Journal of the Chinese Institute of Engineers, 2017 VOL. 40, NO. 2, 126–133 http://dx.doi.org/10.1080/02533839.2017.1294996
A mathematical programming model for constructing the confidence interval of process capability index Cpm in evaluating process performance: an example of five-way pipe Kuen-Suan Chena, Cheng-Fu Huangb and Tsang-Chuan Changa a
Department of Industrial Engineering and Management, National Chin-Yi University of Technology, Taichung, Taiwan, ROC; bDepartment of Business Administration, Feng Chia University, Taichung, Taiwan, ROC
ARTICLE HISTORY
ABSTRACT
The process capability index Cpm can reflect process loss as well as process yield, thus is the most frequently used index for evaluating product quality in manufacturing industries. When evaluating the process performance, confidence intervals are often used for assurance with regard to the critical value of the process capability index. Unfortunately, sampling distributions of Cpm are obtained in a very complex way, which leads to difficulty in calculating the confidence interval of Cpm. Hence, this paper develops a mathematical programming model to construct the (1 − 𝛼) × 100% confidence interval of Cpm. Then for verifying the effectiveness of the proposed approach, the Monte Carlo simulation is used to find the coverage percentage. The proposed mathematical programming model can obtain the (1 − 𝛼) × 100% confidence interval of Cpm without complex statistical computations. Besides, managers can evaluate and monitor the process performance in an easy way. We also provide a case in which a five-way pipe process is presented as an illustration of how the proposed method is implemented.
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1. Introduction The quality of manufacturing processes is important for customers. How to evaluate manufacturing processes in an easier way is a major issue for researchers. Process capability indices (PCIs) are widely used to assess manufacturing processes because PCIs can ensure that the quality of products meet customer’s requirements. In addition, PCIs not only assess performance based primarily on quantitative methods but also provide improved directions of product functions for managers. A review of previous studies on PCIs (Palmer and Tsui 1999; Kotz and Johnson 2002; Spiring et al. 2003; Wu, Pearn, and Kotz 2009; Montgomery 2012; Leiva et al. 2014; Abbasi Ganji and Sadeghpour Gildeh 2016; Perakis and Xekalaki 2016), has indicated that the first process capability index Cp proposed by Juran, Gryna, and Bingham (1974) is a relatively simple ‘precision’ index. This index considers overall variability in processes as they pertain to manufacturing tolerances as a measure of process precision. However, Cp considers only process variability and does not factor in process mean shift. This makes it difficult to determine whether instability in a given process was caused by excessive process variability or a process mean shift from the midpoint of the specification range. Kane (1986) proposed Cpk to compensate for the shortcomings of Cp, both of which are defined as follows:
Cp =
d USL − LSL , = 6𝜎 3𝜎
CONTACT Tsang-Chuan Chang © 2017 The Chinese Institute of Engineers
[email protected]
(1)
Cpk = min
USL − 𝜇 𝜇 − LSL , 3𝜎 3𝜎
Received 4 October 2016 Accepted 10 February 2017 KEYWORDS
Process capability index; confidence interval; mathematical programming model; Monte Carlo; five-way pipe
} =
d − |𝜇 − m| , 3𝜎
(2)
where μ is the process mean, σ is the process standard deviation, USL and LSL are the upper and the lower specification limits, respectively, d = (USL − LSL)∕2 is the half length of the specification interval, and m = (USL + LSL)∕2 is the midpoint in the specification interval. Boyles (1991) described Cp and Cpk as the measurement indices based on process yield. As a result, with a normal process distribution and no shift in the mean, the relationships between ( ) process yield and the two indices are Yield% = 2Φ 3Cp − 1 and ) ) ( ( 2Φ 3Cpk − 1 ≤ Yield% ≤ Φ 3Cpk , where Φ(⋅) is the cumulative distributed function of the standard normal distribution. Cp and Cpk present the complete indications of yield rates, which have led to their wide application in yield-oriented manufacturing for evaluating process performance and potential (Pearn and Chen 1997). Although Cpk is able to detect whether the process mean has deviated from the midpoint of the specification range and thereby identifies the source of process instability, it is unable to reveal the extent of the shift, which can lead to misjudgments (as in cases where process variability is low but the process mean shift is large). From the perspective of Taguchi loss functions, Chan, Cheng, and Spiring (1988) examined the flaws of Cpk and presented the index Cpm:
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(3) The basic idea of the bootstrap approach is to resample the sample data, model the new sample data, and then estimate the sampling distribution of the parameter estimator. However, if the sample is not representative of the population, the unrepresentative sample will cause prediction errors and thus lead to poor decision-making (Franklin and Wasserman 1992; Balamurali and Kalyanasundaram 2002).
Figure 1. Schematic diagram of set region F.
Cpm =
USL − LSL d = √ , √ 2 2 2 6 𝜎 + (𝜇 − T ) 3 𝜎 + (𝜇 − T )2
127
(3)
where T is the target value and 𝜎 2 + (𝜇 − T )2 is the expected value of the Taguchi loss function. In Equation (3), if the process is without a shift (μ = T), Cpm can be given a maximum value. Similarly, the greater the process mean μ shifts from target value T, the greater (𝜇 − T )2 is. At the same time, Cpm is smaller, which means that the expected losses in the process are greater. Clearly, Cpm provides a clear indication as to expected process losses. Ruczinski (1996) and Pearn and Kotz (2006) stated that when process capability is sufficient (Cpm ≥ 1), Cpm has an unequal relationship with process yield (Yield% ≥ 2Φ(3Cpm) − 1). Thus, a higher value for Cpm indicates a reduction in process loss and compliance with the standards of quality under conditions of high yield. As a PCI, Cpm is highly practical. When assessing process performance, most enterprises require assurances as to the minimum value of the process capability index. As a result, manufacturers are constantly seeking lower confidence limits. Kotz and Lovelace (1998) pointed out that the results of PCIs should always be qualified using confidence intervals. Enterprises can use the confidence intervals in assessing process capability in order to meet product quality guarantees and processes at a given yield rate. The formal practice of constructing confidence intervals for PCIs began with Chou, Owen, and Salvador (1990). Since that time, numerous attempts have been made to develop confidence intervals for Cpm (Boyles 1991; Franklin and Wasserman 1992; Zimmer, Hubele, and Zimmer 2001; Balamurali and Kalyanasundaram 2002; Fan and Kao 2004; Perakis and Xekalaki 2004; Daniels et al. 2005; Lin, Pearn, and Yang 2005; Hsu, Wu, and Shu 2008; Parchami and Mashinchi 2013). Although all these approaches seem to be effective in solving the confidence interval of Cpm, there are some disadvantages: (1) The methods for solving the confidence interval of Cpm do not produce a closed form solution, which is less convenient for the evaluation and monitoring of process performance (Zimmer, Hubele, and Zimmer 2001; Perakis and Xekalaki 2004). (2) The sampling distribution of Cpm is usually quite complicated, making the interval estimation very difficult. Therefore, it appears to be difficult to obtain the exact confidence interval of Cpm based on conventional methods in which only an approximate or asymptotic confidence interval is available (Boyles 1991; Fan and Kao 2004; Daniels et al. 2005; Lin, Pearn, and Yang 2005; Hsu, Wu, and Shu 2008; Parchami and Mashinchi 2013).
In view of the above issues, this study develops an alternative method with simplified computation to produce a closed form solution for the determination of confidence intervals for Cpm. In fact, Cpm is a function of δ and γ. The (1 − 𝛼) × 100% joint confidence interval of δ and γ can be derived by Boole’s inequality, where 𝛿 = (𝜇 − T )∕d and 𝛾 = 𝜎∕d. Thus, a mathematical programming model is established by the objective function Cpm (𝛿, 𝛾) and constraints of the (1 − 𝛼) × 100% joint confidence interval of δ and γ. Optimal solutions for the upper and lower limits of the (1 − 𝛼) × 100% confidence interval for Cpm can be derived. Then, the Monte Carlo simulation is employed to evaluate the coverage percentages and prove the credibility of the proposed approach. This proposed approach is different from traditional methods with complicated computational processes. In practical applications, the proposed approach enables enterprises to easily assess process performance and ensure that products meet customer requirements. The remainder of this paper is organized as follows. In Sections 2 and 3, we develop a mathematical programming model to construct the (1 − 𝛼) × 100% confidence interval of Cpm. In Section 4, the Monte Carlo simulation is used to verify the effectiveness of the proposed approach. In Section 5, we present a case to illustrate the applicability of the proposed approach. Conclusions and avenues for future research are summarized in Section 6.
2. Mathematical programming model for constructing the confidence interval of Cpm As mentioned previously, process capability Cpm is sufficient to reflect both the yield and loss of processes and this makes it highly practical as a process capability index. To construct the confidence interval of Cpm, we first convert Cpm into functions of the parameters δ and γ in accordance with the suggestions of Deleryd and Vännman (1999), where 𝛿 = (𝜇 − T )∕d is the accuracy index and 𝛾 = 𝜎∕d is the precision index. Thus, in addition to presenting the process performance of quality characteristics, each (𝛿, 𝛾) can also be used to obtain corresponding index values. Thus, Cpm can be rewritten as follows:
Cpm = Cpm (𝛿, 𝛾) =
1 1 = , √ 2 2 3Z 3 𝛿 +𝛾
Table 1. Corresponding value of Cpm with a process mean shift of 1.5σ. Quality level kσ 7σ 6σ 5σ 4σ 3σ
Cpm 1.2943 1.1094 0.9245 0.7396 0.5547
(4)
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K.-S. CHEN ET AL.
� P
Figure 2. Process capability analysis chart for Cpm at 7σ, 6σ, 5σ, 4σ, and 3σ.
𝛾̂ 𝛾̂ 𝛿̂ − √ t𝛼1∕2 (n − 1) < 𝛿 < 𝛿̂ + √ t𝛼1∕2 (n − 1), n n � � ⎫ � � � (n − 1)̂𝛾 2 ⎪ � (n − 1)̂𝛾 2 0 a2 0.05. In other words, it is reasonable to assume that the process data collected from the factory are normally distributed. For these 150 measurements, the mean and standard deviation of the sample can be calculated as μ = 73.03 and σ = 0.04818, respectively. Thus, we can derive the following: δ = 0.1617, γ = 0.2409, a1 = 0.1004, a2 = 0.2230, b1 = 0.2077, and b2 = 0.2861. A [ 95%]lower and upper confidence limit for Cpm is calculated as CL , CU = [0.9188, 1.4449] using Tables 2 and 3. Likewise, we can use Equation (4) to obtain process performance 1.1488 for the critical quality characteristic (length) of the five-way pipe, indicating that this process of manufacturing has reached the 6σ level of quality. We adopt the hypothesis testing method used in the field of statistics in order to ensure a rigorous evaluation of whether the process performance achieved the level of quality demanded by the customer. Assuming that the customer demanded a 6σ level of quality, statistical hypothesis testing would be performed as follows: { H0 :Cpm = 1.1094 . (14) H1 :Cpm ≠ 1.1094 This enables us to define decision-making rules based on the 95% confidence interval for Cpm. If the confidence interval encompasses the range of the null hypothesis, then we do not have sufficient proof to reject H0. Conversely, if the confidence interval does not encompass the range of the null hypothesis, then we have sufficient proof to reject H0. In this case, the confidence interval [0.9188, 1.4449] encompasses the range assumed for H0, which means that the process performance associated with the critical quality characteristic of the five-way pipe (length) attained the 6σ level of quality demanded by the customer.
6. Conclusions and future research Index Cpm is designed to penalize process drifts deviating from the target value, akin to the idea of squared error loss. It also responds to the process loss as well as process yield, making it a highly practical index. Most enterprises require some sort of assurances in their evaluation of process performance, particularly in determining the critical values in the process capability index. As a result, the issue of determining confidence intervals has become a focus in numerous studies. Unfortunately, the complexity of the probability distribution function of Cpm makes it difficult to derive the real confidence interval easily. To develop a mathematical programming model capable of deriving the (1 − 𝛼) × 100% confidence interval for Cpm, we took advantage of the fact that Cpm is a function of δ and γ and employed Cpm (𝛿, 𝛾) as the objective function with the (1 − 𝛼) × 100% joint confidence intervals of δ and γ as a constraint, respectively. The joint confidence intervals of δ and γ make it possible to determine the upper and lower limits of the confidence interval according to δ and γ within a set of given conditions. This study examined nine
processes and used Monte Carlo simulation to verify the reliability of the proposed model by assessing coverage percentage for nine processes. Our results indicate that deriving the upper and lower limits eliminates the need for the complex computation processes encountered in statistical methods, making it possible for enterprises to evaluate process performance easily and effectively. Moreover, this study mainly aims at constructing the confidence interval of index Cpm to evaluate and monitor the process performance. Cause analysis and improvement measures of poor performance can thus be regarded as further research in the future. On the other hand, Six Sigma quality improvement can effectively enhance quality and reduce costs, such that the procedure has been widely applied in manufacturing to increase product quality. Consequently, the proposed approach in this paper can be applied as the measure step of DMADV (an acronym for Define, Measure, Analyze, Design, Verify) or DMAIC (an acronym for Define, Measure, Analyze, Improve, Control) procedure, and further develop a Six Sigma project to improve the process quality. Finally, the sample data used must be taken from a stable in-control process to ensure an effective quality assessment for process, and one must assume that the data follow a normal distribution. These are the limitations of research design.
Nomenclature a1 a2
𝛿̂ −
𝛾̂ √ 𝛾̂ √
t (n n 𝛼∕4
− 1)
𝛿̂ + n t𝛼∕4 (n − 1) ]1∕2 [ / 2 2 b1 𝜒𝛼 (n − 1) (n − 1)̂𝛾 ∕ ]1∕2 [ / 4 b2 (n − 1)̂𝛾 2 𝜒 2 𝛼 (n − 1) 1− ∕ 4 CL lower confidence limit of Cpm Cpm process capability index CU upper confidence limit of Cpm d half length of the specification interval F set region } { FL a1 ≤ 𝛿 ≤ a2 , 𝛾 = b1 } { FU a1 ≤ 𝛿 ≤ a2 , 𝛾 = b2 Ij indicator function k quality level LSL lower specification limit m midpoint of the specification interval n sample size T target value USL upper specification limit √ Z 𝛿2 + 𝛾 2 α significant level γ precision index δ accuracy index μ process mean σ process standard deviation 𝜎 2 + (𝜇 − T )2 expected value of the Taguchi loss function Φ(⋅) cumulative distributed function of the standard normal distribution
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Disclosure statement No potential conflict of interest was reported by the authors.
References Abbasi Ganji, Z., and B. Sadeghpour Gildeh. 2016. “A Class of Process Capability Indices for Asymmetric Tolerances.” Quality Engineering 28 (4): 441–454. doi:10.1080/08982112.2016.1168524. Balamurali, S., and M. Kalyanasundaram. 2002. “Bootstrap Lower Confidence Limits for the Process Capability Indices Cp, Cpk and Cpm.” International Journal of Quality and Reliability Management 19 (8/9): 1088–1097. doi:10.1108/02656710210442875. Boyles, R. A. 1991. “The Taguchi Capability Index.” Journal of Quality Technology 23: 17–26. Chan, L. K., S. W. Cheng, and F. A. Spiring. 1988. “A New Measure of Process Capability: Cpm.” Journal of Quality Technology 20 (3): 162–175. Chang, T.-C., K.-J. Wang, and K.-S. Chen. 2014a. “Sputtering Process Assessment of ITO Film for Multiple Quality Characteristics with OneSided and Two-sided Specifications.” Journal of Testing and Evaluation 42 (1): 196–203. doi:10.1520/JTE20130054. Chang, T.-C., K.-J. Wang, and K.-S. Chen. 2014b. “Capability Performance Analysis for Processes with Multiple Characteristics Using Accuracy and Precision.” Proceedings of the Institution of Mechanical Engineers Part B – Journal of Engineering Manufacture 228 (5): 766–776. doi:10.1177/0954405413508118. Chen, J.-P., and K. S. Chen. 2004. “Comparing the Capability of Two Process Using Cpm.” Journal of Quality Technology 36 (3): 329–335. Chen, K. S., W. L. Pearn, and P. C. Lin. 2003. “Capability Measures for Processes with Multiple Characteristics.” Quality & Reliability Engineering International 19 (2): 101–110. doi:10.1002/qre.513. Chen, K. S., C. H. Wang, and H. T. Chen. 2006. “A MAIC Approach to TFT-LCD Panel Quality Improvement.” Microelectronics Reliability 46 (7): 1189–1198. doi:10.1016/j.microrel.2005.10.003. Chen, K. S., L. Y. Ouyang, C. H. Hsu, and C. C. Wu. 2009. “The Communion Bridge to Six Sigma and Process Capability Indices.” Quality & Quantity 43 (3): 463–469. doi:10.1007/s11135-007-9123-1. Chou, Y.-M., D. B. Owen, and B. A. Salvador. 1990. “Lower Confidence Limits on Process Capability Indices.” Journal of Quality Technology 22 (3): 223–229. Daniels, L., B. Edgar, R. K. Burdick, and N. F. Hubele. 2005. “Using Confidence Intervals to Compare Process Capability Indices.” Quality Engineering 17 (1): 23–32. doi:10.1081/QEN-200028666. Deleryd, M., and K. Vännman. 1999. “Process Capability Plots – A Quality Improvement Tool.” Quality and Reliability Engineering International 15 (3): 213–227. doi:10.1002/(SICI)1099-1638(199905/06). Fan, S.-K., and C.-K. Kao. 2004. “Lower Bayesian Confidence Limits on the Process Capability Index Cpm: A Comparative Study.” Journal of the Chinese Institute of Industrial Engineers 21 (5): 444–452. doi:10.1080/10170660409509423. Franklin, L. A., and G. S. Wasserman. 1992. “Bootstrap Confidence Limits for Capability Indices.” Journal of Quality Technology 24 (4): 196–210. Hsu, B.-M., C.-W. Wu, and M.-H. Shu. 2008. “Generalized Confidence Intervals for the Process Capability Index Cpm.” Metrika 68 (1): 65–82. doi:10.1007/ s00184-007-0143-6. Huang, C.-F., K.-S. Chen, S.-H. Sheu, and T.-S. Hsu. 2010. “Enhancement of Axle Bearing Quality in Sewing Machines Using Six Sigma.” Proceedings of the Institution of Mechanical Engineers Part B – Journal of Engineering Manufacture 224 (10): 1581–1590. doi:10.1243/09544054JEM1769. Juran, J. M., F. M. Gryna, and R. S. Bingham. 1974. Juran’s Quality Control Handbook. 3rd ed. New York: McGraw-Hill. Kane, V. E. 1986. “Process Capability Indices.” Journal of Quality Technology 18 (1): 41–52.
133
Kotz, S., and N. L. Johnson. 2002. “Process Capability Indices – A Review 1992–2000.” Journal of Quality Technology 34 (1): 2–53. Kotz, S., and C. R. Lovelace. 1998. Process Capability Indices in Theory and Practice. London: Arnold. Leiva, V., C. Marchant, H. Saulo, M. Aslam, and F. Rojas. 2014. “Capability Indices for Birnbaum–Saunders Processes Applied to Electronic and Food Industries.” Journal of Applied Statistics 41 (9): 1881–1902. doi:10.1080/02 664763.2014.897690. Lin, G. H., W. L. Pearn, and Y. S. Yang. 2005. “A Bayesian Approach to Obtain a Lower Bound for the Cpm Capability Index.” Quality and Reliability Engineering International 21 (6): 655–668. doi:10.1002/qre.681. Linderman, K., R. G. Schroeder, S. Zaheer, and A. S. Choo. 2003. “Six Sigma: A Goal-theoretic Perspective.” Journal of Operations Management 21 (2): 193–203. doi:10.1016/S0272-6963(02)00087-6. Montgomery, D. C. 2012. Introduction to Statistical Quality Control. 7th ed. New York: Wiley. Ouyang, L.-Y., K.-S. Chen, C.-M. Yang, and C.-H. Hsu. 2014. “Using a QCAC– Entropy–TOPSIS Approach to Measure Quality Characteristics and Rank Improvement Priorities for All Substandard Quality Characteristics.” International Journal of Production Research 52 (10): 3110–3124. doi:10.10 80/00207543.2013.865092. Palmer, K., and K. L. Tsui. 1999. “A Review and Interpretations of Process Capability Indices.” Annals of Operations Research 87: 31–47. doi:10.1023 /A:1018993221702. Parchami, A., and M. Mashinchi. 2013. “Confidence Interval of Generalized Taguchi Index.” Journal of Intelligent and Fuzzy Systems 25 (3): 577–585. doi:10.3233/IFS-120664. Pearn, W. L., and K.-S. Chen. 1997. “A Practical Implementation of the Process Capability Index Cpk.” Quality Engineering 9 (4): 721–737. doi:10.1080/08982119708919094. Pearn, W. L., and S. Kotz. 2006. Encyclopedia and Handbook of Process Capability Indices: A Comprehensive Exposition of Quality Control Measures. Singapore: World Scientific Publishing. Perakis, M., and E. Xekalaki. 2004. “A New Method for Constructing Confidence Intervals for the Index Cpm.” Quality and Reliability Engineering International 20 (7): 651–665. doi:10.1002/qre.574. Perakis, M., and E. Xekalaki. 2016. “On the Relationship between Process Capability Indices and the Proportion of Conformance.” Quality Technology and Quantitative Management 13 (2): 207–220. doi:10.1080/ 16843703.2016.1169696. Ruczinski, I. 1996. “The Relation between Cpm and the Degree of Includence.” PhD dissertation, University of Würzberg, Würzberg, Germany. Spiring, F., B. Leung, S. Cheng, and A. Yeung. 2003. “A Bibliography of Process Capability Papers.” Quality and Reliability Engineering International 19 (5): 445–460. doi:10.1002/qre.538. Wang, C. C., K. S. Chen, C. H. Wang, and P. H. Chang. 2011. “Application of 6-Sigma Design System to Developing an Improvement Model for Multiprocess Multi-characteristic Product Quality.” Proceedings of the Institution of Mechanical Engineers Part B – Journal of Engineering Manufacture 225 (7): 1205–1216. doi:10.1177/2041297510393464. Wu, C. W., W. L. Pearn, and S. Kotz. 2009. “An Overview of Theory and Practice on Process Capability Indices for Quality Assurance.” International Journal of Production Economics 117 (2): 338–359. doi:10.1016/j.ijpe.2008.11.008. Yu, K. Z., and K. S. Chen. 2016. “Testing and Analysing Capability Performance for Products with Multiple Characteristics.” International Journal of Production Research 54 (21): 6633–6643. doi:10.1080/00207543.2016.12 03469. Zimmer, L. S., N. F. Hubele, and W. J. Zimmer. 2001. “Confidence Intervals and Sample Size Determination for Cpm.” Quality and Reliability Engineering International 17 (1): 51–68. doi:10.1002/qre.380.
