# The Weighted Distance Measure Based Method to Neutrosophic Multiattribute Group Decision Making

Authors: Chunfang Liu; YueSheng Luo
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

Neutrosophic set (NS) is a generalization of fuzzy set (FS) that is designed for some practical situations in which each element has different truth membership function, indeterminacy membership function, and falsity membership function. In this paper, we study the multiattribute group decision making (MAGDM) problems under neutrosophic environment with the incompletely known or completely unknown attribute weight. We first define the single-valued neutrosophic ideal solution (SVNIS) and the weighted distance measure and establish the program models to derive the attribute weights. Then, we give a practical application in the framework of SVNS; the result shows that our method is reasonable and effective in dealing with decision making (DM) problems. Furthermore, we extend the method to interval-valued neutrosophic set (IVNS).


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## Body

## 1. Introduction

Fuzzy set was introduced by Zadeh, which has been widely used in many aspects [1, 2]. On the basis of Zadeh’s work, several high-order fuzzy sets have been proposed as an extension of fuzzy sets, including interval-valued fuzzy set, type-2 fuzzy set, type-n fuzzy set, soft set, rough set, intuitionistic fuzzy set, interval-valued intuitionistic fuzzy set, hesitant fuzzy set, and neutrosophic set (NS) [2–6]. So far, the proposed high-order fuzzy sets have been successfully utilized in dealing with different uncertain problems, such as decision making [7], and pattern recognition [8].As a generalization of fuzzy set, the NS was proposed by Smarandache [5] not only to deal with the decision information which is often incomplete, indeterminate, and inconsistent but also to include the truth membership degree, the falsity membership degree, and the indeterminacy membership degree. For simplicity and practical application, Wang proposed the single-valued NS (SVNS) and the interval-valued NS (IVNS) which are the instances of NS and gave some operations on these sets [8, 9]. Since its appearance, many fruitful results have appeared [10, 11]. On one hand, many researchers have proposed some aggregation operators of SVNS and INS and applied them to MADM problems [12–16]. On the other hand, some researchers have also proposed entropy and similarity measure of the SVNS and IVNS and applied them to MADM and pattern recognition [17, 18]. The above problems that are related to the attribute weights are completely known. However, with the development of the information society and internet technology, the socioeconomic environment gets more complex in many decision areas, such as capital investment decision making, medical diagnosis, and personnel examination. Only one decision maker cannot deal with the complex problems. Accordingly, it is necessary to gather multiple decision makers with different knowledge structures and experiences to conduct a group decision making. In some circumstances, it is difficult for the decision makers to give the information of the attribute weights correctly, which makes the attribute weights incompletely known or completely unknown. How to derive the attribute weights from the given neutrosophic information is an important topic. In intuitionistic fuzzy environments, many researchers have proposed some program models to obtain the incompletely known attribute weights or the completely unknown attribute weights, such as Xu proposed the deviation-based method [19], the ideal solution-based method [20], and the group consensus-based method [21] and Li et al. proposed the consistency-based method [22]. Under the neutrosophic environment, Şahin and Liu proposed the maximizing deviation method [23]. Up to now, we found that there is no research of the weighted distance measure based method to neutrosophic multiattribute group decision making. In this paper, we investigate the MAGDM problems which the information expressed by SVNS or IVNS, and the attribute weights are incompletely known or completely unknown.The rest of the paper is organized as follows. In Section2, we recall the concept of NS, SVNS, INS and their distance measures. In Section 3, we give the weighted distance measure based method to single-valued neutrosophic set (SVNS). Furthermore, we extend the method to interval-valued neutrosophic set (IVNS). Finally, a conclusion is given in Section 4.


## 2. Preliminaries

Definition 1 (see [5]).
Assume thatX is a universe of discourse with a generic element in X denoted by x. A NS A on X is defined by a truth membership function TA(x), an indeterminacy membership function IA(x), and a falsity membership function FA(x). TA(x), IA(x), and FA(x) are defined by (1)TAx:X⟶0-,1+,IAx:X⟶0-,1+,FAx:X⟶0-,1+,where 0-≤sup⁡TAx+sup⁡IAx+sup⁡FA(x)≤3+.For similarity and practical application, Wang proposed the SVNS and IVNS which are the subclasses of NS and preserve all the operations on NS. In the following part, we recall SVNS, IVNS, and their distance measure, respectively.Definition 2 (see [8]).
Assume thatX is a universe of discourse with a generic element in X denoted by x. A single-valued neutrosophic set (SVNS) A on X is defined by a truth membership function TA(x), an indeterminacy membership function IA(x), and a falsity membership function FA(x). TA(x), IA(x), and FA(x) are defined by (2)TAx:X⟶0,1,IAx:X⟶0,1,FAx:X⟶0,1,where TA(x), IA(x), and FA(x) are subsets of [0,1] and satisfy 0≤TA(x)+IA(x)+FA(x)≤3.For similarity, we utilizeA={TA(x),IA(x),FA(x)} to denote a SVNS A in the following part. If X has only one element, for convenience, we call A a single-valued neutrosophic number (SVNN) and denoted by A={TA,IA,FA}.Definition 3 (see [24]).
LetA1={T1,I1,F1}, A2={T2,I2,F2} be two SVNNs; the normalized Hamming distance measure between A1 and A2 is defined by (3)dA1,A2=13T1-T2+I1-I2+F1-F2.Definition 4 (see [9]).
Assume thatX is a universe of discourse with a generic element in X denoted by x, and int⁡[0,1] is the set of all closed subsets of [0,1]. An interval-valued neutrosophic set (IVNS) A on X is defined by a truth membership function TA(x), an indeterminacy membership function IA(x), and a falsity membership function FA(x). TA(x), IA(x), and FA(x) are defined by (4)TAx:X⟶int⁡0,1,IAx:X⟶int⁡0,1,FAx:X⟶int⁡0,1with the condition 0≤sup⁡TAx+sup⁡IAx+sup⁡FA(x)≤3.Here, we denoteTA(x)=[TA-(x),TA+(x)], IA(x)=[IA-(x),IA+(x)], FA(x)=[FA-(x),FA+(x)]. For convenience, we call A an interval-valued neutrosophic number (IVNN) and it is denoted by A={TA-,TA+,[IA-,IA+],[FA-,FA+]}.Definition 5 (see [18]).
LetA1 = {T1-,T1+, [I1-,I1+], [F1-,F1+]}, A2 = {T2-,T2+, [I2-,I2+], [F2-,F2+]} be two IVNNs; the normalized Hamming distance measure between A1 and A2 is defined by (5)dA1,A2=16T1--T2-+T1+-T2++I1--I2-+I1+-I2++F1--F2-+F1+-F2+.


## 3. The Weighted Distance Measure Based Method to Neutrosophic Set

### 3.1. The Weighted Distance Measure Based Method to Single-Valued Neutrosophic Set

LetX={X1,X2,…,Xm} be a set of alternatives, let C={C1,C2,…,Cn} be a set of attributes, let and w={w1,w2,…,wn} be the weight vector of the attribute with wj∈[0,1] and ∑j=1nwj=1. Suppose that there are s decision makers D={D1,D2,…,Ds}, whose corresponding weighted vector is λ={λ1,λ2,…,λs}. Let Ak=(rijk)m×n(k=1,2,…,s) be single-valued neutrosophic decision matrix, where rijk={Tijk,Iijk,Fijk} is the value of the attribute, expressed by SVNNs.In MADM environments, the ideal point is used to help the identification of the best alternative in the decision set. Although the ideal point does not exist in real world, it does provide an effective way to evaluate the best alternative. Now, we suppose that the ideal SVNN isαj∗={t∗,i∗,f∗}={1,0,0}. Based on the ideal SVNN, we define the single-valued neutrosophic positive ideal solution (SVNPIS).Definition 6.
Letαj∗={1,0,0}(j=1,2,…,n) be n ideal SVNNs; then, a SVNPIS is defined by (6)A∗=α1∗,α2∗,…,αn∗.Definition 7.
LetAik={ri1k,ri2k,…,rink}(i=1,2,…,m) be the ith alternative of the kth decision makers (k=1,2,…,s), and let A∗={α1∗,α2∗,…,αn∗} be the SVNPIS; then, the weighted Hamming distance measure (WHDM) between Ai and A∗ is defined by (7)dAi,A∗=∑k=1sλk∑j=1nwjdrijk,αj∗.


#### 3.1.1. Incompletely Known Attribute Weights

In the decision making process, the incomplete information of the attribute weight provided by the decision makers can usually be constructed using several basic ranking forms [25]. Let H be the set of information about the incompletely known attribute weights, which may be constructed in the following forms [26], for i≠j:(a)
A weak ranking:{wi≥wj}.(b)
A strict ranking:{wi-wj≥δi(>0)}.(c)
A ranking with multiples:{wi≥δiwj},0≤δi≤1.(d)
An interval form:{δi≤wi≤δi+εi},0≤δi≤δi+εi.(e)
A ranking of differences:{wi-wj≥wk-wl}, for j≠k≠l.We now establish the following single-objective programming model based on the weighted distance measure method:M1minfw=∑k=1sλk∑i=1m∑j=1nwjdrijk,αj∗s.t.wj∈H,∑j=1nwj=1,wj≥0,j=1,2,…,n,where λk is the weight of the decision maker Dk(k=1,2,…,s) and (8)drijk,αj∗=13Tijk-1+Iijk+Fijk,where d(rijk,αj∗) represents the weighted distance measure between the attribute value rijk and the SVNPIS αj∗. The desirable weight vector w=(w1,w2,…,wn) should make the sum of all weighted distance measures (7) small. So we construct this model to make the overall distance small.By solving the modelM1 with Matlab software, we get the optimal solution w∗=(w1∗,w2∗,…,wn∗), which is considered as the weight of the attributes C1,C2,…,Cn. Then, we utilize d(Ai,A∗) to rank all the alternatives. The smaller the weighted distance measure, the better the alternative.


#### 3.1.2. Completely Unknown Attribute Weights

If the information about the attribute weight is completely unknown, we establish the following programming model:M2minfw=∑k=1sλk∑i=1m∑j=1nwjdrijk,αj∗s.t.∑j=1nwj2=1,wj≥0,j=1,2,…,n.To solve this model, we construct the Lagrange function as follows: (9)Lw,λ=∑k=1sλk∑i=1m∑j=1nwjdrijk,αj∗+λ2∑j=1nwj2-1,where λ is the Lagrange multiplier.Differentiating (9) with respect to wj(j=1,2,…,n) and λ, setting these partial derivatives equal to zero, the following set of the equations are obtained: (10)∂L∂wj=∑k=1sλk∑i=1mdrijk,αj∗+wjλ=0,∂L∂λ=∑j=1nwj2=1.By solving (10), we obtain the weight wj and normalize it with wj∗=wj/∑j=1nwj; then, we get (11)wj∗=∑k=1sλk∑i=1mdrijk,αj∗∑j=1n∑k=1sλk∑i=1mdrijk,αj∗.We get the optimal solution w∗=(w1∗,w2∗,…,wn∗), which is considered as the weight of the attributes C1,C2,…,Cn. Later, we calculate the distance measure (7) and get the most desirable one.


#### 3.1.3. Illustrative Example

Example 1.
Here, we choose the decision making problem adapted from [23]. An automotive company is desired to select the most appropriate supplier for one of the key elements in its manufacturing process. After preevaluation, four suppliers have remained as alternatives for further evaluation. In order to evaluate alternative suppliers, a committee composed of four decision makers has been formed. The committee selects four attributes to evaluate the alternatives: (1) C1: product quality, (2) C2: relationship closeness, (3) C3: delivery performance, and (4) C4: price. Suppose that there are four decision makers, denoted by d1,d2,d3,d4, whose corresponding weight vector is λ=(0.25,0.25,0.25,0.25). The four possible alternatives are to be evaluated under these four attributes and are in the form of SVNNs for each decision maker, as shown in the following single-valued neutrosophic decision matrix: (12)D1=0.4,0.2,0.30.4,0.2,0.30.2,0.2,0.50.7,0.2,0.30.6,0.1,0.20.6,0.1,0.20.5,0.2,0.30.5,0.1,0.20.3,0.2,0.30.5,0.2,0.30.1,0.5,0.20.1,0.4,0.50.7,0.2,0.10.6,0.1,0.20.4,0.3,0.20.4,0.5,0.1,D2=0.1,0.3,0.50.5,0.1,0.50.3,0.1,0.60.4,0.1,0.40.2,0.5,0.40.3,0.4,0.30.2,0.3,0.10.2,0.3,0.50.5,0.2,0.60.2,0.4,0.30.5,0.2,0.50.1,0.5,0.30.2,0.4,0.20.1,0.1,0.30.1,0.5,0.40.5,0.3,0.1,D3=0.3,0.2,0.10.3,0.1,0.30.1,0.4,0.50.2,0.3,0.50.6,0.1,0.40.6,0.4,0.20.5,0.4,0.10.5,0.2,0.40.3,0.3,0.60.4,0.2,0.40.2,0.3,0.20.3,0.5,0.10.3,0.6,0.10.5,0.3,0.20.3,0.3,0.60.4,0.3,0.2,D4=0.2,0.2,0.30.3,0.2,0.30.2,0.3,0.50.4,0.2,0.50.4,0.1,0.20.6,0.3,0.50.1,0.2,0.20.5,0.1,0.20.3,0.5,0.10.2,0.2,0.30.5,0.4,0.30.5,0.3,0.20.3,0.1,0.10.2,0.1,0.40.2,0.3,0.20.3,0.1,0.6.Case 1 (incompletely known attribute weights).
Suppose that the incompletely known information of the attribute weight is given as follows:(13)H=0.18≤w1≤0.2,0.15≤w2≤0.25,0.30≤w3≤0.35,0.3≤w4≤0.4,∑j=14wj=1.Step 1.
By modelM1, we establish the following model: (14)minfw=1.5833w1+1.5038w2+1.825w3+1.625w4s.t.w∈H.Step 2.
By solving this model with Matlab software, we get the weight vector:(15)w1=0.18,w2=0.22,w3=0.30,w4=0.30.Step 3.
Using the distance measure (7), we have (16)dA1,A∗=0.4365,dA2,A∗=0.3618,dA3,A∗=0.4502,dA4,A∗=0.4033.Step 4 (rank the alternatives).
Sinced(A3,A∗) is the biggest, and d(A2,A∗) is the smallest, we rank the alternatives as follows: (17)A2≻A4≻A1≻A3,and A2 is the best alternative.Case 2 (completely unknown attribute weights).Step 1.
By modelM2, we establish the following model: (18)minfw=1.5833w1+1.5038w2+1.825w3+1.625w4s.t.∑j=14wj2=1,wj≥0,j=1,2,3,4.Step 2.
Use (11) to obtain the weight vector of attributes: (19)w1∗=0.18,w2∗=0.22,w3∗=0.30,w4∗=0.30.Step 3.
Using the distance measure (8) and (9), we have (20)dA1,A∗=0.4352,dA2,A∗=0.3613,dA3,A∗=0.4482,dA4,A∗=0.3984.Step 4.
Rank the alternatives. Sinced(A3,A∗) is the biggest, and d(A2,A∗) is the smallest, we rank the alternatives as follows: (21)A2≻A4≻A1≻A3,and A2 is the best alternative.


### 3.2. The Weighted Distance Measure Based Method to Interval-Valued Neutrosophic Set

LetX={X1,X2,…,Xm} be a set of alternatives, let C={C1,C2,…,Cn} be a set of attributes, and let w={w1,w2,…,wn} be the weight vector of the attribute with wj∈[0,1] and ∑j=1nwj=1. Suppose that there are s decision makers D={D1,D2,…,Ds}, whose corresponding weighted vector is λ={λ1,λ2,…,λs}. Let Ak=(rijk)m×n(k=1,2,…,s) be interval-valued neutrosophic decision matrix, where rijk={Tijk,Iijk,Fijk} is the value of the attribute, expressed by IVNNs.Now, we suppose that the ideal IVNN isβj∗={t∗,i∗,f∗}={[1,1],[0,0],[0,0]}. Based on the ideal IVNN, we define the interval-valued neutrosophic positive ideal solution (IVNPIS).Definition 8.
Letβj∗={[1,1],[0,0],[0,0]}(j=1,2,…,n) be n ideal IVNNs; then, a IVNPIS is defined by (22)A∗=β1∗,β2∗,…,βn∗.Definition 9.
LetAi={ri1,ri2,…,rin}(i=1,2,…,m) be the ith alternative, and let A∗={β1∗,β2∗,…,βn∗} be the IVNPIS; then, the weighted Hamming distance measure (WHDM) between Ai and A∗ is defined by (23)dAi,A∗=∑k=1sλk∑j=1nwjdrijk,βj∗.


#### 3.2.1. Incompletely Known Attribute Weights

We now establish the following single-objective programming model based on the weighted distance measure method:M3minfw=∑k=1sλk∑i=1m∑j=1nwjdrijk,βj∗s.t.wj∈H,∑j=1nwj=1,wj≥0,j=1,2,…,n,where λk is the weight of the decision maker Dk(k=1,2,…,s) and (24)drijk,βj∗=16Tij-k-1+Tij+k-1+Iij-k+Iij+k+Fij-k+Fij+k,where d(rijk,βj∗) represents the distance measure between the attribute value rijk and the IVNPIS βj∗. The desirable weight vector w=(w1,w2,…,wn) should make the sum of all weighted distances (23) small. So we construct this model to make the overall distances small. The smaller the WHD, the better the alternative. We use (23) to rank the alternative.By solving the modelM3 with Matlab software, we get the optimal solution w∗=(w1∗,w2∗,…,wn∗), which is considered as the weight of the attributes C1,C2,…,Cn. Then, we utilize d(Ai,A∗) to rank all the alternatives. The smaller the distance, the better the alternative.


#### 3.2.2. Completely Unknown Attribute Weights

If the information about the attribute weight is completely unknown, we establish the following programming model:M4minfw=∑k=1sλk∑i=1m∑j=1nwjdrijk,βj∗s.t.∑j=1nwj2=1,wj≥0,j=1,2,…,n.By Lagrange multiple method, we get the completely unknown weight wj and normalize it with wj∗=wj/∑j=1nwj as follows:(25)wj∗=∑k=1sλk∑i=1mdrijk,βj∗∑j=1n∑k=1sλk∑i=1mdrijk,βj∗,which is considered as the weight of the attributes Cj. Later, we calculate the distance measure d(Ai,A∗) and then get the most desirable one.


#### 3.2.3. Illustrative Example

Example 2.
The decision making problem is adapted from [23]. Suppose that an organization plans to implement ERP system. The first step is to form a project team that consists of CIO and two senior representatives from user departments. By collecting all information about ERP vendors and systems, project team chooses four potential ERP systems Ai(i=1,2,3,4) as candidates. The company employs some external professional organizations (experts) to aid this decision making. The project team selects four attributes to evaluate the alternatives: (1) C1: function and technology, (2) C2: strategic fitness, (3) C3: vendors ability, and (4) C4: vendors reputation. Suppose that there are three decision makers, denoted by D1,D2,D3, whose corresponding weight vector is λ=(1/3,1/3,1/3). The four possible alternatives are to be evaluated under these four attributes and are in the form of IVNNs for each decision maker, as shown in the following interval-valued neutrosophic decision matrix: (26)D1=0.4,0.5,0.2,0.3,0.3,0.50.3,0.4,0.3,0.6,0.2,0.40.2,0.5,0.2,0.6,0.3,0.50.5,0.6,0.3,0.5,0.2,0.50.6,0.7,0.1,0.2,0.2,0.30.1,0.3,0.1,0.4,0.2,0.50.4,0.5,0.2,0.5,0.3,0.70.2,0.4,0.1,0.4,0.3,0.30.3,0.4,0.2,0.3,0.3,0.40.3,0.6,0.2,0.3,0.2,0.50.2,0.7,0.2,0.4,0.3,0.60.2,0.6,0.4,0.7,0.2,0.70.2,0.6,0.1,0.2,0.1,0.20.2,0.5,0.4,0.5,0.1,0.60.3,0.5,0.1,0.3,0.2,0.20.4,0.4,0.1,0.6,0.1,0.5,D2=0.4,0.6,0.1,0.3,0.2,0.40.3,0.5,0.1,0.4,0.3,0.40.4,0.5,0.2,0.4,0.1,0.30.3,0.6,0.3,0.6,0.3,0.60.3,0.5,0.1,0.2,0.2,0.30.3,0.4,0.2,0.2,0.1,0.30.2,0.7,0.3,0.5,0.3,0.60.2,0.5,0.2,0.7,0.1,0.20.5,0.6,0.2,0.3,0.3,0.40.1,0.4,0.1,0.3,0.3,0.50.5,0.5,0.4,0.6,0.3,0.40.1,0.2,0.1,0.4,0.5,0.60.3,0.4,0.1,0.2,0.1,0.30.3,0.3,0.1,0.5,0.2,0.40.2,0.3,0.4,0.5,0.5,0.60.3,0.3,0.2,0.3,0.1,0.4,D3=0.1,0.3,0.2,0.3,0.4,0.50.3,0.3,0.1,0.3,0.3,0.40.2,0.6,0.3,0.5,0.3,0.50.4,0.6,0.3,0.4,0.2,0.30.3,0.6,0.3,0.5,0.3,0.50.3,0.4,0.3,0.4,0.3,0.50.3,0.5,0.2,0.4,0.1,0.50.1,0.2,0.3,0.5,0.3,0.40.4,0.5,0.2,0.4,0.2,0.40.2,0.3,0.1,0.1,0.3,0.40.1,0.4,0.2,0.6,0.3,0.60.4,0.5,0.2,0.6,0.1,0.30.2,0.4,0.3,0.4,0.1,0.30.1,0.4,0.2,0.5,0.1,0.50.3,0.6,0.2,0.4,0.2,0.20.2,0.4,0.3,0.3,0.2,0.6.Case 1 (incompletely known attribute weights).
Suppose that the incompletely known information of the attribute weight is given as follows:(27)H=0.18≤w1≤0.2,0.15≤w2≤0.25,0.30≤w3≤0.35,0.3≤w4≤0.4,∑j=14wj=1.Step 1.
By modelM3, we establish the following model: (28)minfw=1.4278w1+1.7278w2+1.8278w3+1.7667w4s.t.w∈H.Step 2.
By solving this model with Matlab software, we get the weight vector:(29)w1=0.18,w2=0.25,w3=0.20,w4=0.37.Step 3.
Using the distance measure (24) and (25), we have (30)dA1,A∗=0.4204,dA2,A∗=0.4182,dA3,A∗=0.4471,dA4,A∗=0.41.Step 4 (rank the alternatives).
Sinced(A3,A∗) is the biggest, and d(A4,A∗) is the smallest, we rank the alternatives as follows: (31)A4≻A2≻A1≻A3,and A4 is the best alternative.Case 2 (completely unknown attribute weights).Step 1.
By modelM4, we establish the following model:(32)minfw=1.4278w1+1.7278w2+1.8278w3+1.7667w4s.t.∑j=14wj2=1,wj≥0,j=1,2,3,4.Step 2.
Use (25) to obtain the weight vector of attributes: (33)w1∗=0.2115,w2∗=0.2560,w3∗=0.2708,w4∗=0.2617.Step 3.
Using the distance measure (24) and (25), we have (34)dA1,A∗=0.4200,dA2,A∗=0.3776,dA3,A∗=0.4421,dA4,A∗=0.4054.Step 4 (rank the alternatives).
Sinced(A3,A∗) is the biggest, and d(A2,A∗) is the smallest, we rank the alternatives as follows: (35)A2≻A4≻A1≻A3,and A2 is the best alternative.


### 3.3. Comparative Analysis

Considering the proposed method and the maximizing deviation method proposed by Şahin, there exsit some differences. In Şahin’s method, they calculated the distance measure of all the attributes and assign a small weight to the attribute which has a similar effect among the alternatives; then, they used the weighted aggregation operators and the score functions to rank the alternatives; while, the proposed method calculates the distance measure between the attributes and the ideal solution and obtains the weight that make the weighted distance measure small, we then use the weighted distance measure to rank the alternatives which avoid the complex calculation of aggregation operators processing. The two methods are all effective to deal with the incompletely known or completely unknown attribute weight by solve the program models. The advantage of the proposed method is that calculation is simple and convenient, which can deal with the MAGDM problem effectively.


## 3.1. The Weighted Distance Measure Based Method to Single-Valued Neutrosophic Set

LetX={X1,X2,…,Xm} be a set of alternatives, let C={C1,C2,…,Cn} be a set of attributes, let and w={w1,w2,…,wn} be the weight vector of the attribute with wj∈[0,1] and ∑j=1nwj=1. Suppose that there are s decision makers D={D1,D2,…,Ds}, whose corresponding weighted vector is λ={λ1,λ2,…,λs}. Let Ak=(rijk)m×n(k=1,2,…,s) be single-valued neutrosophic decision matrix, where rijk={Tijk,Iijk,Fijk} is the value of the attribute, expressed by SVNNs.In MADM environments, the ideal point is used to help the identification of the best alternative in the decision set. Although the ideal point does not exist in real world, it does provide an effective way to evaluate the best alternative. Now, we suppose that the ideal SVNN isαj∗={t∗,i∗,f∗}={1,0,0}. Based on the ideal SVNN, we define the single-valued neutrosophic positive ideal solution (SVNPIS).Definition 6.
Letαj∗={1,0,0}(j=1,2,…,n) be n ideal SVNNs; then, a SVNPIS is defined by (6)A∗=α1∗,α2∗,…,αn∗.Definition 7.
LetAik={ri1k,ri2k,…,rink}(i=1,2,…,m) be the ith alternative of the kth decision makers (k=1,2,…,s), and let A∗={α1∗,α2∗,…,αn∗} be the SVNPIS; then, the weighted Hamming distance measure (WHDM) between Ai and A∗ is defined by (7)dAi,A∗=∑k=1sλk∑j=1nwjdrijk,αj∗.


### 3.1.1. Incompletely Known Attribute Weights

In the decision making process, the incomplete information of the attribute weight provided by the decision makers can usually be constructed using several basic ranking forms [25]. Let H be the set of information about the incompletely known attribute weights, which may be constructed in the following forms [26], for i≠j:(a)
A weak ranking:{wi≥wj}.(b)
A strict ranking:{wi-wj≥δi(>0)}.(c)
A ranking with multiples:{wi≥δiwj},0≤δi≤1.(d)
An interval form:{δi≤wi≤δi+εi},0≤δi≤δi+εi.(e)
A ranking of differences:{wi-wj≥wk-wl}, for j≠k≠l.We now establish the following single-objective programming model based on the weighted distance measure method:M1minfw=∑k=1sλk∑i=1m∑j=1nwjdrijk,αj∗s.t.wj∈H,∑j=1nwj=1,wj≥0,j=1,2,…,n,where λk is the weight of the decision maker Dk(k=1,2,…,s) and (8)drijk,αj∗=13Tijk-1+Iijk+Fijk,where d(rijk,αj∗) represents the weighted distance measure between the attribute value rijk and the SVNPIS αj∗. The desirable weight vector w=(w1,w2,…,wn) should make the sum of all weighted distance measures (7) small. So we construct this model to make the overall distance small.By solving the modelM1 with Matlab software, we get the optimal solution w∗=(w1∗,w2∗,…,wn∗), which is considered as the weight of the attributes C1,C2,…,Cn. Then, we utilize d(Ai,A∗) to rank all the alternatives. The smaller the weighted distance measure, the better the alternative.


### 3.1.2. Completely Unknown Attribute Weights

If the information about the attribute weight is completely unknown, we establish the following programming model:M2minfw=∑k=1sλk∑i=1m∑j=1nwjdrijk,αj∗s.t.∑j=1nwj2=1,wj≥0,j=1,2,…,n.To solve this model, we construct the Lagrange function as follows: (9)Lw,λ=∑k=1sλk∑i=1m∑j=1nwjdrijk,αj∗+λ2∑j=1nwj2-1,where λ is the Lagrange multiplier.Differentiating (9) with respect to wj(j=1,2,…,n) and λ, setting these partial derivatives equal to zero, the following set of the equations are obtained: (10)∂L∂wj=∑k=1sλk∑i=1mdrijk,αj∗+wjλ=0,∂L∂λ=∑j=1nwj2=1.By solving (10), we obtain the weight wj and normalize it with wj∗=wj/∑j=1nwj; then, we get (11)wj∗=∑k=1sλk∑i=1mdrijk,αj∗∑j=1n∑k=1sλk∑i=1mdrijk,αj∗.We get the optimal solution w∗=(w1∗,w2∗,…,wn∗), which is considered as the weight of the attributes C1,C2,…,Cn. Later, we calculate the distance measure (7) and get the most desirable one.


### 3.1.3. Illustrative Example

Example 1.
Here, we choose the decision making problem adapted from [23]. An automotive company is desired to select the most appropriate supplier for one of the key elements in its manufacturing process. After preevaluation, four suppliers have remained as alternatives for further evaluation. In order to evaluate alternative suppliers, a committee composed of four decision makers has been formed. The committee selects four attributes to evaluate the alternatives: (1) C1: product quality, (2) C2: relationship closeness, (3) C3: delivery performance, and (4) C4: price. Suppose that there are four decision makers, denoted by d1,d2,d3,d4, whose corresponding weight vector is λ=(0.25,0.25,0.25,0.25). The four possible alternatives are to be evaluated under these four attributes and are in the form of SVNNs for each decision maker, as shown in the following single-valued neutrosophic decision matrix: (12)D1=0.4,0.2,0.30.4,0.2,0.30.2,0.2,0.50.7,0.2,0.30.6,0.1,0.20.6,0.1,0.20.5,0.2,0.30.5,0.1,0.20.3,0.2,0.30.5,0.2,0.30.1,0.5,0.20.1,0.4,0.50.7,0.2,0.10.6,0.1,0.20.4,0.3,0.20.4,0.5,0.1,D2=0.1,0.3,0.50.5,0.1,0.50.3,0.1,0.60.4,0.1,0.40.2,0.5,0.40.3,0.4,0.30.2,0.3,0.10.2,0.3,0.50.5,0.2,0.60.2,0.4,0.30.5,0.2,0.50.1,0.5,0.30.2,0.4,0.20.1,0.1,0.30.1,0.5,0.40.5,0.3,0.1,D3=0.3,0.2,0.10.3,0.1,0.30.1,0.4,0.50.2,0.3,0.50.6,0.1,0.40.6,0.4,0.20.5,0.4,0.10.5,0.2,0.40.3,0.3,0.60.4,0.2,0.40.2,0.3,0.20.3,0.5,0.10.3,0.6,0.10.5,0.3,0.20.3,0.3,0.60.4,0.3,0.2,D4=0.2,0.2,0.30.3,0.2,0.30.2,0.3,0.50.4,0.2,0.50.4,0.1,0.20.6,0.3,0.50.1,0.2,0.20.5,0.1,0.20.3,0.5,0.10.2,0.2,0.30.5,0.4,0.30.5,0.3,0.20.3,0.1,0.10.2,0.1,0.40.2,0.3,0.20.3,0.1,0.6.Case 1 (incompletely known attribute weights).
Suppose that the incompletely known information of the attribute weight is given as follows:(13)H=0.18≤w1≤0.2,0.15≤w2≤0.25,0.30≤w3≤0.35,0.3≤w4≤0.4,∑j=14wj=1.Step 1.
By modelM1, we establish the following model: (14)minfw=1.5833w1+1.5038w2+1.825w3+1.625w4s.t.w∈H.Step 2.
By solving this model with Matlab software, we get the weight vector:(15)w1=0.18,w2=0.22,w3=0.30,w4=0.30.Step 3.
Using the distance measure (7), we have (16)dA1,A∗=0.4365,dA2,A∗=0.3618,dA3,A∗=0.4502,dA4,A∗=0.4033.Step 4 (rank the alternatives).
Sinced(A3,A∗) is the biggest, and d(A2,A∗) is the smallest, we rank the alternatives as follows: (17)A2≻A4≻A1≻A3,and A2 is the best alternative.Case 2 (completely unknown attribute weights).Step 1.
By modelM2, we establish the following model: (18)minfw=1.5833w1+1.5038w2+1.825w3+1.625w4s.t.∑j=14wj2=1,wj≥0,j=1,2,3,4.Step 2.
Use (11) to obtain the weight vector of attributes: (19)w1∗=0.18,w2∗=0.22,w3∗=0.30,w4∗=0.30.Step 3.
Using the distance measure (8) and (9), we have (20)dA1,A∗=0.4352,dA2,A∗=0.3613,dA3,A∗=0.4482,dA4,A∗=0.3984.Step 4.
Rank the alternatives. Sinced(A3,A∗) is the biggest, and d(A2,A∗) is the smallest, we rank the alternatives as follows: (21)A2≻A4≻A1≻A3,and A2 is the best alternative.


## 3.1.1. Incompletely Known Attribute Weights

In the decision making process, the incomplete information of the attribute weight provided by the decision makers can usually be constructed using several basic ranking forms [25]. Let H be the set of information about the incompletely known attribute weights, which may be constructed in the following forms [26], for i≠j:(a)
A weak ranking:{wi≥wj}.(b)
A strict ranking:{wi-wj≥δi(>0)}.(c)
A ranking with multiples:{wi≥δiwj},0≤δi≤1.(d)
An interval form:{δi≤wi≤δi+εi},0≤δi≤δi+εi.(e)
A ranking of differences:{wi-wj≥wk-wl}, for j≠k≠l.We now establish the following single-objective programming model based on the weighted distance measure method:M1minfw=∑k=1sλk∑i=1m∑j=1nwjdrijk,αj∗s.t.wj∈H,∑j=1nwj=1,wj≥0,j=1,2,…,n,where λk is the weight of the decision maker Dk(k=1,2,…,s) and (8)drijk,αj∗=13Tijk-1+Iijk+Fijk,where d(rijk,αj∗) represents the weighted distance measure between the attribute value rijk and the SVNPIS αj∗. The desirable weight vector w=(w1,w2,…,wn) should make the sum of all weighted distance measures (7) small. So we construct this model to make the overall distance small.By solving the modelM1 with Matlab software, we get the optimal solution w∗=(w1∗,w2∗,…,wn∗), which is considered as the weight of the attributes C1,C2,…,Cn. Then, we utilize d(Ai,A∗) to rank all the alternatives. The smaller the weighted distance measure, the better the alternative.


## 3.1.2. Completely Unknown Attribute Weights

If the information about the attribute weight is completely unknown, we establish the following programming model:M2minfw=∑k=1sλk∑i=1m∑j=1nwjdrijk,αj∗s.t.∑j=1nwj2=1,wj≥0,j=1,2,…,n.To solve this model, we construct the Lagrange function as follows: (9)Lw,λ=∑k=1sλk∑i=1m∑j=1nwjdrijk,αj∗+λ2∑j=1nwj2-1,where λ is the Lagrange multiplier.Differentiating (9) with respect to wj(j=1,2,…,n) and λ, setting these partial derivatives equal to zero, the following set of the equations are obtained: (10)∂L∂wj=∑k=1sλk∑i=1mdrijk,αj∗+wjλ=0,∂L∂λ=∑j=1nwj2=1.By solving (10), we obtain the weight wj and normalize it with wj∗=wj/∑j=1nwj; then, we get (11)wj∗=∑k=1sλk∑i=1mdrijk,αj∗∑j=1n∑k=1sλk∑i=1mdrijk,αj∗.We get the optimal solution w∗=(w1∗,w2∗,…,wn∗), which is considered as the weight of the attributes C1,C2,…,Cn. Later, we calculate the distance measure (7) and get the most desirable one.


## 3.1.3. Illustrative Example

Example 1.
Here, we choose the decision making problem adapted from [23]. An automotive company is desired to select the most appropriate supplier for one of the key elements in its manufacturing process. After preevaluation, four suppliers have remained as alternatives for further evaluation. In order to evaluate alternative suppliers, a committee composed of four decision makers has been formed. The committee selects four attributes to evaluate the alternatives: (1) C1: product quality, (2) C2: relationship closeness, (3) C3: delivery performance, and (4) C4: price. Suppose that there are four decision makers, denoted by d1,d2,d3,d4, whose corresponding weight vector is λ=(0.25,0.25,0.25,0.25). The four possible alternatives are to be evaluated under these four attributes and are in the form of SVNNs for each decision maker, as shown in the following single-valued neutrosophic decision matrix: (12)D1=0.4,0.2,0.30.4,0.2,0.30.2,0.2,0.50.7,0.2,0.30.6,0.1,0.20.6,0.1,0.20.5,0.2,0.30.5,0.1,0.20.3,0.2,0.30.5,0.2,0.30.1,0.5,0.20.1,0.4,0.50.7,0.2,0.10.6,0.1,0.20.4,0.3,0.20.4,0.5,0.1,D2=0.1,0.3,0.50.5,0.1,0.50.3,0.1,0.60.4,0.1,0.40.2,0.5,0.40.3,0.4,0.30.2,0.3,0.10.2,0.3,0.50.5,0.2,0.60.2,0.4,0.30.5,0.2,0.50.1,0.5,0.30.2,0.4,0.20.1,0.1,0.30.1,0.5,0.40.5,0.3,0.1,D3=0.3,0.2,0.10.3,0.1,0.30.1,0.4,0.50.2,0.3,0.50.6,0.1,0.40.6,0.4,0.20.5,0.4,0.10.5,0.2,0.40.3,0.3,0.60.4,0.2,0.40.2,0.3,0.20.3,0.5,0.10.3,0.6,0.10.5,0.3,0.20.3,0.3,0.60.4,0.3,0.2,D4=0.2,0.2,0.30.3,0.2,0.30.2,0.3,0.50.4,0.2,0.50.4,0.1,0.20.6,0.3,0.50.1,0.2,0.20.5,0.1,0.20.3,0.5,0.10.2,0.2,0.30.5,0.4,0.30.5,0.3,0.20.3,0.1,0.10.2,0.1,0.40.2,0.3,0.20.3,0.1,0.6.Case 1 (incompletely known attribute weights).
Suppose that the incompletely known information of the attribute weight is given as follows:(13)H=0.18≤w1≤0.2,0.15≤w2≤0.25,0.30≤w3≤0.35,0.3≤w4≤0.4,∑j=14wj=1.Step 1.
By modelM1, we establish the following model: (14)minfw=1.5833w1+1.5038w2+1.825w3+1.625w4s.t.w∈H.Step 2.
By solving this model with Matlab software, we get the weight vector:(15)w1=0.18,w2=0.22,w3=0.30,w4=0.30.Step 3.
Using the distance measure (7), we have (16)dA1,A∗=0.4365,dA2,A∗=0.3618,dA3,A∗=0.4502,dA4,A∗=0.4033.Step 4 (rank the alternatives).
Sinced(A3,A∗) is the biggest, and d(A2,A∗) is the smallest, we rank the alternatives as follows: (17)A2≻A4≻A1≻A3,and A2 is the best alternative.Case 2 (completely unknown attribute weights).Step 1.
By modelM2, we establish the following model: (18)minfw=1.5833w1+1.5038w2+1.825w3+1.625w4s.t.∑j=14wj2=1,wj≥0,j=1,2,3,4.Step 2.
Use (11) to obtain the weight vector of attributes: (19)w1∗=0.18,w2∗=0.22,w3∗=0.30,w4∗=0.30.Step 3.
Using the distance measure (8) and (9), we have (20)dA1,A∗=0.4352,dA2,A∗=0.3613,dA3,A∗=0.4482,dA4,A∗=0.3984.Step 4.
Rank the alternatives. Sinced(A3,A∗) is the biggest, and d(A2,A∗) is the smallest, we rank the alternatives as follows: (21)A2≻A4≻A1≻A3,and A2 is the best alternative.


## 3.2. The Weighted Distance Measure Based Method to Interval-Valued Neutrosophic Set

LetX={X1,X2,…,Xm} be a set of alternatives, let C={C1,C2,…,Cn} be a set of attributes, and let w={w1,w2,…,wn} be the weight vector of the attribute with wj∈[0,1] and ∑j=1nwj=1. Suppose that there are s decision makers D={D1,D2,…,Ds}, whose corresponding weighted vector is λ={λ1,λ2,…,λs}. Let Ak=(rijk)m×n(k=1,2,…,s) be interval-valued neutrosophic decision matrix, where rijk={Tijk,Iijk,Fijk} is the value of the attribute, expressed by IVNNs.Now, we suppose that the ideal IVNN isβj∗={t∗,i∗,f∗}={[1,1],[0,0],[0,0]}. Based on the ideal IVNN, we define the interval-valued neutrosophic positive ideal solution (IVNPIS).Definition 8.
Letβj∗={[1,1],[0,0],[0,0]}(j=1,2,…,n) be n ideal IVNNs; then, a IVNPIS is defined by (22)A∗=β1∗,β2∗,…,βn∗.Definition 9.
LetAi={ri1,ri2,…,rin}(i=1,2,…,m) be the ith alternative, and let A∗={β1∗,β2∗,…,βn∗} be the IVNPIS; then, the weighted Hamming distance measure (WHDM) between Ai and A∗ is defined by (23)dAi,A∗=∑k=1sλk∑j=1nwjdrijk,βj∗.


### 3.2.1. Incompletely Known Attribute Weights

We now establish the following single-objective programming model based on the weighted distance measure method:M3minfw=∑k=1sλk∑i=1m∑j=1nwjdrijk,βj∗s.t.wj∈H,∑j=1nwj=1,wj≥0,j=1,2,…,n,where λk is the weight of the decision maker Dk(k=1,2,…,s) and (24)drijk,βj∗=16Tij-k-1+Tij+k-1+Iij-k+Iij+k+Fij-k+Fij+k,where d(rijk,βj∗) represents the distance measure between the attribute value rijk and the IVNPIS βj∗. The desirable weight vector w=(w1,w2,…,wn) should make the sum of all weighted distances (23) small. So we construct this model to make the overall distances small. The smaller the WHD, the better the alternative. We use (23) to rank the alternative.By solving the modelM3 with Matlab software, we get the optimal solution w∗=(w1∗,w2∗,…,wn∗), which is considered as the weight of the attributes C1,C2,…,Cn. Then, we utilize d(Ai,A∗) to rank all the alternatives. The smaller the distance, the better the alternative.


### 3.2.2. Completely Unknown Attribute Weights

If the information about the attribute weight is completely unknown, we establish the following programming model:M4minfw=∑k=1sλk∑i=1m∑j=1nwjdrijk,βj∗s.t.∑j=1nwj2=1,wj≥0,j=1,2,…,n.By Lagrange multiple method, we get the completely unknown weight wj and normalize it with wj∗=wj/∑j=1nwj as follows:(25)wj∗=∑k=1sλk∑i=1mdrijk,βj∗∑j=1n∑k=1sλk∑i=1mdrijk,βj∗,which is considered as the weight of the attributes Cj. Later, we calculate the distance measure d(Ai,A∗) and then get the most desirable one.


### 3.2.3. Illustrative Example

Example 2.
The decision making problem is adapted from [23]. Suppose that an organization plans to implement ERP system. The first step is to form a project team that consists of CIO and two senior representatives from user departments. By collecting all information about ERP vendors and systems, project team chooses four potential ERP systems Ai(i=1,2,3,4) as candidates. The company employs some external professional organizations (experts) to aid this decision making. The project team selects four attributes to evaluate the alternatives: (1) C1: function and technology, (2) C2: strategic fitness, (3) C3: vendors ability, and (4) C4: vendors reputation. Suppose that there are three decision makers, denoted by D1,D2,D3, whose corresponding weight vector is λ=(1/3,1/3,1/3). The four possible alternatives are to be evaluated under these four attributes and are in the form of IVNNs for each decision maker, as shown in the following interval-valued neutrosophic decision matrix: (26)D1=0.4,0.5,0.2,0.3,0.3,0.50.3,0.4,0.3,0.6,0.2,0.40.2,0.5,0.2,0.6,0.3,0.50.5,0.6,0.3,0.5,0.2,0.50.6,0.7,0.1,0.2,0.2,0.30.1,0.3,0.1,0.4,0.2,0.50.4,0.5,0.2,0.5,0.3,0.70.2,0.4,0.1,0.4,0.3,0.30.3,0.4,0.2,0.3,0.3,0.40.3,0.6,0.2,0.3,0.2,0.50.2,0.7,0.2,0.4,0.3,0.60.2,0.6,0.4,0.7,0.2,0.70.2,0.6,0.1,0.2,0.1,0.20.2,0.5,0.4,0.5,0.1,0.60.3,0.5,0.1,0.3,0.2,0.20.4,0.4,0.1,0.6,0.1,0.5,D2=0.4,0.6,0.1,0.3,0.2,0.40.3,0.5,0.1,0.4,0.3,0.40.4,0.5,0.2,0.4,0.1,0.30.3,0.6,0.3,0.6,0.3,0.60.3,0.5,0.1,0.2,0.2,0.30.3,0.4,0.2,0.2,0.1,0.30.2,0.7,0.3,0.5,0.3,0.60.2,0.5,0.2,0.7,0.1,0.20.5,0.6,0.2,0.3,0.3,0.40.1,0.4,0.1,0.3,0.3,0.50.5,0.5,0.4,0.6,0.3,0.40.1,0.2,0.1,0.4,0.5,0.60.3,0.4,0.1,0.2,0.1,0.30.3,0.3,0.1,0.5,0.2,0.40.2,0.3,0.4,0.5,0.5,0.60.3,0.3,0.2,0.3,0.1,0.4,D3=0.1,0.3,0.2,0.3,0.4,0.50.3,0.3,0.1,0.3,0.3,0.40.2,0.6,0.3,0.5,0.3,0.50.4,0.6,0.3,0.4,0.2,0.30.3,0.6,0.3,0.5,0.3,0.50.3,0.4,0.3,0.4,0.3,0.50.3,0.5,0.2,0.4,0.1,0.50.1,0.2,0.3,0.5,0.3,0.40.4,0.5,0.2,0.4,0.2,0.40.2,0.3,0.1,0.1,0.3,0.40.1,0.4,0.2,0.6,0.3,0.60.4,0.5,0.2,0.6,0.1,0.30.2,0.4,0.3,0.4,0.1,0.30.1,0.4,0.2,0.5,0.1,0.50.3,0.6,0.2,0.4,0.2,0.20.2,0.4,0.3,0.3,0.2,0.6.Case 1 (incompletely known attribute weights).
Suppose that the incompletely known information of the attribute weight is given as follows:(27)H=0.18≤w1≤0.2,0.15≤w2≤0.25,0.30≤w3≤0.35,0.3≤w4≤0.4,∑j=14wj=1.Step 1.
By modelM3, we establish the following model: (28)minfw=1.4278w1+1.7278w2+1.8278w3+1.7667w4s.t.w∈H.Step 2.
By solving this model with Matlab software, we get the weight vector:(29)w1=0.18,w2=0.25,w3=0.20,w4=0.37.Step 3.
Using the distance measure (24) and (25), we have (30)dA1,A∗=0.4204,dA2,A∗=0.4182,dA3,A∗=0.4471,dA4,A∗=0.41.Step 4 (rank the alternatives).
Sinced(A3,A∗) is the biggest, and d(A4,A∗) is the smallest, we rank the alternatives as follows: (31)A4≻A2≻A1≻A3,and A4 is the best alternative.Case 2 (completely unknown attribute weights).Step 1.
By modelM4, we establish the following model:(32)minfw=1.4278w1+1.7278w2+1.8278w3+1.7667w4s.t.∑j=14wj2=1,wj≥0,j=1,2,3,4.Step 2.
Use (25) to obtain the weight vector of attributes: (33)w1∗=0.2115,w2∗=0.2560,w3∗=0.2708,w4∗=0.2617.Step 3.
Using the distance measure (24) and (25), we have (34)dA1,A∗=0.4200,dA2,A∗=0.3776,dA3,A∗=0.4421,dA4,A∗=0.4054.Step 4 (rank the alternatives).
Sinced(A3,A∗) is the biggest, and d(A2,A∗) is the smallest, we rank the alternatives as follows: (35)A2≻A4≻A1≻A3,and A2 is the best alternative.


## 3.2.1. Incompletely Known Attribute Weights

We now establish the following single-objective programming model based on the weighted distance measure method:M3minfw=∑k=1sλk∑i=1m∑j=1nwjdrijk,βj∗s.t.wj∈H,∑j=1nwj=1,wj≥0,j=1,2,…,n,where λk is the weight of the decision maker Dk(k=1,2,…,s) and (24)drijk,βj∗=16Tij-k-1+Tij+k-1+Iij-k+Iij+k+Fij-k+Fij+k,where d(rijk,βj∗) represents the distance measure between the attribute value rijk and the IVNPIS βj∗. The desirable weight vector w=(w1,w2,…,wn) should make the sum of all weighted distances (23) small. So we construct this model to make the overall distances small. The smaller the WHD, the better the alternative. We use (23) to rank the alternative.By solving the modelM3 with Matlab software, we get the optimal solution w∗=(w1∗,w2∗,…,wn∗), which is considered as the weight of the attributes C1,C2,…,Cn. Then, we utilize d(Ai,A∗) to rank all the alternatives. The smaller the distance, the better the alternative.


## 3.2.2. Completely Unknown Attribute Weights

If the information about the attribute weight is completely unknown, we establish the following programming model:M4minfw=∑k=1sλk∑i=1m∑j=1nwjdrijk,βj∗s.t.∑j=1nwj2=1,wj≥0,j=1,2,…,n.By Lagrange multiple method, we get the completely unknown weight wj and normalize it with wj∗=wj/∑j=1nwj as follows:(25)wj∗=∑k=1sλk∑i=1mdrijk,βj∗∑j=1n∑k=1sλk∑i=1mdrijk,βj∗,which is considered as the weight of the attributes Cj. Later, we calculate the distance measure d(Ai,A∗) and then get the most desirable one.


## 3.2.3. Illustrative Example

Example 2.
The decision making problem is adapted from [23]. Suppose that an organization plans to implement ERP system. The first step is to form a project team that consists of CIO and two senior representatives from user departments. By collecting all information about ERP vendors and systems, project team chooses four potential ERP systems Ai(i=1,2,3,4) as candidates. The company employs some external professional organizations (experts) to aid this decision making. The project team selects four attributes to evaluate the alternatives: (1) C1: function and technology, (2) C2: strategic fitness, (3) C3: vendors ability, and (4) C4: vendors reputation. Suppose that there are three decision makers, denoted by D1,D2,D3, whose corresponding weight vector is λ=(1/3,1/3,1/3). The four possible alternatives are to be evaluated under these four attributes and are in the form of IVNNs for each decision maker, as shown in the following interval-valued neutrosophic decision matrix: (26)D1=0.4,0.5,0.2,0.3,0.3,0.50.3,0.4,0.3,0.6,0.2,0.40.2,0.5,0.2,0.6,0.3,0.50.5,0.6,0.3,0.5,0.2,0.50.6,0.7,0.1,0.2,0.2,0.30.1,0.3,0.1,0.4,0.2,0.50.4,0.5,0.2,0.5,0.3,0.70.2,0.4,0.1,0.4,0.3,0.30.3,0.4,0.2,0.3,0.3,0.40.3,0.6,0.2,0.3,0.2,0.50.2,0.7,0.2,0.4,0.3,0.60.2,0.6,0.4,0.7,0.2,0.70.2,0.6,0.1,0.2,0.1,0.20.2,0.5,0.4,0.5,0.1,0.60.3,0.5,0.1,0.3,0.2,0.20.4,0.4,0.1,0.6,0.1,0.5,D2=0.4,0.6,0.1,0.3,0.2,0.40.3,0.5,0.1,0.4,0.3,0.40.4,0.5,0.2,0.4,0.1,0.30.3,0.6,0.3,0.6,0.3,0.60.3,0.5,0.1,0.2,0.2,0.30.3,0.4,0.2,0.2,0.1,0.30.2,0.7,0.3,0.5,0.3,0.60.2,0.5,0.2,0.7,0.1,0.20.5,0.6,0.2,0.3,0.3,0.40.1,0.4,0.1,0.3,0.3,0.50.5,0.5,0.4,0.6,0.3,0.40.1,0.2,0.1,0.4,0.5,0.60.3,0.4,0.1,0.2,0.1,0.30.3,0.3,0.1,0.5,0.2,0.40.2,0.3,0.4,0.5,0.5,0.60.3,0.3,0.2,0.3,0.1,0.4,D3=0.1,0.3,0.2,0.3,0.4,0.50.3,0.3,0.1,0.3,0.3,0.40.2,0.6,0.3,0.5,0.3,0.50.4,0.6,0.3,0.4,0.2,0.30.3,0.6,0.3,0.5,0.3,0.50.3,0.4,0.3,0.4,0.3,0.50.3,0.5,0.2,0.4,0.1,0.50.1,0.2,0.3,0.5,0.3,0.40.4,0.5,0.2,0.4,0.2,0.40.2,0.3,0.1,0.1,0.3,0.40.1,0.4,0.2,0.6,0.3,0.60.4,0.5,0.2,0.6,0.1,0.30.2,0.4,0.3,0.4,0.1,0.30.1,0.4,0.2,0.5,0.1,0.50.3,0.6,0.2,0.4,0.2,0.20.2,0.4,0.3,0.3,0.2,0.6.Case 1 (incompletely known attribute weights).
Suppose that the incompletely known information of the attribute weight is given as follows:(27)H=0.18≤w1≤0.2,0.15≤w2≤0.25,0.30≤w3≤0.35,0.3≤w4≤0.4,∑j=14wj=1.Step 1.
By modelM3, we establish the following model: (28)minfw=1.4278w1+1.7278w2+1.8278w3+1.7667w4s.t.w∈H.Step 2.
By solving this model with Matlab software, we get the weight vector:(29)w1=0.18,w2=0.25,w3=0.20,w4=0.37.Step 3.
Using the distance measure (24) and (25), we have (30)dA1,A∗=0.4204,dA2,A∗=0.4182,dA3,A∗=0.4471,dA4,A∗=0.41.Step 4 (rank the alternatives).
Sinced(A3,A∗) is the biggest, and d(A4,A∗) is the smallest, we rank the alternatives as follows: (31)A4≻A2≻A1≻A3,and A4 is the best alternative.Case 2 (completely unknown attribute weights).Step 1.
By modelM4, we establish the following model:(32)minfw=1.4278w1+1.7278w2+1.8278w3+1.7667w4s.t.∑j=14wj2=1,wj≥0,j=1,2,3,4.Step 2.
Use (25) to obtain the weight vector of attributes: (33)w1∗=0.2115,w2∗=0.2560,w3∗=0.2708,w4∗=0.2617.Step 3.
Using the distance measure (24) and (25), we have (34)dA1,A∗=0.4200,dA2,A∗=0.3776,dA3,A∗=0.4421,dA4,A∗=0.4054.Step 4 (rank the alternatives).
Sinced(A3,A∗) is the biggest, and d(A2,A∗) is the smallest, we rank the alternatives as follows: (35)A2≻A4≻A1≻A3,and A2 is the best alternative.


## 3.3. Comparative Analysis

Considering the proposed method and the maximizing deviation method proposed by Şahin, there exsit some differences. In Şahin’s method, they calculated the distance measure of all the attributes and assign a small weight to the attribute which has a similar effect among the alternatives; then, they used the weighted aggregation operators and the score functions to rank the alternatives; while, the proposed method calculates the distance measure between the attributes and the ideal solution and obtains the weight that make the weighted distance measure small, we then use the weighted distance measure to rank the alternatives which avoid the complex calculation of aggregation operators processing. The two methods are all effective to deal with the incompletely known or completely unknown attribute weight by solve the program models. The advantage of the proposed method is that calculation is simple and convenient, which can deal with the MAGDM problem effectively.


## 4. Conclusion

In this paper, we investigate the multiattribute group decision making problems expressed with neutrosophic set and the attribute weights are incompletely known or completely unknown. We first define the single-valued neutrosophic ideal solution (SVNIS) and then establish the optimal models to derive the attribute weight. Furthermore, an approach to MAGDM within the framework of SVNS is developed, and the result shows that our approach is reasonable and effective in dealing with decision making problems. Finally, we extend the method to IVNS.



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*Source: 3145341-2016-05-26.xml*