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Economic Design of Dodge-Romig AOQL Single

Sampling Plans by Variables with the Quadratic Loss

Function

Chung-Ho Chen

Department of Industrial Management, Southern Taiwan University of Technology

Yungkang, Taiwan 710, R.O.C.

Abstract

This article presents the design of integrating Dodge-Romig average outgoing quality limit(AOQL) single sampling plans (SSP) by variables and specification limits. By solving the modified

Kapur and Wang’s model, we not only have the economic specification limits, but also obtain the

optimal inspection policy of Dodge-Romig AOQL SSP by variables.

Key Words: Dodge-Romig Table, Average Total Inspection (ATI), Average Outgoing Quality Limit

(AOQL), Specification Limits, Quadratic Loss Function

1. Introduction

Acceptance sampling plan is a field of statistical

quality control. Tagaras [1] pointed out that in recentyears more emphasis was placed on process control and

off-line quality control methods, but acceptance sam-

pling plan still remained important functions in many

practical quality control systems. Sower et al. [2] pro-

posed an integrated model of statistical process control

(SPC) and acceptance sampling plan. Acceptance sam-

pling plan might be useful in an SPC environment under

the following three situations:

1. When a supplier has not yet achieved statistical

control of the process.

2. When a process may shift from an in-control state

to an out-of-control state.

3. When a customer specifies that a standard accep-

tance sampling plan be used.

Although there certainly exist situations in which

screening inspection (i.e., 100% inspection) are feasible,

there are also many cases where sampling is inevitable,

either because inspection is destructive, or because lot

sizes are large and inspection is expensive, time-consum-

ing, and high error rate.

The classical Dodge-Romig [3] rectifying attrib-utes sampling plans provide the lot tolerance percent

defective (LTPD) on each lot or the average outgoing

quality limit (AOQL) protection for the lots. The de-

sign of Dodge-Romig [3] AOQL single sampling

plans (SSP) by attributes is based on the following four

assumptions: (1) the manufacturing process is nor-

mally in binomial control with a process average frac-

tion defective; (2) inspection is rectifying and rejected

lots are totally inspected; (3) to make sure that the av-

erage quality of his product is satisfactory, the pro-ducer chooses an AOQL value, and considers only

sampling plans satisfying this specification; and (4)

among plans having the specified AOQL, the producer

chooses the one minimizing average total inspection

(ATI) for product of process average fraction defec-

tive. Montgomery [4] pointed out that variables sam-

pling plans usually involve smaller sample size than

attributes sampling plans for the same levels of protec-

tion. Because of economic reasons, Klufa [5,6] pre-

sented the designs of Dodge-Romig LTPD and AOQL

Tamkang Journal of Science and Engineering, Vol. 8, No 4, pp. 313-318 (2005 ) 313

*Corresponding author. E-mail: [email protected]

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SSP by variables.

The traditional concept of conformance to specifi-

cations is that items meet the specification limits.

Taguchi [7] refined the quality of product and presented

the quadratic quality loss function for reducing total

losses to the society. In general, there is an optimal tar-

get value for every measurable quality characteristic.

Any deviation from this target value incurs an economic

loss, even if the value of the quality characteristic lies

within the specification limits. The losses of quadratic

function are expressed in monetary terms and are easy

to understand and to apply in the evaluation of product

or process improvement. The quadratic quality loss

function has been succeeded in the application of SPC,

sampling plans, and specification limits design. Re-

cently, Wu and Tang [8], Li [911], Maghsoodloo and

Li [12], Phillips and Cho [13], Li and Chou [14], Li and

Wu [15], Duffuaa and Siddiqi [16], and Rahim and

Tuffaha [17] addressed the different problems of unbal-

anced tolerance design and optimum manufacturing tar-

get with the quadratic quality loss functions. Tagaras

[1] adopted the quadratic quality loss function for de-

signing the parameters of acceptance sampling plan by

variables.

Kapur and Wang [18], Kapur [19] Kapur and Cho

[20,21] and Chen and Chou [22,23] addressed the prob-

lems of quality loss function applied in the economic

design of specification limits. Kapur and Wang [18,

p.28] pointed out that one of the short-term approaches

to reduce variance of the units shipped to the customer

is to put specification limits on the process and truncate

the distribution by inspection. Chen [24] and Chen and

Chou [25] considered the problems of the integrated de-

signs of Dodge-Romig [3] AOQL SSP by attributes and

specification limits and Dodge-Romig [3] LTPD SSP

by variables and specification limits, respectively. It

turns out that not only the sampling plan parameters,

but also the specification limits have to be looked upon

as design parameters in order to minimize the expected

total cost.

In this paper, we further present the design of inte-

grating Dodge-Romig [3] AOQL SSP by variables and

specification limits. By solving the modified Kapur

and Wang’s [18] model, we not only have the eco-

nomic specification limits, but also obtain the optimal

inspection policy of Dodge-Romig [3] AOQL SSP by

variables. Finally, the comparison of our result with

those of Chen [24] and Chen and Chou [25] will be dis-

cussed.

2. Mathematical Model

For formulating a mathematical programming model,

we have the following three assumptions: (1) quality char-

acteristic, y, follows a normal distribution with parameters

mean and variance 2; (2) a loss is incurred when y devi-

ates from the target value, and this loss is governed by the

quadratic loss function. Further, a quality characteristic is

nominal-the-best type; (3) process mean is centered at the

target value m.

The expected quality loss per unit with quadratic loss

function for 100% inspection of Kapur and Wang’s [18]

model is

(1)

where y is the quality characteristic; z is the coefficient

of specification limit; k is the coefficient of quality

loss; m is the target value; f ( y) is the probability den-

sity function of the truncated normal random variable

( 1

2 1 2

1

2

2

[ ( ) ]

( )

z e

y

, z y z ); ( z )

is the cumulative distribution function for the standard

normal random variable with density function ( z ); z is the lower specification limit for screening product;

+ z is the upper specification limit for screening

product;

(2)

Assume that the process mean () is equal to the tar-

get value (m). From Kapur and Wang [18], we have the

314 Chung-Ho Chen

2

2

2

[ ( )] [( ) ]

( ) ( )

2 ( )[1 ]

2 ( ) 1

z

z

E L y E y m

k y m f y dy

z z k

z

2

21

( ) , - .2

z

z e z

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following two equations

(3)

(4)

where TC 0 is the expected total cost per unit for no in-

spection; TC 1 is the expected total cost per unit for

100% inspection; C r is the scrap cost per unit; C i is the

inspection cost per unit.

Studying the behavior of TC 1 on varying z shows thatTC 1 has a unique minimum for z [0, ). One can adopt

the direct search method for obtaining the optimum z

value which minimizes the expected total cost.

In the Dodge-Romig [3] rectifying inspection plan,

the rejected lot need to take the 100% inspection and the

accepted lot only takes sampling inspection. The ATI de-

notes the average amount of inspection per lot. Hence,

the ratio of the ATI and lot size ( N ) becomes the average

fraction inspected per lot. The expected total cost for

Dodge-Romig [3] AOQL SSP by variables includes the

expected cost for 100% inspection ( ATI

N TC 1) and the

expected cost for no inspection (( ) )1 0 ATI

N TC .

Now, we would like to design the optimal Dodge-

Romig [3] AOQL SSP by variables with the minimum

expected total cost per unit. The non-linear mathematical

programming model of this problem is as follows:

Minimize

(5)

subject to

(6)

where TC s is the expected total cost per unit for Dodge-

Romig [3] AOQL SSP by variables; N is the lot size; p L

is the specified AOQL value; n is the sample size; t is

the critical value; ATI is the average total inspection, 0

ATI / N 1; p is the fraction defective of the process;

L(p; n, t) is the operating characteristic (the probabil-

ity of accepting a submitted lot with fraction defective

p).

From Klufa (1994), we have

(7)

where p is the process average fraction defective; U p1

is a quintile of order (1 p) for a standard normal distri-

bution function ;

(8)

Klufa [5, p. 343] pointed out that for n( 6) and 0 < p

< 0.5, Eq. (7) is convex. Substituting Eq. (7) into Eq. (5),

model (5)(6) can be rewritten as

Minimize

(9)

subject to

(10)

where p = 2(1 ( z )) is the area under the standard nor-

Economic Design of Dodge-Romig AOQL Single Sampling Plans by Variables with the Quadratic Loss Function 315

1 0(1 ) s

ATI ATI TC TC TC

N N

0 p 1max (1 ) ( ; , ) L

n p L p n t p

N

1

( ) ( ; , )

( )[1 ( )] p

ATI N N n L p n t

t U N N n

A

212( 1)

t An n

1

1

1

0

1 2(1 ( ))

1

1 2(1 ( )

( )[1 ( )]

=

( )[1 ( )]

1-

( )[1 ( )]

( )[1 ( )]

1-

s

p

p

z

z

TC

t U N N n

ATC

N

t U N N n

ATC

N

t U N N n

ATC

N t U

N N n A

N

0TC

0 1max (1 ) ( ; , ) L

p

n p L p n t p

N

2

0

TC k

2

1

21 ( )

2 ( ) 1

2(1 ( )) r i

z TC k z

z

z C C

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mal distribution beyond LSL or USL.

For a given n, N , p L, and p, Eq. (10) has one and only

one solution t . For any z 0, n, either TC 1 TC 0 or TC 1

TC 0. Now, suppose that we were able to find the absolute

minimum of ATI

N and that it corresponds to a pair of

values 0 < z * < , 0 < n* N . We have to consider the

following two situations:

(1) Suppose that TC 1 TC 0, then we could further re-

duce the expected total cost per unit by choosing

n* = 0 (0% inspection) and correspond to accep-

tance without control.

(2) Suppose that TC 1 TC 0, then we could further re-

duce the expected total cost per unit by choosingn* = N (100% inspection) and define the eco-

nomic specification limits through the value z *

that minimizes TC 1.

Hence, the optimum inspection policy is either ac-

ceptance without control or 100% inspection. The 100%

inspection arises when the minimum of TC 1 is less than

TC 0. The type of the chosen acceptance sampling proce-

dure becomes irrelevant.

3. Numerical Example

Example 1

Assume that k = 5, = m = 10, 2 = 0.25, C r = 2, and

C i = 0.1. Let N = 2,000 and p L = 0.01 We would like to

find Dodge-Romig [3] AOQL SSP by variables which

minimizes the expected total cost per unit. By solving

Eq. (4), we obtain the z * = 1.3269 which minimizes

TC 1. The optimal inspection policy is to do 100% in-

spection, because the minimum of TC 1 (= 1.0501) is

less than TC 0 (= 1.25). We need to set LSL = 9.335 and

USL = 10.665 to screen the product and ship the product

between the specification limits to the customers.

From Chen and Chou’s [25] model, we also obtain

the 100% inspection policy for the above numerical ex-

ample. Our solution is the same as that of Chen and

Chou’s [25] Dodge-Romig LTPD SSP by variables. Both

of them adopt the 100% inspection for screening the

product. From Chen’s [24] model, the optimum solution

of Dodge-Romig [3] AOQL SSP by attributes is the ac-

ceptance number c* = 0, the sample size n* = 36, and the

coefficient of specification limits z * = 1.3269. Its inspec-

tion policy is different from ours for the above numerical

example.

Example 2

The numerical data is the same as example 1 except

for C i = 0.35. By solving Eq. (4), the minimum TC 1 =

1.30. Because the minimum of TC 1 is less than TC 0 (=

1.25), the optimum policy is to adopt 0% inspection (ac-

ceptance without control).

4. Conclusions

According to Kapur and Wang [18], if we have spec-

ification limits and a process is under control but not ca-

pable of meeting specifications, then inspection in an

on-line quality control system may be a short term ap-

proach to reduce the variance of the units shipped to the

customer. Kapur and Wang [18] have addressed the cost

model which minimizes the loss of society including the

loss of producer (e. g., the inspection cost and the scrap

cost) and the loss of customer (e.g., the loss due to varia-

tion).

This study is an extension of Kapur and Wang’s

[18] work and the proposed model is a generalization of

Kapur and Wang’s [18] one. The optimum inspection

policy is either acceptance without control or 100% in-

spection. The 100% inspection arises when the mini-

mum of the expected total cost per unit for 100% in-

spection is less than the expected total cost per unit for

0% inspection. The type of the chosen acceptance sam-

pling procedure becomes irrelevant. Further study will

extend to an integrated model of Bayesian SSP for vari-

ables and specification limits.

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Economic Design of Dodge-Romig AOQL Single Sampling Plans by Variables with the Quadratic Loss Function 317

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Dodge-Romig LTPD Single Sampling Plans for

Variables Under Taguchi’s Quality Loss Function,”

Total Quality Management ,Vol.12,pp.511 (2001).

Manuscript Received: Sep. 16, 2004

Revision Received: Dec. 16, 2004

Accepted: Apr. 27, 2005

318 Chung-Ho Chen

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