Markov Chain Profit Modelling and Evaluation between Two Dissimilar Systems under Two Types of Failures

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1 Available at Appl. Appl. Math. SSN: Vol., ssue (December 06), pp Applications and Applied Mathematics: An nternational Journal (AAM) Markov Chain Profit Modelling and Evaluation between Two Dissimilar s under Two Types of Failures Saminu. Bala and brahim Yusuf Department of Mathematical Sciences Bayero University Kano, Nigeria saminub@yahoo.com; iyusuf.mth@buk.edu.ng; Received: October, 05; Accepted: July 5, 06 Abstract The present paper deals with profit modelling and comparison between two dissimilar systems under two types of failures based on Markovian Birth-Death process. Type failure is minor in the sense that the work is in a reduced capacity whereas type failure is major because it causes the entire system failure. Both systems consist of four subsystems arranged in series-parallel with three possible states: working with full capacity, reduced capacity and failed state. The systems are attended to by two repairmen in tandem. Through the transition diagrams, systems of differential difference equations are developed and solved recursively to obtain the steady-state availability, busy period of repair men, and profit function. Profit matrices for each subsystem have been developed for different combinations of failure and repair rates. Furthermore, we compare the profit for the two systems and find that system is more profitable than system. Keywords: Profit; Redundant; availability; busy period; modelling; series-parallel MSC 00 No.: 60J0, 90B5. ntroduction The industrial and manufacturing systems comprise of large complex engineering systems arranged either in series, parallel, parallel-series or series-parallel. Examples of these systems are feeding, crushing, refining, steam generation, evaporation, crystallization, fertilizer plant, crystallization unit of a sugar plant, piston manufacturing plant, etc. Reliability, availability 89

2 90 Saminu. Bala and brahim Yusuf and profit are vital in any successful industries and manufacturing settings. Profit may be enhancing using highly reliable system or subsystem. f the reliability and availability of a system is improved, the production and associated profit will also increase. This can be achieved by maintaining reliability and availability at the highest order through maintenance. Large volumes of literature exist on the issue of predicting performance evaluation of various industrial and manufacturing systems. Aggarwal et al. (0) used Markov model for the analysis of urea synthesis of a fertilizer plant. Damcese and Helmy (0) presented reliability study with mixed standby components. Gupta and Tewari (0) analyzed the reliability and availability of thermal power plant. Gupta et al. (007) presented the reliability and availability of serial processes of plastics pipe manufacturing plant. Gupta et al. (005) discussed the mission reliability and availability of flexible polymer powder production system. Gupta et al. (007) presented reliability parameters of a powder generating system. Kadiyan et al. (0) presented the reliability and availability of uncaser system of brewery plant. Khanduja et al. (0) presented the steady-state behaviour and maintenance planning of bleaching system of a paper plant. Kaur (0) discussed the reliability, availability and maintainability of an industrial process. Kaur et al. (0a) discussed the numerical solution of differential difference equations in reliability engineering. Kaur (0c) discussed the use of corrective maintenance data for performance analysis of textile industry. Kaur (0b) discussed the performance analysis of an industrial system under corrective and preventive maintenance. Kumar el al. (0) discussed stochastic modelling of a concrete mixture plant with preventive maintenance. Kumar and Mudgil (0) discussed the availability analysis of the ice cream making unit of a milk plant. Kumar and Tewari (0) discussed the mathematical modelling and performance optimization of CO cooling system of a fertilizer plant. Kumar et al. (0) discussed the performance modelling of furnace draft air cycle in a thermal plant. Kumar and Lata (0) presented the reliability evaluation of condensate system using fuzzy Markov model. Pandey et al. (0) discussed the reliability analysis of a series and parallel network using triangular intuitionistic fuzzy sets. Ram (00) discussed the reliability measures of threestate complex system. Shakuntla (0) presented reliability modelling and analysis of some process industrial systems. Sachdeva et al. (008a) discussed availability modelling of screening system of a paper plant. Sachdeva et al. (008b) studied the behaviour of a biscuit making plant using Markov regenerative modelling. Singh and Goyal (0) presented a methodology to study the steady-state behaviour of repairable mechanical biscuit making plant. Tewari et al. (0) computed the steady-state availability and performance optimization for the crystallization unit of sugar plant using genetic algorithm. Tuteja and Tuteja (99a) studied the cost benefit analysis of a two server two unit system with different types of failure. Tuteja and Tuteja (99b) presented profit evaluation of one server system with partial failure subject to random inspection. Tuteja and Malik (99) presented reliability and profit analysis of two single unit models with three modes and different repair policies. Tuteja et al. (99) analyzed two unit system with partial failure and three types of repairs. Tuteja et al. (99) discussed the stochastic behaviour of a two unit system with two types of repairman and subject to random inspection. The problem considered in this paper is different from those discussed by the authors above. The purpose of this paper is threefold. The first purpose is to develop the explicit expressions for the steady-state availability, busy period of repair men, and profit function. n this paper, we studied two dissimilar systems subject to two types of failures. The second purpose is to compare these systems in terms of their profit. The third is to capture the effect of both failure and repair rates on profit based on assumed numerical values given to the system parameters.

3 AAM: ntern. J., Vol., ssue (December 06) 9 The organization of the paper is as follows. Section presents the model s description and assumptions. Section presents formulations of the models. Numerical examples are presented and discussed in Section. Finally, we make a concluding remark in Section 5. Symbols ndicates the system is in full working state ndicates the system is in failed state ndicates the system in reduced capacity state A, B, C, D: Represent full working state of subsystem a, b, c, d: Represent failed state of subsystem,,, Represent failure rates of subsystems A, B, C, D,,, : represent repair rates of subsystems A, B, C, D Pi ( t), i 0,,,...,6 : Probability that the system is in state S i at time t A k ( ): Steady state availability of the system, k, V k B ( ) P : Steady state busy period of repairman due to minor failure k B ( ) P : Steady state busy period of repairman due to major failure C 0 : Total revenue generated from system using C : Cost incurred due to type failure C : Cost incurred due to type failure P ( ): Profit k F. The Model s Description and Assumptions Model description of The system consists of four dissimilar subsystems arranged in series-parallel as follows:. Subsystem A: t is a single unit and has no standby unit. ts failure is catastrophic and causes complete failure of the system.

4 9 Saminu. Bala and brahim Yusuf. Subsystem B: Consists of two active parallel units. Failure of one unit causes the system to work in reduced capacity. Complete failure occurs when both units fail.. Subsystem C: Consists of two active parallel units. Failure of one unit causes the system to work in reduced capacity. Complete failure occurs when both units fail.. Subsystem D: t is a single unit and has no standby unit. ts failure is catastrophic and causes severe effect on the system performance; that is, a complete failure of the system. Subs. A Subs. B Subs. C Subs. D Subs. B Subs. C Figure. Reliability block diagram of system ABBccD 0 ABBBcCd abbccd AbBccD AbBcCd abbccd 9 ABBcCD AbBcCD abbccd ABBCCD 0 AbBCCD ABBCCd 5 abbccd 6 ABBCCd 7 AbbcCD 5 AbbCCD 8 Figure. Transition diagram of system State 0 indicate full working capacity States indicate reduced capacity (type failure) States 5 indicate failed states (type failure)

5 AAM: ntern. J., Vol., ssue (December 06) 9 Model description of The system consists of four dissimilar subsystems arranged in series-parallel as follows:. Subsystem A: t is a single unit and has no standby unit. ts failure is complete failure of the system.. Subsystem B: Consists of three active parallel units. Failure of one unit, causes the system to work in reduced capacity. Complete failure occurs when both units fail.. Subsystem C: t is a single unit and has no standby unit. ts failure is catastrophic and causes severe effect on the system performance: that is, complete failure of the system.. Subsystem D: t is a single unit and has no standby unit. ts failure is catastrophic and causes severe effect on the system performance; that is, complete failure of the system. Subs. B Subs. A Subs. B Subs. C Subs. D Subs. B Figure. Reliability block diagram of system ABBBCd 5 abbbcd ABBBCD 0 ABBBcD abbbcd 6 abbbcd 7 AbBBCD AbbBCD AbbBcD 8 AbBBCd AbBBcD AbbBCd 0 AbbbCD 9 Figure : Transition diagram of system

6 9 Saminu. Bala and brahim Yusuf State 0 indicates full working capacity. States indicate reduced capacity (type failure). States indicate failed states (type failure). Assumptions The assumptions used in the model s development are as follows:. At any given time the system is either in operating state, reduced capacity or in failed state.. Subsystems/units do not fail simultaneously.. The system is exposed to two types of failures minor and major failure.. Minor failure forces the system to work in reduced capacity states (there is no system failure) whereas major failure brings about system failure. 5. The system is attended to by two repairmen in tandem. 6. Standby units in the same subsystem are of the same nature and capacity as the active units.. Models Formulation Steady State availability, busy periods and profit of The following differential difference equations associated with the transition diagram in Figure of the system are formed using Markov birth-death process: d i P0 ( t) P ( t) P ( t) P ( t) P5 ( t), dt i () d i P ( t) P ( t) P6 ( t) P7 ( t) P8 ( t) P0 ( t), dt i () d i P ( t) P ( t) P9 ( t) P0 ( t) P ( t) P0 ( t), dt i () d i P ( t) P ( t) P ( t) P ( t) P5 ( t) P ( t) P ( t), dt () i d m Pi ( t) mpj ( t), m,,,, dt j 0,,,, i,5,6,7,...,5, (5) with initial conditions i 0, Pt i () 0 i 0. (6)

7 AAM: ntern. J., Vol., ssue (December 06) 95 n the steady state, the derivatives of the state probabilities in Equations 5 are set to zero and solving the resulting equations recursively, we obtained the following steady state probabilities: P ( ) X P0 ( ), P 6 ( ) X X P 0 ( ), P ( ) X X P 0 ( ), P ( ) XP0 ( ), P 7 ( ) X X P 0 ( ), P ( ) X X X P 0 ( ), P ( ) X X P ( ), P ( ) X P ( ), P ( ) X X P ( ), P ( ) XP0 ( ), P 9 ( ) X X P 0 ( ), P ( ) X X X P 0 ( ), P ( ) X P ( ), P ( ) X P ( ), P ( ) X X P ( ) The probability of full working state P 0 ( ) is determined by using the normalizing condition below: P ( ) P( ) P ( ) P ( ) P ( )... P ( ). (7) 0 5 Substituting the values of P( ) - P ( ) in terms of 5 P ( ) 0 into the normalizing condition in (7) below 0 P ( ) X X X X X X X X... X X. (8) P 0( ), d 0 the steady-state availability, busy period due to type and failure and profit function of system are given below: (9) n A ( ) ( ), 0 V Pk k0 d0 n B ( ) ( ), Pk k d0 5 n ( ) Pk ( ), k d0 F 0 V P P B (0) () () P ( ) C A ( ) C B ( ) C B ( ), () where d X X X X X X X X X X X, 0 n X X X X, 0, n X X X X n X X X X X X X X X X X X.

8 96 Saminu. Bala and brahim Yusuf Steady State availability, busy periods and profit of The following differential difference equations associated with the transition diagram in Figure of the system are formed using Markov birth-death process: d dt d i P ( t) P ( t) P6 ( t) P ( t) P ( t) P0 ( t), dt i (5) d i P ( t) P7 ( t) P8 ( t) P9 ( t) P0 ( t) P ( t), dt i (6) d m Pi ( t) mpj ( t), m,,,, j 0,,,, i,,5,6,7,...,. dt (7) i 0, Pt i() 0 i 0. (8) i P0 ( t) P ( t) P ( t) P ( t) P5 ( t), () i n the steady state, the derivatives of the state probabilities in Equations 7 are set to zero and solving the resulting equations recursively we obtained the following steady state probabilities: P ( ) X P0 ( ), P ( ) XP0 ( ), P7 ( ) XX P0 ( ), P0 ( ) X X P0 ( ), P ( ) X P0 ( ), P5 ( ) X P0 ( ), P8 ( ) XX P0 ( ), P ( ) X X P0 ( ), P ( ) X P ( ), P6 ( ) XX P0 ( ), P ( ) X P ( ), P ( ) X XP0 ( ) The probability of full working state P ( ) 0 is determined by using the normalizing condition below: 0 5 P ( ) P( ) P ( ) P ( ) P ( )... P ( ). (9) Substituting the values of P( ) - P ( ) in terms of P ( ) 0 into the normalizing condition in (9) below 0 P ( ) X X X X X X X... X X. (0) P 0( ), () d The steady-state availability, busy period due to type failure, busy period due to type failure and profit function of system are given by n A ( ) ( ), () V Pj j0 d n B ( ) ( ), () Pj j d

9 AAM: ntern. J., Vol., ssue (December 06) 97 5 ( ) Pk ( ), j d F 0 V P P B n P ( ) C A ( ) C B ( ) C B ( ), (5) () where d X X X X X X, n X X, n X X, n X X X X X X, 5 X, X, X.. Numerical Examples and Discussion n this section, we numerically obtained the results for the profit for the systems using the failure and repair rates of Aggarwal et al. (0). For each table the following set of parameter values were fixed: C0 00,000, C 0,000, C 5,000. Table. Profit matrix for subsystem A Table. Profit matrix for subsystem B

10 98 Saminu. Bala and brahim Yusuf Table. Profit matrix for subsystem C Table. Profit matrix for subsystem D Table 5. Optimal values of Profit obtained Subsystem Maximum Profit S/N A B C D Table and Figure 5 present the impact of failure and repair rates of subsystem A against the profit for different values of parameters and for both system and. The failure and repair rates of other subsystems are kept constant as can be seen in the last column of Table. t is clear from Table and Figure that the profit shows increasing pattern with respect to repair rate and decreasing pattern with respect to failure rate in both systems. However, system tends to have more profit than system. Table and Figure 6 present the impact of failure and repair rates of subsystem B against the profit for different values of parameters and for both system and. The failure and repair rates of other subsystems are kept constant as can be seen in the last column of Table. t is clear from Table and Figure 6 that the profit shows increasing pattern with respect to repair rate and decreasing pattern with respect to failure rate in both systems. tend to have more profit than system.

11 AAM: ntern. J., Vol., ssue (December 06) 99 Table and Figure 7 present the impact of failure and repair rates of subsystem C against the profit for different values of parameters and for both system and. The failure and repair rates of other subsystems are kept constant as can be seen in the last column in the Table. t is clear from Table and Figure 7 that the profit shows increasing pattern with respect to repair rate and decreasing pattern with respect to failure rate in both systems. Here, again system has more profit than system. Table and Figure 8 present the impact of failure and repair rates of subsystem D against the profit for different values of parameters and for both system and. The failure and repair rates of other subsystems are kept constant as can be seen in the last column of Table. t is clear from Table and Figure 6 that the profit shows increasing pattern with respect to repair rate and decreasing pattern with respect to failure rate in both systems. However, system tends to have more profit than system. Table 5 helps in determining the subsystem with maximum profit. t is observed that subsystem D has maximum profits of 977 and for system and, respectively. From Table 5, it is observed that the most critical subsystem as far as maintenance is concerned and required immediate attention is subsystem C. 5. Conclusion n this paper, we analyzed two dissimilar systems, each consisting of subsystems A, B, C and D. Explicit expressions for steady-state availability, busy period and profit function for the two systems were derived and comparison between the systems was performed numerically. t is evident from Tables - and Figures 5-8 that the optimal system is system. Models presented in this paper are important to engineers, maintenance managers and plant management for proper maintenance analysis, decision and safety of the system as a whole. The models will also assist engineers, decision makers and plant management to avoid an incorrect reliability assessment and consequent erroneous decision making, which may lead to unnecessary expenditures, incorrect maintenance scheduling and reduction of safety standards. Overall, based on numerical results in the Tables and Figures, it is evident that The revenue obtained decreases with increase in failure rates. The revenue is affected by the number of operating units. The system availability as well as revenue of the system can be increased by adding more redundant units/subsystems, taking more units in the system in cold standby, and by increasing the repair rate. Acknowledgment: The authors are grateful to the handling editor and the reviewer for his constructive comments which have helped to improve the manuscript.

12 500 Saminu. Bala and brahim Yusuf REFERENCES Aggarwal A. K, Kumar S, Singh V, Garg TK. (0). Markov modelling and reliability analysis of urea synthesis system of a fertilizer plant. Journal of ndustrial Engineering nternational,, -, DO /s Damcese, M. A. and Helmy, A.N. (0). Study of reliability with mixed standby components, Application and Applied Mathematics, 7(): Gupta, S and Tewari, P.C. (0). Simulation modelling in availability thermal power plant, Journal Eng. Sci Tech Rev, (), 0-7. Gupta, P., Lal, A.K., Sharma, R.K and Singh, J. (007). Analysis of reliability and availability of serial processes of plastic pipe manufacturing plant A case study, nternational Journal of quality and Reliability Management,, 0-9. Gupta, P., Singh, J and Singh,. (005). Mission reliability and availability prediction of flexible polymer powder production system, OPSEARCH,,5. Gupta, R., Mittal, S and Batra, C. (007). Reliability parameters of a powder generating system with sared load, Journal of Mathematics and Statistics., -9. Kadiyan S, Garg RK, Gautam R. (0). Reliability and availability analysis of uncaser system in a brewery plant. nt J Res MechEng Tech, (), 7-. Khanduja, R., Tewari, PC., Kumar, D. (0). Steady state behaviour and maintenance planning of bleaching system in a paper plant. nternational Journal of ndustrial Engineering,7(): 9-. Kaur, M. (0). Reliability, availability and maintainability analysis of an industrial process, PhD Thesis, School of Mathematics and Computer Application, Thapar University, Patiala, Panjab, ndia. Kaur, M., Lal, A.K., Bhatia, S.S and Reddy, A.S. (0a). Numerical solution of stochastic partial differential difference equation arising in reliability engineering, Applied Mathematics and computation, 9(), Kaur, M., Lal, A.K., Bhatia, S.S and Reddy, A.S. (0b). Reliability distribution of an industrial process, nternational Journal of Research in advent Technology, (5) Kaur, M., Lal, A.K., Bhatia, S.S and Reddy, A.S.(0c). On use of corrective maintenance data for performance analysis of preventively maintained textile industry, Journal of Reliability and Statistical Studies, 6(), 5-6. Kaur, M., Lal, A.K., Bhatia, S.S and Reddy, A.S.(0b). Performance analysis of industrial system under corrective and preventive maintenance, Quality and Reliability Engineering: Recent trends and future directions, Chapter 9; -5 Allied publishers, Bangalore, 0. Kumar, A., Saini, M. and Malik, S. C. (0). Stochastic modelling of a concrete mixture plant with preventive maintenance, Application and Applied Mathematics, 9(): -7. Kumar V and Mudgil V. (0). Availability optimization of ice cream making unit of milk plant using genetic algorithm, ntmanag Bus Stu, (), 7-9. Kumar S, and Tewari PC. (0). Mathematical modelling and performance optimization of CO cooling system of a fertilizer plant. nt J ndusteng Comp,, Kumar R, Sharma AK, Tewari PC. (0). Performance modelling of furnace draft air cycle in a thermal plant. nt J EngSci Tech, (8), Kumar, A and Lata, S. (0). Reliability evaluation of condensate system using fuzzy Markov Model, Annals of Fuzzy Mathematics and nformatics, (), 8-9. Pandey, D., Tyagi, S. K. and Kumar, V. (0). Reliability analysis of a series and parallel network using triangular intuitionistic fuzzy state, Application and Applied Mathematics, 6(); 05-5.

13 AAM: ntern. J., Vol., ssue (December 06) 50 Ram, M. (00). Reliability measures of a three-state complex system: A copula approach, Application and Applied Mathematics, 5(): Shakuntla (0). Reliability modelling and analysis of some process industrial system s, PhD Thesis, School of Mathematics and Computer Application, Thapar University, Patiala, Panjab, ndia. Sachdeva, A, Kumar D and Kumar, P. (008a). Availability modelling of screening system of a paper plant sing GSPN, Journal of Modelling in Management,, 6-9. Sachdeva, A, Kumar D and Kumar, P. (008b). Reliability analysis of pulping system using petri nets, nternational Journal of Quality and Reliability Management, 5, Singh, P and Goyal, A. (0). Behavior analysis of a biscuit making plant using Markov regenerative modelling, nt J. Theor. Appl Res MechEng, (), 8-5. Tewari PC, Khanduja R and Gupta M. (0). Performance enhancement for crystallization unit of sugar plant using genetic algorithm. Journal of ndustrial Engineering nternational 8:: -6. Tuteja, R.K. and Taneja, G. (99a). Cost benefit analysis of a two server, two unit standby system with different types of failure, Microelectronics reliability,, Tuteja, R.K. and Taneja, G. (99b). Profit analysis of a one server one unit system with partial failure subject to random inspection, Microelectronics reliability,, 9-. Tuteja, R.K. and Malik, S. (99). Reliability and profit analysis of two single unit models with three modes and different repair policies of repairmen who appear and disappear randomly, Microelectronics and reliability,, Tuteja, R.K., Arora, R.T and Taneja, G. (99). Analysis of a two unit system with partial failure and three types of repairs. Reliability Engineering ans safety, 99-. Tuteja, R.K., Arora, R.T and Taneja, G. (99). Stochastic behaviour of a two unit system with two types of repairman and subject to random inspection, Microelectronics Reliability,, P r o f i t Series8 0.5 Series6 0. Series 0.5 Series 0.5 Repair rate Failure rate Figure 5. mpact of failure and repair of subsystem A on profit

14 50 Saminu. Bala and brahim Yusuf 9600 P r o f i t Series8 0. Series6 0. Series 0.5 Series 0.05 Repair rate Failure rate Figure 6. mpact of failure and repair of subsystem B on profit Series8 Series6 0.5 Series 0.55 Series 0.5 Figure 7. mpact of failure and repair of subsystem C on profit

15 AAM: ntern. J., Vol., ssue (December 06) Series8 Series6 0. Series 0.5 Series 0.05 Figure 8. mpact of failure and repair of subsystem D on profit