A VSI-EWMA distribution-free control chart for ...

A VSI-EWMA distribution-free control chart for monitoring services quality Phuong Hanh Tran

Kim Phuc Tran

Truong Thu Huong

Department of Economics, Danang Architecture University, VN Division of Artificial Intelligence, Dong A University Research Institute, Dong A University, Da Nang, VN [email protected]

Division of Artificial Intelligence, Dong A University Research Institute, Dong A University, Da Nang, VN LMBA, UMR CNRS 6205, Universite de Bretagne-Sud, Vannes, France [email protected]

Department of Communication Engineering, School of Electronics and Telecommunications Hanoi University of Science and Technology, Hanoi, Vietnam [email protected]

Cédric Heuchenne

Thi Anh Dao Nguyen

Cong Ngon Do

HEC Management School, University of Liège, Liège 4000, Belgium [email protected]

Division of Artificial Intelligence, Dong A University, Danang, Vietnam [email protected]

Division of Artificial Intelligence, Dong A University, Danang, Vietnam [email protected]

ABSTRACT Many data in service quality came from a nonnormal or unknown distribution, hence the commonly-used control charts are not suitable. In this paper, new Arcsine Shewhart Sign and Variable Sampling Interval EWMA (Exponentially Weighted Moving Average) distribution-free control charts are proposed. The procedure does not require the assumption of normal data. A Markov chain method is used to obtain optimal designs and evaluate the statistical performance of the proposed charts. Furthermore, practical guidelines and comparisons with the basic Arcsine EWMA Sign control chart are provided. Results show that the proposed chart is considerably more efficient than the basic Arcsine EWMA Sign control chart. The proposed control charts are illustrated by analysing the service quality of the Vancouver City Call Centre.

CCS CONCEPTS • Mathematics of computing → Markov processes; Nonparametric statistics; • Applied computing → Multi-criterion optimization and decision-making;

KEYWORDS Service quality , Sign statistic, Control chart, EWMA, VSI, Markov chain. ACM Reference Format: Phuong Hanh Tran, Kim Phuc Tran, Truong Thu Huong, Cédric Heuchenne, Thi Anh Dao Nguyen, and Cong Ngon Do. 2018. A VSI-EWMA distributionfree control chart for monitoring services quality. In ICEBA 2018: 2018 International Conference on E-Business and Applications, February 23–25, Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]. ICEBA 2018, February 23–25, 2018, Da Nang, Viet Nam © 2018 Association for Computing Machinery. ACM ISBN 978-1-4503-6368-6/18/02. . . $15.00 https://doi.org/10.1145/3194188.3194191

2018, Da Nang, Viet Nam. ACM, New York, NY, USA, 5 pages. https://doi. org/10.1145/3194188.3194191

1

INTRODUCTION

Statistical Process Control (SPC) is a method of quality control which uses statistical methods in achieving process stability and improving capability through the reduction of variability, see [14]. SPC are widely applied in areas far beyond manufacturing and production. It’s well known that control charts are the fundamental tool for SPC applications. In recent years, many researchers have focused on developing advanced control charts with various applications in manufacturing and service processes, for example, see [9] and [21–23, 27]. There are numerous types of control charts, the most common of charts are the Shewhart control charts, cumulative sum (CUSUM) control charts and the exponentially weighted moving average (EWMA) control charts. The EWMA control charts have a “built in” mechanism for incorporating information from all previous subgroups by means of weights decreasing geometrically with the sample mean age. Thus EWMA type control charts are very effective for small or moderate process shifts, see [24–26] for more details. Unlike the manufacturing sector, the process of providing services includes a number of variables that are difficult to control. According to [7], service is characterised by intangibility, heterogeneity, inseparability, and perishability, all of which make service quality difficult to measure. For this reason, [17] collated 10 dimensions to measure service quality: reliability, responsiveness, competence, access, courtesy, communication, creditability, security, understanding, and tangibles. [18] further explored these dimensions using factor analysis and proposed five major dimensions: tangibility, reliability, responsiveness, assurance, and empathy. These dimensions formed the SERVQUAL questionnaire, which has provided the foundation in a number of investigations on service quality, see [6] for more details. Despite this difficult problem, the literature contains few studies, which discuss the need for innovative control chart methodologies for monitoring services quality. [16] considered a distribution-free phase-II CUSUM control chart to monitor

ICEBA 2018, February 23–25, 2018, Da Nang, Viet Nam the average speed to answer to a customer in a call-centre. Very recently, [15] considered a new nonparametric control chart for monitoring bivariate processes based on a distance measure and its application in monitoring service quality. When the actual distribution of the observations is not that anticipated by the practitioner, the control chart properties can be seriously affected, for example, the false-alarm rate is inflated, see [8]. In this context, the use of a nonparametric (or distribution-free) control chart can be a good solution to the problem as its in-control properties do not depend on any specific parametric distribution. The first work that dealt with a nonparametric control chart was by [13] and [2]. [4] presented an extensive overview of the literature about univariate nonparametric control charts. Then, its properties and design stategies have been thoroughly investigated by many authors. For further details see, [10] [8] [29] [28] [11] [5] and [1]. It is known that, the EWMA-SN control chart suggested by [28] are Fixed Sampling Interval (FSI) control chart. By definition, an adaptive control chart involves varying at least one of the chart’s parameters, such as the sampling interval or sample size. Variable Sampling Interval (VSI) control charts are adaptive control charts where the sampling intervals vary as a function of what is observed from the process. The VSI control charts are demonstrated to detect process changes faster than FSI control charts. The idea is that the time interval until the next sample should be short; if the position of the last plotted control statistic indicates a possible out-of-control situation; and long, if there is no indication of a change. Most work on developing VSI control charts has been done for the problem of monitoring the mean of the process (see [19] and [20]). In this paper, we propose a VSI EWMA-SN control chart as a logical extension of the control chart developed by [28]. The goal of this paper is to show how the VSI behaves with respect to the basic EWMA-SN control chart. In addition, we develop an Arcsine Shewhart Sign control chart for simplicity in application. The rest of this paper proceeds as follows: in Section 2, a brief review of FSI EWMA-SN control chart is provided and Arcsine Shewhart Sign is proposed; Section 3 provides a VSI version of the FSI EWMASN control chart; in Section 4, the computational results of the Arcsine Shewhart Sign, Arcsine EWMA Sign and VSI EWMA-SN charts are presented. Section 5 presents an illustrative example and, finally, some concluding remarks and recommendations are made in Section 6.

2

THE ARCSINE SHEWHART SIGN AND ARCSINE EWMA SIGN CONTROL CHARTS

Let X be a quality characteristic following an unknown continuous distribution with c.d.f. (cumulative distribution function) F X (x |θ ) where θ is the location parameter to be monitored. Without loss of generality, here we consider θ to be the median of the process parameter’s distribution. With reference to the Phase II implementation of the control chart, when the process is in-control we have θ = θ 0 and when the process is out-of-control, we have θ = θ 1 . Let us suppose that, at time i = 1, 2, . . ., we observe subgroup {X i,1 , X i,2 , . . . , X i,n } of size n ≥ 1. By definition, at time i = 1, 2, . . ., the plotting statistic

P.H. Tran et al. SNi is SNi =

n Õ

sign(X i,k − θ 0 )

(1)

k =1

where sign(x) = −1 if x < 0, or x = 0 and sign(x) = 1 if x > 0, respectively. The condition sign(x) = 0 is an occasional event occurring in practice due to rounding-off errors of the measurement system. By definition, the random variable SNi is defined on {−n, −n + 2, . . . , n − 2, n} and its distribution can be easily obtained by considering the relationship SNt = 2D i − n, where D i = #{X i,k > θ 0 , k = 1, . . . , n}, i.e. D t is the number of observations {X i,1 , X i,2 , . . . , X i,n } larger than θ 0 . If the process is in-control, we have P(X i,k < θ 0 |θ = θ 0 ) = P(X i,k > θ 0 |θ = θ 0 ) = p0 = 0.5 and, consequently, D i ∼ Bin(n, 0.5), i.e. a binomial q random variable of Di parameter n and p0 = 0.5. Let Yi = asin n , then distribution √ of Yi would be approximately normal with mean asin( p) and vari1 . The control limits of the Arcsine Shewhart Sign chart are ance 4n simply equal to √ 1 LCL = asin( p0 ) − K √ , (2) 2 n √ 1 (3) UCL = asin( p0 ) + K √ , 2 n where K > 0 is a constant that depends on n and on the desired in-control performance. As with any other Shewhart control chart, the run length of Arcsine Shewhart Sign control chart follows a geometric distribution with parameter p, and Average Run Length (ARL) and the Standard Deviation Run Length (SDRL) can be shown to be 1 , (4) ARL = θ √ 1−θ SDRL = . (5) θ where θ = 2Φ (−K − δ ) . (6) where Φ(.) is the c.d.f.(cumulative distribution function) of the normal (0, 1) distribution and the standardized mean shift δ = √ √ √ 2 n × (asin( p1 ) − asin( p0 )). Let Z 1 , Z 2 , . . . be the EWMA sequence obtained from Y1 , Y2 , . . ., i.e. for i ∈ {1, 2, . . .}, Z i = (1 − λ)Z i−1 + λYi , (7) √ where Z 0 = asin( p0 ) and λ ∈ (0, 1] is a smoothing constant. The control limits of the Arcsine EWMA Sign chart are simply equal to r √ 1 λ LCL = asin( p0 ) − K √ , (8) 2 n 2−λ √ CL = asin( p0 ), (9) r √ 1 λ UCL = asin( p0 ) + K √ , (10) 2 n 2−λ where K > 0 is a constant that depends on n and on the desired in-control performance.

A VSI-EWMA distribution-free control chart for monitoring services quality

3

the 2m + 1 transient states defined above, i.e.

IMPLEMENTATION OF VSI ARCSINE EWMA SIGN CONTROL CHART

In this section, VSI versions of the FSI Arcsine EWMA Sign control chart described in the previous section are presented (denoted as VSI Arcsine EWMA Sign chat). The control statistic Z i for the VSI Arcsine EWMA Sign control chart is given by (7). The upper (UCL) and lower (LCL) control limits of the VSI Arcsine EWMA Sign control chart can be easily calculated:

=

√ 1 asin( p0 ) − K √ 2 n

r

LCL

=

√ 1 asin( p0 ) + K √ 2 n

r

U CL

ICEBA 2018, February 23–25, 2018, Da Nang, Viet Nam

λ , 2−λ

(11)

λ , 2−λ

(12)

where K is a positive constant influencing the width of control interval. For the FSI control chart, the sampling interval is referred to as fixed sampling interval h 0 . As for the VSI control chart, the sampling interval depends on the current value of Z i for VSI Arcsine EWMA Sign control chart. A longer sampling interval h L is used when the control statistic falls within region R L = [LW L, UW L] defined as:

Q · · · Q −m,+m © −m,−m ª .. .. .. ­ ® ­ ® . . . ­ ® ··· Q 0,+m ®® . Q = ­­ Q 0,−m (15) ­ ® . . . ­ ® .. .. .. ­ ® Q · · · Q +m,+m ¬ « +m,−m By definition, we have Q j,k = P(Z i ∈ (Hk − ∆, Hk + ∆]|Z i−1 = H j ) or, equivalently, Q j,k = P(Z i ≤ Hk + ∆|Z i−1 = H j ) − P(Z i ≤ Hk − ∆|Z i−1 = H j ). Replacing Z i = (1 − λ)Z i−1 + λYi , Z i−1 = H j and isolating Yi gives Q j,k = Φ



 Hk + ∆ − (1 − λ)H j −δ λ   Hk − ∆ − (1 − λ)H j −Φ −δ λ

where Φ(.) is the c.d.f.(cumulative distribution function) of the normal (0, 1) distribution and the standardized mean shift δ = √ √ √ 2 n × (asin( p1 ) − asin( p0 )). Let q = (q −m , . . . , q 0 , . . . , qm )T be the (2m + 1, 1) vector of initial probabilities associated with the 2m + 1 transient states, where qj =

LW L UW L

= =

√ 1 asin( p0 ) − W √ 2 n

r

√ 1 asin( p0 ) + W √ 2 n

r

(16)



0 1

if Z 0 < (H j − ∆, H j + ∆] . if Z 0 ∈ (H j − ∆, H j + ∆]

(17)

The AT S can be evaluated through the following expression:

λ , 2−λ

(13)

λ , 2−λ

(14)

where W is the warning limit coefficient of VSI Arcsine EWMA Sign control chart that determines the proportion of times that the control statistic falls within the long and short sampling regions. On the other hand, short sampling interval h S is used when the control statistic falls within the region R S = [LCL, LW L] ∪ [UW L, UCL]. The process is considered out-of-control and action should be taken whenever Z i falls outside the range of the control limits [LCL, U CL]. In order to evaluate the Average Time to Signal (AT S) of the VSI Arcsine EWMA Sign control chart, we follow the discrete Markov chain approach originally proposed by [3] . The Markov chain approach of [3] and [12] is modified to evaluate the Run Length properties of the VSI EWMA-SN chart. This procedure involves dividing dividing the interval [LCL, UCL] into 2m + 1 subintervals (H j − ∆, H j + ∆], j ∈ {−m, . . . , 0, . . . , +m}, centered C L + 2j∆ where 2∆ = U C L−LC L . Each subinterval at H j = LC L+U 2 (2m+1) (H j − ∆, H j + ∆], j ∈ {−m, . . . , 0, . . . , +m}, represents a transient state of a Markov chain. If Z i ∈ (H j − ∆, H j + ∆] then the Markov chain is in the transient state j ∈ {−m, . . . , 0, . . . , +m} for sample i. If Z i < (H j − ∆, H j + ∆] then the Markov chain reached the absorbing state (−∞, LCL] ∪ [UCL, +∞). We assume that H j is the representative value of state j ∈ {−m, . . . , 0, . . . , +m}. Let Q be the (2m + 1, 2m + 1) submatrix of probabilities Q j,k corresponding to

AT S = qT (I − Q)−1 g

(18)

where g is the vector of sampling intervals corresponding to the discretized states of the Markov chain and the jth element дj of the vector g is the sampling interval when the control statistic is in state j (represented by H j ), i.e. дj =



hL hS

if LW L < H j < UW L, otherwise.

(19)

The average sampling interval of the VSI EWMA-SN chart is given as: qT (I − Q)−1 g E(h) = T (20) q (I − Q)−1 1

4

PERFORMANCE STUDY

The performance of the Arcsine Shewhart Sign, FSI and VSI EWMASN control charts, for all combinations of p1 ∈ [0.05, 0.45] and n = {1, 2, . . . , 10}. The sampling interval h F of the FSI charts has been set equal to 1 time unit. The performance of the proposed control chart are presented in the Tables 1, 2 and 3. the results in Tables 1, 2 and 3 clearly indicate that the VSI EWMA-SN chart is superior to the Arcsine Shewhart Sign, FSI EWMA-SN control charts. For instance, when p1 = 0.45, we have ARL 1 = 322.0 for Arcsine Shewhart Sign control chart, we have ARL 1 = 189.4 for Arcsine EWMA Sign control chart and we have AT S 1 = 183.3 for Arcsine VSI EWMA Sign control chart.

ICEBA 2018, February 23–25, 2018, Da Nang, Viet Nam

P.H. Tran et al.

Table 1: The values of out-of-control ARL (ARL 1 ) for n ∈ {1, . . . , 10} and p1 ∈ {0.05, 0.1, . . . , 0.45}, K = 3 ,ARL 0 = 370.4 for Arcsine Shewhart Sign control chart

Figure 1: Arcsine Shewhart Sign control chart for the daily average speed of answer of Vancouver City Call Centre. Arcsine Shewhart Sign chart

0.05 33.3 12.8 6.9 4.5 3.2 2.5 2.1 1.8 1.6 1.4

0.10 52.2 21.9 12.2 7.9 5.6 4.3 3.4 2.8 2.4 2.1

0.15 76.1 35.1 20.5 13.6 9.7 7.4 5.8 4.8 4.0 3.4

0.20 106.9 54.5 33.7 23.1 16.9 12.9 10.3 8.4 7.0 6.0

0.30 194.4 125.4 89.3 67.5 53.1 43.1 35.7 30.1 25.8 22.4

0.35 250.5 185.2 144.5 116.8 96.9 82.0 70.5 61.3 53.9 47.9

0.40 307.7 261.7 226.5 198.7 176.4 158.0 142.6 129.6 118.4 108.7

0.45 352.9 336.8 322.0 308.3 295.5 283.7 272.7 262.5 252.9 243.8

U CL

1.2

1

Yt

n 1 2 3 4 5 6 7 8 9 10

p1 0.25 146.0 83.2 54.9 39.3 29.7 23.3 18.8 15.5 13.1 11.2

0.8

0.6

0.4

Table 2: The values of out-of-control ARL (ARL 1 ) for n ∈ {1, . . . , 10} and p1 ∈ {0.05, 0.1, . . . , 0.45}, K = 0.9533, λ = 0.2, ARL 0 = 370.4 for Arcsine EWMA Sign control chart n 1 2 3 4 5 6 7 8 9 10

0.05 8.1 4.9 3.7 3.1 2.8 2.5 2.3 2.2 2.1 2.0

0.10 11.2 6.4 4.8 3.9 3.4 3.1 2.8 2.6 2.5 2.4

0.15 15.5 8.4 6.1 5.0 4.3 3.8 3.5 3.2 3.0 2.8

0.20 22.1 11.5 8.2 6.5 5.5 4.9 4.4 4.0 3.8 3.5

p1 0.25 33.1 16.9 11.6 9.1 7.6 6.6 5.9 5.3 4.9 4.6

0.30 52.6 26.9 18.1 13.8 11.3 9.7 8.5 7.6 7.0 6.4

0.35 89.3 48.2 32.6 24.6 19.8 16.6 14.4 12.7 11.5 10.4

0.40 160.8 99.6 71.0 54.7 44.3 37.2 32.0 28.0 25.0 22.5

0.45 282.7 227.4 189.4 161.8 140.8 124.4 111.2 100.4 91.4 83.8

Table 3: The values of out-of-control AT S (AT S 1 ) for n ∈ {1, . . . , 10} and p1 ∈ {0.05, 0.1, . . . , 0.45}, W = 0.9,K = 2.8595, h S = 0.5 h L = 1.2806 ,λ = 0.2, AT S 0 = 370.4 for Arcsine VSI EWMA Sign control chart

n 1 2 3 4 5 6 7 8 9 10

5

0.05 6.1 3.8 3.0 2.6 2.3 2.1 2.0 1.9 1.9 1.8

0.10 8.4 4.8 3.7 3.1 2.8 2.5 2.4 2.2 2.1 2.0

0.15 11.7 6.3 4.7 3.9 3.4 3.0 2.8 2.6 2.5 2.4

0.20 17.2 8.6 6.1 5.0 4.3 3.8 3.5 3.2 3.0 2.8

p1 0.25 26.9 12.8 8.7 6.8 5.7 5.0 4.5 4.1 3.8 3.6

0.30 45.1 21.3 13.9 10.4 8.5 7.2 6.4 5.8 5.3 4.9

0.35 81.3 40.9 26.4 19.3 15.2 12.6 10.8 9.6 8.6 7.8

0.40 153.9 91.6 63.1 47.2 37.3 30.6 25.9 22.4 19.7 17.6

0.45 279.9 222.6 183.3 154.9 133.4 116.6 103.3 92.4 83.3 75.7

ILLUSTRATIVE EXAMPLE

In this Section an Arcsine Shewhart Sign control chart to monitor a real quality control problem from the Vancouver City Call Centre data. The context of the example presented here is similar to the one introduced in [16]. The centre statistical data are freely available from the official website (http://vancouver.ca/yourgovernment/311-contact-centre-statistical-data.aspx) of Vancouver. It is important to monitor the daily average speed of answer, specially its median and variability to protect every drop in service

LCL

0.2 1

5

10

15

20

25

30

35

40

45

50

t

quality. Monitoring is recommended at least once a week (n = 7). The data between 1 January 2013 and 15 November 2016 may be considered to be the Phase I or historical data. Then, 350 data points reported in the website of the City of Vancouver between 16 November 2016 and 31 Octorber 2017 is used for phase II. The Yt and the control limits of Arcsine Shewhart Sign control chart are plotted in Figure 1. This figure confirms that there is no abnormality found during this period.

6

CONCLUDING REMARKS

We presented Arcsine Shewhart Sign and Variable Sampling Interval EWMA distribution-free control charts. We have also studied the statistical properties of the Variable Sampling Interval EWMAcontrol chart for several shift sizes. For fixed values of the shift size δ several tables presenting the out-of-control AT S corresponding to many different scenarios. Also, the numerical comparison with the performance of the EWMA-SN control chart proposed by [28] shows that the detection ability of the VSI EWMA-SN control chart are better than the EWMA-SN control chart. Thus, the proposed chart can be used as a best alternative method. We also provide an illustrative example to demonstrate the implementation steps of our proposed charts in the context of monitoring a real quality control problem from the Vancouver City Call Centre data.

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