From the Origin provides a forum for lively discussion of issues of importance to the mathematical community. The Michigan Section-MAA Newsletter solicits opinion pieces for publication in this column from anyone in the Michigan mathematical community. In addition, comments on pieces published in earlier issues are welcomed.
Items for From the Origin should be submitted to the editor by the beginning of October to be considered for inclusion in the December issue and by the beginning of February for the April issue. Main opinion pieces should be at most 1800 words long, and responses at most 400. The editors reserve the right to shorten responses, if necessary, in order to fit as many as possible within the available space.
Students majoring in mathematics might wonder whether they will ever use the mathematics they are learning, once they graduate and get a job. Is any of the analysis, calculus, algebra, numerical methods, combinatorics, math programming, etc. really going to be of value in the real world? An exciting area of applied mathematics called Operations Research (OR) combines mathematics, statistics, computer science, physics, engineering, economics, and social sciences to solve real-world business problems.
Operations Research can be defined as the science of decision-making. It has been successful in providing a systematic and scientific approach to all kinds of government, military, manufacturing, and service operations. Operations Research is a splendid area for graduates of mathematics to use their knowledge and skills in creative ways to solve complex problems and have an impact on critical decisions.
Some key steps in OR that are needed for effective decision-making are:
- Problem Formulation (motivation, short- and long-term objectives, decision variables, control parameters, constraints);
- Mathematical Modeling (representation of complex systems by analytical or numerical models, relationships between variables, performance metrics);
- Data Collection (model inputs, system observations, validation, tracking of performance metrics);
- Solution Methods;
- Validation and Analysis (model testing, calibration, sensitivity analysis, model robustness);
- Interpretation and Implementation (solution ranges, trade-offs, visual or graphical representation of results, decision support systems).
These steps all require a solid background in mathematics and familiarity with other disciplines (such as physics, economics, and engineering), as well as clear thinking and intuition. The mathematical sciences prepare students to apply tools and techniques and use a logical process to analyze and solve problems.
OR became an established discipline during World War II, when the British government recruited scientists to solve problems in critical military operations. Following the end of World War II, interest in OR turned to peacetime applications.
There are now many OR departments in industry, government, and academia throughout the world. Areas where OR has been successful in recent years include:
- Airline Industry (routing and flight plans, crew scheduling, revenue management);
- Telecommunications (network routing, queue control);
- Manufacturing (system throughput and bottleneck analysis, inventory control, production scheduling, capacity planning);
- Healthcare (hospital management, facility design);
- Transportation (traffic control, logistics, network flow, airport terminal layout, location planning).
There are many mathematical techniques that were developed specifically for OR applications. These techniques arose from basic mathematical ideas and became major areas of expertise for industrial operations.
One important area of such techniques is optimization. Many problems in industry require finding the maximum or minimum of an objective function of a set of decision variables, subject to a set of constraints on those variables. Typical objectives are maximum profit, minimum cost, or minimum delay. Frequently there are many decision variables and the solution is not obvious. Techniques of mathematical programming for optimization include linear programming (optimization where both the objective function and constraints depend linearly on the decision variables), non-linear programming (nonlinear objective function or constraints), integer programming (decision variables restricted to integer solutions), stochastic programming (uncertainty in model parameter values) and dynamic programming (stage-wise, nested, and periodic decision-making).
Another area is the analysis of stochastic processes (i.e., processes with random variability), which relies on results from applied probability and statistical modeling. Many real-world problems involve uncertainty, and mathematics has been extremely useful in identifying ways to manage it. Modeling uncertainty is important in risk analysis for complex systems, such as space shuttle flights, large dam operations, or nuclear power generation.
Related to the topic of stochastic processes is queueing theory (i.e., the analysis of waiting lines). Mathematical analysis has been essential in understanding queue behavior and quantifying impacts of decisions. Equations have been derived for the queue length, waiting times, probability of no delay, and other measures. The results have applications in many types of queues, such as customers at a bank or supermarket checkout, orders waiting for production, ships docking at a harbor, users of the Internet, and customers served at a restaurant. Examples of decisions in managing queues are how much space to allocate for waiting customers, what lead times to promise for production orders, and what server count to assign to ensure short waiting times.
An important mathematical problem in manufacturing is the performance analysis of a production line. A typical production line consists of a series of workstations that perform different operations. Jobs flow through the line to be processed at each station. Buffers between stations hold the output of one station and allow it to wait as input to the next. A finite buffer can fill and block output from an upstream station or can empty and starve a downstream station for input. Blocking and starving are key mechanisms of the complex interactions between queues that form in the line. A critical measure of performance is throughput, defined as the number of jobs per unit time that can flow through the line. Throughput is reduced when stations experience random machine failures, a common practical situation. Mathematical modeling is needed to capture the impact on throughput of station reliabilities, as well as processing rates and buffer sizes. A model can support operating decisions, such as how to improve a line to meet a throughput target, how to identify bottlenecks, and how much buffer space to allocate in line design.
OR analysts can model difficult practical problems and offer valuable solutions and policy guidance for decision-makers. Constraints involving budgets, capital investments, and organizational considerations can make the successful implementation of results as challenging as the development of mathematical models and solution methods.
In general, Operations Research requires the use of mathematics to model complex systems, analyze trade-offs between key system variables, identify robust solutions, and develop decision support tools. Students of mathematics can be sure there are plenty of uses for the knowledge and skills they are developing. As the world becomes more complex and more dependent on new technology, mathematics applied to business problems is likely to play an increasingly important role in decision-making in industry.
On a personal note
All three of us developed an interest in the mathematical sciences early on, and took undergraduate degrees in math, or math and physics. We each got into the field of Operations Research as a result of looking for practical ways to use our math training. Below, we each answer the question: “How did you decide on a career in math and decide to join GM?”
Dennis: The math courses I liked best were the ones on applied topics. I found Operations Research an especially appealing subject, since it uses basic mathematical principles in clever ways to solve all kinds of complex problems in everyday life, such as queueing, reliability, scheduling, and optimization. I was intrigued by applications of OR models to traffic flow and congestion, and as a graduate student at University College London I focused on modeling of transportation systems. I continued research on this topic in engineering school faculty positions at Princeton University and University College London. I knew of the traffic studies and other research at GM R&D through meetings and their publications, and was interested in gaining experience in applied research in industry. I joined GM R&D, where I have had the opportunity to work in a variety of research areas, including traffic safety, logistics, inventory control, and production system design, and to see results used in practice. It always impresses me how powerful even simple mathematical models can be in providing insight into system behavior.
Debra: I took a lot of classes in math, computer science, physics, and chemistry, and finally realized I liked sport computing and slick mathematics applied to real world industrial problems. I ended up in Operations Research, which lets me combine my interests in probability, super computing and high performance computing, simulation, and so forth. As a graduate student in the Industrial Engineering/Operations Research Program at Texas A&M University, I found out about working at GM R&D when I was at a technical conference. I decided to interview out of curiosity. I was really surprised and delighted with the people and the caliber of research going on within GM. My first major research project was to explore financial implications of agile machining systems for GM. While working on that project, I was poking around in risk analysis work, and connected with GM Corporate Risk Management, a group that wanted some help with probabilistic modeling of risks. Now I’m working on strategic supply chain risk analysis. I’m examining how to model the GM manufacturing enterprise, exploring the frequency and severity of business interruption events—anything that interrupts production operations—and considering strategic mitigation options that can reduce GM’s risk exposure. What excites me about my research is combining ideas from different subject areas, like math, computer science, statistics, and operations research, to develop novel modeling approaches and solutions for large-scale problems.
Jeff: I pursued areas that I liked, excelled in, and seemed good for a future career. I’m adept at problem solving and math modeling, and I love having a positive impact for people. Operations Research pulls all that together to solve real-world problems—not just mathematical curiosities—and to help people make decisions. It was the right match for me. While I was a graduate student at the University of Michigan, I heard a presentation about the research opportunities at GM R&D. It seemed like a great position, so after the meeting I submitted my resume. About two weeks later I had a job offer. I modeled production systems, including throughput analysis, maintenance systems, production leveling/stability metrics, and manufacturing-related cost-driver studies. I’m now on a two-year rotation in GM Engineering to learn other GM operations, contribute in new ways, and develop new bridges and topic areas for research. My current assignment is leading the development of decision-support tools and methods for engineering issues that include test scheduling, engineering capacity assessment, and engineering process improvements. I like helping people make decisions via decision-support tools, analysis methods, and insights—and then seeing my work produce a positive impact.
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