Chance-constrained programming on sugeno measure space

Abstract

Uncertain programming is a theoretical tool to handle optimization problems under uncertain environment, it is mainly established in probability, possibility, or credibility measure spaces. Sugeno measure space is an interesting and important extension of probability measure space. This motivates us to discuss the uncertain programming based on Sugeno measure space. We have constructed the first type of uncertain programming on Sugeno measure space, i.e. the expected value models of uncertain programming on Sugeno measure space. In this paper, the second type of uncertain programming on Sugeno measure space, i.e. chance-constrained programming on Sugeno measure space, is investigated. Firstly, the definition and the characteristic of α-optimistic value and α-pessimistic value as a ranking measure are provided. Secondly, Sugeno chance-constrained programming (SCCP) is introduced. Lastly, in order to construct an approximate solution to the complex SCCP, the ideas of a Sugeno random number generation and a Sugeno simulation are presented along with a hybrid approach.

Introduction

Uncertainty permeates many branches of decision sciences, engineering, information sciences, system sciences, just to name a few representative disciplines. Because of the factor of uncertainty that is omnipresent in various decision-making systems, we are faced with a genuine challenge when it comes to optimization issues. This has triggered an emergence of different types of methods stemming from uncertain programming. Some of the earliest developments along this line came under the banner of stochastic programming (Charnes and Cooper, 1959, Liu, 1997, Liu, 1999, Liu et al., 2003). In 1970’s, Zadeh (1978) introduced the concept of possibility measure which was further advanced by Dubois and Prade, 1988a, Dubois and Prade, 1998b. Based on probability and possibility measures, Liu (2003) introduced an axiomatic system to study the fuzziness along with a new concept of credibility measure. Fuzzy programming (Liu, 1998, Liu, 1999, Liu and Iwamura, 1998a, Liu and Iwamura, 1998b, Liu and Liu, 2002), random fuzzy programming (Liu, 2002a, Liu, 2002b, Liu and Liu, 2003a), fuzzy random programming (Liu, 2001a, Liu, 2001b, Liu and Liu, 2003b), and rough programming (Liu et al., 2003) are examples of uncertain programming addressing a diversity of existing facets of uncertainty. Gao and Liu, 2005, Liu, 1998, Liu, 1999, Liu, 2001a, Liu, 2001b, Liu, 2002a, Liu, 2002b, Liu, 2003, Liu and Iwamura, 1998a, Liu and Iwamura, 1998b, Liu and Liu, 2002, Liu and Liu, 2003a, Liu and Liu, 2003b, Liu and Liu, 2005, Zheng and Liu, 2006 aimed at the unification of the methods of uncertain programming providing a systematic development framework. Uncertain programming is a theoretical tool to handle optimization under uncertain environment. It has already been applied to system reliability design, project scheduling, vehicle routing, facility location, machine scheduling, inventory problems, and others (Liu, 1999). In spite of the progress being made, some limitations exist. For instance, when dealing with stochastic programming, it is established in the probability space in which we require a satisfaction of the additivity property. In reality, however, this requirement of stochastic programming cannot be easily satisfied or might not be satisfied at all (Ha et al., 2006, Pap, 2002).

When looking at some possible extensions and generalizations of conceptual appeal and practical relevance, Sugeno measure (Sugeno, 1974), which is an important generalization of probability measures (Wang & Klir, 1992) in terms of their non-additive behavior, arises as an interesting alternative. Sugeno measure is a sound representative of non-additive measures which offer an ability to deal with subjective judgment, fuzzy fusion, decision making and non-repeated experimentation, cf. (Basile, 1987, Leszczynski et al., 1985, Melin et al., 2008, Mesiar and Ouyang, 2009, Nather, 1991, Soria-Frisch et al., 2007, Sridhar et al., 2008, Wierzchon, 1983). As an illustration, let us consider an example of choosing a TV set. For convenience, let the universe of discourse consists of two properties characterizing the TV set such as image quality (a) and sound quality (b), say X = {a, b}. Let P(X) denote the power set of X while μ describes an importance degree of various elements of P(X)(which are also referred to as a purchasing possibility). Evidently a TV set with no image and voice will not be purchased. In other words, the purchasing possibility in this case is equal to 0. On the other hand, if we encounter a TV set with excellent quality of image and voice will be purchased hence in this case the purchasing possibility is equal to 1. Usually, the quality of image is more important than the quality of voice, so this might result in purchasing possibilities of 0.4 and 0.2, respectively. In this problem, we could easily encounter more criteria including price, reliability, and others. Considering the subjectivity permeating this problem, one may express it in terms of the following measure

$$\mu (E)=\left\{\begin{array}{cc}0\hfill & E=\varphi \text{,}\hfill \\ 0.4\hfill & E=\{a\}\text{,}\hfill \\ 0.2\hfill & E=\{b\}\text{,}\hfill \\ 1\hfill & E=X\text{.}\hfill \end{array}\right.$$

Evidently, the above measure μ is non-additive as (μ(X) ≠ μ({a}) + μ({b})), that is, μ is not a probability measure. Simultaneously, it is easy to see that μ is neither a possibility measure (Zadeh, 1978), nor a credibility measure (Liu, 2003). We can show that this is a Sugeno measure, with λ = 5 (Ha, Zhang, Pedrycz, & Xing, 2009) (For the sake of readability and ductibility of this paper, we revisit the above statements of Ha et al. (2009)).

We have studied the first type of uncertain programming on Sugeno measure space, i.e. the expected value models of uncertain programming on Sugeno measure space (Ha et al., 2009). The chance-constrained programming is an important part of uncertain programming. This study focuses on the development of models of chance-constrained programming defined in Sugeno measure spaces, which is the second type of uncertain programming on Sugeno measure space.