Complexity

Volume 2017 (2017), Article ID 4376809, 17 pages

https://doi.org/10.1155/2017/4376809

## Fuzzy and Control Charts: A Data-Adaptability and Human-Acceptance Approach

^{1}Department of Industrial Engineering and Management, National Kaohsiung University of Applied Sciences, Kaohsiung 80778, Taiwan^{2}Office of Scientific Research, Lac Hong University, Dong Nai, Vietnam^{3}Department of Industrial Engineering and Management, Cheng Shiu University, Kaohsiung 83347, Taiwan^{4}Dong Nai Technology University, Dong Nai, Vietnam

Correspondence should be addressed to Dinh-Chien Dang

Received 11 October 2016; Revised 20 March 2017; Accepted 27 March 2017; Published 30 April 2017

Academic Editor: Thierry Floquet

Copyright © 2017 Ming-Hung Shu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

For sequentially monitoring and controlling average and variability of an online manufacturing process, and control charts are widely utilized tools, whose constructions require the data to be real (precise) numbers. However, many quality characteristics in practice, such as surface roughness of optical lenses, have been long recorded as fuzzy data, in which the traditional and charts have manifested some inaccessibility. Therefore, for well accommodating this fuzzy-data domain, this paper integrates fuzzy set theories to establish the fuzzy charts under a general variable-sample-size condition. First, the resolution-identity principle is exerted to erect the sample-statistics’ and control-limits’ fuzzy numbers (SSFNs and CLFNs), where the sample fuzzy data are unified and aggregated through statistical and nonlinear-programming manipulations. Then, the fuzzy-number ranking approach based on left and right integral index is brought to differentiate magnitude of fuzzy numbers and compare SSFNs and CLFNs pairwise. Thirdly, the fuzzy-logic alike reasoning is enacted to categorize process conditions with intermittent classifications between in control and out of control. Finally, a realistic example to control surface roughness on the turning process in producing optical lenses is illustrated to demonstrate their data-adaptability and human-acceptance of those integrated methodologies under fuzzy-data environments.

#### 1. Introduction

In nowadays fierce, competitive marketplaces, providing consistent and reliable quality products has been acknowledged as one of the most significant criteria for industrial manufacturers to persist their survival and sustainable growth. As such, establishing effective quality management systems and programs has become their prioritized strategy for lowering the percentage of nonconformities, slashing manufacturing costs, and fulfilling customer satisfaction [1]. Among them, the quality-control scheme has been widely advocated as a powerful control tool for achieving production effectiveness, as well as remaining quality-based competitive advantages [2].

Currently, for practically monitoring and controlling manufacturing processes, Shewhart-type control charts are extensively applied due to their notable capability on genuinely and early revealing process-abnormal conditions so as to ward off mistaken process intervention, prevent excessive product-defects, and eliminate costly scrap or rework of final products [1, 3]. Typically, the control chart constitutes three tracking lines, a center line (CL) and upper and lower control limits (UCL and LCL), whose construction is generally based on moderate numbers of subgroup sample data, each with either equal or unequal sample size, randomly being drawn from the process’s key quality characteristic. For monitoring the process conditions, the control charts function under the situations that when all statistic points of collected sample data fall within the limits and do not exhibit any systematic pattern, the process is classified as in statistical control and no interference is needed; otherwise, the process is being suspected to be affected by some assignable causes that deserve to be comprehensively investigated through well-structured corrective actions [1].

Traditionally, Shewhart control charts are constructed based on random precise data collected from a key quality characteristic. However, in practice, crisp data may fail to describe the nature of several applications, such as the surface roughness of components, the transmission speed of certain lights through a material, and the coating thickness of industrial cutting tools, because they cannot be recorded or measured precisely [4, 5]. Besides, making decisions on whether those products are conforming or nonconforming or judging if a process is in control or out of control usually includes some extent of human subjectivity relating to decision-makers’ intelligence and perceptions. These issues create the vagueness in the measurement system; therefore, the recorded data are considered as fuzzy data [6–9]. With the presence of fuzziness, the variance of normal observations tends to increase [10], and some intermediate decisions indispensably exist in-between the binary classification [6]. Thus, in order to adapt to these fuzzy environments, the traditional control charts with binary classifications are necessarily extended to* “fuzzy control charts”* [7, 11] which are considered as an inevitable and suitable choice in monitoring and controlling a manufacturing process with fuzzy data [7, 12–14]. As such, several fuzzy control charts have been proposed and constructed, for instance, fuzzy and control charts by Senturk and Erginel [7], fuzzy and control charts by Shu and Wu [11], fuzzy and control charts by Nguyen et al. [15], fuzzy MaxGWMA control chart by Shu et al. [4], and fuzzy control chart for multiple objective decision-making problem [16].

Nevertheless, certain problems in the construction and evaluation of fuzzy control charts have been raised. Particularly, Wang and Raz [17] proposed multigrades linguistic terms such as perfect, good, medium, poor, and bad to express the key quality characteristic. However, the underlying probability distribution of the linguistic data was not considered [18]. Hence, Kanagawa et al. [18] suggested estimating the probability distribution existing behind the linguistic data before constructing the control charts. Then, Laviolette et al. [19] and Asai [20] pointed that the probability estimation cannot be easily determined. In addition, as the membership function of linguistic terms is obtained arbitrarily on a given scale regardless of the fuzziness in the judgments of experts [11, 21], the linguistic-based control charts are not firmly validated. Similarly, using defuzzification methods such as fuzzy midrange, fuzzy mode, fuzzy median, and fuzzy average in constructing fuzzy control charts proposed in several researches [7, 17, 22] has also raised a core controversial issue of losing the fuzziness information in the original data as well as misjudgement of the manufacturing process [11, 23, 24], although it allows the control charts to be constructed with binary classifications.

Thus, several scholars have put great effort into preserving the fuzziness of vague data in their approaches. For examples, Grzegorzewski and Hryniewicz [25] utilized the necessity index of strict dominance (NISD); however, Chien et al. [26] claimed that the NISD is content-dependent because the ranking results may change when a new fuzzy number is added. Also, Gülbay and Kahraman [13] came up with an acceptable percentage index called direct fuzzy approach (DFA) which was then found failing in obtaining the fuzzy sample means and variances with the simple using of -cuts [11]. Shu and Wu [11] developed a fuzzy dominance approach (FDA) by extending Yuan’s fuzzy-numbers ranking method [27]. Nevertheless, the FDA approach can only perform nicely at the dominance degree greater than 0.5. Nguyen et al. [28] proposed a detailed procedure to classify a process, but some of their rules were found indistinguishable by Nguyen et al. [15], who later proposed a remedy for a better performance. Though Shu et al. [4] established a thorough system to evaluate manufacturing processes, their classification rules seem quite complicated. Therefore, this paper aims at providing an easier procedure by simplifying Yu and Dat’s ranking method [29]. In addition, our fuzzy and charts can generally deal with variable sample size which is the key advantage of our proposed control charts over those of Nguyen et al. [28].

This paper is organized as follows. Section 2 briefly provides key characteristics of traditional and control charts, playing as the foundation for our detailed procedure to construct fuzzy and control charts presented in Section 3. Also in Section 3, an empirical case in monitoring surface roughness of optical lenses in its turning process is conducted as a paradigm to illustrate the applicability of this new extended approach in building these fuzzy control charts. In order to effectively evaluate them, Section 4 apprises not only our advocated approach by simplifying Yu and Dat’s ranking method [29] but also the elucidatory development of our novel classification mechanism which is then used in the case discussed in Section 3 to provide a thorough controlling and monitoring procedure for the application of our proposed fuzzy control charts in practice. Some concluding remarks make up the last section.

#### 2. Review of Traditional and Control Charts

Literally, process variability must be fully controlled before process mean is monitored because larger variability in manufacturing process always results in higher percentage of nonconforming products, although the process mean is kept unchanged at its target value [1]. The process variability can be monitored with either chart or chart, while chart is used to monitor process mean; thus, in practice, chart usually goes with either chart or chart. Specifically, chart and chart are preferably used if sample size is small (no more than 10), whereas chart and chart are used when sample size is either larger than 10 or variable [2]. In practice, there are several applications where randomly choosing samples with variable sizes is economically preferred [1]. Moreover, many scholars claimed that control charts with variable sample size can detect process shifts markedly faster than the ones with equal sample size [30]. Therefore, this paper investigates samples with variable sizes; correspondingly, only chart and chart are taken into consideration.

Suppose a quality characteristic has a normal distribution with a mean and a standard deviation , that is, . Normally, and are not known in advance. They are usually estimated from initial samples taken from a process that is believed to be in control. Conventionally, 20 or 25 samples are investigated; their grand average is used as the best estimator of , and their average standard deviation can be used to obtain the estimated value of [1].

Let be the size of the sample of the samples investigated. Let denote the value of the quality characteristic in the sample at the observation (). The average of the sample, denoted by , and the grand average of the samples, denoted by , are determined by

Let and denote the standard deviation of the sample and the samples, respectively. They are obtained by

From the values of and , the centerline (CL), upper control limit (UCL), and lower control limit (LCL) of the chart are constructed by where is a constant determined by sample size as shown in several textbooks [1, 2, 31]. And, is the number of standard deviation units (usually called* Sigma*) that are allowed as tolerance; traditionally, is usually used [1, 2].

And the control limits for chart are determined by

The control limits of the control charts determined by (3) and (4) are fluctuated as shown in Figure 1 because each sample has its own value of . By plotting all of and against the variable control limits, we can detect out-of-control signal (if any).