Solution Manual for Reliability Engineering: Probabilistic Models and Maintenance Methods (2nd Edition, 2017) by Joel A. Nachlas

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SOLUTIONS MANUAL

,Exercises Chapter 2

1. Construct the system structure function for the following system



1
4

2 3


1
5

2 3



6



7



( x) 1 1 (1 x1)(1 x2 x3 ) x4 1 (1 x1)(1 x2 x3 ) x5 1 (1 x6 )(1 x7 )



2. Construct the minimum path and minimum cut equivalent structures for a 2-out-of-3 system.

The 2-out of 3 system is the one in which the proper functions of any 2 out of 3 components

implies that the system is functioning properly. The Min-Path sets for this system are P1={1,2}, P2={1,3}

andP3={2,3}. Therefore, the min-path equivalent structure is:




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,The Min-Cut sets for the system are C1= {2,3}, C2= {1,3} and C3= {1,2} so the min-cut equivalent structure

is:




3. Construct the minimum path and minimum cut equivalent structures for a three component series

system.

For a three component series system, there is only one minimum path. The path set is

P1={1,2,3} and the min-path equivalent structure is identical to the system structure. The min cut sets

are C1= {1}, C2= {2} and C3= {3} and the min-cut equivalent structure is also identical to the system

structure.




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,4. Construct the minimum path and minimum cut equivalent structures for a three component parallel

system.

For a three component parallel system, there are three min-paths for which the path sets are

P1={1}, P2={2} and P3={3} so the min path equivalent structure is identical to the system structure. There

is one min cut set C1= {1,2 3} and the min-cut equivalent structure is also identical to the system

structure.



5. Show that the status of any system structure is bounded below by the state of the series system

comprised of the same components and is bounded above by the state of the parallel system
n n

comprised of the same components. That is, show that xi ( x)
i1
∐x .i
i1

n n

If xi 0 it is necessarily less than or equal to ( x) and if xi 1 then xi 1, i so
i1 i1


( x) 1. Thus, the status of the series structure bounds ( x) from below.
n n

If ∐ xi 1 then ( x) does not exceed it and if ∐ xi 0 then xi 0, i so it must be the
i1 i1


case that ( x) 0. Thus, the status of the parallel structure bounds ( x) from above.

6. Construct the minimum path and minimum cut equivalent structures for the following system:




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, 1 2 3




4 5




6 7




The Min-Path sets and the min-path equivalent structure are P1={1,2,3}; P2={1,2,5,7}; P3={1,4,7};

P4={6,7}; P5={3,5,6}; P6={1,3,4,5}; P7={2,3,4,6}




The Min-Cut sets and min-cut equivalent structure are C1={3,7}; C2={1,6}; C3={2,5,7};

C4={2,4,6};C5={1,4,5,7}; C6={6,4,5,3}




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, 1 3


2 2

3 1 4

5 4
5 5
7 6


7 6
7 6




7. Construct the minimum path and minimum cut equivalent structures for the following system:


1

5 7

2


8


3 4 6
9




The Min-Path sets are P1={1,5,7}; P2={2,5,7}; P3={1,5,6,8}; P4={1,5,6,9}; P5={2,5,6,8};

P6={2,5,6,9}; P7={3,4,6,8}, P8={3,4,6,9} and P9={3,4,6,5,7}.

The min-path equivalent structure is:




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, 1 5 7


2 5 7



1 5 6 8



1 5 6 9



2 5 6 8



2 5 6 9



3 4 6 8



3 4 6 9



3 4 5 6 7

The Min-Cut sets are C1={5,3}; C2={5,4}; C3={5,6}; C4={7,6};C5={1,2,3}; C6={1,2,4},

C7={5,8,9} and C8={7,8,9}. The min-cut equivalent structure is:


1 1 7 5
5 5 5 7

2 2 8 8

3 4 6
6
3 4 9 9




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,8. Construct the minimum path and minimum cut equivalent structures for the following system:


2




1 3 5




4




The Min Path sets are P1={2,5}; P2={ 1,4}; P3={1,3,5}; P4={2,3,4} so the min-path equivalent structure is:

1 4



2 5



1 3 5



2 3 4


The Min-Cut sets are C1={4,5}; C2={1,2}; C3={2,3,4}; C4={1,3,5} so the min-cut equivalent structure is:

2 1


1 4
3
3

2 5
4 5




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,9. Construct and compare the minimum path and minimum cut equivalent structures for the following

systems:



1 3
1 3




2 4 2 4




For the system on the left, the Min-Path sets are P1={1,3}; P2={2,4} and the min-path equivalent system is:




The Min-Cut sets are C1={1,2}; C2={3,4}; C3={1,4}; C4={2,3}




For the system on the right, the Min-Path sets are P1={1,3}; P2={1,4}; P3={2,3}; P4={2,4} so the min-path

equivalent structure is




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