Development of the parametric tolerance modeling and optimization ...

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where L1(y) denotes the quality loss when inspection is not implemented, the quality characteristic Y is normally distributed with mean l and variance r 2 , and f(y) denotes the probability density function of Y. 3.2. Process cost To minimize the process bias (the distance between the process mean and the process target) and the process variance, the process parameter settings need to be optimized by achieving the minimum process cost and quality loss. The process cost can be empirically represented using the decision variables. The following empirical linear model is often (Chase and Parkinson, 1991): CM1 ¼ c0 þ c1l þ c2r þ c3lr: ð3Þ 3.3. Total cost The expected total cost for determining the optimal process parameters is then given as E½TC1Š ¼E½L1ðyÞŠ þ E½CM1Š: ð4Þ Using Eqs. (2) and (3), Eq. (4) can be expanded to E½TC1Š ¼ Z 1 1 kðy TÞ 2 f ðyÞdy þðc0 þ c1l þ c2r þ c3lrÞ: ð5Þ Eq. (5) can be written in terms of a standard normal random variable (z) using the transformation z =(y l)/r. Thus, Eq. (5) becomes E½TC1Š ¼ Z 1 1 kfðl þ zrÞ Tg 2 /ðzÞdz þðc0 þ c1l þ c2r þ c3lrÞ: Using the following properties of standard normal random variables: Z 1 1 /ðzÞdz ¼ 1; Z 1 1 z/ðzÞdz ¼ 0; and Z 1 then the expected total cost can be rewritten as 1 ð6Þ z 2 /ðzÞdz ¼ 1; ð7Þ E½TC1Š ¼kfðl TÞ 2 þ r 2 gþc0 þ c1l þ c2r þ c3lr: ð8Þ The optimum values, l* and r*, can be determined by simultaneously equating dE[TC 1]/dl and dE[TC 1]/dr to zero. Then, the closed-form solutions of the optimal process mean and the variance are obtained. The stationary points can be easily verified by satisfying the following condition: d 2 E½TC1Š dl 2 ! d 2 E½TC1Š dr 2 ! > d2 ! 2 E½TC1Š dldr 4. Cost structure for tolerance optimization : ð9Þ A quadratic loss function is used to evaluate a quality loss when an inspection with specification limits is implemented. Besides the loss incurred by the customer, the costs incurred by the manufacturer, such as the rejection and the manufacturing costs, are also included. Depending on customer requirements, lower and upper specification limits (LSL and USL) can be defined respectively as l dr and l + dr, orT dr and T + dr where d represents the number of standard deviations from the middle of a tolerance range to each specification limit and it is always greater than zero. S. Shin et al. / European Journal of Operational Research 207 (2010) 1728–1741 1731 4.1. Quality loss A quality loss is incurred by the customer when the product performance determined by y falls inside the specification limits. By modifying Eq. (1), the expected quality loss E[L2(y)], incurred when the process inspection is implemented at the LSL and USL, can be given as E½L2ðyÞŠ ¼ Z USL LSL LðyÞf ðyÞdy ¼ Z USL LSL kðy TÞ 2 f ðyÞdy: ð10Þ Letting z =(y l*)/r* where l* and r* denote the optimal process mean and standard deviation resulting from the process parameter optimization phase, Eq. (10) can be rewritten as E½L2ðyÞŠ ¼ Z ðUSL l Þ=r ðLSL l Þ=r 4.2. Rejection cost kðzr þ l TÞ 2 /ðzÞdz: ð11Þ A rejection cost is incurred by the manufacturer when the product performance determined by y falls outside the specification limits. Denoting CRas the rejection unit cost incurred when the quality characteristic of a product falls below a lower specification limit or above an upper specification limit, the expected rejection cost E[CR] is defined as E½CRŠ ¼CR Z LSL 1 f ðyÞdy þ Z 1 USL f ðyÞdy : ð12Þ Using z =(y l*)/r*, Eq. (12) can be modified as Z ðUSL l Þ=r ! E½CRŠ ¼CR 1 /ðzÞdz : ð13Þ ðLSL l Þ=r 4.3. Manufacturing cost Additional manufacturing operations, slow processing rates, and improved care on the part of operators in achieving a tight tolerance may increase manufacturing cost. The manufacturing cost usually constitutes a significant portion of the unit production cost, and its exclusion from the tolerance optimization model may result in a suboptimal tolerance. The tolerance allocation models (Speckhart, 1972; Spotts, 1973; Chase et al., 1990; Kim and Cho, 2000b) found in the literature enforce the 3r assumption in order to minimize the manufacturing cost. The manufacturing cost-tolerance relationship proposed in this paper is free of the ad hoc 3r assumption. Even though the middle of a tolerance range is either a process mean (l) or a process target (T), the tolerance range t can be defined in terms of d, l, T and r as t ¼ USL LSL ¼ðlþ drÞ ðl drÞ ¼ðT þ drÞ ðT drÞ ¼2dr: ð14Þ The manufacturing cost (CM2) is often described in literature (Patel, 1980; Bjorke, 1989) as a first-order model CM2 ¼ a0 þ a1t þ e; ð15Þ where e represents the least-squares regression error. The expected manufacturing cost E[CM2], after t is replaced with 2dr from Eq. (14), can be written as E½CM2Š ¼a0 þ 2a1dr: ð16Þ 4.4. Total cost The expected total cost when the process inspection is implemented can be given by E½TC2Š ¼E½L2ðyÞŠ þ E½CRŠþE½CM2Š: ð17Þ
1732 S. Shin et al. / European Journal of Operational Research 207 (2010) 1728–1741 Using the optimal process mean l* and standard deviation r* determined in the process parameter optimization together with Eqs. (11), (13) and (16), the proposed tolerance optimization model is formulated as Minimize E½TC2Š ¼ 5. Proposed model Z ðUSL l Þ=r kðzr þ l TÞ ðLSL l Þ=r 2 /ðzÞdz Z ðUSL l Þ=r ! þ CR 1 ðLSL l Þ=r /ðzÞdz þ a0 þ 2a1dr: ð18Þ When determining optimal tolerances, most research reported in the literature assumes that the levels of the two process parameters, i.e., mean and variance, are typically treated as given constants. However, the arbitrary settings of these parameters in connection with the tolerance optimization often affect the defective rate, material cost, scrap or rework cost, and possibly cause a loss to customers due to the deviation of the product performance from its target value. Once the minimum process variability is achieved by determining the optimal process parameter settings, tolerance optimization can then be implemented. There are three possible cases as shown in Fig. 1. First, when an arbitrary target is considered in a new process design, a typical tolerance optimization scheme focuses on the process mean by specifying USL = l + dr and LSL = l dr. Second, when the process mean is fixed at an arbitrary target, USL = T + dr and LSL = T dr are specified. This case is usually applied to existing processes. The last possible case is when the process can absorb some bias, thereby allowing the process mean to be slightly off the target. The categorized optimization strategy for each case is proposed. Case I: Minimize E[TC1] for the process parameter optimization Minimize E[TC 2] where USL = l + dr and LSL = l dr for the tolerance optimization Case II: Minimize E[TC 1] where l = T for the process parameter optimization Minimize E[TC 2] where USL = l + dr and LSL = l dr for the tolerance optimization Case III: Minimize E[TC 1] for the process parameter optimization Minimize E[TC2] where USL = T + dr and LSL = T dr for the tolerance optimization The models for finding optimal process parameter settings and tolerance range, together with an investigation of the conditions for convexity are presented in each case. 5.1. Case I 5.1.1. Process parameter optimization From Eq. (8), by equating @E[TC1]/@l and @E[TC1]/@r to zero, the closed-form solutions for the optimal process mean l* and deviation r* are obtained as l ¼ 4k2T 2kc1 þ c2c3 4k 2 c2 ; ð19Þ 3 and r ¼ 2kc2 þ 2kc3T c1c3 4k 2 þ c2 : ð20Þ 3 To verify that l* and r* represent the global minima of the E[TC1], the following condition developed from Eq. (9) must be satisfied: ð2kÞð2kÞ > c 2 3 or 4k2 > c 2 3 : 5.1.2. Tolerance optimization Replacing USL and LSL in Eq. (11) with l + dr and l dr, respectively, and using the following results associated with the normal probability density function: Z r 1 Z r 1 /ðzÞdz ¼ UðrÞ; Z r 1 z/ðzÞdz ¼ /ðrÞ; and z 2 /ðzÞdz ¼ UðrÞ r/ðrÞ; ð21Þ where r is a value of the standard normal random variable, the expected quality loss can be determined as E½L2ðyÞŠ ¼ 2k r 2 þðl TÞ 2 n o UðdÞ 2kr 2 d/ðdÞ n o ; ð22Þ k r 2 þðl TÞ 2 and the expected rejection cost determined in Eq. (13) can be written as: Z d E½CRŠ ¼CR 1 /ðzÞdz ¼ 2CR½1 UðdÞŠ: ð23Þ d Substituting Eqs. (22), (23) and (16) into Eq. (17), the expected total cost is calculated as E½TC2Š ¼2k r 2 þðl TÞ 2 n o UðdÞ 2kr 2 d/ðdÞ k r 2 þðl TÞ 2 n o þ 2CRf1 UðdÞg þ a0 þ 2a1dr : ð24Þ The optimum total cost can be derived by minimizing Eq. (24). In order to determine the optimum value d*, the first derivative of E[TC2] with respect to d is calculated as @E½TC2Š @d ¼ 2k/ðdÞ ðl TÞ2 þ d 2 r 2 CR k þ 2a1r : ð25Þ Then, equating Eq. (25) to zero and substituting /(d) with e 1 2 d2= ffiffiffiffiffiffi p 2p, the result can be identified as follows: 1 e 2 2k d2 ! pffiffiffiffiffiffi or r 2 2p ðl TÞ 2 þ d 2 r 2 CR k þ 2a1r ¼ 0; 2k pffiffiffiffiffiffi d 2 " # kðl TÞ2 CR þ 2p kr 2 e 1 2 d2 ¼ 2a1r : ð26Þ Since it is nearly impossible to obtain a closed-form solution to this complex equation, the Lambert W function is employed to obtain such a solution of d effectively. Details are presented in Lemma 1, and in Propositions 1 and 2 of Appendix B. According to Proposition 2, ifg1 v2 þ g2 eg3v2 ¼ g4 where g1, g2, g3, and g4 are not functions of v, then the solution for v can be given by ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Lambert W v ¼ g3g4 e g2g v u 3 u t g1 : ð27Þ g 3 g 2 Therefore, v, g1, g2, g3, and g4 in Proposition 2 can be respectively substituted by d; 2k=r 2 p ffiffiffiffiffiffi 2 2p; kðl TÞ CR =kr 2 ; 1=2, and 2a1r* from Eq. (26). Then, the closed-form solution of d is defined as follows: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ¼ 2Lambert W a1 pffiffiffi ð pe 1 2 Þ kðl TÞ2 CR kr 2 r k ffiffiffi 0 1 B C B @ p C kðl TÞ 2 A 2 CR kr 2 v u ! u ; t ð28Þ
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