Fuzzy multiple deferred state variable sampling plan

Abstract

Traditional multiple deferred state (MDS) sampling plan by variable is not able to decide about lots of products whose proportion parameter (p) is uncertain. In this study, fuzzy MDS (FMDS) variable sampling plan is proposed when p is not precise for the inspection of normally distributed quality characteristics in two cases known and unknown standard deviation. The plan parameters are obtained by the optimization problem, in which the objective function is considered to minimise sum of the fuzzy producer’s and consumer’s risks and the constraints are formulated by use of two-point approach under the specific fuzzy values of the producer’s and consumer’s risk, indexed by fuzzy limiting and acceptable quality levels. The proposed plan includes variable fuzzy single sampling plan (FSSP) as a special case. The obtained results show that the operating characteristic curve of FMDS variable plan seems to be closer to the ideal one with smaller required average sample number compared to variable FSSP. An industrial example is presented to illustrate the proposed plan in real applications.

Keywords

Statistical quality control multiple deferred state sampling plan fuzzy numbers arithmetic variable sampling plan producer’s risk and consumer’s risk

1 Introduction

An acceptance sampling plan is often used to test whether a submitted lot should be accepted or rejected. In general, the acceptance sampling plans are categorized into attribute and variable plans. A major advantage of the variable sampling plan is that it obtains the same operating characteristic (OC) curve with a smaller required sample size. Therefore, variable sampling plan may be appropriate in reducing inspection cost when the inspection is expensive.

Most sampling schemes utilize only the current lot information on sentencing a production lot. Unfortunately, these kinds of plans are not suitable for purposes involving costly or destructive testing, because in practice they often need large sample size in order to provide adequate producer’s and customer’s risks. In order to overcome the mentioned shortcoming, conditional sampling plans were developed in which past, future and current lot information are applied to decide about manufacturing production. Conditional sampling procedures have been considered by many scholars. “Chain sampling” is a conditional plan which was developed by Dodge [13], Dodge and Stephens [14]. Wortham and Mogg [36] studied “dependent state sampling inspection” by attributes in which the past lots information is used to make a decision criterion. The concept of “multiple deferred (dependent) state” (MDS) attribute sampling plan was first introduced by Wortham and Baker [35] in which they used the several successive future (past) lots information. Further studies can be found about MDS attribute sampling plan in Varest [33], Soundararajan and Vijayaraghavan [29], Govindaraju and Subramani [17]. Balamurali and Jun [5] extended MDS attribute sampling plan to MDS variable sampling plan based on approximate sampling distribution for one-sided specification limit under known and unknown variance cases. The variable MDS based on the process yield index was studied by Wu et al. [38] in which they used the conventional variable single sampling plan as a reference plan. Wu et al. [37] presented MDS variable sampling plan based on the normal distribution with two-sided specification limits. Also, Yan et al. [39] used the coefficient of variation of the quality characteristic to design MDS sampling plan. The theory and technique of bayesian MDS sampling plan was presented by Latha and Subbiah [23] applying Gamma prior distribution. A reduction in the sample size was their obtained result by utilize of the bayesian attribute sampling plan without a cost function for a prior distribution. Senthilkumar et al. [28] proposed bayesian repetitive deferred sampling plan by variables in which they used the succeeding lots information under repetitive group sampling plan. The optimal MDS sampling plan was studied under a Gamma-Poisson model by Balamurali et al. [4]. They showed that their proposed plan has a minimum average sample number (ASN) for decision.

One of the usual assumption to design a traditional sampling plan is that the proportion of defective items (p) is assumed crisp. But in real world, there are many cases in which p is not precise. So one can describe it based on fuzzy logic. In this case, sampling plans should be planned in a fuzzy environment by using fuzzy sets theory. Chakraborty [11] studied a class of single sampling plan based on fuzzy optimization. An acceptance sampling plan with fuzzy quality characteristics was discussed by Tong and Wang [31]. Baloui Jamkhaneh et al. [6] introduced fuzzy single sampling (FSSP) by attribute. They showed that their proposed plan is well-defined. A new sequential sampling plan based on sequential probability ratio test for fuzzy hypotheses testing was introduced by Baloui Jamkhaneh and Sadeghpour Gildeh [8]. Fuzzy MDS attribute sampling plan was proposed by Afshari et al. [1] when the inspection is not perfect. Also Afshari and Sadeghpour Gildeh [2] prepared tables to find the parameters of fuzzy MDS attribute plan by use of two-point approach. In order to make more details about fuzzy sampling plan see Kahraman et al. [20], Divya [12], Baloui Jamkhaneh et al. [7], Turanoglu et al. [32], Afshari et al. [3], Tamaki et al. [30], Sampath [27], Grzegorzewski [18, 19], Ohta and Ichihashi [25], Venkateh and Elango [34], Kanagawa and Ohta [21].

Although one of the advantages of the traditional MDS variable sampling plan is that it needs small sample size to decide about the submitted lot, it is not able to sentence the manufacturing productions when the process quality is not precise. So, in this study, fuzzy MDS (FMDS) variable sampling plan is proposed to make decisions in manufacturing productions whose quality characteristics follow normal distribution. We discuss the case as standard deviation is known or unknown. Also, it is shown that the proposed plan is an extention of the existing traditional MDS variable sampling plan and variable FSSP. Moreovere, the comparisons between the proposed plan and variable FSSP are also investigated.

The reminder of this paper is organised as follows. First, the existing traditional MDS plan is briefly reminded in the next section. We fuzzify MDS plan in Section 3. In addition, the tables are prepared to determine the proposed plan parameters in this section. Section 4 deals with the industrial application of the proposed plan. Fuzzy expected disposition time and fuzzy ASN of the proposed plan are given in Sections 5 and 6, respectively. Finally, the proposed plan is compared with variable FSSP in Section 7. Moreover, we consider operating procedure of FSSP by variable in this section. Some conclusions are remarked in last section. In this paper, to solve the nonlinear optimization problem the codes “fmincon” and “fminbnd” in the Matlab R2014a Software are used.

2 Classic MDS variable sampling plan

In this section, MDS variable sampling plan is reviewed in classic case. Then this plan is developed into a fuzzy environment in next section. The following assumptions should be valid to apply MDS variable sampling plan (Soundararajan and Vijayaraghavan [29]):

The serial lots must be generated by a continuing process.

Lots need to have a constant fraction nonconforming.

The quality characteristic of interest follows a normal distribution.

Suppose that the quality characteristic has the upper specification limit U and follows a normal distribution with unknown mean μ and known standard deviation σ. Then the operating procedure in the MDS variable sampling plan is as follows:

Step 1. Select a sample of size n_σ from each submitted lot, (X₁, X₂, ⋯ , X_{n
_σ}) and compute $V_{σ} = \frac{U - \bar{X}}{σ}$ , where $\bar{X} = \frac{1}{n_{σ}} \sum_{i = 1}^{n_{σ}} X_{i}$ .

Step 2. If v_σ ≥ k_aσ then accept the lot, and if v_σ < k_rσ then reject the lot. If k_rσ ≤ v_σ < k_aσ then accept the current lot if all the future m_σ lots in succession are accepted on the condition v_σ ≥ k_aσ (v_σ is an observed value of V_σ. k_aσ and k_rσ are the acceptance numbers under MDS sampling plan and k_rσ < k_aσ. Also m_σ is the number of future lots that a deferred lot has to wait before disposition).

Therefore, MDS variable sampling plan has four parameters, namely k_rσ, k_aσ, m_σ and n_σ. Let p be the fraction nonconforming in a lot which is expressed as

$p = P (X > U ∣ μ) = 1 - Φ (\frac{U - μ}{σ}),$ (1) where Φ (·) is the cumulative distribution function of the standard normal variable.

According to Balamurali and Jun [5], probability of lot acceptance is given by

$\begin{matrix} P_{a} (p) & = & Φ (w_{2}) + (Φ (w_{1}) - Φ (w_{2})) (Φ (w_{2}))^{m_{σ}}, \end{matrix}$ (2) where,

$w_{1} = (z_{p} - k_{r σ}) \sqrt{n_{σ}}, w_{2} = (z_{p} - k_{a σ}) \sqrt{n_{σ}},$ and z_p is the (1 - p)th quantile of normal standard distribution which is definded as

$z_{p} = Φ^{- 1} (1 - p) .$ (3)

Whenever the standard deviation is unknown, the sample standard deviation S is used instead of σ. In this case, the plan operates similar to known-standard deviation case. In the case of unknown-standard deviation, the symbols k_rs, k_as, m_s and n_s are applied instead of k_rσ, k_aσ, m_σ and n_σ, respectively. According to Duncan [16], Balamurali and Jun [5] the lot acceptance probability for the standard deviation unknown case is given by

$P_{a} (p) = Φ (y_{2}) + (Φ (y_{1}) - Φ (y_{2})) (Φ (y_{2}))^{m_{s}},$ (4) where, $\begin{matrix} y_{1} = (z_{p} - k_{rs}) \sqrt{\frac{n_{s}}{1 + k_{rs}^{2} / 2}}, \\ y_{2} = (z_{p} - k_{as}) \sqrt{\frac{n_{s}}{1 + k_{as}^{2} / 2}} . \end{matrix}$

3 Fuzzy MDS variable sampling plan

In this section, firstly we recall one way of ordering fuzzy numbers which is given in Dubois and Prade [15]. Then, MDS variable sampling plan is considered in a fuzzy environment.

Definition 1. Let $\tilde{N} [λ] = [N^{-} [λ], N^{+} [λ]]$ and $\tilde{M} [λ] = [M^{-} [λ], M^{+} [λ]]$ be λ-cut of two fuzzy numbers $\tilde{N}$ and $\tilde{M}$ , respectively. It is said, $\tilde{N} ⪯ \tilde{M}$ if and only if N⁺ [λ] ≤ M^- [λ] for λ ∈ [0, 1], where “⪯” means less than or equal to.

Now, we want to study MDS variable sampling plan when p is imprecise. For simplicity in calculation, let p be a triangular fuzzy number $\tilde{p} = (a_{1}, a_{2}, a_{3})$ . Then ∀ λ ∈ [0, 1], the λ-cut of $\tilde{p}$ is:

$\tilde{p} [λ] = [a_{1} + (a_{2} - a_{1}) λ, a_{3} - (a_{3} - a_{2}) λ] .$ (5)

Let the quality characteristic of interest have the upper specification limit U and follows a normal distribution with unknown mean μ and standard deviation σ. Standard deviation may be known or unknown.

Fig.1

Fuzzy probability of lot acceptance at $\tilde{p} = (0.009, 0.01, 0.011)$ .

3.1 Case of known standard deviation

The operation procedure in known-standard deviation fuzzy multiple deferred state (KSD-FMDS) variable sampling plan is the same steps given in Section 2. So, KSD-FMDS variable sampling plan is determined by four parameters, namely k_rσ, k_aσ, m_σ and n_σ. If k_rσ = k_aσ then the proposed plan reduces to the known standard deviation FSSP (KSD-FSSP) by variable (see Section 7). When the lower specification limit L is required, we use statistic $V_{σ} = \frac{\bar{X} - L}{σ}$ in Step 1.

According to (3) and Buckley’s approach (Buckley [9, 10]) the λ-cut of lot acceptance fuzzy probability is given by

$\begin{matrix} {\tilde{P}}_{a σ} [λ] & = & {Φ (w_{2}) + (Φ (w_{1}) \\ - Φ (w_{2})) (Φ (w_{2}))^{m_{σ}} ∣ p \in \tilde{p} [λ]}, \\ = & [P_{a σ}^{L} [λ], P_{a σ}^{U} [λ]], \end{matrix}$ (6) where, $\begin{matrix} P_{a σ}^{L} [λ] = \min {\tilde{P}}_{a σ} [λ], P_{a σ}^{U} [λ] = \max {\tilde{P}}_{a σ} [λ] . \end{matrix}$

By using the representation theorem of fuzzy numbers given in Klir and Yaun [22], one can obtain the membership function of ${\tilde{P}}_{a σ}$ (OC function) as follows: $\begin{matrix} {\tilde{P}}_{a σ} & = & ⋃_{λ} λ {\tilde{P}}_{a σ} [λ] . \end{matrix}$

3.2 Example 1

Assume the KSD-FMDS plan with parameters m_σ = 1, n_σ = 41, k_rσ = 1.98 and k_aσ = 2.19. Also, let the proportion of defective items is a fuzzy number $\tilde{p} = (0.009, 0.01, 0.011)$ . According to (ref6) we have ${\tilde{P}}_{a σ} [0] = [0.9180, 0.9758]$ . That is, it is expected that for every 10000 lots in such a process, 9180 to 9758 lots will be accepted with at least degree of membership equals zero. Figure 1 shows the plots of $\tilde{p}$ and fuzzy probability of lot acceptance at $\tilde{p}$ .

3.3 Example 2

Consider all the assumptions of Example 1. We want to plot OC curve of this plan. The OC curve plots the probability of lot acceptance against the proportion of defective items (see Montgomery [24]). In order to plot the OC curve of KSD-FMDS plan, $\tilde{p}$ is rewritten by ${\tilde{p}}_{t}$ as follows:

${\tilde{p}}_{t} = (t, b_{2} + t, b_{3} + t),$ (7) where b_i = a_i - a₁ (i = 2, 3) and t ∈ [0, 1 - b₃]. Then for λ ∈ [0, 1] we have:

${\tilde{p}}_{t} [λ] = [t + b_{2} λ, b_{3} + t - (b_{3} - b_{2}) λ] .$ (8)

Let $\tilde{p} = (0.001, 0.002, 0.003)$ , according to (7) we have ${\tilde{p}}_{t} = (t, t + 0.001, t + 0.002)$ , 0 ≤ t ≤ 0.998. Hence for λ ∈ [0, 1], the λ-cut of ${\tilde{p}}_{t}$ is equal to ${\tilde{p}}_{t} [λ] = [t + 0.001 λ, t + 0.002 - 0.001 λ]$ . By applying ${\tilde{p}}_{t}$ instead of $\tilde{p}$ in (6), Fig. 2 is plotted for different values of λ. Figure 2(a), (b), (c) and (d) are plotted when λ = 0, λ = 0.2, λ = 0.6 and λ = 1, respectively. Since the fraction of defective items is vague, the probability of lot acceptance will be imprecise. Thus, as it can be observed in Fig. 2, the OC curve of the proposed plan is a band with lower and upper bounds. Therefore, it should be called λ-cut of fuzzy operating characteristic (FOC) band. By comparing the Fig. 2(a), (b), (c) and (d), it is clear that a less uncertainty value of the proportion nonconforming results in less bandwidth, so that according to 2(d), the lower and upper bounds will become equal as λ = 1. This means that the bandwidth is zero and FOC band will be in crisp state. So, the proposed plan reduces to the MDS variable sampling plan in a nonfuzzy environment, whenever amount of the ambiguity of the proportion parameter is decreased.

Fig.2

The λ-cut of FOC band of variable KSD-FMDS for different values of λ.

Fuzzy sampling plans are usually determined by considering two points on their FOC band, namely $(\tilde{AQL}, 1 - \tilde{α})$ and $(\tilde{LQL}, \tilde{β})$ where $\tilde{AQL}$ and $\tilde{LQL}$ are fuzzy acceptance and limiting quality levels, respectively (Afshari and Sadeghpour Gildeh [2]). Also, $\tilde{α}$ and $\tilde{β}$ are the fuzzy producer’s and consumer’s risk, respectively. A well designed fuzzy sampling plan provide at least lot acceptance probability $1 - \tilde{α}$ when the proportion defective items of process is at $\tilde{AQL}$ level and it must also provide at most probability of acceptance $\tilde{β}$ if the process fraction nonconforming is at $\tilde{LQL}$ level. So, for given $\tilde{AQL}$ and $\tilde{LQL}$ , the parameters of KSD-FMDS variable sampling plan can be found by satisfying the following requirements as:

$1 - \tilde{α} ⪯ {\tilde{P}}_{a σ} (\tilde{AQL}),$ (9) and

${\tilde{P}}_{a σ} (\tilde{LQL}) ⪯ \tilde{β} .$ (10)

One may arrive at a class of KSD-FMDS variable plans satisfying above conditions. Among this class of plans, the authors select a unique plan involving minimum sum of risks for given $\tilde{AQL}$ and $\tilde{LQL}$ . So, the optimization problem to determine the parameters is formulated as follows: $minimize \tilde{α_{σ}^{'}} + \tilde{β_{σ}^{'}}$ subject to, $\begin{matrix} {\tilde{P}}_{a σ} (\tilde{AQL}) = 1 - \tilde{α}, {\tilde{P}}_{a σ} (\tilde{LQL}) ⪯ \tilde{β}, \end{matrix}$

$n_{σ} \geq 1, m_{σ} \geq 1, k_{a σ} > k_{r σ} > 0 .$ (11) where, $\tilde{α_{σ}^{'}} = 1 - {\tilde{P}}_{a σ} (\tilde{AQL})$ and $\tilde{β_{σ}^{'}} = {\tilde{P}}_{a σ} (\tilde{LQL})$ .

Table 1

0 and 1 cuts of parameters of KSD-FMDS with corresponding producer’s and consumer’s risks indexed by $\tilde{AQL}$ and $\tilde{LQL}$ , $(\tilde{α} = (0.0499, 0.05, 0.0501), \tilde{β} = (0.0999, 0.1, 0.1001))$

$\tilde{AQL}$	$\tilde{LQL}$	λ	m _σ	n _σ	k _rσ	k _aσ	$\tilde{α_{σ}^{'}}$	$\tilde{β_{σ}^{'}} [λ]$
(0.0009, 0.001, 0.0011)	(0.0099, 0.01, 0.0101)	1	2	10	2.41	2.74	0.0496	0.0977
		0					[0.0364, 0.0496]	[0.0973, 0.0997]
	(0.0149, 0.015, 0.0151)	1	1	8	2.46	2.65	0.0447	0.0977
		0					[0.0398, 0.0498]	[0.0969, 0.0985]
	(0.0199, 0.02, 0.0201)	1	1	6	2.35	2.61	0.0451	0.0993
		0					[0.0406, 0.0496]	[0.0987, 0.0998]
	(0.0299, 0.03, 0.0301)	1	2	4	2	2.54	0.0452	0.0964
		0					[0.0409, 0.0496]	[0.0961, 0.0967]
	(0.0349, 0.035, 0.0351)	1	3	4	1.93	2.49	0.0497	0.0975
		0					[0.0357, 0.0436]	[0.0974, 0.0980]
(0.0024, 0.0025, 0.0026)	(0.0099, 0.01, 0.0101)	1	1	24	2.23	2.64	0.0492	0.0995
		0					[0.0407, 0.0496]	[0.0953, 0.0997]
	(0.0149, 0.015, 0.0151)	1	2	14	2.21	2.52	0.0485	0.0981
		0					[0.0422, 0.0494]	[0.0975, 0.0995]
	(0.0199, 0.02, 0.0201)	1	2	10	2.10	2.47	0.0490	0.0969
		0					[0.0436, 0.0498]	[0.0966, 0.0979]
	(0.0249, 0.025, 0.0251)	1	1	8	2.11	2.46	0.0470	0.0988
		0					[0.0445, 0.0495]	[0.0983, 0.0994]
	(0.0299, 0.03, 0.0301)	1	3	7	1.94	2.37	0.0495	0.0981
		0					[0.0442, 0.0496]	[0.0977, 0.0985]
(0.0099, 0.01, 0.0101)	(0.0199, 0.02, 0.0201)	1	2	73	1.84	2.21	0.0472	0.0983
		0					[0.0447, 0.0498]	[0.0960, 0.0998]
	(0.0249, 0.025, 0.0251)	1	1	41	1.98	2.19	0.0473	0.0970
		0					[0.0456, 0.0491]	[0.0958, 0.0983]
	(0.0299, 0.03, 0.0301)	1	2	28	1.88	2.13	0.0489	0.0970
		0					[0.0463, 0.0495]	[0.0964, 0.0980]
	(0.0349, 0.035, 0.0351)	1	2	22	1.88	2.09	0.0470	0.0986
		0					[0.0457, 0.0484]	[0.0980, 0.0993]
	(0.0399, 0.04, 0.0401)	1	2	17	1.78	2.07	0.0490	0.0970
		0					[0.0468, 0.0492]	[0.0967, 0.0976]
(0.0149, 0.015, 0.0151)	(0.0299, 0.03, 0.0301)	1	1	65	1.85	2.07	0.0479	0.0976
		0					[0.0464, 0.0495]	[0.0962, 0.0990]
	(0.0399, 0.04, 0.0401)	1	1	31	1.73	2.02	0.0464	0.0989
		0					[0.0453, 0.0474]	[0.0981, 0.0997]
	(0.0449, 0.045, 0.0451)	1	1	25	1.77	1.98	0.0481	0.0988
		0					[0.0471, 0.0490]	[0.0982, 0.0994]
	(0.0499, 0.05, 0.0501)	1	2	21	1.72	1.93	0.0489	0.0981
		0					[0.0480, 0.0498]	[0.0977, 0.0986]

Suppose λ-cut of $\tilde{α_{σ}^{'}}$ , $\tilde{β_{σ}^{'}}$ , $\tilde{α}$ and $\tilde{β}$ for λ ∈ [0, 1] is considered as follows: $\begin{matrix} \tilde{α_{σ}^{'}} [λ] & = & [α_{σ}^{' -} [λ], α_{σ}^{' +} [λ]], \\ \tilde{β_{σ}^{'}} [λ] & = & [β_{σ}^{' -} [λ], β_{σ}^{' +} [λ]], \\ \tilde{α} [λ] & = & [α^{-} [λ], α^{+} [λ]], \\ \tilde{β} [λ] & = & [β^{-} [λ], β^{+} [λ]] . \end{matrix}$

On the other hand, since minimise the sum of errors may increase the sample size, we consider the following condition on the optimization problem: $\begin{matrix} | α^{+} [λ] - α_{σ}^{' -} [λ] | + | β^{+} [λ] - α_{σ}^{' -} [λ] | \leq ɛ, \end{matrix}$ in which, ɛ is a small positive real number near to 0 and | · | shows the absolute value of number. According to Definition 1, the optimization problem (11) is rewritten as follows: $\begin{matrix} minimize α_{σ}^{' +} [λ] + β_{σ}^{' +} [λ], \end{matrix}$ subject to,

$\begin{matrix} α_{σ}^{' +} [λ] \leq α^{-} [λ], β_{σ}^{' +} [λ] \leq β^{-} [λ], \\ | α^{+} [λ] - α_{σ}^{' -} [λ] | + | β^{+} [λ] - α_{σ}^{' -} [λ] | \leq ɛ, \\ n_{σ} \geq 1, m_{σ} \geq 1, k_{a σ} > k_{r σ} > 0 . \end{matrix}$ (12)

The values of parameters k_rσ, k_aσ, m_σ and n_σ in KSD-FMDS variable sampling plan are tabulated in Table 1 for several combinations of $(\tilde{AQL}, \tilde{LQL})$ under $\tilde{α} = (0.0499, 0.05, 0.0501)$ and $\tilde{β} = (0.0999, 0.1, 0.1001)$ . Table 1 gives the parameters with 0 and 1 cuts of corresponding producer’s and consumer’s risks $(\tilde{α_{σ}^{'}}, \tilde{β_{σ}^{'}})$ such that their sum is minimum.

For example if $\tilde{AQL} = (0.0009, 0.001, 0.0011)$ and $\tilde{LQL} = (0.0199, 0.02, 0.0201)$ then according to Table 1, one gets m_σ = 1, n_σ = 6, k_rσ = 2.35, k_aσ = 2.61 with $\tilde{α_{σ}^{'}} [1] + \tilde{β_{σ}^{'}} [1] = 0.1444$ and $\tilde{α_{σ}^{'}} [0] + \tilde{β_{σ}^{'}} [0] = [0.1393, 0.1494]$ which are smaller than $\tilde{α} [1] + \tilde{β} [1] = 0.15$ and $\tilde{α} [0] + \tilde{β} [0] = [0.1498, 0.1502]$ , respectively. For this example, the proposed plan is operated as follows:

From each submitted lot, take a random sample of size 6 and compute $V_{σ} = \frac{U - \bar{X}}{σ}$ . Accept the lot if v_σ ≥ 2.61 and reject the lot if v_σ < 2.35. If 2.35 ≤ v_σ < 2.61, then accept the current lot if forthcoming 1 lot is accepted on the condition v_σ ≥ 2.61.

3.4 Case of unknown standard deviation

The operating procedure in unknown-standard deviation fuzzy multiple deferred state (USD-FMDS) variable sampling plan is same the steps given in Section 2, except that here the sample standard deviation $S = \sqrt{\frac{1}{n_{s} - 1} \sum_{i = 1}^{n_{s}} (X_{i} - \bar{X})^{2}}$ is used instead of σ wherever needed (n_s is the sample size in unknown standard deviation case). Also in the case of unknown-standard deviation, the symbols V_s, k_rs, k_as, m_s and n_s are applied instead of V_σ, k_rσ, k_aσ, m_σ and n_σ, respectively. Thus, the proposed USD-FMDS variable sampling plan is determined by four parameters, namely k_rs, k_as, m_s and n_s. If k_rs = k_as then the proposed scheme reduces to the unknown standard deviation FSSP (USD-FSSP) by variable (see Section 7). When the lower specification limit L is required, we use statistic $V_{s} = \frac{\bar{X} - L}{S}$ in Step 1.

Although the operating procedure of the proposed plan in known and unknown standard deviation cases are similar to each other, the determination of parameters in unknown standard deviation scheme is different from the known standard deviation case. It is clear that $\bar{X} + k_{as} S$ for a large sample size approximately follows (see Duncan [16]):

$\bar{X} + k_{as} S \sim N (μ + k_{as} σ, \frac{σ^{2}}{n} + k_{as}^{2} \frac{σ^{2}}{2 n}) .$ (13)

According to (4) and using Buckley’s approach, the λ-cut of lot acceptance fuzzy probability in USD-FMDS is given by

$\begin{matrix} {\tilde{P}}_{as} [λ] & = & {Φ (y_{2}) + (Φ (y_{1}) - Φ (y_{2})) (Φ (y_{2}))^{m_{s}} \\ ∣ p \in \tilde{p} [λ]}, \\ = & [P_{as}^{L} [λ], P_{as}^{U} [λ]], \end{matrix}$ (14) where, $\begin{matrix} P_{as}^{L} [λ] = \min {\tilde{P}}_{as} [λ], P_{as}^{U} [λ] = \max {\tilde{P}}_{as} [λ] . \end{matrix}$

By applying the representation theorem of fuzzy numbers, one can obtain the membership function of ${\tilde{P}}_{as}$ (OC function) as follows:

${\tilde{P}}_{as} = ⋃_{λ} λ {\tilde{P}}_{as} [λ] .$

If $(\tilde{AQL}, 1 - \tilde{α})$ and $(\tilde{LQL}, \tilde{β})$ are prescribed concepts then for λ ∈ [0, 1], the parameters of USD-FMDS plan can be found by solving the following optimization problem. By applying the same method described for the problems (9) and (10) we have: $\begin{matrix} minimize α_{s}^{' +} [λ] + β_{s}^{' +} [λ], \end{matrix}$ subject to,

$\begin{matrix} α_{s}^{' +} \leq α_{s}^{-} [λ], β_{s}^{' +} \leq β_{s}^{-} [λ], \\ | α^{+} [λ] - α_{s}^{' +} [λ] | + | β^{+} [λ] - β_{s}^{'} - [λ] | \leq ɛ, \\ n_{s} \geq 1, m_{s} \geq 1, k_{as} > k_{rs} > 0 . \end{matrix}$ (15) where, $\tilde{α_{s}^{'}} = 1 - {\tilde{P}}_{as} (\tilde{AQL})$ , $\tilde{β_{s}^{'}} = {\tilde{P}}_{as} (\tilde{LQL})$ , $\tilde{α_{s}^{'}} [λ] = [α_{s}^{' -} [λ], α_{s}^{' +} [λ]]$ and $\tilde{β_{s}^{'}} [λ] = [β_{s}^{' +} β_{s}^{' -} [λ], β_{s}^{' +} [λ]]$ .

The values of the parameters m_s, n_s, k_rs and k_as for USD-FMDS plan are reported in Table 2 for several combinations of $(\tilde{AQL}, \tilde{LQL})$ under $\tilde{α} = (0.0499, 0.05, 0.0501)$ and $\tilde{β} = (0.0999, 0.1, 0.1001)$ . Also, 0 and 1 cuts of corresponding producer’s and consumer’s risks $(\tilde{α_{s}^{'}}, \tilde{β_{s}^{'}})$ are tabulated in Table 2.

For example, if $\tilde{AQL} = (0.0149, 0.015, 0.0151)$ and $\tilde{LQL} = (0.0399, 0.04, 0.0401)$ , then according to Table 2 we have, m_s = 1, n_s = 91, k_rs = 1.76 and k_as = 2.02. Also, it is clear that $\tilde{α_{s}^{'}} [1] + \tilde{β_{s}^{'}} [1] = 0.1470$ and $\tilde{α_{s}^{'}} [0] + \tilde{β_{s}^{'}} [0] = [0.1451, 0.1488]$ which are smaller than $\tilde{α} [1] + \tilde{β} [1] = 0.15$ and $\tilde{α} [0] + \tilde{β} [0] = [0.1498, 0.1502]$ , respectively.

Table 2

0 and 1 cuts of parameters of USD-FMDS with corresponding producer’s and consumer’s risks indexed by $\tilde{AQL}$ and $\tilde{LQL}$ , $(\tilde{α} = (0.0499, 0.05, 0.0501), \tilde{β} = (0.0999, 0.1, 0.1001))$

$\tilde{AQL}$	$\tilde{LQL}$	λ	m _s	n _s	k _rs	k _as	$\tilde{α_{s}^{'}} [λ]$	$\tilde{β_{s}^{'}} [λ]$
(0.0009, 0.001, 0.0011)	(0.0149, 0.015, 0.0151)	1	1	30	2.32	2.73	0.0484	0.0992
		0					[0.0386, 0.0491]	[0.0981, 0.0997]
	(0.0199, 0.02, 0.0201)	1	1	23	2.02	2.73	0.0494	0.0994
		0					[0.0403, 0.0496]	[0.0986, 0.0998]
	(0.0249, 0.025, 0.0251)	1	1	19	1.96	2.69	0.0450	0.0992
		0					[0.0408, 0.0494]	[0.0988, 0.0996]
	(0.0299, 0.03, 0.0301)	1	1	16	1.93	2.66	0.0481	0.0992
		0					[0.0410, 0.0491]	[0.0987, 0.0994]
	(0.0349, 0.035, 0.0351)	1	1	14	2	2.60	0.0460	0.0992
		0					[0.0413, 0.0490]	[0.0988, 0.0994]
(0.0024, 0.0025, 0.0026)	(0.0149, 0.015, 0.0151)	1	2	57	1.92	2.53	0.0437	0.0993
		0					[0.0403, 0.0473]	[0.0978, 0.0999]
	(0.0199, 0.02, 0.0201)	1	1	39	2.08	2.53	0.0468	0.0992
		0					[0.0430, 0.0488]	[0.0982, 0.0997]
	(0.0249, 0.025, 0.0251)	1	1	31	2.01	2.49	0.0478	0.0988
		0					[0.0441, 0.0492]	[0.0984, 0.0995]
	(0.0299, 0.03, 0.0301)	1	1	25	1.88	2.48	0.0458	0.0987
		0					[0.0435, 0.0481]	[0.0982, 0.0991]
	(0.0349, 0.035, 0.0351)	1	1	21	1.85	2.45	0.0458	0.0996
		0					[0.0440, 0.0483]	[0.0985, 0.0996]
(0.0099, 0.01, 0.0101)	(0.0249, 0.025, 0.0251)	1	2	135	1.60	2.17	0.0475	0.0989
		0					[0.0456, 0.0494]	[0.0976, 0.0999]
	(0.0299, 0.03, 0.0301)	1	2	90	1.93	2.13	0.0481	0.0988
		0					[0.0465, 0.0496]	[0.0980, 0.0996]
	(0.0349, 0.035, 0.0351)	1	2	66	1.93	2.10	0.0486	0.0987
		0					[0.0464, 0.0491]	[0.0983, 0.0995]
	(0.0399, 0.04, 0.0401)	1	2	51	1.58	2.08	0.0480	0.0991
		0					[0.0468, 0.0492]	[0.0986, 0.0996]
(0.0149, 0.015, 0.0151)	(0.0399, 0.04, 0.0401)	1	1	91	1.76	2.02	0.0480	0.0990
		0					[0.0469, 0.0491]	[0.0982, 0.0997]
	(0.0449, 0.045, 0.0451)	1	1	70	1.71	2	0.0479	0.0987
		0					[0.0470, 0.0489]	[0.0981, 0.0993]
	(0.0499, 0.05, 0.0501)	1	1	56	1.68	1.98	0.0489	0.0986
		0					[0.0480, 0.0498]	[0.0981, 0.0991]
	(0.0599, 0.06, 0.0601)	1	2	40	1.59	1.90	0.0498	0.0992
		0					[0.0482, 0.0498]	[0.0990, 0.0997]

Table 3

39 observations for flatness (in μm)

10.41	14.40	12.55	11.76	15.09	6.59	15.01	12.18	11.18	15.53
12.60	12.22	8.06	13.80	15.09	9.60	14.40	14.40	14.73	13.22
8.88	4.47	11.64	13.76	11.56	14.44	9.93	9.93	18.41	12.94
10.71	15.53	13.22	12.17	8.73	5.45	8.29	8.33	13.47

4 A numerical example

In this section, real industry data of thin film transistor liquid crystal displays (TFT-LCD) are considered that have also been applied by Yen et al. [40] and the data are reported in Table 3. According to Pearn and Wu [26], flatness is one of the critical quality characteristics. If the flatness of glass is not in control, the TFT-LCD products may result in a certain degree of chromatic aberration. Consider a supplier in manufacturing TFT-LCD products in Taiwan, the production specifications of flatness for a particular model of non-alkali thin-film glass is U = 25 μm. According to a written agreement between the supplier and the consumer, assume the values of $\tilde{AQL}$ , $\tilde{LQL}$ , $\tilde{α}$ and $\tilde{β}$ are set to (0.0024, 0.0025, 0.0026), (0.0199, 0.02, 0.0201), (0.0499, 0.05, 0.0501) and (0.0999, 0.1, 0.1001), respectively. From Table 2, we have m_s = 1, n_s = 39, k_rs = 2.08 and k_as = 2.53.

Note: The authors applied Shapiro-Wilk test of normality to test whether the considered data are from a normally distributed population or not. The obtained result showed that p - value = 0.3322 is greater than 0.05, that is the data came from a normally distributed population. So to implement this plan we selected a sample of size 39 from the current lot and obtained $\bar{X} = 11.91$ , S = 3.03 and $v_{s} = \frac{25 - 11.91}{3.03} = 4.31$ . Then we accept the lot since 4.31 > 2.53.

5 Fuzzy expected disposition time of the proposed plan

The major limitation of the proposed plan is the waiting line which may create until disposition of future lots is determined. Firstly, we discuss about this issue for KSD-FMDS. Then USD-FMDS will be considered. Now, consider the following symbols:

A: Event of unconditional acceptance of a lot,

R: Event of unconditional rejection of a lot,

D: Event of deferring disposition,

U: Event of unconditional acceptance or rejection of a lot,

W: Variable denoting the number of lots that a deferred lot has to wait before disposition,

P_A: Probability that a lot is unconditionally accepted,

P_R: Probability that a lot is unconditionally rejected,

P_D: Probability that a lot disposition is deferred,

P_U: Probability that a lot is unconditionally accepted or rejected i.e. waiting time is equal to zero.

P (W = i): Probability that the waiting time before disposition for a lot is exactly i lots.

Note: The symbols of ${\tilde{P}}_{A}, {\tilde{P}}_{R}, {\tilde{P}}_{D}, {\tilde{P}}_{U}$ and $\tilde{P} (W = i)$ are the previous concepts except they are used when the fraction of defective items is vague.

Wortham and Baker [35] showed the waiting time distribution for traditional MDS attribute sampling plan is given as (i ≥ m)

$\begin{matrix} P (W = i) & = & P_{D} P (W = i - 1) \\ + P_{D} P_{A} P (W = i - 2) \\ + P_{D} P_{A} P_{A} P (W = i - 3) \\ + \dots + P_{D} P_{A}^{m - 1} P (W = i - m), \end{matrix}$ (16) where m is the number of future lots in which conditional acceptance is based.

Now, we want to obtain the expected disposition time of KSD-FMDS variable plan when the proportion parameter is uncertain. Let the proportion parameter be a fuzzy number $\tilde{p} = (a_{1}, a_{2}, a_{3})$ . By taking similar method which was applied by Wortham and Baker [35], the general term for the λ-cut of waiting time distribution in KSD-FMDS variable plan for λ ∈ [0, 1] is given as (i ≥ m_σ):

$\begin{matrix} \tilde{P} (W = i) [λ] & = & {P_{D} P (W = i - 1) \\ + P_{D} P_{A} P (W = i - 2) \\ + P_{D} P_{A} P_{A} P (W = i - 3) \\ + \dots + P_{D} P_{A}^{m_{σ} - 1} P (W = i - m_{σ}) \\ : p \in \tilde{p} [λ]} . \end{matrix}$ (17) where, $P_{A} = Φ ((z_{p} - k_{a σ}) \sqrt{n_{σ}})$ , $P_{R} = 1 - Φ ((z_{p} - k_{r σ}) \sqrt{n_{σ}})$ and $P_{D} = Φ ((z_{p} - k_{r σ}) \sqrt{n_{σ}}) - Φ ((z_{p} - k_{a σ}) \sqrt{n_{σ}})$ .

Hence, according to (17) the λ-cut of fuzzy expected disposition time of KSD-FMDS variable plan designated as ${\tilde{E}}_{σ} (W) [λ]$ is obtained as follows:

$\begin{matrix} {\tilde{E}}_{σ} (W) [λ] & = & {\sum_{i = 0}^{\infty} iP (W = i) p \in \tilde{p} [λ]} \\ = & [{EW}_{σ}^{L}, {EW}_{σ}^{U}], \end{matrix}$ (18) in which ${EW}_{σ}^{L}$ and ${EW}_{σ}^{U}$ are the minimum and maximum of ${\tilde{E}}_{σ} (W) [λ]$ , respectively.

Now, let the standard deviation be unknown. In USD-FMDS variable plan, the general term for λ-cut of waiting time distribution is similar to (17), except that by using (13) and sample standard deviation S instead of σ, the formulas for P_A, P_R and P_D are given as: $\begin{matrix} P_{A} & = & Φ ((z_{p} - k_{as}) \sqrt{\frac{n_{s}}{1 + k_{as}^{2} / 2}}), \\ P_{R} & = & 1 - Φ ((z_{p} - k_{rs}) \sqrt{\frac{n_{s}}{1 + k_{rs}^{2} / 2}}), \\ P_{D} & = & Φ ((z_{p} - k_{rs}) \sqrt{\frac{n_{s}}{1 + k_{rs}^{2} / 2}}) \\ - Φ ((z_{p} - k_{as}) \sqrt{\frac{n_{s}}{1 + k_{as}^{2} / 2}}) . \end{matrix}$

Therefore, the λ-cut of fuzzy expected disposition time of USD-FMDS variable plan designated as ${\tilde{E}}_{s} (W) [λ]$ is obtained as follows:

$\begin{matrix} {\tilde{E}}_{s} (W) [λ] & = & {\sum_{i = 0}^{\infty} iP (W = i) p \in \tilde{p} [λ]} \\ = & [{EW}_{s}^{L}, {EW}_{s}^{U}], \end{matrix}$ (19) in which ${EW}_{s}^{L}$ and ${EW}_{s}^{U}$ are the minimum and maximum of ${\tilde{E}}_{s} (W) [λ]$ , respectively.

Let $\tilde{AQL}$ and $\tilde{LQL}$ be approximately 0.001 and 0.035, respectively. So according to Tables 1 and 2, we have n_σ = 4 and n_s = 14. In order to illustrate the effect of unit increase in the parameter m_σ on the λ-cut of ${\tilde{E}}_{σ} (W)$ , Fig. 3 is plotted by using (18). Figure 3 compares 0 and 1 cuts of ${\tilde{E}}_{σ} (W)$ for unit increase in the parameter m_σ. According to Fig. 3(a) and (b), one can easily see 0 and 1 cuts of the fuzzy expected disposition times increase for each successive increase in m_σ. Also, it is clear that there is not much difference between 0 and 1 cuts of mentiond curve when the value of proportion of defective items is small or large. Because when fraction of nonconforming items is small or large, the probability of lot acceptance or lot rejection increases. It means that the probability of lot deferring disposition decreases.

Fig.3

Comparison of λ-cut of fuzzy expected disposition times of KSD-FMDS for m_σ = 1, 2, 3 as n_σ = 4 under λ = 0, 1.

Also, Fig. 4 shows 0 and 1 cuts of fuzzy expected disposition time of USD-FMDS plan. According to Fig. 4(a) and (b), it can be described that the λ-cut of ${\tilde{E}}_{s} (W)$ curve increases as m_s increases. Also, it is clear that there is not much difference between 0 and 1 cuts of mentioned curve for small and large value of $\tilde{p}$ . Since in this cases, the value of lot deferring disposition probability decreases.

Fig.4

Comparison of λ-cut of fuzzy expected disposition times of USD-FMDS for m_s = 1, 2, 3 as n_s = 14 under λ = 0, 1.

6 Fuzzy average sample number of the proposed plan

Fuzzy average sample number ( $\tilde{ASN}$ ) of the proposed plan indicates the approximate average number of samples taken from several successive lots before making a decision to accept or reject the submitted lot. Moreover the sample size, taken per lot, is fixed in n_σ (in the case of known standard deviation) or n_s (in the case of unknown standard deviation). So, there is a relation between $\tilde{ASN}$ and fuzzy expected disposition time. At first, the authors discuss about fuzzy average sample number of the proposed plan in the known standard deviation case. In the following, the case of unknown standard deviation will be considered.

Assume KSD-FMDS plan with parameters m_σ, n_σ, k_aσ and k_rσ. Let N be a random variable that shows the number of inspected items. By using (17) and fixing the sample size in n_σ, fuzzy probability distribution of N is given as follows:

N	n _σ	2n_σ	3n_σ	…
$\tilde{P} (N = i)$	$\tilde{P} (W = 0)$	$\tilde{P} (W = 1)$	$\tilde{P} (W = 2)$	…

By using Buckley’s approach, λ-cut of fuzzy average sample number of KSD-FMDS plan ( ${\tilde{ASN}}_{σ}$ ) is given as follows:

$\begin{matrix} {\tilde{ASN}}_{σ} \\ = {\sum_{i = n_{σ}}^{\infty} i P (N = i) p \in \tilde{p} [λ]}, \\ = {n_{σ} \sum_{j = 1}^{\infty} j P (W = j - 1) : p \in \tilde{p} [λ]}, \\ = {n_{σ} \sum_{k = 0}^{\infty} k P (W = k) + n_{σ} : p \in \tilde{p} [λ]}, \\ = {n_{σ} E_{σ} (W) + n_{σ} p \in \tilde{p} [λ]}, \\ = [{ASN}_{σ}^{L} [λ], {ASN}_{σ}^{U} [λ]], \end{matrix}$ (20) where ${ASN}_{σ}^{L} [λ]$ and ${ASN}_{σ}^{U} [λ]$ are the minimum and maximum of ${\tilde{ASN}}_{σ} [λ]$ , respectively.

Table 4

0 and 1 cuts of fuzzy ASN in KSD-FMDS when n_σ = 4 and m_σ = 1, 2, 3

	KSD-FMDS				KSD-FMDS
t	m_σ = 1	m_σ = 2	m_σ = 3	t	m_σ = 1	m_σ = 2	m_σ = 3
	$\tilde{{ASN}_{σ}} (W) [λ] \to {\begin{matrix} λ = 1 \\ λ = 0 \end{matrix}$				$\tilde{{ASN}_{σ}} (W) [λ] \to {\begin{matrix} λ = 1 \\ λ = 0 \end{matrix}$
0	4.47	4.98	5.53	0.02	6.17	6.81	6.94
	[4.01, 5.11]	[4.02, 7.38]	[4.29, 9.09]		[6.02, 6.32]	[6.55, 7.10]	[6.74, 7.16]
0.001	4.91	5.96	7.14	0.021	6.11	6.70	6.81
	[4.40, 5.55]	[4.64, 8.79]	[5.25, 11.93]		[5.97, 6.26]	[6.46, 6.96]	[6.63, 7]
0.002	5.28	6.78	8.43	0.022	6.06	6.59	6.68
	[4.77, 5.89]	[5.27, 9.63]	[6.15, 13.44]		[5.93, 6.20]	[6.37, 6.83]	[6.52, 6.86]
0.003	5.59	7.42	9.30	0.023	6.01	6.49	6.57
	[5.10, 6.14]	[5.83, 10.07]	[6.93, 13.83]		[5.88, 6.14]	[6.28, 6.71]	[6.41, 6.73]
0.004	5.84	7.87	9.76	0.024	5.95	6.39	6.46
	[5.38, 6.34]	[6.33, 10.22]	[7.55, 13.52]		[5.83, 6.08]	[6.20, 6.60]	[6.32, 6.60]
0.005	6.04	8.15	9.93	0.025	5.90	6.30	6.35
	[5.62, 6.48]	[6.73, 10.16]	[7.98, 12.89]		[5.79, 6.02]	[6.12, 6.49]	[6.23, 6.49]
0.006	6.19	8.31	9.89	0.026	5.85	6.21	6.26
	[5.81, 6.57]	[7.04, 9.99]	[8.26, 12.17]		[5.74, 5.96]	[6.05, 6.39]	[6.14, 6.38]
0.007	6.30	8.37	9.73	0.027	5.80	6.13	6.17
	[5.97, 6.64]	[7.26, 9.74]	[8.39, 11.47]		[5.69, 5.91]	[5.97, 6.29]	[6.06, 6.28]
0.008	6.39	8.35	9.50	0.028	5.75	6.05	6.08
	[6.10, 6.67]	[7.40, 9.47]	[8.43, 10.82]		[5.65, 5.85]	[5.91, 6.20]	[5.98, 6.19]
0.009	6.44	8.28	9.24	0.029	5.70	5.97	6
	[6.19, 6.68]	[7.48, 9.18]	[8.38, 10.25]		[5.60, 5.80]	[5.84, 6.11]	[5.91, 6.10]
0.01	6.47	8.17	8.97	0.030	5.65	5.90	5.93
	[6.26, 6.67]	[7.51, 8.90]	[8.29, 9.75]		[5.56, 5.74]	[5.78, 6.03]	[5.84, 6.02]
0.011	6.48	8.05	8.71	0.031	5.60	5.83	5.86
	[6.30, 6.65]	[7.50, 8.65]	[8.17, 9.31]		[5.52, 5.69]	[5.72, 5.96]	[5.77, 5.94]
0.012	6.48	7.91	8.45	0.032	5.56	5.77	5.79
	[6.30, 6.64]	[7.42, 8.44]	[7.99, 8.97]		[5.48, 5.64]	[5.66, 5.88]	[5.71, 5.87]
0.013	6.46	7.76	8.21	0.033	5.51	5.71	5.72
	[6.29, 6.63]	[7.31, 8.26]	[7.80, 8.67]		[5.44, 5.60]	[5.60, 5.81]	[5.65, 5.80]
0.014	6.44	7.62	7.99	0.034	5.47	5.65	5.66
	[6.26, 6.61]	[7.19, 8.08]	[7.62, 8.40]		[5.40, 5.55]	[5.55, 5.75]	[5.60, 5.73]
0.015	6.40	7.47	7.78	0.035	5.43	5.59	5.60
	[6.23, 6.57]	[7.08, 7.90]	[7.45, 8.15]		[5.36, 5.50]	[5.50, 5.69]	[5.54, 5.67]
0.016	6.36	7.33	7.59	0.036	5.39	5.54	5.55
	[6.19, 6.53]	[6.97, 7.73]	[7.29, 7.91]		[5.32, 5.46]	[5.45, 5.63]	[5.49, 5.61]
0.017	6.32	7.19	7.41	0.037	5.35	5.49	5.50
	[6.15, 6.48]	[6.86, 7.56]	[7.14, 7.70]		[5.29, 5.42]	[5.41, 5.57]	[5.44, 5.55]
0.018	6.27	7.06	7.24	0.038	5.31	5.44	5.45
	[6.11, 6.43]	[6.75, 7.40]	[7, 7.50]		[5.25, 5.38]	[5.36, 5.52]	[5.40, 5.50]
0.019	6.22	6.93	7.09	0.039	5.28	5.39	5.40
	[6.07, 6.37]	[6.65, 7.24]	[6.87, 7.32]		[5.22, 5.34]	[5.32, 5.47]	[5.35, 5.45]

The values of ${\tilde{ASN}}_{σ}$ in KSD-FMDS variable plan is reported in Table 4 under λ = 0 and λ = 1 when n_σ = 4 and m_σ = 1, 2, 3 for some values of $\tilde{p}$ as $\tilde{AQL} = (0.0009, 0.001, 0.0011)$ , $\tilde{LQL} = (0.0349, 0.035, 0.0351)$ . Table 4, shows that 0 and 1 cuts of ${\tilde{ASN}}_{σ}$ increase for each successive increase in m_σ. One must consider the bracket of reported values in Table 4 because in practice the values of ${\tilde{ASN}}_{σ} [λ]$ are integers. For example according to Table 4, when m_σ changes from 1 to 3, the value of ${\tilde{ASN}}_{σ} [1]$ increases from 6 to 9 for t = 0.007. Now consider USD-FMDS variable plan with parameters n_s, m_s, k_as and k_rs. By taking a similar approach with the known standard deviation case and using (19), λ-cut of fuzzy average sample number of USD-FMDS plan ( ${\tilde{ASN}}_{s}$ ) is obtained as:

$\begin{matrix} {\tilde{ASN}}_{s} [λ] & = & {n_{s} E_{s} (W) + n_{s} p \in \tilde{p} [λ]}, \\ = & [{ASN}_{s}^{L} [λ], {ASN}_{s}^{U} [λ]], \end{matrix}$ (21) where ${ASN}_{s}^{L} [λ]$ and ${ASN}_{s}^{U} [λ]$ are the minimum and maximum of ${\tilde{ASN}}_{s} [λ]$ , respectively.

Table 5

0 and 1 cuts of fuzzy ASN in USD-FMDS when n_s = 14 and m_s = 1, 2, 3

	USD-FMDS				USD-FMDS
t	m_s = 1	m_s = 2	m_s = 3	t	m_s = 1	m_s = 2	m_s = 3
	$\tilde{{ASN}_{s}} (W) [λ] \to {\begin{matrix} λ = 1 \\ λ = 0 \end{matrix}$				$\tilde{{ASN}_{s}} (W) [λ] \to {\begin{matrix} λ = 1 \\ λ = 0 \end{matrix}$
0	17.10	19.47	22.14	0.02	22.83	25.30	25.82
	[15.07, 19.97]	[15.11, 31.32]	[16.36, 40.97]		[22.22, 23.48]	[24.21, 26.52]	[24.98, 26.74]
0.001	19	23.87	29.69	0.021	22.57	24.79	25.23
	[16.79, 21.79]	[17.89, 37.46]	[20.62, 55.38]		[21.99, 23.19]	[23.79, 25.90]	[24.47, 26.05]
0.002	20.58	27.56	35.86	0.022	22.32	24.31	24.68
	[18.39, 23.15]	[20.65, 41.10]	[24.70, 62.87]		[21.77, 22.90]	[23.40, 25.32]	[24, 25.42]
0.003	21.85	30.37	39.92	0.023	22.07	23.86	24.18
	[19.77, 24.16]	[23.15 42.79]	[28.25, 63.92]		[21.55, 22.62]	[23.03, 24.78]	[23.56, 24.85]
0.004	22.85	32.27	41.89	0.024	21.83	23.44	23.71
	[20.94, 24.88]	[25.29, 43.02]	[31.03, 61.11]		[21.34, 22.35]	[22.68, 24.29]	[23.15, 24.32]
0.005	23.61	33.36	42.25	0.025	21.59	23.05	23.28
	[21.90, 25.36]	[27, 42.30]	[32.92, 56.80]		[21.13, 22.08]	[22.35, 23.82]	[22.77, 23.83]
0.006	24.16	33.82	41.59	0.026	21.36	22.68	22.88
	[22.66, 25.65]	[28.25, 41.05]	[33.97, 52.31]		[20.93, 21.82]	[22.04, 23.39]	[22.41, 23.38]
0.007	24.54	33.80	40.37	0.027	21.14	22.34	22.50
	[23.25, 25.78]	[29.08, 39.55]	[34.35, 48.19]		[20.73, 21.58]	[21.74, 22.99]	[22.07, 22.96]
0.008	24.77	33.45	38.89	0.028	20.93	22.01	22.15
	[23.67, 25.80]	[29.54, 37.98]	[34.22, 44.60]		[20.54, 21.34]	[21.46, 22.61]	[21.76, 22.57]
0.009	24.87	32.89	37.35	0.029	20.72	21.70	21.83
	[23.96, 25.72]	[29.70, 36.44]	[33.75, 41.53]		[20.36, 21.11]	[21.20, 22.26]	[21.46, 22.21]
0.01	24.89	32.20	35.83	0.03	20.52	21.42	21.52
	[24.07, 25.66]	[29.50, 35.21]	[32.94, 39.13]		[20.18, 20.89]	[20.94, 21.93]	[21.19, 21.88]
0.011	24.82	31.45	34.40	0.031	20.33	21.14	21.24
	[23.99, 25.62]	[28.93, 34.23]	[31.86, 37.27]		[20.01, 20.67]	[20.71, 21.62]	[20.93, 21.56]
0.012	24.70	30.66	33.07	0.032	20.14	20.89	20.97
	[23.86, 25.52]	[28.35, 33.25]	[30.85, 35.57]		[19.84, 20.47]	[20.48, 21.33]	[20.68, 21.27]
0.013	24.53	29.89	31.84	0.033	19.97	20.64	20.71
	[23.71, 25.36]	[27.77, 32.27]	[29.90, 34.03]		[19.68, 20.27]	[20.27, 21.05]	[20.45, 20.99]
0.014	24.33	29.13	30.73	0.034	19.80	20.42	20.48
	[23.52, 25.15]	[27.20, 31.30]	[29.02, 32.64]		[19.52, 20.09]	[20.06, 20.79]	[20.23, 20.73]
0.015	24.11	28.40	29.71	0.035	19.63	20.20	20.25
	[23.32, 24.91]	[26.64, 30.37]	[28.21, 31.39]		[19.37, 19.90]	[19.87, 20.55]	[20.02, 20.49]
0.016	23.86	27.70	28.79	0.036	19.47	19.99	20.04
	[23.11, 24.65]	[26.11, 29.50]	[27.46, 30.26]		[19.23, 19.73]	[19.69, 20.32]	[19.83, 20.26]
0.017	23.61	27.04	27.94	0.037	19.32	19.80	19.84
	[22.89, 24.36]	[25.60, 28.67]	[26.76, 29.25]		[19.09, 19.57]	[19.51, 20.11]	[19.64, 20.05]
0.018	23.35	26.43	27.17	0.038	19.18	19.61	19.65
	[22.67, 24.07]	[25.11, 27.90]	[26.12, 28.33]		[18.96, 19.41]	[19.34, 19.90]	[19.46, 19.84]
0.019	23.09	25.84	26.47	0.039	19.04	19.44	19.47
	[22.40, 23.78]	[24.65, 27.18]	[25.53, 27.50]		[18.83, 19.26]	[19.19, 19.71]	[19.30, 19.65]

Table 5 displays 0 and 1 cuts of ${\tilde{ASN}}_{s}$ when $\tilde{AQL} = (0.0009, 0.001, 0.0011)$ , $\tilde{LQL} = (0.0349, 0.035, 0.0351)$ , n_s = 14 as m_s changes from 1 to 3 for some values of $\tilde{p}$ . According to Table 5, the bounds of λ-cut of ${\tilde{ASN}}_{s}$ increase when m_s increases. From Table 5, we note that the ${\tilde{ASN}}_{s}$ s in the unknown standard deviation case are greater than ${\tilde{ASN}}_{σ}$ s reported in Table 4 in the known standard deviation case. Figure 5 includes all information given in Table 4 and Table 5. Figure 5(a) shows 0-cuts of

Fig.5

Comparison of λ-cuts of fuzzy ASN of KSD-FMDS and USD-FMDS when m_σ = 1, 2, 3 and m_s = 1, 2, 3 under λ = 0, 1.

{\tilde{ASN}}_{s}

and

{\tilde{ASN}}_{σ}

curves. And 1-cuts of

{\tilde{ASN}}_{s}

and

{\tilde{ASN}}_{σ}

curves are plotted in Fig. 5(b). From Fig. 5, we note following behaviours of fuzzy average sample number:

There is increasing trend in ${\tilde{ASN}}_{σ} [λ]$ under λ = 0, 1 as m_σ increases.

There is increasing trend in ${\tilde{ASN}}_{s} [λ]$ under λ = 0, 1 as m_s increases.

The values of ${\tilde{ASN}}_{s} [λ]$ in USD-FMDS are greater than the values of ${\tilde{ASN}}_{σ} [λ]$ in KSD-FMDS under λ = 0, 1 for each value of m_σ.

There is decreasing trend in ${\tilde{ASN}}_{s} [λ]$ under λ = 0, 1 as the proportion parameter is small or large.

There is decreasing trend in ${\tilde{ASN}}_{σ} [λ]$ under λ = 0, 1 when the fraction of defective items is small or large.

7 Comparison between the proposed plan and FSSP

In this section, we want to make some comparisons between the proposed plan and FSSP by variable from two aspects: comparison of their average sample number and comparison of their FOC band. In order to make these comparisons, we firstly consider the operating procedure of FSSP by variable when the fraction of nonconforming items is a fuzzy number $\tilde{p} = (a_{1}, a_{2}, a_{3})$ in next subsection.

7.1 Algorithm of operating procedure in FSSP by variable

Assume that the quality characteristic has an upper specification limit U and it follows a normal distribution with unknown mean μ and standard deviation σ. When the standard deviation σ is known, the operating procedure is as follows:

Step 1. Select a sample of size n_σ from current lot, (X₁, X₂, ⋯ , X_{n
_σ}) and obtain $V_{σ} = \frac{U - \bar{X}}{σ}$ , where $\bar{X} = \frac{1}{n_{σ}} \sum_{i = 1}^{n_{σ}} X_{i}$ .

Step 2. If v_σ ≥ k_σ then accept the lot, otherwise reject the lot (k_σ is the acceptance number under FSSP in the known standard deviation case. Also v_σ is an observed value of V_σ).

So, variable FSSP in the known standard deviation case has two parameters, namely k_σ and n_σ.

By using Buckley’s approach, λ-cut of fuzzy lot acceptance probability in variable FSSP for the known standard deviation case is given by:

$\begin{matrix} {\tilde{P}}_{σ} [λ] & = & {Φ ((z_{p} - k_{σ}) \sqrt{n_{σ}}) ∣ p \in \tilde{p} [λ]}, \\ = & [P_{σ}^{L} [λ], P_{σ}^{U} [λ]], \end{matrix}$ (22) where $P_{σ}^{L} [λ]$ and $P_{σ}^{U} [λ]$ are minimum and maximum of ${\tilde{P}}_{σ} [λ]$ . Also, Φ (·) and z_p are the previous concepts. When the lower specification limit L is required, the operating procedure will be same as above, except we use statistic $V_{σ} = \frac{\bar{X} - L}{σ}$ in Step 1.

If the standard deviation is unknown, the sample standard deviation S is used instead of σ. In this case, the plan operates similar to known-standard deviation case. In the unknown standard deviation case, the symbols k_s and n_s are applied instead of k_σ and n_σ, respectively. According to Duncan [16], Montgomery [24] and Buckley’s approach, λ-cut of fuzzy lot acceptance probability in unknown standard deviation case is obtained as:

$\begin{matrix} {\tilde{P}}_{s} [λ] & = & Φ ((z_{p} - k_{s}) \sqrt{\frac{n_{s}}{1 + k_{s}^{2} / 2}}) ∣ p \in \tilde{p} [λ]}, \\ = & [P_{s}^{L} [λ], P_{s}^{U} [λ]], \end{matrix}$ (23) where $P_{s}^{L} [λ]$ and $P_{s}^{U} [λ]$ are minimum and maximum of ${\tilde{P}}_{s} [λ]$ .

Table 6

Comparison of fuzzy ASN proposed plan with FSSP in known and unknown variance cases

		$\tilde{{ASN}_{σ}}$ (Known variance case)		$\tilde{{ASN}_{s}}$ (Unknown variance case)
$\tilde{AQL}$	$\tilde{LQL}$	λ	FSSP	FMDS	λ	FSSP	FMDS
(0.0009, 0.001, 0.0011)	(0.0149, 0.015, 0.0151)	1	10	8.59	1	43.6	35.89
		0	[9.52, 10.7]	[4.99, 11.99]	0	[41.39, 45.76]	[17.87, 50.59]
	(0.0199, 0.02, 0.0201)	1	7.97	6.55	1	33.03	28.90
		0	[7.54, 8.39]	[3.89, 8.96]	0	[31.54, 34.49]	[14.86, 39.56]
	(0.0299, 0.03, 0.0301)	1	5.85	4.96	1	22.86	19.99
		0	[5.58, 6.12]	[3.71, 5.94]	0	[21.98, 23.72]	[10.98, 26.51]
(0.0024, 0.0025, 0.0026)	(0.0199, 0.02, 0.0201)	1	15.09	13.94	1	57.97	53.07
		0	[14.66, 15.52]	[11.71, 15.98]	0	[56.55, 59.38]	[40.49, 64.10]
	(0.0249, 0.025, 0.0251)	1	11.94	9.89	1	44.36	41.30
		0	[11.62, 12.25]	[7.60, 11.98]	0	[43.37, 45.34]	[32.62, 49]
	(0.0349, 0.035, 0.0351)	1	8.65	7.12	1	30.49	28.20
		0	[8.45, 8.85]	[5.73, 8.41]	0	[29.91, 31.07]	[23.32, 32.52]
(0.0049, 0.005, 0.0051)	(0.0249, 0.025, 0.0251)	1	22.58	18.86	1	…	…
		0	[22.2, 22.95]	[15.57, 22.04]	0	…	…
	(0.0299, 0.03, 0.0301)	1	17.73	15.42	1	60.05	50.30
		0	[17.45, 18]	[13.99, 16.89]	0	[59.27, 60.83]	[42.62, 57.65]
	(0.0349, 0.035, 0.0351)	1	14.67	12.48	1	48.48	41.35
		0	[14.46, 14.89]	[11.56, 13.42]	0	[47.89, 49.07]	[35.58, 46.84]
(0.0099, 0.01, 0.0101)	(0.0249, 0.025, 0.0251)	1	63.8	49.60	1	…	…
		0	[63.10, 64.49]	[41.68, 57.43]	0	…	…
	(0.0299, 0.03, 0.0301)	1	43.14	35.82	1	136.09	114.73
		0	[42.71, 43.57]	[33.22, 38.58]	0	[134.94, 137.24]	[105.86, 123.94]
	(0.0349, 0.035, 0.0351)	1	32.36	27.04	1	99.51	82.09
		0	[32.06, 32.66]	[25.17, 28.97]	0	[98.73, 100.28]	[75.60, 88.59]
	(0.0399, 0.04, 0.0401)	1	25.84	21.50	1	77.67	67.43
		0	[25.62, 26.07]	[20.23, 22.80]	0	[77.10, 78.24]	[64.03, 71.03]

Note: Similar to the method which was applied to find the parameters of the proposed plan FMDS, the parameters of FSSP in both known and unknown standard deviation cases, are usually determined by considering two points on its OC curve, namely $(\tilde{AQL}, 1 - \tilde{α})$ and ( $\tilde{LQL}, \tilde{β})$ , so that it provides at least lot acceptance probability $1 - \tilde{α}$ when the proportion defective items is at $\tilde{AQL}$ level and it must also provide at most acceptance probability $\tilde{β}$ if the process fraction nonconforming is at $\tilde{LQL}$ level. 0 and 1 cuts of average sample number of FSSP are tabulated in Table 6 for both known and unknown standard deviation cases. One notes that considering the bracket of reported values in Table 6, since the value of average sample number must be integers. For example when $\tilde{AQL} = (0.0009, 0.001, 0.0011)$ and $\tilde{LQL} = (0.0299, 0.03, 0.0301)$ , 1 cut of average sample number in known standard deviation case is equal to 5 while this value increases to 22 in unknown standard deviation case.

7.2 Comparison of average sample numbers

In general, a sampling plan is more desirable as it has a smaller average sample number. Table 6, presents 0 and 1 cuts of fuzzy average sample number of FSSP by variable and the proposed plan in both known and unknown standard deviation cases for some selected combinations of $\tilde{AQL}$ and $\tilde{LQL}$ at $\tilde{α} = (0.0499, 0.05, 0.0501)$ and $\tilde{β} = (0.0999, 0.1, 0.1001)$ . Since in practice the value of average sample number is integers, so one must consider the bracket of reported values in Table 6. For example if $\tilde{AQL} = (0.0099, 0.01, 0.0101)$ and $\tilde{LQL} = (0.0399, 0.04, 0.0401)$ , then according to Table 6, it results in:

In the standard deviation known case we have ${\tilde{ASN}}_{σ} [1] = 25$ and ${\tilde{ASN}}_{σ} [0] = {25, 26}$ with their corresponding membership degree in FSSP, whereas these values are equal to ${\tilde{ASN}}_{σ} [1] = 21$ and ${\tilde{ASN}}_{σ} [0] = [20.23, 22.80]$ in KSD-FMDS plan.

In the standard deviation unknown case we have ${\tilde{ASN}}_{s} [1] = 77$ and ${\tilde{ASN}}_{s} [0] = {77, 78}$ in FSSP while these values equal to ${\tilde{ASN}}_{s} [1] = 67$ and ${\tilde{ASN}}_{s} [0] = [64.03, 71.03]$ in USD-FMDS plan.

According to Table 6, it is clear that the proposed plan in both cases is more efficient than FSSP by variable in terms of average sample number needed to reach the same decision.

7.3 Comparison of FOC bands

In order to make a simple comparison between the proposed plan and the variable FSSP in known standard deviation case, assume that $\tilde{AQL}$ and $\tilde{LQL}$ have been established at (0.0099, 0.01, 0.0101) and (0.0399, 0.04, 0.0401), respectively. 0 and 1 cuts of FOC band of these plans are plotted in Fig. 6. According to Table 1 and using (22), the values of parameters of these plans are as follows:

1.
m_σ = 2, n_σ = 17, k_rσ = 1.78 and k_aσ = 2.07 for KSD-FMDS.

k_σ = 2 and n_σ = 25 for FSSP in the known standard deviation case.

Figure 6(a) and (b) exhibit 0 and 1 cut of FOC bands of the proposed plan and FSSP. In Fig. 6, the plot of dashed and solid curves are FOC bands of FSSP and FMDS plan in known standard deviation case, respectively. According to Fig. 6, one notes that there is no big differences in the probability of lot acceptance between two plans when the proportion parameter is small or large, but at the middle part (where the proportion parameter starts to drop), 0 and 1 cuts of lot acceptance probability in proposed plan is smaller than FSSP. FOC band of the proposed plan seems to be similar to the ideal curve. We should add that the same results were obtained when standard deviation is unknown.

Fig.6
Comparison of λ-cuts of FOC band of KSD-FMDS and FSSP in known standard deviation case under λ = 0, 1.
8 Conclusion

In this paper, in order to enhance the effectiveness of the traditional MDS variable sampling plan for deciding on the manufacturing productions in a fuzzy environment, a new sampling scheme, namely FMDS variable sampling plan, was developed based on fuzzy proportion parameter for the inspection of normally distributed quality characteristics. In the proposed plan both current and forthcoming lots information are applied for decision. For practical use, tables of the plan parameters with several commonly used quality levels and risks were constructed. Also, the authors used fuzzy ASN and FOC band (curve) of the plan as measures to make a comparison between the developed plan and FSSP variable plan. The obtained results showed that the proposed plan is preferable since it is not only significantly reduces ASN compared to FSSP variable plan, but also its FOC band (curve) seems to be closer to the ideal one under the same condition. Also it was shown that the proposed plan reduces to the FSSP variable sampling plan and traditional MDS variable sampling plan under some conditions. For further directions, extensions of FMDS variable sampling plan can be developed for non-normal distributed quality characteristics. Also it can be extended by using of the process capability index on sentencing the submitted lot.

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