Increased market competition means that quality, cost and delivery time are crucial elements of modern production techniques. Taguchi’s robust design is the most powerful method available for reducing product cost, improving quality, and simultaneously reducing development time. Robust design aims to reduce the impact of noise on the product or process quality and leads to greater customer satisfaction and higher operational performance. The objective of robust design is to minimize the total quality loss in products or processes. The PQL model proposed by this paper simultaneously optimizes the static and dynamic problems by minimizing the total quality loss. Using the proposed PQL model and steps for optimization, the method addresses complex parameter design, which varies with the properties and objectives of the experimental data, to improve the product quality. The example of an electron beam surface hardening process is provided to demonstrate the implementation and usefulness of the proposed method.
Quality, cost and delivery time are the main production elements of the modern products due to the stringent market competitiveness. Taguchi’s robust design [
Many publications have addressed multiple static quality characteristics problems (see Derringer and Suich [
In this paper, we propose a PQL index to convert the multiple quality characteristics into a single characteristic problem by minimizing the total PQL value to obtain the optimal parameter conditions.
The quality characteristics can be divided three types according to the target of problem: (1) the smaller-the- better (STB) type for static system; (2) the larger-the-better (LTB) type for static system; and (3) the nominal- the-best (NTB) type, which can be classified as static system and dynamic system.
Taguchi [
where
K = the loss coefficient (constant)
y = a measurable statistic of quality characteristic
m0 = the target value for static NTB quality characteristic
b0 = the slope of ideal function for dynamic NTB quality characteristic
b = the estimated slope of regression for dynamic NTB quality characteristic
The loss coefficient K, m0 and b 0 in
The analysis of means (ANOM) is used to determine the optimal factor levels in robust design. The ANOM is used for estimating the main effects of each parameter, and the effect of a factor level is the deviation it causes from the overall mean response. Let
Suppose there are q control factors,
The relationship between (
Let L be the quality loss for the some parameter conditions and L0 be the quality loss for the starting conditions. The ratio of L to L0 (proportion of quality loss, PQL) is defined as.
Consider the effect of each factor in
Therefore, the optimal parameter conditions
Suppose a product or process has p quality characteristics
If the real quality loss of starting conditions,
quality loss Lj of quality characteristic Yj as the base to find the proportion of quality loss Li of quality characteristic Yi to Yj. Hence, the proportion
, (11)
Therefore, Equation (10) can be rewritten as
To solve the multiple quality characteristics problems, an optimization procedure is proposed as follows.
Step 1. Compute the SN ratio for each quality characteristic and then calculate the main effect of factors for each quality characteristic.
Step 2. Estimate the average SN ratio (h0) under the starting conditions for each quality characteristic.
Step 3. Transform the SN ratios into PQL for each quality characteristic.
Step 4. Estimate the quality loss of starting conditions for each quality characteristic and then program a search module by EXCEL VBA to obtain the optimal parameter conditions.
The case used is that described by Jean and Tzeng [
High energy electron beam is a unique tool for case hardening. The control factors are substrate matrix (factor A), travel speed (factor B), accelerating voltage (factor C), electrical current (factor D), melted width (factor E), beam oscillation (factor F) and post-heat treatment temperature (factor G). The signal factor is electron beam scanning width (factor M). The levels of control and signal factors are listed in
There are two quality characteristics for the process. The first is wear volume (STB type) and the second is microhardness (dynamic type). A L18 orthogonal array is built and the assignment of controls, signal factor, experimental data and the computed SN ratios (h) for all quality characteristics are shown in
The main effect of factors and PQL values for each quality characteristic are shown in
Control factor | Levels | ||
---|---|---|---|
1 | 2 | 3 | |
A. Substrate matrix | Ductile | Gray | |
B. Travel speed, mm∙s−1 | 10 | 20 | 30 |
C. Accelerating voltage, V | 10 | 25 | 50 |
D. Electrical current, mA | 10 | 15 | 20 |
E. Melted width, mm | 5 | 15 | 20 |
F. Beam oscillation | Line | Circle | Ellipse |
G. Post-heat treatment temperature, ˚C | 25 | 150 | 300 |
Signal factor | |||
M. Electron beam scanning width | 5 mm | 10 mm | 20 mm |
a. Starting levels are identified by underscore.
Expt. No. | Factor assignment | Wear volume | Microhardness | SN ratios | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
A | B | C | D | E | F | G | M1 = 5 mm | M2 = 10 mm | M3 = 20 mm | Wear volume | Microhardness | ||||
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 167 | 164 | 171 | 875 | 896 | 921 | −44.473 | −18.311 |
2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 219 | 221 | 228 | 712 | 719 | 698 | −46.954 | −18.952 |
3 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 279 | 289 | 291 | 568 | 546 | 559 | −49.139 | −18.832 |
4 | 1 | 2 | 1 | 1 | 2 | 2 | 3 | 159 | 164 | 167 | 876 | 835 | 868 | −44.263 | −18.757 |
5 | 1 | 2 | 2 | 2 | 3 | 3 | 1 | 174 | 176 | 177 | 889 | 876 | 849 | −44.894 | −19.145 |
6 | 1 | 2 | 3 | 3 | 1 | 1 | 2 | 189 | 199 | 192 | 756 | 732 | 723 | −45.728 | −19.104 |
7 | 1 | 3 | 1 | 2 | 1 | 3 | 2 | 195 | 198 | 197 | 901 | 926 | 893 | −45.875 | −18.887 |
8 | 1 | 3 | 2 | 3 | 2 | 1 | 3 | 178 | 181 | 183 | 789 | 801 | 776 | −45.138 | −18.933 |
9 | 1 | 3 | 3 | 1 | 3 | 2 | 1 | 168 | 172 | 174 | 792 | 786 | 775 | −44.678 | −18.937 |
10 | 2 | 1 | 1 | 3 | 3 | 2 | 2 | 199 | 201 | 206 | 686 | 642 | 613 | −46.108 | −19.652 |
11 | 2 | 1 | 2 | 1 | 1 | 3 | 3 | 226 | 221 | 231 | 621 | 632 | 645 | −47.084 | −18.427 |
12 | 2 | 1 | 3 | 2 | 2 | 1 | 1 | 215 | 221 | 217 | 757 | 723 | 734 | −46.756 | −18.959 |
13 | 2 | 2 | 1 | 2 | 3 | 1 | 3 | 206 | 205 | 203 | 812 | 796 | 772 | −46.221 | −19.177 |
14 | 2 | 2 | 2 | 3 | 1 | 2 | 1 | 202 | 206 | 211 | 768 | 706 | 615 | −46.293 | −20.53 |
15 | 2 | 2 | 3 | 1 | 2 | 3 | 2 | 213 | 208 | 209 | 681 | 723 | 712 | −46.445 | −18.458 |
16 | 2 | 3 | 1 | 3 | 2 | 3 | 1 | 165 | 169 | 167 | 856 | 832 | 841 | −44.455 | −18.865 |
17 | 2 | 3 | 2 | 1 | 3 | 1 | 2 | 175 | 176 | 177 | 845 | 827 | 831 | −44.910 | −18.867 |
18 | 2 | 3 | 3 | 2 | 1 | 2 | 3 | 213 | 217 | 219 | 706 | 675 | 568 | −46.703 | −20.539 |
Quality characteristics | Level | Factor | ||||||
---|---|---|---|---|---|---|---|---|
A | B | C | D | E | F | G | ||
Wear volume (h) | Level 1 | −45.682 | −46.752 | −45.232 | −45.309 | −46.026 | −45.538 | −45.258 |
Level 2 | −46.108 | −45.641 | −45.879 | −46.234 | −45.669 | −45.833 | −46.003 | |
Level 3 | −45.293 | −46.575 | −46.143 | −45.992 | −46.315 | −46.425 | ||
Wear volume (PQL) | Level 1 | 0.906603 | 1.000000 | 0.734119 | 0.808140 | 1.085748 | 1.000000 | 1.000000 |
Level 2 | 1.000000 | 0.774170 | 0.851935 | 1.000000 | 1.000000 | 1.070381 | 1.187204 | |
Level 3 | 0.714623 | 1.000000 | 0.979386 | 1.077194 | 1.195994 | 1.308158 | ||
Microhardness (h) | Level 1 | −18.873 | −18.855 | −18.941 | −18.626 | −19.300 | −18.892 | −19.124 |
Level 2 | −19.275 | −19.195 | −19.142 | −19.277 | −18.821 | −19.561 | −18.987 | |
Level 3 | −19.171 | −19.138 | −19.319 | −19.102 | −18.769 | −19.111 | ||
Microhardness (PQL) | Level 1 | 0.911557 | 1.000000 | 0.956030 | 0.860860 | 1.117014 | 1.000000 | 1.000000 |
Level 2 | 1.000000 | 1.081846 | 1.000734 | 1.000000 | 1.000000 | 1.166878 | 0.969124 | |
Level 3 | 1.075768 | 1.000000 | 1.009621 | 1.067651 | 0.971624 | 0.997091 |
a. Optimal parameter levels for each characteristic are identified by boldface type.
Region | optimal parameter conditions | Predicted SN ratio | |
---|---|---|---|
Wear volume | Microhardness | ||
A1B 3 C 1D1E 2F 1 G 1 | −42.609 db | −18.005 db | |
A1B 1 C 1D1E 2F 1 G 1 | −44.068 db | −17.689 db | |
A1B 1 C 1D1E 2F 1 G 2 | −44.814 db | −17.551 db | |
A1B 1 C 1D1E 2F 3 G 2 | −45.591 db | −17.429 db |
Robust design is used to determine the optimal levels for the control factors in a product or process so that the quality loss is minimized. A real problem in a product or process usually has multiple quality characteristics. This paper presents an effective method based on Taguchi’s quality loss function and SN ratio to simultaneously optimize the robust design involving both static and dynamic quality characteristics. Using the PQL transformed from the factor effects of SN ratios as the quality evaluation, we can convert the static and dynamic multiple quality characteristics into a single characteristic problem to obtain the optimal parameter conditions by minimizing the total PQL value. The implementation and effectiveness of the proposed approach is illustrated through case study.
This research was financially supported by Ministry of Science and Technology (Republic of China) under Contract MOST 104-2221-E-238-003.
Ful-Chiang Wu, (2015) Robust Design of Mixing Static and Dynamic Multiple Quality Characteristics. World Journal of Engineering and Technology,03,72-77. doi: 10.4236/wjet.2015.33C011