RELIABILITY MODELING OF A BUTTER CHURNER AND CONTINUOUS BUTTER MAKING PRODUCTION SYSTEM
Upasana Sharma1 and Drishti 2*
1,2Department Statistics, Punjabi University, Patiala 147002, India usharma@pbi.ac.in, drish2796@gmail.com
Abstract
In the dairy plant, an investigation into the machine that makes butter was subjected to a reliability study in relation to the seasonal demand. In the process of expanding the butter churner into a machine that can make butter continuously, a more reliable operational model was devised. Both the models and the data acquired with MATLAB have been subjected to availability and reliability testing and analysis. In addition, the graphical analysis was carried out with the help of Code Blocks and Excel. A comparison of the two models was then covered as the final topic. It was discovered that (a) the extended model was superior to the current model, (b) the failure rate of the existing line increased, which implies that a new machine needs to be added to the line to share the load, which results in improved production, and (c) the failure rate of the extended model was lower than the failure rate of the existing model. (c) in order to maximise profits while simultaneously minimising losses The effectiveness of the system ought to be enhanced by performing routine maintenance during both the summer and the winter.
Keywords: Butter churner, continuous butter making, seasons, semiMarkov process, profit.
1. Introduction
As a result of high levels of "lifetime" engineering uncertainty, reliability engineering deals with predicting, preventing, and managing engineering failures. Costs of failures caused by equipment failure, parts costs, repairs, and personnel costs are all taken into account when reliability engineering is conducted. Industry engineers now put their effort on efficiency and high quality production. This can be achieved by improving system performance. When it comes to industrial applications on food production lines, ensuring a high level of reliability is highly important; however, reliability itself can be complex, many interconnected variables must be taken into account when guiding and assessing various levels of reliability.
Using maintenance regimes [9] processed site performance improvement in the dairy industry. [8] presented a case study on optimised performance of butter oil production. Based on real data [5] represented generation of wind power and electric power demand. Reliability analysis where operation is effected by temperature conditions was given by [2] and [1]. RAM analysis for modeling complex engineering systems was used by [6].
Introducing redundancy into a system can enhance its reliability. Redundancy with standby (redundant) units refers to the usage of additional units with the primary unit of the system, with the additional unit(s) becoming operational and performing all the desired functions with equivalent parameters upon the failure of the primary unit. Standby redundancy technique was used by several researchers to enchance system performance namely [3], [4], [7] etc. Work on standby units in a dairy industry was done by [10], [11] and [12]. Description of the systems
In model 1, the system which we have considered consists of a churner that works in both the seasons i.e., summer and winter. In winters, due to high demand system is always operating
unless a failure occurs that can be due to electricity hault or any fault in the churner. In summers, due to less demand the system sometimes goes to cold standby state when there is no demand. In model 2, the system consists of churner and continuous butter making. Both the units starts to operate to accomodate the demand in winters, on the failure of any one unit the system works on reduced capacity. In summers, the butter churner is operative and CBM is in cold standby state, it operates on the failure of the churner. The system either goes to cold standby or maintenance state when there is no demand.
Methods
Both the models have been analyzed using semiMarkov process and regenerative point technique probabilistically.
2. Annotations
Table 1:
Notations of the model 1
Notations Descriptions
A Failure rate of the main unit i.e. Churner.
A1 Rate of electricity failure due to which churner stops operating.
Y Rate at which churner goes to down state when demand is less than
production.
5 Rate when churner comes to operative state from a cold standby
state.
a Rate of going from winters to summers.
ft Rate of going from summers to winters.
ch Main unit of the system i.e.ch.
S Summer season.
W Winter season.
Och Main unit of the system is in operating state.
d > p Demand is more than production.
d < p Demand is less than production.
CSch Main unit is in cold standby state.
Frch Main unit is under repair.
HCSch Main unit in cold standby state due to electricity hault.
G(t),g(t) c.d.f. and p.d.f of time to repair of the main unit.
G1 (t),gi(t) c.d.f. and p.d.f of time to repair the electricity hault. G2(t),g2(t) c.d.f. and p.d.f of time to going back to operating state from down state.
3. Transition Probabilities and Mean Sojourn Time
Various states of the system are shown in figure 3.1 called as state transition diagram. Here, the states So, Si, S2 are operating states, S5 is a cold standby state whereas, states S3, S4, S&, S7 are the failed states. Transition Probabilités
• dQ01 (t)= ße(x+ß)(t) dt
• dQ13 (t) = A1 e(A+Ai)(t) dt
• dQ25 (t) = ye(Y+A+Ai)(t)dt
• dQo2 (t) = ae(K+ß)(t) dt
• dQ14 (t) = Ae(A+Ai)(t) dt
• dQ26 (t) = A1 e(Y+A+Ai)(t) dt
• dQ27 (t) = Ae(7+A+A1)(t) dt
The nonzero probabilities p^ are as follows:
• pij=Qij (™) = JT qijdt • po1
• p02=OTP • p13
• p14 = TTX1 • p25
• p26 = Y+XTAT • p27
• p31 = p62 = g *1 (0) • p41 From the above transition probabilities it is verified that:
ß
a+ß
M A+A1
Y
Y+A+Aj A
Y+A+Aj p?2 = g * (0)
poi + p02 = 1
p25 + p26 + p27 = 1
pi3 + pi4 = 1
Figure 1: State Transition Diagram
The unconditional mean time taken by the system to transit for any regenerative state j when time is counted from the epoch of entrance into state i is mathematically state as:
mj = fo°° tdQij(t)dt = q*j(0) mi3 + mi4 = Ц1
moi + mo2 = Ц0 m25 + m26 + m27 = Ц2
The mean sojourn time ^ in the regenerative state iis defined as time of stay in that state before transition to any other state:
Цо
i
a+ß
Ц2 = Y+A+Ai Ц4 = Ц7 = g* (0)
Ц1 = A+Ai
Ц3 = Ц6 = g*(0)
Ц5 = 1
4. Mean Time to System Failure
The average duration between successive system failures, i.e. MTSF is defined as the expected time for which the system is in operation before it completely fails. Mean time to system failure
Upasana Sharma, Drishti RT&A, No 1 (77)
RELIABILITY OF A BUTTER CHURENR AND CBM SYSTEM Volume 19, March 2024
(MTSF) of the system is determined by considering failed state as absorbing state. When the system starts from the state 0, the mean time to system failure is:
To = lim R(s) = lim 1  (s) = N
0 s—>0 v ' s—►0 s D
where,
N=(^0+^1 Poi)(1P25)+(^2+^5 P25)( P02) D=1P25
5. Availability Analysis of the System in Summers
Availability A, (t) is a measure that allows for a system to repair when failure occurs. The availability of the system is defined as the probability that the system is successful at time t. The long run availability of the system is given by
A0 = lims^0[sA0s (s)] = D
where, N1=^2 P02
Dl=^2+^5 P25+^0 P26 + P27
6. Availability Analysis of the System in Winters
Availability A, (t) is a measure that allows for a system to repair when failure occurs. The availability of a system is defined as the probability that the system is successful at time t. The long run availability of the system is given by
AW = lims^0 [sA0w (s)] = §
where, N2=^1 P01
D2=^1+^4 P14+^3 P13
7. Busy Period Analysis for Repair in Summers Busy period B, (t) in summers is defined as the probability that the repairman is busy at time t when the system entered to a regenerative state i. The total time in which the repairman is busy doing repair of the system in steady state is given by:
B0 = lims^0 [sB0s (s)] = §
where,
N3=P02( P26 H6 + P27^7) D1is already defined above.
8. Busy Period Analysis for Repair in Winters
Busy period B,(t) in winters is defined as the probability that the repairman is busy at time t when the system entered to a regenerative state i. The total time in which the repairman is busy doing repair of the system in steady state is given by:
B0w = lims—[sB0w (s)] = §
where,
N4=Poi(W3 P13 + W4 P14 ) D2is already defined above.
9. Expected Number of Repairs in Summers
Let V, (t) be the expected number of repairs in (0, t) given that the system entered into regenerative state i at i = 0. The expected number of repairs during summers in steady state is given by:
Vr = lims^0 sV" (s) =
N5=P02(1  P25)
D1 is already defined above in equation.
10. Expected Number of Repairs in Winters
Let V, (t) be the expected number of repairs in (0, t) given that the system entered into regenerative state i at i = 0. The expected number of repairs during summers in steady state is given by:
Vr = lims^0 sV" (s) = D2
N6=P01
D2 is already defined above in equation.
11. Profit Analysis of the System
Profit incurred to the system model in steady state is given by
p = (C0 A0 + C1 AW)  (C2 B0 + C3 BW + C4V + C5VW)
where,
C0=Revenue per unit up time in summers. C1=Revenue per unit up time in winters.
C2=Cost per unit up time for which the repairman is busy for repair in summers. C3=Cost per unit up time for which the repairman is busy for repair in winters. C4=Cost per repair in summers. C5=Cost per repair in winters.
12. Graphical Analysis and Conclusion
For further numerical and graphical evaluation, let us assume the repair and failure rates to be exponentially distirbuted g(t) = 6ee(t), g1 (t) = QxeQ1(t)
• p01 = a+ft • p02=a+ft
• p13 = A+A1 • p14 = A+A1
• p25 = 7+A+A1 • p26 = y+A+A1
A 1
• P27 = 7+X+AI • P31 = P62 = 1
• P41 = P72 = 1 • F0 = a+ft
• ^ = A+A1 • = 7+A+A1
• № = V6 = ^ • ^4 = ^7 = 1
• № = 1
The parameters obtained using the original data collected from the Verka Milk Plant, Bathinda, Punjab.
Table 2:
Parameters obtained from data collected
Parameters for Values
model 1
A .00045892
Ai .0002563
gi (t) .04213
g(t) .062981
a .0004314
ß .000526
5 .000155
Y .000955
Co 830000
Ci 1030000
C2 10500
C3 12500
C4 12000
C5 15500
System effectiveness measures evaluated are given below:
Table 3:
Parameters obtained from data collected
Parameters for model 1 Values
Mean time to system failure 9453.77 hrs
Availability in summers .8975
Availability in winters .8984
Busy period for repair in summers .000485
Busy period for repair in winters .0004204
Expected number of repairs in summers .000217
Expected number of repairs in winters .000031
MTSF v/s Failure Rate A with varying Failure Rate
4Z50
Figure 2: MTSF v/s Failure Rate
Figure 3: Profit v/s Failure Rate in Summers
Figure 4: Profit v/s Failure Rate in Winters
Figure 5: Profit v/s Failure Rate in Winters
Figure 6: Profit v/s Failure Rate in Winters
Table 4:
Notations of the model
Figures Descriptions
5 Profit P1 increases as the revenue C0 increases. C2=10500; Profit >=< according to C2, when C2 is >=<Rs.275.53, similarly for C2=20500 where cut off point is Rs.163.577 C2=30500; where cut off point is Rs. 452.675
6 Profit P2 increases as the revenue Ci increases. C3=12500; Profit >=< according to C3, when C1 is >=<Rs.251.85, similarly for C3=22500 where cut off point is Rs.140.469. C3=32500; where cut off point is Rs. 429.089
Figure 3 and figure 4 depicts the trend of mean time to system failure and profit v/s the failure rate. It has been observed that as the failure rate A of the system increases mean time to system failure and profit decreases. It also decreases on increasing failure rate Ai. Figure 5,6 states that profit increases as the cost C1 increases as well it increases with increasing profit C3. MODEL 2 Assumptions Model 2 have the following assumptions:
• The system is operating at the initial stage.
• At the initial stage the churner is operating and continuous butter making is in a cold standby state.
• Both the systems operates during winters due to high demand.
• Only one unit is operating during summers due to less demand.
• In summers it also undergoes maintenance.
• The system sometimes goes to cold standby state in case of no demand in summers.
• The repair is done on the failure of the system.
• Repair rates are assumed to have arbitrary distribution.
• Failure rates are taken to be exponentially distributed.
• After repair the system operates as new.
• The system goes to failed state either on the failure of the churner or due to hault in the electricity.
13. Annotations for model 2 Table 5:
Notations of the model 2
Notations Descriptions
A Failure rate of the churner.
A1 Failure rate of the continuous butter making.
Y Rate at which churner goes to down state when demand is less than production.
S Rate when churner comes to operative state from a cold standby state.
a Rate of going to winters.
ft Rate of going to summers.
ch Unit churner of the system.
cbm Unit continuous butter making of the system.
S Summer season.
W Winter season.
Och Churner is in operating state.
Ocbm CBM is in operating state.
d > p Demand is more than production.
d < p Demand is less than production.
CSch Main unit is in a cold standby state.
CScbm CBM is in a cold standby state.
Frch Churner is under repair.
HCSch Churner is in cold standby state due to electricity hault.
G(t),g(t) c.d.f. and p.d.f of time to repair of the churner.
G1 (t),g1 (t) c.d.f. and p.d.f of time to repair of CBM.
G2(t),g2(t) c.d.f. and p.d.f of time to going back to operating state from maintenance.
14. Model 2
15. Annotations for model 2
16. Transition Probabilités and Mean Sojourn Time
Various states of the system are shown in figure 1.5 called as state transition diagram. Here, the states So, S1, S2, S3, S5 are operating states, S4 is a cold standby state whereas, states S9, S10 are the reduced capacity states and rest are failed states.
ae(«+ft)(0 dt
dQ01 (t) = fte(a+ft)(t) dt dQ19 (t) = A1 e(A+A1)(t) dt dQ23 (t) = A2 e(A+A2+Y) dt dQ25 (t) = Ae(A+A2+Y) dt dQ3,13(t) = AeA(t) G(t)dt
dQo2 (t) dQ1,10(t) = Ae(a+ft)(t) dt
dQ24 (t) = 7e(A+A2+Y) dt
dQ32 (t) = g2 (t)eA(t) dt dQ373)(t) = (AeA(t)(c)1)g2 (t)dt
The
dQ42 (t) = öeö(t) dt
dQ56 (t) = Ai eAl (t) G(t)dt
dQ67 (t) = g2(t)dt
dQ/s (t) = AeA(t) Gi(t)dt
dQ9i (t) = gi(t)eA(t)dt
dQ9lio)(t) = (AeA(t)(c)l)gi(t)dt
dQio,ii(t) = AieAi(f) G(t)dt
dQi3,/(t) = g2 (t)dt nonzero probabilities p,j are as follows:
p/=Qij = /o°° j P02 = Pi,io
a a+ß
A
A+Ai p24 = A+A2+7 P32 = g2 (A) P52 = g2*)(Ai ) P72 = gi*)(A) P9i = gi*)(A) Pio,i = g(*)(Ai) From the above transition probabilities it is verified
Poi + Po2 = i P23 + P24 + P25 = i
, (i3) 1 P32 + P37 ) = i
P52+p57) = i
(s)
P72 + p/5) = i
P9i + P9112O) = i
( )
P 1 o,l + Pio,9
dQ52 (t) = g(t)eAl (t) dt
dQ57)(t) = (Ai eAi (t))g(t)dt dQ72 (t)= gi (t)eA(t) dt dQ(S) = (AeA(t)(c)l)gi (t)dt dQ9,i2(t) = AeA(t) Gi(t)dt dQio,i(t)= g(t)eAl(t)(t) dt dQioi9(t) = (Ai eAi (t)(c)l)g(t)dt dQi2,io (t) = gi(t)dt
Poi Pl9
ß
a+ß
Ai
A+Al
A2
P23 = A+A2+7 A
P25
A+A2+7
1  g2 (A)
(13)
P3,13 = P37 )
P56 = p57) = 1  g22)(Ai)
P7S = p7? = 1  g(2) (A) P9,12 = P9ll2o) = 1  gi*)(A)
Pio,ii = Pllol9) = 1  g(2)(Ai) that:
Pl9 + Pl,io = 1 P32 + P3,13 = 1 P52 + P56 = 1 P72 + P7S = 1 P91 + P9,12 = 1 Pio,i + Pio,ii = 1
1
The unconditional mean time taken by the system to transit for any regenerative state j when it (time) is counted from the epoch of entrance into state ¿is mathematically state as:
mj = fo°° tdQij(t)dt = q2(o)
mi9 + mi,io = Ui m32 + №3,13 = U3 m52 + m56 = U5 №72 + №75 = U7 №91 + №9,12 = u9 mio,i + rnio,ii = Uio
rnoi + rno2 = Uo
№23 + №24 + №25 = U2
K2
, (13) №32 + rn37 ;
№52 + rn57) = K
№72 + rn75) = Ki
(12) ix
№91 + rn9,lo = Ki
(11) V
№io,l + №lo,lil = K
Figure 7: Model 2: State Transition Diagram
The mean sojourn time Ui in the regenerative state ¿is defined as time of stay in that state before transition to any other state:
• № = a+ß
• U2 = 7+A+A2
• U = 1
• Fi = s
U1
A+A1
U7 = U9
1g1*)(A)
Uli = /o°° G(t)dt
= ig2 (Ai) U3 = A1
U. = 1g* (Al) U5 = A 1
= (A1) U10 = A1
U12 = /0° Gi(t)dt
17. Mean Time to System Failure for Model 2
The average duration between successive system failures, i.e. MTSF is defined as the expected time for which the system is in operation before it completely fails. Mean time to system failure (MTSF) of the system is determined by considering failed state as absorbing state. When the system starts from the state 0, the mean time to system failure is:
To
limo R*(s)
s—y 0
lim
s—► 0
1  tO* (s)
N D
where,
D=Pl9P23P32P9i  P24  P25P52  Pl9P91  Pl0,1 Pl,10  P23P32 + Pl9P24P91 + Pl9P25P52P91 +
P23 P32 Pl0,1 Pl,10 + P24 Pl0,1 Pl,10 + P25 P52 Pl0,1 Pl,10 + 1
N = U0 (P23P39 + P25P56  Pl9P23P39p91  Pl9P25P56p91  P23P39Pl0,1 Pl,10  P25P56Pl0,1 Pl,10) +
Ul (P91 + P01 P9,12  P23P32P91  P24P42p91  P25P52p91  P02P23P39p91  P02P25P56p91 
P01P23P32P9,12  P01P24^42^9,12  P01P25P52P9,12) + (U2 + U4p24)(¡^2  ¡^2¡P91 
P42P10,1 P1,10  P01P19P42P9,12  P01P42P1,10P10,11) + U3(P02P23  P02P19P23p91 
P02P23P10,1 P1,10) + U5(P02P25  P02P19P25p91  P02P25p10,1 P1,10) + U9(P01P19  P01P19P23P32 
P01P19P24P42  P01P19P25¡M + U10(P01 P1,10  P01P23P32P1,10  P01P24P42p1,10  p01 P25P52P1,10)
A
s
18. Reliability Measures
18.1. Availability Analysis in Summers
Availability A(t) is a measure that allows for a system to repair when failure occurs. The availability of a system is defined as the probability that the system is successful at time t. The long run availability of the system is given by
A0 = lims—^o[sA0s (s)] = D
where,
N=Ho + H2P02 + H3P02P23 + H5P02P25  HoP23P32  HoP24  HoP25P52  HoP§ P75 +
(13) , (6) (13) (6) (8) (6)
H7Po2P23P37 ) + H7Po2P25P57)  HoP23P37 ) P72  H2Po2P57) P75  HoP25P57 P72 +
(13) (8) (6) (8) (6) (8) (13) (8) (6) (8)
H5Po2P23P37 ) P75)  H3Po2P23p57) P75) + HoP23P32p57) P7s  HoP23P37 ) P52p75) + HoP24P57) P75)
D1=(H2 + H4P24)(1  P57)P75)) + H3(P23P72 + P23P52p^ + H5(p75 + P25P72  P23P32p75 
(8)\ , t (6) (6) , (13) (6)N
P24P75)) + H7(P57)  P23P32p57) + P23p37 )P52  P24p5^
18.2. Availability Analysis in Winters when the System Works at Full Capacity
The availability of a system is defined as the probability that the system is successful at time t. The long run availability of the system is given by
A = lims^oMos (s)] = D2
where,
(11) (12) (11) (11) (12) N2 = Ho + H1 Po1  HoP19P91  HoP1o,1 P1,1o  HoP1o 9 P91o  HoP91 p1o 9 P1,1o  H1 Po1 P1o 9 P9 1o (12)
Ho P19 P1o,1 P9,1o;
D2 = H1 (P1o,1 + P91 P1o,9 ) + H9 (P1o,9 + P19 P1o,1) + H1o( P1,1o + P19 P9,1o)
18.3. Availability Analysis in Winters when the System Operates at Reduced
Capacity
Availability of the system when it operates at reduced capapcity is given by
where, N3 = p(
D2 is already defined above.
AW = lims—^0 [sA0w (s)] = g N3 = P01 (^9 P19 + ^10 P1,10 + H9 P^ P1,10 + ^10 P19 p9110))
18.4. Busy Period Analysis for Repair in Summers
Busy period B,(t) in summers is defined as the probability that the repairman is busy at time t when the system entered to a regenerative state i. The total time in which the repairman is busy doing repair of the system in steady state is given by:
B0r = lims—^0 [sB0sr (s)] = N1
where,
N4=P02(^5 P25 + H7 P23 p373) + F7 P25 p57) + F5 P23 p373) p785) )
D2is already defined above.
18.5. Busy Period for Maintenance in Summers
Busy period B, (t) in summers for maintenance is obtained. The total time in which the repairman is busy doing repair of the system in steady state is given by:
B0m = lims—^0 [sB0sm (s)] = g
where,
N5=^3P02P23(P57)P75)  1) D2is already defined above.
Upasana Sharma, Drishti RT&A, No 1 (77) RELIABILITY OF A BUTTER CHURENR AND CBM SYSTEM_Volume 19, March 2024
18.6. Busy Period Analysis for Repair in Winters
Busy period for repair in winters is obtained as given below:
The total time in which the repairman is busy doing repair of the system in steady state is given by:
B0wr = lims—^q [sBQwr (s)] = g
where,
N6 = P01 (^9P19 + ^l0pi,10 + V9P109 P1,10 + ^10P19p91l0) D2is already defined above.
18.7. Expected Number of Repairs in Summers
Let V(t) be the expected number of repairs in (0, t) given that the system entered into regenerative state i at i = 0.
The expected number of repairs during summers in steady state is given by:
Vsr = lims—^0 sVsr (s) = g
N7 = P02 ( P25 P52 + P25 p57 + P23 p373) P72 + P23 p373) p75 + P25 p57 P72 + P25 p57 P75 +
(13) (8) . (13) (6) (8)n P23P37 ) P52P75) + P23^7 ) P57 D1 is already defined above.
18.8. Expected Number of Maintenances in Summers
Let V (t) be the expected number of maintenances. The expected number of repairs during summers in steady state is given by:
Vsm = lims_^0 sVsm (s) = D
g8 = ( P32+P373)) P02 P23( P57) P75) 1) D1 is already defined above.
18.9. Expected Number of Repairs in Winters
Let Vi (t) be the expected number of repairs in winters. The expected number of repairs during summers in steady state is given by:
Vwr = lims_^0 sVwr (s) = g
N9 = P01 ( P19 P91 + P10,1 P1,10 + P10,9 P1,10 + P19 p9ÏÏ + P91 P10,9 P1,10 + P19 P10,1 j^lö + P19 P10,9 J^O +
P10,9 P1,10 P9,10 )
D2 is already defined above.
19. Profit Analysis of the System
Profit incurred to the system model in steady state is given by
P = (CoAO + CiA^ + C2AWr)  (C3B0 + C4BW + C5B0m + C6V?r + C7V + CsVT)
where,
Co=Revenue per unit up time in summers.
C1=Revenue per unit up time in winters when the system operates at full capacity. C2=Revenue per unit up time in winters when the system operates at reduced capacity. C3=Cost per unit up time for which the repairman is busy for repair in summers. C4=Cost per unit up time for which the repairman is busy for repair in winters. C5=Cost per unit up time for which the repairman is busy for maintenance in summers. C6=Cost per repair in summers. C7=Cost per repair in winters. Cs=Cost per maintenance in summers.
20. Graphical Analysis and Conclusion
For further numerical and graphical evaluation, let us assume the repair and failure rates to be exponentially distirbuted
g(t) = 0ee(t), g1 (t) = QxeQ1(t), g2 (t) = 02e02(t)
Poi
P19 P23 P25
ß
a+ß
Ai
A+A1
A2 A+A2+7
A
A+A2+7
P(13) = = 92 p37 = p3,i3 = A1+F52
P57 = p56 = Xf+0
P(8) = Pna = 01 p75 = p78 = ÄT&T
(12)
P9,10 = P9,12 = (12)
P1o,11 = P10,12 :
mu1 = ÂTAT = °2
V3 = A1 (A1 +02)
91 A+01
_ 9 A1 +9
U5 = ^13
^10 = A1(A1+9)
J_
02
F6 = g
P02 = P1,10 P24 = P32 = P52 = P72 = P91 =
P10,1 H0 = V2 = ^4 = V7 = V8 =
a a+ß
A
A+A1 7
A+A2+7
A1 A1+02
A1 A1+9
A A+01
A
A+01

= A1+0
1
a+ß 1
7+A+A2
1
5
^9 = V12
01
A(A+01 ) _ J_
" 01
The parameters obtained using the original data collected from the Verka Milk Plant, Bathinda, Punjab.
Table 6:
Parameters obtained from data collected
Parameters for Values
model 1
A .00045892
A1 .0004567
A2 0.000246572
g1 (t) .06312
g(t) .062981
g2 (t) 0.002628867
a .000562
ß .0004314
5 .000955
7 .000155
C0 830000
C1 1030000
C2 61660
C3 10500
C4 12500
C5 15500
C6 19500
C7 6400
C8 7000
System effectiveness measures evaluated are given below:
Table 7:
Parameters obtained from data collected
Parameters for model 2 Values
Mean time to system failure 99682.28 hrs
Availability in summers 0.985
Availability in winters when system oper .989
ates at full capacity
Availability in winters when system oper .001435
ates at reduced capacity
Busy period for repair in summers .003814
Busy period for maintenance in summers .038744
Busy period for repair in winters .007864
Expected number of repairs in summers .000242
Expected number of maintenances in sum .000120
mers
Expected number of repairs in winters .000499
Figure 8: MTSF v/s Failure Rate
Figure 9: Profit v/s Failure Rate in Summers
Figure 10: Profit v/s Failure Rate in Winters
Figure 11: Profit v/s Failure Rate in Winters
Figure 12: Profit v/s Failure Rate in Winters
Table 8:
Notations of the model
Figures Descriptions
11 Profit P1 increases as the revenue C0 increases. C3=10500; Profit >=< according to C3, when C3 is >=<Rs.645.34, similarly for C3=20500 where cut off point is Rs.573.039 C3=30500; where cut off point is Rs. 500.7389
12 Profit P2 increases as the revenue C1 increases. C4=12500; Profit >=< according to C4, when C1 is >=<Rs.203.345, similarly for C4=22500 where cut off point is Rs.460.203. C4=32500; where cut off point is Rs. 317.061
The MTSF, profit in the summers (P1), and profit in the winters (P2) graphs 8,9,1o exhibit a similar trend with failure rate lambda and A1, which means that as the failure rate rises, the MTSF and profit fall.
21. Conclusion
The significance of implementing dependability in verka milk plant is analysed and concluded upon in this study. Using the parameters laid out in tables above, it has been shown that the second model generates more money after CBM is put into effect. Results from mathematical measurements and graphs showing that MTSF and Profit drop with increasing values of failure rates must be used to gain a more indepth understanding of the essential real influencing elements and, in turn, enhance the reliability model. But the equations derived for MTSF, assessments of the system's functionality, and profit can be used to find alternative cutoff points related to the required rates, costs, and probabilities involved. The formulas for the proposed system can then be generated by plugging in the actual numbers for the relevant rates and costs. Important decisions about the system's dependability and profitability can be made with the help of graphs showing cutoff points for key rates, costs, and revenue.
References
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