Selection of the Most Appropriate Renewable Energy Alternatives by Using a Novel Interval-Valued Neutrosophic ELECTRE I Method

Definition 1.

Let X be a universe of discourse. A single-valued neutrosophic set A in X is:

A = { x , ( T A ( x ) , I A ( x ) , F A ( x ) ) | x ∈ X } , \[ A=\big\{x,\big({T_{A}}(x),{I_{A}}(x),{F_{A}}(x)\big)\hspace{0.1667em}\big|\hspace{0.1667em}x\in X\big\},\]

(1)

where

T A ( x ) ${T_{A}}(x)$

,

I A ( x ) ${I_{A}}(x)$

, and

F A ( x ) ${F_{A}}(x)$

represent the membership functions for truth, indeterminacy, and falsity, respectively. For each point x in X,

T A ( x ) ${T_{A}}(x)$

,

I A ( x ) ${I_{A}}(x)$

,

F A ( x ) ∈ [ 0 , 1 ] ${F_{A}}(x)\in [0,1]$

, and

0 ⩽ T A ( x ) + I A ( x ) + F A ( x ) ⩽ 3 $0\leqslant \hspace{2.5pt}{T_{A}}(x)+\hspace{2.5pt}{I_{A}}(x)+{F_{A}}(x)\leqslant 3$

(Wang et al., 2005a).

Definition 2.

Let X be a universe of discourse. An IVN set N in X is independently characterized by the intervals

T N ( x ) = [ T N ( x ) L , T N ( x ) U ] ⊆ [ 0 , 1 ] ${T_{N}}(x)=[{T_{N(x)}^{L}},{T_{N(x)}^{U}}]\subseteq [0,1]$

,

I N ( x ) = [ I N ( x ) L , I N ( x ) U ] ⊆ [ 0 , 1 ] ${I_{N}}(x)=[{I_{N(x)}^{L}},{I_{N(x)}^{U}}]\subseteq [0,1]$

, and

F N ( x ) = [ F N ( x ) L , F N ( x ) U ] ⊆ [ 0 , 1 ] ${F_{N}}(x)=[{F_{N(x)}^{L}},{F_{N(x)}^{U}}]\subseteq [0,1]$

. They satisfy the condition

0 ⩽ T N L ( x ) + I N L ( x ) + F N L ( x ) ⩽ 3 $0\leqslant {T_{N}^{L}}(x)+{I_{N}^{L}}(x)+{F_{N}^{L}}(x)\leqslant 3$

. Thus, the IVNS N can be denoted as Wang et al. (2005b):

N = [ T N L ( x ) , T N U ( x ) ] , [ I N L ( x ) , I N U ( x ) ] , [ F N L ( x ) , F N U ( x ) ] . \[ N=\left\langle \big[{T_{N}^{L}}(x),{T_{N}^{U}}(x)\big],\big[{I_{N}^{L}}(x),{I_{N}^{U}}(x)\big],\big[{F_{N}^{L}}(x),{F_{N}^{U}}(x)\big]\right\rangle .\]

(2)

We will denote Eq. (2) as

[ T N L , T N U ] $[{T_{N}^{L}},{T_{N}^{U}}]$

,

[ I N L , I N U ] $[{I_{N}^{L}},{I_{N}^{U}}]$

,

[ F N L , F N U ] $[{F_{N}^{L}},{F_{N}^{U}}]$

for short.

Let

a = [ T a L , T a U ] , [ I a L , I a U ] , [ F a L , F a U ] $a=[{T_{a}^{L}},{T_{a}^{U}}],[{I_{a}^{L}},{I_{a}^{U}}],[{F_{a}^{L}},{F_{a}^{U}}]$

and

b = [ T b L , T b U ] , [ I b L , I b U ] , [ F b L , F b U ] $b=[{T_{b}^{L}},{T_{b}^{U}}],[{I_{b}^{L}},{I_{b}^{U}}],[{F_{b}^{L}},{F_{b}^{U}}]$

be two IVN numbers (IVNNs), the relations of them are shown as below (Wang et al., 2005a; Zhang et al., 2015): 1.

a c = ⟨ [ T a L , T a U ] , [ 1 − I a U , 1 − I a U ] , [ F a L , F a L ] ⟩ ${a^{c}}=\langle [{T_{a}^{L}},{T_{a}^{U}}],[1-{I_{a}^{U}},1-{I_{a}^{U}}],[{F_{a}^{L}},{F_{a}^{L}}]\rangle $

.

2.

a ⊆ b $a\subseteq b$

if and only if

T a L ⩽ T b L ${T_{a}^{L}}\leqslant {T_{b}^{L}}$

,

T a U ⩽ T b U ; I a L ⩾ I b L ${T_{a}^{U}}\leqslant {T_{b}^{U}};{I_{a}^{L}}\geqslant {I_{b}^{L}}$

,

I a U ⩾ I b U ; F a L ⩾ F b L ${I_{a}^{U}}\geqslant {I_{b}^{U}};{F_{a}^{L}}\geqslant {F_{b}^{L}}$

,

F a U ⩾ F b U ${F_{a}^{U}}\geqslant {F_{b}^{U}}$

.

3.

a = b $a=b$

if and only if

a ⊆ b $a\subseteq b$

and

b ⊆ a $b\subseteq a$

.

4.

a ⊕ b = ⟨ [ T a L + T b L − T a L T b L , T a U + T b U − T a U T b U ] , [ I a L I b L , I a U I b U ] , [ F a L F b L , F a U F b U ] ⟩ $a\oplus b=\langle [{T_{a}^{L}}+{T_{b}^{L}}-{T_{a}^{L}}{T_{b}^{L}},\hspace{2.5pt}{T_{a}^{U}}+{T_{b}^{U}}-{T_{a}^{U}}{T_{b}^{U}}],[{I_{a}^{L}}{I_{b}^{L}},{I_{a}^{U}}{I_{b}^{U}}],[{F_{a}^{L}}{F_{b}^{L}},{F_{a}^{U}}{F_{b}^{U}}]\rangle $

.

5.

a ⊗ b = ⟨ [ T a L T b L , T a U T b U ] [ I a L + I b L − I a L I b L , I a U + I b U − I a U I b U ] , [ F a h k L + F b L − F a L F b L , F a U + F b U − F a U F b U ] ⟩ $a\otimes b=\langle [{T_{a}^{L}}{T_{b}^{L}},{T_{a}^{U}}{T_{b}^{U}}][{I_{a}^{L}}+{I_{b}^{L}}-{I_{a}^{L}}{I_{b}^{L}},\hspace{2.5pt}{I_{a}^{U}}+{I_{b}^{U}}-{I_{a}^{U}}{I_{b}^{U}}],[{F_{{a_{\phantom{{h_{k}}}}}}^{L}}+{F_{b}^{L}}-{F_{a}^{L}}{F_{b}^{L}},\hspace{2.5pt}{F_{a}^{U}}+{F_{b}^{U}}-{F_{a}^{U}}{F_{b}^{U}}]\rangle $

.

Definition 3.

Let

A ˜ = ⟨ [ T 1 L , T 1 U ] , [ I 1 L , I 1 U ] , [ F 1 L , F 1 U ] ⟩ $\tilde{A}=\langle [{T_{1}^{L}},{T_{1}^{U}}],[{I_{1}^{L}},{I_{1}^{U}}],[{F_{1}^{L}},{F_{1}^{U}}]\rangle $

;

B ˜ = ⟨ [ T 2 L , T 2 U ] , [ I 2 L , I 2 U ] , [ F 2 L , F 2 U ] ⟩ $\tilde{B}=\langle [{T_{2}^{L}},{T_{2}^{U}}],[{I_{2}^{L}},{I_{2}^{U}}],[{F_{2}^{L}},{F_{2}^{U}}]\rangle $

be an interval-valued neutrosophic number where

T 2 L > 0 ${T_{2}^{L}}>0$

;

T 2 U > 0 ${T_{2}^{U}}>0$

;

I 2 L > 0 ${I_{2}^{L}}>0$

;

I 2 U > 0 ${I_{2}^{U}}>0$

;

F 2 L > 0 ${F_{2}^{L}}>0$

;

F 2 U > 0 ${F_{2}^{U}}>0$

. The division operation is given as below

A ∅ B = [ min ( T 1 L T 2 L , T 1 L T 2 L , T 1 U T 2 U , T 1 U T 2 U ) , max ( T 1 L T 2 L , T 1 L T 2 L , T 1 U T 2 U , T 1 U T 2 U ) ] , [ min ( I 1 L I 2 L , I 1 L I 2 L , I 1 U I 2 U , I 1 U I 2 U ) , max ( I 1 L I 2 L , I 1 L I 2 L , I 1 U I 2 U , I 1 U I 2 U ) ] , [ min ( F 1 L F 2 L , F 1 L F 2 L , F 1 U F 2 U , F 1 U F 2 U ) , max ( F 1 L F 2 L , F 1 L F 2 L , F 1 U F 2 U , F 1 U F 2 U ) ] max [ min ( T 1 L T 2 L , T 1 L T 2 L , T 1 U T 2 U , T 1 U T 2 U ) , max ( T 1 L T 2 L , T 1 L T 2 L , T 1 U T 2 U , T 1 U T 2 U ) ] , [ min ( I 1 L I 2 L , I 1 L I 2 L , I 1 U I 2 U , I 1 U I 2 U ) , max ( I 1 L I 2 L , I 1 L I 2 L , I 1 U I 2 U , I 1 U I 2 U ) ] , [ min ( F 1 L F 2 L , F 1 L F 2 L , F 1 U F 2 U , F 1 U F 2 U ) , max ( F 1 L F 2 L , F 1 L F 2 L , F 1 U F 2 U , F 1 U F 2 U ) ] . \[ A\varnothing B=\frac{\left(\substack{\Big[\min \Big(\frac{{T_{1}^{L}}}{{T_{2}^{L}}},\frac{{T_{1}^{L}}}{{T_{2}^{L}}},\frac{{T_{1}^{U}}}{{T_{2}^{U}}},\frac{{T_{1}^{U}}}{{T_{2}^{U}}}\Big),\max \Big(\frac{{T_{1}^{L}}}{{T_{2}^{L}}},\frac{{T_{1}^{L}}}{{T_{2}^{L}}},\frac{{T_{1}^{U}}}{{T_{2}^{U}}},\frac{{T_{1}^{U}}}{{T_{2}^{U}}}\Big)\Big],\\ {} \Big[\min \Big(\frac{{I_{1}^{L}}}{{I_{2}^{L}}},\frac{{I_{1}^{L}}}{{I_{2}^{L}}},\frac{{I_{1}^{U}}}{{I_{2}^{U}}},\frac{{I_{1}^{U}}}{{I_{2}^{U}}}\Big),\max \Big(\frac{{I_{1}^{L}}}{{I_{2}^{L}}},\frac{{I_{1}^{L}}}{{I_{2}^{L}}},\frac{{I_{1}^{U}}}{{I_{2}^{U}}},\frac{{I_{1}^{U}}}{{I_{2}^{U}}}\Big)\Big],\\ {} \Big[\min \Big(\frac{{F_{1}^{L}}}{{F_{2}^{L}}},\frac{{F_{1}^{L}}}{{F_{2}^{L}}},\frac{{F_{1}^{U}}}{{F_{2}^{U}}},\frac{{F_{1}^{U}}}{{F_{2}^{U}}}\Big),\max \Big(\frac{{F_{1}^{L}}}{{F_{2}^{L}}},\frac{{F_{1}^{L}}}{{F_{2}^{L}}},\frac{{F_{1}^{U}}}{{F_{2}^{U}}},\frac{{F_{1}^{U}}}{{F_{2}^{U}}}\Big)\Big]}\right)}{\max \left(\substack{\Big[\min \Big(\frac{{T_{1}^{L}}}{{T_{2}^{L}}},\frac{{T_{1}^{L}}}{{T_{2}^{L}}},\frac{{T_{1}^{U}}}{{T_{2}^{U}}},\frac{{T_{1}^{U}}}{{T_{2}^{U}}}\Big),\max \Big(\frac{{T_{1}^{L}}}{{T_{2}^{L}}},\frac{{T_{1}^{L}}}{{T_{2}^{L}}},\frac{{T_{1}^{U}}}{{T_{2}^{U}}},\frac{{T_{1}^{U}}}{{T_{2}^{U}}}\Big)\Big],\\ {} \Big[\min \Big(\frac{{I_{1}^{L}}}{{I_{2}^{L}}},\frac{{I_{1}^{L}}}{{I_{2}^{L}}},\frac{{I_{1}^{U}}}{{I_{2}^{U}}},\frac{{I_{1}^{U}}}{{I_{2}^{U}}}\Big),\max \Big(\frac{{I_{1}^{L}}}{{I_{2}^{L}}},\frac{{I_{1}^{L}}}{{I_{2}^{L}}},\frac{{I_{1}^{U}}}{{I_{2}^{U}}},\frac{{I_{1}^{U}}}{{I_{2}^{U}}}\Big)\Big],\\ {} \Big[\min \Big(\frac{{F_{1}^{L}}}{{F_{2}^{L}}},\frac{{F_{1}^{L}}}{{F_{2}^{L}}},\frac{{F_{1}^{U}}}{{F_{2}^{U}}},\frac{{F_{1}^{U}}}{{F_{2}^{U}}}\Big),\max \Big(\frac{{F_{1}^{L}}}{{F_{2}^{L}}},\frac{{F_{1}^{L}}}{{F_{2}^{L}}},\frac{{F_{1}^{U}}}{{F_{2}^{U}}},\frac{{F_{1}^{U}}}{{F_{2}^{U}}}\Big)\Big]}\right)}.\]

Definition 4.

Let

a j = ⟨ [ T a j L , T a j U ] , [ I a j L , I a j U ] , [ F a j L , F a j U ] ⟩ ${a_{j}}=\langle [{T_{{a_{j}}}^{L}},{T_{{a_{j}}}^{U}}],[{I_{{a_{j}}}^{L}},{I_{{a_{j}}}^{U}}],[{F_{{a_{j}}}^{L}},{F_{{a_{j}}}^{U}}]\rangle $

,

j = 1 , 2 , … , n $j=1,2,\dots ,n$

be a collection of IVNNs. Based on the weighted aggregation operators of IVNNs, the interval neutrosophic number weighted average operator is given as below (Biswas et al., 2016):

INNWA ( a 1 , a 2 , … , a n ) = ∑ j = 1 n w j a i = [ 1 − ∏ j = 1 n ( 1 − T a j L ) w j , 1 − ∏ j = 1 n ( 1 − T a j U ) w j ] , [ ∏ j = 1 n ( 1 I a j L ) w j , ∏ j = 1 n ( I a j U ) w j ] , [ ∏ j = 1 n ( F a j L ) w j , ∏ j = 1 n ( F a j U ) w j ] , \[\begin{aligned}{}& \mathit{INNWA}({a_{1}},{a_{2}},\dots ,{a_{n}})={\sum \limits_{j=1}^{n}}{w_{j}}{a_{i}}\\ {} & \hspace{1em}=\left\langle \Bigg[1-{\prod \limits_{j=1}^{n}}{\big(1-{T_{aj}^{L}}\big)^{{w_{j}}}},1-{\prod \limits_{j=1}^{n}}{\big(1-{T_{aj}^{U}}\big)^{{w_{j}}}}\Bigg],\Bigg[{\prod \limits_{j=1}^{n}}{\big(1{I_{aj}^{L}}\big)^{{w_{j}}}},{\prod \limits_{j=1}^{n}}{\big({I_{aj}^{U}}\big)^{{w_{j}}}}\Bigg],\right.\\ {} & \hspace{2em}\left.\Bigg[{\prod \limits_{j=1}^{n}}{\big({F_{aj}^{L}}\big)^{{w_{j}}}},{\prod \limits_{j=1}^{n}}{\big({F_{aj}^{U}}\big)^{{w_{j}}}}\Bigg]\right\rangle ,\end{aligned}\]

(3)

where

w j ${w_{j}}$

( j = 1 , 2 , … , n ) $(j=1,2,\dots ,n)$

is the weight of

a j ${a_{j}}$

( j = 1 , 2 , … , n ) $(j=1,2,\dots ,n)$

with

w j ∈ [ 0 , 1 ] ${w_{j}}\in [0,1]$

and

∑ j = 1 n w j = 1 ${\textstyle\sum _{j=1}^{n}}{w_{j}}=1$

.

There are few deneutrosophication methods for comparing neutrosophic numbers (Zhang et al., 2015; Biswas et al., 2016). We propose a new deneutrosophication method in order to compare the interval-valued neutrosophic numbers. It is given in Definition 5. Definition 5.

Let

A = ⟨ ( T L , T U ) , ( I L , I U ) , ( F L , F U ) ⟩ $A=\langle ({T^{L}},{T^{U}}),({I^{L}},{I^{U}}),({F^{L}},{F^{U}})\rangle $

be an interval-valued neutrosophic number. The deneutrosophicated A value (

H ( A ) $\mathfrak{H}(A)$

) is given by Eq. (4):

H ( A ) = ( T L + T U + ( 1 − F L ) + ( 1 − F U ) + T L × T U − ( 1 − F L ) × ( 1 − F U ) ) 4 × ( ( 1 − [ ( I L ) + ( I U ) ] 2 ) − ( ( I L ) × ( I U ) ) ) . \[\begin{aligned}{}\mathfrak{H}(A)& =\frac{({T^{L}}+{T^{U}}+(1-{F^{L}})+(1-{F^{U}})+{T^{L}}\times {T^{U}}\hspace{-0.1667em}\hspace{-0.1667em}-\sqrt{(1-{F^{L}})\times (1-{F^{U}})})}{4}\\ {} & \hspace{1em}\times \bigg(\bigg(1-\frac{[({I^{L}})+({I^{U}})]}{2}\bigg)-\big(\sqrt{\big({I^{L}}\big)\times \big({I^{U}}\big)}\big)\bigg).\end{aligned}\]

(4)

4 Proposed Methodology

IVN ELECTRE I method’s steps are given as follows:

Step 1: Construct the IVN decision matrices (

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Table 1

Scale of IVN decision matrix.

Linguistic term		⟨ [ T L , T U ] , [ I L , I U ] , [ F L , F U ] ⟩ $\langle [{T^{L}},{T^{U}}],[{I^{L}},{I^{U}}],[{F^{L}},{F^{U}}]\rangle $
CB	Certainly bad	⟨ [ 0.1 , 0.2 ] , [ 0.6 , 0.7 ] , [ 0.8 , 0.9 ] ⟩ $\langle [0.1,0.2],[0.6,0.7],[0.8,0.9]\rangle $
VB	Very bad	⟩ [ 0.2 , 0.3 ] , [ 0.5 , 0.6 ] , [ 0.7 , 0.8 ] ⟩ $\rangle [0.2,0.3],[0.5,0.6],[0.7,0.8]\rangle $
B	Bad	⟨ [ 0.3 , 0.4 ] , [ 0.4 , 0.5 ] , [ 0.6 , 0.7 ] ⟩ $\langle [0.3,0.4],[0.4,0.5],[0.6,0.7]\rangle $
BA	Below average	⟨ [ 0.4 , 0.5 ] , [ 0.3 , 0.4 ] , [ 0.5 , 0.6 ] ⟩ $\langle [0.4,0.5],[0.3,0.4],[0.5,0.6]\rangle $
F	Fair	⟨ [ 0.5 , 0.5 ] , [ 0.1 , 0.2 ] , [ 0.5 , 0.5 ] ⟩ $\langle [0.5,0.5],[0.1,0.2],[0.5,0.5]\rangle $
AA	Above average	⟨ [ 0.5 , 0.6 ] , [ 0.3 , 0.4 ] , [ 0.4 , 0.5 ] ⟩ $\langle [0.5,0.6],[0.3,0.4],[0.4,0.5]\rangle $
G	Good	⟨ [ 0.6 , 0.7 ] , [ 0.4 , 0.5 ] , [ 0.3 , 0.4 ] ⟩ $\langle [0.6,0.7],[0.4,0.5],[0.3,0.4]\rangle $
VG	Very good	⟨ [ 0.7 , 0.8 ] , [ 0.5 , 0.6 ] , [ 0.2 , 0.3 ] ⟩ $\langle [0.7,0.8],[0.5,0.6],[0.2,0.3]\rangle $
CG	Certainly good	⟨ [ 0.8 , 0.9 ] , [ 0.6 , 0.7 ] , [ 0.1 , 0.2 ] ⟩ $\langle [0.8,0.9],[0.6,0.7],[0.1,0.2]\rangle $

The illustrated decision matrix (DM) based on Expert j opinions is presented in Table 2.

Step 2: Aggregate the DMs to obtain aggregated IVN decision matrix

( A ) $(A)$

by using Eq. (3) as in Table 3.

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Table 2

DM based on expert j opinions.

Criterion	Type	AL 1 ${\mathrm{AL}_{1}}$	AL 2 ${\mathrm{AL}_{2}}$	…	AL n ${\mathrm{AL}_{\mathbf{n}}}$
C 1 ${\mathbf{C}_{\mathbf{1}}}$	Linguistic cost	⟨ [ T j L , T j U ] , [ I j L , I j U ] , [ F j L , F j U ] ⟩ $\langle [{T_{j}^{L}},{T_{j}^{U}}],[{I_{j}^{L}},{I_{j}^{U}}],[{F_{j}^{L}},{F_{j}^{U}}]\rangle $	⟨ [ T j L , T j U ] , [ I j L , I j U ] , [ F j L , F j U ] ⟩ $\langle [{T_{j}^{L}},{T_{j}^{U}}],[{I_{j}^{L}},{I_{j}^{U}}],[{F_{j}^{L}},{F_{j}^{U}}]\rangle $	⋯ $\cdots \hspace{0.1667em}$	⟨ [ T j L , T j U ] , [ I j L , I j U ] , [ F j L , F j U ] ⟩ $\langle [{T_{j}^{L}},{T_{j}^{U}}],[{I_{j}^{L}},{I_{j}^{U}}],[{F_{j}^{L}},{F_{j}^{U}}]\rangle $
C 2 ${\mathbf{C}_{\mathbf{2}}}$	Numerical cost	⟨ [ T j L , T j U ] , [ I j L , I j U ] , [ F j L , F j U ] ⟩ $\langle [{T_{j}^{L}},{T_{j}^{U}}],[{I_{j}^{L}},{I_{j}^{U}}],[{F_{j}^{L}},{F_{j}^{U}}]\rangle $	⟨ [ T j L , T j U ] , [ I j L , I j U ] , [ F j L , F j U ] ⟩ $\langle [{T_{j}^{L}},{T_{j}^{U}}],[{I_{j}^{L}},{I_{j}^{U}}],[{F_{j}^{L}},{F_{j}^{U}}]\rangle $	⋯ $\cdots \hspace{0.1667em}$	⟨ [ T j L , T j U ] , [ I j L , I j U ] , [ F j L , F j U ] ⟩ $\langle [{T_{j}^{L}},{T_{j}^{U}}],[{I_{j}^{L}},{I_{j}^{U}}],[{F_{j}^{L}},{F_{j}^{U}}]\rangle $
⋮	⋮	⋮	⋮	⋱	⋮
C m ${\mathbf{C}_{\mathbf{m}}}$	Linguistic Benefit	⟨ [ T j L , T j U ] , [ I j L , I j U ] , [ F j L , F j U ] ⟩ $\langle [{T_{j}^{L}},{T_{j}^{U}}],[{I_{j}^{L}},{I_{j}^{U}}],[{F_{j}^{L}},{F_{j}^{U}}]\rangle $	⟨ [ T j L , T j U ] , [ I j L , I j U ] , [ F j L , F j U ] ⟩ $\langle [{T_{j}^{L}},{T_{j}^{U}}],[{I_{j}^{L}},{I_{j}^{U}}],[{F_{j}^{L}},{F_{j}^{U}}]\rangle $	...	⟨ [ T j L , T j U ] , [ I j L , I j U ] , [ F j L , F j U ] ⟩ $\langle [{T_{j}^{L}},{T_{j}^{U}}],[{I_{j}^{L}},{I_{j}^{U}}],[{F_{j}^{L}},{F_{j}^{U}}]\rangle $

Step 3: Calculate the weighted normalized DM (R).

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Table 3

Aggregated DM.

Criterion	Type	AL 1 ${\mathrm{AL}_{1}}$	AL 2 ${\mathrm{AL}_{2}}$	…	AL n ${\mathrm{AL}_{\mathbf{n}}}$
C 1 ${\mathbf{C}_{\mathbf{1}}}$	Linguistic cost	⟨ [ T A L , T A U ] , [ I A L , I A U ] , [ F A L , F A U ] ⟩ $\langle [{T_{A}^{L}},{T_{A}^{U}}],[{I_{A}^{L}},{I_{A}^{U}}],[{F_{A}^{L}},{F_{A}^{U}}]\rangle $	⟨ [ T A L , T A U ] , [ I A L , I A U ] , [ F A L , F A U ] ⟩ $\langle [{T_{A}^{L}},{T_{A}^{U}}],[{I_{A}^{L}},{I_{A}^{U}}],[{F_{A}^{L}},{F_{A}^{U}}]\rangle $	⋯ $\cdots \hspace{0.1667em}$	⟨ [ T A L , T A U ] , [ I A L , I A U ] , [ F A L , F A U ] ⟩ $\langle [{T_{A}^{L}},{T_{A}^{U}}],[{I_{A}^{L}},{I_{A}^{U}}],[{F_{A}^{L}},{F_{A}^{U}}]\rangle $
C 12 ${\mathbf{C}_{\mathbf{12}}}$	Numerical cost	⟨ [ T A L , T A U ] , [ I A L , I A U ] , [ F A L , F A U ] ⟩ $\langle [{T_{A}^{L}},{T_{A}^{U}}],[{I_{A}^{L}},{I_{A}^{U}}],[{F_{A}^{L}},{F_{A}^{U}}]\rangle $	⟨ [ T A L , T A U ] , [ I A L , I A U ] , [ F A L , F A U ] ⟩ $\langle [{T_{A}^{L}},{T_{A}^{U}}],[{I_{A}^{L}},{I_{A}^{U}}],[{F_{A}^{L}},{F_{A}^{U}}]\rangle $	⋯ $\cdots \hspace{0.1667em}$	⟨ [ T A L , T A U ] , [ I A L , I A U ] , [ F A L , F A U ] ⟩ $\langle [{T_{A}^{L}},{T_{A}^{U}}],[{I_{A}^{L}},{I_{A}^{U}}],[{F_{A}^{L}},{F_{A}^{U}}]\rangle $
⋮	⋮	⋮	⋮	⋱	⋮
C m ${\mathbf{C}_{\mathbf{m}}}$	Linguistic benefit	⟨ [ T A L , T A U ] , [ I A L , I A U ] , [ F A L , F A U ] ⟩ $\langle [{T_{A}^{L}},{T_{A}^{U}}],[{I_{A}^{L}},{I_{A}^{U}}],[{F_{A}^{L}},{F_{A}^{U}}]\rangle $	⟨ [ T A L , T A U ] , [ I A L , I A U ] , [ F A L , F A U ] ⟩ $\langle [{T_{A}^{L}},{T_{A}^{U}}],[{I_{A}^{L}},{I_{A}^{U}}],[{F_{A}^{L}},{F_{A}^{U}}]\rangle $	...	⟨ [ T A L , T A U ] , [ I A L , I A U ] , [ F A L , F A U ] ⟩ $\langle [{T_{A}^{L}},{T_{A}^{U}}],[{I_{A}^{L}},{I_{A}^{U}}],[{F_{A}^{L}},{F_{A}^{U}}]\rangle $

Normalization of formulas for benefit (

b i j ${b_{ij}}$

) and cost (

c i j ${c_{ij}}$

) criteria is given in the following, respectively.

r i j = b i j ∑ i = 1 n b i j 2 , \[ {r_{ij}}=\frac{{b_{ij}}}{\sqrt{{\textstyle\textstyle\sum _{i=1}^{n}}{b_{ij}^{2}}}},\]

(5)

where

i = 1 , 2 , … , m $i=1,2,\dots ,m$

and

j = 1 , 2 , … , n $j=1,2,\dots ,n$

.

r i j = 1 c i j ∑ i = 1 n ( 1 c i j ) 2 , \[ {r_{ij}}=\frac{\frac{1}{{c_{ij}}}}{\sqrt{{\textstyle\textstyle\sum _{i=1}^{n}}{(\frac{1}{{c_{ij}}})^{2}}}},\]

(6)

where

i = 1 , 2 , … , m $i=1,2,\dots ,m$

and

j = 1 , 2 , … , n $j=1,2,\dots ,n$

.

Arithmetic operations are carried out with the neutrosophic formulas given in Definition 2, and Definition 3. After calculations, normalized IVN DM

( R ) $(R)$

is illustrated in Table 4.

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Table 4

Normalized decision matrix.

Criterion	Type	AL 1 ${\mathrm{AL}_{1}}$	AL 2 ${\mathrm{AL}_{2}}$	…	AL n ${\mathrm{AL}_{\mathbf{n}}}$
C 1 ${\mathbf{C}_{\mathbf{1}}}$	Linguistic cost	⟨ [ T r L , T r U ] , [ I r L , I r U ] , [ F r L , F r U ] ⟩ $\langle [{T_{r}^{L}},{T_{r}^{U}}],[{I_{r}^{L}},{I_{r}^{U}}],[{F_{r}^{L}},{F_{r}^{U}}]\rangle $	⟨ [ T r L , T r U ] , [ I r L , I r U ] , [ F r L , F r U ] ⟩ $\langle [{T_{r}^{L}},{T_{r}^{U}}],[{I_{r}^{L}},{I_{r}^{U}}],[{F_{r}^{L}},{F_{r}^{U}}]\rangle $	⋯ $\cdots \hspace{0.1667em}$	⟨ [ T r L , T r U ] , [ I r L , I r U ] , [ F r L , F r U ] ⟩ $\langle [{T_{r}^{L}},{T_{r}^{U}}],[{I_{r}^{L}},{I_{r}^{U}}],[{F_{r}^{L}},{F_{r}^{U}}]\rangle $
C 2 ${\mathbf{C}_{\mathbf{2}}}$	Numerical cost	⟨ [ T r L , T r U ] , [ I r L , I r U ] , [ F r L , F r U ] ⟩ $\langle [{T_{r}^{L}},{T_{r}^{U}}],[{I_{r}^{L}},{I_{r}^{U}}],[{F_{r}^{L}},{F_{r}^{U}}]\rangle $	⟨ [ T r L , T r U ] , [ I r L , I r U ] , [ F r L , F r U ] ⟩ $\langle [{T_{r}^{L}},{T_{r}^{U}}],[{I_{r}^{L}},{I_{r}^{U}}],[{F_{r}^{L}},{F_{r}^{U}}]\rangle $	⋯ $\cdots \hspace{0.1667em}$	⟨ [ T r L , T r U ] , [ I r L , I r U ] , [ F r L , F r U ] ⟩ $\langle [{T_{r}^{L}},{T_{r}^{U}}],[{I_{r}^{L}},{I_{r}^{U}}],[{F_{r}^{L}},{F_{r}^{U}}]\rangle $
⋮	⋮	⋮	⋮	⋱	⋮
C m ${\mathbf{C}_{\mathbf{m}}}$	Linguistic benefit	⟨ [ T r L , T r U ] , [ I r L , I r U ] , [ F r L , F r U ] ⟩ $\langle [{T_{r}^{L}},{T_{r}^{U}}],[{I_{r}^{L}},{I_{r}^{U}}],[{F_{r}^{L}},{F_{r}^{U}}]\rangle $	⟨ [ T r L , T r U ] , [ I r L , I r U ] , [ F r L , F r U ] ⟩ $\langle [{T_{r}^{L}},{T_{r}^{U}}],[{I_{r}^{L}},{I_{r}^{U}}],[{F_{r}^{L}},{F_{r}^{U}}]\rangle $	...	⟨ [ T r L , T r U ] , [ I r L , I r U ] , [ F r L , F r U ] ⟩ $\langle [{T_{r}^{L}},{T_{r}^{U}}],[{I_{r}^{L}},{I_{r}^{U}}],[{F_{r}^{L}},{F_{r}^{U}}]\rangle $

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Table 5

Scale for criteria weighting (Karaşan and Kahraman, 2017).

Linguistic term		⟨ [ T L , T U ] , [ I L , I U ] , [ F L , F U ] ⟩ $\langle [{T^{L}},{T^{U}}],[{I^{L}},{I^{U}}],[{F^{L}},{F^{U}}]\rangle $
TI	Trivial importance	⟨ [ 0.06 , 0.22 ] , [ 0.67 , 0.78 ] , [ 0.83 , 1.00 ] ⟩ $\langle [0.06,0.22],[0.67,0.78],[0.83,1.00]\rangle $
UII	Unimportant importance	⟨ [ 0.22 , 0.33 ] , [ 0.56 , 0.67 ] , [ 0.72 , 0.83 ] ⟩ $\langle [0.22,0.33],[0.56,0.67],[0.72,0.83]\rangle $
USI	Unsatisfied importance	⟨ [ 0.33 , 0.44 ] , [ 0.44 , 0.56 ] , [ 0.61 , 0.72 ] ⟩ $\langle [0.33,0.44],[0.44,0.56],[0.61,0.72]\rangle $
LF	Lower than fair	⟨ [ 0.44 , 0.56 ] , [ 0.33 , 0.44 ] , [ 0.50 , 0.61 ] ⟩ $\langle [0.44,0.56],[0.33,0.44],[0.50,0.61]\rangle $
FI	Fair importance	⟨ [ 0.50 , 0.56 ] , [ 0.11 , 0.22 ] , [ 0.44 , 0.50 ] ⟩ $\langle [0.50,0.56],[0.11,0.22],[0.44,0.50]\rangle $
MF	More than fair	⟨ [ 0.50 , 0.61 ] , [ 0.33 , 0.44 ] , [ 0.44 , 0.56 ] ⟩ $\langle [0.50,0.61],[0.33,0.44],[0.44,0.56]\rangle $
SI	Satisfied importance	⟨ [ 0.61 , 0.72 ] , [ 0.44 , 0.56 ] , [ 0.33 , 0.44 ] ⟩ $\langle [0.61,0.72],[0.44,0.56],[0.33,0.44]\rangle $
II	Impactful importance	⟨ [ 0.72 , 0.83 ] , [ 0.56 , 0.67 ] , [ 0.22 , 0.33 ] ⟩ $\langle [0.72,0.83],[0.56,0.67],[0.22,0.33]\rangle $
CII	Certainly impactful importance	⟨ [ 0.83 , 1.00 ] , [ 0.67 , 0.78 ] , [ 0.06 , 0.22 ] ⟩ $\langle [0.83,1.00],[0.67,0.78],[0.06,0.22]\rangle $

Weighted normalized IVN DM (V) is obtained by multiplying the IVN weights vector (

w j ${w_{j}}$

) which is given in Table 5 with the normalized IVN DM (R) as in Eq. (7).

v i j = w j ⊗ r i j , \[ {v_{ij}}={w_{j}}\otimes {r_{ij}},\]

(7)

where,

w j = ⟨ [ T w L , T w U ] , [ I w L , I w U ] , [ F w L , F w U ] ⟩ ${w_{j}}=\langle [{T_{w}^{L}},{T_{w}^{U}}],[{I_{w}^{L}},{I_{w}^{U}}],[{F_{w}^{L}},{F_{w}^{U}}]\rangle $

,

r i j = ⟨ [ T r L , T r U ] , [ I r L , I r U ] , [ F r L , F r U ] ⟩ ${r_{ij}}=\langle [{T_{r}^{L}},{T_{r}^{U}}],[{I_{r}^{L}},{I_{r}^{U}}],[{F_{r}^{L}},{F_{r}^{U}}]\rangle $

, and

v i j = ⟨ [ T v L , T v U ] , [ I v L , I v U ] , [ F v L , F v U ] ⟩ ${v_{ij}}=\langle [{T_{v}^{L}},{T_{v}^{U}}],[{I_{v}^{L}},{I_{v}^{U}}],[{F_{v}^{L}},{F_{v}^{U}}]\rangle $

.

Table 6 presents the weighted normalized IVN DM.

Step 4: Obtain the Concordance and Discordance Indices.

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Table 6

Weightednormalized IVN decision matrix.

Criterion	Type	AL 1 ${\mathrm{AL}_{1}}$	AL 2 ${\mathrm{AL}_{2}}$	…	AL n ${\mathrm{AL}_{\mathbf{n}}}$
C 1 ${\mathbf{C}_{\mathbf{1}}}$	Linguistic cost	⟨ [ T v L , T v U ] , [ I v L , I v U ] , [ F v L , F v U ] ⟩ $\langle [{T_{v}^{L}},{T_{v}^{U}}],[{I_{v}^{L}},{I_{v}^{U}}],[{F_{v}^{L}},{F_{v}^{U}}]\rangle $	⟨ [ T v L , T v U ] , [ I v L , I v U ] , [ F v L , F v U ] ⟩ $\langle [{T_{v}^{L}},{T_{v}^{U}}],[{I_{v}^{L}},{I_{v}^{U}}],[{F_{v}^{L}},{F_{v}^{U}}]\rangle $	⋯ $\cdots \hspace{0.1667em}$	⟨ [ T v L , T v U ] , [ I v L , I v U ] , [ F v L , F v U ] ⟩ $\langle [{T_{v}^{L}},{T_{v}^{U}}],[{I_{v}^{L}},{I_{v}^{U}}],[{F_{v}^{L}},{F_{v}^{U}}]\rangle $
C 2 ${\mathbf{C}_{\mathbf{2}}}$	Numerical cost	⟨ [ T v L , T v U ] , [ I v L , I v U ] , [ F v L , F v U ] ⟩ $\langle [{T_{v}^{L}},{T_{v}^{U}}],[{I_{v}^{L}},{I_{v}^{U}}],[{F_{v}^{L}},{F_{v}^{U}}]\rangle $	⟨ [ T v L , T v U ] , [ I v L , I v U ] , [ F v L , F v U ] ⟩ $\langle [{T_{v}^{L}},{T_{v}^{U}}],[{I_{v}^{L}},{I_{v}^{U}}],[{F_{v}^{L}},{F_{v}^{U}}]\rangle $	⋯ $\cdots \hspace{0.1667em}$	⟨ [ T v L , T v U ] , [ I v L , I v U ] , [ F v L , F v U ] ⟩ $\langle [{T_{v}^{L}},{T_{v}^{U}}],[{I_{v}^{L}},{I_{v}^{U}}],[{F_{v}^{L}},{F_{v}^{U}}]\rangle $
⋮	⋮	⋮	⋮	⋱	⋮
C m ${\mathbf{C}_{\mathbf{m}}}$	Linguistic benefit	⟨ [ T v L , T v U ] , [ I v L , I v U ] , [ F v L , F v U ] ⟩ $\langle [{T_{v}^{L}},{T_{v}^{U}}],[{I_{v}^{L}},{I_{v}^{U}}],[{F_{v}^{L}},{F_{v}^{U}}]\rangle $	⟨ [ T v L , T v U ] , [ I v L , I v U ] , [ F v L , F v U ] ⟩ $\langle [{T_{v}^{L}},{T_{v}^{U}}],[{I_{v}^{L}},{I_{v}^{U}}],[{F_{v}^{L}},{F_{v}^{U}}]\rangle $	...	⟨ [ T v L , T v U ] , [ I v L , I v U ] , [ F v L , F v U ] ⟩ $\langle [{T_{v}^{L}},{T_{v}^{U}}],[{I_{v}^{L}},{I_{v}^{U}}],[{F_{v}^{L}},{F_{v}^{U}}]\rangle $

Let A

L n = { a , b , … , n } ${L_{n}}=\{a,b,\dots ,n\}$

denote a finite set of alternatives.

Concordance index (

C a b ${\mathbb{C}_{ab}}$

) measures are given by Eq. (8):

C a b = { j ∣ x a j ⩾ x b j } . \[ {\mathbb{C}_{ab}}=\{j\mid {x_{aj}}\geqslant {x_{bj}}\}.\]

(8)

Discordance index

( D a b ) ${(\mathbb{D}_{ab}})$

measures the strength of the evidence against the Concordance index by Eq. (9):

D a b = { j ∣ x a j < x b j } . \[ {\mathbb{D}_{ab}}=\{j\mid {x_{aj}}<{x_{bj}}\}.\]

(9)

Step 5: Calculate the Concordance matrix.

The Concordance index

C ( a , b ) $\mathbb{C}(a,b)$

between

AL a ${\mathrm{AL}_{a}}$

and

AL b ${\mathrm{AL}_{b}}$

is determined using Eq. (10):

C ( a , b ) = ∑ j ∈ C ( a , b ) H ( w j ) . \[ \mathbb{C}(a,b)=\sum \limits_{j\in \mathbb{C}(a,b)}\mathfrak{H}({w_{j}}).\]

(10)

Through this calculation for all alternatives, concordance matrix is found as below:

C = − C ( 1 , 2 ) ⋯ C ( 1 , m ) C ( 2 , 1 ) − ⋯ C ( 2 , m ) ⋮ ⋯ ⋱ ⋮ C ( m , 1 ) C ( m , 2 ) ⋯ − . \[ \mathbb{C}=\left[\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}-\hspace{1em}& \mathbb{C}(1,2)\hspace{1em}& \cdots \hspace{1em}& \mathbb{C}(1,m)\\ {} \mathbb{C}(2,1)\hspace{1em}& -\hspace{1em}& \cdots \hspace{1em}& \mathbb{C}(2,m)\\ {} \vdots \hspace{1em}& \cdots \hspace{1em}& \ddots \hspace{1em}& \vdots \\ {} \mathbb{C}(m,1)\hspace{1em}& \mathbb{C}(m,2)\hspace{1em}& \cdots \hspace{1em}& -\end{array}\right].\]

(11)

Step 6: Calculate the Discordance matrix.

Similarly, the Discordance index

D ( a , b ) $\mathbb{D}(a,b)$

between

AL a ${\mathrm{AL}_{a}}$

and

AL b ${\mathrm{AL}_{b}}$

is determined by Eq. (12):

D ( a , b ) = max j ∈ D a b ∣ v a j − v b j ∣ max j ∈ J ; m , n ∈ I ∣ v m j − v n j ∣ , \[ \mathbb{D}(a,b)=\frac{{\max _{j\in {\mathbb{D}_{ab}}}}\mid {v_{aj}}-{v_{bj}}\mid }{{\max _{j\in J;m,n\in I}}\mid {v_{mj}}-{v_{nj}}\mid },\]

(12)

where

v . j ${v_{.j}}$

is obtained by the deneutrosophicated weighted normalized IVN decision matrix based on Eq. (4).

If q and z are used to show the weighted normalized values, the discordance matrix can be given as in Eq. (13).

D = − D ( 1 , 2 ) ⋯ D ( 1 , q ) D ( 2 , 1 ) − ⋯ D ( 2 , q ) ⋮ ⋯ ⋱ ⋮ D ( z , 1 ) D ( z , 2 ) ⋯ − . \[ \mathbb{D}=\left[\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}-\hspace{1em}& \mathbb{D}(1,2)\hspace{1em}& \cdots \hspace{1em}& \mathbb{D}(1,q)\\ {} \mathbb{D}(2,1)\hspace{1em}& -\hspace{1em}& \cdots \hspace{1em}& \mathbb{D}(2,q)\\ {} \vdots \hspace{1em}& \cdots \hspace{1em}& \ddots \hspace{1em}& \vdots \\ {} \mathbb{D}(z,1)\hspace{1em}& \mathbb{D}(z,2)\hspace{1em}& \cdots \hspace{1em}& -\end{array}\right].\]

(13)

Step 7: Determine the threshold value for Concordance and Discordance matrices.

The threshold value for Concordance matrix

( α C ) $({\alpha _{C}})$

is computed by the average of the elements in matrix

C $\mathbb{C}$

via Eq. (14):

α C = ∑ a = 1 m ∑ b = 1 m C ( a , b ) m ( m − 1 ) . \[ {\alpha _{\mathbb{C}}}={\sum \limits_{a=1}^{m}}{\sum \limits_{b=1}^{m}}\frac{\mathbb{C}(a,b)}{m(m-1)}.\]

(14)

The threshold value for Discordance matrix

( α D ) $({\alpha _{D}})$

is computed by the average of the elements in matrix

D $\mathbb{D}$

via Eq. (15):

α D = ∑ a = 1 m ∑ b = 1 m D ( a , b ) m ( m − 1 ) . \[ {\alpha _{\mathbb{D}}}={\sum \limits_{a=1}^{m}}{\sum \limits_{b=1}^{m}}\frac{\mathbb{D}(a,b)}{m(m-1)}.\]

(15)

Step 8: Determine the acceptable relations based on (

α C ${\alpha _{\mathbb{C}}}$

) and (

α D ${\alpha _{\mathbb{D}}}$

). The values equal or larger than

α C ${\alpha _{\mathbb{C}}}$

and the values smaller than

α D ${\alpha _{\mathbb{D}}}$

are simultaneously used to determine the outranking relations in the kernel.

The proposed extension of ELECTRE I method offers ready-made scales to decision makers in order to express their opinions efficiently. It conducts new deneutrosophication and division operators for interval-valued neutrosophic sets. Besides, our model is relatively easy to use and produces robust decisions.

5 Application

Managers of a municipal which is close to sea coast want to invest in renewable energy technologies for self-meeting their energy need. Expert group indicated that they were not sure neither about the exact values of measurable criteria nor about the values of linguistic criteria since the related data depend on uncertain environmental, political, and economic conditions. Hence, we used neutrosophic sets in order to handle the uncertainty and indeterminacy caused by the mentioned conditions. Experts utilized Table 2 and Table 6 to score decision matrices. Figure 1 presents the hierarchy of the application.

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Fig. 1

Hierarchy of the application.

5.1 Renewable Energy Alternatives and Decision Criteria

Renewable energy sources involve biomass energy, geothermal energy, ocean energy, solar energy, wind energy, and hydropower energy. They have an enormous potential to meet energy needs of the world. By doing that, the energy security of the world can be powered by modern conversion technologies by reducing the long-term price of fuels from conventional sources, and decreasing the use of fossil fuels. Using renewable energies does not only impact on reducing the air pollution, safety risks, and greenhouse gas emissions in the atmosphere but also they are recycled in nature. Furthermore, it reduces dependence on imported fuels and creates new jobs and provides regional employment. Taking into consideration the above, we firstly introduce the general characteristics of alternatives, then the selection criteria.

AL1 wave power plant: The immense energy potential of the oceans is being increasingly recognized all over the world (Hammar et al., 2017). This immense energy which is called here ’ocean wave energy’ is the conversion to the energy of wind waves by using generators which are placed on the surface of the ocean. The generated energy is usually used in desalination plants, power plants, and water pumps. The main factors that effect the rate of the energy output are determined by wave height, wave speed, wave length, water density, and temperature (Khan et al., 2017).

AL2 solar power plant: Solar energy is simply an energy source provided by the sun in the form of solar radiation. It is also called photovoltaic energy which means conversion of light into electricity. Starting with the first use, it dates back to the year the 1870s, and this technology is pollution and often noise free. In order to establish a solar energy system, there are some basic criteria to evaluate, such as requirements and properties of installation area, accessibility to that area, the infrastructure of the transmission of the generated electricity, governmental funds for the investment (Onar et al., 2015; Çelikbilek and Tüysüz, 2016).

AL3 biomass energy plant: Biomass energy is widely available, naturally distributed and, most importantly, converts waste into energy that helps to deal with pollution. Four main types of biomass, wood plants and herbaceous plants and grasses are the main types of interest for producing energy, with attention focused on the crops corn or maize, sugarcane, sorghum, and millets, as well as the switchgrass which has been utilized as a source of biofuel (McKendry, 2002; Cebi et al., 2016)]. It has been used as an energy source from the 1800s. Since that time, several characteristics affect the performance of biomass fuel such as the heat value, moisture level, compositions, and size.

AL4 hydro power plant: Hydropower is power derived from the energy of falling water or fast running water. Hydropower plants are used to transform the kinematic energy of falling water to the electricity. A turbine converts the kinetic energy of falling water into mechanical energy. Then a generator uses the mechanical energy generated by the turbine for producing electrical energy. In order to establish a hydropower energy system, there are some basic criteria to evaluate such as efficiency, costs, environmental effects, governmental funds.

AL5 geothermal energy plant: Geothermal energy is produced from the heat generated by Earth’s formation, and subsequent radioactive decay of the earth’s minerals (Wang et al., 2017). Geothermal energy relies on heat from Earth to generate steam and produce electricity. The utilization of geothermal energy depends on the demand for heat or electricity and the distance of the resource from the end consumer, resource temperature, and chemistry of the geothermal fluid (Amoo, 2014).

AL6 wind power plant: Wind energy is a carbon-free energy source depending on average wind speeds, wind turbine hubs, and turbulence intensity. There is a tendency of the use of wind energy that increased rapidly since the 1970s, starting with the oil embargo crisis. Since that time, it is possible to construct a wind energy system depending on country policies, supply chain issues of transmission and integration with the electricity system, compatibility of social and environmental conditions to the investment, economic concerns, regional deployment.

Various criteria for the evaluation of renewable energy alternatives have been used by several researchers (Onar et al., 2015; Çelikbilek and Tüysüz, 2016; Kahraman et al., 2010a, 2010b, 2009; Kaya and Kahraman, 2010; Şengül et al., 2015; Büyüközkan and Güleryüz, 2016; Diemuodeke et al., 2016; Väisänen et al., 2016; Quan and Leephakpreeda, 2015). After analysing these studies, the most common criteria have been listed and sorted for the proposed IVN ELECTRE I method. The evaluation criteria for renewable energy alternatives that will be used in this study are given in Table 7.

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Table 7

Evaluation criteria for renewable energy alternatives.

Main criteria	Sub criteria
C1 system expectations	C11 Reliability
	C12 Land requirements
	C13 Project life
	C14 Sustainability
	C15 Accessibility
C2 regional impacts	C21 Job creation
	C22 Estimated amount of energy produced
	C23 Accident risk and effects
	C24 Infrastructure and transportation facilities
C3 financial factors	C31 Maintenance service
	C32 Efficiency
	C33 Investment cost
	C34 Operation and maintenance cost
	C35 Fuel cost
	C36 Electric cost
	C37 Payback period
C4 environmental factors	C41 Air pollution
	C42 Social acceptability
	C43 Compatibility with political legislative situation
	C44 Compatibility with national energy policy objectives
	C45 Availability of funds

5.2 Problem Solution

Table 8 presents the linguistic evaluations for the sub-criteria collected from the experts. These evaluations are aggregated to obtain the weights of the sub-criteria.

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Table 8

Linguistic evaluations of sub-criteria with respect to experts.

Sub-criteria		DM1	DM2	DM3
C11	Reliability	II	SI	II
C12	Land requirements	LF	FI	FI
C13	Project life	LF	USI	UII
C14	Sustainability	II	TI	USI
C15	Accessibility	LF	USI	TI
C21	Job creation	LF	USI	USI
C22	Estimated amount of energy produced	SI	II	SI
C23	Accident risk and effects	TI	TI	USI
C24	Infrastructure and transportation facilities	USI	USI	TI
C31	Maintenance service	USI	USI	TI
C32	Efficiency	USI	TI	USI
C33	Investment cost	SI	SI	SI
C34	Operation and maintenance cost	SI	SI	MF
C35	Fuel cost	MF	SI	MF
C36	Electric cost	MF	SI	FI
C37	Payback period	MF	MF	FI
C41	Air pollution	MF	FI	FI
C42	Social acceptability	TI	UII	UII
C43	Compatibility with political legislative situation	LF	FI	LF
C44	Compatibility with national energy policy objectives	LF	USI	LF
C45	Availability of funds	CII	II	II

The aggregated weights of the sub-criteria results as interval-valued neutrosophic sets are given in Table 9.

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Table 9

Aggregated weighting results of the sub-criteria.

Criterion	Aggregated results	Criterion	Aggregated results
	⟨ [ T A L , T A U ] , [ I A L , I A U ] , [ F A L , F A U ] ⟩ $\langle [{T_{A}^{L}},{T_{A}^{U}}],[{I_{A}^{L}},{I_{A}^{U}}],[{F_{A}^{L}},{F_{A}^{U}}]\rangle $		⟨ [ T A L , T A U ] , [ I A L , I A U ] , [ F A L , F A U ] ⟩ $\langle [{T_{A}^{L}},{T_{A}^{U}}],[{I_{A}^{L}},{I_{A}^{U}}],[{F_{A}^{L}},{F_{A}^{U}}]\rangle $
C11	⟨ [ 0.693 , 0.806 ] , [ 0.52 , 0.631 ] , [ 0.251 , 0.363 ] ⟩ $\langle [0.693,0.806],[0.52,0.631],[0.251,0.363]\rangle $	C33	⟨ [ 0.611 , 0.722 ] , [ 0.444 , 0.556 ] , [ 0.333 , 0.444 ] ⟩ $\langle [0.611,0.722],[0.444,0.556],[0.333,0.444]\rangle $
C12	⟨ [ 0.478 , 0.556 ] , [ 0.172 , 0.293 ] , [ 0.466 , 0.542 ] ⟩ $\langle [0.478,0.556],[0.172,0.293],[0.466,0.542]\rangle $	C34	⟨ [ 0.581 , 0.693 ] , [ 0.408 , 0.52 ] , [ 0.363 , 0.475 ] ⟩ $\langle [0.581,0.693],[0.408,0.52],[0.363,0.475]\rangle $
C13	⟨ [ 0.351 , 0.463 ] , [ 0.424 , 0.537 ] , [ 0.593 , 0.705 ] ⟩ $\langle [0.351,0.463],[0.424,0.537],[0.593,0.705]\rangle $	C35	⟨ [ 0.536 , 0.648 ] , [ 0.363 , 0.475 ] , [ 0.408 , 0.52 ] ⟩ $\langle [0.536,0.648],[0.363,0.475],[0.408,0.52]\rangle $
C14	⟨ [ 0.479 , 0.62 ] , [ 0.549 , 0.661 ] , [ 0.448 , 0.584 ] ⟩ $\langle [0.479,0.62],[0.549,0.661],[0.448,0.584]\rangle $	C36	⟨ [ 0.536 , 0.634 ] , [ 0.26 , 0.386 ] , [ 0.408 , 0.503 ] ⟩ $\langle [0.536,0.634],[0.26,0.386],[0.408,0.503]\rangle $
C15	⟨ [ 0.312 , 0.438 ] , [ 0.447 , 0.562 ] , [ 0.619 , 0.745 ] ⟩ $\langle [0.312,0.438],[0.447,0.562],[0.619,0.745]\rangle $	C37	⟨ [ 0.5 , 0.595 ] , [ 0.24 , 0.361 ] , [ 0.444 , 0.538 ] ⟩ $\langle [0.5,0.595],[0.24,0.361],[0.444,0.538]\rangle $
C21	⟨ [ 0.38 , 0.492 ] , [ 0.396 , 0.508 ] , [ 0.564 , 0.676 ] ⟩ $\langle [0.38,0.492],[0.396,0.508],[0.564,0.676]\rangle $	C41	⟨ [ 0.5 , 0.579 ] , [ 0.172 , 0.293 ] , [ 0.444 , 0.522 ] ⟩ $\langle [0.5,0.579],[0.172,0.293],[0.444,0.522]\rangle $
C22	⟨ [ 0.648 , 0.762 ] , [ 0.475 , 0.587 ] , [ 0.295 , 0.408 ] ⟩ $\langle [0.648,0.762],[0.475,0.587],[0.295,0.408]\rangle $	C42	⟨ [ 0.159 , 0.291 ] , [ 0.598 , 0.709 ] , [ 0.765 , 0.896 ] ⟩ $\langle [0.159,0.291],[0.598,0.709],[0.765,0.896]\rangle $
C23	⟨ [ 0.149 , 0.297 ] , [ 0.59 , 0.703 ] , [ 0.759 , 0.907 ] ⟩ $\langle [0.149,0.297],[0.59,0.703],[0.759,0.907]\rangle $	C43	⟨ [ 0.462 , 0.556 ] , [ 0.24 , 0.361 ] , [ 0.483 , 0.575 ] ⟩ $\langle [0.462,0.556],[0.24,0.361],[0.483,0.575]\rangle $
C31	⟨ [ 0.26 , 0.385 ] , [ 0.502 , 0.615 ] , [ 0.671 , 0.796 ] ⟩ $\langle [0.26,0.385],[0.502,0.615],[0.671,0.796]\rangle $	C44	⟨ [ 0.413 , 0.525 ] , [ 0.363 , 0.475 ] , [ 0.531 , 0.643 ] ⟩ $\langle [0.413,0.525],[0.363,0.475],[0.531,0.643]\rangle $
C32	⟨ [ 0.26 , 0.385 ] , [ 0.502 , 0.615 ] , [ 0.671 , 0.796 ] ⟩ $\langle [0.26,0.385],[0.502,0.615],[0.671,0.796]\rangle $	C45	⟨ [ 0.774 , 1 ] , [ 0.598 , 0.709 ] , [ 0.128 , 0.283 ] ⟩ $\langle [0.774,1],[0.598,0.709],[0.128,0.283]\rangle $

After the calculation of criteria which is needed for weighted normalized IVN decision matrix and concordance interval matrix in the later parts, we construct the decision matrices with respect to experts. To indicate the types of criteria, C-N, C-L, B-L, and B-N are used as Cost-Numerical, Cost-Linguistic, Benefit-Linguistic, and Benefit-Numerical for short. Experts weights are 0.4 for academicians, and 0.3, for the managers from the energy sector. Table 10 illustrates the linguistic terms which are revealed by these experts.

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Table 10

DM based on expert opinions.

Criterion	Type	E1						E2						E3
		AL1	AL2	AL3	AL4	AL5	AL6	AL1	AL2	AL3	AL4	AL5	AL6	AL1	AL2	AL3	AL4	AL5	AL6
C11	B-N	G	AA	BA	B	VB	G	G	BA	AA	AA	F	G	CB	B	BA	F	BA	CG
C12	C-N	BA	AA	CG	G	AA	F	BA	AA	G	VG	CG	AA	BA	B	VB	B	BA	F
C13	B-N	AA	G	BA	AA	G	BA	AA	G	VG	F	AA	BA	BA	AA	BA	VG	G	CG
C14	B-L	AA	F	AA	G	VG	B	AA	VB	B	B	BA	CG	AA	F	BA	B	B	VB
C15	B-N	AA	F	AA	VB	B	B	AA	F	BA	F	AA	BA	AA	F	BA	VG	G	CG
C21	B-N	AA	G	AA	F	BA	G	F	G	BA	VG	B	VB	AA	G	BA	F	VB	G
C22	B-N	G	BA	G	B	B	AA	AA	G	BA	VG	B	B	G	BA	F	AA	AA	AA
C23	C-L	BA	BA	VG	B	B	G	F	BA	F	CG	G	G	AA	BA	AA	AA	AA	G
C24	C-L	VG	F	CG	G	G	E	VB	CB	B	BA	F	CB	CG	A	CG	B	B	A
C31	C-L	B	B	BA	F	CB	G	B	BA	B	VB	AA	CG	B	BA	B	AA	AA	G
C32	B-L	AA	B	VB	AA	CG	AA	AA	BA	AA	BA	BA	AA	AA	VG	G	CG	BA	AA
C33	C-N	CG	AA	BA	BA	AA	BA	CG	BA	CG	B	B	BA	CG	B	B	VB	B	BA
C34	C-N	AA	BA	AA	AA	AA	G	AA	BA	F	VB	B	G	AA	CB	VB	B	AA	G
C35	C-N	BA	BA	F	VB	F	G	F	F	AA	AA	BA	G	BA	F	BA	F	F	G
C36	C-N	BA	F	AA	AA	F	G	BA	F	VB	B	F	BA	F	VB	B	G	F	G
C37	C-N	VG	F	VB	B	VB	BA	VG	G	CG	AA	F	BA	AA	AA	BA	G	BA	VG
C41	C-N	VB	F	VB	AA	AA	G	B	B	VB	AA	F	F	VB	B	F	BA	BA	VG
C42	B-L	BA	AA	G	AA	F	AA	VG	G	CG	VB	AA	G	CG	AA	F	BA	F	CG
C43	B-L	B	B	G	F	AA	BA	B	B	VB	F	B	BA	VB	CB	B	F	B	BA
C44	B-L	CB	VB	B	B	B	BA	CB	VB	B	G	AA	F	B	BA	B	G	B	VB
C45	B-N	CB	CB	AA	B	B	BA	CB	B	VB	G	F	AA	AA	BA	AA	B	B	BA

The next step is the aggregation of decision matrices given in linguistic terms in Table 10. Aggregated IVN DM (A) is obtained by using Eq. (3) as in Table 11.

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Table 11

Aggregated DM.

Criterion	Type	AL 1 ${\mathrm{AL}_{1}}$	AL 2 ${\mathrm{AL}_{2}}$	…	AL 6 ${\mathrm{AL}_{6}}$
C 11 ${\mathbf{C}_{\mathbf{11}}}$	Linguistic cost	⟨ [ 0.437 , 0.551 ] , [ 0.452 , 0.55 ] , [ 0.39 , 0.51 ] ⟩ $\langle [0.437,0.551],[0.452,0.55],[0.39,0.51]\rangle $	⟨ [ 0.39 , 0.49 ] , [ 0.33 , 0.428 ] , [ 0.46 , 0.557 ] ⟩ $\langle [0.39,0.49],[0.33,0.428],[0.46,0.557]\rangle $	⋯ $\cdots \hspace{0.1667em}$	⟨ [ 0.623 , 0.76 ] , [ 0.45 , 0.553 ] , [ 0.175 , 0.33 ] ⟩ $\langle [0.623,0.76],[0.45,0.553],[0.175,0.33]\rangle $
C 12 ${\mathbf{C}_{\mathbf{12}}}$	Numerical cost	⟨ [ 0.4 , 0.5 ] , [ 0.3 , 0.4 ] , [ 0.45 , 0.55 ] ⟩ $\langle [0.4,0.5],[0.3,0.4],[0.45,0.55]\rangle $	⟨ [ 0.41 , 0.51 ] , [ 0.33 , 0.428 ] , [ 0.44 , 0.541 ] ⟩ $\langle [0.41,0.51],[0.33,0.428],[0.44,0.541]\rangle $	⋯ $\cdots \hspace{0.1667em}$	⟨ [ 0.45 , 0.516 ] , [ 0.14 , 0.25 ] , [ 0.4 , 0.46 ] ⟩ $\langle [0.45,0.516],[0.14,0.25],[0.4,0.46]\rangle $
⋮	⋮	⋮	⋮	⋱	⋮
C 21 ${\mathbf{C}_{\mathbf{21}}}$	Numerical benefit	⟨ [ 0.45 , 0.536 ] , [ 0.216 , 0.32 ] , [ 0.4 , 0.48 ] ⟩ $\langle [0.45,0.536],[0.216,0.32],[0.4,0.48]\rangle $	⟨ [ 0.55 , 0.65 ] , [ 0.4 , 0.5 ] , [ 0.3 , 0.4 ] ⟩ $\langle [0.55,0.65],[0.4,0.5],[0.3,0.4]\rangle $	⋯ $\cdots \hspace{0.1667em}$	⟨ [ 0.47 , 0.57 ] , [ 0.428 , 0.53 ] , [ 0.38 , 0.483 ] ⟩ $\langle [0.47,0.57],[0.428,0.53],[0.38,0.483]\rangle $
⋮	⋮	⋮	⋮	⋱	⋮
C 45 ${\mathbf{C}_{\mathbf{45}}}$	Linguistic benefit	⟨ [ 0.194 , 0.33 ] , [ 0.49 , 0.59 ] , [ 0.62 , 0.755 ] ⟩ $\langle [0.194,0.33],[0.49,0.59],[0.62,0.755]\rangle $	⟨ [ 0.245 , 0.36 ] , [ 0.431 , 0.534 ] , [ 0.59 , 0.7 ] ⟩ $\langle [0.245,0.36],[0.431,0.534],[0.59,0.7]\rangle $	⋯ $\cdots \hspace{0.1667em}$	⟨ [ 0.42 , 0.52 ] , [ 0.3 , 0.4 ] , [ 0.43 , 0.535 ] ⟩ $\langle [0.42,0.52],[0.3,0.4],[0.43,0.535]\rangle $

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Table 12

Weighted normalized DM.

Criterion	Type	AL 1 ${\mathrm{AL}_{1}}$	AL 2 ${\mathrm{AL}_{2}}$	…	AL 6 ${\mathrm{AL}_{6}}$
C 11 ${\mathbf{C}_{\mathbf{11}}}$	Linguistic cost	⟨ [ 0.008 , 0.013 ] , [ 0.623 , 1 ] , [ 0.332 , 0.67 ] ⟩ $\langle [0.008,0.013],[0.623,1],[0.332,0.67]\rangle $	⟨ [ 0.009 , 0.014 ] , [ 0.616 , 1 ] , [ 0.372 , 0.79 ] ⟩ $\langle [0.009,0.014],[0.616,1],[0.372,0.79]\rangle $	⋯ $\cdots \hspace{0.1667em}$	⟨ [ 0.011 , 0.017 ] , [ 0.623 , 1 ] , [ 0.29 , 0.557 ] ⟩ $\langle [0.011,0.017],[0.623,1],[0.29,0.557]\rangle $
C 12 ${\mathbf{C}_{\mathbf{12}}}$	Numerical cost	⟨ [ 0 , 033 , 0.038 ] , [ 1 , 1 ] , [ 0.84 , 0.86 ] ⟩ $\langle [0,033,0.038],[1,1],[0.84,0.86]\rangle $	⟨ [ 0.035 , 0.04 ] , [ 1 , 1 ] , [ 0.87 , 0.89 ] ⟩ $\langle [0.035,0.04],[1,1],[0.87,0.89]\rangle $	⋯ $\cdots \hspace{0.1667em}$	⟨ [ 0.02 , 0.023 ] , [ 1 , 1 ] , [ 0.74 , 0.773 ] ⟩ $\langle [0.02,0.023],[1,1],[0.74,0.773]\rangle $
⋮	⋮	⋮	⋮	⋱	⋮
C 21 ${\mathbf{C}_{\mathbf{21}}}$	Numerical benefit	⟨ [ 0.004 , 0.007 ] , [ 0.48 , 1 ] , [ 0.618 , 0.805 ] ⟩ $\langle [0.004,0.007],[0.48,1],[0.618,0.805]\rangle $	⟨ [ 0.004 , 0.006 ] , [ 0.5 , 1 ] , [ 0.59 , 0.745 ] ⟩ $\langle [0.004,0.006],[0.5,1],[0.59,0.745]\rangle $	⋯ $\cdots \hspace{0.1667em}$	⟨ [ 0.003 , 0.005 ] , [ 0.5 , 1 ] , [ 0.596 , 0.76 ] ⟩ $\langle [0.003,0.005],[0.5,1],[0.596,0.76]\rangle $
⋮	⋮	⋮	⋮	⋱	⋮
C 45 ${\mathbf{C}_{\mathbf{45}}}$	Linguistic benefit	⟨ [ 0.006 , 0.017 ] , [ 0.701 , 1 ] , [ 0.23 , 0.49 ] ⟩ $\langle [0.006,0.017],[0.701,1],[0.23,0.49]\rangle $	⟨ [ 0.009 , 0.02 ] , [ 0.7 , 1 ] , [ 0.23 , 0.5 ] ⟩ $\langle [0.009,0.02],[0.7,1],[0.23,0.5]\rangle $	⋯ $\cdots \hspace{0.1667em}$	⟨ [ 0.02 , 0.04 ] , [ 0.692 , 1 ] , [ 0.23 , 0.502 ] ⟩ $\langle [0.02,0.04],[0.692,1],[0.23,0.502]\rangle $

In the next step, we obtain the weighted normalized DM (R). In Eqs. (5), (6), and (7), all operations are calculated as is given in Definition 2. In Eq. (6), number 1 is represented in interval-neutrosophic set as

⟨ [ 1 , 1 ] , [ 0 , 0 ] , [ 0 , 0 ] ⟩ $\langle [1,1],[0,0],[0,0]\rangle $

. The calculated criteria weights are taken from Table 9. Weighted normalized DM is given in Table 12.

In the next step, we calculate the concordance and discordance indices based on the proposed deneutrosophication method which is given in Definition 5. Due to the excessive number of pairwise comparisons, we will only present Concordance and Discordance indices of Alternative 1 in the following. Concordance indices for Alternative 1:

C 12 = { C 11 , C 13 , C 14 , C 15 , C 22 , C 23 , C 24 , C 32 , C 33 , C 34 , C 37 , C 42 , C 43 , C 45 } , C 13 = { C 11 , C 14 , C 15 , C 22 , C 32 , C 33 , C 34 , C 36 , C 37 , C 42 } , C 14 = { C 11 , C 22 , C 24 , C 33 , C 34 , C 35 , C 37 , C 42 , C 43 } , C 15 = { C 13 , C 24 , C 32 , C 33 , C 37 , C 42 } , C 16 = { C 12 , C 13 , C 22 , C 24 , C 32 , C 33 , C 37 , C 42 } . \[\begin{aligned}{}& {\mathbb{C}_{12}}=\{{C_{11}},{C_{13}},{C_{14}},{C_{15}},{C_{22}},{C_{23}},{C_{24}},{C_{32}},{C_{33}},{C_{34}},{C_{37}},{C_{42}},{C_{43}},{C_{45}}\},\\ {} & {\mathbb{C}_{13}}=\{{C_{11}},{C_{14}},{C_{15}},{C_{22}},{C_{32}},{C_{33}},{C_{34}},{C_{36}},{C_{37}},{C_{42}}\},\\ {} & {\mathbb{C}_{14}}=\{{C_{11}},{C_{22}},{C_{24}},{C_{33}},{C_{34}},{C_{35}},{C_{37}},{C_{42}},{C_{43}}\},\\ {} & {\mathbb{C}_{15}}=\{{C_{13}},{C_{24}},{C_{32}},{C_{33}},{C_{37}},{C_{42}}\},\\ {} & {\mathbb{C}_{16}}=\{{C_{12}},{C_{13}},{C_{22}},{C_{24}},{C_{32}},{C_{33}},{C_{37}},{C_{42}}\}.\end{aligned}\]

Discordance indices for Alternative 1:

D 12 = { D 12 , D 21 , D 31 , D 35 , D 36 , D 41 , D 44 } , D 13 = { D 12 , D 13 , D 21 , D 23 , D 24 , D 31 , D 35 , D 41 , D 43 , D 44 , D 45 } , D 14 = { D 12 , D 13 , D 14 , D 15 , D 21 , D 23 , D 31 , D 32 , D 36 , D 41 , D 44 , D 45 } , D 15 = { D 11 , D 12 , D 14 , D 15 , D 21 , D 22 , D 23 , D 31 , D 34 , D 35 , D 36 , D 41 , D 43 , D 44 , D 45 } , D 16 = { D 11 , D 14 , D 15 , D 21 , D 23 , D 31 , D 34 , D 35 , D 36 , D 41 , D 43 , D 44 , D 45 } . \[\begin{aligned}{}& {\mathbb{D}_{12}}=\{{D_{12}},{D_{21}},{D_{31}},{D_{35}},{D_{36}},{D_{41}},{D_{44}}\},\\ {} & {\mathbb{D}_{13}}=\{{D_{12}},{D_{13}},{D_{21}},{D_{23}},{D_{24}},{D_{31}},{D_{35}},{D_{41}},{D_{43}},{D_{44}},{D_{45}}\},\\ {} & {\mathbb{D}_{14}}=\{{D_{12}},{D_{13}},{D_{14}},{D_{15}},{D_{21}},{D_{23}},{D_{31}},{D_{32}},{D_{36}},{D_{41}},{D_{44}},{D_{45}}\},\\ {} & {\mathbb{D}_{15}}=\{{D_{11}},{D_{12}},{D_{14}},{D_{15}},{D_{21}},{D_{22}},{D_{23}},{D_{31}},{D_{34}},{D_{35}},{D_{36}},{D_{41}},{D_{43}},{D_{44}},{D_{45}}\},\\ {} & {\mathbb{D}_{16}}=\{{D_{11}},{D_{14}},{D_{15}},{D_{21}},{D_{23}},{D_{31}},{D_{34}},{D_{35}},{D_{36}},{D_{41}},{D_{43}},{D_{44}},{D_{45}}\}.\end{aligned}\]

In the next step, we determine the Concordance matrix which is illustrated in Step 5 by using Eq. (10). After the calculations, the concordance and discordance matrices are obtained as in Tables 13–14.

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Table 13

Concordance matrix of deneutrosoficated IVN values.

	0.72	0.51	0.47	0.23	0.35
0.28	–	0.38	0.22	0.19	0.12
0.54	0.62	–	0.49	0.3	0.41
0.47	0.78	0.51	–	0.27	0.35
0.77	0.81	0.7	0.73	–	0.62
0.68	0.88	0.59	0.65	0.38	–

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Table 14

Discordance matrix of deneutrosoficated IVN values.

–	0.02	0.16	0.15	0.8	0.06
0.31	–	0.15	0.13	0.84	0.08
0.05	0.03	–	0.03	0.77	0.11
0.15	0.04	0.06	–	0.76	0.13
1	0.69	0.75	0.69	–	0.7
0.3	0.07	0.22	0.21	0.77	–

Based on the threshold value that is calculated by using Eq. (14), the concordance index matrix is given in Table 15. The threshold value is calculated as 0.595.

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Table 15

Concordance index matrix.

–	1	1	0	0	0
0	–	0	0	0	0
1	1	–	0	0	0
0	1	1	–	0	0
1	1	1	1	–	1
1	1	1	1	0	–

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Table 16

Discordance index matrix.

–	1	1	1	0	1
1	–	1	1	0	1
1	1	–	1	0	1
1	1	1	–	0	1
0	0	0	0	–	0
1	1	1	1	0	–

Based on the threshold value that is calculated by using Eq. (15), the discordance index matrix is given in Table 16. The threshold value is calculated as 0.405.

The last step is the comparison of the Concordance and Discordance index matrices. In Table 15, the value 1 represents that the value of the index is equal to or larger than the threshold. In Table 16, the value 1 represents that the value of the index is smaller than the threshold. If we have the values 1 in the same cells of Tables 15 and 16, it is concluded that the row alternative is better than the column alternative. Through this matching, AL6-Wind Power Plant is determined to be the best alternative that the municipality can invest in. The outranking relations of the alternatives are given in Fig. 2.

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Fig. 2

Outranking relations of the alternatives.

The arrows in Table 2 indicate that AL6-Wind Power Plant is the best among the alternatives, A3-Hydropower Energy Plant is the second best. A1-Wave Power Plant – A3-Biomass Energy Plant are non-comparable among themselves and A5-Geothermal Energy Plant is non-comparable among all alternatives.

5.3 Comparative Analysis

In this section, the proposed IVN ELECTRE I method is compared with IVN TOPSIS method developed by Chi and Liu (2013). Since we apply the same scales which are given in Table 1 and Table 5 for IVN TOPSIS method, the results in Table 12 are used for the next steps of the IVN TOPSIS. After the calculations, positive and negative ideal solutions of IVN TOPSIS are given in Table 17.

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Table 17

Positive and negative solutions of the IVN TOPSIS.

	D + ${D^{+}}$	D − ${D^{-}}$
C11	⟨ [ 0.011 , 0.017 ] , [ 0.606 , 1 ] , [ 0.287 , 0.557 ] ⟩ $\langle [0.011,0.017],[0.606,1],[0.287,0.557]\rangle $	⟨ [ 0.008 , 0.013 ] , [ 0.606 , 1 ] , [ 0.287 , 0.557 ] ⟩ $\langle [0.008,0.013],[0.606,1],[0.287,0.557]\rangle $
C12	⟨ [ 0.02 , 0.023 ] , [ 1 , 1 ] , [ 1 , 1 ] ⟩ $\langle [0.02,0.023],[1,1],[1,1]\rangle $	⟨ [ 0.035 , 0.04 ] , [ 1 , 1 ] , [ 1 , 1 ] ⟩ $\langle [0.035,0.04],[1,1],[1,1]\rangle $
⋮	⋮	⋮
C21	⟨ [ 0.006 , 0.009 ] , [ 0.472 , 1 ] , [ 0.59 , 0.745 ] ⟩ $\langle [0.006,0.009],[0.472,1],[0.59,0.745]\rangle $	⟨ [ 0.002 , 0.004 ] , [ 0.472 , 1 ] , [ 0.59 , 0.745 ] ⟩ $\langle [0.002,0.004],[0.472,1],[0.59,0.745]\rangle $
⋮	⋮	⋮
C45	⟨ [ 0.02 , 0.039 ] , [ 0.685 , 1 ] , [ 0.214 , 0.467 ] ⟩ $\langle [0.02,0.039],[0.685,1],[0.214,0.467]\rangle $	⟨ [ 0.006 , 0.017 ] , [ 0.685 , 1 ] , [ 0.214 , 0.467 ] ⟩ $\langle [0.006,0.017],[0.685,1],[0.214,0.467]\rangle $

Distances to negative and positive ideal solutions are given in Table 18.

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Table 18

Positive and negative ideal solutions of the IVN TOPSIS.

	Positive ideal solution	Negative ideal solution
AL1	⟨ [ 0.166 , 0.168 ] , [ 0.171 , 0.109 ] , [ 0.083 , 0.106 ] ⟩ $\langle [0.166,0.168],[0.171,0.109],[0.083,0.106]\rangle $	⟨ [ 0.177 , 0.174 ] , [ 0.171 , 0.109 ] , [ 0.083 , 0.106 ] ⟩ $\langle [0.177,0.174],[0.171,0.109],[0.083,0.106]\rangle $
AL2	⟨ [ 0.156 , 0.156 ] , [ 0.156 , 0.206 ] , [ 0.261 , 0.234 ] ⟩ $\langle [0.156,0.156],[0.156,0.206],[0.261,0.234]\rangle $	⟨ [ 0.153 , 0.155 ] , [ 0.156 , 0.206 ] , [ 0.261 , 0.234 ] ⟩ $\langle [0.153,0.155],[0.156,0.206],[0.261,0.234]\rangle $
AL3	⟨ [ 0.205 , 0.206 ] , [ 0.207 , 0.167 ] , [ 0.146 , 0.152 ] ⟩ $\langle [0.205,0.206],[0.207,0.167],[0.146,0.152]\rangle $	⟨ [ 0.213 , 0.21 ] , [ 0.207 , 0.167 ] , [ 0.146 , 0.152 ] ⟩ $\langle [0.213,0.21],[0.207,0.167],[0.146,0.152]\rangle $
AL4	⟨ [ 0.161 , 0.159 ] , [ 0.157 , 0.152 ] , [ 0.152 , 0.153 ] ⟩ $\langle [0.161,0.159],[0.157,0.152],[0.152,0.153]\rangle $	⟨ [ 0.159 , 0.159 ] , [ 0.157 , 0.152 ] , [ 0.152 , 0.153 ] ⟩ $\langle [0.159,0.159],[0.157,0.152],[0.152,0.153]\rangle $
AL5	⟨ [ 0.16 , 0.161 ] , [ 0.162 , 0.175 ] , [ 0.194 , 0.188 ] ⟩ $\langle [0.16,0.161],[0.162,0.175],[0.194,0.188]\rangle $	⟨ [ 0.156 , 0.158 ] , [ 0.162 , 0.175 ] , [ 0.194 , 0.188 ] ⟩ $\langle [0.156,0.158],[0.162,0.175],[0.194,0.188]\rangle $
AL6	⟨ [ 0.179 , 0.191 ] , [ 0.207 , 0.663 ] , [ 1.285 , 1.804 ] ⟩ $\langle [0.179,0.191],[0.207,0.663],[1.285,1.804]\rangle $	⟨ [ 0.189 , 0.183 ] , [ 0.176 , 0.152 ] , [ 0.142 , 0.066 ] ⟩ $\langle [0.189,0.183],[0.176,0.152],[0.142,0.066]\rangle $

Closeness coefficient (

c c i $c{c_{i}}$

) of each criterion is calculated as follows: AL1 = 0.4961, AL2 = 0.5010, AL3 = 0.4972, AL4 = 0.5006, AL5 = 0.5016, and AL6 = −6.2766. After the ranking of alternatives according to

c c i $c{c_{i}}$

in ascending order AL6-Wind Power Plant is still the best alternative as in our proposed IVN ELECTRE I method. This comparison shows the reliability of our proposed method.

5.4 Sensitivity Analysis

To demonstrate the stability of the ranking results, a sensitivity analysis (SA) is performed based on the weights of criteria. We develop a pattern to determine the one-at-a-time sensitivity for every sub-criterion. The values of these criteria are every linguistic term which is given in Table 5. After the changes on weights of sub-criteria through this pattern, IVN ELECTRE I operations according to changing weights are re-run and the selection of the best alternative is re-processed. The developed pattern is given in Table 19.

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Table 19

Pattern of the one-at-a-time SA.

		Based on the sub-criteria
		Solution sets	Solution sets	… $\dots \hspace{0.1667em}$	Solution sets
Variables	CLI	Best alternative	… $\dots \hspace{0.1667em}$	… $\dots \hspace{0.1667em}$	Best alternative
	VLI	⋮	… $\dots \hspace{0.1667em}$	⋱	⋮
	LI	⋮	… $\dots \hspace{0.1667em}$	⋱	⋮
	BAI	⋮	… $\dots \hspace{0.1667em}$	⋱	⋮
	AI	⋮	… $\dots \hspace{0.1667em}$	⋱	⋮
	AAI	⋮	… $\dots \hspace{0.1667em}$	⋱	⋮
	HI	⋮	… $\dots \hspace{0.1667em}$	⋱	⋮
	VHI	⋮	… $\dots \hspace{0.1667em}$	⋱	⋮
	CHI	Best alternative	… $\dots \hspace{0.1667em}$	… $\dots \hspace{0.1667em}$	Best alternative

The final results of the SA with the determined best alternative are given in Table 20.

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Table 20

Results of the one-at-the-time sensitivity analysis.

When the results are examined, it seems that the changes made on the 3 criteria which are C12-Land requirements, C33-Investment cost, and C45-Availability of funds have different solutions from the others for finding the best alternative. When the linguistic term is increased from CLI-Certainly Low Importance to CHI-Certainly High Importance for the criterion C12-Land requirements, AL3-Biomass energy plant becomes one of the best alternatives with AL-6 Wind Power Plant since biomass energy plant’s energy production volume is exactly dependent on the area that it is established. When the linguistic term is increased to linguistic term CHI-Certainly High Importance on criterion C33-Investment cost, AL3-Biomass energy plant becomes again one of the best alternatives with AL-6 Wind Power Plant. But, while the linguistic term is CLI-Certainly Low Importance, AL5-Geothermal energy plant is the best alternative. Lastly, when the linguistic term is increased to linguistic term CHI-Certainly High Importance on criterion C45-Availability of funds, AL1-Wave Power Plant becomes the best alternative since it is an area that has not been invested by the private sector enough. In all cases, AL6-Wind Power Plant is the best alternative by a majority and only in special cases that match up with real life our proposed method chooses different alternatives.

6 Conclusion

In multi-criteria decision making (MCDM) problems, the most important problem is the way of handling not only the uncertainty that is the result of lack of knowledge, but also its incompleteness, indeterminacy, and inconsistency level due to the expert groups. The selection of the best renewable energy sources depends not only on alternatives and criteria but also on scoring alternatives with respect to criteria by experts. In this complex environment where the inconsistency arises, the scope of ordinary fuzzy sets is insufficient. To overcome these problems, neutrosophic logic is introduced by Smarandache. In this study, we applied neutrosophic sets to an MCDM model to overcome this impreciseness and indeterminacy in measuring the criteria numerically.

In this paper, an interval-valued neutrosophic multi-criteria group decision making with the ELECTRE method has been proposed for the selection of the most appropriate renewable energy source for a county municipality. Alternatives and criteria have been determined by experts’ ideas and literature review. The importance degrees of the criteria have been determined by aggregation of expert opinions and the scores of alternatives are provided by experts for applying them to interval-valued neutrosophic ELECTRE. In the application, 21 sub-criteria were evaluated using interval-valued neutrosophic sets. According to the obtained results from the aggregation, C45-Availability of Funds is determined as the most efficient sub-criterion. The second and third important sub-criteria are C11-Reliability and C22-Estimated Amount of Energy Produced, respectively. In the interval-valued neutrosophic ELECTRE I method, there are 3 experts who evaluated 6 alternatives with respect to 21 sub-criteria. According to the results, A6-Wave Power Plant is determined as the best alternative for the most appropriate renewable energy source for a municipality. Sensitivity analysis demonstrates that our proposed model is robust. The usage of deneutrosophication operator can be mentioned as a drawback of the proposed method since a complete neutrosophic approach should be preferred.

For further studies, proposed division operation and deneutrosophication method can be extended to singleton neutrosophic sets, triangular neutrosophic sets, or trapezoidal neutrosophic sets. The proposed interval-valued neutrosophic ELECTRE method can be applied to the other areas such as location selection, strategy assessment, construction management, etc. The proposed method can be a basis for other possible neutrosophic outranking methods such as neutrosophic PROMETHEE method and neutrosophic ORESTE method.

Compliance with Ethical Standards

Conflict of Interest: All the authors declare that they have no conflict of interest.

Ethical approval: All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.

Informed consent: Informed consent was obtained from all individual participants included in the study.

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