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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME AND TECHNOLOGY (IJMET) ISSN 0976 – 6340 (Print) ISSN 0976 – 6359 (Online) IJMET Volume 4, Issue 3, May - June (2013), pp. 223-231 © IAEME: www.iaeme.com/ijmet.asp Journal Impact Factor (2013): 5.7731 (Calculated by GISI) ©IAEME www.jifactor.com PROCESS OPTIMIZATION AND PROCESS ADJUSTMENT TECHNIQUES FOR IMPROVING QUALITY IN MANUFACTURING PROCESS Prof. S.Y.Gajjal1, Prof. (Dr) .A.P.S.Gaur2 1 Assistant Professor, SCOE, Pune, 2 Professor & Head, B.I.E.T, Jhansi ABSTRACT Process optimization and process adjustment methods are discussed and combining an EWMA chart with Shewhart chart is traditionally recommended as a mean of providing good protection against both small and large shift in the process mean. Using an EWMA together with Shewhart chart, but we find no performance improvement. In conjunction with some commonly used control chart, these adjustment techniques are then applied on a manufacturing process and Clustering the process adjustment with SPC. 1. INTRODUCTION A Stochastic approximation and optimization method is provided with application to process adjustment and process optimization problem in quality control. They can be unified by a kalman filter. Sample comparisons between these methods and EWMA feedback control are provided [9]. We apply these process adjustment techniques to a classical quality control problem – shifts in the mean value of the quality characteristic of a process. In traditional SPC, it is frequently assumed that an initially in-control process is subject to random shocks, which may shift the process mean to an off-target value [4]. Different types of control charts are then employed to detect such shifts in mean, since the time of the shift is not predictable. However, SPC techniques do not provide an explicit process adjustment method. The lack of adjustments that exists in the SPC applications may cause a large quality off-target cost – a problem of particular concern in a short run manufacturing process. Therefore, it is important to explore some on-line adjustment methods that are able to keep the process quality characteristic on target with relatively little effort. 223 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME 2. PROCESS OPTIMIZATION The KW procedure provides an experimental optimization method for industrial processes, which however has not been used in practice where response surface methods are widely used. It is therefore of interest to investigate if the KW procedure can converge to a stationary point in fewer experiments than a traditional application of RSM. One of the diﬃculties of the KW method is that, given that the gradient of the response needs to be estimated, this result in slower convergence compared to the RM approach. Furthermore, the multivariate KW process studied by Blum requires 2k experiments at each iteration in order to estimate the gradient. A more recent approach by Spall [14], termed”Stochastic perturbation” requires only 2 experiments per iteration. Recent research in this area has increased thanks to the application of KW -like processes to Neural Network”learning” processes. Thinking in this type of application, Darken et al. [12] investigate a KW algorithm with weight sequences. Which equals to η0 for times t<<t. For t>>t, this function behaves as c/t, the traditional RM/KW weights. The idea is to make the algorithm approach a ”good” region rapidly, and only then start the convergence phase thanks to the RM/KW harmonic sequence, which, if used from the beginning, would make convergence too slow. Andradottir [9,10] provides another modiﬁcation of the RM method for use in simulation optimization which appears to converge faster than the original RM process. The convergence rate of KW- like processes can be improved if second- order information is used in the search. Ruppert [6] provides a stochastic version of Newton’s method. The method’s idea is to pre-multiply Yn by some estimate of the inverse of the Hessian of M (θ). This approach is a RM, not a K W, method. It seems that only Fabian [8] has investigated incorporating second order information in KW processes but he did not provide any performance analysis. Nonlinear optimization algorithms such as the BFGS method provide a sequential method for approximating the Hessian of a deterministic function. It seems plausible that such a scheme can be put in a stochastic optimization setting, providing an algorithm that converges faster than traditional gradient- based KW methods. The emphasis of such investigation, however, should be small sample behavior, i.e., to optimize a process with the smallest number of experiments. A diﬀerent area of application of RM and KW processes is using them in certain parts of the traditional RSM framework. Once instance of this area is the use of the RM process to do the line searches needed in the steepest ascent phase of RSM. Investigation and comparison of such approach vs. stopping rules used in steepest ascent are of interest. 3. PROCESS ADJUSTMENT The setup adjustment problem, where a machine is initially oﬀ- target was studied by Grubbs [6]. He ﬁnd an adjustment scheme that is identical to using the RM estimate studied the connections between Grubbs’ procedure, the RM process, and a Kalman ﬁlter approach. Studied the small sample Average Integrated Squared Deviation (AISD) provided by each rule, and concluded that Grubbs’ rule was best in general. However , results from Frees and Ruppert [12] indicate that it is worth exploring the AISD performance of a RM-like rule w here the weights are given by C /(iˆ ß1+ r ) ,where ß1is the slope of M (x ) and C is some constant. These authors did not consider the AISD, but instead looked at E [(x n- θ )2 ] for small n , in an analysis closely related to the aforementioned one by Hodges and Lehmann . For this performance measure, Frees and Ruppert indicate that using C = 1.5 to 2 and r = 1 or 0 works 224 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME best. It is of interest to derive AISD formulae for such weights and try to determine recommendations for C and r , as compared to simply using the harmonic rule {1/n } as used by Grubbs. A different area of application is for those setup adjustment problems in which over adjustment in one direction is much more expensive than under adjusting, i.e .,the underlying loss function is not symmetric. No comparisons of these methods for small or large samples are available. An interesting recent paper is by Chen and Guo [2] who propose to use SA to ﬁnd MSE -optimal EWMA weights in a EWMA controller when there is evidence of a shift, and use a constant EWMA weight otherwise. The idea of coupling Grubbs procedure with a control chart was originally proposed by Del Castillo [7]. It is not clear if always using SA (Grubbs) procedure rather than having a minimum weight value is better or not. How to detect the change point is very important, Chen and Guo used an EWM A control chart to trigger the RM-like adaptation of the weights, unaware that they were using Grubbs’ procedure or SA. 4. MODIFIED PROCESS ADJUSTMENT METHODS The machine tool setup adjustment problem has been discussed by Grubbs (1954, 1983). Suppose there is a random setup error on the machine which can be observed from the quality characteristic of the process output, but cannot be measured directly due to the noise from the process and measurement. However, it is assumed that by varying an input parameter, the error can be eliminated or compensated for. The problem is how to tune the input parameter in face of the uncertainty of the setup error [3]. The process can be formulated as Xt = d+Ut-1+ct ------------------------- (1) Where Xt corresponds to the deviations from target of some quality characteristic of parts produced at discrete point in time, Ut-1 is the level of some controllable factor which has direct impact on subsequent process measurements, d is the setup error and ct represents the randomness from both process and measurements. The simple adjustment rule proposed by Grubbs is given by ∇ Ut-1 = Ut-1 - Ut = Xt --------------- (2) Where {1/t} is a harmonic sequence. This solution is proved to be optimal in the sense that it minimizes the variance of process mean at time t+1. In the following, we will provide a Bayesian formulation to the setup adjustment problem based on a Kalman Filter estimator (for the Bayesian interpretation of a Kalman Filter, see Mein hold and Singpurwalla, 1983). Define d to be a state variable, where d is the setup error d, then equation (1) can be expressed by a state equation and an observation equation as follows: ------------- (3) --------- (4) Given this simple state-space formulation, a Kalman Filter estimator of d can be constructed which is given by 225 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME ------ (5), and --------------- (6) where ,P0 is the variance of the prior distribution of d, i.e., d ~ (d0 , P0), by the Bayesian interpretation. If the adjustments are made according to the simple rule: --------- (7), then ----------- (8) Equations (6) and (8) provide four general cases of interest 5. SAMPLE PERFORMANCE It was shown in the previous section that Grubbs harmonic rule and the Stochastic Approximation of a constant offset are equivalent. This leads to the application of many available stochastic approximation results to the machine setup problem. However, although many asymptotic results exist for stochastic approximation schemes, the emphasis of this section is the performance based on small samples [10]. The performance index we considered herein is the scaled Average Integrated Square Deviation (AISD) over a single potential “production run”, or expected realization, of length n. This is defined as, ----- (9) It is important to point out that Grubb’s extended and harmonic rules, which are optimal for minimizing the variance of the last setup error estimate, are not necessarily optimal for the AISD (n) criterion. The AISD formulae provide a measure of the quadratic cost incurred by a whole process after a shift in process mean occurs at any point in time. When the Kalman Filter scheme (6) and (8) are applied to the process (1), it results in -------------------- (10) And -------- (11), Where C= (d-d0)/σv and D =P0/σ2v, Note that as time passes, E [Xt] tends to zero and Var(X1) tends to the limiting value of . Clearly the scaled AISD (n) performance index captures the transient behavior of both E[X] and Var(X) along one potential time realization of the process of finite duration. To get the corresponding expressions for Grubbs’ rule, one can let C →8 in (12) and (13).For a discrete integral controller, it can be shown that C ----------- (12) and ------ (13) 226 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME The scaled AISD (n) expressions of integral controller allow to study the trade-offs between the sum of variances and the sum of squared expected deviations (squared bias). Table (1) contrasts the scaled AISD performance of Grubbs’ harmonic rule, the discrete integral controller and the Kalman Filter adjusting scheme. The table the values of (n)/ for n=5, 15 and 20. From the table, it can be seen that Grubbs' harmonic rule dominates (or at least equals) the Kalman filter rule for almost all cases, except in the unrealistic case where one is very confident of the a priori offset estimate and this estimate turns out to be quite accurate. Even when the a priori offset estimate is perfect, it might be necessary to adjust, because from the Bayesian point of view we may not be sure about our offset estimate, and this uncertainty in our prior belief is modeled D=P0/ by the variable. Otherwise, a single calibration will be sufficient to bring the process right on the target. Intuitively, if the variance is unknown the performance of the Kalman filter scheme can only worsen. This was confirmed by estimating AISD using simulation. If C=0, it can be seen from (9), (10), and that the AISD indices equal to the average scaled variance since the deviations from target will always be zero on average. In this case, the AISD quantifies the average inflation in variance we will observe for adjusting a process when there was no need to do so. Turning to the discrete integral controller, it can be seen that it also provides a very competitive scheme compared to the Kalman filter scheme. The large parameter γ has the effect of bringing the process back to target more rapidly, but it also causes severe inflation in variance when adjusting a near target process. It seems the value γ =0.2 provides a relatively good trade-off between fast return to target and inflation of variance if the process was close to target. 6. CLUSTERING PROCESS ADJUSTMENT WITH SPC From Figure 1, one can see that the sequential adjustment methods M3 and M4 are superior to the one-step adjustment methods M1 and M2 for almost all shift sizes. Using a CUSUM chart and sequential adjustments M4 has significant advantage over other methods when the shift size is small or moderate, and using a Shewhart chart and sequential adjustments M3 is better for large shifts. Moreover, one-step adjustment methods (Taguchi's method) may deteriorate a process when the shift size is very small [6]. In Figure 2, the probability of random shifts p was decreased to 0.01 and the same simulation as in Figure 1 was conducted. Under these conditions, the EWMA method cannot compete well with the sequential adjustment methods combined with CUSUM or Shewhart chart monitoring. More simulation results for different probabilities of shifts p are listed in Table 2. It is found that the EWMA adjustment method is better for small shifts and M 3 or 4 is better for large shifts when p is large; as p gets smaller (p<0.02), i.e., the process is subject to infrequent random shocks, Method 4 gets harder to beat. Therefore, the proposed cluster methods work better when p is small, which is relevant in the microelectronics industry where process upsets occur very rarely. 227 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME Figure 1. Performance of all controlled processes when p=0.05 80 % of improvement on AISD 70 60 50 40 30 20 10 0 0 0.5 1 1.5 2 2.5 3 3.5 Mean of shift size Figure 2: performance of five controlled process when p=0.01 30 25 % of improvement on AISD 20 Shewhart one 15 Cusum one 10 Shewhaet seq 5 Cusum seq 0 EWMA with -5 lambda = 0.15 -10 0 0.5 1 1.5 2 2.5 3 3.5 Mean of shift size 228 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME Table 1. Comparison of performance of adjustment process n=5 C |D|=0 |D|=1 |D|=2 |D|=3 1/60 1.001 1.836 3.921 8.999 1/2 1.093 1.367 2.456 4.621 1 1.157 1.345 2.345 3.786 γ=0.1 1.045 1.578 2.345 3.234 γ =0.2 1.465 1.556 3.978 5.647 n=15 C |D|=0 |D|=1 |D|=2 |D|=3 1/60 1.003 1.726 3.826 8.562 1/2 1.090 1.360 1.678 3.567 1 1.168 1.293 2.234 3.452 γ=0.1 1.023 1.495 1.236 2.896 γ =0.2 1.523 1.455 2.982 4.689 n = 20 C |D|=0 |D|=1 |D|=2 |D|=3 1/60 1.000 1.788 3.231 8.326 1/2 1.121 1.187 1.506 3.126 1 1.008 1.192 2.211 3.098 γ=0.1 1.039 1.245 1.055 2.405 γ =0.2 1.089 1.389 2.543 3.967 Percentage Mean of shift size improvement 0 1σ 2σ 3σ 4σ P=0.04 M1 5.92(0.24) 24.67(0.380 48.11(0.42) 63.01(0.41) 69.12(0.42) M2 14.21(0.28) 31.02(0.35) 51.09(0.38) 62.01(0.38) 67.00(0.37) IC(γ=0.1) 18.42(0.26) 31.88(0.33) 49.01(0.33) 55.89(0.32) 59.78(0.32) IC(γ=0.2) 17.02(0.31) 32.67(0.35) 51.67(0.41) 62.23(0.43) 65.07(0.41) IC(γ=0.3) 12.98(0.34) 30.65(0.44) 52.01(0.43) 61.21(0.43) 66.04(0.45) P=0.02 M1 1.59(0.22) 12.01(0.33) 29.02(0.41) 42.21(0.44) 47.97(0.44) M2 8.11(0.22) 17.76(0.30) 33.01(0.38) 42.01(0.41) 46.98(0.42) IC(γ=0.1) 10.41(0.23) 19.12(0.31) 31.21(0.38) 38.01(0.37) 41.00(0.401) IC(γ=0.2) 7.44(0.26) 17.09(0.34) 31.60(0.42) 39.66(0.44) 43.77(0.43) IC(γ=0.3) 2.21(0.26) 13.01(0.37) 29.01(0.43) 18.21(0.46) 43.02(0.46) P=0.004 M1 -3.43(0.19) -1.04(0.24) 3.67(0.29) 9.03(0.36) 12.87(0.38) M2 1.56(0.13) 3.76(0.18) 8.01(0.24) 12.10(0.31) 14.87(0.36) IC(γ=0.1) -0.41(0.14) 1.67(0.13) 5.87(0.24) 7.97(0.28) 10.02(0.32) IC(γ=0.2) -5.97(0.18) -3.01(0.22) 1.76(0.29) 5.12(0.34) 7.43(0.35) IC(γ=0.3) -11.8(0.21) -8.77(0.26) -3.41(0.36) 0.46(0.38) 3.46(0.41) Table 2. Performance of integrated schemes and EWMA scheme when varying the probability of a shift 229 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME 7. CONCLUSION Important connection between Grubbs harmonic rule and stochastic approximation was made. The Kalman filter adjustment scheme allows easy generalization to the multivariate case. The performance indices for each of these cases consider precision and accuracy over all observations, as opposed to Grubbs' criterion which is based on looking only at the last of a series of observations. In conjunction with some commonly used process monitoring charts, the harmonic adjustment method was applied for controlling against shifts in the process mean. It is shown that sequential adjustments are superior to single adjustment strategies for almost all types of process shifts and magnitudes considered. RM and KW methods have grown to a very considerable size over the years. Recent application to simulation optimization, process control, and Neural Network learning has gene rated a renewed interest in these methods. A CUSUM chart used together with a simple sequential adjustment scheme can reduce the average squared deviations of a shifted process more than any other combined scheme we studied when the shift size is not very large. REFERENCE [1] MacGregor, J.F. (1990), “A Difference View of the Funnel Experiment”, Journal of Quality Technology, 22: 255-259. [2] Montgomery, D.C. (2001), Introduction to Statistical Quality Control, 4th edition, John Wiley & Sons, Inc. [3] Patel, N.S. and Jenkins, S.T. (2000), “Adaptive Optimization of Run-to-Run Controllers: the EWMA Example”, IEEE Transactions on Semiconductor Engineering, 13(1): 97-107. [4] Box, G.E.P. and Luceno, A. (1997), Statistical Contorl by Monitoring and Feedback Adjustment, John Wiley & Sons, New York, NY. 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