Pareto efficiencyPareto efficiency or Pareto optimality is a situation where no action or allocation is available that makes one individual better off without making another worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian civil engineer and economist, who used the concept in his studies of economic efficiency and income distribution. The following three concepts are closely related: Given an initial situation, a Pareto improvement is a new situation where some agents will gain, and no agents will lose.
Linear programmingLinear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints.
Integer programmingAn integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objective function and the constraints (other than the integer constraints) are linear. Integer programming is NP-complete. In particular, the special case of 0-1 integer linear programming, in which unknowns are binary, and only the restrictions must be satisfied, is one of Karp's 21 NP-complete problems.
Production–possibility frontierIn microeconomics, a production–possibility frontier (PPF), production possibility curve (PPC), or production possibility boundary (PPB) is a graphical representation showing all the possible options of output for two goods that can be produced using all factors of production, where the given resources are fully and efficiently utilized per unit time. A PPF illustrates several economic concepts, such as allocative efficiency, economies of scale, opportunity cost (or marginal rate of transformation), productive efficiency, and scarcity of resources (the fundamental economic problem that all societies face).
Linear programming relaxationIn mathematics, the relaxation of a (mixed) integer linear program is the problem that arises by removing the integrality constraint of each variable. For example, in a 0–1 integer program, all constraints are of the form The relaxation of the original integer program instead uses a collection of linear constraints The resulting relaxation is a linear program, hence the name.
Welfare economicsWelfare economics is a field of economics that applies microeconomic techniques to evaluate the overall well-being (welfare) of a society. This evaluation is typically done at the economy-wide level, and attempts to assess the distribution of resources and opportunities among members of society. The principles of welfare economics are often used to inform public economics, which focuses on the ways in which government intervention can improve social welfare.
Fundamental theorems of welfare economicsThere are two fundamental theorems of welfare economics. The first states that in economic equilibrium, a set of complete markets, with complete information, and in perfect competition, will be Pareto optimal (in the sense that no further exchange would make one person better off without making another worse off). The requirements for perfect competition are these: There are no externalities and each actor has perfect information. Firms and consumers take prices as given (no economic actor or group of actors has market power).
Passenger railroad carA passenger railroad car or passenger car (American English), also called a passenger carriage, passenger coach (British English and International Union of Railways), or passenger bogie (Indian English) is a railroad car that is designed to carry passengers. The term passenger car can also be associated with a sleeping car, a baggage car, a dining car, railway post office and prisoner transport cars. The first passenger cars were built in the early 1800s with the advent of the first railroads, and were small and little more than converted freight cars.
Cutting-plane methodIn mathematical optimization, the cutting-plane method is any of a variety of optimization methods that iteratively refine a feasible set or objective function by means of linear inequalities, termed cuts. Such procedures are commonly used to find integer solutions to mixed integer linear programming (MILP) problems, as well as to solve general, not necessarily differentiable convex optimization problems. The use of cutting planes to solve MILP was introduced by Ralph E. Gomory.
Deadweight lossIn economics, deadweight loss is the difference in production and consumption of any given product or service including government tax. The presence of deadweight loss is most commonly identified when the quantity produced relative to the amount consumed differs in regards to the optimal concentration of surplus. This difference in the amount reflects the quantity that is not being utilized or consumed and thus resulting in a loss.
Allocative efficiencyAllocative efficiency is a state of the economy in which production is aligned with consumer preferences; in particular, the set of outputs is chosen so as to maximize the wellbeing of society. This is achieved if every good or service is produced up until the last unit provides a marginal benefit to consumers equal to the marginal cost of production. In economics, allocative efficiency entails production at the point on the production possibilities frontier that is optimal for society.
Travelling salesman problemThe travelling salesman problem (TSP) asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?" It is an NP-hard problem in combinatorial optimization, important in theoretical computer science and operations research. The travelling purchaser problem and the vehicle routing problem are both generalizations of TSP.
Knapsack problemThe knapsack problem is the following problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine which items to include in the collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. It derives its name from the problem faced by someone who is constrained by a fixed-size knapsack and must fill it with the most valuable items.
Indian RailwaysIndian Railways (IR) is a statutory body under the ownership of the Ministry of Railways, Government of India that operates India's national railway system. It manages the fourth largest national railway system in the world by size, with a total route length of , running track length of and track length of . of all the gauge routes are electrified with 25 kV 50 Hz AC electric traction . In 2020, Indian Railways carried 808.6 crore (8.086 billion) passengers and in 2022, Railways transported 1418.
Public transportPublic transport (also known as public transportation, public transit, mass transit, or simply transit) is a system of transport for passengers by group travel systems available for use by the general public unlike private transport, typically managed on a schedule, operated on established routes, and that charge a posted fee for each trip. There is no rigid definition of which kinds of transport are included, and air travel is often not thought of when discussing public transport—dictionaries use wording like "buses, trains, etc.
Passenger trainA passenger train is a train used to transport people along a railroad line. These trains may consist of unpowered passenger railroad cars (also known as coaches or carriages) hauled by one or more locomotives, or may be self-propelled; self propelled passenger trains are known as multiple units or railcars. Passenger trains stop at stations or depots, where passengers may board and disembark. In most cases, passenger trains operate on a fixed schedule and have priority over freight trains.
Decision problemIn computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values. An example of a decision problem is deciding by means of an algorithm whether a given natural number is prime. Another is the problem "given two numbers x and y, does x evenly divide y?". The answer is either 'yes' or 'no' depending upon the values of x and y. A method for solving a decision problem, given in the form of an algorithm, is called a decision procedure for that problem.
Financial marketA financial market is a market in which people trade financial securities and derivatives at low transaction costs. Some of the securities include stocks and bonds, raw materials and precious metals, which are known in the financial markets as commodities. The term "market" is sometimes used for what are more strictly exchanges, organizations that facilitate the trade in financial securities, e.g., a stock exchange or commodity exchange.
Rail transportRail transport (also known as train transport) is a means of transport that transfers passengers and goods on wheeled vehicles running on rails, which are incorporated in tracks. In contrast to road transport, where the vehicles run on a prepared flat surface, rail vehicles (rolling stock) are directionally guided by the tracks on which they run. Tracks usually consist of steel rails.Rolling stock in a rail transport system generally encounters lower frictional resistance than rubber-tyred road vehicles, so passenger and freight cars (carriages and wagons) can be coupled into longer trains.
Market (economics)In economics, a market is a composition of systems, institutions, procedures, social relations or infrastructures whereby parties engage in exchange. While parties may exchange goods and services by barter, most markets rely on sellers offering their goods or services (including labour power) to buyers in exchange for money. It can be said that a market is the process by which the prices of goods and services are established. Markets facilitate trade and enable the distribution and allocation of resources in a society.
