Introduction
Robust design for uncertainty
Experimental designs with RSM
Process capability index as response values for statistical analysis
The application for product design and process planning
Component j | Lower level | Middle level | Upper level |
---|---|---|---|
\(U_{1}\)
| 4.9654 mm | 5.0000 mm | 5.0346 mm |
\(U_{2}\)
| 8.4740 mm | 8.5000 mm | 8.5260 mm |
\(U_{3}\)
| 3.9654 mm | 4.0000 mm | 4.0346 mm |
\(U_{4}\)
| 8.3311 mm | 8.3600 mm | 8.3889 mm |
\(U_{5}\)
| 17.6740 mm | 17.7000 mm | 17.7260 mm |
\(t_{1}\)
| $10.7702 (0.03 mm) | $10.7653 (0.06 mm) | $10.7543 (0.09 mm) |
\(t_{2}\)
| $12.7705 (0.02 mm) | $12.7618 (0.04 mm) | $12.7437 (0.06 mm) |
\(t_{3}\)
| $10.7702 (0.03 mm) | $10.7653 (0.06 mm) | $10.7543 (0.09 mm) |
\(t_{4}\)
| $5.7009 (0.04 mm) | $5.6971 (0.06 mm) | $5.6879 (0.08 mm) |
\(t_{5}\)
| $7.1243 (0.03 mm) | $7.1227 (0.05 mm) | $7.1185 (0.07 mm) |
Response surface for variable \(C_{pmc}\)
| |||||
---|---|---|---|---|---|
Response mean | 0.006908 | ||||
Root MSE | 0.000000181 | ||||
R-square | 0.9979 | ||||
Coefficient of variation | 0.4648 |
Regression | DF | S S | R-Square | F value |
\(\text{ Pr }>F\)
|
---|---|---|---|---|---|
Linear | 10 | 4.8437E\(-\)9 | 0.0424 | 14865.9 |
\(<\)0.0001 |
Quadratic | 10 | 3.1978E\(-\)8 | 0.2797 | 98144.2 |
\(<\)0.0001 |
Crossproduct | 45 | 7.74886E\(-\)8 | 0.6779 | 52848.7 |
\(<\)0.0001 |
Total Model | 65 | 0.000000114 | 1.0000 | 53973.7 |
\(<\)0.0001 |
Factor | Degrees of freedom | Sum of squares | Mean square | Prob \(>\) F | |
---|---|---|---|---|---|
\(U_{1}\)
| 11 | 2.6443291E\(-\)8 | 2.4039356E\(-\)9 |
\(<\)0.0001 | |
\(U_{2}\)
| 11 | 4.3891891E\(-\)8 | 3.9901719E\(-\)9 |
\(<\)0.0001 | |
\(U_{3}\)
| 11 | 3.2048017E\(-\)8 | 2.913456E\(-\)9 |
\(<\)0.0001 | |
\(U_{4}\)
| 11 | 2.8242178E\(-\)8 | 2.5674708E\(-\)9 |
\(<\)0.0001 | |
\(U_{5}\)
| 11 | 2.3253698E\(-\)8 | 2.1139726E\(-\)9 |
\(<\)0.0001 | |
\(t_{1}\)
| 11 | 1.170191E\(-\)10 | 1.06381E\(-\)11 |
\(<\)0.0001 | |
\(t_{2}\)
| 11 | 3.1594869E\(-\)9 | 2.872261E\(-\)10 |
\(<\)0.0001 | |
\(t_{3}\)
| 11 | 1.233381E\(-\)10 | 1.121255E\(-\)11 |
\(<\)0.0001 | |
\(t_{4}\)
| 11 | 7.152263E\(-\)10 | 6.502057E\(-\)11 |
\(<\)0.0001 | |
\(t_{5}\)
| 11 | 2.986217E\(-\)11 | 2.714743E\(-\)12 |
\(<\)0.0001 |
Factors | F-ratio | F-ranking |
---|---|---|
\(U_{1}\)
| 73778.80 | 3 |
\(U_{2}\)
| 122462.00 | 1 |
\(U_{3}\)
| 89416.40 | 2 |
\(U_{4}\)
| 78797.80 | 4 |
\(U_{5}\)
| 64879.60 | 5 |
\(t_{1}\)
| 326.49 | 9 |
\(t_{2}\)
| 8815.21 | 6 |
\(t_{3}\)
| 344.12 | 8 |
\(t_{4}\)
| 1995.54 | 7 |
\(t_{5}\)
| 83.32 | 10 |
Operation number | Working machine | Reference surface | Processed surface | Lower process capability | Upper process capability |
---|---|---|---|---|---|
\(i\)
|
\(t_{Li}\) (mm) |
\(t_{Ui}\) (mm) | |||
1 | Turret lathe | F | E | 0.04 | 0.14 |
2 | Turret lathe | F | D | 0.04 | 0.14 |
3 | Turret lathe | F | C | 0.05 | 0.13 |
4 | Turret lathe | F | B | 0.05 | 0.13 |
5 | Turret lathe | B | A | 0.05 | 0.13 |
\(r\)
| Blueprint dimensions | Dimension chain | Dimensions chain vector | Design tolerance | Target value | Quality loss coef. |
---|---|---|---|---|---|---|
\(r\)
|
\(A_{re}\)
|
\(S_{r}\)
|
\(T_{r}\)
|
\(K_{r}\)
| ||
1 | A–F | 5–4 | [1,1] | 0.25 | 200 | 6,000 |
2 | C–E | 3–1 | [1,\(-\)1] | 0.26 | 41 | 10,500 |
3 | C–D | 3–2 | [1,\(-\)1] | 0.26 | 25 | 9,000 |
4 | B–C | 4–3 | [1,\(-\)1] | 0.12 | 79 | 1,500 |
5 | A–B | 5 | [1] | 0.24 | 42 | 10,500 |
Operation \(i\)
| Lower level | Middle level | Upper level |
---|---|---|---|
\(U_{1}\)
| 38 mm | 39 mm | 40 mm |
\(U_{2}\)
| 55 mm | 56 mm | 57 mm |
\(U_{3}\)
| 79 mm | 81 mm | 82 mm |
\(U_{4}\)
| 157 mm | 160 mm | 163 mm |
\(U_{5}\)
| 42 mm | 43 mm | 44 mm |
\(t_{1}\)
| $93.0028 (0.04) mm | $71.1156 (0.09) mm | $63.0939 (0.14) mm |
\(t_{2}\)
| $93.7241 (0.04) mm | $71.3709 (0.09) mm | $63.0717 (0.14) mm |
\(t_{3}\)
| $78.5737 (0.05) mm | $56.9872 (0.09) mm | $50.5266 (0.13) mm |
\(t_{4}\)
| $79.4912 (0.05) mm | $57.2412 (0.09) mm | $50.5153 (0.13) mm |
\(t_{5}\)
| $79.4991 (0.05) mm | $57.2688 (0.09) mm | $50.5201 (0.13) mm |
Response surface for variable \(C_{pmc}\) | |||||
---|---|---|---|---|---|
Response mean | 0.007992 | ||||
Root MSE | 0.001249 | ||||
\(R^{2}\) | 0.9624 | ||||
Coefficient of variation | 15.6241 |
Regression | DF | S S | R-square | F value |
\(\text{ Pr } > F\)
|
---|---|---|---|---|---|
Linear | 10 | 0.000584 | 0.1356 | 37.47 |
\(<\)0.0001 |
Quadratic | 10 | 0.003526 | 0.8183 | 226.14 |
\(<\)0.0001 |
Crossproduct | 45 | 0.0000363 | 0.0084 | 0.52 | 0.9927 |
Total Model | 65 | 0.004147 | 0.9624 | 40.91 |
\(<\)0.0001 |
Factors | Degrees of freedom | Sum of squares | Mean square | Prob \(>\) F | |
---|---|---|---|---|---|
\(U_{1}\)
| 11 | 0.000010044 | 0.000000913 | 0.8368 | |
\(U_{2}\)
| 11 | 0.000015449 | 0.000001404 | 0.5425 | |
\(U_{3}\)
| 11 | 0.000127 | 0.000011517 |
\(<\)0.0001 | |
\(U_{4}\)
| 11 | 0.000276 | 0.000025111 |
\(<\)0.0001 | |
\(U_{5}\)
| 11 | 0.000010136 | 0.000000921 | 0.8325 | |
\(t_{1}\)
| 11 | 0.000001496 | 0.000000136 | 1.0000 | |
\(t_{2}\)
| 11 | 0.000002247 | 0.000000204 | 0.9996 | |
\(t_{3}\)
| 11 | 0.000013530 | 0.000001230 | 0.6506 | |
\(t_{4}\)
| 11 | 0.000042951 | 0.000003905 | 0.0078 | |
\(t_{5}\)
| 11 | 0.000094262 | 0.000008569 |
\(<\)0.0001 |
Summary
Factors | F-ratio | F-ranking |
---|---|---|
\(U_{1}\)
| 0.59 | 7 |
\(U_{2}\)
| 0.90 | 5 |
\(U_{3}\)
| 7.39 | 2 |
\(U_{4}\)
| 16.10 | 1 |
\(U_{5}\)
| 0.59 | 8 |
\(t_{1}\)
| 0.09 | 10 |
\(t_{2}\)
| 0.13 | 9 |
\(t_{3}\)
| 0.79 | 6 |
\(t_{4}\)
| 2.50 | 4 |
\(t_{5}\)
| 5.50 | 3 |
-
Step 3: Use the combined levels of inputs, \(U\) and \(t\), as the arrangement of an experimental design matrix to find: a) \(K[\sigma ^2 + (U-T)^2]\), \(\sigma ^2 =(t/P)^{2}\), normally \(P = 3,\, K\) is \(C_A /S^{2}\), b) \(C_M (t)\) is \(a+be^{-ct}\) or tolerance cost with given tolerance level (see Table 1 or Table 6). Then have the results from items a and b fed into Eq. (7) to find \(C_{pmc}\).
-
Step 4: Perform RSM with SAS software to obtain the best values of \(U\) and \(t\) for the maximization of \(C_{pmc}\) and to find Cpmc prediction functions by regression analysis. See second-order model \(C_{pmc}\) predicting equation in Examples 1 and 2. Then, employ ANOVA to rank the important parameters (see Tables 3, 8).
-
Step 6: If improvement is needed, then the above steps can be repeated based on the suggestions made in Step 5.