Vehicular Fleet Reliability Estimation: A Case Study

Transcription

1 International Journal of Performability Engineering, Vol.5, No. 3, April, 2009, pp RAMS Consultants Printed in India Vehicular Fleet Reliability Estimation: A Case Study B. HARI PRASAD *, S. B. SINGH and M. S. BHAT School of Management Studies, Jawaharlal Nehru Technological University, Hyderabad (Received on May 26, 2008; Final Revision on November 1, 2008) Abstract: Mission reliability is the probability of successful completion for a stated mission by a group of people holding a set of a given population of equipment deployed in the prevailing operating environment. Thus, while reliability is considered for a single piece of equipment, mission reliability on the other hand is for a group of equipment having mixed vintage which could be deployed as a combined force for a mission. Public transport companies, goods transport organizations, police, paramilitary and military units maintain a large fleet of vehicles to carry out their stated missions. Estimation and prediction of mission reliability is of paramount importance for any operational success. For this purpose, reliability prediction of equipment is estimated from field failure data obtained from their history records. This paper discusses the technique of mission reliability prediction using the weibull probability plots of identified failure distribution. The identified failure distribution is validated using Chi-Square test. It is followed by identifying the weak links in the system using Pareto Analysis. Keywords: Weibull distribution, chi-square test, Pareto analysis, mission reliability 1. Introduction Success of an organisational mission depends upon a variety of factors. One of the most important factors is the preparedness of the machine or the weapon system to optimal performance level. History is full of the examples where failure of these machines has led to premature aborting of the missions besides loss of precious human lives. It is for this reason that the system hardware ought to have a very high degree of inherent and achieved reliability. It is a well-known fact that a hundred percent reliable machine is impossible to design and manufacture. Moreover, majority of the system hardware is of mechanical type, which deteriorates at varying rates with its usage, under different operating conditions. Thus, the system hardware is also bound to fail at some point of time. This failure however, must be avoided at all costs, especially during the execution of any mission. It is necessary to evaluate mission reliability of vehicles before launching it in an operation. A transport vehicle fleet commander is interested to know the mission reliability of his fleet of vehicles/weapon systems along with logistics specific requirements for its all time success. Estimation of mission reliability helps in not only evaluating the mission reliability of the fleet but also provides valuable inputs about weak link areas which can be improved upon by employing various maintenance techniques. Notation MTBF N mean time between failures total fleet of vehicle population *Corresponding author s hari_newdelhi@yahoo.com 235

2 236 B. Hari Prasad, S.B. Singh and M.S. Bhat n n s R(t) R(t+t m ) n i F i n t n f t t m η β γ χ 2 c i number of vehicles participating during the mission number of vehicles likely to survive during the mission reliability at the beginning of mission reliability at the end of mission number of failures cumulative number of failures theoretical frequency number of vehicles likely to fail during the mission mid class interval time in kms mission duration in kms weibull distribution location parameter weibull distribution shape parameter weibull distribution scale parameter chi-square intercept on y axis rank order 2. Mission Reliability The basic purpose of reliability prediction from field failure data is to provide valuable inputs for pushing the future missions towards success. The equipment mix deployed in a mission may have a homogeneous composition or a heterogeneous composition. Here the study is restricted to a fleet of vehicles having a homogeneous composition. However the individual vehicle participating in the mission may not have the same vintage. While the new vehicle will have low probability of failure others being old will have high rate of failure. The mission reliability is estimated from the ratio of total number of survivals after the mission to the total number of equipment fielded into the mission. This calculation is of importance as this data may be of use in planning future missions. However, in absence of such data, mission reliability and in turn the expected number of failures can be computed using the distribution parameters of the probability distribution that best fits the failure data of the equipment. 3. Estimation of Fleet Mission Reliability The probability of vehicles survival for the mission duration is given by conditional probability or reliability function as, R( t / t) = R( t t ) / R( t) (1) m + Assuming that the failures of the vehicles follows weibull distribution with η =0, β and γ, which are estimated from the plot on a standard graph. The slope of the line would yield the value of β and e -c/β will give the value of γ. Equation (1) then can be expressed as, β β R( t / t) = exp[ ( t + t ) / ν ] / exp[ t / ν ] (2) m m The equation (2) can be evaluated if t and the weibull parameters are known. The value of t m is known to the transport vehicle fleet commanders and the value of t can be obtained from the logbooks of the vehicles. Then n n R( t / t) = s i m f i s n s= = n n and Mission reliability = m 17 s= 1 n / N The suggested mission reliability model will be a decision support tool that could be applied by the vehicle fleet commanders to predict the mission reliability and anticipated s

3 Vehicular Fleet Reliability Estimation: A Case Study 237 failures due to technical reasons, prior to inducting them into a mission. This will further assist in deciding the quantum of the additional resources at the outset of a mission. 4. Case Study An example to illustrate the methodology described above is discussed below. The grouped data indicating the failure frequency in various kilometer-age intervals for a vehicle fleet is given in Table 1. The sorting and grouping of the raw data has been carried out as per Sturge s empirical formula. N in this case is Probability Plot to establish Failure Distribution The probability plot for weibull distribution has been used for finding the distribution parameters. The expected frequencies of failure are then calculated using the distribution parameters to estimate transport vehicle fleet mission reliability. Table 1 shows the calculation of the x and y co-ordinates from the grouped data. The value of i has been calculated using Bernard s empirical formula. The x and y co-ordinates, assuming η=0, have been calculated. S. No. Class Interval Table 1: Probability plot calculations t n i F i x = ln t i= n i /2+ F i-1 Q(t)= (i- 0.3)/(N+0.4) y = ln ln [1/(1-Q(t))] The Fig. 1 is plotted using the values of ln t and ln ln [1/(1-Q(t))] from Table 2 which is used for evaluating the values of β and γ for the weibull distribution. From the Fig. 1 β = 1.92, c = and γ is: γ = e -c/β = e -(-14.2)/1.92 = kms

4 238 B. Hari Prasad, S.B. Singh and M.S. Bhat ln ln 1/(1 - Q(t)) ln t Fig. 1: Probability Plot From Fig. 1, it can be seen that probability plot has given a very good fit. However the goodness of fit can only be ascertained using Chi-Square goodness of fit test. 4.2 Chi-Square Goodness of Fit Test The test has been conducted by evaluating the expected frequencies of failure of vehicle fleet using the values of β and γ at different values of t as shown in Table 2. Table 2: Chi-Square Test S No. t n i F(t) = 1 exp [-(t/ γ) β ] P i = F(t i ) - F(t i-1 ) n t = N P i χ 2 = (n i - n t ) 2 / n t Σχ 2 = 17.41

5 Vehicular Fleet Reliability Estimation: A Case Study 239 Table 3: Observed and Weibull Reliabilities S No. t Reliability Weibull/Estimated Kilometers Observed Fig. 2: Weibull & Observed Reliabilities Number of parameters estimated in the present study is two. Therefore, the number of degrees of freedom for the test is = 14.The critical value of χ 2 at 95% significance level and 14 degrees of freedom is 23.69, which is more than the calculated value. Hence it is concluded that weibull distribution fits the data. The MTBF and variance can be obtained using the following equations: MTBF = ν Γ[( 1/ β ) + 1] where, Γ value is taken from gamma function tables Variance = Weibull Observed Reliability Reliability ν Γ( 2/ β ) + 1] { νγ[(1/ β ) + 1]} Observed and estimated reliabilities are given in Table 3. Estimated reliabilities have been calculated using weibull distribution reliability expression R(t) = exp (t/γ) β, with parameters γ and β determined above and the Observed reliabilities have been calculated from grouped data by dividing the number of survivors from the total number of vehicles used in mission i.e The reliability values of weibull and Observed do not differ much as can be seen from the Table 3 and Fig Computation of Mission Reliability Let duration of the mission is 100, 150, 200, 250, 300, 350 and 400 kms and total number of serviceable vehicles at the beginning of the mission is 357. Table 4 shows the calculations carried out for finding the mission reliability for the duration 100 kms. 2

6 240 B. Hari Prasad, S.B. Singh and M.S. Bhat Table 4: Probable number of failures for 100 kms mission S No. Class Interval t n i R(t) R (t + tm) R(tm/t) n s n f Similarly the percentage mission reliability for mission duration of 150, 200, 250, 300, 350 and 400 kms are computed and are give below in Table 5. Table 5: Mission duration and % Mission Reliability S No. Mission duration No. of failures % mission reliability of the fleet From the Table 5 it is observed that the mission reliability is not sufficient for present day operations where mission is likely to be more than 400 kms. Hence there is a requirement to increase mission reliability and some measures needs to be initiated for improving the reliability of the transport vehicle fleet participating in the mission. 5. Pareto Analysis Total number of defects of the transport vehicle system-wise has been established from the data collected. The system wise defects are tabulated in the decreasing order. Systemwise defects, their frequency and relative frequency percentage has been worked out in Table 6.

7 Vehicular Fleet Reliability Estimation: A Case Study 241 S No. System Table 6: Pareto Analysis No. of failures Cumulative failures % failures % Cumulative failures 1 Pneumatic System Running Gear Electrical System Fuel system Engine Instrument system Cooling system Transmission system Lubrication system Chassis Engine Exhaust system Engine Induction system Suspension system The Pareto analysis has revealed that the first three systems contribute for about 66% of the total defects of vehicle. We have thus been able to separate Vital few from Trivial many. It is relevant for us to pay particular attention to these three systems, then paying equal attention to all the systems. 6. Enhancing Mission Reliability It is clear from above results, that the suggested mission reliability of % for 400 kms is very low which needs to be improved. Pareto analysis is used to identify the systems which are contributing to maximum number of failures. By studying these systems in detail, using Reliability Logic Diagram, Fault Tree Analysis and Failure Mode Effect & Criticality Analysis we can identify the weak links in these systems. By applying latest maintenance techniques such as TPM and CBM to the identified weak links the failure rates of these systems can be drastically reduced. This will enhance the reliability of vehicle substantially, which in turn will improve the mission reliability. 7. Conclusion The approach to mission reliability estimation and prediction allows us to assess the fleet reliability of transport vehicles during in-service phase of equipment life cycle. This approach provides the structure for capturing and evaluating the mission success explicitly. It helps to integrate the field data with the existing engineering knowledge/ experience so that transport vehicle fleet mission reliability can be predicted. This structured and disciplined approach will help the fleet commanders to decide as to what extent the mission objectives can be obtained. It also helps in identifying the critical bottlenecks which need their focus to resolve. Finally with this approach, communication amongst all units and personnel can be improved. In the present era of decreasing budget and personnel assets, this approach can be used as a crucial management tool and support system to provide us the cutting edge advantage over our rivals.

8 242 B. Hari Prasad, S.B. Singh and M.S. Bhat References [1] Aggarwal, K. K., Reliability Engineering, Kluwer Academic Publishers, Netherlands, [2} Bossche, A, et. al., Fault propagation models for reliability assessment of production systems, SRE symposium, Reliability: A Competitive Edge, Arnhem, Netherlands, pp , [3] Ebeling, C. E., Introduction to Reliability and Maintainability Engineering, Tata Mc Graw Hill, New Delhi, [4] Garni, A. Z. A., et. al., Reliability analysis of aeroplane brakes, Quality and Reliability Engineering International, Vol. 15, pp , [5] Ghose, M.K., and S. Rajagopalan, Bayesian assessment for discrete data : A Case Study, Reliability Engineering, Vol. 11, pp , [6] Kocza, G., and A. Bossche, Application of the integrated reliability analysis system, Reliability Engineering and System Safety, Vol. 64, pp , [7] Kumar, K., Effect of ranking on reliability prediction using Weibull Analysis, Reliability Engineering and System Safety, Vol. 24(3), pp , [8] Meyers, R, and L. Stern, Criteria for identification of a probability distribution based on analysis of life test data, Convention Trans., pp , [9] Misra, K. B., Reliability Analysis and Prediction, Elsevier Science Publishers, Netherlands, [10] Patterson, A., and D. L. Iverson, An integrated approach to system design, reliability and diagnosis, NASA Technical memorandum, [11] Prasad, M., Reliability estimation and RCM application to tracked combat vehicle using field data, M. Tech Thesis, IIT, Delhi, [12] Sols, A., and J. A. Nachlas, Availability of multifunctional systems, Reliability Engineering and System Safety, Vol. 47, pp , [13] Young, S. S., et.al., Random effects for the reliability management of modules of a fighter aircraft, Reliability Engineering and System Safety, Vol. 91, pp , B. Hari Prasad is pursuing his Ph.D. in Reliability Engineering from Jawaharlal Nehru Technological University (JNTU), Hyderabad, India. He received his B. Tech. from Punjab Engineering College, Chandigarh and M.Tech. from Indian Institute of Technology (IIT) Delhi, India. Working under the supervision of Prof. Prem Vrat, his M. Tech. project was adjudged as the best. The officer is serving in Defence Forces since last 22 years. S. B. Singh is the Head of the Department of Industrial Engineering at Military College of Electronics and Mechanical Engineers, Secunderabad. He has done his M. Tech. and Ph.D. from IIT Kharagpur and Delhi respectively. The officer has 27 years of industrial, training and administrative experience. The officer has published several papers in national and international journals. M. S. Bhat is the Chairman, Board of Studies and Board of Management Studies of JNTU, Hyderabad. He has more than 50 research articles to his credit and co-author of two books. He did his Ph.D. from Osmania University, Hyderabad and M. Tech from JNTU. He has been Vice Chairman of Institution of Electronics and Telecommunication Engineers, Hyderabad Centre, India. He is on the panel of members of expert committee of All India Council for Technical Education and served as examiner of universities, Union Public Service Commission, Andhra Pradesh Public Service Commission and University Grant Commission.