Optimal Tolerance Design by Response Surface Methodology

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int . j. prod . res ., 1999, vol . 37, no . 14, 32753288

Optimal tolerance design by response surface methodology

ANGUS JEANG {

Response surface methodology (RSM) is applied to data analysis for experi-mental models to determine the optimal tolerance design in an assembly. RSMis a combination of mathematical and statistical techniques, which providesdesigners with not only optimal tolerance values but also the critical componentsof an assembly. This feature is very important during the design activities as itenables designers to have feedback and suggestions for design improvement. Todevelop an economical and quality product, the response variable is the total cost

which consists of quality loss and tolerance cost in this study. Most of the litera-ture on this subject assumes that the assembly function is known before a toler-ance design problem is analysed. With the current development in CAD(computer-aided design) software, design engineers can proceed with tolerancedesign problems without knowing assembly functions in advance. In this study,the Monte Carlo simulation is employed using VSA-3D/Pro software to obtainexperimental data, followed by RSM to optimize and analyse computer results.Consequently, a tolerance design for quality improvement and cost reduction canbe achieved for any complex assembly at the early stages of design.

1. IntroductionQuality engineering employs one of the robust design methods, the experimental

design approach, to improve product quality by reducing the e ects of variation(Montgomery 1991). Design by experimental methods plays a major role in engin-eering design activities in which new products are developed and existing ones areimproved. These experimental methods have broad applications in many industries.The purpose of experimentation is to develop an appropriate product design whichwill enhance functionality, quality, cost e ectiveness, and which will shorten designand development time. Quite often, optimal values exceed available experimentalresources; therefore, the method adopted for the experimental approach must becapable of obtaining a better understanding of the entire feasible range. In addition,it is more convenient to proceed with the design process if a mathematical relation-ship can be built between particular quality characteristics and the set of designfactors. Generally, if the relationship is too complex or physically unknown, anexperimental approach is necessary. Furthermore, the dispersion of particular qual-

ity characteristics for given design inputs needs to be known, because decisions madeduring design activities are dynamic and uncertain in nature (Myers 1991). As isknown, response surface methodology (RSM) is a combination of mathematical andstatistical techniques which can resolve the arguments mentioned above. Also, thismethod has the ability to produce an approximating function using a smaller number

International Journal of Production Research ISSN 00207543 print/ISSN 1366588X online # 1999 Taylor & Francis Ltdhttp://www.tandf.co.uk/JNLS/prs.htm

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Revision received December 1998.{ Department of Industrial Engineering, Feng Chia University, Taichung, Taiwan,

Republic of China. e-mail: [email protected]


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of experimental runs (Myers and Montgomery 1995). Therefore, this study intro-duces RSM to ll the need for solving the tolerancing problem.

The matter of tolerancing a single-component product is not di cult. One onlyneeds to look at the process distribution for the component to determine the propertolerance. However, a nished product consists of many components created bydi erent processes; thus, the question becomes how to determine which combinationof component tolerances is best. Therefore, in addition to tolerance design for asingle component of a product, most design problems should allocate componentstolerances so that the assembly dimensions of a nished product fall into acceptablequality and functionality ranges. The assembly dimensions of a nished product area combination of dimensions of several components from assembling chains. Thiscombination of dimensions causes the overall assembly dimensions to vary within adistributed range. Tolerance analysis of assembly dimensions relates the variation of

total assembly tolerances (resultant tolerances) to the variation of component toler-ances (Chase et al . 1990).

To analyse a tolerancing problem properly, functional relationships betweenassembly dimensions and component dimensions should be identied and knownprior to the analysis. These functions will be referred to as assembly functions intolerance design. However, these functions are usually not known, are di cult toobtain, or are in very complex form. In addition, designers prefer to have as manyfeasible designs as possible to allow for changes as the design team encounterscomplexities (Andreasen and Hein 1987, Gross 1989). Hence, it is necessary todene various assembly functions before conducting tolerance analysis. Thisbecomes an impractical task due to the hundreds or even thousands of calculationsneeded for obtaining assembly functions to process design activities. Fortunately,with recent developments in software, these tasks are workable. As is known, manyproducts are now routinely designed with the aid of computer models (William andJerome 1991). These models sometimes replace physical experiments, reducing thecost of experimentation, and perhaps more importantly, speeding up product devel-opment. In this study, VSA-3D/Pro model, a three-dimensional tolerance analysisintegrated with Pro/E, is used to analyse the tolerancing problem (VSA-3D/Pro1996). Given the input which consists of component tolerances, the software gen-erates experimental assembly data through computer simulation. Then, the RSMapproach is employed to nd the optimal tolerance design by means of statisticalanalysis and mathematical optimization. Thus, a tolerance design system via com-puter simulation becomes a practical tool.

This paper is divided into the following sections: section 2 contains relevantbackground information; section 3 provides an application example to demonstrate

the presented approach; section 4 is a discussion; and section 5 contains the conclu-sions.

2. BackgroundBefore developing the model, it is necessary to introduce relevant background

information regarding tolerancecost function, quality loss function and RSM.

2.1. Tolerancecost functionManufacturing cost usually increases as the tolerance of quality characteristics in

relation to the ideal value is reduced, due to the need for more rened and preciseoperations, while the acceptable ranges of output are reduced. Conversely, large

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tolerances are less costly to achieve as they require less precise manufacturingprocesses; but they usually result in poor performance, premature wear and partrejection. Various tolerancecost functions for machining are presented in Chaseet al .s work (1990). One of the examples is C M t a be ct . The parameters inC M t are found from a regression analysis based on empirical data.

2.2. Quality loss functionVariability in the production process is unavoidable due to inconsistency in tool,

workpiece, material and process parameters. Taguchi (1989) suggests that given anideal target value, an evaluation function associated with deviations from the targetcan be developed. In this study it is referred to as the quality loss function. This lossfunction is a quadratic expression for measuring the cost of the average value versusthe target value and the variability of product characteristics in terms of the mone-tary loss due to product failure in the eyes of the consumers. The loss function is:

L X K X T X 2: 1

This loss function represents a quality loss at one particular quality value X. Theexpected value of loss function, L (X), can be written as:

E L X K U X T X 2

2X : 2

Equation (2) contains two components. The rst term, U X T X 2, is the result of

the average value of X, U X , from the target value, T X . The second term, 2X , is thevariance of X.

2.3. Response surface methodologyThis study uses RSM because of its ability to produce an approximate function

using a smaller number of experimental runs. RSM is essentially a combination of mathematical and statistical techniques used in the empirical study of relationshipsand optimization, in which a large number of variables in the system inuence adependent variable or response (Myers and Montgomery 1995). Usually this rela-tionship is too complex or unknown, and the use of an empirical method is necess-ary. The RSM solves this problem. It is assumed that the experimenter is concernedwith a system involving some response variable Y which depends on the input vari-ables Xi . It is also assumed that Xi is continuous and controllable. The functionalrelationship between the response and the levels of inputs can be written as:

Y RX1; X2; . . . ; Xn : 3

The function is unknown and perhaps complicated. So, the rst step in RSM is tond a suitable approximation R by a low-order polynomial in some region of theindependent variables. If the approximate function is linear in the variables, a rst-order polynomial can be used and written in terms of the design variables.

Y a0 a1X1 a2X2 anXn: 4

Otherwise, a second-order polynomial can be used. This determines whether or not amechanistic model for such a relationship is known.

Y a0 X

n

i 1a i Xi

Xn

i 1bi X2i

Xn

i 1

Xn

j 1cij Xi X j : 5

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The frequent use of second-order polynomial models is justied by the fact that theyreect the non-linear behaviour of the system (Montgomery 1991, Myers 1991).Experimental designs for tting a second-order response surface must involve atleast three levels of each variable so that the coe cients in the model can be esti-mated. Rotatability property is desirable for response surface models because theorientation of the design with respect to surface is unknown. Hence, the orientationof the design is an important factor with regards to the response surface, and willa ect the collection of data and the tting of the response face. There are twoexperimental designs in the class of 3 k factorial design that can be used for ttingthe second-order model to response surfaces. These designs are the central compositedesign (CCD) and BoxBehnken design (BBD) (Myers and Montgomery 1995).Both designs are a fraction of 3 k factorial design, which are rotatable. This propertyis important for the experimental design of three factorial 3k design. This study

uses the BBD because it allows e cient estimation of the coe cients in equations (4)and (5). Using this experimental design, the levels of each factor are assumed to beequally spaced. The least squares estimate is used to estimate the coe cients inapproximating polynomials. The response surface analysis then proceeds in termsof the tted surface. For the purpose of analysis, we often plot the contours of theresponse surface as shown in gure 7 to visualize the shape of the three-dimensionalresponse surface. Each contour represents a specic value for the response Y undercombinations of the levels of selected factors. If the tted surface is an adequateestimation of the true functional relationship, then the analysis of the tted responsewill be nearly equivalent to the analysis of the studied problem. The eventual goal of RSM is to determine optimal factor levels in the system. The general proceduretaken in the development of a function relationship with RSM can be stated asfollows:

(1) determine the mathematical model;(2) set up the experimental design and perform the experiment;(3) estimate the coe cients in the mathematical model; and

(4) analyse the results.All the costs incurred in a product life cycle can be divided into two main

categories: manufacturing costs which occur before the product is sold to thecustomer; and quality loss which occurs after the product is sold (Phadke 1989).A loose tolerance (low manufacturing cost) indicates that the variability of productquality characteristics will be great (high quality loss). On the other hand, a tighttolerance (high manufacturing cost) shows that the variability of product qualitycharacteristics will be small (small quality loss). Hence, there is a need to adjust the

design tolerances to reach an economic balance between quality loss and manu-facturing costs for product tolerance design (Jeang 1994, 1995, Cheng andMaghsoodloo 1995). Therefore, the response variable in this study is the sum of tolerance cost and quality loss. The total cost function is:

TC e X

q

r 1kr U er T r

2 2er

Xm

j 1C M tej : 6

where: m is the total number of components from q assembly dimensions in anished product; U er and er are the r-th resultant dimensions and the r-th resultantvariances of statistical data from the e-th experimental results; T r is the design target

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value for the r-th assembly dimension; and the tolerance costs C M tej are based onthe j -th component tolerance tej established in the e-th experiment.

There are two types of tolerance: dimensional tolerance and geometrical toler-ance (Ligget 1993). Dimensional tolerance is dened as the permissible variation inthe dimensions of a part, e.g. the features of height, depth, diameter and angles.Generally, the bilateral limits of tolerance specication with equal values of plus andminus are applied to these features. However, geometrical tolerances are either atolerance band between two parallel lines or between two parallel surfaces, or athree-dimensional zone shaped like its feature. Basically they are used to controlthe form, prole, orientation, location and run-out of a product. For convenience,the maximum acceptable specication in the type of dimensional tolerance isexpressed as tU ; similarly, the maximum allowable band or zone in the type of geometrical tolerance is referred to as tD . Because the rotatability property is of

critical importance for response surface models, the levels of each factor in experi-mental design must be established in an equal space. That is, the tolerance tej estab-lished in the e-th experiment is equal to one of the levels: tU =3, 2tU =3 and tU for thedimensional tolerance; and tD =3, 2tD =3 and tD for geometrical tolerance.

3. An applicationThis application is related to motor assembly which consists of an x-base, crank,

shaft and motor base. Figures 16 are graphic representations of the motor assembly

with dimensioning and tolerancing schemes. Table 1 provides some relevant infor-mation for these gures. The ordering number in the rst row of table 1 is also givenin gures 26 for the purpose of easy association. The objective is to determine anappropriate tolerance allocation so that there is su cient clearance between thecrank and x-base, which is depicted in gure 1. As indicated in table 1, x-baseatness, motor base atness, motor shaft size and motor shaft perpendicularityare the features which have an e ect on clearance measurement. Hence, q 1 andm 4 in this example. For convenience in the presented tolerancing analysis, these

four features will be called factors X1 (motor shaft size), X2 (motor shaft perpendi-cularity), X3 (x-base) and X4 (motor base atness). The number of levels for each


Figure 1. A motor assembly drawing.


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factor is three. Table 3 shows the factors and levels of an experiment with 27 runs.Each run is repeated 400 times to obtain adequate samples for accurate results(Evans 1975). The total cost which consists of quality loss and tolerance cost isconsidered as a response value for the presented tolerance analysis. These two costfactors will be dened in the following illustration. The value of K 1 is the loss due to

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Figure 2. X-base.

Figure 3. Motor base.


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failure, divided by the functional limits of the product (Phadke 1989). K 1 18 500 inthis example. U e1 is the process mean and 2e1 is the process variance for the e-thexperimental data which is summarized in table 3. T 1 is the target value for the

dimension measurement at clearance, which is 0.89 cm. C M

tej

is the tolerancecost in dollars, where tej is the tolerance level for factor j under experiment e, andthe subscript j represents factors 1, 2, 3 and 4, respectively. These tolerance costs areprovided in table 2. The combinations for various tolerance levels established inexperiments are shown in table 3. The total cost for the e-th experiment, TC e, canbe expressed as k1 U e1 T 1

2 2e1

P 4 j 1 C M tej . The results are also shown in

the last column of table 3. They are considered to be response values in the RSMapproach. To determine the relative magnitude of the e ect of each factor on theresponse values, TC e, the F values are used to rank the factors in order of importance(Montgomery 1991). The results are shown in table 4. The larger the value of F , themore important that factor is in inuencing the response values, TC e, in an assembly


Figure 4. Shaft.


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design. In studying TC e, the e ect on factors 1, 2, 3 and 4 is signicant at the 5%condence level. Hence, all the factors should be carefully monitored to reducethe total cost, TC , with special attention to factor 3 (x-base). In case a designimprovement is needed, X3 is the rst priority for attention. By referring totable 4, the approximating function is 821 :9010 214:3417X1 1105:7800X2 9048:5867X3 1468:5458X4 1239:8500X21 6813:2000X 22 42 633:0000X23 9412:8125X24 . The optimal decision in tolerance allocation for factors 1, 2, 3 and 4is 0.141 81, 0.080 894, 0.098 334 and 0.064308 cm, respectively. Total cost is $228.206with a standard deviation of 0.603 55. Then, the predicting process should result inan estimation of the probability distribution of the variable TC . This permits risk tobe objectively incorporated into the decision-making process. It may not be easy tosolve the above systems manually; therefore, computer soft codes, e.g. SAS or S ,can be applied to nd optimal solutions using the RSM approach without di culty.Consequently, via computer simulation and the RSM approach, a robust tolerancedesign resulting in high quality and cost-e ective products can be achieved for anyassembly during the early stages of design.

The Taguchi-type experiment conducts matrix experiments using specialmatrices, called orthogonal arrays, allowing the experiments to be performed

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Figure 5. Motor.


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e ciently (Phadke 1989). For example, only nine experiments are needed forthe above application. However, the trade-o between approaches obtained byRSM or the Taguchi method is the degree of accuracy required, and the time ore ciency of the experiment involved. If time or e ciency is concerned, as is oftenthe case during initial design evaluation and prototyping, a technique, e.g. theTaguchi experiment can be considered. When more specic product designs areto be evaluated, accuracy may be necessary and the technique of RSM would beconsidered.

4. DiscussionBy simply inputting designable engineering parameters and parameters

representing manufacturing process conditions, the computer model generates aproducts quality characteristics. Then, a standard statistical analysis is performedbased on these computer outputs. To improve quality, one must choose the


Figure 6. Crank.


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designable engineering parameters which ensure that the product qualitycharacteristics are uniformly good in product performance. These computermodels sometimes replace physical experiments, reducing the cost of experi-mentation, and more importantly, speeding up product development.

An assembly is designed to full certain mechanical functions. Themanufacturing process for assembling products consists of putting together thevarious components and subassemblies to form a nished product which hasa certain expected functionality. The accurate analysis and proper allocationof tolerances among the assembly components are important to ensure thatfunctionality and design quality requirements are met. Despite the fact thatmany CAD softwares are currently available to assist designers, the tasks of component tolerance allocation, conrmation and analysis are still performedmanually through an interactive mode with the graphic system. For better results,

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Figure 7. The surface and contour plot of TC on component tolerances t1 and t2.


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these manually performed tasks may all be completed through the integration of CAD softwares, tolerance analysis and allocation model in one system.With an integration of VSA-3D/Pro and the RSM approach, component

tolerance is allocated optimally by an approximated function, and suggestionsfor possible design improvement are obtained by referring to the results of thestatistical analysis. Should the results be unsatisfactory, the engineer can alterthe component tolerance and assembly tolerance, and proceed using thepresented computer model again. All these activities can be accomplished verye ectively under the current circumstances. Furthermore, potential problemswhich might be expensive to change along the product life cycle can beeliminated.

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Factor i Lower level Middle level Upper level

X1 $18.065 (0.100) $13.626 (0.150) $12.815 (0.200)X2 $35.178 (0.050) $24.681 (0.075) $21.903 (0.100)X3 $279.612 (0.050) $170.394 (0.075) $108.574 (0.100)

X4 $29.874 (0.040) $19.622 (0.060) $17.983 (0.080)Table 2. Tolerance costs for factors X1, X2, X3 and X4 at various levels.

Expt.number Mean Std Total

e te1 te2 te3 te4 U e1 2e1 TC e

1 0.150 0.075 0.100 0.080 0.8884 0.0588 228.9252 0.150 0.075 0.100 0.040 0.8884 0.0582 239.5173 0.150 0.075 0.050 0.080 0.8892 0.0370 361.2714 0.150 0.075 0.050 0.040 0.8893 0.0369 373.0205 0.150 0.100 0.075 0.080 0.8887 0.0477 266.0736 0.150 0.100 0.075 0.040 0.8888 0.0475 277.6057 0.150 0.050 0.075 0.080 0.8889 0.0465 277.2438 0.150 0.050 0.075 0.040 0.8889 0.0464 288.9639 0.150 0.100 0.100 0.060 0.8883 0.0591 228.449

10 0.150 0.100 0.050 0.060 0.8892 0.0377 361.10011 0.150 0.050 0.100 0.060 0.8885 0.0587 240.83612 0.150 0.050 0.050 0.060 0.8893 0.0363 372.45313 0.200 0.075 0.075 0.080 0.8887 0.0513 274.63314 0.200 0.075 0.075 0.040 0.8887 0.0512 286.33515 0.100 0.075 0.075 0.080 0.8890 0.0440 266.99316 0.100 0.075 0.075 0.040 0.8890 0.0439 278.72117 0.200 0.075 0.100 0.060 0.8883 0.0617 236.27718 0.200 0.075 0.050 0.060 0.8891 0.0423 369.88019 0.100 0.075 0.100 0.060 0.8886 0.0558 228.62720 0.100 0.075 0.050 0.060 0.8894 0.0329 362.03721 0.200 0.100 0.075 0.060 0.8886 0.0519 274.64822 0.200 0.050 0.075 0.060 0.8888 0.0507 290.88123 0.100 0.100 0.075 0.060 0.8889 0.0445 266.67924 0.100 0.050 0.075 0.060 0.8891 0.0435 278.31425 0.150 0.075 0.075 0.060 0.8888 0.0481 271.19226 0.150 0.075 0.075 0.060 0.8884 0.0465 268.42327 0.150 0.075 0.075 0.060 0.8896 0.0474 269.911

Table 3. Full factorial experiment design and response data.


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5. ConclusionsWith the RSM approach, critical component tolerances can be identied and

optimal component tolerance values can be determined based on a statistical analysisand mathematical optimization. This approach enables designers to have immediatefeedback and suggestions for design improvement. Using the presented computermodel, a high quality, cost-e ective assembly design can be achieved during the earlystages of design.

AcknowledgementsThis research was carried out at the Department of Industrial Engineering at

Feng Chia University in the Design, Quality and Productivity Laboratory (DQPL),with support from the Engineering School at Feng Chia University, the EducationDepartment of the Republic of China, and the National Science Council of theRepublic of China under grant no. NSC 88-2213-E-035-022. I would like to expressmy deep appreciation to the editor and referees for their suggestions in the revisionof this paper. I would also like to thank my research assistants, Mr Jinny-ShoungFang and Mr Eric Liu, graduate students in the I.E. Department.

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Optimal Tolerance Design by Response Surface Methodology