**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 suc‐ cessive 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 inspec‐ tion 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:


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

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

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

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

*U* : The upper control threshold

( )

*m Y x*

l

; 1;

*xx m p*

( )

*z x z*

*m x* −1

= =

å å

dd

1

*p p*

l


1

å

*ΔCx* =(*mc* + *npc* '−*R*)(

**Conforming Items [3]**

*N* : The number of items in the batch

*I*: The cost of inspecting one item

*R*: The cost of rejecting the batch

*c*: The cost of one nonconforming item

*p*: The proportion of nonconforming items in the batch

*x Min e c*

60 Practical Concepts of Quality Control

( ( )) ( )

> - ç ÷ - + è ø

*m m z z*

*Y x m x x*

= + -

practice the rejection cost *R* is usually big enough so that, we overlooked that case.

**3. Acceptance Sampling Policy Based on Number of Successive**

are aimed. The notations required to model the problem at hand are given as:

1

*Y x m p p R mc npc <sup>x</sup>*

1 1

*x m x Y x*


è ø - - +

*m Y x*

æ ö G -+ ç ÷ -> >

*x R mc npc*

( ) ( ) ( )

*c*

å

ì ü > + ï ï

1 '

= í ý ï ï G -+ æ ö

1 '

( ) ( )

1,1

When*mc* + *npc* '>*R*, It is concluded that Eq. (16) is positive for all values of *x*so*x* =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 *R* is less than the total cost of inspecting *m* items and the cost of defective items, hence rejecting the batch would be the optimal decision. However, in

) *p <sup>x</sup>*−1(1− *p*)*m*−(*x*−1)

In a typical acceptance-sampling plan, when the number of conforming items between suc‐ cessive 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 inspec‐ tion 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

æ ö æ ö ç ÷ -£ -£ ç ÷ î þ è ø è ø

*m x m z m z z z*

 a

1 1 1 2 2 0

( ( ))

 dd

+ *c*∑ *Y* =*x*

*<sup>m</sup> e* <sup>−</sup>*λλ <sup>Y</sup>* <sup>−</sup>*<sup>x</sup>*


*m Y x*

l

l


*e*

1

 b

*<sup>Γ</sup>*(*<sup>Y</sup>* <sup>−</sup> *<sup>x</sup>* <sup>+</sup> 1) (16)

(15)

*L* : The lower control threshold

Consider an incoming batch of *N* items with a proportion of nonconformities*p*, 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 inspec‐ tion continues. The expected total cost associated with this inspection policy can be ex‐ pressed using Eq. (17).

$$E\text{ (TC)} = E\text{ (AC)} + E\text{ (RP)} + E\text{ (I)}\tag{17}$$

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

State 1: *Yi* falls within two control thresholds L, i.e.,*L* <*Yi* <*U* , thus the inspection proc‐ ess continues.

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

State 3: *Yi* is 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=*p*<sup>11</sup> =Pr{*L* <*Yi* <*U* }

Probability of accepting the batch=*p*<sup>12</sup> =Pr{*Yi* ≥*U* }

Probability of rejecting the batch=*p*<sup>13</sup> =Pr{*Yi* ≤ *L* }

where the probabilities can be obtained based on the fact that the number of conforming items between the successive (*i* −1) *th* and *<sup>i</sup> th* nonconforming items, *Yi* , follows a geometric distribution with parameter*p*, i.e., Pr(*Yi* <sup>=</sup>*r*) =(1<sup>−</sup> *<sup>p</sup>*)*<sup>r</sup> <sup>p</sup>*;*<sup>r</sup>* =0, 1, 2, ...Then, the transition proba‐ bility matrix is expressed as follows:

$$\mathbf{P}\_{-2} \begin{bmatrix} p\_{11} & p\_{12} & p\_{13} \\ \mathbf{0} & 1 & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & 1 \end{bmatrix} \tag{18}$$

absorption occurs. Knowing that in each visit to transient state, the average number of inspec‐

Therefore, the expected cost for acceptance-sampling policy can be expressed as a function

*<sup>E</sup>*(*I*)= *<sup>I</sup>*

*E*(*TC*)=*cNp f* <sup>12</sup> + *R f* <sup>13</sup> +

<sup>1</sup> <sup>−</sup> *<sup>p</sup>*<sup>11</sup> <sup>+</sup> *<sup>R</sup>*(1<sup>−</sup> *<sup>p</sup>*<sup>12</sup>

1 − *p*<sup>11</sup> ) + *I <sup>p</sup>* ( <sup>1</sup> 1 − *p*<sup>11</sup>

Eq. (25) can be solved numerically using search algorithms to find *L* and *U* that minimize the expected total cost. The objective function, *E*(*TC*), should be minimized regarding two con‐ straints 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 nonconform‐ ing proportion of the batch is not acceptable. Then, in one hand, if*p* = *AQL* , the probability of rejecting the batch should be less than*α*. On the other hand, in case where*p* = *LQL* , the probability of accepting the batch should be less than*β* where *α* and *β*are the probabilities of

<sup>1</sup>−Pr{*<sup>L</sup>* <sup>&</sup>lt;*Yi* <sup>&</sup>lt;*<sup>U</sup>* } <sup>≥</sup>1−*<sup>α</sup>*

The optimum values of *L* and *U* among a set of alternative values are determined solving the model given in (25), numerically, where the probabilities are obtained using the geometric

**4. Acceptance Sampling Policy Using the Minimum Angle Method based**

The practical performance of any sampling plan is determined through its operating charac‐ teristic 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

} <sup>1</sup>−Pr{*<sup>L</sup>* <sup>&</sup>lt;*Yi* <sup>&</sup>lt;*<sup>U</sup>* } <sup>≥</sup>1−*<sup>β</sup>*

Substituting for *f* 12 and*m*11, the expected cost equation can be rewritten as:

*p*12

*<sup>p</sup>* <sup>=</sup> *AQL* <sup>→</sup> Pr{*Yi* <sup>≥</sup>*<sup>U</sup>* }

*<sup>p</sup>* <sup>=</sup> *LQL* <sup>→</sup> <sup>1</sup><sup>−</sup> Pr{*Yi* <sup>≥</sup>*Ui*

**on Number of Successive Conforming Items [5]**

*E*(*TC*)= *Npc*

*<sup>p</sup>* (the mean of the geometric distribution), the expected inspection cost is given by

*I*

*<sup>p</sup> m*<sup>11</sup> (23)

*<sup>p</sup> m*<sup>11</sup> (24)

New Models of Acceptance Sampling Plans http://dx.doi.org/10.5772/50835 63

) (25)

(26)

tions is <sup>1</sup>

of *f* 12, *f* 13 and *m*11as follows:

Type-I and Type-II errors, hence,

distribution.

As it can be seen, the matrix *P* is an absorbing Markov chain with states 2 and 3 being ab‐ sorbing and state 1 being transient.

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

$$
\begin{bmatrix}
\mathbf{A} & \mathbf{O} \\
\mathbf{R} & \mathbf{Q}
\end{bmatrix}
\tag{19}
$$

Rearranging the *P* matrix yields the following matrix:

$$\begin{aligned} \,\_3^2\left[ \begin{array}{cccc} 1 & 0 & 0\\ 0 & 1 & 0\\ p\_{12} & p\_{13} & p\_{11} \end{array} \right] \end{aligned} \tag{20}$$

Then, the fundamental matrix *M* can be obtained as follows [4],

$$M = \eta \mathbf{u}\_{11} = (I - \mathbf{Q})^{-1} = \frac{1}{1 - p\_{11}} = \frac{1}{1 - \Pr[L \le Y\_i \le U]} \tag{21}$$

Where *I* is the identity matrix and *m*11 denotes 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],

$$F = \mathbf{M} \times \mathbf{R} = \mathbf{1} \begin{bmatrix} p\_{12} & p\_{13} \\ \hline 1 - p\_{11} & 1 - p\_{11} \end{bmatrix} \tag{22}$$

The elements of the *F* matrix, *f* <sup>12</sup>, *f* <sup>13</sup>, 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., *f* <sup>12</sup>). The expected rejection cost is the rejection cost (*R*) multiplied by the probability of the batch being rejected (i.e., *f* <sup>13</sup>). Moreover, *m*11is 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 inspec‐ tions is <sup>1</sup> *<sup>p</sup>* (the mean of the geometric distribution), the expected inspection cost is given by

$$E\left(I\right) = \frac{I}{p}m\_{11} \tag{23}$$

Therefore, the expected cost for acceptance-sampling policy can be expressed as a function of *f* 12, *f* 13 and *m*11as follows:

$$E\{TC\} = cNpf\_{12} + Rf\_{13} + \frac{l}{p}m\_{11} \tag{24}$$

Substituting for *f* 12 and*m*11, the expected cost equation can be rewritten as:

1 2 3 1 11 12 13

é ù ê ú ê ú ê ú ë û

*ppp*

010 001

As it can be seen, the matrix *P* is an absorbing Markov chain with states 2 and 3 being ab‐

To analyze the above absorbing Markov chain, the transition probability matrix should be

*A O*

2 3 1

100 010 *ppp*

é ù ê ú ê ú ê ú ë û

<sup>1</sup> 12 13 11

<sup>1</sup> <sup>−</sup> *<sup>p</sup>*<sup>11</sup> <sup>=</sup> <sup>1</sup>

2 3

*p*13 1− *p*<sup>11</sup>

Where *I* is the identity matrix and *m*11 denotes 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],

1− *p*<sup>11</sup>

The elements of the *F* matrix, *f* 12, *f* 13, denote the probabilities of the batch being accepted or

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., *f* <sup>12</sup>). The expected rejection cost is the rejection cost (*R*) multiplied by the probability of the batch being rejected (i.e., *f* <sup>13</sup>). Moreover, *m*11is the expected long-run number of times the transient state 1 is occupied before

2 3

Then, the fundamental matrix *M* can be obtained as follows [4],

*<sup>M</sup>* <sup>=</sup>*m*<sup>11</sup> =(*<sup>I</sup>* **-** *<sup>Q</sup>*)−<sup>1</sup> <sup>=</sup> <sup>1</sup>

*<sup>F</sup>* **<sup>=</sup>***<sup>M</sup>* **<sup>×</sup>** *<sup>R</sup>* =1 *<sup>p</sup>*<sup>12</sup>

**P** (18)

*<sup>R</sup> <sup>Q</sup>* (19)

<sup>1</sup> <sup>−</sup> Pr{*<sup>L</sup>* <sup>&</sup>lt; *Yi* <sup>&</sup>lt; *<sup>U</sup>* } (21)

(20)

(22)

2 3

=

sorbing and state 1 being transient.

62 Practical Concepts of Quality Control

rearranged in the following form:

rejected, respectively.

Rearranging the *P* matrix yields the following matrix:

$$E\{TC\} = Npc \frac{p\_{12}}{1 - p\_{11}} + R\left(1 - \frac{p\_{12}}{1 - p\_{11}}\right) + \frac{I}{p} \left(\frac{1}{1 - p\_{11}}\right) \tag{25}$$

Eq. (25) can be solved numerically using search algorithms to find *L* and *U* that minimize the expected total cost. The objective function, *E*(*TC*), should be minimized regarding two con‐ straints 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 nonconform‐ ing proportion of the batch is not acceptable. Then, in one hand, if*p* = *AQL* , the probability of rejecting the batch should be less than*α*. On the other hand, in case where*p* = *LQL* , the probability of accepting the batch should be less than*β* where *α* and *β*are the probabilities of Type-I and Type-II errors, hence,

$$\begin{aligned} p = AQL &\rightarrow \frac{\Pr\{Y\_i \ge U\}}{1 - \Pr\{L \le Y\_i \le U\}} \ge 1 - \alpha\\ p = LQL &\rightarrow 1 - \frac{\Pr\{Y\_i \ge U\_i\}}{1 - \Pr\{L \le Y\_i \le U\}} \ge 1 - \beta \end{aligned} \tag{26}$$

The optimum values of *L* and *U* among a set of alternative values are determined solving the model given in (25), numerically, where the probabilities are obtained using the geometric distribution.
