relibmss 0.6.3

A Python package for binary/multi state systems

relibmss

A Python package for binary/multi state systems with BDD/MDD.

Installation

pip install relibmss

Usage

Calculate the probability of a fault tree

import relibmss as ms

# Create a binary system (fault tree)
bss = ms.BSS()

# Define events (This version only supports repeated events)
A = bss.defvar('A')
B = bss.defvar('B')
C = bss.defvar('C')

# Make a tree
top = A & B | C # & is AND gate, | is OR gate

# Set probabilities
prob = {
    'A': 0.1,
    'B': 0.2,
    'C': 0.3
}

# Calculate the probability
print(bss.prob(top, prob))

# Set the interval of the probability
prob = {
    'A': (0.1, 0.2),
    'B': (0.2, 0.3),
    'C': (0.3, 0.4)
}

# Calculate the probability
print(bss.prob_interval(top, prob))

Obtain the minimal cut sets

import relibmss as ms

# Create a binary system
bss = ms.BSS()

# Define events (This version only supports repeated events)
A = bss.defvar('A')
B = bss.defvar('B')
C = bss.defvar('C')

# Make a system
top = bss.kofn(2, [A, B, C]) # k-of-n gate

# Obtain the minimal path vectors
s = bss.mpvs(top) # s is a set of minimal path vectors (ZDD representation)

# Convert the ZDD representation to a list of sets
# Convert the ZDD representation to a list of sets
min_path = s.extract()
print('The number of minimal path vectors:', len(min_path))
for x in min_path:
    print(x)

Draw a BDD

import relibmss as ms

# Create a binary decision diagram
bss = ms.BSS()

# Define variables
A = bss.defvar('A')
B = bss.defvar('B')
C = bss.defvar('C')

# Make a tree
top = A & B | C

# Draw the BDD
bdd = bss.getbdd(top)
source = bdd.dot() # source is a string of the dot language
print(source)

# Example: Display the BDD in Jupyter Notebook
from graphviz import Source
from IPython.display import Image, display
Image(Source(source).pipe(format='png'))

An example of a large fault tree

## This is an example of a large fault tree
## Computational time may be long (about 1 minute)

import relibmss as ms

bss = ms.BSS()
c = [bss.defvar("c" + str(i)) for i in range(61)]

g62 = c[0] & c[1]
g63 = c[0] & c[2]
g64 = c[0] & c[3]
g65 = c[0] & c[4]
g66 = c[0] & c[5]
g67 = c[0] & c[6]
g68 = c[0] & c[7]
g69 = c[0] & c[8]
g70 = g62 | c[9]
g71 = g63 | c[10]
g72 = g64 | c[11]
g73 = g65 | c[12]
g74 = g62 | c[13]
g75 = g63 | c[14]
g76 = g64 | c[15]
g77 = g65 | c[16]
g78 = g62 | c[17]
g79 = g63 | c[18]
g80 = g64 | c[19]
g81 = g65 | c[20]
g82 = g62 | c[21]
g83 = g63 | c[22]
g84 = g64 | c[23]
g85 = g65 | c[24]
g86 = g62 | c[25]
g87 = g63 | c[26]
g88 = g64 | c[27]
g89 = g65 | c[28]
g90 = g66 | c[29]
g91 = g68 | c[30]
g92 = g67 | c[31]
g93 = g69 | c[32]
g94 = g66 | c[33]
g95 = g68 | c[34]
g96 = g67 | c[35]
g97 = g69 | c[36]
g98 = g66 | c[37]
g99 = g68 | c[38]
g100 = g67 | c[39]
g101 = g69 | c[40]
g102 = g66 | c[41]
g103 = g68 | c[42]
g104 = g67 | c[43]
g105 = g69 | c[44]
g106 = bss.kofn(3, [g70, g71, g72, g73])
g107 = bss.kofn(3, [g74, g75, g76, g77])
g108 = bss.kofn(3, [g78, g79, g80, g81])
g109 = bss.kofn(3, [g82, g83, g84, g85])
g110 = bss.kofn(3, [g86, g87, g88, g89])
g111 = bss.kofn(3, [g94, g95, g96, g97])
g112 = bss.kofn(3, [g98, g99, g100, g101])
g113 = g90 & g92
g114 = g91 & g93
g115 = g102 & g104
g116 = g103 & g105
g117 = g113 | c[45]
g118 = g114 | c[46]
g119 = g107 | g108 | c[51]
g120 = g109 | g110
g121 = g66 | g117 | c[47]
g122 = g68 | g118 | c[48]
g123 = g67 | g117 | c[49]
g124 = g69 | g118 | c[50]
g125 = bss.kofn(2, [g121, g123, g122, g124])
g126 = g111 | g112 | g125 | c[52]
g127 = g115 & g120
g128 = g116 & g120
g129 = g62 | g127 | c[53]
g130 = g63 | g128 | c[54]
g131 = g64 | g127 | c[55]
g132 = g65 | g128 | c[56]
g133 = g62 | g129 | c[57]
g134 = g63 | g130 | c[58]
g135 = g64 | g131 | c[59]
g136 = g65 | g132 | c[60]
g137 = bss.kofn(3, [g133, g134, g135, g136])
g138 = g106 | g119 | g137
g139 = g62 | g66 | g117 | g129 | c[47]
g140 = g63 | g68 | g118 | g130 | c[48]
g141 = g64 | g67 | g117 | g131 | c[49]
g142 = g65 | g69 | g118 | g132 | c[50]
g143 = g139 & g140 & g141 & g142
g144 = g111 | g112 | g143 | c[52]
top = g126 & g138 & g144

bdd = bss.getbdd(top)
print(bdd.size()) # The numbers of nodes and edges in the BDD

s = bdd.mpvs() # Obtain the minimal path vectors (minimal cut sets) from the BDD directly

min_path = s.extract()
print('The number of minimal path sets:', len(min_path))

print('Example: 100 minimal path sets')
from itertools import islice
for x in islice(min_path, 0, 100):
    print(x)

Importance analysis

Compute the Birnbaum importance for each event as the first order derivative of the top event probability with respect to the probability of the event. This is the Birnbaum importance in the case where event occurrences are independent.

import relibmss as ms

# Create a binary system (fault tree)
bss = ms.BSS()

# Define events (This version only supports repeated events)
A = bss.defvar('A')
B = bss.defvar('B')
C = bss.defvar('C')

# Make a tree
top = A & B | C # & is AND gate, | is OR gate

# Set probabilities
prob = {
    'A': 0.1,
    'B': 0.2,
    'C': 0.3
}

# top = 1-(1-pa*pb)*(1-pc) = pa*pb+pc-pa*pb*pc
# top / pa = pb - pb*pc = 0.2 - 0.2*0.3 = 0.14
# top / pb = pa - pa*pc = 0.1 - 0.1*0.3 = 0.07
# top / pc = 1 - pa*pb = 1 - 0.1*0.2 = 0.98

print(bss.bmeas(top, prob))

# Set the interval of the probability
prob = {
    'A': (0.1, 0.2),
    'B': (0.2, 0.3),
    'C': (0.3, 0.4)
}

print(bss.bmeas_interval(top, prob))

### Structure importance measure can be calculated as follows

prob = {
    'A': 0.5,
    'B': 0.5,
    'C': 0.5
}

print(bss.bmeas(top, prob))

TODO for fault tree analysis

• FTA with MCS
• Importance analysis
• Sensitivity analysis
• Uncertainty analysis; etc.

Multi-state system

Definition of Gate

MSS does not have default gates. Users need to define gates by themselves. The operation that can be used in the definition of a gate is as follows:

• Arithmetic operations: +, -, *, /
• Comparison operations: ==, !=, >, <, >=, <=
• Logical operations:
  • mss.And: AND gate
  • mss.Or: OR gate
  • mss.Not: NOT gate
  • mss.switch: Switch-case structure
  • mss.case: Case structure

import relibmss as ms

# def for a gate with switch-case structure
def gate1(mss, x, y):
    return mss.switch([
        mss.case(cond=mss.And([x == 0, y == 0]), then=0),
        mss.case(cond=mss.Or([x == 0, y == 0]), then=1),
        mss.case(cond=mss.Or([x == 2, y == 2]), then=3),
        mss.case(cond=None, then=2) # default
    ])

Example of a multi-state system

import relibmss as ms

# Define gates
def gate1(mss, x, y):
    return mss.switch([
        mss.case(cond=mss.And([x == 0, y == 0]), then=0),
        mss.case(cond=mss.Or([x == 0, y == 0]), then=1),
        mss.case(cond=mss.Or([x == 2, y == 2]), then=3),
        mss.case(cond=None, then=2) # default
    ])

def gate2(mss, x, y):
    return mss.switch([
        mss.case(cond=x == 0, then=0),
        mss.case(cond=None, then=y)
    ])

mss = ms.MSS() # Context for the multi-state system

# Define variables

A = mss.defvar('A', 2) # 2 states
B = mss.defvar('B', 3) # 3 states
C = mss.defvar('C', 3) # 3 states

# Define a multi-state system
sx = gate1(mss, B, C)
ss = gate2(mss, A, sx)

# Define probabilities
prob = {
    'A': [0.1, 0.9],
    'B': [0.2, 0.3, 0.5],
    'C': [0.3, 0.4, 0.3]
}

# Calculate the probability
print(mss.prob(ss, prob, [0,1,2])) # compute probability that the system is in state 0, 1, 2

Draw an MDD

import relibmss as ms

# Define gates
def gate1(mss, x, y):
    return mss.switch([
        mss.case(cond=mss.And([x == 0, y == 0]), then=0),
        mss.case(cond=mss.Or([x == 0, y == 0]), then=1),
        mss.case(cond=mss.Or([x == 2, y == 2]), then=3),
        mss.case(cond=None, then=2) # default
    ])

def gate2(mss, x, y):
    return mss.switch([
        mss.case(cond=x == 0, then=0),
        mss.case(cond=None, then=y)
    ])

mss = ms.MSS()

A = mss.defvar('A', 2)
B = mss.defvar('B', 3)
C = mss.defvar('C', 3)

# Define the order of variables
# this should be done before making MDD
mss.set_varorder({"A": 2, "B": 1, "C": 0})

sx = gate1(mss, B, C)
ss = gate2(mss, A, sx)

mdd = mss.getmdd(ss)
source = mdd.dot()

from graphviz import Source
from IPython.display import Image, display
Image(Source(source).pipe(format='png'))

Obtain the minimal vector sets

import relibmss as ms

# Define gates
def gate1(mss, x, y):
    return mss.switch([
        mss.case(cond=mss.And([x == 0, y == 0]), then=0),
        mss.case(cond=mss.Or([x == 0, y == 0]), then=1),
        mss.case(cond=mss.Or([x == 2, y == 2]), then=3),
        mss.case(cond=None, then=2) # default
    ])

def gate2(mss, x, y):
    return mss.switch([
        mss.case(cond=x == 0, then=0),
        mss.case(cond=None, then=y)
    ])

mss = ms.MSS()

A = mss.defvar('A', 2)
B = mss.defvar('B', 3)
C = mss.defvar('C', 3)

sx = gate1(mss, B, C)
ss = gate2(mss, A, sx)

mdd = mss.mpvs(ss)
print(mdd.dot())

TODO

• Add more examples
• Add more functions for fault tree analysis
• Add more functions for multi-state system analysis

License

MIT License