Title:
Mathematical methods in defense analyses
Personal Author:
Series:
AIAA education series
Publication Information:
Reston, VA : American Institute of Aeronautics and Astronautics, 2000
ISBN:
9781563473975
Available:*
Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
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Searching... | 30000010127350 | U104 P79 2000 | Open Access Book | Book | Searching... |
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Summary
Summary
A presentation of the various mathematical methods used in military operations research. It provides the calculations necessary for analyzing all aspects of defence operations, from weapon performance to combat modelling. Included is an updated version of Defense Analyses Software.
Author Notes
Dr. Przemieniecki was born in Poland and earned his Ph.D. and D.Sc. degrees from the University of London. In 1999, he received an honorary doctorate Honoris Causa from the Warsaw University of Technology
Table of Contents
Preface | p. xix |
Chapter 1. Scientific Methods in Military Operations | p. 1 |
1.1 Introduction | p. 1 |
1.2 Mathematical Methods | p. 4 |
Theory of Combat | p. 5 |
Decision Theory | p. 6 |
Linear Programming | p. 6 |
Queuing Theory | p. 6 |
Network Analysis | p. 7 |
Game Theory | p. 7 |
Differential Games | p. 7 |
War-Gaming and Simulation | p. 7 |
Future Directions of Research | p. 8 |
1.3 Quantitative Competence | p. 8 |
References | p. 9 |
Problems | p. 10 |
Chapter 2. Characteristic Properties of Weapons | p. 11 |
2.1 Weapon Performance Data | p. 11 |
2.2 Empirical Averages | p. 15 |
Arithmetic Mean (Average) | p. 15 |
Median | p. 16 |
2.3 Scatter or Dispersion | p. 16 |
Range | p. 16 |
Interquartile Range | p. 16 |
Mean Deviation | p. 17 |
Standard Deviation | p. 17 |
Coefficient of Variation | p. 18 |
2.4 Mean Point of Impact: Systematic Error | p. 19 |
Problems | p. 19 |
Chapter 3. Passive Targets | p. 21 |
3.1 Binomial Distribution | p. 21 |
3.2 Normal (Gaussian) Distribution | p. 25 |
3.3 Poisson Distribution | p. 26 |
3.4 Probability Density Distribution: Linear Error Probable (LEP) | p. 28 |
3.5 Single-Shot Probability of Hit for a Rectangular Target | p. 31 |
3.6 Probability of Target Kill or Hit for Multiple Shots | p. 32 |
3.7 Probability of Destruction of a Point Target: Circular Error Probable (CEP) | p. 34 |
Case 1 Zero Offset and Equal Variances | p. 36 |
Case 2 Offset Distribution and Equal Variances | p. 38 |
Case 3 Zero Offset and Unequal Variances | p. 41 |
Case 4 Offset Distribution and Unequal Variances | p. 42 |
3.8 Probability of Hit of a Rectangular Target: The Polya-Williams Approximation | p. 42 |
Case 1 Equal Variances: Square Target | p. 42 |
Case 2 Nonequal Variances: Rectangular Target | p. 44 |
3.9 Probability of Hit of an Elliptic Target with Unequal Variances | p. 45 |
3.10 Probability of Destruction of a Point Target in Space | p. 46 |
3.11 Linear, Circular, and Spherical Error Probables | p. 49 |
3.12 Expected Fractional Damage of a Uniform-Valued Target | p. 50 |
3.13 Damage Functions for a Point Target in a Plane | p. 56 |
Gaussian Damage Function | p. 57 |
Exponential Damage Function | p. 58 |
Inclined Step Damage Function | p. 58 |
Log-Normal Damage Function | p. 59 |
3.14 Shrapnel Damage Functions | p. 61 |
Surface Targets | p. 61 |
Space Targets | p. 63 |
3.15 Weapon Effective Radius (R[subscript w]) for Surface Targets | p. 64 |
Gaussian Damage Function | p. 66 |
Exponential Damage Function | p. 66 |
Inclined Step Damage Function | p. 66 |
Log-Normal Damage Function | p. 66 |
Shrapnel Damage Function | p. 67 |
3.16 Weapon Effective Radius (R[subscript w]) for Space Targets | p. 68 |
References | p. 69 |
Problems | p. 69 |
Chapter 4. Deterministic Combat Models | p. 71 |
4.1 Effective Firing Rate: Attrition Rate Coefficients | p. 71 |
4.2 Markovian Attrition Rates: Probabilistic Rates | p. 72 |
4.3 Lanchester Model for Directed Fire: Square Law | p. 78 |
4.4 Lanchester Model for Area Fire: Linear Law | p. 82 |
4.5 Guerrilla Warfare Model: Mixed Law | p. 87 |
4.6 Autonomous Fire Model: Logarithmic Law | p. 90 |
4.7 Geometric Mean Model: Linear Law | p. 91 |
4.8 Helmbold Models: Size Effects | p. 93 |
4.9 Autonomous/Directed Fire Model | p. 95 |
4.10 Force Parity | p. 95 |
4.11 Battle Disengagement: Force Breakpoints | p. 96 |
4.12 Variable Attrition Coefficients: Mobile Attack Model | p. 97 |
4.13 Force Reinforcements in Combat | p. 101 |
Arbitrary Variation of P(t) | p. 102 |
4.14 Mixed Models | p. 104 |
4.15 Iwo Jima Battle | p. 105 |
4.16 Casualty Exchange Ratio | p. 108 |
Directed Fire | p. 108 |
Area Fire | p. 110 |
References | p. 111 |
Problems | p. 112 |
Chapter 5. Probabilistic Combat Models | p. 113 |
5.1 Sequential Combat Duel: Time-Independent Combat | p. 113 |
5.2 Continuous Combat Duel: Time-Dependent Combat | p. 116 |
5.3 Continuous Combat States: Directed Fire Model | p. 121 |
5.4 Continuous Combat States: Area Fire Model | p. 127 |
5.5 Many-on-Many Engagements | p. 134 |
Uniform Assignment of Targets | p. 135 |
Random Assignment of Targets | p. 137 |
Shoot-Look-Shoot Assignment of Targets | p. 138 |
References | p. 141 |
Problems | p. 141 |
Chapter 6. Strategic Defense | p. 143 |
6.1 Strategic Defense Initiative: Layered Defense | p. 143 |
6.2 Layered Defense Against MIRVed Attack | p. 149 |
6.3 Antiballistic Missile (ABM) Defense: Game Theory | p. 153 |
Game Theory: Payoff Matrix | p. 154 |
Two-Person Zero-Sum Game: Pure and Mixed Strategies | p. 155 |
ABM Defense Analysis | p. 158 |
6.4 Optimal Penetration Routes Through Air Defenses: Threat Function | p. 165 |
6.5 Defense Against a Penetrating Aircraft | p. 169 |
Direct Penetration of the Defended Missile Site | p. 169 |
Offset Penetration in Relation to SAM Site | p. 173 |
6.6 Sufficiency Models for Strategic Forces | p. 177 |
Equivalent Megatonnage (EMT) | p. 177 |
Counter Military Potential (CMP) | p. 179 |
Comparison of Strategic Nuclear Forces | p. 180 |
References | p. 182 |
Problems | p. 182 |
Chapter 7. Theater Missile Defense | p. 185 |
7.1 Concept for the Theater Missile Defense | p. 185 |
7.2 Probability of Penetration of Missiles (or Tanks) | p. 186 |
Types of Defensive Tactics | p. 186 |
Random Assignments of Targets | p. 187 |
Uniform Assignments of Targets | p. 190 |
Shoot-Look-Shoot Assignments of Targets | p. 193 |
7.3 Probabilities of Zero penetration | p. 195 |
References | p. 198 |
Problems | p. 198 |
Chapter 8. Tactical Engagements of Heterogeneous Forces | p. 199 |
8.1 Directed Fire "Many-on-Many" Engagements: Numerical Solutions | p. 199 |
8.2 Aggregated Forces | p. 202 |
8.3 Superiority Parameters | p. 203 |
8.4 Aggregated Force Solution | p. 208 |
8.5 Numerical Example: "One-on-Two" Tactical Engagement | p. 209 |
8.6 Comments on the Directed Fire Solution | p. 211 |
8.7 Area Fire "Many-on-Many" Engagements | p. 212 |
8.8 Guerrilla Warfare "Many-on-Many" Engagements | p. 215 |
8.9 Positioning of Defense Forces | p. 217 |
Front Line Segment Defense | p. 217 |
Optimal Mobile Defense: Single Segment | p. 218 |
Case 1: v[subscript a] = v[subscript d] | p. 220 |
Case 2: v[subscript a] [ v[subscript d] | p. 222 |
Case 3: v[subscript a] ] v[subscript d] | p. 224 |
Optimal Mobile Defense: Multiple Segments | p. 226 |
References | p. 228 |
Problems | p. 229 |
Chapter 9. Reliability of Operations and System | p. 231 |
9.1 Reliability of Series Operations | p. 231 |
9.2 Reliability of Parallel (Redundant) Operations | p. 233 |
9.3 Reliability of Combined (Series and Parallel) Operations | p. 235 |
9.4 Example of an Air-to-Air Engagement | p. 236 |
9.5 Reliability Variation with Time | p. 236 |
Exponential Distribution | p. 239 |
Normal Distribution | p. 239 |
Log-Normal Distribution | p. 243 |
Weibull Distribution | p. 243 |
9.6 Derivation of Reliability from Probabilistic Considerations | p. 245 |
9.7 Hazard Function h(t) | p. 251 |
9.8 Computation of a Reliability Function from Experiment | p. 253 |
The Probability Graph Method | p. 255 |
The Kolmogorov-Smirnov (K-S) Test Method | p. 255 |
9.9 Maintainability of Weapon Systems | p. 255 |
9.10 Operational Availability of Systems | p. 257 |
References | p. 258 |
Problems | p. 258 |
Chapter 10. Target Detection | p. 261 |
10.1 Intermittent Glimpses | p. 261 |
10.2 Continuous Search | p. 264 |
10.3 Variation of Detection Rate with Range: Inverse Square and Cube Laws of Detection | p. 266 |
Low Altitude Detection: r ]] h | p. 267 |
High Altitude Detection: h ]] r (Space Surveillance) | p. 267 |
10.4 Probability of Detection in Search of a Given Area | p. 268 |
Exhaustive Search | p. 268 |
Random Search | p. 269 |
Inverse Cube Law Search | p. 270 |
10.5 One-Dimensional Search | p. 272 |
10.6 Constant Velocity Target | p. 274 |
10.7 Detection of Electromagnetic Radiation from a Target | p. 278 |
References | p. 280 |
Problems | p. 280 |
Chapter 11. Optimization Methods | p. 283 |
11.1 Mathematical Optimization | p. 283 |
Classical Programming | p. 283 |
Classical Programming: Unconstrained Optimization | p. 286 |
Nonlinear Programming | p. 288 |
Linear Programming | p. 289 |
11.2 Application of the Lagrange Multiplier Method: A Cluster Bomb | p. 290 |
11.3 Examples of Linear Programming | p. 292 |
References | p. 296 |
Problems | p. 297 |
Chapter 12. Modeling | p. 299 |
12.1 Models | p. 299 |
12.2 Modeling of Military Operations | p. 301 |
Combat Mission | p. 302 |
Equipment | p. 303 |
General Models | p. 304 |
References | p. 308 |
Appendix A. Probability Tables | p. 309 |
A.1 Cumulative Binomial Probabilities | p. 309 |
A.2 Cumulative Poisson Probabilities | p. 309 |
A.3 The Normal (Gaussian) Probabilities | p. 310 |
A.4 Error Function erf(z) | p. 312 |
A.5 Single-Shot Probability of Hit on a Rectangular Target | p. 312 |
Appendix B. Derivation of the Characteristic Function [Phi subscript N](s) | p. 317 |
Reference | p. 319 |
Appendix C. Analytical Solution of Equations of Combat | p. 321 |
C.1 General Solution of X = CX | p. 321 |
C.2 Right and Left Generalized Eigenvectors | p. 324 |
C.3 Examples of the Dominant Left Eigenvectors | p. 325 |
C.4 Computation of the Dominant Left Eigenvectors and Their Eigenvalues | p. 327 |
References | p. 327 |
Appendix D. Calculation of the Average Probability of No Detection | p. 329 |
Appendix E. Defense Analysis Software | p. 331 |
E.1 Introduction | p. 331 |
E.2 Subroutine Instructions | p. 331 |
E.3 Passive Targets | p. 333 |
E.3.1 Binomial Probabilities | p. 333 |
E.3.2 Poisson Probabilities | p. 335 |
E.3.3 Normal (Gaussian) Probabilities | p. 337 |
E.3.4 Error Function | p. 339 |
E.3.5 Probability of Hit of a Rectangular Target | p. 340 |
E.3.6 Probability of Destruction of a Point Target with Offset Distribution | p. 342 |
E.3.7 Probability of Destruction of a Space Point Target with Offset Distribution | p. 343 |
E.3.8 Expected Fraction of Damage of a Circular Target | p. 344 |
E.3.9 Probability of Destruction of a Point Target with the Exponential Damage Characteristics | p. 344 |
E.3.10 Probability of Destruction of a Surface Point Target with the Inclined Step Damage Function | p. 345 |
E.3.11 Probability of Destruction of a Surface Point Target with the Log-Normal Damage Characteristics | p. 346 |
E.3.12 Weapon Effective Radius for Normal (Gaussian) Damage Function | p. 347 |
E.3.13 Weapon Effective Radius for Exponential Damage Function | p. 348 |
E.3.14 Weapon Effective Radius for Inclined Step Damage Function | p. 349 |
E.3.15 Weapon Effective Radius for Log-Normal Damage Function | p. 350 |
E.3.16 Weapon Effective Radius for Shrapnel Damage Function | p. 351 |
E.4 Deterministic Combat Models | p. 352 |
E.4.1 Directed Fire Lanchester Deterministic Model | p. 352 |
E.4.2 Area Fire Lanchester Deterministic Model | p. 354 |
E.4.3 Guerrilla Warfare Deterministic Model | p. 355 |
E.5 Probabilistic Combat Models | p. 357 |
E.5.1 Probabilistic Directed Fire Model | p. 357 |
E.5.2 Probabilistic Area Fire Model | p. 358 |
E.6 Strategic Defense | p. 359 |
E.6.1 SAM Defense for Direct Penetration | p. 359 |
E.6.2 SAM Defense for Offset Penetration | p. 361 |
E.7 Theater Missile Defense | p. 362 |
E.7.1 Missile Penetration for Random Assignments of Targets; 1-Layer Defense | p. 362 |
E.7.2 Missile Penetration for Random Assignments of Targets; 2-Layers Defense | p. 363 |
E.7.3 Missile Penetration for Uniform Assignments of Targets; 1-Layer Defense | p. 364 |
E.7.4 Missile Penetration for Uniform Assignments of Targets; 2-Layer Defense | p. 366 |
E.7.5 Probability of Zero Missile Penetrations for Specified Number of Warheads W | p. 367 |
E.8 Tactical Engagements of Heterogeneous Forces | p. 368 |
E.8.1 Heterogeneous Force Levels in Tactical Engagements (Directed Fire Model) | p. 368 |
E.8.2 Heterogeneous Force Levels in Tactical Engagements (Area Fire Model) | p. 371 |
E.8.3 Heterogeneous Force Levels in Tactical Engagements (Guerrilla Warfare Model) | p. 373 |
E.9 Reliability of Operations and Systems | p. 376 |
E.9.1 Normal Probability Density Function and MTTF | p. 376 |
E.9.2 Normal Reliability Function | p. 377 |
E.9.3 Normal Hazard Function | p. 379 |
E.9.4 Log-Normal Density Function and MTTF | p. 380 |
E.9.5 Log-Normal Reliability Function | p. 381 |
E.9.6 Log-Normal Hazard Function | p. 382 |
E.9.7 Weibull Probability Density Function and MTTF | p. 383 |
E.9.8 Weibull Reliability Function | p. 384 |
E.9.9 Weibull Hazard Function | p. 386 |
E.10 Target Detection and Search | p. 387 |
E.10.1 Search for Constant Velocity Target | p. 387 |
E.11 Miscellaneous | p. 388 |
E.11.1 Dominant Eigenvalue and Left Eigenvector | p. 388 |
References | p. 390 |
Index | p. 391 |