AEWMA t Control Chart for Short Production Runs

1 Introduction

Control charts are effective tools in statistical process control (SPC) for process monitoring and quality improvement. Usually, SPC manuals associate the Shewhart, EWMA and CUSUM control charts implementation with manufacturing processes which operate indefinitely. In fact, resource capacity within manufacturing systems characterized by a high degree of flexibility and variety of produced items is usually scheduled to manufacture short runs of productions. This kind of phenomenon is common in job shop manufacturing of small series of mechanical parts. Because the traditional control charts have to assume that the process mean and deviation are known previously or estimate the process parameters by taking many samples, the short run property leads to many difficulties in implementing control charts, such as there are not enough samples to estimate distribution parameters.

Usually, at least 25 ~ 30 samples having size n are taken in Phase I control chart to correctly estimate the distribution parameters of the monitored statistic. In particular, to get estimated limits close to the true chart limits, Quesenberry^[1] has suggested that at least 400/ (n – 1) samples should be taken. Short production runs allowing for a limited number of scheduled inspections, results in a number of samples which is too small. To overcome this problem, many researchers have investigated the procedure to reduce the samples in Phase I chart. Neduraman and Pigniatiello^[2] and Tsai et al.^{[3, 4]} have proposed different procedures based on evaluating statistics having a t distribution to construct the prospective control limits with a few initial subgroups. These procedures still need a few of initial subgroups, even the initial sample numbers are decreased; thus, they are more effect in medium or large run production. Quesenberry^{[5, 6]} has proposed self-starting charts by defining a set of sequential Q statistics to detect the shifts in process mean or variance. Castillo and Montgomery^[7] have proposed some enhancements for Q chart: An exponentially weighted moving average (EWMA) method and an adaptive Kalman filtering method. He, et al.^[8] has indicated that Q charts are biased with unknown variances. For some specific mean shift, the Q charts have a large out-of-control average run length than the expected number of false alarms; this problem is particularly obvious if the shift occurs at the early stage of sampling, which is known as ‘masking of the shift’ problem. Celano, et al.^[9] have demonstrated that the modified Q charts proposed by He, et al.^[8] still present some bias and difficulty of implementation in practice.

Recently, t control chart has been proposed by Zhang, et al.^[10], and the performances of Shewhart t chart and EWMA t chart for long runs have been investigated against the errors in estimating the process standard deviation. Kazemzadeh^[11] have investigated the variable sampling interval (VSI) EWMA t chart. As an efficient quality control tool for short production runs, t chart has been investigated as a self-starting control chart. Nenes and Tagaras^[12] have proposed two proper statistical performance measurement. One of them is truncated average run length (TARL), which is defined as the mean number of samples until a signal or until the completion of the process, whichever occurs first. The other on is the probability of having at least one signal during an out-of-control production run. Utilizing these measures, Celano^[9] have investigated the short run Shewhart t chart and EWMA t chart with perfect and imperfect set-up conditions, and the self-starting property of t chart has been investigated. Celano^[13] have investigated the performance of Shewhart, EWMA and CUSUM t charts with unknown shift size. Besides, the economic performances of Shewhart and CUSUM t chart for short production runs have been investigated by Celano^{[14, 15]}, and the economic model for short production runs has been specifically designed.

However, a single EWMA control chart can not perform well for small and large shifts simultaneously^[16]. The small size of smoothing parameter λ will lead to the potential delay in detecting sudden large shift, which is known as the ‘inertia problem’. To overcome the ‘inertia problem’ of EWMA control chart, the adaptive varying smoothing parameter EWMA (AEWMA) control chart that weights the past observations of the monitored process using a suitable function of current error has been proposed by Capizzi and Masarotto^[16]. Woodall and Mahmoud^[17] have defined a new measure of inertia, the signal resistance (SR). The value of SR shows that AEWMA chart has much better inertial properties than the ordinary EWMA chart. Shu^[18] has investigated the property of AEWMA control chart for monitoring process dispersion, and proposed another optimization method for AEWMA control chart. Saleh, et al.^[19] have denoted that the AEWMA chart outperform the EWMA chart to monitor the process mean when process parameters are estimated, especially when a small number of Phase I samples is available and the shift size is unpredictable.

To overcome the inertia problem of EWMA t control chart, we propose an AEWMA t control chart for short production runs in this paper. The Markov chain method is implemented to calculate the TARL of AEWMA t control chart. The joint effect of parameters on the TARL performance is investigated. Based on the investigation, a new optimization algorithm of AEWMA control chart is proposed. The simulation results show that the performance of AEWMA t control chart is superior to the EWMA t control chart.

The rest of paper is organized as follow. In Section 2, the monitored T statistic is introduced. In Section 3, the AEWMA t control chart is proposed, and the Markov chain methods used to for computing the statistical measures are developed. In Section 4, optimization statistical design procedure of AEWMA t control chart are introduced based in the analysis of joint effects of parameters λ and γ. In Section 5, considering the two property measures, the performance of AEWMA t control chart is compared with EWMA t control chart. Finally, some conclusions are given.

2 The T Statistic

Assuming that, N parts during a production horizon H are scheduled to produce in a manufacturing process. The normal distributed quality character Χ ~ N (μ₀ + δσ₀, τσ₀) with the target value equaling to M should be monitored, where μ₀ and σ₀ are the in-control population mean and standard deviation, respectively. The shift of the mean and deviation are measured as the multiples δ and τ of the in-control population standard deviation σ₀, respectively. Let Is denotes the number of scheduled inspections within the production horizon H. A subgroup {X_i,1, X_i,2, · · · , X_i,n}, i = 1, 2, · · · , Is of n measures is collected at ith inspection epoch. The sample size n is subject to n ≤ N/Is. The subgroup mean X̅_i and standard deviation S_i are computed as:

At the beginning of the production, the in-control mean μ₀ should be as close as possible to the target value M. However, the process begin with an initial setup error is common. The setup error is measured as the multiple Δ of σ₀. That is to say, Δ = 0 denotes that the process is perfect setup, μ₀ = M; otherwise, an initial setup error Δσ₀ will shift the in-control population mean μ₀ = M to μ₀ = M + Δσ₀. When parameter values are Δ ≠ 0, δ ≠ 0 and τ > 1, the statistic T_i can be computed:

where σ₁ = τ · σ₀ represents the out-of-control process deviation; Z=nX¯i−μ1nX¯i−μ1σ1σ1 is a standard normal random variable; V = (n – 1)(S_i/σ₁)² is chi-squared distributed with n – 1 degrees of freedom. The standard deviation of T_i is:

where n should be greater than 3. The process is in-control meaning that Δ = 0, δ = 0 and τ = 1, the statistic T_i obeys t distribution. We define the notation F (·|v) as the t cumulative distribution function with ν degrees of freedom. Otherwise, when the process is out-of-control, the statistic T_i obeys a non-central t distribution for Δ ≠ 0 or δ ≠ 0. We define the notation G (·|v, ψ) as the non-central t cumulative distribution function with ν degrees of freedom, and ψ is the non-central parameter.

3 AEWMA t Control Chart

The EWMA t control chart has been investigated by many researchers with the form:

where λ ∈ (0,1] is the smoothing constant. For EWMA t chart, the smoothing parameter λ will remain the same during the whole process. The small value of smoothing parameter λ will lead to large potential delay in detecting a sudden large shift, which is known as the “inertia problem”. To overcome the inertial problem, Capizzi and Masarotto^[16] have proposed an alternative form of EWMA control chart, which is known as AEWMA chart. We utilize the AEWMA chart to monitor the T statistic, which is defined as AEWMA t chart. The AEWMA t chart is described as:

where e_i = T_i – Y_i-1 is the estimation error of ith epoch; ϕ (e_i) is a score function. By defining ω (e_i) = ϕ (e_i) /e_i, the equation (6) can be rewritten as follow:

[16] has given three score functions, the function ϕ_hu(e) has been widely used^{[18, 19, 21]}. The score function ϕ_hu (e) is defined as follow:

The control limits of AEWMA t chart can be defined as the multiple of σ_T. The control limits of AEWMA t control chart equal to:

A useful statistic measure of control chart is the average run length (ARL), which is defined as the expected number of samples until the chart triggers a signal. For the short run case, where Is is limited, the process may end without the chart having issued any out-of-control signal. Therefore, truncated ARL (TARL) is a proper statistical measure for short production runs. If the process ends without any signal during the Is samples, the truncated run length is Is + 1.

To calculate the TARL of AEWMA t control chart, the Markov chain theory is implemented. The interval from LCL to UCL is partitioned into 2m+1 subintervals I_j, j = –m, · · · , 0, · · · , m. The width of each subinterval is d = (UCL – LCL)/(2m + 1). The statistic Y_i is said to be in state j at time i which means that the inequality S_j – d/2 < Y_i < S_j + d/2 is satisfied, where S_j is the midpoint of subinterval I_j. Assuming that Y_i = S_j, whenever it falls in subinterval I_j. The absorbing state is denoted as m + 1 state, which means that Y_i < LCL or Y_i > UCL. The transition probability matrix P is as follow:

where the generic element p_j,k of P represents the probability that statistic Y_i goes from state j to state k in one step. The value of p_j,k can be calculated:

where ϕhu−1v is the inverse function of ϕ_hu (e). The form of ϕhu−1v is

where, λ and γ are two decision parameters of AEWMA chart.

The value of P_j,m+1 denotes that Y_i goes from in-control state j to the absorbing state m+1, and P_j,m+1 can be calculated:

When process operates in out-of-control states, the out-of-control transition probability matrix P̃ can be calculated in the same manner. The generic element p̃_j,k is obtained:

The l-step transition probability matrix P^l is the result of multiplication of P by itself l times. The matrix P^l can be rewritten in partitioned form:

where the matrix W^l is defined as

The in-control TARL denoted by TARL₀ can be calculated:

The out-of-control TARL denoted by TARL₁ can be calculated in the same manner by replacing P̃ to P in Equation (15).

Another useful performance measure is q(Is, δ, τ), which is defined as the probability of getting a signal within Is inspections. When the process is in-control, q(Is, 0, 1) means that the probability that the AEWMA t control chart signals a false alarm within Is inspections.

where p0,m+1Is is the in-control transition probability from the transient state 0 to absorbing state m + 1 of the Markov chain after Is steps. When the process is out-of-control, q (Is, δ, τ) represents the probability that AEWMA t control chart signals a true alarm within Is samples. In the same manner, the value of q (Is, δ, τ) can be calculated as the manner of equation (18).

The statistical design of AEWMA t control chart is more complex than EWMA t control chart. The out-of-control TARL at two mean shift sizes δ₁, δ₂ (δ₁ < δ₂) have to be minimized subject to the in-control TARL equals to Is.

where θ = (h, λ, γ) is the parameter set of AEWMA t chart.

4 The Optimal Procedure of AEWMA t Control Chart

The statistical design of an AEWMA t control chart is a 3-dimensional optimization problem, the parameter h, λ and γ are computed to satisfy equation (19). [16, 20] have investigated the effect of λ and γ alone on the chart performance. The joint effect of λ and γ on the run length performance of AEWMA t control chart have been investigated by [18, 21]. However, the joint effect of λ and γ on the TARL performance for short production run has never been investigated. To better understand the overall effect, figure 1 shows contour plots of the out-of-control TARL values of AEWMA t chart as a function of λ and γ for n = 10 Is = 10, 30, δ = 0.5, 2.0 when zero-state in-control TARL is Is.

Figure 1

Contour plots of TARL for different Is and δ

From Figures 1(a), 1(b) and 1(d), when detecting a small increase (δ = 0.5) in the process mean, the TARL decreases as λ decreases or γ increases. From Figure 1(b), the TARL values are almost unchanged for γ < 3.5. However, contour plots become more complex in figure 1(c). The trend of TARL changing with λ and γ is no more obvious.

Two optimization procedures have been proposed by [16, 18], respectively. However, the procedure proposed by [18] suggests specifying an value of λ to give the minimum value of ARL of an EWMA chart at small shift size δ₁, firstly. Then, the optimal γ value having the minimum ARL at δ₂ is searched to satisfy the conditions that the percentage increase in the out-of-control ARL at δ₁ is no more than a small positive numberα. However, Capizzi and Masarotto^[16] have suggested to find the optimal solution θ₁ = (h, λ, γ) having the minimum TARL₁ (δ₂, θ₁), firstly. Then, the optimal parameter θ₂ = (h, λ, γ) having minimum TARL₁ (δ₁,θ₂) is selected. The parameter θ₂ = (h, λ, γ) has to be subject to the condition that the percentage increases of TARL₁ (δ₂, θ₁) is at most of a small positive number α. The procedure proposed by Capizzi and Masarotto^[16] is more complex than the procedure produced by Shu^[18], but the solutions of the method proposed by Capizzi and Masarotto^[16] are more efficient than that of Shu^[18].

Based on the analysis of joint effect of λ and γ, a new two-stage optimization algorithm for short run AEWMA t control chart is proposed. Details of the two-stage design procedure are summarized below:

• 1) Specify the value of sample size n, the interval [λ₁, λ₂] of λ and the interval [γ₁, γ₂] of λ; choose n₁ values of λ from the interval of [λ₁, λ₂]; choose n₂ value of γ from the interval of [λ₁, λ₂]; calculate the control limits h for n₁ × n₂ combinations of (λ, γ) subject to TARL₀ = Is.

• 2) Based on the solution θ = (h, λ, γ) of Step 1), calculate the TARL values of AEWMA control chart for different shift sizes δ₁, δ₂(δ₁ < δ₂); find the optimum solution θ = (h, λ, γ) having the minimum out-of-control TARL at δ₁, δ₂, which aredenoted as TARL1δ1,θ1∗ and TARL1δ2,θ2∗, respectively.

• 3) Choose two small positive constants α₁, α₂ (e.g. α₁ = 0.005, α₂ = 0.05); find the parameter θ subject to the following condition:

  (20)TARL1(δ1,θ)<1+α1TARL1(δ1,θ1∗),TARL1(δ2,θ)<1+α2TARL1(δ2,θ2∗).

• 4) If non of the θ is found, increase the parameter α₁, α₁, and repeat Step 3) until the θ is found.

• 5) If only one θ is found, this is the final solution that can get the out-of-control TARL nearly minimum; otherwise, two or more θ (i.e., θ_j ,i = 1, 2, · · · ) are found, subdivide the interval [δ₁, δ₂] into l parts, calculate the TARL values of these shift sizes (i.e., δ_i, i = 1, 2,···,l) for each θ; find the θ subject to the condition:

  (21)minθ⁡∑i=1lTARL1δi,θk

The ranges of λ and γ is another important problem. Shu^[18] has suggested that λ falls into the range [0.1, 0.4], and the value of γ falls into the range [1.5σ_T, 3σ_T]. However, the λ and γ getted by Capizzi and Masarotto[16] have exceeded the ranges proposed by Shu^[18]. From figure 1(b) and 1(d), it is declared that the up-limit of γ should be greater than 3δ_T. In this paper, the parameters λ and γ are within the range (0, 1) and (0, 10), respectively.

5 Numerical Simulations

To illustrate the performance of AEWMA t chart, both the perfect and imperfect initial setup conditions are discussed. The performances of AEWMA t chart are compared with the EWMA t chart optimized at the shift sizes δ = 0.5 and δ = 2.0.

5.2 Imperfect Setup Condition

When the process begins with an initial setup error (Δ ≠ 0), the distribution function of Τ is same as the perfect setup condition of δ = Δ. Namely, the initial setup error changes the value of δ. However, the mean shift size δ = –Δ should be specifically noted. From Table 1, we know that ψ equals to zero with the form of δ+Δnandδ+Δδ+Δττnforδ=−Δ. Mathematically speaking, the imperfect setup process is in-control for δ = –Δ. The TARL of AEWMA chart for n = 10 is shown in Figure 2.

Figure 2

TARL values of AEWMA t chart with the imperfect initial setup condition

The out-of-control TARL and q (Is, δ, τ) are calculated for n = 5, 10, 25, 50; Is = 10; τ = 1 and Δ = –1. The results are shown in Table 7 and Table 8. Obviously, each row is symmetrical about (δ = 1, due to δ + Δ = 0. Other properties are same as the perfect-setup condition. From Table 7, the ratio of AEWMA chart gets the optimal TARL is 78.57%(66/84). Almost all the non-optimal TARL are very close to the optimal values (the deviations are less than 0.0072), except n = 5, δ = 0.75, 1.25 (the deviations are 5.6%(0.18/3.17), 34.43%(l.57/4.56), 31.4%(l.84/5.56) for τ = 1, 1.5, 2.0 , respectively). From Table 8, the ratio of AEWMA chart gets the optimal q (Is, δ, τ) is 88.10%(74/84).

Table 7

TARL values of EWMA and AEWMA t control chart for Δ = –1, Is = 10

		EWMA					AEWMA
		δ					δ
n	τ	0.00	0.50	1.00	1.50	2.00	0.00	0.50	1.00	1.50	2.00
	1.00	2.77	5.22	10.00	5.22	2.77	2.79	5.21	10.00	5.21	2.79
		3.22	7.31	10.00	7.31	3.22
5	1.50	3.99	7.13	10.00	7.13	3.99	4.00	7.10	10.00	7.10	4.00
		5.69	8.74	10.00	8.74	5.69
	2.00	5.22	8.20	10.00	8.20	5.22	5.21	8.18	10.00	8.18	5.21
		7.31	9.29	10.00	9.29	7.31
	1.00	1.66	3.29	10.00	3.29	1.66	1.67	3.28	10.00	3.28	1.67
		1.58	5.30	10.00	5.30	1.58
10	1.50	2.42	5.16	10.00	5.16	2.42	2.42	5.15	10.00	5.15	2.42
		3.33	7.64	10.00	7.64	3.33
	2.00	3.29	6.68	10.00	6.68	3.29	3.28	6.66	10.00	6.66	3.28
		5.30	8.64	10.00	8.64	5.30
	1.00	1.01	1.76	10.00	1.76	1.01	1.01	1.76	10.00	1.76	1.01
		1.01	1.97	10.00	1.97	1.01
25	1.50	1.27	3.14	10.00	3.14	1.27	1.27	3.14	10.00	3.14	1.27
		1.27	4.23	10.00	4.23	1.27
	2.00	1.76	4.78	10.00	4.78	1.76	1.76	4.78	10.00	4.78	1.76
		1.97	6.21	10.00	6.21	1.97
	1.00	1.00	1.15	10.00	1.15	1.00	1.00	1.15	10.00	1.15	1.00
		1.01	1.84	10.00	1.84	1.01
50	1.50	1.01	1.96	10.00	1.96	1.01	1.01	1.97	10.00	1.97	1.01
		1.41	2.48	10.00	2.48	1.41
	2.00	1.15	3.17	10.00	3.17	1.15	1.15	3.35	10.00	3.35	1.15
		1.84	3.17	10.00	3.28	1.84

Table 8

q values of EWMA and AEWMA t control chart for Δ = -1, Is = 10

		EWMA					AEWMA
		δ					δ
n	τ	0.00	0.50	1.00	1.50	2.00	0.00	0.50	1.00	1.50	2.00
	1.00	1.000	0.965	0.222	0.965	1.000	1.000	0.967	0.227	0.967	1.000
		0.987	0.617	0.178	0.617	0.987
5	1.50	0.998	0.779	0.222	0.779	0.998	0.998	0.787	0.227	0.787	0.998
		0.819	0.396	0.178	0.396	0.819
	2.00	0.965	0.605	0.222	0.605	0.965	0.967	0.614	0.227	0.614	0.967
		0.617	0.304	0.178	0.304	0.617
	1.00	1.000	0.998	0.201	0.998	1.000	1.000	0.999	0.203	0.999	1.000
		1.000	0.828	0.177	0.828	1.000
10	1.50	1.000	0.931	0.201	0.931	1.000	1.000	0.933	0.203	0.933	1.000
		0.969	0.547	0.177	0.547	0.969
	2.00	0.998	0.779	0.201	0.779	0.998	0.999	0.782	0.203	0.782	0.999
		0.828	0.399	0.177	0.399	0.828
	1.00	1.000	1.000	0.182	1.000	1.000	1.000	1.000	0.182	1.000	1.000
		1.000	0.999	0.177	0.999	1.000
25	1.50	1.000	0.991	0.182	0.991	1.000	1.000	0.991	0.182	0.991	1.000
		1.000	0.918	0.177	0.918	1.000
	2.00	1.000	0.909	0.182	0.909	1.000	1.000	0.909	0.182	0.909	1.000
		0.999	0.732	0.177	0.732	0.999
	1.00	1.000	1.000	0.177	1.000	1.000	1.000	1.000	0.179	1.000	1.000
		1.000	1.000	0.234	1.000	1.000
50	1.50	1.000	1.000	0.177	1.000	1.000	1.000	1.000	0.179	1.000	1.000
		1.000	1.000	0.234	1.000	1.000
	2.00	1.000	0.976	0.177	0.976	1.000	1.000	0.975	0.179	0.975	1.000
		1.000	1.000	0.234	1.000	1.000

6 Conclusion

When SPC is performed for short production runs, the number of inspections is limited. It is impossible for quality practitioner to get a correct estimation of the population parameters μ₀ and δ₀ for quality characteristic. The Τ statistic is adopted as a tentative solution to overcome this problem, because the preliminary Phase I estimation of the quality character population parameters is not needed. In this paper, the AEWMA t control chart is proposed. Based on the investigation of the joint effect of parameters λ and γ, a new optimization algorithm is proposed for the statistical design of AEWMA t control chart. Two statistical performance measures have been calculated by implementing the Markov chain theory. Simulations are performed to compare the performance of EWMA t control chart and AEWMA t control chart. The results show that the AEWMA t control chart can overcome the inertia problem of EWMA t control chart. Future research will be focused on the investigation of the evaluation of the economic impact of AEWMA t control chart.