Quadratic Quality Loss Function Research Articles

Process machining allowance for reliability analysis of mechanical parts based on hidden quality loss

The machining allowance variation is significant for the reliability of a part during the machining process. Usually, when the machining allowance of a part increases, the machining and production cost also increase. When the machining allowance decreases, the machining surface will have defects. The parts will produce many scraps and reliability will decrease. The machining allowance of a part consists of multiple process machining allowances. To analyze the impact caused by machining allowance variation, the hidden quality loss and process machining allowance are combined through the process capability index (PCI). Then the asymmetric quadratic quality loss function (AQF) and quadratic exponential function (QEF) are used to analyze them. A prediction model of hidden quality loss of process machining allowance is proposed. On the premise that the quality characteristic value obeys normal function distribution, a numerical model is given and used to obtain process machining allowance-inherent reliability of the product. The actual case is used to compare and verify the two models.

Optimum specification limit coefficient setting based on the smaller-the-better quality characteristics

In this work, the authors propose the application of type I continuous sampling plan (CSP-1 plan) with expected inspection cost, expected replacement cost, expected warranty cost, expected quality loss for conforming product, and expected quality loss for nonconforming product per item. The quality characteristics of product is considered as the normal distribution with smaller-the-better characteristics. The quality loss function of product addresses the quadratic quality loss function. The CSP-1 plan is applied in the process control. The optimal combination of parameters is obtained from the minimum expected total cost per item produced under the specified average outgoing quality limit (AOQL) protection. One needs to have the optimal clearance number, sampling frequency, and specification limit coefficient.

The integrated application of acceptance sampling in the quality control of manufacturing process and quality assurance of shipment product

In this work, the authors propose the integrated application of CSP-1 plan and Dodge-Romig rectifying inspection plan with expected inspection cost, expected replacement cost, expected warranty cost, expected quality loss for conforming product, and expected quality loss for non-conforming product per item. The quality characteristics of product is considered as the normal distribution with nominal-the-best and symmetric quadratic quality loss function. The CSP-1 plan is firstly applied in the process control. The optimal combination of parameters is obtained from the minimum expected total cost per item produced under the specified average outgoing quality limit (AOQL) protection. One needs to have the optimal clearance number, sampling frequency, and specification limit coefficient. Then Dodge-Romig rectifying inspection plan is applied in the product improvement. The optimal sample size, clearance number, and quality investment are obtained by minimizing the expected total cost per lot under the specified consumer’s risk.

Optimum specification limit setting under the larger-the-better quality characteristics

In this paper, the optimal clearance number and sampling frequency of type I continuous sampling plan (CSP-1 plan) and specification limit coefficient of product are determined with the minimum expected total cost per item under the specified average outgoing quality limit (AOQL) protection. The expected total cost per item includes the expected inspection cost, the expected replacement cost, the expected warranty cost, the expected quality loss for conforming product, and the expected quality loss for non-conforming product per item. The quality characteristic of product is considered as the normal distribution with larger-the-better characteristics. The quality loss function of product addresses the quadratic quality loss function. The sensitivity analysis of numerical example will be provided for illustration.

Quality control for production and shipment process of product

In this article, the authors propose the integrated application of type I continuous sampling plan (CSP-1 plan) and Dodge-Romig rectifying inspection plan with quality loss and product cost. The normal quality characteristic, nominal-the-best, and asymmetric quadratic quality loss function of product are considered. The CSP-1 plan is firstly applied in the process control. Then Dodge-Romig rectifying inspection plan is applied in the product improvement. The optimal combination of parameters of CSP-1 plan and asymmetric specification limit coefficients are obtained under the specified average outgoing quality limit (AOQL) protection. The optimal combination of parameters of Dodge-Romig rectifying inspection plan and quality investment are determined under the specified consumer’s risk. Finally, the sensitivity analysis of numerical example will be provided for illustration.

Reliability analysis for mechanical parts considering hidden cost via the modified quality loss model

AbstractThe improvement of mechanical parts inherent reliability has an impact on the reputation and performance of the company. To estimate the inherent reliability of products more conveniently and economically, a hidden quality cost‐production cost (HQC‐PC) reliability prediction model is put forward. To estimate the hidden quality cost (HQC) of products more accurately, a quadratic exponential quality loss function model is established, which is different from Taguchi's quadratic quality loss function (QLF) and the modified QLFs. In the new quality loss model, the growth rate of quality loss on both sides of the target value is considered. Under the condition that the quality characteristic value obeys normal distribution, the general estimation formulas of HQC in the tolerance range is obtained considering sampling error and the numerical model of inherent reliability is established. The effect of different parameters on the inherent reliability of products is discussed with practical case, such as design and production parameters. Then, the appropriate process capability index (PCI) is selected according to different production processes. The relationship between the HQC‐PC reliability prediction model and PCI is derived by a numerical model of inherent reliability. A new analysis method of inherent reliability is proposed.

Optimal process mean setting with asymmetric quadratic quality loss and process adjustment cost based on exponential product characteristic

In this study, the author presents the economic selection of process mean with the quality loss and process adjustment cost. Assume that the quality characteristic is exponentially distributed with parameter (λ). The product quality loss is measured by adopting the asymmetric quadratic quality loss function. The adjustment cost of process mean is assumed to be proportional to μ 2 and the adjustment cost of process variance is assumed to be proportional to 1/σ 2. The objective is to obtain the optimal process mean under the minimization of the expected total cost of product including the process adjustment cost and product quality loss. The numerical example and sensitivity analysis of some parameters are provided for illustration.

Optimal process mean selection by considering trade-off model with product quality loss and process adjustment cost

In this study, the author considers the problem about economic process mean selection of product with the quality loss and process adjustment cost. Assume that the quality characteristic is normally distributed with process mean (μ) and variance (σ 2). The product quality loss is measured by adopting the asymmetric quadratic and line quality loss function. The adjustment cost of process mean is assumed to be proportional to μ 2 and the adjustment cost of process variance is assumed to be proportional to 1/σ 2. One can obtain the optimal process mean by minimizing the expected total cost of product including the product quality loss and process adjustment cost. Finally, the numerical example and sensitivity analysis of parameters are provided for illustration.

Economic manufacturing quantity model with machine breakdown and deteriorating production process

In this study, the author proposes a jointly setting model for determining the process mean and economic manufacturing quantity (EMQ) under the machine breakdown and deteriorating production process. The system addresses the corrective maintenance, preventive maintenance, and allowable shortage. The quality loss of conforming product is considered and Taguchi’s asymmetric quadratic quality loss function is used for evaluating the product quality. The optimal process mean and economic manufacturing quantity are jointly determined by minimizing the total expected cost of product per unit time including the set-up cost, holding cost, corrective maintenance cost, preventive maintenance cost, shortage cost. A solution procedure is devised to obtain the optimal solution and the sensitivity analysis of key parameters is conducted to investigate the effect on the optimal solution.

Integration of the variance of quadratic loss for evaluating process performance

This article interprets Taguchi’s quadratic quality loss function from a different viewpoint by considering the variance of quadratic loss as well as the mean of quadratic loss. The behavior of the variance of quadratic loss is characterized by the kurtosis and variance of the quality characteristic. To evaluate the location and dispersion performances of the quadratic loss simultaneously, a distance method linked with a Pareto front approach is proposed for process performance evaluation.

A cubic quality loss function and its applications

AbstractThe traditional quality evaluation method believed that when the product quality characteristic was within the specification limit, no loss was produced. Taguchi proposed that even if the characteristic was within the range of users' demand, the fluctuation of quality characteristic would still cause loss to users and society. Therefore, he proposed a quadratic quality loss function to describe this loss. The function was established based on the Taylor expansion. It neglected terms with powers higher than 2, which would cause a certain deviation between the calculated and true value. Moreover, the tolerance and loss in the quadratic loss function must satisfied specific relationship, which limited its use. In this paper, the Taylor expansion items are raised to third order. The quality loss coefficients are discussed and analyzed so that the cubic quality loss function is established. In addition, the method of calculating hidden quality cost is given by using the cubic loss function. The hidden quality cost is affected by the quality loss coefficients and should be a range. The cubic quality loss function studies the inapplicability of quadratic loss function, so it widens the scope of application of quality loss function.

Optimal process mean setting under the quality and cost

In this study, the author considers the optimal process mean setting problem with process adjustment cost and product quality. Assume that the process adjustment cost is proportional to the square of process mean and reciprocal of variance. Taguchi’s asymmetric quadratic quality loss function is adopted for measuring the product quality. The uniform and triangular distributions of quality characteristic are addressed in the formulation of mathematical model. The optimal process mean will be obtained by minimizing the expected total cost of product. Finally, the numerical example and sensitivity analysis of parameters are provided for illustration.

Process Capability and Average Roughness in Abrasive Water Jet Cutting Process of Stainless Steel

Process capability analysis is frequently employed to evaluate if a product or a process can meet the customer’s requirement. In general, process capability analysis can be represented by using the process capability index. Until now, the process capability index was frequently used for manufacturing processes with quantitative characteristics. However, for a process with qualitative characteristic like cutting surface, the data’s type and single specification caused limitations of using the process capability index. Taguchi developed a surface quality by abrasive water jet cutting or quadratic quality loss function to address such issues. In this study, we intend to construct a measurable index which incorporates the process capability index philosophy concept to analyze the process capability with the consideration of the qualitative surface roughness. The manufacturers can employ the proposed index to self-assess the process capability. The objective of this study was to examine the effects of abrasive water jet machining variables like cutting speed of the stainless steel material. The roughness of the varied surface through the cut depth was also measured and determined as a process capability index of 3 zones machined surface.

Hidden quality cost function of a product based on the cubic approximation of the Taylor expansion

The-Nominal-The-Best (N-type) loss function is established based on the Taylor expansion. Results become more accurate as more Taylor expansion items are retained. N-type loss function neglects terms with powers higher than two, which inevitably leads to a certain deviation between the calculated result and the true value. In this paper, Taylor expansion is retained to the third-order, and the quality loss function is extended to three items. The quality loss coefficients of each item are determined, and the asymmetric piecewise cubic quality loss function is established. The deviation between the cubic and quadratic functions is evaluated. The formula for calculating the hidden quality cost of a product is derived by choosing an appropriate density distribution function and using process capability. Two cases are utilised to analyse and discuss the quality loss and hidden quality cost of a product using the cubic quality loss and quadratic quality loss functions. This paper provides a more accurate approach for the study of product quality management.

The modified optimum profit model with/without product warranty period under the imperfect quality, single supplier, and multi-retailers

In this study, the author proposes a modified Chen and Liu’s [1] pure procurement model with/without product warranty period under the imperfect quality, single supplier and multi-retailers. The 100% inspection for the supplier’s product is executed. The conformance product is sold to the retailers and the non-conformance one is scrapped and sold at a lower price. Taguchi’s quadratic quality loss function is applied in measuring the product quality. The optimum supplier’s wholesale price, process mean, and product warranty period and each retailer’s economic order quantity will be jointly determined under the maximum expected total profit including the supplier and retailers.

Imperfect production process model with product rework, multi-delivery, and quality loss

Krishnamoorthi and Panayappan developed an imperfect manufacturing system with multi-delivery policy and rework for determining the optimal number of deliveries and production run. However, their model did not consider the quality cost of conforming products. In this article, the authors presents a modified Krishnamoorthi and Panayappan’s model for obtaining the optimal process mean (process quality level), production run, and the number of deliveries based on the minimization of the expected total cost of product per unit time. Taguchi’s asymmetric quadratic quality loss function is considered for measuring the product quality.

Determining the optimum process mean in a two-stage production system based on conforming run length sampling method

We develop a profit optimization model based on Markovian approach to determine the optimum process mean in the two-stage serial production system. If quality characteristics of the items fall within the specification limits then the items are accepted. Our inspection approach in sampling plan is that when the number of conforming items between successive defective items falls below the lower control threshold, then the batch is rejected, and if it falls above the upper control threshold, then the batch is accepted and if it falls within the control thresholds then the process of inspecting the items continues. Taguchi’s symmetric quadratic quality loss function is applied for evaluating the product quality. Then a numerical example is provided to demonstrate the application of proposed model. Also a sensitivity analysis of the model is provided.

Proposing a two-criterion quality loss function using critical process capability indices

PurposeAverage quadratic quality loss function (QQLF) measures quality of a given process using mean shift from its target value and variance. While it has a target parameter for the mean, it lacks a target for the variance revisable for counting any progress of the process across different quality levels, above/below the standard level; thus, it appears too general. Hence, in this research, it was initially supposed that all processes are located at two possible quality spaces, above/below the standard level. The purpose of this paper is to propose a two-criterion QQLF, in which each criterion is specifically proper to one of the quality spaces.Design/methodology/approachSince 1.33 is a literarily standard or satisfactory value for two most important process capability indices Cp and Cpk, its upper/lower spaces are assumed as high-/low-quality spaces. Then the indices are integrated into traditional QQLF, of type nominal the best (NTB), to develop a two-criterion QQLF, in which each criterion is more suitable for each quality space. These two criteria have also been innovatively embedded in the plan-do-check-act (PDCA) cycle to help continuous improvement. Finally, the proposed function has been examined in comparison with the traditional one in Feiz Hospital in the province of Isfahan, Iran.FindingsResults indicate that the internal process of the studied case is placed on the lower quality space. So the first criterion of revised QQLF gives a more relevant evaluation for that process, compared with the traditional function. Moreover, this study has embedded both proposed criteria in the PDCA cycle as well.Research limitations/implicationsFormulating the two-criterion QQLF only for observations of normal and symmetric distributions, and offering it solely for NTB characteristics are limitations of this study.Practical implicationsTwo more relevant quality loss criteria have been formulated for each process (service or manufacturing). However, in order to show the comprehensiveness of the proposed method even in service institutes, emergency function of Feiz Hospital has been examined.Originality/valueThe traditional loss function of type NTB merely and implicitly targets zero defect for variance. In fact, it calculates quality loss of all processes placed on different quality spaces using a same measure. This study, however, provides a practitioner with opportunity of targeting excellent or satisfactory targets.

The joint determination of specification limits, process mean, and economic manufacturing quantity

This study proposes a modified economic manufacturing quantity (EMQ) model with deteriorating production process. Assume that the quality characteristic of the product is normally distributed. Consider the unstable production process with positive shift and an age-dependent positive drift while process variance increases. The target value of product characteristic is considered to be equal to its specification center and Taguchi’s symmetric quadratic quality loss function is applied to evaluate the product quality. The specification limits, process mean, and EMQ are jointly determined by minimizing the total expected cost of product per unit time under the specified process capability index Cpm value. A solution procedure is devised to find the optimal solution for modified EMQ model and the sensitivity analysis of key parameters is conducted to investigate the effect on the optimal solution. Numerical results show that the expected total cost of product per unit time is significantly influenced by the process capability index, the quality loss coefficient, the production rate, the demand rate, and process standard deviation.

Optimal tolerance design of hierarchical products based on quality loss function

Taguchi’s loss function has been used for optimal tolerance design, but the traditional quadratic quality loss function is inappropriate in the tolerance design of hierarchical products, which are ubiquitous in industrial production. This study emphasizes hierarchical products and extends the traditional quality loss function on the basis of Taguchi’s quadratic loss function; the modified formulas are subsequently used to establish quality loss function models of the nominal-the-best, larger-the-better, and smaller-the-better characteristics of hierarchical products. An example is presented to demonstrate the application of the extended smaller-the-better characteristic loss function model to the optimal tolerance design of hierarchical products. Furthermore, the problem associated with selecting materials of various grades in the design process is discussed. The results show that the extended quality loss function model demonstrates good operability in the tolerance design of hierarchical products.