Major Fields of Study
Management Science/Operations Research
Management science is concerned with the development of mathematical models to support managerial decision-making in a wide variety of areas. Typical applications are the design and operation of distribution and production systems, financial planning models, marketing media mix decisions, and the deployment of emergency services.
The program is administered through the Interdepartmental Committee on Operations Research, which consists of faculty from the Graduate School of Business, the School of Engineering and Applied Science, and the Graduate School of Arts and Sciences.
The primary objective of the program is to develop a student's ability to do original research. Thus a major portion of the program is the preparation of a doctoral dissertation, which is written under the close supervision of faculty members. Preparatory course work is used to develop the necessary methodological tools as well as to provide an understanding of problems and issues in the application area. Candidates with a relevant master's degree can expect to complete the program in three to four years. Those with a bachelor's degree will normally take four years.
Students concentrating in management science take methodological courses in optimization and stochastic processes, courses in the methodology of operations management and may take electives from other divisions and departments. The subject of the dissertation may be a contribution to the methodology of the field, an application to a significant applied problem or a combination of the two. The format and requirements of this program are somewhat different from those of other programs in the Business School.
Requirements of the Degree
Applications for admission are accepted from qualified applicants for entrance in September and should be received no later than January 1. An application fee of $100 made by credit/debit card must accompany the application materials. In special circumstances the fee may be waived; a request for a waiver must be made to the Doctoral Office.
A bachelor's degree or the equivalent, representing a four-year course of study in an accredited college or university, is required for admission. A master's degree is not required, although many applicants have a master's degree in such fields as economics, engineering and statistics. Superior academic performance is expected. Those who hold American PhD or ScD degrees or their foreign equivalents are not normally eligible for admission. (Occasionally, an exception to this rule can be arranged.) All candidates for admission must take either the Graduate Management Admissions Test (GMAT) or preferably the Graduate Record Examination (GRE) given by Education Testing Service at numerous centers throughout the U.S. and abroad. Applicants are strongly encouraged to take the GRE rather than the GMAT. For further details on taking these tests, applicants should address inquiries to the Graduate Management Admissions Test, Educational Testing Service, CN 6103, Princeton, New Jersey 08540-6103, or to the Graduate Record Examination, Educational Testing Service, CN 6000, Princeton, New Jersey 08541-6000. Applicants should arrange to take the GMAT or GRE exam prior to the end of the calendar year, to ensure our receipt of the test score by the January 1 deadline. Score reports typically arrive at the school up to six weeks following the test date.
For those whose native language is not English, demonstration of a high degree of competence in both written and spoken English is required. It is strongly recommended that applicants take either the test of Spoken English (TSE-A*) or the Test of English as a Foreign Language (TOEFL) (code 2141), preferably the TSE-A*. Arrangements to take these examinations should be made by writing directly to TOEFL/TSE, P.O. Box 6151, Princeton, New Jersey 08541-6151, USA. Applicants who have received a prior degree in an English-speaking country may request an exemption from the TSE and TOEFL.
Admission interviews are possible upon completion of the application, but not required. To make arrangements to speak with faculty or staff members, applicants should contact the Doctoral Office at firstname.lastname@example.org.
A strong background and ability in mathematics is essential for successful completion of the program. The entering student must be able to use mathematics as a language, translating verbal descriptions of phenomena into the appropriate mathematical framework. While there are no specific mathematical prerequisites for the doctoral program, students with some background in linear algebra, real analysis, the elements of optimization and probability and statistics will be better prepared for required course work.
Enrollment Commitment Deposit
Students admitted to the program are required to pay a deposit to reserve a place in the entering class. This commitment deposit can be waived for an applicant who is already a registered student at Columbia University. For those students receiving financial aid, this deposit is refunded after the successful completion of one term in the program; it is not refundable in the event that the applicant fails to register in the School.
Residence and Course-Credit Requirements
To maintain good standing as a PhD candidate, a student must be registered for two terms out of three during the trimester academic year, although once involved in research towards a thesis, students are generally expected to be in residence in all three terms. Satisfactory completion of twenty courses is required for the PhD degree. The exact number and selection of courses is determined by the student in consultation with faculty members in the major field. Normal course loads during the autumn and spring terms are four to five courses per term; summer-term course loads are generally lighter with heavier emphasis on research work. While taking course work a student on financial aid is expected to serve as a teaching or research assistant; a maximum of two assistantships per term is allowed. PhD candidates are allowed, but not encouraged, to take one in five courses for R (Registration) credit. The other courses must be taken for letter grade credit (H = Honors; HP = High Pass; P = Pass; LP = Low Pass; F = Fail). Students are expected to take courses in their major fields for letter-grade credit.
In addition, registration for ten credits or Doctoral Research Instruction, B9000, is required.
Up to ten courses of relevant graduate work completed at Columbia or at another university may be credited toward the course requirement for the degree. If a student has completed more than twenty graduate courses at Columbia before entrance into the PhD program, he or she may be granted a maximum of fifteen courses of advanced standing. Advanced standing granted depends on the assessment by the Doctoral Committee of the quality and relevance of the work. Credit for advanced standing is not conferred until the PhD student has successfully completed at least one term in the program.
Students must satisfy requirements in the following areas:
- Breadth Requirements
Students entering the doctoral program with only a bachelor's degree are required to take two courses from the other divisions of the Business School: Accounting, Finance and Economics, Management and Marketing. For students entering the program with a graduate level degree, other than an MBA or its graduate equivalent, this requirement may be waived, subject to approval of the Doctoral Coordinator. Students holding an MBA or its equivalent are automatically exempt from the breadth requirement.
- Field Examination
The student demonstrates proficiency in the major field by passing a comprehensive written examination. A detailed description of the exam is given in the section titled The Field Examination.
- Oral Examination
In the oral examination a student's preparation to do independent research is evaluated, in-depth knowledge within the major field is examined, and the ability to relate basic knowledge of other areas to the major field is tested. The student presents a dissertation proposal and must answer questions not only on the proposal but also on relevant background.
The oral examination is normally held within one year after the written major field examination. The oral examination committee is usually composed of three faculty members. At least two committee members are chosen from the student's major field.
Master of Philosophy
The Master of Philosophy (MPhil) degree is a prerequisite to the PhD degree and is awarded upon successful completion of all requirements of the PhD degree as stated above, except for the preparation and defense of the dissertation. Application for the MPhil degree is normally made after passing the oral examination and is made through the Doctoral Office. The MPhil Degree is awarded on the University degree-conferral date immediately following the completion of the other degree requirements.
A student without an approved dissertation proposal is given the MPhil extra muros, a provision which formally severs the candidate's ties with the University but permits the PhD degree to be awarded on the basis of a defense of original scholarly material published by the candidate within ten years of receiving the MPhil Degree. For further details, see the bulletin of the Graduate School of Arts and Sciences.
The Field Examination
The Field Exam is administered on the last two Fridays of May. It is given in two separate sections one week apart from each other: one covers Optimization, and the other covers Stochastic Models. A student who does not earn a sufficiently high score on either section must retake that section at the end of the second year. Passing both sections by the end of the second year is a requirement for continuing in the program. In some cases, a student may be given a Conditional Pass requiring that the student take an additional course in a specified topic to develop greater proficiency. A required grade in the course is usually specified in such cases. The purpose of the Field Exam is to ensure that students master course material before undertaking research. The faculty of the Division try to ensure that all students are well-prepared for the exam. Studying for the exam is important, but a student who does well in course work should not have difficulty passing the exam by the end of the second year.
The following is a list of topics commonly covered in the Field Exam along with indicative references. The specific content of the exam may vary slightly from year to year. Students should talk to the Division's Doctoral Coordinator in the Spring for updated information.
Linear Programming LP duality; Sensitivity analysis, parametric programming and economic interpretation of duality. Simplex and interior point algorithms; Dantzig-Wolfe decomposition.
(Reference: Bertsimas and Tsitsiklis, Introduction to Linear Optimization.)
Foundations of Optimization Classical optimization and nonlinear programming: unconstrained optimization; Lagrange multipliers; Karush-Kuhn-Tucker theorem. Duality theory. Deterministic Continuous-Time Optimal control: Hamilton-Jacobi-Bellman equation; Pontryagin’s maximum principle.
(References: Sundaram, A First Course in Optimization Theory, Bertsekas, Nonlinear Programming, Sethi and Thompson, Optimal Control Theory.)
Network Flows Shortest paths; maximum flows; minimum cost flows. Assignments. Matchings; Minimum spanning trees.
(References: Ahuja, Magnanti and Orlin, Network flows, and Bertsimas and Tsitsiklis, Introduction to Linear Optimization.)
II. Stochastic Models
Stochastic Processes Poisson processes, discrete and continuous-time Markov chains. Renewal processes, semi-Markov processes, regenerative processes. Elementary Markov Decision processes. Convergence concepts, SLLN, CLT, martingales, stopping times, optional stopping.
(Reference: Ross, Stochastic Processes.)
Queueing Theory Markovian queues; M/G/l; priority queues. Stability of queues; random walks associated with G/G/1 queues; Lindley's recursion; Little's law, PASTA. GI/GI/1 queue in heavy traffic.
(References: Gross and Harris, Fundamentals of Queuing Theory, and Bertsekas and Gallager, Data Networks.)
Inventory Theory Stochastic demand, single-item, constant leadtime models
(Reference: Zipkin, Foundations of Inventory Management.)
Additional topics may be included in these exams depending on the specific course offerings on that year. Examples include Integer programming and combinatorial optimization and Simulation.
During the months of July-August of their 1st year in the program (after the qualifier), students are required to participate in the summer research project with one or more faculty members.
A dissertation embodying original research constitutes the major requirement of the program. The dissertation is written under the guidance of a faculty sponsor. The dissertation is defended before a University committee (the composition of which is described below under Dissertation, Defense and Deposit).
As a student progresses on the dissertation proposal, it is wise to establish a working relationship not only with the sponsor but also with other faculty members who will serve as readers of the dissertation at the time of the defense.
Each student must submit a written proposal (about 10 pages) detailing the dissertation
plan to the Doctoral Coordinator for approval by January 1st of their 3rd
year in the program. The proposal must
be accompanied by an
approval of dissertation proposal form signed by the sponsor and
by two other
faculty members, usually the chairman and the third reader, indicating
their willingness to supervise the dissertation. Approval forms are
the Doctoral Office.
By June 1st of their 3rd year in the program, students are required to make an oral presentation to their proposal committee on the topic of their dissertation research. This seminar serves as an Oral Examination: the student must answer questions about both the thesis proposal and the related literature.
Candidates can expect to spend at least one year of full-time work on the dissertation.
Dissertation, Defense, and Deposit
Students should plan to complete the dissertation within three terms (one year) after passing the oral examination.
When a dissertation is completed to the satisfaction of the sponsor, it is recommended for the defense. Through the Doctoral Office the student files an Application for the Final Examination for the PhD (Dissertation Defense) with the Office of the Dean of the Graduate School of Arts and Sciences, 108 Low Library.
Candidates should acquaint themselves thoroughly with the pamphlet. The PhD Dissertation: Research Proposal, Sponsorship and Defense, available from the Doctoral office or the Dissertation Secretary, 108 Low Library, 854-2854.
The dissertation is defended before a University-wide committee of five, including the sponsor (the major advisor), the chairperson (a member of the Faculty of Business), a third inside reader (who must be a member of the Faculty of Business) and two outside examiners (who must be Graduate School of Arts and Sciences faculty members from any department other than the candidate's). In some cases, a variation is allowed and one of the outside examiners may be associated with an institution other than the Graduate School of Arts and Sciences. The student's sponsor, not the student, is responsible for selecting the committee, which is then subject to the approval of the Assistant Dean of the Graduate School of Arts and Sciences. The Doctoral Office assists the sponsor in selecting the committee members, getting committee approval, and scheduling the defense.
When the defense committee has met and examined the candidate on the dissertation, it evaluates and judges the dissertation defense to fall within one of three categories: Pass, Incomplete or Fail. The following are definitions of each category:
Pass:The dissertation is deemed acceptable subject to minor revisions. Upon completion of the required revision, the candidate is recommended for the degree. All revisions must be completed and the dissertation deposited no later than six months from the date of the dissertation defense.
Incomplete:The dissertation is deemed acceptable subject to major revisions. Upon completion of the required revisions, the candidate is recommended for the degree, The defense committee shall delegate the responsibility for the outcome of the revisions to a subcommittee made up of one to three members of the original committee. All revisions must be completed and the dissertation deposited no sooner than three months and no later than one year from the date of the dissertation defense.
Fail:The dissertation is deemed unacceptable, and the candidate is not recommended for the degree. Subject to the rules of the Graduate School of Arts and Sciences, however, the candidate, within ten years of the award of the MPhil degree, may submit to the Dean a body of published, independent, and original scholarly material. If that material is deemed acceptable, the candidate is permitted to sit for another examination.
In the case of a Pass or Incomplete dissertation, when the required revisions have been made and duly certified by the sponsor or the revisions subcommittee, the candidate is ready to make the final deposit of the dissertation with the Dissertation Secretary, 108 Low Library. The dissertation may be deposited in microfilm or printed form. The pamphlet, The PhD Dissertation: Research Proposal, Sponsorship and Defense, available from the Doctoral Office and the from the Dissertation Secretary, outlines requirements and gives full instructions for the deposit under both options.
With the deposit of the dissertation, all requirements for the PhD degree are satisfied, assuming that all Bursar's accounts are clear. The degree is awarded on the University degree-conferral date immediately following the deposit of the dissertation. Information about the annual Commencement exercises held each year in May is mailed by the Secretary of the University in March.
The Doctoral Committee reviews each student's performance every year and requires students to demonstrate a satisfactory rate of progress toward completion of the program requirements. Such progress is measured in terms of course work performance, completion of the qualifying exam within a reasonable period of time, meeting specific requirements set by the faculty members in the student's major field and proceeding steadily toward the completion of all other requirements for the PhD degree. Among specific factors taken into account are course grades, scores on the qualifying exam, work in the major field, evaluation of the student's progress and potential (especially as it relates to the ability to perform independent, original research) and the student's evaluation of his or her own progress, time in the program and potential for completion of the program. A formal evaluation both by faculty members in the student's major field and by the Doctoral Coordinator is made each summer.
As a satisfactory rate of progress is required at all times, the Doctoral Committee has an absolute right to terminate a student's candidacy at any time for academic reasons or to deny the student readmission to the program.
Delays in Completing Degree Requirements
Candidates who hold a master's degree should plan to complete their work for the degree within a four-year period. Students entering with a bachelor's degree should complete work for the degree within four and a half years.
A prospective candidate must complete the requirements, including the defense of the dissertation, within six years if courses of advanced standing have been granted, and within seven years without advanced standing. Exceptions may be made on the recommendation of the School to the Dean of the Graduate School of Arts and Sciences when special circumstances warrant. Exceptions are rare, far from automatic and place the burden of proof for the extension on the student. A student must satisfy the academic requirements in force at the time the degree is awarded. Accordingly, if more than the normal time to complete requirements has been taken, the student may have to satisfy increased or changed requirements.
A student whose studies must be interrupted for a compelling reason such as maternity leave, military service or sustained ill health may be granted a leave of absence for a stated period, usually not to exceed one year. Forms requesting leaves of absence can be obtained from the Doctoral Office. The period of a leave of absence is not counted as part of the time allowed for the completion of degree requirements.
If as student leaves the University without obtaining a leave of absence, application must be made for readmission to candidacy. If readmitted, the student becomes subject to all current requirements.
If a student interrupts work and seeks readmission more than five years after his or her latest residence, all earlier academic work is reassessed and the dissertation subject reconsidered, either before the student is formally readmitted or during the first term of renewed residence. The amount of course credit granted the student is determined by the Doctoral Office in consultation with members of the faculty.