Modelling and analysis of selective assembly using Taguchi's loss function

An assembly is the integrative process of joining components to make a completed product. It brings together the upstream process of design, engineering and manufacturing processes. The functional performance of an assembled product and its manufacturing cost are directly affected by the individual component tolerances. But, the selective assembly method can achieve tight assembly tolerance through the components manufactured with wider tolerances. The components are segregated by the selective groups (bins) and mated according to a purposeful strategy rather than being at random, so that small clearances are obtained at the assembly level at lower manufacturing cost. In this paper, the effect of mean shift in the manufacturing of the mating components and the selection of number of groups for selective assembly are analysed. A new model is proposed based on their effect to obtain the minimum assembly clearance within the specification range. However, according to Taguchi's concept, manufacturing a product within the specification may not be sufficient. Rather, it must be manufactured to the target dimension. The concept of Taguchi's loss function is applied into the selective assembly method to evaluate the deviation from the mean. Subsequently, a genetic algorithm is used to obtain the best combination of selective groups with minimum clearance and least loss value within the clearance specification. The effect of the ratio between the mating part quality characteristic's dimensional distributions is also analysed in this paper.     

Economical operation of ( x,s ) control chart indexed by Taguchi's loss function

Assume that lot quality characteristics obey a normal distribution. Kanagawa et al. have proposed the ( x, s ) control chart which enable the user to monitor both changes of the mean and variance simultaneously. Further, the results of Watakabe and Arizono enable the user to evaluate the performance for out-of-control state in the case of using the ( x, s ) control chart. On the other hand, Taguchi has presented an approach to quality improvement in which reduction of deviation from the target value is the guiding principle. In this approach, the loss is expressed as a quadratic form with respect to the difference between the measured value x of a product characteristic X and the target value T of a product quality characteristic. Then, we can evaluate the process quality based on the Taguchi's loss criterion. We consider here the economical operation of the ( x, s ) control chart in conformity with the expected total operation cost function based on the sampling cost and the loss due to derivation in the process quality. First, we consider the economical operation of the ( x, s ) control chart in the situation that the loss to be considered is known. Further, the economical operation of the ( x, s ) control chart is also discussed under any loss instead of a known loss. Then, the relationship between the two economical operations proposed here corresponds to the relationship between the lot tolerance per cent defective plans under the fixed fraction defective and the average outgoing quality limit plans under any fraction defective in rectifying inspection plans.     

Analysis of dynamic robust design experiments

Robust design is an important method for improving product or manufacturing process design by making the output response insensitive (robust) to diDcult-tocontrol variations (noise). Most of the robust design research in the literature focuses on problems with static responses. This paper investigates robust design problems with dynamic responses. For analysing robust design experiments with dynamic systems, Taguchi (1986) proposes a two-step procedure to identify the 'optimal' factor settings that minimize the average quadratic loss. In this paper we show that Taguchi's procedure is only appropriate under a multiplicative model. We develop the appropriate two-step procedure for dynamic systems under an additive model. The procedure reduces the dimension of the optimization problem and allows for future changes of the target slope without re-optimization. We illustrate the proposed procedure and Taguchi's procedure with real examples. We also discuss future research and extensions to general classes of models.     

Combined parameter and tolerance design optimization with quality and cost

The need to remain competitive for survival in the current world market has led manufacturing sectors to consider the low cost and high quality of product and process design. Production of quality products at low cost in today's manufacturing industry requires simultaneous consideration of product design and process planning, particularly in the early stage of design and planning. During product design, parameter design determines the design target (design setting) and tolerance design determines design tolerance. During process design, the parameter design determines the process mean (process setting) and tolerance design determines process tolerance. This study provides a mathematical relationship to link the elements of design target, design tolerance, process mean and process tolerance in one equation. By following this equation, manufacturability for all possible combinations of product and process design can be ensured, which increases the flexibility of both product design and process planning. With this in mind, an analysis model that includes manufacturing cost and quality loss simultaneously has been developed to determine the optimal values of design tolerance, process mean and process tolerance. The proposed model provides a method of combining the optimization of parameter and tolerance design over product/process in the early stage of design.     

Sequential optimization of parameter and tolerance design for multiple dynamic quality characteristics

System design, parameter design and tolerance design are the three stages of design process as presented by G. Taguchi. Systems design identifies the basic elements of the design to provide new or improved products to customers. Parameter design determines the optimal parameter settings, which will minimize variation from the target performance of the product. Tolerance design finally identifies the components of the design, which are sensitive in terms of affecting the quality of the product, and establishes tolerance limits that will give the required level of variation in the design. Most studies have focused primarily on optimizing the parameter design or tolerance design for multiple static quality characteristics. In this paper, a mathematical formula corresponding to the model is derived from Taguchi's quadratic quality loss function to minimize the expected total cost for the parameter design of multiple dynamic quality characteristics. When the optimal parameter design is not sufficient to reduce the output variation, the first-order Taylor series expansion is then used to analyse the variations of noise factors for optimizing the tolerance design. It concludes with an example demonstrating this approach.     

Economic parameter design for ultra-fast laser micro-drilling process

The basic requirement in this type of micro-drilling process is to achieve high product quality with the minimum machining cost, which can be realised through parameter design. In this paper, we propose a new economic parameter design under the framework of Bayesian modelling and optimisation. First of all, the Bayesian seemingly unrelated regression (SUR) models are utilised to develop the relationship models between input factors and output responses in the laser micro-drilling process. After that, simulated response values which reflect the real laser micro-drilling process are obtained by using the Gibbs sampling procedure. Moreover, a novel rejection cost function and a quality loss function are constructed based on the simulated responses. Finally, an optimisation scheme integrating the rejection cost (i.e. rework cost and scrap cost) function and the quality loss function is implemented by using multi-objective genetic algorithm to find feasible economic parameter settings for laser micro-drilling process.     

Economic design of X̄ and R charts under Weibull shock models

This paper considers the problem of a continuous production process where both the mean and variance are simultaneously monitored by an X̄ and R chart respectively, and generalizes the model of Costa (IIE Transactions 1993; 25 (6):27–33). The product variable quality characteristic is assumed to be normally distributed and the process is subject to two independent assignable causes (such as tool wear‐out, overheating or vibration). One cause changes the process mean and the other changes the process variance. However, the occurrence of one kind of assignable cause does not preclude the occurrence of the other. It is also assumed that the occurrence times of the assignable causes are described by Weibull distributions with increasing failure rates. A cost model is developed and a non‐uniform sampling interval scheme is adopted. A two‐step search procedure is employed to determine the optimal design parameters. The relative contribution of the paper over the results obtained by Costa is addressed. A sensitivity analysis of the model is conducted and the cost savings associated with the use of non‐uniform sampling intervals instead of constant sampling intervals are evaluated. The economic design model is then extended to an economic–statistical design model for achieving desired levels of statistical performance while minimizing the expected cost. Performances of purely economic design and economic–statistical design are compared. Copyright © 2000 John Wiley & Sons, Ltd.     

Using analytic hierarchy process to determine process economics in multivariate loss functions

Performance of a product usually depends on several responses (quality characteristics) which must meet all of the specifications simultaneously. This could be achieved by applying of (the) robust design methodology to problems with multiple characteristics. In the literature, several works have been published concerning multi-response optimization methods, which aim to achieve the best possible robustness. One of the approaches for multi-response optimization is Loss Function Approach which allows the practitioner to include variance–covariance structure of the responses, prediction quality and the economic importance of the responses relevant to the product or process. In this paper, we propose utilizing Analytic Hierarchy Process, a multi-criteria decision making tool, to determine the economic importance matrix in the multivariate loss function. An example of the suggested method is presented on a study conducted for a company producing water proof polymer roofing materials.     

Taguchi methods in practice: An analysis of Goh's paradox

The background of ‘Goh's paradox’ in connection with the application of the common version of Genichi Taguchi's parameter design routine is examined. A detailed case study is used to show how the paradox can be resolved, as well as to assess the probability of success of marginal analysis in Taguchi's prescribed procedure for quality and reliability improvement.     

Economic–statistical Design of Maximum Exponentially Weighted Moving Average Control Charts

The maximum exponentially weighted moving average (MaxEWMA) control chart effectively combines the two EWMA charts into one chart and monitors both increases and decreases in the mean and/or variability. In this paper, we develop the economic–statistical design of the MaxEWMA control chart in which the Taguchi's quadratic loss function is incorporated into the Duncan's economical model. Numerical simulations are executed to minimize the expected total cost model and determine the optimal decision variables, including the sample size, sampling interval, control limit width, and the smoothing constant of the MaxEWMA control chart. It is shown that the optimal control limit width and smoothing constant increase as the optimal cost value increases and that both the optimal sample size and sampling interval always decrease as the magnitudes of mean and/or variance shifts increase. Copyright © 2012 John Wiley & Sons, Ltd.     

Robust design for multiple dynamic quality characteristics using data envelopment analysis

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 individual 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 analysis. 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.     

Robust design with dynamic characteristics using stochastic sequential quadratic programming

Robust design with dynamic characteristics is an important off-line quality engineering technique for improving product quality over a range of input conditions by reducing variations caused by uncontrolled factors. Since several studies have indicated that there are important limitations to Taguchi's S/N ratio analysis, the solution procedure for dynamic systems deserves further investigation. This paper proposes a stochastic optimization modeling procedure to overcome the difficulty in Taguchi's method to accommodate dynamic characteristics. The main idea underlying the proposed method is to minimize the total variations on quality characteristics while attaining the target performance over a range of input conditions. Due to the nonlinear nature of the stochastic optimization model, two stochastic versions of sequential quadratic programming respectively embedded with a Monte Carlo simulation and numerical approximations are devised to solve the problem. In the robust design of a temperature control circuit often discussed in dynamic problems, the proposed method performs efficiently and effectively. Compared with the Taguchi method, the design solved in this paper has smaller variations, indicating that the proposed method is a promising technique for dynamic-characteristic robust design.     

A comparison of deterministic and statistical sampling techniques for quality analysis of integrated circuits

There has been a great amount of publicity about Taguchi methods which employ deterministic sampling techniques for robust design. Also given wide exposition in the literature is tolerance design which achieves similar objectives but employs random sampling techniques. The question arises as to which approach—random or deterministic—is more suitable for robust design of integrated circuits. Robust design is a two-step process and quality analysis—the first step—involves the estimation of ‘quality factors’, which measure the effect of noise on the quality of system performance. This paper concentrates on the quality analysis of integrated circuits. A comparison is made between the deterministic sampling technique based on Taguchi's orthogonal arrays and the random sampling technique based on the Monte Carlo method, the objective being to determine which of the two gives more reliable (i.e. more consistent) estimates of quality factors. Results obtained indicated that the Monte Carlo method gave estimates of quality which were at least 40 per cent more consistent than orthogonal arrays. The accuracy of prediction of quality by Taguchi's orthogonal arrays is strongly affected by the choice of parameter quantization levels—a disadvantage—since there is a very large number (theoretically infinite) of choices of quantization levels for each parameter of an integrated circuit. The cost of the Monte Carlo method is independent of the dimensionality (number of designable parameters), being governed only by the confidence levels required for quality factors, whereas the size of orthogonal array required for a given problem is partly dependent on the number of circuit parameters. Two integrated circuits—a 7-parameter CMOS voltage reference and a 20-parameter bipolar operational amplifier—were employed in the investigation. Quality factors of interest included performance variability, acceptability (relative to customer specifications) and deviation from target.     

Optimization of product and process design for quality and cost

Robust product and process design is an important technique for achieving high quality at low cost. It involves making the product's function much less sensitive to various sources of noise such as manufacturing variation, environmental variation and deterioration. This is a problem in optimization involving minimization of the mean square loss resulting from the deviation of the product's function from its target. Here we show that the optimization can be carried out in two steps: first maximize a quantity called signal-to-noise ratio (S/N) and then bring the performance on target by special adjustment parameters. The two-step procedure works for a wide variety of product functions and makes the optimization process more efficient and practical compared to the direct minimization of the quadratic loss function.     

Process parameters determination for precision manufacturing

Conventional process planning in manufacturing operations presents fixed process means and process tolerances for all operations and allows actual outputs to be distributed around these fixed values, as long as the final outputs fall within acceptable specifications. Some approaches attempt to maximize the process tolerances of all manufacturing operations for part production. Other approaches intend to minimize the tolerance cost or quality loss based on known functions. Most of them consider process mean and process tolerance as independent decision variables in process planning, with the condition that the resultant working dimensions are equal to the design target values of blueprint dimensions. These approaches assume that there is no process drift or deterioration. However, these conventional approaches are inappropriate for small‐volume, high‐value and precision processing, particularly of a complex part. Hence this study introduces an alternative approach to the tolerance‐balancing problem that does not provide specific objective functions, which determines process means and process tolerances simultaneously and adjusts them sequentially. Copyright © 2000 John Wiley & Sons, Ltd.     

Robust Parameter Design: A Review

Parameter design is an engineering methodology intended as a cost‐effective approach for improving the quality of products and processes. The assumption is that there are both controllable factors (control variables) and uncontrollable/difficult to control factors (noise variables) that operate on the quality characteristic of a process. The goal of parameter design is to choose the levels of the control variables that optimize a defined quality characteristic while minimizing the variation imposed on the process via the noise variables. Parameter design was popularized in the mid 1980s by Japanese quality consultant Genichi Taguchi. A panel discussion edited by Nair summarized important responses to Taguchi's ideas and methodology. In the last decade, there have been many applications and new developments in this important area. This review paper focuses largely on the work done since 1992, but a historical perspective of parameter design is also given. Copyright © 2003 John Wiley & Sons, Ltd.     

The use of a generalized signal-to-noise ratio to identify adjustment and dispersion factors in taguchi experiments

A signal-to-noise ratio proposed by Taguchi for ‘nominal the best’ can be made more flexible and used additionally for the cases ‘smaller the better’ and ‘larger the better’. The main feature which distinguishes this generalized ratio from Taguchi's signal-to-noise ratios is that its precise form is determined by the experimental data. Taken together with a transformation of the mean response it effectively identifies adjustment (signal) factors and dispersion factors. Examples are given to illustrate the routine operational procedure and also to demonstrate that Taguchi's signal-to-noise ratios can lead to inefficient, and sometimes incorrect, identification of design factors. Two important considerations are to preserve the evident appeal of the Taguchi method to engineers and also to provide a theoretical justification which is acceptable to statisticians. The principal objective of the joint transformation is to achieve approximate functional independence between the mean and variance in the new metric. This, in turn, leads to efficient identification of adjustment and dispersion factors. A comparison is made with a similar approach using Box and Cox transformations.     

Unification of robust design and goal programming for multiresponse optimization—a case study

Fixing the levels of input process parameters to meet a required specification of output is a common process quality control problem. Especially when the output has many quality characteristics, and each of these quality characteristics has to satisfy a given specification, difficulties may arise. One such problem was encountered in an injection moulding process. This process was optimized using Taguchi's Robust Design methodology. Details of the process, problems encountered and outcome of optimization are presented in this paper. The optimization study using Taguchi's methodology revealed that the optimum conditions obtained for one response are not completely compatible with those of other responses. So trade-offs were made in selection of levels for factors using engineering judgement. This increases the uncertainty in the decision making process. In this paper, an approach is presented to optimize multiresponses simultaneously using goal programming in conjunction with Taguchi's methodology. Details of modelling, analysis and inferences obtained with relevance to the case are presented. This study revealed that the optimum conditions obtained using goal programming in conjuction with Taguchi's methodology have better goal attainment properties compared to Robust design. To understand goal attainment behaviour of output characteristics for various process conditions, a detailed sensitivity analysis was also conducted. The outcome of this analysis is also discussed in this paper. © 1997 John Wiley & Sons, Ltd.     

An intelligent approach to robust multi-response process design

In order to meet strict customer demands in a global highly-complex industrial sector, it is necessary to design manufacturing processes based on a clear understanding of the customer's requirements and usage of a product, by translating this knowledge into the process parameter design. This paper presents an integrative, general and intelligent approach to the multi-response process design, based on Taguchi's method, multivariate statistical methods and artificial intelligence techniques. The proposed model considers process design in a general case where analytical relations and interdependency in a process are unknown, thus making it applicable to various types of processes, and incorporates customer demands for several (possible correlated) characteristics of a product. The implementation of the suggested approach is presented on a study that discusses the design of a thermosonic copper wire bonding process in the semiconductor industry, for assembly of microelectronic devices used in automotive applications. The results confirm the effectiveness of the approach in the presence of different types of correlated product quality characteristics.     

Integrating statistical process control,engineering process control and Taguchi's quality engineering

In this paper, a scheme is proposed to integrate statistical process control (SPC), engineering process control (EPC) and Taguchi's quality engineering (TQE). Then, two models are proposed to implement the proposed scheme. The models employ the concept of Taguchi's quadratic loss function to determine whether to take an EPC action by comparing the cost of the action and the cost of quality. A case study is used to compare these two models with the model in the literature where SPC and EPC have been integrated. The results have shown that the first model resulted in about a 25% saving and the second model resulted in even greater saving of about 30% for the case under consideration.