Published online 2 May 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/qre.921
Research
Robust Design for Multiple
Dynamic Quality Characteristics
Using Data Envelopment Analysis
Lee-Ing Tong1, Chung-Ho Wang2and Chih-Wei Tsai1, ∗,†1Department of Industrial Engineering and Management, National Chiao Tung University, 1001 Ta Hsueh Road,
Hsinchu, Taiwan 30050, Republic of China
2Department of Power Vehicle and Systems Engineering, Chung Cheng Institute of Technology, National Defense
University, Taoyuan, Taiwan, Republic of China
The Taguchi method is extensively adopted in various industries to continuously improve product design in response to customer requirements. The dynamic system of the Taguchi method is frequently implemented to design products with flexible applications. However, Taguchi’s dynamic system can be employed only for indi-vidual quality characteristic, and the relationship between the quality characteristic and the signal factor is assumed to be linear. Because of these restrictions, Taguchi’s dynamic system is ineffective for multiple quality characteristics or when the quality characteristic has a nonlinear relationship with the signal factor. This study describes a novel procedure for optimizing a dynamic system based on data envelopment anal-ysis. The proposed procedure overcomes the limitations of Taguchi’s dynamic system. Two cases are analyzed to demonstrate the effectiveness of the proposed procedure. The results show that the proposed procedure can enhance multiple dynamic quality characteristics. Copyright©2008 John Wiley & Sons, Ltd.
Received 30 November 2006; Revised 4 February 2008; Accepted 28 February 2008
KEY WORDS: Taguchi method; dynamic system; data envelopment analysis; multiple quality characteristics; robust design
1.
INTRODUCTION
T
he Taguchi method is extensively adopted to improve industrial/manufacturing procedures and product quality. The Taguchi method uses parameter design to determine a robust, optimal factor-level combination that can intervene in noise factors. Noise factors produce product/process variances. These factors are difficult or costly to control. The Taguchi method includes static and dynamic systems. A dynamic system contains signal factors for expressing the intended output. The relationship between a quality characteristic and a signal factor is assumed to be linear. In other words, the quality characteristic is a linear function of the signal factor. However, product/process design tends to be rather complex to comply with constantly changing customer requirements and production technology. Multiple responses∗Correspondence to: Chih-Wei Tsai, Department of Industrial Engineering and Management, National Chiao Tung University, 1001
Ta Hsueh Road, Hsinchu, Taiwan 30050, Republic of China.
must be evaluated simultaneously to determine overall product/process quality. Accordingly, the linear relationship is in fact rarely achieved in the current complex product/process design. Such a scenario limits the application of the conventional Taguchi method. Product/process design with a dynamic system offers the flexibility needed to satisfy customer requirements than that of a static system and can enhance a manufacturer’s competitiveness, particularly in an extremely competitive market. Therefore, quality improvement involving a dynamic system is an important issue.
Recent studies of dynamic system optimization, including Miller and Wu1, Miller and Wu2, Wasserman3, Lunani et al.4, McCaskey and Tsui5, Chen6, and Bae and Tsui7, have noted some of the limitations of Taguchi’s dynamic system. Wasserman3 observed that the factor-level combination of a dynamic system using Taguchi’s signal-to-noise ratio (SNR) might not be optimal. Lunani et al.4noted that using SNR as a quality performance measure might produce inaccuracies due to a biased dispersion effect, thus making it impossible to minimize quality loss. Accordingly, Lunani et al.4 combined half-normal graph schemes with analysis of variance to replace SNR to alleviate the drawbacks of the Taguchi method. Lunani et al.4 thereby determined the vital factors affecting the dispersion effect in a dynamic system. Chen6developed a stochastic optimization modeling procedure that incorporated a sequential quadratic programming technique to determine the optimal factor-level combination in a dynamic system. The above studies focused on optimizing single-response problems in a dynamic system.
Given that multiple responses must be simultaneously evaluated to enhance product/process quality, several studies have recommended procedures for optimizing multiple responses in a dynamic system. For instance, Tong et al.8adopted principal component analysis and technique for order preference by similarity to ideal solution to optimize a dynamic system with multiple quality characteristics. Hsieh et al.9applied regression analysis and desirability function to optimize a dynamic system with multiple quality characteristics. Su et al.10 also employed a desirability function to optimize dynamic multiple quality characteristics. Additionally, determining the optimal factor-level combination with the Taguchi method is restricted to the initially designed factor levels. However, an optimal factor-level combination might fall in the interval of factor levels when the factor is continuous. Vining and Mylers11discussed optimization of a product/process involving continuous factors by employing a response surface methodology. Tong et al.12employed a dual response surface methodology to optimize a dynamic multi-response problem with continuous factors.
While focusing mainly on a dynamic system with multiple quality characteristics, this study develops a novel optimization procedure using data envelopment analysis (DEA). The proposed procedure does not require any specific relationship between quality characteristics and signal factors, allowing it to efficiently determine an optimal factor-level combination in a dynamic system. In other words, the developed procedure is more practical than procedures that require certain assumptions. The relative efficiencies of the location effect and dispersion effect for a dynamic system are initially estimated using DEA. Therefore, overall relative efficiencies are obtained as an overall quality performance (OQP) of a dynamic system with multiple responses. A prediction model of OQP on design factors is then established, and the optimal factor-level combination in the feasible region of design factors is thus determined. Finally, two cases are utilized to demonstrate the effectiveness of the proposed procedure.
2.
DYNAMIC SYSTEM OF THE TAGUCHI METHOD
Without advanced or costly equipment, the Taguchi method can obtain an optimized parameters setting, which is robust to interference with noise factors. Moreover, experiments allocating Taguchi’s orthogonal array can reduce the quantity of experiments to lower the experimental cost. The parameter design of the Taguchi method includes both a static system and a dynamic system. These two systems differ in that the dynamic system includes signal factors to express intended outputs. In a dynamic system, a quality characteristic is valued according to the levels of the signal factor. Taguchi’s dynamic system assumes a linear relationship between quality characteristics and signal factors. The linear function can be expressed as
where Y denotes the quality characteristic of a dynamic system, M represents the signal factor, is the slope or sensitivity of the system, and denotes the random error.
Based on the linear function assumption, the Taguchi method uses linear regression to obtain, the slope between the quality characteristic and the signal factor. Additionally, the deviation from the regression line of the quality characteristic represents the variability of the dynamic system (namely2e). Both and 2e can be obtained using the following equations, respectively13:
= m i=1 n j=1(yi jMi) m i=1 n j=1Mi2 (2) 2 e= 1 mn−1 m i=1 n j=1 (yi j−Mi)2 (3)
where represents the slope between the quality characteristic and the signal factor, and 2e indicates the variability of a dynamic system. Mi denotes the i th signal factor, i=1,...,m. yi j is the quality characteristic
with the combination of the i th signal factor, i=1,...,m, and the jth noise factor, j =1,...,n.
Taguchi applies the SNR as a quality performance measure. The SNR of a dynamic system is defined as in the following equation. A higher SNR implies a higher quality.
=10log2
2 e
(4)
3.
DATA ENVELOPMENT ANALYSIS
DEA can clarify the complex relationship between inputs and outputs. Therefore, DEA is effective for assessing the relative efficiency in a group of decision-making units (DMUs)14. Charnes et al.15introduced a DEA model based on the concept of weighted virtual inputs and outputs, referred to as the CCR model. Subsequently, Banker et al.16 released the limitation of constant returns to scale in the CCR model and presented an alternative DEA model, referred to as the BCC model. Both models are widely used in practical applications.
The DEA model evaluates the relative efficiency of each DMU by using the ratio of weighted virtual output to weighted virtual input. Weights in the DEA model are obtained from the data rather than given in advance. Therefore, each DMU is assigned the optimum set of weights with values that may vary from one DMU to another. By selecting the weight that maximizes the relative efficiency value of the evaluated DMU, the relative efficiency value of the kth DMU can obtained using the following equation:
hk= rurYkr iviXki (5) where hk denotes the relative efficiency value of the kth DMU, Xki represents the i th input of the kth
DMU, andvi is the corresponding weight. Additionally, Ykr denotes the r th output of the kth DMU, and ur
represents the corresponding weight.
The CCR model maximizes the relative efficiency value of the kth DMU by analyzing a reference set of DMUs. The CCR model is represented as
Max hk= rurYkr iviXki s.t. rurYqr iviXqi ≤1, q =1,..., L ur≥, vi≥ (6)
where hk denotes the relative efficiency value of the kth DMU; Ykr and Xki represent the r th output and
the i th input, respectively, for the kth evaluated DMU; Yqr and Xqi are the r th output and the i th input,
respectively, for the qth DMU, q=1,..., L; ur denotes the r th output weight;vi represents the i th input
weight, and is a non-Archimedean constant, representing an extremely small positive value14. The first constraint in Equation (6) ensures that the relative efficiency value hk does not exceed 1, whereas the second
constraint restricts the CCR model to yield a positive weight.
As a nonlinear programming model may cause a complex procedure to derive a solution for Equation (6), Equation (6) can be transformed into a linear programming model, as shown in the following equation, to simplify the calculation:
Max hk= r urYkr s.t. i viXki=1 r urYqr− i viXqi≤0, q =1,..., L ur≥, vi≥ (7)
According to Equation (7), the maximum relative efficiency value of the kth DMU (denoted by h∗k) can be determined. A high h∗k value corresponds to a superior performance among a set of DMUs, and the value of h∗k is between 0 and 1. When h∗k=1, the kth DMU locates on the efficient frontier, which is a dominant solution for all evaluated DMUs.
4.
PROPOSED METHOD
While focusing mainly on a dynamic system with multiple quality characteristics, this study presents a novel optimization procedure for enhancing product/process quality using DEA. Initially, the designed experiments for a dynamic system based on an orthogonal array are determined using the Taguchi method. Experiments are thereby conducted to obtain the response values. The proposed optimization procedure comprises the following six steps:
Step 1: Design experiments for each quality characteristic. This study uses an orthogonal array to design
a dynamic system experiment. Control factors are located on the inner orthogonal array, and the signal factor level and noise factor level combinations are located on the outer orthogonal array. Accordingly, all experimental runs are conducted, and the corresponding response values are collected.
Step 2: For each quality characteristic, estimate the relative efficiency of the location effect for each DMU. Each treatment in the orthogonal array is regarded as a DMU when applying DEA. Additionally,
experimental observations in each DMU are set as outputs, revealing location measures in a dynamic system, whereas is set as the input. As (i.e. input) is identical in its evaluation of the relative efficiency for each competing DMU, the returns to scale effect on relative efficiency is a constant. Therefore, this study uses the CCR model to determine relative efficiency for evaluating quality performance in a dynamic system. The relative efficiency of the location effect for the kth DMU can be calculated as follows:
Max hkr= m i=1 n j=1 ui jryi j kr s.t. vrXkr=1 m i=1 n j=1 ui jryi j qr−vrXqr≤0, q =1,..., L ∀Xqr∈{}, q =1,..., L ui jr≥, vr≥ (8)
where yi j qr is an experimental observation of the r th quality characteristic, r=1,...,s, for the qth DMU
with the combination of the i th signal factor level, i=1,...,m, and the jth noise factor level, j =1,...,n. Additionally, yi j kr denotes a specific value of yi j qr under the condition of q=k. hkr represents the relative
efficiency value of the location effect of the r th quality characteristic for the kth DMU (i.e. the kth treatment);
Xkr is the corresponding input value; ui jr denotes an output weight of the r th quality characteristic with
the combination of the i th signal factor level and the j th noise factor level;vr represents an input weight
of the r th quality characteristic; and Xqr is an input of the r th quality characteristic for the qth DMU.
Step 3: For each quality characteristic, estimate the relative efficiency of the dispersion effect for each DMU. Taguchi’s dynamic system assumes a linear relationship between a quality characteristic and a signal
factor. Accordingly, the predicted error is utilized as a measure of the dispersion effect. However, in actual practice, the relationship between a quality characteristic and a signal factor is never linear. The Taguchi method thus erroneously estimates dispersion effects. Therefore, this study applies the range of observations in each treatment (i.e. each DMU) as a dispersion effect of a dynamic system. The inverse dispersion effect is defined as dqr= m i=1 max j {yi j qr}−minj {yi j qr} −1 , q =1,..., L (9)
where dqr represents the inverse dispersion effect of the qth DMU under the r th quality characteristic. A
large dqr value produces a low dispersion effect in a dynamic system. Accordingly, the relative efficiency
for the dispersion effect of the r th quality characteristic under the kth DMU can be determined based on the constructed CCR model as follows:
Max hkr=urdkr s.t. vrXkr=1 urdqr−vrXqr≤0, q =1,..., L ∀Xqr∈{}, q =1,..., L ur≥, vr≥ (10)
where hkr is the relative efficiency value of the dispersion effect of the r th quality characteristic for the kth DMU (i.e. the kth treatment). dkr denotes the inverse dispersion effect of the r th quality characteristic under
the specific kth DMU; Xkr represents the corresponding input (i.e.); and dqr is the inverse dispersion effect
of the r th quality characteristic under the qth DMU. Xqr is an input of the r th quality characteristic under
the qth DMU; and ur and vr are the output and input weights of the dispersion effect for the kth DMU,
respectively.
Step 4: Calculate the OQP for each DMU of the system. Robust product/process design attempts to
minimize the variance of a response and to bring the mean response close to the target. Therefore, the location effect and the dispersion effect representing the mean performance and variance of a response, respectively, are utilized to establish the OQP of multiple dynamic quality systems. The formula of OQP is as follows: Oq= s r=1 ((hqrhqr)1/2)r 1/sr=1r , q =1,..., L (11)
where Oq denotes the OQP value of the qth DMU in the dynamic system, hqr represents a location effect
of the qth DMU obtained from Step 2, and hqr indicates a dispersion effect of the qth DMU obtained from Step 3. Additionally,ris a weight of the r th quality characteristic, indicating the relative importance among
multiple quality characteristics. A high OQP value yields a superior performance in a multiple dynamic quality system.
Step 5: Establish a prediction model of OQP on design factors. According to the OQP value (i.e. Oq) of
each DMU obtained in Step 4, an appropriate prediction model of OQP on design factors can be determined as follows: OQP= f (Cz)=b0+ g z=1 bz,w(Cz)w (12)
where b0 represents the intercept; Cz represents the level condition of a control factor, z=1,..., g, for a
certain factor-level combination;(Cz)wdenotes Czto thewth power, w =1,2,...; and bz,wis the regression
coefficient of(Cz)w.
Step 6: Determine the optimal product/process condition. Because a large OQP value corresponds to a high
OQP, the optimal product/process condition can be derived from the feasible region of design factor-level settings in the following equation:
Max OQP= f (Cz)
s.t. 0≤OQP≤100%
when Czimplies a categorical level:
Cz= g z=1 zDz g z=1 Dz=1 (13) Dz=0 or 1
when Czimplies a continuous level:
1≤Cz≤Max{z}
Cz≥0, z =1,..., g
where Dz is a dummy variable, andz is all initial designed factor-level conditions for a control factor,
z=1,..., g.
5.
ILLUSTRATIONS
The following two cases demonstrate the effectiveness of the proposed procedure.
Case 1: Optimizing the process condition of a temperature control circuit. Wu and Yeh17 simulated
experimental observations for a temperature control circuit introduced by Tomishima18. The dynamic system involves two equally important quality characteristics, RT-ON and RT-OFF. Four control factors
( A, B,C, and D), each with three levels, are located on the inner orthogonal array. A signal factor and a
noise factor, each with three levels, are located on the outer orthogonal array. The L18orthogonal array was used to design the experiments. Further details of this case can be found in Tomishima18and Wu and Yeh17. These experimental data for the temperature control circuit were reanalyzed using the proposed optimization procedure. The optimization results were compared with those of the conventional Taguchi method and Wu and Yeh17. Initially, the relative efficiencies of location effects(hkr) and dispersion effects (hkr) with respect to each quality characteristic RT-ON and RT-OFF were determined using Equations (8) and (10). Table I lists the calculations. Accordingly, the OQP of DMUs was derived using Equation (11). For example, the response values of the first experimental run for the first quality characteristic (i.e. RT-ON), yi j 11,
Table I. The efficiencies of location effects and dispersion effects with respect to RT-ON and RT-OFF
Control factors RT-ON (y1) RT-OFF(y2)
No. A B C D hkr (%) hkr (%) hkr (%) hkr (%) 1 1 1 1 1 41.51 24.74 44.45 49.81 2 2 2 2 2 36.61 33.45 44.45 49.81 3 3 3 3 3 33.81 40.70 44.45 49.81 4 1 1 2 3 31.76 49.30 47.15 44.44 5 2 2 3 1 36.29 41.68 47.15 44.44 6 3 3 1 2 51.48 19.32 37.00 72.77 7 1 2 1 2 58.73 21.89 61.70 38.98 8 2 3 2 3 52.51 28.14 61.70 38.98 9 3 1 3 1 16.28 100.00 21.85 92.32 10 1 3 3 2 76.05 20.10 100.00 22.14 11 2 1 1 3 23.86 66.97 29.63 74.72 12 3 2 2 1 29.95 42.06 29.63 74.72 13 1 2 3 3 46.34 38.67 70.73 29.62 14 2 3 1 1 100.00 7.12 55.49 48.50 15 3 1 2 2 16.67 96.90 20.96 100.00 16 1 3 2 1 100.00 10.38 92.55 25.98 17 2 1 3 2 21.71 81.70 32.77 61.51 18 3 2 1 3 26.77 52.91 27.42 87.68
Table II. The OQP value of each DMU Control factors No. A B C D Oq (%) 1 1 1 1 1 38.831591 2 2 2 2 2 40.578505 3 3 3 3 3 41.778897 4 1 1 2 3 42.559429 5 2 2 3 1 42.193214 6 3 3 1 2 40.452926 7 1 2 1 2 41.933218 8 2 3 2 3 43.418346 9 3 1 3 1 42.569682 10 1 3 3 2 42.891203 11 2 1 1 3 43.369040 12 3 2 2 1 40.865673 13 1 2 3 3 44.017941 14 2 3 1 1 37.205669 15 3 1 2 2 42.895592 16 1 3 2 1 39.746897 17 2 1 3 2 43.483652 18 3 2 1 3 42.957454
Wu and Yeh17. The relative efficiency value of the location effect of RT-ONfor the first DMU, h11=41.51%, can be obtained using Equation (8). Similarly, h11=24.74%, h12=44.45%, and h12=49.81% can be obtained using Equations (8) and (10), respectively. Finally, Oq=38.831591% for the first experimental
run in Table II is obtained using Equation (11). The weights of responses RT-ON and RT-OFF were set to 1 (i.e. in Equation (11),1=1 and 2=1), as the two quality characteristics are equally important. Table II lists the OQP values of each DMU, indicating an overall quality measure in a dynamic multi-response system.
According to the OQP values in Table II, the prediction model of OQP on design control factors was established by employing the stepwise regression approach as
OQP= 0.386672414171712+0.0042290508948704× A
−0.000421877582588344× A2+0.0165591668577828× B
−0.00616582342127432× B2+0.00503301213499012×C × D (14) Given that a large OQP value yields a superior quality performance for a dynamic system, the optimal factor-level combination is determined as A3B1C3D3 using Equations (13) and (14). Additionally, and 2e of
the dynamic system were derived using Equations (2) and (3). Table III lists and 2e values. These values were used to establish the quadratic regression models of and 2e on designed factors, respectively, as
For response RT-ON(y1): ˆ1= 4.007057−0.385862× A+0.04482× A2+2.838296× B +0.436832× B2−1.774904×C +0.233345×C2−1.800993× D +0.328771× D2−0.625026× AB +0.117892× AC −0.36615× BC +0.138594× AD−0.548080× B D+0.442152×C D (15) log(ˆ21e) = −0.620458−0.234226× A−0.030201× A2+0.576154× B +0.045826× B2−0.561984×C +0.035376×C2−0.570838× D +0.056997× D2+0.02669× AB +0.004556× AC −0.055985× BC +0.011346× AD−0.09495× B D+0.143968×C D (16)
Table III. The values ofand2e for each quality characteristic
Control factors RT-ON (y1) RT-OFF(y2)
No. A B C D 1 21e 2 22e 1 1 1 1 1 3.096 0.055 1.667 0.003 2 2 2 2 2 2.699 0.030 1.667 0.003 3 3 3 3 3 2.475 0.020 1.667 0.003 4 1 1 2 3 2.308 0.014 1.765 0.004 5 2 2 3 1 2.642 0.019 1.765 0.004 6 3 3 1 2 3.839 0.090 1.380 0.001 7 1 2 1 2 4.318 0.070 2.308 0.005 8 2 3 2 3 3.828 0.042 2.308 0.005 9 3 1 3 1 1.181 0.003 0.817 0.001 10 1 3 3 2 5.534 0.083 3.751 0.014 11 2 1 1 3 1.733 0.007 1.111 0.001 12 3 2 2 1 2.205 0.019 1.111 0.001 13 1 2 3 3 3.345 0.022 2.648 0.008 14 2 3 1 1 7.434 0.665 2.069 0.003 15 3 1 2 2 1.211 0.004 0.785 0.001 16 1 3 2 1 7.461 0.312 3.462 0.010 17 2 1 3 2 1.568 0.005 1.225 0.002 18 3 2 1 3 1.956 0.012 1.026 0.001
For response RT-OFF (y2): ˆ2= 1.706315−0.781251× A+0.158126× A2+0.451129× B +0.101454× B2+0.175218×C −0.008988×C2+0.101215× D −0.018349× D2−0.2249× AB −0.067128× AC +0.094456× BC +0.007511× AD−0.001094× B D−0.027667×C D (17) log(ˆ22e) = −2.551005−0.359034× A+0.000429× A2+0.236008× B −0.02061× B2+0.127796×C −0.020476×C2−0.004788× D +0.000532× D2−0.001766× AB +0.001689× AC +0.038189× BC +0.001946× AD+0.00172× B D−0.00165×C D (18) Instead of2e, log(2e) is employed to establish a regression model to satisfy the assumptions of a regression model. Accordingly, the predicted values of and 2e and the corresponding SNR under full factorial level combinations can be obtained. As the responses RT-ONand RT-OFFare equally important, the average SNR is determined as an overall quality measure for the temperature control circuit system. Table AI in Appendix lists these calculations. The factor-level combination A3B1C3D3was determined as an optimal one based on Table AI in Appendix. Notably, this result is the optimum solution among all factor-level combinations of a full factorial design and is identical to the optimal factor-level combination obtained from the proposed procedure. Table IV compares the proposed procedure using DEA with the conventional Taguchi method, and that of Wu and Yeh17. The optimization of the proposed procedure is consistent with the results of Wu and Yeh17, thus verifying the effectiveness of the proposed procedure.
Case 2: Biological reduction in ethyl 4-chloro acetoacetate to produce an optically pure compound. A
dynamic system with multiple responses from Tong et al.12involves optimizing a procedure for biologically reducing ethyl 4-chloro acetoacetate to produce an optically pure compound. The responses S-CHBE and R-CHBE, which are two reactions produced by this procedure, are used as two quality characteristics. A large value is desired for response S-CHBE, whereas a small value is desired for response R-CHBE. Eight control factors ( A, B,C, D, E, F,G, and H), one signal factor with three levels, and one noise factor with two levels were designed. Each control factor has three levels, except for factor A, which has two levels.
The response values of S-CHBE and R-CHBE were analyzed using the proposed procedure. Accordingly, the location effects, dispersion effects, and OQP values associated with each treatment in dynamic experi-ments were obtained using Equations (8), (10), and (11). The weights of the responses were set to equal 1 (namely1=1 and 2=1) as both S-CHBE and R-CHBE are equally important. Table V shows the location effects and dispersion effects, and Table VI lists the OQP values. Based on the OQP values in Table VI, the prediction model of OQP on eight design factors obtained by the stepwise regression approach is
Table IV. Comparisons of different optimal approaches
RT-ON(y1) RT-OFF(y2)
Method ˆ1 ˆ21e SNR1(1) ˆ2 ˆ22e SNR2(2) Average SNR Taguchi method 2.653600 0.012226 27.603794 1.846381 0.004068 29.233044 28.418419 Wu and Yeh17 2.529299 0.003358 32.799792 0.741390 0.000818 28.271109 30.535451 Proposed procedure 2.529299 0.003358 32.799792 0.741390 0.000818 28.271109 30.535451
Table V. The efficiencies of location effects and dispersion effects with respect to S-CHBE and R-CHBE
Control factors S-CHBE(y1) R-CHBE(y2)
No. A B C D E F G H hkr (%) hkr (%) hkr (%) hkr (%) 1 1 1 1 1 1 1 1 1 100.00 51.13 86.11 13.17 2 1 1 2 2 2 2 2 2 92.88 31.54 100.00 26.82 3 1 1 3 3 3 3 3 3 93.59 28.04 94.23 100.00 4 1 2 1 1 2 2 3 3 100.00 25.62 100.00 7.88 5 1 2 2 2 3 3 1 1 100.00 42.35 80.81 11.68 6 1 2 3 3 1 1 2 2 78.10 43.31 100.00 10.37 7 1 3 1 2 1 3 2 3 84.92 27.67 100.00 6.03 8 1 3 2 3 2 1 3 1 85.36 57.99 92.63 29.98 9 1 3 3 1 3 2 1 2 100.00 32.02 68.13 31.44 10 2 1 1 3 2 2 2 1 88.02 31.24 92.18 18.33 11 2 1 2 1 1 3 3 2 91.96 32.31 73.35 16.15 12 2 1 3 2 3 1 1 3 85.03 39.44 100.00 21.46 13 2 2 1 2 3 1 3 2 95.12 29.86 86.11 43.74 14 2 2 2 3 1 2 1 3 72.29 48.59 100.00 12.09 15 2 2 3 1 2 3 2 1 100.00 34.51 72.77 29.94 16 2 3 1 3 2 3 1 2 72.25 52.44 86.15 18.40 17 2 3 2 1 3 1 2 3 92.84 27.00 61.21 49.93 18 2 3 3 2 1 2 3 1 91.90 100.00 90.70 10.56 given as OQP= 0.728689051908353−0.108979662055521× A−0.303333300226461× B +0.0530479728745114× B2+0.0477032647622895×C −0.00267205243659638×C2−0.0448906596024196× D +0.0157781619586083× D2+0.0555060917557161× A× B +0.0312508718770257× E ×G −0.0159688120501931×G × H (19) The optimal factor-level combination A1B1C3D1E3F1G3H1was determined using Equations (13) and (19). Additionally, the values of and 2e were calculated using Equations (2) and (3). Table VII lists and
2
e values. These values were used to establish the regression models of and log(2e) on designed factors
as follows:
For response S-CHBE(y1):
ˆ1= 0.565615−0.135774× A+0.077371× B −0.034239× B2 +0.045958×C +0.022482×C2+0.097202× D+0.002716× D2 −0.124232× E +0.031141× E2+0.049376× F −0.014238× F2 −0.0563×G +0.020129×G2−0.067481× H +0.014431× H2 +0.027852× AB −0.068517×C D (20) log(ˆ21e) = −0.174523+0.623949× A+0.335465× B −0.011088× B2 −0.97924×C +0.097114×C2−0.888545× D+0.029706× D2 −0.308284× E +0.103635× E2+0.333492× F −0.049443× F2 +0.185438×G −0.063798×G2−0.00995× H −0.032182× H2 −0.242662× AB +0.283379×C D (21)
Table VI. The OQP value of each DMU Control factors No. A B C D E F G H Oq (%) 1 1 1 1 1 1 1 1 1 49.071438 2 1 1 2 2 2 2 2 2 52.943258 3 1 1 3 3 3 3 3 3 70.517866 4 1 2 1 1 2 2 3 3 37.694357 5 1 2 2 2 3 3 1 1 44.713672 6 1 2 3 3 1 1 2 2 43.276736 7 1 3 1 2 1 3 2 3 34.501193 8 1 3 2 3 2 1 3 1 60.890242 9 1 3 3 1 3 2 1 2 51.175320 10 2 1 1 3 2 2 2 1 46.427242 11 2 1 2 1 1 3 3 2 43.313885 12 2 1 3 2 3 1 1 3 51.794626 13 2 2 1 2 3 1 3 2 57.190449 14 2 2 2 3 1 2 1 3 45.395505 15 2 2 3 1 2 3 2 1 52.364539 16 2 3 1 3 2 3 1 2 49.504355 17 2 3 2 1 3 1 2 3 52.610292 18 2 3 3 2 1 2 3 1 54.468658
Table VII. The values ofand2e for each quality characteristic
Control factors S-CHBE(y1) R-CHBE(y2)
No. A B C D E F G H 1 21e 2 22e 1 1 1 1 1 1 1 1 1 0.453 0.171 0.104 0.018 2 1 1 2 2 2 2 2 2 0.422 0.047 0.122 0.011 3 1 1 3 3 3 3 3 3 0.408 0.070 0.112 0.011 4 1 2 1 1 2 2 3 3 0.461 0.116 0.108 0.010 5 1 2 2 2 3 3 1 1 0.455 0.138 0.097 0.003 6 1 2 3 3 1 1 2 2 0.376 0.053 0.140 0.016 7 1 3 1 2 1 3 2 3 0.396 0.063 0.127 0.065 8 1 3 2 3 2 1 3 1 0.395 0.024 0.106 0.007 9 1 3 3 1 3 2 1 2 0.508 0.101 0.074 0.001 10 2 1 1 3 2 2 2 1 0.405 0.067 0.106 0.004 11 2 1 2 1 1 3 3 2 0.400 0.172 0.068 0.026 12 2 1 3 2 3 1 1 3 0.361 0.097 0.108 0.024 13 2 2 1 2 3 1 3 2 0.438 0.076 0.103 0.002 14 2 2 2 3 1 2 1 3 0.350 0.034 0.137 0.012 15 2 2 3 1 2 3 2 1 0.469 0.121 0.081 0.003 16 2 3 1 3 2 3 1 2 0.347 0.026 0.113 0.013 17 2 3 2 1 3 1 2 3 0.368 0.056 0.060 0.002 18 2 3 3 2 1 2 3 1 0.427 0.051 0.104 0.010
For response R-CHBE(y2):
ˆ2= 0.070285−0.04084× A+0.033234× B −0.01369× B2
−0.009693×C +0.00839×C2+0.099493× D−0.014809× D2 +0.0089× E −0.006109× E2−0.012399× F +0.001442× F2 −0.022989×G +0.005393×G2−0.004835× H +0.003962× H2
Table VIII. Comparative analysis of optimal factor-level combination S-CHBE(y1) R-CHBE(y2) Method ˆ1 ˆ21e SNR1(1) ˆ2 ˆ22e SNR2(2) Average SNR Initial 0.383167 0.074819 2.927631 0.092342 0.013715 −2.063979 0.431826 Taguchi method 0.356916 0.050206 4.043736 0.060674 0.001191 4.899849 4.471793 Tong et al.12 0.533061 0.120447 3.727584 0.063975 0.000558 8.654087 6.190836 Proposed procedure 0.637323 0.049053 9.180519 0.094251 0.000196 16.566749 12.873634 log(ˆ22e) = −1.29594+0.862421× A−0.931252× B +0.352817× B2 −1.306126×C +0.082039×C2+0.340233× D−0.323628× D2 +1.216981× E −0.387937× E2−0.702966× F +0.21409× F2 −0.45689×G +0.071235×G2+0.387471× H −0.057581× H2 −0.324895× AB +0.507063×C D (23)
To further verify the effectiveness of the proposed procedure, the optimization result was compared with that of Taguchi’s single-response method and the procedure developed by Tong et al.12. Table VIII lists these comparisons. According to Table VIII, the average SNR of the proposed procedure is 12.873634 dB, i.e. higher than that of the Taguchi method with an SNR value of 4.471793 dB, and the procedure of Tong
et al.12with an SNR value of 6.190836 dB. The effectiveness of the proposed procedure was thus further
verified. Therefore, the proposed optimization procedure using DEA obtains a more robust dynamic system than available procedures do.
6.
CONCLUSIONS
This study describes a novel optimization procedure for a dynamic system with multiple responses using DEA. The relative efficiencies of location effects and dispersion effects resulting from DEA are used as quality performance measures for the product/process mean and variance, respectively. Therefore, capable of simultaneously accounting for the location effects and the dispersion effects for multiple quality character-istics, OQP can also be treated as an overall quality measure for a dynamic system with multiple responses. Consequently, a prediction model of OQP on design factors can be established to determine the optimal process condition.
In contrast with the conventional Taguchi method, the proposed procedure is not only effective for multiple quality characteristics, but also requires no assumptions between quality characteristics and signal factors. Moreover, the optimal process condition obtained from the proposed procedure is not restricted to the initially designed factor-level combinations. Therefore, the proposed procedure can determine the optimal process condition more efficiently than the Taguchi method and other optimization procedures can.
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APPENDIX A
As the responses RT-ON and RT-OFF are equally important, the average SNR is determined as an overall quality measure for the temperature control circuit system. Table AI lists these calculations. The factor-level combination A3B1C3D3was determined as an optimal one based on Table AI.
Table AI. Predicted SNR for all factor-level combinations of a full factorial design for Case 1
A B C D ˆ1 ˆ21e SNR1(1) ˆ2 ˆ22e SNR2(2) Average SNR 1 1 1 1 3.087 0.054 22.467 1.666 0.003 29.944 26.206 1 1 1 2 2.305 0.025 23.325 1.691 0.003 30.085 26.705 1 1 1 3 2.180 0.015 25.097 1.679 0.003 30.025 27.561 1 1 2 1 2.206 0.023 23.182 1.814 0.004 29.637 26.409 1 1 2 2 1.866 0.015 23.683 1.811 0.004 29.652 26.668 1 1 2 3 2.184 0.012 25.864 1.772 0.004 29.478 27.671 1 1 3 1 1.791 0.012 24.300 1.944 0.004 29.601 26.951 1 1 3 2 1.894 0.011 25.298 1.913 0.004 29.509 27.404 1 1 3 3 2.654 0.012 27.604 1.846 0.004 29.233 28.418 1 2 1 1 5.696 0.210 21.895 2.290 0.005 30.584 26.239
Table AI. Continued. A B C D ˆ1 ˆ21e SNR1(1) ˆ2 ˆ22e SNR2(2) Average SNR 1 2 1 2 4.366 0.077 23.930 2.314 0.005 30.668 27.299 1 2 1 3 3.694 0.037 25.681 2.301 0.005 30.604 28.142 1 2 2 1 4.449 0.080 23.942 2.532 0.006 30.030 26.986 1 2 2 2 3.561 0.041 24.913 2.529 0.006 30.028 27.470 1 2 2 3 3.331 0.027 26.097 2.488 0.006 29.888 27.992 1 2 3 1 3.669 0.036 25.752 2.757 0.008 29.749 27.750 1 2 3 2 3.223 0.026 26.092 2.725 0.008 29.676 27.884 1 2 3 3 3.435 0.024 26.970 2.657 0.008 29.473 28.221 1 3 1 1 9.179 1.006 19.229 3.117 0.007 31.550 25.390 1 3 1 2 7.301 0.297 22.535 3.140 0.007 31.591 27.063 1 3 1 3 6.081 0.114 25.100 3.126 0.007 31.519 28.309 1 3 2 1 7.566 0.337 22.303 3.454 0.010 30.632 26.468 1 3 2 2 6.130 0.139 24.329 3.449 0.010 30.613 27.471 1 3 2 3 5.352 0.074 25.864 3.407 0.010 30.491 28.178 1 3 3 1 6.420 0.133 24.921 3.773 0.014 29.999 27.460 1 3 3 2 5.426 0.076 25.875 3.740 0.014 29.934 27.904 1 3 3 3 5.089 0.057 26.594 3.671 0.014 29.771 28.182 2 1 1 1 2.467 0.028 23.342 1.075 0.001 29.694 26.518 2 1 1 2 1.823 0.013 23.999 1.107 0.001 29.945 26.972 2 1 1 3 1.838 0.008 26.207 1.103 0.001 29.893 28.050 2 1 2 1 1.704 0.012 23.715 1.155 0.002 29.261 26.488 2 1 2 2 1.502 0.008 24.465 1.160 0.002 29.306 26.885 2 1 2 3 1.959 0.007 27.470 1.128 0.002 29.061 28.266 2 1 3 1 1.407 0.006 24.935 1.218 0.002 29.068 27.001 2 1 3 2 1.648 0.006 26.710 1.195 0.002 28.928 27.819 2 1 3 3 2.547 0.007 29.751 1.136 0.002 28.498 29.125 2 2 1 1 4.451 0.116 22.309 1.474 0.002 30.332 26.320 2 2 1 2 3.260 0.044 23.834 1.505 0.002 30.489 27.162 2 2 1 3 2.726 0.022 25.371 1.500 0.002 30.423 27.897 2 2 2 1 3.322 0.045 23.915 1.649 0.003 29.863 26.889 2 2 2 2 2.573 0.024 24.485 1.653 0.003 29.874 27.180 2 2 2 3 2.481 0.016 25.821 1.620 0.003 29.679 27.750 2 2 3 1 2.660 0.020 25.422 1.806 0.004 29.619 27.521 2 2 3 2 2.352 0.015 25.708 1.782 0.004 29.511 27.609 2 2 3 3 2.703 0.014 27.125 1.722 0.004 29.207 28.166 2 3 1 1 7.309 0.594 19.539 2.076 0.003 31.613 25.576 2 3 1 2 5.570 0.180 22.359 2.106 0.003 31.697 27.028 2 3 1 3 4.488 0.071 24.523 2.099 0.003 31.617 28.070 2 3 2 1 5.814 0.201 22.259 2.345 0.005 30.848 26.553 2 3 2 2 4.517 0.085 23.806 2.348 0.005 30.832 27.319 2 3 2 3 3.877 0.047 25.080 2.314 0.005 30.668 27.874 2 3 3 1 4.785 0.080 24.567 2.597 0.006 30.316 27.441 2 3 3 2 3.930 0.047 25.158 2.572 0.006 30.223 27.690 2 3 3 3 3.732 0.036 25.870 2.510 0.006 29.992 27.931 3 1 1 1 1.936 0.013 24.666 0.800 0.001 30.676 27.671 3 1 1 2 1.432 0.006 25.211 0.839 0.001 31.072 28.141 3 1 1 3 1.584 0.004 28.119 0.843 0.001 31.068 29.593 3 1 2 1 1.291 0.006 24.689 0.813 0.001 29.743 27.216 3 1 2 2 1.229 0.004 25.985 0.825 0.001 29.863 27.924 3 1 2 3 1.823 0.003 30.003 0.801 0.001 29.580 29.792 3 1 3 1 1.113 0.003 26.231 0.809 0.001 29.027 27.629 3 1 3 2 1.492 0.003 29.068 0.793 0.001 28.866 28.967 3 1 3 3 2.529 0.003 32.800 0.741 0.001 28.271 30.535 3 2 1 1 3.296 0.056 22.858 0.974 0.001 30.300 26.579 3 2 1 2 2.243 0.022 23.633 1.013 0.001 30.595 27.114 3 2 1 3 1.848 0.011 24.926 1.015 0.001 30.558 27.742
Table AI. Continued. A B C D ˆ1 ˆ21e SNR1(1) ˆ2 ˆ22e SNR2(2) Average SNR 3 2 2 1 2.285 0.022 23.777 1.082 0.001 29.752 26.765 3 2 2 2 1.674 0.012 23.753 1.093 0.001 29.814 26.783 3 2 2 3 1.721 0.008 25.530 1.067 0.001 29.570 27.550 3 2 3 1 1.740 0.010 24.806 1.172 0.002 29.396 27.101 3 2 3 2 1.571 0.008 25.158 1.155 0.002 29.261 27.210 3 2 3 3 2.060 0.007 27.610 1.102 0.002 28.830 28.220 3 3 1 1 5.529 0.305 20.007 1.351 0.001 31.467 25.737 3 3 1 2 3.928 0.095 22.105 1.388 0.001 31.645 26.875 3 3 1 3 2.985 0.038 23.646 1.390 0.001 31.579 27.613 3 3 2 1 4.152 0.104 22.180 1.553 0.002 30.837 26.509 3 3 2 2 2.993 0.045 22.965 1.563 0.002 30.848 26.906 3 3 2 3 2.492 0.026 23.860 1.537 0.002 30.643 27.252 3 3 3 1 3.241 0.042 23.983 1.738 0.003 30.378 27.181 3 3 3 2 2.524 0.025 24.000 1.720 0.003 30.262 27.131 3 3 3 3 2.465 0.020 24.842 1.666 0.003 29.944 27.393 Authors’ biographies
Lee-Ing Tong is a Professor in the Department of Industrial Engineering and Management, National Chiao
Tung University, Taiwan, Republic of China. She has published several journal articles in the areas of quality engineering.
Chung-Ho Wang is a Professor in the Department of Power Vehicle and Systems Engineering, Chung
Cheng Institute of Technology, National Defense University, Taiwan, Republic of China. He has published several journal articles in the areas of quality engineering.
Chih-Wei Tsai is pursuing his PhD degree in the Department of Industrial Engineering and Management,
National Chiao Tung University, Taiwan, Republic of China. He received an MBA in Industrial and Infor-mation Management at National Cheng Kung University, Taiwan, Republic of China.
