Operations Research with Engineering
Degrees Offered
- Master of Science in Operations Research with Engineering (Non-Thesis)
- Doctor of Philosophy in Operations Research with Engineering
Program Description
Operations Research (OR) involves mathematically modeling physical systems (both naturally occurring and man-made) with a view to determining a course of action for the system to either improve or optimize its functionality. Examples of such systems include, but are not limited to, manufacturing systems, chemical processes, socio-economic systems, mechanical systems (e.g., those that produce energy), and mining systems.
Program Director
Alexandra Newman, Professor, Mechanical Engineering
Program Requirements
Master of Science in Operations Research with Engineering (Non-Thesis)
| Core Courses | 18.0 | |
| ORWE courses not taken as core courses | 12.0 | |
| Total | 30.0 | |
All Masters students are required to take a set of core courses (18 credits) that provides basic tools for the more advanced and specialized courses in the program as specified below.
| MATH324 | STATISTICAL MODELING | 3.0 |
| MATH530 | INTRODUCTION TO STATISTICAL METHODS | 3.0 |
| ORWE586 | LINEAR OPTIMIZATION | 3.0 |
| or ORWE585 | NETWORK MODELS | |
| MATH438 | STOCHASTIC MODELS | 3.0 |
| or MATH538 | STOCHASTIC MODELS | |
| or EBGN526 | STOCHASTIC MODELS IN MANAGEMENT SCIENCE | |
| ORWE587 | NONLINEAR OPTIMIZATION | 3.0 |
| or ORWE588 | INTEGER OPTIMIZATION | |
| MEGN502 | ADVANCED ENGINEERING ANALYSIS | 3.0 |
| or CSCI406 | ALGORITHMS | |
| or CEEN405 | NUMERICAL METHODS FOR ENGINEERS | |
| or CEEN505 | NUMERICAL METHODS FOR ENGINEERS | |
The remaining 12 credits of coursework can be completed with any ORWE-labeled course not taken as core. Or, specialty tracks can be added in areas, for example, including: (i) operations research methodology; (ii) systems engineering; (iii) computer science; (iv) finance and economics; and (v) an existing engineering discipline that is reflected in a department name such as electrical, civil, environmental, or mining engineering.
Students who do not wish to specialize in a track mentioned in the table below and do not wish to complete 12 additional credits of ORWE-labeled coursework can "mix and match" from the ORWE coursework and coursework mentioned in the tables below in consultation with and approval from their academic advisers.
Examples of specialty tracks from various departments across campus are given below:
Energy Systems within Mechanical Engineering Track (12 credits from the course list below)
| MEGN461 | THERMODYNAMICS II | 3.0 |
| MEGN567 | PRINCIPLES OF BUILDING SCIENCE | 3.0 |
| MEGN583/AMFG501 | ADDITIVE MANUFACTURING | 3.0 |
| MEGN570 | ELECTROCHEMICAL SYSTEMS ENGINEERING | 3.0 |
| MEGN560 | DESIGN AND SIMULATION OF THERMAL SYSTEMS | 3.0 |
Additive Manufacturing Track (12 credits from the course list below)*
*Subject to approval by graduate council
| AMFG511 | DATA DRIVEN ADVANCED MANUFACTURING | 3.0 |
| MEGN583/AMFG501 | ADDITIVE MANUFACTURING | 3.0 |
| AMFG531 | MATERIALS FOR ADDITIVE MANUFACTURING | 3.0 |
| AMFG421/521 | DESIGN FOR ADDITIVE MANUFACTURING | 3.0 |
Applied Mathematics and Statistics Track (12 credits from the course list below)
| MATH500 | LINEAR VECTOR SPACES | 3.0 |
| MATH532 | SPATIAL STATISTICS | 3.0 |
| MATH536 | ADVANCED STATISTICAL MODELING | 3.0 |
| MATH537/538 | MULTIVARIATE ANALYSIS | 3.0 |
| MATH438/538 | STOCHASTIC MODELS | 3.0 |
| MATH551 | COMPUTATIONAL LINEAR ALGEBRA | 3.0 |
| EENG511 | CONVEX OPTIMIZATION AND ITS ENGINEERING APPLICATIONS | 3.0 |
Economics Track (12 credits from the course list below)
| EBGN509 | MATHEMATICAL ECONOMICS | 3.0 |
| EBGN510 | NATURAL RESOURCE ECONOMICS | 3.0 |
| EBGN530 | ECONOMICS OF INTERNATIONAL ENERGY MARKETS | 3.0 |
| EBGN535 | ECONOMICS OF METAL INDUSTRIES AND MARKETS | 3.0 |
| EBGN590 | ECONOMETRICS I | 3.0 |
| EBGN645 | COMPUTATIONAL ECONOMICS | 3.0 |
| CSCI555 | GAME THEORY AND NETWORKS | 3.0 |
Business Track (12 credits from the course list below)
| ORWE559 | SUPPLY CHAIN MANAGEMENT | 3.0 |
| EBGN560 | DECISION ANALYTICS | 3.0 |
| EBGN571 | MARKETING ANALYTICS | 3.0 |
| EBGN562 | STRATEGIC DECISION MAKING | 3.0 |
Computer Science Track (12 credits from the course list below)
| CSCI542 | SIMULATION | 3.0 |
| CSCI562 | APPLIED ALGORITHMS AND DATA STRUCTURES | 3.0 |
| CSCI571 | ARTIFICIAL INTELLIGENCE | 3.0 |
| CSCI575 | ADVANCED MACHINE LEARNING | 3.0 |
| CSCI555 | GAME THEORY AND NETWORKS | 3.0 |
Civil Engineering - Geotechnics Track (12 credits from the course list below)
| CEEN506 | FINITE ELEMENT METHODS FOR ENGINEERS | 3.0 |
| CEEN510 | ADVANCED SOIL MECHANICS | 3.0 |
| CEEN519 | RISK ASSESSMENT IN GEOTECHNICAL ENGINEERING | 3.0 |
| CEEN511 | UNSATURATED SOIL MECHANICS | 3.0 |
| CEEN512 | SOIL BEHAVIOR | 3.0 |
| CEEN515 | HILLSLOPE HYDROLOGY AND STABILITY | 3.0 |
Civil Engineering-Structures Track (12 credits from the course list below)
| CEEN506 | FINITE ELEMENT METHODS FOR ENGINEERS | 3.0 |
| CEEN530 | ADVANCED STRUCTURAL ANALYSIS | 3.0 |
| CEEN531 | STRUCTURAL DYNAMICS | 3.0 |
| CEEN533 | MATRIX STRUCTURAL ANALYSIS | 3.0 |
| CEEN543 | CONCRETE BRIDGE DESIGN BASED ON THE AASHTO LRFD SPECIFICATIONS | 3.0 |
| CEEN545 | STEEL BRIDGE DESIGN | 3.0 |
Nuclear Engineering Track (12 credits from the course list below)
| NUGN506 | NUCLEAR FUEL CYCLE | 3.0 |
| NUGN510 | INTRODUCTION TO NUCLEAR REACTOR PHYSICS | 3.0 |
| NUGN520 | INTRODUCTION TO NUCLEAR REACTOR THERMAL-HYDRAULICS | 3.0 |
| NUGN580 | NUCLEAR REACTOR LABORATORY | 3.0 |
| NUGN590 | COMPUTATIONAL REACTOR PHYSICS | 3.0 |
| NUGN585/586 | NUCLEAR REACTOR DESIGN I | 2.0 |
Electrical Engineering-Antennas and Wireless Communications Track (12 credits from the course list below)
| EENG525 | ANTENNAS | 3.0 |
| EENG527 | WIRELESS COMMUNICATIONS | 3.0 |
| EENG530 | PASSIVE RF & MICROWAVE DEVICES | 3.0 |
| EENG526 | ADVANCED ELECTROMAGNETICS | 3.0 |
| EENG528 | COMPUTATIONAL ELECTROMAGNETICS | 3.0 |
Electrical Engineering-Energy Systems and Power Electronics Track (12 credits from the course list below)
| EENG570 | ADVANCED HIGH POWER ELECTRONICS | 3.0 |
| EENG580 | POWER DISTRIBUTION SYSTEMS ENGINEERING | 3.0 |
| EENG581 | POWER SYSTEM OPERATION AND MANAGEMENT | 3.0 |
| EENG583 | ADVANCED ELECTRICAL MACHINE DYNAMICS | 3.0 |
Electrical Engineering-Information and Systems Sciences Track (12 credits from the course list below)
| EENG509 | SPARSE SIGNAL PROCESSING | 3.0 |
| EENG511 | CONVEX OPTIMIZATION AND ITS ENGINEERING APPLICATIONS | 3.0 |
| EENG515 | MATHEMATICAL METHODS FOR SIGNALS AND SYSTEMS | 3.0 |
| EENG517 | THEORY AND DESIGN OF ADVANCED CONTROL SYSTEMS | 3.0 |
| EENG519 | ESTIMATION THEORY AND KALMAN FILTERING | 3.0 |
| EENG527 | WIRELESS COMMUNICATIONS | 3.0 |
| EENG589 | DESIGN AND CONTROL OF WIND ENERGY SYSTEMS | 3.0 |
| MEGN544 | ROBOT MECHANICS: KINEMATICS, DYNAMICS, AND CONTROL | 3.0 |
Mining and Earth Systems Track (12 credits from the course list below)
| MNGN502 | GEOSPATIAL BIG DATA ANALYTICS | 3.0 |
| MNGN512 | SURFACE MINE DESIGN | 3.0 |
| MNGN516 | UNDERGROUND MINE DESIGN | 3.0 |
| MNGN536 | OPERATIONS RESEARCH TECHNIQUES IN THE MINERAL INDUSTRY | 3.0 |
| MNGN539 | ADVANCED MINING GEOSTATISTICS | 3.0 |
Doctor of Philosophy in Operations Research with Engineering
The ORWE PhD allows students to complete an interdisciplinary doctoral degree in Operations Research with Engineering by taking courses and conducting research in eight departments/divisions: Applied Mathematics and Statistics, Electrical Engineering, Computer Sciences, Civil and Environmental Engineering, Economics & Business, Mining Engineering, Mechanical Engineering, and Metallurgical & Materials Engineering.
Specialty Requirements
Doctoral students develop a customized curriculum to fit their needs. The degree requires a minimum of 72 graduate credits that includes coursework and a thesis. Coursework is valid for nine years towards a PhD degree; any exceptions must be approved by the Director of the ORWE program and by the student's adviser.
Credit requirements
| Core Courses | 24.0 | |
| Area of Specialization Courses | 12.0 | |
| Any Combination of Specialization Courses or Research | 12.0 | |
| Research Credits | 24.0 | |
| Total Semester Hrs | 72.0 | |
Research Credits
Students must complete at least 24.0 research credits. The student's faculty adviser and the doctoral thesis committee must approve the student's program of study and the topic for the thesis.
Qualifying Examination Process and Thesis Proposal
Upon completion of the appropriate core coursework, students must pass Qualifying Exams I (written, over four courses) and II (oral, consisting of a report and research presentation) to become a candidate for the PhD, ORWE specialty. Qualifying Exam I is generally taken no later than three semesters after entry into the PhD program, and Qualifying Exam II follows no more than two semesters after having passed Qualifying Exam I. The proposal defense should be completed within ten months of passing Qualifying Exam II.
Transfer Credits
Students may transfer up to 24.0 credits of graduate-level coursework from other institutions toward the PhD degree subject to the restriction that those courses must not have been used as credit toward a Bachelor's degree. The student must have achieved a grade of B or better in all graduate transfer courses and the transfer must be approved by the student's doctoral thesis committee and the Director of the ORWE program.
Although most doctoral students will only be allowed to transfer up to 24 credits, with approval from the student’s doctoral committee, exceptions may be made to allow students who have earned a specialized thesis-based master’s degree in operations research or other closely related field from another university to transfer up to 36 credits in recognition of the degree. Students should consult with their academic advisors and ORWE director for details.
Unsatisfactory Progress
In addition to the institutional guidelines for unsatisfactory progress as described elsewhere in this bulletin: Unsatisfactory progress will be assigned to any full-time student who does not pass the following prerequisite and core courses in the first three semesters of study:
| CSCI262 | DATA STRUCTURES | 3.0 |
| ORWE586 | LINEAR OPTIMIZATION | 3.0 |
| CSCI406 | ALGORITHMS | 3.0 |
| EBGN526 | STOCHASTIC MODELS IN MANAGEMENT SCIENCE | 3.0 |
Unsatisfactory progress will also be assigned to any students who do not complete requirements as specified in their admission letters. Any exceptions to the stipulations for unsatisfactory progress must be approved by the ORwE committee. Part-time students develop an approved course plan with their advisor.
Prerequisites
Students must complete the following undergraduate prerequisite courses with a grade of B or better:
| CSCI261 | PROGRAMMING CONCEPTS | 3.0 |
| CSCI262 | DATA STRUCTURES | 3.0 |
Required Course Curriculum
All PhD students are required to take a set of core courses that provides basic tools for the more advanced and specialized courses in the program.
| Core Courses | ||
| CSCI406 | ALGORITHMS | 3.0 |
| MEGN502 | ADVANCED ENGINEERING ANALYSIS | 3.0 |
| ORWE586 | LINEAR OPTIMIZATION | 3.0 |
| MATH530 | INTRODUCTION TO STATISTICAL METHODS | 3.0 |
| MATH438 | STOCHASTIC MODELS | 3.0 |
| ORWE585 | NETWORK MODELS | 3.0 |
| ORWE588 | INTEGER OPTIMIZATION | 3.0 |
| ORWE587 | NONLINEAR OPTIMIZATION | 3.0 |
| Total Semester Hrs | 24.0 | |
Students are required to take four courses from the following list:
| Area of Specialization Courses | ||
| CSCI555 | GAME THEORY AND NETWORKS | 3.0 |
| CSCI562 | APPLIED ALGORITHMS AND DATA STRUCTURES | 3.0 |
| EBGN509 | MATHEMATICAL ECONOMICS | 3.0 |
| EBGN528 | INDUSTRIAL SYSTEMS SIMULATION | 3.0 |
| or MATH542 | SIMULATION | |
| or CSCI542 | SIMULATION | |
| EBGN560 | DECISION ANALYTICS | 3.0 |
| EBGN575 | ADVANCED MINING AND ENERGY ASSET VALUATION | 3.0 |
| EENG517 | THEORY AND DESIGN OF ADVANCED CONTROL SYSTEMS | 3.0 |
| MATH531 | THEORY OF LINEAR MODELS | 3.0 |
| MATH532 | SPATIAL STATISTICS | 3.0 |
| MATH537 | MULTIVARIATE ANALYSIS | 3.0 |
| MATH582 | STATISTICS PRACTICUM | 3.0 |
| MEGN592 | RISK AND RELIABILITY ENGINEERING ANALYSIS AND DESIGN | 3.0 |
| MNGN536 | OPERATIONS RESEARCH TECHNIQUES IN THE MINERAL INDUSTRY | 3.0 |
| MNGN538 | GEOSTATISTICAL ORE RESERVE ESTIMATION | 3.0 |
| ORWE688 | ADVANCED INTEGER OPTIMIZATION | 3.0 |
| ORWE686 | ADVANCED LINEAR OPTIMIZATION | 3.0 |
| 5XX/6XX | Special Topics (Requires approval of the advisor and OrwE program director) | 3.0 |
Mines’ Combined Undergraduate / Graduate Degree Program
Students enrolled in Mines’ combined undergraduate/graduate program may double count up to six credits of graduate coursework to fulfill requirements of both their undergraduate and graduate degree programs. These courses must have been passed with “B-” or better, not be substitutes for required coursework, and meet all other University, Department, and Program requirements for graduate credit.
Students are advised to consult with their undergraduate and graduate advisors for appropriate courses to double count upon admission to the combined program.
Courses
ORWE481. OPTIMIZATION MODELS IN MANUFACTURING. 3.0 Semester Hrs.
We address the mathematical formulation and solution of optimization models relevant in manufacturing operations. The types of deterministic optimization models examined include: (i) network models; (ii) linear programs; (iii) integer programs; and, (iv) nonlinear programs. Application areas include scheduling, blending, design, equipment replacement, logistics and transportation, among other topics. Students learn not only how to mathematically formulate the models, but also how to solve them with a state-of-the-art modeling language (AMPL) and appropriate solver (e.g., CPLEX or Minos). Algorithms for each problem class will be briefly discussed.
ORWE559. SUPPLY CHAIN MANAGEMENT. 3.0 Semester Hrs.
(II) Due to the continuous improvement of information technology, shorter life cycle of products, rapid global expansion, and growing strategic relationships, supply chain management has become a critical asset in today?s organizations to stay competitive. The supply chain includes all product, service and information flow from raw material suppliers to end customers. This course focuses on the fundamental concepts and strategies in supply chain management such as inventory management and risk pooling strategies, distribution strategies, make-to-order/make-to-stock supply chains, supplier relationships and strategic partnerships. It introduces quantitative tools to model, optimize and analyze various decisions in supply chains as well as real-world supply chain cases to analyze the challenges and solutions. 3 hours lecture; 3 semester hours.
ORWE585. NETWORK MODELS. 3.0 Semester Hrs.
(I) We examine network flow models that arise in manufacturing, energy, mining, transportation and logistics: minimum cost flow models in transportation, shortest path problems in assigning inspection effort on a manufacturing line, and maximum flow models to allocate machine-hours to jobs. We also discuss an algorithm or two applicable to each problem class. Computer use for modeling (in a language such as AMPL) and solving (with software such as CPLEX) these optimization problems is introduced. 3 hours lecture; 3 semester hours.
ORWE586. LINEAR OPTIMIZATION. 3.0 Semester Hrs.
(I) We address the formulation of linear programming models, linear programs in two dimensions, standard form, the Simplex method, duality theory, complementary slackness conditions, sensitivity analysis, and multi-objective programming. Applications of linear programming models include, but are not limited to, the areas of manufacturing, energy, mining, transportation and logistics, and the military. Computer use for modeling (in a language such as AMPL) and solving (with software such as CPLEX) these optimization problems is introduced. Offered every other year. 3 hours lecture; 3 semester hours.
ORWE587. NONLINEAR OPTIMIZATION. 3.0 Semester Hrs.
(I) This course addresses both unconstrained and constrained nonlinear model formulation and corresponding algorithms (e.g., Gradient Search and Newton's Method, and Lagrange Multiplier Methods and Reduced Gradient Algorithms, respectively). Applications of state-of-the-art hardware and software will emphasize solving real-world engineering problems in areas such as manufacturing, energy, mining, transportation and logistics, and the military. Computer use for modeling (in a language such as AMPL) and solving (with an algorithm such as MINOS) these optimization problems is introduced. Offered every other year. 3 hours lecture; 3 semester hours.
ORWE588. INTEGER OPTIMIZATION. 3.0 Semester Hrs.
(I) This course addresses the formulation of integer programming models, the branch-and-bound algorithm, total unimodularity and the ease with which these models are solved, and then suggest methods to increase tractability, including cuts, strong formulations, and decomposition techniques, e.g., Lagrangian relaxation, Benders decomposition. Applications include manufacturing, energy, mining, transportation and logistics, and the military. Computer use for modeling (in a language such as AMPL) and solving (with software such as CPLEX) these optimization problems is introduced. Offered every other year. 3 hours lecture; 3 semester hours.
ORWE686. ADVANCED LINEAR OPTIMIZATION. 3.0 Semester Hrs.
(II) As an advanced course in optimization, we expand upon topics in linear programming: advanced formulation, the dual simplex method, the interior point method, algorithmic tuning for linear programs (including numerical stability considerations), column generation, and Dantzig-Wolfe decomposition. Time permitting, dynamic programming is introduced. Applications of state-of-the-art hardware and software emphasize solving real-world problems in areas such as manufacturing, mining, energy, transportation and logistics, and the military. Computers are used for model formulation and solution. Offered every other year. Prerequisite: MEGN586. 3 hours lecture; 3 semester hours.
ORWE688. ADVANCED INTEGER OPTIMIZATION. 3.0 Semester Hrs.
(II) As an advanced course in optimization, we expand upon topics in integer programming: advanced formulation, strong integer programming formulations (e.g., symmetry elimination, variable elimination, persistence), in-depth mixed integer programming cuts, rounding heuristics, constraint programming, and decompositions. Applications of state-of-the-art hardware and software emphasize solving real-world problems in areas such as manufacturing, mining, energy, transportation and logistics, and the military. Computers are used for model formulation and solution. Prerequisite: MEGN588. 3 hours lecture; 3 semester hours. Offered every other year.