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Proposal of the “Total Error Minimization Method” for robust design
Engineering Science and Technology, an International Journal xxx (xxxx) xxx
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Proposal of the ‘‘Total Error Minimization Method” for robust design Shin-ichi Inage ⇑ Hitachi Ltd. Social Infrastructure information Sysytem Division, Hitacchi Omori 2nd Bldg., 27-18, Minami Oi 6-chome, Shinagawa-ku, Tokyo, 140-8572, Japan Tokyo Institute of Technology, Solution Research Laboratory, International Research Center of Advanced Energy Systems for Sustainability, 2-12-1-16-25 Ookayama, Meguro-ku, Tokyo 152-8550, Japan
a r t i c l e
i n f o
Article history: Received 18 July 2018 Revised 6 November 2018 Accepted 6 November 2018 Available online xxxx Keywords: Robust design Design optimization Orthogonal array Multi performances
a b s t r a c t We propose a new approach for the robust design of a product with single or multiple performances/outputs without axioms. When there are no theoretical models, experimental methods (e.g., the popular and robust Taguchi and Nakazawa design methods) with an orthogonal array are important. Meanwhile, if mathematical or physical models are available, other robust design approaches using, for example, a genetic algorithm, are applicable. The proposed approach is compatible with both experimental and theoretical approaches. The approach is proposed through the minimization of the total difference (i.e., error) between the required specification and experimental output for different variables with noise. We applied the approach to an electric circuit problem to obtain the required specification and target cost. The optimal results are the same as those obtained using the Nakazawa method. Additionally, a genetic-algorithm-based design with a proposed metric, namely the optimal design under contentious variables, is presented. The obtained performance/output better matches the required specification when using the proposed approach than when using the discrete approach. Employing the proposed method, we expect a united robust design without an axiom. Ó 2018 Karabuk University. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction This paper proposes a new approach for the robust design of a product having single or multiple performances/outputs. Since the 1980s, robust design has contributed to improvements in the quality of products. G. Taguchi proposed a concept of robust design having two stages [15–19], focusing on the effect of the noise factor on variations in performance/output and introducing the signal-to-noise (SN) ratio as a metric with which to measure the effect. Even today, his approach is applied in many fields of engineering [5–7]). His concept of the SN ratio is based on direct consideration of the effect of the noise factor on the performance/ output of a product, and his rational and systematic approach of using an orthogonal array and a design of experiments are remarkable [18]. In particular, the Taguchi method is effective when the target product has one performance/output. In the case of multiple performances/outputs, the design navigation method of Nakazawa is well known [8]. This approach introduces the reciprocal of satisfaction (Recsat) as a metric and is based on the application of infor⇑ Address: Tokyo Institute of Technology, Solution Research Laboratory, International Research Center of Advanced Energy Systems for Sustainability, 2-12-1-1625 Ookayama, Meguro-ku, Tokyo 152-8550, Japan. E-mail address: [email protected]
mation theory developed by Suh [10]. Suh defined an axiom by stating, ‘‘The optimal design is to minimize the required information content to achieve the required performance/output of a product.” According to his definition, information content is equivalent to the resource or effort required to achieve the required performance/output of the product. Nakazawa used Recsat as information content. Following his approach, robust design is achieved for the minimum Recsat of the product system. If this axiom can be accepted, design navigation (i.e., the Nakazawa method) can be used to decide the best combination of variables in terms of the performance/output of the product. The Taguchi and Nakazawa methods use different metrics to measure the sensitivity to variation according to different concepts. Meanwhile, the two methods provide similar solutions for robust design, suggesting that we can establish a more united approach. The Taguchi and Nakazawa methods are discrete methods and are especially effective when using an experimental design with an orthogonal array and the product has no mathematical or physical model. Meanwhile, for the product that has a mathematical or physical model, with the remarkable progress of optimization algorithms (e.g., genetic algorithms), contentious robust design is becoming a powerful tool [11]. The present paper focuses on finding a new approach for robust design that is compatible with the Taguchi and Nakazawa methods
Peer review under responsibility of Karabuk University. https://doi.org/10.1016/j.jestch.2018.11.005 2215-0986/Ó 2018 Karabuk University. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article as: S.-i. Inage, Proposal of the ‘‘Total Error Minimization Method” for robust design, Engineering Science and Technology, an International Journal, https://doi.org/10.1016/j.jestch.2018.11.005
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and more general approaches based on, for example, genetic algorithms. 2. Brief summary of conventional methods 2.1. Taguchi method The Taguchi method is a well-known and popular method of robust design (Genichi Taguchi, 1980). The method has two basic principles.
This SN ratio is a metric used to estimate the sensitivity of the performance of the product to noise; i.e., the SN ratio is high for low sensitivity and low for high sensitivity. In this paper, to demonstrate how to apply the Taguchi method, we applied the method to the simple electric circuit having two transistors and resistances shown in Fig. 1 (as used by [19]. We traced the analysis of this circuit with Taguchi method by Hayashi et al. [14] [19]. For an input voltage Ei, the output Eo can be estimated as
Eo ¼ 1) A limited set of experiments is conducted using an orthogonal array to estimate the sensitivity of the quality measures to various processing and observable and controllable design parameters and to noise from uncontrollable variation. 2) The product is designed so that the sensitivity of the quality measurements to noise is minimized. In other words, the product/process is designed so that the performance/output is in a regime that minimizes the effects of uncontrollable process variations. In the case of the Taguchi method, ‘‘noise” is introduced intentionally to estimate the sensitivity of the performance of a product to uncontrollable variations. Noise N1, which results in the performance highest, and noise N2, which results in the lowest performance, are usually considered. With orthogonal arrays, a limited set of experiments is performed for different combinations of controllable variations with noise N1 and N2. The following are then estimated for the performance y of the product. Performance results: Under a combination of controllable variations with N1 and N2: For N1: performance y1 For N2: performance y2
ðP 2 þ P3 Þ P1
ð4Þ
where
ðR1 þ R2 =2Þ 1 þ hfe1 þ R4 hfe2 1 þ hfe1 þ R4 P1 ¼ ðR1 þ R2 =2Þ 1 þ hfe1 P2 ¼ Ei V BE1 þ
P3 ¼
ð5Þ
ðV BE2 þ V Z ÞR4 hfe2 ðV BE2 þ V Z ÞR4 hfe2 þ R1 þ R22 R3 þ R22
ð6Þ
ðV BE2 þ V Z ÞR4 Io R4 ðR1 þ R2 =2Þ ðR1 þ R2 =2Þ 1 þ hfe1
ð7Þ
The experimental levels of design variables are given in Table 1. These variables are dispatched to an L18 orthogonal array as shown in Table 2. In the circuit, the resistance Ro is controllable and the current Io is varied from 0 to 800 mA. In the L18 array, the first and last rows are treated as dummy variables. Noise N1 and noise N2 are considered for each variable as shown in Table 3. As defined, noise N1 results in the highest performance/output Eo while noise N2 results in the lowest Eo. The other parameters VBE1, VBE2, and Vz are set at a constant 0.6 V. For this problem, the target performance/output (i.e., specification) is set at 10 ± 1 V (i.e., 9 V < Eo < 11 V) under any current Io. Table 4 summarizes the
Average performance:
Pn
i¼1 yi
y¼
n
¼
1 ðy þ y2 Þ 2 1
SN ratio:
y2 g ¼ 10log 2 s
ð1Þ
! ð2Þ
where
s ¼ 2
Pn 2 i¼1 yi y
n1 2 2 1 is dispersion y1 y þ y1 y ¼ 21
Fig. 1. Target electronic circuit [19].
ð3Þ
Table 1 Experimental levels of design variables taken from Hayashi et al. [14]. Design variables
The first level
The second level
The third level
R1 R2 R3 R4 hfe1 hfe2
2.2 kX 0.11 kX 0.3 kX 0.08 kX 18 50
11 kX 0.56 kX 1.5 kX 0.42 kX 35 100
56 kX 2.7 kX 7.5 kX 2.2 kX 70 200
Table 2 Applied L18 orthogonal table. No.
Dummy
R1
R2
R3
R4
hfe1
hfe2
Dummy
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2
1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3 2 3 1 3 1 2 2 3 1 3 1 2
1 2 3 2 3 1 3 1 2 2 3 1 1 2 3 3 1 2
1 2 3 2 3 1 3 1 2 2 3 1 1 2 3 3 1 2
1 2 3 3 1 2 2 3 1 2 3 1 3 1 2 1 2 3
1 2 3 3 1 2 3 1 2 1 2 3 2 3 1 2 3 1
Please cite this article as: S.-i. Inage, Proposal of the ‘‘Total Error Minimization Method” for robust design, Engineering Science and Technology, an International Journal, https://doi.org/10.1016/j.jestch.2018.11.005
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S.-i. Inage / Engineering Science and Technology, an International Journal xxx (xxxx) xxx
performance calculations for different combinations of controllable variations based on the L18 orthogonal array. Details of calculations are given in the appendix. For each of cases 1 to 18, two perfor-
mance results of products for N1 and N2 and the average of the two results and the SN ratio can be estimated. On the basis of these results and employing the concept of experimental design, the
Table 3 Consideration of noise N1 and N2. Variables
R1 (%)
R2 (%)
R3 (%)
R4 (%)
hfe1 (%)
hfe2 (%)
Ei (%)
I0 (mA)
N1 N2
10 10
10 10
10 10
10 10
30 30
30 30
10 10
800 0
Table 4 Results obtained using the Taguchi method. Case
Variablues
R1 kX
R2 kX
R3 kX
R4 kX
hfe1 –
hfe2 –
Ei V
I0 A
Calculation results
1
Base N1 N2
2.2 2.2484 2.249465
0.11 0.110121 0.110121
0.3 0.3009 0.300903
0.08 0.080064 0.080064
18 21.24 21.8232
50 75 87.5
20 24 24.8
– 0.8 0
E0 9.19 15.33
Average 12.26
Despersion 18.81551
SN ratio 9.02
2
Base N1 N2
2.2 2.2484 2.249465
0.56 0.560616 0.560617
1.5 1.5045 1.504514
0.42 0.420336 0.420336
35 41.3 42.434
100 150 175
20 24 24.8
– 0.8 0
E0 2.73 4.59
Average 3.66
Despersion 1.723907
SN ratio 8.91
3
Base N1 N2
2.2 2.2484 2.249465
2.7 2.70297 2.702973
7.5 7.5225 7.522568
2.2 2.20176 2.201761
70 82.6 84.868
200 300 350
20 24 24.8
– 0.8 0
E0 1.52 1.96
Average 1.74
Despersion 0.094812
SN ratio 15.03
4
Base N1 N2
11 11.242 11.24732
0.11 0.110121 0.110121
0.3 0.3009 0.300903
0.42 0.420336 0.420336
35 41.3 42.434
200 300 350
20 24 24.8
– 0.8 0
E0 29.39 42.80
Average 36.10
Despersion 90.01358
SN ratio 11.61
5
Base N1 N2
11 11.242 11.24732
0.56 0.560616 0.560617
1.5 1.5045 1.504514
2.2 2.20176 2.201761
70 82.6 84.868
50 75 87.5
20 24 24.8
– 0.8 0
E0 5.86 11.84
Average 8.85
Despersion 17.91223
SN ratio 6.41
6
Base N1 N2
11 11.242 11.24732
2.7 2.70297 2.702973
7.5 7.5225 7.522568
0.08 0.080064 0.080064
18 21.24 21.8232
100 150 175
20 24 24.8
– 0.8 0
E0 8.40 15.64
Average 12.02
Despersion 26.24386
SN ratio 7.41
7
Base N1 N2
56 57.232 57.2591
0.11 0.110121 0.110121
1.5 1.5045 1.504514
0.08 0.080064 0.080064
70 82.6 84.868
100 150 175
20 24 24.8
– 0.8 0
E0 19.43 24.37
Average 21.90
Despersion 12.1753
SN ratio 15.95
8
Base N1 N2
56 57.232 57.2591
0.56 0.560616 0.560617
7.5 7.5225 7.522568
0.42 0.420336 0.420336
18 21.24 21.8232
200 300 350
20 24 24.8
– 0.8 0
E0 3.84 16.49
Average 10.16
Despersion 80.07242
SN ratio 1.11
9
Base N1 N2
56 57.232 57.2591
2.7 2.70297 2.702973
0.3 0.3009 0.300903
2.2 2.20176 2.201761
35 41.3 42.434
50 75 87.5
20 24 24.8
– 0.8 0
E0 12.45 38.93
Average 25.69
Despersion 350.6769
SN ratio 2.75
10
Base N1 N2
2.2 2.2484 2.249465
0.11 0.110121 0.110121
7.5 7.5225 7.522568
2.2 2.20176 2.201761
35 41.3 42.434
100 150 175
20 24 24.8
– 0.8 0
E0 1.13 1.92
Average 1.53
Despersion 0.310778
SN ratio 8.74
11
Base N1 N2
2.2 2.2484 2.249465
0.56 0.560616 0.560617
0.3 0.3009 0.300903
0.08 0.080064 0.080064
70 82.6 84.868
200 300 350
20 24 24.8
– 0.8 0
E0 6.67 9.85
Average 8.26
Despersion 5.0376
SN ratio 11.32
12
Base N1 N2
2.2 2.2484 2.249465
2.7 2.70297 2.702973
1.5 1.5045 1.504514
0.42 0.420336 0.420336
18 21.24 21.8232
50 75 87.5
20 24 24.8
– 0.8 0
E0 1.62 6.29
Average 3.95
Despersion 10.92804
SN ratio 1.55
13
Base N1 N2
11 11.242 11.24732
0.11 0.110121 0.110121
1.5 1.5045 1.504514
2.2 2.20176 2.201761
18 21.24 21.8232
200 300 350
20 24 24.8
– 0.8 0
E0 6.15 11.95
Average 9.05
Despersion 16.83545
SN ratio 6.87
14
Base N1 N2
11 11.242 11.24732
0.56 0.560616 0.560617
7.5 7.5225 7.522568
0.08 0.080064 0.080064
35 41.3 42.434
50 75 87.5
20 24 24.8
– 0.8 0
E0 11.21 17.79
Average 14.50
Despersion 21.66127
SN ratio 9.87
15
Base N1 N2
11 11.242 11.24732
2.7 2.70297 2.702973
0.3 0.3009 0.300903
0.42 0.420336 0.420336
70 82.6 84.868
100 150 175
20 24 24.8
– 0.8 0
E0 9.22 14.71
Average 11.97
Despersion 15.07748
SN ratio 9.78
16
Base N1 N2
56 57.232 57.2591
0.11 0.110121 0.110121
7.5 7.5225 7.522568
0.42 0.420336 0.420336
70 82.6 84.868
50 75 87.5
20 24 24.8
– 0.8 0
E0 10.41 19.46
Average 14.94
Despersion 40.94777
SN ratio 7.36
17
Base N1 N2
56 57.232 57.2591
0.56 0.560616 0.560617
0.3 0.3009 0.300903
2.2 2.20176 2.201761
18 21.24 21.8232
100 150 175
20 24 24.8
– 0.8 0
E0 64.35 110.71
Average 87.53
Despersion 1074.961
SN ratio 8.53
18
Base N1 N2
56 57.232 57.2591
2.7 2.70297 2.702973
1.5 1.5045 1.504514
0.08 0.080064 0.080064
35 41.3 42.434
200 300 350
20 24 24.8
– 0.8 0
E0 16.71 22.91
Average 19.81
Despersion 19.1949
SN ratio 13.10
Please cite this article as: S.-i. Inage, Proposal of the ‘‘Total Error Minimization Method” for robust design, Engineering Science and Technology, an International Journal, https://doi.org/10.1016/j.jestch.2018.11.005
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effects of the experimental levels of design variables in Table 1 are obtained as shown in Fig. 2. The figure plots the relationships between levels 1–3 of variables and the average output and SN ratio. The Taguchi method has two stages. The first stage is the determination of the levels of variables according to the SN ratio. To begin, using the SN ratio graph in Fig. 2, the variables that most affect the SN ratio are checked. In this case, the variables R4 and hfe1 have the maximum SN ratio. As previously mentioned, the SN ratio is a metric of the sensitivity of the performance/output to variation. The levels of variables that have the maximum SN ratio can therefore be used to reduce variations of the performance/output of the product passively. The second stage is the determination of the level of variables from the SN ratio and average performance. In the first stage, the levels of variables that have higher SN ratios are found. In the second stage, the variables that have no relationship or only a weak relationship with the SN ratio are chosen and the optimal level of each variable is chosen to adjust the required performance/output of the product. To adjust the performance to a target value of 10 V, the variables R1 and R3 are useful as they hardly affect the SN ratio yet strongly affect the performance itself (see Fig. 2a). Through the two-stage design, the product can provide the required performance/output with robustness against variation due to noise. This is the typical process of the Taguchi method. From the average response graph of the SN ratio in Fig. 2b, R4 = 0.08 kX and hfe1 = 70 are chosen as they have the highest SN ratios. Good robustness against variation is expected using these values of R4 and hfe1.
due to noise using an orthogonal array. In particular, the method performs well in obtaining multiple required performances/outputs of a product. A product usually has several required performances. In design navigation, the Recsat metric is defined as
Rc ¼ ln
System range Common range
ð8Þ
The system range is the range of variation of the performance/ output, the design range is the allowance of the required performance/output of the product, and the common range is the range of overlap of the system and design ranges. The relationship among the ranges is shown in Fig. 3. Recsat is estimated as follows. Average performance:
y¼
Pn
i¼1 yi
n
ð9Þ
2.2. Design navigation method (Nakazawa method) Design navigation (i.e., the Nakazawa method) can confirm the required performance/output with robustness against variation
Fig. 3. Definitions of design, system, and common ranges in the Nakazawa method.
Fig. 2. Response graphs of the performance test.
Please cite this article as: S.-i. Inage, Proposal of the ‘‘Total Error Minimization Method” for robust design, Engineering Science and Technology, an International Journal, https://doi.org/10.1016/j.jestch.2018.11.005
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Dispersion:
s2 ¼
Pn 2 i¼1 yi y
ð10Þ
n1
The system range SR is defined as
pffiffiffiffiffi SRu ¼ y þk s2
ð11Þ
pffiffiffiffiffi SRt ¼ y k s2
ð12Þ
where the subscript u indicates the maximum range and the subscript t the minimum range. It follows that
Systemrange ¼ ðSRu SRt Þ
ð13Þ
The common range depends on the allowance of the required performance/output; e.g., in the case of Eo in Fig. 1, the allowance is 9 V < Eo < 11 V and the common range is 11–9 = 2 V. Recsat can finally be written as
pffiffiffiffiffi ! 2k s2 Rc ¼ ln Y max Y min
ð14Þ
where the weight k is, for example, 1.5; Y is the target performance/ output (a required specification); and the subscript max indicates the maximum allowance of the output while the subscript min indicates the minimum allowance of the output. As previously mentioned, the Nakazawa method focuses on finding the optimal design in terms of the multiple performances/outputs of a product. The term ‘‘optimal” refers to the minimum variation of the multiple performances/outputs of the product. For each performance/output, Recsat is experimentally measured using an orthogonal array and the Taguchi method. The total Recsat can then be calculated as
Rc;t ¼
X
ð15Þ
Rc;i
where Rc,t is the total Recsat and Rc,i is the Recsat of performance i.
The Nakazawa method assumes the axiom that the optimal design is achieved when the total Recsat is a minimum for a combination of variables. Nakazawa proposed the axiom on the basis of information theory and Shannon’s concept of information entropy. As it is an axiom, we cannot prove its validity. The Nakazawa method is applied to the electric circuit problem shown in Fig. 1. Meanwhile, the Nakazawa method is applicable to multiple-performance design. Not only the output voltage Eo but also the cost of the circuit is therefore considered. The cost estimation is assumed to be given by the simple equation Hayashi et al. [14].
C ¼ 5R2 þ hfe1 þ f fe2
ð16Þ
where R2, hef1, and hfe2 are given at the levels listed in Table 1. The total Recsat for each variable level estimated from the calculated Eo and the average output given in Table 4 is given in Table 5. In this case, the allowable range of Eo is from 9 to 11 V while the allowed cost is less than 135. Details of the calculation of total Recsat are given in the appendix. Fig. 4 shows the estimated Recsat for each variable. The optimal levels of variables are chosen according to the minimum Recsat. The optimal combination is therefore that given in Table 6. The optimal performance is 10.12 V (which is within the range of 9–11 V) while the cost is 133.5 (which is less than 135). The required performance is therefore achieved employing the Nakazawa method.
3. Proposal of the total Error minimization method This paper proposes a new approach of achieving a robust design of a product having single and multiple performances/outputs without axioms. In the case of design without theoretical models, an experimental approach, such as the adoption of the Taguchi or Nakazawa method with an orthogonal array, is important. Meanwhile, if mathematical or physical models are available, another robust design using, for example, a genetic algorithm is applicable (12). The new approach unifies these experimental and theoretical approaches.
Table 5 Estimation of Recsat for each variable. Average
Despersion
Sru
STt
Ro
Cost
Despersion
Sru,c
STt,c
Ro,c
R1
Rt
R1-1 R1-2 R1-3
5.23 15.41 30.00
17.72 59.04 607.02
11.55 26.94 66.96
1.08 3.89 6.95
1.84 2.44 3.61
163.28 163.28 163.28
8705.48 3406.48 3690.14
303.24 250.83 254.40
23.33 75.74 72.16
0.919 1.083 1.065
2.20 11.00 56.00
2.762 3.528 4.675
R2-1 R2-2 R2-3
Average 15.96 22.16 12.53
Despersion 55.15 264.31 51.32
Sru 27.10 46.55 23.27
STt 4.82 2.23 1.78
Ro 2.41 3.19 2.37
Cost 158.22 160.47 171.17
Despersion 3949.07 5009.07 6729.07
Sru,c 252.48 266.63 294.21
STt,c 63.95 54.30 48.12
Ro,c 0.976 0.967 1.041
R2 0.11 0.56 2.70
Rt 3.386 4.161 3.416
R3-1 R3-2 R3-3
Average 30.30 11.20 9.15
Despersion 563.40 33.27 21.24
Sru 65.90 19.86 16.06
STt 5.30 2.55 2.23
Ro 3.57 2.16 1.93
Cost 163.28 163.28 163.28
Despersion 6520.24 3869.18 5412.68
Sru,c 284.41 256.59 273.64
STt,c 42.16 69.98 52.93
Ro,c 0.959 1.054 0.989
R3 0.30 1.50 7.50
Rt 4.532 3.212 2.923
R4-1 R4-2 R4-3
Average 14.79 13.46 22.40
Despersion 25.65 73.90 263.18
Sru 22.39 26.36 46.73
STt 7.19 0.57 1.94
Ro 2.03 2.56 3.19
Cost 163.28 163.28 163.28
Despersion 6998.18 3620.68 5183.24
Sru,c 288.77 253.54 271.28
STt,c 37.80 73.03 55.29
Ro,c 0.949 1.069 0.997
R4 0.08 0.42 2.20
Rt 2.976 3.626 4.189
hfe1-1 hfe1-2 hfe1-3
Average 22.50 16.88 11.28
Despersion 348.90 84.15 15.45
Sru 50.51 30.64 17.17
STt 5.52 3.12 5.38
Ro 3.33 2.62 1.77
Cost 140.28 157.28 192.28
Despersion 4333.30 4716.30 5065.30
Sru,c 239.03 260.30 299.04
STt,c 41.54 54.27 85.53
Ro,c 0.748 0.937 1.462
hfe1 18.00 35.00 70.00
Rt 4.081 3.559 3.237
hfe2-1 hfe2-2 hfe2-3
Average 13.37 23.10 14.19
Despersion 73.55 218.95 48.27
Sru 26.23 45.30 24.61
STt 0.50 0.90 3.76
Ro 2.55 3.10 2.34
Cost 96.62 146.62 246.62
Despersion 471.18 605.68 725.24
Sru,c 129.18 183.53 287.01
STt,c 64.06 109.70 206.22
Ro,c (0.086) 1.071 #NUM!
hfe2 50.00 100.00 200.00
Rt 2.469 4.171 #NUM!
Please cite this article as: S.-i. Inage, Proposal of the ‘‘Total Error Minimization Method” for robust design, Engineering Science and Technology, an International Journal, https://doi.org/10.1016/j.jestch.2018.11.005
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Fig. 4. Recsat curve for Rc,t.
Table 6 Results of the confirmation test. Parameter
R1 (kX)
R2 (kX)
R3 (kX)
R4 (kX)
hfe1
hfe2
Output Voltage (V)
Cost
Value
2.2
2.7
7.5
0.08
70
50
10.12
133.5
3.1. Fundamental concept
3.2. Validation of the proposed method
As discussed, the Taguchi and Nakazawa methods use an original metric, like the SN ratio or Recsat, to measure the sensitivity to variations. These metrics are defined by the average or target performance/output of the product and the dispersion. The Taguchi method uses the average performance for different levels of variables. It therefore requires a two-stage design to perform well after the confirmation of robustness against variations using the SN ratio. Meanwhile, the Nakazawa method uses the target performance (i.e., specification) with its allowance to estimate the dispersion, estimating the original metric Recsat. Additionally, finding the minimum Recsat allows the robust combining of variables. Taking this approach, we use the new metric
The proposed approach is applied to the electric circuit problem presented in Fig. 1. The total error for each variable and level of the variable obtained from the results of Eo and average output in Table 4 and Eqs. (17)–(19) are presented in Table 7. In this case, the target performance of Eo is 10 V While the target cost is 135. The detail of the calculation of total error is presented in the appendix. Table 8 gives the average response of the total error while Fig. 5 presents the average response graphs of total error. Table 9 gives the results of the confirmation test. The response trend in Fig. 5 is similar to that for the Nakazawa method. The results of the confirmation test are therefore the same as those for the Nakazawa method. These validation tests confirm the proposed approach. Additionally, as discussed in Section 2.2, the Taguchi method recommends R4-1 and hfe1-3 as the optimal levels from the point of view of the SN ratio. The proposed approach also recommends these levels. The proposed approach can be applied to a genetic algorithm directly. The genetic algorithm is well known as a powerful multi-point search algorithm that is based on the simulation of the heredity and evolution of living things [1,13,9,20,2]. The proposed approach is expected to have the following advantages.
E¼
s2
0:5 ð17Þ
Y2
where
Pn s2 ¼
Y Þ2 n1
i¼1 ðyi
ð18Þ
and Y is the target output (i.e., specification). In fact, E defined by Eq. (17) is the error relative to the target performance/output of the product. For Eq. (18), yi is estimated by conducting experiments using an orthogonal array with noise N1 and N2 as with the Taguchi or Nakazawa method. As this approach uses the target performance/output directly, it does not require the two-stage design. In the case of multiple performances/outputs, Ei is estimated for each performance i. For the system, the total error is then defined as
Et ¼
X
Ei
ð19Þ
If the combination of variable levels minimizes the total error in Eq. (19), the product has the multiple required performances/outputs. This is not an axiom but a universal truth. Therefore, using this concept, without axioms, robust design can be achieved. Additionally, if there are mathematical or physical models with which to design the product, the metric in Eq. (17) can be used as an objective function of genetic algorithms. In this case, without using an orthogonal array, the optimal combination of variables is obtained.
a) A genetic algorithm can be applied to problems even in a discrete search space. b) An optimal solution can be determined even when the landscape of the objective function is multi-modal. c) In multi-objective optimization, the genetic algorithm, which is a multi-point search algorithm, can determine a Pareto-optimal set using one trial to check the trade-off relations among multiple objectives. In the case of the combination of discrete variables mentioned above, robust design with an orthogonal array is the only alternative. Meanwhile, if contentious variables can be accepted, using a genetic algorithm or another optimization algorithm, a solution better than that of the discrete approach is obtained. As an example of optimization with multi object functions, Hasan Koten et al. coupled one dimensional engine code and multi objective optimization code and evaluated about 15,000 cases to define the proper boundary conditions to optimize the parameters on compressed biogas-diesel dual-fuel engine and ignition delay
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S.-i. Inage / Engineering Science and Technology, an International Journal xxx (xxxx) xxx Table 7 Estimation of error in performance using an orthogonal array. R1 kX
R2 kX
R3 kX
R4 kX
hfe1 –
hfe2 –
Ei V
I0 A
Calculation results
1
2.2 1.98 2.42
0.11 0.121 0.099
0.3 0.33 0.27
0.08 0.072 0.088
18 12.6 23.4
50 65 35
20 18 22
– 0.8 0
E0 9.19 15.33
Average 12.26
Despersion 10.217
Error 0.32
Cost 68.55
2
2.2 1.98 2.42
0.56 0.616 0.504
1.5 1.65 1.35
0.42 0.378 0.462
35 24.5 45.5
100 130 70
20 18 22
– 0.8 0
E0 2.73 4.59
Average 3.66
Despersion 80.367
Error 0.90
Cost 137.8
3
2.2 1.98 2.42
2.7 2.97 2.43
7.5 8.25 6.75
2.2 1.98 2.42
70 49 91
200 260 140
20 18 22
– 0.8 0
E0 1.52 1.96
Average 1.74
Despersion 136.506
Error 1.17
Cost 283.5
4
11 9.9 12.1
0.11 0.121 0.099
0.3 0.33 0.27
0.42 0.378 0.462
35 24.5 45.5
200 260 140
20 18 22
– 0.8 0
E0 29.39 42.80
Average 36.10
Despersion 1362.017
Error 3.69
Cost 235.55
5
11 9.9 12.1
0.56 0.616 0.504
1.5 1.65 1.35
2.2 1.98 2.42
70 49 91
50 65 35
20 18 22
– 0.8 0
E0 5.86 11.84
Average 8.85
Despersion 2.644
Error 0.16
Cost 122.8
6
11 9.9 12.1
2.7 2.97 2.43
7.5 8.25 6.75
0.08 0.072 0.088
18 12.6 23.4
100 130 70
20 18 22
– 0.8 0
E0 8.40 15.64
Average 12.02
Despersion 8.162
Error 0.29
Cost 131.5
7
56 50.4 61.6
0.11 0.121 0.099
1.5 1.65 1.35
0.08 0.072 0.088
70 49 91
100 130 70
20 18 22
– 0.8 0
E0 19.43 24.37
Average 21.90
Despersion 283.191
Error 1.68
Cost 170.55
8
56 50.4 61.6
0.56 0.616 0.504
7.5 8.25 6.75
0.42 0.378 0.462
18 12.6 23.4
200 260 140
20 18 22
– 0.8 0
E0 3.84 16.49
Average 10.16
Despersion 0.053
Error 0.02
Cost 220.8
9
56 50.4 61.6
2.7 2.97 2.43
0.3 0.33 0.27
2.2 1.98 2.42
35 24.5 45.5
50 65 35
20 18 22
– 0.8 0
E0 12.45 38.93
Average 25.69
Despersion 492.466
Error 2.22
Cost 98.5
10
2.2 1.98 2.42
0.11 0.121 0.099
7.5 8.25 6.75
2.2 1.98 2.42
35 24.5 45.5
100 130 70
20 18 22
– 0.8 0
E0 1.13 1.92
Average 1.53
Despersion 143.638
Error 1.20
Cost 135.55
11
2.2 1.98 2.42
0.56 0.616 0.504
0.3 0.33 0.27
0.08 0.072 0.088
70 49 91
200 260 140
20 18 22
– 0.8 0
E0 6.67 9.85
Average 8.26
Despersion 6.053
Error 0.25
Cost 272.8
12
2.2 1.98 2.42
2.7 2.97 2.43
1.5 1.65 1.35
0.42 0.378 0.462
18 12.6 23.4
50 65 35
20 18 22
– 0.8 0
E0 1.62 6.29
Average 3.95
Despersion 73.119
Error 0.86
Cost 81.5
13
11 9.9 12.1
0.11 0.121 0.099
1.5 1.65 1.35
2.2 1.98 2.42
18 12.6 23.4
200 260 140
20 18 22
– 0.8 0
E0 6.15 11.95
Average 9.05
Despersion 1.798
Error 0.13
Cost 218.55
14
11 9.9 12.1
0.56 0.616 0.504
7.5 8.25 6.75
0.08 0.072 0.088
35 24.5 45.5
50 65 35
20 18 22
– 0.8 0
E0 11.21 17.79
Average 14.50
Despersion 40.461
Error 0.64
Cost 87.8
15
11 9.9 12.1
2.7 2.97 2.43
0.3 0.33 0.27
0.42 0.378 0.462
70 49 91
100 130 70
20 18 22
– 0.8 0
E0 9.22 14.71
Average 11.97
Despersion 7.728
Error 0.28
Cost 183.5
16
56 50.4 61.6
0.11 0.121 0.099
7.5 8.25 6.75
0.42 0.378 0.462
70 49 91
50 65 35
20 18 22
– 0.8 0
E0 10.41 19.46
Average 14.94
Despersion 48.757
Error 0.70
Cost 120.55
17
56 50.4 61.6
0.56 0.616 0.504
0.3 0.33 0.27
2.2 1.98 2.42
18 12.6 23.4
100 130 70
20 18 22
– 0.8 0
E0 64.35 110.71
Average 87.53
Despersion 12021.851
Error 10.96
Cost 120.8
18
56 50.4 61.6
2.7 2.97 2.43
1.5 1.65 1.35
0.08 0.072 0.088
35 24.5 45.5
200 260 140
20 18 22
– 0.8 0
E0 16.71 22.91
Average 19.81
Despersion 192.390
Error 1.39
Cost 248.5
issue [3,4]. Taking this approach, the metrics in Eqs. (17) and (19) are applied as the objective functions of the genetic algorithm. The variables R1, R2,. . ., hfe2 are treated as design parameters. In the calculation, the ranges of variables are those listed in Table 10. The present paper applies the neighborhood cultivation genetic algorithm (NCGA) [12], [11]. The NCGA has a neighborhood crossover mechanism in addition to the mechanisms of general genetic algorithms. The NCGA has advantages over other genetic algo-
rithms from the point of view of the solution survey in a discrete search space. On this calculation, population size and numbers of generation are 50 and 50, respectively. Table 11 gives the results of the optimal combination of contentious variables and its output Eo and cost. In the table, result 1 is the calculation for the target E0 = 10 V and cost = 135. Eo is closer to the target here than in the discrete case reported in Table 9. Meanwhile, the cost slightly exceeds 135. Therefore, result 2 for the target E0 = 10 V and
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S.-i. Inage / Engineering Science and Technology, an International Journal xxx (xxxx) xxx
cost = 134 is obtained. Using the lower target cost, both allowable Eo and an allowable cost can be obtained. Fig. 6 shows the relationship between the errors in Eo and cost and the number of generations of a genetic algorithm. Both errors decrease with an increasing number of generations. The proposed approach is applicable to robust design when either taking the discrete approach with an orthogonal array or using contentious variables.
Table 8 Estimation of total error. Levels
Average
Error on performance
Cost
Error on cost
Total error
R1-1 R1-2 R1-3
5.23 15.41 30.00
0.78 0.86 2.83
163.28 163.28 163.28
0.21 0.21 0.21
0.99 1.07 3.04
R2-1 R2-2 R2-3
15.96 22.16 12.53
1.29 2.15 1.03
158.22 160.47 171.17
0.17 0.19 0.27
1.46 2.34 1.30
R3-1 R3-2 R3-3
30.30 11.20 9.15
2.95 0.85 0.67
163.28 163.28 163.28
0.21 0.21 0.21
3.16 1.06 0.88
R4-1 R4-2 R4-3
14.79 13.46 22.40
0.76 1.07 2.64
163.28 163.28 163.28
0.21 0.21 0.21
0.97 1.28 2.85
hfe1-1 hfe1-2 hfe1-3
22.50 16.88 11.28
2.10 1.67 0.71
140.28 157.28 192.28
0.04 0.17 0.42
2.14 1.84 1.13
hfe2-1 hfe2-2 hfe2-3
13.37 23.10 14.19
0.82 2.55 1.11
96.62 146.62 246.62
0.28 0.09 0.83
1.10 2.64 1.93
4. Discussion Table 12 compares the Taguchi and Nakazawa methods and the proposed approach. The major differences are the definitions of performance and dispersion. The Taguchi and Nakazawa methods introduce a logarithmic form to the metric. This difference is not important as we can apply the log or ln function to the total error and the tendency of the total error response graph does not change. Additionally, the strategy based on minimum total error has clear physical meaning, which means there are minimum discrepancies between the real performance/output and required specification. This approach therefore does not require axioms.
Fig. 5. Average response graph of total error.
Table 9 Results of the confirmation test for the proposed method. Parameter
R1 (kX)
R2 (kX)
R3 (kX)
R4 (kX)
hfe1
hfe2
Output Voltage (V)
Cost
Value
2.2
2.7
7.5
0.08
70
50
10.12
133.5
Table 10 Range of contentious variables. Variables
R1 (kX)
R2 (kX)
R3 (kX)
R4 (kX)
hfe1
hfe2
Range
2.28–56
0.11–2.7
0.3–7.5
0.08–2.7
18–70
50–200
Table 11 Results of the optimal combination of contentious variables. Variables
R1 (kX)
R2 (kX)
R3 (kX)
R4 (kX)
hfe1
hfe2
Eo (V)
Cost
Discrete Result-1 Result-2
2.2 29.22 55.00
2.7 2.35 1.57
7.5 3.71 7.03
0.08 0.81 1.07
70 45.71 46.98
50 77.78 79.91
10.12 10.02 10.02
133.5 135.26 134.75
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9
Fig. 6. Errors in Eo and cost versus the number of generations of a genetic algorithm.
Table 12 Comparison of the three methods. Methods
Taguchi Method
Metrics
Average Value SN ratio
Definition of metrics
Unit of Metrics Strategy Noise for output Applications
Average Value: Pn yi y ¼ i¼1 SN ratio: n 2 g ¼ 10log ys2 where, Pn 2 ðy yÞ s2 ¼ i¼1n1i db Two-stage design Considered Stand alone
Axioms
–
Design Navigation
Proposal
Recsat (Reciprocal of satisfaction) Recsat: pffiffiffiffi 2k s2 Rc ¼ ln Y max Y min where, Pn 2 ðy yÞ k = 1.5 (for example) s2 ¼ i¼1n1i Y: Target output (Specification)
Total error
– Direct design
–
Minimum Recsat means the optimal product
5. Conclusions A new approach for the robust design of a product with single or multiple performances/outputs was proposed based on a new metric which means total error and validated. The method was proposed through minimization of the total difference (i.e., error) between the required specification and experimental output for different variables with noise. This approach was applied to obtain the required specification and target cost for an electric circuit
Error: 2 0:5 E ¼ Ys 2 where, Pn ðy Y Þ2 2 i¼1 i Y: Target output (Specification) s ¼ n1
1) Stand alone 2) with Genetic Algorithms –
problem. The optimal results were the same as those obtained using the Nakazawa method and met the required specification. The proposed approach is compatible with experimental and theoretical approaches. A genetic-algorithm-based design with a proposed metric, namely optimal design with contentious variables, was established. Relative to the result of the discrete approach, the performance/output achieved with the proposed method was closer to the required specification. Employing the proposed method, we expect a united robust design without an axiom.
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S.-i. Inage / Engineering Science and Technology, an International Journal xxx (xxxx) xxx
Average SN ratio for level 1 of R1:
Acknowledgment We thank Glenn Pennycook, MSc, from Edanz Group for editing a draft of this manuscript.
9:02 þ 8:89 þ 15:03 þ 8:74 þ 11:32 þ 1:55 ¼ 9:09ðdbÞ 6
Appendix
Finally, the response graph of the SN ratio in Fig. 2 can be plotted.
A-1 calculation using the Taguchi method
A-2 calculation using the Nakazawa method
An example of the calculation of the Taguchi method is described here. In Table 4, for case 1, the levels of variables R1– hfe2 and Ei and I0 with noise are listed in Table A1. From Eqs. (1)–(3): Average Eo:
We use the results given in Table 4. Through the concept of experimental design, for each level of a variable, the average Eo and dispersion can be estimated. Average Eo for level 1 of R1:
Pn
i¼1 yi
y¼
n
¼
1 1 ðy þ y2 Þ ¼ ð9:19 þ 15:33Þ ¼ 12:26 2 1 2
ðA-1Þ
12:26 þ 3:66 þ 1:74 þ 1:53 þ 8:26 þ 3:95 ¼ 5:23ðVÞ 6
Dispersion s2:
s2 ¼
n1
¼
SN ratio:
g ¼ 10log
y2 s2
! ¼ 10log
12:262 18:81
ðA-7Þ
Recsat for the performance/output of the product in case 1 from Table 4 is as follows.
2 2 1 y1 y þ y1 y 21
¼ ð9:19 12:26Þ2 þ ð15:33 12:26Þ2 ¼ 18:81
ðA-6Þ
Average dispersion s2 for level 1 of R1:
ð12:26 5:23Þ2 þ ð3:66 5:23Þ2 þ ð1:74 5:23Þ2 þ ð1:53 5:23Þ2 þ ð8:26 5:23Þ2 þ ð3:95 5:23Þ2 ¼ 17:72 61
Pn 2 i¼1 yi y
ðA-5Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffi!
systemrange 1:5 17:72 ¼ ln ¼ 1:843 ¼ ln commonrange 11 9
Rc;E1 ðA-2Þ
ðA-8Þ
Recsat for cost: Average Eo for level 1 of R1:
! ¼ 9:02
ðA-3Þ
Fig. 4 can be plotted from these calculations. Through the concept of experimental design, for each level of a variable, the aver-
68:55 þ 137:88 þ 283:5 þ 135:55 þ 272:8 þ 81:5 ¼ 163:28 6 ðA-9Þ 2
Average dispersion s for level 1 of R1:
ð68:55 163:28Þ2 þ ð137:88 163:28Þ2 þ ð283:5 163:28Þ2 þ ð135:55 163:28Þ2 þ ð272:8 163:28Þ2 þ ð81:5 163:28Þ2 ¼ 8705:48 61 ðA-10Þ
age Eo and SN ratio can be estimated: Level 1 of R1: Average Eo for level 1 of R1:
System range:
12:26 þ 3:66 þ 1:74 þ 1:53 þ 8:26 þ 3:95 ¼ 5:23ðVÞ 6
ðA-4Þ
pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SRu ¼ y þk s2 ¼ 163:28 þ 1:5 8705:48 ¼ 303:2
ðA-11Þ
pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SRt ¼ y k s2 ¼ 163:28 1:5 8705:48 ¼ 23:3
ðA-12Þ
Table A1 Conditions for Case 1 in Table 4. Case
1
Variables
Base N1 N2
R1
R2
R3
R4
hfe1
hfe2
Ei
I0
kX
kX
kX
kX
–
–
V
mA
2.2 1.98 2.42
0.11 0.121 0.099
0.3 0.33 0.27
0.08 0.072 0.088
18 12.6 23.4
50 65 35
20 18 22
– 800 0
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S.-i. Inage / Engineering Science and Technology, an International Journal xxx (xxxx) xxx
Therefore, Recsat Rc of cost is
systemrange 303:2 23:3 ¼ ln ¼ ln ¼ 0:919 commonrange 135 23:3
Rc;c
References
ðA-13Þ
Here, as the cost should be less than 135, the common range is defined by 135 and SRt. The total Recsat is therefore
Rt;1 ¼ Ro;1 þ Rt;1 ¼ 1:843 þ 0:919 ¼ 2:762
ðA-14Þ
Employing the same procedure for other levels and variables, the Recsat response graph in Fig. 4 is finally obtained. A-3 calculation using the proposed method As mentioned in Section A-1, for case 1 in Table 7, the dispersion is estimated as follows. Average Eo:
Pn
y¼
i¼1 yi
n
¼
1 1 ðy þ y2 Þ ¼ ð9:19 þ 15:33Þ ¼ 12:26 2 1 2
ðA-15Þ
Dispersion:
s ¼ 2
Pn 2 i¼1 yi y n1
¼ ð12:26 10Þ2 ¼ 10:217
ðA-16Þ
The target performance/output is 10 V and the error is therefore as follows. Error in performance/output
Ep ¼
s2
0:5 ¼
Y2
0:5 10:217 102
¼ 0:32
ðA-17Þ
Employing the concept of experimental design, the error in the performance of each variable is obtained as listed in Table 8. The error in the cost is estimated as follows. Dispersion of cost for level 1 of R1:
s2 ¼
Pn 2 i¼1 yi y n1
¼ ð163:28 135Þ2 ¼ 799:7584
ðA-18Þ
Error in cost for level 1 of R1:
Ec ¼
s2
Y2
0:5 ¼
0:5 799:7584 1352
¼ 0:209
ðA-19Þ
The total error for level 1 of R1 is therefore
Et ¼ Ep þ Ec ¼ 0:78 þ 0:209 ¼ 0:99
11
ðA-20Þ
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Proposal of the ‘‘Total Error Minimization Method” for robust design Shin-ichi Inage ⇑ Hitachi Ltd. Social Infrastructure information Sysytem Division, Hitacchi Omori 2nd Bldg., 27-18, Minami Oi 6-chome, Shinagawa-ku, Tokyo, 140-8572, Japan Tokyo Institute of Technology, Solution Research Laboratory, International Research Center of Advanced Energy Systems for Sustainability, 2-12-1-16-25 Ookayama, Meguro-ku, Tokyo 152-8550, Japan
a r t i c l e
i n f o
Article history: Received 18 July 2018 Revised 6 November 2018 Accepted 6 November 2018 Available online xxxx Keywords: Robust design Design optimization Orthogonal array Multi performances
a b s t r a c t We propose a new approach for the robust design of a product with single or multiple performances/outputs without axioms. When there are no theoretical models, experimental methods (e.g., the popular and robust Taguchi and Nakazawa design methods) with an orthogonal array are important. Meanwhile, if mathematical or physical models are available, other robust design approaches using, for example, a genetic algorithm, are applicable. The proposed approach is compatible with both experimental and theoretical approaches. The approach is proposed through the minimization of the total difference (i.e., error) between the required specification and experimental output for different variables with noise. We applied the approach to an electric circuit problem to obtain the required specification and target cost. The optimal results are the same as those obtained using the Nakazawa method. Additionally, a genetic-algorithm-based design with a proposed metric, namely the optimal design under contentious variables, is presented. The obtained performance/output better matches the required specification when using the proposed approach than when using the discrete approach. Employing the proposed method, we expect a united robust design without an axiom. Ó 2018 Karabuk University. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction This paper proposes a new approach for the robust design of a product having single or multiple performances/outputs. Since the 1980s, robust design has contributed to improvements in the quality of products. G. Taguchi proposed a concept of robust design having two stages [15–19], focusing on the effect of the noise factor on variations in performance/output and introducing the signal-to-noise (SN) ratio as a metric with which to measure the effect. Even today, his approach is applied in many fields of engineering [5–7]). His concept of the SN ratio is based on direct consideration of the effect of the noise factor on the performance/ output of a product, and his rational and systematic approach of using an orthogonal array and a design of experiments are remarkable [18]. In particular, the Taguchi method is effective when the target product has one performance/output. In the case of multiple performances/outputs, the design navigation method of Nakazawa is well known [8]. This approach introduces the reciprocal of satisfaction (Recsat) as a metric and is based on the application of infor⇑ Address: Tokyo Institute of Technology, Solution Research Laboratory, International Research Center of Advanced Energy Systems for Sustainability, 2-12-1-1625 Ookayama, Meguro-ku, Tokyo 152-8550, Japan. E-mail address: [email protected]
mation theory developed by Suh [10]. Suh defined an axiom by stating, ‘‘The optimal design is to minimize the required information content to achieve the required performance/output of a product.” According to his definition, information content is equivalent to the resource or effort required to achieve the required performance/output of the product. Nakazawa used Recsat as information content. Following his approach, robust design is achieved for the minimum Recsat of the product system. If this axiom can be accepted, design navigation (i.e., the Nakazawa method) can be used to decide the best combination of variables in terms of the performance/output of the product. The Taguchi and Nakazawa methods use different metrics to measure the sensitivity to variation according to different concepts. Meanwhile, the two methods provide similar solutions for robust design, suggesting that we can establish a more united approach. The Taguchi and Nakazawa methods are discrete methods and are especially effective when using an experimental design with an orthogonal array and the product has no mathematical or physical model. Meanwhile, for the product that has a mathematical or physical model, with the remarkable progress of optimization algorithms (e.g., genetic algorithms), contentious robust design is becoming a powerful tool [11]. The present paper focuses on finding a new approach for robust design that is compatible with the Taguchi and Nakazawa methods
Peer review under responsibility of Karabuk University. https://doi.org/10.1016/j.jestch.2018.11.005 2215-0986/Ó 2018 Karabuk University. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article as: S.-i. Inage, Proposal of the ‘‘Total Error Minimization Method” for robust design, Engineering Science and Technology, an International Journal, https://doi.org/10.1016/j.jestch.2018.11.005
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and more general approaches based on, for example, genetic algorithms. 2. Brief summary of conventional methods 2.1. Taguchi method The Taguchi method is a well-known and popular method of robust design (Genichi Taguchi, 1980). The method has two basic principles.
This SN ratio is a metric used to estimate the sensitivity of the performance of the product to noise; i.e., the SN ratio is high for low sensitivity and low for high sensitivity. In this paper, to demonstrate how to apply the Taguchi method, we applied the method to the simple electric circuit having two transistors and resistances shown in Fig. 1 (as used by [19]. We traced the analysis of this circuit with Taguchi method by Hayashi et al. [14] [19]. For an input voltage Ei, the output Eo can be estimated as
Eo ¼ 1) A limited set of experiments is conducted using an orthogonal array to estimate the sensitivity of the quality measures to various processing and observable and controllable design parameters and to noise from uncontrollable variation. 2) The product is designed so that the sensitivity of the quality measurements to noise is minimized. In other words, the product/process is designed so that the performance/output is in a regime that minimizes the effects of uncontrollable process variations. In the case of the Taguchi method, ‘‘noise” is introduced intentionally to estimate the sensitivity of the performance of a product to uncontrollable variations. Noise N1, which results in the performance highest, and noise N2, which results in the lowest performance, are usually considered. With orthogonal arrays, a limited set of experiments is performed for different combinations of controllable variations with noise N1 and N2. The following are then estimated for the performance y of the product. Performance results: Under a combination of controllable variations with N1 and N2: For N1: performance y1 For N2: performance y2
ðP 2 þ P3 Þ P1
ð4Þ
where
ðR1 þ R2 =2Þ 1 þ hfe1 þ R4 hfe2 1 þ hfe1 þ R4 P1 ¼ ðR1 þ R2 =2Þ 1 þ hfe1 P2 ¼ Ei V BE1 þ
P3 ¼
ð5Þ
ðV BE2 þ V Z ÞR4 hfe2 ðV BE2 þ V Z ÞR4 hfe2 þ R1 þ R22 R3 þ R22
ð6Þ
ðV BE2 þ V Z ÞR4 Io R4 ðR1 þ R2 =2Þ ðR1 þ R2 =2Þ 1 þ hfe1
ð7Þ
The experimental levels of design variables are given in Table 1. These variables are dispatched to an L18 orthogonal array as shown in Table 2. In the circuit, the resistance Ro is controllable and the current Io is varied from 0 to 800 mA. In the L18 array, the first and last rows are treated as dummy variables. Noise N1 and noise N2 are considered for each variable as shown in Table 3. As defined, noise N1 results in the highest performance/output Eo while noise N2 results in the lowest Eo. The other parameters VBE1, VBE2, and Vz are set at a constant 0.6 V. For this problem, the target performance/output (i.e., specification) is set at 10 ± 1 V (i.e., 9 V < Eo < 11 V) under any current Io. Table 4 summarizes the
Average performance:
Pn
i¼1 yi
y¼
n
¼
1 ðy þ y2 Þ 2 1
SN ratio:
y2 g ¼ 10log 2 s
ð1Þ
! ð2Þ
where
s ¼ 2
Pn 2 i¼1 yi y
n1 2 2 1 is dispersion y1 y þ y1 y ¼ 21
Fig. 1. Target electronic circuit [19].
ð3Þ
Table 1 Experimental levels of design variables taken from Hayashi et al. [14]. Design variables
The first level
The second level
The third level
R1 R2 R3 R4 hfe1 hfe2
2.2 kX 0.11 kX 0.3 kX 0.08 kX 18 50
11 kX 0.56 kX 1.5 kX 0.42 kX 35 100
56 kX 2.7 kX 7.5 kX 2.2 kX 70 200
Table 2 Applied L18 orthogonal table. No.
Dummy
R1
R2
R3
R4
hfe1
hfe2
Dummy
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2
1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3 2 3 1 3 1 2 2 3 1 3 1 2
1 2 3 2 3 1 3 1 2 2 3 1 1 2 3 3 1 2
1 2 3 2 3 1 3 1 2 2 3 1 1 2 3 3 1 2
1 2 3 3 1 2 2 3 1 2 3 1 3 1 2 1 2 3
1 2 3 3 1 2 3 1 2 1 2 3 2 3 1 2 3 1
Please cite this article as: S.-i. Inage, Proposal of the ‘‘Total Error Minimization Method” for robust design, Engineering Science and Technology, an International Journal, https://doi.org/10.1016/j.jestch.2018.11.005
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performance calculations for different combinations of controllable variations based on the L18 orthogonal array. Details of calculations are given in the appendix. For each of cases 1 to 18, two perfor-
mance results of products for N1 and N2 and the average of the two results and the SN ratio can be estimated. On the basis of these results and employing the concept of experimental design, the
Table 3 Consideration of noise N1 and N2. Variables
R1 (%)
R2 (%)
R3 (%)
R4 (%)
hfe1 (%)
hfe2 (%)
Ei (%)
I0 (mA)
N1 N2
10 10
10 10
10 10
10 10
30 30
30 30
10 10
800 0
Table 4 Results obtained using the Taguchi method. Case
Variablues
R1 kX
R2 kX
R3 kX
R4 kX
hfe1 –
hfe2 –
Ei V
I0 A
Calculation results
1
Base N1 N2
2.2 2.2484 2.249465
0.11 0.110121 0.110121
0.3 0.3009 0.300903
0.08 0.080064 0.080064
18 21.24 21.8232
50 75 87.5
20 24 24.8
– 0.8 0
E0 9.19 15.33
Average 12.26
Despersion 18.81551
SN ratio 9.02
2
Base N1 N2
2.2 2.2484 2.249465
0.56 0.560616 0.560617
1.5 1.5045 1.504514
0.42 0.420336 0.420336
35 41.3 42.434
100 150 175
20 24 24.8
– 0.8 0
E0 2.73 4.59
Average 3.66
Despersion 1.723907
SN ratio 8.91
3
Base N1 N2
2.2 2.2484 2.249465
2.7 2.70297 2.702973
7.5 7.5225 7.522568
2.2 2.20176 2.201761
70 82.6 84.868
200 300 350
20 24 24.8
– 0.8 0
E0 1.52 1.96
Average 1.74
Despersion 0.094812
SN ratio 15.03
4
Base N1 N2
11 11.242 11.24732
0.11 0.110121 0.110121
0.3 0.3009 0.300903
0.42 0.420336 0.420336
35 41.3 42.434
200 300 350
20 24 24.8
– 0.8 0
E0 29.39 42.80
Average 36.10
Despersion 90.01358
SN ratio 11.61
5
Base N1 N2
11 11.242 11.24732
0.56 0.560616 0.560617
1.5 1.5045 1.504514
2.2 2.20176 2.201761
70 82.6 84.868
50 75 87.5
20 24 24.8
– 0.8 0
E0 5.86 11.84
Average 8.85
Despersion 17.91223
SN ratio 6.41
6
Base N1 N2
11 11.242 11.24732
2.7 2.70297 2.702973
7.5 7.5225 7.522568
0.08 0.080064 0.080064
18 21.24 21.8232
100 150 175
20 24 24.8
– 0.8 0
E0 8.40 15.64
Average 12.02
Despersion 26.24386
SN ratio 7.41
7
Base N1 N2
56 57.232 57.2591
0.11 0.110121 0.110121
1.5 1.5045 1.504514
0.08 0.080064 0.080064
70 82.6 84.868
100 150 175
20 24 24.8
– 0.8 0
E0 19.43 24.37
Average 21.90
Despersion 12.1753
SN ratio 15.95
8
Base N1 N2
56 57.232 57.2591
0.56 0.560616 0.560617
7.5 7.5225 7.522568
0.42 0.420336 0.420336
18 21.24 21.8232
200 300 350
20 24 24.8
– 0.8 0
E0 3.84 16.49
Average 10.16
Despersion 80.07242
SN ratio 1.11
9
Base N1 N2
56 57.232 57.2591
2.7 2.70297 2.702973
0.3 0.3009 0.300903
2.2 2.20176 2.201761
35 41.3 42.434
50 75 87.5
20 24 24.8
– 0.8 0
E0 12.45 38.93
Average 25.69
Despersion 350.6769
SN ratio 2.75
10
Base N1 N2
2.2 2.2484 2.249465
0.11 0.110121 0.110121
7.5 7.5225 7.522568
2.2 2.20176 2.201761
35 41.3 42.434
100 150 175
20 24 24.8
– 0.8 0
E0 1.13 1.92
Average 1.53
Despersion 0.310778
SN ratio 8.74
11
Base N1 N2
2.2 2.2484 2.249465
0.56 0.560616 0.560617
0.3 0.3009 0.300903
0.08 0.080064 0.080064
70 82.6 84.868
200 300 350
20 24 24.8
– 0.8 0
E0 6.67 9.85
Average 8.26
Despersion 5.0376
SN ratio 11.32
12
Base N1 N2
2.2 2.2484 2.249465
2.7 2.70297 2.702973
1.5 1.5045 1.504514
0.42 0.420336 0.420336
18 21.24 21.8232
50 75 87.5
20 24 24.8
– 0.8 0
E0 1.62 6.29
Average 3.95
Despersion 10.92804
SN ratio 1.55
13
Base N1 N2
11 11.242 11.24732
0.11 0.110121 0.110121
1.5 1.5045 1.504514
2.2 2.20176 2.201761
18 21.24 21.8232
200 300 350
20 24 24.8
– 0.8 0
E0 6.15 11.95
Average 9.05
Despersion 16.83545
SN ratio 6.87
14
Base N1 N2
11 11.242 11.24732
0.56 0.560616 0.560617
7.5 7.5225 7.522568
0.08 0.080064 0.080064
35 41.3 42.434
50 75 87.5
20 24 24.8
– 0.8 0
E0 11.21 17.79
Average 14.50
Despersion 21.66127
SN ratio 9.87
15
Base N1 N2
11 11.242 11.24732
2.7 2.70297 2.702973
0.3 0.3009 0.300903
0.42 0.420336 0.420336
70 82.6 84.868
100 150 175
20 24 24.8
– 0.8 0
E0 9.22 14.71
Average 11.97
Despersion 15.07748
SN ratio 9.78
16
Base N1 N2
56 57.232 57.2591
0.11 0.110121 0.110121
7.5 7.5225 7.522568
0.42 0.420336 0.420336
70 82.6 84.868
50 75 87.5
20 24 24.8
– 0.8 0
E0 10.41 19.46
Average 14.94
Despersion 40.94777
SN ratio 7.36
17
Base N1 N2
56 57.232 57.2591
0.56 0.560616 0.560617
0.3 0.3009 0.300903
2.2 2.20176 2.201761
18 21.24 21.8232
100 150 175
20 24 24.8
– 0.8 0
E0 64.35 110.71
Average 87.53
Despersion 1074.961
SN ratio 8.53
18
Base N1 N2
56 57.232 57.2591
2.7 2.70297 2.702973
1.5 1.5045 1.504514
0.08 0.080064 0.080064
35 41.3 42.434
200 300 350
20 24 24.8
– 0.8 0
E0 16.71 22.91
Average 19.81
Despersion 19.1949
SN ratio 13.10
Please cite this article as: S.-i. Inage, Proposal of the ‘‘Total Error Minimization Method” for robust design, Engineering Science and Technology, an International Journal, https://doi.org/10.1016/j.jestch.2018.11.005
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effects of the experimental levels of design variables in Table 1 are obtained as shown in Fig. 2. The figure plots the relationships between levels 1–3 of variables and the average output and SN ratio. The Taguchi method has two stages. The first stage is the determination of the levels of variables according to the SN ratio. To begin, using the SN ratio graph in Fig. 2, the variables that most affect the SN ratio are checked. In this case, the variables R4 and hfe1 have the maximum SN ratio. As previously mentioned, the SN ratio is a metric of the sensitivity of the performance/output to variation. The levels of variables that have the maximum SN ratio can therefore be used to reduce variations of the performance/output of the product passively. The second stage is the determination of the level of variables from the SN ratio and average performance. In the first stage, the levels of variables that have higher SN ratios are found. In the second stage, the variables that have no relationship or only a weak relationship with the SN ratio are chosen and the optimal level of each variable is chosen to adjust the required performance/output of the product. To adjust the performance to a target value of 10 V, the variables R1 and R3 are useful as they hardly affect the SN ratio yet strongly affect the performance itself (see Fig. 2a). Through the two-stage design, the product can provide the required performance/output with robustness against variation due to noise. This is the typical process of the Taguchi method. From the average response graph of the SN ratio in Fig. 2b, R4 = 0.08 kX and hfe1 = 70 are chosen as they have the highest SN ratios. Good robustness against variation is expected using these values of R4 and hfe1.
due to noise using an orthogonal array. In particular, the method performs well in obtaining multiple required performances/outputs of a product. A product usually has several required performances. In design navigation, the Recsat metric is defined as
Rc ¼ ln
System range Common range
ð8Þ
The system range is the range of variation of the performance/ output, the design range is the allowance of the required performance/output of the product, and the common range is the range of overlap of the system and design ranges. The relationship among the ranges is shown in Fig. 3. Recsat is estimated as follows. Average performance:
y¼
Pn
i¼1 yi
n
ð9Þ
2.2. Design navigation method (Nakazawa method) Design navigation (i.e., the Nakazawa method) can confirm the required performance/output with robustness against variation
Fig. 3. Definitions of design, system, and common ranges in the Nakazawa method.
Fig. 2. Response graphs of the performance test.
Please cite this article as: S.-i. Inage, Proposal of the ‘‘Total Error Minimization Method” for robust design, Engineering Science and Technology, an International Journal, https://doi.org/10.1016/j.jestch.2018.11.005
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Dispersion:
s2 ¼
Pn 2 i¼1 yi y
ð10Þ
n1
The system range SR is defined as
pffiffiffiffiffi SRu ¼ y þk s2
ð11Þ
pffiffiffiffiffi SRt ¼ y k s2
ð12Þ
where the subscript u indicates the maximum range and the subscript t the minimum range. It follows that
Systemrange ¼ ðSRu SRt Þ
ð13Þ
The common range depends on the allowance of the required performance/output; e.g., in the case of Eo in Fig. 1, the allowance is 9 V < Eo < 11 V and the common range is 11–9 = 2 V. Recsat can finally be written as
pffiffiffiffiffi ! 2k s2 Rc ¼ ln Y max Y min
ð14Þ
where the weight k is, for example, 1.5; Y is the target performance/ output (a required specification); and the subscript max indicates the maximum allowance of the output while the subscript min indicates the minimum allowance of the output. As previously mentioned, the Nakazawa method focuses on finding the optimal design in terms of the multiple performances/outputs of a product. The term ‘‘optimal” refers to the minimum variation of the multiple performances/outputs of the product. For each performance/output, Recsat is experimentally measured using an orthogonal array and the Taguchi method. The total Recsat can then be calculated as
Rc;t ¼
X
ð15Þ
Rc;i
where Rc,t is the total Recsat and Rc,i is the Recsat of performance i.
The Nakazawa method assumes the axiom that the optimal design is achieved when the total Recsat is a minimum for a combination of variables. Nakazawa proposed the axiom on the basis of information theory and Shannon’s concept of information entropy. As it is an axiom, we cannot prove its validity. The Nakazawa method is applied to the electric circuit problem shown in Fig. 1. Meanwhile, the Nakazawa method is applicable to multiple-performance design. Not only the output voltage Eo but also the cost of the circuit is therefore considered. The cost estimation is assumed to be given by the simple equation Hayashi et al. [14].
C ¼ 5R2 þ hfe1 þ f fe2
ð16Þ
where R2, hef1, and hfe2 are given at the levels listed in Table 1. The total Recsat for each variable level estimated from the calculated Eo and the average output given in Table 4 is given in Table 5. In this case, the allowable range of Eo is from 9 to 11 V while the allowed cost is less than 135. Details of the calculation of total Recsat are given in the appendix. Fig. 4 shows the estimated Recsat for each variable. The optimal levels of variables are chosen according to the minimum Recsat. The optimal combination is therefore that given in Table 6. The optimal performance is 10.12 V (which is within the range of 9–11 V) while the cost is 133.5 (which is less than 135). The required performance is therefore achieved employing the Nakazawa method.
3. Proposal of the total Error minimization method This paper proposes a new approach of achieving a robust design of a product having single and multiple performances/outputs without axioms. In the case of design without theoretical models, an experimental approach, such as the adoption of the Taguchi or Nakazawa method with an orthogonal array, is important. Meanwhile, if mathematical or physical models are available, another robust design using, for example, a genetic algorithm is applicable (12). The new approach unifies these experimental and theoretical approaches.
Table 5 Estimation of Recsat for each variable. Average
Despersion
Sru
STt
Ro
Cost
Despersion
Sru,c
STt,c
Ro,c
R1
Rt
R1-1 R1-2 R1-3
5.23 15.41 30.00
17.72 59.04 607.02
11.55 26.94 66.96
1.08 3.89 6.95
1.84 2.44 3.61
163.28 163.28 163.28
8705.48 3406.48 3690.14
303.24 250.83 254.40
23.33 75.74 72.16
0.919 1.083 1.065
2.20 11.00 56.00
2.762 3.528 4.675
R2-1 R2-2 R2-3
Average 15.96 22.16 12.53
Despersion 55.15 264.31 51.32
Sru 27.10 46.55 23.27
STt 4.82 2.23 1.78
Ro 2.41 3.19 2.37
Cost 158.22 160.47 171.17
Despersion 3949.07 5009.07 6729.07
Sru,c 252.48 266.63 294.21
STt,c 63.95 54.30 48.12
Ro,c 0.976 0.967 1.041
R2 0.11 0.56 2.70
Rt 3.386 4.161 3.416
R3-1 R3-2 R3-3
Average 30.30 11.20 9.15
Despersion 563.40 33.27 21.24
Sru 65.90 19.86 16.06
STt 5.30 2.55 2.23
Ro 3.57 2.16 1.93
Cost 163.28 163.28 163.28
Despersion 6520.24 3869.18 5412.68
Sru,c 284.41 256.59 273.64
STt,c 42.16 69.98 52.93
Ro,c 0.959 1.054 0.989
R3 0.30 1.50 7.50
Rt 4.532 3.212 2.923
R4-1 R4-2 R4-3
Average 14.79 13.46 22.40
Despersion 25.65 73.90 263.18
Sru 22.39 26.36 46.73
STt 7.19 0.57 1.94
Ro 2.03 2.56 3.19
Cost 163.28 163.28 163.28
Despersion 6998.18 3620.68 5183.24
Sru,c 288.77 253.54 271.28
STt,c 37.80 73.03 55.29
Ro,c 0.949 1.069 0.997
R4 0.08 0.42 2.20
Rt 2.976 3.626 4.189
hfe1-1 hfe1-2 hfe1-3
Average 22.50 16.88 11.28
Despersion 348.90 84.15 15.45
Sru 50.51 30.64 17.17
STt 5.52 3.12 5.38
Ro 3.33 2.62 1.77
Cost 140.28 157.28 192.28
Despersion 4333.30 4716.30 5065.30
Sru,c 239.03 260.30 299.04
STt,c 41.54 54.27 85.53
Ro,c 0.748 0.937 1.462
hfe1 18.00 35.00 70.00
Rt 4.081 3.559 3.237
hfe2-1 hfe2-2 hfe2-3
Average 13.37 23.10 14.19
Despersion 73.55 218.95 48.27
Sru 26.23 45.30 24.61
STt 0.50 0.90 3.76
Ro 2.55 3.10 2.34
Cost 96.62 146.62 246.62
Despersion 471.18 605.68 725.24
Sru,c 129.18 183.53 287.01
STt,c 64.06 109.70 206.22
Ro,c (0.086) 1.071 #NUM!
hfe2 50.00 100.00 200.00
Rt 2.469 4.171 #NUM!
Please cite this article as: S.-i. Inage, Proposal of the ‘‘Total Error Minimization Method” for robust design, Engineering Science and Technology, an International Journal, https://doi.org/10.1016/j.jestch.2018.11.005
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Fig. 4. Recsat curve for Rc,t.
Table 6 Results of the confirmation test. Parameter
R1 (kX)
R2 (kX)
R3 (kX)
R4 (kX)
hfe1
hfe2
Output Voltage (V)
Cost
Value
2.2
2.7
7.5
0.08
70
50
10.12
133.5
3.1. Fundamental concept
3.2. Validation of the proposed method
As discussed, the Taguchi and Nakazawa methods use an original metric, like the SN ratio or Recsat, to measure the sensitivity to variations. These metrics are defined by the average or target performance/output of the product and the dispersion. The Taguchi method uses the average performance for different levels of variables. It therefore requires a two-stage design to perform well after the confirmation of robustness against variations using the SN ratio. Meanwhile, the Nakazawa method uses the target performance (i.e., specification) with its allowance to estimate the dispersion, estimating the original metric Recsat. Additionally, finding the minimum Recsat allows the robust combining of variables. Taking this approach, we use the new metric
The proposed approach is applied to the electric circuit problem presented in Fig. 1. The total error for each variable and level of the variable obtained from the results of Eo and average output in Table 4 and Eqs. (17)–(19) are presented in Table 7. In this case, the target performance of Eo is 10 V While the target cost is 135. The detail of the calculation of total error is presented in the appendix. Table 8 gives the average response of the total error while Fig. 5 presents the average response graphs of total error. Table 9 gives the results of the confirmation test. The response trend in Fig. 5 is similar to that for the Nakazawa method. The results of the confirmation test are therefore the same as those for the Nakazawa method. These validation tests confirm the proposed approach. Additionally, as discussed in Section 2.2, the Taguchi method recommends R4-1 and hfe1-3 as the optimal levels from the point of view of the SN ratio. The proposed approach also recommends these levels. The proposed approach can be applied to a genetic algorithm directly. The genetic algorithm is well known as a powerful multi-point search algorithm that is based on the simulation of the heredity and evolution of living things [1,13,9,20,2]. The proposed approach is expected to have the following advantages.
E¼
s2
0:5 ð17Þ
Y2
where
Pn s2 ¼
Y Þ2 n1
i¼1 ðyi
ð18Þ
and Y is the target output (i.e., specification). In fact, E defined by Eq. (17) is the error relative to the target performance/output of the product. For Eq. (18), yi is estimated by conducting experiments using an orthogonal array with noise N1 and N2 as with the Taguchi or Nakazawa method. As this approach uses the target performance/output directly, it does not require the two-stage design. In the case of multiple performances/outputs, Ei is estimated for each performance i. For the system, the total error is then defined as
Et ¼
X
Ei
ð19Þ
If the combination of variable levels minimizes the total error in Eq. (19), the product has the multiple required performances/outputs. This is not an axiom but a universal truth. Therefore, using this concept, without axioms, robust design can be achieved. Additionally, if there are mathematical or physical models with which to design the product, the metric in Eq. (17) can be used as an objective function of genetic algorithms. In this case, without using an orthogonal array, the optimal combination of variables is obtained.
a) A genetic algorithm can be applied to problems even in a discrete search space. b) An optimal solution can be determined even when the landscape of the objective function is multi-modal. c) In multi-objective optimization, the genetic algorithm, which is a multi-point search algorithm, can determine a Pareto-optimal set using one trial to check the trade-off relations among multiple objectives. In the case of the combination of discrete variables mentioned above, robust design with an orthogonal array is the only alternative. Meanwhile, if contentious variables can be accepted, using a genetic algorithm or another optimization algorithm, a solution better than that of the discrete approach is obtained. As an example of optimization with multi object functions, Hasan Koten et al. coupled one dimensional engine code and multi objective optimization code and evaluated about 15,000 cases to define the proper boundary conditions to optimize the parameters on compressed biogas-diesel dual-fuel engine and ignition delay
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S.-i. Inage / Engineering Science and Technology, an International Journal xxx (xxxx) xxx Table 7 Estimation of error in performance using an orthogonal array. R1 kX
R2 kX
R3 kX
R4 kX
hfe1 –
hfe2 –
Ei V
I0 A
Calculation results
1
2.2 1.98 2.42
0.11 0.121 0.099
0.3 0.33 0.27
0.08 0.072 0.088
18 12.6 23.4
50 65 35
20 18 22
– 0.8 0
E0 9.19 15.33
Average 12.26
Despersion 10.217
Error 0.32
Cost 68.55
2
2.2 1.98 2.42
0.56 0.616 0.504
1.5 1.65 1.35
0.42 0.378 0.462
35 24.5 45.5
100 130 70
20 18 22
– 0.8 0
E0 2.73 4.59
Average 3.66
Despersion 80.367
Error 0.90
Cost 137.8
3
2.2 1.98 2.42
2.7 2.97 2.43
7.5 8.25 6.75
2.2 1.98 2.42
70 49 91
200 260 140
20 18 22
– 0.8 0
E0 1.52 1.96
Average 1.74
Despersion 136.506
Error 1.17
Cost 283.5
4
11 9.9 12.1
0.11 0.121 0.099
0.3 0.33 0.27
0.42 0.378 0.462
35 24.5 45.5
200 260 140
20 18 22
– 0.8 0
E0 29.39 42.80
Average 36.10
Despersion 1362.017
Error 3.69
Cost 235.55
5
11 9.9 12.1
0.56 0.616 0.504
1.5 1.65 1.35
2.2 1.98 2.42
70 49 91
50 65 35
20 18 22
– 0.8 0
E0 5.86 11.84
Average 8.85
Despersion 2.644
Error 0.16
Cost 122.8
6
11 9.9 12.1
2.7 2.97 2.43
7.5 8.25 6.75
0.08 0.072 0.088
18 12.6 23.4
100 130 70
20 18 22
– 0.8 0
E0 8.40 15.64
Average 12.02
Despersion 8.162
Error 0.29
Cost 131.5
7
56 50.4 61.6
0.11 0.121 0.099
1.5 1.65 1.35
0.08 0.072 0.088
70 49 91
100 130 70
20 18 22
– 0.8 0
E0 19.43 24.37
Average 21.90
Despersion 283.191
Error 1.68
Cost 170.55
8
56 50.4 61.6
0.56 0.616 0.504
7.5 8.25 6.75
0.42 0.378 0.462
18 12.6 23.4
200 260 140
20 18 22
– 0.8 0
E0 3.84 16.49
Average 10.16
Despersion 0.053
Error 0.02
Cost 220.8
9
56 50.4 61.6
2.7 2.97 2.43
0.3 0.33 0.27
2.2 1.98 2.42
35 24.5 45.5
50 65 35
20 18 22
– 0.8 0
E0 12.45 38.93
Average 25.69
Despersion 492.466
Error 2.22
Cost 98.5
10
2.2 1.98 2.42
0.11 0.121 0.099
7.5 8.25 6.75
2.2 1.98 2.42
35 24.5 45.5
100 130 70
20 18 22
– 0.8 0
E0 1.13 1.92
Average 1.53
Despersion 143.638
Error 1.20
Cost 135.55
11
2.2 1.98 2.42
0.56 0.616 0.504
0.3 0.33 0.27
0.08 0.072 0.088
70 49 91
200 260 140
20 18 22
– 0.8 0
E0 6.67 9.85
Average 8.26
Despersion 6.053
Error 0.25
Cost 272.8
12
2.2 1.98 2.42
2.7 2.97 2.43
1.5 1.65 1.35
0.42 0.378 0.462
18 12.6 23.4
50 65 35
20 18 22
– 0.8 0
E0 1.62 6.29
Average 3.95
Despersion 73.119
Error 0.86
Cost 81.5
13
11 9.9 12.1
0.11 0.121 0.099
1.5 1.65 1.35
2.2 1.98 2.42
18 12.6 23.4
200 260 140
20 18 22
– 0.8 0
E0 6.15 11.95
Average 9.05
Despersion 1.798
Error 0.13
Cost 218.55
14
11 9.9 12.1
0.56 0.616 0.504
7.5 8.25 6.75
0.08 0.072 0.088
35 24.5 45.5
50 65 35
20 18 22
– 0.8 0
E0 11.21 17.79
Average 14.50
Despersion 40.461
Error 0.64
Cost 87.8
15
11 9.9 12.1
2.7 2.97 2.43
0.3 0.33 0.27
0.42 0.378 0.462
70 49 91
100 130 70
20 18 22
– 0.8 0
E0 9.22 14.71
Average 11.97
Despersion 7.728
Error 0.28
Cost 183.5
16
56 50.4 61.6
0.11 0.121 0.099
7.5 8.25 6.75
0.42 0.378 0.462
70 49 91
50 65 35
20 18 22
– 0.8 0
E0 10.41 19.46
Average 14.94
Despersion 48.757
Error 0.70
Cost 120.55
17
56 50.4 61.6
0.56 0.616 0.504
0.3 0.33 0.27
2.2 1.98 2.42
18 12.6 23.4
100 130 70
20 18 22
– 0.8 0
E0 64.35 110.71
Average 87.53
Despersion 12021.851
Error 10.96
Cost 120.8
18
56 50.4 61.6
2.7 2.97 2.43
1.5 1.65 1.35
0.08 0.072 0.088
35 24.5 45.5
200 260 140
20 18 22
– 0.8 0
E0 16.71 22.91
Average 19.81
Despersion 192.390
Error 1.39
Cost 248.5
issue [3,4]. Taking this approach, the metrics in Eqs. (17) and (19) are applied as the objective functions of the genetic algorithm. The variables R1, R2,. . ., hfe2 are treated as design parameters. In the calculation, the ranges of variables are those listed in Table 10. The present paper applies the neighborhood cultivation genetic algorithm (NCGA) [12], [11]. The NCGA has a neighborhood crossover mechanism in addition to the mechanisms of general genetic algorithms. The NCGA has advantages over other genetic algo-
rithms from the point of view of the solution survey in a discrete search space. On this calculation, population size and numbers of generation are 50 and 50, respectively. Table 11 gives the results of the optimal combination of contentious variables and its output Eo and cost. In the table, result 1 is the calculation for the target E0 = 10 V and cost = 135. Eo is closer to the target here than in the discrete case reported in Table 9. Meanwhile, the cost slightly exceeds 135. Therefore, result 2 for the target E0 = 10 V and
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S.-i. Inage / Engineering Science and Technology, an International Journal xxx (xxxx) xxx
cost = 134 is obtained. Using the lower target cost, both allowable Eo and an allowable cost can be obtained. Fig. 6 shows the relationship between the errors in Eo and cost and the number of generations of a genetic algorithm. Both errors decrease with an increasing number of generations. The proposed approach is applicable to robust design when either taking the discrete approach with an orthogonal array or using contentious variables.
Table 8 Estimation of total error. Levels
Average
Error on performance
Cost
Error on cost
Total error
R1-1 R1-2 R1-3
5.23 15.41 30.00
0.78 0.86 2.83
163.28 163.28 163.28
0.21 0.21 0.21
0.99 1.07 3.04
R2-1 R2-2 R2-3
15.96 22.16 12.53
1.29 2.15 1.03
158.22 160.47 171.17
0.17 0.19 0.27
1.46 2.34 1.30
R3-1 R3-2 R3-3
30.30 11.20 9.15
2.95 0.85 0.67
163.28 163.28 163.28
0.21 0.21 0.21
3.16 1.06 0.88
R4-1 R4-2 R4-3
14.79 13.46 22.40
0.76 1.07 2.64
163.28 163.28 163.28
0.21 0.21 0.21
0.97 1.28 2.85
hfe1-1 hfe1-2 hfe1-3
22.50 16.88 11.28
2.10 1.67 0.71
140.28 157.28 192.28
0.04 0.17 0.42
2.14 1.84 1.13
hfe2-1 hfe2-2 hfe2-3
13.37 23.10 14.19
0.82 2.55 1.11
96.62 146.62 246.62
0.28 0.09 0.83
1.10 2.64 1.93
4. Discussion Table 12 compares the Taguchi and Nakazawa methods and the proposed approach. The major differences are the definitions of performance and dispersion. The Taguchi and Nakazawa methods introduce a logarithmic form to the metric. This difference is not important as we can apply the log or ln function to the total error and the tendency of the total error response graph does not change. Additionally, the strategy based on minimum total error has clear physical meaning, which means there are minimum discrepancies between the real performance/output and required specification. This approach therefore does not require axioms.
Fig. 5. Average response graph of total error.
Table 9 Results of the confirmation test for the proposed method. Parameter
R1 (kX)
R2 (kX)
R3 (kX)
R4 (kX)
hfe1
hfe2
Output Voltage (V)
Cost
Value
2.2
2.7
7.5
0.08
70
50
10.12
133.5
Table 10 Range of contentious variables. Variables
R1 (kX)
R2 (kX)
R3 (kX)
R4 (kX)
hfe1
hfe2
Range
2.28–56
0.11–2.7
0.3–7.5
0.08–2.7
18–70
50–200
Table 11 Results of the optimal combination of contentious variables. Variables
R1 (kX)
R2 (kX)
R3 (kX)
R4 (kX)
hfe1
hfe2
Eo (V)
Cost
Discrete Result-1 Result-2
2.2 29.22 55.00
2.7 2.35 1.57
7.5 3.71 7.03
0.08 0.81 1.07
70 45.71 46.98
50 77.78 79.91
10.12 10.02 10.02
133.5 135.26 134.75
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S.-i. Inage / Engineering Science and Technology, an International Journal xxx (xxxx) xxx
9
Fig. 6. Errors in Eo and cost versus the number of generations of a genetic algorithm.
Table 12 Comparison of the three methods. Methods
Taguchi Method
Metrics
Average Value SN ratio
Definition of metrics
Unit of Metrics Strategy Noise for output Applications
Average Value: Pn yi y ¼ i¼1 SN ratio: n 2 g ¼ 10log ys2 where, Pn 2 ðy yÞ s2 ¼ i¼1n1i db Two-stage design Considered Stand alone
Axioms
–
Design Navigation
Proposal
Recsat (Reciprocal of satisfaction) Recsat: pffiffiffiffi 2k s2 Rc ¼ ln Y max Y min where, Pn 2 ðy yÞ k = 1.5 (for example) s2 ¼ i¼1n1i Y: Target output (Specification)
Total error
– Direct design
–
Minimum Recsat means the optimal product
5. Conclusions A new approach for the robust design of a product with single or multiple performances/outputs was proposed based on a new metric which means total error and validated. The method was proposed through minimization of the total difference (i.e., error) between the required specification and experimental output for different variables with noise. This approach was applied to obtain the required specification and target cost for an electric circuit
Error: 2 0:5 E ¼ Ys 2 where, Pn ðy Y Þ2 2 i¼1 i Y: Target output (Specification) s ¼ n1
1) Stand alone 2) with Genetic Algorithms –
problem. The optimal results were the same as those obtained using the Nakazawa method and met the required specification. The proposed approach is compatible with experimental and theoretical approaches. A genetic-algorithm-based design with a proposed metric, namely optimal design with contentious variables, was established. Relative to the result of the discrete approach, the performance/output achieved with the proposed method was closer to the required specification. Employing the proposed method, we expect a united robust design without an axiom.
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S.-i. Inage / Engineering Science and Technology, an International Journal xxx (xxxx) xxx
Average SN ratio for level 1 of R1:
Acknowledgment We thank Glenn Pennycook, MSc, from Edanz Group for editing a draft of this manuscript.
9:02 þ 8:89 þ 15:03 þ 8:74 þ 11:32 þ 1:55 ¼ 9:09ðdbÞ 6
Appendix
Finally, the response graph of the SN ratio in Fig. 2 can be plotted.
A-1 calculation using the Taguchi method
A-2 calculation using the Nakazawa method
An example of the calculation of the Taguchi method is described here. In Table 4, for case 1, the levels of variables R1– hfe2 and Ei and I0 with noise are listed in Table A1. From Eqs. (1)–(3): Average Eo:
We use the results given in Table 4. Through the concept of experimental design, for each level of a variable, the average Eo and dispersion can be estimated. Average Eo for level 1 of R1:
Pn
i¼1 yi
y¼
n
¼
1 1 ðy þ y2 Þ ¼ ð9:19 þ 15:33Þ ¼ 12:26 2 1 2
ðA-1Þ
12:26 þ 3:66 þ 1:74 þ 1:53 þ 8:26 þ 3:95 ¼ 5:23ðVÞ 6
Dispersion s2:
s2 ¼
n1
¼
SN ratio:
g ¼ 10log
y2 s2
! ¼ 10log
12:262 18:81
ðA-7Þ
Recsat for the performance/output of the product in case 1 from Table 4 is as follows.
2 2 1 y1 y þ y1 y 21
¼ ð9:19 12:26Þ2 þ ð15:33 12:26Þ2 ¼ 18:81
ðA-6Þ
Average dispersion s2 for level 1 of R1:
ð12:26 5:23Þ2 þ ð3:66 5:23Þ2 þ ð1:74 5:23Þ2 þ ð1:53 5:23Þ2 þ ð8:26 5:23Þ2 þ ð3:95 5:23Þ2 ¼ 17:72 61
Pn 2 i¼1 yi y
ðA-5Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffi!
systemrange 1:5 17:72 ¼ ln ¼ 1:843 ¼ ln commonrange 11 9
Rc;E1 ðA-2Þ
ðA-8Þ
Recsat for cost: Average Eo for level 1 of R1:
! ¼ 9:02
ðA-3Þ
Fig. 4 can be plotted from these calculations. Through the concept of experimental design, for each level of a variable, the aver-
68:55 þ 137:88 þ 283:5 þ 135:55 þ 272:8 þ 81:5 ¼ 163:28 6 ðA-9Þ 2
Average dispersion s for level 1 of R1:
ð68:55 163:28Þ2 þ ð137:88 163:28Þ2 þ ð283:5 163:28Þ2 þ ð135:55 163:28Þ2 þ ð272:8 163:28Þ2 þ ð81:5 163:28Þ2 ¼ 8705:48 61 ðA-10Þ
age Eo and SN ratio can be estimated: Level 1 of R1: Average Eo for level 1 of R1:
System range:
12:26 þ 3:66 þ 1:74 þ 1:53 þ 8:26 þ 3:95 ¼ 5:23ðVÞ 6
ðA-4Þ
pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SRu ¼ y þk s2 ¼ 163:28 þ 1:5 8705:48 ¼ 303:2
ðA-11Þ
pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SRt ¼ y k s2 ¼ 163:28 1:5 8705:48 ¼ 23:3
ðA-12Þ
Table A1 Conditions for Case 1 in Table 4. Case
1
Variables
Base N1 N2
R1
R2
R3
R4
hfe1
hfe2
Ei
I0
kX
kX
kX
kX
–
–
V
mA
2.2 1.98 2.42
0.11 0.121 0.099
0.3 0.33 0.27
0.08 0.072 0.088
18 12.6 23.4
50 65 35
20 18 22
– 800 0
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S.-i. Inage / Engineering Science and Technology, an International Journal xxx (xxxx) xxx
Therefore, Recsat Rc of cost is
systemrange 303:2 23:3 ¼ ln ¼ ln ¼ 0:919 commonrange 135 23:3
Rc;c
References
ðA-13Þ
Here, as the cost should be less than 135, the common range is defined by 135 and SRt. The total Recsat is therefore
Rt;1 ¼ Ro;1 þ Rt;1 ¼ 1:843 þ 0:919 ¼ 2:762
ðA-14Þ
Employing the same procedure for other levels and variables, the Recsat response graph in Fig. 4 is finally obtained. A-3 calculation using the proposed method As mentioned in Section A-1, for case 1 in Table 7, the dispersion is estimated as follows. Average Eo:
Pn
y¼
i¼1 yi
n
¼
1 1 ðy þ y2 Þ ¼ ð9:19 þ 15:33Þ ¼ 12:26 2 1 2
ðA-15Þ
Dispersion:
s ¼ 2
Pn 2 i¼1 yi y n1
¼ ð12:26 10Þ2 ¼ 10:217
ðA-16Þ
The target performance/output is 10 V and the error is therefore as follows. Error in performance/output
Ep ¼
s2
0:5 ¼
Y2
0:5 10:217 102
¼ 0:32
ðA-17Þ
Employing the concept of experimental design, the error in the performance of each variable is obtained as listed in Table 8. The error in the cost is estimated as follows. Dispersion of cost for level 1 of R1:
s2 ¼
Pn 2 i¼1 yi y n1
¼ ð163:28 135Þ2 ¼ 799:7584
ðA-18Þ
Error in cost for level 1 of R1:
Ec ¼
s2
Y2
0:5 ¼
0:5 799:7584 1352
¼ 0:209
ðA-19Þ
The total error for level 1 of R1 is therefore
Et ¼ Ep þ Ec ¼ 0:78 þ 0:209 ¼ 0:99
11
ðA-20Þ
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