neutrosophic approach for image segmentation.

1064-1246/14/$27.50 © 2014 – IOS Press and the authors. All rights reserved

J. Ye / A multicriteria decision-making method using aggregation operators

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sets (SNSs), which can be described by three real numbers in the real unit interval [0, 1], and some operational laws for SNSs and to propose two aggregation operators, including a simplified neutrosophic weighted arithmetic average operator and a simplified neutrosophic weighted geometric average operator. Then, a multicriteria decision-making method using the two aggregation operators of SNSs is established in which the evaluation information of alternatives with respect to criteria is given by truth-membership degree, indeterminacy-membership degree, and falsitymembership degree under the simplified neutrosophic environment. And then the ranking order of alternatives is performed through the cosine similarity measure between an alternative and the ideal alternative and the best choice can be obtained according to the measure values. However, the main advantage of the proposed simplified neutrosophic multicriteria decision-making method can handle not only incomplete information but also the indeterminate information and inconsistent information which exist commonly in real situations. The rest of paper is organized as follows. Section 2 introduces the some concepts of neutrosophic sets. SNSs and some operational laws are defined and two simplified neutrosophic weighted aggregation operators are proposed in Section 3. The two simplified neutrosophic weighted aggregation operators and cosine similarity measure for SNSs are applied to a multicriteria decision-making problem under the simplified neutrosophic environment and through the cosine similarity measure between each alternative and the ideal alternative, the ranking order of alternatives and the best one(s) can be obtained in Section 4. In Section 5, a numerical example demonstrates the application of the proposed decision-making method. Finally, some final remarks and future research are offered in Section 6.

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In any multicriteria decision-making problem, the final solution must be obtained from the synthesis of performance degrees of criteria [2]. To do this, the aggregation of information is fundamental. Therefore, many researchers have developed a variety of aggregation operators with assessment on [0, 1] [12–15, 20–22], proportional assessment on [1/9, 9] [2], and linguistic assessment [16, 17]. Then two of the most common operators for aggregating arguments are the weighted arithmetic average operator and the weighted geometric average operator [20–22], which have been wildly applied to decision-making problems. Therefore, these aggregation operators are important tools for aggregating fuzzy information, intuitionistic fuzzy information, interval-valued fuzzy information, and interval-valued intuitionistic fuzzy information in the decision-making problems. Whereas the neutrosophic set generalizes the above mentioned sets from philosophical point of view. From scientific or engineering point of view, the neutrosophic set and set-theoretic operators need to be specified. Otherwise, it will be difficult to apply it to the real applications. Therefore, Wang et al. [4] proposed interval neutrosophic sets (INSs) and some operators of INSs. Then, Ye [7] defined the Hamming and Euclidean distances between INSs and developed the similarity measures between INSs based on the relationship between similarity measures and distances and a multicriteria decision-making method using the similarity measures between INSs in interval neutrosophic setting, in which criterion values with respect to alternatives are evaluated by the form of INSs. Recently, Wang et al. [5] proposed a single valued neutrosophic set (SVNS), which is an instance of the neutrosophic set, and provide the settheoretic operators and various properties of SVNSs. Furthermore, Ye [8] presented the information energy of SVNSs, correlation of SVNSs, correlation coefficient of SVNSs, and weighted correlation coefficient of SVNSs based on the extension of the correlation of intuitionistic fuzzy sets and demonstrated that the cosine similarity measure is a special case of the correlation coefficient in single valued neutrosophic setting, and then applied them to single valued neutrosophic decision-making problems. Meanwhile, motivated by some intuitionistic fuzzy aggregation operators with assessment on [0, 1] [20–22], we can also extend them to neutrosophic sets. Thus, it will be necessary to develop some aggregation operators for aggregating neutrosophic information in the decision-making applications. To do so, the main purposes of this paper are to define the concept of simplified neutrosophic

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2. Some concepts of neutrosophic sets Neutrosophic set is a part of neutrosophy, which studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra [1], and is a powerful general formal framework. Neutrosophic set permits one to incorporate indeterminacy, hesitation and/or uncertainty independent of the membership and non-membership information. Thus the notion of neutrosophic set is a generalization of fuzzy, intuitionistic fuzzy, and interval-valued sets. Smarandache [1] gave the following definition of a neutrosophic set.

J. Ye / A multicriteria decision-making method using aggregation operators

Deﬁnition 3. [1] A neutrosophic set A is contained in the other neutrosophic set B, A ⊆ B if and only if inf TA (x)≤inf TB (x), sup TA (x)≤sup TB (x), inf IA (x)≥inf IB (x), sup IA (x)≥sup IB (x), inf FA (x)≥inf FB (x), and sup FA (x)≥sup FB (x) for every x in X.

A = {x, TA (x), IA (x), FA (x) |x ∈ X}

which is called a SNS. It is a subclass of neutrosophic sets. In this paper, we shall use the SNS whose TA (x), IA (x) and FA (x) values are single points in the real standard [0, 1] instead of subintervals/subsets in the real standard [0, 1]. Thus, each SNS can be described by three real numbers in the real unit interval [0, 1]. Therefore, the sum of TA (x ) ∈ [0, 1], IA (x ) ∈ [0, 1] and FA (x ) ∈ [0, 1] satisfies the condition 0 ≤ TA (x ) + IA (x ) + FA (x ) ≤ 3. For the sake of simplicity, the SNS A = {x, TA (x), IA (x), FA (x) |x ∈ X} is denoted by the simplified symbol A = TA (x), IA (x), FA (x) . In this case, we can give the following definitions.

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Deﬁnition 4. [1] The union of two neutrosophic sets A and B is a neutrosophic set C, written as C = A∪B, whose truth-membership, indeterminacy membership and false-membership functions are related to those of A and B by TC (x)=TA (x) ⊕ TB (x) TA (x) TB (x), IC (x) = IA (x) ⊕ IB (x) IA (x) IB (x), and FC (x) = FA (x) ⊕ FB (x) FA (x) FB (x) for any x in X.

Deﬁnition 6. Let X be a space of points (objects), with a generic element in X denoted by x. A neutrosophic set A in X is characterized by a truth-membership function TA (x), a indeterminacy-membership function IA (x) and a falsity-membership function FA (x). If the functions TA (x), IA (x) and FA (x) are singleton subintervals/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]. Then, a simplification of the neutrosophic set A is denoted by

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Deﬁnition 2. [1] The complement of a neutrosophic c (x ) = set A is denoted by Ac and is defined as TA c + + {1 } TA (x ), IA (x ) = {1 } IA (x ), and FA c (x ) = {1+ } FA (x ) for every x in X.

of SNSs, which are a subclass of neutrosophic sets, and the operational laws of SNSs, then propose two weighted aggregation operators of SNSs for the information aggregation in neutrosophic decision making problems.

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Deﬁnition 1. [1] Let X be a space of points (objects), with a generic element in X denoted by x. A neutrosophic set A in X is characterized by a truth-membership function TA (x), an indeterminacy-membership function IA (x) and a falsity-membership function FA (x). The functions TA (x), IA (x) and FA (x) are real standard or nonstandard subsets of ]0– , 1+ [, that is 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 IA (x)+sup FA (x)≤3+ .

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Deﬁnition 5. [1] The intersection of two neutrosophic sets A and B is a neutrosophic set C, written as C = A ∩ B, whose truth-membership, indeterminacy-membership and false-membership functions are related to those of A and B by TC (x) = TA (x) TB (x), IC (x) = IA (x) IB (x), and FC (x) = FA (x) FB (x) for any x in X.

3. Operational relations of SNSs Smarandache [1] has provided a variety of real-life examples for possible applications of his neutrosophic sets, which are comprised of three subsets of the nonstandard interval ]0– , 1+ [, i.e., TA (x) ⊆]0– , 1+ [, IA (x) ⊆]0– , 1+ [, and FA (x) ⊆]0– , 1+ [. However, it is difficult to apply neutrosophic sets to practical problems. Therefore, we will reduce neutrosophic sets of nonstandard intervals into a kind of SNSs of standard intervals that will preserve the operations of the neutrosophic sets. In this section, we introduce the concept

Deﬁnition 7. The SNS A is contained in the other SNS B, A ⊆B if and only if TA (x)≤TB (x), IA (x)≥IB (x), and FA (x)≥FB (x) for every x in X. Deﬁnition 8. Let A, B are two SNSs. Operational relations are defined by A + B = TA (x) + TB (x) − TA (x)TB (x), IA (x) + IB (x) − IA (x)IB (x), FA (x) + FB (x) − FA (x)FB (x)

(1)

A · B = TA (x)TB (x), IA (x) IB (x), FA (x)FB (x)

(2)

λA = 1 − (1 − TA (x))λ , 1 − (1 − IA (x))λ , 1 − (1 − FA (x))λ , λ > 0 (3)

J. Ye / A multicriteria decision-making method using aggregation operators

Aλ = TAλ (x), IAλ (x), FAλ (x) , λ > 0

(4)

Based on the above operational laws, we can propose the following weighted arithmetic aggregation operator and weighted geometric aggregation operator for SNSs. Deﬁnition 9. Let Aj (j = 1, 2, . . . , n) be a SNS. The simplified neutrosophic weighted arithmetic average operator is defined by Fw (A1 , A2 , . . . , An ) =

n

wj Aj

(5)

Proof. The proof of Equation (7) can be done by means of mathematical induction. (1) When n = 2, then, w1 A1 = 1 − (1 − TA1 (x))w1, 1 − (1 − IA1 (x))w1 , 1 − (1 − FA1 (x))w1 , w2 A2 = 1 − (1 − TA2 (x))w2, 1 − (1 − IA2 (x))w2 , 1 − (1 − FA2 (x))w2 . Thus, Fw (A1 , A2 ) = w1 A1 + w2 A2 = 2 − (1 − TA1 (x))w1 − (1 − TA2 (x))w2

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j=1

−(1 − (1 − TA1 (x))w1 )(1 − (1 − TA2 (x))w2 ),

j=1

Especially, Assume W = (1/n, 1/n, . . . , 1/n), then Gw is called as an arithmetic average operator for SNSs.

Gw (A1 , A2 , . . . , An ) =

n

w Aj j

(6)

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j=1

where W = (w1 , w2 , . . . , wn ) is the weight vector of Aj n (j = 1, 2, . . . , n), wj ∈ [0, 1] and wj = 1. Especially, j=1

−(1 − (1 − IA1 (x))w1 )(1 − (1 − IA2 (x))w2 ), 2 − (1 − FA1 (x))w1 − (1 − FA2 (x))w2

−(1 − (1 − FA1 (x))w1 )(1 − (1 − FA2 (x))w2 ) = 1 − (1 − TA1 (x))w1 (1 − TA2 (x))w2 ,

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Deﬁnition 10. Let Aj (j = 1, 2, . . . , n) be a SNS. The simplified neutrosophic weighted geometric average operator is defined by

2 − (1 − IA1 (x))w1 − (1 − IA2 (x))w2

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where W = (w1 , w2 , . . . , wn ) is the weight vector n of Aj (j = 1, 2, . . . , n), wj ∈ [0, 1] and wj = 1.

1 − (1 − IA1 (x))w1 (1 − IA2 (x))w2 , 1 − (1 − FA1 (x))w1 (1 − FA2 (x))w2

j=1

1−

1−

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n

(1 − TAj (x))wj ,

j=1

1−

n

1−

n

(1 − IAj (x)) , (1 − FAj (x))

wj

1−

k

(1 − FAj )wj

(7)

where W=(w1 , w2 , . . . , wn ) is the weight vector of Aj n (j = 1, 2, . . . , n), wj ∈ [0, 1] and wj = 1.

(9)

j=1

(3) When n = k+1, by applying Equations (8) and (9), we can get

j=1

j=1

j=1

(1 − IAj )wj ,

Fw (A1 , A2 , . . . , Ak+1 ) k = 1− (1 − TAj (x))wj

wj

j=1

k j=1

Theorem 1. For a SNS Aj (j = 1, 2, . . . , n), we have the following result by use of Equation (5): Fw (A1 , A2 , . . . , An ) =

(8)

(2) When n = k, by applying Equation (7), we get k Fw (A1 , A2 , . . . , Ak ) = 1 − (1 − TAj )wj ,

Assume W = (1/n, 1/n, . . . , 1/n), then Gw is called as an geometric average operator for SNSs.

+(1 − (1 − TAk+1 (x))wk+1 ) −(1 −

k

(1 − TAj (x))wj )

j=1

(1 − (1 − TAk+1 (x))wk+1 ),

J. Ye / A multicriteria decision-making method using aggregation operators k

1−

Theorem 2. For a SNS Aj (j = 1, 2, . . . , n), we have the following result by applying Equation (6):

(1 − IAj (x))wj

j=1

Gw (A1 , A2, . . . , An ) n n n wj wj wj = TAj (x), IAj (x), FAj (x) (11)

+(1 − (1 − IAk+1 (x))wk+1 ) −(1 −

k

j=1

(1 − IAj (x))wj )

(1 − (1 − IAk+1 (x)) 1−

),

j=1

By a similar proof manner, we can give the proof of Theorem 2 (omitted). It is obvious that the Gw operator has the following properties:

(1 − IAj (x))wj

j=1

+(1 − (1 − FAk+1 (x))wk+1 ) k j=1

=

(1) Idempotency: Let Aj (j = 1, 2, ..., n) be a collection of SNSs. If Ai (j = 1, 2,..., n) is equal, i.e. Aj = A for j = 1, 2,..., n, then Gw (A1 , A2 , . . . , An ) = A. (2) Boundedness: Let Aj (j = 1, 2, ..., n) be a collection of SNSs and let A− =

(1 − FAj (x))wj )

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−(1 −

(1 − (1 − FAk+1 (x))wk+1 )

1−

k+1

(1 − TAj (x))wj , (1 − IAj (x))wj ,

j=1

1−

k+1

(1 − FAj (x))

j=1

wj

(10)

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k+1

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j=1

1−

Therefore, considering the above results, we have Equation (7) for any n. This completes the proof.

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It is obvious that the Fw operator has the following properties:

(1) Idempotency: Let Aj (j = 1, 2,..., n) be a collection of SNSs. If Ai (j = 1, 2,..., n) is equal, i.e. Aj = A for j = 1, 2,..., n, then Fw (A1 , A2 , . . . , An ) = A. (2) Boundedness: Let Aj (j = 1, 2, ..., n) − be a collection of SNSs and let A = min TAj (x), max IAj (x), max FAj (x) and j j j A+ = max TAj (x), min IAj (x), min FAj (x) j

j

j=1

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k

j=1

where W = (w1 , w2 , . . . , wn ) is the weight vector of Aj n (j = 1, 2, . . . , n), wj ∈ [0, 1] and wj = 1.

j=1 wk+1

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j

for j = 1, 2,..., n, then A− ⊆ Fw (A1 , A2 , . . . , An ) ⊆ A+ . (3) Monotonity: Let Aj (j = 1, 2, ..., n) be a collection of SNSs. If Aj ⊆Aj ∗ for j = 1,

2,..., n, then Fw (A1 , A2 , . . . , An ) ⊆ Fw A∗1 , A∗2 , . . . , A∗n .

min TAj (x), max IAj (x), max FAj (x) and j j j A+ = max TAj (x), min IAj (x), min FAj (x) j

j

for j = 1, 2, ..., n, then A2 , . . . , An ) ⊆ A+ . (3) Monotonity: Let Aj (j = 1, a collection of SNSs. If j = 1, 2, ..., n, then Gw (A1 , Gw A∗1 , A∗2 , . . . , A∗n .

j

A− ⊆ Gw (A1 , 2, ..., n) be Aj ⊆Aj ∗ for A2 , . . . , An ) ⊆

The aggregation results Fw and Gw are still SNSs. Obviously, there are different focal points between Equations (7) and (11). The weighted arithmetic average operator emphasizes group’s major points, and then the weighted geometric average operator emphasizes personal major points.

4. Decision-making method based on the simpliﬁed neutrosophic weighted aggregation operators In this section, we present a handling method for multicriteria decision-making problems by means of the two aggregation operators and cosine similarity measure for SNSs under the simplified neutrosophic environment. Let A = {A1 , A2 , . . . , Am } be a set of alternatives and let C = {C1 , C2 , . . . , Cn } be a set of criteria. Assume that the weight of the criterion Cj (j = 1, 2, . . . , n),

J. Ye / A multicriteria decision-making method using aggregation operators

entered by the decision-maker, is wj , wj ∈ [0, 1] and n j=1 wj = 1. In the decision process, the evaluation information of the alternative Ai on the criteria is represented by the form of a SNS: Ai =

Cj , TAi (Cj ), IAi (Cj ), FAi (Cj ) |Cj ∈ C ,

where 0 ≤ TAi (Cj ) + IAi (Cj ) + FAi (Cj ) ≤ 3, TAi (Cj ) ≥ 0, IAi (Cj ) ≥ 0, FAi (Cj ) ≥ 0, j = 1, 2, . . . , n, and i = 1, 2, . . . , m. For convenience, the value of SNSs is denoted by αij = tij , iij , fij (i = 1, 2, . . . , m; j = 1, 2, . . . , n). Therefore, we can get a simplified neutrosophic decision matrix D = (αij )m×n :

Step 1: Calculate the weighted arithmetic average values by using Equation (7) or the weighted geometric average values by using Equation (11) Step 2: Calculate the cosine similarity measure between each alternative and the ideal alternative by using Equation (12). Step 3: Give the ranking order of the alternatives from the obtained measure values, and then get the best choice. Step 4: End.

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⎤ t11 , i11 , f11 t12 , i12 , f12 · · · t1n , i1n , f1n ⎥ ⎢ t ⎢ 21 , i21 , f21 t22 , i22 , f22 · · · t2n , i2n , f2n ⎥ ⎥ ⎢ =⎢ ⎥. .. .. .. .. ⎥ ⎢ . . . . ⎦ ⎣ tm1 , im1 , fm1 tm2 , im2 , fm2 · · · tmn , imn , fmn

D = (αij )m×n

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⎡

Si (αi , α∗ ) = =

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Then, the aggregating simplified neutrosophic value αi for Ai (i = 1, 2, . . . , m) is αi = ti , ii , fi = Fiw (αi 1 , αi 2 , . . . , αi n ) or αi = ti , ii , fi = Giw (αi 1 , αi 2 , . . . , αi n ) is obtained by applying Equations (7) or (11) according to each row in the simplified neutrosophic decision matrix D = (aij )m ×n . To rank alternatives in the decision-making process, we define an ideal SNS value as the ideal alternative ␣∗ = 1, 0, 0 , and then based on the cosine similarity measure between SVNSs proposed by Ye [8], the cosine similarity measure between SNSs ␣i (i = 1, 2,..., m) and ␣∗ can be defined as follows:

ti ti∗ + ii i∗i + fi fi∗

2 2 2 2 2 2 ti + i i + f i ti∗ + i∗i + fi∗ ti

(12)

ti2 + i2i + fi2

Then, the bigger the measure value Si (␣i , ␣∗ ) (i = 1, 2,..., m) is, the better the alternative Ai is, because the alternative Ai is close to the ideal alternative ␣∗ . Through the cosine similarity measure between each alternative and the ideal alternative, the ranking order of all alternatives can be determined and the best one can be easily identified as well. In summary, the decision procedure for the proposed method can be summarized as follows:

5. Numerical example

In this section, an example for a multicriteria decision-making problem of engineering alternatives is used as a demonstration of the application of the proposed decision-making method in a realistic scenario, as well as the application and effectiveness of the proposed decision-making method. Let us consider the decision-making problem adapted from [8]. There is an investment company, which wants to invest a sum of money in the best option. There is a panel with four possible alternatives to invest the money: (1) A1 is a car company; (2) A2 is a food company; (3) A3 is a computer company; (4) A4 is an arms company. The investment company must take a decision according to the following three criteria: (1) C1 is the risk; (2) C2 is the growth; (3) C3 is the environmental impact. Then, the weight vector of the criteria is given by W = (0.35, 0.25, 0.4), which is adopted from the literature [8]. For the evaluation of an alternative Ai (i = 1, 2, 3, 4) with respect to a criterion Cj (j = 1, 2, 3), it is obtained from the questionnaire of a domain expert. For example, when we ask the opinion of an expert about an alternative A1 with respect to a criterion C1 , he or she may say that the possibility in which the statement is good is 0.4

J. Ye / A multicriteria decision-making method using aggregation operators

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Step 1: We can obtain the weighted arithmetic average value (aggregating simplified neutrosophic value) ␣i for Ai (i = 1, 2, 3, 4) by using Equation (7): ␣1 = 0.3268, 0.2000, 0.3881 , ␣2 = 0.5627, 0.1414, 0.2000 , ␣3 =0.4375, 0.2416, 0.2616 , and ␣4 = 0.5746, 0.1555, 0.1663 . Step 2: By applying Equation (12), we can compute each cosine similarity measure Si (␣i , ␣∗ ) (i = 1, 2, 3, 4) as follows: S1 (␣1 , ␣∗ ) = 0.5992, S2 (␣2 , ␣∗ ) = 0.9169, S3 (␣3 , ␣∗ ) = 0.7756, and S4 (␣4 , ␣∗ ) = 0.9297. Step 3: From the measure value Si (␣i , ␣∗ ) (i = 1, 2, 3, 4) between an alternative and the ideal alternative, the ranking order of four alternatives is A4 A2 A3 A1 .

Obviously, we can see that the above two kinds of ranking orders and the best alternative are the same. The method proposed in this paper differs from existing approaches for fuzzy multi-criteria decision making not only due to the fact that the proposed method uses the SNS concept and two simplified neutrosophic aggregation operators, but also due to the consideration of the indeterminacy information besides truth and falsity information in the evaluation of the alternative with respect to criteria, which makes it have more feasible and practical than other traditional decision making methods in real decision-making problems. Therefore, its advantage is easily reflecting the ambiguous nature of subjective judgments because SNSs are suitable for capturing imprecise, uncertain, and inconsistent information in the multicriteria decision analysis.

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The proposed method is applied to solve this problem according to the following computational procedure:

Step 2’: By using Equation (12), we can compute each cosine similarity measure Si (␣i , ␣∗ ) (i = 1, 2, 3, 4) as follows: S1 (␣1 , ␣∗ ) = 0.5863, S2 (␣2 , ␣∗ ) = 0.9188, S3 (␣3 , ␣∗ ) = 0.7696, and S4 (␣4 , ␣∗ ) = 0.9601. Step 3’: The ranking order of four alternatives is A4 A2 A3 A1 . Thus, we can see that the alternative A4 is still the best choice among all the alternatives.

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and the statement is poor is 0.3 and the degree in which he or she is not sure is 0.2. For the neutrosophic notation, it can be expressed as ␣11 = 0.4, 0.2, 0.3 . Thus, when the four possible alternatives with respect to the above three criteria are evaluated by the expert, we can obtain the following simplified neutrosophic decision matrix D: ⎡ ⎤ 0.4, 0.2, 0.3 0.4, 0.2, 0.3 0.2, 0.2, 0.5 ⎢ ⎥ ⎢0.6, 0.1, 0.2 0.6, 0.1, 0.2 0.5, 0.2, 0.2 ⎥ ⎢ ⎥. D=⎢ ⎥ ⎣0.3, 0.2, 0.3 0.5, 0.2, 0.3 0.5, 0.3, 0.2 ⎦ 0.7, 0.0, 0.1 0.6, 0.1, 0.2 0.4, 0.3, 0.2

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Therefore, we can see that the alternative A4 is the best choice among all the alternatives. On the other hand, we can also utilize the weighted geometric average operator as the following computational procedure: Step 1’: We compute the weighted geometric average values by applying Equation (11) for Ai (i = 1, 2, 3, 4), each aggregating simplified neutrosophic value ␣i (i = 1, 2, 3, 4) is as follows: ␣1 = 0.3031, 0.2000, 0.3680 , ␣2 = 0.5578, 0.1320, 0.2000 , ␣3 = 0.4181, 0.2352, 0.2551 , and ␣4 = 0.5385, 0, 0.1569 .

6. Conclusion This paper introduced the concept of SNSs, which are a subclass of neutrosophic sets, and defined some operational laws of SNSs. Then, we proposed two aggregation operators for SNSs, including a simplified neutrosophic weighted arithmetic average operator and a simplified neutrosophic weighted geometric average operator. The two aggregation operators were applied to multicriteria decision-making problems under the simplified neutrosophic environment, in which criterion values with respect to alternatives are evaluated by the form of simplified neutrosophic values and the criterion weights are known information. We utilized the cosine similarity measure between an alternative and the ideal alternative to rank the alternatives and to determine the best one(s) according to the measure values. Finally, a numerical example is provided to illustrate the application of the developed approach. The proposed simplified neutrosophic multicriteria decision-making method is more suitable for real scien-

J. Ye / A multicriteria decision-making method using aggregation operators

tific and engineering applications because it can handle not only incomplete information but also the indeterminate information and inconsistent information which exist commonly in real situations. The techniques proposed in this paper can provide more useful way for decision-makers. In the future, we shall deal with group decision making problems with incomplete decision contexts and preference relations performed in the selection process under the simplified neutrosophic environment and apply the simplified neutrosophic aggregation operators to solve practical applications in other areas such as expert system, information fusion system, and medical diagnoses.

[9] [10] [11] [12]

[13]

[14]

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F. Smarandache, A unifying ﬁeld in logics. Neutrosophy: Neutrosophic probability, set and logic, American Research Press, Rehoboth 1999. F. Herrera, E. Herrera-Viedma and F. Chiclana, A study of the origin and uses of the ordered weighted geometric operator in multicriteria decision making, International Journal of Intelligent Systems 18(6) (2003), 689–707. H.D. Cheng and Y. Guo, A new neutrosophic approach to image thresholding, New Mathematics and Natural Computation 4(3) (2008), 291–308. H. Wang, F. Smarandache Y.Q. Zhang et al., Interval neutrosophic sets and logic: Theory and applications in computing, Hexis, Phoenix, AZ 2005. H. Wang, F. Smarandache Y.Q. Zhang, et al., Single valued neutrosophic sets, Multispace and Multistructure 4 (2010), 410–413. I. Turksen, Interval valued fuzzy sets based on normal forms, Fuzzy Sets and Systems 20 (1986), 191–210. J. Ye, Similarity measures between interval neutrosophic sets and their applications in multicriteria decision-making, Journal of Intelligent & Fuzzy Systems, DOI: 10.3233/IFS-120724 (2013). J. Ye, Multicriteria decision-making method using the correlation coefficient under single-valued neutrosophic environment,

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International Journal of General Systems 42(4) (2013), 386–394. K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), 87–96. K. Atanassov and G. Gargov, Interval valued intuitionistic fuzzy sets, Fuzzy Sets and Systems 31 (1989), 343–349. L.A. Zadeh, Fuzzy Sets, Information and Control 8 (1965), 338–353. P.D. Liu, A weighted aggregation operators multi-attribute group decision-making method based on interval-valued trapezoidal fuzzy numbers, Expert Systems with Applications 38(1) (2011), 1053–1060. P.D. Liu and F. Jin, A multi-attribute group decision-making method based on weighted geometric aggregation operators of interval-valued trapezoidal fuzzy numbers, Applied Mathematical Modelling 36(6) (2012), 2498–2509. P.D. Liu, X. Zhang and F. Jin, A multi-attribute group decisionmaking method based on interval-valued trapezoidal fuzzy numbers hybrid harmonic averaging operators, Journal of Intelligent & Fuzzy Systems 23(5) (2012), 159–168. P.D. Liu, Some Hamacher aggregation operators based on the interval-valued intuitionistic fuzzy numbers and their application to group decision making, IEEE Transactions on Fuzzy systems, (2013). DOI: 10.1109/TFUZZ.2013.2248736 P.D. Liu, Some generalized dependent aggregation operators with intuitionistic linguistic numbers and their application to group decision making, Journal of Computer and System Sciences 79(1) (2013), 131–143. P.D. Liu and F. Jin, Methods for aggregating intuitionistic uncertain linguistic variables and their application to group decision making, Information Sciences 205 (2012), 58–71. Y. Guo, H.D. Cheng, Y. Zhang and W. Zhao, A new neutrosophic approach to image denoising, New Mathematics and Natural Computation 5(3) (2009), 653–662. Y. Guo and H.D. Cheng, New neutrosophic approach to image segmentation, Pattern Recognition 42 (2009), 587–595. Z.S. Xu and R.R. Yager, Some geometric aggregation operators based on intuitionistic fuzzy sets, International Journal of General System 35 (2006), 417–433. Z.S. Xu, Intuitionistic fuzzy aggregation operators, IEEE Transactions on Fuzzy Systems 15(6) (2007), 1179– 1187. Z.S. Xu, Methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making, Control and decision 22(2) (2007), 215–219.

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[18]

[19] [20]

[21]

[22]

Journal of Intelligent & Fuzzy Systems 26 (2014) 2459–2466 DOI:10.3233/IFS-130916 IOS Press

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A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets Jun Ye∗

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Department of Electrical and Information Engineering, Shaoxing University, Shaoxing, Zhejiang Province, P.R. China

OR

Abstract. The paper introduces the concept of simplified neutrosophic sets (SNSs), which are a subclass of neutrosophic sets, and defines the operational laws of SNSs. Then, we propose some aggregation operators, including a simplified neutrosophic weighted arithmetic average operator and a simplified neutrosophic weighted geometric average operator. Based on the two aggregation operators and cosine similarity measure for SNSs, a multicriteria decision-making method is established in which the evaluation values of alternatives with respective to criteria are represented by the form of SNSs. The ranking order of alternatives is performed through the cosine similarity measure between an alternative and the ideal alternative and the best one(s) can be determined as well. Finally, a numerical example shows the application of the proposed method.

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1. Introduction

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Keywords: Neutrosophic set, simplified neutrosophic set, operational laws, aggregation operator, cosine similarity measure, multicriteria decision-making

Intuitionistic fuzzy sets [9] and interval-valued intuitionistic fuzzy sets [10] can only handle incomplete information but not the indeterminate information and inconsistent information which exist commonly in real situations. Then, the neutrosophic set proposed by Smarandache is a powerful general formal framework which generalizes the concept of the classic set, fuzzy set [11], interval valued fuzzy set [6], intuitionistic fuzzy set [9], interval-valued intuitionistic fuzzy set [10], paraconsistent set [1], dialetheist set [1], paradoxist set [1], and tautological set [1]. So the notion of neutrosophic sets is more general and overcomes the ∗ Corresponding author. Jun Ye, Department of Electrical and Information Engineering, Shaoxing University, 508 Huancheng West Road, Shaoxing, Zhejiang Province 312000, P.R. China. Tel.: +86 575 88327323; E-mail: [email protected].

aforementioned issues. In the neutrosophic set, indeterminacy is quantified explicitly and truth-membership, indeterminacy-membership, and false-membership are independent. This assumption is very important in many applications such as information fusion in which the data are combined from different sensors. Recently, neutrosophic sets have been applied to image thresholding, image denoise applications, and image segmentation. Cheng and Guo [3] proposed a thresholding algorithm based on neutrosophy, which could select the thresholds automatically and effectively. Guo et al. [18] defined some concepts and operators based on neutrosophic sets and applied them for image denoising, which can process not only noisy images with different levels of noise, but also images with different kinds of noise well. Guo and Cheng [19] applied neutrosophic sets to process the images with noise and proposed a novel neutrosophic approach for image segmentation.

1064-1246/14/$27.50 © 2014 – IOS Press and the authors. All rights reserved

J. Ye / A multicriteria decision-making method using aggregation operators

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sets (SNSs), which can be described by three real numbers in the real unit interval [0, 1], and some operational laws for SNSs and to propose two aggregation operators, including a simplified neutrosophic weighted arithmetic average operator and a simplified neutrosophic weighted geometric average operator. Then, a multicriteria decision-making method using the two aggregation operators of SNSs is established in which the evaluation information of alternatives with respect to criteria is given by truth-membership degree, indeterminacy-membership degree, and falsitymembership degree under the simplified neutrosophic environment. And then the ranking order of alternatives is performed through the cosine similarity measure between an alternative and the ideal alternative and the best choice can be obtained according to the measure values. However, the main advantage of the proposed simplified neutrosophic multicriteria decision-making method can handle not only incomplete information but also the indeterminate information and inconsistent information which exist commonly in real situations. The rest of paper is organized as follows. Section 2 introduces the some concepts of neutrosophic sets. SNSs and some operational laws are defined and two simplified neutrosophic weighted aggregation operators are proposed in Section 3. The two simplified neutrosophic weighted aggregation operators and cosine similarity measure for SNSs are applied to a multicriteria decision-making problem under the simplified neutrosophic environment and through the cosine similarity measure between each alternative and the ideal alternative, the ranking order of alternatives and the best one(s) can be obtained in Section 4. In Section 5, a numerical example demonstrates the application of the proposed decision-making method. Finally, some final remarks and future research are offered in Section 6.

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TH

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In any multicriteria decision-making problem, the final solution must be obtained from the synthesis of performance degrees of criteria [2]. To do this, the aggregation of information is fundamental. Therefore, many researchers have developed a variety of aggregation operators with assessment on [0, 1] [12–15, 20–22], proportional assessment on [1/9, 9] [2], and linguistic assessment [16, 17]. Then two of the most common operators for aggregating arguments are the weighted arithmetic average operator and the weighted geometric average operator [20–22], which have been wildly applied to decision-making problems. Therefore, these aggregation operators are important tools for aggregating fuzzy information, intuitionistic fuzzy information, interval-valued fuzzy information, and interval-valued intuitionistic fuzzy information in the decision-making problems. Whereas the neutrosophic set generalizes the above mentioned sets from philosophical point of view. From scientific or engineering point of view, the neutrosophic set and set-theoretic operators need to be specified. Otherwise, it will be difficult to apply it to the real applications. Therefore, Wang et al. [4] proposed interval neutrosophic sets (INSs) and some operators of INSs. Then, Ye [7] defined the Hamming and Euclidean distances between INSs and developed the similarity measures between INSs based on the relationship between similarity measures and distances and a multicriteria decision-making method using the similarity measures between INSs in interval neutrosophic setting, in which criterion values with respect to alternatives are evaluated by the form of INSs. Recently, Wang et al. [5] proposed a single valued neutrosophic set (SVNS), which is an instance of the neutrosophic set, and provide the settheoretic operators and various properties of SVNSs. Furthermore, Ye [8] presented the information energy of SVNSs, correlation of SVNSs, correlation coefficient of SVNSs, and weighted correlation coefficient of SVNSs based on the extension of the correlation of intuitionistic fuzzy sets and demonstrated that the cosine similarity measure is a special case of the correlation coefficient in single valued neutrosophic setting, and then applied them to single valued neutrosophic decision-making problems. Meanwhile, motivated by some intuitionistic fuzzy aggregation operators with assessment on [0, 1] [20–22], we can also extend them to neutrosophic sets. Thus, it will be necessary to develop some aggregation operators for aggregating neutrosophic information in the decision-making applications. To do so, the main purposes of this paper are to define the concept of simplified neutrosophic

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2. Some concepts of neutrosophic sets Neutrosophic set is a part of neutrosophy, which studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra [1], and is a powerful general formal framework. Neutrosophic set permits one to incorporate indeterminacy, hesitation and/or uncertainty independent of the membership and non-membership information. Thus the notion of neutrosophic set is a generalization of fuzzy, intuitionistic fuzzy, and interval-valued sets. Smarandache [1] gave the following definition of a neutrosophic set.

J. Ye / A multicriteria decision-making method using aggregation operators

Deﬁnition 3. [1] A neutrosophic set A is contained in the other neutrosophic set B, A ⊆ B if and only if inf TA (x)≤inf TB (x), sup TA (x)≤sup TB (x), inf IA (x)≥inf IB (x), sup IA (x)≥sup IB (x), inf FA (x)≥inf FB (x), and sup FA (x)≥sup FB (x) for every x in X.

A = {x, TA (x), IA (x), FA (x) |x ∈ X}

which is called a SNS. It is a subclass of neutrosophic sets. In this paper, we shall use the SNS whose TA (x), IA (x) and FA (x) values are single points in the real standard [0, 1] instead of subintervals/subsets in the real standard [0, 1]. Thus, each SNS can be described by three real numbers in the real unit interval [0, 1]. Therefore, the sum of TA (x ) ∈ [0, 1], IA (x ) ∈ [0, 1] and FA (x ) ∈ [0, 1] satisfies the condition 0 ≤ TA (x ) + IA (x ) + FA (x ) ≤ 3. For the sake of simplicity, the SNS A = {x, TA (x), IA (x), FA (x) |x ∈ X} is denoted by the simplified symbol A = TA (x), IA (x), FA (x) . In this case, we can give the following definitions.

TH

OR

Deﬁnition 4. [1] The union of two neutrosophic sets A and B is a neutrosophic set C, written as C = A∪B, whose truth-membership, indeterminacy membership and false-membership functions are related to those of A and B by TC (x)=TA (x) ⊕ TB (x) TA (x) TB (x), IC (x) = IA (x) ⊕ IB (x) IA (x) IB (x), and FC (x) = FA (x) ⊕ FB (x) FA (x) FB (x) for any x in X.

Deﬁnition 6. Let X be a space of points (objects), with a generic element in X denoted by x. A neutrosophic set A in X is characterized by a truth-membership function TA (x), a indeterminacy-membership function IA (x) and a falsity-membership function FA (x). If the functions TA (x), IA (x) and FA (x) are singleton subintervals/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]. Then, a simplification of the neutrosophic set A is denoted by

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Deﬁnition 2. [1] The complement of a neutrosophic c (x ) = set A is denoted by Ac and is defined as TA c + + {1 } TA (x ), IA (x ) = {1 } IA (x ), and FA c (x ) = {1+ } FA (x ) for every x in X.

of SNSs, which are a subclass of neutrosophic sets, and the operational laws of SNSs, then propose two weighted aggregation operators of SNSs for the information aggregation in neutrosophic decision making problems.

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Deﬁnition 1. [1] Let X be a space of points (objects), with a generic element in X denoted by x. A neutrosophic set A in X is characterized by a truth-membership function TA (x), an indeterminacy-membership function IA (x) and a falsity-membership function FA (x). The functions TA (x), IA (x) and FA (x) are real standard or nonstandard subsets of ]0– , 1+ [, that is 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 IA (x)+sup FA (x)≤3+ .

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Deﬁnition 5. [1] The intersection of two neutrosophic sets A and B is a neutrosophic set C, written as C = A ∩ B, whose truth-membership, indeterminacy-membership and false-membership functions are related to those of A and B by TC (x) = TA (x) TB (x), IC (x) = IA (x) IB (x), and FC (x) = FA (x) FB (x) for any x in X.

3. Operational relations of SNSs Smarandache [1] has provided a variety of real-life examples for possible applications of his neutrosophic sets, which are comprised of three subsets of the nonstandard interval ]0– , 1+ [, i.e., TA (x) ⊆]0– , 1+ [, IA (x) ⊆]0– , 1+ [, and FA (x) ⊆]0– , 1+ [. However, it is difficult to apply neutrosophic sets to practical problems. Therefore, we will reduce neutrosophic sets of nonstandard intervals into a kind of SNSs of standard intervals that will preserve the operations of the neutrosophic sets. In this section, we introduce the concept

Deﬁnition 7. The SNS A is contained in the other SNS B, A ⊆B if and only if TA (x)≤TB (x), IA (x)≥IB (x), and FA (x)≥FB (x) for every x in X. Deﬁnition 8. Let A, B are two SNSs. Operational relations are defined by A + B = TA (x) + TB (x) − TA (x)TB (x), IA (x) + IB (x) − IA (x)IB (x), FA (x) + FB (x) − FA (x)FB (x)

(1)

A · B = TA (x)TB (x), IA (x) IB (x), FA (x)FB (x)

(2)

λA = 1 − (1 − TA (x))λ , 1 − (1 − IA (x))λ , 1 − (1 − FA (x))λ , λ > 0 (3)

J. Ye / A multicriteria decision-making method using aggregation operators

Aλ = TAλ (x), IAλ (x), FAλ (x) , λ > 0

(4)

Based on the above operational laws, we can propose the following weighted arithmetic aggregation operator and weighted geometric aggregation operator for SNSs. Deﬁnition 9. Let Aj (j = 1, 2, . . . , n) be a SNS. The simplified neutrosophic weighted arithmetic average operator is defined by Fw (A1 , A2 , . . . , An ) =

n

wj Aj

(5)

Proof. The proof of Equation (7) can be done by means of mathematical induction. (1) When n = 2, then, w1 A1 = 1 − (1 − TA1 (x))w1, 1 − (1 − IA1 (x))w1 , 1 − (1 − FA1 (x))w1 , w2 A2 = 1 − (1 − TA2 (x))w2, 1 − (1 − IA2 (x))w2 , 1 − (1 − FA2 (x))w2 . Thus, Fw (A1 , A2 ) = w1 A1 + w2 A2 = 2 − (1 − TA1 (x))w1 − (1 − TA2 (x))w2

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j=1

−(1 − (1 − TA1 (x))w1 )(1 − (1 − TA2 (x))w2 ),

j=1

Especially, Assume W = (1/n, 1/n, . . . , 1/n), then Gw is called as an arithmetic average operator for SNSs.

Gw (A1 , A2 , . . . , An ) =

n

w Aj j

(6)

TH

j=1

where W = (w1 , w2 , . . . , wn ) is the weight vector of Aj n (j = 1, 2, . . . , n), wj ∈ [0, 1] and wj = 1. Especially, j=1

−(1 − (1 − IA1 (x))w1 )(1 − (1 − IA2 (x))w2 ), 2 − (1 − FA1 (x))w1 − (1 − FA2 (x))w2

−(1 − (1 − FA1 (x))w1 )(1 − (1 − FA2 (x))w2 ) = 1 − (1 − TA1 (x))w1 (1 − TA2 (x))w2 ,

OR

Deﬁnition 10. Let Aj (j = 1, 2, . . . , n) be a SNS. The simplified neutrosophic weighted geometric average operator is defined by

2 − (1 − IA1 (x))w1 − (1 − IA2 (x))w2

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where W = (w1 , w2 , . . . , wn ) is the weight vector n of Aj (j = 1, 2, . . . , n), wj ∈ [0, 1] and wj = 1.

1 − (1 − IA1 (x))w1 (1 − IA2 (x))w2 , 1 − (1 − FA1 (x))w1 (1 − FA2 (x))w2

j=1

1−

1−

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n

(1 − TAj (x))wj ,

j=1

1−

n

1−

n

(1 − IAj (x)) , (1 − FAj (x))

wj

1−

k

(1 − FAj )wj

(7)

where W=(w1 , w2 , . . . , wn ) is the weight vector of Aj n (j = 1, 2, . . . , n), wj ∈ [0, 1] and wj = 1.

(9)

j=1

(3) When n = k+1, by applying Equations (8) and (9), we can get

j=1

j=1

j=1

(1 − IAj )wj ,

Fw (A1 , A2 , . . . , Ak+1 ) k = 1− (1 − TAj (x))wj

wj

j=1

k j=1

Theorem 1. For a SNS Aj (j = 1, 2, . . . , n), we have the following result by use of Equation (5): Fw (A1 , A2 , . . . , An ) =

(8)

(2) When n = k, by applying Equation (7), we get k Fw (A1 , A2 , . . . , Ak ) = 1 − (1 − TAj )wj ,

Assume W = (1/n, 1/n, . . . , 1/n), then Gw is called as an geometric average operator for SNSs.

+(1 − (1 − TAk+1 (x))wk+1 ) −(1 −

k

(1 − TAj (x))wj )

j=1

(1 − (1 − TAk+1 (x))wk+1 ),

J. Ye / A multicriteria decision-making method using aggregation operators k

1−

Theorem 2. For a SNS Aj (j = 1, 2, . . . , n), we have the following result by applying Equation (6):

(1 − IAj (x))wj

j=1

Gw (A1 , A2, . . . , An ) n n n wj wj wj = TAj (x), IAj (x), FAj (x) (11)

+(1 − (1 − IAk+1 (x))wk+1 ) −(1 −

k

j=1

(1 − IAj (x))wj )

(1 − (1 − IAk+1 (x)) 1−

),

j=1

By a similar proof manner, we can give the proof of Theorem 2 (omitted). It is obvious that the Gw operator has the following properties:

(1 − IAj (x))wj

j=1

+(1 − (1 − FAk+1 (x))wk+1 ) k j=1

=

(1) Idempotency: Let Aj (j = 1, 2, ..., n) be a collection of SNSs. If Ai (j = 1, 2,..., n) is equal, i.e. Aj = A for j = 1, 2,..., n, then Gw (A1 , A2 , . . . , An ) = A. (2) Boundedness: Let Aj (j = 1, 2, ..., n) be a collection of SNSs and let A− =

(1 − FAj (x))wj )

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−(1 −

(1 − (1 − FAk+1 (x))wk+1 )

1−

k+1

(1 − TAj (x))wj , (1 − IAj (x))wj ,

j=1

1−

k+1

(1 − FAj (x))

j=1

wj

(10)

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k+1

OR

j=1

1−

Therefore, considering the above results, we have Equation (7) for any n. This completes the proof.

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It is obvious that the Fw operator has the following properties:

(1) Idempotency: Let Aj (j = 1, 2,..., n) be a collection of SNSs. If Ai (j = 1, 2,..., n) is equal, i.e. Aj = A for j = 1, 2,..., n, then Fw (A1 , A2 , . . . , An ) = A. (2) Boundedness: Let Aj (j = 1, 2, ..., n) − be a collection of SNSs and let A = min TAj (x), max IAj (x), max FAj (x) and j j j A+ = max TAj (x), min IAj (x), min FAj (x) j

j

j=1

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k

j=1

where W = (w1 , w2 , . . . , wn ) is the weight vector of Aj n (j = 1, 2, . . . , n), wj ∈ [0, 1] and wj = 1.

j=1 wk+1

2463

j

for j = 1, 2,..., n, then A− ⊆ Fw (A1 , A2 , . . . , An ) ⊆ A+ . (3) Monotonity: Let Aj (j = 1, 2, ..., n) be a collection of SNSs. If Aj ⊆Aj ∗ for j = 1,

2,..., n, then Fw (A1 , A2 , . . . , An ) ⊆ Fw A∗1 , A∗2 , . . . , A∗n .

min TAj (x), max IAj (x), max FAj (x) and j j j A+ = max TAj (x), min IAj (x), min FAj (x) j

j

for j = 1, 2, ..., n, then A2 , . . . , An ) ⊆ A+ . (3) Monotonity: Let Aj (j = 1, a collection of SNSs. If j = 1, 2, ..., n, then Gw (A1 , Gw A∗1 , A∗2 , . . . , A∗n .

j

A− ⊆ Gw (A1 , 2, ..., n) be Aj ⊆Aj ∗ for A2 , . . . , An ) ⊆

The aggregation results Fw and Gw are still SNSs. Obviously, there are different focal points between Equations (7) and (11). The weighted arithmetic average operator emphasizes group’s major points, and then the weighted geometric average operator emphasizes personal major points.

4. Decision-making method based on the simpliﬁed neutrosophic weighted aggregation operators In this section, we present a handling method for multicriteria decision-making problems by means of the two aggregation operators and cosine similarity measure for SNSs under the simplified neutrosophic environment. Let A = {A1 , A2 , . . . , Am } be a set of alternatives and let C = {C1 , C2 , . . . , Cn } be a set of criteria. Assume that the weight of the criterion Cj (j = 1, 2, . . . , n),

J. Ye / A multicriteria decision-making method using aggregation operators

entered by the decision-maker, is wj , wj ∈ [0, 1] and n j=1 wj = 1. In the decision process, the evaluation information of the alternative Ai on the criteria is represented by the form of a SNS: Ai =

Cj , TAi (Cj ), IAi (Cj ), FAi (Cj ) |Cj ∈ C ,

where 0 ≤ TAi (Cj ) + IAi (Cj ) + FAi (Cj ) ≤ 3, TAi (Cj ) ≥ 0, IAi (Cj ) ≥ 0, FAi (Cj ) ≥ 0, j = 1, 2, . . . , n, and i = 1, 2, . . . , m. For convenience, the value of SNSs is denoted by αij = tij , iij , fij (i = 1, 2, . . . , m; j = 1, 2, . . . , n). Therefore, we can get a simplified neutrosophic decision matrix D = (αij )m×n :

Step 1: Calculate the weighted arithmetic average values by using Equation (7) or the weighted geometric average values by using Equation (11) Step 2: Calculate the cosine similarity measure between each alternative and the ideal alternative by using Equation (12). Step 3: Give the ranking order of the alternatives from the obtained measure values, and then get the best choice. Step 4: End.

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⎤ t11 , i11 , f11 t12 , i12 , f12 · · · t1n , i1n , f1n ⎥ ⎢ t ⎢ 21 , i21 , f21 t22 , i22 , f22 · · · t2n , i2n , f2n ⎥ ⎥ ⎢ =⎢ ⎥. .. .. .. .. ⎥ ⎢ . . . . ⎦ ⎣ tm1 , im1 , fm1 tm2 , im2 , fm2 · · · tmn , imn , fmn

D = (αij )m×n

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⎡

Si (αi , α∗ ) = =

AU

TH

OR

Then, the aggregating simplified neutrosophic value αi for Ai (i = 1, 2, . . . , m) is αi = ti , ii , fi = Fiw (αi 1 , αi 2 , . . . , αi n ) or αi = ti , ii , fi = Giw (αi 1 , αi 2 , . . . , αi n ) is obtained by applying Equations (7) or (11) according to each row in the simplified neutrosophic decision matrix D = (aij )m ×n . To rank alternatives in the decision-making process, we define an ideal SNS value as the ideal alternative ␣∗ = 1, 0, 0 , and then based on the cosine similarity measure between SVNSs proposed by Ye [8], the cosine similarity measure between SNSs ␣i (i = 1, 2,..., m) and ␣∗ can be defined as follows:

ti ti∗ + ii i∗i + fi fi∗

2 2 2 2 2 2 ti + i i + f i ti∗ + i∗i + fi∗ ti

(12)

ti2 + i2i + fi2

Then, the bigger the measure value Si (␣i , ␣∗ ) (i = 1, 2,..., m) is, the better the alternative Ai is, because the alternative Ai is close to the ideal alternative ␣∗ . Through the cosine similarity measure between each alternative and the ideal alternative, the ranking order of all alternatives can be determined and the best one can be easily identified as well. In summary, the decision procedure for the proposed method can be summarized as follows:

5. Numerical example

In this section, an example for a multicriteria decision-making problem of engineering alternatives is used as a demonstration of the application of the proposed decision-making method in a realistic scenario, as well as the application and effectiveness of the proposed decision-making method. Let us consider the decision-making problem adapted from [8]. There is an investment company, which wants to invest a sum of money in the best option. There is a panel with four possible alternatives to invest the money: (1) A1 is a car company; (2) A2 is a food company; (3) A3 is a computer company; (4) A4 is an arms company. The investment company must take a decision according to the following three criteria: (1) C1 is the risk; (2) C2 is the growth; (3) C3 is the environmental impact. Then, the weight vector of the criteria is given by W = (0.35, 0.25, 0.4), which is adopted from the literature [8]. For the evaluation of an alternative Ai (i = 1, 2, 3, 4) with respect to a criterion Cj (j = 1, 2, 3), it is obtained from the questionnaire of a domain expert. For example, when we ask the opinion of an expert about an alternative A1 with respect to a criterion C1 , he or she may say that the possibility in which the statement is good is 0.4

J. Ye / A multicriteria decision-making method using aggregation operators

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Step 1: We can obtain the weighted arithmetic average value (aggregating simplified neutrosophic value) ␣i for Ai (i = 1, 2, 3, 4) by using Equation (7): ␣1 = 0.3268, 0.2000, 0.3881 , ␣2 = 0.5627, 0.1414, 0.2000 , ␣3 =0.4375, 0.2416, 0.2616 , and ␣4 = 0.5746, 0.1555, 0.1663 . Step 2: By applying Equation (12), we can compute each cosine similarity measure Si (␣i , ␣∗ ) (i = 1, 2, 3, 4) as follows: S1 (␣1 , ␣∗ ) = 0.5992, S2 (␣2 , ␣∗ ) = 0.9169, S3 (␣3 , ␣∗ ) = 0.7756, and S4 (␣4 , ␣∗ ) = 0.9297. Step 3: From the measure value Si (␣i , ␣∗ ) (i = 1, 2, 3, 4) between an alternative and the ideal alternative, the ranking order of four alternatives is A4 A2 A3 A1 .

Obviously, we can see that the above two kinds of ranking orders and the best alternative are the same. The method proposed in this paper differs from existing approaches for fuzzy multi-criteria decision making not only due to the fact that the proposed method uses the SNS concept and two simplified neutrosophic aggregation operators, but also due to the consideration of the indeterminacy information besides truth and falsity information in the evaluation of the alternative with respect to criteria, which makes it have more feasible and practical than other traditional decision making methods in real decision-making problems. Therefore, its advantage is easily reflecting the ambiguous nature of subjective judgments because SNSs are suitable for capturing imprecise, uncertain, and inconsistent information in the multicriteria decision analysis.

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The proposed method is applied to solve this problem according to the following computational procedure:

Step 2’: By using Equation (12), we can compute each cosine similarity measure Si (␣i , ␣∗ ) (i = 1, 2, 3, 4) as follows: S1 (␣1 , ␣∗ ) = 0.5863, S2 (␣2 , ␣∗ ) = 0.9188, S3 (␣3 , ␣∗ ) = 0.7696, and S4 (␣4 , ␣∗ ) = 0.9601. Step 3’: The ranking order of four alternatives is A4 A2 A3 A1 . Thus, we can see that the alternative A4 is still the best choice among all the alternatives.

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and the statement is poor is 0.3 and the degree in which he or she is not sure is 0.2. For the neutrosophic notation, it can be expressed as ␣11 = 0.4, 0.2, 0.3 . Thus, when the four possible alternatives with respect to the above three criteria are evaluated by the expert, we can obtain the following simplified neutrosophic decision matrix D: ⎡ ⎤ 0.4, 0.2, 0.3 0.4, 0.2, 0.3 0.2, 0.2, 0.5 ⎢ ⎥ ⎢0.6, 0.1, 0.2 0.6, 0.1, 0.2 0.5, 0.2, 0.2 ⎥ ⎢ ⎥. D=⎢ ⎥ ⎣0.3, 0.2, 0.3 0.5, 0.2, 0.3 0.5, 0.3, 0.2 ⎦ 0.7, 0.0, 0.1 0.6, 0.1, 0.2 0.4, 0.3, 0.2

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Therefore, we can see that the alternative A4 is the best choice among all the alternatives. On the other hand, we can also utilize the weighted geometric average operator as the following computational procedure: Step 1’: We compute the weighted geometric average values by applying Equation (11) for Ai (i = 1, 2, 3, 4), each aggregating simplified neutrosophic value ␣i (i = 1, 2, 3, 4) is as follows: ␣1 = 0.3031, 0.2000, 0.3680 , ␣2 = 0.5578, 0.1320, 0.2000 , ␣3 = 0.4181, 0.2352, 0.2551 , and ␣4 = 0.5385, 0, 0.1569 .

6. Conclusion This paper introduced the concept of SNSs, which are a subclass of neutrosophic sets, and defined some operational laws of SNSs. Then, we proposed two aggregation operators for SNSs, including a simplified neutrosophic weighted arithmetic average operator and a simplified neutrosophic weighted geometric average operator. The two aggregation operators were applied to multicriteria decision-making problems under the simplified neutrosophic environment, in which criterion values with respect to alternatives are evaluated by the form of simplified neutrosophic values and the criterion weights are known information. We utilized the cosine similarity measure between an alternative and the ideal alternative to rank the alternatives and to determine the best one(s) according to the measure values. Finally, a numerical example is provided to illustrate the application of the developed approach. The proposed simplified neutrosophic multicriteria decision-making method is more suitable for real scien-

J. Ye / A multicriteria decision-making method using aggregation operators

tific and engineering applications because it can handle not only incomplete information but also the indeterminate information and inconsistent information which exist commonly in real situations. The techniques proposed in this paper can provide more useful way for decision-makers. In the future, we shall deal with group decision making problems with incomplete decision contexts and preference relations performed in the selection process under the simplified neutrosophic environment and apply the simplified neutrosophic aggregation operators to solve practical applications in other areas such as expert system, information fusion system, and medical diagnoses.

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