Efficacy of fuzzy MADM approach in Six Sigma analysis phase in automotive sector
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Abstract
Six Sigma is a strategy for achieving process improvement and operational excellence within an organization. Decisions on critical parameter selection in analysis phase are always very crucial; it plays a primary role in successful execution of Six Sigma project and for productivity improvement in manufacturing environment and involves the imprecise, vague and uncertain information. Using a case study approach; the paper demonstrates a tactical approach for selection of critical factors of machine breakdown in center less grinding (CLG) section at an automotive industry using fuzzy logic based multi attribute decision making approach. In this context, we have considered six crucial attributes for selection of critical factors for breakdown. Mean time between failure is found to be the pivotal selection criterion in CLG section. Having calculated the weights pertinent to criteria through two methods (fuzzy VIKOR and fuzzy TOPSIS) critical factors for breakdown are prioritized. Our results are in strong agreement with the perceptions of production and maintenance department of the company.
Keywords
Six Sigma Analytical hierarchy process Fuzzy logic MADM Center less grinding Automotive industryIntroduction
Companies are continuously facing the resistance to settle into the ever changing technological environment. Six Sigma has been recognized for many years as an efficient strategy and has helped several companies to rise to this challenge. This is one of the most important and popular developments in the field of process improvement. It has saved large amounts of money and improved the processes for a large number of manufacturing organizations worldwide (Neuman and Cavanagh 2000; Snee and Hoerl 2003; Harry and Schroeder 2005). Six Sigma has gone through a considerable evolution since the early exposition. Initially it was a quality improvement methodology based on statistical concepts. Then it transformed to a disciplined process improvement technique. In its current existence; it is commonly presented as ‘a breakthrough strategy’ of best in class. It is accepted that in current scenario Six Sigma is applicable to various environments such as service, manufacturing, process, software industry regardless of the size of the business, and, if successfully implemented, it may lead to nearly perfect solutions and services (Banuelas et al. 2005; Antony et al. 2006; Chakrabarty and Tan 2007). Six Sigma has enormous potential to reduce breakdown costs, improve performance, grow revenue, strengthen focus, and empower resources (Snee and Hoerl 2004). It is a commanding strategy that employs a regimented approach to undertake process variability using the application of statistical and non‐statistical tools and techniques in an accurate way (Jiju 2004). This teaches everyone in the organization to become more effective and efficient (Eckes 2003). Business leaders must be aware that successful implementation of Six Sigma requires not only technical understanding, but also behavioral awareness (Linderman et al. 2003). Most of the SMEs (Small and mediumsized enterprises) are not aware of Six Sigma and many not have the proper resources to execute Six Sigma projects (Jiju et al. 2005). In comparison with conventional approaches of quality and process improvement, Six Sigma is the most effective approach because of the interrelation between planning, organizational structures, measures, tools and techniques (Tilo et al. 2004; Zu et al. 2008). Six Sigma is a process improvement strategy which includes various phases logically related with each other acronym DMAIC methodology (define, measure, analyze, improve and control) is used for continuous improvement in any system or processes (Amer et al. 2008). It is the strategy of achieving key improvements in the process by applying DMAIC methodology through elimination of causes. Manufacturing units can put into action such strategies to enhance productivity of their manufacturing processes (Singh and Singh 2014).
In this case study, we are focusing on the analysis phase of DMAIC methodology through which all Six Sigma projects are executed. In the analysis phase, all measurements will be analyzed by understanding them and to make basic problem easier. The idea is to search for the factors having the biggest impact on process performance and determine the roots causes. In this case we have identified various factors for breakdown/failure in center less grinding section of machine shop in an automotive industry. The aim of study is to prioritize the critical breakdown factors in CLG section for further improvement. In this context, the key attributes/impacts were identified that depended on the views of various decision makers (such as machine operators, maintenance experts, production manager, technical and financial experts, etc.) and there are no crystalline themes among the views of these decision makers. Therefore it turned out to be necessary to forecast the excellent solution in terms of selecting critical factors for such problems using a decisionmaking technique. Such problems can be attempted with multiple attribute decision making (MADM) approach. MADM models are used to select best alternative from the large number of alternatives for a set of selection criteria. This approach has been effectively applied in broad range of decisionmaking problems in engineering and scientific fields (Perego and Rangone 1998; Pahlavani 2010). A variety of methods are reported under MADM category in literature (Chen et al. 1992; Tönshoff et al. 2007). MADM approach includes analytic hierarchy process (AHP) (Saaty 2014), graph theory and matrix approach (GTMA) (Rabbani et al. 2014), VlseKriterijumska Optimisacija I Kompromisno Resenje (VIKOR) (Liu et al. 2014; Singh and Kumar 2014), technique for order preference by similarity to ideal solution (TOPSIS) (Chu 2002; Chu and Lin 2003; Khanna et al. 2011; Dey et al. 2014), simple additive weighting (SAW) (Afshari et al. 2010) multiplicative analytical hierarchy process (MAHP) (Cheng and Mon 1994), weighted product method (WPM), Group decision making (GDM) (Chen 2000) and many others. These techniques have been successfully applied to various fields of engineering and among these, VIKOR and TOPSIS are outstanding multiple attribute decision making (MADM) approaches. These have been applied to various problems ranging from advanced manufacturing (Kulak and Kahraman 2005), production planning (Chen and Liao 2003), supplier selection (Azar et al. 2011), decision making (Sanayei et al. 2010), machine tool selection (Nguyen et al. 2014), supply chain management (Wei et al. 2007) and many more (Vats and Vaish 2013; Ding and Kamaruddin 2014; Tahriri et al. 2014; Tiwary et al. 2014; Vats and Vaish 2014a, b). These approaches work on crisp value of attributes/impacts. The aim of present study is to select critical factors for breakdown/failure in CLG section under fuzzy environment using fuzzy VIKOR and fuzzy TOPSIS methodology using AHP weights. The present study is one of the first efforts to evaluate failure parameters using fuzzy MADM approach in Six Sigma analysis phase in Indian automotive sector.
Evaluation criteria
Attributes/impact  Symbol  Depiction 

Ease of maintenance  C _{1}  It describes the ease with which a machine can be maintained in order to correct defects or their causes. Ease of Maintenance is the means whereby the Project Team confirms whether equipment can be maintained inservice and meets the maintainability and ease of maintenance criteria within the maintenance strategy 
Safety  C _{2}  There are common hazards associated with the use of machine shop equipment and tools. Working safely is the first thing because the safe way is the correct way. The costs of accidents and ill health to engineering machine shops may be disproportionately high. Many employees are ‘key’ workers whose losses through injury or ill health severely disrupt production and lowers productivity and profitability 
MTBF  C _{3}  It is the prime factor for selecting critical reasons of breakdown in machine shops. MTBF is stated as the average time between system failures of the entire machine shop. It defines of how reliable a component is. It shows the failure rate of each parameter responsible for breakdown in CLG section 
Cost  C _{4}  It is also a key factor for investigating critical reasons of breakdown. It includes the cost of breakdown, maintenance, repair and all activities necessary to meet all its functional requirements throughout the service life. This becomes a critical to estimate such costs 
Green effect  C _{5}  Green Effects go beyond just energy efficiency and attempt to rate an effort with regard to the total environmental stewardship of a machine shop. It includes minimum wastage, low energy consumption and user friendly environment. In this regard green effects are significantly more encompassing than just energy. An energy efficient shop floor may not be a green shop floor, but a true shop floor will be energy efficient 
Repair time  C _{6}  It is the Portion of breakdown time during which one or more experts are working on a system to effect a repair. Repair time includes preparation time, fault detection time, fault correction time and final bind up time 
Methods
As discussed in previous section, the present study emphasizes on finding out critical factors responsible for breakdown time in CLG’s to improve their availability and to enhance company profit. This is done by first optimizing the parameters using AHP and then using VIKOR and TOPSIS with fuzzy logic to sum up the result.
Analytical hierarchy process (AHP)
Fuzzy logic
VIKOR
Opricovic (2011) developed VIKOR, the Serbian name: VlseKriterijumska Optimizacija I Kompromisno Resenje; method to determine the compromise solution for a set of alternatives. Compromise solution is a feasible solution closest to the ideal solution for a MADM problem. The compromise solutions could be the basis for agreements, involving the decision maker’s preferences by criteria weight. This method focuses on ranking and selecting from a set of alternatives, and determines compromise solutions for a problem with conflicting criteria, which can help the decision makers to reach a final decision (Sanayei et al. 2010). VIKOR algorithm determines the weight stability intervals for the obtained compromise solution with the input weights given by the experts.
TOPSIS
TOPSIS (Technique for order preference by similarity to an ideal solution) method was presented by Hwang and Yoon (Yoon and Hwang 1995). TOPSIS uses different weighting schemes and distance metrics to compares results of different sets of weights applied to set of multiple criteria data (Olson 2004; Önüt and Soner 2008). The basic principle is that the chosen alternative should have the shortest distance from the ideal solution and the farthest distance from the negative ideal solution. The ideal solution is a solution that maximizes the benefit criteria and minimizes the cost criteria, whereas the negative ideal solution maximizes the cost criteria and minimizes the benefit criteria. Benefit criteria is for maximization, while the cost criteria is for minimization. The best alternative is the one, which is closest to the ideal solution and farthest from the negative ideal solution (Wang and Elhag 2006).
Methodology used
 Step 1

Calculation of AHP weights.
As discussed in “Analytical hierarchy process (AHP)” section, AHP weights (W _{ j }) are calculated for all breakdown parameters. This provides the weights of different criteria.
 Step 2

Define linguistic terms, relevant membership function and corresponding fuzzy numbers.
A set of fuzzy rates is required in order to compare all the alternatives for each criterion. These fuzzy terms are assigned by the decision makers and responsible for intra criterion comparisons of the alternatives.
 Step 3

Decision matrix formation.
Let p be the parameters and q be the alternative. For k number of decision makers in the proposed model for the aggregated fuzzy rating for C _{ j } criterion is represented as x _{ ijk =} {x _{ ijk1} , x _{ ijk2} , x _{ ijk3} , x _{ ijk4}}. For i = 1, 2,… p; j = 1,2,… q; k = 1,2,… k, x _{ ijk } is calculated as (Kahraman et al. 2003; Kwong and Bai 2003):$$\left\{ \begin{aligned} x_{ij1} = \mathop {\hbox{min} }\limits_{k} \left\{ {b_{ijk1} } \right\} \hfill \\ x_{ij2} = \frac{1}{k}\sum {b_{ijk2} } \hfill \\ x_{ij3} = \frac{1}{k}\sum {b_{ijk3} } \hfill \\ x_{ij4} = \mathop {\hbox{max} }\limits_{k} \left\{ {b_{ijk4} } \right\} \hfill \\ \end{aligned} \right.$$(4)Thus the obtained decision matrix (M) is shown as:$$M = \left[ {\begin{array}{*{20}l} {x_{11} } \hfill & {x_{12} } \hfill & \cdots \hfill & {x_{1p} } \hfill \\ {x_{21} } \hfill & {x_{22} } \hfill & \cdots \hfill & {x_{2p} } \hfill \\ \vdots \hfill & \vdots \hfill & \ddots \hfill & \vdots \hfill \\ \vdots \hfill & \vdots \hfill & \ddots \hfill & \vdots \hfill \\ {x_{q1} } \hfill & {x_{q2} } \hfill & \cdots \hfill & {x_{qp} } \hfill \\ \end{array} } \right]$$  Step 4

Defuzzification.
Defuzzification is performed to obtain the crisp values for each criterion corresponding to each alternative. This provides a quantitative value for the linguistic variables and fuzzy numbers assigned based on the verbal reasoning of the decision makers. Following equation lead to the crisp values:$$\begin{aligned} f_{ij} & = {\text{Defuzz}}\left( {x_{ij} } \right) \, = \frac{{\int {\mu \left( x \right) \cdot x{\text{d}}x} }}{{\int {\mu (x) \cdot {\text{d}}x} }} \\ & = \frac{{\int_{{x_{ij1} }}^{{x_{ij2} }} {\left\{ {\left( {x  x_{ij1} } \right)/(x_{ij2}  x_{ij1} )} \right\} \cdot x{\text{d}}x} + \int_{xij2}^{{x_{ij3} }} {x{\text{d}}x} + \int_{{x_{ij3} }}^{{x_{ij4} }} {\left\{ {(x_{ij4}  x)/(x_{ij4}  x_{ij3} )} \right\} \cdot x{\text{d}}x} }}{{\int_{{x_{ij1} }}^{{x_{ij2} }} {\left\{ {(x  x_{ij1} )/(x_{ij2}  x_{ij1} )} \right\}{\text{d}}x + \int_{xij2}^{{x_{ij3} }} {{\text{d}}x} + \int_{{x_{ij3} }}^{{x_{ij4} }} {\left\{ {(x_{ij4}  x)/(x_{ij4}  x_{ij3} )} \right\} \cdot x{\text{d}}x} } }} \\ & = \frac{{  x_{ij1} x_{ij2} + x_{ij3} x_{ij4} + (1/3)(x_{ij4}  x_{ij3} )^{2} + (1/3)(x_{ij2}  x_{ij1} )^{2} }}{{  x_{ij1}  x_{ij2}  x_{ij3} + x_{ij4} }} \\ \end{aligned}$$(5)The crisp values, thus obtained are integrated with AHP weights to calculate final ranking usingVIKOR and TOPSIS approach as discussed below.
VIKOR approach steps
 Step 5

Determination of ideal and negative ideal solutions;
The ideal solution f* and negative ideal solution f ^{−} are determined as$$f^{*} = \, \{ \hbox{max} f_{ij} \}$$(6)$$f^{  } = \left\{ {\hbox{min} f_{ij} } \right\}$$(7)  Step 6
 Calculation of utility and regret measures$$S_{i} = \sum\limits_{j = 1}^{n} {W_{j} \frac{{\left( {f_{j}^{*}  f_{ij} } \right)}}{{\left( {f_{j}^{*}  f_{j}^{  } } \right)}}} ;\quad \forall i$$(8)where S_{ i } and R_{ i } represent the utility and regret measures, respectively and W _{ j } is the relative weight assigned to the jth parameter using AHP.$$R_{i} = {\text{Max}}_{j} \left[ {W_{j} \frac{{\left( {f_{j}^{*}  f_{ij} } \right)}}{{\left( {f_{j}^{*}  f_{j}^{  } } \right)}}} \right];\quad \forall i$$(9)
 Step 7
 Calculation of VIKOR indexwhere Q _{ i } represents ith alternatives VIKOR value, v is the group utility weight, it is generally considered as 0.5 (unsupervised) and;$$Q_{i} = v\left[ {\frac{{S_{i}  S^{*} }}{{S^{  }  S^{*} }}} \right] + \left( {1  v} \right)\left[ {\frac{{R_{i}  R^{*} }}{{R^{  }  R^{*} }}} \right];\quad \forall i$$(10)$$S^{*} = \min_{i} \left( {S_{i} } \right);$$(11)$$S^{  } = \max_{i} \left( {S_{i} } \right);$$(12)$$R^{*} = \, \min_{i} \left( {R_{i} } \right);$$(13)$$R^{  } = \max_{i} \left( {R_{i} } \right);$$(14)
Breakdown factor with least value of VIKOR index Q _{ i } is preferred.
TOPSIS approach steps
 Step 5
 Normalized the matrix as given below:$$r_{ij} = \frac{{f_{ij} }}{{\sqrt {\sum\nolimits_{i = 1}^{m} {\left( {f_{ij} } \right)^{2} } } }};\quad \forall_{j}$$(15)
 Step 6
 Calculate the weighted normalized decision matrix as given:$$V_{ij} = \left[ {r_{ij} } \right]_{ \, m \times n} \times \left[ {W_{j} } \right]_{ \, n \times m}^{\text{diagonal}}$$(16)
 Step 7

Calculate the positive ideal and negative ideal solution:
The positive ideal solution V _{ j } ^{+} and negative ideal solution V _{ j } ^{−} are as given below:$$V_{j}^{ + } = \left\{ {\left( {\hbox{max} V_{ij} ,j \in J_{1} } \right),\left( {\hbox{min} V_{ij} ,j \in J_{2} } \right),i = 1,2,3 \ldots m} \right\};\quad \forall j$$(17)where J _{1} and J _{2} represents higher best and lower best criteria respectively.$$V_{j}^{  } = \left\{ {\left( {\hbox{min} V_{ij} ,j \in J_{1} } \right),\left( {\hbox{max} V_{ij} ,j \in J_{2} } \right),i = 1,2,3 \ldots m} \right\};\quad \forall j$$(18)  Step 8
 Calculate the distance d _{ i } ^{+} and d _{ i } ^{−} from the positive ideal solution and negative ideal solution respectively$$d_{i}^{+} = \left[{\sum\limits_{j = 1}^{n} {\left({V_{ij}  V_{j}^{+}} \right)^{2}}} \right]^{0.5},\quad i = 1,2,3, \ldots m$$(19)$$d_{i}^{} = \left[{\sum\limits_{j = 1}^{n} {\left({V_{ij}  V_{j}^{}} \right)^{2}}} \right]^{0.5},\quad i = 1,2,3,\ldots m$$(20)
 Step 9
 Calculation of TOPSIS rank index:$$C_{i}^{ + } = \frac{{d_{i}^{  } }}{{d_{i}^{  } + d_{i}^{ + } }}$$(21)
Breakdown factor with highest rank index C _{ i } ^{+} are preferred.
Results and discussion
Subjective weights of the evaluation criteria calculated using AHP
Attributes/impact  C _{1}  C _{2}  C _{3}  C _{4}  C _{5}  C _{6}  Weights  Rank 

Ease of maintenance (C _{1})  1  5  0.11  0.14  5  0.14  0.0768  4 
Safety (C _{2})  0.20  1  0.11  0.14  3  0.14  0.0381  5 
MTBF (C _{3})  9  9  1  9  9  9  0.4945  1 
Cost (C _{4})  7  7  0.11  1  7  7  0.2187  2 
Green effect (C _{5})  0.20  0.33  0.11  0.14  1  0.14  0.0239  6 
Repair time (C _{6})  7  7  0.11  0.14  7  1  0.1478  3 
Linguistic variables and corresponding fuzzy numbers
Linguistic variable  Fuzzy number 

Absolutely high (AH)  (0.8, 0.9, 1.0, 1.0) 
Very high (VH)  (0.7, 0.8, 0.8, 0.9) 
High (H)  (0.5, 0.6, 0.7, 0.8) 
Above average (AA)  (0.4, 0.5, 0.5, 0.6) 
Below average (BA)  (0.2, 0.3, 0.4, 0.5) 
Very poor (VP)  (0.1, 0.2, 0.2, 0.3) 
Absolutely poor (AP)  (0.0, 0.0, 0.1, 0.2) 
Linguistic decision matrix of factors for breakdown in CLG’s for all evaluation criteria
Evaluation Criteria (attribute/impact)  Breakdown factors in Centre Less Grinding’s (alternatives)  

Conveyor malfunction (F _{1})  Loader failure (F _{2})  Gear box fault (F _{3})  Coolant pump malfunction (F _{4})  Hydraulic motor not working (F _{5})  Hydraulic oil leakage (F _{6})  Slide failure (F _{7})  Spindle jam (F _{8})  CWD unit fault (F _{9})  Electrical faults (F _{10})  Sensor faults (F _{11})  Grinding wheel fault (F _{12})  Improper lubrication (F _{13})  
C _{1}  AP  VP  BA  AP  BA  VP  AP  VP  VP  VP  H  BA  VH 
C _{2}  VP  BA  VP  BA  BA  VP  VP  BA  AP  BA  H  AA  AA 
C _{3}  AP  VP  BA  VP  AA  VP  VP  VP  VP  VP  H  AA  AA 
C _{4}  AH  VH  H  VH  AA  VH  AH  VH  AH  H  BA  AA  VP 
C _{5}  VH  VH  AA  VH  BA  H  VH  VH  H  VH  VP  H  VH 
C _{6}  AH  AH  H  AH  VH  AH  AH  AH  AH  VH  VP  H  AP 
Calculated crisp values for assigned fuzzy rates
Evaluation criteria  Breakdown factors in CLG’s (alternatives)  

F _{1}  F _{2}  F _{3}  F _{4}  F _{5}  F _{6}  F _{7}  F _{8}  F _{9}  F _{10}  F _{11}  F _{12}  F _{13}  
C _{1}  0.0778  0.2333  0.3667  0.0778  0.3667  0.2333  0.0778  0.2333  0.2333  0.2333  0.6667  0.3667  0.8333 
C _{2}  0.3833  0.3667  0.2333  0.3667  0.3667  0.2333  0.2333  0.3667  0.0778  0.3667  0.6667  0.5333  0.5333 
C _{3}  0.1778  0.2333  0.3667  0.2333  0.5333  0.2333  0.2333  0.2333  0.2333  0.2333  0.6667  0.5333  0.5333 
C _{4}  0.9444  0.8333  0.6667  0.8333  0.5333  0.8333  0.9444  0.8333  0.9444  0.6667  0.3667  0.5333  0.2333 
C _{5}  0.9833  0.8333  0.5333  0.8333  0.3667  0.6667  0.8333  0.8333  0.6667  0.8333  0.2333  0.6667  0.8333 
C _{6}  0.9444  0.9444  0.6667  0.9444  0.8333  0.9444  0.9444  0.9444  0.9444  0.8333  0.2333  0.6667  0.0778 
Calculated VIKOR and TOPSIS ranking
Evaluation criteria  Breakdown factors in CLG’s (alternatives)  

F _{1}  F _{2}  F _{3}  F _{4}  F _{5}  F _{6}  F _{7}  F _{8}  F _{9}  F _{10}  F _{11}  F _{12}  F _{13}  
VIKOR rank index  0  0.09  0.297  0.08  0.485  0.088  0.054  0.09  0.061  0.133  0.75  0.503  0.648 
VIKOR ranks  1  7  9  4  10  5  2  6  3  8  13  11  12 
TOPSIS ranks  1  7  9  4  10  5  2  6  3  8  13  11  12 
TOPSIS rank index  0.9987  0.9806  0.7315  0.9825  0.3599  0.9811  0.9912  0.9806  0.9894  0.9333  0.0172  0.2982  0.0574 
Conclusions
 (a)
The study helps to highlight the importance of ‘Analysis Phase’ for successful implementation of Six Sigma project.
 (b)
The study has also helped to prove that FuzzyMADM approach can be effectively used to select most critical CTQs, which can be further improved to achieve better sigma rating.
 (c)
Within MADM, the study has successfully explored the efficacy of AHP, VIKOR and TOPSIS methods to prioritize the CTQs which are highly important for execution of Six Sigma project.
 (d)
The study will help the managers and practitioners to implement Six Sigma more effectively and more scientifically, using MADM approaches instead of using conventional statistical tools.
Notes
Authors’ contribution
All authors made substantial contribution to conception or design of the work, data collection, data analysis and interpretation, drafting the article, critical revision of the article and final approval of the version to be published.
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