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)

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(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

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(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

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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.

R EFERENCES

[1] Bhatia Pooja and Sharma Shweta(2019) Reliability analysis of two unit standby system for high pressure die casting machine. Arya Bhatta Journal of Mathematics and Informatics, 11(1):19-28.

[2] Chaudhari Amit and Vasude van Hari(2020) A Review of the Reliability Techniques Used in the Case of Casting Process Optimization. Arya Bhatta Journal of Mathematics and Informatics, 6(4):309-316.

[3] Kumar Amit, Varshne y A and Ram Mange y(2015) Sensitivity analysis for casting process under stochastic modelling. International Journal o/Industrial Engineering Computations, 6(3):419-432.

[4] Kumar Ashish, Baweja Sonali and Barak M(2015) Stochastic behavior of a cold standby system with maximum repair time. Decision Science Letters, 4(4):569-578.

[5] Laukli, Hans Ivar(2004) High Pressur e Die Casting of Aluminium and Magnesium Alloys: Grain Structur e and Segregation Characteristics. Fakultet/or naturvitenskap og teknologi, 19(4):150-164.

[6] Müller Sebastian, Müller Anke,Rothe Felix, Dilger Klaus and Droder Klaus(2018) An Initial Study of a Lightw eight Die Casting Die Using a Modular Design Appr oach. International Journal o/Metalcasting, 12(4):870-883.

[7] Sharma Satpal(2014) Modeling and Optimization of Die Casting Process for ZAMAK Alloy. Journal o/Engineering & Technology, 4(2):217-229.

[8] Sharma Upasana and Kaur Jaswinder (2016). Cost benefi analysis of a compressor standby system with preference of service, repair and replacement is given to recently failed unit. International Journal of Mathematics Trends and Technology, 30(2):104-108.

[9] Sharma Upasana and Sharma Gunjan(2018) Stochastic modelling of a cold standby unit working in a power plant system. international Journal of Innovative Knowledge Concepts, 6(8):49-54.

[10] Sriniv asan(2006) Reliability analysis of a three unit warm standb y redundant system with repair. Annals of Operations Research, 143(1):227-239.

[11] Timelli Giulio(2010) Constitutiv e and stochastic models to predict the effect of casting defects on the mechanical properties of High Pressure Die Cast AlSi9Cu3 (Fe) alloys. Metallurgical Science and Tecnology, 28(2):117-129.

[12] Yourui Tao, Shuyong Duan and Xujing Yang (2016). Reliability modeling and optimization of die-casting existing epistemic uncertainty . international Journal on Interactive Design and Manufacturing (IJIDeM), 10(2):51-57.