# Operations Research

What problems does it solve?

Operations Research (OR) is a scientific method of providing executive departments with a quantitative basis for decisions regarding the operations under their control. A great deal of operations research today deals with determining the optimum way to achieve a goal based on some mathematical or statistical model of a situation. The task of the operations research worker is to present the quantitative aspect in an intelligible form and to point out, if possible, the non-quantitative aspects that may need consideration by the executive before reaching decisions.

Operations Research as we know it today is primarily an out-growth of military research in World War II that sought optimal ways to allocate scarce resources. This included questions such as "How should patrol aircraft be deployed to maximize the expected number of enemy submarines detected in a limited number of hours of search?" and "How should a limited inventory of spare parts be distributed among units in the field, advance depots, and distribution warehouses to minimize equipment down-time due to parts shortages?" At the end of the war, most operations researchers moved into industry, where similar questions in budgeting, planning, marketing, decision-making, and other aspects of management were in need of answers.

A sample problem might be the optimum operation of toll booths on a bridge or turnpike. The problem is to achieve the best balance between having idle attendants during slack hours and too much delay during rush-hour. One would have to determine the statistics of the traffic flow, construct a mathematical model of the queuing system, determine the expected number of idle attendants and the expected delay as a function of the number of attendants. The resultant expression is then analyzed to determine the optimum performance, given any other restraints imposed.

The solution to the toll-booth problem is well understood, but there are other OR problems that are mathematically challenging and still unsolved. One that is far from solved in general was originally described as the following traveling salesman problem. Sales representatives of a corporation have customers in each of a list of cities. The goal is to find the shortest tour enabling them to visit all their customers exactly once. This can be translated into a problem in graph theory - how to characterize the shortest path joining a certain number of nodes. It can also be translated into a problem of minimizing a certain function subject to constraints. Linear and integer programming are among the techniques utilized in attempting to solve problems of this type.

What should one study in college?

The major mathematical tools of OR are vector calculus, linear algebra, differential and difference equations, probability, statistics, and computer programming. Other courses particularly relevant to this field include number theory, abstract algebra, graph theory and combinatorics. Still other relevant courses may be given in or outside of the mathematics department, e.g., linear programming, control theory, integer programming, dynamic programming, game theory, and queuing theory, as well as computer science courses and simulation.

An individual with a bachelor's degree in mathematics and an applied minor can possibly obtain direct employment in operations research, but a masters degree in OR is the credential preferred by most employers. A good working knowledge of economics, finance, and organization theory is also valuable, and that is something that a mathematics undergraduate can pursue, e.g., as a minor.