Development of the parametric tolerance modeling and optimization ...

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with multiple quality characteristics. The second area considers
more complex manufacturing situations involving a sequential
production system. Using a multi-stage production system, Bowling
et al. (2004) proposed a method for determining the optimum
process mean for each stage by integrating a Markovian model to
represent the production system, while Kang et al. (2004) applied
robust economic optimization concepts to optimize the process
mean and the robustness measure for a chemical process within
a multiobjective optimization framework. Jeong and Kim (2009)
also proposed a multi-response optimization method by using an
interactive desirability function incorporating decision maker’s
preference information.
2.2. Tolerance optimization
Tolerance, the second important focus in this research area, involves
the problem of determining optimal specification limits
from the viewpoint of cost reduction and functional performance.
For example, early work by Speckhart (1972), Spotts (1973), Chase
et al. (1990) and Kim and Cho (2000b) considered the reduction of
manufacturing cost in a tolerance allocation problem whereas
Fathi (1990), Phillips and Cho (1998) and Kim and Cho (2000a)
studied the issue of tolerance design from the viewpoint of functional
performance, expressing it in monetary terms using the
Taguchi quality loss concept. In an integrated study considering
the effect of both cost reduction and functional performance together,
Tang (1988) developed an economic model for selecting
the most profitable tolerance in situations where inspection cost
is a linear function. Tang and Tang (1989) then extended this investigation
to include screening inspection for multiple performance
variables in a serial production process. Also, in 1994, they comprehensively
addressed the concerns inherent in the design of such
screening procedures (Tang and Tang, 1994).
When integrating both viewpoints, i.e., cost and functional performance,
a trade-off is often necessary. Along this line, Jeang
(1997) considered the simultaneous optimization of manufacturing
cost, rejection cost, and quality loss using a process capability
index to establish a relationship between tolerance and standard
deviation. In an attempt to achieve a more realistic basis for use
in industrial settings, Kapur and Cho (1996) investigated tolerance
optimization problems using truncated normal, Weibull, and multivariate
normal distributions, respectively. However, in situations
where the historical data of losses are available, regression analysis
can be applied, as exemplified by Phillips and Cho (1998), who
studied the minimization of the quality loss and the rejection costs.
They developed optimization models using the first-order and second-order
empirical loss functions.
Research in the past several years has focused on an increasingly
more complex manufacturing environment. By Plante
(2002), trade-offs are considered for multivariate cases when conducting
parametric and nonparametric tolerance allocation. A genetic
algorithm is utilized to determine complex optimization
problems encountered by the selection of design and manufacturing
tolerances under different stack-up conditions (Singh et al.,
2003). Manarvi and Juster (2004) developed an integrated tolerance
synthesis model for tolerance allocation in assembly design.
Wang and Liang (2005) studied the tolerance optimization problem
in the context of machining processes, such as milling, turning,
drilling, reaming, boring, and grinding. Shin and Cho (2007) studied
two separate process parameters, such as process mean and
variability, in the bi-objective optimization framework. Recently,
Prabhaharan et al. (2007) and Peng et al. (2008) considered optimal
process tolerances for mechanical assemblies as a combinatorial
optimization problem by considering stack-up conditions. Lee
et al. (2007) proposed a method to find the process mean maximizing
the expected profit for a multi-product production process.
Hong and Cho (2007) suggested a joint optimization method associated
with the process target and tolerance limits based on measurement
errors. Jeang and Chung (2008) developed an
optimization model for product quality performance to determine
optimal use time, initial settings, a process mean, and a process tolerance
that simultaneously minimize total cost, quality loss, failure
cost, and tolerance cost. Chen and Kao (2009) proposed a process
control model for the canning/filling industry to find the process
mean and screening limits that minimize the expected total costs
by utilizing the concept of the surrogate variable which is assumed
to be highly correlated by the performance variable.
3. Cost structure for process parameter optimization
Loss to customers due to quality performance is zero when the
performance of all key quality characteristics occurs at their customer-identified
target values. However, due to the inherent variability
associated with the characteristics, the loss is always
incurred by the customer and it can be reduced by minimizing
the deviation from the target value of each quality characteristic.
However, it is often the case that process costs often increase as
we try to achieve a smaller deviation from the process target value.
An optimization of process parameter settings involves determining
the optimal process mean and standard deviation which simultaneously
minimize the loss incurred by the customer and the
process cost incurred by the manufacturer.
When modeling a problem to optimize the process parameter
settings, the customer loss and the process cost need to be expressed
in terms of the decision variables (process mean and
standard deviation). In the following sections, the loss incurred
by the customer is written in terms of these decision variables
by using a quadratic loss function, whereas the relationship between
the process cost and the decision variables is empirically
determined.
3.1. Loss due to process variability
A quality loss function (QLF) can be employed to quantify the
product quality on a monetary scale when its performance deviates
from customer-identified target value(s) in terms of one or more
key characteristics. The quality loss includes long-term losses related
to reliability and the cost of warranties, excess inventory,
customer dissatisfaction, and eventual loss in market share. A popular
QLF is a quadratic QLF which is a quasi-convex function with
many desirable mathematical properties including unimodality,
non-negative loss, zero loss due to target values, and accommodation
of discontinuities (Cho and Leonard, 1997). The quadratic QLF
can be applied to the situations either where there is little or no
information about the functional relationship between quality
and cost, or where there is no direct evidence to refute a quadratic
representation. Compared to other QLFs, such as step-loss or piecewise
linear loss functions, the quadratic QLF may be a good approximation
of measuring the quality of a product, particularly over the
range of characteristic values in the neighborhood of the target
values.
Assuming L(y) to be a measure of losses associated with the
quality characteristic y whose target value is T, the quadratic loss
function is given by
LðyÞ ¼kðy TÞ 2 ; ð1Þ
where k is a positive coefficient, which can be determined from the
information on losses relating to exceeding the tolerance given by
the customer. The expected loss is then defined as
E½L1ðyÞŠ ¼
Z 1
1
LðyÞf ðyÞdy ¼
Z 1
1
kðy TÞ 2 f ðyÞdy; ð2Þ