An Extended Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) with Maximizing Deviation Method Based on Integrated Weight Measure for SingleValued Neutrosophic Sets
Abstract
:1. Introduction
2. Preliminaries
 (i)
 $A$is contained in$B,$if${T}_{A}\left(x\right)\le {T}_{B}\left(x\right),{I}_{A}\left(x\right)\ge {I}_{B}\left(x\right),$and${F}_{A}\left(x\right)\ge {F}_{B}\left(x\right),$for all$x\in U.$This relationship is denoted as$A\subseteq B.$
 (ii)
 $A$and$B$are said to be equal if$A\subseteq B$and$B\subseteq A.$
 (iii)
 ${A}^{c}=\left(x,\left({F}_{A}\left(x\right),1{I}_{A}\left(x\right),{T}_{A}\left(x\right)\right)\right),$for all$x\in U.$
 (iv)
 $A\cup B=\left(x,\left(\mathrm{max}\left({T}_{A},{T}_{B}\right),\mathrm{min}\left({I}_{A},{I}_{B}\right),\mathrm{min}\left({F}_{A},{F}_{B}\right)\right)\right),$for all$x\in U.$
 (v)
 $A\cap B=\left(x,\left(\mathrm{min}\left({T}_{A},{T}_{B}\right),\mathrm{max}\left({I}_{A},{I}_{B}\right),\mathrm{max}\left({F}_{A},{F}_{B}\right)\right)\right),$for all$x\in U.$
 (i)
 $x\u2a01y=\left({T}_{x}+{T}_{y}{T}_{x}\ast {T}_{y},{I}_{x}\ast {I}_{y},{F}_{x}\ast {F}_{y}\right)$
 (ii)
 $x\u2a02y=\left({T}_{x}\ast {T}_{y},{I}_{x}+{I}_{y}{I}_{x}\ast {I}_{y},{F}_{x}+{F}_{y}{F}_{x}\ast {F}_{y}\right)$
 (iii)
 $\lambda x=\left(1{\left(1{T}_{x}\right)}^{\lambda},{\left({I}_{x}\right)}^{\lambda},{\left({F}_{x}\right)}^{\lambda}\right),$where$\lambda >0$
 (iv)
 ${x}^{\lambda}=\left({\left({T}_{x}\right)}^{\lambda},1{\left(1{I}_{x}\right)}^{\lambda},1{\left(1{F}_{x}\right)}^{\lambda}\right),$where$\lambda >0$.
 (i)
 The Hamming distance between$A$and$B$are defined as:$${d}_{H}\left(A,B\right)={\displaystyle \sum}_{i=1}^{n}\left\{\left{T}_{A}\left({x}_{i}\right){T}_{B}\left({x}_{i}\right)\right+\left{I}_{A}\left({x}_{i}\right){I}_{B}\left({x}_{i}\right)\right+\left{F}_{A}\left({x}_{i}\right){F}_{B}\left({x}_{i}\right)\right\right\}$$
 (ii)
 The normalized Hamming distance between$A$and$B$are defined as:$${d}_{H}^{N}\left(A,B\right)=\frac{1}{3n}{\displaystyle \sum}_{i=1}^{n}\left\{\left{T}_{A}\left({x}_{i}\right){T}_{B}\left({x}_{i}\right)\right+\left{I}_{A}\left({x}_{i}\right){I}_{B}\left({x}_{i}\right)\right+\left{F}_{A}\left({x}_{i}\right){F}_{B}\left({x}_{i}\right)\right\right\}$$
 (ii)
 The Euclidean distance between$A$and$B$are defined as:$${d}_{E}\left(A,B\right)=\sqrt{{\displaystyle \sum}_{i=1}^{n}\left\{{\left({T}_{A}\left({x}_{i}\right){T}_{B}\left({x}_{i}\right)\right)}^{2}+{\left({I}_{A}\left({x}_{i}\right){I}_{B}\left({x}_{i}\right)\right)}^{2}+{\left({F}_{A}\left({x}_{i}\right){F}_{B}\left({x}_{i}\right)\right)}^{2}\right\}}$$
 (iv)
 The normalized Euclidean distance between$A$and$B$are defined as:$${d}_{E}^{N}\left(A,B\right)=\sqrt{\frac{1}{3n}{\displaystyle \sum}_{i=1}^{n}\left\{{\left({T}_{A}\left({x}_{i}\right){T}_{B}\left({x}_{i}\right)\right)}^{2}+{\left({I}_{A}\left({x}_{i}\right){I}_{B}\left({x}_{i}\right)\right)}^{2}+{\left({F}_{A}\left({x}_{i}\right){F}_{B}\left({x}_{i}\right)\right)}^{2}\right\}}$$
3. A TOPSIS Method for SingleValued Neutrosophic Sets
3.1. Description of Problem
3.2. The Maximizing Deviation Method for Computing Incomplete or Completely Unknown Attribute Weights
 (i)
 The deviation value of all alternatives to other alternatives for the parameter ${e}_{j}\in A,$ denoted by ${D}_{j}\left({w}_{j}\right),$ is defined as:$${D}_{j}\left({w}_{j}\right)={\displaystyle \sum}_{i=1}^{m}{D}_{ij}\left({w}_{j}\right)={\displaystyle \sum}_{i=1}^{m}{\displaystyle \sum}_{k=1}^{m}{w}_{j}d\left({x}_{ij},{x}_{kj}\right),$$
 (ii)
 The total deviation value of all parameters to all alternatives, denoted by $D\left({w}_{j}\right),$ is defined as:$$\left({w}_{j}\right)={\displaystyle \sum}_{j=1}^{n}{D}_{j}\left({w}_{j}\right)={\displaystyle \sum}_{j=1}^{n}{\displaystyle \sum}_{i=1}^{m}{\displaystyle \sum}_{k=1}^{m}{w}_{j}d\left({x}_{ij},{x}_{kj}\right),$$
 (iii)
 The individual objective weight of each parameter ${e}_{j}\in A,$ denoted by ${\theta}_{j},$ is defined as:$${\theta}_{j}=\frac{{{\displaystyle \sum}}_{i=1}^{m}{{\displaystyle \sum}}_{k=1}^{m}d\left({x}_{ij},{x}_{kj}\right)}{{{\displaystyle \sum}}_{j=1}^{n}{{\displaystyle \sum}}_{i=1}^{m}{{\displaystyle \sum}}_{k=1}^{m}d\left({x}_{ij},{x}_{kj}\right)}$$
3.3. TOPSIS Method for MADM Problems with Incomplete Weight Information
3.3.1. The Proposed TOPSIS Method for SVNSs
3.3.2. Attribute Weight Determination Method: An Integrated WEIGHT MEASure
Algorithm 1. (based on a modified TOPSIS approach). 
Step 1. Input the SVNS $A$ which represents the information pertaining to the problem. Step 2. Input the subjective weight ${h}_{j}$ for each of the attributes ${e}_{j}\in A$ as given by the decision makers. Step 3. Compute the objective weight ${\theta}_{j}$ for each of the attributes ${e}_{j}\in A,$ using Equation (9). Step 4. The integrated weight coefficient ${w}_{j}$ for each of the attributes ${e}_{j}\in A,$ is computed using Equation as follow:
$${w}_{j}=\frac{{h}_{j}{\theta}_{j}}{{{\displaystyle \sum}}_{j=1}^{n}{h}_{j}{\theta}_{j}}$$
Step 6. The difference between each alternative and the RNPIS, ${D}^{+}$ and the RNNIS ${D}^{}$ are computed using Equations (12) and (13), respectively. Step 7. The relative closeness coefficient ${C}_{i}$ for each alternative is calculated using Equation (14). Step 8. Choose the optimal alternative based on the principal of maximum closeness coefficient. 
4. Application of the Topsis Method in a Made Problem
4.1. Illustrative Example
Algorithm 2. (based on the modified TOPSIS approach). 
Step 1. The SVNS $A$ constructed for this problem is given in tabular form in Table 2 Step 2. The subjective weight ${h}_{j}$ for each attribute ${e}_{j}\in A$ as given by the procurement team (the decision makers) are $h=\left\{{h}_{1}=0.15,{h}_{2}=0.15,{h}_{3}=0.22,{h}_{4}=0.25,{h}_{5}=0.14,{h}_{6}=0.09\right\}.$ Step 3. The objective weight ${\theta}_{j}$ for each attribute ${e}_{j}\in A$ is computed using Equation (9) are as given below: $\theta =\{{\theta}_{1}=0.139072,{\theta}_{2}=0.170256,{\theta}_{3}=0.198570,{\theta}_{4}=0.169934,{\theta}_{5}=0.142685,$ ${\theta}_{6}=0.179484\}.$ Step 4. The integrated weight ${w}_{j}$ for each attribute ${e}_{j}\in A$ is computed using Equation (15). The integrated weight coefficent obtained for each attribute is: $w=\{{w}_{1}=0.123658,\text{}{w}_{2}=0.151386,\text{}{w}_{3}=0.258957,\text{}{w}_{4}=0.251833,{w}_{5}=0.118412,$ ${w}_{6}=0.0957547\}.$ Step 5. Use Equations (10) and (11) to compute the values of ${b}^{+}$ and ${b}^{}$ from the neutrosophic numbers given in Table 2. The values are as given below: ${b}^{+}=\{{b}_{1}^{+}=\left[0.7,0.2,0.1\right],\text{}{b}_{2}^{+}=\left[0.9,\text{}0,\text{}0.1\right],\text{}{b}_{3}^{+}=\left[0.8,\text{}0,\text{}0\right],\text{}{b}_{4}^{+}=\left[0.9,\text{}0.3,\text{}0\right],$ ${b}_{5}^{+}=\left[0.7,\text{}0.2,\text{}0.2\right],\text{}{b}_{6}^{+}=[0.8,0.20.1\}$ and ${b}^{}=\{{b}_{1}^{}=\left[0.5,0.8,0.5\right],\text{}{b}_{2}^{}=\left[0.6,\text{}0.8,\text{}0.5\right],\text{}{b}_{3}^{}=\left[0.1,\text{}0.7,\text{}0.5\right],\text{}{b}_{4}^{}=\left[0.3,\text{}0.8,\text{}0.7\right],$ ${b}_{5}^{}=\left[0.5,\text{}0.8,\text{}0.7\right],\text{}{b}_{6}^{}=\left[0.5,0.8,0.9\right]\}.$ Step 6. Use Equations (12) and (13) to compute the difference between each alternative and the RNPIS and the RNNIS, respectively. The values of ${D}^{+}$ and ${D}^{}$ are as given below: ${D}^{+}=\{{D}_{1}^{+}=0.262072,\text{}{D}_{2}^{+}=0.306496,\text{}{D}_{3}^{+}=0.340921,\text{}{D}_{4}^{+}=0.276215,\text{}{D}_{5}^{+}=0.292443,$ ${D}_{6}^{+}=0.345226,\text{}{D}_{7}^{+}=0.303001,\text{}{D}_{8}^{+}=0.346428,\text{}{D}_{9}^{+}=0.271012,\text{}{D}_{10}^{+}=0.339093\}.$ and ${D}^{}=\{{D}_{1}^{}=0.374468,\text{}{D}_{2}^{}=0.307641,\text{}{D}_{3}^{}=0.294889,\text{}{D}_{4}^{}=0.355857,\text{}{D}_{5}^{}=0.323740$ ${D}_{6}^{}=0.348903,\text{}{D}_{7}^{}=0.360103,\text{}{D}_{8}^{}=0.338725,\text{}{D}_{9}^{}=0.379516,\text{}{D}_{10}^{}=0.349703\}.$ Step 7. Using Equation (14), the closeness coefficient ${C}_{i}$ for each alternative is: ${C}_{1}=0.0133,\text{}{C}_{2}=0.3589,\text{}{C}_{3}=0.5239,\text{}{C}_{4}=0.1163,\text{}{C}_{5}=0.2629,$ ${C}_{6}=0.3980,\text{}{C}_{7}=0.2073,\text{}{C}_{8}=0.4294,\text{}{C}_{9}=0.0341,\text{}{C}_{10}=0.3725$. Step 8. The ranking of the alternatives obtained from the closeness coefficient is as given below:
$${x}_{1}>{x}_{9}>{x}_{4}>{x}_{7}>{x}_{5}>{x}_{2}>{x}_{10}>{x}_{6}>{x}_{8}>{x}_{3}.$$

4.2. Adaptation of the Algorithm to NonIntegrated Weight Measure
4.2.1. ObjectiveOnly Adaptation of Our Algorithm
4.2.2. SubjectiveOnly Adaptation of Our Algorithm
5. Comparatives Studies
5.1. Comparison of Results Obtained Through Different Methods
5.2. Discussion of Results
 (i)
 The method proposed in this paper uses an integrated weight measure which considers both the subjective and objective weights of the attributes, as opposed to some of the methods that only consider the subjective weights or objective weights.
 (ii)
 Different operators emphasizes different aspects of the information which ultimately leads to different rankings. For example, in [40], the GSNNWA operator used is based on an arithmetic average which emphasizes the characteristics of the group (i.e., the whole information), whereas the GSNNWG operator is based on a geometric operator which emphasizes the characteristics of each individual alternative and attribute. As our method places more importance on the characteristics of the individual alternatives and attributes, instead of the entire information as a whole, our method produces the same ranking as the GSNNWG operator but different results from the GSNNWA operator.
5.3. Analysis of the Performance and Reliability of Different Methods
Analysis
 (i)
 The algorithms in [10,11,39] all use the data given below as inputs$$S=\left\{\begin{array}{c}\left[0.4,0.2,0.3\right],\left[0.4,0.2,0.3\right],\left[0.2,0.2,05\right]\\ \left[0.6,0.1,0.2\right],\left[0.6,0.1,0.2\right],\left[0.5,0.2,0.2\right]\\ \left[0.3,0.2,0.3\right],\left[0.5,0.2,0.3\right],\left[0.5,0.3,0.2\right]\\ \left[0.7,0.0,0.1\right],\left[0.6,0.1,0.2\right],\left[0.4,0.3,0.2\right]\end{array}\right\}$$All the five algorithms from papers [10,11,39] yields either one of the following rankings:$${A}_{4}>{A}_{2}>{A}_{3}>{A}_{1}\mathrm{or}{A}_{2}{A}_{4}{A}_{3}{A}_{1}$$
 (ii)
 The method proposed in [44] also uses the data given in $S$ above as inputs but ignores the opinions of the decision makers as it does not take into account the subjective weights of the attributes. The algorithm from this paper yields the ranking of ${A}_{4}>{A}_{2}>{A}_{3}>{A}_{1}.$To fit this data into our algorithm, we randomly assigned the subjective weights of the attributes as ${w}_{j}=\frac{1}{3}$ for $j=1,2,3$. A ranking of ${A}_{4}>{A}_{2}>{A}_{3}>{A}_{1}$ was nonetheless obtained from our algorithm.
 (iii)
 The methods introduced in [14,43,45] all use the data given below as input values:$$S=\left\{\begin{array}{c}\left[0.5,0.1,0.3\right],\left[0.5,0.1,0.4\right],\left[0.7,0.1,02\right],\left[0.3,0.2,0.1\right]\\ \left[0.4,0.2,0.3\right],\left[0.3,0.2,0.4\right],\left[0.9,0.0,0.1\right],\left[0.5,0.3,0.2\right]\\ \left[0.4,0.3,0.1\right],\left[0.5,0.1,0.3\right],\left[0.5,0.0,0.4\right],\left[0.6,0.2,0.2\right]\\ \left[0.6,0.1,0.2\right],\left[0.2,0.2,0.5\right],\left[0.4,0.3,0.2\right],\left[0.7,0.2,0.1\right]\end{array}\right\}$$
 (iv)
 It can be observed that for the methods introduced in [10,11,39,44], we have $0.8\le {T}_{ij}+{I}_{ij}+{F}_{ij}\le 1$ for all the entries. A similar trend can be observed in [14,43,45], where $0.6\le {T}_{ij}+{I}_{ij}+{F}_{ij}\le 1$ for all the entries. Therefore, we are not certain about the results obtained through the decision making algorithms in these papers when the value of ${T}_{ij}+{I}_{ij}+{F}_{ij}$ deviates very far from 1.
 $T:$ the track record of the suppliers that is approved by the committee
 $I:$ the track record of the suppliers that the committee feels is questionable
 $F:$ the track record of the suppliers that is rejected by the committee
6. Conclusions
 (i)
 A novel TOPSIS method for the SVNS model is introduced, with the maximizing deviation method used to determine the objective weight of the attributes. Through thorough analysis, we have proven that our algorithm is compliant with all of the three tests that were discussed in Section 5.3. This clearly indicates that our proposed decisionmaking algorithm is not only an effective algorithm but one that produces the most reliable and accurate results in all the different types of situation and data inputs.
 (ii)
 Unlike other methods in the existing literature which reduces the elements from singlevalued neutrosophic numbers (SVNNs) to fuzzy numbers, or interval neutrosophic numbers (INNs) to neutrosophic numbers or fuzzy numbers, in our version of the TOPSIS method the input data is in the form of SVNNs and this form is maintained throughout the decisionmaking process. This prevents information loss and enables the original information to be retained, thereby ensuring a higher level of accuracy for the results that are obtained.
 (iii)
 The objective weighting method (e.g., the ones used in [10,11,14,39,40,43,45]) only takes into consideration the values of the membership functions while ignoring the preferences of the decision makers. Through the subjective weighting method (e.g., the ones used in [42,44]), the attribute weights are given by the decision makers based on their individual preferences and experiences. Very few approaches in the existing literature (e.g., [15]) consider both the objective and subjective weighting methods. Our proposed method uses an integrated weighting model that considers both the objective and subjective weights of the attributes, and this accurately reflects the input values of the alternatives as well as the preferences and risk attitude of the decision makers.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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$\mathit{U}$  ${\mathit{e}}_{1}$  ${\mathit{e}}_{2}$  $\mathbf{\dots}$  ${\mathit{e}}_{\mathit{n}}$ 

${x}_{1}$  $\left({T}_{11},{I}_{11},{F}_{11}\right)$  $\left({T}_{12},{I}_{12},{F}_{12}\right)$  $\dots $  $\left({T}_{1n},{I}_{1n},{F}_{1n}\right)$ 
${x}_{2}$  $\left({T}_{21},{I}_{21},{F}_{21}\right)$  $\left({T}_{22},{I}_{22},{F}_{22}\right)$  $\dots $  $\left({T}_{2n},{I}_{2n},{F}_{2n}\right)$ 
$\vdots $  $\vdots $  $\vdots $  $\ddots $  $\vdots $ 
${x}_{m}$.  $\left({T}_{m1},{I}_{m1},{F}_{m1}\right)$  $\left({T}_{m2},{I}_{m2},{F}_{m2}\right)$  $\dots $  $\left({T}_{mn},{I}_{mn},{F}_{mn}\right)$ 
$\mathbf{U}$  ${\mathbf{e}}_{1}$  ${\mathbf{e}}_{2}$  ${\mathbf{e}}_{3}$ 

${x}_{1}$  $\left(0.7,0.5,0.1\right)$  $\left(0.7,0.5,0.3\right)$  $\left(0.8,0.6,0.2\right)$ 
${x}_{2}$  $\left(0.6,0.5,0.2\right)$  $\left(0.7,0.5,0.1\right)$  $\left(0.6,0.3,0.5\right)$ 
${x}_{3}$  $\left(0.6,0.2,0.3\right)$  $\left(0.6,0.6,0.4\right)$  $\left(0.7,0.7,0.2\right)$ 
${x}_{4}$  $\left(0.5,0.5,0.4\right)$  $\left(0.6,0.4,0.4\right)$  $\left(0.7,0.7,0.3\right)$ 
${x}_{5}$  $\left(0.7,0.5,0.5\right)$  $\left(0.8,0.3,0.1\right)$  $\left(0.7,0.6,0.2\right)$ 
$\mathrm{U}$  ${\mathrm{e}}_{1}$  ${\mathrm{e}}_{2}$  ${\mathrm{e}}_{3}$ 
${x}_{6}$  $\left(0.5,0.5,0.5\right)$  $\left(0.7,0.8,0.1\right)$  $\left(0.7,0.3,0.5\right)$ 
${x}_{7}$  $\left(0.6,0.8,0.1\right)$  $\left(0.7,0.2,0.1\right)$  $\left(0.6,0.3,0.4\right)$ 
${x}_{8}$  $\left(0.7,0.8,0.3\right)$  $\left(0.6,0.6,0.5\right)$  $\left(0.8,0,0.5\right)$ 
${x}_{9}$  $\left(0.6,0.7,0.1\right)$  $\left(0.7,0,0.1\right)$  $\left(0.6,0.7,0\right)$ 
${x}_{10}$  $\left(0.5,0.7,0.4\right)$  $\left(0.9,0,0.3\right)$  $\left(1,0,0\right)$ 
$\mathrm{U}$  ${\mathrm{e}}_{4}$  ${\mathrm{e}}_{5}$  ${\mathrm{e}}_{6}$ 
${x}_{1}$  $\left(0.9,0.4,0.2\right)$  $\left(0.6,0.4,0.7\right)$  $\left(0.6,0.5,0.4\right)$ 
${x}_{2}$  $\left(0.6,0.4,0.3\right)$  $\left(0.7,0.5,0.4\right)$  $\left(0.7,0.8,0.9\right)$ 
${x}_{3}$  $\left(0.5,0.5,0.3\right)$  $\left(0.6,0.8,0.6\right)$  $\left(0.7,0.2,0.5\right)$ 
${x}_{4}$  $\left(0.9,0.4,0.2\right)$  $\left(0.7,0.3,0.5\right)$  $\left(0.6,0.4,0.4\right)$ 
${x}_{5}$  $\left(0.7,0.5,0.2\right)$  $\left(0.7,0.5,0.6\right)$  $\left(0.6,0.7,0.8\right)$ 
$\mathrm{U}$  ${\mathrm{e}}_{4}$  ${\mathrm{e}}_{5}$  ${\mathrm{e}}_{6}$ 
${x}_{6}$  $\left(0.4,0.8,0\right)$  $\left(0.7,0.4,0.2\right)$  $\left(0.5,0.6,0.3\right)$ 
${x}_{7}$  $\left(0.3,0.5,0.1\right)$  $\left(0.6,0.3,0.6\right)$  $\left(0.5,0.2,0.6\right)$ 
${x}_{8}$  $\left(0.7,0.3,0.6\right)$  $\left(0.6,0.8,0.5\right)$  $\left(0.6,0.2,0.4\right)$ 
${x}_{9}$  $\left(0.7,0.4,0.3\right)$  $\left(0.6,0.6,0.7\right)$  $\left(0.7,0.3,0.2\right)$ 
${x}_{10}$  $\left(0.5,0.6,0.7\right)$  $\left(0.5,0.2,0.7\right)$  $\left(0.8,0.4,0.1\right)$ 
Method  The Final Ranking  The Best Alternative 

Ye [39] (i) WAAO * (ii) WGAO **  ${x}_{1}>{x}_{4}>{x}_{9}>{x}_{5}>{x}_{7}>{x}_{2}>{x}_{10}>{x}_{8}>{x}_{3}>{x}_{6}$ ${x}_{10}>{x}_{9}>{x}_{8}>{x}_{1}>{x}_{5}>{x}_{7}>{x}_{4}>{x}_{2}>{x}_{6}>{x}_{3}$  ${x}_{1}$ ${x}_{10}$ 
Ye [10] (i) Weighted correlation coefficient (ii) Weighted cosine similarity measure  ${x}_{1}>{x}_{4}>{x}_{5}>{x}_{9}>{x}_{2}>{x}_{8}>{x}_{7}>{x}_{3}>{x}_{6}>{x}_{10}$ ${x}_{1}>{x}_{9}>{x}_{4}>{x}_{5}>{x}_{2}>{x}_{10}>{x}_{8}>{x}_{3}>{x}_{7}>{x}_{6}$  ${x}_{1}$ ${x}_{1}$ 
Ye [11]  ${x}_{1}>{x}_{9}>{x}_{4}>{x}_{7}>{x}_{5}>{x}_{2}>{x}_{8}>{x}_{6}>{x}_{3}>{x}_{10}$  ${x}_{1}$ 
Huang [14]  ${x}_{1}>{x}_{9}>{x}_{4}>{x}_{5}>{x}_{2}>{x}_{7}>{x}_{8}>{x}_{6}>{x}_{3}>{x}_{10}$  ${x}_{1}$ 
Peng et al. [40] (i) GSNNWA *** (ii) GSNNWG ****  ${x}_{9}>{x}_{10}>{x}_{8}>{x}_{6}>{x}_{1}>{x}_{7}>{x}_{4}>{x}_{5}>{x}_{2}>{x}_{3}$ ${x}_{1}>{x}_{9}>{x}_{4}>{x}_{5}>{x}_{7}>{x}_{2}>{x}_{8}>{x}_{3}>{x}_{6}>{x}_{10}$  ${x}_{9}$ ${x}_{1}$ 
Peng & Liu [15] (i) EDAS (ii) Similarity measure  ${x}_{1}>{x}_{4}>{x}_{6}>{x}_{9}>{x}_{10}>{x}_{3}>{x}_{2}>{x}_{7}>{x}_{5}>{x}_{8}$ ${x}_{10}>{x}_{8}>{x}_{7}>{x}_{4}>{x}_{1}>{x}_{2}>{x}_{5}>{x}_{9}>{x}_{3}>{x}_{6}$  ${x}_{1}$ ${x}_{10}$ 
Maji [41]  ${x}_{5}>{x}_{1}>{x}_{9}>{x}_{6}>{x}_{2}>{x}_{4}>{x}_{3}>{x}_{8}>{x}_{7}>{x}_{10}$  ${x}_{5}$ 
Karaaslan [42]  ${x}_{1}>{x}_{9}>{x}_{4}>{x}_{5}>{x}_{7}>{x}_{2}>{x}_{8}>{x}_{3}>{x}_{6}>{x}_{10}$  ${x}_{1}$ 
Ye [43]  ${x}_{1}>{x}_{9}>{x}_{4}>{x}_{5}>{x}_{7}>{x}_{2}>{x}_{8}>{x}_{3}>{x}_{6}>{x}_{10}$  ${x}_{1}$ 
Biswas et al. [44]  ${x}_{10}>{x}_{9}>{x}_{7}>{x}_{1}>{x}_{4}>{x}_{6}>{x}_{5}>{x}_{8}>{x}_{2}>{x}_{3}$  ${x}_{10}$ 
Ye [45]  ${x}_{9}>{x}_{7}>{x}_{1}>{x}_{4}>{x}_{2}>{x}_{10}>{x}_{5}>{x}_{8}>{x}_{3}>{x}_{6}$  ${x}_{9}$ 
Adaptation of our algorithm (objective weights only)  ${x}_{9}>{x}_{1}>{x}_{4}>{x}_{10}>{x}_{7}>{x}_{6}>{x}_{5}>{x}_{8}>{x}_{3}>{x}_{2}$  ${x}_{9}$ 
Adaptation of our algorithm (subjective weights only)  ${x}_{1}>{x}_{9}>{x}_{4}>{x}_{7}>{x}_{5}>{x}_{2}>{x}_{6}>{x}_{10}>{x}_{8}>{x}_{3}$  ${x}_{1}$ 
Our proposed method (using integrated weight measure)  ${x}_{1}>{x}_{9}>{x}_{4}>{x}_{7}>{x}_{5}>{x}_{2}>{x}_{10}>{x}_{6}>{x}_{8}>{x}_{3}$  ${x}_{1}$ 
Paper  Test 1 Compliance  Test 2 Compliance  Test 3 Compliance  

Ye [39]  WAAO *  Y  Y  N 
WGAO *  N  Y  N  
Ye [10]  Weighted correlation coefficient  Y  Y  N 
Weighted cosine similarity measure  N  Y  N  
Ye [11]  Y  Y  N  
Huang [14]  Y  Y  N  
Peng et al. [40]  GSNNWA **  Y  Y  N 
GSNNWG **  Y  Y  N  
Peng & Liu [15]  EDAS  Y  Y  N 
Similarity measure  N  Y  Y  
Maji [41]  N  N  N  
Karaaslan [42]  Y  Y  N  
Ye [43]  Y  Y  N  
Biswas et al. [44]  Y  N  Y  
Ye [45]  Y  Y  N  
Adaptation of our proposed algorithm (objective weights only)  Y  N  Y  
Adaptation of our proposed algorithm (subjective weights only)  Y  Y  N  
Our proposed algorithm  Y  Y  Y 
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Selvachandran, G.; Quek, S.G.; Smarandache, F.; Broumi, S. An Extended Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) with Maximizing Deviation Method Based on Integrated Weight Measure for SingleValued Neutrosophic Sets. Symmetry 2018, 10, 236. https://doi.org/10.3390/sym10070236
Selvachandran G, Quek SG, Smarandache F, Broumi S. An Extended Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) with Maximizing Deviation Method Based on Integrated Weight Measure for SingleValued Neutrosophic Sets. Symmetry. 2018; 10(7):236. https://doi.org/10.3390/sym10070236
Chicago/Turabian StyleSelvachandran, Ganeshsree, Shio Gai Quek, Florentin Smarandache, and Said Broumi. 2018. "An Extended Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) with Maximizing Deviation Method Based on Integrated Weight Measure for SingleValued Neutrosophic Sets" Symmetry 10, no. 7: 236. https://doi.org/10.3390/sym10070236