Application of improved fuzzy best worst analytic hierarchy process on renewable energy

Introduction

Decision-making suggests the mining of appropriate different from a collection of indicators (Plous, 1993). DM is termed as Multi-Criteria Decision-Making (MCDM) that is applied in several fields, like management, economics, and engineering (Stewart, 1992; Zavadskas & Turskis, 2011; Wallenius et al., 2008). Since 1960, more than a few MCDM Techniques have been developed, proposed and implemented successfully in many application areas. The MCDM ways conjointly utilized in the fuzzy surroundings as a result of indistinctness in human thinking, complexity, therefore the high aptitude of fuzzy data to replicate decision data (Guo & Zhao, 2017). Thus, researchers are paying attention to use fuzzy-based MCDM tools for example fuzzy Data Envelopment Analysis (DEA) (Rufuss et al., 2018; Jafarzadeh, Akbari & Abedin, 2018), fuzzy AHP (Rufuss et al., 2018; Nazari et al., 2018; Jeou-Shyan et al., 2011; Ahmed & Kilic, 2019), fuzzy VIKOR (VlseKriterijumska Optimizcija I Kaompromisno Resenje in Serbian) (Yager, 2017; Mandal et al., 2015; Tadić, Zečević & Krstić, 2014), fuzzy ELECTRE (Elimination and Choice Translating algorithm) (Tadić, Zečević & Krstić, 2014; Peng et al., 2015), fuzzy ANP (Analytic Network Process) (Tadić, Zečević & Krstić, 2014; Mikhailov & Singh, 2003) and fuzzy TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) (Han & Trimi, 2018; Dwivedi, Srivastava & Srivastava, 2018; Chen & Hong, 2014).

A very well known MCDM method called as BWM was presented in 2015 (Rezaei, 2015). The importance of inconsistency assists the decision-makers (DMs) to experience if the last choices are complete properly or not, and also to find out how consistent the opinions are (Pohekar & Ramachandran, 2004). The easiness, correctness and fewer redundant make BWM more useful (Rezaei, 2015). Meanwhile in 2017, fuzzy-based MCDM methods known as fuzzy BWM have been proposed by Guo & Zhao (2017). Recently, various authors showed their interest on BWM method in fuzzy environment such as Ghoushchi, Yousefi & Khazaeili (2019), Maghsoodi et al. (2019) and Tian, Wang & Zhang (2018). Ahmed et al. (2017) utilized the fuzzy partitioning method in the Crone’s disease classification. Azar et al. (2013) used fuzzy linguistic hedges for the selection for differential diagnosis of Erythemato-Squamous diseases. Balas & Balas (2008) explained the usefulness of giving controllers with various amounts of world knowledge: general knowledge on system theory, specific information on the processes, etc.

A triangular or trapezoidal fuzzy number is typically focussed to provide the decision groups perception of indicators regarding every criteria. In fact, a triangular fuzzy number could be a special case of a trapezoidal fuzzy number (Zheng et al., 2012). Once the two important values are an equivalent number, the trapezoidal fuzzy number becomes a triangular fuzzy number (Zheng et al., 2012). From these literatures, it can be said that a trapezoidal fuzzy number is most suitable than triangular fuzzy number.

Literature review

Making decision brings out the suitable alternative from a group of alternatives. It is done on the basis of various criteria, named as MCDM, appropriate to various fields, like engineering, economics, management, etc. (Stewart, 1992; Zavadskas & Turskis, 2011; Wallenius et al., 2008). The accustomed method of the MCDM strategies depends on the ranking of all the alternatives and selecting the superior one by an approach in the light of some criteria. There are a variety of MCDM techniques which have been used by scientists, economists, mathematicians to solve the real life problems. One of such technique is AHP which is used for frequent applications in multiple sectors of economics, politics, and engineering that was designed by Saaty (1980). Regardless of its widespread applicability, the AHP method has various limitations: firstly, it uses a big number of pairwise comparisons to come to a decision. This condition clogs its application to the most essential issues. Secondly, it has a lack of consistency. So, it is required to check the CR as the CR must be less than 1 for the method to be consistent.

The technique BWM is considered to prevail over the difficulties of AHP. Youssef (2020) proposed an approach that incorporates TOPSIS and the BWM for the ranking CSPs using evaluation criteria characterizing their services. Knowing of the fewer pairwise comparison and also the excessive consistency of the matrix of pairwise comparison within the BWM than that in AHP, the BWM will be as well-liked as AHP almost immediately following. Compared to AHP, BWM only performs reference comparisons, meaning that it only has to determine the priority of the best criterion over all the other criteria and the priority of all the criteria over the worst criterion. Thus BWM significantly increase the overall consistency of the problem compared to AHP. Moreover, this technique has some disadvantages such as vagueness in human opinions and uncertainty in the information related to the criteria (Zhao & Guo, 2015).

Trapezoidal fuzzy best-worst analytic hierarchy process

The proposed method namely Trapezoidal fuzzy Best Worst Analytic Hierarchy (TrFBWAHP) consists of two phases respectively phase I and phase II. Phase I is used to evaluate PV of criteria and Phase II is utilized to determine the PV of indicators. Figure 1 represents computational process of proposed method.

Phase-1: Best–Worst Method (BWM) is a novel method to evaluate the PV of criteria. In this study, a modified fuzzy BWM approach namely Trapezoidal Best-West Method (TrFBWM) to evaluate the PV of each criterion is proposed. The TrFBWM techniques consist of five steps. The evaluation steps are given in below:

Step-1: Define decision making Criteria: In MCDM techniques criteria are significant for selection of most important indicator. Let $C=\left\{c_1,c_2,...,c_n\right\}$ and $A=\left\{A_1,A_2,...,A_p\right\}$ denote the sets of criteria and indicator respectively. The performance of each indicator is depending on the set of elements of C.

So, $PV(A_i)=f_i\left(c_1,c_2,...,c_n\right);\mathrm{\forall }i=1,2,...,p$

Step-2: Selection of Best and Worst Criteria: Next step decision maker (DM) selects the best and worst criteria among the sets of criteria. DM may be selecting these best and worst criteria with help different survey like expert, literature, media survey etc. Let $c_B$ and $c_w$ be the best and worst criteria respectively. Then $c_B,c_w\in C$.

Step-3: Trapezoidal Fuzzy Pairwise Comparison: Total Pairwise comparison matrix is not requiring. In Step-3 pairwise comparison are divided into two parts which are described in step 3.1 and 3.2. In Table 1, Trapezoidal fuzzy pairwise comparisons between criteria have done by using Linguistic measures of importance. Thus convert each Linguistic term into trapezoidal fuzzy measures of importance which is shown in below Table 1.

Step-3.1: Trapezoidal Fuzzy Comparison of each Criterion with respect to Best criteria: In this step we compare each criteria of set C with respect to selected best criteria $c_B\in C$ and this is described by Table 2 in below. Here ${\hat{x}}_{Bi}(i=1,2,...,n)$ takes any one of trapezoidal fuzzy measure of importance from Table 1 and ${\hat{x}}_{BB}=\left(1,1,1,1\right)$.

Step-3.2: Trapezoidal Fuzzy Comparison of each Criterion with respect to Worst criteria: In this step we compare the worst criteria $c_w\in C$ with respect to every criteria of setC and this is depicted by Table 3 in below. Here, ${\hat{x}}_{iw}(i=1,2,...,n)$ takes any one of trapezoidal fuzzy measure of importance from Table 1 and ${\hat{x}}_{ww}=\left(1,1,1,1\right)$.

Step-4: Optimal Trapezoidal Fuzzy PV: In this step determine the optimal trapezoidal fuzzy PV. Let ${\hat{w}}_1,{\hat{w}}_2,...,{\hat{w}}_n$ be the PV of criteria $c_1,c_2,...,c_n$ respectively and ${\hat{w}}_1,{\hat{w}}_2,...,{\hat{w}}_n$ are the decision variable of the optimization techniques. Let ${\hat{w}}_B$ and ${\hat{w}}_w$ be the priority values (PVs) of the criteria $c_B$ and $c_w$ respectively.

For optimal trapezoidal fuzzy PV two ratios ${\displaystyle \frac{{\hat{w}}_B}{{\hat{w}}_i}}$ and ${\displaystyle \frac{{\hat{w}}_i}{{\hat{w}}_w}}$ $\left(i=1,2,...,n\right)$ satisfies ${\displaystyle \frac{{\hat{w}}_B}{{\hat{w}}_i}-{\hat{x}}_{Bi}=0}$ and ${\displaystyle \frac{{\hat{w}}_i}{{\hat{w}}_w}-{\hat{x}}_{iw}=0}$, $\mathrm{\forall }i=1,2,...,n$ and it is possible for all $i$, if it should evaluate the minimum of maximum gaps of $\left|{\displaystyle \frac{{\hat{w}}_B}{{\hat{w}}_i}-{\hat{x}}_{Bi}}\right|$ and $\left|{\displaystyle \frac{{\hat{w}}_i}{{\hat{w}}_w}-{\hat{x}}_{iw}}\right|$, $\mathrm{\forall }i=1,2,...,n$. Here all this PV in TrFBWAHP are Trapezoidal fuzzy sets (TrFS). Some time, we suggest to use ${\hat{w}}_i=\left(l_{{\hat{w}}_i},m_{{\hat{w}}_i},n_{{\hat{w}}_i},s_{{\hat{w}}_i}\right)$, $i=1,2,...,n$ for optimal indicator selection. Thus the optimization problem can be written as

(1) $$\begin{array}{c}min\underset{i}{max}\left\{\left|{\displaystyle \frac{{\hat{w}}_B}{{\hat{w}}_i}-{\hat{x}}_{Bi}}\right|,\left|{\displaystyle \frac{{\hat{w}}_i}{{\hat{w}}_w}-{\hat{x}}_{iw}}\right|\right\}\\ \mathrm{S}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}\sum _{i=1}^{n}R\left({\hat{w}}_i\right)=1\\ l_{{\hat{w}}_i}\le m_{{\hat{w}}_i}\le n_{{\hat{w}}_i}\le s_{{\hat{w}}_i},\mathrm{\forall }i=1,2,\dots ,n\\ l_{{\hat{w}}_i}\ge 0,\mathrm{\forall }i=1,2,\dots ,n\end{array}\}$$where ${\hat{w}}_B=\left(l_{{\hat{w}}_B},m_{{\hat{w}}_B},n_{{\hat{w}}_B},s_{{\hat{w}}_B}\right)$, ${\hat{w}}_i=\left(l_{{\hat{w}}_i},m_{{\hat{w}}_i},n_{{\hat{w}}_i},s_{{\hat{w}}_i}\right)$, ${\hat{w}}_w=\left(l_{{\hat{w}}_w},m_{{\hat{w}}_w},n_{{\hat{w}}_w},s_{{\hat{w}}_w}\right)$, ${\hat{x}}_{Bi}=\left(l_{{\hat{x}}_{Bi}},m_{{\hat{x}}_{Bi}},n_{{\hat{x}}_{Bi}},s_{{\hat{x}}_{Bi}}\right)$ and ${\hat{x}}_{Wi}=\left(l_{{\hat{x}}_{Wi}},m_{{\hat{x}}_{Wi}},n_{{\hat{x}}_{Wi}},s_{{\hat{x}}_{Wi}}\right)$.

Next we convert the optimization problem (1) into a non-linear optimization. The transformed nonlinear form of the Eq. (1) is of the form

(2) $$\begin{array}{c}min\hat{\zeta }\\ \mathrm{S}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}\left|{\displaystyle \frac{{\hat{w}}_B}{{\hat{w}}_i}-{\hat{x}}_{Bi}}\right|\le \hat{\zeta }\\ \left|{\displaystyle \frac{{\hat{w}}_i}{{\hat{w}}_w}-{\hat{x}}_{iw}}\right|\le \hat{\zeta }\\ \sum _{i=1}^{n}R\left({\hat{w}}_i\right)=1\\ l_{{\hat{w}}_i}\le m_{{\hat{w}}_i}\le n_{{\hat{w}}_i}\le s_{{\hat{w}}_i},\mathrm{\forall }i=1,2,\dots ,n\\ l_{{\hat{w}}_i}\ge 0,\mathrm{\forall }i=1,2,\dots ,n\end{array}\}$$where, $\hat{\zeta }=\left(l_{\hat{\zeta }},m_{\hat{\zeta }},n_{\hat{\zeta }},s_{\hat{\zeta }}\right)$

where $\hat{\zeta }$ is a trapezoidal fuzzy number thus $l_{\hat{\zeta }}\le m_{\hat{\zeta }}\le n_{\hat{\zeta }}\le s_{\hat{\zeta }}$.

Now we consider the equality condition among all the components of the trapezoidal fuzzy number to minimize the value of the objective function. This equal value is defined as $\lambda (\ge 0)$, where $\lambda =l_{\hat{\zeta }}=m_{\hat{\zeta }}=n_{\hat{\zeta }}=s_{\hat{\zeta }}$

Therefore $\hat{\zeta }=\left(\lambda ,\lambda ,\lambda ,\lambda \right)$. Thus the transformed equation of (2) is

(3) $$\begin{array}{c}min\left(\lambda ,\lambda ,\lambda ,,\lambda \right)\hfill \\ \mathrm{S}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}\left|{\displaystyle \frac{\left(l_{{\hat{w}}_B},m_{{\hat{w}}_B},n_{{\hat{w}}_B},s_{{\hat{w}}_B}\right)}{\left({\mu }_{{\hat{w}}_i},{\nu }_{{\hat{w}}_i},{\pi }_{{\hat{w}}_i}\right)}-\left(l_{{\hat{x}}_{Bi}},m_{{\hat{x}}_{Bi}},n_{{\hat{x}}_{Bi}},s_{{\hat{x}}_{Bi}}\right)}\right|\le \left(\lambda ,\lambda ,\lambda ,\lambda \right)\hfill \\ \left|{\displaystyle \frac{\left(l_{{\hat{w}}_i},m_{{\hat{w}}_i},n_{{\hat{w}}_i},s_{{\hat{w}}_i}\right)}{\left(l_{{\hat{w}}_w},m_{{\hat{w}}_w},n_{{\hat{w}}_w},,s_{{\hat{w}}_w}\right)}-\left(l_{{\hat{x}}_{Wi}},m_{{\hat{x}}_{Wi}},n_{{\hat{x}}_{Wi}},s_{{\hat{x}}_{Wi}}\right)}\right|\le \left(\lambda ,\lambda ,\lambda ,\lambda \right)\hfill \\ \sum _{i=1}^{n}R\left({\hat{w}}_i\right)=1\hfill \\ l_{{\hat{w}}_i}\le m_{{\hat{w}}_i}\le n_{{\hat{w}}_i}\le s_{{\hat{w}}_i},\mathrm{\forall }i=1,2,...,n\hfill \\ l_{{\hat{w}}_i}\ge 0,\mathrm{\forall }i=1,2,...,n\hfill \end{array}\}$$

Solving the Eq. (3) we get the optimal trapezoidal fuzzy PV $\left({{\hat{w}}_1}^{\ast },{{\hat{w}}_2}^{\ast },...,{{\hat{w}}_n}^{\ast }\right)$ where, ${{\hat{w}}_i}^{\ast }=\left(l_{{{\hat{w}}_i}^{\ast }},m_{{{\hat{w}}_i}^{\ast }},n_{{{\hat{w}}_i}^{\ast }},s_{{{\hat{w}}_i}^{\ast }}\right),\mathrm{\forall }i=1,2,...,n$ of the criteria.

Step-5: Check Consistency ratio for Trapezoidal fuzzy BWM (TrFBWM): Consistency ratio (CR) is the most significant value to examine the degree of consistency in a pairwise comparison. In Trapezoidal fuzzy BWM, a pairwise comparison is consistent matrix if ${\hat{x}}_{Bi}\times {\hat{x}}_{iW}-{\hat{x}}_{BW}=0$. If ${\hat{x}}_{Bi}\times {\hat{x}}_{iW}-{\hat{x}}_{BW}>or<0$ then the pairwise comparison matrix is inconsistent. When ${\hat{x}}_{BW}$ is equal to both ${\hat{x}}_{Bi}$ and ${\hat{x}}_{iW}$ then the inequality will reach the greatest, which output in $\hat{\zeta }$.Here we consider the existence of inequality, as per the relation $\left({\displaystyle \frac{{\hat{w}}_B}{{\hat{w}}_i}}\right)\times \left({\displaystyle \frac{{\hat{w}}_i}{{\hat{w}}_w}}\right)-\left({\displaystyle \frac{{\hat{w}}_B}{{\hat{w}}_w}}\right)=0$, we can obtain the following Eq. (4) as

(4) $$\left({\hat{x}}_{Bi}-\hat{\zeta }\right)\times \left({\hat{x}}_{iW}-\hat{\zeta }\right)-\left({\hat{x}}_{BW}-\hat{\zeta }\right)=0$$

For the maximum trapezoidal fuzzy inconsistency ${\hat{x}}_{Bi}={\hat{x}}_{iW}={\hat{x}}_{BW}$, thus the equation transformed into

(5) $${\hat{\zeta }}^2-\left(1+2{\hat{x}}_{BW}\right)\hat{\zeta }+\left({{\hat{x}}^2}_{BW}-{\hat{x}}_{BW}\right)=0$$where, $\hat{\zeta }=\left(l_{\hat{\zeta }},m_{\hat{\zeta }},n_{\hat{\zeta }},s_{\hat{\zeta }}\right)$ is a TrFS and ${\hat{x}}_{BW}=\left(l_{BW},m_{BW},n_{BW},s_{BW}\right)$

The maximum possible trapezoidal fuzzy value ${\hat{x}}_{BW}=\left(l_{BW},m_{BW},n_{BW},s_{BW}\right)$ is $\left(8,{\displaystyle \frac{17}{2},9,9}\right)$, thus $l_{BW}=8,m_{BW}={\displaystyle \frac{17}{2},n_{BW}=9\mathrm{a}\mathrm{n}\mathrm{d}{s}_{BW}=9}$. It is clear that $max\left\{l_{BW},m_{BW},n_{BW},s_{BW}\right\}$ cannot exceed the value 9, thus the consistency index (CI) for TrFS is 13.77 (using Eq. (5)). Similarly using the same process we get the CI for TrFS. Table 4 shows CI value of TrFS.

Step-6: Crisp PV: In the final step, we convert optimal trapezoidal fuzzy PV $\left({{\hat{w}}_1}^{\ast },{{\hat{w}}_2}^{\ast },...,{{\hat{w}}_n}^{\ast }\right)$ where, ${{\hat{w}}_i}^{\ast }=\left(l_{{{\hat{w}}_i}^{\ast }},m_{{{\hat{w}}_i}^{\ast }},n_{{{\hat{w}}_i}^{\ast }},s_{{{\hat{w}}_i}^{\ast }}\right),\mathrm{\forall }i=1,2,...,n$ into crisp value using the formula (6).

(6) $$R({{\hat{w}}_i}^{\ast })={\displaystyle \frac{l_{{{\hat{w}}_i}^{\ast }}+2m_{{{\hat{w}}_i}^{\ast }}+2n_{{{\hat{w}}_i}^{\ast }}+s_{{{\hat{w}}_i}^{\ast }}}{6},\mathrm{\forall }i=1,2,...,n}$$

Step-7: Normalized PVs: Normalized PVs of criteria are calculated by using Eq. (7).

(7) $${{\hat{w}}_i}^{\ast }={\displaystyle \frac{R({{\hat{w}}_i}^{\ast })}{{\sum }_{i=1}^{n}R({{\hat{w}}_i}^{\ast })},\mathrm{\forall }i=1,2,...,n}$$

Phase-2: After determining the PV of each criterion (by TrFBWM), Trapezoidal Fuzzy Analytic Hierarchy Process (TrFAHP) is used to find the local PV of each indicator with respect to each criterion. The proposed TrFAHP techniques consist of five steps. The evaluation steps are shown as follows:

Step-1: Trapezoidal fuzzy judgement matrices: In step 1 build trapezoidal fuzzy pairwise comparison matrix of indicators with respect to each criterion $C_k\left(k=1,2,...,n\right)$ is $A_k={\left[{\hat{f}}_{ij}\left(k\right)\right]}_{p\times p}$ where ${\hat{f}}_{ij}\left(k\right)=\left(l_{ij}\left(k\right),m_{ij}\left(k\right),n_{ij}\left(k\right),s_{ij}\left(k\right)\right)$ is trapezoidal fuzzy measures of importance of ith indicator with jth indicator with help of Table 1 for kth criteria. For checking consistency of each trapezoidal fuzzy judgement matrices, we convert each trapezoidal fuzzy judgement into score indices. The SI has calculated by the Eq. (6). We convert trapezoidal fuzzy judgement into score indices and then the pairwise comparison matrix will be transform into as normal crisp pairwise comparison matrix of AHP. So consistency ratio (CR) of pairwise comparison matrix calculates normal AHP process. If CR < 1.0 of transform trapezoidal fuzzy judgement matrices by SI then there corresponding trapezoidal fuzzy judgement matrices is consistent otherwise inconsistent.

Step-2: Local PVs: Now calculate the local PVs of indicators with the help of the Eq. (8).

$$a_j\left(k\right)={\left[\prod _{j=1}^p{l}_{ij}\left(k\right)\right]}^{{\displaystyle \frac{1}{p}}},\mathrm{\forall }j=1,2,...,p$$

$$b_j\left(k\right)={\left[\prod _{j=1}^p{m}_{ij}\left(k\right)\right]}^{{\displaystyle \frac{1}{p}}},\mathrm{\forall }j=1,2,...,p$$

$$d_j\left(k\right)={\left[\prod _{j=1}^p{n}_{ij}\left(k\right)\right]}^{{\displaystyle \frac{1}{p}}},\mathrm{\forall }j=1,2,...,p$$

$$e_j\left(k\right)={\left[\prod _{j=1}^p{s}_{ij}\left(k\right)\right]}^{{\displaystyle \frac{1}{p}}},\mathrm{\forall }j=1,2,...,p$$And

$$a\left(k\right)=\sum _{j=1}^p{a}_j\left(k\right)$$

$$b\left(k\right)=\sum _{j=1}^p{b}_j\left(k\right)$$

$$d\left(k\right)=\sum _{j=1}^p{d}_j\left(k\right)$$

$$e\left(k\right)=\sum _{j=1}^p{e}_j\left(k\right)$$Then the triangular fuzzy PV of jth indicator with respect to kth criteria is defined as follows:

(8) $${\hat{w}}_{A_j}\left(k\right)=\left(a_j^{-1}e,b_j^{-1}d,d_j^{-1}b,e_j^{-1}a\right),\mathrm{\forall }j=1,2,\dots ,p$$In this way find the PVs of each indicator $A_j\left(j=1,2,...,p\right)$ as per each criterion $C_k\left(k=1,2,...,n\right)$.

Step-3: Crisp PV: Using Eq. (6), converts each fuzzy PV of each indicator $A_j\left(j=1,2,...,p\right)$ into crisp PV.

Step-4: Normalized PVs: Using Eq. (7), converts each crisp PV of each indicator $A_j\left(j=1,2,...,p\right)$ into normalized PV.

Step-5: Global PVs: Global indicators PVs are calculated by multiplying the local PV of the indicator with local PV of criteria.

Step-6: Normalized Global PVs: Finally global PVs convert into normalized form using Eq. (7).

Methodology

The objective is studying a hybrid MCDM method namely TrFBWAHP based on trapezoidal fuzzy sets. This novel approach is used on the identification of the most significant indicators of efficiency of any HPP under certain uncertainties like climate change. Currently, production of power from hydropower plants is affected because of the consequence of climatic change. The hydrology is exaggerated which sequentially changes the performance efficiency of hydropower plants. Changes in the climatic pattern are the result of boost in atmospheric concentrations of greenhouse gases occurring from human activities. A lot of greenhouse gases, such as carbon dioxide ( $C{O}_2$), present in the nature and maintain hotness of the earth, locking the heat in the atmosphere. Enlarged reservoir evaporation (Allawi et al., 2020), amount change (Hidalgo et al., 2020) and changes in pattern of river runoff (Dandapat, Gnanaseelan & Parekh, 2020) will have numerous impacts on the hydroelectric power production due to changes in the climate. These cause bad effects on other energy sectors (Akram et al., 2020), financial effects (Ma, Rogers & Zhou, 2020) and system operation (Bento, 2020).

Climate change is one of the great challenges of the 21st century. Hydropower is the central renewable and small carbon energy source generating 15% of total world electrical energy. On a global scale, the effect of climate change on hydropower energy production will have an impact, and this impact will be strong differences between the dry and wet regions. In areas where hydropower generation will decrease due to climate change impacts, entire nations may find themselves without a reliable source of electricity, which could be compensated by new power plants, an increase of their efficiency, and better water management.

Performance efficiency of a hydro power plant is defined as the efficiency with which the hydro power plant produces the power converting the mechanical energy of the water of the river flowing on the turbine. Hydropower energy is extensively utilized all over the world. This is the one renewable energy, currently used on the huge amount. For keeping the hydropower plant in excellent state, the power plant performance requires to be examined continuously (Zulkifli et al., 2015). Briongos et al. (2020) calculated the operating efficiency of a hybrid wind–hydro power plant situated in El Hierro Island. A new technique is presented to determine the round trip energy efficiency of hydropower plants connected to pumped storage along with underground lower reservoir (Menéndez et al., 2020).

Recently MCDM finds its huge applications in various decision making problems of renewable energy (Kumar et al., 2017). Majumder, Majumder & Saha (2018) utilized harmonic mean hierarchy process (HMHP) and measuring attractiveness by a categorical-based evaluation technique (MACBETH) methods as multi criteria decision-making techniques and polynomial neural networks to forecast the function which will represent the present position of the power plant. Majumder & Saha (2019) suggested a hybridized model using Decision Making Trial and Evaluation Laboratory (DEMATEL) with the Analytic Hiera-rchy Process (AHP) for estimating the plant efficiency of hydro power plant. Various types of multi-criteria decision making methods namely Fuzzy analytic hierarchy process, Fuzzy weighted sum model, Fuzzy analytic network process were used in detection of the significant indicator from a group of indicators for analyzing the performance reliability in hydropower plants (Majumder et al., 2019).

According to the literature review, it was found that the factors, Storage capacity (A₁) (Majumder, 2016), Turbine efficiency (A₂) (Hammid, Sulaiman & Abdalla, 2018), Rate of flow or discharge (A₃) (Majumder, 2016; Majumder & Saha, 2016), Capacity factor (A₄) (Sözen, Alp & Kilinc, 2012), Utilization factor (A₅) (Dinkar & Morankar, 2015), Hydraulic head (A₆) (Majumder, 2016; Baede et al., 2001) of efficiency of HPP are impacted with some climate parameters like Temperature (C₁) (Scavia et al., 2002; Farmer, 2015), Precipitation (C₂) (Trenberth, 2014; Cubasch, Voss & Mikolajewicz, 2000) and Evapo-transpiration (C₃) (Abtew & Melesse, 2013). Now we discuss about all the criteria and indicators.

Storage capacity: The HPPs with impoundments like reservoir or dams provide the facility of storage of water that can be used for electricity generation. Due to precipitation watershed runoff carries and deposit sediments of different size into the reservoir which reduces the storage capacity of the reservoir. Again increase in temperature or drought like conditions will reduce the water level and increase the salinity of the stored resource which may cause corrosion in the turbines and thereby the HPP will be enforced to work at below capacity level. That is why, more the storage capacity less will be the vulnerability and vice-versa (Majumder, 2016).

Turbine efficiency: The turbine being the core component of hydropower systems performs the function of converting the hydraulic energy into electrical energy with the help of mechanical energy (Hammid, Sulaiman & Abdalla, 2018). Thus the mechanical efficiency largely influences the efficiency of hydro power plants. That is why even if there is a large flow rate available in the river due to reduced efficiency of turbines HPP will operate below capacity level. Excess precipitation carries sediments of different sizes from the catchment. The quality of water also get effected due to the flushed out pesticides and fertilizers. The suspended sediments of specific size can damage the turbine blades. Mixing of salt in the water will increase the salinity which may corrode the blades. Both of this phenomena will decrease the turbine efficiency. That is why more the precipitation more will be vulnerability to plant production due to the probability of decreased operating efficiency of the turbines.

Rate of flow or discharge: The rate of flow or discharge of water to the turbine largely affects the power generation of a HPP. Change in the climatic conditions results in the variation of annual precipitation and thus regular pattern of stream flow also changes. The discharge depends upon rainfall and evapo-transpiration (Majumder, 2016). And climate change brings changes in precipitation and evapo-transpiration. High temperature may lead to high evaporation and thereby reducing the availability of water whereas increased precipitation may increase the discharge. Thus climate change also affects this parameter (Majumder, 2016; Majumder & Saha, 2016).Therefore climate change creates a direct impact on capacity of a hydro power plant. Thus vulnerability of HPP is inversely proportional to discharge of the river on which the HPP is installed.

Capacity factor: Capacity factor is the ratio of the energy generated with a time interval to the total power possible to be generated from HPP (Sözen, Alp & Kilinc, 2012). Climate change may result into drought and reduces the potential energy of water. Again an increased precipitation increases the water level which can be converted to produce kinetic energy in the river. A plant working at below capacity level indicates that it is working below potential and vice-versa. So, capacity factor varies inversely with vulnerability of the HPP.

Utilization factor: Utilization Factor can be defined as the ratio of power that was generated to the power that can be generated. This factor depicts the efficiency of the powerplant in utilizing the available power potential. More the utilization less will be the vulnerability. Rather than the climatic factors the efficiency of penstocks, turbines and generators effect the magnitude of utilization factor of a HPP (Dinkar & Morankar, 2015).

Hydraulic head: Hydraulic head is the measurement of liquid pressure of any static water column in terms of height of water level above any datum. The performance and electricity generation of a hydropower plant directly depends on this parameter. More the hydraulic head more will be the potential energy which can be converted to flowing water and less will be the vulnerability of HPP and vice versa (Majumder, 2016; Baede et al., 2001).

Temperature: The global temperature of earth is referred here which is a significant factor of climate change in the present world. “A rise in global average surface temperatures is the best-known indicator of a warming climate change” (Scavia et al., 2002; Farmer, 2015). Not only global temperature the regional temperature and also local heat islands can also change the climate of a location. Climatic impact can be best represented by this parameter as temperature is responsible for occurrence of precipitation, change in wind speed, humidity and even evapo-transppiration. Thus this parameter was included in the decision hierarchy as Secondary criteria which can represent the characteristic of the climatic abnormality.

Precipitation: Precipitation is a form of water dropping on the earth’s surface from the atmosphere that includes rain, snow, hail, dew, fog etc. Although it is mainly a function of temperature but the impact of precipitation in the availability of water has such a significance which merit the selection of this parameter as a Secondary criteria (Trenberth, 2014). Abrupt change in the climatic conditions is characterized by abnormal precipitation due to which water level gets altered and thereby affecting the global hydro cycle (Cubasch, Voss & Mikolajewicz, 2000). It is a parameter that effects the climate change and itself also gets modified by changing climate (Cubasch, Voss & Mikolajewicz, 2000).

Evapo-transpiration: Evapo-transpiration is defined as the movement of water from the earth’s surface to the atmosphere through evaporation and transpiration in plants. The magnitude of Evapo-transpiration is effected by variation in both temperature and rate of transpiration. Evapo-transpiration also effect the net water balance of a catchment. The energy available in the flowing water also depends on the net water availability in the reservoir or in the catchment. The rise of temperature and decrement in the rainfall increases the evapo-transpiration rate and reduces the water availability (Abtew & Melesse, 2013). Thus evapo-transpiration takes part in climate change and is also affected by the changing climate (Abtew & Melesse, 2013). That is why this parameter is also included in the decision hierarchy as secondary criteria.

In this present study, all these factors are considered as indicators and all climate parameters are considered as criteria for applying the proposed new decision-making method. Figure 2 represent the hierarchical structure of the considering decision problem. Mathematically the objective can be written by the Eq. (9).

(9) $$e=f(A,W_A)$$where, e represents efficiency of HPP, $A=\left\{A_1,A_2,A_3,A_4,A_5,A_6\right\}$ and W_A represents the set of indicator and their corresponding PV or PV of each indicator respectively.

(10) $$A_i=F\left(C,W_c\right),i=\{1,2,3,4,5,6\}$$where, A_i represents indicators, $C=\left\{C_1,C_2,C_3\right\}$ and W_C represents the set of criteria and their corresponding PV of each criteria respectively.

For finding the PV of set of criteria (W_C) and indicators (W_A), we use new MCDM approach and computational process of the new MCDM technique discussed in “Application of TrFBWAHP MCDM”. Figure 3 shows the schematic diagram of detailed methodology.