fuzzy multinomial control chart with variable sample size

A. Pandurangan et al. / International Journal of Engineering Science and Technology (IJEST)

FUZZY MULTINOMIAL CONTROL CHART WITH VARIABLE SAMPLE SIZE A. PANDURANGAN Professor and Head Department of Computer Applications Valliammai Engineering College, Kattankulathur, Chennai – 603 203, India. e-mail id : [email protected]

R. VARADHARAJAN Assistant Professor in Statistics Department of Management Studies VELS University, Pallavaram, Chennai – 600 117, India e-mail id: [email protected]

Abstract: The control chart technique is being widely used in industries to monitor a process for quality improvement. One of the basic charts for attributes is the p-chart. For a p – chart each item is classified as either nonconforming or conforming to the specified quality characteristic. In Some cases, an item may be classified in more than two categories such as “bad”, “medium”, “good”, and “excellent”. Based on this concept, Amirzadeh et al. [1] have developed Fuzzy Multinomial chart (FM-chart) with the fixed sample size (FSS). In this paper a Fuzzy Multinomial process with Variable Sample Size (VSS) is proposed and the control limits for the FM chart have been obtained using multinomial distribution. The proposed method is compared with the conventional p – chart. It is seen that FM chart with VSS performs better than the conventional chart. Key words: Multinomial distribution; FM-chart; p–chart; Variable sample size; Linguistic variable; Membership function; Fuzzy statistics. 1. Introduction: Statistical Process Control (SPC) is used to monitor the process stability which ensures the predictability of the process. The power of control charts lies in their ability to detect process shift and to identify abnormal condition in the process. In 1924, Walter Shewhart designed the first control chart as follows: Let w be a sample statistic that measures some quality characteristic of interest and suppose that the mean of w is µw and the standard deviation of w is σw. Then the center line (CL), the upper control limit (UCL) and the lower control limit (LCL) are defined as UCL = µw + d σw

CL = µw

LCL = µw – d σw

where d is the “distance” of the control limits from the center line, expressed in standard deviation units. A single measurable quality characteristic such as dimension, weight or volume is called a variable. In such cases, control charts for variables are used. These include the X -chart for controlling the process average and the R -chart (or S -chart) for controlling the process variability. If the quality-related characteristics such as characteristics for appearance, softness, color, taste, etc., attributes control charts such as p-chart, c-chart are used to monitor the production process. Some times the product units are classified as either "conforming" or" nonconforming", depending upon whether or not they meet some specifications. The p -chart is used to monitor the process based upon the fraction of nonconforming units.

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2. Fuzzy logic and Linguistic variables: The concept of fuzzy logic plays a fundamental role in formulating quantitative fuzzy variables. These are variables whose states are fuzzy members. The members represent linguistic concepts, such as very small, small, medium and so on, as interpreted in a particular context. The resulting constructs are usually called linguistic variables. The linguistic terms are commonly used in industry to express properties or characteristics of a particular product. The conformity to specifications on a quality standard is evaluated onto a two-state scale, for example, acceptable or unacceptable, good or bad, and so on. In some situations the binary classification might not be suitable, where product quality can assume more intermediate states. The assignment of weights, to reflect the degree of severity of product nonconformity has been adopted in many circumstances. When the products are classified into mutually exclusive linguistic categories, fuzzy control charts are used. Different procedures are proposed to construct these charts. Raz and Wang [7,10] developed fuzzy control charts for linguistic data which are mainly based on membership and probabilistic approaches. In this paper, a fuzzy multinomial control chart (FM chart) for linguistic variables with variable sample size is proposed. The FM – chart deals with a linguistic variable which is classified into more than two categories. The FM – chart with VSS found to be more effective than p – chart for studying the shift in process mean. 3. Methodology:

~

Based upon Fuzzy set theory, a linguistic variable L is characterized by the set of k mutually exclusive members {l1 , l 2 ...l k } . We attach a weight mi to each term l i that reflects the degree of membership in the set. Then it can be written by a fuzzy set as

~ L = {(l1 , m1 ), (l2 , m2 ),...(lk , mk )}

. . . . . . . . . . . . . . . . . . . . . . . . . (1).

To monitor the out of control signal in the production process take independent samples of different sizes. The size of the sample to be drawn each time is decided by choosing a member randomly from

{n1 , n2 ,...ns }.

4. Fuzzy Multinomial Control Chart: In this section a new approach for construction of Fuzzy multinomial control chart based on variable sample size is proposed. The statistical principles underlying the fuzzy multinomial control chart (FM chart) with variable sample size are based on the multinomial distribution. As defined in (1),

~ L

is a linguistic variable which can take k mutually exclusive members {l1 , l 2 ...l k } .

Assume that the production process is operating in a stable manner and pi is the probability that an item is l i , i = 1, 2 …k. and successive items produced are independent. Suppose that a random sample of size n r units of the product is selected and let X i , i = 1, 2 …k, be the number of items of the product that are l i , i = 1, 2 …k. Then

{X 1 , X 2 ... X k } has a multinomial distribution with parameters

n r and p1 , p 2 ... p k . It is known that

each X i , i = 1, 2 …k, marginally has a binomial distribution with the mean n r pi and variance n r pi (1 − pi ) , i = 1, 2 …k, respectively. The weighted average of the linguistic variable k

~ L=

X m i

with sample size n r is defined by

k

X m

i

i =1 k

X

~ L

i

=

i

i =1

nr

,

nr ∈ {n1 , n2, ...ns }

. . . . . . . . . . . . . . . . . . . (2)

i

i =1

The control limits for FM – chart are

~

~ var(L ) , where d is the ~ ~ distance of the control limits from the center line. The procedure for computing E[L ] and var(L ) for each UCL = E[L ] + d

~ ~ ~ var(L ) ; CL = E[L ] ; and LCL = E[L ]

d

sample is given in the following theorem.

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5. Theorem: Let

~ L = {(l1 , m1 ), (l2 , m2 ),...(lk , mk )} be a linguistic variable such that pi

is the probability that an

item is l i , i = 1, 2, …k. If X i , i = 1, 2 …k is the number of units of the product that are l i , i = 1, 2 …k in a sample of size n r , then k

~

pm

(i) E[L ] =

i

i

i =1

{

k k  1 k 2 − − m p ( 1 p ) 2 mi m j p i p j ) , nr ∈ n1 , n2, ...ns  i i   i n r  i =1 i =1i < j j =1  where n1 , n 2, ...n s are pre-determined sample sizes.

~

(ii) var( L ) =

}

Proof: In a sample of n r units, X i has a binomial distribution with the mean n r pi and variance

nr pi (1 − pi ) , i = 1, 2, …k and Cov ( X i , X j ) = − nr pi p j , if i ≠ j and then

(i)

 k  k m X   i i   mi E ( X i ) ~  = i =1 The mean is: E[L ] = E  i =1 = nr  nr   

k

m n i

r

pi

i =1

nr

 p m , nr ∈ {n1 , n2, ...ns } . . . . . . . . . . . . (3) k

=

i

i

i =1

(ii)

  k mi X i    k 1   ~  = 2  var( mi X i ) , The variance is: var( L ) = var  i =1  nr  nr  i =1    1 = 2 [var(m1 x1 + m2 x 2 + ... + mk x k )] nr

k k  1 k 2 m var( X ) 2 mi m j cov( X i , X j ) +     i i 2 n r  i =1 i =1i < j j =1  k k k  1  2 = 2  mi var( X i ) + 2   mi m j cov( X i , X j )  n r  i =1 i =1i < j j =1  k k k  1  2 = 2  mi n r p i (1 − p i ) + 2   mi m j ( − n r p i p j )  n r  i =1 i =1i < j j =1  k k k  1  2 =  mi pi (1 − pi ) − 2   mi m j p i p j ) , where nr ∈ n1 , n2, ...ns . . . . (4) n r  i =1 i =1i < j j =1  ~ ~ The results for E[L ] and var(L ) become same as that of the results due to Amirzadeh et al. [1], when the sample size is fixed. That is, when n r = n for all r, the mean and variance derived in (3) and (4) for a

=

{

}

linguistic variable with variable sample size will be equal to the mean and variance of the linguistic variable for fixed sample size.

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6.

Choice of Sample size:

Many authors, for example [2], [8] have recommended variable sample sizes (VSS) for the construction of control charts for variables as well as attributes. Later, the Markov dependent sample size (MDSS) was proposed by Sivasamy et al [9], and Pandurangan [6] for the advantage of economic sampling inspection. Now, to construct FM control chart the sample size for each draw can be randomly chosen from the pre – determined set

{n , n 1

2,

...ns }. The advantage of taking variable sample size lies in ASN and consequently the costs of

sampling inspection. 7.

Numerical Example:

On a production line, a visual control of the aluminum die-cast of a lighting component might have the following assessment possibilities 1. "reject" if the aluminum die-cast does not work; 2. "poor quality" if the aluminum die-cast works but has some defects; 3. "medium quality" if the aluminum die-cast works and has no defects, but it has some aesthetic flaws; 4. "good quality" if the aluminum die-cast works and has no defects, but has few aesthetic flaws; 5. "excellent quality" if the aluminum die-cast works and has neither defects nor aesthetic flaws of any kind. To monitor the quality of this product, 25 samples of different sizes are selected. The degrees of membership for the above assessment are taken as 1, 0.75, 0.5, 0.25 and 0 respectively. The data with

pˆ i

~ L i and

are given in Table – 1. The value of

~ L i can be calculated to the following ways k

~ Li =

k

X m X m i

i

i =1 k

X

i

=

i =1

nr

i

,

nr ∈ {n1 , n2, ...ns }

i

i =1

~ {(12 ×1) + (10 × 0.75) + (12 × 0.5) + (54 × 0.25) + (12 × 0)} L1 = = 0.390 100 ~ {(8 ×1) + (7 × 0.75) + (9 × 0.5) + (48× 0.25) + (8 × 0)} L2 = = 0.372 80 ~ {(6 ×1) + (11× 0.75) + (12× 0.5) + (43× 0.25) + (8 × 0)} L3 = = 0.388 80 ˆ i , the control limits for p – charts can be calculated as follows And so on, and the value of p

{

}

Di , nr ∈ n1 , n2 , ...n s nr D D 12 8 pˆ 1 = 1 = = 0.120 , pˆ 2 = 2 = = 0.100, n1 100 n2 80 pˆ i =

pˆ 3 =

D3 6 = = 0.075 ; n3 80

and so

on.

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A. Pandurangan et al. / International Journal of Engineering Science and Technology (IJEST) Table – 1

Sample No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Sample size 100 80 80 100 110 110 100 100 90 90 110 120 120 120 110 80 80 80 90 100 100 100 100 90 90

Poor Quality(PQ) 10 7 11 7 16 5 12 22 8 5 13 13 12 9 10 5 8 13 7 11 8 9 12 13 10

Reject® 12 8 6 9 10 12 11 10 10 6 20 15 9 8 6 8 10 7 5 8 5 8 10 6 9

Medium Quality(MQ) 12 9 12 13 18 17 13 18 13 14 23 20 22 20 19 12 12 10 14 14 16 15 14 17 14

Good Quality(GQ) 54 48 43 53 54 60 50 45 50 51 47 58 64 61 61 47 40 42 54 50 58 51 50 45 46

Excellent Quality(EQ) 12 8 8 18 12 16 14 5 9 14 7 14 13 22 14 8 10 8 10 17 13 17 14 9 11

~ Li

pˆ i

0.390 0.372 0.388 0.340 0.405 0.357 0.390 0.468 0.389 0.328 0.482 0.410 0.375 0.333 0.348 0.369 0.400 0.403 0.342 0.358 0.335 0.350 0.385 0.394 0.389

0.120 0.100 0.075 0.090 0.091 0.109 0.110 0.100 0.111 0.067 0.182 0.125 0.075 0.067 0.055 0.100 0.125 0.088 0.056 0.080 0.050 0.080 0.100 0.067 0.100

The control limits for p – chart is obtained as follows k

D

i

p=

i =1 s

n

=

234 = 0.096 2450

r

r =1

UCL1 = p + d

For Sample 1:

p(1 − p) 0.096(1 − 0.096) = 0.096 + 3 = 0.184 nr 100

CL1 = p = 0.096; LCL1 = p − d For Sample 2:

UCL2 = p + d

0.096(1 − 0.096) p(1 − p) = 0.096 − 3 = 0.008 , 100 nr p(1 − p) 0.096(1 − 0.096) = 0.096 + 3 = 0.195 nr 80

CL2 = p = 0.096; LCL2 = p − d

0.096(1 − 0.096) p(1 − p) = 0.096 − 3 = −0.003, 80 nr

and so on.

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Fig. 1: The p – chart for the 25 samples.

In Figure 1, the out of control signal is seen corresponding to 11th sample, for which the sample size is 110. From the p – chart, out of 2450 sample observations, 1070 sample observations were needed to get the signal. The corresponding center line and control limits are as under CL = 0.096, UCL = 0.180 and LCL = 0.012 To construct the FM – chart, the UCL and LCL values are computed for each sample as follows For Sample 1:

~ ~ UCL1 = E[ L1 ] + d var(L1 ) k k  1 k 2  mi pi (1 − pi ) − 2   mi m j pi p j ) n1  i =1 i =1i < j j =1 i =1  = 0.3750 + 3 0.0008214 = 0.4609 k ~ CL1 = E[ L1 ] =  pi m = 0.3750 k

=  p i mi + d

i =1

~ ~ LCL1 = E[ L1 ] − d var(L1 ) k k  1 k 2 m p ( 1 − p ) − 2 mi m j p i p j )   i i   i n1  i =1 i =1 i =1i < j j =1  = 0.3750 − 3 0.0008214 = 0.2891 k

=  p i mi - d

For Sample 2:

~ ~ UCL2 = E[ L2 ] + d var(L2 )

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k k  1 k 2  mi pi (1 − pi ) − 2   mi m j pi p j ) n2  i =1 i =1 i =1i < j j =1  = 0.3863 + 3 0.001234 = 0.4917 k ~ CL2 = E[ L2 ] =  pi m = 0.3863 k

=  p i mi + d

i =1

~ ~ LCL2 = E[ L2 ] − d var( L2 ) k k  1 k 2 m p ( 1 − p ) − 2 mi m j p i p j )   i i   i n2  i =1 i =1 i =1i < j j =1  = 0.3863 − 3 0.001234 = 0.2809 , and so on. k

=  p i mi - d

FM – chart for the 25 samples is given in Figure 2.

Fig. 2: The FM – chart for the 25 samples.

Figure 2 shows that the process is out of control at samples 8 and 11, the respective sample sizes are 100 and 110 and the corresponding center lines and control limits are (i) CL = 0.3750, UCL = 0.4609 and LCL = 0.2891 for sample size 100. ( PR = 0.10, PPQ = 0.22, PMQ = 0.18, PGQ = 0.45, PEQ = 0.05) (ii)

CL = 0.3898, UCL = 0.4710 and LCL = 0.3086 for sample size 110. ( PR = 0.1818, PPQ = 0.1181, PMQ = 0.2091, PGQ = 0.4273, PEQ = 0.0636)

From figure 2, it is seen that the FM chart gives the first signal corresponding to 8th sample. Whereas, in p – chart the first signal for the existence of assignable causes is seen only at the 11th sample. That is, in the case of FM – chart, only 780 samples are inspected to get the first out of control signal. But, 1070 samples are to

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be inspected to get the alarm with the help of a p – chart. Thus, the FM is more economical and more sensitive in identifying any shift in the specified quality level. Conclusion: FM – chart has been proposed for linguistic data set. To draw the chart, samples of varying sizes are chosen randomly from a pre–determined set. The FM-chart has been compared with the conventional p–chart with VSS. It is shown that the FM – chart is more economical and more sensitive in giving the alarm for shift in the specified quality level. This work can be extended for Markov dependent sample sizes. REFERENCES [1]

Amirzadeh, V.; M. Mashinchi; M.A. Yaghoobi. (2008). Construction of Control Charts Using Fuzzy Multinomial Quality. Journal of Mathematics and Statistics 4 (1): pp.26-31.

[2] [3]

Costa A.F.B, (1994). “ X -chart with variable sample size”. Journal of Quality Technology, 26, pp. 155-163. Franceschini, F. and D. Romano, (1999). Control chart for linguistic variables: A method based on the use of linguistic quantifiers. International Journal of Production Research. 37: pp. 3791-3801. Gulbay, M., C. Kahraman and D. Ruan, (2004). α - cut fuzzy control charts for linguistic data. International Journal of Intelligent Systems. 19: pp.1173-1196. Kanagawa, A., F. Tamaki and H. Ohta, (1993).Control charts for process average and variability based on linguistic data. International Journal of Production Research. 31:pp. 913-922. Pandurangan A., (2002). “Some Applications of Markov Dependent Sampling Scheme in SQC”, Unpublished Ph.D Thesis, Bharathidasan University, Tiruchirappalli, Raz, T. and J.H. Wang, (1990). Probabilistic and membership approaches in the construction of control charts for linguistic data. Production Planning and Control. J. Math. & Stat. 4 (1): 26-31, 2008 31.

[4] [5] [6] [7]

Sawlapurkar, U, Reynolds, M.R.Jr, Arnold, J.C (1990). “Variable Sample Size X -charts”. Presented at the winter conference of the American Statistical Association, Orlando, FL. [9] Sivasamy, R, Santhakumaran, A and Subramanian, C (2000). “Control charts for Markov Dependent Sample Size”. Quality Engineering, 12(4), pp. 596-601. [10] Wang, J.H. and T. Raz, (1990). On the construction of control charts using linguistic data. International Journal of Production Research. 28: 477-487.

[8]

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