straightness and flatness evaluation using data envelopment analysis

7/21/2019 Straightness and Flatness Evaluation Using Data Envelopment Analysis

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ORIGINAL ARTICLE

Straightness and flatness evaluation using data

envelopment analysis

Sohyung Cho &Joon-Young Kim

Received: 17 May 2011 /Accepted: 10 January 2012 /Published online: 11 February 2012# Springer-Verlag London Limited 2012

Abstract In today's world of precision engineering, ro-

bustness and accuracy in the evaluation of the formtolerances are considered as competitive advantages for

manufacturing enterprises. Amongst various methods for

accurate and robust evaluation, which have been stud-

ied, nonlinear optimization methods based on operation-

al research have proved to be successful as far as they

can ensure unique and global convergence in practical

applications. However, it is well known that ensuring

the convergence is the most difficult thing to deal with

for a nonlinear optimization technique because the per-

formance is in general highly sensitive to parameter

setting. Therefore, this paper introduces a robust linear

programming formulation-based algorithm in which the

performance is not dependent on the quality of param-

eters. Interestingly, in this algorithm, the data envelop-

ment analysis technique is used to form a convex hull

that decides the minimum enclosed zone in a robust

manner. From the computational experiments, it is

shown that the proposed algorithm can be a promising

alternative to the traditional nonlinear optimization

method for straightness and flatness evaluation.

Keywords Form tolerance . Robust algorithm . Data

envelopment analysis

1 Introduction

The geometric form of manufactured components deviates

from its nominal design to a certain extent due to errors in

manufacturing, wear, etc. that can be confirmed by measure-

ment and evaluation. While uncertainty in measurement has

been significantly reduced by recent technological advance-

ment in inspection methods, evaluation of geometric toler-

ances is still considered as a major challenge because of its

nonlinear nature. There are numerous algorithms that can

evaluate various geometric tolerances by modeling the eval-

uation of geometric tolerances as nonlinear optimization

problems and then solving the problems optimally (or close

to optimal). However, it is known that the convergence of

nonlinear optimization techniques is highly sensitive to set-

ting of parameters such as starting conditions. It should be

pointed out that in today's world of precision engineering,

enhancing robustness as well as accuracy in the evaluation

of the tolerances is greatly required as important competitive

advantages for manufacturing enterprises.

The purpose of geometric tolerance evaluation is to pro-

vide the exact value of the error in measured data points

from manufactured parts conforming to a standard, for ex-

ample, ANSI Y14.5M [1]. Specifically, ANSI and ISO

promote the use of the minimum zone form tolerance that

is based on minimizing the maximum deviation of the

inspected surface from a reference feature. The objective

of the minimum zone criterion is to fit the data points with

minimum peak-to-valley deviation to the fitting line. The

studies show that the minimum zone criterion yields more

accurate fitting results because: (1) it yields a zone value

S. Cho (*)

Industrial and Manufacturing Engineering,

Southern Illinois University,

Edwardsville, IL 62026-1805, USA

e-mail: [email protected]

J.-Y. Kim

Division of Marine Equipment Engineering,

Korea Maritime University,

Youngdo-gu,

Busan 606-791, South Korea

Int J Adv Manuf Technol (2012) 63:731740

DOI 10.1007/s00170-012-3925-6


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smaller than other criteria, and (2) it is more consistent with

the standard definition of physical fittings [2]. The mini-

mum zone form tolerance can be evaluated either using

specific computational geometry techniques [3, 4] or by

solving non-linear optimization problems [57]. However,

this optimization function has numerous local minima, and

most solution methods need a set of good starting points to

obtain the global minimum solution. In addition, the formu-lation for the tolerance zone is typically non-differentiable,

which limits the use of conventional gradient-based optimi-

zation techniques. Recently, evolution or machine-learning

algorithm-based evaluation methods have been studied in

order to solve nonlinear optimization problems more effi-

ciently [810]. Examples of evolution algorithms include

particle swarm, genetic algorithm, simulated annealing, and

support vector machine learning. It has been reported that

these methods can solve the problems faster than conven-

tional nonlinear optimization methods [10]. However, it is

known that the performance of these methods in terms of

computational efficiency and accuracy depends on the qual-ity of initial conditions and other parameters such as specific

kernel functions in the case of support vector machine

learning.

Straightness is a condition where an element of a surface or

an axis is a straight line. A straightness tolerance specifies a

tolerance zone within which the considered element or axis

must lie. Each element of the surface must lie between two

parallel lines separated by the amount of the prescribed

straightness tolerance [1]. For discrete measurement, the

straightness error is defined as the minimum separation be-

tween two parallel lines within which all the data points must

lie. The region bounded by two parallel lines with minimum

separation is defined as the minimum zone of the data points.

The definition that confines the straightness error to the min-

imum zone is called minimum zone criterion. Flatness is the

condition of a surface having all elements in one plane. A

flatness tolerance specifies a tolerance zone defined by two

parallel planes within which the surface must lie [1]. Similar to

the straightness problems, evaluation of flatness under the

minimum zone criterion is to find two parallel planes that

bound all measured data with minimum separation.

Focusing on the aforementioned issue of computational

efficiency and accuracy in the evaluation of form tolerances,

this paper introduces a novel algorithm to evaluate straight-

ness and flatness using the data envelopment analysis (DEA)

approach, which is based on the linear programming (LP)

model and used to assess an organization's operational effi-

ciency. Due to its nature of LP model, the algorithm intro-

duced in this paper expects to be robust (stable) in terms of

computational time and efficiency. Interestingly, the first ap-

proach to find the minimum zone using not only efficient

frontiers that are typical output of DEA but also inefficient

frontiers obtained from modified DEA is introduced and

tested. This paper is organized as follows: Section 2 reviews

related research works. An overview of DEA is provided in

Section 3. In Section 4, traditional DEA is modified to be used

for evaluation of form tolerances, particularly straightness and

flatness. In Section5, performance of the proposed algorithm

in this paper is compared to other algorithms such as least

square, nonlinear programming, and other LP-based algo-

rithm, respectively. The robustness of the proposed algorithmis addressed in the same section, and the conclusion of this

paper is provided in Section6.

2 Literature review

Research works for the evaluation of form tolerances, par-

ticularly straightness and flatness, focus on one of the fol-

lowing techniques:

1. Least squares method (LSM): Among several techni-

ques existing for evaluation of straightness and flatnesstolerances, LSM is a widely accepted technique used in

industry for the evaluation of form errors. This tech-

nique finds the minimum sum of squared errors of the

measured points from the nominal feature and thus

provides only an approximation solution quickly [11].

More specifically, LSM does not ensure precise values

of straightness and flatness errors due to a different

objective function from the minimum zone criterion.

2. Computational geometric approach (CGA): Lai and

Wang [12] and Traband et al. [3] proposed computa-

tional geometry-based techniques for calculating exact

values of the minimum zone straightness and flatness.

More specifically, these research works proposed the

minimum solution for straightness and flatness prob-

lems in accord with the minimum zone criterion. The

basic principle of this technique is to find the minimum

zone of the convex hull that encloses all the measured

points. The minimum zone was determined by a pair of

parallel supporting lines parallel to an edge of the con-

vex hull. Based on the same concept, Lee [13] presented

an approach, called convex-hull edge method, which

guarantees the minimum zone solution of flatness eval-

uation problems. Samuel and Shunmugam [4] also used

similar computational geometric techniques to solve the

minimum zone and function-oriented solutions for the

straightness and the flatness problems.

3. Operational research techniques (OR): Wang [5] used

optimization techniques for minimum zone evaluation

of form errors. Form errors were formulated as nonlin-

ear optimization problems, and solution procedures

were developed to solve the nonlinear optimization

problems. Kanada and Suzuki [14] studied an applica-

tion of some nonlinear optimization techniques for

732 Int J Adv Manuf Technol (2012) 63:731740


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minimum zone flatness. They also applied various op-

timization algorithms to calculate the minimum zone

straightness. Carr and Ferreira [15] developed algo-

rithms to verify minimum zone straightness and flat-

ness. The proposed model searched for two parallel

supporting planes so that all measured points were

below one plane and above the other while both planes

were as close together as possible. The model was aconstrained nonlinear programming problem, but the

objective function and all but one constraint were linear.

Hence, the successive linear programming algorithm

was used to solve a sequence of linear programs that

converge to the solution of the nonlinear problem. Cher-

aghi et al. [6] formulated straightness and flatness errors

as nonlinear optimization problems and then developed

a linear search method that reduces the nonlinear opti-

mization problem to a linear programming problem with

only two constraints. It was reported that the exact error

values are quickly found by an iterative search proce-

dure. Weber et al. [16] presented a unified linear ap-proximation technique for the evaluation of various

form tolerances. They formulated a linear program mod-

el using a Taylor expansion of the minimum zone tol-

erance function.

4. Evolution algorithms (EAs): Malyscheff et al. [9] mod-

eled minimum zone straightness and flatness by modi-

fying a support vector classification that is based on

statistical learning. In this research work, a gradient

ascent approach was introduced to sequentially solve a

formulated non-convex optimization problem, noting

that straightness and flatness share some similarities

their functions are in 2D and 3D linear forms, respec-

tively, which are also the basic functions used in support

vector machines. In Kovvur et al. [10], particle swarm

optimization, which is one of the most recent and pop-

ular EAs, has been shown to perform well for the

evaluation of minimum zone form tolerance such as

circularity, cylindricity, sphericity, flatness, and straight-

ness form tolerances. In this approach, each particle of

the population, called the swarm, progresses toward the

optimal solution by adjusting its trajectory toward its

own previous best position and toward the previous best

position attained by any member of the population. The

advantage of using PSO over other optimization meth-

ods is the relatively insignificant impact of the starting

conditions.

The following summary can be drawn from the re-

view of existing research: some reported algorithms in

CGA, such as convex hull/polygon method, minmax

method, control geometric feature rotation scheme, and

median approach, have been proved to be successful in

view of minimum zone evaluation. However, none of

these CGA algorithms satisfies the universality require-

ment to be able to evaluate various form tolerances. In

fact, the mathematical definition of minimum zone form

errors in the ANSI standards yields a nonlinear discon-

tinuous optimization problem. To tackle this problem,

various nonlinear optimization techniques and OR meth-

ods have been proposed and tested. Among them are

Chebyshev approximation [17], downhill simplex meth-od [14], and HookeJeeve direct search method [18].

Among these approaches, few have proved to be suc-

cessful if unique and global convergence is ensured in

practical applications. However, it is known that ensur-

ing the convergence is the most difficult thing to deal

with for a nonlinear optimization technique because the

performance is in general highly sensitive to parameter

setting. Therefore, a robust algorithm of which perfor-

mance is not dependent on the quality of parameters is

h i gh l y r e qu i re d , a nd t hi s p a pe r i n tr o du c es s uc h

algorithms.

3 Overview of the DEA method

Examples of commonly used efficiency measurement

methods include ordinary least squares (OLS), corrected

ordinary least squares (COLS), stochastic frontier anal-

ysis (SFA), and DEA. Figure 1 illustrates these methods

with their unique nature. In the figure, dots represent

producers that use a set of inputs to produce a set of

outputs. For instance, consider a set of banks. Inputs

can be a square footage of space and a number oftellers, etc., and outputs can be a number of checks

cashed and number of transactions, etc. OLS, COLS,

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Input

Output OLS

COLS

DEASFA

Fig. 1 Comparison of efficiency assessment methods including the

DEA method

Int J Adv Manuf Technol (2012) 63:731740 733


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and SFA are parametric in their nature requiring an

assumption for a functional form while DEA is non-

parametric and thus requires no assumption of the func-

tional form [19]. In the parametric approach, any mis-

specification of the functional form can result in

catastrophic results. Among these methods, SFA and

DEA are the two competing paradigms on efficiency

analysis [20].Since the introduction by Charnes, Cooper, and Rho-

des [21], DEA has proven to be an attractive method

for assessing and evaluating the efficiency and perfor-

mance of decision-making units (DMUs) within organ-

izations producing multiple outputs from multiple

inputs. The linear programming model for the input-

oriented constant return to scale (CRS) model was in-

troduced by Charnes et al. [21], also referred to as the

CCR model, is given by:

min jPn

j1

ljxij jx ij i 1; 2; . . . ; m

Pn

j1

ljyrj yrj r 1; 2; . . . :;s

lj 0 j 1; 2; . . . :; n

1

wherejis the efficiency score for the jth DMU; where there

aren systems or DMUs, the jth DMU represents one of then

DMU under evaluation, xijand yrjare the ith input and rth

output for the jth DMU, respectively, and s are dual varia-

bles. The CCR model (or CRS model) assumes constant

return to scale economies, which means that doubling output

exactly doubles inputs. The CCR model was extended to the

variable return to scale (VRS) model by Banker, Charnes, and

Cooper [22], referred to as the BCC model. The VRS input-

oriented model is the same as the CRS model except for the

fact that the sum ofs is equal to 1 and is written as:

min jPn

j1

ljxij jx ij i 1; 2; . . . ; m

Pn

j1

ljyrj yrj r 1; 2; . . . :;s

Pn

j1lj 1; lj 0 j 1; 2; . . . :n

2

wherejis the efficiency score for the jth DMU; where there

aren systems or DMUs, the jth DMU represents one of then

DMU under evaluation, xijand yrjare the ith input and rth

output for the jth DMU, respectively, and s are dual varia-

bles. It should be emphasized here that the DEA model

assumes monotonicity [23].

DEA applications for assessing performance and efficiency

have been utilized in many industries and organizations, such

as hospitals, restaurants, US Air Force wings, universities,

cities, courts, business firms, and more [24]. There is, howev-

er, another competing paradigm on how to construct frontiers

to evaluate and assess the performance and efficiency of

DMUs, which is the stochastic frontier function approach

(SFA) [25]. DEA utilizes mathematical programming techni-ques, whilst the SFA approach utilizes econometric regression

theory. The major advantage of the DEA approach is that no

assumption has to be made about the functional form other

than the concavity of the frontier functions. To the contrary,

the SFA approach imposes an explicit and possibly over-

restrictive, functional form for the data.

4 DEA-based algorithm for tolerance evaluation

This paper introduces the first-time approach that uses tra-

ditional DEA for identifying inefficient frontiers such that

both efficient and inefficient frontiers are used to form a

convex hull that encloses all the data points measured inside

as shown in Fig. 2.

4.1 Straightness evaluation

First, the problem modeled by Eq. 2 is solved for

measured data points, setS0(xj, yj), and data points that

satisfy the condition of j01 are identified as efficient

frontiers. Note that other DMUs (data points) with j1. Note that the efficient and ineffi-

cient frontiers generate a convex hull that contains the

se t S. F in ally, the d ata p oints form e fficien t a nd



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inefficient frontiers that are used to compute two paral-

lel lines with minimum distance (d) that contains the set

S. The minimum distance d is straightness. The algo-

rithm to find the minimum distance d is as follows:



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As an example, consider a problem instance with 5 points

(n05) as shown in the following table:

n05 Point 1 Point 2 Point 3 Point 4 Point 5

Xvalues 1.0 2.0 3.0 4.0 5.0

Yvalues 1.1 1.9 3.3 4.0 4.8

Steps 1 and 2 identify points 1, 3, and 5 as efficient frontiers

and points 1, 2, and 5 as inefficient frontiers. Step 3 gives deff,i0

max(dk)0[0.4709 0.5600], and step 4 gives deff,i0[0.4685

0.3116]. Finally, step 5 provides d00.3116. The following

figure shows the efficient and inefficient frontiers and the

convex hull generated for this problem instance (Fig.3).

4.2 Flatness evaluation

Figure4 illustrates a problem instance randomly generated

for the flatness evaluation (n0100). For these data points,problem formulation for straightness by Eq. 2 is modified

for 3D flatness evaluation as follows:

min jPn

j1

ljxij xij i 1; 2; . . . ; m

Pn

j1

ljyrj yrj r 1; 2; . . . :;s

Pn

j1

ljztj jztj i 1; 2; . . . ; u

Pn

j1

lj 1; lj 0 j 1; 2; ::::n

4

0 5 10 15 20 25 300

10

20

30

40

50

60

70

80

90

x values

yv

alues

Fig. 2 Efficient frontiers

(points) and inefficient frontiers

(points) out ofn030 (one of the

parallel lines to enclose all the

data points passes the point

marked by *)

1 1.5 2 2.5 3 3.5 4 4.5 51

1.5

2

2.5

3

3.5

4

4.5

5

X-values

Y-

values

Fig. 3 Efficient frontiers

(points) and inefficient frontiers

(points) out ofn05



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Note thatz-axis values are considered as input in this approach

in order to form a 3D convex hull instead of a 2D convex hull.

Here, data points that satisfy the condition ofj01 are identi-

fied as efficient frontiers. Note that other DMUs (data points)

with j1.

Note that both efficient and inefficient frontiers are used to

form a convex hull that includes the setS0(xi,yi,zi). Finally,

the data points that form efficient and inefficient frontiers

02

46

810

0

5

100

5

10

15

20

25

30

xy

z

Fig. 4 Randomly generated data points for flatness evaluation (n0

100, scale in millimeters)

Table 1 Example problem instances from Traband et al. [3]

n05 n010 n015

xi yi xi yi xi yi xi yi

2 3 1 2.428 0.3952 0.1010 3.0662 0.1067

1 5 2 2.891 0.6953 0.1026 3.2165 0.1025

0 2 3 3.445 0.9669 0.1000 3.4217 0.1068

1 1 4 2.931 1.2762 0.1014 3.6179 0.1069

2 2 5 3.895 1.5797 0.1005 3.8185 0.1089

6 4.196 1.8593 0.1035

7 4.497 2.1333 0.1032

8 4.662 2.4197 0.1049

9 4.545 2.6001 0.1049

n020 n025

xi yi xi yi xi yi xi yi xi yi

0.05 0.1288 0.55 0.1000 0.2845 0.0006 3.6307 0.0011 6.2962 0.0021

0.10 0.1309 0.60 0.1837 0.6600 0.0008 3.9237 0.0011 6.5240 0.0021

0.15 0.204 0.65 0.1712 1.2041 0.0010 4.2647 0.0012 6.7114 0.0023

0.20 0.1841 0.70 0.2304 1.4994 0.0005 4.5122 0.0012 6.9960 0.0021

0.25 0.1329 0.75 0.1753 1.8494 0.0004 4.8150 0.0013 7.2076 0.0023

0.30 0.1570 0.80 0.2107 2.2261 0.0015 5.1334 0.0013

0.35 0.2608 0.85 0.1820 2.5724 0.0012 5.3603 0.0010

0.40 0.2588 0.90 0.1730 2.9076 0.0014 5.6534 0.0008

0.45 0.2237 0.95 0.2723 3.2548 0.0009 5.9058 0.0020

0.50 0.1891 1.00 0.1949 3.4142 0.0009 6.0774 0.0021

Table 2 Accuracy comparison for examples in Table1

n05 n010 n015 n020 n025

DEA 2.1213203 0.8578577 0.0051857 0.1645859 0.0013113

OTZ 2.1213204 0.8578577 0.0051857 0.1645859 0.0013113

NLP 2.1213204 0.8578577 0.0051857 0.1645859 0.0013113

LSM 2.5860858 0.8877250 0.0053790 0.1705494 0.0014633



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are used to compute two parallel planes with minimum

distance (d) that includes the setS, and the minimum dis-

tance d is flatness. The algorithm to find the minimum

distance dis as follows:

In step 3, an open source code developed by Kessler [26]

in Matlab is used to generate a convex hull formed by only

efficient and inefficient frontiers.

5 Computational experiments

Straightness First, the performance of the proposed DEA-

based algorithm in this paper has been compared to other

approaches such as LSM, nonlinear programming approach

(NLP) from Malyscheff et al. [9], and optimization tech-

nique zone (OTZ) from Cheraghi et al. [6]. Problem instan-

ces are shown in Table1, and performance comparison for

these test problems is provided in Table2.

Next, randomly generated problem instances have been

used to test the performance of the DEA algorithm. Here,

problems have been generated based on the following linear

relationship with disturbances (): yi b0 b1xi " . Itshould be pointed out here that the performance of other

algorithms highly depends on the quality of parameters such

as incremental angle parameter in the case of OTZ and

initial condition in the case of NLP. Table 3and Fig.5show

performance improvement (percent) by the DEA algorithm

against LSM and computational effort of the DEA compared

to OTZ and NLP. Note that each output in Table 3has been

obtained as an average of ten replications, particularly for

NLP with different initial conditions.

The results from the computational experiments show

that the DEA algorithm finds the optimal solution for all

the test problem instances without the fine tuning of any

parameters. In other words, while the accuracy and compu-

tational time of NLP and OTZ depend on the quality of

initial conditions and other parameters, the DEA algorithm

can evaluate the straightness and flatness in a more robust

manner due to its nature of LP model.

Flatness First, the performance of the DEA algorithm has

been tested for problem instances from the literature [6]. For

these instances shown in Table4, the DEA finds the optimal

values of 1.9612 (n015) and 4.8573 (n025) like other

algorithms such as NLP.

Next, the performance of the DEA has been tested for

randomly generated problem instances and Table 5 shows

that the accuracy of the DEA is identical to the performance

Table 3 Straightness performance comparison for randomly generated

data points

n 10 50 100 500 1,000

Improvement by DEA 13.8 4.5 5.2 10.1 9.2

CPU-DEA 1.05 2.69 5.26 6.28 7.93

CPU-OTZ 5.46 5.48 5.47 6.93 8.64

CPU-NLP 1.53 3.62 5.96 11.79 35.50



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of NLP for these problem instances. In the table, each data

field has been obtained as an average of ten replications. In

the flatness evaluation, DEA reduces the number of points

that form a convex hull such that in the case ofn0196 only

14% of the data (27 points) are required to form a convex

hull with which the flatness can be accurately evaluated.

This implies that the other 86% of the data (169 points) are

enclosed in the convex hull.

6 Conclusion

This paper has introduced a robust algorithm for the evalu-

ation of form tolerances, especially straightness and flatness

of manufactured parts. Specifically, this paper has devel-

oped a DEA approach-based algorithm that can form a

convex hull from both efficient and inefficient frontiers,

which enclose all of the measured data points. Computa-

tional experiments showed that this algorithm is more robust

than other existing methods in terms of the accuracy and

computational effort because it is based on LP formulation.

Fig. 5 Performance

comparison for straightness

evaluation (n01,000) by

various algorithms

Table 4 Problem instances for flatness evaluation from Cheraghi et al.

[6]

n015 n025

xi yi zi xi yi zi xi yi zi

2 1 5 0 0 2 75 0 7

1 1 4 0 25 5 75 25 7

0 1 1 0 50 6 75 50 6

1 1 2 0 75 8 75 75 7

2 1 2 0 100 9 75 100 9

2 0 4 25 0 5 100 0 7

1 0 3 25 25 7 100 25 6

0 0 3 25 50 8 100 50 6

1 0 2 25 75 9 100 75 6

2 0 2 25 100 12 100 100 8

2 1 3 50 0 6

1 1 4 50 25 7

0 1 2 50 50 8

1 1 1 50 75 9

2 1 2 50 100 11

Table 5 Flatness performance comparison for randomly generateddata points

n DEA/LSM

(% improvement)

DEA/NLP

(% improvement)

n (nn)/n (%)

25 9.2 0 15 40.0

49 4.2 0 17 65.3

100 3.8 0 21 79.0

144 3.4 0 25 82.6

196 3.5 0 27 86.2



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Interestingly, in the case of flatness evaluation with about 200

points, the required number of points to form a convex hull is

reduced by 86% that ensures considerable saving in compu-

tational effort. Future research may focus on the application of

the DEA algorithm for circularity and sphericity tolerances.

Acknowledgments The authors acknowledge partial support for this

research from the (Unmanned Technology Research Center (UTRC) at

the Korea Advanced Institute of Science and Technology (KAIST),

originally funded by DAPA, ADD.

References

1. ANSI/ASME Y14.5 (1994) National standard on dimensioning

and tolerancing. ASME, New York

2. Choi W, Kurfess TR, Cagan J (1988) Sampling uncertainty in

coordinate measurement data analysis. Precis Eng 22(3):153163

3. Traband MT, Joshi S, Wysk RA, Cavalier TM (1988) Evaluation

of straightness and flatness tolerance using the minimum zone.

Manuf Rev 2:189

1954. Samuel GL, Shunmugam MS (1999) Evaluation of straightness

and flatness error using computational geometric techniques. Com-

put Aided Des 31(13):829843

5. Wang Y (1992) Minimum zone evaluation of form tolerances.

Manuf Rev 5(3):213220

6. Cheraghi SH, Lim HS, Motavalli S (1996) Straightness and flat-

ness tolerance evaluation: an optimization approach. Precis Eng 18

(1):3037

7. Zhu LM, Ding H, Xiong YL (2003) A steepest descent algorithm

for circularity evaluation. Comput Aided Des 35:255265

8. Hong LJ, Shultes BC, Anand S (2001) Robust evaluation of

straightness and flatness tolerance using simulated annealing.

Transactions of NAMRI. 533540

9. Malyscheff AM, Trafalis TB, Raman S (2002) From support

vector machine learning to the determination of the minimum

enclosing zone. Comput Ind Eng 42(1):5974

10. Kovvur Y, Ramaswami H, Anand RB, Anand S (2008) Minimum

zone form tolerance evaluation using particle swarm optimization.

Int J Intell Syst Technol Appl 4(1/2):7996

11. Murthy TSR, Abdin SZ (1980) Minimum zone evaluation of

surfaces. Int J Mach Tool Des Res 20(2):123136

12. Lai K, Wang J (1988) A computational geometry approach to geo-

metric tolerancing, 16th North American Manufacturing Research

Conference. 376379

13. Lee MK (1997) A new convex-hull based approach to evaluating

flatness tolerance. Comput Aided Des 29(12):861

86814. Kanada T, Suzukit S (1993) Evaluation of minimum zone flatness

by means of nonlinear optimization techniques and its verification.

Precis Eng 15(2):9399

15. Carr K, Ferreira P (1995) Verification of form tolerances part I:

basic issues, flatness, and straightness. Precis Eng 17:131143

16. Weber T, Motavalli S, Fallahi B, Cheraghi SH (2002) A unified

approach to form error evaluation. Precis Eng 26:269278

17. Shunmugam MS (1986) Establishing reference figures for form

evaluation of engineering surfaces. J Manuf Syst 10(4):314321

18. Elmaraghy WH, Elmaraghy HA, Wu Z (1990) Determination of

actual geometric deviations using coordinate measuring machine

data. Manuf Rev 3(1):3239

19. Sarafidis V (2002) An assessment of comparative efficiency mea-

surement techniques, Europe Economics: Occasional Paper 2,

London, UK, October

20. Wang S (2003) Adaptive non-parametric efficiency frontier analysis:

a neural-network-based model. Comput Oper Res 30:279295

21. Charnes A, Cooper WW, Rhodes E (1981) Evaluating program

and managerial efficiency: an application of data envelopment

analysis to program follow-through. Manag Sci 27(6):668697

22. Banker RD, Charnes A, Cooper WW (1984) Models for the

estimation of technical and scale inefficiencies in data envelopment

analysis. Manag Sci 30:10781092

23. Pendharkar PC (2005) A data envelopment analysis-based approach for

data preprocessing. IEEE Trans Knowl Data Eng 17(10):13791388

24. Cooper WW, Seiford LW, Zhu J (2004) Handbook of data envelop-

ment analysis. Kluwer Academic Publishers, Boston

25. Bauer PW (1990) Recent developments in the econometric esti-

mation of frontiers. J Econ 46(1/2):3956

26. Kessler P (2011) Notes on flatness on the web: http://www.

mechanicaldust.com/Documents/flatness.pdf

http://www.mechanicaldust.com/Documents/flatness.pdfhttp://www.mechanicaldust.com/Documents/flatness.pdfhttp://www.mechanicaldust.com/Documents/flatness.pdfhttp://www.mechanicaldust.com/Documents/flatness.pdf