Robust Tolerance Design With the Integer ...

Chang-Xue (Jack) Feng Hintz Manufacturing Technology Laboratory, Penn State University Berks Campus, P.O. Box 7009, Reading, PA 19610-6009

A. Kusiak Professor Intelligent Systems Laboratory, Department of Industrial Engineering, The University of Iowa, Iowa City, IA 52242-1527

Robust Tolerance Design With the Integer Programming Approach The quality loss function incorporates the cost of tolerances, however, it does not consider the manufacturing cost and design constraints. In this paper, a stochastic integer programming (SIP) approach is presented for simultaneous selection of tolerances and manufacturing processes. A direct link between the minimum manufacturing cost and the required level of manufacturing yield is established through the process capability index Cpk. As the tolerances in SIP are discrete, the solution generated is acceptable for manufacturing. It is shown that the integer programming models are applicable in the quality loss function and six sigma design approaches. The SIP approach is illustrated with a classical example of nonlinear tolerance design. The comparison of the proposed SIP approach, the Taguchi method, and the conventional mathematical models in tolerance synthesis is presented.

1

Introduction Design of tolerances impacts the manufacturing cycle time, quality, and cost of a product. A brief review of the tolerance design literature is provided in Chase and Parkinson (1992), Voelcker (1993), and Wu et al. (1988). Evans (1974) and Zhang and Huq (1993) reviewed the literature on probabilistic (statistical) tolerance design. This research evaluates the suitability of the Taguchi method, integer programming, and six sigma quality design for solving the probabilistic tolerance synthesis problem. The three approaches are illustrated with the example proposed by Fortini (1967), and reexamined in Krishnaswami and Mayne (1994) and Greenwood and Chase (1988). The process shift has not been considered in the previous models of tolerance synthesis, and it is incorporated in the approaches presented in this paper. The manufacturing variation is also considered as a constraint that replaces the quality loss as to ensure the required level of manufacturing yield. A comparative study of the above three approaches is conducted using Fortini's (1967) example. This paper addresses the issue of achieving quality by robust tolerance design and features the following contributions. First, it extends the Taguchi quality loss function to multi dimensional chains. Second, it considers discrete tolerances rather than continuous. Third, the stochastic integer programming (SIP) approach is used to model the relationship between manufacturing cost, manufacturing yield, and discrete tolerances. The SIP model follows the general approach of concurrent engineering as it considers manufacturability, cost, and quality at the product design stage. Fourth, it reveals that the quality loss might be a small portion of the total cost. The process capability index Cpk in the SIP model is a better indicator of quality level or manufacturing yield than the quality loss coefficient A in the quality loss function approach. Fifth, it is shown that the SIP model is applicable to the quality loss function and six sigma quality design approaches. The paper is organized as follows. The remainder of this section introduces the assumptions used in this paper. Sections 2 extends the Taguchi quality loss function to multi dimensional chains in tolerance synthesis. The SIP approach is presented in Section 3. Section 4 shows the relationship between the six sigma quality design and the SIP approach. In Section 5, the

three approaches are compared using Fortini's example. Section 6 concludes the paper. Assumptions The following assumptions are made in this paper: (1)

(2) (3)

2

The normal distribution law. The processes used to generate each tolerance follow the normal (Gaussian) distribution for a large number of identical items (see Harry and Steward, 1988). The process independence law. The processes used to generate each tolerance are independent. The value of the standard deviation a of each process is known.

The Quality Loss Function

The applications of the quality loss function (QLF) advocated by Taguchi (1986) to tolerance synthesis have been broadly presented in the literature, e.g., Askin and Goldberg (1988), Jeang (1994), Kapur (1993), Soderberg (1994), and Vasseur et al. (1992). A closely related research is discussed in Krishnaswami and Mayne (1994), and the example presented in Fortini (1967) is used in this paper. The Taguchi robust engineering design method (Taguchi, 1986) can be implemented in the form of: 1) quality loss function, and 2) Taguchi's experimental design (e.g., Askin and Goldberg 1988, Gadallah and Elmaraghy 1993). This section extends the quality loss function approach to tolerance synthesis of multi dimensional chains. 2.1 The Function. The quality loss function was derived from the Taylor series (Taguchi et al., 1989). Let /J, be the nominal dimension (target value), t a small deviation from this nominal value, and L(x) the quality loss function resulted from x. Then the Taylor series about the target value p can be written as follows L(x) = L(p + x — p)

(1)

L{x) = L(p) + ^ ^ - (x - p)

Contributed by the Manufacturing Engineering Division for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received

April 1995; revised May 1996. Associate Technical Editor: J. W. Sutherland.

Journal of Manufacturing Science and Engineering

+ ^(x-p)>

+ ... (2)

NOVEMBER 1997, Vol. 1 1 9 / 6 0 3

Copyright © 1997 by ASME Downloaded From: http://manufacturingscience.asmedigitalcollection.asme.org/ on 11/02/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use

single side tolerance of this dimension. Denote t = x - \i if x > =\x, and — t = x — \i if x < fj.. Assuming the quality loss cost is A for ta = \ xQ - ji |, then

L(x)n Repair or scrap cost

A

A = ati

(5)

Altl

(6)

Therefore, (4) is rewritten as li-t

n

n+t

L(t) = ~2t2 to

x

Fig. 1 The quality loss function

Single Component Tolerances. By ignoring the terms of order (power) higher than two, Eq. (2) becomes a convex quadratic function (see Figure 1). By definition, L(x) = 0 when x = ii, and the minimum value of the loss function is obtained at this point. As a result, its first derivative with respect to x is L'iiX) = 0. Thus (2) is rewritten as (Taguchi et al., 1989)

Ux) = ^M (x - v)2

(3)

Taguchi (Taguchi et al., 1989) suggested that different values of A be used to determine tolerances for a customer and a manufacturer. It is suggested that A be the repair or scrap cost associated with the product if one uses it to determine the manufacturing tolerance (the three sigma normal variation of the manufacturing process). However, the value of A is the quality loss after the product is delivered to the customer if one uses it to determine the customer tolerance. For a normal distribution of many items of the same component, the quality loss is the expected value of L(x) in (4), i.e., QL = E[L(x)]

2

L(x) = a(x - ii) where a = L"(x)/2\

(4)

is constant. Observe that \x — fj,\ is a

(7)

= a I

= E[a(x - /i)2]

(x — ^)2p(x)dx

= aa2

(8)

Nomenclature Parameters i = component tolerance index I = total number of components with tolerance requirements Ik = total number of component tolerances in dimension chain I j = process index Jt = total number of processes available for producing component tolerance i k = dimensional chain index K — total number of dimensional chains Cy = manufacturing cost of generating dimension i with process j fx = value of the centered process mean or design nominal fis = value of the skewed process mean M(f/) = manufacturing cost of dimension j' L(x) = quality loss resulted from x QL = total quality loss cost for the entire component or assembly p{x) = probability density function U = smallest tolerance value for dimension i which can be obtained by the available technology A = estimated quality loss coefficient

USLj = upper specification limit of Tk = single side functional dimension (' tolerance stackup limit for LSL, = lower specification limit of dimensional chain k dimension »' A* = full range functional tolerance stackup limit for Variables dimensional chain k x = dimension obtained from a C„ = process capability index manufacturing process, x = \i ± t Cpk = extension of Cp which t = single side tolerance value considers the process mean y = total of the manufacturing and shift in relation to the design quality loss cost nominal (mean) value ty = three sigma value of process ;' for ay = standard deviation' of generating dimension *' with a process j used to generate centered process dimension i tSIJ = three sigma value of process j for cr, = standard deviation of generating dimension / with a dimension i skewed process o> = required standard deviation 6y = six sigma value of process j for of dimensional chain k generating dimension ;' with a