Zhijun Li
e-mail: zhijunli@umich.edu
Michael Kokkolaras1
e-mail: mk@umich.edu
Panos Papalambros
e-mail: pyp@umich.edu
S. Jack Hu
e-mail: jackhu@umich.edu
Department of Mechanical Engineering,
University of Michigan,
2350 Hayward Street,
Ann Arbor, MI 48109-2125
Product and Process Tolerance
Allocation in Multistation
Compliant Assembly Using
Analytical Target Cascading
Tolerance allocation is the process of determining allowable dimensional variations in
products (parts and subassemblies) and processes (fixtures and tools) in order to meet
final assembly quality and cost targets. Traditionally, tolerance allocation is conducted
by solving a single optimization problem. This “all-in-one” (AIO) approach may not be
desirable or applicable for various reasons: the assembler of the final product may not
have access to models and/or data to compute appropriate tolerance values for all subassemblies and parts in the case of outsourcing; optimization algorithms may face numerical difficulties when solving very large-scale, simulation-based nonlinear problems;
interactions are often obscured in AIO models and trade-offs may not be quantifiable
readily. This paper models multistation compliant assembly as a hierarchical multilevel
process and proposes the application of analytical target cascading for formulating and
solving the tolerance allocation problem. Final product quality and cost targets are
translated into tolerance specifications for incoming parts, subassemblies, and station
fixtures. The proposed methodology is demonstrated using a vehicle side frame assembly
example. Both quality- and cost-driven tolerance allocation problems are formulated. A
parametric study with respect to budget is conducted to quantify the cost-quality tradeoff. We believe that the proposed multilevel optimization methodology constitutes a valuable new paradigm for tolerance design in multistation assembly involving a large number of parts and stations, and creates research opportunities in this area.
关DOI: 10.1115/1.2943296兴
Keywords: product and process tolerance allocation, multi-station compliant assembly,
hierarchical multi-level optimization, analytical target cascading
1
Introduction
Tolerance is the allowable range of variation from design intent
in a dimension. Tolerance allocation is a decision-making process
performed early in the product development cycle, before parts
are produced and tools are made. In this process, both product
共parts and subassemblies兲 and process 共fixtures and tools兲 tolerances are determined with the aid of manufacturing models and
design rules in order to meet final assembly quality and cost targets.
Tolerance allocation 共also referred to as tolerance synthesis兲
consists of three key ingredients: tolerance analysis; modeling of
the relations among variation, tolerance, and cost; and optimization. Tolerance analysis is conducted using variation propagation
models that compute how part, subassembly, and process variations propagate to final product variation, which is related to product quality. Variations are typically translated to tolerances using
statistical principles, and analytical models are then used to estimate cost as a function of tolerance. Finally, optimization problems are formulated to minimize cost and/or final product variation subject to geometric and operational constraints.
The tolerance allocation optimization problem can be linear or
nonlinear, with continuous or discrete variables. Accordingly, integer programming, sequential quadratic programming, or simulated annealing algorithms are used to solve it. Spotts 关1兴 devel1
Corresponding author.
Contributed by the Design for Manufacturing Committee for publication in the
JOURNAL OF MECHANICAL DESIGN. Manuscript received August 17, 2006; final manuscript received April 16, 2008; published online August 8, 2008. Review conducted
by Jeffrey Herrmann.
Journal of Mechanical Design
oped a nonlinear model that was solved using the method of
Lagrange multipliers. Lee and Woo 关2兴 suggested an integer programming formulation and solved the associated problem using
the branch and bound method. Chase et al. 关3兴 developed a simple
nonlinear model and used exhaustive search, univariate search,
sequential quadratic programming, and branch and bound to solve
the tolerance allocation problem. Zhang and Wang 关4兴 developed a
nonlinear integer model to allocate tolerances while minimizing
cost. The design problem was nonconvex, possessed multiple local minima, and was solved using simulated annealing. Extensive
reviews of existing approaches and tolerance allocation problem
formulations can be found in Refs. 关5–7兴.
In general, two types of variation propagation models exist,
depending on whether the components are considered rigid or
compliant. Rigid body-based assembly models consider geometric
and kinematic errors only. Compliant assembly models consider
deformations due to clamping and joining in addition to geometric
and kinematic errors. Finite element analysis models are necessary to take into consideration the stiffness of parts and subassemblies, and the forces applied by each tool, thus evaluating the
sensitivity matrices in variation propagation models.
Allocation of tolerances for rigid assembly processes has been
addressed extensively 关8兴. Ding et al. 关9–11兴 proposed and demonstrated a framework for process-oriented tolerance synthesis for
rigid multistation assembly systems, where process tolerances are
optimally allocated by solving a nonlinear constrained optimization problem. The tolerance-variation relation was based on pinhole fixture mechanisms in multistation assembly processes. Pro-
Copyright © 2008 by ASME
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cess degradation such as tool wear was also incorporated into the
framework, which provided the ability to design tolerances for the
whole life cycle of a production system.
Tolerance allocation methods developed for rigid body assembly may not be applicable to compliant assembly systems because
of required finite element analysis and consideration of out-ofplane errors 关12兴. Previous research efforts on tolerance analysis
for compliant processes 共e.g., Refs. 关13–16兴兲 do not consider tolerance allocation even though compliant assemblies are widely
used in industries such as automotive, aerospace, and electronics.
For example, 37% of all stations in automotive body structure
manufacturing assemble compliant parts 关17兴. Shiu et al. 关18兴 presented a tolerance allocation methodology for compliant beam
structures in automotive and aerospace assembly processes. Based
on a beam structure model, the method minimized manufacturing
cost associated with tolerances of product functional requirements
subject to constraints related to process requirements. The simplified beam structure models, however, limit further applications
and extension of this methodology to multistation compliant assembly systems.
Traditionally, tolerance allocation is conducted by solving a
single optimization problem, where all the tolerance values of all
the parts that comprise the product are computed using one large
lumped model. This “all-in-one” 共AIO兲 approach may not be desirable or applicable for various reasons.
•
•
•
•
Original equipment manufacturers 共OEMs兲 in many industry branches are sometimes mere assemblers of the final
product. In these cases, the design and manufacturing of the
parts and subassemblies that form the final product are outsourced to suppliers. The OEM does not have access to
models and/or data to compute appropriate tolerance values
for all subassemblies and parts since the suppliers protect
their business by not releasing relevant proprietary information.
Even when an OEM produces a product without any outsourcing, the design, development, and manufacturing process is typically distributed among departments and/or divisions of the OEM according to expertise. The tolerance
allocation problem appears already in a naturally decomposed form. The program manager of a specific product
needs to determine early on in the design process appropriate tolerance values for the subassemblies that will be delivered from different parts of the organization.
Tolerance allocation is a nonlinear programming problem;
optimization algorithms may face numerical difficulties
when solving very large-scale nonlinear problems, especially when the required analyses are based on simulations
where gradients may not be computable or trusted. Solving
a number of smaller problems may be much easier.
Interactions are often obscured in AIO models and trade-offs
may not be quantifiable readily. Decomposed problems can
offer more insights by enabling investigations into the coupling structure of the original problem.
In this paper, we propose a methodology for product and process tolerance allocation in multistation compliant assembly. We
model multistation compliant assembly as a hierarchical multilevel process, employ recently developed models that are appropriate for multistation variation propagation, and propose an optimization strategy to translate final product quality and cost targets
to tolerance specifications for incoming parts, subassemblies, and
station fixtures. We show that the multilevel optimization approach enables the generation of results that reveal whether final
product quality is more sensitive to some parts and subassemblies
than others. We also quantify cost-quality trade-offs, so that we
can allocate resources accordingly for both purchasing incoming
parts and improving the precision of the stations that process the
more sensitive subassemblies. We believe that the proposed multilevel optimization methodology constitutes a valuable new para091701-2 / Vol. 130, SEPTEMBER 2008
digm for tolerance allocation in multistation assembly involving a
large number of parts and stations, thus creating new research
opportunities. It should be emphasized that we do not consider the
target setting problem, i.e., assigning final product quality and cost
targets. We assume that these are given 共specified a priori兲; we are
concerned with translating these targets to specifications for the
assembly process and components.
The paper is organized as follows. We first present the variation
propagation model and the variation, tolerance, and cost relations
used in this work. We then introduce analytical target cascading
共ATC兲, which is used to model multistation tolerance allocation as
a hierarchical multilevel optimization process. Finally, we demonstrate the proposed methodology using a vehicle side frame assembly example: We consider both quality- and cost-driven optimization formulations, conduct parametric studies with respect to
budget, and discuss the obtained results.
2
Variation Propagation Model
We begin with a single station variation propagation model that
is based on the method of influence coefficients. Considering the
compliant nature of sheet metal parts, Liu and Hu 关19兴 proposed a
model to analyze the effects of component deviations and assembly springback on assembly variation by applying linear mechanics and statistics. The linear model is
vw = Svu
共1兲
where vw and vu are the dimensional variation vectors of the key
product characteristics 共KPCs兲 of the assembly and its components, respectively. The sensitivity matrix S establishes the linear
relationship between the incoming part deviation and the output
assembly deviation.
For multistation assembly processes, it is necessary to define an
appropriate variation representation in order to track the variation
propagation from station to station. The variation propagation process is sequential, i.e., to estimate the variation at station k, it is
necessary to know the variation at the previous station 共k − 1兲.
Moreover, there is a station-to-station interaction introduced by
the release of holding fixtures in the current station and the use of
new fixtures in subsequent stations.
A multistation assembly process can be considered as a sequential discrete-time dynamic system, where the time index in the
traditional state space model is replaced by a station index. Therefore, a state space representation can be used to illustrate stationto-station variation propagation in multistation assembly processes 关20兴.
Based on the single station model 共1兲, Camelio et al. 关21兴 proposed the use of a state space model to represent the dimensional
variation propagation in a multistation compliant assembly process. The model identifies three sources of variation in a compliant assembly 共part, fixture, and joining tool variation兲 and computes variation propagation as
xk = 共Sk − Pk + I兲共I + Mk兲xk−1 − 共Sk − Pk + I兲Mkuk1
− 共Sk − Pk兲共uk2 + uk3兲 + wk
共2兲
where xk is the state vector that contains dimensional variations at
measurement points. For a N-2-1 fixture configuration, the input
vectors include the dimensional variation state vectors of 共a兲 the
part KPCs xk−1, 共b兲 the “3-2-1” locating fixtures uk1, 共c兲 the 共N
− 3兲 additional holding fixtures uk2, and 共d兲 the assembly tool
共e.g., welding gun兲 uk3; wk is the noise vector. If the fixture
scheme is 3-2-1 and welding guns are perfect, Eq. 共2兲 is simplified
to
xk = 共Sk − Pk + I兲共I + Mk兲xk−1 − 共Sk − Pk + I兲Mkuk1 + wk
共3兲
= Akxk−1 + Bkuk1 + wk
In order to obtain matrices Ak and Bk of Eq. 共3兲, we compute the
relocation 共or reorientation兲 matrix Mk, the part deformation maTransactions of the ASME
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trix Pk, and the sensitivity matrix Sk. The matrix Mk represents
how the state vector changes due to the change of the locating
scheme from the previous station to the current station. The matrix
Pk represents part deformation during assembly. The method of
influence coefficients is applied for each station in order to obtain
the deformation matrix Pk and the sensitivity matrix Sk 关19兴. The
relocation matrix Mk is defined using homogeneous transformation and the positions of the locators. In this work, wk is assumed
to be equal to zero.
A compliant assembly variation analysis 共CAVA兲 model has
been developed for computing the aforementioned matrices 关22兴.
The model relates the dimensional deviations of an assembly to
individual component deformations, which can be varied statistically according to variations in noncomponent factors 共machining,
fixturing, and tooling兲. It also considers the springback, the force
information, the N-2-1 locating principle, and additional variations arising from the configuration of the assembly system 共series, parallel, or hybrid lines兲.
The CAVA input file includes information on the finite element
model, measurement points, welding points, fixture locating
points, and fixture release points. CAVA calls MSC. Nastran to
conduct the finite element analysis, and outputs matrices Ak and
B k.
3
Variation, Tolerance, Quality, and Cost Relations
Under the assumption of statistically independent variations for
parts and fixtures, the covariance can be derived from Eq. 共3兲 as
⌺xk = Ak⌺xk−1ATk + Bk⌺uk BTk
共4兲
1
The vectors ␴x2 = diag共⌺xk兲 and ␴u2 = diag共⌺uk 兲 include subask1
k
1
sembly KPC variances and fixture key control characteristic
共KCC兲 variances, respectively. In this work, it is assumed that the
process capability C p is equal to 1, so that tolerance t is related to
standard deviation ␴x according to
t = 6␴x
共6兲
Given a tolerance vector t of some assembly, the associated
quality q is given by
q = 储t储⬁
共7兲
Note that any norm can be used to translate final product dimensional variations into a single quality measure, but it seems that
the infinity norm is preferred by industrial practitioners 关11兴.
Using tight tolerances is costly for manufacturers 共tight tolerances may make assembly easier and less expensive, but do require high-quality, and thus expensive, incoming parts兲, while
loose tolerances may lead to reduced product quality 关24兴. Cost
models take into account product and process tolerances and
evaluate total assembly cost through cost-tolerance relations. Most
cost models consider fixed costs and manufacturing costs for each
Journal of Mechanical Design
共8兲
c共t兲 = k1 + k2e−k3t
where k1 represents fixed costs, k2 is the cost of producing a single
component dimension to a specified tolerance t, and k3 describes
how sensitive the process cost is to changes in tolerance specifications. For simplicity, we assume that k1 = 0, k2 = 1, and k3 = 3,
and that process tolerances are subject to the same cost model. For
a vector of product tolerances with dimension n and a vector of
process tolerances with dimension m, the particular cost-tolerance
model becomes
n
共5兲
In the optimization, the actual value of C p does not influence the
results as long as the same process capability is assumed for all
components, subassemblies, and final assembly. Note that this
work does not consider geometric variance. Even though one of
the authors has done some work in variation analysis taking into
consideration geometric covariance 关23兴, there is no established
work or procedure in translating a given geometric tolerance 共tolerance zone兲 into variation of the part for variation analysis. The
current practice in industry is to digitize the tolerance zone. As a
result, the correlation among these digitized locations is usually
ignored in variation analysis.
Product tolerances t are associated with KPCs, while process
tolerances ␶ are associated with KCCs. Based on Eqs. 共4兲 and 共5兲,
the product tolerance vector of a subassembly at station k depends
on the product tolerance vectors of the parts that comprise the
subassembly and the process tolerance vector of the station:
tk = Ftk共tk−1, ␶k兲
component of the assembly 关25兴. Fixed costs include setting up,
fixturing, loading and unloading, handling, tooling, and other preprocessing operations. Manufacturing costs are associated with
tolerance specifications of incoming parts, fixtures, and process
tools. Manufacturing costs represent the costs of producing a
single component dimension to a specified tolerance.
The most widely used cost-tolerance models were developed by
Speckhart 关26兴. He proposed an exponential cost-tolerance model
for machining, and suggested an allocation model for both linear
and nonlinear design functions using worst-case or statistical approaches. Wu et al. 关27兴 reviewed several cost models and concluded that models that define cost as a combined 共exponential/
reciprocal power兲 function of tolerances are the most accurate,
followed by models based on exponential relations, and models
based on reciprocal relations. Other similar functions can be applied to describe the cost-tolerance relations. A good survey of
cost models can also be found in Ref. 关3兴. Reciprocal and exponential tolerance-cost models are the most popular ones 关26–30兴.
Since manufacturing cost is both site and process dependent,
cost is usually calculated based on empirical relations and data. If
data are not available, choosing an appropriate cost model depends on a comprehensive understanding of the specific manufacturing system. Reciprocal and exponential cost functions are good
alternatives, offering decent data fit and simple function structures. In this work, the exponential function of Ref. 关26兴 is chosen:
c = Fc共t, ␶兲 =
兺e
m
−3ti
i=1
4
+
兺e
−3␶i
共9兲
i=1
Multilevel Optimization for Tolerance Allocation
Tolerance allocation is based on the trade-off between cost and
quality. This trade-off is quantified by formulating and solving the
biobjective optimization problem
min 兵c共t, ␶兲,q共t, ␶兲其
t,␶
subject to g共t, ␶兲 ⱕ 0
共10兲
where t is the product tolerance vector, ␶ is the process tolerance
vector, c is total assembly cost, and q is final product quality. The
constraints g共t , ␶兲 艋 0 include geometric, operational, and bound
constraints.
The Pareto set of solutions of Problem 共10兲 can be generated by
solving a series of cost-driven problems, Problem 共11兲, with different qsi bound values or a series of quality-driven problems,
Problem 共12兲, with different bi bound values:
min c共t, ␶兲
t,␶
subject to q共t, ␶兲 ⱕ qsi
g共t, ␶兲 ⱕ 0
共11兲
min q共t, ␶兲
t,␶
subject to c共t, ␶兲 ⱕ bi
SEPTEMBER 2008, Vol. 130 / 091701-3
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Fig. 1 Variation models as an ATC process
g共t, ␶兲 ⱕ 0
where qsi is the quality requirement and bi is the available budget.
As mentioned in the Introduction, we model multistation assembly as a hierarchical multilevel process 共see Fig. 1兲, and use
ATC, a multilevel optimization methodology 关31兴, to formulate
and solve the tolerance allocation problem.
In ATC, top-level design targets are cascaded to subsystem and
component design specifications by exploiting the multilevel hierarchy of a decomposed system. Specifically, ATC operates by formulating and solving a minimum target deviation optimization
problem for each element of the multilevel hierarchy. Assuming
that inputs of higher level elements are the outputs of lower-level
elements, it aims at minimizing the gap between what upper-level
elements “want” and what lower-level elements “can.” The objective is to identify early in the design process the relations and
possible trade-offs among elements, and to determine specifications that yield consistent system design with minimized deviation
from design targets. ATC has been applied successfully to a variety of engineering design problems 共e.g., Refs. 关32–36兴兲, and has
proven theoretical convergence properties for certain coordination
strategies 关37兴.
As shown in Fig. 1, in a multistation assembly process, variations from incoming parts 共at level N兲 and fixture variations 共from
level 共N − 1兲 to level 0兲 are propagated level by level to variations
of the final product. At the top level 共the final station兲, where all
the subassemblies are joined to form the final assembly, target
values can be assigned for product and process tolerances and cost
of the final assembly. Using the introduced variation propagation
model, tolerance-variation relation, and cost-tolerance model,
these target values can be cascaded level-by-level down through
the subassemblies, all the way to the bottom level, and then rebalanced up to enable negotiation among levels. The cost associated
with each station is accounted for during the process.
The general mathematical formulation of the ATC process is
presented in detail in the aforementioned references. Here, the
specific formulation is presented as it applies to the tolerance allocation problem in multistation compliant assembly systems. The
mathematical formulation of the tolerance allocation subproblem
at station j of level i is
M ij
min 储tij − tijH储22 +
zij
兺 储t
共i+1兲k
− t共Li+1兲k储22 + 共cij − cijH兲2
k=1
M ij
+
兺 共c
共i+1兲k
− c共Li+1兲k兲2
k=1
subject to gij共t共i+1兲1, . . . ,t共i+1兲M ij, ␶ij,c共i+1兲1 . . . ,c共i+1兲M ij兲 ⱕ 0
091701-4 / Vol. 130, SEPTEMBER 2008
hij共t共i+1兲1, . . . ,t共i+1兲M ij, ␶ij,c共i+1兲1 . . . ,c共i+1兲M ij兲 = 0
共12兲
共13兲
where M ij denotes the number of “children” stations of station j,
t共i+1兲1 , . . . , t共i+1兲M ij represent product tolerance vectors of parts or
subassemblies at the children stations, ␶ij is the process tolerance
vector of fixture locators at the current station, and
c共i+1兲1 , . . . , c共i+1兲M ij are costs accumulated up to the children stations. The vector of optimization variables zij consists of
t共i+1兲1 , . . . , t共i+1兲M ij, c共i+1兲1 , . . . , c共i+1兲M ij, and ␶ij. Note that tH
ij and
cH
are
quantities
computed
at
the
level
above,
while
ij
L
L
L
L
, . . . , t共i+1兲M
and c共i+1兲1
, . . . , c共i+1兲M
are quantities computed
t共i+1兲1
ij
ij
at the level below.
The ATC process is applicable because of the existence of the
functional dependencies tij = Ftij共t共i+1兲1 , . . . , t共i+1兲M ij , ␶ij兲 and cij
= Fcij共t共i+1兲1 , . . . , t共i+1兲M ij , ␶ij , c共i+1兲1 , . . . , c共i+1兲M ij兲, according to
Eqs. 共6兲 and 共9兲. The total cost cij of a station j at level i is defined
as the sum of costs accumulated up to the children stations plus
the cost of the station process, which depends on both product and
process tolerances. Note that the top-level station does not have a
“parent” station. Tolerance and cost targets at top-level station
共parameters with superscript H兲 are typically dictated by the management. Similarly, bottom-level stations do not have children stations. The second and fourth terms of the objective function in
Problem 共13兲 are not included in the formulation of bottom-level
subproblems.
5
Vehicle Side Frame Assembly
The proposed methodology for tolerance allocation in compliant multistation assembly is demonstrated using a vehicle side
frame assembly example. The assembly process consists of three
stations and four incoming parts, which include a motor rail 共Part
1兲, two door rings 共Parts 2 and 3兲, and a rear quarter 共Part 4兲. The
hierarchy is shown in Fig. 2. Parts 1 and 2 are assembled at station
I to form Subassembly 1. Parts 3 and 4 are joined together at
Station II to form Subassembly 2. At Station III, the top station,
Subassemblies 1 and 2 are joined to form the final assembly. Thus,
we have the following tolerance and cost functional dependencies:
t01 = Ft01共t11,t12, ␶01兲,
t11 = Ft11共t21,t22, ␶11兲,
t12 = Ft12共t23,t24, ␶12兲
c01 = Fc01共t11,t12, ␶01兲,
c11 = Fc11共t21,t22, ␶11兲,
共14兲
c12 = Fc12共t23,t24, ␶12兲
where t21, t22, t23, and t24 are tolerance vectors for Parts 1, 2, 3,
and 4, respectively, at level 2, t11 is the tolerance vector for SubTransactions of the ASME
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Fig. 2 Fixture layout in the multistation compliant assembly system
assembly 1 at Level 1, t12 is the tolerance vector for Subassembly
2 at Level 1, t01 is the final assembly product tolerance vector at
the top station, and ␶01, ␶11, and ␶12 are the process tolerance
vectors at Stations I, II, and III, respectively.
Figure 2 also depicts the finite element models used to compute
the stiffness of parts and subassemblies and the forces applied by
each tool. A variety of data describe the process on each station.
These data include material properties, boundary conditions, and
locations of measurement points, fixture locating points, and
welding points. All this information is shared between the finite
element and the variation propagation models. In this example,
the length and thickness for each incoming part are 700 mm and
2 mm, respectively. The material is mild steel with Young’s
modulus E = 2.06 GPa, and Poisson’s ratio ␯ = 0.3. Locations of
measurement points, fixture locating points, and welding point are
shown on Fig. 2. The typical 3-2-1 fixture layout is used on every
station, consisting of two locating pins, P4way and P2way, and three
NC blocks. As a 3-2-1 fixture layout can be denoted by
兵P4way , P2way , NCk , k = 1 , 2 , 3其, the fixture layout changes in the
multistation compliant assembly process are represented as follows:
Station I: 兵兵P1, P2,NC1,NC2,NC3其,兵P3, P4,NC4,NC5,NC6其其
Station II: 兵兵P5, P6,NC7,NC8,NC9其,兵P7, P8,NC10,NC11,NC12其其
Station III: 兵兵P1, P4,NC1,NC5,NC13其,兵P6, P7,NC8,NC10,NC14其其
5.1 Quality-Driven Tolerance Allocation. We apply the ATC
methodology to a quality-driven optimization formulation that
considers product tolerances only, and conduct a parametric analysis with respect to available budget in order to quantify costquality trade-offs. Two budget scenarios are considered, one with
individual local budget constraints and one with a single global
budget constraint.
5.1.1 Local Budget Constraints. One way to evaluate the costquality trade-off is to constrain the money supply for each station.
In this scenario, local budget constraints ensure that money spent
to purchase each incoming part is equal and will not exceed the
available budget bl. Product quality targets 共final product tolerH
ances tH
01兲 are assigned by management. In this work, t01 is set
Journal of Mechanical Design
equal to 0. The mathematical formulation of the ATC subproblems
for Stations I, II, and III is
min
t11,t12
2
L 2
L 2
储t01共t11,t12兲 − tH
01储2 + 储t11 − t11储2 + 储t12 − t12储2
subject to
min
t21,t22
t11,t12 ⱖ 0
共15兲
2
储t11共t21,t22兲 − tH
11储2
subject to
c21共t21兲 艋 bl
c22共t22兲 艋 bl
0.01 mm 艋 t21,t22 艋 2 mm
min
t23,t24
and
共16兲
2
储t12共t23,t24兲 − tH
12储2
subject to
c23共t23兲 艋 bl
c24共t24兲 艋 bl
0.01 mm 艋 t23,t24 艋 2 mm
共17兲
respectively. The tolerance of a clearance is usually larger than
0.01 mm. Thus, in this work, product tolerance is chosen from the
interval 关0.01, 2兴 mm, i.e., 0.01 mmⱕ t ⱕ 2 mm. For process tolerance, the upper bound is assumed to be one-third of the product
2
tolerance upper bound, so that 0.01 mmⱕ ␶ ⱕ 3 mm.
The optimization subproblems are solved using the MATLAB
implementation of the sequential quadratic programming 共SQP兲 algorithm. For this bilevel hierarchy, the top-level problem is solved
first. The optimal values are then cascaded as targets for the
bottom-level problems, which are then solved independently to
match them. Bottom-level optimal values are then passed up to
the top-level problem to be used as parameters in the consistency
terms of the objective function. This completes one ATC iteration.
This process is repeated until the optimization variable values do
not change significantly after successive ATC iterations.
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min
t21,t22
2
H 2
储t11共t21,t22兲 − tH
11储2 + 共c11共t21,t22兲 − c11兲
subject to
min
t23,t24
0.01 mm 艋 t21,t22 艋 2 mm
and
共19兲
2
H 2
储t12共t23,t24兲 − tH
12储2 + 共c12共t23,t24兲 − c12兲
subject to
0.01 mm 艋 t23,t24 艋 2 mm
共20兲
respectively.
Fig. 3 Reduction in final product variation as a result of increased budget for both scenarios
5.1.2 Global Budget Constraints. In this scenario, the total
cost accumulated from bottom levels is required to be less than
m
the global budget bg = 兺i=1
bli, where m is the number of incoming
parts at bottom levels, and bli is the local budget for purchasing
the ith incoming part. Note that local budgets are not necessarily
equal. In ATC, the purchasing costs for incoming parts are treated
as design variables and are determined by solving the top-level
problem. The mathematical formulation of the ATC subproblems
for Stations I, II, and III is
min
t11,t12,c11,c12
2
L 2
L 2
储t01共t11,t12兲 − tH
01储2 + 储t11 − t11储2 + 储t12 − t12储2
+ 共c11 − cL11兲2 + 共c12 − cL12兲2
subject to
c01共c11,c12兲 = c11 + c12 艋 bg
t11,t12,c11,c12 ⱖ 0
共18兲
5.1.3 Results. For both considered scenarios, a parametric
study was conducted with respect to available budget. Since the
only cost incurred is from purchasing incoming parts, the idea is
to investigate the effects of local budgets for incoming parts on
the final product quality. The parametric study is conducted to
quantify the cost-quality trade-off, i.e., determine how much quality improves with increased budgets of incoming parts.
The percentage improvement of final product quality 共variation
reduction兲 is depicted in Fig. 3. It is observed that budget increases result in improvement of final product quality in both
scenarios. The second scenario, however, yields greater variation
reduction. This can be explained by inspecting Fig. 4, where percentage reduction in variations of incoming parts are depicted. In
the first scenario, the budget is increasing equally for each incoming part, so variation reduction is the same for all parts, according
to the cost-tolerance relations and tolerance-variation models. In
the second scenario, a global budget constraint must be satisfied.
The latter allows flexibility and results in allocating resources to
more sensitive parts 共in this case Parts 3 and 4兲, which in turn
results in better final product quality.
5.2 Cost-Driven Tolerance Allocation. Here, we apply the
ATC methodology to a cost-driven optimization formulation. We
first consider product tolerance allocation only, and then we also
conduct process tolerance allocation in order to compare the two
sets of results.
Fig. 4 Reduction in part product variation as a result of increased budget for both
scenarios
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5.2.1 Product-Related Tolerance Allocation Only. This scenario focuses on the total cost incurred by purchasing incoming
parts. Dimensional variation is affected mainly by the variability
of parts, fixtures, and joining methods at each of the multiple
stations. As a first step, only the variability of parts is taken into
account. The fixture locators are assumed to have no out-of-plane
error and to be positioned at their nominal positions.
To avoid trivial results, upper bounds must be given for tolerances at final product KPCs. In current industrial practice, final
products typically have six-sigma tolerance target values of qs
= 1.5 mm. The quality constraint ensures that final product quality
will satisfy the quality requirement qs. Once again, we assume
that the total cost target cH
01 is assigned by management. In this
work, we set cH
01 equal to 0. The mathematical formulation of the
ATC subproblems for Stations I, II, and III is
min
t11,t12,c11,c12
L 2
L 2
2
共c01共c11,c12兲 − cH
01兲 + 储t11 − t11储2 + 储t12 − t12储2
+ 共c11 − cL11兲2 + 共c12 − cL12兲2
subject to
储t01共t11,t12兲储⬁ 艋 qs
t11,t12,c11,c12 ⱖ 0
min
t21,t22
min
6
2
H 2
储t11共t21,t22兲 − tH
11储2 + 共c11共t21,t22兲 − c11兲
0.01 mm 艋 t21,t22 艋 2 mm
subject to
t23,t24
共21兲
and
共22兲
2
H 2
储t12共t23,t24兲 − tH
12储2 + 共c12共t23,t24兲 − c12兲
0.01 mm 艋 t23,t24 艋 2 mm
subject to
共23兲
respectively.
5.2.2 Product- and Process-Related Tolerance Allocation.
This scenario considers both product and process tolerance design
variables. The changes in tolerance values for fixture locators result in extra costs for the station. Therefore, process tolerances
and associated costs must be included in the problem formulation.
The mathematical formulation of the ATC subproblems for Stations I, II, and III is
L 2
2
共c01共c11,c12, ␶01兲 − cH
01兲 + 储t11 − t11储2
min
t11,t12,␶01,c11,c12
+ 储t12 − tL12储22 + 共c11 − cL11兲2 + 共c12 − cL12兲2
subject to
储t01共t11,t12, ␶01兲储⬁ 艋 qs
t11,t12,c11,c12 ⱖ 0
0.01 mm 艋 ␶01 艋
min
t21,t22,␶11
共24兲
0.01 mm 艋 t21,t22 艋 2 mm
0.01 mm 艋 ␶11 艋
min
mm
2
H 2
储t11共t21,t22, ␶11兲 − tH
11储2 + 共c11共t21,t22, ␶11兲 − c11兲
subject to
t23,t24,␶12
2
3
2
3
mm
and
共25兲
2
H 2
储t12共t23,t24, ␶12兲 − tH
12储2 + 共c12共t23,t24, ␶12兲 − c12兲
subject to
0.01 mm 艋 t23,t24 艋 2 mm
0.01 mm 艋 ␶12 艋
2
3
mm
共26兲
respectively.
5.2.3 Results. Figure 5 compares the two sets of results 共one
obtained by omitting and one obtained when including process
tolerance allocation in the problem formulation兲. The quality reJournal of Mechanical Design
quirement 共maximum variation兲 is satisfied in both cases, but individual variations at all but two measurement points of the final
assembly increase when considering tolerance allocation of fixture
locators. This is because the assumption of perfect assembly is of
course not valid in reality. In the presence of nonperfect process
tolerances, product tolerances of incoming parts must decrease
accordingly in order to satisfy the requirements for final product
quality.
Specifically, while product tolerances of Subassembly 1 increase, product tolerances of Subassembly 2 decrease. The reason
is that final product quality is more sensitive to Parts 3 and 4, and
thus Subassembly 2, than to Parts 1 and 2 共and thus Subassembly
1兲. Under the same budget, it is more effective to reduce tolerances of Parts 3 and 4 to achieve better final assembly quality. The
increased product tolerances of Subassembly 1 in Scenario 4 are a
consequence of both Parts 1 and 2 having reached their upper
bounds and nonzero fixture variation. Note that the results are not
symmetric even though the parts are. This is due to the fact that
the measurement points on the final assembly are not symmetric
共careful inspection of Fig. 2 required兲. This asymmetry in the
location of the measurement points has an impact to the tolerances
of the final assembly.
Concluding Remarks
We presented a multilevel optimization methodology for product 共parts and subassemblies兲 and process 共fixtures兲 tolerance allocation in multistation compliant assembly. We modeled multistation compliant assembly as a hierarchical multilevel process
and employed appropriate compliant variation propagation model
and popular cost-tolerance relations to compute final product quality and cost given part, subassembly, and fixture variations. We
then used analytical target cascading to formulate and solve the
multilevel tolerance allocation optimization problem. This
decomposition-based approach translates final product quality and
cost targets to tolerance specifications for incoming parts, subassemblies, and station fixtures and quantifies cost-quality tradeoffs, so that we can allocate resources accordingly for both purchasing incoming parts and improving the precision of the stations
that process critical subassemblies.
We emphasize that the proposed decomposition-based approach
is not an end in itself. If tolerance allocation 共or any other optimization problem兲 can be conducted using an AIO approach, then
it probably should. We consider here cases where the tolerance
allocation problem either appears already in a decomposed form
due to the structure of the organization that designs and manufactures the product or must be decomposed because 共i兲 it cannot be
solved using an AIO approach because the OEM assembles outsourced parts and/or subassemblies and thus has no access to
models and/or data to solve the tolerance allocation problem, 共ii兲
numerical algorithms for nonlinear programming fail to solve the
large-scale, simulation-based problem, or 共iii兲 interactions are
hard to identify and trade-offs cannot be quantified readily. We
cannot provide any conclusive statements relative to the required
computational effort for solving ATC problems. This depends
largely on the coupling of the subproblems. However, we can
expect the efficiency of the methodology to increase with problem
size, i.e., ATC 共and decomposition approaches, in general兲 may be
more desirable for large-scale problems. Finally, even if the computational cost is high, it may be the only alternative to failing
numerical optimization of large-scale, simulation-based AIO problems.
We demonstrated the proposed methodology using a vehicle
side frame assembly example. We first considered quality-driven
product tolerance allocation only with both local individual and
global budget constraints for purchasing incoming parts, and conducted a parametric study with respect to available budget in order
to quantify the cost-quality trade-off. The results showed that,
under the same available budget, the multilevel optimization strategy can take advantage of local information and yields larger
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Fig. 5 Comparison of allocated tolerance values for KPCs of parts, subassemblies, and
final assembly, with and without process tolerance allocation
quality improvements when the global budget constraint formulation is used. We then considered a cost-driven formulation and
conducted tolerance allocation first for parts and subassemblies
only and then for parts, subassemblies, and fixtures. The results
confirmed that product tolerances need to be tightened to account
for nonzero fixture variations. The example demonstrated that the
multilevel optimization strategy is able to identify parts and subassemblies that affect final product quality more than others, so
that resources can be allocated accordingly to improve the precision of the assembly process at the appropriate stations.
We conclude that the proposed multilevel optimization methodology offers a promising new paradigm for tolerance allocation in
multistation assembly involving a large number of parts and stations, and that it opens research opportunities in this area.
091701-8 / Vol. 130, SEPTEMBER 2008
Acknowledgment
The authors would like to acknowledge the help and contributions of Dr. Jianpeng Yue and the partial support of this work by
the General Motors Collaborative Research Laboratory at the University of Michigan.
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