CC BY
11
Rahul, Mohit Yadav, Hemant Kumar RT&A, No 1 (77) PROFIT ANALYSIS OF REPAIRABLE JUICE PLANT_Volume I9, March 2024
PROFIT ANALYSIS OF REPAIRABLE JUICE PLANT
Rahul1, Mohit Yadav2* and Hemant Kumar3
^Department of Mathematics, University Institute of Sciences Chandigarh University, Mohali, Chandigarh 3SOET, Raffles University, Neemrana, Rajasthan
*Corresponding Author 22msm40223@cuchd.in, mohit.e15793@cumail.in, hemantkumar@rafflesuniversity.edu.in
Abstract
Juice is a non-fermented beverage that is obtained by squeezing fruits to increase immunity. Generally, juice contains calcium, vitamin, iron, etc. to give the refresh tests. There are multiple steps to store the juice at large levels such as storing, grinding pasteurization, etc. In this paper, the performance and reliability measures of a juice plant are discussed. The juice plant has three distinct units. Unit A has washing and storage tank, unit B has grinding, blending, evaporation and pasteurization, and unit C has bottling, labeling and packing units. If any unit partially fails then the system works to a limited extent. A technician is always available to repair the failed unit. The system fails when one unit completely fails. In this paper, the failure time and repair time follow general distributions. The regenerative point graphical technique is used to explore the reliability measures.
Keywords: Reliability measures, juice plant, evaporation and pasteurization.
I. Introduction
Manufacturers must constantly innovate their products in order to keep up with the rising demand for their products, which is made feasible by optimizing their manufacturing processes. The MTSF, availability and profitability of a juice factory with priority in repair are discussed in this study by utilizing the regenerating point graphical technique under specific circumstances.
Barlow et al. [2] investigated the reliability theory with redundancy and system availability while taking into account the significance of individual system components. The reliability study of a single unit system with non-repairable spare units and its optimization applications was covered by Nakagawa and Osaki [12]. Balagurusamy [1] described the terms related to the system's meantime, failure, repair, redundancy, maintainability, availability, etc. Tuteja and Malik [16] examined the dependability of two distinct single-unit models with three operating modes and various repair procedures applied to the repairman. Malik [11] examined a single-unit system with a server under inspection. Pawar et al. [13] threw light on an operating system under different climates having repair at varying levels of damages subject to inspection.
Gupta [4] talked about employing a base state to analyze a single-unit system. The reliability analysis of a one-unit system with finite vacations was examined by Liu and Liu [10]. The dependability metrics of a repairable stochastic model on the production of printed circuit boards were given by Kumar and Batra [9]. Chaudhary et al. [3] studied the valuable parameters for the nature of the distillery system having three distinct units and a single server facility using the
Rahul, Mohit Yadav, Hemant Kumar RT&A, No 1 (77) PROFIT ANALYSIS OF REPAIRABLE JUICE PLANT_Volume I9, March 2024
regenerative point graphical technique. Kumar et al. [8] threw light on the preventive maintenance
of a sustainable one unit system under degradation facilities. Sharma and Goel [15] described the
nature of whole-grain flour mills having two units using base state and regenerative point
techniques. Kumar and Saini [5] described the fault detection concept in stochastic computing
device under repair and replacement by an expert repairman. A redundant system with a first
come, first served repair policy was examined by Kumar et al. [7] under different weather. Sengar
and Mangey [14] analyzed the reliability measures of a complex manufacturing system with an
inspection facility. Kumar et al. [6] analyzed the reliability and performance of two unit system
under inspection facility.
II. System Assumptions
To describe the juice plant, there are following assumptions
• The juice plant consists of three distinct units A, B and C.
• 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.
• Unit A has washing and storage tank.
• Unit B has grinding, blending, evaporation and pasteurization.
• Unit C has bottling, labeling and packing units.
• Failure rate and repair rate are generally distributed and are independent.
• The repaired unit functions just like a brand-new one.
III. System Notations To explain the juice plant, there are following notations
• Sr . rth directed simple path from state 'i' to state 'j' where 'r' takes the positive
' J
integral values for different directions from state 'i' to state 'j'. ^ sffyj A directed simple failure free path from state 4 to state 'i'.
m — cycle A circuit (may be formed through regenerative or non regenerative / failed
state) whose terminals are at the regenerative state 'm'. m — cycle A circuit (may be formed through the unfailed regenerative or non
regenerative state) whose terminals are at the regenerative 'm' state. Ufck Probability factor of the state 'k reachable from the terminal state 'k of 'k
cycle.
U— The probability factor of state 'k reachable from the terminal state 'k of
k cycle.
j Mean sojourn time spent in the state 'i' before visiting any other states.
f. Total unconditional time spent before transiting to any other regenerative state while the system entered regenerative state 'i' at t=0.
q Expected waiting time spent while doing a job given that the system entered to
the regenerative state 'i' at t=0. A / A / a First unit is in the operative state/reduced state/failed state.
B / B / b Second unit is in the operative state/reduced state/failed state.
C / C / c Third unit is in the operative state/reduced state/failed state.
\ , Fixed partial failure rate of the unit A/B/C respectively.
, Jl5 Fixed complete failure rate of the unit A/B respectively.
Rahul, Mohit Yadav, Hemant Kumar RT&A, No 1 (77) PROFIT ANALYSIS OF REPAIRABLE JUICE PLANT_Volume 19, March 2024
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 Primary, secondary and tertiary circuits are used to find the base state such that
Table 1: Circuit Descriptions
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, = ABC , S = ABC, S2 = ABC S3 = ABC, S4 = aBC, S5 = AbC
w1, w2, w3 w4, w5 Q G> □
V. Transition Probabilities There are following transition probabilities
po,1 = /(ll + l2 + l), p0,2 = l2/(l1 +l2 + l3) , p0,3 = l /(ll +12 + ) pi,0 = W1 /(wi + I4 ), pi,4 = I4 /(wi + I4 ), pi,0 = w2 /(w2 + l )
p2,5 = l5 /(w2 + l5 ), P3,0 = p4,0 = P5,0 = 1 (1)
It has been conclusively established that
poi + p03 =1, pi0 + pi2 + pi4 = 1, p21 + p27 = 1 p31 + ?38 = 1 p41 + p45 = 1 , p56 = P76 = P86 =1, p31 + p3lg(65f = 1
pi0 + pi2 + pi1.4 + ^1.4^ = 1, p21 + p2i.7(65) n = 1 (2)
VI. Mean Sojourn Time
X
Time taken by a system in a particular state becomes, f = X mt, j = j P(T > t) dt.
J 0
f0 = l/(ll + l2 +l3)
fl = l/(w +I4) , f2 = l/(w2 +I5)
f3 (t) = l/(w3), f 4 = l/(w4) , f 5 = l/(w5) (3)
VII. Evaluation of Parameters
Using the circuit table, '0' is used as the base state to calculate the reliability 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,l,0) + (0,2,0) + (0,3,0) = l, U01 = l
Â\ + Â2 + A3
U 0,2 =
A
A
— , U03 = -A1 + A2 + +A3 ' A + A2 + A3 )
U0,4 =-
AA
1A4
(A1 + A2 + A3)(w1 + A4)
, U 0,5 =
A2A5
(Ai +A2 + A3 )(^2 +A5)
I. Mean Time to System Failure
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 4=0) then MTSF becomes
T0 =
3
z Sri = 0
pr(0-
i)U
n* - 0 i1 - Vklh
1-ZSr
pr(0 Sr(sf ) > 0)
nk2 - 0 i1 - Vk2k2
>
Tn = =-
(wi + A4 )(w2 + A5 )(W3 + A3 ) + W3 A (W2 + A5 ) + ^2 (w\ + A4 )] W3 [A + ^2 + A3 )(Wi + A4 )(W2 + A5 )
(4)
AlWi(W2 + A5) -^2 W2 (Wi + A4)]
II. Availability of the system
The system is available for use at regenerative states j=0, 1, 2, 3 with 4=0 then the availability of system is defined as
pr(0 Sr > i)
if Sr -+j)\fj .»j
3 \pr(0 5
A0 = z Sr > z Sr
j = 0 n k ki * 01 i Vkiki J , i = 0
Ao =
n
^ * 011 - Vk2k2
w4 w5[( w1 + A4)(w2 + A5)(w3 + A3)
+ w3 {Ai (w2 + ¿5 ) + a2 (wi + a4 )}]
w4 w5 (wi + A4 )(w2 + A5 )(w3 + A3) + Aiw3 w5 (w2 + A5 )(w4 + A4) + A2 w3 w4 (wi + A4 )(w5 + A5)
(5)
>
III. Busy Period of the Technician
The Technician is busy due to repair of the failed unit at regenerative states j= 1, 2, 3, 4, 5 with 4 = 0 then the fraction of time for which the server remains busy is defined as
Bo =
5
z Sr j = i
pr(0 -
Sr
n k * 0 ii -
Bn =
i - L kiki
w4 w5A3(wi +A4)(w2 +A5) + Aiw3 w5 (w2 + A5 )(w4 + A4) + A2 w3 w4 (wi + A4 )(w5 + A5)
5
> z Sr
i = 0
w4 w5 (wi + A4 )(w2 + A5 )(w3 + A3) + Aiw3 w5 (w2 + A5 )(w4 + A4) + A2 w3 w4 (wi + A4 )(w5 + A5)
pr(0 —i)
n ^ * 0 ii - Vkk
2 k 2
(6)
IV. Estimated number of visits made by the Technician
The technician visits at regenerative states j= 1, 2, 3 with 4=0 then the number of visits by the repairman is defined as
V0 =
3
z Sr j = 1
pr(0 —Sr^ j )
n * - 0 i1 -
Vn =
1 - - L kiki w4 w5A3 (w1 + A4 )(w2 + A5)) + A1w3 w4 w5(w2 +A5) + A2 w3 w4 w5(w1 +A4)
5
> z Sri
i = 0
_
Pr(0 ——0 V.u.
n - 0i1 - Vk2 k2
w4 w5(w1 + A4 )(w2 + A5 )(w3 +A3) + A1w3 w5 (w2 + A5 )(w4 + A4 ) + A2 w3 w4 (w1 + A4 )(w5 + A5 )
(7)
V. Profit Analysis
The profit function may be used to do a profit analysis of the system and it is given by
P = Eo AO — Ei Bo — E2VO (8)
where, Eo = 25000 (Revenue per unit uptime of the system)
E = 500 (Cost per unit time for which technician is busy due to repair) E2 = 200 (Cost per visit of the technician)
VIII. Discussion
Tables 2, 3 and 4 described the nature of mean time to system failure, availability and profit values
Table 2: MTSF vs. Repair Rate (w)
w2 1 Ai=0.3, A2=0.4 A3=0.25, A4=0.35 A5=0.5, wi=0.4 w3=0.5, w4=0.5 w5=0.6 Ai=0.4 A2=0.5 A3=0.3
0.4 3.628692 3.333333 3.090278 3.037974
0.45 3.675035 3.368794 3.132184 3.062553
0.5 3.720609 3.403509 3.173516 3.086409
0.55 3.765432 3.4375 3.214286 3.109568
0.6 3.809524 3.47079 3.254505 3.197278
0.65 3.852901 3.503401 3.294183 3.153901
0.7 3.895582 3.535354 3.333333 3.175126
0.75 3.937583 3.566667 3.371965 3.195751
0.8 3.97892 3.59736 3.410088 3.2158
0.85 4.019608 3.627451 3.447712 3.235294
Table 3: Availability vs. Repair Rate (w)
W2 1 Xi=0.3, X2=0.4 Xa=0.25, X4=0.35 X5=0.5, wi=0.4 w3=0.5, w4=0.5 w5=0.6 Xi=0.4 X2=0.5 X3=0.3
0.4 0.623324 0.604782 0.542904 0.58624
0.45 0.628307 0.609813 0.547959 0.591319
0.5 0.633159 0.614717 0.552904 0.596275
0.55 0.637887 0.619499 0.557741 0.601111
0.6 0.642494 0.624164 0.562476 0.605834
0.65 0.646985 0.628716 0.56711 0.610447
0.7 0.651365 0.633159 0.571646 0.614953
0.75 0.655637 0.637497 0.576089 0.619357
0.8 0.659806 0.641734 0.58044 0.623662
0.85 0.663876 0.645873 0.584703 0.62787
Table 4: Profit vs. Repair Rate (w)
W2 1 ^1=0.3, X2=0.4 X3=0.25, X4=0.35 X5=0.5, w1=0.4 w3=0.5, w4=0.5 w5=0.6 X1=0.4 X2=0.5 X3=0.3
0.4 2438.338 2386.076 2019.52 2333.814
0.45 2467.262 2415.876 2049.467 2364.49
0.5 2495.431 2444.927 2078.759 2394.423
0.55 2522.874 2473.257 2107.417 2423.64
0.6 2549.618 2500.892 2135.461 2452.166
0.65 2575.691 2527.857 2162.912 2480.023
0.7 2601.117 2554.178 2189.787 2507.239
0.75 2625.919 2579.875 2216.104 2533.831
0.8 2650.121 2604.972 2241.881 2559.823
0.85 2673.744 2629.49 2267.135 2584.836
of the juice plant having an increasing trend corresponding to repair rate (w2). In these tables, the values of parameters Xi=0.3, ta=0.4, ta=0.25, ta=0.35, ta=0.5, wi=0.4, w3=0.5, w4=0.5, w5=0.6 respectively taking as constant for the simplicity. When Xi=0.3 changing into Xi=0.4; ^2=0.4 changing into ^2=0.5 and ta=0.25 changing into ta=0.3 then MTSF, availability and profit values have decreasing trends.
IX. Conclusion
The performance of the juice plant is discussed using the regenerative point graphical technique. 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 a system.
X. Future Scope
It is analyzed that the role of the regenerative point graphical technique for the juice plant will be beneficial and also used by the management, manufacturers and the persons engaged in reliability engineering and working on analyzing the nature and performance analysis of the system.
References
[1] Balagurusamy, E. (1984). Reliability Engineering. Tata McGraw-Hill Education.
[2] Barlow, R. E., Proschan, F. and Hunter, L. C. (1965). Mathematical Theory of Reliability John Wiley and Sons Inc. New York, 4, 927-929.
[3] Chaudhary, N., Goel, P. and Kumar, S. (2013). Developing the reliability model for availability and behaviour analysis of a distillery using Regenerative Point Graphical Technique. International Journal of Informative and Futuristic Research, 1(4), 26-40.
[4] Gupta, V. K. (2011). Analysis of a single unit system using a base state. Aryabhatta Journal of Mathematics & Informatics, 3(1), 59-66.
[5] Kumar, A. and Saini, M. (2018). Stochastic modeling and cost-benefit analysis of computing device with fault detection subject to expert repair facility. International Journal of Information Technology, 10, 391-401.
[6] Kumar, A., Garg, R., & Barak, M. S. (2023). Reliability measures of a cold standby system subject to refreshment. International Journal of System Assurance Engineering and Management, 14(1), 147-155.
[7] Kumar, A., Pawar, D. and Malik, S. C. (2020). Reliability analysis of a redundant System with 'FCFS' repair policy subject to weather conditions. International Journal of Advanced Science and Technology, 29(3), 7568-7578.
[8] Kumar, J., Kadyan, M. S., Malik, S. C. and Jindal, C. (2014). Reliability measures of a single-unit system under preventive maintenance and degradation with arbitrary distributions of random variables. Journal of Reliability and Statistical Studies, 77-88.
[9] Kumar, R. and Batra, S. (2012). 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(10), pp. 432-435.
[10] Liu, R. and Liu, Z. (2011). Reliability analysis of a one-unit system with finite vacations. In MSIE, 248-252.
[11] Malik, S. C., Chand, P. and Singh, J. (2008). Stochastic analysis of an operating system
with two types of inspection subject to degradation. Journal of Applied Probability and
Statistics, 3(2), 227-241.
[12] Nakagawa, T. and Osaki, S. (1976). Reliability analysis of a one unit system with unrepairable spare units and its optimization applications. Journal of the Operational Research society, 27(1), 101-110.
[13] Pawar, D., Malik, S. C. and Bahl, S. (2010). Steady state analysis of an operating system with repair at different levels of damages subject to inspection and weather conditions. International Journal of Agriculture and Statistical Sciences, 6(1), 225-234.
[14] 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.
[15] Sharma S. and Goel, P. (2015). Behavioral Analysis of Whole Grain Flour Mill Using RPGT. International Journal of Engineering Technology, Management and Applied Sciences, 3, 194-201.
[16] Tuteja, R. K. and Malik, S. C. (1992). Reliability and profit analysis of two single unit models with three modes and different repair policies of repairmen who appear and disappear randomly. Microelectronics Reliability, 32(3), 351-356.
