Grey Relational Analysis–Based MADM Strategy Under Possibility Environment and Its Application in the Identification of Most Important Parameter Affecting Climate Change and the Impact of Urbanization on Hydropower Plants

Abstract

This paper aims to establish the notion of Hamming distance under the possible environment and propose a multi-attribute decision-making (MADM) strategy based on grey relational analysis (GRA) under the Possibility environment. This study identified the most critical parameter or factors affected by climate change, urbanization, and machine failures. Furthermore, we validate the proposed MADM strategy by solving a real-world numerical example, namely, “identification of the most important parameters affecting climate change and the impact of urbanization on hydropower plants.”

Introduction

Two sorts of frequent uncertainty in the virtual environment are unpredictability and fuzziness. Consequently, to cope with such forms of contradiction, there are two different hypotheses: one is the Probability theory, while the other is the Possibility theory. In 1965, Zadeh (Zadeh 1965) grounded the fuzzy set (FS) notion, where every element has a membership value between 0 and 1. Later on, Zadeh (Zadeh 1978) developed the notion of the Possibility theory in 1978. In 1979, Baldwin and Pilsworth (Baldwin and Pilsworth 1979) presented the fuzzy truth definition of the Possibility measure for the decision classification. Afterward, Dubois and Prade (Dubois and Prade 1988) further studied the Possibility theory. In 1999, De Campos and Huete (Campos and Huete 1999) presented the independence concepts of the Possibility theory. Later on, Carlsson and Fuller (2001) introduced the notion of Possibilistic mean value and variance of fuzzy members. In 2017, Kocalerchuk (Kocalerchuk 2017) studied the relationships between the Probability theory and the Possibility theory. So many researchers around the globe have already proposed different models of MADM for FSs and their extensions. Notable contributions in the field of further development by Biswas and Pramanik are the fuzzy ranking method for assignment problems with fuzzy costs (Biswas and Pramanik 2011a), a fuzzy approach to replacement problems with the value of money changing with time (Biswas and Pramanik 2011b), multi-objective assignment problems with fuzzy costs in the case of military affairs (Biswas and Pramanik 2011c), and the priority-based fuzzy goal programming method under a fuzzy environment (Pramanik and Biswas 2011). In the same era, Dey et al. proposed the neutrosophic soft multi-attribute decision-making based on GRA (Dey et al. 2016a), a GRA-based MADM strategy under the intuitionistic fuzzy set environment (Dey et al. 2015a), and a MADM strategy based on extended GRA under the interval neutrosophic set environment (Dey et al. 2015b). The MADM strategy based on extended GRA under the neutrosophic uncertain linguistic set environment (Dey et al. 2016b), established the neutrosophic soft MADM approach based on GRA under the neutrosophic soft set environment (Dey et al. 2016c). Parallelly, Pramanik and Dey introduced a multi-objective quadratic programming problem under fuzzy goal programming (Pramanik and Dey 2011a), a multi-objective linear fractional programming problem under fuzzy goal programming (Pramanik and Dey 2011b), and, subsequently, a bi-level multi-objective programming problem with fuzzy parameters (Pramanik and Dey 2011c). In 2014, Md. Saad et al. (Md Saad et al. 2014) proposed a Hamming distance method with subjective and objective weights for personnel selection. In 2015, Biswas et al. (2014) presented an entropy-based GRA method for MADM strategy under the neutrosophic environment in 2014. Collected from the conclusion, the proposed MADM strategy can also be used in the area of decision-making problems such as weaver selection (Dey et al. 2015a, 2015b), personnel selection (Md Saad et al. 2014), and teacher selection (Pramanik and Dey 2011c).

Furthermore, the proposed MADM-strategy will open up a new avenue of research in the possibility environment.

Climate change and the rate of urbanization influence and unbalance the equilibrium between the demand and supply of energy resources. Hydropower plants are among the most vulnerable systems. But not all the operational parameters get influenced by this change. There have been no studies to date that attempt to determine the most significant parameters that are most affected by climate and urbanization uncertainties. As a result, in this study, we establish the notion of Hamming distance under the Possibility environment and propose a MADM strategy based on GRA to determine the parameters influencing climate change and the impact of urbanization on hydropower plants.

The rest of this paper is divided into the following sections:

The “Preliminaries” section briefly explains the preliminary concerning fuzzy set, Hamming distance, Possibility set, and their different properties. The “Hamming Distance Under Possibility Environment” section gives the notion of Hamming distance between two Possibility values. The “GRA-Based MADM Strategy Under Possibility Environment” section presents a new MADM strategy based on GRA under the Possibility environment. The “Validation of the Proposed MADM Strategy” section presents an illustrative numerical example, namely, “identification of the most important parameters affecting climate change and the impact of urbanization on hydropower plants,” to show the applicability and effectiveness of the proposed MADM strategy. The “Conclusions” section summarizes the results and discusses the future scope of research.

Preliminaries

In this section, we provide some existing definitions and results that will help in the preparation of the article’s significant results.

Definition 2.1. A fuzzy set H over a fixed set U is defined as follows:

$$H = \left\{\langle x, {\upmu }_{H}\left(x\right)\rangle : x\in U\right\}$$

where ${\mu }_{H}$(x) $\in$ [0, 1] is the membership value of x over H (Zadeh 1965).

Definition 2.2. Let G be a fuzzy subset of U, and let ∏_x be a possibility distribution associated with the variable X, which takes the values in U. The possibility measure, π(G), of G is defined by.

$$\mathrm{Poss }\left\{X \in G\right\}\triangleq\uppi \left(G\right)\triangleq {\mathrm{Sup}}_{u\in U}\left\{\upmu_{G}\left(u\right)\wedge\uppi_{x}\left(u\right)\right\}$$

where μ_G is the membership function for G (Zadeh 1978).

Definition 2.3. The possibility distribution function with respect to X (or the possibility distribution function of ∏_x) is denoted by π_x and is defined numerically equal to the membership function of fuzzy set F, i.e., π_G $\triangleq$ μ_F, where π_x(u) is the possibility that x = μ, is equal to the values of membership μ_F(u) (Zadeh 1978).

Definition 2.4. Given two fuzzy subsets of H₁ and H₂ with a reference set, $H=\left\{{h}_{1}, {h}_{2}, \dots , {h}_{n}\right\}$ and membership functions ${\upmu }_{{H}_{1}}$ and ${\upmu }_{{H}_{2}}$. Then, the Hamming distance between H₁ and H₂ is defined as (Grzegorzewski 2004).

$${d}_{HD}\left({H}_{1}, {H}_{2}\right)={\sum }_{i=1}^{n}\left|{\upmu }_{{H}_{1}}\left({h}_{i}\right)-{\upmu }_{{H}_{1}}\left({h}_{i}\right)\right|$$

Definition 2.5. The weighted Hamming distance of dimension n is a mapping ${d}_{WHD}:{\left[\mathrm{0,1}\right]}^{n}\times {\left[\mathrm{0,1}\right]}^{n}\to \left[\mathrm{0,1}\right]$ that is associated with weighted vector $\mathcal{W}$ of dimension n with $\mathcal{W}=\sum_{i=1}^{n}{w}_{i}=1,$ and ${w}_{i}$ ∈ [0, 1]. Then, the weighted Hamming distance between two H₁ and H₂ corresponding to the membership ${\upmu }_{{H}_{1}}$ and ${\upmu }_{{H}_{2}}$ is defined as (Merigo and Gil-Lafuente 2012).

$${d}_{WHD}\left({H}_{1}, {H}_{2}\right)={\sum }_{i=1}^{n}{w}_{i}.\left| {\upmu }_{{H}_{1}}\left({h}_{i}\right)-{\upmu }_{{H}_{1}}({h}_{i})\right|$$

Hamming Distance Under Possibility Environment

In this section, we procure the notion of Hamming distance under the Possibility environment and formulate several interesting results on it.

Definition 3.1. Assume that X = {(α_i, ${\mu }_{X}$(α_i)): α_i ∈Ψ, i = 1, 2,…, n} and Y = {(α_i, ${\mu }_{Y}$(α_i)): α_i ∈Ψ, i = 1, 2,…, n}, where ${\mu }_{X}$(α_i) and ${\mu }_{Y}$(α_i) are the Possibility values of α_i with respect to X and Y, respectively, be two Possibility sets over a non-empty set Ψ. Then, the Hamming distance between X and Y is defined as follows:

$${H}_{d}(X, Y)={\sum }_{i=1}^{n}|{\mu }_{X}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{Y}\left({\mathrm{\alpha }}_{i}\right)|$$

Here, $0\le {H}_{d}(X, Y) \le 1$.

Example 3.1. Suppose that X = {(α₁, 0.5), (α₂, 0.6)} and Y = {(α₁, 0.6), (α₂, 0.7)} be two Possibility sets over a non-empty set Ψ = {α₁, α₂}. Then, the Hamming distance between X and Y is H_d(X, Y) = 0.2.

Theorem 3.1. The Hamming distance between two Possibility sets is bounded.

Proof. Let X = {(α_i, ${\mu }_{X}$(α_i)): α_i ∈Ψ} and Y = {(α_i, ${\mu }_{Y}$(α_i)): α_i ∈Ψ}, where ${\mu }_{X}$(α_i) and ${\mu }_{Y}$(α_i) (i = 1, 2,…, n) are the Possibility values of α_i with respect to X and Y. Therefore, H_d(X, Y) = $\sum_{i=1}^{n}|{\mu }_{X}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{Y}\left({\mathrm{\alpha }}_{i}\right)|$.

Now, we have 0 ≤ ${\mu }_{X}\left({\mathrm{\alpha }}_{i}\right)$, ${\mu }_{Y}\left({\mathrm{\alpha }}_{i}\right)$ ≤ 1, for all α_i ∈Ψ, i = 1, 2,…, n

$$\Rightarrow 0 \le \left|{\mu }_{X}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{Y}\left({\mathrm{\alpha }}_{i}\right)\right| \le 1,\text{ for all }{\mathrm{\alpha }}_{i} \in\Psi , i = 1, 2,\dots , n$$

$$\Rightarrow0\leq{\textstyle\sum_{i=1}^n}\left|\mu_X\left({\mathrm\alpha}_i\right)-\mu_Y\left({\mathrm\alpha}_i\right)\right|\leq n$$

$$\Rightarrow 0\le {H}_{d}(X, Y) \le n$$

Therefore, the Hamming distance between two Possibility sets is bounded.

Theorem 3.2. Suppose that P = {(α_i, ${\upmu }_{P}$(α_i)): α_i ∈Ψ, i = 1, 2,…, n}, Q = {(α_i, ${\upmu }_{Q}$(α_i)): α_i ∈ Ψ, i = 1, 2,…, n} and R = {(α_i, ${\upmu }_{R}$(α_i)): α_i ∈Ψ, i = 1, 2,…, n} be three Possibility sets over a fixed set Ψ, where cardinality of Ψ is n. If P ⊆ Q ⊆ R, then H_d(P, Q) ≤ H_d(P, R) and H_d(Q, R) ≤ H_d(P, R).

Proof. Let P = {(α_i, ${\upmu }_{P}$(α_i)): α_i ∈Ψ, i = 1, 2,…, n}, Q = {(α_i, ${\upmu }_{Q}$(α_i)): α_i ∈ Ψ, i = 1, 2,…, n} and R = {(α_i, ${\upmu }_{R}$(α_i)): α_i ∈Ψ, i = 1, 2,…, n} be three Possibility sets over a fixed set Ψ, where cardinality of Ψ is n. Therefore,

$${H}_{d}(P, Q)={\sum }_{i=1}^{n}\left|{\mu }_{P}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{Q}\left({\mathrm{\alpha }}_{i}\right)\right|$$

$${H}_{d}(Q, R)={\sum }_{i=1}^{n}\left|{\mu }_{Q}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{R}\left({\mathrm{\alpha }}_{i}\right)\right|$$

$${H}_{d}(P, R)={\sum }_{i=1}^{n}\left|{\mu }_{P}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{R}\left({\mathrm{\alpha }}_{i}\right)\right|$$

Now, H_d(P, Q)

$$={\sum }_{i=1}^{n}|{\mu }_{P}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{Q}\left({\mathrm{\alpha }}_{i}\right)|$$

$$\le {\sum }_{i=1}^{n}\begin{array}{cc}\left|{\mu }_{P}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{R}\left({\mathrm{\alpha }}_{i}\right)\right|& [\mathrm{since }P \subseteq Q \subseteq R]\end{array}$$

$$={ H}_{d}(P, R)$$

$$\Rightarrow {H}_{d}\left(P, Q\right)\le {H}_{d}\left(P, R\right)$$

Further, H_d(Q, R)

$$={\sum }_{i=1}^{n}|{\mu }_{Q}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{R}\left({\mathrm{\alpha }}_{i}\right)|$$

$$\le {\sum }_{i=1}^{n}\begin{array}{cc}\left|{\mu }_{P}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{R}\left({\mathrm{\alpha }}_{i}\right)\right|& \left[\mathrm{since }P \subseteq Q \subseteq R\right]\end{array}$$

$$={H}_{d}\left(P, R\right)$$

$$\Rightarrow {H}_{d}\left(Q, R\right)\le {H}_{d}\left(P, R\right)$$

Hence, H_d(P, Q) ≤ H_d(Q, R) and H_d(Q, R) ≤ H_d(P, R), whenever P ⊆ Q ⊆ R.

Definition 3.2. Assume that X = {(α_i, ${\mu }_{X}$(α_i)): α_i ∈Ψ, i = 1, 2,…, n} and Y = {(α_i, ${\mu }_{Y}$(α_i)): α_i ∈Ψ, i = 1, 2,…, n}, where ${\mu }_{X}$(α_i) and ${\mu }_{Y}$(α_i) are the Possibility values of α_i with respect to X and Y, respectively, be two Possibility sets over a non-empty set Ψ. Suppose that ${w}_{i}$(i = 1, 2,…, n) are the corresponding weights of α_i ∈ Ψ. Then, the weighted Hamming distance between X and Y is defined as follows:

$${WH}_{d}\left(X, Y\right)={\sum }_{i=1}^{n}{w}_{i}\left|{\mu }_{X}\left({\alpha }_{i}\right)-{\mu }_{Y}\left({\alpha }_{i}\right)\right|$$

Here, 0 ≤ WH_d(X, Y) ≤ 1.

Example 3.2. Let us consider two Possibility sets over Ψ as shown in Example 3.1. Suppose that ${w}_{1}$=0.4 and ${w}_{2}$=0.6 are the corresponding weights of X and Y. Then, WH_d(X, Y) = 0.1.

Theorem 3.3. The weighted Hamming distance between two Possibility sets is bounded.

Proof. Let X = {(α_i, ${\mu }_{X}$(α_i)): α_i ∈Ψ} and Y = {(α_i, ${\mu }_{Y}$(α_i)): α_i ∈Ψ}, where ${\mu }_{X}$(α_i) and ${\mu }_{Y}$(α_i) (i = 1, 2,…, n) are the Possibility values of α_i with respect to X and Y. Suppose that ${w}_{i}$(i = 1, 2,…, n) are the corresponding weights of α_i ∈ Ψ. Therefore, WH_d(X, Y) = $\sum_{i=1}^{n}{w}_{i}. |{\mu }_{X}\left({\alpha }_{i}\right)-{\mu }_{Y}\left({\alpha }_{i}\right)|$.

Now, we have 0 ≤ ${\mu }_{X}\left({\mathrm{\alpha }}_{i}\right)$, ${\mu }_{Y}\left({\mathrm{\alpha }}_{i}\right)$ ≤ 1, for all α_i ∈Ψ, i = 1, 2,…, n

$$\Rightarrow 0\le \left|{\mu }_{X}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{Y}\left({\mathrm{\alpha }}_{i}\right)\right| \le 1,\text{ for all }{\mathrm{\alpha }}_{i} \in\Psi , i = 1, 2,\dots , n$$

$$\Rightarrow 0 \le {w}_{i}. \left|{\mu }_{X}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{Y}\left({\mathrm{\alpha }}_{i}\right)\right|\le 1,\text{ for all }{\mathrm{\alpha }}_{i} \in\Psi , i = 1, 2,\dots , n$$

$$\Rightarrow 0\le {\sum }_{i=1}^{n}{w}_{i}.\left|{\mu }_{X}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{Y}\left({\mathrm{\alpha }}_{i}\right)\right|\le n$$

$$\Rightarrow 0\le {WH}_{d}\left(X, Y\right)\le n$$

Therefore, the weighted Hamming distance between two Possibility sets is bounded.

Theorem 3.4. Suppose that P = {(α_i, ${\mu }_{P}$(α_i)); α_i ∈Ψ, i = 1,2,…,n}, Q = { (α_i, ${\mu }_{Q}$(α_i)); α_i ∈Ψ, i = 1,2,…,n} and R = { (α_i, ${\mu }_{R}$(α_i)); α_i ∈Ψ, i = 1,2,…,n} be three Possibility set over Ψ, where cardinality of Ψ is n. If P ⊆ Q ⊆ R, then WH_d(P, Q) ≤ WH_d(Q, R) and WH_d(Q, R) ≤ WH_d(P, R).

Proof. Let P = {(α_i, ${\upmu }_{P}$(α_i)): α_i ∈Ψ, i = 1, 2,…, n}, Q = {(α_i, ${\upmu }_{Q}$(α_i)): α_i ∈ Ψ, i = 1, 2,…, n} and R = {(α_i, ${\upmu }_{R}$(α_i)): α_i ∈Ψ, i = 1, 2,…, n} be three Possibility sets over a fixed set Ψ, where cardinality of Ψ is n. Therefore,

$${WH}_{d}(P, Q)={\sum }_{i=1}^{n}{w}_{i}.|{\mu }_{P}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{Q}\left({\mathrm{\alpha }}_{i}\right)|$$

$${WH}_{d}(Q, R)={\sum }_{i=1}^{n}{w}_{i}.|{\mu }_{Q}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{R}\left({\mathrm{\alpha }}_{i}\right)|$$

$${WH}_{d}(P, R)={\sum }_{i=1}^{n}{w}_{i}.|{\mu }_{P}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{R}\left({\mathrm{\alpha }}_{i}\right)|$$

Now, WH_d(P, Q)

$$={\sum }_{i=1}^{n}{w}_{i}.|{\mu }_{P}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{Q}\left({\mathrm{\alpha }}_{i}\right)|$$

$$\le \sum_{i=1}^{n}{w}_{i}.\begin{array}{cc}|{\mu }_{P}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{R}\left({\mathrm{\alpha }}_{i}\right)|& [\mathrm{since }P \subseteq Q \subseteq R]\end{array}$$

$$={WH}_{d}(P, R)$$

$$\Rightarrow {WH}_{d}(P, Q)\le {WH}_{d}(P, R)$$

Further, WH_d(Q, R)

$$={\sum }_{i=1}^{n}{w}_{i}.|{\mu }_{Q}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{R}\left({\mathrm{\alpha }}_{i}\right)|$$

$$\le \sum_{i=1}^{n}{w}_{i}.\begin{array}{cc}|{\mu }_{P}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{R}\left({\mathrm{\alpha }}_{i}\right)|& [\mathrm{since }P \subseteq Q \subseteq R]\end{array}$$

$$={WH}_{d}(P, R)$$

$$\Rightarrow {WH}_{d}\left(Q, R\right)\le {WH}_{d}\left(P, R\right)$$

Hence, WH_d(P, Q) ≤ WH_d(Q, R) and WH_d(Q, R) ≤ WH_d(P, R), whenever P ⊆ Q ⊆ R.

Definition 3.3. Let X = {(α_i, ${\mu }_{X}$(α_i)): α_i ∈Ψ, i = 1, 2,…, n} and Y = {(α_i, ${\mu }_{Y}$(α_i)): α_i ∈Ψ, i = 1, 2,…, n}, where ${\mu }_{X}$(α_i) and ${\mu }_{Y}$(α_i) are the Possibility values of α_i with respect to X and Y respectively, be two Possibility sets over a non-empty set Ψ. Then, the normalized Hamming distance between X and Y is defined as follows:

$${NH}(X, Y)=\frac{1}{2n}{\sum }_{i=1}^{n}|{\mu }_{X}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{Y}\left({\mathrm{\alpha }}_{i}\right)|$$

Here, 0 ≤ NH_d(X, Y) ≤ 1.

Example 3.3. Let us consider two Possibility sets over Ψ as shown in Example 3.1. Then, the normalized Hamming distance between X and Y is NH_d(X, Y) = 0.05.

Theorem 3.5. The normalized Hamming distance between two possibility sets is bounded.

Proof. Let X = {(α_i, ${\mu }_{X}$(α_i)): α_i ∈Ψ} and Y = {(α_i, ${\mu }_{Y}$(α_i)): α_i ∈Ψ}, where ${\mu }_{X}$(α_i) and ${\mu }_{Y}$(α_i) (i = 1, 2,…, n) are the Possibility values of α_i with respect to X and Y. Therefore, NH_d(X, Y) = $\frac{1}{2n}\sum_{i=1}^{n}|{\mu }_{X}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{Y}\left({\mathrm{\alpha }}_{i}\right)|$.

Now, we have 0 ≤ ${\mu }_{X}\left({\mathrm{\alpha }}_{i}\right)$, ${\mu }_{Y}\left({\mathrm{\alpha }}_{i}\right)$ ≤ 1, for all α_i ∈Ψ, i = 1, 2,…, n

$$\Rightarrow 0\le |{\mu }_{X}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{Y}\left({\mathrm{\alpha }}_{i}\right)| \le 1,\mathrm{ for all }{\mathrm{\alpha }}_{i} \in\Psi , i = 1, 2,\dots , n$$

$$\Rightarrow 0\le {\sum }_{i=1}^{n}|{\mu }_{X}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{Y}\left({\mathrm{\alpha }}_{i}\right)|\le n$$

$$\Rightarrow 0\le \frac{1}{2n}{\sum }_{i=1}^{n}|{\mu }_{X}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{Y}\left({\mathrm{\alpha }}_{i}\right)|\le \frac{1}{2}$$

$$\Rightarrow 0 \le {NH}_{d}\left(X, Y\right)\le \frac{1}{2}$$

Therefore, the normalized Hamming distance between two Possibility sets is bounded.

Theorem 3.6. Suppose that P = {(α_i, ${\upmu }_{P}$(α_i)): α_i ∈Ψ, i = 1, 2,…, n}, Q = {(α_i, ${\upmu }_{Q}$(α_i)): α_i ∈ Ψ, i = 1, 2,…, n} and R = {(α_i, ${\upmu }_{R}$(α_i)): α_i ∈Ψ, i = 1, 2,…, n} be three Possibility sets over a fixed set Ψ, where cardinality of Ψ is n. If P ⊆ Q ⊆ R, then ${NH}_{d}$(P, Q) ≤ ${NH}_{d}$(Q, R) and ${NH}_{d}$(Q, R) ≤ ${NH}_{d}$(P, R).

Proof. Let P = {(α_i, ${\upmu }_{P}$(α_i)): α_i ∈Ψ, i = 1, 2,…, n}, Q = {(α_i, ${\upmu }_{Q}$(α_i)): α_i ∈ Ψ, i = 1, 2,…, n} and R = {(α_i, ${\upmu }_{R}$(α_i)): α_i ∈Ψ, i = 1, 2,…, n} be three Possibility sets over a fixed set Ψ, where cardinality of Ψ is n. Therefore,

$${NH}_{d}(P, Q)=\frac{1}{2n}{\sum }_{i=1}^{n}|{\mu }_{P}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{Q}\left({\mathrm{\alpha }}_{i}\right)|$$

$${NH}_{d}(Q, R)=\frac{1}{2n}{\sum }_{i=1}^{n}|{\mu }_{Q}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{R}\left({\mathrm{\alpha }}_{i}\right)|$$

$${NH}_{d}(P, R)=\frac{1}{2n}{\sum }_{i=1}^{n}|{\mu }_{P}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{R}\left({\mathrm{\alpha }}_{i}\right)|$$

Now, NH_d(P, Q)

$$=\frac{1}{2n}{\sum }_{i=1}^{n}|{\mu }_{P}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{Q}\left({\mathrm{\alpha }}_{i}\right)|$$

$$\le \frac{1}{2n}\begin{array}{cc}\sum_{i=1}^{n}|{\mu }_{P}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{R}\left({\mathrm{\alpha }}_{i}\right)|& [\mathrm{since }P \subseteq Q \subseteq R]\end{array}$$

$$={NH}_{d}(P, R)$$

$$\Rightarrow {NH}_{d}(P, Q)\le {NH}_{d}(P, R)$$

Further, NH_d(Q, R)

$$=\frac{1}{2n}{\sum }_{i=1}^{n}|{\mu }_{Q}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{R}\left({\mathrm{\alpha }}_{i}\right)|$$

$$\le \frac{1}{2n}\begin{array}{cc}\sum_{i=1}^{n}|{\mu }_{P}\left({\mathrm{\alpha }}_{i}\right)-{\mu }_{R}\left({\mathrm{\alpha }}_{i}\right)|& [\mathrm{since }P \subseteq Q \subseteq R]\end{array}$$

$$={NH}_{d}(P, R)$$

$$\Rightarrow {NH}_{d}(Q, R)\le {NH}_{d}(P, R)$$

Hence, NH_d(P, Q) ≤ NH_d(Q, R) and NH_d(Q, R) ≤ NH_d(P, R), whenever P ⊆ Q ⊆ R.

GRA-Based MADM Strategy Under Possibility Environment

For a decision-maker (DM), selecting an alternative from a set of feasible alternatives based on some attributes is challenging. To do so, the DM should design a MADM approach for making the decision. Assume that α = {α₁, α₂,…, α_p} be the collection of p alternatives and S = {S₁, S₂,…, S_q} be the family of q attributes. The DM gives evaluation information in terms of Possibility numbers for each option α_i (i = 1, 2,…, p) depending on the attribute S_j (j = 1, 2, …, q). As a result, a decision matrix can express the entire evaluation information of all possibilities.

The following are the steps in the suggested MADM strategy:

• Step-1: Construct the decision matrix using the Possibility number.

The whole evaluation assessment of each alternative α_i (i = 1, 2,…, p) against the attributes S_j (j = 1, 2,…, q) is presented in terms of Possibility set ${E}_{{\mathrm{\alpha }}_{i}}$ = {(${s}_{j}$, ${sup}_{s\le {\widetilde{H}}_{{\mathrm{\alpha }}_{i}}}({\mu }_{\widetilde{H}}({s}_{j})$): ${s}_{j}$∈S}, which is the evaluation assessment of alternative α_i (i = 1, 2,…, p) over the attribute S_j (j = 1, 2, …, q).

Then, the decision matrix (DM) is given by:

DM	S1	S2	…	…	Sq
α1	${\mu }_{{\alpha }_{1}}({s}_{1})$	${\mu }_{{\alpha }_{1}}({s}_{2})$	.	…	${\mu }_{{\alpha }_{1}}({s}_{q})$
α2	${\mu }_{{\alpha }_{2}}({s}_{1})$	${\mu }_{{\alpha }_{2}}({s}_{2})$	.	…	${\mu }_{{\alpha }_{2}}({s}_{q})$
. . .	. . .	. . .	. . .	. . .	. . .
αp	${\mu }_{{\alpha }_{p}}({s}_{1})$	${\mu }_{{\alpha }_{p}}({s}_{2})$	.	…	${\mu }_{{\alpha }_{p}}({s}_{q}$)

• Step-2: Determine the weights for the attributes.

The weights of the attributes play a crucial part in making decisions in every MADM strategy. Assume that the decision-makers have no idea of all attributes’ information weights, then use the compromise function below. In that instance, the decision-maker can determine the attribute weights.

Compromise Function: The compromise function is defined as follows:

$${\upxi }_{j}=\sum_{i=1}^{p}{\mu }_{{\alpha }_{i}}\left({s}_{j}\right)$$

(1)

Then, the weight of the j^th attribute is defined as follows:

$${w}_{j}=\frac{{\upxi }_{j}}{\sum_{j=1}^{q}{\upxi }_{j}}$$

(2)

Here, ${\sum }_{j=1}^{q}{w}_{j}$=1.

• Step-3: Construct the ideal possibility estimates reliability solution (IPERS) and Ideal possibility estimates un-reliability solution (IPEURS) for the decision matrix.

The IPERS for the decision matrix is presented as:

$${R}^{+}=\left[{\mu }_{X}^{+}\left({\alpha }_{1}\right),{\mu }_{X}^{+}\left({\alpha }_{2}\right),\dots , {\mu }_{X}^{+}({\alpha }_{q})\right]$$

(3)

where ${\mu }_{X}^{+}\left({\alpha }_{j}\right)$= max{${\mu }_{X}$(α_i): i = 1, 2, 3, ….., p}.

The IPEURS for the decision matrix is presented as:

$${R}^{-}=\left[{\mu }_{X}^{-}\left({\alpha }_{1}\right),{\mu }_{X}^{-}\left({\alpha }_{2}\right),\dots , {\mu }_{X}^{-}({\alpha }_{q})\right]$$

(4)

where ${\mu }_{X}^{-}\left({\alpha }_{j}\right)$= min{${\mu }_{X}$(α_i): i = 1, 2, 3, ….., p}.

• Step-4: Determination of each alternative’s possibility grey relational coefficient (PGRC) from IPERS and IPEURS.

The PGRC of each alternative from IPERS is presented as:

$${G}_{ij}^{+}=\frac{\underset{i}{\mathrm{min}}\underset{j}{\mathrm{min}}{\Delta }_{ij}^{+} + \underset{i}{\mathrm{k }.\mathrm{ max}}\underset{j}{\mathrm{max}}{\Delta }_{ij}^{+}}{{\Delta }_{ij}^{+}\underset{i}{+\mathrm{ k }.\mathrm{ max}}\underset{j}{\mathrm{max}}{\Delta }_{ij}^{+}}$$

(5)

where ${\Delta }_{ij}^{+}$ = H_d(${\mu }_{X}^{+}\left({\alpha }_{j}\right), {\mu }_{X}\left({\alpha }_{j}\right)$), i = 1, 2,…, p, j = 1, 2,…, q and $k\in$[0, 1].

The PGRC of each alternative from IPEURS is given below:

$${G}_{ij}^{ -}=\frac{\underset{i}{\mathrm{min}}\underset{j}{\mathrm{min}}{\Delta }_{ij}^{ -} +\underset{i}{\mathrm{ k }.\mathrm{ max}}\underset{j}{\mathrm{max}}{\Delta }_{ij}^{ -}}{{\Delta }_{ij}^{ -}\underset{i}{+\mathrm{ k }.\mathrm{ max}}\underset{j}{\mathrm{max}}{\Delta }_{ij}^{ -}}$$

(6)

where ${\Delta }_{ij}^{-}$= H_d (${\mu }_{X}\left({\alpha }_{j}\right), {\mu }_{X}^{-}\left({\alpha }_{j}\right)$), i = 1, 2, …, p, j = 1, 2, …, q and $k\in$[0, 1].

Here, ${G}_{ij}^{+}$ and ${G}_{ij}^{ -}$ are the identification coefficient used to adjust the range of the comparison environment and to control the level of differences of the relation coefficients. The comparison environment remains unchanged when k = 1, and the comparison environment disappears when k = 0. If the identification coefficient is smaller, then the range of grey relational coefficient will become so large. Generally, k = 0.5 is considered for decision-making situation.

• Step-5: Determine the PGRC.

The PGRC of each alternative from IPERS and IPEURS are defined as follows:

$${G}_{i}^{+}= \sum_{j=1}^{q}{w}_{j}{G}_{ij}^{+}$$

(7)

and

$${G}_{i}^{-}=\sum_{j=1}^{q}{w}_{j}{G}_{ij}^{-}$$

(8)

where i = 1, 2, …, p.

• Step-6: Determine the Possibility of relative relational degree.

The Possibility relative relational degree of each alternative can be defined as follows:

$${\mathfrak{R}}_{i}=\frac{{G}_{i}^{+}}{{G}_{i}^{+}+{G}_{i}^{-}}$$

(9)

where i = 1, 2, …, p.

• Step-7: Rank the alternatives.

The ascending order of the Possibility relative relational degree can be used to determine the ranking order of all alternatives. The alternative with the highest ℜ_i value is the best alternative.

The flow chart of the proposed MADM strategy is given below see Fig. 1:

Fig. 1

Flow chart of the proposed MADM strategy

Validation of the Proposed MADM Strategy

In this section, we provide a numerical example to demonstrate the effectiveness of the proposed MADM method.

Example 5.1. “Identification of the most important parameters affecting climate change and the impact of urbanization on hydropower plants.”

Changes in climate and the rate of urbanization influence and disrupt the consistency between demand and supply of energy resources. Hydropower plants are one of the systems that have been hit the worst. However, this change does not influence all operating parameters. Our goal was to find an indicator that may indicate the operational status of hydropower facilities in the face of climate and urbanization changes; thus, we looked into the essential alternatives most influenced by climatic and urbanization uncertainties. Even though the list of indications is not exhaustive, the features described are regularly reviewed before deciding on the operational status of the power plant. According to the research, generator productivity was the most important factor. The operation of hydropower plants have been damaged in recent years due to changing weather patterns. The river’s hydrology has changed, influencing the performance of the process of hydroelectric units connected to it. The expected effect of human-caused increases in greenhouse gas emissions is a changing climate known as global warming. Many greenhouse gases, such as carbon dioxide (CO₂), are naturally created and retain a lot of heat, keeping the Earth warm. Changes in the quantity and timing of river flows, as well as increased reservoir evaporation, will all have an impact on hydroelectric power output as a result of climate change. Water-based energy sources, as well as traditional power plants, are harmed as a result. Urban water sources have been used in residential, industrial, and commercial areas and for urban facilities like city cleaning, firefighting, lake and pool maintenance, and recreational land irrigation. Another issue that affects hydropower plant output is the breakdown of machines utilized to assist in power generation. Turbines, generators, transformers, pipelines, and switch gears are all parts of the power generation process. An essential part of the research is measuring and forecasting hydropower unit performance decline. The accuracy of the performance decrease projection is used to build the unit’s maintenance strategy. Hydropower unit can operate in a relatively safe and predictable environment with high-accuracy forecasting. Forecast accuracy has a direct effect on hydropower plant economic advantage; as a consequence, performance degradation assessments and projections, as well as various research investigations, are required further to verify the stability of the hydropower processing step.

This study aimed to identify the most critical parameter or factors affected by climate change, urbanization, and machine failures. Existing techniques demand that parameters be evaluated based on their impact on decision-making (Majumder and Saha 2019). Many mathematicians have utilized a range of techniques to establish the relative relevance of one attribute over another in terms of criteria or objectives. This way of comparison is not written in stone. This research proposes a method for determining the relative relevance of one parameter to another using a list of conditions (attributes) or objectives. This strategy will not use a scale or any other mechanism to rate the parameter relevance separately.

Identifying the most significant parameter or set of characteristics most impacted by climate change, urbanization, and machine failure could be a big help. These several characteristics can then be monitored to control the impact on hydropower plant performance. In reality, the strategy can be put into practice by employing specific advanced smooth algorithms that are used to find the essential characteristics fairly and objectively. As a significant subject in the domain of decision theory, MADM processes strive to create a structured and justified framework for analyzing major approaches in order to aid decision-makers in making an appropriate decision with reference to their preferences.

There are various types of investigations available. It is hard to pick the essential aspect influencing climate change and urbanization’s impact on hydropower plants. The investigative reporter can select specific attributes that will serve as the foundation for their inquiry or process improvement authority and other requirements.

The following are the different attributes taken for the study:

• (i) Climatic impact (Z₁)

Hydropower plant efficiency falls as climate change worsens due to the impacts of climate change on activities, financial ramifications, and consequences on other energy industries.

• (ii) Urbanization impact (Z₂)

As per the research, the economic, electrical supply, and environmental effects of urbanization on hydropower plants are significantly negatively correlated to hydropower plants’ efficiency.

• (iii) Impact of machine failure (Z₃)

This attribute is counterproductive to hydropower plant efficiency because of the economic implications of machine breakdown on a hydropower plant.

The qualities used in this study were selected after analyzing the literature. To determine the fundamental components regulating expenditure in hydropower plants, we refer (Banerjee et al. 2017; Biswas et al. 2014; Dey et al. 2016a, 2016b, 2016c). When most of these requirements and attribute data are standardized, these factor data become unit less. After that, the mean of each data point is utilized to standardize the entire data set. Table 1 provides an outline of the factors.

Table 1 The tabular representation of the information of parameters P₁, P₂, P₃, and P₄ against the attributes Z₁, Z₂, and Z₃

Assume that the decision-maker selects four alternatives after the initial screening. Suppose that Ű = {P₁, P₂, P₃, P₄}, where P₁ = efficiency of penstock, P₂ = efficiency of turbine, P₃ = efficiency of generator, P₄ = efficiency of transformer, be the family of four parameters from which the decision-maker will select the most important parameter affecting climate change and the impact of urbanization on hydropower plants. Let Z = {Z₁, Z₂, Z₃} be the collection of attributes based on which the decision-maker will select the most appropriate parameter.

By using Eq. (1) and Eq. (2), we get the weights of the attributes as follows: w₁ = 0.3155080214, w₂ = 0.3262032086, w₃ = 0.3582887701.

From Table 2 is obtained by Tables 3, 4, 5, 6, 7, and 8, it is clear that ℜ₁ < ℜ₂ < ℜ₃ < ℜ₄. Therefore, ${P}_{4}$; i.e., efficiency of transformer is the most appropriate parameter affecting climate change and the impact of urbanization on hydropower plants.

Table 2 The Possibility relative relational degree of each alternative P_i (i = 1, 2, 3, 4) are presented

Table 3 The IPERS (${R}^{+}$) and IPEURS (${R}^{-}$) for the decision matrix

Table 4 The PGRC of each alternative from IPERS for the decision matrix

Table 5 The PGRC of each alternative from IPEURS for the decision matrix

Table 6 Determination of each alternative’s Possibility grey relational coefficient (PGRC) from IPERS

Table 7 Determination of each alternative’s Possibility grey relational coefficient (PGRC) from IPEURS

Table 8 The PGRCs ${G}_{i}^{+}$ and ${G}_{i}^{-}$ of each alternative P_i (i = 1, 2, 3, 4) are presented

The comparative analysis with VNS-based MADM strategy under Possibility environment techniques is listed in Table 9. In VNS-based MADM strategy under Possibility environment techniques, we have to find out the ranking of objective function; hence, finding the minimum alternatives will be the best suitable alternatives. Therefore, the proposed model is more versatile and can quickly solve complications problem. The similarity measure established for possibility becomes better than the existing similarity measure for MCDM.

Table 9 Comparative study

Conclusions

In this article, a GRA-based MADM strategy under a Possibility environment is developed. The proposed MADM strategy is validated by solving an illustrative MADM problem to demonstrate the effectiveness of the MADM strategy.

The proposed MADM strategy can also be used in the area of decision-making problems such as weaver selection, personnel selection, and teacher selection.

Furthermore, the proposed MADM strategy will open up a new avenue of research in the possibility environment.

The possibility function is used in this work to evaluate the idea of a GRA-based MADM strategy in a possibility environment with the identification of the most important parameters influencing climate change and the impact of urbanization on hydropower plants. This will be beneficial not only on its own but will also enable motivated researchers to find solutions to other uncertainty-related problems using analogous methods. In the subsequent paper, a novel approach to calculating the possibility value was demonstrated using examples of real-world decision-making problems such as weaver selection, personnel selection, and teacher selection. When making decisions is the primary goal, this technique will be proven to be quite effective in many real-life circumstances.