A DEA-based multi-response fusion model in the context of Taguchi method

Taguchi methods have been widely used to optimize machining parameters of a manufacturing process so that the product performances or responses are close to the desired targets and not sensitive to external noises. For single-response problems, the optimal machining parameters can be obtained based on Taguchi signal-to-noise ratio or response surface models. For multi-response problems, a multi-objective optimization approach or a fusion model is needed before obtaining the optimal parameters; and the latter is preferred due to its simplicity. This paper proposes a fusion model to combine multiple responses into a composite response variable. The proposed model first transforms the response data obtained from Taguchi experiments into the data with smaller-the-better or larger-the-better quality characteristics, and then combines the transformed data into the composite response variable in the way similar to Data Envelopment Analysis (DEA). Once the fusion model is built, the problem reduces into a single-response problem, which can solved using existing approaches. The proposed model can be easily implemented using a spreadsheet program; and one real-world example is included to illustrate its appropriateness.


Introduction
Taguchi method aims to obtain an optimal combination of design parameter levels so that the performances (or responses) of a product are close to the desired targets and not sensitive to external noises [1-2]. Due to its excellent idea, Taguchi method has been widely recognized and used in all phases of product life cycle [3].
For single-response problems, the optimal design parameters can be obtained based on Taguchi signal-to-noise ratio (SNR). Some drawbacks resulting from the SNR have been identified (e.g., see [4][5]) and various improvements have been developed in two main directions: modifications of the SNR (e.g., [6]) and development of response surface models (e.g., [7][8]). As a result, there are a number of methods or models can be used to solve the single-response problem.
However, many practical optimization problems in the context of Taguchi method involve multiple response variables [9][10][11][12][13]. In this case, it is not easy to obtain the optimal design variables; and a multi-objective optimization approach or a performance fusion model is needed before obtaining the optimal parameters. The fusion approach is preferred due to its simplicity. Through a fusion model, multiple response variables are combined into a single composite response variable so that a multiresponse problem reduces into a single-response problem.
A typical method to build a fusion model is the generalized mean model, which includes the weighted sum models (e.g., see [14][15]  2 approach needs to specify some cost parameters, and this may be a hard task. The idea of Data Envelopment Analysis (DEA) also has been used to build a fusion model. DEA uses a weighted sum model to combine smaller-the-better (SB) characteristic variables into an input and to combine largerthe-better (LB) characteristic variables into an output. An efficiency measure is defined as the ratio of the output and input (e.g., [17][18]). However, a problem with the DEA model is that it cannot directly include nominal-the-best (NB) characteristic variables in the model. To address this issue, and a new transformation method can be introduced to transform a NB characteristic variable into a SB characteristic variable so that the transformed variable can be viewed as an input.
The use of the weighted sum model involves data transformation and determination of the weight parameters. The weight parameters can be are mathematically determined from the experiment data. For example, Emovon et al. determine the weights under the assumption that the weight is proportional to the sample variance [19]; Jiang et al. determine the weights under the assumption that the weight of a variable (or criterion) is proportional to the coefficient of variation (cv) of its sample [20]. However, for some data transformation methods, the sample variance or cv changes after the transformation. This raises another problem with the DEA model: how the compatibility between the data transformation method and the method to determine the weight is ensured. To address this issue, and a proportional transformation is proposed to normalize the SB or LB response data. The proportional transformation has two important features. The first is that the transformed response variables have the same average and hence the transformation is called the equal-magnitude transformation. This feature ensures that the weight is not distorted by the "magnitudes" of the response variables. The second is that the cv values of a response variable before and after the transformation are the same. Th is feature ensures the above-mentioned compatibility. Based on these two transformations, a DEA-based fusion model is proposed in this paper. The proposed model can be easily implemented using a spreadsheet program; and one real-world example is included to illustrate its appropriateness.
The paper is organized as follows. The proposed model is presented in Section 2 and illustrated in Section 3. The paper is concluded in Section 4.
Taguchi method is based on orthogonal array experiments; and the array L N is selected by m and the number of levels of X l , where N is the number of experiments. For each experiment, there are M trials; and the total number of trials is . The process to construct the proposed fusion model includes four steps, and specific details are outlined as follows.

Step 1: transform NB responses into SB responses
For a NB response variable Y * , let T denote its target value. The following transformation makes it into a SB variable: 1 * ⁄ . (1)

2.2.
Step 2: normalize the experiment data by proportional transformation Without loss of generality, suppose that n performance measures, (P 1 , P 2 , …, P n ), are SB. Let p i denote the measurement value of P i for a trial. The overall performance of the trial can be evaluated by ∑ , 0 1, ∑ 1.
(2) Where w i 's are the weight parameters, and can be determined under the assumption that the weight is proportional to cv; / , where μ i is the sample mean of P i . Clearly, q i is dimensionless and can be viewed as a realization of Q i , given by ⁄ .
(4) Equation (4) implies that the transformation is of equal-magnitude and constant-cv. These properties are particularly desired due to the above-mentioned reasons.

Step 3: aggregation of responses with the same characteristic
The proportionally transformed SB variables are aggregated into a composite SB indicator using the weighted sum model and denoted as Z S . Similarly, the proportionally transformed LB variables are aggregated into a composite LB indicator, which is denoted as Z L .
The NB variables are first transformed by Equation (1), and then the transformed NB variables are normalized by Equation (3). Finally, the normalized NB variables are aggregated into a composite NB indicator using the weighted sum model, which is denoted as Z N .

Step 4: construction of fusion model
) denote the normalized output parameter values and ( , 1 ) denote the normalized input parameter values. DEA defines the relative efficiency of a decision-making unit as [17] ∑ ∑ .
Where u i 's are the weights of outputs and v i 's are the weights of inputs. For the multi-response optimization problem considered in this paper, the input is S Z + N Z and the output is L Z . Thus, the composite response variable (i.e., fusion model) can be defined as ⁄ .
(6) This definition is slightly different from Equation (5) so that Z is smaller-the-better. Once the fusion model is obtained, the optimal combination of design parameter levels can be determined using a proper Taguchi method.

Problem and earlier results
The problem deals with optimizing the welding parameters of flux-cored arc welding of stainless-steel cladding process [16]. Input variables are  wire feed rate (X 1 ),  voltage (X 2 ),  welding speed (X 3 ), and  the distance from the contact tip to the work piece (N or X 4 ).
The parameter levels of input variables are presented in Table 1. Responses are  bead width (Y 1 , LB);  penetration (Y 2 , SB);  reinforcement (Y 3 , LB);  dilution percentage (Y 4 , SB), which is equal to (penetration area of the weld)/(total area of the weld);  percentage of productivity (Y 5 , LB); and  electric current (Y 6 , SB), which will be used to calculate the energy costs of the process.
The experiment design is shown in the first five columns of Table 2; and the results of the experiments are shown in the last six columns.   [16]. Based on a principal component analysis, the composite response variable is related to the welding parameters and two second order polynomial response surface models with 15 terms (including constant term) are fitted. From the fitted models yield the optimal parameter combinations, which are shown in the second and third columns of Table 3. As seen, the optimal combination obtained from Model 1 is close to the one of Experiment 21; and the optimal MEIE 2021 Journal of Physics: Conference Series 1983 (2021) 012108 IOP Publishing doi:10.1088/1742-6596/1983/1/012108 5 combination obtained from Model 2 is close to the one of Experiment 10. Clearly, the two solutions are not quite consistent, particularly for X 3 , which is the most important parameter, as shown later. These imply that the response surface models are somehow complex and the results are not quite reliable.

Fusion model
Using the fusion model proposed in the above section, we reanalyze the data. Table 4 shows the model parameters. A large cv value implies that the corresponding response variable is sensitive to the change of controllable variables and hence is a key factor. Clearly, the most important response variable is Y 2 , the second important response variable is Y 4 , and the least important response variable is Y 5 . The values of composite response variable are computed from the resulting fusion model, and the results are shown in Figure 1. Clearly, among all the 31 experiments, Experiment 21 has the best overall performance and Experiment 10 has the second best overall performance. Their combination of parameter levels are shown in the fourth and fifth columns of Table 3. It can be noted that the combination for Experiment 21 is similar to the one of Model 1 in [16] and the combination for Experiment 10 is similar to the one of Model 2 in [16]. Therefore, Model 1 may be better than Model 2.
Where "z k " corresponds to those values of Z from the experiments with Level j for X i . Table 5 shows the values of SNR, along with the values of K. A large range of SNR implies that the response is sensitive to the corresponding input parameter and hence this input parameter is important. The last row of Table 5 shows the importance ranking. As seen, X 3 is the most important input parameter and X 1 is the least important input parameter. Since the parameter levels of X 3 associated with Models 1 and 2 of [16] are significantly different (see Table 4), the response surface models of Almeida et al. are questionable.  Figure 2 displays the plots of SNRs vs. levels of input parameters. It clearly shows that the optimal levels of X 2 and X 3 are Level 1. The optimal levels of X 1 and X 4 may be Level 4 and Level 5, respectively. This optimal combination is shown in the last column of Table 3. However, the optimal levels of X 1 and X 4 need to be confirmed. This is because the left-most and the right-most points in Figure 2 are less reliable since their K values are one. Therefore, we use a weighted least squared method (with K values as weights) to fit the data of X 1 and X 4 in Table 5 to the following relation:  The fitted curves are shown in Figure 3. According to the fitted model, the optimal level of X 1 is 4.262, which corresponds to X 1 = 10.39. This confirms that the optimal levels of X 1 is really close to Level 4. For X 4 , the SNR increases with parameter level. Therefore, the optimal level 5 is confirmed. Figure 3. Plots of fitted SNR curves for X 1 and X 4 .

Conclusions
This paper has dealt with the Taguchi experiment optimization problem with multiple responses. A multi-response fusion model has been proposed, and illustrated by a real-world example. The proposed model has the following features or advantages:  Similar to but slightly different from DEA, the proposed fusion model includes not only SB and LB characteristics but also NB characteristics.  The NB characteristics are transformed into SB characteristics by a binomial transformation; and an equal-magnitude and constant-cv transformation is introduced to normalize the experiment data.  The proposed model is simple and can be easily implemented using a spreadsheet program.
The example analyzed in this paper does not involve the NB characteristics and more examples will be carried out to further verify its appropriateness and usefulness.