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int. j. prod. res., 1999, vol. 37, no. 17, 39954006

Statistical tolerance synthesis using distribution function zones

CHUCK ZHANG{*, JUN LUO{ and BEN WANG{

Tolerance is one of the most important parameters in design and manufacturing.Tolerance synthesis has a signicant impact on manufacturing cost and productquality. In the international standards community two approaches for statisticaltolerancing of mechanical parts are being discussed: process capability indices anddistribution function zone. The distribution function zone (DFZone) approachdenes the acceptability of a population of parts by requiring that the distributionfunction of relevant values of the parts be bounded by a pair of specied distri-

bution functions. In order to apply this approach to statistical tolerancing, oneneeds a method to decompose the assembly level tolerance specication to obtaintolerance parameters for each component in conjunction with a correspondingtolerance-cost model. This paper introduces an optimization-based statistical tol-erance synthesis model based on the DFZone tolerance specications. A newtolerance-cost model is proposed and the model is illustrated with an assemblyexample.

1. Introduction

Tolerance is one of the most important parameters in production and processdesign. It is dened as the maximum deviation from a nominal specication withinwhich the part is acceptable for its intended purpose. Tolerance is usually expressedas lower and upper deviations from a nominal value. Two types of tolerance areoften used: design tolerance and manufacturing tolerance. Design tolerances arerelated to the functional requirements of an assembly or of a component.Manufacturing tolerances are mainly devised as a process plan for fabricating a

part. Manufacturing tolerance must ensure the realization of design tolerance.Design tolerance has an impact on the manufacturing cycle time, quality and costof a product (Zhang and Wang 1993a).

Tolerance analysis involves the identication of related tolerances in an assemblyor a mechanism and the calculation of the total stack-up of these related tolerances.In tolerance analysis, basically, the process of tolerance accumulation is modeled andthe resultant tolerance is veried and checked against design specication. If designrequirements are not met, tolerances are adjusted and the stack-up is recalculated.

Tolerance synthesis is a process of allocating tolerance values associated with designrequirements in terms of functionality or assemblability among identied design ormanufacturing tolerances. Well-allocated tolerances can ensure that a product isproduced with high quality at low cost.

International Journal of Production Research ISSN 00207543 print/ISSN 1366588X online # 1999 Taylor & Francis Ltd

http://www.tandf.co.uk/JNLS/prs.htm

http://www.taylorandfrancis.com/JNLS/prs.htm

Revision received January 1999.{ Department of Industrial Engineering, Florida A&M University-Florida State

University, College of Engineering, 2525 Pottsdamer Street, Tallahassee, FL 32310-6046,USA.

* To whom correspondence should be addressed. e-mail: [email protected]

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A tolerancing problem can be solved with a conventional worst-case approach ora statistical approach. In case of mass production, product characteristics are moreoften statistically distributed than in a continuous manner, and the parameters ofstatistical distribution (e.g. mean, standard deviation) are the true quality indicators.Therefore, for mass production, it is more appropriate for a designer to use statisticaltolerance for each part dimension rather than a conventional worst-case tolerance.Statistical tolerances characterize the statistical distributions of part characteristicsin order to determine distributions of the assembly characteristic (Bjorke 1989).

In the tolerancing literature, a considerable amount of research has been con-ducted for a wide range of tolerance analysis and synthesis problems using variousapproaches (Zhang and Wang 1993b, Vasseur et al. 1997, Chase and Greenwood1988, Turner and Guilford 1992, Zhang and Wang 1992, Zhang 1996, Feng andKusiak 1997). An important aspect of tolerancing, statistical tolerancing, has also

received much attention. Statistical tolerance analysis was rst proposed byMansoor (1963) . His method is based on the assumption that the component dimen-sions follow normal distribution and their resultant assembly tolerance can beobtained by using the root sum of square (RSS) method. Parkinson later generalizedthis technique as

r Z

3

21

22

2n

q

If the tolerance range i; i 1; 2; . . . ; n was assigned to be equal to 3 times the

standard deviation of the normal distribution (i.e. Z

3), 99.73% of the resultantdimensions would fall in the range mr r=2, where mr is the mean of the resultantdimension. Bjorke (1989) developed a method similar to the above model based onbeta distribution. However, he assumed that the resultant dimension would follow anormal distribution. This assumption can cause inaccuracies if the number of com-ponents in the assembly is small. Chase and Greenwood (1988) introduced a uniedtolerance analysis method based on the estimated mean shift. This was accomplishedby selecting a shift factor fi for each component between 0.0 and 1.0. The resultingtolerance had the form

r Xni 1

fi i

Xni 1

1 fi2

2i

s

This method is eective in dealing with mean-shift, if the shift factor can be esti-mated accurately. However, if the component dimensions are not normally distrib-uted, this method will give inaccurate results. Turner and Guilford (1992) proposed avector space approach for tolerance specication and implemented the scheme

within a system called GEOTOL. This scheme can represent a few of the geometrictolerances such as location and orientation. However, analysis based on this schemeis computationally intensive and not readily adaptable for statistical toleranceanalysis. Varghese et al. (1996) proposed a statistical tolerance analysis approachusing the nite range probability density function and numerical convolution. Usingthis method, statistical analysis can be conducted with less computations and betteraccuracy.

OConnor and Srinivasan (1997) proposed a new statistical tolerancing scheme,based on the specications of tolerancing, using the distributed function zone

(DFZone). This scheme provides a promising solution to statistical tolerancing.The new approach has been applied to a case of statistical tolerance analysis

3996 C. Zhang et al.

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(OConnor and Srinivasan 1997, Srinivasan 1997). However, the statistical tolerancesynthesis using DFZone specication has not been reported.

In this study, an optimization based statistical tolerance synthesis model is pro-posed based on the DFZone tolerance specication. This model is demonstratedwith an assembly example. The following sections provide details of the modeland the example.

2 . M e th od o lo gy

2.1. Distribution function zone (DFZone)In 1974 the West German Standards Organization made the rst serious attempt

in the direction of codifying the statistical tolerancing by modifying the specicationof a worst-case tolerance interval. The designer could specify a central subinterval ofthe worst-case interval and a percentage. It was interpreted as requiring a population

of parts such that the values of all of the parts would lie in the worst-case interval.The percentage of values in the central subinterval would be at least the speciedpercentage, and the percentage of values in either of the remaining subintervalswould be no more than half the remaining percentage.

This means that in any acceptable population of parts, at least 86% of thediameters shall be within 10:14 0:03, at most 7% shall be within 10:14

0:03 0:05, and

at most 7% shall be within 10:14 0:05 0:03 . In addition, these parts shall be produced

under a state of statistical control.

To begin the development of the DFZone-based tolerancing model, severaldenitions should be recalled (OConnor and Srinivasan 1997). A distributionfunction, denoted DF, is a non-decreasing, right-continuous function Fmapping F : X 2 < ! Fx 2 0; 1 with limx! 1 Fx 0 and limx! 1 Fx 1.Distribution functions are of interest in probability theory, because for any realrandom variable X, the function dened for all X 2 < by Fxx Pr x Xf g is aDF, which contains all probabilistic information about x. Conversely, any DF is theDF of some random variables.

IfL and Uare two DFs with LX UX for all X, then we write L U. IfFis

a collection of DFs, then we denote by ZL; U; F the collection of DFs bounded byL and U, that is

ZL; U; F f 2 F : L f Uf g

We refer to ZL; U; F as the DFZone in F bounded by L and U. The choice of Frestricts attention to a relevant family of distributions. When F is the collection ofDFs, we simplify the notation to ZL; U, and refer to ZL; U as the DFZonebounded by L and U.

The tolerance specications considered above are readily described in theseterms. If the worst-case tolerance interval is t1; t2, the central subinterval iss1; s2, and the percentage is p, then let

LX

0 for X < s21 p

2for s2 X t2

1 for t2 X

and UX

0 f or X < t11 p

2f or t1 X s1

1 f or s1 X

8>>>:

8>>>:

Now if x is the random variable of values of the part and Fx is its DF, then the

tolerance specication is that Fx 2 ZL; U. This can also be expressed graphicallyby saying that the curve of Fx lies between the graphs of L and U. Figure 2

3997Statistical tolerance synthesis

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depicts the tolerance specication for the example in Figure 1. In principle, theDFZone model allows the unilateral tolerance specications. However, the bilateralspecications with equal bounds are preferred for the simplicity of specication andmodeling. In the cases of unilateral tolerance specications, a conversion may beapplied to transform them into the bilateral specications.

The actual value of a diameter is denoted as x. Any population of diameterswhose distribution function falls within the shaded zone is acceptable. This

DFZone-based statistical tolerance specication is under consideration by theInternational Organization of Standards (ISO) to become an internationalstandard. Detailed descriptions of the DFZone are available in OConnor andSrinivasan (1997).

As stated earlier, tolerance synthesis is a process of allocating tolerance valuesassociated with design requirements among identied design or manufacturing tol-erances. The proposed DFZone standard mainly focuses on part level considera-tions. In manufacturing practice, a designer usually starts with an assembly level

design budget. The tolerancing problem is then tackled by several iterations oftolerance analysis, where part variations are composed to determine the assemblyvariation. In general, this can be extremely dicult, even in the case of traditionalworst-case tolerance synthesis, if the parts are linked in geometrically complex ways.However, if the parts are linked in a one-dimensional way, leading to the commonstack up analysis/synthesis and a linear gap function, the analysis is much simpler.Often this is the case, and when it is not, it is often still possible to obtain usefulresults by linearizing the problem. In statistical tolerance analysis and synthesis anadded complexity arises if the parts are statistically dependent. Again it is customary

to assume away the problem, and in fact it is often reasonable to do so by assumingstatistical independence (OConnor and Srinivasan 1997).


F 10.140.05 ST 0.03 P86%

Figure 1. DFZone syntax.

X

0.93

0.07

1

Pr{x < X}

10.09 10.11 10.17 10.19

Figure 2. DFZone semantics.

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2.2. Cost-tolerance modelIt is known that in manufacturing a higher precision level part (with tighter

tolerance) usually requires a higher manufacturing cost. There is a simple monotonicdecreasing relationship between manufacturing cost and tolerance in a certain range.Several models were reported to describe this cost tolerance relationship, such as theSutherland function, reciprocal function, reciprocal square function and exponentialfunction. Among them, the exponential function is found to be relatively simple andaccurate (Wu et al. 1988, Dong and Soom 1990). Its mathematical representation is

g Ae B 0 g0

a < < b

In this model A, d0 and g0 determine the position of the cost-tolerance curve,

whereas B controls the curvature of it (see Figure 3). These parameters can bederived using a curve-tting approach based on experimental data. a and bdene a region in which the tolerance is economically feasible (Zhang and Wang1993).

Based on the above understanding of the cost-tolerance relationship, a compre-hensive cost versus statistical tolerance model is developed for statistical tolerancesynthesis that relates manufacturing cost to yield, mean and process specicationranges. The new cost-tolerance model is dened as

Cost K1 K2P2

2 1

B1

D

where K1 is the coecient of xed cost; K2 is the coecient of cost related totolerance parameters; is the nominal dimension; Bis the worst-case interval dimen-sion; D is the central sub-interval dimension; P is the percentage of D.

As in the traditional cost-tolerance models, the parameters/constants in the newcost-tolerance model can be obtained with curve-tting/regression from the empiri-

cal or production data. In this model the manufacturing cost of each component inthe assembly is made of two parts. One is the xed cost and the other is the tolerance


Tolerance (d)

d0 da db

g0

A+g0

Cost (g(d))

Figure 3. A cost tolerance model.

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cost. The xed cost is related to machine set up, machine operation and other factorsthat are not determined by design parameters. The tolerance cost is mainly aected

by the tolerance parameters. According to the monotonic decreasing relationshipbetween manufacturing cost and tolerance, if the two tolerance limits in DFZonespecication become larger, the total manufacturing cost decreases. It is also obviousthat if a higher quality of product is required (P is larger), the manufacturing costwould increase.

This model is then veried by examining the relationship among cost and two ofthe three DFZone statistical tolerance parameters when the third is xed.

Figure 4 shows how the cost varies with the two tolerance limits when the prob-ability P of central sub-interval tolerance D is xed. The monotonic decreasingcost-tolerance relationship is obvious.

In gure 5 the central-limit tolerance D is xed. This 3D mesh shows that costand the worst-case tolerance B still keep the monotonic decreasing relationship.Cost will increase faster when a higher P is chosen.

A similar cost-tolerance relationship mesh exists when the worst-case toleranceB is xed and the other two tolerance parameters vary (see gure 6). The higher theprobability, the higher the cost and the larger the tolerance limit, the lower the cost.

By combining the cost-tolerance relationship and DFZone statistical tolerance

specication, the tolerance synthesis problem can be stated as follows: for anassembly design requirement given in the DFZone format, allocate the assembly


0.1 0.6 0.9 1.3 1.61.1

1.4

1.7

2

0

10

20

30

40

50

60

70

80

9080-90

70-80

60-70

50-60

40-50

30-40

20-30

10-20

0-10

Cost

D

B

Figure 4. Costtolerance relation (P is xed).

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

0.55

0.65

0.75

0.85

0.95

0

10

20

30

40

50

60

70

80

90

100

90-100

80-90

70-80

60-70

50-60

40-50

30-40

20-30

10-20

0-10

B

P

Cost

Figure 5. Costtolerance relation (D is xed).

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`budget tolerance to all the components of this assembly statistically and optimally,that is, the tolerance synthesis should result in a minimum manufacturing cost.

2.3. Optimization modelAs stated above, the tolerance synthesis problem using the DFZone specication

represents a well-dened optimization problem. A general problem formation isgiven in section 2.3.1. In section 2.3.2 a mathematical representation of the optimi-zation model is presented. A grinder head assembly example is used to illustrate theeectiveness of the method. Discussions and conclusions are given in sections 3 and 4.

2.3.1. Problem formationThe tolerance synthesis problem using the DFZone specication can be

formulated as follows.

Given: nominal dimension i of all the parts in an assemblyworst-case interval Bt of the critical characteristic in an assemblycentral sub-interval Dt of the critical characteristic in an assemblypercentage Pt of DImachine processing capability Idesign limit i

Find:

worst-case interval dimension Bi for component icentral sub-interval dimension Di for component ipercentage Pi ofDi for component i

2.3.2. Optimization modelThe tolerance synthesis problem under the DFZone specication represents a

well-dened optimization problem. Critical assembly tolerance design requirementsand machining capacities form the constraints of the problem. The tolerance par-ameters of each component in the assembly are the design variables to be optimized

so that the total manufacturing cost of the whole assembly is optimal.A mathematical representation of the optimization model is as follows:


0.1 0.4 0.6 0.7 0.9 1.1 1.3 1.5 1.6 1.8

0.55

0.65

0.75

0.85

0.95

0

10

20

30

40

50

60

70

80

90

100 90-100

80-90

70-80

60-70

50-60

40-50

30-40

20-30

10-20

0-10

P

D

Cost

Figure 6. Costtolerance relation (B is xed).

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Min Cost Xni 1

K1i K2iP

2i

2i 1 i

Bi1

i

Di

264

375 i 1; 2; . . . ; n

subject to:pi Pi 1; Di Bi

Bi i; Di i iXni 1

Di Dt

Xn

i 1

Bi Bt

0:51 Pi 0:51 Pt

In this model Pi, Di and Bi are the design variables to be optimized. pi is thepercentage design limit of each component. i is the machine capability. i isthe dimension design limit set by designers for each component. The rst group ofconstraints are derived from a part level DFZone. The relationshipi < Di < Bi < i must be ensured. The second group of constraints reects

DFZone relationships within an assembly-level DFZone.It has been proved that, for an assembly described by a linear gap function of

statistically independent values of certain dimensions on the constituent parts of theassembly, the composition of the DFZone of each component can be achieved underconvolution of these DFs (OConnor and Srinivasan 1997). The property of theresulted DFZone bounded by the constituent DFZones is preserved. For two suchDFs, the convolution can be expressed as follows:

F GX F; YG; Z

Y 2 JF

Z 2 JG

Y Z X

in which Fand Gare DFZones. F; Y and G; Z denote the values of the jump ofFat Yand Gat Z, respectively. Jis the resulting DFZone that is also a step functionwith nite jumps. In decomposing a desired DFZone, a reverse derivation procedure

is followed to obtain the constraints in the optimization model. For a DFZonedened by Dt, Bt and Pt, the properties of DFZones of its components can bepreserved if the following conditions hold true:

Xni 1

Di Dt

Xn

i

1

Bi Bt

0:51 Pi 0:51 Pt


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3. Example and results

The proposed model is illustrated with a tolerance synthesis problem of a grinderhead assembly. The technical drawings of the assembly and its components areshown in gure 7.

There are 19 components in the grinder head assembly. One of the crucialdimensions in the assembly is the gap between the cap (right) and the head bodyas it determines the proper sealing of the assembly. If it were too small, there wouldnot be sucient space for the seal installation. If it were too large, there could be aleak even after the seal is installed. Therefore the gap needs to be controlled in acertain range. A dimension chain is formed according to its technical drawing (see


(a)

gap

(b)

Figure 7. Example assembly: (a) component drawing; (b) assembly drawing.

(a)

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gure 8). The gap is specied in the DFZone-based statistical tolerance format. Anoptimizations procedure is then carried out to nd the DFZone statistical toleranceof each component.

To simplify the problem, symmetric tolerance zones are used. The input par-ameters of each component and tolerance design requirement of the gap are listedin table 1.

The example problem was solved using the Solver Tools provided in Microsoft

Excel. The tolerance allocation results are shown in table 2. Four tolerance alloca-tion results are listed. The rst three are randomly generated, non-optimum toler-ance allocation results which satisfy all the constraints. The last one is the optimumtolerance allocation result. It can be seen from the table that 13% of the manu-facturing cost reduction can be achieved when the optimum tolerance allocation isapplied instead of a non-optimum tolerance allocation (Tolerance Design #2).

In the optimum tolerance allocation result, the dimension of each component isspecied in a DFZone-based statistical tolerance format. For example, part 1 is

specied as 8 0:056 ST 0:043 P95:95%. According to the syntax of theDFZone specication, it indicates that a batch of part 1 is acceptable, if at least95.95% of the dimensions are within 8 0:043, at most 2.025% within 8

0:043 0:056 and at

most 2.025% within 8 0:056 0:043.

The optimum statistical tolerance allocation results can be used to assist inprocess/machine selection to produce components for an assembly. They will leadto the most economic manufacturing for component production.

4 . C on clu sion s

Statistical tolerancing has been a centre of extensive research for many years, dueto its importance in design and manufacturing. This paper has presented an


X4

d

X1

X2X3

X7

X6

X5

Figure 8. Dimension chain of the grinder head assembly.

Components m l K1 K2 pl l l

X1 8 5 4.00 0.85 8.20 0.01

X2 23 8 3.50 0.80 23.30 0.03X3 5 5 4.00 0.85 5.10 0.01X4 18 7 3.50 0.85 18.20 0.03X5 60 12 3.00 0.75 60.30 0.05X6 5 5 4.00 0.85 5.10 0.01X7 118 18 3.00 0.75 118.50 0.10

Gap design Pt Dt Bt Tol. Spec. 1 0.44 ST 0.34 P80%requirements 0.80 0.34 0.44

Table 1. Input parameters of the optimization model.

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approach and an optimization model to solve the statistical tolerance synthesis prob-lems based on the DFZone tolerance specication. A cost-tolerance model wasintroduced based on the new tolerance specication. The constraints in the optimi-zation model were derived from the relationship between the assembly-level DFZoneand the part-level DFZone. This toleancing approach and optimization model wereillustrated with a grinder head assembly example. Excel solver was employed as theoptimization tool for the test problem. A comparison between the optimum and anon-optimum tolerance allocations has revealed the eectiveness of the proposedstatistical tolerance allocation approach and the model. As the DFZone-based sta-

tistical tolerance specication is being considered by the International Organizationof Standards (ISO) to become an international standard, more research work can be


Design Nominal dim. Sub-interval Worst-case interval Percentagevalue m D-m B-m P

Design gap 1 0.340 0.440 80.00%

Component 1 8 0.040 0.050 97.35%Component 2 23 0.043 0.052 94.84%Component 3 5 0.040 0.050 93.60%Component 4 18 0.039 0.050 96.61%Component 5 60 0.047 0.057 92.36%Component 6 5 0.040 0.050 89.57%Component 7 118 0.040 0.080 87.60%

Tolerancedesign 1

8>>>>>>>>>>>>>:

Total manufacturing cost: 85.04 unit value

Component 1 8 0.034 0.050 94.30%

Component 2 23 0.035 0.049 99.00%Component 3 5 0.051 0.058 92.80%Component 4 18 0.042 0.053 98.61%Component 5 60 0.037 0.047 85.35%Component 6 5 0.041 0.055 90.41%Component 7 118 0.030 0.070 93.76%

Tolerancedesign 2

8>>>>>>>>>>>>>:


Component 1 8 0.040 0.056 94.25%Component 2 23 0.032 0.048 98.34%

Component 3 5 0.041 0.054 93.00%Component 4 18 0.044 0.056 98.61%Component 5 60 0.044 0.051 99.00%Component 6 5 0.042 0.050 90.54%Component 7 118 0.080 0.120 87.60%

Tolerancedesign 3

8>>>>>>>>>>>>>:


Component 1 8 0.043 0.056 95.95%Component 2 23 0.044 0.058 99.12%Component 3 5 0.042 0.054 92.61%Component 4 18 0.044 0.058 98.61%Component 5 60 0.044 0.061 100.00%Component 6 5 0.042 0.054 92.61%Component 7 118 0.080 0.100 83.57%

Optimal

design

8>>>>>>>>>>>>>:


Table 2. DFZone optimal design tolerances.

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expected in this area. The authors plan to conduct the following future researchwork: application of the proposed statistical tolerancing approach to the non-linear assembly; and renement and validation of the cost-tolerance model basedon DFZone tolerance specications.

References

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approach. ASME Transactions, Journal of Manufacturing Science and Engineering, 119,603610.

MANSOOR

, E. M., 1963, The application of probability to tolerances used in engineeringdesign. Proceedings of the Institute of Mechanical Engineers, 178, 2951.OCONNOR, M. A. and SRINIVASAN, V., 1997, Composing distribution function zones for

statistical tolerance analysis. Proceedings of 5th CIRP International Seminar onComputer-Aided Tolerancing (CAT), pp. 1324.

VASSEUR, H., KURFESS, T. R. and CAGAN, J., 1997, Use of quality loss function to selectstatistical tolerances. Transactions of the ASME, Journal of Manufacturing Scienceand Engineering, 11 9, 410415.

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SRINIVASAN, V., 1997, ISO deliberates statistical tolerancing. Proceedings of 5th CIRPInternational Seminar on Computer-Aided Tolerancing (CAT), pp. 2535.

TURNER, J. and GUILFORD, J., 1992, Representing geometric tolerances in solid models.ASME Symposium on Computer in Engineering, Vol. 1, pp. 319327.

WU, Z., ELMARAGHY, W. H. and ELMARAGHY, H. A., 1988, Evaluation of cost-tolerancealgorithms for design tolerance analysis and synthesis. Manufacturing Review, 1,168179.

ZHANG, C. and WANG, H. P., 1992, Simultaneous optimization of design and manufacturingtolerances with process (machine) selection. Annals of the CIRP, 41 , 569572; 1993a,Integrated tolerance optimization with simulated annealing. International Journal ofAdvanced Manufacturing Technology, 8, 167174; 1993b, The discrete tolerance optimi-zation problem. Manufacturing Review, 6, 6071.

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