Latest revision as of 06:37, 4 May 2022

Abstract

Robust design has received a great deal of attention from quality researchers in recent years, and a number of optimization methodologies based on the dual response format have been proposed. The majority of existing bi-objective optimization models concentrate on the trade-offs between the process mean and variability functions without investigating the interactions between control factors and quality characteristics. The primary objective of this research is to integrate the Stackelberg leadership model into the robust design procedure and propose a Stackelberg game-based robust design (SGRD) method to determine appropriate control factor settings by minimizing the values of desired optimization targets based on an analysis of possible combinations of input and output quality parameters. Herein, first, a bi-objective robust design optimization problem is formulated as a dual response model using response surface methodology (RSM). Second, the proposed SGRD model is developed via decomposition into two leader-follower game models. Finally, the mean square error (MSE) criterion is applied to evaluate models and select non-dominated solutions in various situations. Numerical examples are used to demonstrate that the proposed method provides significant solutions in cases containing unidentified priorities between the dual responses and undiscovered correlations among several inputs and outcomes. In addition, according to the case study analysis, the proposed method is more efficient than the conventional dual response approach when dealing with bi-objective robust design optimization problems.

Keywords: Robust Parameter Design, Bi-objective Optimization, Dual Response Approach, Game Theory, Stackelberg Competition model

1. Introduction

The subject of quality management and improvement has received widespread and consistent attention from researchers in the last century. Many studies on literature have shown that a high-efficiency and cost-efficient system plays a vital role in realistic production processes. Taguchi [1] introduced a procedure for experimental design, which is also known as robust design, and that improves the robustness of quality characteristics by minimizing the response to noise factors. The main objective of robust design is to determine the optimal design settings for a product that can minimize the process variability and target deviation. Based on Taguchi's philosophy, subsequent experimental design methods have been developed to identify controlled parameter settings and, thus, improve the robustness of the production process.

Conventional robust design methods can be preliminarily classified as sequential and simultaneous optimization models. Vining and Myers [2] investigated the application of a dual response approach to minimize the process variability by assigning the process mean at the target with defining the priority criterion to separately optimize the estimated functions for the process mean and the standard deviation response based on RSM. Following the prioritized dual response format, Del Castillo and Montgomery [3] and Copeland and Nelson [4] presented improved optimization techniques for the situations in which one objective is achieved in advance (i.e., the process bias or variance is customer-specified). Instead of assigning a specific value to the optimization objective, the bias-specified and variability-specified robust design methods reached the goal of robust design by setting upper bounds for the process mean and standard deviation, respectively [5,6]. Priority-based methods reduce the complexity of problems by transforming multi-objective models into constrained single-objective models. However, existing prioritization schemes are limited because the majority of feasible solutions in the solution space are removed once the first prioritized objective has finished optimization. In this case, the reliability of efficient solutions in objective space is significantly decreased.

The MSE method proposed by Lin and Tu [7], which is based on the relaxed zero-bias assumption, deals with the dual response problem in a simultaneous mode. In terms of the MSE model and its simultaneous optimization order, the concept of weighted sum model is used by Tang and Xu [8], Koksoy and Doganaksoy [9] and,, Ding et al. [10] to investigate the trade-offs between process bias and variance on different weights, instead of the default equivalent weight terms used by Lin and Tu [7]. To find non-dominated solutions and non-convex Pareto frontier, Shin and Cho [11] introduced the lexicographic weighted Tchebycheff method based on the L-infinity norm. The weight-based simultaneous models show good accuracy and efficiency when used to search for the optimal solutions of a robust design problem. Nevertheless, it is difficult to identify the actual weight terms in simultaneous models by quality operators in practice.

The majority of existing robust design optimization methodologies focus on the trade-offs between process bias and variability. However, the interaction between control variables and quality characteristics has always been ignored. To this end, we incorporate the Stackelberg game theory philosophy into classical sequential optimization methods that can consider the impact of control factors on output performance. In the proposed SGRD model in this paper, one object is optimized in advance as the leader, given the constraint of another object (the follower)'s best response function. The leader's objective function may be adjusted according to the follower's best response function, which is variable due to the presence of various controlled factors. Therefore, it is worth discovering situations with permutations and combinations among controlled factors to analyze the correlations between input factors and quality characteristics. Doing so means that every possible interaction between the preferred objective and non-preferred target are taken into account. In addition, in real-world production procedures, quality engineers are often confronted with scenarios in which the importance between process characteristics may be undiscovered or specified by customers. The numerical examples in Section 4 include two cases; the first has unknown priority for quality characteristics, while the second is based on customer-specified bias-prioritized and variance-prioritized situations.

Section 2 presents a literature review and background information regarding the robust parameter design model and Stackelberg leader-follower model. A description of the proposed SGRD model is given in Section 3. Two numerical examples are discussed and drawn in Section 4. Section 5 includes the conclusion and discussion of this paper. Figure 1 shows an overview of the proposed bi-objective SGRD optimization procedure.

Figure 1. Optimization procedure of the proposed SGRD model

2. Literature review

2.1 Robust parameter design

The robust parameter design introduced by Genichi Taguchi is an experimental design technique that investigates the interactions between noise factors and outputs [12]. Taguchi proposed a two-step model using the signal theory that considered both mean and standard deviation of quality characteristics. The goal of this method is to minimize the effects of uncontrolled factors by finding the optimal settings for the controlled factors, process bias, and variance, more accurately. Despite the ongoing debate about the accuracy of the Taguchi method for modeling noise factors, the basic philosophy of robust design has generally been applied in the quality management area.

From the perspective of optimization order, the robust design process can be said to consist of simultaneous and sequential optimization procedures. In simultaneous optimization models, the weight term is considered a crucial criterion on which to evaluate the trade-off among objectives. Lin and Tu [7] integrated the MSE model, which takes both process bias and variance into account simultaneously, to solve the large variance problem in [2]. As illustrated by Lin and Tu [7], the MSE model's estimation is based on the assumption that the two objectives have the same weight. To identify more flexible and appropriate weight terms, Cho et al. [13], Tang and Xu [8], Koksoy and Doganaksoy [9] and Ding et al. [10] suggested using the concept of weighted sum based on the MSE model. In terms of the L-infinity norm, Kim and Lin [14] used fuzzy logic and membership functions to optimize process mean and variance simultaneously. Instead of the weighted sum model, Shin and Cho [11] suggested a lexicographical weighted Tchebycheff method to assign weight terms and find non-dominated solutions for a bi-objective optimization problem.

A sequential optimization procedure is also known as a priority-based performance method. RSM with the priority criterion was used in the first attempt at integration into the dual-response method in [2]. To achieve bi-objective optimization, Vining and Myers [2] assigned the mean value of the dual-response model at a customer-specified target and individually minimized the performance variance. Del Castillo and Montgomery [3] and Copeland and Nelson [4] incorporated the generalized reduced gradient method and Nelder-Mead simplex method into the dual-response model to guarantee a locally optimal solution. Kim and Cho [15] and Tang and Xu [8] improved the dual-response model using the goal programming method with a prioritized model. Shin and Cho [5, 6] introduced bias-specified and variability-specified robust design models using the e-constraint method for customer-specified bias and variance upper bounds. Combined with the bias-specified model, Steenackers and Guillaume [16] used the MSE model to determine the optimal design points. Regarding the variance-preference case, Kovach et al. [17] introduced a variance-prioritized multi-response robust design model that contains multiple quality characteristics via goal programming.

2.2 The Stackelberg leadership model

Game theory is the study of mathematical models of strategic interaction among rational decision-makers [18]. Modern Game Theory is considered to have begun with the 1944 book "Theory of Games and Economics Behavior" by John von Neumann and Oskar Morgenstern. Game Theory was later fleshed out by John Forbes Nash to mathematically represent strategic interactions among rational agents. It has been wildly used in all social science domains, and its applications have been extended to the majority of other fields associated with decision making.

The Stackelberg model is an economic strategy game described by Heinrich Freiherr von Stackelberg in 1934. This leadership game holds that, in a market, a leader firm and follower firm compete for quantity and take actions in sequence. Chen and Cruz [19] and Simaan and Cruz [20] extended the static two-person non-cooperative nonzero-sum Stackelberg game to a dynamic game format with an asymmetric information criterion. Shimizu and Aiyoshi [21] introduced a general bi-level Stackelberg game format and proposed an interior penalty method (i.e., barrier method) to solve the Stackelberg game-based non-linear min-max optimization problem. The penalty method was used in a static linear non-convex Stackelberg game model by Anandalingam and White [22]. In addition, Anandalingam and White [22] introduced and integrated the follower's rational reaction set into the leader's objective function. However, Lai [23] pointed out that the use of the penalty method may restrict the leader's power to make decisions and applied membership functions in extended hierarchical/multi-level Stackelberg game models. A fuzzy approach was combined with the membership function and used for multi-level programming in [24-26]. To search the Stackelberg equilibrium in a multi-level optimization problem, Liu [27], D'Amato et al. [28], and Hong et al. [29] discussed the application of a genetic algorithm. The Stackelberg leader-follower competition model has been extensively applied in the security defense and supply chain domains, as discussed in [30].

3. Models and methods

3.1 Extensive-form game model

Game theory is widely recognized as an analytical tool to exploit the interactions among several participants; it has been applied to many fields. In terms of the properties of the game, a game can be classified as cooperative/non-cooperative, simultaneous/sequential, perfect information/imperfect information, symmetric/asymmetric, and zero-sum/non-zero-sum. In this paper, we concentrate mostly on a non-cooperative sequential game model with perfect information.

In a general sequential game, which is also known in game theory as an extensive-form game, players take their actions in a specified sequence. Usually, the player who moves first is the front player, and the player who moves later is the rear player. The front player can make his decision in advance according to the anticipation of the rear player's movement. Then, the rear player can move based on the actions of the front player. Similar to a general sequential game, in a Stackelberg game, the player who makes the first decision is called the game leader, and the player who moves after the leader is known as the game follower. A particular extensive-form game with infinite action spaces can be used to represent the Stackelberg competition model, which is illustrated in Figure 2. The potential actions of each player are shown fanning out from the player's decision node. Players can select and make decisions from the action space, which may be constrained by boundaries.

Figure 2. An extensive-form Stackelberg game with infinite action space

In sequential games, a backward induction methodology is applied to search for the sub-game perfect Nash equilibrium, where the consideration order is reversed so that it flows from the follower to the leader. Concerning the Stackelberg leadership model, a first partial derivative method is applied to analyze the later player's best response functions located in the follower's action space.

3.2 The Stackelberg equilibrium: a general formulation

We identify two players in a non-cooperative sequential Stackelberg game that separately control the optimization objectives and control factors. Player 1's objective is to minimize his performance function ${\textstyle {f}_{1}(x,y)}$ and considers only factor ${\textstyle x}$. Correspondingly, the objective of player 2 is to minimize his performance function ${\textstyle {f}_{2}(x,y)}$ and considers only factor ${\textstyle y}$. Both functions are based on the factors ${\textstyle x}$ and ${\textstyle y}$, which means that the strategies of two players have interacted. A variable set ${\textstyle (x,y)\in {(S}_{x},\,{S}_{y})}$ refers to a pair of rational strategies played by the two players, where ${\textstyle {S}_{x}}$ and ${\textstyle {S}_{y}}$ are the rational strategies set of the two players, respectively. The game ends when the game leader has optimized his performance function, which also means that an equilibrium is found. The Stackelberg equilibrium point and the associated Stackelberg equilibrium solution are denoted as ${\textstyle \left({x}^{\ast },{y}^{\ast }\right)}$ and ${\textstyle \lbrace {f}_{1}\left({x}^{\ast },{y}^{\ast }\right),{f}_{2}\left({x}^{\ast },{y}^{\ast }\right)\rbrace }$ , separately. If player 1 acts as the leader and player 2 acts as the follower, based on the Stackelberg game rules, player 1 optimizes his objective ${\textstyle {f}_{1}(x,y)}$ first, and player 2 optimizes ${\textstyle {f}_{2}(x,y)}$ later. The Stackelberg equilibrium point ${\textstyle ({x}^{\ast },{y}^{\ast })}$ can be found in the objective function of player 1 if:

where ${\textstyle {f}_{1}(x,{\tilde {y}})}$ represents the objective function of player 1 that considers the best response of player 2 with the related controlled factor ${\textstyle {\tilde {y}}}$, which can be found in the objective function of player 2 by:

Accordingly, if player 2 is the leader and player 1 is the follower, the Stackelberg equilibrium solution can be obtained in the objective function of player 2 if:

where ${\textstyle {f}_{2}({\tilde {x}},y)}$ denotes the objective function of player 2 that takes the best response of player 1 with the corresponding controlled factor ${\textstyle {\tilde {x}}}$ into account, which can be found in the objective function of player 1 by:

3.3 Bi-objective robust design model

In a bi-objective robust design model, the optimization target is to minimize both the process bias ${\textstyle {({\hat {\mu }}\left({\boldsymbol {\mathrm {x} }}\right)-\tau )}^{2}}$ and the process variance ${\textstyle {\hat {\sigma }}^{2}({\boldsymbol {\mathrm {x} }})}$, where ${\textstyle {\boldsymbol {\mathrm {x} }}}$ is denoted as the vector of input factors. Then, a bi-objective robust design optimization problem can be integrated and formulated as follows:

where ${\textstyle \tau }$ refers to the target value of the process mean. ${\textstyle X}$ represents the feasible solution set, where ${\textstyle {R}^{n}\supseteq X=}$${\displaystyle \left\{{\boldsymbol {\mathrm {x} }}\in {R}^{n}:\,{g}_{m}\left({\boldsymbol {\mathrm {x} }}\right)\leq 0,\forall m\right\}}$, and ${\textstyle {g}_{m}\left({\boldsymbol {\mathrm {x} }}\right)}$ denotes the mth constraint. One possible solution ${\textstyle {\boldsymbol {\mathrm {x} }}_{1}\in X}$ is claimed that can dominate another solution ${\textstyle {\boldsymbol {\mathrm {x} }}_{2}\in X}$, if ${\textstyle {\left\{{\hat {\mu }}\left({\boldsymbol {\mathrm {x} }}_{1}\right)-\tau \right\}}^{2}\leq {\left\{{\hat {\mu }}\left({\boldsymbol {\mathrm {x} }}_{2}\right)-\tau \right\}}^{2}}$ and ${\textstyle {\hat {\sigma }}^{2}({\boldsymbol {\mathrm {x} }}_{1})\leq {\hat {\sigma }}^{2}({\boldsymbol {\mathrm {x} }}_{2})}$. An efficient solution ${\textstyle {\boldsymbol {\mathrm {x} }}^{\ast }\in X}$ (with the corresponding outcomes ${\textstyle {\left\{{\hat {\mu }}\left({\boldsymbol {\mathrm {x} }}^{\ast }\right)-\tau \right\}}^{2}}$ and ${\textstyle {\hat {\sigma }}^{2}({\boldsymbol {\mathrm {x} }}^{\ast })}$) is called the non-dominated solution if it cannot be dominated by any other solution in the model; it can then be regarded as an efficient trade-off between the objectives.

The conventional RSM is used to model and analyze response functions for estimating the process mean ${\textstyle {\hat {\mu }}\left({\boldsymbol {\mathrm {x} }}\right)}$ and standard deviation ${\textstyle {\hat {\sigma }}\left({\boldsymbol {\mathrm {x} }}\right)}$, which are:

where

and

where

In these two second-order equations, parameters ${\displaystyle {\bf {a}}}$ and ${\displaystyle {\bf {A}}}$ are the vector and matrix of regression coefficients for the estimated process mean; parameters ${\displaystyle {\bf {b}}}$ and ${\displaystyle {\bf {B}}}$ are the vector and matrix of regression coefficients for estimated process variance. The ordinary least square method is used to obtain the values of ${\displaystyle {\bf {a}}}$ and ${\displaystyle {\bf {b}}}$.

For an intuitive comparison among the Stackelberg solutions based on various situations, an MSE model is introduced as an evaluation criterion to determine the non-dominated solution, which is formulated as:

3.4 Proposed SGRD model

The integration procedure of the proposed bi-objective SGRD optimization problem from a Stackelberg game model is shown in Figure 3. The process bias and variance can be identified as the two game players of the proposed SGRD model. Then, the objective functions introduced in the Stackelberg leader-follower model, ${\textstyle {f}_{1}\left(x,y\right)}$ and ${\textstyle {f}_{2}\left(x,y\right)}$ , can be replaced by the estimated functions of process bias ${\textstyle {({\hat {\mu }}\left({\boldsymbol {\mathrm {x} }}\right)-\tau )}^{2}}$ and variance ${\textstyle {\hat {\sigma }}^{2}({\boldsymbol {\mathrm {x} }})}$, respectively. The optimization objectives of the bi-objective optimization problem ${\textstyle \left\{{({\hat {\mu }}\left({\boldsymbol {\mathrm {x} }}\right)-\tau )}^{2},\,{\hat {\sigma }}^{2}({\boldsymbol {\mathrm {x} }})\right\}}$ and the input factors ${\textstyle {\boldsymbol {\mathrm {x} }}}$ are separately controlled by the process bias and variance. We can define the input factors that are controlled by the process bias are denoted by ${\textstyle \,{\boldsymbol {\mathrm {x} }}_{\mu }}$, and ${\textstyle \,{\boldsymbol {\mathrm {x} }}_{\mu }\subseteq {\boldsymbol {\mathrm {x} }}}$. Correspondingly, as the complement of ${\textstyle {\boldsymbol {\mathrm {x} }}_{\mu }}$, the input factors that are controlled by the process variance are denoted by ${\textstyle {\boldsymbol {\mathrm {\,x} }}_{\sigma }}$,where ${\textstyle {\boldsymbol {\mathrm {x} }}_{\sigma }=}$${\displaystyle {{\boldsymbol {\mathrm {x} }}_{\mu }}^{c}\subseteq {\boldsymbol {\mathrm {x} }}}$. Similar to the Stackelberg game model, the response functions of process bias and variance are depended on the controlled factors ${\textstyle {\boldsymbol {\mathrm {x} }}_{\mu }}$ and ${\textstyle {\boldsymbol {\mathrm {x} }}_{\sigma }}$. Thus, the objective functions for the process bias and variance can be written as ${\textstyle {({\hat {\mu }}\left({\boldsymbol {\mathrm {x} }}_{\mu },{\boldsymbol {\mathrm {x} }}_{\sigma }\right)-\tau )}^{2}}$ and ${\textstyle {\hat {\sigma }}^{2}({\boldsymbol {\mathrm {x} }}_{\mu },{\boldsymbol {\mathrm {x} }}_{\sigma })}$, respectively. Based on the Stackelberg game rules, the target of process bias is to minimize ${\textstyle {({\hat {\mu }}\left({\boldsymbol {\mathrm {x} }}_{\mu },{\boldsymbol {\mathrm {x} }}_{\sigma }\right)-\tau )}^{2}}$ and considers only factor ${\textstyle {\boldsymbol {\mathrm {x} }}_{\mu }}$. Accordingly, the target of process variance is to minimize ${\textstyle {\hat {\sigma }}^{2}({\boldsymbol {\mathrm {x} }}_{\mu },{\boldsymbol {\mathrm {x} }}_{\sigma })}$ and considers only factor ${\textstyle {\boldsymbol {\mathrm {x} }}_{\sigma }}$.