AN ABSTRACT OF THE THESIS OF John Anthony Battilega for the DOCTOR OF PHILOSOPHY (Name) (Degree) in Mathematics presented on May 4, 1973 (Major) (Date) Title: COMPUTATIONAL IMPROVEMENTS TO BENDERS DECOMPOSITION FOR GENERALIZED FIXED CHARGE PROBLEMS Abstract approved Redacted for privacy Professor Donald Guthrie, Jr. A computationally efficient algorithm has been developed for determining exact or approximate solutions for large scale gener- alized fixed charge problems. This algorithm is based on a relaxa- tion of the Benders decomposition procedure, combined with a linear mixed integer programming (MIP) algorithm specifically designed to solve the problem associated with Benders decomposition and a com- putationally improved generalized upper bounding (GUB) algorithm which solves a convex separable programming problem by generalized linear programming. A dynamic partitioning technique is defined and used to improve computational efficiency.All component algor- ithms have been theoretically and computationally integrated with the relaxed Benders algorithm for maximum efficiency for the gener- alized fixed charge problem. The research was directed toward the approximate solution of a particular class of large scale generalized fixed charge problems, and extensive computational results for problemsof this type are given.As the size of the problem diminishes, therelaxations can be enforced, resulting in a classical Bendersdecomposition, but with special purpose sub-algorithmsand improved convergence pro- perties. Many of the results obtained apply to the sub-algorithmsinde- pendently of the context in which theywere developed. The proce- dure for solving the associated MIP is applicableto any linear 0/1 problem of Benders form, and the techniquesdeveloped for the linear program are applicable to any large scale generalized GUB implemen- tation. Computational Improvements to Benders Decomposition for Generalized Fixed Charge problems by John Anthony Battilega A THESIS submitted to Oregon State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy June 1973 APPROVED: Redacted for privacy Professor of Statistics in charge of major Redacted for privacy Ch rman of Department of Mathematics Redacted for privacy Dean of Graduate School Date thesis is presented May 4, 1973 Typed by Janice Wilson for John Anthony Battilega ACKNOWLEDGEMENTS I am indebted to the following people who directlyorindir- ectly contributed to the results of thisresearch: Dr. R. E. Gaskell, formerly of Oregon State University, whoserved as my major professor during the period 1963-1966;Dr. R. E. D. Woolsey, Colorado School of Mines, for encouragement andincentive; Mr. Warren V. Garrison, formerly of Martin-Marietta Corporation,for basic programming assistance with the first versionof the linear program; Commander D. D. Swain, United States Navy,, for suggestions in the formulation of the original problem andfor sponsorship of the research; Mr. Paul L. Groover, Martin-MariettaCorporation, for assistance and motivation during the initial stages ofthe research, and for listening to my explanations ofmany unsuccessful sorties; Mr. William L. Cotsworth, Martin-Marietta Corporation,for sugges- tions in the area of computational efficiency (particularlyin the implementation of the generalized linear program), forsignificant assistance in the overall programming task, and formany nights jointly spent testing the algorithmon large problems; Dr. Donald Guthrie, Oregon State University, for continuedunderstanding and support as a thesis advisor; and most of all to JosephF. Bozovich, Martin-Marietta Corporation, without whose continuousencouragement, support, and advice this work would not have been possible. 1 am also grateful to Oregon State Universityfor the award of a National Defense Education Act Fellowship for the period 1963-1966. The necessary background and the opportunity to realizethis work are a direct result of manyyears of sacrifice and encourage- ment from my parents, Ercole and Odelia Battilega. My final gratitude, for ten years of unselfish support, and for typing several drafts of this work, is tomy wife, Nancy. This research was supported under contract from the Office of the Assistant Secretary of Defense (Systems Analysis)to Martin- Marietta Corporation, Denver, Colorado. TABLE OF CONTENTS Introduction Problem Formulation 11 20 Problem Solution and Reduction . .... 35 Dynamic Partitioning . Relaxed Decomposition Procedures . 43 Motivation 44 Relaxations of the Benders Procedure 45 Dual Variable Computations . 52 Determination ofy0- ...... .... 56 Alternate Constraint Computations 60 Stability Considerations 72 Integrated Large Scale Generalized Linear Programming 77 GLIB Computational Efficiency 77 Generalized Linear Programming and Dynamic Partitioning 89 Relaxed Benders Algorithm Integration 97 A Starting Basis Algorithm . 100 Scaling Procedures 109 Bootstrap Mixed Integer Programming 112 Computational Improvements to Balas Codes 115 Computational Improvements for Problems of Benders Form120 The Bootstrap Algorithm 121 Relaxed Benders Algorithm integration 126 Stability Modifications 127 Statement of the Algorithm . ..... 131 Computational Experience .... ... 133 Problem Descriptions........... 135 Phases of Development 139 Generalized Linear Program Results .... 139 Associated Mixed Integer Program Results . 146 Relaxed Benders Algorithm Results 147 Stability Modification 1 . .......... 159 Test Problem Convergence ..... .... 160 Relaxation Parameters 161 Summary and Directions for Further Research 170 Summary 170 Directions for Further Research 173 Bibliography 176 Appendix I: Benders Decomposition 190 Appendix II: Generalized Upper Bounding . 195 Appendix III: Balas Additive Algorithm . 202 Appendix IV: A Weapon Allocation Application 208 Appendix V: A Test Problem 217 COMPUTATIONAL IMPROVEMENTS TO BENDER'S DECOMPOSITION FOR GENERALIZED FIXED CHARGE PROBLEMS I. INTRODUCTION This research was motivated by a particular class of resource allocation problems, further described in Appendix IV. The first result obtained was a. mathematical programming model which described - the class of allocation problems. This model is one of the class of problems known as fixed charge problems (Cooper and Drebes (1967), Steinberg (1970), Jones and Soland (1969), Murty (1968), Balinski and Spielberg (1969), and Hirsh and Dantzig (1968).) Simplification of the Model yields the fixed charge transpor- tation or location problems of the general.variety discussed in (1969), Gavett and Plyter (1966), Toregas, Swain, Revelle, and Bergmen (1971), Stroup (1967); and Kuhn and BauMol (1962), although these formulations are pure integer programs. Specifically, the mathematical programming problem of interest is a 0/1 mixed integer programming problem of the following form. - maximize f(x) subject to the constraints _ - Ax +By b (1) x 0 y.=0,1 where f(x) is a concave separable function. A special case of (1) is a linear 0/1 mixed integer program of the form maximize c subject to the constraints - Ax +By 5 b (2) x, 0 . y. 0,1 Problem (1) is a fixed charge problem which directly general- izes the problem described by Miercort and:Soland (1971). Problem (2) generalized the fixed-charge transportation/location problems discussed by Balinski (1961), Hadley (1964), Spielberg (1964), Efroymson and Ray (1966), Gray (1967 and 1971), and Khumawala (1972). (In, this work, Problem (2) will also be used as a part of the algor- ithm for solving (1)..) The dimensionality of the problems of interest classify (1) as a large-scale 0/1 mixed integer program. (100-2000/1 vari- ables, 700-1000 constraint rows, and several hundred thousand con- tinuous variables (related in a special way).) When this research was initiated, it was not clear that (1) could be solved in the desired time (30,-60 minutes, CDC 6400 cpu time). Even though the algorithm was to be developed using the CDC 6400, the procedure must not use machine-dependent features (as in the OPHELIE-II(1970) System, for example)-in an essential way to achieve computational efficiency. Furthermore, published compu- tational results for mixed integer algorithms were not encouraging: most of the problems considered were smaller than (1), solution times were dependent on problem structure, c, solution times Were dependent on the problem data within a given structure, solution times were dependent on the computer, compiler, and code used to implement the algorithm. Nearly all references in the bibliography contain data on one or more of these items. However, summary results citing computation times can be found in Trauth and Woolsey (1969), Mears and Dawkins (1968), Geoffrion and Marsten (1972), Garfinkel and Nemhauser (1972), and Spielberg (1972). The importance of problem structure is empha- sized in Balinski (1965), Balinski and Spielberg (1969), and Geoffrion and Marsten (1972). (These three papers, along with the work of Garfinkel and Nemhauser (1972) give a survey of the state of the art of mixed integer and integer programming.)The effects of algorithm implementation on its efficiency are discussed in Mears and Dawkins (1968) and Klingman, Glover, Karney, and Napier (1971). Because of initial uncertainties involved with the large scale (1), the following additional computational objectives were estab- lished; To solve the problem by generating a sequence of "good" feasible solutions. To be able to bound the deviation of the current solution from the theoretical optimum. If possible, to provide an