Plane Thermoelastic Waves in Infinite Half-Space Caused Decision Making: Applications in Management and Engineering Vol. 5, Issue 2, 2022, pp. 30-45. ISSN: 2560-6018 eISSN: 2620-0104 DOI: https://doi.org/10.31181/dmame0305102022d * Corresponding author. E-mail addresses: djeniseles11@gmail.com (Jeni Seles Martina. D), deepa.g@vit.ac.in (Deepa. G*) THE ENERGY OF ROUGH NEUTROSOPHIC MATRIX AND ITS APPLICATION TO MCDM PROBLEM FOR SELECTING THE BEST BUILDING CONSTRUCTION SITE Jeni Seles Martina Donbosco1 and Deepa Ganesan1* 1 Department of Mathematics, VIT University, Vellore, India Received: 10 August 2022; Accepted: 26 September 2022; Available online: 5 October 2022. Original scientific paper Abstract: An approach to data processing for relational databases is called a rough set theory. It is an interesting area of uncertainty mathematics that is mainly related to fuzzy theory. Rough set theory and neutrosophic set theory can be joined to create a powerful tool for dealing with indeterminacy. Neutrosophic matrices help decision-makers deal with multi-criteria decision- making by providing them with more useful and practical when we apply the concept of matrix energy. In this paper, we defined a Rough neutrosophic matrix and its energy. Some propositions, lower and upper limits of the rough neutrosophic matrix's energy were derived. The proposed energy of the rough neutrosophic matrix was applied in multi-criteria decision-making problems. The problem is to select the best place for constructing the school building. Applying the energy method to the MCDM problem became more relatable and produced good results. Keywords: Rough set, rough neutrosophic set, rough neutrosophic matrix, energy of rough neutrosophic matrix, multi-criteria decision making. 1. Introduction Fuzzy sets, fuzzy membership functions, and fuzzy logic were first introduced (Zadeh 1965). Fuzzy Matrix Theory, which focused on the convergence of fuzzy matrices’ powers, was first presented by Michael G. Thomason in 1977. It can be applied in several circumstances. It is commonly known that the matrix representation offers an additional advantage in resolving the issue. Intuitionistic fuzzy matrices were first introduced (Khan et al., 2002). It is difficult to determine the value of membership or non-membership as a point, though. The Neutrosophic set was first introduced (Smarandache 1998). He put out the concepts of Neutrosophic Set, Probability, and Logic to specifically address the issue of indeterminacy. He also created hesitant and dual hesitant neutrosophic sets, single and interval valued mailto:djeniseles11@gmail.com mailto:deepa.g@vit.ac.in The energy of rough neutrosophic matrix and its application to MCDM problem for selecting … 31 neutrosophic sets, and multi-valued neutrosophic sets. After that the introduction of fuzzy relational maps and neutrosophic relational maps was presented (Smarandache et al., 2004). In this, they included square neutrosophic matrices. The neutrosophic matrix and associated algebraic operations were created (Smarandache et al., 2014). Christi DiStefano and colleagues introduced the idea of matrix energy in 2009. They devised the equation for the matrix’s energy. A generalization of the energy of a graph is the energy of a matrix. A paper titled Energy of Matrixes was Proposed (Bravo et al., 2017). They produced a number of theorems on matrix energy as well as upper and lower bounds. The notion of matrix energy does not hold true in a neutrosophic setting or for MCDM problems, however, the notion of graph energy will gain in popularity. We, therefore, examine energy in neutrosophic matrices and how it might be used in MCDM in this paper. The idea of a rough set was first introduced by Pawlak (1982). Its foundation is the approximation of sets by a pair of sets known as the lower and upper approximations of a set. In this instance, the equivalence relation is the basis for those approximations. Then he compares the fuzzy set with the rough set concept (Pawlak 1985). Then the rough set techniques for incomplete information systems were presented, along with fuzzy rough sets (Kryszkiewicz 1998). The Rough Intuitionistic fuzzy set was proposed (Rizvi et al., 2002) They defined the Rough Intuitionistic fuzzy set and its properties. The Generalized fuzzy rough sets were introduced (Wei-Zhi Wu et al., 2003). This paper studies fuzzy rough sets using both constructive and axiomatic approaches. Then the rough set and fuzzy rough set on interval-valued fuzzy was proposed (Gong et al., 2008). Both axiomatic and constructive methods to develop a complete framework for the study of interval type-2 rough fuzzy sets are used (Zhang 2012). A concept of the Rough Fuzzy set model for a set-valued ordered fuzzy decision system was presented (Bao et al., 2014). In order to create decision rules for long-term forecasting of air passengers suggested a novel hybrid method based on rough set theory (Sharma et al., 2018). Then they used a combination technique to evaluate India’s sugarcane production based on the rough set approach (Sharma et al., 2021). They also provide a basic decision-making process based on a set theory for assessing the performance of Delhi hotels (Sharma et al., 2022). They develop a rough set theory to offer a set of decision rules and significant feature sets. Rough set theory and neutrosophic set theory will both be useful methods for handling incomplete, ambiguous, uncertain, and incorrect data. (Said Broumi et al., 2014) introduced the idea of the Rough Neutrosophic Set. They outlined the rough neutrosophic sets and their operations in this study. Then they proposed the Interval Valued Neutrosophic Rough Set (Said Broumi et al., 2015). A rough grey relational analysis-based strategy for neutrosophic multi-attribute decision-making is illustrated (Kalyan Mondal et al., 2015). In this work, the rough neutrosophic decision matrix is defined, and an MCDM issue is solved using this matrix. A number of authors offered various ideas for a rough neutrosophic field (Alias et al., 2017, Yang et al., 2017, Pramanik et al., 2017). They studied MCDM in rough neutrosophic sets with coefficient correlation, rough single-valued neutrosophic sets, and rough neutrosophic multisets. On novel multi-granulation neutrosophic rough set on a single value and its uses was discussed (Bo et al., 2018). Rough Neutrosophic Set is used in medical diagnosis (Samuel et al., 2018). This paper discusses the use of medical diagnostics to identify the patient’s health. An article with the title medical diagnosis focused on single- valued neutrosophic uncertain rough multisets over two universes was published in the same year (Zhang et al., 2018). Rough Neutrosophic Sets Pi-Distance for Medical Diagnosis was presented (Samuel et al., 2019). The objective of the study is to establish a causal relationship between the illness and the patient’s symptoms and to examine Jeni et al./Decis. Mak. Appl. Manag. Eng. 5 (2) (2022) 30-45 32 the patient’s state using a rough neutrosophic set. The notions of the neutrosophic soft set with rough set theory (Das et al., 2021), neutrosophic single-valued rough sets, and including topology (Jin et al., 2021; Almathkour et al.,, 2022) will be further developed. The rough set is important in every field of the neutrosophic environment as result. Multi-Criteria Decision Making (MCDM) is a key and quickly developing subject in operations research. Indeterminacy should be handled in the modeling approach of challenges since MCDM problems are well addressed in fuzzy. The development of the MCDM field in a fuzzy environment led to the proposal of the Neutrosophic Fuzzy MCDM, which was used in numerous methods (Otay 2022, Wang et al., 2022). There are some more different methods in MCDM to select the best alternatives. Recently, several researchers work on many types of methods (Gorcun et al., 2021, Arora et al., 2022). In this study, we added a new step to the method for resolving MCDM issues with a rough neutrosophic matrix by determining its energy. In section 2, The fundamental definitions are provided. In section 3, we introduced the energy of the rough neutrosophic matrix together with its hypotheses, and upper and lower bounds. A new strategy named the Rough Neutrosophic Energy Method was introduced in section 4 and described in detail. The numerical example of the suggested method was resolved in section 5. Then, a conclusion was given. 2. Preliminaries Definition 2.1. Rough set (Pawlak, 1982) Let U be the universal set and R be an equivalence relation on U (this is called an indiscernibility relation). The collection of all equivalence classes of U under 𝑅 is defined as 𝐴 = 𝑈/𝑅, which is called an approximation space. Let 𝑋 ⊆ 𝑈 be a subset of U. We define lower and upper approximation of X in A, denoted 𝐴(𝑋) and 𝐴(𝑋) respectively, as follows 𝐴(𝑋) = {𝑎 ∈ 𝑈 ∶ [𝑎]𝑅 ⊆ 𝑋} 𝐴(𝑋) = {𝑎 ∈ 𝑈 ∶ [𝑎]𝑅 ∩ 𝑋 ≠ ∅} where [𝑎]𝑅 denotes the equivalence class of R containing an element a. The pair 𝐴(𝑋) = (𝐴(𝑋), 𝐴(𝑋)) is called the rough set of X in A. Definition 2.2. Neutrosophic Set (Smarandache, 1998) Let U be the universal set and every element 𝑎 ⊆ 𝑈 has a degree of True, Indeterminacy, False membership in neutrosophic set. It is denoted by S. Then it can be written as 𝑆 = { ⟨ 𝑎,𝑇𝑆(𝑎), 𝐼𝑆(𝑎),𝐹𝑆(𝑎)⟩ ∶ 𝑎 ∈ 𝑈} where, 0 ≤ 𝑇𝑆(𝑎) + 𝐼𝑆(𝑎) + 𝐹𝑆(𝑎) ≤ 3 and Truth Membership function 𝑇𝑆: 𝑈 → [0,1], Indeterminacy Membership function 𝐼𝑆: 𝑈 → [0,1], False Membership function 𝐹𝑆: 𝑈 → [0,1]. Definition 2.3. Rough Neutrosophic Set (Said Broumi et al., 2014) Let U be the universal set and every element 𝑎 ∈ 𝑈. Let R be an equivalence relation on U and S be the neutrosophic set in U with truth membership function 𝑇𝑆, indeterminacy function 𝐼𝑆 and false membership function 𝐹𝑆. The lower and upper The energy of rough neutrosophic matrix and its application to MCDM problem for selecting … 33 approximations of 𝑆 in 𝑈/𝑅 is denoted by 𝑁(𝑋) and 𝑁(𝑋) and they are defined as follows, 𝑁(𝑆) = {⟨ 𝑎,𝑇𝑁(𝑆)(𝑎),𝐼𝑁(𝑆)(𝑎),𝐹𝑁(𝑆)(𝑎)⟩: 𝑏 ∈ [𝑎]𝑅,𝑎 ∈ 𝑈} 𝑁(𝑆) = {⟨ 𝑎,𝑇𝑁(𝑆) (𝑎),𝐼𝑁(𝑆) (𝑎),𝐹𝑁(𝑆) (𝑎)⟩ ∶ 𝑏 ∈ [𝑎]𝑅,𝑎 ∈ 𝑈} where, 𝑇𝑁(𝑆)(𝑎) = ⋀ 𝑇𝑆(𝑏) 𝑏∈ [𝑎]𝑅 𝑇𝑁(𝑆)(𝑎) = ⋁ 𝑇𝑆(𝑏) 𝑏∈ [𝑎]𝑅 𝐼𝑁(𝑆)(𝑎) = ⋁ 𝐼𝑆(𝑏) 𝑏∈ [𝑎]𝑅 𝐼𝑁(𝑆)(𝑎) = ⋀ 𝐼𝑆(𝑏) 𝑏∈ [𝑎]𝑅 𝐹𝑁(𝑆)(𝑎) = ⋁ 𝐹𝑆(𝑏) 𝑏∈ [𝑎]𝑅 𝐹𝑁(𝑆)(𝑎) = ⋀ 𝐹𝑆(𝑏) 𝑏∈ [𝑎]𝑅 where, 0 ≤ 𝑇𝑁(𝑆)(𝑎) + 𝐼𝑁(𝑆)(𝑎) + 𝐹𝑁(𝑆)(𝑎) ≤ 3 and 0 ≤ 𝑇𝑁(𝑆)(𝑎) + 𝐼𝑁(𝑆)(𝑎) + 𝐹𝑁(𝑆)(𝑎) ≤ 3. Where, ⋁ means ‘max’ and ⋀ means ‘min’ and 𝑇𝑆(𝑎),𝐼𝑆(𝑎),𝐹𝑆(𝑎) are truth, indeterminacy, false membership function of a on S. Therefore 𝑁(𝑆) and 𝑁(𝑆) are two neutrosophic sets in U. The pair (𝑁(𝑆),𝑁(𝑆)) is called the Rough Neutrosophic set in U/R. If 𝑁(𝑆) = 𝑁(𝑆) for any 𝑎 ∈ 𝑈, then S is called definable neutrosophic set. Definition 2.4. Energy of Matrix (Bravo et al., 2017) Let 𝑀𝑛(ℂ) denote the space of 𝑛 × 𝑛 matrices with entries in ℂ and P be a matrix in 𝑀𝑛(ℂ). We define the energy of A as 𝐸(𝑃) = ∑|𝜆𝑖 − 𝜇| 𝑛 𝑖=1 where, 𝜆1,𝜆2,… 𝜆𝑛 are the eigenvalues of P and 𝜇 is the mean of eigenvalues. If 𝜇 = 0 or P is the adjacency matrix of a graph G then E(P) is precisely the energy of the graph G. Definition 2.5. Energy of Neutrosophic Matrix Let P(N) be the Neutrosophic matrix with the order of 𝑛 × 𝑛 (square matrix). It can be expressed as three matrices, the first matrix contains the entries 𝑎𝑖𝑗 as truth membership values, the second contains the entries 𝑏𝑖𝑗 as indeterminacy membership values and the third matrix contains the entries 𝑐𝑖𝑗 as false membership values. It is denoted as 𝑃(𝑁) = ⟨ 𝑃(𝑇𝑖𝑗),𝑃(𝐼𝑖𝑗),𝑃(𝐹𝑖𝑗)⟩𝑛× 𝑛 and 𝑎𝑖𝑗 ∈ 𝑃(𝑇𝑖𝑗)𝑛× 𝑛 ,𝑏𝑖𝑗 ∈ 𝑃(𝐼𝑖𝑗)𝑛× 𝑛 𝑎𝑛𝑑 𝑐𝑖𝑗 ∈ 𝑃(𝐹𝑖𝑗)𝑛× 𝑛 The energy of a neutrosophic matrix is defined as 𝐸[𝑃(𝑁)] = ⟨ 𝐸[𝑃(𝑇𝑖𝑗)],𝐸[𝑃(𝐼𝑖𝑗)],𝐸[𝑃(𝐹𝑖𝑗)]⟩ = 〈∑|𝜆𝑖 − 𝜇| 𝑛 𝑖=1 ,∑|𝜁𝑖 − 𝜇| 𝑛 𝑖=1 ,∑|𝜂𝑖 − 𝜇| 𝑛 𝑖=1 〉 Jeni et al./Decis. Mak. Appl. Manag. Eng. 5 (2) (2022) 30-45 34 where, 𝜆𝑖,𝜁𝑖 and 𝜂𝑖, (𝑖 = 1,2,… 𝑛) are the eigenvalues of Truth, Indeterminacy, and False membership values respectively and 𝜇𝜆, 𝜇𝜁, and 𝜇𝜂 are the mean values of 𝜆𝑖,𝜁𝑖 and 𝜂𝑖 respectively. 3. Energy of Rough Neutrosophic Matrix Definition 3.1. Energy of Rough Neutrosophic Matrix Let 𝐷(𝑁) = ⟨𝐷(𝑁𝑖𝑗(𝑆)),𝐷(𝑁𝑖𝑗(𝑆))⟩ be the Rough Neutrosophic matrix with the order 𝑛 × 𝑛. where, 𝐷(𝑁𝑖𝑗(𝑆)) and 𝐷(𝑁𝑖𝑗(𝑆)) are a lower and upper approximation of the neutrosophic set S. The rough neutrosophic matrix can be expressed as 6 matrices, first 3 matrices are under lower approximation which contains the elements 𝑎𝑖𝑗, 𝑏𝑖𝑗, 𝑐𝑖𝑗, another 3 matrices are under upper approximation which contains the elements 𝑎𝑖𝑗,𝑏𝑖𝑗, 𝑐𝑖𝑗. where, 𝑎𝑖𝑗,𝑎𝑖𝑗 are truth membership values, 𝑏𝑖𝑗,𝑏𝑖𝑗 are indeterminacy membership values and 𝑐𝑖𝑗, 𝑐𝑖𝑗 are false membership values. which is denoted as, 𝐷(𝑁) = ⟨ 𝐷(𝑁𝑖𝑗(𝑆)),𝐷(𝑁𝑖𝑗(𝑆))) ⟩ = ⟨ (𝐷(𝑇𝑖𝑗(𝑆)),𝐷(𝐼𝑖𝑗(𝑆)) ,𝐷(𝐹𝑖𝑗(𝑆))),(𝐷(𝑇𝑖𝑗(𝑆)) ,𝐷(𝐼𝑖𝑗(𝑆)) ,𝐷(𝐹𝑖𝑗(𝑆)))⟩ where the elements, 𝑎𝑖𝑗 ∈ 𝐷(𝑇𝑖𝑗(𝑆)) ,𝑏𝑖𝑗 ∈ 𝐷(𝐼𝑖𝑗(𝑆)),𝑐𝑖𝑗 ∈ 𝐷(𝐹𝑖𝑗(𝑆)), 𝑎𝑖𝑗 ∈ 𝐷(𝑇𝑖𝑗(𝑆)), 𝑏𝑖𝑗 ∈ 𝐷(𝐼𝑖𝑗(𝑆)),𝑐𝑖𝑗 ∈ 𝐷(𝐹𝑖𝑗(𝑆)). Then the energy of Rough Neutrosophic matrix defined as 𝐸[𝐷(𝑁)] = ⟨ (𝐸[𝐷(𝑇𝑖𝑗(𝑆))],𝐸[ 𝐷(𝐼𝑖𝑗(𝑆))],𝐸[𝐷(𝐹𝑖𝑗(𝑆))]), (𝐸[𝐷(𝑇𝑖𝑗(𝑆))],𝐸[𝐷(𝐼𝑖𝑗(𝑆))],𝐸[𝐷(𝐹𝑖𝑗(𝑆))])⟩ 𝐸[𝐷(𝑁)] = 〈 (∑|𝜆𝑖 − 𝜇𝜆| 𝑛 𝑖=1 ,∑|𝜁𝑖 − 𝜇𝜁| 𝑛 𝑖=1 ,∑|𝜂𝑖 − 𝜇𝜂| 𝑛 𝑖=1 ) , (∑|𝜆𝑖 − 𝜇𝜆| 𝑛 𝑖=1 ,∑|𝜁 𝑖 − 𝜇 𝜁 | 𝑛 𝑖=1 ,∑|𝜂 𝑖 − 𝜇𝜂| 𝑛 𝑖=1 ) 〉 where, 𝜆𝑖, 𝜁𝑖, 𝜂𝑖 are the eigenvalues of truth, indeterminacy, and false values of lower approximation matrices and 𝜆𝑖, 𝜁𝑖, 𝜂𝑖 are the eigenvalues of truth, indeterminacy, and false values of upper approximation matrices. 𝜇𝜆, 𝜇𝜁, 𝜇𝜂, 𝜇𝜆, 𝜇𝜁 and 𝜇𝜂 are mean values of the eigen values 𝜆𝑖, 𝜁𝑖, 𝜂𝑖 , 𝜆𝑖, 𝜁𝑖 and 𝜂𝑖 respectively. Example: Let D be the Rough Neutrosophic Matrix with the order of 3× 3. 𝐷 = [ ⟨(.8, .6, .7),(.9, .3, .4)⟩ ⟨(.4, .6, .5),(.7, .3, .2)⟩ ⟨(.3, .5, .6),(.7, .2, .1)⟩ ⟨(.5, .7, .7),(.6, .4, .5)⟩ ⟨(.6, .6, .7),(.8, .5, .3)⟩ ⟨(.2, .7, .8),(.9, .2, .3)⟩ ⟨(.1, .4, .5),(.5, .2, .3)⟩ ⟨(.5, .7, .8),(.6, .1, .2)⟩ ⟨(.4, .6, .9),(.7, .4, .3)⟩ ] 𝑛×𝑛 D can be expressed as 6 matrices. The energy of rough neutrosophic matrix and its application to MCDM problem for selecting … 35 𝐷(𝑇𝑖𝑗) = ( 0.8 0.5 0.1 0.4 0.6 0.5 0.3 0.2 0.4 ) 𝐷(𝐼𝑖𝑗) = ( 0.6 0.7 0.4 0.6 0.6 0.7 0.5 0.7 0.6 ) 𝐷(𝐹𝑖𝑗) = ( 0.7 0.7 0.5 0.5 0.7 0.8 0.6 0.8 0.9 ) 𝐷(𝑇𝑖𝑗) = ( 0.9 0.6 0.5 0.7 0.8 0.6 0.7 0.9 0.7 ) 𝐼𝑖𝑗(𝑆) = ( 0.3 0.4 0.2 0.3 0.5 0.1 0.2 0.2 0.4 ) 𝐹𝑖𝑗(𝑆) = ( 0.4 0.5 0.3 0.2 0.3 0.2 0.1 0.3 0.3 ) Energy of D matrix, 𝐸(𝐷) = ⟨(1.4749,2.4105,2.6262),(2.6387,0.9544,1.0003)⟩ Theorem 3.3. Let D(N) be the Rough neutrosophic matrix. If 𝜆𝑖, 𝜁𝑖, 𝜂𝑖 , 𝜆𝑖, 𝜁𝑖 and 𝜂𝑖, (𝑖 = 1,2,… 𝑛) are the eigenvalues of lower approximation of Truth 𝐷(𝑇𝑖𝑗), Indeterminacy 𝐷(𝐼𝑖𝑗), and False 𝐷(𝐹𝑖𝑗) and upper approximation of Truth 𝐷(𝑇𝑖𝑗), Indeterminacy 𝐷(𝐼𝑖𝑗), and False 𝐷(𝐹𝑖𝑗) membership values respectively. 1)∑|𝜆𝑖 − 𝜇𝜆| 𝑛 𝑖=1 = ∑|𝑎𝑖𝑖 − 𝜇𝜆| 𝑛 𝑖=1 = ∑|𝜆𝑖 − 𝜇𝜆| 𝑛 𝑖=1 = ∑|𝑎𝑖𝑖 − 𝜇𝜆| 𝑛 𝑖=1 = 0 ∑|𝜁𝑖 − 𝜇𝜁| 𝑛 𝑖=1 = ∑|𝑏𝑖𝑖 −𝜇𝜆| 𝑛 𝑖=1 = ∑|𝜁 𝑖 − 𝜇 𝜁 | 𝑛 𝑖=1 = ∑|𝑏𝑖𝑖 − 𝜇𝜁| 𝑛 𝑖=1 = 0 ∑|𝜂𝑖 − 𝜇𝜂| 𝑛 𝑖=1 = ∑|𝑐𝑖𝑖 − 𝜇𝜂| 𝑛 𝑖=1 = ∑|𝜂 𝑖 − 𝜇𝜂| 𝑛 𝑖=1 = ∑|𝑐𝑖𝑖 − 𝜇𝜂| 𝑛 𝑖=1 = 0 2) ∑(𝜆𝑖 − 𝜇𝜆) 2 = 𝑛 𝑖=1 ∑𝑎𝑖𝑖 2 + 2 ∑ 𝑎𝑖𝑗𝑎𝑗𝑖 − 𝑛 1≤𝑖<𝑗≤𝑛 𝜇𝜆 2 𝑛 𝑖=1 , ∑(𝜆𝑖 −𝜇𝜆) 2 = 𝑛 𝑖=1 ∑𝑎𝑖𝑖 2 + 2 ∑ 𝑎𝑖𝑗𝑎𝑗𝑖 − 𝑛 1≤𝑖<𝑗≤𝑛 𝜇 𝜆 2 𝑛 𝑖=1 ∑(𝜁𝑖 −𝜇𝜁) 2 = 𝑛 𝑖=1 ∑𝑎𝑖𝑖 2 + 2 ∑ 𝑎𝑖𝑗𝑎𝑗𝑖 − 𝑛 1≤𝑖<𝑗≤𝑛 𝜇𝜁 2 𝑛 𝑖=1 , ∑(𝜁 𝑖 − 𝜇 𝜁 ) 2 = 𝑛 𝑖=1 ∑𝑎𝑖𝑖 2 + 2 ∑ 𝑎𝑖𝑗𝑎𝑗𝑖 −𝑛 1≤𝑖<𝑗≤𝑛 𝜇 𝜁 2, 𝑛 𝑖=1 ∑(𝜂𝑖 − 𝜇𝜂) 2 = 𝑛 𝑖=1 ∑𝑎𝑖𝑖 2 + 2 ∑ 𝑎𝑖𝑗𝑎𝑗𝑖 − 𝑛 1≤𝑖<𝑗≤𝑛 𝜇𝜂 2 𝑛 𝑖=1 , ∑(𝜂 𝑖 − 𝜇𝜂) 2 = 𝑛 𝑖=1 ∑𝑎𝑖𝑖 2 + 2 ∑ 𝑎𝑖𝑗𝑎𝑗𝑖 − 𝑛 1≤𝑖<𝑗≤𝑛 𝜇𝜂 2 𝑛 𝑖=1 Theorem 3.4. Let 𝐷(𝑁) = ⟨(𝐷(𝑇𝑖𝑗),𝐷(𝐼𝑖𝑗),𝐷(𝐹𝑖𝑗)) ,(𝐷(𝑇𝑖𝑗),𝐷(𝐼𝑖𝑗),𝐷(𝐹𝑖𝑗))⟩ be the Rough neutrosophic matrix. Then the lower and upper bound of each energy is as follows Jeni et al./Decis. Mak. Appl. Manag. Eng. 5 (2) (2022) 30-45 36 𝑖)√(∑|𝜆𝑖 − 𝜇𝜆| 𝑛 𝑖=1 ) 2 − 2 ∑ |𝜆𝑖 − 𝜇𝜆||𝜆𝑗 − 𝜇𝜆| 1≤𝑖<𝑗≤𝑛 + 𝑛(𝑛 − 1)[|𝐷 − 𝜇𝜆|] 2 𝑛 ≤ 𝐸(𝐷(𝑇𝑖𝑗)) ≤ √2[(∑|𝜆𝑖 − 𝜇𝜆| 𝑛 𝑖=1 ) 2 − 2 ∑ |𝜆𝑖 − 𝜇𝜆||𝜆𝑗 − 𝜇𝜆| 1≤𝑖<𝑗≤𝑛 ] 𝑖𝑖)√(∑|𝜁𝑖 −𝜇𝜁| 𝑛 𝑖=1 ) 2 − 2 ∑ |𝜁𝑖 − 𝜇𝜁| |𝜁𝑗 − 𝜇𝜁| 1≤𝑖<𝑗≤𝑛 +𝑛(𝑛 − 1)[|𝐷 − 𝜇𝜁|] 2 𝑛 ≤ 𝐸(𝐷(𝐼𝑖𝑗)) ≤ √2[(∑|𝜁𝑖 − 𝜇𝜁| 𝑛 𝑖=1 ) 2 − 2 ∑ |𝜁𝑖 − 𝜇𝜁| |𝜁𝑗 − 𝜇𝜁| 1≤𝑖<𝑗≤𝑛 ] 𝑖𝑖𝑖)√(∑|𝜂𝑖 − 𝜇𝜂| 𝑛 𝑖=1 ) 2 − 2 ∑ |𝜂𝑖 −𝜇𝜂| |𝜂𝑗 − 𝜇𝜂| 1≤𝑖<𝑗≤𝑛 + 𝑛(𝑛 − 1)[|𝐷 −𝜇𝜂|] 2 𝑛 ≤ 𝐸(𝐷(𝐹𝑖𝑗)) ≤ √2[(∑|𝜂𝑖 −𝜇𝜂| 𝑛 𝑖=1 ) 2 − 2 ∑ |𝜂𝑖 −𝜇𝜂| |𝜂𝑗 − 𝜇𝜂| 1≤𝑖<𝑗≤𝑛 ] 𝑖𝑣)√(∑|𝜆𝑖 − 𝜇𝜆| 𝑛 𝑖=1 ) 2 − 2 ∑ |𝜆𝑖 − 𝜇𝜆||𝜆𝑗 − 𝜇𝜆| 1≤𝑖<𝑗≤𝑛 +𝑛(𝑛 − 1)[|𝐷 − 𝜇𝜆|] 2 𝑛 ≤ 𝐸(𝐷(𝑇𝑖𝑗)) ≤ √2[(∑|𝜆𝑖 −𝜇𝜆| 𝑛 𝑖=1 ) 2 − 2 ∑ |𝜆𝑖 − 𝜇𝜆||𝜆𝑗 − 𝜇𝜆| 1≤𝑖<𝑗≤𝑛 ] 𝑣)√(∑|𝜁 𝑖 −𝜇 𝜁 | 𝑛 𝑖=1 ) 2 − 2 ∑ |𝜁 𝑖 − 𝜇 𝜁 | |𝜁 𝑗 − 𝜇 𝜁 | 1≤𝑖<𝑗≤𝑛 + 𝑛(𝑛 − 1)[|𝐷 −𝜇𝜁|] 2 𝑛 ≤ 𝐸(𝐷(𝐼𝑖𝑗)) ≤ √2[(∑|𝜁𝑖 − 𝜇𝜁| 𝑛 𝑖=1 ) 2 − 2 ∑ |𝜁 𝑖 −𝜇 𝜁 | |𝜁 𝑗 − 𝜇 𝜁 | 1≤𝑖<𝑗≤𝑛 ] The energy of rough neutrosophic matrix and its application to MCDM problem for selecting … 37 𝑣𝑖)√(∑|𝜂 𝑖 −𝜇𝜂| 𝑛 𝑖=1 ) 2 − 2 ∑ |𝜂 𝑖 − 𝜇𝜂| |𝜂𝑗 − 𝜇𝜂| 1≤𝑖<𝑗≤𝑛 + 𝑛(𝑛 − 1)[|𝐷 − 𝜇𝜂|] 2 𝑛 ≤ 𝐸(𝐷(𝐹𝑖𝑗)) ≤ √2[(∑|𝜂𝑖 − 𝜇𝜂| 𝑛 𝑖=1 ) 2 − 2 ∑ |𝜂 𝑖 − 𝜇𝜂| |𝜂𝑗 − 𝜇𝜂| 1≤𝑖<𝑗≤𝑛 ] 4. The Rough Neutrosophic Energy Method In this section, we present a new approach to multi-criteria decision-making for selecting the best alternative using Rough Neutrosophic matrix energy. Determine the set of k alternatives over m criteria. The alternatives are evaluated by n decision makers. So, we set 𝐴 = {𝐴1,𝐴2,… 𝐴𝑘}, 𝐶 = {𝐶1,𝐶2,… 𝐶𝑚} and 𝐷𝑀 = {𝐷𝑀1,𝐷𝑀2,… 𝐷𝑀𝑛} Step 1: The rating values of each alternative on every criterion and the weighted values of m criteria were given by each decision-maker. We take each alternative rating and weight value as a matrix. Consider the ratings of m criteria given by n decision-makers as a 𝑚 × 𝑛 matrix for weight W. ( 𝐷𝑀1 𝐷𝑀2 … 𝐷𝑀𝑛 𝐶1 ⟨ 𝛼{11} ,𝛽{11},𝛾{11}⟩ ⟨ 𝛼{12} ,𝛽{12},𝛾{12}⟩ … ⟨ 𝛼{1𝑛} ,𝛽{1𝑛},𝛾{1𝑛}⟩ 𝐶2 ⟨ 𝛼{21} ,𝛽{21},𝛾{21}⟩ ⟨ 𝛼{22} ,𝛽{22},𝛾{22}⟩ … ⟨ 𝛼{2𝑛} ,𝛽{2𝑛},𝛾{2𝑛}⟩ ⋮ ⋮ 𝐶𝑚 ⟨ 𝛼{𝑚1} ,𝛽{𝑚1},𝛾{𝑚1}⟩ ⟨ 𝛼{𝑚2} ,𝛽{𝑚2},𝛾{𝑚2}⟩ …⟨ 𝛼{𝑚𝑛} ,𝛽{𝑚𝑛},𝛾{𝑚𝑛}⟩ ) (1) Consider the ratings of n decision-makers over n criteria as a 𝑛 × 𝑚 matrix in alternative 𝐴1. ( 𝐶1 𝐶2 … 𝐶𝑚 𝐷𝑀1 ⟨ 𝑎{11} ,𝑏{11}, 𝑐{11}⟩ ⟨ 𝑎{12} ,𝑏{12}, 𝑐{12}⟩ …⟨ 𝑎{1𝑚} ,𝑏{1𝑚}, 𝑐{1𝑚}⟩ 𝐷𝑀2 ⟨ 𝛼{21} ,𝛽{21},𝛾{21}⟩ ⟨ 𝛼{22} ,𝛽{22},𝛾{22}⟩ …⟨ 𝛼{2𝑚} ,𝛽{2𝑚},𝛾{2𝑚}⟩ ⋮ ⋮ 𝐷𝑀𝑛 ⟨ 𝛼{𝑛1} ,𝛽{𝑛1},𝛾{𝑛1}⟩ ⟨ 𝛼{𝑛2} ,𝛽{𝑛2},𝛾{𝑛2}⟩ … ⟨ 𝛼{𝑛𝑚} ,𝛽{𝑛𝑚},𝛾{𝑛𝑚}⟩ ) (2) Step 2: Determine the weights of decision makers. Let 𝐷𝑀1,𝐷𝑀2,… 𝐷𝑀𝑛 be the decision makers, they have individual’s weights. Consider 𝐷𝑀1 = ⟨ 𝑥1,𝑦1,𝑧1⟩ , 𝐷𝑀2 = ⟨ 𝑥2,𝑦2,𝑧2⟩, …, 𝐷𝑀𝑛 = ⟨ 𝑥𝑛,𝑦𝑛,𝑧𝑛⟩ Step 3: Determine Rough Neutrosophic Matrix for criteria and alternatives. The relation between the weight of decision makers and criteria is formed as a Rough neutrosophic matrix for criteria. 𝑊(𝐶1𝐷𝑀1) = ⟨ (𝑚𝑖𝑛(𝑥1,𝛼11),𝑚𝑎𝑥(𝑦1,𝛽11),𝑚𝑎𝑥(𝑧1,𝛾11)),(𝑚𝑎𝑥(𝑥1,𝛼11),𝑚𝑖𝑛(𝑦1,𝛽11),𝑚𝑖𝑛(𝑧1,𝛾11))⟩ 𝑊(𝐶1𝐷𝑀1) = ⟨ (𝛼11,𝛽11,𝛾11),(𝛼11,𝛽11,𝛾11)⟩ (3) Jeni et al./Decis. Mak. Appl. Manag. Eng. 5 (2) (2022) 30-45 38 𝑊 = ( 𝐷𝑀1 … 𝐷𝑀𝑛 𝐶1 ⟨ (𝛼11,𝛽11,𝛾11),(𝛼11,𝛽11,𝛾11)⟩ … ⟨ (𝛼1𝑛,𝛽1𝑛,𝛾1𝑛),(𝛼1𝑛,𝛽1𝑛,𝛾1𝑛)⟩ 𝐶2 ⟨ (𝛼21,𝛽21,𝛾21),(𝛼21,𝛽21,𝛾21)⟩ … ⟨ (𝛼2𝑛,𝛽2𝑛,𝛾2𝑛) ,(𝛼2𝑛,𝛽2𝑛,𝛾2𝑛)⟩ ⋮ ⋮ 𝐶𝑚 ⟨ (𝛼𝑚1,𝛽𝑚1,𝛾𝑚1),(𝛼𝑚1,𝛽𝑚1,𝛾𝑚1)⟩ … ⟨ (𝛼𝑚𝑛,𝛽𝑚𝑛,𝛾𝑚𝑛),(𝛼𝑚𝑛,𝛽𝑚𝑛,𝛾𝑚𝑛)⟩ ) The relation between the weight of criteria and alternatives is formed as a Rough neutrosophic matrix for alternative. 𝐴1(𝐷𝑀1𝐶1) = ⟨ (𝑚𝑖𝑛(𝛼11,𝑎11),𝑚𝑎𝑥(𝛽11,𝑏11),𝑚𝑎𝑥(𝛾11 , 𝑐11)), (𝑚𝑎𝑥(𝛼11,𝑎11),𝑚𝑖𝑛(𝛽11,𝑏11),𝑚𝑖𝑛(𝛾11, 𝑐11))⟩ 𝐴1(𝐷𝑀1 𝐶1) = ⟨ (𝑎11,𝑏11, 𝑐11),(𝑎11,𝑏11, 𝑐11)⟩ (4) 𝐴1 = ( 𝐶1 … 𝐶𝑚 𝐷𝑀1 ⟨ (𝑎11,𝑏11, 𝑐11),(𝑎11,𝑏11, 𝑐11)⟩ … ⟨ (𝑎1𝑚,𝑏1𝑚, 𝑐1𝑚),(𝑎1𝑚,𝑏1𝑚, 𝑐1𝑚)⟩ 𝐷𝑀2 ⟨ (𝑎21,𝑏21, 𝑐21),(𝑎21,𝑏21, 𝑐21)⟩ … ⟨ (𝑎2𝑚,𝑏2𝑚, 𝑐2𝑚),(𝑎2𝑚,𝑏2𝑚, 𝑐2𝑚)⟩ ⋮ ⋮ 𝐷𝑀𝑛 ⟨ (𝑎𝑛1,𝑏𝑛1, 𝑐𝑛1),(𝑎𝑛1,𝑏𝑛1, 𝑐𝑛1)⟩ … ⟨ (𝑎𝑛𝑚,𝑏𝑛𝑚,𝑐𝑛𝑚),(𝑎𝑛𝑚,𝑏𝑛𝑚, 𝑐𝑛𝑚)⟩ ) Step 4: In this step, we convert the non-square matrix into a square matrix. From the above W, the matrix is expressed as 6 matrices which are truth, indeterminacy, false matrix of lower approximation and truth, indeterminacy, false matrix of upper approximation Which are denoted by 𝑊(𝑇 𝑖𝑗 ), 𝑊(𝐼 𝑖𝑗 ), 𝑊(𝐹 𝑖𝑗 ) and 𝑊(𝑇𝑖𝑗), 𝑊(𝐼𝑖𝑗), 𝑊(𝐹𝑖𝑗). Similarly, 𝐴1 matrix expressed as 𝐴1(𝑇𝑖𝑗), 𝐴1(𝐼𝑖𝑗), 𝐴1(𝐹𝑖𝑗) and 𝐴1(𝑇𝑖𝑗), 𝐴1(𝐼𝑖𝑗), 𝐴1(𝐹𝑖𝑗). 𝐴1 (𝑇𝑖𝑗)𝑛×𝑚 ∗ 𝑊(𝑇 𝑖𝑗 ) 𝑚×𝑛 = ( 𝛼𝑎 11 𝛼𝑎 21 ⋯ 𝛼𝑎 𝑛1 ⋮ ⋱ ⋮ 𝛼𝑎 1𝑛 𝛼𝑎 2𝑛 ⋯ 𝛼𝑎 𝑛𝑛 ) 𝑛×𝑛 (5) Step 5: Using the definition of Rough Neutrosophic matrix energy, calculate the energy of the matrix. We got six energies for truth, indeterminacy, and false matrices of lower and upper approximation for one alternative. 𝐸(𝐴1) = ⟨ (𝐸(𝐴1(𝑇)),𝐸(𝐴1(𝐼))𝐸(𝐴1(𝐹))),(𝐸(𝐴1(𝑇)) ,𝐸(𝐴1(𝐼))𝐸(𝐴1(𝐹)))⟩ (6) Step 6: Continue this process for k alternatives. For each alternative, we got Rough Neutrosophic matrix energies of 𝐸(𝐴1),𝐸(𝐴2)… 𝐸(𝐴𝑘). Step 7: For ranking the energy values we determine the average values of lower and upper approximation values. Then we get, The energy of rough neutrosophic matrix and its application to MCDM problem for selecting … 39 𝐸(𝐴1) = ⟨ 𝐸(𝐴1(𝑇)),𝐸(𝐴1(𝐼)),𝐸(𝐴1(𝐹)) ⟩ 𝐸(𝐴2 ) = ⟨ 𝐸(𝐴2 (𝑇)),𝐸(𝐴2(𝐼)),𝐸(𝐴2(𝐹)) ⟩ ⋮ 𝐸(𝐴𝑘) = ⟨ 𝐸(𝐴𝑘(𝑇)),𝐸(𝐴𝑘(𝐼)),𝐸(𝐴𝑘(𝐹))⟩ Finally, we rank the alternatives according to their truth values. The alternative that has the highest truth energy value will be the best. 5. Numerical Example We solve the problem by using our proposed method to choose the best place to construct the school building in a particular town. In this problem, the decision makers are Project manager (𝐷𝑀1), Approval officer (𝐷𝑀2), Engineer (𝐷𝑀3), and Public representative (𝐷𝑀4). The following are the criteria for deciding where to build: 𝐶1- Land Clearance, 𝐶2- Land Title, 𝐶3- Zonal Clearance, 𝐶4- Cost, 𝐶5- Transport Facility, and 𝐶6- Building Plan. The decision-makers choose the best place from the following alternatives based on the above criterion, Place A, Place B, Place C, Place D, Place E, and Place D. The decision makers give their ratings in terms of linguistic variables. IT is shown in Table 1. Table 1. Linguistic variable for SVNN S.No Linguistic Variable Neutrosophic numbers 1 Very Poor (VP)/ Very low (VL) ⟨ 0.1,0.8,0.9 ⟩ 2 Poor (P)/ Low (L) ⟨ 0.35,0.6,0.7⟩ 3 Medium (M)/ Fair (F) ⟨ 0.5,0.4,0.45 ⟩ 4 Good(G)/ High (H) ⟨ 0.8,0.2,0.15⟩ 5 Very Good (VG)/ Very High (VH) ⟨0.9,0.1,0.1 ⟩ Step: 1 The decision makers evaluate the criteria and each alternative by the linguistic variable. It is shown in Table 2 and 3 respectively. Table 2. Weights of Criteria Criteria 𝐷𝑀1 𝐷𝑀2 𝐷𝑀3 𝐷𝑀4 𝐶1 𝑉𝐺⟨ .9, .1, .1⟩ 𝐺⟨ .8, .2, .15 ⟩ 𝑉𝐺⟨ .9, .1, .1⟩ 𝑀⟨ .5, .4, .45 ⟩ 𝐶2 𝑀⟨ .5, .4, .45 ⟩ 𝐺⟨ .8, .2, .15 ⟩ 𝑀⟨ .5, .4, .45 ⟩ 𝐺⟨ .8, .2, .15 ⟩ 𝐶3 𝐺⟨ .8, .2, .15 ⟩ 𝑉𝐺⟨ .9, .1, .1⟩ 𝐺⟨ .8, .2, .15 ⟩ 𝐺⟨ .8, .2, .15 ⟩ 𝐶4 𝐻⟨ .8, .2, .15 ⟩ 𝐻⟨ .8, .2, .15 ⟩ 𝐹⟨ .5, .4, .45 ⟩ 𝑉𝐻⟨ .9, .1, .1 ⟩ 𝐶5 𝑀⟨ .5, .4, .45⟩ 𝐺⟨ .8, .2, .15 ⟩ 𝐺⟨ .8, .2, .15 ⟩ 𝑀⟨ .5, .4, .45 ⟩ 𝐶6 𝑉𝐺⟨ .9, .1, .1⟩ 𝑉𝐺⟨ .9, .1, .1⟩ 𝐺⟨ .8, .2, .15 ⟩ 𝐺⟨ .8, .2, .15 ⟩ Table 3. Ratings in terms of Linguistic variables for each alternative Alt 𝐷𝑀 𝐶1 𝐶2 𝐶3 𝐶4 𝐶5 𝐶6 A 𝐷𝑀1 𝐺⟨.8, .2, .15⟩ 𝑉𝐺⟨.9, .1, .1⟩ 𝐺⟨.8, .2, .15⟩ 𝐹⟨.5, .4, .45⟩ 𝑀⟨.5, .4, .45⟩ 𝐺⟨.8, .2, .15⟩ 𝐷𝑀2 𝑀⟨.5, .4, .45⟩ 𝐺⟨.8, .2, .15⟩ 𝑀⟨.5, .4, .45⟩ 𝑉𝐻⟨.9, .1, .1⟩ 𝑉𝐺⟨.9, .1, .1⟩ 𝑃⟨.35, .6, .7⟩ Jeni et al./Decis. Mak. Appl. Manag. Eng. 5 (2) (2022) 30-45 40 𝐷𝑀3 𝑃⟨.35, .6, .7⟩ 𝑀⟨.5, .4, .45⟩ 𝐺⟨.8, .2, .15⟩ 𝐻⟨.8, .2, .15⟩ 𝑀⟨.5, .4, .45⟩ 𝑉𝐺⟨.9, .1, .1⟩ 𝐷𝑀4 𝑀⟨.5, .4, .45⟩ 𝑃⟨.35, .6, .7⟩ 𝑀⟨.5, .4, .45⟩ 𝐻⟨.8, .2, .15⟩ 𝑉𝐺⟨.9, .1, .1⟩ 𝐺⟨.8, .2, .15⟩ B 𝐷𝑀1 𝑀⟨.5, .4, .45⟩ 𝐺⟨.8, .2, .15⟩ 𝑃⟨.35, .6, .7⟩ 𝐿⟨.35, .6, .7⟩ 𝐺⟨.8, .2, .15⟩ 𝑀⟨.5, .4, .45⟩ 𝐷𝑀2 𝑉𝑃⟨.1, .8, .9⟩ 𝑀⟨.5, .4, .45⟩ 𝐺⟨.8, .2, .15⟩ 𝐿⟨.35, .6, .7⟩ 𝑀⟨.5, .4, .45⟩ 𝐺⟨.8, .2, .15⟩ 𝐷𝑀3 𝑃⟨.35, .6, .7⟩ 𝑃⟨.35, .6, .7⟩ 𝐺⟨.8, .2, .15⟩ 𝐹⟨.5, .4, .45⟩ 𝑃⟨.35, .6, .7⟩ 𝑀⟨.5, .4, .45⟩ 𝐷𝑀4 𝐺⟨.8, .2, .15⟩ 𝑉𝑃⟨.1, .8, .9⟩ 𝐺⟨.8, .2, .15⟩ 𝐹⟨.5, .4, .45⟩ 𝑉𝑃⟨.1, .8, .9⟩ 𝑃⟨.35, .6, .7⟩ C 𝐷𝑀1 𝑉𝐺⟨.9, .1, .1⟩ 𝐺⟨.8, .2, .15⟩ 𝑉𝐺⟨.9, .1, .1⟩ 𝐹⟨.5, .4, .45⟩ 𝐺⟨.8, .2, .15⟩ 𝐺⟨.8, .2, .15⟩ 𝐷𝑀2 𝑀⟨.5, .4, .45⟩ 𝑉𝐺⟨.9, .1, .1⟩ 𝐺⟨.8, .2, .15⟩ 𝐻⟨.8, .2, .15⟩ 𝑉𝐺⟨.9, .1, .1⟩ 𝑀⟨.5, .4, .45⟩ 𝐷𝑀3 𝐺⟨.8, .2, .15⟩ 𝑀⟨.5, .4, .45⟩ 𝐺⟨.8, .2, .15⟩ 𝐹⟨.5, .4, .45⟩ 𝑀⟨.5, .4, .45⟩ 𝐺⟨.8, .2, .15⟩ 𝐷𝑀4 𝑉𝐺⟨.9, .1, .1⟩ 𝑉𝐺⟨.9, .1, .1⟩ 𝑀⟨.5, .4, .45⟩ 𝐻⟨.8, .2, .15⟩ 𝐺⟨.8, .2, .15⟩ 𝑀⟨.5, .4, .45⟩ D 𝐷𝑀1 𝐺⟨.8, .2, .15⟩ 𝑀⟨.5, .4, .45⟩ 𝐺⟨.8, .2, .15⟩ 𝐿⟨.35, .6, .7⟩ 𝑃⟨.35, .6, .7⟩ 𝑉𝑃⟨.1, .8, .9⟩ 𝐷𝑀2 𝑃⟨.35, .6, .7⟩ 𝑉𝑃⟨.1, .8, .9⟩ 𝐺⟨.8, .2, .15⟩ 𝐹⟨.5, .4, .45⟩ 𝑉𝐺⟨.9, .1, .1⟩ 𝑃⟨.35, .6, .7⟩ 𝐷𝑀3 𝑀⟨.5, .4, .45⟩ 𝑃⟨.35, .6, .7⟩ 𝑃⟨.35, .6, .7⟩ 𝐻⟨.8, .2, .15⟩ 𝑃⟨.8, .2, .15⟩ 𝑉𝑃⟨.1, .8, .9⟩ 𝐷𝑀4 𝑃⟨.35, .6, .7⟩ 𝑀⟨.5, .4, .45⟩ 𝑉𝑃⟨.1, .8, .9⟩ 𝐻⟨.8, .2, .15⟩ 𝑀⟨.5, .4, .45⟩ 𝑀⟨.5, .4, .45⟩ E 𝐷𝑀1 𝐺⟨.8, .2, .15⟩ 𝑃⟨.35, .6, .7⟩ 𝑃⟨.35, .6, .7⟩ 𝐹⟨.5, .4, .45⟩ 𝑀⟨.5, .4, .45⟩ 𝐺⟨.8, .2, .15⟩ 𝐷𝑀2 𝑃⟨.35, .6, .7⟩ 𝐺⟨.8, .2, .15⟩ 𝑉𝐺⟨.9, .1, .1⟩ 𝐻⟨.8, .2, .15⟩ 𝑀⟨.5, .4, .45⟩ 𝐺⟨.8, .2, .15⟩ 𝐷𝑀3 𝑀⟨.5, .4, .45⟩ 𝑃⟨.35, .6, .7⟩ 𝑉𝑃⟨.1, .8, .9⟩ 𝐿⟨.35, .6, .7⟩ 𝐺⟨.8, .2, .15⟩ 𝑃⟨.35, .6, .7⟩ 𝐷𝑀4 𝑀⟨.5, .4, .45⟩ 𝑃⟨.35, .6, .7⟩ 𝐺⟨.8, .2, .15⟩ 𝐿⟨.35, .6, .7⟩ 𝑀⟨.5, .4, .45⟩ 𝑃⟨.35, .6, .7⟩ Step: 2 Weights of decision makers. 𝐷𝑀1 = 𝐺 ⟨ 0.8,0.2,0.15 ⟩ , 𝐷𝑀2 = 𝑉𝐺⟨ 0.9,0.1,0.1⟩, 𝐷𝑀3 = 𝑀 ⟨ 0.5,0.4,0.45 ⟩ and 𝐷𝑀4 = 𝐺 ⟨ 0.8,0.2,0.15 ⟩ Step: 3 Determine Rough Neutrosophic Matrix for criteria. Table 4 shows that the relation between the weight of decision makers and criteria are formed as Rough neutrosophic matrix for criteria 𝑊(𝐶1𝐷𝑀1) = ⟨ (min (0.8,0.9),max (0.2,0.1),max (0.15,0.1)), (𝑚𝑎𝑥(0.8,0.9),𝑚𝑖𝑛(0.2,0.1),𝑚𝑖𝑛(0.15,0.1))⟩ 𝑊(𝐶1𝐷𝑀1) = ⟨ (0.8,0.2,0.15),(0.9,0.1,0.1)⟩ Table 4. Rough Neutrosophic Matrix of Criteria C 𝐷𝑀1 𝐷𝑀2 𝐷𝑀3 𝐷𝑀4 𝐶1 ⟨(.8, .2, .15),(.9, .1, .1) ⟩ ⟨(.8, .2, .15),(.9, .1, .1) ⟩ ⟨(.5, .4, .45),(.9, .1, .1)⟩ ⟨(.5, .4, .45),(.8, .2, .15)⟩ 𝐶2 ⟨(.5, .4, .45),(.8, .2, .15)⟩ ⟨(.8, .2, .15),(.9, .1, .1) ⟩ ⟨(.5, .4, .45),(.5, .4, .45)⟩ ⟨(.8, .2, .15),(.8, .2, .15)⟩ 𝐶3 ⟨(.8, .2, .15),(.8, .2, .15)⟩ ⟨(.9, .1, .1),(.9, .1, .1)⟩ ⟨(.5, .4, .45),(.8, .2, .15)⟩ ⟨(.8, .2, .15),(.8, .2, .15)⟩ 𝐶4 ⟨(.8, .2, .15),(.8, .2, .15)⟩ ⟨(.8, .2, .15),(.9, .1, .1) ⟩ ⟨(.5, .4, .45),(.5, .4, .45)⟩ ⟨(.8, .2, .15),(.9, .1, .1)⟩ 𝐶5 ⟨(.5, .4, .45),(.8, .2, .15)⟩ ⟨(.8, .2, .15),(.9, .1, .1) ⟩ ⟨(.5, .4, .45),(.8, .2, .15)⟩ ⟨(.5, .4, .45),(.8, .2, .15)⟩ 𝐶6 ⟨(.8, .2, .15),(.9, .1, .1)⟩ ⟨(.9, .1, .1),(.9, .1, .1)⟩ ⟨(.5, .4, .45),(.8, .2, .15)⟩ ⟨(.8, .2, .15),(.8, .2, .15)⟩ Table 5 shows that the relation between the weight of criteria and alternatives are formed as Rough neutrosophic matrix for alternative 𝐴1(𝐷𝑀1𝐶2) = ⟨ (𝑚𝑖𝑛(0.5,0.9),𝑚𝑎𝑥(0.4,0.1),𝑚𝑎𝑥(0.45,0.1)), (𝑚𝑎𝑥(0.5,0.9),𝑚𝑖𝑛(0.4,0.1),𝑚𝑖𝑛(0.45,0.1))⟩ 𝐴1(𝐷𝑀1𝐶2) = ⟨ (0.5,0.4,0.45),(0.9,0.1,0.1) ⟩ The energy of rough neutrosophic matrix and its application to MCDM problem for selecting … 41 Table 5. Rough neutrosophic matrix of Alternative 1 DM Rough Neutrosophic values of each criterion for 𝐴1 𝐷𝑀1 𝐶1⟨(.8, .2, .15),(.9, .1, .1)⟩ 𝐶2⟨(.5, .4, .45),(.9, .1, .1)⟩ 𝐶3⟨(.8, .2, .15),(.8, .2, .15)⟩ 𝐶4⟨(.5, .4, .45),(.8, .2, .15)⟩ 𝐶5⟨(.5, .4, .45),(.5, .4, .45) ⟩ 𝐶6⟨(.8, .2, .15),(.9, .1, .1)⟩ 𝐷𝑀2 𝐶1⟨(.5, .4, .45),(.8, .2, .15)⟩ 𝐶2⟨(.8, .2, .15),(.8, .2, .15)⟩ 𝐶3⟨(.5, .4, .45),(.9, .1, .1)⟩ 𝐶4⟨(.8, .2, .15),(.9, .1, .1)⟩ 𝐶5⟨(.8, .2, .15),(.9, .1, .1)⟩ 𝐶6⟨(.35, .6, .7),(.9, .1, .1)⟩ 𝐷𝑀3 𝐶1⟨(.35, .6, .7),(.9, .1, .1)⟩ 𝐶2⟨(.5, .4, .45),(.5, .4, .45)⟩ 𝐶3⟨(.8, .2, .15),(.8, .2, .15)⟩ 𝐶4⟨(.5, .4, .45),(.8, .2, .15)⟩ 𝐶5⟨(.5, .4, .45),(.8, .2, .15)⟩ 𝐶6⟨(.8, .2, .15),(.9, .1, .1)⟩ 𝐷𝑀4 𝐶1⟨(.5, .4, .45),(.5, .4, .45)⟩ 𝐶2⟨(.1, .8, .9),(.8, .2, .15)⟩ 𝐶3⟨(.5, .4, .45),(.8, .2, .15)⟩ 𝐶4⟨(.8, .2, .15),(.9, .1, .1)⟩ 𝐶5⟨(.5, .4, .45),(.9, .1, .1)⟩ 𝐶6⟨(.8, .2, .15),(.8, .2, .15)⟩ Step 4: We convert the non-square matrix into a square matrix. From table 4 and 5, we expressed the both matrices into 6 matrices. Now we consider the truth lower approximation matrix of both tables 𝐴1 (𝑇𝑖𝑗)𝑛×𝑚 = [ 0.8 0.5 0.8 0.5 0.5 0.8 0.5 0.8 0.5 0.8 0.8 0.35 0.35 0.5 0.8 0.5 0.5 0.8 0.5 0.1 0.5 0.8 0.5 0.8 ] 𝑊(𝑇 𝑖𝑗 ) 𝑚×𝑛 = [ 0.8 0.5 0.8 0.8 0.5 0.8 0.8 0.8 0.9 0.8 0.8 0.9 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.8 0.8 0.8 0.5 0.8 ] 𝐴1 (𝑇𝑖𝑗)𝑛×𝑚 ∗ 𝑊(𝑇 𝑖𝑗 ) 𝑚×𝑛 = [ 2.82 3.28 1.95 2.73 2.52 3.085 1.875 2.61 2.46 2.92 1.725 2.505 2.38 2.69 1.6 2.26 ] Step 5: Calculating the Energy of Rough Neutrosophic Matrix Eigen values of the above matrix 9.8836, 0.0176 + 0.0579i, 0.0176 - 0.0579i, - 0.0287 and mean of the eigen values is 2.4725. 𝐸(𝑇 𝑖𝑗 ) = |9.8836 − 2.4725| + |0.0176 + 0.0579𝑖 − 2.4725| + |0.0176 − 0.0579𝑖 − 2.4725| + |−0.0287 − 2.4725| = 14.8235 Step 6: Similarly, we can find the energy for all the matrices of lower and upper approximation of truth, indeterminacy, and false matrices. Energy of Rough neutrosophic Matrices Place A = [(14.8236, 3.6111, 3.8090), (23.7794, 1.1667, 0.9430)] Place B = [(11.4882, 4.8871, 5.5057), (23.2477, 1.3475, 1.1150)] Place C = [(16.1570, 2.9861, 3.0163), (24.2484, 1.1979, 1.0370)] Place D = [(11.0676, 4.9059, 5.4638), (22.5909, 1.3974, 1.1322)] Place E = [(13.4987, 4.3102, 4.6900), (22.1196, 1.4901, 1.2752)] Jeni et al./Decis. Mak. Appl. Manag. Eng. 5 (2) (2022) 30-45 42 Step 7: Calculate the average value of the lower and upper approximation of energy of Rough Neutrosophic sets, then the ranking of alternatives will be decided by the truth values. Average energy of each Alternative is given below Place A = [(19.3015, 2.3889, 2.376)] Place B = [(17.3679, 3.1173, 3.3103)] Place C = [(20.2027, 2.092, 2.0266)] Place D = [(16.8292, 3.1516, 3.298)] Place E = [(17.8091, 2.9001, 2.9826)] The Average energy of truth and ranking order of alternatives presented in Table 6. Table 6. Ranking order Alternatives Truth Energy Ranking Order Place A 19.3015 II Place B 17.3679 IV Place C 20.2027 I Place D 16.8292 V Place E 17.8091 III The ranking order of the alternatives is C > A > E > B > D. Place C is the best location to build the school construction in the town. 6. Conclusion The energy of the matrix helps to determine the matrix's weight. We apply this idea of energy to the Rough neutrosophic matrix. The Energy of Rough Neutrosophic Matrix contains truth indeterminacy and false energy for the lower and upper approximations of each matrix. The final energy was determined by averaging the lowest and upper approximations of each energy. In that, the ranking of alternatives is evaluated using truth value. In our taken problem, the decision-maker chooses the perfect spot for the construction of the school. The building should be constructed at Place C. It satisfies all requirements. As a result, the Energy of Rough Neutrosophic Matrix will be used in every situation and our proposed energy method helps to solve the multi-criteria decision-making problems. Compared to other MCDM methods our presented method simplifies the work and also give more effective result. Further, we will extend the Rough neutrosophic energy concept to other types of Rough neutrosophic matrices such as interval-valued, multi-valued and so on. Author Contributions: Conceptualization, J.S.M. and D.G.; methodology, J.S.M. and D.G.; software, J.S.M. and D.G.; validation, J.S.M. and D.G.; formal analysis, J.S.M. and D.G.; investigation, D.G.; writing—original draft preparation, J.S.M; writing—review and editing, D.G.; visualization, J.S.M. and D.G.; supervision, D.G. All authors have read and agreed to the published version of the manuscript. The energy of rough neutrosophic matrix and its application to MCDM problem for selecting … 43 Funding: This research received no external funding. 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