Presentaion

1. RESEARCH PROPOSAL PRESENTATION
On
“Mathematical Assessment and Reliability
Optimization of Redundant Complex
System”
Presented by:
Taruna
(Registration No. PC2333001013001)
Under the Supervision of
Dr. Sohan Tyagi
1
April 10, 2023

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Introduction
• Development in technology and the needs of the modern
society are racing against each other. Now a days, people
almost depend on technologies as a result of which the
complexity of industrial systems is increasing and modern
technology is improving commendably.
• Though the systems are being improved day by day, yet their
failure cannot be prevented. However, the failure rates may be
decreased in numerous ways including proper knowledge
about the failures of the system, designing of the system and
more understanding about its components and the measures
that affect the reliability of a system.

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Reliability
• Reliability of a system/device is the probability of a system/device
performing its anticipated purpose adequately for the intended
period of time under the given operating conditions
• This definition comprises of five words: probability, intended
function, adequately, time and operating conditions. Occurrence
of an event can’t express with certainty. Thus, probability is a tool
to measure the prediction of happening an event. Intended
function is the work which is to be done by the system under
study. Adequately means a system performing its function exactly
the same way for which it is constructed. Time is crucial factor on
which reliability depends. As time increases, reliability of a system
decreases. Operating conditions means the environment
conditions such as humidity, temperature, shock, altitude,
vibration, pressure, voltage etc.

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Reliability
• “Quantitatively, reliability of a device in time‘t’ is the probability
that it will not fail in a given environment before time t. If T is a
random variable representing the time till the failure of the device
starting with an initial operable condition at t = 0, then reliability R
(t) of device is given by
R (t) = P [T > t] = 1 -P [T ≤ t] = 1-F (t)
Thus, reliability is always a function of time. It also depends on
environmental conditions which may or may not vary with time.”
“Following assumptions are made with regard to reliability of a
system:
• (i) R (0) = 1 since the device is assumed to be operable at t = 0.
• (ii) R (∞) = 0 since no device can work forever without failure.
• (iii) R (t) is non-increasing function between limits 0 and 1.”

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System Configurations
• A system, whether a hardware or an industrial process, is
composed of many components or subsystems. The reliability of a
system is directly dependent on reliability of these components
and hence its can be measured on the basis of reliability of its
components.
• While modelling systems for estimation of their reliability
parameters first step is to understand the composition of system.
• The system is divided into a combination or structure of its
subsystems on the basis of its dependencies on components. The
two important structures while calculating these dependencies
are Series Structures and Parallel Structures. These structures are
represented by Reliability Block Diagrams. Accordingly, system
configuration is prepared in the form of a “Reliability Block
Diagram”.

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Series Configuration
• Suppose in a system there are ‘n’ components and its function is
executed successfully only if each of the ‘n’ components of system
is operative, in other words failure of any of one the components
results in complete failure of system then the components are
said to be in series configuration. They are represented by
Reliability Block Diagram given below
• If R1(t), R2(t), … Rn(t) are reliability or Survivor functions of
components 1, 2, ..n respectively, then Reliability function of
system RS(t) is represented by product of reliability functions of
components i.e.
𝑅𝑆(𝑡) = 𝑅1(𝑡)𝑅2(𝑡) … 𝑅𝑛(𝑡)

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Parallel Configuration
• Suppose in a system there are ‘n’ components and its function is
executed successfully even if one of the ‘n’ components of system
is operative, in other there is a complete failure of system if and
only if all the ‘n’ components fail, then the components are said to
be in parallel configuration. They are represented by Reliability
Block Diagram given in Figure
• If R1(t), R2(t), … Rn(t) are reliability or Survivor
functions of components 1, 2, ..n respectively,
then Reliability function of system RS(t) is
represented by product of reliability functions
of components i.e.

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Complex system – A combination of series
parallel system Configuration
• Some systems are made up of combinations of several series
and parallel configurations. The way to obtain system reliability
in such cases is to break the total system configuration down
into homogeneous subsystems. Then, consider each of these
subsystems separately as a unit, and calculate their reliabilities.
Finally, put these simple units back (via series or parallel
recombination) into a single system and obtain its reliability.

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Type of Systems
On the basis of repair point of view, the systems can be classified
as:
• Non-repairable system and Repairable system
Non-Repairable System
This type of system operates only once. Such systems have an
instantaneous life requirement.
e.g. fuses, missiles, flash bulbs. Reliability is the important criteria
to calculate the effectiveness of non-repairable system.

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Repairable System
In order to increase the system reliability, failed units be replaced
by new ones. However when this proves to be very expensive,
resort is made to repair the failed units. On failure, a unit is sent
to a repair facility. If the repair facility is not free, failed unit queue
up for repair. The life time of unit while online, while in standby
and the repair time are all independent random variables.
Different random variables can form the basis of research such as
i. Availability and reliability
ii. Time necessary for repair
iii. Number of repair that can be handled
iv. Switch over time to and from the repair facilities etc.

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Continuously operating System
This type of system once put in operation continues to operate till
its failure or the system is stopped for planned maintenance. e.g.
nuclear furnaces, earth satellites etc.
Once on and off operating system
This type of system is characterized by the fact that it can be
operated and re-operated when desired e.g. turbines, pumps,
computer, etc.
Intermittently operating system
In this case, the system is always in operational readiness, but is
required to operate intermittently e.g. telephone, radar, etc.

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Redundancy
“Redundancy is a device to improve reliability of a system. In a
redundant system, more units are made available than which are
necessary.”

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Redundancy
Active Redundancy
In this case of redundancy, the system has a positive probability
of failure even when it is not in operation. This may happen due
to the effect of temperature, environment condition etc. Active
redundancy can further be classified as hot redundancy and
warm redundancy:-
(i) If the off-line unit can fail and is loaded in exactly the same
way as the operating unit, it is called hot standby unit.
(ii) If the off-line unit can fail and can diminish the load, it is
called warm standby unit. The probability of failure for a warm
standby is less than that of failure for operative unit.

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Redundancy
(b) Passive or Cold Standby Redundancy: This is that form of
redundancy in which the offline unit cannot fail and is
completely unloaded. Reliability R (t) of an n-unit standby system
at any time instant t is given by
where Ti is the life time of ith unit and all the n-units are
independent.

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Mathematical Model
A quantitative description of a natural phenomenon is called a
mathematical model of that phenomenon. There are two types
of mathematical models-
• Deterministic model
• Stochastic model.
Deterministic model
A deterministic model predicts a single outcome from a given set
of circumstances. Deterministic models assume that known
average rates with no random deviations are applied to large
populations. For example if 10,000 individuals each have a 95%
chance of surviving 1 year, then we can be reasonably certain
that 9500 of them will indeed survive.

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Stochastic model
Stochastic model (process) used for the description of a system
operation over time. A stochastic model predicts a set of possible
outcomes weighted by their likelihoods, or probabilities. The word
"stochastic" derives from the Greek to aim, to guess) and means
"random" or "chance." In Reliability engineering stochastic model is
used to describing a system operation with respect to time. The
component failure and repair time typically becomes random variable.
Stochastic modeling develops a mathematical or financial model to
derive all possible outcomes of a given problem or scenarios using
random input variables. It focuses on the probability distribution of
possible outcomes. Examples are Monte Carlo Simulation, Regression
Models, and Markov-Chain Models.

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Literature Review
• The reliability analysis of an industry can help the management in
taking timely decision for its smooth functioning. In 1960, first text book
on reliability by Dummer and Griffen appeared in literature. Since then
a number of research papers have been published in the field of
reliability.
• Singh (1976) used reliability technology to analyse the working of
production system.
• Dhillon and Singh (1981) discussed the basic theory of reliability in their
book entitled “Engineering reliability-new technique and applications”
• Kumar et al. (1989, 1990, and 1992) calculated the availability for
number of systems in process industries.
• Gupta et al. (2005) discussed the reliability and availability analysis of
serial processes of butter oil plant and behavior analysis of the cement
industry.

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Literature Review
• Agnihotri et al. (2008) have studied the reliability analysis of boiler used
in readymade garment industry.
• Mokadies et al. (2010) discussed the comparison between two cold and
warm standby outdoor electric power systems in changing weather.
• The standby redundancy allocation in series and parallel systems was
discussed by Misra et al. (2011).
• Chib et al. (2016) studied the analysis of a two non-identical unit cold
standby system with partial and total failure
• Dąbrowska, E. (2020) used Monte Carlo simulation approach for
reliability analysis of complex systems.
• Saberzadehet al., (2022) studied the Reliability of degrading complex
systems with two dependent components per element.

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Proposed Tentative Objectives
Mathematical Assessment and Reliability Optimization of
Redundant Complex System.
• To develop reliability models for systems comprising one or
more operative units and no or some standby units.
• To determine the optimum number of standby units for the
standby systems.
• To obtain the lower/upper bounds for some parameters with
regard to the profitability of the systems.
• To make a comparative study between the models to decide
as to whether the cold or warm standby units should be used.

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Proposed Methodology
• Understanding the selected industrial process/system.
• Review of literature for understanding reliability assessment
modeling of the system/process.
• Conceptualization of reliability assessment model.
• Collection of relevant primary and secondary data on the
system/process in question.
• Review of models and equations commonly used in the
mathematical assessment and optimization of complex
systems.
• Transforming the conceptual model into quantitative model.

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References
• Barlow, R. E. and Hunter, L. C., (1960), Reliability analysis of one unit
system. Operation Research, vol.9, pp. 200-208.To determine the
optimum number of standby units for the standby systems.
• Singh, J., (1976), Some problems on Queues and Reliability. Ph.D.
Thesis Kurukshetra University Kurukshetra
• Dhillon, B. S. and Singh, C., (1981), Engineering Reliability –New
Techniques and Applications. John Wiley, New York.
• Kumar, D., Singh, J. and Pandey, P. C., (1989), Maintenance planning
for the pulping system in paper industry. Reliability Engineering and
System Safety. vol. 25 (4), pp. 293- 303.
• Kumar, D., Singh, J., and Pandey, P. C., (1990), Cost analysis of a
multi-component Screening system in paper industry.
Microelectronic and Reliability, vol. 30 (3), pp. 457- 461.

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References
• Kumar, D., Singh, J. and Pandey, P. C., (1992), Behavior analysis of
appear production system with different policies. Microelectronic and
Reliability, vol. 31(1), pp. 47-51.
• Gupta, P., Lal, A. K., Sharma R. K. and Singh, J., (2005), Numerical
analysis of reliability and availability of the serial processes in butter oil
processing plant. International Journal of Quality and Reliability
Management, vol. 22(3), pp. 303-316.
• Mokaddis, G. S. and Matta, C. H., (2010), Cost analysis of a two
dissimilar-unit cold standby redundant system subject to inspection and
random change in units. Journal of Mathematics and Statistics, vol. 6(3),
pp. 306-31.
• Misra, N., Amit Kumar, and Ishwari Dutt Dhariyal, (2011), Standby
redundancy allocations in series and parallel systems. Journal of Applied
Probability, 48, pp. 43-55.

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References
• Chib, R., J P Singh Joorel and Vikas Sharma (2016): Analysis of a two
non-identical unit cold standby system with partial and total failure
and priority, Proceedings of the 10th INDIACom-2016; IEEE
Conference ID: 37465 International Conference on “Computing for
Sustainable Global Development”, 6304-6308 (2016)
• Dąbrowska, E. (2020). Monte Carlo simulation approach to reliability
analysis of complex systems. Journal of KONBiN, 50(1), 155-170.
• Saberzadeh, Z., & Razmkhah, M. (2022). Reliability of degrading
complex systems with two dependent components per
element. Reliability Engineering & System Safety, 222, 108398.

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THANKS