International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
  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
Journal Impact Factor (2013): 5.7731 (Calculated by GISI)               ©IAEME


                           Prof. S.Y.Gajjal1, Prof. (Dr) .A.P.S.Gaur2
                                    Assistant Professor, SCOE, Pune,
                                    Professor & Head, B.I.E.T, Jhansi


           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.


           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.

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 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.


         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
         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

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.


        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

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


       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)

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


        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.

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

                                % of improvement on AISD

                                                                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


     % of improvement on AISD

                                20                                                                                                    Shewhart one

                                15                                                                                                    Cusum one

                                                                                                                                      Shewhaet seq
                                                                                                                                      Cusum seq
                                                                                                                                      EWMA with
                                                                                                                                      lambda = 0.15
                                                           0          0.5       1       1.5             2         2.5       3             3.5
                                                                                    Mean of shift size

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

                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
                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

                                                       Mean of shift size
                                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

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME


        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.


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6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME

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