Computers, Materials & Continua DOI:10.32604/cmc.2021.016973
Article

Emergency Decision-Making Based on q-Rung Orthopair Fuzzy Rough Aggregation Information

Ahmed B. Khoshaim1, Saleem Abdullah2, Shahzaib Ashraf3,* and Muhammad Naeem4

1Department of Mechanical Engineering, King Abdulaziz University, Jeddah, Saudi Arabia
2Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, 23200, Pakistan
3Department of Mathematics and Statistics, Bacha Khan University, Charsadda, 24420, Pakistan
4Deanship of Combined First Year, Umm Al-Qura University, Makkah, Saudi Arabia
*Corresponding Author: Shahzaib Ashraf. Email: shahzaibashraf@bkuc.edu.pk
Received: 17 January 2021; Accepted: 07 May 2021

Abstract: With the frequent occurrences of emergency events, emergency decision making (EDM) plays an increasingly significant role in coping with such situations and has become an important and challenging research area in recent times. It is essential for decision makers to make reliable and reasonable emergency decisions within a short span of time, since inappropriate decisions may result in enormous economic losses and social disorder. To handle emergency effectively and quickly, this paper proposes a new EDM method based on the novel concept of q-rung orthopair fuzzy rough (q-ROPR) set. A novel list of q-ROFR aggregation information, detailed description of the fundamental characteristics of the developed aggregation operators and the q-ROFR entropy measure that determine the unknown weight information of decision makers as well as the criteria weights are specified. Further an algorithm is given to tackle the uncertain scenario in emergency to give reliable and reasonable emergency decisions. By using proposed list of q-ROFR aggregation information all emergency alternatives are ranked to get the optimal one. Besides this, the q-ROFR entropy measure method is used to determine criteria and experts’ weights objectively in the EDM process. Finally, through an illustrative example of COVID-19 analysis is compared with existing EDM methods. The results verify the effectiveness and practicability of the proposed methodology.

Keywords: q-rung orthopair fuzzy rough set; q-ROFR entropy measure; aggregation information; emergency decision making

1  Introduction

Catastrophic events such as earthquakes, hurricanes, flooding, and droughts, among others lead to mass destruction such as a large number of deaths, infrastructure damage, and adverse social instability and public security consequences [1]. For example, the 2005 Kashmir earthquake in Pakistan destroyed more than 780,000 buildings and killed 87,350 humans and over a billion animals. In 2019, the Super Typhoon Lekima brought catastrophic damages to mainland China and the direct economic losses amounted to approximately 52 billion yuan. The corona virus disease 2019 (COVID-19) spread to over 200 countries and about 370,000 people have died so far. While emergency response and immediate measures play a key role in addressing such situations, the implementation of emergency decision-making (EDM) with outdated procedures will ultimately lead to possible failures in emergency decisions. Therefore, the EDM process is a vital and essential part of the whole emergency response [2–4]. In fact, inappropriate information on decision-making and tight time pressures in the context of the unforeseen environment of decision-making make it hard for decision-makers to make an effective and reasonable choice [5]. Therefore, to detect the optimum solution for the EDM procedure in order to reduce the economic losses and casualties, it is very important to develop systematic and scientific EDM techniques [6]. Therefore, tackling EDM quickly and effectively has become an important research topic in recent years [7].

Nowadays, the information management and decision-making have become much more important because of increasing emergency situations. With the increasing complexity of the data, new and more accurate tools are necessary because they handle human inaccuracy or ambiguous knowledge more effectively when compared to the classical tools. Zadeh [8] introduced the concept of fuzzy sets (FSs) to deal with uncertain information in real-life situations. Atanassov [9] proposed in 1986 the notion of intuitionistic FSs (IFSs) by generalizing the well-known theory of FSs. Although IFSs are successful in a wide range of applications, they still have some limitations because of the restriction that the sum of membership grade and that of non-membership grade must not exceed 1. To handle this issue, Yager [10] further extended the theory of IFSs and proposed the notion of Pythagorean FSs (PFSs) for modeling the higher-level imprecise and vague information. After Yager’s pioneering work, several researchers initiated the study in the field of PFS theory to show its applications in various disciplines. Khan et al. [11] introduced the Dombi norm based on PFSs and discussed their applications in decision-making problems (DMPs). Yager et al. [12] presented a link between Pythagorean fuzzy membership grades and complex numbers. Batool et al. [13] extended the PFSs to Pythagorean probabilistic hesitant FSs and elaborated their applications in DMPs. Peng et al. [14] established the division and subtraction operations under the Pythagorean fuzzy environment and studied their properties in detail. Ashraf et al. [15] proposed the novel approach using the sine function under Pythagorean fuzzy settings. Zhang [16] defined a similarity measure for DMPs under the Pythagorean fuzzy environment.

Let us assume that the value of membership grade (MG) is set to 0.8 and that of non-membership grade (NMG) is set to 0.9. From the available information, it is clear that MG+NMG>1 and MG2+NMG2>1, which does not satisfy the fundamental condition of IFSs as well as PFSs. To resolve this issue, Yager [17] proposed a more general concept, called q-rung orthopair FS (q-ROFS). The prominent feature of the q-ROFS is that the sum of qth power of the DM and DNM should be less than or equal to 1, which gives more flexibility to the decision-makers for providing MG and NMG more comprehensively. Note that the space of acceptable orthopair of membership grades increases as the value of q increases. A q-ROFS is reduced into the IFS and the PFS, respectively, when we take q = 1 and q = 2. Yager et al. [18] presented the approximate reasoning with q-ROFSs by defining the concepts of possibility and certainty. Khan et al. [19] proposed the q-ROFS-based knowledge measures and discussed their applications in decision-making. Hussain et al. [20] presented the aggregation operators under q-ROF soft sets and elaborated their applications to tackle the real-life DMPs. Liu et al. [21] defined some q-ROF weighted arithmetic/geometric aggregation operators and used them for real-world DMPs. Khan et al. [22] presented the novel ranking methodology under q-ROF environments. Joshi et al. [23] established some novel aggregation methods for q-ROF information by considering the confidence levels of the experts. Verma [24] introduced the order-q-ROF divergence and entropy measures with their application in multi-attribute group decision-making (MAGDM).

Pawlak [25] initiated the important notion of rough set theory in 1982, which handles imprecise and ambiguous data more effectively. Investigation into the rough set, both theoretically and practically, in the recent era has made tremendous progress. The notion of rough sets has been enhanced in various ways by several scholars. Dubois et al. [26] established the structure of fuzzy rough sets (FRSs). Zhang et al. [27] established the decision-making methodology using FRSs to tackle the uncertain information in DMPs. Khan et al. [28] established the novel notion of probabilistic hesitant FRSs and discussed their applications in DMPs. Chinram et al. [29] proposed the evaluation based on distance from average solution methodology under intuitionistic FRSs to tackle the multi-attribute group decision-making. Zhou et al. [30] established the generalized approximation operators under intuitionistic FRSs.

In some real-life circumstances, decision-makers (DMs) have a strong point of view about the ranking or rating of plans, projects, or political statements of a government. For example, the construction of a cricket ground by a university to render its accomplishment and performance. The members of the university administration may rate their project highly by assigning a DM (μ=0.9), whereas others may rate the same project as a wastage of money and try to defame it by providing strong opposite points of view. So, they assign a DNM (v = 0.7). In this situation, μ+ν>1 and μ2+ν2>1, but μq+νq<1 for q > 3 so that (μ,ν) is neither IFN nor PFN but it is q-ROFN. Thus, Yager’s q-ROFNs are more efficient to deal with uncertainty in the data. q-rung orthopair fuzzy rough sets (q-ROFRSs), a hybrid intelligent structure of rough sets, and q-ROFS are advanced classification strategies that address ambiguous and incomplete data. We conclude from the analysis that in decision-making, aggregation operators have a significant role to play in aggregating the collective data from different sources to a single value. In accordance with the best available knowledge to date, the development of aggregation operators with the hybridization of q-ROFS with a rough set is not observed in the q-ROPF setting. Therefore, this motivates the current work of q-ROF rough study. Furthermore, we will investigate aggregation operators based on rough information that are q-rung orthopair fuzzy rough weighted averaging, order weighted averaging, hybrid weighted averaging, weighted geometric, and order weighted geometric and hybrid weighted geometric aggregation operators under t-norm and t-conorm.

This paper is organized as follows: In Section 2, we review some concepts related to q-ROFSs and rough sets. In Section 3, we proposed the novel notion of q-ROFRS and discussed its basic operations. In Section 4, we proposed the list of averaging/geometric aggregation operators for q-ROPFR information. In Section 5, we present the entropy measure and decision-making methodology. In Section 6, we demonstrate the numerical example of the public health emergency problem to show the applicability and effectiveness of the proposed methodology. Finally, Section 8 concludes the paper, illustrating achievements and setting future directions.

2  Preliminaries

In this section, we resolve the essential knowledge about q-ROFS and rough sets.

Definition 1 ( [17]). Let M be a non-empty set. A q-ROFS Z of a set M is a set having the form

where the values μz(δ)∈[0,1] and νz(δ)∈[0,1] are called the positive and negative membership grades of the element δ, subject to (μz(δ))q+(νz(δ))q≤1 with q>2∀δ∈M.

For simplicity, Z={(δ,μz(δ),νz(δ)):δ∈M} is represented by Z=(μz,νz), if there is no confusion and is called q-rung orthopair number (q-ROFN). The collections of all q-ROFNs in M will be represented by q-ROPFN(N).

Definition 2 ( [25]). Let M be a non-empty set and I∈M×M be any arbitrary relation on the set M. A mapping I*:M→P(M) is defined as I*(b)={∂∈M:(b,∂)∈I}, for b∈M, where I*(b) is the successor neighborhood of an object b w.r.t. I. The pair (M,I) is said to be crisp approximation space. For any x⊆M, the lower and upper approximation of x w.r.t. approximation space (M,I) is denoted and defined by I(x)={b∈M:I*(b)⊆x} and I¯(x)={b∈M:I*(b)∩x≠φ}, respectively. Therefore, (I(x),I¯(x)) is known as a rough set and I(x),I¯(x):P(M)→P(M) are upper and lower approximation operators.

3  q-Rung Orthopair Fuzzy Rough Aggregation Information

The aggregation information plays an important role in combining data into one format and addressing the DMP. In this section, we propose a list of novel aggregation information.

3.1 q-Rung Orthopair Fuzzy Rough Averaging Aggregation operators

Definition 3. Let us consider (M,I) be the q-ROF approximation space. Suppose I(xg)=(I(xg),I¯(xg))q-ROFRS(M)∈(g∈N). Then, the weighted averaging aggregation operator can be defined as

where (β1,β2,…,βn)T is the weight information of (I(x1),I(x2),…,I(xn)), that is, βg≥0 and ∑g=1nβg=1.

Theorem 1. Suppose (M,I) be a q-ROF approximation space. Suppose I(xg)=(I(xg),I¯(xg))∈q-ROFRS(M)(g∈N) and (β1,β2,…,βn)T is weight information of (I(x1),I(x2),…,I(xn)) such that βg≥0 and ∑g=1nβg=1. Then, the WA aggregation operator is a mapping Dn→D such that

In algebraic-strict Archimedean t-norm and t-conorm, if we assign values to generators t and s, then we obtain two algebraic operations for q-ROFRVs:

Proof: The proof is straightforward by using mathematical induction.

Definition 4. Suppose (M,I) be a q-ROF approximation space. Suppose I(xg)=(I(xg),I¯(xg))∈q-ROFRS(M)(g∈N). Then, the ordered weighted averaging aggregation operator is defined as

where β(g) is denoted the order according to (β(1),β(2),β(3),…,β(n)) and (β1,β2,…,βn)T is the weight information of (I(x1),I(x2),…,I(xn)), that is, βg≥0 and ∑g=1nβg=1.

Theorem 2. Suppose (M,I) be a q-ROF approximation space. Suppose I(xg)=(I(xg),I¯(xg))∈q-ROFRS(M)(g∈N) and (β1,β2,…,βn)T is the weight information of (I(x1),I(x2),…,I(xn)) such that βg≥0 and ∑g=1nβg=1. Then, OWA aggregation operator is a mapping Dn→D such that

In algebraic-strict Archimedean t-norm and t-conorm, if we assign values to generators t and s, then we have algebraic operations for q-ROFRVs:

Proof: The proof is straightforward by using mathematical induction.

Definition 5. Suppose (M,I) be a q-ROF approximation space and suppose I(xg)=(I(xg),I¯(xg))∈q-ROFRS(M)(g∈N). Then, the hybrid weighted averaging aggregation operator is defined as

where β(g) is the order according to (β(1),β(2),β(3),…,β(n)) such that I(xβ(g)′)(I(xβ(g)′)=nβgI(xβ(g)):g∈N), I¯(xβ(g)′)(I¯(xβ(g)′)=nβgI¯(xβ(g)):g∈N), and (β1,β2,…,βn)T is the weight information of (I(x1),I(x2),…,I(xn)), that is, βg≥0 and ∑g=1nβg=1. Also, (η1,η2,…,ηn)T is the associated weight information of (I(x1),I(x2),…,I(xn)), that is, ηg≥0 and ∑g=1nηg=1.

Theorem 3. Suppose (M,I) be a q-ROF approximation space. Suppose I(xg)=(I(xg),I¯(xg))∈q-ROFRS(M)(g∈N) and (β1,β2,…,βn)T be the weight information of (I(x1),I(x2),…,I(xn)) such that βg≥0 and ∑g=1nβg=1. Then, HWA aggregation operator is a mapping Dn→D with associated weight information (η1,η2,…,ηn)T, that is, ηg≥0 and ∑g=1nηg=1, such that

In algebraic-strict Archimedean t-norm and t-conorm, if we assign values to generators t and s, then we have two algebraic operations for q-ROFRVs:

Proof: The proof is straightforward by using mathematical induction.

4  Development of q-ROFR Entropy Measure

To calculate the differences between two q-ROFRVs, this segment developed the generalized and weighted generalized distance measures of q-ROFR information. To measure the fuzziness of q-ROFRVs, we propose entropy measures for q-ROFRS based on the developed distance operators.

Definition 6. Suppose (M,I) be a q-ROF approximation space. Suppose I(xg)=(I(xg),I¯(xg)),K(xg)=(K(xg),K¯(xg))∈q-ROFRS(M)(g∈N). Then, the generalized distance measure (GDM) is described for any ℘>0(∈R) as

Definition 7. Suppose (M,I) be a q-ROF approximation space. Suppose I(xg)=(I(xg),I¯(xg)),K(xg)=(K(xg),K¯(xg))∈q-ROFRS(M)(g∈N). Then, the weighted generalized distance measure (WGDM) is described for any ℘>0(∈R) as

where βg(g∈N) are the weight information such that βg≥0 and ∑g=1nβg=1.

Definition 8. Suppose (M,I) be q-ROF approximation space. Suppose I(xg)=(I(xg),I¯(xg))∈q-ROFRS(M)(g={1,2}∈N). Then, the GDM is reduced to

Novel entropy measure for q-ROFRVs is developed in this segment.

Definition 9. Suppose (M,I) be a q-ROF approximation space and I(xg)=(I(xg),I¯(xg))∈q-ROFRS(M)(g∈N). Then, the q-ROFR entropy measure is described as

where υI(xg) is the indeterminacy of I(xg).

For q-ROF approximation space (M,I), suppose I(xg)=(I(xg),I¯(xg)),K(xg)=(K(xg),K¯(xg))∈q-ROFRS(M)(g∈N). Then, the q-ROFR entropy measure satisfies the following properties:

(1)   E(I(xg))=0 iff I(xg) is the crisp set,

(2)   E(I(xg))≤E(I(xg)c), and

(3)   E(I(xg))≤E(K(xg)) if I(xg)≤K(xg), that is, I(xg)≤K(xg) and I¯(xg)≤K¯(xg).

5  Algorithm for DMPs

Here, we have developed a framework for addressing uncertainty in DM under q-rung orthopair fuzzy rough information. Consider a DM problem with {λ1,λ2,…,λg} as a set of m alternatives and a set of attributes {τ1,τ2,…,τh} with (β1,β2,…,βh)T being the weights, that is, βt∈[0,1] and ∑t=1hβt=1. To test the reliability of kth alternative λk under the tth attribute τt, let a set of DMs {D1,D2,…,Dĵ} and (η1,η2,…,ηĵ)T be DM weights such that ηs∈[0,1] and ∑s=1ĵηs=1.

Step 1. The expert’s evaluation matrices are constructed. Ej=[I(xiIĵ)]g×h=(I(xiIĵ),I¯(xiIĵ)), where I(xiI)={(b,μI(xiI)(b),νI(xiI)(b)):b∈M} and I¯(xiI)={(b,μI¯(xiI)(b),νI¯(xiI)(b)):b∈M} such that 0≤(μI(xiI)(b))q+(νI(xiI)(b))q≤1 and 0≤(μI¯(xiI)(b))q+(νI¯(xiI)(b))q≤1 are the q-ROF rough values.

Step 2(a). The expert ideal matrix (EIM) is calculated using a q-ROFRWA aggregation operator, which is closer to each expert information. EIM=(EI11EI12¡tex-math¿Math˙135¡/tex-math¿…EI1hEI21EI22¡tex-math¿Math˙136¡/tex-math¿…EI2hMMOMEIg1EIg1¡tex-math¿Math˙137¡/tex-math¿…EIgh), where

Step 2(b). Compute the expert right ideal matrix (ERIM) and expert left ideal matrix (ELIM) as follows:

Step 2(c). The distance of NiI(k) with EIM, ERIM, and ELIM is evaluated as DEIM, DERIM, and DELIM, respectively, as follows: