J.-C Malela-Majika

S C Shongwe

P Castagliola

One-sided precedence monitoring schemes for unknown shift sizes using generalized 2-of-(h+1) and w-of-w improved runs-rules

Keywords: Distribution-free, Order statistic, Precedence statistic, Runs-rules, Case U, Markov chain, Phase I, Phase II

Parametric monitoring schemes are expected to perform better than their nonparametric counterparts when the assumption of a specific form of a distribution such as the normal is met. In practice, such assumption is often violated. Consequently, the performance of parametric schemes deteriorates considerably. To solve this problem, more efficient and flexible monitoring schemes based on nonparametric tests are recommended. In this paper, efficient and robust one-sided monitoring schemes are developed using generalized {1-of-1 or 2-of-(h+1)} and {1-of-1 or w-of-w} improved runs-rules (IRR) without any distributional assumption in the zero-and steady-states modes. Moreover, the zero-and steady-states run-length properties of the resulting one-sided IRR schemes are formulated and empirically evaluated using the Markov chain technique. Comparisons with other well-known one-sided Shewhart-type nonparametric schemes (e.g. basic precedence scheme and precedence scheme with standard runs-rules) indicate that the proposed schemes have a better performance. Real data are used to demonstrate the design and implementation of the one-sided improved runs-rules precedence schemes. Finally, a discussion on the two-sided IRR precedence schemes is also provided.

Introduction

Statistical process monitoring (SPM) is a collection of statistical techniques and tools used in industrial and non-industrial processes to distinguish between a process that is operating under chance causes of variation (i.e. the process is said to be in-control (IC)) or under assignable causes of variation (i.e. the process is said to be out-of-control (OOC)). A monitoring scheme is the main tool in SPM used to generally distinguish among the different types of variation, see [START_REF] Montgomery | Introduction to Statistical Quality Control[END_REF].

Monitoring schemes that are generally based on the assumption of normality or any other specific form of a distribution (e.g. exponential, Poisson, Student's t-distribution, etc.) are called parametric monitoring schemes. However, in most of the cases, the underlying distribution of the quality process is unknown, or there is not enough information to justify the assumption of normality; then, in such cases, monitoring schemes that do not depend on a particular distributional assumption are preferred.

Nonparametric or distribution-free monitoring schemes can serve this broader purposesee a more thorough discussion of this important class of monitoring schemes in [START_REF] Qiu | Introduction to Statistical Process Control[END_REF] and [START_REF] Chakraborti | Nonparametric (Distribution-free) control charts: An updated overview and some results[END_REF]. A key advantage of nonparametric schemes is that their IC run-length distribution remains the same for all continuous process distributions. This means that, for example, the IC average run-length (𝐴𝑅𝐿 0 ) or the false alarm rate (FAR) of a nonparametric scheme is the same across all continuous distributions. Therefore, it is said that nonparametric schemes are IC robust. This is not true for parametric monitoring schemes in general and consequently, their IC robustness can be a matter of legitimate concern. Moreover, nonparametric charts are often more robust and efficient under some heavy-tailed and skewed underlying distributions. The drawback of nonparametric charts is that they are relatively less sensitive than their parametric counterparts when the underlying distribution of the quality process is known.

When the design parameter(s) of interest are known or specified, this is referred to parameters known (hereafter, Case K); however, when the design parameter(s) are unspecified or unknown, this is referred to parameters unknown (hereafter, Case U). Case K and Case U nonparametric monitoring schemes have received a lot of attention in recent SPM literature. For Case K, see for example, [START_REF] Gibbons | Nonparametric Shewhart-type sign control charts based on runs[END_REF], [START_REF] Khilare | A nonparametric synthetic control chart using sign statistic[END_REF], [START_REF] Kritzinger | Improved Shewhart-type runs-rules nonparametric sign charts[END_REF], [START_REF] Khilare | Steady-state behavior of nonparametric control charts using sign statistic[END_REF], [START_REF] Patil | Economic design of nonparametric sign control chart[END_REF], [START_REF] Pawar | Nonparametric moving average control charts using sign and signed-rank statistics[END_REF], etc. For Case U, see for example, [START_REF] Chakraborti | A nonparametric control chart based on the Mann-Whitney statistic[END_REF], Chakraborti et al. (2009a), [START_REF] Albers | Minimum control charts[END_REF], [START_REF] Graham | Design and implementation issues for a class of distribution-free Phase II EWMA exceedance control charts[END_REF], [START_REF] Malela-Majika | Distribution-free precedence schemes with a generalized runs-rule for monitoring unknown location[END_REF], etc. In Case U scenario, it is well-known and accepted that there are two distinct phases or stages, namely Phase I (or retrospective phase) and Phase II (or prospective or monitoring phase). Phase I involves the preliminary (including statistical) analysis which includes planning, administration, design of the study, data collection, data management, exploratory work (including graphical and numerical analysis, goodness-of-fit analysis, etc.) to ensure that the process is in a state of IC, and a search for appropriate control limits from the historical data. For more details on control charts for Phase I applications, readers are referred to Chakraborti et al. (2009b) and [START_REF] Jones-Farmer | An overview of phase I analysis for process improvement and monitoring[END_REF]. In Phase II, monitoring schemes are implemented prospectively to continuously monitor any departures from an IC state using the parameters estimated in Phase I. That is, in Phase II, at each sampling time, smaller samples (i.e. test sample) are observed, and a decision is made on each of them to know whether the process is IC or notsee [START_REF] Jensen | Effects of parameter estimation on control chart properties: A literature review[END_REF] and Psarakis et al. (2013) more details on Phase II monitoring schemes.

Shewhart-type charts are the most popular charts because of their simplicity, ease of construction and use, and the fact that they are quite efficient in detecting moderate to large shifts. Hence, Chakraborti, van der Laan and van de [START_REF] Chakraborti | A class of distribution-free control charts[END_REF] proposed a class of nonparametric Shewhart-type control charts called the precedence charts, using some order statistic of a Phase II sample as the charting statistic and the control limits are constructed from a Phase I (or reference) sample. The latter paper and the manner in which control charts are formulated have led to a renewed interest in nonparametric methods based on order statistics in industrial statistics and medical researches, particularly in the area of process monitoring and the choice of best treatment, see for example, Chakraborti et al. (2009a), [START_REF] Balakrishnan | A distribution-free control chart based on order statistics[END_REF][START_REF] Balakrishnan | One-sided control charts based on precedence and weighted precedence statistics[END_REF], Malela-Majika et al. (2016b), [START_REF] Triantafyllou | Nonparametric control charts based on order statistic: Some advances[END_REF][START_REF] Triantafyllou | A new distribution-free control scheme based on order statistics[END_REF], [START_REF] Koutras | A general class of nonparametric control charts[END_REF], to count a few. While the majority of attention has been paid to two-sided precedence monitoring schemes, [START_REF] Balakrishnan | One-sided control charts based on precedence and weighted precedence statistics[END_REF] argued that there are many real-life situations where it makes more sense to monitor either an increase only, or a decrease only, in a process characteristic(s) of interest. Consequently, following a similar line of argument as in Thus, this paper contributes to the SPM literature by:  Investigating the IC and OOC performances of the one-sided IRR 2-of-(h+1) and IRR w-of-w precedence schemes for any integers ℎ > 0 and 𝑤 > 1.  Studying both zero-and steady-state run-length performances of the proposed schemes.

 Investigating both specific shifts as well as overall performances of the proposed schemes for a range of shifts in the location parameter.

The remainder of this paper is organized as follows: in Section 2, the one-sided 1-of-1 (i.e. basic) median precedence monitoring scheme is introduced and the operation of the proposed schemes are outlined. The general form of the transition probability matrices (TPMs), zero-state and steady-state run-length characteristics of the proposed one-sided IRR 2-of-(h+1) and IRR w-of-w precedence monitoring schemes are developed in Section 3. Section 4 discusses the IC and OOC zero-and steady-state performances and compares their performances to other one-sided Shewhart-type counterparts. A real-life application of the proposed schemes is given in Section 5. Section 6 gives an extension of the IRR 2-of-(h+1) and IRR w-of-w precedence monitoring schemes to the two-sided version. Finally, Section 7 provides concluding remarks and some recommendations.

Precedence monitoring schemes with improved runs-rules

Basic and SRR precedence monitoring schemes

Let 𝑋 = {𝑋 1 , 𝑋 2 , … , 𝑋 𝑚 } be a Phase I (or reference) sample of size 𝑚 available from an IC process with an unknown continuous c. Assume that the location model is given by 𝐺(𝑡) = 𝐹(𝑡 -𝛿), for all t, where 𝛿 is the location difference (or shift in the location parameter). The process is IC in Phase II when 𝐺 ≡ 𝐹 (i.e. 𝛿 = 0).

The precedence monitoring scheme is a general class of nonparametric monitoring schemes that uses the 𝑗 𝑡ℎ order statistic in the Phase II sample of size n, i.e. 𝑌 (𝑗:𝑛) . The 𝑗 𝑡ℎ order statistic can be any quartile, decile, percentile, etc.; note though, the most used order statistics are the minimum, lower quartile, median, upper quartile and maximum. In the case of the one-sided Shewhart-type precedence scheme, the charting statistic 𝑌 (𝑗:𝑛) is compared separately to either the lower control limit (LCL) or the upper control limit (UCL). The control limits of the lower and upper one-sided precedence scheme, i.e. LCL and UCL, are given by the 𝑎 𝑡ℎ and 𝑏 𝑡ℎ order statistics of the Phase I sample, respectively, where 1 ≤ a < m and 1 < b ≤ m. When 𝑛 is odd, say, 𝑛 = 2𝑟 + 1, then 𝑗 = 𝑟 + 1 corresponds to the unique test sample median and the corresponding precedence scheme is called median precedence scheme (or, simply called precedence scheme in this paper).

Let 𝑝 denotes the probability that the precedence scheme does not signal, i.e. 𝑝 = 𝑃(𝑌 (𝑗:𝑛) ≥ 𝑋 (𝑎:𝑚) )

and 𝑝 = 𝑃(𝑌 (𝑗:𝑛) ≤ 𝑋 (𝑏:𝑚) ) for the lower and upper one-sided charts, respectively. To check whether the parameter of interest has shifted, we use 𝐿𝐶𝐿 =𝑋 (𝑎:𝑚) and 𝑈𝐶𝐿 =𝑋 (𝑏:𝑚) , where 𝑎 and 𝑏 are found from 𝑃(𝑌 (𝑗:𝑛) ≥ 𝑋 (𝑎:𝑚) ) ≤ 1 -𝑝 and 𝑃(𝑌 (𝑗:𝑛) ≤ 𝑋 (𝑏:𝑚) ) ≤ 1 -𝑝 for the lower and upper one-sided precedence schemes, respectively. The control limit is found by setting 𝑝 (or 1 -𝑝) to some desired high (or low) significant values, say 0.9973 (or 0.0027). Equivalently, 𝑎 and 𝑏 are found such that the IC ARL (𝐴𝑅𝐿 0 ) is as close as possible to some high desired values such as 370, 500 and 1000. In

Phase II, the 𝑗 𝑡ℎ order statistic from a uniform (0,1) distribution follows a beta distribution with parameters 𝑗 and 𝑛 -𝑗 + 1 (see e.g. [START_REF] Gibbons | Nonparametric Shewhart-type sign control charts based on runs[END_REF].

Therefore, for an upper and lower one-sided precedence charts, the conditional probabilities that the plotting statistic plots in region A, B, C and D (see Figure 1(a)) are given by 𝑝 𝐴 = 𝑃(𝑌 (𝑗:𝑛) ≥ 𝑋 (𝑏:𝑚) |𝑋 (𝑏:𝑚) = 𝑥 (𝑏:𝑚) ) = 𝐼(𝐺𝐹 -1 (𝑈 (𝑏:𝑚) ), 𝑗, 𝑛 -𝑗 + 1),

(1) 𝑝 𝐵 = 1 -𝑃(𝑌 (𝑗:𝑛) ≥ 𝑋 (𝑏:𝑚) |𝑋 (𝑏:𝑚) = 𝑥 (𝑏:𝑚) ) = 1 -𝐼(𝐺𝐹 -1 (𝑈 (𝑏:𝑚) ), 𝑗, 𝑛 -𝑗 + 1), 𝑝 𝐶 = 𝑃(𝑌 (𝑗:𝑛) ≥ 𝑋 (𝑎:𝑚) |𝑋 (𝑎:𝑚) = 𝑥 (𝑎:𝑚) ) = 𝐼(𝐺𝐹 -1 (𝑈 (𝑎:𝑚) ), 𝑗, 𝑛 -𝑗 + 1), 𝑝 𝐷 = 1 -𝑃(𝑌 (𝑗:𝑛) ≥ 𝑋 (𝑎:𝑚) |𝑋 (𝑎:𝑚) = 𝑥 (𝑎:𝑚) ) = 1 -𝐼(𝐺𝐹 -1 (𝑈 (𝑎:𝑚) ), 𝑗, 𝑛 -𝑗 + 1), respectively, where 𝐼(. , . , . ) denotes the incomplete beta function and 𝑈 (𝑒:𝑙) represents the 𝑒 𝑡ℎ order statistic of a sample of size 𝑙 from the Uniform(0,1) distribution. All the above expressions depend on the c.d.f. 𝐹 and 𝐺 only through the transformation function 𝛹 = 𝐺𝐹 -1 . It is important to know that the process is IC if 𝐺 𝐹. In this case, 𝛹(𝑢) = 𝐺𝐹 -1 (𝑢) = 𝑢 for any 𝑢 ∈ (0,1).

To improve the detection ability of the basic Shewhart-type monitoring schemes towards small and moderate shifts, the SPM literature recommends the use of supplementary runs-rules. The SRR 2-of-(h+1) schemes need at least two plotting statistics to decide if the process is IC or OOC and it uses the charting regions in Figure 1(a). The design of the SRR 2-of-(h+1) schemes is summarized as follows: For a specific integer value of h, take a sample of size n and compute the charting statistic. If at some random time 𝑡 the charting statistic plots on region A (region D) for the first time, then keep track of the charting regions that the scheme plots on from time 𝑡 + 1 until time 𝑡 + ℎ, or alternatively, until the second charting statistic plots on region A (region D), respectively. The SRR w-of-w monitoring scheme needs exactly w consecutive plotting statistics to decide if the process is IC or OOC and it also uses the charting regions in Figure 1(a). The design of the SRR w-of-w scheme is summarized as follows: For a specific integer value w, take a sample of size n and compute the charting statistic. If at some random time 𝑡 the charting statistic plots on region A (region D) for the first time, then the SRR w-of-w scheme will give an OOC signal, if and only if, the consecutive charting statistics from time (𝑡 +1) to (𝑡 + 𝑤-1) plot on region A (region D), respectively.

The charting regions corresponding to the one-sided basic and SRR schemes are as follows:

 Upper one-sided basic and SRR schemes: Regions A and B (see Figure 1 The IRR 2-of-(h+1) scheme is the combination of the SRR 2-of-(h+1) scheme and the basic (i.e. 1-of-1) scheme (discussed in Section 2.1). Therefore, the upper (lower) one-sided IRR 2-of-(h+1) scheme gives a signal when either a single point plots on or above (below) the UCL (LCL) or 2 out of ℎ + 1 successive points plot on or above (below) the upper (lower) warning limit. Similarly, the IRR w-of-w is the combination of the SRR w-of-w scheme and the 1-of-1 scheme. The upper (lower) one-sided IRR w-of-w scheme gives a signal when either a single point plots on or above (below) the UCL (LCL) or 𝑤 successive points plot on or above (below) the upper (lower) warning limit.

The charting regions which are separated by UCL or LCL and the upper or lower warning limit (UWL or LWL) corresponding to each one-sided IRR 2-of-(h+1) and IRR w-of-w schemes are as follows:  Upper one-sided scheme: Regions 1, 2 and 3 (see Figure 1 where regions 1 and 6 are OOC regions, whereas regions 2 and 5 are nonconforming regions and regions 3 and 4 are conforming regions. Note that a sample plots on a conforming region when it is under the influence of common causes of variation only; however, when it plots on a nonconforming region, it implies it has some assignable causes of variation present.

The conditional probabilities that the plotting statistic plots in region 1, 2, 3, 4, 5 and 6 of the onesided IRR 2-of-(h+1) and IRR w-of-w precedence schemes (see Figure 1(b)) are given by 𝑝 1 = 𝑃(𝑌 (𝑗:𝑛) ≥ 𝑋 (𝑏 2 :𝑚) |𝑋 (𝑏 2 :𝑚) = 𝑥 (𝑏 2 :𝑚) ) = 𝐼(𝐺𝐹 -1 (𝑈 (𝑏 2 :𝑚) ), 𝑗, 𝑛 -𝑗 + 1),

𝑝 2 = 𝐼(𝐺𝐹 -1 (𝑈 (𝑏 2 :𝑚) ), 𝑗, 𝑛 -𝑗 + 1) -𝐼(𝐺𝐹 -1 (𝑈 (𝑏 1 :𝑚) ), 𝑗, 𝑛 -𝑗 + 1), 𝑝 3 = 𝑃(𝑌 (𝑗:𝑛) ≤ 𝑋 (𝑏 1 :𝑚) |𝑋 (𝑏 1 :𝑚) = 𝑥 (𝑏 1 :𝑚) ) = 1 -𝐼(𝐺𝐹 -1 (𝑈 (𝑏 1 :𝑚) ), 𝑗, 𝑛 -𝑗 + 1), 𝑝 4 = 𝑃(𝑌 (𝑗:𝑛) ≥ 𝑋 (𝑎 1 :𝑚) |𝑋 (𝑎 1 :𝑚) = 𝑥 (𝑎 1 :𝑚) ) = 𝐼(𝐺𝐹 -1 (𝑈 (𝑎 1 :𝑚) ), 𝑗, 𝑛 -𝑗 + 1), 𝑝 5 = 𝐼(𝐺𝐹 -1 (𝑈 (𝑎 1 :𝑚) ), 𝑗, 𝑛 -𝑗 + 1) -𝐼(𝐺𝐹 -1 (𝑈 (𝑎 2 :𝑚) ), 𝑗, 𝑛 -𝑗 + 1), 𝑝 6 = 𝑃(𝑌 (𝑗:𝑛) ≤ 𝑋 (𝑎 2 :𝑚) |𝑋 (𝑎 2 :𝑚) = 𝑥 (𝑎 2 :𝑚) ) = 1 -𝐼(𝐺𝐹 -1 (𝑈 (𝑎 2 :𝑚) ), 𝑗, 𝑛 -𝑗 + 1).

Note that 𝑏 1 and 𝑏 2 (𝑎 1 and 𝑎 2 ) are computed such that the attained 𝐴𝑅𝐿 0 is as close as possible to the nominal 𝐴𝑅𝐿 0 (see Section 4.1 for more details).

The operational procedure of the one-sided IRR 2-of-(h+1) and IRR w-of-w precedence schemes is as summarized in Table 1, where CRL (i.e. conforming run-length) is the number of conforming samples that fall in between any two consecutive nonconforming samples.

<Insert Table 1> 3. Run-length properties of the one-sided IRR 2-of-(h+1) and IRR w-of-w precedence scheme

In this section, mathematical foundations and necessary notations that are needed to derive the runlength (RL) properties of the one-sided Shewhart-type IRR 2-of-(h+1) and IRR w-of-w precedence schemes using Markov chain technique are given.

Transition probability matrix (TPM)

To illustrate how to construct the TPMs, one can follow a similar procedure implemented in Fu and Lou (2003, Chapter 4) to formulate general TPMs for one-sided schemes that are valid for all integer values of h > 0 and w > 1.

Firstly, since in general, the TPM is of the form,

𝐏 = ( 𝐐 | 𝐫 --- 𝟎 ′ | 1 ) ( 3
)
where 𝐏 is the (𝜏 + 1) × (𝜏 + 1) matrix, 𝐐 is the 𝜏 × 𝜏 essential TPM, 𝐫 = 𝟏 -𝐐𝟏 is a 𝜏 × 1 vector so that each row sum to unity, with 𝟏 = (1 1 … 1) ′ and 𝟎 = (0 0 … 0) ′ . Then using the Markov chain procedure discussed in Champ (1992), Shongwe et al. (2019a, b), it can be shown that 𝜏 = (h+1) for the IRR 2-of-(h+1) schemes whereas 𝜏 = w for the IRR w-of-w schemes. Consequently, it follows that for any integer value of h > 0, the TPM of the lower or upper one-sided IRR 2-of-(h+1) schemes is, in general,

given by Equation (4).

𝜙 𝜂 2 𝜂 3 𝜂 4 ⋯ 𝜂 ℎ 𝜂 ℎ+1 OOC 𝜙 𝜋 # 𝜋 * 0 0 ⋯ 0 0 𝜋 𝜆 𝜂 2 0 0 𝜋 # 0 ⋯ 0 0 𝜋 * + 𝜋 𝜆 𝜂 3 0 0 0 𝜋 # ⋯ 0 0 𝜋 * + 𝜋 𝜆 ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ 𝜂 ℎ-1 0 0 0 0 ⋯ 𝜋 # 0 𝜋 * + 𝜋 𝜆 𝜂 ℎ 0 0 0 0 ⋯ 0 𝜋 # 𝜋 * + 𝜋 𝜆 𝜂 ℎ+1 𝜋 # 0 0 0 ⋯ 0 0 𝜋 * + 𝜋 𝜆 OOC 0 0 0 0 ⋯ 0 0 1 (4)
As explained in Shongwe et al. (2019b)

where 𝑝 1 , 𝑝 2 , 𝑝 3 , 𝑝 4 , 𝑝 5 and 𝑝 6 are defined in Equation (2).

Similarly, it follows that for any integer value of w > 1, the TPM of the lower or upper one-sided IRR w-of-w schemes is, in general, given by Equation ( 6), with the probability elements 𝜋 # , 𝜋 * and 𝜋 𝜆 as defined in Equation ( 5), with 𝜂 𝑟 (𝑟=2,3,…, 𝑤).

𝜙 𝜂

2 𝜂 3 𝜂 4 ⋯ 𝜂 𝑤-1 𝜂 𝑤 OOC 𝜙 𝜋 * 𝜋 # 0 0 ⋯ 0 0 𝜋 𝜆 𝜂 2 𝜋 * 0 𝜋 # 0 ⋯ 0 0 𝜋 𝜆 𝜂 3 𝜋 * 0 0 𝜋 # ⋯ 0 0 𝜋 𝜆 ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ 𝜂 𝑤-2 𝜋 * 0 0 0 ⋯ 𝜋 # 0 𝜋 𝜆 𝜂 𝑤-1 𝜋 * 0 0 0 ⋯ 0 𝜋 # 𝜋 𝜆 𝜂 𝑤 𝜋 * 0 0 0 ⋯ 0 0 𝜋 # +𝜋 𝜆 OOC 0 0 0 0 ⋯ 0 0 1 (6)

General run-length characteristics

The characteristics of the run-length (RL) distribution reveal important information about the shortterm and long-term performance of a monitoring scheme. Note that once Equation ( 3) is determined (see the respective TPMs in Equations ( 4) and ( 6)), then important properties of the RL can be determined via an appropriate Markov chain technique discussed in [START_REF] Fu | Distribution Theory of Runs and Patterns and Its Applications: A Finite Markov Chain Imbedding Approach[END_REF]. That is, the conditional average run-length (CARL = E(RL)) defined as

CARL = 𝝃 T 𝐑 (7)
where 𝝃 denotes either the zero-or steady-state initial (𝜏 ×1) probability vector discussed in Sections 3.3 and 3.4; and

𝐑 = (𝐈 -𝐐) -1 𝟏 (8)
i.e. the 𝜏 × 𝜏 vector, with ARL values of being in each separate 𝜏 transient or non-absorbing states and 𝐈 is the 𝜏 × 𝜏 identity matrix.

From Equation ( 8), with 𝐐 extracted from Equation (4), using basic algebraic matrix manipulation, it follows that for any integer value h > 0, the ARL vector of the lower or upper one-sided IRR 2-of-(h+1)

scheme is given by

𝐑 = ( 𝜁 1 𝜁 2 𝜁 3 𝜁 4 ⋮ 𝜁 ℎ-1 𝜁 ℎ 𝜁 ℎ+1 ) = 1 (1 -𝜋 # )(1 -𝜋 # -𝜋 * 𝜋 # ℎ ) ( 1 -𝜋 # -𝜋 * 𝜋 # ℎ + 𝜋 * 𝜋 # 0 1 -𝜋 # -𝜋 * 𝜋 # ℎ + 𝜋 * 𝜋 # ℎ 1 -𝜋 # -𝜋 * 𝜋 # ℎ + 𝜋 * 𝜋 # ℎ-1 1 -𝜋 # -𝜋 * 𝜋 # ℎ + 𝜋 * 𝜋 # ℎ-2 ⋮ 1 -𝜋 # -𝜋 * 𝜋 # ℎ + 𝜋 * 𝜋 # 3 1 -𝜋 # -𝜋 * 𝜋 # ℎ + 𝜋 * 𝜋 # 2 1 -𝜋 # -𝜋 * 𝜋 # ℎ + 𝜋 * 𝜋 # ) . (9)
Similarly, using 𝐐 extracted from Equation (5), the ARL vector of the lower or upper one-sided IRR w- of-w scheme is given by

𝐑 = ( 𝜁 1 𝜁 2 𝜁 3 𝜁 4 ⋮ 𝜁 𝑤-2 𝜁 𝑤-1 𝜁 𝑤 ) = 1 1 -𝜋 * -𝜋 # (1 -𝜋 * 𝑤 ) ( 1 -𝜋 * 𝑤 1 -𝜋 * 𝑤-1 1 -𝜋 * 𝑤-2 1 -𝜋 * 𝑤-3 ⋮ 1 -𝜋 * 3 1 -𝜋 * 2 1 -𝜋 * )
.

(10)

The unconditional ARL (UARL) is given by

𝑈𝐴𝑅𝐿 = ∫ ∫ 𝐶𝐴𝑅𝐿 𝑓 𝑏 1 𝑏 2 𝑡 0 1 0 (𝑢, 𝑡)𝑑𝑢 𝑑𝑡 (11)
where

𝑓 𝑏 1 𝑏 2 (𝑢, 𝑡) = 𝑚! (𝑏 1 -1)! (𝑏 2 -𝑏 1 -1)!(𝑚-𝑏 2 )! 𝑡 𝑏 1 -1 (𝑡 -𝑢) 𝑏 2 -𝑏 1 -1 (1 -𝑡) 𝑚-𝑏 2 which is the joint pdf of
the 𝑏 1 𝑡ℎ and 𝑏 2 𝑡ℎ order statistics in a sample of size m from the Uniform (0,1) distribution. For more details readers are also referred to Chakraborti et al. (2009a).

Zero-state run-length characteristics

In the zero-state mode, we have 𝝃 T = q T = (1,0,0,…,0) where the unique "1" is located at the 1 st position so that the zero-state ARL (ZSARL) is given by 𝑍𝑆𝐴𝑅𝐿 = 𝐪 T 𝐑. That is, using Equation ( 9), it follows that for any integer value h > 0, the conditional 𝑍𝑆𝐴𝑅𝐿 (𝐶𝑍𝑆𝐴𝑅𝐿) of the lower or upper onesided IRR 2-of-(h+1) scheme is given by

𝐶𝑍𝑆𝐴𝑅𝐿 = 1 -𝜋 # -𝜋 * 𝜋 # ℎ + 𝜋 * 𝜋 # 0 (1 -𝜋 # )(1 -𝜋 # -𝜋 * 𝜋 # ℎ ) . (12)
with 𝜋 # and 𝜋 * defined in Equation ( 5). Thus, the unconditional 𝑍𝑆𝐴𝑅𝐿 (𝑈𝑍𝑆𝐴𝑅𝐿) for any integer value h > 0, is then defined by

𝑈𝑍𝑆𝐴𝑅𝐿 = ∫ ∫ [ 1 -𝜋 # -𝜋 * 𝜋 # ℎ + 𝜋 * 𝜋 # 0 (1 -𝜋 # )(1 -𝜋 # -𝜋 * 𝜋 # ℎ ) ] 𝑓 𝑏 1 𝑏 2 𝑡 0 1 0 (𝑢, 𝑡)𝑑𝑢 𝑑𝑡. (13)
However, using Equation ( 10), it follows that for any integer value w > 1, the 𝐶𝑍𝑆𝐴𝑅𝐿 of the lower or upper one-sided IRR w-of-w scheme is given by

𝐶𝑍𝑆𝐴𝑅𝐿 = 1 -𝜋 * 𝑤 1 -𝜋 * -𝜋 # (1 -𝜋 * 𝑤 ) , (14)
with 𝜋 # and 𝜋 * defined in Equation ( 5). Thus, the 𝑈𝑍𝑆𝐴𝑅𝐿 for any integer value w > 1, is then defined by

𝑈𝑍𝑆𝐴𝑅𝐿 = ∫ ∫ [ 1 -𝜋 * 𝑤 1 -𝜋 * -𝜋 # (1 -𝜋 * 𝑤 ) ] 𝑓 𝑏 1 𝑏 2 𝑡 0 1 0 (𝑢, 𝑡)𝑑𝑢 𝑑𝑡. ( 15
)

Steady-state run-length characteristics

In the steady-state mode, we have 𝝃 T = s T = (s 1 , s 2 , … , s τ ), i.e. a non-zero initial probability vector, which is obtained by dividing each element of 𝐐, when 𝛿 = 0, by its corresponding row sum, so that the modified essential TPM is called the conditional essential TPM, denoted by 𝐐 𝟎 . That is, 𝐐 𝟎 is the altered version of 𝐐 so that the 'new' essential TPM is ergodic, see some recent discussions by [START_REF] Knoth | The case against the use of synthetic control charts[END_REF] and Shongwe et al. (2019a). That is, 𝐬 T 𝐐 𝟎 = 𝐬 T subject to ∑ 𝑠 𝑗 𝜏 𝑗=1

= 1.

Using matrix 𝐐 in Equation ( 4) and following the procedure described above, it follows that the initial probability vector of the lower or upper one-sided IRR 2-of-(h+1) precedence scheme for any value of h > 0 is given by

𝐬 T = (s 1 , s 2 , … , s τ ) = 1 1 + ℎ𝜃 (1 𝜃 𝜃 … 𝜃) with 𝜃 = 𝜋 * 𝜋 # + 𝜋 * . (16)
Similarly, using matrix 𝐐 in Equation ( 5), the initial probability vector of the lower or upper onesided IRR w-of-w precedence scheme for any value of w > 1 is given by

𝐬 T = (s 1 , s 2 , … , s τ ) = 1 -𝜋 * 1 -𝜋 * 𝑤 (𝜋 * 0 𝜋 * 𝜋 * 2 𝜋 * 3 … 𝜋 * 𝑤-2 𝜋 * 𝑤-1 ). ( 17
)
The conditional 𝑆𝑆𝐴𝑅𝐿 (𝐶𝑆𝑆𝐴𝑅𝐿) of the lower or upper one-sided IRR 2-of-( h+1) precedence scheme for any value of h > 0 is given by

𝐶𝑆𝑆𝐴𝑅𝐿 = 𝐬 T 𝐑 = 1 1 + ℎ𝜃 𝜁 1 + 𝜃 1 + ℎ𝜃 ∑ 𝜁 𝑖 ℎ+1 𝑖=2 , ( 18
)
where 𝜃 is defined in Equation ( 16) and 𝜁 𝑖 (𝑖 = 1,2, … , ℎ + 1) are defined in Equation ( 9). Thus, the unconditional 𝑆𝑆𝐴𝑅𝐿 (𝑈𝑆𝑆𝐴𝑅𝐿) for any integer value h > 0, is then defined by

𝑈𝑆𝑆𝐴𝑅𝐿 = ∫ ∫ [ 1 1 + ℎ𝜃 𝜁 1 + 𝜃 1 + ℎ𝜃 ∑ 𝜁 𝑖 ℎ+1 𝑖=2 ] 𝑓 𝑏 1 𝑏 2 𝑡 0 1 0 (𝑢, 𝑡)𝑑𝑢 𝑑𝑡. (19)
However, using Equation ( 10), it follows that for any integer value w > 1, the 𝐶𝑆𝑆𝐴𝑅𝐿 of the lower or upper one-sided IRR w-of-w scheme is given by

𝐶𝑆𝑆𝐴𝑅𝐿 = 𝐬 T 𝐑 = ∑ 𝑠 𝑖 • 𝜁 𝑖 𝑤 𝑖=1 , (20)
with 𝑠 𝑖 defined in Equation ( 17) and 𝜁 𝑖 (𝑖 = 1,2, … , ℎ + 1) are defined in Equation ( 10). Thus, the 𝑈𝑆𝑆𝐴𝑅𝐿 for any integer value w > 1, is then defined by

𝑈𝑆𝑆𝐴𝑅𝐿 = ∫ ∫ [∑ 𝑠 𝑖 • 𝜁 𝑖 𝑤 𝑖=1 ] 𝑓 𝑏 1 𝑏 2 𝑡 0 1 0 (𝑢, 𝑡)𝑑𝑢 𝑑𝑡. ( 21
)

Overall performance measurement

Many studies in SPM use the ARL values to assess the performance of schemes [START_REF] Li | The computation of average run length and average time to signal: an overview[END_REF]. This measure evaluates the performance of a control chart for a specific shift. Therefore, schemes which are designed on the basis of a specified optimal shift (say, 𝛿 𝑜𝑝𝑡 ) will perform poorly if the shift is actually different from 𝛿 𝑜𝑝𝑡 . When researchers are interested in measuring the chart's performance for a range of shifts, 𝛿 𝑚𝑖𝑛 ≤ 𝛿 ≤ 𝛿 𝑚𝑎𝑥 , it is recommended to use measures of the overall performance (see [START_REF] Machado | A side-sensitive synthetic chart combined with an 𝑋 ̅ chart[END_REF]; where 𝛿 𝑚𝑖𝑛 and 𝛿 𝑚𝑎𝑥 are lower and upper bound of 𝛿, respectively.

In this paper, we make use of one of the characteristics of the quality loss function (QLF), the average extra quadratic loss (AEQL) value, in order to investigate the overall performance of the proposed schemes. A QLF describes the relationship between the size of the shift and the quality impact. More recently, a number of researchers tend to supplement the results of the ARL with that of QLF because users tend not to know beforehand what exact shift value(s) is targeted for a specific process; for more discussions about this, see [START_REF] Wu | Optimisation designs of the combined Shewhart-CUSUM control charts[END_REF], [START_REF] Reynolds | An evaluation of GLR control chart combined with an 𝑋 ̅ chart[END_REF], [START_REF] Tran | On the performance of Shewhart median chart in the presence of measurement errors[END_REF], [START_REF] Rakitzis | An overview of synthetic-type control charts: Techniques and Methodology[END_REF], etc. Defining 𝑓(𝛿) as the probability density function (pdf) of a uniform distribution with parameters 𝛿 𝑚𝑖𝑛 and 𝛿 𝑚𝑎𝑥 , i.e. shifts occur with equal probability, the unconditional AEQL (denoted as 𝑈𝐴𝐸𝑄𝐿) may be given by:

𝑈𝐴𝐸𝑄𝐿 = 1 𝛿 𝑚𝑎𝑥 -𝛿 𝑚𝑖𝑛 ∫ (𝛿 2 × 𝑈𝐴𝑅𝐿(𝛿)) 𝛿 𝑚𝑎𝑥 𝛿 𝑚𝑖𝑛 𝑑𝛿. ( 22
)
where 𝑈𝐴𝑅𝐿(𝛿) is the unconditional ARL given in Equations ( 13), ( 15), ( 19) and ( 21). When comparing several schemes (with the charting constants computed while the process is IC and the ARL 0 is approximately equal to the nominal ARL 0 ), the scheme with the minimum 𝑈𝐴𝐸𝑄𝐿 value is considered to be the most efficient.

To investigate the difference between the zero-and steady state 𝑈𝐴𝐸𝑄𝐿 values of a monitoring scheme, the percentage difference (denoted as %Diff) of the AEQL values is calculated using the following formula

%𝐷𝑖𝑓𝑓 = ( 𝑈𝑍𝑆𝐴𝐸𝑄𝐿 -𝑈𝑆𝑆𝐴𝐸𝑄𝐿 𝑈𝑆𝑆𝐴𝐸𝑄𝐿 ) × 100, (23)
where 𝑈𝑍𝑆𝐴𝐸𝑄𝐿 and 𝑈𝑆𝑆𝐴𝐸𝑄𝐿 represents the unconditional zero-and steady-state AEQL values, respectively. Note that the percentage difference of the ARL values can also be calculated in a similar way. and 𝑈𝐶𝐿 = 𝑋 (𝑏 2 :𝑚) (𝐿𝑊𝐿 = 𝑋 (𝑎 1 :𝑚) and 𝐿𝐶𝐿 = 𝑋 (𝑎 2 :𝑚) ) where 𝑏 1 < 𝑏 2 (𝑎 1 > 𝑎 2 ), respectively. Due to space restriction, only the IC design and OOC performance of the upper one-sided IRR 2-of-(h+1) and IRR w-of-w precedence schemes will be presented in details.

Zero

The zero-state charting constants of the upper one-sided IRR 2-of-(h+1) and IRR w-of-w precedence schemes are determined using Equations ( 12) and ( 14), respectively; whereas, the steady-state charting constants of the upper one-sided IRR 2-of-(h+1) and IRR w-of-w precedence schemes are found using Equations ( 18) and ( 20), respectively. For instance, when h = 1 for the IRR 2-of-(h+1) scheme or 𝑤 = 2 for the IRR w-of-w scheme, i.e. IRR 2-of-2 scheme, the couple (𝑏

IC robustness of the proposed precedence schemes

The 𝐴𝑅𝐿 0 of the nonparametric monitoring schemes does not depend on the underlying process distribution. This statement can be verified by checking their IC robustness. To check the robustness property of the proposed monitoring schemes, various distributions are considered and these are: (i)

The standard normal distribution, i.e. N(0,1), to investigate the effect of symmetric distributions, (ii)

The Student's t-distribution, with degree of freedom 𝑣, i.e. t(𝑣), to study the effect of heavy tails, and

(iii) The gamma distribution, i.e. GAM(𝛼,𝛽), to investigate the effect of skewness. Note that when 𝑣 ≥ 30, the 𝑡(𝑣) distribution approximate the normal distribution. Thus, to properly investigate the effect of the heavy tail, 𝑣 is set to a small value. Thus, in this paper we used 𝑣 = 5. For the 𝐺𝐴𝑀(𝛼,𝛽), when the shape parameter 𝛼 converges towards infinity when the scale parameter 𝛽 remains fixed, the gamma distribution reduces to the normal distribution. Therefore, to study the effect of skewness, in this paper both shape and scale parameters are fixed to small integer values such that 𝛼 = 𝛽 = 1 which is practically equivalent to an exponential distribution with parameter 𝛽 = 1 (i.e. 𝐸𝑋𝑃(1)). Since the performance of the proposed schemes depends on the Phase I and Phase II probabiltity distributions only through the transformation function, 𝛹(𝑢), it is very important to show how to find 𝛹(𝑢). Table 3 gives the IC and OOC transformation functions for the N(0,1), t(5) and GAM(1,1) distributions. It can be observed that when the process is IC, 𝜓(𝑢) = 𝑢 regardless of the nature of the p.d.f. under consideration. This property confirms the IC robustness of the precedence schemes. For more details on how to derive the transformation functions, readers are referred to the Appendix.

<Insert Table 3> From Table 4, it can also be seen that, as expected, for all continuous distributions under consideration, the proposed schemes yield the same IC characteristics (i.e., 𝐶𝑍𝑆𝐴𝑅𝐿 0 or 𝐶𝑆𝑆𝐴𝑅𝐿 0 ).

For instance, the upper control and warning limits, attained 𝐶𝑍𝑆𝐴𝑅𝐿 0 and 𝐶𝑆𝑆𝐴𝑅𝐿 0 values of the IRR 2-of-3 precedence scheme (i.e. when h = 2) are computed under the N(0,1), t(5) and GAM(1,1)

distributions for m = 500, n = 5 and a nominal 𝐴𝑅𝐿 0 value of 500. It is observed that (𝑈𝑊𝐿 ̂, 𝑈𝐶𝐿 ̂) = (𝑋 (460:500) , 𝑋 (469:500) ) yields the attained 𝐶𝑍𝑆𝐴𝑅𝐿 0 and 𝐶𝑆𝑆𝐴𝑅𝐿 0 values of 500.61 and 500.60, respectively, for all the distributions under consideration. This shows that the proposed precedence control schemes are IC robust. Therefore, the charting constants and characteristics of the IC RL distribution do not depend on the nature of the underlying distribution.

<Insert Table 4>

OOC Performance

Since the proposed precedence monitoring schemes are IC robust, it is of interest to compare their performance when the process is OOC. For a specific shift, 𝛿 ≠ 0, the monitoring scheme with a small OOC 𝑈𝑍𝑆𝐴𝑅𝐿 (𝑈𝑍𝑆𝐴𝑅𝐿 𝛿 ) or small OOC 𝑈𝑆𝑆𝐴𝑅𝐿 (𝑈𝑆𝑆𝐴𝑅𝐿 𝛿 ) value is considered to be more sensitive. When comparing the overall performance of several monitoring schemes, the scheme with the smallest 𝑈𝐴𝐸𝑄𝐿 value is preferred. The 𝑈𝑍𝑆𝐴𝑅𝐿 𝛿 values of the upper one-sided IRR 2-of-(h+1) and IRR w-of-w precedence schemes are determined using Equations ( 13) and (15), respectively; whereas, the 𝑈𝑆𝑆𝐴𝑅𝐿 𝛿 values of the upper one-sided IRR 2-of-(h+1) and IRR w-of-w precedence schemes are found using Equations ( 19) and ( 21), respectively. The 𝑈𝑍𝑆𝐴𝐸𝑄𝐿 and 𝑈𝑆𝑆𝐴𝐸𝑄𝐿 values are computed using Equation ( 22). Given that 𝛿 𝑚𝑖𝑛 = 0, note that, when 𝛿 𝑚𝑎𝑥 = 0.7, the UAEQL value gives the measure of the overall performance for small shifts only. When 𝛿 𝑚𝑎𝑥 = 1.5, the UAEQL value measures the overall performance covering small to moderate shifts. For 𝛿 𝑚𝑎𝑥 = 2.5, the UAEQL value measures the overall performance covering small to large shifts.

Tables 5 and6 display the unconditional zero-and steady-state performance results of the upper onesided IRR 2-of-(h+1) and IRR w-of-w monitoring schemes, respectively. The UAEQL and the %𝐷𝑖𝑓𝑓 of the UAEQL values are computed using Equations ( 22) and ( 23), respectively. In these tables, a grey shaded cell shows that the proposed scheme performs better in that particular situation. Table 5 presents the unconditional zero-and steady-state (in brackets) OOC ARL performances of the proposed upper one-sided IRR 2-of-(h+1) precedence monitoring scheme under the N(0,1), t(5) and GAM(1,1) distributions for ℎ = 1, 2, 5 and 10. However, Table 6 displays the unconditional zero-and steady-state (in brackets) OOC ARL performances of the proposed upper one-sided IRR w-of-w precedence monitoring scheme under the N(0,1), t(5) and GAM(1,1) distributions for w = 2, 5, 10 and 15. The results in Table 5 shows that the zero-and steady-states performances are almost similar. For both IRR 2-of-(h+1) and IRR w-of-w schemes, the percentage difference between the 𝑈𝑍𝑆𝐴𝑅𝐿 and 𝑈𝑆𝑆𝐴𝑅𝐿 values doest not exceed 1%. In terms of the overall performance, the percentage difference between the 𝑈𝑍𝑆𝐴𝐸𝑄𝐿 and 𝑈𝑆𝑆𝐴𝐸𝑄𝐿 values is between 0.02% and 0.09% for the IRR 2-of-(h+1) scheme and between 0.02% and 0.8% for the IRR w-of-w scheme. For small shifts in the location parameter, in terms of the 𝐴𝑅𝐿 values, the enhanced IRR 2-of-(h+1) and IRR w-of-w precedence schemes perform better under the GAM(1,1) distribution followed by N(0,1) distribution when 𝛿 ≤ 0.5 for both zero-and steadystate modes. For moderate and large shifts, the proposed schemes perform better under the t( 5)

distribution regardless of the value of h or w. In both zero-and steady-state modes, for small shifts, the proposed IRR 2-of-(h+1) precedence scheme is more sensitive under the N(0,1) distribution than the t( 5) distribution when 𝛿 ≤ 0.4.

A thorough examination of the results shows that when 𝛿 ∈ (0, 1.1) the IRR w-of-w scheme is more sensitive for large values of w. However, when 𝛿 ∈ (1.1, 1.7), the IRR w-of-w scheme is more sensitive for small values of w. When 𝛿 ≥ 1.7, the proposed IRR w-of-w precedence scheme is sensitive regardless of the value of w. The proposed IRR schemes (i.e. both IRR 2-of-(h+1) and IRR w-of-w schemes) are relatively inefficient under the GAM(1,1) distribution for moderate and large shifts. Unlike the SRR w- of-w scheme, the OOC ARL of the IRR w-of-w scheme converges toward one for large shifts.

Consequently, the IRR w-of-w control scheme performs better than the SRR w-of-w control scheme for large shifts in the location parameter.

When comparing the zero-and steady state performances of the proposed schemes, it is observed that the steady-state ARL values are slightly smaller than the zero-state ARL values; so that their corresponding AEQL have a difference of no more than 1% (see Tables 5 and6). Since the zero-and steady performances of the proposed precedence schemes are almost similar and because of the page restriction, in Figure 2, the proposed precedence schemes are compared to the basic precedence scheme in terms of the 𝑍𝑆𝐴𝐸𝑄𝐿 values under symmetrical, heavy-tailed and skewed distributions.

Thus, in terms of the 𝑍𝑆𝐴𝐸𝑄𝐿 values (i.e., overall performance), for 𝛿 𝑚𝑎𝑥 = 0.7, the IRR precedence schemes are more sensitive under skewed distributions (see Figure 2). For 𝛿 𝑚𝑎𝑥 = 1.5 and 𝛿 𝑚𝑎𝑥 = 2.5, the proposed monitoring schemes are more sensitive under heavy-tailed distributions followed by symmetric distributions and relatively insensitive under skewed distributions. It can also be seen that the proposed precedence schemes are superior to the basic precedence scheme regardless of the nature of the underlying distribution.

<Insert Tables 5 and6> Table 7 compares the zero-and steady state AEQL values of the proposed IRR 2-of-(h+1) precedence scheme with the SRR 2-of-(h+1) and basic precedence schemes. However, Table 8 compares the zero-and steady state AEQL values of the proposed IRR w-of-w precedence scheme with the SRR w-of-w and basic precedence monitoring schemes. In Tables 7 and8, the best precedence scheme is shaded in grey under different probability distributions.

The results in Tables 7 and8 as well as Figure 2 yield the following findings:

1. In terms of the AEQL values with 𝛿 𝑚𝑎𝑥 = 0.7 and 1.5 (i.e. for "small" and "small to moderate" shifts),  The IRR precedence schemes outperform the SRR and basic precedence schemes regardless of the value of h or w for both zero-and steady-state modes.  The overall performance of the IRR w-of-w precedence scheme is an increasing function of w; which means that for "small" as well as "small to moderate" shifts, the larger the value of w, the more sensitive the proposed IRR w-of-w precedence scheme (see Figure 2(a) -(b)).  The IRR 2-of-(h+1) and IRR w-of-w precedence monitoring schemes perform worse under GAM(1,1) distribution except for "small" shifts in the process location regardless of the value of h (see Figure 2(b)-(c)).  For ℎ > 5, the performance of the SRR 2-of-(h+1) schemes deteriorate as h increases.

However, the IRR 2-of-(h+1) precedence scheme performs uniformly better regardless of the value of h. Therefore, the proposed IRR 2-of-(h+1) precedence scheme outperforms the SRR 2-of-(h+1) precedence monitoring scheme in terms of the overall performance regardless of the value of ℎ (see Table 7).  The SRR w-of-w precedence scheme is more efficient under the t(5) distribution regardless of the value of w. The sensitivity of the SRR w-of-w precedence control schemes increases slowly as h increases. Whereas, the sensitivity of the IRR w-of-w control schemes increases rapidly as w increases in the interval [2,5] and for 𝑤 > 5 the overall performance of the IRR w-of-w control schemes increases slowly.  Under small shifts, the proposed IRR 2-of-(h+1) precedence scheme performs better than the basic precedence scheme regardless of the value of ℎ. Under skewed distributions, the IRR 2-of-(h+1) precedence scheme is more sensitive followed by the basic precedence scheme regardless of the value of ℎ. However, the IRR w-of-w precedence scheme is more sensitive under skewed distributions when 𝑤 ∈ {2, 3} and for 𝑤 > 3 the IRR w-of-w precedence scheme is more sensitive under heavy-tailed distributions regardless of the value of 𝑤 (see Figure 2(a)).  Under "small to moderate" shifts, the proposed IRR schemes perform worse under skewed distributions. However, the IRR precedence schemes outperform the SRR and basic precedence schemes.

2. In terms of the "small to large" shifts, AEQL, with 𝛿 𝑚𝑎𝑥 = 2.5,  Both SRR and IRR schemes perform uniformly better in steady-state mode for small values of h or w. However, as the value of w increases, the performance of the SRR w-of-w precedence monitoring scheme deteriorates rapidly. Nevertheless, the SRR 2-of-(h+1) and IRR 2-of-(h+1) as well as the IRR w-of-w precedence schemes preserve their sensitivity regardless of the value of h or w; note though, the IRR 2-of-(h+1) and IRR w-of-w schemes outperform the SRR 2-of-(h+1) , SRR w-of-w and basic schemes.  For the proposed IRR 2-of-(h+1) precedence scheme, the steady-state overall performance is slightly smaller than the zero-state overall performance by less 1%.  Both upper one-sided IRR 2-of-(h+1) and IRR w-of-w schemes are more sensitive under light and heavy-tailed distributions followed by symmetric distributions. They are relatively insensitive under skewed distributions.  The SRR w-of-w control scheme performs worst for large values of w; however, IRR w-of-w and IRR 2-of-(h+1) monitoring schemes perform uniformly better regardless of the value of 𝑤 and h, respectively, and outperform the basic precedence scheme from small to large shifts in the location parameter.  The proposed IRR schemes are more sensitive under heavy-tailed distributions regardless of the value of ℎ and outperform the basic precedence scheme. <Insert Tables 7 and8> <Insert Figure 2> Note that to confirm the results found in Tables 5678, Monte Carlo simulations with 50000 replications were used. The discrepancy between the results found using exact formulas and simulations is within 1% which means these results are almost similar. Because of the similarity of the results, the simulations results are not displayed in this paper.

Illustrative example

In this section, an illustrative example on the design and implementation of the proposed IRR monitoring schemes is given using a well-known dataset from Montgomery (2005, page 223; Tables 5.2 and5.3). The data are the inside diameters of piston rings manufactured by a forging process. The data given in Table 5.2 contains fifteen Phase II samples, each of size 𝑛 = 5. Table 5.3 of [START_REF] Montgomery | Introduction to Statistical Quality Control[END_REF] contains 125 Phase I observations, that were collected when the process was considered IC (𝑚 = 125). These data are considered to be the Phase I (or reference) observations for which a goodness of fit test for normality is not rejected. In this example, the detection ability of the proposed IRR precedence schemes is compared to the detection ability of the SRR and basic schemes.

For both zero-and steady-state modes, for a nominal 𝐴𝑅𝐿 0 of 500, the zero-state and steady-state UWL and UCL of the upper one-sided IRR 2-of-(h+1) precedence scheme for h = 2 (i.e., the UWL and UCL of the IRR 2-of-3 scheme) are given by the 110 th and 117 th order statistics, that is 𝑈𝑊𝐿 ̂= 𝑋 (110:125) = 74.013 and 𝑈𝐶𝐿 ̂= 𝑋 (117:125) = 74.015, respectively. However, for both zero-and steady state modes, the UCLs of the upper one-sided SRR 2-of-(h+1) and basic precedence schemes for h = 2 (i.e., the UCLs of SRR 2-of-3 and RR 1-of-1 schemes) are given by the 115 th and 122 th order statistics, that is 𝑈𝐶𝐿 ̂ = 𝑋 (115:125) = 74.015 and 𝑈𝐶𝐿 ̂ = 𝑋 (122:125) = 74.02, respectively. A plot of the IRR 2-of-3 , SRR 2- of-3 and basic precedence (i.e. median) charting statistics for both cases is shown in Figure 3. It is seen that the IRR 2-of-3 precedence scheme signals for the first time on the 9 th sample in the prospective phase (Phase II); whereas, the SRR 2-of-3 and basic precedence schemes signal for the first time on the 13 th and 14 th samples, respectively. However, for both zero-and steady-state modes, for a nominal 𝐴𝑅𝐿 0 of 500, the zero-state and steady-state UWL and UCL of the upper one-sided IRR w-of-w precedence scheme for w = 3 (i.e., the UWL and UCL of the IRR 3-of-3 scheme) are given by the 99 th and 117 th order statistics, that is 𝑈𝑊𝐿 ̂= 𝑋 (99:125) = 74.009 and 𝑈𝐶𝐿 ̂= 𝑋 (117:125) = 74.015, respectively. Whereas, for both zero-and steady state modes, the UCLs of the upper one-sided SRR w-of-w (for w = 3) and basic precedence schemes (i.e., the UCLs of SRR 3-of-3 and RR 1-of-1 schemes) are given by the 107 th and 122 th order statistics; that is, 𝑈𝐶𝐿 ̂ = 𝑋 (107:125) = 74.012 and 𝑈𝐶𝐿 ̂ = 𝑋 (122:125) = 74.02, respectively. A plot of the IRR 3-of-3 , SRR 3-of-3 and basic precedence (i.e. median) charting statistics for both cases is shown in Figure 4. It is seen that the IRR 3-of-3 precedence scheme signals for the first time on the 9 th sample in the prospective phase; whereas, both the SRR 3-of-3 and basic precedence schemes signal for the first time on the 14 th sample in the prospective phase.

<Insert Figures 3 and4> The above example shows that the addition of runs-rules improves the basic precedence scheme and the IRR precedence schemes outperform both the SRR and basic schemes.

Extension of the IRR 2-of-(h+1) and IRR w-of-w precedence schemes to two-sided scenario

TPMs and run-length properties

In this section, a brief summary of the extension to the two-sided version of the IRR 2-of-(h+1) and IRR w- of-w precedence schemes, with charting regions shown in Figure 5, is presented. The IRR 2-of-(h+1) discussed here is an improved version of the two-sided RR 2-of-(h+1) presented in [START_REF] Malela-Majika | Distribution-free precedence schemes with a generalized runs-rule for monitoring unknown location[END_REF].

<Insert Figure 5> The two-sided IRR 2-of-(h+1) scheme signals when either a single point falls on Region 1 or 5; or when 2 out of ℎ + 1 successive samples fall on Region 2 (Region 5) which are separated by at most ℎ -1 samples falling on Region 3 or 4 (Region 2 or 3), respectively. Following a similar procedure as in Section 3.1, it follows that the TPM of the two-sided IRR 2-of-(h+1) precedence scheme for any value of h is given by

𝜂 1 𝜂 2 ⋯ 𝜂 ℎ-3 𝜂 ℎ-2 𝜂 ℎ-1 𝜂 ℎ 𝜙 𝜂 ℎ+2 𝜂 ℎ+3 𝜂 ℎ+4 𝜂 ℎ+5 ⋯ 𝜂 2ℎ-1 𝜂 2ℎ 𝜂 2ℎ+1 OOC 𝜂 1 𝑞 3 𝑞 4 𝑞 2 +𝑞 1 +𝑞 5 𝜂 2 𝑞 3 𝑞 4 𝑞 2 +𝑞 1 +𝑞 5 𝜂 3 𝑞 3 𝑞 4 𝑞 2 +𝑞 1 +𝑞 5 ⋮ ⋱ ⋮ ⋮ 𝜂 ℎ-2 𝑞 3 𝑞 4 𝑞 2 +𝑞 1 +𝑞 5 𝜂 ℎ-1 𝑞 3 𝑞 4 𝑞 2 +𝑞 1 +𝑞 5 𝜂 ℎ 𝑞 3 𝑞 4 𝑞 2 +𝑞 1 +𝑞 5 𝜙 𝑞 2 𝑞 3 𝑞 4 𝑞 1 +𝑞 5 𝜂 ℎ+2 𝑞 2 𝑞 3 𝑞 4 +𝑞 1 +𝑞 5 𝜂 ℎ+3 𝑞 2 𝑞 3 𝑞 4 +𝑞 1 +𝑞 5 𝜂 ℎ+4 𝑞 2 𝑞 3 𝑞 4 +𝑞 1 +𝑞 5 ⋮ ⋮ ⋱ ⋮ 𝜂 2ℎ-2 𝑞 2 𝑞 3 𝑞 4 +𝑞 1 +𝑞 5 𝜂 2ℎ-1 𝑞 2 𝑞 3 𝑞 4 +𝑞 1 +𝑞 5 𝜂 2ℎ 𝑞 2 𝑞 3 𝑞 4 +𝑞 1 +𝑞 5 𝜂 2ℎ+1 𝑞 2 𝑞 3 𝑞 4 +𝑞 1 +𝑞 5 OOC 1
where

𝑞 1 = 𝑃(𝑌 (𝑗:𝑛) ≥ 𝑋 (𝑏 2 :𝑚) |𝑋 (𝑏 2 :𝑚) = 𝑥 (𝑏 2 :𝑚) ) = 𝐼(𝐺𝐹 -1 (𝑈 (𝑏 2 :𝑚) ), 𝑗, 𝑛 -𝑗 + 1), (24)
𝑞 2 = 𝐼(𝐺𝐹 -1 (𝑈 (𝑏 2 :𝑚) ), 𝑗, 𝑛 -𝑗 + 1) -𝐼(𝐺𝐹 -1 (𝑈 (𝑏 1 :𝑚) ), 𝑗, 𝑛 -𝑗 + 1), 𝑞 3 = 𝐼(𝐺𝐹 -1 (𝑈 (𝑏 1 :𝑚) ), 𝑗, 𝑛 -𝑗 + 1) -𝐼(𝐺𝐹 -1 (𝑈 (𝑎 1 :𝑚) ), 𝑗, 𝑛 -𝑗 + 1), 𝑞 4 = 𝐼(𝐺𝐹 -1 (𝑈 (𝑎 1 :𝑚) ), 𝑗, 𝑛 -𝑗 + 1) -𝐼(𝐺𝐹 -1 (𝑈 (𝑎 2 :𝑚) ), 𝑗, 𝑛 -𝑗 + 1), 𝑞 5 = 𝑃(𝑌 (𝑗:𝑛) ≤ 𝑋 (𝑎 2 :𝑚) |𝑋 (𝑎 2 :𝑚) = 𝑥 (𝑎 2 :𝑚) ) = 1 -𝐼(𝐺𝐹 -1 (𝑈 (𝑎 2 :𝑚) ), 𝑗, 𝑛 -𝑗 + 1).

Similarly, the TPM of the two-sided IRR w-of-w precedence scheme is given by

𝜂 1 𝜂 2 ⋯ 𝜂 𝑤-3 𝜂 𝑤-2 𝜂 𝑤-1 𝜑 𝜂 𝑤+1 𝜂 𝑤+2 𝜂 𝑤+3 ⋯ 𝜂 2𝑤-2 𝜂 2𝑤-1 OOC 𝜂 1 𝑞 3 𝑞 4 𝑞 2 +𝑞 1 +𝑞 5 𝜂 2 𝑞 2 𝑞 3 𝑞 4 𝑞 1 +𝑞 5 𝜂 3 𝑞 2 𝑞 3 𝑞 4 𝑞 1 +𝑞 5 ⋮ ⋱ ⋮ ⋮ ⋮ 𝜂 𝑤-2 𝑞 2 𝑞 3 𝑞 4 𝑞 1 +𝑞 5 𝜂 𝑤-1 𝑞 2 𝑞 3 𝑞 4 𝑞 1 +𝑞 5 𝜑 𝑞 2 𝑞 3 𝑞 4 𝑞 1 +𝑞 5 𝜂 𝑤+1 𝑞 2 𝑞 3 𝑞 4 𝑞 1 +𝑞 5 𝜂 𝑤+2 𝑞 2 𝑞 3 𝑞 4 𝑞 1 +𝑞 5 ⋮ ⋮ ⋮ ⋱ ⋮ 𝜂 2𝑤-3 𝑞 2 𝑞 3 𝑞 4 𝑞 1 +𝑞 5 𝜂 2𝑤-2 𝑞 2 𝑞 3 𝑞 4 𝑞 1 +𝑞 5 𝜂 2𝑤-1 𝑞 2 𝑞 3 𝑞 4 +𝑞 1 +𝑞 5 OOC 1
with 𝑞 1 , …, 𝑞 5 defined in Equation ( 24). For more thorough discussions on these two-sided IRR 2-of- where 𝑎 2 < 𝑎 1 < 𝑏 1 < 𝑏 2 and

𝑓 𝑎 2 𝑎 1 𝑏 1 𝑏 2 (𝑟, 𝑠, 𝑢, 𝑡) = 𝑚! (𝑎 2 -1)!(𝑎 1 -𝑎 2 -1)!(𝑏 1 -𝑎 1 -1)!(𝑏 2 -𝑏 1 -1)!(𝑚-𝑏 2 )! 𝑟 𝑎 2 -1 (𝑠 -𝑟) 𝑎 1 -𝑎 2 -1 (𝑢 - 𝑠) 𝑏 1 -𝑎 1 -1 (𝑡 -𝑢) 𝑏 2 -𝑏 1 -1 (1 -𝑡) 𝑚-𝑏 2
is the joint pdf of the 𝑎 2 𝑡ℎ , 𝑎 1 𝑡ℎ , 𝑏 1 𝑡ℎ and 𝑏 2 𝑡ℎ order statistics in a sample of size 𝑚 from the Uniform (0,1) distribution, 𝑟, 𝑠, 𝑢 and 𝑡 are random variables from the Uniform (0,1) distribution, with

𝐶𝑍𝑆𝐴𝑅𝐿 = (1 + 𝑞 2 ( 1 -𝑞 3 ℎ 1 -𝑞 3 )) (1 + 𝑞 4 ( 1 -𝑞 3 ℎ 1 -𝑞 3 )) 1 -𝑞 3 -𝑞 2 𝑞 3 ℎ -𝑞 4 𝑞 3 ℎ -𝑞 2 𝑞 4 ( 1 -𝑞 3 2ℎ 1 -𝑞 3 )
and

𝐶𝑆𝑆𝐴𝑅𝐿 = 𝑠 ℎ+1 𝜍 ℎ+1 + ∑ 𝑠 𝑖 × (𝜍 𝑖 + 𝜍 (2ℎ+2)-𝑖 ) ℎ 𝑖=1
.

Note that

𝐬 = ( s 1 s 2 s 3 ⋮ s ℎ-2 s ℎ-1 s ℎ s ℎ+1 s ℎ+2 s ℎ+3 s ℎ+4 ⋮ s 2ℎ-1 s 2ℎ s 2ℎ+1 ) ′ = 1 2 ( 1 -𝛾 1 ℎ 1 -𝛾 1 ) + 2𝛾 1 ℎ (1 -𝛾 2 ) -1 ( 𝛾 1 ℎ-1 𝛾 1 ℎ-2 𝛾 1 ℎ-3 ⋮ 𝛾 1 2 𝛾 1 1 2𝛾 1 ℎ (1 -𝛾 2 ) -1 1 𝛾 1 𝛾 1 2 ⋮ 𝛾 1 ℎ-3 𝛾 1 ℎ-2 𝛾 1 ℎ-1 ) ′ with 𝛾 1 = 𝑞 3 𝑞+𝑞 3 , 𝛾 2 = 𝑞 3 2𝑞+𝑞 3
and 𝑞 = 𝑞 2 = 𝑞 4 (since 𝐬 is computed while the process is IC). In addition,

𝐑 = ( ς 1 ς 2 ⋮ ς ℎ-3 ς ℎ-2 ς ℎ-1 ς ℎ ς ℎ+1 ς ℎ+2 ς ℎ+3 ς ℎ+4 ς ℎ+5 ⋮ ς 2ℎ ς 2ℎ+1 ) = 1 1 -𝑞 3 -𝑞 2 𝑞 3 ℎ -𝑞 4 𝑞 3 ℎ -𝑞 2 𝑞 4 ( 1 -𝑞 3 2ℎ 1 -𝑞 3 ) ( (1 + 𝑞 2 Ν 1 )(1 + 𝑞 4 N 0 ) (1 + 𝑞 2 Ν 2 )(1 + 𝑞 4 N 0 ) ⋮ (1 + 𝑞 2 Ν ℎ-3 )(1 + 𝑞 4 N 0 ) (1 + 𝑞 2 Ν ℎ-2 )(1 + 𝑞 4 N 0 ) (1 + 𝑞 2 Ν ℎ-1 )(1 + 𝑞 4 N 0 ) (1 + 𝑞 4 N 0 ) (1 + 𝑞 2 Ν 0 )(1 + 𝑞 4 N 0 ) (1 + 𝑞 2 Ν 0 ) (1 + 𝑞 4 N ℎ-1 )(1 + 𝑞 2 Ν 0 ) (1 + 𝑞 4 N ℎ-2 )(1 + 𝑞 2 Ν 0 ) (1 + 𝑞 4 N ℎ-3 )(1 + 𝑞 2 Ν 0 ) ⋮ (1 + 𝑞 4 N 2 )(1 + 𝑞 2 Ν 0 ) (1 + 𝑞 4 N 1 )(1 + 𝑞 2 Ν 0 ) ) . with Ν 𝑖 = 𝑞 3 𝑖 ( 1 -𝑞 3 ℎ-𝑖 1 -𝑞 3 ).
Similarly, the 𝑈𝑍𝑆𝐴𝑅𝐿 and 𝑈𝑆𝑆𝐴𝑅𝐿 of the two-sided IRR w-of-w scheme for any value of w are given by Equations ( 25) and ( 26), respectively; however, the

𝐶𝑍𝑆𝐴𝑅𝐿 = (1 -𝑞 2 𝑤 )(1 -𝑞 4 𝑤 ) 𝐷 and 𝐶𝑆𝑆𝐴𝑅𝐿 = s 𝑤 ς 𝑤 + ∑ s 𝑗 × (ς 𝑗 + ς 2𝑤-𝑗 ) 𝑤-1 𝑗=1 ;
where,

𝐷 = (1 -𝑞 3 )((1 -𝑞 2 )(1 -𝑞 4 )) -𝑞 3 ((1 -𝑞 2 )(𝑞 4 -𝑞 4 𝑤 ) + (1 -𝑞 4 )(𝑞 2 -𝑞 2 𝑤 )) -(1 + 𝑞 3 )((𝑞 2 -𝑞 2 𝑤 )(𝑞 4 -𝑞 4 𝑤 )).
Note that

𝐑 = ( ς 1 ς 2 ς 3 ⋮ ς 𝑤-2 ς 𝑤-1 ς 𝑤 ς 𝑤+1 ς 𝑤+2 ⋮ ς 2𝑤-3 ς 2𝑤-2 ς 2𝑤-1 ) = 1 𝐷 ( (1 -𝑞 2 )(1 -𝑞 4 𝑤 ) (1 -𝑞 2 2 )(1 -𝑞 4 𝑤 ) (1 -𝑞 2 3 )(1 -𝑞 4 𝑤 ) ⋮ (1 -𝑞 2 𝑤-2 )(1 -𝑞 4 𝑤 ) (1 -𝑞 2 𝑤-1 )(1 -𝑞 4 𝑤 ) (1 -𝑞 2 𝑤 )(1 -𝑞 4 𝑤 ) (1 -𝑞 4 𝑤-1 )(1 -𝑞 2 𝑤 ) (1 -𝑞 4 𝑤-2 )(1 -𝑞 2 𝑤 ) ⋮ (1 -𝑞 4 3 )(1 -𝑞 2 𝑤 ) (1 -𝑞 4 2 )(1 -𝑞 2 𝑤 ) (1 -𝑞 4 )(1 -𝑞 2 𝑤 ) )
, and

𝐬 = ( s 1 s 2 s 3 ⋮ s 𝑤-2 s 𝑤-1 s 𝑤 s 𝑤+1 s 𝑤+2 ⋮ s 2𝑤-3 s 2𝑤-2 s 2𝑤-1 ) = ( 1 -𝑞 1 -𝑞 𝑤 ) ( 𝑞 𝑤-1 𝑞 𝑤-2 𝑞 𝑤-3 ⋮ 𝑞 2 𝑞 1 -( 𝑞 -𝑞 𝑤 1 -𝑞 ) 𝑞 𝑞 2 ⋮ 𝑞 𝑤-3 𝑞 𝑤-2 𝑞 𝑤-1 )
.

For more thorough discussions of these expressions, see Shongwe et al. (2019a) and Shongwe (2020).

IC and OOC performance analysis

In this section, a brief analysis of the performance of the two-sided IRR 2-of-(h+1) and IRR w-of-w precedence monitoring schemes is presented. Table 9 displays the zero-state performances of the twosided IRR 2-of-(h+1) and IRR w-of-w precedence monitoring schemes for a nominal 𝐴𝑅𝐿 0 of 500 under the N(0,1), t(5) and GAM(1,1) distributions when (𝑚,𝑛) = (500,5), h∈{1,5}, w∈{2,5} and (𝛿 𝑚𝑖𝑛 ,𝛿 𝑚𝑎𝑥 ) = (0,2). At each shift value or range of shift values, the distribution with the best performance is boldfaced. From Table 9, it is observed that under heavy-tailed distributions, the two-sided precedence schemes present better ARL and AEQL results as compared to the normal distribution.

Moreover, under skewed distributions, the two-sided precedence scheme has the worst performance for small shifts in the process location as well as in terms of the AEQL. Similar results are observed for the corresponding steady-state mode.

<Insert Table 9> 7. Summary and recommendations [START_REF] Malela-Majika | Distribution-free precedence schemes with a generalized runs-rule for monitoring unknown location[END_REF] proposed two-sided SRR 2-of-(h+1) precedence monitoring schemes to monitor both the increase and decrease in the location process parameter. In this paper, one-sided IRR 2-of-(h+1) and IRR w-of-w precedence monitoring schemes are proposed to monitor either an increase or decrease in the location process parameter without any distributional assumption. The sensitivity and robustness of the these schemes are investigated using zero-and steady-state properties of the runlength distribution through the Markov chain technique. It is found that the proposed one-sided IRR 2- of-(h+1) and IRR w-of-w precedence schemes are IC robust and superior in performance to the one-sided SRR 2-of-(h+1) , SRR w-of-w and basic precedence schemes. The IRR w-of-w precedence scheme is more sensitive than the IRR 2-of-(h+1) precedence scheme when compared head to head (i.e. when 𝑤 = ℎ + 1) for all ℎ > 1 and 𝑤 > 2.

Table 1. Operation of the Phase II one-sided IRR 2-of-( h+1) and IRR w-of-w precedence schemes

Step Operation 1 Specify the desired value of h or w. -For the IRR 2-of-(h+1) scheme, calculate CRL, and if CRL < h, go to Step 8; otherwise return to Step 3. 8

Issue an OOC signal. Take necessary corrective action to find and remove the assignable causes and return to Step 3.

  d.f. (cumulative distribution function), 𝐹(𝑥). Let 𝑌 𝑗 𝑘 (with 𝑗 = 1, 2, …, 𝑛 and 𝑘 = 1, 2, …) denote the 𝑘 𝑡ℎ test sample of size 𝑛 𝑘 , 𝑛 𝑘 = 𝑛 ∀𝑘, since it is assumed that the Phase II (or test) samples are all of the same size. For instance, 𝑌 𝑗 1 = {𝑌 1 , 𝑌 2 , … , 𝑌 𝑛 } is the first test sample of size 𝑛. Let 𝐺 𝑘 (𝑦) denote the c.d.f. of the distribution of the 𝑘 𝑡ℎ Phase II sample and let 𝐺 𝑘 (𝑦) = 𝐺(𝑦) ∀𝑘, since the Phase II samples are all assumed to be i.i.d. (independent and identically distributed).

  (a)),  Lower one-sided basic and SRR schemes: Regions C and D (see Figure 1(a)); where regions B and C are IC regions for schemes (i.e. basic and SRR), whereas regions A and D are OOC and nonconforming regions for the basic and SRR schemes, respectively. <Insert Figure 1> 2.2 Operation of the one-sided IRR 2-of-(h+1) and IRR w-of-w precedence schemes

  (b)),  Lower one-sided scheme: Regions 4, 5 and 6 (see Figure 1(b));

  -state and steady-state performance of the proposed precedence schemes 4.1 IC design of the proposed monitoring schemes One of the most important steps in the design and implementation of a monitoring scheme is the computation (or search) of the control limits. The first step in the design of the upper (lower) onesided IRR 2-of-(h+1) and IRR w-of-w schemes is based on the determination of the 𝑈𝑊𝐿 and 𝑈𝐶𝐿 (LWL and LCL), respectively. The 𝑈𝑊𝐿 and 𝑈𝐶𝐿 (LWL and LCL) are defined by the 𝑏 1 𝑡ℎ and 𝑏 2 𝑡ℎ (𝑎 1 𝑡ℎ and 𝑎 2 𝑡ℎ ) order statistics, also known as charting constants of the Phase I sample, which means 𝑈𝑊𝐿 = 𝑋 (𝑏 1 :𝑚)

(

  h+1) and IRR w-of-w TPMs, see for instance[START_REF] Malela-Majika | Shewhart control schemes with supplementary 2-of-(h+1) side-sensitive runs-rules under the Burr-type XII distribution[END_REF],Shongwe et al. (2019a) and[START_REF] Shongwe | On the design of nonparametric runs-rules schemes using the Markov chain approach[END_REF].Next, the 𝑈𝑍𝑆𝐴𝑅𝐿 and 𝑈𝑆𝑆𝐴𝑅𝐿 of the two-sided IRR 2-of-(h+1) scheme for any value of h are defined by 𝑈𝑍𝑆𝐴𝑅𝐿 = ∫ ∫ ∫ ∫ 𝐶𝑍𝑆𝐴𝑅𝐿 𝑓 𝑎 2 𝑎 1 𝑏 1

23-

  Determine: 𝑈𝐶𝐿 = 𝑋 (𝑏 2 :𝑚) / 𝑈𝑊𝐿 = 𝑋 (𝑏 1 :𝑚) or 𝐿𝐶𝐿 = 𝑋 (𝑎 2 :𝑚) / 𝐿𝑊𝐿 = 𝑋 (𝑎 1 :𝑚) , see Figure1(b). At each inspection point 𝑖, collect a Phase II sample of size n and compute 𝑌 (𝑗:𝑛) For the IRR w-of-w scheme, if 𝑌 (𝑗:𝑛) 𝑖+1 , 𝑌 (𝑗:𝑛) 𝑖+2 , … , 𝑌 (𝑗:𝑛) 𝑖+𝑤-1 ∈ Region 2, go to Step 8; otherwise return to Step 3. For the IRR 2-of-(h+1) scheme, calculate CRL, and if CRL < h, go to Step 8; otherwise return to Step 3. (b) -For the IRR w-of-w scheme, if 𝑌 (𝑗:𝑛) 𝑖+1 , 𝑌 (𝑗:𝑛) 𝑖+2 , … , 𝑌 (𝑗:𝑛) 𝑖+𝑤-1 ∈ Region 5, go to Step 8; otherwise return to Step 3.

Figure 1 .

 1 Figure 1. Different charting regions of the basic, SRR 2-of-(h+1) , SRR w-of-w , IRR 2-of-(h+1) and IRR w-of-w onesided precedence schemes

  1 , 𝑏 2 ) = (457, 469) yields the attained 𝐶𝑍𝑆𝐴𝑅𝐿 0 and 𝐶𝑆𝑆𝐴𝑅𝐿 0 values of 500.51 and 500.41, respectively, so that (𝑈𝑊𝐿 ̂, 𝑈𝐶𝐿 ̂) = (𝑋 (457: 500) , 𝑋 (469: 500) ). The charting constants (i.e. 𝑏 1 and 𝑏 2 ) and attained 𝐶𝑍𝑆𝐴𝑅𝐿 0 and 𝐶𝑆𝑆𝐴𝑅𝐿 0 values of the upper one-sided IRR 2-of-(h+1) and IRR w-of-w schemes are given in Table2for nominal 𝐴𝑅𝐿 0 values of 370 (first row) and 500 (second) when h ∈{1, 2, 5, 10} and w ∈{2, 5, 10}with n ∈ {5, 7} and 𝑚 ∈ {100, 200, 500}. As expected, it can be observed that for both IRR 2-of-(h+1)and IRR w-of-w schemes, for large sample sizes, the attained 𝐶𝑍𝑆𝐴𝑅𝐿 0 and 𝐶𝑆𝑆𝐴𝑅𝐿 0 values are much closer to the nominal 𝐴𝑅𝐿 0 value. For fixed values of h, 𝑤, and m, and a selected nominal 𝐴𝑅𝐿 0 value (e.g. nominal 𝐴𝑅𝐿 0 = 370 or 500), when n increases/decreases, the charting constants 𝑏 1 and 𝑏 2 decreases/increases. Moreover, when h, 𝑤 and m are kept constant, as the n increases (decreases), the magnitudes of both charting constants 𝑏 1 and 𝑏

2 decrease (increase). Note that, for m, n, h and 𝑤 fixed, if the charting constant 𝑏 1 increases (decreases), the attained 𝐶𝑍𝑆𝐴𝑅𝐿 0 (or 𝐶𝑆𝑆𝐴𝑅𝐿 0 ) value increases (decreases). A key observation from Table

2

is that for a given nominal 𝐴𝑅𝐿 0 values, the zero-and steady-state charting constants are the same and the attained 𝐶𝑍𝑆𝐴𝑅𝐿 0 and 𝐶𝑆𝑆𝐴𝑅𝐿 0 values are approximately equal. When 𝑏 1 = 𝑏 2 , the upper one-sided IRR precedence scheme is equivalent to the upper one-sided SRR precedence scheme. From Table

2

it can also be seen that for large Phase I sample sizes, the attained 𝑍𝑆𝐴𝑅𝐿 0 and 𝑆𝑆𝐴𝑅𝐿 0 are much closer to the nominal 𝐴𝑅𝐿 0 as compared to small Phase I sample sizes.

<Insert Table

2>

Table 2 .

 2 Charting constants (𝑏 1 , 𝑏 2 ) and the corresponding attained 𝐶𝑍𝑆𝐴𝑅𝐿 0 (𝐶𝑆𝑆𝐴𝑅𝐿 0 -in brackets) values of the upper one-sided IRR 2-of-(h+1) and IRR w-of- w precedence schemes when n ∈{5,7}, j ∈{3,4} h ∈{1,2,5,10}, 𝑤 ∈ {2,5,10} for nominal 𝐴𝑅𝐿 0 of 370 (first row) and 500 (second row) for each m

	∈{100,200,500}

Table 3 .

 3 IC and OOC transformation function under the N(𝜇, 𝜎), GAM(𝛼,𝛽) and t(𝜈) distributions

	Distribution	Parameters	𝐹(𝑥)	𝐹 -1 (𝑥)	𝐺(𝑥)		𝜓(𝑢) (OOC case)	𝜓(𝑢) (IC case)
	N(𝜇, 𝜎) GAM(𝛼, 𝛽) t(𝜈)	𝜇 = 0 and 𝜎 = 1 𝛼 = 𝛽 = 1 𝜈 = 5	Φ(𝑥) 𝑥 ∈ (-∞, ∞) 1 -exp (-𝑥) 𝑥 ∈ [0, ∞) 𝐹 𝜈 (𝑥) 𝑥 ∈ (-∞, ∞)	Φ -1 (𝑥) -ln (1 -𝑥) 𝐹 𝜈 -1 (𝑥)	Φ(𝑥 -𝛿) 1 -exp ( -𝑥 𝛿 + 1 𝐹 𝜈 (𝑥 -√2𝛿)	)	Φ(-𝛿 + Φ -1 (𝑢)) 1 -exp ( 1 ln (1 -𝑢)) 𝛿 + 1 𝐹 𝜈 (-√2𝛿 + 𝐹 𝜈 -1 (𝑢))	𝑢 𝑢 𝑢

Table 4

 4 Charting constants (𝑏 1 , 𝑏 2 ) and attained 𝐶𝑍𝑆𝐴𝑅𝐿 0 (𝐶𝑆𝑆𝐴𝑅𝐿 0in brackets) values of the upper one-sided IRR 2-of-(h+1) and IRR w-of-w precedence schemes for different values of h and w when m = 500 and n = 5

	IRR 2-of-(h+1)

Table 5

 5 The OOC unconditional zero-state (and steady-state, in brackets) ARL and AEQL performance of the upper one-sided IRR 2-of-(h+1) precedence scheme for h∈{1,2,5,10} when m = 500 and n = 5 for a nominal 𝑍𝑆𝐴𝑅𝐿 0 (𝑆𝑆𝐴𝑅𝐿 0 ) of 500

	h		1			2			5			10
	Shift(δ)	N(0,1)	t(5)	GAM(1,1))	N(0,1)	t(5)	GAM(1,1))	N(0,1)	t(5)	GAM(1,1))	N(0,1)	t(5)	GAM(1,1))
	0.1	282.78(282.76) 294.48(294.47)	235.52(235.51)	282.83(282.82) 294.48(294.55)	235.53(235.52)	282.89(282.87)	294.67(294.65)	235.55(235.54)	282.08(282.04)	293.77(293.73)	234.86(234.82)
	0.2	164.40(164.39) 173.38(173.38)	126.64(126.64)	164.41(164.40) 173.45(173.44)	126.62(126.61)		173.56(173.54)	126.62(126.60)	163.81(163.77)	172.78(172.74)	126.13(126.10)
	0.3	98.35 (98.35)	102.41(102.41)	75.43 (75.43)	98.34 (98.34)	102.47(102.46)	75.40 (75.39)	98.37 (98.36)	102.58(102.57)	75.39 (75.38)	97.88 (97.85)	101.96(101.92)	75.05 (75.02)
	0.4	60.56 (60.56)	60.89 (60.89)	48.68 (48.68)	60.54 (60.54)	60.94 (60.93)	48.65 (48.64)	60.57 (60.56)	61.06 (61.04)	48.64 (48.63)	60.22 (60.19)	60.60 (60.56)	48.40 (48.38)
	0.5	38.39 (38.39)	36.60 (36.59)	33.49 (33.48)	38.37 (38.37)	36.64 (36.63)	33.46 (33.45)	38.40 (38.39)	36.76 (36.75)	33.46 (33.45)	38.17 (38.15)	36.47 (36.44)	33.30 (33.28)
	0.6	25.06 (25.06)	22.36 (22.36)	24.26 (24.26)	25.05 (25.05)	22.40 (22.39)	24.23 (24.23)	25.09 (25.08)	22.53 (22.52)	24.24 (24.24)	24.95 (24.93)	22.37 (22.35)	24.14 (24.12)
	0.7	16.86 (16.86)	13.99 (13.99)	18.34 (18.33)	16.85 (16.85)	14.03 (14.02)	18.32 (18.31)	16.90 (16.89)	14.16 (14.15)	18.33 (18.32)	16.82 (16.81)	14.10 (14.08)	18.26 (18.25)
	0.8	11.69 (11.69)	9.03 (9.03)	14.36 (14.35)	11.69 (11.68)	9.06 (9.06)	14.34 (14.34)	11.74 (11.73)	9.19 (9.18)	14.36 (14.35)	11.71 (11.70)	9.19(9.18)	14.32 (14.30)
	0.9	8.36 (8.35)	6.06 (6.06)	11.57 (11.57)	8.36 (8.35)	6.09 (6.09)	11.56 (11.56)	8.41 (8.40)	6.20 (6.20)	11.58 (11.58)	8.41 (8.40)	6.23 (6.22)	11.56 (11.55)
	1.0	6.16 (6.15)	4.25 (4.25)	9.57 (9.57)	6.16 (6.16)	4.28 (4.28)	9.56 (9.56)	6.21 (6.21)	4.38 (4.38)	9.58 (9.58)	6.23 (6.22)	4.41 (4.40)	9.57 (9.56)
	1.1	4.67 (4.67)	3.13 (3.12)	8.08 (8.07)	4.68 (4.68)	3.15 (3.15)	8.07 (8.07)	4.73 (4.73)	3.22 (3.22)	8.09 (8.09)	4.75 (4.74)	3.26 (3.25)	8.09 (8.08)
	1.2	3.66 (3.65)	2.41 (2.41)	6.94 (6.94)	3.67 (3.66)	2.43 (2.43)	6.94 (6.94)	3.71 (3.70)	2.49 (2.48)	6.96 (6.96)	3.73 (3.72)	2.51 (2.50)	6.96 (6.96)
	1.3	2.94 (2.94)	1.95 (1.95)	6.06 (6.06)	2.95 (2.95)	1.96 (1.96)	6.06 (6.06)	2.99 (2.98)	2.00 (2.00)	6.08 (6.08)	3.00 (3.00)	2.01 (2.01)	6.09 (6.08)
	1.4	2.43 (2.43)	1.64 (1.64)	5.36 (5.36)	2.44 (2.44)	1.65 (1.65)	5.36 (5.36)	2.47 (2.47)	1.67 (1.67)	5.39 (5.38)	2.48 (2.48)	1.68 (1.68)	5.39 (5.39)
	1.5	2.06 (2.06)	1.43 (1.43)	4.80 (4.80)	2.07 (2.07)	1.44 (1.44)	4.80 (4.80)	2.09 (2.09)	1.45 (1.45)	4.83 (4.82)	2.10 (2.10)	1.46 (1.46)	4.83 (4.83)
	1.6	1.79 (1.79)	1.29 (1.29)	4.34 (4.34)	1.80 (1.79)	1.30 (1.30)	4.35 (4.34)	1.81 (1.81)	1.30 (1.30)	4.37 (4.36)	1.82 (1.82)	1.31 (1.30)	4.38 (4.37)
	1.7	1.59 (1.58)	1.19 (1.19)	3.97 (3.97)	1.59 (1.59)	1.20 (1.20)	3.97 (3.97)	1.60 (1.60)	1.20 (1.20)	3.99 (3.99)	1.61 (1.60)	1.20 (1.20)	4.00 (3.99)
	1.8	1.44 (1.43)	1.13 (1.13)	3.65 (3.65)	1.44 (1.44)	1.13 (1.13)	3.65 (3.65)	1.45 (1.44)	1.13 (1.13)	3.67 (3.67)	1.45 (1.45)	1.13 (1.13)	3.68 (3.68)
	1.9	1.32 (1.32)	1.08 (1.08)	3.38 (3.38)	1.32 (1.32)	1.08 (1.08)	3.39 (3.38)	1.33 (1.33)	1.08 (1.08)	3.40 (3.40)	1.33 (1.33)	1.08 (1.08)	3.41 (3.41)
	2.0	1.23 (1.23)	1.05 (1.05)	3.15 (3.15)	1.23 (1.23)	1.05 (1.05)	3.16 (3.16)	1.24 (1.24)	1.05 (1.05)	3.17 (3.17)	1.24 (1.24)	1.05 (1.05)	3.18 (3.18)
	2.1	1.17 (1.17)	1.03 (1.04)	2.96 (2.96)	1.17 (1.17)	1.03 (1.03)	2.96 (2.96)	1.17 (1.17)	1.03 (1.03)	2.98 (2.97)	1.17 (1.17)	1.03 (1.00)	2.98 (2.98)
	2.2	1.12 (1.12)	1.02 (1.02)	2.79 (2.79)	1.12 (1.12)	1.02 (1.02)	2.79 (2.79)	1.12 (1.12)	1.02 (1.02)	2.80 (2.80)	1.12 (1.12)	1.02 (1.00)	2.81 (2.81)
	2.3	1.08 (1.08)	1.01 (1.01)	2.64 (2.64)	1.08 (1.08)	1.01 (1.01)	2.64 (2.64)	1.08 (1.08)	1.01 (1.01)	2.66 (2.65)	1.08 (1.08)	1.01 (1.01)	2.66 (2.66)
	2.4	1.06 (1.06)	1.01 (1.01)	2.51 (2.51)	1.06 (1.06)	1.01 (1.00)	2.51 (2.51)	1.06 (1.06)	1.01 (1.01)	2.52 (2.52)	1.06 (1.06)	1.01 (1.01)	2.53 (2.53)
	2.5	1.04 (1.04)	1.00 (1.00)	2.39 (2.39)	1.04 (1.04)	1.00 (1.00)	2.40 (2.40)	1.04 (1.04)	1.00 (1.00)	2.41 (2.41)	1.04 (1.04)	1.00 (1.00)	2.41 (2.41)
	AEQL	61.15 (61.13)	52.10(52.09)	102.66 (102.64)	61.21 (61.20)	52.22 (52.20)	102.69 (102.66)	61.45 (61.43)	52.53 (52.50)	103.05 (103.00)	61.41 (61.36)	52.48 (52.44)	103.09 (103.00)
	%Diff	0.03%	0.02%	0.02%	0.02%	0.04%	0.03%	0.03%	0.06%	0.05%	0.08%	0.08%	0.09%

Table 6

 6 The OOC unconditional zero-state (and steady-state, in brackets) ARL and AEQL values of the upper one-sided IRR w-of-w precedence scheme for w∈{2,5,10,15} when m = 500 and n = 5 for a nominal 𝑍𝑆𝐴𝑅𝐿 0 (𝑆𝑆𝐴𝑅𝐿 0 ) of 500

	w		2			5			10			15
	Shift(δ)	N(0,1)	t(5)	GAM(1,1))	N(0,1)	t(5)	GAM(1,1))	N(0,1)	t(5)	GAM(1,1))	N(0 ,1)	t(5)	GAM(1,1) )
	0.1	282.78(282.76) 294.48(294.47)	235.52(235.51)	280.96(280.93) 290.45(290.41)	234.66(334.64)	279.41(279.34)	286.46 (286.37)	234.38(234.33)	278.21(278.19)	284.26(284.09)	234.26(234.18)
	0.2	164.40(164.39) 173.38(173.38)	126.64(126.64)	161.24(161.21) 164.53(164.47)	125.53(125.51)	158.19(158.09)	154.93 (154.75)	125.38(125.17)	156.34(156.16)	150.53(150.20)	125.19(125.09)
	0.3	98.35 (98.35)	102.41(102.41)	75.43 (75.43)	94.39 (94.34)	90.06 (89.99)	74.30 (74.27)	90.56 (90.42)	78.98 (78.73)	74.06 (74.00)	88.57 (88.32)	75.62 (75.19)	74.10 (74.00)
	0.4	60.56 (60.56)	60.89 (60.89)	48.68 (48.68)	56.40 (56.35)	48.20 (48.11)	47.63 (47.61)	52.83 (52.67)	40.53 (40.25)	47.49 (47.43)	51.44 (51.15)	39.65 (39.17)	47.60 (47.51)
	0.5	38.39 (38.39)	36.60 (36.59)	33.49 (33.48)	34.56 (34.51)	26.20 (26.11)			22.66	32.53 (32.46)	31.36 (31.07)	23.31 (22.87)	32.69 (32.59)
	0.6	25.06 (25.06)	22.36 (22.36)	24.26 (24.26)	21.88 (21.83)	15.11 (15.03)	23.49 (23.46)	20.33 (20.17)	14.21 (13.97)	23.52 (23.46)	20.40 (20.12)	15.37 (15.01)	23.71 (23.62)
	0.7	16.86 (16.86)	13.99 (13.99)	18.34 (18.33)	14.42 (14.37)	9.49 (9.42)	17.70 (17.67)	13.75 (13.60)	9.82 (9.63)	17.79 (17.74)	14.14 (13.90)	10.95 (10.68)	17.99 (17.91)
	0.8	11.69 (11.69)	9.03 (9.03)	14.36 (14.35)	9.94 (9.90)	6.49 (6.43)	13.84 (13.81)	9.84 (9.71)	7.24 (7.09)	13.97 (13.91)	10.33 (10.15)	8.08 (7.91)	14.16 (14.09)
	0.9	8.36 (8.35)	6.06 (6.06)	11.57 (11.57)	7.18 (7.14)	4.76 (4.71)	11.16 (11.13)	7.38 (7.28)	5.50 (5.41)	11.31 (11.26)	7.82 (7.70)	6.02 (5.93)	11.48 (11.42)
	1.0	6.16 (6.15)	4.25 (4.25)	9.57 (9.57)	5.41 (5.38)	3.68 (3.64)	9.23 (9.21)	5.73 (5.65)	4.23 (4.18)	9.39 (9.34)	6.05 (5.98)	4.49 (4.43)	9.55 (9.50)
	1.1	4.67 (4.67)	3.13 (3.12)	8.08 (8.07)	4.24 (4.21)	2.93 (2.91)	7.81 (7.79)	4.55 (4.50)	3.27 (3.25)	7.97 (7.93)	4.75 (4.71)	3.38 (3.36)	8.11 (8.07)
	1.2	3.66 (3.65)	2.41 (2.41)	6.94 (6.94)	3.42 (3.40)	2.39 (2.38)	6.73 (6.71)	3.68 (3.65)	2.57 (2.56)	6.89 (6.86)	3.78 (3.77)	2.60 (2.59)	7.00 (6.97)
	1.3	2.94 (2.94)	1.95 (1.95)	6.06 (6.06)	2.84 (2.82)	1.99 (1.98)	5.90 (5.88)	3.01 (3.00)	2.06 (2.06)	6.04 (6.02)	3.06 (3.05)	2.07 (2.07)	6.14 (6.11)
	1.4	2.43 (2.43)	1.64 (1.64)	5.36 (5.36)	2.40 (2.39)	1.69 (1.68)	5.24 (5.22)	2.50 (2.50)	1.71 (1.71)	5.37 (5.35)	2.52 (2.52)	1.71 (1.71)	5.45 (5.43)
	1.5	2.06 (2.06)	1.43 (1.43)	4.80 (4.80)	2.07 (2.06)	1.47 (1.47)	4.70 (4.69)	2.12 (2.12)	1.47 (1.47)	4.83 (4.81)	2.13 (2.13)	1.48 (1.48)	4.89 (4.87)
	1.6	1.79 (1.79)	1.29 (1.29)	4.34 (4.34)	1.81 (1.80)	1.31 (1.31)	4.27 (4.26)	1.83 (1.83)	1.31 (1.31)	4.38 (4.36)	1.84 (1.84)	1.31 (1.31)	4.43 (4.42)
	1.7	1.59 (1.58)	1.19 (1.19)	3.97 (3.97)	1.61 (1.60)	1.20 (1.21)	3.91 (3.90)	1.62 (1.62)	1.20 (1.20)	4.01 (3.99)	1.62 (1.62)	1.21 (1.21)	4.04 (4.04)
	1.8	1.44 (1.43)	1.13 (1.13)	3.65 (3.65)	1.45 (1.45)	1.13 (1.13)	3.61 (3.60)	1.46 (1.46)	1.13 (1.13)	3.69 (3.68)	1.46 (1.46)	1.13 (1.13)	3.72 (3.72)
	1.9	1.32 (1.32)	1.08 (1.08)	3.38 (3.38)	1.33 (1.33)	1.08 (1.08)	3.35 (3.34)	1.33 (1.33)	1.08 (1.09)	3.43 (3.42)	1.33 (1.33)	1.09 (1.09)	3.45 (3.44)
	2.0	1.23 (1.23)	1.05 (1.05)	3.15 (3.15)	1.24 (1.24)	1.05 (1.05)	3.13 (3.13)	1.24 (1.24)	1.05 (1.06)	3.20 (3.19)	1.24 (1.24)	1.05 (1.05)	3.22 (3.21)
	2.1	1.17 (1.17)	1.03 (1.04)	2.96 (2.96)	1.17 (1.17)	1.03 (1.03)	2.95 (2.94)	1.17 (1.17)	1.03 (1.03)	3.00 (3.00)	1.17 (1.17)	1.03 (1.02)	3.01 (3.01)
	2.2	1.12 (1.12)	1.02 (1.02)	2.79 (2.79)	1.12 (1.12)	1.02 (1.02)	2.78 (2.77)	1.12 (1.12)	1.02 (1.02)	2.83 (2.83)	1.12 (1.12)	1.02 (1.02)	2.84 (2.84)
	2.3	1.08 (1.08)	1.01 (1.01)	2.64 (2.64)	1.09 (1.09)	1.01 (1.01)	2.64 (2.63)	1.08 (1.08)	1.01 (1.01)	2.70 (2.68)	1.09 (1.09)	1.01 (1.01)	2.69 (2.68)
	2.4	1.06 (1.06)	1.01 (1.01)	2.51 (2.51)	1.06 (1.06)	1.01 (1.01)	2.51 (2.51)	1.06 (1.06)	1.01 (1.01)	2.55 (2.54)	1.06 (1.06)	1.01 (1.01)	2.55 (2.55)
	2.5	1.04 (1.04)	1.00 (1.00)	2.39 (2.39)	1.04 (1.04)	1.00 (1.00)	2.40 (2.39)	1.04 (1.04)	1.00 (1.00)	2.43 (2.43)	1.03 (1.03)	1.00 (1.00)	2.43 (2.43)
	AEQL	61.15 (61.13)	52.10(52.09)	102.66 (102.64)	57.92 (57.79)	46.45 (46.32)	100.96 (100.74)	57.66 (57.41)	45.98 (45.73)	102.59 (102.30)	58.11 (57.81)	46.74 (46.40)	103.09 (103.00)
	%Diff	0.03%	0.02%	0.02%	0.22%	0.28%	0.22%	0.44%	0.55%	0.28%	0.52%	0.73%	0.09%

Table 7

 7 The unconditional zero-state (and steady-state, in brackets) AEQL values of the upper one-sided IRR 2-of-(h+1) , SRR 2-of-(h+1) and basic precedence schemes for the N(0,1), t(5) and GAM(1,1) distributions under different types shifts when h ∈{1,2,5,10,15} Note that 𝛿 ∈ [𝛿 𝑚𝑖𝑛 , 𝛿 𝑚𝑎𝑥 ] with 𝛿 𝑚𝑖𝑛 =0; 𝛿 𝑚𝑎𝑥 =0.7 for 'small only', 𝛿 𝑚𝑎𝑥 =1.5 for 'small to moderate', and 𝛿 𝑚𝑎𝑥 =2.5 for 'small to large'.

	N(0,1)	t(5)	GAM(1,1)

#

Table 8

 8 The unconditional zero-state (and steady-state, in brackets) AEQL values of the upper one-sided IRR 2-of-(h+1) , SRR 2-of-(h+1) and basic precedence schemes for the N(0,1), t(5) and GAM(1,1) distributions under different types shifts when w ∈{2,5,10,15}

	N(0,1)	t(5)	GAM(1,1)

Table 9

 9 The unconditional ZSARL profiles of the two-sided side-sensitive IRR 2-of-(h+1) and IRR w-of-w precedence schemes for h∈{1,5}, w∈{2,5}and (𝛿 𝑚𝑖𝑛 ,𝛿 𝑚𝑎𝑥 )=(0,2) when m = 500 and n = 5 for a nominal 𝑍𝑆𝐴𝑅𝐿 0 of 500𝑎 1 = 80, 𝑏 1 = 421, 𝑏 2 = 427 𝑎 2 = 53, 𝑎 1 = 60, 𝑏 1 = 441, 𝑏 2 = 448 𝑎 2 = 145, 𝑎 1 = 163, 𝑏 1 = 338, 𝑏 2 = 356

			𝒉 =1 (i.e. 𝒘 = 2)		h=5			w=5
	𝜹	N(0,1)	t(5)	GAM(1,1) N(0,1)	t(5)	GAM(1,1)	N(0,1)	t(5)	GAM(1,1)
	0.0	503.16 503.16	503.16	517.51	517.51	517.51	499.55	499.55	499.55
	0.2	231.30 211.94	418.48	217.16	215.24	458.66	120.33	84.52	223.71
	0.4	64.50	50.79	145.55	53.38	49.13	160.23	14.44	7.33	22.17
	0.6	21.18	14.60	52.85	16.53	13.63	59.45	3.21	1.97	3.01
	0.8	8.48	5.47	20.47	6.73	5.18	23.76	1.62	1.30	1.40
	1.0	4.28	2.73	8.40	3.53	2.71	10.33	1.26	1.12	1.10
	1.2	2.51	1.71	3.87	2.25	1.77	5.19	1.13	1.06	1.02
	1.4	1.74	1.31	2.03	1.67	1.36	3.01	1.06	1.02	1.01
	1.6	1.35	1.12	1.28	1.36	1.16	1.95	1.03	1.00	1.00
	1.8	1.17	1.05	1.04	1.19	1.07	1.41	1.01	1.00	1.00
	2.0	1.08	1.02	1.00	1.09	1.03	1.13	1.00	1.00	1.00
	AEQL	27.75	21.74	50.38	24.61	21.59	59.19	12.10	10.27	14.41
	Charting	𝑎 2 = 74,
	constant

# Note that 𝛿 ∈ [𝛿 𝑚𝑖𝑛 , 𝛿 𝑚𝑎𝑥 ] with 𝛿 𝑚𝑖𝑛 =0; 𝛿 𝑚𝑎𝑥 =0.7 for 'small only', 𝛿 𝑚𝑎𝑥 =1.5 for 'small to moderate', and 𝛿 𝑚𝑎𝑥 =2.5 for 'small to large'.

Acknowledgements

The authors would like to convey their gratitude to the two reviewers as well as the editorial team for taking their valuable time to evaluate and give us constructive suggestions to improve the manuscript.

Practitioners and operaters in industrial or non-industrial environments are adviced to use the proposed one-sided IRR 2-of-(h+1) and IRR w-of-w precedence schemes over the one-sided SRR 2-of-(h+1) , SRR w-of-w and basic precedence schemes regardless of the situation. Thus, for "small" shifts in the location process parameter, we recommend h = 1 & w = 3 or 4 under symmetrical distributions and h = 1 & w = 3, 4 or 5 under heavy-tailed and skewed distributions. For "small and moderate" shifts as well as "small to large shifts", we recommend h = 1 and w = 2, 3 or 4) regardless of the nature of the underlying distribution. Since the SPM literature recommends the use of small values of h or w for simplicity in the design of monitoring schemes supplemented with runs-rules, it is recommended to use the IRR 2-of-(h+1) or IRR w-of-w precedence scheme two reasons, which are: (i) simplicity in the design and implementation of monitoring schemes, (ii) higher efficiency in monitoring quality processes regardless of the size of shifts and nature of the underlying process distribution. For the sake of completeness, a discussion on the two-sided IRR 2-of-(h+1) and IRR w-of-w precedence schemes is provided.

In future, the performance of the one-sided 2-of-(h+1) and w-of-w precedence schemes will be investigated using improved modified runs-rules and the improved modified side-sensitive design for synthetic schemes will be investigated.

Appendix: Transformation functions

Recall that in Sections 2 and 4 it is stated that for the precedence scheme, its performance in terms of the characteristics of the RL such as 𝐴𝑅𝐿 depends on the Phase I and Phase II probability distributions only through the transformation function 𝛹(𝑢). Once the latter is specified, the ARL can be calculated for specified values of 𝑛, 𝑗, 𝑏 1 and 𝑏 2 (𝑎 1 and 𝑎 2 ). In this Appendix, we show how to get the transformation function for the underlying process distributions considered in this paper.

Conversion function under the normal distribution

If 𝐹~N(0,1) and 𝐺~𝑁(𝛿, 1). Then, ѱ(𝑢) can be determined as follows

where 𝜇 1 and 𝜇 2 (with 𝜇 2 = 𝜇 1 + 𝛿 and 𝜇 1 = 0) represent the location parameters (or means) of the Phase I and Phase II samples, respectively.

Since ѱ(𝑢) = 𝐺𝐹 -1 (𝑢) it follows that

Thus,

where 𝛿 represents a shift in the process mean. Therefore, for an IC process 𝛿 = 0 and it follows that

Conversion function under the gamma distribution

If 𝐹~𝐺𝐴𝑀(1,1)and 𝐺~𝐺𝐴𝑀(1, 𝛽) with 𝛽 ≠ 1. Then, ѱ(𝑢) can be determined as follows

and

Then,

A shift in the mean is given by 𝛿 = 𝜇 2 -𝜇 1 , that is, 𝛽 -1 = 𝛿, then 𝛽 = 𝛿 + 1. Therefore,

ln(1 -𝑢)).

(A.9)

For the IC process, 𝛿 = 0. Then,

Note that the transformation functions for other probability distributions can be derived in a similar way.

Conversion function under the Student's t-distribution distribution

If 𝐹~𝐹 𝜈 (𝑥) and 𝐺~𝐹 𝜈 (𝑥 -√2𝛿) where

and

if 𝜈 is even, (A.11) where

) 𝑐 𝑗-1 ; 𝑐 0 = 1 and 𝑑 𝑗 = ( 2𝑗-1 2𝑗

) 𝑑 𝑗-1 ; 𝑑 0 = 1 and 𝜈 is a positive integer.

It can be shown without loss of generality that ѱ(𝑢) is given as follows ѱ(𝑢) = 𝐹 𝜈 (-√2𝛿 + 𝐹 𝜈 -1 (𝑢)).

(A.12)

For the IC process,