Design of generalized fuzzy multiple deferred state sampling plan for attributes

Abstract

In industry, for the quality inspection processes, acceptance sampling plans proved to be economically viable, but the unpredictability of the plan’s characteristics made the use of conventional acceptance sampling plans less reliable. The generalized fuzzy multiple deferred state sampling plan (GFMDSSP) is suggested in this study for qualities that consider the difficulty in calculating the precise value of the percentage of defectives in a batch. The strategy is created with a minimal average sample size in mind and the performance measures have already been determined. An analysis of the current fuzzy acceptance sampling plans for characteristics is conducted, and an important conclusion is drawn regarding the effectiveness of the proposed scheme. Analysis of the impact of inspection errors on the sampling process reveals a decline in plan acceptance standards that is correlated with escalating inspection errors. Finally, some numerical examples are provided to support the findings.

Keywords

Fuzzy acceptance sampling plans average sample number acceptable quality level limiting quality level inspection errors

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 plans 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.

Table 1
Abbreviations used

AQL Acceptable quality level

ASN Average sample number

ATI Average total inspection

DSP Double sampling plan

FOC Fuzzy operating characteristic curve

FSSP Fuzzy single sampling plan

FDSP Fuzzy double sampling plan

FMDSSP Fuzzy multiple deferred state sampling plan

GMDSSP Generalized multiple deferred state sampling plan

GFMDSSP Generalized fuzzy multiple deferred state sampling plan

LQL Limiting quality level

MDSSP Multiple deferred state sampling plan

A crucial foundational step towards a globalized quality outcome was set by statistical quality control. The early stages of the development cycle, when new goods are conceived, old product designs are enhanced, and manufacturing processes are optimized, are frequently crucial to the success of the final product. The information in those data is utilized to regulate and enhance the process and the product, and statistical methods play a crucial part in quality control and improvement by serving as the primary tool for sampling, testing, and evaluation of a product. In order to achieve the necessary quality for the output of the product or service, these strategies are utilized in the industrial and service sectors. Plans for acceptance sampling have been widely praised for their simplicity and effectiveness. The improvement in quality engineering using acceptance sampling can be seen in [34]. Economic aspect of implementing these plans was studied further and new plans were developed [38]. These plans were widely used in different fields including biomedical engineering, supply chain and logistics, inventory management, food processing, and warehouse management.

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 [40]), which later evolved through the design of multiple deferred state sampling plans (MDSSP) as seen in [39]. The stages of attribute MDSSP 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 MDSSP is studied in [32, 35]. The determination of plan parameters concerning the acceptable quality level (AQL) and the limiting quality level (LQL) is studied in [26]. Later, the MDSSP 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 [11 , 42]).

The crisp nature of the fraction of non-conformity (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 SSP [16]. Risks and quality attributes were considered as fuzzy quantities to design an acceptance sampling plan by attributes [27]. Later, a fuzzy SSP is developed where the plan parameters were considered as fuzzy numbers [24]. Further developments in fuzzy acceptance sampling plans for attributes can be seen in [28 , 37]. Consequently, the uncertainty in the fraction of defectives in the MDSSP for attributes is addressed in [1, 3]. A variable MDSSP 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, [7 , 25]). The effect of measurement errors on fuzzy acceptance sampling plans was also studied [29]. The case with the MDSSP was analyzed in [4] and its fuzzy counterpart was studied in [6]. The applications of sampling plans to quality engineering problems were also increased. The supply chain and inventory management utilized sampling schemes with fuzziness and inspection errors for the quality management [36]. The results proved a notable drop in the probability of acceptance when the errors are identified.

When the limitations of MDSSP were identified, the assumptions of the MDS sampling scheme were generalized to consider measurement data [13]. 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 MDSSP 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 GMDSSP 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 organization of this paper is as follows. The forthcoming section carries some preliminaries of fuzzy sets. The assumptions and a systematic design of the GFMDSSP are 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 generalized 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 are mentioned in Section 8.

2 Some Preliminaries

This section presents some fundamentals related to fuzzy numbers.

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

$\exists x_{0} \in ℝ$ such that $μ_{\tilde{X}} (x_{0}) = 1$

$\forall x_{1}, x_{2} \in ℝ$ and ∀α ∈ [0, 1] implies that $μ_{\tilde{X}} [α x_{1} + (1 - α) x_{2}] \geq min {μ_{\tilde{X}} (x_{1}), μ_{\tilde{X}} (x_{2})}$

for each given $x \in ℝ$ and ɛ > 0, ∃ δ > 0 s.t. $\forall z \in ℝ, | x - z | < δ$ implies that $μ_{\tilde{X}} (z) < μ_{\tilde{X}} (x) + ɛ$

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

Definition 2. [21] The ν cut of a fuzzy number $\tilde{X} (ν \in [0, 1])$ is a crisp set defined as $\tilde{X} [ν] = {x \in ℝ, μ_{\tilde{B}} (x) \geq ν}$ usually denoted as $\tilde{X} [ν] = [X_{I} [ν], X_{II} [ν]]$ where $X_{I} [ν] = min {x \in ℝ, μ_{\tilde{B}} (x) \geq ν}$ and $X_{II} [ν] = max {x \in ℝ, μ_{\tilde{B}} (x) \geq ν}$

Definition 3. [15] Let A_i be a partition of sample space X, say $A = {x_{1}, x_{2}, . . ., x_{k}}, 1 \leq k \leq n, \tilde{P} (A_{i}) = \tilde{a_{i}}$ . Then for α ∈ [0, 1] we get $\tilde{P} (A) [ν] = {\sum_{i = 1}^{k} a_{i} : S}$ where $S = {a_{i} \in \tilde{a_{i}} [ν], i \in 1, 2, . . ., n, \sum_{i = 1}^{n} a_{i} = 1}$ . This is the restricted fuzzy arithmetic.

Definition 4. [14] The probability of obtaining r success in m experiments where R is a fuzzy binomial random variable is: $\tilde{P} (r) [ν] = {m r p^{r} q^{m - r} | S}, \forall 0 \leq ν \leq 1$ where $S = {p \in \tilde{p} [ν], q \in \tilde{p} [ν], p + q = 1}$

Definition 5. [14] For a random variable X following fuzzy Poisson distribution, the probability of obtaining X = x is $\tilde{P} (x) [ν] = {\frac{λ^{x} e^{- λ}}{x!} | λ \in \tilde{λ} [ν]}, \forall 0 \leq ν \leq 1$

3 Design of Generalized Fuzzy Multiple Deferred State Attribute Sampling Plan

This section aims to review the existing generalized MDSSP 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 misclassification. The following are assumptions required for the GMDSSP:

Continuous inspection is assumed for a series of successive lots.

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

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 GFMDSSP are:

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.

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

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

An overview of the GMDSSP 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 ≤ c₁ and rejected if d > c₂. If c₁ < d ≤ c₂ the lot under inspection is accepted provided k lots out of m previous lots have been accepted with the condition that d ≤ c₁. The lot is rejected otherwise. The GMDSSP is differentiated by the parameters n, k, m, c₁, and c₂. The probability of acceptance of the GMDS sampling plan for attributes is given by $\begin{matrix} P_{a} (p) = P (d \leq c_{1}) + P (c_{1} < d \leq c_{2}) \\ \sum_{j = k}^{m} [(\begin{array}{l} m \\ j \end{array}) [P (d \leq c_{1})]^{j} [1 - P (d \leq c_{1})]^{m - j}] \end{matrix}$ (1) When the fraction of defectives is fuzzy in nature, the existing plan can be extended with fuzzy parameter $\tilde{p}$ .

3.1 Probability of Acceptance of GFMDSSP

The GFMDSSP 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 GFMDSSP is given by: $\begin{matrix} {\tilde{P}}_{a} (\tilde{p}) = \tilde{P} (d \leq c_{1}) + \tilde{P} (c_{1} < d \leq c_{2}) \\ \sum_{j = k}^{m} [(\begin{array}{l} m \\ j \end{array}) [\tilde{P} (d \leq c_{1})]^{j} [1 - \tilde{P} (d \leq c_{1})]^{m - j}] \end{matrix}$

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 c₁, if d ≤ c₁, the lot is accepted. The probability of acceptance will be $\tilde{P} (d \leq c_{1})$ . For the conditional acceptance number c₂, if c₁ < d ≤ c₂, 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 $\sum_{j = k}^{m} [(\begin{array}{l} m \\ j \end{array}) [\tilde{P} (d \leq c_{1})]^{j} [1 - \tilde{P} (d \leq c_{1})]^{m - j}]$ The fuzzy probability of conditional acceptance can be obtained as: $\begin{matrix} \tilde{P} (c_{1} < d \leq c_{2}) \sum_{j = k}^{m} [(\begin{array}{l} m \\ j \end{array}) [\tilde{P} (d \leq c_{1})]^{j} \\ [1 - \tilde{P} (d \leq c_{1})]^{m - j}] \end{matrix}$ Hence, the fuzzy probability of acceptance of the current lot under the proposed GFMDSSP is: $\begin{matrix} {\tilde{P}}_{a} (\tilde{p}) = \tilde{P} (d \leq c_{1}) + \tilde{P} (c_{1} < d \leq c_{2}) \\ \sum_{j = k}^{m} [(\begin{array}{l} m \\ j \end{array}) [\tilde{P} (d \leq c_{1})]^{j} [1 - \tilde{P} (d \leq c_{1})]^{m - j}] \end{matrix}$ (2) Under fuzzy Binomial probability distribution, equation (2) can be written as: $\begin{matrix} \tilde{P_{a}} (\tilde{p}) & = \sum_{d = 0}^{c_{1}} [(\begin{array}{l} n \\ d \end{array}) {\tilde{p}}^{d} (1 - \tilde{p})^{n - d}] + \\ \sum_{d = c_{1} + 1}^{c_{2}} [(\begin{array}{l} n \\ d \end{array}) {\tilde{p}}^{d} (1 - \tilde{p})^{n - d}] * \\ \sum_{j = k}^{m} (\begin{array}{l} m \\ j \end{array}) {\sum_{d = 0}^{c_{1}} [(\begin{array}{l} n \\ d \end{array}) {\tilde{p}}^{d} (1 - \tilde{p})^{n - d}]}^{j} * \\ {1 - \sum_{d = 0}^{c_{1}} [(\begin{array}{l} n \\ d \end{array}) {\tilde{p}}^{d} (1 - \tilde{p})^{n - d}]}^{m - j} \end{matrix}$ (3) Under fuzzy Poisson probability distribution, equation (2) can be written as: $\begin{matrix} \tilde{P_{a}} (\tilde{p}) & = \sum_{d = 0}^{c_{1}} [\frac{e^{- n \tilde{p}} (n \tilde{p})^{d}}{d!}] + \sum_{d = c_{1} + 1}^{c_{2}} [\frac{e^{- n \tilde{p}} (n \tilde{p})^{d}}{d!}] * \\ \sum_{j = k}^{m} (\begin{array}{l} m \\ j \end{array}) {\sum_{d = 0}^{c_{1}} [\frac{e^{- n \tilde{p}} (n \tilde{p})^{d}}{d!}]}^{j} \\ {1 - \sum_{d = 0}^{c_{1}} [\frac{e^{- n \tilde{p}} (n \tilde{p})^{d}}{d!}]}^{m - j} \end{matrix}$ (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 ν- 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 ν- cut of $\tilde{p}$ and $\tilde{q}$ is defined as $\begin{matrix} \tilde{p} [ν] = [p_{1} + (p_{2} - p_{1}) ν, p_{3} - (p_{3} - p_{2}) ν], \\ \forall 0 \leq ν \leq 1 \end{matrix}$ (5) and $\begin{matrix} \tilde{q} [ν] = [q_{1} + (q_{2} - q_{1}) ν, q_{3} - (q_{3} - q_{2}) ν], \\ \forall 0 \leq ν \leq 1 \end{matrix}$ (6) Also let $S = {p \in \tilde{p} [ν], q \in \tilde{q} [ν], p + q = 1}$ (7)

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

$\tilde{P_{a}} (\tilde{p}) [ν] = [{\tilde{P}}_{aI} (\tilde{p}) [ν], {\tilde{P}}_{aII} (\tilde{p}) [ν]]$ (8) where ${\tilde{P}}_{aI} (\tilde{p}) [ν] = min {{\tilde{P}}_{a} (\tilde{p}) : S}$ (9) and ${\tilde{P}}_{aII} (\tilde{p}) [ν] = max {{\tilde{P}}_{a} (\tilde{p}) : S}$ (10)

3.2 Average Sample Number for GFMDSSP

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: $ASN = n$

3.3 Average Total Inspection for GFMDSSP

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 GFMDSSP 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 GFMDSSP.

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 $\tilde{p} [ν]$ 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 $\begin{matrix} \tilde{ATI} (\tilde{p}) & = n \tilde{P_{a}} (\tilde{p}) + (1 - \tilde{P_{a}} (\tilde{p})) N \\ = n + (1 - \tilde{P_{a}} (\tilde{p})) (N - n) \end{matrix}$ The ν- cut for $\tilde{ATI} (\tilde{p})$ can be defined according to ([15]): $\tilde{ATI} (\tilde{p}) [ν] = [{ATI}_{I} (\tilde{p}) [ν], {ATI}_{II} (\tilde{p}) [ν]]$ 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 c₁ = c₂ = c, k = m and m→ ∞, the proposed GFMDS plan reduces to the fuzzy single sampling plan for attributes defined by [31].

4 Determination of Optimal Parameters for GFMDSSP

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 ([33]). In the GMDSSP designed by [8], the ASN is minimized to obtain the plan parameters (say n, k, m, c₁ and c₂) for a given pair of (p_α, 1 - α) and (p_β, β). Fuzzy ASP are also determined using the above approach. Two points on the FOC band are considered, $(\tilde{p_{α}}, 1 - \tilde{α})$ and $(\tilde{p_{β}}, \tilde{β})$ corresponding to fuzzy AQL (denoted by $\tilde{AQL}$ ) and fuzzy LQL (denoted by $\tilde{LQL}$ ) respectively. $\tilde{α}$ is the fuzzy producer’s risk and $\tilde{β}$ is the fuzzy consumer’s risk. The above parameters can be represented as triangular fuzzy numbers as $\tilde{p}$ , say, $\begin{matrix} \tilde{p_{α}} = & (p_{1 α}, p_{2 α}, p_{3 α}) \\ \tilde{p_{β}} = & (p_{1 β}, p_{2 β}, p_{3 β}) \\ \tilde{α} = & (α_{1}, α_{2}, α_{3}) \\ \tilde{β} = & (β_{1}, β_{2}, β_{3}) \end{matrix}$ Also, the ν cut can be obtained as in Equations (6) as: $\begin{matrix} \tilde{p_{α}} [ν] = & [p_{1 α} + (p_{2 α} - p_{1 α}) ν, p_{3 α} - (p_{3 α} - p_{2 α}) ν], \\ \forall 0 \leq ν \leq 1 = [p_{α I}, p_{α II}] \\ \tilde{p_{β}} [ν] = & [p_{1 β} + (p_{2 β} - p_{1 β}) ν, p_{3 β} - (p_{3 β} - p_{2 β}) ν], \\ \forall 0 \leq ν \leq 1 = [p_{β I}, p_{β II}] \\ \tilde{α} [ν] = & [α_{1} + (α_{2} - α_{1}) ν, α_{3} - (α_{3} - α_{2}) ν], \\ \forall 0 \leq ν \leq 1 = [α_{I}, α_{II}] \\ \tilde{β} [ν] = & [β_{1} + (β_{2} - β_{1}) ν, β_{3} - (β_{3} - β_{2}) ν], \\ \forall 0 \leq ν \leq 1 = [β_{I}, β_{II}] \end{matrix}$ 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: ${\tilde{ASN}}_{I} (\tilde{p}) = {\begin{matrix} min & ASN (\tilde{p}) = \tilde{n} \\ subject to & \tilde{P_{aI}} (\tilde{p_{1}}) \geq 1 - \tilde{α_{I}} \\ \tilde{P_{aI}} (\tilde{p_{2}}) \leq \tilde{β_{I}} \end{matrix}$ (11) ${\tilde{ASN}}_{II} (\tilde{p}) = {\begin{matrix} min & ASN (\tilde{p}) = \tilde{n} \\ subject to & \tilde{P_{aII}} (\tilde{p_{1}}) \geq 1 - \tilde{α_{II}} \\ \tilde{P_{aII}} (\tilde{p_{2}}) \leq \tilde{β_{II}} \end{matrix}$ (12) When the value of ν = 1, optimization problems (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: $\begin{matrix} min & ASN (p) & = n \\ subject to & P_{a} (p_{1}) & \geq 1 - α \end{matrix}$ (13) $P_{a} (p_{2}) \leq β$

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 (2) and fuzzy Poisson probability model is given in Table (3).

Table 2

Optimal parameters of GFMDSSP for fuzzy Binomial distribution when $\tilde{α} [1] = 5 %$ and $\tilde{β} [1] = 10 %$

$\tilde{p_{1}}$	$\tilde{p_{2}} [1]$	n	m	k	c ₁	c ₂
0.001	0.03	82	1	1	0	1
0.001	0.035	70	1	1	0	1
0.0025	0.04	62	1	1	0	1
0.0025	0.05	49	1	1	0	1
0.005	0.08	30	1	1	0	1
0.005	0.1	24	1	1	0	1
0.0075	0.12	20	1	1	0	1
0.0075	0.15	16	1	1	0	1
0.01	0.17	14	1	1	0	1
0.01	0.2	12	1	1	0	1

Table 3

Optimal parameters of GFMDSSP for fuzzy Poisson distribution when $\tilde{α} [1] = 5 %$ and $\tilde{β} [1] = 10 %$

$\tilde{p_{1}}$	$\tilde{p_{2}} [1]$	n	m	k	c ₁	c ₂
0.001	0.03	84	1	1	0	1
0.001	0.035	72	1	1	0	1
0.0025	0.04	63	1	1	0	1
0.0025	0.05	50	1	1	0	1
0.005	0.08	32	1	1	0	1
0.005	0.1	25	1	1	0	1
0.0075	0.12	21	1	1	0	1
0.0075	0.15	17	1	1	0	1
0.01	0.17	15	1	1	0	1
0.01	0.2	13	1	1	0	1

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 GFMDSSP is carried out for screening of the lots with c₁ = 0, c₂ = 3, k = 1, m = 5 and n = 87. For 0 ≤ ν ≤ 1, the ν- cut of $\tilde{p}$ is: $\tilde{p} [ν] = [0.01 + 0.01 ν, 0.03 - 0.01 ν]$ $\tilde{p} [0] = [0.01, 0.03]$ By Equations (5-10), the ν- cut for $\tilde{P_{a}} (\tilde{p})$ can be obtained as: $\begin{matrix} \tilde{P_{a}} (\tilde{p}) [ν] = {\tilde{P} (d \leq 0) + \tilde{P} (0 < d \leq 3) \\ \sum_{j = 1}^{5} [(\begin{array}{l} 5 \\ j \end{array}) [\tilde{P} (d \leq 0)]^{j} [1 - \tilde{P} (d \leq 0)]^{5 - j}] : S} \end{matrix}$ For ν = 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 ν = 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 ν = 0.3. In this case, we obtain $\tilde{p} [ν] = [0.013, 0.027]$ and $\tilde{P_{a}} (\tilde{p}) [ν] = [0.36, 0.87]$ . 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, c₁ = 1, c₂ = 4, k = 1, m = 5. For 0 ≤ ν ≤ 1, the ν- cut of $\tilde{p}$ is given by: $\tilde{p} [ν] = [0.02 + 0.01 ν, 0.04 - 0.01 ν]$ For ν = 0, $\tilde{p} [ν] = [0.02, 0.04]$ . Using Equations (7-10), for ν = 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 ν = 1, the plan reduces to the classical MDSSP 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 (4). 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 θ to identify the variations of $\tilde{p}$ (see [3]). Now $\tilde{p} = (a_{1}, a_{2}, a_{3})$ can be written as $\tilde{p_{θ}}$ as follows: $\tilde{p_{θ}} = (θ, b_{2} + θ, b_{3} + θ)$ $\tilde{p_{θ}} [ν] = [θ + b_{2} ν, b_{3} + θ - (b_{3} - b_{2}) ν], \forall 0 \leq ν \leq 1$ where b_j = a_j - a₁, (j = 2, 3) and θ ∈ [0, 1 - b₃]. The probability of acceptance for $\tilde{p_{θ}}$ can be obtained by replacing p in Equations (4) by p_θ. For $\tilde{p} = (0.02, 0.03, 0.04)$ , we have $\tilde{p_{θ}} = (θ, 0.01 + θ, 0.02 + θ)$ and $\begin{matrix} \tilde{p_{θ}} [0] & = (θ + 0.01 * 0, 0.02 + θ - 0.01 * 0], \\ = [θ, 0.02 + θ] \forall 0 \leq θ \leq 0.98 \end{matrix}$ The fuzzy probability of acceptance for different values of ν and θ is given in Table (4).

Fig. 1

FOC bands for GFMDSSP for different values of ν.

Table 4

The fuzzy probability of lot acceptance for GFMDS plan without inspection errors corresponding to different values of ν and θ

θ	$\tilde{p_{θ}}$	$\tilde{p_{θ}}$	$\tilde{P_{a}} (\tilde{p_{θ}}) [0]$	$\tilde{p_{θ}} [0.3]$	$\tilde{P_{a}} (\tilde{p_{θ}}) [0.3]$	$\tilde{p_{θ}} [0.7]$	$\tilde{P_{a}} (\tilde{p_{θ}}) [0.7]$
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 GFMDSSP by showing that the ASN is the minimum for the proposed plan in comparison with the existing fuzzy acceptance sampling plans.

5 Probability of acceptance of GFMDSSP in terms of generalized fuzzy number

Throughout this paper, the fraction of defectives is assumed to be a triangular fuzzy number. However, this can be generalized 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 $ℙ$ denote the crisp proportion of defectives and crisp probability measure respectively, we can express $\tilde{p}$ as $\tilde{p} = {(p, μ_{\tilde{p}} (p) | p \in ℙ}$ where $μ_{\tilde{p}} (p)$ denotes the membership function of $\tilde{p}$ . The ν- cut or ν level sets of z_i is $\begin{matrix} \tilde{z_{i}} [ν] & = [min_{z_{i}} {z_{i} \in S (\tilde{z_{i}}) | μ_{\tilde{z_{i}}} (z_{i}) \geq ν}, \\ max_{z_{i}} {z_{i} \in S (\tilde{z_{i}}) | μ_{\tilde{z_{i}}} (z_{i}) \geq ν}] \\ = [\tilde{z_{iI}} [ν], \tilde{z_{iII}} [ν]] \end{matrix}$ where $S (\tilde{z_{i}}) = {z_{i} \in \tilde{z_{i}} [ν], \sum_{i = 1}^{n} z_{i} = 1}$ .

The upper and lower bound of ν- cut of $\tilde{P_{a}} (\tilde{p})$ can be found by solving the following pair of optimization problems: $\tilde{P_{aI}} (\tilde{p}) [ν] = {\begin{matrix} min P_{a} (p) \\ s . t . \tilde{z_{iI}} [ν] \leq \tilde{z_{i}} \leq \tilde{z_{iII}} [ν] \end{matrix}$ and $\tilde{P_{aII}} (\tilde{p}) [ν] = {\begin{matrix} max P_{a} (p) \\ s . t . \tilde{z_{iI}} [ν] \leq \tilde{z_{i}} \leq \tilde{z_{iII}} [ν] \end{matrix}$ where P_a (p) denotes the conventional probability of acceptance when p is crisp. Thus ν- cut of $\tilde{P_{a}} (\tilde{p})$ is $\tilde{P_{a}} (\tilde{p}) [ν] = [\tilde{P_{aI}} (\tilde{p}) [ν], \tilde{P_{aII}} (\tilde{p}) [ν]]$ According to [43], the ν- cuts form a nested interval for different values of ν. For 0 < ν₁ < ν₂ ≤ 1, we have $\tilde{P_{aI}} (\tilde{p}) [ν_{2}] \geq \tilde{P_{aI}} (\tilde{p}) [ν_{1}]$ and $\tilde{P_{aII}} (\tilde{p}) [ν_{1}] \geq \tilde{P_{aII}} (\tilde{p}) [ν]$ .

Example 4.Consider the proportion of defectives having a pentagonal membership function and following a fuzzy binomial distribution. The ν- cut for a pentagonal fuzzy number $\tilde{x} = (x_{1}, x_{2}, x_{3}, x_{4}, x_{5})$ is [2ν (x₂ - x₁) + x₁, - 2ν (x₅ - x₄) + x₅] for ν ∈ [0, 0.5] and [2ν (x₃ - x₂) +2x₂ - x₃, 2ν (x₄ - x₃) - x₄ + 2x₃] for ν ∈ [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, c₁ = 0, c₂ = 3, k = 1, m = 5. Then, $\tilde{p} [ν] = {\begin{matrix} \begin{matrix} [0.02 + 0.02 ν, 0.08 - 0.02 ν], & ν \in [0, 0.5] \\ [0.01 + 0.04 ν, 0.17 - 0.04 ν], & ν \in [0.5, 1] \end{matrix} \end{matrix}$ The probability of acceptance when p is a crisp variable is given in Equation (1)

Case 1: ν ∈ [0, 0.5]

To find ν- cut for $\tilde{P_{a}} (\tilde{p})$ , consider the following pair of optimization problems: $\tilde{P_{aI}} (\tilde{p}) = {\begin{matrix} min P_{a} (p) \\ s . t . 0.02 + 0.02 ν \leq p \leq 0.08 - 0.02 ν \end{matrix}$ and $\tilde{P_{aII}} (\tilde{p}) = {\begin{matrix} max P_{a} (p) \\ s . t . 0.02 + 0.02 ν \leq p \leq 0.08 - 0.02 ν \end{matrix}$ Solving the above problems, the ν- cut of fuzzy probability of acceptance for say ν = 0.3 is obtained as: $\tilde{P_{a}} (\tilde{p}) [0.3] = [0.002, 0.393]$

Case 2: ν ∈ [0.5, 1]

To find ν- cut for $\tilde{P_{a}} (\tilde{p})$ , consider the following pair of optimization problems: $\tilde{P_{aI}} (\tilde{p}) = {\begin{matrix} min P_{a} (p) \\ s . t . 0.01 + 0.04 ν \leq p \leq 0.17 - 0.04 ν \end{matrix}$ and $\tilde{P_{aII}} (\tilde{p}) = {\begin{matrix} max P_{a} (p) \\ s . t . 0.01 + 0.04 ν \leq p \leq 0.17 - 0.04 ν \end{matrix}$ Solving the above problems, the ν- cut of fuzzy probability of acceptance for say, ν = 0.8 is obtained as: $\tilde{P_{a}} (\tilde{p}) [0.8] = [0, 0.27]$

6 Efficiency of GFMDSSP

The limitation of classical MDSSP 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 and in recent works, it is shown that a GMDSSP overcomes the above-said shortcomings and safeguards the interests of the producer and consumer [8]. They also showed that the GMDSSP 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 GFMDSSP over the existing fuzzy acceptance sampling plans for attributes. We compare the GFMDSSP with fuzzy SSP and fuzzy DSP proposed by [31] and fuzzy MDSSP 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 (8) and for fuzzy Poisson distribution is shown in Table (9). From the tables, it is clear that the ASN has been considerably reduced when the GFMDSSP is employed which proves the economic advantage of GFMDSSP over the existing fuzzy acceptance sampling plans.

Table 5
ASN of FSSP, FDSP, FMDSSP and GFMDSSP for fuzzy binomial distribution

$\tilde{p_{1}}$ $\tilde{p_{2}} [1]$ FSSP FDSP FMDSSP GFMDSSP

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

$\tilde{p_{1}}$	$\tilde{p_{2}} [1]$	FSSP	FDSP	FMDSSP	GFMDSSP
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 6

ASN of FSSP, FDSP, FMDSSSP and GFMDSSP for fuzzy Poisson distribution

$\tilde{p_{1}}$	$\tilde{p_{2}} [1]$	FSSP	FDSP	FMDSSP	GFMDSSP
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 7

Comparison of fuzzy probability of lot acceptance for GFMDSSP with and without inspection errors for ν = 0

θ	$\tilde{p_{θ}}$	$\tilde{P_{a}} (\tilde{p_{θ}}) [0]$	$\tilde{p_{δ θ}} [0]$	$\tilde{P_{a}} (\tilde{p_{δ θ}}) [0]$
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]

Table 8

ASN of FSSP, FDSP, FMDSSP and GFMDSSP for fuzzy Binomial distribution

$\tilde{p_{1}}$	$\tilde{p_{2}} [1]$	FSSP	FDSP	FMDSSP	GFMDSSP
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 9

ASN of FSSP, FDSP, FMDSSP and GFMDSSP for fuzzy Poisson distribution

$\tilde{p_{1}}$	$\tilde{p_{2}} [1]$	FSSP	FDSP	FMDSSP	GFMDSSP
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

7 GFMDSSP 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 δ₁ and δ₂). The GFMDSSP with inspection error is designed by considering the possible events of inspection process:

A₁: The item is imperfect.

A₂: The item is perfect.

B: The item is classified as imperfect post-inspection.

B|A₂: A perfect item is classified as imperfect.

B^c|A₁: 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_{δ}}

. Then,

\tilde{p} = \tilde{P} (A_{1})

and

\tilde{p_{e}} = \tilde{P} (B)

. Also, δ₁ = P (B|A₂) and δ₂ = P (B^c|A₁). By Definition 2, for ν ∈ [0, 1] ,

\tilde{p_{δ}} = \tilde{p} (1 - δ_{2}) + \tilde{q} δ_{1}

(14) The ν- cut is given by

\tilde{p_{δ}} [ν] = {\tilde{p} (1 - δ_{2}) + \tilde{q} δ_{1} : S}

(15) where S is defined by:

S = {p \in \tilde{p_{δ}} [ν], q \in \tilde{q_{δ}} [ν], p_{δ} + q_{δ} = 1}

(16) The fuzzy probability of acceptance when inspection errors are considered can be obtained by replacing

\tilde{p}

\tilde{p_{δ}}

in Equations (4) for binomial and Poisson models respectively for S given in Equation (16). The ν- cut is obtained similarly which is represented as

\tilde{P_{a}} (\tilde{p_{δ}}) [ν] = [{\tilde{P}}_{eI} [ν], {\tilde{P}}_{eII} [ν]]

(17) throughout this article.

Example 5. Suppose that the inspection process carried out in the production system in Example (4) is found to be imperfect. The inspection errors are identified as δ₁ = 0.01 and δ₂ = 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: $\tilde{p_{δ}} = (0.0191, 0.0282, 0373)$ $\tilde{p} [ν] = [0.01 + 0.01 ν, 0.03 - 0.01 ν]$ $\tilde{p_{δ}} = [0.0191 + 0.0091 ν, 0.0373 - 0.0091 ν]$ When ν = 0, $\tilde{p} [0] = [0.01, 0.03] {and \tilde{p_{δ}} [0] = [0.0191, 0.0373]$ Using Equations 17), the 0- cut of fuzzy probability of lot acceptance with inspection errors is: $\tilde{P_{a}} (\tilde{p_{δ}}) [0] = [0.13, 0.66]$ 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 4, 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 (4).

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 4, a facilitator parameter θ can be used to plot the FOC bands. Consider the assumptions in Example 7. $\tilde{p_{δ}} = (c_{1}, c_{2}, c_{3})$ can be represented using the facilitator parameter θ as: $\tilde{p_{δ θ}} = (θ, d_{2} + θ, d_{3} + θ)$ The ν- cut of $\tilde{p_{δ θ}}$ is: $\tilde{p_{δ θ}} [ν] = [θ + d_{2} + θ, d_{3} + θ - (d_{3} - d_{2}) α],$ $\forall 0 \leq ν \leq 1$ where d_i = c_i - c₁ (i = 2, 3) and θ ∈ [0, 1 - d₃]. For finding the probability of acceptance, the fuzzy proportion of defectives in equations can be replaced with $\tilde{p_{δ θ}}$ . For $\tilde{p_{δ}} = (0.0191, 0.0282, 0.0373)$ , $\tilde{p_{δ θ}} = [θ, 0.0091 + θ, 0.0182 + θ]$ . The ν cut of $\tilde{p_{δ θ}}$ is: $\tilde{p_{δ θ}} [ν] = [θ + 0.0091 ν, 0.0182 + θ - 0.0091 ν]$ For ν = 0, $\tilde{p_{δ θ}} [0] = [θ, 0.0182 + θ] \forall 0 \leq θ \leq 0.9818$ The values of $\tilde{p_{δ θ}} [0]$ and $\tilde{P_{a}} (\tilde{p_{δ}}) [0]$ are shown in Table (10) for different values of θ. Also, the FOC band for GFMDSSP 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.

Table 10

Comparison of fuzzy probability of lot acceptance for GFMDSSP with and without inspection errors for ν = 0

θ	$\tilde{p_{θ}}$	$\tilde{P_{a}} (\tilde{p_{θ}}) [0]$	$\tilde{p_{δ θ}} [0]$	$\tilde{P_{a}} (\tilde{p_{δ θ}}) [0]$
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]

Fig. 2

FOC bands for GFMDS plan with and without inspection errors.

8 Conclusion

The generalized multiple deferred state sampling strategy was expanded in this study to include the fuzzy environment. The fuzzy set theory is used to deal with the uncertainty in calculating the precise value of defective fraction. When p is precise and the plan is thus well-defined, the GFMDSSP reduces to a traditional GMDSSP. The numerical examples show how the proposed strategy is constructed, advantageous, and how flaws affect it. By contrasting the average sample size with that of single, double, and multiple stage acceptance sampling plans in the fuzzy environment, the efficiency of GFMDSSP is determined. To address the flawed measuring procedure, the plan is revised to include inspection faults. When sample errors are found, a decrease in the chance of acceptance is observed. The outcomes demonstrate that this strategy outperformed the shortcomings of the previous fuzzy acceptance sampling strategies. For fuzzy binomial and fuzzy poisson probability distributions, the suggested strategy is defined. Fuzzy inspection errors can be taken into account in further research. In this context, neutrosophic statistics may also be used.

Footnotes

Acknowledgement

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

References

Afshari

and Gildeh

B.S.

, Construction of fuzzy multiple deferred state sampling plan, 2017 Joint 17th World Congress of International Fuzzy Systems Association and 9th International Conference on Soft Computing and Intelligent Systems (IFSA-SCIS) IEEE, 2017a, pp. 1–7.

Afshari

and Gildeh

B.S.

, Modified sequential sampling plan using fuzzy sprt, 2017 5th Iranian Joint Congress on Fuzzy and Intelligent Systems (CFIS), IEEE, 2017b, pp. 116–121.

Afshari

and Gildeh

B.S.

, Designing a multiple deferred state attribute sampling plan in a fuzzy environment, American Journal of Mathematical and Management Sciences 36(4) (2017), 328–345.

Afshari

and Gildeh

B.S.

, The effects of misclassification errors on multiple deferred state attribute sampling plan, Journal of Industrial and Systems Engineering 11(2) (2018a), 31–46.

Afshari

and Gildeh

B.S.

, Fuzzy multiple deferred state variable sampling plan, Journal of Intelligent & Fuzzy Systems 34(4) (2018b), 2737–2752.

Afshari

, Gildeh

B.S.

and Sarmad

, Fuzzy multiple deferred state attribute sampling plan in the presence of inspection errors, Journal of Intelligent & Fuzzy Systems 33(1) (2017), 503–514.

Anderson

M.T.

, Greenberg

B.S.

and Stokes

S.L.

, Acceptance sampling with rectification when inspection errors are present, Journal of Quality Technology 33(4) (2001), 493–505.

Aslam

, Balamurali

and Jun

C.H.

, Determination and economic design of a generalized multiple dependent state sampling plan, Communications in Statistics-Simulation and Computation 50(11) (2021a), 3465–3482.

Aslam

, Balamurali

and Jun

C.H.

, A new multiple dependent state sampling plan based on the process capability index, Communications in Statistics-Simulation and Computation 50(6) (2021b), 3465–3482.

10.

Aslam

, Nazir

and Jun

C.H.

, A new attribute control chart using multiple dependent state sampling, Transactions of the Institute of Measurement and Control 37(4) (2015), 3465–3482.

11.

Aslam

, Yen

C.-H.

, Chang

C.-H.

and Jun

C.H.

, Multiple dependent state variable sampling plans with process loss consideration, The International Journal of Advanced Manufacturing Technology 71(5) (2014), 1337–1343.

12.

Balamurali

and Jun

C.H.

, Multiple dependent state sampling plans for lot acceptance based on measurement data, European Journal of Operational Research 180(3) (2007), 1221–1230.

13.

Bhattacharya

and Aslam

, Generalized multiple dependent state sampling plans in presence of measurement data, IEEE Access 8 (2020), 162775–162784.

14.

Buckley

J.J.

, Fuzzy probabilities: new approach and applications, Vol. 115, Springer Science & Business Media, 2005.

15.

Buckley

J.J.

, Fuzzy Probability and Statistics, Berlin: Springer, Vol. 196, 2006.

16.

Chakraborty

, A class of single sampling plans based on fuzzy optimisation, Quality Control and Applied Statistics 37(7) (1992), 359–362.

17.

Collins

R.D.

Jr , Case

K.E.

and Bennett

G.K.

, The effects of inspection error on single sampling inspection plans, International Journal of Production Research 11(3) (1973), 289–298.

18.

Dodge

, Chain sampling inspection plans, Journal of Quality Technology 9(3) (1977), 139–142.

19.

Dodge

H.F.

and Romig

H.G.

, A method of sampling inspection, The Bell System Technical Journal 8(4) (1929), 613–631.

20.

Dodge

H.F.

and Romig

H.G.

, Single sampling and double sampling inspection tables, The Bell System Technical Journal 20(1) (1941), 1–61.

21.

Dubois

and Prade

, Operations on fuzzy numbers, International Journal of Systems Science 9(6) (1978), 613–626.

22.

Fard

N.S.

and Kim

J.J.

, Analysis of two stage sampling plan with imperfect inspection, Computers & Operations Research 25(1–4) (1993), 453–456.

23.

Ferrell

W.G.

Jr and Chhoker

, Design of economically optimal acceptance sampling plans with inspection error, Computers & Operations Research 29(10) (2002), 1283–1300.

24.

Gildeh

B.S.

, Jamkhaneh

B.E.

and Yari

, Acceptance single sampling plans with fuzzy parameters, Iranian Journal of Fuzzy Systems 8(2) (2011), 47–55.

25.

Govindaraju

, Inspection error adjustment in the design of single sampling attributes plan, Quality Engineering 19(3) (2007), 227–233.

26.

Govindaraju

and Subramani

, Selection of multiple deferred (dependent) state sampling plans for given acceptable quality level and limiting quality level, Journal of Applied Statistics 20(3) (1993), 423–428.

27.

Grzegorzewski

, Acceptance sampling plans by attributes with fuzzy risks and quality levels, in: Frontiers in Statistical Quality Control 6, Springer 20(3) (2001), 36–46.

28.

Isik

and Kaya

, Design and analysis of acceptance sampling plans based on intuitionistic fuzzy linguistic terms, Iranian Journal of Fuzzy Systems 18(6) (2021), 101–118.

29.

Jamkhaneh

B.E.

, Gildeh

B.S.

and Yari

, Inspection error and its effects on single sampling plans with fuzzy parameters, Structural and multidisciplinary Optimization 43(4) (2011), 555–560.

30.

Kahraman

, Bekar

E.T.

and Senvar

, A fuzzy design of single and double acceptance sampling plans, Intelligent decision making in quality management, Springer (2016), 179–211.

31.

Kahraman

and Kaya

, Fuzzy acceptance sampling plans, in Production Engineering and Management Under Fuzziness, Springer, 2010, pp. 457–481.

32.

Kuralmani

and Govindaraju

, Selection of multiple deferred (dependent) state sampling plans, Communications in Statistics-Theory and Methods 21(5) (1992), 1339–1366.

33.

Montgomery

D.C.

, Statistical quality control, Volume 7, Wiley New York, 2009.

34.

Schilling

E.G.

and Neubauer

D.V.

, Acceptance sampling plan in quality control, Marcel Dekker, Inc,NewYork, 1992.

35.

Soundararajan

and Vijayaraghavan

, Construction and selection of multiple dependent (deferred) state sampling plan, Journal of Applied Statistics 17(3) (1990), 397–409.

36.

Thomas

J.T.

and Kumar

, Design of fuzzy economic order quantity model in the presence of inspection errors in single sampling plans, Journal of Reliability and Statistical Studies 15(01) (2022), 211–228.

37.

Turanoglu

, Kaya

and Kahraman

, Fuzzy acceptance sampling and characteristic curves, International Journal of Computational Intelligence Systems 5(1) (2012), 13–29.

38.

Wetherill

G.B.

and Chiu

W.K.

, A review of acceptance sampling scheme with emphasis on the economic aspect, International Statistical Review 43(02) (1975), 191–210.

39.

Wortham

and Baker

, Multiple deferred state sampling inspection, The International Journal of Production Research 14(6) (1976), 719–731.

40.

Wortham

and Mogg

, Dependent stage sampling inspection, The International Journal of Production Research 8(4) (1970), 385–395.

41.

C.W.

and Wang

Z.H.

, Developing a variables multiple dependent state sampling plan with simultaneous consideration of process yield and quality loss, The International Journal of Production Research 55(8) (2017), 2351–2364.

42.

Yan

, Liu

and Dong

, Designing a multiple dependent state sampling plan based on the coefficient of variation, Springer Plus 5(1) (2016), 1–13.

43.

Zimmermann

H.J.

, Fuzzy set theory—and its applications, Springer Science & Business Media (2001).

AQL	Acceptable quality level
ASN	Average sample number
ATI	Average total inspection
DSP	Double sampling plan
FOC	Fuzzy operating characteristic curve
FSSP	Fuzzy single sampling plan
FDSP	Fuzzy double sampling plan
FMDSSP	Fuzzy multiple deferred state sampling plan
GMDSSP	Generalized multiple deferred state sampling plan
GFMDSSP	Generalized fuzzy multiple deferred state sampling plan
LQL	Limiting quality level
MDSSP	Multiple deferred state sampling plan