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In the final stages of product development, dimensional tolerances are specified by designers to ensure high functionality at low costs. A traditional approach to this decision-making process is to minimize economic losses to the manufacturer and the consumer. This paper presents a new approach for tolerance allocation optimization that considers sustainability not only from economic costs but also ecological costs. The framework is formulated as a multi-objective optimization problem and explored with a case study on the design of an automotive body panel. Results of the case study include Pareto frontiers of non-dominated optimal solutions along with a parametric study to explore the influence of material choice on the results.
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Tolerance Specification Optimization for Economic and Ecological Sustainability, p. 1, 2013.
© Springer-Verlag Berlin Heidelberg 2013
Tolerance Specification Optimization for Economic and
Ecological Sustainability
Steven Hoffenson, Andreas Dagman, Rikard Söderberg
Department of Product and Production Development, Chalmers University of Technology,
Gothenburg, Sweden
Abstract. In the final stages of product development, dimensional tolerances
are specified by designers to ensure high functionality at low costs. A tradition-
al approach to this decision-making process is to minimize economic losses to
the manufacturer and the consumer. This paper presents a new approach for tol-
erance allocation optimization that considers sustainability not only from eco-
nomic costs but also ecological costs. The framework is formulated as a multi-
objective optimization problem and explored with a case study on the design of
an automotive body panel. Results of the case study include Pareto frontiers of
non-dominated optimal solutions along with a parametric study to explore the
influence of material choice on the results.
Keywords: tolerance allocation, variation propagation, sustainability, cost min-
imization, multi-objective optimization
1 Introduction
The quality of products is partially determined when designers select tolerances to
specify in engineering drawings. Tolerance selection is a necessary step in product
design since no product dimension can be manufactured with perfect accuracy, and a
tradeoff exists because tighter tolerances increase manufacturing costs and looser
tolerances often inhibit the assemblability and functionality of the product. Poor as-
semblability can result in a high number of defective parts and products being dis-
carded as well as time wasted. Poor functionality, which is typically associated with
consumer perception of quality, can result in products being discarded and replaced
early in the expected life cycle along with customer dissatisfaction and lowered brand
reputation. Discarding parts and products is not only an economic concern for manu-
facturers and consumers, but it is also an ecological issue, since resource and energy
usage in the production and disposal of replacement parts can be significant.
This article presents a multi-objective modeling and optimization framework for
considering the economic and ecological ramifications of tolerance selection in the
final stages of product design. The design approach is demonstrated through a case
study that considers the tolerance specifications for an automotive body panel, where
specified tolerances influence a critical dimension that affects the product’s assem-
bled functionality. Results include Pareto frontiers as well as a parametric study to
introduce the influence of material choice on the optimization results.
The ensuing section discusses previous literature on tolerance analysis and varia-
tion propagation, previous efforts in tolerance-cost minimization, and techniques for
environmental impact assessment. Section 3 presents the framework, methodology,
and models used in the case study. Optimization results for the tolerances in the au-
tomotive panel are delivered in Section 4 and discussed in Section 5. The final section
offers conclusions and discusses the broader implications of the work.
2 Background
Dimensional tolerance allocation is a subject currently addressed at several confer-
ences and journals, and there is abundant literature discussing years of investigation
on variation propagation and cost optimization in the domain of tolerance specifica-
tion. Sections 2.1 and 2.2 discuss the state of the art in these two areas, respectively.
The novelty of the present approach is the addition of ecological sustainability as an
objective for tolerance allocation, and Section 2.3 provides a brief background of
research on ecological sustainability and how it is managed in general applications.
2.1 Tolerance Allocation and Variation Propagation
Tolerances of product dimensions must be chosen carefully due to the aforementioned
tradeoff between quality and cost, and they are often chosen based on the ways that
each dimension influences the functional requirements of the part or product. Vari-
ance in one measurement of a part or product is said to propagate and influence the
resulting variance in other measurements. Techniques to calculate the relationships
and propagation between specified tolerances and critical dimension variations are
often referred to as tolerance analysis and synthesis [1].
Designers typically set tolerances based on either worst-case design scenarios or
statistical analyses of tolerance probability distributions and the associated critical
dimension variations. This is most simply calculated using linear or linearized toler-
ance accumulation models, where the tolerances of components assembled in series
are summed or root-sum-squared to predict the variation across the length of the
whole [2]. A more common method is statistical tolerancing, where researchers and
practitioners typically assume normal distributions for prescribed tolerances and cal-
culate the distribution of the critical dimension [3]. When this calculation is impracti-
cal due to complex geometries, Monte Carlo methods are employed, in which a num-
ber of randomly-selected tolerances are generated as inputs to measure the resulting
output variations [4].
All of these methods of analyzing and synthesizing tolerances have been imple-
mented in a variety of commercial and proprietary computer software packages to aid
in tolerance allocation and, in some cases, optimization [5,6].
2.2 Tolerance-Cost Minimization
Typical tolerance optimization is conducted with the objective of minimizing manu-
facturing costs, seeking to improve the manufacturer’s economic sustainability while
preserving product functionality. The two challenges in doing so are in understanding
the relationships between specified tolerances and critical dimension variations, as
discussed in the previous section, and in modeling the relationships between specified
tolerances and costs. Data linking manufacturing costs and tolerances depend on a
number of environmental factors and are typically proprietary, so researchers often
use simple mathematical functions to describe these relationships for various manu-
facturing processes [7]. In many cases, the production of a component has more than
one eligible manufacturing process, and piecewise functions or discrete tolerance-cost
points can then be used for optimization [8,9]. Curves are generally fit to a set of cost-
tolerance data when available, but it is common for researchers to present methods
using assumed or generic cost-tolerance curves due to either an absence of data or
unwillingness to publish proprietary data. One popular function is the reciprocal func-
tion, shown in Equation (1) where c is cost, t is tolerance, and a and b are parameters
fit to match actual or estimated costs [10-12].
= a + b/
Some previous work in this area treats costs as losses to the manufacturer, where
the loss is the difference between the cost to manufacture a certain tolerance and the
minimum possible manufacturing cost, which is typically associated with loose toler-
ances [13-15]. This removes fixed costs from the equation and allows comparison of
financial costs with functionality losses due to loose tolerances as described by
Taguchi et. al [16]. The process capability index is commonly used in this work to
measure process performance, normalized to three standard deviations from the mean
of a tolerance distribution [17,18]. Söderberg [13] extends the loss function technique
with an additional objective representing loss to the customer, as parts with looser
tolerances are more likely to fail early during the use phase of a product.
2.3 Ecological Sustainability Metrics and Assessment Tools
In addition to influencing manufacturing costs, product quality for the manufacturer,
and quality to the customer, tolerance selection affects the ecological impact of a
product. Factors including choices of manufacturing processes, time and electricity
requirements, material usage, and rates of defective parts produced link tolerances
with ecological sustainability.
Ecological, or environmental, sustainability has been increasingly studied and de-
bated in recent decades, particularly in the context of Life Cycle Assessment (LCA)
or Life Cycle Engineering (LCE) techniques, which specifically target the cradle-to-
grave impact of products and processes [19]. A number of environmental impact da-
tabases, standards, and software packages have been proposed to aid in designing for
the health of the planet. These measurement tools include extensive environmental
impact databases that associate ecological impacts with various human actions, and
many of them include user interfaces to aid in identifying inputs and analyzing out-
puts [20,21]. Since the outputs typically fall under different categories of impact, such
as resource depletion, greenhouse gas emissions, air pollution, water pollution, and
landfill use, the tools use different techniques for presenting the results in a managea-
ble format. These include equating ecological impacts with monetary values [21] or
normalizing them against an average consumer’s annual impact [22].
3 Methodology
The present study combines tools and techniques from the literature to present a
framework for analysis and optimization of tolerances to minimize costs and ecologi-
cal impacts. The approach is illustrated in Figure 1.
Given VariablesParameters Outcomes
Defective parts
(%) discarded
Batch size
Materials Manufact.
Fig. 1. Framework of product tolerance-cost-environmental-impact relationships
Here, the late-stage design problem is considered where the product function, ar-
chitecture, and geometry have been decided upon, and batch size, locator positions,
materials and manufacturing processes are parameters. Parameters are not allowed to
vary during optimization, but they may be modified between optimizations to demon-
strate how they influence the solution; this is referred to as a parametric study, and
Section 4.2 shows the results of such a study on material choice. The ensuing subsec-
tions describe models that link the input parameters and variables to the outcomes in
Figure 1, and with them a bi-objective optimization problem is formulated.
3.1 Modeling
The following sub-sections describe the models used to calculate the links in Figure 1
between tolerance specifications and quality, manufacturing parameters and cost, and
manufacturing parameters and environmental impact. The models are built around the
case study of a D-pillar in an automotive vehicle body, which is comprised of two
stamped sheet metal components.
Variation Propagation. To model the propagation of variation from prescribed
tolerances to critical functional dimensions, the software package RD&T is used [13].
This program is designed specifically for the purpose of analyzing variation propaga-
tion in complex geometries and includes a graphical user interface for creating models
and visualizing results. The D-pillar used in this paper is shown in Figure 2, and it is
subject to manufacturing variations in places where the parts are supported by other
frame components and at the mating surface of the two parts. The functional require-
ment is that the three-dimensional position of the upper-rear corner, denoted with
black squares in Figure 2, be located near the nominally designed coordinates.
The allowed variation for these critical dimensions is unclear and depends to some
extent on the judgment of the designer and the design of the connecting parts. In this
case, the variation in all three coordinate dimensions is allowed within one millimeter,
but parametric studies are recommended to understand the sensitivity of the results to
this decision.
Fig. 2. D-pillar model in RD&T
Cost Modeling. As previously discussed, cost data on manufacturing processes are
not widely available or publishable. In this study, like in much of the literature, a
reciprocal manufacturing cost function is assumed with the form of Equation (1). As
in Ding, the equation parameters are set at a = 0, b = 1 [11].
Ecological Impact Modeling. Since the objective of this article is to discuss the
tradeoffs among costs and ecological impacts when making late-stage design deci-
sions, a monetary-based rating system is used to present environmental impacts. Envi-
ronmental Priority Strategies in product design (EPS) is one such metric for ecologi-
cal sustainability, and it contains an extensive database that links materials and pro-
cesses with environmental impacts, quantified in terms of the Environmental Load
Unit (ELU) [21]. In this rating system, one ELU is equivalent to an environmental
damage cost of one Euro (€).
Environmental impacts for a component such as the D-pillar of a vehicle come in
the production phase, the use phase, and the end-of-life. EPS provides production
impacts for creating various materials and manufacturing with different processes, as
well as end-of-life impacts for different scenarios, including disposal in a landfill,
combustion (for combustible materials), and reuse. The reuse scenario, which in-
volves disassembling and recycling the components, is typically assigned negative
ELUs, since reusing a material reduces future needs to produce usable materials from
raw minerals. This study reports results for the landfill and reuse scenarios, demon-
strating the benefits of designing with disassembly and reuse in mind.
The use phase of an automotive part is almost entirely dependent on mass, as high-
er-mass parts will require more fuel combustion over the life of the vehicle. Studies
consistently indicate that a 10% mass reduction on a vehicle results in a 7% reduction
in fuel consumption [23]. This is extrapolated and scaled for every percentage of mass
change in this study. Using the assumptions that the full vehicle mass is 1800 kilo-
grams, the baseline fuel consumption is 10 liters per 100 kilometers, and the baseline
calculation that the D-pillar weighs 4.78 kilograms, the impact of the D-pillar on fuel
consumption is calculated. The use phase impact is then calculated using the assump-
tion that an average vehicle drives 300,000 kilometers over its lifetime and the EPS
specification that unleaded petroleum costs the environment 1.12 ELU per kilogram.
The environmental impact of the use phase is thus calculated by the amount of fuel
required by the weight of the D-pillar over the life of the vehicle, compared to how
much fuel would be consumed if the vehicle had no right-side D-pillar.
3.2 Optimization Framework
A multi-objective optimization formulation has been developed around the use of
these three modeling tools. To simplify the optimization problem, the response sur-
face method (RSM) is used on the computationally-intensive model of variation prop-
agation in RD&T, where each simulation uses the Monte Carlo method with 5000
points. Based on 3600 complete simulations with full-factorial sampling of the two
input tolerances on a range of 0.05 to 3.00 millimeters, a polynomial response surface
was fit using linear regression to generate a closed-form mathematical function for the
probability of unacceptable critical dimensions in the part as a function of the input
tolerances. This surrogate model is given as Equation (2), where is the percentage
of unacceptable parts and
are the input tolerances for the mating sur-
faces and support points, respectively. The model fits the data with a 0.9985 coeffi-
cient of determination, suggesting that the structure of the model is acceptable.
   
 
  (2)
Using this surrogate model, a cost function is formulated with respect to tolerances
and the percentage of discarded parts, shown in Equation (3) where C is the economic
cost in Euros and Cmat is the cost of the materials in Euros, taken from [24]. The final
term on the right side of the equation    represents a multiplier to account for
discarded products, as a larger percentage of discarded products raises the effective
cost of producing acceptable products.
  
   (3)
An additional model for ecological impact is formulated using the percentage of
discarded parts and each of the factors from EPS. This is given as Equation (4), where
E is ecological impact in ELUs and is a function of mass m and ecological factors Ei
relating to material production, manufacturing process, the use phase, and end-of-life.
          (4)
Combining these models, a bi-objective optimization problem can be solved, for-
mulated as Equation (5).
   (5)
Here, the two components of the objective function are economic cost C measured
in Euros, and ecological impact E measured in ELUs, each with its associated
weighting parameter w. Optimization is performed with respect to the tolerance input
variables t1 and t2.
4 Results
Multi-objective optimization results are commonly presented as Pareto frontiers,
where each point on a curve represents a Pareto-optimal design that cannot be im-
proved for one objective without a sacrifice to the other. Results for the D-pillar case
study are presented in this section, starting with the baseline scenario and extending
with a parametric study on material choice.
4.1 D-pillar Design Costs and Ecological Impact
The baseline D-pillar is constructed with mild steel sheet metal and an allowance of
one millimeter of variance in all three critical dimensions. The Pareto frontier demon-
strating the tradeoff between cost and environmental impact is presented in Figure 3.
Fig. 3. Pareto frontier for baseline D-pillar, end-of-life landfill disposal
Each Pareto-optimal solution is represented in Figure 3 by a box, the dimensions of
which indicate the optimizing tolerances. The horizontal width of the box represents t1
(a wider box indicates a wider tolerance), and the vertical height represents t2. Here,
the upper-left solution corresponds with a purely cost-minimizing objective, and the
lower-right solution minimizes environmental impact. The results shown here corre-
spond with end-of-life disposal in a landfill; when reusing the materials is possible,
the optimal economic costs stay the same while the environmental impacts are low-
ered by 8%. Combustion of mild steel is not feasible.
4.2 D-pillar Design with Parametric Variation of Material Choice
The choice of material in this case study is important to the optimization solutions, as
it affects outcomes such as part mass, and therefore the use phase ecological impact,
as well as ecological sustainability impacts of extraction, production, and end-of-life
disposal. Further, the material thickness is adjusted for each material based on the
yield strength to ensure that the part can withstand the same compressive forces as a
steel component (e.g., for a rollover/roof-crush test). Data on the strength, density,
cost, and ecological impacts of five common automotive body materials were found
in [24] and [21], and the resulting Pareto frontiers from optimizing the D-pillar using
these materials are given in Figure 4, assuming end-of-life disposal in a landfill.
Fig. 4. Pareto frontiers of D-pillar varying material
Stainless steel has a much higher ecological impact of production than the other
materials, and so the entire curve is off the visible chart in an environmental impact
range of 340 to 380 ELUs. Like in Figure 3, Figure 4 presents data points as boxes
where the widths and heights represent the corresponding optimal t1 and t2 values.
Changing the end-of-life behavior from landfill disposal to reuse reduces the ecologi-
cal impact by 14%, 63%, 27%, and 29% for the latter four materials, respectively.
5 Discussion
The convexity of the bi-objective optimization results for the D-pillar tolerances
demonstrates a clear tradeoff between economic costs to the manufacturer and ecolog-
ical impact to society. The lower-right corner of Figure 3 shows that at high manufac-
turing costs, the tightest tolerances are associated with the lowest environmental im-
pact. As the weighting of the objective function shifts towards economic costs, t1 is
observed to increase, followed by an increase in t2. Even when the environmental
impact does not affect the objective function, t2 never reaches its maximum allowed
value, shown by the box in the upper-left corner of the plot. This is attributed to the
economic cost of discarding a large number of faulty parts when t2 is high. Designing
parts in which the materials can be reused rather than discarded is also beneficial, as it
reduces the environmental impact by up to 63%.
Parametrically varying the material choice shows that, for cost-minimizing firms,
mild steel is the best option for the D-pillar. In cases where the environmental impact
is more important to the manufacturer and costs are more flexible, magnesium be-
comes a better choice with tighter tolerances. In a scenario where the use phase be-
comes more important, e.g., if the number of kilometers driven increases substantially
from the assumption of 300,000, lighter materials such as aluminum and magnesium
may become more attractive than the mild steel. Likewise, in cases where corrosion of
the part is a concern, galvanized or stainless steels may become better choices. When
the design allows for reuse of the material, magnesium becomes the best choice.
6 Conclusions
While the specific results presented in this paper rely on assumptions built into the
models, the framework provides new and important insights into how late-stage de-
sign choices should be considered with respect to internal manufacturing costs and
external ecological costs. The automotive panel case shows a substantial tradeoff
between economic and ecological costs resulting from tolerance and material choices,
and further research is planned to reveal the impacts of additional sustainability deci-
sions by designers and policymakers. As legislation and rising consumer interests in
ecological sustainability continue to affect the market, this design approach will be-
come more common for firms seeking to maximize profits in a competitive market.
Acknowledgments. The authors acknowledge valuable intellectual and modeling
contributions from Kristina Wärmefjord and Lars Lindkvist of the Department of
Product and Production Development at Chalmers. This work, carried out at the
Wingquist Laboratory VINN Excellence Centre within the Area of Advance Pro-
duction at the Chalmers University of Technology, in Gothenburg, Sweden, was sup-
ported by the Swedish Governmental Agency for Innovation Systems (VINNOVA).
That support is gratefully acknowledged.
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... Much of this research seeks to minimize production costs within some variation threshold; however, economic sustainability for the company also relies on revenues, and it is not well-understood how product quality influences consumer demand and sales. Moreover, recent studies have shown that tolerance decisions can serve as tradeoffs between economic and ecological objectives (Hoffenson et al., 2012), both of which factor into the idea of sustainability (Holling, 2001). Manufacturers, though, do not have the same incentives to design for ecological sustainability as they do for economic sustainability, and therefore a stronger understanding of how ecological sustainability influences manufacturer profits is necessary. ...
... Due to concerns such as global climate change, air and water pollution, and declining reserves of natural resources, environmental sustainability has risen to become a top priority of research and policy initiatives. Research has shown how tolerance selection can influence environmental impacts through electricity usage and the amount of faulty parts or products discarded early in the lifecycle (Hoffenson et al., 2012). Other, often concurrent, choices such as material selection, locator positions, and the end-of-life strategy can also have profound impacts on a product's ecological footprint. ...
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... The selection of the model type and its fitting parameters, however, depends on the amount and type of data influenced by the given manufacturing process, machine and its settings, feature type, etc. [8,14]. Besides, numerous research activities focus on the quantification and representation of further, hardly measurable or intangible cost impacts of tolerancing, such as the present worth of loss by product degradation influencing the satisfaction and loyalty of the customer [31,32] or ecological and social costs [33,34], as well as their consideration in the optimization process, mainly by specific quality loss functions [35]. ...
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... Finally, recycling cost is also a candidate for optimization variables as considered in [36]. More generally, [37] proposes the minimization both of economic and ecological costs, illustrated on automotive industry and materiel choice. ...
In the manufacturing process of a product, various assembly steps are necessary. Several types of requirements have to be met at each level and involve considerations about dimensional uncertainties on the parts to be assembled. Tolerancing is the activity in charge of the management of these uncertainties and takes place both in the product development phase and in the series production phase. In the context of the aeronautics industry, in particular with regards to tolerancing on aerostructures, specificities have to be taken into account in the development of adequate methods and tools. Prior to production, one of the main issues of tolerancing amounts to allocate tolerance limits suited to a given acceptable scrap rate. The aim is to allow the actors concerned with tolerance intervals to agree on a consistent and robust tolerance value. A statistical methodology based on a Chernov bound approach applied to a sum of uniform distributions is proposed. In the production phase, the availability of measurement data allows to refine the statistical tolerancing approach. The linear model often considered can be corrected to serve new approaches. A methodology to manage acceptance criteria on tolerance values is proposed, basing the decision support on risk concepts pertinently defined for industrial actors. Within the framework of the revision of tolerance sharing in an assembly, an optimization problem is formulated with appropriate industrial costs in order to propose the optimal tolerance re-sharing in a stack chain. Finally, the proposed methodologies are implemented in tools allowing industrial processing and end-to-end management of tolerances from elementary parts to final product assembly, thus contributing to the elaboration of the product virtual twin.
... Sustainability has become the concern of tolerance design and make or buy decision. For example, a research by [12] developed a model to include not only economic consideration, but also ecological cost to determine optimal tolerance allocation. They formulated the model in a bi-objective optimization. ...
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Tolerance has been recognized as the connector between engineering design and manufacturing stages. It plays an important role in both stages and must be optimally determined to minimize several objectives such as manufacturing cost and quality loss. Considering the engineering design, the tolerance should be set tightly in order to get better quality. Conversely, considering manufacturing stage, the tolerance should be set loosely in order to make the manufacturing process less costly. Process and technological capabilities as well as limited production capacities made manufacturing companies to outsource some of their needed components to their suppliers. Hence, the manufacturing companies should decide which components have to make using its own production facilities and which one that must be purchased from the suppliers. These decisions can be made simultaneously with the determination of optimal tolerance. Due to the intense concerns on environment, sustainability gained more attention in recent years and attracted many researchers including in the field of make or buy decisions. We attempt to integrate the make or buy decisions with sustainability in the form of remanufacturing to determine the optimal component tolerance. We found that the Bass model, a widely used model to predict new product demand, can be integrated to make or buy decisions model specifically in determining the expected number of returned product to be remanufactured. We identify some problems in the integration such as the lead time which comes from the product useful life and time to collect the return product and also the representation of remanufacturing cost function in its relation with tolerance assignment, and considerations of single or multi manufacturing generations.
... Some of these uncertainties are geometric part deviations, which are inevitably observed on every manufactured artefact due to the axioms of manufacturing imprecision and measurement uncertainty (Srinivasan, 2006). It has been reported, that these geometric part deviations not only influence the product quality and cost (Wartzack et al., 2011), but also affect the economic and ecological sustainability (Hoffenson et al., 2013a(Hoffenson et al., , 2013b. Thus, there exists a strong need for the management of geometric part and product variations throughout the whole product lifecycle in order to ensure that the final product meets customer demands regarding not only product quality and cost but also sustainability. ...
Conference Paper
... Though not the main focus in aero engine manufacture, powertrain development etc., and thus not in this paper, other research is exploring incorporating softer requirements such as styling in design automation (Reid, et al., 2012). There are also methods for combining optimization of manufacturing constraints with soft requirements such as economic and social sustainability (Hoffenson, et al., 2013). Weber, et al. (2003) highlight the issue of the vast flora of models used to represent a mechanical design. ...
Conference Paper
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Development platforms are automated software tools used to synthesize new designs. They are prevalent in the embedded system design domain, with applications ranging from integrated circuits, circuit boards, electro-mechanical controls, and entire networked systems. Historically, this has enabled rapid and error-free design of very complex embedded software and electronics hardware, even those that control mechanical systems such as aerospace and automotive controls through automation of the design process. The state of mechanical design automation has far less commercial adoption or industrial demonstration of development platforms in mechanical design. This paper elaborates on what challenges mechanical design automation faces to reach the level of design automation in the embeded systems domain. Given a design library approach, it is concluded that uncertainty management is a key issue for future research, including model uncertainty for mechanical design modeling. These issues are then contextualized using a case from the aerospace industy.
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It is widely acknowledged that the allocation of part tolerances is a highly responsible task due to the complex repercussions on both product quality and cost. As a consequence, since its beginnings in the 1960s, least-cost tolerance allocation using optimization techniques, i.e. tolerance-cost optimization, was continuously in focus of numerous research activities. Nowadays, increasing cost and quality pressure, availability of real manufacturing data driven by Industry 4.0 technologies, and rising computational power result in a continuously growing interest in tolerance-cost optimization in both research and industry. However, inconsistent terminology and the lack of a classification of the various relevant aspects is an obstacle for the application of tolerance-cost optimization approaches. There is no literature comprehensively and clearly summarizing the current state of the art and illustrating the relevant key aspects. Motivated to overcome this drawback, this article provides a comprehensive as well as detailed overview of the broad research field in tolerance-cost optimization for both beginners and experts. To facilitate the first steps for readers who are less familiar with the topic, the paper initially outlines the fundamentals of tolerance-cost optimization including its basic idea, elementary terminology and mathematical formulation. These fundamentals serve as a basis for a subsequent detailed discussion of the key elements with focus on the different characteristics concerning the optimization problem, tolerance-cost model, technical system model and the tolerance analysis model. These aspects are gathered and summarized in a structured mind map, which equips the reader with a comprehensive graphical overview of all the various facets and aspects of tolerance-cost optimization. Beside this, the paper gives a retrospect of the past fifty years of research in tolerance cost-optimization, considering 290 relevant publications. Based thereon, current issues and future research needs in tolerance-cost optimization were identified.
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Involving customer values in the design process in necessary for improving the total quality of a product. This paper presents the basic ideas for a method that allows tolerances to be assigned to dimensions in a tolerance chain with regard to both customer and manufacturer objectives. The method uses as extended 'quality loss function' to consider customer objectives. The total life of a component is here focused, representing one important aspect of quality. A minimum manufacturing cost function for the tolerance of a critical dimension, dependent on a number of manufactured components, is determined. This function is used to consider manufacturers' objectives. Based on the customer's total loss function and the minimum manufacturing cost function, the optimal tolerance limits of a critical dimension are determined. These are the tolerances that simultaneously satisfy the customer and the manufacturer as much as possible. The ideas behind the method are described using a roller bearing application as an example.
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Contents Concern for the damaging effect of human activity on the environment prompts efforts to analyze and correct them. The focus of this report is on the role of materials and processes in this, and on data, methods and supporting software to support design to minimize the damage. The approach is a broad one, seeking to develop a resource that, although approximate, has wide applicability. This is only possible if prerequisites of strict procedure and exactitude are relaxed. The method allows greater rigor and precision to be incorporated as the data to enable them becomes available. The report introduces the problem, describes the CES Eco Selector software that incorporates the method, and the Eco MaterialUniverse data that supports it. The data and a version of the software appropriate for teaching are also available as the CES EduPack Eco Design Edition. Here we document sources of the data, the ways in which estimates have been made for the numerous materials for which no eco-data are available, and illustrate the use of the system for selection.
The capability indices Cp, CPU, CPL, k and Cpk are presented and related to process parameters. These indices are shown to form a complementary system of measures of process performance, and can be used with bilateral and unilateral tolerances, with or without target values. A number of Japanese industries currently use the five indices and the U.S. automotive industry has started using these measures in a number of areas. Various applications of the indices are discussed along with statistical sampling considerations.
In multistage manufacturing systems, quality of final products is strongly affected not only by product design characteristics but also by key process design characteristics. However, historically, tolerance research has primarily focused on allocating tolerances based on product design characteristics for each component. Currently, there is no analytical approach for multistage manufacturing processes to optimally allocate tolerances to integrate product and process characteristics at minimum cost. One of the major obstacles is that the relationship between tolerances of process and product characteristics is not well understood and modeled. Under this motivation, this paper aims at presenting a framework addressing the process-oriented (rather than product-oriented) tolerancing technique for multistage manufacturing processes. Based on a developed state space model, tolerances of process design characteristics at each fabrication stage are related to the quality of final product. All key elements in the framework are described and then derived for a multistage assembly process. An industrial case study is used to illustrate the proposed approach.
An optimization model has been built with consideration of required reliability, minimum machining cost and quality loss. The normal and the lognormal distributions of the tolerances that depend on the production types of components are used in the reliability model. Cost-tolerance data obtained from Bjørke is used to calculate the machining cost. The asymmetric quadratic quality loss model is used to calculate the quality loss caused by the deviation and the mean-shift of distributions. The tolerance allocation of a sliding vane compressor is optimized for reliability, cost and quality loss, and the optimum tolerances of components are recommended. The results show that high accuracies of the slot length, the vane thickness and the slot width are required. Hence, their tolerances are smaller than other components. If the correlation coefficient of the bottom cover plate and the top cover plate and the correlation coefficient of the front cover plate and the rear cover plate are equal, the effect of correlation coefficients on the cost is insignificant.
This paper describes a general, rigorous approach for robust optimal design. The method allows a designer to explicitly consider and control, as an integrated part of the optimization process, the effects of variability in design variables and parameters on a design. Variability is defined in terms of tolerances which bracket the variation of fluctuating quantities. A designer can apply tolerances to any model input and can analyze how the tolerances affect the design using either a worst case or statistical analysis. As part of design optimization, the designer can apply the method to find an optimum that will remain feasible when subject to variation, and/or the designer can minimize or constrain the effects of tolerances as one of the objectives or constraints of the design problem.
Reliability-based design optimization (RBDO) has a probabilistic constraint that is used for evaluating the reliability or safety of the system. In modern engineering design, this task is often performed by a computer simulation tool such as finite element method (FEM). This type of computer simulation or computer experiment can be treated a black box, as its analytical function is implicit. This paper presents an efficient sampling strategy on learning the probabilistic constraint function under the design optimization framework. The method is a sequential experimentation around the approximate most probable point (MPP) at each step of optimization process. Our method is compared with the methods of MPP-based sampling, lifted surrogate function, and nonsequential random sampling. We demonstrate it through examples. [DOI: 10.1115/1.4005597]