The impact of machines on our daily lives can be felt in every aspect of our lives. Industries have played a huge role in modern economies. The two determinants of success are quality and availability, which are crucial in an environment of fierce domestic and international competition. Production downtime can have dramatic repercussions. Using actual field data, Vališ et al. (2020)
The paper discusses a powder encapsulation system. It consists of one drying chamber with three components: The first component is the heating tank, the second component is the condenser, and the third component is the concentrate. They are all in operation when the system begins to process material. During the winter months, a high level of milk production keeps the system operating, but the system goes into cold standby during the summer and is maintained. In the event that one of the three units fails, the entire system fails.
SemiMarkov process and regenerative point technique are used to obtain the following measures of system effectiveness in steady-state:
• Transition probabilities and mean sojourn times in different states. MTSF of the system.
• Steady-state availability for the system.
• A busy period for the repairman.
• Expected number of repairs.
• Additionally, the system's profit potential is analyzed graphically.
• The system is initially operative at state 0.
• Time to failure of each unit is assumed to follow an exponential distribution, whereas repair time distribution is taken to be arbitrary.
• There are similarities and statistical independence between the units.
• After each repair, the system works as well as new.
• λ → Failure rate of the drying chamber unit.
• λ1, λ2, λ3 → Failure rate of units one, two, and three respectively.
• λ4 → Maintenance rate of a powder system.
• α → Rate of going to summer.
• β → Rate of going to winter.
• S → Summer.
• W → Winter.
• O→ Operative state.
• cs → Cold Standby state.
• um → Under maintenance.
• Fr → Failure under repair.
• dc → Drying chamber of powder plant.
• u1, u2, u3 → Units 1,2,3 respectively.
• Odc → Drying chamber of the powder system is operating.
• ou1, ou2, ou3 → Units 1,2,3 is operating.
• cs dc → Drying chamber of the powder system is in the cold standby state.
• cs1, cs2, cs3 → Units 1,2,3 is in cold standby state.
• G(t), g(t) → c.d.f. and p.d.f of time to repair the drying chamber unit.
• G1(t), g1(t) → c.d.f. and p.d.f of time to repair unit one.
• G2(t), g2(t) → c.d.f. and p.d.f of time to repair of unit two.
• G3(t), g3(t) → c.d.f. and p.d.f of time to repair of unit three.
• G4(t), g4(t) → c.d.f. and p.d.f of maintenance of the powder plant.
The non-zero elements pij can be represented as below:
pij = Qj (∞) =
• p01 = β /β+α
• p02 = α /β+α
• p14 = λ3 /λ+λ1+λ2+λ3
• p15 = λ /λ+λ1+λ2+λ3
• p16 = λ2 /λ+λ1+λ2+λ3
• p17 = λ1 /λ+λ1+λ2+λ3
• p32 = g4∗(0)
• p41 = g3∗(0)
• p51 = g∗(λ1 + λ2 + λ3)
• p58 = p54(8) = (λ3 /λ1+λ2+λ3) (1 − g∗(λ1 + λ2 + λ3))
• p59 = p57(9) = (λ1 /λ1+λ2+λ3) (1 − g∗(λ1 + λ2 + λ3))
• p5,10 = p56(10) = (λ2 /λ1+λ2+λ3) (1 − g∗(λ1 + λ2 + λ3))
• p61 = g2∗(0)
• p71 = g1∗(0)
• p84 = p97 = p10,6 = g∗(0)
By these transition probabilities it is also verified that
• p01 + p02 = 1
• p14 + p15 + p16 + p17 = 1
• p51 + p58 + p59 + p5,10 = 1
• p51 + p54(8) + p57(9) + p56(10) = 1
• p23 = p32 = p41 = p61 = p71 = p84 = p97 = p10,6 = 1
The unconditional meantime is taken by the system to transit for any regenerative state ‘j’ when t (time) is counted from the epoch of entrance into state ‘i’ is mathematically stated as:
• m01 + m02 = µ0
• m14 + m15 + m16 + m17 = µ1
• m51 + m58 + m59 + m5,10 = µ5
• m51 + m54(8) + m57(9) + m56(10) = K
where,
The mean sojourn time (µi) in the regenerative state ‘i ’is defined as time of stay in that state before transition to any other state:
• µ0 = 1/ α + β
• µ1 = 1/ λ + λ1 + λ2 + λ3
• µ2 = 1 /λ4
• µ3 =
• µ5 = (1 /λ1 + λ2 + λ3) [1 − g∗(λ1 + λ2 + λ3)]
• µ4 =
• µ6 =
• µ7 =
• µ8 = µ9 = µ10 =
Mean time to system failure (MTSF) of the system is determined by considering the failed state as an absorbing state. By probabilistic arguments, we obtain the following recursive relations for
ϕ₀(t)= Q01(t) ⋆ ϕ1(t) + Q02(t) ⋆ ϕ2(t)
ϕ1(t)= Q14(t) +Q15(t) ⋆ ϕ5(t)+Q16(t)+Q17(t)
ϕ2(t)=Q23(t)
ϕ5(t)= Q51(t) ⋆ ϕ1(t) + Q58(t) +Q59(t) +Q5,10(t)
Taking Laplace-Stieltjes Transform (L.S.T.) of the relations given by above equation and solving them for
The reliability R(t) of the system at time t is given as
R(t) = Inverse Laplace transform of
The mean time to system failure (MTSF), when the system started at the beginning of state 0 is:
MTSF =
Using L’ Hospital rule and putting the value of φ0**(s) we get
Let A0(t) be the probability that the system is available in winters at a given time instant t, given that it already entered state “i” at time t. BR0(t) be the probability that the repairman is busy for repair in winters at a time instant t, given that it already entered state “i” at time t. BM0(t) be the probability that the repairman is busy for maintenance in summers at a time instant t, given that it already entered state “i” at time t. VR0(t) be the expected number of repairs in winters at (0, t). VM0(t) be the expected number of maintenances in summers at (0, t).
As in the case of MTSF, we have derived these measures of system effectiveness (A0, BR0, BM0, VR0, VM0) using the same probabilistic arguments as with MTSF, except that here the failed state is not considered the absorbing state.
The costs defined are as follows:
C0=Revenue per unit up time.
C1=Cost per unit up time for which the repairman is busy for repair.
C2=Cost per unit up time for which the repairman is busy for maintenance in winters.
C3=Cost per repair.
C4=Cost per maintenance.
N1 = p01(µ1 + µ5p15)
D1 = µ1 + µ4(p14 + p15p54(8)) + Kp15 + µ6(p16 + p15p56(10)) + µ7(p17 + p15p57(9)) ... (1)
N2 = W4p01(p14 + p15p54(8)) + W5p01p15 + W6p01(p16 + p15p56(10)) + W7p01(p17 + p15p57(9))
D1 is defined in equation 1.
N4 = p01(1 + p15p54(8) + p15p57(9) + p15p56(10))
D1 is already defined in equation 1.
N3 = W2p01p12
D2 is defined in equation 2.
N5 = p01p12
D2 is defined in equation 2.
Here Wi is the probability that the system is under repair at time “t” before transiting any other state.
For graphical representation we assume the particular case of exponential distribution, likewise:
g(t) = θe-θ(t), g1(t) = θ1e -θ₁(t), g2(t) = θ2e -θ₂(t), g3(t) = θ3e -θ₃(t), g4(t) = θ4e −θ₄(t)
for numerical analysis various cut-off points are evaluated from the information gathered at the Verka milk plant
λ₂=.00002898, λ₃=.0000246, λ₄=.000292675,
• Mean time to system failure = 7046.117188 hrs
• Availability in winters = 0.471440
• Busy period for repair =0.498495
• Busy period for maintenance =0.007057
• Expected number of repairs =0.000152
• Expected number of maintenances =0.000007
• Profit = Rs. 46143.08
The numerical values are obtained from the original data collected from the Verka Milk Plant, Bathinda, Punjab. From the data collected various parameters were calculated which include failure and repair rates of the chamber, unit 1, 2, and 3. From these failure rates values were evaluated of the system effectiveness measures using MATLAB and Code Blocks.
λ₁=0.00024589 |
λ₁=0.0005 |
λ₁=0.0024589 |
λ₃ |
801.214172 |
730.630981 |
670.514893 |
0.002 |
641.27887 |
570.695557 |
510.57959 |
0.003 |
561.311157 |
490.727783 |
430.611786 |
0.004 |
513.330505 |
442.747223 |
382.631195 |
0.005 |
481.343414 |
410.760132 |
350.644135 |
0.006 |
458.495483 |
387.912201 |
327.796204 |
0.007 |
441.359558 |
370.776245 |
310.660248 |
0.008 |
428.031616 |
357.448303 |
297.332306 |
0.009 |
417.369263 |
346.78595 |
286.669952 |
0.01 |
408.645508 |
338.062195 |
277.946198 |
0.011 |
λ₁=0.00024589 |
λ₁=0.0005 |
λ₁=0.0024589 |
λ₃ |
2055.256592 |
1865.224609 |
829.78479 |
0.002 |
1833.494507 |
1643.462524 |
608.022583 |
0.003 |
1696.155029 |
1506.123047 |
470.683136 |
0.004 |
1602.739014 |
1412.707031 |
377.26712 |
0.005 |
1535.079956 |
1345.047974 |
309.608154 |
0.006 |
1483.81616 |
1293.78418 |
258.344299 |
0.007 |
1443.63208 |
1253.600098 |
218.16024 |
0.008 |
1411.285889 |
1221.253906 |
185.814041 |
0.009 |
1384.68859 |
1194.656616 |
159.216736 |
0.01 |
1362.432251 |
1172.400269 |
136.960449 |
0.011 |
Data of profit and failure rate λ₃ can graphically be analyzed as given in
C₁=6700 |
C₁=27000 |
C₁=47000 |
C₀ |
-405.397888 |
-750.289734 |
-1129.555786 |
10000 |
-203.339981 |
-525.161255 |
-904.427429 |
20000 |
-1.28206 |
-300.032715 |
-679.29895 |
30000 |
200.775894 |
-74.904243 |
-454.17041 |
40000 |
402.833771 |
150.224167 |
-229.042007 |
50000 |
604.891724 |
375.352722 |
-3.913464 |
60000 |
806.949585 |
600.481079 |
221.214951 |
70000 |
1009.007568 |
825.609619 |
446.343506 |
80000 |
1211.06543 |
1050.738159 |
671.471924 |
90000 |
1413.123413 |
1275.866577 |
896.600342 |
100000 |
1615.181274 |
1500.995239 |
1121.729126 |
110000 |
Upward trend in
Using line of regression y = a x + b where a is the slope and b is the intercept more values can be evaluated for the various measures i.e., mean time to system failure, profit, availability etc. some of them have been stated below to save space.
For λ₁=0.0024589 in Profit and revenue the y intercept will b= -2404.74635 and the slope a= 0.29463713 from this line of regression we can find more values with different failure rates.
For λ₁=0.0024589 in MTSF the b = 5003.582 and a = -2251640 we can find more values with different failure rates.
Known Parameter |
Varying Parameter |
MTSF
|
|||
Decreasing λ≥.0011 |
Increasing λ<.0011 |
||||
λ₂=.00002898, λ₃=.0000246, λ₄=.000292675, 𝜃=.00392166, 𝜃=.0003165, 𝜃₂=.000298, 𝜃₃=.000896, 𝜃₄=.00125363, α=.00023148, β=.00023148, C0= 105000, C1=6700, C2=2550, C3=1500, C4=1000 |
λ=.0002458 λ=.0005 λ=.002458
|
408.645508 338.062195 277.941698
|
801.214172 730.6300981 670.514893 |
||
Known Parameter |
Varying Parameter |
Profit |
|
||
Decreasing |
Increasing |
||||
λ₂=.00002898, λ₃=.0000246, λ₄=.000292675, 𝜃=.00392166, 𝜃=.0003165, 𝜃₂=.000298, 𝜃₃=.000896, 𝜃₄=.00125363, α=.00023148, β=.00023148, C0= 105000, C1=6700, C2=2550, C3=1500, C4=1000 |
λ=.0002458 λ=.0005 λ=.002458 |
λ<.0011 1362.432251 1172.40026 136.96044 |
λ≥ .0011 2055.2565 1865.2246 829.78479
|
||
Known Parameter |
Varying Parameter |
Cost |
|||
Positive |
Negative |
Zero |
|||
λ₂=.00002898, λ₃=.0000246, λ₄=.000292675, 𝜃=.00392166, 𝜃=.0003165, 𝜃₂=.000298, 𝜃₃=.000896, 𝜃₄=.00125363, α=.00023148, β=.00023148, C0= 105000, C1=6700, C2=2550, C3=1500, C4=1000 |
C1=6700 C1=27000 C1=47000 |
C0>35000 C0>45000 C0>6500 |
C0<35000 C0<45000 C0<650000 |
C0=35000 C0=45000 C0=65000 |
The parameters shown in the
Cut-off points in the graphical analysis shows at what point the values will start decreasing/increasing. Cut-off points vary widely and by knowing the cut-off points the engineers can know the level of gains or losses, which can be seen in
In case of mean time to system failure (MTSF) as the λ increases the MTSF decreases
We can see the cut-off values at λ=.0011 and λ<.0011 in
Similar is the case with profit v/s the failure rate where the cut-off values is graphically presented in
Profit v/s revenue depicts the cut off points as when
i. C1=6700 the profit depends on C0=35000,
ii. C1=27000 the profit depends on C0=45000,
iii. C1=47000 the profit depends on C0=650000.
This paper analyses the reliability of a powder system taking into account its seasonal effect. Previously, the reliability analysis of a powder system was carried out without addressing the seasonal effect; The present research will help improve system performance at the Verka Milk Plant, thereby reducing production losses and increasing profit. It will also be helpful in all the other systems with similar working conditions and help understand the seasonal variation in the working of the system. The originality of this study lies in the method of analyzing the system in terms of season-dependent expressions. This gives a better, more accurate view of the whole picture in terms of the system analysis.