Open access peer-reviewed chapter

# The Optimal System for Complex Series-Parallel Systems with Cold Standby Units: A Comparative Analysis Approach

Written By

Ibrahim Yusuf and Ismail Muhammad Musa

Submitted: July 8th, 2020 Reviewed: November 28th, 2020 Published: June 16th, 2021

DOI: 10.5772/intechopen.95274

From the Edited Volume

## Practical Applications in Reliability Engineering

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## Abstract

The purpose of this research is to propose three reliability models (configurations) with standby units and to study the optimum configuration between configurations analytically and numerically. The chapter considered the need for 60 MW generators in three different configurations. Configuration 1 has four 15 MW primary units, two 15 MW cold standby units and one 30 MW cold standby unit; Configuration 2 has three 20 MW primary units, three 20 cold standby units; Configuration 3 has two 30 MW primary units and three 30 MW cold standby units. Some reliability features of series–parallel systems under minor and complete failure were studied and contrasted by the current. Failure and repair time of all units is assumed to be exponentially distributed. Explanatory expressions for system characteristics such as system availability, mean time to failure (MTTF), profit function and cost benefits for all configurations have been obtained and validated by performing numerical experiments. Analysis of the effect of different system parameters on the function of profit and availability has been carried out. Analytical comparisons presented in terms of availability, mean time to failure, profit function and cost benefits have shown that configuration 3 is the optimal configuration. This is supported by numerical examples in contrast to some studies where the optimal configuration of the system is not uniform as it depends on some system parameters. Graphs and sensitivity analysis presented reveal the analytical results and accomplish that Configuration 3 is the optimal in terms of design, reliability physiognomies such as availability of the system, mean time to failure, profit and cost benefit. The study is beneficial to engineers, system designers, reliability personnel, maintenance managers, etc.

### Keywords

• optimality
• availability
• standby
• partial
• complete failure
• MTTF

## 1. Introduction

Systems or configurations are designed with intention of meeting the optimal designed that has the reliability requirement at satisfaction of the buyers or customers usually studied with intention to the increase their reliability characteristics in terms such as mean time to failure (MTTF), busy period of repairman, availability, generated revenue as well as profit. Reliability models are vital in measuring the overall performance of system in ensuring quality of products. Achieving a high level of reliability through redundancy is often an essential requisite.

Literature on the reliability of comparative analysis of systems with standby units is numerous, and here we study previous papers on the issues of systems with standby units. Due to their significance in education, communication, military, industry and economics, many researchers have done excellent work in the field of reliability and performance analysis of serial systems by studying and constructing mathematical models to test their performance under different operating conditions. For instance; Singh et al. [1, 2] used copula to study the performance analysis of the complex system in the series configuration under different failure and repair discipline. Lado and Singh [3] recently discussed the cost assessment of complex repairable systems consisting two subsystems in series configuration using Gumbel Hougaard family copula. Yusuf [4] presented the availability modeling and evaluation of repairable system subject to minor deterioration under imperfect repairs. Singh and Ayagi [5] provided a frame work to analyze the performance of a complex system under preemptive resume repair policy using copula. Niwas and Garg [6] discussed the availability, reliability and profit of an industrial system based on cost free warranty policy. Monika et al. [7] provided a complex system having two subsystems in series configuration under k-out-of-n: G, policy. The k-out-of-n works if and only if at least k of the n components works. Gahlot et al. [8] analyzed the performance of repairable system in series configuration under different types of failure and repair policies using copula linguistics. Singh.,V.V and Singh, N. P [9] analyzed the performance of three-unit redundant system with switch and human failure. Saini and Kumar [10] discussed the performance evaluation of evaporation system in sugar industry using RAMD analysis. Malik and Tewari [11] presented performance modeling and maintenance priorities decision for water flow system of a coal based thermal power plant. Lado et al. [12] discussed the performance and cost assessment of repairable complex system with two subsystems connected in series configuration.

Researchers in the past have presented excellent works on reliability analysis of complex repairable systems and proclaimed better performance of the repairable system by their operations. Chen et al. [13] dealt with reliability analysis of a cold standby system with imperfect repair and under poisson shocks. Corvaro et al. [14] presented RAM analysis on reciprocating compressors. Garg [15] analyzed the reliability of industrial system using fuzzy kolmogrov’s differential equations. Garg [16] presented an approach for analyzing the reliability of series–parallel system using credibility theory and different types of intuitionistic fuzzy numbers. Garg and Sharma [17] discussed two phase approach for reliability and maintainability analysis of an industrial system. Garg [18] presented RAM analysis of industrial systems using PSO and fuzzy methodology. Kakkar et al. [19] analyzed the reliability of two-unit parallel repairable industrial system. Kakkar [19] discussed the reliability of two dissimilar parallel unit repairable system with failure during preventive maintenance. Niwas and Kadyan [20] dealt with reliability modeling of a maintained system with warranty and degradation. Negi and Singh [21] analyzed the reliability of non-repairable complex system with weighted subsystems connected in series. Patil et al. [22] presented the reliability analysis of CNC turning center based on the assessment of trends in maintenance data. Tsarouhas [23] dealt with RAM analysis for wine packaging production line. Wang et al. [24] analyzed the reliability of two-dissimilar-unit warm standby repairable system with priority in use. Wu [25] analyzed the reliability of a cold standby system attacked by shocks. Wu and Wu [26] analyzed the reliability of two-unit cold standby repairable systems under Poisson shocks. Garg [27] analyzed the reliability of industrial system using fuzzy kolmogrov’s differential equations. Kakkar et al. [28] analyzed the reliability of two dissimilar parallel unit repairable system with failure during preventive maintenance. Kumar and Malik [29, 30] dealt with reliability measures of a computer system with priority to PM over the H/W repair activities subject to MOT and MRT. Kumar and Lather [31] analyzed the reliability of a robotic system using hybridized technique. Kumar et al. [32] dealt with availability and cost analysis of an engineering system involving subsystems in series configuration. Suleiman et al. [33] dealt with comparative analysis between four dissimilar solar panel configurations.

Still, a further study om serial system of the new type of models with a justified and satisfactory assessment is required. For this reason, this chapter has three goals. The first goal is, to develop explicit expressions describing mean time to failure. The second is to compare the four configurations in terms of their mean time to failure. The third is to perform a parametric investigation of various system parameters with the mean time to failure, as well as to capture their effect on the mean time to failure. Analytical and numerical computations are presented to compare their mean time to failure (MTTF). Cost/benefit measure have been obtained for all configurations, where the benefit is mean time to failure.

The rest of the paper is organized as follows. Section 2 presents the notation used. Section 3 gives a description of the system. Section 4 deals with derivation of the models. Analytical comparison between configurations are presented in Section 4. The results of our numerical simulations are presented and discussed in Section 5. The paper is concluded in Section 6.

## 2. Notations

α0/β0: Unit failure/Repair rate.

pit: Probability that Configuration 1/Configuration 2/Configuration 3 is in state i at time t.

Pt: Probability row vector.

Qn/ATn/MTTFn,n=1,2,3: Transition matrix/steady state Availability/Mean time to failure for the Configuration 1/Configuration 2/Configuration 3.

C1/C2/C3: cost for Configuration 1/Configuration 2/Configuration 3.

k0/k1/k2: Revenue generated/cost due to repair of partial failure/cost due to repair of complete failure.

## 3. Description of the systems

The present paper considered the requirement of 60 MW generators in following configurations: Configuration 1 has four 15 MW primary units, two 15 MW cold standby units and one 30 MW cold standby unit; Configuration 2 has three 20 MW primary units, three 20 cold standby units; Configuration 3 has two 30 MW primary units and three 30 MW cold standby units. It is assumed that units fail independent of the other (Table 1). It is also assumed that switching from standby to operation is automatic. Primary unit fails with exponential failure time distribution with parameter α0 and immediately the cold standby is switch to operation. Also, unit fails independent of the other with exponential failure time distribution with parameter α0. Both units have exponential repair time distribution with parameter β0. The systems (Configurations) are depicted in Figures 13 below.

## 4. Reliability models formulation

### 4.1 Models formulation for configuration 1

The corresponding set of differential-difference equations for Configuration 1 as follows:

ddtp0t=4α0p0t+β0p1t+β0p2tddtp1t=4α0+β0p1t+2α0p0t+β0p3t+β0p5tddtp2t=4α0+β0p2t+2α0p0t+β0p3t+β0p4tddtp3t=4α0+β0p3t+2α0p1t+2α0p2t+β0p6t+β0p7tddtp4t=β0p4t+2α0p2tddtp5t=β0p5t+2α0p1tddtp6t=β0p6t+2α0p3tddtp7t=β0p7t+2α0p3tE1

With initial conditions

pk0=1,k=00,k=1,2,3,,7E2

Eq. (1) can be expressed in the form as:

ddtpt=Q1ptE3

With

Q1=4α0β0β0000002α04α0+β00β00β0002α004α0+β0β0β000002α02α04α0+2β000β0β0002α00β000002α0000β0000002α000β000002α0000β0

Expression of availability, probability of partial and complete failure for configuration 1 are given by

AT1=p0+p1+p2+p3E4
BP1=p1+p2+p3E5
BP2=p4+p5+p6+p7E6

To obtained (4), the procedure is to compute the states probabilities pk,k=0,1,2,,7 by setting (3) to zero to give

Q1PtT=0E7

and using the following normalizing condition

p0+p1+p2+p3+p4+p5+p6+p7=1E8

to give

4α0β0β0000002α04α0+β00β00β0002α004α0+β0β0β000002α02α04α0+2β000β0β0002α00β000002α0000β0000002α000β0011111111p0p1p2p3p4p5p6p7=00000001E9

By solving the system of equations in (9) using MATLAB package for the solution of pk give in Table 2 below.

ConfigurationNumber of Primary unitsNumber of standby unitsCost of Configuration
1Four primary 15 MWTwo cold standby 15 MW unitsC1=48,000,000
2Three primary 20 MWThree cold standby 20 MWC2=42,000,000
3Two primary 30 MWThree cold standby 30 MWC3=39,000,000

### Table 1.

Size of configurations and their corresponding cost.

 p0∞=β0316α03+12α02β0+4α0β02+β03 p4∞=4α02β016α03+12α02β0+4α0β02+β03 p1∞=2α0β0216α03+12α02β0+4α0β02+β03 p5∞=4α02β016α03+12α02β0+4α0β02+β03 p2∞=2α0β0216α03+12α02β0+4α0β02+β03 p6∞=8α0316α03+12α02β0+4α0β02+β03 p3∞=4α02β016α03+12α02β0+4α0β02+β03 p7∞=8α0316α03+12α02β0+4α0β02+β03

### Table 2.

Steady state probabilities of configuration 1.

(4), (5) and (6) are now expressed as:

AT1=β03+4α0β02+4α02β016α03+12α02β0+4α0β02+β03E10
BP1=4α0β0α0+β016α03+12α02β0+4α0β02+β03E11
BP2=8α022α0+β016α03+12α02β0+4α0β02+β03E12

Profit = total revenue generated – cost incurred by the repair man due to partial failure – cost incurred by the repair man due complete failure.

PF1=k0AT1k1BP1k2BP2E13

Using the method adopted in Wang and Kuo [34], Wang and Pearn [35], Wang et al. [36] and Yen, T,-S and Wang, K.–H [37], the mathematical model of mean time to failure for Configuration 1 is derived using the relation

MTTF1=P0M111111T=20α02+8α0β0+β028α024α0+β0E14

Where P0=1000 and

M1=4α02α02α00β04α0+β002α0β004α0+β02α00β0β04α0+2β0

obtained by transposing Q1 and deleting rows and columns of failure states.

### 4.2 Models formulation for configuration 2

Applying similar description in 4.1 above, the differential-difference equations for Configuration 2 are expressed in the form:

ddtpt=Q2ptE15

where

Q2=3α0β000β0β00000000α0y0β0000β0β0000000α0y0β00000β0β0000000y0000000β0β0β0α0000β000000000α00000β000000000α00000β00000000α000000β00000000α000000β0000000α0000000β0000000α0000000β000000α00000000β00000α000000000β0

and y0=3α0+β0.

With initial conditions

P0=1000000000000E16

Expression for system availability, probability of partial and complete failure for Configuration 2 are given by

AT2=p0+p1+p2+p3E17
BP3=p1+p2+p3E18
BP4=p4+p5+p6+p7++p12E19

Setting (15) to zero to give

Q2p=0E20

The normalizing condition for this analysis is

j=012pj=1E21

Combining (20) and (21) to give system of equations

3α0β000β0β00000000α0y0β0000β0β0000000α0y0β00000β0β0000000y0000000β0β0β0α0000β000000000α00000β000000000α00000β00000000α000000β00000000α000000β0000000α0000000β0000000α0000000β000000α00000000β001111111111111p0p1p2p3p4p5p6p7p8p9p10p11p12=0000000000001E22

Solving the system of equations in (22) for the state probabilities pk,k=0,1,2,,12, using MATLAB package, to give states probabilities in Table 3 below.

 p0∞=β043α04+3α03β0+3α02β02+3α03β0+β04 p7∞=α02β023α04+3α03β0+3α02β02+3α03β0+β04 p1∞=α0β033α04+3α03β0+3α02β02+3α03β0+β04 p8∞=α03β03α04+3α03β0+3α02β02+3α03β0+β04 p2∞=α02β023α04+3α03β0+3α02β02+3α03β0+β04 p9∞=α03β03α04+3α03β0+3α02β02+3α03β0+β04 p3∞=α03β03α04+3α03β0+3α02β02+3α03β0+β04 p10∞=α043α04+3α03β0+3α02β02+3α03β0+β04 p4∞=α03β03α04+3α03β0+3α02β02+3α03β0+β04 p11∞=α043α04+3α03β0+3α02β02+3α03β0+β04 p5∞=α03β03α04+3α03β0+3α02β02+3α03β0+β04 p12∞=α043α04+3α03β0+3α02β02+3α03β0+β04 p6∞=α02β023α04+3α03β0+3α02β02+3α03β0+β04

### Table 3.

Steady state probabilities of configuration 2.

Expressions for the system availability, probability of partial and complete failure for configuration 2 in (17) to (19) as well as profit function are now

AT2=β04+2α0β03+α02β02+α03β03α04+3α03β0+3α02β02+3α03β0+β04E23
BP3=α0β03+α03β0+α02β023α04+3α03β0+3α02β02+3α03β0+β04E24
BP4=3α04+2α0β03+2α03β0+2α02β023α04+3α03β0+3α02β02+3α03β0+β04E25
PF2=k0AT2k1BP3k2BP4E26

Mathematical model of mean time to failure for Configuration 2 is derived using the relation

MTTF2=P0M211111T=40α03+27α02β0+8α0β02+β03α081α03+54α02β0+16α0β02+2β03E27

Where P0=1000 and

M2=3α0α000β03α0+β0α000β03α0+β0α000β03α0+β0

M2 is obtained from Q2 using similar argument above.

### 4.3 Models formulation for configuration 3

Following similar argument in 4.1 above, the differential-difference equations obtained for Configuration 3 are expressed in the form:

ddtpt=Q3ptE28

where

Q3=2α0β0β000000000α0y10β0β0000000α00y1000000β0β00α00y10β0β900000α000y100β0β000000α00β000000000α000β000000000α000β00000000α0000β00000α0000000β0000α00000000β0

Where y1=2α0+β0.

With initial conditions

P0=10000000000E29

Expression for system availability, probability of partial and complete failure for Configuration 3 are given by

AT3=p0+p1+p2+p3+p4E30
BP5=p1+p2+p3+p4E31
BP6=p5+p6+p7+p8+p9+p10E32

Setting (28) to zero to give

Q3p=0E33

The normalizing condition for this analysis is

j=010pj=1E34

Combining (33) and (34) to give system of equations

2α0β0β000000000α0y10β0β0000000α00y1000000β0β00α0y10β0β900000α000y100β0β000000α00β000000000α000β000000000α000β00000000α0000β00000α0000000β0011111111111p0p1p2p3p4p5p6p7p8p9p10=00000000001E35

Solving the system of equations in (35) for the state probabilities pk,k=0,1,2,,10, presented in Table 4 below.

 p0∞=β034α03+4α02β0+2α0β02+β03 p6∞=α034α03+4α02β0+2α0β02+β03 p1∞=α0β024α03+4α02β0+2α0β02+β03 p7∞=α034α03+4α02β0+2α0β02+β03 p2∞=α0β024α03+4α02β0+2α0β02+β03 p8∞=α034α03+4α02β0+2α0β02+β03 p3∞=α02β04α03+4α02β0+2α0β02+β03 p9∞=α02β04α03+4α02β0+2α0β02+β03 p4∞=α02β04α03+4α02β0+2α0β02+β03 p10∞=α02β04α03+4α02β0+2α0β02+β03 p5∞=α034α03+4α02β0+2α0β02+β03

### Table 4.

Steady state probabilities of configuration 3.

Expressions for the system availability, probability of partial and complete failure for configuration 3 in (30) to (32) as well as profit function are now

AT3=β03+2α0β02+2α02β04α03+4α02β0+2α0β02+β03E36
BP5=2α0β0α0+β04α03+4α02β0+2α0β02+β03E37
BP6=2α022α0+β04α03+4α02β0+2α0β02+β03E38
PF3=k0AT3k1BP5k2BP6E39

Mathematical model of mean time to failure for Configuration 3 is derived using the relation

MTTF`3=P0M3111111T=20α03+16α02β0+6α0β02+β032α028α02+4α0β0+β02E40

Where P0=10000

M3=2α0α0α000β02α0+β00α0α0β002α0+β0000β002α0+β000β0002α0+β0

M3 is obtained from Q3 using similar argument above.

## 5. Results and discussion

### 5.1 Analytical comparison

In this section, the configurations are compared analytically in terms of their availability and mean time to failure to determine the optimal configuration by taking the difference between mean time to failure (MTTF) and availability for the configurations using MAPLE software package.

MTTF3MTTF1=16α04+192α03β0+100α02β02+28α0β03+3β048α024α0+β08α02+4α0β0+β02E41
MTTF3>MTTF1α0,β0>0
MTTF3MTTF2=980α06+1624α05β0+1246α04β02+567α03β03+158α02β04+26α0β05+2β062α028α02+4α0β0+β0281α03+54α02β0+16α0β02+2β03E42
MTTF3>MTTF2α0,β0>0
MTTF1MTTF2=340α05+544α04β0+361α03β02+126α02β03+24α0β04+2β058α024α0+β081α03+54α02β0+16α0β02+2β03E43
MTTF1>MTTF2α0,β0>0

Using mean time to failure models of configurations, it is clear from (41)(43) that

MTTF3>MTTF1>MTTF2
AT3AT1=2α02β04α02+4α0β0+3β0216α03+12α02β0+4α0β02+β034α03+4α02β0+2α0β02+β03E44
AT3>AT1α0,β0>0
AT3AT2=α0β02α05+4α04β0+α03β02+2α0β04+β054α03+4α02β0+2α0β02+β033α04+3α03β0+3α02β02+3α03β0+β04E45
AT3>AT2α0,β0>0
AT2AT1=α0β02α04+α03β0+10α02β02+4α0β03β043α04+3α03β0+3α02β02+3α03β0+β0416α03+12α02β0+4α0β02+β03E46
AT2>AT1α0,β0>0

Using availability models of configurations, it is clear from (44)(46) that

AT3>AT2>AT1

### 5.2 Comparison based on ranking of the configurations

Tables 5 and 6 depict the ranking of configuration base on their availability and mean time to failure. It clear from Table 5 that configuration 3 is the optimal configuration whenever 0β01. Thus, verifying the above analytical claim that AT3>AT2>AT1 and MTTF3>MTTF1>MTTF2α0,β0>0 .

CaseParameter RangeResultConstant parameters
10<α0<0.2AV3>AV2>AV1MTTF3>MTTF1>MTTF2β0=0.6
0.2<α0<0.4AV3>AV2>AV1MTTF3>MTTF1>MTTF2
0.4<α0<0.6AV3>AV2>AV1MTTF3>MTTF1>MTTF2
0.6<α0<0.8AV3>AV2>AV1MTTF3>MTTF1>MTTF2
0.8<α0<1.0AV3>AV2>AV1MTTF3>MTTF1>MTTF2

### Table 5.

Ranking of configurations based on their availability and MTTF for α001.

CaseParameter RangeResultConstant parameters
20<β0<0.2AV3>AV2>AV1MTTF3>MTTF1>MTTF2α0=0.2
0.2<β0<0.4AV3>AV2>AV1MTTF3>MTTF1>MTTF2
0.4<β0<0.6AV3>AV2>AV1MTTF3>MTTF1>MTTF2
0.6<β0<0.8AV3>AV2>AV1MTTF3>MTTF1>MTTF2
0.8<β0<1.0AV3>AV2>AV1MTTF3>MTTF1>MTTF2

### Table 6.

Ranking of configurations based on their availability and MTTF for β001.

### 5.3 Comparison based on availability, profit and mean time to failure

In this section, β0=0.8, k0=50,000,000, k1=1250 and k2=2150 are fixed and vary α0 from 0.1 to 1 in Figures 46 and α0=0.018, k0=50,000,000, k1=1250 and k2=2150 are fixed and vary β0 from 0.1 to 1 in Figures 79 and obtained the following results.

Simulations in Figures 46 compare the steady state availability, profit and MTTF with respect to α0 for all the three configurations considered. From these figures, availability, profit and MTTF decreases as α0 increases for any configuration. Furthermore, Configuration 3 seems to be most effective and reliable configuration among all the three configurations. From these figures, it is clear that Configuration 3 produces more availability, profit and MTTF than the other configurations. Thus, Configuration 3 is the optimal configuration in this study. On the other hand, simulations in Figures 79 compare the steady state availability, profit and MTTF with respect to β0 for all the three configurations. It is evident from these figures that availability, profit and MTTF increases as β0 increases for any configuration. Similar to Figures 46, Configuration 3 seems to be most effective and reliable configuration among all the three configurations and hence is the optimal configuration.

### 5.4 Comparison based on cost benefit

In this section, the configurations are compared based on their Ck/Avk and Ck/MTTFk using MATLAB software. The following parameters values are used for the purpose of analysis:

β0=0.8, C1=48,000,000, C2=42,000,000, C3=39,000,000 (Yen, T,-S and Wang, K.–H [37]) are fixed and vary α0 between 0.1 and 1 in Figures 10 and 11.

α0=0.018,, C1=42,000,000, C1=39,000,000 and vary β0 between 0.1 to 1 in Figures 12 and 13 and obtained the following results:

Figures 10 and 11 depict the results of Ck/Avk and Ck/MTTFk for each configuration i i=123 with respect to α0. From these figures, it is evident that Ck/Avk and Ck/MTTFk increase as α0 increases for each configuration. It is evident from these Figures that the optimal configuration using both Ck/Avk and Ck/MTTFk is Configuration 3.

From Figures 12 and 13, it is clear that Ck/Avk and Ck/MTTFk decrease as β0 increases. It is clear from these Figures that the optimal configuration using both Ck/Avk and Ck/MTTFk is again Configuration 3. Configurations 1 and 2 tend to have more Ck/Avk and Ck/MTTFk that Configuration 3. From the result presented in this study, it is clear that the survival of manufacturing and industrial systems depends upon its design and reliability characteristics. Through the system design and reliability characteristics, management can tend to realize whether such systems operate at minimum cost of maintenance, quality of the product, production output as well as revenue mobilization.

### 5.5 Sensitivity analysis

Sensitivity analysis presented in Tables 7 and 8 depict the change in availability, MTTF and profit of the three configurations with respect to failure rate α0 for different values of β0. It is clear from these tables that availability, MTTF and profit of the three configurations decreases as α0 increase. Availability, MTTF and profit tend to be higher for the three configurations for whenever β0=0.9. This sensitivity analysis implies that maintenance action such as inspection, preventive maintenance, etc. should be invoke to reduce the occurrence of failure in order to attain maximum value of availability, MTTF and profit. From these tables it is evident that Configuration 3 has higher values of availability, MTTF and profit than configurations 1 and 2 for different values of β0. On the other hand, Sensitivity analysis in depicted in Tables 9 and 10, displayed the variation of availability, MTTF and profit with respect to β0 for different values of α0. It is evident from the tables that availability, MTTF and profit increases as β0 increases for different values of α0. Increase in the values of α0 decrease the availability, MTTF and profit as shown in the tables. This sensitivity analysis suggest that perfect repair, preventive maintenance, inspection should be invoke at early failure to restore the system to its position as good as new. It is also clear from these tables that Configuration III has higher values of availability, MTTF and profit than configurations 1 and 2.

 α0 AV1∞ AV2∞ AV3∞ MTTF1 MTTF2 MTTF3 β0=0.3 0.1 0.6522 0.7313 0.8361 9.4643 4.9887 25.6897 0.2 0.3962 0.5287 0.6084 3.8920 2.4878 9.3654 0.3 0.2727 0.3846 0.4545 2.4167 1.6558 5.5128 0.4 0.2050 0.2902 0.3565 1.7475 1.2405 3.8699 0.5 0.1632 0.2286 0.2912 1.3674 0.9916 2.9717 β0=0.6 0.1 0.8571 0.8538 0.9494 13.0000 4.9965 40.2941 0.2 0.6522 0.7313 0.8361 4.7321 2.4943 12.8448 0.3 0.5000 0.6230 0.7143 2.7778 1.6605 7.0000 0.4 0.3962 0.5287 0.6084 1.9460 1.2439 4.6827 0.5 0.3243 0.4494 0.5227 1.4923 0.9942 3.4809 β0=0.9 0.1 0.9252 0.8989 0.9764 16.6346 4.9985 55.1600 0.2 0.7852 0.8112 0.9154 5.6066 2.4970 16.4662 0.3 0.6522 0.7313 0.8361 3.1548 1.6629 8.5632 0.4 0.5445 0.6576 0.7537 2.1531 1.2459 5.5391 0.5 0.4609 0.5899 0.6768 1.6224 0.9959 4.0169

### Table 7.

Availability and MTTF sensitivity as function of α0 for different values of β0.

α0Profit 107
β0=0.3β0=0.6β0=0.9
PF1PF2PF3PF1PF2PF3PF1PF2PF3
05.00005.00005.00005.00005.00005.00005.00005.00005.0000
0.13.06663.53044.04204.16594.19684.69344.55464.44324.8555
0.21.80562.45782.83933.06663.53044.04203.76923.96434.4934
0.31.22981.74232.08502.30442.94933.38373.06663.53044.0420
0.40.92051.29821.62181.80562.45782.83932.52323.13353.5928
0.50.73161.01721.31901.46822.05862.41432.11472.77513.1881
0.60.60540.83041.10851.22981.74232.08501.80562.45782.8393
0.70.51560.69950.95471.05421.49391.82701.56732.18182.5441
0.80.44860.60350.83780.92051.29821.62181.38001.94462.2952
0.90.39680.53050.74610.81571.14261.45561.22981.74232.0850

### Table 8.

Profit sensitivity as function of α0 for different values of β0.

 α0 AV1∞ AV2∞ AV3∞ MTTF1 MTTF2 MTTF3 α0=0.012 0.1 0.9150 0.8915 0.9726 131.8506 41.6518 432.21 0.2 0.9749 0.9432 0.9929 217.2939 41.6638 778.30 0.3 0.9883 0.9615 0.9968 303.5201 41.6657 1125.2 0.4 0.9933 0.9708 0.9982 390.0050 41.6662 1472.4 0.5 0.9956 0.9765 0.9989 476.6069 41.6664 1819.5 α0=0.015 0.1 0.8784 0.8671 0.9583 92.0139 33.3142 290.60 0.2 0.9623 0.9299 0.9890 146.3675 33.3294 511.70 0.3 0.9821 0.9523 0.9950 201.3889 33.3319 733.60 0.4 0.9896 0.9638 0.9972 256.6425 33.3327 955.70 0.5 0.9933 0.9708 0.9982 312.0040 33.3330 1177.90 α0=0.018 0.1 0.8399 0.8434 0.9418 69.2650 27.7550 211.6955 0.2 0.9479 0.9168 0.9843 106.7765 27.7726 364.8292 0.3 0.9749 0.9432 0.9929 144.8626 27.7759 518.8338 0.4 0.9854 0.9569 0.9960 183.1581 27.7769 673.0276 0.5 0.9904 0.9652 0.9974 221.2852 27.7773 827.2852

### Table 9.

Availability and MTTF sensitivity as function of β0 for different values of α0.

β0Profit 107
α0=0.012α0=0.015α0=0.018
PF1PF2PF3PF1PF2PF3PF1PF2PF3
0−0.0002−0.0002−0.0002−0.0002−0.0002−0.0002−0.0002−0.0002−0.0002
0.14.64364.50754.88814.48504.39594.82894.31564.28754.7597
0.24.89694.74314.97114.84444.68254.95524.78394.62304.9359
0.34.95214.82604.98714.92684.78424.97994.89694.74314.9711
0.44.97254.86844.99274.95774.83664.98874.94004.80514.9837
0.54.98224.89424.99534.97254.86844.99274.96094.84294.9895
0.64.98754.91164.99684.98074.88994.99494.97254.86844.9927
0.74.99084.92404.99764.98574.90544.99634.97964.88684.9946
0.84.99294.93344.99824.98904.91704.99724.98434.90074.9959
0.94.99444.94074.99864.99134.92614.99774.98754.91164.9968

### Table 10.

Profit sensitivity as function of β0 for different values of α0.

## 6. Conclusion

In this paper, three different standby serial systems each supplying 60 MW are considered. The expressions for the reliability characteristics such as system availability, busy period of repairman due to partial and complete failure as well as profit functions for all the configurations have been obtained and validated by performing analytical and numerical experiments. Analysis of the effect of various system parameters on mean time to failure, profit function and availability was performed. These are the main contributions of this study. On the basis of the numerical results obtained in Figures and Tables for a particular case, it is evident that the optimal system configuration is configuration 3. This is supported from analytical comparison presented in terms of the availability and mean time to failure models obtained in which configuration III is the optimal configuration for all α0,β0>0 contrary to some studies where the optimality among the system configuration is not uniform as it depends on some system parameters. The contributions of this paper are as follows.

1. Failure is categorized into partial and complete failure

2. Analytical comparison between the configuration in terms of their availability and mean time to failure is performed

3. Optimal configuration in analytical comparison agrees with that of numerical comparison

4. Optimal configuration is unique for all parameter values

## Conflict of interest

There are no conflicts of interest to this chapter.

AMS (2010) subject classification: 90B25

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Written By

Ibrahim Yusuf and Ismail Muhammad Musa

Submitted: July 8th, 2020 Reviewed: November 28th, 2020 Published: June 16th, 2021