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Srinivasa Rao Velampudi, Sridhar Akiri, Pavan Kumar Subara, Yadavalli VSS
A COMPREHENSIVE CASE STUDY ON IRR MODEL RT&A, No 1 (72) USING k OUT OF n CONFIGURATION_Volume 18, March 2023
A COMPREHENSIVE CASE STUDY ON INTEGRATED REDUNDANT RELIABILITY MODEL USING k-out-of-n CONFIGURATION
Srinivasa Rao Velampudi
Department of Sciences and Humanities, Raghu Institute of Technology, Visakhapatnam, Andhra
Pradesh, India, vsr.rit@gmail.com
2Sridhar Akiri *
Department of Mathematics, GITAM School of Science, GITAM (Deemed to be University), Visakhapatnam, Andhra Pradesh, India, sakiri@gitam.edu
3Pavan Kumar Subbara
Department of Mathematics, GITAM School of Science, GITAM (Deemed to be University), Bangalore Campus, Karnataka, India, psubbar@gitam.edu
4Yadavalli V S S
Department of Industrial & Systems Engineering, Pretoria University, 0002 Pretoria, South Africa,
sarma.yadavalli@up.ac.za
Abstract
Designers may introduce a system with multiple technologies in series to improve system efficiency. The configuration can be applied to k out of n systems if each technology contains k out of n factors. The k out of n configuration method is successful until every component of the system is successful. The efficiency of the entire system is more in amount than that of a single system factor in a k out of n shape. An Integrated Reliability Model (IRM) for the k out of n, here, an additional system is suggested to account for both the efficiencies of the factors and the number of factors in every phase and the different constraints to optimize the efficiency of the system. To enhance system efficiency, the authors employed the numerous methods of Lagrangean approach to determine the numbers and efficiency of the factors as well as the reliabilities of the phase under different parameters namely load, size, and cost. The dynamic programming approach and simulation method have been adapted to attain an integer result as well as to see the values real.
Keywords: Reliability Theory, IRM, Lagrangean Approach, Stage Efficiency, D P Approach, System Efficiency
1. Introduction
The structure's reliability [1] can be improved by either placing superfluous units, applying the element of greater reliability or by adopting the two methods at a time and both of them use extra resources. Optimizing structure reliability, and conditions to resource availability viz. size, value, load, are examined. In general, reliability is tested as an element of value; But, when tested with
Srinivasa Rao Velampudi, Sridhar Akiri, Pavan Kumar Subara, Yadavalli VSS
A COMPREHENSIVE CASE STUDY ON IRR MODEL RT&A, No 1 (72)
USING k OUT OF n CONFIGURATION_Volume 18, March 2023
real-world problems, the invisible effect of other restraints such as load, size [4], etc. has a special effect on improving structural reliability. The specific functionality of the over-reliability model with several limitations to optimize the recommended setup was examined to maximize the recommended setup. The problem examines the unknowns that is, various elements (Xej), the element reliability (re$), and the stage reliability (Rs$) at a specific point for disposing of multiple restraints to magnify the structure reliability that is described as a [14] United Reliability Model (URM). In literature, United Reliability Models [8] are enhanced by applying value restraints where there is a fixed association between value and reliability. A unique pattern of planned work is a deliberation of the load and size as supplementary restraints along with value to form and improve the superfluous reliability system for [15] k out of n structure composition [6, 7].
2. Methods
2.1. Assumptions and Notations:
• Each stage's elements are believed to be identical, i.e., all elements have the same level of reliability.
• All elements are supposed to be statistically independent, meaning that their failure has no bearing on the performance of other elements in the structure.
Rsr - Structure Efficiency
Rsj - Efficiency of phase 'sf, 0 < Rs$ < 1
rej - Efficiency of each component in phase 'ej' ; 0 < rej < 1
%ej - Number of components in phase '' ej'
r - ej Worth coefficient of each component in phase ' ef
Lej - Load coefficient of each component in phase ' ef
ej Size coefficient of each component in phase ' ef
- Greatest allowable structure - Value
Leo - Greatest allowable structure - Load
$e0 - Greatest allowable structure - Size
LMM Lagrangean Multiplier Method
DPA Dynamic Programming Approach
IRRM Integrated Redundant Reliability Model
Cj, dj, i$, kj, m$, n$ are Constants.
2.2 Mathematical Analysis:
The efficiency of the system to the provided worth function
Rsr = Z?=i B(m,i)pi(1-p)+
(1)
The following relationship between worth and efficiency is used to calculate the worth coefficient of each unit in phase 'ef .
rej = sinh
dj
(2)
Therefore Similarly,
Cej = f$ sinh [r(
ej!
L#j = pj sinh [rt
ej
Se$ = q$ sinh [rej] ni Since value-components are linear in ej,
^j =1 Cej. %ej — CeO Similarly load-components and size-components are also linear in ej,
Yij = 1 ^ej. %ej — L#O
^- = 1 Sej. %ej — $eO
Substituting (2) in (3)
?n=1 fj sinh [Tej] 2 . Xej - Ceo —0 ^=1 Pj Sinh [Tej ] 3 . Xej - Leo —0 zn=i qj Sinh [Tej] ni . Xej - Seo —0
The transformed equation through the relation Xe$ =
InRsj
ej lnr
- r .)ej-k
Where RSj = Y-=2 B (ej, k) (rei)3 (1 - )
Subject to the constraints
Z-=! fj sinh [rt
ej
YIj=/Pj sinh [r#
ej
l5Rs ■ lnr„
lnfis lnrp
■ - Ce) <0
Z-=!qj sinh [Tej] -j ■
lnfis lnrp
■ - Le) <0
■ - Se) <0
(2a) (2b) (2c)
(3a)
(3b) (3c)
(4a) (4b) (4c)
(5)
(6)
(7a) (7b) (7c)
Positivity restrictions ej >0 A Lagrangean function is defined as
Lf = R&r + öi
Z-=- fj sinh [Tej] 2 ■
d. IHR" - _ \nr„, e
+ ô7
Z-=! Pj sinh [Tej] k ■
k. InlRs" - j
i ^e
In rpi e
Z-=! qj sinh [Tej] -j
In's ■ lnr„
- Se
+ (8)
The Lagrangean function can be used to find the ideal point and separating it by Rs$, rej, &i, 62 and 63.
:r( =1+ô/E-u fj sinh [rt
03
dip drei
07 03
d'sR
Z-=! qj sinh [Tej] -j ■ = 0/
ej
-]+ Ô7
Z-.! Pj sinh [rt
ej
' lnrej RsjJ
] dj ln'sj"
ej] j ■ lnrej
4n'sj>
■ lnrej\ \U] ■
In'sjl
lnrej\ \U] ■ '
+ (9)
irei--;—
ej rei .Inre
+
hl~ei--:-
j rej4nrej
IT- — ---
e j rei.lnre
+
= Z-=! fj sinh [re
In Rs
■ - r
IJo,
= Z- n.sinh lr ■] kj - j
d b2 Ll = ! Pj sinh ^ j ■ lnrej Le
(10) (11) (12)
d
k
lnrej Rsj
Srinivasa Rao Velampudi, Sridhar Akiri, Pavan Kumar Subara, Yadavalli VSS
A COMPREHENSIVE CASE STUDY ON IRR MODEL RT&A, No 1 (72) USING k OUT OF n CONFIGURATION_Volume 18, March 2023
% =H=i u s^\rej]-J ^L - Se0 CL3)
Where 61; &2 and 53 are Lagrangean multipliers.
The number of elements in each phase (Xe$), the best element reliability (rej), the phase reliability (RSj) and the structure reliability (Rsr)are derived by using the Lagrangean method [12]. This method provides a real (valued) solution concerning worth, load, and size.
2.3 Case Problem
To derive the multiple parameters of a given mechanical system using optimization techniques [9], where all the assumptions like value, weight, and volume are directly proportional to system reliability has been considered in this research work. The same logic may not be true in the case of electronic systems. Hence, the optimal element accuracy (re$), phase reliability (Rs$), Number of elements in each phase (Xe$), and structure accuracy (Rsr ) can be evaluated in any given mechanical system [10]. In this work, an attempt has been made to evaluate the Structure accuracy [13] of a special purpose machine that is used for single phase industrial power generators assembly.
The machine is used for the assembly of 3 or 4 components on the base of the power generator. The machine's approximate worth was $3000, which is considered a structure value, the load of the machine is 120 pounds which is the load of the structure, and the space occupied by the machine is 100 cm3, which is the volume or size of the structure. To attract the authors from different cross sections, the authors attempted to use hypothetical numbers, which can be changed according to the environment.
2.4 Constants
The data required for the constants for the case problem are provided in Table 1.
Tablel: Worth, Load and Size Pre-fixed Constant Values
Phase Worth Constants Load Constants Size Constants
fj dj Pj kj Vj nj
1 2200 0.85 100 0.92 100 0.94
2 2400 0.88 80 0.88 90 0.89
3 2600 0.91 60 0.91 80 0.86
The efficiency of each factor, phase, and number of factors in each stage, as well as the structural efficiency [2, 3], are shown in the tables below.
2.4.1 The Details of Component-Worth Constraint by using Lagrangean Multiplier Method without Rounding-Off
The value-related efficiency design is described in the Table 2.
Table2: Worth Constraint Analysis by using Lagrangean Multiplier Method
Phase fj dj rej Log rej Rsj Log R%j %ej r . uej Cej ■ %ej
01 2200 0.85 0.8741 -0.0584 0.6777 -0.1690 2.89 2233 6456
02 2400 0.88 0.8445 -0.0734 0.6487 -0.1880 2.56 2334 5977
03 2600 0.91 0.8456 -0.0728 0.5461 -0.2627 3.61 2516 9077
Final Worth 21510
2.4.2 The Details of Component-Load Constraint by using Lagrangean Multiplier Method without Rounding-Off
The equivalent results for the load are shown in the Table 3.
Table3: Load Constraint Analysis by using Lagrangean Multiplier Method
Phase Pj kj rej Log re$ Rsj Log R%j %ej Lej Lej .%ej
01 100 0.92 0.8741 -0.0584 0.6777 -0.1690 2.89 100 290
02 80 0.88 0.8445 -0.0734 0.6487 -0.1880 2.56 78 199
03 60 0.91 0.8456 -0.0728 0.5461 -0.2627 3.61 58 209
Final Load 698
2.4.3 The Details of Component-Size Constraint by using Lagrangean Multiplier Method without Rounding-Off
Equivalent results for size are described in the Table 4.
Table4: Size Constraint Analysis by using Lagrangean Multiplier Method
Phase 1j nj rej Log rej Rsj Log R%j %ej ç . uej $ej . ^ej
01 100 0.94 0.8741 -0.0584 0.6777 -0.1690 2.89 100 289
02 90 0.89 0.8445 -0.0734 0.6487 -0.1880 2.56 87 224
03 80 0.86 0.8456 -0.0728 0.5461 -0.2627 3.61 78 282
Final Size 795
3. Efficiency Design by using Lagrangean Multiplier Method
The efficiency design [11] summarizes the e$ values as integers (rounding the value of e$ to the nearest integer), and the acceptable outcomes for the worth, load, and size are listed in the tables. Calculate variance due to worth, load, size, and construction capacity (before and after rounding off e$ to the nearest integer) to obtain information.
3.1 Efficiency Design by using Lagrangean Multiplier Method Concerning Worth, Load and Size with Rounding-Off
Table5: Efficiency design relating to Worth, Load and Size Constraint Analysis by using Lagrangean Multiplier Method with Rounding Off is shown in the following table
Phase rej Rsj %ej r . uej Cej . %ej Lej Lej .%ej ^ej^ej $ej . ^ej
01 0.8741 0.6777 3 2233 6699 100 300 100 300
02 0.8445 0.6487 3 2334 7002 78 234 87 261
03 0.8456 0.5461 4 2516 10066 58 232 78 312
Total Worth, Load and Size 23767 766 873
Structure Efficiency (Rsr) 0.9987
Var'at'on 'n "Worth = Total Worth with rounding off—Total Worth without rounding off
Total Value without rounding off
10.49%
Variation in Load
Total Load with rounding off—Total Load without rounding off Total Load without rounding off
09.72%
Variation in Size
Total Size with rounding off—Total Size without rounding off Total Size without rounding off
05.00%
Variation in Efficiency
Effciency with rounding off—Efficiency without rounding off Structure Efficiency without rounding off
■ 10.06%.
4. Dynamic Programming Approach
Using the Lagrangean technique [5], which has a number of drawbacks, such as having to provide the amount of components needed at each stage (Xej('ej')) in real numbers, which may be difficult to apply. The generally used approach of rounding down the value of results in changes in worth, load, and size, affecting system reliability and having a significant impact on the model's efficiency design. This flaw could be considered, for which the author suggests a substitute empirical implementation that uses the dynamic programming method to obtain an integer solution by using the solutions produced from the Lagrangean approach as parameters for the proposed dynamic programming method.
Table6: Phase I of the Dynamic Programming
Phase-I('e/') Phase - I - Reliability (Rsj)
01 0.5789
02 0.7183
03 0.8577
04 0.9265
05 0.9605
06 0.9945
Table7: Phase II of the Dynamic Programming
Phase - II ('ej') Phase - II - Reliability (Rsj)
04 0.4370 0.4490 0.4540
05 0.4666 0.4916 0.4978 0.5238
06 0.4962 0.5342 0.5416 0.6125 0.3991
07 0.5258 0.5768 0.5854 0.7012 0.5496 0.4285
08 0.5406 0.5981 0.6073 0.7899 0.7571 0.6614 0.6734 0.4523
09 0.5554 0.6194 0.6292 0.8786 0.8936 0.7258 0.7254 0.4835
10 0.5628 0.6407 0.6511 0.9229 0.9187 0.9021 0.8852 0.7264
Table8: Phase III of the Dynamic Programming
Phase - III ('ej') Phase - III - Reliability (Rsj)
09 0.4932 0.6235 0.6461 0.5938 0.5814 0.4235 0.2824 -
10 0.5383 0.7045 0.7044 0.6451 0.6287 0.4516 0.3125 0.1818
11 0.5824 0.7436 0.7654 0.6821 0.7345 0.5216 0.4514 0.2345
12 0.6265 0.7874 0.7952 0.7514 0.7841 0.6834 0.5387 0.3678
13 0.6706 0.7951 0.8164 0.8354 0.8547 0.7893 0.6868 0.4864
14 0.7147 0.8356 0.8573 0.9125 0.9571 0.8514 0.7256 0.6454
Srinivasa Rao Velampudi, Sridhar Akiri, Pavan Kumar Subara, Yadavalli VSS
A COMPREHENSIVE CASE STUDY ON IRR MODEL RT&A, No 1 (72) USING k OUT OF n CONFIGURATION_Volume 18, March 2023
5. Results
The application of the Lagrangean multiplier technique yielded a real-valued solution for the suggested Integrated Redundant Reliability Systems for the k-out-of-n configuration mathematical models under investigation, as well as the much-required integer solution. The author obtained new phase reliability (RSj) by using a dynamic programming approach. The new values for stage reliability (RSj) are (0.9445, 0.9229, and 0.9571). The Dynamic Programming Approach is utilised, and the results for the given mathematical function are outlined in tables 9, 10, and 11 that follow in order to derive the required conclusions.
5.1 The Details of Component-Worth Constraint by using Dynamic Programming Approach
The value-related efficiency design is described in the Table 9.
Table9: The Details of Component-Worth constraint by using Dynamic Programming Approach
Phase h dj rei Log r#j Rsj Log Rsj r . uej Cej ■ Xej
01 2200 0.85 0.9982 -0.0008 0.9945 -0.0024 3 2580 7740
02 2400 0.88 0.9736 -0.0116 0.9229 -0.0348 3 2735 8205
03 2600 0.91 0.9891 -0.0048 0.9571 -0.0190 4 3016 12064
Final Worth 28009
Mutation in Worth -Component = 30.21%
Mutation in Structure Efficiecy = 01.23%
5.2 The Details of Component-Load Constraint by using Dynamic Programming Approach
The equivalent results for the load are shown in the Table 10.
Table10: The Details of Component-Load constraint by using Dynamic Programming Approach
Phase Pi rej Log re$ Rsj Log Rsj Lej Lej ej
01 100 0.92 0.9982 -0.0008 0.9945 -0.0024 3 117 351
02 80 0.88 0.9736 -0.0116 0.9229 -0.0348 3 91 273
03 60 0.91 0.9891 -0.0048 0.9571 -0.0190 4 70 280
Final Load 904
Mutation in Load-Component = 29.32%
Mutation in Structure Efficiecy = 01.23%
5.3 The Details of Component-Size Constrain by using Dynamic Programming Approach
The equivalent results for size are described in the Table 11.
Tablell: The Details of Component-Size constraint by using Dynamic Programming Approach
Phase 1j n$ rej Log rej Rsj Log R%j %ej ej $ej • ^ej
01 100 0.94 0.9982 -0.0008 0.9945 -0.0024 3 117 351
02 90 0.89 0.9736 -0.0116 0.9229 -0.0348 3 103 309
03 80 0.86 0.9891 -0.0048 0.9571 -0.0190 4 93 372
Final Size 1032
Structure Efficiency (Rsr ) 0.9864
Mutation in Size-Component = 29.81%
Mutation in Structure Efficiecy = 01.23%
5.4 Comparison of Optimization of Integrated Redundant Reliability k out of n systems - LMM with rounding-off and Dynamic Programming approach for Worth
Table12: Results Correlated LMM with rounding off approach and Dynamic programming approach for Worth
With Rounding Off Dynamic Programming
Phase %ej rej Rsj r . uej ^ej • %ej rej Rsj r . uej ^ej • %ej
01 3 0.8741 0.6777 2233 6699 0.9982 0.9945 2580 7740
02 3 0.8445 0.6487 2334 7002 0.9736 0.9229 2735 8205
03 4 0.8456 0.5461 2516 10066 0.9891 0.9571 3016 12064
Total Worth 23767 28009
Structure Efficiency Using With LMM Approach (Rsr) 0.9987 Using DP Approach (Rsr ) 0.9999
5.5 Comparison of Optimization of Integrated Redundant Reliability k out of n systems - LMM with rounding-off and Dynamic Programming approach for Load
Table13: Results Correlated with LMM rounding off approach and Dynamic programming approach for Load
With Rounding Off Dynamic Programming
Phase %ej rej Rsj Lej Lej • %ej rej Rsj Lej Lej • %ej
01 3 0.8741 0.6777 100 300 0.9982 0.9945 117 352
02 3 0.8445 0.6487 78 234 0.9736 0.9229 91 273
03 4 0.8456 0.5461 58 232 0.9891 0.9571 70 278
Total Load 767 904
Structure Efficiency Using With LMM Approach (R&r ) 0.9987 Using DP Approach (Rsr ) 0.9999
5.6 Comparison of Optimization of Integrated Redundant Reliability k out of n systems - LMM with rounding-off and Dynamic Programming approach for Size
Table14: Results Correlated LMM with rounding off approach and Dynamic programming approach for Size
With Rounding Off Dynamic Programming
Phase %ej rej Rsj ç . uej $ej • ^ej rej Rsj ç . uej $ej • ^ej
01 3 0.8741 0.6777 100 300 0.9982 0.9945 117 352
02 3 0.8445 0.6487 87 261 0.9736 0.9229 103 307
03 4 0.8456 0.5461 78 312 0.9891 0.9571 93 372
Total Size 873 1031
Structure Efficiency Using With LMM Approach (R&r ) 0.9987 Using DP Approach (Rsr ) 0.9999
6. Discussion
This work proposes an integrated reliability model for a k out of n configuration system with many efficiency criteria. When the data are discovered to be in reals, the Lagrangean multiplier approach is used to compute the number of components (Xe$), component efficiencies (re$), phase efficiencies (Rs$), and system efficiency (Rsr). To obtain practical applicability, the dynamic way of programming approach is employed to construct an integer solution using the inputs from the Lagrangean method.
This work proposes an integrated reliability model for a k out of n configuration system with many efficiency criteria, when the data are discovered to be in real solution. The Lagrangean multiplier approach is used to compute the number of components (Xe$) and the respective component efficiencies (rej) are 0.8741, 0.8445 & 0.8456, stage reliabilities (Rsj) are 0.6777, 0.6487 & 0.5461, and structure reliability (Rsr) is 0.9987. To obtain practical applicability, Dynamic programming approach is employed to construct an integer solution whereas component reliabilities (rej) are 0.9982, 0.9736 & 0.9891, stage reliabilities (Rsj) are 0.9945, 0.9229 & 0.9571, and the system reliability (Rsr ) is 0.9999, Using the inputs from the Lagrangean method. Finally, we observed that the worth, load and size components changed slightly, but compare with stage reliability, resulting in increased system reliability.
The IRM generated in this manner is quite valuable, particularly in real-world settings when a k from n configuration IRM with reliability engineer redundancy is required. In circumstances where the system value is low, the proposed model is especially valuable for the dependability design engineer to build high-quality and efficient materials.
In future study, the authors recommend utilizing a unique approach that limits the minimum and maximum component reliability values while maximizing system dependability using any of the current heuristic processes to build similar IRMs with redundancy.
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