Application of Uncertain Variables to Knowledge-Based Resource Distribution

Application of Uncertain Variables to Knowledge-Based Resource Distribution

Donat Orski (Wroclaw University of Technology, Poland)
Copyright: © 2009 |Pages: 22
DOI: 10.4018/978-1-59904-576-4.ch004
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The chapter concerns a class of systems composed of operations performed with the use of resources allocated to them. In such operation systems, each operation is characterized by its execution time depending on the amount of a resource allocated to the operation. The decision problem consists in distributing a limited amount of a resource among operations in an optimal way, that is, in finding an optimal resource allocation. Classical mathematical models of operation systems are widely used in computer supported projects or production management, allowing optimal decision making in deterministic, well-investigated environments. In the knowledge-based approach considered in this chapter, the execution time of each operation is described in a nondeterministic way, by an inequality containing an unknown parameter, and all the unknown parameters are assumed to be values of uncertain variables characterized by experts. Mathematical models comprising such two-level uncertainty are useful in designing knowledge-based decision support systems for uncertain environments. The purpose of this chapter is to present a review of problems and algorithms developed in recent years, and to show new results, possible extensions and challenges, thus providing a description of a state-of-the-art in the field of resource distribution based on the uncertain variables.
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Among many theories of uncertainty (Klir, 2006) developed for different applications the uncertain variables introduced by Bubnicki (2001a, 2001b) may be considered as a useful tool for modeling expert’s knowledge in knowledge-based decision systems. In the definition of the uncertain variable 978-1-59904-576-4.ch004.m01 we consider two soft properties: “978-1-59904-576-4.ch004.m02” which means “978-1-59904-576-4.ch004.m03 is approximately equal to 978-1-59904-576-4.ch004.m04” or “978-1-59904-576-4.ch004.m05 is the approximate value of 978-1-59904-576-4.ch004.m06,” and “978-1-59904-576-4.ch004.m07” which means “978-1-59904-576-4.ch004.m08 approximately belongs to the set 978-1-59904-576-4.ch004.m09” or “the approximate value of 978-1-59904-576-4.ch004.m10 belongs to 978-1-59904-576-4.ch004.m11.” The uncertain variable978-1-59904-576-4.ch004.m12 is defined by a set of values 978-1-59904-576-4.ch004.m13 (real number vector space), the function 978-1-59904-576-4.ch004.m14 (i.e., the certainty index that 978-1-59904-576-4.ch004.m15, given by an expert) and the following definitions for 978-1-59904-576-4.ch004.m16:


The function 978-1-59904-576-4.ch004.m22 is called a certainty distribution. Let us consider a plant with the input vector 978-1-59904-576-4.ch004.m23 and the output vector 978-1-59904-576-4.ch004.m24, described by a relation 978-1-59904-576-4.ch004.m25 (relational knowledge representation) where the vector of unknown parameters 978-1-59904-576-4.ch004.m26 is assumed to be a value of an uncertain variable described by the certainty distribution 978-1-59904-576-4.ch004.m27 given by an expert. If the relation 978-1-59904-576-4.ch004.m28 is not a function, then the value 978-1-59904-576-4.ch004.m29 determines a set of possible outputs 978-1-59904-576-4.ch004.m30. For the requirement 978-1-59904-576-4.ch004.m31 given by a user, we can formulate the following decision problem: For the given 978-1-59904-576-4.ch004.m32, 978-1-59904-576-4.ch004.m33 and 978-1-59904-576-4.ch004.m34 one should find the decision 978-1-59904-576-4.ch004.m35 maximizing the certainty index that the set of possible outputs approximately belongs to 978-1-59904-576-4.ch004.m36 (i.e., belongs to 978-1-59904-576-4.ch004.m37 for an approximate value of 978-1-59904-576-4.ch004.m38). Then

where 978-1-59904-576-4.ch004.m40. It is easy to see that 978-1-59904-576-4.ch004.m41 maximizes 978-1-59904-576-4.ch004.m42 where 978-1-59904-576-4.ch004.m43 is a set of all 978-1-59904-576-4.ch004.m44 such that the implication 978-1-59904-576-4.ch004.m45 is satisfied. The uncertain variables are dedicated to analysis and decision problems (Bubnicki, 2002, 2004a) in a class of systems containing a decision plant described by a relational knowledge representation with unknown parameter characterized by an expert.

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