An Integer Programming Approach to the Construction oftrend-free Experimental Plans on Split-Plot Designs

An Integer Programming Approach to the

Construction of Trend-Free Experimental Plans

on Split-Plot Designs

And.... L Carrano(alceleOrlt.ed~, BrianK.Thorn, andGuillermoLopez,Industrial and Systems Engineering Dept., Rochester InstiMe of Technology, Rochester, New York, USA

Abstract

Inmanyexperimental designs, the standard procedure in-volves randomization of the factlevel combination run or-der. There arecases,however, where it Isknownthata time or positiontrend thatcan seriously compromisetheresults of the

experimentmaybepresent. ThesetrendsInclude wear of tool-IngandeqUipment, leaming curves, changeIntemperatures, andso on, andthetrends may show up as linear, quadratic, or even higher order trends. All previously publishedwork has dealtwithvarious methods ofconstructingtrend-resistant run orderplanson full and fractional factorialdesigns.These pre-viouseffortshavenot addressed any additional~in thetrendsthatemerge when using hierarchicaldesignssuch as spIlt-piotplans.These designs are commoninmany manu-facturing experiments where complete randomization is not possibleoristooexpensive tobepractical.TheobjectIYeof thisworkisto establish the foundations of a method for con-structing linear and quadratic trend-resistant plans In two-Iev-eI spIlt-piot designs that addresses the two-dimensional trends that may occur. The methodology Involves deIIeIopment of a hybrid approach using the foldover method in each of the dimensionsofInterest and embedding these in a nonlinear Integer programming model Inthesearch for a feasible solu-tion. Feasibility of this approach is shownfor the particular case of a spilt-plot design (25 whoIe-pIot factorsand3' x 2' split-plot factors)performedon abrasive machining. In this case study, an experimental plan thatIs robust against all linear trends and most quadratic trends was achieved.

K~: Split-Plot Experimentsl Design, Integer Mathe-matical Programming, Trend-Resistant Designs, Foldover Methods

Introduction

Often in manufacturing or design, experimental design techniques are applied to understand input! output relationships or to develop recommended pa-rameter settings for a product or process. In these applications. a batch of experiments that follows an experimental array is perfonned. Common practice in an experimental setting includes randomization of the run order. The analytic procedures used to

analyze the results from experimentation usually re-quire that the experimental observations (or errors) be independent random variables. The practice of randomization helps in making this assumption valid. Additionally. by properly randomizing the experi-ment. the potential effects of extraneous factors may be "averaged out" as well. Unfortunately. experi-mental settings that impose restrictions on random-ization are routinely encountered during industrial experimentation. Batch processing. expensive equip-ment setups. and other factors often conspire to com-promise the experimenter's ability to randomize the order ofexperimentation. Techniques for dealing with such restrictions include the use of split-plot designs and hierarchical or nested factorial designs.

Furthennore. in some experimental situations. it is known that there will betime or position trends that can seriously compromise results. For example. in an experiment on a complex multi-stage manufacturing process (e.g.• a semiconductor fabrication facility). there might be a certain learning process thatoccurs over time as a result of making level changes in the factors being studied. or there could be progressive changes or drift in the temperature at which process equipment operates. or there might be systematic changes in the concentration of important chemical reagents. or there couldbewear issues associated with some of the process equipment. among others. Also. the nature of discrete manufacturing. where material flow occurs in batches (in the previous example. sili-con wafers). imposes a restriction on randomization that must be considered when perfonning experiments to improve the manufacturing processes or the pro-cess output. In situations such as those described above. it is important to consider systematic (non-ran-dom ron order) designs in which the estimates of the important factors of interest are trend resistant.

Previous work in this area includes Cox (1958), Hill(1960), Daniel and Wl1coxon (1966), Draper and Stoneman (1968), Joiner and Campbell (1976), Cheng and Jacroux (1988), Cheng (1990), and Jacroux (1990). These efforts have resulted in the develop-ment of experidevelop-mental designs that are resistant to lin-ear and quadratic trends for various

2'(

full factorial and2k-pfractional factorial designs. Concerning two-dimensional trends, there has been limited work in this area. Cox (1979) stated that row x column de-signs, specifically Latin squares, were subjected to row x column trend interactions and systematically generated 4 x 4 and 5 x 5 designs that resulted in a significant reduction in the variance ofthe experiment. This research brought forth the knowledge that the trend interactions can be influential in row x column designs. Edmonson (1993) extended this work to 2n

x 2mdesigns as well as other Latin-square designs. Mandeli (1999) introduced linear and quadratic row x column trend interactions and systematically gen-erated F-square and F-rectangle designs with these trend interactions. As shown, current archival litera-ture does not address the two-dimensional trends nor the trend interactions in split-plot designs.

The work proposed here extends the development of trend-resistant designs to other experimental set-tings beyond the

2"

and

2";>

families. Specifically, this effort will provide the foundations for a gener-alizable method that can be used to control for the effects of linear and quadratic trends in multiple di-mensions for commonly encountered split-plot and nested factorial design scenarios.

Methodology

and Case Study

The methodology proposed for building trend re-sistance into split-plot experiments is a hybrid ap-proach based on the foldover method (Cheng and Jacroux 1988), combined with integer nonlinear mathematical programming. This approach was motivated by work in abrasive wood machining ap-plications where tool wear, time, and location trends were significantinthe experiment (Carrano, Taylor, aDOLeID~ler

ZOOZ)_.

The underlying idea is that there is a trend in the experiment and that it can be modeled by a low-order polynomial. The objective is to assign the treatment combinations in such an order that the contrasts of major interest (main effects, in most cases) are or-thogonaltothe trend.Ingeneral, a contrast with

coef-ficients (a" a:!, ...,an) is considered orthogonal to a k-order trend ifthe responses are uncorrelated, have common variance, and meet the following:

(1) To illustrate with an example, consider a full rep-licate of the 23in standard Yates order [(1), a, b, ab, c, ac, be, abe], with equal intervals in~e or space. The A contrast (-1, + 1, -1,

+

1, -1, + 1, -1, + 1)is not resistant to a linear trend, where k

=

1, because the inner product:

(-1, +1, -1, +1, -1, +1, -1, +1)(1,2, 3, 4,5,6,7,8)'

'*

0 However, the ABC contrast (-1, +1, +1, -1, +1,-1, -+1,-1, +1) is resistant to both linear(k=I)

(-1, +1, +1, -1, +1, -1. -1. +1)(1.2.3.4,5,6,7.8)'

=

0 and quadratic(k=2) trends:

(-I. +1, +1. -I. +1.-1,-I. +1) (I. 4. 9. 16.25.36.49.64)'

=

0 For an explanation of the proposed approach and to show initial feasibility, an example that explores a particular case is presented. This consists of a seven-factor, two-level split-plot factorial with five whole-plot factors (A. B, C, D. and E) and two split-whole-plot factors (F at two levels and G at three levels).Inthis particular case, the five whole-plot factors determined a

i'i

factorial for 32 runs containing all possible treat-ment combinations of the 31x 21

=

6 split-plot blocks. This example was developed for an abrasive wood machining operation where all six specimens form-ing the split-plot were exposed to the same machin-ing pass, and therefore conditions, via customized fixturing. A complete description of this experiment with its corresponding results can be found in archi-val literature (Carrano, Taylor, and Lemaster 2(02). The factors involved were: (1) inputs or indepen-dent variables: wood species, grit size, depth of cut, tooling resilience, feed rate, spindle speed, and grain orientation, and (2) response: surface roughness as an indicator of final quality. Table 1 shows the set-tings for the two-level factors.

The wood species factor (G) had three levels de-nominated as: Eastern white pine (Pinus strobus L.), Hard maple (Acer saccharum Marsh), and White oak (Quercus albaL.).The experimental design was a tra-ditional split-plot. This type of design restricts ran-domization by imposing a variant ofa nested structure: all treatment-level combinations of a subgroup of

fac-tors are to be tested within each treatment-level com-bination of the remaining factors. The rationale be-hind this choice was due to some factors being easier

tovary within one machining

run

thanothers; that is, for a given setup, it was convenient to maintain spindle speed, feed rate, depth of cut, grit size, and spindle type (rigid versus resilient), while it was possible to machine different wood species and grain orientations in the same pass, with the aid of designed fIxturing. Also, there were some economies attained with nested designs when compared to full factorial designs.

Provided this structure, the so-called whole plot factors were: spindle speed (A), feed rate (B), depth of cut(C), grit size (D),and tooling resilience (E), allof them fIxed for

a

given experimental run. The split-plot factors were the remaining grain orienta-tion (F) and wood species(0). A fixture conducive to this experimental design was custom built. This is shown inFigure J.

Figure 2 shows the setup for one experimental run (machining pass) containing all combination of split-plot factors and using the resilient sanding head. The size of the experiment was determined by the factorial explosion and the experimental unit size. If treated as a full factorial, the total number of treatment combinations would be given by 3' x

26

=

192. Because all treatment combinations of the split-plot factors were treated together, the ex-perimental unit size was 3' x 21

=

6 specimens (species*orientation). This yielded 32 blocks or experimental runs per replication.

In this model, with fIve whole-plot factors, it is necessary to search for fIve independent generators plus the control tuple. These tuples are vectors of zeros and ones representing high and low levels of the treatment combinations obtained from the fIve factors. The six tuples chosen (for the whole-plot factors) were defined as shownin Table 2.

At this point, it is necessary to define the operator

EB

thatdescribes the combination between two or more 1'IJble 1

Setala.. for Two-Level Factors

Factor Code Spindlespeed A Feedrate 8 Depthof cut C Gritsiz.e 0 Tooling resilience E Grainorientation F Low-Level Setting(-) 1000 rpm 200inJmin. 0.004in. 150 Resilient Endgrain High-Level Setting(+) lSOOrpm 4OOinJmin. 0.008 in. 220 Rigid Parallelgrain FlrMn I WorkholcUal Fistare Flpre 2

Experimental Raa (MachlalnlPus)withRaiDent Saaellnl Rnd

tuples by adding their components in modulo 2. This is, by letting two tuples of N-order bere~sentedas

.(1) ( . (I) • (I) • (I) ) d.(2) ( . (2) • (2) • (2)

1

=

1" ₁₂ , ••• , In an 1

=

I, ,12 , ••• , In

), where the it) and i/2) represent zeroes and ones, then i(1)

EB

i(2)

=

if)>, where i}3)

=

i/1

)

+

i?) (modulo 2)

forj

=

I, 2, ..., n. This operator can also be used to show that none of the generators above is a linear combination of the five remaining tuples. As shown in Table3, the factorial plan for the whole-plot fac-tors (25

=

32 treatment combinations) is then gener-atedby combining the tuples in Yates standard order. By following this methodology, it was possible to generate a run sequence for the 32 runs that yielded resistance against linear trends for all fIve whole-plot factors main effects (A, B, C, D, E) and resis-tance against second-order (quadratic) trends for four of the fIve whole-plot factors (A, C, D, E). This

ad-Fi,,,re 3

Arrangemeat of Spllt·Plot Factors 10 Experimental Setup

Table 4

Orthogonality 01 MaiD Effects to LJDear aDd Quadratic TreDds

6 5 4 3 2 POsmON IN FIXTURE

-x

y

-z x z

-y

11 12 13 14 1 Ii 16

Y

X

-y

-X

-z

Z

21 22 23 24 25 26 2 32 • •

Main Effect l : .\ _{l:a/ jl} ai' i-I i-I A 0 0 8 0 -512 C 0 0 0 0 0 E 0 0

time trends across the runs but also against positional and time trends within the same machining pass. This scales the problem to two dimensions, and no method that addresses this issue is found in the available lit-erature. Available research would have implied meth-ods only considering the trend in the sub-plots dimension, while completely ignoring the trend in the whole-plot dimension. The sub-plots are nested under the whole-plots, meaning that the trends af-fecting the whole-plots will also affect the sub-plots. This produces a row x column trend interaction or two-dimensional trend.

The proposed methodology, as applied to this particular case, approaches the two-dimensional foldover by nonlinear integer mathematical program-ming.Figure3 illustrates the problem by coding the levels (low, medium, and high) for factor G as [Xl, [Y], and [Z]; and the levels (high and low) for factor F as [+] and [-]. Under this coding scheme,-Z rep-resents the treatment GhighFlow,the coding

+

X

repre-sents the treatment

V

l6iV

Flijjilf and

so

fanh,

In Figure 3, the experimental runs (1, 2, .. " 32) are performed in the order dermed in Table 3. As mentioned before, this problem could be modeled as a nonlinear integer model. One proposed devel-opment takes the following form:

Table Z

IDdepeDdeDt GeDenton (TlIpla)

A B C D E j<Ol

=

_{(0 0 0 0 0)} j<l)

=

( l 0 I I 0) jIll

=

_{(0 I 0 I 0)} j<l)

=

( 1 0 0 0 I ) jlf'

=

_{(0 0 I 0 I )} j(5)

=

( I I I I I ) Table 3

PluSeqaeDdag GeDentloD by Foldover Method

Foldover Generator Treatment Order Method Combination Combination

I jlOI _{(00000) (I)}

2 j<ll _{(101l0) acd}

3 jIll (01010) bd

4 jlll~jlll _{(l1l00) abc}

5 jill _{(10001) ae}

6 j<llejl]1 _{(oolll) cde}

7 jI21~jll' (11011) abde

8 jIll~jl2l~jIll _{(01l01) bee}

9 j'fl _{(00101) ce}

10 jlll~j'fl (Iooll) ade

11 jillejef, _{(01111) bcde}

12 jll)~jilt~jefl _{(11001) abe}

13 j'l)~ilfl _{(10100) ac}

14 jill~jell~iefl _{(00010) d}

15 ill)~i<l'~j14, _{(1l110) abed}

16 jIll~jIll~jll.~il41 _{(01000) b}

17 jiSt (1l111) abcde 18 jlll~jIS' (01001) be 19 i'21~i's, (10101) ace 20 i 'll e jll.~j'5' (0001l) de 21 i'·\'eiISI (01110) bed 22 ill. e jf], e j'S' _{(11000) ab} 23 ill) e j<l' e i 's, _{(00100) c} 24 ill I~jll'~jll'~jiS' _{(10010) ad}

25 i'41~i lsi _(11010) abd

26 jill e jff. e j'5' _{(0 Il 00) be}

27 jIll e jlf)~j'S) _{(10000) a}

28 ill I~ifI)~j14'~j'S' _{(00110) cd}

29 jt\,~jlf'~ils, _{(01011) bde}

30 ill I~jIll~j'f' e j'S' _{(11101) abce}

31 i llt~ill)~i14,~jlSI _{(00001) e}

32 jll' e jll' e jOt e j'f' e jl5' (1011l) acde

dresses the trends that arise in the dimension defined by the whole-plot factors such as time and learning ttends. Table 4 shows which main effects are or-thogonal to the linear and quadratic ttends.

Six treatments accounted for the two split-plot fac-tors: F at two levels and G at three levels. The allo-cation of the six specimens within each run poses a different problem. as they are embedded into a hier-archical design structure, The problem is such that not only was it desired to build resistance against

XU'~j'ZuE

{-1.0.1}

P"P

2~o

where:

Xij : Glowinexperimental run i (i

=

1.2•...• 32)

and fIxture positionj (j = 1. 2... 6) YIj: Gmediumin experimental run i (i

=

1, 2...

32) and fIxture positionj (j

=

I. 2. ...• 6)

Zlj: G_biP in experimental run i (i

=

1. 2•...• 32) and fIxture positionj (j

=

1. 2. ...• 6)

PI' P2:

Dummy variables in the objective function The objective function is. in this case. composed of dummy variables because the main interest is in the search offeasibility.Ifa feasible solution set were found. then the resultant plan will be resistant to lin-ear trends. The first three constraints enforce the two-dimensional foldover requirement for each of the three levels of factor G. The remaining constraints

MIN

PI +P

2

S.t.

~{t(Xu)j

J

=0

~{t(Yu)j

)=0

~{t(Zu)j

)=0

IXul+IYul+IZul

=

1

'Vij combination 6

1:Xu =0

'Vi

j-I 6

1:Yu =0

'Vi

j-I 6

1:Zu =0

'Vi

j-I 6

1:IX

ul=2

'Vi

j-I

6

1:IYul=2

'Vi

j-I

6

1:I

Z

ul=2

'Vi

j-I

(2)

made sure that, within each run. all combinations of F and G were properly assigned. Following the ter-minology established beforehand,ifthe feasible so-lution yielded X₁₃

=

+

1. this meant that a 0low specimen in combination with Fblpwas to be placed

in thethirdposition of the fIxture during experimen-tal run#1. Similarly. if the solution yielded

Zn

4

=-I, then a GhI&h specimen in combination with Flow was allocated in the fourth position in the fIXture during run #23. For this particular problem. the non-linear system was converted into standard fonn and solved by using Lingo version 4.0. The model had 3,850 linear constraints, 1,152 integer variables, and solvedafter18,758 iterations. The runtimewas 1'25" with a 400

MHz

processor. A feasible solution was found, and a run sequence that is robust against lin-ear trends acrossall32 runs for F and G was conse-quently obtained. The complete coded solution given

by

the mQdel i§ pre§ented in

Table

j;

The factor resistance against linear trends in this two-dimensional scenario can be verifIed by Eq. (3) for any run sequencing and factor.

(3) wherej=1, 2, ... , 6 represents the spatial location of the specimen in the fIXture; i

=

I, 2, ... ,

32

repre-sents the run number, and aijis the integer

assign-ment value (from the set 1,0,-1)obtained from the integer mathematical programming solution. This equation is applied independently for X,Y, and Z.

Finally, a comparison between the solution set ob-tained with the proposed approach against traditional randomization was developed. A pseudo-random generation procedure was used to generate 500,000 seeds and an equal number of random designs. Equa-tion (3) was used to evaluate the resistance of each one of these random plans against identical trends. The best random design produced a value of 6, the worst 2,202, and the average ofall 500,000 plans was 680. None of the random designs produced the ideal value of 0 (highest trend resistance).

Conclusions and Future Work

In this work, feasibility of a procedure for con-structing two-dimensional, trend-free experimental plans on two-level. split-plot designs has been es-tablished. The proposed method utilizes results ob-tained from widely understood methods for

Table S

RIIII SequeDCIIII for SpUt.P1ot Factors

Run

Fixture Position

# I 2 3 4 5 6 1 +y -y +X -z -X +z 2 +y +X -z

-x

-y +z 3 +y +X -y -X -z +z 4 +y -z +X -X +z -y 5 +y -y +X -z

-x

+z 6 +X -X +y +z -y -z 7 +X +y +z -X -z -y 8 +X +y +z -X -z -y 9 +y +X -z +z -X -y 10 +X +y -X -z -y +z 11 -X +y +z +X -Y -z 12 +X +y -y -X +z -z 13 +z +X -X +Y -z -Y 14 +y +X +z -X -z -y 15 +y -z -X +z -Y +X 16 +y +X -X -z -y +z 17 +X +z -X -Y +Y -z 18 -z +y -X +z -y +X 19 +X -X +y -y -z +z 20 +X -X +z -z +y -y 21 +X -y -z -X +y +z 22 +X +z -y -X -z +y 23 -X +Y +X -z +z -Y 24 -y +X +z -z +y -X 25 +y -y +z +X -X -z 26 -X +z -y -z +X +y 27 +X -X -y +z -z +y 28 -y -X +X +y -z +z 29 -y +X +z +y -z -X 30 -X -z -y +y +X +z 31 -X -y -z +X +z +y 32 -X -z +Y +z -Y +X where X Glow + F_hlih y Gmodlum - Flow Z G_blih

constructing trend-free run orders in2kand2k-p

sce-narios (the foldover method or, if preferred, the Daniel-Wilcoxon method) and combines them with an integer programming formulation of the problem to generate trend-resistant designs under split-plot or nested factorial environments. The mathematical approach works well for the particular case stated but has not been extended to the general case where different hierarchical designs and different combi-nations of nested and crossed factors are present. Also, the ramifications of the integer nonlinear pro-gramming model are yet tobeexplored. A thorough analysis when more than one feasible solution is found as well as development of a metric to evalu-atesuch feasible solutions is needed. Finally, in the

case when no feasible solution is found, a detailed methodology for constraint relaxation must be de-veloped. Future efforts willbeallocated in the afore-mentioned areas as well as in developing a generalizable method that canbeused to control for the effects of trends in multiple dimensions for com-monly encountered split-plot, split-split-plot, and other nested factorial design scenarios.

References

Carrano. A.L.; Taylor. J.B.; and LemllSter. R.L. (2002). "Parametric characterization of peripheral sanding," Fo"st Produclll Journal (v52, n9). pp44-50.

Cheng. C. and Jacroux. M. (1988). "The construction of trend-free run orders of two-level factorial designs," Jounwl of the

Ameri-can Statistical Association (v83, n404), Theory and Methods.

ppI152-1158. .

Cheng, C. (1990). "Construction of run orders of factorial designs,"

Statistical Design and Analysis of Experiments. Subir Ghosh. ed.•

Ch. 14, pp423-439.

Cox, D.R. (1958). Planning of Experiments. New York: John Wiley and Sons. Inc.

Cox, D.R. (1979)."Aremark on systematic Latin squares." Journal of

Royal Statistical Society. Series B (Methodologil"al) (v41. n3).

pp388-389.

Daniel. C. and Wl1coxon,F.(1966). "Factorial 2.... plans robust against linear and quadratic trends." Technometrics (v8. n2). pp259-278. Draper. N. and Stoneman. D. (1968). "Factor changes and linear trends in eight-run two-level factorial designs." Technometrics

(vlO, n2).

Edmonson. R.N. (1993). "Systematic row-and-column designs bal-anced for low order polynomial interactions between rows and columns." Journal of the Royal Statistical Society. Series B

(Meth-adological) (v55. n3). pp707-723.

Hill, H. (1960). "Experimental designs to adjust for time trends."

Technometrics (v2. nO. pp67-82.

Jacroux. M. (1990). "Methods for constructing trend-resistant run orders of 2-level factorial experiments," Statistical Design and

Analysis of Experiments. Subir Ghosh. ed.• Ch. 15. pp441-457.

Joiner. B. and Campbell. C. (1976). "Designing experiments when run order is important," 7'echnonlt'trics (vI8. n3).

Mandeli. J.P. (1999). "Construction of systematic row-column de-signs with treatments unconfounded with linear row x column and quadratic row x column interactions." Statistics& Probability

utter"(v42). pp229-237.

Authors' Biographies

Andres L. Carrano is an IlSsociate professor in theIndustrial and Systems Engineering Dept.atRochester Institute of Technology (Roch-ester. NY). He received a PhD in industrial engineering from North Carolina State University. His research interests include design forthe

environment. sustainable product design, and surface metrology. He is a senior member of SME and founding chair of SME Chapter S317.

Brian K. Thorn is an associate prQfessor in the Industrial and Systems Engineering Dept. at Rochester Institute of Technology (Roch-ester. NY).Hereceived a BS in industrial engineering from Rochester Institute of Technology and an MS and PhD from Georgia Tech. His research interests include applied statistical methods. sustainable prod-uct and process design.&.,well as life cycle analysis.

Guillermo Lopez is a graduate research assistant and master of science candidate inthe Industrial and Systems Engineering Dept. at Rochester Institute of Technology (Rochester, NY). He received a BS in industrial engineering from Pontificia Universidad Catolica Madre yMaestra (Santo Domingo. Dominican Republic).