Complexity measures for assembly sequences

Our work focuses on various complexity measures for two-handed assembly sequences. For many products, there exist an exponentially large set of valid sequences, and a natural goal is to use automated systems to select wisely from the choices. Since assembly sequencing is a preprocessing phase for a long and expensive manufacturing process, any work towards nding a better assembly sequence is of great value when it comes time to assemble the physical product in mass quantities. Although there has been a great deal of algorithmic success for nding feasible assembly sequences, there has been very little success towards optimizing the costs of sequences. We attempt to explain this lack of progress, by proving the inherent di culty in nding optimal, or even near-optimal, assembly sequences. We begin by introducing a formal framework for studying the optimization of complexity measures for assembly sequencing. Based on the previous work of both researchers and practitioners, we collect a list of various cost measures, goals, and restrictions that others have considered desirable. Together with this, we de ne a graph-theoretic problem that is a generalization of assembly sequencing, focusing on the combinatorial aspect of the family of feasible assembly sequences, while temporarily separating out the speci c geometric assumptions inherent to assembly sequencing. In this virtual assembly sequencing model, we are still able to explain the success of previous algorithms in nding feasible sequences e ciently. At this point, we begin to consider the approximability of the various cost measures for sequencing, and we examine the lack of success in optimizing the costs. We examine several intuitive, yet unsuccessful, heuristics for optimizing the cost of sequences, giving constructions which result in worst case performance, while also testing these heuristics experimentally on several products, previously used as a test bed for research in assembly sequencing. Because of this lack of success in designing approximation algorithms, we continue by examining the source of di culty in these problems. We show how these problems capture the combined di culty of several covering, scheduling, and sequencing problems from the literature. We iv use techniques common to the theory of approximability to prove the hardness of nding even near{optimal sequences for most cost measures in our generalized framework. Our strongest lower bounds prove that nding any solution within a 2log1 n-factor of the optimal cost solution is quasi-NP-hard for any > 0. As a special case, we prove similar, strong inapproximability results for the problem of scheduling with and/or precedence constraints. Of course, hardness results in our generalized framework do not immediately carry over to the original geometric problems. It may be that hard instances of our graph-theoretic problem do not arise from the geometric settings originally part of the assembly sequencing problems. Therefore, we re-introduce the geometry, and continue by realizing several of these hardness results in rather simple geometric settings. We are able to show strong inapproximability results in far simpler settings than the domain of most assembly sequencers, for example using an assembly consisting solely of unit disks in the plane. In the face of our results, the overwhelming open problem is to design any non-trivial approximation algorithms for minimizing the cost of assembly sequencing. In our general setting, we give strong lower bounds, however there is still a gap between the trivial upper bound, and so it would be interesting to develop an algorithm that is able to match the lower bounds. Furthermore, it would be quite valuable to identify any practical geometric settings not considered by us, that would allow for approximation-algorithms which are able to break our generalized lower bounds. v Acknowledgments I dedicate this dissertation to my parents, Marilyn and Bob, as they deserve the credit for the person whom I have become. Their parenting has instilled in me a sense of accomplishment, a sense of caring, a sense of responsibility, a sense of morality, and a sense of curiosity. When I become a parent myself, I only hope I can continue in their style. Additionally, I would like to thank my siblings for encouraging my intellectual curiosity, beginning with early math lessons on the blackboard in our basement, and continuing with endless hours spent playing game after game while growing up. To my anc e, Susan, I wish to express my heartfelt gratitude for her constant support in my life. In many ways, the completion of this thesis symbolizes an end to a period in my life; however a new chapter is about to begin with our marriage next month. I look forward to a life with Susan as my partner. In my academic life, I wish to acknowledge not only those whose help has been so bene cial to by thesis, but also those who have been a part of my life over the years. With four years at Brown, followed by six years at Stanford, the list has grown quite rapidly. To begin with, I would like to thank all the members of my reading committee. Always with an open door, Rajeev Motwani has served as both my advisor and also as a role model. Rajeev's enthusiasm for research seemed unlimited, and his commitment to quality teaching was remarkable. I often found myself sitting in his classes, admiring his organization, preparation and explanations; these lessons I take with me as I become a teacher myself. Leo Guibas, who served as my rst advisor, has been enlightening, with both his vision of the eld as well as his ability to explain his ideas to others. I would also like to thank Leo for the many times I found myself enjoying dinner, being entertained by personal stories about vi the history of computer science, the people involved, and about life in general. Finally, JeanClaude Latombe has been instrumental in the creation of this thesis. His work in assembly planning is what brought me to studying this problem. In the same regard, I would like to acknowledge all of the members of the Stanford assembly planning group, namely Leo Guibas, Jean-Claude Latombe, Danny Halperin, Cyprien Godard, G. Ramkumar, and Bruce Romney. In particular, I am especially grateful for my frequent meetings with Danny Halperin while at Stanford, as his interactions were invaluable. Additionally, there were several other people in the assembly planning community whom I have met over the years. Randy Wilson's thesis has been the springboard for a good deal of this work. Although he graduated from Stanford shortly after my arrival, continued conversations with Randy have been a key in shaping my thesis. Additionally, I would like to thank both Randy, Jack Snoeyink, and Russell Brown, for sharing various models of assemblies for use in the experiments of Section 4.5. Finally, I would like to thank both Ken Goldberg and Jan Wolter for several interesting conversations in the halls of robotics conferences. A large part of my experience as a student has been in learning to be a teacher. A signi cant part of this lesson has been my exposure to the styles and techniques of so many di erent professors for whom I have served as a teaching assistant. These relationships have provided me with many ideas and thoughts in shaping my own style and image as a teacher. For this reason I would like to thank, Andrei Broder, Todd Feldman, Leo Guibas, Philip Klein, Leora Morgenstern, Vaughn Pratt, Lyle Ramshaw, Roberto Tamassia, and Je Vitter. The computer science students at Stanford have been such a constant part of my life for so long, it is di cult to thank each of them properly. but I would like to thank the following for the role they have played in my daily life at work: Donald Aingworth, Julien Basch, Craig Becker, Jerry Cain, Moses Charikar, Chandra Chekuri, Scott Cohen, Steve Cousins, Harish Devarajan, Kathleen Fisher, Cyprien Godard, Ashish Goel, Sudipto Guha, Piotr Indyk, David Karger, Robert Kennedy, Sanjeev Khanna, Daphne Koller, AndrewKosoresow, Je rey Oldham, Steven Phillips, G. Ramkumar, Bruce Romney, Craig Silverstein, Eric Torng, Eric Veach, and Suresh Venkatasubramanian. I am also quite grateful to have kept in such close contact, even to this day, with friends from Brown, most notably, Ken Avenoso, David Luks, and David Pawson. It is a testament to the value of email, that I feel as if these people are still just down the hall fromme. Finally, I would like to thank those who helped me so much in enjoying the time, while delaying my vii thesis. My bridge playing has improved, much in thanks to Greg Crawford, Jesse David, Rajendra Gangadean, Brian Murphy, and Jason Scott. Financial support for this work has been given by a grant from the Stanford Integrated Manufacturing Association (SIMA), by NSF/ARPA Grant IRI-9306544, by NSF Grant CCR-9215219, by ARO MURI Grant DAAH04-96-1-0007 and by NSF Award CCR-9357849, with matching funds from IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. viii