Performance Evaluation of Nano-coated Balls Being Used in Rzeppa Constant Velocity Joint by using MCDM

Performance Evaluation of Nano-coated Balls Being Used in Rzeppa Constant Velocity Joint by using MCDM

Arshpreet Kaur1, *

1Department of Mechanical Engineering, Punjabi University, Patiala, India; Jasminder Singh Dureja1

Department of Mechanical Engineering, Punjabi University, Patiala, India; Jasmaninder Singh Grewal2

2Department of Mechanical and Production Engineering, Guru Nanak Dev Engineering College, Ludhiana, 141006, Punjab, India

*Corresponding: arshpandher89@gmail.com

Abstract Rzeppa constant velocity joint plays an important role in the automotive industry because of its primary function to transmit power from the prime-mover to the front wheel-axle. The main element that helps in the smooth movement of the joint are balls that are enclosed inside a cage-like assembly. These balls deteriorate after their continuous use and to avoid these protective coatings are implanted on the surface of balls. The coatings used in this analysis are Latuma and Alcrona Pro deposited on the substrate (EN-19) with the help of PVD-DC Magnetron Sputtering technique. The setup used to investigate the tribological performance in this work is a ball-on-disk apparatus based on the ASTM G-133 standard. The four different experimental conditions explored are with and without lubrication at room temperature as well as at high temperature. To select the best optimum working condition of a rzeppa joint an operation research-based multi-criteria decision- making tool is utilized. The Weightage importance is calculated with the help of entropy weight method and four different ranking techniques utilized are VIKOR, TOPSIS, EDAS, and PROMETHEE-II. Latuma with lubrication at room temperature ranked one in three out of four ranking techniques used in this work.

Keywords Nanocoatings, Tribology, Rzeppa joint, MCDM.

1. Introduction

Bearings are important components in different mechanical systems that provide support and reduce friction and wear between two mating parts [1]. Furthermore, it acts as a supporting element for various machine components such as in the case of automobiles in axles, wheels, and various constant velocity joints like rzeppa joint, etc. The elements that act as rotating elements are balls, cylindrical, or taper rollers [2]. It can find a wide range of applications in various fields such as aero-engines, automobiles, high-speed precision spindles, turbine gearboxes, etc. [1]. The Rzeppa constant velocity joint (CVJ) is a homokinetic ball-type joint widely used in automotive drive- train applications that are used for steering movements. Different components of the Rzeppa joint are inner rings, outer rings, balls, and boots [3]. In which inner rings are connected to the shaft, the outer rings act as housings as well and balls are placed in between the outer and inner rings in the cage and rubber boots help to protect the joint from failure and increase the life of the joint [4]. There are many reasons for the failure of bearings and damage to the Rzeppa joint such as cracks and fractures due to bearing overloading because of sudden shock loads, and failures because of cavities, and cuts in the track due to entering of foreign elements inside the joint that leads to surface fatigue or abrasion. Moreover, the wear of

balls inside the cage due to boot damage, and inadequate lubrication leads to abrasive wear [5]. There are various methods that help to improve the life of components such as hard-facing, hard-chromium plating, diffusion heat treatment, lubrication films, and surface coatings [6]. The coating technologies include gradient, metastable, multicomponent, and multi-layer coatings [7]. Thin layer coatings have been utilized extensively in various technical applications, including internal combustion engines (ICEs), cutting tools, hard disk drives, automobiles, etc., to increase the wear resistance and performance of mechanical components [8]. Different techniques i.e. physical vapor deposition, thermal spraying, or chemical vapor deposition have produced nano-coatings that have been widely used in recent years to enhance the characteristics and functionality of a wide range of materials [9]. These coatings are also supportive for reducing the wear of friction reducing elements like balls, rollers, cylinders etc. in bearings [10]. These techniques are quite effective in case of ball bearing to protect the balls from wear and improve their life.

The ball performance that is used in rzeppa joint by using multi- body Kinematic model which enables simulation of ball angular

1 velocity [11]. Researchers studied the film thickness of

different lubricating greases, oils using optical

interferometry on a ball-on-disc setup which are in agreement

with base oil film thickness measurements according to Hamrocks equation [12]. The behavior of the Rzeppa joint at distinct parameters by developing different mathematical tools and models was examined and it was concluded that the shape of the contact force is mainly determined from the geometry of the system [13]. The correlation between ball-on-disc and gear test friction measurement was investigated and found to be reasonably good in terms of absolute values [14]. Based on different applications and component designs coating processes are selected. For instance, protecting the surface from oxidation, wear, corrosion, and different tribological issues. Different coating methods are used to increase the life of components and parts. Different coating processes have different coating thicknesses such as Electrochemical (~10- 75µm), PVD Sputtering, Evaporation, Ion Implantation, Ion Plating (~1-100µm) etc [15]. The various thermal spray coating processes are applied to structural steels to prevent surface erosion caused by high temperatures or corrosion and erosion combined [16]. In addition to this different physical vapor deposition, chemical vapor deposition, and spray coating methods are used to enhance mechanical and surface properties [17]. The selection criteria of coatings depend upon various factors such as metallurgy of substrate material, availability of required coating material as well as their adaptability to respective coating techniques and accessibility of coating equipment. In addition to this, functional requirements, size, shape, and degradation also contributed to the selection of coating techniques [18]. Nanotechnology focuses on the application of materials having a size of less than 100 nm. Furthermore, nanomaterials have a high surface area to volume ratio, thus can give physical and/or chemical features that are not offered by micro- and macro-sized materials [17]. Nano coatings are composed of crystalline amorphous nano-phase mixtures and have recently increased their value in the field of research and development because of their unique properties against surface protection and tribological issues [19]. A layer of protective coatings formed naturally, synthetically, and deposited over the surface of the material to obtain the required properties. [17]. Moreover, coatings can be divided into two different categories such as metallic and non-metallic coatings [15]. In recent trends, sudden peaks have been observed in the case of physical vapor deposition-based applications. Contemporarily, PVD has vast utilization in actual practice in industries for high-temperature applications. [20]. A nano- coating shows excellent results against wear, corrosion, and oxidation, some of the phases in these coatings provide super- hardness whereas others offer high electrical and or thermal conductivity [21].

TiAlN-coated carbide inserts by PVD method are used for studying the friction and wear mechanisms in the dry sliding condition against steel and cast iron, by using pin-on-disc apparatus under normal lod and varied sliding speed [22]. In earlier years, TiAlN coatings were popularly used for wear-

resistant properties but nowadays these are used for high- temperature erosion, corrosion, and wear utilizations in different industrial applications [23]. By using a vacuum cathodic arc ion plating method, a layer of TiAlCrN coatings was prepared on substrate alloy and research concluded that these coatings showed superior results against oxidation at a temperature range of 700 to 800ºC whereas, it also reflected improved wear resistance at ambient temperature [24]. TiAlN coatings were deposited by cathodic arc ion plating to study the effect of Al concentration on wear and friction properties and research figured out that with increasing aluminium content in the layer, nano-hardness decreased, and no correlation formed between coefficient of friction and test temperature. At a high temperature of 673K width of the wear track also gets reduced. [25]. Dou Özkan et.al studied the different coatings such as TiAlN, AlTiN, and AlCrN with a thickness of ~3.5 µm prepared on the substrate by cathodic arc physical vapor deposition process and they concluded that AlCrN exhibits better wear behaviour than other coatings. [26]. Furthermore, the Sliding tribological performance of AlCrN coating was researched and they revealed that AlCrN undergoes good wear resistant behaviour [27]. Adam Gilewicz et.al tried to probe out the physical configuration and properties of protective coating AlCrN affixed with the help of CAE-PVD technique [28].

MCDM is a kind of numerical tool used by professional decision makers for the material selection problems in which selection is based on multiple criterias [29]. Eshlaghy & Homayonfar presented are review paper that consists of 628 publications of highly reputed journals in which MCDM is utilized for the purpose of material selection [30]. Ghoushchi et.al used MCDM techniques in transportation systems to predict climatic conditions due to environmental changes [31]. Stojcic et.al reviews the various research publications in which MCDM is utilized to improve the sustainability in the field of engineering [32]. Emovon reviewed various research articles from different platforms for the understanding of various MCDM techniques that are utilized in material selection processes [33]. Rahim presented an enormous number of papers in a single publication to review the different approaches used in MCDM for choosing and supporting material selection methods [34].

1. Experimental & Methodology

   1. Substrate material

      EN-19 has been selected as substrate material being used in steel balls for rzeppa constant velocity joint. EN-19 is medium carbon steel with high hardenability and good fatigue, abrasion and impact resistance. The chemical composition of the substrate is given in Table 1. Spectroscopy analysis was done on Q8 MAGLLAN spark emission spectrometer at the R&D Centre of the bicycle and sewing machine in Ludhiana to confirm the grade and elemental structure of the substrate materials.

      Table 1. Chemical Composition (wt %) for EN-19

      EN-19	C	S	P	Si	Mn	Cr
      Nominal	0.35-0.45	0.050	0.050	0.10-0.35	0.50-0.80	0.90-1.20
      Actual	1.010	0.010	0.012	0.250	0.44	1.420

      2

   2. Selection and deposition of coatings

      In this research work two nano-coatings i.e., Latuma (AlTiN based monolayer) and Alcrona Pro (AlCrN based monolayer) by Oerlikon Balzers, were selected. PVD-DC Magnetron Sputtering method is used for deposition of coatings on EN-19 stainless steel balls being used in Rzeppa joint. The process parameters for LATUMA and ALCRONA PRO as shown in Table 2.

   3. Characterization of coatings

      The surface roughness of the deposited coatings was measured with center line average (CLA) method. The coating thickness values were measured from SEM back scattered micrographs. SEM/EDAX and XRD analysis has been done to know the elemental structure of the coatings. SEM/EDAX analyses were performed at IIT (ICC) Roorkee. Porosity of the as-coated was observed less 0.5 % and measured at Guru Nanak Dev Engineering College Ludhiana.

      Table 2. Process parameters for LATUMA and ALCRONA PRO

      Parameters	BALINIT-LATUMA	ALCRONA PRO
      Layer Type	Mono	Mono
      Machine Used	Standard Balzers Rapid Coating System(RCS) Machine	Standard Balzers Rapid Coating System(RCS) Machine
      Make	Oerlikon Balzers, Swiss	Oerlikon Balzers, Swiss
      Deposition Pressure	3.5 Pa	3.5 Pa
      Number of targets	Ti(02), Ti50Al50(04)	Al70Cr30 (06)
      Reactive Gas	Nitrogen	Nitrogen
      Coating Thickness	4µm ± 1 µm	4µm ± 1 µm
      No. of Target	8	8
      Targets Composition	Ti, Ti50Al50	Al70Cr30
      Max Service Temp. (ºC)	1000	1100
      Coating Hardness HIT(GPa)	35±3	36±3
      Substrate bias voltage	-40V to -170 V	-40V to -170 V
      Note: – Target material: Positive biasing. Substrate material: Negative biasing.

   4. Wear studies

      In this investigation the work has been focused to compare the sliding wear behavior of the nano coated specimens under different environmental conditions, i.e. at room temperature and high temperature (400C) without lubrication and with lubrication. In first set of experiments the sliding velocity was kept constant i.e. 1m/sec and the normal loads of 30N, 40N and 50N was applied whereas in the second set of experiments normal loads was kept constant and the sliding velocity of 0.5 m/sec, 1 m/sec and 2 m/sec was used. The balls were made to slide against the rotating disc of EN-31 material having hardness of 62-65 HRC. The wear loss and coefficient of friction data was generated with the help of WINDUCOM 2010 software. A ball-on-disc apparatus was used to perform wear testing based on ASTM G- 133 standard.

2. Multi Criteria Decision Making (MCDM) techniques A systematic way to select the best option when several performance criteria are at play is called multi-criteria decision making, or MCDM. It is frequently used in engineering applications where multiple factors, including

   surface roughness, coating thickness, wear loss, and coefficient of friction, must be considered simultaneously, such as material selection, coating evaluation, and process optimization. It offers an organized approach for ranking several options from best to worst and determining the optimum course of action. MCDM is used in this work to identify the best nano-coating condition, allowing for a thorough evaluation of tribological performance and enhancing the dependability and effectiveness of Rzeppa joint components.

   1. VIKOR technique

      Cavallini et al. [35] used VIKOR-MCDM method to choose the best protective coating out of nine available coatings deposited on Al-7075 aloy. Chauhan and Vaish [36] applied VIKOR and TOPSIS to find out the best soft as well as hard magnetic material. Rai et al. [37] proposed a mathematical MCDM based model in which VIKOR played a vital role in selecting a highly suitable material for a flywheel. Jahan et

      al. [38] tried to find the foremost material for hip joint prosthesis with the help of the VIKOR-MCDM technique. Giorgetti et al. [39] took the help of C-VIKOR to select an optimal material for a valve seat in an IC engine. The Algorithm of VIKOR-MCDM technique is presented in a

      flowchart manner given below figure 1:

      Fig. 1. Flowchart showing the algorithm of VIKOR-MCDM technique [35]

   2. TOPSIS technique

      Tian et al. [40] utilized TOPSIS and grey correlation to select the best green decorative material for interior design. Yadav et al. [41] applied TOPSIS-PSI to select the best composition of hybrid aluminum nanocomposite for marine engineering applications. Chen [42] tried to deduce an MCDM model with the EWM, AHP, and TOPSIS for the selection of building material suppliers. Bhowmik et al. [43] worked on the selection of energy-efficient material with the help of

      MCDA-EWM-TOPSIS. Dhanalakshmi et al. [44] tried to develop a material selection tool for pyrolysis-based applications with the help of FAHP-TOPSIS-EDAS. The Algorithm of TOPSIS-MCDM technique is presented in a flowchart manner given below figure 2.

      Fig. 2. Flowchart showing the algorithm of TOPSIS-MCDM technique [40]

   3. EDAS technique

      Yalcin and Uncu [45] tried to solve the Robert selection problem faced at the industrial level with the help of the EDAS-MCDM technique. Rashid et al. [46] also worked on the robot selection problem by combining three different MCDM techniques TOPSIS-VIKOR-EDAS. Chatterjee et al. [47] worked on gear material selection problem with the help of DoE-EDAS. Nordan

      and Cengiz [48] applied EDAS for selecting wind turbines and FAHP was used for calculating criteria importance. Kahraman et al. [49] used fuzzy-EDAS for the allocation of solid waste disposal sites. The Algorithm of EDAS-MCDM technique is presented in a flowchart manner given below figure 3.

      Fig. 3. Flowchart showing the algorithm of EDAS-MCDM technique [45]

   4. PROMETHEE-II technique

      Maity et al. [50] applied PROMETHEE-II to find out the optimal tool steel from the set of ten materials and nine criteria are involved in this study. Vinodh and Girubha [51] used

      PROMETHEE-II for the implementation of the best sustainable approach in manufacturing industries. Gul et al.

      [52] used fuzzy-PROMETHEE for the selection of material

      for the automotive instrument panel. The above-calculated results are validated with the other three fuzzy- MCDM techniques fuzzy-TOPSIS, fuzzy-VIKOR, and fuzzy- ELECTRE. Golam and Afruna [53] utilized FAHP- PROMETHEE for the selection of femoral component

      material in TKR and sensitivity analysis is also performed. The Algorithm of PROMETHEE-II -MCDM technique is presented in a flowchart manner given below figure 4.

      Sr.No.	Coating Condition	Surface Roughness (micron)	Coating Thickness (micron)
      1	AlCrN with Lubrication at high temperature	0.08	2.01
      2	AlCrN with Lubrication at room temperature	0.08	2.01
      3	AlCrN without Lubrication at high temperature	0.08	2.01
      4	AlCrN without Lubrication at room temperature	0.08	2.01
      5	TiAlN with Lubrication at high temperature	0.027	3.5
      6	TiAlN with lubrication at room temperature	0.027	3.5
      7	TiAlN without Lubrication at high temperature	0.027	3.5
      8	TiAlN without Lubrication at room temperature	0.027	3.5

      Fig. 4. Flowchart showing the algorithm of PROMETHEE-II MCDM technique [50] Table 3. Results of Surface roughness, coating thickness for various samples

      Table 4. Results of Wear in terms of weight loss and coefficient of friction of worn-out samples

      Sr.No.	Coating Condition	Surface Roughness (micron)	oating Thickness (micron)
      1	AlCrN with Lubrication at high temperature	0.00382	0.0144
      2	AlCrN with Lubrication at room temperature	0.0026	0.0126
      3	AlCrN without Lubrication at high temperature	0.0282	0.124
      4	AlCrN without Lubrication at room temperature	0.0098	0.1564
      5	TiAlN with Lubrication at high temperature	0.0082	0.0126
      6	TiAlN with Lubrication at room temperature	0.0042	0.006
      7	TiAlN without Lubrication at high temperature	0.03	0.1218
      8	TiAlN without Lubrication at room temperature	0.056	0.1308

3. Selection of experimental conditions

   The different alternatives and criteria selected in the decision-making matrix for nano-coatings combinations are tabulated in table 5 under different conditions such as: –

   Table 5. Notations for different Nano coatings combinations

   Notation	Experimental Condition
   AlCrN (1)	Coating with lubrication at High Temperature.
   AlCrN (2)	Coating with lubrication at Room Temperature.
   AlCrN (3)	Coating without lubrication at High Temperature.
   AlCrN (4)	Coating without lubrication at Room Temperature.
   TiAlN (1)	Coating with lubrication at High Temperature.
   TiAlN (2)	Coating with lubrication at Room Temperature.
   TiAlN (3)	Coating without lubrication at High Temperature.
   TiAlN (4)	Coating without lubrication at Room Temperature.

   Sr.No.	Notation of criteria	Criteria
   1	1p	Surface Roughness
   2	2p	Coating Thickness
   3	3p	Weight Loss
   4	4p	Coefficient of friction

   Table 6. Criteria for allocation of Nano coatings

   As mentioned in Table 6 evaluation criteria are selected and are elaborated as follows:

   1. Surface Roughness: – It plays an important role in choosing nano-coatings for surface engineering applications. In the case of lubrication, a lower magnitude of surface roughness leads to less wear as well as minimum energy loss while higher values of surface roughness help to keep the lubricants intact and lubrication efficiency [54].

   2. Coating Thickness: – The role of coating thickness majorly affects two properties corrosion and wears resistance. A layer of thick coating helps to reduce the corrosion which subsequently increases the life of the substrate and it also helps to decrease the wear and tear of the substrate [55].

   3. Wear in terms of weight Loss: -Wear rate plays a vital role while selecting a coating and quantitatively provides coating robustness. It is quite helpful in dropping material degradation [56].

   4. Coefficient of friction: – It is quite beneficial in selecting coatings for numerous applications. It helps in finding out wear resistance and increases performance enhancement [57].

   Table 7. Decision matrix of Nano-coatings combinations

   Condition	Surface Roughness	Coating thickness	Wear in terms of weight Loss	Coefficient of friction
   AlCrN (1)	0.08	2.01	0.0038	0.0144
   AlCrN (2)	0.08	2.01	0.0026	0.0126
   AlCrN (3)	0.08	2.01	0.0282	0.124
   AlCrN (4)	0.08	2.01	0.0098	0.1564
   TiAlN (1)	0.027	3.5	0.0082	0.0126
   TiAlN (2)	0.027	3.5	0.0042	0.006
   TiAlN (3)	0.027	3.5	0.03	0.1218
   TiAlN (4)	0.027	3.5	0.056	0.1308

   Table 7 represents the decision matrix of nano-coatings AlCrN and TiAlN for evaluating the performance under different conditions such as surface roughness, coating thickness, wear

   in terms of weight loss and coefficient of friction. As indicated in the table AlCrN coatings undergoes better wear resistance as compared to TiAlN coatings.

4. Results and discussion

   In this MCDM analysis, while selecting the best optimum coating combination the weightage importance is calculated with the help of the entropy weight method, the calculation of EWM has been shown in the appendix from Tables A1 and A2. After computation, the Weightage for surface roughness, coating thickness, and wear in terms of weight loss and coefficient of friction are 12.40, 3.58, 42.84, and 41.69 respectively. The ranking techniques involved in this work are TOPSIS, VIKOR, EDAS and PROMETHEE-II. For TOPSIS,

   the computation work is shown in appendix B from Tables B1 and B2. The ranking order for these techniques is: AlCrN (1) > TiAlN (2) > TiAlN (1) > AlCrN (2) > AlCrN (3) > TiAlN (3) >

   AlCrN (4) > TiAlN (4). For VIKOR, the computation work is

   shown in appendix C from Tables C1, C2 and C3. The ranking order for these techniques is: TiAlN (2) > TiAlN (1)> AlCrN

   (2) > AlCrN (1) > AlCrN (4) > TiAlN (3) > AlCrN (3) > TiAlN

   (4). For EDAS, the computation work is shown in appendix D from Tables D1 to D6. The ranking order for these techniques is: TiAlN (2) > AlCrN (2) > AlCrN (1) > TiAlN (1) > AlCrN

   (4) > TiAlN (3) > AlCrN (3) > TiAlN (4). For PROMETHEE-

   II, the computation work is shown in appendix E from Tables E1 to E5. The ranking order for these techniques is: TiAlN (2)

   > TiAlN (1)> AlCrN (2) > AlCrN (1) > AlCrN (4) > AlCrN (3)

   TiAlN (3) > TiAlN (4). The comparison of results obtained from all the four ranking techniques are tabulated in Table 8.

   Table 8. Comparison of results of all ranking techniques

   CONDITION	TOPSIS	VIKOR	EDAS	PROMETHEE-II
   AlCrN (1)	1	4	3	4
   AlCrN (2)	4	3	2	3
   AlCrN (3)	5	7	7	6
   AlCrN (4)	7	5	5	5
   TiAlN (1)	3	2	4	2
   TiAlN (2)	2	1	1	1
   TiAlN (3)	6	6	6	7
   TiAlN (4)	8	8	8	8

5. Conclusions

The ball bearings are used in various industrial and automobile applications. The Balls used in rzeppa joints gets deteriorated

1. 1. A nano-coating TiAlN with lubrication at room temperature obtained from VIKOR, EDAS and PROMETHEE-II.

   after their regular usage in automotives. To enhance the life of2. 2. It is observed that in all four ranking techniques, the first four

   balls different protective coatings are utilized. In this research work, experimental and MCDM methods are used to figure out which coating gives better results under various conditions. The following are the conclusions: –

   best coating combinations are in lubrication conditions.

   1. The result of TOPSIS is completely odd as compared to other three techniques.

   2. TiAlN coating without lubrication at room temperature is ranked last in all four MCDM techniques.

Acknowledgements

The authors would like to thank Mr. Jitendra Dabral, DGM operations, Oerlikon Balzers Coatings India Pvt. Limited for his

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Appendix A

j

j =

=1

j

j = j ln(j) , j = 1,2,3, ,

=1

, =

1

ln

Table 9. Normalization of decision matrix

Table10. Entropy Computations

Appendix B

j

j =

=1

2

j

j = j × wj

{+, +, +, , +} = { | = 1, , }

1 2 3 j

{, , , , } = { | = 1, , }

1 2 3 j

+ = (j +)2

j j

=1

= (j )2

j j

=1

=+

+

Table 11. Normalization of decision matrix for TOPSIS

Table 12. Weighted Normalized decision matrix for TOPSIS

Appendix C

j

j =

=1

2

j

j

+ = ( , j = 1, 2, 3 , )

j

j

= ( , j = 1, 2, 3 , )

j

j = wj

j

(+ _)

(+ j)

=1 j j

(+ j)

j = [wj ] j

(+ _)

j j

j _ j _

= [ j ] + (1 ) [

+ _

j

]+ _

j j j j

Table 13. Best and Worst Values for all attribute

Table 14. Normalization of decision matrix for VIKOR

Table 15. Calculation of Ranks from VIKOR

Appendix D

j j

= [ ]

1×

j

j =

=1

j

= [ ]

×

j

= [ ]

×

j = (0,(jj (For Beneficial Criteria)

))

j

j = (0,(jj (For Beneficial Criteria)

))

j

j = (0,(jj (For Non-Beneficial Criteria)

))

j

j = (0,(jj (For Non-Beneficial Criteria)

))

j

= wj × j

=1

= wj × j

=1

= ( )

= 1 ( )

1

= 2 ( + )

Where, 0 1

Table 16. Decision matrix for EDAS

Table 17. Positive distance from Average for EDAS

Table 18. Weighted Sum of Positive distance from Average for EDAS

Table 19. Negative distance from Average for EDAS

0.495327103

Table 20. Weighted Sum of Negative distance from Average for EDAS

Table 21. Calculation of Ranks from EDAS

Appendix E

[j=(j)]

j = (For Beneficial Criteria)

[(j)j]

j = (For Non-Beneficial Criteria)

j(, ) = 0, j j

j(, ) = j j, j j

j=1

jj(, )

(, ) = w

j=1 j

+ =

1

1

(, ) , ( )

=1

1

=

1

(, ),

=1

() = + _

Table 22.Maximum and Minimum Values for all attributes (PROMETHEE-II)

Table 23. Normalized Decision Matrix for PROMETHEE-II

Table 24. Calculation of preference functions for all pair of combinations

Table 25. Pair-wise comparison of all pairs

Table 26. Calculation of ranks from PROMETHEE-II