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Linear programming (LP), also called linear optimization, is a method to achieve the best outcome in a mathematical model whose requirements and objective are represented by linear relationships. Linear programming is a special case of mathematical programming.
Mathematical optimization 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 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 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.
Multiple-criteria decision-making (MCDM) or multiple-criteria decision analysis (MCDA) is a sub-discipline of operations research that explicitly evaluates multiple conflicting criteria in decision making. It is also known as multiple attribute utility theory, multiple attribute value theory, multiple attribute preference theory, and multi-objective decision analysis.
Model predictive control (MPC) is an advanced method of process control that is used to control a process while satisfying a set of constraints. It has been in use in the process industries in chemical plants and oil refineries since the 1980s. In recent years it has also been used in power system balancing models and in power electronics. Model predictive controllers rely on dynamic models of the process, most often linear empirical models obtained by system identification. The main advantage of MPC is the fact that it allows the current timeslot to be optimized, while keeping future timeslots in account. This is achieved by optimizing a finite time-horizon, but only implementing the current timeslot and then optimizing again, repeatedly, thus differing from a linear–quadratic regulator (LQR). Also MPC has the ability to anticipate future events and can take control actions accordingly. PID controllers do not have this predictive ability. MPC is nearly universally implemented as a digital control, although there is research into achieving faster response times with specially designed analog circuitry.
Adaptive control is the control method used by a controller which must adapt to a controlled system with parameters which vary, or are initially uncertain. For example, as an aircraft flies, its mass will slowly decrease as a result of fuel consumption; a control law is needed that adapts itself to such changing conditions. Adaptive control is different from robust control in that it does not need a priori information about the bounds on these uncertain or time-varying parameters; robust control guarantees that if the changes are within given bounds the control law need not be changed, while adaptive control is concerned with control law changing itself.
Info-gap decision theory seeks to optimize robustness to failure under severe uncertainty, in particular applying sensitivity analysis of the stability radius type to perturbations in the value of a given estimate of the parameter of interest. It has some connections with Wald's maximin model; some authors distinguish them, others consider them instances of the same principle.
Robust optimization is a field of mathematical optimization theory that deals with optimization problems in which a certain measure of robustness is sought against uncertainty that can be represented as deterministic variability in the value of the parameters of the problem itself and/or its solution. It is related to, but often distinguished from, probabilistic optimization methods such as chance-constrained optimization.
Multi-objective optimization or Pareto optimization is an area of multiple-criteria decision making that is concerned with mathematical optimization problems involving more than one objective function to be optimized simultaneously. Multi-objective is a type of vector optimization that has been applied in many fields of science, including engineering, economics and logistics where optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives. Minimizing cost while maximizing comfort while buying a car, and maximizing performance whilst minimizing fuel consumption and emission of pollutants of a vehicle are examples of multi-objective optimization problems involving two and three objectives, respectively. In practical problems, there can be more than three objectives.
A supply-chain network (SCN) is an evolution of the basic supply chain. Due to rapid technological advancement, organizations with a basic supply chain can develop this chain into a more complex structure involving a higher level of interdependence and connectivity between more organizations, this constitutes a supply-chain network.
Stochastic control or stochastic optimal control is a sub field of control theory that deals with the existence of uncertainty either in observations or in the noise that drives the evolution of the system. The system designer assumes, in a Bayesian probability-driven fashion, that random noise with known probability distribution affects the evolution and observation of the state variables. Stochastic control aims to design the time path of the controlled variables that performs the desired control task with minimum cost, somehow defined, despite the presence of this noise. The context may be either discrete time or continuous time.
IOSO is a multiobjective, multidimensional nonlinear optimization technology.
The fuzzy pay-off method for real option valuation is a method for valuing real options, developed by Mikael Collan, Robert Fullér, and József Mezei; and published in 2009. It is based on the use of fuzzy logic and fuzzy numbers for the creation of the possible pay-off distribution of a project. The structure of the method is similar to the probability theory based Datar–Mathews method for real option valuation, but the method is not based on probability theory and uses fuzzy numbers and possibility theory in framing the real option valuation problem.
Channel coordination aims at improving supply chain performance by aligning the plans and the objectives of individual enterprises. It usually focuses on inventory management and ordering decisions in distributed inter-company settings. Channel coordination models may involve multi-echelon inventory theory, multiple decision makers, asymmetric information, as well as recent paradigms of manufacturing, such as mass customization, short product life-cycles, outsourcing and delayed differentiation. The theoretical foundations of the coordination are based chiefly on the contract theory. The problem of channel coordination was first modeled and analyzed by Anantasubramania Kumar in 1992.
The scenario approach or scenario optimization approach is a technique for obtaining solutions to robust optimization and chance-constrained optimization problems based on a sample of the constraints. It also relates to inductive reasoning in modeling and decision-making. The technique has existed for decades as a heuristic approach and has more recently been given a systematic theoretical foundation.
Portfolio optimization is the process of selecting an optimal portfolio, 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 to intangible.
OptiY is a design environment software that provides modern optimization strategies and state of the art probabilistic algorithms for uncertainty, reliability, robustness, sensitivity analysis, data-mining and meta-modeling.
AIMMS is a prescriptive analytics software company with offices in the Netherlands, United States and Singapore.
Algebraic modeling languages like AIMMS, AMPL, GAMS, MPL and others have been developed to facilitate the description of a problem in mathematical terms and to link the abstract formulation with data-management systems on the one hand and appropriate algorithms for solution on the other. Robust algorithms and modeling language interfaces have been developed for a large variety of mathematical programming problems such as linear programs (LPs), nonlinear programs (NPs), mixed integer programs (MIPs), mixed complementarity programs (MCPs) and others. Researchers are constantly updating the types of problems and algorithms that they wish to use to model in specific domain applications.
Soft computing is an umbrella term used to describe types of algorithms that produce approximate solutions to unsolvable high-level problems in computer science. Typically, traditional hard-computing algorithms heavily rely on concrete data and mathematical models to produce solutions to problems. Soft computing was coined in the late 20th century. During this period, revolutionary research in three fields greatly impacted soft computing. Fuzzy logic is a computational paradigm that entertains the uncertainties in data by using levels of truth rather than rigid 0s and 1s in binary. Next, neural networks which are computational models influenced by human brain functions. Finally, evolutionary computation is a term to describe groups of algorithm that mimic natural processes such as evolution and natural selection.
References
- 1 2 3 Pishvaee, M. S.; Razmi, J.; Torabi, S. A. (2012-11-01). "Robust possibilistic programming for socially responsible supply chain network design: A new approach". Fuzzy Sets and Systems. Theme : Operational Research. 206: 1–20. doi:10.1016/j.fss.2012.04.010.
- 1 2 3 Pishvaee, Mir Saman; Fazli Khalaf, Mohamadreza (2016-01-01). "Novel robust fuzzy mathematical programming methods". Applied Mathematical Modelling. 40 (1): 407–418. doi: 10.1016/j.apm.2015.04.054 .
- ↑ Zahiri, Behzad; Tavakkoli-Moghaddam, Reza; Pishvaee, Mir Saman (2014-08-01). "A robust possibilistic programming approach to multi-period location–allocation of organ transplant centers under uncertainty". Computers & Industrial Engineering. 74: 139–148. doi:10.1016/j.cie.2014.05.008.
- ↑ Mousazadeh, M.; Torabi, S. A.; Zahiri, B. (2015-11-02). "A robust possibilistic programming approach for pharmaceutical supply chain network design". Computers & Chemical Engineering. 82: 115–128. doi:10.1016/j.compchemeng.2015.06.008.
- ↑ Bairamzadeh, Samira; Pishvaee, Mir Saman; Saidi-Mehrabad, Mohammad (2015-12-22). "Multiobjective Robust Possibilistic Programming Approach to Sustainable Bioethanol Supply Chain Design under Multiple Uncertainties". Industrial & Engineering Chemistry Research. 55 (1): 237–256. doi:10.1021/acs.iecr.5b02875.
- ↑ Zhou, Y.; Li, Y.P.; Huang, G.H. (2015-12-15). "A robust possibilistic mixed-integer programming method for planning municipal electric power systems". International Journal of Electrical Power & Energy Systems. 73: 757–772. doi:10.1016/j.ijepes.2015.06.009.
- ↑ Xu, Ye; Huang, Guohe (2015-10-15). "Development of an Improved Fuzzy Robust Chance-Constrained Programming Model for Air Quality Management". Environmental Modeling & Assessment. 20 (5): 535–548. doi:10.1007/s10666-014-9441-3.