grey relational analysis of Multi objective problem
- 1. M-, Manufacturing division, Mechanical Dept, SRMIST, KTR Grey Relation Analysis (GRA) Calculations 1 Normalization: Exp No – 1 Surface Roughness y1(K) = 3.2886 µm Max Surface Roughness max(K) = 9.1723 µm Min Surface Roughness min(K) = 3.2886 µm For Ra smaller the better formula is used 𝑇𝑖 (k) = 𝑚𝑎𝑥 𝑘 − 𝑦𝑖 𝑘 𝑚𝑎𝑥 𝑘 − 𝑚𝑖𝑛(𝑘) = 9.1723 −3.2886 9.1723 −3.2886 = 1 Exp No – 2 Surface Roughness y1(K) = 4.3808 µm Max Surface Roughness max(K) = 9.1723 µm Min Surface Roughness min(K) = 3.2886 µm For Ra smaller the better formula is used 𝑇𝑖 (k) = 𝑚𝑎𝑥 𝑘 − 𝑦𝑖 𝑘 𝑚𝑎𝑥 𝑘 − 𝑚𝑖𝑛(𝑘) = 9.1723 −4.3808 9.1723 −3.2886 = 0.81
- 2. M-, Manufacturing division, Mechanical Dept, SRMIST, KTR Grey Relation Analysis (GRA) 2 In Normalization, we have 3 options in which we have to decide whether we need Smaller the better or Larger the better so for Surface Roughness (Ra) Smaller the better, Thrust Force (Fz) Smaller the better, and Material removal rate (MRR) Higher the better is the best option so by the help of MRR formula we calculated MRR and all other options are calculated and Each experimental result is normalized and tabulated below. 2. Grey relational Co-efficient and grey relational grade The next step is to find Grey relational co-efficient of the normalized values. The formula which is used to find co-efficient is given below 𝐺𝑅𝐶𝑗 = ∆ 𝑚𝑖𝑛 + 𝛿∆𝑚𝑎𝑥 (∆ 𝑚𝑎𝑥 −∆𝑗)+𝛿∆𝑚𝑎𝑥 Where 𝐺𝑅𝐶𝑛𝑗 – Grey Relational Co-efficient for a corresponding value ∆𝑚𝑖𝑛 – Minimum value of corresponding parameters ∆𝑚𝑖𝑛= 0 ∆𝑚𝑎𝑥 – Maximum value of corresponding parameters ∆𝑚𝑎𝑥= 1 𝛿 - The quality loss factor,
- 3. M-, Manufacturing division, Mechanical Dept, SRMIST, KTR 3 Grey Relation Analysis (GRA) Calculations Grey relational Co-efficient and grey relational grade: ∆𝑚𝑖𝑛 – Minimum value of corresponding parameters ∆𝑚𝑖𝑛= 0 ∆𝑚𝑎𝑥 – Maximum value of corresponding parameters ∆𝑚𝑎𝑥= 1 In this project value is considered as 0.4 for Ra, 0.3 for Fz and MRR 𝐺𝑅𝐶1 = ∆ 𝑚𝑖𝑛 + 𝛿∆𝑚𝑎𝑥 (∆ 𝑚𝑎𝑥 −∆𝑗)+𝛿∆𝑚𝑎𝑥 = 0 +0.4∗1 1−∆𝑗 +0.4∗1 = 0.4 1−∆𝑗 +0.4 Exp No – 1 Normalized value ∆𝑗 = 1 𝐺𝑅𝐶1 = 0.4 1−1 +0.4 = 1 Exp No – 2 Normalized value ∆𝑗 = 0.81 𝐺𝑅𝐶1 = 0.4 1−0.81 +0.4 = 0.68
- 4. M-, Manufacturing division, Mechanical Dept, SRMIST, KTR 4 Grey Relation Analysis (GRA) Calculations Grey relational grade: for GRA Grade we take the average of 3 terms i.e. (Ra + Fz + MRR) /3 Grade = 𝑅𝑎+𝐹𝑧+𝑀𝑅𝑅 3 Exp No. 1 Grade = 1+0.42+0.26 3 = 0.56 Exp No. 1 Grade = 0.68+0.29+0.35 3 = 0.44 After getting Grade for all the 18 experiment ranking was done which is shown in next slide table
- 5. M-, Manufacturing division, Mechanical Dept, SRMIST, KTR Grey Relation Analysis (GRA) 5 In this, The Value of 𝛿 is 0.4 for Ra, 0.3 for Fz, and 0.3 for MRR. After this, the average is taken out to find the Grade i.e. (Ra + Fz + MRR). After that Rank is Calculated. Which is shown in below table NORMALIZATION GREY REALTION CO-EFFICIENT Exp No. Ra Fz MRR Ra Fz MRR Grade Rank 1 1.00 0.58 0.15 1.00 0.42 0.26 0.56 5 2 0.81 0.26 0.45 0.68 0.29 0.35 0.44 13 3 0.32 0.00 0.71 0.37 0.23 0.51 0.37 18 4 0.46 0.95 0.00 0.43 0.87 0.23 0.51 8 5 0.49 0.83 0.57 0.44 0.64 0.41 0.50 9 6 0.00 0.46 0.71 0.29 0.36 0.51 0.38 17 7 0.44 0.95 0.08 0.42 0.87 0.25 0.51 7 8 0.68 0.77 0.32 0.55 0.56 0.31 0.47 12 9 0.88 0.61 0.64 0.78 0.43 0.45 0.55 6 10 0.65 0.74 0.15 0.53 0.53 0.26 0.44 14 11 0.26 0.60 0.51 0.35 0.43 0.38 0.39 16 12 0.74 0.64 0.89 0.61 0.46 0.73 0.60 3 13 0.25 0.84 0.28 0.35 0.65 0.29 0.43 15 14 0.40 1.00 0.71 0.40 1.00 0.51 0.64 2 15 0.29 0.54 0.89 0.36 0.39 0.73 0.50 10 16 0.53 0.99 0.28 0.46 0.95 0.29 0.57 4 17 0.47 0.83 0.57 0.43 0.65 0.41 0.49 11 18 0.87 0.77 1.00 0.76 0.56 1.00 0.77 1 Normalization and Grey relation co-efficient table
- 6. M-, Manufacturing division, Mechanical Dept, SRMIST, KTR Results 6 Control Parameter Results Rank Tool Point Angle 135 Spindle Speed 3500 1 Feed Rate 350 Pecking Depth 1.5 Input Para level 1 level 2 level 3 Delta Rank Spindle Speed 0.467 0.492 0.563 0.096 1 Feed Rate 0.504 0.488 0.530 0.042 3 Pecking Depth 0.475 0.515 0.530 0.055 2 Optimized combination of parameters Response Table for Grey Relational Grade With the help of the ranking system, we get our optimum combination of parameters with the method GRA The Response table for Grey relational Garde is also shown in side table which shows that the spindle speed most affecting the Hole quality and Surface roughness.
- 7. M-, Manufacturing division, Mechanical Dept, SRMIST, KTR Data Envelopment Analysis based Ranking (DEAR) Method 7 The DEAR method means Data Envelopment Analysis Based Ranking which is the optimization technique of machined process, to do optimization with this technique there are 3 steps which are as follows • In the first step, we have to find out Normalized values, and also in this, we have 2 options “Larger the better” and “Smaller the better”. The formula of both the option is given below. Options Larger the Better Smaller the Better Formulas 𝐍𝐢𝐣 = 𝐎𝐢𝐣 𝐢=𝟏 𝐧𝐬 𝐎𝐢𝐣 𝐍𝐢𝐣 = 𝟏 𝐎𝐢𝐣 𝟏 𝐢=𝟏 𝐧𝐬 𝐎𝐢𝐣 • After getting the normalized value we need to transform it to the weighted response by multiplication of objective data and normalized data as given in the formula 𝐴𝑖𝑗 = 𝑁𝑖𝑗 * 𝑂𝑖𝑗
- 8. M-, Manufacturing division, Mechanical Dept, SRMIST, KTR Data Envelopment Analysis based Ranking (DEAR) Method 8 • In the third step, we need to find out the sum of the objective weighted response, since we have only one output to maximize that is MRR so 𝐴𝑚𝑎𝑥 = 𝐴𝑀𝑅𝑅 and two output to minimize and that is Ra, Fz so 𝐴𝑚𝑖𝑛 = 𝐴𝑅𝑎 + 𝐴𝐹𝑧. Calculations for DEAR method Normalization Ex No. 1 Step 1 For Surface Roughness Smaller the better formula is used to find out Normalized value 𝑁𝑅𝑎1 = 1 𝑂𝑖𝑗 1 𝑖=1 𝑛𝑠 𝑂𝑖𝑗 = 1 3.2886 1 108.889 = 33.11 Where 𝑂𝑖𝑗 = Objective Matrix 𝑁𝑖𝑗 = Normalized value of 𝑂𝑖𝑗
- 9. M-, Manufacturing division, Mechanical Dept, SRMIST, KTR Data Envelopment Analysis based Ranking (DEAR) Method 9 For Thrust Force Fz Smaller the better formula is used to find out Normalized value 𝑁𝐹𝑧1 = 1 𝑂𝑖𝑗 1 𝑖=1 𝑛𝑠 𝑂𝑖𝑗 = 1 667.4 1 10367.7 = 15.53446 For Material Removal Rate (MRR) Larger the better formula is used to find out Normalized value 𝑁𝑀𝑅𝑅1 = 𝑂𝑖𝑗 𝑖=1 𝑛𝑠 𝑂𝑖𝑗 = 162.49 4190.873 = 0.038772
- 10. M-, Manufacturing division, Mechanical Dept, SRMIST, KTR Data Envelopment Analysis based Ranking (DEAR) Method 10 Exp No. Ra Fz MRR N Ra N Fz N MRR 1 3.2886 667.4 162.49 33.11 15.53446 0.038772 2 4.3808 945.1 224.40 24.86 10.96995 0.053545 3 7.2619 1175 277.20 14.99 8.823574 0.066144 4 6.4643 343.2 130.90 16.84 30.20892 0.031235 5 6.3045 450.4 248.02 17.27 23.01887 0.059181 6 9.1723 773.3 277.20 11.87 13.40709 0.066144 7 6.5676 342.5 147.26 16.58 30.27066 0.035139 8 5.1836 506 196.35 21.01 20.48953 0.046852 9 3.9712 643.9 261.80 27.42 16.10141 0.062469 10 5.3345 531.4 162.50 20.41 19.51016 0.038774 11 7.6702 654.6 235.62 14.20 15.83822 0.056222 12 4.808 614.9 314.16 22.65 16.86079 0.074963 13 7.6969 444.9 188.50 14.15 23.30344 0.044978 14 6.8295 302.8 277.20 15.94 34.23943 0.066144 15 7.4424 705.2 314.16 14.63 14.70179 0.074963 16 6.0465 315.4 188.50 18.01 32.87159 0.044978 17 6.4164 446.8 248.02 16.97 23.20434 0.059181 18 4.0498 504.9 336.60 26.89 20.53417 0.080317 SUM 108.889 10367.7 4190.873157
- 11. M-, Manufacturing division, Mechanical Dept, SRMIST, KTR Data Envelopment Analysis based Ranking (DEAR) Method 11 Weight Response Calculations For weight response calculation we have to find out 𝐴𝑖𝑗 for all 3 outputs i.e. Ra, Fz, and MRR 𝐴𝑖𝑗 = 𝑁𝑖𝑗 X 𝑂𝑖𝑗 𝐴𝑅𝑎1 = 𝑁𝑅𝑎1 * 𝑂𝑅𝑎1 = 33.11 * 3.2886 = 108.889 𝐴𝐹𝑧1 = 𝑁𝐹𝑧1 * 𝑂𝐹𝑧1 = 15.53446 * 667.4 = 10367.7 𝐴𝑀𝑅𝑅1 = 𝑁𝑀𝑅𝑅1 * 𝑂𝑀𝑅𝑅1 = 0.038772 * 162.49 = 6.3 ARa AFz AMRR 108.889 10367.7 6.300119119 108.889 10367.7 12.01548177 108.889 10367.7 18.33504311 108.889 10367.7 4.088601434 108.889 10367.7 14.67819241 108.889 10367.7 18.33504311 108.889 10367.7 5.17463619 108.889 10367.7 9.199353227 108.889 10367.7 16.35440574 108.889 10367.7 6.30062718 108.889 10367.7 13.24706865 108.889 10367.7 23.55034426 108.889 10367.7 8.478123934 108.889 10367.7 18.33504311 108.889 10367.7 23.55034426 108.889 10367.7 8.478123934 108.889 10367.7 14.67819241 108.889 10367.7 27.03483397 Exp No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Step 2
- 12. M-, Manufacturing division, Mechanical Dept, SRMIST, KTR Data Envelopment Analysis based Ranking (DEAR) Method 12 Step 3 After this we have To determine the ratio between the sum of maximum objective weighted responses to sum of minimum objective responses which is also known as Multiple Response Performance Index (MRPI). In this step we need to find out Amax and Amin. From 3 outputs we have to maximize MRR and Minimize Ra and Fz, So values are 𝐴𝑚𝑎𝑥 = 𝐴𝑀𝑅𝑅 and 𝐴𝑚𝑖𝑛 = 𝐴𝑅𝑎 + 𝐴𝐹𝑧. 𝐴𝑚𝑎𝑥1 = 𝐴𝑀𝑅𝑅1 = 6.30 𝐴𝑚𝑖𝑛1 = 𝐴𝑅𝑎1 + 𝐴𝐹𝑧1 = 108.889 + 10367.7 = 10476.59 MRPI = 𝐴𝑚𝑎𝑥 𝐴𝑚𝑖𝑛
- 13. M-, Manufacturing division, Mechanical Dept, SRMIST, KTR Data Envelopment Analysis based Ranking (DEAR) Method 13 MRPI = 𝐴𝑚𝑎𝑥 𝐴𝑚𝑖𝑛 = 6.300119 10476.59 = 0.000601 With the MRPI we get rank for all 18th experiments as shown in below table ARa AFz AMRR Amax Amin MRPI Rank 108.889 10367.7 6.300119119 6.300119 10476.59 0.000601 16 108.889 10367.7 12.01548177 12.01548 10476.59 0.001147 11 108.889 10367.7 18.33504311 18.33504 10476.59 0.00175 6 108.889 10367.7 4.088601434 4.088601 10476.59 0.00039 18 108.889 10367.7 14.67819241 14.67819 10476.59 0.001401 8 108.889 10367.7 18.33504311 18.33504 10476.59 0.00175 4 108.889 10367.7 5.17463619 5.174636 10476.59 0.000494 17 108.889 10367.7 9.199353227 9.199353 10476.59 0.000878 12 108.889 10367.7 16.35440574 16.35441 10476.59 0.001561 7 108.889 10367.7 6.30062718 6.300627 10476.59 0.000601 15 108.889 10367.7 13.24706865 13.24707 10476.59 0.001264 10 108.889 10367.7 23.55034426 23.55034 10476.59 0.002248 3 108.889 10367.7 8.478123934 8.478124 10476.59 0.000809 13 108.889 10367.7 18.33504311 18.33504 10476.59 0.00175 5 108.889 10367.7 23.55034426 23.55034 10476.59 0.002248 2 108.889 10367.7 8.478123934 8.478124 10476.59 0.000809 14 108.889 10367.7 14.67819241 14.67819 10476.59 0.001401 9 108.889 10367.7 27.03483397 27.03483 10476.59 0.00258 1 Exp No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
- 14. M-, Manufacturing division, Mechanical Dept, SRMIST, KTR Results 14 Control Parameter Results Rank Tool Point Angle 135 Spindle Speed 3500 1 Feed Rate 350 Pecking Depth 1.5 Optimized combination of parameters Response Table for MRPI With the help of the ranking system, we get our optimum combination of parameters with the DEAR method The Response table for MRPI is also shown in side table which shows that the Pecking depth is most affecting the Hole quality and Surface roughness. Input Para level 1 level 2 level 3 Delta Rank Spindle Speed 0.001 269 0.001 391 0.001 287 0.000 122 3 Feed Rate 0.000 618 0.001 307 0.002 023 0.000 141 2 Pecking Depth 0.001 247 0.001 292 0.001 409 0.000 162 1