Some group inspection based multi-attribute control charts ... - CiteSeerX

c Heldermann Verlag
ISSN 0940-5151

Economic Quality Control Vol 20 (2005), No. 2, 191 – 204

Some Group Inspection Based Multi-Attribute Control Charts to Identify Process Deterioration M.P. Gadre and R.N. Rattihalli

Abstract: In a production process, when quality of the product depends on more than one characteristic, ‘Multivariate Quality Control’ (M QC) techniques are used. Many M QC techniques have been developed to control multivariate variable processes, but not much work has been reported on multivariate attribute processes. In this article, we propose some group inspection based multi-attribute control charts to identify process deterioration. The charts proposed are the ‘Multi-Attribute np’ (M A − np) chart, the ‘Multi-Attribute Synthetic’ (M A − Syn) chart and the ‘Multi-Attribute Group Runs’ (M A − GR) chart. The charts are developed by using MP-test based on the exact distribution. It is numerically illustrated that, M A − GR chart performs better than the other two charts. Also in steady state M A − GR chart performs better than M A−np and M A−Syn control charts. A procedure of identifying the attributes causing signal is also proposed. Keywords: Average time to signal, Group runs chart, Multi-attribute control chart, Multivariate attribute process, np chart, Synthetic chart.

1

Introduction

Statistical process control is used for maintaining quality of the product. Quality of the product depends on one or more characteristics. If it is only classified in conforming and nonconforming items, a common practice to monitor the process is to use the np chart. Many other attribute control charts have been discussed in the literature (Refer Montgomery [10]). Wu et al. [11] proposed the synthetic control chart for detecting increases in fraction non-conforming by combining the np chart and the ’Conforming Run Length’ (CRL) chart (Bourke [1]). Let d be the number of nonconforming items in a group (sample) of size n, then d has Bi(n, p) distribution. If c is the acceptance number, the group is declared as non-acceptable if d > c. Further, for g > 1, Yg (the gth group based CRL) is defined as the number of accepted groups inspected between (g-1)th (if one such exists) and the gth non-acceptable group including the gth non-acceptable group. The synthetic chart declares the process as out of control as soon as Yg < Ls (the lower control limit of the chart) for some g ≥ 1. As an illustration let Ls = 3, Y1 = 4, Y2 = 5, Y3 = 2. In this case the synthetic chart gives a signal after observing the third non-acceptable group. It is to be noted that a signal may be due to a shift in the process mean (a correct signal) or may be due to the natural variability (a false alarm). Therefore, when a signal is

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received, it is desirable to monitor the process further to identify the cause for the signal. Therefore, Gadre and Rattihalli [5] proposed the ’Group Runs’ (GR) control chart, which is an extension of the synthetic control chart. A GR chart declares the process as out of control, if Y1 < Lg , (the lower control limit of the chart) or for some g(≥ 2) Yg and Yg+1 are less than Lg for the first time. Due to a very high competition in today’s world market, it becomes essential for the producer to take sufficient care to maintain the quality of the product. This can be achieved by considering more than one relevant characteristic and using multivariate quality control (MQC) techniques. Lowery and Montgomery [8] have shown that a MQC procedure is more sensitive in monitoring a multivariate process than a procedure based on univariate control charts. Nowadays, due to technological advancement, MQC techniques have become admissible for application and, consequently, many control charts, such as multivariate CUSUM chart (for a review one may refer Lowery and Montgomery [8]), multivariate EWMA chart (Lowery et al. [9]) have been developed in the last two decades. All these charts are related to multivariate variable processes. In many production processes, the counts related data are sufficient (see Kamansky et al.[6]). A process where product quality is based on counts with respect to more than one characteristic is called a multivariate attribute process and actually not much work has been reported on this topic. For testing quality of the product based on m attributes Gadre and Rattihalli [4] have developed ’Exact Multi-Attribute Control Chart’ (E-MACC) by using the exact joint distribution and the MP-test. The chart is of np type. In other words, it declares the process as out of control, if the group of fixed size n satisfies a certain condition. In case of single attribute Gadre and Rattihalli [5] have illustrated that the GR chart performs better than the np chart as well as the synthetic chart and the same is expected in the multiattribute case, too. Therefore, in this article, we develop three multi-attribute control charts, namely, the ’Multi-Attribute np’ (MA-np) chart, the ’Multi-Attribute Synthetic’ (MA-Syn) chart and the ’Multi-Attribute Group Runs’ (MA-GR) chart for detecting process deterioration and illustrate that MA-GR performs better than the other two. The remainder of the paper is organized as follows. The required notations and terms are listed in Section 2. Numerical illustrations and the comparison of the charts are contained in Section 3. The steady state performance of these charts is compared in the subsequent section. When MACC gives a signal, it just indicates that the process has gone out of control, but it does not readily identify the attributes causing the signal. Therefore, in Section 5, a procedure is described and illustrated for identifying the responsible attributes for a signal. Some concluding remarks are given in the last section.

Some Group Inspection Based Multi-Attribute Control Charts

2

193

Notations and Terms

In this section, we introduce the notations used to develop the ’Multi-Attribute Control Charts’ (MACCs) and describe the procedure of their implementation. 2.1

Notations

Let the quality of an item be assessed on the basis of m attributes, A1 , . . . , Am . Let Xjk ( j = 1, . . . , m; k = 1, . . . , n) be 0 or 1, according as the kth item in the sample being conforming or non-conforming with respect to (w.r.t.) the jth attribute. Thus, each Xjk is a Bernoulli random variable and hence the vector X k = (X1k , . . . , Xmk ) is a m-dimensional binary random response of the kth item having 2m states. For each binary vector x, we associate a number r (r = 0, 1, . . . , q = 2m − 1) given by m
xs 2s−1 (1) r= s=1

As an illustration, consider the problem of two attributes. Then the possible four states are (0, 0), (0, 1), (1, 0), (1, 1) and the related values of r are 0, 1, 2, 3. Here x = (0, 1) indicates that the corresponding unit is nonconforming w.r.t. the second characteristic and is conformimg w.r.t. the first characteristic. Let q
with πr = 1 (2) P (X k = x) = πr r=0

If π = (π0 , π1 , . . . , πq ), then following the notations of Ekholm et al. [3], the vectors X k , k = 1, . . . , n, are independent random vectors having m-dimensional Bernoulli distribution denoted by Bm (π) For r = 0, 1, . . . , q, let Zr be the number of observations in the group of size n having path x and r be the associated value. Here Z = (Z0 , Z1 , . . . , Zq ) is sufficient for π and has multinomial distribution with parameters (n, π). Thus, in the above illustrating example, n = 10 and Z = (7, 2, 1, 0) indicate that among the 10 units in the group, there are 7 conforming, 2 are non-conforming w.r.t. second attribute only and the remaining unit is non-conforming only w.r.t. the first attribute. Let π = (π0 , π ∗ ), where π ∗ = (π1 , . . . , πq ). For r = 1, . . . , q, let 1. π0r be the maximum permissible value of r, when the process is running satisfactorily and 2. π1r (> π0r ) be the minimum value of r, essential to indicate that the process has gone out of control. Define π ∗0 = (π01 , . . . , π0q ), π ∗1 = (π11 , . . . , π1q ) with π ∗0 < π ∗1 (componentwise) and π 0 = (π00 , π ∗0 ), π 1 = (π10 , π ∗1 ), where π ∗0 < π ∗1 implies π00 > π10 . A group of n units in MACCs is declared as conforming (non-conforming) accordingly if π ∗ ≤ π ∗0 (π ∗ > π ∗1 ). It can be shown that (refer to Lehmann [7]) a non-randomised MP test for testing

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M.P. Gadre and R.N. Rattihalli

π ∗ = π ∗0

is given by ψS (X) =



1 0

against

if if

H11 :

π ∗ = π ∗1

(3)

Tn (Z) > k Tn (Z) ≤ k

where X = (X11 , X21 , . . . , Xmn ) and Tn (Z) =    q   π1r 1 − r=1 π0r     hr = ln  q    π0r 1 − π1r

(4) q r=1

hr Zr and for r = 1, . . . , q

(5)

r=1

The constant k ≥ 0 is to be chosen suitably and is discussed later. A M A − np chart is based on n observations and declares the process as out of control or under control according to Tn (Z) > k or Tn (Z) ≤ k. The quantities (n, k) are called the control parameters of the M A − np chart. In case of a M A − Syn chart and a M A − GR chart the status of a group of size n is decided by using a procedure similar to that of the M A − np chart. That is a group of fixed size n is declared as non-conforming or conforming according as Tn (Z) > k or Tn (Z) ≤ k. Similar to Wu et al. [11] and Gadre and Rattihalli [5], a M A − Syn chart declares the process as out of control when Yg < Ls for the first time and a M A − GR chart declares the process as out of control if Y1 < Lg or if for some g > 2, the quantities Yg and Yg+1 are both less than Lg for the first time. Here, the index of L denotes the respective M ACC. The quantities (ns , ks , Ls ) and (ng , kg , Lg ) are called the control parameters of the corresponding M ACCs. Let P (π) be the probability of a group being declared non-conforming when the process level is π. Thus, q


P (π) = P hr Zr > k (6) r=1

The following example shall serve for illustrating the computation of P (π). Example 1: Consider the production of engineering components, where among others tapping and drilling are important operations. Denote these attributes by A1 and A2 , and let π 0 = (0.96166, 0.00654, 0.0109, 0.0209) (7) π 1 = (0.8297, 0.0426, 0.0535, 0.0742) For these values of π 0 and π 1 , we get

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h1 = 2.021514 h2 = 1.738516 h3 = 1.414612

(8)

Here n = 2 and the value of k is the one among the possible values of are given by the following set of numbers:  0, 1.414612, 1.738516, 2.021514, 2.829224,  3.153128, 3.436126, 3.477032, 3.76003, 4.043028

q r=1

hr Zr which

(9)

with the corresponding Z vectors: (2, 0, 0, 0), (1, 0, 0, 1), (1, 0, 1, 0), (1, 1, 0, 0), (0, 0, 0, 2), (0, 0, 1, 1), (0, 1, 0, 1), (0, 0, 2, 0), (0, 1, 1, 0), (0, 2, 0, 0) As an illustration, set k = 1.414612 and π = π 0 . Then q

q



hr Zr > 1.414612 = 1 − P hr Zr ≤ 1.414612 P (π) = P r=1

= 1−P

q


hr Zr = 0

−P

r=1

q


r=1

hr Zr = 1.414612

r=1

= 1 − P (Z = (2, 0, 0, 0)) − P (Z = (1, 0, 0, 1)) = 1 − 0.92479 − 0.040197 = 0.035013

(10) •

For notational convenience let P = P (π), P0 = P (π 0 ), P1 = P (π 1 ) and AT S = AT S(π), AT S0 = AT S(π 0 ), AT S1 = AT S(π 1 ). As a desirable property AT S0 should not be smaller than a fixed number τ . The quantities (π 0 , π 1 , τ ) are the input parameters of the chart. Similar to the expressions of AT S(p) given in Wu et al. [11] and Gadre and Rattihalli [5], the expressions for AT S(π) for various M ACCs considered here are as below. • M A − np Chart: AT S =

n P

(11)

• M A − Syn Chart: AT S =

n 1 P 1 − (1 − P )L−1

• M A − GR Chart: −2  1 n AT S = P 1 − (1 − P )L−1

(12)

(13)

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The procedures of obtaining values of the control parameters are similar to those as in case of single attribute. For the sake of completeness, in the following, we describe the procedure related to M A − GR chart in brief.

3

Stepwise Procedure to Find Optimum Values of (n, k, L)

Optimization is performed in three steps. First, for fixed (n, k), the optimal level of L is determined, second, for fixed n, the optimal value of k and finally the optimal value of n. Before discussing the optimization procedure, it is essential to show the existence of a combination of (n, k, L) with AT S0 ≥ τ . We call such a point (n, k, L) a feasible point. Notice that the point (τ, 0, 2) is a feasible point as for this point AT S0 = AT S1 =

τ τ )3 (1−π00 τ τ )3 (1−π10

Hence, the search procedure is started by taking the value of AT S1 as 3.1

(14) τ τ )3 . (1−π10

Optimization of L

For given (n, k) compute P0 and P1 . From (13) it is seen that for given (n, k, L), AT S(π) is a decreasing function of π ∗ . Therefore, if the condition AT S0 ≥ τ is satisfied for some L (≥ 2), then it is satisfied for eacg L = 2, 3, . . . , L . Thus, for fixed (n, k), the optimal value L∗n,k that minimizes AT S1 is the largest value meeting the constraint AT S0 ≥ τ . Note that    ln 1 − Pn0 τ ˆ= L +1 ln(1 − P0 ) is the solution for AT S0 = τ and, hence, the optimum value L∗n,k of L becomes      ln 1 − Pn0 τ   L∗n,k =  + 1 ln(1 − P0 )

(15)

(16)

where [a] denotes the largest integer not greater than a. Example 1 (Cont.): Along with the above values of π 0 and π 1 , let τ = 200. Further, to start the calculation, let (n, k) be (1, 0). For these values, we get, P0 = 0.0383 and therefore from (13) and (16) we have L∗1,0 = 12 with AT S0 = 213.5086 and AT S1 = 7.7272. •

Some Group Inspection Based Multi-Attribute Control Charts

197

Remark1: Note that for given (n, k) if L∗n,k < 2 then, for any value of L, (n, k, L) is not a feasible point. Compute AT S1 at (n, k, L∗(n,k) ) if it is a feasible point; else, if permissible, increase the value of k suitably; else, increase the value of n by unity if necessary; otherwise terminate the search procedure. 3.2

Optimization of k

For fixed n, let k1 , k2 , . . . , ktn be the distinct possible values of the chart statistic Tn (Z). Consider the domain     Dn = (n, k)  k ∈ {k1 , k2 , . . . , ktn }, AT S0 ≥ τ, L = L∗n,k ≥ 2 (17) and Dn =

n 

Dt

(18)

t=1

One can search for the optimum point at which AT S1 is minimum over Dn . Let the minimum value of AT S1 attained so far be denoted by mn . Example 1 (Cont.): Here

 D1 = D1 = {(1, k) | k ∈ {0, 1.414612, 1.738516, 2.021514}, AT S0 ≥ τ, L = L∗1,k ≥ 2 (19)

The minimum values of AT S1 corresponding to the first three values of k are respectively • 7.7272, 10.6813 and 23.4742. Thus m1 = 7.7272. 3.3

Optimization of n

Notice that for M A − GR chart, AT S(P ) is not smaller than the group size n. This fact is used to find the optimum value of n. Start with n = 1 and compute mn . Since mn is ˜ just exceeding bounded above by (1−πτ r )3 , and is non-increasing in n, there exists a value n 1 mn . Continue the search by increasing n by unity at every stage so long as mn ≥ n. Then the search will be terminated when n becomes (˜ n − 1) and the optimum value of (n, k, L) is the one that corresponds to AT S1 = mn˜ −1 . Example 1 (Cont.): Taking n = 2, it can observed that the minimum value of AT S1 over D2 is 8.9807 implying m2 = 7.7272. Continuing in this way it is easily seen that ˜ = 8 and, therefore, it is not necessary to consider a group size n m7 = 7.7272. Here n larger than 7. The resulting control parameters are (ng , kg , Lg ) = (1, 0, 12) with AT S1 = • mn˜ −1 = m7 = 7.7272. For further details one may refer to Gadre and Rattihalli [5]. In the next section, we compare the performance of the three proposed control charts with the help of various numerical examples.

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Numerical Examples

Example-1 (Cont.): For the given input parameters, values of the control parameters of the above mentioned M ACCs along with the respective AT S0 and AT S1 are displayed in Table 1. Table 1: Design Parameters and values of AT S0 , AT S1 for various charts. Control Chart n M A − np chart 9 M A − Syn chart 1 M A − GR chart 1

k L AT S0 AT S1 2.0215 – 203.4976 19.1712 1.4146 20 201.7935 12.1941 0 12 213.5086 7.7272

Table 1 clearly indicates that AT S1 of M A − GR chart is significantly smaller as for the other two charts. Thus, we conclude that in this example M A − GR is better than the other two charts. Table 2: Values of AT S(π) for various values of π. Value of π M A − np Chart M A − Syn Chart M A − GR Chart (0.95, 0.01, 0.015, 0.025) 126.3843 104.7508 107.5654 (0.935, 0.015, 0.02, 0.03) 80.2451 58.0935 56.3418 (0.92, 0.02, 0.025, 0.035) 56.8465 38.1125 34.6803 (0.88, 0.03, 0.04, 0.05) 30.4964 19.095 14.6224 (0.81, 0.05, 0.06, 0.08) 16.8843 10.2059 6.4758 (0.74, 0.07, 0.09, 0.10) 12.4473 6.4862 4.1425 (0.60, 0.10, 0.15, 0.15) 9.6831 4.017 2.5182 Table 2 is useful to study the behavior of the three charts corresponding to changes in the π value. From Table 2 it is observed that except for the first value of π, AT S(π) of M A − GR is much smaller than those of two other charts. For the first value of π, AT S(π) of M A − GR chart is less than the one of M A − np chart and not much larger than the AT S(π) of the M A − Syn chart. Thus, one may conclude that M A − GR is superior to M A − np chart and for most of the range M A − GR chart is preferred to M A − Syn chart. Example 2: Here, we consider nine combinations of π 0 , π 1 and τ to compare the performance of the three charts. The values considered are. A : π 0 = (0.96166, 0.00654, 0.0109, 0.0209), π 1 = (0.8297, 0.0426, 0.0535, 0.0742) B : π 0 = (0.95066, 0.00654, 0.0109, 0.0319), π 1 = (0.8097, 0.0426, 0.0535, 0.0942) (20) C : π 0 = (0.8, 0.07, 0.10, 0.03), π 1 = (0.4, 0.30, 0.20, 0.10) τ : (a)200, (b)2000, (c)20000 Table 3 below contains the values of the control parameters along with the respective AT S1 values corresponding to the three control charts. The table shows that, for all 9

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cases, AT S1 of the M A − GR chart is less than that for the other two charts. Thus, we conclude that M A − GR chart is ‘better’ than the other two charts. Table 3: Control parameters of the three charts along with corresponding AT S1 values. Ex. Comb. 1 2 3 4 5 6 7 8 9

5

(A, a) (A, b) (A, c) (B, a) (B, b) (B, c) (C, a) (C, b) (C, c)

n 9 23 28 9 23 26 5 11 16

M A − np k AT S1 2.0215 19.1712 5.6584 39.9865 8.3278 66.7202 2.4866 19.5191 5.9894 39.9608 8.4761 67.5119 5.4318 9.51 10.6042 16.0433 15.9227 23.2893

n 1 7 17 1 7 18 2 5 9

M A − Syn k L AT S1 1.4146 20 12.1941 2.0215 6 23.4956 4.8507 6 37.4691 1.2433 20 12.1941 2.4866 6 24.3615 5.0291 7 37.1933 2.1484 8 5.8111 5.4318 6 9.7438 8.4558 4 13.4230

n 1 10 10 1 9 12 3 4 5

M A − GR k L AT S1 0 12 7.7272 2.0215 7 19.3621 3.1531 7 28.2767 0 8 8.8211 2.0344 5 18.5646 3.5028 7 27.5772 2.1484 5 4.7751 3.7942 4 7.2371 5.1805 4 9.9596

Runs Rule Representation and Steady State AT S Performance of the three M ACCs

Gadre and Rattihalli [5] have discussed the steady state AT S performance of the GR control chart identifying increases in fraction non-conforming depending on the np based procedure. Here, we discuss that of the M A − GR chart. The steady state AT S measures average time to signal, when the effect of head start has been faded away. Let the groups be classified as ‘0’ or ‘1’ according as conforming or nonconforming and for illustration purpose assume that L = 4. Thus the M A − GR chart will produce a signal if Y1 or two successive Yr s are less than 4 for the first time. A Markov chain representation in this situation can be developed by using the 14 initial states and the transition probability matrix. . Table 4: The Initial States and their Labels. State Label Initial State State Label Initial State 1 000 8 000110 2 0001 9 0001010 3 00010 10 00010010 4 000100 11 0001100 5 00011 12 00010100 6 000101 13 000100100 7 0001001 14 Signal

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In Table 4, the initial state ‘000’ indicates the sequence of at least 3 (=L − 1) conforming groups. The other initial states can be described analogously. The transition probability matrix related to L = 4 and the 14 states is as follows.   Q P 0 0 0 0 0 0 0 0 0 0 0 0  0 0 Q 0 P 0 0 0 0 0 0 0 0 0     0 0 0 Q 0 P 0 0 0 0 0 0 0 0     Q 0 0 0 0 0 P 0 0 0 0 0 0 0     0 0 0 0 0 0 0 Q 0 0 0 0 0 P     0 0 0 0 0 0 0 0 Q 0 0 0 0 P     0 0 0 0 0 0 0 0 0 Q 0 0 0 P     0 0 0 0 0 0 0 0 0 0 Q 0 0 P     0 0 0 0 0 0 0 0 0 0 0 Q 0 P     0 0 0 0 0 0 0 0 0 0 0 0 Q P     Q 0 0 0 0 0 0 0 0 0 0 0 0 P     Q 0 0 0 0 0 0 0 0 0 0 0 0 P     Q 0 0 0 0 0 0 0 0 0 0 0 0 P  0 0 0 0 0 0 0 0 0 0 0 0 0 1

For given L, the Markov chain has the following non absorbing initial states. 1. A sequence containing at least (L - 1) zeros. 2. A Sequence in (1) followed by 1 and further appended by at most (L - 2) zeros. There are (L - 1) such sequences. 3. Each of the sequences in (2) followed by 1 and is further appended by a sequence of at most (L - 2) zeros. The total number of such sequences is (L − 1)2 . The total number of non absorbing states is 1 + (L − 1) + (L − 1)2 = L(L − 1) + 1. The matrix R1 of non absorbing initial states is a square matrix of order L(L − 1) + 1. Note that the (i, j)th element of R1 is  Q if the ith initial state leads to jth initial state, and     jth initial state corresponds to the sequence ending with ‘0’  P if the ith initial state leads to jth initial state, and Ri (i, j) = (21)   jth initial state corresponds to the sequence ending with ‘1’    0 otherwise Let η be a (L(L − 1) + 1)-dimensional row vector corresponding to the stationary probability distribution of the Markov chain for each of the non absorbing states, which is conditioned on no signal. Then, as shown in Brooke and Evans [2], the ARL-vector is given by (I − R)−1 1. Further, as mentioned in Brooke and Evans [2], the steady state ARL is given by η × ARL and the steady state AT S of M A − GR chart is the product of ng and the steady state ARL.

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201

It is to be noted that for any run length based control chart, the steady state AT S is not smaller than zero state AT S. If the Markov chain representation of the chart has only one non-absorbing state, both AT Ss are same. To compare the steady state AT S performance of the charts, the steady state AT S0 of the two charts must be equal. For the purpose, we compute adjusted steady state AT S of Chart II with respect to the Chart I as [S.S.AT S(δ)]II [S.S.AT S(0)]I (22) [Adj.S.S.AT S(δ)]II = [S.S.AT S(0)]II In Table 5 we compare the steady state performance of the three M ACCs developed in this article for various values of π corresponding to Ex. 7 in Example 2. Ex.7 in Example-2 (Cont.): Table 5 contains SSAT S and Adj.SSAT S values corresponding to various values of π for three M ACCs. Table 5: Values of steady state AT S(π) related to various values for the three charts.

(0.8, 0.07, (0.74, 0.10, (0.66, 0.15, (0.58, 0.20, (0.48, 0.25, (0.4, 0.3, (0.25, 0.35, (0.1, 0.4,

0.1, 0.03) 0.12, 0.04) 0.13, 0.06) 0.15, 0.07) 0.18, 0.09) 0.2, 0.1) 0.25, 0.15) 0.3, 0.2)

M A − np 221.946 94.632 39.3059 21.7322 13.0953 9.51 6.5074 5.2381

S.S.AT S(π) Adj(S.S.AT S(π)) M A − Syn M A − GR M A − Syn M A − GR 244.6599 326.4099 221.946 221.946 100.1579 108.1776 90.8594 73.5566 42.3373 40.3233 38.4068 27.4183 22.3449 20.6742 20.2704 14.0577 12.135 11.7007 11.0084 7.9560 8.2165 8.5090 7.4537 5.7858 4.5929 5.7807 4.1665 3.9103 2.9097 4.6883 2.6396 3.1879

From Table 5, we conclude that for small to moderate shifts in the process level, the steady state performance of M A − GR is better than that of the other two charts. Remark 2: Very rarely the process level shoots up to a quantity greater than (π  )∗ (> π ∗1 ) (in this example π  = (0.25, 0.35, 0.25, 0.15)) and the producer would like to detect small to moderate increases in π ∗ . In all such situations, M A − GR chart is superior to the other two M ACCs. A macro in MATLAB is written to compute SSAT S of M A − Syn and M A − GR chart. When the process goes out of control, a natural query would be to search for responsible attributes. In any M ACC, the signal indicates only that the process is being running out of control, but it does not readily identify the responsible attributes. In the following section, for M A − GR chart, we propose a procedure to identify the attributes responsible for causing a signal.

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Procedure to Detect the Attributes Responsible for Lack of Control

Consider a non-conforming group indicating a signal and the preceding non-conforming group of size ng (if one such exists). • Case I: If a preceding non-conforming group does not exist then based on the nonconforming group of ng units indicating the signal, compute pˆj , j = 1, . . . , m, i.e., the proportion of non-conforming units with respect to the jth attribute as, pˆj =


zr ng r∈B

(23)

j

Define Dj =

pˆj − p0j p1j − p0j

for j = 1, . . . , m

(24)

In a sense the quantity Dj indicates the degree of responsibility of the jth attribute for the process going out of control. We call the jth attribute responsible for the signal if Dj > 1 id at least one such attribute exists. Otherwise the attribute with largest value of Dj is considered the only responsible attribute. • Case II: Suppose there is a non-conforming group preceding the group indicating the signal. Find the attributes responsible based on the group indicating the signal as discussed in Case I. Let these attributes be A1 , . . . , Au . Combine the preceding non-conforming group of size ng and the actual group indicating the signal to get an enlarged group of 2ng units. Based on this group of 2ng units, compute pˆj and Dj , j = 1, . . . , m. Define D = min Di 1≤i≤u

(25)

The jth attribute is also considered responsible, if Dj > D. Remark 3: For M A − np and M A − Syn charts, the responsible attributes can be identified by using the procedure discussed in Case I above. Ex.7 in Example-2 (Cont.): Let the control parameters be (n, k, L) = (3, 2.1484, 5) and the process level π ˜ = (0.73, 0.10, 0.15, 0.02). The realisation of the process was generated by means the corresponding multinomial distribution with parameters (3, π ˜ ).

Some Group Inspection Based Multi-Attribute Control Charts

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The M A − GR chart gave a signal after the 9th group was generated with Z vectors corresponding to the 8th and the 9th groups given by (1, 0, 2, 0) and (1, 0, 2, 0), respectively. Based on the non-conforming group indicating the signal, we obtain D1 = 0 and D2 = 3.156863. This indicates that A2 is the only responsible attribute. As stated in Case II, based on the group of 2ng units, the values of D1 and D2 are 0 and 3.156863, respectively. Thus, also the combined group indicates A2 as the only responsible attribute. •

7

Conclusions

Three different group inspection based M ACCs namely the M A − np chart, the M A − Syn chart and the M A − GR chart are proposed and the zero state and steady state performances are studied. It is shown that M A − Syn chart performs better than MA-np chart, and M A − GR chart performs much better than M A − np chart. Particularly, for detecting small to moderate increases in the vector of fraction non-conforming, M A − GR chart is better than M A − Syn chart. Moreover, the computation and implementation of the M A − GR chart is very simple. Thus M A − GR chart is widely applicable M QC technique in industries. The procedure of identifying the attributes responsible for the signal is simple to understand and easy to apply. Although the proposed chart is based on 100 % inspection, it can be used in non-100 % inspection cases if uniform sampling is used. Acknowledgement: We are thankful to the Referee and the Regional Editor for valuable comments, which helped to improve the presentation of the manuscript significantly.

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M. P. Gadre Department of Statistics Mudhoji College, Phaltan 415523 India R. N. Rattihalli Department of Statistics Shivaji University, Kolhapur 416004 India