MultiCriteria Group DecisionMaking Using an mPolar Hesitant Fuzzy TOPSIS Approach
Abstract
:1. Introduction
2. $\mathit{m}$Polar Hesitant Fuzzy Set Model
 Empty set: ${\hslash}_{m}^{e}=\left({\left\{0\right\}}_{m}\right)$.
 Full set: ${\hslash}_{m}^{f}=\left({\left\{1\right\}}_{m}\right)$.
 Complete ignorance: (All values are possible) ${\hslash}_{m}=[\mathbf{0},\mathbf{1}]$, where $\mathbf{0}=(0,0,\cdots ,0)$ and $\mathbf{1}=(1,1,\cdots ,1)$.
 Nonsense set: $\mathsf{\Phi}$.
2.1. Basic Operations
 Lower bound:$${\hslash}_{m}^{}\left(z\right)=\left(\right)open="("\; close=")">inf\left\{{\zeta}_{h}\right{\zeta}_{h}\in {p}_{1}\circ {\hslash}_{m}\left(z\right)\},inf\left\{{\zeta}_{h}\right{\zeta}_{h}\in {p}_{2}\circ {\hslash}_{m}\left(z\right)\},\cdots ,inf\left\{{\zeta}_{h}\right{\zeta}_{h}\in {p}_{m}\circ {\hslash}_{m}\left(z\right)\}$$
 Upper bound:$${\hslash}_{m}^{+}\left(z\right)=\left(\right)open="("\; close=")">sup\left\{{\zeta}_{h}\right{\zeta}_{h}\in {p}_{1}\circ {\hslash}_{m}\left(z\right)\},sup\left\{{\zeta}_{h}\right{\zeta}_{h}\in {p}_{2}\circ {\hslash}_{m}\left(z\right)\},\cdots ,sup\left\{{\zeta}_{h}\right{\zeta}_{h}\in {p}_{m}\circ {\hslash}_{m}\left(z\right)\}$$
 Complement:$${\hslash}_{m}^{c}\left(z\right)=\left(\right)open="("\; close=")">\{1{\zeta}_{h}{\zeta}_{h}\in {p}_{1}\circ {\hslash}_{m}\left(z\right)\},\{1{\zeta}_{h}{\zeta}_{h}\in {p}_{2}\circ {\hslash}_{m}\left(z\right)\},\cdots ,\{1{\zeta}_{h}{\zeta}_{h}\in {p}_{m}\circ {\hslash}_{m}\left(z\right)\}$$
 Union:$$({\hslash}_{m}^{\left({H}_{1}\right)}\cup {\hslash}_{m}^{\left({H}_{2}\right)})\left(z\right)=\left(\right)open="("\; close=")">\{{\zeta}_{h}\in {p}_{i}\circ {\hslash}_{m}^{\left({H}_{1}\right)}\left(z\right)\cup {p}_{i}\circ {\hslash}_{m}^{\left({H}_{2}\right)}\left(z\right){\zeta}_{h}\ge sup\{{\hslash}_{m}^{\left({H}_{1}\right)}\left(z\right),{\hslash}_{m}^{\left({H}_{2}\right)}\left(z\right)\}\}$$
 Intersection:$$({\hslash}_{m}^{\left({H}_{1}\right)}\cap {\hslash}_{m}^{\left({H}_{2}\right)})\left(z\right)=\left(\right)open="("\; close=")">\{{\zeta}_{h}\in {p}_{i}\circ {\hslash}_{m}^{\left({H}_{1}\right)}\left(z\right)\cap {p}_{i}\circ {\hslash}_{m}^{\left({H}_{2}\right)}\left(z\right){\zeta}_{h}\le inf\{{\hslash}_{m}^{\left({H}_{1}\right)+}\left(z\right),{\hslash}_{m}^{\left({H}_{2}\right)+}\left(z\right)\}\}$$
 Direct sum:$$({\hslash}_{m}^{\left({H}_{1}\right)}\oplus {\hslash}_{m}^{\left({H}_{2}\right)})\left(z\right)=\left(\right)open="("\; close=")">\{{\zeta}_{{h}_{1}}+{\zeta}_{{h}_{2}}{\zeta}_{{h}_{1}}{\zeta}_{{h}_{2}}{\zeta}_{{h}_{1}}\in {p}_{i}\circ {\hslash}_{m}^{\left({H}_{1}\right)}\left(z\right),{\zeta}_{{h}_{2}}\in {p}_{i}\circ {\hslash}_{m}^{\left({H}_{2}\right)}\left(z\right)\}$$
 Direct product:$$({\hslash}_{m}^{\left({H}_{1}\right)}\otimes {\hslash}_{m}^{\left({H}_{2}\right)})\left(z\right)=\left(\right)open="("\; close=")">\left\{{\zeta}_{{h}_{1}}{\zeta}_{{h}_{2}}\right{\zeta}_{{h}_{1}}\in {p}_{i}\circ {\hslash}_{m}^{\left({H}_{1}\right)}\left(z\right),{\zeta}_{{h}_{2}}\in {p}_{i}\circ {\hslash}_{m}^{\left({H}_{2}\right)}\left(z\right)\}$$
 1.
 Lower bound:$$\begin{array}{ll}{\hslash}_{m}^{\left({H}_{1}\right)}\left({z}_{1}\right)& =\left(\right)open="("\; close=")">inf\{0.2,0.3\},inf\{0.4,0.5,0.6\},inf\{0.4,0.6\}\end{array}$$$$\begin{array}{ll}{\hslash}_{m}^{\left({H}_{2}\right)}\left({z}_{3}\right)& =\left(\right)open="("\; close=")">inf\{0.3,0.4\},inf\{0.2,0.4,0.6\},inf\{0.5,0.7\}\end{array}$$
 2.
 Upper bound:$$\begin{array}{ll}{\hslash}_{m}^{\left({H}_{1}\right)+}\left({z}_{2}\right)& =\left(\right)open="("\; close=")">sup\{0.3,0.5\},sup\{0.4,0.6\},sup\{0.7,0.8\}\end{array}$$$$\begin{array}{ll}{\hslash}_{m}^{\left({H}_{2}\right)+}\left({z}_{3}\right)& =\left(\right)open="("\; close=")">sup\{0.3,0.4\},sup\{0.2,0.4,0.6\},sup\{0.5,0.7\}\end{array}$$
 3.
 Complement:$$\begin{array}{ll}{\hslash}_{m}^{\left({H}_{1}\right)c}\left({z}_{1}\right)& =\left(\right)open="("\; close=")">\{10.2,10.3\},\{10.4,10.5,10.6\},\{10.4,10.6\}\end{array}$$$$\begin{array}{ll}{\hslash}_{m}^{\left({H}_{2}\right)c}\left({z}_{2}\right)& =\left(\right)open="("\; close=")">\{10.5,10.6\},\{10.2,10.3,10.4\},\{10.3,10.5,10.8,10.9\}\end{array}$$
 4.
 Union:$$\begin{array}{ll}({\hslash}_{m}^{\left({H}_{1}\right)}\cup {\hslash}_{m}^{\left({H}_{2}\right)})\left({z}_{1}\right)& =max\left(\right)open="\{"\; close="\}">(0.2,0.4,0.4),(0.4,0.6,0.7)=(0.4,0.6,0.7)\end{array}$$$$\begin{array}{ll}({\hslash}_{m}^{\left({H}_{1}\right)}\cup {\hslash}_{m}^{\left({H}_{2}\right)})\left({z}_{3}\right)& =max\left(\right)open="\{"\; close="\}">(0.1,0.5,0.7),(0.3,0.2,0.5)=(0.3,0.5,0.7)\end{array}$$
 5.
 Intersection:$$\begin{array}{ll}({\hslash}_{m}^{\left({H}_{1}\right)}\cap {\hslash}_{m}^{\left({H}_{2}\right)})\left({z}_{2}\right)& =max\left(\right)open="\{"\; close="\}">(0.5,0.6,0.8),(0.6,0.4,0.9)=(0.5,0.4,0.8)\end{array}$$$$\begin{array}{ll}({\hslash}_{m}^{\left({H}_{1}\right)}\cap {\hslash}_{m}^{\left({H}_{2}\right)})\left({z}_{3}\right)& =max\left(\right)open="\{"\; close="\}">(0.2,0.7,0.8),(0.4,0.6,0.7)=(0.2,0.6,0.7)\end{array}$$
 6.
 Direct sum:$$\begin{array}{ll}({\hslash}_{m}^{\left({H}_{1}\right)}\oplus {\hslash}_{m}^{\left({H}_{2}\right)})\left({z}_{1}\right)=(& \{0.52,0.68,0.76,0.79,0.58,0.72\},\{0.76,0.82,0.8,0.85,0.84,0.88\},\\ & \{0.88,0.82,0.92\}).\end{array}$$
 7.
 Direct product:$$\begin{array}{ll}({\hslash}_{m}^{\left({H}_{1}\right)}\otimes {\hslash}_{m}^{\left({H}_{2}\right)})\left({z}_{3}\right)=(& \{0.03,0.04,0.06,0.08\},\{0.1,0.2,0.3,0.12,0.24,0.36,0.14,0.28,0.42\},\\ & \{0.35,0.49,0.40,0.56\}).\end{array}$$
 1.
 Commutativity:
 (i)
 $({\hslash}_{m}^{(1)}\cup {\hslash}_{m}^{(2)})(z)=({\hslash}_{m}^{(2)}\cup {\hslash}_{m}^{(1)})(z),$
 (ii)
 $({\hslash}_{m}^{(1)}\cap {\hslash}_{m}^{(2)})(z)=({\hslash}_{m}^{(2)}\cap {\hslash}_{m}^{(1)})(z).$
 2.
 Associativity:
 (i)
 $(({\hslash}_{m}^{(1)}\cup {\hslash}_{m}^{(2)})\cup {\hslash}_{m}^{(3)})(z)=({\hslash}_{m}^{(1)}\cup ({\hslash}_{m}^{(2)}\cup {\hslash}_{m}^{(3)}))(z),$
 (ii)
 $(({\hslash}_{m}^{(1)}\cap {\hslash}_{m}^{(2)})\cap {\hslash}_{m}^{(3)})(z)=({\hslash}_{m}^{(1)}\cap ({\hslash}_{m}^{(2)}\cap {\hslash}_{m}^{(3)}))(z).$
 3.
 Idempotency:
 (i)
 $({\hslash}_{m}^{(1)}\cup {\hslash}_{m}^{(1)})(z)={\hslash}_{m}^{(1)}(z),$
 (ii)
 $({\hslash}_{m}^{(1)}\cap {\hslash}_{m}^{(1)})(z)={\hslash}_{m}^{(1)}(z).$
 1.
 ${\left({\hslash}_{m}^{c}\right)}^{}\left(z\right)=\mathbf{1}{\hslash}_{m}^{+}\left(z\right),$
 2.
 ${\left({\hslash}_{m}^{c}\right)}^{+}\left(z\right)=\mathbf{1}{\hslash}_{m}^{}\left(z\right).$
 1.
 $({\hslash}_{m}\cup {\hslash}_{m}^{f})(z)={\hslash}_{m}^{f}(z)$ and $({\hslash}_{m}\cap {\hslash}_{m}^{f})(z)={\hslash}_{m}(z),$
 2.
 $({\hslash}_{m}\cup {\hslash}_{m}^{e})(z)={\hslash}_{m}(z)$ and $({\hslash}_{m}\cap {\hslash}_{m}^{e})(z)={\hslash}_{m}^{e}(z).$
2.2. Comparison Laws of mHFEs
 If $s({\hslash}_{m}^{(1)})>s({\hslash}_{m}^{(2)})$, then ${\hslash}_{m}^{(1)}$ is superior to (or finer than) ${\hslash}_{m}^{(2)}.$
 If $s({\hslash}_{m}^{(1)})<s({\hslash}_{m}^{(2)})$, then ${\hslash}_{m}^{(1)}$ is inferior to (or weaker than) ${\hslash}_{m}^{(2)}.$
 If $s({\hslash}_{m}^{(1)})=s({\hslash}_{m}^{(2)})$, then ${\hslash}_{m}^{(1)}$ is indifferent to ${\hslash}_{m}^{(2)}.$
 If none of the above are true, then ${\hslash}_{m}^{(1)}$ is totally different from ${\hslash}_{m}^{(2)}.$
 If $\Delta ({\hslash}_{m}^{(1)})>\Delta ({\hslash}_{m}^{(2)})$, then ${\hslash}_{m}^{(1)}$ is superior to (or finer than) ${\hslash}_{m}^{(2)}.$
 If $\Delta ({\hslash}_{m}^{(1)})<\Delta ({\hslash}_{m}^{(2)})$, then ${\hslash}_{m}^{(1)}$ is inferior to (or weaker than) ${\hslash}_{m}^{(2)}.$
 If $\Delta ({\hslash}_{m}^{(1)})=\Delta ({\hslash}_{m}^{(2)})$, then ${\hslash}_{m}^{(1)}$ is indifferent to ${\hslash}_{m}^{(2)}.$
 If none of the above are true, then ${\hslash}_{m}^{(1)}$ is completely different from ${\hslash}_{m}^{(2)}.$
3. The $\mathit{m}$Polar Hesitant Fuzzy TOPSIS Approach
Algorithm 1 Algorithm of the proposed approach for multicriteria group decisionmaking (MCGDM). 

3.1. Selection of a Perfect Brand Name
 “Articulate core identity”, which may include the following features:
 The “Vision”, or why your company exists;
 the “Mission”, or what your company does;
 the “Value”, or how you do what you do; and
 the “Direction”, or where it goes on.
 “Brainstorm”, which may include the following features:
 The “Founder”, a name based on a real or fictional person;
 the “Description”, a name that describes what you do or make;
 the “Magic spell”, a name that is a portmanteau (two words together) or a real word with a madeup spelling; and
 the “Fabricated”, a totally madeup name or word.
 “Test”, which may include the following features:
 “Sounds good”, it is good to hear;
 “Not confusing”, it is not linked with other brand names;
 “Not mispronounced”, it is easy to pronounce; and
 “Related publicity”, it focuses on a targeted group of customers.
3.2. Selection of Suitable Product Design for a Company
 The “Appearance" of a product design may include the following features:
 “Contrast and symmetry";
 “Color and shade"; and
 “Body texture and surface".
 The “Material" of a product design may include the following features:
 “Fine quality";
 “Low cost"; and
 “Reversibility".
 The “Dimensions and Tolerances" of a product design may include the following features:
 “Size and functions";
 “Flexibility"; and
 “Nominal geometry".
 The “Performance Standards" of a product design may include the following features:
 “Market value";
 “Customer satisfaction"; and
 “Availability and evaluating report".
4. Comparison Analysis of Proposed Approach
 All previously proposed TOPSIS methods for decisionmaking were not suitable for such situations, where the alternatives are assessed depending on hesitant situations of decisionmakers, under the conditions of huge data with multipolar information. An mHFTOPSIS method is able to deal with these situations, having such kinds of multipolar data under hesitant situations. This method is also preferable, because it is able to deal with both pessimistic and optimistic decisions, in which the decisionmakers are free from any external conditions and requirements. In this method, all aspects related to alternatives, according to the preferences of the decisionmakers, are discussed. The proposed approach is able to provide more flexible and precise results, in order to choose the best alternative considering multipolar information under hesitancy. Although its calculations are complex and difficult to handle, we have generated a computer programming code to make these complex calculations easier.
 An mF linguistic TOPSIS method is also considered as a flexible approach, as compared to various other extensions of TOPSIS, but this approach is limited, as a linguistic variable and its values are considered as fixed criteria for the evaluation and ranking of alternatives. This approach is valid only when the alternatives have linguistic variables and corresponding values. In this method, the alternatives are assessed depending on the linguistic values of a variable, which are further classified by m different characteristics. This approach is only able to observe and recognize expertise about the linguistic variable and the values of alternatives, in the form of words and sentences having multipolar information. It is unable to provide any information about the hesitant situation of a decision. This approach is unable to discuss general cases, other than those with linguistic values and variables.
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A
MATLAB Computer Programming Code of Proposed Approach 

1. clc 
2. m=input(‘enter the total number of poles’); 
3. H=input(‘enter the r×m decision matrix in each entry’); 
4. w=input(‘enter the weights as dimension 1×q’); 
5. [u,q]=size(w); 
6. [p,v]=size(H); 
7. $r=v/(q\ast m)$; 
8. if sum(w,2)==1 
9. W=zeros(p,v); 
10. for j=1:p 
11. for k=1:q 
12. for v1=k*r*m(r*m1):k*r*m 
13. W(j,v1)=w(1,k).*H(j,v1); 
14. end 
15. end 
16. end 
17. W 
18. mHPIS=zeros(1,v); mHNIS=ones(1,v); 
19. for j=1:p 
20. for v1=1:v 
21. mHPIS(1,v1)=max(mHPIS(1,v1),W(j,v1)); 
22. mHNIS(1,v1)=min(mHNIS(1,v1),W(j,v1)); 
23. end 
24. end 
25. mHPIS 
26. mHNIS 
27. Y=zeros(p,v); Z=zeros(p,v); 
28. for j=1:p 
29. for v1=1:v 
30. Y(j,v1)=(W(j,v1)mHPIS(1,v1)). $\widehat{\phantom{\rule{3.33333pt}{0ex}}}$ 2; 
31. Z(j,v1)=(W(j,v1)mHNIS(1,v1)). $\widehat{\phantom{\rule{3.33333pt}{0ex}}}$ 2; 
32. end 
33. end 
34. D_p=zeros(p,q);D_n=zeros(p,q); 
35. for j=1:p 
36. for k=1:q 
37. for v1=k*r*m(r*m1):k*r*m 
38. D_p(j,k)=D_p(j,k)+Y(j,v1); 
39. D_n(j,k)=D_n(j,k)+Z(j,v1); 
40. end 
41. end 
42. end 
43. D=[sqrt(sum(D_p,2)./(r*m)) sqrt(sum(D_n,2)./(r*m))] 
44. E=D(:,2)./sum(D,2) 
45. end 
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Alternatives  Criteria’s  

${\mathit{c}}_{\mathbf{1}}$  ${\mathit{c}}_{\mathbf{2}}$  ⋯  ${\mathit{c}}_{\mathit{q}}$  
${a}_{1}$  ${\hslash}_{m}^{11}$  ${\hslash}_{m}^{12}$  ⋯  ${\hslash}_{m}^{1q}$ 
${a}_{2}$  ${\hslash}_{m}^{21}$  ${\hslash}_{m}^{22}$  ⋯  ${\hslash}_{m}^{2q}$ 
⋮  ⋮  ⋮  ⋮  ⋮ 
${a}_{p}$  ${\hslash}_{m}^{p1}$  ${\hslash}_{m}^{p2}$  ⋯  ${\hslash}_{m}^{pq}$ 
Alternatives  Criteria’s  

${\mathit{c}}_{\mathbf{1}}$  ${\mathit{c}}_{\mathbf{2}}$  ⋯  ${\mathit{c}}_{\mathit{q}}$  
Weights  
${\mathit{w}}_{\mathbf{1}}$  ${\mathit{w}}_{\mathbf{2}}$  ⋯  ${\mathit{w}}_{\mathit{q}}$  
${a}_{1}$  ${\hslash}_{m}^{{11}^{\prime}}$  ${\hslash}_{m}^{{12}^{\prime}}$  ⋯  ${\hslash}_{m}^{1{q}^{\prime}}$ 
${a}_{2}$  ${\hslash}_{m}^{{21}^{\prime}}$  ${\hslash}_{m}^{{22}^{\prime}}$  ⋯  ${\hslash}_{m}^{2{q}^{\prime}}$ 
⋮  ⋮  ⋮  ⋮  ⋮ 
${a}_{p}$  ${\hslash}_{m}^{p{1}^{\prime}}$  ${\hslash}_{m}^{p{2}^{\prime}}$  ⋯  ${\hslash}_{m}^{p{q}^{\prime}}$ 
Brand Names  Articulate Core Identity  
Vision  Mission  Value  Direction  
${B}_{n1}$  {0.40, 0.50}  {0.30, 0.60, 0.70}  {0.20, 0.70}  {0.30, 0.70, 0.80} 
${B}_{n2}$  {0.60, 0.70}  {0.30, 0.70}  {0.50, 0.60}  {0.40, 0.60, 0.80} 
${B}_{n3}$  {0.40}  {0.40, 0.50, 0.80}  {0.20, 0.30, 0.50}  {0.60, 0.80} 
${B}_{n4}$  {0.70, 0.80}  {0.60, 0.80}  {0.50}  {0.70, 0.80, 0.90} 
${B}_{n5}$  {0.40, 0.60}  {0.55, 0.70}  {0.40, 0.50, 0.70}  {0.75, 0.80} 
Brand Names  Brainstorm  
Founder  Descriptive  Magic spell  Fabricated  
${B}_{n1}$  {0.30, 0.70}  {0.40, 0.50, 0.80}  {0.60, 0.80}  {0.70, 0.80} 
${B}_{n2}$  {0.10, 0.20, 0.30}  {0.50, 0.60}  {0.10, 0.50}  {0.60, 0.80} 
${B}_{n3}$  {0.10, 0.15}  {0.20, 0.50}  {0.40}  {0.70, 0.80, 0.90} 
${B}_{n4}$  {0.40, 0.50}  {0.65, 0.70}  {0.40, 0.70}  {0.70, 0.80} 
${B}_{n5}$  {0.45, 0.50}  {0.50, 0.70}  {0.10, 0.20}  {0.50, 0.60, 0.70} 
Brand Names  Test  
Sounds good  Not confusing  Not mispronounced  Related publicity  
${B}_{n1}$  {0.40, 0.60}  {0.70, 0.80}  {0.30, 0.50}  {0.60, 0.70, 0.90} 
${B}_{n2}$  {0.30, 0.40, 0.60}  {0.20}  {0.40, 0.70}  {0.20, 0.30, 0.50} 
${B}_{n3}$  {0.30, 0.50, 0.70}  {0.50, 0.80}  {0.60, 0.90}  {0.50, 0.70} 
${B}_{n4}$  {0.20, 0.50}  {0.10, 0.25}  {0.60, 0.80}  {0.50, 0.60, 0.80} 
${B}_{n5}$  {0.60, 0.80, 0.90}  {0.50, 0.60}  {0.70}  {0.20, 0.30, 0.35} 
Brand Names  Articulate Core Identity  
Vision  Mission  Value  Direction  
${B}_{n1}$  {0.40, 0.50, 0.50}  {0.30, 0.60, 0.70}  {0.20, 0.70, 0.70}  {0.30, 0.70, 0.80} 
${B}_{n2}$  {0.60, 0.70, 0.70}  {0.30, 0.70, 0.70}  {0.50, 0.60, 0.60}  {0.40, 0.60, 0.80} 
${B}_{n3}$  {0.40, 0.40, 0.40}  {0.40, 0.50, 0.80}  {0.20, 0.30, 0.50}  {0.60, 0.80, 0.80} 
${B}_{n4}$  {0.70, 0.80, 0.80}  {0.60, 0.80, 0.80}  {0.50, 0.50, 0.50}  {0.70, 0.80, 0.90} 
${B}_{n5}$  {0.40, 0.60, 0.60}  {0.55, 0.70, 0.70}  {0.40, 0.50, 0.70}  {0.75, 0.80, 0.80} 
Brand Names  Brainstorm  
Founder  Descriptive  Magic spell  Fabricated  
${B}_{n1}$  {0.30, 0.70, 0.70}  {0.40, 0.50, 0.80}  {0.60, 0.80, 0.80}  {0.70, 0.80, 0.80} 
${B}_{n2}$  {0.10, 0.20, 0.30}  {0.50, 0.60, 0.60}  {0.10, 0.50, 0.50}  {0.60, 0.80, 0.80} 
${B}_{n3}$  {0.10, 0.15, 0.15}  {0.20, 0.50, 0.50}  {0.40, 0.40, 0.40}  {0.70, 0.80, 0.90} 
${B}_{n4}$  {0.40, 0.500.50}  {0.65, 0.70, 0.70}  {0.40, 0.70, 0.70}  {0.70, 0.80, 0.80} 
${B}_{n5}$  {0.45, 0.50, 0.50}  {0.50, 0.70, 0.70}  {0.10, 0.20, 0.20}  {0.50, 0.60, 0.70} 
Brand Names  Test  
Sounds good  Not confusing  Not mispronounced  Related publicity  
${B}_{n1}$  {0.40, 0.60, 0.60}  {0.70, 0.80, 0.80}  {0.30, 0.50, 0.50}  {0.60, 0.70, 0.90} 
${B}_{n2}$  {0.30, 0.40, 0.60}  {0.20, 0.20, 0.20}  {0.40, 0.70, 0.70}  {0.20, 0.30, 0.50} 
${B}_{n3}$  {0.30, 0.50, 0.70}  {0.50, 0.80, 0.80}  {0.60, 0.90, 0.90}  {0.50, 0.70, 0.70} 
${B}_{n4}$  {0.20, 0.50, 0.50}  {0.10, 0.25, 0.25}  {0.60, 0.80, 0.80}  {0.50, 0.60, 0.80} 
${B}_{n5}$  {0.60, 0.80, 0.90}  {0.50, 0.60, 0.60}  {0.70, 0.70, 0.70}  {0.20, 0.30, 0.35} 
Brand Names  Articulate Core Identity  
Vision  Mission  Value  Direction  
${B}_{n1}$  {0.0920, 0.1150, 0.1150}  {0.0690, 0.1380, 0.1610}  {0.0460, 0.1610, 0.1610}  {0.0690, 0.1610, 0.1840} 
${B}_{n2}$  {0.1380, 0.1610, 0.1610}  {0.0690, 0.1610, 0.1610}  {0.1150, 0.1380, 0.1380}  {0.0920, 0.1380, 0.1840} 
${B}_{n3}$  {0.0920, 0.0920, 0.0920}  {0.0920, 0.1150, 0.1840}  {0.0460, 0.0690, 0.1150}  {0.1380, 0.1840, 0.1840} 
${B}_{n4}$  {0.1610, 0.1840, 0.1840}  {0.1380, 0.1840, 0.1840}  {0.1150, 0.1150, 0.1150}  {0.1610, 0.1840, 0.2070} 
${B}_{n5}$  {0.0920, 0.1380, 0.1380}  {0.1265, 0.1610, 0.1610}  {0.0920, 0.1150, 0.1610}  {0.1725, 0.1840, 0.1840} 
Brand Names  Brainstorm  
Founder  Descriptive  Magic spell  Fabricated  
${B}_{n1}$  {0.1020, 0.2380, 0.2380}  {0.1360, 0.1700, 0.2720}  {0.2040, 0.2720, 0.2720}  {0.2380, 0.2720, 0.2720} 
${B}_{n2}$  {0.0340, 0.0680, 0.1020}  {0.1700, 0.2040, 0.2040}  {0.0340, 0.1700, 0.1700}  {0.2040, 0.2720, 0.2720} 
${B}_{n3}$  {0.0340, 0.0510, 0.0510}  {0.0680, 0.1700, 0.1700}  {0.1360, 0.1360, 0.1360}  {0.2380, 0.2720, 0.3060} 
${B}_{n4}$  {0.1360, 0.1700, 0.1700}  {0.2210, 0.2380, 0.2380}  {0.1360, 0.2380, 0.2380}  {0.2380, 0.2720, 0.2720} 
${B}_{n5}$  {0.1530, 0.1700, 0.1700}  {0.1700, 0.2380, 0.2380}  {0.0340, 0.0680, 0.0680}  {0.1700, 0.2040, 0.2380} 
Brand Names  Test  
Sounds good  Not confusing  Not mispronounced  Related publicity  
${B}_{n1}$  {0.1720, 0.2580, 0.2580}  {0.3010, 0.3440, 0.3440}  {0.1290, 0.2150, 0.2150}  {0.2580, 0.3010, 0.3870} 
${B}_{n2}$  {0.1290, 0.1720, 0.2580}  {0.0860, 0.0860, 0.0860}  {0.1720, 0.3010, 0.3010}  {0.0860, 0.1290, 0.2150} 
${B}_{n3}$  {0.1290, 0.2150, 0.3010}  {0.2150, 0.3440, 0.3440}  {0.2580, 0.3870, 0.3870}  {0.2150, 0.3010, 0.301} 
${B}_{n4}$  {0.0860, 0.2150, 0.2150}  {0.0430, 0.1075, 0.1075}  {0.2580, 0.3440, 0.3440}  {0.2150, 0.2580, 0.3440} 
${B}_{n5}$  {0.2580, 0.3440, 0.3870}  {0.2150, 0.2580, 0.2580}  {0.3010, 0.3010, 0.3010}  {0.0860, 0.1290, 0.1505} 
Product Design  Appearance  
Contrast and Symmetry  Color and Shade  Body Texture and Surface  
${P}_{d1}$  {0.25, 0.45, 0.47}  {0.30, 0.31, 0.36}  {0.20, 0.25, 0.26} 
${P}_{d2}$  {0.46, 0.48, 0.49}  {0.47, 0.49}  {0.55, 0.60, 0.61, 0.63} 
${P}_{d3}$  {0.51, 0.53, 0.57, 0.60}  {0.46, 0.52, 0.70}  {0.29, 0.30, 0.51, 0.52} 
${P}_{d4}$  {0.39, 0.41, 0.43}  {0.60, 0.68, 0.71, 0.73}  {0.50, 0.67, 0.69} 
Product Design  Material  
Fine Quality  Low Cost  Reversibility  
${P}_{d1}$  {0.45, 0.49, 0.51, 0.59}  {0.67, 0.68, 0.71}  {0.50, 0.56, 0.63, 0.64} 
${P}_{d2}$  {0.49, 0.50}  {0.71, 0.74, 0.79}  {0.35, 0.59, 0.61, 0.65} 
${P}_{d3}$  {0.71, 0.73, 0.77}  {0.46, 0.52, 0.70}  {0.29, 0.30, 0.51, 0.52} 
${P}_{d4}$  {0.53, 0.54, 0.56, 0.58}  {0.60, 0.63, 0.73, 0.79}  {0.40, 0.47, 0.49} 
Product Design  Dimension and Tolerance  
Size and Functions  Flexibility  Nominal Geometry  
${P}_{d1}$  {0.85, 0.86, 0.87}  {0.53, 0.59, 0.66}  {0.72, 0.75, 0.76, 0.78} 
${P}_{d2}$  {0.66, 0.68, 0.69}  {0.47, 0.50, 51, 0.64}  {0.65, 0.66, 0.81} 
${P}_{d3}$  {0.51, 0.55}  {0.66, 0.68, 0.75, 0.76}  {0.39, 0.40, 0.58, 0.62} 
${P}_{d4}$  {0.59, 0.61, 0.73, 0.74}  {0.26, 0.38, 0.41, 0.43}  {0.51, 0.77} 
Product Design  Performance Standards  
Market Value  Customer Satisfaction  Availability/Evaluating Report  
${P}_{d1}$  {0.55, 0.65}  {0.40, 0.48, 0.60, 0.61}  {0.80, 0.85, 0.86} 
${P}_{d2}$  {0.54, 0.58, 0.59, 0.61}  {0.77, 0.79, 0, 84}  {0.55, 0.60, 0.68} 
${P}_{d3}$  {0.81, 0.83, 0.87}  {0.56, 0.62, 0.70}  {0.69, 0.70, 0.76, 0.82} 
${P}_{d4}$  {0.37, 0.48, 0.49, 0.59}  {0.26, 0.38, 0.41, 0.43}  {0.60, 0.67} 
Product Design  Appearance  
Contrast and Symmetry  Color and Shade  Body Texture and Surface  
${P}_{d1}$  {0.25, 0.25, 0.45, 0.47}  {0.30, 0.30, 0.31, 0.36}  {0.20, 0.20, 0.25, 0.26} 
${P}_{d2}$  {0.46, 0.46, 0.48, 0.49}  {0.47, 0.47, 0.47, 0.49}  {0.55, 0.60, 0.61, 0.63} 
${P}_{d3}$  {0.51, 0.53, 0.57, 0.60}  {0.46, 0.46, 0.52, 0.70}  {0.29, 0.30, 0.51, 0.52} 
${P}_{d4}$  {0.39, 0.39, 0.41, 0.43}  {0.60, 0.68, 0.71, 0.73}  {0.50, 0.50, 0.67, 0.69} 
Product Design  Material  
Fine Quality  Low Cost  Reversibility  
${P}_{d1}$  {0.45, 0.49, 0.51, 0.59}  {0.67, 0.67, 0.68, 0.71}  {0.50, 0.56, 0.63, 0.64} 
${P}_{d2}$  {0.49, 0.49, 0.49, 0.50}  {0.71, 0.71, 0.74, 0.79}  {0.35, 0.59, 0.61, 0.65} 
${P}_{d3}$  {0.71, 0.71, 0.73, 0.77}  {0.46, 0.46, 0.52, 0.70}  {0.29, 0.30, 0.51, 0.52} 
${P}_{d4}$  {0.53, 0.54, 0.56, 0.58}  {0.60, 0.63, 0.73, 0.79}  {0.40, 0.40, 0.47, 0.49} 
Product Design  Dimension and Tolerance  
Size and Functions  Flexibility  Nominal Geometry  
${P}_{d1}$  {0.85, 0.85, 0.86, 0.87}  {0.53, 0.53, 0.59, 0.66}  {0.72, 0.75, 0.76, 0.78} 
${P}_{d2}$  {0.66, 0.66, 0.68, 0.69}  {0.47, 0.50, 0.51, 0.64}  {0.65, 0.65, 0.66, 0.81} 
${P}_{d3}$  {0.51, 0.51, 0.51, 0.55}  {0.66, 0.68, 0.75, 0.76}  {0.39, 0.40, 0.58, 0.62} 
${P}_{d4}$  {0.59, 0.61, 0.73, 0.74}  {0.26, 0.38, 0.41, 0.43}  {0.51, 0.51, 0.51, 0.77} 
Product Design  Performance Standards  
Market Value  Customer Satisfaction  Availability/Evaluating Report  
${P}_{d1}$  {0.55, 0.55, 0.55, 0.65}  {0.40, 0.48, 0.60, 0.61}  {0.80, 0.80, 0.85, 0.86} 
${P}_{d2}$  {0.54, 0.58, 0.59, 0.61}  {0.77, 0.77, 0.79, 0, 84}  {0.55, 0.55, 0.60, 0.68} 
${P}_{d3}$  {0.81, 0.81, 0.83, 0.87}  {0.56, 0.56, 0.62, 0.70}  {0.69, 0.70, 0.76, 0.82} 
${P}_{d4}$  {0.37, 0.48, 0.49, 0.59}  {0.26, 0.38, 0.41, 0.43}  {0.60, 0.60, 0.60, 0.67} 
Product Design  Appearance  
Contrast and Symmetry  Color and Shade  Body Texture and Surface  
${P}_{d1}$  {0.0503, 0.0503, 0.0905, 0.0946}  {0.0604, 0.0604, 0.0624, 0.0724}  {0.0402, 0.0402, 0.0503, 0.0523} 
${P}_{d2}$  {0.0926, 0.0926, 0.0966, 0.0986}  {0.0946, 0.0946, 0.0946, 0.0986}  {0.1107, 0.1207, 0.1227, 0.1268} 
${P}_{d3}$  {0.1026, 0.1066, 0.1147, 0.1207}  {0.0926, 0.0926, 0.1046, 0.1408}  {0.0583, 0.0604, 0.1026, 0.1046} 
${P}_{d4}$  {0.0785, 0.0785, 0.0825, 0.0865}  {0.1207, 0.1368, 0.1429, 0.1469}  {0.1006, 0.1006, 0.1348, 0.1388} 
Product Design  Material  
Fine Quality  Low Cost  Reversibility  
${P}_{d1}$  {0.1017, 0.1107, 0.1152, 0.1333}  {0.1514, 0.1514, 0.1536, 0.1604}  {0.1129, 0.1265, 0.1423, 0.1446} 
${P}_{d2}$  {0.1107, 0.1107, 0.1107, 0.1129}  {0.1604, 0.1604, 0.1672, 0.1785}  {0.0791, 0.1333, 0.1378, 0.1468} 
${P}_{d3}$  {0.1604, 0.1604, 0.1649, 0.1739 }  {0.1039, 0.1039, 0.1175, 0.1581}  {0.0655, 0.0678, 0.1152, 0.1175} 
${P}_{d4}$  {0.1197, 0.1220, 0.1265, 0.1310}  {0.1355, 0.1423, 0.1649, 0.1785}  {0.0904, 0.0904, 0.1062, 0.1107} 
Product Design  Dimension and Tolerance  
Size and Functions  Flexibility  Nominal Geometry  
${P}_{d1}$  {0.2236, 0.2236, 0.2263, 0.2289}  {0.1394, 0.1394, 0.1552, 0.1736}  {0.1894, 0.1973, 0.2000, 0.2052} 
${P}_{d2}$  {0.1736, 0.1736, 0.1789, 0.1815}  {0.1237, 0.1316, 0.1342, 0.1684}  {0.1710, 0.1710, 0.1736, 0.2131} 
${P}_{d3}$  {0.1342, 0.1342, 0.1342, 0.1447}  {0.1736, 0.1789, 0.1973, 0.2000}  {0.1026, 0.1052, 0.1526, 0.1631} 
${P}_{d4}$  {0.1552, 0.1605, 0.1921, 0.1947}  {0.0684, 0.1000, 0.1079, 0.1131}  {0.1342, 0.1342, 0.1342, 0.2026} 
Product Design  Performance Standards  
Market Value  Customer Satisfaction  Availability/Evaluating Report  
${P}_{d1}$  {0.1704, 0.1704, 0.1704, 0.2014}  {0.1239, 0.1487, 0.1859, 0.1890}  {0.2478, 0.2478, 0.2633, 0.2664} 
${P}_{d2}$  {0.1673, 0.1797, 0.1828, 0.1890}  {0.2385, 0.2385, 0.2447, 0.2602}  {0.1704, 0.1704, 0.1859, 0.2107} 
${P}_{d3}$  {0.2509, 0.2509, 0.2571, 0.2695}  {0.1735, 0.1735, 0.1921, 0.2169}  {0.2138, 0.2169, 0.2354, 0.2540} 
${P}_{d4}$  {0.1146, 0.1487, 0.1518, 0.1828}  {0.0805, 0.1177, 0.1270, 0.1332}  {0.1859, 0.1859, 0.1859, 0.2076} 
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Akram, M.; Adeel, A.; Alcantud, J.C.R. MultiCriteria Group DecisionMaking Using an mPolar Hesitant Fuzzy TOPSIS Approach. Symmetry 2019, 11, 795. https://doi.org/10.3390/sym11060795
Akram M, Adeel A, Alcantud JCR. MultiCriteria Group DecisionMaking Using an mPolar Hesitant Fuzzy TOPSIS Approach. Symmetry. 2019; 11(6):795. https://doi.org/10.3390/sym11060795
Chicago/Turabian StyleAkram, Muhammad, Arooj Adeel, and José Carlos R. Alcantud. 2019. "MultiCriteria Group DecisionMaking Using an mPolar Hesitant Fuzzy TOPSIS Approach" Symmetry 11, no. 6: 795. https://doi.org/10.3390/sym11060795