Mathematical Models of Target Coverage and Missile Allocation

261 Pages·2012·12.92 MB·English

Mathematical Modes
of Target Coverage and
Missile Allocation
1%1% A. ROSS ECKLER
Lf STEFAN A. BURR
DTI-0-
PUBLSHED BY OCT 1 0 19184
"',. Reviewed & Edited by the MORS
Standing Publications Committee
DR. JAMES K ARIMA CDR JAMES J MARTIN, USN
DR. WALTER DEEMER MR. SIDNEY MOGLEWER, Chairman
DR. STANLEY L. DOLINS MR. ALFRED S. RHODE
D(cid:127)R. ROBERT J. LUNDEGARD DR. JACOB A STOCKFISCH
Z23 Sponsors
I The Chief of Naval Research
The Chief of Research & Development, U.S. Army
The Assistant Chief of Staff, Studies & Analysis, U.S. Air Force
(cid:127) 0'2' "i 1309
MORS Board of Directors
LTC Richard Wm Anson. USA
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Mr Norman Farrell
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Mrs Joann K Langston
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Mr Sidney Moglewer
CAPT Edward D Napier. USN
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I
I-4
Models
Mathematical
of Tarqet Coverage andA
Missile Allocation
A. ROSS ECKLER
STEFAN A. BURR
OnC T.. -1r I-3
PUBLISHED BY ..
.. 101..,,
,.
SRep roduction in whole or in part is permitted
fcr any purpose of the U.S Government.
1972
I
Copyright By ©
Military Operations Research Society
1972
1.
I
HI STOR ICAL NOTE
The art of projecting missiles is very old, dating back at
least to the Roman ballista, but it was placed on a scientific
footing until the sixteenth century, when the Italian mathematician
Niccolo Fontana Tartaglia studied the trajectories of missiles fired
from weapons ranging from pistols to cannon. As the first mathe-
matician to optimize the aim of a weapon one might call him the
prototype of the modern missile analyst. Yet his knowledge was
purchased at a price, as revealed by the following passage toward
the end of the dedication in his Nova Scientia Inventa (1537):
"But then in reflecting one day it struck me as blame-
worthy, infamous, and cruel, and meriting no small
punishment before God, to wish to refine an art so
injurious to one's fellow men - a vile destroyer of
the human race, and especially of Christians in their
incessant warfare."
Similar misgivings about the social consequences of scientific work
devoted to war have been expressed ever since, culminating in the
angst of the atomic scientists after World War II. The authors of
this monograph are not immune: but our concern has been tempered
by the hope that a quantitative understanding of missile defense
strategies may actually reduce the probability of intcrnational con-
flict. At the very least, this monograph should discourage any
naive belief that a perfect defense is possible.
.............. 'n o r
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11'
FOREWORD
The publication of this monograph represents a new venture
in our continuing effort to broaden the services that the Military
Operations Research Society (MORS) offers to the professional
military operations research analyst. It is our hope that the re-
sponse generated by this publication will encourage a continuing
series of monographs of special interest to our society. In particu-
lar, we place lugh value on the encouragement to authors that such
a series might offer and the consequent enlargement of the litera-
ture of military operations research.
The MORS is extremely fortunate and proud to have Dfs.
Ecker and Burr's monograph as our first publication. A first pub-
lication always sets a standard for others to follow. As such, this
monograph represents the highest standards of both technical excel-
lence and relevance to military operations research.
I also vwant to recognize the MORS committee on publication
and its Chairman, Mr. Sid Moglewer, who conceived of this project
-
and carried it through to the very successful conclusion. Particu-
lar recognition should go to Dr. Walter Deemer, who as a member
of that Committee identified the original manuscript and gave gener-
ously of his time to its publication.
ROBERT H. STEVENS
President 1971-72
AI
I(cid:127)
PREFACE
It is commonplace for the authors of a survey monograph to
invite readers to submit additions or corrections for a possible
later edition. We are keenly aware that we are guilty of sins of
omission, for the literature on the target coverage and missile al-
location problem is widely dispersed, and much of it is virtually
inaccessible to the layman. We are interested in giving credit for
priority associated with each methodology: however, our major in-
terest is not in tracing the historical thread of a development but in
making sure that the most. important ideas have been brought to-
gether and systematically compared.
In a book with two authors, questions inevitably arise concern-
ing the nature of each one's contributions. The senior author (A. R.
Eckler) has been responsible for searching the literature for rele-
"vant material, for deciding upon the basic structure of the book, and
(for the most part) for writing up results in a form intelligible to
the non-mathematical reader. The junior author (S. A. Burr) has
been responsible for correcting, clarifying and occasionally develop- A
ing in detail the mathematics, as well as improving the organization
and exposition of the monograph in many sections. This division of
responsibility may help the reader decide to whom any criticisms,
additions or inquiries should be addressed.
Many sections of this monograph were originally developed
by Bell Telephone I,aboratories colleagues of the authors during
the years 1965 through 1970; their work has materially enhanced
the scope of this book One of the motivations for writing this book
was to bring their excellent work to the attention of a wider audience.
These contributors were:
D. J. Brown* J . A. Hooke M. J. Spahn*
J. Eilbott S. Horing C. W. Spofford
M. L. Eubanks* D. Jagerman R. E. Thomas
D. Guthrie* H. Polowy* M. S. Waterman*
H. Heffes W. L. Roach F . M. Worthington*
S. A. Smith
An asterisk after the name indicates that the author is no longer as-
sociated with Bell Telephone Laboratories. It is hoped that sufficient
details of their work have been given to satisfy the needs of most
readers interested in missile allocation strategies. However, the
occasional reader who requires more detailed information about
these models may telephone the senior author at Bell Telephone
Laboratories, Holmdel, New Jersey.
I
A. Ross Eckler A
May 7, 1972 Stefan A. Burr
TABLE OF CONTENI S
1. An Outline of Objectives and Some Underlying Assumptions
1.1 The Choice of a Criterion of Effectiveness
1.2 Some Comments on the Scope of the Monograph
1.3 Another Survey of the Missile Allocation Problem
1.4 A Survey of Mathematical Techniques
1.5 Some Comments on Terminology and Notation
1.6 Summary A
2. Point and Area Targets in the No-Defense Case
2.1 Survival/Destruction Probabilities for One or More Point
Targets
2.1.1 Equal Variances, Distribution Centered at Origin
2.1.2 Equal Variances, Offset Distribution
2.1.3 Unequal Variances, Distribution Centered at Origin
2.1.4 Unequal Variances, Offset Distribution
2.1.5 Targets With More Than One Point
2.1.6 Models of Aiming Error Associated With a Salvo
2.2 Expected Fractional Damage of a Uniform-Valued Circular
Target
2.2.1 One Weapon Impact, Gaussian Aiming Error
2.2.2 Multiple Weapon Impacts, Gaussian Aiming Error
2.2.3 Offense Can Place All Weapons Exactly
2.3 Expected Fractional Damage of a Gaussian Target
2.3.1 One Weapon Impact, Gaussian Aiming Error
2.3.2 Multiple Weapon Impacts, Gaussian Aiming Error
2.3.3 Non-Gaussian Aiming Error
2.3.4 Offense Can Place All Weapons Exactly
2.3.5 A Generalization of the Gaussian Target
2.4 The Diffused Gaussian Damage Function
2.4.1 Alternative Damage Functions
2.4.2 One Weapon Impact, Various Target Charactoristics
2.4.3 Gaussian Target, More Than One Impact
2.4.4 Uniform Circular Target
2.5 Matching the Attack Dispersion to an Area Target
2.5.1 Multiple Aiming-Points
2.5.2 The Maximum Expected Damage if the Attacker
Selects the Variance of the Weapon Impact-Points
2.5.3 An Upper Bound to the Expected Damage if the
Attacker Selects Any Probability Density Function of
Weapon Impact- Points
2.5.4 An Asymptotic Probability Density Function of
Weapon Impact-Points for a Gaussian-Valued Target
2.6 Estimating the Probability of Survival/Destruction From
Impact-Point Data
2.6.1 Estimation of the Probability of impact Within a Circle
2.6.2 Estimation of thc Radius of a Circle Corresponding
to a Given Impact Probability
2.6.3 Estimation of the Parameters of a Diffused Gaussian
Damage Function
2.7 Offensive Shoot-Adjust-Shoot Strategies
2.8 Attack Evaluation by Detense Using Rada," Information
2.9 Summary
3. Defense of a Target of Unspecified Structure
3.1 Defense Strategies A-ainst Weapons -I Unknown Lethal
Radius
3.2 Defense Strategies Against a Sequential Attack of Unknown
Size
3.2.1 Maximizing the Expected Rank of the First Penetrator
3.2.2 An Exact Procedure for Maximizing the Expected Rank
3.2.3 A Constant Value Decrement Criterion
3.2.4 Known Distribution on Attack Size
.. 2.5 The Selection of an Attack Distribution
3.3 Defense Strategies Against a Sequential Attack by Weapons
of Unknown Lethal Radius
3.3.1 Maximizing the Probability of Intercepting the Nearest
Weapon
3.3.2 Maximizing the Total Score of the Intercepted
Weapons
3.4 Defense Strategies Against a Sequential Attack Containing
Exactly One Weapon Mixed With Decoys
3.5 Shoot- Look-Shoot Defense Strategies
3.5.1 A Two-Stage Shoot- Look-Shoot Strategy
3.5.2 Time-Limited Shoot-Look-Shoot Strategies
3.6 Defenses Limited by Traffic-Handling Capab-ility
3.7 Sum mary
4. Offense and Defense Strategies for a Group of Identical Targets
4.1 Preliminaries Concerning Preallocation Strategies
4.2 Offense- Last-Move and Defense- Last-Move Strategies
4.3 Strategies When Neither Side Knows the Other's Allocation
4.3.1 An Explicit Solution to the Preallocation Problem
4.3.2 A Simplified Problem: Perfect Offensive Weapons
and Defensive Missiles
4.3.3 Arriving at Integral Allocations
4.3.4 Generalizations of the Preallocation Problem
4.3.5 The Variation in the Number of Targets Surviving
in a Matheson Game
4.3.6 Other Models of Preallocation Offense and Defense
4.4 Some Nonpreallocation Strategies
4.4.1 A Group(cid:127) Preferential Defense Strategw; Against an
Offense That Can Valy His Attack Size
4.4.2 A Gi oup Preferential Defense Strategsy Against an
Offense of Fixed Size
4.4.3 A More General Class of Nonpreallocation Strategies
4.4.4 A Nonpreallocation Strategy Involving a Stockpile of
Defensive Missiles Held in Reserve
4.5 Defense Dama-e Assessment Strategies
4.5.1 Damage Assessment Strategies When the Attack is
4.5.2 KDnamowagn- et oA tshsee sDsmefieennste. Strategies Against Attacks of
Size
Unknown
4.6 Attacher-Oriented Defense Strategies
4.6.1 Neither Side Knows the Other's Allocation
4.6.2 Offense Knows How Defense Will Assign All Missiles
4.7 Offensive Damage Assessment Strategies
4.7.1 Strategies if Targets are Soft and Defensive Missiles
are Reliable
4.7.2 Strategies if Targets are Hard and Defensive Missiles
are Reliable
4.7.3 Strategies if Defensive Missiles are Unreliable
4.7.4 Damage Assessment ior Unconstrained Offenuivc
Weapon Stockpiles
4.8 Summary
5. Offense and Defense Strategies for a Group of Targets With
Different Values
5.1 Offense Allocation to a Group of Targets in the No-Defense
Case
5.2 Two General Techniques for One-Sided Allocation Problems
5.2.1 Dynamic Programming
5.2.2 Lagrange Multipliers
5.3 General Methods for Constructing Two-Sided Offense-Last-
Move Strategies
5.3.1 A Lagrangian Approach to Max-Min Problems
5.3.2 A Dynamic Programming Approach to Max-Min Problems
5.4 Two-Sided Offense-Last-Move Strategies Using a Special
Payoff Function
5.4.1 A Partial Solution in an Idealized Case
5.4.2 An Approximate Solution in a Limiting Case
5.5 Two-Sided Offense- Last-Move Strategies for Reliable
Missiles
5.5.1 A Lagrangian Approach to a Specific Max-Min Problem
5.5.2 A Special Case: Reliable Weapons