# Taguchi’s Loss Function Applied to Economic Process Control Essay

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Get Access**TAGUCHI’S LOSS FUNCTION APPROACH TO ECONOMIC DESIGN OF RECTIFYING SINGLE SAMPLING Plan BY VARIABLE WITH ASSYMETRIC QUALITY COST.**

**Abstraction**

*The quadratic Taguchi ( quality ) loss map has been given broad attending in the application of statistical ( Economic ) procedure control, which in most instances is ever aimed at obtaining optimum scene for a procedure. This article is an extension of Chung-Ho Chen ( 2005 ) and focuses on the usage of Taguchi loss map with abbreviated normal chance denseness map in planing optimum review policy and economic specification bounds for rectifying individual sampling program by variable with unequal quality costs (**) .*

**Keywords:**Rectifying Single Sampling program ( RSSP ) , Taguchi’s Loss map,

Target value, Specification limits, Cost of bit () , Cost of rework () , Cost of Inspection.

**Introduction**

Acceptance sampling is the procedure of indiscriminately inspecting a sample of goods and make up one’s minding whether to accept the full batch based on the consequences of assessment of the sample. It determines whether a batch of goods should be accepted or rejected.

In the current fabrication universe, the way from natural stuffs to i¬?nal merchandise frequently take topographic point over multiple subdivision of the makers and across multiple provinces of the state.

A fabrication company may probably sub-contracted different parts to different sub-contractors. For case, the assemblying of a merchandise might take topographic point in yet another works and the wadding and transportation, from yet another location. In order to guarantee a certain quality degree, companies use review at the different supply concatenation phases.

Sampling is used everyplace in quality assureance. It should be noted that review at different supply concatenation involve trying some of the points produced. Even if all points produced or services performed are inspected for conformity to specifications, we are still trying from the existence of all similar points or service that have been or will be produced.

Rectifying review program arises when a batch is rejected and so necessitate 100 % review and the recognized batch merely take trying inspection.The statistical theory of trying review by variables has been treated extensively in many standard statistical quality control text editions and by many writers such as Lieberman and Resnikoff [ 1955 ] , and D. C. Montgomery ( 1991 ) .

Economic design on the type of review is to choose the type of review that will minimise the amount of production costs, credence costs and unsatisfactory merchandise costs influenced by the determination sing review. Certain conditions are apparent that are favourably to no ( 0 % ) review, 100 % review and to trying review.

The traditional construct of conformity to specifications is that points meet the specification bounds. Prior to Taguchi ( 1986 ) , manufacturers of merchandise believe that any point produced within specifications is every bit good as any other. Taguchi argues that best quality is achieved by minimising the divergence from the mark value. The cost of quality so can be measured as a map of procedure divergence from mark value. The basic doctrine is that, whenever an point is produced, a cost is imposed on the society. Part of that cost is borne by the manufacturer and portion is borne by the consumer as a consequence of utilizing the merchandise or services. If these costs are plotted as a map of “quality” , manufacturer costs tend to increase with increased “quality” ; client costs tend to diminish because of greater efficiency, less breakdowns etc.

Harmonizing to Chandra ( 2001 ) , a merchandise is said to be scrapped if its quality features under survey is less than the lower specification bound. Hence, Cost of bit is the loss incurred when an point is produced below the specification bounds. It is relative to the quality degree of the point produced and can be obtain as follows ;

Where ;

is the cost of bit

is the cost of bring forthing the point.

is any sum received by disposing the point as a “secondary” merchandise.

Leavenworth and Grant ( 1999 ) stated that ;*the higher the per centum of unsatisfactory merchandise ( bit ) submitted and the higher the unsatisfactory merchandise cost per unit of such merchandise, the more favorable the status for 100 % review as compared with no ( 0 % ) review.*This can be shown by the figure below ;

Figure 1.0 Relationship between ETC* and.

So besides from Chandra ( 2001 ) , merchandise is said to be reworked if its quality features under survey is greater than the upper specification bound. Hence, the cost of rework is the excess payments made to do unacceptable merchandise acceptable.

A merchandise is inspected if 100 % or trying review is required. Hence, cost of review is the clip that the review forces spend in measuring the proficient public presentation of the merchandise ( A.V. Feigenbaum ( 1983 ) ) .

Leavenworth and Grant ( 1999 ) stated that ; “the*higher the unit cost of review, so the less the effectivity of 100 % review and this give a more favourable status for no inspection.”*This can be represented by the figure below.

Figure 2.0 relationship between ETC* and.

In this paper, an extension was attempted to Taguchi loss map attack to economic design of rectifying individual sampling program by variable with asymmetric quality cost. () . Just like Chung-Ho Chen ( 2005 ) , the economic specification bounds and optimum review policy was obtained.

**MATHEMATICAL MODEL AND ASSUMPTIONS**

Recall, for a variable sampling program with two specification bounds and known discrepancy ; the batch is rejected ifand if, hence,. Where, m*=*is the allowance on either side of the nominal size, therefore USL =+and LSL =, where.

Harmonizing to Leavenworth and Grant ( 1999 ) , no economic comparing of alternate programs for credence review is possible without doing premises sing the quality of the merchandise submitted for credence and the decision of any survey will depend on these premises. Hence, for explicating the mathematical theoretical account, the undermentioned premises are made ;

- The quality features y follows a normal distribution ( Truncated normal distribution ) with parametric quantities meanand discrepancy
^{2}. - A loss is incurred when the quality features y, deviates from the mark value T, and this loss is governed by the quadratic loss map.
- The quality features y is nominal-the-best type.
- The procedure mean, is centered at the mark value T.
- Cost of bit () is non equal to Cost of rework () .
- It is assumed that when Y & lt ; LSL, the loss L ( Y ) = C
_{s}, and when Y & gt ; USL, the associated loss L ( Y ) = C_{tungsten} - Discrepancies of the two groups are equal, therefore the demand for pooled discrepancy.

From Chandra ( 2001 ) , the general signifier of the loss map for unequal quality cost is given as ;

Now the expected value of the entire loss map is defined as ;

Where ;

( Constant )

T is the mark value,is the coefficient of the quality loss for observations,is the coefficient of the quality loss for observations,is the coefficient of the specification bounds,

are p.d.f of a abbreviated normal random variable for observationsandseverally.

are the several lower and upper specification bounds for testing the merchandise.

is the coefficient of the specification bounds.

is the cumulative distribution map for the standard normal random variable with denseness map.

.

**EXPECTED MATHEMATICAL DECISION FUNCTION**

When 0 % review is performed, the expected entire cost per unit is due to fluctuation in the procedure. Hence the expected sum cost per unit for 0 % review is ;

But for 100 % review, we have to take into consideration the expected entire loss per unit for 100 % review, expected cost per unit due to trash, expected cost per unit due to make over and expected cost per unit due to review. Hence, the expected entire cost per unit for 100 % review is obtained as ;

Whereis the expected sum cost per unit for no ( 0 % ) review,is the expected sum cost per unit for 100 % review,is the bit cost per unit,is rework cost per unit,is the cost of review.

is the estimation of the pooled discrepancy.

Chung-Ho Chen ( 2005 ) shows that a alone minimal value exist foron changing, which implies an optimum value ofat the lower limit cost for. This can be obtained by the usage of MATLAB package in minimisingto obtain the optimum.

To obtain the optimal review policy, the rejected batch demand to take the 100 % review and the recognized batch merely takes trying review. The ATI denotes the mean entire review per batch. Leavenworth and Grant ( 1999 ) show that in analysing and measuring assorted trying program that minimizes the ATI, it is convenient to province the job in footings of Average Fraction Inspected ( AFI ) which is obtain as ;

Where ;

N is the sample size, N is the batch size and P_{a}is the chance of credence and

From Chung-Ho Chen ( 2005 ) , for& gt ; 0,, eitheror. If the minimal AFI is obtained and correspond to a brace of values, and, so we have the undermentioned decision as respect the optimum review policy ;

- If the min AFI = 0, and corresponds to min m & gt ; 0 and n=0,

Hence,

Therefore,

But AFI = 0, therefore we have 1-P_{a}=0

Implies, and this corresponds to no ( 0 % ) review.

- If min AFI = 1, and corresponds to minand n=N, so, ATI=N

And this besides correspond to n=N, i.e. 100 % review.

Hence, the optimal review is either 0 % review or 100 % review. The economic specification bounds is so specify through the optimum m value that minimizes, which will be used to test the merchandise that will be shipped to client when 100 % review is adopted.

**Numeric EXAMPLE**

Assuming the undermentioned parametric quantities. where, T is the mark value,is the procedure mean,is the coefficient of the quality loss for observations,is the coefficient of the quality loss for observations,is the coefficient of the specification bounds,is the bit cost per unit,is rework cost per unit,is the cost of review. By minimising, we obtain. The optimal determination is to execute 100 % review because min( 0.287 ) & lt ;and the economic specification bounds to test the merchandise is set as.

By increasing the cost of review in the illustration above every bit mentioned by Grant and Leavenworth, we observe a favorable status for 0 % review. i.e. presuming all the parametric quantities above are the same exceptwe obtain min, therefore 0 % review is performed.

**Decision**

This survey is a reappraisal of an economic design of Chung-Ho Chen ( 2005 ) which was a generalisation of Kapur and Wang’s survey.

An extension of the survey was attempted to when the cost of non run intoing the mark value is non changeless and when the bit and rework cost per point are non equal i.e..

The optimal review policy is either no ( 0 % ) review i.e. credence without control or 100 % review. The 100 % review arises when the lower limit of the expected entire cost per unit for 100 % review () is less than the expected sum cost per unit for 0 % review () and no review if otherwise.

implies the cost ( loss ) per unit due to fluctuation in the procedure.

implies the loss per unit due to non run intoing the mark value, unsatisfactory merchandise ( scrap or rework ) cost and cost of review

The economic ( optimum ) specification bounds are defined through the optimum value ofthat minimizes, which is so used to obtain Lower and upper specification bounds.

Since this is besides an economic design, the type of rectifying individual sampling program becomes irrelevant Chung-Ho Chen ( 2005 ) .

**REFRENCES**

- Bisgaard, S. Hunter, G. and Pallesen, G. ( 1984 ) :

Economic Choice of Quality of Manufactured Product. TECHNOMETRICS vol. 26 No 1: 9 – 18.

- Chandra, M. J. ( 2001 ) :

Statistical Quality Control. CRC imperativeness LLC: 52 – 79.

- Chung-Ho Chen ( 2005 ) :

Economic Design of Dodge-Romig AOQL Single Sampling Plans by Variables with the Quadratic Loss Function. Tamkang Journal of Science and Engineering, Vol. 8, No 4: 313 – 318.

- Feigenbaum, A. V. ( 1983 ) :

Entire Quality Control ( 3^{rd}edition ) . McGraw-Hill: 109 – 119.

- Feng, Q. and Kapur, K. C. ( 2006 ) :

Economic development of specifications for 100 % review based on asymmetric quality loss maps. IIE Transactions. 38: 659 – 669.

- Grant, E. L. and Leavenworth, R. S. ( 1999 ) :

Statistical Quality Control ( International edition ) . McGraw-Hill: 631 – 643.

- Lieberman, G. J. and Resnikoff, J. ( 1955 ) :

Sampling Plans for Inspection by variables. Journal of the American Statistical Association, Vol. 50, issue 270: 457 – 516.

- Montgomery, D. C. ( 2009 ) :

Introduction To Statistical Quality Control, ( 6^{Thursday}edition ) John Wiley, New York, NY, U.S.A. 631 – 667.

- Osanaiye, P. A. ( 1989 ) :

An Economic pick of trying review programs under changing procedure quality. Appl. Statist. Vol. 38, No 2: 301 – 308.

- Ross, S. M. ( 2004 ) :

Introduction to Probability and Statistics for Engineers and Scientists ( 3^{rd}edition ) . Elsevier Academic imperativeness.

- Tagaras, G. ( 1992 ) :

Economic Acceptance Sampling by Variables with Quadratic Quality Costs, IIE Transactions.

- Wen-Hsien, T. ( 1998 ) :

Quality cost measuring under activity-based costing. International Journal of Quality & A ; Reliability Management, Vol. 15 No. 7: 719 – 752.

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