Mathematical Methods in Defense Analyses

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 243
Case 3: 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