Analysis on sustainable manufacturing criteria of automotive industry

In this paper, criteria within the most significant domain of manufacturing in automotive industry in Malaysia is analysed and prioritized. This paper aims to propose a multi criteria decision making (MCDM) procedure using concept of fuzzy theory, graph theory and Laplacian matrix approach which is embedded into Graph Theory and Matrix approach (GTMA). The manufacturing domains are ranked using Laplacian energy while the criteria within the most significant domain is prioritized using method of graph drawing whereby the criteria’s is ranked based on the average distance among other criteria in Cartesian coordinate system. The result indicates that cost, quality and service in economic domain are among significant criteria’s that practitioners should give an attention in developing a quality sustainable manufacturing framework so that the industry could sustain for a longer period of time.


Introduction
Sustainable manufacturing is defined as the creation of manufactured products that minimize negative environmental impacts, conserve energy and natural resources, are safe for employees, consumers, and communities, are economically sound [1]. This era, sustainable manufacturing has become a very important issue among governments and industries around the world. There is now a well-recognized need for achieving overall sustainability in industrial activities, arising due to several established and emerging causes: diminishing non-renewable resources, stricter regulations related to environment and occupational safety/health, increasing consumer preference for environmentally friendly products, etc. [2].
As for automotive industry, according to the Malaysian Automotive Association (MAA), a cooling domestic economy and uncertainty over the pace of global recovery slow the Malaysia's automotive industry, which show a decline in sales or production in year-on-year. Vehicle sales declined by 12.4% to 529,434 units in 2020 from 604,281 units of the previous year. The collapse in Malaysia's car market can be attributed to the unpopular passenger car model in the local market. In order not to make the situation worse, the government should seize the opportunity to implement sustainable manufacturing practices to improve the effectiveness, efficiency, and performance of automotive industry. Sustainable manufacturing practices have seen as an effective solution to the automotive industry whereby it supports the continued growth and expansion of the industry [3].
The practice involves development of standard framework. The framework takes into consideration of three domains namely, economic, environmental, and social. Manufacturing industry is considered as the most complex-oriented industries since it utilized many resources from other industries which may leads to practitioners to considering many criteria from each of the domain in developing a good manufacturing framework for the industry to sustain. Because of that reason, researchers come out with different method or approach in order to help practitioners in the selection of the criteria. Some of the MCDM approaches that are used to solve any problem involving decision making with many criteria's in the automotive industry are Fuzzy Analytical Hierarchy Process (FAHP) [4,5], Analytical Hierarchy Process (AHP) [4,5,6], Fuzzy Technique for Order Preference by Similarity to Ideal Solution (FTOPSIS) [4], Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) [4],

Graph theory and matrix approach (GTMA)
Graph theory and matrix approach (GTMA) is a new technique of decision making [16], which is reasonable and systematic [11,17]. The matrix is useful in analysing diagraph models in easy way which explains the system and problems in numerous science and technology [11]. GTMA methodology has several advantages but the main advantage is it can provide information of interrelationship between criteria by using pair wise comparison matrix which later the interrelationship between criteria is visualised on diagraph representation. The graph could give information of interdependencies of the criteria through visualisation while matrix representation is useful in analysing the directional graphs particularly when numbers of nodes are large, and graphs become complex to visualize.
The fact is that human perception always contains a certain degree of vagueness and ambiguity, makes the traditional GTMA fails to perceive these traits. Therefore, fuzzy set theory is used to deal with the uncertainty and imprecision associated with information on every different parameter, and the situation where only partial information is available. Fuzzy theory has been described as a problem-  [14]. Generally, a fuzzy set is defined by its membership function, which represents the degree to which any element x of X has partial membership in M. The degree to which an element belongs to the set is defined by a value ranging from 0 to 1 [18].
One representation of fuzzy number is called triangular fuzzy number (TFN) which can be simply written as ( , , ) where is the lowest possible value, is the moderate possible value and is the upper possible value describing the fuzzy event. A fuzzy number ̃= ( , , ) is called triangular fuzzy number if its membership function is given by,

Fuzzy Graph
Fuzzy graph was another extension of fuzzy theory's application in its relation to graph theory. This is because fuzzy theory was designed as a mathematical formalized tool in dealing with imprecise information. Thus, the emergence of fuzzy graph theory could be very beneficial in dealing with graph-theoretic problems which dealing with uncertainties. Definition of fuzzy graph given by Rosenfeld in 1975 is introduced as follows.
Definition 1 Fuzzy Graph [19] Fuzzy graph G = (σ, μ) is a pair of functions σ: S → [0,1] and μ: S * S → [0,1], ∀x, y ∈ S we have μ(x, y) ≤ σ(x)^ σ(y). Rosenfeld (1975) has generalized the version of graph fuzziness in which he has considered fuzzy graph to consist both fuzzy sets of vertices as well as for the edges. Yeh and Bang (1975), on the other hand, have taken a particular case of fuzzy graph initiated by Rosenfeld (1975) into new definition as follows: Definition 2 Fuzzy Graph [20] A fuzzy graph G is a pair (V, R) whereby V is a set of vertices and R is a fuzzy set of edges.

Combinatorial Laplacian Matrix of FACS
According to Mohar et al. (1991), Laplacian is considered as an interesting matrix since its spectrum is much more natural and important than adjacency matrix spectrum. Koren (2005) used Laplacian spectrum in graph drawing since the Laplacian spectrum is more fundamental than adjacency spectrum. Further investigation on the Laplacian matrix leads to the formation of Directed Laplacian matrix. Directed Laplacian matrix for aperiodic strongly connected graph was firstly defined by Chung in 2006. The definition is given as follows.

Definition 3
Laplacian of a directed graph. [23] The Laplacian of a directed graph G is defined by where is a diagonal matrix with entries ( , ) = ( ) and P denotes a transition matrix while denotes the conjugated transpose of .
where ( * ) denotes the transpose of * . The combinatorial Laplacian of FACS is defined as: where ( * ) denotes the transpose of * and is the diagonal matrix of the stationary distribution, i.e. = ( 1 , … , ). In this study, combinatorial Laplacian of FACS is adopted since the proposed graph is sharing a similar structure with FACS graph.

Graph Embedding Method
The procedure to transform the graph into 2D-Euclidean space is adopted from Carmel et al. (2002) which is based on Laplacian matrix and solving a unique one-dimensional optimization problem in determining coordinates of each node. In this study, definition of balanced vector and optimal arrangement of directed graph [23] is adopted to transform the proposed graph into coordinated graph, Definition of balance vector and optimal arrangement for the study are presented as follow.
Consequently, = ( 1 , … , ) is determined by using concept of minimization of hierarchy energy function where it is equivalent to solving an optimal arrangement [26]. In this study, the same concept is modified and redefined as follows.

Definition 7
Let ( , * ) be a digraph with Laplacian, and balance, . Its optimal arrangement, * , is the solution for = , subject to the constraint = 0. The optimal arrangement * is the vector for -coordinates of criteria and is solved by using Conjugate Gradient method. On the other hand, = ( 1 , … , ) representing the -coordinates of criteria are solved using minimization of edge squared lengths via Tutte-Hall energy function, According to Carmel et al., (2002), minimization of this energy function is the Fiedler vector, which is the eigenvector of the Laplacian associated with the smallest positive eigenvalue. In this case, the Fiedler vector of combinatorial directed Laplacian is obtained by solving its eigenvalue problem.

The Proposed Method and Its Application
Multi-Criteria Decision Making (MCDM) has become one of the most important and active fields of operations research or management science. MCDM represents the process of determining the best solution based on established criteria and common problems in daily life. The MCDM method has been successfully utilized in the sustainable manufacturing and solving the prioritizing problems related to enablers, issues, and indicators [27]. This study provides decision makers with a MCDM for evaluating sustainable manufacturing in the automotive industry. In order to rank automotive criteria using GTMA, a research framework is developed as shown in Figure 1. Following the research framework shown in Figure 1 the goal of ranking the criteria is determined. The criteria's are ranked based on experts' opinions using surveys. Experts were asked to perform pair wise comparison of the criteria based on the importance scale. The following steps show the development of an GTMA based model for sustainable manufacturing criteria evaluation in automotive industry. In this study, GTMA approach which consists of a diagraph and its associated weighted matrix particularly Laplacian matrix is developed. Laplacian energy value is then calculated. The methodology which involved nine steps is presented as follows.   Step 1: Identification the sustainable manufacturing criteria. This study begins by listing all possible criteria that may be involved in the sustainable manufacturing automotive industry through literature review and selection of the most frequent criteria among the selected influential studies in Malaysia. The criteria must be constructed by adopting the triple bottom line of sustainability consisting of economic, environmental, and social domains. As a result, the sustainable manufacturing criteria consist of three domains divided into twenty four criteria were identified as shown in Table 1. Table 1. Sustainable manufacturing criteria in automotive industries.
Step 2: Establish fuzzy pairwise comparison matrices of each criterion in every domain. Here, the general form of the fuzzy pairwise comparison will be as follows: Fuzzy pair wise comparison matrices of each criteria are assigned by the expert using linguistic scale of importance. The linguistics scale and its reciprocal scales are represented by triangular fuzzy number as shown in Table 2.   (7,8,9) (1/9, 1/8, 1/7) Absolutely important (AI) (9,9,9) (1/9, 1/9, 1/9) Step 3: Calculate the fuzzy relative importance or the fuzzy weights for each criteria in every domain. The fuzzy relative important or the fuzzy weights for each criteria are calculated using arithmetic mean method given by = off-diagonal element of . = the number of decision maker.
= the relative importance value given by decision maker which based on scale in Table 2. , is referring to left, moderate and upper value of triangular fuzzy number.
There are three fuzzy relative importance matrix for three domain in this study. The fuzzy relative importance of criteria is calculated using a modified approach which is the arithmetic mean in Equation (8) in order to obtain the off-diagonal element of matrix A.
For example (Economic Domain), the off-diagonal elements of < , , > for i = 2 for Delivery (DL), j = 1 for Flexibility (FX) The arithmetic mean aggregate is (1.7037, 2.0370, 2.3704). Step 4: Develop the weighted directed graph of relative importance among criteria. The weighted directed graph is developed based on the matrix A obtained in Step 3. The weighted directed graph is to show the visualization of relative importance among the criteria. The digraph model presents a graphical representation of interrelationship among criteria where the graph consists of nodes V= {v i } for = 1,2,3, … and set of directed edges E={e ij } for i,j=1,2,3,…m. The number of criteria equal to the number of nodes in digraph model and the edges e ij represent the relative importance among the criteria. The digraph representation of importance of the criteria to each other is illustrated in Figure 2.
Step 5: Normalized the weights. The weights of criteria obtained in Step 3 are normalized using Definition 8 in order to ensure that the weights are effectively compared.   [29] The normalized adjacency matrix is = −1/2 −1/2 (9) where is the adjacency matrix of and = { } is the degree matrix and −1/2 as follow, For example (Economic Domain): i) The elements of < , , > for i = 2 for Delivery (DL), j = 1 for Flexibility (FX) The normalized the weights is (0.1287, 0.1256, 0.1225).
Therefore, the normalized the weights in economic can be expressed in the matrix form as follows: Therefore, the defuzzify the Fuzzy Combinatorial Laplacian in economic can be expressed in the matrix form as follows: Table 3. Defuzzification of Fuzzy Combinatorial Laplacian matrix of the graph for economic domain.  Step 8: Determine the Laplacian Energy. The calculation of Laplacian energy must include eigenvalue of Fuzzy Combinatorial Laplacian using Definition 9.
Definition 9 Eigenvector and eigenvalue [30] An eigenvector of a square matrix is a non-zero vector that, when the matrix is multiplied by , yields a constant multiple of , the multiplier being commonly denoted by . That is: The number is called the eigenvalue of and is said to be an eigenvector corresponding to . It can rewrite the condition as Ranking domains are determined with the highest the Laplacian energy value, the more preferred the domain. This rank will order the highest to lowest. Table 4 shows that the economic domain is the first 12 rank followed by social and the last is environmental. Next, the economic domain more preferred than analysed to determine which criteria are preferred in the economic domain. Step 9: Embeds the nodes into 2D-Euclidean space and determine the rank of criteria based on Euclidean distance. The image of x i embedded into k domanial space is given by y * = [v 2 * , … , v k+1 * ] [30] . Based on Definition 6 and Definition 7, -coordinates of the nodes criteria of ( , ) is given by y * . Herewith, solve eigenvalue problem of to get the Fiedler vector, = ( 1 , … , ) which representscoordinates of the nodes criteria [26]. It solved by using eigenvector of the Laplacian associated with the smallest positive eigenvalue. Positioned the nodes in 2D-Euclidean space using ( , y) coordinate obtained and draw its corresponding edges.
The Euclidean distance between criteria is calculated using Equation (15). The Euclidian distance matrix shows the distance between pair of criteria. By definition, criteria's distance from itself, which is shown in the main diagonal of the matrix is zero. [30]. A Euclidean matrix on the real dimensional vector space is the usual distance matrix. Therefore, the Euclidean distance between criteria in economic as follows:

Result and Discussion
Pairwise comparison for each criterion is established using combination of three mathematical concept that is fuzzy theory, graph theory and Laplacian matrix approach based on GTMA. The ranking domain it can be identified that economic sustainability is the first rank followed by social and the lastly is environmental. Next, the economic domain more preferred then analysed to determine which criteria are preferred in the economic domain. In this study, for each criterion can be transformed into 2D-Euclidean space as in Figure 3. Every edge of the graph in Figure 2 is associated to a membership value for fuzzy edge connectivity whereby no information on location of each node in this graph. In contrast, every edge of the graph in Figure 3 provides not only information on membership value, it also gives the information of its length. Here, every node has its own coordinate, and the location of every node is given in Table 6.    Table 7 shows the lowest average Euclidean distance are 0.2831 which representing Cost criteria. This is show that Cost criteria is the most preferred followed by Quality criteria. Service is ranked in third place followed by the Flexibility, Technology, Delivery and Production capacity criteria based on the preference. The larger Euclidean distance are 0.9866 refers to technical capability. Figure 3 indicates that ( 5 ) and ( 8 ) which are positioned at considerable distance away from the other variables and ( 6 ), ( 4 ), ( 1 ), ( 3 ), ( 2 ) and ( 7 ) which is closely located to each other in the graph. This information indicates that outlying variables shown in the graph can be determined from the largest value of the criteria in the Euclidean distance. Since larger value of the entries in the Euclidean distance is correspond to the least importance variable. Subsequently, the outlying variable of coordinated indicate the 'most depleting' variables. The sequences of ranking the criteria using Euclidean distance is Cost > Quality > Service > Flexibility > Technology > Delivery > Production capacity > Technical capability.

Conclusion
This paper has proposed a new multi criteria decision making using fuzzy theory, graph theory and Laplacian matrix approach based on GTMA method in the evaluation of sustainable manufacturing criteria in the automotive industry. The network relationship model is constructed using the GTMA method. The weights of the criteria sustainable manufacturing are assigned through pairwise comparison and calculated using fuzzy combinatorial Laplacian matrix approach. The combination of the three mathematical concept of fuzzy theory, graph theory and Laplacian matrix approach based on GTMA method will provide a better understanding of the interrelationships between the criteria and help solve a complex evaluation problem, so that it can enhance the quality of decision making. The model enables and assists companies to give an opportunity for the researcher and decision maker to do an analysis of criteria that related with automotive sustainability. This study also shown that cost, quality and service in economic domain are among significant criteria's that a company should give priority in developing and formulating a comprehensive standard framework of planning, improving and thus make the company a sustainable manufacturing company in future.