72-4533 JONES Jr., Chester George, 1936- ATTRITION OF BOMBERS AS A FUNCTION OF WEAPON LOAD AND GROUPING OF THE AIRBORNE DEFENSE. The Ohio State University, Ph.D., 1971 Engineering, industrial University Microfilms, A XEROX Company , Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED ATTRITION OF BOMBERS AS A FUNCTION OF WEAPON LOAD AND GROUPING OF THE AIRBORNE DEFENSE DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Chester George Jones Jr., B.M.E., M.S.E.E. ****** The Ohio State University 1971 Approved by Adviser Department of Industrial Engineering ACKN CWLEDGMEN TS The author wishes to express his appreciation to all those who encouraged his continuation of this effort I am especially appreciative of the support given to me by Mr. Paul Nosker and Mr. Duane Dunlap who were my supervisors at Wright Patterson Air Force Base during this effort. I also wish to express my gratitude to two of my co-workers at Wright Patterson Air Force Base Dr. Clifford Fawcett and Mr. Vladimir Vukmir. It was my association with Dr. Fawcett and Mr. Vukmir that provided the initial catalyst for the work presented in this volume. I wish to thank my advisor, Professor Albert B. Bishop for his help and patience during this effort. His many suggestions have added greatly to the content of this volume. I also wish to thank the other members of the Industrial Engineering Department for their many invaluable contributions to my education. Finally, I wish to thank my wife, Clara, and children, Pamela, Daniel, and Michael for their understanding and patience. My wife’s help in preparing this manuscript greatly exceeded what one would normally expect. In addition to typing the drafts, her many suggestions greatly improved the content of this volume. VITA May 14, 1936 ......... Born - Utica, New York 1960 ............ B.M.E., Syracuse University 1960-1961 ............ Design Engineer, Remington Rand Corp., Norwalk, Conn. 1961-1962 ...... Manufacturing Engineer, Remington Rand Corp., Ilion, New York 1964 .... ......... M.S.E.E., University of Illinois 1964-1966 ...... Design Engineer, NCR Company Dayton, Ohio 1966-1971 ............ Operations Research Analyst, Aeronautical Systems Division, WPAFB, Ohio PUBLICATIONS “Effectiveness Determination of Bombers Penetrating Through an Air-to-Air Defense", Co-Author, Operations Research, Vol. 18, No. 3, May - June, 1970, pp 516-525. "Further Thoughts on the Formulation of the Bomber Penetration Problem", ASBES Working Paper Number 67-10, Nov. 1967. iv "Sensitivity Analysis of Parameters Affecting Bomber Penetration Through an Air-to-Air Defense", SES 67-3 July, 1967, "Generalized Aircraft Performance Model", Co-Author, SES 67-2, May, 1967, FIELDS OF STUDY Major Field: Operations Research Optimization and Control Theory. Professor Albert B. B ishop Operations Research, Professor Walter C. Giffin Mathematical Programming. Professor Richard L. Francis Decision Theory and Management Information Systems, Professor William T. Morris Experimental Design. Professor John B. Newhardt v TABLE OF CONTENTS P age ACKNOWLEDGMENTS...................................... ii VITA ........................................ iv LIST OF TABLES ..................................... LIST OF FIGURES ..................................... CHAPTER I. INTRODUCTION ............................. 1 The Options Available to the Offense The Options Available to the Defense II. HISTORICAL R E V I E W ........................ 19 Lanchester's Formulations More Exact Solutions Thomas1 Formulation A More Recent Approach III. ESTABLISHING A LOWER BOUND ............ 44 Interpretation of Defensive Weapon E f fects Derivation of the Model IV. ESTABLISHING AN UPPER B O U N D ............ 71 V. AN EXPECTED VALUE SOLUTION ............ 89 Model Description Markovian Nature of the Model Probabilistic Versus Deterministic Weapon Load vi Page VI. SENSITIVITY ANALYSES ................... 130 Deviations From the Solution Found in Chapter V Variation of the Problem Formulation Uncertainty Remaining After the Bounding Analysis VII. MIXED DEFENSIVE WEAPON LOADS ......... 155 Determination of the Number of Possible Weapon Loadings A Lower Bound on Bomber Survivability for Mixed Defensive Weapon Loadings An Upper Bound for Mixed Defensive Weapon Loadings An Expected Value Solution for Mixed Defensive Weapon Loadings VIII. ALLOCATION OF OFFENSIVE WEAPONS . 175 Problem Definition The Bomb Allocation Model IX. CONCLUDING REMARKS ..................... 195 Limitations of the Solution Procedure APPENDIX A ............................................... 207 211 C ............................................... 223 ............................................... 238 LIST OF R E F E R E N C E S ................................. 241 vii LIST OF TABLES Tabic Page 1. Description of Defensive Weapons .......... 65 2. Lower Bounds on Bomber Effectiveness . • . 67 3. Upper Bounds on Bomber Effectiveness When Decoys Are U s e d ................. 81 4. Upper Bounds on Bomber Effectiveness When Defensive Missiles Are Used . 82 5. Remaining Candidates for the Optimum Solution ...... ................... 87 6. Analysis of One Battle Cycle .......... * . 94 7. Probability of Bomber Survival ...... 101 8. Number of Surviving Bombers ....... 102 9. Transition Matrix for the First Two Transitions ............. 106 10. Transition Matrix for the Third and All Succeeding Transitions ................ 107 11. Comparison of Deterministic and Probabilistic Weapon Load Models . 110 12. Probability of Attack of ith Bomber . 115 13. Number of Offensive Weapons Delivered as a Function of Defensive Weapons Carried and Interceptor Group Size . 125 14. Analysis of One Battle Cycle ....... 173 viii Table Page 15. Optimum Allocation of Bombs for Various Weapon Loadings ....................... 189 16. Upper and Lower Bounds on Reaching T a r g e t s ................................. 191 17. Optimum Allocation of Bombs for Bounding and Average Solutions ........ 191 18. Upper Bound on the Number of Offensive Weapons Delivered as a Function of the Number of Number Six Defensive Missiles Carried and Interceptor Group S i z e ....................... 213 19. Expected Number of Offensive Weapons Delivered as a Function of the Number of Number Six Defensive Missiles Carried and Interceptor Group S i z e ....................... 217 20. Reduced Payoff Matrix for the Two-Player Zero-Sum Game Problem ........... 219 21. Summary of Solutions for the Three Situations Considered ........ 222 22. Probability Distribution of Available Defensive Weapons at the Start of Each C y c l e ................ ....... 233 23. Expected Number of Bombers Surviving Cycle One as a Function of Defensive Weapons Carried and Interceptor Group Size ......... 234 24. Expected Number of Bombers Surviving Cycle Two as a Function of Defensive Weapons Carried and Interceptor Group Size ..................... 235 25. Expected Number of Bombers Surviving Cycle Three as a Function of Defensive Weapons Carried and Interceptor Group Size ....... 236 26. Upper Bounds on Bomber Effectiveness When Defensive Missiles Are Used . 239 ix LIST OF FIGURES Figure Page 1. MODEL OVERVIEW ............................ 92 2. MAXIMUM AREA SEARCHED ................... 121 3. AN UNREACHABLE BOMBER ................... 121 4. THE EFFECT OF THE NUMBER OF DEFENSIVE WEAPONS CARRIED ON THE NUMBER OF BOMBS DELIVERED ..................... 134 5. THE EFFECT OF INTERCEPTOR GROUP SIZE (NjJ ON THE NUMBER OF BOMBS DELIVERED ............................ 136 6. THE EFFECT OF MISSILE SIZE ON BOMBER EFFECTIVENESS . ................... 142 7. THE EFFECT OF DECOY SIZE ON BOMBER EFFECTIVENESS . ..................... 143 8. THE EFFECT OF MISSILE EFFECTIVENESS ON THE NUMBER OF BOMBS DELIVERED . 144 9. THE EFFECT OF INTERCEPTOR FORCE SIZE ON THE NUMBER OF BOMBS DELIVERED . 145 10. THE EFFECT OF INTERCEPTOR KILL PROBABILITY (Pu^) ON THE NUMBER OF BOMBS DELIVERED ................ 146 11. THE EFFECT OF THE INTERCEPTORS' ABILITY TO REACH BOMBERS ON THE NUMBER OF BOMBS DELIVERED ..................... 147 X Figure Page 12. UNCERTAINTY IN THE BOUNDING ANALYSIS VS THE NUMBER OF DECOYS/BOMBER .... 150 13. UNCERTAINTY IN THE BOUNDING ANALYSIS VS THE NUMBER OF ANTI-INTERCEPTOR m i s s i l e s /Bo m b e r ....................... 151 14. UNCERTAINTY IN THE BOUNDING ANALYSIS VS THE NUMBER OF ANTI-MISSILE m i s s i l e s /Bo m b e r ....................... 152 15. STEP ONE OF THE M O D E L ........................ 226 16. STEP TWO OF THE M O D E L ........................ 228 17. STEP THREE OF THE M O D E L ......................230 CHAPTER I INTRODUCTION A large portion of today's military resources are devoted to manned bomber and interceptor aircraft. Despite the growth of long range missile capabilities, it is generally considered likely that bombers will continue to be an important attack weapon in the future. Since interceptor aircraft appear to offer one of the best means of defending against bombers, it is also probable that interceptors will remain to represent a significant portion of the world's military resources. Modern aircraft, both bombers and interceptors, have become very complex and therefore increasingly expensive. No nation can afford to purchase such expensive weapons without a justification of the expenditure. It is also apparent that the worth of such systems is dependent upon the weapons that will be carried in a typical mission. Any analysis of these