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The Doctoral Program in Decision, Risk, and Operations (DRO) at Columbia Business School is designed to lead a small group of outstanding students to successful research careers in academia and industry. Our recent graduates have accepted appointments at the leading institutions of business education, such as Wharton, MIT, Kellogg, Carnegie Mellon, Cornell, Duke, and NYU, as well as in leading research and development positions in industry.
The Decision, Risk, and Operations Division provides an exceptional research environment for PhD studies. The division’s research focus involves the development and analysis of quantitative models motivated by business problems. These models are used to support decision making, to measure and manage risks, and to enhance understanding of business practices. Such problems are analyzed using tools from mathematical programming, game theory, probability, and statistics. The division has particular expertise and maintains a diverse portfolio of ongoing research projects in the following application areas:
- Supply Chain Management
- Revenue Management, Auctions, and Game Theory
- Financial Engineering and Risk Management
- Logistics, Production Planning and Scheduling, and Management of Service Systems
- Stochastic Models, Simulation, Processing, and Queuing Networks
Early student participation in research is strongly encouraged. The division hosts a weekly seminar series, which introduces students to cutting-edge research and provides a forum for faculty-student interactions. Interdepartmental collaborations give students an opportunity to work with their counterparts from the Departments of Industrial Engineering and Operations Research, and Statistics.
Admission to the program is highly competitive. All applicants are required to have a bachelor’s degree or the equivalent, representing a four-year course of study in an accredited college or university. Superior academic performance is expected, and a strong background and ability in mathematics are essential for successful completion of the program. Applicants are strongly encouraged to take the GRE rather than the GMAT. The program is full-time only and is typically completed in five years.
Admitted students are awarded a four-year fellowship that covers their tuition and fees and provides a monthly stipend. During the course of their PhD career, students can receive supplemental support as teaching and/or research assistants. Funding for the fifth year is merit based and determined by the Department.
The Field Exam is administered shortly after the end of the spring semester. It is given in two separate sections one week apart from each other: one covers deterministic optimization, and the other covers stochastic models. Students take both sections at the end of their first year of study. 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, which requires 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 members 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.
I. Deterministic Optimization
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)
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)
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)
- Linear Programming
- Foundations of Optimization
- Network Flows
II. Stochastic Models
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)
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)
Stochastic demand, single-item, constant leadtime models
(Reference: Zipkin, Foundations of Inventory Management)
- Stochastic Processes
- Queueing Theory
- Inventory Theory
Additional topics may be included in these exams depending on the specific course offerings that year. Examples include integer programming and combinatorial optimization and simulation.
During the course of study, students receive rigorous training that includes methodological courses in optimization and stochastic processes; courses in methodology of operations and risk management; and a broad range of courses from the Engineering School and the economics, mathematics, and statistics departments.
For more information, visit the DRO Division.
Sample Decision, Risk, and Operations courses: