On the rectifying multiple deferred state plan in the presence of uncertain parameter

Abstract

The quality of manufactured products plays a very important role in increasing consumer satisfaction. One approach to improving outgoing lot quality is applying screening method. In this paper, a multiple deferred state sampling plan is presented for a unilateral-univariate normal process with imprecise process quality in the presence of the rectifying inspection. To assess the performance of the proposed plan, a mathematical model is derived for calculating the fuzzy average total inspection ( $\tilde{ATI}$ ) under the operation of the proposed plan. The obtained conclusions indicate that the proposed plan is more economical than the existing plans in terms of $\tilde{ATI}$ measure. A numerical example is given to demonstrate how to apply the introduced plan in the real world.

Keywords

Statistical quality control fuzzy multiple deferred state sampling plan average total inspection fuzzy numbers arithmetic

1 Introduction

Acceptance sampling plans have been common quality control methods applied in industry to test whether a submitted lot should be accepted or not. They are divided to attribute and variable plans. A main benefit of the variable plans is that they usually provide more information about the manufacturing production than attribute schemes. Each sampling plan has some parameters. The plan parameters are usually determined so that its operating characteristic curve passes through two points (AQL, 1 - α) and (LQL, β). AQL and LQL are acceptance quality level and limiting quality level of the lot with acceptance probabilities 1 - α and β, respectively (see Montgomery [25]).

Most plans often employ the running lot information on sentencing a submitted lot. Unfortunately, such schemes mainly require large sample size to satisfy both consumer and producer. In order to prevail the pointed disadvantage, conditional plans were extended that are designed based on the current, past or future lots information. Dependent state sampling plan was studied by Mogg and Wortham [23] in which the past lots information is utilized to decide about the submitted lot. Wortham and Baker [32] introduced a plan named multiple deferred state (MDS) attribute sampling plan in which they apply various consecutive forthcoming lots information. Later, Balamurali and Jun [11] developed MDS attribute plan to the state where quality characteristics are variable. The variable MDS plan based on the process yield index was presented by Wu et al. [34] in which the conventional variable single sampling plan is considered as a reference plan. Wu et al. [35] studied MDS sampling plan by variable as the specification limits are two-sided. Sentilkumar et al. [27] proposed bayesian repetitive deferred sampling plan in which the succeeding lots information are used under repetitive group sampling plan. Also, Yan et al. [37] used the coefficient of variation of the quality characteristic to design MDS sampling plan. The optimal MDS sampling plan was studied under a Gamma-Poisson model by Balamurali et al. [10]. They showed that their proposed plan has a minimum average sample number for decision. Aslam et al. [9] represented MDS plan by considering the process loss function. Wu et al. [36] developed MDS plan based on a capability index as specification limit is unilateral. Aslam et al. [8] designed a plan based on MDS concept for linear profiles. Ahmadi Nadi et al. [6] introduced a modified variable repetitive group sampling plan including MDS idea by using overal yield of manufacturing process. Further studies about MDS sampling plan can be found in Vaerst [31], Soundararajan and Vijayaraghavan [28] and Govindaraju and Subramani [20].

The proportion of nonconforming items (p) is generally assumed to be precise in classic plans. But in practice, there are many cases in which p is not estimated accurately. In this case, the sampling plans should be planned based on fuzzy logic. Chakraborty [16] addressed a class of single sampling plan based on fuzzy optimization. Also, Divya [17] studied about quality interval acceptance single sampling plan when p is not crisp by applying poisson distribution. Baloui Jamkhaneh et al. [12] discussed the fuzzy attribute single sampling plan. They showed that their proposed plan is well-defined. Fuzzy double sampling plan (FDSP) investigated by Afshari et al. [3] for variable quality characteristic with upper specification limit. Tong and Wang [30] considered an acceptance sampling plan as the quality characteristic is vague. A new sequential sampling plan based on sequential probability ratio test was introduced by Baloui Jamkhaneh and Sadeghpour Gildeh [14]. Fuzzy MDS (FMDS) sampling plan by attribute has been studied by Afshari et al. [4]. Later, Afshari and Sadeghpour Gildeh [1] have extended FMDS attribute plan to the state where the interested quality characteristic is variable, and they also introduced variable fuzzy single sampling plan (FSSP) to compare their proposed schemes. Also, FMDS attribute plan was considered in the presence of misclassification errors by Afshari et al. [5].

Rectifying inspection is one of the approaches to refine the quality level of lot during the inspection. In one way of rectifying inspection, if the lot is rejected, 100% inspecting is applied and all the observed nonconforming items are substituted with conforming ones. On the other hand, if the lot is accepted, the defective items in the sample will be replaced with conforming units. Rectifying attribute single sampling plan was determined by Guenther [21] under minimising average total inspection (ATI). Dodge and Roming [18] presented a method to select a single sampling plan indexed through the average outgoing quality limit (AOQL) by minimising the ATI. Measure of AOQL was studied under the operation of MDS plan in a fuzzy environment by Afshari and Sadeghpour Gildeh [2]. They introduced a method to calculate the mentioned measure as defective proportion is imprecise. Razmkhah et al. [26] used the ranked set sampling method to design rectifying double acceptance sampling. A new meaning of maximum allowable average outgoing quality (AOQ) was introduced by Suresh and Ramkumar [29] in rectifying single sampling plan. Wu [33] presented an algorithm in which the likelihood function is applied to estimate the AOQ in rectifying inspection. Baloui Jamkhaneh and Sadeghpour Gildeh [13] discussed about rectifying double sampling plan in a fuzzy environment. The concept of attribute rectifying bayesian MDS sampling scheme under weighted poisson model was presented by Latha and Subbiah [22] in the presence of Gamma prior. In order to make more details about rectifying inspection see Montgomery [25] and Duncan [19].

As discussed above, the appealing feature of the concept of MDS method motivated us to develop this concept in the presence of rectifying inspection as the process quality is reported an imprecise value. So, in this paper, we present rectifying FMDS plan on making a judgment about manufacturing productions when the interesting quality characteristic follows normal distribution with upper specification limit. The proposed plan is discussed in two cases known and unknown standard deviation. Moreover, to asses the introduced plan, ATI measure is considered and derived a mathematical expression for the ATI based on the fuzzy numbers arithmetic. Also the advantages of the proposed plan are studied over the existing plans.

The outlines of this paper are as follows. In the next section, we recall traditional MDS variable sampling plan. The rectifying FMDS variable sampling plan is offered in Section 3. Section 4 deals with an important measure fuzzy ATI of the proposed plan. In Section 5, advantages of the proposed plan are discussed. To show the application of the proposed plan, an industrial example is provided in Section 6. Some conclusions are summarized in Section 7. In this paper, the R software (version x64 3.3.0) is used to compute and plot the graphs wherever is needed. A part of this paper has been presented in 6th Iranian Joint Congress on Fuzzy and Intelligent Systems (CFIS) (Afshari and Sadeghpour Gildeh [2]), and this research article is a development version of it.

2 Traditional MDS variable sampling plan

In this section, MDS sampling plan is recalled when the usual inspection is done. The following assumptions should be valid to use variable MDS sampling plan (Soundararajan and Vijayaraghavan [28]):

The serial lots must be produced by a continuing process.

Lots need to have a constant proportion of defective items.

The quality characteristic of interest follows a normal distribution.

Assume 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 method in the known standard deviation MDS (KSD-MDS) variable sampling plan is as follows:

Step 1. Take a sample of size n_σ from each submitted lot, (Y₁, Y₂, ⋯ , Y_{n
_σ}) and compute $V_{σ} = \frac{U - \bar{Y}}{σ}$ , where $\bar{Y} = \frac{1}{n_{σ}} \sum_{i = 1}^{n_{σ}} Y_{i}$ (V_σ is an index to sentence the submitted lot by using the sample information with size n_σ which is taken in Step 1 for known standard deviation case).

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. Also, k_aσ and k_rσ are the acceptance and rejection numbers, respectively, so that k_rσ < k_aσ, and m_σ is the number of forthcoming lots for known standard deviation case).

Let, p be the fraction nonconforming in a lot, is expressed as

$p = P (Y > U ∣ μ) = 1 - Φ (\frac{U - μ}{σ}),$ where Φ (·) is the cumulative distribution function of the standard normal variable. Suppose the standardized quality characteristic corresponding to the proportion of nonconforming p is defined as z_p = Φ^-1 (1 - p), in which Φ^-1 (.) denotes the inverse function of Φ (.). Then according to Balamurali and Jun [11], probability of lot acceptance for KSD-MDS plan is given by

$\begin{matrix} P_{a σ} (p) & = & Φ (w_{2}) + (Φ (w_{1}) - Φ (w_{2})) (Φ (w_{2}))^{m_{σ}}, \end{matrix}$ (1) in which, $w_{1} = (z_{p} - k_{r σ}) \sqrt{n_{σ}}$ and $w_{2} = (z_{p} - k_{a σ}) \sqrt{n_{σ}}$ , so that z_p is the (1 - p)th quantile of normal standard distribution.

Whenever the standard deviation is unknown, the sample standard deviation S is applied for σ. In this situation, the operating approach is alike to known standard deviation case, and the symbols V_s, k_rs, k_as, m_s and n_s are used for V_σ, k_rσ, k_aσ, m_σ and n_σ, respectively. According to Duncan [19] and Balamurali and Jun [11], the lot acceptance probability for the unknown standard deviation MDS (USD-MDS) plan is given by

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

3 Rectifying FMDS variable sampling plan

Here we present rectifying FMDS variable sampling plan in the presence of screening approach as p is imprecise. Assume that the size of each incoming lot to the inspection activity is equal to N with the same parameter $\tilde{p}$ . For simpleness in computation, let $\tilde{p}$ be a triangular fuzzy number $\tilde{p} = (b_{1}, b_{2}, b_{3})$ . Then ∀ α ∈ [0, 1], the α-cut of $\tilde{p}$ is

$\tilde{p} [α] = [b_{1} + (b_{2} - b_{1}) α b_{3} - (b_{3} - b_{2}) α] .$ (3) Let the desired quality characteristic has the upper specification limit U and follows a normal distribution with unknown mean μ and standard deviation σ. The rectifying operating algorithm in known standard deviation FMDS (KSD-FMDS) scheme is presented as follows:

Step 1. Choose a sample of size n_σ from each submitted lot, (Y₁, Y₂, ⋯ , Y_{n
_σ}), and calculate $V_{σ} = \frac{U - \bar{Y}}{σ}$ , where $\bar{Y} = \frac{1}{n_{σ}} \sum_{i = 1}^{n_{σ}} Y_{i}$ (V_σ is an index to sentence the submitted lot by using the sample information with size n_σ which is taken in Step 1 for known standard deviation case).

Step 2. If v_σ ≥ k_aσ then the lot is accepted and the observed damaged items in the sample, are exchanged with conforming objects. If v_σ < k_rσ then the lot is rejected and 100% inspecting is done and all the observed defective elements are replaced with perfect components. If k_rσ ≤ v_σ < k_aσ, then the lot falls in deferring disposition and one cannot decide immediately about it, and inspection has to be continued to accept or reject the current lot. If all the forthcoming m_σ lots in sequence are accepted under the circumstance v_σ ≥ k_aσ then the current lot is accepted and nonconforming products again are substituted with perfect items in the samples derived per lot. On the other side, 100% inspection is done and all the seen imperfect items per lot are changed with conforming ones (v_σ is an observed value of V_σ. Also, k_aσ and k_rσ are the acceptance and rejection numbers, respectively, so that k_rσ < k_aσ, and m_σ is the number of forthcoming lots for known standard deviation case).

So, the proposed rectifying KSD-FMDS plan is determined by four parameters, namely k_rσ, k_aσ, m_σ and n_σ.

According to (1) and applying Buckley’s approach (Buckley [15]), the α-cut of lot acceptance fuzzy probability in rectifying KSD-FMDS is given by

$\begin{matrix} {\tilde{P}}_{a σ} (\tilde{p}) [α] & = & {Φ (w_{2}) + (Φ (w_{1}) \\ - & Φ (w_{2})) (Φ (w_{2}))^{m_{σ}} ∣ p \in \tilde{p} [α]}, \\ = & [P_{a σ}^{-} [α], P_{a σ}^{+} [α]], \end{matrix}$ (4) where, $P_{a σ}^{-} [α] = \min {\tilde{P}}_{a σ} (\tilde{p}) [α]$ and $P_{a σ}^{+} [α] = \max {\tilde{P}}_{a σ} (\tilde{p}) [α]$ . Now, let the standard deviation be unknown. The rectifying operating procedure in the unknown standard deviation FMDS (USD-FMDS) plan, is accurately like to rectifying KSD-FMDS plan but the sample standard deviation $S = \sqrt{\frac{1}{n_{s} - 1} \sum_{i = 1}^{n_{s}} (y_{i} - \bar{Y})^{2}}$ is used instead of σ in Step 1. Further, the symbols V_s, v_s, k_rs, k_as, m_s and n_s are used instead of V_σ, v_σ, k_rσ, k_aσ, m_σ and n_σ, respectively.

By knowing that $\bar{Y} + k_{as} S \sim N (μ + k_{as} σ \frac{σ^{2}}{n} + k_{as}^{2} \frac{σ^{2}}{2 n})$ (see Duncan [19]). According to (2) and using Buckley’s approach, the α-cut of lot acceptance fuzzy probability in rectifying USD-FMDS is given by

$\begin{matrix} {\tilde{P}}_{as} (\tilde{p}) [α] & = & {Φ (x_{2}) + (Φ (x_{1}) \\ - & Φ (x_{2})) (Φ (x_{2}))^{m_{s}} ∣ p \in \tilde{p} [α]}, \\ = & [P_{as}^{-} [α], P_{as}^{+} [α]], \end{matrix}$ (5) where, $P_{as}^{-} [α] = \min {\tilde{P}}_{as} (\tilde{p}) [α]$ and $P_{as}^{+} [α] = \max {\tilde{P}}_{as} (\tilde{p}) [α]$ .

Note 1: When the lower specification limit L is required, the rectifying operating procedure of the proposed plan will be same as above, except that one uses statistics $V_{σ} = \frac{\bar{Y} - L}{σ}$ and $V_{s} = \frac{\bar{Y} - L}{S}$ in Step 1 for known and unknown standard deviation cases, respectively.

3.1 Example 1

Assume the rectifying KSD-FMDS plan with parameters m_σ = 1, n_σ = 41, k_rσ = 1.98 and k_aσ = 2.19. Also, let $\tilde{p} = (0.009, 0.01, 0.011)$ . Hence according to (ref7) it results ${\tilde{P}}_{a σ} (\tilde{p}) [0] = [0.9180, 0.9758]$ and ${\tilde{P}}_{a σ} [1] = 0.9527$ . That is, it is expected that for every 10000 lots in such a process, approximately 9527 lots will be accepted as the quality of production process is approximately 0.01. Figure 1 shows the membership function plots of $\tilde{p}$ and fuzzy probability of lot acceptance at $\tilde{p}$ .

Fig. 1

Plots of membership functions of $\tilde{p}$ and fuzzy probability of lot acceptance.

3.2 Example 2

Consider rectifying KSD-FMDS plan as given in Example 1. We want to plot the operating characteristic (OC) curve of this plan. The OC curve plots the probability of lot acceptance against the proportion of defective items (Montgomery [25]). In order to plot the OC curve of rectifying KSD-FMDS plan, $\tilde{p}$ is rewritten by ${\tilde{p}}_{l}$ as follows

${\tilde{p}}_{l} = (l, c_{2} + l, c_{3} + l),$ (6) where c_i = b_i - b₁ (i = 2, 3) and l ∈ [0, 1 - c₃]. Then for α ∈ [0, 1]

${\tilde{p}}_{l} [α] = [l + c_{2} α, c_{3} + l - (c_{3} - c_{2}) α] .$ (7) Let $\tilde{p} = (0.009, 0.01, 0.011)$ , by using (6) it results ${\tilde{p}}_{l} = (l, l + 0.001, l + 0.002)$ , 0 ≤ l ≤ 0.998. Hence for α ∈ [0, 1], the α-cut of ${\tilde{p}}_{l}$ is equal to ${\tilde{p}}_{l} [α] = [l + 0.001 α, l + 0.002 - 0.001 α]$ . By applying ${\tilde{p}}_{l}$ instead of $\tilde{p}$ in (4), Fig. 2 is plotted for different values of α. 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 fuzzy operating characteristic (FOC) band. By comparing the Fig. 2(a)–(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 become equal as α = 1. This means that the bandwidth is zero and FOC band is in a crisp state.

Fig. 2

FOC band of rectifying KSD-FMDS plan for some values of α.

4 Fuzzy average total inspection of rectifying FMDS plan

Average total inspection (ATI) is an important criterion to evaluate the performance of the rectifying plans. So in this section, we consider this measure to find a regular model for calculating it in two known and unknown standard deviation cases for the proposed plan. We show that since the proportion of nonconforming items is considered to be vague, the ATI is imprecise. So it should be named fuzzy ATI which is shown by symbol $\tilde{ATI}$ .

At first, one way of ordering fuzzy numbers is recalled as the following defenition (see Momeni and Sadeghpour Gildeh [24]):

Definition 1. Let $\tilde{A} [α] = [A_{α}^{-} A_{α}^{+}]$ and $\tilde{B} [α] = [B_{α}^{-} B_{α}^{+}]$ be α-cut of two fuzzy numbers $\tilde{A}$ and $\tilde{B}$ , respectively. The D_p,q-distance, indexed by parameters 1≤ p ≤ ∞ and 0 ≤ q ≤ 1, between two fuzzy numbers $\tilde{A}$ and $\tilde{B}$ is a nonnegative function given as follows:

$\begin{matrix} D_{p, q} (\tilde{A}, \tilde{B}) = \\ {\begin{matrix} {(1 - q) \int_{0}^{1} | A_{α}^{-} - B_{α}^{-} |^{p} d α \\ + q \int_{0}^{1} | A_{α}^{+} - B_{α}^{+} |^{p} d α}^{\frac{1}{p}}, & p < \infty \\ (1 - q) sup_{0 < α \leq 1} (| A_{α}^{-} - B_{α}^{-} |) \\ + q inf_{0 < α \leq 1} (| A_{α}^{+} - B_{α}^{+} |), & p = \infty \end{matrix} \end{matrix}$ (8) in which ∣ .∣ is the symbol for absolute value.

Note 2: For simplicity in calculation, we consider p = 2 and $q = \frac{1}{2}$ throughout the paper. One of the methods to order the fuzzy numbers is to compare their distance from another fuzzy number. For example, let $\tilde{A} = (a_{1}, a_{2}, a_{3})$ , $\tilde{B} = (b_{1}, b_{2}, b_{3})$ and $\tilde{C} = (c, c, c)$ are three fuzzy numbers so that c < min {a₁, b₁}. It is said $\tilde{A} \leq \tilde{B}$ if $D_{2, \frac{1}{2}} (\tilde{A}, \tilde{C}) \leq D_{2, \frac{1}{2}} (\tilde{B}, \tilde{C})$ . In this study, to order the fuzzy numbers, we consider c = 0.

It is necessary to mention that the major limitation of the proposed rectifying FMDS plan is the waiting line which may be created to determine the disposition of future lots. That is, when the submitted lot is not accepted or rejected unconditionally, one must wait until disposition of m future lots among i future products are determined. The value of i is defined as an agreement between the consumer and the manufacturer. Therefore, it is necessary to consider this issue in the following discussions.

4.1 Mathematical model for

{\tilde{ATI}}_{σ}

in known standard deviation case

Here 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,

W: Variable denoting the number of lots that a deferred lot has to wait before determining their disposition (length of waiting time)

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,

TI: The symbol to display the total inspection,

TI_(j): The symbol to display TI at the jth moment,

ATI_(j): The symbol to display ATI at the jth moment.

Note 3: Wortham and Baker [32] showed ∀ (j ≥ m), the waiting time distribution is given as

$\begin{matrix} P (W = j) & = & P_{D} P (W = j - 1) \\ + & P_{D} P_{A} P (W = j - 2) \\ + & P_{D} P_{A} P_{A} P (W = j - 3) \\ + & \dots + P_{D} P_{A}^{m - 1} P (W = j - m), \end{matrix}$ (9) where m is the number of future lots in which conditional acceptance is based.

In the following theorem, we derive a general formula to calculate $\tilde{ATI}$ of the proposed rectifying KSD-FMDS plan which is shown by ${\tilde{ATI}}_{σ}$ .

Theorem 4.1. Let the rectifying KSD-FMDS plan with parameters k_rσ, k_aσ, m_σ, n_σ be applied to inspect the submitted lot of size N. If the quality characteristic of interest follows normal distribution N (μ, σ²) (known σ) with upper specificatin limit U, then the α-cut of $\tilde{{ATI}_{σ}}$ until ith moment is

$\begin{matrix} {\tilde{ATI}}_{σ} (i) [α] & = & {{ATI}_{(i)} + \sum_{j = 0}^{i - 1} ({ATI}_{(j)} \\ - & {ATI}_{(i)}) P (W = j), p \in \tilde{p} [α]} \\ = & [{ATI}_{σ}^{-} [α], {ATI}_{σ}^{+} [α]], \end{matrix}$ (10)in which ${ATI}_{σ}^{-} [α] = min {\tilde{ATI}}_{σ} (i) [α]$ , ${ATI}_{σ}^{+} [α] = max {\tilde{ATI}}_{σ} (i) [α]$ ,

$\begin{matrix} {ATI}_{(j)} & = & {P_{D} {ATI}_{(j - 1)} + P_{D} P_{A} {ATI}_{(j - 2)} \\ + & \dots + P_{D} P_{A}^{(m_{σ} - 1)} {ATI}_{(j - m_{σ})}} \\ + & {(N + n_{σ}) P_{D} P (W = j - 1) \\ + & (2 N + 2 n_{σ}) P_{D} P_{A} P (W = j - 2) \\ + & \dots + (m_{σ} N \\ + & m_{σ} n_{σ}) P_{D} P_{A}^{m_{σ} - 1} P (W = j - m_{σ})} \\ - & {P_{D} (NP (A ∣ W = j - 1) \\ + & n_{σ} P (R ∣ W = j - 1)) \\ + & 2 P_{D} P_{A} (NP (A ∣ W = j - 2) \\ + & n_{σ} P (R ∣ W = j - 2)) \\ + & \dots + m_{σ} P_{D} P_{A}^{(m_{σ} - 1)} (NP (A ∣ W = j - m_{σ}) \\ + & n_{σ} P (R ∣ W = j - m_{σ}))} (j \geq m_{σ}) \end{matrix}$ (11)P_A = Φ (w₂), P_D = Φ (w₁) - Φ (w₂), P_R = 1 - Φ (w₁) and P (W = j) is given in (9) except that m is replaced by m_σ. Also, the symbols P (A ∣ W = j) and P (R ∣ W = j) are P_A and P_R when W = j, respectively.

Proof. See Appendix A.1.■

The procedure to find a regular formula for computing $\tilde{ATI}$ of rectifying USD-FMDS plan which is displayed by ${\tilde{ATI}}_{s}$ is exactly similar to what presented in known standard deviation case, except that one uses the sample standard deviation S instead of σ wherever it is needed. Moreover, the values of P_R, P_A and P_D are replaced with 1 - Φ (x₁), P_A = Φ (x₂) and P_D = Φ (x₁) - Φ (x₂), because $\bar{Y} + k_{as} S$ approximately follows $N (μ + k_{as} σ \frac{σ^{2}}{n} + k_{as}^{2} \frac{σ^{2}}{2 n})$ (see Duncan [19]).

4.2 Example 3

Assume the following plans with their corresponding parameters: KSD-FMDS plan: m_σ = 2, n_σ = 22, k_rσ = 1.88, k_aσ = 2.09, USD-FMDS plan: m_s = 2, n_s = 66, k_rs = 1.93, k_as = 2.10.

Let N and i be equal to 10000 and 4, respectively. Suppose that the proportion of defective items is a fuzzy number $\tilde{p} = (0.014, 0.015, 0.016)$ . In this example, we want to compare the values of $\tilde{{ATI}_{σ}}$ and $\tilde{{ATI}_{s}}$ under the operation of the proposed method. By using Theorem 4.1, it results that

$\begin{matrix} {\tilde{ATI}}_{σ} [0] = [161.45, 1438], {\tilde{ATI}}_{σ} [1] = 780.57, \\ {\tilde{ATI}}_{s} [0] = [407.93, 1893.64], {\tilde{ATI}}_{s} [1] = 1126.81 . \end{matrix}$ That is, if the incoming quality is approximately 0.015, the average total inspection is approximately 780 items to make a final decision about the submitted lot under the operation of rectifying KSD-FMDS plan. While one approximately needs 1126 items under the operation of rectifying USD-FMDS plan at the same value of incoming quality. One notes that considering the bracket of the reported values of $\tilde{ATI}$ , since the value of average total number must be integers.

Figure 3 displays membership function plots of $\tilde{{ATI}_{σ}}$ and $\tilde{{ATI}_{s}}$ at $\tilde{p} = (0.014, 0.015, 0.016)$ . According to Fig. 3, it can be seen that the value of ${\tilde{ATI}}_{σ}$ is less than the value of ${\tilde{ATI}}_{s}$ . Moreover, it is visible that both ${\tilde{ATI}}_{σ}$ and ${\tilde{ATI}}_{s}$ are pseudo-triangular fuzzy numbers.

Fig. 3

Membership function plots of ${\tilde{ATI}}_{σ}$ and ${\tilde{ATI}}_{s}$ at $\tilde{p} = (0.014, 0.015, 0.016)$ .

By using Theorem 4.1, relation (8) and writing a computer program in R software, the values of ${\tilde{ATI}}_{σ}$ and ${\tilde{ATI}}_{s}$ are reported in Table 2 in Appendix B as well as their distance from zero for some values of l (or ${\tilde{p}}_{l}$ ). According to Table 2 and by comparing the reported distances, it results $D_{2, \frac{1}{2}} ({\tilde{ATI}}_{(σ) l} (i), 0) \leq D_{2, \frac{1}{2}} ({\tilde{ATI}}_{(s) l} (i), 0)$ for each value of l (or ${\tilde{p}}_{l}$ ). That is, the value of ${\tilde{ATI}}_{s}$ is always more than the value of ${\tilde{ATI}}_{σ}$ for each value of incoming lot quality. For instance, when the incoming quality is approximately 0.01, the average total inspection is approximately 183 items in the known standard deviation case. While it increases to 311 items as the standard deviation is unknown. Also, Table 2 displays that the amount of $\tilde{ATI}$ increases as incoming quality reduces in both known and unknown standard deviation cases. Since by decreasing incoming quality, a sampling inspection is converted into a 100% inspection.

Similar to what explained to plot the FOC band, $\tilde{p}$ is again rewritten by ${\tilde{p}}_{l}$ as given in (ref8). By applying ${\tilde{p}}_{l}$ instead of $\tilde{p}$ in Theorem 4.1, a computer program was written to obtain the α-cut of $\tilde{ATI}$ in both known and unknown standard deviation cases. Figure 4(a) and (b), show the curves of ${\tilde{ATI}}_{σ}$ and ${\tilde{ATI}}_{s}$ , respectively, for some values of α. Since incoming quality is considered imprecise, the $\tilde{ATI}$ curve is a band with lower and upper bounds. Hence, it should be named the $\tilde{ATI}$ band. According to Fig. 4, it results that by decreasing the amount of ambiguity of the proportion parameter, the bandwidth of band will be narrower than before. Consequently, the bandwidth is zero when α = 1 and $\tilde{ATI}$ band will be in a crisp state in both known and unknown standard deviation cases. Also, Fig. 4 shows that the amount of the average total inspection is always between the sample size and lot size in both rectifying KSD-FMDS and USD-FMDS plans. It means that if the quality of manufacturing process is good, the average total inspection is close to the sample size taken from per lot in both known and unknown standard deviation cases. Since in such a situation, the probability of lot acceptance increases unconditionly and a sampling inspection is only done. On the other hand, the probability of lot acceptance reduces with decreasing process quality. So a sampling inspection is replaced by a 100% inspection.

Fig. 4

Plots of $\tilde{ATI}$ band in the proposed plan for α = 0, 0.3, 0.7, 1.

5 Comparison of the proposed plan with the existing plans

The ATI function, which displays the relationship between entering quality and the number of items required to be inspected, is widely used as a criterion to referee between sampling plans. In this section, a comparative discussion is conducted between the proposed plan, existing FSSP (introduced in Afshari and Sadeghpour Gildeh [1]) and existing FDSP (introduced in Afshari et al. [3]) based on $\tilde{ATI}$ . Tables 3 and 4 in Appendix B present some $\tilde{ATI}$ values of the above mentioned plans as well as their corresponding distance from zero for known and unknown standard deviation cases, respectively, based on the requirements in Afshari and Sadeghpour Gildeh [1] as N = 10000, $\tilde{α} = (0.0499, 0.05, 0.0501)$ and $\tilde{β} = (0.0999, 0.1, 0.1001)$ . According to Tables 3 and 4, the proposed plan implements better than the other schemes in terms of $\tilde{ATI}$ . For example, from Table 4, when $\tilde{AQL} = (0.0099, 0.01, 0.0101)$ and $\tilde{LQL} = (0.0299, 0.03, 0.0301)$ , then the $\tilde{ATI}$ with FSSP is equal to (425.6, 623.1, 888.9) with corresponding distance 654.1 from zero, but (378.4, 545.9, 768.7) with distance 571 from zero under the operation of FDSP, while the $\tilde{ATI}$ under the proposed plan is equal to (104.6, 235.5, 404.2) with distance 259.8 from zero. By comparing the obtained distances (259.8 < 571 < 654.1) and considering Note 2, it results there is a reduction in the value of $\tilde{ATI}$ under the operation of the introduced plan in comparison with other existing plans, because its corresponding distance from zero (259.8) is the least among the reported distances. This trend can be easily observed for the other sets of requirements in both known and unknown standard deviation. Figure 5 displays 0 and 1-cuts of FOC bands for these plans according to the demands given above. The FOC bands illustrate that each of the three schemes prepares similar support for both manufacturer and purchaser. Specially, the proposed plan provides the same support with less $\tilde{ATI}$ , which remarkably influences in the inspection cost. Hence, these conclusions indicate that the proposed plan is a powerful scheme from sights of both customer and producer since the final decision at the same level of protection about the submitted lot comes from a small sample size.

Fig. 5

FOC bands of the proposed plan, FSSP and FDSP.

6 Industrial application

In this section, we display how to apply the proposed plan in a real world. To do this, we use the introduced plan to make a decision about acceptance or rejection of the colour STN (super-twisted nematic) displays studied by Aslam et al. [7]. Since the standard deviation is unknown, we take the proposed rectifying USD-FMDS to make a decision about the lot of productions. Colour STN displays are created by mixing the traditional monochrome and colour filters where each pixel is divided into R, G and B sub-pixels. In this study, the quality characteristic is the membrane thickness of each pixel. The upper specification limit of quality characteristic is U = 12500A°. Suppose that one would like to decide about the mentioned manufacturing products under the agreed conditions as the values of $\tilde{AQL}$ , $\tilde{LQL}$ , $\tilde{α}$ and $\tilde{β}$ are approximately set to 0.01, 0.03, 0.05 and 0.1, respectively. Then according to the reported parameter tables by Afshari and Sadeghpour Gildeh [1], it results m_s = 2, n_s = 90, k_rs = 1.93, k_as = 2.13. Let the size of lot and waiting time length be set N = 10000 and i = 3, respectively. Before applying the proposed plan, the Shapiro-Wilk’s test of normality is applied to test whether the considered data are normally distributed or not. These data are reported in Table 5 in Appendix B. The mentioned test showed that p - value = 0.2321 is greater than 0.05. So the considered industrial data came from a normally distributed population.

To apply the proposed plan a sample of size 90 is taken from the submitted lot and obtained $\bar{y} = 11713.38$ , s = 43.58 and $v_{s} = \frac{12500 - 11713.38}{43.58} = 18.05$ . Based on our decision rule, a lot could be accepted since v_s = 18.05 > 2.13 and all the observed defective items in the sample of size 90 are substituted with conforming ones before delivery of the lot to the customers.

Finally, it is worth to compare the needed values of $\tilde{ATI}$ to sentence STN displays by applying FSSP and FDSP with the introduced plan under the same conditions. From the given parameter tables in Afshari and Sadeghpour Gildeh [1] and Afshari et al. [3], it concludes

FSSP plan: n_s = 136, k_as = 2.07, FDSP plan: n_s = 98, k_rs = 2.02, k_as = 2.2.

Figure 6 shows the $\tilde{ATI}$ bands of the proposed plan, FSSP and FDSP for α = 0, 1. From Fig. 6, the following notes are obtained:

If incoming quality is good, then the amount of the average total inspection is close to the related sample size under the operation of all three plans. Since as the process quality is good, the probability of lot acceptance increases, then a sampling inspection is only done. So that, one can see that the value of ATI is approximately equal to 90, 98 and 136 in the proposed plan, FDSP and FSSP, respectively, as the incoming quality is good.

By decreasing the process quality, the value of the $\tilde{ATI}$ value starts to increase in each plan. Because by reducing the process quality, a sampling inspection is converted into a 100% inspection.

The value of $\tilde{ATI}$ in the proposed plan is more less than the value of $\tilde{ATI}$ in both FSSP and FDSP for each value of the proportion of nonconforming items. Since in the proposed plan, future lot information is used to decide about the submitted lot as well as current lot information. While in both FSSP and FDSP, current lot information is only applied to accept or reject the submitted lot.

Fig. 6

$\tilde{ATI}$ bands of the proposed plan, FSSP and FDSP.

As a result, the proposed scheme is more economical than existing plans due to saving the lot inspection time and cost.

7 Conclusion

In this paper, a rectifying fuzzy multiple deferred state variable sampling plan was proposed for the univariate normal process whose quality is uncertain. To evaluate the efficiency of the proposed plan, we discussed the average total inspection (ATI) under the operation of the introduced plan and concluded mathematical formula to calculate ATI in both known and unknown standard deviation cases. To investigate and display the use of the obtained model in a real world, an industrial example was demonstrated. Also we discussed the advantages of the proposed plan in comparison with the existing plans which use only the current information of process for making a decision about the manufacturing productions in a fuzzy environment. The obtained results showed that applying the proposed scheme is economical and time saving to inspect the manufacturing products. Development of the proposed plan based on multivariate quality characteristics as well as based on process capability indices can be considered in the future research. Furthermore, developing of other types of MDS-based plans such as modified and overall-yield-based repetitive group sampling plans can be beneficial for future directions.

Footnotes

Appendix

Acknowledgment

We would like to thank the reviewers for their useful comments and suggestions, which have considerably improved the paper. This research was supported by a grant from Ferdowsi University of Mashhad, Iran; No. 2/51994.

References

Afshari

and Sadeghpour Gildeh

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

Afshari

and Sadeghpour Gildeh

, Rectifying Fuzzy Multiple Deferred State Variable Sampling Plan, 6th Iranian Joint Congress on Fuzzy and Intelligent Systems (CFIS) (17th Conference on Fuzzy Systems and 15th Conference on Intelligent Systems), Kerman, Iran, indexed in IEEE, 2018.

Afshari

, Sadeghpour Gildeh

and Ahmadi Nadi

, Fuzzy Double Variable Sampling Plan, 15th Iran International Industrial Engineering Conference, Yazd, Iran, indexed in IEEE, 2019.

Afshari

, Sadeghpour Gildeh

and Sarmad

, Multiple deferred state sampling plan with fuzzy parameter, International Journal of Fuzzy Systems 20(2) (2018), 549–557.

Afshari

, Sadeghpour Gildeh

and Sarmad

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

Ahmadi Nadi

, Sadeghpour Gildeh

and Afshari

, Optimal design of overall yield-based variable repetitive sampling plans for processes with multiple characteristics, Applied Mathematical Modelling 81 (2020), 194–210.

Aslam

, Azam

and Jun

C.-H.

, A new lot inspection procedure based on exponentially weighted moving average, International Journal of Systems Science 46(8) (2015), 1392–13400.

Aslam

, Wang

F.-K.

, Khan

and Jun

C.-H.

, A multiple dependent state repetitive sampling plan for linear profiles, Journal of the Operational Research Society 69(3) (2018), 467–473.

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(5–8) (2014), 1337–1343.

10.

Balamurali

, Jeyadurga

and Usha

, Designing of bayesian multiple deferred state sampling plan based on Gamma Poisson distribution, American Journal of Mathematical and Management Sciences 35(1) (2016), 77–90.

11.

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.

12.

Baloui Jamkhaneh

, Sadeghpour Gildeh

and Yari

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

13.

Baloui Jamkhaneh

and Sadeghpour Gildeh

, Acceptance double sampling plan using fuzzy poisson distribution, World Applied Sciences Journal 16(11) (2012), 1578–1588.

14.

Baloui Jamkhaneh

and Sadeghpour Gildeh

, Sequential sampling plan using fuzzy SPRT, Journal of Intelligent and Fuzzy Systems 25(3) (2013), 785–791.

15.

Buckley

J.-J.

, Fuzzy Probability and Statistics, Springer-Verlag, Berlin Heidelberg, (2006).

16.

Chakraborty

T.-K.

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

17.

Divya

P.-R.

, Quality interval acceptance single sampling plan with fuzzy parameter using poisson distribution, International Journal of Advancements in Research and Technology 1(3) (2012), 115–125.

18.

Dodge

H.-F.

and Roming

H.-G.

, Sampling inspection tables, 2th ed, NewYork, Wiley, 1959.

19.

Duncan

A.-J.

, Quality control and industrial statistics, 5th ed., Homewood, IL: Irwin, 1986.

20.

Govindaraju

and Subramani

, Selection of multiple deferred state mds-1 sampling plans for given acceptable quality level and limiting quality level involving minimum risks, Journal of Applied Statistics 17(3) (1990), 427–434.

21.

Guenther

W.-C.

, Dtermination of rectifying inspection plans for single sampling by attributes, Journal of Quality Technology 16(1) (1984), 56–63.

22.

Latha

and Subbiah

, Selection of bayesian multiple deferred state (BMDS-1) sampling plan based on quality regions, International Journal of Recent Scientific Research 6(5) (2016), 3864–3867.

23.

Mogg

J.-M.

and Wortham

A.-W.

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

24.

Momeni

and Sadeghpour Gildeh

, Nonparametric tests for median in fuzzy environment, International Journal of Fuzzy Systems 18(1) (2016), 130–139.

25.

Montgomery

D.-C.

, Introduction to statistical quality control, Wiley, New York, 2012.

26.

Razmkhah

, Sadeghpour

and Ahmadi

, An economic design of rectifying double acceptance sampling plans Via maxima nomination sampling, Stochastics and Quality Control 32(2) (2017), 99–104.

27.

Senthilkumar

, Ramya

S.-R.

and Raffie

B.-E.

, Construction and selection of repetitive deferred variables sampling (RDVS) plan indexed by quality levels, Journal of Academia and Industrial Research 3(10) (2015), 497–503.

28.

Soundararajan

and Vijayaraghavan

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

29.

Suresh

K.-K.

and Ramkumar

T.-B.

, Selection of a sampling plan indexed with maximum allowable average outgoing quality, Journal of Applied Statistics 23(6) (1996), 645–654.

30.

Tong

and Wang

, Fuzzy acceptance sampling plans for inspection of geospatial data with ambiguity in quality characteristics, Computers and Geosciences 48(11) (2012), 256–266.

31.

Vaerst

, A procedure to construct multiple deferred state sampling plan, Methods of Operations Research 37 (1982), 477–485.

32.

Wortham

A.-W.

and Baker

R.-C.

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

33.

, Using the likelihood function to estimate the average outgoing quality, Quality Engineering 10(1) (1997), 59–64.

34.

C.-W.

, Liu

S.-W.

and Lee

, Design and construction of a variable multiple dependent state sampling plan based on process yield, European Journal Industrial Engineering 9(6) (2015), 819–838.

35.

C.-W.

, Lee

and Chen

, A novel lot sentencing method by variables inspection considering multiple dependent state, Quality and Reliability Engineering International 32(3) (2016), 985–994.

36.

C.-W.

, Lee

A.-H.

and Chien

C.-C.C.

, A variables multiple dependent state sampling plan based on a one-sided capability index, Quality Engineering 29(4) (2017), 719–729.

37.

Yan

, Liu

and Dong

, Designing a multiple dependent state sampling plan based on the coefficient of variation, SpringerPlus 5(1) (2016), 1447.