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Chance Constrained Programming
Chance Constrained Programming (CCP) is a mathematical optimization approach used to handle problems under uncertainty. It was first introduced by Charnes and Cooper in 1959 and further developed by Miller and Wagner in 1965. CCP is widely used in various fields, including finance, engineering, and operations research, to optimize decision-making processes where certain constraints need to be satisfied with a specified probability. Theoretical Background Chance Constrained Programming involves the use of probability and confidence levels to handle uncertainty in optimization problems. It distinguishes between single and joint chance constraints: * Single Chance Constraints: These constraints ensure that each individual constraint is satisfied with a certain probability. * Joint Chance Constraints: These constraints ensure that all constraints are satisfied simultaneously with a certain probability. Mathematical Formulation A general chance constrained optimization problem can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Optimization
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 maxima and minima, maximizing or minimizing a Function of a real variable, real function by systematically choosing Argument of a function, input values from within an allowed set and computing the Value (mathematics), value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. Optimization problems Opti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Programming
In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints are not linear equalities or the objective function is not a linear function. An optimization problem is one of calculation of the extrema (maxima, minima or stationary points) of an objective function over a set of unknown real variables and conditional to the satisfaction of a system of equalities and inequalities, collectively termed constraints. It is the sub-field of mathematical optimization that deals with problems that are not linear. Definition and discussion Let ''n'', ''m'', and ''p'' be positive integers. Let ''X'' be a subset of ''Rn'' (usually a box-constrained one), let ''f'', ''gi'', and ''hj'' be real-valued functions on ''X'' for each ''i'' in and each ''j'' in , with at least one of ''f'', ''gi'', and ''hj'' being nonlinear. A nonlinear programming problem is an optimization problem of the form : \begin \text & f(x) \\ \text ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Portfolio Optimization
Portfolio optimization is the process of selecting an optimal portfolio (asset distribution), out of a set of considered portfolios, according to some objective. The objective typically maximizes factors such as expected return, and minimizes costs like financial risk, resulting in a multi-objective optimization problem. Factors being considered may range from tangible (such as assets, liabilities, earnings or other fundamentals) to intangible (such as selective divestment). Modern portfolio theory Modern portfolio theory was introduced in a 1952 doctoral thesis by Harry Markowitz, where the Markowitz model was first defined. The model assumes that an investor aims to maximize a portfolio's expected return contingent on a prescribed amount of risk. Portfolios that meet this criterion, i.e., maximize the expected return given a prescribed amount of risk, are known as efficient portfolios. By definition, any other portfolio yielding a higher amount of expected return must also h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chance-constrained Portfolio Selection
Chance-constrained portfolio selection is an approach to portfolio selection under loss aversion. The formulation assumes that (i) investor's preferences are representable by the expected utility of final wealth, and that (ii) they require that the probability of their final wealth falling below a survival or safety level must to be acceptably low. The chance-constrained portfolio problem is then to find: :Max \sum_wjE(Xj), subject to Pr(\sum_ wjXj < s) ≤ , wj = 1, wj ≥ 0 for all j, ::where s is the survival level and is the admissible probability of ruin; w is the weight and x is the value of the ''jth'' asset to be included in the portfolio. The original implementation is based on the seminal work of Abraham Charnes and [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Production Planning
Production planning is the planning of Production (economics), production and manufacturing modules in a company or industry. It utilizes the resource allocation of activities of employees, raw material, materials and production capacity, in order to serve different customers.Fargher, Hugh E., and Richard A. Smith. "Method and system for production planning." U.S. Patent No. 5,586,021. 17 Dec. 1996. Different types of production methods, such as single item manufacturing, batch production, mass production, continuous production etc. have their own type of production planning. Production planning can be combined with production control into production planning and control, or it can be combined with enterprise resource planning. Overview Production planning is the future of production. It can help in efficient manufacturing or setting up of a production site by facilitating required needs. A production plan is made periodically for a specific time period, called the planning ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Process Engineering
Process engineering is a field of study focused on the development and optimization of industrial processes. It consists of the understanding and application of the fundamental principles and laws of nature to allow humans to transform raw material and energy into products that are useful to society, at an industrial level. By taking advantage of the driving forces of nature such as pressure, temperature and concentration gradients, as well as the law of conservation of mass, process engineers can develop methods to synthesize and purify large quantities of desired chemical products. Process engineering focuses on the design, operation, control, optimization and intensification of chemical, physical, and biological processes. Their work involves analyzing the chemical makeup of various ingredients and determining how they might react with one another. A process engineer can specialize in a number of areas, including the following: * Agriculture processing * Food and dairy pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chemical Engineering
Chemical engineering is an engineering field which deals with the study of the operation and design of chemical plants as well as methods of improving production. Chemical engineers develop economical commercial processes to convert raw materials into useful products. Chemical engineering uses principles of chemistry, physics, mathematics, biology, and economics to efficiently use, produce, design, transport and transform energy and materials. The work of chemical engineers can range from the utilization of nanotechnology and nanomaterials in the laboratory to large-scale industrial processes that convert chemicals, raw materials, living cells, microorganisms, and energy into useful forms and products. Chemical engineers are involved in many aspects of plant design and operation, including safety and hazard assessments, process engineering, process design and analysis, modeling and simulation, modeling, control engineering, chemical reaction engineering, nuclear engineering, biologi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Renewable Energy
Renewable energy (also called green energy) is energy made from renewable resource, renewable natural resources that are replenished on a human lifetime, human timescale. The most widely used renewable energy types are solar energy, wind power, and hydropower. Bioenergy and geothermal power are also significant in some countries. Some also consider Nuclear power proposed as renewable energy, nuclear power a renewable power source, although this is controversial, as nuclear energy requires mining uranium, a nonrenewable resource. Renewable energy installations can be large or small and are suited for both urban and rural areas. Renewable energy is often deployed together with further electrification. This has several benefits: electricity can heat pump, move heat and Electric vehicle, vehicles efficiently and is clean at the point of consumption. Variable renewable energy sources are those that have a fluctuating nature, such as wind power and solar power. In contrast, ''contro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propagation Of Uncertainty
In statistics, propagation of uncertainty (or propagation of error) is the effect of variables' uncertainties (or errors, more specifically random errors) on the uncertainty of a function based on them. When the variables are the values of experimental measurements they have uncertainties due to measurement limitations (e.g., instrument precision) which propagate due to the combination of variables in the function. The uncertainty ''u'' can be expressed in a number of ways. It may be defined by the absolute error . Uncertainties can also be defined by the relative error , which is usually written as a percentage. Most commonly, the uncertainty on a quantity is quantified in terms of the standard deviation, , which is the positive square root of the variance. The value of a quantity and its error are then expressed as an interval . However, the most general way of characterizing uncertainty is by specifying its probability distribution. If the probability distribution of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Programming
Linear 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 and objective are represented by linear function#As a polynomial function, 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 mathematical optimization, optimization of a linear objective function, subject to linear equality and linear inequality Constraint (mathematics), constraints. Its feasible region is a convex polytope, which is a set defined as the intersection (mathematics), intersection of finitely many Half-space (geometry), half spaces, each of which is defined by a linear inequality. Its objective function is a real number, real-valued affine function, affine (linear) function defined on this polytope. A linear programming algorithm finds a point in the po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abraham Charnes
Abraham Charnes (September 4, 1917 – December 19, 1992) was an American mathematician who worked in the area of operations research. Charnes published more than 200 research articles and seven books, including ''An Introduction to Linear Programming''. His works influenced the development of Data envelopment analysis (DEA) method. In his 1953 article with William W. Cooper he developed the chance constrained programming method for solving optimization problems in the presence of uncertainty. Charnes received his bachelor's degree in 1938, master's degree in 1939, and PhD degree (with a thesis entitled ''Wing-Body Interaction in Linear Supersonic Flow'') in 1947 from the University of Illinois. Charnes taught at the Carnegie Institute of Technology, Purdue University, Northwestern University, and at the University of Texas at Austin since 1968. In 1975 Charnes was shortlisted for the Nobel Prize in economics. In 1982 he was awarded (jointly with William W. Cooper and Richard ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Optimization
Stochastic optimization (SO) are optimization methods that generate and use random variables. For stochastic optimization problems, the objective functions or constraints are random. Stochastic optimization also include methods with random iterates. Some hybrid methods use random iterates to solve stochastic problems, combining both meanings of stochastic optimization. Stochastic optimization methods generalize deterministic methods for deterministic problems. Methods for stochastic functions Partly random input data arise in such areas as real-time estimation and control, simulation-based optimization where Monte Carlo simulations are run as estimates of an actual system, and problems where there is experimental (random) error in the measurements of the criterion. In such cases, knowledge that the function values are contaminated by random "noise" leads naturally to algorithms that use statistical inference tools to estimate the "true" values of the function and/or make sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
