S Srikrishna

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IJARSE, Vol. No.4, Special Issue (02), February 2015                                     ISSN-2319-8354(E) 
673 | P a g e  
 
ACCEPTANCE SINGLE SAMPLING PLAN USING 
VAGUE PARAMETERS 
S. Srikrishna  
 Department of Mathematics, SSN College of Engineering, Chennai, (India) 
ABSTRACT 
 
The purpose of this paper is to present the acceptance single sampling plan when the fraction of nonconforming 
items is a vague number. We have shown that the operating characteristic (oc) curves of the plan whose vague 
values depends on the ambiguity proportion parameter in the lot when that sample size and acceptance numbers 
is fixed having a higher and lower bounds. Finally we have concluded the discussion by a numerical example, 
whose values are represented by a vague triangular set. 
Keywords: Statistical Quality Control, Acceptance Single Sampling, Vague Number. 
 
I INTRODUCTION 
 
In the classical set theory introduced by Cantor, German mathematician values of elements in a set are only one 
of 0 and 1. That is, for any element there are only two possibilities in or not in the set. The theory cannot handle 
the data with ambiguity and uncertainty. 
Zadeh proposed fuzzy theory in 1965. The most important feature is that fuzzy set A is a class of objects that 
satisfy a certain property each object x has a membership degree of A, denoted by . This membership 
function has the following characteristics: The single degree contains the evidence for both supporting and 
opposing x. It cannot only represent one of the two evidences but it cannot represent both at the same time too. 
In order to deal with this problem, Gau and Buehrer proposed the concept of vague set in 1993 by replacing the 
values of an element in a set with a subinterval of [0, 1]. Namely a true membership function (x) and false 
membership function (x) are used to describe the boundaries of membership degree. These two boundaries 
form a subinterval ( (x), 1- (x)) of [0, 1]. The vague set theory improves description of the object in real 
world, becoming a promising tool to deal with inexact, uncertain or vague knowledge. Many researchers have 
applied this theory to many situation such as fuzzy control decision making, knowledge discovery. And the tool 
has presented more challenging than with the fuzzy set theory in applications. 
Statistical quality control is an efficient method of improving a firm and process quality of production. Sampling 
for acceptance or rejection of a lot is an important field in statistical quality control. 
Acceptance single sampling is one of the sampling methods for acceptance or rejection which is long with 
classical attribute quality characteristic. In sampling plans, the fraction of defined items is considered as a crisp 
value but in practice, the fraction of defined items value must be known exactly. Many times these values are 
estimated or it is provided by experiment. The vagueness present in the value of p with personal judgement 
experiment or estimated may be treated by means of vague set theory. As known vague set theory is a powerful 
International Journal of Advance Research In Science And Engineering       http://www.ijarse.com 
IJARSE, Vol. No.4, Special Issue (02), February 2015                                     ISSN-2319-8354(E) 
674 | P a g e  
 
mathematical tool for modelling uncertainty with the evidence of both supporting and opposing, we define the 
imprecise proportion parameter as a vague number. With this definition, the number of non conforming items in 
the sample following a poisson distribution will have the vague parameter. 
Classical acceptance sampling plans have been studied by many researchers. They are thoroughly selaborated by 
Schilling (1982). Single sampling by attributes with relaxed requirements were discussed by Ohta and Ichihashi 
(1988),and Grzegorzewski(1998,2001b). Grzegrozewski (2000b,2002) also considered sampling plan by 
variables with fuzzy requirements. Sampling plan by attributes for vague data were considered by Hrniewicz 
(1992). 
The paper is organized as follows: section 2 gives the basic concepts of vague set theory and the arithmetic 
operations of triangular vague set. In section 3, we discuss about acceptance sampling plan with vague 
parameter. Section 4 deals with OC bands with vague parameter. Section 5 illustrates the above discussions with 
numerical examples and the conclusion is discussed in section 6. 
 
II PRELIMINARIES AND DEFINITIONS 
1.1 Basic Concepts of Vague Set.  
Let U be the universe of discourse U {u1, u2, ..., un}. A vague set  [Chen and Shiy-Ming (2003), Lu.A and 
Nu.W (2004,2005)] in U is characterized by a truth membership function   : U  and a false 
membership function   : U , where  is a lower bound of the grade of membership of ui 
derived from the evidence for ui.  is a lower bound on the negation of grade of membership of ui derived 
from the evidence against  ui such that . The grade of membership of ui in the vague set  is 
bounded by a subinterval [ . For example, a vague set  in the universe of discourse U is 
shown in the figure 1: 
      
Fig 1: Vague Set 
 
 
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Definition 2.2.  
Let  be a vague set of the universe of discourse U with truth membership function   and the false 
membership function  respectively. The vague set  is convex [Chen and Shiy-Ming (2003), Lu.A and Nu.W 
(2004,2005)], if and only if for every ui in U, 
 
, where   [0, 1]. 
Definition 2.3.  
A vague set  of universe of discourse U is called a normal vague set [Chen and Shiy-Ming (2003), Lu.A and 
Nu.W (2004,2005)], if ui  U, . That is )=0. 
Definition 2.4.  
A vague number [Chen and Shiy-Ming (2003), Lu.A and Nu.W (2004,2005)] is a vague subset in the universe 
of discourse U that is both convex and normal. 
In the following, we present the arithmetic operation of triangular vague set. 
 
2.5 Arithmetic Operation of Triangular Vague sets. 
Let us consider the triangular vague set , where the triangular vague set  can be parameterized by a tuple 
, where µ1 is the truth membership for (a,b,c) and µ2 is the negation of false 
membership for (a,b,c). For convenience, the tuple can also be abbreviated into , where 
0≤µ1≤µ2≤µ3≤µ4≤1 are as follows: 
 
                                                          
 
                                                          
Based on the above two operations, we now define the multiplication of two vague matrices. Let  and  be the 
two matrices whose entries are triangular vague sets represented as , where  is 
the truth membership of  ,  is the negation of false membership of  and 
 , where  is the truth membership of  and  is the negation of 
false membership of  respectively given by 
                                . 
Then the multiplication of  and  is defined by  
 
 
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Definition 2.6. 
Let x be a random variable having the poisson mass function P(x) stands for the probability that x X, then 
P(x)= , x=0, 1, 2, ... and the parameter >0. Now if  for  is a fuzzy number, then (x) to be 
fuzzy probability that X=x, we can find α-cut of this fuzzy number as  for every α[0, 1]. 
Let X be a random variable having the fuzzy probability distribution and  in the definition (1) be small, which 
means that all p [α] are sufficiently small, then [a, b][α] is given by 
  
 
III ACCEPTANCE SAMPLING PLAN WITH VAGUE PARAMETER 
Suppose that we want to inspect a lot with a large size of N. First take a randomized sample of size „n‟ from the 
lot, then inspect all items in the sample and the number of defective items (d) will be count down. If the number 
of observed defective items is lesser than equal to acceptance number then the lot will be accepted, otherwise 
the lot rejection. If the size of lot be large the random variable d had a binomial distribution with parameter n 
and p in which p indicates the lot s defective items. However there exists the size of sample be large and p is 
small then random variable„d‟ has a Poisson distribution with =np. So, the probability for the number of 
defective items to be exactly equal to d is 
 
 and the probability of acceptance of the lot  is 
. 
 Suppose we want to inspect a lot with the large size of N, such that the proportion of damaged item is not 
known precisely, so we represent this parameter with a vague number  with the evidence of both favourable 
and non favourable as follows: . 
A single sampling plan with a vague parameter if defined by the sample size n and acceptance number c and if 
the number of observation defective product is less than or equal to c, the lot will be acceptance. If N is a large 
number, then the number of defective items in this sample (d) has a binomial distribution with a vague value and 
if  is a small, then random variable (d) has a vague value whose parameter =n  and so the vague probability 
for number of defective items in a sample size that is exactly equal to d is defined by means of α-cut given by 
(d-defective)= , where  and  
 
and the vague acceptance probability is given by 
 
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Where is a triangular number  defined by . 
This triangular number  can be determined by means of α-cut,  
 and 
 
 
 
IV OC-BAND WITH VAGUE PARAMETER 
 
Operating characteristic curve is one of the important criteria in the sampling plan. By this curve one could be 
determined the probability of acceptance or rejection of a lot having some specific defective items. The OC 
curve represents the performance of the acceptance sampling plans by plotting the probability of acceptance a 
lot versus its production quality which is expressed by the proportion of non conforming items in the lot [B.P.M. 
Duate, P.M. Saraiva (2008)]. OC curve aids in selection of plans that are effective in reducing risk and indicates 
discriminating power of the plan. 
Suppose that the event A is the event of acceptance of a lot. Then the vague probability of acceptance a lot in 
terms of fraction of defective items having vague values represented as a triangular vague value. The uncertainty 
degree of a proportion parameter is one of the factors that bandwidth depends on that. Knowing the uncertainty 
degree of proportion parameter with the evidence of acceptance membership value and non acceptance 
membership value and the variation of its position as horizontal axis we have different vague number 
 and hence we will have different proportion (P) which the OC bands are plotted in terms 
of it. To achieve this aim, we consider the structure of  as follows: 
, where  which can be obtained by using α-cuts. 
Let = . Then  with which variation k in 
the domain of the interval between 0 and 1- a3. The  OC band is plotted according to the following calculation: 
The triangular number can be obtained by using α-cut as follows: 
 
                                        
                          and  are obtained as  
                                                                          using α-cuts, 
, where      
                                                                    
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IJARSE, Vol. No.4, Special Issue (02), February 2015                                     ISSN-2319-8354(E) 
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V EXAMPLE 
Example 5.1. 
Let us assume that experience of a management company shows that half percent are ill-packed. Major 
customers choose and inspect 60 items of this product available in a large store to buy them. If the number of 
non conforming items in this sample equals zero or one, the customers will buy all products in the store. If the 
non confirming increases, the customer will not buy them. Because of the proportion of defective products has 
explained linguistically we can consider that as a vague number  
Therefore the probability of purchasing will be described in the following: 
If n=60, c=1, then             
                                           = ([0.3α,0.6-0.3α],0.85,0.8) and 
                                         . 
Using 0-cut and 1-cut, we get  means that having more evidence of 
acceptance memberships, it is expected that for every 100 lots in such a process 88 to 100 lots will be accepted.   
In this example, if we take c = 1 for a2 = 0.005, a3 = 0.01, then we have at 0 - cut 
 and 
 
Table 5.1: Vague Probability of Acceptance 
k 
  
0 <(0,0.05,0.1);0.9;0.85> <(0.8781,0.939,1);0.9;0.8> 
0.01 <(0.01,0.015,0.02);0.9;0.85> <(0.6626,0.7735,0.8781);0.9;0.85> 
0.02 <(0.02,0.025,0.03);0.9;0.85> <(0.4628,0.5627,0.66260;0.9;0.8> 
0.03 <(0.03,0.035,0.04);0.9;0.85> <(0.3084,0.3856,0.4628);0.9;0.8> 
0.04 <(0.04,0.045,0.05);0.9;0.85> <(0.1991,0.2575,0.4628);0.9;0.8> 
0.05 <(0.05,0.055,0.06);0.9;0.85> <(0.1257,0.16240.1991);0.9;0.8> 
 
Fig 2 shows the OC band of the example. This figure represents that then the OC band having high evidence of 
grade of membership and negation of grade of membership against the evidence, when the process quality 
decrease from a very good state to a moderate state. 
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Fig 2: OC Band for a single Sampling Plan with Vague Parameter 
 
Example 5.2: Suppose that c = 0, n = 20 in example 5. 1, then we have a2 = 0.005, a3 = 0.01 
 and , for 0 ≤ k ≤ 0.99. 
Therefore OC curve in terms of vague probabilities is as 
follows:  
 
Table 5.2: Vague Probability of Acceptance 
k 
 
0 <(0.8787,0.9094,1);0.85;0.9> 
0.01 <(0.6703,0.7445,0.8787);0.85;0.9> 
0.02 <(0.5488,0.6095,0.6703;0.85;0.9> 
0.03 <(0.4493,0.4991,0.5488);0.85;0.9> 
 
 
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Fig 3: OC Band for a single Sampling Plan with Vague Parameter 
 
The above figure shows OC band for n = 20 indicating that 0.85 is the lower bound for the membership of 
evidence for the OC bands and 0.9 is the upper bound for the negation of membership against the evidence for 
the vague probability of acceptance for proportion of defective items whose flow follows reduction of values 
and it will be more the increase of n. 
 
VI CONCLUSION 
 
In this paper, we have proposed a method for designing acceptance single sampling plan with vague 
quality characteristic using Poisson distribution represented as triangular vague probabilities. As it 
was shown that OC curves of the plan is like a band having a higher and lower bounds, we had shown 
that we made an attempt to capture the vagueness of the plan of OC bands. 
 
REFERENCES 
 
[1] E.G. Schiling, Acceptance Sampling Quality Control  Dekker, New York 1982. 
[2] H. Ohta ,H. Ichihashi Determination of single-sampling attribute plans based on membership function, Int. J. 
Prod, Res 26, 1477-1485,1998,. 
[3] P. Grzegorzewski A soft design of acceptance sampling by attributes, in: proceedings of the VIth 
international workshop on intelligent statistical quality control Wurzburg, September 14-16 , PP.29-38, 
1988. 
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IJARSE, Vol. No.4, Special Issue (02), February 2015                                     ISSN-2319-8354(E) 
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[4] P. Grzegorzewski  Acceptance sampling plans by attributes with fuzzy risks and quality levels, in: Frontiers 
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[5] P. Grzegorzewski  A soft design of acceptance sampling plans by variables, in: technologies for contrueting 
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[6] O. Hryniewisz,  statistical acceptance sampling with uncertain information from a sample and fuzzy quality 
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[7] Chen., and Shyi-Ming., “Analysing Fuzzy System Reliability using Vague Set Theory”, International 
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[8] Lu.A and Ng.W., “ Managing Merged Data by Vague Functional Dependencies”, Proceeding of Conceptual 
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[9] Lu. A and Ng.W., “ Vague Sets or Intuitionistic Fuzzy Sets for Handling Vague Data which one is better”, 
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[10] B.P.M. Duate, P.M. Saraiva, “An optimization basd approach for designing attribute acceptance sampling 
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ACCEPTANCE SINGLE SAMPLING PLAN USING VAGUE PARAMETERS

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ACCEPTANCE SINGLE SAMPLING PLAN USING VAGUE PARAMETERS

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