Availability analysis considering two subsystem failures simultaneously via markov modeling

This paper discusses the performance analysis of the alloy wheel plant (AWP). Till now researchers (1, 5, 9, 14, and 25) took assumption that only one failure at one time but this paper considered two failures at one time. The AWP mainly consists of four subsystems in series (Gdc Machine, Cutting Machine, Heat Treatment Machine and Shot Blasting Machine). The failure and repair rates are implicit to be constant and statistically independent. A transition diagram developed form the interrelationship among the working and failed state. Mathematical modeling of this system is done by applying markov- birth – death processes. These equations are solved by recursive method and optimized reliability is calculated for failure and repair rates. The obtaining result is helpful to improve the overall performance of alloy wheel plant.

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Introduction

In today's scenario changes in technology are very rapid, every industry trying to make a more reliable product, which is having better life, better performance. In this paper authors try to improve reliability of AWP and for optimizing reliability failure and repair rates are given. Singh and Goyal (2013) [2] discussed the Behavior analysis of a biscuit making plant using Markov Regenerative modeling. The methodology of the system based on Markov model which is used to find the availability of the system and transient behavior of repairable mechanical biscuit shaping system pertaining to a biscuit manufacturing plant. Garg, Kumar and Singh (2009) [2] discussed about the Availability analysis of a cattle feed plant using matrix method. This plant contains seven subsystems arrange in series. The mathematical model has been developed by using Markov birth-death process and makes a transition diagram. Tewari and Kumar (2016) [3] the paper describes the availability analysis of milling system in a rice milling plant. The failure and repairs rates are assumed to be exponential distribution and problem is formulated using Markov birth-death process. Kumar (2014) [4] discussed the availability analysis of a thermal power plant boiler air circulation system using Markov approach. In this plant the system contains four subsystems with full working state, reduced capacity state and failed state. Rajbala & Garg (2019) [6] discussed about the steady state and time dependent availability analysis of a manufacturing plant. Rajbala & Garg (2019) [6] discussed about the Behaviour analysis of alloy wheel plant. Rajbala & Kumar (2021) [7] discussed about an article on the system reliability and availability analysis using RPGT-A general approach. Kumar and Garg (2019) [19] have discussed the reliability technology theory and its applications. Kumar et al. (2018) [10] have studied behaviour analysis of a bread making system. Kumar et al. (2019) [19] analyzed sensitivity analysis of a cold standby framework with priority for preventive maintenance consist two identical units with server failure utilizing RPGT. Present paper consists two units one of which is online while other is kept is cold standby mode. Online & cold standby unit are indistinguishable in nature & have just two modes one is good and other is totally failed. Rajbala, et al. (2019) [6] have studied the system modeling and analysis: a case study EAEP manufacturing plant. Kumar et al. (2017) [14] have studied behavior analysis in the urea fertilizer industry.

Kumar et al. (2017) [14] have examined the mathematical modeling & profit analysis of an edible oil refinery plant. Kumar et al. (2019) [13] studied mathematical modeling & behavioral analysis of a washing unit in paper mill. Kumar et al. (2018) [10] paper analyzed sensitivity analysis of 3:4: good system plant. Priya et al. (2021) [15] Vedic mathematics in derivatives and integration, differential equations and partial differential equations. Kumari et al. (2021) [19] studied the constrained problems using PSO. Kumari et al. (2021) [17] discussed the profit analysis of an agriculture thresher plant in steady state using RPGT. Anchal et al. (2021) [18] discussed the SRGM model using differential equation has been proposed, in which two categories of faults: simple and hard with respect to time in which these occur for isolation and removal after their detection has been presented.

System Description

The description of the following subsystems are described as below:

 Gdc Machine: In the Gravity die casting machine, casting is produced with the help of gravity pressure.  Cutting Machine: Cutting machine is used to cut the riser.  Heat Treatment Machine: In heat treatment machine hardness of casting is improved. In heat treatment machine there are two types of hot chamber in which wheel is hold for 10 hours. i. Solutionizing Chamber-In this chamber wheel is hold for 6 hours and then wheel is cooled quickely with the help of water. ii. Azing Chamber-After solutiuonizing wheel is forward to azing chamebr for 4 hours.  Shotblasting Machine: In this machine small steel shots are bumbared on casting to make it rough. Steel balls are bumbarded with the help of high air pressure.

Transition Diagram

The transition diagram is used to express the behaviour of the all the subsystems.  p I (t), probability of the system is in i th state at time

Differential equation of transition diagram P₁'(t)+ (α₁+α₂+α₃+α₄ + (1-(1-α₁) (1-α₂)) + (1-(1-α₁) (1-α₃)) + (1-(1-α₁) (1-α₄)) + (1-(1-α₂) (1-α₃)) + (1-(1-α₂) (1-α₄)) + (1-(1-α₃) (1-α₄)) P₁(t) = β₁ P₂(t) + β₂ P₃(t) + β₃ P₄(t) + β₄ P₅(t).

After simplification this equation can be written as: p₁' (t) + (4(α₁+α₂+α₃+α₄) -(α₁ α₂+α₁ α₃+α₁ α₄+α₂ α₃+α₂ α₄+α₃ α₄)) p₁ (t) = β₁ p₂ (t) + β₂ p₃ (t) + β₃ p₄ (t) + β₄ p₅ (t) p₂' (t) + β₁ p₂ (t) =α₁ p₁ (t) p₃' (t) + β₂ p₃ (t) = α₂ p₁ (t) + β₂ p₆ (t)

p₄' (t) + β₃ p₄ (t) = α₃ p₁ (t) + β₁ p₇ (t) + β₂ p₉ (t) p₅' (t) + β₄ p₅ (t) = α₄ p₁ (t) + β₁ p₈ (t) + β₄ p₁₀ (t) + β₃ p₁ ₁ (t) p₆' (t) + β₂ p₆ (t) = (1-(1-α₁) (1-α₂)) p₁ (t) p₇' (t) + β₁ p₇ (t) = (1-(1-α₁) (1-α₃)) p₁ (t) p₈' (t) + β₁ p₈ (t) = (1-(1-α₁) (1-α₄)) p₁ (t) p₉' (t) + β₂ p₉ (t) = (1-(1-α₂) (1-α₃)) p₁ (t) p₁₀' (t) + β₄ p₁₀ (t) = (1-(1-α₂) (1-α₄)) p₁ (t) p₁₁' (t) + β₃ p₁₁ (t) = (1-(1-α₃) (1-α₄)) p₁ (t)

After solving these equations, we get the following equations p₂ (t) = z₀ p₁ (t); Where z₀ = α₁/β₁ p₃ (t) = z₁ p₁ (t); Where z₁ = (2α₂+α₁-α₂α₁ /β₂) p₄ (t) = z₂ p₁ (t); Where z₂ = (3α₃+α₁+α₂-α₃α₁-α₃α₂ / β₃) p₅ (t) =z₃ p₁ (t); Where z₃= (4α₄+α₁+α₂+α₃-α₄α₁-α₄α₂-α₄α₃/β₄) p₆ (t) = z₄ p₁ (t); Where z₄ = (α₂+α₁-α₁α₂/β₂) p₇ (t) = z₅ p₁ (t); Where z₅ = (α₃+α₁-α₃α₁/β₁) p₈ (t) = z₆ p₁ (t); Where z₆ = (α₄+α₁-α₄α₁/β₁) p₉ (t) = z₇ p₁ (t); Where z₇ = (α₃+α₂-α₃α₂/β₂) p₁₀ (t) = z₈ p₁ (t); Where z₈ = (α₄+α₂-α₄α₂/β₄) p₁₁ (t) = z₉ p₁ (t); Where z₉= (α₄+α3-α₄α₃/β₃) The Probability pᵢ (t) is determined by using normalizing condition ∑ 𝑝ᵢ 11 𝑖=0 = 1 p₁ + p ₂+ p₃ + p₄ + p ₅+ p₆ + p₇ + p₈ + p₉ + p₁₀ + p₁₁ = 1 p₁+z₀ p₁ (t) +z₁ p₁ (t) + z₂ p₁ (t) + z₃ p₁ (t) + z₄ p₁ (t) + z₅ p₁ (t) + z₆ p₁ (t) + z₇ p₁ (t) + z₈ p₁ (t) + z₉ p₁ (t) =1 (1 + z₀ + z₁ + z₂ + z₃ + z₄ + z₅ + z₆ + z₇ + z₈ + z₉) p₁ (t) = 1 p₁ (t) * V = 1 p₁ (t) = 1/V Where, V = (1 + z₀ + z₁ + z₂ + z₃ + z₄ + z₅ + z₆ + z₇ + z₈ + z₉) Aᵥ = P₁

Performance Analysis

The performances of AWP mostly affected by failure and repair rates of each subsystem. The transition diagram includes all possible state (Working state and failed state) that is, failure rate is αᵢ and repair rate is βᵢ. The table 1 to 4 represents the availability decision for AWP.

α₁→ β₁↓	0.002	0.004	0.006	0.008	0.010	Constant value
0.02	0.269	0.264	0.260	0.198	0.194	α₂=0.001 β₂=0.01 α₃=0.003 β₃=0.03 α₄=0.005 β₄=0.05
0.04	0.298	0.269	0.269	0.199	0.198
0.06	0.299	0.272	0.274	0.205	0.203
0.08	0.324	0.279	0.289	.212	0.213
0.10	0.349	0.288	0.291	0.224	0.220

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Availability for different values of failure and repair rate of GDC Machine

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Conclusion

The repair rate of GDC machine affect availability of system more effectively as comparison to another system. This modal is more realistic as compared to those models in which one failure is considered at one time. As in practical situations any number of failures can happen at one time. Further, we will extend our work to any random failures in random time without putting any assumption on number of failures. Subsystem name and their rank based on their "repair rates effect availability".

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Transition diagram of the states

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α₂→ β₂↓	0.001	0.002	0.003	0.004	0.005	Constant value
0.01	0.269	0.268	0.267	0.254	0.251	α₁=0.002 β₁=0.02 α₃=0.003 β₃=0.03 α₄=0.005 β₄=0.05
0.02	0.272	0.270	0.269	0.256	0.253
0.03	0.281	0.276	0.274	0.265	0.256
0.04	0.284	0.277	0.275	0.269	0.257
0.05	0.289	0.278	0.278	0.270	0.268

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Availability for different value of failure and repair rate of CUTTING Machine

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α₃→ β₃↓	0.003	0.005	0.007	0.009	0.011	Constant value
0.03	0.269	0.267	0.263	0.260	0.258	α₁=0.002 β₁=0.02 α₂=0.001 β₂=0.01 α₄=0.005 β₄=0.05
0.05	0.275	0.273	0.270	0.268	0.264
0.07	0.279	0.278	0.275	0.273	0.270
0.09	0.281	0.279	0.277	0.274	0.269
0.11	0.284	0.283	0.280	0.277	0.275

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Availability for different value of failure and repair rate of HEAT TREATMENT Machine

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Variation of failure and repair rate of GDC Machine

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Variation of failure and repair rate of Cutting Machine

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α₄→ β₄↓	0.005	0.007	0.009	0.011	0.013	Constant value
0.05	0.269	0.266	0.260	0.257	0.253	α₁=0.002 β₁=0.02 α₂=0.001 β₂=0.01 α₃=0.003 β₃=0.03
0.07	0.276	0.272	0.270	0.268	0.266
0.09	0.281	0.277	0.275	0.272	0.270
0.11	0.286	0.284	0.282	0.280	0.278
0.13	0.290	0.288	0.286	0.284	0.280

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Value of availability for different value of failure and repair rate of SHOTBLASTING Machine

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Variation of failure and repair rate of Heat Treatment Machine

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Variation of failure and repair rate of SHOT BLASTING Machine

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1. Singh P, Goyal A. Behavior analysis of a biscuit making plant using Markov Regenerative modeling, International Journal on theoretical and applied research in mechanical engineering. 2013;2(3):2103.
2. Garg D, Kumar K, Singh J. Availability analysis of a cattle feed plant using matrix method. International Journals of Engineering. 2010;3(2):201-19.
3. Tewari PC, Kumar P. Availability analysis of milling system in a rice milling plant, World Academic of science, Engineering and technology. 2016 Sep 4;10(8):1600-9.
4. Kumar. Availability analysis of thermal power plant boiler air circulation system using Markov approach, Decision Science letters, 2014;3(1):65-72.
5. Rajbala, Arun Kumar, Deepika Garg. Systems Modeling and Analysis: A Case Study of EAEP Manufacturing Plant. International Journal of Advanced Science and Technology. 2019;28(14):08-18.
6. Rajbala, Deepika Garg. Behaviour analysis of alloy wheel plant. International Journal of Engineering and Advanced Technology (IJEAT). 2019;9(2):319-327. ISSN 2249-8958.
7. Rajbala, Kumar A. Article on the system reliability and availability analysis using RPGT-A general approach, Galaxy International Interdisciplinary Research Journal. 2021;9(5):371-375.
8. Kumar A, Garg D, Goel P. Mathematical modeling and behavioral analysis of a washing unit in paper mill, International Journal of System Assurance Engineering and Management. 2019;1(6):1639-1645.
9. Kumar A, Garg D, Goel P. Sensitivity analysis of a cold standby system with priority for preventive maintenance. Journal of Advance and Scholarly Researches in Allied Education. 2019;16(4):253-258.
10. Kumar A, Goel P, Garg D. Behaviour analysis of a bread making system. International Journal of Statistics and Applied Mathematics. 2018;3(6):56-61.
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