New Models of Acceptance Sampling Plans

1. Introduction

Acceptance sampling is a procedure used for sentencing incoming batches. Sampling plan consist of a sample size and a decision making rule. The sample size is the number of items to sample or the number of measurements to take. The decision making rule involves the acceptance threshold and a description of how to use the sample result to accept or reject the lot. Acceptance sampling plans are also practical tools for quality control applications, which involve quality contracting on product orders between the vendor and the buyer. Those sampling plans provide the vendor and the buyer rules for lot sentencing while meeting their preset requirements on product quality. Scientific sampling plans are the primary tools for quality and performance management in industry today. In an industrial plant, sampling plans are used to decide either to accept or reject a received batch of items. With attribute sampling plans, these accept/reject decisions are based on a count of the number of defective items. The sample size is assumed constant in traditional sampling plans.

In this section, several new decision making policies for the acceptance sampling problem are introduced. The objective of these models is to find constant control thresholds for lot sentencing problem.

The single stage acceptance sampling plan based on the control threshold policy is presented in section 2, the acceptance sampling policy based on number of successive conforming items is presented in section 3, and acceptance sampling policy using the minimum angle method is presented in sections 4. Acceptance sampling policy based on cumulative sum of conforming Items run lengths comes in section 5 and acceptance sampling policy based on Bayesian inference comes in section 6. Finally the chapter is concluded in section 7.

2. Single Stage Acceptance Sampling Plan based on the Control Threshold Policy [1]

We suppose a batch of size nis received which its proportion of the defectives items is equal top. For a batch of sizen, random variable Yis defined as the number of inspected items and zis defined as the number of items classified as 'defective' after inspection. The number of inspected items has an upper threshold equal tom. For Y=1,2,...,minspected items (m≤n) the batch will be rejected if x≤zwhere xis the upper control level for batch acceptance. In the other words, when the number of defective items in the inspected items gets more than the control threshold xthen decision making process stops and the batch is rejected.

The probability distribution function of Yis determined by the following equations,

In Eq. (1), Y=mindicates that all items are inspected therefore, the number of defective items has been less than xor xthdefective item has been mthinspected item. For the casex≤Ym, xthdefective item has been Ythinspected item thus, the probability distribution function of Yfollows a negative binomial distribution. The expected mean of the number of inspected items is determined as follows:

Since Pr{Y}=(Y−1x−1)px(1−p)Y−xx≤Ymis a negative binomial distribution thus using the approximation method of estimating negative binomial probabilities with Poisson distribution [2], following is concluded,

where λ=x1−ppis the parameter of Poisson distribution. In order to improve the accuracy of this approximation, mand xshould be sufficiently large numbers. Using the above approximation method, following is concluded,

Now, let Pxdenotes the probability of rejecting the batch. The batch is rejected if the number of defective items is more than or equal to xthus the value of Pxis determined by the following equation,

In order to calculate the total cost, including the cost of rejecting the batch, the cost of inspection and the cost of defective items, assume Ris the cost of rejecting the batch, cis the inspection cost of one item and c'is the cost of one defective item, so the total cost, Cx, is determined by conditioning Cxon two events of rejecting or accepting the batch, thus the objective function is written as follows:

Thus we have,

In Eq. (7), cE[Y]xis the total cost of inspection and npc'is the total cost of defective items. The optimal value of xis determined by minimizing the value of objective functionCx. Using the optimization methods, it is concluded that,

To evaluate above equation, following equality is considered,

Since mis a sufficiently large number thus the value of me−λλm−(x−1)Γ(m−(x−1)+1)is approximately equal to zero therefore it is concluded that,

To ensure that xminimizes the objective function (7), it is necessary to find the value of xthat satisfies following inequalities:

Hence,

Now Ifmc+npc'<R, then,

Since with increasing the value of xthe value of binomial distribution with parameters mandpdecreases thus according to the properties of binomial distribution, it is concluded thatx>(m+1)ptherefore, the optimal value of xis determined using the following formula,

Also The objective function, Cx, should be minimized regarding two constraints on Type-I and Type-II errors associated with the acceptance sampling plans. Type-I error is the probability of rejecting the batch when the nonconformity proportion of the batch is acceptable. Type-II error is the probability of accepting the batch when the nonconforming proportion of the batch is not acceptable. Then, in one hand, ifp=δ1, the probability of rejecting the batch should be less thanα. On the other hand, in case wherep=δ2, the probability of accepting the batch should be less thanβwhere δ1is the AQL (Accepted Quality Level ) and δ2is the LQL (Limiting Quality Level) andαis the probability of Type-I error and βis the probability of Type-II error in making a decision, therefore, the optimal value of xis determined using the following formula,

Whenmc+npc'>R, It is concluded that Eq. (16) is positive for all values of xsox=0. In this case, if one defective item is found in an inspected sample then the batch would be rejected. In this case, the rejection cost Ris less than the total cost of inspecting mitems and the cost of defective items, hence rejecting the batch would be the optimal decision. However, in practice the rejection cost Ris usually big enough so that, we overlooked that case.

3. Acceptance Sampling Policy Based on Number of Successive Conforming Items [3]

In a typical acceptance-sampling plan, when the number of conforming items between successive nonconforming items is more than an upper control threshold, the batch is accepted, and when it is less than a lower control threshold, the batch is rejected otherwise, the inspection process continues. This initiates the idea of employing a Markovian approach to model the acceptance-sampling problem. As a result, in this method, a new acceptance-sampling policy using Markovian models is proposed, in which determining the control thresholds are aimed. The notations required to model the problem at hand are given as:

1. N: The number of items in the batch

2. p: The proportion of nonconforming items in the batch

3. I: The cost of inspecting one item

4. c: The cost of one nonconforming item

5. R: The cost of rejecting the batch

6. E(TC): The expected total cost of the system

7. E(AC): The expected total cost of accepting the batch

8. E(RP): The expected total cost of rejecting the batch

9. E(I): The expected total cost of inspecting the items of the batch

10. U: The upper control threshold

11. L: The lower control threshold

Consider an incoming batch of Nitems with a proportion of nonconformitiesp, of which items are randomly selected for inspection and based on the number of conforming items between two successive nonconforming items, the batch is accepted, rejected, or the inspection continues. The expected total cost associated with this inspection policy can be expressed using Eq. (17).

Let Yibe the number of conforming items between the successive (i−1)thand ithnonconforming items, Uthe upper and Lthe lower control thresholds. Then, if Yi≥Uthe batch is accepted, if Yi≤Lthe batch is rejected. Otherwise, if L<Yi<Uthe process of inspecting items continues. The states involved in this process can be defined as follows.

State 1: Yifalls within two control thresholds L, i.e.,L<Yi<U, thus the inspection process continues.

State 2: Yiis more than or equal the upper control threshold, i.e.,Yi≥U, hence the batch is accepted.

State 3: Yiis less than or equal the lower control threshold, i.e.,Yi≤L, hence the batch is rejected.

The transition probabilities among the states can be obtained as follows.

Probability of inspecting more items=p11=Pr{L<Yi<U}

Probability of accepting the batch=p12=Pr{Yi≥U}

Probability of rejecting the batch=p13=Pr{Yi≤L}

where the probabilities can be obtained based on the fact that the number of conforming items between the successive (i−1)thand ithnonconforming items, Yi, follows a geometric distribution with parameterp, i.e., Pr(Yi=r)=(1−p)rp;r=0,1,2,...Then, the transition probability matrix is expressed as follows:

As it can be seen, the matrix Pis an absorbing Markov chain with states 2 and 3 being absorbing and state 1 being transient.

To analyze the above absorbing Markov chain, the transition probability matrix should be rearranged in the following form:

Rearranging the Pmatrix yields the following matrix:

Then, the fundamental matrix Mcan be obtained as follows [4],

Where Iis the identity matrix and m11denotes the expected long-run number of times the transient state 1 is occupied before absorption occurs (i.e., accepted or rejected), given that the initial state is 1. The long-run absorption probability matrix, F, is calculated as follows [4],

The elements of the Fmatrix, f12,f13, denote the probabilities of the batch being accepted or rejected, respectively.

The expected cost can be obtained using Eq. (17) containing the batch acceptance, rejection, and inspection costs. The expected acceptance cost is the cost of nonconforming items (Npc) multiplied by the probability of the batch being accepted (i.e.,f12). The expected rejection cost is the rejection cost (R) multiplied by the probability of the batch being rejected (i.e.,f13). Moreover, m11is the expected long-run number of times the transient state 1 is occupied before absorption occurs. Knowing that in each visit to transient state, the average number of inspections is1p(the mean of the geometric distribution), the expected inspection cost is given by

Therefore, the expected cost for acceptance-sampling policy can be expressed as a function of f12,f13and m11as follows:

Substituting for f12andm11, the expected cost equation can be rewritten as:

Eq. (25) can be solved numerically using search algorithms to find Land Uthat minimize the expected total cost. The objective function, E(TC), should be minimized regarding two constraints on Type-I and Type-II errors associated with the acceptance sampling plans. Type-I error is the probability of rejecting the batch when the nonconformity proportion of the batch is acceptable. Type-II error is the probability of accepting the batch when the nonconforming proportion of the batch is not acceptable. Then, in one hand, ifp=AQL, the probability of rejecting the batch should be less thanα. On the other hand, in case wherep=LQL, the probability of accepting the batch should be less thanβwhere αand βare the probabilities of Type-I and Type-II errors, hence,

The optimum values of Land Uamong a set of alternative values are determined solving the model given in (25), numerically, where the probabilities are obtained using the geometric distribution.

4. Acceptance Sampling Policy Using the Minimum Angle Method based on Number of Successive Conforming Items [5]

The practical performance of any sampling plan is determined through its operating characteristic curve. When producer and consumer are negotiating for designing sampling plans, it is important especially to minimize the consumer risk. In order to minimize the consumer’s risk, the ideal OC curve could be made to pass as closely through[AQL,1−α],[AQL,β]. One approach to minimize the consumers risks for ideal condition is proposed with minimization of angle ϕbetween the lines joining the points[AQL,1−α], [AQL,β]and[AQL,1−α],[LQL,β]. Therefore in this case, the value of performance criteria in minimum angle method will be [6],

where Pra(LQL),Pra(AQL)is the probability of accepting the batch when the proportion of defective items in the batch is respectivelyLQL,AQL. Assume Ais the point[AQL,1−α], Bis the point [AQL,β]and Cis the point [LQL,β]thus the smaller value ofTan(ϕ), the angle ϕapproaching zero, and the chord ACapproachingAB, the ideal condition.

The values of Pra(LQL),Pra(AQL)are determined as follows,

Since the values of LQL,AQLare constant andLQLAQLtherefore the objective function is determined as follows,

Another performance measure of acceptance sampling plans is the expected number of inspected items. Since sampling and inspecting usually has cost, therefore designs that minimizes this measure and satisfy the first and second type error inequalities are considered to be optimal sampling plans. Since the proportion of defective items is not known in the start of process, in order to consider this property in designing the acceptance sampling plans, we try to minimize the expected number of inspected items for acceptable and not acceptable lots simultaneously. Therefore the optimal acceptance sampling plan should have three properties, first it should have a minimized value in the objective function of the minimum angle method that is resulted from the ideal OC curve and also it should minimize the expected number of inspected items either in the decisions of rejecting or accepting the lot. Therefore the second objective function is defined as the expected number of items inspected. The value of this objective function is determined based on the value ofm11(p)where m11(p)is the expected number of times in the long run that the transient state 1 is occupied before absorption occurs, since in each visit to transient state, the average number of inspections is1p, consequently the expected number of items inspected is given by1pm11(p). Now the objective functions WandZare defined as the expected number of items inspected respectively in the acceptable condition(p=AQL)and not acceptable condition(p=LQL).

Now one approach to optimize the objective functions simultaneously is to define control thresholds for objective functions Z,Wand then trying to minimize the value of objective functionV. For example if parameters Z1,W1are defined as the upper control thresholds for Z,Wthen the optimization problem can be defined as follows,

Optimal values of L,Ucan be determined by solving above nonlinear optimization problem using search procedures or other optimization tools.

5. Acceptance Sampling Policy Based on Cumulative Sum of Conforming Items Run Lengths [7]

In an acceptance-sampling plan, assume Yiis the number of conforming items between the successive (i−1)thand ithdefective items. Decision making is based on the value of Sithat is defined as,

The proposed acceptance sampling policy is defined as follows,

1. If Si≥Uthen the batch is accepted

2. If Si≤Lthe batch is rejected

3. If L<Si<Uthe process of inspecting the items continues

where Uis the upper control threshold and Lis the lower control threshold.

In each stage of the data gathering process, the index of different states of the Markov model,j, is defined as:

1. j=1represents the state of rejecting the batch. In this state Si≤Lthus the batch is rejected.

2. j=Yi+2 where Yi=0,1,2...,U−1represents the state of continuing data gathering. In this state, L<Si=Yi+Yi−1<Uthus the inspecting process continues.

3. j=U+2represents the state of accepting the batch. In this state Si≥Uhence the batch is accepted.

In other word, the acceptance-sampling plan can be expressed by a Markov model, in which the transition probability matrix among the states of the batch can be expressed as:

where, pjkis probability of going from state jto state kin a single step and Yi+1denotes the number of conforming items between the successive defective items and Pr(Yi+1=r)=(1−p)rpr=0,1,2,...where pdenotes the proportion of defective items in the batch.

The values of pjkare determined based on the relations among the states, for example where U+2>j>1,L≥j−2,k=1then according to the definition ofj, it is concluded that j=Yi+2 and transition probability of going form state jto state k=1is equal to the probability of rejecting the batch that is evaluated as follows,

In the other case where, U+2>j>1,U+2>k>1,U>j+k−4>L, based on the definition ofj, we havej=Yi+2 thus it is concluded that

In the other case where, U+2>j>1,k=U+2, then according to the definition ofj, we havej=Yi+2 thus it is concluded that,

In the other case where, U+2>j>1,U+2>k>1,j+k−4≥U , then according to the definition of j, we havej=Yi+2 thus it is concluded that,

As a result, when L=1and U=3for example, the transition probability matrix among the states of the system can be expressed as:

And it can be seen the matrix Pis an absorbing Markov chain with states 1 and 5 being absorbing and states 2, 3, and 4 being transient.

Analyzing the above absorbing Markov chain requires to rearrange the single-step probability matrix in the following form:

whereAis the identity matrix representing the probability of staying in a state that is defined as follows

Ois the probability matrix of escaping an absorbing state (always zero) that is defined as follows

Qis a square matrix containing the transition probabilities of going from a non-absorbing state to another non-absorbing state that is defined as follows

And Ris the Matrix containing all probabilities of going from a non-absorbing state to an absorbing state (i.e., accepted or rejected batch) that is defined as follows

Rearranging the Pmatrix in the latter form yields the following:

Bowling et. al. [4] proposed an absorbing Markov chain model for determining the optimal process means. According to their method, matrix Mthat is the fundamental matrix containing the expected number of transitions from a non-absorbing state to another non-absorbing state before absorption occurs can be obtained by the following equation,

For the above numerical example, i.e., when L=1andU=3, the fundamental matrix Mcan be obtained as:

where Iis the identity matrix.

Since mjjrepresents the expected number of the times in the long-run the transient state jis occupied before absorption occurs (i.e., before accepted or rejected), and matrix Fis the absorption probability matrix containing the long run probabilities of the transition from a non-absorbing state to an absorbing state. The long-run absorption probability matrix, F, can be calculated as follows:

Again when L=1andU=3, the elements of F(fjk   ;   j=2,3,4   ;   k=1,5)represent the probabilities of the batch being accepted and rejected, respectively, given that the initial state isj=2,3,4. In this case, the probability of accepting the batch is obtained as:

Also the expected number of inspected items will be determined as follows,

This new acceptance-sampling plan should satisfy two constraints of the first and the second types of errors. The probability of Type-I error shows the probability of rejecting the batch when the defective proportion of the batch is acceptable. The probability of Type-II error is the probability of accepting the batch when the defective proportion of the batch is not acceptable. Then on the one hand ifp=AQL, the probability of rejecting the batch will be less than αand on the other hand, in case wherep=LQL, the probability of accepting the batch will be less thanβwhere αand βare the probabilities of Type-I and Type-II errors. Hence,

From the inequalities in (50), the proper values of the thresholds Land Uare determined and among the feasible ones, we select one that has the least value for expected number of inspected items that is obtained using Eq. (49).

6. A New Acceptance Sampling Design Using Bayesian Modelling and Backwards Induction [8]

In this research, a new selection approach on the choices between accepting and rejecting a batch based on Bayesian modelling and backwards induction is proposed. The Bayesian modelling is utilized to model the uncertainty involved in the probability distribution of the nonconforming proportion of the items and the backwards induction method is employed to determine the sample size. Moreover, when the decision on accepting or rejecting a batch cannot be made, we assume additional observations can be gathered with a cost to update the probability distribution of the nonconforming proportion of the batch. In other words, a mathematical model is developed in this research to design optimal single sampling plans. This model finds the optimum sampling design whereas its optimality is resulted by using the decision tree approach. As a result, the main contribution of the method is to model the acceptance-sampling problem as a cost optimization model so that the optimal solution can be achieved via using the decision tree approach. In this approach, the required probabilities of decision tree are determined employing the Bayesian Inference. To do this, the probability distribution function of nonconforming proportion of items is first determined by Bayesian inference using a non-informative prior distribution. Then, the required probabilities are determined by applying Bayesian inference in the backward induction method of the decision tree approach. Since this model is completely designed based on the Bayesian inference and no approximation is needed, it can be viewed as a new tool to be used by practitioners in real case problems to design an economically optimal acceptance-sampling plan. However, the main limitation of the proposed methodology is that it can only be applied to items not requiring very low fractions of nonconformities.