A proposed satisfaction function model to optimize process performance with multiple quality responses in the Taguchi method

Abstract

The Taguchi method has been found effective for optimizing a single quality response. In reality, customers are concerned about multiple quality responses of a product. Although several approaches have been proposed to deal with this issue, however they ignored engineers’ satisfaction regarding process factor settings. This research, therefore, proposes an approach for optimizing multiple responses in the application of the Taguchi method. The mathematical relationships between each response quality and process factors are first formulated. Then, a proper satisfaction function is selected to represent each response and process factor. A complete optimization model is developed. Three case studies are provided for illustration; in all of which the proposed approach provides the largest improvement percentages while considering process engineers’ satisfaction about process factors. Compared to previous approaches in literature, such as gray analysis and fuzzy-gray analysis, the proposed approach provides optimal solution within factor setting ranges, relies on mathematical relationships between response and process factors, and considers engineers’ preference regarding process factor settings. Definitely, the proposed approach shall provide practitioners a great assistance in optimizing performance with multiple responses while considering their preferences about responses as well as process factor settings.

Introduction

In today’s escalating competition, customers are concerned about more than one quality characteristic of a product or process. The Taguchi ¹ method utilizes fractional factorial designs, or so-called orthogonal arrays, to determine the combination of factor levels that optimize a quality response of main interest. There are three main types of quality responses, including the smaller-the-better (STB), the larger-the-better (LTB), and the nominal-the-best (NTB). The Taguchi method is reported to be only efficient in optimizing business application with a single quality response. ² Therefore, optimization of multiple responses has received an increasing research attention.^3–17 Nevertheless, most of the proposed approaches in previous literature relied on nonparametric methods and ignored the satisfaction of a decision maker about process settings due to cost and time constraints or the preference on desired response values.

The concept of satisfaction functions ¹⁸ explicitly integrates the decision maker’s preferences in goal programming models in an uncertain environment and evaluates the impact of the deviations from the decision maker’s aspiration level. The general shape of the satisfaction function is presented in Figure 1, where F_i(δ_i) is the satisfaction function associated with deviations δ_i, α_id is the indifference threshold, α_io is the dissatisfaction threshold, and α_iv is the veto threshold.

Figure 1.

The general shape of the satisfaction functions.

In Figure 1, the F_i(δ_i) in the interval [0, α_id] indicates total satisfaction. Then, satisfaction decreases within the interval (α_id, α_io]. The F_i(δ_i) in the interval (α_io, α_iv] indicates dissatisfaction. Otherwise, the solution is completely unacceptable. The concept of satisfaction functions has been utilized in different applications.^19–22

The concept of satisfaction functions has been rarely reported in optimizing business performance. This research, therefore, aims at optimizing multiple responses in manufacturing applications on the Taguchi method utilizing satisfaction function model. This approach is of great importance to practitioners seeking improving a product or process performance while considering desired response targets and engineer’s satisfaction about factor settings. The remaining of this article including the introduction is organized as follows. Section “The proposed optimization approach” introduces the proposed approach. Section “Illustrations of the proposed optimization approach” provides three case studies for illustration. Section “Optimization results” summarizes research results. Finally, conclusions are made in section “Conclusion”.

The proposed optimization approach

Typically, the Taguchi method conducts I experiments with different level combinations of J controllable process factors to optimize K responses. The proposed optimization approach is outlined in the following steps:

• Step 1. Formulate mathematical relationships between the kth response and the J process factors, that is

$$y_k=f(x_1,x_2,\dots ,x_j),k=1,\dots ,K\text{and}j=1,\dots ,J$$

(1)

where f is assumed to be a linear function that relates the kth response, y_k, with the J process factors, x_j.

• Step 2. Let S, L, and N denote the number of responses of STB, LTB, and NTB, respectively. Define a proper satisfaction function to represent each response, y_k, as follows:

• 1. For the STB-type response, the proper satisfaction function is depicted in Figure 2 and is expressed as ¹³

$$F_{y_s}^{+}\left({\delta }_{y_s}^{+}\right)=\{\begin{array}{cc}\hfill 1,\hfill & \hfill 0\le {\delta }_{y_s}^{+}\le {\alpha }_{y_{s}d}^{+}\hfill \\ \multicolumn{1}{c}{\frac{{\alpha }_{y_s\upsilon }^{+}-{\delta }_{y_s}^{+}}{{\alpha }_{y_s\upsilon }^{+}-{\alpha }_{y_{s}d}^{+}},}& \hfill {\alpha }_{y_{s}d}^{+}<{\delta }_{y_s}^{+}\le {\alpha }_{y_s\upsilon }^{+}\hfill \end{array}$$

(2)

where ${\delta }_{y_s}^{+}$ denotes the positive deviation from the imprecise fuzzy value (aspiration level) of the sth response, g_ys. The $F_{y_s}^{+}({\delta }_{y_s}^{+})$ denotes the satisfaction function associated with ${\delta }_{y_s}^{+}.$ The ${\alpha }_{y_{s}d}^{+}$ and ${\alpha }_{y_s\upsilon }^{+}$ are the indifferent threshold and the veto threshold, respectively. Typically, a process engineer has a tendency to reduce the STB response to a target value of 0, thus the g_ys is usually set a value of 0 or at the lower bound of the y_s interval.

Figure 2.

The satisfaction function for STB-type response.

Furthermore, the corresponding constraints will be

$$y_s-{\delta }_{y_s}^{+}=g_{y_s}$$

(3)

$$0\le {\delta }_{y_s}^{+}\le {\alpha }_{y_s\upsilon }^{+}$$

(4)

Equation (3) indicates that the value of y_s is larger than g_ys by a value of ${\delta }_{y_s}^{+}$ . Let ${\beta }_{{y_s}_1}$ and ${\beta }_{{y_s}_2}$ be two binary variables defined as

$$\begin{array}{cc}\multicolumn{1}{c}{{\beta }_{{y_s}_1}}& =\{\begin{array}{cc}\hfill 1,\hfill & \hfill 0\le {\delta }_{y_s}^{+}\le {\alpha }_{y_{s}d}^{+}\hfill \\ \multicolumn{1}{c}{0,}& \hfill \text{Otherwise}\hfill \end{array}\text{and}\hfill \\ \multicolumn{1}{c}{{\beta }_{{y_s}_2}}& =\{\begin{array}{cc}\hfill 1,\hfill & \hfill {\alpha }_{y_{s}d}^{+}<{\delta }_{y_s}^{+}\le {\alpha }_{y_s\upsilon }^{+}\hfill \\ \multicolumn{1}{c}{0,}& \hfill \text{Otherwise}\hfill \end{array}\hfill \end{array}$$

(5)

Then, the $F_{y_s}^{+}\left({\delta }_{y_s}^{+}\right)$ given in equation (2) can be rewritten as

$$F_{y_s}^{+}\left({\delta }_{y_s}^{+}\right)={\beta }_{{y_s}_1}+\left(\frac{{\alpha }_{y_s\upsilon }^{+}}{{\alpha }_{y_s\upsilon }^{+}-{\alpha }_{y_{s}d}^{+}}\right){\beta }_{{y_s}_2}-\left(\frac{1}{{\alpha }_{y_s\upsilon }^{+}-{\alpha }_{y_{s}d}^{+}}\right){\beta }_{{y_s}_2}{\delta }_{y_s}^{+}$$

(6)

Since the process engineer seeks the largest degree of satisfaction, the objective function is to maximize $F_{y_s}^{+}\left({\delta }_{y_s}^{+}\right)$ . Still in equation (6), the formulation is nonlinear because ${\beta }_{{y_s}_2}$ and ${\delta }_{y_s}^{+}$ are variables. Hence, a linearization procedure is needed to obtain a linear term. Let

$${\zeta }_{y_s}=\left(\frac{1}{{\alpha }_{y_s\upsilon }^{+}-{\alpha }_{y_{s}d}^{+}}\right){\beta }_{{y_s}_2}{\delta }_{y_s}^{+}$$

(7)

Since maximize $-{\zeta }_{y_s}$ equals minimize ${\zeta }_{y_s}$ , then ${\zeta }_{y_s}$ is given by ²³

$${\zeta }_{y_s}\ge \left(\frac{1}{{\alpha }_{y_s\upsilon }^{+}-{\alpha }_{y_{s}d}^{+}}\right){\delta }_{y_s}^{+}-\left(\frac{{\alpha }_{y_s\upsilon }^{+}}{{\alpha }_{y_s\upsilon }^{+}-{\alpha }_{y_{s}d}^{+}}\right)(1-{\beta }_{{y_s}_2})$$

(8-1)

where ${\beta }_{{y_s}_2}=\{0,1\},{\delta }_{y_s}^{+}\ge 0$ . The constraint (8-1) can be rewritten as

$$\begin{array}{c}\multicolumn{1}{c}{\left(\frac{1}{{\alpha }_{y_s\upsilon }^{+}-{\alpha }_{y_{s}d}^{+}}\right){\delta }_{y_s}^{+}+\left(\frac{{\alpha }_{y_s\upsilon }^{+}}{{\alpha }_{y_s\upsilon }^{+}-{\alpha }_{y_{s}d}^{+}}\right)}\\ {\beta }_{{y_s}_2}-{\zeta }_{y_s}\le \left(\frac{{\alpha }_{y_s\upsilon }^{+}}{{\alpha }_{y_s\upsilon }^{+}-{\alpha }_{y_{s}d}^{+}}\right)\hfill \end{array}$$

(8-2)

Knowing that, when ${\delta }_{y_s}^{+}$ falls within the interval [0, ${\alpha }_{y_{s}d}^{+}$ ], the values of ${\beta }_{{y_s}_1}$ and ${\beta }_{{y_s}_2}$ are equal to 1 and 0, respectively. Conversely, when ${\delta }_{y_s}^{+}$ is within the interval ( ${\alpha }_{y_{s}d}^{+}$ , ${\alpha }_{y_s\upsilon }^{+}$ ], the values of ${\beta }_{{y_s}_1}$ and ${\beta }_{{y_s}_2}$ equal to 0 and 1, respectively. Since

$${\beta }_{{y_s}_1}+{\beta }_{{y_s}_2}=1$$

(9)

Utilizing equation (5), additional two constraints are formulated

$${\alpha }_{y_{s}d}^{+}{\beta }_{{y_s}_2}-{\delta }_{y_s}^{+}\le 0$$

(10)

$${\delta }_{y_s}^{+}-{\alpha }_{y_{s}d}^{+}{\beta }_{{y_s}_1}-{\alpha }_{y_s\upsilon }^{+}{\beta }_{{y_s}_2}\le 0$$

(11)

Finally, the objective function is to maximize the satisfaction function and is expressed as follows

$$\text{Maximize}{z}_{y_s}={\beta }_{{y_s}_1}+\left(\frac{{\alpha }_{y_s\upsilon }^{+}}{{\alpha }_{y_s\upsilon }^{+}-{\alpha }_{y_{s}d}^{+}}\right){\beta }_{{y_s}_2}-{\zeta }_{y_s}$$

(12)

• 2. For the LTB-type response, let ${\delta }_{y_l}^{-}$ denote the negative deviation from the imprecise fuzzy value of the lth response, g_yl. The ${\alpha }_{y_{l}d}^{-}$ and ${\alpha }_{y_l\upsilon }^{-}$ are the indifferent threshold and the veto threshold associated with ${\delta }_{y_l}^{-}$ , respectively. The satisfaction function, $F_{y_l}^{-}({\delta }_{y_l}^{-})$ , for negative deviations for LTB is shown in Figure 3 and is represented as ¹³

$$F_{y_l}^{-}\left({\delta }_{y_l}^{-}\right)=\{\begin{array}{cc}\hfill 1,\hfill & \hfill 0\le {\delta }_{y_l}^{-}\le {\alpha }_{y_{l}d}^{-}\hfill \\ \multicolumn{1}{c}{\frac{{\alpha }_{y_l\upsilon }^{-}-{\delta }_{y_l}^{-}}{{\alpha }_{y_l\upsilon }^{-}-{\alpha }_{y_{l}d}^{-}},}& \hfill {\alpha }_{y_{l}d}^{-}<{\delta }_{y_l}^{-}\le {\alpha }_{y_l\upsilon }^{-}\hfill \end{array}$$

(13)

Figure 3.

The satisfaction function for LTB-type response.

Typically, a process engineer aims to obtain the upper desired value of LTB response. Let g_yl be the upper bound of the y_l. The associated constraints will be

$$y_l+{\delta }_{y_l}^{-}=g_{y_l}$$

(14)

$$0\le {\delta }_{y_l}^{-}\le {\alpha }_{y_l\upsilon }^{-}$$

(15)

Similar to STB, two binary variables, ${\beta }_{{y_l}_1}$ and ${\beta }_{{y_l}_2}$ , are required to represent $F_{y_l}^{-}\left({\delta }_{y_l}^{-}\right)$ . The ${\beta }_{{y_l}_1}$ and ${\beta }_{{y_l}_2}$ variables are defined as

$$\begin{array}{cc}\multicolumn{1}{c}{{\beta }_{{y_l}_1}}& =\{\begin{array}{cc}\hfill 1,\hfill & \hfill 0\le {\delta }_{y_l}^{-}\le {\alpha }_{y_{l}d}^{-}\hfill \\ \multicolumn{1}{c}{0,}& \hfill \text{Otherwise}\hfill \end{array}\text{and}\hfill \\ \multicolumn{1}{c}{{\beta }_{{y_l}_2}}& =\{\begin{array}{cc}\hfill 1,\hfill & \hfill {\alpha }_{y_{l}d}^{-}<{\delta }_{y_l}^{-}\le {\alpha }_{y_l\upsilon }^{-}\hfill \\ \multicolumn{1}{c}{0,}& \hfill \text{Otherwise}\hfill \end{array}\hfill \end{array}$$

(16)

Then, the $F_{y_l}^{-}\left({\delta }_{y_l}^{-}\right)$ can be rewritten as

$$\begin{array}{c}\multicolumn{1}{c}{F_{y_l}^{-}\left({\delta }_{y_l}^{-}\right)={\beta }_{{y_l}_1}+\left(\frac{{\alpha }_{y_l\upsilon }^{-}}{{\alpha }_{y_l\upsilon }^{-}-{\alpha }_{y_{l}d}^{-}}\right)}\\ {\beta }_{{y_l}_2}-\left(\frac{1}{{\alpha }_{y_l\upsilon }^{-}-{\alpha }_{y_{l}d}^{-}}\right){\beta }_{{y_l}_2}{\delta }_{y_l}^{-}\hfill \end{array}$$

(17)

Equation (17) includes nonlinear terms. A linear form can be expressed as

$$\begin{array}{c}\multicolumn{1}{c}{\left(\frac{1}{{\alpha }_{y_l\upsilon }^{-}-{\alpha }_{y_{l}d}^{-}}\right){\delta }_{y_l}^{-}+\left(\frac{{\alpha }_{y_l\upsilon }^{-}}{{\alpha }_{y_l\upsilon }^{-}-{\alpha }_{y_{l}d}^{-}}\right)}\\ {\beta }_{{y_l}_2}-{\zeta }_{y_l}\le \left(\frac{{\alpha }_{y_l\upsilon }^{-}}{{\alpha }_{y_l\upsilon }^{-}-{\alpha }_{y_{l}d}^{-}}\right)\hfill \end{array}$$

(18)

Knowing that the sum of ${\beta }_{{y_l}_1}$ and ${\beta }_{{y_l}_2}$ equals to 1, then

$${\alpha }_{y_{l}d}^{-}{\beta }_{{y_l}_2}-{\delta }_{y_l}^{-}\le 0$$

(19)

$${\delta }_{y_l}^{-}-{\alpha }_{y_{l}d}^{-}{\beta }_{{y_l}_1}-{\alpha }_{y_l\upsilon }^{-}{\beta }_{{y_l}_2}\le 0$$

(20)

Finally, the objective function will be to maximize the satisfaction function or mathematically

$$\text{Maximize}{z}_{y_l}={\beta }_{{y_l}_1}+\left(\frac{{\alpha }_{y_l\upsilon }^{-}}{{\alpha }_{y_l\upsilon }^{-}-{\alpha }_{y_{l}d}^{-}}\right){\beta }_{{y_l}_2}-{\zeta }_{y_l}$$

(21)

• 3. For the NTB-type response, let ${\delta }_{y_n}^{-}$ and ${\delta }_{y_n}^{+}$ denote the negative and positive deviations from the imprecise fuzzy value of the nth response, g_yn, respectively. Then, the corresponding satisfaction functions for negative and positive deviations, $F_{y_n}^{-}({\delta }_{y_n}^{-})$ and $F_{y_n}^{+}({\delta }_{y_n}^{+})$ , respectively, are depicted in Figure 4 and are expressed, respectively, by ¹⁴

$$F_{y_n}^{-}\left({\delta }_{y_n}^{-}\right)=1-\frac{{\delta }_{y_n}^{-}}{{\alpha }_{y_n\upsilon }^{-}},0\le {\delta }_{y_n}^{-}\le {\alpha }_{y_n\upsilon }^{-}$$

(22)

$$F_{y_n}^{+}\left({\delta }_{y_n}^{+}\right)=1-\frac{{\delta }_{y_n}^{+}}{{\alpha }_{y_n\upsilon }^{+}},0\le {\delta }_{y_n}^{+}\le {\alpha }_{y_n\upsilon }^{+}$$

(23)

The ${\alpha }_{y_n\upsilon }^{-}$ and ${\alpha }_{y_n\upsilon }^{+}$ are the veto thresholds associated with ${\delta }_{y_n}^{-}$ and ${\delta }_{y_n}^{+}$ , respectively.

Figure 4.

The satisfaction functions for the NTB-type response: (a) negative deviation and (b) positive deviation.

Hence, the model constraints are expressed as

$$y_n+{\delta }_{y_n}^{-}-{\delta }_{y_n}^{+}=g_{y_n}$$

(24)

$$0\le {\delta }_{y_n}^{-}\le {\alpha }_{y_n\upsilon }^{-}$$

(25)

$$0\le {\delta }_{y_n}^{+}\le {\alpha }_{y_n\upsilon }^{+}$$

(26)

However, at least one of ${\delta }_{y_n}^{-}$ and ${\delta }_{y_n}^{+}$ must have a zero value. Two binary variables, ${\beta }_{{y_n}_1}$ and ${\beta }_{y{n}_2}$ , are needed to express $F_{y_n}^{-}({\delta }_{y_n}^{-})$ and $F_{y_n}^{+}({\delta }_{y_n}^{+})$ , which are defined as

$$\begin{array}{cc}\multicolumn{1}{c}{{\beta }_{{y_n}_1}}& =\{\begin{array}{cc}\hfill 1,\hfill & \hfill 0\le {\delta }_{y_n}^{-}\le {\alpha }_{y_n\upsilon }^{-}\hfill \\ \multicolumn{1}{c}{0,}& \hfill \text{Otherwise}\hfill \end{array}\text{and}\hfill \\ \multicolumn{1}{c}{{\beta }_{{y_n}_2}}& =\{\begin{array}{cc}\hfill 1,\hfill & \hfill 0\le {\delta }_{y_n}^{+}\le {\alpha }_{y_n\upsilon }^{+}\hfill \\ \multicolumn{1}{c}{0,}& \hfill \text{Otherwise}\hfill \end{array}\hfill \end{array}$$

(27)

Accordingly, two satisfaction functions are obtained and can be rewritten as

$$F_{y_n}^{-}\left({\delta }_{y_n}^{-}\right)={\beta }_{{y_n}_1}(1-\frac{{\delta }_{y_n}^{-}}{{\alpha }_{y_n\upsilon }^{-}})={\beta }_{{y_n}_1}-\left(\frac{1}{{\alpha }_{y_n\upsilon }^{-}}\right){\beta }_{{y_n}_1}{\delta }_{y_n}^{-}$$

(28)

$$F_{y_n}^{+}\left({\delta }_{y_n}^{+}\right)={\beta }_{{y_n}_2}(1-\frac{{\delta }_{y_n}^{+}}{{\alpha }_{y_n\upsilon }^{+}})={\beta }_{{y_n}_2}-\left(\frac{1}{{\alpha }_{y_n\upsilon }^{+}}\right){\beta }_{{y_n}_2}{\delta }_{y_n}^{+}$$

(29)

However, the variables ${\beta }_{{y_n}_1}$ , ${\beta }_{{y_n}_2}$ , ${\delta }_{y_n}^{-}$ , and ${\delta }_{y_n}^{+}$ of the functions in equations (28) and (29) result in nonlinear satisfaction functions. Thus, a linearization procedure is needed. Let

$${\zeta }_{y_{n1}}=\left(\frac{1}{{\alpha }_{y_n\upsilon }^{-}}\right){\beta }_{{y_n}_1}{\delta }_{y_n}^{-}$$

(30)

Since maximize $-{\zeta }_{y_{n1}}$ equals minimize ${\zeta }_{y_{n1}}$ , then

$${\zeta }_{y_{n1}}\ge \left(\frac{1}{{\alpha }_{y_n\upsilon }^{-}}\right){\delta }_{y_n}^{-}-(1-{\beta }_{{y_n}_1})$$

(31)

Equivalently

$$\left(\frac{1}{{\alpha }_{y_n\upsilon }^{-}}\right){\delta }_{y_n}^{-}+{\beta }_{{y_n}_1}-{\zeta }_{y_{n1}}\le 1$$

(32)

Similarly, let

$${\zeta }_{y_{n2}}=\left(\frac{1}{{\alpha }_{y_n\upsilon }^{+}}\right){\beta }_{{y_n}_2}{\delta }_{y_n}^{+}$$

(33)

Since maximize $-{\zeta }_{y_{n2}}$ equals minimize ${\zeta }_{y_{n2}}$ , then

$${\zeta }_{y_{n2}}\ge \left(\frac{1}{{\alpha }_{y_n\upsilon }^{+}}\right){\delta }_{y_n}^{+}-(1-{\beta }_{{y_n}_2})$$

(34)

or

$$\left(\frac{1}{{\alpha }_{y_n\upsilon }^{+}}\right){\delta }_{y_n}^{+}+{\beta }_{{y_n}_2}-{\zeta }_{y_{n2}}\le 1$$

(35)

In a similar manner, the following constraints are formulated

$${\delta }_{y_n}^{-}-{\alpha }_{y_n\upsilon }^{-}{\beta }_{{y_n}_1}\le 0$$

(36)

$${\delta }_{y_n}^{+}-{\alpha }_{y_n\upsilon }^{+}{\beta }_{{y_n}_2}\le 0$$

(37)

Also

$${\beta }_{{y_n}_1}+{\beta }_{{y_n}_2}=1$$

(38)

In the end, the objective function will be to maximize the sum of the satisfaction functions; equivalently

$$\text{Maximize}{z}_{y_n}=\sum _{\forall \mathrm{yn}}({\beta }_{{y_n}_1}+{\beta }_{{y_n}_2}-{\zeta }_{y_{n1}}-{\zeta }_{y_{n2}})$$

(39)

• Step 3. In Taguchi method, a process engineer has no information about the target value of x_j. Thus, the satisfaction functions for negative and positive deviations shown in Figure 5 are used to describe the jth process factor. The imprecise fuzzy value, g_xj, of x_j, is usually preferred at interval midrange. Let ${\delta }_{x_j}^{-}$ and ${\delta }_{x_j}^{+}$ denote the negative and positive deviations from g_xj, respectively. Then, the satisfaction functions for negative and positive deviations $F_{x_j}^{-}({\delta }_{x_j}^{-})$ and $F_{x_j}^{+}({\delta }_{x_j}^{+})$ , respectively, are defined as ¹⁴

$$F_{x_j}^{-}\left({\delta }_{x_j}^{-}\right)=1,0\le {\delta }_{x_j}^{-}\le {\alpha }_{x_j\upsilon }^{-}$$

(40)

$$F_{x_j}^{+}\left({\delta }_{x_j}^{+}\right)=1,0\le {\delta }_{x_j}^{+}\le {\alpha }_{x_j\upsilon }^{+}$$

(41)

Figure 5.

The satisfaction functions for the process variables: (a) negative deviation and (b) positive deviation.

The ${\alpha }_{x_j\upsilon }^{-}$ and ${\alpha }_{x_j\upsilon }^{+}$ are the veto thresholds associated with ${\delta }_{x_j}^{-}$ and ${\delta }_{x_j}^{+}$ , respectively. Let $g_{x_j}^l$ and $g_{x_j}^u$ denote the lower and the upper limits of x_j, respectively. Then, the ${\delta }_{x_j}^{-}$ and ${\delta }_{x_j}^{+}$ are, respectively, calculated as

$${\alpha }_{x_j\upsilon }^{-}=g_{x_j}-g_{x_j}^l$$

(42)

and

$${\alpha }_{x_j\upsilon }^{+}=g_{x_j}^u-g_{x_j}$$

(43)

Furthermore, the constraints of x_j are formulated as

$$x_j+{\delta }_{x_j}^{-}-{\delta }_{x_j}^{+}=g_{x_j}$$

(44)

$$0\le {\delta }_{x_j}^{-}\le {\alpha }_{x_j\upsilon }^{-}$$

(45)

$$0\le {\delta }_{x_j}^{+}\le {\alpha }_{x_j\upsilon }^{+}$$

(46)

Similarly, two binary variables are required to represent the x_j satisfaction functions, which are defined as

$$\begin{array}{cc}\multicolumn{1}{c}{{\beta }_{{x_j}_1}}& =\{\begin{array}{c}\hfill 1,0\le {\delta }_{x_j}^{-}\le {\alpha }_{x_j\upsilon }^{-}\hfill \\ \multicolumn{1}{c}{0,\text{Otherwise}}\end{array}\text{and}\hfill \\ \multicolumn{1}{c}{{\beta }_{{x_j}_2}}& =\{\begin{array}{c}\hfill 1,0\le {\delta }_{x_j}^{+}\le {\alpha }_{x_j\upsilon }^{+}\hfill \\ \multicolumn{1}{c}{0,\text{Otherwise}}\end{array}\hfill \end{array}$$

(47)

Thus, the satisfaction functions, equations (46) and (47), can be rewritten as

$$F_{x_j}^{-}\left({\delta }_{x_j}^{-}\right)={\beta }_{{x_j}_1}$$

(48)

$$F_{x_j}^{+}\left({\delta }_{x_j}^{+}\right)={\beta }_{{x_j}_2}$$

(49)

Similarly, the following constraints can be formulated

$${\delta }_{x_j}^{-}-{\alpha }_{x_j\upsilon }^{-}{\beta }_{{x_j}_1}\le 0$$

(50)

$${\delta }_{x_j}^{+}-{\alpha }_{x_j\upsilon }^{+}{\beta }_{{x_j}_2}\le 0$$

(51)

Also

$${\beta }_{{x_j}_1}+{\beta }_{{x_j}_2}=1$$

(52)

Finally, the objective function is expressed as

$$\text{Maximize}{z}_{x_j}=\sum {\beta }_{{x_j}_1}+{\beta }_{{x_j}_2}$$

(53)

• Step 4. Assign weights for the objectives of the satisfaction functions, w_ys, w_yl, w_yn, and w_xj, according to their relative importance to a process or product. For a combination of sth, lth, and nth responses with J process factors, the general optimization model is formulated as

$$\begin{array}{c}\multicolumn{1}{c}{\text{Maximize}\sum _{s=1}^S{w}_{y_s}({\beta }_{{y_s}_1}+\left(\frac{{\alpha }_{y_s\upsilon }^{+}}{{\alpha }_{y_s\upsilon }^{+}-{\alpha }_{y_{s}d}^{+}}\right){\beta }_{{y_s}_2}-{\zeta }_{y_s})}\\ +\sum _{l=1}^L{w}_{y_l}({\beta }_{{y_l}_1}+\left(\frac{{\alpha }_{y_l\upsilon }^{-}}{{\alpha }_{y_l\upsilon }^{-}-{\alpha }_{y_{l}d}^{-}}\right){\beta }_{{y_l}_2}-{\zeta }_{y_l})\hfill \\ +\sum _{n=1}^N{w}_{y_n}({\beta }_{{y_n}_1}+{\beta }_{{y_n}_2}-{\zeta }_{y_{n1}}-{\zeta }_{y_{n2}})\hfill \\ +\sum _{j=1}^J{w}_{x_j}({\beta }_{{x_j}_1}+{\beta }_{{x_j}_2})\hfill \end{array}$$

Subject to

$$y_s=f(x_1,x_2,\dots ,x_j)$$

$$y_s-{\delta }_{y_s}^{+}=g_{y_s}$$

$$0\le {\delta }_{y_s}^{+}\le {\alpha }_{y_s\upsilon }^{+}$$

$$\left(\frac{1}{{\alpha }_{y_s\upsilon }^{+}-{\alpha }_{y_{s}d}^{+}}\right){\delta }_{y_s}^{+}+\left(\frac{{\alpha }_{y_s\upsilon }^{+}}{{\alpha }_{y_s\upsilon }^{+}-{\alpha }_{y_{s}d}^{+}}\right){\beta }_{{y_s}_2}-{\zeta }_{y_s}\le \left(\frac{{\alpha }_{y_s\upsilon }^{+}}{{\alpha }_{y_s\upsilon }^{+}-{\alpha }_{y_{s}d}^{+}}\right)$$

$${\alpha }_{y_{s}d}^{+}{\beta }_{{y_s}_2}-{\delta }_{y_s}^{+}\le 0$$

$${\delta }_{y_s}^{+}-{\alpha }_{y_{s}d}^{+}{\beta }_{{y_s}_1}-{\alpha }_{y_s\upsilon }^{+}{\beta }_{{y_s}_2}\le 0$$

$${\beta }_{{y_s}_1}+{\beta }_{{y_s}_2}=1$$

$$y_l=f(x_1,x_2,\dots ,x_j)$$

$$y_l+{\delta }_{y_l}^{-}=g_{y_l}$$

$$0\le {\delta }_{y_l}^{-}\le {\alpha }_{y_l\upsilon }^{-}$$

$$\left(\frac{1}{{\alpha }_{y_l\upsilon }^{-}-{\alpha }_{y_{l}d}^{-}}\right){\delta }_{y_l}^{-}+\left(\frac{{\alpha }_{y_l\upsilon }^{-}}{{\alpha }_{y_l\upsilon }^{-}-{\alpha }_{y_{l}d}^{-}}\right){\beta }_{{y_l}_2}-{\zeta }_{y_l}\le \left(\frac{{\alpha }_{y_l\upsilon }^{-}}{{\alpha }_{y_l\upsilon }^{-}-{\alpha }_{y_{l}d}^{-}}\right)$$

$${\alpha }_{y_{l}d}^{-}{\beta }_{{y_l}_2}-{\delta }_{y_l}^{-}\le 0$$

$${\delta }_{y_l}^{-}-{\alpha }_{y_{l}d}^{-}{\beta }_{{y_l}_1}-{\alpha }_{y_l\upsilon }^{-}{\beta }_{{y_l}_2}\le 0$$

$${\beta }_{{y_l}_1}+{\beta }_{{y_l}_2}=1$$

$$y_n=f(x_1,x_2,\dots ,x_j)$$

$$y_n+{\delta }_{y_n}^{-}-{\delta }_{y_n}^{+}=g_{y_n}$$

$$0\le {\delta }_{y_n}^{-}\le {\alpha }_{y_n\upsilon }^{-}$$

$$0\le {\delta }_{y_n}^{+}\le {\alpha }_{y_n\upsilon }^{+}$$

$$\left(\frac{1}{{\alpha }_{y_n\upsilon }^{-}}\right){\delta }_{y_n}^{-}+{\beta }_{{y_n}_1}-{\zeta }_{y_{n1}}\le 1$$

$$\left(\frac{1}{{\alpha }_{y_n\upsilon }^{+}}\right){\delta }_{y_n}^{+}+{\beta }_{{y_n}_2}-{\zeta }_{y_{n2}}\le 1$$

$${\delta }_{y_n}^{-}-{\alpha }_{y_n\upsilon }^{-}{\beta }_{{y_n}_1}\le 0$$

$${\delta }_{y_n}^{+}-{\alpha }_{y_n\upsilon }^{+}{\beta }_{{y_n}_2}\le 0$$

$${\beta }_{{y_n}_1}+{\beta }_{{y_n}_2}=1$$

$$x_j+{\delta }_{x_j}^{-}-{\delta }_{x_j}^{+}=g_{x_j},\forall j$$

$$0\le {\delta }_{x_j}^{-}\le {\alpha }_{x_j\upsilon }^{-},\forall j$$

$$0\le {\delta }_{x_j}^{+}\le {\alpha }_{x_j\upsilon }^{+},\forall j$$

$${\delta }_{x_j}^{-}-{\alpha }_{x_j\upsilon }^{-}{\beta }_{{x_j}_1}\le 0,\forall j$$

$${\delta }_{x_j}^{+}-{\alpha }_{x_j\upsilon }^{+}{\beta }_{{x_j}_2}\le 0,\forall j$$

$${\beta }_{{x_j}_1}+{\beta }_{{x_j}_2}=1,\forall j$$

$${\beta }_{{y_s}_1},{\beta }_{{y_s}_2}=\{0,1\},{\zeta }_{y_s},y_s\ge 0$$

$${\beta }_{{y_l}_1},{\beta }_{{y_l}_2}=\{0,1\},{\zeta }_{y_l},y_l\ge 0$$

$${\beta }_{{y_n}_1},{\beta }_{{y_n}_2}=\{0,1\},{\zeta }_{y_{n1}},{\zeta }_{y_{n2}},y_n\ge 0$$

$${\beta }_{{x_j}_1},{\beta }_{{x_j}_2}=\{0,1\},x_j\ge 0,\forall j$$

• Step 5. Solve the optimization model to determine the values of the decision variables x_j and y_k. Validate or compare the obtained results with those obtained using other approaches.

Illustrations of the proposed optimization approach

Three case studies are employed to illustrate the proposed procedure.

Optimizing the process conditions of an injection-molded thermoplastic part

Chiang ⁴ aimed at optimizing the process conditions of an injection-molded thermoplastic part, polycarbonate/acrylonitrile butadiene styrene (PC/ABS) cell phone shell, by utilizing the gray relational analysis and fuzzy logic. Eight principle machining parameters are investigated simultaneously utilizing L₁₈ (2¹× 3⁷) orthogonal array, including the opening mold time (x₁, s), mold temperature (x₂, °C), melt temperature (x₃, °C), filling time (x₄, s), filling pressure (x₅, MPa), packing time (x₆, s), packing pressure (x₇, MPa), and cooling time (x₈, s). Three responses are of main interest, including the strength of welding line (y₁, kg/cm², LTB), shrinkage (y₂, STB), and difference of forming distributive temperature (y₃, °C, STB). Each experiment was repeated three times. The level values of the process controllable parameters and the average responses are displayed in Table 1.

Table 1.

The experimental results with L₁₈ array for thermoplastic injection process.

Experiment I	Controllable parameters								$y_{1i}$	$y_{2i}$	$y_{3i}$
	x 1	x 2	x 3	x 4	x 5	x 6	x 7	x 8
1	5	40	220	0.65	100	1	100	6	70.2	0.013	8.32
2	5	40	240	0.75	120	1.5	120	7	75.4	0.012	8.51
3	5	40	260	0.85	150	2	140	8	72.1	0.008	8.23
4	5	50	220	0.65	120	1.5	140	8	87.1	0.015	9.47
5	5	50	240	0.75	150	2	100	6	95.2	0.015	8.67
6	5	50	260	0.85	100	1	120	7	96.3	0.016	9.13
7	5	60	220	0.75	100	2	120	8	114.4	0.021	9.66
8	5	60	240	0.85	120	1	140	6	114.4	0.021	10.18
9	5	60	260	0.65	150	1.5	100	7	107	0.019	9.53
10	8	40	220	0.85	150	1.5	120	6	78.2	0.013	8.48
11	8	40	240	0.65	100	2	140	7	87.4	0.012	9.13
12	8	40	260	0.75	120	1	100	8	73.3	0.008	8.43
13	8	50	220	0.75	150	1	140	7	86.8	0.016	9.66
14	8	50	240	0.85	100	1.5	100	8	93.8	0.017	8.92
15	8	50	260	0.65	120	2	120	6	97	0.016	8.77
16	8	60	220	0.85	120	2	100	7	113.8	0.023	9.66
17	8	60	240	0.65	150	1	120	8	116.8	0.019	10.17
18	8	60	260	0.75	100	1.5	140	6	107.6	0.021	9.82

The proposed procedure was then implemented to solve the three responses for the injection molding process and is described as follows:

• Step 1. The multiple linear regression equations for y₁, y₂, and y₃ are estimated as

$$\begin{array}{c}\multicolumn{1}{c}{y_1=-5.18941+0.83827x_1+1.81111x_2}\\ +0.01181x_3+2.50000x_4-0.04386x_5\hfill \\ +3.68889x_6+0.00903x_7-0.43056x_8\hfill \\ \multicolumn{1}{c}{y_2=0.0108502+0.0001728x_1+0.0004778x_2}\\ -0.0000514x_3+0.0033333x_4\hfill \\ -0.0000273x_5+0.0001667x_6-0.0000069x_7\hfill \\ -0.0008889x_8\hfill \\ \multicolumn{1}{c}{y_3=6.07176+0.04938x_1+0.06597x_2-0.00558x_3}\\ -0.66111x_4-0.00090x_5\hfill \\ -0.29556x_6+0.01235x_7+0.05278x_8\hfill \end{array}$$

• Step 2. For y₁ (LTB type), the value of g_y1 equals to 130 kg/cm², and the values of ${\alpha }_{y_{1}d}^{-}$ and ${\alpha }_{y_1\upsilon }^{-}$ equal to 30 and 50, respectively. The satisfaction function of y₁ is represented as

$$F_{y_1}^{-}\left({\delta }_{y_1}^{-}\right)=\{\begin{array}{cc}\hfill 1,\hfill & \hfill 0\le {\delta }_{y_1}^{-}\le 30\hfill \\ \multicolumn{1}{c}{2.5-0.05{\delta }_{y_1}^{-},}& \hfill 30<{\delta }_{y_1}^{-}\le 50\hfill \end{array}$$

or

$$F_{y_1}^{-}\left({\delta }_{y_1}^{-}\right)={\beta }_{{y_1}_1}+2.5{\beta }_{{y_1}_2}-0.05{\beta }_{{y_1}_2}{\delta }_{y_1}^{-}$$

Similarly, for y₂ (STB type), the value of g_y2 equals 0, the target of the shrinkage response. The ${\alpha }_{y_{2}d}^{+}$ and ${\alpha }_{y_2\upsilon }^{+}$ are assumed to be equal to 0.01 and 0.012, respectively. The satisfaction function of y₂ is represented as

$$F_{y_2}^{+}\left({\delta }_{y_2}^{+}\right)=\{\begin{array}{cc}\hfill 1,\hfill & \hfill 0\le {\delta }_{y_2}^{+}\le 0.01\hfill \\ \multicolumn{1}{c}{6-500{\delta }_{y_2}^{+},}& \hfill 0.01<{\delta }_{y_2}^{+}\le 0.012\hfill \end{array}$$

or

$$F_{y_2}^{+}\left({\delta }_{y_2}^{+}\right)={\beta }_{{y_2}_1}+6{\beta }_{{y_2}_2}-500{\beta }_{{y_2}_2}{\delta }_{y_2}^{+}$$

Finally, for y₃ (STB type) response, the value of g_y3 equals 0, the target of the temperature difference which is a measure of warpage. Assume the values of ${\alpha }_{y_{3}d}^{+}$ and ${\alpha }_{y_3\upsilon }^{+}$ equal 8 and 8.5, respectively. The satisfaction function of y₃ is represented as

$$F_{y_3}^{+}\left({\delta }_{y_3}^{+}\right)=\{\begin{array}{cc}\hfill 1,\hfill & \hfill 0\le {\delta }_{y_3}^{+}\le 8\hfill \\ \multicolumn{1}{c}{17-2{\delta }_{y_3}^{+},}& \hfill 8<{\delta }_{y_3}^{+}\le 8.5\hfill \end{array}$$

or

$$F_{y_3}^{+}\left({\delta }_{y_3}^{+}\right)={\beta }_{{y_3}_1}+17{\beta }_{{y_3}_2}-2{\beta }_{{y_3}_2}{\delta }_{y_3}^{+}$$

• Step 3. The satisfaction functions for negative and positive deviations are used to describe the eight process factors. For example, the satisfaction function of x₁ is obtained as follows. The value of g_x1 equals to 6.5, which is the midpoint of x₁ range. The values of ${\alpha }_{x_1\upsilon }^{-}$ and ${\alpha }_{x_1\upsilon }^{+}$ are both equal to 1.5. For example, the x₁ satisfaction functions are defined as

$$F_{x_1}^{-}\left({\delta }_{x_1}^{-}\right)={\beta }_{{x_1}_1}$$

$$F_{x_1}^{+}\left({\delta }_{x_1}^{+}\right)={\beta }_{{x_1}_2}$$

• Step 4. Assume that the relative importance of the satisfaction functions of responses and factors is assigned equal weights. Then, the complete model is formulated as

$$\begin{array}{l}\text{Maximize}Z=0.1\times \left({{\beta }_{y_1}}_1+20{{\beta }_{y_1}}_2-{\zeta }_{y_1}\right)+0.4\times \left({{\beta }_{y_2}}_1+6{{\beta }_{y_2}}_2-{\zeta }_{y_2}\right)+\\ 0.4\times \left({{\beta }_{y_3}}_1+17{{\beta }_{y_3}}_2-{\zeta }_{y_3}\right)+{\displaystyle \sum _{j=1}^8\left({{\beta }_{x_j}}_1+{{\beta }_{x_j}}_2\right)}\end{array}$$

Subject to:

$$\begin{array}{c}\multicolumn{1}{c}{y_1=-5.18941+0.83827x_1+1.81111x_2}\\ +0.01181x_3+2.50000x_4-0.04386x_5\hfill \\ +3.68889x_6+0.00903x_7-0.43056x_8\hfill \\ \multicolumn{1}{c}{y_2=0.0108502+0.0001728x_1+0.0004778x_2}\\ -0.0000514x_3+0.0033333x_4\hfill \\ -0.0000273x_5+0.0001667x_6-0.0000069x_7\hfill \\ -0.0008889x_8\hfill \\ \multicolumn{1}{c}{y_3=6.07176+0.04938x_1+0.06597x_2-0.00558x_3}\\ -0.66111x_4-0.00090x_5\hfill \\ -0.29556x_6+0.01235x_7+0.05278x_8\hfill \end{array}$$

$$\begin{array}{l}{y}_1+{\delta }_{y_1}^{-}=135,0\le {\delta }_{y_1}^{-}\le 50,0.05{\delta }_{y{,}_1}^{-}+20{\beta }_{y_{1,2}}-{\zeta }_{y_1}\le 20,38{\beta }_{y_{1,2}}-{\delta }_{y_1}^{-}\le 0\\ {\delta }_{y_1}^{-}-38{\beta }_{y_{1,1}}-40{\beta }_{y_{1,2}}\le 0,y_2-{\delta }_{y_2}^{+}=0,0\le {\delta }_{y_2}^{+}\le 0.012,500{\delta }_{y_2}^{+}+6{\beta }_{y_{22}}-{\zeta }_{y_2}\le 6\\ 0.01{\beta }_{y_{22}}-{\delta }_{y_2}^{+}\le 0,{\delta }_{y_2}^{+}-0.01{\beta }_{y_{21}}-0.012{\beta }_{y_{22}}\le 0,y_3-{\delta }_{y_3}^{+}=0,0\le {\delta }_{y_3}^{+}\le 8.5\\ 2{\delta }_{y_3}^{+}+17{\beta }_{y_{32}}-{\zeta }_{y_3}\le 17,8{\beta }_{y_{32}}-{\delta }_{y_3}^{+}\le 0,{\delta }_{y_3}^{+}-8{\beta }_{y_{31}}-8.5{\beta }_{y_{32}}\le 0\\ x_1+{\delta }_{x_1}^{-}-{\delta }_{x_1}^{+}=6.5;0\le {\delta }_{x_1}^{-,+}\le 1.5,{\delta }_{x_1}^{-}-1.5{\beta }_{y_{1,1}}\le 0,{\delta }_{x_1}^{+}-1.5{\beta }_{x_{1,2}}\le 0\\ x_2+{\delta }_{x_2}^{-}-{\delta }_{x_2}^{+}=50,0\le {\delta }_{x_2}^{-,+}\le 10,{\delta }_{x_2}^{-}-10{\beta }_{x_{21}}\le 0,{\delta }_{x_2}^{+}-10{\beta }_{x_{22}}\le 0\\ x_3+{\delta }_{x_3}^{-}-{\delta }_{x_3}^{+}=280,0\le {\delta }_{x_3}^{-,+}\le 20,{\delta }_{x_3}^{-}-20{\beta }_{x_{31}}\le 0,{\delta }_{x_3}^{+}-20{\beta }_{x_{32}}\le 0\\ x_4+{\delta }_{x_4}^{-}-{\delta }_{x_4}^{+}=0.75,0\le {\delta }_{x_4}^{-,+}\le 0.1,{\delta }_{x_4}^{-}-0.1{\beta }_{x_{41}}\le 0,{\delta }_{x_4}^{+}-0.1{\beta }_{x_{42}}\le 0\\ x_5+{\delta }_{x_5}^{-}-{\delta }_{x_5}^{+}=140,0\le {\delta }_{x_5}^{-,+}\le 25,{\delta }_{x_5}^{-}-25{\beta }_{x_{51}}\le 0,{\delta }_{x_5}^{+}-25{\beta }_{x_{52}}\le 0\\ x_7+{\delta }_{x_7}^{-}-{\delta }_{x_7}^{+}=120,0\le {\delta }_{x_7}^{-,+}\le 20,{\delta }_{x_7}^{-}-20{\beta }_{x_{71}}\le 0,{\delta }_{x_7}^{+}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{-20{\beta }_{{x_7}_1}\le 0,{\delta }_{x_7}^{+}-20{\beta }_{{x_7}_2}\le 0}\\ x_8+{\delta }_{x_8}^{-}-{\delta }_{x_8}^{+}=7,0\le {\delta }_{x_8}^{-,+}\le 1,{\delta }_{x_8}^{-}\hfill \\ -{\beta }_{{x_8}_1}\le 0,{\delta }_{x_8}^{+}-{\beta }_{{x_8}_2}\le 0\hfill \\ {\beta }_{{y_k}_1}+{\beta }_{{y_k}_2}=1,{\beta }_{{y_k}_1},{\beta }_{{y_k}_2}=\{0,1\},{\zeta }_{y_k},y_k\ge 0,\forall k\hfill \\ {\beta }_{{x_j}_1}+{\beta }_{{x_j}_2}=1,{\beta }_{x_j{,}_1},{\beta }_{{x_j}_2}=\{0,1\},x_j\ge 0,\forall j\hfill \end{array}$$

By solving the model, the resulted optimal factor settings are then estimated as shown in Table 2. It is found that the optimal settings are x₁ = 5, x₂ = 51.56, x₃ = 300, x₄ = 0.85, x₅ = 180, x₆ = 2, x₇ = 100, and x₈ = 8.

Table 2.

Optimal factor settings for thermoplastic injection process.

	Process factors settings
	x 1	x 2	x 3	x 4	x 5	x 6	x 7	x 8
Initial	5	40	240	0.75	120	1.5	120	7
Gray-fuzzy	5	40	260	0.85	120	2	100	8
Optimal	5	51.56	300	0.85	180	2	100	8

Optimizing film coating process for polymer blends

Kuo and Wu ³ investigated the influence of major processing parameters on the morphological properties of the material surface of a film coating process for polymer blends using the gray analysis. Two STB-type responses are studied: the root mean square roughness, (y₁, nm), and the roughness average (y₂, nm). Four 3-level controllable process factors are investigated, including the silicone concentration ratio (x₁, wt%), cure temperature (x₂, °C), vacuum degree (x₃, cm Hg), and cure time (x₄, min), utilizing the L₉ (3⁴) orthogonal array shown in Table 3.

Table 3.

The experimental results for film coating process.

Experiment I	Controllable variables				$y_{1i}$	$y_{2i}$
	x 1	x 2	x 3	x 4
1	0.95	140	0	60	1.721	1.225
2	0.95	150	40	80	1.786	1.192
3	0.95	160	76	100	4.961	3.958
4	0.90	140	40	100	2.115	1.574
5	0.90	150	76	60	3.440	2.760
6	0.90	160	0	80	4.557	3.346
7	0.85	140	76	80	1.959	1.457
8	0.85	150	0	100	3.161	2.371
9	0.85	160	40	60	3.576	2.592

The multiple linear regression equations for y₁ and y₂ are estimated as

$$y_1=-15.6745-0.76x_1+0.1217x_2+0.0037x_3+0.0125x_4$$

$$y_2=-12.7698-0.15x_1+0.0940x_2+0.0051x_3+0.0111x_4$$

Since y₁ and y₂ are STB-type responses, the values of g_y1 and g_y2 equal 0. For y₁, the values of ${\alpha }_{y_{1}d}^{+}$ and ${\alpha }_{y_1\upsilon }^{+}$ values are assumed to be equal to 2 and 2.6, respectively. While, the values of ${\alpha }_{y_{2}d}^{+}$ and ${\alpha }_{y_2\upsilon }^{+}$ for y₂ are equal to 1.0 and 1.8, respectively. Similarly, the values of ${\alpha }_{x_{j}d}^{+}$ , ${\alpha }_{x_{j}d}^{-}$ , and g_xj are assumed for all process factors. The combinations (initial, optimal) of the process factor settings are silicone concentration ratio (0.95, 0.95 wt%), cure temperature (15 °C, 140 °C), vacuum degree (0, 0 cm Hg), and cure time (60, 60 min). It is found that the satisfaction for the two responses and the four process factor settings are all 100%.

Optimizing plastic pipes extrusion process

This cases study is conducted to validate the efficiency of the proposed approach in optimizing plastic extrusion process with two main pipes’ responses: diameter (y₁, NTB, mm) and thickness (y₂, NTB, mm). Based on the process knowledge, three process factors are selected, including temperature (x₁, °C), motor speed (x₂, r/min), and vacuum (x₃, bar). The $L_{18}(2^1\times 3^7)$ array is utilized for providing experimental format as shown in Table 4.

Table 4.

The experimental results for plastic extrusion process.

Experiment I	x 1	x 2	x 3	y 1	y 2
1	1	1	1	16.269	2.330
2	1	2	2	16.130	2.101
3	1	3	3	16.070	2.091
4	2	1	1	16.118	2.149
5	2	2	2	16.081	2.009
6	2	3	3	16.126	2.148
7	3	1	2	16.105	2.216
8	3	2	3	16.131	2.166
9	3	3	1	16.228	2.131
10	1	1	3	16.114	2.175
11	1	2	1	16.185	2.188
12	1	3	2	16.191	2.086
13	2	1	2	16.148	2.155
14	2	2	3	16.110	2.061
15	2	3	1	16.089	2.103
16	3	1	3	16.100	2.201
17	3	2	1	16.105	2.019
18	3	3	2	16.124	2.079

The multiple regression equations for y₁ and y₂ are estimated, respectively, as

$$\begin{array}{c}\multicolumn{1}{c}{y_1=53.9-0.46x_1+0.114x_2+1.7x_3+0.0014x_1^2}\\ +0.0014x_2^2+0.000582x_1{x}_2+0.118x_1{x}_3\hfill \\ \multicolumn{1}{c}{y_2=39.3-0.4x_1-0.184x_2-2.1x_3+0.0012x_1^2}\\ +0.001x_2^2+0.00027x_1{x}_2+0.015x_1{x}_3\hfill \end{array}$$

The value of g_y1 equals to 16 mm, and the values of ${\alpha }_{y_{1}v}^{+}\text{and}{\alpha }_{y_{1}v}^{-}$ are equal to 0.02 and 0.3, respectively. The minimum acceptable value of y₁ is 15.98. The values of g_y2, ${\alpha }_{y_{2}v}^{+},\text{and}{\alpha }_{y_{2}v}^{-}$ equal to 2.0 mm, 0.02 and 0.4, respectively. The minimum acceptable value of y₂ is 1.98. The g_xj, ${\alpha }_{\mathrm{xjv}}^{+},\text{and}{\alpha }_{\mathrm{xjv}}^{-}$ are determined and then inserted in the model. The completed model for optimizing the overall satisfaction functions is formulated as

$$\begin{array}{c}\multicolumn{1}{c}{\text{Minimize}Z=0.35(\beta y_{11}+\beta y_{12}-{\zeta }_{y_{11}}-{\zeta }_{y_{12}}}\\ +\beta y_{21}+\beta y_{22}-{\zeta }_{y_{21}}-{\zeta }_{y_{22}})+\hfill \\ 0.1(\beta x_{11}+\beta x_{12}+\beta x_{21}+\beta x_{22}+\beta x_{31}+\beta x_{32})\hfill \\ y_1=53.9-0.46x_1+0.114x_2+1.7x_3+0.0014x_1^2\hfill \\ +0.0014x_2^2+0.000582x_1{x}_2+0.118x_1{x}_3\hfill \\ y_2=39.3-0.4x_1-0.184x_2-2.1x_3+0.0012x_1^2\hfill \\ +0.001x_2^2+0.00027x_1{x}_2+0.015x_1{x}_3\hfill \\ y_1+{\delta }_{y_1}^{-}-{\delta }_{y_1}^{+}=16,0\le {\delta }_{y_1}^{-}\le 0.02,0\le {\delta }_{y_1}^{+}\le 0.3\hfill \\ y_2+{\delta }_{y_2}^{-}-{\delta }_{y_2}^{+}=2,0\le {\delta }_{y_2}^{-}\le 0.02,0\le {\delta }_{y_2}^{+}\hfill \\ \le 0.4,{\delta }_{y_1}^{-}-0.02\beta y_{11}\le 0\hfill \\ {\delta }_{y_1}^{+}-0.3\beta y_{12}\le 0,{\delta }_{y_2}^{-}-0.02\beta y_{21}\le 0,{\delta }_{y_2}^{+}\hfill \\ -0.4\beta y_{22}\le 0\hfill \\ (1/0.02){\delta }_{y_1}^{-}+\beta y_{11}-{\zeta }_{y_{11}}\le 1,(1/0.3){\delta }_{y_1}^{+}\hfill \\ +\beta y_{12}-{\zeta }_{y_{12}}\le 1\hfill \\ (1/0.02){\delta }_{y_2}^{-}+\beta y_{21}-{\zeta }_{y_{21}}\le 1,(1/0.4){\delta }_{y_2}^{+}\hfill \\ +\beta y_{22}-{\zeta }_{y_{22}}\le 1\hfill \\ \beta y_{11}+\beta y_{12}=1,\beta y_{21}+\beta y_{22}=1,\beta x_{11}+\beta x_{12}=1\hfill \\ \beta x_{21}+\beta x_{22}=1,\beta x_{31}+\beta x_{32}=1,x_1+{\delta }_{x_1}^{-}\hfill \\ -{\delta }_{x_1}^{+}=140\hfill \\ x_1+{\delta }_{x_1}^{-}-{\delta }_{x_1}^{+}=140,x_2+{\delta }_{x_2}^{-}-{\delta }_{x_2}^{+}=70,\hfill \\ x_3+{\delta }_{x_3}^{-}-{\delta }_{x_3}^{+}=2\hfill \\ 0\le {\delta }_{x_1}^{-}\le 5,0\le {\delta }_{x_1}^{+}\le 5,0\le {\delta }_{x_2}^{-}\le 10,0\le {\delta }_{x_2}^{+}\hfill \\ \le 10,0\le {\delta }_{x_3}^{-}\le 0.5,0\le {\delta }_{x_3}^{+}\le 0.5\hfill \\ {\delta }_{x_1}^{-}-5\beta x_{11}\le 0,{\delta }_{x_1}^{+}-5\beta x_{12}\le 0,{\delta }_{x_2}^{-}-10\beta x_{21}\hfill \\ \le 0,{\delta }_{x_2}^{+}-10\beta x_{22}\le 0,{\delta }_{x_3}^{-}-0.5\beta x_{31}\le 0\hfill \\ {\delta }_{x_3}^{+}-0.5\beta x_{32}\le 0,\beta y_{11},\beta y_{12},\beta y_{21},\beta y_{22},\beta y_{31},\hfill \\ \beta y_{32}=\{0,1\}\hfill \\ y_1,y_2,y_3,x_1,x_2,x_3,{\zeta }_{y_{11}},{\zeta }_{y_{12}},{\zeta }_{y_{21}},{\zeta }_{y_{22}},{\zeta }_{y_{31}},{\zeta }_{y_{32}}>0\hfill \end{array}$$

The optimal process settings are found as follows: the temperature (x₁ = 140.0 °C), motor speed (x₂ = 63.7 r/min), and vacuum (x₃ = 2.5 bar).

Optimization results

The improvement analysis for the three case studies is presented in the following.

Results for thermoplastic injection process (case study I)

A comparison of the optimal results with the initial values and the gray-fuzzy analysis results ⁴ are shown in Table 5. In Table 5, it is obvious that y₁ (LTB) is significantly improved from 75.4 (at initial settings) to 95 (optimal settings), while y₂ and y₃, which are both STB responses, are improved slightly from 0.012 and 8.51 to 0.011 and 8.39, respectively. Comparing the anticipated improvements between gray-fuzzy and proposed approach, it is obvious that the proposed approach (= 35.74%) provides much larger total anticipated improvement than gray-fuzzy analysis (= 24.88%).

Table 5.

Anticipated improvements for thermoplastic injection process

Response	Initial condition	Optimal conditions		Anticipated improvement		Percentage improvement
		Gray-fuzzy analysis (GFA)	Satisfaction model (SM)	GFA	SM	GFA	SM
y 1 (LTB)	75.4	76.215	95.000	0.815	19.60	1.08	25.99
y 2 (STB)	0.012	0.010	0.011	−0.002	−0.001	16.67	8.33
y 3 (STB)	8.51	7.903	8.39	−0.607	−0.12	7.13	1.41
Total improvement percentage						24.88	35.74

STB: smaller-the-better; LTB: larger-the-better.

Results for film coating process (case study II)

Table 6 provides a comparison among the obtained optimal results, the initial values, and the gray analysis results. ³ In Table 6, it is obvious that y₁ and y₂ are improved significantly by setting process factors at optimal levels. That is, using the proposed approaches y₁ and y₂, which are both STB responses, are decreased from 2.6 and 1.85 to 1.39 and 0.91, respectively. Furthermore, comparing between gray analysis (=59.65%) and proposed approach (= 97.35%), it is clear that the proposed approach provides larger total percentage of anticipated improvements.

Table 6.

Comparison between the optimal results of the film coating process.

Parameter	Initial condition	Optimal conditions		Anticipated improvement (%)
		Gray analysis (GA)	Satisfaction model (SM)	GA	SM
x 1	0.95	0.95	0.95
x 2	150	140	140
x 3	0	40	0
x 4	60	80	60
y 1 (STB)	2.60	1.78	1.39	−0.82 (31.54%)	1.21 (46.54%)
y 2 (STB)	1.85	1.33	0.91	−0.52 (28.11%)	0.94 (50.81%)
Total improvement percentage				59.65	97.35

Results for plastic pipes extrusion process (case study III)

Table 7 displays a comparison between quality responses at initial and optimal factor settings for plastic pipes extrusion process. The obtained initial (optimal) response values are found as the diameter 16.2139 mm (16.0579 mm) and thickness 2.1758 mm (2.0 mm). The percentages of improvement in y₁ (NTB) and y₂ (NTB) are 0.96% and 8.08%, respectively. The total improvement percentage is 9.04%. The improvements in the three case studies indicate the efficiency of the proposed approach in improving process performance while considering engineer’s preferences about process factor settings.

Table 7.

Improvement analysis for plastic extrusion process.

Parameter	Initial	Optimal	Improvements
x 1	145.0	140.0
x 2	80.0	63.7
x 3	2.5	2.5
y 1 (NTB)	16.2139	16.0579	0.1561 (0.96%)
y 2 (NTB)	2.1758	2.0000	0.1758 (8.08%)
Total improvement percentage			9.04

NTB: nominal-the-best.

Conclusion

This research illustrates a proposed approach for optimizing multiple responses in the application of the Taguchi method. The relationships between each response quality and process factors are first estimated. Then, a proper satisfaction function is selected to represent each response and process factor. A complete optimization model is developed. Three case studies are provided for illustration; in all of which the proposed approach provides the largest improvement percentages while considering process engineers’ satisfaction about process factors. The proposed approach shall provide practitioners a great assistance in optimizing performance with multiple responses while considering their preferences about responses as well as process factor settings.