Introduction

Fossil-based energy sources (coal, natural gas, oil) are used as primary sources to meet rising energy demand with population growth and industrialization. The emission of harmful gases (carbon dioxide, nitrous oxide) caused by the combustion of fossil-based resources causes the greenhouse effect. Greenhouse effect-based global warming and climate change reveal environmental, economic and social problems on a global scale. Renewable energy (solar, wind, hydro, geothermal, biomass) is the most important energy source that is proposed as an alternative to fossil-based energy sources to remove the environmental and social hazards. The sun, which is the main source of the energies on earth (except tidal, nuclear, geothermal), emerges as the most important renewable energy source with its unlimited, equal distribution and economic advantages. Renewable energy sources, especially solar energy, should be included in the global energy strategy to meet the growing energy demand.

Solar rays are used to obtain heat and electricity from direct and indirect ways by active and passive methods. The sun’s rays are converted into electrical energy by directly or using thermal methods with high-tech flat plate and concentrated photovoltaic (PV) and concentrated solar power (CSP) solar energy systems. Technological developments in recent years have increased the efficiency values of solar energy systems and have ensured the spread of independent and grid related solar energy systems. Although solar energy systems have the advantages of environmental sensitivity and low operating costs, the rapid advances in solar energy technologies could not prevent the high initial cost problem. Therefore, high-tech PV and CSP solar power systems should be installed in the right place and on the right conditions. Hence, investors need to decide on the most optimum project by evaluating alternative solar energy projects balancing social, ecological, technical and economic aspects.

The average daily radiation values and the number of sunny days are important starting criteria for the development of solar energy systems. Countries that meet these leading criteria are developing policies to direct and control solar energy investments. Investors should make the right choice among alternative solar projects under these natural and artificial factors. Investors face complex investment decision problems due to uncertainties and variations in the factors affecting the plant installation process. Direct or indirect interaction between factors complicates the decision-making process and causes a great risk for investors. Investors start the decision-making process for solar energy investment projects with the principle of low risk and high profitability.

This study deals with the application of the AHP (Analytic Hierarchy Process) method in determining the most suitable project according to the defined criteria among alternative PV solar power plant projects proposed for solar energy investment. Decision makers make pairwise evaluations among criteria and alternative solar power plants based on their knowledge and experience. The uncertainties in the evaluations are transformed into more stable results using hesitant linguistic expressions and included in the decision-making process.

The contributions of the study to the literature are as follows: (1) the practicality of fuzzy logic based approaches used in AHP decision-making method is once again demonstrated with HFLTS. (2) Factors affecting solar power plant selection are examined in detail using HFLTS and included in the decision-making process. The application of selecting solar power plant using HFLTS based AHP model is the first study in the literature. The study explains in detail the decision makers’ evaluation in hesitant conditions and the effects of the decisions made on the alternative selection based on the original AHP decision-making model. Linguistic expressions are used to enable decision makers to reflect their knowledge and experience naturally and clearly. The fact that single linguistic terms (such as low, medium, high) are insufficient to meet the evaluations in the decision-making process requires the use of hesitant fuzzy linguistic terms. In addition, detailed explanation of the study provides opportunity for the application and development of the method easily by academicians and sector employees at all levels.

The continuation of the paper is organized under the following headings. “Overview of AHP methodology” describes the purpose and application steps of basic AHP, Fuzzy AHP based on fuzzy logic decision-making methods and hesitant fuzzy sets and linguistic terms. “Fuzzy AHP based on hesitant fuzzy linguistic evaluation” discusses hesitant fuzzy linguistic expressions, the process of transforming these expressions into trapezoidal fuzzy numbers and incorporating them into the AHP model. “An application: solar energy plant project selection” describes the application of the proposed model among five alternative solar projects under thirteen sub-criteria. In “Comparison and sensitivity analyses”, the validity of the methods and factors is questioned by making a comparison and sensitivity analysis. In “Conclusions”, the findings are evaluated and recommendations are made for future studies.

Overview of AHP methodology

AHP model

MCDM (Multiple Criteria Decision Making) method is applied to make the most suitable choice among the alternatives offered under multiple criteria and different targets [1]. The AHP technique developed in 1980 to solve unstructured problems by Saaty is an important MCDM method which is used to determine the most appropriate option among alternative options under multiple criteria and different targets. Saaty proposed the AHP method as a decision-making tool for the solution of unstructured MCDM problems in 1980 [2]. AHP is a powerful and useful MCDM technique that can solve complex decision-making problems requiring high flexibility and reliability. AHP provides flexibility and reliability to solve complex MCDM problems with its usable structure [3]. In AHP method, pairwise comparisons are made according to the indices and options determined by the decision tree created by the decision maker. The AHP method, which offers comprehensive and rational solutions, is widely used in decision-making problems for the development of renewable energy systems [4,5,6]. The criteria and alternatives defined in the hierarchical decision tree created by the decision maker are operated to the pairwise comparison process in the AHP method. This method comprehensively carries out the decision-making process by prioritizing the criteria and evaluating the alternatives according to the criteria separately. The basic steps of the AHP approach are as follows [2]:

  • The problem, including decision goals, alternatives and criteria for evaluating alternatives is modeled hierarchically.

  • Priorities are defined among the elements in the hierarchy by making pairwise comparisons.

  • Evaluations based on pairwise comparisons are synthesized to define general priorities for the hierarchy.

  • Consistency of evaluations is checked.

  • Alternatives are evaluated and the most appropriate decision is obtained under this process.

Fuzzy AHP

Fuzzy logic is used to include uncertainty and hesitancy in calculations in the AHP decision-making method that compares and evaluates alternatives on the basis of criteria. Subjective uncertainties in the decision-making process can be better reflected in the pairwise comparisons by fuzzy numbers. Although there are various types of fuzzy numbers, fuzzy triangular numbers (TFNs) and fuzzy trapezoidal numbers (TrFNs) are more commonly used to reflect the linguistic evaluations of experts.

The impact weights for each criterion in the decision-making process are determined either directly by experts based on their knowledge and experience or by pairwise comparisons of criteria [4]. In this study, experts make their evaluations with linguistic expressions using linguistic term set defined in Table 1.

Table 1 The linguistic terms and fuzzy number representation for pairwise comparisons [7]
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The linguistic evaluations of the experts for pairwise comparisons are transformed into triangular or trapezoidal fuzzy numbers and a comparison matrix which reflects the fuzzy weights is obtained. Different approaches are proposed for the evaluation of the fuzzy weighted matrix and for the weighting of the criteria and alternatives in the AHP decision model [8,9,10,11]. The main process steps in the Fuzzy AHP method are as follows:

  • Generation of a comparison matrix of the criteria and the alternatives with respect to the criteria using linguistic terms.

  • Conversion of linguistic expressions in the comparison matrix to fuzzy numbers.

  • Checking the consistency values of the pairwise comparison matrices.

  • Calculation of weights (priorities) of criteria and alternatives with respect to each criterion.

  • Calculating the overall priority values of the alternatives based on the criteria and sorting the alternatives according to these values.

If there are \(n\) criteria in the decision problem, \({C}_{i}\) refers to the ith criterion (\(i=1, 2, 3,\dots ., n\)). Experts compare the criteria with the linguistic evaluations and generate the comparison matrix (\(nxn\)). Comparative linguistic expressions of experts are converted into triangular or trapezoidal fuzzy numbers and the new numeric matrix is obtained as \(\stackrel{\sim }{P}\).

The relative importance between the two criteria ith and jth is expressed by the \({\stackrel{\sim }{p}}_{ij}=({l}_{ij},{m}_{ij},{n}_{ij},{u}_{ij})\) trapezoidal fuzzy number, and also the inverse relative importance relationship between them is shown as \({\stackrel{\sim }{p}}_{ji}=\frac{1}{{\stackrel{\sim }{p}}_{ij}}={\left({\stackrel{\sim }{p}}_{ij}\right)}^{-1}=({u}_{ij}^{-1},{n}_{ij}^{-1},{m}_{ij}^{-1},{l}_{ij}^{-1})\) (\(i,j=\mathrm{1,2},\dots ,n\)) [12].

Pairwise comparison matrix (\(\stackrel{\sim }{P}\)) is represented as;

The primary stage of the calculation process is to check the consistency of the comparison matrix. Consistency calculations are made by converting the elements of the comparison matrix created by trapezoidal fuzzy numbers into crisp values by defuzzification method [11]. The transformation of a trapezoidal fuzzy number \(\stackrel{\sim }{A}=(a,b,c,d)\) into a crisp value with defuzzification method is as follows [13]:

$$\mathrm{d}\mathrm{e}\mathrm{f}(\stackrel{\sim }{A})=\frac{(a+2b+2c+d)}{6}$$
(1)

The form of the fuzzy comparison matrix, \(\stackrel{\sim }{P}\) converted to crisp values is shown as traditional \(P\) comparison matrix:

The consistency of the \(P\) matrix reflects the consistency state of the fuzzy comparison matrix, \(\stackrel{\sim }{P}\). The consistency check steps for the comparison matrix, \( P\) are as follows:

  • Determination of the largest eigenvalue of the matrix [14]:

    $$\mathrm{P}\mathrm{w}={\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}w,$$
    (2)

    where \(w\) represents the eigenvector of the comparison matrix, \(P\).

  • The consistency of the comparison matrix is evaluated according to the calculated consistency ratio (CR) based on consistency index (\(CI\)) and random index (\(RI\)) in Table 2. As a general acceptance, if the consistency ratio is less than 0.1 (\(CR<0.1\)), the comparison matrix is considered appropriate, otherwise pairwise comparisons should be reassessed by experts.

    $$\mathrm{C}\mathrm{R}=\mathrm{C}\mathrm{I}/\mathrm{R}\mathrm{I},$$
    (3)
    $$\mathrm{C}\mathrm{I}=({\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}-n)/(n-1),$$
    (4)

    where \(n\) refers to the size of the matrix.

Table 2 The random index (RI) scale [15]
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In the next step, weights are calculated for the pairwise comparison matrices that are valid according to the consistency index. The weights are calculated through the fuzzy weight matrix, \( \stackrel{\sim }{W}\) that is obtained on the basis of fuzzy pairwise comparison matrix, \(\stackrel{\sim }{P}\) [11, 16]. The steps for calculating fuzzy weight vectors are as follows:

  • A geometric mean is calculated for each boundary value of trapezoidal fuzzy numbers.

    $${a}_{i}={\left[\prod_{j=1}^{n}{l}_{ij}\right]}^{1/n}, {b}_{i}={\left[\prod_{j=1}^{n}{m}_{ij}\right]}^{1/n},{c}_{i}={\left[\prod_{j=1}^{n}{n}_{ij}\right]}^{1/n},{d}_{i}={\left[\prod_{j=1}^{n}{u}_{ij}\right]}^{1/n}, \left(i=\mathrm{1,2},\dots ,n\right).$$
    (5)

  • The geometric mean values calculated for the comparison headers are added under each boundary value of the trapezoidal number.

    $$a=\sum_{j=1}^{n}{a}_{j},b=\sum_{j=1}^{n}{b}_{j},c=\sum_{j=1}^{n}{c}_{j},d=\sum_{j=1}^{n}{d}_{j}.$$
    (6)

  • The boundary values of the trapezoid fuzzy number are divided into the sum of the boundary values in inverse ranking, and a fuzzy weight vector is obtained for each comparison value as in Eq. 7. The weight vectors are defuzzified (\(\mathrm{def}\left({\stackrel{\sim }{w}}_{i}\right)\)) according to equation Eq. 1 and these values are normalized over the overall sum of the weights as in Eq. 8. Normalized weight vector (\(W\)) refers the weight value for each comparison value.

    $${\stackrel{\sim }{w}}_{i}=\left({a}_{i}{d}^{-1},{b}_{i}{c}^{-1},{c}_{j}{b}^{-1},{d}_{j}{a}^{-1}\right)\left(i=\mathrm{1,2},\dots ,n\right),$$
    (7)
    $${w}_{i}=\frac{\mathrm{def}\left({\stackrel{\sim }{w}}_{i}\right)}{\sum_{i=1}^{n}\mathrm{def}\left({\stackrel{\sim }{w}}_{i}\right)},$$
    (8)
    $$W=\left[{w}_{1} {w}_{2} {w}_{3}\dots {w}_{n}\right].$$
    (9)

In addition, experts evaluate the criteria individually using linguistic expressions in Table 3 and determine the impact values of the criteria for use in the decision-making process [11].

Table 3 Linguistic terms and their trapezoidal fuzzy numbers for evaluation [17]
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Decision makers evaluate the factors linguistically to determine the impact strength of the criteria. Fuzzy evaluation vectors of criteria are calculated with decision makers, \({D}_{a}\)\((a=\mathrm{1,2},\dots ,m)\) and criteria, \({e}_{i}\)\((i=\mathrm{1,2},\dots ,n)\) using the linguistic term set, \(\{VL, L, M, H, VH\}\). The fuzzy evaluation value of the criteria defined by decision makers’ linguistic assessments is calculated as follows:

$${\stackrel{\sim }{e}}_{i}=\frac{1}{m}\left({\stackrel{\sim }{e}}_{i}^{1}+{\stackrel{\sim }{e}}_{i}^{2}+\dots +{\stackrel{\sim }{e}}_{i}^{m}\right),$$
(10)

where decision maker’s (\(m\)) evaluation value for the criteria \({e}_{i}\) is defined with trapezoidal fuzzy number \({\stackrel{\sim }{e}}_{i}=(l,m,n,u)\) with respect to Table 3 [11]. The fuzzy evaluating matrix for all criteria is shown in Eq. 11:

$$\stackrel{\sim }{E} = \left[{\stackrel{\sim }{e}}_{1} {\stackrel{\sim }{e}}_{2}\dots {\stackrel{\sim }{e}}_{n}\right].$$
(11)

Impact values of the criteria are calculated according to the process followed in weight calculation process. The Fuzzy evaluating matrix (\(\stackrel{\sim }{E}\)) is operated following the Eqs. 5–9 process and the normalized evaluating matrix (\(E\)) is obtained. The weight values (\(W\)) obtained from the pairwise comparison of the criteria are combined with the impact values of the criteria (\(E\)) to calculate the actual weight values (\(A\)) for the criteria.

$${a}_{i}={w}_{i}{e}_{i}.$$
(12)

The weights of the alternatives obtained from the pairwise comparison with respect to the each criterion are combined with the weights of the criteria (\(A\)). The priority values of the alternatives are calculated and the alternative with the highest priority value is preferred.

Hesitant Fuzzy Sets and linguistic terms

Hesitant Fuzzy Sets (HFS) was introduced by Torra [18]. The concept of hesitant fuzzy sets reflects the possible values of membership values that belong to the fuzzy set. HFSs, which are extension of fuzzy sets, have been used in different scientific studies along with other fuzzy logic extensions (type-2, intuitionistic, neutrosophic). The use of HFS in various interdisciplinary research fields (Automation control system, Computer science, Engineering, Mathematics, Operations research management science) proves that it is an important application tool [19,20,21].

HFSs which are extended fuzzy clusters, define the membership degrees of an element in the range [0,1] [22]. \({h}_{E}(x)\) (\(h={h}_{E}(x)\), hesitant fuzzy element (HFE), [23]) represents the membership degrees of the element \(x\in X\) where \(X\) is a fixed set. HFS is shown mathematically as follows:

$$E=\left\{\langle x,{h}_{E}(x)\rangle |x\in X\right\}.$$

Linguistic information is an important concept related to HFS.

Real-life problems include non-numerical properties and evaluation criteria as well as numerical values. Linguistic variables defined by Zadeh [24] are used in problems involving the grading and evaluation of qualitative features. Linguistic expressions that provide freedom of evaluation in uncertain and uncertain conditions are developed with new approaches (decision-making approaches, aggregation operators, linguistic intuitionistic, linguistic interval-valued intuitionistic) [22, 25, 26].

\(S=\left\{{s}_{i}|i=\mathrm{0,1},\dots ,g\right\}\) is defined a finite Linguistic term sets (LTS) where \(g\) is a positive even number and \({s}_{i}\) is a linguistic term. Accordingly, LTS can be defined as [25]:

  • \({s}_{i}\le {s}_{j}\), where \(i<j\) (LTS is ordered set)

  • \(\mathrm{N}\mathrm{e}\mathrm{g}\left({s}_{i}\right)={s}_{j}\), where \(j=g-i\) (negation of linguistic term)

  • \(\mathrm{m}\mathrm{a}\mathrm{x}\left({s}_{i},{s}_{j}\right)={s}_{i}\), where \({s}_{i}\ge {s}_{j}\)

  • \(\mathrm{m}\mathrm{i}\mathrm{n}\left({s}_{i},{s}_{j}\right)={s}_{i}\), where \({s}_{i}\le {s}_{j}\)

Addition and multiplication of any two linguistic terms (\({s}_{i},{s}_{j}\in {S}_{[0,g]}\)) takes place on the basis of t-norm and t conorm.

  • \({s}_{i}\oplus {s}_{j}={s}_{j}\oplus {s}_{i}={s}_{g{\left(\frac{i}{g}, \frac{j}{g}\right)}_{t-\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}},}\)

  • \({s}_{i}\otimes {s}_{j}={s}_{j}\otimes {s}_{i}={s}_{g{\left(\frac{i}{g}, \frac{j}{g}\right)}_{t-\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}},}\)

The HFLTSs (\({H}_{S}\)) proposed by Rodriguez and colleagues [27] aim to overcome the hesitations of decision makers in the evaluation process. \({H}_{S}\) is defined as a ordered and finite subset of the linguistic term set \(S=\{{s}_{0},{s}_{1},\dots ,{s}_{g}\}\). Linguistic evaluations expressed by decision makers using context-free grammar are converted into HFLTSs with the transformation function. HFSs developed with linguistic expressions allow decision makers to make their evaluations with a wide set of linguistic expressions. Thus, HFLTS provides an opportunity to be used in various scientific fields with different approaches [26, 28,29,30].

Fuzzy AHP based on hesitant fuzzy linguistic evaluation

The pairwise comparisons made by the decision makers with the crisp values defined by Saaty [2] limits the decision makers to make an accurate assessment in the AHP technique. The use of linguistic expressions [31] in pairwise comparisons allows the decision makers to make a more realistic and natural assessments. Nevertheless, the linguistic form of evaluations based on the knowledge and experience of the decision maker involves a hesitant situation. Therefore, pairwise comparisons are defined with hesitant fuzzy linguistic term sets (HFLTS) [32] such as “at most strongly more important”, “between equally important and very strongly more important”, and “lower that moderately more important” to better reflect uncertain and hesitant evaluations of decision makers.

The fuzzy envelope method is implemented using the Ordered Weighted Average (OWA) aggregation operator [33] which aggregates the hesitant fuzzy expressions. The fuzzy envelop form of HFLTS represents a linguistic interval defined by the linguistic upper bound and lower bound. The application steps of the AHP decision-making technique based on hesitant linguistic evaluations are as follows;

  • The decision goal, the criteria for the goal, and the alternatives to be evaluated according to the criteria is defined in the hierarchical model. Criteria and alternatives based on criteria are evaluated in the pairwise comparison matrices. Evaluations are expressed linguistically based on the knowledge and experience of decision makers. Hesitant linguistic evaluations that are words in a natural language are generated by context-free grammar,\({ G}_{H}\). A context-free grammar is defined based on the extension of the Backus-Naur Form [32]. HFLTS is defined as a finite subset of consecutive linguistic terms in the linguistic term set S = {equally important (EI), low important (LI), medium important (MI), strongly important (SI), and absolute important (AI)}. Membership functions of linguistic terms defined in triangular form are normal and their fuzzy values are shown in Table 1 with respect to linguistic terms respectively. Sample hesitant linguistic expressions generated by \({G}_{H}\) are defined as; at least low important, greater than equally important, and between medium important and absolute important. Experts specify the inverse relations in pairwise evaluations by putting the word “(–)” symbol before the linguistic expression such that (–) lower than EI, (–) between EI and MI (Table 4).

  • Comparative linguistic evaluations of experts are converted to HFLTS (Hesitant Fuzzy Linguistic Term Sets, \({H}_{S}\)) using transformation functions (\({E}_{{G}_{H}}\)) to include in decision-making calculations [32]. The functions used to transform hesitant linguistic expressions to HFLTS are as follows [32] (Table 5):

$${E}_{{G}_{H}}\left({s}_{i}\right)=\left\{{s}_{i}|{s}_{i}\in S\right\}$$
$$ \begin{aligned}&{E}_{{G}_{H}}\left(\mathrm{a}\mathrm{t} \;\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{t}\; {s}_{i}\right)=\left\{{s}_{j}|{s}_{j}\in S \mathrm{a}\mathrm{n}\mathrm{d}\; {s}_{j}\ge {s}_{i}\right\}\; \mathrm{o}\mathrm{r}\; \\&{E}_{{G}_{H}}\left(\mathrm{a}\mathrm{t} \mathrm{m}\mathrm{o}\mathrm{s}\mathrm{t}\; {s}_{i}\right)=\left\{{\mathrm{s}}_{j}|{s}_{j}\in \;S\; \mathrm{a}\mathrm{n}\mathrm{d}\; {s}_{j}\le {s}_{i}\right\}\end{aligned} $$
$$ \begin{aligned}&{E}_{{G}_{H}}\left(\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}\; \mathrm{t}\mathrm{h}\mathrm{a}\mathrm{n}\; {s}_{i}\right)=\left\{{s}_{j}|{s}_{j}\in \;S \;\mathrm{a}\mathrm{n}\mathrm{d}\; {s}_{j}<{s}_{i}\right\}\; \mathrm{o}\mathrm{r} \;\\ &{E}_{{G}_{H}}\left(\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}\; \mathrm{t}\mathrm{h}\mathrm{a}\mathrm{n}\; {s}_{i}\right)=\left\{{s}_{j}|{s}_{j}\in\; S \;\mathrm{a}\mathrm{n}\mathrm{d}\; {s}_{j}>{s}_{i}\right\}\end{aligned} $$
$${E}_{{G}_{H}}\left(\mathrm{b}\mathrm{e}\mathrm{t}\mathrm{w}\mathrm{e}\mathrm{e}\mathrm{n}\; {s}_{i}\; \mathrm{a}\mathrm{n}\mathrm{d}\; {s}_{j}\right)=\left\{{s}_{k}|{s}_{k}\in \;S \;\mathrm{a}\mathrm{n}\mathrm{d} \;{s}_{i}\le {s}_{k}\le {s}_{j}\right\}$$

where \(S =\{{s}_{0} : EI, {s}_{1} : LI, {s}_{2}: MI, {s}_{3}:SI, {s}_{4} :\mathrm{A}I\}\) is a linguistic term and sample transformation operations are as follows:

$$ \begin{aligned}& {E}_{{G}_{H}}(\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}\; \mathrm{t}\mathrm{h}\mathrm{a}\mathrm{n} \;\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{u}\mathrm{m} \;\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}) \\ &\quad= \{\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{l} \mathrm{i}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}, \mathrm{l}\mathrm{o}\mathrm{w} \;\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\}\end{aligned} $$
$$ \begin{aligned}&{E}_{{G}_{H}}(\mathrm{a}\mathrm{t} \;\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{t}\; \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{g}\; \mathrm{i}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t})\\ &\quad = \{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{g}\; \mathrm{i}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}, \mathrm{a}\mathrm{b}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{e}\; \mathrm{i}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\}\end{aligned} $$
$$ \begin{aligned}&{E}_{{G}_{H}}(\mathrm{b}\mathrm{e}\mathrm{t}\mathrm{w}\mathrm{e}\mathrm{e}\mathrm{n}\; \mathrm{l}\mathrm{o}\mathrm{w} \;\mathrm{a}\mathrm{n}\mathrm{d} \;\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{g}\; \mathrm{i}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t})\\ &\quad = \{\mathrm{l}\mathrm{o}\mathrm{w}\; \mathrm{i}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}, \mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{u}\mathrm{m}\; \mathrm{i}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}, \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{g} \;\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\}\end{aligned} $$

  • Envelope of HFLTS (\(\mathrm{e}\mathrm{n}\mathrm{v}\left({H}_{S}\right)\)), which defines linguistic expressions with linguistic intervals, allows for easier comparison of HFLTS. The enveloping method, expressed by the lower (\({H}_{{S}^{-}}\)) and upper (\({H}_{{S}^{+}}\)) limits of the linguistic expression, is shown as follows:

    $$\mathrm{e}\mathrm{n}\mathrm{v}\;\left({H}_{S}\right)=\left[{H}_{{S}^{-}},{H}_{{S}^{+}}\right], {H}_{{S}^{-}}\le {H}_{{S}^{+}}$$
Table 4 Sample linguistic pairwise comparison matrix
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Table 5 Sample HFLTS of linguistic pairwise comparisons
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In this study, fuzzy envelopes of HFLTS are defined using trapezoidal fuzzy membership functions, \(\stackrel{\sim }{A}=(a, b, c, d)\). The parameters of the trapezoidal fuzzy number are determined using OWA aggregation operations [33, 34]. Actually, the OWA aggregation operator is used to find intermediate parameters (b and c) of the trapezoidal fuzzy membership function (Table 6). The parameters of the trapezoidal membership functions obtained by the OWA aggregation process represent the fuzzy envelope of the linguistic expression.

  • The obtained trapezoidal membership functions are defuzzified and crisp values are obtained for weight calculations. The consistency of the decisions is controlled according to the consistency ratio (CR) of the crisp pairwise comparison matrices of the criteria and alternatives with respect to the criteria (Table 7).

  • The priority values (weights) for the criteria and alternatives are calculated depending on the trapezoidal fuzzy numbers and defuzzified values in the pairwise comparison tables. Alternatives’ weights based on the criteria are synthesized with weights of the criteria and general weight (composition impact) values are obtained for the alternatives. The overall composition impact value is calculated for each alternative and the alternatives are sorted according to their impact values. The alternative with the highest combined effect value is preferred as the most suitable solution.

Table 6 Sample fuzzy envelope of HFLTS of linguistic pairwise comparisons
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Table 7 Sample defuzzification of trapezoidal fuzzy envelope using Eq. 1
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In the AHP decision model, the hesitant fuzzy linguistic term set is used as an assessment tool to enable decision makers to reflect their thoughts more comfortably in evaluations. Since the study was developed based on the ordinary AHP model, the applied method has a similar mathematical operation process as current studies [35, 36]. In contrast, the proposed set of terms for describing hesitant fuzzy linguistic expressions differs from existing methods. In this study, alternative PV solar energy projects are evaluated by AHP method based on hesitant fuzzy linguistic evaluations and the most suitable project is preferred.

An application: solar energy plant project selection

Determining the most suitable project among alternative PV solar energy installation projects is discussed as the main goal of the decision problem in the application phase. The decision group was formed by eight academicians and seven sector managers working on renewable energy and solar energy issues. Thirteen sub-criteria are defined by three basic criteria as technical (T), economic (E) and political (P). Operation management (OM,), technical infrastructure (TI), solar irradiance (SI), system technology (ST), plant location (PL), contingency plan (CP) and maintenance and repair (MR) are technical criterion; energy price change (EPC), value change of money (VCM), power demand (PD) and project finance (PF) are economic criterion; and permission (PM) and energy policy (EP) are technical criterion [37,38,39,40] (Table 8). The problem is modeled hierarchically based on the defined goal, criteria and alternatives (Fig. 1).

Table 8 Main and sub-criteria that affect to selection of PV solar energy plant
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Fig. 1
figure 1

Hierarchy model for project selection

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Decision makers evaluate the criteria and alternative projects with respect to sub-criteria linguistically by pairwise comparison according to Table 8 to collect the evaluation data. Criteria defined for PV power plant projects are evaluated by experts using linguistic expressions in pairwise comparison matrix. For example, the comparison between PM and EP criteria is linguistically defined as “at least SI” and the fuzzy envelope is obtained in the form of a trapezoidal fuzzy membership function as \(\stackrel{-}{A}\)=(1, 1.67, 5.67, 9) using OWA operators. The linguistic comparison evaluations for criteria and alternative projects are transformed into a trapezoidal fuzzy membership function. Then the trapezoidal fuzzy forms are defuzzified into crisp values using the weighted average method and the AHP decision is applied. The hesitant linguistic evaluations, trapezoidal fuzzy numbers of HFLTS, defuzzified representations, and consistency control of pairwise comparison matrices for criteria and alternative projects with respect to criteria are given between Tables 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, and 25. While the criteria are evaluated comparatively, it is desirable to know the impact weights of the criteria. The criteria are evaluated linguistically by 15 experts according to Table 3 and the number of linguistic evaluations for each criterion is shown in Table 26.

Table 9 Hesitant linguistic pairwise comparison matrix of criteria
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Table 10 Trapezoidal membership functions of HFLTS of hesitant linguistic comparison
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Table 11 Defuzzified form of trapezoidal membership functions of comparison matrix and consistency
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Table 12 Weight calculation of comparison matrix of criteria and normalized form of weight
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Table 13 Hesitant linguistic pairwise comparison matrix, trapezoidal membership functions of HFLTS, fuzzy weight vector, consistency and normalized weights of alternatives based on permission criterion
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Table 14 Hesitant linguistic pairwise comparison matrix, trapezoidal membership functions of HFLTS, fuzzy weight vector, consistency and normalized weights of alternatives based on energy policy criterion
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Table 15 Hesitant linguistic pairwise comparison matrix, trapezoidal membership functions of HFLTS, fuzzy weight vector, consistency and normalized weights of alternatives based on energy price change criterion
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Table 16 Hesitant linguistic pairwise comparison matrix, trapezoidal membership functions of HFLTS, fuzzy weight vector, consistency and normalized weights of alternatives based on value change of money criterion
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Table 17 Hesitant linguistic pairwise comparison matrix, trapezoidal membership functions of HFLTS, fuzzy weight vector, consistency and normalized weights of alternatives based on power demand criterion
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Table 18 Hesitant linguistic pairwise comparison matrix, trapezoidal membership functions of HFLTS, fuzzy weight vector, consistency and normalized weights of alternatives based on project finance criterion
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Table 19 Hesitant linguistic pairwise comparison matrix, trapezoidal membership functions of HFLTS, fuzzy weight vector, consistency and normalized weights of alternatives based on operation management criterion
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Table 20 Hesitant linguistic pairwise comparison matrix, trapezoidal membership functions of HFLTS, fuzzy weight vector, consistency and normalized weights of alternatives based on technical infrastructure criterion
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Table 21 Hesitant linguistic pairwise comparison matrix, trapezoidal membership functions of HFLTS, fuzzy weight vector, consistency and normalized weights of alternatives based on solar irradiance criterion
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Table 22 Hesitant linguistic pairwise comparison matrix, trapezoidal membership functions of HFLTS, fuzzy weight vector, consistency and normalized weights of alternatives based on system technology criterion
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Table 23 Hesitant linguistic pairwise comparison matrix, trapezoidal membership functions of HFLTS, fuzzy weight vector, consistency and normalized weights of alternatives based on plant location criterion
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Table 24 Hesitant linguistic pairwise comparison matrix, trapezoidal membership functions of HFLTS, fuzzy weight vector, consistency and normalized weights of alternatives based on contingency plan criterion
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Table 25 Hesitant linguistic pairwise comparison matrix, trapezoidal membership functions of HFLTS, fuzzy weight vector, consistency and normalized weights of alternatives based on maintenance and repair criterion
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Table 26 Experts’ evaluation for criteria and their impact weights
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The consistency values of the comparison matrices are calculated before starting the evaluation process. Since the consistency ratio based on consistency indexes is smaller than 0.1 (\(\mathrm{C}\mathrm{R}<0.1\)) as shown in the tables, all comparison matrices are said to be consistent.

The pairwise comparison matrix of the criteria defined by the hesitant fuzzy linguistic expressions is converted into trapezoidal fuzzy numbers with HFLTS and OWA operations respectively. These trapezoidal fuzzy numbers are converted to real numbers using the defuzzification method defined in Eq. 1. The fuzzy weight vector and weights of the criteria are calculated based on the defuzzified comparison matrix (Table 12).

While the most effective criteria are system technology, energy policy, solar irradiance and energy price change respectively, the weakest criteria are maintenance and repair, contingency plan and power demand respectively according to the normalized weight of the criteria.

Alternative solar energy projects are evaluated using hesitant fuzzy linguistic expressions in the pairwise comparison matrix for each criterion. Hesitant linguistic expressions are converted to HFLTS and trapezoidal fuzzy numbers with the help of the OWA aggregation process respectively.

Trapezoidal fuzzy numbers are converted to actual values and the consistency of the comparison matrix of the projects is measured. The comparison matrices of the projects based on criteria appear to be consistent, and the calculation results are shown in Tables 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 and 25. The fuzzy weight vectors are calculated by the trapezoidal fuzzy numbers obtained from the comparison matrix of projects with respect to the criteria and the weight values of the alternative projects are determined for each criterion. The weight values obtained for alternative projects under the criteria are combined with the weight values of the criteria and the general priority values are calculated for alternative projects (Table 27). The priorities of the projects differ according to the criteria, for example, the first project has the highest priority value (0.489) according to the CVM criterion, whereas it has the lowest priority value (0.063) according to the PD criterion.

Table 27 Calculation of alternative projects’ overall priority and order the alternatives
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In addition to the comparative evaluation of the criteria, the independent impact weights of the criteria are also calculated. Decision makers evaluate criteria linguistically using linguistic expressions as defined in Table 26 and the number of linguistic values defined for each criterion is shown in Table 26. The fuzzy evaluation vectors are obtained for the criteria based on the number of decision makers’ evaluations defined for each criterion and the trapezoidal fuzzy number of linguistic expressions from Table 3. The weight values obtained from the pairwise comparison matrix are combined with the fuzzy evaluation vectors according to the weight calculation process and the general fuzzy weight vectors are obtained for the criteria (Table 26). The ranking of the weight values of the criteria obtained from the evaluation process and the weight values of the criteria obtained from pairwise comparison matrix are the same. The general weight values obtained from evaluation process of criteria are used in the general priority calculations of alternative projects.

In the final stage of the decision-making process, the impact values of the criteria and the priority values of alternative solar energy projects for each criterion are brought together. Calculations are made separately for each project. The priority value of a project by a criterion and the weight value of that criterion are multiplied and this process is repeated under all criteria for the defined project. The values obtained are summed and the priority value (composition impact) is calculated for the defined project as shown in Table 27. The obtained AHP output defines the general priority values of the PV project alternatives as 0.189, 0.23, 0.287, 0.135, 0.158 respectively. The third project option with the highest overall priority value is selected as the most appropriate project.

In summary, alternative solar power plant projects are evaluated using the AHP model based on HFLTS. Factors used in the selection of the solar power plant are compared by experts using HFLTSs. System technology (0.18), Energy Policy (0.15), Energy price change (0.13), Solar irradiance (0.13) and Technical infrastructure (0.11) factors are defined as the most effective factors and Maintenance and repair (0.012), Contingency plan (0.016), Power demand (0.02) and Operation management (0.03) are defined as the weakest factors according to expert evaluations. Taking into consideration the effective and ineffective factors, the latent activities that direct managerial activities and investments can be defined as follows: technological developments increase system efficiency, government supports and incentives direct investors, price changes in alternative energy sources affect consumer behavior, geographical location and environmental factors maximize solar radiation reaching the system, technical infrastructure competence provides ease of system installation and energy distribution, high-tech system ignores maintenance, repair, operational management and emergency activities and the prevalence of alternative energy sources gives consumers choice flexibility. Alternative solar energy projects are evaluated under each factor by pairwise comparison and the project that prevails over other projects under effective factors is preferred in the overall evaluation (Table 27).

Comparison and sensitivity analyses

A comparison and sensitivity analysis is performed to evaluate the suitability of the HFLTS based AHP decision-making model and the consistency of the solar power plant selection application. Decision makers evaluate alternative projects under the same criteria according to Buckley’s AHP and Chang’s extent analysis methods. Expert evaluations are processed under Buckley’s AHP and Chang’s extent analysis methods, and the general priority values of alternative projects for each method are shown in Fig. 2.

Fig. 2
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General priority values of alternative projects according to AHP based methods

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The order of alternative projects is the same in all methods and the third project is preferred as the best alternative (Fig. 2). Chang’s extent analysis method reveals the distinctions among alternative projects more clearly. HFL-AHP and Buckley’s AHP methods show more similar breaks and differences among alternative projects are clearly reflected in these methods. The fact that the order of preference of alternative projects is the same in all methods indicates that AHP decision-making method is an effective and consistent decision-making tool. The difference among the priority values of the projects is due to the different evaluation scales, pairwise comparison methods and aggregation tools.

Sensitivity analysis is performed to observe the effect degrees of the factors used in the selection of the solar energy project (Fig. 3). While the weight values of PM, EP, PD, PF, ST factors increase, the preference order of alternative projects does not change, and the 3rd and 2nd alternatives are more clearly distinguished from other alternatives.

Fig. 3
figure 3

Sensitivity analysis reflecting the effects of factors

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As the weight values of EPC and VCM factors increase, the order of choice of alternative projects differs. These factors have a significant impact on project 1, and change is more limited in projects 2 and 5. Increasing the weight value of OM, TI, SI, MR factors causes significant changes on the order of preference of 1st, 5th and 4th projects. These factors do not have a significant impact on the preferences of the 3rd and 2nd projects. As the weight value of PL and CP factors increases, alternative projects 3 and 2 appear more prominently, and the order of alternatives 1 and 5 does not change. These results show that the identified factors have important effects in evaluating alternative projects. Therefore, factors should be correctly defined and comparative evaluations should be done carefully by experts.

Conclusions

In this study, it is aimed to select the most suitable PV solar energy plant project among the alternative projects affected by economic, political and technical factors. The evaluation of alternative projects under different criteria has led to the preference of the AHP MCDM model. Criteria and alternatives cannot be evaluated with exact numerical values, so evaluations are made linguistically by experts. Because evaluations based on the knowledge and experience of the experts involve uncertainty and hesitancy, evaluations are made on the basis of hesitant fuzzy linguistic terms.

The hesitant fuzzy linguistic expressions used in the comparison of criteria and alternative projects are converted to HFLTS and to the trapezoidal fuzzy numbers through the OWA aggregation operation. The consistency of the comparison matrices is measured by defuzzifying the trapezoidal fuzzy numbers derived from the hesitant linguistic terms. Trapezoidal fuzzy numbers are also converted to fuzzy weight vectors to calculate the weight values of criteria and alternative projects. Criteria are evaluated singularly with a set of linguistic expressions and trapezoidal fuzzy numbers are generated. These trapezoidal fuzzy numbers are combined with the comparative weight values of the criteria and the overall weight values are generated for the criteria. The priority value of the alternative projects according to the criteria and the weight vector of the criteria are evaluated together and the overall priority values (composition impact) of the alternative projects are calculated. The project with the highest overall priority value is preferred.

According to this study, while the effect of “system technology”, “energy policy”, “solar irradiance” and “energy price change” criteria is strong, the effect of “power demand”, “contingency plan” and “maintenance and repair” criteria is weak in the selection of solar energy projects. The projects have different priorities according to the criteria. According to the criteria, the third project has the highest priority value with five times, while the other projects get the highest priority value with two times. However, the first project has the lowest priority value with seven times, while the fourth project that is the last in the overall evaluation does not have the lowest priority value. In the general evaluation, while the third project preferred according to the criteria with high impact values is the first in the ranking, the fourth project preferred according to the criteria with low impact values is the last. This result demonstrates the importance of the impact weights of criteria on the decision-making process. In future studies, it is planned to develop the defined criteria and to develop decision-making models based on linguistic evaluations for renewable energy projects.

Factors affecting the selection of solar energy system obtained by literature review are an important source for academic and sector studies. However, the impact values of the factors may change due to geographical location, environmental, social, political and economic differences. Therefore, the factors and expert evaluations defined in the application section cannot be used for each problem. In addition, the usefulness of the HFLTS evaluation method, the validity of AHP decision-making tools and the success of using these tools together are gained in literature. The fact that a large number of decision makers are involved in the decision-making process and that the scope of HFLTS is insufficient in the evaluation process are important limitations for the study.

In the future studies, the new method should be developed according to the size of the decision makers and the decision-making criteria used by the decision makers. The study applied for solar energy systems can be extended to other renewable energy systems. Group decision-making methods based on linguistic interval-valued intuitionistic fuzzy set and hybrid models can be used in the development process [21, 25, 41]. The developed methods can be used in the selection of the most suitable renewable energy systems and the validity of the methods can be verified with comparative analysis.