CC BY
11
Ramanpr eet Kaur, Upasana Sharma
MEASURES TO ENSURE THE RELIABILITY RT&A, No 3 (69)
& USING REFRIGERA TION Volume 17, September 2022
MEASURES TO ENSURE THE RELIABILITY OF WATER SUPPLY IN THE MLDB SYSTEM USING REFRIGERATION
Ramanpr eet Kaur **, Upasana Sharma2 •
*1,2Department of Statistics,Punjabi Univ ersity ,Patiala *1 rkgill9192@gmail.com 2ushar ma@pbi.ac.in
Abstract
Various components work together to form a system's overall structure. Last but not least, how well each component functions affects how the system functions. Both a functioning and failing state are possible for a system built from components. Failure has a big effect on the way systems work in industry. So, in order to enhance system performance, it is essential to get rid of these errors. The aim of this research is to assess the scope of water supply concerns in the MLDB (Multi-Level Die Block) system at the Piston Foundry Plant.The MLDB system, which consists of a robotic key unit that works with the water supply, is the subject of this research. Robotic failure and a lack of water supply cause the system to fail.A reliability model is created in order to calculate MTSF (mean time to system failure), availability, busy times for repair, and profit evaluation. The abovementioned measurements were computed numerically and graphically using semi-Markov processes and the regenerating point technique. The results of this study are novel since no previous research has concentrated on the critical function of water delivery in the MLDB system in piston foundries. According to the discussion, the findings are both highly exciting and beneficial for piston manufacturing businesses who use the MLDB system. For companies that make pistons and use the MLDB system, the conclusions, according to the debate, are particularly beneficial.
Keywords: MLDB, MTSF, availability , semi-Marko v process, regenerating point technique.
1. Introduction
Many study articles on reliability exist in the literatur e and many estimations such as reliability , availability , engagement length and other factors for standb y system have been taken. Reliability principles have been utilised in different manufacturing and technological areas throught the last 45 years. Previously, researchers examined the various ways to standby systems such as: Srinivasan [10] gave an examination of warm standb y system dependability for a repair facility. The stochastic standby system behaviour with repair time was handled by Kumar et al. [4]. Sharma and Kaur [8] conducted a cost-benefi analysis of a compressor standby system. A power plant system's cold standby unit was stochastically modelled by Sharma and Sharma[9].
Some authors provided an overview of the different reliability modelling methodologies used in die casting systems such as: High Pressur e Grain structur e and segr egation in die casting of magnesium and aluminium alloys Characteristics mentioned by Laukli [5]. High pressur e die cast AlSi9Cu3 (Fe) alloys are provided by Timelli [11] using constitutiv e and stochastic models to anticipate the impact of casting fl ws on the mechanical properties. Die Casting Process Modeling and Optimization for ZAMAK Alloy given by Sharma [7]. Existing epistemic uncertainty in die-casting is modelled for reliability and optimised by Yourui et al.[ 12]. Sensitivity study for the casting method provided by Kumar [3]. An Early Investigation of a Lightw eight provided by Muller et al. [6] Die Casting Die Using a Modular Design Appr oach. High pressur e die casting machine reliability analysis of two unit standby system offered by Bhatia and Sharma
[1]. The Casting Process Optimization Case Study: A Review of the Reliability Techniques used by Chaudhari and Vasude van [2]. According to the discussion above, every researcher has addr essed reliability analysis of the die casting method used in piston foundries. Resear ch finding pertaining to the MLDB system in piston foundries have not been discovered. A few of them, though, have gather ed and analysed real data. There are a variety of systems in piston foundr y operations that must be analysed using real data at various rates and costs. Our efforts are closing this gap by gathering genuine data from a company called Federal-Mogul Powertrain, India Limited, which is based in Bahadur garh, Punjab, near Patiala. Federal-Mogul is the world's leading maker of world-class pistons, piston rings and cylinder linears, with products for two-and three -wheelers, vehicles and tractors, among other applications.
The purpose of this research is to assess the MLDB system's water supply problems. For the MLDB system in the piston plant, a reliability model has been established. The MLDB system is an enhanced version of the die casting technology that was introduced to raise the piston foundr y's output rate. For the operation of the MLDB system in the piston plant, ther e is one main unit, which is robotic and two sub-units. Water is supplied to the system via a fan (WSF). The system fails due to a lack of water supply . We create a novel reliability model to overcome the failure in water supply, which differ from the present approach in the piston plant. A main robotic unit that works with the water supply through a refrigeration(WSR) is required for the operation of this new model, the MLDB system. To run the entire system, both the robotic and the WSR units must be operational. Water supply from fan(WSF) is utillised as a cold standb y unit for better working conditions. System failur e occurs due to robotic failur e and a lack of water supply.
For the model, there are a few assumptions that need to be made:
• S0 is the starting state of the system.
• The main unit, i.e. robotic, receives priority for repair.
• All failur e and repair times were calculated using an exponential distribution.
• After each repair in the states, the system perfor ms a new function.
• A repair man is dispatched as soon as a unit fails.
2. Methods
The follo wing are the materials and methods that were used to complete this resear ch:
Semi-Marko v processes and regenerating point techniques are employed in order to tackle the challenges. Many system effectiv eness metrics have been acquir ed, including mean time to system breakdo wn, system availability , busy period for repair and predicted number of repairs. The profit are also made. Using C++, Python and MS Excel programming , graphical analyses are created for a specifi situation.
3. Notations and States for the Model
Rb ^ Main unit of the MLDB system i.e. Robotic.
O(Rb) ^ Main unit of the MLDB system is in operating state.
WSR ^ Water supply refrigerator for the system.
WSF ^ Water supply fan for the system.
O (WSR) ^ Water supply refrigerator is in operating state.
O (WSF) ^ Water supply fan is in operating state.
CS(WSF) ^ WSF is in cold standb y state.
A, Ai, A2 ^ Failure rates of the main unit i.e. Robotic, WSF and WSR respectiv ely.
Fr(Rb) ^ Failures of the main unit i.e. Robotic under repair.
Fr(WSR)y Fr(WSF) ^ Failures of the WSR and WSF are under repair respectiv ely. FR(WSF), FR(WSR) ^ Repair is continuing from previous state for WSF and WSR respectiv ely. Fwr(WSF), Fwr(WSR) ^ Failed WSF and WSR are waiting for repair respectiv ely. G(t),g(t) ^ c.d.f. and p.d.f of repair time for Robotic. Gi (t),gi (t) ^ c.d.f. and p.d.f of repair time for WSR. G2(t),g2(t) ^ c.d.f. and p.d.f of repair time for WSF.
4. The System's Reliability Measur es 4.1. Transition Probabilities
The various phases of the system are depicted in a transition diagram (see in Fig.1).
Figure 1: State Transition Diagram
The epochs of entry into states S0, Si, S2, S3, S5 and S6 are regenerativ e states, while the rest are non-regenerativ e stages. The operational states are S0, S2 and S5, while the failing states are S1, S3, S4, S6 and S7. The transition probabilities are:
dQ0i (t) = Ae-(A+Ai )tdt dQ02 (t) = Ai e-(A+Ai )tdt
dQi0 (t) = gi (t)dt dQ20 (t) = gi (t)e-(A+Al )tdt
dQ23 (t) = Ae-(A+A2 )tGi'(t)dt dQ24 (t) = A2 e-(A+A2 )tGi'(t)dt
dQ24)(t) = [A2 e-(A+A2 )t©i ]gi (t)dt dQ50 (t) = g2 (t)e-(A+Ai )tdt
dQ56 (t) = Ae-(A+Ai )tG2~(t)dt dQ57 (t) = Ai e-(A+Ai )tG2~(t)dt
dQ^it) = [Ai e-(A+Ai )t©i ]g2 (t)dt dQ72 (t) = g2 (t)dt
dQ45 (t) = gi (t)dt dQ65 (t) = g(t)dt
dQ32 (t) = g(t)dt
(1)
The non-zero elements pij can be represented as below:
A
P01 =
A + Ai
P02 =
a1
A + Ai
P20 = g* (A + A2 )
P24 = P25
(4) _ A2 [1 - gl (A + A2 )]
(A + A2)
p = A[1 - g2 (A + Ai)]
P56 = (Ä+ÄT)
P10 = p32 = p65 = g* (0) = 1 P72 = g2 (0) = 1
It is also verifie that:
p23
A[1 - g1 (A + A2)]
(A + A2)
P50 = g2 (A + A1) p = p(7)= A1 [1 - g2 (A + A1)]
p57 = p52 = "
(A + A1)
p45 = g1 (0) = 1
(2)
p01 + p02 = 1
(4)
p20 + p23 + p(5) (7)
p50 + p56 + P52
p20 + p23 + p24 = 1 p50 + p56 + p57 = 1 p10 = p32 = p45 = p65
p72
(3)
When it (time) is calculated from the epoch of arrival into state 'j', the unconditional mean time taken by the system to transit for each regeneration state 'i'is mathematically define as:
mi
f TO
J tdQij(t) = -q**(0)
it is also verifie that
m01 + m02
Vo
m2o + m23 + m25) = K
(7)
m5o + m56 + m52) = K2 m32 = V3 m65 = V6
m20 + m23 + m24 m50 + m56 + m57
m1o = V1
m45 = V4 m72 = V7
V2 V5
wher 1
m01 = /•TO / tAe-(A+A1 )tdt Jo m02 = TO / tA1 e-(A+A1 )fdt J0 JTO
m20 = /■ to / g1 (t)te-(A+A )fdt J0 JTO m23 = / Ate-(A+A )tG1~(t)dt J0 JTO
m24 = / A2 te-(A+A2 )tG1(t)dt J0 (4) m25 = / t[A2 e-(A+A2)t ©1 ]g1 (t)dt J0 JTO
m50 = /TO g2 (t)te-(A+A1 )fdt J0 JTO m56 = / Ate-(A+A1 )fG2_(t)dt J0 JTO
m57 = / A1 te-(A+A1 )fG2(t)dt 0J JTO (7) <2 = / t[A1 e-(A+A1 )f©1 ]g2 (t)dt J0 JTO
m10 = m32 = m65 = tg(t)tdt J0 JTO m45 = / tg1 (t)tdt J0 JTO
m72 = / tg2 (t)tdt J0 JTO K1 = / G1(t)dt 0
K2 = / G2(t)dt 0
(4)
(5)
(6)
The mean sojourn time (Vi) in the regenerativ e state 'i'is define as the period of time spent in that state befor e transitioning to any other state:
f TO
Vi = E(Ti) = P(Ti > t)
0
1
As we get
V0
V5
A + A1
1 - g2 (A + Ai ) A + A1
V4 = -gl (0)
V2
1 - gl (A + A2 )
A + A2
Vi = V3 = V6 = -g* (0) V7 = -g2 (0)
4.2. Mean Time To System Failur e
The failed states of the system ar e consider ed absorbing to deter mine the mean time to system failure (MTSF) of the system. The following recursiv e relation for fa(t) is obtained with probabilities arguments:
$0 (t) = Qoi (t) + Q02 (t)©fa (t)
$2 (t) = Q20 (t)©fa (t) + Q23 (t) + Q24 (t)
(9)
Taking Laplace Stieltje Transfor ms(L.S.T) of these relations in equations(9) and solving for fa^* (s) we obtain
$*** (s)
N (s) D(s)
wher e
N(s) = Q0* (s) + Q0* (s)[Q2* (s) + Q2* (s)] D(s) = [l - Q0*(s)Q2*]
(10)
(11) (12)
Now the mean time to system failur e (MTSF) , when the system started at the beginning of state S0 is
T = lim 1 - *(S)
s—S
Using L' Hospital rule and putting the value of fa**(s) from equation(13), we have
N
T0 = D
(13)
(14)
wher e
N = V0 + V2 [ p02 ] (15)
D = 1 - P02 P20 (16)
4.3. Availability Analysis
Let Aj(t) be the probability that the system is in the up state at instant t, given that the system entered the regenerativ e state i at t=0. The following recursive relations are satisfie by the availability Aj(t):
A0(t A1 (t
A2 (t A3 (t A5 (t A6(t
M0 (t) + 901 (t)© A1 (t) + 902 (t)© A2 (t) q 10 (t)© A0 (t)
M2 (t) + 920 (t)©A0(t) + 923 (t)© A3 (t) + q24)(t)© A5 (t) 932 (t)© A2 (t)
M5 (t) + 950 (t)© A0 (t) + 956 (t)© A6 (t) + 9572) (t)© A2 (t) 965 (t)© A5 (t)
1
Ramanpr eet Kaur, Upasana Sharma
MEASURES TO ENSURE THE RELIABILITY RT&A, No 3 (69)
& USING REFRIGERA TION Volume 17, September 2022
wher e
Mo (t) = e-(A+A1 )f M2 (t) = e-(x+x2 )tG1(t)
M5 (t) = e-(A+Al )tG7(t) (18)
Taking Laplace Transformation of the above equation(18) and letting s —>• 0, we get
M0 (0) = Fo M* (0) = m
M5 (0) = ¥5 (19)
Taking Laplace transfor m of the above equations(17) and solving them for
A0 <s)= "NO <20)
wher e
N1 (s) = M*(s)[l - q*23(s)fa(s) - q56(s)q65(s) - №*(s)q{£*(s) + (s^2(s)-
q*56(s)q65 (s)]+ M2(s)q*02 (s)[1 - ^6(s)q65 (s)]+ M*(s)^ (s)q{£*(s) (21)
D1 (s) = [1 - q*56(s)q*65 (s) - q*23 (s)q3*2 (s) + q*23 (s)q3*2 (s^6(s)q*65 (s) - q2f(s)q5f(s)]-
q01 (skw(s)[1 - q*56(s)q*65(s) - fa(s)q32(s) + fa№2^fa^q'65(s)-q(45)*(s)q572)*(s)] - q*02 (s)q20(s) + fa (s)fa(s)q*56(s)fa (s) - fa (s)fa(s)q2£*(s) (22)
In steady state, system availability is given as
A0 = lim sA0 (s) = N (23)
s-^0 D1
wher e
N1 = ¥0 [1 - P23 - P56 + P23 P56 - pi P^] + ¥2 [P02 (1 - P56 )] + ¥5 [P02 P^] (24)
D1 = ¥0 [1 - P23 - P56+P23 P56 - p24) p5?]+¥1P01 [1 - P23 - P56+P23 P56 - p25 p5?]
+ K1P02 [1 - P56] + KP02 pi + ¥6 [P02 P23 P56] (25)
4.4. Busy Period Analysis of the Repair man
Let BRj(t) be the probability that the repairman is busy at time t given that the system entered regenerativ e state i at i=0. The recursiv e relation for BRj(t) are as follows:
BR0 (t) = q01 (t)© BR1 (t)+ q02 (t)© BR2 (t) BR1 (t) = W1 (t) + q 10 (t)© BR0 (t)
BR2 (t) = W2 (t) + q20 (t)© BR0 (t)+ q23 (t)© BR3 (t)+ q(45)(t)© BR5 (t) BR3 (t) = W3 (t) + q32 (t)© BR2 (t)
BR5 (t) = W5 (t) + q50 (t)© BR0 (t) + q56 (t)© BR6 (t) + q^^(t)© BR2 (t)
BR6 (t) = W6 (t) + q65 (t)© BR5 (t) (26)
wher 1
W1 (t) = G(t) W2 (t) = e-(A+A2 ]tG1(t) W3 (t) = G( t)
W5 (t) = e-(A+A1) G2(t)dt W6 (t) = G(t) (27)
Taking Laplace Transformation of the above equation(27) and letting s —>• 0, we get
W (0) = rn
W* (o)
W (o)
Hi
W* (0) w* (o)
H2 H6
Taking Laplace transfor m of the above equations(26) and solving them for
N2 (s)
BR0 (s)
Di (s)
(28)
(29)
wher e
n2(s) = W*(s)q0i(s)[l - q2s(s) - (s)q6s(s) + <fe*300^6(s)q6s(s)] + W2 (s)q02 (s)[l - q*6 (s)q6* (s)] + W* (s)q02 (s)[<& (s) - q*6 (s)q6* (s)] + W* (s)q02 (s)q2f (s) + W* (s)«& (s)q2f (*)<& (s)
The value of D1 (s) is already define in equation(22).
System total fraction of the time when it is under repair in steady state is given by
N2
BRo = lim sBR0 (s) = tt
s—y 0 Dl
(30)
(31)
wher 1
N2 = Hi [P01 (1 - P23 - P56 + P23P56 - p24)P(7))] + H2 [P02 (1 - P56)]
+ H3 [P02 (P23 - P56 )] + H5 [P02 j^] + H6 [p02P56P^]
,«1
(32)
The value of D1 is already define in equation(25).
4.5. Expected Number of Repairs
Let ER,(t) be the expected no. of repairs in (0,t] given that the system entered regenerativ e state i at i=0. The recursiv e relations for ER,(t) are as follows:
((t)©[1 + ER1 (t)] + Q02 (t)©[1 + ER2 (t)]
,(t)©ER0 (t)
>(t)©ER0 (t) + Q23 (t)©[1 + ER3 (t)] + Q°5)(t)©[1 + ER5 (t)] ,(t)©ER2 (t)
, (t)©ER0 (t) + Q56 (t)©[1 + ER6 (t)] + Q(2) (t)©ER2 (t) ¡(t)©ER5 (t) (33)
Taking L.S.T.of above relations and obtain the value of VR0 *(s), we get
N3 (s)
ER0(t) = Q01
ER1 (t) = Q10
ER2 (t) = Q20
ER3 (t) = Q32
ER5 (t) = Q50
ER6(t) = Q65
ER0 (s)
D1 (s)
(34)
wher e
N3(s) = (Q0*(s) + Q0*(s))[1 - Q23*(s)Q32(s) - Q55*(s)Q6*(s) - q25) **(s)q52)* * + Q2*(s)Q32(s) - Q23*(s)Q22(s) - Q55*(s)Q6*(s)] + (Q2*(s) + Q2f *(s)) [1 - Q55* (s)Q6* (s) + Q55* (s)q25) * *(s)]
The value of D1 (s) is already define in equation(22).
(s)
For system steady state, the number of repairs per unit time is given by
N3
ERo = lim sER*0* (s) = -3
s—^o Di
wher i
The value of D1 is already define in equation(25).
5. Profi Analysis
The pr ofi incurr ed by the system model in steady state is calculated as follo ws:
wher i
P = Zo Ao - Zi BRo - Z2ERo - Z
P = Profit
Zo = Revenue per unit up time.
Z1 = Cost per unit up time for which the repair man is busy for repair. Z2 = Cost per repair. Z3 = Installation Cost.
(36)
N3 = [1 - P23 - P56 + P23P56 - p25P?] + Po2 (1 - P2o )[1 - P56 + P56P^] (37)
(38)
6. Particular Cases
For the particular case, the failure rates and repair rates are exponentially distributed as follows:
g(t) = ae-at g2 (t) = 1x2 e-X21
As we get,
Po1 =
A
Po2
A + A1
Ai A + A1
A
P23 (A + A2 + «1) a2
P50 A + A1 + a2
= (7) = A1
P57 = P52 = (A + A1 + a2) 1
Uo =
U5
A + A1 1
A + A1 + a2 1
U4 = K1 = — a1
g1 (t) = «1 e-a11
p2o
a1
A + A2 »1
(4)
P24 = P25
P56
A2
(A + A2 + «1)
A
(A + A1 + a2)
P1o = P32 = P65 = P45 = P72 = 1
_ 1
U2 = (A + A2 + a) 1
U1 = U3 = U6 = -a
U7 = K2 = —
a.2
Based on the facts received i.e.,
3
Ramanpr eet Kaur, Upasana Sharma
MEASURES TO ENSURE THE RELIABILITY RT&A, No 3 (69)
& USING REFRIGERA TION Volume 17, September 2022
Table 1: Information Gathered
Description Notation Rate(/hr)
Failur e Rate of robotic A 0.001378336 / hr
Failur e Rate of WSF A2 0.000117273 / hr
Repair Rate of robotic a 0.20271061 / hr
Repair Rate of WSF 0.005767389 / hr
The remaining values are assumed and are listed in Table 2:
Table 2: Assumed Values
Description Notation Rate(/hr)
Failur e Rate of WSR A1 0.000018325 / hr
Repair Rate of WSR a1 0.003728205 / hr
Revenue per unit uptime(per month) Z0 Rs.10, 80, 000
Cost per unit uptime, when repair man is busy for repair(per month) Z1 Rs.12, 466
Cost per reapir(per month) Z2 Rs.18, 350
Various measur es of system effectiveness are shown in Table 3:
Table 3: Results
Description Notation Rate(/hr)
Mean Time to System Failur e T0 714.866577 / hrs
Availability of the system A0 0.909847
Busy period of Repair man BR0 0.21322
Expected no. of Repairs ER0 0.002053
Profi P Rs.14, 21,955
7. Graphical Repr esentation
This study has prepar ed graphs for the MTSF (as shown in Figur e.2), Profi as a result of failur e rate of main unit( A)(Figure 3.) and revenue ( uptime of the system per unit) (Z0) for various estimates of repair man cost for busy work in (Z1) is shown in Figur e 4.
Figure 2: MTSF v/s Fa/lure Rate
PROFIT v/s RATE OF FAILURE OF WSF<Ju) FOR DIFFERENT VALUES OF RATE OF FAILURE OF MAIN UWT().[
20C000D 1SCOOOD
FAILURE RATEIM >
Figure 3: Profit v/s Failure Rate
socooo PROFIT v/s REVENUE PER UNIT UP TEVIE(Zo) FOR DIFFERENT VALUES OF COST OF REPAIRMAN IS BUSY FOR REPAIR (Zi)
6OCOOO - V»
400000 -^ 200000 —Z,= 12,456 -«-Z, = 19,466 A Z, - 21,466
1 -200000 ^ P R -400000
F -600000
1 T -SOCOOO ^^ 1 -»
-lOOOOOO
-1200000 -
Figure 4: Profit v/s Revenue
8. Discussion
Discussion for the FAILURE RATE v/s MTSF and PROFIT v/s FAILURE RATE in the Table 4.
Table 4: Results
Variation Effect
A/ A1 increasing (t) MTSF decr eases (D
A/ A1 increasing (t) Profi decreases (D
As shown in above table, the behaviour of MTSF and Profi w.r.t. rate of failur e of Main unit for the different values of the rate of failur e of WSF. It clear from the table that MTSF and Profi gets decreased with increase in values of rate of failur e of Main unit i.e. A. Also MTSF and Profi decreases as failure rate of WSF i.e. A1 increases.
Discussion for the PROFIT v/s REVENUE in the Table 5. as belo w:
Table 5: Results
Variation Effect
Z0 increasing (t) Profi increases (t)
Z1 = 12, 466; Profi >= < according as z0 when Z0 is > = < 8,00, 000
Z1 = 19, 466; Profi >= < according as z0 when it Z0 > = < 9,25, 525
Z1 = 21,466; Profi >= < according as z0 when it Z0 > =< 9, 98, 980
Above table depicts the behaviour of the profi w.r.t. revenue per unit uptime of the system (Z0) for different values of cost of repairman is busy under repair (Z1). The graph exhibits that there is inclination in the trend of profi increases with increases in the values of Z0. Also, following conclusion can be drawn from the discussion for Profi v/s Revenue :
For Z1 = 12,466, the profi is positiv e or zero or negativ e according as Z0 is > = < 8,00,000. Hence, for this case the revenue per unit up time should be fixed equal or greater than 8,00,000. Similarly , discussion for other values of Z1.
9. Conclusion
The conclusion is based on data from Feder al-Mogul Powertrain. By using various parameters in the existing model at piston plant, the numerical value of profi is calculated as Rs. 10,45,838 and profi for current resear ch is Rs. 14,21,955. From numerical values it has been shown that profi for new model is greater as compar e to existing model, when referigerator facility is used. The finding of this study are novel since no previous research has highlighted the critical function of water supply for the MLDB system in piston foundries. The discussion reveal that the results analysed are quite interesting and beneficia for piston manufacturing businesses who use the MLDB system. In the same way, system designers might apply the escommended strategy to their own sectors. The generated equations can be used to figu e out how practical different mechanism-type systems are.
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