Weighted signal-to-noise ratio robust design for a new double sampling npx chart

Journal Pre-proofs Weighted signal-to-noise ratio robust design for a new double sampling npx

chart

Wenhui Zhou, Zixuan Wang, Wei Xie PII: DOI: Reference:

S0360-8352(19)30593-5 https://doi.org/10.1016/j.cie.2019.106124 CAIE 106124

To appear in:

Computers & Industrial Engineering

Received Date: Revised Date: Accepted Date:

17 January 2019 7 August 2019 10 October 2019

Please cite this article as: Zhou, W., Wang, Z., Xie, W., Weighted signal-to-noise ratio robust design for a new double sampling npx chart, Computers & Industrial Engineering (2019), doi: https://doi.org/10.1016/j.cie. 2019.106124

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Weighted signal-to-noise ratio robust design for a new double sampling npx chart Wenhui Zhou School of Business Administration, South China University of Technology, Guangzhou, 510640, China. [email protected].

Zixuan Wang∗ School of Business Administration, South China University of Technology, Guangzhou, 510640, China. [email protected].

Wei Xie School of Business Administration, South China University of Technology, Guangzhou, 510640, China. [email protected].

Corresponding author(Email: [email protected])

Weighted signal-to-noise ratio robust design for a new double sampling npx chart Abstract Recently, to monitor the mean shifts of a process by using attribute inspection, a new np chart, called npx chart, was proposed to synthesize the advantages of attribute and variable charts. Attracted by the easy-implementation property and good performance of this chart, in this paper, we propose a double-sampling (DS) npx chart to improve the efficiency of the single-sampling npx chart. We build up a process cost model to compare the performance of DS npx chart and the traditional npx chart. To minimize the process cost with uncertainty, we introduce a robust design method based on the “weighted signal-to-noise ratio (WSNR)”. Specifically, we take into account the weighted expectation on multiple scenarios and regard signal-to-noise ratio as a response to variance. Compared to the traditional designs, the WSNR robust design not only preserves the statistical strengths and economic efficiency, but also shows the advantages of flexibility and adaptability. Then, numerical experiments are conducted to measure the performance of our model. The results demonstrate that the DS npx chart is superior to the npx chart in controlling the ARL1 and the process cost. By analyzing the process cost, we find that the larger the scenario range is, the better the WSNR robust design performs. In addition, the performance of the proposed robust design presents clear advantages to the existing robust measures (e.g., absolute robustness, robust deviation, and relative robustness), for which the WSNR will be a promising approach for the practitioners. Keywords: Double sampling, npx chart, robust economic design, Taguchi’s method, SN ratio.

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1

Introduction

Statistical process control (SPC) has been verified empirically as an efficient technique that can be used to evaluate and monitor a variety of manufacturing processes. Given specific requirements for products or services, the SPC can help establish and maintain the acceptability and stability of the associated processes. As a powerful tool in SPC, the control chart can be constructed to monitor the performance of a production process from a statistical approach. Specifically, an attribute control chart, e.g., np chart, is adopted to monitor the attribute characteristics, while a variable control chart, e.g., X chart, is implemented to monitor the variable characteristics. From different perspectives, both the attribute chart and variable chart can be used to detect the process shifts. Montgomery (2009) compares the efficiency of X chart and np chart for a specified level of protection against process shift. To achieve the same detection capability, the sample size of np chart should be much larger than that of the X chart. Hence, this huge difference may affect the applicability of the np chart in practice. To address this issue, a novel npx chart is proposed by Wu et al. (2009) to compete with the traditional X chart. In particular, compared with the np chart, npx chart can monitor the mean shifts (which is the advantage of variable charts). For implementation, the specification limits of np chart are fixed. However, for npx chart, its limits are replaced by warning limits, which can be set and are associated with mean of a variable. In addition, compared with X chart, npx shows advantages in detection effectiveness (e.g., larger sample size with equal inspection cost) and simpler instruments (using attribute inspection). Hence, due to the fact that the npx chart is more convenient to implement to indicate the cause, it becomes more attractive than the X chart in practice (Wu et al., 2009). With the development of the manufacturing industry, the requirement of production flexibility becomes an important criterion, for which quick and economic SPC solutions become appealing to the manufacturers. Double sampling (DS) is most efficient at detecting process shifts in non-conforming proportions. Compared with single-sampling,

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DS not only reduces the number of required inspections, but also shows a better detection performance for moderate to large shifts (Daudin, 1992). Hence, DS is commonly used to improve the performance of traditional Shewhart control charts. Besides the performance of control charts, the manufacturers often care about the costs of running the experiments (Duncan, 1956; Xie et al., 2001; Serela, 2008; Zhang et al., 2011). The main objective of an economic design is to maximize the profits or minimize the costs associated with the charts’ operations. For economic design, it is often difficult to accurately estimate the associated costs and process parameters (Lorenzen and Vance, 1986). For example, if the set of parameters is designed for a specific scenario and a different scenario is finally realized, such mismatch may increase the associated cost. To address this problem, robust design approaches are proposed to confront the uncertainties involved in the estimation of process parameters (Linderman and Choo (2002)). In this paper, we implement the DS scheme on npx chart to propose a new chart, called DS npx chart, to further improve the efficiency of the single-sampling npx chart (Wu et al., 2009). Specifically, a genetic algorithm (GA) is utilized to optimize the parameters of the DS npx control chart. Numerical analysis shows that the DS npx chart offers a remarkable improvement in the ARL1 performance and has significant impacts on the process cost saving for the large shift. This paper aim at proposing a new robust design approach, called “weighted signalto-noise ratio (WSNR)” robust design, to deal with the uncertain environment where the cost parameters can not be estimated accurately. Different from the previous robust design, which mainly consider optimizing the worst case or minimizing the relative deviation, we focus on the mean and variance of the process cost under multiple scenarios. To improve the measurement ability, we adopt the signal-to-noise (SN) ratio as the response of variance. In particular, the weighted expectation on multiple scenarios is applied in our method to reflect the relative frequency of different scenarios. Therefore, the main contribution of this work is twofold: 1) we propose a new control chart by implementing the DS scheme to the npx chart; and 2) we construct a robust

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economic design model by using the SN ratio to deal with the multi-objective decisionmaking problem (minimize the weighted process cost while controlling its variance). By conducting the numerical experiments, we show that the DS npx chart offers a remarkable improvement in the performance of ARL1 and has significant impacts on the cost saving for a large shift. On the other hand, compared to the pervious robust methods, the newly proposed WSNR robust design can significantly improve the economic performance of the DS npx chart and is superior in stability and flexibility when the uncertainty becomes large. In addition, we numerically verify that the WSNR design is still robust when there is a slightly asymmetry in the process distribution. The rest of the paper is organized as follows. Section 2 briefly reviews the related literature. In Section 3, we present an overview of the npx chart. Section 4 is devoted to the modelling framework of DS npx chart, and the associated process cost model is proposed in Section 5. In Section 6, we discuss the robust economic design of the new chart. Numerical experiments are conducted in Section 7 to verify the applicability of DS npx chart. Finally, Section 8 concludes the work.

2

Literature Review

Our work is closely-related to two streams of research, i.e., the application of DS scheme in control charts and the robust economic design of control charts. DS is one of the most important sampling schemes used in designing control chart. Compared to a classic Shewhart chart, the DS scheme shows its advantages in improving the performance without increasing the (in-control) average number of items inspected per time unit. In addition, the implementation of DS scheme can dramatically reduce the average sample size (e.g., the size reduction can be as high as 50% when the process is in control). In the literature, Croasdale (1974) makes the first attempt to study the DS scheme based on average production run length with Shewhart charts. Daudin (1992) studies the DS X chart which offers better statistical efficiency (in terms of the average run length) than the Shewhart X chart without

4

increasing the sampling efforts. Faraz et al. (2015) investigate the economic statistical design of the DS T 2 chart and find that more benefits can be achieved then the classical T 2 . He and Grigoryan (2002) construct a DS s chart to monitor the process variation caused by the standard deviation, which can be regarded as a complementary to the DS X chart. Rodrigues et al. (2011) propose a DS scheme for the np chart and compare the performance among the traditional np chart and other classic control charts. The result shows that the DS scheme is the most efficient one in detecting process shifts of non-conforming proportions. Chong et al. (2014) present a synthetic DS np chart which consists of the DS np and conforming run length (CRL) sub-charts. In terms of the zero state performance, the synthetic DS np chart presents the best performance, from an overall viewpoint, by comparing with the standard np, synthetic np, DS np, VSS np, EWMA np, CUSUM np and CUSUM np charts. However, most of the existing research is initially focused on the DS scheme for variables and the DS scheme for attributes has received little attention. Because npx chart has a high efficiency in detecting mean shift of attribute inspection, we propose a new DS npx chart, which is a natural extension of the DS X to incorporate the associated attribution. Up to now, to the best of our knowledge, we are among the first attempts in constructing the DS npx to fill a gap in the literature. To reduce the costs of implementing an SPC, the economic process design by optimizing the relevant cost parameters has attracted considerable attentions from the researchers. As a seminal work, Duncan (1956) proposes to investigate the process costs of a static X chart. In his work, the associated cost parameters are formulated and jointly optimized. After that, the research in improving effectiveness and cost saving of the Shewhart chart has been extensively conducted (Saniga, 1989; Ho, 1994; Castillo and Montgomery, 1996; Al-Oraini and Rahim, 2002). However, for economic design, it is often difficult to accurately estimate the associated costs and process parameters (Lorenzen and Vance, 1986). To address this problem, Pignatiello and Tsai (1988) introduce a design method in which user confidence is induced to realize the design procedure and the losses of operation can be controlled even the process pa-

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rameters are not accurate. Consequently, by considering the process variability, robust economic design becomes a promising approach to confronting the uncertainties. Linderman and Choo (2002) are the first to introduce the robust optimization approach to economic design of control charts, which can reduce the monetary losses resulted from the parameter variations and lead to a conservative solution. In their paper, three discrete robustness measures are employed, i.e., absolute robustness, robust deviation, and relative robustness. These three criteria are then widely-adopted to realize the robust design for other control charts, e.g., EWMA chart and T 2 chart (Amiri et al., 2015; Chalaki et al., 2015). Vommi and Seetala (2007) propose a risk-based robust design with the principle of minimization of maximum risk. The uncertain parameter range is taken as a set and the risk is defined as the proportion cost of selecting non-optimal parameters. Vommi et al. (2007) introduce a simple statistic to deal with process with multiple scenarios. Mortarino (2010) focuses on the interactions among changes of different cost parameters under uncertain environments. Safaei et al. (2015) propose a robust design considering the uncertainty in budgets. Their model enables the decision makers to set parameter ranges with risk preferences. Since it is difficult to adopt a specific form of process distribution in the real world, robust algorithms of nonparametric process control charts gained lots of momentum in recent years (see Li et al. (2016a), Li et al. (2016b), and Li et al. (2016c)) Evidentally, the above-mentioned papers are restricted to a single design for the worst-case (absolute robustness) or the smallest deviation from the best possible performance (robust deviation and relative robustness) across all scenarios. However, the mean and variance of the cost in multiple scenarios have not been studied. Unlike the previous research, we adopt the SN ratio as a response to variance and consider the application of weighted expectation for different scenarios, which contributes to the literature by proposing the WSNR robust economic design model.

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3

Overview of npx Chart

The npx chart was introduced by Wu et al. (2009) to compete with the X chart in detecting the variable mean shifts. The parameter set of the npx chart includes the lower and upper warning limits (RL and RU ), the sample size (nnp ), the sampling interval (hnp ), and the upper control limit (U CLnp ). One distinctive feature of the npx chart is that, to classify the conforming or nonconforming units, it adopts statistical warning limits to replace the specification limits. In particular, a sample nnp is collected every hnp hours. Denote Dnp as the number of non-conforming units. When Dnp > U CLnp , the process is defined as out-of-control. Otherwise, the process is assumed to be in control.

Figure 1: Progress gauges (Wu et al., 2009)

For attribute detection, the main concern is to determine whether an item is conforming (without knowing the actual value of a variable). Thus, the “Go/No Go” ring gauge is adopted as a tool to check whether the shaft diameter x exceeds the relevant limits (Kennedy and Bond, 1987). Assume the calibrated size of the ring gauge is made equal to the upper warning limit RU . Figure 1 shows the designs of a double-end gauge (part (a)) and a progressive gage (part (b)). They can be used to detect the non-conforming item with only one test. Moreover, because the attribute inspection requires less cost in most applications, when the inspection costs are the same, the npx chart can use a larger sample size and/or sampling frequency, which implies a higher detection effectiveness.

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4

The Implementation of DS npx Chart

Recall that the DS np chart has been employed to detect a quality characteristic following a binomial distribution with parameters n and p. On the other hand, by dichotomously classifying items (conforming or nonconforming), the npx chart is proposed to monitor the process mean, in which the number of nonconforming items is used for evaluation. Motivated by the DS np chart constructed by Rodrigues et al. (2011), we implements DS method to the npx chart, called DS npx chart, to further improve the efficiency of the single-sampling npx chart (Wu et al., 2009). The new chart comprises eight parameters, that is, the lower and upper warning limits (RL and RU ), the warning limit to detect the nonconforming items (W L), the upper control limits (U CL1 and U CL2 ), the sampling interval (h), the size of the first sample (n1 ), and the size of the second sample (n2 ). Assume µ0 and σ are the target values of process mean and standard deviation, respectively, and the process variance remains unchanged. When a process shift occurs, the mean value µ0 will shift to µ1 = µ0 + δσ0 , where δ is the magnitude of a mean shift in terms of σ0 . In what follows, we describe the procedure of constructing and operating the DS npx chart in detail. Step 1. To classify conforming and nonconforming items, we apply the statistical warning limits RL and RU to substitute the related specification limits. Note that the warning limits RU and RL are two symmetric variables based on the mean of in-control state µ0 . Thus, one has RL = µ0 − kw σ0 ,

(1)

RU = µ0 + kw σ0 ,

(2)

and

where kw is the warning limit coefficient. Step 2. Determine the non-integer values of U CL1 , U CL2 , and W L to avoid the ambiguity that may rise (note that the values of limits can be set as integers. However, if the observed number of non-conforming items equal to the value of the

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control limits, we need to discuss whether the process is out-of control or in control. See Rodrigues et al. (2011) and Chong et al. (2014)). Step 3. Collect a sample of n1 from the process and use an attribute inspection to evaluate the sample items. If the quality characteristic exceeds RU or RL , the item is decided as non-conforming (otherwise, it passes the inspection). Then, we use d1 to count the number of non-conforming items. Step 4. If d1 < W L, the process is in-control and the scheme goes to Step 3. When d1 > U CL1 , the process is out-of-control and the scheme moves to Step 6. If W L < d1 < U CL1 , the scheme proceeds to Step 5. Step 5. An addition sample of size n2 is taken from the process and the ring gage is utilized to classify the items. Let d2 denote the number of non-conforming units in this sample. If (d1 + d2 ) < U CL2 , the process is defined as in-control and the scheme goes to Step 3. Otherwise, the process reveals out-of-control signal and the scheme proceeds to Step 6. Step 6. Take measures to investigate and remove the assignable cause. Then, the scheme returns to Step 3. The entire procedure of the DS npx control chart, from Step 1 to Step 6, is graphically displayed in Figure 2.

Figure 2: Demonstration of the implementation of DS npx chart

From the figure, one can see that the warning limit and upper control limits together partition the chart area into three regions as I1 = [0, W L], I2 = [W L, U CL1 ], and I3 = [U CL1 , n1 ].

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Let Q1 be the probability of d1 < W L and Q2 be the probability of d1 +d2 < U CL2 . Then Q1 and Q2 can be used to represent the in-control probabilities for the first and the second stages, respectively. Given a value of x, ⌊x⌋ stands for the integer part of x and ⌈x⌉ represents (⌊x⌋ + 1). Then, Q1 and Q2 can be calculated as ⌊W L⌋

Q1 = Pr{d1 ≤ ⌊W L⌋} =





∑  n1  d n −d   p 1 (1 − p) 1 1 , d1 d1 =0

(3)

and Q2 = Pr{⌊W L⌋ < d1 < ⌈U CL1 ⌉ and d1 + d2 ≤ ⌊U CL2 ⌋} = Pr{⌊W L⌋ < d1 < ⌈U CL1 ⌉} × Pr{d1 + d2 ≤ ⌊U CL2 ⌋|⌊W L⌋ < d1 < ⌈U CL1 ⌉}       ⌈U CL ⌉−1 ⌊U CL ⌋−d 2 1 ∑1 ∑  n1  d1   n2  d2  =   p (1 − p)n1 −d1    p (1 − p)n2 −d2  . d2 =0 d1 =⌊W L⌋+1 d1 d2 (4) Then the probability that the process is indicated as “in control” is

Q = Q1 + Q2 .

(5)

The expression of p can be written as follows. p = 1 − Φ( RU −(µσ00 +δσ0 ) ) + Φ( RL −(µσ00+δσ0 ) )

(6)

= 1 − Φ(kw − δ) + Φ(−kw − δ), where Φ(·) is the cumulative distribution function of the standard normal distribution. p0 (p1 ) represent the in-control (out-of-control) value. Let Pr{d1 ∈ I2 } be the probability that d1 falls into region I2 . Because the width of warning region denotes the probability of taking the second sample, the expected

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sample size E(n|δ) can be computed as E(n|δ) = n1 + n2 Pr{d1 ∈ I2 } = n1 + n2 ×

⌈U∑ CL1 ⌉ d1 =⌊W L⌋

(

n1

) pd1 (1 − p)n1 −d1 .

(7)

d1

Note that E(n|δ = 0) (E(n|δ ̸= 0)) is the expected sample size for the in-control (out-of-control) state. To evaluate the efficiency of a control chart, we can examine its speed of monitoring a process shift or disturbance that affects the value of p. In general, this indicator can be measured as follows. ARL =

1 . 1−Q

(8)

The in-control ARL and out-of-control ARL can be computed by Equation (8) with p = p0 and p = p1 (fraction of nonconforming units), respectively.

5

The Cost Model

In this section, we adopt the model proposed by Lee et al. (2012) and construct a cost function based on the design optimization of DSVSI (Lorenzen and Vance, 1986). For economic design models, it is normal to view the operations of a production process as a series of independent production cycles. The production cycle time includes three periods, i.e., the in-control period, the out-of-control period, and the time spent on finding the assignable cause, which can be illustrated by Figure 3. The production period will begin when the process starts operating in the initial state. Then it continues running until an assignable cause occurs. Once the control chart delivers an out-ofcontrol signal, the root cause must be corrected to make the process return to the in-control state. Hence, the cycle time of a DS npx chart should include the following components: 1. For the in-control period, we assume the initial process is in-control and the time

11

at which an assignable cause occurs is exponentially distributed with mean 1/λ. 2. For the out-of-control state, if the assignable cause occurs, the average number of sampling before detecting the process mean shift is ARL1 . Suppose the sampling interval is h. Let τ be the average time from the last in-control sample to the shift, one has τ=

eλh − 1 − λh . λ(eλh − 1)

(9)

Then, the expected time elapsed from detecting an assignable cause to issuing an out-of-control signal is h × ARL1 − τ . 3. The time for searching and correcting the assignable cause is assumed to be a constant D. Then, the expected production cycle time of DS npx chart, denoted by T , can be calculated as T = 1/λ + h × ARL1 − τ + D.

(10)

Figure 3: Diagram for an in-control period and an out-of-control period in a cycle.

In this work, we consider five cost components to construct the cost function for the DS npx chart: (1) the cost spent on the in-control state, (2) the cost spent on the out-of-control state, (3) the cost spent on sampling, inspection, and plotting per cycle, (4) the cost spent on detecting and eliminating an assignable cause, and (5) the cost of false alarms. (1) If the process is in-control, let CI be the expected cost spent on the in-control state per cycle, C0 be the loss cost per hour, and 1/λ be the expected time staying in 12

the in-control state. Therefore, the expected cost of an in-control state is given by

CI = C0 /λ.

(11)

(2) If the process is out-of-control, let CO as the expected cost spent on the outof-control state per cycle, M be the loss rate per hour when the process is in an outof-control condition, and T − 1/λ be the expected length of the out-of-control period, then CO = M × (T − 1/λ).

(12)

(3) Let CS be the cost spent on sampling, inspection, and plotting (we aggregate these three costs into the sampling cost for convenience) per cycle. It is obvious that the expected numbers of sampling and sample size are different. Let a3 and a4 be the fixed and variable costs for sampling respectively. Then, for in-control state, the sampling cost is a3 + a4 × E(n|δ = 0). It is easy to know that the expected number of in-control sampling is

e−λh . 1−e−λh

Then, the sampling cost for the in-control state is [a3 +a4 ×E(n|δ =

e−λh

0)] × 1−e−λh . When the process mean shifts, the cost becomes a3 + a4 × E(n|δ ̸= 0) and the expected cost spent on the out-of-control state is [a3 +a4 ×E(n|δ ̸= 0)]×[T −1/λ]/h. Thus, the total cost of sampling, inspection, and plotting CS is given by

CS =

[a3 + a4 × E(n|δ = 0)] × e−λh [a3 + a4 × E(n|δ ̸= 0)] × [T − 1/λ] + . 1 − e−λh h

(13)

(4) Assume the cost of detecting and eliminating the assignable cause is a2 . (5) The probability of false alarm is 1/ARL0 . Let CF be the cost spent on the false alarms per cycle and a1 be the investigation cost of a false alarm. We adopt the expected sample number of in-control state (e−λh /(1 − e−λh )) derived by Duncan (1956), then the false alarm cost can be expressed as

CF = a1 ×

1 e−λh × . ARL0 1 − e−λh

13

(14)

Finally, the total cost per cycle T C could be calculated as

T C = CI + CO + CS + CF + a2 ,

(15)

and the expected cost per hour is given by the ratio of the expected cost per cycle to the expected cycle time as AC = T C/T.

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(16)

The WSNR Robust Economic Design

It is well-known that robust economic design takes the parameter uncertainties into consideration. In the previous studies, most of the robust approaches adopt the criteria proposed by Linderman and Choo (2002), which are focused on the worst-case optimization (e.g., the maximum value or the maximum deviation). To the best of our knowledge, only Vommi et al. (2007) apply the weighted expectation to address the variation from multiple scenarios. However, they have not discussed the mean and variance of the process cost (AC) from different scenarios or minimized the risk of mismatch, which are important factors to the method’s performance. In this part, we propose a new approach, i.e., WSNR robust design, to control the scenario variance and to minimize the weighted expectation of the ACs from different scenarios. First, to characterize different scenarios of the parameter sets, we apply Taguchi’s orthogonal arrays to conduct an experimental design. Linderman and Choo (2002), Vommi et al. (2007), and Vommi and Seetala (2007) set the hourly penalty costs of running the process in out-of-control as 100, 200, 300 at process shifts of 0.5, 1.0 and 2.0, respectively. However, this assumption can not accurately reflect the changes of actual parameters, because the possibility of excessive fluctuations in parameters are relatively small. Thus, we adopt the fractional factorial design to study the parameters and robust criteria. According to a set of benchmark parameters, one can use orthogonal tables to design experiments with multiple factors and multiple levels. For

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the experiments, each factor is discussed in three levels with 27 degrees of freedom. Then, an L27 (38 ) matrix can be constructed in Table 1. To minimize the weighted process cost function in n = 27 runs, the method starts by searching a coarse grid to find an approximated minimum cost. These tables ensure that the experimental design is straightforward and consistent. With the Taguchi’s approach, the number of analytical explorations can be significantly reduced, for which the testing time and costs are minimized. Table 1: Taguchi’s L27 array Experiment

Factor 1

Factor 2

Factor3

Factor 4

Factor 5

Factor 6

Factor 7

Factor 8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3

1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3

1 1 1 2 2 2 3 3 3 2 2 2 3 3 3 1 1 1 3 3 3 1 1 1 2 2 2

1 1 1 2 2 2 3 3 3 3 3 3 1 1 1 2 2 2 2 2 2 3 3 3 1 1 1

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

1 2 3 1 2 3 1 2 3 2 3 1 2 3 1 2 3 1 3 1 2 3 1 2 3 1 2

1 2 3 1 2 3 1 2 3 3 1 2 3 1 2 3 1 2 2 3 1 2 3 1 2 3 1

1 2 3 2 3 1 3 1 2 1 2 3 2 3 1 3 1 2 1 2 3 2 3 1 3 1 2

Then, we adopt the SN ratio as the response of variance (Taguchi’s method uses the SN ratio as the quality characteristics of choices). The SN ratio is defined metaphorically as the ratio of useful information to false or irrelevant data in the process. The production factors can be divided into three categories, i.e., the “control factors” (which are the decision variables in the process, e.g., (n1 , n2 , U CL1 , U CL2 , W L, kw )), 15

“noise factors” (which affect the process variability measured by the SN ratio, e.g., (M, c0 , a1 , a2 , a3 , a4 , 1/λ, δ)), and “fixed factors“ (which do not affect the SN ratio, e.g., h and D). For example, the first experiment in L27 (i.e., the combination “11111111“ in Table 1) requires all noise factors to be kept at their base levels when measuring the production performance. Hence, AC1 (M, c0 , a1 , a2 , a3 , a4 , 1/λ, δ) is the response to row 1. Considering the weighted expectation on ACs from different scenarios, we denote the weighted SN ratio as SNw . Then the SNw can be calculated as SNw = −10 lg

1 ∑ ( wi ACi2 ), n

(17)

where ACi represents the expected cost per hour of the ith experiment. To achieve a good stability, it is better for the quality characteristics to be close to the target values and have a good resistance to the noise interference. Because the SN ratio not only takes into account the average level of the cost characteristics but also considers its fluctuation range, it is reasonable and comprehensive to use SN ratio to evaluate the cost level. Finally, by considering the possibilities of different scenarios, we quantify the weighted sum of all scenarios. We adopt the weight of ith scenario proposed by Vommi et al. (2007) as wi =

∑λi , i λi

where λ is the rate of occurrence of assignable cause. Thus, the

problem can be formulated as: min

n ∑

wi ACi

i=1

s.t

SNw > κ, RU > RL , U CL1 > W L, kw , RU , RL , U CL1 , U CL2 , W L > 0, n1 , n2 ∈ N,

where κ is a constant that controls the low bound of SNw .

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7

Numerical Experiments

7.1

Comparison of DS npx chart and npx chart

In this subsection, compared with the npx , we first demonstrate the effectiveness of the proposed DS npx chart. Recall the five parameters of the npx chart, i.e., (nnp , hnp , RL , RU , U CL). It is obvious that the npx chart does not need a lower control limit compare to the conventional np chart. We implement the cost model proposed in Section 3 and the parameters from Montgomery (2009) to detect the shifts for a set of δ values ranging from 0.1 to 1.0 (GA is used to find the corresponding solutions). For both charts, Table 2 exhibits the results for ARL0min = 370.4 and presents the values of ARL0 and ARL1 and the optimal design parameters. Let Cr be the relative percentage of cost reduction for DS npx chart, compared to the npx chart, one has

Cr =

ACDS

npx

− ACnpx

ACnpx

× 100%.

(18)

For example, when δ = 0.7 and ARL0min = 370.4, the table shows that Cr = ((56.1441 − 69.1989)/69.1989) × 100% = −18.86%. This implies that, relative to the npx chart, the DS npx chart can approximately reduce 18.86% of the associated costs. Since Wu et al. (2009) have shown that the npx chart generally outperforms the X chart for the same inspection cost, the DS npx chart is also more efficient than the X chart. Table 2: Optimal values of ARL1 and the associated Cr when ARL0min = 370.4. npx control chart

Double sampling npx control chart

δ

n

kw

U CL

n1

0.1

6

1.9313

2.5668

0.2

9

1.7907

3.5061

0.3

7

1.6402

0.4

8

0.5 0.6

ARL1

AC

n2

kw

WL

U CL1

U CL2

npx

DS npx

npx

DS npx

Cr

4

8

1.8618

1.1423

2.0666

3.2413

347.3375

345.1470

96.7796

97.5352

0.78

5

16

1.9283

1.7688

2.1753

4.1874

280.5469

272.8136

96.0096

96.8676

0.89

3.2181

5

8

1.9437

1.8073

2.5925

3.003

228.0183

201.2834

94.4652

94.7204

0.27

1.7252

3.5055

4

13

1.8149

1.8149

2.6501

4.2825

143.8227

132.9748

94.4090

91.0473

-0.39

13

1.9763

3.9954

9

19

1.9315

1.5929

3.7500

5.7628

71.4012

50.4017

83.0029

79.4409

-4.29

12

1.9379

3.3138

8

15

1.8505

1.8942

3.9374

5.2905

44.6301

31.3031

74.9723

68.0987

-9.16

0.7

9

1.7907

3.6052

6

26

1.9112

1.0154

3.5686

5.7578

30.7236

16.2379

69.1989

56.1441

-18.86

0.8

12

1.9379

3.9316

5

18

1.8685

1.7917

3.7339

5.634

19.2714

10.0100

57.4159

46.1726

-19.85

0.9

13

1.9762

3.7374

9

13

1.9782

1.9876

3.2106

4.1125

10.4429

5.4377

44.8938

37.0495

-17.74

1

12

1.9125

3.9206

8

22

1.934

1.6079

3.3304

5.5459

6.8785

3.5575

37.4343

31.3758

-16.18

17

From Table 2, we have the following observations.

First, with the constraint

ARL0min = 370.4, the values of ARL1 based on the DS npx chart are smaller than that of the npx chart. When δ is small (δ = 0.1, 0.2, 0.3), the values of ARL1 for the two charts are close to each other. As δ increases, the DS npx chart outperforms the npx chart by reducing the value of ARL1 . For example, the ARL1 of DS npx chart is almost half of that of the npx chart. In other words, when the shift is relatively large, the DS npx chart can significantly reduce the average run length of the out-of-control state. In particular, we can find that the AC of the npx chart is relatively smaller than the DS npx chart when δ < 0.3. As the shift becomes larger, the DS npx chart shows more advantages in AC reduction. Compared to the npx chart, applying the DS scheme can help reduce the associated cost by up to 19.85%. Furthermore, the optimal kw , W L, U CL1 , U CL2 change very little as long as δ is fixed. In conclusion, compared with the traditional npx chart, the DS npx chart offers a remarkable improvement in the ARL1 performance and has significant impacts on the AC savings for a large shift.

7.2

The economic performance of WSNR robust design

In this section, we first show the limitations of traditional economic design under an uncertain environment. Then, the comparison of WSNR robust design and traditional economic design is investigated. Finally, we compare the new robust design approach with the previous robust design methods (e.g., absolute robustness design, robust deviation design, and relative robustness design) to show its advantages. First, for the economic design of DS npx , we adopt the parameters used by Montgomery (2009) for the economic design of X charts, where λ = 0.05, a1 = 50, a2 = 25, a3 = 1, a4 = 0.1, D = 1, M = 100, c0 = 10, h = 1 and δ = 0.5. Substituting the abovementioned parameters into Equation (16) to construct the model and use GA to find the optimal solutions. By repeating the tests, the optimal parameters of DS npx chart can be obtained as ∗ n∗1 = 9, n∗2 = 19, kw = 1.9315, W L∗ = 1.5929, U CL∗1 = 3.7500, U CL∗2 = 5.7628,

18

and the associated statistical performance can be observed by

E(n|δ = 0) = 34.8451, E(n|δ ̸= 0) = 37.5271, ARL0 = 370.6423, ARL1 = 50.4017. The optimal expected total cost is AC ∗ = 79.4409. The efficacy of the economic design relies on the accuracy of parameter estimations. However, it is usually difficult to obtain such precise data in practice. The input parameters used in the economic design include cost parameters (e.g., cost of sampling) and some procedure parameters (e.g., process loss rate). In addition, traditional economic design assumes that the process is subject to a single scenario. But, in reality, the control charts should be able to handle multiple different scenarios. With these in mind, the economic design of a control chart should be robust to variations in parameters and scenarios. To illustrate the need of robust economic design, we examine additional scenarios for different levels of shifts which will result in various ACs. The following table shows the effects of parameters on the traditional economic design model. Table 3: Effects of parameters on the economic design M

AC

c0

AC

δ

AC

λ

AC

80 90 100 110 120 a1 45 47.5 50 52.5 55

65.8814 73.1622 80.443 87.7239 95.0047 AC 80.4395 80.4413 80.443 80.4448 80.4566

8 9 10 11 12 a2 20 22.5 25 27.5 30

79.8992 80.1711 80.443 80.715 80.6699 AC 80.3751 80.4091 80.443 80.477 80.5117

0.3 0.4 0.5 0.6 0.7 a3 0.8 0.9 1.0 1.1 1.2

94.7204 91.0473 79.4409 68.0987 56.1441 AC 80.2444 80.3437 80.443 80.5424 80.6417

0.03 0.04 0.05 0.06 0.07 a4 0.08 0.09 0.1 0.11 0.12

70.6159 76.2424 80.443 83.5758 86.0021 AC 79.9339 80.0884 80.443 80.7997 81.1524

Table 3 provides the solutions for different parameter settings with the traditional economic design model. From the table, it can be seen that the variations of design can have significant impacts across different situations. For instance, the changes of M , δ, and λ will deliver quite different ACs. In particular, if λ increases from 0.03 to 0.07, the AC will vary from 70.6159 to 86.0021, which makes the traditional economic design difficult to be implemented in an uncertain environment. Next, we will propose the robust economic design model to deal with the situations that the parameters may 19

not be estimated accurately or the production environment exists uncertainty. As aforementioned, the economic model of DS npx chart requires 10 input parameters to determine the corresponding cost. We first investigate the robustness of input parameters on the optimal design. In particular, the WSNR robust design is performed by varying six cost parameters (M, c0 , a1 , a2 , a3 , a4 ) and two key process parameters (1/λ, δ), which are noise factors in this experiment (cannot be known exactly and only can be estimated with different degrees of precision). We consider the 8 noise parameters at three degree levels and remain the other parameters unchanged for each experimental run. The experiment is conducted to find an efficient robust solution for the design of control chart at three shift scenarios. The shift scenarios (5%, 10%, and 15%) based on three degree status (base, low, and high) for the variables are presented in Table 4. In particular, the base status will remain the parameters unchanged, the low status will decrease 5% (10% or 15%) of the parameters from the base status, and the high status will increase 5% (10% or 15%) of the parameters from the base one. Thus, the Taguchi’s L27 (38 ) array can be used to study the robustness of the 8 factors in Table 4. For the Taguchi’s array shown in Table 1, we use “1” (“2” or “3”) to represent the base status (the low status or the high status). Table 4: Model parameters for the different shift scenarios Noise Parameter λ C0 M a1 a2 a3 a4 δ

5% shift scenario low base high 0.0475 9.5 95 47.5 23.75 0.95 0.095 0.475

0.05 10 100 50 25 1 0.1 0.5

0.0525 10.5 105 52.5 26.25 1.05 0.105 0.525

10% shift scenario low base high 0.045 9 90 45 22.5 0.9 0.09 0.45

0.05 10 100 50 25 1 0.1 0.5

0.055 11 110 55 27.5 1.1 0.11 0.55

15% shift scenario low base high 0.0425 8.5 85 42.5 21.25 0.85 0.085 0.425

0.05 10 100 50 25 1 0.1 0.5

0.0575 11.5 115 57.5 28.75 1.15 0.115 0.575

Table 5: Design of DS npx chart based on three noise scenarios Noise Scenarios

n1

n2

kw

WL

U CL1

U CL2

5% 10% 15%

7 6 8

14 22 20

1.7966 1.9887 1.9123

1.4058 1.5402 1.6787

3.0904 3.7587 3.8816

5.8477 5.717 5.2541

From Table 8, one can see that the design of DS npx chart is based on three 20

scenarios. The values of kw , W L, and U CL1 have no substantial differences for the scenarios 5% to 15%, while n1 (n2 ) ranges from 6 (14) to 8 (22). Thus, when the change of noise parameter becomes larger, the sample size increases. For each shift scenario, we calculate the mean, variance and maximum of the results for both traditional design (TD) and the proposed WSNR robust design. Similar to Vommi et al. (2007), we set equal weight to each situation to complete the numerical analysis. In addition, we evaluate the rate of change of the three measures mentioned above. For example, the relative change of the mean(variance, maximum) of AC from TD to WSNR robust design can be calculated as

Relative Change (M ean/V ariance/M aximum) =

W SN R − T D × 100%. TD

(19)

Compared to the previous research (Pignatiello and Tsai, 1988), e.g., variations from 10% to 50%, we assume the ranges of parameters will change from 5% to 15%, which is a more realistic situation. In this case, we can take slighter change of noise parameters into consideration by using SN ratio and multi-objection decision making techniques. Table 6: Comparison of statistics for the results of TD and WSNR Noise Scenario 5%

10%

15%

Type of Design

Mean

Variance

Maximum

TD WSNR Relative change TD WSNR Relative change TD WSNR Relative change

79.8203 73.0699 -8.04% 82.9212 73.9963 -10.76% 84.0136 74.3157 -11.54%

18.4979 9.80650 -46.98% 56.0163 37.6105 -32.85% 114.1124 85.5149 -25.06%

87.0634 78.9052 -9.37% 96.2293 85.7955 -12.16% 105.4052 92.3586 -11.81%

Table 6 records the above-mentioned statistics for the TD and WSNR robust design models. From the table, one can see that, for all of the shift scenarios, the means, variances and maximums of ACs for WSNR robust design are smaller than that of TD. In particular, the gap of relative change in means between TD and WSNR robust design becomes large as the shift level increases, and the rate of change in variance of WSNR robust design is always lower than that of TD. This demonstrates that the 21

Table 7: Comparison of statistics for the results of WSNR, AR, RD and RR Noise Scenario

Robust design

Mean

Variance

Maximum

5%

WSNR AR RD RR WSNR AR RD RR WSNR AR RD RR

73.0699 73.0182 73.2119 73.7503 73.9963 74.1234 74.7855 74.4275 74.3157 74.4411 76.2218 75.5914

9.8065 9.8792 10.692 9.5849 37.6101 40.7408 41.7224 39.3962 85.5149 92.2027 91.4578 87.0803

78.9052 78.3932 79.1556 79.4572 85.7955 85.4291 87.293 86.6463 92.3586 92.7392 94.2891 93.1582

10%

15%

WSNR robust design enhances the stability of the process cost. Besides, the maximum AC indicates the worst-case performance in the design of a control chart across all scenarios. We can see that the maximum AC of WSNR robust design is always smaller than that of TD, which means robust design can minimize the worst case scenario. Moreover, by observing that the relative change decreases (increases) as the degree of shift enlarges, it is interesting to find that the efficiency of the WSNR robust design increases as the shift level increases. To further evaluate the performance of WSNR robust design, comparisons are made with the discrete robustness measures proposed by Linderman and Choo (2002), i.e., absolute robustness (minimizes the worst-case scenario), robust deviation (minimizes the deviation from the optimal solutions), and relative robustness (minimizes the percentage deviation from the best performance). Table 7 compares the statistical indicators of the four robust design approaches. Under the 5% shift scenario, the differences among the traditional design, absolute robustness (AR) design, robust deviation (RD) design and relative robustness (RR) design are not significant. According to Table 6, we know that the WSNR robust design shows slight advantage to the traditional design. However, as the uncertainty increases, i.e., under the 10% and 15% shift scenarios, Table 7 demonstrates that the WSNR robust design has the best performance in most cases. Compared with other robust design, the WSNR robust design has the smallest mean and variance. This result indicates that the WSNR robust design is advanced in minimizing the mean cost

22

and controlling the degree of data variability. In addition, across all of the scenarios, the maximum value of WSNR robust design never be the worst one. In other words, the WSNR robust design can minimize the worst possible situation while controlling data fluctuations. Hence, we can conclude that, compared to the pervious methods, the proposed WSNR robust design approach can significantly improve the economic performance of the DS npx chart and is more stable and flexible when the uncertainty becomes large. In addition, to evaluate the cost difference of WSNR robust design and the other three robust measures, we calculate the absolute change of AR, RD, RR to WSNR robust design based on the value of WSNR robust design. Take AR as an example, we have Absolute Change = AR − W SN R.

(20)

The results are shown in Figure 4, 5, and 6. According to the absolute change, it can be seen that the WSNR robust design can save costs under most circumstances. In particular, when the shift is small (5% shift scenario), the absolute change is not significant, i.e., the number of positive absolute changes is close to the number of ACs of negative absolute changes. However, as the uncertainty increases, i.e., the shift becomes larger, the absolute changes increase and remain positive under most of the cases, which implies that the WSNR robust design performs well as the uncertainty increases. Therefore, from the above performance analysis, the WSNR robust design of DS npx chart is a promising substitute to the existing robust measures for minimizing the risk of uncertainty involved in the design parameters.

23

Figure 4: The absolute changes in AC under 5% shift scenario

Figure 5: The absolute changes in AC under 10% shift scenario

24

Figure 6: The absolute changes in AC under 15% shift scenario

7.3

Robustness of the DS npx control chart to non-normality

The DS npx control chart is designed under the assumption that the observations are generated from a normal distribution. However, it is difficult to accurately estimate whether the distribution is normally distributed as a priori (Li et al., 2016a,b). In this subsection, we will discuss a case that the distribution is slightly asymmetric and unknown. Assume that the mean and standard deviation are known. Take the Gamma distribution as an example. A Gamma-distributed random variable X with shape parameter k and scale parameter θ can be denoted by X ∼ Γ(k, θ). The probability density function (PDF) using the shape-scale parametrization is xk−1 e− θ , f or x > 0 and k, θ > 0. θk Γ(k) x

f (x; k, θ) =

Hence, the mean and variance can be expressed as E[X] = kθ, V ar(x) = kθ2 The skewness of the Gamma distribution only depends on its shape parameter k as

√2 , k

e.g., a larger k indicates a closer approximation of normal distribution.

25

Tables 8 and 9 present the ARL0 s and ARL1 s for the normal distribution and various Gamma distributions. For simplicity, we use the superscript G and N to represent Gamma distribution and normal distribution, respectively. The growth rate N of ARLG 0 s relative to ARL0 s is shown in the parenthesis. Table 8 indicates that the N ARLG 0 s are comparable to the ARL0 s when the skewness is small, i.e., the distribution

is slightly asymmetric. While ARLG 0 s increase with the skewness. From Table 9, we can observe that the ARLG 1 s increase about 1% when the skewness is 0.1. For shifts that are greater or equal to 0.6, the growth rate of ARLG 1 s is up to 3.99%. G It is interesting to note that ARLN 0 s and ARL1 s both become larger as the skew-

ness increase, which indicates that the false alarm α has dropped. However, previous research shows that the number of false alarms under Gamma distribution would be significantly higher than that under the normal distribution. Then, we will illustrate that our results also support this conclusion. When the skewness is small, Figure 7 presents the false alarms of Gamma and normal distributions. When applying the same warning limits RU and RL , we can observe that the PDF of Gamma distribution rises faster on the left side, which leads to the false alarm αG < αN . When the skewness √ is large (2 2), Gamma distribution is monotone decreasing. We can obtain ARLG 0 as 42, which is consistent with the result of Borror et al. (1999). The results show that the skewness of Gamma distribution indeed affects the value of the αG . Table 8: ARL0 for the DS npx control chart and Gamma Distribution 0.3

0.4

δ 0.5

0.6

0.7

370.1157 373.1539(0.01%) 370.2685(0.04%) 371.0715(0.25%) 373.9472(1.03%) 378.7676(2.33%) 385.5712(4.17%) 394.4053(6.56%)

370.0681 370.1118(0.01%) 370.2428(0.04%) 371.1611(0.29%) 374.4593(1.18%) 380.0211(2.68%) 387.9443(4.83%) 398.3669(7.64%)

370.2704 370.3245(0.01%) 370.4867(0.05%) 371.6238(0.36%) 375.7063(1.47%) 382.5843(3.32%) 392.3663(5.96%) 405.1980(9.43%)

370.1611 370.2160(0.01%) 370.3806(0.06%) 371.5351(0.37%) 376.4150(1.69%) 382.7033(3.38%) 392.7339(6.09%) 405.9880(9.67%)

370.3150 370.4667 370.6358 371.8209 376.4907 383.2690 393.9538 407.0313

skewness Normal Gamma(4002 , 1/200) Gamma(1002 , 1/100) Gamma(402 , 1/40) Gamma(202 , 1/20) Gamma(1600/9, 3/40) Gamma(100, 1/10) Gamma(64, 1/8)

0.01 0.02 0.05 0.10 0.15 0.20 0.25

26

(0.04%) (0.08%) (0.41%) (1.67%) (3.49%) (6.38%) (9.91%)

Table 9: ARL1 for the DS npx control chart and Gamma Distribution 0.3

0.4

δ 0.5

0.6

0.7

201.2834 202.1632(0.43%) 203.0221(0.86%) 205.8332(2.26%) 211.3736(5.01%) 218.1331(8.37%) 226.3149(12.44%) 236.1649(17.32%)

132.9748 133.0968(0.09%) 133.2377(0.19%) 133.7773(0.60%) 135.0931(1.59%) 136.9826(3.01%) 139.5166(4.91%) 142.7805(7.37%)

50.4017 50.5431(0.28%) 50.7596(0.71%) 51.4606(2.10%) 52.8201(4.79%) 54.4588(8.04%) 56.4312(11.96%) 58.8038(16.67%)

31.3031 31.3116(0.03%) 31.3228(0.06%) 31.3738(0.23%) 31.5221(0.69%) 31.7594(1.45%) 32.0984(2.54%) 32.5535(3.99%)

16.2485 16.2493(0.05%) 16.2611(0.12%) 16.3026(0.38%) 16.4048(1.01%) 16.5183(1.71%) 16.6914(2.77%) 16.8858(3.97%)

skewness Normal Gamma(4002 , 1/200) Gamma(1002 , 1/100) Gamma(402 , 1/40) Gamma(202 , 1/20) Gamma(1600/9, 3/40) Gamma(100, 1/10) Gamma(64, 1/8)

0.01 0.02 0.05 0.10 0.15 0.20 0.25

PDFs for Gamma(16,0.25) and N(4,1)

0.5

Normal distribution Gamma distribution 0.4

PDF

0.3

0.2 RU= µ0 +kw σ 0

RL = µ0 -kw σ 0 0.1

αN αG

0 0

1

2

3

4

5

6

7

8

x

Figure 7: PDFs for the normal and gamma distribution with small skewness

Next, we utilize the WSNR robust design to compute the AC of the DS npx control chart with Gamma distribution. From Table 10, one can see that the WSNR robust design performs well when skewness is small for all shift scenarios. Specifically, the actual AC G is within 4% with the normal distribution. Table 10: The AC of the DS npx control chart under WSNR design skewness Normal Gamma(4002 , 1/200) Gamma(1002 , 1/100) Gamma(402 , 1/40) Gamma(202 , 1/20) Gamma(1600/9, 3/40) Gamma(100, 1/10) Gamma(64, 1/8)

0.01 0.02 0.05 0.10 0.15 0.20 0.25

5%

shift scenario 10%

15%

73.0699 73.1275(0.07%) 73.1707(0.14%) 73.3562(0.39%) 73.8110(1.01%) 74.1107(1.42%) 74.4073(1.83%) 75.2428(2.97%)

73.9963 74.0549(0.07%) 74.0591(0.13%) 74.2753(0.37%) 74.5916(0.80%) 74.9802(1.32%) 75.4922(2.02%) 76.3047(3.11%)

74.3157 74.3852(0.09%) 74.4224(0.14%) 74.6111(0.39%) 75.0060(0.92%) 75.3941(1.45%) 76.4022(2.80%) 76.6968(3.20%)

In conclusion, when the distribution is slightly asymmetric, the ARL0 and ARL1 are not very sensitive to the assumption of normality. However, the performance of control chart decreases as the skewness increases. 27

The case discussed above is that the distribution is slightly asymmetric and unknown. When the asymmetric distribution is known, it is necessary to redesign the control limits based on the corresponding distribution. After that, the robust design method proposed in this paper can also be applied to the economic design of control charts with non-normal distributions.

8

Conclusion

In this paper, we develop a DS npx chart to monitor the process mean and variance. By conducting comparisons, we find that the performance of the proposed DS npx chart is better than that of the npx chart in terms of ARL and the expected cost per hour. From the cost-saving perspective, we establish an economic design model for the proposed DS npx chart to reduce the process costs. To address the issues involved in parameter measure errors, production environment uncertainties, and other technical difficulties, we implement a robustness framework based on the SN ratio to improve the economic design performance. The new robust design considers the weighted expectation on multiple scenarios and takes the SN ratio as the response to variance. To compare the performance of the previous robust methods with our approach, we conduct numerical experiments to demonstrate the advantages of the WSNR robust design. Furthermore, by analyzing the expected cost per hour, the performance of robust design model is superior to the traditional design according to the associated statistics, e.g., mean, variance, and maximum value. When the shifts of the noise parameters become larger, the effect of robust design is more pronounced. This advantage can assist the user to select the best combination design of control chart according to the corresponding cost and/or statistical performance under an uncertain environment. In addition, we numerically verify that the WSNR design is still robust when there is a slightly asymmetry in the process distribution. However, this research still exists some limitations. For example, we only consider the application of double sampling scheme in npx chart. For future research, it is

28

meaningful to explore the combination of npx chart and other chart to further enhance the performance of the attribute chart, e.g., CUSUM chart or CRL chart. Besides, studies that consider the correlation involved in the observations are also worthwhile to be carried out.

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The WSNR robust design has the smallest mean and variance than traditional design.

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The WSNR robust design can minimize the worst case and control data fluctuations.

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The new design outperforms than other robust designs as the uncertainty increases.

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The DS npx chart is better than the npx chart in controlling the process cost.