A DETRMINISTIC RELIABILITY BASED MODEL FOR PROCESS CONTROL by iaemedu

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									International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383(Print),
International Journal of Production Technology
ISSN 0976 – 6391(Online) Volume 1, Number 1, July - Aug (2010), © IAEME
and Management (IJPTM), ISSN 0976 – 6383(Print)                              IJPTM
ISSN 0976 – 6391(Online) Volume 1                                        ©IAEME
Number 1, July - Aug (2010), pp. 32-44
© IAEME, http://www.iaeme.com/ijptm.html


  A DETRMINISTIC RELIABILITY BASED MODEL FOR
                               PROCESS CONTROL
                                        Jose K Jacob
                                  Ph.D Research Scholar,
                           Anna University Coimbatore, Tamilnadu
                              E-mail: jkjsolutions@yahoo.com

                                    Dr. Shouri P.V.
               Dept. of Mechanical Engineering, Model Engineering College
                                Cochin, Kerala- 689603
                             E-mail: pvshouri@yahoo.com

ABSTRACT
        Reliability of process plants is assuming more importance than ever before for
improving the effectiveness of the production process to meet the ever increasing
demands of the society. Process reliability and availability study gives us the necessary
information for improving the production and optimizing the cost of production and
maintenance. The reliability of a process unit is the probability that the process performs
its required function, in terms of quality and quantity, for a given time period under given
conditions. Process reliability analysis can provide a true indication of the process
potential and can also bring out various aspects of design that require more attention.
Statistical control charts is widely used process monitoring tool in the manufacturing
industry. An algorithm to develop a control chart is presented in this paper. A case study
based on Chocolate manufacturing company is also presented.
Key words : Process reliability, Availability, Mean Time Between Failure (MTBF),
Control chart, Failure rate




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International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383(Print),
ISSN 0976 – 6391(Online) Volume 1, Number 1, July - Aug (2010), © IAEME


1.      INTRODUCTION
1.1 Reliability and Availability
        Reliability is the probability of the equipment or process functioning without
failure, when operated as prescribed for a given interval of time, under stated conditions
[Kumar et al,1992]. High costs motivate seeking engineering solutions to reliability
problems for reducing financial expenditures, enhancing reliability, satisfying customers
with on-time deliveries through increased equipment availability, and by reducing costs
and problems arising from products that fail easily [Barringer,2000]. Reliability in power
plants is affected by operating periods, i.e. between scheduled outages; budget periods;
and peak-production periods [Warburton,1998]. Measuring the reliabilities of plant and
equipment by quantifying the annual cost of unreliability incurred by the facility puts
reliability into a business context. Higher-plant reliability reduces equipment failure
costs. Failure decreases production and limits gross profits [Warburton, 1998]. Failure is
a loss of function when that function is needed – particularly for meeting finance goals.
Failure requires a clear definition for organizations striving to make reliability
improvements [Barringer, 2000]. Improving reliability involves reducing the frequency
of failure. Reliability is a measure of the probability for failure-free operation, i.e. it is
often expressed as:
             −t 
R (t ) = exp        = exp(− λt )                                                  (1)
             MTBF 
         Long failure-free periods result in increased productivity; fewer spare parts need
to be stocked and less manpower employed in maintenance activities, and hence lower
costs. To the supplier of a product, high reliability is realized by completing a failure-free
warranty period under specified operating condition with few failures during the
subsequent design life of the product [Barringer & Weber, 1996]. Increased availability,
decreased down-time, smaller maintenance costs and lower secondary-failure
expenditures result in bigger profits.
        While general calculations of reliability are based on considerations of the initial
failure-mode, which may be termed ‘‘infant’’ mortality (decreasing failure rates then with
time) or wear-out mode (i.e. increasing failure rates then with time). Key parameters
describing reliability are mean time to failure, mean time between/before repairs, mean


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International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383(Print),
ISSN 0976 – 6391(Online) Volume 1, Number 1, July - Aug (2010), © IAEME


life of components, failure rate and the maximum number of failures in a specific time-
interval.
reliability + unreliability = 1                                                             (2)

                                    Weibull Distribution




                              Exponential Distribution




                                    Figure 1 Bath Tub Curve
1.2 Control Chart
        The main purpose statistical process control is to improve quality and
productivity. One of the instrument that form quality tool set is the control chart. Control
charts are efficient instruments for checking the changes or variations in the process.
Through out this tool it is intended to get a model, which detects better the variations in
the average, when it is necessary to control more sensitive process. Failure process
monitoring is an important issue for complex or repairable systems. It is also a common
problem for a fleet of systems such as equipment or vehicles of the same type in a
company. Statistical control chart is a widely used process monitoring tool in the
manufacturing industry; and it can be used in this type of failure process monitoring. This
is usually done by plotting the number of failures or breakdowns per unit time such as
week or month. Standard c-chart or u-chart, which is for the monitoring of the number of
defects in a sample, can then be used for this purpose. However, this procedure requires a




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International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383(Print),
ISSN 0976 – 6391(Online) Volume 1, Number 1, July - Aug (2010), © IAEME


large number of failures and it is not appropriate for application to a highly reliable
system
         When there is an excessive number of a failure, the chart will signal in such an
out-of-control situations. Although the anticipated false alarm probability, the probability
that the process is not changed when the plot shows an alarm, is 0.27% by a traditional
chart, it could be much higher because when the number of failures is Poisson distributed,
normal distribution which is used, is not a good approximation when the average number
of failures is small. Moreover, the lower control limit (LCL) is usually set at zero, which
is not useful because then process improvement cannot be detected. Chan et al. [1]
recently proposed a procedure based on the monitoring of cumulative production quantity
between the observations of two defects in a manufacturing process. This approach has
shown to have a number of advantages: it does not involve the choice of a subjective
sample size; it raises fewer false alarms; it can be used in any environment irrespective of
whether the process is of high quality; and it can detect further process improvement.
Since the quantity produced between the observations of two defects is related to the time
between failures in reliability study, this approach can be readily adopted for process
monitoring in reliability and maintenance.
         The use of cumulative quality is a different and new approach, which is of
particular advantage in reliability and maintenance engineering. We further extend the
procedure to the case when cumulative time to the rth failure is used to monitor the failure
process. The implementation and interpretations are provided and numerical examples
are used to illustrate the application procedure.
2. METHODOLOGY
2.1      t- Chart
         For a system in normal operation, failures are random event caused by, for
example, sudden increase of stress and human error. The failure occurrence process can
usually be modeled by a homogeneous Poisson process with a certain intensity. Hence,
our aim here is to monitor the failure process and detect any change of the intensity
parameter. The procedure in Ref. [1] is based on the monitoring of the cumulative
production quantity between observing two defects in a manufacturing process. This



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International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383(Print),
ISSN 0976 – 6391(Online) Volume 1, Number 1, July - Aug (2010), © IAEME


cumulative Production quantity is an exponential random variable that describes the
length till the occurrence of next defects in a Poisson process.
        When process failures can be described by a Poisson process, the time between
failures will be exponential and the same procedure can be used in reliability monitoring.
Here, we briefly describe the procedure for reliability monitoring. Since time is our
preliminary concern, the control chart will be termed t-chart in this paper.

        The distribution function of exponential distribution with parameter λ is given by
                          t
      1         −
F  t,    = 1 − e MTBF
   MTBF                                     t ≥0                                           (3)

The distribution function of Weibull distribution is



                                                                                            (4)

Where:




and:

    •   β is the shape parameter, also known as the Weibull slope
    •   η is the scale parameter
    •   γ is the location parameter

        The control limits for t-chart are defined in such a manner that the process is
considered to be out of control when the time to observe exactly one failure is less than
LCL, TL, or greater than upper control limit (UCL), TU. When the process is normal,
there is a chance for this to happen and it is commonly known as false alarm. The
traditional false alarm probability is set to be 0.27% although any other false alarm
probability ‘α ‘can be used.




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International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383(Print),
ISSN 0976 – 6391(Online) Volume 1, Number 1, July - Aug (2010), © IAEME




                              Figure 2 Normal Distribution Curve

        The actual acceptable false alarm probability should in fact depend on the actual
product or process. Assuming an acceptable probability of false alarm of α, the control
limits can be obtained as:
                                                         −1
      1                                     1 
              −1
                     1
TL =        ln                        TU =        ln (2 / α )
      MTBF     1 − α / 2 and                MTBF                                        (5)

        The median of the distribution is usually called the center line (CL), TC, and it can
be computed as
                 −1                        −1
      1                  1 
TC =        ln 2 = 0.693      
      MTBF               MTBF                                                           (6)

        These control limits can then be utilized to monitor the failure times of
components. After each failure, the time can be plotted on the chart. If the plotted point
falls between the calculated control limits, it indicates that the process is in the state of


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International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383(Print),
ISSN 0976 – 6391(Online) Volume 1, Number 1, July - Aug (2010), © IAEME


statistical control and no action is warranted. If the point falls above the UCL, it indicates
that the process average, or the failure occurrence rate, may have decreased which
resulted in an increase in the time between failures. This is an important indication of
possible process improvement. If this happens, the management should look for possible
causes for this improvement and if the causes are discovered then action should be taken
to maintain them. If the plotted point falls below the LCL, it indicates that the process
average, or the failure occurrence rate, may have increased which resulted in a decrease
in the failure time. This means that process may have deteriorated and thus actions should
be taken to identify and remove them.

        In either case, the people involved can know when the reliability of the system is
changed and by a proper follow up they can maintain and improve the reliability. Another
advantage of using the control chart is that it informs the maintenance crew when to leave
the process alone, thus saving time and resources. It can be noted here that the parameter
λ should normally be estimated with the data from the failure process.

2.2 tr Chart

        Monitoring the failure occurrence process using the t chart is straightforward.
However, since the decision is based on only one observation, it may cause many false
alarms or it is insensitive to process shift if the control limits are wide (with small value
of a ). To deal with this problem, we can monitor using the time between r failures.
Denote the time to observing the rth failure by tr .

        To monitor the system based on the time between the occurrences of r failures, we
need a distribution to model the cumulative time till the rth failure, tr : It is well known
that the sum of r exponentially distributed random variables is the Erlang distribution. An
Erlang random variable is defined as the length until the occurrence of r defects (failures)
in a Poisson process.

        Then the probability density function of Tr is given as:
                                 r
                          1  r −1
                                 tr
             1    MTBF               1  
f  tr , r , 
                    =
                                     exp−      t r 
             MTBF      (r − 1)!        MTBF                                          (7)


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International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383(Print),
ISSN 0976 – 6391(Online) Volume 1, Number 1, July - Aug (2010), © IAEME


        The cumulative Erlang distribution is

                                    1  k
                                  (      t r )
             1           r −1
                                    MTBF            1  
F  t r , r, 
                    = 1 − ∑
                                                exp−      t r 
             MTBF        k =0      k!             MTBF                               (8)

        To calculate the control limits of the tr-chart, the exact probability limits will be
used. If α is the accepted false alarm risk then the upper control limit, UCLr, the center
line, CLr, and the lower control limit, LCLr, can be easily calculated by using Eq. (6,7&8)
in the following manner:

                                     1           k
                                   (      UCLr )
              1          r −1
                                     MTBF               1         
  
F UCLr , r ,        = 1 − ∑
                                                    exp−      UCLr  = 1 − α / 2
              MTBF        k =0        k!              MTBF                           (9)
                                                    k
              1            r −1
                                   1             1          
F (CLr , r ,       ) = 1 − ∑   MTBF CLr  exp −  MTBF CLr  = 0.5
                                                                 
              MTBF         k =0                                                    (10)

                                      1            k
                              r −1          LCLr )
                                   (
               1                  MTBF                 1          
F  LCLr , r , 
                      = 1− ∑
                                                      exp −       LCLr  = α / 2
               MTBF       k =0         k!                MTBF                       (11)

        The decision-making procedure for the tr-chart remains same as the t-chart. A
point plotted below the LCL signifies deterioration of process and warrants corrective
action, while a point plotted above the UCL denoted process improvement and action
should be taken to identify the cause of improvement and to maintain it. From reliability
point of view, a point plotted below the control limit points out that the time between
failures may have decreased while a point plotted above the control limit indicates that
the time between failures may have increased.

3. ALGORITHM FOR CONTROL CHART DEVELOPMENT
Steps

    1. Draw Reliability Block Diagram (RBD)

    2. Measure MTBF of each component.




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International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383(Print),
ISSN 0976 – 6391(Online) Volume 1, Number 1, July - Aug (2010), © IAEME


    3. Calculate steady state availability of each component using the formula
                   MTBF
         Ai =
                MTBF + MTTR

                                                                               U i = 1 − Ai
    4. Calculate unavailability of each component using the formula

    5. Calculate percentage unavailability of each component.

    6. Identify the critical component – one which has the maximum% of unavailability.
        Control chart is to be prepared for the critical component.
                                                                                                  t
                                                         1         −
                                                   F  t,    = 1 − e MTBF
    7. Calculate LCL, CL and UCL for t chart using    MTBF 
                                                                                  −1
              1                                                        1 
                      −1
                           1
        TL =        ln                                           TU =        ln (2 / α )
              MTBF     1−α / 2                                         MTBF 

                                                  −1                    −1
                                 1                  1 
                           TC =        ln 2 = 0.693      
                                 MTBF               MTBF 

    8. Calculate         LCL,         CL               and   UCL        for     tr-chart       using
                                            1  k
                                          (      t r )
                     1          r −1
                                            MTBF            1  
        F  t r , r, 
                            = 1 − ∑
                                                        exp−      t r 
                     MTBF        k =0      k!             MTBF  
                                            1 
                                          (      UCLr )
                                                          k

                      1                r −1
                                            MTBF               1         
        F UCLr , r , 
                             = 1− ∑
                                                           exp−      UCLr  = 1 − α / 2
                      MTBF       k =0        k!              MTBF      
                                                             k
                      1            r −1
                                            1                  1        
        F (CLr , r ,       ) = 1 − ∑    MTBF CLr  exp −  MTBF CLr  = 0.5
                                                                             
                      MTBF         k =0                               
                                                1            k
                                          r −1        LCLr )
                                              (
                       1                    MTBF                1          
        F  LCLr , r , 
                              = 1− ∑
                                                                exp−       LCLr  = α / 2
                       MTBF          k =0        k!               MTBF       

    9. Mark LCL, CL and UCL and plot the time between failures on the chart.

4. MODEL APPLIED IN A PROCESS INDUSTRY
        Table 1 shows the mean time between failures of a critical component of a
chocolate manufacturing company. First 40 are the values are MTBF before modification


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International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383(Print),
ISSN 0976 – 6391(Online) Volume 1, Number 1, July - Aug (2010), © IAEME


and the second 40 values are that after modification. Figure 3, 4 & 5 shows the t – chart,
t2 – chart and t3 – chart respectively.

                                                         Table 1 Failure Data




          7000
          6000

          5000
          4000
   MTBF




                                                                                                                 Before Modification
          3000                                                                                                   After Modification

          2000

          1000
            0
                 1
                     3
                         5
                             7
                                 9
                                     11
                                          13
                                               15
                                                    17
                                                         19
                                                               21
                                                                    23
                                                                         25
                                                                              27
                                                                                   29
                                                                                        31
                                                                                             33
                                                                                                  35
                                                                                                       37
                                                                                                            39




                                                    Failure no.


                                                              Figure 3 t Chart




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International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383(Print),
ISSN 0976 – 6391(Online) Volume 1, Number 1, July - Aug (2010), © IAEME



          6000                                                                             LCL = 14
                                                                                            CL = 453
          5000                                                                             UCL = 2405

          4000
   MTBF




                                                                                                Before Modification
          3000
                                                                                                After Modification
          2000

          1000

            0
                 1

                     3

                             5


                                     7

                                         9

                                               11

                                                        13

                                                                 15

                                                                       17

                                                                                 19
                             Cumulative failure(2 failures) no.


                                             Figure 4 t2 Chart

                                                                                           LCL = 57.211
          6000
                                                                                            CL = 711.83
          5000                                                                             UCL = 2937.709

          4000
   MTBF




                                                                                                Before Modification
          3000
                                                                                                After Modification
          2000

          1000

            0
                 1

                     2

                         3

                                 4

                                     5

                                         6

                                              7

                                                    8

                                                             9

                                                                 10

                                                                      11

                                                                            12

                                                                                      13




                             Cumulative failure(3 failure) no.


                                             Figure 5 t3 Chart
5. CONCLUSIONS
           Until now, statistical control charts have been mostly used to monitor production
processes. Although reliability monitoring, especially that for complex equipment or fleet
of systems is an important subject. In this study, the focus was on the use of control
charting technique to monitor the failure of components. To deal with the problems
suffered by conventional quality control charts, the monitoring procedure specifically
based on exponential distribution was used. A new procedure based on the monitoring of
time to observe r failures, referred to as tr chart was proposed and it can be more
appropriate for reliability monitoring. The tr chart differs from the t chart in the sense that
it plots the cumulative time until observing r failures. This procedure is useful and more



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International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383(Print),
ISSN 0976 – 6391(Online) Volume 1, Number 1, July - Aug (2010), © IAEME


sensitive when compared with the t-chart although it will wait until r failures for a
decision. These charts can be regarded as powerful tools for reliability monitoring. tr
gives more accurate results than t – chart. The control chart for Weibull distribution can
also be established in the same way and can be used for monitoring the early life and
wear out life time of the process. Before conducting the modifications the changes can be
simulated to analyze the failure pattern so that appropriate decisions regarding plant
modifications can be made.
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 [4].   L.S.Srinath, “Reliability Engineering”, Affiliated East-West Press Pvt. Ltd., 1991.
 [5].   Dale H Besterfield “ Quality control” Prentice – Hall, Inc, 1990.
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 [8].   Shouri PV and Sreejith PS.Algorithm for Breakeven Reliability Allocation in
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International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383(Print),
ISSN 0976 – 6391(Online) Volume 1, Number 1, July - Aug (2010), © IAEME


[11].   Barringer PE. Reliability engineering principles. Barringer & Associates, Humble,
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        4; 1996.




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