**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 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 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 angle method will be [6],

$$\operatorname{Tan}(\phi) = \begin{pmatrix} \operatorname{LQL} & \operatorname{-} A \operatorname{QL} \\ \hline \operatorname{Pr}\_a(\operatorname{AQL} \text{ } \text{ }) - \operatorname{Pr}\_a(\operatorname{LQL} \text{ }) \end{pmatrix} \tag{27}$$

functions *W* and*Z*are defined as the expected number of items inspected respectively in the

<sup>11</sup> ,

*W Min m AQL AQL*

1

ì ü <sup>=</sup> í ý î þ

ì ü <sup>=</sup> í ý î þ

Now one approach to optimize the objective functions simultaneously is to define control thresholds for objective functions *Z*, *W* and then trying to minimize the value of objective function*V* . For example if parameters *Z*1, *W*1 are defined as the upper control thresholds for

1 1

Optimal values of *L* , *U* can be determined by solving above nonlinear optimization prob‐

**5. Acceptance Sampling Policy Based on Cumulative Sum of Conforming**

*th* and *<sup>i</sup> th* defective items. Decision making is based on the value of *Si*

is the number of conforming items between the

*Si* =*Yi* + *Yi*−<sup>1</sup> (32)

,

*Z ZW W* < <

<sup>11</sup> ,

{ } ,

. .

*S t*

*L U Min V*

*Z Min m LQL LQL*

1

( )

(30)

65

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

(31)

that is

( )

acceptable condition(*p* = *AQL* ) and not acceptable condition(*p* = *LQL* ).

*L U*

*L U*

*Z*, *W* then the optimization problem can be defined as follows,

lem using search procedures or other optimization tools.

The proposed acceptance sampling policy is defined as follows,

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‐

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

**Items Run Lengths [7]**

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

If *Si* ≤ *L* the batch is rejected

el, *j* , is defined as:

successive (*i* −1)

defined as,

In an acceptance-sampling plan, assume *Yi*

where Pr*<sup>a</sup>* (*LQL* ), Pr*<sup>a</sup>* (*AQL* )is the probability of accepting the batch when the proportion of 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 angle *ϕ* approaching zero, and the chord *AC* approaching*AB*, the ideal condition.

The values of Pr*<sup>a</sup>* (*LQL* ), Pr*<sup>a</sup>* (*AQL* ) are determined as follows,

$$\begin{aligned} p &= AQL \ \ -\Pr\_a(AQL \ ) = f\_{12}(AQL \ ) = \frac{\Pr\{U \le Y\_i\}}{1 - \Pr\{U \ge Y\_i > L\}} \\ p &= LQL \ \ -1 - \Pr\_a(LQL \ ) = 1 - f\_{12}(LQL \ ) = 1 - \frac{\Pr\{U \le Y\_i\}}{1 - \Pr\{U > Y\_i > L\}} \end{aligned} \tag{28}$$

Since the values of *LQL* , *AQL* are constant and*LQL AQL* therefore the objective function is determined as follows,

$$V = \underset{L, \mathcal{U}}{\text{Min}} \{ \text{Pr}\_a(LQL \ ) - \text{Pr}\_a(AQL \ ) \}\tag{29}$$

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 occurs, since in each visit to transient state, the average number of inspections is <sup>1</sup> *<sup>p</sup>* , conse‐ quently the expected number of items inspected is given by <sup>1</sup> *<sup>p</sup> m*11(*p*). Now the objective functions *W* and*Z*are defined as the expected number of items inspected respectively in the acceptable condition(*p* = *AQL* ) and not acceptable condition(*p* = *LQL* ).

$$\begin{aligned} W &= \underset{L,U}{\text{Min}} \left\{ \frac{1}{AQL} \, \boldsymbol{m}\_{\text{11}} \left( AQL \right) \right\} \\ Z &= \underset{L,U}{\text{Min}} \left\{ \frac{1}{LQL} \, \boldsymbol{m}\_{\text{11}} \left( LQL \right) \right\} \end{aligned} \tag{30}$$

Now one approach to optimize the objective functions simultaneously is to define control thresholds for objective functions *Z*, *W* and then trying to minimize the value of objective function*V* . For example if parameters *Z*1, *W*1 are defined as the upper control thresholds for *Z*, *W* then the optimization problem can be defined as follows,

$$\begin{aligned} \underset{L,U}{\text{Min}} \{V\} \\ \text{S.t.} \\ Z < Z\_1, W < W\_1 \end{aligned} \tag{31}$$

Optimal values of *L* , *U* can be determined by solving above nonlinear optimization prob‐ lem using search procedures or other optimization tools.
