2. Construcción teórica
2.2 Diversidad de uso de suelo 1 Estudio de mercado
2.2.3 Espacios urbanos
As it was mentioned in the introduction to Chap. 3, there are two basic rea- sons of the uncertainty in the decision making:
1. The plant with a fixed input and output (a state) is non-deterministic. 2. There is no full information of the plant.
Ad 1. Let us consider a static plant with the input vector u∈U and the output vector y∈Y. We say that the plant with the input u and the output y acts (or behaves) in deterministic way, or shortly, the plant is determinis- tic if the value u determines (i.e. uniquely defines) the value y. It means that in the same conditions the decision u gives always the same effect y,
or that the plant is described by a function y = Φ(u).
For example, let y denote the amount of a product in one cycle of a production process and u denote the amount of a resource (e.g. the amount of a raw material), and y = ku. It means that if the value u will be the same in different cycles then in each cycle we obtain the same amount of the product y = ku. If the parameter k varies in successive cycles and un,
yn, kn denote the values in the n-th cycle then yn = knun, which means that the amount of the product yn is uniquely determined by the amount of the raw material un and the value of the coefficient kn, or – when the se- quence kn is determined in advance – by the index of a cycle. The coeffi- cient kn may be treated as a second input, i.e. a disturbance zn. In the case when the disturbances occur in the description of the plant, we can say that the plant with the fixed input (u, z) and the output y is deterministic if the values (u, z) uniquely determine the value y, i.e. the plant is described by a function y = Φ(u, z). Such plants have been considered in Sects. 3.1 and 4.1. As it has been shown above, a non-deterministic plant with the input u and the output y may be proved to be a deterministic one if other inputs which also have an influence on y are taken into account. In practice to in- troduce or even to call them may be impossible. Consequently, for the fixed u only a set of possible outputs may be given. In the example con- sidered, the amount of the product may depend not only on the amount of the raw material but also on many other variables and for the fixed u, only the set of possible values of y may be given, e.g. in the form of the inequal- ity c1u≤y≤c2u with the given values c, which means that c1≤k≤c2. In different production cycles one may obtain different values yn for the
same value un. Hence, the different pairs (un, yn) are possible in our plant. A set of examples of such points is illustrated in Fig. 6.1 where the shaded domain is a set of all possible points. Of course, the figure concerns a gen- eral plant of this kind, in which negative values u and y are possible, i.e. the plant described by the inequalities
u c y u c1 ≤ ≤ 2 for u≥0 and u c y u c2 ≤ ≤ 1
for 0u≤ , under the assumption that c1,c2 >0.
Ad 2. The plant is deterministic but the function y = Φ(u) is unknown or is not completely known. If in the known form of the function Φ some parameters are unknown, we speak about parametric uncertainty.
6.1 Uncertainty and Relational Knowledge Representation 119 y u u c y= 2 u c y= 1
Fig. 6.1. Illustration of the relationship between u and y in the example under consideration
Using terms known, unknown, uncertain etc. we must determine a subject they are concerned with (who knows or does not know?, who is not certain or rather not sure?). It is convenient to distinguish in our considerations three subjects: an expert as a source of the knowledge, a designer and an executor of the decision algorithm (controlling device, controlling com- puter). The uncertainty caused by an incomplete knowledge of the plant concerns the expert who formulates the knowledge, and consequently is transferred to the designer which uses this knowledge to design a decision algorithm.
Both reasons of the uncertainty (points 1. and 2. listed at the beginning of this section) concern the designer: the designer’s uncertainty may be caused by the non-deterministic behaviour of the plant (an objective uncer- tainty as a consequence of the non-deterministic plant) or by an incomplete information on the plant given by an expert (a subjective uncertainty or the expert’s uncertainty). In the first case, the sets of possible values y which may occur in the plant for the fixed u are exactly defined by the expert. In the example considered above it means that the values c1 and c2 are known.
In the case of the second kind of uncertainty y = ku. The expert, not knowing exactly the value k, may give its estimation in the form of the inequality c1≤k≤c2. Formally, the designer’s uncertainty is the same as in the first case, i.e. the designer knows the set of possible values y: c1u≤y≤c2u for u≥0. However, the interpretation is now different: pos- sible points (un, yn) lie on the line y = ku located between the lines y = c1u and y = c2u in Fig. 6.1.
Both kinds of uncertainty may occur together. It means that for the fixed u, different values y may occur in the plant and the expert does not know exactly the sets of possible values y, e.g. does not know the values c1 and c2 introduced in our example when the first kind of uncertainty was con- sidered. In both cases of uncertainty described above we shall shortly speak about an uncertain plant, remembering that in fact, an uncertainty is not necessary to be a feature of a plant but it may be an expert’s uncer- tainty. In a similar sense we speak generally about an uncertain algorithm (uncertain decision maker, uncertain controller) and an uncertain system. These names are used for different formal descriptions of an uncertainty, not only for the relational description considered in this chapter.
Let us denote by
Dy(u) ⊆Y
the set of all possible values y for the fixed u ∈U. In the example consid- ered above
Dy(u) = { y: c1u≤y≤c2u }
independently from the different interpretations of this set. The formula- tion of the sets Dy(u) for all values u which may occur in the plant means the determination of the set of all possible pairs (u, y) which may appear. This is a subset of Cartesian product U×Y, i.e. the set of all pairs (u, y) such that u∈U and y∈Y. Such a subset is called a relation
u ρ y ∆= R(u, y) ⊆U×Y. (6.1) In a special case for the deterministic plant, this is the function y = Φ(u), i.e.
R(u, y) = {(u, y)∈U×Y: (u∈Du) ∧ y = Φ(u)}
where Du denotes the set in which the function is defined (in particular Du = U). For simplicity, the plant described by the relation (6.1) we shall call a relational plant, remembering that in fact the relational description does not have to be a feature of the plant but is a form of the uncertainty description. In the further considerations we shall assume that the relation describing the plant is not reduced to a function, i.e. the plant is uncertain. Usually, the relation (6.1) is defined by a property ϕ(u, y) concerning u and y, which for fixed values of the variables u and y is a proposition in two-valued logic. Such a property is called a predicate. The relation R de- notes a set of all pairs (u, y) for which this property is satisfied, i.e.
6.1 Uncertainty and Relational Knowledge Representation 121
R(u, y) = {(u, y)∈U×Y: w[ϕ(u, y)] = 1}∆={(u, y)∈U×Y: ϕ(u, y)}
where w[ϕ(u, y)]∈{0,1} is a logical value (0 or 1 means that a sentence is false or true, respectively). Usually the property ϕ(u, y) is directly called a relation and instead saying that (u, y) belongs to the relation R, we say that it satisfies this relation. In our example ϕ(u, y) = “c1u≤y≤c2u”. The rela- tion R(u, y) has often a form of a set of equalities and (or) inequalities concerning the vectors u and y. Below, four examples of the description of a relational plant are given:
1. p = 2, l = 3 (two inputs, three outputs) u(1) + 2u(2) – y(1) + 5y(2) + y(3) = 0, 3u(1) – u(2) + y(1) – 2y(2) + y(3) = 4. 2. p = 2, l = 2 u(1) + ( y(2))2 = 4, u(1) + u(2)u(1) + y(1)≤ 0, u(2) + y(2)≥ 1. 3. p = l = 1 (u(1))2 + (u(2))2 = 4. 4. p = l = 1 (u(1))2 + (u(2))2 ≤ 4.
It is easy to show that none of the above relations is a function.
If the disturbances z∈Z act on the plant then they appear in the descrip- tion of the plant by a relation
R(u, y, z) ⊆U×Y×Z (6.2)
or the relation R(u, y; z) ⊆U ×Y for the fixed z. In many cases an expert presents the knowledge on the plant in the form of a set of relations
Ri(u, w, y, z), i = 1, 2, ..., k (6.3) where w∈W denotes the vector of additional auxiliary variables appearing in the knowledge description. The set (6.3) may be reduced to one relation by eliminating the variable w:
R(u, y, z) = {(u, y, z)∈U×Y: W w∈ [(u, w, y, z)∈
∩
k i i u w y z R 1 ) , , , ( = ]}. (6.4) It is then the set of all triplets (u, y, z) for which there exists w such that (u, w, y, z) satisfies all relations Ri. The formal description of the knowl-edge of the plant differing from a traditional model (for the static plant it is a functional model y = Φ(u)) is sometimes called a knowledge representa- tion of the plant. More generally, we speak about the knowledge represen- tation as a description of the knowledge given by an expert and concerning a determined part of a reality, a domain, a system, a way of acting etc. In the computer implementation, the knowledge representation is called a knowledge base which may be treated as a generalization of a traditional data base. The term knowledge representation is defined and understood in different ways (not always precisely). That is why, independently of the names used, it is so important to formalize precisely terms occurring in concrete considerations and to formalize concrete problems based on these terms. In the case considered in this chapter the knowledge representation is the relation (6.1) or (6.2), and more generally – the set of relations (6.3). This is a relational knowledge representation of the static plant under consideration [24, 52]. In the next sections analysis and decision making problems based on the relational knowledge representation will be de- scribed.