Figure 1.
Overrunning clutch assembly.
Figure 1.
Overrunning clutch assembly.
Figure 2.
Univariate search method to select the best process for each component. X1-P1 indicates that X1 component is produced by the P1 process; TC1113 = $26.56 indicates that components X1, X2, X3, and X4 are manufactured using processes P1, P1, P1, and P3, respectively, and the total cost to manufacture the same will be $26.56.
Figure 2.
Univariate search method to select the best process for each component. X1-P1 indicates that X1 component is produced by the P1 process; TC1113 = $26.56 indicates that components X1, X2, X3, and X4 are manufactured using processes P1, P1, P1, and P3, respectively, and the total cost to manufacture the same will be $26.56.
Figure 3.
Implementation of HSA.
Figure 3.
Implementation of HSA.
Figure 4.
Iteration number vs. NoA for various bin numbers—the 3213 process combination (a) NoA for ±0.5°, (b) NoA for ±1°, (c). NoA for ±1.5° and (d) NoA for ±2°.
Figure 4.
Iteration number vs. NoA for various bin numbers—the 3213 process combination (a) NoA for ±0.5°, (b) NoA for ±1°, (c). NoA for ±1.5° and (d) NoA for ±2°.
Figure 5.
NoA for various bin numbers and assembly specifications for the 3213 process combination (a) NoA for ±0.5°, (b) NoA for ±1°, (c). NoA for ±1.5° and (d) NoA for ±2°.
Figure 5.
NoA for various bin numbers and assembly specifications for the 3213 process combination (a) NoA for ±0.5°, (b) NoA for ±1°, (c). NoA for ±1.5° and (d) NoA for ±2°.
Figure 6.
NoA vs. bin number for various assembly specifications for the 3213 process combination.
Figure 6.
NoA vs. bin number for various assembly specifications for the 3213 process combination.
Figure 7.
Assembly specification vs. NoA for various bin numbers—the 3213 process combination.
Figure 7.
Assembly specification vs. NoA for various bin numbers—the 3213 process combination.
Figure 8.
Assembly specification vs. NoA for various bin numbers—the 1213 process combination.
Figure 8.
Assembly specification vs. NoA for various bin numbers—the 1213 process combination.
Figure 9.
Assembly specification vs. NoA for various bin numbers—the 2213 process combination.
Figure 9.
Assembly specification vs. NoA for various bin numbers—the 2213 process combination.
Figure 10.
Maximum NoA produced for different assembly specifications and various process combinations.
Figure 10.
Maximum NoA produced for different assembly specifications and various process combinations.
Table 1.
Manufacturing details of overrunning clutch assembly.
Table 1.
Manufacturing details of overrunning clutch assembly.
C.No. | Nominal Dimension (Di) (mm) | tai (mm) | Ps | P.No. | Ai | Bi | tmin (mm) | tmax (mm) |
---|
X1 | 55.29 | 0.179806 | 3 | P1 | 10.0 | 0.015 | 0.015 | 0.08 |
P2 | 5.0 | 0.500 | 0.060 | 0.15 |
P3 | 3.5 | 0.750 | 0.120 | 0.25 |
X2 | 22.86 | 0.165358 | 2 | P1 | 8.0 | 0.250 | 0.020 | 0.15 |
P2 | 3.0 | 0.650 | 0.080 | 0.30 |
X3 | 22.86 | 0.120132 | 1 | P1 | 2.5 | 0.300 | 0.040 | 0.20 |
P2 | 5.0 | 0.045 | 0.120 | 0.25 |
X4 | 101.69 | 0.200581 | 3 | P1 | 4.0 | 0.560 | 0.080 | 0.12 |
P2 | 6.0 | 0.160 | 0.150 | 0.25 |
P3 | 0.5 | 0.880 | 0.200 | 0.40 |
Table 2.
Manufacturing cost of overrunning clutch assembly for tai, tmin, and tmax.
Table 2.
Manufacturing cost of overrunning clutch assembly for tai, tmin, and tmax.
C.No. | Cai ($) | P.No. | Cmin ($) | Cmax ($) |
---|
X1 | 7.67 | P1 | 11.00 | 10.19 |
P2 | 13.33 | 8.33 |
P3 | 9.75 | 6.50 |
X2 | 6.93 | P1 | 20.50 | 9.67 |
P2 | 11.13 | 5.17 |
X3 | 5.00 | P1 | 10.00 | 4.00 |
P2 | 5.38 | 5.18 |
X4 | 4.89 | P1 | 11.00 | 8.67 |
P2 | 7.07 | 6.64 |
P3 | 4.90 | 2.70 |
Table 3.
Mean and standard deviation of components for different process combinations.
Table 3.
Mean and standard deviation of components for different process combinations.
C.No. | µi (mm) | PC1–1213 | PC2–2213 | PC3–3213 |
---|
P.No. | tmi (mm) | σi = tmi/3 | P.No. | tmi (mm) | σi = tmi/3 | P.No. | tmi (mm) | σi = tmi/3 |
---|
X1 | 55.29 | P1 | 0.08 | 0.02667 | P2 | 0.15 | 0.05 | P3 | 0.25 | 0.08333 |
X2 | 22.86 | P2 | 0.3 | 0.1 | P2 | 0.3 | 0.1 | P2 | 0.3 | 0.1 |
X3 | 22.86 | P1 | 0.25 | 0.08333 | P1 | 0.25 | 0.08333 | P1 | 0.25 | 0.08333 |
X4 | 101.69 | P3 | 0.4 | 0.13333 | P3 | 0.4 | 0.13333 | P3 | 0.4 | 0.13333 |
Table 4.
Representation of variables in HSA.
Table 4.
Representation of variables in HSA.
HSA Parameters | Equivalent Term in Optimization Problem | Equivalent Term in Present Work | Example |
---|
Musical Instrument | Decision Variable | No. of components | X1, X2, X3, and X4 |
Pitch Range | Value Range | Number of bins | 5 |
Harmony | A Solution Vector | A combination of bins for each component | (X1) 13245 (X2) 43251 (X3) 42135 (X4) 52143 |
Aesthetics | Objective Function | Number of accepted assemblies | 985 |
Practice | Iteration | No. of iterations | 150 |
Experience | Memory Matrix | Storing of the best solution | |
Harmony Memory Size | Size of Solution Vector | No. of initial solutions | 20 |
HMCR (0.7–0.99) | Harmony Memory Considering Rate | 0.7 | 0.7 |
PAR | Pitch Adjusting Rate | 0.3 | 0.3 |
Table 5.
Technical terms and their meanings.
Table 5.
Technical terms and their meanings.
Terms | Meaning |
---|
Trimming | Remove the components which do not satisfy LLi ≤ di ≤ ULi |
Partitioning | Arranging the components in ascending order based on its di and placing an equal number of components in each bin starting from bin number 1 to nb |
Mating | Selecting one component (X1, X2, X3, and X4) randomly from the bin based on the bin combination of cbi and making an assembly. Check the assembly’s specification (within LSL and USL); if it meets the requirement, then treat it as good assembly (NoA); otherwise, treat it as a surplus part |
normrnd | Matlab function will generate the required number of normally distributed random numbers based on the given mean and standard deviation value |
randperm (nb) | Matlab function will generate a permutation combination for the given bin number nb |
Table 6.
Initial harmony memory.
Table 6.
Initial harmony memory.
H.No.(k) | cbX1 | cbX2 | cbX3 | cbX4 | NoAk |
---|
1 | 2 5 1 4 3 | 2 5 3 1 4 | 5 3 4 1 2 | 3 1 2 4 5 | 308 |
2 | 4 2 1 5 3 | 4 2 5 1 3 | 4 2 5 3 1 | 5 3 4 2 1 | 649 |
3 | 4 2 1 5 3 | 1 5 2 3 4 | 3 4 2 5 1 | 3 1 5 4 2 | 407 |
4 | 4 3 1 5 2 | 3 2 4 5 1 | 5 2 3 4 1 | 3 5 2 4 1 | 508 |
5 | 1 5 4 3 2 | 1 3 4 5 2 | 3 2 4 5 1 | 2 4 1 3 5 | 326 |
6 | 2 1 3 4 5 | 3 2 4 5 1 | 3 1 5 2 4 | 3 5 2 4 1 | 557 |
7 | 5 2 3 1 4 | 2 5 3 1 4 | 1 2 3 4 5 | 1 5 4 2 3 | 354 |
8 | 1 2 4 3 5 | 1 3 5 2 4 | 5 2 4 3 1 | 5 3 4 2 1 | 397 |
9 | 2 3 1 4 5 | 3 1 2 5 4 | 2 5 1 4 3 | 5 4 2 1 3 | 212 |
10 | 4 3 1 2 5 | 4 5 3 2 1 | 3 2 1 4 5 | 3 2 4 5 1 | 458 |
Table 7.
Harmonies after sorting based on NoA.
Table 7.
Harmonies after sorting based on NoA.
H.No.(k) | cbX1 | cbX2 | cbX3 | cbX4 | NoAk |
---|
1 | 4 2 1 5 3 | 4 2 5 1 3 | 4 2 5 3 1 | 5 3 4 2 1 | 649 |
2 | 2 1 3 4 5 | 3 2 4 5 1 | 3 1 5 2 4 | 3 5 2 4 1 | 557 |
3 | 4 3 1 5 2 | 3 2 4 5 1 | 5 2 3 4 1 | 3 5 2 4 1 | 508 |
4 | 4 3 1 2 5 | 4 5 3 2 1 | 3 2 1 4 5 | 3 2 4 5 1 | 458 |
5 | 4 2 1 5 3 | 1 5 2 3 4 | 3 4 2 5 1 | 3 1 5 4 2 | 407 |
6 | 1 2 4 3 5 | 1 3 5 2 4 | 5 2 4 3 1 | 5 3 4 2 1 | 397 |
7 | 5 2 3 1 4 | 2 5 3 1 4 | 1 2 3 4 5 | 1 5 4 2 3 | 354 |
8 | 1 5 4 3 2 | 1 3 4 5 2 | 3 2 4 5 1 | 2 4 1 3 5 | 326 |
9 | 2 5 1 4 3 | 2 5 3 1 4 | 5 3 4 1 2 | 3 1 2 4 5 | 308 |
10 | 2 3 1 4 5 | 3 1 2 5 4 | 2 5 1 4 3 | 5 4 2 1 3 | 212 |
Table 8.
Best combination of bins for the first iteration.
Table 8.
Best combination of bins for the first iteration.
bcbitr | NoAitr |
---|
cbX1 | cbX2 | cbX3 | cbX4 |
---|
4 2 1 5 3 | 4 2 5 1 3 | 4 2 5 3 1 | 5 3 4 2 1 | 649 |
Table 9.
RHMCR and rcb values.
Table 9.
RHMCR and rcb values.
H.No. | RHMCR | rcb |
---|
X1 | X2 | X3 | X4 | rcbX1 | rcbX2 | rcbX3 | rcbX4 |
---|
1 | 0.0697 | 0.6581 | 0.1267 | 0.0176 | 1 | 6 | 7 | 7 |
2 | 0.1398 | 0.9349 | 0.9424 | 0.3936 | 8 | 0 | 0 | 1 |
3 | 0.4975 | 0.3145 | 0.218 | 0.8001 | 1 | 9 | 3 | 0 |
4 | 0.0639 | 0.7984 | 0.1875 | 0.442 | 7 | 0 | 3 | 4 |
5 | 0.9388 | 0.0765 | 0.605 | 0.1321 | 0 | 8 | 1 | 1 |
6 | 0.2711 | 0.4453 | 0.7794 | 0.9148 | 9 | 3 | 0 | 0 |
7 | 0.6946 | 0.413 | 0.216 | 0.0049 | 1 | 6 | 6 | 1 |
8 | 0.0204 | 0.9048 | 0.0267 | 0.7212 | 1 | 0 | 8 | 0 |
9 | 0.0013 | 0.4189 | 0.1216 | 0.8414 | 1 | 6 | 1 | 0 |
10 | 0.6254 | 0.8232 | 0.0301 | 0.6189 | 2 | 0 | 8 | 7 |
Table 10.
Harmony after HMCR.
Table 10.
Harmony after HMCR.
H.No.(k) | cbX1 | cbX2 | cbX3 | cbX4 |
---|
1′ | 4 2 1 5 3 | 1 3 5 2 4 | 1 2 3 4 5 | 1 5 4 2 3 |
2′ | 1 5 4 3 2 | 3 2 4 5 1 | 3 1 5 2 4 | 5 3 4 2 1 |
3′ | 4 2 1 5 3 | 2 5 3 1 4 | 5 2 3 4 1 | 3 5 2 4 1 |
4′ | 5 2 3 1 4 | 4 5 3 2 1 | 5 2 3 4 1 | 3 2 4 5 1 |
5′ | 4 2 1 5 3 | 1 3 4 5 2 | 4 2 5 3 1 | 5 3 4 2 1 |
6′ | 2 5 1 4 3 | 3 2 4 5 1 | 5 2 4 3 1 | 5 3 4 2 1 |
7′ | 4 2 1 5 3 | 1 3 5 2 4 | 5 2 4 3 1 | 5 3 4 2 1 |
8′ | 4 2 1 5 3 | 1 3 4 5 2 | 3 2 4 5 1 | 2 4 1 3 5 |
9′ | 4 2 1 5 3 | 1 3 5 2 4 | 4 2 5 3 1 | 3 1 2 4 5 |
10′ | 2 1 3 4 5 | 3 2 4 5 1 | 3 2 4 5 1 | 1 5 4 2 3 |
Table 11.
RPAR, r1, and r2 values.
Table 11.
RPAR, r1, and r2 values.
H.No. | RPAR | r1 | r2 |
---|
X1 | X2 | X3 | X4 | r1X1 | r1X2 | r1X3 | r1X4 | r2X1 | r2X2 | r2X3 | r2X4 |
---|
1′ | 0.0917 | 0.0738 | 0.4457 | 0.8808 | 3 | 1 | 0 | 0 | 1 | 2 | 0 | 0 |
2′ | 0.2529 | 0 | 0 | 0.6409 | 1 | 0 | 0 | 0 | 3 | 0 | 0 | 0 |
3′ | 0.5235 | 0.009 | 0.344 | 0 | 0 | 1 | 0 | 0 | 0 | 4 | 0 | 0 |
4′ | 0.6034 | 0 | 0.3208 | 0.6256 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
5′ | 0 | 0.1307 | 0.599 | 0.4411 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 0 |
6′ | 0.5166 | 0.4696 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
7′ | 0.5402 | 0.5435 | 0.8627 | 0.7751 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
8′ | 0.3204 | 0 | 0.7643 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
9′ | 0.1678 | 0.277 | 0.0321 | 0 | 1 | 3 | 1 | 0 | 3 | 3 | 3 | 0 |
10′ | 0.2417 | 0 | 0.8962 | 0.4639 | 4 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
Table 12.
Harmony after HMCR and PAR.
Table 12.
Harmony after HMCR and PAR.
H.No. | After HMCR and PAR | NoA |
---|
cb1X1 | cb1X2 | cb1X3 | cb1X4 |
---|
1′ | 1 2 4 5 3 | 3 1 5 2 4 | 1 2 3 4 5 | 1 5 4 2 3 | 709 |
2′ | 4 5 1 3 2 | 2 5 4 3 1 | 4 1 5 3 2 | 5 3 4 2 1 | 608 |
3′ | 4 2 1 5 3 | 1 5 3 2 4 | 5 2 3 4 1 | 3 4 1 5 2 | 671 |
4′ | 5 2 3 1 4 | 1 5 2 3 4 | 5 2 3 4 1 | 3 2 4 5 1 | 555 |
5′ | 2 1 5 3 4 | 1 3 4 5 2 | 4 2 5 3 1 | 5 3 4 2 1 | 503 |
6′ | 2 5 1 4 3 | 3 2 4 5 1 | 2 1 5 4 3 | 1 3 5 2 4 | 312 |
7′ | 4 2 1 5 3 | 1 3 5 2 4 | 5 2 4 3 1 | 5 3 4 2 1 | 519 |
8′ | 4 2 1 5 3 | 2 4 5 3 1 | 3 2 4 5 1 | 5 3 4 2 1 | 404 |
9′ | 1 2 4 5 3 | 1 3 5 2 4 | 5 2 4 3 1 | 3 2 5 1 4 | 460 |
10′ | 4 1 3 2 5 | 1 4 3 5 2 | 3 2 4 5 1 | 1 5 4 2 3 | 549 |
Table 13.
Selection of harmonies for next iteration.
Table 13.
Selection of harmonies for next iteration.
Before Sorting | After Sorting | SHMNo. |
---|
HMNo. | NoA | HMNo. | NoA |
---|
1 | 649 | 1′ | 709 | 1″ |
2 | 557 | 2′ | 671 | 2″ |
3 | 508 | 1 | 649 | 3″ |
4 | 458 | 3′ | 608 | 4″ |
5 | 407 | 2 | 557 | 5″ |
6 | 397 | 4′ | 555 | 6″ |
7 | 354 | 5′ | 549 | 7″ |
8 | 326 | 6′ | 519 | 8″ |
9 | 308 | 3 | 508 | 9″ |
10 | 212 | 7′ | 503 | 10″ |
1′ | 709 | 8′ | 460 | NS |
2′ | 671 | 4 | 458 | NS |
3′ | 608 | 5 | 407 | NS |
4′ | 555 | 9′ | 404 | NS |
5′ | 549 | 6 | 397 | NS |
6′ | 519 | 7 | 354 | NS |
7′ | 503 | 8 | 326 | NS |
8′ | 460 | 10′ | 312 | NS |
9′ | 404 | 9 | 308 | NS |
10′ | 312 | 10 | 212 | NS |
Table 14.
Selected harmonies for next iteration.
Table 14.
Selected harmonies for next iteration.
IHMno | HMNo | X1 | X2 | X3 | X4 | NoA |
---|
1″ | 1′ | 1 2 4 5 3 | 3 1 5 2 4 | 1 2 3 4 5 | 1 5 4 2 3 | 709 |
2″ | 2′ | 4 2 1 5 3 | 1 5 3 2 4 | 5 2 3 4 1 | 3 4 1 5 2 | 671 |
3″ | 1 | 4 2 1 5 3 | 4 2 5 1 3 | 4 2 5 3 1 | 5 3 4 2 1 | 649 |
4″ | 3′ | 4 5 1 3 2 | 2 5 4 3 1 | 4 1 5 3 2 | 5 3 4 2 1 | 608 |
5″ | 2 | 2 1 3 4 5 | 3 2 4 5 1 | 3 1 5 2 4 | 3 5 2 4 1 | 557 |
6″ | 4′ | 5 2 3 1 4 | 1 5 2 3 4 | 5 2 3 4 1 | 3 2 4 5 1 | 555 |
7″ | 5′ | 4 1 3 2 5 | 1 4 3 5 2 | 3 2 4 5 1 | 1 5 4 2 3 | 549 |
8″ | 6′ | 4 2 1 5 3 | 1 3 5 2 4 | 5 2 4 3 1 | 5 3 4 2 1 | 519 |
9″ | 3 | 4 3 1 5 2 | 3 2 4 5 1 | 5 2 3 4 1 | 3 5 2 4 1 | 508 |
10″ | 7′ | 2 1 5 3 4 | 1 3 4 5 2 | 4 2 5 3 1 | 5 3 4 2 1 | 503 |
Table 15.
Best bin combinations for various assembly specifications and process combinations.
Table 15.
Best bin combinations for various assembly specifications and process combinations.
AS | PC3–3213 | PC2–2213 | PC1–1213 |
---|
X1 | X2 | X3 | X4 | NoA | X1 | X2 | X3 | X4 | NoA | X1 | X2 | X3 | X4 | NoA |
---|
±0.5° | 1 | 2 | 3 | 2 | 3 | 1 | 2 | 3 | 1 | 1 | 3 | 2 | 674 | 3 | 2 | 1 | 1 | 2 | 3 | 2 | 3 | 1 | 1 | 2 | 3 | 658 | 2 | 1 | 3 | 3 | 2 | 1 | 3 | 1 | 2 | 3 | 1 | 2 | 681 |
±1° | 1 | 2 | 3 | 3 | 2 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 967 | 1 | 3 | 2 | 3 | 1 | 2 | 2 | 1 | 3 | 2 | 1 | 3 | 985 | 2 | 1 | 3 | 2 | 1 | 3 | 3 | 2 | 1 | 3 | 1 | 2 | 994 |
±1.5° | 1 | 2 | 3 | 3 | 2 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 996 | 2 | 1 | 3 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 996 | 2 | 1 | 3 | 3 | 2 | 1 | 3 | 2 | 1 | 3 | 2 | 1 | 999 |
±2° | 1 | 3 | 2 | 2 | 1 | 3 | 2 | 3 | 1 | 1 | 3 | 2 | 996 | 2 | 1 | 3 | 1 | 2 | 3 | 3 | 2 | 1 | 2 | 1 | 3 | 996 | 1 | 3 | 2 | 1 | 3 | 2 | 1 | 3 | 2 | 1 | 3 | 2 | 999 |