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Assembly Automation Emerald Article: A review on dimensional tolerance synthesis: paradigm shift from product to process Sandipan Karmakar, Jhareswar Maiti

Article information: To cite this document: Sandipan Karmakar, Jhareswar Maiti, (2012),"A review on dimensional tolerance synthesis: paradigm shift from product to process", Assembly Automation, Vol. 32 Iss: 4 pp. 373 - 388 Permanent link to this document: http://dx.doi.org/10.1108/01445151211262438 Downloaded on: 27-10-2012 References: This document contains references to 79 other documents To copy this document: [email protected]

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Research article

A review on dimensional tolerance synthesis: paradigm shift from product to process Sandipan Karmakar and Jhareswar Maiti Industrial Engineering and Management, Indian Institute of Technology, Kharagpur, India Abstract Purpose – The purpose of this paper is to present a state-of-the-art review of dimensional tolerance synthesis and to demonstrate the evolution of tolerance synthesis from product to process-oriented strategy, as well as to compare the same for single stage and multistage manufacturing systems (MMS). The main focus is in delineating the different approaches, methods and techniques used with critical appraisal of their uses, applicability and limitations, based on which future research directions and a generic methodology are proposed. Design/methodology/approach – Starting with issues in tolerancing research, the review demonstrates the critical aspects of product and process-oriented tolerance synthesis. The aspects considered are: construction of tolerance design functions; construction of optimization functions; and use of optimization methods. In describing the issues of process-oriented tolerance synthesis, a comparative study of single and multistage manufacturing has been provided. Findings – This study critically reviews: the relationship between the tolerance variables and the variations created through manufacturing operations; objective functions for tolerance synthesis; and suitable optimization methods based upon the nature of the tolerance variables and the design functions created. Research limitations/implications – This study is limited to dimensional tolerance synthesis problems and evolution of process-oriented tolerance synthesis to counteract dimensional variation problems in assembly manufacturing. Originality/value – The paper provides a comprehensive and step-by-step approach of review of dimensional tolerance synthesis. Keywords Assembly, Manufacturing systems, Dimensional tolerance, Key product characteristics, Key control characteristics, Single and multistage manufacturing, Rigid body, Compliant body, Variation propagation, Stream of variation Paper type Literature review

Nomenclature KPC KCC WCA RSS Dit, Dot, Dg and Db

¼ ¼ ¼ ¼ ¼

dit, dot and dg

¼

m, mT

¼

Xi Y, f xi:itk ti:itk Tf

¼ ¼ ¼ ¼ ¼

Cp Cpi:Cp of itk

key product characteristics key control characteristics worst case analysis root sum of squares diameter of inner track, outer track, groove and balls of a ball bearing tolerances on the inner track, outer track and groove of a ball bearing radial clearance and its tolerance for a ball bearing tolerance variable design function component dimension component tolerance tolerance on the final KPC

z Rc Rf Ra RT RS RR x
i b, b2 Pr( *) F Li, Ui

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0144-5154.htm

T, T2 CT ( * ) FQ ðT Þ

Assembly Automation 32/4 (2012) 373– 388 q Emerald Group Publishing Limited [ISSN 0144-5154] [DOI 10.1108/01445151211262438]

~ Y 373

¼ process capability index ¼ component based on standard deviation (si) of itk component product process ¼ number of standard deviation ¼ common intersection region ¼ feasible region ¼ acceptable region ¼ tolerance zone ¼ safe region ¼ reliable region ¼ mean of random deviations ¼ actual and threshold reliability of the process ¼ probability function ¼ multivariate probability desnity function ¼ lower and upper limit of ith dimension ¼ feasible and optimum process or KCC tolerance ¼ cost function of attaining feasible tolerances ¼ quality measure of attaining the feasible tolerance T ¼ KPC vector

Dimensional tolerance synthesis: shift from product to process

Assembly Automation

Sandipan Karmakar and Jhareswar Maiti

Volume 32 · Number 4 · 2012 · 373 –388

D U( *) N( *) d a Var( *) Cov( *) s2s SU diag( *) k · kk’ :ktk ~a ~ b; K K P~ b ; P~ a ~ ba C ~sb ; ~sa K

¼ operator depicting deviation from nominal ¼ uniform distribution ¼ normal distribution ¼ actual gap or clearance ¼ arbitrary orientation angle ¼ variance operator ¼ covariance operator ¼ specified variance ¼ covariance matrix of process variation ¼ diagonal matrix ¼ norm ¼ component stiffness matrix before and after welding in compliant assembly ¼ holding force and springback force in compliant assembly ¼ sensitivity matrix ¼ vector of random deviations before and after compliant assembly ¼ stiffness coefficient of a body

manufacturing which makes dimensional tolerancing methods immensely popular and widespreaded applicable. Traditionally, there have been two ways to entrapthe manufacturing uncertainties through tolerance design. One way is to analyze the proper functionality of a product, taking into account the variabilities of individual parts in a bottom up approach, termed as “tolerance analysis” or “variation analysis”. On contrary, allocating tolerances on the component features in a top down approach, maintaining proper functionality of the final product,termed as “tolerance synthesis” (Figure 2). Typically, tolerance synthesis is a “constrained optimization problem” which makes a trade-off between the allocated tolerances and the economy of attaing those tolerances in terms of manufacturing cost and/or quality measures. The sole aim of tolerance synthesis is to design and manufacture “interchangeable” components necessitating judicious allocation of some margins on the key characteristics or KCs (a term coined by Thornton (1999)) of the products as well as the manufacturing processes. KCs defining the product functionality and quality are termed as key product characteristics (KPC) whereas, those characterizing the process features to achievethe KPCs are termed as key control characteristics (KCC) (Shi, 2007). In todays’ mass manufacturing scenario, it is customary for industries to outsource the components instead of manufacturing them internally, requiring benchmarking of suppliers’ manufacturing processes. This has shifted the focus of interchangeability from product to process. Ding et al. (2005) first integrated process related information in tolerance synthesis based on stream of variation (SOV) modeling (Ding et al., 2000) in multistage manufacturing systems (MMS). Moreover, in MMS, realization of product and process interchangeabilities is much more complex, as opposed to single stage system sowing to “cascading effect” of variation propagation. In the last few decades, researches on tolerance synthesis have seen an unprecedented and tremendous progress. So, it is highly needful at this juncture, to provide an exhaustive review on synthesizing tolerances focussingevolution from product to process interchangeabilities. This state-of-the-art reviewwill definitely help both researchers and industry professionals to know the extent of progress of the research in this field. Specifically the scope of this review is limited to: . Dimensional tolerancing. . Evolution of process oriented tolerance synthesis problem with focus on rigid and compliant assemblies.

1. Introduction Tolerances provide a reasonable leeway to counteract the manufacturing uncertainties without downgrading the required performance of products. Tolerance can be defined as the “specified amount a feature is allowed to vary from the nominal or target, which may include location, size, form or orientation of the feature as applicable” (Fischer, 2004). Based on the types of product features, tolerances can be of two types: dimensional and geometric (Figure 1). These two schemes of tolerance systems have some inherent differences as: . features on which limits are assigned to (basic dimensions like length, radius, etc. in dimensional case, but forms, orientations, locations, run-outs and profiles in geometric case); . assumption of tolerance zones (rectangular and cylindrical for dimensional and geometric cases, repectively); and . considerations of datum and locations tolerances with sequence of datum specification (only in geometric case) (Cogorno, 2006). In comparison with the dimensional tolerancing, geometric tolerancing is quite complex and difficult to put through in Figure 1 Dimensional vs geometric tolerance on a circular shape

Figure 2 Difference between tolerance analysis and tolerance synthesis

Tolerance Zone

TOL

ER

AN

CE

Components

D±d

Tolerance Analysis TOLERANCE

Assembly

TOLERANCE

Assembly

Tolerance Synthesis (a) Parametric or Dimensional

(b) Geometric TOL

Source: Adapted from Fischer (2004)

374

ER

AN

CE

Components

.

Dimensional tolerance synthesis: shift from product to process

Assembly Automation

Sandipan Karmakar and Jhareswar Maiti

Volume 32 · Number 4 · 2012 · 373 –388

2. Construction of tolerance synthesis design functions

Extension of process motivated tolerance synthesis from single stage to MMSs.

Essentially tolerance synthesis problem revolves around three issues. These are: 1 Construction of design function(s). 2 Construction of the objective function(s). 3 Selection of suitable optimization methods.

The design functions establish the functional relationships between the individual components’ tolerances and that of the final product. If there are Xi (i ¼ 1,2, . . . n) tolerance related decision variables so that a equation Y ¼ f(X1,X2, . . . , Xn) is to be satisfied to ensure the proper functionality of the final product then, Y is called as “design function”. The design functions used in the earlier research works can be classified in the following categories: . Allocation based on heuristics. . Worst case analysis (WCA). . Root sum of squares (RSS). . Set space based models. . Probabilistic yield models. . Surrogate models (SMs). . Application oriented models.

The framework of this review is shown in Figure 3. An example of a ball bearing A ball bearing has primarily three components: inner race (IR), outer race (OR) and balls (Figure 4). IR and OR are characterized by KPCs, Dit and Dot with tolerances ^dit and ^dot, respectively. The final KPC is “radial clearance” (m) which is a function of Dit, Dot and Db. Tolerance ^ mT on m then, is mT ¼ f(dit, dot, Db). This is the product oriented design function. Now, the KCCs, namely different machining parameters and the mechanical press assembling the IR and OR and balls, affect the designed KPCs, i.e. Dit, Dot as well as m. So, consolidating these, the process oriented design function becomes mT ¼ F{f(dit, dot, Db), KCC}. Again, dit, dot and other KCC tolerances, affect the manufacturing objectives, namely cost of manufacturing as well as the quality measures of products. So, a trade-off is required between tolerances to be assigned to component KPCs (product oriented) or to both KPCs and KCCs (process oriented) and cost of attaining those tolerances and/or the quality loss of not attaining those tolerances.

2.1 Allocation based on heuristics This approach do not utilize the process knowledges, rather it is a trial and error based approach. For example, Peters (1970) assumed the component tolerances are: . directly proportional to individual component dimensions; . directly proportional to the variabilities of individual dimensions; and . inversely proportional to the cost of attainingthe component tolerances.

Figure 3 Framework of review

Constraint Design Functions

• Allocation Based on Heuristics • Worst Case Analysis(WCA) • Root Sum of Squares(RSS) • Set Space Based Method • Probabilistic Yield Models • Process Capability Based Surrogate Models • Application Oriented Models Issues of Tolerance Synthesis

Paradigmatic Evolution in Tolerance Synthesis from Product to Process Interchangeability

Objective Functions

• Cost-tolerance Functional Relationship • Quality Loss Functional relationship

Optimization Methods

• Traditional Optimization Methods • Quality Engineering and Statistical Methods • Stochastic Population Based Search Methods

375

Comparison of Tolerance Synthesis of Single Stage v/s Multistage Manufacturing Systems

Dimensional tolerance synthesis: shift from product to process

Assembly Automation

Sandipan Karmakar and Jhareswar Maiti

Volume 32 · Number 4 · 2012 · 373 –388

Figure 4 An example of ball bearing

OR

OR

OR

Db

Dg ± dg

Track

Balls

IR

IR

IR

IR

IR

IR

OR

OR

OR

(a) Inner race(IR) and outer race (OR) without balls inserted

(b) Inner race(IR) and outer race (OR) with balls inserted

(c) Side view of same ball bearing

(Bjorke, 1989). Later, both linear and non linear RSS approach based design functions are proposed by Choi et al. (2000). A modified version of RSS approach based on Cp of individual components, proposed by Motorola’s Six Sigma Program, is expressed in the equation (3) (Chase and Parkinson, 1991):

On the other hand, Ji et al. (2000) assumed the component tolerances are: . same for each component dimension; . directly proportional to the cubic root of the nominal size; . same influence method (considering the sensitivity of assigning tolerances); and . proportional scaling method.

" Z

Few other attempts have also been made incorporating the sensitivity of tolerances on individual component dimensions into the design functions (Speckhart, 1972; Caro et al., 2005; Prabhaharan et al., 2005).

 n  X ›f 2 i¼1

›xi

ti C pi ð1 2 mi Þ

#1=2 # Tf

ð3Þ

2.4 Set space based models Bandler (1974) introduced a set space based design function (Rc), where Rc is intersection “acceptable space” (Ra) and “feasible space” (Rf), i.e. Rc ¼ Rf > Ra. Later, this concept is extended using WCA on feasible spaces (Turner, 1993). Some researchers considered “reliability index based method” (Lee and Woo, 1989, 1990) based on Bandler (1974). The tolerance zone (RT) is partitioned into safe region (RS) and reliable region (RR) and the accumulated tolerance acts as the divider between RS and RR. Mathematically:

2.2 Worst case analysis WCA assumes all the dimensions at their worst possible values, i.e. with absolute maximum possible variation. This approach can be mathematically described as in equation (1):
n

X

›f
t i # T f ð1Þ
›x
i i¼1 This method was first proposed by Speckhart (1972). Few variants of WCA are also proposed like Lagranges’ multiplier based WCA (Spotts, 1972) and linear or non-linear decomposition based WCA.

{XjðRT > RS Þ n
o
RR ¼ X
>ni¼1 ð
xi 2 gsi # xi # x
i þ gsi Þ>m ðF ðXÞÞ j j¼1 ð4Þ

2.3 Root sum of squares Due to the inherent drawbacks of WCA, researchers attempted RSS approach. The primary assumption of this approach is that the component dimensions follow normal distribution and their resultant assembly tolerance can be obtained by using equation (2):
2 n

X

›f
t 2 # T 2 ð2Þ f
›x
i i i¼1

Consequently, RR is converted into the reliability index of the process as, b ø PrðRR Þ. The design function is then reduced to b $ b2 and 1 2 d ¼ F(b2 ). 2.5 Probabilistic yield models Here, the probability density of manufacturing yield is considered as the design function expressed in equation (5) (Lee and Johnson, 1993):

In tolerance synthesis, RSS was first adopted by Speckhart (1972). Variants of RSS based design functions are also proposed that consider the component tolerances as beta distributed and final product tolerances as normal distributed random variables

Y ¼ R x1u x1i

376

···

R xnu xni

Pr>m i¼1 ðLi # xi # U i Þ qðx1 ; . . . ; xn ÞFðx1 ; . . . ; xn Þdx1 · · ·dxn

ð5Þ

Dimensional tolerance synthesis: shift from product to process

Assembly Automation

Sandipan Karmakar and Jhareswar Maiti

Volume 32 · Number 4 · 2012 · 373 –388

The test function q can be formulated as: ( ) 1 if F i ðx1 ; x2 ; . . . ; xn Þ $ 0 qðx1 ; x2 ; . . . ; xn Þ ¼ 0 if F i ðx1 ; x2 ; . . . ; xn Þ , 0

3.2 Quality loss functional relationship Taguchi (1986) introduced the philosophy of quality loss function, later utilized by many researchers to construct the objective function for tolerance synthesis problem. An example of loss function curve has been shown in Figure 5, describingthe relationship between KPC tolerances and corresponding losses incurred when attained KPC falls away from the target at y0. D’Errico and Zaino (1988) is most probably the first who used quality loss function as objective function for tolerance synthesis. Other research studies in the same line are Kapur (1993), Choi et al. (2000) and Jeang (2001). In some cases, an asymmetric loss function is assumed to find out the optimal process center where the quality loss function is minimum (Li, 2000; Maghsoodloo, 2000). Few studies have considered a bi-objective function of manufacturing cost and quality loss for tolerance synthesis (Haq et al., 2005).

ð6Þ

where, Fi(x1,x2, . . . , xn) is called the ith design function. Following this, later Zhang et al. (1999) developed statistical distribution function zone based design function approach. 2.6 Surrogate models In complex multivariate tolerance synthesis problems, construction of the design function becomes difficult because, often the design space becomes higher dimensional space with multiple local optima, non-convex and disconnected as well (Huang et al., 2010). So, the complex nature of the design spaces often renders the conventional design principles ineffective or even invalid. Recent advancements in surrogate modeling techniques through computer experiments, can take up the challenges in formulating the design functions using “space-filling” methods (Fang and Li, 2006). One such attempt has been made to construct the design functions based on process capability (Huang et al., 2009).

4. Tolerance synthesis optimization methods To solve the tolerance synthesis problems, the utilized optimization methods can be classified into following three categories: 1 Traditional optimization methods. 2 Quality engineering and statistical methods. 3 Stochastic population based search methods.

2.7 Application oriented models These models typically consider application issues of different manufacturing scenarios. For example, Chase and Greenwood (1988) proposed an approach based on estimated mean shift (a shift factor, f i’ [ ½0:0; 1:0
). An approach based on machining datum selection in a machining process for construction of design function was studied by Bai et al. (2000). Again, Chase et al. (1990) proposed design function based on alternative manufacturing process selection. It is evident that, unlike the previous ones, these design functions are too specific to the domain of application.

A brief summary of these methods is given in Table II. 4.1 Traditional optimization methods The traditional optimization methods are defined in the sense that they consider the gradient based directional search methods. They have been in widespread use owing easy implementation and comprehensibility. But, as the product design is becoming more and more complex, researchers face difficulties inperforming computations to reach the optimality because, the search space can become discrete, non-convex, disjointed and multimodal. On the other hand, it is required to employ the probabilistic random behavior of tolerance, whereas these optimization principles work for deterministic decision variables.These issues pose restrictions on traditional optimization algorithms to be implemented.

3. Construction of objective functions Objective functions depict the manufacturing objectives to be accomplished by setting the tolerances. Based on the requirements, the tolerance synthesis problem can be a single or multi-objective optimization problem. The types of objective functions prevail in literature are: 1 cost of manufacturing; 2 quality loss; 3 manufacturing yield; 4 process capability; or 5 any combination of the above.

4.2 Quality engineering and statistical methods In mid-1990s, there has been a new trend in solving tolerance synthesis optimization problem by quality engineering and statistical methods. Quality engineering aims at integrating the KPCs and KCCs in such a way that every product coming out of the system has minimum KPC deviation. Design of experiments (DOE) coupled with analysis of variance (ANOVA) are especially used for this purpose. Quality engineering is also assumed synonymous to “robust engineering” where the main aim is to design the system less or insensitive to the environmental factors that affects their performance. The “design of orthogonal array lies” at the heart of Taguchi’s quality engineering principles. Though designing this array is not very difficult, but working with it, for complex product structure needs highly efficient computing facilities. On the other hand, the traditional statistical methods in solving tolerance synthesis problems suffer from distributional assumptions of tolerance variables. In almost every cases, the assumed distributions are either normal or beta distribution. A small violation in the distributional assumption leads to inability in correctly capturing the true random behavior

These are described in the following sections. 3.1 Cost/tolerance functional relationship Most of the literatures have utilized cost-tolerance relationship as the objective function (Hong and Chang, 2002). These relationships can be exponential, reciprocal, or a combination of them. A summary of the cost/tolerance functional relationships is given in Table I. They are generally estimated from historical data. Strictly speaking, in any manufacturing process, very little information is available to formulate the cost/ tolerance relationship. Addressing this issue, a recent study reports a cost modeling based on the variability of equipment utilization (Sanz-Lobera et al., 2010). To encounter these problems, a different approach based on Taguchi’s quality loss philosophy has been considered as objective function. 377

Dimensional tolerance synthesis: shift from product to process

Assembly Automation

Sandipan Karmakar and Jhareswar Maiti

Volume 32 · Number 4 · 2012 · 373 –388

Table I Different types of proposed cost/tolerance functions Cost model type

Cost model

Reference

Linear Reciprocal Reciprocal squared Reciprocal powered

CðTÞ ¼ a 2 b T CðTÞ ¼ a þ ðb=TÞ CðTÞ ¼ a þ ðb þ T 2 Þ CðTÞ ¼ a þ ðb=T k Þ CðTÞ ¼ b=T k CðTÞ ¼ b expð2TÞ CðTÞ ¼ b ðexpð2mTÞ=T k Þ CðTÞ ¼ ai 2 bi Ti Discrete points CðTÞ ¼ a0 exp{ 2 ai ðT 2 a2 Þ} þ a3 CðTÞ ¼ a0 b1 4 T þ a1 b2 4 T þ a2 b3 4 T þ a3 b 44 T þ a 4 b5 4 T CðTÞ ¼ a0 þ a1 T þ a2 T 2 þ . . . þ an T n P C ¼ ni¼1 ðcio =ai Þ

Edel and Auer (1964) Chase and Greenwood (1988) Spotts (1972) Wu et al. (1988) Wu et al. (1988) Wilde and Prentice (1975) Wu et al. (1988) Bjorke (1989) Lee and Woo (1989) Dong et al. (1994) Dong et al. (1994) Singh et al. (2004) Sanz-Lobera et al. (2010)

Exponential Exponential/reciprocal power Piecewise linear Empirical Modified exponential B-spline curve Polynomial Special function considering ctt 5 production rate within designated tolerance interval

.

Figure 5 Loss function curve in conjunction with normal curve

QualityLoss/Normal Probability Value

0.7

In spite of the advantages of these methods, it is quite challenging for tolerance synthesis problem. First, the computational complexity of these methods is really a major concern. Second, as these methods start working with a randomly generated population base, improper representation of the candidate solutions may lead to completely erroneous results. Finally, these methods do not always guarantee the global optima as these methods have a tendency to get stuck up at the local optima and many-a-times requires a very efficient algorithms to get free from the local optima. Keeping in view the optimization methods adopted in solving tolerance synthesis problems, Table III shows a comparative study of applicability of different optimization methods to different types of tolerance design spaces.

0.6 Quality Loss Curve Normal Curve

0.5

0.4

0.3

0.2

0.1

0 –3

the direction of optimization is selected randomly unlike traditional optimization methods.

5. Evolution of tolerance synthesis –2 yUSL

–1 yL

0

1 yU

2

yLSL

3

Traditionally, in tolerance synthesis, the process information are implicitly defined in terms of process capability and availability of manufacturing facility. For this reason, the traditional tolerance synthesis is termed as “product oriented”. But to focus on the process interchangeability, it is of utmost importance to explicitly include the process information. The revolutionary study of Ceglarek et al. (1994) states that, a significant proportion of dimensional variation is contributed by the inadequate tooling maintenance issues (37 percent) and installation problems (13 percent) (Figures 6 and 7). So, it is highly recommendedto incorporate the tolerances of KCCs in design functions. To incorporate process information in tolerance allocation, “tolerance charting” was developed in the early 1950s (Lehtihet et al., 2000). But, tolerance charting is more focussed on process planning and alternate manufacturing process selection without explicte inclusion of KCC information. The theory behind the effects of process variables on product quality attributes was propounded by Taguchi (1986) through DOE and robust parameter design. Being experimental in nature,this needs to examine the process variables at each operation which is a challenging task to map the KCCs on KPCs. In process oriented tolerance synthesis, the process information needs to be explicitly included in creation of tolerance design function and objective functions.

4

Quality Attribute/KPC (y)

of tolerance variables and that renders erroneous solutions to the problem considered (Kotz and Johnson, 1993; Sommerville and Montgomery, 1996). 4.3 Stochastic population based search methods Many a times, even for a small product, the design space becomes irregular and computationally intractable. Such irregular design spaces include multi-modal, discrete, disjoint and heavy or flat tailed non normally distributed spaces. Conventional optimization as well as statistical methods are found insufficient to tackle these issues. This type of problems can be solved using stochastic population based search methods such as simulated annealing, genetic algorithms, ant colony optimization algorithms and particle swarm optimization. Advantages using these optimization methods are as follows: . optimization starts with an initial population of candidate solutions and always maintain the size of the population; and 378

Dimensional tolerance synthesis: shift from product to process

Assembly Automation

Sandipan Karmakar and Jhareswar Maiti

Volume 32 · Number 4 · 2012 · 373 –388

Table II Different types of optimization methods used in tolerance synthesis Method category

Method sub category

Reference

Traditional optimization/methods

Exhaustive search Univariate search Linear programming Non linear programming Lagrange’s multiplier

Chase et al. (1990) Chase et al. (1990) Bjorke (1989) Bandler (1974) and Lee and Woo (1990) Speckhart (1972); Spotts (1972) and Chase and Greenwood (1988) Wilde and Prentice (1975) Ostwald and Huang (1977) Loof et al. (2007) Peters (1970) Chase et al. (1990); Li et al. (2008a, b) and Huang et al. (2009) Singhal and Pinel (1981) Chase and Greenwood (1988)

Geometric programming Binary integer programming Branch and bound Graphical method Sequential quadratic programming Quality engineering and statistical methods Parametric sampling based yield maximization Estimated mean Shift for other than 6a tolerance distribution Taguchi method ANOVA, design of experiments and FFD

Evolutionary optimization methods

D’Errico and Zaino (1988) Bisgaard et al. (2000); Feng and Kusiak (2000) and Gupta and Feng (2000) Mixture Experiments Shehirlioglu and Ozlar (2008) Quality loss function Kapur (1993) Monte Carlo simulation Bowman (2009) Robust design Caro et al. (2005) Process capability for truncated normal distribution Sweet and Tu (2006) Genetic algorithms Lee and Johnson (1993); Chen and Fischer (2000) and Singh et al. (2004) Neural networks Kopardekar and Anand (1995) Simulated annealing Zhang and Wang (1993) Fuzzy logic based simulated annealing Dupinet et al. (1996) Ant colony optimization algorithm Prabhaharan et al. (2005) Fuzzy logic based quality loss Cao et al. (2006) Fuzzy set weight center Wang et al. (2007) Scatter search Prabhaharan et al. (2007) SQP and neighborhood cultivation GA Li et al. (2008a, b) Pareto based multi-objective Particle swarm optimization Forouraghi (2009) Interval constraint propagation Wilhelm and Lu (1992) and Yang et al. (2000)

Table III Applicability of optimization methods to different types of tolerance design spaces Tolerance design space type Convex and unimodal Convex and multimodal Non-convex and unimodal Non-convex and multimodal Discrete

Traditional optimization methods

Suggested optimization methods Quality engineering based methods

X

X

X

X X X X X

objective function and FQ(T) is some quality measure of the process which must be lying below (lower the better) or above (higher the better) a threshold, C or in between (nominal the best) C1 and C2.

Mathematically, a process oriented tolerance synthesis problem can be expressed as: T * ¼ minCT ðT Þ T

Stochastic population based search methods

ð7Þ 5.1 Process oriented tolerance design functions Ding et al. (2005) first described the problem of allocating optimal tolerances on KCCs in order to achieve the manufacturing objectives. Before going to describe the

subjected to; FQ ðT Þ # or $ C or C 1 # FQ ðT Þ # C 2 Here T and T2 represent the feasible and optimum tolerance allocated on KCCs, CT(T) represents the manufacturing 379

Dimensional tolerance synthesis: shift from product to process

Assembly Automation

Sandipan Karmakar and Jhareswar Maiti

Volume 32 · Number 4 · 2012 · 373 –388

Figure 6 Root cause classification for dimensional variation

Figure 8 A typical 3-2-1 fixture layout NC3

Full Volume production phase

LP2way NC2

LP4way NC1

One Shift production phase

workpiece y x

Launch phase z

Source: Adapted from Shi (2007)

production phase

Figure 9 Design of two-sheets assembly

0% 20% 40% 60% 80% 100% 120%

y1

Figure 7 Attribution of fixture faults in full volume production phase

y2

A

B

Z P4,A

P2,A

y3

P4,B

P2,B

X

In a hypothetical manufacturing scenario, if for example the locator pin P2B becomes faulty due to wear and tear or misaligned, it fails to serve its intended purpose. As a result KPCs will deviate from the nominal, as shown in Figure 10 where the sheet B rotates by an angle of a due to faulty P2B locator.

process oriented tolerance design function an exemplary case study of sheet metal assembly is explained here. 5.1.1 An example of sheet metal assembly A sheet metal assembly considering rigid as well as compliant parts are described here. The difference between the rigid and the compliant parts is that the body deformation is not accountedfor rigid parts, whereasit is considered for compliant parts. Rigid body assembly. Sheet metal assembly is especially prevailed in automotive body-in-white (BIW) assembly, aircraft fuselage assembly, etc. The sheets are placedon locating elements to constrain the possible degrees of freedom (DOFs) of the sheets. In BIW assembly, typical 3-2-1 locating system is used to constrain the six DOFs of each sheet (Figure 8). In 3-2-1 locating principle, two locating pins, LP4way and LP2way constrain the motion in X-Z and Zdirections, respectively, whereas the NC-blocks (NCi, i ¼ 1,2,3) constrain the motion in Y-axis only. A 2-D sheet metal assembly design is shown in Figure 9, where sheet A is to be joined with sheet B by spot welding operation at points y1, y2, y3 which constitute the KPCs. P4A and P2A are the four-way and two-way locator pins on part A and P4A and P2A are that on part B, respectively, which constitute the KCCs. In order to produce this assembly as per design, tolerances on ~ ¼ ½y1 y2 y3
T . the quality attribute vector form the KPC ¼ Y To synthesize the process tolerances, KPC tolerances are translatedto the KCC tolerances.

Compliant body assembly. For compliant bodies, more than three NC-blocks are required and the fixture layout is defined as n-2-1, (LP4way, LP2way, NCi, i ¼ 1,2, . . . , n). During the assemblies of components in a single station, the components undergo four sub-operations which are described below: 1 Part loading and locating operation in a station (Figure 11(a)). 2 Part holding operation (Figure 11(b)). 3 Part joining operation by spot welding, rivetting, etc. (Figure 11(c)). 4 Part unclamping operation causingspringback in the subassembly (Figure 11(d)). 5.1.2 Formulation of design function (rigid body case) In rigid body case, the scenario is widely modeled as a 2-D assembly system excluding the variations occurred due to the NC-blocks. The KCCs for this assembly are the information related to the pin-hole locator pairs (Figure 12). The tolerance design function in this case depicts the relationship between the KPC and KCC tolerances using the clearance tolerances of locators in corresponding holes (Figure 13). The gap or actual clearance d [ ½0; T i =2
, is the distance between the center of the locator pin and the center of the hole. This distance (d) is assumed as a normal random variable, i.e. d~NðT i =2; T 2i =36Þ. As d for a four-way locator 380

Dimensional tolerance synthesis: shift from product to process

Assembly Automation

Sandipan Karmakar and Jhareswar Maiti

Volume 32 · Number 4 · 2012 · 373 –388

Figure 10 Sheet assembly in single stage manufacturing

y1

A

y1 A

B

y2

B y2

Z

Z P4,A

P2,A

P4,A

y3

P4,B

P2,A

y3

P2,B

P'4,B a

P'2,B

X

X

Actual Assembly due to Process Error at Fixture locators on Part B

Assembly as per Design

Figure 11 Compliant sheet assembly Clamping force Pb Part 2

Part 2

sb

Part 1

Part 1

(b) Part 2 clamped with clamping force Pb at design nominal

(a) Part 2 deviated from design nominal with distance sb from nominal Pb

Pa Part 2

Part 2

Part 1

Part 1

(c) Part 2 being welded with Part 1 with the clamping force Pb

sa

(d) Subassembly springbacks a distance sa from nominal with force Pa when clamp is released

Figure 12 Pin-hole locating pair Z Ti

dpin

dhole

Z

X

ð8Þ

DZ ¼ d sin u

ð9Þ

Given that d and u are independent to each other, the mean and variances of the deviations can be computed as:

X Ti

dpin

Ti

dhole Ti (a) 4-way locator

DX ¼ d cos u

E½DX
¼ E½d cos u


ð10Þ

E½DZ
¼ E½d sin u


ð11Þ

(b) 2-way locator

Source: Adapted from Shi (2007) pin-hole pair can be assumed symmetrical in all directions, the orientation angle u can be uniform random variable, u , U(0,2p) (Figure 13(a)). The deviations in X- and Z-directions are expressed as:

2 sx;4way ¼ Var½DX
¼ E½DX 2
¼ E½d2 cos2 u
¼

5T 2i 36

ð12Þ

2 sZ;4way ¼ Var½DZ
¼ E½DZ 2
¼ E½d2 sin2 u
¼

5T 2i 36

ð13Þ

Cov½DX; DZ
¼ E½DX; DZ
¼ E½d2
E½sin u cos u
¼ 0 381

ð14Þ

Dimensional tolerance synthesis: shift from product to process

Assembly Automation

Sandipan Karmakar and Jhareswar Maiti

Volume 32 · Number 4 · 2012 · 373 –388

Figure 13 Deviations created from pin-hole clearance Z

P'2

Z

P'2

a

q X

Z d

dz

d

a

X

P'1 P1

dx

X

P2

P2

d

(a) -way locator

(b) -way locator 4

(c) eviations in 2-way locator

Source: Adapted from Ding et al. (2005) So, it is evident that the deviations in both X- and Z-directions in case of a four-way locator are having zero mean and same variances. More importantly, they are uncorrelated in nature. But in case of a two-way locator pin, the clearance is not symmetrical in the hole as the hole is not circular (Figure 13(b) and (c)). Due to this reason the deviations in X- and Z-directions are expressed as: DX ¼ d sin a:u

ð15Þ

DZ ¼ 2d cos a:u

ð16Þ

.

.

~b} ~ b
{S {P~ b } ¼ ½K

ð23Þ

E½DX
¼ E½d sin a:u
¼ 0

ð17Þ

The parts are joined together while the force P~ b or {P~ b } are still being applied to hold the parts in nominal position. When the joining and clamping tools are removed the the assembly will spring back. the spring back is determined assuming a force of magnitude P~ a or {P~ a } same as that of P~ b or {P~ b } but acting in opposite directions. If assembly ~ a then springback variation can be stiffness matrix is K determined as:

E½DZ
¼ E½d cos a:u
¼ 0

ð18Þ

~a} ~ a
{S {P~ a } ¼ ½K

ð24Þ

{P~ b } ¼ {P~ a }

ð25Þ

.

Here a is the arbitrary orientation angle and u is a binary random variable having value 1 if the pin touches top edge of pin hole and 2 1 if the pin touches bottom edge of pin hole. The variables d and u are independent to each other. The mean, variances and the covariance of the deviations in X- and Z-directions can be computed as:

s2X;2way ¼ Var½DX
¼ E½d2 sin2 au2
¼ s2Z;2way

The components located using n-2-1 fixturing principle ~ b from the nominal will have their upstream deviation S part position. ~ b or {S ~ b } a force P~ b or {P~ b } is To nullify the deviation S applied by the clamping tools or/and welding or riveting ~ b represents the component stiffness matrix then tool. If K ~ u is expressed as: F

5T 2i sin a 18

5T 2i ¼ Var½DZ
¼ E½d2 cos2 au2
¼ cos a 18

Cov½DX; DX
¼ E½d2 sin a cos au2
¼

5T 2i sin a cos a 18

.

ð19Þ

~ a } ¼ ½K ~ a
{S

21

~b} ~ b
{S ½K

~b} ~ a } ¼ ½½C

{S {S ba

ð20Þ

ð26Þ ð27Þ

[Cba] can be determined by the method of influence coefficients (Liu and Hu, 1997) using the finite element methods (FEM) based simulations (Li et al., 2010; YanFeng, 2009).

ð21Þ

So, it implies that the deviations in X- and Z-directions in case of two-way locator pin are correlated and in order to eliminate this correlation, the orientation angle a is assumed 08 (horizontal) or 908 (vertical). The elements of equations (12)-(14) and (19)-(21) constitute the process variation matrix (SU). Naturally for the four-way locators and assuming uncorrelated deviations in both X- and Z-directions for the two-way locators, SU becomes a diagonal matrix. Accordingly, the process oriented design function can be described as:  X    ð22Þ FQ ðT Þ ¼ s2s 2 diagð U Þ $ 0; ;T i . 0

Based on [Cba] formulation, the process variation is modeled next. In compliant body assemblies two types of process variation sources affect the quality of the assemblies. These are welding gun and fixtures errors. Their effects are modeled in the following ways. Welding gun variation. Welding gun variation has a deep impact on the final assembly qualitydepending on the types of welding guns like, position controlled, equalized and force controlled welding guns. In this work, only effect of position controlled welding guns is described. A position controlled welding gun is used to weld two parts assuming that the gun can apply a sufficient force over the two parts to close the gap between the parts and the welding electrodes. Figure 14 shows an exemplary situation where a position controlled welding gun is used to welding parts 1 and 2. The force required to close the gap between the parts will be:

1

5.1.3 Formulation of design function (compliant body case) As discussed in Section 5.1.1 and shown in Figure 11 the components undergo four sub-operations in a single station. The mathematical formulation of each of the sub-operation are described below: 382

Dimensional tolerance synthesis: shift from product to process

Assembly Automation

Sandipan Karmakar and Jhareswar Maiti

Volume 32 · Number 4 · 2012 · 373 –388

Figure 14 Position controlled welding gun

SU ¼ S1 þ S2 þ S3 ~ 2 Q} ~ 2 Q} ~ T S ~ g {C ~ þM ~ T S ~3 M ~ ¼ {C

Welding Gun Part 1

s2 s1

U T

The elements of SU represent the deviations of the locator elements and joining tools. Going the same way, as in equation (22) establishes the design function for process oriented tolerance synthesis in case of compliant assemblies. Shiu et al. (2003) studied the case of compliant body assembly for tolerance synthesis problem.

Source: Adapted from Shi (2007)

P 1 ¼ K 1 ðs1 2 sg Þ

ð28Þ

P 2 ¼ K 2 ðs2 2 sg Þ

ð29Þ

5.2 Construction of objective functions and optimization methods To construct the objective functions in most of the cases, the negative exponential or reciprocal or combinations of both models have been employed. These kind of objective functions have been described in depth in Section 3.1. After the objective function and the constraint design functions are set then a suitable optimization routine is deployed to allocate the best possible tolerances on the process elements. The optimization methods have been described in greater details in Section 4.

K1 and K2 are the stiffness coefficients of parts 1 and 2. The resulting force Ps which will be the result of springback with stiffness coefficient Ks will be, the sum of P1 and P2. The springback can be computed as: Ps K1 K1 K1 þ K2 ¼ S1 þ s2 2 sg Ks Ks Ks Ks

ð30Þ

6. Process oriented tolerance synthesis in MMS Process oriented tolerance synthesis in MMSs is a challenging task as it requires the establishment of relationship between the tolerances of process elements across multiple stages and the variation of final product. An MMS is highly complex in terms of three types of the interactions between the quality attributes (Figure 15). The X-axis represents autocorrelation between the same attribute (M2) between different stages of manufacturing, Y-axis represents the temporal autocorrelation owing to the variation in system like wear and tear of production machines, tool degradation and Z-axis represents cross correlation between different attributes (M1,M2, . . . , Mm) in all the stages. Here it is to be mentioned that MMSs can be regarded as an extension of the single stage manufacturing systems because in later the X-axis representing the stages of manufacturing is absent. The variation flows in MMSs, in a cascading manner, incorporation of variation propagation thus poses a serious challenge to the designers. Extending the case described in Figures 9 and 10, if one more sheet is to be joined with the subassembly, made up of sheet A and B, then the case can be described as in Figure 16. Here in stage 1, sheet A and B are joined keeping their motion constrained on fixture locators P4A,P2A for sheet A and P4B, P2B for sheet B. After the welding operation is done in presence of the faulty locator P2B, the

Using the sensitivity matrix definition vs can be computed as: " # " # " # v1 2 vg vg v1 ~ ¼c 2c ss ¼ C v2 2 vg vg v2

ð31Þ

~ g the variance Finally writing the welding gun deviation as U k due to this can be rewritten as: ~ 2 Q} ~ 2 Q} ~ T S ~ {C ~ S1 ¼ {C Ug

ð32Þ

~ is the part deformation matrix defined as the Here Q deformation in the part carried from the upstreamoperations. Fixture variation. The variation model for compliant assemblies requires the fixturing variation to be decomposed ~ 3 for 3-2-1 locating principle and into two components, U n23 ~ U for n 2 3 additional holding fixtures. So, considering ~ 3 can be the locating fixture variation the variance due to U written as: ~ T S ~3 M ~ S2 ¼ M U

ð33Þ

~ is the reorientation matrix which is defined as effect of Here M component re-orientation occurring when the variation due to earlier operations in current operation is reset to zero.Additional to this, the extra n 2 3 holding clamps can be analyzed as extra position controlled welding guns and assuming them as additional sources of variation the method of influence coefficients can be applied to compute the variation. So, considering the holding fixture variations, state space model can be rewritten as: ~ 2 Q} ~ 2 Q} ~ T S ~ n23 {C ~ S 3 ¼ {C U

ð35Þ

~ 2 Q} ~ 2 Q} ~ S ~ n23 {C ~ þ {C U

Part 2

sg

ss ¼

U

Figure 15 Complex data relationship in MMS

ð34Þ

Combining the above results, the total variance created in two sheets assembly operation is given as:

Source: Adapted from Shi (2007) 383

Dimensional tolerance synthesis: shift from product to process

Assembly Automation

Sandipan Karmakar and Jhareswar Maiti

Volume 32 · Number 4 · 2012 · 373 –388

Figure 16 Sheet assemply in multiple stages y1

y1

A

Z P4,A

P2,A

y2

C y2

A B

y3 P'4,B

Z

P4,C

P2,C

B

P4,A y3 P'4,B

P'2,B

P'2,B

X

X (a) Actual Assembly after operation at Stage 1

(b) Actual Assembly after reorientation and operation at Stage 2

(2004, 2008a) the objective function consists of a norm-based measure of the difference of tolerances (product and processes) within the levels (considering the hierarchical structure of the assembly) along with the linear cost function of attaining the tolerances. In Li et al. (2008b), a bi-objective function is used to minimize the cost of attaining product and process tolerances and the corresponding process sensitivity index.

subassembly (A-B) is welded at wrong place which introduces part variation. When this A-B subassembly is shifted to stage 2 for another sheet C to be joined, the A-B subassembly needs to be located at P4B, P2B and sheet C to be located on P4C, P2C. This new positioning of subassembly A-B will introduce a “reorientation error” and if even anyone of these four locators has any fault in them will again introduce a part variation in stage 2. For carrying out process oriented tolerance synthesis the modeling of variation propagation through the stages of manufacturing is also to be addressed. There are several types of models on variation propagation available in literature, for example, auto-regressive model of first order (Lawless et al., 1999), integrated system model (Suri and Otto, 1999, 2001), state transition model (Mantripragada and Whitney, 1999; Whitney, 2004), and SOV model (Ding et al., 2000; Huang et al., 2007). Among these, SOV modeling are explored in much greater depth due to its certain implicit advantages: . the complicated stage-wise interaction is handled automatically in this model through the state transition; and . its linear structure.

7. Future scope of research This article establishes some promising research gaps found from the review. A structured procedure for conducting tolerance synthesis is provided as a framework in Figure 17. The critical issues that need to be addressed in future are listed below: . The implementation of tolerance synthesis in industry gives enormous advantages in attaining manufacturing objectives. But its use is fairly limited, because of the use of the superficial process knowledge mostly from the machinist handbook. The process knowledge of this level of abstraction does not usually provide the designer with meaningful tolerance values unless the analysis is carried out on how the parts are actually manufactured, such as set-ups, fixturing and control of the tools. So, there is an urgent need of in-depth study regarding integration of the process related information in designing tolerances. . The WCA approach assumes that both the KPCs and KCCs occur simultaneously at their worst limits. Correspondingly, as the number of components in the assembly increases, the KPC as well KCC tolerances must be greatly reduced in order to meet the assembly limits, requiring higher production costs. So, the WCA approach proves to be inefficient in synthesizing economic tolerances in real manufacturing scenario for complex assemblies with large number of components. Available research works does not explore this open problem. . In the RSS approach, the low probability of occurrence of the worst case combination of KPC and KCC tolerances is taken into account statistically, assuming a normal or gaussian distribution for component variations. For example, the approach proposed by (Bjorke, 1989), fails when the numbers of components are less. RSS approach generally predicts too few rejects when compared to real assembly processes. Further research can be done to overcome this problem. . The target or nominal values of the KPCs and KCCs may also shift from the midpoint of the tolerance range. Bjorke (1989) developed a method based on beta and normal

It is worthwhile to mention that the creation of design function in case of MMSs is an extension of that in single stage manufacturing systems discussed in Section 5.1. In case of rigid assemblies, Huang et al. (2009) constructed the tolerance design function as a process capability based SM for an MMS. The design space is compressed using a partitioning strategy along the diagonal of the rectangular design space, to identify the candidate design space for tolerance selection. Then a three step methodology has been adopted for finding an approximate yield model or SM. These are: . sampling the compressed design space by space filling methods; . transforming the generated sample points into normal random variables by a multivariate distribution transformations (MDT); and . fitting a regression model to the generated candidate design points and their corresponding yields. Finally, this yield model is used as the design function for the process oriented tolerance synthesis problem. In case of compliant assembly tolerance allocation problems, there are few works such as Li et al. (2004, 2008a, b) which have considered different objective functions rather than the conventional cost/tolerance objective functions. In Li et al. 384

Dimensional tolerance synthesis: shift from product to process

Assembly Automation

Sandipan Karmakar and Jhareswar Maiti

Volume 32 · Number 4 · 2012 · 373 –388

Figure 17 Proposed generic methodology to carry out process oriented tolerance synthesis START

Identify KPC and KCC variables

Identify objective function in terms of either manufacturing cost or manufacturing yield or quality loss function based on the designed KPCs

Modeling of variation of KPCs interms of variation in KCCs

Optimize the objective function subjected to design functions

Identify Final KPC Tolerance Design Space in terms of variation STOP

Convert KPC Tolerance Design Space into KCC Tolerance Design Space using variation model

Use the functional relationship between KPC and KCC tolerance space as design function

.

.

.

data driven SMs can be an attractive alternative to the models derived from first principles (Fang and Li, 2006; Huang and Kong, 2010). Again in these models, there is no explicit mention of mixed type data handling and modeling stochasticity, which needs to be explored in greater depth.

distributed random variables for components and resultant dimensions, repectively. This assumption can cause inaccuracies if the number of components in the assembly is small. Process oriented tolerance synthesis for MMS with target-shift can go to a very high complexity and erroneous results. So, there must be some in depth research in this area. Generally the complex functional requirements of a product establishes complex tolerance design space. For example, the gap or flushness control in auto body manufacturing consists of having control over a irregular tolerance region. This type of irregular region consists of several dimensional variables following wide range of mixture of probability distributions. This gives another research direction called “function oriented process capability and tolerance design” (Huang et al., 2009) which is also least explored in current research pool. Although cases with symmetric tolerances dominate in research studies, how to tackle the issue of assymetric tolerance synthesis in reality is still unclear. In case of assymetric tolerances, if the tolerance window and distribution of tolerances is not known a-priori to the designer, then generating random samples through computer simulations is still an area to be explored (Pahwa and Huang, 2009). There is a challenge of computation complexity in terms of searching optimal design parameters in multi dimensions, non convex, and discontinuous tolerance design space. This makes many available algorithms ineffective or even invalid. In order to model such complex tolerance design spaces,

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Assembly Automation

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Dimensional tolerance synthesis: shift from product to process

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Dimensional tolerance synthesis: shift from product to process

Assembly Automation

Sandipan Karmakar and Jhareswar Maiti

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Zhang, C., Luo, J. and Wang, B. (1999), “Statistical tolerance synthesis using distribution function zones”, International Journal of Production Research, Vol. 37 No. 17, pp. 3995-4006.

Further reading Santner, T. and Williams, B.J. (2003), The Design and Analysis of Computer Experiments, Springer, New York, NY.

Corresponding author Jhareswar Maiti can be contacted at: jhareswar.maiti@ gmail.com

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