Ph.D. in Operations Research

The Tepper School's doctoral program in operations research (OR) is designed to encourage students to make contributions toward basic scientific knowledge in the area. This knowledge can take several forms including:

• The derivations of fundamental results of an analytical or mathematical nature that lead to the development of algorithms for aiding decision-making
• The development of new analytical models appropriate for management science applications in areas such as Marketing, Operations, and Finance
• Controlled experimentation that leads to empirical results that make efficiency comparisons possible among algorithms

A major goal of the program is to train students to recognize operations research problems in real-world situations, and to give them the opportunity to learn about the deployment of operations research models in one or more of these substantive areas. Towards this goal, the program provides the opportunity to develop knowledge of functional areas of business to which optimization can be applied such as Marketing, Operations and Finance. There is a rich tradition of graduates from the program going on to successful careers in these areas both in academia (in business schools, engineering schools in IE and OR departments as well as in Math and Computer Science departments) and industry. 

Course of Study

The basic operations research courses offered include: linear, nonlinear, integer and dynamic programming; graph theory and network optimization; convex optimization and convex analysis; and stochastic models. Each course is taught by a faculty member who is actively pursuing research in the subject area. Since classes are usually small, students frequently meet informally with their instructors. The third semester competence examination is based on the areas covered in these courses.

Research

The research papers assigned for the first and second summers of graduate study are designed to give students an early introduction to research work. The paper may be done individually or jointly with other students or faculty members. Easy interaction in the Tepper School with researchers in the other areas of business and economics and in such related areas as computer science, machine learning, and statistics encourages the application of operations research in imaginative new directions.

In many cases, work on these papers leads to the work on the Ph.D. dissertation, which can begin as soon as the student has passed the third-semester qualifying examination.

Almost invariably, by the end of their second year, if not earlier, students have already worked on professional problems with some of the faculty. For this reason, student working papers written in collaboration with a faculty member are common.

Our History

Carnegie Mellon has pioneered several important developments in both theoretical and applied operations research. Geometric programming, chance constrained programming, and the applications of linear programming to capital budgeting and cost management were among the accomplishments of the '50s and early '60s. Since 1968, when the doctoral program in operations research was started, the Tepper School has initiated several new developments in integer and non-convex programming, enumerative methods, cutting plane theory, disjunctive programming, constraint programming, network design, algorithm design, machine learning, data mining, and scheduling models. 

Recently, the group has pioneered advances in Approximation Algorithms for Network Design, as well as theory and applications of Modern Convex Optimization. Examples on the Selected Research Topics page illustrate the basic research currently in progress, and examples of new operations research applications can be found elsewhere on the Doctoral Program website.

Research Topics

• Mixed-Integer Programming
• Convex Optimization
• Benders Decomposition
• Branch and Price
• Approximation and Online Algorithms
• Network Design
• Analytical Models in Marketing and Operations
• Connections with Artificial Intelligence
• Interplay between Estimation and Optimization
• Bayesian Optimization
• Massively Distributed and Parallel Algorithm Design
• Machine Learning
• Cultural Factors
• Ethics of Artificial Intelligence

Many of our students are very active in the Carnegie Mellon INFORMS Student Chapter.

To learn about the joint PhD program in Algorithms, Combinatorics and Optimization, please visit the webpage. 

Please visit our Ph.D. Student Profiles page to view the profiles of our current doctoral candidates.

Requirements

The requirements for the Ph.D. in Operations Research include:

• Course work
• Third semester qualifying examinations
• First and second year papers
• Ph.D. dissertation

A normal course load involves taking 120 units during the first two years, including core Operations Research courses and a minor area of concentration. Courses that run over a mini are worth 6 units.

The purpose of the minor requirement is to broaden the students’ knowledge to make them capable of teaching a wider range of courses and to enhance their research capabilities. The needs and preferences of individuals are recognized; therefore, the courses they take can vary. Examples of areas of concentration for the minor requirement include:

• Statistics
• Computer Science
• Operations Management
• Finance
• Marketing
• Machine Learning

Students with appropriate preparation prior to their entry to the program may take the qualifying exams prior to the third semester point if they choose, but they must take the entire set of qualifiers in the Operations Research area of study. The qualifying exams consist of eight questions, five of which are mandatory while the other three are chosen from a list of six.

Core Courses

• Linear Programming
• Network Optimization I
• Integer Programming
• Stochastic Processes (e.g. Performance Modeling)
• Modern Convex Optimization

Plus three from the following list: 

• Network Optimization II
• Convex Analysis
• Constraint Programming
• Advanced Linear Programming
• Advanced Integer Programming
• Nonlinear Programming

Additional Electives in Operations Research at the Tepper School

• Combinatorial Optimization
• Convex Polytopes
• Dynamic Programming
• Open Source Software for Optimization
• Social, Economic and Information Networks

Students are encouraged to also take electives in other departments on campus, such as Computer Science or Mathematical Sciences.

Example of a Course Sequence

Year 1 

M1: Linear Programming, Network Optimization I, Performance Modeling

M2: Network Optimization II OR Advanced LP, Constraint Programming OR Convex Analysis, Performance Modeling

M3: Integer Programming, Convex Optimization, course in minor area

M4: Advanced IP, Nonlinear Programming, course in minor area

Year 2

M1: Elective, course in minor area

M2: Network Optimization II OR Advanced LP, Constraint Programming OR Convex Analysis

M3: Elective, course in minor area

M4: Elective, elective