Theory and Practice of Uncertain Programming

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.

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.

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.

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].

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.

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.

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.

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.

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).

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.

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].

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.

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.

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.

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.

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.

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.

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.

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.

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.