Construction and Optimality of a Special Class of Balanced Designs

QUALITY AND RELIABILITY ENGINEERING INTERNATIONAL

Qual. Reliab. Engng. Int. 2006; 22:507–515

Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/qre.757

Special Issue Construction and Optimality of aSpecial Class of Balanced DesignsStefano Barone∗ ,† and Alberto LombardoDipartimento di Tecnologia Meccanica, Produzione e Ingegneria Gestionale, University of Palermo, Italy

The use of balanced designs is generally advisable in experimental practice.In technological experiments, balanced designs optimize the exploitation ofexperimental resources, whereas in marketing research experiments they avoiderroneous conclusions caused by the misinterpretation of interviewed customers.In general, the balancing property assures the minimum variance of first-ordereffect estimates. In this work the authors consider situations in which all factorsare categorical and minimum run size is required. In a symmetrical case, it is oftenpossible to find an economical balanced design by means of algebraic methods.Conversely, in an asymmetrical case algebraic methods lead to expensive designs,and therefore it is necessary to adopt heuristic methods. The existing methodsimplemented in widespread statistical packages do not guarantee the balancingproperty as they are designed to pursue other optimality criteria. To deal withthis problem, the authors recently proposed a new method to generate balancedasymmetrical designs aimed at estimating first- and second-order effects. To reducethe run size as much as possible, the orthogonality cannot be guaranteed. However,the method enables designs that approach the orthogonality as much as possible(near orthogonality). A collection of designs with two- and three-level factors andrun size lower than 100 was prepared. In this work an empirical study was conductedto understand how much is lost in terms of other optimality criteria when pursuingbalancing. In order to show the potential applications of these designs, an illustrativeexample is provided. Copyright c© 2006 John Wiley & Sons, Ltd.

Received 29 July 2005; Revised 21 November 2005

KEY WORDS: balancing; interaction estimability; asymmetrical (mixed-level) designs; nearly orthogonalarrays; optimality; two- and three-level designs

1. INTRODUCTION

When designing experiments aimed at improving products, processes and services, it is often requiredto analyse not only first-order effects of factors, but also their interactions. If all factors have eithertwo or three levels, the design problem has many solutions proposed in the literature (Box et al.1;

∗Correspondence to: Stefano Barone, Dipartimento di Tecnologia Meccanica, Produzione e Ingegneria Gestionale, Universita of Palermo,Viale delle Scienze, 90128 Palermo, Italy.†E-mail: [email protected]

Contract/grant sponsor: National Interest Research Project (PRIN 2005)

Copyright c© 2006 John Wiley & Sons, Ltd.

508 S. BARONE AND A. LOMBARDO

Montgomery2; Wu and Hamada3). Conversely, when asymmetrical (mixed-level) designs are needed, differentproblems arise based on the type of factors involved. If all factors are continuous, optimal solutions can be foundvia response surface methods (Box and Draper4; Myers and Montgomery5). If categorical factors are present,comprehensive solutions are still not available (see Xu6, Huwang et al.7 and Nguyen8).

In this work a critical review of asymmetrical designs properties is presented, with particularly emphasison orthogonality and balancing. We focus on asymmetrical designs that allow an estimation of both mainand interaction effects and ensure the balancing property so defined: ‘for each factor its levels appear equallyoften in the design’. Henceforth, this property will be called ‘I-grade balancing’. It provides several advantages(Section 2). Unfortunately, the I-grade balancing does not guarantee the orthogonality of first-order effects,unlike the proportional-frequencies case (Kendall et al.9).

In some cases, depending on the number of levels of experimental factors, a large number of runs mightbe required in order to ensure both orthogonality and balancing. In such cases, many authors favour thefirst property that is easily obtainable by means of the collapsing technique (Montgomery2). This consists inredefining the levels of one or more factors present in a more economical known design. For instance, in amixed two-level and four-level design, a four-level factor can be collapsed in a three-level factor by mergingtwo levels. However, these designs are usually proposed only for first-order effect estimation, because second-order effect estimation is not of interest.

Barone and Lombardo10 proposed a heuristic method for constructing balanced designs as economicallyas possible that enable an estimation of first- and second-order effects and that ensure the orthogonality(in the above-specified sense) as near as possible. The desired properties of a new class of experimentaldesigns (balanced asymmetrical nearly orthogonal designs aimed at estimating first- and second-order effects,BANOD2) are stated and discussed in Section 2. These properties are based on needs and constraints usuallyencountered in the experimental practice. The procedure for generating the required designs is describedextensively in Section 3. As the most common experimental situations of asymmetrical designs involve two-and three-level factors, a complete collection of BANOD2 with run size lower than 100, was prepared forthe readers’ convenience. The results of an empirical study of BANOD2 optimality is presented in Section 5.An applicative example is presented in Section 6. Some comments conclude the paper.

2. REQUIRED PROPERTIES FOR A NEW CLASS OF DESIGNS (BANOD2)

In any experimental situation, the experimenter sets out several requirements. These usually are estimability offirst- and second-order effects, balancing, minimum run size, and optimality.

2.1. Estimability of first- and second-order effects

Generally, the estimability of interactions is required after the screening phases of an experimental research orwhen, based on a priori knowledge, it is expected that confounding first- and second-order effects can induceerroneous conclusions.

Whenever possible, it is advisable at least to ensure a clear estimability of two-factor interaction effects.This is obtained by using regular designs, in which any main effect is estimated independently from allinteraction effects (clear); or, any two-factor interaction is estimated independently from all the others (stronglyclear). Such conditions can lead to a large run size, especially in the mixed-level case. Accordingly, it becomesnecessary to relax the condition of clear estimability in favour of a simple estimability as clearly as possible,as defined below.

Given the design matrix having N rows (run size) and m columns (factors), the matrix of contrasts, X, willhave N rows and a number of columns M given by

M = 1 +m∑

i=1

(si − 1) +m−1∑i=1

m∑j=i+1

(si − 1)(sj − 1) (1)

Copyright c© 2006 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2006; 22:507–515DOI: 10.1002/qre

CONSTRUCTION AND OPTIMALITY OF BALANCED DESIGNS 509

equal to the degrees of freedom necessary for the estimation of mean, first-order and second-order effects.If X is non-singular, then it is possible to deduce that all the first- and second-order effects are simplyestimable.

Details about the coding for the construction of the matrix X are provided in Appendix A.

2.2. I-grade balancing

The I-grade balancing property means that for each factor all levels appear equally often in the design.Its advantages are:

(1) it relieves the experimenter from the arbitrariness of choosing the factor levels to which to assign more/lessexperimental runs;

(2) it allows optimization of the use of experimental resources (tools, rough material, time, etc.) intechnological experiments;

(3) it avoids biased results due to erroneous perceptions when the experimental response is based on humanjudgement, as occurs in marketing research experiments (Kuhfeld et al.11);

(4) it ensures minimum variance of first-order effect estimates (Wu and Hamada3, p. 322).

2.3. Minimum run size

This is conditional on the properties described in Sections 2.1 and 2.2.

2.4. Optimality

The classical criteria for defining design optimality are based on some characteristics of the matrix XTX:

A-optimality max tr(XTX) ≡ min tr[(XTX)−1] ≡ minM∑i=1

λi

D-optimality max det(XTX) ≡ min det[(XTX)−1] ≡ minM∏i=1

λi

E-optimality min maxi

λi

where λi(i = 1, . . . , M) are the eigenvalues of (XTX)−1.General comments on the above criteria were given by Barone and Lombardo. There, a new optimality

criterion is proposed, based on the idea of making the estimates of the main effects as orthogonal as possible.Assuming that the I-grade balancing property is satisfied (for each factor h, its levels ih = 1, . . . , sh appear

equally often nih = N/sh times), the ‘II-grade balancing’ property was there defined as: ‘for each pair offactors, h and k, all possible combinations of levels (ih = 1, . . . , sh; ik = 1, . . . , sk) appear equally oftennihik = N/(shsk) times’. This property implies the orthogonality between the main effects estimates.

Based on the above definitions, the proposed criterion is

B-optimality minm−1∑h=1

m∑k=h+1

sh∑ih=1

sk∑ik=1

|int[nihik − N/(shsk)]| (2)

It means approaching the II-grade balancing as closely as possible.The new class of experimental designs, denoted as BANOD2, is characterized by the properties stated and

discussed in Sections 2.1–2.4.



3. THE PROCEDURE FOR GENERATING BANOD2

The procedure for generating BANOD2 develops in the following three steps.

Step 1. Determine the run size N based on the needed degrees of freedom M and the least common multiple of(s1, s2, . . . , sm) in order to meet the I-grade balancing.

Step 2. Set a balanced column as the first column of the design.Step 3. The following columns are added, one at time, starting from a balanced column generated randomly or

by any other method. Then all possible swaps between non-equal elements are tried until the objectivefunction, based on the chosen optimization criterion (A-, B-, D-, E-optimality), is maximized.

The procedure ensures that the I-grade balancing is automatically satisfied.The estimability of first- and second-order effects—if not verified by the column initially chosen—is usually

met during the swapping and it holds until the end of the procedure.The B-optimality criterion is preferred because, when it is possible to obtain the perfect II-grade balancing

(N is a multiple of shsk for any h and k), an absolute minimum of this objective function exists, occurring whenall pairs of design matrix columns are orthogonal. Moreover, this criterion makes the optimization procedurefaster because the computation of eigenvalues is not necessary. For this reason, when the run size is moderatelyhigh (N > 70), the B-optimality criterion becomes the only viable one not to suffer a heavy computationalburden.

In the authors’ experience no more than a few dozen iterations are needed to optimize a column with anychosen criterion. Hence, the computation time is very limited (few minutes for N < 200) with a normal PC.

4. BANOD2 WITH TWO- AND THREE-LEVEL FACTORS

The procedure for generating the experimental plans is not wholly automatic. Hence, it appeared usefulto prepare a collection of experimental designs for the most common situations met in practice, in whichcategorical two- and three-level factors are present. The provided collection includes all designs with a run sizelower than 100 and multiples of two and three. Table I shows a summary of these designs (they are available onrequest from the authors). Each cell of Table I corresponds to a specific combination of numbers of two- andthree-level factors. The following are reported in each cell.

• The run size of the proposed design (in bold).• The degrees of freedom (in brackets).• The run size of the designs already available in the literature (McLean and Anderson12; in italic).

Currently, such designs seem to be the only possible competitors of the BANOD2. They areimplemented in the statistical software STATISTICA.

The notation ‘FF’ indicates that a reduction of the full factorial is not possible. The symbol ‘>’ indicates thatthe required run size exceeds 100.

If the degrees of freedom were an exact multiple of six, it was preferred to construct a design with run sizeequal to the next highest multiple, in order to have at least one degree of freedom available for error varianceestimation.

5. EMPIRICAL EVALUATION OF BANOD2 OPTIMALITY

An empirical study was made to evaluate the optimality properties of the BANOD2. This investigation wasaimed at understanding how much is lost in terms of D- and E-optimality, when the B-optimality criterion ispursued. For this purpose, the BANOD2 were compared to experimental plans generated by pursuing only theD-optimality criterion. To make a fair comparison, the two candidates must have the same run size and the sameestimation capability. Therefore, for every BANOD2 it was necessary to generate a D-optimal design having thesame run size, the same number of factors and levels, and able to estimate all first- and second-order effects.



Table I. Characteristics of the collection of BANOD2 for two- and three-level factors (fromBarone and Lombardo10)

Factors at three levelsFactors attwo levels 1 2 3 4 5 6

1 (6) FF (14) FF 30 (26) 48 (42)81 66 (62)162 90 (86)2432 (10) FF 24 (20) 36 (34)54 54 (52)162 78 (74)162 >

3 18 (15) 30 (27)36 48 (43)72 66 (63)162 90 (87)216 >

4 24 (21)36 36 (35)72 54 (53)108 78 (75)162 > >

5 30 (28)48 48 (44)72 66 (64)144 90 (88)216 > >

6 42 (36)48 60 (54)96 78 (76)288 > > >

7 48 (45)96 66 (65)144 90 (89)432 > > >

8 60 (55)96 78 (77)144 > > > >

9 72 (66)128 96 (90) > > > >

10 84 (78) > > > > >

11 96 (91) > > > > >

The algorithms for generating the D-optimal designs are well established and implemented in widespreadstatistical packages. Among them the authors preferred Design-Expert (v.6.0.10, Stat-Ease Inc.), which has aspecific section implementing the standard procedure for generating D-optimal designs.

The results of this investigation are summarized in Table II. In each row the table reports, the design labelin the first column (e.g. (18)2331 is an 18 run design with three two-level factors and one three-level factor),and the degrees of freedom (DOF) needed for the estimation of first- and second-order effects in the secondcolumn. In the following columns, the det(XTX) (DB and DE in the table), the maximum and the minimumeigenvalues of XTX (MB and mB, ME and mE in the table), the value of B-optimality calculated accordingto Equation (2) (BB and BE in the table) are reported, respectively, for the proposed BANOD2 and for theD-optimal design proposed by Design-Expert. The ratios between the previously listed quantities are reportedin the following three columns of the table. As det(XTX) is equal to the product of the eigenvalues, in order tomake fair comparisons, it was opportune to refer to the geometrical mean of eigenvalues, raising both DB andDE to the power of 1/DOF.

Finally, the table provides further information by indicating if the D-optimal design is equal to the BANOD2,and (if not) if it is at least I-grade balanced.

6. AN ILLUSTRATIVE EXAMPLE

A first example, based on the authors’ experience of a technological problem concerning the optimization of amarble cutting process was presented by Barone and Lombardo10.

The example presented here is taken from the healthcare literature (Ratcliffe et al13). It concerns a study toinvestigate patient preferences for attributes associated with the efficacy and side-effects of a clinical treatment.Five attributes (A, joint aches; B, joint pains; C, mobility; D, risk of mild/moderate side effects; E, risk ofserious side-effects) and three levels (0, low; 1, medium; 2, high) were chosen for each of them. This situationis common in many experimental fields.

The authors of the study were concerned to reduce the run size as much as possible. Therefore, they limited theinference to the estimation of main effects only. Consequently, the chosen experimental design was an OMEPconsisting of 16 scenarios. It is immediately noticeable that such a plan cannot be balanced. In fact, in order toobtain plans of this type, it is usual to adopt the collapsing method (Wu and Hamada3), by merging two levelsof a factor, starting from a four-level fractional factorial design.

In this case we obtained an alternative balanced plan consisting of 15 runs, as shown in Table III.If the estimation of some or all two-factor interactions was necessary, a BANOD2 could also be proposed.

Obviously, it would be more expensive in terms of run size.



Tabl

eII

.Com

pari

son

betw

een

BA

NO

D2

and

D-o

ptim

alde

sign

s

BA

NO

D2

Des

ign-

Exp

ert

Plan

DO

FD

BM

Bm

BB

BD

EM

Em

EB

E(D

B/D

E)1/

DO

FM

E/M

Bm

B/m

ER

esul

tI-

grad

eba

lanc

e

(18)

2331

151.

40×

10−1

1.34

3.33

×10

−10

1.56

×10

−11.

353.

34×

10−1

20.

9928

1.00

50.

997

�=Y

es(2

4)22

3220

9.48

×10

−21.

563.

70×

10−1

01.

08×

10−1

1.54

2.58

×10

−13

0.99

330.

991

1.43

1�=

Yes

(24)

2431

217.

02×

10−1

1.33

6.67

×10

−10

7.02

×10

−11.

336.

67×

10−1

01.

000

1.00

01.

000

BA

NO

D2

Yes

(30)

2133

261.

34×

10−2

1.81

2.28

×10

−10

1.87

×10

−21.

812.

27×

10−1

180.

9872

0.99

71.

002

�=N

o(3

0)23

3227

8.03

×10

−31.

941.

20×

10−1

06.

50×

10−2

1.55

2.33

×10

−17

0.92

551.

254

0.51

4�=

No

(30)

2531

285.

46×

10−3

1.82

1.39

×10

−10

6.91

×10

−21.

953.

46×

10−1

180.

9134

0.93

30.

401

�=N

o(3

6)22

3334

2.26

×10

−21.

663.

39×

10−1

02.

26×

10−2

1.66

3.39

×10

−10

1.00

01.

000

1.00

0B

AN

OD

2Y

es(3

6)24

3235

2.39

×10

−62.

038.

37×

10−3

02.

73×

10−3

1.94

1.23

×10

−130

0.81

781.

048

0.06

8�=

No

(42)

2631

366.

67×

10−3

1.87

1.77

×10

−10

1.39

×10

−21.

942.

02×

10−1

490.

9798

0.96

30.

878

�=N

o(4

8)21

3442

3.22

×10

−52.

066.

28×

10−2

04.

19×

10−3

1.83

1.79

×10

−123

0.89

051.

123

0.35

1�=

No

(48)

2532

441.

68×

10−5

2.03

4.66

×10

−20

2.39

×10

−42.

041.

53×

10−1

550.

9414

0.99

60.

304

�=N

o(4

8)23

3343

1.00

×10

−52.

178.

06×

10−2

01.

39×

10−3

1.98

1.55

×10

−143

0.89

171.

099

0.52

1�=

No

(48)

2731

452.

23×

10−6

2.28

6.55

×10

−20

5.47

×10

−52.

231.

32×

10−1

860.

9314

1.02

20.

497

�=N

o(5

4)22

3452

1.44

×10

−10

2.39

2.28

×10

−20

1.34

×10

−62.

241.

01×

10−1

790.

8389

1.06

80.

227

�=N

o(5

4)24

3353

7.33

×10

−82.

113.

08×

10−2

04.

84×

10−7

2.26

5.39

×10

−278

0.96

500.

937

0.57

2�=

No

(60)

2632

549.

18×

10−9

2.26

1.88

×10

−20

2.35

×10

−52.

071.

23×

10−1

710.

8648

1.09

20.

153

�=N

o(6

0)28

3155

7.51

×10

−12

2.37

1.18

×10

−30

3.80

×10

−62.

317.

78×

10−2

180

0.78

751.

030

0.01

5�=

No

(66)

2533

649.

33×

10−1

12.

371.

40×

10−2

04.

99×

10−9

2.44

5.05

×10

−211

40.

9397

0.97

10.

278

�=N

o(6

6)21

3562

2.23

×10

−12

2.50

6.05

×10

−30

4.70

×10

−82.

237.

61×

10−2

340.

8516

1.12

00.

080

�=N

o(6

6)23

3463

2.60

×10

−13

2.51

5.13

×10

−30

1.64

×10

−82.

386.

88×

10−2

870.

8391

1.05

30.

075

�=N

o(6

6)27

3265

1.06

×10

−14

2.39

5.54

×10

−30

4.39

×10

−92.

435.

33×

10−2

950.

8195

0.98

40.

104

�=N

o(7

2)29

3166

5.06

×10

−92.

225.

01×

10−2

06.

95×

10−8

2.36

9.87

×10

−225

80.

9611

0.94

00.

507

�=N

o(7

8)22

3574

7.02

×10

−21

2.72

1.62

×10

−30

2.67

×10

−10

2.38

7.14

×10

−210

60.

7195

1.14

30.

023

�=N

o(7

8)24

3475

7.05

×10

−20

2.66

1.88

×10

−30

6.35

×10

−11

2.54

6.07

×10

−211

60.

7596

1.04

50.

031

�=N

o



Table III. An alternative balanced de-sign for the clinical treatment example

Run A B C D E

1 0 2 0 2 02 0 2 0 0 13 0 1 2 1 04 0 0 1 1 15 0 1 1 0 26 1 1 0 1 17 1 2 1 1 08 1 1 2 2 29 1 0 2 0 1

10 1 0 0 2 211 2 0 2 2 012 2 0 0 0 013 2 2 2 1 214 2 2 1 0 215 2 1 1 2 1

7. CONCLUDING REMARKS

This article underlines the importance of balancing and estimability of second-order effects in asymmetricaldesigns for categorical factors.

The B-optimality criterion is valuable for the experimental practice, because it leads to (I-grade balanced)designs that optimize experimental resources, avoid biased conclusions and relieve the experimenter fromarbitrary choices. It is valuable from a scientific point of view, because it allows a deeper knowledge of thephenomena under study (estimability of first- and second-order effects). It is valuable from a methodologicalperspective, because it assures the minimum variance of first-order effect estimates and a particular optimalityof the designs.

The analysis of the proposed BANOD2 in terms of D-optimality shows that in two of the examined cases theD-optimal designs are equal to the BANOD2; in two cases the D-optimal design is different from the BANOD2and I-grade balanced; in all the other cases, the D-optimal designs are not I-grade balanced.

In terms of maximum eigenvalues, the two procedures lead to almost equivalent plans. In terms of minimumeigenvalues, the BANOD2 are increasingly less efficient as the run size increases.

Details about the FORTRAN code can be requested from the authors.

Acknowledgements

This research was supported by the National Interest Research Project (PRIN 2005) Progettazione Stabisticadell’innovazione ‘continua’ di prodotto. The authors, who contributed to this work on equal terms, express theirgratitude to the guest editor of this issue.

REFERENCES

1. Box GEP, Hunter WG, Hunter JS. Statistics for Experimenters. Wiley: New York, 1978.2. Montgomery DC. Design and Analysis of Experiments (5th edn). Wiley: New York, 2000.3. Wu CFJ, Hamada M. Experiments. Wiley: New York, 2000.4. Box GEP, Draper NR. Empirical Model Building and Response Surfaces. Wiley: New York, 1987.5. Myers RH, Montgomery DC. Response Surface Methodology. Wiley: New York, 1995.



6. Xu H. An algorithm for constructing orthogonal and nearly-orthogonal arrays with mixed levels and small runs.Technometrics 2002; 44:356–368.

7. Huwang L, Wu CFJ, Yen CH. The idle column method: design construction, properties and comparisons. Technometrics2002; 44:347–355.

8. Nguyen N-K. A note on the construction of near-orthogonal arrays with mixed levels and economic run size.Technometrics 1996; 38:279–283.

9. Kendall M, Stuart A, Ord JK. The Advanced Theory of Statistics. Griffin: Irvine, CA, 1983.10. Barone L, Lombardo A. Balanced asymmetrical nearly orthogonal designs for first and second order effect estimation.

Journal of Applied Statistics 2006; 33:373–386.11. Kuhfeld WF, Tobias RD, Garratt M. Efficient experimental design with marketing research applications. Journal of

Marketing Research 1994; 31:545–557.12. McLean RA, Anderson VL. Applied Factorial and Fractional Designs. Marcel Dekker: New York, 1984.13. Ratcliffe J, Buxton M, McGarry T, Sheldon R, Chancellor J. Patients’ preferences associated with treatments for

osteoarthritis. Rheumatology 2004; 43:337–345.

APPENDIX A. CONSTRUCTION OF THE CONTRAST MATRIX

For constructing the contrast matrix X it is possible to follow Kendall and Stuart9 (Vol. 3, Ch. 35). The block of Xconcerning the main effects can be constructed starting from a matrix with labels (0, 1), where the presence (1)and the absence (0) of a specific level are indicated for each column, and subsequently imposing that eachcolumn sums to zero (contrast definition). Alternatively, this block can be constructed by using orthogonalpolynomials (Montgomery2, Appendix X). In both cases each factor generates as many contrast columns as thenumber of its levels minus 1.

The remaining part of the contrast matrix, related to the interaction effects, can be constructed by multiplyingthe columns related to the main effects.

It is useful to show the block representation of XTX matrix for a design with only two factors:

XTX =

N 0 0 00 A D 00 DT B 00 0 0 C

where the first row and the first column are related to the mean contrast, the second row and the second columnare related to the main effect contrasts of the first factor, the third row and the third column are related to themain effect contrasts of the second factor, and the fourth row and the fourth column are related to the effectcontrasts of their interaction.

It is possible to see that, in the case of parameterization a la Kendall and Stuart, the sub-matrix C isalways orthogonal to the sub-matrices A and B; furthermore it is possible to demonstrate that A and B aremutually orthogonal in the proportional-frequencies case (D = 0, orthogonal condition). However, A and B arenot diagonal. Conversely, in the case of parameterization through orthogonal polynomials, A and B are alsodiagonal in the I-grade balanced case. This is the reason why this parameterization was here preferred.

The B-optimality criterion is independent from the coding of the contrasts because it takes into account onlythe number of occurrences of a level for each factor. This is particularly in agreement with the nature of thefactors that are here thought to be categorical.

Authors’ biographies

Stefano Barone is Assistant Professor of ‘Applied Statistics’ at the Faculty of Engineering of the Universityof Palermo (Italy). He received his Laurea degree in Aeronautical Engineering in 1996 and his PhD in TotalQuality Management in 2000 from the University of Naples. In 2000–2001 he received a post-doctoral grantfrom ELASIS (FIAT Research Center) and in 2001–2002 a post doctoral position at the University of Naples.



In 2003–2004 he was visiting researcher at the Chalmers University of Technology, Sweden. He teachesStatistical Methods for Six Sigma and Statistics and Probability. He is a member of the Italian Statistical Societyand of the scientific council of the European Network for Business and Industrial Statistics.

Alberto Lombardo received his Laurea degree in Statistics in 1980. He works at the University of Palermo(Italy), where currently he is full Professor in Statistics for Experimental and Technological Research at theDepartment of Technology, Production and Managerial Engineering and teaches statistics and probability anddesign of experiments at the Faculty of Engineering. He was a member of the Scientific Board of the ItalianStatistical Society in 1998–2002. His current research interests are design of experiments, service quality: designand assessment, and environmental statistics.


Construction and Optimality of a Special Class of Balanced Designs

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