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Key words: Adaptive control chart, Cost function, Optimal design, Multiple assignable ... Shewhart control charts constitute an effective on-line process control ...
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Economic Quality Control Vol 22 (2007), No. 2, 273 – 293

Economic Design of A Modified Variable ¯ Chart Sample Size and Sampling Interval X Shashibhushan B. Mahadik and Digambar T. Shirke

Abstract: A comprehensive economic cost function is derived for a modified variable sample size and sampling interval X (MVSSI) chart. As a variable sample size and sampling interval, a variable sample size, a variable sampling interval, and a static X charts are particular cases of an MVSSI X chart, the model is directly applicable to these charts, too. The cost function can be used to obtain optimal design parameters of these charts that minimize the process control related cost per hour for given process and cost parameters. It can also be used to compare the economic performances of these charts. A numerical study shows that the optimal adaptive X charts differ in their performance only marginally from each other. However, both the statistical and economic performances of an optimal MVSSI X chart are superior to that of the other adaptive X charts. A sensitivity analysis for a particular example reveals that the optimal design obtained using the cost function is relatively insensitive to errors in specifying various process and cost parameters. Key words: Adaptive control chart, Cost function, Optimal design, Multiple assignable causes, Statistical process control

1

Introduction

Shewhart control charts constitute an effective on-line process control technique used to monitor and control the variability in a production process. The implementation of the control chart for a particular process requires determination of the three chart parameters, viz, the sampling interval, the sampling size, and the control limit(s). These parameters are called the design parameters of the control chart and are generally specified according to statistical or/and economic criteria. The statistical performance of a control chart generally refers to the time required by it to detect an out-of-control condition and the economic performance refers to the process control related cost due to the chart. In a statistically designed control chart the design parameters are chosen, such that the chart meets some statistical performance requirements, while the minimization of the net sum of all costs involved yields an economic design. For a economic-statistical design the parameters are chosen to minimize the costs subject to some constraints on the statistical performance. For a traditional control chart, all the design parameters are fixed during the entire period of monitoring. In that sense the traditional chart is static. A control chart is termed to be

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adaptive, if at least one of the three design parameters may change its value for the next sample depending on the location of the current sample point on the chart. Reynolds, Amin, Arnold, and Nachlas [22] were the first to introduce the idea of an adaptive control chart. They proposed a variable sampling interval (VSI) X chart, under the assumption of known and constant variance. Further, Prabhu, Runger, and Keats [20] and Costa [8] independently proposed a variable sample size (VSS) X chart. Prabhu, Montgomery, and Runger [19] proposed a variable sample size and sampling interval (VSSI) X chart. Costa [9] proposed a variable parameter (VP) X chart. Mahadik and Shirke [15] proposed a modified variable sample size and sampling interval (MVSSI) X chart. In all of these papers, the statistical performances of the proposed charts have been extensively studied and it has been shown that adapting design parameters of a control chart improves the statistical performance significantly. A VSSI X chart proposed by Prabhu et al. [19] uses two sampling intervals, two sample sizes, and a single set of threshold limits to switch between pairs of the adaptive parameters. A threshold limit is between a control limit and the centerline on each side of the centerline. If the current sample point falls in a warning region, that is, between a threshold limit and a control limit, the large sample size and the shorter sampling interval is used to take the next sample. On the other hand, if the current sample point falls in the central region, the small sample size and the longer sampling interval is used to take the next sample. A VSSI X chart that uses two different sets of threshold limits, one to switch between the sampling intervals and the other to switch between the sample sizes has also been studied in the literature. However, the performance of such a chart does not differ significantly from the VSSI X chart that uses a single set of threshold limits for both the parameters (see, e. g., Park and Reynolds [18]). In this paper, by a VSSI X chart we refer to the one that uses a single set of threshold limits. Among the VSI, VSS, and VSSI X charts, a VSI X chart is superior in detecting large shifts in the process mean, while a VSSI X chart is superior in detecting small shifts with respect to the statistical performance. Both the VSI and VSSI X charts perform almost equally for detecting moderate shifts. Neither of the two is uniformly superior to the other to detect a shift of any magnitude over a wide range. An MVSSI X chart is an effective alternative to the VSI and VSSI X charts when it is needed to detect quickly both small as well as large shifts (see, Mahadik and Shirke [15]). It detects small shifts as quickly as a VSSI X chart does, and large shifts as quickly as a VSI X chart does. Further, it is as quick as the VSI and VSSI X charts for detecting moderate shift. However, it is also necessary to investigate, whether these advantages of an MVSSI X chart also yield a better economic performance. Hence, in this paper, we develop an economic objective function for an MVSSI X chart that can be used to evaluate the economic performance of the chart. The objective function can also be used to obtain the economically optimal set of the design parameters of the chart. A large number of researchers have studied the economic design of the static control charts (see Montgomery [16], Collani [3, 4, 6] and Ho and Case [12]). Lorenzen and Vance [13] provided a unified approach to obtain the economic design of static Shewhart control chart. Collani [5, 7] developed another unified approach, which unifies different

¯ Chart Economic Design of A Modified Variable Sample Size and Sampling Interval X

275

theories, reduces significantly the number of input variables, results in simpler objective functions, and permits approximation algorithms to be used. A good amount of work has also been done on the economic designs of the adaptive control charts. Park and Reynolds [17] studied the economic design of a VSS X chart. Das, Jain, and Gosavi [11], Bai and Lee [1], and Chen [2] investigated the economic performance of a VSI X chart with a single assignable cause model. Yu and Hou [25] considered the economic design of a VSI X chart with multiple assignable causes. Prabhu, Montgomery, and Runger [21] studied the constrained economic-statistical design of a VSSI X chart with a single assignable cause model. The constraints ensure matching of the compared static and VSSI X charts with respect to the sampling requirements when the process is in control and thereby fix two of the six design parameters. Park and Reynolds [18] considered the economic design of a VSSI X chart with multiple assignable causes. Costa and Rahim [10] investigated the economic performance of a VP X chart with a single assignable cause model using a Markov chain approach. Further, Magalhaes, Costa, and Epprecht [14] studied the economic-statistical design and the economic design of a VP X chart with a single assignable cause model. Tagaras [24] has given an extensive survey of the research work on adaptive control charts published until late 1997. The economic objective function developed in this paper is essentially a generalization of the unified approach proposed by Lorenzen and Vance [13] towards the economic design of a static control chart with a single assignable cause to an adaptive X chart with multiple assignable causes. Section 2 describes an MVSSI X chart. The economic objective function is developed in Section 3. In Section 4, we discuss the optimization procedure to determine the optimal design parameters and also compare the performances of the optimal static, VSI, VSS, VSSI, and MVSSI X charts. It may be difficult to specify some of the process and cost parameters in the function for a particular process. Hence, sensitivity analysis is done in Section 5 to study the sensitivity of the optimal design parameters and the sensitivity of the minimum expected cost per hour to the errors in specifying the process and cost parameters. Conclusions are summarized in Section 6.

2

An MVSSI X CHART

We assume the normal approximation with mean µ and a known and constant standard deviation σ for the quality characteristic to be monitored. Let µ0 be the target value of µ. Let X i − µ0 (1) Zi = √σ n(i)

be the control chart statistic used to monitor the process mean, where X i is the mean of ith subgroup with size n(i), i = 1, 2, . . .. Let t(i) be the sampling interval between (i − 1)st and ith subgroups.

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Shashibhushan B. Mahadik and Digambar T. Shirke

An MVSSI X chart uses two sampling intervals t1 and t2 , such that tmax ≥ t1 ≥ t2 ≥ tmin ; three sample sizes n1 , n0 , and n2 , such that nmin ≤ n1 ≤ n0 ≤ n2 ≤ nmax ; and two sets of threshold limits (−ω1 , ω1 ) and (−ω2 , ω2 ) such that 0 < ω1 ≤ ω2 < L, where tmin and tmax are respectively the minimum and maximum feasible sampling interval lengths, nmin and nmax are respectively the minimum and maximum feasible sample sizes, and −L and L are respectively the fixed lower and upper control limits. The threshold limits partition the in-control area of the chart into three regions as follows: I1 = [−ω1 , ω1 ] I2 = [−ω2 , −ω1 ) ∪ (ω1 , ω2 ] I3 = (LCL, −ω2 ) ∪ (ω2 , U CL)

(2) (3) (4)

When (i − 1)st sample point falls within the control limits, the decision to choose the values of the adaptive parameters for the ith sample, i = 2, 3, . . ., depends on the location of (i − 1)st sample point. The values are chosen according to the adaptive parameter function,   (n1 , t1 ) , if Zi−1 ∈ I1 (n0 , t2 ) , if Zi−1 ∈ I2 (n(i), t(i)) = (5)  (n2 , t2 ) , if Zi−1 ∈ I3 The process is treated to be out-of-control, when Zi falls outside the control limits. Figure 1 shows a typical MVSSI X chart. A pair of values of the adaptive parameters that indicated for a particular region applies to the next sample when the current sample point falls in that region. At start-up the pair of values of adaptive parameters for the first sample is chosen using the in-control (that is, when µ = µ0 ) probability distribution of (n(i), t(i)), as no prior sample point is available.

Figure 1: An MVSSI X chart We note that, when ω1 = ω2 , an MVSSI X chart matches a VSSI X chart; when the sample size is fixed, it matches a VSI X chart; when the sampling interval is fixed and

¯ Chart Economic Design of A Modified Variable Sample Size and Sampling Interval X

277

ω1 = ω2 , it matches a VSS X chart, and when both the sampling interval and the sample size are fixed, it matches a static X chart. Thus, an MVSSI X chart provides a unified approach to determine the properties of the VSSI, VSS, VSI, and static X charts. Below we develop a comprehensive economic objective function for this chart.

3

The Economic Objective Function

Define the quality cycle as the time between the starts of successive in-control periods. There are costs during the in-control period due to sampling, nonconformities produced, and false alarms. Suppose there are m possible assignable causes A1 , A2 , . . . , Am that can occur and the occurrence of assignable cause Aj , j = 1, 2, . . . , m, produces a shift in the process mean of magnitude δj σ. We assume that when an assignable cause Aj occurs and shifts the process to an out-of control state, the process remains in that state and can not be disturbed by another assignable cause until the chart signals the out-of control condition and Aj is removed. External intervention is required to remove Aj . Suppose from historical data it is known that given an assignable cause has occurred, the conditional probability that it is Aj is pj . Thus, when an assignable cause occurs, the magnitude of the shift is δj σ with probability pj , j = 1, 2, . . . , m. During the out-of-control period there are costs due to sampling, increased level of nonconformities produced, searching for the cause, repairing the system, and downtime, if any has occurred. Upon repairing the system, one quality cycle has been completed and the next cycle begins. An entire cycle is represented in Figure 2.

Cycle starts t

Assignable cause occurs t

in-control  -

chart signals

Assignable cause detected

t

t

out-of-control

Assignable cause removed t -

Figure 2: Quality Cycle For the quality cycle introduced above, we derived the expected cost per hour, when an MVSSI X chart is applied to the process. Using a result from renewal reward process theory (see, Ross [23]), the expected cost per hour is obtained as the ratio of the expected cost per cycle to the expected cycle time in hours. Below, we first derive an expression for the expected cycle time in hours. The cycle time is the sum of the following time components. a) the time until an assignable cause occurs (i.e. the in-control period),

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Shashibhushan B. Mahadik and Digambar T. Shirke

b) the time until the chart gives an out-of-control signal, c) the time to analyze the sample that produces the out-of-control signal (signaling sample) and chart the result, and d) the time to discover the assignable cause and repair the process. As usual, the exponential probability distribution with mean λ1 is assumed for the incontrol period implying that if the process is continued during the search state, the expected time until the occurrence of an assignable cause is simply λ1 . If the production ceases during the search state, the expected time until the occurrence of an assignable cause is λ1 plus the expected time spent in searching false alarms. Let T0 be the average search time for a false alarm. Then the expected time spent in searching during false alarms is αsT0 , where α = 2[1−Φ(L)] is the probability of a false alarm, Φ is the standard normal distribution function, L is the UCL of the chart, and s is the expected number of samples during the in-control period. Thus, if the production ceases during the search state, the expected time equals λ1 + αsT0 . The expression for s is derived in the Appendix. Let 1 = 1, if production continues during searches and 1 = 0, if production ceases during searches. Then the expected time until an assignable cause occurs is 1 (6) + (1 − 1 ) αsT0 λ Let T be the expected value of the time from the time of a shift until the chart signals an out-of-control condition. then T is given by: m   Tj pj (7) T = j=1 

where Tj is the conditional expected value of the time from the time of a shift until the chart signals given that the magnitude of the shift is δj σ, j = 1, 2, . . . , m. Expression for  Tj can be derived using the Markov chain approach. It is given by: −1    Tj = b I − Qδj t−τ 

where b = (b1 , b2 , b3 ), bj is the in-control conditional probability that Zi falls in Ij , given that it falls within the control limits, j = 1, 2, 3, I is the identity matrix of order  3,t = (t1 , t2 , t2 ), and   δ δj δj j p11 p12 p13   Qδj =  pδ21j pδ22j pδ23j  δ δ δ p31j p32j p33j δ

where plkj = P rδj (Zi ∈ Ik | Zi−1 ∈ Il ) l, k = 1, 2, 3, and τ is as given in the Appendix. Let e be the expected time to sample and analyze one item. Then the expected time to analyze the signaling sample and chart the result is given by

¯ Chart Economic Design of A Modified Variable Sample Size and Sampling Interval X

279

ns e, where ns is the expected size of the signaling sample. An expression for ns is derived in the Appendix. Let T1 be the expected time to detect an assignable cause and T2 be the expected time to repair the process. Thus, the expected time from the time of a shift until the cycle ends equals: T + ns e + T1 + T2 Hence, from (6) and (8), the expected cycle time ECT is given by: 1 ECT = + (1 − 1 )αsT0 + T + ns e + T1 + T2 λ

(8)

(9)

Next, we derive an expression for the expected cost per cycle. Let C0 and C1 be the average costs due to nonconformities produced per hour during the in-control and outof-control periods, respectively. Let 2 = 1, if production continues during repair and 2 = 0, if production ceases during repair. Then, the expected cost per cycle due to nonconformities produced is given by: C0 (10) + C1 (T + ns e + 1 T1 + 2 T2 ) . B1 = λ Let Y be the average cost per false alarm. This includes the cost of searching and testing for the causes plus the cost of downtime, if production ceases during the search. Let W be the average cost for locating and repairing an assignable cause when one exists. Again W includes any downtime that is appropriate. Then the average cost for false alarms and locating and repairing a true assignable cause is given by B2 = Y αs + W

(11)

Let Cδj be the conditional expected cost for sampling per cycle given the magnitude of a shift is δj σ, j = 1, 2, . . . , m. Then the expected cost per cycle for sampling is given by: B3 =

m 

Cδj pj

(12)

j=1

An expression for Cδj is derived in the Appendix. Thus, the total expected cost per cycle is B1 + B2 + B3 and the expected cost per hour C is given by: C=

(B1 + B2 + B3 ) ECT

(13)

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Shashibhushan B. Mahadik and Digambar T. Shirke

The economic objective function derived above is more general than those developed for the adaptive charts earlier. For example, the objective function of Prabhu et al. [21] and Magalhaes et al. [14] includes almost all the process and cost parameters considered here but refers to one single assignable cause only. Further, the objective function of Park and Reynolds [18] does not allow considering a situation where the production is ceased during searching signals. It does not consider the cost due to nonconformities produced during the in-control period and the cost of locating an assignable cause and repairing the process. Moreover, the cycle considered in their objective function does not contain the time to analyze the signaling sample and the time to detect and eliminate an assignable cause. We note that the developed economic objective function is also applicable to VSI, VSS, VSSI, and static X charts, as these are the particular cases of an MVSSI X chart. Hence, the application of the objective function is two-fold. It can be used to obtain the optimal designs of these charts and also to compare the performances of the charts for given process and cost parameters. In the following section, we discuss the optimization procedure to obtain the optimal designs followed by the comparisons of the optimal static, VSI, VSS, VSSI, and MVSSI X charts.

4

Optimization and Comparison of Different Schemes

The optimal MVSSI X chart can be designed for given process and cost parameters by minimizing the expected cost per hour C in (13) with respect to the components of the  design vector x = (n1 , n0 , n2 , t1 , t2 , ω1 , ω2 , L) subject to the following constraints: tmin ≤ t2 ≤ t1 ≤ tmax , nmin ≤ n1 ≤ n0 ≤ n2 ≤ nmax , 0 ≤ ω1 ≤ ω2 ≤ L 0