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.
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 we consider two soft properties: “” which means “ is approximately equal to ” or “ is the approximate value of ,” and “” which means “ approximately belongs to the set ” or “the approximate value of belongs to .” The uncertain variable is defined by a set of values (real number vector space), the function (i.e., the certainty index that , given by an expert) and the following definitions for :
The function is called a certainty distribution. Let us consider a plant with the input vector and the output vector , described by a relation (relational knowledge representation) where the vector of unknown parameters is assumed to be a value of an uncertain variable described by the certainty distribution given by an expert. If the relation is not a function, then the value determines a set of possible outputs . For the requirement given by a user, we can formulate the following decision problem: For the given , and one should find the decision maximizing the certainty index that the set of possible outputs approximately belongs to (i.e., belongs to for an approximate value of ). Then
. It is easy to see that
is a set of all
such that the implication
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.