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.
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. 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. 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.
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.
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? 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. 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. 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? 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.
For the next decade, I researched production systems looking at throughput, maintenance, leveling, stability, agility, and cost-drivers. We wish to thank Dr. Allen, A. Academic Press, Orlando, Florida. Toggle navigation. Mathematics in Industry. View Extract. Description Editor Bio In this book, a wide range of problems concerning recent achievements in the field of industrial and applied mathematics are presented.
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