Development of the parametric tolerance modeling and optimization ...

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Fig. 21. Plots of the effect of jl Tj on E[TC 2], E[C R], E[L 2], E[C M2], and d* when r = r*. 6.3. Case III 6.3.1. Determining optimal settings of process parameters Since the closed-form solutions of l* and r* for this case are the same as those of Case I, see the first part of Section 6.1 for more details. 6.3.2. Determining optimal tolerance Although the closed-form solution of d* for this case cannot be formulated, d* can be generated by minimizing E[TC 2] given in Eq. (44). By using MATLAB software package, the minimum E[TC2] is obtained at 150.4008 when d is 1.0269, defined as d*. This optimal value can be verified from the plots shown in Figs. 11 and 12. The former shows that the first derivative of E[TC2] is zero at d* and the latter shows that the second derivative of E[TC2] atd* is greater than zero. The change of d to E[TC2], E[CM2], E[CR] and E[L2] is depicted in Fig. 10. From d*, the optimum LSL and USL are set at 48.9959 and 51.0041, respectively. The sensitivity analysis of the effect of r on d* is provided in Table 8 and Fig. 20. It can be observed that r has inverse variation in d* but it has direct variation in E[C R]. Similarly, the sensitivity analysis of the effect of the process bias (or jl Tj) ond* is provided in Table 9 and Fig. 21. It shows that the process bias directly affect d* and E[CR]. For the 100% inspection of Y, E[L2] is slightly affected by r and jl Tj, whereas E[C R] is greatly affected by both r and jl Tj. 7. Conclusions and further study Very often, engineers face the problem of determining the optimal tolerance in many industrial settings. This paper proposed an integrated scheme for determining the optimal process parameters and tolerances, the algorithm to quantify the cost of achieving optimal process parameters and tolerances, together with a procedure to incorporate Lambert W function to generate closed-form solutions for tolerance optimization problems. The proposed methodologies are superior to earlier models that determine optimal tolerance assuming fixed settings for the process parameters. Moreover, the proposed models and solutions might be appealing to engineers because the function is found in most standard optimization software packages. Continuous research efforts over the past few decades in the area of tolerance optimization has led to the formulation of effective optimization models that are free of the restrictive assumptions imposed by earlier ones. Along these lines, a fruitful area for future research is to extend the modeling S. Shin et al. / European Journal of Operational Research 207 (2010) 1728–1741 1739 methodology presented in this paper to a generalized tolerance optimization problem involving multiple quality characteristics. Further, the consideration of interactions between quality characteristics and the use of a surrogate variable may be potential topics for future investigations. Acknowledgements This work was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea Government (MOST) (No. R01-2007-000-21070-0). This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF- 2008-331-D00686). Appendix A Key notations, used in the model developments, are summarized and explained as follows: Y the quality characteristic Y of the product y the observed value of Y f(y) the density function of Y l the process mean of Y l* the optimum process mean of Y r the standard deviation of Y r* the optimum process standard deviation of Y r2 the variance of Y T the target value of Y s the target value of Y in the standard normal distribution domain, s =(y l)/r k the quality loss coefficient L(y) the quality loss associated with Y L1(y) the quality loss associated with Y and incurred when process inspection is not implemented L2(y) the quality loss associated with Y and incurred when process inspection is implemented QLF quality loss function CM1 the manufacturing unit cost incurred when process inspection is not implemented c0 the constant in the CM1 regression function c1 the coefficient of l in the CM1 regression function c2 the coefficient of r in the CM1 regression function c3 the coefficient of lr in the CM1 regression function d the number of standard deviations where the specification limit is located from the middle of its tolerance range LSL the lower specification limit associated with Y USL the upper specification limit associated with Y E[ ] the expected value CR the rejection unit cost /( ) the standard normal probability density function U(.) the standard normal cumulative distribution function z the standard normal random variable t the tolerance range CM2 the manufacturing unit cost incurred when process inspection is implemented a0 the constant in the CM2 regression function a1 the coefficient of t in the CM2 regression function e the least-squares regression error TC1 the process total cost incurred when the process inspection is not implemented TC2 the total cost incurred when the process inspection is implemented Appendix B Lemma 1. Suppose v 2 R 1 and a mapping g:v ? Risg = ve v , then the solution for v is given by v = LambertW(g).
1740 S. Shin et al. / European Journal of Operational Research 207 (2010) 1728–1741 Proof. See Corless et al. (1996). h Proposition 1. If g2 =(v + g1)e v where g1 and g2 are not functions of v, then v ¼ Lambert W g2eg ð 1Þ g1 . Proof. Consider the following equation: g 2 ¼ðv þ g 1 Þe v : ðA-1Þ Let v + g1 = w so Eq. (A-1) becomes g 2 ¼ we w g 1: ðA-2Þ To convert Eq. (A-2) in the standard form g = ve v , we first modify Eq. (A-2) as follows: g 2 e g 1 ¼ we w : ðA-3Þ Next, substitute g 2 e g 1 ¼ x to the right hand side of Eq. (A-3) then we obtain x = we w . From Lemma 1, w is given by: w ¼ Lambert WðxÞ: ðA-4Þ Therefore, v is obtained as v ¼ Lambert W g2e g ð 1Þ g1 : Proposition 2. If g4 ¼ g1 v2 þ g2 eg3v2 , where g1, g2, g3, and g4 are not functions of v, then v ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Lambert W g3g4 e g2g v u 3 u t g1 : g 3 g 2 Proof. In order to prove Proposition 2 using Lemma 1, we first consider the equation g4 ¼ g1 v2 þ g2 e g3v2 : ðA-5Þ g 3g 4 g 1 Multiplied both sides with g3/g1, Eq. (A-5) becomes ¼ g3 v2 þ g2 e g3v2 : ðA-6Þ Let w = g 3(v 2 + g 2), Eq. (A-6) can be written as g 3g 4 e g 2g 3 g 1 ¼ we w : ðA-7Þ Further, substituted g3g4 e g2g3 g1 dard form g = ve v . ¼ x, Eq. (A-7) is now in the stan- w ¼ Lambert WðxÞ: ðA-8Þ Replaced x ¼ g3g4 e g2g3 g back into Eq. (A-8), the standard form in 1 Eq. (A-8) can be written as w ¼ Lambert W g3g4 e g2g3 g . Eq. (A-9) 1 can be expanded to express the solution in terms of v by replacing w with g3(v + g2). Thus, the solution for v is given by ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Lambert W v ¼ g3g4 e g2g v u 3 u t g1 : ðA-9Þ References g 3 g 2 Al-Fawzan, M.A., Rahim, M.A., 2001. 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