Mathematical Problems in Engineering

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Application of Operations Research Tools for Solving Sustainable Engineering Problems

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Research Article | Open Access

Volume 2021 |Article ID 1239505 | https://doi.org/10.1155/2021/1239505

Amin Vafadarnikjoo, Marco Scherz, "A Hybrid Neutrosophic-Grey Analytic Hierarchy Process Method: Decision-Making Modelling in Uncertain Environments", Mathematical Problems in Engineering, vol. 2021, Article ID 1239505, 18 pages, 2021. https://doi.org/10.1155/2021/1239505

A Hybrid Neutrosophic-Grey Analytic Hierarchy Process Method: Decision-Making Modelling in Uncertain Environments

Academic Editor: Zeljko Stevic
Received07 May 2021
Revised01 Jun 2021
Accepted08 Jun 2021
Published18 Jun 2021

Abstract

The analytic hierarchy process (AHP) is recognised as one of the most commonly applied methods in the multiple attribute decision-making (MADM) literature. In the AHP, encompassing uncertainty feature necessitates using suitable uncertainty theories, since dealing efficiently with uncertainty in subjective judgements is of great importance in real-world decision-making problems. The neutrosophic set (NS) theory and grey systems are two reliable uncertainty theories which can bring considerable benefits to uncertain decision-making. The aim of this study is to improve uncertain decision-making by incorporating advantages of the NS and grey systems theories with the AHP in investigating sustainability through agility readiness evaluation in large manufacturing plants. This study pioneers a combined neutrosophic-grey AHP (NG-AHP) method for uncertain decision-making modelling. The applicability of the hybrid NG-AHP method is shown in an illustrative real-case study for agility evaluations in the Iranian steel industry. The computational results indicate the effectiveness of the proposed method in adequately capturing uncertainty in the subjective judgements of decision makers. In addition, the results verify the significance of the research in group decision-making under uncertainty. The practical outcome reveals that, to become a more sustainable agile steel producer in the case country, they should first focus on the “organisation management agility” as the most significant criterion in the assessment followed by “manufacturing process agility,” “product design agility,” “integration of information system,” and “partnership formation capability,” respectively.

1. Introduction

In recent years, corporations have moved to the centre of focus in the sustainability debate. The reason for this is that they are considered to be responsible for enormous negative impacts on the environment and society [1]. Sustainability aims to produce a dynamic balance between the three sustainability dimensions: environmental, economic, and social over time [2]. In the contexts of sustainable operations and agility, agile manufacturing (AM) should first be defined. AM has the property of robustness, which means that AM systems must be able to tolerate changes and interruptions within the given demand requirements. Therefore, AM operations can be seen as inherently sustainable. In other words, agility and sustainability are interconnected in this sense, and an agile system can have the potential capability to work as a sustainable system. This link has not been adequately researched in the literature. In this study, we have considered the general definition of sustainability taking into account all three pillars of sustainability in the steel industry.

In addition to the predicted increase in the global population up to 9.7 billion people by 2050 [3], the steel requirement per capita is also expected to increase by 2050 to 11.8 tons. Steel production is responsible for around 7% of global greenhouse gas (GHG) emissions. Bearing in mind the increasing production volume per capita and the necessity to decrease our global greenhouse gas emissions to tackle climate change, it is evident that the steel industry has to shift to more sustainable processes [4]. Steel manufacturing needs more attention from the sustainable development perspective particularly in developing countries. Iskanius et al. [5] investigated the leading forces and abilities for agility in Finnish steel product manufacturing and concluded that the need for agility is clearly recognised in the traditional steel industry and it has to be considered in the long-term strategy planning of steel manufacturing enterprises. Thus, the research question to be addressed is as follows:

RQ: how can agility readiness impact sustainable engineering decisions under uncertain decision-making environment in steel manufacturing?

Decision support tools such as multiple attribute decision-making (MADM) can now be identified as invaluable business analytic methods for helping large organisations to move forward towards providing sustainable operations by developing agility in their manufacturing processes. In the MADM literature, there are methods that have simple implementation and flexibility such as analytic hierarchy process (AHP), best-worst method (BWM) [6], level-based weight assessment (LBWA) [7], and full consistency method (FUCOM) [8]. The AHP is one of the most commonly practised MADM methods [9] mainly because of its ease of application and its flexibility for integration with various methods. An abundance of studies in the literature is focused specifically on applications of AHP such as traffic accessibility [10], advertising media selection [11], selecting e-purse smart card technology [12], and topic popularity selection [13], to name a few. AHP has been utilised to assess complicated multiattribute alternatives by collecting opinions of a group of decision makers (DMs). The feature of inclusion of subjective factors has been considered as one of the AHP’s advancements compared to other MADM methods [14]. Many studies have focused on the fuzzy set (FS)-based extension of AHP, namely, fuzzy AHP (F-AHP), so as to capture uncertainty [1518]. However, few studies have considered the extension of AHP simultaneously with other uncertainty theories such as grey systems and neutrosophic set (NS) theories, which are able to enhance decision-making process under uncertain environment. There are only a few recent developments and applications of AHP and NS theory in the literature [1925]. Furthermore, only limited research, which is directly related to grey AHP method, has been carried out [2633]. This gap has motivated the current research to develop AHP under hybrid grey and NS decision-making environments to deal with uncertainty embedded in human subjective judgements which can incorporate the advantage of both in one decision-making model. The NS theory is able to independently quantify the indeterminacy membership function values. Unlike the FS theory, the NS theory has the capability to express the information about rejection. There are growing applications of the NS theory in the decision-making literature [6, 24, 3440]. Smarandache [41] introduced the NS theory and in [42] thoroughly elaborated on the distinctions between NS and intuitionistic fuzzy set (IFS) theories by providing explanatory examples. It has improved the IFS theory which was initially introduced by Atanassov [43] as an extension of Zadeh’s FS theory [44]. Besides the NS theory, Pythagorean fuzzy set (PFS) theory was introduced by Yager [45] and has been a recent extension of IFS theory which is drawing the attention of researchers in the realm of decision-making under uncertainty [4648]. D-numbers are also introduced to deal with uncertainty in decision-making [49, 50]. In addition to NS theory, grey system theory compared to many mainstream uncertainty theories, such as FS theory, has appreciable features, particularly when it is necessary to deal with uncertain data, and lack of information such as (1) generating satisfactory results utilising a relatively small data volume; (2) producing robust results regarding the noise, and lack of modelling information; and (3) yielding fairly flexible, nonparametric assumptions, and a general way to integrate fuzziness into a problem [16]. Smarandache [51] discussed the NS theory and grey systems all together. In most grey AHP studies, the utilised whitenisation functions cause information loss by converting grey information into crisp values. Moreover, calculating the consistency ratio (CR) to check the consistency among evaluations of DMs in pairwise comparison matrices is another cause of concern. Additionally, in several studies, the integration method is utilised as a combination of grey relational analysis (GRA), and AHP or grey incidence analysis (GIA) and AHP [5256]. The GRA and GIA are characterised under the grey system theory concept as two distinct MADM methods. The main common feature of these studies is that AHP is applied for calculating criteria weights, and then either GRA or GIA is used for evaluation of alternatives. This category of studies should not be mixed up with the grey-based AHP method, because GRA/GIA-AHP methods do not apply AHP in combination with grey systems theory.

In this paper, we take advantage of operational research (OR) tools from the realm of MADM to evaluate agility readiness of Iranian steel manufacturing corporations with the aim of developing sustainable operations. Two methods, namely, G-AHP (i.e., grey AHP) and N-AHP (i.e., neutrosophic AHP), are combined, and the application of the model is demonstrated as a hybrid method NG-AHP (i.e., neutrosophic-grey AHP) in an agility evaluation case in the Iranian steel industry. It is believed that based on the provided method, the uncertainty of DMs can be best handled via hybrid neutrosophic-grey uncertainty theories. The N-AHP approach is an integration of the NS theory with the AHP, in which the single-valued trapezoidal neutrosophic numbers (SVTNNs) are utilised in the AHP calculations [19]. The proposed G-AHP method has furthered the existing grey AHP methods in two major ways. Firstly, it preserves the grey characteristics of grey numbers during calculation steps by reducing information loss, specifically by omitting the need for whitenisation function deployment. Secondly, it ensures assessment consistency in pairwise comparisons by introducing two importance rating scales and constructing the pairwise comparisons based on the suggested procedure. It also obtains the aggregated opinions of DMs efficiently, while also handling the inherent ambiguity in the subjective judgements of DMs through preserving grey values.

To the best of our knowledge, OR tools such as MADM methods have not been applied extensively for sustainable development and there is a gap in the literature in this matter [57]. Moreover, only a few studies have explored sustainability through the agility perspective in manufacturing settings. In other words, achieving sustainable operations in engineering through AM or sustainable agility has not been a well-researched topic in the literature. It is explained that agility and sustainability are closely connected, meaning that studying agility in manufacturing enterprises can thus lead to better understanding of sustainable development. To bridge this gap, we are exploring agility readiness in the steel manufacturing business by applying a novel combined MADM method. The current research features the following three specific contributions:(1)Investigating sustainable engineering by agility readiness evaluation in an Iranian steel manufacturing setting.(2)Extending group AHP to a grey environment (G-AHP) while preserving the grey characteristics of the judgements with fully consistent evaluations in the pairwise comparison matrices. The first characteristic of the proposed G-AHP is that no whitenisation function is needed unlike most other grey AHP methods which would utilise whitenisation functions to get crisp values. The second feature is that no CR calculation is required in pairwise comparison matrices due to the way they are established, which also helps save time and cost. To the best of our knowledge, no grey AHP method in the literature has these two traits simultaneously and with a straightforward procedure.(3)Integrating N-AHP [19] with G-AHP (i.e., NG-AHP) in a real-world agility evaluation case in the Iranian steel industry to illustrate its capability and versatility in one hybrid methodological application. The application of the hybrid methodology (i.e., NG-AHP) was shown to reveal the benefits of both methods in one single framework, while also emphasising the synergistic effects of the two in one single framework and also overcoming their drawbacks at the same time.

In Figure 1, a generic hierarchical structure is shown in which the applied levels of proposed methods are presented. The calculation steps of each method are shown in Figure 2.

The proposed hybrid NG-AHP is comprised of two separate methods including integration of the N-AHP (Section 4), and G-AHP (Section 5). The criteria weights are calculated by the N-AHP, and the importance weights of alternatives are obtained by G-AHP. Ultimately, the weights are integrated, and alternatives are ranked to calculate the total weights of alternatives in the final decision matrix (Section 6). In Section 7, findings are discussed, and the paper is concluded in Section 8.

2. Sustainability and Agility

Agility is characterised as the ability to react to and handle unpredictable changes and encompasses cost reduction, quality improvement, delivery, and service improvement. Agility lies in the domain of AM which is the ability to meet volatile business requirements with adaptability and has been developed in response to lean manufacturing (LM) systems [58, 59]. Leanness aims at maximising profit through cost reduction, while agility tries to maximise profit by providing precisely what a customer needs [60]. Agility is also considered as the interface between the company and the market [61].

Sustainability generally concentrates on protecting natural resources against exploitation via productivity and competitiveness by manufacturing and service organisations. However, the concept of sustainability includes two key aspects other than the environmental aspect, which are economic and social [62, 63]. Thereby, the three dimensions of sustainability (i.e., environmental, economic, and social) have to be considered and treated equally. Gunasekaran and Spalanzani [62] investigated sustainable business development (SBD) in manufacturing and services, which has been regarded as a critical issue due to many causes such as climate change and natural disasters. Sustainability efforts can be included in all stages of a supply chain from product design and manufacturing to the product end-of-life stage such as remanufacturing [64]. Rostamzadeh et al. [65] investigated sustainability issues in the supply chain risk management domain by applying an integrated fuzzy MCDM based on TOPSIS and criteria importance through intercriteria correlation (CRITIC). Ivory and Brooks [66] offered a conceptual framework illuminating the strategic agility metacapabilities (resource fluidity, collective commitment, and strategic sensitivity) and related practices/processes that firms use to effectively deal with corporate sustainability with a paradoxical lens.

There would be an intuitive possible connection between agility and sustainability because more efficient and improved quality production by being quick and flexible in agile manufacturing potentially would lead to less production waste and carbon emissions and ultimately to more sustainable production. Carvalho et al. [60] recognised the trade-offs between lean, agile, resilient, and green (LARG) management systems as a probable pathway towards a more sustainable system. It is also indicated that agility and sustainability are regarded as performance measures for contemporary enterprises. In the current manufacturing scenario, agility needs to be matched with sustainability [67]. Pham and Thomas [68] suggested that for firms to be competitive, they should achieve an effective level of leanness, agility, and sustainability that associates with change and uncertainty in an operational system and the individual business environment. Flumerfelt et al. [59] investigated theories and practices of agile and lean manufacturing systems to gain an understanding of whether these employ sustainability or not. They recognised AM operations are sustainable because sustainability means ability to endure, and AM systems must be robust which means they are capable to endure alterations under various demand circumstances.

3. Preliminaries

3.1. Neutrosophic Set Theory

Some basic definitions of NS theory are provided in this section [69].

Definition 1. (see [41]). Let be a finite set of objects and let signify a generic element in . The NS in is characterised by a truth-membership function , an indeterminacy-membership function , and a falsity-membership function . , , and are the elements of . It can be shown asNote that .

Definition 2. (see [70]). Let be a finite set of elements, and let signify a generic element in . A single-valued neutrosophic set (SVNS) in is defined by a truth-membership function , an indeterminacy-membership function , and a falsity-membership function . , , and are the elements of . It can be shown asNote that .
For convenience, an SVNS is sometimes shown as a called simplified form.

Definition 3. (see [71]). An SVTNN , , , and is a particular single-valued neutrosophic number (SVNN) whose , , and are presented as the following equations, respectively:

Definition 4. (see [72]). Given , and , and , , , , and , and then equations (6) and (7) are true:When , then equations (8) and (9) are true:

Definition 5. (see [72]). Given and . Then, the score function of can be calculated in accordance with the following equation:

Definition 6. (see [72]). In order to compare two SVTNNs , and where , then according to equation (10), the score functions will be computed, and if , then ; if , then .

Definition 7. Let be a set of SVTNNs, then a trapezoidal neutrosophic weighted arithmetic averaging (TNWAA) operator is computed on the basis of [72]where is the weight of while , and .

3.2. Subtraction, Division, and Inverse of SVTNNs

The subtraction and division of simplified SVNNs (or single-valued neutrosophic values) and SVNSs are introduced by Smarandache [73] and Ye [74], respectively. Rani and Garg [75] also studied subtraction and division operations on interval neutrosophic sets. In this section, subtraction, division, and inverse of SVTNNs in general nonsimplified form are defined.

3.2.1. Subtraction of SVTNNs

Let , and be two SVTNNs, and with the restrictions that , , , and , , and , then the subtraction of the two SVTNNs is shown in

Note: for a negative value, replace it with zero. For a value of over one, replace it with one.

Proof. Let us consider equation (13) whereBy adding neutrosophically, to the sides of equation (13)–(16) results,Then,andIt is concluded that , noting the remark above.

3.2.2. Division of SVTNNs

Let and be two SVTNNs where , , , and with the restrictions that , , and , then the division of the two SVTNNs is shown in

Note: for a negative value, replace it with zero. For a value of over one, replace it with one.

Proof. Let us consider equation (18) whereBy multiplying neutrosophically, to the sides of equation (18)–(21) is obtained:Then,and

3.2.3. Inverse of an SVTNN

Let be an SVTNN where , , and , then the inverse of is represented in

Note: for a negative value, replace it with zero. For a value of over one, replace it with one.

Proof. Let us consider equation (23), where , and ,Then, based on the division rule of two SVTNNs referring to equation (17), the proof is provided.

3.3. Grey System Theory

In this section, some basic definitions of grey systems theory are provided [69].

Definition 8. A grey number is defined as an interval with known upper, and lower bounds which are shown by and , respectively, but there is no known distribution information for [76, 77], as represented in

Definition 9. Given and are two grey numbers, then the basic operations of grey numbers can be defined as follows [78, 79]:

Definition 10. The length of a grey number is defined as

Definition 11. Comparison of grey numbers [80].
Given and are two grey numbers, the possibility degree of can be defined as follows:where .
There are four possible cases on the real number axis to determine the relationship between and :(1)If and , then . Thus, (2)If , then . Thus, (3)If , then . Thus, (i)(4-a) If , then (ii)(4-b) If , then

Definition 12. (see [81]). Whitenised (whitened or crisp value) of a grey number is a deterministic number with its value between the upper and lower bounds of a grey number . The whitenised value can be defined as equation (31) in which is whitening coefficient, and :For , equation (32) will be resulted:

Definition 13. (see [81, 82]). Given and are two grey numbers, then the distance between and can be calculated as signed difference between their centres as shown in

Definition 14. (see [79]). Given is a grey number, and ; then, equation (34) is resulted:

4. The N-AHP Method

The N-AHP method follows the steps below as introduced in [19].  Step 1(hierarchical structure): it is an essential step to establish a hierarchy, representing the goal, criteria, and alternatives because it makes the problem more comprehensible.  Step 2(pairwise comparison matrix): the DMs evaluate elements (i.e., alternatives or criteria), using the Saaty rating scale Table 1. In the experts’ judgements questionnaire, DMs choose a linguistic phrase representing the importance degree of each element in comparison to others.Given signify the elements, and shows a quantified evaluation on a pair of elements, and by DM . This leads to a pairwise comparison matrix as represented in [84, 85]  Step 3(calculating CR): referring to Saaty’s suggestion [86], a consistency test has to be conducted to differentiate the consistent comparisons from the inconsistent comparisons. See equation (36) and Table 2. If the value , then the DMs have to do a revision in their evaluations [88]:  Step 4(replacing the linguistic information with the SVTNNs): the elements of the pairwise comparison matrices are replaced with the corresponding SVTNNs using the scale shown in Table 3 (see Section 3.2.3 for calculating inverse of an SVTNN).  Step 5(aggregating the opinions of DMs in SVTNNs): to aggregate the opinions of DMs, the TNWAA operator is used, as shown in equation (11).  Step 6(neutrosophic synthetic values): the neutrosophic synthetic value of each element is computed based onwhere is the number of elements and is the element of the aggregated pairwise comparison matrix.  Step 7(determining the final importance weights): this is calculated based on equation (38), and the final importance weights are shown by which are in SVTNNs. In order to compare weights, equation (10) is used:


Numerical ScaleVerbal Scale

1Equal importance
2Weak importance
3Moderate importance
4Moderate plus importance
5Strong importance
6Strong plus importance
7Very strong importance
8Very very strong importance
9Extreme importance


n12345678910

RI000.580.91.121.241.321.411.451.49


Numerical scaleSVTNNsScore function

0.11
0.12
0.14
0.16
0.19
0.24
0.29
0.49
0.5
1.15
1.28
2.78
4.49
4.66
6.75
7.15
9

5. The G-AHP Method

The proposed G-AHP is inspired by the fuzzy Delphi method in [84, 85]. The main characteristics of the proposed G-AHP method compared to other similar grey AHP methods in the literature are as follows: (1) no whitenisation function is used; all the calculations from the beginning to the end are in grey numbers, and in accordance with basic grey operations rules (Section 3.3). This preserves the grey characteristics of the values and judgements and helps reach a more valid outcome; (2) no consistency calculation is needed; the pairwise comparison matrices are constructed in a way that CR values of any pairwise comparison matrix are zero, and evaluations are fully consistent; (3) two judgement scales are introduced. Here, the 5-point judgement or importance scale (Table 4) is utilised by DMs to show the significance of each element individually. The 9-point relative importance scale (Table 5) is constructed for obtaining an importance comparison of each element compared to other elements in pairwise comparison matrices.


Numerical scaleLinguistic term

2Poor (P)
3Fairly poor (FP)
4Moderate (M)
5Fairly good (FG)
6Good (G)


Numerical valueVerbal term

0.33Extremely less important
0.50Very strongly less important
0.67Strongly less important
0.83Moderately less important
1.00Equally important
1.20Moderately more important
1.50Strongly more important
2.00Very strongly more important
3.00Extremely more important

It is assumed that there are DMs, and let be the importance weight vector of the DMs where . It is also given that the decision-making model includes two finite sets of alternatives and criteria which are shown by and , respectively. The steps of the grey weights’ calculation applied in the proposed G-AHP method are represented as follows:(i)Step 1 (constructing the hierarchical structure): at this initial step, the hierarchical structure of the decision-making problem including goal, criteria, alternatives, or subalternatives will be constructed.(ii)Step 2 (asking for experts’ opinions): the DMs are asked to evaluate elements (criteria or alternatives) on the basis of their significance. The DMs determine the relative importance of each alternative over or each criterion over by using the importance scale (Table 4).(iii)Step 3 (pairwise comparison matrices): according to each DM’s opinion, the pairwise comparison matrices are constructed utilising the numerical scale (Table 4). As shown in equation (39), in the case of comparing criteria, should be replaced with . The represents the relative significance of element over element from the viewpoint of the DM:(iv)Step 4 (weighted pairwise comparison matrices): is the importance weight of the DM which belongs to the interval , and the greater the weight value, the more significant the DM’s opinion is. According to each DM’s importance weight and elements of matrices of equation (39), the values can be calculated based on(v)In the case of equal importance weights of DMs, there is no need to calculate equations (40) and (41), and simply equation (39) can be used.(vi)Step 5 (grey number calculation): in order to calculate grey numbers , all evaluations are taken into account, considering the importance weight of each DM as equations (42)–(44), where :(vii)According to the aforementioned explanations, the weighted grey pairwise comparison matrix for alternatives is defined in equations (45) and (46). In the case of criteria, should be replaced with in equations (45) and (46), where , and :(vii)Step 6 (grey weight calculation): grey weight of each alternative (i.e., ) can be calculated using equations (47) and (48). For criteria, should be replaced with in the following equations:

6. The Case Application

The proposed NG-AHP method was applied to agility evaluations in the Iranian steel industry. Five agility evaluation criteria were observed as evaluation criteria and were applied to four steel companies [89]. The criteria were organisation management agility (C1), product design agility (C2), manufacturing process agility (C3), partnership formation capability (C4), and integration of information system (C5). Based on the data collected from the chosen experts, the aim was to identify the most relevant agility criteria. Subsequently, the steel enterprises were ranked according to the agility readiness criteria. Four steel manufacturing companies were investigated in the present research, and their names were anonymized as SC1, SC2, SC3, and SC4.

Here, the expert selection process and their importance weight assignment task were carried out based on the experts’ knowledge and expertise in the related steel industry. Six DMs who were steel industry experts and were available to provide insights on agility readiness criteria evaluation, as well as being independent from the four steel companies, were selected. Brief profiles of the experts are represented in Table 6. The importance weight of each DM is provided as regarding their knowledge and experience.


DMsExpertiseDepartmentImportance weights

DM1Industrial engineering (MSc)Selling0.15
DM2Accounting (MA)Finance0.30
DM3Industrial engineering (MSc)Procurement0.10
DM4Metallurgy engineering (BSc)Manufacturing0.25
DM5Scientific assistant (MBA)R&D0.15
DM6Industrial engineering (BSc)HR0.05

The experts were initially contacted to participate in the study by completing two types of questionnaires for N-AHP and G-AHP, based on the scales provided in Tables 3 and 4. The acquired data are presented in Tables 7 and 8. The hierarchical structure of this problem is depicted in Figure 3.


C1C2C3C4C5

120.5043
0.5010.5042
22154
0.250.250.2010.50
0.330.500.2521

13232
0.3310.5022
0.502123
0.330.500.5010.33
0.500.500.3331

12485
0.501365
0 250.33132
0.130.170.3310.50
0.200.200.5021

15284
0.2010.5032
0.502162
0.130.330.1710.25
0.250.500.5041

10.170.2530.50
61243
40.50162
0.330.250.1710.50
20.330.5021

16223
0.1710.200.200.17
0.505132
0.5050.3312
0.3360.500.501


DM1DM2DM3DM4DM5DM6

C1SC1G (6)G (6)FP (3)G (6)G (6)FG (5)
C2M (4)FP (3)M (4)M (4)G (6)FG (5)
C3FG (5)FP (3)FG (5)G (6)G (6)FG (5)
C4M (4)FG (5)G (6)FG (5)FG (5)G (6)
C5G (6)M (4)FP (3)G (6)G (6)G (6)

C1SC2M (4)FG (5)M (4)FG (5)G (6)FP (3)
C2FP (3)M (4)FP (3)M (4)M (4)FP (3)
C3FP (3)FP (3)M (4)M (4)M (4)FP (3)
C4FP (3)M (4)M (4)FG (5)M (4)FG (5)
C5M (4)M (4)M (4)M (4)M (4)FP (3)

C1SC3FG (5)G (6)M (4)FG (5)FG (5)FG (5)
C2FG (5)M (4)FG (5)M (4)G (6)FG (5)
C3M (4)M (4)FG (5)FG (5)FG (5)FG (5)
C4M (4)FG (5)M (4)FG (5)M (4)G (6)
C5FG (5)M (4)FP (3)M (4)M (4)G (6)

C1SC4FP (3)FG (5)M (4)M (4)M (4)FG (5)
C2P (2)M (4)M (4)M (4)FG (5)FG (5)
C3P (2)M (4)FP (3)FG (5)M (4)FG (5)
C4P (2)FP (3)M (4)M (4)FP (3)G (6)
C5FP (3)M (4)M (4)FG (5)M (4)FG (5)

The proposed N-AHP was applied so as to obtain weights for five criteria. These weights were used later in the G-AHP method to acquire the final ranking of steel companies. In Table 7, the initial pairwise comparison matrices based on the opinions of six DMs using the NS rating scale (Table 3) are shown .

The calculated CRs for each pairwise comparison matrix were 2.23%, 7.66%, 2.36%, 3.99%, 6.56%, and 7.57%, respectively; they were all below 10% indicating cardinal output-based consistency. The aggregation neutrosophic matrix was calculated based on TNWAA operator, and then by applying equations (37) and (38), final weights were estimated (Table 9).


CriteriaSVTNN weightsCrispNormalisedRank

C10.74520.32271
C20.50710.21963
C30.63320.27422
C40.11450.04965
C50.30920.13394

Through G-AHP, opinions of DMs were obtained for the evaluation of each steel company (SC1, SC2, SC3, and SC4) against criteria based on the scale provided in Table 4. The numerical values in Table 4 then were substituted for linguistic phrases (Table 8).

Here, only the weight computations of four steel companies based on C1 (organisation management agility) are presented to show how the G-AHP method works. The resulted weights then make up the first column of the final decision matrix as shown in Table 10. The pairwise comparison matrices of four steel companies based on C1 (organisation management agility) according to opinions of six DMs are denoted as , , , , , and as presented in Table 11. All the CRs for comparative matrices will be equal to zero due to the applied method of acquiring opinions of DMs. The interpretation of the values in linguistic terms can be figured out based on the scale represented in Table 5. These values range from 0.33 with the corresponding verbal term extremely less important to 3 with the corresponding verbal term extremely more important.


C1C2C3C4C5
0.32270.21960.27420.04960.1339

SC1[0.0013, 0.0695][0.0013, 0.0521][0.0013, 0.0633][0.0013, 0.0870][0.0013, 0.0706]
SC2[0.0202, 0.7395][0.0290, 0.6110][0.0196, 0.5976][0.0218, 1.0392][0.0194, 0.7444]
SC3[0.0519, 2.8128][0.0525, 2.5580][0.0556, 2.5279][0.0580, 2.6429][0.0427, 3.1695]
SC4[0.0607, 3.8360][0.0662, 3.4875][0.0741, 3.4512][0.0511, 3.7932][0.0702, 3.5025]


SC1SC2SC3SC4

SC11.001.501.202.00
SC20.671.000.801.33
SC30.831.251.001.67
SC40.500.750.601.00

SC11.001.201.001.20
SC20.831.000.831.00
SC31.001.201.001.20
SC40.831.000.831.00

SC11.000.750.750.75
SC21.331.001.001.00
SC31.331.001.001.00
SC41.331.001.001.00

SC11.001.201.201.50
SC20.831.001.001.25
SC30.831.001.001.25
SC40.670.800.801.00

SC11.001.001.201.50
SC21.001.001.201.50
SC30.830.831.001.25
SC40.670.670.801.00

SC11.671.001.001.00
SC20.601.000.600,60
SC31.671.001.001.00
SC41.671.001.001.00

In order to obtain weighted pairwise comparison matrices of four steel companies, equations (40) and (41) were utilised considering importance weights vector as, and , , , , , and were obtained as shown in Table 12.


SC1SC2SC3SC4

SC10.1500.2250.1800.300
SC24.4440.1500.1200.200
SC35.5568.3330.1500.250
SC43.3335.0004.0000.150

SC10.3000.3600.3000.360
SC22.7780.3000.2500.300
SC33.3334.0000.3000.360
SC42.7783.3332.7780.300

SC10.1000.0750.0750.075
SC213.3330.1000.1000.100
SC313.33310.0000.1000.100
SC413.33310.00010.0000.100

SC10.25003000.3000.375
SC23.3330.2500.2500.313
SC33.3334.0000.2500.313
SC42.6673.2003.2000.250

SC10.1500.1500.1800.225
SC26.6670.1500.1800.225
SC35.5565.5560.1500.188
SC44.4444.4445.330.150

SC10.0500.0830.0500.050
SC212.0000.0500.0300.030
SC320.00033.3330.0500.050
SC420.00033.33320.0000.050

The weighted grey pairwise comparison matrix for steel companies was defined according to equations (45) and (46) as follows: