A.P.Yeremeyev , D.A.Tikhonov

Moscow Power Engineering Institute (Technical University)

Applied Mathematics Department

At representation of knowledge of the experts a problem of allocation in space of problem area of the important and essential elements is frequently put. So, for intelligent decision support systems (IDSS) for management of nuclear power object such elements are the basic systems of object as the main circulating pump (MCP), condenser installation and etc. In turn, for objects the most essential are the meanings of their base parameters, for example, for MCP it is meaning of pressure, for the information on pressure it is knowledge, connected to critical intervals of meanings. And such allocation of concepts is possible for any level of hierarchy of space of problem area.

The most natural and structured way of necessary representation consists in representation of knowledge by fuzzy semantic networks, top and the arches of which are weighed as fuzzy and gravity of the presence. Taking into account nondeterminacy of the circuit of search of the decisions at management of complex systems as nuclear power objects and dynamics of change of a condition of problem area, base of knowledge (KB) IDSS conveniently to represent as production system. Use of offered representation at the account of the relative importance of fuzzy elements of space of problem area assumes integration of a number of concepts from area of fuzzy sets [1-3]:

• Fuzzy inclusion of fuzzy sets.

Let A and B are fuzzy sets in Х, then the degree of inclusion v(A, B) of fuzzy set A in fuzzy set B is under the formula

v(A,B) = & _{xI X}(m _A(x)® m _B(x)),

there are m _A(x), m _B(x) the degrees of a fitting А и В, & - operation of conjunction (m _A(x)& m _B(x) =min(m _A(x),m _B(x)), which undertakes on all xI X, "® " - operation of implication of the fuzzy sets (m _A(x) ® m _B(x) =min(1-m _A(x),m _B(x)).

By use of priorities of elements of set Х it is possible partitioning of the given set on to classes of equivalence on meaning of a priority: Х₁={х| Р(х)=y(1)},…, Х_К={х| Р(х)=y(К)}, there Р(х) is a priority of an element х, y(к) - priority of elements of a class кI К, y(1), y(К) - the greatest and least meanings of priorities of classes. Then we shall define concept of a degree of inclusion in view of a priority v_пр(A,B) of fuzzy set A in fuzzy set B, which is under the formula

v_пр(A,B) = & _{kI K} (y(1)/y(k))* *(& _{xI Xk}(m _A(x)® m _B(x))) (1)

If vпр(A,B) ? 0.5, we shall believe, that the set A fuzzy joins in view of a priority in set B, and to designate AI _прB. If vпр(A,B) <0.5, we shall count, that the set does not join fuzzy in view of a priority in set, and to designate AE _прB. It is clear, that the determined concept of fuzzy inclusion of fuzzy sets in view of a priority is integration of concept of inclusion of fuzzy sets.

• Fuzzy equality of fuzzy sets.

u (A,B) = & _{xI X}(m _A(x)« m _B(x)),

where & is operation of conjunction, which undertakes on all x, "« " is operation of equivalence of the fuzzy sets, which is defined as follows: m _A(x)« m _B(x) U (m _A(x)® m _Q(x))& (m _Q(x)® m _A(x)).

By use of priorities of elements of set Х, determined as is described above, we shall enter concept of a degree of equality in view of a priority u_пр(A,B)of fuzzy set to fuzzy set, which is under the formula

u_пр(A,B)= & _{kI K}(y(1)/y(k))*

*( & _{xI Xk}(m _A(x)« m _Q(x))) (2).

If uпр(A,B) ? 0.5, we shall believe, that the sets A and B are fuzzy equal in view of a priority and to designate A » _прB. If uпр(A,B) <0.5, we shall count, that the sets A and B fuzzy are not equal in view of a priority and to designate A ? _прB. A case, when uпр(A,B)=0.5 sets A and B are fuzzy equal and are not equal in view of a priority. We shall name them mutually indifferent and we shall designate A ~ _прB. Easily to see, that the considered concept of fuzzy equality of fuzzy sets in view of a priority is integration of concept of equality of fuzzy sets.

We shall emphasize, that degrees of inclusion and the equality with a priority can be determined for any two fuzzy sets and to accept any meaning from 0 up to 1.

We shall present expression (2), using definition of operation of equivalence of fuzzy sets [2], as

u_пр(A,B)= & _{kI K}(y(1)/y(k))* *(& _{xI Xk}((m _A(x)® m _Q(x))& (m _Q(x)® m _A(x))));

Taking into account commutativity of conjunction we receive u_пр(A,B)= (& _{kI K}(y(1)/y(k))*(& _{xI Xk}((m _A(x)® m _Q(x)))& (& _{kI K}(y(1)/y(k))(m _Q(x)® m _A(x)))));

that according to (1), results to

u_пр(A,B) = v_пр(A,B)& v_пр(B,A).

We receive, that the degree of equality of fuzzy sets in view of a priority is equal minimum of degrees of their mutual inclusion.

• "Fuzzy" situation in view of a priority of making its(her) elements.

Let Y={ y₁,y₂,...,y_p } is a set of attributes, the meanings of which describe condition of object of management, K is a number of classes of priorities of attributes of fuzzy situation, d (k) - meaning of a priority of class of number k, d (1) - meaning of the greatest priority. Each attribute y_i (iI J={1,2,…,p}) is described by a structural part appropriate linguistic variable < y_i, T_i, D_i >, where T_i={Ti1,Ti2,...,Tir_i} is the term - set of linguistic variable y_i (set of linguistic meanings of an attribute y_i, r_i - number of meanings of an attribute); D_i - base set of an attribute y_i. At the description of terms Tij (jI L = {1,2,...,ri }), appropriate to meanings of an attribute y_i, are used fuzzy variable <Tij,D_i ,Cij>, i.e. the meaning Tij is described by fuzzy set Cij in base set D_i:

Cij = {< m cij (d)/d>}, dI Di.

And Ti1,Ti2,...,Tir also have the various relative importance.

Fuzzy situation is a fuzzy set of the second level [1]: s={<m _S(yi)/yi>}, yiI Y,

where m _S(yi)= {<m m _s(yi) (Tij)/Tij>}, jI L, iI J.

We shall bring an example of an fuzzy situation, arising at management of MCP:

{ << 0.8/ "large" >, < 0.3/ "average" >, < 0.1/ "small" > / " pressure 1 " >,

<< 0.2/ "large" >, < 0.8/ "average" >, < 0.5/ "small" > / " temperature 1 " >,

<< 0.7/ " very large " >, < 0.4/ "large" >, < 0.2/ "average" >, < 0.05/ "small" > / " temperature 2 " >}, where the relative importance " pressure 1 ", " temperature 1 ", " temperature 2 " - 9, 4, 8 accordingly (on data of the experts), and relative importance " very large ", "large", "average", "small" - 3, 1, 1, 1 (also on data of the experts) accordingly.

For work with fuzzy situations it is necessary to set some attitudes(relations), thus setting some degrees of their affinity.

• Fuzzy inclusion of situations in view of a priority of making elements.

The concept of a degree of inclusion of a situation in view of a degree of a priority of making its(her) elements is based on definition(determination) of a degree of inclusion in view of a priority of fuzzy sets.

Let s_i={<m _Si(y_i)/y_i>},s_j={<m _Sj(y_i)/y_i>}, (y_iI Y) are some situations, K is a number of classes of priorities of attributes of fuzzy situation, d(k) – meaning of the priority of class number k, d(1) - meaning of the greatest priority.

The degree of inclusion in view of a priority of making elements of a situation s_i in a situation s_j is designated v_пр(s_i ,s_j ) and is defined by expression

v_пр(s_i,s_j)=& _{kI K}(d(1)/d(k))* *(& _{yI Yk}v_пр(m s_i(y),m s_j(y))).

Here the size v_пр(m s_i(y),m s_j(y)) is defined by expression (1) and is a degree of inclusion in view of a priority of fuzzy set m s_i(y) in fuzzy set m s_j(y).

We shall consider, that the situation s_i fuzzy joins in view of a priority in a situation s_j(s_i I _прs_j), if a degree of inclusion s_i in s_j there is less some threshold of inclusion t_inc from an interval [0.6,1], determined by conditions of management.

Or else, the situation s_i fuzzy joins in view of a priority of making its elements in a situation s_j, if the fuzzy meanings of attributes of a situation s_i fuzzy join in view of a priority in fuzzy meanings of attributes of a situation s_j.

• Fuzzy equality of situations in view of a priority of making elements

If s_i I _прs_j и s_j I _прs_i, then the situations s_i,s_j we shall perceive as one situation s_i,j= s_i E s_j= s_j= s_i. Actually existence of two mutual inclusions of situations s_i and s_j means, that at the given threshold of inclusion t_inc the situations si and sj are about similar. Such similarity of situations shall name as fuzzy equality in view of a priority, thus degree of fuzzy equality u_пр(s_i,s_j) of the situations s_i and s_j is determined as follows:

u_пр(s_i ,s_j)= v_пр(s_i ,s_j ) & v_пр(s_j ,s_i ).

It is well visible, that

u_пр(s_i,s_j)=& _{kI K}(d(1)/d(k))* *(& _{yI Yk}u_пр(m s_i(y),m s_j(y))).

• Fuzzy (p-q) generality of situations in view of a priority of making their attributes.

Fuzzy (p-q) as a generality of situations in view of a priority of making their attributes we shall name such similarity of situations, when the fuzzy meanings of attributes in view of their priorities are fuzzy equal, except fuzzy meanings not more, than q of attributes. If the situations s_i and s_j are described р by attributes, for them (p-q) generality in view of a priority of their attributes it is sufficiently of fuzzy equality in view of a priority of p-q of attributes from set Y. For representation of concept (p-q) generality of situations in view of a priority of their attributes we shall enter concept of a degree (p-q) generality of situations in view of a priority of their attributes.

Let s_i={<m _Si(y_i)/y_i>}, s_j={<m _Sj(y_i)/y_i>}, (y_iI Y) are the fuzzy situations, K - number of classes of priorities of attributes of fuzzy situation, d (k) - meaning of a priority of class number k, d (1) - meaning of the greatest priority.

The degree of (p-q) generality kp-q пр(s_i,s_j) for the situations s_i and s_j in view of a priority of making their attributes is determined by expression

kp-qпр(s_i,s_j)=& _{kI K}(d(1)/d(k))* *(& _{yI Yk\Yq}u_пр(m s_i(y),m s_j(y))),

where |Yq|? q and ymI Yq, if u_пр(m s_i(y_m),m s_j(y_m))< t_inc.

Well appreciably, that at definition of (p-q) generality of the situations s_i and s_j in view of a priority of making their attributes the attributes, which have fuzzy unequal meanings in view of a priority in s_i and s_j (if number of these attributes does not exceed q) are not taken into account, hence, at Yq =? the situations s_i and s_j are fuzzy equal in view of a priority of making their attributes.

Is similar to definition of fuzzy equality with a priority shall count situations s_i and s_j have (p-q) generality of situations in view of a priority of making their attributes, if kp-q пр (s_i ,s_j,)? t_inc.

The entered definitions create base for processing fuzzy production-network model of representation of knowledge [4]. So the determination of set of the activization rules of KB for the current situation s₀ occurs by finding of all rules in KB, the situation in the left part of which fuzzy joins in view of a priority of making its attributes in current situation, i.e. finding all i/L_{iI пр} s₀, where L_i - left part of i-th fuzzy network production.

For determination of in parallel feasible subsets of set of activization entered concept (p-q) generality of a situation in view of a priority of making their attributes is used. I.e. dependence of production on an input, output and input - output is investigated on the basis of them (p-q) generality in view of a priority of making their attributes.

We shall describe peculiarities of work of IDSS, taking into account the relative importance and fuzzy of elements of problem area.

At a stage of expert interrogation at creation of system are determined:

• space of problem area (list of all used concepts and relations);

• representation of fuzzy information in entered space (list of all attributes with possible meanings of linguistic variable);

• priorities of elements of entered space with their thresholds of comparison, which are used at determination of set of activization of production;
• - production rules with priorities (factors of utility, factors of labour input of execution), teams to the persons, accepting the decisions and possible restrictions for generalized rules;
• - description of the subset of normal condition of problem area is possible.
• On an operation phase of system are determined:
• current situation with fuzzy factors of making its attributes;
• the target condition, if not is given a subset of normal condition.

1. Zadeh L.A. Concept linguistic variable and its application to acceptance of the approached decisions. - М.: the World, 1976, 165p, in Russian.

3. Melikhov A.N., Bershtein L.S., Korovin S.YA. “Situational consulting systems with fuzzy logic summary”.- М.: a Science, Physico-mathematical lit.,1990, 272p, in Russian.

4. Yeremeyev A.P., Tikhonov D.A., Nikolaeva T.A. " Dynamic model of representation of fuzzy knowledge on the basis of network production " MFI- 97/ Thesis of the reports of an international conference " Information means and technologies ".- М.: МАI, 1997, pp.142-148, in Russian.