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Thesis and Dissertation Collection
1976
An example of phased mission reliability
analysis for a hypothetical naval weapons system.
Pilnick, Steven E.
Monterey, California. Naval Postgraduate School
http://hdl.handle.net/10945/17766
Downloaded from NPS Archive: Calhoun
AN EXAMPLE OF PHASED MISSION
RELIABILITY ANALYSIS FOR A HYPOTHETICAL
NAVAL WEAPONS SYSTEM
Steven E. Pilnick
NAVAL POSTGRADUATE SCHOOL
Monterey, California
THESIS
AN EXAMPLE OF PHASED MISSION
RELIABILITY ANALYSIS FOR A HYPOTHETICAL
NAVAL WEAPONS SYSTEM
by
Steven E. Pilnick
December 197 6
Thesis Advisor:
J.
D.
Esary
Approved for public release; distribution unlimited.
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An Example of Phased Mission Reliability
Analysis for a Hypothetical Naval Weapons
System
Steven
E.
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Master's Thesis
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Naval Postgraduate School
Monterey, CA 93940
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Naval Postgraduate School
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Reliability analysis; phased mission
ABSTRACT
20.
(Contlmto on rooormo
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11
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In a phased mission, the functional organization of the system changes
during consecutive time periods, which introduces analysis complexities not
present with just a single phase.
This occurs since the performance of a
particular component in one phase of the mission is not independent of its
performance in another phase.
In this paper, an example is analyzed,
largely with graphical techniques and diagrams, so as to avoid the complicated
mathematics which characterize much of the existing methodology.
DD
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OBSOLETE
UNCLASSIFIED
SECURITY CLASSIFICATION OF THIS PAGE (Wnon Dim
Mntoro*)
AN EXAMPLE OF PHASED MISSION RELIABILITY ANALYSIS FOR A
HYPOTHETICAL NAVAL WEAPONS SYSTEM
by
Steven
B.E.
,
E.
Bilnick
Lietrtenant, United States Navy
State University of New York at Stony Brook,
Submitted in partial fulfillment of the
requirements for the degree of
MASTER
OF;
SCIENCE IN OPERATIONS RESEARCH
from the
NAVAL POSTGRADUATE SCHOOL
December 1976
1970
own Not mum
ABSTRACT
In
a
phased mission, the functional organization
of the system changes during consecutive time periods,
which
introduces
analysis
complexities
occurs
not present
with just a
single
performance
of a particular component in one phase of
phase.
This
the mission is not independent of its
another phase.
In this paper,
since
performance
avoid
the
in
an example is analyzed,
largely with graphical techniques and diagrams,
to
the
complicated
mathematics
characterize much of the existing methodology.
so
as
which
Mi MM
TABLE OF CONTENTS
I.
II.
III.
INTRODUCTION..
7
THE SLBM SYSTEM
PROBLEM ANALYSIS
. ...
8
„
11
A.
PHASE SEQUENCE DIAGRAM
11
B.
PHASE BLOCK DIAGRAMS
11
C.
1.
Cut Cancellation...
2.
Transformation.....
14
»
17
RELIABILITY OF PSEUDO-COMPONENTS..
22
Conditional Component Phase Reliability..
21
24
D.
Unconditional Component Reliability......
OBJECTIVE RELIABILITY
27
E.
Objective Block Diagrams
Objective Reliability Evaluation
2.
MISSION SUCCESS
1.
2.
1.
25
27
30
1.
Successful Mission Outcomes..............
36
2.
Mission Success Levels
36
3.
Expected Success
39
LIST OF REFERENCES
INITIAL DISTRIBUTION LIST
LIST OF FIGURES
41
;
_
..
42
6
LIST OF FIGURES
1.
Phase Sequence Diagram
12
2.
Phase Block Diagrams
13
3.
Cut Cancellations
15
4.
Phase Block Diagrams After Cut Cancellation
16
5.
Candidates for Pseudo-Component Expansion...........
18
6.
Transformed Phase Block Diagrams
21
7.
Transformed and Simplified Phase Block Diagrams.-... 23
8.
Pseudo-Component Reliabilities
26
9.
Relevant Phases..
28
10.
Block Diagram of Objective
11.
Pivotal Decomposition
12.
Reduction Into Modules......
35
13.
Successful Mission Outcomes.
38
14.
Mission Success Levels......
40
29
2
-
33
INTRODUCTION
In
system
a
phased mission, the functional organization of the
changes
examples
of
mission are
consecutive
during
Reliability analysis for
mission encounters complexities
since
phase,
of
of
for
the
analysis
of
include an operational readiness
the
functions
system
to
missions.
systems
is
and
phased
a
not
just
a
particular
necessarily
phase.
In a
extended
existing
missions,
so as to
phase,
(OR)
during
which
solely to maintain its readiness for
later phases of active operation.
extended
of
phased
a
with
another
in
modified
recent paper. Bell [1975]
methods
mission
a
performance
its
present
not
performance
the
component in one phase
independent
Some
periods.
systems which are required to perform a phased
space vehicles,
public safety systems, and
military weapons systems.
single
time
The
results
were
then
which perform complex multi-objective
This type of model is particularly applicable
to
strategic weapons systems.
The work in this paper follows very closely the woifk
Bell [1975],
An example is analyzed,
techniques and diagrams, so
mathematics
which
were
as
to
of
largely with graphical
avoid the complicated
required in the development of the
methodology of phased mission reliability analysis.
THE
II.
A
hypothetical
system (SLBM)
Ballistic
,
motivated
was
three
missiles,
Navy*s
the
by
missile
Fleet
introduced by Bell [1975].
extended in this example to
system,
was
That hypothetical system is
include
ballistic
submarine- launched
which
Missile
SLBM SYSTEM
which
of
each
has
different
a
objective.
The SLBM system consists of the following components:
the
submarine
stability, power,
the
and:
(S)
which
provides
propulsion,
household services.
inertial
navigation
subsystem
which
(N)
provides information on platform position and orientation.
the communication subsystem
(C)
provides
which
the link between the submarine and its command center.
the fire control subsystem
(FC)
provides
which
trajectory information to each missile guidance computer.
the ejection subsystems (E1,E2,E3), one for
each
which launch the missiles from the submarine while
missile,
the latter is submerged.
the guidance component of each missile
which computes and
control
(G1,G2,G3)
rocket
engines the
commands required to maintain the trajectory stored
transmits
the
to
within its memory, and triggers stage separation.
two
internal power sources for each missile (VP1
and VS1, VP2 and VS2, VP3 and VS3)
a
the
each missile
•
first-
(RF1
two
and
second-stage
rocket engines of
and BS1, RF2 and RS2, RF3 and RS3)
first-stage
igniters
and IS1, IP2 and IS2,; IP3 and IS3)
for each missile (IP1
.
the second-stage igniter of each missile
(J1, J2,
J3)
.
the warhead of each missile
(W1
#
W2,
W3)
.
The SLBM system has an operational readiness (OR)
followed
active
five
by
phases
phase,
for each objective.
operational characteristics of the system can be
The
summarized
as follows:
During the OR phase
submarine
the
patrols
its
assigned area, maintaining current position information with
Should the inertial
the inertial navigation subsystem.
component
then
fail,
periodically from
a
position information can be obtained
navigation satellite, which provides the
necessary for calibration after repairs are completed.
The communication subsystem is used continually during this
data
phase
ship-shore
routine
for
message
control subsystem is exercised periodically
the
checked
sources
power
missile
routine
through
order
the
for
OR
phase.
the
system
operations,
must
it
Other
have
be
if
found to be
ready
to
commence
In
active
services and current
submarine
must
be
able
to
the launch command via the communication subsystem.
The fire control phase for
commences when
a
the
first
objective
launch command is received. All maintenance
actions cease, and launch preparations commence.
control
are
failed
failures go undetected.
navigation information available, and it
receive
OR
All components which are
tests.
to
the
components
guidance
and
monitored can be repaired or replaced
during
during
Similarly, the performance of
tc monitor its status.
phase
The fire
traffic.
The
fire
subsystem transmits trajectory data to the guidance
component
positioned
of
the
for
first
missile,
launch.
The
and
fire
the
control
submarine
phases
is
for
subseguent objectives proceed in succession.
a During
the Launch phase for each missile, the
submarine is held stable while the aissile is ejected,
severing its link with the platform and causing it to switch
to internal power.
although activated.
The power sources,
are not required to supply power during this phase.
The booster phase
engine
ignites.
starts
when
first-stage
the
This occurs as each missile breaks through
the surface of the water.
The missile is then boosted along
The port igniter can be powered only by the
port power source, and the starboard igniter only by the
its trajectory.
starboard
power
fire the engine.
power
source,
The guidance
either
from
one igniter is sufficient to
but
component,
must
source,
which
function
can
take
throughout the
phase.
During
the
flight
phase,
guidance
igniter, second-stage engine,
the
second-stage
component,
and
at
least one power source must function.
m
beginning
Shutdown of the
of
the
terminal
second-stage
engine
marks
the
phase during which the warhead
follows a ballistic trajectory to its target.
10
III.
There
two
are
goals in this example.
The first is to
determine the reliability of the SLBM system with respect to
objective,
particular
each
each target is destroyed.
that
is,
the probability that
The secc-nd goal is to analyze the
overall success of the total mission.
A.
PHASE SEQUENCE DIAGRAM
As a tool in the analysis of
sequence
diagram
is
used.
sequence diagram is shown in
organizational
the
For the SLBM system,
Fig
1.
of
phase
the
2,
the phase
It graphically depicts
For example, the
the first missile, which has objective
active phase
phase
sequence and numbering of phases.
phase is numbered with two digits.
phase
the
example,
this
is numbered phase
sequence
branch for objective
12.
1,
Each
launch
and is
Following phase 12,
simultaneously up the first
and along the trunk for successive
continues
1,
objectives.
B.
PHASE BLOCK DIAGRAMS
The functional organization of
phase
of
diagram.
correspond
components
within
each
the mission is graphically represented by a block
Fcr
each
to
characteristics of
the
th;e
phase,
the
description
block
of
diagrams
the
operational
SLBM system are shown in Fiq. 2.
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Cut Cancel latioa
1 .
simplify the
block diagrams by a procedure, suggested by Rubin [1964] and
The
Heisburg and Schmidt [1966], called cut cancellation.
cancellations which are permitted by this technique are
A simple rationale for these cancellations
shown in Fig 3.
It is desireable, at this
can be illustrated referring to
be seen that component G1
objective
(phase
1
14)
point,
Fig 3.
to
For example, it can
is required in the flight phase of
and that if it is not functioning
,
1
objective
can not be
It can be reasoned that since the requirement
through the end of
accomplished.
phase,
that
that G1 functions in phase 14 includes the requirement
it
does
fail
not
eliminated
phases.
in phases
or 13,
1 1
that
G1 can therefore be
consideration in those earlier
It is said that G1 first becomes relevant in phase
further
from
14.
A
somewhat more complex situation occurs in the case
of the submarine
three
(component
S)
,
which is
required
for
all
respect to objective 1, S first
becomes relevant in phase 12; with respect to objective 2,
in
phase 22; and with respect to objective 3, in phase 32.
objectives.
With
Cut cancellation is thus
phases 00,
The
11,
permitted,
for
component
S,
in
21, and 31.
resulting
blocl»_
diagrams,
cancellation, are shown in Fig. 4.
14
following
cut
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Transf oraa tion of
2.
Mult i-phase Objective
the.
analysis of phased mission
reliability lies in the technique, suggested by Esary and
Ziehms [1975], of transforming a system with several phases
single-phase system.
into an equivalent, synthetic,
This
essence
The
procedure
makes
the
of
possible
it
compute
to
reliability
by
standard methods.
In-as-much as the performance of each component in
particular phase is dependent on the performance in
transformation
the
phases,
involves
pseudo-components
a
earlier
original
replacing
components
with
performance
in each phase independent of performance in all
represent
which
other phases.
shortcoming
A
technique
this
of
is
that
the
number
transformation
generates
a
large
of
pseudo-components, which may be unwieldy.
Recognizing this.
reduction
Bell [1975] suggested
a
procedure
pseudo-components
in
a
manner which retains the desirable
characteristics of
th;e
transformation.
The
procedures
[1975] are modified
follows:
a.
which
resulted
the
direct
for
phase
the
graphical
application
SLBM
and 7S3.
subsequent
system,
as
cancellation,
performed
cut
block
diagrams
of
Fig 4,
identify components which appear in more than one phase.
the
of
of Esary and Ziehms [1975] and Bell
Having already
in
for
In
they are FC, S, VP1, VS1, VP2, VS2, VP3,
circle the second and
These
of each of those components.
Now, as shown in
appearances
Fig 5,
are the candidates for pseudo-component expansion.
17
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Replace each of the ancircled components with
b.
a
pseudo-component
numbered with the phase number where it
For example, replace the blocks
first appears.
E1
in phase 12, with the blocks
E12
S12
Each
of
these
pseudo-component
blocks
which
specifically
now
-
represents
a
is required to survive only through
the end of the phase with which it is numbered.
Replace
c.
each of the circled components with
independent
arrangement
of
pseudo-components.
Use lower case letters plus phase
numbers to represent a pseudo-component which is required to
function only during one phase.
This step in the procedure,
an
equivalent
referred
to
as
series
pseudo-component
illustrated by the following example:
expansion,
The block
EC
is replaced in phase 21 by
FC11
fc12
19
fc21
is
best
It
should
be
clear
that
the series arrangement of FC11,
which functions only through the
which
functions
only
during
end
phase
of
phase
and
12,
11,
fc12,
fc21, which
functions only during phase 21, is the equivalent, in phase
21, to the original component FC functioning through the end
of
that
phase.
pseudo-components
It
are
should
also
be
independent
of
that
the
another.
The
clear
one
results of transforming the phase block diagrams
in
Fig 6.
20
are
shown
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Siaplif icatioa
•
The phase block diagrams can
each
other r
to
series,
in
connected to
create synthetic, equivalent,
now
be
single-phase systems. However, before proceeding,
it is
desireable to simplify the block diagrams utilizing a
procedure properly known as ideopotent
can
be
done
whereever
series with itself.
a
cancellation.
This
particular block will appear in
Graphically, the idempotent
law
says,
for example, that
S12
S12
S12
The procedure for simplification is as follows:
a.
Referring
Fig 6,
to
identify
those
groups
of
pseudo-components, which would be in series with themselves
successive
if
transformed
phase block diagrams were
connected
in
series.
These
are
candidates
for
pseudo-components,
simplification.
and
since
it
Note that VP13, for example, is not a
is in distinct and different groups of
pseudo-components in phases
b.
particular
In the SLBM system, they are S12, s21, s22,
FC11, fc12, and fc21.
candidate
those
13
and 14.
Remove the second and subsequent appearances
of each of the candidates.
The
resulting
simplified phase block diagrams are shown in
22
transformed
Fig 7.
and
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It
be noted that following this simplification,
should
phase
transformed
are
diagrams
block
equivalents
phase-by-phase,
exact
block diagrams.
However, when combined
done
in
subsequent
a
systems
single-phase
step,
are
no
series,
in
resulting
the
equivalent
longer,
original phase
the
of
the
to
as
is
synthetic,
the
original
multi-phase systems.
C.
RELIABILITY OF PSEUDO-COMPONENTS
The first goal in this SLBM example, as stated
is
determine the reliability of the system with respect
to
that is, the probability
to each particular objective;
target is destroyed.
each
earlier,
necessary
In order to proceed, it is first
discuss
to
that
reliability
the
of
the
pseudo-components.
Conditi onal Compon ent Ph ase Reliability
1 •
Represented
by
either
upper
letters, the two different types of
reliabilities
which
case
or
lower case
pseudo-components
have
are of different and distinct natures.
For the pseudo-components represented by lower case letters,
the reliability is the probability that the pseudo-component
will function only daring the
Thought
that
the
of
another
original
way,
particular
phase
indicated.
it is the conditional probability
component
will
function
during
that
particular phase, given that it was functioning at the start
of that phase.
This conditional probability is referred
as conditional component phase reliability.
24
to
2-
Unconditiona l C oap onent Reliabilit y
The reliability of the pseudo-components represented
by
upper
letters
case
pseudo-component
phase
that
the
survives through the end of the particular
indicated.
unconditional
probability
the
is
This
equivalent
the
is
of
the
original component
survives through the end of the first phase in which it is
relevant.
probability
that
the
This unconditional probability is referred to as
It might be noted that
unconditional coaponent reliability.
the availability
of each component at the commencement of
active operations is included in the unconditional component
reliability.
Hypothetical values for the reliability of
the
pseudo-components
are
shown
respective block.
25
in
Fig
8
each
of
next to each
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D.
OBJECTIVE RELIABILITY
The reliability of the SLBH system with respect to
can be determined by working with the transformed
objective
phase
simplified
and
each
diagrams
block
consists of two parts:
procedure
Fig 8.
of
The
the first is construction
of the synthetic, equivalent, single-phase block diagram
pseudo-components
for
evaluation of the
objective
each
objective;
of
the second part is
reliability
the
using
block
diagram and given values for pseudo-component reliabilities.
1 •
Objective Blgck Dia gram s
The
procedure
diagrams is actually
a
for constructing the objective block
continuation
of
transformation
the
process, and consists of the following steps:
a.
diagram
Identify the sections of the
phase
which are relevant to each objective.
sequence
This step is
illustrated in Fig 9, the shaded portions of the
sequence diagrams representing the relevant phases.
b.
block
diagrams
Connect, in
series,
the
transformed
phase
phase
of all of the identified phases to form the
objective block diagram.
For example, the block diagram
objective 2 is shown in Fig. 10.
27
of
Objective
Phase Sequence Diagram
3NKK
1
v»NV
JHHhh
2
WW
3
S8
Figure
9
-
RELEVANT PHASES
28
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Objective Relia bility Evaluation
2.
Due
the presence of some pseudo-components more
to
than once (for example,
objective
series
of
numerical
diagrams
block
independent
evaluation,
and
VP23
can
7S23
in
Fig
10)
,
the
simply be treated as a
which
would have
made
not
modules,
straight-forward.
As an alternative,
it is possible to employ a well-known graphical technique of
structural reliability, which is based on
a
procedure called
pivotal decomposition (see, for example, Barlow and Proschan
[1975])
.
This technique is best illustrated by
example,
and
consists of the following steps:
a.
front
of
the
Pull all of the
block
through E22, RF23, and
independent
diagram.
G2'4
through
In
blocks
to
Fig 10, these are COO
W25.
This
leaves
dependent sections, which in the case of objective
2
VP23
IP23
r— VP23
vp24—
VS23
IS23
!— VS23
vs2U
b.
Pivot
the
the
are
on one of the pseudo-components which
cause the dependence,; say 7P23.
split
To accomplish this,
the block diagram into two branches by considering the block
diagrams which result if
30
(1)
VP23 always functions:
vp24
IP23
L
-VS23
VS23
IS23
(2)
vs24 —
7P23 always fails:
VS2>3
vs24
IS23
dependence still remains in the "VP23
always functions" branch, pivot again, this time on VS23.
c.
Split
the
Since
"VP23 always functions" branch into two branches
by considering
the
block
diagrams
which
addition to VP23 always functioning
(1)
7S23 always functions:
IP23
r—
vp2U—
vs24
IS23
31
result
if,
in
(2)
VS23 always fails:
vp24
IP23
Since all branches now contain independent modules, no
pivoting
is
required-
The
result
of
these
steps
more
is
As shown in the figure, just beyond
illustrated in Fig 11.
each pivot pcint and just below the branch lines, label each
branch by a description of the event along that branch, such
as "VP23 functions."
32
(.950)
(.950)
COO
(.950)
NOO
(.980)
fcl2
fc21
(.990)
(.900)
-
(.950)
S12
E12
FC11
(1.00)
Figure 11
(.950)
(.990)
s21
(.950)
(.990)
s22
(.990)
(.950)
E22
(.950)
PIVOTAL DECOMPOSITION OF OBJECTIVE
BLOCK DIAGRAM
33
2
Above each branch line, in parentheses, write
the probability of the occurance of the event along that
For example,* the probability that VP23 functions is
branch.
d.
just its reliability, and the probability that it
fails
is
one minus the reliability.
Above each pseudo-component block, write the
of that block functioning' (the reliability of
e.
probability
the pseudo-component)
f.
The
blocks
Fig
in
11
can be reduced into
modules to simplify subsequent calculations.
in
This is
shown
Fig 12, where modules are indicated by dotted lines, and
the reliability of each module is indicated in
g.
Multiply
all
of
parentheses.
probabilities
the
(reliabilities) leading to the end of each branch, and write
down
the
product.
This
is illustrated in
Fig
12
by the
numbers in brackets.
h.
R2,
The
reliability
of objective 2, denoted by
can now be written down by summing the probabilities
of
all of the branches. Thus
R2 = .553
Similarly,
obtained.
reliabilities
The results are:
the
H1
of objectives
= .606
R3 = .505
34
1
and
3
can be
(.5621)
r
coo
NOO
FC11
fcl2
fc21
s21
s22
E22
RF23
G24
J24
RS24
W25
S12
E12
J
(.9950)
"I
(.9975)
I
(.9975)
1
i
i
IP23
vp24
o
o
vs24 -
1
J
LJ
(.9025)
,
(.8574)
,1
VS23
Figure 12
-
IS23
vs24
REDUCTION INTO MODULES
35
'*
S
o
^o
E.
MISSION SUCCESS
Having determined!
reliability
the
or
probability
of
accomplishing each objective, the final goal is to analyze
This aspect of
the overall success of the total mission.
the
analysis
may
by answering the following
achieved
be
questions:
(1)
What is the probability that any particular
number of objectives are accomplished on a mission?
(2)
What
the expected or average number of
is
objectives that will be accomplished per mission?
1 •
Successful Mission O utco mes
To
possible
begin
successful
considered.
*Ine
these
answering
outcomes
of
a
mission
accomplished objectives could
objective
1
alone
objective
2
alone
objective
3
alone
objectives
1
and
2
objectives
objectives
1
and
3
2
and
3
objectives
1,
2,
first
questions,
and
must
the
be
be:
3
The probability of accomplishment,
or reliability of
each objective alone has already been determined.
Following
the same procedures which produced R1, R2,
the joint
objective
reliabilities
can
be found.
36
and R3,
It should be noted
that in
set theory terminology, the
objectives
joint
are
the intersections of the events which are single objectives.
Determining joint objective reliabilities
consists
of
the
following steps:
Construct the joint objective block diagram
the transformed and simplified
connecting, in series,
a.
by
phase block diagrams of the relevant phases,
case
of joint objective
1
which
in
the
and 2, for example, are indicated
by shading in the phase sequence diagram as follows:
Evaluate
b.
the
joint
objective
which is denoted in the case of joint objective
reliability
1
and
2,
for
example, as R12.
successful
outcomes
and
the
corresponding phase sequence diagrams are summarized in
Fig 13.
A
convenient tool from set theory,
the Venn
The
diagram,
is
possible
also
shown
for
values for objective and joint
this example are also shown.
37
each outcome.
objective
The numerical
reliabilities
in
Objectives
Accompli shed
Phase Sequence Diagram
Rel iabi
1 i
(Probabi
1 i
Symbol
ty
ty
Numeri cal
Value
.606
V77/A
m
.553
V//////%
V\
.505
V///////A
1,2
12
1,3
2,3
.433
13
.395
*23
.395
VZWs
1,2,3
123
Figure 13
-
.309
SUCCESSFUL MISSION OUTCOMES
38
Venn
Di agram
2.
Missio n Succes s Levels
The
probability
that
any
objectives are accomplished on
directly with the aid of the
These
quantities,
which
a
particular
number
of
mission can be written down
corresponding
correspond
to
Venn
diagrams.
varying levels of
mission success, are tabulated in Fig 14.
It is noted that
in a more general application, where Venn diagrams might not
a PPly*
other straight-forward approaches may
Bell [1975], for example}.
39
be
used
(see
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en
3
Expe cted Success
•
Osing the probability of each mission success level,
the
average or expected number of successes per mission can
be determined by weighing each number by its probability
occurance
then
and
This
adding-
of
illustrated in the
is
following result:
Expected number
of objectives
accomplished
=
(0)
+
(.250)
(2)
+
(.296)
(1) (.145)
+
(3)
(.309)
1^66
Alternative measures of
examined
could
be
in
a
similar
weighted
as
mission
manner.
success
be
Particularly, objectives
importance,
to
could
and
specific
combinations of accomplished objectives could be considered.
Bell [1975], for example, suggested an approach which could
be used in the case of a complex mission success criteria.
41
LIST OF REFERENCES
1.
Barlow, R.
E.
Reliab ility
Wilson,
2.
and Proschan, F.
Holt,
T heory
of
Rinehart, and
1975.
Merlin
3ell,
Testing .
Life
and
Stat istica l
r
Maintained
G.,
Stanjdby
Multi,-Phase Mission Reliability of
NPS55Ey75121
Systems.
Naval
.
Postgraduate School, Monterey, Ca., 1975.
3.
Esary,
J.
P hased
D.
and Ziehms,
Missi ons.
Analysi s.
R.
E.
Rubin,
J.
C,
Barlow,
Proceedings
The
of
Reliability
,
Reliability
In
Singpurwalla, editors.
4.
H.
J.
Tree
Fa ult
and
of
N.
D.
1975.
Reliability
the
and
Fussell,
B.
SIAM.,
Analysis
Complex
of
Net work s.
Reliability
Aero-Space
and
Maintainability Conference, 1964.
5.
Weisburg,
S.
A.
and Schmidt,
J.
for Est imatin g Siystem Re lia bility.
1966.
Compute r Technique
H.,
Proceedings of
Annual Symposium on Reliability.
42
IEEE 7C26,
the
1966.
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No.
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Copies
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2.
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3.
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2
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Naval Postgraduate School
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4.
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D.
Esacy, Code 55Ey
1
Department of Operations Research
Naval Postgraduate School
Monterey, California 93940
5.
Professor
H.
B.
Kline, Code 54Kx
1
Department of Administrative Sciences
Naval Postgraduate School
Monterey, California 93940
6.
LT Steven E.
Piinick, OSN
RD 4, Box 60
Middletown, New York 10940
43
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