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Research article

An empirical assessment of Tukey combined extended exponentially weighted moving average control chart

  • Received: 21 December 2024 Revised: 14 February 2025 Accepted: 24 February 2025 Published: 26 February 2025
  • MSC : 62G05, 62P30

  • Statistical process control (SPC) is a quality control method that enables the monitoring of processes using statistical methodologies. Nonparametric control charts, including the Tukey control chart (TCC), are a robust and effective instrument to assess a method since the actual distribution of the quality characteristic in question is indeterminate. The extended exponentially weighted moving average (EEWMA) control chart was employed to monitor the mean process because of its rapid detection of shifts. To maximize the benefits of both control charts, we developed a method known as EEWMA-TCC, which combines EEWMA with TCC. The efficacy of the proposed chart was evaluated under symmetrical distribution using various individual and aggregate performance metrics based on average run length (ARL) and percentage reduction in ARL (PDARL). Our findings indicated that the suggested chart outperforms control charts, including the TCC chart, the EWMA chart, the EEWMA chart, and the EWMA-TCC (mixed exponentially weighted moving average-Tukey) chart, in the quick identification of shifts. An application of the proposed designs in the crucial dimension of machined part data is demonstrated. The results indicated that they were consistent with the research findings. On the other hand, nonparametric control charts provide an alternate way to track the mean process.

    Citation: Khanittha Talordphop, Yupaporn Areepong, Saowanit Sukparungsee. An empirical assessment of Tukey combined extended exponentially weighted moving average control chart[J]. AIMS Mathematics, 2025, 10(2): 3945-3960. doi: 10.3934/math.2025184

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  • Statistical process control (SPC) is a quality control method that enables the monitoring of processes using statistical methodologies. Nonparametric control charts, including the Tukey control chart (TCC), are a robust and effective instrument to assess a method since the actual distribution of the quality characteristic in question is indeterminate. The extended exponentially weighted moving average (EEWMA) control chart was employed to monitor the mean process because of its rapid detection of shifts. To maximize the benefits of both control charts, we developed a method known as EEWMA-TCC, which combines EEWMA with TCC. The efficacy of the proposed chart was evaluated under symmetrical distribution using various individual and aggregate performance metrics based on average run length (ARL) and percentage reduction in ARL (PDARL). Our findings indicated that the suggested chart outperforms control charts, including the TCC chart, the EWMA chart, the EEWMA chart, and the EWMA-TCC (mixed exponentially weighted moving average-Tukey) chart, in the quick identification of shifts. An application of the proposed designs in the crucial dimension of machined part data is demonstrated. The results indicated that they were consistent with the research findings. On the other hand, nonparametric control charts provide an alternate way to track the mean process.



    The linear complexity and the k-error linear complexity are important cryptographic characteristics of stream cipher sequences. The linear complexity of an N-periodic sequence s={su}u=0, denoted by LC(s), is defined as the length of the shortest linear feedback shift register (LFSR) that generates it [1]. With the Berlekamp-Massey (B-M) algorithm [2], if LC(s)N/2, then s is regarded as a good sequence with respect to its linear complexity. For an integer k0, the k-error linear complexity LCk(s) is the smallest linear complexity that can be obtained by changing at most k terms of s in the first period and periodically continued [3]. The cryptographic background of the k-error linear complexity is that some key streams with large linear complexity can be approximated by some sequences with much lower linear complexity [2]. For a sequence to be cryptographically strong, its linear complexity should be large enough, and its k-error linear complexity should be close to the linear complexity.

    The relationship between the linear complexity and the DFT of the sequence was given by Blahut in [4]. Let m be the order of 2 modulo an odd number N. For a primitive N-th root βF2m of unity, the DFT of s is defined by

    ρi=N1u=0suβiu0iN1. (1.1)

    Then

    LC(s)=WH(ρ0,ρ1,,ρN1), (1.2)

    where WH(A) is the hamming weight of the sequence A. The polynomial

    G(X)=N1i=0ρiXiF2m[X] (1.3)

    is called the Mattson-Solomon polynomial (M-S polynomial) of s [5]. It can be deduced from Eqs (1.2)and (1.3) that the linear complexity of s is equal to the number of the nonzero terms of G(X), namely

    LC(s)=|G(X)|. (1.4)

    By the inverse DFT,

    su=N1i=0ρiβiu=G(βu)0uN1. (1.5)

    There are many studies about two-prime generators. In 1997–1998, Ding calculated the linear complexity and the autocorrelation values of binary Whiteman generalized cyclotomic sequences of order two [6,7]. In 2013, Li defined a new generalized cyclotomic sequence of order two of length pq, which is based on Whiteman generalized cyclotomic classes, and calculated its linear complexity [8]. In 2015, Wei determined the k-error linear complexity of Legendre sequences for k=1,2 [9]. In 2018, Hofer and Winterhof studied the 2-adic complexity of the two-prime generator of period pq [10]. Alecu and Sălăgean transformed the optimisation problem of finding the k-error linear complexity of a sequence into an optimisation problem in the DFT domain, by using Blahut's theorem in the same year [11]. In 2019, in terms of the DFT, Chen and Wu discussed the k-error linear complexity for Legendre, Ding-Helleseth-Lam, and Hall's sextic residue sequences of odd prime period p [12]. In 2020, Zhou and Liu presented a type of binary sequences based on a general two-prime generalized cyclotomy, and derived their minimal polynomial and linear complexity [13]. In 2021, the autocorrelation distribution and the 2-adic complexity of generalized cyclotomic binary sequences of order 2 with period pq were determined by Jing [14].

    This paper is organized as follows. Firstly, we present some preliminaries about Whiteman generalized cyclotomic classes and the linear complexity in Section 2. In Section 3, we give main results about the linear complexity of Whiteman generalized cyclotomic sequences of order two. In Section 4, we give the 1-error linear complexity of these sequences. At last, we conclude this paper in Section 5.

    Let p and q be two distinct odd primes with gcd(p1,q1)=2, and N=pq, e=(p1)(q1)/2. By the Chinese Remainder Theorem, there is a fixed common primitive root g of both p and q such that ordN(g)=e. Let x be an integer satisfying

    x=g(modp)x=1(modq).

    Then the set

    Di={gsximodN:s=0,1,,e1}

    for i=0,1 is called a Whiteman generalized cyclotomic class of order two [15].

    As pointed out in [14], the unit group of the ring ZN is

    ZN={a(mod N):gcd(a,N)=1}={ip+jq(mod N):1iq11jp1}.

    Let P={p,2p,,(q1)p}, Q={q,2q,,(p1)q} and R={0}. Then ZN=ZNPQR. The sequence s(a,b,c)={su}u=0 over F2 is defined by

    su={c,if u=0,a,if uP,b,if uQ,12(1(up)(uq)),if uZN,

    where () denotes the Legendre symbol and a,b,cF2 [14].

    Lemma 2.1. [7] 1D1, if |pq|/2 is odd; and 1D0, if |pq|/2 is even.

    Lemma 2.2. [6]

    (1)Ifp±1(mod8),q±1(mod8)orp±3(mod8),q±3(mod8),then2D0.(2)Ifp±1(mod8),q±3(mod8)orp±3(mod8),q±1(mod8),then2D1.

    Lemma 2.3. [6] (1) If aP, then aP=P and aQ=R.

    (2) If aQ, then aP=R and aQ=Q.

    (3) If aDi, then aP=P, aQ=Q, and aDj=D(i+j)mod2, where i,j=0,1.

    It was shown in [6] that, for a primitive N-th root βF2m of unity, we have

    iPβi=1,iQβi=1,

    and

    iD0βi+iD1βi+iPβi+iQβi=1. (2.1)

    Lemma 2.4. [6]

    uDjβiu={p12(mod2),ifiP,q12(mod2),ifiQ.

    Actually, if p1(mod8) or p3(mod8), then (p1)/2=1; if p1(mod8) or p3(mod8), then (p1)/2=0. By symmetry, if q1(mod8) or q3(mod8), then (q1)/2=1; if q1(mod8) or q3(mod8), then (q1)/2=0.

    Lemma 2.5. Define

    Di(X)=uDiXuF2[X],i=0,1.

    Then for β, we have D0(β)=0 and D1(β)=1 if 2D0; D0(β)=ω and D1(β)=1+ω if 2D1, where ωF4F2.

    Proof. (1) If 2D0, by Lemma 2.3 we have

    [Di(β)]2=Di(β2)=2uDiβ2u=Di(β)F2.

    (2) If 2D1, by Lemma 2.3 we have

    [Di(β)]2=Di(β2)=2uDi+1β2u=Di+1(β),[Di(β)]4=[Di(β)2]2=[Di+1(β)]2=Di+1(β2)=2uDiβ2u=Di(β).

    Hence Di(β)F4F2.

    And by Eq (2.1), we have D0(β)D1(β) and D0(β)+D1(β)=1. Assume that D0(β)=0, D1(β)=1 for 2D0, and D0(β)=ω, D1(β)=1+ω for 2D1, where ωF4F2.

    Let LC(s(a,b,c)) be the linear complexity of s(a,b,c), and the other symbols be the same as before.

    Theorem 3.1. Let pv(mod8) and qw(mod8), where v,w=±1,±3. Then the linear complexity of s(a,b,c) respect to different values of p and q is as shown as Table 1.

    Table 1.  The linear complexity of s(a,b,c).
    s(0,0,0) s(0,0,1) s(0,1,0) s(0,1,1) s(1,0,0) s(1,0,1) s(1,1,0)) s(1,1,1)
    (1,3) or (3,1) pqp pqq+1 pq1 pqpq+2 pqpq+1 pq pqq pqp+1
    (1,3) or (3,1) pq1 pqpq+2 pqp pqq+1 pqq pqp+1 pqpq+1 pq
    (1,1) or (3,3) pqp+q12 pq+pq+12 pq+p+q32 pqpq+32 pqpq+12 pq+p+q12 pq+pq12 pqp+q+12
    (1,1) or (3,3) pq+p+q32 pqpq+32 pqp+q12 pq+pq+12 pq+pq12 pqp+q+12 pqpq+12 pq+p+q12
    (3,1) or (1,3) pqq pqp+1 pqpq+1 pq pq1 pqpq+2 pqp pqq+1
    (1,1) or (3,3) pq+pq12 pqp+q+12 pqpq+12 pq+p+q12 pq+p+q32 pqpq+32 pqp+q12 pq+pq+12

     | Show Table
    DownLoad: CSV

    Proof. We provide the process of calculating LC(s(0,0,0)) when v=1 and w=3, and can prove other cases in a similar way.

    By Lemmas 2.1–2.3 and Eq (1.1), we have 1D1, 2D1, then

    ρi=N1u=0suβiu=uD1βiu=uD0βiu,

    and ρ0=0. By Eq (1.3), we have

    G(X)=N1i=0ρiXi=iD0ρiXi+iD1ρiXi+iPρiXi+iQρiXi+ρ0=iD0uD0βiuXi+iD1uD0βiuXi+iPuD0βiuXi+iQuD0βiuXi.

    Let t=iu. Then by Lemmas 2.3–2.5, we have

    G(X)=iD0tD0βtXi+iD1tD1βtXi+iPp12Xi+iQq12Xi=D0(β)D0(X)+D1(β)D1(X)+iPXi=ωD0(X)+(1+ω)D1(X)+iPXi.

    By Eq (1.4) we can get the linear complexity of s(0,0,0) as

    LC(s(0,0,0))=|G(X)|=pqp.

    Actually, the linear complexity of s(1,0,0) was studied by Ding in [6] with its minimal polynomial.

    Let LCk(s(a,b,c)) be the k-error linear complexity of s(a,b,c), ˜s={˜su}u=0 be the new sequence obtained by changing at most k terms of s, that ˜s=s+e, where e={eu}u=0 is an error sequence of period N. Ding has provided in [2] that, the k-error linear complexity of a sequence can be expressed as

    LCk(s)=minWH(e)k{LC(s+e)}. (4.1)

    It is clearly that LC0(s)=LC(s) and

    NLC0(s)LC1(s)LCl(s)=0,

    where l=WH(s).

    Let G(X), Gk(X) and ˜G(X) be the M-S polynomials of s, e and ˜s respectively. Note that

    G(X)=N1i=0ρiXi, Gk(X)=N1i=0ηiXi, ˜G(X)=N1i=0ξiXi, (4.2)

    where ρi, ηi and ξi are the DFTs of s, e and ˜s respectively. By Eqs (1.5), (4.1) and (4.2), we have ˜G(X)=G(X)+Gk(X), then

    ξi=ρi+ηi. (4.3)

    Assume that eu0=1 for 0u0N1 and eu=0 for uu0 in the first period of e. Then the DFT of e is

    ηi=N1u=0euβiu=βiu00iN1.

    Specially, if u0=0, then ηi=1 for all 0iN1; otherwise, η0=1 and the order of ηi is N for 1iN1.

    Theorem 4.1. Let pv(mod8) and qw(mod8), where v,w=±1,±3, and the other symbols be the same as before. Then the 1-error linear complexity of s(a,b,c) is as shown as Table 2.

    Table 2.  The 1-error linear complexity of s(a,b,c).
    s(0,0,0) and s(0,0,1) s(0,1,0) and s(0,1,1) s(1,0,0) and s(1,0,1) s(1,1,0)) and s(1,1,1)
    (1,3) or (3,1) (1) pqp, if p>q;
    (2) pqq+1, if p<q.
    pqpq+2 pqpq+1 (1) pqp+1, if p>q;
    (2) pqq, if p<q.
    (1,3) or (3,1) pqpq+2 (1) pqp, if p>q;
    (2) pqq+1, if p<q.
    (1) pqp+1, if p>q;
    (2) pqq, if p<q.
    pqpq+1
    (1,1) or (3,3) (1) pqp+q12, if p>q;
    (2) pq+pq+12, if p<q.
    pqpq+32 pqpq+12 (1) pqp+q+12, if p>q;
    (2) pq+pq12, if p<q.
    (1,1) or (3,3) pqpq+32 (1) pqp+q12, if p>q;
    (2) pq+pq+12, if p<q.
    (1) pqp+q+12, if p>q;
    (2) pq+pq12, if p<q.
    pqpq+12
    (3,1) or (1,3) (1) pqp+1, if p>q;
    (2) pqq, if p<q.
    pqpq+1 pqpq+2 (1) pqp, if p>q;
    (2) pqq+1, if p<q.
    (1,1) or (3,3) (1) pqp+q+12, if p>q;
    (2) pq+pq12, if p<q.
    pqpq+12 pqpq+32 (1) pqp+q12, if p>q;
    (2) pq+pq+12, if p<q.

     | Show Table
    DownLoad: CSV

    Proof. We consider the case v=1,w=3 for LC1(s(0,0,0)). By Lemmas 2.1–2.5 and Eq (1.1), we have 1D1, 2D1 and

    ξi=ρi+ηi=uD0βiu+βiu0={ω+βiu0,if iD0,1+ω+βiu0,if iD1,1+βiu0,if iP,βiu0,if iQ,1,if i=0.

    Then by Eq (4.2), we can get

    ˜G(X)=N1i=0ξiXi=iD0(ω+βiu0)Xi+iD1(1+ω+βiu0)Xi+iP(1+βiu0)Xi+iQβiu0Xi+1.

    According to Lemma 2.3, we can get the following results.

    (1) If u0=0, then

    ˜G(X)=iD0(ω+1)Xi+iD1ωXi+iQXi+1,|˜G(X)|=pqq+1.

    (2) If u0Q, then

    ˜G(X)=iD0(ω+βiu0)Xi+iD1(1+ω+βiu0)Xi+iQβiu0Xi+1,|˜G(X)|=pqq+1.

    (3) If u0D0 or u0D1 or u0P, then

    ˜G(X)=iD0(ω+βiu0)Xi+iD1(1+ω+βiu0)Xi+iP(1+βiu0)Xi+iQβiu0Xi+1,|˜G(X)|=pq.

    Compare the results of Cases (1)–(3) with LC(s(0,0,0))=pqp. If p>q, then pqp<pqq+1<pq; if p<q, then pqq+1<pqp<pq. Hence

    LC1(s(0,0,0))={pqp,if p>q,pqq+1,if p<q.

    Similarly we can prove the other cases for LC1(s(a,b,c)).

    All results of LC(s(a,b,c)) and LC1(s(a,b,c)) in Sections 3 and 4 have been tested by MAGMA program.

    The purpose of this paper is to determine the linear complexity and the 1-error linear complexity of s(a,b,c). In most of the cases, s(a,b,c) possesses high linear complexity, namely LC(s(a,b,c))>N/2, consequently has decent stability against 1-bit error. Notice that the linear complexity of some of the sequences above is close to N/2. Then the sequences can be selected to construct cyclic codes by their minimal generating polynomials with the method introduced by Ding [16].

    This work was supported by Fundamental Research Funds for the Central Universities (No. 20CX05012A), the Major Scientific and Technological Projects of CNPC under Grant (No. ZD2019-183-008), the National Natural Science Foundation of China (Nos. 61902429, 11775306) and Shandong Provincial Natural Science Foundation of China (ZR2019MF070).

    The authors declare that they have no conflicts of interest.



    [1] W. A. Shewhart, Economic control of quality of manufactured product, D. Van Nostrand Company Inc., 1931.
    [2] S. W. Roberts, Control chart tests based on geometric moving averages, Technometrics, 1 (1959), 239–250. https://doi.org/10.1080/00401706.1959.10489860 doi: 10.1080/00401706.1959.10489860
    [3] M. B. C. Khoo, A moving average control chart for monitoring the fraction non-conforming, Qual. Reliab. Eng. Int., 20 (2004), 617–635. https://doi.org/10.1002/qre.576 doi: 10.1002/qre.576
    [4] N. Abbas, Homogeneously weighted moving average control chart with an application in substrate manufacturing process, Comput. Ind. Eng., 120 (2018), 460–470. https://doi.org/10.1016/j.cie.2018.05.009 doi: 10.1016/j.cie.2018.05.009
    [5] A. K. Patel, J. Divecha, Modified exponentially weighted moving average (EWMA) control chart for an analytical process data, J. Chem. Eng. Mater. Sci., 2 (2011), 12–20. https://doi.org/10.5897/JCEMS.9000014 doi: 10.5897/JCEMS.9000014
    [6] N. Khan, T. Yasmin, M. Alsam, C. H. Jun, On the performance of modified EWMA charts using resampling schemes, Oper. Res. Decis., 28 (2018), 29–43. https://doi.org/10.5277/ord180303 doi: 10.5277/ord180303
    [7] S. E. Shamma, A. K. Shamma, Development and evaluation of control charts using double exponentially weighted moving averages, Int. J. Qual. Reliab. Manag., 9 (1992), 18–25. https://doi.org/10.1108/02656719210018570 doi: 10.1108/02656719210018570
    [8] M. A. Mahmoud, W. H. Woodall, An evaluation of the double exponentially weighted moving average control chart, Commun. Stat. Simul. Comput., 39 (2010), 933–949. https://doi.org/10.1080/03610911003663907 doi: 10.1080/03610911003663907
    [9] V. Alevizakos, K. Chatterjee, C. Koukouvinos, The triple exponentially weighted moving average control chart, Qual. Technol. Quant. Manag., 18 (2021), 326–354. https://doi.org/ 10.1080/16843703.2020.1809063 doi: 10.1080/16843703.2020.1809063
    [10] M. Naveed, M. Azam, N. Khan, M. Aslam, Design of a control chart using extended EWMA statistic, Technologies, 6 (2018), 108. https://doi.org/10.3390/technologies6040108 doi: 10.3390/technologies6040108
    [11] N. Abbas, M. Riaz, R. J. M. M. Does, Mixed exponentially weighted moving average—cumulative sum charts for process monitoring, Qual. Reliab. Eng. Int., 29 (2013), 345–356. https://doi.org/10.1002/qre.1385 doi: 10.1002/qre.1385
    [12] B. Zaman, M. Riaz, N. Abbas, R. J. M. M. Does, Mixed cumulative sum—exponentially weighted moving average control charts: an efficient way of monitoring process location, Qual. Reliab. Eng. Int., 31 (2015), 1407–1421. https://doi.org/10.1002/qre.1678 doi: 10.1002/qre.1678
    [13] S. Sukparungsee, Y. Areepong, R. Taboran, Exponentially weighted moving average—moving average charts for monitoring the process mean, PLoS One, 15 (2020), e0228208. https://doi.org/10.1371/journal.pone.0228208 doi: 10.1371/journal.pone.0228208
    [14] R. Taboran, S. Sukparungsee, Y. Areepong, Mixed moving average-exponentially weighted moving average control charts for monitoring of parameter change, In: Proceedings of the International MultiConference of Engineers and Computer Scientists (IMECS 2019), Hong Kong, 2019.
    [15] K. Talordphop, S. Sukparungsee, Y. Areepong, New modified exponentially weighted moving average-moving average control chart for process monitoring, Connect. Sci., 34 (2022), 1981–1998. https://doi.org/10.1080/09540091.2022.2090513 doi: 10.1080/09540091.2022.2090513
    [16] M. Naveed, M. Azam, M. S. Nawaz, M. Saleem, M. Aslam, M. Saeed, Design of moving average chart and auxiliary information based chart using extended EWMA, Sci. Rep. , 13 (2023), 5562. https://doi.org/10.1038/s41598-023-32781-4 doi: 10.1038/s41598-023-32781-4
    [17] S. F. Yang, J. S. Lin, S. W. Cheng, A new nonparametric EWMA sign control chart, Expert Syst. Appl., 38 (2011), 6239–6243. https://doi.org/10.1016/j.eswa.2010.11.044 doi: 10.1016/j.eswa.2010.11.044
    [18] S. F. Yang, S. W. Cheng, A new nonparametric CUSUM mean chart, Qual. Reliab. Eng. Int., 27 (2011), 867–875. https://doi.org/10.1002/qre.1171 doi: 10.1002/qre.1171
    [19] M. Aslam, M. A. Raza, M. Azam, L. Ahmad, C. H. Jun, Design of a sign chart using a new EWMA statistic, Commun. Stat. -Theor. M., 49 (2020), 1299–1310. https://doi.org/10.1080/03610926.2018.1563163 doi: 10.1080/03610926.2018.1563163
    [20] K. Talordphop, S. Sukparungsee, Y. Areepong, Design and analysis of extended exponentially weighted moving average signed-rank control charts for monitoring the process mean, Mathematics, 11 (2023), 4482. https://doi.org/10.3390/math11214482 doi: 10.3390/math11214482
    [21] K. Talordphop, S. Sukparungsee, Y. Areepong, Design of nonparametric extended exponentially weighted moving average—sign control chart, Appl. Sci. Eng. Prog., 18 (2025), 7272. https://doi.org/10.14416/j.asep.2023.12.001 doi: 10.14416/j.asep.2023.12.001
    [22] F. Alemi, Tukeyʼs control chart, Qual. Manag. Health Care, 13 (2004), 216–221. https://doi.org/10.1097/00019514-200410000-00004 doi: 10.1097/00019514-200410000-00004
    [23] Q. U. A. Khaliq, M. Riaz, S. Ahmad, On designing a new Tukey-EWMA control chart for process monitoring, Int. J. Adv. Manuf. Tech., 82 (2016), 1–23. https://doi.org/10.1007/s00170-015-7289-6 doi: 10.1007/s00170-015-7289-6
    [24] M. Riaz, Q. U. A. Khaliq, S. Gul, Mixed Tukey EWMA-CUSUM control chart and its applications, Qual. Technol. Quant. Manag., 14 (2017), 378–411. https://doi.org/10.1080/16843703.2017.1304034 doi: 10.1080/16843703.2017.1304034
    [25] R. Taboran, S. Sukparungsee, Y. Areepong, A new nonparametric Tukey MA-EWMA control charts for detecting mean shifts, IEEE Access, 8 (2020), 207249–207259. https://doi.org/10.1109/ACCESS.2020.3037293 doi: 10.1109/ACCESS.2020.3037293
    [26] R. Taboran, S. Sukparungsee, Y. Areepong, Design of a new Tukey MA-DEWMA control chart to monitor process and its applications, IEEE Access, 9 (2021), 102746–102757. https://doi.org/10.1109/ACCESS.2021.3098172 doi: 10.1109/ACCESS.2021.3098172
    [27] S. Phantu, S. Sukparungsee, A mixed double exponentially weighted moving average—Tukey's control chart for monitoring of parameter change, Thail. Stat. , 18 (2020), 392–402.
    [28] K. Talordphop, S. Sukparungsee, Y. Areepong, Performance of new nonparametric Tukey modified exponentially weighted moving average—moving average control chart, PLoS One, 17 (2022), e0275260. https://doi.org/10.1371/journal.pone.0275260 doi: 10.1371/journal.pone.0275260
    [29] S. Sukparungsee, Asymmetric Tukey's control chart robust to skew and non-skew process observation, Conference: World Academy of Science, Engineering and Technology International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, 7 (2013), 1275–1278. https://doi.org/10.5281/ZENODO.1086603
    [30] Q. U. A. Khaliq, M. Riaz, I. A. Arshad, S. Gul, On the performance of median-based Tukey and Tukey-EWMA charts under rational subgrouping, Sci. Iran. , 28 (2021), 547–556. https://doi.org/10.24200/sci.2019.5470.1289 doi: 10.24200/sci.2019.5470.1289
    [31] Y. Mahmood, M. B. C. Khoo, S. Y. The, S. Saha, On designing TEWMA-Tukey control charts for normal and non-normal processes using single and repetitive sampling schemes, Comput. Ind. Eng., 170 (2022), 108343. https://doi.org/10.1016/j.cie.2022.108343 doi: 10.1016/j.cie.2022.108343
    [32] W. Peerajit, Developing average run length for monitoring changes in the mean on the presence of long memory under seasonal fractionally integrated MAX model, Math. Stat., 11 (2023), 34–50. https://doi.org/10.13189/ms.2023.110105 doi: 10.13189/ms.2023.110105
    [33] W. Peerajit, Accurate average run length analysis for detecting changes in a long-memory fractionally integrated MAX process running on EWMA control chart, WSEAS Trans. Math., 22 (2023), 514–530. https://doi.org/10.37394/23206.2023.22.58 doi: 10.37394/23206.2023.22.58
    [34] M. Riaz, A. S. Abbasi, M. Abid, K. A. Hamzat, A new HWMA dispersion control chart with an application to wind farm data, Mathematics, 8 (2020), 2136. https://doi.org/10.3390/math8122136 doi: 10.3390/math8122136
    [35] D. C. Montgomery, Introduction to statistical quality control, 8 Eds., New York: John Wiley and Sons Inc., 2019.
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