An acceptance sampling plan (ASP) is essential to ensure that product quality conforms to standards. Generally, producers decide to accept or reject lot sentencing which depends on a single sampling plan (SSP) that is often used in various industries. An SSP is easy to use but usually results in a larger sample size than other sampling plans [1]. Sometimes, an SSP cannot decide whether a lot will be accepted or rejected. Moreover, when using an SSP, the producer is often at a psychological disadvantage because the rejected lots are not given a second chance [2]. In these cases, the producer should apply a double sampling plan (DSP) for the inspection process. If the results of the first sample are not definitive for acceptance or rejection, a second sample is taken, which then leads to a decision on the disposition of the lot. For this reason, a DSP is more economical and reliable than an SSP for lots with very low or very high proportions of defects because a decision can be made after taking the first sample [3]. Producers can also apply more effective sampling plans, such as the multiple dependent state sampling plan (MDSSP), to assist in lot acceptance decisions. Wortham et al. [4] introduced the MDSSP to deliver lots for serial inspection in a continuous production process. Sample size can be reduced by implementing the MDSSP since the decision regarding the disposition of the current lot is made using the results of samples drawn from both current and previous lots. The MDSSP has been used by numerous researchers in a variety of situations. Govindaraju et al. [5] proposed an MDSSP design to minimize the sum of producer and consumer risks within the acceptable quality level limits, while Balamurali et al. [6] investigated the MDSSP under normal distribution using a variable sampling plan. The Bayesian approach was employed by Balamurali et al. [7] to analyze the MDSSP. Some studies [8–12] applied MDSSP concepts to design control charts. Rao et al. [13] presented a generalization of the MDSSP called the GMDSSP and discovered that this was more efficient in lowering sample size. The mean lifetime of products using a GMDSSP was investigated by Aslam et al. [14] for the gamma, Burr type XII and Birnbaum-Saunders distributions, while Aslam et al. [15] developed the MMDSSP, a modified version of the MDSSP that they claimed was more adaptable and effective than the existing MDSSP in terms of sample size and inspection cost over time truncated life. Charongrattanasakul et al. [16] proposed a novel adaptive version of the MDSSP that accepted the current lot if the quality of the product was excellent, good or moderate, while existing sampling plans only operated at two levels. They claimed that their proposed sampling plan was more flexible and efficient in terms of average sample number than the MDSSP and MMDSP. Other studies [17–19] applied the Bayesian approach to design a group chain sampling plan that considered inspection based on preceding and succeeding lots.

Nowadays, many products, including electronic devices and electrical appliances, are highly reliable and testing each item does not ensure the mean lifetime of the product. Therefore, life testing is used to determine product lifetime under specified conditions. Several studies presented an ASP for the truncated life test under various lifetime distributions. Tripathi et al. [20] presented an SSP for generalized half-normal distribution, where the lifetime experiment was truncated at a specified time, while Rao et al. [21] presented an MDSSP that decreased the life of the product under exponentiated half-logistic distribution. A novel ASP for length-biased weighted Lomax distribution was created by Al-Omari et al. [22] based on a truncated life test, while an attribute-modified chain sample inspection plan was created by Tripathi et al. [23] based on a time-truncated life test under the Darna distribution. Abushal et al. [24] developed an ASP for the power-inverted Topp-Leone distribution, which is a truncated life test that takes advantage of the median life of products. Life testing analyzes failure times of test units under normal operating conditions.

Generally, products are manufactured using high-quality processes to ensure long lifetimes. Therefore, collecting failure statistics for these products under use conditions is very difficult. As a result, the accelerated life test (ALT) is becoming more popular as this gives information on a highly reliable product lifetime in a short time. The ALT analyzes a product under conditions that are more extreme than general use (temperature, strain, stress, etc.), thus forcing the product to fail faster. These tests speed up the detection of various product defects and failure types. Producers now realize that the ALT plays an essential role in rapidly inspecting finished products. Several authors have proposed sampling plans under the ALT. Kim et al. [25] suggested an SSP for ALT under the Weibull distribution. They considered the case where the life test was hybrid censored, while Gao et al. [26] discussed the design of an ALT sampling plan for an exponential distribution using time-censoring. Some studies suggested other sampling plans. Aslam et al. [27] developed an SkSP-V sampling plan for ALT when product lifetime followed the Weibull distribution, while Aslam et al. [28] proposed a group skip-lot sampling plan using ALT resampling when the product lifetime followed the Weibull distribution. Statistical inference for ALTs under various types of stress (constant-stress, step-stress and progressive-stress) has also been proposed for censoring data under various lifetime distributions. For more details, see [29–33].

The economic design of the inspection process should consider all costs associated with implementing the plan such as inspection costs, internal failure costs and outgoing failure costs. Changes in these cost parameters will affect the total cost of the inspection. Several authors studied the economic designs of various ASPs using different approaches. Hsu et al. [2] proposed an economic model for an SSP to determine the minimum appropriated cost for both producer and consumer, while Aslam et al. [34] proposed an economic design of a group ASP to ensure the lifetime of products following the Weibull distribution using a Bayesian approach. Fallahnezhad et al. [35] proposed repeating group ASPs that included give-away cost per unit of extra sold material and inspection error, while Balamurali et al. [36] presented an economic design of a quick switching sampling system to minimize total cost while meeting the risk requirements of both producers and consumers. Finally, Hakamipour [37] compared constant-stress and step-stress predictors under a cost constraint for progressive Type I censoring.

This study focused on designing sampling plans for ALT based on continuous production processes and delivering lots for serial inspection. To the best of our knowledge, an ASP design for the ALT using the MDSSP has not been previously presented. Many ALT studies used only a single sample to decide whether to accept or reject a lot. In some situations, producers cannot decide to accept or reject a lot based on a single sampling plan because the quality level of the first sample can be both good and bad. Therefore, here, a novel adaptive version of the MDSSP (AMDSSP) for the ALT was used to decrease sample size when the mean lifetime followed the Weibull distribution. This novel sampling plan was designed based on the concept of the DSP together with the existing MDSSP. An economic model of the AMDSSP for ALT was also developed. The proposed sampling plan was compared with existing sampling plans in terms of average sample number, probability of current lot acceptance and total cost of inspection.

The remainder of this paper is arranged as follows. A brief explanation of the ALT and the Weibull distribution is provided in Section 2, with the proposed operating method and design of the AMDSSP for the ALT given in Section 3. An economic design of the AMDSSP for the ALT is presented in Section 4, with a numerical illustration and application of two real datasets in Section 5. A discussion and conclusions drawn are provided in Section 6.

Wortham et al. [4] presented the multiple dependent state sampling plan (MDSSP) as an attribute of the inspection process. This sampling plan requires continuous sampling from both the current and previous lots and decides whether to accept or reject the current lot. This approach results in reduced sample size and is often used when manufacturing is continuous with multiple lots, and each lot is submitted sequentially for inspection. However, the MDSSP considers data from previous lots using a minimum sample size to eliminate a current lot of moderate quality. The existing MDSSP will accept or reject the current lot under a single sampling but in some cases, a single sampling may not be sufficient to accept or reject the current lot. Therefore, this presents an opportunity to increase the producer’s risk and reduce the consumer’s risk.

This study put the MDSSP and the DSP concepts into practice using a novel adaptation of the MDSSP called the AMDSSP. If the quality of the first sample is undecided, the second sample must be inspected before deciding whether to accept the model. This proposed sampling plan can reduce the sample size of the MDSSP by recognizing that it will accept the current lot if it is of good or moderate quality. Therefore, the proposed sampling plan is more flexible than the existing MDSSP, with reduced sample size. Suppose the AMDSSP operates under the same conditions and procedures as the existing MDSSP by considering the following conditions. The product inspected consists of serial lots produced by continuous processes. Every lot inspected should be of the same quality. A certain number of samples are taken from each lot. The current lot will be of the same quality as the previous lot, and customers trust that the manufacturer is being honest.

This research used the AMDSSP to design an optimal sampling plan for the ALT when the lifetime of the product followed the Weibull distribution. This proposed plan consisted of five parameters including n₁, n₂, c₁, c₂ and m. The AMDSSP for ALT has the following operational steps:

Step 1. Choose the first random sample size n₁ for the known AF under accelerated conditions at time 0 from the current lot. Sample items should be put through the life test under accelerated conditions. Count the nonconforming items before the censoring time τ_A, which is denoted by d₁.

Step 2. The current lot is accepted as being of good quality if d₁ ≤ c₁, and it is rejected if d₁ > c₂ or when the censoring time τ_A is reached, whichever comes first. Otherwise, go to Step 3.

Step 3. Choose the second sample size n₂ if c₁ < d₁ ≤ c₂. Sample items should be put through the life test under accelerated conditions. Count the nonconforming items which expire before the censoring time τ_A and denote as d₂. If d₁ + d₂ ≤ c₂ and the remaining m previous lots are of good quality, consider the current lot to be of moderate quality. Otherwise, reject the current lot.

Let c₁ be the maximum acceptable number of nonconforming items for unconditional acceptance c₁ ≥ 0 and c₂ be the maximum acceptable number of additional nonconforming items for conditional acceptance c₂ > c₁. We can summarize the above steps in a flow chart, as presented in Fig. 1.

The probability of current lot acceptance when it is of good quality without considering the quality of m previous lots is denoted by P_I (p) and given as follows:(8) PI(p)=P(d1≤c1).

The probability of current lot acceptance when it is of moderate quality provided previous lots are of good quality (d₁ ≤ c₁) is denoted by P_II (p) and obtained as follows:(9) PII(p)=P(c1<d1≤c2)P(d1+d2≤c2)[P(d1≤c1)]m.

As a result, the operating characteristic (OC) function is described by(10) Pa(p)=PI(p)+PII(p)=P(d1≤c1)[1+P(c1<d1≤c2)P(d1+d2≤c2)[P(d1≤c1)]m−1].

The binomial distribution can be used to derive the OC function from Eq. (11) as follows:(11) Pa(p)=∑d1=0c1(n1d1)pd1(1−p)n1−d1×(1+(∑d1=c1+1c2(n1d1)pd1(1−p)n1−d1)⋅(∑d2=0c2−d1(n2d2)pd2(1−p)n2−d2)⋅(∑d1=0c1(n1d1)pd1(1−p)n1−d1)m−1).

Another important point to note is that we can switch AMDSSPs to SSPs and DSPs by considering m → ∞, the AMDSSP is reduced to SSP with an acceptance number c₁, while at m → 0, the AMDSSP is reduced to DSP with acceptance numbers c₁ and c₂. The average sample number (ASN) of the AMDSSP is derived by:(12) ASN=n1+n2(1−PI); PI=P(d1≤c1)+P(d1>c2)=n1+n2∑d1=c1+1c2(n1d1)pd1(1−p)n1−d1

Sample size and costs are necessary for the inspection process to provide good quality electronic products. Effective economic sampling plans can reduce the sample size and cost of the inspection process. The AMDSSP for the ALT is designed to obtain a lower ASN and total cost of the inspection process than existing sampling plans. This proposed sampling plan ensures that the mean lifetime (µ) and the ratio between true mean lifetime and specified mean lifetime (µ₀) or mean ratio are essential. It was observed that µ/µ₀ ≥ 1. If µ/µ₀ is increased, then the true mean lifetime is longer than the specified mean lifetime.

In quality control studies, there is no explicit guarantee when using a sampling method that all products will be of good quality. Two types of risks can occur. The producer’s risk (α) denotes the probability that a good lot will be rejected as performing unsatisfactorily, while the consumer’s risk (β) denotes the probability of accepting a lot of poor quality. The producer’s risk can be controlled directly, while the consumer’s risk depends on the sample size used in the test. A larger sample size gives a more negligible consumer’s risk. The consumer’s risk is often difficult to control because of the lack of flexibility in choosing the sample size.

The mean ratio, which affects the quality level of the product, is related to failure probability. The acceptable quality level (AQL or p₁) and the limiting quality level (LQL or p₂) are taken into consideration when determining the requirements for producer’s risks (alpha) and consumer’s risks (beta). The AMDSSP for the ALT is practical for two points (AQL, 1−α) and (LQL, β) and is considered for changes in the OC curve. A producer expects that the probability of current lot acceptance should be greater than 1−α at p₁. On the other hand, a customer expects that the probability of current lot acceptance should be less than β at p₂.‏ The nonlinear optimization technique is used to determine the optimal parameters resulting in reduced size of the ASN and total cost at p₁ under the ALT.

This section presents an economic design of an AMDSSP for the ALT following the concepts of Hsu et al. [2] and Hakamipour [37]. Performance indicators of the proposed sampling plan, such as P_I(p), P_II(p) and P_a(p) are given in Eqs. (8)–(10) and average total inspection (ATI) values are used, where ATI is defined as [39]:

(13) ATI=n1PI(p)+(n1+n2)PII(p)+N(1−Pa(p))

Three cost components are considered in the AMDSSP for ALT under the current lot inspection as the cost of the accelerated life test, the expected cost of internal failure per lot and the expected cost of external failure per lot.

First component: Let C_A be the cost of the accelerated life test for each lot of products, as shown in Eq. (14).(14) CA=Cs+CuATI+CoτA where C_s represents the fixed cost for setting up an accelerated life test experiment for the time interval [0, τ_A], C_u represents the sampling cost per unit and C_o represents the operating cost of conducting an accelerated life test per unit time.

Second component: Let C_I be the cost of internal failure for each lot of products, as shown in Eq. (15).(15) CI=Cd(ATI⋅p+(1−Pa(p))(N−ATI)p) where C_d represents the cost of replacement per unit, N represents the lot size and ATI · p + (1−P_A(p))(N − ATI) p represents the expected number of nonconforming items detected per lot.

Third component: Let C_E be the cost of external failure for each lot of products, as shown in Eq. (16).(16) CE=Cnd⋅(N−ATI)p⋅Pa(p) where C_nd represents the cost of an outgoing nonconforming per unit and (N − ATI) p · P_a(p) represents the expected number of nonconforming items not detected per lot. Therefore, the total cost for inspection of the products per lot under the AMDSS plan for ALT is given by:(17) TC=CA+CI+CE=Cs+CuATI+CoτA+Cd(ATI⋅p+(1−Pa(p))(N−ATI)p)+Cnd⋅(N−ATI)p⋅Pa(p)

In this research, the nonlinear optimization technique was used to determine the optimal parameters for reduced size of the ASN and total cost at p₁ under the ALT. The genetic algorithm (GA) method with nonlinear optimization was applied using the MATLAB program. The procedure for applying the GA method to determine the optimal parameters is shown as a flow chart in Fig. 2.

This paper proposed an adaptive multiple dependent state sampling plan (AMDSSP) for ALT when the lifetime of the product followed the Weibull distribution. We developed the proposed sample plan using the DSP concept together with the existing MDSSP. Under the accelerated condition, the effect of temperature on the electronic device was considered using the acceleration factor (AF) of the Arrhenius model. The nonlinear optimization technique determined the optimal plan parameters to satisfy consumer’s risk and producer’s risk simultaneously. Tables for optimal plan parameters are presented for values of δ, β, a, AF, µ/µ₀ and τ_A. The ASN value was used to judge the performance of the AMDSSP for ALT. Studies showed that the scale parameters were associated with the AF under accelerated conditions. Higher AF as higher temperature tests, increased the ASN. The illustrative example showed operational process for the AMDSSP for ALTs. An economic design of AMDSSP for ALT was also proposed under three costs for the current lot inspection as cost of the accelerated life test, the expected cost of internal failure per lot and the expected cost of external failure per lot. Optimal proposed plan parameters to achieve the lowest total cost of inspection gave values of a, AF, µ/µ₀ and τ_A. The ATI and TC for inspection determined the effectiveness of the proposed economic model. As AF increased and τ_A decreased, the ATI and the TC for inspection also increased. Efficacy studies of AMDSSP, MDSSP and SSP for ALT were compared when the lifetime of the product followed the Weibull distribution. Results showed that ASN and TC reduced under accelerated conditions using the AMDSSP rather than MDSSP and SSP. The application considered two real datasets on failure time of electronic devices under temperature stress. These datasets were applied to demonstrate the usability and utility of the proposed sampling plan for ALT. We concluded that the AMDSSP was more flexible, efficient and economical than MDSSP and SSP for ALT following the Weibull distribution. Future studies will consider the AMDSSP under various types of stress for accelerated testing techniques.