Operations research in the context of Optimization

A mathematical model is an abstract description of a concrete system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in many fields, including applied mathematics, natural sciences, social sciences and engineering. In particular, the field of operations research studies the use of mathematical modelling and related tools to solve problems in business or military operations. A model may help to characterize a system by studying the effects of different components, which may be used to make predictions about behavior or solve specific problems.

Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.

In the more general approach, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics.

Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service.

Queueing theory has its origins in research by Agner Krarup Erlang, who created models to describe the system of incoming calls at the Copenhagen Telephone Exchange Company. These ideas were seminal to the field of teletraffic engineering and have since seen applications in telecommunications, traffic engineering, computing, project management, and particularly industrial engineering, where they are applied in the design of factories, shops, offices, and hospitals.

A white-collar worker is a person who performs knowledge-based, aptitude-based, managerial, or administrative work generally performed in an office or similar setting. White-collar workers include job paths related to banking, finance, compliance, legal, risk management, internal audit, data privacy, cybersecurity, insurance, government, consulting, academia, accountancy, business and executive management, customer support, design, economics, science, technology, engineering, market research, human resources, operations research, marketing, public relations, real estate, information technology, networking, healthcare, architecture, and research and development.

In contrast, blue-collar workers perform manual labor or work in skilled trades; pink-collar workers work in care, health care, social work, or teaching; green-collar workers specifically work in the environmental sector; and grey-collar jobs combine manual labor and skilled trades with non-manual or managerial duties.

Project management is the process of supervising the work of a team to achieve all project goals within the given constraints. This information is usually described in project documentation, created at the beginning of the development process. The primary constraints are scope, time and budget. The secondary challenge is to optimize the allocation of necessary inputs and apply them to meet predefined objectives.

The objective of project management is to produce a complete project which complies with the client's objectives. In many cases, the objective of project management is also to shape or reform the client's brief to feasibly address the client's objectives. Once the client's objectives are established, they should influence all decisions made by other people involved in the project– for example, project managers, designers, contractors and subcontractors. Ill-defined or too tightly prescribed project management objectives are detrimental to the decisionmaking process.

Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combinatorial optimization problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem. In many such problems, such as the ones previously mentioned, exhaustive search is not tractable, and so specialized algorithms that quickly rule out large parts of the search space or approximation algorithms must be resorted to instead.

Combinatorial optimization is related to operations research, algorithm theory, and computational complexity theory. It has important applications in several fields, including artificial intelligence, machine learning, auction theory, software engineering, VLSI, applied mathematics and theoretical computer science.

The Mathematical Sciences are a group of areas of study that includes, in addition to mathematics, those academic disciplines that are primarily mathematical in nature but may not be universally considered subfields of mathematics proper.

Statistics, for example, is mathematical in its methods but grew out of bureaucratic and scientific observations, which merged with inverse probability and then grew through applications in some areas of physics, biometrics, and the social sciences to become its own separate, though closely allied, field. Theoretical astronomy, theoretical physics, theoretical and applied mechanics, continuum mechanics, mathematical chemistry, actuarial science, computer science, computational science, data science, operations research, quantitative biology, control theory, econometrics, geophysics and mathematical geosciences are likewise other fields often considered part of the mathematical sciences.

In cybernetics, the term variety denotes the total number of distinguishable elements of a set, most often the set of states, inputs, or outputs of a finite-state machine or transformation, or the binary logarithm of the same quantity. Variety is used in cybernetics as an information theory that is easily related to deterministic finite automata, and less formally as a conceptual tool for thinking about organization, regulation, and stability. It is an early theory of complexity in automata, complex systems, and operations research.

In mathematics, computer science, and network science, network theory is a part of graph theory. It defines networks as graphs where the vertices or edges possess attributes. Network theory analyses these networks over the symmetric relations or asymmetric relations between their (discrete) components.

Network theory has applications in many disciplines, including statistical physics, particle physics, computer science, electrical engineering, biology, archaeology, linguistics, economics, finance, operations research, climatology, ecology, public health, sociology, psychology, and neuroscience. Applications of network theory include logistical networks, the World Wide Web, Internet, gene regulatory networks, metabolic networks, social networks, epistemological networks, etc.; see List of network theory topics for more examples.

In planning and policy, a wicked problem is a problem that is difficult or impossible to solve because of incomplete, contradictory, and changing requirements that are often difficult to recognize. It refers to an idea or problem that cannot be fixed, where there is no single solution to the problem; "wicked" does not indicate evil, but rather resistance to resolution. Another definition is "a problem whose social complexity means that it has no determinable stopping point". Because of complex interdependencies, the effort to solve one aspect of a wicked problem may reveal or create other problems. Due to their complexity, wicked problems are often characterized by organized irresponsibility.

The phrase was originally used in social planning. Its modern sense was introduced in 1967 by C. West Churchman in a guest editorial he wrote in the journal Management Science. He explains that "The adjective 'wicked' is supposed to describe the mischievous and even evil quality of these problems, where proposed 'solutions' often turn out to be worse than the symptoms". In the editorial, he credits Horst Rittel with first describing wicked problems, though it may have been Churchman who coined the term. Churchman discussed the moral responsibility of operations research "to inform the manager in what respect our 'solutions' have failed to tame his wicked problems." Rittel and Melvin M. Webber formally described the concept of wicked problems in a 1973 treatise, contrasting "wicked" problems with relatively "tame", solvable problems in mathematics, chess, or puzzle solving.

In social choice and operations research, the utilitarian rule (also called the max-sum rule) is a rule saying that, among all possible alternatives, society should pick the alternative which maximizes the sum of the utilities of all individuals in society. It is a formal mathematical representation of the utilitarian philosophy, and is often justified by reference to Harsanyi's utilitarian theorem or the Von Neumann–Morgenstern theorem.

In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable guarantees on the distance of the returned solution to the optimal one. Approximation algorithms naturally arise in the field of theoretical computer science as a consequence of the widely believed P ≠ NP conjecture. Under this conjecture, a wide class of optimization problems cannot be solved exactly in polynomial time. The field of approximation algorithms, therefore, tries to understand how closely it is possible to approximate optimal solutions to such problems in polynomial time. In an overwhelming majority of the cases, the guarantee of such algorithms is a multiplicative one expressed as an approximation ratio or approximation factor i.e., the optimal solution is always guaranteed to be within a (predetermined) multiplicative factor of the returned solution. However, there are also many approximation algorithms that provide an additive guarantee on the quality of the returned solution. A notable example of an approximation algorithm that provides both is the classic approximation algorithm of Lenstra, Shmoys and Tardos for scheduling on unrelated parallel machines.

The design and analysis of approximation algorithms crucially involves a mathematical proof certifying the quality of the returned solutions in the worst case. This distinguishes them from heuristics such as annealing or genetic algorithms, which find reasonably good solutions on some inputs, but provide no clear indication at the outset on when they may succeed or fail.