Design of SkSP-R Variables Sampling Plans

In this paper, we present the designing of the skip-lot sampling plan including the re-inspection called SkSP-R .The plan parameters of the proposed plan are determined through a nonlinear optimization problem by minimizing the average sample number satisfying both the producer’s risk and the consumer’s risks. The proposed plan is shown to perform better than the existing sampling plans in terms of the average sample number. The application of the proposed plan is explained with the help of illustrative examples.


Resumen
En este artículo, se presenta el diseño de un plan de muestreo de lotes incluyendo reinspección llamado SkSP-R.Los parámetros del plan propuesto se determinan a través de un problema de optimización no lineal que minimiza el número de muestras promedio óptimo que satisface el riesgo del productor a un nivel de calidad aceptable y el riesgo del consumidor a un nivel de calidad límite.El plan propuesto se desempeña mejor que otros planes de muestreo existentes en términos del número de muestras promedio.Se presenta una aplicación del plan propuesto con la ayuda de tabulados.

Introduction
Acceptance sampling is an important tool of statistical quality control.This tool is used to enhance the quality of the product through the inspection from the raw stage to the final stage.Without the proper inspection or testing the product may cause the bad reputation of the company in the global market.Good products sent to the market after the inspection increase the demand and alternately increase the profit of the company.Therefore, sampling plans have received the attention of the industrial engineers.Various sampling plans have been widely used in many industries including the electronic industry (Deros, Peng, Ab Rahman, Ismail & Sulong 2008), medical industry Fu, Tsai, Lin & Wei (2004) and construction industry Gharaibeh, Liu & Wani (2012).
Acceptance sampling plans are basically divided into two major categories namely; attribute sampling plans and variables sampling plans.The attribute sampling plans are used when the quality characteristic is just classified as good or bad.The variable sampling plans are used when the quality characteristic of interest can be measurable on numerical scale.The attribute sampling plans are easy to apply, however the variable sampling plans are generally more informative than the attribute sampling plans.Collani (1990) in one of his articles, criticized the variable sampling plans and, at the same time, Seidel (1997) proved that variable sampling plans are more optimal than the attribute sampling plans.
The single sampling plan (SSP) is one of the widely used sampling plans in the industries for the inspection of the finished products.This sampling plan is easy to apply and the industrial engineers can reach a decision quickly using this sampling plan.But, there are some other sampling schemes which are considered more efficient than the single sampling plan.As the cost of inspection is directly proportional to the sample size required for the acceptance or rejection decision, a large sample size incurs a large cost for the inspection which is not favorable for the producer and consumer.Therefore, some other sampling schemes such as double sampling, multiple sampling, sequential sampling and skip-lot sampling plans have been developed in order to save the cost and time of the inspection.
A re-inspection procedure can be used when the experimenters need to inspect the product again if they cannot make a decision on the basis of the original inspection.Govindaraju & Ganesalingam (1997) originally developed the sampling plan for resubmitted lots for the application of inspection of attribute quality characteristics.Recently, Aslam, Balamurali, Jun & Ahmad (2013) and Wu et al. (2012) proposed variable sampling plans using a process capability index for the inspection of resubmitted lots.By incorporating the idea of the re-inspection concept of Govindaraju & Ganesalingam (1997), Balamurali, Aslam & Jun (2014) introduced a new skip-lot sampling system designated as SkSP-R for attributing quality characteristics.
By exploring the literature of acceptance sampling, we note that there is no development on SkSP-R plan available for the inspection of measurable quality characteristics.So, in this paper, we will focus on the development of the SkSP-R sampling plan for the variables inspection by assuming that the quality characteristic of interest follows a normal distribution with standard deviation a known or unknown.We will present the designing methodology, application and the efficiency of the proposed plan.We show that the proposed plan performs better than the existing sampling plan.The rest of the paper is organized as follows: the SkSP-R plan under variables inspection is proposed in Section 2, the designing methodology of the SkSP-R plan under variables inspection for the known standard deviation (sigma) case is given in Section 3, the designing methodology of the SkSP-R plan for the unknown standard deviation case is given in Section 4, a comparison of the SkSP-R plan under variables inspection with existing sampling plans is given in Section 5 and certain concluding remarks are given in the last section.

Execution of SkSP-R Plan
As pointed out earlier, Balamurali et al. (2014) developed a new system of skip-lot sampling plan designated as SkSP-R, which is based on the principles of both continuous sampling plans and the re-inspection scheme of Govindaraju & Ganesalingam (1997) for the quality inspection of the continuous flow of bulk products.The SkSP-R plan uses the concept of reference plan similar to the SkSP-2 plan of Perry (1970).In this paper, the SkSP-R plan uses the variables single sampling plan as the reference plan.
Suppose that the quality characteristic of interest has the upper specification limit U and follows a normal distribution with unknown mean µ and known standard deviation σ.The operating procedure of the SkSP-R plan with variables sampling plan as the reference plan is explained below.
1. Start with the normal inspection by applying the variables single sampling plan as the reference plan.During the normal inspection, lots are inspected one by one in order of being submitted.
2. From each lot submitted for inspection, take a random sample of size n σ and measure the quality characteristics(X 1 , X 2 , . . ., X nσ ).Compute V = (U− X) σ , where X = 1 nσ nσ i=1 X i .Accept the lot if v ≥ k σ and reject the lot if v < k σ .(Note: In case of lower specification limit, the component v will be computed σ and other calculations are the same).
3. When i consecutive lots are accepted based on the reference plan under normal inspection, discontinue the normal inspection and switch to the skipping inspection.
4. During the skipping inspection, inspect only a fraction f of lots selected at random by applying the variables single sampling plan as the reference plan.The skipping inspection is continued until a sampled lot is rejected.
5. When a lot is rejected after s consecutively sampled lots have been accepted, then go for re-inspection procedure for the immediate next lot as in step (5) given below.
6.During re-inspection procedure, perform the inspection using the reference plan.If the lot is accepted, then continue the skipping inspection.On nonacceptance of the lot, re-inspection is done for m times and the lot is rejected if it has not been accepted on (m-1)st resubmission.
7. If a lot is rejected on the re-inspection scheme, then we immediately revert to the normal inspection in Step (1).
8. Replace or correct all the non-conforming units found with conforming units in the rejected lots.
The proposed plan involves the reference plan and four parameters, namely f (0<f <1), the fraction of lots inspected in skipping inspection mode, i, the clearance number of normal inspection, s, the clearance number for re-inspection procedure and m, the number of time the lots are submitted for re-inspection.In general, i, s and m are positive integers.So, the plan is designated as SkSP-R (i, f, s, m).The operation of the proposed plan is depicted by a flow diagram as shown in Figure 1.

Known Sigma Variables SkSP-R Plan Design
Under variables sampling inspection, an item is classified as non-conforming if it exceeds the upper specification limit U .So, the fraction non-conforming in a lot based on normal distribution will be defined as In the case of lower specification limit L, the fraction non-conforming is determined as whereΦ(y)is the normal cumulative distribution function and is given by According to Balamurali et al. (2014), the operating characteristic (OC) function of the SkSP-R system, which gives the proportion of lots that are expected to be accepted for specified fraction non-conforming (product quality) p is given by whereP is the probability of acceptance of the reference plan, i.e, the probability of accepting the lot under the variables single sampling plan with parameters (n σ , k σ ) and Q = 1 − P .Here P is given by In general, any sampling plan can be designed based on two points on the OC curve approach.A well-designed sampling plan can significantly reduce the difference between the required and the actual existing quality of the products.The producer usually would focus on a specific level of product quality, called acceptable quality level (AQL), which would yield a high probability for accepting a lot.Alternatively, the consumer would also focus on another point at the other end of the OC curve, called limiting quality level (LQL).So, the producer wants the probability of acceptance at AQL to be larger than his confidence level (1 − α) and the consumer desires that the lot acceptance probability at LQL should be less than his risk β.That is, the acceptance sampling plan must have its OC curve passing through those two designated points (AQL, 1 − α) and (LQL, β).Generally the AQL is denoted by p 1 and the LQL is denoted by p 2 .
The OC function of the SkSP-R variables plan at the AQL (= p 1 ) and LQL (= p 2 ) satisfying the corresponding producer's risk α and consumer's risk β are respectively given as and where P 1 = Φ(w 1 ), which is the probability of acceptance of the reference plan at AQL, Q = 1 − Φ(w 1 ), P 2 = Φ(w 2 ), which is the probability of acceptance of the reference plan at LQL and Here w 1 is the value of w at p = p 1 , w 2 is the value of w at p = p 2 .That is, wherev 1 is the value of v at AQL and v 2 is the value of v at LQL.
For given AQL or LQL, the values of i, f, s, m, k σ and the sample size n σ are determined by formulating a nonlinear optimization problem.Throughout this paper, we consider s = i and m = 2 as suggested by Govindaraju & Ganesalingam (1997) in order to reduce the number of parameters.The average sample number (ASN), by definition, means the expected number of sampled units required for making a decision about the lot.It is also known that the ASN of the known sigma SkSP-R plan is given as (see Balamurali et al. 2014) The ASN at AQL and LQL respectively of the SkSP-R plan when s=i are given as and The ASN given above can be used as an objective function to be minimized in a nonlinear optimization problem since there are several choices for the objective function, it is considered here to minimize ASN at LQL given in (9) because it is larger than the ASN at AQL.Therefore, the problem will be reduced to the following nonlinear optimization problem.
Subject to To solve the above problem finding of optimal parameters of (i, f, n σ , k σ ), we use a grid search procedure.The parameters (i, f, n σ , k σ ) for the known sigma plan are determined by six combinations of (α, β) namely (0.05, 0.1), (0.01,0.1),(0.01,0.05), which are reported in Tables 1-3.Suppose that a quality characteristic has the upper specification limit U and the lower specification limit L and that an item having the quality characteristic beyond these limits is declared as nonconforming.The nominal-best quality characteristics usually have double specification limits.It is to be pointed out that in the case of double specification limits, the designing methodology is slightly different.However, a one sided case serves as a reasonable approximation.The sampling plans based on double specification limits have been investigated by many authors (see for example Lee, Aslam & Hun, 2012).Example 1.For example, if p 1 = 0.005, p 2 = 0.01, α = 0.05 and β = 0.10, Table 1 gives the optimal parameters as n σ =49, k σ = 2.51998, i =3 and f =0.05.Hence the optimal parameters of the SkSP-R plan for the specified requirements are i =3, f =0.05, s=3, m=2, n σ =49 and k σ = 2.51998.For this plan, the probability of acceptance at AQL is 0.95259 and ASN at LQL is 48.382.

Designing of Unknown Sigma SkSP-R Plan
Whenever the standard deviation is unknown, we should use the sample standard deviation S instead of σ.In this case, the operation of the reference plan is as follows.
Step 1: From each submitted lot, take a random sample of size n S and measure the quality characteristics X 1 , X 2 , . . ., X n S Step 2: Computev = (U− X)

S
, where The operation of SkSP-R plan for unknown sigma case is exactly the same as in the known sigma case, but the only difference is that the reference plan will be operated as mentioned above.
Thus, the unknown sigma variable SkSP-R plan has the parameters namely, i, s, m along with the sample size n S , and the acceptable criterion k S .The OC function for the unknown sigma case is different from the known sigma case.It is known that X ± k S Sis approximately normally distributed with mean µ ± k S E(S) and variance σ 2 n S + k S V ar(S) (see Duncan 1986, Balamurali & Jun 2006).That is, Therefore, the probability of accepting a lot is given by , then the probability of accepting a lot is considered as Φ(w S ).
Hence the lot acceptance probability for the sigma unknown case of SkSP-R should satisfy the following two inequalities at AQL and LQL: and where Here w 1S is the value of w at p=p 1 , w 2S is the value of w at p=p 2 .That is, wherev 1 is the value of v at AQL and v 2 is the value of v at LQL.In this case, the nonlinear optimization problem becomes We may determine the parameters of the unknown sigma SkSP-R plan by solving the nonlinear problem given in (14).For given AQL or LQL, the values of i, f, s, m, k S and the sample size n S are determined by using a search procedure.The parameters (i, f, s, m, n S , k S ) for the unknown sigma plan are determined for six combinations of (α, β) namely (0.05, 0.1), (0.01, 0.1), (0.01, 0.05), which are reported in Tables 4-6.Example 2. For example, i fp 1 = 0.005, p 2 = 0.01, α = 0.05 and β = 0.10, Table 4 gives the optimal parameters as n S = 204, k S = 2.51998, i = 3 and f = 0.05.
Hence the optimal parameters of the SkSP-R plan for the specified requirements are i = 3, f = 0.05, s = 3, m = 2, n S = 204, k S = 2.51998.For this plan, the probability of acceptance at AQL is 0.95251 and ASN at LQL is 201.403.

Comparison
In this section we compare the variables SkSP-R plan with the variables single sampling plan.For this purpose we provide Table 7 which gives the ASN values at LQL of both sampling plans with α = 5% and β = 10% for various combinations of AQL and LQL.For the comparison, we have considered both known and unknown standard deviation sampling plans.From this table, it is clearly understood that the ASN of variables SkSP-R plan is considerably smaller as compared to the variables single sampling plan for any combinations of AQL and LQL.For example, if p 1 = 0.01 and p 2 = 0.03, Table 7 gives the ASN of the variables single sampling plan and variables SkSP-R plan as 44 and 14.807 for the known sigma case.It indicates that the variables SkSP-R plan achieves a reduction of over 66% in ASN compared to the ASN of the s, the ASN values are obtained from Table 7 as 137 and 52.352 respectively for the variables single sampling plan and the variables SkSP-R plan under the unknown sigma case.By comparing these values, we conclude that the variables SkSP-R plan achieves over a 61% reduction in ASN over the variables single sampling plan.However it is to be pointed out that the SkSP-R plan does not offer the same protection as the variables single sampling plan except under the stationary conditions of the underlying Markov chain requiring a higher number of lots of the same quality to achieve conditions.Under periods of changing quality, like the onset of a problem, the protection offered by SkSP-R plan is considerably lesser than represented by AQL and LQL.In contrast, the variables single sampling plan maintains the protection represented by the AQL and LQL under all transitive conditions of changing quality.

Conclusions
The SkSP-R sampling plan is designed for the variable data in this paper.The necessary measures of the proposed plan for known and unknown standard deviation of normal distribution have been derived.The proposed plan can be used in the industry when the quality of interest follows the normal distribution.The efficiency of the proposed plan over the existing plan is studied.The proposed plan performs better than the existing variables single sampling plan in terms of minimum ASN.The application of the proposed plan in the industry can reduce the inspection cost.The extensive tables have been developed for various combinations of AQL and LQL and various producer and consumer risks are provided for this purpose.The proposed plan for non-normal distributions will be considered as future research.The current study only considers the case of constant process fraction non-conforming.The performance of the proposed plan should be evaluated for the case of shifted fraction non-conforming in a future study.

Table 1 :
Optimal parameters of variables SkSP-R plan for known standard deviation with k = i and m = 2 with α = 0.05 and β = 0.10.

Table 2 :
Optimal parameters of variables SkSP-R plan for known standard deviation with k = i and m = 2 with α = 0.01 and β = 0.05.

Table 3 :
Optimal parameters of variables SkSP-R plan for known standard deviation with k = i and m = 2 with α = 0.01 and β = 0.05.

Table 4 :
Optimal parameters of variables SkSP-R Plan for unknown standard deviation with k = i and m = 2 with α = 0.05 and β = 0.10.

Table 5 :
Optimal parameters of variables SkSP-R plan for unknown standard deviation with k = i and m = 2 with α = 0.01 and β = 0.10.

Table 6 :
Optimal Parameters of Variables SkSP-R Plan for Unknown Standard Deviation with k = i and m = 2 with α = 0.01 and β=0.05.

Table 7 :
ASN comparison of the proposed plan with variables single sampling plan with α = 0.05 and β = 0.10.