Mathematical Methods in Defense Analyses.

Cover -- Title -- Copyright -- Forword -- Table of Contents -- Preface -- Chapter 1. Scientific Methods in Military Operations -- 1.1 Introduction -- 1.2 Mathematical Methods -- Theory of Combat -- Decision Theory -- Linear Programming -- Queuing Theory -- Network Analysis -- Game Theory -- Differential Games -- War-Gaming and Simulation -- Future Directions of Research -- 1.3 Quantitative Competence -- References -- Problems -- Chapter 2. Characteristic Properties of Weapons -- 2.1 Weapon Performance Data -- 2.2 Empirical Averages -- Arithmetic Mean (Average) -- Median -- 2.3 Scatter or Dispersion -- Range -- Interquartile Range -- Mean Deviation -- Standard Deviation -- Coefficient of Variation -- 2.4 Mean Point of Impact: Systematic Error -- Problems -- Chapter 3. Passive Targets -- 3.1 Binomial Distribution -- 3.2 Normal (Gaussian) Distribution -- 3.3 Poisson Distribution -- 3.4 Probability Density Distribution: Linear Error Probable (LEP) -- 3.5 Single-Shot Probability of Hit for a Rectangular Target -- 3.6 Probability of Target Kill or Hit for Multiple Shots -- 3.7 Probability of Destruction of a Point Target: Circular Error Probable (CEP) -- Case 1: Zero Offset and Equal Variances -- Case 2: Offset Distribution and Equal Variances -- Case 3: Zero Offset and Unequal Variances -- Case 4: Offset Distribution and Unequal Variances -- 3.8 Probability of Hit of a Rectangular Target:The Polya-Williams Approximation -- Case 1: Equal Variances: Square Target -- Case 2: Nonequal Variances: Rectangular Target -- 3.9 Probability of Hit of an Elliptic Target with Unequal Variances -- 3.10 Probability of Destruction of a Point Target in Space -- 3.11 Linear, Circular, and Spherical Error Probables -- 3.12 Expected Fractional Damage of a Uniform-Valued Target -- 3.13 Damage Functions for a Point Target in a Plane -- Gaussian Damage Function.

Exponential Damage Function -- Inclined Step Damage Function -- Log-Normal Damage Function -- 3.14 Shrapnel Damage Functions -- Surface Targets -- Space Targets -- 3.15 Weapon Effective Radius (R(sub[w])) for Surface Targets -- Gaussian Damage Function -- Exponential Damage Function -- Inclined Step Damage Function -- Log-Normal Damage Function -- Shrapnel Damage Function -- 3.16 Weapon Effective Radius (R(sub[w])) for Space Targets -- References -- Problems -- Chapter 4. Deterministic Combat Models -- 4.1 Effective Firing Rate: Attrition Rate Coefficients -- 4.2 Markovian Attrition Rates: Probabilistic Rates -- 4.3 Lanchester Model for Directed Fire: Square Law -- 4.4 Lanchester Model for Area Fire: Linear Law -- 4.5 Guerrilla Warfare Model: Mixed Law -- 4.6 Autonomous Fire Model: Logarithmic Law -- 4.7 Geometric Mean Model: Linear Law -- 4.8 Helmbold Models: Size Effects -- 4.9 Autonomous/Directed Fire Model -- 4.10 Force Parity -- 4.11 Battle Disengagement: Force Breakpoints -- 4.12 Variable Attrition Coefficients: Mobile Attack Model -- 4.13 Force Reinforcements in Combat -- Arbitrary Variation of P(t) -- 4.14 Mixed Models -- 4.15 Iwo Jima Battle -- 4.16 Casualty Exchange Ratio -- Directed Fire -- Area Fire -- References -- Problems -- Chapter 5. Probabilistic Combat Models -- 5.1 Sequential Combat Duel: Time-Independent Combat -- 5.2 Continuous Combat Duel: Time-Dependent Combat -- 5.3 Continuous Combat States: Directed Fire Model -- 5.4 Continuous Combat States: Area Fire Model -- 5.5 Many-on-Many Engagements -- Uniform Assignment of Targets -- Random Assignment of Targets -- Shoot-Look-Shoot Assignment of Targets -- References -- Problems -- Chapter 6. Strategic Defense -- 6.1 Strategic Defense Initiative: Layered Defense -- 6.2 Layered Defense Against MIRVed Attack -- 6.3 Antiballistic Missile (ABM) Defense: Game Theory.

Game Theory: Payoff Matrix -- Two-Person Zero-Sum Game: Pure and Mixed Strategies -- ABM Defense Analysis -- 6.4 Optimal Penetration Routes Through Air Defenses:Threat Function -- 6.5 Defense Against a Penetrating Aircraft -- Direct Penetration of the Defended Missile Site -- Offset Penetration in Relation to SAM Site -- 6.6 Sufficiency Models for Strategic Forces -- Equivalent Megatonnage (EMT) -- Counter Military Potential (CMP) -- Comparison of Strategic Nuclear Forces -- References -- Problems -- Chapter 7. Theater Missile Defense -- 7.1 Concept for the Theater Missile Defense -- 7.2 Probability of Penetration of Missiles (or Tanks) -- Types of Defensive Tactics -- Random Assignments of Targets -- Uniform Assignments of Targets -- Shoot-Look-Shoot Assignments of Targets -- 7.3 Probabilities of Zero Penetration -- References -- Problems -- Chapter 8. Tactical Engagements of Heterogeneous Forces -- 8.1 Directed Fire "Many-on-Many" Engagements: Numerical Solutions -- 8.2 Aggregated Forces -- 8.3 Superiority Parameters -- 8.4 Aggregated Force Solution -- 8.5 Numerical Example: "One-on-Two" Tactical Engagement -- 8.6 Comments on the Directed Fire Solution -- 8.7 Area Fire "Many-on-Many" Engagements -- 8.8 Guerrilla Warfare "Many-on-Many" Engagements -- 8.9 Positioning of Defense Forces -- Front Line Segment Defense -- Optimal Mobile Defense: Single Segment -- Case 1: v(sub[a]) = v(sub[d]) -- Case 2: v(sub[a]) &lt -- v(sub[d]) -- Case 3: v(sub[a]) &gt -- v(sub[d]) -- Optimal Mobile Defense: Multiple Segments -- References -- Problems -- Chapter 9. Reliability of Operations and Systems -- 9.1 Reliability of Series Operations -- 9.2 Reliability of Parallel (Redundant) Operations -- 9.3 Reliability of Combined (Series and Parallel) Operations -- 9.4 Example of an Air-to-Air Engagement -- 9.5 Reliability Variation with Time -- Exponential Distribution.

Normal Distribution -- Log-Normal Distribution -- Weibull Distribution -- 9.6 Derivation of Reliability from Probabilistic Considerations -- 9.7 Hazard Function h(t) -- 9.8 Computation of a Reliability Function from Experiment -- The Probability Graph Method -- The Kolmogorov-Smirnov (K-S) Test Method -- 9.9 Maintainability of Weapon Systems -- 9.10 Operational Availability of Systems -- References -- Problems -- Chapter 10. Target Detection -- 10.1 Intermittent Glimpses -- 10.2 Continuous Search -- 10.3 Variation of Detection Rate with Range: Inverse Square and Cube Laws of Detection -- Low Altitude Detection: r » h -- High Altitude Detection: h » r (Space Surveillance) -- 10.4 Probability of Detection in Search of a Given Area -- Exhaustive Search -- Random Search -- Inverse Cube Law Search -- 10.5 One-Dimensional Search -- 10.6 Constant Velocity Target -- 10.7 Detection of Electromagnetic Radiation from a Target -- References -- Problems -- Chapter 11. Optimization Methods -- 11.1 Mathematical Optimization -- Classical Programming -- Classical Programming: Unconstrained Optimization -- Nonlinear Programming -- Linear Programming -- 11.2 Application of the Lagrange Multiplier Method: A Cluster Bomb -- 11.3 Examples of Linear Programming -- References -- Problems -- Chapter 12. Modeling -- 12.1 Models -- 12.2 Modeling of Military Operations -- Combat Mission -- Equipment -- General Models -- References -- Appendix A. Probability Tables -- A.1 Cumulative Binomial Probabilities -- A.2 Cumulative Poisson Probabilities -- A.3 The Normal (Gaussian) Probabilities -- A.4 Error Function erf(z) -- A.5 Single-Shot Probability of Hit on a Rectangular Target -- Appendix B. Derivation of the Characteristic Function Φ(sub[N])(s) -- References -- Appendix C. Analytical Solution of Equations of Combat -- C.1 General Solution of X = CX.

C.2 Right and Left Generalized Eigenvectors -- C.3 Examples of the Dominant Left Eigenvectors -- C.4 Computation of the Dominant Left Eigenvectors and Their Eigenvalues -- References -- Appendix D. Calculation of the Average Probability of No Detection -- Appendix E. Defense Analyses Software -- E.1 Introduction -- E.2 Subroutine Instructions -- E.3 Passive Targets -- E.3.1 Binomial Probabilities -- E.3.2 Poisson Probabilities -- E.3.3 Normal (Gaussian) Probabilities -- E.3.4 Error Function -- E.3.5 Probability of Hit of a Rectangular Target -- E.3.6 Probability of Destruction of a Point Target with Offset Distribution -- E.3.7 Probability of Destruction of a Space Point Target with Offset Distribution -- E.3.8 Expected Fraction of Damage of a Circular Target -- E.3.9 Probability of Destruction of a Point Target with the Exponential Damage Characteristics -- E.3.10 Probability of Destruction of a Surface Point Target with the Inclined Step Damage Function -- E.3.11 Probability of Destruction of a Surface Point Target with the Log-Normal Damage Characteristics -- E.3.12 Weapon Effective Radius for Normal (Gaussian) Damage Function -- E.3.13 Weapon Effective Radius for Exponential Damage Function -- E.3.14 Weapon Effective Radius for Inclined Step Damage Function -- E.3.15 Weapon Effective Radius for Log-Normal Damage Function -- E.3.16 Weapon Effective Radius for Shrapnel Damage Function -- E.4 Deterministic Combat Models -- E.4.1 Directed Fire Lanchester Deterministic Model -- E.4.2 Area Fire Lanchester Deterministic Model -- E.4.3 Guerrilla Warfare Deterministic Model -- E.5 Probabilistic Combat Models -- E.5.1 Probabilistic Directed Fire Model -- E.5.2 Probabilistic Area Fire Model -- E.6 Strategic Defense -- E.6.1 SAM Defense for Direct Penetration -- E.6.2 SAM Defense for Offset Penetration -- E.7 Theater Missile Defense.

E.7.1 Missile Penetration for Random Assignments of Targets.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.