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Linear Optimization: The Simplex Workbook (Teacher's Edition, a.k.a Book & Instructor's Notes & Solution Manual) (Solutions) PDF

399 Pages·2009·3.024 MB·English
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Preview Linear Optimization: The Simplex Workbook (Teacher's Edition, a.k.a Book & Instructor's Notes & Solution Manual) (Solutions)

Dedication To Karen, for her constant love and support. To Calvin and Kate, for patiently waiting for Daddy to play. To Uncle Frank, who would have loved to read it. To Mom, my teaching model. Preface ? This course is intended primarily for upper-level undergraduate mathematics majors, although math education, economics, computer science, and other majors regularly frequent my course, including some graduate students (in fact, many of the challenges and projects are aimed at graduatestudents). Butthefocus,asyouwillreadabout,iscenteredmoreontheunderstanding of the mathematics itself (and mathematical applications) than on the industrial applications and use of relevant software to solve large problems. That doesn’t mean one can’t use this text for such other purposes, and I hope you’ll see that you can, but instead that we want to do mathematics first and foremost. There are plenty of books that follow the applied model for teaching this subject, and too few that follow the path we outline below. We do assume that students have passed the usual introductory courses in linearalgebra, computer algorithms,and proof writing — typical requirements of all math majors. The Subject A little explanation is in order for our choice of the title Linear Optimization1 (and corresponding terminology) for what has traditionally been called Linear Programming. The word programming in this context can be confusing and/or misleading to students. Linear programming problems are referred to as optimization problems but the general term linear programming remains. This can cause people unfamiliar with the subject to think that it is aboutprogrammingin the sense of writing computer code. It isn’t. This workbook is about the beautiful mathematics underlying the ideas of optimizing linear functions subject to linear constraints and the algorithms to solve such problems. In particular, much of what we discuss is the mathematics of Simplex Algorithm for solving such problems, developed by George Dantzig in the late 1940s. The word program in linear programming is a historical artifact. When Dantzig first developed the Simplex Algorithm to solve what are now called linear programming problems, his initial model was a class ofresourceallocationproblemsto be solvedforthe U.S.Air Force. The decisionsaboutthe allocationswere called‘Programs’by the Air Force,andhence the term. Dantzig’sarticle2 is a fascinatingdescriptionof the origins of this subject written by the person who originated many of the ideas. Included is a description of how Tjalling Koopmans (who won a Nobel Prize in economics for his work in decision science) suggested shortening Dantzig’s description ‘programming in a linear structure’ to ‘linear programming’during a walk onthe beachwith Dantzig. Also includedis anote that, at thetime, ‘code’ wasthe wordusedforcomputer instructions and not ‘program’. To be clear to potential and current students that this is a mathematics course requiring a background in writing proofs and not a computer coding class, we prefer the terminology linear optimization. We do look atcomputer algorithmsbut focus onthe underlying mathematics. A smallamountofcomputer coding (forexample,simple MAPLE)programswillbeveryuseful, butwritingcodeisnotthe centralpurposeofthe course. The shorthand LP has been used to refer to both the generalsubject of linear programming as well as specific instances of linear programs. To distinguish, in our notation, LO (linear optimization) refers to 1I’mnotthatoriginal—thereareatleastadozenbooksthatusethisterminology. 2G.Dantzig,LinearProgramming,Operations Research50(2002), 42–47. viii Preface the general class of optimizing linear functions subject to linear constraints while LOP (linear optimization problem) refers to specific instances of such problems. Furthermore, using the optimization term brings the subject in line with other, closely related fields that are increasingly called Optimization (Nonlinear, Quadratic, Convex, Integer, Combinatorial). ? Of course, if you are more inclined to maintain the status quo, feel free to continue verbalizing “ell-pee” in place of “lop”; beware, however, that the extra syllable spoken repeatedly over the course of an entire semester can take its toll on your energy. The Simplex Algorithm is the focus of study in this book. In particular, we do not discuss Karmarkar’s AlgorithmorKhachian’sEllipsoidAlgorithmormoregeneralinterior-pointapproachestosolvingLOPs. The main reason for this is that, as I hope you’ll experience, the Simplex Algorithm leads to richer connections with linear algebra, geometry, combinatorics, game theory, probability, and graph theory. Furthermore, in the post-optimality analysis that occurs in economic modeling and in Integer Optimization, the Simplex Algorithm plays a central role. ? For a superb and extensive treatment of interior methods, let me suggest [[14]]. Why Teach This? This is a reasonable question, and there are better answers than, “Why not?” and, “Because it’s cool!” One of the areas in which Mathematics departments lag behind those of, say, Physics and, most notably these days, Biology, is in the delivery of up-to-date discoveries in the subject (galaxies, big bang,genomics,medicine,etc.). Today’scollegemathematicscurriculumis virtuallyunchangedfrom 50 years ago — go into your library and look at the degree requirements from 1960, or 1940, for that matter—and, in orderto competefor majors,departmentsneed to be perceived bystudents as far more relevant to the modern world. One of the main recommendations of the CUPM Guide,3 in fact, is awareness of connections to other subjects and introducing contemporary topics to enhance student perceptions of the vitality and importance of mathematics in the modern world. Forexample,theSimplexAlgorithmiswidelyrecognized4 asoneofthetoptenalgorithmsofthe20th century for its widespread use, its powerful applications to industry and to mathematics, it’s elegant simplicity, and it’s fascinating mathematical underpinnings, and yet, the percentage of departments offering LO in 2000–01 was only 13%.5 Given the revolution that the subject has brought to the business world alone, it is difficult to argue against having the subject available to math majors at virtually every school in the country. The case for LO increases when we realize that most of the students in mathematics programs do not go on to graduate school, but instead to jobs in a wide range of industries. In fact, 92% of studentsearningbachelor’sdegreesinthe mathematicalsciencesfrom 1994–96wentdirectlyintothe workforce,many of whom ended up in management.6 How nice it would be for them to have such a tool as Simplex, or at least to know of the benefits of optimization to their enterprise. In addition, a significant portion (26% in 20007) of students taking upper division courses are preservice secondary mathematics teachers. This is relevant because Discrete Mathematics is now in state curriculum 3Undergraduate Programs and Courses in the Mathematical Sciences: CUPM Guide 2004, www.maa.org/cupm. 4J. Dongarra and F. Sullivan, Top ten algorithms of the century, Computing Science and Engineering, January/February 2000. 5D.J.Lutzer,J.W.MaxwellandS.R.Rodi,StatisticalAbstractof Undergraduate Programs intheMathematicalSciences inthe UnitedStates: Fall 2000 CBMS Survey,AMS,2002. 6National Survey of Recent College Graduates, NationalScienceFoundation, 1997,www.nsf.gov/sbe/srs/nsf01337. 7Statistical Abstract,op. cit. Preface ix standards throughout the country, and LO is listed in that category in many states.8 How nice it would be for our high school teachers to understand more of the applicability of the algebra they teach. Moreover, our undergraduate curricula compartmentalizes topics. Consequently, most math majors graduatewitha sensethat manyoftheircourses havelittle ornothingtodowith eachother. Aclass in LO can do much to counter this misimpression,blending linearalgebra, geometry, modeling,prob- ability,gametheory,algorithms,combinatorics,graphtheory,computerprogramming,andtheoretical computer science into one big, tasty LOG.9 Finally, why not teach it? It’s cool! Terminology Besides the usage of LOP and ILOP, we introduce other quirks into the language, mostly for handiness and consistency and occasionally for fun. For example, we discuss four kinds of linear combinations, based on whether or not the extra affine and conic conditions hold, so it makes sense to use the similar notations lspan, aspan, nspan, and vspan for linear, affine, conic, and convex (both affine and conic) combinations, respectively. In particular, lspan makes more sense in this scheme than does span, the more common term found in linear algebra texts. Geometric hulls get the same treatment, with lhull, ahull, nhull, and vhull, respectively. For fun we use the term FLOP (Fractional LOP) when we need to distinguish a LOP from being an ILOP. Indeed, the first step in solving an ILOP is to relax the integer constraints to allow for rationals and find the resulting floptimal solution, which is used as a first approximation to the iloptimal solution. The term BLOP refers to an ILOP whose variables are binary (either 0 or 1).10 Also, when we discuss game theory, we talk about the GLOP (Game LOP) derived from a game. We don’t go too much farther down the self-parody road — hopefully there is no SLOP in the book. I think we’re also the first LO book to use the term parameter in place of nonbasic variable. That must be worthsomekindofaward,right? Actually,I stoleitfromvirtuallyeverylinearalgebratexteverwritten. Chapter Flow Outlandish as it may seem, someone studying from this text will need to start with Chapter 1, followed by Chapter 2. After that, there are many directions of travel. ? Each chapter includes an initial commentary that elaborates further on the following — which parts of the chapter need what prerequisites and which are prerequisites for others. If and when you wish to study geometry, you’ll want to learn Chapter 3 before Chapter 8, but you can pretty much learn them whenever you want. No other chapter uses Chapter 3 explicitly, although it does offer very beneficial intuition that permeates almost every other chapter. This is why it is placed so early. Chapter8,however,isn’tnecessaryformuch(unlessonecontinuesontostudy graduateleveloptimization), but does use material from Chapter 7 (and Section 8.4 needs Chapter 6) — and is kind of fun. ? Chapter3canbe replacedbyChapter0forthosewholiketodevelopstudents’theorem-proving abilities right off the bat. I sometimes enjoy using Chapter 0 as a theoretical, rather than industrial motivation for developing the Fundamental Theorem of Linear Optimization 2.9.1. 8In fact, this course covers 8 of the 10 NCTM standards (Algebra, Geometry, Data Analysis and Probability, Problem Solving,ReasoningandProof,Communication,Connections, andRepresentation)—onesummerItaughtaspecialsectionof thecoursetohighschoolteachers whobenefitted greatlyfromsuchbreadth. 9LinearOptimizationGoulash. 10ThusitisquitenaturalformanyacademicstobeinterestedinBO. x Preface Chapter 4 is really the heart of the course. Everything feeds off of duality. This is why we derive the dual immediately in Section 1.1. Spending extra time here pays dividends later, as everything thereafter depends critically upon it. The material of Chapter 5 is useful for learning the tip of the how-this-is-done-in-the-real-worldiceberg. Theabilitytostateandworkwitheverythinginmatrixformisaveryusefulskillingeneral,andinparticular comes in handy in Chapter 12. Thus it does not need to be studied before any other chapter (if at all, as it is not essential material otherwise — in fact, Chapter 5 is only crucial to proving Theorem 12.1.4). Chapter 6 puts Chapter 4 in general context, and is required for all subsequent chapters. ? Itispossible,however,tocoverthematerialofChapter6moreinformally,asneeded. Infact,as with most chapters, one can decide to present onlyits first two sections(maybe plus its Section 3 Practice) in an effort to cover a broader spectrum of material. This text is designed for such a degree of flexibility. Chapter 7 contains material that is necessary for Chapter 8. Chapter 9 is not needed by anyone who doesn’twanttohaveagoodtime. Chapter10isrequiredbyChapter11,andSection12.4iskeyforChapter 13. Chapters 7–13 offer the greatest flexibility for studying your favorite topics within a semester’s time. Of course, you could slow down, spend extra time on the exercises, and complete the whole book in two semesters. Have at it! A visual description of the above dependencies is given below. Book Format and Usage It is not difficult to notice that this text is different from most, and not just because I can’t resist even the lamest of jokes,11 so it may be worth some discussion on why this is so, what benefits this may have for you, and how best to take advantage of the new format. First and foremost, a great deal of information is missing, information that typically is included in mathematics texts. ? Mosttextsprohibitdiscoverystylelearningbyexplainingeverythinginsightonthepage,leaving little room for instructor influence, whereas the premise of the discovery method is to withhold information! The commonly reported benefits of the approach (increased understanding, reten- tion, enjoyment, motivation, confidence,etc.) motivate this format, from which you can decide how much more you wish to share with your students. Having spent most of your life reading from such texts, you may be used to being fed facts and algorithms, andwell usedto memorizingthem in orderto reproducethem on exams. But I believe that youare capable ofmuchmore: ofderivingresults,infact,discoveringthemthroughexperimentation,ofmakingconjectures, of proving theorems, of solving problems and checking them yourself, and of asking creative questions. 11Mostoftheserequiremyagetounderstandanyway.

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