Behavioral study of piston manufacturing plant through stochastic models

Lal, Arvind K.; Kaur, Manwinder; Lata, Sneh Article Behavioral study of piston manufacturing plant through stochastic models Journal of Industrial Engineering International Provided in Cooperation with: Islamic Azad University (IAU), Tehran Suggested Citation: Lal, Arvind K.; Kaur, Manwinder; Lata, Sneh (2013) : Behavioral study of piston manufacturing plant through stochastic models, Journal of Industrial Engineering International, ISSN 2251-712X, Springer, Heidelberg, Vol. 9, pp. 1-10, https://doi.org/10.1186/2251-712X-9-24 This Version is available at: https://hdl.handle.net/10419/147165 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/2.0/ CASE STUDY Open Access Behavioral study of piston manufacturing plant through stochastic models Arvind K Lal 1 , Manwinder Kaur 1* and Sneh Lata 2 Abstract Piston plays a vital role in almost all types of vehicles. The present study discusses the behavioral study of a piston manufacturing plant. Manufacturing plants are complex repairable systems and therefore, it is difficult to evaluate the performance of a piston manufacturing plant using stochastic models. The stochastic model is an efficient performance evaluator for repairable systems. In this paper, two stochastic models and computation algorithm of the piston manufacturing plant are illustrated using the state-space transition diagram and availability parameter is used for its behavioral study. Finally, the conclusion is discussed based on the resulting computations. Keywords: Piston manufacturing plant; Behavioral study; Stochastic model; Time-dependent availability Introduction Piston is a cylindrical shaped component tightly fitted within another cylinder. Piston plays an important role in all vehicles and thus, the performance of the piston manufacturing process is equally important as that of piston quality and needs to be evaluated. The present study discusses the behavioral study of a piston manufacturing plant. Manufacturing plants are complex repairable systems, and therefore, it is difficult to evaluate their performance metric including reliability and availability. The stochastic process is an efficient performance evaluator model for repairable systems. Several authors have considered such models to analyze the reliability and availability of manufacturing plants such as plastic manufacturing plant (Gupta et al. 2007), cement industries (Gupta et al. 2005b), butter oil manufacturing plant (Gupta et al. 2005c), and thermal power plant (Gupta and Tiwari 2009) for performance evaluation. Gupta et al. (2005a) discussed the performance of the polymer powder production process using the supplementary variable method. In some of above studies, analytical approaches like Laplace transforms (Gupta 2003), matrix method (Gupta et al. 2007), and Lagrange’s method (Mahajan and Singh 1999) are used to discuss transient state of the resulting mathematical expression in order to study the performance metrics including reliability and time-dependent availability. However, the computational part and analysis have not been effectively performed using these analytic methods and thus thesewerenotusedintheanalysisofsteadystatebehavior of the process in these studies. Further, various other techniques such as genetic algorithm (Kumar et al. 2010), GABLT (Sharma and Kumar 2010), and stochastic reward petri nets (Sachdeva et al. 2009) have also been used to analyze the steady state behavior of the systems. However, these analytic methods are not useful to obtain the solution of the transient state of complex manufacturing system. Therefore, we have used a numerical method to solve such a complex mathematical problem relating to the piston manufacturing plant. In this paper, two stochastic models and computation algorithm of the piston manufacturing plant are illustrated using the state transition diagram. The objective of this study is to help the industry management by developing a mathematical model for piston manufacturing plant to evaluate its performance results based on availability. In the subsequent section, we will discuss the piston manufacturing process and its further division into subsystems in order to develop the stochastic models. * Correspondence: [email protected] 1 School of Mathematics and Computer Applications, Thapar University, Patiala, PB 147004, India Full list of author information is available at the end of the article © Lal et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Lal et al. Journal of Industrial Engineering International 2013 2013, 9:24 http://www.jiei-tsb.com/content/9/1/24 The process and its systems and subsystems In the piston manufacturing plant, a gravity die casting process is carried out in the foundry to make the piston with the help of semiautomatic die machines without external pressures. The resultant pistons are sent for cleaning up the surface of the piston from the risers and runners and later, the strength, hardness, and other foundry defects are evaluated. Then a bunch of pistons is placed in the piston blank for its machining operations after which a fixture seat machine is used. Then, two rough grooves are made on the piston by using a carbide tool tip to fit a pin in them. Subsequently, another hole is drilled on the piston for oil passage and then finishing is given to the rough grooves using a diamond tool. Later, the piston is shaped into an oval to ensure its smooth movement within the cylinder using the carbide tool. In the next process, finish crown and cavity machines are used to put a rough cavity first and then a finish cavity on the piston. Further, valves are made on the piston using the diamond tool followed by a rounding-off operation on the corners of piston. Then, circlip grooves are made on the piston in the absence of any coolant and air pressure. After that, deburring, cleaning, and surface treatment operations are performed. Final inspections are carried out on the manufactured piston under some operational tests before packing. In order to reduce the complexity for the effective behavioral (or performance) analysis, the piston manufacturing process is categorized into two systems namely, S1 and S2. The system which formed the piston into oval shape is denoted by S1 and the system which produces the final product as piston is denoted by S2. The flow chart of the piston manufacturing process is given in Figure 1. Further, system S1 is divided into six subsystems, namely, A, B, C, D, E, and F. The brief description of these subsystems is as follows: 1. Subsystem A is a fixture seat machining operation which is performed for clamping of pistons. 2. Subsystem B is a machining operation of rough grooving and turning operation. 3. Subsystem C is rough pin hole boring machine used to make pin holes on the piston to give proper size to the holes, which are used to fit the pin by which a rod is connected to crankshaft. 4. An oil hole is drilled on the piston for oil passage and the operating machine; oil hole drilling is considered as sub-system D. 5. Subsystem E is the finishing grooving machine used to give finishing to the rough grooves. 6. Subsystem F is the finish profile turning operation in which the piston is formed into oval shape in order to overcome expansion problems during working of the piston at high temperatures. Similarly, system S2 is divided into nine subsystems namely G, H, I, J, K, L, M, N, and O which are described as follows: 1. Subsystem G is a finish pin hole boring machine. 2. Finish crown and cavity is subsystem H, which is operated to give finishing to the crown of piston, i.e., the upper part of the piston. Figure 1 Flow chart of piston manufacturing plant. Lal et al. Journal of Industrial Engineering International Page 2 of 10 2013, 9:24 http://www.jiei-tsb.com/content/9/1/24 3. Subsystem I is a valve milling machine to make the valve recession of the piston. 4. Subsystem J is the chamfering or radiusing machine used to round off the corners of the piston to ensure smooth run in the cylinder. 5. Circlip grooves are made on the piston through subsystem K, a circlip grooving machine. 6. Subsystem L is the deburring machine. The deburring or brushing operation is performed on the piston using this machine. 7. Subsystem M is the cleaning machine which helps to clean the inside and outside of the piston. 8. Subsystem N is the surface treatment operation to coat the piston with some mixture. 9. Subsystem O is the final inspection of the manufactured product though operational tests before being packed. In addition to above discussion, we have considered two kinds of failures for each subsystem, which are major and minor. Major failures are those in which the system shows complete breakdown while the minor failures are those which can be repaired during working conditions (reduced state). Now, we proceed with each system description and its subsystem. The subsystems A, B, C, D, F, F, I, J, and M are subject to major failure while C,E,G,andKaresubjecttoboth major and minor failures. Subsystems L, N, and O are considered to have no failure. Next, the state space of stochastic process for both systems is described in a diagrammatic form, known as transition diagram. Transition diagrams Transition diagrams for system S1 and S2 are shown in Figures 1 and 2, respectively. In the transition diagrams, we consider system S1 or system S2 in good state when all of its subsystems show good conditions, in reduced state when any of its subsystem is in reduced state, and in failed state when any of its subsystem fails. The corresponding subsystem state notations are described in the Section ‘Notations’. Notations In this section, the notations are presented as follows: 1. The good condition of each subsystem is represented by capital letters, as A, B, C, D, E, and F for subsystems in S1 and G, H, I, J, K, and M for subsystems in S2, respectively. 2. Lowercase letters, including a, b, c, d, e, and f and g, h, i, j, k, and m, represent the failed state of subsystems A, B, C, D, E, and F, and G, H, I, J, K, and M, respectively. 3. The reduced working state of subsystems C, E, and G is indicated by  C,  E, and  G, respectively. 4. The constant failure rates of subsystems A, B, D, F,  C, E, C, and E are represented by α i (i= 1 to 8), respectively, in Figure 2. This parameter α i (i=1to 7) also represents the respective constant failure rates of subsystems H, I, J, K, M,  G, and G in Figure 3. 5. The parameter β i represents the respective constant repair rates of subsystems A, B, D, F,  C; E, C, and E, for i= 1 to 8 in Figure 2. In Figure 3,it Figure 2 State transition diagram for system S 1 .The constant failure rates of subsystems A, B, D, F,  C,  E, C, and E are represented by α i (i= 1 to 8), respectively. The parameter β i represents the respective constant repair rates of subsystems A, B, D, F,  C,  E, C, and E, for i= 1 to 8. Lowercase letters, including a, b, c, d, e, and f represent the failed state of subsystems A, B, C, D, E, and F respectively. Lal et al. Journal of Industrial Engineering International Page 3 of 10 2013, 9:24 http://www.jiei-tsb.com/content/9/1/24 represents respective constant repair rates of subsystems H, I, J, K, M,  G, and G for i= 1 to 7. 6. Symbol P j (t) represents the probability that the system is in jth state at time t(j= 1 to 24 for system S1 and j=1 to 13 for system S2). Symbol P j '(t) represent the rate of change of probability of jth state with respect to time t. (j= 1 to 24 for system S1 and j= 1 to 13 for system S2). Assumptions The following assumptions are used to model the performance analysis of the process for both systems S1 and S2: 1. Failure and repair rates are independent with each other and their unit is per hour. 2. There are no simultaneous failures among the subsystems. 3. Subsystems C, E, and G fail only through reduced states. 4. Repair of C, E, and G in the reduced state does not make these subsystems completely functional. 5. Switching to standby components is perfect. 6. Subsystems L, N, and O are considered as subsystems that never failed. Stochastic models and computation algorithm of piston manufacturing plant Systems S1 and S2 have been formulated by following the mnemonic rule discussed in the work of (Gupta et al. 2005a, b) using transition diagrams. The resulting ChapmanKolmogorov first-order differential-difference equations for both systems under probabilistic considerations of each state discussed in state transition diagrams (Figures 2 and 3) are obtained as shown in the succeeding sections. Mathematical formulation of system S 1 In this section, the mathematical formulation of system S1 is performed, with the help of transition diagram shown in Figure 2, as follows: P0 1tðÞþλ1P1tðÞ¼X 4 i¼1 βiP4þitðÞþX 3 i¼2 β3þiPitðÞ þX 8 i¼7 βiP10þitðÞ;ð1Þ P0 2tðÞþλ2P2tðÞ¼X 4 i¼1 βiP8þitðÞþβ8P20 tðÞþα5P1tðÞ;ð2Þ P0 3tðÞþλ3P3tðÞ¼X 4 i¼1 βiP12þitðÞþβ7P19 tðÞþα6P1tðÞ;ð3Þ P0 4tðÞþλ4P4tðÞ¼X 4 i¼1 βiP20þitðÞþα5P3tðÞþα6P2tðÞ;ð4Þ P0 4þitðÞþβiP4þitðÞ¼αiP1tðÞ;i¼1;2;3;4ð5Þ P0 8þitðÞþβiP8þitðÞ¼αiP2tðÞ ð6Þ P0 12þitðÞþβiP12þitðÞ¼αiP3tðÞ ð7Þ P0 16þitðÞþβ7P16þitðÞ¼α7PjtðÞ;i¼1;2;j¼2;4ð8Þ Figure 3 State transition diagram for system S 2 .Parameter α i (i= 1 to 7) represents the respective constant failure rates of subsystems H, I, J, K, M,  G, and G. Parameter β i represents the respective constant repair rates of subsystems H, I, J, K, M,  G, and G for i= 1 to 7. Lowercase letters, including g, h, i, j, k, and m, represent the failed state of subsystems G, H, I, J, K, and M, respectively. Lal et al. Journal of Industrial Engineering International Page 4 of 10 2013, 9:24 http://www.jiei-tsb.com/content/9/1/24 P0 18þitðÞþβ8P18þitðÞ¼α8PjtðÞ;i¼1;2j¼3;4ð9Þ P0 20þitðÞþβiP20þitðÞ¼αiP4tðÞ;i¼1;23;4;ð10Þ where λ1¼α1þα2þα3þα4þα5þα6ð11Þ λ2¼α1þα2þα3þα4þα6þα7þβ5ð12Þ λ3¼α1þα2þα3þα4þα5þα8þβ6ð13Þ λ4¼α1þα2þα3þα4þα7þα8ð14Þ with initial conditions P 1 (0) = 1 and 0, otherwise. The time-dependent availability A 1 (t) of the system (S1) is A1tðÞ¼X 4 i¼1 PitðÞ:ð15Þ Mathematical formulation of system S2 In this section, the mathematical formulation of system S2 is performed, with the help of transition diagram shown in Figure 3, as follows: P0 1tðÞþλ1P1tðÞ¼X 5 i¼1 βiP2þitðÞþβ6P2tðÞ þβ7P13 tðÞ ð16Þ P0 2tðÞþλ2P2tðÞ¼X 5 i¼1 βiP7þitðÞþα6P1tðÞ ð17Þ P0 2þitðÞþβiP2þitðÞ¼αiPjtðÞ;ð18Þ where j¼1 fori¼1;2;…;5 2fori¼6;7;…;13 ;λ1¼α1þα2þα3þα4þα5þα6  ð19Þ λ2¼α1þα2þα3þα4þα5þα7þβ6ð20Þ with initial conditions P10ðÞ¼1 andPi0ðÞ¼0;i¼2−15:ð21Þ The time-dependent availability A 2 (t) of the system (S2) is A2tðÞ¼X 2 i¼1 PitðÞ:ð22Þ System S1 completes its target only when system S2 starts, with the output of the previous system for further operation. Thus, S1 and S2 are in series which helps us to compute the time-dependent availability of the process industry as APtðÞ¼A1tðÞA2tðÞ:ð23Þ AsthesystemofdifferentialEquations(1,2,3,4,5, 6, 7, 8, 9, and 10 and 16, 17, and 18) is very complex, it is difficult to find its analytical solution, particularly using Laplace transformation method. Therefore, a numerical procedure is used to obtain the computation for the availability of the process, which is discussed next. Computational algorithm In order to obtain the availability parameter of the process industry, the results are approximated for the following transient probabilities: p1itðÞ¼PX tðÞ¼iX 0ðÞ¼1ð24Þ j ½ using initial distribution P10ðÞ¼1;Pj0ðÞ¼0;forj¼2;…;24 ð25Þ to study the effect of failure and repair rate variations on the plant performance. Also, as the transition matrix changes with respect to the change in transition rates, the complete transition matrix data for this case study was, thus, not evaluated for both the systems under different failure and repair rate variations. The algorithm for solving the system of equations (1,2,3,4,5,6,7,8,9,and10)thatcorrespondstosystem S1 is as follows: 1. Input initial time, step size, transition rates (i.e., failure and repair rates) of each state. 2. Set initial distribution P 1 (0) = 1, P j (0) = 0, for j=2,…, 24. 3. Apply Runge–Kutta fourth-order method on differential equations (1,2,3,4,5,6,7,8,9, and 10) to obtain probabilities p 1i (t), p 12 (t), …,p 1n (t), n= 24. 4. Compute time-dependent availability using Equation 15. Further, in order to compute the second row of transition matrix, replace step 2 with initial distribution, P 2 (0) = 1, P j (0) = 0, for j=1,3,…, 24 for computing the following transition probabilities: p2itðÞ¼PX tðÞ¼iX 0ðÞ¼2 j ½ð26Þ and proceed this way to analyze the results to compute each row of transition matrix. Similarly, results for S2 systems were computed. We have performed 72,000 iterations on the system of differential-difference equations (1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 and 16, 17, and 18) together with conditions (Equations 14 and 21) to solve them using the above algorithm, taking step size t= 0.005 with the assumption Lal et al. Journal of Industrial Engineering International Page 5 of 10 2013, 9:24 http://www.jiei-tsb.com/content/9/1/24 that it equals to 1 h, considering the fact discussed (Gupta et al. 2007). This leads us to computations of time-dependent availability from 30 to 360 h. The resulting computations are shown in Table 1. The availability results are also simulated for various fluctuations of repair as well as failure rates of each subsystem (Tables 2,3,4,5,6,7,8,9,10,11,12,13). The simulated data for various fluctuating environment of failure and repair rates can be referred from Tables 2,3,4,5,6,7 for system S1 and from Tables 8,9,10,11,12,13 for system S2. Based on these results, the behavior study of both the system is discussed next. Behavior study of both systems Results are obtained for the overall time-dependent availability of the process, using Equation 23, with empirical data collected from industry. The data has been taken from industry in the form of operating hours, number of failures, number of repairs, and corresponding time for each repair for all the subsystems of the process industry. This data is composed into parameterized form of exponentially distributed failure rate and repair rates and given as α 1 = 0.0014, α 2 = 0.0006, α 3 = 0.0009, α 4 = 0.0009, α 5 = 0.0208, α 6 = 0.0208, α 7 = 0.0009, and α 8 =0.0039andβ 1 =1.08,β 2 = 0.043, β 3 =0.5,β 4 =0.286,β 5 =0.154,β 6 =0.25,β 7 = 0.059 and β 8 = 0.087 for system S1, respectively; and α 1 = 0.0014, α 2 = 0.0003, α 3 = 0.0001, α 4 = 0.0003, α 5 = 0.0001, α 6 = 0.0208, and α 7 =0.004andβ 1 = 0.33, β 2 =0.5, β 3 =0.67,β 4 =0.035,β 5 =3.03,β 6 =0.222,andβ 7 =0.125 for system S2, respectively. The computations of overall time-dependent availability of the process industryareshowninTable1. The tabular form of computations helps us to conclude that the process is 96% reliable for operation at 30 h. Further, we found that the time-dependent availability of system S1 is more affected by fixture seat machining (A) in comparison to the effect of other systems as observed from the simulated results shown in Tables 2,3,4,5,6,7. The time-dependent availability of the system S1 decreases by 1.02% from the empirical results (shown in Table 1), while it decreases by approximately 0.4% with the increase in time from 30 to 360 h. However, other subsystems’failure rate affects the system availability by 0.2% (for rough grooving cum turning and oil hole drilling), 0.06% (rough pin hole boring), 0.01% (for finish grooving), and 0.03% (finish profile turning) in comparison to fixture seat Table 1 Time-dependent availability of the systems S 1 and S 2 and process industry Value of parameters described in first column of this table Time (h) 30 60 90 120 150 180 210 240 270 300 330 360 A 1 (t) 0.9777 0.9705 0.9663 0.9634 0.9613 0.9596 0.9582 0.9572 0.9564 0.9557 0.9552 0.9548 A 2 (t) 0.9914 0.9893 0.9887 0.9885 0.9884 0.9883 0.9883 0.9883 0.9883 0.9883 0.9883 0.9883 A p (t) 0.9692 0.9601 0.9554 0.9523 0.9501 0.9484 0.9471 0.9460 0.9452 0.9446 0.9440 0.9436 Table 2 Fixture seat machining’s failure (α 1 ) and repair rate (β 1 ) effect on time-dependent availability of S 1 Time in hours α 1 value β 1 value 0.0054 0.0104 0.154 0.0204 1.08 1.1800 1.2800 1.3800 30 0.9742 0.9698 0.9655 0.9612 0.9777 0.9778 0.9779 0.9780 60 0.9670 0.9627 0.9585 0.9543 0.9705 0.9706 0.9707 0.9707 90 0.9629 0.9587 0.9545 0.9503 0.9663 0.9664 0.9665 0.9666 120 0.9600 0.9558 0.9517 0.9475 0.9634 0.9635 0.9636 0.9637 150 0.9579 0.9537 0.9495 0.9454 0.9613 0.9614 0.9615 0.9615 180 0.9562 0.9520 0.9479 0.9438 0.9596 0.9597 0.9598 0.9598 210 0.9549 0.9507 0.9466 0.9425 0.9582 0.9583 0.9584 0.9585 240 0.9538 0.9497 0.9456 0.9415 0.9572 0.9573 0.9574 0.9575 270 0.9530 0.9489 0.9447 0.9406 0.9564 0.9565 0.9566 0.9566 300 0.9524 0.9482 0.9441 0.9400 0.9557 0.9558 0.9559 0.9560 330 0.9518 0.9477 0.9436 0.9395 0.9552 0.9553 0.9554 0.9554 360 0.9514 0.9473 0.9432 0.9391 0.9548 0.9549 0.9550 0.9550 Values of other parameters taken, in respect to varying values of α 1 of S1 are as follows: α 2 =0.0006,α 3 =0.0009,α 4 =0.0009,α 5 = 0.0208, α 6 = 0.0208, α 7 = 0.0009, α 8 = 0.0039; β 1 =1.08,>β 2 = 0.043, β 3 =0.5,β 4 = 0.286, β 5 = 0.154, β 6 =0.25,β 7 = 0.059, β 8 = 0.087, respectively. Values of other parameters taken, in respect to varying values of β 1 of S1 are as follows: α 1 =0.0014,α 2 =0.0006,α 3 =0.0009,α 4 =0.0009,α 5 = 0.0208, α 6 = 0.0208, α 7 = 0.0009, α 8 = 0.0039; β 2 =0.043,β 3 =0.5,β 4 = 0.286, β 5 = 0.154, β 6 =0.25,β 7 = 0.059, β 8 =0.087,respectively. Lal et al. Journal of Industrial Engineering International Page 6 of 10 2013, 9:24 http://www.jiei-tsb.com/content/9/1/24 machining when compared to empirical results of system S1 availability, A 1 (Table 1). Similarly, repair rate fluctuations have been studied for the effect of maintenance on time-dependent availability of the system. We found that the time-dependent availability of system S1 shows 0.001% increase for fixture seat machining and rough pin hole boring, and for other subsystems, time-dependent availability shows 0.01% (for rough grooving cum tuning and finish profile turning), 0.02% (for rough oil drilling), and 0.025% Table 3 Rough grooving and turning’sfailure(α 2 )and repair rate (β 2 ) effect on time-dependent availability of S1 Time in hours α 2 value β 2 value 0.0007 0.0008 0.0009 0.0010 0.044 0.045 0.046 0.047 30 0.9761 0.9744 0.9728 0.9712 0.9778 0.9779 0.9780 0.9781 60 0.9684 0.9664 0.9644 0.9624 0.9707 0.9709 0.9711 0.9713 90 0.9642 0.9621 0.9600 0.9579 0.9666 0.9669 0.9671 0.9673 120 0.9613 0.9592 0.9571 0.9550 0.9637 0.9640 0.9643 0.9645 150 0.9591 0.9570 0.9549 0.9528 0.9616 0.9618 0.9621 0.9624 180 0.9575 0.9553 0.9532 0.9511 0.9599 0.9601 0.9604 0.9607 210 0.9561 0.9540 0.9519 0.9498 0.9585 0.9588 0.9591 0.9593 240 0.9551 0.9530 0.9509 0.9488 0.9575 0.9578 0.9580 0.9583 270 0.9543 0.9522 0.9501 0.9480 0.9567 0.9569 0.9572 0.9574 300 0.9536 0.9515 0.9494 0.9473 0.9560 0.9563 0.9565 0.9568 330 0.9531 0.9510 0.9489 0.9468 0.9555 0.9558 0.9560 0.9563 360 0.9527 0.9506 0.9485 0.9464 0.9551 0.9553 0.9556 0.9559 Values of other parameters taken, in respect to varying values of α 2 of S1 are as follows: α 1 =0.0014,α 3 = 0.0009, α 4 = 0.0009, α 5 = 0.0208, α 6 =0.0208,α 7 = 0.0009, α 8 =0.0039;β 1 =1.08,β 2 = 0.043, β 3 = 0.5, β 4 = 0.286, β 5 =0.154,β 6 = 0.25, β 7 = 0.059, β 8 = 0.087, respectively. Values of other parameters taken, in respect to varying values of β 2 of S1 are as follows: α 1 = 0.0014, α 2 =0.0006,α 3 = 0.0009, α 4 =0.0009,α 5 =0.0208,α 6 = 0.0208, α 7 =0.0009,α 8 = 0.0039; β 1 = 1.08, β 3 = 0.5, β 4 = 0.286, β 5 =0.154,β 6 =0.25,β 7 = 0.059, β 8 =0.087,respectively. Table 4 Rough pin hole boring failure rate (α 7 ) and repair rate (β 7 ) effect on time-dependent availability of S1 Time in hours α 7 value β 7 value 0.0010 0.0011 0.0012 0.0013 0.060 0.061 0.062 0.063 30 0.9775 0.9773 0.9771 0.9770 0.9777 0.9777 0.9777 0.9777 60 0.9702 0.9699 0.9696 0.9693 0.9705 0.9705 0.9706 0.9706 90 0.9659 0.9656 0.9652 0.9648 0.9664 0.9664 0.9665 0.9665 120 0.9630 0.9626 0.9621 0.9617 0.9635 0.9636 0.9636 0.9637 150 0.9608 0.9603 0.9599 0.9594 0.9613 0.9614 0.9615 0.9616 180 0.9591 0.9586 0.9581 0.9576 0.9597 0.9597 0.9598 0.9599 210 0.9577 0.9572 0.9567 0.9562 0.9583 0.9584 0.9585 0.9586 240 0.9567 0.9562 0.9557 0.9552 0.9573 0.9574 0.9575 0.9576 270 0.9558 0.9553 0.9548 0.9543 0.9565 0.9566 0.9567 0.9568 300 0.9552 0.9547 0.9542 0.9536 0.9558 0.9559 0.9560 0.9561 330 0.9547 0.9541 0.9536 0.9531 0.9553 0.9554 0.9555 0.9556 360 0.9543 0.9537 0.9532 0.9527 0.9549 0.9550 0.9551 0.9552 Values of other parameters taken, in respect to varying values of α 7 of S1 are as follows: α 1 =0.0014,α 2 = 0.0006, α 3 = 0.0009, α 4 = 0.0009, α 5 =0.0208,α 6 = 0.0208, α 8 =0.0039;β 1 =1.08,β 2 = 0.043, β 3 = 0.5, β 4 = 0.286, β 5 =0.154,β 6 = 0.25, β 7 = 0.059, β 8 = 0.087, respectively. Values of other parameters taken, in respect to varying values of β 7 of S1 are as follows: α 1 = 0.0014, α 2 =0.0006,α 3 = 0.0009, α 4 =0.0009,α 5 =0.0208,α 6 = 0.0208, α 7 =0.0009,α 8 = 0.0039; β 1 = 1.08, β 2 =0.043, β 3 = 0.5, β 4 = 0.286, β 5 = 0.154, β 6 =0.25,β 8 = 0.087, respectively. Table 5 Oil hole drilling’s failure (α 3 ) and repair (β 3 ) rate on time-dependent availability of S1 Time in hours α 3 value β 3 value 0.0010 0.0011 0.0012 0.0013 0.6 0.7 0.8 0.9 30 0.9775 0.9773 0.9771 0.9769 0.9780 0.9782 0.9783 0.9784 60 0.9703 0.9701 0.9699 0.9697 0.9708 0.9710 0.9711 0.9712 90 0.9661 0.9660 0.9658 0.9656 0.9666 0.9668 0.9670 0.9671 120 0.9633 0.9631 0.9629 0.9627 0.9637 0.9639 0.9641 0.9642 150 0.9611 0.9609 0.9607 0.9605 0.9615 0.9617 0.9619 0.9620 180 0.9594 0.9592 0.9590 0.9588 0.9599 0.9600 0.9602 0.9603 210 0.9581 0.9579 0.9577 0.9575 0.9585 0.9587 0.9589 0.9590 240 0.9570 0.9568 0.9567 0.9565 0.9575 0.9577 0.9578 0.9579 270 0.9562 0.9560 0.9558 0.9556 0.9566 0.9568 0.9570 0.9571 300 0.9555 0.9553 0.9552 0.9550 0.9560 0.9562 0.9563 0.9564 330 0.9550 0.9548 0.9546 0.9545 0.9555 0.9557 0.9558 0.9559 360 0.9546 0.9544 0.9542 0.9541 0.9551 0.9552 0.9554 0.9555 Values of other parameters taken, in respect to varying values of α 3 of S1 are as follows: α 1 =0.0014,α 2 = 0.0006, α 4 = 0.0009, α 5 = 0.0208, α 6 =0.0208,α 7 = 0.0009 α 8 =0.0039;β 1 =1.08,β 2 = 0.043, β 3 = 0.5, β 4 = 0.286, β 5 = 0.154, β 6 = 0.25, β 7 = 0.059, β 8 = 0.087, respectively. Values of other parameters taken, in respect to varying values of β 3 of S1 are as follows: α 1 = 0.0014, α 2 =0.0006,α 3 = 0.0009, α 4 =0.0009,α 5 =0.0208,α 6 = 0.0208, α 7 = 0.0009 α 8 = 0.0039; β 1 =1.08, β 2 = 0.043, β 4 =0.286,β 5 =0.154,β 6 = 0.25, β 7 =0.059,β 8 =0.087,respectively. Table 6 Finish grooving’s failure (α 8 ) and repair rate (β 8 ) effect on time-dependent availability of S1 Time in hours α 8 value β 8 value 0.0040 0.0041 0.0042 0.0043 0.097 0.107 0.117 0.127 30 0.9776 0.9775 0.9773 0.9772 0.9780 0.9783 0.9785 0.9788 60 0.9703 0.9701 0.9699 0.9697 0.9712 0.9717 0.9722 0.9727 90 0.9661 0.9659 0.9656 0.9654 0.9673 0.9681 0.9688 0.9693 120 0.9632 0.9629 0.9627 0.9624 0.9646 0.9656 0.9664 0.9671 150 0.9610 0.9607 0.9605 0.9602 0.9626 0.9637 0.9647 0.9655 180 0.9593 0.9590 0.9587 0.9585 0.9611 0.9623 0.9633 0.9642 210 0.9580 0.9577 0.9574 0.9571 0.9598 0.9612 0.9622 0.9632 240 0.9569 0.9566 0.9564 0.9561 0.9589 0.9603 0.9614 0.9624 270 0.9561 0.9558 0.9555 0.9553 0.9581 0.9595 0.9607 0.9617 300 0.9554 0.9552 0.9549 0.9546 0.9575 0.9590 0.9602 0.9612 330 0.9549 0.9546 0.9544 0.9541 0.9570 0.9585 0.9598 0.9608 360 0.9545 0.9542 0.9540 0.9537 0.9567 0.9582 0.9595 0.9605 Values of other parameters taken, in respect to varying values of α 8 of S1 are as follows: α 1 =0.0014,α 2 = 0.0006, α 3 = 0.0009, α 4 = 0.0009, α 5 =0.0208,α 6 = 0.0208, α 7 =0.0009;β 1 =1.08,β 2 = 0.043, β 3 = 0.5, β 4 = 0.286, β 5 =0.154,β 6 = 0.25, β 7 = 0.059, β 8 = 0.087, respectively. Values of other parameters taken, in respect to varying values of β 8 of S1 are as follows: α 1 = 0.0014, α 2 =0.0006,α 3 = 0.0009, α 4 =0.0009,α 5 =0.0208,α 6 = 0.0208, α 7 =0.0009,α 8 =0.0039; β 1 =1.08, β 2 = 0.043, β 3 =0.5,β 4 =0.286,β 5 =0.154,β 6 = 0.25, β 7 = 0.059. Lal et al. Journal of Industrial Engineering International Page 7 of 10 2013, 9:24 http://www.jiei-tsb.com/content/9/1/24 increase (for finish grooving) with corresponding increase in repair rates as observed from Tables 2,3,4,5,6,7. So the weak subsystem was found to be the fixture seat machining which is due to the effect maintenance practice as well. Likewise, the results were simulated for systems in S2 and can be referred from Tables 8,9,10,11,12,13. We observed that the time-dependent availability of this system is more affected by circlip grooving (K), in comparison to other subsystems. It is observed that the Table 7 Finish profile turning’s failure (α 4 ) and repair (β 4 ) rate effect on time-dependent availability of S1 Time in hours α 4 value β 4 value 0.0010 0.0011 0.0012 0.0013 0.296 0.306 0.316 0.326 30 0.9774 0.9770 0.9767 0.9764 0.9778 0.9779 0.9780 0.9781 60 0.9701 0.9698 0.9695 0.9692 0.9706 0.9707 0.9708 0.9708 90 0.9660 0.9657 0.9654 0.9650 0.9664 0.9665 0.9666 0.9667 120 0.9631 0.9628 0.9625 0.9622 0.9635 0.9636 0.9637 0.9638 150 0.9609 0.9606 0.9603 0.9600 0.9614 0.9615 0.9615 0.9616 180 0.9593 0.9589 0.9586 0.9583 0.9597 0.9598 0.9599 0.9599 210 0.9579 0.9576 0.9573 0.9570 0.9583 0.9584 0.9585 0.9586 240 0.9569 0.9566 0.9562 0.9559 0.9573 0.9574 0.9575 0.9575 270 0.9561 0.9557 0.9554 0.9551 0.9565 0.9566 0.9566 0.9567 300 0.9554 0.9551 0.9548 0.9544 0.9558 0.9559 0.9560 0.9561 330 0.9549 0.9546 0.9542 0.9539 0.9553 0.9554 0.9555 0.9555 360 0.9545 0.9541 0.9538 0.9535 0.9549 0.9550 0.9551 0.9551 Values of other parameters taken, in respect to varying values of α 4 of S1 are as follows: α 1 =0.0014,α 2 = 0.0006, α 3 =0.0009,α 5 = 0.0208, α 6 = 0.0208, α 7 = 0.0009, α 8 =0.0039;β 1 =1.08,β 2 = 0.043, β 3 = 0.5, β 4 = 0.286, β 5 =0.154,β 6 = 0.25, β 7 = 0.059, β 8 = 0.087, respectively. Values of other parameters taken, in respect to varying values of β 4 of S1 are as follows: α 1 = 0.0014, α 2 =0.0006,α 3 = 0.0009, α 4 =0.0009,α 5 =0.0208,α 6 = 0.0208, α 7 =0.0009,α 8 = 0.0039; β 1 = 1.08, β 2 = 0.043, β 3 =0.5,β 5 =0.154,β 6 =0.25,β 7 = 0.059, β 8 =0.087,respectively. Table 8 Effect of failure rate (α 1 ) and repair rate (β 1 )of finish crown and cavity on time-dependent availability of S2 Time in hours α 1 value β 1 value 0.0054 0.0104 0.154 0.0204 0.34 0.35 0.36 0.37 30 0.9905 0.9894 0.9883 0.9873 0.9914 0.9914 0.9914 0.9914 60 0.9885 0.9874 0.9864 0.9854 0.9894 0.9894 0.9894 0.9894 90 0.9878 0.9868 0.9858 0.9848 0.9887 0.9887 0.9887 0.9887 120 0.9876 0.9866 0.9855 0.9846 0.9885 0.9885 0.9885 0.9885 150 0.9875 0.9865 0.9855 0.9845 0.9884 0.9884 0.9884 0.9884 180 0.9875 0.9864 0.9854 0.9845 0.9884 0.9884 0.9884 0.9884 210 0.9875 0.9864 0.9854 0.9844 0.9883 0.9884 0.9884 0.9884 240 0.9875 0.9864 0.9854 0.9844 0.9883 0.9884 0.9884 0.9884 270 0.9875 0.9864 0.9854 0.9844 0.9883 0.9883 0.9884 0.9884 300 0.9875 0.9864 0.9854 0.9844 0.9883 0.9883 0.9884 0.9884 330 0.9875 0.9864 0.9854 0.9844 0.9883 0.9883 0.9884 0.9884 360 0.9875 0.9864 0.9854 0.9844 0.9883 0.9883 0.9884 0.9884 Values of other parameters taken, in respect to varying values of α 1 of S2 are as follows: α 2 =0.0003,α 3 = 0.0001, α 4 =0.0003,α 5 = 0.0001, α 6 = 0.0208, α 7 = 0.004; β 1 =0.33,β 2 = 0.5, β 3 = 0.67, β 4 =0.035,β 5 =3.03,β 6 =0.222,β 7 = 0.125, respectively. Values of other parameters taken, in respect to varying values of β 1 of S2 are as follows: α 1 = 0.0014, α 2 =0.0003,α 3 = 0.0001, α 4 =0.0003,α 5 = 0.0001, α 6 =0.0208,α 7 =0.004,β 2 = 0.5, β 3 =0.67,β 4 = 0.035, β 5 =3.03,β 6 = 0.222, β 7 = 0.125, respectively. Table 9 Effect of failure rate of valve milling (α 2 )on time-dependent availability of the system S2 Time in hours α 2 value β 2 value 0.0004 0.0005 0.0006 0.0007 0.6 0.7 0.8 0.9 30 0.9913 0.9913 0.9913 0.9913 0.9914 0.9914 0.9914 0.9914 60 0.9893 0.9893 0.9893 0.9893 0.9893 0.9894 0.9894 0.9894 90 0.9887 0.9887 0.9886 0.9886 0.9887 0.9887 0.9887 0.9887 120 0.9884 0.9884 0.9884 0.9884 0.9885 0.9885 0.9885 0.9885 150 0.9884 0.9883 0.9883 0.9883 0.9884 0.9884 0.9884 0.9884 180 0.9883 0.9883 0.9883 0.9883 0.9884 0.9884 0.9884 0.9884 210 0.9883 0.9883 0.9883 0.9883 0.9883 0.9883 0.9884 0.9884 240 0.9883 0.9883 0.9883 0.9883 0.9883 0.9883 0.9883 0.9884 270 0.9883 0.9883 0.9883 0.9883 0.9883 0.9883 0.9883 0.9884 300 0.9883 0.9883 0.9883 0.9883 0.9883 0.9883 0.9883 0.9884 330 0.9883 0.9883 0.9883 0.9883 0.9883 0.9883 0.9883 0.9884 360 0.9883 0.9883 0.9883 0.9883 0.9883 0.9883 0.9883 0.9884 Values of other parameters taken, in respect to varying values of α 2 of S2 are as follows: α 1 = 0.0014, α 2 =0.0003,α 3 =0.0001,α 4 = 0.0003, α 5 = 0.0001, α 6 =0.0208,α 7 = 0.004, β 1 =0.33,β 2 = 0.5, β 3 = 0.67, β 4 = 0.035, β 5 =3.03,β 6 = 0.222, β 7 = 0.125, respectively. Values of other parameters taken, in respect to varying values of β 2 of S2 are as follows: α 1 = 0.0014, α 2 = 0.0003, α 3 =0.0001, α 4 =0.0003,α 5 = 0.0001, α 6 =0.0208,α 7 =0.004;β 1 =0.33,β 3 =0.67,β 4 =0.035, β 5 =3.03,β 6 =0.222,β 7 = 0.125, respectively. Table 10 Effect of failure rate (α 3 ) and repair rate (β 3 )of chamfering or radiusing on time-dependent availability of S2 Time in hours α 3 value β 3 value 0.0002 0.0003 0.0004 0.0005 0.68 0.69 0.70 0.71 30 0.9912 0.9911 0.9909 0.9908 0.9914 0.9914 0.9914 0.9914 60 0.9892 0.9891 0.9889 0.9888 0.9893 0.9893 0.9893 0.9894 90 0.9885 0.9884 0.9882 0.9881 0.9887 0.9887 0.9887 0.9887 120 0.9883 0.9882 0.9880 0.9879 0.9885 0.9885 0.9885 0.9885 150 0.9882 0.9881 0.9879 0.9878 0.9884 0.9884 0.9884 0.9884 180 0.9882 0.9881 0.9879 0.9878 0.9883 0.9884 0.9884 0.9884 210 0.9882 0.9880 0.9879 0.9878 0.9883 0.9883 0.9883 0.9883 240 0.9882 0.9880 0.9879 0.9878 0.9883 0.9883 0.9883 0.9883 270 0.9882 0.9880 0.9879 0.9878 0.9883 0.9883 0.9883 0.9883 300 0.9882 0.9880 0.9879 0.9878 0.9883 0.9883 0.9883 0.9883 330 0.9882 0.9880 0.9879 0.9878 0.9883 0.9883 0.9883 0.9883 360 0.9882 0.9880 0.9879 0.9878 0.9883 0.9883 0.9883 0.9883 Values of other parameters taken, in respect to varying values of α 3 of S2 are as follows: α 1 =0.0014,α 2 = 0.0003, α 4 =0.0003,α 5 = 0.0001, α 6 = 0.0208, α 7 = 0.004; β 1 =0.33,β 2 = 0.5, β 3 = 0.67, β 4 =0.035,β 5 =3.03,β 6 = 0.222, β 7 = 0.125, respectively. Values of other parameters taken, in respect to varying values of β 3 of S2 are as follows: α 1 = 0.0014, α 2 =0.0003,α 3 = 0.0001, α 4 =0.0003,α 5 = 0.0001, α 6 =0.0208,α 7 =0.004,β 1 =0.33,β 2 = 0.5, β 4 = 0.035, β 5 =3.03,β 6 = 0.222, β 7 = 0.125, respectively. Lal et al. Journal of Industrial Engineering International Page 8 of 10 2013, 9:24 http://www.jiei-tsb.com/content/9/1/24