Cover image for Mathematical methods in defense analyses
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

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30000010127350 U104 P79 2000 Open Access Book Book
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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

Prefacep. xix
Chapter 1. Scientific Methods in Military Operationsp. 1
1.1 Introductionp. 1
1.2 Mathematical Methodsp. 4
Theory of Combatp. 5
Decision Theoryp. 6
Linear Programmingp. 6
Queuing Theoryp. 6
Network Analysisp. 7
Game Theoryp. 7
Differential Gamesp. 7
War-Gaming and Simulationp. 7
Future Directions of Researchp. 8
1.3 Quantitative Competencep. 8
Referencesp. 9
Problemsp. 10
Chapter 2. Characteristic Properties of Weaponsp. 11
2.1 Weapon Performance Datap. 11
2.2 Empirical Averagesp. 15
Arithmetic Mean (Average)p. 15
Medianp. 16
2.3 Scatter or Dispersionp. 16
Rangep. 16
Interquartile Rangep. 16
Mean Deviationp. 17
Standard Deviationp. 17
Coefficient of Variationp. 18
2.4 Mean Point of Impact: Systematic Errorp. 19
Problemsp. 19
Chapter 3. Passive Targetsp. 21
3.1 Binomial Distributionp. 21
3.2 Normal (Gaussian) Distributionp. 25
3.3 Poisson Distributionp. 26
3.4 Probability Density Distribution: Linear Error Probable (LEP)p. 28
3.5 Single-Shot Probability of Hit for a Rectangular Targetp. 31
3.6 Probability of Target Kill or Hit for Multiple Shotsp. 32
3.7 Probability of Destruction of a Point Target: Circular Error Probable (CEP)p. 34
Case 1 Zero Offset and Equal Variancesp. 36
Case 2 Offset Distribution and Equal Variancesp. 38
Case 3 Zero Offset and Unequal Variancesp. 41
Case 4 Offset Distribution and Unequal Variancesp. 42
3.8 Probability of Hit of a Rectangular Target: The Polya-Williams Approximationp. 42
Case 1 Equal Variances: Square Targetp. 42
Case 2 Nonequal Variances: Rectangular Targetp. 44
3.9 Probability of Hit of an Elliptic Target with Unequal Variancesp. 45
3.10 Probability of Destruction of a Point Target in Spacep. 46
3.11 Linear, Circular, and Spherical Error Probablesp. 49
3.12 Expected Fractional Damage of a Uniform-Valued Targetp. 50
3.13 Damage Functions for a Point Target in a Planep. 56
Gaussian Damage Functionp. 57
Exponential Damage Functionp. 58
Inclined Step Damage Functionp. 58
Log-Normal Damage Functionp. 59
3.14 Shrapnel Damage Functionsp. 61
Surface Targetsp. 61
Space Targetsp. 63
3.15 Weapon Effective Radius (R[subscript w]) for Surface Targetsp. 64
Gaussian Damage Functionp. 66
Exponential Damage Functionp. 66
Inclined Step Damage Functionp. 66
Log-Normal Damage Functionp. 66
Shrapnel Damage Functionp. 67
3.16 Weapon Effective Radius (R[subscript w]) for Space Targetsp. 68
Referencesp. 69
Problemsp. 69
Chapter 4. Deterministic Combat Modelsp. 71
4.1 Effective Firing Rate: Attrition Rate Coefficientsp. 71
4.2 Markovian Attrition Rates: Probabilistic Ratesp. 72
4.3 Lanchester Model for Directed Fire: Square Lawp. 78
4.4 Lanchester Model for Area Fire: Linear Lawp. 82
4.5 Guerrilla Warfare Model: Mixed Lawp. 87
4.6 Autonomous Fire Model: Logarithmic Lawp. 90
4.7 Geometric Mean Model: Linear Lawp. 91
4.8 Helmbold Models: Size Effectsp. 93
4.9 Autonomous/Directed Fire Modelp. 95
4.10 Force Parityp. 95
4.11 Battle Disengagement: Force Breakpointsp. 96
4.12 Variable Attrition Coefficients: Mobile Attack Modelp. 97
4.13 Force Reinforcements in Combatp. 101
Arbitrary Variation of P(t)p. 102
4.14 Mixed Modelsp. 104
4.15 Iwo Jima Battlep. 105
4.16 Casualty Exchange Ratiop. 108
Directed Firep. 108
Area Firep. 110
Referencesp. 111
Problemsp. 112
Chapter 5. Probabilistic Combat Modelsp. 113
5.1 Sequential Combat Duel: Time-Independent Combatp. 113
5.2 Continuous Combat Duel: Time-Dependent Combatp. 116
5.3 Continuous Combat States: Directed Fire Modelp. 121
5.4 Continuous Combat States: Area Fire Modelp. 127
5.5 Many-on-Many Engagementsp. 134
Uniform Assignment of Targetsp. 135
Random Assignment of Targetsp. 137
Shoot-Look-Shoot Assignment of Targetsp. 138
Referencesp. 141
Problemsp. 141
Chapter 6. Strategic Defensep. 143
6.1 Strategic Defense Initiative: Layered Defensep. 143
6.2 Layered Defense Against MIRVed Attackp. 149
6.3 Antiballistic Missile (ABM) Defense: Game Theoryp. 153
Game Theory: Payoff Matrixp. 154
Two-Person Zero-Sum Game: Pure and Mixed Strategiesp. 155
ABM Defense Analysisp. 158
6.4 Optimal Penetration Routes Through Air Defenses: Threat Functionp. 165
6.5 Defense Against a Penetrating Aircraftp. 169
Direct Penetration of the Defended Missile Sitep. 169
Offset Penetration in Relation to SAM Sitep. 173
6.6 Sufficiency Models for Strategic Forcesp. 177
Equivalent Megatonnage (EMT)p. 177
Counter Military Potential (CMP)p. 179
Comparison of Strategic Nuclear Forcesp. 180
Referencesp. 182
Problemsp. 182
Chapter 7. Theater Missile Defensep. 185
7.1 Concept for the Theater Missile Defensep. 185
7.2 Probability of Penetration of Missiles (or Tanks)p. 186
Types of Defensive Tacticsp. 186
Random Assignments of Targetsp. 187
Uniform Assignments of Targetsp. 190
Shoot-Look-Shoot Assignments of Targetsp. 193
7.3 Probabilities of Zero penetrationp. 195
Referencesp. 198
Problemsp. 198
Chapter 8. Tactical Engagements of Heterogeneous Forcesp. 199
8.1 Directed Fire "Many-on-Many" Engagements: Numerical Solutionsp. 199
8.2 Aggregated Forcesp. 202
8.3 Superiority Parametersp. 203
8.4 Aggregated Force Solutionp. 208
8.5 Numerical Example: "One-on-Two" Tactical Engagementp. 209
8.6 Comments on the Directed Fire Solutionp. 211
8.7 Area Fire "Many-on-Many" Engagementsp. 212
8.8 Guerrilla Warfare "Many-on-Many" Engagementsp. 215
8.9 Positioning of Defense Forcesp. 217
Front Line Segment Defensep. 217
Optimal Mobile Defense: Single Segmentp. 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 Segmentsp. 226
Referencesp. 228
Problemsp. 229
Chapter 9. Reliability of Operations and Systemp. 231
9.1 Reliability of Series Operationsp. 231
9.2 Reliability of Parallel (Redundant) Operationsp. 233
9.3 Reliability of Combined (Series and Parallel) Operationsp. 235
9.4 Example of an Air-to-Air Engagementp. 236
9.5 Reliability Variation with Timep. 236
Exponential Distributionp. 239
Normal Distributionp. 239
Log-Normal Distributionp. 243
Weibull Distributionp. 243
9.6 Derivation of Reliability from Probabilistic Considerationsp. 245
9.7 Hazard Function h(t)p. 251
9.8 Computation of a Reliability Function from Experimentp. 253
The Probability Graph Methodp. 255
The Kolmogorov-Smirnov (K-S) Test Methodp. 255
9.9 Maintainability of Weapon Systemsp. 255
9.10 Operational Availability of Systemsp. 257
Referencesp. 258
Problemsp. 258
Chapter 10. Target Detectionp. 261
10.1 Intermittent Glimpsesp. 261
10.2 Continuous Searchp. 264
10.3 Variation of Detection Rate with Range: Inverse Square and Cube Laws of Detectionp. 266
Low Altitude Detection: r ]] hp. 267
High Altitude Detection: h ]] r (Space Surveillance)p. 267
10.4 Probability of Detection in Search of a Given Areap. 268
Exhaustive Searchp. 268
Random Searchp. 269
Inverse Cube Law Searchp. 270
10.5 One-Dimensional Searchp. 272
10.6 Constant Velocity Targetp. 274
10.7 Detection of Electromagnetic Radiation from a Targetp. 278
Referencesp. 280
Problemsp. 280
Chapter 11. Optimization Methodsp. 283
11.1 Mathematical Optimizationp. 283
Classical Programmingp. 283
Classical Programming: Unconstrained Optimizationp. 286
Nonlinear Programmingp. 288
Linear Programmingp. 289
11.2 Application of the Lagrange Multiplier Method: A Cluster Bombp. 290
11.3 Examples of Linear Programmingp. 292
Referencesp. 296
Problemsp. 297
Chapter 12. Modelingp. 299
12.1 Modelsp. 299
12.2 Modeling of Military Operationsp. 301
Combat Missionp. 302
Equipmentp. 303
General Modelsp. 304
Referencesp. 308
Appendix A. Probability Tablesp. 309
A.1 Cumulative Binomial Probabilitiesp. 309
A.2 Cumulative Poisson Probabilitiesp. 309
A.3 The Normal (Gaussian) Probabilitiesp. 310
A.4 Error Function erf(z)p. 312
A.5 Single-Shot Probability of Hit on a Rectangular Targetp. 312
Appendix B. Derivation of the Characteristic Function [Phi subscript N](s)p. 317
Referencep. 319
Appendix C. Analytical Solution of Equations of Combatp. 321
C.1 General Solution of X = CXp. 321
C.2 Right and Left Generalized Eigenvectorsp. 324
C.3 Examples of the Dominant Left Eigenvectorsp. 325
C.4 Computation of the Dominant Left Eigenvectors and Their Eigenvaluesp. 327
Referencesp. 327
Appendix D. Calculation of the Average Probability of No Detectionp. 329
Appendix E. Defense Analysis Softwarep. 331
E.1 Introductionp. 331
E.2 Subroutine Instructionsp. 331
E.3 Passive Targetsp. 333
E.3.1 Binomial Probabilitiesp. 333
E.3.2 Poisson Probabilitiesp. 335
E.3.3 Normal (Gaussian) Probabilitiesp. 337
E.3.4 Error Functionp. 339
E.3.5 Probability of Hit of a Rectangular Targetp. 340
E.3.6 Probability of Destruction of a Point Target with Offset Distributionp. 342
E.3.7 Probability of Destruction of a Space Point Target with Offset Distributionp. 343
E.3.8 Expected Fraction of Damage of a Circular Targetp. 344
E.3.9 Probability of Destruction of a Point Target with the Exponential Damage Characteristicsp. 344
E.3.10 Probability of Destruction of a Surface Point Target with the Inclined Step Damage Functionp. 345
E.3.11 Probability of Destruction of a Surface Point Target with the Log-Normal Damage Characteristicsp. 346
E.3.12 Weapon Effective Radius for Normal (Gaussian) Damage Functionp. 347
E.3.13 Weapon Effective Radius for Exponential Damage Functionp. 348
E.3.14 Weapon Effective Radius for Inclined Step Damage Functionp. 349
E.3.15 Weapon Effective Radius for Log-Normal Damage Functionp. 350
E.3.16 Weapon Effective Radius for Shrapnel Damage Functionp. 351
E.4 Deterministic Combat Modelsp. 352
E.4.1 Directed Fire Lanchester Deterministic Modelp. 352
E.4.2 Area Fire Lanchester Deterministic Modelp. 354
E.4.3 Guerrilla Warfare Deterministic Modelp. 355
E.5 Probabilistic Combat Modelsp. 357
E.5.1 Probabilistic Directed Fire Modelp. 357
E.5.2 Probabilistic Area Fire Modelp. 358
E.6 Strategic Defensep. 359
E.6.1 SAM Defense for Direct Penetrationp. 359
E.6.2 SAM Defense for Offset Penetrationp. 361
E.7 Theater Missile Defensep. 362
E.7.1 Missile Penetration for Random Assignments of Targets; 1-Layer Defensep. 362
E.7.2 Missile Penetration for Random Assignments of Targets; 2-Layers Defensep. 363
E.7.3 Missile Penetration for Uniform Assignments of Targets; 1-Layer Defensep. 364
E.7.4 Missile Penetration for Uniform Assignments of Targets; 2-Layer Defensep. 366
E.7.5 Probability of Zero Missile Penetrations for Specified Number of Warheads Wp. 367
E.8 Tactical Engagements of Heterogeneous Forcesp. 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 Systemsp. 376
E.9.1 Normal Probability Density Function and MTTFp. 376
E.9.2 Normal Reliability Functionp. 377
E.9.3 Normal Hazard Functionp. 379
E.9.4 Log-Normal Density Function and MTTFp. 380
E.9.5 Log-Normal Reliability Functionp. 381
E.9.6 Log-Normal Hazard Functionp. 382
E.9.7 Weibull Probability Density Function and MTTFp. 383
E.9.8 Weibull Reliability Functionp. 384
E.9.9 Weibull Hazard Functionp. 386
E.10 Target Detection and Searchp. 387
E.10.1 Search for Constant Velocity Targetp. 387
E.11 Miscellaneousp. 388
E.11.1 Dominant Eigenvalue and Left Eigenvectorp. 388
Referencesp. 390
Indexp. 391