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Über dieses Buch

Real-life decisions are usually made in the state of uncertainty (randomness, fuzziness, roughness, etc.). 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. It includes numerous modeling ideas, hybrid intelligent algorithms, and various applications in transportation problem, inventory system, facility location & allocation, capital budgeting, topological optimization, vehicle routing problem, redundancy optimization, and scheduling. 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

Fundamentals

Frontmatter

Chapter 1. Mathematical Programming

Abstract
As one of the most widely used techniques in operations research, mathematical programming is defined as a means of minimizing (or maximizing) a quantity (sometimes multiple quantities), known as objective function, subject to a set of constraints represented by mathematical relationships such as equations and inequalities.
Baoding Liu

Chapter 2. Genetic Algorithms

Abstract
Genetic algorithms (GAs) are a stochastic search method for optimization problems based on the mechanics of natural selection and natural genetics (i.e., survival of the fittest). GAs have 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 GAs is just able to obtain the global optimal solution fairly. In addition, GAs do not require the specific mathematical analysis of optimization problems, which makes GAs themselves easily coded by users who are not necessarily good at mathematics and algorithms.
Baoding Liu

Chapter 3. Neural Networks

Abstract
Neural networks (NNs), inspired by the current understanding of biological NNs, are a class of adaptive systems consisting of a number of simple processing elements, called neurons, that are interconnected to each other in a feedforward or recurrent way. Although NNs can perform some human brain-like tasks, there is still a huge gap between biological and artificial NNs.
Baoding Liu

Stochastic Programming

Frontmatter

Chapter 4. Random Variables

Abstract
Probability theory has been studied since the 17th century, and applied in a wide variety of areas of engineering, science, and management. As an important tool, stochastic simulation is defined as a technique of performing sampling experiments on the models of stochastic systems. It is heavily based on sampling random variables from probability distributions. Stochastic simulation is also referred to as Monte Carlo simulation. Although simulation is an imprecise technique which provides only statistical estimates rather than exact results and is also a slow and costly way to study problems, it is indeed a powerful tool dealing with complex problems without analytic techniques.
Baoding Liu

Chapter 5. Stochastic Expected Value Models

Abstract
The first type of stochastic programming is the so-called expected value model (EVM), which optimizes some expected objective functions subject to some expected constraints, for example, minimizing expected cost, maximizing expected profit, and so forth.
Baoding Liu

Chapter 6. Stochastic Chance-Constrained Programming

Abstract
As the second type of stochastic programming developed by Charnes and Cooper [41], chance-constrained programming (CCP) offers a powerful means of modeling stochastic decision systems with assumption that the stochastic constraints will hold at least a of time, where a is referred to as the confidence level provided as an appropriate safety margin by the decision-maker.
Baoding Liu

Chapter 7. Stochastic Dependent-Chance Programming

Abstract
In practice, there usually exist multiple events in a complex stochastic decision system. Sometimes, the decision-maker wishes to maximize the chance functions of these events (i.e., the probabilities of satisfying the events). In order to model this type of stochastic decision system, Liu [160] provided the third type of stochastic programming, called dependent-chance programming (DCP), in which the underlying philosophy is based on selecting the decision with maximal chance to meet the event.
Baoding Liu

Fuzzy Programming

Frontmatter

Chapter 8. Fuzzy Variables

Abstract
Since its introduction in 1965 by Zadeh [304], fuzzy set theory has been well developed and applied in a wide variety of real problems. The term fuzzy variable was first introduced by Kaufmann [122], then it appeared in Zadeh [306][307] and Nahmias [225]. Possibility theory was proposed by Zadeh [307], and developed by many researchers such as Dubois and Prade [62][63].
Baoding Liu

Chapter 9. Fuzzy Expected Value Models

Abstract
Assume that x is a decision vector, ξ is a fuzzy vector, f (x, ξ) is a return function, and g j (x, ξ) are constraint functions, j = 1, 2, ⋯, p.
Baoding Liu

Chapter 10. Fuzzy Chance-Constrained Programming

Abstract
Analogous to stochastic chance-constrained programming (CCP), fuzzy CCP provides a means of allowing the decision-maker to consider objectives and constraints in terms of the possibility of their attainment.
Baoding Liu

Chapter 11. Fuzzy Dependent-Chance Programming

Abstract
Following the idea of dependent-chance programming (DCP) in stochastic environment, Liu [175] provided a fuzzy DCP theory in which the underlying philosophy is based on selecting the decision with maximum possibility to meet the event.
Baoding Liu

Chapter 12. Fuzzy Programming with Fuzzy Decisions

Abstract
Traditionally, mathematical programming models produce crisp decision vectors such that some objectives achieve the optimal values. However, for practical purposes, sometimes we should provide a fuzzy decision rather than a crisp one. Bouchon-Meunier et al [26] surveyed various approaches to maximizing a numerical function over a fuzzy set. Buckley and Hayashi [32] presented a fuzzy genetic algorithm (GA) for maximizing a real-valued function by selecting an optimal fuzzy set.
Baoding Liu

Rough Programming

Frontmatter

Chapter 13. Rough Variables

Abstract
Rough set theory, initialized by Pawlak [233], has been proved to be an excellent mathematical tool dealing with vague description of objects. A fundamental assumption in rough set theory is that any object from a universe is perceived through available information, and such information may not be sufficient to characterize the object exactly. One way is the approximation of a set by other sets. Thus a rough set may be defined by a pair of crisp sets, called the lower and the upper approximations, that are originally produced by an equivalence relation (reflexive, symmetric, and transitive1).
Baoding Liu

Chapter 14. Rough Programming

Abstract
By rough programming we mean the optimization theory in rough environments. This chapter will introduce rough expected value model (EVM), rough chance-constrained programming (CCP) and rough dependent-chance programming (DCP). Since an interval number can be regarded as a special rough variable, we also obtain a spectrum of interval programming as a byproduct.
Baoding Liu

Fuzzy Random Programming

Frontmatter

Chapter 15. Fuzzy Random Variables

Abstract
Fuzzy random variables are mathematical descriptions for fuzzy stochastic phenomena, and are defined in several ways. Kwakernaak [142][143] first introduced the notion of fuzzy random variable. This concept was then developed by several researchers such as Puri and Ralescu [242], Kruse and Meyer [139], and Liu and Liu [188].
Baoding Liu

Chapter 16. Fuzzy Random Expected Value Models

Abstract
By fuzzy random programming we mean the optimization theory in fuzzy random environments. For the optimization problems with fuzzy random information, we need fuzzy random programming to model them.
Baoding Liu

Chapter 17. Fuzzy Random Chance-Constrained Programming

Abstract
This chapter will introduce a general framework of fuzzy random chance-constrained programming (CCP) initialized by Liu [179]. Although fuzzy random simulations are able to compute uncertain functions, in order to speed up the process of handling uncertain functions, we train a neural network (NN) to approximate them based on the training data generated by fuzzy random simulations. Finally, we integrate fuzzy random simulations, NN and genetic algorithm (GA) to produce a more powerful and effective hybrid intelligent algorithm for solving fuzzy random CCP models, and illustrate its effectiveness by some numerical examples.
Baoding Liu

Chapter 18. Fuzzy Random Dependent-Chance Programming

Abstract
Following the idea of dependent-chance programming (DCP), Liu [180] introduced the concepts of uncertain environment, event, and chance function for fuzzy random decision problems, and constructed a theoretical framework of fuzzy random DCP, in which the underlying philosophy is based on selecting the decision with maximum chance to meet the event. Liu [180] also integrated fuzzy random simulation, neural network (NN) and genetic algorithm (GA) to produce a hybrid intelligent algorithm for solving fuzzy random DCP models.
Baoding Liu

Random Fuzzy Programming

Frontmatter

Chapter 19. Random Fuzzy Variables

Abstract
Liu [184] initialized the concept of random fuzzy variable. The primitive chance measure of random fuzzy event was defined by Liu [184] as a function from [0,1] to [0,1]. The expected value operator of random fuzzy variable was given by Liu and Liu [189]. Random fuzzy simulations will also play an important role in solving random fuzzy programming models.
Baoding Liu

Chapter 20. Random Fuzzy Expected Value Models

Abstract
Liu and Liu [189] defined an expected value operator of random fuzzy variable, and introduced a spectrum of random fuzzy expected value model (EVM). A random fuzzy simulation was also designed to estimate the expected value of random fuzzy variable. In order to solve general random fuzzy EVM, we integrated random fuzzy simulation, neural network (NN) and genetic algorithm (GA) to produce a hybrid intelligent algorithm, and illustrated its effectiveness via some numerical examples.
Baoding Liu

Chapter 21. Random Fuzzy Chance-Constrained Programming

Abstract
This chapter introduces the random fuzzy chance-constrained programming (CCP) proposed by Liu [184]. We also integrate random fuzzy simulation, neural network (NN) and genetic algorithm (GA) to produce a hybrid intelligent algorithm for solving random fuzzy CCP models, and illustrate its effectiveness by some numerical examples.
Baoding Liu

Chapter 22. Random Fuzzy Dependent-Chance Programming

Abstract
Liu [185] presented a spectrum of random fuzzy dependent-chance programming (DCP) in which the underlying philosophy is based on selecting the decision with maximum chance to meet the event. This chapter introduces the theory of random fuzzy DCP, and integrates random fuzzy simulation, neural network (NN) and genetic algorithm (GA) to produce a hybrid intelligent algorithm for solving random fuzzy DCP models.
Baoding Liu

General Principle

Frontmatter

Chapter 23. Multifold Uncertainty

Abstract
In this book, we have discussed three basic types of uncertainty (randomness, fuzziness, and roughness), where a stochastic variable is a measurable function from a probability space to the real line, a fuzzy variable is a function from a possibility space to the real line, and a rough variable is a measurable function from a rough space to the real line.
Baoding Liu

Chapter 24. Uncertain Programming

Abstract
Uncertain programming was defined by Liu [171] as the optimization theory in generally uncertain environments. From the viewpoint of optimization theory, there is no difference among these uncertainties except for the arithmetical operations on them.
Baoding Liu

Backmatter

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