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JPRS L/9569
24 February 1981
Translation
ANALYTICAL DESIGN OF SHIPS
By
L.Yu. Khudyakov
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h;
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JPRS L/9569
24 February 1981
ANALYTICAL DESIGN OF SHIPS
Leningrad ISSLEDOVATEL'SKOYE PROYEKTIROVANIYE KORABLEY (Analytical
Design of Ships) in Russian 1980 signed to press 26 Mar 80 pp 1-240
[Book by L.Yu. Khudyakov, Sudostroyeniye, 2,200 copies, 240 pages,
UDC 629.12.001.21
CONTENTS
Annotation 1
Scientific Editor's Preface 2
1'ntroduction 4
CHAPTER T. Basic Problems of the Theory of Analytical Design of Ships
�1.1. Subject of Analytical Design Theory 7
g1.2. Basic Problems of AD Theory and the Genera]. Characteristic
of the Methods of Solving Them 9
�1.3. Generai Statement of the Problem of Optimizing a 5hip's
Tactical-Techn,ical Characteristics as a Problem of Mathema.tical
Programmixg 14
CHAPTER 2. Solution of the Problems of the Ship Engineering Design Block
�2.1. Characteristics of the Problems af the Engineering Design
Block 18
�2.2. Three Basic MetYiods of Determining Displacement and Principal
Dimensions of a Ship 21
�2.3. Equations of the Analytical Method of Ship Design 26
�2.4. Use of Similarity and Dimensionality Theories When Compiling
the Equations of the Analytical Method of Ship Design 56
�2.5. Application of Mathematical Statistics When Compiling the
Equations of the Analytical Method of Ship Design 64
CHAPTER 3. Methods of Estima.ting the Efi`ectiveness of Ships 74
�3.1. Basic Concepts. General View of the Effectiveness Indexes 75
g3.2. Calculation of Effectiveness Indexes 81
�3.3. Mathematical Model of Estimating the Effectiveness of
Independent Combat Operations of Ships of Side A Against
Ships of Side B 93
- a - [I - USSR - G FOUO]
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�3.4.
Markov Mathematical Models of Tao-Way Duels
100
g3.5.
Mathematical Model of Estimating the Effectiveness of
Operations of Ships Patrolling Given Part of the Sea in
_
the Presence of Enemy Counteraction
113
CHAPTER 4. Methods of Estimating Economic Indexes
121
�4.1.
Determining the Construction Cost of Ships
122
�4.2.
Determining the Construction Cost of a Series of Ships.
Consideration of the Size of the Series
12$
�4.3.
Determination of the Cost of Maintaining Ships
130
�4.4.
General Formula for Determining the Cost of Building and
Maintaining a Fleet of Ships. The Cost of Solving the
Problem as One of the Military-Economic Indexes
134
g4.5.
Power Correlation-Regression Models of Estimating the Cost
Service Indexes
137 ~
CHAPTER 5. Optimization of Tactical-Technical Characteristics of Ships
139
g5.1.
'Cost-Effectiveness' Type Criteria and Their General
Properties
139
�5.2.
Example of Optimizing the Characteristics of Ship When
the Service Index Is the Displacement
149
- �5.3.
Optimization of TTE and TDP os a Ship by the Method of
Comparative Evaluation of Versions
152
g5.4.
Search for the Optimal Version of a Ship ia the Space
of the TTE and the TDP
153
�5.5.
Example of Optimizing the Mass of Hauled Cargo and the
5hip's Speed by the Provisional Gradient Method
162
�5.6.
Optimization of TTE and TDP of a Ship Considering
Several Effectiveness Indexes
166 -
�5.7.
Optimization of the TTE and the TDP of a Ship Under
Conditions of Indeterminacy With Respect to Some
Initial Data
172
95.8.
Example of Optimizing th TTC of a Ship by the Method
,
of Comparative Evaluation of Versions
176
�5.9.
Use of the LP-Search Method When Optimizing the TTC of
Ships
181
CHAPTER 6. Some Special Problems Connected With the Analytical
~ Design of Ships 190
�6.1. Consideration of the Reliability of Technical Means When
s Estimating the Effectiveness of Ships 190
�6.2. Optimization of the Service Life of Ships 207
Conclusion 21$
Appendix 1. Solution of the Three-Term Maes and Volume Equations
Using the Auxiliary Function Table and Obtaining an
Approximate Solution in Explicit Form 220
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Appendix 2. Coordinates of the Ten-Dimensional Sobol' Points (With -
Rounding to the Third Place) 222
Appendix 3. Some Expressions Used in the Initial Stagea of Ship Design 223
- Bibliography 227
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ANNOTATION
[Text] A study is made of the basic content and methods of solving problems in
the theory of analytical design of ships a new area in general design theory
- connected with scientific substantiation of design assignments for ships of
various classes and types. Methods of evaluating the technical characteristics
of ships in the analytical design phase, their effectiveness indexes, economic
indexes, in particular, the cost of building and maintaining the ship as part of
the fleet, the military-econonic optimization of the tactical-technical character-
istics of the ship are dis cussed. The book is intended for ship design special-
ists.
There are 8 tables, 61 illustrations and 67 references.
,
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SCIENTIFIC EDITOR' S PREFACE
The scientific substantiation of the development of the naval fleet, the studies
of the laws and principles of its construction touch on a broad class of problems
pertaining to determining the expedient direction of development of the fleet,
the nature of its balance, the deadlines and sequence for building ships, weapons,
ac cessory equipment and engineering devices, and their operation and maintenance.
The solutton of the indicated problems is en,:ountering serious difficulties in
connection with the unprecedented technica]. complication, rise in coat, and shor.t
de adlines for building ships, their weapons, accessory equipment and equipment.
Ac cordingly, generation of the assignment to build a new ship under modern condi-
ti ons requires profound, multivariant scientific research develoi:ldents, comprehen-
sive analysis of possible alternative solutions and, on the basis of this analysis,
synthesis of the design of a ship with optimal tactical-technical characteristics.
This approach to the process of building a ship has ta'ken the shape of a special
ph ase called analytical design.
Th is book by L. Yu. Khudyalcov is one of the first attempts to present a systematic
- dis cussion of the procedural problems of the theory of the analytical design of
' ships. Along with three basic aspects (operative-tactical, technical and
e conomic) which provide the basis for analytical design, significant attention is
given to the problem of optimizing the tactical-technical characteristics of the
designed ship. The book has been written on a very high scientific level�using
the corresponding mathematical apparatus and illustrative examples.
The circle of prob lems considered during the analytical design of ships is quite
b road, and it is very difficult to discuss each of them in sufficient detail.
Traditional courses in ship design theory are devoted to the problems of determining
displacement and the principal dimensions and also the planned provision for the
basic shipbuilding properties of the ship as a floating facility. At the same time
the approximate solution of these problems is only one of the steps (blocks in
the terminology of the author) of analyticsl design the technical development
phase of the versions of the ship. In the plan for writing the book there is a
mo re complex prob lem of the development of the methodology of analytical design
encompassing analysis of the broad spectrum of combat and engi.neering characteristics
of a ship interrelated with the indexes of military-economic effectiveness.
Th erefore from the point of view of completeness and, above all, detail of the
di s cussion of a11 problems considered in the analytical design of ships, the book
by L. Yu. Khudyakov cannot claim to be the "last word" in solving the stated prob-
lem. For examgle, the book does not consider the specific problems of planned
2
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provision for various charaeteristics of the ships in the analytical.design
phase, the complete algorithms for calculating the effectiveness indexes,
quaLitative conclusions with.'respect to the optimal combination of Eactical-
technical characteristics of ships of different classes, axid so on.
However, the purpose of this book 3s only.to introduce the reader to a group of
basic ideas and methods of the theory of analytical design of ships, and from
this point of view it has been written entirely satisfactorily and will be useful
for a broad class of specialists involved with the analyti�al design of shtps.
The book is especially useful for specialis ts who have begun to study analytical
design inasmuch as after reading it they'.wi 11 be able to delve more purposefully
into their field.
The value of the book also lies in the fact that it is a useful scientific start
for further development and improvement of analytical design using the achieve-
ments of computer engineering and its capab ilities for the creation of automated
- design.
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L
INTRODUCTIAN
The military potential of the naval fleet is determined by many factors, and one
of them is the makeup of tihe fleet containing various types and classes of
submarines, surface ships, and support ships in the nunober and combinations
corresponding to the missions facing the fleet and the balance of it [151.
This paper has as its purpose to offer a brief systemattc discussion of the
problems pertaining to the method of developing ship design assignments and also
design research connected with substantiating various solutions in shipbuilding.
The effectiveness of a ship arises from a number of its properties such as
accessory equipment, shielding, survival probability, reliability, seakeeping,
and so on. These qualities are quantitatively characterized by values called
tactical- technical elements (TTE). In turn, the TTE depend on a number of
characteristics of the weapons, accessory equipment and technical means the
technical design p arameters (TDP).
In the consolidated plan, the insurance of high effectiveness of a ship is
connected with solving two basic problems: the development and the creation of
improved types of weapons, accessory equipment and technical means the "bricks"
from which the "building" of the ship is constructed and the substantiation
of an efficient combination of primary characteristics of the ship, that is, its
~ intent and the specific values of the TTE and the TDP.
The solution of the first of the indicated prob lems involves machine building,
electrical engineering, instrument making, material sciences, ship's theory and
strsctural mechanics and also a number of other areas of science and engineering.
The second problem is within the competence of ship desi;,n theory. At the same
time, it must be noted that the two indicated problems cannot be aolved separately
f rom a third pertaining to the conditions of combat use of the ship as an element '
of the system of mixed forces of the fleet called on to solve 'one -combat prbblem
or anothe r. This problem pertains to the operative-tactical area, but ifi must
be considered in its interrelation to the formation of the tngineering desigr_ of
the ship.
The creation and the introduction of nuclear missiles, nuclear pawer engineering,
radio electronics and other achievements of scientific and.technical progress on
board ships have led to a qualitative jump in improvement of the effectiveness
of the ships. Under these conditions and in connection with sharp complication
of ships from the tactical point of view, increased cost of b uilding and maintain-
ing them and also the more complex nature of the combat use of ships, the prot-?em
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of optimizing the TTE and the TDP has acquired special urgency. At the present
time it is insufficient to consider only the operative-tactical aspects in-the
earliest stages of compiling the assignments for bui lding ships. A complex study
is needed which takes into account the technical and economic factors.
This problem is solved by design developments with b road variation of the entire
set of tactical-technical characteristics (TTC) of the ships. Such developments
aimed at the discovery of the optimal combination of TTC of a designed ship have
come to be called analytical design (AD). Its basi c practical "output" is the
scientifically substantiated data used to develop shipbuilding plans and also the
ship design assignment.
In this book a study is made of the basic content an d methods of solving the
prob lems or AD theory as one of the basic areas of general ship design theory.
T"ne class of problems arising during the process of the analytical design of
ships is very b road. The statement and the methods of solving a number of AD
problems, for example, estimating the effectiveness and optimization of the TTE
and TDP do not enter into the traditional courses in ship design theory [4, 26, 351.
The book by V. V. Ashik [1] which contains a chapter on the optimization of the
characteristics of the designed ship, published in 1975, constitutes an exception.
However, the problems of the methodology of the development of ship design assign-
ments are not considered in this book. It is also necessary to note the book by
A. A. Narusbayev [34] which is very similar with respect to content to the
problematics of the theory of analytical design, b ut it does not fully encompass
the class of analytical design problems. In contrast to the paper by
A. A. NaruGbayev, the present book contains a more detailed investigation, for
example, of the problems of constructing and calculating the effectiveness
indexes, evaluating the economd c indexes and certain other problems. Therefore,
the author hopes that the material of this book wil 1 help the reader to have a
more complete view of the analytical design problem.
Analytical design as a system and an independent step in the multistep process of
ship design has now found its place in the work practice of specialized organiza-
tions. The given book pursues the goal of familiarizing the reader only with the
fundamentals of inethodology of analytical design in asmuch as this methodology and
the theory corresponding to it are continuing to b e developed and improved
intensely. Accordingly, the author has tried to give attention to-a ninnber of
procedural:.difficulties and prob lems and also to s ay more about the prob lems which
previously were not included in traditional courses on ship design theory.
When discussing the material of the book, the author used the SI system. In
particular, the condition of equilibrium of a floating ship without way is
expressed not in the form of a direct equality of the weight of the ship (the
pressure of the ship in the water caused by gravity) to the buoyancy force (the
total force of hydrostatic pressure of the water against the hull of the ship),
but in the form of equality of the mass of the ship to the mass of the water -
displaced by it.
Many comrades gave the author a great deal of assistance in working on the book.
Among them it is necessary especially to mention honored scientists and engineers
of the RSFSR, Doctors of Technical Sciences, Prof essors V. N. Burov and
L. A. Korshunov, Candidates of Technical Sciences B. A. Kolyzayev,
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A. P. Makkaveyev and Ya. V. Gakh. The author expresses deep appreciation to them
for their assistance and support. The improvement of the book has been greatly
promoted by the counsel and advice of the reviewers: H onored Scientist an,l
Engineer of the RSFSR, Doctor of Naval Sciences, Professor S. K. Svirin and
Doctor of Technical Sciences, Professor V. T. Tomashevskiy. The author is also
grateful to N. M. Ivanova, M. G. Tepina and G. D. Karanatova for assistance in
laying out the manuscript of this b ook.
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CHAPTER 1. BASIC PROBLEMS OF THE THEORY OF ANALYTICAL DESIGN OF SHIPS
�1.1. Snbject of Analytical Design Theory -
The subject of AD theory is the set of TTCl of a ship investigated from the
operative-tactical, technical and economic points of view to select the version
(or versions) of the ship most preferable in the �sense of effectiveness of
the solution of the stated problems and the possibilities of building and main-
taining the ship as part of the f leet.
This definition can be extended also to civilian-ships if by the TTC we mean the
operating-technical characteristics of the ships, and by the operative-tactical
point of view, the investigation of the effectiveness of their use for the
corresponding purposes (transport, fishing, scientific research, and so on).
At the present time AD is conducted to solve the following problems f acing ship-
building: the substantiation of the directions of development of ships for the
future considering the achievements of scientific and technical progress; the
-development of ship design assignments; the discovery of areas in which the general
requi:rements on ship design and substantiation of these requirements must be
developed; substantiation of the directions of development of ship equipment
and for warships also weapons and accessory equipment in the future.
The necessity for AD in the interests of the two last-mentioned problems arises
from the fact that in the general case the deficiency of individual technical
solutions can be established only by estimating their effect on the overall complex
of TTC of the ship and through this complex, on the effectiveness and the possi-
bility of building and maintaining the ship. This evaluation of partial technical
solutions sometimes is called "evaluation in terms of the ship." Although in
practice "evaluation in terms of the ship" freqnently encounters significant diffi-
culties, it is the most objective.
- Thus, AD is characterized by the general procedural solution of a number of
problems connected with building ships, by weapons, accessory equipment and
technical means. The basis of this methodology is a11 around investigation of the
TTC of the ship as a complex technical system and search for the optimal version
considering the use of the ship within the soyedineniye and taken together with
1Hereafter the TTC will refer to the set of TTE and TDP.
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the combat materiel of other types of forces. Apparently, the term "analytical
design" was introduced for the first time in Soviet literature by V. L. Pozdyunin,
who defined it as follows [38]:
"Analytical or experimental design takes into account the existing production
and operating conditions from the point of view of replacement of them by new
= conditions or alteration of them to create new structural models of the ships
_ or their component elements corresponding either to entirely new goals of water
transportation and defense of the country or certain alterations of the goals."
What are the objective causes of isolation of AD theory in an independent area
of general ship design theory and broad application of AD in the activity of
scientific and design organizations?
A brief answer to this question is that the objective requirement for a scientific
solution to the basic AD problems has begun to correspond to the general
sci.entific possibilities of the creation of the corresponding theory.
The objective requirement of a scientific solution to the problems of developing
the requirements on designed ships, their weapons, accessory equipment and
technical means has existed for a long time, but during the periods of basically
evolution ary development of shipbuilding when for long time intervals the TTC of
ships of different classes and types changed comparatively little, the problem
of substantiation of the assignmants for the design of new ships was not very
urgent. The solution of this problem on the basis of intuition and experience
of the designer considering information about the TTC of ships already built an d
information about their practical use, as a rule, did not lead to serious errors
- although in a number of cases errors did occur.
The demands for objective scientific methods of substantiating the requirements
on the designed ships has increased as the ships have become more complicated
and various alternative technical conditions have appeared, none of which could
be recognized as clearly preferable without the correspon ding quantitative analysis.
In 1908, A. N. Krylov proposed [25] a quantitative method of comparative evaluation
of different designs of battleships developed for competition. ^'his method was
based on the intuitive constructi on of a criterion whi ch depends on the TTC of
the ships. At that time there were no general scientific possibilities for
stricter solution of the prob lems of optimizing the TTC pf ships. V. L. Pozdyunin
also indicated the clearly intuitive method of solving the problem of opttmi.zing
the TTC of sh ips, talking about the necessity of "collectivP investigation of b oth
assignments and designs in various design stages at specially organized scientific
and engineering conferences..." [38]. Although at this time the so-called expert
evaluations is used, the basis for it is the special foxmal-logical methods of
processing opinions and estimates of the experts which were developed comparatively
re cen t ly .
The necessity for careful development of assignments for desigri of modern ships,
optimization of their TTC and the directions of development is obvious at the
present time. This arises from the complication of ships from the technical point
of view, a sharp increase in the numb er of posoible alternative technical so lutions
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and versions of operative-tactical use and also an increase in cost of building
and maintaining the ships. There are examples where the ships have become
obsolete during the design process or they have turned out to be ineffective as a
result not cnly of the high rates of scientific and technical progress, but also
the insufficiently complete and comprehensive substantiation of the ship assign-
ments. Generally speaking under the conditions of acceleration of the rates of
scientific and-�technical progress, the rate at which the ships become obsolete is
increasing and, consequentlq, the optimal times for them to be part of the fleet
are decreasing. This fact, along with an increase in cost of building and main-
taining the ships, is leading to an increase in the price of errors and miscalcula-
tians when designing them.
It is clear that for sufficiently accurate optimization of the TTC of a ship it is
already necessary in the design stage to estimate the effectiveness c.r solving the
missions assigned to the ship. At the same time the ship is a complex engineering
dynamic system, the characteristics of which can, generally speaking, vary randomly
- with time as a result of interaction with the external environment and the enemy.
The mathematical expectation of the processes of the functioning of such systems is
encountering serious difficulties, including difficulties of a computational nature.
Similar difficulties also exist when solving the problems of optimization occurring
when designing complex engineering objects. Here, first of all, it is necessary
to note the complexity of the optiudz;�ble funetionals which, as a rule, can be
given only by sufficiently complex algorithms, ambiguity of the optimization cri-
teria, comp lexity and variety of the limiting conditions, and so on.
The above-indicated difficulties in solving the basic AD problem optimization
of the TTC of the ship were essentially, although not final'y, overcome after
World War II whPn high speed camputer engineering came into being, and a number
of new mathematical methods of optimization and also mathematical simulation of
the structure and functioning of comp lex objects and processes were developed.
One of the first studies by the new mathematical methods of optimization must be
considered the works af L. V. Kantrovich with respect to statement and solution
of the problems of linear programming [22).
_ Thus, the increasing demands for scientific solution of the basic problems of
the analytical design of ships has to a known degree begun to correspond to the
general scientific possibilities of the theoretical formation of AD as a new
applied scientific area. This is also an objective cause for scientific formula-
tion of AD theory as an independent branch of general design theory.
- �1.2. Basic Problems of AD Theory and the General Characteristic of the Methods
of Solving Them
In accordan ce with the final goal of AD optimization of the TTC of the ship on
the basis of analysis of the operative-tactical, technical and economic aspects
connected with building and using it in the fleets it is possible to isolate
the following procedural problems sub3ect to solution in the AD process.
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1. Determination of the sets of TTC of the.versiflns of the ship subject to
comparison for optimization the so-called engineering.design blbck.l
2. Determination of the effectivenesss index of the ship with different sets of
TTC when solving the prob lems invested in it the effectivenesB evaluation
block.
3. Estimation of the service indexes, that is, the expenditures of different
types of reserves,2 including economic, required for the building and maintenance
of the ship the reserve block.
4. The optimization of the TTC of a ship, including the selection and construc-
tion of the optimization criteria the optimization block.
These problems, the b lock diagram of the solution of which is shawn in Figure 1.13
- are characteristic of the AD process not only for ships, but also any accessory
equipment system [51), civilian vessels and, in general, any technical systems
with the corresponding interpretation of the concept of effectiveness.
In the subsequent chapters a study will be made of the content, and a characteris-
tic will be presented for the methods of solving the problems of each of the
_ above-indicated AD b locks, and meanwhile we shall make a few preliminary remarks.
In essence, the problem of the engineering design block which wil7. be called the
technical block for brevity, coincides with the basic prob lem of ship design theory
in the understanding established up to now. Actually, we are talking about
determining the set of TTC of a ship which satisfies certain values of variable
ch aracteristics. The b asic content of design theory is defined in this way in
the existing literature. The methods of solving different problems connected with
technical development of the designed ship are discussed in a number of courses
and monographs on design theory [1, 4, 26, 351.
The necessity for investigating a large number of versions of the designed ship
with limited information with respect to a number of initial and intermediate
data, including those pertaining to the geometric shape of the ship and its
iHere and hereafter the term "engineering design"�reflects.:the essence of the
problem solved in the given stage determination of technically compatible
sets of TTC of the ship (in contrast to the engineering design-stage in the
general p-rocess of b uilding the ship). 2In the given case the term "reserve" is understood in the broader sense in
contrast to the application of this term, for example, for denoting the endurance
index in reliability theory.
3Hereafter the step of developing the intent of the ship will not be considered
inasmuch as basically unformalized methods and approaches are used in this step.
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Figure 1.1. Block diagram of the process of analytical design of a ship
Key:
l. Intent of the ship
2. Variable tactical-technical characteristics
3. Engineering design block
4. Initial data for engineering design ,
5. Effectiveness evaluation block
6. Reserve block
7. Initial data for estimating effectiveness
8. Optimization block
9. Initial data for technical-economic evaluation
10. Optimal version of the ship
component elements is characteristic for AD. Therefore in the AD stage b road use
is made of the so-called analytical methods-o� design (see Chapter 2). Inasmuch
as during the AD process quite frequently it is necessary to cunsider versions
of the ship not having designed or constructed analog, it is necessary for the
specialists in AD operatively to develop relatively simple approximate procedures
and approaches for quantitative evaluation of a number of characterisfics. Here,
just as when solving other AD problems, much depends on the skill, experience
and intuition of the researcher, although the decisive role unconditionally belongs
to knowing the basic principles and achievements of the corresponding special
sciences. An important methodological procedure of AD theory is also the
application of the methods of similarity and mathematfcal statis.tics which permit
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determination of the characteristics of the designed ship by analoguus
characteristics of analogs already built or designed.
From the point of view of the technical development of a ship, the.design process
- usually is divided into the following steps: technical developments when prepar-
ing and substantiating the operative-tactical intent of building the ship (as a
rule, they are satisfied within the framework of the research with respect to
shaping the prospective shipbuilding programs); technical developments for the
preparation and substantiation of the tactical-technical assignment for ship design
(the conceptua.l design), the preliminaYy design, contact design, development of
wcrking drawings (detail design). The technical development of the versions af
the ship during AD is carried out on the level of the first two of the above-
indicated steps.
1n order to have the possibility of quantitatively evaluating the effectiveness
of the designed ship, it is necessary first of all to select the index or indexes
of effectiveness and, secondly, to construct the algorithm making it possible to
calculate their values as a function of the TTC of the ship.
The theoretical basis for solving the indicated prob lems is a comparatively new
applied science formed after World War II operations research which studies
the methods of quantitative estimation and optimization of various purposeful
effects. The basis for this science is mathematical simulation of the functioning
of the investigated system 4nd optimiz,ation of various parameters on which this
functioning depends. The theory of operations research uses a large arsenal of
mathematical methods connected with probability theory, random process theory,
mathematical statistics, mathematical methods of optimization, and so on. The solu-
tion of the majority of practical problems in the field of operations research
requires the application of a computer. There is a broad literature on the theory
of operations research and its practical applications [12, 13, 19, 24, 31, 45, 591.
It must be noted that individual problems of operations research, in particular,
shooting theory, have already been resolved for a long time both in the Soviet Union
and abroad. For example, the general principles of estimating effectiveness as
applied to the problems of shooting theory are investigated by A. N. Kolmogorov [22].
Very frequently, during AD, cost is used as a united measure of the expenditures
of different types of resources required for building ships and maintaining them
in the fleet. In order to estimate this index in indIvidual design stages, the
methods of similarity and statistics are used permitting evaluation of the cost
of the designed ship with respect to given analogs. If cost is used as the basic
service index, then optimization of the TTC of a ship in the AD process is called
military-economic optimization, and the research aimed at solving this problem is
called military-economic analysis.
The military-economic approach when optimizing the TTC of ships and other accessory
equipment systems, combat complexes, and so on has found broad application at the
present time [18, 5:11. In the United.States this approach.is called "cost-
effectiveness analysis." The optimization of the operating-technical characteris-
tics by cost-effectiveness analyeis is the basic problem of technical-economic
analysis for civilian ships [34].
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At the same time, the military-economic approach is not the only one for
optimizing the TTC and individual technical sy.stems of the ships. The nature of
the reserves which must be considered in-each specific case can be highly varied
(displacement and dimensions of the ship, the conditions of.its combat and baee
support, the deadlines for building, the production capabilities of industry, and
so on). The problems of optimizing the.geometric composition of a ship are of a
specific nature, that is, the placement of a weapons, accessory equipment,
technical means-and personnel on the ship. Such prob lems sometimes must be solved
in the AD stage.
What has been stated above- gives rise to the different forms and hierarchical levels
of the optimization probleins arising during analytical design of ships. As for
the purely mathematical nature of these problems, they most frequently are formu-
lated as mathematical program problems. At the same time it must be noted that
many of the problems of design optimization of ships' characteristics still do not
have strict mathematical statement.
In connection with the complexity of adequate mathematical models of the blocks of
the problem of technical development of versions and evaluation of their effective-
ness, the problems of optimizing the TTC of designed ships frequently lose visibil-
ity and the correct statement, analysis and interpretation of the optimization
results reQuire a great deal of experience and intuition on the part of the
researcher. The solution of the design problems of optimization is also impossib 1e
without great mathematical skill inasmuch as very frequently it is necessary to
use individual mathematical methods in different combinations.
It is necessary to consider that AD theoiy with respect to the optimization of the
TTC of ships only generates recommendations and the necessary information for
adopting design solutions. The final adoption of these solutions remains within
the competence of the directing agents and people responsible for the solutions.
Such people can be, in particular, the customer representative confirming the
design assignment and the chief designer heading up the entire subsequent process
of developing the ship's design.
From what has been discussed it is
scientific area at the 3unction of
ship-building sciences, operations
methods of optimization, and so on
one of the characteristic features
applied sciences.
obvious thaC AD theory is an example of the
a number of scientific fields, in particular,
research, applied economics, mathematical
The occurrence of such areas, as is known, is
of the modern phase of development of basic and
In AD theory, just as in general ship design theory, there is still no united and
sufficiently stable terminology with respect to a number of prob lems and concepts.
Therefore the terminology used in the given book to a great extent reflects the
personal point of view of the author and also the viekTs of specialists known to
him, and it does not claim to final editing. In particular-, the follawing terms
will be used many times in the text of the book: "weapons," "accessory equipment,"
"means of combat;" "combat supplies on hand," "technical means." As applied to
the ship, unless specially stipulated, these terms can be understood as follows:
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Means af combat are.means (rockets, torpedoes,mines, bombs, missiles, and so on)
having a direct damaging effect on the enemy;
Combat supplies on hand are the means of comb at at storage lccatidns and in the
installations for use;
Weapons are the set of ineans of combat and devices for the direct use of them
(rocket launchers and artillery mounts, torpedo tubes, mortar tubes,;and so on
plus the above-indicated means of comb at and comb at supplies on hand);

Accessory equipment is the set of ineans provided for ship handling, communications
and also the use of weapons for target detection, target indication, determination
of location coordinates and the plane of the true meridian, and so on;
Technical means are various types of machinery, devices, sy:,tesis and other equip-
ment giving the ship maneuverability, its seakeeping qualities, steaith, invulner-
ability and habitability.
It is also necessary to consider that the terminology of AD theory is highly.fluid
inasmuch as it must take into account the terminological char,ges in a number of
scientific disciplines and branches of industry, new -classes and types of ships,
new architectural and structural features, and so on. Therefore the author hopes
that when evaluating this book from the point of view of the terminology used the
reader will consider the above-mentioned f acts.
�1.3. General Statement of the Problem of Optimizing a Ship's Tactical-Technical
Characteristics as a Problem of Mathematical Progratmning
With respect to the nature of their influence on comb at effectiveness, the entire
set of TTC of a ship can be b roken only into already known groups: TTE and TDP.
By TTE we mean the characteristics which directly influence the combat effective-
ness of the ship, that is, directly enter into the algorithms for calculating the
combat effectiveness indexes as initial data. For example, the TTE include the
following: the composition of the weapons and accessory equipment considering
their natural characteristics, maximum speed of the ship, its sea endurance,
cruising range, principal dimensions, handling characteristics, stealth character-
istics, protection, reliability, invulnerab3lity, and so on.
By the TDP we mean the characteristics indirectly influencing the effectiveness
indexes through the TTE which depend on the TDP. For example, the TDP include
the following: the structural design and architectural class of the hull, the
class and characteristics of the power plant, the electric power system, the
type and characteristics of the ship's hull material, and so on.
When optimizing the set of TTE and TDP of a ship, a set-of scalar characteristics
are always isolated which are subjected to optimization, and during the process
of this optimization they can vary independently of each other.- Let us designate
these characteristics by the vector X with the components xl,..., x. This means
that each set of values of the variable characteristfcs can be placed in
correspondence to some point of an n-dimensional euclidian space. The
coordinates of this point are the values of the individual variable characteristics.
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With respect to its mathematical nature, the optimizable eharacteristics of xi,
n can.be continuous and discrete values, that is, in the general case
the variables xi can be divided into two groups: continuously variable and
discretely variable. This separation is connected with the fact that for optimiza-
tion with respect to continuous and discrete values in tl:e general case different
mathematical methods are used. Sometimes.certain discrete characteristics can be -
approzcimately considered as conicinuous or vice versa.
Along with the characteristics of the ship defined by the vector X, the effective-
ness indexes and resource indexes can depend on a number of other parameters which
do not vary in each specific investigation. The values of these parameters are
- fixed by the designer either in accordance with the general and special requirements
on the design of the ship, its mechanisms and systems or on the basis of analysis
of prototypes or, finally, intuitive arguments. Let us designate these characteris-
tics by the vector 0. For example, frequently the parameters of the shape of the
ship's hull, the reserve buoyancy, the habitability standards, the general layout,
and so on are fixed in the AD stage. In special cases, individual characteristics
mentioned above as components of the vector X can also be fixed.
In addition to the characteristics defined by the vectors X and 0, there is a
third group of so-called dependent characteristics, the values of which are com-
pletely defined by giving the vectors X and 0. Let us denote these characteristics
by the vector Y with the components yl,..., y. This group includes displacement
and principal dimensions of the ship, its loawcomponents, seakeeping and handling
characteristics, the parameters of ttte provisional laws of damage under the effect
of various forms of weapons and ammunition, stealth and protection characteristics.
The establishment of the relation between the vectors X, 0 and Y in the form
Y - f (X, 6)
is the primary goal of the engineering design block.
(1.1)
In the general case the relation between the vectors X, 0 and Y is not fully
formalized, and it is established by the graphoanalytical method, for the determina-
tion of a number of ship's characteristics requires graphical development of the
design solutions. However, inasmuch as in AD it is necessary to consider a large _
number of versions, an effort is made to use approximate analytical methods with a
minimum of highly tedious graphical operations.
Recently special automated design computer systems (SAPR) for ships and other
engineering projects have begun to be developed and used [18, 30, 671. The basis
for these systems is computers and specialized electronic devices for graphical
data input and output and also the performance of certain operations of making
drawings. The SAPR combine the great computing capabilities of the computer with
the possibility for the human designer operatively to inf luence the design process,
including making and evaluating a number of the construetion and other decisions
which at the present time cannot be fully algorithmized. For example, such
decisions include the principles of the general placement of accessory equipment,
machinery, equipment and personnel on the ship, architectural features of the hull,
and so on. On the whole, the development and improvement of SAPR is a prospective
area fcr t;ie improvement of AD methodology.
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- In the effectiveness evaluatiori block relations are estab].ished for the effective-
ness indexes 3t, l, as a function o12 vectors X, 0 and*Y:
(X> e, Yl, 1= 1,
where L is the number of investigated effectiveness indexes.
Inas much as the vectors X, 0 and Y are relatied to each other by tfie expressidn
(1.1), the relations (1.2) can be represented in the form
31 = qDc (X, 0), l = 1, . . L. (1.3)
In accordance with (1.3), the-functianal ft must ba-given in the set of possible
values of the vectors X and 0.
In the block for estimating the serviee indexes, the relation-is established
between the service indexes Sk, k=1, K and the vectors X, 0 and Y. Consider-
ing expression (1.1) these relations have the form
Sk = 'Pk (X, E)), k = 1, . . . , K, (1. 4)
where K is the number of investigated service indexes.
The functionals 'pk must be given in the set of possible values of the vectors
X and 0.
An optimalness criterion including the effectiveness indexes (1.3), service indexes
(1.4) and in the general case, the X and Y vector components is primarily con-
structed in the optimization block. The criterion is a system of conditions
imposed on the above-indicated indexes and components. These conditions must be
satisfied by the optimal vector X or the optimal set of vectors X if the solution
of the problem is not unique.
Most frequently the optimalness criterion is farmulated as an extremal problem in
a closed bounded set of vectors X which is called the admissible set N' . The
optimal set of vectors X is a subset of the set and in case of a unique solu-
tion the optimal vector X is a point of the set 'T .
The procedure for giving the set 3E. has great significance. Tao basic cases are
of the greatest interest here: the set X is given in the form of a discrete finite
set of vectors X, that is, a finite number of versions of the ship characterized
by the vectors X1,..., ,Ym is given, where ;1c' is the number of investigated
versions; the set ..t is given in the form of a set of continuous Closed bounded
sets of values of continuous variable components of the vector X for each of the
possible sets of values of discretely variable characteristics which are also
components of the vector X.
In the first case in the n-dimensional euclidian space (n is the number of components
of the vector X) the set 2E is represented in the form of the set of discrete
points (Figure 1.2, a). In the second case the set X can be represented in the
form of a combination of (n-nl)-dimensional sets p-1, yn where q)
is the number of possible sets of values of discretely variable components os the
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vector X, and nl is the number of these.components (n1;n). Figure 1.2, b gives
the form of the sets 2Ev schematically, for the three-dimensional case when
there is one discrete component x3 which assumes two values x3 and:x3 and two
continuous components xl and x2.
Q)
x
Figure 1.2. Methods of givfng the set Y
2
For the disarete method of giving the set N the optimization of the ship's
characteristics is called comparative evaluation of versions; in the second case
it is called optimization in space of the TTE and the TDP of the ship.
17
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CHAPTER 2. SOLUTION OF THE PROBLEMS OF THE SHIP ENGINEERING DESIGN BLOCK
�2.1. Characteristics of the Problems of the Engineering Design Block
The basic prob lem of the engineering design block in the general AD scheme is establishment of the relation between the independently variable characteristics
of the shil Q,ubject to optimization and the dependent characteristics which
influence t:.e effectiveness and service indexes. Mathematically this problem
reduces to expansion of the function (1.1). Here the central problem is determina-
tion of the displacement, principal dimensions and load elements of the ship.
As is knawn, the load is the set of data on the distribution of all loads on the
ship. For convenience of the calculations it is divided into a number of sections
and items combining uniform loads. Hawever, the load b�reakdawn is to a defined
degree arbitrary.
Along with displacement and the principal dimensions, a.number of other characteris-
tics which influence the effectiveness and service indexes are also determined
during the AD process. For example, it is possible to select the engine power as
the optimized characteristic instead of the speed of the ship. In this case
after determining the displacement and the principal dimensions it'is necessary
to calculate the maximum speed.
The basic characteristics of the problem of expansion of the function (1.1)
consist in the following.
1. In the general case the function (1.1) is algorithmic inasmuch as the relation
between the majority of the ship's characteristics cannot be expressed in
explicit analytical form.
2. The determination of the number of the ship's characteristics requires graphical
development of the design solutions connected with the general placement of the
machinery, equipment and accessory equipment and the placement of personnel and
also the shape of the hull and its architectural features. The greater part of
these problems cannot be formalized at the present time; therefore creative partic-
ipation of the designer is necessary.
3. The expansion of the function (1.1) usually requires adoption of compromise
solutions (partial optimization with respect to individual indexes) inasmuch as
insurance of individual qualities of the ship, as a rule, is connected with satis-
fying contradictory requirements, and consideration of the latter in the general
optimization model greatly complicates the problem.
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4. FYom the mathematical point of view the function (1.1) is given in implicit
form, for individual components of the vector Y can depend not only on the
components of the vectors X and 0, but also eaeh other. In particular, the masses
of some of the structural elements of the ship, its technical means and
accessory equipment depend on the displacement and the principal dimensions
which, in turn, depend on the above-indicated masses of individual elements.
For example, the engine power required to insure given maximum speed of the ship '
depends on the displacement and the principa2 dimensions of the ship at the same
time as the displacement and the principal dimensions themselves depend on the _
weight and volume of the power plant and, consequently, its power. Analogously,
the weights of the hull, certain systems and devices, the eQuipment of the com-
partments depend on displacement an3 directions of the ship, and the displacement
and dimensions dQpend on the above-indicated weights. Accordingly, the expansion
of the function (1.1) is connected with the necessity for compiling and solving
the system of equations given algorithmically in the general case.l-
5. The prob lem of the expansion of the function (1.1), as a rule, is undefined
in asmuch as the number of equations relating the individual components of the
vector Y to the components of the vectors % and 0 is usually less than the number
of desired characteristics (components of the vector Y). In particular, this
fact gives rise to the possibility of using compromise solutions mentioned above.
6. The expansion of the function (1.1) requires an iterative approach (successive
approximations) both because of the algorithmic assignment vf the corresponding
- equations and in connection with the possibility of compiling the complete system
of equations in which all requirements on the ship would be considered
sufficiently exactly. Therefore the satisfaction of a significant nt,mber of the
requirements is established by check calculations aiter developing tYie drawings
of individual structural elements of the ship and determining a number of its
elements.
The check calculations usually reveal defined lack of correspondence of the
obtained characteristics of the ship to the required characteristics. In order to
elimtnate these divergences the designer takes the corresponding measures after
which the check calculations are repeated. The process continues until all dis-
crepancies are eliminated (Figure 2.1).
lBy algorithmic we mean the giving of equations in which their individual parts
are defined by the algorithms for calculating the values of the corresponding
functions of the desired and given (fixed) arguments.
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i
( Pactier eoaoFianie:ueNnA H ocfaueiibia nFoehTHb;x x3pahrep11crux
(9) sopa6.7A - KOAf00HEHTbI dexropa .
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(11~ 6aox oueNxii BapHaxr nopa6nA - B 6.nox oueHhH (12)
~ 602BOF1 340eN71IDNOCTH ~10~TOPbI X, Y, e T@XHHKO-3HOHOMN1leC1iHX ~
I70Ka33TE.72I1
Figure 2.1. Schematic diagram of the solution of the problems of the
engineering design block during the AD process
_ Key:
1. Assignment requirements
2. Independent variable of the problem components of the vector X and their
ranges of variation
3. Formation of the initial data, restrictions and assumptions components of
the vector 0 4. Investigation of alternative concepts of the ccnstruction of the ship
5. Preliminary estimation of the possible range of values of the principal. ship-
building characteristics components of the vector Y
6. General and special requirements on the design of a ship of the given class
7. Design on the level of the engineering subsystems of the ship and their elements
8. Functionally compatible sets of discrete variables with respect to engineering
subsystems
9. Calculation of displacemeiit and the basic des3gn characteristics of the ship
components of the vector Y
16. Version of the ship vectors X, Y, 0
11. To the combat effectiveness evaluation block
12. To the.technical-economic indexes evaluation block
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�2.2. Three Basic Methods of Determining Dispiacement and Principal Dimensiona
of a Ship
At present, usually three basic methods of solving the problem conriected with
determining disptacement and the principal dimensions of a ship are isolated as
applied to AD: the graphical method, analytical and graphoanaly tical.
In the graphical method the drawings of the general placement of weapons,
accessory equipment, machinery, fittings, personnel, storage and other objects on
the ship are made first. The shape of the ship's hull is given graphically in the
form of a theoretical drawir.g. The degree of detail of the drawings depends on
the stage of development of the design. The design is guided by the general
qualitative arguments and some local indexes connected with ins uring various
properties of the ship. Oa the b asis of the qualitative nature of the above-
indicated arguments and local nature of the partial indexes, as a rule, the ship
corresponding to the developed drawings does not meet a number of the requirements
and conditions. This fact is established by a set of test_calculations which are
performed using the above-indicated drawings. For example, the weight of the ship
can fail to correspond to the floating volume. The trimming conditions, stability,
unsinkability, and so on may fail to be satisfied. The designer eliminates these
discrepancies by altering the dimensions and shape of the hull, the location of
the weapons, accessory equipment, technieal means, and so on. The corresponding
changes are introduced into the drawings, the test calculations are again performed,
" and the matching process continues.
The nomenclature of the check calculations depends on the stage of development
of the design, and it continues to expand as we go deeper into the developed stages.
Thus, in the earliest stages (conceptual design) the load, buoyancy, initial
- stabiliLy, trim, propulsive performance and cruising range are calculated. Then the
strength, unsinkability stability at large angles of inclination, steerability,
- roll, and so on are calculated. As a result of several anproximations it is
possib le to satisfy all of the requirements and obtain the desired characteristics -
(components of the vector Y). Here the number of approximations depends on the
experience of the designer. The intuition and experience of th e designer to a
significant degree also determine the optimalness of the obtained version of the
ship with respect to certain partial criteria, for example, minimum displacement.
It must be noted that the solution of the problem of determining the set of TTC of
a ship satisfying the given requirements does not always exist for limited ranges
of possible values of the TDP.
The highly tedious nature of the graphical method has led to the development of
methods of determining the elements of a designed ship b ased on analytical or
algorithmic representa.tion of the relations between the desired and the given
characteristics. These relations can have the form of equations, formulas,
_ algorithms and so on. In spite of the fact that-until recently it was not possib le
- to develoB a sufficiently accurate analytical representation of-many of the
relations connected with the graphical repres+entation of a ship, analytical
methods play an important role in the early stages of design when a laige fltutber of
versions are developed and it is necessary to lower the requirements on accuracy
of determining the desired characteristics. Here it is necess ary to consider
that the accuracy of the method of determining the developments of a ship in the
early design stages must correspond to the relatively low accuracy of the initial
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_ data for these stages. (The basic equations of the analytical method of
determining displacement and the principal dimensions of a ship will be considered
in �2.3.)
By the graphical analytical method frequently we mean the method which iG referred
_ to as graphical above. Here we must consider that in the graphical method
check calculations are made which are taken into account by the analytical part
of this method. However, this representation must b e recognized as not completely
accurate inasmuch as the basic problem determination of the displacemerLt and
the principal dimensions of the ship is solve6. graphically in the graphical
method.
More strictly, it is necessaxy to consider the graphoanalytical method to be the
method in which the ship's characteristics are determined by solving the
corresponding system of equations but graphical drawings of the ship and
individual structural assemb lies of it are used to compile these equations. At
the present time this method is the most typical of the typical AD stage.
Among the above-investigated meth ods the most exact is the graphical method
essentially the only one in the final design stages. At the same time, the
graphical method is the most lsb or-consuming. Even when a computer is used a
significant amount of time is s�pent on developing the drawings, preparing the
required initial data based on thtw and input of this data to the computer. There-
fore one of the prospective areas has :ome to be automation of the graphical opera-
tions and input of the required initial nzta from the drawings to tne computer for
performing' the corresponding calculations. T'he accuraey of the development of the
versions of the ship is improved as a result of broader use of graphical procedures..
As for the analytical method, it retains its value as the first approximation and
as a method of recalculating the ship's characteristics with few versions of them.
As is known, the analytical method is connected with using a large number of
initial data in accordance with the previously constructed and designed prototypes;
therefore usually it is impossib le to determine the elements of ships with new
structural solutions by the purely analytical method. Actually, when compiling
the equations of the analytical method for submarines it is necessary (see �2.3)
- to knaw the weight of a unit volume of the pressure hull as a function of such
parameters as the maximum depth of submersion, the type of hull material, the
material strength characteristics, and so on. In solving this problem the methods
of similarity statistics are used which permit the prob lem to be solved on the
basis of data on greviously constructed and designed hulls. However, if the hull
of the designed submarine has essentially new structural features, this approach
leads to a large error and can turn out to be unacceptab le. In this case it is
first necessary to develop a new hull design, determine the mass characteristics
of it as a function of the above-indicated variables and then use this function in
the equations of the analytical method. However, wi`th this approach the designer
is already using the graphoanaly tical method.
_ As an example of the structural:features of the pressure hu11s of submarines it
is possible to present the so-calle-3 "figure eight" design which was used in the
vicinities of compartments II and Yv on the XXI series German submarines built
at the end of World War II. The design, which resembled a figure eight in
transverse cross section was two horizontal cylinders j oined by a common spacer
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- platform which made it possible to increase the usable volume of the storage
battery comFartments (battery tanks) on the small and medium displacemer'- diesel
electric submarines, th at is, with relatively small hull diameters. At the same
time, the application of the "figure eight," generally speaking, increased the
- weight of the hull (by comparison with a hull made up of one circular cylinder)
as a result of installation of the spacer platform and -fastenings in the transition
areas from the "figure eight" to the circular cylinder.
- The structural features of the hulls of.heavy surface ships include armor and under-
_ water protection (UWP) systems. In particular, the on-board UWP systems of battle-
ships, heavy cruisers and aircraft carriers built and designed bef ore World War II
and afteYward were distinguished by great variety of specific realization. The
].eading shipbuilding powers proposed and realized their awn UWP systems (Fig 2.2).
~ Although a detailed discussion of the advantages and disadvantages of the various
~ UWP systems is beyond the scope of ttiis book, let us make a few comments.
Among the foreign countries probably Germany achleved the greatest success in
improving the UWP before World War II. The tJWP systems oi -:.e German b attleships
of the "Bismarck" class and "N" design (not built) had com.,yaratively high effective-
ness. The underwater protection of the "Bismarck" class battleships had a two-
chamber design with one basic longitudinal armored bulkhead as far as possible
from the outer hull of the ship. This system (with one--basic longitudinal armored
bulkhead) was used on the Japanese "Yamato" class ships.l
On the heavy Soviet ships designed and laid up before World War II and afterward,
the UWP systems, just as a number of other basic elements, received significant
development and were greatly improved [15]. It is possible to state with certainty
that our sh ips were the best from the potnt of view of the UWP system. The design
of these ships was supported by broad experimental and theoretical research of
various UWP systemg considering their effectiveness, the mass-dimensional
characteristic, and so on.
'E. Ye. Gulyayev proposed underwater protection for a ship by installing a nimnber
of Iongitudinal watertight bulkheads connected to each other by framing which, as
they collapsed, gradually absorbed tne energy of the blast, in 1900. This prin-
ciple formed the basis for all of the UWP systems proposed and built later.
- The idea and the first structural proposal to introduce cylindrical bulkheads into
the UWP system were advanced by the prominent Russian shipb uilder I. G. Bubnov.
Later similar b ulkheads (cylindrical and curved) appeared in the Italian Pualese
design (1920) and in the American Hovgard design (1940).
The proposal to introduce an internal armored bulkhead into the UWP was made by
N. Ye. Kuteynikov on the basis of experience in the Russian-Japanese War.
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c)
Figure 2.2. UWP systems for surface ships: a-- Russian system
(Gulyayev type); b-- Italian "cylindrical" system; c-= German system
with filtration bulkhead
Figure 2.3. Architectural types of submarines
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Finally, it is necessary to mention the original architectural class of
submarine ("Minoga," "Bars") proposed in his time by I. G. Bubnov and called the
Russian class (Figure 2.3, a). On these single-hull submarines the main ballast
tanks were located only in the extremities outside the pressure hixll and in the
midsection of the superstructure (the deck tank). Subsequently the diesel sub-
marines were developed and improved, the double-hull architectural class with
reserve buoyancy became predominant. It insured the condition of "single-compart-
ment" surface unsinkability (Figure 2.3, b). The series XXI, XXIII and XXVI
German submarines built at the end of World War II constituted an exception.
Surface unsinkability was not provided for on these submarines.
The architectural class of modern American atomic submarines is very close to the
single-hull class. Surface unsinkability is not structurally provided for in these
submarines and, consequently, the possibilities of rescuing the submarine from
the submerged position when water gets into the pressure hull are extremely
limited (Figure 2.3, c). It can be stated that theore*_ically the architectural
class of American atomic submarines is very similar to the Russian class of
I. G. Bubnov (single hull with location of the main ballast tanks outside the
pressure hull), although this theoretical similarity is concealed by the absence
of a developed superstructure and external hu11 design of the submarines insuring
the best hydrodynamic qualities for underwater crulsing. (Let us note that the
outside lines and the developed superstructure on the Bubnov submarines were the
result of the necessity for insuring satisfactory seakeeping qualities on the
surface and placement of outside lattice torpedo tubes in the superstructure.)
After the loss of the "Thresher" (1963) and "Scorpion" (1968) atomic submarines
and also considering the fact that from 1900 to 1971, 21 American submarines had
been lost as a result of various accidents and 431 people had died, the prob lem of
improving the degree of insurance of unsinkability and rescue of personnel from
sunken submarines was subjected to further investigation and discussion in the
United States. In the opinion of the specialists, the navy could equip its nuclear-
powered submarines with all of the necessary devices insuring complete safety of
them, but then the submarines would not be suitab le for anything. This statement
indicates that significant reduction of the requirements on insuring invulnerabil-
ity of American itomic submarines was in the interests of improving other combat
characteristics.
Ending this brief investigation of the three b asic methods of determining the
engineering characteristics of a designed ship from the point of view of their
use in analytical design, let us again emphasize that the purely analytical
method, as a rule, is connected with carryover of the majority of structural and
technical solutions realized on previously built and designed ships to the ship
being designed. In this sense the purely analytical method has highly-Iimited
possibilities, and when designing ships with new engineering solutions, sometimes
it turns out, in general, to be useless. Another serious-deficiency of the
purely analytical method, as was already noted previously, is the difficulty of
1'J. Gorz, POD"YEM ZATONWSHIKH KORABLEY [Raising Sunken Ships), Leningrad,
Sudostroyeniye, 1978.
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suff iciently accurate consideration within the framawork of this method of such
factors as the shape of the ship's hull and the placement of weapons, accessory
equipment, technical means and personnel on it and also the stability conditions,
unsinkability, trimming, inwlnerability and.so on connected with these factora.
�2.3. Equations of the Analytical Method of Ship Design
In th is section a study is made of the principles of compiling the basic equations
of the analytical method of ship design. The basic equations include the equations
of mass, buoyancy, volume, initial transverse stability and power (speed) of the
ship.
1. Mass Equation. The mass equation is an analytical expressian of the condition
of equality of the mass of tiie ship to the sum of the masses entering into the
load, that is, the sum of the masses of the component elements of the ship.
Inasmuch as, according to Archimedes law, the mass of a ship floating without
way must always be equal to the mass displacement, that is, the mass of water
disp laced by the ship, we have the equality
Drn = ~ mi' (2.1)
where Dm is the mass displacement, mi is the mass of the individual load
components.
Expression (2.1) is an equation on the basis of the fact that the individual
mass mi can depend on the displacement Dm.
The expansion of the mass equation consists in establishing the dependence of mi
on TTE and TDP of the ship (the components of the vectors X, Y and 0) including
the displacement and the principal dimensions. The number of terms in the right-
hand side of equation (2.1) depends on the adopted load breakdown. The expedient
load breakdown can be determined from arguments of accuracy of calculating the
displacement [2].
Frequently in the initial design stages the mass equation is used expressed as a
function of displacement, that is, the relative mass in the righthand si:de of
equation (2.1) is considered dependent on the displacement Dm. Here equation
(2.1) is represented in the form
mc (Dm) T lllc~ (2.2)
r
where mz is the sum of the masses which do not depend on the displacement;.
For establishment of the dependence of the masses mi on displacement Dm and the
other investigated characteristics of the ship, the methods of similaritq and
mathematical statistics are used.. The application of these methods arises:from
the complexity of an exact mathematical description of the investigated relations
and also the absence of a number of initial data in the early design stages.
For these conditions recalculation of various characteristics and relations from
prototype ships already designed�-or has important significance. If various
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characteristics of the designed ship are recalculated from different prototypes,
the latter are called partial prototypes. By.the argumer.ts of calculation
- accuracy, as a rule, it is necessary to select a ship that has been developed no
leas than the contract design level as only a designed, but.not built prototype.
When using prototypes the similarity methods establish the dimensionless relations
(similarity criteria) serving for recalculation of the corresponding characteris-
tics, and the methods of mathematical statistics are used to find the values of
the parameters of the funetional relations between the masses mi and the components
of the vectors X, Y and 0, including the displacement D. as one of the components
of the vector Y(see �2.4, 2.5).
The simplest form of the mass equation expressed as a function of displacement is
the so-called three-term mass equation. When compiling this equation the masses
of the hull, the number of the ship's systems and devices, decks, hatch covers,
ladders, hull fittings, and so on are considered proportional to displacement to
the first power, that is, the mass of the ship. The masses of the power plant,
the fuel, water and oil reserves required for operation-.of the power plant are
taken proportional to the displacement to the 2/3 pawer, that is, the surface of
the loaded part of the ship's hull.l The masses of the weapons, accessory equip-
ment, payload (for transport ships), systems and devices servicing the weapons and
accessory equipment or providing for storage, loading and unloading the payload,
are considered independent of the displacement. Thus, all of the loads making up
the ship's load are divided into three groups, and the mass equation acquires t-he
form
AD~ BDm3 C= Dm, (2.3)
where ADm, BD~/3 are the sums of the masses proportional to Dm to the first power -
and to the 2/3 pawer; C is the sum of the masses which do not depend on the
displacement.
In some cases a term proportional to Dm/3, that is, the linear dimension of the
ship, fo r example, the length, is introduced into the mass equation. It is
possible to include the mass of the electric pawer lines, the pipes of certain
systems, and so on in this group For surface ships sometimes it is necessary to
isolate a term proportional to Dm~3 characterizing the mass of the long3tudinal
couplings of the ship's hull [2]. In general, as S. A. Bazilevskiy demonstrated
[2], the form of the mass equation (the necessity for considering the terms pro-
portional to different powers of DM) can be substantiated beginning with arguments
of calculation accuracy. The coefficients A, B and C depend on the TTE and TDP
of the ship (the vectors X, Y and 0), and, consequently, solving this equation,
we ob tain the displacement Dm as a function of the TTE and TDP.
1The mass displacement is proportional to the volume of the ship submerged in
the water, and the surface of the loaded part of the hull is proportional to
this volume to the 2/3 power.
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There are several procedures for solving equarion (2.3). For example, by
substitution of Dm73=z, this equation is.reduced to the cubic equation
- (1 - .4) z3 Bz2 C = 0, (2. 4)
which is solved by the numerical method or by the Cardan formulas.l The
numerical method of solving the direct equation (2.3) can also be used.
The simplest solution is the solution of equation (2.3) bq the methocl-of successive
approximations based on the knawn principle of a stationary point from~.functional
analysis.[60]. Let us denote the lefthand side of equation (2.3) by f(D) (for
simplicity of notation the index on Dm will be omitted). The function f(D) can be
considered as the special case of.the�operator given in the set of values D with
values again in this set. It is obvious that for C>0 the displacement found from
the mass equation is always greater than D-*-(B/(1-A))3. Consequently, the
desired disnlacement lies in the range (D*, It is easy to check that for
D; ID*, the values of the function f(D) are in the same interval. In
addition for D E(D*, oo) the inequality f' (D) - 0,
-0,013 9- 0,01 Ozl 1,16z.: a 0,
0,200 - zl U, 0,200 - z., a 0,
-0,106 0,669z, 0,'?79z.. :>0,
0,201 O,OI Ozi - 1,16zz :>0.
The purpose function of the problem expressed in terms of zl and z2 has the form
f=0.306+0.331z1+0.721z2. The graphical solution of the problem is illustrated in
Figure 2.4 where the restriction z3;10 is not shown in the figure, for all points
of the square defined by the conditions Ocz1;0.200 and 0;z2;0.200 satisfy the
indicated restriction on z3. The admissible region of values of zl and z2 is
crosshatched in Figure 2.4, and the crosshatching on the individual lines indicates
the part of the plane in which the corresponding restrictions (inequalities)
are satisfied.
The conditions of constancy of the purpose function f define the family of
straight lines, two of which (for f=0.406 and f=0.356) are shown in the figure.
The minimum value of f in the admissible region of values of zl and z2 is
achieved at the point with the coordinates zl opt=0.152; z2 oPt=0.013; and the
values of zg, z4 and f corresponding to this point will be z3 opt-0.200;
z4 oPt=O; fopt=0.356. .
Returning to the initial variables, tae find (dL)opt=0.052; (dB)oPt=-0.087;
(dT)oPt=0.10; (dH)aPt=-0.10.
Thus, in the investigated example with an increase in relative mass of the constant
loads by 10% it is necessary to increase the length by 5.2% and the draught of
the ship by 10%, to decrease the breadth by 8.7%, and the hull height by 10%.
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Here the increase in displacement will be minimal and equal to "6.5X1.
s ,
6
Figure 2.4. Graphical solution of the problem (2.28)
2. Buoyancy Equation. When investigating the mass equation expressed as a func-
tion of the principal dimensions, the expression
Dm = pBLBT (2.29)
was used which can be considered as the buoyaney equation. For this reason, the
above-presented different forms of the mass equation expressed as a function of
the principal dimensions can be interpreted as a result of the joint investigation
of the mass and buoyancy equations. The buoyancy equation in differential form
is written in the form
dDm a"` d8 -f - dL B"` dB ~"'dT . (2.30)
The b uoyancy equation expresses the fact that the mass of the water in the body of
the submerged part of the ship's hull must be equal to the mass of the ship, that
is, its mass displacement.
In the general case the necessity for using the buoyancy equation is connected
with the fact that the masses of the individual load items can depend on different
components of the floating volume of the ship, where these components themselves
depend on the unknown load items and displacement. Let us explain this fact as
applied to submarines-:.
As will be demonstrated in the follawing section, under certain conditions the
mass of the pressure hull of the submarine is considered proportional to its
volume, which is a b asic, but not unique component part of the constant floating
volume the normal displacement: Along with the normal displacement Dm the
In the given case inasmuch as a decrease in the hull height is
permitted.
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volume of the pressure hull V,rk can be considered as the desired unknawn. Knawing
Virk is highly significant inasmuch as the basic technical means, accessory equip-
ment and personnel are placed in the pressure hull.
Isolating the mass of the pressure hull in the three-term mass equation expressed
as a function of the normal displacement, we obtain the equation
9rtKV.K + A'DM -I- BD'ml' -I- C = Dm, (2.31)
where q~k is the mass of a unit volume of the pressure hull; A'Dm is the sum of
the masses proportional to the normal displacement.
Very frequently the value of Virk is cansidered proportional to the displacement:
VrtK = ~11K P'^ 1 (2.32)
where to,,K is a dimensionless constant called the relative volume of the pressure
hull.
In this case the equation (2.31) is reduced to the ordinary three-term mass equa-
tion, and expression (2.32) will be the simplest form of the buoyancy equation.
Aftex substitution of expression (2.32) in equation (2.31) we obtain
- r4mcmnK /{,`I pm BD~i' -i- C = Dm� (2 . 33)
~ P i
With this rep resentation the equality of mu to the given value can also be
considered as one of the simplest forms of the buoyancy equation.
In the more general case the b uoyancy equation for a submarine can be represented
in the form
PVnK -f- P~ Vi = Dm, (2.34)
t
where EVi is the total impermeable volume with the exception of the pressure hull.
i
The equations (2.31) and (2.34) can be considered as a system with respect to the
two unknowns Dm and Vwk. After substitution of the value of V7k from (2.34) in
equation (2.31) we obtain
( in-K ~ A') Dm -'r BDm' C- QnK ~ V i= Dm~
. i
This equation is already the result of joint investigation of the mass and
buoyancy equations. After its solution with respect to Dm the value of Vwk is
J found by the formula
Dm _ ~
VnK � V i.
P ~
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3. Volume Equation. The displacement and the principgl dimensions of the ship
found by joint solution of the mass and buoyancy equations cannot in the general
case be considered final inasmuch as the eorresponding volume of the hull can be
insufficient for placement of the weapons, accessory equipment, technical means
and personnel. The most exact testing of the conditions of the general plan is
realized by the graphical method by compiling the corresponding general arrangement
plans. One of the approximate analytical methods of considering the general
arrangement conditions is compiling and solving the volume equation proposed by
V. L. Pozdyunin [38].
This equation expresses the fact that the volume of Che ship's hull and super-
structures (for surf ace ships) must be equal to the sum of the volumes required
for placement of the weapons, accessory equipment, technical means and personnel,
and for transport vessels, the hauled cargo and passengers. Thus, the volume
equation has the general form
i I
K I ZC', N - N- Vr,
r
(2.35)
where Vk is the voltune of the ship's hull with respect to the upper deck; wH is
the volume of the superstructures; Vi are the volumes required-for placement
of technical means, weapons, accessory equipment and personnel.
The exp ansion of equation (2.35) consists in estab lishing the dependence of the
left and righthand sides on the hull volume, principal dimensions, displacement
and other investigated characteristics. In order to compile the volume equation
the same methods are used as for the mass equation, that is, the methods of
similarity and statistics. In the first approximation it is possible to propose
that part of the volumes is proportional to the displacement, part is proportional
to the hull volume Vk, and part does not depend on the displacement or the hull
volume.
The volumes Vi required for placement of the paaer plant, fuel, oil and water,
just as the masses of these components are expressed in the first approxima:Jon
using the admiralty formula
~ - Umax
i 3y = m3Y C
,
. ~
QT t'9h
~'T = - Rp Di+ , (2.36) PT CtC' 3K
� ~
QM ~'SF
R'1"D'j�''
ti' = p~~ Ct0 3K ,
~
98 t'3K Rn!�'
~~n =
F's C1iK
where q*T, q* M, Q* are the specific consimption of the fuel, oil and water
respectively, tons~(kilawatt-hour); tu3y is the specific volume of the power
plant, m3/kilowatt; DV is-t'Lie volumetric displacement, m3; PT, pM, PB are the
specific mass of the fuel, oil and water. (The specific mass pB of water used
for operation of the power plant can differ from the specific mass of seawater
for which the ship's displacement is determined.)
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If the hull volume and displacement are represented in the form Vk=dV LBH,
DV=dLBT, where H is the hull height of the ship; 8v-is the coefficient of full-
ness of displacement with respect to the upper.deck, then
VK - aS 7 Dv.
Considering this relation and the above-presented expressions for V3y, y7' v"
and VB, we obtain the equation (2.35) in more expanded form:
b~ T(1 Dv = Av, S T Dv ;,4,�.:D~. '
.
`~mar 9i 2K '(2.37)
; n tv,r Dv Cv,
(-'W ;K !'t
� i-P. m. e where w=w H/Vk is the relative volume of the superstructures; AV1 is the coefficient
of volumes proportional to Vk; AV2 is the coefficient of volumes proportional
to Dp; CV is the sum of the volumes not dependent on Vk and DV.
The displacement DV found from the volume equation must be compared with the dis-
placement Dm obtained by the method of joint solution of the mass and buoyancy
equations. If Dm3pDV, the final displacement of the ship will be Dm. In this
case, from the buoy ancy conditions the volume of the ship's hull will be greater
than the volume required by the general plan conditions or equal to it. For
DmpDV the hull volume
remains "free," equal to (1/p)(Dd-pDV), can be used for improvement of some of the
ship's characteristics connected with the necessity for increasing the volumes
without a significant increase in mass. (For example, there is a possibility of
disposition of additional personnel, certain forms of stores, improvement of the
habitability conditions, and so on.)
For submarines the volume equation is represented in three-term form (entirely
analogous to the three-term mass equation):
- ..~~,p~, g~D~!~ C,,.= D~�,
where AV, BV, CV are the corresponding coefficients.
(2.38)
It is possib le to consider the volume equations expressed as a function of the
- principal diinensions analogously to how this was done for the mass equation.
The volume equation is appreciably more approximate by comparison with the mass
equation. First, it does not take into account the specific geametric configura-
tion of the arranged objects and numerous, basically unformalized requ3:rements
on their individual and mutual arrangements, and it operates only with their
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volumes. Secandly, finding the coefficients of the volune equation by process-
ing the characteristics of prototypes is.very difficult as a result of complexity
of isolating the volume not connected with placement conditions--and caused by
requirements on buoyancy, stability,.other seakeeping qualities, trim, and so on.
~ Q) b ) ,
4 A='q p A=A
ti
~ Ti� ~ � I �o
A ,q
� C -
~ ~ . .
Figure 2.5. Diagrams of the placement of torpedo tubes and spare
- torpedoes on submarines
The use of the volume equation, just as other equations of the analytical design
method is connected with carrying over the structural solution used on prototypes
to the designed ship; therefore when designing ships with new structural solutions
compiling the volume equation requires graphical developments in the general case
of at least basic new structural assemblies.
For example, it is knawn [10] that on foreign submarines two arrangements of the
torpedo tubes are used: in the baw, horizontal and parallel to the diametral
plane, with cutting of the foYward bulkhead of the pressure hull (Figure 2.5, a)
and in the midse^tion along the length of the ship, horizontally at an angle to
the diametral plane, with cutting of the pressure hull performed in this area in
the form of a"circular" cone (Figure 2.5, b).
The first arrangement is traditional; it has been used in practice on all sub-
marines since World War I. The second arrangement is realized on the series XXVI
German submarinesl built at the end of World War II, azd at the present time it is
used on the modern American atomic submarines [10]. Each of these arrangements
are characterized by defined advantages and disadvantages, the discussion of which
is beyond the scope of this book. From the point of view of the problem investi-
gated here of compiling the volume equation it is important to note that for the
two indicated arrangements of the torpedo tubes (considering placement of the
spare torpedoes)the volumetric characteristics, for example, the volume per spare
torpedo, turn out to be different.
It is clear that the designer having prototypes only for the first arrangement of
the torpedo tubes at his disposal and deciding to use-the second arrangement,
cannot obtain sufficiently exact initial data for compiling the volume equation
without graphical development. In this case it-is necessary first to develop the

_ lOn the series XXVI submarines the torpedo tubes were aimed aft in accordance
with the tactic of "diving" under the target.
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structural assembly of the placement of the torpedo tubes and spare torpedoes
graphically (for different numbers of tubes and spare torpedoes), to define the
corresponding volumetric characteristics on-the basis of this development and then
use them when compiling the volwne equation.
It must be noted that in some cases, even with graphical method of design, first
the most important original structural assembly is developed, and then the entire
ship is developed. This approach has come to be called "assembly design."
Now let us propose that the designer has all of the data required for compiling
the volume equation. In this case the cause of occurrence of errors in determining
the volumetric displacement can be the above-mentioned f ailure to consider the
configuration of the placed ob3ects and the requifements on their individual and
mutual arrangement on the ship. For example, when designing submarines armed
with ballistic missiles (submarine missile carriers), it is highly desirable (by
stability and propulsive performance arguments) to select the diameter of the
pressure hull in the vicinity of the missile compartment so as to insure minimum
possible protrusions of the missile tubes beyond the pressure hull considering
restrictions with respect to draught, technological possibilities, and so on.
This structural solution is applicab le,-in partic lar, on the American nuclear-
powered submarine missile carriers [10]. The diameter of the pressure hull
selected in this way frequently becomes defining for all other compartments, that
is, the minimum required diameters of the other compartments turn out to be
smaller than the missile compartment diameter.
Selecting the diameter and length (considering the number of missile tubes) of
the missile compartment, the designer proceeds with laying out the other compart-
ments, in particular, the primary control station.(PCS).compartment located
forward of tI.e missile compartment. The minimum required length of Che PCS
conpartment in some cases is determined by the conditions of placement of the
telescopic devices (Periscopes, radio and radio technical antennas, and so on).
For the adopted length and diameter equal to the diameter of the missile compart-
ment, the volume of the PCS compartment can turn out to be extremely large by
comparison with the volume required for the placement of equipment and facilities
- which must be located in the PCS compartment. As a result, "free" space appears
in this compartment. The elimination of this "free" space by decreasing the
diazneter of the PCS compartment is undesirable inasmuch as a sharp change in
diameter leads to an increase in hull weight, volume of the interside space, and
so on. Theoretically the designer can try to fill the "free" space with equipment
and facilities from other compartments, but restrictions on the individual and
mutual arrangement of the objects on the ship crnne into play. For example, it is
_ undesirab le to put main power plant-elements in the PCS compartment, and so on [10].
The above-indicated peculiarities leading to the appearance of "free" spaces when
composing the general arrangement of the weapons, accebsory equipment, technical
means and personnel on the ship are not taken into account by the volume equation
inasmuch as it considers only the sum of the volumes required for placement of
each of the objects individually without considering their configuration and
requirements on arrangement by compartments.
Analogous examples can also be presented for surface ships. However, here the
problem of considering the volumes required by the placement conditions is not
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so urgent as for submarines inasmuch as by the conditions insuring unsinkability
of surf ace ships it is necessary to have a comparatively large (usually more than
100%) reserve buoyancy which can be iised for combat supplies on hand, elemenCs o�
Sccessory equipment, technical means and personnel. The volume equation has
more significance for transport ships. Sometimes the so-called-capacity equation
is used for transport ships instead of the volume equation [35]. "
The author has consid.:~red it necessary to present a speeial discussion of the
above-indicated peculiarities of compiling and using the volume equation so as to
emphasize once again the importance of graphical development in the process of
designing ships, including during analytical design.
4. Initial Transverse Stability Equation. The initial transverse stability of a
ship is characterized by the metacentric height
It .=zc+.o-zc,
(2.39)
where zC, zG are the y-coordinates of the center of buayancy and center of gravity;
p is the transverse metacenCric radius.
The value of zG is determined using the general arrangement plans, and the values
of zC and p, using the lines plan. The expansion of equation (2.39) consists in
establishing the analytical relations of the variables�entering into it as a func-
tion of the principal dimensions of the ship, the lines coefficients and other
characteristics.
In order to find the value of zC in the initial design stages when the lines plan
is unavailab le, analytical representations are used for the curve of waterplane
areas. As applied to surface ships these are most frequently parabolic curves.
The expression for zC is represented in general.form:
ZcT,
(2.40)
where 01(6/a) is a function of the ratio d/a. In reference [35] the following
expressions are presented for 01(d/a) obtained on the basis of processing the -
statistical data by V. V. Ashik
'Ti ( u ) - 0,858 - 0,370 a ,
V. G. Vlasov
and L. M. Nogid
Tl (a)= 0,372 + 0,168 a, 14 ff1`al 2 \d 1",
The approximate analytical expression for the transverse metacentric radi,us p
can be obtained under the condition of analytical assignment of the shape of
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the construction waterline. For surface ships frequently the parabolic.representa-
tion of the waterline shape is used. The general expression for p is assumed in -
the form _
o b ) e' C2..41,
_ =4'z(a T
, a , .
where, according to A. P. Van der Fleet [1, 351,
Tn ( Z i OG~ = k s , k= I l.4't0 11.7�
If we express the y-axis of the centgr- of gravity zG in fractions of a hull
height,theri considering (2.40) and (2.41) we obtain the fornrula for h in the
form
ItTal T -'H' (2.42)
where C is a statistical coefficient.
Formula (2.42) is also written in the follawing equivalent forms:
1 a a H T
e-`~'~al e Ta~T -t T B' (2.43)
s
h b , /b \e
T=~i(a)T~'sl aI al T: T.
Expressions (2.42) or (2.43) can be considered as the equations with respect to
the principal dimensions B, H and T. The stability equation must be considered
jointly with the equations of masses, volumes and buoyancy. If the displacement
and the principal dimensions of the ship are defined in advance, then using
expressions (2.42) or (2.43), it is possible to find the metacentric height.
When considering the stability cociditions in the initial design stages it is
necessary to consider the possible simplifications of the problem connected with
the peculiarities of the ship's hull shape, the general arrangement system, and so -
on. Let us i llustrate this principle in the example of submarines.
Figure 2.6. Mutual arrangement of the transverse metacenter and center
of gravity of a submarine with hull in the shape of a solid of revolution
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For atomic submarines of the majority of traditional classes and types, the shape
mf the outer hull is very close to a solid of revolution [10]. 'With this hull
shape, for any draught, including underwater, the metacenter is approximately on
the longitudinal axis of symmetry of the hull. In this case the insurance of
initial stability reduces to the placement of the basic.loads for which the center
of gravity of the ship will be belaa the longitudinal axis of symmetry of the hull
(Figure 2.6). Here it is necessary to consider that the' center of gravity of the
hull of investigated shape can be considered approximately on the axis-of rotation
(the axis of symmetry). Considering the above-indicated peculiarities, the
lnitial'�stab3,lity of a submarine both'-on the surface and under water basically is
determined by the values (masses) and arrangement with respect to height of only
the follawing loads (Figure 2.7): the pawer plant (MPP), the missile and torpedo
weaponry (MW and TW), and the solid ballast (SB).
For fixed diaplacement D=const the metacentric height will be
h-_ m3y Az;y mTO ~zTO mlPn AzPo - rnT;, Az,a (2.44)
D '
Key: 1. power plant; 2. torpedoes; 3. missile; 4. missiles; 5. solid ballast
where mMpp, , mSB, mlm are the masses of the pawer plant, the torpedo weaponry,
the solid ball~ast and one missile tube with missile; OzpP, AzTW, Az~ , AzsB
are the distances of the centers of gravity of the corresponding loads from the
axis of symmetry of the hull; n is the number of missile tubes; D is the normal
displacement.
)
c
Key:
Figure 2.7. Diagram of the arrangement of b asic loads determining the
transverse stability of a missile submarine
1. main power plant
2. AzMW
3' AZSB
4. Az
5. soTW
lid ballast
6. Az.,PP
ic
Figure 2.8. Diagram of the arrangement of rocket tubes for Ltube 1,
where S' =B ~w e= Bw and AB/2 is the width of the interside space.
For fixed values of Az.MpP, AzTW and AzgB the maximum admissible number of tubes is
reached under the condition of the minimum value of AzMW. From the formulas
(2.46) it follows that the minimum of AzMW is reached for B'=1 when the length
of the tube is approximately equal to the hull diameter. Figure 2.9 shaws the
values of 2 Az /Ltube as a function of B' for a=0, 0.5 and 1.0. Tn practice the
value of (AzSB~increases with an increase in the hull diameter, but inasmuch as
usually mlmn>B, the conclusion of the expediency of insuring the equality
BIZLtube is ma ntained also considering this fact.
2r1
~
~
0
B'
Figure 2.9. Dependence of 2 OzMW/Ltube on B'
Key: 1. 2(Ozj4W/Ltube)
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In the investigated example the stability conditions can be taken into.account
directly in the mass equation. Let us propose that the stability will always be
insured with an increase in the mass.of the . solid ballast. 'From the formula
(2..44) it follows (for AzTW.O)' that
Dh m9y A.z3Y mlp nAzpo
mr6 OZT6
or considering AzSB Y* (X, rh),
~
- where y*(X, ,M) is the deterministic value of rj under the condition that ,JV ships
- with TTE characterized by the vector X are in operation.
In accordance with the expression (3_3), the effectiveness index will be defined
by the expression 3 (X, .M) = u(y*(X, r~l).
~y1 b (41
i
I
- ~
p y 0 !lo !l
-~i
c)
U(y1
r ~
I I
I ~
~ i
J yo y 4 .~o .'i r 
Figu-re 3.1. 3ome typical utility functions
Let us consider some special cases of effectiveness indexes follvwing from the
~
general form. Here let us limir ourselves only to a uniform fleet, since for a
dissimilar fleet the corresnonding expressions ;are obtained analogously.
l. Let u(y)=y for all y, that is, u(y) be a linear function (Figure 3.1, a) for
which the effect of the operation increases uniformly as the achieved value of n
increases. From expression (3.3) it follows that
9 (X, M) = J y dF (y/X, rY') = En (X, ,M), (3.5)
y
wh^re En(',Y, X) is the mathematical expectation of the value of 71 with the dis-
tribution function F (y/X, ,M).
Thus, the effectiveness index ic zhe mathem2tical expectation of the random
variable n-- tlie parame*::r determining the outcome of the operation.
Inasmuch as for y-*-- the linear function u(y) is not valid, for this utility func-
tion the effectiveness index in the form of (3.3) does not always exist. For
example, for the random variables distributed by Pascal's law there is no
- mathematical expectation.
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2. Let u(y) have the form of a Heuiside unit function or step function (Figure
3.1, b). Then
p f or y< yo,
u(y)1 for yayo,
where yo is a given value.
- For this function u(y) for solution of the stated problem.it is necessary to
insure that the value of Tj no less than yo is achieved (exceeding this value does
not give an additional effect). Substituting the expression far u(y) in (3.3),
we obtain
3(x, .nn)= J dF (y/X, .1r) = p lTj:~.,yo'X. A'}.
y>y. (3.6)
Consequently, here the effectiveness index will be Lhe prob ability of the event
n3y0, and the value of P{n>,y0} is the probability of solving the stated problem.
. Let us note th at the two indicated cases were investigated by A. N. Kolmogorov
[23], and at the present time they are the most frequently used in practical
research.
3. Let the function u(y) have the form shown in Figure 3.1, c. The analytical
expression for this fi.uictfon called the linear function with saturation will be
. Q f or' y< 0,
u(y) = y,'yo for 0< y> - 3 I/ff1 :;P-0 be satisfied.
The condition Eil - 31/Y-11 3 0 defines the corresponding minimum value of ./Y' .
For the formulas (3.15) we have
1 - h.od -f- ( m )n
.M :;0- 9 kom
and for the formulas (3.16) and (3.17), respectively,
.M ~ 9 1 - koxP .
komP
Concluding this general survey of inethods of calculating thR effectiveness indexes,
let us note some peculiarities of constructing the operative-tac:tical and the
corresponding mathematical models as applied to the problems of analyLical design
of ships.
1. The process of constructing the mathematical model of estimating the
- effectiveness of a ship in the AD stages can be provisionally divided into two
steps: construction of the model containing mmplex operative-tactical
characteristics (handling, intensity of detection, range of detection means, target
indication, communication; probability.of detection and destruction, and so on);
construction of the model containing explicitly the TTE and TDP of the ship by
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expansion of the relations for the above-indicated complex characteristics as
functions of TTE and TDP
2. The individual TTE and TDP of a ship are manifested to a different degree at
different hierar.chicallevels of investigation of the combat operations. For
example, the sea endurance and cruising range are purely operative TTE. In the
maj ority of cases it is also possible to include the combat supplies on hand in
this graup. At the same time the speed of the ships is exhibited simultaneously
on the operative level (deploymenc in the combat zones) and the tactical level
(attacking, avoiding the enemy weapons, after salvo maneuvering). The
characteristics of steerability and maneuverability are among the purely tactical
qualities. The construction of models which arE critical with respect to the
entire set of TTE and TDP is very difficult and, as a rule, leads to in practice
unexecutable algorithms.
In the models of different hierarcbie&1_1euels, different mathematical methods are
used. Thus, on the operative level a more important role is played by the
analytical methods, and on the tactical level, frequently it is necessary to use
the method of statistical simulation. It is highly significant that many of the
tactical characteristics are weakly manifested in the models on the operative level.
The most typical methods of overcoming the indicated difficulties are the follaw-
ing: selection of the series of TTE and TDP by optimizing them on the basis of
the investigation of special tactical situations and statistical simulation of
the special tactical situations with subsequent approximation and use of the
analytical relations for the individual tactical characteristics as a function of
the TTE and TDP of the ship in models of a higher hierarchical level. (The
second of the indicated approaches sometimes is called the principle of general-
ized information use [511).
3. When developing the algorithms for estimating the effectiveness of various
versions of the ship very frequently it is necessary to optimize the procedures
for using the forces and means. For example, it is npcessary to optimize the
speed of the submarines during target search. In the problems of estimating the
effectiveness of using weapons it is necessary to optimize the firing method as a
function of accuracy of the target indication means, the weapons dispersion
characteristics, and so on.l
Thus, in the model designed for optimizing the TTE and the TDP of a ship, and this
is the primary goal of AD, "internal" opttmization problems appear. This compli-
cates the algorithm, the more so in that in the general case for different
versions of the ship the optimal tactical operating procedures are differPnt.
- Theoretically it is necessary to perform the above-indicated "internal" optimiza-
tion in each step of the movement toward the optimum with respect to the TTE
lIn the general case the optimization of the methods of using forces and means
must be two-way, that is , it is carried out both for our awn forces and for the
forces of the eneiny. This leads to the necessity for solving the corresponding
gamE problems. There is an applied discussion of the game theory as applied to
naval thematics in the book by V. G. Suzdal' TEORI`IA IGR DLYA FLOTA [Game Theory
- for the Fleet], Moscow, Voyenizdat, 1976.
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- and the TDP. However, this method is frequently impossible from the computational
point of view, and the majority of tactical procedures (except the inost signifi-
cant ones) must be rigidly fixed for all versions of the ship in the AD models.
4. From the general point of view, the TTE and the TDP of a ship can be optimized
on the basis of simulating an entire war as a whole and even postwar problems.
The construction of such a model, as a rule, is impossible. Only a single typical
operation or sequence of typical operations is simulated, as a result of which it
is necessary to make the achieved effect commensurate with the losses of ships.
It is only admissible not to consider the losses for ships meant to be used one
time. At the same time, none of the ships from the classes or subclasses can be
considered as ships meant for one-time use.
Thus, in practice, it i.s always necessary to consider at leaet two indexes: the
achieved effect and the losses, which in time lead to the optimization problems
considering several effectiveness indexes. As will be demonstrated in Chapter 5,
at the present time such problems are not subject to strict formalization.
93.3. Mathematical Model of Estimating the Effectiveness of Independent Combat
Operations of Ships of Side A Against Ships of Side B
Let us assume that targets -(ships of side B) are dist_ibuted in an unlimited part
of the ocean. The destrucCion of these ships is the mission of the ships of
side A: The number of ships of side B is considered unlimited. The ships on both
sides operate individually, independently of each other. (?his system can be
characteristic for certain types of combat operations of submarines:) The ships
of side A search for the ships of side B. The search process is considered
Poisson [19].
After detectic+n there is a duel between the ships which lasts until one of the
sides is destroyed. The duration of the duels will be neglected by comparison
with the total time the ships of side A are in the combat zone. The mutual
destruction of both sides in one duel is considered impossible (see �3. 4.
The consumption of the combat supplies on hand in each duel, according to the
quasiregularity principle, is taken equal'-tbrits mathematical expectation until
destruction of the target.
The process of combat operations of each ship of side A on one voyage halts
either on expiration of a given time (the residual sea endurance Pcombat) or
after a given quantity or combat supplies on hand (the supplies allocated for
operations in the zone) are expended or as a result of destruction by the enemy.
Under the assumptions that have been made, the functioning of each ship of side A
can be described by a Markov process (Figure 3.5).
Let us introduce the following notation: y the intensity of detections of
ships on side B by ships on side A; Pdes A" the probability of destruction of
- a ship of side A in one duel; Pdes B-- the same for a ship of side B; M is the
niunber of ships of side B which can be destroyed by one ship of side A with
respect tn the quantity of cambat supplies on hand. The value of M is equal to
the total quantity of cmnbat supplies on hand allocated for operations in the zone
divided by the mean consimption of combat supplies in one duel.
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A ship of side A functions in the stgtes with the numbers 0, 1, M 1,
destroying 0, 1, M-1 ships of side B, respectively. In the state M a ship
of side A stops combat operations as a result of expending the combat supplies
on hand, destroying M ships of side B at this ~ time. In the states -.M+l, p:+2, , 2M
a ship of s ide A was destroyed by the enemy after destruction of 0, 1, 2,..., M-1
ships of side B, respectively.
As the parameter defining the outcome of the cambat operations of the ship _
of side A, it is possible to take the random number of destroyed targets on one
a voyage (without considering counteraction in transit to the combat zone and back).
Let us designate this value by nl. The distribution function of this value is
completely defined by the probabilities
~ p 0 po (T aA) + p,H+l (T 6.z) ,
P{ ill = l~= pl(T s,a) -F- p:H+s 76A) ~
ci>
P ~11= k 1= Pa (T 6A) + p,u+k+, (T aA), k= 0, 1, ...,.M - 1,
P { 711= M } = P. (Taa),
Key: (1) combat
(3.19)
where the values entering into the righthand side are the probabilities that the
introduced Markov process will be in the corresponding states at 'the time
t=T comb at ,
'
YDYM6 vpYN! YpYN./ YpYNb \ YPyn.6
~ ~ r � � � M
YprN~ YP(?) PYK~
A. ~ e,~. ~ � � � 1M
Figure 3.5. Graph of the process of functioning of a ship on side A.
Key: (1) des A
(2) des B
The probability that a ship of side A would be lost in the combat zone is defined
by the formula
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M -
pYe. 6A pMIk (T6A) �
Key: 1. des. combat
or
. M
py.. an =1 - ~ Pk (Ta~�
The probability distribution (3.19) makes it possible to find any numerical
- characteristics of the random variable nl. For example, the mathematical
expectation nl will be
M-I
E%= M PM (T 6~ -I- kF.~+ k[ pk MoA) + pM+x+i (T an)] �
(3.20).
The mathematical expect3tion of the number of targets destroyed by one ship of
side A before its awn deatruction conaidering counteraction of the enemy on the
crossings, can be found t+y the formula
(1 P~,,, i)
E~i = E~1i 1-0 - Py�� aA) (1 - Py� 1) (1-
where n is a random number of targets destroyed by one ship of side A before its
own destruction on a series of voyages; Pdes 1, Pdes 2aTe the probabilities of
destruction of a ship of side A, respectively, in transit to the comb?t_ zone
and back.
1 The value of 1- (1-Py,,,bA) (1-Py�1) (1-Py�=) is the mathematical expectation of
the number of voyages of a ship of side A before its destruction.
If a study is.made of the comb at operations oP.a fleet of.M ships of side A, then the average number of targets destroyed by the fleet before its awn
destruction will be Erl,e = E'1X� Here the value of En depends on the vector X
characterizing the TTE of the ships of side A.
Thus, for complete description of the investigated model it is necessaryto have
the expressions for the probabilities Pk(Tcombat) and pM+k(Tcombat), k=0, 1,...., M
_ or the algorithm for calculating them.
Differential Equations for Finding the Probabilities Pk(Tcambad and
PM+k(Tcombat)� In accordance with the theory of Markov random processes the
probability of the states of the process characterized by the graph in Fig 3.5
satisfies~:the system of ordinary difierential equations
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dPo (t) .
dl _Ypo (t)~
dPt (t) = Y pYH bPA,_1 (t) - YPk M, k = 1, . . . ,
dt M - 1,
dP,y (t) _ Yprx spM_l (t)'
~3. 21)
di
dPm.a (t) _ 'YPYB Apk-1 k = 1, . . . ~ M,
dt
where t is the current time of the process.
When writing equations (3.21) it is considered that pdes A+ Pde B�19 for by
a'ssumption the duel lasts until one of the sides is destroyed an~ the mutual
destruction of the sides is possible.1 `
As the initial conditions for the system (3.21) it is exepdient to take
po (0) = 1, pk (0) _ 0, p.tir+k (0) = 0, k = 1, 2, . . M. (3.22)
After introduction of the dimensionless time T=yt the system (3.21) assunes the
form
dPo (T) - -Po

dPk (-s) dz - _ (3.23)
dv - pyd bpll pk (i), k - 1, . . . , iYt - 1
. dpM (T) _
dt Y pH 6P~H-1 lZ~r
�
dPM+k (Z)
dz _ pyu A Pk.1(Z), k = 1, . . . , M.
The probabiLities Pk(t) and PM+k(t)s k=0, 1, M, found from the system (3.21)
for t=T ~b are equal to the corresponding probabilities found from the
system ~5.215 for T-yTcombat'
In the system (3..23) let us replace the variables, setting Pk(T)=e Tl1k(T),
k=0, l, 2M, where uk(T) are new variables. As a result, we obtain the
system of equations with respect to uk(T):
. duo (i) _ 0
dz '
du~=t) = pYe buk_1(Z), k= 1, M- 1, (3.24)
dum (z) _ uM (T) PYR 6UM-1 (Z),
ds
duM.k (T) _ r
dT - uM+k (T) pYN Auk_i lZ)r k = 1, . . M
under the initial conditions
. UO (0) =1 ~ uk (0) = 4, uM+x (0) = 0, k - 1, . . . , M.
(3.25)
1This is an example of fixing the method of operations o� sides in a given model.
In the more general statement it is possible to find fihe opfimal duration of the
duel taking inCo account the fact.-that in a number of cases the targets can
stop the duel by their discretion.
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Performing the corresponding calculations, it is possible to see that the
solution of (3.24) under the conditions (3.25) has the form
. uo (T) -1, . .
k uk.(~) - Pkl~ Tk' k= 1, M-
M-1
tiy 'c
u/y (T) =pyMxb I -C i ~ vl 8
~:a0
k-1
Tv
uM+k (T) = PyH qPyNlb 1- e"t ~ v, et, k= 1, A~.
,_o
Returning to the variables Pk and dimensionless time t, we obtain the solution of
tlie system (3.21) under the conditions (3.22):
pk !t` _ (Yt)k pyd be k = 0, . . . , M - 1, .
~ ) k!
M-1
M (3.26)
PM (t) _~'yH b 1- e
vao . _
k-1
_ p pk_1 1 -yr (vt)y k _ 1 , . . . , M.
PM+k (t)yH A ys 6 - e E yl '
v~o
The probabilities Pk(Tcombadsk-0, 1, 2M corresponding to the point in time
t=Tcombat, Which permit the effectiveness itidexes Enj,,- and En to be found, itre
found by formulas (3.26).
Asymptotic Solutions of the System of Equations (3.21). In a number of cases the
asymptotic solutions of the system of equations (3.21) for YTcombat''�� and M''��
are of interest. 'Phis corresponds to unlimited residual sea enflurancz and'
unlimited combat supplies on hand.
1� YTcombat', M is finite. From formula (3.26) it follows that:
Pk=limPk(t)=0, k.-4, M-1,
yt- ao
PN -1im P� (t) = Py b, ,
yf4w
pM+k =1im PM:k (t) = pyx APyRlg~ k= l, M.
y!- o0
Hence, we obtain
M-1
lim E~l = MPy b+ pyx A F~ kPvH 6=
yT6Ai m ka0 .
= MP^' pYx 6 M 1 M Pyx 6 (3.27)
yx b-1- PYA A~ 1- Pyx 6~-~M - 1) Pyx 6= pyH A~ 1- p/yyx 6) �
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When deriving.formulas (3.27) the condition pdes A+ Pdes Bm 1 and the formula
for the sum of a decrease in arithmetic-geometric progression are-~used j431:
M-1 /
k_ (M - 1) QM 4 11 -.9M-,)
~kq 1-q + (l-q)' ' Q\1.
Just as should be expected, the conditian
y.x b_
2bi 5f i M k-i ~N , 1- Pni
~ Pk = P'rx b- p~N ~ prN s= pyH s ~ pvx a P 1
_ k=0 k=1 YH b
is satisfied (let us remember�that here Pdes A71-pdes B)'
It is also easy to obtain the formulas
M
yT
Ii111 PYx 6A - pYH A Fi P,k,;,'6 = 1- Py b,
- aa
lim pYx 6 (1 hf 1-PyH 1
vT6A'0� PYe A\ - PyH 6~ j- Pye b~~- PyH lPy� 2)
M*(PVK PYPYInS
YPvN.e rPyHb
. . . k . . .
YPYKA
`
. . . k ~
Figure 3.6. Graph of the process of functioning of a ship of
side A with unlimited combat supplies on hand
If in formulas (3.27)-(3.29) we also to the limit for M+-, then
lim Eli - - P'"' 6
p ,
yH A
lim Ev1= py. 6(1 - Py.H~),
rr a
limPY�6R=
(YTaa 00, /YI oo).
(3.28)
(3.29)
(3.30)
If the counteraction in transit to the combat zone is absent (Pdes 1^0), we
have
lim E-q = Py" s, (3. 31)
yT6Ai�� Yx A
M-*m.
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that.fs, the average number of.destroyed targets is equal to the probabilfty
ratio of desCrdction of the ships of the'sides in one duel.
2� M'"�,. YTcombat are infinite. In this case the functioning of the ship of
- side A can be described by a Markov process with--an infittite- denwnerable number
of states (Figure 3.6). Here in the-states 0, 1, k, ....a ship of side A
' functions, destroying 0, 1, k, targets, respectively. In-the states-
01, 11, k', `a- ship of side A is destroyed, destroying 0, 1, k,
targets, respectively, before this.
The distribution of nl is given by the prob abilities
- p 1711 = 01 = po (T6A) -f- po (T6a),
P {,ii = 1 } = pi (T6A) -I- I'i (Tan)-
p(T1, =k) =pk(Twi)+ pk7an), k= 0, 1,
where pk(Tcombad' k=01 11 are the probabilities of the states 0, 1, k,...
for t=Tcombat; P*k(TcBmbat). k=0, 1, are the probabilities of the atates
0' 1' k'
> > >
Considering the system of equations similar-to-(3.21) and the initial conditions
PD (0)=1, P*p (0)=0, Pk(0)=P*k(0)=0, k=1, 2, for the probabilities
Pk(t) and P*k(t), it is possible to obtain the formulas
(k'!k py� be ''r , k = 0, 1, . . .
k
y .
pk (t) = PYm ApY~ 6 1- e yt v~; , k= 0, 1, .
vao
In the given case for the value of lim En we have
M4_
i
lim E,l _~i k IPk (Ta,a) -I- Pk (Trsn)I�
1y-m k=0
After the corresponding calculations, it is possible to obtain
limEli,= ~,YHA jl - e p~AyT6Al. (3.32)
M- o. J
It is natural that for YTcombat-'�� the formula (3.32) coincides with the
corresponding formula (3.30).
Analogously, it is possible to find
lim PYH a.a = 1- e PY" AyT6n
M-ae �
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Hence, it follaws that without considering counteraction in transit
lim En = pro 6 (3.33)
,
JN-w yti A
qT~-r
that is, we again obtain formula (3.31).
Concluding Remarks. In order that the above-described model of cambat operationa
be able to be used during AD of the ships of side A, it is neceasary to express
_ the operative- tact i cal values of Y, Pdes A, Pdes B and M in terms of the TTE of
the ships of both sides and also the characteristics of their detection, target
indication means and weapons. This problem requires specific definition of class
and type of ships of the sides and also the type and characteristics of their
weapons and accessory equipment. Here it is necessary also to fix the method of
using the weapons (for example, firing a salvo of -n fixed number of combat means
at the present location of the target, and so on). Here, as has already been
' pointed out above, the problems of "internal" optimdzation, for example, of the
speed of the ships'during search (if the range of the detection-means depends on
the speed of the ship), the number of combat means in one salvo, the firing
method, and so on, arise.
93.4. Markov Mathematical Models of Two-Way Duels
When construczing the mathematical mndels of the combat opexations of different
weapon carriers, including ships, frequently it is necessary to coneider the
results of two-way duels with the enemy. In the general case the models of duel
situations are sgecial modules of more general models of the comb at operations.
One of the examples of such a general model was investigated in �3.3. Models of
duel situations can also have independent significance when estimating measures
Lo improve the characteristics of the weapons (for example, firing rate), target
indicating and laying means, protection of the ships, and so on.
Otie of the most effective analytical methods:-:of describing duel situations is the
use of the theory of Markov random processes with a discrete number of states.
Belaw the two simplest models of two-way duels between single targets are con-
sidered as examples.
The duel always presupposes the action taken by the enemies on each other by a
weapon. Therefore in models of duels the effectiveness of the application of the
weapon must be taken into account. The problems of eatimating the effectiveness
of the firing and also the effectiveness of appJ.ying a weapon are an independent
division of operations research [12, 19].
The estimate of the firing effectiveness is based on knawledge of two groups of
characteristics: the characteristics deflning the probability that the weaponl
will.-hit a vulnerable part of the target, and the characteristics of the

lIn this section the terms "weapons" and."combat means" are identical (see
Chapter l), that is, bq a weapon we.meari combat means.
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damaging effect of the weapon, including vulnerability of the target with
respect to various types of ammunition.
For example, when estimating the effectiveness of firing at a single target [12],
the A. N. Kolmogorov formula is used:
n
prf)~n) _ ~ pn, mG
Key: l. damage
(3.34)
where Pdamage(n) is the probability of damaging the target in n rounds fired;
pm n is the probability of m hits for n rounds; G(m) is the probability of
damaging the target in m hits. The function G(m) characterizing the vulnerability
of the target is called the provisional damage law (PDL).
depend on the method and the accuraey of firing (the
The probabilities Pn m
scattering character~stics of the shells, the accuraey of target indication, and '
so on), the damaged space of the target determined by its,dimensions, and for
proximity weapons, the charactertstics of the physical field to which the sighting
system of the weapon and the proximity fuse react. The probabilities Pn, m also
depend on the nature of the maneuvering of the target, that is, on its maneuvering
characteristics, the use of ineans of counteracting the weapon, and so on.
The function G(m) has the simplest form in the absence of tize effect of accumula-
tion of loss by the target where eacxt shell, hitting the damaged space, damages
the target independently of the others. In this case G(m)al-(1-1/w)m, where
w is the average number of required hits `o damage the target.
If in this case
ability of hits,
[23]
Key: 1. damage
the individual rounds are independent in the sense of the prob-
then from the A. N. Kolmogorov formula we have the expression
n
P,ap(ll) - l-n11- ~1
(1) r_l `
where Pi is the probability of a hit on the i-th round.
(3.35)
Let us note that establishment of the PDL is a very complex problem, especially
for large surface ships having relatively high invulnerability. This arises,
first of all, from the complexity and ambiguity (depending on the nature of the
solved problem) of determining the event equivalent to damaging the ship.
Secondly, when using nonnuclear weapons large surface ships have the effect of
accumulation of loss. At the present time the basic method of calculating the
PDL is the method of statistical simulation in which it is necessary to knaw the
location of the basic damage of all targets on the ship and also their protection
characteristics. As a rule, this does not permit the use of the method of
statistical simulation in the AD stage; therefore the problem of developing the
analytical methods of calculating the PDL is highly urgent.
,
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By investigating duel situations, the problem is complicated also by the fact
that during the process of a duel the ship can go from state to state in which
Che possibility of the enemy`s influencing it changes, as it is hit by the shells.
In the examples of duels presented belaw, the PDL assumes the simplest form, and
it is called the unit PDL. This law is given by the expression
Ofor m=0,
G(m) 1 tor m:~-- 1.
For the unit PDL, the target is damaged on the first hit.�- This situation occurs
for targets with loa invulnerability or high-pawer shells.
We sliall also assume that during the process of the duels the targets exchange
independent rounds (salvos) with respect to the probability of hitting the target.
Duel with Successive Exchange of Salvos. Considering the assumptions made above,
a two-way duel with successive exchange of salvos can be ddscribed by a Markov
process with a discrete number of states and discrete time.
The graph of the states and the probabilities of the transitions in one salvo
(step) of the process is illustrated in Figure 3.7. In states 1 and 2 both
targets are functioning; in state 1 the next salvo is by target A, and in
state 2, target B. In state 3 target B is destroyed; in state 4, target A. The
probability of damaging target B by one salvo of target A is pA, and target A
by one salvo of target B, pB respectively.
Figure 3.7. Graph of the process of a two-way duel with successive
exchange of salvos
Let us denote by Pk(n), k=1, 4, the prob abilities of the states of the process
after n salvos, and by P(n), the column vector of these probabilities. The duel
is fully chatacterized by the initial vector P(0) and the matri x of probabilities
of transitions in one salvo which in the given case has the form
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0 4a Pa 0
9s 0 0 Pe
Q1 - 0 0 1 0
0 0 0 1
The vectors�P(n), n=1, 2, satisfy the recurrent expression P(n)=Q1*P(n-1,
n=1, 2, from which we have
p(n) _ (Qi)0 p(0), (3.36)
where Qi is the n-th degree of the matrix Q1, * is the transposition sign.
In the general case the prob abilities pA and pg can vary during the duel process,
for example, as a result of variation of the dis tance between targets or as a
result of the accumulation of damage in the targets. Then we have
~ IIQI;)�P (0), (3.37)
P (n) _
where Qli is the matrix of transition probabilities for the i-th s alvo.
The most typical initial conditions are
P* (0) =(a, 1- a, 0, 0),
(3.38)
where O,a; l is the probability that the duel begins with a salvo of target A.
The probability a frequently is identified with the pr~bability of leading in
detection. This is possible, for example, when firing range by the weapon is
, greater than the detection range.
Let us propose that the ranges of the detection means of the targo-ts are indepen-
dent random normally distributed variab les with mathematical expectation dA and
dB and dispersions v2A and cs2B. Under these conditions th.e probability a is
calculated .by the formula -dS
6 ~
a= 2 1 ; ~ Y2dA V aA+ a2
where
x
2 je-"di.
Yn In some cases it is possible to consider that aA/dA=a B/dB-k-const, that is, the
_ ratio of the mean square deviation of the detection range to its mathematical
- expectation. is a constant which is identical for both� -nargets. Under this
assumption a depends only on dA/dg and lc '(Figure 3.8).
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Figure 3.8. Probability of leading in detection a as a function of
the values of dA/dB and k
The foranulas (3.36) or (3.37) permit calculation of the probabilities of destruc-
tion of each of the targets after any given number of salvos. Sometimes the
outcome of the duel is of intereat for an infinite number of salvos, that is,
vector p(00)'._ J;m p(n).It is obvious that in such a duel P1(-)=P2(-)=0, and
naw
Pg(-) and P4(-) are the prob abilities of destruction of each of the sides.
For calculation of P(O�) we shall consider a duel with successive exchange of
salvos, but with continuous time, assuming that the time between successive
salvos is distributed by exponential law with parameter X. The value of 1/a
is the average time b etween successive salvos. In such a duel, the prob abilities
of states Pi(t), i=1, 4, as a function of time t satisfy the system of
3ifferential equations - dPl (t) _ - n.Pl (t) -f- XqsPz (t),
dt
dP2 (t) _ XqAp, (f) - %P: (t), (3.39)
dt
dF9 _ XPApi (t)+
dt
XP6p2 (t)
dt
under the initial conditions
Pl (0) = a, P2 (0) =1 - a, P, (0) - Py (0) = 0. (3.40)
It is obvious that we have the equality P(oo) = lim P(n) = lim p(t), where p(t)
- rt-s~ t-�
is the probability vector Pi(t), i=1, 4.
Using the integral Laplace transformation
~ .
p; (S) Pr (t) e :r dt, i = 1, . . . , 4
it is possible to reduce the system of differential":equations (3.39) to a system
of linear algebraic equaCions with respect to Pi(s):
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SP1(S) - Pl (0) _ - XPi (S) + X9A (S),
5P2 (s) - P$ (0) _ ;-qApi (S) - XPa (S) ~
~ sP, (s) - P, (0) _ XPaPI (S)+
sP4 (s) - P4 (0) _ xP6pa (S) �
- Solving this system, we find
P (s) - (5 + Pl (0) X96Pa (0)
i (S + 19qAqs '
~4qP1(0) (s X),ps (O)
pz ~5) - (S + X)s - xaqAq6 ,
pa ~S) = 1 pa (0) ~Pp (s Pl (U) a96Pz (U)
s . s (s + )L)2 -X2qAqs
1 ~'P5 ~9.tPi (0) (S A) Pa (0)
~S) p~ t0) S (S y 42 -112qAqs .
Using� the property
lim Pj (t) =1im sP, (s), i- 1, 4,
I-Go f +0
considering the initial conditions (3.40), we obtain the formulas for the
components of the desired vector 'p (oo)- -
� P, (00) = 0,
Pq(oo) 0, (3.41)
Pq PqQ6
ps = a qAqs (1 - a) 1_ QAQ6 '
. P~ ~�O) = a 1 P QA a) 1 pQ .
- Ay6 Aa6
In �3.3 it was demonstrated
average number of targets B
If we set pA=pB=p, that*is,
the sides identical, we obt
that the ratio P' = p''" 6) is the
P. ( ) yxA �O
destroyed by one target A before its awn destruction.
consider the effectiveness of the weapon of each of
ain
~
Pe (00) _ 1 -0 - (X) P (3.42)
P4 (oo) � 1 - ap
where
p9 ~ _ 1 - ( I - a) P and Pi (oc) - 1 ap .
2-P 2-P
Graphically, this function is represented in Figure 3.9.
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The problem can arise of the required relation b etween pA, pB and a insuring
predominance of one of the sides, for example, side A, whicli conesponds to the
condition P3(06)?P4(-). Considering the formulas (3.41), thib condition assumes
tho- form
PA>P6I 1 .
+(2�-1)P6 (3.43)
The condition (3.43) is presented graphically in Figure 3.10 in which the regions
of values of PA and pB' satisfying (3.43) are located aboue the corresponding
curves for each of the fixed values of a. From Figure 3.10 3t is obvious, in
particular, that for p$>0.5 reliable lead at the beginning of the duel by the
target B over the target A(a=0) cannot be campensated for by any increase in
effectiveness of the weapon of side A, that is, the prob ability pA.
The investigated model of a duel makes it possib le to consider the lead-not only
in the first salvo, but also in several first salvos. Let, for example, the
target A lead the target B with a probability one in the first np salvos. Then
wh en calculating F'(.OOT it is only necessary to vary the initial conditions,
setting P1(0)=(1-pA)"0-1, p3(0)=1-(1-pA)n0-1, P2(0)=P4(0)=0. - Let us set a=1
and calculate the salvos for target A from the value of no-1.
N.6 (1)
pyN. A
5,0
4,6
9, L
2,L
1,0
p=1,0
0
8
,
0,6
0 � 0
'f
0, 4
,
0,6
0,8
1, 0
-
0 0,2 0,4 0,6 0,6 a
Figure 3.9. Ratio of the probabilities of destroying the targets as
a function of the values of a and p in a duel of unlimited duration
with successive exchange of salvos (n--, pA=pB p)�
Key:
1' Pdes B (�D)/Pdes A(-)
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AA
I,0
O
~
tt p
4.~
0,8
0,6
0,4
0
2
,
v u,z u,.r Up u,o ps
Figure 3.10. Regions of values of the probabilities pA and pB insuring
predominance of side A in the duel of unlim3.ted duration with
successive exchange of salvos
Tlze more general problem can also be considered where the number of leading
salvos is a random variable with distribution function F(np). If Pi(n, n0),
i=1, 4 are the probabilities of states for the number of leading salvos
equal to n0, then the total probabilities defined by the expressions
p; (n) = f pc (n, no)df (no), t= 1, 4.
no
Duel with Random Exchange of Salvos and Continuous Time. Let us assume that each
of the targets A and B produces salvos independently of each other at random points
in time where the time between the next salvos of each of the sides is distributed
by an exponential law with the parameters, aA(t) and 7lg (t) which can, generally
speaking, depend on the time reckoned from the beginning of the duel. The time of
movement of the weapons of both targets toward the target will be neglected by
comparison with the time between the next salvos. Parameters J1A:(t) and ag(t)
are the firing rates (the number of salvos per unit time) of targets A and B,
respectively. The probabilities of damaging the targets by one salvo, just as
before, will be designated by pA(t) and pB(t), considering their possible
dependence on time.
The investigated duel can be described by a Markov process (Figure 3.11). The
targets A and B are destroyed in states 1 and 2, reapectively. In state 3 both
targets function. Let us denote by. aA(t) and X B(t) the intensities of the flaws of so-called
damaging rounds:
%iq PA Wi T6(t) P6 (t) �
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1- b(t) / \!-A(t)
f ) t 2
Figure 3.11. Graph of the process of a two-way duel with random
exchange of salvos
The weapon reserve of each of the sides will be coneidered unlimited.
The probability Pi (t) , i=1, 2, 3, of the states of the process in the given case
satisfies the system of differenttal equations
dPl (i) _
dt ks (t) pa (t),
dP= (t) _ XA (t) ps (t)' (3.44)
dt
dPs (t) _ [i~A (t) -L ks ~t)l 1's (t)�
The solution of the system otd~equations (3.44) gives
. t t l
pi (t) = p, (0) pa (0) Ts (z) eCP - ~ I~A ~6 15~~ a51 di,
p, (t) = p2 (0) -f- p3 (0) jt ~.A (ti)exPt jI- j dz,
o
t
P3 (t) = pa (0) eCP - ~ [~.a M ?.s (`s)1 d~ ,
where Pi(0), i=1, 2, 3, are the initial values of the probabilities of atates.
In the case where aA(t)=71A and J1g(t)=J~B, that is, when the intensities of the
flows of damaging salvos do not depend on time, we obtain
PI ~t) = pl (0) -f- pa (0) 11A + 1s ` exp (X,a -I- X6) t] I,
Pz (t) = pa (0) T p9 (d) X.4 T- !A ~g (1 - exp -I- Ts) t) I ~ (3.45)
- P, (t) = p3(0) exP CXa -f- T6) tl �
_ For a duel of infinite duraCion (t-~-) from the formulas (3.45) we have
~s
p, (00) = pi (0) + ps (0) "A + X6 ~
pi (�O) = P, (0) -f- P, (0) _ XA .
l 'A + I6
Ps(oo)=0.
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On derivation of the formulas (3.45) and (3.46) it was proposed that directly
at the time of beginning of the duel (far f=0) the targets do not produce saZvos,
ettd therefore damage to the targets is impossible.
Using the initial conditions Pi(0)9 i=l, 2, 3, various versions~~of the beginriing
of the duel can be taken into account. [Hereafter it-is assumed everywhere that
the probabilities pA(t) and pB(t) and also. XA(t) and X B(t) do not depend on time.]
1. The duel begins with simultaneous exchange of salvos at the initial point in
time (t=0). In this case the damage at the time t=0 to one of the targets or
their mutual destruction is possible. The probabilities of destroying the targets
in such a duel are calculated by the formulas
pya a(t) = PaPs ~-~1 - paPs) Pi (t),
(3.47)
pya s~t) = P,tPs -f- (1 - P.aPe) Pz (t), where the probabilities P1(t) and PZ(t) are found from the solution of the system
of equations (3.44) with the initial conditions
Pl (0) = p~ P P6) + p2 (0) = Pl PqPg ) ' ps (0) _ ~ ~ lPa)~Ap6 P5) .
For t4- the formulas (3.47) assume the form
s
pyx A(~) = PE (1- P.4) (1- Ps) ~A+ Is~
pyx s (O�) = pa i- (1 - Pa) 0 - Ps) ~A + lb .
In the given case Pdes A(��) + pdes B(��)'1+PAPB>11 for the probabilities
pdes AH and Pdes B(��) take into account the possibility of mutual damabe of the
targets at the time t=0.
2. The duel begins with successive exchange of salvos at the initial point in
time t=0. Here the probabilities of destruction of the targets are found by
solution of equations (3.44) under the initial conditions
pl (0) =,aPs ~ 1- PA) Ps,
PZ (0) = aPa (1 = a) (1 - P6) AA'
- ps (0) = 1- PA - P6 -f- PAP6+
~ where a is the prob ability that at the time ta0 the salvo of target A will be
made first.
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After simple calculations for t-- we obtain
~s
pYN n (00) = p~ (C*) = P6 - aPAP6 + 0 - PA - As + PaPs) ~A
. Ps,,, s (�O) = Ps (00.5 (v>0) the inequality > ISo/si is satisfied
and consequently, the value of t is positive. If the examination of the region lasts a determinate time t, then the mathematical
expectation (E,Mnor) and the dispersion (Z,/Y'nor) of the number of destroyed ships
are equal .
Eti"nor = Y'n (I r' -fnor = .Mne l' (1 - e xr
Substituting the expression for t, we obtain
E.Mnor = 0 ~ ~~'ror = ~SI� (1 - ~S ) �
We again emphasize that the value of ,M� in the formula for the dispersion is
defined by the expression (3.56).1
Just as in all of the examples of the given chapter, for use of the above-
- described model during analytical design it is necessary to express the tactical
parameters a, d, v(in the special case dd~ e and Pdes.k) in terms of the TTE
- and the TDP of the ships performing the opera~ion of reconnaissance of the
region.
lIt is interesting that the mathematical expectation of the number of destroyed
ships does not depend on the *,olygnn f leet of ships
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CHAPTER 4. METHODS OF ESTIMATING ECONOMIC INDEXES
During analytical design the expenditures of various types of resources connected
with b uilding and maintaining ships are considered. Quantitatively, theae
expenditures are characterized by the service indexes which, together with the
effectiveness indexes, are used to construct:.the optimization criteria of the TTE
and the TDP of ships.
The nature of the service indexes is always varied. For example, if the displace-
ment and the principal dimensions of the ship have great signii'icance (the possi-
bility of building, maintaining, transporting), L'heri these values are considered
as service-indexes. The service indexes iuclude the design and building time, the
number of personnel, and so on. Just as the effectiveness indexes, the service-
indexes can be of a random nature. However, in contrast to the effectiveness
indexes where the random factors constitute the essence of the combat operations
and the functioning of the ship, the random nature of the service indexes b asically
is connected with unreliability of the initfal data and imperfection of the cal-
culation techniques. These prob lems have still been insufficiently investigated;
- therefore the service indexes at the present time in practice are always con-
sidered as determinate values. Here the researchers knaw that the determinate
values of the service indexes are only the mathematical expectations of the
corresponding random variables.. The above-indicated determinate approach is
uaderstood in the sense that when constructing the optimalness criteria, the
distrib ution functions of the service indexes are not used.
One of the most important forms of the service indexes is the economic index.
The cost of building and maintaining the ship is the generalized economic index.
In this chapter the methods of determining only the cost service indexes, in
particular, the cost of building and maintaining the ships in the fleet are con-
sidered.
The cost of building any product is made up of three basic parts: the cost of the
consumed production means, th-at is, the cost of raw materials, fuel, materials
and the fraction of the cost of the means of labor corresponding to their wear
during the production process; the cost of the personal use fund of the workers
performing the labor in the form of wages; the cost of the social use fund
connected with expansion of production, -.the maintenance of the state admi.nistra-
tive agencies, public health, education, and so on.
Inasmuch as the determination of the third component of the cost is very difficult,
the cost including only the first two cost components is used as the generalized
economic index. Thus, only the cost of building and iaaintaining ships wi1.l be
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considered although instead of the terin "complete cost" we shall use the term
"cost."
a The cost of building a ship in the general case includes the cost of construction
and the cost of the planning and design work. Hawever, considering the design
cost is connected with defined labor arising first of all from the inadequate
study of the relations between the TTE and the TDP and the cost of designing the
ship and, secondly, by the fact thgt the design cost must be distributed to the
entire series of ships of a given design being built--at the same time as in the
AD stage the volume of the series usually is unknowm. (The methods of determining
the ship design cos t are not considered in this book.)
The same general requirements can b e imposed on the service indexes as on the
effectiveness indexes, that is, criticalness with respect to the 1TE and the TDP,
calculatability and simplicity. Here criticalness also has important significance,
for in subsequent optimization it is necessary that the service indexes simul-
taneously be critical with respect to the entire set of TTE and TDP selected for
optimization.
54.1. Detercnining the Construction Cost of Ships
The cost of building a ship is determined by the sum of the expenditures with
respect to the following costing items: materials, intermediate products and
finished products; the wages of the production workers; the overhead (shop and
general plant); deliveries and operations by other contracting parties; other
direct expenditures (individual expenditures).
The expenditures on materials, wages, deliveries by other contracting parties
and also individual expenditures are direct expenses which pertain to each ship
being built. The shop and the general plant overhead are distributed to all of
the ships built proportionally to the wages. These are the indirect expenses.
There are [27] three basic methods of calculating the co8t of buildin$ a ship
corresponding to the various design stages: by the consolidated mass'-normatives
(for the conceptual design and the preliminary design); with respect to the
consolidated statistical normatives (for the engineering design); with respect
to the costing nor~tives (for the detailed design).
It is natural that in AD it is expedient to use the first of the above-indicated
methods. Here the initial data are the mass load items of the ship compiled by
the structural breakdawn groups. Inasmuch as in the engineering design b lock
(see Chapter 2) the relation is established for the individual load items as
a function of the TTE and the TDP of the ship, the method of calculating the cost
- by the consolidated mass indexes permits establishment of the relation between
the cost of b uilding the ship and its TTE and-.TDP. Let us remember that the
cost of the comb at supplies on hand (missiles, torpedoes, artillexy supplies)
and variable loads (the fuel and oil reserves, the ZIP[spare parts, tools and
accessories] and other equipment) is not considered. The cost of equipment is

1The mass normative (or index) is the ratio of the labor cons wnption of
manufacturing the product to its mass.
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taken into account in the cost of maintaining the ship, and the.necessity for
considering the cost of the combat supp lies on hand arises in cases where ships
with different types and forms of weapons are cansidered during the AD process,
that is, the type and form of weapon enter into the optimizable parameters.
In essence the method of calculating the construction cost by the consolidated
mass indexes is based on similarity arguments, in accordance with which single-
type structural assemb lies of the ships are-considered similar to each other,
and the coet of creating them, proportional to their mass. Just-as when
calculating the load elements, the similarity methods can be used jointly with
statistical methods.
In accordance with the costing items presented at the beginning of this section,
the cost of building the ship is represented in the form
S=(1 T k2) (spe T S. + SKn T SnP), (4.1)
where S is the cost of building the ahip; Spg, SI�I, SK7~, S,~ are the costs of the '
shipyard operations, materials, deliveries by other contracting agents and other
direct expenditures; k2 is the coefficient taking into account the trading over-
heads and planned deductions.
The determination of the labor consumption with respect to individual types of
operations can be used as the basis for calculating the values of SpB and S7t :
building the hull, installing accessory equipment, machinery and equipment at
the shipyard. The labor consumption of the operations is determined as a function
of the masses of the individual load items, the type of materials, and so on.
In order to estab lish the type of dependence of the labor consumption on the
masses, prototype data or the data from statistical processing of several proto-
types are used. Thus,
Spe = (1 + kl) ~ Sx4 rTt ~ (1 + kl) ~ SHa tft (mc), (4.2)
t=1 _ -
Key: 1. overhead
where Ti, i=1, n, is the labor consumptian with respect to individual
_ structural-technological assembliea; Ti=fi (mi), i=1, n, is the dependence
- of the labor consumption on the corresponding masses; Soverhead i, i=1, n,
is the cost of a unit of labor consumption (norm-hour) for individual assemblies;
kl is the overhead coefficient; n is the number of structural-technological
assemblies.
Frequently formula (4.2) is simplified by introducing the average cost of a norm-
hour with respect to all assemblies
" (4.3)
SPB = (1 -f- ki) SHa i ifr (mt)+
where soqerhead is the mean cost of a norm-hour for the iiefined builder.
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In order to find the functions fi(mi), as-a rule, statistical procesting of the
data with respect to several prototypes by the methods of regression analysis
is ueed. Here, usually linear models are coneidered, in accordance with which
the functions fi(mi) are represented in the form
fr (mr) = almr To1, i = 1, . . . , n,
(4.4)
where ai, Toi, i=1, n, are statistical coefficients which depend on the
TDP of the ship, for example, the peculiarities of the architecture and the
material of the hull, the type of pawer plant, and so on. These parameters can
be optimizab le discrete variables.
If the data from one prototype are used, then the regression line passes through
the origin of coordinates and the point corresponding to the prototype, then
T(nP)
Ti - mlOp) m:, 1 = 1, n,
where /n,�P), T~,np1, i= l, n, are the variables pertainfng to the prototypes;
Ti/mi=ti are the specific labor consumptions defined by the prototype, or the
statistical data of several prototypes.
When using the specific labor consumptions, the formula for SPg can be written
as follaws:
n
~
Sps kl) ~ Sxai ttmi �
r~t
(4.5)
The cost of other direct expenses S7rp usually is found by the statistical data as
a f unction of the total labor consumption n Ti, cansidering S,P proportional
~=1
to the total labor consumption. The cost of materials depends on the mass of the individual assemblies
n
SI,q=E sMimi, where sMi is the specific cost of materials with respect to the
i=1
i-th assembly (the cost of materials per ton).
In the AD stage the most complex is determination of the cost of the deliveries
by other contracting agents Skir which is a significant part of the total cost
of building ships. According to the data of [47], the deliveries and operations
by other contracting agents for civilian maritime ships amount to 25 to 35% of
the total cost of the ship. In warships this proportion increases as a result
of greater saturation with radio electronics and automation means, the presence
- of weapons use systems, the application of more complex power plants, and so on.
Some idea of the cost of deliveries and operations by other contracting agents
for warships can be obtained on the basis of the data on the American nuclear-
powered submarines (Tables 4.1, 4.2).
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Table 4.1
Cost of Shipyard Operations for tWe-American Nuclear-Pawered Submarines
(according to the.�data of [65]). . .
.

Torpedo'suBmAtiries
. .
" " ~ 'Missile 'submariries
Cost,
.
ffillioris of dollars
"Lip$ comti" "Sturgeon" "Los 'AriAeles" "Lafavefte"
' "Trident"
Total (construction)
178 81.3
233 127
780
Shipyard operations
80 44
83 22
285
The same, %
45 54.2
35 17.3
36.5
Table 4.2
Components
of the Construct3aa Cost
of American Nuclear-Powered
Submarines (according to the data. of � [1,0
.
1Y.
~ ~ � ~

Shipyard
Operations
and
Type of submarine
operations ~ .
. Materials deliveries
by other
.
.
. ddrittdaing
agerits
Torpedo , 43-46 14-15 40-42
Missile 33038 15-16 4&650
In the engineering and detailed design stages�the problem of determining the cost
of deliveries by other contracting agents does not come up for the ship designer
inasmuch as in these stages, as a rule, there are already technical specifications .
for the delivery of products by other contracting agents with the application of
the cost determined by the suppliars. During AD, when it is necessary to know tiie
cost as a function of a number of characteristics of the products of other
contracting agents, the problem is more complicated, for the products delivered
by other contracting agents are created by many branches of industry, each of
which has its own characteristic features arising from the specific nature of the
built p roducts and production facilities. Theae peculiarities are not always
known to the ship designers or they cannot be taken into account sufficiently
completely and with sufficiently high quality. Although the method of determining
the construction cost by the consolidated mass indexes�is generally accepted in
the metals industry, it usually cannot be used for radio electronic equipment,
navigation and communications system, information control systems, and so on.
Beginning with what has been said, the cost of the enumerated products and systems
is calculated by the data of the other contracting agents considering the
compositinn and the type of accessory equipment. With this approach the optimizable
(during the AD process) characteristics can turn out to be only the type and
quantitative composition of the accessory equipment, where the type is a
discrete variable, and the quantitative composition can.be approximately con-
sidered a continuous variable.
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For all the remaining products of other contracting agents (except the accessory
equipment), the ovezwhelming majority of which is made up of inechanical and
electrotechnical products, theoretically it is possibie to use the metliod of
calculating the cost by the consolidated mass indexes in terms of labor consump-
tion with respect to the known specific labor consumption of the structural-
technblogical assemblies of the corresponding praducts and also the data on the
cost of a norm-hour at the manufacturing plants. Obtaining the indicated data
is difficult as a result of the large nomenclature of products, their structural
differences and the large nwnber of manufacturing plants. Under these conditions
it is possib le to recommend the method of determining the cost directly'by the
mass indexes without intermediate determination of the labor consumption. With
this approach the cost (S'kff) of the products of other contracting agents in
the investigated group is defined by the formula
t
Sxn = 1:~sIm� (4.6)
where mi is the mass of the product of the J-th type of the other contracting
agents; sj is the specific cost (cost per ton of mass) of the j-th type product;
k-is the number of types of products of other contracting agents in the
investigated group.
The values of s. arE found by the data from prototypes considering the type and
the characterisiic features of the corresponding products. For example, the
specific cost of various types of power plants (nuclear steam turbine, non-
nuclear steam turbine, diesel, gas turbine, and so on) must be determined
separately. Here the type and layout of the power plant can be discrete optimiza-
ble variables. Inasmuch as the mass mJ depends on the characteristics of the
corresponding products and the ship as a whole, the function (4.6) determines the
relation of S'k7r to the TTE and the TDP of the ship.
Thus, the cost of building the ship is defined by the general expression
n n
S=( 1 ~ ks) t ~ k~) ~'SH4 i!, (mr) -i~ SMrmi I
n ~ (4.7)
~ krtP L I i(mt) -r L S/m; rt SsoopJ ~
Key: 1. accessory equipment = acc
where 5acc is the cost of the accessory equipment; k7rp is the coefficient
of other direct expenses.
Correspondingly, when using the average cost of a norm-hour and the specific
labor consumption we have
n n I
S = (1 -t- A~) [/Z ~ trm, ~ SNinti ?.~S,m; -L SB~PI ~ (4. 8)
where k-(1 -f- k,) SH4 T knp�
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The specific expansion of expressions �(4.7) or (4.8) as firactioas of TTE and
TDP.depends:�.on the type and structural characteristics of the' designed ship.
Expression (4.8) can be represented'in the form
~ n
s (1 -I- kz) [k ~~trmr J:,l&tmi -I- ~ S;m~ D (1 k$) Sewp, (4.9)
where D is the displacement of the ship; mi are the relative masses (mi-ni/D)
of the structural-technological assemblies.
For ships that are similar to each other, the values entering into the
coefficient in front of the displacement in formula (4.9) are approximately
identical. This makes it possible to propose another more approximate formula
for calculating the cost of buildiag the ship
S = spD + (1 - + kz) S.ooP,
(4.10)
where sD is the cost of all expEnses (in addition tio the cost of accessory equip-
ment) per ton of displacement.
For civilian ships the value of 3acc basically including the navigational and
comunications means is relatfvely law and, according to the data of
B. M. Smirnov [47], it amounts to 2-3X of the total cost of the ship. Therefore
formula (4.10) for civilian shipe can be represented in the form
S = soD.
(4.11)
The cost (in thousands of dollars) of a ton of standard displacement of U.S. Navy
ships according to the data of [10] is:
Submarines :
Diesel electric
12-14
Nuclear-powered torpedo submarines of the
"Skipj ack" type
16.5-21.5
"Thresher" type
13.5-16
Nuclear-pawered missile submarines
"George Washington" type _
16-20
"Lafayette" type
16-17
Trawlers
4-5
Destroyers, frigates
7-11
Aircraft carriers
3-4
Nuclear-pawered frigates
15-19
Nuclear-powered cruisers
23
Nuclear-powered aircraft carriers
5
It is possible to more precisely define formulas (4.10) and (4.11) somewhat by
dividing the load into two basie component parts: the.hull with equipment and
the. power plant. In this case, instead of (4.10) and '(!+.11) we obtain
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S = (SKOmKa + S3ym9y) D, (4.12)
S = (SKOmKO S9ym9y) D -F SawPl (4.13)
where ~~iKO, myy are the � relative masses of the hull with equipment and power
plants; Sho. S3y are the'specific cost per unit mass of the hull with equipment
and pacaer plant.l
�4.2. Determining the Construction Cost of a.Series of Shipa. Consideration
of the Size of the Series
In some cases it is necessary to consider the so-called "series factor" in the
calculations of determining the cost of buildirig a fleet of ships. This factor
is connected with a decrease in coastruction cost with an increase in the order
number of tlie ship in the series. As is knawn, with an� increase in the number of
ships in series the materials are used more efficiently, the labor consumption
goes dawn as a result of improvement of technology, introduction of specialization
and so on. These factors also lead to a decrease in cost of deliveries by other
contracting agents.
In the general case the cost S(A) of construction of N ships is equal to the
- sum of the cost of the individual ships:
S(,M) - ~sj, (4.14)
. rm, .
where Si is the cost of a ship with the i-th order number.
The dependence of the cost Si on the order number i is characte'rized by the follaw-
ing properties: far i=1, that is, for the prototype in the series, the cost is
maximal; for i-~- the values of Si approach some limit the cost of the series
ship. The exact establishment of the dependence of Si on i requires a detailed
analysis of the reduction in expenditures with respect to each of the component
parts of the total cost. In the initial design stages this analysis is difficult;
r therefore the statistical data with respect to the ship as a whole are used. The
following dependence of Si on i satisfying the above-indicated general properties
can b e proposed for stattstical analysis:
Sr =,S� (b " (4.15)
where S., a, b are statistically determined values.
From expression (4.15) it follaws that
S! 1 rb a \
Sl - a-~- h\ + 1 1'
-V. L. Pozdyunin [38] indicated the� -possibility of approximate determination of
the construction cost by formula (4.12).
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The last expression permits determination of the~-coefficients a and b on the
basis of statistical data on the values of Si/S1.
The algorithm for caleulating the cost of the ship uaua3.lp�is constructed as
applied to same fixed order number n. In this case, it is necessary in
formula (4.15) to set
Sn..
S. b a/n '
Thus, for the cost of building ,M ships, we obtain
S (.M) = aS.(p (X), (4.16)
where X
~Y') _ ~ 1 + bi b = b .
1 ' a
ra~
The function (Y) satisfiea- the required relations
q) ~t') = cp ~1 b, (4.17)
which permits quite sfmple calculation of (P(.M)for small values of X. . For
sufficiently large .M from (4.17) we obtain equation
d(; (.h�) 1
dX - b + 71
the solution of which gives
~ (A*) = bJY' + II7 J{� 1 E,
where e=0.577 is the Euler constant. The last formula facilitates the calculation
of (p (.M) for large .M.1
It is necessary to turn attention to the correctness of considering the series
factor on the procedural level inasmuch as the fleets of ships consfdered during
military and economic studies frequently are only provisional values and do not
have direct bearing on the actual volume of the construction series of the ships.
This situation usually occurs when optimi2ing the TTE and the TDP of the ship by
the criterion of minimum cost of the fleet solving a defined problem with given
level of effectiveness. Considering the increase in cost for the first ship of
the series becomes meaningless in this case, and it is necessary to take the cost
of the series ship in Qie calculation. The cost of building the fleet in this
case is a linear function of the number of ahips.
1For example, for b=1 this foanula gives an error of'3_2% for .M -3 , and for
,M - 5 the error does not exceed 4% for any values of'b. 129
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At the'same time using the criterion of maxLmum effectiveness for the given
budget for building azid maintaining the f leet-the series f actor muat be considered
if the budget corresponds to its actual value.
94.3. Determination of the Cost of Maintaining Ships
The cost of maintaining a ship in the fleet is made up of two parts: the cost of
operation and maintenance end the cost of building and maintaining the base
support means (the basing cost).1
The problems of determining the b asing cost are considered in this book. The
base support system in the general case contains shore and floating f acilities.
In order to determine the cost of building and maintaining the floating
facilities the same procedure can be used as for ships. Caleulating the cost of
shore facilities requires the application of special methods.
It is necessary to note the complexity of the problem of the basing cost distribu-
tion for individual ships or forces of ships. Similar difficulties arise also
when considering the cost of other types of shore and floating forces and means
of supporting the fleet operations such as comanunications, navigation, and so on.
For these reas-ons it is expedient to consider the cost only of that part of the
specific means which are needed to support the activity of the class and type of
ships considered in the AD process. For example, if ships with their awn detection
and target indication means are comp ared with ships using external sources nf
information, the cost of the latter means must be taken into account.
Analogously, when comparing ships with nuclear and nonnuclear power plants, the
cost of the special means of shore support of the ships with nuclear power plants
is taken into account.
The cost of operating and maiataining a ship is made up of the basic components
connected with repairs arid planned replacement of equipment; material and techni-
cal supply, including fuels and lubricants and also consummable training combat
supplies; maintenance of personnel.
The operating and maintenance cost usually pertains to ane year of service life
of the ship. For calculation of_it a time graph is constructed (see Figure 4.1)
for the use of the ship between successive deliveries of 3.t to the repair yard
(considering the duration of one yard repair).
For ships with nuclear pawer plants, periodic recharging of the reactive cores
must be taken into account, which can be combined with yard repair.
If the time parameters A, TM,ff and T ard repair-.are known, where A is the
sea endurance, TM,~ is the time betw~en voyages, Tyard repair is the yard repair
1The cost of maintaining a ship includes a number of other expenses (training of
personnel, maintenance of central administra`Cion and so on)., but the expenditures
on operation, maintenance and basing are primary and direct.
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_ time, then the mean annual operating maintenacUCe cost (S;) can be found by the
formula
. ~ s(l) ~ l ) S`wn�p + S9p
- S9 = s�cnnc -t- A- ~  (4.18)
R \ 1 T,,,, T,p
where R is the reserve of the basic ship~s equipment before the yard repair;
S(�1,;, is thP cost of ona repair between voyages; S,p) is the cost of one
yard repair; SN',n is the cost of material and technical supply for one voyage;
Snc is the average cost of maintaining one crewman per unit time (for a year);
n,,,c is the number of crew.
_ q TMn A . Tsr'n T L
JP
Figure 4.1. Time graph of the use of ahips between yard repairs
The parameters 7'MR, ,T,p and costs SKl~p, S~, S~r depend on the TTE and wro
the TDP of the ship.
Theoretically the cost of the annual operatfon and maintenance of a ship must
increase with time, at least for the subsequent periods��between successive yard
repairs. The reason for this is the physical wear of the michinery, equipment
and hull, which increases the frequeney and volume of repairs. In cases where
obtaining the indicated function is difficult, the annual operating and maintenance
cost can be assumed invariar_t for the entire service life of the ship.
In spite of the external simplicity of the formula (4.18), the expansion of the
relations for the variables entering into it as a function of the TTE and TDP of
the ship presents defined difficulties, especiallq in the initial design stages.
Accordingly, the annual operating and maintenance cost of the ships sometimes is
expressed as a fraction of the construction cost. For example, for civilian ships,
according to the data of [47], the depreciation deductions are taken equal to
5-6% of the construction cost, the cost of each current and medium repair, 3-3.5%,
res.pectively, and the expenditures on material and technical support (except
fuel), 0.5%. The indirect expenditures (general operation and administrative-
technical expenditures on steamship lines), are taken equal to 25% of the cost
of the crew maintenance according to the statistical data.
Thus, for approximate calculation it is possible to propose the formula
s3 = c,S,
(4.19)
where S is the construction cost; cs is the coefficient defined by the prototype
(or by several prototypes).
In �4.2.it was demonstrated that in a number'of cases the eonstruetion cost can
be approximately considered proportional to the ship displacement. Consequently,
_ when using formula (4.19) it is also poasible to consider the annual operating
and maintenance cost proportional to the displacement. F'or this reason, for a
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long time during the period of evolutionary development o� shipbuilding the
displace.ment was used as the economic service index. At the present time when
ships are equipped with expensive accessory systems, power plants and special
systems, the use of displacement as the economic service index is insuffisient
inasmuch as - a more detailed expansion of the dependenee of. the' cost of building
and maintaining the ship on its TTE and the TDP is reqnixed.
During the military-economic studies it is necessary to know the cost of
operating and maintaining a f leet of ships for the time of its actual operation
in the fleet. When determining this cost usually a functian is used that is
linear with respect to the number of ships ,M
S3 (X, T) = SBi (T) X,
(4.20)
where S, (1*, T) is the cost of operating and maintaining .M ships for the time T
(the time the ship is in the fleet); S31 (T) is the operating and maintenance cost
of one ship for the time T.
Let us consider the peculiarities of calculating 5_31 (T) beginning with the assump-
tion of invariability of the annual operating and maintenance cost 4 for the
entire service life of the ship.
Qne of the peculiarities �of determining the total operating and maintenance cost
of a ship for the entire time it is in the sea consists in the necessity for
reducing the expenditures at different times to a single point in time, for
example, the time of completion of construction, inasmuch as the expenditures
which will be made t years after the current pofnt in time must be decreased by
(l+a)t times, where a is the so-called discount rate or the normative investment
effectiveness coefficient [27). Introduction of th+e-discount rate is explained
by the fact that one ruble invested in the natianal economy today will bring a
profit after t years and become equiva].ent to (l+a) t rubles. Therefore the
expenditures of 1 ruble today are more aignificant than the expenditures of the
same ruble after t years.
Thus, under the condition of constancy in time of the value of S3 the operating
and maintenance cost of a ship for T years is calculated by the formula
T -
So (T) = s3 ~ 1a~~ = s3 a [1 - +a~T,� (4.21)
Formula (4.21) asstmmes that the annual expenditures pertain to the end of each year; therefore during the first year of operativn the reduced expenditures will
be not 83, , but s3 1+ a. . If the annual expenditures are reduced to the
beginning of the.year, then
r
I ' t + a ~ (4.22)
S3I (T lf ) - 53 + S3 a I 1 - (I L x)T -
For sufficiently small values o� a formulas (4.21) and (4.22) give close results.
In practice a does not exceed 0.08-0.10.
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It is characteristic that for T-m the formulas� .(4.21) and (4.22) give finite
values of s3 a and sy (1 �r a), a , respectively. Generally speaking, this is an
unexpected result.
For the fixed time T and sufficiently small values of a the formula (4.21) has
the form
S9, (T) 5,7',
(4.23)
inasmuch as for small a the approximate equality (l+a)-TZ1-aT is valid. In this
case the discount rate in practice is not considered.
For final determination of the operating-and maintenance cost of the ship it is
necessary to establish the length of time for which the azinual operating and
maintenance cost must be summed. For civiliati ships this period can be considered
to coincide with the service life of the ship, and the latter, in turn, is
determined by the nature of variation in time of the obsolescence and physical
wear of the ship. (Iri Chapter 6 some approaches to solving the problem of
substantiating the service life of the ships will be cansidered.)
The operating time of warships in the fleet ia equal to the aum of the op erating
time in peacetime and the lifetime (before destruction) during a war. Both of
these values are random variables which depend on many factors.
Let us consider one of the approaches to the determination of the operating time
of warships.
The random time T of operation of the ship can be represented in the form
T=TNB+'C$ Bp where Tl�B, TBB are the, random operating time of the ship during peace-
time and during wartime, respectively.
For a fixed service life Tservice the average operating time of the ship during
peacetime can be found by the formula
T,ja = TNeI' (Tc.-j) Tcn [ 1- F(Tcn )1, (4.24)
f 1.~
Key: 1. service
where F(t) is the time distribution function before the beginning of the war after
construction of the ship; T~~ is the provisional mathematical expectation of
the value of Te under the condition TM xiu() -x~u) ~
where xi0i) are componentis of the vector X~Oi).
Interesting problems with respect to the optimal location of the points X(Oi) in
the vicinity of the initial point X(0) occur when the calculation of the responses
of the purpose function 4D is connected with the appearance of random errors caused,
for example, by the presence of unalgorithmized operations of decision making by
the designer or specialists with respect to estimating the effectiveness and the
service indexes in the individual AD blocks. In this case errors will appear in
the grad en vector components. They can be influenced by the location of the
points Xt~i~ Thus, it is possible to demonstrate [33] that the arrangement of the
points X(Oi) .will not be optimal if they differ from the initial point X(O) by only
coordinate.
The problems of studying and optimizing various proces$es and systems, the internal
structure of which is {inknown or is quite complex, and obtaining data by experimental
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(physical or numerical) meana is connected with randmn errors, are the subject of
a new area of mathematical statisti,cs - experiment planning theory [32, 33, 54].
4. After calculating M). vector gzad ~(X(0)) along this direction a transition ia
made to the new point The direction of movement along the gradient vector is
determined by the meaning, of the operator extr 0 (maximum or minimum). In the
general case the vector X(1) is defined by the expression;
Xc" _ Xco> + t gradN(X(")�
where t is a scalar vslue defining the step along the gradient.
If max 4D is found, then t> 0, and if min 0, then t< 0.
5. For the point XM and subsequent points, the operations describe4l~.n ie.ms 1-4
are repeated. If during the process of the calculations the points X~ J, X(2)
always belong to the set K. then the process continues to the point Xo t for
which the condition grad ~(X) a 0 is satisfied the necessary and suf~icient con-
dition for the minimum of the convex and raaximum of the concave function 0(X12) The
latter is convex if the following inequality is satisf ied for any XM and X
~ q) [aX", i (1 - a) X121] >) = Dc'' -4- aD lx=x(,)
am~, ct2, = c1, , ;aD ' At~
D D -r ac, Ix=x(,)
'
where, as was demonstrated in Chapter 2, the derivatives am are def ined by
the expressions �
an i
,
8rn~ 2 9gy ta
1-A- 3
3D`;,;j2 m
OD
dc~ 2 93y
I- A~ 3 Cw p' i,
1The f irst compotLent of the vector X is the carrying capacity of the ship, and the
second, its speed.
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Performing the coxzesponding calculat~onsl we f~.nd v(11) m 17,400 tons, D(12) _
16,400 tons, ~(11) = 9.48 ¬s; ~~�2 m 10,35 knots.
On the basis of these data we obtain the approximate values of the components of
the gradient vector at the point X(l)
.ft 9,48 -9,45 _ 0,03 knots/thousand tons,
dm0 1X=X0) ~ 1
c3m IN 10,35 -9,45 _ 0,45.
2
3. Let us solve the linear problem 00 rriax imo- ;-a-
00
mo v L d/7!p lx=x(') &j 1"(-X('J'
5< mo 1/2.
Thus, the interval [0, 11, which is the initial set of values of the undefined pa-
rameter w is broken down into two subintervals [0, 1/2] and [1/2, 1]. If w E[0,
1/2J, then nopt = 1, and if w `[1/2, 1], thp.r. noPt = 2. It is natural that it is
simpler to obtain the data to answer the question as to which of the inequalities
, (w < 1/2 or w> 1/2) occurs in the specific problem than to find the exact value of
w.
' g 5.8. Example of Optimizing the TTC of a Ship by the Method of Comparative
Evaluation of Versions
As is known, the optimization of the TTC of a ship must be carried out on the basis
_ oF investigation of a system of usually mixed forces designed to solve the given
problem. Cases are also possible where a different-type fleet of ships of one
class and type of forces of the fleet turns out to be optimal when solving the same
~ problem.
+ Below, an example of optimizing the TTC of a ship by the method of comparative
:.A evaluation of versions is considered which takes into account the ef.fect of a fleet
J of ships of one class, but with different TTC. Hereafter such a fleet will be
called mixed-type. (The basis for this example was borrowed from reference
[64]-)
i
_ Let us propose that f.or the solution of some problem ships of n types (versions)
are used. The application of mixe:d-type fleets is permitted. The ships operate
independently of each other, and the volumes of the solution of the general problem
by one ship of each of the types are characterized by random variables r11i, i= l,
' n, the distribution functions of which have the form
0 f or Yu < 0,
1- koN)p; f or 04 yla Qic,
~ F(1i,: f
l 1 for y;; > C~li, (5.42)
where yli are the values of the random variables 71ir; Ql;, koH; p; are given values
which depend on the TTC of the ships of the investigated types.
Thus, it is proposed that each ship of the ith type solves a problem in the volume
Ql~ with the probability p; = ko;,~pr, and with a probabi:lity 1- p the volume of solu-
t3or~ of the problem is zero. The value of k;;; is the operative use coefficient of
the ith type ship and, as is demonstrated in g 3.2, characterizes the probability
that the ship will be in the combat zone at the time of solving tha problem. The
value of pi~ is the proba.bility of solving the problem in the volume Q under the
condition ~hat the ship is in the zone. (This system of combat operations was in-
vestigated in � 3.2).
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Let us asswne that duri,ng operations oE the fleet of sipst iucludi,ng mixed--type,
the total volume of solution of the problem def ined by the ra.ndom variable tl is -
equal to tlie sum of the voltmes of aolution of the problem by each of the ships of
Che f 1.eet : .
n
(5.43)
_ where ~N, is the number of ith type shipa in the fleet.
Let us consider the problem of selecting the optimal fleet of ships begi.nning with
the military-economic argumenCs considering the assumptions made above. Here we
shall adhere to the previously discussed general optimization schemE including the
construction of the effectivenessand cost indexea, the selection of the optimalness
criterion and, especially, finding the optimal solution.
The parameter characterizing the outcome of the combat operations in the given
example is the random variable tl. As was demonstrated in Chapter 3, for construc-
- tion of the effectiveness index first of all it is necessary to have the utility
function u(y), where y is the values of n, and, secondly, to define the distribu-
tion functi.on F(c~/X) of the value r1 for solution of the problem by the fleet of
ships defined by the vector X _(X1, rS'�). Let us consider two forms of the
function u(y): linear u(y) = y and step
0 for y~, Q~,
u~y)I for J>,Q`',
where QE is a given value of the total volume of solution of the problem.
- For a linear utility function the effectiveness index is the mathematical expecta-
tion ,VI,1 (.,N) of the variable n;
rt
31(.;Y') ptQir1Y'i - ;vIn (5.44)
- It is not necessary to find the specific form of the distribution function F(~~X)
- here.
With the step utility function the effectiveness index is the probability of solv=
ing the problem in the volume equal to Q. or exceeding it:
~
(y%~~) = p , QZ rN; . (5. 45)
f dF
In the given case it is necessary to def ine the specif ic form of
If the total number of ships in the operating fleet is quite large (in practice it
. �
- is sufficient that .ti';'> 5 to 6), then under the assumptions made regarding inde-
r-~
- pendence of the operations of the individual ships and correctness of the formula '
(5.43) the function F(yi)r) will approximately have the form of a normal probabil-
ity Liistribution law
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- ~
ft-Mn12
F(y-W) = 1 er 2an ctta
lf 2ao~ ,
(5.46)
n n .
0 2 - n
where r1l
Ptqqc = 1- pi.
;
- Substituting (5.46) in (5.45) and replacing variables y QA1'~ = t, we obtain the
expression for 9:
M-n-Qp,
a~
(t) dt, (5.47)
where (p (t) ; ~2n c ~ .
As the economic index let us take the cost S(,M) of building and maintaining the
fleet of ships, and we shall consider the f.unction S(.M) linear with respect to
n
S (Jr) _ ~1 Srf'r ; (5.48)
here si is the cost of builcling and maintaining one ship of the ith type.
The next step is selection of the optimalness criterion for finding the optima.l
vector .Qna,,, defining the optimal composition of tr.e forces in quantitative and
qualitative respects. The versions of the ship will be optima.l which correspond to
the nonzero components in the vector ft:tn : but the optimalness of these versions
in the general case is retained only they are used in the mixed-type fleet.
- As the military-economic optimalness criterion let us take the condition of maxi.mi-
zing the effective index with limited cost of building.and maintaining the fleet
of f orces
max 9 (r1n),
.ff
S (.M) < S, (5.49)
X; > 0, i = 1, . . n.
where SE isthe given allocated budget.
Let us consider the solution of the problem (5.49) for the adopted two forms of the
utility function: linear and step. Sere we use the fact that the ma.ximizing of
the index in accordance with expression (5.47) is equivalent to maximizing
the value ot w-(;VI,~ - QEY6,n�
- In the case of the linear utility function the problem (5.49) has the form
_ max ~ pAt'Xi,
si.M; < S-_1
(5.50)
0, n.
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The solut3,on of this type of pxoblem o� ljnear pxagxamwing has already bsen con-
~ sidered in & 5.1. The optiaial vector r1'opc ti3,11 contain only one aonzero component
,rNk, for which the condition
max p`Q" ,
i sr (5.51)
is satisfied, that is, the optimal fleet is single-type and the optimal veraion is
found by the criterion of combat economicalness (5.53.). ,
For a step utility function the problem (5.49) is equivalent to the prohlem
n _
2] piQlrxi - QE
max
L~ pi4lQ2i l'M!
(5.52)
n
Si.Mi 4 SEr
i=l �
.MjZ?-0, i c 1 , . . . , Il.
The cost restriction in the form of an inequality can be replaced here by the
equality inasmuch as from the physical arguments it is clear that the optimal value
_ of the index 3z (rt') does not decrexse with an increase in SE.
Af ter replacement of the variables z; _(s;. ss) rr;, ic, and introduction of
the rPlative values of Qij- Q1j1QE, s; = s,;SE, i= 1, n, we obtain
R
~ aj21- l
niax i=t ,
z (5.53)
R
V biZt
i=1
n
L2i=1~ Zl~0, i =1, !I,
i=1
where Z=(zl, zn) is the vector of the new variables, and ai = p;Ql;is;,
~ -
bc = prqrQir'st, i= 1. , rl. .
Solving problem (5.53), that is, finding the optimal vector Zopt, by going to the
old variables we f ind the optimal vector Alopt. .
!11 � 6Z n
.
Let us note that 4~ ucz;. ''QE_ b;z;.
,_1 rai
The problem (5.53) can be solved by numerical methods, for example, the provisional
, gradient method investigated in � 5.4. Here the auxiliary problem (5.29) will be
the elementary probldm o� linear programming.
~
At the same ti.me the prob'lem (5.53) can be solved also by sorting using the fact
that the optimal vector A-opt will contain no more than two nor.zero components. The
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correctness of the given statement follows from the following arguments. The pur-
pose function in the problem (5.53) has the form co (a, = a-1 ' where a=
n _
arz�
r~i ~=1
This function increases with respect to a for each f ixed value of S. Consequently,
for any fixEd a the optimal vector Z is the solution of the problem of linear pro-
gramming
n n ..n
max Y a,z� bizt =69 El zj = 1,
z r-i t-1 r=i
zi:;o~ 0, i=1, n.
(5.54)
Inasmuch as in any problem of linear programming the number of nonzero components
of the optimal vector does not exceed the number of restrictions (besides the con-
ditions zi > U, i= 1, n), the optimal vector in the problem (5.54) can con-
tain no more than two nonzero components. Hence, it follows that also in the
initial problem (5.53) the number of nonzero components of the optimal vector
will not exceed two.
The above-presented result is valid not only for the step functions, but also for
any nondecreasing bounded utility function u(y). Actually, for an arbitrary func-
tion u(y) and normal distribution law of the value of n, the effectiveness index
.MaDt has the form
OD
11(lll,~ Cit.
(5.55)
If u(y) is a nondecreasing and bounded_function of y, then the integral in the
expression (5-55) converges and 3: (:14n, an) is a nondecreasing function of M.r, for
' each f ixed value of 6t~. The latter means that 3 z(.~VI,~, an) has the same property as
the function w(ct, S) 3.n the problem (5.53). Consequently, for any nondecreasing
bounded utility function the optimal fleet of forces in tlie investigated example
can consist of ships of no more than two different types (versions).
Einally, it is possible to state [64] that this conclusion zextains effective not
only for the linear, but also f.qr any concave cost functionS (~V). Let us remember
that for a concave function S(.,Ir) for any two vectors Ar, and ,m,, and any number
O< b< 1, the equality J' (8.5'1 (1 - d) rh.,) j SS (-Irl) +0 - s) s(-/rZ). is"satisfied.
In particular, the cost function S(.Y) will be concave when considering the series
- factor (see � 4.3). Let us return to this division of the problem (5.53). From
what has been discussed above it follows that it is sufficient to solve this prob-
lem for all possible pairs of investigated versions of the ship and then to compare
the corresponding values of the purpose function w(Z). The pair for which the
value of the purpose functiorc turns out to be the highest will give the optimal
vector of the problem (5.53). For n= 2, that is, f or two versions of the ship
- characterized by the parameters (al, bl) and (a2, b2), the problem (5.53) reduces
to optimizing a function of one variable
max a2 -'-(51 -a2)z,-1
11 62 (bi -5z) Z-1 - (5.56)
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In the given case the vector Z has the form Z- (zl, 1- zl).
The problem (5.56) can be solved by elementary methods. For the presence of a maxi-
mum iaside the interval [0, 1], and not at its boundaries (for zl = 0 or zl = 1)
the conditions
dc) (2i) V z,zp 0' I~
(5.57)
should be satisfied, where w(zl) is the purpose function in the problem (5.56).
The maximum inside the interval [0, 1] means that on investigation of the given pair
of versions of the ship the f leet including both of these versions is optimal. If
the maximwn is reached at zl = 1, then the optimal fleet consists of ships of ouly
the first version; for zl = 0 the optimal fleet consists of ships of only the
second version.
It is possible to demonstrate [64] that for the step utility function the mixed-
type optimal fleet of ships will occur only in the case of relatively small values
of the allocated budget (SE) by comparfson with the required volume (QE) of solution
of the combat problem when the probability of its solution (the event r > Q) by
a single-type f leet of each of the versions is less than 0.5. Here the matiematical
expectation of the volume of solution of the.problem by the single-type fleet of
each of the versions is less than QE. In practice the situation musr be recognized
as more an exception than the rule inasmuch as the values of the probability of
solution of the combat less than 0.5 usually are considered unacceptable.
Consequently, for real values of QE and SE the optimal version will be, as a rule,
single-type. The optimal version of the ship does not necessarily insure maximum
mathematical expectation of the volume of the solution of the stated combat prob-
lem.
The physical meaning of the theoretical possibility of the appearance of optimal
mixed-type fleets in the investigated example is very simple. For small budgets
where MY, < QE sometime5 it turns out to be more advantageous to use the mixed-type
fleet iiisuring a smaller value of ~j (by comparison with the best single-type
fleet), but having a larger value of the dispersion Qz which, in turn, can give a
larger value of the probability P{n > QZ}. For Mn QE and equality of the mathemati-
c~l expectations the best fleet will be the one insuring the minimum dispersion
6n�
g 5.9. Use of the LP-Search Method When Optimizing the TTC of Ships
When solving the complex problems of optimization where the purpose functions and
the admissible regions of optimizable parameters do not have such properties as
unimodality, convexity, concavity, connectedness, and so on or when checking the
satisfaction of the indicated conditions is difficult (the corresponding functions
are given by complex algorithms), the necessity arises for investigating the
space of the optimized parameters at a finitte number of discrete points. For
example, in pure form this approach is used When optimizing the TTC of ships by
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the method of comparative estimation of versions. As was pointed out in g 1.3 and
5.3, this method consists in calculating the values of the purpose function of the
optimalness criterion for a given number of versions clarifying the limiting condi-
tions, with subsequent selection of the optimal versions in accordance with the
value of the purpose function.
The discrete method of investigating the space of optimizable parameters at indi-
vidual points locateu in the entire admissible region or some part of it can be
used (in combination with the methods of directional search) for selection of the
initial point, and falling into a valley or falling on a peak of the purpose func-
tion, and so on. .
The following problem arises: how are the points (sets of TTC of the versions of
the ship) selected �or the discrete method of examining the space of optimizable
parameters?
In the case of practical design, this problem frequently is solved on the basis of
the experience and intuition of the designer designating both the ranges of varia-
tion of the optimizable TTC and the specif ic TTC of the individual investigated
versions of the ship. If the solution of the indicated problem on the basis of
experience and intuition is diff icult, the uniform coverage of the region of values
of the continuously variable TTC by random or nonrandom points, from which the best
is selected with respect to the value of the purpose function, is used. This
~ coverage must theoretically be accomplished for each of the�possible sets of
discretely variable TTC which in the f inal analysis permits the optimal version to
be approximately f ound by the variables of both types.
The LP-search method [50] is based on covering the admissible region of optimizable
parameters by a nonrandom grid of points. The latter is constructed using the
so-called LPT-series of points uniformly distributed in a unit n-dimensional cube
where n is the number of continuously variable parameters. This series was con-
structed by I. M. Sobol' [49].
The LP-search method sometimes turns out to be quite effective when searching for
extremal values of complex functions of many variables in the case of complex
nature of the admissiUle regions from the assignment. The LP-search method can be
used also for the statement and solution of the problems of vector optimization
(see � 5.6) although the method of analyzing the tables of values of the purpose
and the bounding functions discussed below is applicable for any discrete examina-
tion of the admissible region of values of optimizable parameters.
Let us first consider the basic principles of the LP-search method when solving
- optimization problems with respect to one index. The general form of these prob-
lems is:
extr F kX),
x
C't < fI (�'C) C't*, l= 1, L,
_ x< Cl and x; xl' f or all R and
J; X=(xl, xn) is the vector of continuously variable parameters.
Usually the restrictions directly on the values of xi are called parametric, and on
fk(X), functional. For properness of the problem (5.58) from the mathematical
point of view the above-indicated restrictions must define some nonempty, bounded
,and closed set X. in the n-dimensional space of vectors X. Geometirically this set
is the intersection of an n-dimensional parallelepiped (7rn) def ined by the parame-
,tric restrictions with the set defined bp the functional restrictions. The func-
tions F(X) and f Q(%), SC = 1, L in practical problems, as a rule, are piece-
wise-continuous and bounded at all points of the parallelepiped 7rn. However, the
set } can have a highly complex form. Thus, it can be not collected, convex, and
so on. These peculiarities greatly complicate the direct application of the methods
of directional search, for example, the gradient methods, for solution of the prob-
lem (5.58).
When optimizing the TTC of a ship by h the method of comparative evaluation of ver-
sions in the uarallelepiped 7rn, :tio points X,,, X M_1 are selected. Let J1"
points (.�k" < .,Y') satisfy the functional restrictions. The point out of the ..1"; paints
for which extr F(X) is achieved is considered optimal.
If there is no a priori information about the nature of the set ~ and the behavior
or the function F(X) in it, then it is necessary to strive for more unif orm probing
by the points X,, j= 0, .M - 1, of the entire parallelepiped 7rn. It is also
obvious that the points Xj must ue selected so that for .M -.oo the probability of
incidence of at least one point in ':he vicinity as small as one might like of the
unknown optimal point XoPt will approach one.
For random search the indicated requirements are satified if independent random
points uniformly distributed in nn are selected as Xj . The probing of the paralle-
lepiped Trn can also be carried out by deterministic, uniformly distributed points.
However, not all of the uniform distributions of points in "n have identical uni-
formity in the def ined sense. for a quantitative evaluation of this property it is
necessary to introduce the corresponding measure.
Let us note above all that in order to introduce the new variables
.
z1 - zj
xt' - xi
the parallelepiped:Trn can be reduced to a unit n-dimensional cube Kn and, correspond-
ingly, it is possible to consider the degree of uniformity of arrangement of the
points in Kn. As the measure of nonuniformity of arrangement of the points
,go, ���.kS�-t, belonging to the cube Kn, the value
~a E xp (~a) - 4�up
can be selected which is called the deviation.
(5.59)
In formula (5.59) 7rp is a parallelepiped with sides parallel to the coordinate
axes, the diagonal OP and located in the cube K.n; vp is the volume of the parallele-
piped 7 p; ~.3,(:TP) is the number of points (out of the total number X), falling in
ir P (Figure
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The series of points is called uniformly distributed in Ka if the limiting relation
is satisfied
lim ~ = 0.
~'y�- s
~z
~
0
k'igure 5.11. Position of the parallelepiped w P in the cube n
for the two-dimensional case (n = 2).
(5.60)
Most frequently the points of cubic grids are selected as uniformly distributed
points, for which
M1 - 0.S ~ m� - U.3
I . . . , .tn m,, . . . , mR <
Figure 5.12, a shows such a grid for M= 4. The total number of points in the uni-
form cubic grid is equal to Mn.
For the cubic grid sup in the formula (5.59) is achieved for the selection of
points ~Yo t~5 - E, 1, 1~, where e is a small value. In this case (~P)
\
i
:b'( _ ~h" and for E-Y 0 the deviation will be A,= 0,5,,,Pl- n
` ,i�i -
a)
y~
1
0
6)_
Xj
13
x7~
2
x
x
x 4
1�
x
~
"9
f2
Y
ti
6
3
x10
5
x
'S
0
x
� 0
~
Figure 5.12. Arrangement of 16 uniformly distributed points in a
two-dimensional unit cube K2: a-- for the cubic grid; b-- for
points of the Sobol' series.
Correspondingly, for the ratio A';r we obtain the estimate t
JY'
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:When investigating the random uniform arrangement of points in the cube Kn the
,wandom variable
`.,y' P) - .h�cP
1; JY'UP (I
.for -sufficiently large ,A' is distributed by .a normal law with math.,matical expec- ~
itation.equal to zero and disperaion equal to one. Hence, it follows that
p j~ ~X (11P) - .MVP 2i1 r'MaP (1 - VP)} = 1 - (D (21j), (5.61)
= r= '
where cli(z) - 2 fe=' dt is the probability integral; n is a given positive num-
iber.
The expression under the square root sign in (5.61) reaches a maximum for vp =1/2.
Conaequently, selecting the point Xp =(1/2, 1, 1), we obtain
P {A a 111 TI 1 - (D (2,l).
For a sufficiently small value of ti = til with probability close to one the inequal-
ity ill I '.T, is valid. On the other hand, for a sufficiently large value of
11 - 'q 2~< '92 1,, m flence, it follows that Il, (�'-,M < A "g92 V,`M and, consequently,
0 1
j7~ �
Thus, the random uniform grid for n> 3 has an advs.ntage over the cubic grid in the
sense of asymptotic (for -A' oo) uniformity of arrangement of the points in the n-
dimensional unit cube.
However, there are series with better indexes of uniformity of.arrangement of the
points for A� oo, These series include the Sobol' LPT-series. The Sobol' points
are found by the tollowing algorithm.
Let the binary representation of the number of the jth Sobol' point Xi have the
form 1= C,�~"'-1 ; e~21 e,2�, where e, s= 1. m are the binary numbers
assuming values of 0 or l. Then the coor~inates ;i! (i = 1, n; j= 0, .M- 1)
of the n-dimensioSobol~2~oints are calculateit by the formulas xio = 0, i= 1,
. . . , r:, xi~ = eiV * e2V i * . . . e~Vi, i = 1, . . . , n, j = l, 2, . . . Here
Vi~s' are socalled directional numbers, and the opemion * denotes bit-by-bit
mod 2 addition in the binary sy$tem. The numbers Vi are fractions, the numera-
tors of which are in the tablesl, and the denominators of all numbers are 2s.
The table of directional numbers permits calculation of the coordinates of the
lI. M. Sobol', Yu. L. Levitan, "Obtaining Points Uniformly Arranged in a Multi--
dimensional Cube," Institute of Appli.ed Mathematics of the USSR Academy of Sciences,
Moscow, 1976, preprint No 40.
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n-dimensional points with the numbers j< 221 for dimensionality n< 31. In this
paper for .M = 2", v= 1, 2, the deviazion is estimated as A < C(n) 1nR-',M,
where C(n) is a constant which depends on n.
In appendix 2 a table of coordinates of the first 33 ten-dimensional Sobol' pointa
is presented. In order to obtain points of lower dimensionality it is necessary
to select the corresponding number of�first coordinates of each point (Figure 5.12,
b).
Thus, for X - oo the series of Sobol' points has the best uniformity of arrangement
in a unit cube by comparison with tYie cubic and random uniform grids. The series
with the best uniformity are unknown. Sowever, it is necessary to consider that
this conclusion is of an asymptotic nature (rN- 00� -
It does not appear possible to give a unique answer to the problem of the required
number of Sobol' points for the solution of problem (5.58) inasmuch as this depends
on the nature of the function F(X), the set X and the required accuracy of finding
the optimum. I. M. Sobol' presents the f ollowing estimate for the required number
of points:
.M = 22+ 1 Y-n .1 (5.62)
Here [vln- ] is the integral part of the number rn (n, just as before, is the dimen-
sionality of the vector X).
The estimate of (5.62) was obtained on the basis af solving simple problems for
n< 12. In the above-cited paper [50] it is pointed out that when solving prac-
tical problems most frequently a value ofal~ = 256 is used.
Thus, the basic idea of the LP-search method consists in examination of the admis-
sible region of values of optimized parameters at discr2te points corresponding
to the Sobol' points unif ormly distributed in a unit cube. It is obvious that
even when selecting the number of points by the recommendation of (5.62) this
number turns out to be quite large. The possibility of practical use of the LP-
search method, just as the other methods, based on discrete nondirectional sorting,
essentially depends on the time f or calculation of the values of the functions
f(X) and ft (X) ,t = 1, . . . , L. Therefore the application of the LP-search method
in pure form f or optimizing the characteristics of such complex objects as a ship
as a whole is hardly expedient. Obviously it is more eff icient to use the LP-
search in combination with the directional search methods, considering that the
series of Sobob' points has quite good uniformity and with sma.ll numbers of probing
points. In particular, if we break down the series of Sobol' points into sections
of length 2n (that is, into groups of 2n points following each other) and corre-
spondingly break up the cube into 2n "octants" by a unif orm cubic grid, then for
each n< 16 the points of eac of the sections of the series are distributed with
respect to the different "octants." An analoga-,is groperty, but already only for
each n< 6, occurs also in the case of breakdowc'1 of the series into sect ions of
length 4n and, correspondingly, the cube into 4n "octants.
Now let us consider how the properness of the statement of the problem k5.58) is
analyzed using the LP-search method. Selecting a number',,tin of uniformly distri-
_ buted points in the parallelepiped ffn, it is possible to expect that onl3 .4"
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poiats will fall into the region X+ in which the functional restrictions are satis-
fied. The value X', and, more precisely, its mathematical expectation, can be
n
.
f ound by the f ormula .M1 _.M where vTr = n(xr- rr ) is the volume of the
parallelepiped vy is the volume of the region X.
If ;J,i = 0, the solution of problem (5.58) does not exist. However, direct establish-
ment of this f act for the complex f orm of the functions f R(X) , f or example without
rhythmic assignment of them, is very diff icult. In addition, in essence the para-
meters C* and C~* in the functional restrictions. cannot have the nature of absolu-
tely rigidly given values. In a number of cases the researcher can relax the re-
strictions or, on the contrary, make them more rigid. This means that some func-
tions f9;(X) are of a criterial nature and, consequently, the problem (5.58) can be
considered as a vector optimization problem which is reduced to the problem of
optimization with respect to one index by imposing restrictions on the other in-
dexes f Q (X) .
The iaea uf analyzing properness and more preciae definition of the statement of
the prob:Lem (5.58) consists in investigation of it as a vector optimization problem
as a result of conversion of the nonrigid functional restrictions to optimizable
indexes. A:Eter this transformation of t:ze problem (5.58) for all of the selected
uniformly distributed points in 7tn, a table of values of the functions F(X) and
fR(X), t = 1, L, is constructed. For sufficiently large h� the analysis of
such tables makes it possible to j udge whether at least one of the points falls
into the admissible region X and haw variation of the nonrigid restrictions in-
fluences the admissible region i and the choice of the optimal point.
Let two nonrigid restrictions on the functions fl(X) and f2(X) and one rigid re-
striction on the function f 3(X) exist in the problem (5.58). Let us assume that
the table of values of the functions F(X) and fQ(X), k= 1, 2, 3 has the fox�m or
Table 5.1 in which the admissible values of the function f3(X) are noted. Let' us
conaider that the values of the functions are arranged in.increasing order, and
the index F(X) must be maximized.
Table 5.1. Values of the functions F(X) and fR(X) at eight points
F(.1)
~
Fo ~
Fa
F3
Fe
I Fl
F,
F_
F7
ito
fL4
it:
f1a
~ fll
i1s
f13
/17
%u I
Iss
l:a
i lsi
I
~
I:s
%s_
Cs;
la
Ia;
I/ia
I fsi
~ f3a
I
f.
I
I(ss
fss
lao
Note. F. is the value of
the
function
F(X)
at
the
point X.~ f is
the
value of
the
ICth
function
fQ(X)
at the pointjX3.
From Table
pointa X1,
is X2.
5.1 it is obvious that beginning with the rigid requirement of f(X) the
X2, X3 and X4 will f all in tU a admissible region of I,. The besi point
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If, for example, the one-way restrictions fl(X) ~ G1* and f2(X) < C2* are imposed
on the functions f1(X) and f 2(X) , where f 12 > Ci* > f 14 and C2* > f 24, then the
opCimal point will be the point X4 wliich in the given case will be the only admis-
sible point. At the same time if the nonrigid restrictions are relaxed somewhat,
setting Ci* = f12 and C** = f22, then 2:points X2 and X4 will fall into the admissible
region, of which the better will be X2. Analogously, if Ci* < f14, then no point
will fall into the admissible region. In this case it is necessary to try to in-
crease the number of points or relax the restriction on f 1(X).
The correction of the nonrigid restrictions is made not only in the direction of
relaxation. For example, if all of the trial points turn out to be admissible, then
the researcher can narrow the admissible region (make the restrictions more rigid),
on grounds following from the essence of the solved problem,
The above-described analysis of the tables of trial points in the space of optimi-
zable parameters is of an informa.l nature and requires the use of certain heuris-
tic arguments of the researcher. However, this analysis frequently turns out to
be highly useful for proper formulation and solution of complex problems of optimi-
zation. When realizing a computer-aided LP-search, the step in which the table of
trial points is analyzed is expediently performed using the researcher-computer
dialog mode.
An analysis of the tables of trial points is highly useful for the formulation and
solution of the problems of vector optimization. Such an ana.lysis permits, for
eY.,ample, detection of indexes that vary insignif icantly little, the discovery of
dependent indexes simultaneously increasing or decreasing, facilitation of the
solut ion of the scalarization problem (see � 5.6), and so on.
It must be considered that on the basis of the LP-search it is possible to construct
iterative procedures in which the trial points are selected by series with gradual
narrowing of the search region with respect to the best of the points of the pre-
ceding series. This approach is considered as a deterministic analog of the direc-
tional random search (see � 5.4).
Let us consider the problem (5.34) of optimizing the mass of hauled cargo and the
speed of a transport ship as a numerical example of using the LP-search method.
This problem was solved in � 5.5 by the provisional gradient method. Let us remem-
ber that in � 5.5 the solution of this pr oblem was not obtained as a result of the
presence of a singularity in the purpose function in the form of an ascending peak.
As the first series of trial points in the LP-search method let us take 16 points
in a rectangle def ined by the parametric res*rictions [functional restrictions are
absent in problem (5. 34) The coordinates oi the paints and the values of the
purpose function for the initial data adopted in � 5.5 are presented in Table 5.2.
In this table xlj and 3i 23 will be coordinates of the Sobol' points in a unit two-
- dimens ional cube. From Table 5.2 it follows that the best points are those with
the numbers 7, 11 and 13 which are in the upper right-hand corner of the initial
rectangle Tr2.
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.
Now it is possible to narrow the initial search region and select a second series
mf trial pointe. As was stated earlier, joint application of the methods of
directional search and LP-search ca.n turn out to be eff ective. In g 5.5, the point
mp d 17,000 tons, v= 25.5 knots for which D= 34,800 tons and 12.45 knots,
w$s obtained as a result of two steps by the provisional gradient method. Here
a4)/amo > 0 and 80/av < 0. Using these data, let us apply the LP-search method in
a rectangle defined by the conditions 17 < mo < 20 and 20 < v< 25.
Table 5.2. Probing the initial region of optimizable parameters
. m0 and v at 15 Sobol` points.
I
i
'
x~l ( xsl I
mpi
7NC. TWI
y3 b
~i
I TWC. T1
(D 1
)'3 (bl
0
i
0 ~ 0
5,00
10,0
i,80
6,40
1
0.500 I 0,500
12,5
20.0
.111,9
11,4
I
2
0.250 0,750
8,75
25.0
19,5
11,2
3
0.750 ! 0,250
16.2
15.0
25.2
9,65
4
0,125 ; 0,625 '
6,90
22,5
14,0
11,1
5
0.623 I 0,125
14,4
12.5
21,8
8,27
6
0.37- 0,375
10,4
17,5
17,3
10,5
;
U.8;5 ~ 0,875
18.0
27,5
40,0
12,4
I 8
0.062 j 0,937
5,90
26.7
18.2
9,30
9
0.562 I 0.937
~ 13.5
18.7
22,6
11,1
10
0.312 ~ 0,187
9.70
13.7
15.9
8,35
11
i 0.812 ~ 0,687
17,2
23,7
32,8
12.4
12
0,187 0,312
7.80
16.3
I 12,9
9.80
I 13
0,687 0,812
16,3
26.2
34,0
12.35
14
I 0,437 0,562
11,5
2!,2
+ 21,0
11,6
I 15
I 0,937 0,062
19,0
11.2
I 28,3
I 7,45
Key: a. thousands of tons b. knots
Table 5.3 Probing a restricted region of optimizable parameters
- m0 and v at eight Sobol' points
i I
X 1j
I A 2i
mal
I TWC. T(a)
1 ' ~
}r3 (b
D 1 (a ~ 01
T6fC. T y01
0
0
0
�17,0
20,0
29,0
11,7
1
0,~00
0,500
18,5
22,5
33,5
12,4
2
0,250
0,750
17,8
23,8
34,0
12,5
3
0,750
0,250
19,3
21,3
33,7
12,2
4
0,125 1
0,625
17,4
23,0
32,4
12,35
5
0,625
0,125
18,9
20,6
32.5
12,0
6
0,375
0,375
18,1
21,9
3:;,;
37
5
12,2
12
6
7
0,875
i
0,875
19,6
I 24.4
,
,
1
~
Key: a. thousands of tons b. knots
In Table 5.3 the results are presented from probing the indicated region by eight
Sobol' points. From this table it is obvious that the eighth point turns out to
be the best, which consider~ng rounding of the coordinates coincides with the
optimal point (see � 5.5).
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CHAPTER 6. SOME SPECIAL PROBI.EMS CONNECTED WITH TIiE ANALYTICAL DESIGN OF SH.IPS
One of the peculiarities of the procedural approach of ship design theory, including
AD theory, consists in the necessity for partial optimization of some debign solu-
tions in order to decrease the number of variables optimized "in terms of the ship"
as a whole as much as possible. The reason for this is complexity of the construc-
tion of the mathematical models of optimization which take into account all of the
variety of characteristics of the ship simultaneously which are manifested on
different hierarchical levels of investigation of its combat use. It is very com-
plicated, for example, to perf orm a quantitative evaluation and optimize the t ech-
nical solutions in the general model connected with insuring certain operating
qualities of the ships. At the same time,neglecting the latter can lead to very
serious consequences. As has already been pointed out, the TDA of a ship inf luence
its effectiveness indirectly, through the TTE, and sometimes it is very difficult to
trace this ef f ect to the end (to indexes of a suf f iciently high h3erarchical level).
It is expedient to perform a partial optimization with respect to the TTC which
are relatively stable on variation of the conditions of combat use of the ship or
~ to a known degree secondary with respect to their influence on effecti~reness and
_ expenditures of resources.
- In the given chapter some problems connected with the above-indicated partial opti-
mization are investigated as examples.
- � 6.1. Consideration of the Reliability of Technical Means When Estimating the
Ef fectiveness of Shipsl
When estimating the effectiveness of a ship in the general case it is necessary to `
consider the process of functioning of its individual technical subsysteuBwhich
"insure" diff erent TTE of the ship. The possible operating failures, combat and
emergency damage to the subsystems lead to the fact that the TTE vector of the
ship can vary randomly during the performance of the stated mission by the ship.
In addition, in the case of systematic operations the volume and nature of the
failures and damage have an influence on the repair time between voyages and, con-
sequently, the operative stress coefficient of tt'.e ship.
Let us remember that the capacity of the ship (or its individual technical subsys-
tems) to retain the properties required ior a given purpose in time, for given
lZn this section results axe presented which were obtaiiLied by the author jointly
with V. A. Usachev [37].
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operat3ng conditions is called tlie reliability, and the capacity cf the ship (or
i.ts individual subsystems) to retain the properties required for the given pur-
pose, in the presence of effects (explosions, fire, flooding, and so on) not
prdv3.ded for by normal operating coaditions is called"zhe invulnerability or sur-
vival probability [44].
From the presented def initions it follows that the difference of the concepts af
reliabiZity and invulnerability is highly provisional and is connected with the .
nature of external operating conditions. Consideration of the properties of relia-
bility and invulnerability requires investigation of the process (or processes) of
transition of ;Lndividual technical subsystems of the ship from state to state as
a resu}t of possible appearance of operating failures and also combat and emergency
damage . This process will be called the process of internal functioning def ining
the set of values of the TTE of the ship which can be insured by the technical sub-
systems at each point in time.
_ By analogy with the process of internal functioning it is possible to introduce the
process of eacternal functioning into the investigation which at ~each point in time
determines the set of values of TTE of the ship which are required for the perfor-
mance of the corresponding stage of the operation. -
The joint investigation of the processes of external and internal fun.ctioning permits
esti.mation of the eff ectiveness of the ship considering its reliability or, on the
contrary, estimation of ttie reliability using the effectiveness indexes. Here two
basic cases are possible: estim.ation of the effectiveness of the ship considering
the reliability of its technical subsystems and estimation of the reliability of
the ship or its individual subsystems with a fixed external functioning process.
In the given section, a study is made of the two indicated cases in the example of
the power plant (PP).
EstimatiQn of the reliability of the ship as a whole or estimation of its effective-
ness considering the reliability of individual technical subsystems in the general
case leads to highly complex and awkward mathematical models which basically is
caused by a large number of states in which the technical subsystems can be and
the presence of a mutual relation between the functioning processes of the indivi-
dual subsystems. Accordingly, it is necessary to mak.e a series of additional
simplifying assumptions. Frequently it is assumed [9] that each of the TTE of the
ship is insured by only one subsystem (each subsystem insures only one of the TTE).
For exanple, L-he speed is insured only by the PP, and so on. In addition, the
assumption is made that individual subsystems function independently of each other.
Inasmuch as the methods of estimating the reliability of the ship',s technical sub-
systems and the ships as a whole are discussed in a number of special sources [77,
341, only examples illustrating the methods of considering relia.bility and Markov
- models of estimating the effectiveness of ships will be presented below.
Let us consider the problem of estimating the reliabiliCy of a single ship when
solving the problem of finding a target in a given part of the sea. We shall con-
sider the following assumptions valid: the search process is Poisson [19]; in the
I Hereaf ter, we shall consider only ihP evaluation of reliability; the investigated
examples are of a procedural nature. "
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search process the target does not go outside the given region, and it does not
avoid detection; the range of the target detection means by the ship does not de-
pend on the speed of the ship; the speed of the ship i:, determined during the
search only by the condition of the PP [power plant], which can vary with time as
a result of failures of the elements of the power plant; the PP is considered a
system that is not repairable at sea. The run time of the individual elements
per 'Lailure is a random variable with exponential distribution luw; the elements
of tne power plant are the follcwing: the nuclear reactor (P), the ma.in turbogear
assembly (T) and the line of shaf ting (B).
~ A study is made of two power plant systems under the condition of insuring identi-
cal maximum speed of the ship: the power plant (1, 1, 1) consisting of one reactor,
one main turbogear assembly GTZA and one line of shaf ting (Figure 6.1,a) and the
power plant (2, 2, 2) consisting of two reactors, two GTZA a.aid two lines of shafting
(Figure 6.1,b). The power plant (2, 2, 2) is, according to the diagram, a redun-
- dant power plant (l, 1, 1). A study is made of two subversions of the power plant
- (2, 2, 2): without a connector between the rea.ctors (Figure 6.1,b) and with a
connecter (Figure 6.1, c). The reliability of the connector is considered ideal.
Each of the power plants of the (1, 1, 1) type entering into the power plant (2, 2,
2) is provisionally called the power plant of one side. The reliability character-
istics of the individual elements for the above-indicated two power systems are
assumed to be identinal and independent of the power at which the elements operate.
4r)
P r B
b:),
P T B
P T B
'85 .
R r T B
Figure 6.1. Schematics of power plants.
The probability of target detection with unlimited search time is taken as the
effectiveness index.
Power Plant (1, 1, 1). Under the assumptions that are made, the functioning of the
ship with power plant (1, 1, 1) can be described by a Markov random process, the
graph of states and transitions of which is presented in Figure 6.2,a. The
states of this process are the following: 1, 0-- the power plant is in a state
of good repair, and the target is not detected; 1, 1-- the power plant is in a
state of good repair, and the target is detected; 0, 0-- the power plant has
failed, and the target is not detected.
The value of X in Figure 6.2,a is the total intensity of failures of the reactor,
tr.e turbine and the line of shaf ting, and Y1 is the d$tection intensity. On the
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FOR OFFICT I - 7i; i. l 2k - k�h;'k
1- k,ll; ti z y,; 4 � ,
and Pdetection(-) is the probability of detection for the power plant (1, 1, 1)
defined by the formula (6.2).
The coefficient ~1(k, y Z/a) characterizes the advantage of the power plant (2, 2, 2)
without connector over the power plant (1, 1, 1) with respect to the limiting value
at t-~- - of the probability of target detection (Figure 6.3).
Let us now consider the mode of use of the power plant (2, 2, 2) without a connec-
tor in which the search is always conducted with operation of the power plant on
one side, and the power plant on the other side is in unloaded reserve, tha*_ is,
it goes into operation af ter failure of the operating power plar.t. Let us assume
that the reserve power plant does not fail during the waiting period. In the
terms of reliability theory [39], the above investigated mode of simultaneous use
of the power plants on both sides is called the ready reserve mode.
The functioning of power plant (2, 2, 2) using it in the unloaded reserve mode of
one of the sides is described by the same process as in the case of the ready re-
serve, but the intensity of transitions from the state (2, 0) to the state (1, 0)
is equal to a instead of 2X, and the intensity of transitions from the state (2, 0)
to the state (i, 1) is equal Co Y2 insiead of yl.
Performing the corresponding calculations, we find
Po!H k1'i --kYl) -r-kyi).
(1) (i. + kYi)2
Key: 1. detection
or
po6e ~O0) = P0(6x (oo) V'2
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(6.6)
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A,E~re
% (k, 71//�) = k 1-f- Yi/% 2-F- kYi/4
1 + kYi/% 1 -i- kYi/)� ~
Ttie coefficient 02(k, Y1/X) characterizes the advantage of power plant (2, 2, 2)
operating in tlie unloaded reserve mode of one of the sides over the power plant (l,
1,, 1). CortesporidingZy, the difference 02 -0l characterizes the effect of the
ftansition to the unloaded reserve mode for power plant (2, 2, 2) without connector
(see Figure 6.3).
The value of yl/a is the ratio of the average time before a failure of power plant
type (1, 1, 1) to the average time to detection of the target when searching at
iiaximum speed.
The limitiiig equations exist
=1im (Px (k, Yi/X) = 1 ~
vlia-m
tliat is, with high reliability of the elements which corresponds to X 0 and Y1/X}
OD, none of the investigated power plants has advantages, which is entirely natural.
The advantage of power plant (2, 2, 2) by comparison with power plant (1, 1, 1)
increases with a decrease in the reliability of the elements. At the limit when
Yi /a + 0, this gain is 1.3 and 1.6 times f or the ready and unloaded reserve modes,
respectively.
Let us assume that the probability of fail-safe operation of power plant (1, 1, 1)
for the time 1/yl (.for a time equal to the average time until detection of the tar-
get when searching at maximum speed) is equal to no less than the given value of P3.
In this case eX/Y1 > P3, hence y l/a >-1/ln P3. For P3 = 0.9, we obtain yl/x > 10
and, consequently, 0 l< 1.07 and 0 2 (t) c~ 10,5; 11 f or 0< t 0.5 and vice versa take place at random points.in time,
-
forming the simplest flows of events with transition intensities v12 and v21, -
respectively. This means that the external functioning process is a Markov pro-
cess.
Let us consider the power plants of the types (1, 1, 1) and (2, 2, 2), where the
power plant (2, 2, 2) does not have a connector, and on intermediate powers it
operates in the ready reserve mode. The failure intensity of the power plant (1, 1,
, 1) when operating at more than 50% power will be denoted by X100' and when opera-
ting at less than 50% power, as0 . Let us assume that these f ailure intensities
will occur also for each of the sides in the power plant (2, 2, 2).
Power Plant (1, 1, 1). Under the assumptions made, the functioning of the power
- plant (1, 1, 1) considering the processes of its external and internal functioning
is described by a Markov random process (Figure 6.5, a). The states of this pro-
cess are denoted as f ollows (the f irst f igure indicates the number of the internal
state, and the second f igure, the external state): 2, 2-- all elements of the _
power plant are in a state of good repair; more than 50% power is required; 2, 1
' all of the elements of the power plant are in a state of good repair, a power
of lsss than 50�6 is required; 0-- the power plant has failed.
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Q) y:f
2,2 Z�~
yri
~;1pp O ~!p
6)
Figure 6.5. Graphs of the processea of joint internal and external
�unctioning: a-- for power plant (1, 1, 1); b-- for power plant
_ (2, 2, 2) without connector for the ready reserve mode.
The probabilities of states of the process satisfy the system of differential equa-
tions
= dP~: (t)
dt --(v:i + Xioo) p== (t) 7' vl-pzi (t)t
aP21 (t) (,,ix -r- ~-so) p2i (t) t ~'upzz (t),
dt - (6.19)
dP, (t) _
dt %sop:1 (t) X10oP:2 (t)
under the initial conditions
P:i (0) = 1 Pz: (0) = Po (0) _ 0 (6.20)
or
pz: (0) = 1 pu (0) = pu (0) _ 0 (6.21)
Solving the system of equations (6.19) under the conditions (6.20) or (6.21), it
is possible to obtain the formula for the probability PD(t) of failure of the powe
plant in the time t. If X 100 ax 50 aX, then independently of v12, v21 and the
initial conditions
- p'O','X = po (6.22)
(t) = 1 - e"~.
(1)
Key: 1. failure
The average t3ine before failure is defined by the expression
a ~
Tl t dPo (t) - J[ 1- Po (t)) dt.
0 0
" T'Or X 100 a x 50 ~ x
T, = 10, or T1= (6.23)
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whexe 'rl =~Tl is the dimensionlesa magaitude of the average time of fail-eafe
o,per.ation.
Po,wer Plant (2, 2, 2). The fun.ctioning of the power plant (2, 2, 2) is described
- :by a Markov process (Figure 6.5, b). The states 2, 2 and 2, 1 have the same mean-
ing as for the power plant (1, 1, 1), and the other states are denoted as follows:
- 1, 1-- the power plant of one side is in a state of good repair, less than 50%
. -power is required; 1, 2-- the power plaat of one side is in a state of good re-
pair, a power of more than 50% is required; 0-- the power plants on both sides
have f ailed .
The probabilities of the states of the process satisfy the system of differential
equations
_ dps9 (i) (v2l -f- 2Rloo) P22 (t) + yizpsi WI
dt
dP:i (1) (yis + 2?.ao) pz1 (f ) + vz1pr: (t),
dt
dPjj (1)
dt ~vis + Xao) pI, (t) 2i,boPu (t),
dPl: (t) dt - vlzp,l (t) 2).ioopsz (t), (6.24)
dPo (t) ~ ~aoPii ~t)
dt
under the initial conditions
pzi (0) = 1 1 pzz (0) = pll (0) = pU (0) = po (0) = 0 (6.25)
or pzs (0)= 1, P21(0) = p11(0) = pjs (0) = Po (0) = 0. (6.26)
The probability of f ailure of the power plant in the time t is equal to the sum of
the probabilities P0 (t) and P12(t). These probabilities correspond to the states
for which the available power of the power plant is less than required.
Let us consider the special case where v12 = v21 = v and a100 , x 50 X� The solu-
tion of the system of equations (6.24) under the conditions (6.25) gives
p1z (t) = e- e_z~,e _ I_ e_z
1 -v 1 -v 1 -}-v '
_ po(t) 2 e- u
-e'-
=~1-v2)~~-}-v) -~t*v l _ 2(1 ~ v)
1
- 1 z[1-e'z(1 +v) ~r~
2(1 1 '
where v = vA.
The probability of failure of the power plant will be
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P(z) t 1 2 e_ (1+v) it _ V_ e-zxr e-= (t+"v) u
I.
1-}-v ~
Key; 1. failure
~ Calculating the integral f tdP((t), we find the average time of fail-safe operation
OTK
0
3 - 2v - v1 l
T~ 2(1 -v)(1+-v)z
or in dimensionless form
(6.27)
3-2v -v3 '
tt 2(1- v) (1 "v)= . (6.28)
Thus, the average operating time before failure of the power plant (2, 2, 2) depends
on v, that is, the switching frequency from the mode N< 0.5 to themode N> 0.5,
and vice versa. Let us note that for vl2 = v21 = v the power plamt must operate on
the average an identical amount of time 3n the modes N< 0.5 and N,~ > 0.5, but the
operating time T2 decreases with an increase in v for ~i = const. From Figure 6.6
it is obvious that for 0< v< 0.6 the power plant (2, 2, 2) has the advantage with
respect to operating time before failure, for in this region T2 > 1. Correspond-
ingly, for v> 0.6 the power plant (1, 1, 1) for which T1(v) = 1 has the advan-
tage. The maximum advantage of the power plant (2, 2, 2) over power plant (1, 1, 1)
i8 T2/T = 1.5, and it is reached for v= 0. In this case the required power re-
mains al all times less than 50% inasmuch as this power is required for t a 0, and
the switching frequency f rom mode-to-mode is zero. The maximum advantage of the
power plant (1, 1, 1) is 't /'t2 = 2, and it is reached for -v which corresporids
to the case where the requ~red power is always greater than 50%.
From the investigated example, in particular, it follows that the assignment of
only the stationary probability distribution of different levels of required.poXt.er
is insuff icient for estimation of the reliability of the power plant inasmuch as
this distribution does not uniquely define the intensities of the transitions be-
tween the required operating modes of the power plant with respect to power. In
the above-investigated example the stationary probability distribution of the
required power has the f orru
hjoo = 'v12 , ltro = 1 - hl00 = y?' .
yl: '7" V21 v19 T V21
However, assignment of the probabilities h120V and h5p permits determination only of
the relation between the intensities vl2 an21, and not the values of these
intensities which determine the reliab3Iity characteristics of the power plant.
In the given section a study was made of the simplest method of estimating the re-
liability of cmnplex engineering systems of the ship based on using the theory of
LXarkov random processes with a discrete set of states and continuous time. This
method is connected with a quite strong assumption with respect to the nature of
the flows of failures and repairs of the system elemeats (the flows must be ordi-
nary and without af tereffect). At the present time the methods of theoretical
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) 2.
1,6
1,2
0,6
0,
F3.gure 6. 6. Graphs of the functions '[1(v) and 't20).
evalua�tion of the system relisbility have been developed for less rigid requirements
on the na:ture of the flows of failures and repairs of the elements. These methods
are based�on the theory of semimarkov processes and the use of logical-probability
ranS I.ogical-sta-tistical models. In the la.tter case the structure of the system and
;the :peculiarities of its functioning are described by the methods of mathematical
logYc, and the quantitative estimate of the reliability ie made by the methods of
,probability theory or statistical simulation [44]. The logical-probabi].ity and
logicas-statistical methods find broad application when estimating the structural
~-eliabilit.y of engineering systems.
1.6.2. Qptimization of the Service Life of Ships
,Ef,f icient aervice lif e of the ship as part of a f leet depends on the r a t e s of its
physical wear and obsolescence. At the present time the def ining factor is
obsolescence by which we mean a reduction in effectiveness of the ship as a result
of an 3ncrease in the effectiveness of the forces and means of the enemy. As a
resul-t of sharp acceleration of the rates of scientif ic and technical progress this
�form,of wear 1as become basic for warships.
Figure 6.7. Nature.of the variation of 9 (e) in time. 1-- without
repairs and modifications; 2-- with two repairs at the times t p1
and tp2; 3-- for two repairs and one modification at the time
tM-.under the condition that ~the modif ication completely compensates
for physical wear and obsolescence.
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In the problems of substantiating optimal service lives, ships are considered as
potential means of waging war. Therefore the effect o" finding the ship in the
fleet can be characterized by an integral index with respect to time which depends
on the effectiveness of the ship at each current point in time. A value called
_ the combat potential is uaed as th3s index:
r
II (7') = J 3(t) dt, (6.29)
where,9 (t) is the current value of the effectiveness index under the condition
that the combat operations begin at the time t; T is the service life of the ship.
The indexes of the type (6.29) are also used for civilian ships [34, 37]. In the
latter case the value of R(T) is called the service effect of the ship [34].
Because of obsolescence and physical wear the function 3 (1), generally speaking
decreases with an increase in t. An increase in 3(t) can occur only after repairs
and modifications. (Here it is considered that the repairs are prima.rily aimed at
elimination of physical wear, and modifications paratially compensate also.for
obsolescence).
Instead of 3(t) it is possible to consider the relative effectiveness 3(t) = 9(t)19
(0), where 3(0) is the effectiveness corresponding to the time the ship becomes
part of the fleet (Figure 6.7). When considering modif ication the values of the
function 3(t) do not necessarily fall in the interval [0, 1], as occurs when con-
sidering only certain repairs. Theoretically af ter modification the effectiveness
of the ship can exceed its initial value 3(0). When using the function 3(f) the
r
expressian for the combat potential will be n(T) = J3 (1) di� For convenience of
U
calculations the discontinuous function g(t) in the presence of repairs and modi-
fications (including consideration of their duration) is replaced by a continuous
function from the condition of equality of combat potentials [34].
In a number of cases the so-called residual cost of the ship serving for t years is
of interest (see g 4.4). It is natural to assume that the residual cost S0 (t) is
proportional to the residual cambat potential:
T
f 3 (T) dT
So~~~-sr _SII(T)-TI(f)
T n ~T~ ' (6. 30)
(t) di
where S is the initial cost (the huilding cu3t) of the ship;
r
n (t) - f ~9 (ti) dT.
0
Here it is assumed that S(T) = 0, that is, the residual cost of
the entire service life is zero. In accordance with expression
potential plays the role of the "production" to which the cost
carried over during its service.
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the ship serving
(6.30) the combat
of the ship is
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The above-preseated expxessions for the combat potential, which are integrals of
the functions 3(t) or 3(t), are vali,d ia the case where the importance of iasuring
some level of effectiveness at various poiats in time is identical. If the
effectiveness, insured at various points in time is not equivalent, then the,combat
r
potential can be def ined as II (T) - 13 (t) w (t) dt, where w(t) is the "weight" function
U
characterizing the relative significance of the effectiveness at various points in
time. Later we shall consider the case w(t) = 1.
_ The simplest atatement of the problem of optimizing the service life consists in
maximizing the combat potential per unit expenditure connected with the building
and maintenance of the ship during its service life. Let us represent the cost of
T
building and maintaining the ship in the form SE (T) = S-}- f s; (t) dt, Where SE (T) is
a
the total cost of building and maintenance during the service life; S is the build-
ing coet; s3(t) is the cost of maintenance per unit time at the time t(T).
The above-atated problem of optimizing the aervice life reduces to maximization of
the function
r
f 3 (t) dt
TI (T) b
gz (r) - r
S +f s3 (t) dt
0
with respect to T.
Differentiating this function with respect to T and equa.ting the derivative to zero,
we obtain the equation for finding the optimal service life "
r
(3 (t) dt
3(T) ` b (6.31)
s3(T) T . S ~ s3 (t) dt
in accordance with this equation the optimal service lif e is characterized by the
fact that at the time t- Topt the ratio of the current value of the effectiveness
to the expenditures on maintaining the ship per unit time is equal to the ratio of
the values of the combat potential and the total expenditures that accumulated at
this tiate.
As an illustration let us consider the solution of the equation (6.31) under the
followinh assumptions: the cost of maintaining the ship per unit time (44) does not
depend on time and, correspondingly, the value fo SE(T) is calculated by the formula
SZ (T)= S + s;T, (6.32)
the function 3(t) is exponeatial;
3 (t) � a', (6.33)
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where 0 4 a-~ 1 is a given value.
Introducing the notation v= -ln a, let us represent the exponential function by
' tihe exponent
e v:, v> 0.
(6.34)
The exponential f orm of the function 3(t) does not always corresuond well to the
actual process of obsolescence of ships. With this form of 3(t) the obsolescence
- rate is proportional to the current value of thP effectiveness and, consequently,
the entirely "new" ship has the greatest obsolescence rate. At the same time, from
the general arguments it is possible to conclude that during the process of obso-
lescence there must be a time delay when the "new" ship becomes obsolescent compara-
tively slowly. Af ter this period, an increase in the obsolescence rate takes place
as a result of the appearance of new scientific and technical possibilities for
improving the characteristics of the ships aad gradual introduction of the ships
into operation on which the indicated possibilities have been implemented. Finally,
in the third period the obsolescence rate again decreases, and the effectiveness
stabilizes at some, as a rule, very low level.
The above indicated general properties are satisfied by the so-called logistic law
of aging (obsolescence) [34] given by the function
0 + 0) e'vt
1 pe-vl '
where v, ~ are the given parametexs.
Hereafter for simplicity we shall consider only the exponential function 3(f).
Under the given assumptions with respect to the form of the functions 3(t) and
SE (T), equation (6.31) reduces to the form
ee = 6 -I- B,
where B=(1 = b).'b, b= s3l(vS), 0= vT.
(6.35)
The solution of equation (6.35), that is, the value of OoPt depends on the dimen-
sionless parameter s3'(%o5). The optimal service life (ToPt) is found from the expres-
sion Topt -0 opt/v. Let, for example, v= 0,05 liroli, s3 = 1� 108 rubles/year, S=
50�106 rubles. For these initial data we obtain b= 0.40, OoPt = 1.64 and ToPt =
32.8 years.
Let us show that introducing the discount rate when determining the operating cost
(see Chapter 4) increases the value of Topt. On introducing the discount rate the
cost of building and maintaining the ship is calculated by the formula
SE(T)=S-r a I
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where a is the discount rate. Setting (on the basis of smallness of a) ln (1 + a) - a, it is pdasible to abtain
ttie equation for finding Oopt
e
ee = A a (I a)'' - A (1 - a It
) (6.36)
where q= aS l 1-4- s
a
S3
Let us propose the same values of the initial data (S, s3, exist as in the preceding
example, and a= 0.03 1/year. For these initial data it is possible to reduce equa-
tion (6.36) to the form in (ee -i- 0,67) = 1,64 + 0,68. Its solution gives OoPt = 3.5,
atnd, consequently, Topt = 70 yeara instead of 32.8 years as occurred without intro-
ducing the discount rate.
It is obvious that modifications also increase the optimal service lif e, for they
decrease the average obsolescence rate (for an exponential law of obsolescence
the mean value of v).
The most complex problem for the above-investigated approach `.o determining the
optimal service life consists in establishing the form of the function 3(t) or the
parameter v when approximating 3 (t) by an exponentia.l function. The solution of
this problem requires prediction ot the development of forces and means of the
enemy in qualitative and quantitative respects for comparatively long time inter-
vals. The forecasting problem on the methodological and.vractical levels is in
the initial stage of development. Finding the function 3(t) is also connected with
cementing significant computational difficulties. Actually, finding the values of
3(t) for each fixed t requires execution of a block of the effectiveness estimation
algorithms. In Chapter 3 it is pointed out that even a single execution of the
algorithms for estimating the effectiveness of a ship frequently turns out to be
a problem of quite large computational volume. In these ca.ses it is expedient not
to include the service life among the optimizable variables during AD of the ships
but to substantiate them individually. This approach is based on the following
arguments.
Let us represent the effective index3 (X, t).which depe4ds on the vector X(optimiz-
able TTE and TDP) and time t in the form 3(X, t) = 3(X, 0) 3(X, t), where 3(X, 0)
is the value of the effectiveness index on completion of construction of the ship
(t = 0); 3(X,t) is the function characteriziag the obsolescence of the ship.
In this case for the combat potential we have the expression
T
II (X, T) = 3(X, 0) f 3(X, t) dt.
0
- Let us assume that the function 3(X, 0 does not depend on the vector X. For the
exponential representation where3 (X,t) = e vt,this means that the parameter v does
not-depend on X. As was demonstratea suove, the optimal service life found by the
criterion (6.31) depends only on the ratio s9,,s and the parameter v. If the ratio
Zll
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S3'S does not depend on the vector X or depends weakly on it, then optimization of
the service life can be brought about independently of optimization of the vector
X. In this case the optimal service life found by the above--indicated method must
hereafter be considered when determining the cost of building and maintaining the _
ship in the optimization criteria for the vectoX X.
If the function 3(X, t) depends on X, then optiinization of the service life must be
carried out jointly with optimization of the vector X. The service lif e in this
case is included in the vector X as one of the componeazts, and the function ~ (X)
is considered as the effectiveness index.
The optimal values of Xopt and ToPt can also be found successively. Initially, the
problem
r
~ 3 (X, t) dt
b
mrX S (X) -}-s; (X) T
is considered.
The solution of this problem f or each f ixed X def ines the function T t(X) which
subsequently must be used when constructing the criteria of optimizaffon of the
- vector X. It is obvious that in the procedures for searching for the optimal ver-
sion of the ship (the vector Xo t) by the method of successive approximations there
is no necessity for finding theP function T (X) in advance. In this case, it is
necessary to include the solution of the par�ila1 problem of optimization of the
service life for fixed vector X in the set of algorithms for the calculations per-
formed in each step of the process. This service life is assumed for calculation
of the cost of building and maintaining the version of the ship corresponding to
the given vector X.
Now with certain alterations let us consider the approach to optimization of the
service lif e of technical means contained in the Manne article [63].
It is proposed that the fixed budget can be spent in three areas: the development
of new, improved types of ships of a given class, the building of ships, maintenance
of ships as part of the fleet.
Let us make the following asswnptions:
1. The expenditures on maintaining ships per unit time are proportional to the
expenditures on building them. If the expenditures on building in the time t are
equal to xt, where x are the expenditures on building per unit time, then the
expenditures on maintenance per unit time of all the ships in operation ;aill be
cxt, where c is a constant having dimensiona.lity the inverse of time;
2. The introduction of each new type of ship into construction requires certain
constant expenditures S connected with the development and introduction of a new
series into production;P
3. The total ef�ectiveness 3E of all shipa in operation at each point in time is
an additive function of the effectiveness of individual ships, that is, is
the sum of the effectivenesses of individual ships;
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4. The real obaolescence functi,on 3(i) considering repaixs and possible modifica-
F:ti,ons is .appxoximated by a monotonically nonincreasing function of timet that is, ,
d3 (t),:dt 4 0 for all t> 0. The time t s 0 is considered to be the present (cur-
~reitt) ,po3,nt in time, and the time t='r is 't years in the past from t= 0(the
reckoning nf time.goes in the opposite direction).
;If ne,w types of ships are introd,uced every v years, and the old types are talcen out
of the fleet after nz years, where n is an iuteger, then Y. can .be represented in
,the form
n
3E (X, n, z) _ R .Mk3 (kz), (6.37)
where .Mk is the number of ships conatructed in the time intervals [kz, (k - 1) z],
,.k = l, n. The lengths of these intervals are identical and equal to z years.
Figure 6.8. Geometric meaning of the index (6.38).
If independently of the type of ship the expenditures on building it are equal to
S, then ,Mk = xz/S, k= l, n. The expression (6.37) is written as follows in
this case:
1
3E (X, n, z) = S xa 2] ~(kz).
Inasmuch as S= const, during optimiaation it is possible to use the index
n
XZ Fj 3 (kZ),
k=L
(6.38)
which must be maximized with respect to the variables x, n and z. Geometrically
the index (6.38) is a value propor.tional to the area of the rectangles crosshatched
in Figure 6.8.
Let us denote the total expenditures on development, buiTding and maintenance of
ships in the time z by zsE, where s. is the total expenditures per unit time, for
example, for a year.
Obviously the equality
xz cxnz' So = zsE.
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(6.39)
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The problem of maxitnizing the total ef f ectivenese of the ships in operation
r a
max I xz F. 3 (kz) , .
~ k=l
z T cxnz + Z p = sm, . (6.40)
x>0, na0, z>0.
~ Let us assume that the function is exponential, that is 3(t) = at, 0 0; C> 0).
By introducing a new variable x= D/C and the notation a- B/C1/3, B= B/(1 - A),
_ C= C/ (1 - A) the equation (II.1.1) is reduced to the form
ax�/. + 1 = z.
Table 11. 1. Values of the function f(a)
a (
f(a) I
o I
1(0) I
Q.. I !(a)
a I F(a)
0110
1,107
0,60
1,930
1,10
3,57
1,60
6,67
0,15
1,166
0,65
2,05
1,15
3,80
1,65
7,09
0,20
1,229
0,10
2,18
1,20
4,05
1,70
7,53
"0,25
1,297
0,75
2,31
1,25
4,31
1,75
8.00
0,30
1,370
0,80
2,46
1,30
4,59
1,80
8,49
0,35
1,448
0,85
2,61
1,35
4,89
1,85
9,01
0,40
1,531
0,90
2,78
1,40
5,20
1,90
9,56
0,45
1,621
0,95
2,96
1,45
5,54
1,95
10,1
0150
1,717
1,00
3,15
1,50
5,89
2,00
10,7
0,55
1,820
1,05
3,35
( 1,55
I
6,27
2,05
11,4
11.1.2)
If the solution of the equation (II.1.2) is found as a function of the parameter a,
that is, the function x- f(a) is defined, then the solution of the initial equa-
tion (lI. 1. 1) will be D= Cf (a) or in expanded form
D= C ( B 1
A f 1(I - A)'I, Cll, ~ (II .1..3)
The values of the function f(a) for 0.1 < a< 2.05 with step Aa. = 0.05 are presented '
in Table II.1. If it is necessary to obtain the intermediate values of f(a) the
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lineas interpolation ia carried outl. The Sraphs of the function f(a) for 0< a<
~ and the function
7 (a) =1+ 2,3a',
which apgroximates f(a) well in the range of 0.6 K a< 2 are illustrated in Figure
A. 1.
fta),
to
e
fYal
.
/
r
10
/
~
f
.
~
p
4 0,8
,2
,6
6
4
2
a
Figure II.1. The graphs of the functions f(a) and f(a).
In accordance with the function (II=:1.4) the solution of equation (Il.l.l) can be
found by the approximate formula
D 1 C A~1 -~-2'3 I g/~ ]2(1 - A).C'/.
In the ranSe of values
0,6 G(I _ A~l. C'1. ~ 2,0
the relative errOr of formula (1I.1.5) does not exceed 5-6% by comparison with the
euact solution of equation (ll.l.l).
The method of solving the three-term mass and volume equati.ons using the table of
values of the auxiliary function f(a) and also by formula (II.1.5) is convenient for
cal.culating estima.tes without using a computer. The given method of solving the
mase sud volume equations, including formula (II.1.5) are applicable only for
C#0 (C> 0).
1The table.of values of thP functioa f(a) was calculazed by A. M. Ivanov.
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APPENDIX 2. COORDINATES OF THE TEN-DIMENSIONAL SOBOL' POINTS (WITH ROUNDING TO
THE THIRD PLACE)
I
~
/
I 1
I 2
I 3
I 4
I 5
I 6
I 7
8
9
I 10
� 0
0
0
Q
0
0
0
0
I
0
0
0
I
0,500
0,500
0,500
0,500
0,500
0,500
0,500
0,500
0,500
0,500
2
0,250
0,750
0,250
0,750
0,250
0,750
0,250
0,750
0,250
0,750
3
0.750
0,250
0,750
0,250
0,750
0,250
0,750
0,250
0,750
0,250
.4
0,125
0,625
0,875
0,875
0,625
0,125
0.375
0,375
0,875
0,625
5
0,625
0,125
0,375
0,375
0,125
0,625
0,875
0,875
0,375
0,125
6
0,375
0,375
0,625
0,125
0,875
0,875
0,125
0,625
0,125
0,875
7
0,875
0,875
0,125
0,625
0,375
0,375
0,625
0,125
0,625
0,375
8
0,063
0,938
0,688
0,313
0,188
0,063
0,438
0,563
0.813
0,688
9
0,563
0,438
0,188
0,813
0,688
0,563
0,938
0,063
0,313
0,188
10
0,313
0,188
0,938
0,563
0,438
0,813
0,188
0,313
0,063
0,938
11
0,813
0,688
0,438
0,063
0,938
0,313
0,688
0,813
0,563
0,438
12
0,188
0,313
0,313
0,688
0,563
0,188
0,063
0,938
0,188
0,063
13
0,438
0,563
0,063
0,438
0,813
0,938
0,313
0,188
0,938
0,313
14
0,938
0,063
0,563
0,938
0,313
0,438
0,813
0,688
0,438
0,813
15
0,031
0,531
0,406
0,219
0,469
0,281
0,969
0,281
0,094
0,844
16
0,531
0,031
0,906
0,719
0,969
0,781
0,469
0,781
0,594
0,344
17
0,281
0,281
0,156
0,969
0,219
0,531
0,719
0,531
0,844
0,594
18
0,781
0,781
0,656
0,469
0,719
0,031
0,219
0,031
0,344
0,094
19
0,156
0,156
0,531
0,844
0,844
0,406
0,594
0,156
0,969
0,469
20
0,656
0,656
0,031
0,344
0,344
0,906
0,094
0,656
0,469
0,969
21
0,406
0,906
0,781
0,094
0,594
0,656
0,844
0,906
0,219
0,219
22
0,906
0,406
0,281
0,594
0,094
0,156
0,344
0,406
0,719
0,719
23
0.094
0,469
0,844
0,406
0,281
0,344
0,531
0,844
0,781
0,406
24
0,594
0,969
0,344
0,906
0,781
0,844
0,031
0,344
0,281
0,906
25
0,844
0,219
0,094
0,156
0,531
0,094
0,281
0,594
0,531
0,656
26
0,219
0,814
0,219
0,531
0,906
0,469
0,906
0,719
0,156
0,781
27
0,719
0,344
0,719
0,031
0,406
0,969
0,406
0,219
0,656
0,281
28
0,469
0,094
0,469
0,281
0,656
0,719
0,656
0,469
0,906
0,531
29
0,969
0,594
0,969
0,781
0,156
0,219
0,156
0,969
0,406
0,031
30
0,016
0,797
0,953
0,673
0,797
0,922
0,734
0,890
0,547
0,828
31
0,516
0,297
0,453
0,172
0,297
0,422
0,234
0,390
0,047
0,328
32
0,266
0,047
0,103
0,422
0,547
0,172
0,984
0,141
0,297
0,578
I
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APPENDIX 3. SOME EXPRESSIONS USED IN THE INITIAL STAGES OF SSIP DESIGN
l. For invariant values of the engine power (N) and the admiralty coefficient (Cw)
tbe maxfmum apeed.(v) of a ship is inversely proportional to the displacement D:
v = ( NCm) 1/30-2/9 ^-D-2/9.
2. For invariant values of the specif ic mass of the power plant (qP the co-
efficients C and A(a is the coefficient of the masa or volume equafion) the maxi-
mum speed ofwthe ship increases with an increase in the given displacement (DD).
According to the mass equation we have
D = (Cm/93Y)1l3 / 1 _ A _ Emc/Dal 113D01/9 (n. 3. 2)
1
- Key: 1. PP = power plant
When using the volume equation instead of A it is necessary to substitute AQ, and
instead of Em; the value of r V; (the sum of the volumes which do not depend on
diaplacement). If, in addition ~ ml/D;, = idem, then v- D~19, that is, the speed
~
increases.very slowly.
3. The variatian of the maximum speed of a ship with an increase in power as a re-
sult of an increase in the number of echelone of the power plant (the power of each
echelon is fixed, the number of screws is equal to the number of echelons) is:
un 1/3 1/3 1- A l2~9
vl n kcn rI _ A~� (kmn _ 1) mYyJ f (R. 3. 3)
~
where kcn =Cmn,C1, kmn =m3y/myy, m'sy ="~y/D~ and the index n pertains to the n-
echelon system, and the index 1, to the single-echeloa system.
The formula (II.3.3) corresponds to the case where the displacement is defined by
the masa equation and buoyancy (Dm > pDV). If the displacement is defined by the
v?Jyme equation, then ins e d of A it is aecessary to substitute~ , and instead of
mpp, thevalue of Vpp~. V~pIDl. Let us consider the example of us3ng expression
(IE.3.3) for submarines. Let n= Z, k~i2 ~ 29 kc2 = 0.7, A= 0.59, m.pp0.16 to 0.30
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[10]. For theae initial data we havel v2/vl m 0.99 to 1.04, that is, the applica-
tion of the two^echelon power plant o� double power in the given case in practice
does not give a gain in speed. Inasmuch as the cofactor in brackets in expression
(1I,3.3) increases with a decrease in A, setting AV $3 0 with a margin and approxima-
tely setting m.Pp) , VPP), we obtain v2/vl 1.03 to 1.09. It is possible to ob-
tain a noticeable increase in the speed (by 12-20%) only for k ~ 1, that is, when
insuring identical propulsive qualities of the single-shaft an82 two-shaf t propul-
sion systems. However, here it is necessary to consider that the gain in speed is
_ connected with doub}ing the power of the power plant and increasing the displacement
by 1.4 to 1.7 times .
4. The formulas f or recalculating the lines plan curves for linear transformation
of a prototype: D= DokLkBlT (displacement), z= zOk,r (the y-coordinate of the
center of buoyancy), p= polcB/k.r (the transverse metacentric radius), x= xolcL
(the x-axis of the center of buoyancy and the center of flotation), R= ROkY/kT
(the longitudinal metacentric radius). Here kL, kB, lc,r are the coeff icients of
linear transformation of the principal dimensions L, B, T.
5. The limits of variation of the lines coeff icients and the ratios L/B and B/T
for surface ships~:
Class of
shi s
a
a
s
c.le
~ e/r
Cruisers
0,45-0,60
0,69-0,73
0,76-0,90
8,5-11,3
2,6-4,2
Destroyers
0,44-0,53
; 0,68-0,73
0,75-0,86
9,2-I 1,9
2,5-4, I
Patrol
vessels
0,40-0,55
0,75-0,85
0,75-0,85
8,3-10,1
2,6-4,0
Trawlers
0,50-0,60
0,65-0,80
0,75-0,95
6,4-7,5
3,5-4,3
6. The relative wetted surface of suhmarines underwater [10]: with "stem" lines
w= 6.05 + 0.26L/B; with hulls in the form of solids of revolution w= 5.60 +
0.26L/B.
7. Values of the total drag coefficient and the proppulsive coeff icient of modern
American submarines underwater [10]: single-shaft ~ s(3.0 to 3.2)�10 3, n= 0.80
to 0. 85; two-ehaf t(3. 2 to 3.5) � 10"'3, 11 = 0.65 to 0.75.
8. The heel period ('re) and pi,tch period (T~) o� ships in calm water
-When selecting the number of echelons of
consider the reliability factor which can
the f inal decision.
ZA. V..Gerasimov, A. I. Pastukhov, V.I, So
tals of Ship Theory), Moscow, Voyenizdat,
the power plant i,t is also necessary to
turn out to be signif icant when making
lpv'yev,QSNQpY TEORII KORABLYA (Fundamen-
1958.
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2n B 2n�
ze = V8 ke v,p a-~-Ily1I~T~,
Vg
where B= T axe the beam and draf t of the ship, he is the transverse metacentric
heigh.t, g is the gravitational acceleration, ke, k* are dimensionless coeff icients.
9. Tfie steady turning radius of a surface ship
R4 = (kiik2) (D/Sp).
(1)
(II.3.5)
Keq; 1. turning
where D is the displacement, m 3 ; Sp is the rudder area, m 2 ; kl, k2 are dimensianless
coeff icients.
The coefficient kl is def ined by the value of (S/e)(B/L) (L, B are the length and
beam of the ship, 8 is the blodc coefficieat, e is the coeff icient of f ineness of
the submerged part of the longitudinal center plane):
(S/e) (B!L) . . . . . . . . . . 0,05 0,075 0,10 0,125 0,15
kl . . . . . . . . . . . . . . 1,5 0,71 0,46 0,33 0,29
The coeff icient k2 depends on the angle of helm ap:
kp, degrees . . . . . . . . . . . . . . 0 t0 20 30
9 0 0,61 0,89 1,0
The speed drop in the turning circle can be approximately defined by the formula of
G. A. Firsov
v Ru
vu - th 2,45L '
(1)
Key: 1. turn
(II.3.6)
10. The distribution (in percentage of the normal displaceffient) of the mass load
of nonnuclear surface ships with artillery armament [35]:
Class of I
Load
1'"K' m6~ '"e% "'M~ mT~
~
ID368e8
Of the hull~
armor;
1~8-~
meat; machinery,
fuel,
water
and oil;
equ~.pment,
consum-
able materials and crew.
ships mK I
m6
I ma
l1tN
nr
mc
Battleships
28-40
22-40
12-22
6-12
3-12
1-2
Heavy cruisers
28-34
14-33
13-15
10-20
11-15
2-3
Light cruisers
34-38
15-22
10-15
12-28
7-10
2-3
Destroyers
32-36
0-2
10-15
32-38
10-16
3-4
225
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11. Distxi.but3,on (3,n percentage of normal displacement) of the mass load of
American nuclear-power submarines 110J;
Load divisions
Subclass of submarine
Torpedo
Missile
ull
38-42
40-44
chinery
26-40
14-18
lectrical equipment
4-6
3-5
Radioelectronic equipment
2-3
1-2
Systems and devices
7-8
5-7
,Equipment
2-3
2-3
Armament
2-4
10-14
Fuel and drinking water
3-4
2-3
Crew and supplies
3-4
2-3
Reserve displacement
2-3
2-3
Solid ballast
2-3
2-3
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COPYRIGHT: Izdate7.'stvo "Sudostroyeniye", 1980 [8144/0419-10845]
- 10845
CSO: 8144/0419
~
END
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