Mathematical Methods in Defense Analyses, Third Edition (Aiaa Education Series)

Annotation This text presents the various mathematical methods used in military operations research in one easy-to-use reference volume. The reader will find the calculations necessary to analyze all aspects of defense operations, from weapon performance to combat modeling. The text is so clearly written and organized that even newcomers to the field will find it useful. Included with the text is an updated version of Defense Analyses Software, a compendium of software subroutines that allow the reader to compute numerical values for functions or tables derived in the text. Each subroutine is provided with a detailed reference to the equation from which it was derived to ensure that its intended application is consistent with the assumptions used in the derivation. The third edition has a new chapter on theater missile defense based on the concept of layered defense with different strategies of allocating defense interceptors against short- or mid-range ballistic missiles Cover......Page 1 Title......Page 4 Copyright......Page 5 Forword......Page 10 Table of Contents......Page 12 Preface......Page 20 1.1 Introduction......Page 22 1.2 Mathematical Methods......Page 25 Theory of Combat......Page 26 Queuing Theory......Page 27 War-Gaming and Simulation......Page 28 1.3 Quantitative Competence......Page 29 References......Page 30 Problems......Page 31 2.1 Weapon Performance Data......Page 32 Arithmetic Mean (Average)......Page 36 Interquartile Range......Page 37 Standard Deviation......Page 38 Coefficient of Variation......Page 39 Problems......Page 40 3.1 Binomial Distribution......Page 42 3.2 Normal (Gaussian) Distribution......Page 46 3.3 Poisson Distribution......Page 47 3.4 Probability Density Distribution: Linear Error Probable (LEP)......Page 49 3.5 Single-Shot Probability of Hit for a Rectangular Target......Page 52 3.6 Probability of Target Kill or Hit for Multiple Shots......Page 53 3.7 Probability of Destruction of a Point Target: Circular Error Probable (CEP)......Page 55 Case 1: Zero Offset and Equal Variances......Page 57 Case 2: Offset Distribution and Equal Variances......Page 59 Case 3: Zero Offset and Unequal Variances......Page 62 Case 1: Equal Variances: Square Target......Page 63 Case 2: Nonequal Variances: Rectangular Target......Page 65 3.9 Probability of Hit of an Elliptic Target with Unequal Variances......Page 66 3.10 Probability of Destruction of a Point Target in Space......Page 67 3.11 Linear, Circular, and Spherical Error Probables......Page 70 3.12 Expected Fractional Damage of a Uniform-Valued Target......Page 71 3.13 Damage Functions for a Point Target in a Plane......Page 77 Gaussian Damage Function......Page 78 Inclined Step Damage Function......Page 79 Log-Normal Damage Function......Page 80 Surface Targets......Page 82 Space Targets......Page 84 3.15 Weapon Effective Radius (R(sub[w])) for Surface Targets......Page 85 Log-Normal Damage Function......Page 87 Shrapnel Damage Function......Page 88 3.16 Weapon Effective Radius (R(sub[w])) for Space Targets......Page 89 Problems......Page 90 4.1 Effective Firing Rate: Attrition Rate Coefficients......Page 92 4.2 Markovian Attrition Rates: Probabilistic Rates......Page 93 4.3 Lanchester Model for Directed Fire: Square Law......Page 99 4.4 Lanchester Model for Area Fire: Linear Law......Page 103 4.5 Guerrilla Warfare Model: Mixed Law......Page 108 4.6 Autonomous Fire Model: Logarithmic Law......Page 111 4.7 Geometric Mean Model: Linear Law......Page 112 4.8 Helmbold Models: Size Effects......Page 114 4.10 Force Parity......Page 116 4.11 Battle Disengagement: Force Breakpoints......Page 117 4.12 Variable Attrition Coefficients: Mobile Attack Model......Page 118 4.13 Force Reinforcements in Combat......Page 122 Arbitrary Variation of P(t)......Page 123 4.14 Mixed Models......Page 125 4.15 Iwo Jima Battle......Page 126 Directed Fire......Page 129 Area Fire......Page 131 References......Page 132 Problems......Page 133 5.1 Sequential Combat Duel: Time-Independent Combat......Page 134 5.2 Continuous Combat Duel: Time-Dependent Combat......Page 137 5.3 Continuous Combat States: Directed Fire Model......Page 142 5.4 Continuous Combat States: Area Fire Model......Page 148 5.5 Many-on-Many Engagements......Page 155 Uniform Assignment of Targets......Page 156 Random Assignment of Targets......Page 158 Shoot-Look-Shoot Assignment of Targets......Page 159 Problems......Page 162 6.1 Strategic Defense Initiative: Layered Defense......Page 164 6.2 Layered Defense Against MIRVed Attack......Page 170 6.3 Antiballistic Missile (ABM) Defense: Game Theory......Page 174 Game Theory: Payoff Matrix......Page 175 Two-Person Zero-Sum Game: Pure and Mixed Strategies......Page 176 ABM Defense Analysis......Page 179 6.4 Optimal Penetration Routes Through Air Defenses:Threat Function......Page 186 Direct Penetration of the Defended Missile Site......Page 190 Offset Penetration in Relation to SAM Site......Page 194 Equivalent Megatonnage (EMT)......Page 198 Counter Military Potential (CMP)......Page 200 Comparison of Strategic Nuclear Forces......Page 201 Problems......Page 203 7.1 Concept for the Theater Missile Defense......Page 206 Types of Defensive Tactics......Page 207 Random Assignments of Targets......Page 208 Uniform Assignments of Targets......Page 211 Shoot-Look-Shoot Assignments of Targets......Page 214 7.3 Probabilities of Zero Penetration......Page 216 Problems......Page 219 8.1 Directed Fire "Many-on-Many" Engagements: Numerical Solutions......Page 220 8.2 Aggregated Forces......Page 223 8.3 Superiority Parameters......Page 224 8.4 Aggregated Force Solution......Page 229 8.5 Numerical Example: "One-on-Two" Tactical Engagement......Page 230 8.6 Comments on the Directed Fire Solution......Page 232 8.7 Area Fire "Many-on-Many" Engagements......Page 233 8.8 Guerrilla Warfare "Many-on-Many" Engagements......Page 236 Front Line Segment Defense......Page 238 Optimal Mobile Defense: Single Segment......Page 239 Case 1: v(sub[a]) = v(sub[d])......Page 241 Case 2: v(sub[a]) v(sub[d])......Page 245 Optimal Mobile Defense: Multiple Segments......Page 247 References......Page 249 Problems......Page 250 9.1 Reliability of Series Operations......Page 252 9.2 Reliability of Parallel (Redundant) Operations......Page 254 9.3 Reliability of Combined (Series and Parallel) Operations......Page 256 9.5 Reliability Variation with Time......Page 257 Normal Distribution......Page 260 Weibull Distribution......Page 264 9.6 Derivation of Reliability from Probabilistic Considerations......Page 266 9.7 Hazard Function h(t)......Page 272 9.8 Computation of a Reliability Function from Experiment......Page 274 9.9 Maintainability of Weapon Systems......Page 276 9.10 Operational Availability of Systems......Page 278 Problems......Page 279 10.1 Intermittent Glimpses......Page 282 10.2 Continuous Search......Page 285 10.3 Variation of Detection Rate with Range: Inverse Square and Cube Laws of Detection......Page 287 High Altitude Detection: h » r (Space Surveillance)......Page 288 Exhaustive Search......Page 289 Random Search......Page 290 Inverse Cube Law Search......Page 291 10.5 One-Dimensional Search......Page 293 10.6 Constant Velocity Target......Page 295 10.7 Detection of Electromagnetic Radiation from a Target......Page 299 Problems......Page 301 Classical Programming......Page 304 Classical Programming: Unconstrained Optimization......Page 307 Nonlinear Programming......Page 309 Linear Programming......Page 310 11.2 Application of the Lagrange Multiplier Method: A Cluster Bomb......Page 311 11.3 Examples of Linear Programming......Page 313 References......Page 317 Problems......Page 318 12.1 Models......Page 320 12.2 Modeling of Military Operations......Page 322 Combat Mission......Page 323 Equipment......Page 324 General Models......Page 325 References......Page 329 A.2 Cumulative Poisson Probabilities......Page 330 A.3 The Normal (Gaussian) Probabilities......Page 331 A.5 Single-Shot Probability of Hit on a Rectangular Target......Page 333 Appendix B. Derivation of the Characteristic Function Φ(sub[N])(s)......Page 338 References......Page 340 C.1 General Solution of X = CX......Page 342 C.2 Right and Left Generalized Eigenvectors......Page 345 C.3 Examples of the Dominant Left Eigenvectors......Page 346 References......Page 348 Appendix D. Calculation of the Average Probability of No Detection......Page 350 E.2 Subroutine Instructions......Page 352 E.3.1 Binomial Probabilities......Page 354 E.3.2 Poisson Probabilities......Page 356 E.3.3 Normal (Gaussian) Probabilities......Page 358 E.3.4 Error Function......Page 360 E.3.5 Probability of Hit of a Rectangular Target......Page 361 E.3.6 Probability of Destruction of a Point Target with Offset Distribution......Page 363 E.3.7 Probability of Destruction of a Space Point Target with Offset Distribution......Page 364 E.3.9 Probability of Destruction of a Point Target with the Exponential Damage Characteristics......Page 365 E.3.10 Probability of Destruction of a Surface Point Target with the Inclined Step Damage Function......Page 366 E.3.11 Probability of Destruction of a Surface Point Target with the Log-Normal Damage Characteristics......Page 367 E.3.12 Weapon Effective Radius for Normal (Gaussian) Damage Function......Page 368 E.3.13 Weapon Effective Radius for Exponential Damage Function......Page 369 E.3.14 Weapon Effective Radius for Inclined Step Damage Function......Page 370 E.3.15 Weapon Effective Radius for Log-Normal Damage Function......Page 371 E.3.16 Weapon Effective Radius for Shrapnel Damage Function......Page 372 E.4.1 Directed Fire Lanchester Deterministic Model......Page 373 E.4.2 Area Fire Lanchester Deterministic Model......Page 375 E.4.3 Guerrilla Warfare Deterministic Model......Page 376 E.5.1 Probabilistic Directed Fire Model......Page 378 E.5.2 Probabilistic Area Fire Model......Page 379 E.6.1 SAM Defense for Direct Penetration......Page 380 E.6.2 SAM Defense for Offset Penetration......Page 382 E.7.1 Missile Penetration for Random Assignments of Targets; 1-Layer Defense......Page 383 E.7.2 Missile Penetration for Random Assignments of Targets; 2-Layer Defense......Page 384 E.7.3 Missile Penetration for Uniform Assignments of Targets; 1-Layer Defense......Page 385 E.7.4 Missile Penetration for Uniform Assignments of Targets; 2-Layer Defense......Page 387 E.7.5 Probability of Zero Missile Penetrations for Specified Number of Warheads W......Page 388 E.8.1 Heterogeneous Force Levels in Tactical Engagements (Directed Fire Model)......Page 389 E.8.2 Heterogeneous Force Levels in Tactical Engagements (Area Fire Model)......Page 392 E.8.3 Heterogeneous Force Levels in Tactical Engagements (Guerrilla Warfare Model)......Page 394 E.9.1 Normal Probability Density Function and MTTF......Page 397 E.9.2 Normal Reliability Function......Page 398 E.9.3 Normal Hazard Function......Page 400 E.9.4 Log-Normal Density Function and MTTF......Page 401 E.9.5 Log-Normal Reliability Function......Page 402 E.9.6 Log-Normal Hazard Function......Page 403 E.9.7 Weibull Probability Density Function and MTTF......Page 404 E.9.8 Weibull Reliability Function......Page 405 E.9.9 Weibull Hazard Function......Page 407 E.10.1 Search for Constant Velocity Target......Page 408 E.11.1 Dominant Eigenvalue and Left Eigenvector......Page 409 References......Page 411 C......Page 412 E......Page 413 I......Page 414 M......Page 415 P......Page 416 R......Page 417 T......Page 418 Z......Page 419

This text presents the various mathematical methods used in military operations research in one easy-to-use reference volume. The reader will find the calculations necessary to analyze all aspects of defense operations, from weapon performance to combat modeling. The text is so clearly written and organized that even newcomers to the field will find it useful.

Included with the text is an updated version of Defense Analyses Software, a compendium of software subroutines that allow the reader to compute numerical values for functions or tables derived in the text. Each subroutine is provided with a detailed reference to the equation from which it was derived to ensure that its intended application is consistent with the assumptions used in the derivation.

The third edition has a new chapter on theater missile defense based on the concept of layered defense with different strategies of allocating defense interceptors against short- or mid-range ballistic missiles.

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.