Process Yield Improvement Through Optimum Design of ......Fixture Layouts in 3D Multistation...

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T. Phoomboplab

D. Ceglarek

Warwick Digital Laboratory, WMGUniversity of Warwick,

Coventry CV4 7AL, UK;Department of Industrial and Systems

Engineering,University of Wisconsin–Madison,

Madison, WI 53706

Process Yield ImprovementThrough Optimum Design ofFixture Layouts in 3D MultistationAssembly SystemsFixtures control the positions and orientations of parts in an assembly process. Inaccu-racies of fixture locators or nonoptimal fixture layouts can result in the deviation of aworkpiece from its design nominal and lead to overall product dimensional variabilityand low process yield. Major challenges involving the design of a set of fixture layouts formultistation assembly system can be enumerated into three categories: (1) high-dimensional design space since a large number of locators are involved in the multista-tion system, (2) large and complex design space for each locator since the design spacerepresents the area of a particular part or subassembly surfaces on which a locator isplaced, (here, the design space varies with a particular part design and is further ex-panded when parts are assembled into subassemblies), and (3) the nonlinear relationsbetween locator nominal positions and key product characteristics. This paper presents anew approach to improve process yield by determining an optimum set of fixture layoutsfor a given multistation assembly system, which can satisfy (1) the part and subassemblylocating stability in each fixture layout and (2) the fixture system robustness againstenvironmental noises in order to minimize product dimensional variability. The proposedmethodology is based on a two-step optimization which involves the integration of ge-netic algorithm and Hammersley sequence sampling. First, genetic algorithm is used fordesign space reduction by estimating the areas of optimal fixture locations in initialdesign spaces. Then, Hammersley sequence sampling uniformly samples the candidatesets of fixture layouts from those predetermined areas for the optimum. The process yieldand part instability index are design objectives in evaluating candidate sets of fixturelayouts. An industrial case study illustrates and validates the proposed methodology.�DOI: 10.1115/1.2977826�

Keywords: fixture layout, multistation assembly, process yield, instability index, geneticalgorithm, Hammersley sequence sampling

IntroductionFixture design is one of the most important design tasks during

rocess design for a new product development since it involvesefining the locations and orientations of parts during assemblyrocesses as well as providing physical support, which can greatlyffect product dimensional variations and process yield. Gener-lly, fixture design process can be divided into three stages whichre �1� fixture planning, �2� fixture configuration, and �3� fixtureonstruction �1�. In the fixture planning stage, issues related to theumber of fixtures needed, the type of fixtures, the orientation ofxture corresponding to orientation, and the joining or machiningperations, which fixtures have to handle are identified. The fix-ure configuration stage determines the layout of a set of locatorsnd clamps on a workpiece surface such that the workpiece isompletely restrained. Finally, the fixture construction stage in-olves constructing fixture components and then installing them toupport the workpiece. Specifically for complex assemblies suchs an automotive body, a ship hull, and an aircraft fuselage, fixtureayout design, which falls under the domain of the fixture plan-ing and fixture configuration stages, is a primary concern and it

Contributed by the Manufacturing Engineering Division of ASME for publicationn the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript receivedugust 17, 2007; final manuscript received July 18, 2008; published online October0, 2008. Review conducted by Shivakumar Raman. Paper presented at the 2007nternational Conference on Manufacturing Science and Engineering �MSEC2007�,
tlanta, GA, October 15–17, 2007.
ournal of Manufacturing Science and EngineeringCopyright © 20

ded 10 Oct 2008 to 137.205.175.119. Redistribution subject to ASM

involves adjusting the design nominalof locator positions in orderto eliminate mean shifts and reduce variations of all key productcharacteristics �KPCs�.

However, research on fixture layout design in a multistationassembly process is limited because of the lack of a methodologyto predict product dimensional variations and a process yield dur-ing the product/process design phase �2–4�. The absence of such amethodology poses special challenges in assessing a performanceof a fixture layout design and its impact on a process yield. Cur-rently, researchers have developed a variation propagation modelfor multistation assembly processes using a state-space represen-tation �5–7�, which allows prediction of KPC variations underknown statistical characterizations of key control characteristics�KCCs�. The extension of this work to three-dimensional �3D�rigid body assembly processes has led the development of theso-called stream-of-variation �SOVA� methodology �8,9�.

Nevertheless, the fixture layout design for multistage assemblyprocesses continues to pose various challenges. For example, mul-tistage assembly processes usually involve a large number of lo-cators since these processes consist of a large number of partsassembled in many assembly stations. For instance, a typical au-tomotive body assembly consists of 200–250 sheet metal partsassembled in 60–100 assembly stations with 1700 to 2100 loca-tors �4,10�. In addition, the locating positions used in one stationmay be reused in the different stations. Thus, a fixture layoutdesign methodology optimizing a fixture layout independently for
each assembly station may not necessarily lead to a good design
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olution because of the interdependencies between the fixture lay-uts in assembly systems. The challenges in designing a set ofxture layouts for multistation assembly system can be summa-ized into three categories as follows.

�1� High-dimensional design space. Multistation automotiveassembly processes have a very large number of locatorssince the number of locators increases proportionally to thenumber of parts and assembly stations.

�2� Large and complex design space for each locator. The de-sign space of a locator can be defined as any allowed posi-tion of the locator on a corresponding workpiece surface.Thus, the design space of each locator varies with the sizeand surface shape of the workpiece. In addition, the designspace is further expanded since fixtures need to be designednot only for individual parts but also for all intermediatesubassemblies.

�3� Nonlinear relations between locator nominal position andKPCs. Deviations of locators from their nominal positionshave nonlinear effects on KPC variations.

This paper addresses the aforementioned challenges by propos-ng a methodology for optimizing fixture layouts in all assemblytations simultaneously. This is to ensure that the variation propa-ation throughout the whole assembly line is considered tochieve the dimensional quality threshold of the final product. TheOVA model as presented in Refs. �8,9� is applied to assess theerformance of fixture layout design on product dimensionalariation. The proposed methodology is based on the integrationf the genetic algorithm �GA� with Hammersley sequence sam-ling �HSS�. The instability index is also incorporated into theroposed methodology to ensure that a fixture layout design meetshe locating stability requirement. The methodology is conductedn two steps. First, the genetic algorithm is used for design spaceeduction by estimating the areas of optimal fixture locations innitial design spaces. Then, Hammersley sequence sampling uni-ormly samples the candidate sets of fixture layouts from thoseredetermined areas for the optimum. GA is selected in the firsttep since it can handle the nonlinearity between the KPC andocator positions. The shortcoming of GA as a time-consumingeuristic optimization technique is alleviated by incorporating thenstability index and discretizing the continuous design spaces ofll locators to improve convergence to an optimal solution. Thenstability index helps to expedite the search capability by elimi-ating those fixture layouts that do not meet the locating stabilityequirement. Discretization of continuous design spaces reduceshe initial candidate design space of locator positions, which GAan select. HSS is conducted in the second step to compensate forny potentially missed optimum solutions by uniformly samplinghe candidate locator positions in the areas defined by conductingA search in the first step. The uniform sampling of HSS evenly

elects the representatives of fixture layouts in all assembly sta-ions, which increase the probability to select near-optimal locatorositions. The rest of this paper is organized as follows. The state-f-the-art in fixture layout design is reviewed in Sec. 2. The prob-em formulation, as well as the descriptions of a process yieldalculation based on the SOVA model and instability index, isescribed in Sec. 3. The proposed multifixture layout design meth-dology is presented in Sec. 4. A case study illustrating the appli-ation of the proposed methodology on an automotive underbodyssembly and a comparison of optimization algorithms are pre-ented in Sec. 5. Finally, conclusions are drawn in Sec. 6.

Literature ReviewIn general, the fixture layout design has to satisfy four func-

ional requirements, which are �i� locating stability; �ii� determin-stic workpiece location; �iii� clamping stability; and �iv� totalestraint �1,11–13�. Locating stability is related to the design of axture layout that can provide static equilibrium of a workpiece
hen it is placed on fixtures. Second, the fixture should provide
61005-2 / Vol. 130, DECEMBER 2008


the deterministic location for the workpiece to ensure positionaccuracy during operation. Third, clamping stability involves de-termining the sequence of clamping and its layout that does notdisturb the stability and position accuracy of a workpiece estab-lished by locators in the previous two functional requirements.Last, clamps should completely restrain the workpiece to with-stand any forces and couples to maintain the workpiece in anaccurate position. Past research in fixture layout design has pro-posed approaches to address these functional requirements�14–24�.

Locating stability is one of the most important requirements infixture design since a workpiece has to satisfy this requirementbefore achieving other functional requirements. Locating stabilityis mainly concerned with static equilibrium under the given fix-turing condition in the presence of manufacturing forces. In addi-tion, the fixture layout design has to ensure that all locators main-tain contact with the workpiece throughout the manufacturingoperation. Issues involving locating stability begin when theworkpiece is placed on locators as these locators provide a supportagainst gravity forces until the workpiece is processed. Thus, lo-cating stability also involves fixture force and kinematic analysisto estimate the necessary clamping forces to maintain a workpiecein equilibrium. Roy and Liao �14� presented a quantitative evalu-ation of the part location stability based on screw theory. Themethod proposed by Roy and Liao �14� can help in designing afixture configuration in automated fixture design environment.

Deterministic workpiece location involves designing the locatorpositions or a fixture layout to provide a unique and accurateposition and orientation of a workpiece with respect to its fixturereference frame �15�. Common challenges involving the fixturelayout design that will meet this functional requirement includethe positioning accuracy, which is subject to a random manufac-turing error of fixture elements, geometric variability of the work-piece, and workpiece positioning errors induced by fixture posi-tion. In general, the position variability of the workpiece can bepredicted from the statistical characterization of the dimensioningand tolerancing scheme assigned to the fixtures and their contactpoints on the workpiece. Thus, determining the fixture layout,which is not sensitive to these variation sources, can minimize theworkpiece positional variability. Researchers have responded tothis challenge by proposing various methods and sensitivity indi-ces in order to determine the optimal locator positions in a fixturelayout. For example, Cai et al. �16� proposed a variational methodto design a robust fixture layout using the Euclidean norm of thefixture sensitivity index. Wang �15� and Wang and Pelinescu �17�determined the optimal fixture layout by maximizing the determi-nant of the information matrix �D-optimality�. Carlson �18� as-sessed the fixture locating scheme in terms of workpiece positionerrors by using the quadratic sensitivity equation. However, allaforementioned methodologies are limited to fixture layout designinvolving a single workpiece.

Clamping stability and total restraint are functional require-ments that are related to determining the clamping positions andforces, which do not affect the part locating stability and positionaccuracy of workpiece provided by the locators. Clamps applyforces on the workpiece against any external force to ensure totalrestraint �11�. The challenge in designing clamping locations is tominimize the workpiece deformations under clamping and exter-nal forces. Many studies have addressed these challenges byadopting the finite element method �FEM� to design locator andclamp layouts. For instance, Menassa and DeVries �19� proposedthe integration of Broyden–Fletcher–Goldfarb–Shanno optimiza-tion algorithm and FEM simulation to determine the fixture layoutthat can minimize deflection of the workpiece. Cai et al. �20�proposed the fixture layout optimization for deformable sheetmetal parts based on nonlinear programming and FEM analysis.In a similar vein, Krishnakumar and Melkote �21� employed GAto optimize a fixture layout that can minimize the deformation of
the machined surface due to clamping and machining forces. In
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ddition, there are also several approaches proposed to determinehe clamping design. For instance, Li and Melkote �22� proposedn approach to minimize the maximum positional errors by con-idering the workpiece dynamics during machining. DeMeter �23�roposed a technique to determine the optimal locator layoutased on min-max loading criteria. Marin and Ferreira �24� pre-ented the method to optimize the 3-2-1 fixture layout based oncrew theory. Nevertheless, these studies mainly focus on designf fixture layout for a single workpiece. Currently, most of thexture layout design methodologies are limited to a single work-iece fixture layout design. Table 1 summarizes the current state-f-the-art research in the area of fixture layout design and pre-ented in the context of fixture functional requirements.

Current research related to fixture layout design for multistationssembly processes is limited because of the challenges in devel-ping a variation propagation model and computational complex-ty. Recently, Camelio et al. �25� presented the fixture layout de-ign for compliant part assembly by considering the effects of partariation, tooling variation, and assembly springback. Kim anding �26� proposed a methodology to design multifixture layouts

n multistation assembly based on a station-indexed state-spaceodel �5–7,28�. The extension of Kim and Ding �26� in designingxture layouts for a product family can be found in the work ofzquierdo et al. �27�. Kim and Ding �26� involved determining aet of fixture layouts for all assembly stations that are insensitiveo the variations of random manufacturing errors of fixture ele-

ents, geometric variability of the workpiece, and workpiece po-itioning errors induced by fixturing position. The methodologyeveloped by Kim and Ding �26� uses E-optimality to minimizehe eigenvalue of the information matrix and exchange algorithmEA�, first proposed by Cook and Nachtsheim �29�, to determinehe optimal set of fixture layouts. The design objective of the

ethodology proposed by Kim and Ding �26� is similar to theesign objective of this paper in designing a set of multiple fixtureayouts, which is robust to environmental noise. However, therere fundamental differences between both methodologies in termsf two-dimensional �2D� versus 3D problem formulation as wells in specifics of the developed methodologies as elaborated be-ow.

The design problem addressed by Kim and Ding �26� is limitedo 2D assembly processes while this paper focuses on 3D assem-ly processes. Designing fixture layout in 2D leads to three sim-lifications, as follows: �i� the locating stability functional require-ent is not taken into consideration, �ii� the variations caused byating joints between the two parts are not included in the 2Dodel, and �iii� the design space dimensionality, as well as non-

inearity between the KCC locator positions and KPCs in 2D fix-ure layout design is significantly less than those in the 3D fixtureayout design problems since no out-of-plane variation is includedn the 2D model. These three simplifications significantly limit the

Table 1 State-of-the-a

Locatingstability w

Fixturelayoutdesign

complexity

Singleworkpiece 3D Roy and

Liao �14�

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ndustrial application of 2D fixture layout design methodology

ournal of Manufacturing Science and Engineering


since �i� the locating stability consideration is mandatory and mustbe achieved before meeting other functional requirements and �ii�there is a variety of part-to-part joints in the assembly processessuch as lap, butt, and T-joints, which have different impacts onproduct dimensional variations. This paper considers the fixturelayout design for 3D assembly processes, which addresses theaforementioned challenges in 2D problems.

Furthermore, the methodology proposed in this paper also dif-fers from Kim and Ding’s �26� approach with respect to the fol-lowing specifics.

�1� An evaluation index to assess fixture layout design. Kimand Ding �26� minimized the sensitivity index of informa-tion matrix in their approach while the approach proposedin this paper integrates the following two objectives: �i�maximize the percentage of KPCs conforming to specifica-tions and �ii� minimize part locating instability. Evaluationindices used in both methodologies have different advan-tages and disadvantages. The sensitivity index used by Kimand Ding �26� is less computationally intensive than thecalculation ratio of KPCs conforming to design specifica-tions by using simulation techniques such as the MonteCarlo approach. However, sensitivity index is difficult tointerpret and to provide explicit relations between the indexand product quality while the percentage of KPCs conform-ing to design specifications can explicitly indicate productdimensional quality. Moreover, part locating stability isused in this paper to ensure that fixture layout design satis-fies necessary fixturing functional requirement, which is notconsidered by Kim and Ding �26�.

�2� Searching algorithm to select optimal locator positions.Kim and Ding �26� used EA with enhanced computationalcapabilities done by increasing the exchange rate in eachiteration and reducing the candidate fixture locations byusing an experience-based approach. Since 2D fixture lay-out design optimizes only the layout of two locators perpart, four-way and two-way locating pins, the reduction ofdesign space based solely on experience is feasible. On theother hand, the methodology proposed in this paper is ana-lytically based and conducts two-step optimization: �i� ini-tial reduction of design space by using GA approach and�ii� uniform sampling for the optimum multiple fixture lay-outs by using HSS approach. Instead of relying on the de-signers’ experience, in the first step, GA is employed toreduce the size of all locator design spaces, especially im-portant for the design spaces of NC blocks, which havehighly nonlinear relations with KPCs. The first step inte-grates GA with the part instability index, which further en-hances search performance by eliminating the candidatefixture layouts that do not meet the location stability re-

fixture layout design

ture design functional requirements

terministicpiece location

Clampingstability Total restraint

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Menassa and DeVries �19�Cai et al. �20�Krishnakumar and Melkote �21�Li and Melkote �22�DeMeter �23�Marin and Ferreira �24�

lio et al. �25�nd Ding �26�rdo et al. �27�

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mine areas that have a higher probability of containing op-timal positions for each locator. Then, HSS conductsuniform sampling in the areas around the locator positionidentified by GA for optimal locator positions. Two-stepoptimization, which integrates GA and HSS, is very benefi-cial in a large design space problem with nonlinear rela-tions between locator nominal positions and KPCs, espe-cially when determining the locations of NC blocks in 3Dfixture layout design. A comparative analysis of the meth-odologies proposed in this paper with the approach pro-posed by Kim and Ding �26� is summarized in Table 2.

Problem DescriptionLet us denote that there are N locators in a given multistation

ssembly process. A locator position in the assembly system isescribed as

rk = �x,y,z�k, k = 1,2, . . . ,N �1�

here �x ,y ,z�k represents the Cartesian coordinates of a locator rkIn this paper, a binary assembly process is taken into consider-

tion where two parts are assembled at each station. A fixtureayout L in an assembly station includes information about locatorositions, which are used for locating two parts: a root part �S�nd a mating part �M�. Thus, a fixture layout L is represented ascollection of two distinct sets.

L = �S,M� �2�

S = �ri�, i = 1, . . . ,n �3�

M = �r j�, j = 1, . . . ,n − m �4�

here n=6, if the corresponding part is a rigid body, n�6, if aorresponding part is a compliant part, and m is a number ofegree-of-freedom �DOF�, which the part-to-part joint constrainsmating part.Let � represents a set of fixture layouts for a given multistation

ssembly process with p assembly stations. A set of fixture lay-uts, �, is expressed as

Table 2 Comparative analysis of Kim an

Kim and Ding �26�

Designproblem

2D fixture layout design• Functional requirement:

Deterministic workpiece locatio

• Fixture elements: only four-wayway locating pins

• Mating joint: only 2D lap-jointdesign between parts

• Design space: 2D design spacewith moderate nonlinearity betwKCCs and KPCs

Multifixturelayout designmethodologies

Exchange algorithm

• Evaluation index: Minimizingsensitivity index based on E-optimality approach

• Enhanced capability in searchingoptimal locator positions: Increanumber of exchanges per iteratiand reduce the candidate spaceexperience-based approach

� = �L1,L2, . . . ,Lp� �5�

61005-4 / Vol. 130, DECEMBER 2008


The above relations of locator positions, fixture layouts, and aset of fixture layouts are illustrated in Fig. 1.

In a single assembly station, all locator positions are required toprovide the locating stability for workpieces. Additionally, the setof fixture layouts in a multistage assembly process has to be ro-bust to environment noise, which results in minimum dimensionalvariations of the final product. In this paper, a process yield and aninstability index are used as criteria for determining an optimalfixture system. The design parameters are the locator positions,rk,in a set of fixture layout �. Therefore, the optimization scheme isexpressed as

maximize� yield��

subject to �� 0 �6�

where yield �·� is a function for calculating process yield, and ��·�is a function for assessing the instability index for a set of fixturelayouts in an assembly system.

The remaining part of Sec. 3 is organized as follows. The re-view of design evaluation indices and method to calculate processyield based on SOVA model are presented in Sec. 3.1. To assessand compare the location stability between the two sets of fixturelayouts, this paper adopts the instability index based on screwtheory proposed by Roy and Liao �14�, which is presented in Sec.3.2.

3.1 Design Evaluation Indices. A fundamental aspect of fix-ture layout performance is its robustness against environmental

ing †26‡ and the proposed methodology

Proposed in this paper

3D fixture layout design• Functional requirement: Locating

stability and deterministic workpiecelocation

two- • Fixture elements: four-way locatingpins, two-way locating pins, and NCblocks

• Mating joint: lap joints, butt joints, T-joints, and mixed joints in 3D assembly

• Design space: 3D design space andhigh nonlinearity between KCCs andKPCs

Two-step optimization whichintegrates GA and HSS

• Evaluation indices: �i� Maximizingpercentages of KPCs conforming tospecifications and �ii� minimizing theinstability index

• Enhanced capability in searchingoptimal locator positions: Using GAand instability index as strategic designspace reduction and using HSS toconduct local search

Fig. 1 Fixture layout representation for a multistage assembly

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oise to minimize product dimensional variability, which is char-cterized by KPC variations. In assembly processes, the variationsf KPCs are caused by two main factors, which are �1� variabilityf locator positions and �2� part-to-part interactions �9�. The po-ition variability of a locator depends on the dimensioning andolerancing scheme assigned to locators and geometrical shaperrors of a workpiece. Product dimensional variability caused byart-to-part interactions stem from errors related to part-to-partating features, which characterize variation propagation between

arts �10�.The SOVA model is used in this paper as an assembly response

unction to assess the robustness of a designed set of fixture lay-uts against the aforementioned variation sources. The SOVAodel allows evaluating KPC variations since it defines relations

etween KPC and KCC vectors, as shown in Eq. �7�. The detailsn formulating the SOVA model are discussed in Refs. �8,9�.

�KPC1

KPC2

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a21 a22 . . . a2,n

] ] � ]

am,1 am,2 . . . am,n

�m�n

SOVA matrix

�KCC1

KCC2

]

KCCn

�n�1

or KPC = A�� · KCC �7�

here A�� is the SOVA matrix, which is formulated based onhe dimensional relationships among fixture layout design, part-o-part mating features, and the assembly sequence. In practical,he relations between KCC variations, including locator positionariations, and KPC variations are non-linear. Huang et al. �8,9�roposed the technique to approximate the relations into linearodel, as shown in Eq. �7�. Each element in the SOVA matrix,

i,j, represents as sensitivity of KCCj on KPCi, which consistsnformation regarding to nominal positions of locators and part-o-part joint. Locator position adjustment while conducting fixtureayout optimization results in different structure of SOVA matrix,hich then can be used to compare the performance of each set ofxture layouts. If the variations of KPCs illustrated by variancenalysis or simulation are reduced after a fixture layout is ad-usted, it indicates that the fixture layout increases in its robust-ess. We use the SOVA model as the assembly responseunction,A, to evaluate the impact of KCC variations, u, on thePC variations, y. The relationship in Eq. �7� can be expressed as

y = Au �8�In general, KPC variations can be reduced by �i� tightening the

olerances of KCCs, u or �ii� optimizing the assembly responseunction represented as a SOVA matrix, A, to be robust to KCCariations. Tightening KCC tolerances is the most straightforwardpproach to reducing KPC variations. However, its trade-off in-olves increasing tooling cost in order to produce tooling at higherrecision. Optimizing the assembly process is more appealing inractice since KCC tolerance ranges can be increased and thessembly process is still able to achieve the same KPC variationevels. Relaxing KCC tolerances usually leads to lower productionost. Fixture layout design is one approach that can increase as-embly process robustness.

In a fixture layout design, we aim to determine the locatorominal positions, which can minimize the KPC variations whilehe tolerances that control variations of KCCs, u, are constant.ifferent locator positions contribute to the alteration of elements

n the SOVA matrix, A. The robustness of the SOVA matrix re-ulting from altering the locator positions can be assessed by twopproaches: �i� a loss function based on sensitivity indices and �ii�n estimated percentage of nonconforming items.

In the sensitivity index approach, the product dimensional qual-ty is measured by the variations of yTy=uTATAu. To minimizehe variations of yTy, the robustness of the SOVA matrix, A, has
o be improved in order to be insensitive to the KCC variation


inputs, u. The sensitivity index can be defined as the variations ofoutput signals to input noise �26�, which can be expressed as

S =yTy

uTu=

uTATAu

uTu�9�

The sensitivity index, S, has to be minimized such that thesignificant variations of uTu contribute to minor variations of yTy.If the KCC variations of vector u are constant, the KPC variationsdepend on the assembly response function A. The challenge is toselect the design index to assess ATA. Several measures are pro-posed based on optimality criteria in experimental design. Kimand Ding �26� provided the analysis of the three optimality criteriain fixture layout design which are �i� D-optimality �mindet�ATA��, �ii� A-optimality �min tr�ATA��, and �iii� E-optimality�min �max�ATA�; �max is the extreme eigenvalue�. The advantagesand disadvantages of these three optimality criteria for fixture lay-out design are discussed below:

D-optimality is to minimize the determinant of a matrix ATA,�min det�ATA��. The advantage of D-optimality in fixture layoutdesign is that it minimizes both the variances and the covarincesof matrix ATA. It is equivalent to minimizing the overall processvariations; min det �ATA�=min i=1

m �i, where �i is an eigenvalue.D-optimality is very effective to evaluating the design problems,which inherent highly nonlinear relationships such as fixture lay-out design. However, the singularity of matrix ATA is a majorobstacle to the use of D-optimality in multistage fixture layoutdesign.

A-optimality is to minimize the trace of matrix ATA, mintr�ATA�, which is the summation of sensitivities of all KCC-KPCpairs in the assembly processes. Nevertheless, A-optimality doesnot consider the dimensional variation impact from covarianceswithin matrix ATA. Thus, A-optimality does not imply that thepercentage of nonconforming items will be reduced since the co-variances among the locator nominal positions on KPC variationsare high.

E-optimality is to minimize the extreme eigenvalue of matrixATA, min �max�ATA�. E-optimality is similar to D-optimality,which considers both variances and covariances of all pairs ofKCC-KPC, but E-optimality considers only �max�ATA�. Thus,E-optimality can avoid the singularity of matrix ATA during com-putation, and it is aligned with the Pareto principle in qualityengineering �26�. However, minimizing only the maximum eigen-value, �max�ATA� cannot guarantee that overall variations, i=1

m �i,of the new set of fixture layouts design are decreased. It leaves thepossibility that several principle components dominate the overallvariations of matrix ATA, and the summations of these eigenval-ues can contribute to larger variations even though its extremeeigenvalue is lower than the previous fixture layout design. There-fore, it is difficult to decide that process increases its robustnessby assessing only the extreme eigenvalue.

On the other hand, the percentage of nonconformance items canbe used to evaluate the performance among fixture layout designsby maintaining constant u. In general, process performance ismeasured by process capability indices, Cp or Cpk, where Cp canbe defined as �USL−LSL� /6�; USL and LSL are the upper andlower specification limits, respectively, and � is the standardvariation of a single KPC variable. In multivariate cases, the KPCtolerance/specification region in multivariate m space is the vol-ume of the hyper-rectangular cube �30�, which can be defined as

i=1

m

�USLi − LSLi� �10�

The KPC variations of a multivariate process can be assessedby using chi-square distance defined as

�2 = �y − ��−1

�y − �� 11�
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However, Cp in evaluating multivariate normality KPC varia-ions cannot be obtained directly by dividing the volume of thePC hyper-rectangular cube specification shown in Eq. �10� with

he actual process chi-square distance expressed in Eq. �11� be-ause KPC tolerances/specifications are a hypercube while thehi-square distance has elliptical probability region. Thus, to de-ermine Cp, it is necessary to estimate the KPC tolerance regionnto an ellipsoid shape. As a result, when the process is centered athe target and Cp=1, this implies that 99.73% of the process varia-ions are inside the estimated KPC tolerance ellipsoid. Taam et al.30� proposed an approach to calculate Cp by approximating thePC tolerance hypercube with the largest ellipsoid that can lie

nside the KPC tolerance hypercube. However, to estimate theargest ellipsoid volume is difficult in the case where m�3.

The process yield proposed in this paper is similar to Cp in aultivariate process. However, instead of focusing on estimating

he KPC tolerance ellipsoid shape, the process yield defines therobability that the KPC variation vector, y, lies in KPC toleranceypercube, as illustrated in Fig. 2�b�. The process yield providesn understandable design criterion for design engineers to evalu-te their process design. Moreover, the process yield does notepend on the multivariate normality distribution assumption. Toenchmark the robustness of any two processes in the case that00% probability of KPC variation vectors, y, lies in the KPColerance hypercube can also be performed by integrating conceptf sensitivity indices to process yield. A-optimality and-optimality can be used to analyze the variances or principleomponents of interest.

In this paper, the process yield is used as the quality index inssessing the performance of a set of fixture layouts. Yield isefined as a function of KPCs, which represents the probability ofll KPCs simultaneously being within their respective specifica-ion ranges as shown

yield �� = Pr��i=1

m

LSLi � KPCi � USLi� �12�

here LSLi and USLi are the lower and upper specification limitsor KPCi, respectively.

Yield can be estimated by using Monte Carlo technique byimulating k KCC vectors, KCC1, KCC2 , . . . ,KCCk, whereCCi= �KCC1. . .KCCn�i

T. A variation of each KCC expressed inartesian coordinate, �x ,y ,z�, is randomly generated based on

ts statistical characterizations. Then, the KCCi ; i=1, . . . ,k, isubstituted into Eq. �7� to obtain a vector of, KPC variations,PC1 , . . . ,KPCk, where KPC= �KPC1. . .KPCm�T.�KPCi� is a function to provide a response whether all KPCs

re in-specification windows. If all KPC variations are withinpecification windows; LSL�KPCi�USL, then �KPCi�=1;therwise �KPCi�=0. Thus, yield can be expressed as:

yield �� =

i=1

k

�KPCi�

� 100% �13�

Fig. 2 KPC variations compared witimizing fixture layouts „b… after opti

k

61005-6 / Vol. 130, DECEMBER 2008


3.2 Instability Index. The instability index, �, is adoptedfrom Roy and Liao �14� to compare the locating stability betweentwo fixture layouts. In this paper, a binary assembly process istaken into consideration where at each station a root part is lo-cated on the fixture layout S, and then a mating part located byfixture layout M, and a part-to-part joint is assembled to the rootpart. Thus, the root part has to be in static equilibrium and is fullyconstrained by fixture layout S before being assembled to a mat-ing part. The static equilibrium of a root part located by fixturelayout S, as shown in Fig. 3, can be expressed in matrix form as

�0 0 0 − 1 − 1 0

0 0 0 0 0 − 1

1 1 1 0 0 0

r1y r2y r3y 0 0 r6z

− r1x − r2x − r3x − r4z − r5z 0

0 0 0 r4y r5y − r6x

��F1

F2

F3

F4

F5

F6

� + �0

0

− Wg

− Wgrgy

Wgrgx

0

� = 0

�14�

where Fi, i=1,2 , . . . ,6 represent supporting and locating forces;r1x ,r1y , . . . ,r6z represent the x ,y ,z coordinates of six locators inthe fixture layout; rgx ,rgy ,rgz represent the x ,y ,z coordinates ofthe center of gravity of the workpiece; Wg represents the weight ofthe workpiece.

If there are only supporting forces from three NC blocks, F1,F2, and F3, against the weight of a root part, Wg, in order tomaintain static equilibrium, Eq. �14� can be simplified to

� 1 1 1

r1y r2y r3y

− r1x − r2x − r3x��F1

F2

F3� + � − Wg

− Wgrgy

Wgrgx� = 0 �15�

To calculate the instability index, the following information isrequired: �1� a new wrench, wd, of external forces and couples torebalance the static equilibrium in an adjusted fixture layout and�2� twist caused by root part weight, tmg. The wrench and twistcan be obtained as described below.

�1� A new wrench, wd, of external forces and couples. When

PC tolerance region: „a… before op-ing fixture layouts

th Kmiz

Fig. 3 3-2-1 fixture layout for prismatic workpiece



wo

t

Ft

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the position of any NC block used to locate a root part isadjusted while conducting fixture layout optimization, theprevious static equilibrium condition, as shown in Eq. �15�,is altered and equilibrium needs to be determined again.For example, if the position of NC block No.1 is changedfrom �x1 ,y1� to �x1� ,y1�, the new equilibrium equation witha new external force and moment vector �we� to balance theroot part locating stability can be expressed as in Eq. �16�where the force vector, �F1 F2 F3�T, acting at the NCblock is unchanged.

� 1 1 1

ry1 ry2 ry3

− rx1� − rx2 − rx3��F1

F2

F3� + � − Wg

− Wgrgy

Wgrgx� + � fz

Mx

My�

we

= �0

0

0��16�

Both the force and moments of we can be obtained fromEq. �16�, and they can be presented in a wrench form as

wd = �0 0 fz Mx My 0�T �17��2� Twist caused by root part weight, tmg: To obtain the twist

tmg of a root part caused by its weight, let us assume thatthe workpiece undergoes an infinitesimal movement causedby gravity force Wg, as shown in Fig. 4. This movementcan be expressed as a twist about the origin of coordinates:

tmg = ��x �y �z vx vy vz� �18�

where �x ,�y ,�z are the components of part angular dis-placement, and vx ,vy ,vz are components the of part trans-lational displacement. For example, the twist of gravityforce, Wg, about the origin of coordinates, as shown in Fig.4, is

tmg = �− 1/rgy 1/rgx 0 0 0 − 1� �19�

The instability index represents the virtual work done by therench wd against the twist tmg. The instability index, �, can bebtained by

� = fxvx + fyvy + fzvz + Mx�x + My�y + Mz�z �20�The instability index defined by using virtual work can be in-

erpreted as follows.

�1� ��0 represents the positive work done by wrench wd inaccomplishing twist tmg. This positive virtual work impliesthat the adjustment of a locator position in a new fixturelayout reduces the workpiece stability. Therefore, in theproposed methodology in this paper, it is concluded that theadjusted fixture layout is worse than the pre-adjusted one.

�2� �=0 represents that no work is done by wrench wd inaccomplishing twist tmg. The virtual work �=0 can be in-terpreted that there is no improvement in locating stabilitycondition after the adjustment of a locator position.

�3�

ig. 4 External wrench, wd, and twist caused by gravity force,mg

��0 represents the negative work done by wrench wd in



accomplishing twist tmg. The negative virtual work impliesthat the adjustment of locator position into a new fixturelayout increases the workpiece stability. Therefore, in theproposed methodology in this paper, it is concluded that theadjusted fixture layout is better than the pre-adjusted one.

4 MethodologyIn keeping with the previous discussion, multiple fixture lay-

outs of all assembly stations are required to be designed simulta-neously results in a high-dimensional optimization problem.Moreover, the design space of each locator is large, and the loca-tor nominal positions have nonlinear relationships with KPCs. Inorder to address these challenges, this paper proposes a two-stepoptimization approach based on genetic algorithm for designspace reduction and Hammerley sequence sampling for directsearch optimization scheme in a design space predetermined byGA. Yield and instability index are incorporated to the proposeddesign approaches as design objectives. The procedure of the pro-posed methodology in designing multiple fixture layouts is shownin Fig. 5. The proposed methodology is based on the followingassumptions.

1. All parts are rigid body.2. The Locator-part constraint is characterized by frictionless

Step 1: Fixture Planning1.1 Define the number and types of locators required in the

assembly processOutput: { }p1 ,,, LLLΛ 2 K=

1.2 Define the design spaces for N locatorsOutput: { }Nℜℜℜ=ℜ K21system

O

1.3 Discretize design space, iℜ , into t nodes

Outputs: ℜ { }itii OOO K21=→

{ }Nsystem OOOO ,,, 21 K=

Step 2: Design space reduction by Genetic Algorithm (GA)2.1 Design space reduction by GA

0)(subject to)Yield(maximize

<ΛΛΛ

ϖ

Output: { }NGAGAGAGA OOO ,,, 21 K=O

2.1 Define design spaces of interest, iA , around GA candidatenodesOutput: { }NAAA K21new =ℜ

Step 3: Local search by Hammersley Sequence Sampling(HSS)3.1 Projection of design space of interest, , into 2D planeiA

Output: { }ND BBB K212new =ℜ

3.2 Uniform sampling sets of fixture layouts by HSSOutput: nΛΛΛ ,,, K1 2

Step 4: Evaluation of sampled sets of fixture layouts4.1 Evaluation of sets of fixture layout generated by HSS

against design objective

0)(subject to)Yield(maximize

<ΛΛΛ

ϖOutput: an optimum set of fixture layouts optimumΛ

Fig. 5 The optimization procedure of the proposedmethodology

point contact.

DECEMBER 2008, Vol. 130 / 061005-7


t

adldapcasal

0

Downloa

3. The part-to-part joints always maintain full contact to eachother.

4. Locators are not only to determine the positions and orien-tations of the part but also function as clamps in constrainingparts.

5. Only fixture and part-to-part joint errors are considered,which are considerably small compared with part and as-sembly dimensions.

Step 1. Fixture planning.Step 1.1. Define the number and types of locators required in

he assembly process.The number and types of locators required to constrain root, S,

nd mating, M, parts in each assembly station �L= �S ,M�� areefined in this step. It also can be observed that the number ofocators in each assembly station depends on the part-to-part jointesign and assembly sequence. To illustrate this, let us assume anssembly process of two rigid parts, each with 6 DOFs. The firstart positioned in the assembly station called a root part is fullyonstrained by fixture locators. The second part positioned in thessembly station is called a mating part and has its 6 DOFs con-trained by part-to-part joint and fixture locators. The potentialllocation of mating part 6 DOFs to be constrained by fixtureocators �M� and by part-to-part joint is shown in Fig. 6.

Fig. 6 Mating part degrees of freedoand locators

Fig. 7 3-2-1 fixture layout for

61005-8 / Vol. 130, DECEMBER 2008


Furthermore, the assembly sequence in selecting of root andmating part in each station directly affects the fixture layout de-sign. An example of fixture layouts for S and M parts is illustratedin Fig. 7. The root part in Fig. 7�a� is located by a typical 3-2-1fixture layout. The locators for the root part S consist of twolocating pins, four-way pin �P4way� and two-way pin �P2way�, andthree net contact blocks �NC1–3�. Two locating pins constrain 3DOFs in the X-Y plane, where P4way controls the part translationin the X and Y axes, and P2way controls the rotation of the root partabout Z axis. Three NC blocks constrain the remaining 3 DOFs,which are translation in Z axis and rotations about the X and Yaxes. Thus, the locators required for the root part in 3-2-1 fixturelocating scheme can be defined as

S = �P4wayr ,P2way

r ,NC1r ,NC2

r ,NC3r� �21�

Then, the mating part is located by part-to-part joint and theremaining of DOFs are constrained by fixture locators, M. TheDOFs of a part-to-part joint, which constrain the mating part,affect the fixture planning, as shown in Fig. 6. The part-to-partjoint shown in Fig. 7�b� constrains the 3 DOFs of the mating part:translation in Z axis and rotations about X and Y axes. Therefore,the remaining DOFs of the mating part are constrained by P4wayand P2way fixture locators.

allocation between part-to-part joint
m
a single station assembly



p�jfialr

p

ojsacs

wlaf

pla

Fa

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In an automotive body assembly, the following three types ofart-to-part joints are widely used: �1� lap joint, �2� butt joint, and3� T-joint �4,10�, as shown in Fig. 8. In this paper, a part-to-partoint is assumed to constrain 3 DOFs of a mating part. A set ofxture locators required for the mating part M where part-to-partsreap joint, butt joint, and T-joint can be defined as follows,espectively:

Mlap = �P4waym ,P2way

m � �22�

Mbutt = �P2waym ,NC1

m,NC2m� �23�

MT = �P2waym ,NC1

m,NC2m� �24�

Thus, the fixture layout for the single assembly station with aredetermined part-to-part joint �Mjoint� can be defined as

L = �S,M� = �P4wayr ,P2way

r ,NC1r ,NC2

r ,NC3r ,Mjoint� �25�

Step 1.2. Define the design space for each locator.The design space, Ri, of each locator, ri, is defined as an area

n a workpiece that the locator can be placed. Design engineersustify the locator design space by considering other design con-traints in the subsequent fixture construction stage such as allow-ble maximum deformation of workpiece or the potential diffi-ulty in locator installation and calibration. The continuous designpace of a locator ri can be defined as

Ri = ��x,y,z��, x � �lx,ux�, y � �ly,uy� z = f i�x,y� �26�

here �x ,y ,z� represents the Cartesian coordinate of a locator ri;

x,y and ux,y are the lower and upper boundaries in the x and yxes, respectively; and z= f i�x ,y� is the workpiece surface shapeunction.

For example, in Fig. 9 the design space Ri covers the wholeart. Let us assume that a set of fixture layouts � consists of Nocators in a given assembly system. Then, the design space of thessembly process can be expressed as

ig. 8 Example of part-to-part joints used in automotive bodyssembly

Fig. 9 Illustration of the propose



Rsystem = �R1 R2 . . . RN� �27�

Step 1.3 Discretize design space, Ri, into nodes.To reduce the computational effort of GA, the continuous de-

sign space, Ri, for each locator is discretized into nodes. Thesenodes represent the candidate design space to be considered in thenext step. The design space of locator ri discretized into t nodes,as shown in Fig. 9, can be expressed as

Oi = �O1 O2 . . . Ot�i where Oj = �xj,yj,zj� � Ri

and j = 1, . . . ,t �28�

The continuous design spaces �R1 , . . . ,RN� of N locators in agiven assembly system after the discretization into nodes can beexpressed as

Osystem = �O1,O2, . . . ,ON� �29�Step 2. Design space reduction by genetic algorithm (GA).Step 2.1: Design space reduction by GA.Since the design space for each locator can be relatively large

and its position has the nonlinear relations with multiple KPCs, itis necessary to reduce the design space to the area that potentiallycontains the optimal locator position. The design space reductionis conducted by using the GA approach. In each iteration, GAselects one node from each candidate space Oi; i=1, . . . ,N, toformulate a candidate set of fixture layouts, �= �Oj

1 , . . . ,OjN�. The

GA optimization scheme is expressed as determining a set of fix-ture layout �, which maximizes the process yield subjected to aconstraint of instability index,��, as shown below:


subject to �� 0 �30�In general, the genetic algorithm adopted in this paper involves

four major steps, as illustrated in Fig. 10. In the first step, theCartesian coordinates of locator positions, which are aimed to beoptimized, are modeled into a chromosome vector. In the secondstep, the initial population size in each generation is defined. Ini-tial populations in this paper are selected randomly with uniformdistribution function. Then, the process yield of individual popu-lation is evaluated subjected to locating stability requirement, asshown in Eq. �30�. In the last step, the chromosome of populationis improved by selecting the best individual to reproduce in thenext generations. The reproduction process to improve the chro-mosome involves two functions: �i� crossover function and �ii�mutation function. Crossover function is to combine two individu-als, or parent, to produce a new individual. In this paper, crossoverfunction randomly selects chromosomes from parents by generat-ing binary vector, which have a length equal to a number of chro-mosome in a population. If an element of a binary vector is 1, achromosome is selected from the first parent. On the other hand, achromosome is selected from the second parent if an element of

d methodology Steps 1.2–3.1

DECEMBER 2008, Vol. 130 / 061005-9


bctptditpe

n

dAcsSlT

wc=

p

fmTtpss

woy

e

0

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inary vector is 0. Mutation function is to create small randomhanges in individual population, which help to prevent GArapped in local optima. GA is terminated when there is no im-rovement in a process yield or the number of iterations reacheshe maximum number of generations designated in the GA proce-ure. Denote the �� as the improvement in the instability index,.e., ��=��i�−��i−1�. Figure 9 shows a node OGA

i , which ishe optimal position of locator ri selected by GA. The optimalositions of all locators in the system selected by GA can bexpressed as

OGA = �OGA1 ,OGA

2 , . . . ,OGAN � �31�

Step 2.2. Define design space of interest around GA candidateodes.

The area around each node predetermined by GA in Eq. �31� isefined as the new design space so-called design space of interest,i. The design space of interest covers the area that GA did not

onsider, which might contain the optimal locator position. Theize of design space of interest is related to the grid size defined intep 1. It can be seen in Fig. 9 that design space of interest of

ocator ri is significantly smaller than an initial design space, Ri.he new design space of the assembly system can be expressed as

Rnew = �A1 A2 . . . AN� �32�

Ai = ��x,y,z��, x � �lx�,ux��, y � �ly�,uy��, z = f i�x,y� �33�

here lx,y� and ux,y� are the lower and upper boundaries of x and yoordinates in design space of interest Ai, respectively; zfi�x ,y� is the workpiece surface shape function.Step 3. Local search by Hammersley sequence sampling (HSS).Step 3.1. Projection of design space of interest, Ai, into 2D

lane.The direct sampling of potential locator positions on part sur-

ace is very complex and time consuming since geometrical infor-ation of part surface has to be included in the HSS algorithm.herefore, the sampling procedure can be simplified by selecting

he locator positions in 2D plane. The 2D plane is obtained byrojecting design space of interest Ai into a given plane. Designpace of interest in 3D space, Ai, is projected into 2D space, Bi, ashown in Fig. 9. The 2D space, Bi, can be expressed as follows:

Bi = ��x,y��, x � �lx�,ux�� and y � �ly�,uy�� 34�

here Bi is the projection area in 2D plane of Ai, which consistsf a set of points within the lower and upper boundaries of x andcoordinates.Design spaces of interest for all locators in 2D plane can be

Fig. 10 Step 2 design space reduction by genetic algorithm

xpressed as

61005-10 / Vol. 130, DECEMBER 2008


Rnew2D = �B1 B2 . . . BN� �35�

Step 3.2. Uniform sampling sets of fixture layouts by HSS.The optimal positions of all locators are searched by using

HSS. To sample the locator positions, first the transformed 2Ddesign spaces �B1 B2 . . . BN� are formulated intoN-dimensional hypercube design space. The example hypercubewith three 2D plane design spaces �B1–3� is shown in Fig. 11�a�.The samples are selected uniformly in the hypercube shown inFig. 11�a�. Then, the sample points are projected onto each facetof the hypercube, which is a 2D plane design space, as shown inFig. 11�b�. Finally, the sampling locator positions in the 3D designspace �A1 A2 . . . AN� can be obtained by substituting thecoordinates of samples in the 2D design spaces into correspondingpart surface functions; z= f i�x ,y� ; i=1, . . . ,N, as shown in Fig.11�c�.

The transformation of locator position sampled in 2D plane, Bi,into 3D design space, Ai, is mathematically expressed as

Bi:�xj,yj� → Ai:�xj,yj,zj = f�xj,yj�� = ri, i = 1, . . . ,N �36�

� = �r1, . . . ,rN� �37�Step 4. Evaluation of sampled sets of fixture layouts.The n sets of fixture layouts; �1 ,�2 , . . . ,�n, generated by us-

ing HSS are evaluated by �i� formulating SOVA models, A��, foreach set of fixture layouts; and �ii� conducting Monte Carlo simu-lation to obtain a process yield. The optimization problem is for-mulated as follows:


subject to �� 0 �38�

5 Case Study: Floor Pan AssemblyThe developed methodology is illustrated and validated by ap-

plying it to automotive underbody assembly process. Floor Pansubassembly consists of four parts; floor pan left �FPL� and right�FPR�, and bracket left �BrktL� and right �BrktR�, assembled inthree stations, as shown in Fig. 12. The dimensional quality of thefloor pan assembly is evaluated by 12 KPCs and reported as pro-cess yield. The manufacturer aims to design the fixture layouts inthree assembly stations to improve the robustness of the floor panassembly process by increasing the process yield from a currentlevel of 85% without tightening any KCC tolerances, i.e., withoutincreasing tooling cost. The process yield is assessed by MonteCarlo simulation in which the variations of fixture locators andpart-to-part joints are set according to their tolerances. Tolerancesof locators are assumed to be 0.3 mm. while part-to-part jointtolerances are assumed to be 0.5 mm. for linear variation and 0.25 degree for angular variation. The proposed multifixturelayout optimization methodology is applied to this case study. Thecomparative study in terms of optimization performances amongthe proposed methodology and other fixture layout optimizationalgorithms are also shown in this section.

Step 1: Fixture planning.Step 1.1. Define the number and types of fixtures required in

Fig. 11 Sets of fixture layouts generation by HSS

the assembly process.



tlt

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The assembly sequence, part-to-part joints, and required loca-ors in each assembly station are shown in Table 3. The fixtureayouts for three assembly stations and a set of fixture layouts forhe system can be formulated as follows

L1 = �S1,M1�, L2 = �S2,M2� and L3 = �S3,M3� �39�

Fig. 12 Floor pan assembly

Table 3 Required locato

Station No. Parts

Station 1 FPL �root part�FPR �mating part�

Station 2 Subassembly �FPL+FPR�Bracket left �mating part�

Station 3 Subassembly�FPL+FPR+BrktL�

Bracket right �matingpart�

Table 4 Design space

Station 1Locators

Root par

P4way1s P2way

1s NC11s

Designspaces

R1 R2 R3


Station 2Locators

Root par

P4way2s P2way

2s NC12s

Designspaces

R8 R9 R10


Station 3Locators

Root par

P4way3s P2way

3s NC13s

Designspaces

R15 R16 R17



� = �L1,L2,L3� �40�Step 1.2. Define the design space for each locator.The design spaces of all locators in all assembly stations are

defined in Tables 4–6. The design space for each locator is shownin Figs. 13–18. The coordinates of center of gravity and weight ofeach root part in each assembly station are shown in Table 7.

Step 1.3. Discretize design space, Ri, into nodes.The design spaces of all four-way and two-way locating pins

are discretized into grid of 50�50 mm2. Similarly, the designspaces of NC blocks are discretized into grid of 100�110 mm2.Figure 19 shows one example of dicretization of design space R3representing NC block NC1

1s into 60 nodes. All 21 discretizeddesign spaces in the system can be expressed as

Osystem = �O1,O2, . . . ,O21� �41�Step 2. Design space reduction by GA.Step 2.1. Design space reduction by GA.In each iteration, GA selects one node from each of 21 candi-

date design spaces in Eq. �41� to formulate a set of fixture layouts�, which is then evaluated by using a process yield and an insta-bility index, �. The optimization scheme is expressed, as shownin Eq. �30�. The genetic algorithm configurations are selected asfollows: �1� chromosomes are formulated by Cartesian coordi-nates of 21 locators; �2� population size has 50 individuals in eachgeneration; �3� selection of parents for reproduction is based onranking for the most fitness and stochastic uniform selection with

n each assembly station

Joints Required locators

ap joint S1= �P4way1s , P2way

1s ,NC11s ,NC2

1s ,NC31s�

M1= �P4way1m , P2way

1m �


2s ,NC12s ,NC2

2s ,NC32s�


2m �


3s ,NC13s ,NC2

3s ,NC33s�


3m �

f locators in Station 1

Mating Part

NC21s NC3

1s P4way1m P2way

1m

R4 R5 R6 R7


Mating Part

NC22s NC3

2s P4way2m P2way

2m

R11 R12 R13 R14


Mating part

NC23s NC3

3s P4way3m P2way

3m

R18 R19 R20 R21

rs i

L

L

L

s o

t

s o

t

s o

t

DECEMBER 2008, Vol. 130 / 061005-11


aGbg5a

n

bewltPt

Fp

Fp

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crossover fraction of 0.8; �4� mutation function is based on theaussian distribution; �5� crossover function is based on using ainary vector as criterion; and �6� the GA is terminated at 5000enerations or there is no improvement in a process yield within0 consecutive generations. The locator positions selected by GAre shown in Tables 8–10 and Figs. 20–22.

Step 2.2. Define design space of interest around GA candidateodes.

The area around each node predetermined by GA is defined toe the design space of interest, Rnew= �A1 A2 . . . A21 �. How-ver, in this case study, some locators are required to be definedith additional constraints. First, both bracket left and right have

imited areas to place four-way and two-way locating pins. Thus,he positions of four-way and two-way pins for both parts �P4way

2m ,

2way2m , P4way

3m , and P2way3m � are assigned to be the same as the loca-

ions selected by GA. Second, to minimize the need for locating

ig. 13 Locator design spaces on floor pan left „FPL, rootart… in Station 1

ig. 14 Locator design spaces on floor pan right „FPR, matingart… in Station 1

ig. 15 Locator design spaces of the root part „FPL+FPR… in
tation 2
61005-12 / Vol. 130, DECEMBER 2008


holes on the parts, some locators are reused and have the sameposition in different assembly stations. These locators are �1�P4way

1s , P4way2s , and P4way

3s ; �2� P2way1s , P2way

2s , and P2way3s ; �3� NC3

1s,NC1

2s, and NC13s; �4� NC1

1s, NC22s, and NC2

3s; and �5� NC32s and

NC33s. Therefore, the design spaces of interest in each group of

these locators are equivalent to the integration of areas aroundeach node within the group. The design spaces of interest for all

Fig. 16 Locator design spaces of bracket left „BrktL, matingpart… in Station 2

Fig. 17 Locator design spaces of the root part in Station 3 „thecoordinates of the boundaries are the same, as shown in Fig.15…

Fig. 18 Locator design spaces of bracket right „BrktR, matingpart… in Station 3

Table 7 Center of gravity coordinates of a root part in threeassembly stations

Center of gravitycoordinates

Station 1�FPL�

Station 2Subassembly�FPL+FPR�

Station 3Subassembly

�FPL+FPR+BrktL�

X 1250 1250 1340Y −290 0 −110Z 0 0 0

Weight �kg� 60 120 150

Fig. 19 An example of design space discretization



la

p

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J

Downloa

ocators are shown in Fig. 23. This case study uses 2�2 grid sizeround the nodes to define design space of interest.

Step 3. Local search by HSS.Step 3.1. Projection of design space of interest, Ai, into 2D

lane.The boundaries of the 2D planes, B, are shown in Tables 11–13.Step 3.2. Uniform sampling sets of fixture layouts by HSS.One thousand sets of fixture layouts, �i ; i=1, . . . ,1000, are

ampled from the hypercube of 21 2D plane design spaces, B1–21y using HSS. The sampled locator positions in 2D plane designpaces, B1–21, are projected onto part surfaces to obtain the actual

Table 8 Locator positions

Station 1Coordinates

Root pa

P4way1s P2way

1s NC11s

X 900.00 1700.00 1650.0Y −250.00 −250.00 −180.0Z 0.00 0.00 0.5

Table 9 Locator positions


Root par

P4way2s P2way

2s NC12s

X 750.00 1700.00 1350.00Y −300.00 −300.00 −620.00Z 0.00 0.00 25.34

Table 10 Locator position


Root pa

P4way3s P2way

3s NC13s

X 750.00 1700.00 1450.00Y −300.00 −300.00 −620.00Z 0.00 0.00 25.34

Fig. 20 Locator positions selected by GA in Station 1




positions in 3D design space, A1–21 by using CAD software toacquire the coordinates on workpiece.

Step 4. Evaluation of sampled sets of fixture layouts.One thousand sets of fixture layouts are evaluated for their pro-

cess yield and stability. The set of fixture layouts that has thehighest process yield and satisfy stability requirement is consid-ered to be the optimum set of fixture layouts. The locator positionsof the optimum set of fixture layouts are shown in Tables 14–16and Figs. 24 and 25. The process yield of the optimum set offixture layouts is 96.16% while the process yield of the initialindustrial design is around 85%.

The performance of the proposed optimization algorithm is

lected by GA in Station 1

Mating part

NC21s NC3

1s P4way1m P2way

1m

750.00 1250.00 1700.00 800.00−400.00 −510.00 231.35 231.35

0.00 8.94 0.00 0.00

lected by GA in Station 2

Mating part

NC22s NC3

2s P4way2m P2way

2m

1550.00 1050.00 1556.00 1556.00−290.00 191.35 −154.00 −551.00

0.00 1.39 90.00 90.00

elected by GA in Station 3

Mating part

NC23s NC3

3s P4way3m P2way

3m

1550.00 1150.00 1556.00 1556.00−290.00 301.35 135.00 532.00

0.00 1.39 90.00 90.00


se

rt

005

se

t

s s

rt

Fig. 23 Locator design spaces of interest

DECEMBER 2008, Vol. 130 / 061005-13


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enchmarked with other optimization algorithms in terms of �i�ethodology ffectiveness measured by closeness of its solution to

he global optimum and �ii� computational efficiency measured byime to converge to optimum solutions. However, it is consider-bly difficult to determine the global optimum in nonlinear opti-ization problems by all available optimization algorithms with-

ut an exhaustive search �26�. Instead of searching for the globalptimum, this optimization algorithm proposed in this paper pro-ides an optimum set of fixture layouts, which contributes to aignificant improvement in the process yield and workpiece sta-

Table 11 The upper and lower boundary coordinat

Locator P4way1s P2way

1s NC11s

Design space ofinterest

A1 A2 A3

Projected 2D planedesign space

B1 B2 B3

Upper boundary of2D plane �x,y�

�950,−250� �1750,−250� �1750,−1

Lower boundary of2D plane �x,y�

�750,−350� �1650,−350� �1450,−4



2s NC12

esign space of interest A8�=A1� A9�=A2� A10�=AProjected 2D plane

design spaceB8 B9 B10

Upper boundary of 2Dplane �x,y�

�950,−250� �1750,−250� �1550,−

Lower boundary of 2Dplane �x,y�

�750,−350� �1650,−350� �1150,−



3s NC13s

Design space ofinterest

A15�=A1� A16�=A2� A17�=A5

Projected 2Dplane design

space

B15 B16 B17

Upper boundaryof 2D plane �x,y�

�950,−250� �1750,−250� �1550,−40

Lower boundaryof 2D plane �x,y�

�750,−350� �1650,−350� �1150,−62

Table 14 The optimal lo


Root pa

P4way1s P2way

1s NC11

X 800.00 1700.00 1743.0Y −270.00 −270.00 −205.0Z 0.00 0.00 1.4



Root par

P4way2s P2way

2s NC12

X 800.00 1700.00 1308.00Y −270.00 −270.00 −611.00Z 0.00 0.00 29.62

61005-14 / Vol. 130, DECEMBER 2008


bility with a minimal computational effort. The methodology ef-fectiveness and computational efficiency of the proposed optimi-zation algorithm are compared with those of sequential quadraticprogramming �SQP�, simplex search, genetic algorithm, and EAused by Kim and Ding �26�. The performances of optimizationalgorithms are summarized in Table 17.

A gradient-based search such as sequential quadratic program-ming is used in determining an optimal fixture layout design�16,19�. The disadvantage of a gradient-based search is that thedesign solutions are easily entrapped in a local optimum since its

f the projected 2D plane design space in Station 1

NC21s NC3

1s P4way1m P2way

1m

A4 A5 A6 A7

B4 B5 B6 B7

�850,−300� �1550,−400� �1750,281� �850,281�

�750,−500� �1150,−620� �1650,231� �750,231�


NC22s NC3

2s P4way2m P2way

2m

A11�=A3� A12 A13 A14

B11 B12 B13 B14

� �1750,−180� �1250,191� �1556,−154� �1556,−551�

� �1450,−400� �950,401� �1556,−154� �1556,−551�


NC23s NC3

3s P4way3m P2way

3m

A18�=A3� A19�=A12� A20 A21

B18 B19 B20 B21

�1750,−180� �1250,191� �1556,135� �1556,532�

�1450,−400� �950,401� �1556,135� �1556,532�

or positions in Station 1

Mating part

NC21s NC3

1s P4way1m P2way

1m

777.00 1308.00 1700.00 800.00−275.00 −611.00 240.00 240.00

0.00 29.62 0.00 0.00


Mating part

NC22s NC3

2s P4way2m P2way

2m

1743.00 1099.00 1556.00 1556.00−205.00 299.00 −154.00 −551.00

1.42 0.00 90.00 90.00

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400

620

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earching algorithm is based on the steepest ascent/descent direc-ion. Moreover, it is difficult for the gradient-based search to ob-ain the derivative information where a process yield and an in-tability index are the quality measures. To illustrate this, therocess yield is multivariate probability density function of ran-om KPC variables, which depend on locator position variations.herefore, to formulate the explicit model showing relationshipsetween the process yield and nominal locator positions, whichllow obtaining derivative information, is infeasible in the high-imensional problems. In addition, the gradient-based search doesot consider the workpiece stability because of the difficulty inormulating the model that can represent instability index andield measure simultaneously.

A simplex search can be used to determine the optimal fixtureayouts. Although a simplex search is a direct search method,hich does not require gradient derivative information, the design

olutions obtained from a simplex search also easily converges toocal optimum. A simplex search was applied on the floor panubassembly case study by using fminsearch function available inATLAB. The design spaces are defined as continuous and the



Root pa

P4way3s P2way

3s NC13

X 800.00 1700.00 1308.00Y −270.00 −270.00 −611.00Z 0.00 0.00 29.62

Fig. 24 The optimal locator positions selected by HSS

ig. 25 The optimal locator positions selected by HSS on „a…racket left and „b… bracket right

Table 17 Comparison of optimization methods

Methodologyeffectiveness

Computationalefficiency

ptimizationethodologies Process yield

Workpiecestability

consideration�Yes/No�

Computationaltime �s�

radient-based search — No —implex search 88.24% Yes 3600enetic algorithm 95.40% Yes 8103xchange algorithm — No —roposed methodology 96.16% Yes 2062



searching operation is terminated in 1 h. The sets of fixture lay-outs sampled by the simplex search are validated for workpiecestability first, and then the process yields are calculated. The resultshows that the process yield of the optimal set of fixture layoutsincreases only 2.1% from an initial design and 7.9% lower thanprocess yield of the optimum fixture layouts obtained from theproposed method in this paper.

The performance of genetic algorithm in determining the opti-mal set of fixture layouts was also conducted. Although geneticalgorithm can avoid the design solutions converging to local op-tima, it takes considerably computational time in continuous de-sign spaces. For example, genetic algorithm was applied on thefloor pan subassembly case study to determine the optimum fix-ture layouts. It takes 8103 s in computational time to obtain a setof fixture layouts that can meet 95% yield and satisfy workpiecestability requirement, while the proposed methodology canachieve the same result within 2062 s.

The performances of the exchange algorithm proposed by Kimand Ding �26� is also studied and compared with that of the pro-posed methodology. In terms of algorithm effectiveness, both ex-change algorithm and the proposed methodology can provide op-timal sets of fixture layouts, which meet the dimensional qualityrequirement. However, the optimal result obtained from EA issometimes practically infeasible since the workpiece stability isnot incorporated in its design criteria, sensitivity index. Therefore,comparison between the proposed methodology and EA in termsof the computational efficiency is difficult because both methodsuse the different assessment indices.

In addition, the proposed multi-fixture layout optimizationmethodology can be used to eliminate the limitation of KCC tol-erance optimization in improving the assembly process robust-ness. In some design problems, a tolerance optimization method-ology might not be able to identify a set of KCC tolerances whichcan achieve the process yield requirement �33�. The frameworkwhich can integrate multiple design synthesis tasks �e.g., multi-fixture layout optimization and KCC tolerance optimization� inoptimizing assembly process design can be found in Phoomboplaband Ceglarek �34�. Additionally, the proposed methodology can beextended into fixture workspace synthesis for reconfigurable as-sembly �35�.

6 ConclusionsThis paper presents a methodology to improve a process yield

by optimizing the locator positions in a multistation assembly sys-tem. The performance of fixture layouts is assessed by a processyield, which represents the robustness of fixturing system in termsof a final product dimensional quality. In addition, fixture locatingstability is taken into consideration to ensure that the design offixture layouts is feasible in practical. The variation sources in realindustrial assembly processes, which are locator positions andpart-to-part joint variations are also taken into consideration. Theproposed methodology is based on two-step optimization, whichintegrates heuristic algorithm �GA� with a low-discrepancy sam-pling technique �HSS�. The application of the proposed method-ology is illustrated through a case study using an automotive un-derbody assembly where process yield greatly increases from 85%to 96% after optimizing the locator positions with no increase of


Mating part

NC23s NC3

3s P4way3m P2way

3m

1743.00 1099.00 1556.00 1556.00−205.00 299.00 135.00 532.00

1.42 0.00 90.00 90.00

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tooling cost.

DECEMBER 2008, Vol. 130 / 061005-15


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cknowledgmentThe authors gratefully acknowledge the financial support of the

K EPSRC Star Award EP/E044506/1 and US NSF-CAREERward DMII-0239244. The authors also appreciate the fruitfuliscussions with Professor W. Huang, University of Massachu-etts, Professor James Kong, Oklahoma State University and Dr.ing Zhou from Dimensional Control Systems, Inc.

omenclatureKPC � key product characteristicKCC � key control characteristic

ri � vector defining the location of a locatorLi � fixture layout for a single stationSi � fixture layout for a root part

Mi � fixture layout for a mating part�i � a set of fixture layouts for a particular assem-

bly systemP4way

s � four-way pin in a fixture layout for a root part�Si�

P2ways � two-way pin in a fixture layout for a root part

�Si�NCi

s � NC block in a fixture layout for a root part�Si�

P4waym � four-way pin in a fixture layout for a mating

part �Mi�P2way

m � 2-way pin in a fixture layout for a mating part�Mi�

NCim � NC block in a fixture layout for a mating part

�Mi�Fi � supporting and locating force of a locator

Rx,y,z � the resultant forceMx,y,z � moment of resultant force

Wg � weight of a partwd � external wrench to balance locating stability

f � force in external wrench wdm � moment in external wrench wd

lx,y� and ux,y� � lower and upper boundary of x and y coordi-nated in design space of interest Ai

tmg � twist of a workpiece due to weight� � angular displacement in twist tmgv � translation displacement in twist tmg

� � instability index�A�� SOVA matrix

LSL, USL � lower and upper specification limitsRi � design space for a locatorOi � a set of nodes discretized from design space

RiOi � a Cartesian coordinate of a discretized node

OGA � a node selected by GAz= f i�x ,y� � surface function of a workpiece

Ai � design space of interestBi � 2D projection of design space of interest �Ai�

ppendix: Hammersley Sequence SamplingHSS is used in Step 3 of the proposed methodology. The chal-

enge in the local search step is high-dimensional design space,hich usually requires a large number of iterations or samples.SS is a sampling technique that selects samples uniformly in aypercube design space, which requires fewer samples to con-erge to the solution within desired variance compared with otherampling technique or space filling technique such as Number–heoretical Net �31� or Latin hypercube �32�. Kalagnanam andiwekar �32� provided a procedure for selecting N Hammersleyoints in k-dimensional hypercube. Any integer n�n�1,2 , . . . ,N�� can be written in radix-R notation �R is a prime

umber� as follows:

61005-16 / Vol. 130, DECEMBER 2008


n nm · nm−1 . . . n2 · n1 · n0 = n0 + n1R + n2R2 + ¯ + nmRm

where R1 ,R2 , . . . ,Rk−1 are the first k−1 prime numbers and m= �logR n�= �ln n / ln R� �the square brackets denote the integerpart�. A unique fraction between 0 and 1 called the inverse radixnumber can be constructed by reversing the order of the digits ofp around the decimal point as follows:

�R�n� = n0n1n2 ¯ nm = n0R−1 + n1R−2 + ¯ + nqR−m−1

The Hammersley points on a k-dimensional cube are given bythe following sequence:

zk�n� = � n

N,�R1

�n�,�R2�n�, . . . ,�Rk−1

�n��, n = 1,2, . . . ,N

The Hammersley points generated in a unit hypercube are

xk�n� = 1 − zk�n�

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