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

In an acceptance-sampling plan, assume *Yi* is the number of conforming items between the successive (*i* −1) *th* and *<sup>i</sup> th* defective items. Decision making is based on the value of *Si* that is defined as,

$$S\_i = Y\_i + Y\_{i-1} \tag{32}$$

The proposed acceptance sampling policy is defined as follows,

If *Si* ≥*U* then the batch is accepted

If *Si* ≤ *L* the batch is rejected

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 minimi‐ zation of angle *ϕ* between 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

(*AQL* )−Pr*<sup>a</sup>*

defective items in the batch is respectively*LQL* , *AQL* . Assume *A* is the point *AQL* , 1−*α* , *B*is the point *AQL* , *β* and *C* is the point *LQL* , *β* thus the smaller value of*Tan*(*ϕ*), the

(*AQL* ) are determined as follows,

(*AQL* )= *<sup>f</sup>* <sup>12</sup>(*AQL* )= Pr{*<sup>U</sup>* <sup>≤</sup>*Yi*

(*LQL* )=1<sup>−</sup> *<sup>f</sup>* <sup>12</sup>(*LQL* )=1<sup>−</sup> Pr{*<sup>U</sup>* <sup>≤</sup>*Yi*

Since the values of *LQL* , *AQL* are constant and*LQL AQL* therefore the objective function

(*LQL* )−Pr*<sup>a</sup>*

occurs, since in each visit to transient state, the average number of inspections is <sup>1</sup>

quently the expected number of items inspected is given by <sup>1</sup>

Another performance measure of acceptance sampling plans is the expected number of inspected items. Since sampling and inspecting usually has cost, therefore designs that min‐ imizes 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 prop‐ erties, 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 of*m*11(*p*)where *m*11(*p*) is the expect‐ ed number of times in the long run that the transient state 1 is occupied before absorption

angle *ϕ* approaching zero, and the chord *AC* approaching*AB*, the ideal condition.

(*AQL* )is the probability of accepting the batch when the proportion of

}

1−Pr{*U* >*Yi* > *L* }

}

(*AQL* )} (29)

(28)

*<sup>p</sup>* , conse‐

*<sup>p</sup> m*11(*p*). Now the objective

1−Pr{*U* >*Yi* > *L* }

(*LQL* ) ) (27)

*Tan*(*ϕ*)=( *LQL* - *AQL* Pr*<sup>a</sup>*

angle method will be [6],

64 Practical Concepts of Quality Control

(*LQL* ), Pr*<sup>a</sup>*

(*LQL* ), Pr*<sup>a</sup>*

*p* = *AQL* →Pr*<sup>a</sup>*

*p* = *LQL* →1−Pr*<sup>a</sup>*

*V* =*Min L* ,*U* {Pr*<sup>a</sup>*

where Pr*<sup>a</sup>*

The values of Pr*<sup>a</sup>*

is determined as follows,

If *L* <*Si* <*U* the process of inspecting the items continues

where *U* is the upper control threshold and *L* is the lower control threshold.

In each stage of the data gathering process, the index of different states of the Markov mod‐ el, *j* , is defined as:

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

In the other case where, *U* + 2> *j* >1, *k* =*U* + 2, then according to the definition of *j*, we have

In the other case where, *U* + 2> *j* >1, *U* + 2>*k* >1, *j* + *k* −4≥*U* , then according to the defini‐

As a result, when *L* =1and *U* =3 for example, the transition probability matrix among the

( ) ( ) ( ) ( ) ( ) ( )

1 2 3 4 5

10000 Pr 1 0 0 Pr 2 Pr 3 Pr 0 0 Pr 1 0 Pr 2 0 Pr 0 0 0 Pr 1 00001

é ù ê ú £ = <sup>³</sup> £ =³

*Y Y Y Y YY*

ë û

And it can be seen the matrix *P* is an absorbing Markov chain with states 1 and 5 being ab‐

Analyzing the above absorbing Markov chain requires to rearrange the single-step probabil‐

where*A*is the identity matrix representing the probability of staying in a state that is defined

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

*<sup>P</sup>* <sup>=</sup> *<sup>A</sup> <sup>O</sup>*

*<sup>A</sup>*<sup>=</sup> <sup>1</sup> <sup>0</sup>

( ) ( )

**P** (38)

= ³

*<sup>R</sup> <sup>Q</sup>* (39)

<sup>0</sup> <sup>1</sup> (40)

*Y Y*

*p jk* =Pr(*L* <*Si*+1 =*Yi*+1 + *Yi* <*U* , *Yi*+1 =*k* −2, *j* + *k* −4≥*U* )

=Pr(*L* < *j* −2 + *Yi*+1 <*U* , *Yi*+1 =*k* −2, *j* + *k* −4≥*U* )

=Pr(*L* < *j* + *k* −4<*U* , *j* + *k* −4≥*U* )=0

*p jU* +2 =Pr(*Si*+1 =*Yi*+1 + *Yi* ≥*U* ) =Pr(*Yi*+1 + *j* −2≥*U* ) =Pr(*Yi*+1 ≥*U* − *j* + 2) (36)

(37)

67

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

*j* =*Yi* + 2 thus it is concluded that,

tion of j, we have *j* =*Yi* + 2 thus it is concluded that,

states of the system can be expressed as:

sorbing and states 2, 3, and 4 being transient.

=

ity matrix in the following form:

as follows

follows

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

*j* =*U* + 2 represents the state of accepting the batch. In this state *Si* ≥*U* hence 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:

$$\begin{aligned} \Pr\begin{bmatrix} 1 & j=k=1\\ 0 & j=1, k>1\\ \Pr\left(Y\_{i+1}\le L-j+2\right) & U+2>j>1, L\ge j-2, k=1\\ 0 & U+2>j>1, Lj>1, U+2>k>1, j+k-4\le L\\ 0 & U+2>j>1, U+2>k>1, j+k-4\ge U\\ \Pr\left(Y\_{i+1}=k-2\right) & U+2>j>1, U+2>k>1, U>j+k-4>L\\ 1 & j=k=U+2\\ 0 & j=U+2, kj>1, k=U+2 \end{aligned} \tag{33}$$

where, *p jk* is probability of going from state *j*to state *k*in a single step and *Yi*+1denotes the number of conforming items between the successive defective items and Pr(*Yi*+1 <sup>=</sup>*r*) =(1<sup>−</sup> *<sup>p</sup>*)*<sup>r</sup> <sup>p</sup> <sup>r</sup>* =0, 1, 2, ... where *p*denotes the proportion of defective items in the batch.

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

$$\Pr\_{j} p\_{j} = \Pr\{L \ge S\_{i+1} = Y\_{i+1} + Y\_{i}\} = \Pr\{L \ge Y\_{i+1} + j - 2\} = \Pr\{Y\_{i+1} \le L \mid -j + 2\} \tag{34}$$

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

$$\begin{aligned} \Pr\_{jk} &= \Pr\{L \quad \le S\_{i+1} = Y\_{i+1} + Y\_i \le \ell L \mid Y\_{i+1} = k - 2\} = \\ &\Pr\{L \quad < j - 2 + Y\_{i+1} \le \ell L \mid Y\_{i+1} = k - 2\} = \\ &\Pr\{L \quad < j - 2 + k - 2 \le \ell L \mid Y\_{i+1} = k - 2\} = \Pr\{L \quad < j + k - 4 < \ell L \mid Y\_{i+1} = k - 2\} \end{aligned} \tag{35}$$

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

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

state, *L* <*Si* =*Yi* + *Yi*−<sup>1</sup> <*U* thus the inspecting process continues.

( <sup>1</sup> )

( <sup>1</sup> )

( <sup>1</sup> )

0

ì ï ï ï ï ï ï ï í ï ï ï

66 Practical Concepts of Quality Control

1

*jk*

=

*p*

batch.

*i*

*i*

the batch that is evaluated as follows,

*p <sup>j</sup>*<sup>1</sup> =Pr(*L* ≥*Si*+1 =*Yi*+1 + *Yi*

*j*, we have *j* =*Yi* + 2 thus it is concluded that

*p jk* =Pr(*L* <*Si*+1 =*Yi*+1 + *Yi* <*U* , *Yi*+1 =*k* −2) =

Pr(*L* < *j* −2 + *Yi*+1 <*U* , *Yi*+1 =*k* −2) =

+

1 1 0 1, 1

accepted.

*j* =*Yi* + 2 where *Yi* =0, 1, 2..., *U* −1 represents the state of continuing data gathering. In this

*j* =*U* + 2 represents the state of accepting the batch. In this state *Si* ≥*U* hence the batch is

In other word, the acceptance-sampling plan can be expressed by a Markov model, in which

*U j Lj k*

*Y k* <sup>+</sup> *U j U k U jk L*

where, *p jk* is probability of going from state *j*to state *k*in a single step and *Yi*+1denotes the number of conforming items between the successive defective items and Pr(*Yi*+1 <sup>=</sup>*r*) =(1<sup>−</sup> *<sup>p</sup>*)*<sup>r</sup> <sup>p</sup> <sup>r</sup>* =0, 1, 2, ... where *p*denotes the proportion of defective items in the

The values of *p jk* are determined based on the relations among the states, for example where *U* + 2> *j* >1, *L* ≥ *j* −2, *k* =1 then according to the definition of *j*, it is concluded that *j* =*Yi* + 2 and transition probability of going form state *j*to state *k* =1is equal to the probability of rejecting

In the other case where, *U* + 2> *j* >1, *U* + 2>*k* >1, *U* > *j* + *k* −4> *L* , based on the definition of

Pr(*L* < *j* −2 + *k* −2<*U* , *Yi*+1 =*k* −2) =Pr(*L* < *j* + *k* −4<*U* , *Yi*+1 =*k* −2)

= - +>> +>> >+->

 2 1, 2 1, 4 0 2 1, 2 1, 4 Pr 2 2 1, 2 1, 4

+>> <- =

*U j U k jk L U j U k jk U*

+>> +> > +-£ +>> +> > +-³

) =Pr(*L* ≥*Yi*+1 + *j* −2) =Pr(*Yi*+1 ≤ *L* − *j* + 2) (34)

(33)

(35)

the transition probability matrix among the states of the batch can be expressed as:

Pr - 2 2 1, 2, 1 0 2 1, 2, 1

2

*jkU jU kU*

0 2, 2 Pr 2 2 1, 2 *<sup>i</sup>*

ï == + <sup>ï</sup> =+ <+ <sup>ï</sup> <sup>ï</sup> ³ -+ +> > = + <sup>î</sup>

*Y U j U j kU* <sup>+</sup>

*Y Lj U j L j k*

*j k j k*

= = = > £ + +>> ³- =

$$\Pr\_{j \mid \mathcal{U} \approx 2} \Pr\{S\_{i+1} = Y\_{i+1} + Y\_i \ge \mathcal{U}\} = \Pr\{Y\_{i+1} + j - 2 \ge \mathcal{U}\} = \Pr\{Y\_{i+1} \ge \mathcal{U} - j + 2\} \tag{36}$$

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

$$\begin{aligned} \text{Pr}\_{jk} &= \text{Pr} \{ L \prec S\_{i+1} = Y\_{i+1} + Y\_i \preccurlyeq L \mid Y\_{i+1} = k-2, \ j+k-4 \ge lI \mid \} \\ &= \text{Pr} \{ L \prec j-2 + Y\_{i+1} \preccurlyeq L \mid Y\_{i+1} = k-2, \ j+k-4 \ge lI \} \\ &= \text{Pr} \{ L \prec j+k-4 \le lI, \ j+k-4 \ge lI \} = 0 \end{aligned} \tag{37}$$

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

$$\begin{array}{c c c c c c c} & 1 & 2 & 3 & 4 & 5 \\ & 2 & \Pr\left(Y \le 1\right) & 0 & 0 & \Pr\left(Y = 2\right) & \Pr\left(Y \ge 3\right) \\ & \Pr\left(Y \le 0\right) & 0 & \Pr\left(Y = 1\right) & 0 & \Pr\left(Y \ge 2\right) \\ & 4 & 0 & \Pr\left(Y = 0\right) & 0 & 0 & \Pr\left(Y \ge 1\right) \\ & 5 & 0 & 0 & 0 & 1 \end{array} \tag{38}$$

And it can be seen the matrix *P* is an absorbing Markov chain with states 1 and 5 being ab‐ sorbing and states 2, 3, and 4 being transient.

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

$$P = \begin{bmatrix} A & \mathbf{O} \\ \mathbf{R} & \mathbf{Q} \end{bmatrix} \tag{39}$$

where*A*is the identity matrix representing the probability of staying in a state that is defined as follows

$$A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \tag{40}$$

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

$$\mathbf{O}\_{-\frac{1}{2}}\begin{bmatrix} \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \end{bmatrix} \tag{41}$$

( )

where *I*is the identity matrix.

can be calculated as follows:

∑ *j*=2 *∞*

> =∑ *j*=2 4

rors. Hence,

∑ *j*=2 *U* +1 (

Probability of accepting the batch=

*f <sup>j</sup>*5Pr(*Y* = *j* −2) + Pr(*Y* ≥3)

Since *mjj*

( )


New Models of Acceptance Sampling Plans http://dx.doi.org/10.5772/50835 69

*Y*

*F* **=***M* **×** *R* (47)

*j*=2 *U* +1

( *j* −2)*mjj*

(48)

(49)

( )

*Y*

**-1 M = I-Q** (46)


represents the expected number of the times in the long-run the transient state *j*is

occupied before absorption occurs (i.e., before accepted or rejected), and matrix *F* is the ab‐ sorption 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* ,

Again when *L* =1and*U* =3, the elements of *F* ( *f jk* ; *j* =2, 3, 4 ; *k* =1, 5) represent the probabilities of the batch being accepted and rejected, respectively, given that the initial

state is *j* =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,

(the number of inspected items in state j) (the number of visits to state j) ) <sup>=</sup> <sup>∑</sup>

Expected number of inspected items =

Pr(Accepting the batch|the initial state is *j*)×Pr(the initial state is *j*)

This new acceptance-sampling plan should satisfy two constraints of the first and the sec‐ ond 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 if*p* = *AQL* , the probability of rejecting the batch will be less than *α* and on the other hand, in case where*p* = *LQL* , the probability of accepting the batch will be less than*β* where *α* and *β*are the probabilities of Type-I and Type-II er‐

2 1 0 Pr 2 3 0 1 Pr 1 0 4 Pr 0 0 1

é ù - = ê ú = -=

1 2 3 4

( )

*Y*

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

$$\begin{bmatrix} \mathbf{Q} \cdot \mathbf{\hat{i}} \\ \mathbf{Q} \cdot \mathbf{\hat{j}} \\ \mathbf{\hat{k}} \end{bmatrix} \begin{bmatrix} \mathbf{0} & \mathbf{0} & \mathbf{Pr} \left( \mathbf{\hat{i}} = \mathbf{2} \right) \\ \mathbf{0} & \mathbf{Pr} \left( Y = \mathbf{1} \right) \\ \mathbf{Pr} \left( Y = \mathbf{0} \right) & \mathbf{0} & \mathbf{0} \end{bmatrix} \tag{42}$$

And *R*is 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

$$
\begin{bmatrix}
\Pr\_2\left[\Pr\left(\stackrel{!}{Y}\le 1\right) & \Pr\left(\stackrel{!}{Y}\ge 3\right)\right] \\
\mathbf{R} \ge 3 \\
\mathbf{A}\left[\begin{array}{c}
\Pr\left(Y\le 0\right) & \Pr\left(Y\ge 2\right) \\
\mathbf{0} & \Pr\left(Y\ge 1\right)
\end{array}\right]
\end{bmatrix}
\tag{43}
$$

Rearranging the *P* matrix in the latter form yields the following:

$$\begin{array}{c c c c c c c} \text{1} & 1 & 3 & 2 & 3 & 4 \\ s & 0 & 1 & 0 & 0 & 0 \\ \text{P} & ^2 & \text{Pr} \left( Y \le 1 \right) & \text{Pr} \left( Y \ge 3 \right) & 0 & 0 & \text{Pr} \left( Y = 2 \right) \\ s & \text{Pr} \left( Y \le 0 \right) & \text{Pr} \left( Y \ge 2 \right) & 0 & \text{Pr} \left( Y = 1 \right) & 0 \\ s & 0 & \text{Pr} \left( Y \ge 1 \right) & \text{Pr} \left( Y = 0 \right) & 0 & 0 \\ \end{array} \tag{44}$$

Bowling et. al. [4] proposed an absorbing Markov chain model for determining the optimal process means. According to their method, matrix *M* that is the fundamental matrix contain‐ ing 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,

$$M = (I - \mathbb{Q})^{\cdot 1} \tag{45}$$

For the above numerical example, i.e., when *L* =1and*U* =3, the fundamental matrix *M* can be obtained as:

$$\mathbf{M} = \left(\mathbf{I} - \mathbf{Q}\right)^{-1} = \mathbf{3} \begin{bmatrix} 2 & 3 & 4 & 4 \\ 1 & 0 & -\Pr\left(Y = 2\right) \\ 0 & 1 - \Pr\left(Y = 1\right) & 0 \\ -\Pr\left(Y = 0\right) & 0 & 1 \end{bmatrix}^{-1} \tag{46}$$

where *I*is the identity matrix.

2 3 4

000 000

é ù ê ú ë û

*Q*is a square matrix containing the transition probabilities of going from a non-absorbing

( )

And *R*is the Matrix containing all probabilities of going from a non-absorbing state to an

( ) ( ) ( ) ( )

1 5

Pr 1 Pr 3 Pr 0 Pr 2 0 Pr 1

é ù £ ³ ê ú £ ³

³ ë û

1 5 2 3 4

10000 01000 Pr 1 Pr 3 0 0 Pr 2 Pr 0 Pr 2 0 Pr 1 0 0 Pr 1 Pr 0 0 0

é ù ê ú

( ) ( ) ( )

£ ³ =

*Y Y Y*

³ = ë û

Bowling et. al. [4] proposed an absorbing Markov chain model for determining the optimal process means. According to their method, matrix *M* that is the fundamental matrix contain‐ ing the expected number of transitions from a non-absorbing state to another non-absorbing

For the above numerical example, i.e., when *L* =1and*U* =3, the fundamental matrix *M* can

( ) ( ) ( ) ( ) ( )

£³ =

*YY Y Y Y*

state before absorption occurs can be obtained by the following equation,

*Y Y Y Y*

( )

**P** (44)

**R** (43)

*M* **=**(*I* **-** *Q*)**-1** (45)

*Y*

*Y*

0 0 Pr 2

2 3 4

é ù <sup>=</sup> ê ú <sup>=</sup>

= ë û

( )

*Y*

(41)

(42)

1 5

O <sup>=</sup>

state to another non-absorbing state that is defined as follows

2 3 4

=

( )

absorbing state (i.e., accepted or rejected batch) that is defined as follows

*Y*

2 3 4

=

Rearranging the *P* matrix in the latter form yields the following:

68 Practical Concepts of Quality Control

=

be obtained as:

Q 0 Pr 1 0 Pr 0 0 0 Since *mjj* represents the expected number of the times in the long-run the transient state *j*is occupied before absorption occurs (i.e., before accepted or rejected), and matrix *F* is the ab‐ sorption 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:

$$F = M \times R \tag{47}$$

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

Probability of accepting the batch=

$$\begin{aligned} &\sum\_{j=2}^{4} \text{Pr(Accepting the batch } | \text{ the initial state is } j) \times \text{Pr(the initial state is } j) \\ &= \sum\_{j=2}^{4} f\_{j, \text{j}} \text{Pr(Y = j-2)} + \text{Pr(Y \ge 3)} \end{aligned} \tag{48}$$

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

$$\begin{aligned} &\text{Expected number of inspected items} = \\ &\sum\_{j=2}^{U+1} \left\langle \text{the number of inspected items in state j} \right\rangle = \sum\_{j=2}^{U+1} (j-2)m\_{j\bar{j}} \end{aligned} \tag{49}$$

This new acceptance-sampling plan should satisfy two constraints of the first and the sec‐ ond 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 if*p* = *AQL* , the probability of rejecting the batch will be less than *α* and on the other hand, in case where*p* = *LQL* , the probability of accepting the batch will be less than*β* where *α* and *β*are the probabilities of Type-I and Type-II er‐ rors. Hence,

$$\begin{aligned} p &= AQL \ \ \ \text{\textquotedblleft Probability of accepting the batch} \ \mathbf{1} \ \ \alpha \\ p &= LQL \ \ \text{\textquotedblright} \end{aligned} \tag{50}$$
  $p = LQL \ \ \text{\textquotedblleft Probability of accepting the batch} \ \mathbf{4} \ \ \mathbf{8}$ 

mined with certainty. However, the probability distribution function of the random variable

; *j* =0, 1, 2, ..., *i*}denotes the outcomes of experiment *ei*

Cost function: The function *u*(*e*, *z*, *a*, *p*) on *E* ×*Z* × *A*×*P* denotes the cost associated with per‐

Consider a batch of size *N* with an unknown percentage of nonconforming *p* and assume *m* items are randomly selected for inspection. Based on the outcome of the inspection process in terms of the observed number of nonconforming items, the decision-maker desires to ac‐ cept the batch, reject it, or to perform more inspections by taking more samples. As Raiffa & Schlaifer [9] stated "the problem is how the decision maker chose *e*and then, having ob‐ served*z*, choose *e*such that *u*(*e*, *z*, *a*, *p*) is minimized. Although the decision maker has full control over his choice of *e*and*a*, he has neither control over the choices of *z*nor*p*. However, we can assume he is able to assign probability distribution function over these choices." They formulated this problem in the framework of the decision tree approach, the one that

For a nonconforming proportion*p*, referring to Jeffrey's prior (Nair et al. [10]), we first take a Beta prior distribution with parameters *v*<sup>0</sup> =0.5 and *u*<sup>0</sup> =0.5 to model the absolute uncertainty. Then, the posterior probability density function of *p* using a sample of *v* + *u* inspected items

*<sup>Γ</sup>*(*<sup>v</sup>* <sup>+</sup> 0.5)*Γ*(*<sup>u</sup>* <sup>+</sup> 0.5) *<sup>p</sup> <sup>v</sup>*−0.5(1<sup>−</sup> *<sup>p</sup>*)

*<sup>f</sup>* (*p*)= *Beta*(*<sup>v</sup>* <sup>+</sup> 0.5, *<sup>u</sup>* <sup>+</sup> 0.5)= *<sup>Γ</sup>*(*<sup>v</sup>* <sup>+</sup> *<sup>u</sup>* <sup>+</sup> 1)

*p*and consequently to update the probability distribution of*p*. Further, *ei*

forming experiment*e*, observing*z*, making decision*a*, and finding*p*.

.

;*i* =1, 2, ...}is the set of experiments to gather more information on

is defined an ex‐

shows

71

where *zj*

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

*<sup>u</sup>*−0.5 (51)

*p* can be obtained using Bayesian inference.

the number of nonconforming items in*ei*

*N* : The total number of items in a batch

*C*: The cost of one nonconforming item

is partially adapted in this research as well.

*R*: The cost of rejecting a batch

*S*: The cost of inspecting one item

**6.2. Problem Definition**

**6.3. Bayesian Modelling**

is

periment in which *i*items of the batch are inspected.

*n*: An upper bound on the number of inspected item

Set of experiments: *E* ={*ei*

Sample space: *Z* ={*zj*

From the inequalities in (50), the proper values of the thresholds *L* and *U* are 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).
