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Chain Sampling Plan Using Fuzzy Probability Theory



Ezzatallah Baloui Jamkhaneh and Bahram Sadeghpour Gildeh
 
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ABSTRACT

The acceptance sampling plan problem is one of the most important components in statistical quality control. One of these methods is chain sampling. This paper extends the concept of chain sampling plan when the proportion defective products are a trapezoidal fuzzy number for wider application. This paper has provided the definition, calculation and draw of the Operating Characteristics (OC) curve by using the concept of fuzzy probability. It is shown that the Operating Characteristic (OC) curve of the plan is like a band having high and low bounds, their width depends on the ambiguity proportion parameter in the lot and parameter i. Finally, many examples were used and then compared the OC bands for some value of i and for Poisson and binomial distributions.

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  How to cite this article:

Ezzatallah Baloui Jamkhaneh and Bahram Sadeghpour Gildeh, 2011. Chain Sampling Plan Using Fuzzy Probability Theory. Journal of Applied Sciences, 11: 3830-3838.

DOI: 10.3923/jas.2011.3830.3838

URL: https://scialert.net/abstract/?doi=jas.2011.3830.3838
 
Received: October 30, 2011; Accepted: December 10, 2011; Published: January 03, 2012



INTRODUCTION

Statistical Quality Control (SQC) is the most popular application of statistical methods. Acceptance sampling plan is a method of measuring random samples of products applied in SQC and improvement. Acceptance sampling plan is an important part of quality management for industrial and business purposes helping decision making process. It is used to improve quality of products during production. Several sampling procedures are available in the literature of acceptance sampling for the application of attribute quality characteristics (Schiling, 1982). Chain sampling plan (Chsp) is one of the sampling methods for acceptance or rejection with classical attributes of quality characteristics.

For situation in which testing is destructive or very expensive, sampling plans with small sample sizes are usually selected. These small sample size plans often have acceptance numbers of zero. Plans with zero acceptance numbers are often undesirable. However, in that their OC curves are convex throughout. This means that the probability of lot acceptance begins to drop very rapidly as the lot proportion defective becomes greater than zero. This is often unfair to the producer and in situation where rectifying inspection is used, it can require the consumer to screen a large number of lots that are essentially of acceptable quality (Montgomery, 1991). Dodge (1955) suggested an alternate procedure, known as chain sampling that might be a substitute for ordinary single sampling plans with zero acceptance numbers in certain circumstances. This is especially desirable in a situation in which small samples are demanded because of economic or physical difficulties for obtaining a sample. The chain sampling plan is characterized by the parameters n and i, where n is the sample size and i is the number of preceding samples with zero defective. Chain sampling plan will be useful when testing is costly or destructive. Chain sampling plan allows significant reduction in sample size under conditions of a continuing succession of lot from a stable producer. Chain sampling plan, in traditional form, is based on the crispness of parameter but problems of sampling plan have both random and fuzzy nature. Fuzzy acceptance sampling procedures mentioned in the previous section have been proposed for working with imprecise parameter. Karwowski and Evans (1986) identified three key reasons why fuzzy theory is relevant to production management, that are as follows:

Imprecision and vagueness are inherent to the decision maker's mental model
In the production management environment, the information required formulating a model's objective, decision variables, constraints and parameters may be vague or not precisely measurable
Imprecision and vagueness as a result of personal bias and subjective opinion may further dampen the quality and quantity of available information

In traditional sampling plan, the proportion of defective is generally assumed to be a crisp value. However, real parameters are usually vague and the assumptions are too rigid so working by such traditional methods is inaccurate. Recently, neural networks, genetic algorithms and fuzzy logic have attracted more attention and have been successfully employed in manufacturing.

Using the OC curve, Dodge (1955) has studied the properties of the chain sampling plan. Clark (1960) has presented additional OC curves which cover most of the situations. Soundararajan (1978) has described procedures and tables for construction and selection of chain sampling plans indexed by specified parameters. In early research, Fuzzy Logic was mainly used in acceptance sampling and statistical process control (Ohta and Ichihashi, 1998; Chakraborty, 1992, 1994; Kanagawa and Ohta, 1990; Kanagawa et al.,1993; Wang and Chen, 1995; Tamaki et al., 1991; Kanagawa and Ohta, 1990; Grzegorzewski, 1998, 2001; Grzegorzewski, 2001, 2002) also considered sampling plans by variables with fuzzy requirements. Sampling plans by attributes for vague data were considered by Hryniewisz (1992, 1994). Hryniewisz. (2008) provided a short overview of basic problems of statistical quality control that have been solved by using of the probability theory and the fuzzy sets theory. Statistical Quality control considered by Al-Nasser and Al-Rawwash. (2007), Mahabubuzzaman et al. (2002), (Subramaniam and Arumugam (2006) Srinivasa Rao. (2011). Finally the properties of sampling plan under situations involving both impreciseness and randomness by using the theory of chance was studied by Sundaram (2009). Jamkhaneh et al. (2011a and b) considered acceptance sampling plan under the conditions of the fuzzy parameter. They showed that the OC curve of the plan for every alpha-cut are like a band having high and low bounds whose width depends on the ambiguity proportion parameter. The aim of this paper is to generalize the classical framework of chain sampling plan by attributes to chain sampling plan based on fuzzy quality. In the acceptance sampling plan, it is well known that the probability distribution plays a crucial role. Here, the fuzzy probability theory attributed to Buckley (2003) is used to explore the possibility of introducing a suitable sampling plan for a situation having impreciseness and the operating characteristic of a chain sampling plan calculating have been by using the concept of fuzzy probability. According to Buckley's definition, the number of defective items in the sample with fuzzy parameter Image for - Chain Sampling Plan Using Fuzzy Probability Theory has a fuzzy binomial probability mass function.

DEFINITIONS

Let X = {χ1,..., χn} be a finite set and Ρ be a probability function defined on all subsets of X with

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

and:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

X together with Ρ is a discrete (finite) probability function. If B be a subset of X, we have:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

In practice all the Ki 's values must be known exactly. Many times these values are estimated, or they are provided by experts. We now assume that the Ki 's values are uncertain and we will model this uncertainty using fuzzy set theory (Buckley, 2003).

Definition 1: (Dubis and Prade, 1978): The fuzzy subset Image for - Chain Sampling Plan Using Fuzzy Probability Theory of real line R, with the membership function Image for - Chain Sampling Plan Using Fuzzy Probability Theory is a fuzzy number if and only if (a) Image for - Chain Sampling Plan Using Fuzzy Probability Theory is normal, (b) Image for - Chain Sampling Plan Using Fuzzy Probability Theory is convex (c) Image for - Chain Sampling Plan Using Fuzzy Probability Theory is upper semi continuous and (d) supp (Image for - Chain Sampling Plan Using Fuzzy Probability Theory ) is bounded.

Definition 2: (Dubis and Prade, 1978): The α-cut of a fuzzy number Image for - Chain Sampling Plan Using Fuzzy Probability Theory is a non-fuzzy set defined as Image for - Chain Sampling Plan Using Fuzzy Probability Theory. Hence, we have Image for - Chain Sampling Plan Using Fuzzy Probability Theorywhere Image for - Chain Sampling Plan Using Fuzzy Probability Theory, Image for - Chain Sampling Plan Using Fuzzy Probability Theory.

Definition 3: (Dubis and Prade, 1978; Hatami-Marbini et al., 2009): A trapezoida fuzzy number is a fuzzy number that its membership function defined by four values, a1≤a2≤a3≤a4 where the base of the trapezoid is the interval [a1,a4] and its top (where the membership equals one) is over [a2,a3] such that we can describe a membership function in the following manner:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
(1)

Trapezoidal fuzzy numbers with a2 = a3 are called triangular fuzzy numbers.

Definition 4: (Buckley, 2003): Let Image for - Chain Sampling Plan Using Fuzzy Probability Theory's, i=1,...,n are fuzzy numbers and probability of event X = x is uncertain, then random variable X together with the Image for - Chain Sampling Plan Using Fuzzy Probability Theory value is a discrete fuzzy probability function. We write Image for - Chain Sampling Plan Using Fuzzy Probability Theory for fuzzy Ρ and Image for - Chain Sampling Plan Using Fuzzy Probability Theory , where the reference of Image for - Chain Sampling Plan Using Fuzzy Probability Theory is [0,1]. Let B = {x1, ...., xl}be a subset of X and then define:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
(2)

where, stands for the statement:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

this is our restricted fuzzy arithmetic.

Definition 5: (Buckley, 2003): Let p be the probability of a "success" in each Bernoulli trial, is not known precisely. We substitute Image for - Chain Sampling Plan Using Fuzzy Probability Theory instead of Ρ and Image for - Chain Sampling Plan Using Fuzzy Probability Theory for q, so that there is a Image for - Chain Sampling Plan Using Fuzzy Probability Theory and a Image for - Chain Sampling Plan Using Fuzzy Probability Theory with p + q = 1. Now let Image for - Chain Sampling Plan Using Fuzzy Probability Theory be the fuzzy probability of r successes in m independent trials of the experiment. Under our restricted fuzzy algebra the fuzzy binomial probability mass function is defined as:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
(3)

where, now S is the statement, Image for - Chain Sampling Plan Using Fuzzy Probability Theory

if Image for - Chain Sampling Plan Using Fuzzy Probability Theory then:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
(4)

And if Image for - Chain Sampling Plan Using Fuzzy Probability Theory be the fuzzy probability of χ successes so that a ≤ χ ≥ b, then:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

then:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
Image for - Chain Sampling Plan Using Fuzzy Probability Theory

where, S is the same with past case.

Definition 6: (Buckley, 2006): Let χ be a random variable having the Poisson mass function. If P (χ) stands for the probability that X = χ, then:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

for χ = 0,1,2,... and parameter λ > 0.

Now substitute fuzzy number for λ to produce the fuzzy Poisson probability mass function. Let to be the fuzzy probability that X = χ. Then we find α-cut of this fuzzy number as:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

Let X be a random variable having the fuzzy binomial distribution and Image for - Chain Sampling Plan Using Fuzzy Probability Theory in the definition 5 be small which means that all Image for - Chain Sampling Plan Using Fuzzy Probability Theory are sufficiently small. Let Image for - Chain Sampling Plan Using Fuzzy Probability Theory be the fuzzy probability that a≤X≤b. Also set Image for - Chain Sampling Plan Using Fuzzy Probability Theory using the fuzzy Poisson approximation. Then:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
(5)

Example 1: Suppose that m = 4, Image for - Chain Sampling Plan Using Fuzzy Probability Theory("about 0.18-0.22") for Ρ and Image for - Chain Sampling Plan Using Fuzzy Probability Theory for q. Now we will calculate the fuzzy probabilities Image for - Chain Sampling Plan Using Fuzzy Probability Theory and Image for - Chain Sampling Plan Using Fuzzy Probability Theory, where B = {0,1}. By using the equations of Definition 5, we have:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

where:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

Since:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

on:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

We obtain:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
Fig. 1: The fuzzy probabilities Image for - Chain Sampling Plan Using Fuzzy Probability Theory

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

And then:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

Figure 1 shows the membership function of fuzzy probabilities Image for - Chain Sampling Plan Using Fuzzy Probability Theory in Example 1 by using fuzzy binomial distribution. We have also:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

where:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

Since:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

on Image for - Chain Sampling Plan Using Fuzzy Probability Theory we obtain:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

when α = 0, we obtain Image for - Chain Sampling Plan Using Fuzzy Probability Theory.

Suppose that m = 40 in Example 1 then by using fuzzy Poisson distribution, we have:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

where:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

since, Image for - Chain Sampling Plan Using Fuzzy Probability Theory, we obtain:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

Chain sampling plan with fuzzy parameter: In this section, first we introduce the chain sampling plan for classical attributes characteristics. Suppose that we want to inspect a lot with a size of N. First, we choose and inspect a random sample of size n and then the number of defective items (D) will be counted. The procedure of Chsp-1 is as follows:

If the number of observed defective items (d) is equal zero, then the lot will be accepted
If the number of observed defective items is equal two or more, then the lot will be rejecting
If the number of observed defective items is equal one, accept the lot provided there have been no defective items in the previous i lots. In practice, values of i are usually between three and five

If the size of the lot is very large, the random variable D has a binomial distribution with parameters n and p, in which p indicates the proportion of the defective items in the lot. So, the probability for the number of defective items to be exactly equal to d is:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
(6)

and the probability of acceptance of the lot is:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
(7)

If p be small and n be great, then the random variable D has a Poisson distribution with parameter λ = np:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

Suppose that we want to inspect a lot of size of N, where the proportion of defective items or the probability of defectiveness is not known precisely and which has some uncertain value. So we represent this parameter with a trapezoidal fuzzy number Image for - Chain Sampling Plan Using Fuzzy Probability Theory as follows:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
(8)

A chain sampling plan with a fuzzy parameter is defined by the sample size n and i that i is the number of preceding samples with zero defective. If conditions first and third is established then the lot will be accepted. If N is a large number, then the number of defective items in this sample has a fuzzy binomial probability distribution (Buckley, 2006). So the fuzzy probability acceptance is:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
(9)

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
(10)

for 0≤α≤1, where, S stands for the statement:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
(11)

and if Image for - Chain Sampling Plan Using Fuzzy Probability Theory be small, then the random variable D has a fuzzy Poisson distribution with fuzzy parameter Image for - Chain Sampling Plan Using Fuzzy Probability Theory (Buckley, 2006). Then the probability of acceptance of the lot is:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
(12)

Example 2: A Company is packaging its products in boxes. In each box there are twenty products. The management of this firm believes that approximately between 1 and 2% of the products have problems packaging. A customer who wants to buy one of these lots of boxes, must choose one box randomly and then investigates, if all the products of that box are good then he will buy the lot or if the number of observed defective items is equal one, accept the lot provided there have been no defective items in the previous 3 lots, otherwise this lot will be rejected. Because the proportion of defective products was explained linguistically, we can consider that as a fuzzy number Image for - Chain Sampling Plan Using Fuzzy Probability Theory= (0,0.01,0.02,0.03) and accordingly we will have Image for - Chain Sampling Plan Using Fuzzy Probability Theory= (0.97,0.98,0.99,1). Then the α-cut of the fuzzy probability of lot acceptance is:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
Fig. 2: The fuzzy probability of lot acceptance for a Chsp with fuzzy parameter Image for - Chain Sampling Plan Using Fuzzy Probability Theory

Under α = 0 we obtain Image for - Chain Sampling Plan Using Fuzzy Probability Theory (3)[0] = [0.5979,1], i.e., it is expected that for every 100 lots in such a process, 60 to 100 lots will be accepted. And under α = 1 we obtain Image for - Chain Sampling Plan Using Fuzzy Probability Theory(3)[1] = [0.7487,0.9083]. Figure 2 shows the membership function of fuzzy probability of lot acceptance in Example 2 by using fuzzy binomial distribution. By using fuzzy Poisson distribution the α-cut of the fuzzy probability of lot acceptance is:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

under α = 0 we obtain Image for - Chain Sampling Plan Using Fuzzy Probability Theory (3)[0] = [0.6032, 1], i.e., it is expected that for every 100 lots in such a process, 60 to 100 lots will be accepted. And under α = 1 we obtain Image for - Chain Sampling Plan Using Fuzzy Probability Theory (3)[0] = [0.7511, 0.9085].

Fuzzy operating characteristic (FOC) band: Operating characteristic curve plots the probability of accepting the lot (Y-axis) versus the lot fraction defectives (X-axis) under the specified sampling plan considered during management of a project. The OC curve is the primary tool for displaying and investigating the properties of a sampling plan. Other applications of the OC curve are:

Operating characteristic curve describes how well an acceptance plan discriminates between good and bad lots
Operating characteristic curve aids in the selection of plans that are effective in reducing risks
The critical points, producer's risk and customer's risk are determined by the OC curve

Suppose that the event B is the event of lot acceptance. Then the fuzzy probability of lot acceptance in terms of fuzzy proportion of defective items would be a band with upper and lower bounds. Hence, we call it Fuzzy Operating Characteristic (FOC) band. The uncertainty value of a proportion parameter is one of the factors that the bandwidth depends on. The less uncertainty value results in less bandwidth and if the proportion parameter gets a crisp value, the lower and upper bounds will become equal and the OC curve will be in classical state. Knowing the uncertainty value of the proportion parameter (a1, a 2, a 3, a 4) and the variation of its position on the horizontal axis, we have a different fuzzy number (Image for - Chain Sampling Plan Using Fuzzy Probability Theory)on which the FOC band is plotted in terms of it. To achieve this aim we consider the structure of Image for - Chain Sampling Plan Using Fuzzy Probability Theory as follows:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
(13)

where, bi = ai - a1, i = 2, 3, 4 and kε[0,1-b4]. The α-cut of FOC band is plotted according to the values of the following fuzzy probability:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
(14)

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
(15)

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
(16)

Example 3: In Company related to Example 2, we have assumed that Image for - Chain Sampling Plan Using Fuzzy Probability Theory =(0,005,0.01,0.15,0.02) then:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

and:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

Image for - Chain Sampling Plan Using Fuzzy Probability Theory


Image for - Chain Sampling Plan Using Fuzzy Probability Theory
Fig. 3: FOC band for a chain sampling plan with fuzzy parameter

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
Fig. 4: FOC band for a chain sampling plan with fuzzy binomial and Poisson distribution

By using fuzzy Poisson distribution the α-cut of the fuzzy probability of lot acceptance is:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

Figure 3 compared α-cut of FOC bands of fuzzy binomial distribution for Chsp-1 and SSP. Table 1 shows α-cut (α = 0) of the fuzzy probability of lot acceptance for some value of i and for Poisson and binomial distributions.

Figure 4 and 5 compared α-cut of FOC bands of fuzzy binomial distribution and fuzzy Poisson distribution for different i. Table 2 shows α-cut (α = 0) of the fuzzy probability of lot acceptance for some value of Image for - Chain Sampling Plan Using Fuzzy Probability Theory and for Poisson and binomial distributions.

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
Fig. 5: FOC band for a chain sampling plan with fuzzy binomial and Poisson distribution

Table 1: Fuzzy probability of acceptance with k = 0.005, n = 5 and α = 0
Image for - Chain Sampling Plan Using Fuzzy Probability Theory

Table 2: Fuzzy probability of acceptance with I = 3, n = 5 and α = 0
Image for - Chain Sampling Plan Using Fuzzy Probability Theory

Table 1 and 2 shows that OC band with using from fuzzy Poisson distribution optimal approximate for OC band with using from fuzzy binomial distribution for the proportion of defective items with small fuzzy numbers. With regard this; such plan can be designed based on OC fuzzy Poisson distribution.

However, with increasing Image for - Chain Sampling Plan Using Fuzzy Probability Theory this approximation is weaker and it will be more with the increase of i.

Rectifying chain sampling inspection plan: Rectifying inspection serve to improve lot quality. Under rectifying sampling inspection plan whenever we accept a lot we replace all the defective items encountered in the sample by good items. Whereas rejected lot are sent for 100% inspection and all the defectives encountered are replaced by good items.

Fuzzy average outgoing quality: The random variable, Outgoing Quality (OQ), is defined as the ratio of the number of nonconforming items in the output of this process after sampling inspection and rectification to the total number of items in the lot. The fuzzy probability distribution of OQ is then:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
(17)

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
(18)

where:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
(19)

The fuzzy mathematical expectation of OQ is obtained from Eq. 17 as:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
(20)

The fuzzy mathematical expectation of OQ is named fuzzy average outgoing quality (FAOQ).

Example 4: Suppose that the size of lot be equal to 100 and n = 5, i = 3, Image for - Chain Sampling Plan Using Fuzzy Probability Theory = (0, 0.01, 0.02, 0.03), then fuzzy average outgoing quality is as follows:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

under α = 1 we obtain:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

Thus, rectifying chain sampling inspection plan changes the quality of the lots in percent defective from "about 1 to 2 percent" to "about 0.94 to 1.85 percent" on the average. Figure 6 shows the membership function of FAOQ in Example 4 by using fuzzy binomial distribution.

Fuzzy average total inspection: The random variable, Total Inspection (TI), is defined as the number of inspection items in the output of this process after sampling inspection and rectification in the lot (Fig. 7). The fuzzy probability distribution of TI is then:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
(21)

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
Fig. 6: Fuzzy average outgoing quality with N=100, n=5, i=3, Image for - Chain Sampling Plan Using Fuzzy Probability Theory = (0 ,0.01, 0.02, 0.03)

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
Fig. 7: Fuzzy average total inspection with N=200, n=5, i=3, Image for - Chain Sampling Plan Using Fuzzy Probability Theory= (0, 0.01, 0.02, 0.03)

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
(22)

The fuzzy mathematical expectation of TI is obtained from Eq. 22 as:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory
(23)

The fuzzy mathematical expectation of TI is named fuzzy average total inspection (FATI).

In Example 4, fuzzy average total inspection is as follows:

Image for - Chain Sampling Plan Using Fuzzy Probability Theory

under α = 0 we obtain FATI [0] = [5, 10.43] and FATI [1] = [5.7316, 7.6558].

CONCLUSION

We proposed a chain acceptance sampling plan based on a fuzzy parameter by using fuzzy probability theory. We modeled this parameter using trapezoidal fuzzy number. The main result is that this plan yields a fuzzy environment consistent with classical views thereof. We calculated the operating characteristic curve of a chain sampling plan and the acceptance probability by using the concept of fuzzy probability. Finally, we have shown that in our plan, the α-cut of FOC band is like a band having high and low bounds.

REFERENCES

1:  Al-Nasser, A.D. and M. Al-Rawwash, 2007. A control chart based on ranked data. J. Applied Sci., 7: 1936-1941.
CrossRef  |  Direct Link  |  

2:  Buckley, J.J., 2003. Fuzzy Probability: New Approach and Application. Physica-Verlage, Heidelberg, Germany

3:  Buckley, J.J., 2006. Fuzzy probability and Statistics. Springer-Verlag, Heidelberg. Berlin, Germany, Pages: 270

4:  Clark, C.R., 1960. OC Curve for ChSP-1 chain sampling plans. Ind. Qual. Control, 17: 10-12.

5:  Chakraborty, T.K., 1992. A class of single sampling plans based on fuzzy optimization. Opsearch, 29: 11-20.

6:  Chakraborty, T.K., 1994. Possibilistic parameter single sampling inspection plans. Opsearch, 31: 108-126.

7:  Dodge, H.F., 1955. Chain sampling inspection plans. Ind. Qual. Control, 11: 10-13.

8:  Grzegorzewski, P., 1998. A soft design of acceptance sampling by attributes. Proceedings of the 6th International Workshop on Intelligent Statistical Quality Control, September 14-16, 1998, Wurzburg, pp: 29-38

9:  Grzegorzewski, P., 2001. Acceptance Sampling Plans by Attributes with Fuzzy Risks and Quality Levels. In: Frontiers in Statistical Quality Control, Wilrich, P. Th and H.J. Lenz (Eds.). Springer, Heidelberg, pp: 36-46

10:  Grzegorzewski, P., 2002. A Soft Design of Acceptance Sampling Plans by Variables. In: Technologies for Constructing Intelligent Systems, Bouchon-Meunier, B. (Ed.). Vol. 2, Physica-Verlag GmbH Heidelberg, Germany, pp: 275-286

11:  Hatami-Marbini, A., S. Saati and A. Makui, 2009. An application of fuzzy numbers ranking in performance analysis. J. Applied Sci., 9: 1770-1775.
CrossRef  |  Direct Link  |  

12:  Hryniewisz, O., 1992. . Statistical acceptance sampling with uncertain information from a sample and fuzzy quality criteria. Working Paper of SRI PAS, Warsaw, (In Polish).

13:  Hryniewisz, O., 1994. Statistical Decisions with Imprecise Data and Requirements. In: Systems Analysis and Decisions Support in Economics and Technology, Kulikowski, R., K. Szkatula and J. Kacprzyk (Eds.)., Omnitech Press, Warszawa, pp: 135-143

14:  Hryniewisz, O., 2008. Statistics with fuzzy data in statistical quality control. Soft Comput., 12: 229-234.
CrossRef  |  

15:  Kanagawa, A. and H. Ohta, 1990. . A design for single sampling attribute plan based on fuzzy set theory. Fuzzy Sets Syst., 37: 173-181.
CrossRef  |  Direct Link  |  

16:  Kanagawa, A., F. Tamaki and H. Ohta, 1993. Control Charts for processing average and variability based on linguistics data. Int. J. Product. Res., 31: 913-922.

17:  Mahabubuzzaman, A.K.M., L.B. Lutfar, M.K. Kabir and Z. Ahmed, 2002. A comparative study on the quality control of jute yarn conventional drawing method vs modern drawing method. Asian J. Plant Sci., 1: 646-647.
CrossRef  |  Direct Link  |  

18:  Montgomery, D.C., 1991. Introduction to Statistical Quality Control. Wiley, New York, USA., Pages: 678

19:  Ohta, H. and H. Ichihashi, 1998. Determination of single-sampling-attribute plans based on membership functions. Int. J. Product. Res., 26: 1477-1485.
CrossRef  |  Direct Link  |  

20:  Sundaram, S., 2009. Hybrid single sampling plan. World Applied Sci. J., 6: 1685-1690.
Direct Link  |  

21:  Schiling, E.G., 1982. Acceptance sampling quality control. Dekker, New York. USA

22:  Soundararajan, V., 1978. Producer's and tables for the construction and selection of chain sampling plan. J. Qual. Technol., 10: 56-60.

23:  Srinivasa Rao, G., 2011. A hybrid group acceptance sampling plans for lifetimes based on generalized exponential distribution. J. Applied Sci., 11: 2232-2237.
CrossRef  |  

24:  Subramaniam, R. and N. Arumugam, 2006. Sequential sampling technique for decision making in the management of cotton aphids: Aphids gossypii Glover. J. Entomol., 3: 254-260.
CrossRef  |  Direct Link  |  

25:  Tamaki, F., A. Kanagawa and H. Ohta, 1991. A fuzzy design of sampling inspection plans by attributes. Jpn. J. Fuzzy Theory Syst., 3: 315-327.

26:  Jamkhaneh, E.B., B. Sadeghpour-Gildeh and G. Yari, 2011. Acceptance single sampling plan with fuzzy parameter. Iran. J. Fuzzy Syst.,

27:  Jamkhaneh, E.B., B. Sadeghpour-Gildeh and G. Yari, 2011. Inspection error and its effects on single sampling plans with fuzzy parameters. Struct. Multidisciplinary Optimizat., 43: 555-560.
CrossRef  |  

28:  Karwowski, W. and G.W. Evans, 1986. Fuzzy concepts in production management research: A review. Int. J. Prod. Res., 24: 129-147.
CrossRef  |  Direct Link  |  

29:  Wang, R.C. and C.H. Chen, 1995. Economic statistical np-control chart designs based on fuzzy optimisation. Int. J. Qual. Reliability Manage., 12: 82-92.
CrossRef  |  Direct Link  |  

30:  Dubois, D. and H. Prade, 1978. Operations on fuzzy numbers. Int. J. Syst. Sci., 9: 613-626.
CrossRef  |  Direct Link  |  

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