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2009 | Buch

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Theory and Practice of Uncertain Programming

verfasst von: Baoding Liu

Verlag: Springer Berlin Heidelberg

Buchreihe : Studies in Fuzziness and Soft Computing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Real-life decisions are usually made in the state of uncertainty such as randomness and fuzziness. How do we model optimization problems in uncertain environments? How do we solve these models? In order to answer these questions, this book provides a self-contained, comprehensive and up-to-date presentation of uncertain programming theory, including numerous modeling ideas, hybrid intelligent algorithms, and applications in system reliability design, project scheduling problem, vehicle routing problem, facility location problem, and machine scheduling problem. Researchers, practitioners and students in operations research, management science, information science, system science, and engineering will find this work a stimulating and useful reference.

Inhaltsverzeichnis

Frontmatter

Mathematical Programming

Abstract

As one of the most widely used techniques in operations research, mathematical programming is defined as a means of maximizing a quantity known as objective function, subject to a set of constraints represented by equations and inequalities. Some known subtopics of mathematical programming are linear programming, nonlinear programming, multiobjective programming, goal programming, dynamic programming, and multilevel programming.

It is impossible to cover in a single chapter every concept of mathematical programming. This chapter introduces only the basic concepts and techniques of mathematical programming such that readers gain an understanding of them throughout the book.

Baoding Liu

Genetic Algorithms

Abstract

Genetic algorithm (GA) is a stochastic search method for optimization problems based on the mechanics of natural selection and natural genetics (i.e., survival of the fittest). GA has demonstrated considerable success in providing good solutions to many complex optimization problems and received more and more attentions during the past three decades. When the objective functions to be optimized in the optimization problems are multimodal or the search spaces are particularly irregular, algorithms need to be highly robust in order to avoid getting stuck at a local optimal solution. The advantage of GA is just able to obtain the global optimal solution fairly. In addition, GA does not require the specific mathematical analysis of optimization problems, which makes GA easily coded by users who are not necessarily good at mathematics and algorithms.

Baoding Liu

Neural Networks

Abstract

Neural network (NN), inspired by the current understanding of biological NN, is a class of adaptive systems consisting of a number of simple processing elements, called neurons, that are interconnected to each other in a feedforward way. Although NN can perform some human brain-like tasks, there is still a huge gap between biological and artificial NN. An important contribution of NN is the ability to learn to perform operations, not only for inputs exactly like the training data, but also for new data that may be incomplete or noisy. NN has also the benefit of easy modification by retraining with an updated data set. For our purpose, the significant advantage of NN is the speed of operation after it is trained.

Baoding Liu

Stochastic Programming

Abstract

With the requirement of considering randomness, different types of stochastic programming have been developed to suit the different purposes of management. The first type of stochastic programming is the expected value model, which optimizes the expected objective functions subject to some expected constraints. The second, chance-constrained programming, was pioneered by Charnes and Cooper [37] as a means of handling uncertainty by specifying a confidence level at which it is desired that the stochastic constraint holds. After that, Liu [174] generalized chance-constrained programming to the case with not only stochastic constraints but also stochastic objectives. In practice, there usually exist multiple events in a complex stochastic decision system. Sometimes the decision-maker wishes to maximize the chance functions of satisfying these events. In order to model this type of problem, Liu [166] provided a theoretical framework of the third type of stochastic programming, called dependent-chance programming.

This chapter will give some basic concepts of probability theory and introduce a spectrum of stochastic programming. A hybrid intelligent algorithm is also documented.

Baoding Liu

Fuzzy Programming

Abstract

Fuzzy programming offers a powerful means of handling optimization problems with fuzzy parameters. Fuzzy programming has been used in different ways in the past. Liu and Liu [184] presented a concept of expected value operator of fuzzy variable and provided a spectrum of fuzzy expected value models which optimize the expected objective functions subject to some expected constraints. In addition, Liu and Iwamura [168][169] introduced a spectrum of fuzzy maximax chance-constrained programming, and Liu [171] constructed a spectrum of fuzzy minimax chance-constrained programming in which we assume that the fuzzy constraints will hold with a given credibility level. Liu [172] provided a fuzzy dependent-chance programming theory in order to maximize the chance functions of satisfying some events.

Baoding Liu

Hybrid Programming

Abstract

In many cases, fuzziness and randomness simultaneously appear in a system. In order to describe this phenomena, a fuzzy random variable was introduced by Kwakernaak [142] as a random element taking “fuzzy variable” values. By fuzzy random programming we mean the optimization theory in fuzzy random environments. Liu and Liu [198] presented a spectrum of fuzzy random expected value model (EVM), Liu [179] initialized a general framework of fuzzy random chance-constrained programming (CCP), and Liu [180] introduced the concepts of uncertain environment and chance function for fuzzy random decision problems, and constructed a theoretical framework of fuzzy random dependent-chance programming (DCP).

Baoding Liu

Uncertain Programming

Abstract

Uncertainty theory, founded by Liu [189] in 2007, is a branch of mathematics based on normality, monotonicity, self-duality, and countable subadditivity axioms. By uncertain programming we mean the optimization theory in generally uncertain environments. This chapter introduces the concept of uncertain variable and provides a general framework of uncertain programming.

Baoding Liu

System Reliability Design

Abstract

One of the approaches to improve system reliability is to provide redundancy for components in a system. There are two ways to provide component redundancy: parallel redundancy and standby redundancy. In parallel redundancy, all redundant elements are required to operate simultaneously. This method is usually used when element replacements are not permitted during the system operation. In standby redundancy, one of the redundant elements begins to work only when the active element fails. This method is usually employed when the replacement is allowable and can be finished immediately.

The system reliability design problem is to determine the optimal number of redundant elements for each component so as to optimize some system performance.

Baoding Liu

Project Scheduling Problem

Abstract

Project scheduling problem is to determine the schedule of allocating resources so as to balance the total cost and the completion time. Uncertainty always exists in project scheduling problem due to the vagueness of project activity duration times. This chapter will introduce some optimization models for uncertain project scheduling problems.

Baoding Liu

Vehicle Routing Problem

Abstract

Vehicle routing problem (VRP) is concerned with finding efficient routes, beginning and ending at a central depot, for a fleet of vehicles to serve a number of customers.

Baoding Liu

Facility Location Problem

Abstract

Facility location problem is to find locations for new facilities such that the conveying cost from facilities to customers is minimized. Facility location problem has been studied for half a century because of its widely practical application backgrounds.

In practice, some factors such as demands, allocations, even locations of customers and facilities are usually changing. In an uncapacitated facility location problem, the customers are supplied by the nearest factory. However, in a capacitated problem, the customers may not be supplied by the nearest factory only. In order to solve this type of problem, this chapter introduces some optimization models for uncertain capacitated facility location problem.

Baoding Liu

Machine Scheduling Problem

Abstract

Machine scheduling problem is concerned with finding an efficient schedule during an uninterrupted period of time for a set of machines to process a set of jobs. Much of research work has been done on this type of problem during the past five decades.

Baoding Liu

Backmatter

Titel: Theory and Practice of Uncertain Programming
verfasst von: Baoding Liu
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-540-89484-1
Print ISBN: 978-3-540-89483-4
DOI: https://doi.org/10.1007/978-3-540-89484-1

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