CC BY
5
STOCHASTIC ANALYSIS OF JUICE PLANT SUBJECT TO REPAIR FACILITY
Amit Kumar1 and Pinki Kumari2
department of Mathematics, Govt. College, Satnali, Haryana 2Department of Physics, Lord University, Chikani, Alwar, Rajasthan [email protected], [email protected]
Abstract
The performance of a juice plant is analyzed by using the base state and the regenerative point graphical technique. The juice plant under consideration consists of three distinct units. It is considered that units A and B may be in a complete failed state through partial failure mode but unit C is in only partially failed state. If one of the units A or B or C partially fails then the system works in a reduced state. When any unit is completely failed then the system is in failed state and no unit can fail further when the system is in a failed state. A technician is always available to repair the failed unit. In this paper, the failure time and repair time follow general distributions. Tables are used to describe the reliability measures such as mean time to system failure, availability and profit values of juice plant.
Keywords: Reliability, juice plant, repair time, mean sojourn time and profit.
I. Introduction
Nowadays, manufacturers have to produce their products continuously to meet the increasing demands of their products which are possible by making their productions as efficient as possible. This paper discusses the MTSF, availability and profit of a juice plant with priority in repair using the regenerative point graphical technique under specified conditions. A large amount of research work has been done on repairable systems such that Bao and Mays [2] analyzed the hydraulic reliability of water distribution systems under demand, pipe roughness and pressure head. Gnedenko and Igor [7] explored reliability and probability studies for engineering purposes. Jack and Murthy [9] discovered the role of limited warranty and extended warranty for the product. Wang and Zhang [19] examined the repairable system of two non identical components under repair facility using geometric distributions. Diaz et al. [5] threw light on the warranty cost management system. Kumar and Goel [15] explored the idea of an imperfect switch on redundant systems in banking industry. Goyal [8] described the availability and behavior of single unit system under preventive maintenance and degradation after complete failure using RPGT. Kumar and Goel [14] analyzed the preventive maintenance in two unit cold standby system under general distributions. Malik and Rathee [17] threw light on the two parallel units system under preventive maintenance and maximum operation time. Kashid and Kumar [11] examined the availability of two unit system under degradation and subject to the repair facility. Kumar et al. [12] evaluate the effects of washing unit in the paper industry by using the regenerative point graphical technique. Levitin et al. [16] explored the results of optimal preventive replacement of failed units in a cold standby system by using the poisson process. Agarwal et al. [1] analyzed the performance and reliability of water treatment plant under repair facility. Barak et al. [3] threw light on the
availability and profit values of milk plant under repair facility. Kumar et al. [13] described the cold standby redundant system under repair and refreshment facilities subject to inspection. Chaudhary and Sharma [4] explored the parallel non identical units system that gives priority to repair over preventive maintenance. Garg and Garg [6] analyzed the reliability and profit values of briquette machine under neglected faults like sound and overheating. Jia et al. [10] explored the two unit system under demand and energy storage techniques. Sengar and Mangey [18] examined the performance of complicated systems under inspection using copula methodology.
II. System Assumptions
There are following system assumptions:
• The juice plant consists of three distinct units.
• Unit A consists of a washing and storage tank.
• Unit B has grinding, blending, evaporation and pasteurization.
• Unit C has bottling, labeling and packing units.
• It is considered that units A and B may be in a complete failed state through partial failure mode but unit C is in only partially failed state.
• Failure and repair times follow general distributions.
• The failed unit works like a new unit after repair.
III. System Notations
There are following system notations:
Sr
I -
ssff
m - cycle
>7
m - cycle
Uk ,k
A M'i
V,
A / A / a B / B / b C / C / c
Äy , Ä2 ,
wy, w2, w3 w4, w5 O
rth directed simple path from state 'i' to state f where V takes the positive integral values for different directions from state 'i' to state j'. A directed simple failure free path from state 4 to state 'i'.
A circuit (may be formed through regenerative or non regenerative / failed state) whose terminals are at the regenerative state 'm'.
A circuit (may be formed through the unfailed regenerative or non regenerative state) whose terminals are at the regenerative 'm' state.
Probability factor of the state 'k' reachable from the terminal state 'k' of 'k' cycle.
The probability factor of state 'k' reachable from the terminal state 'k' of k cycle . Mean sojourn time spent in the state 'i' before visiting any other states.
Total unconditional time spent before transiting to any other regenerative state while the system entered regenerative state 'i' at t=0.
Expected waiting time spent while doing a job given that the system entered to the regenerative state 'i' at t=0.
System's first unit is in the operative state/reduced state/failed state. System's second unit is in the operative state/reduced state/failed state. System's third unit is in the operative state/reduced state/failed state. The constant partial failure rate of the unit A/B/C respectively. The constant complete failure rate of the unit A/B respectively. Fixed repair rate of the unit A/B/C after partial failure respectively. Fixed repair rate of unit A/B after the complete failure respectively. Upstate/ reduced state/ failed state
IV. Circuits Descriptions
The individual circuit description is given by the table 1:
Table 1: Circuit Descriptions
Primary, Secondary and Tertiary Circuit at the vertex (/)
i (C1) (C2) (C3)
0 (0,1,0), (0,2,0), (0,3,0) (0,1,4,0), (0,2,5,0) Nil Nil
1 (1,0,1) (0,2,0), (0,3,0) Nil
2 (2,0,2) (0,1,0), (0,3,0) Nil
3 (3,0,3) (0,1,0), (0,2,0) Nil
4 (4,0,1,4) (0,1,0), (0,2,0) (0,3,0), (1,0,1) (2,0,2), (3,0,3)
5 (5,0,2,5) (0,1,0), (0,2,0) (0,3,0), (2,0,2) (1,0,1), (3,0,3)
Figure 1: State Transition Diagram
where, S0 = ABC , S1 = ABC, S2 = ABC, S3 = ABC, S4 = aBC, S5 = AbC
V. Transition Probabilities The transition probabilities are following
Po,1 = /(^1 + ^2 + ¿3^ P 0,2 = ^2 /(^1 + ^2 + ^3) , P 0,3 = ^3 /(^1 + ^2 + Pl,0 = W /(W + ^4 ) , P1,4 = ^4 /(W + ^4 ) , P2,0 = W2 /(w2 + )
p2,5 = ¿5 /(w2 + ¿5 ) , p3,0 = p4,0 = p5,0 = 1 It has been conclusively established that
P01 + P03 = 1 , P10 + p12 + P14 = 1 , P21 + P27 = 1 P31 + P38 = 1 P41 + P45 = 1 , P56 = P76 = P86 =1, P31 + P318(65)« = 1
P10 + P12 + P114 + P11.4(56)W = 1' P21 + P21.7(65)» = 1
(1)
(2)
VI. Mean Sojourn Time
Let /Ui represents the mean sojourn time. Mathematically, the time taken by a system in a particular state becomes
Pi = Z mi j = J P(T > t) dt. j 0
and jU0 = + A2 +^3), Ml = 1/(wl +^4) , M2 = 1/(w2 +^5) ¿MO = 1/(ws), M4 = 1/(w4 ), M5 = 1/(w5)
(3)
VII. Evaluation of Parameters
All reliability parameters (such as mean time to system failure, availability, busy period of the server and expected number of visits) are determined by using the regenerative point graphical technique. The probability factors of all the reachable states from the base state '0' are given below
U0 0 = (0,1,0) + (0,2,0) + (0,3,0) = 1, U0 j =
Àj + ^2 +
U 0,2 =
^2
U 0,3 =-
A3
U 0,4 =■
(A1 + A2 + A3) ' (A1 + A2 + A3)(w1 +A4)
I. Mean Time to System Failure (MTSF)
U 0,5 =-
(4 + ^ +^3)(W2 + A5)
The regenerative un-failed states (¿=0, 1, 2, 3) to which the system can transit (with initial state 0) before entering to any failed state (using base state ^=0) then MTSF becomes
T0 =
T0 =
2 Sri = 0
pr(0 SrW >i)\.M.
n k - 0 i1 " Vkkt
n
1~ZSr
pr (0 ^&) > 0)
n
k2 * 011 "
2 2
(W1 + ^4 )(W2 + )(W3 + A3 ) + W3 [^1 (W2 + ^5 ) + ( W1 + ^4 )]
W3[(^1 + A2 + ^3)(W1 + ^4)(W2 + ^5) - A1W1 (w2 + ^5) - A2W2(W1 + ^4)]
(4)
II. Availability of the system
The system is available for use at regenerative states j=0, 1, 2, 3 with ^=0 then the availability of system is defined as
A0 =
3 Sr j = 0
pr (0-
Sr
>j)f -Mj
n k, - o i1 - Vk~k
ri
2 Sr i = 0
pr(0
Sr
>i) r-Mi
n k, - 0 Í1 - fc
"2
2 2
A =
w4 w5[(wi + Ä4 )(w2 + )(w3 +^3) + W3(^i( W2 +^5) + ^2(^1 +^,4)}]
W4 W5 (W1 + ^.4 )(W2 + ^5 )(W3 + 23) + W3 W5 (w2 + ^5 )(W4 + )
+ ^2 W3 W4(W1 +^4)(W5 +^5)
(5)
III. Busy Period of the Server
The server is busy due to repair of the failed unit at regenerative states j= 1, 2, 3, 4, 5 with £ = 0 then the fraction of time for which the server remains busy is defined as
B0 =
2 Sr-
j=1
pr (0-
Sr
■>j) \-Vj
n k * 0 i1 - Vuk
1
11
2 Sri = 0
pr(0
Sr
■>i) >-Mi
n k„ - 0 i1 - VkZk
2
2ft2
B0 =
W4 W5^3(W1 +^4)(W2 +^5) + ^1W3 W5(W2 + ^5)^4 + ^.4)
+ ^-2 W3 W4W +^4)(w5 +^.5)
W4 W5(W1 +^4)(W2 + ^5 )(W3 +^3) + ^W3 W5(W2 + Ä5 )(W4 + ^.4)
+ ^-2 W3 W4 (W1 +^4)(w5 +^.5)
(6)
IV. Estimated number of visits made by the server
The repairman visits at regenerative states j= 1, 2, 3 with £=0 then the number of visits by the repairman is defined as
V0 =
3
2 Sr-
j=1
pr(0 Sr > j)
n k * 0 i1 - Vkkk
1
11
5
2 Sri = 0
pr(0 Sr > i) k^.
n k, * 0 i1 - fe
2
22
V0 =
W4 W5 À3 (W1 +^4)(W2 + ^5)) + ^1W3 W4 W5 (W2 + ^5)
+ ^2 W3 W4 W5 (W1 + ^.4 )
W4W5 (W1 + ^4 )(W2 + ^5 )(W3 + ^3) + À1W3 W5 (W2 + ^5 )(W4 + ^4)
+ ^2 W3 W4 (W1 +^4)(W5 +^.5)
(7)
V. Profit Analysis
The profit function may be used to do a profit analysis of the system and it is given by
P = EoA0 - ExB0 - E2V0
where, E0 = 5000 (Pay per unit uptime of the system)
E1 = 1000 (Charge per unit time for which server is busy due to repair) E2 = 500 (Charge per visit of the server)
VIII. Discussion
Table 2 describes the nature of mean time to system failure of the juice plant having an increasing trend corresponding to increment in repair rate (wi). In this table, the values of parameters Xi=0.2, ^2=0.3, ^3=0.15, ^4=0.25, ^5=0.4, wi=0.35, W3=0.4, W4=0.45, W5=0.5 respectively taking as constant for the simplicity. When Xi=0.2 changing into Xi=0.25; ^2=0.3 changing into ^2=0.35 and ta=0.15 changing into ^3=0.2 then MTSF values have decreasing trends.
Table 2: MTSF vs. Repair Rate
W2 1 Xi=0.2, ^2=0.3 ta=0.15, ^4=0.25 Xs=0.4, w1=0.35 w3=0.4, w4=0.45 w5=0.5 ^1=0.25 ^2=0.35 ^3=0.2
0.1 4.0647311 3.8613371 3.218144 3.775294
0.2 4.1010786 3.8894957 3.235483 3.806505
0.3 4.1312067 3.9129156 3.250148 3.833119
0.4 4.1565858 3.9327001 3.262713 3.856081
0.5 4.1782569 3.9496349 3.273599 3.876094
0.6 4.1969772 3.9642941 3.283122 3.893693
0.7 4.2133109 3.9771076 3.291522 3.90929
0.8 4.2276872 3.9884034 3.298987 3.923207
0.9 4.240438 3.998436 3.305664 3.935703
1 4.2518242 4.007406 3.311673 3.946984
Table 3 explores the increasing trends of availability corresponding to increments in repair rate (w2) where the system's other parameters possess constant values. When the failure rate of unit changes (Xi=0.2 to 0.25), (^2=0.3 to 0.35) and (^3=0.15 to 0.2) then the availability of system declines.
Table 3: Availability vs. Repair Rate
W2 1 ^1=0.2, ^2=0.3 ^3=0.15, ^4=0.25 Xs=0.4, w1=0.35 w3=0.4, w4=0.45 w5=0.5 ^1=0.25 ^2=0.35 ^3=0.2
0.1 0.25028 0.214076 0.185918 0.190417
0.2 0.364639 0.319498 0.281347 0.293845
0.3 0.427923 0.381465 0.338725 0.358019
0.4 0.467056 0.421888 0.376681 0.401314
0.5 0.493101 0.450148 0.403461 0.432266
0.6 0.511375 0.470912 0.423258 0.455361
0.7 0.524719 0.48675 0.438421 0.473168
0.8 0.534774 0.499191 0.450366 0.487264
0.9 0.542545 0.509198 0.45999 0.498663
1 0.548679 0.517404 0.46789 0.508046
Table 4 explores the trend of profit values with respect to repair rate (w2) and its value increase corresponding to increments in repair rate (wi) where the system's other parameters possess constant values. When the failure rate of unit changes (Xi=0.2 to 0.25), (ta=0.3 to 0.35) and (ta=0.15 to 0.2) then the profit of system declines.
Table 4: Profit vs. Repair Rate
1 Xi=0.2, ^2=0.3 ta=0.15, ^4=0.25 Xs=0.4, W1=0.35 W3=0.4, W4=0.45 w5=0.5 ^1=0.25 ^2=0.35 ^3=0.2
0.1 2692.9 2046.417 1588.805 1690.491
0.2 4428.411 3640.771 3027.839 3256.171
0.3 5389.1 4576.626 3891.418 4226.372
0.4 5983.344 5186.276 4461.578 4880.064
0.5 6378.977 5611.935 4863.085 5346.797
0.6 6656.653 5924.286 5159.353 5694.6
0.7 6859.482 6162.259 5385.877 5962.46
0.8 7012.365 6348.978 5563.998 6174.235
0.9 7130.568 6498.993 5707.276 6345.293
1 7223.9 6621.895 5824.711 6485.948
IX. Conclusion
The performance of the juice plant is discussed using the regenerative point graphical technique (RPGT). The above tables explore that when the repair rate increases then the MTSF, system's availability and profit values also increase but when the failure rate increases then the MTSF, availability and profit values decrease. It is clear that RPGT is helpful for industries to analyze the behaviour of the products and components of system.
References
[1] Agrawal, A., Garg, D., Kumar, A. and Kumar, R. (2021). Performance analysis of the water treatment reverses osmosis plant. Reliability Theory and Applications, 16(3): 16-25.
[2] Bao, Y. and Mays, L. W. (1990). Model for water distribution system reliability. Journal of Hydraulic Engineering, 116(9): 1119-1137.
[3] Barak, M. S., Garg, R. and Kumar, A. (2021). Reliability measures analysis of a milk plant using RPGT. Life Cycle Reliability and Safety Engineering, 10(3): 295-302.
[4] Chaudhary, P. and Sharma, A. (2022): A two non-identical unit parallel system with priority in repair and correlated life times. Reliability Theory and Application, 17(1): 113-122.
[5] Diaz, V. G., Gomez, J. F., Lopez, M., Crespo, A. and Leon, P. M. (2009). Warranty cost models State-of-Art: A practical review to the framework of warranty cost management. ESREL: 20512059.
[6] Garg, D. and Garg, R. (2022). Reliability analysis and profit optimization of briquette machine by considering neglected faults. Reliability Theory and Application, 17(67): 210-222.
[7] Gnedenko B. and Igor A.U. (1995). Probabilistic reliability engineering. John Wiley and Sons.
[8] Goyal, V. (2015). Availability modeling and behavioral analysis of a single unit system under preventive maintenance and degradation after complete failure using RPGT. Arya Bhatta Journal of Mathematics and Informatics, 7(2): 381-390.
[9] Jack, N. and Murthy, D. P. (2007). A flexible extended warranty and related optimal strategies. Journal of the Operational Research Society, 58(12): 1612-1620.
[10] Jia, H., Peng, R., Yang, L., Wu, T., Liu, D. and Li, Y. (2022). Reliability evaluation of demand based warm standby systems with capacity storage. Reliability Engineering and System Safety, 218: (108132)
[11] Kashid, D. U. and Kumar, R. (2017). Availability modeling of two units system subject to degradation when intermediate repair is feasible using RPGT. International Multidisciplinary Conference on Commerce, Management, Technology, Engineering and Environmental Sciences & International Conference on Humanities and Social Sciences, (Andheri) Mumbai.
[12] Kumar, A., Garg, D. and Goel, P. (2019). Mathematical modeling and behavioral analysis of a washing unit in paper mill. International Journal of System Assurance Engineering and Management, 10: 1639-1645.
[13] Kumar, A., Garg, R. and Barak, M. S. (2021). Reliability measures analysis of a cold standby system subject to refreshment. International Journal of System Assurance Engineering and Management, 14: 147-155.
[14] Kumar, J. and Goel, M. (2016). Availability and profit analysis of a two-unit cold standby system for general distribution. Cogent Mathematics, 3(1): 1262937.
[15] Kumar, S. and Goel, P. (2014). Availability analysis of two different units system with a standby having imperfect switch over device in banking industry. Arya Bhatta Journal of Mathematics and Informatics, 6(2): 299-304.
[16] Levitin, G., Finkelstein, M. and Xiang, Y. (2020). Optimal preventive replacement for cold standby systems with reusable elements. Reliability Engineering and System Safety, 204: (107135).
[17] Malik, S. C. and Rathee, R. (2016). Reliability modelling of a parallel system with maximum operation and repair times. International Journal of Operational Research, 25(1): 131-142.
[18] Sengar S. and Mangey R. (2022). Reliability and performance analysis of a complex manufacturing system with inspection facility using copula methodology. Reliability Theory & Applications, 17(71): 494-508.
[19] Wang, G. J. and Zhang, Y. L. (2007). An optimal replacement policy for repairable cold standby system with priority in use. International Journal of Systems Science, 38(12): 1021-1027.
