RELIABILITY AND PROFIT ANALYSIS OF A WATER PROCESS SYSTEM CONSIDERING NONE SWITCHING OF REDUNDANT
GRAPHICAL INTERPRETATION AND CONCLUSIONS
For graphical analysis purpose, following particular case is considered: t
g(t)= βe
−β 1t 1 1g (t)= βe
−β 1t 1 1g (t)= βe
−β 3(t) 3 3g (t)= βe
−β 1t 1 1 i (t)= αe−α 2t 2 2 i (t)= αe−α 2 01 1 2 p = λ λ + λ 1 02 1 2 p = λ λ + λ 13 p =x p14=y p20=1 p30=p1 32 1 p =q p45=a p46=b p50 =p2 52 2 p =q p60 =p3 p62 =q3 0 1 2 1 µ = λ + λ 1 1 1 µ = α 2 1 µ = β 1 3 1 2 q µ = -p λ 4 2 1 µ = α 2 5 2 2 q µ = -p λ 3 6 3 2 q µ = -p λVarious graphs are plotted for MTSF, expected uptimes with full/reduced capacity and profit of the system taking different values of rates of different faults (λ1, λ2), repair rates (β, β1, β2, β3), inspection rates (α1, α2) and various probabilities (a, x, p1, p2, p3).
Following is concluded from the various plotted graphs:
Fig. 2 gives the graph between MTSF (T0) and rate of minor faults (λ2) for different values of rate of major faults (λ1). It is concluded that the MTSF decreases with increase in the values of the rates of major/minor faults.
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Fig.3
The curves in fig. 3 show the behaviour of MTSF (T0) with respect to probability of proper repair of a Type-II redundant subsystem (p3) for different values of probability of fault in non-redundant subsystem (x). The graph reveals that the MTSF increases with increase in the values of the probability of proper repair of a Type-II redundant subsystem and MTSF decreases with increase in the values of the probability of fault in non-redundant subsystem.
Reliability and profit analysis of a water process system considering none switching of redundant units and proper/ improper repairs of minor faults
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Fig. 4 presents the graph between expected uptime with full capacity of the system (AF0) and rate of minor faults (λ2) for different values of rate of major faults (λ1). It can be concluded that the expected uptime with full capacity of the system decreases with increase in the values of the rates of minor/ major faults.
Fig.5
The curves in fig. 5 present the behaviour of expected uptime with full capacity of the system (AF0) with respect to probability of proper repair of a Type-I redundant subsystem (p2) for different values of inspection rate (α2). It is concluded that the expected uptime with full capacity of the system increases with increase in the values of the probability of proper repair of a Type-I redundant subsystem and has higher values for higher values of inspection rate.
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The graph in fig. 6 gives patterns of expected uptime with full capacity of the system (AF0) with respect to probability of proper repair of a Type-II redundant subsystem (p3) for different values of probability of fault in non- redundant subsystem (x). From the graph it can be concluded that the expected uptime with full capacity of the system increases with increase in the values of the probability of proper repair of a Type-II redundant subsystem and has lower values for higher values of probability of fault in non-redundant subsystem.
Fig.7
Fig. 7 presents the graph between expected uptime with reduced capacity of the system (AR0) and rate of major faults (λ1) for different values of probability of fault in Type-I redundant subsystem (a). The graph reveals
Reliability and profit analysis of a water process system considering none switching of redundant units and proper/ improper repairs of minor faults
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that the expected uptime with reduced capacity of the system decreases with increase in the values of rate of major faults and has lower values for higher values of probability of fault in Type-I redundant subsystem.
Fig.8
The curves in fig. 8 reveal the behaviour of expected uptime with reduced capacity of the system (AR0) with respect to probability of proper repair of a non-redundant subsystem (p1) for different values of inspection rate (α1). It can be concluded from the graph that the expected uptime with reduced capacity of the system increases with increase in the values of the probability of proper repair of an non-redundant subsystem and has higher values for higher values of the inspection rate.
Fig.9
Fig. 9 gives the patterns of the profit incurred from the system (P) with respect to the rate of minor faults (λ2) for the different values of rate of major faults (λ1). It is concluded that the profit decreases with the increase in the values of the rate of minor faults and has lower values for higher values of the rate of major faults when other parameters remain fixed. From the fig. 9, it can also be observed that for λ1 = 0.001, the profit is positive or zero or
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negative according as λ2 is < or = or > 0.0016. Thus, in this case, the system is profitable whenever λ2 < 0.0016. Similarly, for λ1 = 0.0012 and λ1 = 0.0014, the system is profitable whenever λ2 < 0.0018 and 0.002 respectively.
Fig.10
The curves in the fig. 10 show the behaviour of profit of the system (P) with respect to the probability of proper repair of a non-redundant subsystem (p1) for different values of inspect rate (α1). It is evident from the graph that the profit increases with the increase in the values of probability of proper repair of a non-redundant subsystem and has lower values for higher values of inspection rate when other parameters remain fixed. From the fig. 10, it can also be observed that for α1 = 1.5, the profit is negative or zero or positive according as pis < or = or > 0.694030. Thus, in this case, the system is profitable whenever p1 > 0.694030. Similarly, for α1 = 1.7 and α1 = 1.9, the system is profitable whenever p1 > 0.532147 and 0.385979 respectively.
Reliability and profit analysis of a water process system considering none switching of redundant units and proper/ improper repairs of minor faults
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Fig. 11 highlights the behaviour of profit incurred from the system (P) with respect to probability of proper repair of a Type-I redundant subsystem (p2) for different values of inspection rate (α2). It can be concluded from the graph that the profit increases with the increase in the values of probability of proper repair of a Type-I redundant subsystem and has lower values for higher values of inspection rate when other parameters remain fixed. From the fig.11, it can also be observed that for α2 = 2.2, the profit is negative or zero or positive according as pis < or = or > 0.717649. Thus, in this case, the system is profitable whenever p2 > 0.717649. Similarly, for α2 = 2.6 and α2 = 3.0, the system is profitable whenever p2 > 0.517551 and 0.328838 respectively.
Fig.12
The curves in the fig.12 show the behaviour of profit with respect to revenue per unit uptime with full capacity of the system (C0) for different values of probability of proper repair of an non-redundant subsystem (p1). It is concluded from the graph that the profit increases with the increase in the values of revenue per unit up time with full capacity and has higher values for higher values of probability of proper repair of a non-redundant subsystem when other parameters remain fixed. From the fig.12 it can also be observed that for p1 = 0.1, the profit is negative or zero or positive according as C0 is < or = or > 11716.28. Thus in this case, the system is profitable whenever C0 > 11716.28. Similarly, for p1 = 0.5 and p1 = 0.9, the system is profitable whenever p2 > 9581.08 and 7445.88 respectively.
REFERENCES
1. Garg, R. C. and Kumar, A., “A complex system with two types of failure and repair”, IEEE Trans. Reliability, 26, 299-300 (1977).
2. Anita Teneja “Reliabity and profit evaluation of a two unit cold stand system with inspection & chances of replacement”, Aryabhatta J. of Maths & Info. Vol. 6, (1) pp 211-218. (2014)
3. Kumar, R. and Batra, S., “Economic and reliability analysis of a stochastic model on printed circuit boards manufacturing system considering two-types of repair facilities”, International Journal of Electrical Electronics and Telecommunication Engineering, 43, No.10, pp.432-435 (2012)
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4. Kumar, R. and Bhatia, P., “Reliability and cost analysis of one unit centrifuge system with single repairman and inspection”, Pure and Applied Mathematika Science, LXXIV, No. 1-2, 113-121 (2011).
5. Kumar, R. and Kapoor, S., “Reliability and performance analysis of a base transceiver system considering software based hardware and common cause failures”, International Journal of Statistika and Mathematika, Vol. 11, No.1, pp.29-43 (2014).
6. Kumar, R. and Rani, S., “Cost-benefit analysis of a reliability model on water process system having two types of redundant subsystems, International Journal of Applied Mathematical Research, 2, No. 2, 293-302 (2013).
7. Kumar, R., Sharma, M. K. and Mor, S. S., “Profit evaluation of a warranted sophisticated system with three operational stages”, J. Indian Soc. Stat. Opers. Res., XXX, 53-61 (2009).
8. Kumar, R., Vashistha, U. and Tuteja, R. K., “A two-unit redundant system with degradation and replacement”, Pure and Applied Mathematika Science, LIV, No. 1-2, 27-38 (2001).
9. Mahmoud, M.A.W., “Reliability Study of a Two unit cold standby redundant system with two types of failure and preventive maintenance”, Microelectron Reliability, 29, No.6, 1061-1068 (1989).
10. Murari, K. and Goyal, V., “Reliability system with two types of repair facilities”, Micro-electron Reliability, 23(6), 1015-1025 (1983).
11. Srinivasan, S.K. and Gopalan, M.N., “Probabilistic analysis of a two-unit system with a warm standby and single repair facility.” Operations Research, 21, 748-754 (1973).
12. Taneja, G., Tayagi, V.K and Bhardwaj: “Profit analysis of single unit programmable logic controller”, Pure and Applied Mathematica Sciences, LX, No.1-2, 55-71 (2004).
13. Dalip Singh & G. Taneja “Reliability analysis of a power generating system through Gas & Stem Turbines with scheduled Inspection”. Aryabhatta J. of Maths & Info. Vol. 5, (2) pp 373-380. (2013)
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Aryabhatta Journal of Mathematics & Informatics Vol. 7, No. 1, Jan-June, 2015 ISSN : 0975-7139
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