# Design of generalized fuzzy multiple deferred state (GFMDS) sampling plan
for attributes

Julia Thampy Thomas  Mahesh Kumar Corresponding author. E-mail:
<EMAIL_ADDRESS>Department of Mathematics, National Institute of Technology,
Calicut, India 673601
E-mail<EMAIL_ADDRESS>Department of Mathematics, National
Institute of Technology, Calicut, India 673601
E-mail<EMAIL_ADDRESS>
###### Abstract

A sampling plan is a pilot tool for a supply and demand chain quality check
strategy. These plans proved to be economically viable for the quality
inspection processes but the uncertainty in the plan parameters challenged the
reliability of the application of traditional acceptance sampling plans. This
study proposes a generalized fuzzy multiple deferred state (GFMDS) sampling
plan for attributes that consider the ambiguity in determining the exact value
of the percentage of defectives in a batch. The performance measures have been
derived and the plan is designed in terms of a minimum average sample number.
A comparison study is done over the existing fuzzy acceptance sampling plans
for attributes and a pertinent observation is made regarding the efficiency of
the GFMDS scheme. The effect of inspection errors on the sampling procedure is
analyzed and the drop in the acceptance criteria of the plan is observed
corresponding to the intensified inspection errors. Several numerical examples
are presented to validate the results.

###### keywords:

Fuzzy acceptance sampling plans, average sample number, AQL, LQL, inspection
errors

and

##  1 Introduction

Logistics and supply chain management are integral parts of manufacturing
systems and the quality assurance of the products is often challenging.
Acceptance sampling plans are carried out extensively in industrial quality
assurance processes. The lower cost of inspection and minimal risks makes
acceptance sampling plans more attractive. The quantitative measurements of
the inspection process are handled by acceptance sampling plan for variables
whereas the qualitative nature is inspected by attribute sampling plans.
Lowering inspection costs depends on the proportion of defectives and the
sample size. Facilitating an acceptance sampling plan with the lowest average
samples inspected is an efficient way to ensure quality control from an
economic perspective.

The ease of application of attribute sampling plans in industries made them
more popular than variable sampling plans for which more information is
needed. The history of acceptance sampling plans for attributes dated long
back when a method of sampling inspection was introduced [19]. Single sampling
plans (SSP) and double sampling plans (DSP) were the pioneers in acceptance
sampling schemes (see, [20]) and eventually the limitations of these sampling
schemes were identified and rectification is carried out. The information
furnished by the current sample was the determining factor in the sampling
plans designed initially. The chain sampling scheme used cumulative
information of several samples to accept or reject a lot [18]. The complexity
of chain sampling is relaxed when a new acceptance sampling scheme was
designed where the stages of inspection were inter-dependent (see [36]), which
later evolved through the design of multiple deferred state (MDS) sampling
plans as seen in [35]. The stages of attribute MDS acceptance plan are
acceptance, rejection, or conditional acceptance of the lot. The advantage of
MDS plans over SSP and DSP is the reduced sample size. In the MDS plan, the
sample information from the previous samples is employed to conclude the
decision on the lot submitted. The selection of optimal parameters for MDS
plans is studied in [33] and [30]. The determination of plan parameters
concerning the acceptable quality level (AQL) and the limiting quality level
(LQL) is studied in [25]. Later, the MDS plan is used to develop an attribute
control chart (see [10]). Variable sampling schemes using the MDS method were
also developed and gained relevance over the existing plans (see [12], [11],
[37], [38] and [8]).

The crisp nature of the fraction of non-conformity’s ($p$) is not assured in
real-world supply chain models. This ambiguity often arises when a linguistic
value is used to mention the number of defectives. The development of
acceptance sampling plans on the ground of fuzzy logic and applications proved
more applicable to these cases. Fuzzy set theory is used for the mathematical
modeling of epidemiology, biostatistics, reliability engineering, statistical
quality control, and so on. Fuzzy optimization is used to develop a single
sampling plan [16]. Risks and quality attributes were considered as fuzzy
quantities to design an acceptance sampling plan by attributes [26]. A fuzzy
SSP is developed by [32] where the plan parameters were considered as fuzzy
numbers. Further, developments in fuzzy acceptance sampling plans for
attributes can be seen in [34], [28] and [27]. Consequently, the uncertainty
in the fraction of defectives in the MDS sampling plan for attributes is
addressed in [1] and [3]. A variable MDS plan with fuzzy parameters is
developed in [5].

The chances of inspection errors are often ignored in the quality control
processes. However, no sampling schemes proved perfect when applied to a
supply chain where the assumptions of sampling schemes undergo fluctuations
from their theoretical framework. The effect of classification errors in
acceptance sampling schemes was studied in [17] where the SSP was modified in
this respect. Later, many plans were rephrased by considering the chances of
misclassifications of the submitted lot for attribute sampling plans (see,
[22], [23], [24], and [7]). The effect of measurement errors on fuzzy
acceptance sampling plans were also studied [13]. The case with the MDS plan
was analyzed in [4] and its fuzzy counterpart was studied in [6]. The results
proved a notable drop in the probability of acceptance when the errors are
identified.

When the limitations of MDS sampling plans were identified, the assumptions of
the MDS sampling scheme were generalized to consider measurement data [14].
The efficiency of this plan over the existing MDS plan versions was discussed.
The economic advantage of the plan was considered in [8] which significantly
contributed to the reduction in total cost for supply chain models. The
reduction in average sample number is the key to obtaining optimal parameters
for generalized MDS plans which eventually contributed to low inspection
costs. This research is highly motivated by the extensive applications of
fuzzy logic to the probability theory and quality assurance models. The
objective of this article is to develop a generalized fuzzy multiple deferred
state acceptance sampling plan where the limitations of the crisp nature of
$p$ in a generalized MDS plan can be resolved. The advantage of the proposed
plan is discussed by comparing the average sample number of the existing fuzzy
acceptance sampling plans. The organisation of this paper is as follows. The
forthcoming section carries some preliminaries of fuzzy sets. The assumptions
and a systematic design of the GFMDS plan is given in Section 3 where the
measures of performance are derived. Section 4 deals with the determination of
suitable parameters for the GFMDS plan and the plan is extended to generalised
fuzzy numbers as seen in Section 5. Section 6 discusses the efficiency of the
sampling plan under consideration. Further, the GFMDS plan is modified with
consideration of inspection errors and presented in Section 7. Conclusions is
mentioned in the Section 8.

##  2 Some Preliminaries

This section presents some fundamentals related to fuzzy numbers.

######  Definition 2.1.

[21] The fuzzy subset $\tilde{X}$ of real line $\mathbb{R}$ with membership
function $\mu_{\tilde{X}}:\mathbb{R}\rightarrow[0,1]$ is a fuzzy number if and
only if

  1. 1.

$\exists x_{0}\in\mathbb{R}s.t.\mu_{\tilde{X}}(x_{0})=1$

  2. 2.

$\forall x_{1},x_{2}\in\mathbb{R}$ and $\forall\lambda\in[0,1]$ implies that
$\mu_{\tilde{X}}[\alpha
x_{1}+(1-\alpha)x_{2}]\geq\min\\{\mu_{\tilde{X}}(x_{1}),\mu_{\tilde{X}}(x_{2})\\}$

  3. 3.

for each given $x\in\mathbb{R}$ and $\epsilon>0,\exists\delta>0$ s.t. $\forall
z\in\mathbb{R},|x-z|<\delta$ implies that
$\mu_{\tilde{X}}(z)<\mu_{\tilde{X}}(x)+\epsilon$

  4. 4.

support of the fuzzy subset, $\\{x|\mu_{\tilde{X}}(x)>0\\}$ is bounded.

######  Definition 2.2.

[21] The $\nu$ cut of a fuzzy number $\tilde{X}(\nu\in[0,1])$ is a crisp set
defined as $\tilde{X}[\nu]=\\{x\in\mathbb{R},\mu_{\tilde{B}}(x)\geq\nu\\}$
usually denoted as $\tilde{X}[\nu]=[X_{I}[\nu],X_{II}[\nu]]$ where
$X_{I}[\nu]=\min\\{x\in\mathbb{R},\mu_{\tilde{B}}(x)\geq\nu\\}$ and
$X_{II}[\nu]=\max\\{x\in\mathbb{R},\mu_{\tilde{B}}(x)\geq\nu\\}$

######  Definition 2.3.

[15] Let $A_{i}$ be a partition of sample space
$X={x_{1},x_{2},...,x_{n}},\tilde{P}(A_{i})=\tilde{a_{i}}$ and $D$ is an
event. Then for $\alpha\in[0,1]$ we get

$\tilde{P}(D)[\nu]=\left\\{\sum_{i=1}^{n}a_{i}P(D|A_{i}):S\right\\}$

where
$S=\\{a_{i}\in\tilde{a_{i}}[\nu],i\in{1,2,...,n},\sum_{i=1}^{n}a_{i}=1\\}$

##  3 Design of Generalized Fuzzy Multiple Deferred State (GFMDS) Attribute
Sampling Plan

This section aims to review the existing generalized MDS sampling plan for
attributes and extend the same in a fuzzy environment. The inspection errors
are considered in the next stage and the plan is further extended to inspect
the chances of mis-classification. The following are assumptions required for
the GMDS as discussed by [8]:

  1. 1.

Continuous inspection is assumed for a series of successive lots.

  2. 2.

The quality of lots under consideration is assumed to be uniform.

  3. 3.

The destructive nature of testing and the higher cost of inspection makes a
small sample size desirable.

Apart from the above-stated assumptions, the essential conditions which favor
the implementation of the GFMDS plan are:

  1. 1.

The quality level of the lots undergoing examination can be represented as a
fuzzy number. In most of the real-time experiments, the determination of exact
value is often impossible and fuzzy linguistic quantifiers are used for the
representation.

  2. 2.

The reliability of the manufacturing process gives assurance to the consumer.
The criteria for acceptance are based on satisfying pre-defined parameters.

  3. 3.

The proportion of defectives is observed to follow fuzzy binomial or fuzzy
Poisson distribution.

##  4 Design of Generalized Fuzzy Multiple Deferred State (GFMDS) Attribute
Sampling Plan

This section aims to review the existing generalized MDS sampling plan for
attributes and extend the same in a fuzzy environment. The inspection errors
are considered in the next stage and the plan is further extended to inspect
the chances of mis-classification. The following are assumptions required for
the GMDS as discussed by

  1. 1.

Continuous inspection is assumed for a series of successive lots.

  2. 2.

The quality of lots under consideration is assumed to be uniform.

  3. 3.

The destructive nature of testing and higher cost of inspection makes a small
sample size desirable.

Apart from the above-stated assumptions, the essential conditions which favor
the implementation of the GFMDS plan are:

  1. 1.

The quality level of the lots undergoing examination can be represented as a
fuzzy number. In most of the real-time experiments, the determination of exact
value is often impossible and fuzzy linguistic quantifiers are used for the
representation.

  2. 2.

The reliability of the manufacturing process gives assurance to the consumer.
The criteria for acceptance are based on satisfying pre-defined parameters.

  3. 3.

The proportion of defectives is observed to follow fuzzy binomial or fuzzy
Poisson distribution.

An overview of the GMDS plan is given as:
Step 1. From each lot, a sample of size $n$ is chosen randomly and tested
against the required criteria. The percentage of defectives is denoted by $d$.
Step 2. The lot is accepted if $d\leq c_{1}$ and rejected if $d>c_{2}$. If
$c_{1}<d\leq c_{2}$ the lot under inspection is accepted provided $k$ lots out
of $m$ previous lots have been accepted with the condition that $d\leq c_{1}$.
The lot is rejected otherwise. The GMDS plan is differentiated by the
parameters $n,k,m,c_{1}$, and $c_{2}$. The probability of acceptance of the
GMDS sampling plan for attributes is given by

$\begin{split}P_{a}(p)=P(d\leq c_{1})+P(c_{1}<d\leq c_{2})\\\
\sum_{j=k}^{m}\left[{m\choose{j}}[P(d\leq c_{1})]^{j}[1-P(d\leq
c_{1})]^{m-j}\right]\end{split}$ (1)

When the fraction of defectives is fuzzy in nature, the existing plan can be
extended with fuzzy parameter $\tilde{p}$.

###  4.1 Probability of Acceptance of GFMDS

The GFMDS plan can be designed so that the proportion of defectives is fuzzy
in nature. The assumptions and operating process remains the same except that
the random variable $d$ follows a fuzzy probability distribution.

Theorem 1
The acceptance probability under the GFMDS sampling plan is given by:

$\begin{split}\tilde{P}_{a}(\tilde{p})=\tilde{P}(d\leq
c_{1})+\tilde{P}(c_{1}<d\leq c_{2})\\\
\sum_{j=k}^{m}\left[{m\choose{j}}[\tilde{P}(d\leq
c_{1})]^{j}[1-\tilde{P}(d\leq c_{1})]^{m-j}\right]\end{split}$

Proof: The number of non-conforming items in a sample is the random variable
$d$, which is a crisp value and the fraction of defectives is a triangular
fuzzy number, $\tilde{p}=(p_{1},p_{2},p_{3})$. A sample of size $n$ is
randomly chosen and inspected. For the unconditional acceptance number
$c_{1}$, if $d\leq c_{1}$, the lot is accepted. The probability of acceptance
will be $\tilde{P}(d\leq c_{1})$. For the conditional acceptance number
$c_{2}$, if $c_{1}<d\leq c_{2}$, the current lot is accepted provided the $k$
lots out of $m$ previous lots are accepted in the first step. Therefore, the
fuzzy probability of acceptance of $k$ lots out of $m$ lots is determined by

$\sum_{j=k}^{m}\left[{m\choose{j}}[\tilde{P}(d\leq
c_{1})]^{j}[1-\tilde{P}(d\leq c_{1})]^{m-j}\right]$

The fuzzy probability of conditional acceptance can be obtained as:

$\begin{split}\tilde{P}(c_{1}<d\leq
c_{2})\sum_{j=k}^{m}\left[{m\choose{j}}[\tilde{P}(d\leq c_{1})]^{j}\right.\\\
\left.[1-\tilde{P}(d\leq c_{1})]^{m-j}\right]\end{split}$

Hence, the fuzzy probability of acceptance of the current lot under the
proposed GFMDS sampling plan is:

$\begin{split}\tilde{P}_{a}(\tilde{p})=\tilde{P}(d\leq
c_{1})+\tilde{P}(c_{1}<d\leq c_{2})\\\
\sum_{j=k}^{m}\left[{m\choose{j}}[\tilde{P}(d\leq
c_{1})]^{j}[1-\tilde{P}(d\leq c_{1})]^{m-j}\right]\end{split}$ (2)

Under fuzzy Binomial probability distribution, equation (2) can be written as:

$\begin{split}\tilde{P_{a}}(\tilde{p})&=\sum_{d=0}^{c_{1}}\left[{n\choose{d}}\tilde{p}^{d}(1-\tilde{p})^{n-d}\right]+\\\
&\sum_{d=c_{1}+1}^{c_{2}}\left[{n\choose{d}}\tilde{p}^{d}(1-\tilde{p})^{n-d}\right]*\\\
&\sum_{j=k}^{m}{m\choose{j}}\left\\{\sum_{d=0}^{c_{1}}\left[{n\choose{d}}\tilde{p}^{d}(1-\tilde{p})^{n-d}\right]\right\\}^{j}*\\\
&\left\\{1-\sum_{d=0}^{c_{1}}\left[{n\choose{d}}\tilde{p}^{d}(1-\tilde{p})^{n-d}\right]\right\\}^{m-j}\end{split}$
(3)

Under fuzzy Poisson probability distribution, equation (2) can be written as:

$\begin{split}\tilde{P_{a}}(\tilde{p})&=\sum_{d=0}^{c_{1}}\left[\dfrac{e^{-n\tilde{p}}(n\tilde{p})^{d}}{d!}\right]+\sum_{d=c_{1}+1}^{c_{2}}\left[\dfrac{e^{-n\tilde{p}}(n\tilde{p})^{d}}{d!}\right]*\\\
&\sum_{j=k}^{m}{m\choose
j}\left\\{\sum_{d=0}^{c_{1}}\left[\dfrac{e^{-n\tilde{p}}(n\tilde{p})^{d}}{d!}\right]\right\\}^{j}\\\
&\left\\{1-\sum_{d=0}^{c_{1}}\left[\dfrac{e^{-n\tilde{p}}(n\tilde{p})^{d}}{d!}\right]\right\\}^{m-j}\end{split}$
(4)

Here, $\tilde{P_{a}}(\tilde{p})$ is the fuzzy probability of acceptance for a
sample of size $n$ and proportion of defectives $\tilde{p}$ is a fuzzy number.
The $\nu$\- cut for $\tilde{P_{a}}(\tilde{p})$ can be defined according to
([15]):
Let $\tilde{p}=(p_{1},p_{2},p_{3})$ and $\tilde{q}=(q_{1},q_{2},q_{3})$ be a
triangular fuzzy numbers. The $\nu$\- cut of $\tilde{p}$ and $\tilde{q}$ is
defined as

$\tilde{p}[\nu]=[p_{1}+(p_{2}-p_{1})\nu,p_{3}-(p_{3}-p_{2})\nu],\forall
0\leq\nu\leq 1$ (5)

and

$\tilde{q}[\nu]=[q_{1}+(q_{2}-q_{1})\nu,q_{3}-(q_{3}-q_{2})\nu],\forall
0\leq\nu\leq 1$ (6)

Also let

$S=\\{p\in\tilde{p}[\nu],q\in\tilde{q}[\nu],p+q=1\\}$ (7)

The $\nu$\- cut for $\tilde{P_{a}}(\tilde{p})$ can be represented as

$\tilde{P_{a}}(\tilde{p})[\nu]=[\tilde{P}_{aI}(\tilde{p})[\nu],\tilde{P}_{aII}(\tilde{p})[\nu]]$
(8)

where

$\tilde{P}_{aI}(\tilde{p})[\nu]=\min\\{\tilde{P}_{a}(\tilde{p}):S\\}$ (9)

and

$\tilde{P}_{aII}(\tilde{p})[\nu]=\max\\{\tilde{P}_{a}(\tilde{p}):S\\}$ (10)

###  4.2 Average Sample Number for GFMDS

The average sample number of a sampling plan is the average number of units of
sample per lot upon which the acceptance or rejection of the lot is made. The
average sample number of the proposed GFMDS plan is given as:

$ASN=n$

###  4.3 Average Total Inspection for GFMDS

Average total inspection is the average number of units per lot which
underwent inspection based on the sample size for accepted lots and all units
per lot which underwent inspection based on the sample size for rejected lots.
ATI of the proposed model can be derived by the following theorem:

Theorem 2
The ATI for GFMDS plan is given by

$ATI(\tilde{p})=n+{1-\tilde{P_{a}}(\tilde{p})}(N-n)$

where $\tilde{P_{a}}(\tilde{p})$ is the fuzzy probability of acceptance given
by Equation (2) and $N$ is the lot size.
Proof: The lots submitted are subject to a rectifying inspection process
according to the GFMDS plan.
Case 1. If there are no non-conformities in the submitted lots, then the whole
lot will be accepted. In this case, the number of inspection per lot will be
the sample size $n$.
Case 2. If the number of non-conformities in the submitted lot are 100%, then
every lot undergoes a rectifying inspection and in that case, the inspected
lots will be $N$.
Case 3. When the proportion of defectives $\tilde{p}$ is a fuzzy number such
that $\tilde{p}[\nu]$ is defined by Equation (5), the average number of
inspected units per lot, ATI will vary between sample size $n$ and lot size
$N$.
When the fuzzy probability of acceptance for the submitted lots is
$\tilde{P_{a}}(\tilde{p})$ defined by Equation (2), the ATI per lot will be

$\displaystyle\tilde{ATI}(\tilde{p})$
$\displaystyle=n+\tilde{P_{a}}(\tilde{p})+{1-\tilde{P_{a}}(\tilde{p})}N$
$\displaystyle=N+{1-\tilde{P_{a}}(\tilde{p})}(N-n)$

The $\nu$\- cut for $\tilde{ATI}(\tilde{p})$ can be defined according to
([15]):

$\tilde{ATI}(\tilde{p})[\nu]=[ATI_{I}(\tilde{p})[\nu],ATI_{II}(\tilde{p})[\nu]]$

where

$ATI_{I}(\tilde{p})=\min\\{\tilde{ATI}(\tilde{p}):S\\}$

and

$ATI_{II}(\tilde{p})=\max\\{\tilde{ATI}(\tilde{p}):S\\}$

where $S$ as in Equation (7).

Remark 1 The performance measures of the proposed GFMDS plan, say acceptance
probability, ASN and ATI will reduce to the performance measures of FMDS plan
defined by [3] when $k=m$.

Remark 2 When $c_{1}=c_{2}=c,k=m$ and $m\rightarrow\infty$, the proposed GFMDS
plan reduces to the fuzzy single sampling plan for attributes defined by [29].

##  5 Determination of Optimal Parameters for GFMDS Plan

The traditional approach for designing an acceptance sampling plan is based on
Acceptable Quality Level (AQL) and Limiting Quality Level (LQL), two points on
the operating characteristic curve. Such a plan would meet the producer’s and
consumer’s quality requirements. AQL is the worst level of quality for which
the consumer would accept the process at an average and LQL is the poorest
level of quality that the consumer is willing to accept in an individual lot
([31]). In the GMDS plan designed by [8], the ASN is minimized to obtain the
plan parameters (say $n,k,m,c_{1}$ and $c_{2}$) for a given pair of
$(p_{\alpha},1-\alpha)$ and $(p_{\beta},\beta)$. Fuzzy ASP are also determined
using the above approach. Two points on the FOC band are considered,
$(\tilde{p_{\alpha}},1-\tilde{\alpha})$ and
$(\tilde{p_{\beta}},\tilde{\beta})$ corresponding to fuzzy AQL (denoted by
$\tilde{AQL}$) and fuzzy LQL (denoted by $\tilde{LQL}$) respectively.
$\tilde{\alpha}$ is the fuzzy producer’s risk and $\tilde{\beta}$ is the fuzzy
consumer’s risk. The above parameters can be represented as triangular fuzzy
numbers as $\tilde{p}$, say,

$\displaystyle\tilde{p_{\alpha}}=$
$\displaystyle(p_{1\alpha},p_{2\alpha},p_{3\alpha})$
$\displaystyle\tilde{p_{\beta}}=$
$\displaystyle(p_{1\beta},p_{2\beta},p_{3\beta})$
$\displaystyle\tilde{\alpha}=$
$\displaystyle(\alpha_{1},\alpha_{2},\alpha_{3})$
$\displaystyle\tilde{\beta}=$ $\displaystyle(\beta_{1},\beta_{2},\beta_{3})$

Also, the $\nu$ cut can be obtained as in Equations (5) and (6) as:

$\begin{split}\tilde{p_{\alpha}}[\nu]=&[p_{1\alpha}+(p_{2\alpha}-p_{1\alpha})\nu,p_{3\alpha}-(p_{3\alpha}-p_{2\alpha})\nu],\\\
&\quad\forall 0\leq\nu\leq 1=[p_{\alpha I},p_{\alpha II}]\\\
\tilde{p_{\beta}}[\nu]=&[p_{1\beta}+(p_{2\beta}-p_{1\beta})\nu,p_{3\beta}-(p_{3\beta}-p_{2\beta})\nu],\\\
&\quad\forall 0\leq\nu\leq 1=[p_{\beta I},p_{\beta II}]\\\
\tilde{\alpha}[\nu]=&[\alpha_{1}+(\alpha_{2}-\alpha_{1})\nu,\alpha_{3}-(\alpha_{3}-\alpha_{2})\nu],\\\
&\quad\forall 0\leq\nu\leq 1=[\alpha_{I},\alpha_{II}]\\\
\tilde{\beta}[\nu]=&[\beta_{1}+(\beta_{2}-\beta_{1})\nu,\beta_{3}-(\beta_{3}-\beta_{2})\nu],\\\
&\quad\forall 0\leq\nu\leq 1=[\beta_{I},\beta_{II}]\end{split}$

Our model assumes that the proportion of defectives is the only parameter of
the distribution that is imprecise in nature. All the other parameters are
treated according to the classical probability theory. The advances in the
design of new sampling schemes intends in diminishing ASN and our proposed
plan adopts the same methodology. The solution of the following pair of non-
linear optimization problem provides the minimal ASN:

$\tilde{ASN}_{I}(\tilde{p})=\begin{cases}\min&ASN(\tilde{p})=\tilde{n}\\\
\text{subject to}&\tilde{P_{aI}}(\tilde{p_{1}})\geq 1-\tilde{\alpha_{I}}\\\
&\tilde{P_{aI}}(\tilde{p_{2}})\leq\tilde{\beta_{I}}\end{cases}$ (11)
$\tilde{ASN}_{II}(\tilde{p})=\begin{cases}\min&ASN(\tilde{p})=\tilde{n}\\\
\text{subject to}&\tilde{P_{aII}}(\tilde{p_{1}})\geq 1-\tilde{\alpha_{II}}\\\
&\tilde{P_{aII}}(\tilde{p_{2}})\leq\tilde{\beta_{II}}\end{cases}$ (12)

When the value of $\nu=1$, Equations (11) and (12) reduces to its crisp
counterpart provided by [8]. The ASN becomes a crisp parameter. The results
agree with the parameters of the GMDS plan in this case. The pair of non-
linear optimization problem reduces to the following problem:

$\displaystyle\min$ $\displaystyle ASN({p})$ $\displaystyle=n$ subject to
$\displaystyle P_{a}({p_{1}})$ $\displaystyle\geq 1-\alpha$ $\displaystyle
P_{a}({p_{2}})$ $\displaystyle\leq\beta$

Throughout this paper, we assume ASN to be crisp in nature. Since our proposed
plan consider the plan parameters as crisp values, the ASN will be minimized
for a fuzzy proportion of defectives also. Optimal parameters of GFMDS plan
for given $\tilde{AQL}$ and $\tilde{LQL}$ for fuzzy binomial probability
distribution is given in Table 1 and fuzzy Poisson probability model is given
in Table 2.

Table 1: Optimal parameters of GFMDS plan for fuzzy Binomial distribution when $\tilde{\alpha}[1]=5\%$ and $\tilde{\beta}[1]=10\%$ $\tilde{p_{1}}[1]$ | $\tilde{p_{2}}[1]$ | $n$ | $m$ | $k$ | $c_{1}$ | $c_{2}$
---|---|---|---|---|---|---
0.001 | 0.01 | 261 | 2 | 1 | 0 | 1
0.0025 | 0.05 | 49 | 1 | 1 | 0 | 1
0.005 | 0.10 | 24 | 1 | 1 | 0 | 1
0.0075 | 0.08 | 32 | 2 | 1 | 0 | 1
0.010 | 0.04 | 87 | 5 | 1 | 0 | 3
0.015 | 0.05 | 76 | 9 | 1 | 0 | 3
0.02 | 0.08 | 43 | 5 | 1 | 0 | 3
0.05 | 0.15 | 32 | 4 | 1 | 1 | 4
Table 2: Optimal parameters of GFMDS plan for fuzzy Poisson distribution when $\tilde{\alpha}[1]=5\%$ and $\tilde{\beta}[1]=10\%$ $\tilde{p_{1}}[1]$ | $\tilde{p_{2}}[1]$ | $n$ | $m$ | $k$ | $c_{1}$ | $c_{2}$
---|---|---|---|---|---|---
0.001 | 0.01 | 262 | 2 | 1 | 0 | 1
0.0025 | 0.05 | 50 | 1 | 1 | 0 | 1
0.005 | 0.10 | 25 | 1 | 1 | 0 | 1
0.0075 | 0.08 | 33 | 2 | 1 | 0 | 1
0.010 | 0.04 | 91 | 6 | 1 | 0 | 3
0.015 | 0.05 | 79 | 10 | 1 | 0 | 3
0.02 | 0.08 | 46 | 6 | 1 | 0 | 3
0.05 | 0.15 | 35 | 6 | 1 | 1 | 4

Example 1 Consider a manufacturing system with continuous production. Let the
proportion of defectives be a triangular fuzzy number,
$\tilde{p}=(0.01,0.02,03)$, which follows a fuzzy Binomial distribution. The
GFMDS plan is carried out for screening of the lots with
$c_{1}=0,c_{2}=3,k=1,m=5$ and $n=87$. For $0\leq\nu\leq 1$, the $\nu$\- cut of
$\tilde{p}$ is:

$\tilde{p}[\nu]=[0.01+0.01\nu,0.03-0.01\nu]$ $\tilde{p}[0]=[0.01,0.03]$

By Equations (5-10), the $\nu$\- cut for $\tilde{P_{a}}(\tilde{p})$ can be
obtained as:

$\begin{split}\tilde{P_{a}}(\tilde{p})[\nu]=\left\\{\tilde{P}(d\leq
0)+\tilde{P}(0<d\leq 3)\right.\\\ \left.\sum_{j=1}^{5}\left[{5\choose
j}[\tilde{P}(d\leq 0)]^{j}[1-\tilde{P}(d\leq
0)]^{5-j}\right]:S\right\\}\end{split}$

For $\nu=0$, we get

$\tilde{P_{a}}(\tilde{p})[0]=[0.28,0.95]$

This means for all 100 lots underwent an inspection process, 28 to 95 lots
will be accepted with at least degree of membership zero. Consider the case
when $p$ is crisp. The acceptance probability in this case can be evaluated by
taking $\nu=1$, which gives $p=0.02$ and $P_{a}=0.62$. When the rate of
uncertainty is expressed in terms of linguistic fuzzy parameters, say the
chance that the value of $p=0.02$ is 30%, we can express this in terms of a
fuzzy number by considering $\nu=0.3$. In this case, we obtain
$\tilde{p}[\nu]=[0.013,0.027]$ and
$\tilde{P_{a}}(\tilde{p})[\nu]=[0.36,0.87]$. That is, for all 100 lots
underwent an inspection process, 36 to 87 lots will be accepted as per the
inspection policy.

Example 2 The proposed GFMDS plan can be applied to a manufacturing system
where the proportion of non conformity’s follow a fuzzy Poisson distribution.
Let the fraction of defectives be a triangular fuzzy number, say
$\tilde{p}=(0.02,0.03,0.04)$ and the sampling plan parameters are:
$n=86,c_{1}=1,c_{2}=4,k=1,m=5$. For $0\leq\nu\leq 1$, the $\nu$\- cut of
$\tilde{p}$ is given by:

$\tilde{p}[\nu]=[0.02+0.01\nu,0.04-0.01\nu]$

For $\nu=0$, $\tilde{p}[\nu]=[0.02,0.04]$. Using Equations (7-10), for
$\nu=0$, we get

$\tilde{P_{a}}(\tilde{p})[0]=[0.57,0.95]$

That is, for every 100 lots in the inspection process, it is expected that 57
to 95 lots are accepted with the degree of membership zero. When $\nu=1$, the
plan reduces to the classical MDS plan with fraction of defectives $p=0.03$
(Since $\tilde{p}[1]=[0.02+0.01*1,0.04-0.01*1]=0.03$). In this case, the
probability of acceptance will be $P_{a}(p)=0.79$.

Example 3 In this example, the FOC band is obtained for the parameters in
Example (2). The assumptions remains unchanged. Since $\tilde{p}$ is a fuzzy
random variable, the probability of acceptance $\tilde{P_{a}}$ is also fuzzy.
The fuzzy OC band is obtained in this case (Figure 1) unlike the OC curve when
$p$ and $P_{a}$ are crisp values. In order to construct the FOC band, we
consider a facilitator parameter $\theta$ to identify the variations of
$\tilde{p}$ (see [3]). Now $\tilde{p}=(a_{1},a_{2},a_{3})$ can be written as
$\tilde{p_{\theta}}$ as follows:

$\tilde{p_{\theta}}=(\theta,b_{2}+\theta,b_{3}+\theta)$
$\tilde{p_{\theta}}[\nu]=[\theta+b_{2}\nu,b_{3}+\theta-(b_{3}-b_{2})\nu],\quad\forall
0\leq\nu\leq 1$

where $b_{j}=a_{j}-a_{1},\quad(j=2,3)$ and $\theta\in[0,1-b_{3}]$. The
probability of acceptance for $\tilde{p_{\theta}}$ can be obtained by
replacing $p$ in Equations (2) and (4) by $p_{\theta}$. For
$\tilde{p}=(0.02,0.03,0.04)$, we have
$\tilde{p_{\theta}}=(\theta,0.01+\theta,0.02+\theta)$ and

$\begin{split}\tilde{p_{\theta}}[0]&=(\theta+0.01*0,0.02+\theta-0.01*0],\\\
&=[\theta,0.02+\theta]\quad\forall 0\leq\theta\leq 0.98\end{split}$

The fuzzy probability of acceptance for different values of $\theta$ when
$\nu=0$ is given in Table (3).

Figure 1: FOC bands for GFMDS plan for different values of $\nu$ Table 3: The fuzzy probability of lot acceptance for GFMDS plan without inspection errors corresponding to different values of $\nu$ and $\theta$ $\theta$ | $\tilde{p_{\theta}}$ | $\tilde{p_{\theta}}[0]$ | $\tilde{P_{a}}(\tilde{p_{\theta}})[0]$ | $\tilde{p_{\theta}}[0.3]$ | $\tilde{P_{a}}(\tilde{p_{\theta}})[0.3]$ | $\tilde{p_{\theta}}[0.7]$ | $\tilde{P_{a}}(\tilde{p_{\theta}})[0.7]$
---|---|---|---|---|---|---|---
0 | (0,0.01,0.02) | [0,0.02] | [0.952,1.00] | [0.003,0.017] | [0.1411,0.7391] | [0.007,0.013] | [0.8781,0.9876]
0.01 | (0.01,0.02,0.03) | [0.01,0.03] | [0.755,0.998] | [0.013,0.027] | [0.3608,0.8781] | [0.017,0.023] | [0.5011,0.7391]
0.02 | (0.02,0.03,0.04) | [0.02,0.04] | [0.461,0.952] | [0.023,0.037] | [0.1352,0.5011] | [0.027,0.033] | [0.2049,0.3608]
0.03 | (0.03,0.04,0.05) | [0.03,0.05] | [0.227,0.755] | [0.033,0.047] | [0.0444,0.2049] | [0.037,0.043] | [0.0698,0.1352]
0.04 | (0.04,0.05,0.06) | [0.04,0.06] | [0.098,0.461] | [0.043,0.057] | [0.0138,0.0698] | [0.047,0.053] | [0.0221,0.0444]
0.05 | (0.05,0.06,0.07) | [0.05,0.07] | [0.039,0.227] | [0.053,0.067] | [0.0042,0.0221] | [0.057,0.063] | [0.0068,0.0138]
0.06 | (0.06,0.07,0.08) | [0.06,0.08] | [0.015,0.098] | [0.063,0.077] | [0.0013,0.0068] | [0.067,0.073] | [0.0022,0.0042]
0.07 | (0.07,0.08,0.09) | [0.07,0.09] | [0.006,0.039] | [0.073,0.087] | [0.0005,0.0022] | [0.077,0.083] | [0.0006,0.0013]
0.08 | (0.08,0.09,0.10) | [0.08,0.10] | [0.002,0.015] | [0.083,0.097] | [0,0.0006] | [0.087,0.093] | [0.0002,0.0005]
0.09 | (0.09,0.10,0.11) | [0.09,0.11] | [0.001,0.006] | [0.093,0.107] | [0,0.0002] | [0.097,0.103] | [0,0.0002]
0.10 | (0.10,0.11,0.12) | [0.10,0.12] | [0.001,0.002] | [0.103,0.117] | [0,0] | [0.107,0.113] | [0,0]

The next section validates the efficiency of the GFMDS plan by showing that
the ASN is minimum for the proposed plan on comparison with the existing fuzzy
acceptance sampling plans.

##  6 Probability of acceptance of GFMDS Plan in terms of generalised fuzzy
number

Throughout this paper, the fraction of defectives is assumed to be a
triangular fuzzy number. However, this can be generalised to a fuzzy number
with a different membership function. Let $f_{z_{i}}(\tilde{p})$ be the
probability density function for fuzzy proportion of defectives $\tilde{p}$
for parameters $z_{i}$ where $i=1,2,...,n$. If $p$ and $\mathbb{P}$ denotes
the crisp proportion of defectives and crisp probability measure respectively,
we can express $\tilde{p}$ as

$\tilde{p}=\\{(p,\mu_{\tilde{p}}(p)|p\in\mathbb{P}\\}$

where $\mu_{\tilde{p}}(p)$ denotes the membership function of $\tilde{p}$. The
$\nu$\- cut or $\nu$ level sets of $z_{i}$ is

$\displaystyle\tilde{z_{i}}[\nu]$
$\displaystyle=\left[\underset{z_{i}}{\min}\\{z_{i}\in
S(\tilde{z_{i}})|\mu_{\tilde{z_{i}}}(z_{i})\geq\nu\\},\right.$
$\displaystyle\left.\underset{z_{i}}{\max}\\{z_{i}\in
S(\tilde{z_{i}})|\mu_{\tilde{z_{i}}}(z_{i})\geq\nu\\}\right]$
$\displaystyle=[\tilde{z_{iI}}[\nu],\tilde{z_{iII}}[\nu]]$

where
$S(\tilde{z_{i}})=\\{z_{i}\in\tilde{z_{i}}[\nu],\sum_{i=1}^{n}z_{i}=1\\}$.
The upper and lower bound of $\nu$\- cut of $\tilde{P_{a}}(\tilde{p})$ can be
found by solving the following pair of optimization problem:

$\tilde{P_{aI}}(\tilde{p})[\nu]=\begin{cases}\min P_{a}(p)\\\
\text{s.t.}\tilde{z_{iI}}[\nu]\leq\tilde{z_{i}}\leq\tilde{z_{iII}}[\nu]\end{cases}$

and

$\tilde{P_{aII}}(\tilde{p})[\nu]=\begin{cases}\max P_{a}(p)\\\
\text{s.t.}\tilde{z_{iI}}[\nu]\leq\tilde{z_{i}}\leq\tilde{z_{iII}}[\nu]\end{cases}$

where $P_{a}(p)$ denotes the conventional probability of acceptance when $p$
is crisp. Thus $\nu$\- cut of $\tilde{P_{a}}(\tilde{p})$ is

$\tilde{P_{a}}(\tilde{p})[\nu]=[\tilde{P_{aI}}(\tilde{p})[\nu],\tilde{P_{aII}}(\tilde{p})[\nu]]$

According to [39], the $\nu$\- cuts form a nested interval for different
values of $\nu$. For $0<\nu_{1}<\nu_{2}\leq 1$, we have
$\tilde{P_{aI}}(\tilde{p})[\nu_{2}]\geq\tilde{P_{aI}}(\tilde{p})[\nu_{1}]$ and
$\tilde{P_{aII}}(\tilde{p})[\nu_{1}]\geq\tilde{P_{aII}}(\tilde{p})[\nu]$

Example 4 Consider the proportion of defectives having a pentagonal membership
function and following a fuzzy Binomial distribution. The $\nu$\- cut for a
pentagonal fuzzy number $\tilde{x}=(x_{1},x_{2},x_{3},x_{4},x_{5})$ is
$[2\nu(x_{2}-x_{1})+x_{1},-2\nu(x_{5}-x_{4})+x_{5}]$ for $\nu\in[0,0.5]$ and
$[2\nu(x_{3}-x_{2})+2x_{2}-x_{3},2\nu(x_{4}-x_{3})-x_{4}+2x_{3}]$ for
$\nu\in[0.5,1]$. Let the fuzzy proportion of defectives be
$\tilde{p}=(0.02,0.03,0.05,0.07,0.08)$ and plan parameters be
$n=87,c_{1}=0,c_{2}=3,k=1,m=5$. Then,

$\tilde{p}[\nu]=\begin{cases}\begin{aligned} ,&\quad\nu\in[0,0.5]\\\
[0.01+0.04\nu,0.17-0.04\nu],&\quad\nu\in[0.5,1]\end{aligned}\end{cases}$

The probability of acceptance when $p$ is a crisp variable is given in
Equation (1)
Case 1: $\nu\in[0,0.5]$
To find $\nu$\- cut for $\tilde{P_{a}}(\tilde{p})$, consider the following
pair of optimization problems:

$\tilde{P_{aI}}(\tilde{p})=\begin{cases}\min\quad P_{a}(p)\\\ \text{s.t.}\quad
0.02+0.02\nu\leq p\leq 0.08-0.02\nu\end{cases}$

and

$\tilde{P_{aII}}(\tilde{p})=\begin{cases}\max\quad P_{a}(p)\\\
\text{s.t.}\quad 0.02+0.02\nu\leq p\leq 0.08-0.02\nu\end{cases}$

Solving the above problems, the $\nu$\- cut of fuzzy probability of acceptance
for say $\nu=0.3$ is obtained as:
$\tilde{P_{a}}(\tilde{p})[0.3]=[0.002,0.393]$
Case 2: $\nu\in[0.5,1]$
To find $\nu$\- cut for $\tilde{P_{a}}(\tilde{p})$, consider the following
pair of optimization problems:

$\tilde{P_{aI}}(\tilde{p})=\begin{cases}\min\quad P_{a}(p)\\\ \text{s.t.}\quad
0.01+0.04\nu\leq p\leq 0.17-0.04\nu\end{cases}$

and

$\tilde{P_{aII}}(\tilde{p})=\begin{cases}\max\quad P_{a}(p)\\\
\text{s.t.}\quad 0.01+0.04\nu\leq p\leq 0.17-0.04\nu\end{cases}$

Solving the above problems, the $\nu$\- cut of fuzzy probability of acceptance
for say, $\nu=0.8$ is obtained as: $\tilde{P_{a}}(\tilde{p})[0.8]=[0,0.27]$

##  7 Efficiency of GFMDS Plan

The limitation of classical MDS plans is that the lots are being placed in a
state subject to certain conditions for an indefinite time. The traditional
single sampling plan has the limitation that a lot is rejected as soon as a
defective item is identified [8] showed that a GMDS plan overcomes the above-
said shortcomings and safeguards the interests of the producer and consumer.
They also showed that the GMDS plan is designed with minimum ASN and ATI
compared to the existing sampling schemes. In this section, the authors would
like to investigate the efficiency of the proposed GFMDS plan over the
existing fuzzy acceptance sampling plans for attributes. We compare the GFMDS
plan with fuzzy SSP and fuzzy DSP proposed by [29] and fuzzy MDS plan designed
by [3]. The ASN for each of the sampling schemes is obtained for a combination
of $\tilde{AQL}$ and $\tilde{LQL}$. The results obtained for fuzzy Binomial
distribution are shown in Table (4) and for fuzzy Poisson distribution is
shown in Table (5). From the tables, it is clear that the ASN has been
considerably reduced when the GFMDS plan is employed which proves the economic
advantage of GFMDS plan over the existing fuzzy acceptance sampling plans.

Table 4: ASN of FSSP, FDSP, FMDS and GFMDS for fuzzy Binomial distribution $\tilde{p_{1}}[1]$ | $\tilde{p_{2}}[1]$ | FSSP | FDSP | FMDS | GFMDS
---|---|---|---|---|---
0.001 | 0.010 | 531 | 363.55 | 388 | 261
0.001 | 0.015 | 258 | 188.11 | 165 | 165
0.0025 | 0.005 | 4948 | 4942.89 | 3197 | 1822
0.005 | 0.010 | 2473 | 2471.44 | 1597 | 910
0.010 | 0.030 | 390 | 364.49 | 275 | 170
0.010 | 0.050 | 132 | 103.16 | 87 | 62
Table 5: ASN of FSSP, FDSP, FMDS and GFMDS for fuzzy Poisson distribution $\tilde{p_{1}}[1]$ | $\tilde{p_{2}}[1]$ | FSSP | FDSP | FMDS | GFMDS
---|---|---|---|---|---
0.001 | 0.010 | 533 | 364.97 | 390 | 262
0.001 | 0.015 | 260 | 190.59 | 167 | 167
0.0025 | 0.005 | 4952 | 4946.86 | 3200 | 1824
0.005 | 0.010 | 2476 | 2473.43 | 1600 | 912
0.010 | 0.030 | 393 | 366.33 | 277 | 172
0.010 | 0.050 | 134 | 127.76 | 108 | 63
Table 6: Comparison of fuzzy probability of lot acceptance for GFMDS plan with and without inspection errors for $\nu=0$ $\theta$ | $\tilde{p_{\theta}}[0]$ | $\tilde{P_{a}}(\tilde{p_{\theta}})[0]$ | $\tilde{p_{\delta\theta}}[0]$ | $\tilde{P_{a}}(\tilde{p_{\delta\theta}})[0]$
---|---|---|---|---
0 | [0,0.2] | [0.952,1.00] | [0,0.0182] | [0.6917,1.00]
0.01 | [0.01,0.03] | [0.755,0.998] | [0.01,0.0282] | [0.3243,0.95]
0.02 | [0.02,0.04] | [0.461,0.952] | [0.02,0.0382] | [0.1190,0.6193]
0.03 | [0.04,0.06] | [0.227.0.755] | [0.03,0.0482] | [0.0386,0.2745]
0.04 | [0.05,0.07] | [0.098,0.461] | [0.04,0.0582] | [0.0119,0.0978]
0.05 | [0.06,0.08] | [0.039,0.227] | [0.05,0.0682] | [0.0037,0.0313]
0.06 | [0.07,0.09] | [0.015,0.098] | [0.06,0.0782] | [0.0012,0.0096]

##  8 GFMDS plan with Inspection Errors

One of the blind assumptions in acceptance sampling plans is the absence of
any measurement error. The traditional sampling plans were carried out in this
belief which in turn affects the system reliability as a whole. The possible
errors in an inspection process is of two types: a perfect item is classified
as an imperfect item and vice versa. These errors are named as Type I error
and Type II error respectively (hereafter denoted by $\delta_{1}$ and
$\delta_{2}$). The GFMDS plan with inspection error is designed by considering
the possible events of inspection process:
$A_{1}$: The item is imperfect
$A_{2}$: The item is perfect
$B$: The item is classified as imperfect post-inspection
$B|A_{2}$: A perfect item is classified as imperfect
$B^{c}|A_{1}$: An imperfect item is classified as perfect
The actual and experimental values of fuzzy proportion of defectives are
denoted by $\tilde{p}$ and $\tilde{p_{\delta}}$. Then,
$\tilde{p}=\tilde{P}(A_{1})$ and $\tilde{p_{e}}=\tilde{P}(B)$. Also,
$\delta_{1}=P(B|A_{2})$ and $\delta_{2}=P(B^{c}|A_{1})$. By Definition 2.3,
for $\nu\in[0,1],$

$\tilde{p_{\delta}}=\tilde{p}(1-\delta_{2})+\tilde{q}\delta_{1}$ (14)

The $\nu$\- cut is given by

$\tilde{p_{\delta}}[\nu]=\\{\tilde{p}(1-\delta_{2})+\tilde{q}\delta_{1}:S\\}$
(15)

where $S$ is defined by:

$S=\\{p\in\tilde{p_{\delta}}[\nu],q\in\tilde{q_{\delta}}[\nu],p_{\delta}+q_{\delta}=1\\}$
(16)

The fuzzy probability of acceptance when inspection errors are considered can
be obtained by replacing $\tilde{p}$ by $\tilde{p_{\delta}}$ in Equations (3)
and (4) for Binomial and Poisson models respectively for $S$ given in Equation
(16). The $\nu$\- cut is obtained similarly which is represented as

$\tilde{P_{a}}(\tilde{p_{\delta}})[\nu]=[\tilde{P}_{eI}[\nu],\tilde{P}_{eII}[\nu]]$
(17)

throughout this article.

Example 5 Suppose that the inspection process carried out in the production
system in Example (2) is found to be imperfect. The inspection errors are
identified as $\delta_{1}=0.01$ and $\delta_{2}=0.08$ from the previous lots
which were subjected to the same inspection process. If all the process
parameters remains unchanged, then the experimental fuzzy proportion of
defectives were obtained as:

$\tilde{p_{\delta}}=(0.0191,0.0282,0373)$
$\tilde{p}[\nu]=[0.01+0.01\nu,0.03-0.01\nu]$
$\tilde{p_{\delta}}=[0.0191+0.0091\nu,0.0373-0.0091\nu]$

When $\nu=0$,

$\tilde{p}[0]=[0.01,0.03]\quad\text{and}\quad\tilde{p_{\delta}}[0]=[0.0191,0.0373]$

Using Equations (15 \- 17), the 0- cut of fuzzy probability of lot acceptance
with inspection errors is:

$\tilde{P_{a}}(\tilde{p_{\delta}})[0]=[0.13,0.66]$

It is expected that for every 100 lots, 13 to 66 lots are accepted with a
least degree of membership zero. Comparing with GFMDS plan as in Example 2, it
is seen that the number of accepted lot reduces when inspection errors are
considered. The fuzzy probability of acceptance for different values of
inspection errors are shown in Table 3.

Example 6 The effect of inspection errors on GFMDS plan can be studied by
constructing and comparing FOC bands for GFMDS plan with and without
inspection errors. As in Example 2, a facilitator parameter $\theta$ can be
used to plot the FOC bands. Consider the assumptions in Example 8.
$\tilde{p_{\delta}}=(c_{1},c_{2},c_{3})$ can be represented using the
facilitator parameter $\theta$ as:

$\tilde{p_{\delta\theta}}=(\theta,d_{2}+\theta,d_{3}+\theta)$

The $\nu$\- cut of $\tilde{p_{\delta\theta}}$ is:

$\tilde{p_{\delta\theta}}[\nu]=[\theta+d_{2}+\theta,d_{3}+\theta-(d_{3}-d_{2})\alpha],\quad\forall
0\leq\nu\leq 1$

where $d_{i}=c_{i}-c_{1}\quad(i=2,3)$ and $\theta\in[0,1-d_{3}]$. For finding
the probability of acceptance, the fuzzy proportion of defectives in equations
can be replaced with $\tilde{p_{\delta\theta}}$. For
$\tilde{p_{\delta}}=(0.0191,0.0282,0.0373)$,
$\tilde{p_{\delta\theta}}=[\theta,0.0091+\theta,0.0182+\theta]$. The $\nu$ cut
of $\tilde{p_{\delta\theta}}$ is:

$\tilde{p_{\delta\theta}}[\nu]=[\theta+0.0091\nu,0.0182+\theta-0.0091\nu]$

For $\nu=0,$

$\tilde{p_{\delta\theta}}[0]=[\theta,0.0182+\theta]\quad\forall
0\leq\theta\leq 0.9818$

The values of $\tilde{p_{\delta\theta}}[0]$ and
$\tilde{P_{a}}(\tilde{p_{\delta}})[0]$ are shown in Table (6) for different
values of $\theta$. Also, the FOC band for GFMDS plan with and without
inspection errors is constructed (see, Figure 2). The results shows that the
fuzzy probability of acceptance with and without considering the measurement
errors is falling when the 0- cut of the experimental and actual values of
fuzzy proportion of defectives are rising.

Figure 2: FOC bands for GFMDS plan with and without inspection errors

##  9 Conclusion

This study extended the generalized MDS sampling plan to the fuzzy
environment. The uncertainty in determining the exact value of the fraction of
defectives is dealt by the fuzzy set theory. The design of GFMDS plan is
reduced to a classical GMDS plan when $p$ is crisp and thus the plan is well-
defined. The numerical examples illustrates the construction, advantage and
effect of errors on the proposed plan. The efficiency of GFMDS is established
by comparing the average sample number with that of single, double, and
multiple deferred plans in the fuzzy environment. The plan is rephrased
including inspection errors to deal with the imperfect measurement process.
The drop in the probability of acceptance is identified when sampling errors
are identified. The obtained results show this plan overcame the limitations
of the existing fuzzy acceptance sampling plans.

### Acknowledgement

The authors would like to thank DST, Govt. of India for extending the
laboratory support under the project (SR/FST/MS-1/2019/40) of Department of
Mathematics, NIT Calicut. The first author also like to thank CSIR, Govt. of
India for extending financial support (09/874(0039)/2019-EMR-I).

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