Selection of the most appropriate alternative of renewable energy source is one of the most important key principles for a sustainable and clean environment. This selection process focuses to choose the best location for an anticipated purpose. Since the use of exhaustible resources in cities and industries cause air pollution, safety risks, and greenhouse gas emissions in the atmosphere, the building of a renewable energy plant becomes a more important issue for a sustainable city than ever. Furthermore, renewable energy alternative selection problem involves many criteria such as operational, environmental, social, and economic, and each of them addresses the problem in a broader and different perspective.

In multi-criteria decision making (MCDM) problems, the most important problem is the way of handling uncertainty (Mardani et al., 2015a, 2015b, 2017, 2018). In this type of problems, the criteria can be tangible or intangible. Having more intangible criteria than tangible criteria causes a harder evaluation process for experts. In this environment, MCDM methods need to determine the evaluation criteria, a set of possible alternatives, and collect the appropriate information about alternatives with respect to criteria, and to evaluate them for the decision makers’ purposes (Tzeng and Huang, 2011). To handle these difficulties, many models have been proposed and among these models, Analytic Hierarchy Process (AHP) (Saaty, 1980), Analytic Network Process (ANP) (Saaty, 1996), Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) (Hwang and Yoon, 1981; Zavadskas et al., 2016), ELimination Et Choix Traduisant la REalité (ELECTRE) (Roy, 1991; Govindan and Jepsen, 2016), VIKOR (Opricovic, 1998; Mardani et al., 2016), linguistic models (Cabrerizo et al., 2017, 2018; Morente-Molinera et al., 2019; Zhang et al., 2018; Liu et al., 2018) are the most used ones.

When considering real life conditions, things are not often precise, and they cannot be described by crisp or deterministic models. Thus, the capacity of making precise statements is quite challenging. In order to handle these vague and imprecise events, Zadeh (1965) introduced fuzzy sets together with degrees of membership of elements to these sets (Zadeh, 1965). Since that time, ordinary fuzzy sets have been extended to type-2, intuitionistic, hesitant, orthopair fuzzy sets, and neutrosophic sets. Zadeh introduced type-2 fuzzy sets to define the uncertainty of membership functions to reduce vagueness (Zadeh, 1975). Then, (Zadeh, 1975; Grattan-Guiness, 1975; Jahn, 1975; Sambuc, 1975) introduced interval-valued fuzzy sets (IVFSs), independently from each other. Atanassov introduced intuitionistic fuzzy sets (IFSs) to express decision maker’s opinions more freely by using not only membership functions but also non-membership functions of the elements in a fuzzy set (Atanassov, 1986). Hesitant fuzzy sets (HFSs) were introduced by Torra (2010) in order to operate with a set of possible membership values for an element in a fuzzy set (Torra, 2010). Despite all these extensions, fuzzy sets could not handle all types of uncertainties such as indeterminate and inconsistent information.

In order to tackle this inadequacy, Smarandache (1995) introduced neutrosophic logic and neutrosophic sets (Smarandache, 1999). A neutrosophic set is composed of three subsets which are degree of truthiness (T) , degree of indeterminacy (I) , and degree of falsity (F) . These subsets are between ]−0,1+[ non-standard unit interval (Rivieccio, 2008). Thus, a membership function of a neutrosophic number is represented by truth sub-set; non-membership function is represented by falsity sub-set; and hesitancy is represented by indeterminacy sub-set. These features constitute the superiority of neutrosophic sets over the other extensions of fuzzy sets. We utilize neutrosophic sets since their main advantage is the capability in distinguishing relativity and absoluteness of decision makers’ preferences.

The first ELECTRE method, ELECTRE I, was introduced by Roy (1968). The roots of its development are based on the construction of a contradictory and very heterogeneous set of criteria, and quantitative and qualitative consequences which are not only associated with numerical ordinal scales but also are attached with imprecise, uncertain, and ill-determined knowledge of data (Roy, 1968). The other types of ELECTRE methods are ELECTRE IS, ELECTRE II, ELECTRE III, ELECTRE IV, and ELECTRE TRI. These methods mainly hold ELECTRE I characteristics and link to its basic idea. Briefly, ELECTRE I and ELECTRE IS are introduced for the selection problems; ELECTRE TRI, for the assignment problems, and ELECTRE II, III and IV, for the ranking problems (Leyva López, 2005). Since our problem’s characteristics hold heterogeneous and multi-criteria structure; qualitive and quantitative consequences build on interval-number scales; uncertain and indeterminate knowledge, it is proper to apply ELECTRE I method.

In this study, an IVN ELECTRE I method is developed and applied to a renewable energy sources alternative selection for a county municipality. The originality of this paper can be explained with 3 basics. Firstly, we develop a novel interval-valued neutrosophic ELECTRE I method and apply it to a renewable energy source selection. Secondly, a new and an efficient deneutrosophication and division operation is proposed. Finally, for validating the proposed method, we compare the results with the interval-valued neutrosophic (IVN) TOPSIS method. A sensitivity analysis with an explanatory pattern is also performed to demonstrate the stability of the ranking results of the IVN ELECTRE I method. We believe that our study can guide the researchers who are interested in decision making under imprecise and indeterminate environment.

The remainder of this paper is prepared as follows. Section 2 reviews the literature related to the neutrosophic MCDM. In Section 3, renewable energy source alternatives and criteria are determined. In Section 4, preliminaries for IVN neutrosophic sets are given. In Section 5, an application for a county municipality to determine the best renewable energy source is introduced by using the proposed IVN ELECTRE I method. The paper ends with the conclusions and suggestions for further research.

Neutrosophic sets have been used in several single and multiple decision-making methods in recent years. The number of studies on neutrosophic sets and its applications has increased dramatically since 2010. The types of neutrosophic sets in the literature are singleton, interval-valued, triangular, or trapezoidal neutrosophic sets. These studies have been briefly summarized in the following.

Bausys et al. (2015) applied neutrosophic sets to COPRAS method for the decision making problems by using uncertain data. Sun et al. (2015) extended Choquet integral operator with interval neutrosophic numbers for multi-criteria decision making problems. Bausys and Zavadskas (2015) extended VIKOR method with interval valued neutrosophic sets for the multi-criteria decision making problems. Zavadskas et al. (2015) applied WASPAS method with single-valued neutrosophic set for the assessment of locations for the construction site of a waste incineration plant. Ye (2016b) introduced correlation coefficients of interval neutrosophic hesitant fuzzy sets for the MCDM problems. Li et al. (2016) introduced Some single valued neutrosophic number heronian mean operators for the MCDM problems. Ye studied the cross-entropy of single valued neutrosophic sets (SVNSs) as an extension of the cross entropy of fuzzy sets. The practical example showed the effectiveness of the proposed cross entropy method for MCDM techniques (Ye, 2014). Ye developed a multiple attribute group decision-making method by using the neutrosophic linguistic numbers weighted arithmetic average (NLNWAA) and neutrosophic linguistic numbers weighted geometric average (NLNWGA) operators (Ye, 2016a). Ma et al. studied a problem of time-aware trustworthy service selection which is formulated as MCDM problem of creating ranked services list using cloud service interval neutrosophic set (CINS) (Ma et al., 2016). It was solved by developing a CINS ranking method. The developed model which is based on a real-world dataset shows the practicality and effectiveness of the proposed approach. Huang defined a new distance measure between two SVNSs (Huang, 2016). The study figured out that neutrosophic sets are very effective to define incompleteness, indeterminacy and inconsistency information. Baušys and Juodagalvienė studied a location selection problem of the garage at the parcel of a single-family residential house by using an integrated method of AHP and Weighted Aggregated Sum-Product Assessment (WASPAS) with SVNSs (Baušys and Juodagalvienė, 2017). They reveal that the application of SVNSs allows to model uncertainty of the initial information clearly. Peng et al. presented an MCDM problem based on the Qualitative Flexible Multiple Criteria (QUALIFLEX) method where the criteria values are addressed by multi-valued neutrosophic information (Peng et al., 2017a). Stanujkic et al. (2017) extended MULTIMOORA method to single valued neutrosophic sets. Zavadskas et al. (2017) applied single-valued neutrosophic MAMVA method for the sustainable market evaluation of buildings. The assessments showed that the given decisions by the proposed method are robust and can be applied to various application areas. Akram & Sitara studied an SVN graph structure which is a generalization of fuzzy graph structures (Akram and Sitara, 2017). Applications of SVN graph structures in decision-making problems showed that using neutrosophic sets allow to clarify uncertainty of the complex environments. The illustrative example of choosing the best manufacturing alternative in the flexible manufacturing system indicated the robustness of the proposed methodologies. Şahin presented prioritized aggregation operators for aggregating the normal neutrosophic information and then extended these operators to the generalized prioritized weighted aggregation operators (Şahin, 2017). The results of the study indicated that normal neutrosophic sets are a powerful tool for handling incompleteness, indeterminacy, and inconsistency of evaluation information. Ye (2017) introduced weighted aggregation operators of trapezoidal neutrosophic numbers for the MCDM problems. Liang et al. (2017) applied single-valued trapezoidal neutrosophic DEMATEL method for the evaluation of e-commerce websites. Zhang et al. proposed a restaurant decision support model using social information for tourists on TripAdvisor.com by using interval-valued neutrosophic numbers (Zhang et al., 2017). The paper found out that the new model considers the active, neutral and passive information in online reviews all at once and takes the inter-dependency among criteria into consideration on tourists’ decision-making. Peng et al. presented a new outranking approach for MCDM problems, which are developed in the context of a simplified neutrosophic environment by singleton subsets in [0,1] (Peng et al., 2017b). The comparison analysis showed that the use of neutrosophic sets is an influential way of handling indeterminacy. Ji et al. (2018) applied multi-valued neutrosophic TODIM method for the selection of personnel under the consideration of risk preference decision-makers. Abdel-Basset et al. (2018) applied an integrated AHP–SWOT analysis by considering neutrosophic logic for the decision making of strategic planning. Feng et al. (2018) applied an integrated methodology consisting of DEMATEL and ELECTRE III methods by considering neutrosophic set environment for the shopping mall photovoltaic plan selection.

In this paper, unlike the above papers, an interval-valued neutrosophic ELECTRE I is presented for alternative energy source selection problem of a county municipality. Since evaluation criteria include not only uncertain but also incomplete, indeterminate and inconsistent characteristics in the evaluation process, interval-valued neutrosophic sets are used to handle this handicap. In the literature, multi-valued neutrosophic ELECTRE III and single valued neutrosophic ELECTRE methods have been developed by Peng et al. (2016, 2014), Feng et al. (2018). The models in Peng et al. (2016) and (2014) are based on single valued neutrosophic sets and do not present a flexible definition area for T, I, F values to decision makers. Feng et al. (2018) developed an interval valued neutrosophic ELECTRE III method for the shopping mall photovoltaic plan selection. However, our proposed method includes novel deneutrosophication and division operators together with comprehensive linguistic scales in order to weigh the criteria and to assess the alternatives. In this paper, we develop ELECTRE I method by using interval-valued neutrosophic sets to a new deneutrosophication method and a new division operator for interval-valued neutrosophic sets. We believe this paper can guide researchers to the application of interval-valued neutrosophic sets to other MCDM techniques.

Smarandache introduced neutrosophic sets as an extension of intuitionistic fuzzy sets. They have become very popular in recent years and have been used in many MCDM methods such as neutrosophic AHP (Radwan et al., 2016); neutrosophic TOPSIS (Chi and Liu, 2013; Nădăban and Dzitac, 2016); neutrosophic EDAS (Peng and Liu, 2017).

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=⟨(TL,TU),(IL,IU),(FL,FU)⟩ be an interval-valued neutrosophic number. The deneutrosophicated A value ( H(A) ) is given by Eq. (4): (4) H(A)=(TL+TU+(1−FL)+(1−FU)+TL×TU−(1−FL)×(1−FU))4×((1−[(IL)+(IU)]2)−((IL)×(IU))).

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

Step 1: Construct the IVN decision matrices ( Dj ) based on experts’ opinions (j) by using the scale which is given in Table 1 (Karaşan and Kahraman, 2017).

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) by using Eq. (3) as in Table 3.

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

Normalization of formulas for benefit ( bij ) and cost ( cij ) criteria is given in the following, respectively. (5) rij=bij∑i=1nbij2, where i=1,2,…,m and j=1,2,…,n . (6) rij=1cij∑i=1n(1cij)2, where i=1,2,…,m and j=1,2,…,n .

Arithmetic operations are carried out with the neutrosophic formulas given in Definition 2, and Definition 3. After calculations, normalized IVN DM (R) is illustrated in Table 4.

Weighted normalized IVN DM (V) is obtained by multiplying the IVN weights vector ( wj ) which is given in Table 5 with the normalized IVN DM (R) as in Eq. (7). (7) vij=wj⊗rij, where, wj=⟨[TwL,TwU],[IwL,IwU],[FwL,FwU]⟩ , rij=⟨[TrL,TrU],[IrL,IrU],[FrL,FrU]⟩ , and vij=⟨[TvL,TvU],[IvL,IvU],[FvL,FvU]⟩ .

Table 6 presents the weighted normalized IVN DM.

Step 4: Obtain the Concordance and Discordance Indices.

Let A Ln={a,b,…,n} denote a finite set of alternatives.

Concordance index ( Cab ) measures are given by Eq. (8): (8) Cab={j∣xaj⩾xbj}. Discordance index (Dab) measures the strength of the evidence against the Concordance index by Eq. (9): (9) Dab={j∣xaj<xbj}.

Step 5: Calculate the Concordance matrix.

The Concordance index C(a,b) between ALa and ALb is determined using Eq. (10): (10) C(a,b)=∑j∈C(a,b)H(wj). Through this calculation for all alternatives, concordance matrix is found as below: (11) C=−C(1,2)⋯C(1,m)C(2,1)−⋯C(2,m)⋮⋯⋱⋮C(m,1)C(m,2)⋯−.

Step 6: Calculate the Discordance matrix.

Similarly, the Discordance index D(a,b) between ALa and ALb is determined by Eq. (12): (12) D(a,b)=maxj∈Dab∣vaj−vbj∣maxj∈J;m,n∈I∣vmj−vnj∣, where 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). (13) D=−D(1,2)⋯D(1,q)D(2,1)−⋯D(2,q)⋮⋯⋱⋮D(z,1)D(z,2)⋯−.

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

The threshold value for Concordance matrix (αC) is computed by the average of the elements in matrix C via Eq. (14): (14) αC= ∑a=1m ∑b=1mC(a,b)m(m−1). The threshold value for Discordance matrix (αD) is computed by the average of the elements in matrix D via Eq. (15): (15) αD= ∑a=1m ∑b=1mD(a,b)m(m−1).

Step 8: Determine the acceptable relations based on ( αC ) and ( αD ). The values equal or larger than αC and the values smaller than αD 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.

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.