## Mathematics and Operations Research in Industry

*By Dennis E. Blumenfeld, Debra A. Elkins, and Jeffrey M. Alden*
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 combines mathematics, statistics, computer science, physics, engineering, economics, and social sciences to solve real-world business problems. Numerous companies in industry require Operations Research professionals to apply mathematical techniques to a wide range of challenging questions.

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.

The term ?Operations Research? is known as ?Operational Research? in Britain and other parts of Europe. Other terms used are ?Management Science,? ?Industrial Engineering,? and ?Decision Sciences.? The multiplicity of names comes primarily from the different academic departments that have hosted courses in this field. The subject is frequently referred to simply as ?OR?, and includes both the application of past research results and new research to develop improved solution methods.

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 (optimization, stochastic processes, simulation, heuristics, and other mathematical techniques);

Validation and Analysis (model testing, calibration, sensitivity analysis, model robustness); and

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. Mathematical methods were developed to determine the most effective use of radar and other new defense technologies at the time. OR groups were later formed in the U.S. to meet needs of wartime operations, such as the optimal movement of troops, supplies, and equipment.

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. Examples of where OR has been successful in recent years are the following:

Airline Industry (routing and flight plans, crew scheduling, revenue management);

Telecommunications (network routing, queue control);

Manufacturing Industry (system throughput and bottleneck analysis, inventory control, production scheduling, capacity planning);

Healthcare (hospital management, facility design); and

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 (non-linear 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). A common example is the single-server queue in which customer arrivals and service times are random. Figure 1 illustrates the queue, and the curve shows how sensitive the average queue length becomes under high traffic intensity conditions. Mathematical analysis has been essential in understanding queue behavior and quantifying impacts of decisions. Equations have been derived for the queue length, waiting times, and 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.

*Figure 1: Single-Server Queueing System*
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 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.

Another real-world mathematical problem, common to many industries, is the distribution of material and products from plants to customers. For a network of origins and destinations, there are many shipping alternatives, including choices of transportation mode (e.g., road, rail, air) and geographical routes. Some key decisions are routing options over the network, and shipping frequencies on network links. As shown in Figure 2, routing options involve shipping direct, via a terminal or distribution center, and by a combination of routes. These options affect distances traveled and times in transit, which in turn affect transportation and inventory costs. Shipping frequency decisions also affect these costs. Transportation costs favor large infrequent shipments, while inventory costs favor small frequent shipments. Trade-offs between these costs are complex for large networks, and finding the optimal solution is a challenging mathematical problem. In addition to decisions for operations of a given network, there are major strategic decisions, such as the selection and location of distribution centers.

*Figure 2: Network Routing Options*
Other OR topics requiring mathematical analysis are inventory control (when to reorder material to avoid shortages under demand uncertainty), manufacturing operations (what size of production run will minimize sum of inventory and production setup costs), location planning (where to locate the hub to serve markets with minimal travel distances), and facility layout (how to design airport terminals to minimize walking distances, maximize number of gates, allow for future expansion, and conform to government regulations).

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 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 Blumenfeld:** 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 to gain experience of 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 Elkins:** 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 Alden**: I basically pursued areas that I liked, excelled in, and seemed good for a future career. Since I really enjoy problem solving, math modeling, and helping people make better decisions, I naturally migrated to Operations Research. So I was sure what I wanted to do, but not sure where to work. Well, I attended a presentation about the research opportunities at GM R &D given by Larry Burns (now a VP at GM). It seemed like an ideal place to work, so I gave him my resume and soon accepted a research position at GM R&D. For the next decade, I researched production systems looking at throughput, maintenance, leveling, stability, agility, and cost-drivers. I?m now on a two-year rotation in GM Engineering managing development of decision-support tools and methods for engineering in a variety of areas that include test-scheduling, warranty cost drivers, and work load planning.

We wish to thank Dr. Devadatta Kulkarni at General Motors R&D Center for valuable comments and suggestions.

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