# Design of Variable Acceptance Sampling Plan for Exponential Distribution
under Uncertainty

M. KUMAR and A. M. MATHAI
National Institute of Technology Calicut

###### Abstract

In an acceptance monitoring system, acceptance sampling techniques are used to
increase production, enhance control, and deliver higher-quality products at a
lesser cost. It might not always be possible to define the acceptance sampling
plan parameters as exact values, especially, when data has uncertainty. In
this work, acceptance sampling plans for a large number of identical units
with exponential lifetimes are obtained by treating acceptable quality life,
rejectable quality life, consumer’s risk, and producer’s risk as fuzzy
parameters. To obtain plan parameters of sequential sampling plans and
repetitive group sampling plans, fuzzy hypothesis test is considered. To
validate the sampling plans obtained in this work, some examples are
presented. Our results are compared with existing results in the literature.
Finally, to demonstrate the application of the resulting sampling plans, a
real-life case study is presented

###### category:

62K05 Statistics Design of statistical experiments

###### keywords:

Optimal statistical designs

###### category:

62K25 Statistics Design of statistical experiments

###### keywords:

Fuzziness and design of statistical experiments

###### category:

62K99 Statistics Design of statistical experiments

###### category:

62P30 Statistics Applications of statistics

###### keywords:

Applications of statistics in engineering and industry; control charts

###### keywords:

Exponential distribution, fuzzy acceptance sampling plan, fuzzy hypothesis
testing, fuzzy optimization.

††terms: Design; Reliability.

Author’s address: M. Kumar and A. M. Mathai, Department of Mathematics,
National Institute of Technology, Calicut, Kerala, India, PIN - 673601.
E-mail address<EMAIL_ADDRESS>(M. Kumar) (Corresponding author) and
<EMAIL_ADDRESS>(A. M. Mathai).

##  1 Introduction

Acceptance sampling plan (ASP) for life testing has a unique role in the field
of reliability and quality control. In the field of manufacturing, there is a
need for reliable plan which is economical in terms of time and manufacturing
costs. The classical sampling plans mainly depend upon the accuracy of data,
the test of hypotheses, and appropriate decision rules. At the same time,
existing literature has shown the existence of situations where the above
assumptions are rather unrealistic or vague (see [Watanabe and Imaizumi
(1993), Gildeh et al. (2011)]). The uncertainties of parameters in the
sampling plans are due to season, trend, and random variations. In such cases
of sampling inspection, the data will be recorded as linguistic variables, and
the fuzzy set theory proposed by klir1996fuzzy is used to analyze uncertain
data. sherman1965 explained an acceptance sampling plan for repetitive group
sampling, in which the decision of accepting and rejecting the lot depends on
the number of defectives. Probabilities for long-run acceptance, rejection,
and the number of samples to get an acceptance decision are also formulated.
Different methods have been considered by many authors in the literature to
design acceptance sampling plans and most of them are concentrating on the
reduction of sample size but not the cost (see [Sherman (1965), Aslam et al.
(2012)]). MK designed a new optimal acceptance sampling plan by variables for
a lot of units having exponentially, distributed lifetime based on the time
between successive failures. They considered both the methods of sequential
sampling plan and repetitive group sampling plans under different criteria. In
all these, an optimization problem is performed to minimize the expected
testing cost under Type I and Type II errors to obtain the optimal parameters.

All the above sampling plans in the literature assume that data do not have
uncertainties. In other words, these sampling plans are designed for crisp
data, which are not practical when real-life problems are concerned where
uncertainties in data are unavoidable. torabi2009 redefined the concepts of
fuzzy hypotheses testing by defining the weighted probability distribution
function. They introduced the sequential probability ratio test (SPRT) by
attributes for fuzzy hypotheses testing and illustrated several examples.
jameel2012 had proposed a sequential sampling plan by variables using SPRT for
fuzzy hypothesis in the case of a lot of units having a normally distributed
lifetime. According to their plan, the width of the continuation of testing in
the graph of the sampling plan narrowed as the sample size increased.

Fuzzy hypothesis testing was studied via various methods by several authors
[delgado1985, watanabe1993, arnold1995, arnold1996, taheri2001,
torabi2007likelihood, jamkhaneh2010]. jameel2012, sakawa2013 and trappey1988
discussed the optimization of fuzzy non-linear programming problem (NLPP) by
converting it to crisp NLPP in the cases of fuzzy constraints.

In previous literature, fuzzy hypothesis testing is done using SPRT and the
likelihood ratio test. The decision for accepting and rejecting the lot is
made by the cumulative averages. The optimization was concentrated on sample
size and failure time. In this paper, we consider a lot of identical units
having exponential lifetime distribution. Here we have obtained sampling plans
based on the time between successive failures, which is so far not considered
in the literature. It is noted that the design of ASP by considering the time
between successive failures has an advantage in bringing down the test cost
involved in ASP (for example see MK). Further, the obtained ASP guarantees
minimum testing cost which is novel in the design of fuzzy ASP.

This paper is organized as follows: In Section 2, the sequential sampling plan
is discussed with fuzzy parameters. A fuzzy optimization problem is
constructed and solved to minimize the testing cost subject to fuzzy
constraints of Type I and II errors. In Section 3, three different methods of
repetitive group sampling for fuzzy hypothesis testing are considered. In the
first method, the lot is accepted or rejected by checking the minimum of n
samples under the fuzzy hypothesis. In the second criterion, we consider the
maximum of n samples to make a decision. Finally, an ASP is derived using data
obtained from Type I censoring with the fuzzy hypothesis is discussed. The
fuzzy optimization problem to minimize the testing cost subject to fuzzy Type
I and Type II errors is solved to get optimal parameters of sampling plans in
all three methods. Moreover, in all these cases, a comparative study is done
with the existing crisp methods. In Section 4, the de-fuzzification of all
fuzzy parameters using the center of gravity method is discussed and it is
observed that the weighted probability distribution method is better than de-
fuzzification. Section 5 depicts a comparison of all sampling plans using
various numerical examples and a real-life case study.

##  2 Sequential Sampling Plan with Fuzzy Parameters

In this section, we define a sequential sampling plan (SSP), for fuzzy
hypothesis and for that consider a lot of units with exponential failure time
with probability density function given by

$f(x)=\begin{cases}\lambda\exp\left(-\lambda x\right),&x>0,\lambda>0,\\\
0,&\text{ otherwise. }\end{cases}$ (1)

Let $X_{i}$ denote the $i^{th}$ failure time, $i\in\mathbb{N}$. The acceptance
sampling plan is based on the time between two consecutive failures (see MK)
denoted by $Y_{i+1}=X_{i+1}-X_{i},\ i=1,2,\ldots$ and $Y_{1}=X_{1}$. Also,
$Y_{i}$, $i=1,2,3,\ldots$ follows exponential distribution with same parameter
$\lambda$. The following is the ASP: At any point of failure,

continue the process, if $\quad t_{1}\leq Y_{i}<t_{2}$,

accept the lot, if $\quad Y_{i}\geq t_{2}$,

reject the lot, if $\quad Y_{i}<t_{1}.$

The following are the hypotheses for SSP as a statistical test, for accepting
or rejecting the lot:

$\left\\{\begin{array}[]{l}H_{0}:\text{The lot is of acceptable quality life
(AQL)},\\\ H_{1}:\text{The lot is of rejectable quality life
(RQL)}.\end{array}\right.$

The probability of Type I and II errors $\alpha$ and $\beta$ for the
acceptance and rejection of the lot of SSP in crisp case is described in MK as
follows:

$\begin{split}&P\left(\text{Reject the
lot}\mid\frac{1}{\lambda}\geq\lambda_{0}\right)\leq\alpha,\\\
&P\left(\text{Accept the
lot}\mid\frac{1}{\lambda}\leq\lambda_{1}\right)\leq\beta,\end{split}$ (2)

where $\alpha$ is the producer’s risk and $\beta$ is the consumer’s risk.
According to wald2004, by the definition of the ASP used here, the
inequalities defined in (2) will be equivalent to the following

$\begin{split}\qquad\qquad\qquad\qquad\quad&P\left(\text{Reject the
lot}\mid\frac{1}{\lambda}=\lambda_{0}\right)\leq\alpha,\\\
&P\left(\text{Accept the
lot}\mid\frac{1}{\lambda}=\lambda_{1}\right)\leq\beta.\qquad\qquad\qquad\quad\end{split}$

In most real-life situations, while handling the data, the parameters, AQL,
RQL, $\alpha$, and $\beta$ will be expressed as linguistic variables and they
can be converted to fuzzy parameters. Let $\lambda_{0}$ denote the acceptable
quality life (AQL) and $\lambda_{1}$ the rejectable quality life (RQL) of an
item in the lot.

Let $\lambda_{0}$, $\lambda_{1}$, $\alpha$, and $\beta$ be fuzzy numbers. Then
the fuzzy hypothesis is given by:

$\begin{split}H_{0}&:\text{average life is approximately AQL},\\\
H_{1}&:\text{average life is approximately RQL}.\end{split}$ (3)

$\left\\{\begin{array}[]{l}H_{0}:\frac{1}{\lambda}\approx\lambda_{0},\\\
H_{1}:\frac{1}{\lambda}\approx\lambda_{1}.\end{array}\right.\qquad\quad$

That is,

$\left\\{\begin{array}[]{l}H_{0}:\lambda\approx\frac{1}{\lambda_{0}},\\\
H_{1}:\lambda\approx\frac{1}{\lambda_{1}},\end{array}\right.\qquad\quad$
$\left\\{\begin{array}[]{l}H_{0}:\lambda\quad\textrm{is}\quad
H_{0}(\lambda),\\\ H_{1}:\lambda\quad\textrm{is}\quad
H_{1}(\lambda),\end{array}\right.$

where the membership function, $H_{j}(\lambda),\ j=0,1$ (see Figure 1) is
given by [palaniappan2005fuzzy, olunloyo2009design]. That is,

$H_{j}(\lambda)=\left\\{\begin{array}[]{ll}\frac{1+cos(\textit{a}\pi(\lambda-\frac{1}{\lambda_{j}}))}{2}&,\quad\frac{1}{\lambda_{j}}-\frac{1}{\textit{a}}\leq\lambda\leq\frac{1}{\lambda_{j}}+\frac{1}{\textit{a}},\\\
0&,\quad\text{ otherwise, }\qquad\qquad\qquad j=0,1.\end{array}\right.$
$a>\lambda_{j},j=0,1\quad\textrm{and}\quad a>0.$

Figure 1: Membership function of $\frac{1}{\lambda_{j}},\ j=0,1.$

The following probability requirements will be used to determine whether to
accept or reject the lot:

$P\left(\text{Reject the
lot}\mid\lambda\approx\frac{1}{\lambda_{0}}\right)\leq\alpha,$ (4)
$P\left(\text{Accept the
lot}\mid\lambda\approx\frac{1}{\lambda_{1}}\right)\leq\beta.$ (5)

Our goal is to find an optimal pair ($t_{1},t_{2}$) such that the total cost
(TC) of testing is minimized subject to the Type I and Type II errors given by
(4) and (5) when all the parameters, namely, AQL, RQL, $\alpha$, and $\beta$
are fuzzy.

The overall cost of testing is calculated by multiplying the testing cost of
an item per unit time by the entire duration of testing. The failure time of
an item is the random variable $Y_{i}$, then the duration of the test is
$Y_{i}$ times the number of failed items. Note that, the overall cost of
testing (TC) is a random variable.

Let $p_{a}$, $p_{r}$ and $p_{c}$ denote the probability of acceptance,
rejection of the lot and continuation of testing respectively, while the
failure of an item is observed. The overall probability of acceptance,
rejection are determined by
$P_{A}=\frac{P_{a}}{1-p_{c}},P_{R}=\frac{p_{r}}{1-p_{c}}$ and expected number
of failed items by $N=\frac{1}{1-p_{c}}$ (see sherman1965).

Under the hypothesis $H_{j},j=0,1$ the weighted probability density function
(WPDF) (see torabi2009) of $Y_{i}$ is given by,

$\displaystyle\begin{split}f_{j}(y)=\biggr{\\{}\left[\left(\textit{{a}}-\lambda_{j}\right)y^{3}+3\textit{{a}}y^{2}\lambda_{j}+\pi^{2}\textit{{a}}^{2}\left(\textit{{a}}-\lambda_{j}\right)y+\pi^{2}\textit{{a}}^{3}\lambda_{j}\right]e^{\frac{2y}{\textit{{a}}}}-(\lambda_{j}+\textrm{{a}})y^{3}\\\
-3\textit{{a}}y^{2}\lambda_{j}-\pi^{2}\textit{{a}}^{2}(\textrm{{a}}+\lambda_{j})y-\pi^{2}\textit{{a}}^{3}\lambda_{j}\biggr{\\}}\frac{\pi^{2}\textit{{a}}^{2}e^{-\frac{y}{\textit{{a}}}-\frac{y}{\lambda}_{j}}}{2y^{2}\left(y^{2}+\pi^{2}\textit{{a}}\right)^{2}\lambda_{j}}.\end{split}$
$p_{c}=P\left(t_{1}\leq
Y_{i}<t_{2}\right)=\frac{\pi^{2}\textit{a}^{3}}{2}\left[\frac{\left(e^{\frac{2t_{1}}{\textit{a}}}-1\right)e^{-\frac{t_{1}}{\lambda_{j}}-\frac{t_{1}}{\textit{a}}}}{t_{1}^{3}+\pi^{2}\textit{a}^{2}t_{1}}-\frac{\left(e^{\frac{2t_{2}}{\textit{a}}}-1\right)e^{-\frac{t_{2}}{\lambda_{j}}-\frac{t_{2}}{\textit{a}}}}{t_{2}^{3}+\pi^{2}\textit{a}^{2}t_{2}}\right],$
(6) $p_{\textit{{a}}}=P\left(Y_{i}\geq
t_{2}\right)=\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2t_{2}}{\textit{a}}}-1\right)e^{-\frac{t_{2}}{\lambda_{j}}-\frac{t_{2}}{\textit{a}}}}{2t_{2}^{3}+2\pi^{2}\textit{a}^{2}t_{2}},$
(7)
$p_{r}=P\left(Y_{i}<t_{1}\right)=1-\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2t_{1}}{\textit{a}}}-1\right)e^{-\frac{t_{1}}{\lambda_{j}}-\frac{t_{1}}{\textit{a}}}}{2t_{1}^{3}+2\pi^{2}\textit{a}^{2}t_{1}}.$
(8)

###  2.1 Model development and fuzzy optimization problem

Our aim is to minimize TC such that conditions (4) and (5) are satisfied. We
note that the optimization problem with a random objective for TC leads to an
intractable problem, and cannot be solved analytically by satisfying the
constraints in (4) and (5). Hence, as in MK, we consider a problem of
minimizing the expected testing cost (ETC) satisfying (4) and (5). Considering
$C$ as the cost for testing an item for unit time and $E_{0}(Y_{i})$ as the
expectation of $Y_{i}$ under $H_{0}$, we have

$\mathrm{Total\ ETC}=\frac{C\times E_{0}(Y_{i})}{1-p_{c}},$

where $E_{0}(Y_{i})$ is given by,

$E_{0}(Y_{i})=\frac{\textrm{{a}}}{2}\operatorname{ln}\left[\frac{\textrm{{a}}+\lambda_{0}}{\textrm{{a}}-\lambda_{0}}\right]+\frac{\textrm{{a}}}{2}\left[\cos\left(\frac{\textrm{{a}}\pi}{\lambda_{0}}\right)\displaystyle\int\limits_{\frac{\textrm{{a}}\pi}{\lambda_{0}}-\pi}^{\frac{\textrm{{a}}\pi}{\lambda_{0}}+\pi}\frac{\cos
u}{u}du+\sin\left(\frac{\textrm{{a}}\pi}{\lambda_{0}}\right)\displaystyle\int\limits_{\frac{\textrm{{a}}\pi}{\lambda_{0}}-\pi}^{\frac{\textrm{{a}}\pi}{\lambda_{0}}+\pi}\frac{\sin
u}{u}du\right].$

The inequality in the constraints (4) and (5) will be more relaxed than the
crisp case and the tilda sign denotes the fuzziness in the parameters $\alpha$
and $\beta$. According to ASP, $\lambda_{0}\geq t_{2}$ as it is the AQL. Thus,
the optimization problem under fuzziness in parameters is formulated as,

$\displaystyle P_{1}(t_{1},t_{2}):$
$\displaystyle\underset{\left(t_{1},t_{2}\right)}{\operatorname{Min}}\left(\frac{CE_{0}(Y_{i})}{1-p_{c}}\right)\text{subject
to}$ (9) $\displaystyle P\left(\text{Reject the
lot}\mid\lambda\approx\frac{1}{\lambda_{0}}\right)\stackrel{{\scriptstyle\sim}}{{\leq}}\alpha,$
$\qquad\qquad\quad\qquad P\left(\text{Accept the
lot}\mid\lambda\approx\frac{1}{\lambda_{1}}\right)\stackrel{{\scriptstyle\sim}}{{\leq}}\beta,$
(10)

where $t_{1},t_{2}>0$ and $t_{2}>t_{1}$. That is,

$\begin{split}\quad&\underset{\left(t_{1},t_{2}\right)}{\operatorname{Min}}\left(\frac{CE_{0}(Y_{i})}{1-p_{c}}\right)\text{
subject to}\\\
&\left(\frac{p_{r}}{1-p_{c}}\mid\lambda\approx\frac{1}{\lambda_{0}}\right)\stackrel{{\scriptstyle\sim}}{{\leq}}\alpha,\\\
&\left(\frac{p_{a}}{1-p_{c}}\mid\lambda\approx\frac{1}{\lambda_{1}}\right)\stackrel{{\scriptstyle\sim}}{{\leq}}\beta,\end{split}$

where $t_{1},t_{2}>0$ and $t_{2}>t_{1}$. Substituting for $p_{c},p_{a}$ and
$p_{r}$ from (6), (7) and (8), the fuzzy optimization problem,
$P_{1}(t_{1},t_{2})$, becomes problem $P_{1}^{\prime}(t_{1},t_{2})$:

$P_{1}^{\prime}(t_{1},t_{2}):\qquad\underset{\left(t_{1},t_{2}\right)}{\operatorname{Min}}\quad\frac{CE_{0}(Y_{i})}{1-\frac{\pi^{2}\textit{a}^{3}}{2}\left[\frac{\left(e^{\frac{2t_{1}}{\textit{a}}}-1\right)e^{-\frac{t_{1}}{\lambda_{0}}-\frac{t_{1}}{\textit{a}}}}{t_{1}^{3}+\pi^{2}\textit{a}^{2}t_{1}}-\frac{\left(e^{\frac{2t_{2}}{\textit{a}}}-1\right)e^{-\frac{t_{2}}{\lambda_{0}}-\frac{t_{2}}{\textit{a}}}}{t_{2}^{3}+\pi^{2}\textit{a}^{2}t_{2}}\right]}$
such that
$\displaystyle\frac{1-\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2t_{1}}{\textit{a}}}-1\right)e^{-\frac{t_{1}}{\lambda_{0}}-\frac{t_{1}}{\textit{a}}}}{2t_{1}^{3}+2\pi^{2}\textit{a}^{2}t_{1}}}{1-\frac{\pi^{2}\textit{a}^{3}}{2}\left[\frac{\left(e^{\frac{2t_{1}}{\textit{a}}}-1\right)e^{-\frac{t_{1}}{\lambda_{0}}-\frac{t_{1}}{\textit{a}}}}{t_{1}^{3}+\pi^{2}\textit{a}^{2}t_{1}}-\frac{\left(e^{\frac{2t_{2}}{\textit{a}}}-1\right)e^{-\frac{t_{2}}{\lambda_{0}}-\frac{t_{2}}{\textit{a}}}}{t_{2}^{3}+\pi^{2}\textit{a}^{2}t_{2}}\right]}\stackrel{{\scriptstyle\sim}}{{\leq}}\alpha,$
(11)
$\displaystyle\frac{\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2t_{2}}{\textit{a}}}-1\right)e^{-\frac{t_{2}}{\lambda_{1}}-\frac{t_{2}}{\textit{a}}}}{2t_{2}^{3}+2\pi^{2}\textit{a}^{2}t_{2}}}{1-\frac{\pi^{2}\textit{a}^{3}}{2}\left[\frac{\left(e^{\frac{2t_{1}}{\textit{a}}}-1\right)e^{-\frac{t_{1}}{\lambda_{1}}-\frac{t_{1}}{\textit{a}}}}{t_{1}^{3}+\pi^{2}\textit{a}^{2}t_{1}}-\frac{\left(e^{\frac{2t_{2}}{\textit{a}}}-1\right)e^{-\frac{t_{2}}{\lambda_{1}}-\frac{t_{2}}{\textit{a}}}}{t_{2}^{3}+\pi^{2}\textit{a}^{2}t_{2}}\right]}\stackrel{{\scriptstyle\sim}}{{\leq}}\beta,$

where $t_{1},t_{2}>0$, $t_{2}>t_{1},\textrm{{a}}>0\text{ and
}\textrm{{a}}>\lambda_{j},j=0,1.$ Note that, the integrals in $E_{0}(Y_{i})$
do not have a closed-form expression. Hence, we consider two different
approaches of solving the problem $P_{1}^{\prime}(t_{1},t_{2})$:

  * Case (i): First, we evaluate the integrals in $E_{0}(Y_{i})$ numerically, by using the numerical integration method called composite Simpson’s rule (see richard2011) and get an approximate value for $E_{0}(Y_{i})$, say $E_{0}(Y_{i})^{*}$.

  * Case (ii): We find an upper bound for $E_{0}(Y_{i})$, and solve a new problem $Q_{1}(t_{1},t_{2})$, as follows:

Minimize $ETC_{UB}$ such that conditions (9) and (10) are satisfied,

$\text{where,}\ ETC_{UB}=\frac{C\times E_{0}(Y_{i})^{**}}{1-p_{c}},$ (12)

and

$\left|E_{0}(Y_{i})\right|\leq\frac{\textrm{{a}}}{2}\operatorname{ln}\left(\frac{\textrm{{a}}+\lambda_{0}}{\textrm{{a}}-\lambda_{0}}\right)\left[1+\left|\cos\left(\frac{\textrm{{a}}\pi}{\lambda_{0}}\right)\right|+\left|\sin\left(\frac{\textrm{{a}}\pi}{\lambda_{0}}\right)\right|\right]=E_{0}(Y_{i})^{**}.$

After the substitution of expressions for $p_{c},p_{a}$ and $p_{r}$ from (6),
(7) and (8), the fuzzy optimization problem, $Q_{1}(t_{1},t_{2})$, becomes
problem $Q_{1}^{\prime}(t_{1},t_{2})$:

$\begin{split}Q_{1}^{\prime}(t_{1},t_{2}):\qquad\underset{\left(t_{1},t_{2}\right)}{\operatorname{Min}}\quad&\frac{CE_{0}(Y_{i})^{**}}{1-\frac{\pi^{2}\textit{a}^{3}}{2}\left[\frac{\left(e^{\frac{2t_{1}}{\textit{a}}}-1\right)e^{-\frac{t_{1}}{\lambda_{0}}-\frac{t_{1}}{\textit{a}}}}{t_{1}^{3}+\pi^{2}\textit{a}^{2}t_{1}}-\frac{\left(e^{\frac{2t_{2}}{\textit{a}}}-1\right)e^{-\frac{t_{2}}{\lambda_{0}}-\frac{t_{2}}{\textit{a}}}}{t_{2}^{3}+\pi^{2}\textit{a}^{2}t_{2}}\right]}\\\
\text{ such
that}\quad&\frac{1-\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2t_{1}}{\textit{a}}}-1\right)e^{-\frac{t_{1}}{\lambda_{0}}-\frac{t_{1}}{\textit{a}}}}{2t_{1}^{3}+2\pi^{2}\textit{a}^{2}t_{1}}}{1-\frac{\pi^{2}\textit{a}^{3}}{2}\left[\frac{\left(e^{\frac{2t_{1}}{\textit{a}}}-1\right)e^{-\frac{t_{1}}{\lambda_{0}}-\frac{t_{1}}{\textit{a}}}}{t_{1}^{3}+\pi^{2}\textit{a}^{2}t_{1}}-\frac{\left(e^{\frac{2t_{2}}{\textit{a}}}-1\right)e^{-\frac{t_{2}}{\lambda_{0}}-\frac{t_{2}}{\textit{a}}}}{t_{2}^{3}+\pi^{2}\textit{a}^{2}t_{2}}\right]}\stackrel{{\scriptstyle\sim}}{{\leq}}\alpha,\\\
&\frac{\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2t_{2}}{\textit{a}}}-1\right)e^{-\frac{t_{2}}{\lambda_{1}}-\frac{t_{2}}{\textit{a}}}}{2t_{2}^{3}+2\pi^{2}\textit{a}^{2}t_{2}}}{1-\frac{\pi^{2}\textit{a}^{3}}{2}\left[\frac{\left(e^{\frac{2t_{1}}{\textit{a}}}-1\right)e^{-\frac{t_{1}}{\lambda_{1}}-\frac{t_{1}}{\textit{a}}}}{t_{1}^{3}+\pi^{2}\textit{a}^{2}t_{1}}-\frac{\left(e^{\frac{2t_{2}}{\textit{a}}}-1\right)e^{-\frac{t_{2}}{\lambda_{1}}-\frac{t_{2}}{\textit{a}}}}{t_{2}^{3}+\pi^{2}\textit{a}^{2}t_{2}}\right]}\stackrel{{\scriptstyle\sim}}{{\leq}}\beta,\end{split}$

where $t_{1},t_{2}>0$, $t_{2}>t_{1},\textrm{{a}}>0\text{ and
}\textrm{{a}}>\lambda_{j},j=0,1.$

####  2.1.1 Fuzzy optimization algorithm

The optimization problem $P_{1}^{\prime}(t_{1},t_{2})$ and
$Q_{1}^{\prime}(t_{1},t_{2})$ are fuzzy nonlinear programming problems (NLPP)
and they are converted to crisp nonlinear programming problems
$P_{1}^{\prime\prime}(t_{1},t_{2})$ and $Q_{1}^{\prime\prime}(t_{1},t_{2})$,
respectively by the following steps as in jameel2012 with fuzzy parameters
$\stackrel{{\scriptstyle\sim}}{{\alpha}}\
=\left\\{\left(x,\sigma_{\alpha}(x)\right):x\in\mathbb{R}\right\\}$ and
$\stackrel{{\scriptstyle\sim}}{{\beta}}\
=\left\\{\left(x,\sigma_{\beta}(x)\right):x\in\mathbb{R}\right\\}$ having
membership functions (see Figure 2),

$\sigma_{\alpha}(x)=\begin{cases}1,&\text{if }x<\alpha,\\\
\frac{\alpha+b_{1}-x}{b_{1}},&\text{if }\alpha\leq x<\alpha+b_{1},\\\
0,&\text{if }x\geq\alpha+b_{1},\end{cases}$ (13)
$\sigma_{\beta}(x)=\begin{cases}1,&\text{if }x<\beta,\\\
\frac{\beta+b_{2}-x}{b_{2}},&\text{if }\beta\leq x<\beta+b_{2},\\\ 0,&\text{if
}x\geq\beta+b_{2},\end{cases}$ (14)

where $b_{i}\in[0,1],i=1,2$.

Figure 2: Membership function of $\alpha$.

  1. Step 1: First, solve the following NLPP,

$\displaystyle Z_{1}(t_{1},t_{2}):$
$\displaystyle\underset{\left(t_{1},t_{2}\right)}{\operatorname{Min}}\quad\text{ETC*}$
(15) such that $\displaystyle g_{1}(t_{1},t_{2})\leq\alpha,$ $\displaystyle
h_{1}(t_{1},t_{2})\leq\beta,$ $\begin{split}\text{where}\ \
\text{ETC*}&=\frac{CE_{0}(Y_{i})^{*}}{1-\frac{\pi^{2}\textit{a}^{3}}{2}\left[\frac{\left(e^{\frac{2t_{1}}{\textit{a}}}-1\right)e^{-\frac{t_{1}}{\lambda_{0}}-\frac{t_{1}}{\textit{a}}}}{t_{1}^{3}+\pi^{2}\textit{a}^{2}t_{1}}-\frac{\left(e^{\frac{2t_{2}}{\textit{a}}}-1\right)e^{-\frac{t_{2}}{\lambda_{0}}-\frac{t_{2}}{\textit{a}}}}{t_{2}^{3}+\pi^{2}\textit{a}^{2}t_{2}}\right]},\\\
g_{1}(t_{1},t_{2})&=\frac{1-\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2t_{1}}{\textit{a}}}-1\right)e^{-\frac{t_{1}}{\lambda_{0}}-\frac{t_{1}}{\textit{a}}}}{2t_{1}^{3}+2\pi^{2}\textit{a}^{2}t_{1}}}{1-\frac{\pi^{2}\textit{a}^{3}}{2}\left[\frac{\left(e^{\frac{2t_{1}}{\textit{a}}}-1\right)e^{-\frac{t_{1}}{\lambda_{0}}-\frac{t_{1}}{\textit{a}}}}{t_{1}^{3}+\pi^{2}\textit{a}^{2}t_{1}}-\frac{\left(e^{\frac{2t_{2}}{\textit{a}}}-1\right)e^{-\frac{t_{2}}{\lambda_{0}}-\frac{t_{2}}{\textit{a}}}}{t_{2}^{3}+\pi^{2}\textit{a}^{2}t_{2}}\right]},\end{split}$
$h_{1}(t_{1},t_{2})=\frac{\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2t_{2}}{\textit{a}}}-1\right)e^{-\frac{t_{2}}{\lambda_{1}}-\frac{t_{2}}{\textit{a}}}}{2t_{2}^{3}+2\pi^{2}\textit{a}^{2}t_{2}}}{1-\frac{\pi^{2}\textit{a}^{3}}{2}\left[\frac{\left(e^{\frac{2t_{1}}{\textit{a}}}-1\right)e^{-\frac{t_{1}}{\lambda_{1}}-\frac{t_{1}}{\textit{a}}}}{t_{1}^{3}+\pi^{2}\textit{a}^{2}t_{1}}-\frac{\left(e^{\frac{2t_{2}}{\textit{a}}}-1\right)e^{-\frac{t_{2}}{\lambda_{1}}-\frac{t_{2}}{\textit{a}}}}{t_{2}^{3}+\pi^{2}\textit{a}^{2}t_{2}}\right]},$

$t_{1},t_{2}>0$, $t_{2}>t_{1},\textrm{{a}}>0,\textrm{{a}}>\lambda_{j},j=0,1$
and $b_{i}\in[0,1],i=1,2$.

  2. Step 2: Then, solve the NLPP,

$\displaystyle Z_{2}(t_{1},t_{2}):$
$\displaystyle\underset{\left(t_{1},t_{2}\right)}{\operatorname{Min}}\quad\text{ETC*}$
(16) such that $\displaystyle g_{1}(t_{1},t_{2})\leq\alpha+b_{1},$
$\displaystyle h_{1}(t_{1},t_{2})\leq\beta+b_{2},$

where $t_{1},t_{2}>0$,
$t_{2}>t_{1},\textrm{{a}}>0,\textrm{{a}}>\lambda_{j},j=0,1$ and
$b_{i}\in[0,1],i=1,2$.

  3. Step 3: After solving the NLPPs given by (15) and (16), we get the optimal values for $Z_{1}(t_{1},t_{2})$ and $Z_{2}(t_{1},t_{2})$. Then, find

$\begin{split}Z_{U}&=\max\left\\{Z_{1}(t_{1},t_{2}),Z_{2}(t_{1},t_{2})\right\\},\\\
Z_{L}&=\min\left\\{Z_{1}(t_{1},t_{2}),Z_{2}(t_{1},t_{2})\right\\}.\end{split}$

  4. Step 4: Finally, we get the corresponding crisp alternative of the fuzzy NLPP $P_{1}^{\prime}(t_{1},t_{2})$ as

$\begin{split}P_{1}^{\prime\prime}(t_{1},t_{2}):\qquad&\underset{\left(t_{1},t_{2}\right)}{\operatorname{Max}}\quad\phi\\\
\text{ such
that}\qquad&\phi-\left(\frac{\text{ETC*}-Z_{L}}{Z_{U}-Z_{L}}\right)\leq 0,\\\
&\phi-\left(\frac{\alpha+b_{1}-g_{1}(t_{1},t_{2})}{b_{1}}\right)\leq 0,\\\
&\phi-\left(\frac{\beta+b_{2}-h_{1}(t_{1},t_{2})}{b_{2}}\right)\leq 0,\\\
\end{split}$

where $t_{1},t_{2}>0$,
$t_{2}>t_{1},\textrm{{a}}>0,\textrm{{a}}>\lambda_{j},j=0,1,b_{i}\in[0,1],i=1,2$
and $\phi\in[0,1]$.

  5. Step 5: Similarly, by interchanging ETC* in (15) and (16) with $ETC_{UB}$ given by (12) and by repeating the above Steps 1 to 4 with $ETC_{UB}$, we get the corresponding crisp NLPP, $Q_{1}^{\prime\prime}(t_{1},t_{2})$, of $Q_{1}^{\prime}(t_{1},t_{2})$ given by,

$\begin{split}Q_{1}^{\prime\prime}(t_{1},t_{2}):\qquad&\underset{\left(t_{1},t_{2}\right)}{\operatorname{Max}}\quad\phi\\\
\text{ such
that}\qquad&\phi-\left(\frac{\text{ETC}_{UB}-Z_{L}}{Z_{U}-Z_{L}}\right)\leq
0,\\\ &\phi-\left(\frac{\alpha+b_{1}-g_{1}(t_{1},t_{2})}{b_{1}}\right)\leq
0,\\\ &\phi-\left(\frac{\beta+b_{2}-h_{1}(t_{1},t_{2})}{b_{2}}\right)\leq
0,\\\ \end{split}$

where $t_{1},t_{2}>0$,
$t_{2}>t_{1},\textrm{{a}}>0,\textrm{{a}}>\lambda_{j},j=0,1,b_{i}\in[0,1],i=1,2$
and $\phi\in[0,1]$.
The optimal pair $(t_{1},t_{2})$ is tabulated in Table 5.1 using fmincon
solver in MATLAB in both of these methods for different values of fuzzy
parameters. In Table 5.1, the results of MK is presented. Next, a comparative
study is made using Table 5.1 and 5.1.

##  3 Repetitive Group Sampling Plan with Fuzzy Parameters

In this section, we develop three different fuzzy acceptance sampling plans
using repetitive group sampling plans (RGSP) based on minimum of $n$ samples,
maximum of $n$ samples, and Type I censoring. The lifetime of units is assumed
as exponentially distributed with pdf given by (1). The decision on rejection
and acceptance of the lot is made using fuzzy inequalities given by (9) and
(10) with fuzzy parameters AQL, RQL, $\alpha$, and $\beta$. In the following
subsections, we describe these sampling plans, and the corresponding results
are compared.

###  3.1 Acceptance sampling plan based on minimum of n samples

Here, the acceptance sampling plan is based on the time of the first failure
of an item in a random sample of size $n$. Suppose independent and identically
distributed random variables $X_{i},\ i=1,2,3,\cdots,n$ denotes the lifetime
of an $i^{th}$ item in a random sample of size $n$.
Let $Y_{min}=\min\left\\{X_{1},X_{2},\ldots,X_{n}\right\\}$ and it follows the
exponential distribution with mean $\frac{1}{n\lambda}$. The acceptance
sampling plan procedure is as follows:

  1. 1.

Fix the values of AQL and RQL at $\alpha$ and $\beta$.

  2. 2.

Take a random sample of size $n$ and obtain $Y_{min}$.

  3. 3.

Reject the lot, if $Y_{min}<t_{1}$ or accept the lot, if $Y_{min}\geq t_{2}$,
else, move to Step 4.

  4. 4.

If $t_{1}\leq Y_{min}<t_{2}$, then Step 2 is repeated by taking a new sample
from the lot.

The aim is to compute the optimal value of $(n,t_{1},t_{2})$ which minimizes
the cost of testing subject to the fuzzy hypothesis (3) with $\alpha$ and
$\beta$ taken as fuzzy numbers. Under the hypothesis $H_{j},j=0,1$, the
weighted probability density function of $Y_{min}$ is defined by

$\displaystyle\begin{split}f_{j}(y)=\biggr{\\{}\left[\left(\textit{{a}}-\lambda_{j}\right)n^{3}y^{3}+3\textit{{a}}n^{2}y^{2}\lambda_{j}+\pi^{2}\textit{{a}}^{2}\left(\textit{{a}}-\lambda_{j}\right)ny+\pi^{2}\textit{{a}}^{3}\lambda_{j}\right]e^{\frac{2ny}{\textit{{a}}}}-(\lambda_{j}+\textrm{{a}})n^{3}y^{3}\\\
-3\textit{{a}}n^{2}y^{2}\lambda_{j}-\pi^{2}\textit{{a}}^{2}(\textrm{{a}}+\lambda_{j})ny-\pi^{2}\textit{{a}}^{3}\lambda_{j}\biggr{\\}}\frac{\pi^{2}\textit{{a}}^{2}e^{-\frac{ny}{\textit{{a}}}-\frac{ny}{\lambda}_{j}}}{2ny^{2}\left(n^{2}y^{2}+\pi^{2}\textit{{a}}\right)^{2}\lambda_{j}}.\end{split}$

The probability of acceptance, rejection of the lot, and continuation of the
sampling process under the hypothesis $H_{j},j=0,1$ are given respectively as:

$p_{\textit{{a}}}=P\left(Y_{min}\geq
t_{2}\right)=\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2nt_{2}}{\textit{a}}}-1\right)e^{-\frac{nt_{2}}{\lambda_{j}}-\frac{nt_{2}}{\textit{a}}}}{2n^{3}t_{2}^{3}+2nt_{2}\pi^{2}\textit{a}^{2}},$
(17)
$p_{r}=P\left(Y_{min}<t_{1}\right)=1-\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2nt_{1}}{\textit{a}}}-1\right)e^{-\frac{nt_{1}}{\lambda_{j}}-\frac{nt_{1}}{\textit{a}}}}{2n^{3}t_{1}^{3}+2nt_{1}\pi^{2}\textit{a}^{2}},$
(18) $p_{c}=P\left(t_{1}\leq
Y_{min}<t_{2}\right)=\frac{\pi^{2}\textit{a}^{3}}{2n}\left[\frac{\left(e^{\frac{2nt_{1}}{\textit{a}}}-1\right)e^{-\frac{nt_{1}}{\lambda_{j}}-\frac{nt_{1}}{\textit{a}}}}{n^{2}t_{1}^{3}+\pi^{2}\textit{a}^{2}t_{1}}-\frac{\left(e^{\frac{2nt_{2}}{\textit{a}}}-1\right)e^{-\frac{nt_{2}}{\lambda_{j}}-\frac{nt_{2}}{\textit{a}}}}{n^{2}t_{2}^{3}+\pi^{2}\textit{a}^{2}t_{2}}\right].$
(19)

Note that, the total cost of testing is the product of the total testing time
and the testing cost of an item for unit time. But the total time for
concluding the testing process is the random variable since $Y_{min}$ and
sample number are random variables. Hence, we compute the expected testing
cost similar to the previous section by assuming $C$ as the testing cost of an
item for unit time. Let $E_{j}(Y_{min})$ denotes the expectation of $Y_{min}$
under $H_{j},j=0,1.$ Then expected testing cost (ETC) is given by,

$\mathrm{ETC}=\frac{C\times E_{j}(Y_{min})}{1-p_{c}}.$ (20)

We observe that minimizing the random total cost subject to constraints (9)
and (10) will lead to an intractable optimization problem. Hence, in this
case, we minimize ETC subjecting to the constraints (9) and (10) and obtain
optimal values of $(n,t_{1},t_{2})$. The next subsection will describe the
formulation of the fuzzy optimization problem and its solution procedure.

####  3.1.1 Fuzzy optimization problem and solution procedure

The expectation of $Y_{min}$ under null hypothesis $H_{0}$ is taken for
calculating expected cost since it is a function of unknown parameter
$\lambda$. Thus, the expectation of $Y_{min}$ under $H_{0}$ is given by,

$E_{0}(Y_{min})=\frac{\textrm{{a}}}{2n}\left[\operatorname{ln}\left[\frac{\textrm{{a}}+\lambda_{0}}{\textrm{{a}}-\lambda_{0}}\right]+\cos\left(\frac{\textrm{{a}}\pi}{\lambda_{0}}\right)\displaystyle\int\limits_{\frac{\textrm{{a}}\pi}{\lambda_{0}}-\pi}^{\frac{\textrm{{a}}\pi}{\lambda_{0}}+\pi}\frac{\cos
u}{u}du+\sin\left(\frac{\textrm{{a}}\pi}{\lambda_{0}}\right)\displaystyle\int\limits_{\frac{\textrm{{a}}\pi}{\lambda_{0}}-\pi}^{\frac{\textrm{{a}}\pi}{\lambda_{0}}+\pi}\frac{\sin
u}{u}du\right].$

It can be seen that the integrals in the above equation do not have an
elementary anti-derivative without which one cannot proceed to solve the
optimization problem. Hence, we resort to numerical integration to compute
approximate value for $E_{0}(Y_{min})$. Composite Simpson’s rule is applied to
obtain $E_{0}(Y_{min})$ numerically (say, $E_{0}(Y_{min})^{*}$). Thus, we form
the following nonlinear fuzzy optimization problem $P_{2}(n,t_{1},t_{2})$
given by

$\displaystyle P_{2}(n,t_{1},t_{2}):$
$\displaystyle\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Min}}\left(\frac{CE_{0}(Y_{min})^{*}}{1-p_{c}}\right)\quad\text{such
that}$ (21) $\displaystyle P\left(\text{Reject the
lot}\mid\lambda\approx\frac{1}{\lambda_{0}}\right)\stackrel{{\scriptstyle\sim}}{{\leq}}\alpha,$
$\qquad\qquad\qquad\quad P\left(\text{Accept the
lot}\mid\lambda\approx\frac{1}{\lambda_{1}}\right)\stackrel{{\scriptstyle\sim}}{{\leq}}\beta,$
(22)

where $t_{1},t_{2}>0,t_{2}>t_{1}$ and $\lambda_{0}\geq t_{2}$ (by the
definition of ASP). That is,

$\begin{split}&\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Min}}\left(\frac{CE_{0}(Y_{min})^{*}}{1-p_{c}}\right)\\\
\text{ such
that}\qquad&\left(\frac{p_{r}}{1-p_{c}}\mid\lambda\approx\frac{1}{\lambda_{0}}\right)\stackrel{{\scriptstyle\sim}}{{\leq}}\alpha,\\\
&\left(\frac{p_{a}}{1-p_{c}}\mid\lambda\approx\frac{1}{\lambda_{1}}\right)\stackrel{{\scriptstyle\sim}}{{\leq}}\beta,\end{split}$

where $t_{1},t_{2}>0$ and $t_{2}>t_{1}$. Substituting for $p_{a}$, $p_{r}$ and
$p_{c}$ from Eqs (17), (18) and (19), the problem $P_{2}(n,t_{1},t_{2})$
becomes $P_{2}^{\prime}(n,t_{1},t_{2})$,

$\displaystyle P_{2}^{\prime}(n,t_{1},t_{2}):$
$\displaystyle\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Min}}\text{ETC}^{*}$
(23) such that $\displaystyle
g_{2}(n,t_{1},t_{2})\stackrel{{\scriptstyle\sim}}{{\leq}}\alpha,$
$\displaystyle
h_{2}(n,t_{1},t_{2})\stackrel{{\scriptstyle\sim}}{{\leq}}\beta,$
$\text{where}\ \
\text{ETC}^{*}=\frac{CE_{0}(Y_{min})^{*}}{1-\frac{\pi^{2}\textit{a}^{3}}{2n}\left[\frac{\left(e^{\frac{2nt_{1}}{\textit{a}}}-1\right)e^{-\frac{nt_{1}}{\lambda_{0}}-\frac{nt_{1}}{\textit{a}}}}{n^{2}t_{1}^{3}+\pi^{2}\textit{a}^{2}t_{1}}-\frac{\left(e^{\frac{2nt_{2}}{\textit{a}}}-1\right)e^{-\frac{nt_{2}}{\lambda_{0}}-\frac{nt_{2}}{\textit{a}}}}{n^{2}t_{2}^{3}+\pi^{2}\textit{a}^{2}t_{2}}\right]},$
$\begin{split}g_{2}(n,t_{1},t_{2})&=\frac{1-\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2nt_{1}}{\textit{a}}}-1\right)e^{-\frac{nt_{1}}{\lambda_{0}}-\frac{nt_{1}}{\textit{a}}}}{2n^{3}t_{1}^{3}+2nt_{1}\pi^{2}\textit{a}^{2}}}{1-\frac{\pi^{2}\textit{a}^{3}}{2n}\left[\frac{\left(e^{\frac{2nt_{1}}{\textit{a}}}-1\right)e^{-\frac{nt_{1}}{\lambda_{0}}-\frac{nt_{1}}{\textit{a}}}}{n^{2}t_{1}^{3}+\pi^{2}\textit{a}^{2}t_{1}}-\frac{\left(e^{\frac{2nt_{2}}{\textit{a}}}-1\right)e^{-\frac{nt_{2}}{\lambda_{0}}-\frac{nt_{2}}{\textit{a}}}}{n^{2}t_{2}^{3}+\pi^{2}\textit{a}^{2}t_{2}}\right]},\\\
h_{2}(n,t_{1},t_{2})&=\frac{\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2nt_{2}}{\textit{a}}}-1\right)e^{-\frac{nt_{2}}{\lambda_{1}}-\frac{nt_{2}}{\textit{a}}}}{2n^{3}t_{2}^{3}+2nt_{2}\pi^{2}\textit{a}^{2}}}{1-\frac{\pi^{2}\textit{a}^{3}}{2n}\left[\frac{\left(e^{\frac{2nt_{1}}{\textit{a}}}-1\right)e^{-\frac{nt_{1}}{\lambda_{1}}-\frac{nt_{1}}{\textit{a}}}}{n^{2}t_{1}^{3}+\pi^{2}\textit{a}^{2}t_{1}}-\frac{\left(e^{\frac{2nt_{2}}{\textit{a}}}-1\right)e^{-\frac{nt_{2}}{\lambda_{1}}-\frac{nt_{2}}{\textit{a}}}}{n^{2}t_{2}^{3}+\pi^{2}\textit{a}^{2}t_{2}}\right]},\end{split}$

$t_{1},t_{2}>0$, $t_{2}>t_{1},\textrm{{a}}>0\text{ and
}\textrm{{a}}>\lambda_{j},j=0,1.$
The membership functions given in (13) and (14) are used for the parameters
$\alpha$ and $\beta$ in $P_{2}^{\prime}(n,t_{1},t_{2})$. Next, the crisp
nonlinear optimization problem, $P_{2}^{\prime\prime}(n,t_{1},t_{2})$,
corresponding to $P_{2}^{\prime}(n,t_{1},t_{2})$ is obtained by applying the
steps of the fuzzy optimization algorithm discussed in Subsection 2.1.1 to the
problem $P_{2}^{\prime}(n,t_{1},t_{2})$ and
$P_{2}^{\prime\prime}(n,t_{1},t_{2})$ is given by,

$\begin{split}P_{2}^{\prime\prime}(n,t_{1},t_{2}):\qquad&\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Max}}\quad\phi\\\
\text{ such
that}\qquad&\phi-\left(\frac{\text{ETC*}-Z_{L}}{Z_{U}-Z_{L}}\right)\leq 0\\\
&\phi-\left(\frac{\alpha+b_{1}-g_{2}(n,t_{1},t_{2})}{b_{1}}\right)\leq 0,\\\
&\phi-\left(\frac{\beta+b_{2}-h_{2}(n,t_{1},t_{2})}{b_{2}}\right)\leq 0,\\\
\end{split}$

$\text{where}\quad
Z_{U}=\max\left\\{Z_{1}(n,t_{1},t_{2}),Z_{2}(n,t_{1},t_{2})\right\\},\
Z_{L}=\min\left\\{Z_{1}(n,t_{1},t_{2}),Z_{2}(n,t_{1},t_{2})\right\\},$
$t_{1},t_{2}>0$, $t_{2}>t_{1},\textrm{{a}}>0,\
\textrm{{a}}>\lambda_{j},j=0,1,\ b_{i}\in[0,1],i=1,2$ and $\phi\in[0,1]$.

The optimal solution of $P_{2}^{\prime\prime}(n,t_{1},t_{2})$ is obtained by
solving the nonlinear optimization problem using fmincon solver in MATLAB and
results are tabulated in Table 5.1 for various values of
$\textit{a},b_{1},b_{2},\lambda_{1},\lambda_{2},\alpha\text{ and }\beta$ and
compared with crisp case given in Table 5.1.

Observe that the problem $P_{2}^{\prime}(n,t_{1},t_{2})$ consists of a
objective function ETC, which involves $E_{0}(Y_{min})$. Note that this
$E_{0}(Y_{min})$ is approximated by replacing the integrals in it with the
numerical equivalent obtained by the composite Simpson’s rule. Next, we
consider another problem $Q_{2}(n,t_{1},t_{2})$ of minimizing the upper bound
for ETC subject to constraints (21) and (22). This will again form a fuzzy
optimization problem where the aim is to find optimal values for $n$, $t_{1}$,
and $t_{2}$. It is clear that the actual total cost will be lower than the
upper bound for ETC ($ETC_{UB}$), where $ETC_{UB}$ is given by,

$ETC_{UB}=\frac{C\times E_{0}(Y_{min})^{**}}{1-p_{c}}$ (24)
$\text{and}\qquad\left|E_{0}(Y_{min})\right|\leq\frac{\textrm{{a}}}{2n}\operatorname{ln}\left(\frac{\textrm{{a}}+\lambda_{0}}{\textrm{{a}}-\lambda_{0}}\right)\left[1+\left|\cos\left(\frac{\textrm{{a}}\pi}{\lambda_{0}}\right)\right|+\left|\sin\left(\frac{\textrm{{a}}\pi}{\lambda_{0}}\right)\right|\right]=E_{0}(Y_{min})^{**}.$

Thus, we consider the following fuzzy optimization problems:

$\begin{split}Q_{2}(n,t_{1},t_{2}):\qquad&\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Min}}\left(\frac{CE_{0}(Y_{min})^{**}}{1-p_{c}}\right)\\\
\text{ such
that}\qquad&\left(\frac{p_{r}}{1-p_{c}}\mid\lambda\approx\frac{1}{\lambda_{0}}\right)\stackrel{{\scriptstyle\sim}}{{\leq}}\alpha,\\\
&\left(\frac{p_{a}}{1-p_{c}}\mid\lambda\approx\frac{1}{\lambda_{1}}\right)\stackrel{{\scriptstyle\sim}}{{\leq}}\beta,\end{split}$

where $t_{1},t_{2}>0$ and $t_{2}>t_{1}$. Substituting for $p_{a}$, $p_{r}$ and
$p_{c}$ from Eqs (17), (18) and (19), the problem $Q_{2}(n,t_{1},t_{2})$
becomes $Q_{2}^{\prime}(n,t_{1},t_{2})$,

$\begin{split}Q_{2}^{\prime}(n,t_{1},t_{2}):\qquad&\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Min}}\frac{CE_{0}(Y_{min})^{**}}{1-\frac{\pi^{2}\textit{a}^{3}}{2n}\left[\frac{\left(e^{\frac{2nt_{1}}{\textit{a}}}-1\right)e^{-\frac{nt_{1}}{\lambda_{0}}-\frac{nt_{1}}{\textit{a}}}}{n^{2}t_{1}^{3}+\pi^{2}\textit{a}^{2}t_{1}}-\frac{\left(e^{\frac{2nt_{2}}{\textit{a}}}-1\right)e^{-\frac{nt_{2}}{\lambda_{0}}-\frac{nt_{2}}{\textit{a}}}}{n^{2}t_{2}^{3}+\pi^{2}\textit{a}^{2}t_{2}}\right]}\\\
\text{ such
that}\qquad&g_{2}(n,t_{1},t_{2})\stackrel{{\scriptstyle\sim}}{{\leq}}\alpha,\\\
&h_{2}(n,t_{1},t_{2})\stackrel{{\scriptstyle\sim}}{{\leq}}\beta,\end{split}$

where $t_{1},t_{2}>0$, $t_{2}>t_{1},\textrm{{a}}>0\text{ and
}\textrm{{a}}>\lambda_{j},j=0,1.$

The fuzzy nonlinear optimization problem, $Q_{2}^{\prime}(n,t_{1},t_{2})$, is
converted to the corresponding crisp nonlinear optimization problem, by
following the fuzzy optimization algorithm in Subsection 2.1.1 for
$Q_{2}^{\prime}(n,t_{1},t_{2})$. The crisp nonlinear problem,
$Q_{2}^{\prime\prime}(n,t_{1},t_{2})$, hence obtained is given by,

$\begin{split}Q_{2}^{\prime\prime}(n,t_{1},t_{2}):\qquad&\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Max}}\quad\phi\\\
\text{ such
that}\qquad&\phi-\left(\frac{\text{ETC}_{UB}-Z_{L}}{Z_{U}-Z_{L}}\right)\leq
0,\\\ &\phi-\left(\frac{\alpha+b_{1}-g_{2}(n,t_{1},t_{2})}{b_{1}}\right)\leq
0,\\\ &\phi-\left(\frac{\beta+b_{2}-h_{2}(n,t_{1},t_{2})}{b_{2}}\right)\leq
0,\\\ \end{split}$ $\begin{split}\text{where}\ \
\text{ETC}_{UB}&=\frac{CE_{0}(Y_{min})^{**}}{1-\frac{\pi^{2}\textit{a}^{3}}{2n}\left[\frac{\left(e^{\frac{2nt_{1}}{\textit{a}}}-1\right)e^{-\frac{nt_{1}}{\lambda_{0}}-\frac{nt_{1}}{\textit{a}}}}{n^{2}t_{1}^{3}+\pi^{2}\textit{a}^{2}t_{1}}-\frac{\left(e^{\frac{2nt_{2}}{\textit{a}}}-1\right)e^{-\frac{nt_{2}}{\lambda_{0}}-\frac{nt_{2}}{\textit{a}}}}{n^{2}t_{2}^{3}+\pi^{2}\textit{a}^{2}t_{2}}\right]},\end{split}$

$Z_{U}=\max\left\\{Z_{1}(n,t_{1},t_{2}),Z_{2}(n,t_{1},t_{2})\right\\},\
Z_{L}=\min\left\\{Z_{1}(n,t_{1},t_{2}),Z_{2}(n,t_{1},t_{2})\right\\},$
$t_{1},t_{2}>0$, $t_{2}>t_{1},\textrm{{a}}>0,\
\textrm{{a}}>\lambda_{j},j=0,1,b_{i}\in[0,1],i=1,2$ and $\phi\in[0,1]$.

The crisp nonlinear problem $Q_{2}^{\prime\prime}(n,t_{1},t_{2})$, is solved
using fmincon solver in MATLAB and optimal values of $n,t_{1}$ and $t_{2}$ are
obtained. Results are tabulated in Table 5.1 for different values of
$\textit{a},b_{1},b_{2},\lambda_{1},\lambda_{2},\alpha\text{ and }\beta$ and
compared with crisp case given in Table 5.1.

###  3.2 Acceptance sampling plan based on maximum of n samples

This section deals with the maximum lifetime among the $n$ samples taken at a
time. Let $X_{i},i=1,2,3,\cdots,n$ be the independent and identically
distributed exponential random variables which denotes the lifetime of
$i^{th}$ item with p.d.f given by (1) and
$Y_{max}=\max(X_{1},X_{2},\cdots,X_{n})$. $P(Y_{max}>t)=1-P(Y_{max}\leq
t)=1-\left(1-e^{-t\lambda}\right)^{n}$. The acceptance sampling plan procedure
(see MK) is as follows:

  1. 1.

Fix the values of AQL and RQL at specified producer’s risk $\alpha$ and
consumer’s risk $\beta$.

  2. 2.

Take a random sample of size $n$ and find the value of $Y_{max}$.

  3. 3.

Reject the lot, if $Y_{max}<t_{1}$ or accept the lot, if $Y_{max}\geq t_{2}$,
else, move to Step 4.

  4. 4.

If $t_{1}\leq Y_{max}<t_{2}$, then Step 2 is repeated by taking a new sample
from the lot.

As per our plan, the goal is to find the optimal values of $(n,t_{1},t_{2})$
which minimizes the total testing cost subject to the fuzzy hypothesis (3)
with all the parameters, namely, AQL, RQL, $\alpha$ and $\beta$ are fuzzy
numbers. We get the required probabilities, probability of continuation of the
sampling process, probability of acceptance and rejection of lot under the
hypothesis, $H_{j}$ for the given acceptance sampling plan respectively as,

$p_{c}=P\left(t_{1}\leq
Y_{max}<t_{2}\right)=\left(1-\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2t_{2}}{\textit{a}}}-1\right)e^{-\frac{t_{2}}{\lambda_{j}}-\frac{t_{2}}{\textit{a}}}}{2t_{2}^{3}+2\pi^{2}\textit{a}^{2}t_{2}}\right)^{n}-\left(1-\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2t_{1}}{\textit{a}}}-1\right)e^{-\frac{t_{1}}{\lambda_{j}}-\frac{t_{1}}{\textit{a}}}}{2t_{1}^{3}+2\pi^{2}\textit{a}^{2}t_{1}}\right)^{n},$
(25) $p_{\textit{{a}}}=P\left(Y_{max}\geq
t_{2}\right)=1-\left(1-\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2t_{2}}{\textit{a}}}-1\right)e^{-\frac{t_{2}}{\lambda_{j}}-\frac{t_{2}}{\textit{a}}}}{2t_{2}^{3}+2\pi^{2}\textit{a}^{2}t_{2}}\right)^{n},$
(26)
$p_{r}=P\left(Y_{max}<t_{1}\right)=\left(1-\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2t_{1}}{\textit{a}}}-1\right)e^{-\frac{t_{1}}{\lambda_{j}}-\frac{t_{1}}{\textit{a}}}}{2t_{1}^{3}+2\pi^{2}\textit{a}^{2}t_{1}}\right)^{n}.$
(27)

Here, the overall testing cost is the product of the cost of testing a sample
for unit time (C) and the total testing time. Note that, $Y_{max}$ and the
number of samples are random quantities and hence the total testing time is a
random quantity. Therefore, the expected testing cost (ETC) is calculated as
the total testing cost is a random quantity. Let $E_{j}(Y_{max})$ denotes the
expectation of $Y_{max}$ under the hypothesis $H_{j},j=0,1.$. Then the
expected testing cost (ETC) is given by,

$\mathrm{ETC}=\frac{C\times E_{j}(Y_{max})}{1-p_{c}}.$ (28)

We note that the minimization of random costs subject to (9) and (10) leads to
an intractable optimization problem. Thus, we minimize ETC such that
conditions (9) and (10) are satisfied to obtain the optimal values of
$(n,t_{1},t_{2})$. The fuzzy optimization problem formation and solution
procedure are explained in the next subsection.

####  3.2.1 Fuzzy optimization problem and solution procedure

The expectation of $Y_{max}$ under the null hypothesis $H_{0}$
($E_{0}(Y_{max})$) is obtained since ETC is a function of the unknown quantity
$\lambda$ and is given by,

$E_{0}(Y_{max})=\frac{\textrm{{a}}}{2}\sum\limits_{i=1}^{n}\frac{1}{i}\left[\operatorname{ln}\left[\frac{\textrm{{a}}+\lambda_{0}}{\textrm{{a}}-\lambda_{0}}\right]+\cos\left(\frac{\textrm{{a}}\pi}{\lambda_{0}}\right)\displaystyle\int\limits_{\frac{\textrm{{a}}\pi}{\lambda_{0}}-\pi}^{\frac{\textrm{{a}}\pi}{\lambda_{0}}+\pi}\frac{\cos
u}{u}du+\sin\left(\frac{\textrm{{a}}\pi}{\lambda_{0}}\right)\displaystyle\int\limits_{\frac{\textrm{{a}}\pi}{\lambda_{0}}-\pi}^{\frac{\textrm{{a}}\pi}{\lambda_{0}}+\pi}\frac{\sin
u}{u}du\right].$

Hence, we consider the fuzzy optimization problem of minimizing ETC under
$H_{0}$. Note that, the integrals in the equation of $E_{0}(Y_{max})$ do not
have a closed-form expression. In order to solve the optimization problem, the
integrals in $E_{0}(Y_{max})$ are computed numerically using composite
Simpson’s rule and obtain $E_{O}(Y_{max})^{*}$. Thus, we formulate the
following nonlinear fuzzy optimization problem $P_{3}(n,t_{1},t_{2})$ given
by,

$\displaystyle P_{3}(n,t_{1},t_{2}):$
$\displaystyle\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Min}}\left(\frac{CE_{0}(Y_{max})^{*}}{1-p_{c}}\right)\qquad\text{such
that}$ (29) $\displaystyle P\left(\text{Reject the
lot}\mid\lambda\approx\frac{1}{\lambda_{0}}\right)\stackrel{{\scriptstyle\sim}}{{\leq}}\alpha,$
$\qquad\qquad\qquad P\left(\text{Accept the
lot}\mid\lambda\approx\frac{1}{\lambda_{1}}\right)\stackrel{{\scriptstyle\sim}}{{\leq}}\beta,$
(30)

where $t_{1},t_{2}>0$, $t_{2}>t_{1}$, and $t_{2}\leq\lambda_{0}$. That is,

$\begin{split}&\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Min}}\left(\frac{CE_{0}(Y_{max})^{*}}{1-p_{c}}\right)\\\
\text{such
that}\qquad&\left(\frac{p_{r}}{1-p_{c}}\mid\lambda\approx\frac{1}{\lambda_{0}}\right)\stackrel{{\scriptstyle\sim}}{{\leq}}\alpha,\\\
&\left(\frac{p_{a}}{1-p_{c}}\mid\lambda\approx\frac{1}{\lambda_{1}}\right)\stackrel{{\scriptstyle\sim}}{{\leq}}\beta,\qquad\qquad\qquad\qquad\end{split}$

where $t_{1},t_{2}>0$, $t_{2}>t_{1}$, and $t_{2}\leq\lambda_{0}$. Substituting
for $p_{c}$, $p_{a}$ and $p_{r}$ from (25), (26) and (27), the problem
$P_{3}(n,t_{1},t_{2})$ becomes $P_{3}^{\prime}(n,t_{1},t_{2})$,

$\displaystyle P_{3}^{\prime}(n,t_{1},t_{2}):$
$\displaystyle\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Min}}\quad\text{ETC}^{*}$
(31) such that $\displaystyle
g_{3}(n,t_{1},t_{2})\stackrel{{\scriptstyle\sim}}{{\leq}}\alpha,$
$\displaystyle
h_{3}(n,t_{1},t_{2})\stackrel{{\scriptstyle\sim}}{{\leq}}\beta,$

where

$\begin{split}\text{ETC}^{*}&=\frac{CE_{0}(Y_{max})^{*}}{\left(1-\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2t_{2}}{\textit{a}}}-1\right)e^{-\frac{t_{2}}{\lambda_{0}}-\frac{t_{2}}{\textit{a}}}}{2t_{2}^{3}+2\pi^{2}\textit{a}^{2}t_{2}}\right)^{n}-\left(1-\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2t_{1}}{\textit{a}}}-1\right)e^{-\frac{t_{1}}{\lambda_{0}}-\frac{t_{1}}{\textit{a}}}}{2t_{1}^{3}+2\pi^{2}\textit{a}^{2}t_{1}}\right)^{n}},\\\
g_{3}(n,t_{1},t_{2})&=\frac{\left(1-\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2t_{1}}{\textit{a}}}-1\right)e^{-\frac{t_{1}}{\lambda_{0}}-\frac{t_{1}}{\textit{a}}}}{2t_{1}^{3}+2\pi^{2}\textit{a}^{2}t_{1}}\right)^{n}}{\left(1-\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2t_{2}}{\textit{a}}}-1\right)e^{-\frac{t_{2}}{\lambda_{0}}-\frac{t_{2}}{\textit{a}}}}{2t_{2}^{3}+2\pi^{2}\textit{a}^{2}t_{2}}\right)^{n}-\left(1-\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2t_{1}}{\textit{a}}}-1\right)e^{-\frac{t_{1}}{\lambda_{0}}-\frac{t_{1}}{\textit{a}}}}{2t_{1}^{3}+2\pi^{2}\textit{a}^{2}t_{1}}\right)^{n}},\end{split}$
$h_{3}(n,t_{1},t_{2})=\frac{1-\left(1-\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2t_{2}}{\textit{a}}}-1\right)e^{-\frac{t_{2}}{\lambda_{1}}-\frac{t_{2}}{\textit{a}}}}{2t_{2}^{3}+2\pi^{2}\textit{a}^{2}t_{2}}\right)^{n}}{\left(1-\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2t_{2}}{\textit{a}}}-1\right)e^{-\frac{t_{2}}{\lambda_{1}}-\frac{t_{2}}{\textit{a}}}}{2t_{2}^{3}+2\pi^{2}\textit{a}^{2}t_{2}}\right)^{n}-\left(1-\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2t_{1}}{\textit{a}}}-1\right)e^{-\frac{t_{1}}{\lambda_{1}}-\frac{t_{1}}{\textit{a}}}}{2t_{1}^{3}+2\pi^{2}\textit{a}^{2}t_{1}}\right)^{n}},$

$t_{1},t_{2}>0$, $t_{2}>t_{1},\textrm{{a}}>0,t_{2}\leq\lambda_{0},\text{ and
}\textrm{{a}}>\lambda_{j},j=0,1.$

The membership functions of the fuzzy parameters $\alpha$ and $\beta$ in
$P_{3}^{\prime}(n,t_{1},t_{2})$ are taken as in (13) and (14). The steps of
the fuzzy optimization algorithm described in the Subsection 2.1.1 are applied
to the fuzzy nonlinear optimization problem $P_{3}^{\prime}(n,t_{1},t_{2})$
and we get the corresponding crisp nonlinear optimization problem
$P_{3}^{\prime\prime}(n,t_{1},t_{2})$, given by,

$\begin{split}P_{3}^{\prime\prime}(n,t_{1},t_{2}):\qquad&\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Max}}\quad\phi\\\
\text{ such
that}\qquad&\phi-\left(\frac{\text{ETC*}-Z_{L}}{Z_{U}-Z_{L}}\right)\leq 0,\\\
&\phi-\left(\frac{\alpha+b_{1}-g_{3}(n,t_{1},t_{2})}{b_{1}}\right)\leq 0,\\\
&\phi-\left(\frac{\beta+b_{2}-h_{3}(n,t_{1},t_{2})}{b_{2}}\right)\leq 0,\\\
\end{split}$

$\text{where}\
Z_{U}=\max\left\\{Z_{1}(n,t_{1},t_{2}),Z_{2}(n,t_{1},t_{2})\right\\},\
Z_{L}=\min\left\\{Z_{1}(n,t_{1},t_{2}),Z_{2}(n,t_{1},t_{2})\right\\},$
$t_{1},t_{2}>0$, $t_{2}>t_{1},\ \textrm{{a}}>0,\ t_{2}\leq\lambda_{0},\
\phi\in[0,1],\ \textrm{{a}}>\lambda_{j},j=0,1$, and $b_{i}\in[0,1],i=1,2$.

The above crisp nonlinear optimization problem is solved to get the required
optimal value of $(n,t_{1},t_{2})$ using the fmincon solver in MATLAB, and
examples for different values of parameters in case of both crisp and fuzzy
values are tabulated in Table 5.1 and Table 5.1, respectively to verify the
computation.

Next, we consider another optimization problem of minimizing an upper bound
for ETC subject to the constraints (29) and (30) because of the presence of
integrals with no closed-form expression in the objective function ETC. Hence,
we formulate an optimization problem $Q_{3}(n,t_{1},t_{2})$ of minimizing the
upper bound for ETC such that the fuzzy inequalities in (29) and (30) are
satisfied and our aim is to get the optimal values for $n$, $t_{1}$ and
$t_{2}$. Note that, the overall cost of testing will be less than the cost
obtained through ETC upper bound ($ETC_{UB}$), where $ETC_{UB}$ is derived as,

$ETC_{UB}=\frac{C\times E_{0}(Y_{max})^{**}}{1-p_{c}}$ (32)
$\text{and}\qquad\left|E_{0}(Y_{max})\right|\leq\frac{\textrm{{a}}}{2}\sum\limits_{i=1}^{n}\frac{1}{i}\operatorname{ln}\left(\frac{\textrm{{a}}+\lambda_{0}}{\textrm{{a}}-\lambda_{0}}\right)\left[1+\left|\cos\left(\frac{\textrm{{a}}\pi}{\lambda_{0}}\right)\right|+\left|\sin\left(\frac{\textrm{{a}}\pi}{\lambda_{0}}\right)\right|\right]=E_{0}(Y_{max})^{**}.$

Then the corresponding fuzzy nonlinear optimization problem is given by,

$\begin{split}Q_{3}(n,t_{1},t_{2}):\qquad&\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Min}}\left(\frac{CE_{0}(Y_{max})^{**}}{1-p_{c}}\right)\\\
\text{such
that}\qquad&\left(\frac{p_{r}}{1-p_{c}}\mid\lambda\approx\frac{1}{\lambda_{0}}\right)\stackrel{{\scriptstyle\sim}}{{\leq}}\alpha,\\\
&\left(\frac{p_{a}}{1-p_{c}}\mid\lambda\approx\frac{1}{\lambda_{1}}\right)\stackrel{{\scriptstyle\sim}}{{\leq}}\beta,\end{split}$

where $t_{1},t_{2}>0$, $t_{2}>t_{1}$, and $t_{2}\leq\lambda_{0}$. Substituting
for $p_{c}$, $p_{a}$ and $p_{r}$ from (25), (26) and (27), the problem
$Q_{3}(n,t_{1},t_{2})$ becomes $Q_{3}^{\prime}(n,t_{1},t_{2})$,

$\displaystyle Q_{3}^{\prime}(n,t_{1},t_{2}):$
$\displaystyle\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Min}}\quad\text{ETC}_{UB}$
(33) such that $\displaystyle
g_{3}(n,t_{1},t_{2})\stackrel{{\scriptstyle\sim}}{{\leq}}\alpha,$
$\displaystyle
h_{3}(n,t_{1},t_{2})\stackrel{{\scriptstyle\sim}}{{\leq}}\beta,$
$\textrm{where}\ \
\text{ETC}_{UB}=\frac{CE_{0}(Y_{max})^{**}}{\left(1-\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2t_{2}}{\textit{a}}}-1\right)e^{-\frac{t_{2}}{\lambda_{0}}-\frac{t_{2}}{\textit{a}}}}{2t_{2}^{3}+2\pi^{2}\textit{a}^{2}t_{2}}\right)^{n}-\left(1-\frac{\pi^{2}\textit{a}^{3}\left(e^{\frac{2t_{1}}{\textit{a}}}-1\right)e^{-\frac{t_{1}}{\lambda_{0}}-\frac{t_{1}}{\textit{a}}}}{2t_{1}^{3}+2\pi^{2}\textit{a}^{2}t_{1}}\right)^{n}},\qquad\qquad\qquad\qquad\qquad$

$t_{1},t_{2}>0$, $t_{2}>t_{1},\textrm{{a}}>0,t_{2}\leq\lambda_{0}\text{ and
}\textrm{{a}}>\lambda_{j},j=0,1.$

By following the steps of the fuzzy optimization algorithm in Subsection 2.1.1
for $Q_{3}^{\prime}(n,t_{1},t_{2})$, the fuzzy nonlinear optimization problem,
$Q_{3}^{\prime}(n,t_{1},t_{2})$, is converted to the corresponding crisp
nonlinear optimization problem. The crisp nonlinear problem,
$Q_{3}^{\prime\prime}(n,t_{1},t_{2})$, is given by,

$\begin{split}Q_{3}^{\prime\prime}(n,t_{1},t_{2}):\qquad&\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Max}}\quad\phi\\\
\text{ such
that}\qquad&\phi-\left(\frac{\text{ETC}_{UB}-Z_{L}}{Z_{U}-Z_{L}}\right)\leq
0,\\\ &\phi-\left(\frac{\alpha+b_{1}-g_{3}(n,t_{1},t_{2})}{b_{1}}\right)\leq
0,\\\ &\phi-\left(\frac{\beta+b_{2}-h_{3}(n,t_{1},t_{2})}{b_{2}}\right)\leq
0,\\\ \end{split}$

$\text{where}\
Z_{U}=\max\left\\{Z_{1}(n,t_{1},t_{2}),Z_{2}(n,t_{1},t_{2})\right\\},\
Z_{L}=\min\left\\{Z_{1}(n,t_{1},t_{2}),Z_{2}(n,t_{1},t_{2})\right\\},$
$t_{1},\ t_{2}>0$, $t_{2}>t_{1},\ t_{2}\leq\lambda_{0}$, $\textrm{{a}}>0$,
$\textrm{{a}}>\lambda_{j},j=0,1$, $b_{i}\in[0,1],i=1,2$ and $\phi\in[0,1]$.

The crisp optimization problem, $Q_{3}^{\prime\prime}(n,t_{1},t_{2})$ is
solved by the fmincon solver in MATLAB, and examples for various values of
fuzzy and crisp parameters are illustrated in Table 5.1 and Table 5.1,
respectively to verify the computation and also for a comparative study.

###  3.3 Acceptance sampling plan under Type I censoring

In this ASP, from the given lot, a random sample of size $n$ is taken and
tested up to a predetermined time. Acceptance and rejection of the lot are
decided upon the fuzzy hypothesis given in (3). Our ASP, in this case, is as
follows:

  1. 1.

Take a random sample of size $n$ and run the test up to time $\tau$. Let $q$
be the number of failures obtained with corresponding lifetimes
$X_{1,n},X_{2,n},\ldots,X_{q,n}$.

  2. 2.

The maximum likelihood estimate of $\lambda$ is given by,
$\hat{\lambda}=\frac{\sum\limits_{i=i}^{q}X_{i,n}+(n-q)\tau}{q}.$

  3. 3.

If $t_{1}\leq\hat{\lambda}<t_{2}$, repeat Steps 1 and 2; else go to Step 4.

  4. 4.

If $\hat{\lambda}<t_{1}$, reject the lot. If $\hat{\lambda}\geq t_{2}$, accept
the lot.

But $\hat{\lambda}$ is asymptotically normal with mean $\lambda$ and standard
deviation
$\frac{\lambda}{\sqrt{n\left(1-\exp\left(\frac{-\tau}{\lambda}\right)\right)}}$
(see bartholomew1963sampling). By taking all the parameters as fuzzy numbers,
we get the required probabilities for the acceptance sampling plan after
simplification under $H_{j},j=0,1$ as follows:

$p_{c}=p\left(t_{1}\leq\hat{\lambda}<t_{2}\right)=p\left(\frac{t_{1}-\lambda_{j}}{\lambda_{j}}\sqrt{n\left(1-\exp\left(\frac{-\tau}{\lambda_{j}}\right)\right)}\leq
Z<\frac{t_{2}-\lambda_{j}}{\lambda_{j}}\sqrt{\left.n\left(1-\exp\left(\frac{-\tau}{\lambda_{j}}\right)\right)\right)}\right.,$
(34)
$p_{a}=p\left(t_{2}\leq\hat{\lambda}\right)=p\left(Z\geq\frac{t_{2}-\lambda_{j}}{\lambda_{j}}\sqrt{n\left(1-\exp\left(\frac{-\tau}{\lambda_{j}}\right)\right)}\right),$
(35)
$p_{r}=p\left(\hat{\lambda}<t_{1}\right)=p\left(Z<\frac{t_{1}-\lambda_{j}}{\lambda_{j}}\sqrt{n\left(1-\exp\left(\frac{-\tau}{\lambda_{j}}\right)\right)}\right),$
(36)

where $Z$ is the standard normal random variable. Our problem is to find
$\left(n,t_{1},t_{2}\right)$ with minimum total testing cost. Since total
testing cost is a random quantity, we minimize the expected testing cost at
$\lambda_{0}$ subject to fuzzy hypothesis given by (3) and obtain the optimal
values of $(n,t_{1},t_{2})$. Since there is a truncation in total testing time
$\tau$, two alternatives exist such as the testing can be terminated at $\tau$
or testing can be stopped, if all the test items fail before time $\tau$.
Assume that $\tau_{0}$ represents the time when every test item for a specific
sample has failed. Then, $\tau_{0}\leq\tau$. The total number of samples that
need to be tested is $\frac{1}{1-p_{c}}$. Because of this, the overall testing
time is $\frac{\tau}{1-p_{c}}$ or less. The total testing expense will be less
than or equal to $\frac{C\tau}{1-p_{c}}$, which is a random amount, given that
$C$ represents the cost of testing a sample for a unit of time. However, the
actual cost can be less. Since it is difficult to minimize the random cost, we
consider minimizing the expected value of the upper bound for random testing
cost, which is denoted by $\mathrm{ETC}_{\mathrm{ub}}$ and is given by

$\mathrm{ETC}_{\mathrm{ub}}=\frac{C\times\tau}{1-p_{c}}.$

####  3.3.1 Fuzzy optimization problem and solution procedure

The optimal strategy for minimising $\mathrm{ETC}_{\mathrm{ub}}$ will depend
on $\lambda$ because $\mathrm{ETC}_{\mathrm{ub}}$ is a function of $\lambda$.
Since there is no prior distribution assumed, therefore $\lambda$ is unknown.
To get beyond this issue, we take $\mathrm{ETC}_{\mathrm{ub}}$ under $H_{0}$.
Our basic objective is to reduce $\mathrm{ETC}_{\mathrm{ub}}$ subject to fuzzy
Type I and Type II errors based on the fuzzy hypothesis given by (3). Since
$\lambda_{0}$ is the AQL, we have $t_{2}\leq\lambda_{0}$ (applying the ASP
definition provided in Section 3.3). Thus we have the fuzzy optimization
problem $P_{4}(n,t_{1},t_{2})$:

$\begin{split}P_{4}(n,t_{1},t_{2}):\qquad&\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Min}}\left(\frac{C\times\tau}{1-p_{c}}\right)\\\
\text{such that}\qquad&P\left(\text{Type I error}\mid
H_{0}\right)\stackrel{{\scriptstyle\sim}}{{\leq}}\alpha,\\\ &P\left(\text{Type
II error}\mid
H_{1}\right)\stackrel{{\scriptstyle\sim}}{{\leq}}\beta,\end{split}$

where $\alpha$, $\beta$ are fuzzy and $t_{1},t_{2}>0$ and $t_{2}>t_{1}$. That
is,

$\begin{split}&\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Min}}\left(\frac{C\times\tau}{1-p_{c}}\right)\text{under}\
H_{0}\\\ \text{ such that}\qquad&\left(\frac{p_{r}}{1-p_{c}}\mid
H_{0}\right)\stackrel{{\scriptstyle\sim}}{{\leq}}\alpha,\\\
&\left(\frac{p_{a}}{1-p_{c}}\mid
H_{1}\right)\stackrel{{\scriptstyle\sim}}{{\leq}}\beta,\qquad\qquad\qquad\end{split}$

where $t_{1},t_{2}>0$ and $t_{2}>t_{1}$. Substituting for $p_{c},p_{a}$ and
$p_{r}$ from (34), (35) and (36), the fuzzy optimization problem
$P_{4}(n,t_{1},t_{2})$ becomes $P_{4}^{\prime}(n,t_{1},t_{2})$, given by,

$\displaystyle P_{4}^{\prime}(n,t_{1},t_{2}):$
$\displaystyle\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Min}}\mathrm{ETC}_{\mathrm{ub}}$
(37) such that $\displaystyle
g_{4}(n,t_{1},t_{2})\stackrel{{\scriptstyle\sim}}{{\leq}}\alpha,$
$\displaystyle
h_{4}(n,t_{1},t_{2})\stackrel{{\scriptstyle\sim}}{{\leq}}\beta,$
$\begin{split}\textrm{where}\ \
\mathrm{ETC}_{\mathrm{ub}}&=\frac{C\times\tau}{1-p\left(\frac{t_{1}-\lambda_{0}}{\lambda_{0}}\sqrt{n\left(1-\exp\left(\frac{-\tau}{\lambda_{0}}\right)\right)}\leq
z<\frac{t_{2}-\lambda_{0}}{\lambda_{0}}\sqrt{\left.n\left(1-\exp\left(\frac{-\tau}{\lambda_{0}}\right)\right)\right)}\right.},\qquad\qquad\\\
g_{4}(n,t_{1},t_{2})&=\frac{p\left(z<\frac{t_{1}-\lambda_{0}}{\lambda_{0}}\sqrt{n\left(1-\exp\left(\frac{-\tau}{\lambda_{0}}\right)\right)}\right)}{1-p\left(\frac{t_{1}-\lambda_{0}}{\lambda_{0}}\sqrt{n\left(1-\exp\left(\frac{-\tau}{\lambda_{0}}\right)\right)}\leq
z<\frac{t_{2}-\lambda_{0}}{\lambda_{0}}\sqrt{\left.n\left(1-\exp\left(\frac{-\tau}{\lambda_{0}}\right)\right)\right)}\right.},\qquad\qquad\qquad\qquad\\\
h_{4}(n,t_{1},t_{2})&=\frac{p\left(z\geq\frac{t_{2}-\lambda_{1}}{\lambda_{1}}\sqrt{n\left(1-\exp\left(\frac{-\tau}{\lambda_{1}}\right)\right)}\right)}{1-p\left(\frac{t_{1}-\lambda_{1}}{\lambda_{1}}\sqrt{n\left(1-\exp\left(\frac{-\tau}{\lambda_{1}}\right)\right)}\leq
z<\frac{t_{2}-\lambda_{1}}{\lambda_{1}}\sqrt{\left.n\left(1-\exp\left(\frac{-\tau}{\lambda_{1}}\right)\right)\right)}\right.},\qquad\qquad\qquad\qquad\\\
t_{1},t_{2}>0\text{ and }&t_{2}>t_{1}.\end{split}$

The fuzzy nonlinear optimization problem, $P_{4}^{\prime}(n,t_{1},t_{2})$, is
solved by changing it to its crisp alternative
$P_{4}^{\prime\prime}(n,t_{1},t_{2})$ by applying the steps in the algorithm
for fuzzy optimization explained in the Subsection 2.1.1., by taking the
membership functions (2.14 and 2.15) of $\alpha$ and $\beta$. The resulting
crisp optimization problem $P_{4}^{\prime\prime}(n,t_{1},t_{2})$ is given by,

$\begin{split}P_{4}^{\prime\prime}(n,t_{1},t_{2}):\qquad&\underset{\left(n,t_{1},t_{2}\right)}{\operatorname{Max}}\quad\phi\\\
\text{ such
that}\qquad&\phi-\left(\frac{\mathrm{ETC}_{\mathrm{ub}}-Z_{L}}{Z_{U}-Z_{L}}\right)\leq
0,\\\ &\phi-\left(\frac{\alpha+b_{1}-g_{4}(n,t_{1},t_{2})}{b_{1}}\right)\leq
0,\\\ &\phi-\left(\frac{\beta+b_{2}-h_{4}(n,t_{1},t_{2})}{b_{2}}\right)\leq
0,\\\ \end{split}$

$\text{where }\
Z_{U}=\max\left\\{Z_{1}(n,t_{1},t_{2}),Z_{2}(n,t_{1},t_{2})\right\\},\
Z_{L}=\min\left\\{Z_{1}(n,t_{1},t_{2}),Z_{2}(n,t_{1},t_{2})\right\\},t_{1},\
t_{2}>0$, $t_{2}>t_{1},\ b_{i}\in[0,1],\ i=1,2$ and $\phi\in[0,1]$.

The optimal values of required parameters $n$, $t_{1}$ and $t_{2}$ are found
by solving $P_{4}^{\prime\prime}(n,t_{1},t_{2})$ using fmincon solver in
MATLAB. In Table 5.1, examples for various parameter values are tabulated.
Also, a comparative study with corresponding crisp values of parameters is
also recorded in Table 5.1.

##  4 De-fuzzification Methods and their limitations in Acceptance Sampling
Plans

De-fuzzification is choosing a meaningful crisp value in relation to a fuzzy
set. Many applications employ standard de-fuzzification methods like the
center of gravity to generate crisp output. It is observed that most of the
standard de-fuzzification methods give the same crisp output for the fuzzy
numbers $H_{j}(\lambda),j=0,1,\alpha$ and $\beta$ as that of the method of
center of gravity. Thus, for the purpose of applying de-fuzzification methods
to the acceptance sampling plans discussed in previous sections, a widely used
de-fuzzification method, namely center of gravity method for the fuzzy number
$\lambda_{j},j=0,1$ with membership function given by,

$H_{j}(\lambda)=\left\\{\begin{array}[]{ll}\frac{1+cos(\textit{a}\pi(\lambda-\lambda_{j}))}{2}&,\quad\lambda_{j}-\frac{1}{\textit{a}}\leq\lambda\leq\lambda_{j}+\frac{1}{\textit{a}},\\\
0&,\quad\text{ otherwise, }\qquad\qquad\qquad j=0,1.\end{array}\right.$

In this method, the corresponding crisp value is computed as,

$\lambda_{j}^{*}=\frac{\displaystyle\int_{\lambda_{j}-\frac{1}{a}}^{\lambda_{j}+\frac{1}{a}}H_{j}(\lambda)\cdot\lambda\
d\lambda}{\displaystyle\int_{\lambda_{j}-\frac{1}{a}}^{\lambda_{j}+\frac{1}{a}}H_{j}(\lambda)\
d\lambda}=\lambda_{j}.$ (38)

So this de-fuzzification method in designing acceptance sampling plans
produces results similar to those discussed in MK. Hence, while designing
acceptance sampling plans, the uncertainty in the data is not taken care of by
de-fuzzification methods such as the center of gravity method. Therefore, it
is evident from (38) that the acceptance sampling plan obtained using the
weighted probability density function dealt with fuzzy hypotheses in a better
way than any de-fuzzification methods.

##  5 Numerical Results, Comparison and Case Study

In this section, the computational outcomes of all the above sections are
discussed. The numerical results for various sampling plans with different
parameter values in the case of fuzzy parameters are presented in Tables 5.1,
5.1, 5.1, and 5.1 and their corresponding results in crisp case are tabulated
in Tables 5.1, 5.1, 5.1, and 5.1. In all the cases, the expected testing cost
for fuzzy parameters is compared with the crisp case. It is observed from the
tables that, the cost of testing in the fuzzy case approaches that of the
crisp case as the fuzzy parameter values come close to the crisp value, which
validates the sampling plans. For example, in Table 5.1 of sequential sampling
plan, we have “$\lambda_{0}$ is approximately 300”, “$\lambda_{1}$ is
approximately 50”, “$\alpha$ is close to 0.05”, “$\beta$ is close to 0.05” for
$a=1500$, our sampling plan (for $C=1$) gives expected testing cost of
$665.7614$, and for $a=15000$, ETC is reduced to $661.2965$ whereas in the
crisp case, i.e, $\lambda_{0}=300,\lambda_{1}=50,\alpha=0.05$ and
$\beta=0.05$, the corresponding ETC is 654.1617 which shows the convergence of
ETC in fuzzy case to crisp as the fuzzy numbers approaches to exact value.
Table 5.1 illustrates the comparison of ETCs associated with all acceptance
sampling plans with fuzzy parameters. Consider the following example (in Table
5.1) for the set of fuzzy parameters: $\lambda_{0}=500,\ \lambda_{1}=150,\
\alpha=0.05,\ \beta=0.05,\ a=15000$, the ETC obtained for the acceptance
sampling plans, namely, sequential sampling plan, RGSP based on minimum of $n$
observations, maximum $n$ observations, and RGSP based on Type I censoring are
3866.4515, 491341.2, 2003.4707 and 99.9722, respectively. These comparisons
show that the RGSP with Type I censoring has the lowest testing cost of all
the acceptance sampling plans discussed here for fuzzy parameters.

###  5.1 Case study: A real example

To investigate the different acceptance sampling plans employed in this work
using fuzzy hypotheses, consider a real-world case study discussed in
alizadeh2015testing. The data presented here come from an experiment of
testing new compact electrical appliance models. The lifespans shown here are
the number of cycles of use that have been completed before the appliances
break down, the appliances were operated repeatedly by an automatic testing
system. This data follows exponential distribution.

$170,2694,3034,3214,2400,2471,329,13403,381,708,7846,1062,1594,1925,1990,2223,2327,2451,2551,$
$2565,2568,2702,11,35,2761,2831,3059,3112,3478,3504,4329,6367,6976,1167,49,958.$

Let the plan parameters be given in linguistic variables as $``\lambda_{0}$ is
nearly $3000"$, $``\lambda_{1}$ is close to $600"$, $\alpha$ and $\beta$ are
$``$approximately $0.05"$. These uncertainties are handled by taking them as
fuzzy numbers with appropriate fuzzy number representations. The optimization
problems $P_{1}^{\prime\prime}(t_{1},t_{2})$,
$P_{2}^{\prime\prime}(n,t_{1},t_{2})$, $P_{3}^{\prime\prime}(n,t_{1},t_{2})$,
and $P_{4}^{\prime\prime}(n,t_{1},t_{2})$ are solved for the plan parameters
$t_{1}$ and $t_{2}$ with $a=18000$ and $b_{1}=b_{2}=0.05$. The various
acceptance sampling plans discussed in this article produce the following
results for the aforementioned data:

  1. 1.

The sequential sampling plan gives $t_{1}=41$ and $t_{2}=3159$. Based on the
data set, we accept the lot, at the point of $8^{th}$ failure point, as
$Y_{i},i=1,2,\cdots 7$ is in between $t_{1}$ and $t_{2}$ and $Y_{8}$ is
greater than $t_{1}=41$.

  2. 2.

The RGSP results in $t_{1}=4$, $t_{2}=141$ and $n=20$ based on minimum of $n$
observations. We then take into account, the first 20 observations from the
preceding data. The minimum of these 20 observations in this instance exceeds
141. So we can accept the lot.

  3. 3.

Based on the RGSP, using the maximum of $n$ observations, we obtain
$t_{1}=203,\ t_{2}=2630$ and $n=2$. The first two observations from the above
data are then taken into consideration and their maximum is 2694, which is
higher than 2630. Therefore, we accept the lot.

  4. 4.

Based on Type I censoring, the RGSP yields the values $t_{1}=1219$,
$t_{2}=1990$ and $n=13$. We then take into account, the first 13 items using
the data provided above. Find the average lifespan, when the number of cycles
is truncated at $T=2000$. The 13 observations in the data have an average
lifespan of $3040.6667$, which is higher than $t_{2}=1990$. Therefore, we
accept the lot.

350pt[!htb]  $\lambda_{0}$ $\lambda_{1}$ $\alpha$ $\beta$ $a$
$Q_{1}^{\prime\prime}(t_{1},t_{2})$ $P_{1}^{\prime\prime}(t_{1},t_{2})$
$t_{1}$ $t_{2}$ $ETC_{UB}$ $t_{1}$ $t_{2}$ $ETC^{*}$ 300 50 0.05 0.05 1500
7.0971 247.9307 1317.112 5.8231 251.1178 665.7614 15000 6.2579 254.3787
1336.622 6.4907 251.617 661.2965 0.05 0.1 1500 8.271 205.0141 1141.6933 8.1902
205.6224 567.5001 15000 8.2609 205.0045 1127.7786 7.6285 204.1645 564.5195 300
70 0.1 0.1 2100 12.4249 268.0108 1341.6862 10.8498 289.1994 720.7401 15000
13.6293 274.7578 1349.7204 13.5624 275.2881 676.2045 0.05 0.05 2100 4.2697
393.4824 2126.2055 3.8761 412.3522 1128.9831 15000 4.0084 406.8951 2215.1938
3.9803 407.1574 1107.8074 Sequential sampling plan with fuzzy parameters for
C=1 and $b_{1}=b_{2}=0.05$.

230pt  $\lambda_{0}$ $\lambda_{1}$ $\alpha$ $\beta$ $t_{1}$ $t_{2}$ $ETC$ 300
50 0.05 0.05 6.9591 275.6183 654.1617 0.05 0.1 8.0812 204.9714 564.3796 300 70
0.1 0.1 13.6317 275.0377 675.3388 0.05 0.05 4.0960 406.8418 1106.1365
Sequential sampling plan with crisp parameters for C=1. 400pt  $\lambda_{0}$
$\lambda_{1}$ $\alpha$ $\beta$ $a$ $Q_{2}^{\prime\prime}(n,t_{1},t_{2})$
$P_{2}^{\prime\prime}(n,t_{1},t_{2})$ $t_{1}$ $t_{2}$ $n$ $ETC_{UB}$ $t_{1}$
$t_{2}$ $n$ $ETC^{*}$ 300 200 0.05 0.05 1500 0.00001 81.5216 42 863623.0499
0.000001 79.3124 50 202277.1689 15000 0.0000037 70.1102 48 888411.9581
0.0000041 65.3243 50 308804.9568 0.05 0.1 1500 0.0005 283.8829 10 531916.8655
0.0008 283.9992 9 111118.3155 15000 0.0004 320.7604 9 850862.4562 0.0005
284.0415 9 155357.7843 500 300 0.05 0.05 1500 0.00011 158.5504 30 240876.2939
0.0006 177.8714 23 46681.3855 15000 0.0004 137.4607 29 93348.8731 0.0004
134.4211 30 49062.1925 0.05 0.1 1500 0.0002 203.9995 17 44642.5866 0.0027
204.2658 17 20676.9203 15000 0.0021 203.7636 17 55787.4515 0.0018 224.5448 15
26779.9451 RGSP by minimum of $n$ observations with fuzzy parameters for C=1
and $b_{1}$ = $b_{2}$ = $0.05$.

260pt  $\lambda_{0}$ $\lambda_{1}$ $\alpha$ $\beta$ $t_{1}$ $t_{2}$ $n$ $ETC$
300 200 0.05 0.05 0.00001 65.801 50 316930.5172 0.05 0.1 0.00012 284.172 10
370615.1316 500 300 0.05 0.05 0.0003 134.4514 30 50246.7326 0.05 0.1 0.0015
204.3024 17 29028.4419 RGSP by minimum of $n$ observations for crisp
parameters for C=1. 400pt  $\lambda_{0}$ $\lambda_{1}$ $\alpha$ $\beta$ $a$
$Q_{3}^{\prime\prime}(n,t_{1},t_{2})$ $P_{3}^{\prime\prime}(n,t_{1},t_{2})$
$t_{1}$ $t_{2}$ $n$ $ETC_{UB}$ $t_{1}$ $t_{2}$ $n$ $ETC^{*}$ 300 50 0.05 0.05
1500 141.864 326.2226 9 1703.2061 130.9470 338.7602 11 921.1108 15000 135.3401
332.8650 9 1678.8201 130.6584 338.9876 12 916.1062 0.05 0.1 1500 84.0375
290.9444 5 1510.1843 176.205 197.1019 5 634.73 15000 51.922 322.6559 3
1287.7343 176.3513 196.9506 5 631.3610 500 150 0.05 0.05 1500 285.8609
887.9414 4 3811.4313 155.5672 997.4723 4 2039.2569 15000 310.9016 864.9856 6
3437.3812 224.4646 938.2128 4 2003.4707 0.05 0.1 1500 200.0049 621.0167 3
2290.9583 199.9814 621.0203 3 1366.8428 15000 197.0036 621.0157 3 2223.5466
196.998 621.0034 3 1111.5077 RGSP by maximum of $n$ observations with fuzzy
parameters for C=1 and $b_{1}=b_{2}=0.05$.

260pt  $\lambda_{0}$ $\lambda_{1}$ $\alpha$ $\beta$ $t_{1}$ $t_{2}$ $n$ $ETC$
300 50 0.05 0.05 239.5461 228.7969 5 620.609 0.05 0.1 192.6274 181.4727 4
545.6562 500 150 0.05 0.05 441.8257 756.6208 6 1399.2133 0.05 0.1 200.7733
621.8946 3 1110.014 RGSP by maximum of $n$ observations with crisp parameters
for C=1. 270pt  $\lambda_{0}$ $\lambda_{1}$ $T$ $\alpha$ $\beta$ $t_{1}$
$t_{2}$ $n$ $ETC_{ub}$ 300 200 50 0.01 0.01 236.8898 236.8898 33 49.9885 0.05
0.05 245.5488 245.5488 30 49.9887 300 200 100 0.01 0.01 235.182 235.182 28
99.9730 0.01 0.05 227.6252 227.6252 46 99.9722 RGSP with Type I censoring with
fuzzy parameters for C=1 and $b_{1}=b_{2}=0.01.$

270pt  $\lambda_{0}$ $\lambda_{1}$ $T$ $\alpha$ $\beta$ $t_{1}$ $t_{2}$ $n$
$ETC_{ub}$ 300 200 50 0.01 0.01 237.1385 237.1385 33 50 0.05 0.05 246.520
246.5220 30 50 300 200 100 0.01 0.01 234.6239 234.6239 28 100 0.01 0.05
227.449 227.449 46 100 RGSP with Type I censoring with crisp parameters for
C=1. 300pt  $\lambda_{0}$ $\lambda_{1}$ $\alpha$ $\beta$ $A$ $B$ $C$ $D$ 300
50 0.05 0.05 661.2965 $4.065681\cdot 10^{4}$ 916.1062 49.8877 300 200 0.05
0.05 1.8562$\cdot 10^{4}$ $3.088049\cdot 10^{5}$ 4606.3980 49.9887 500 150
0.05 0.05 3866.4515 $4.913412\cdot 10^{5}$ 2003.4707 99.9722 Comparison of
expected testing costs in the sequential sampling plan (column A), RGSP based
on minimum of $n$ observations (column B), maximum of $n$ observations (column
C) and RGSP under Type I censoring (column D) for fuzzy parameters with C=1,
$a=15000$ and $b_{1}=b_{2}=0.05.$

##  6 Conclusion

The majority of acceptance sampling plans available in the existing literature
are based on the assumption that data are crisp and assume that there is no
uncertainty in the data. Thus, no sampling plans in the literature address the
need of including ambiguous parameter information in the design of acceptance
sampling plans. In this study, we designed acceptance sampling plans by
incorporating uncertainties in data and solving a series of fuzzy optimization
problems to obtain the total expected cost of testing when all parameters in
the sampling plan are expressed in linguistic variables. Note that, all the
parameters AQL, RQL, $\alpha$, and $\beta$ are taken as fuzzy numbers. Observe
that, the objective function considered in optimization problems is the ETC
and which is minimized, satisfying the constraints of Type I and Type II
errors. This was necessary, due to the reason that the actual costs involved
in all the sampling plans are random in nature and thereby the corresponding
optimization problems are intractable. However, in practice, the actual cost
involved in testing may be less than the ETC reported here. For example, in
RGSP based on minimum of $n$ observations for fuzzy parameters (in Table 5.1),
when “$\lambda_{0}$ is approximately 300”, “$\lambda_{1}$ is approximately
200”, “$\alpha$ is close to 0.05”, “$\beta$ is close to 0.05” for $a=1500$ and
$C=1$, we get ETC = $202277.1689$ and for $a=15000$, ETC becomes
$308804.9568$, but in the crisp case, i.e,
$\lambda_{0}=300,\lambda_{1}=200,\alpha=0.05$ and $\beta=0.05$, the
corresponding ETC is 316930.5172. Next, consider another example in Table 5.1
of sequential sampling plan for fuzzy parameters, when “$\lambda_{0}$ is
approximately 300”,“$\lambda_{1}$ is approximately 50”, “$\alpha$ is close to
0.05”, “$\beta$ is close to 0.05” for $a=1500$, our ASP (for $C=1$) gives ETC
of $665.7614$ and for $a=15000$, ETC is reduced to $661.2965$ whereas in the
crisp case, i.e, $\lambda_{0}=300,\lambda_{1}=50,\alpha=0.05$ and
$\beta=0.05$, the corresponding ETC is 654.1617 which shows the convergence of
ETC in fuzzy case to crisp as the fuzzy numbers approaches to exact value. We
observe from our computational experience that, some of the fuzzy acceptance
sampling plans have reported slightly bigger costs than that in the existing
ASP (see MK). On the other hand, the corresponding cost is lower than that of
crisp ASP. The lower cost in crisp ASP may be due to the assumption that the
parameters are crisp, which is not true in practice, it can be vague, which is
to be taken into account while designing ASP. So in this work, we have solved
fuzzy optimization problems to design acceptance sampling plans based upon
uncertainties in the parameters which is more practical and real. In Section
4, we discussed, de-fuzzification methods and their limitations in ASP, it is
noted that certain de-fuzzification methods such as the center of gravity
method, do not handle the fuzziness in the parameters, thereby the results
obtained exactly coincide with those results obtained with crisp case (see
MK). Hence, we have used the weighted probability density function to
incorporate the uncertainties in the parameter effectively. Table 5.1 compares
the testing costs for the acceptance sampling plans with fuzzy parameters. For
the set of parameters
$\lambda_{0}=500,\lambda_{1}=150,\alpha=0.05,\beta=0.05,a=15000$, the expected
cost obtained for the acceptance sampling plans, namely, sequential sampling
plan, RGSP using minimum of $n$ observations, maximum $n$ observations and
RGSP based on Type I censoring are 3866.4515, 491341.2, 2003.4707, and
99.9722, respectively. These comparisons demonstrate that the RGSP based on
Type I censoring has the lowest testing cost. To illustrate the actual use of
our acceptance sampling plans, a real-world example is also presented. The
results obtained in this work can also be extended for more general class of
lifetime distributions such as Weibull distribution based upon any other
censoring schemes such as hybrid censoring.

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