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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 32 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 33 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 34 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 35 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. 36 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 37 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) 38 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. 39 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 40 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 41 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 42 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. REFERENCES [1]. Chan L.Y, Xie M, Goe TN. Cumulative control charts for monitoring production process. Int J Prod Res 2000; 38(2): 397-408. [2]. 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