Regarding education quality management problems as dynamic system theory problems.

Sandru, Ovidiu Ilie ; Sandru, Ioana Maria Diana

1. INTRODUCTION

Given the new state of social development characterized by a dynamic and expansive economy based on knowledge, the quality enhancement and reform of and in the field of education based on a lifelong learning approach have become major priorities for all forms of social organization. Due to complex factors involved in these processes, the organization of educational networks is no longer achievable by traditional means. Within the present century, the use of scientific methods that exploit the progress achieved in the fields of informatics and mathematics has become a necessity. The main objective of this paper is to show how the social command regarding the continuous reform of education and demarche to ensure required quality standards can be abstractized through the general notion of dynamic system endowed with the "cost" function (a thorough presentation of the notion of dynamic system and of the mathematical apparatus used in this field is given in (Arrow et al, 1958, Kalman et al., 1969)). In general terms, by the notion of dynamic system endowed with the function "cost" we understand an "entity" described by a set of possible "states" that react to certain stimuli received by the external environment when it evolves to a new state and it generates a certain answer whose intensity can be evaluated by a given system of measurement, named cost function.

The possibility to express concrete coordination activities related to educational networks in the mathematical format mentioned above is important due to the opportunity to project optimal managerial policies to pursue established goals.

We shall try, further on, to indicate how we can achieve this transcription and what ways we have to follow in order to solve the problems studied.

2. MATHEMATICAL MODELING OF EDUCATIONAL SYSTEMS

The starting point of the model we intend to present is given by a recent study undertaken by the authors in (Sandru & Sandru, 2009), where we applied the theory of dynamic systems in order to model and solve some specific marketing problems.

Let A be a set of goals regarding the future socio-economic development of any form of social organization (of a country, for example), and let B be the education network meant to contribute to the achievement of these goals by training and / or re-qualifying individuals. In order to meet the development related goals, society's representatives will establish certain tasks specific to the educational system. In this respect, the decision factors of the educational network will design their own ways of action. In order to precisely describe the way the educational system interacts with current social commands, we define the following set of mathematical objects: a set T whose elements will quantify different moments in time to which the events occurred within the implementation process of the goals specified by set A, will be referred to; a set X whose elements represent various states in the interaction process between A and B; a set [V.sup.A] whose elements will describe possible requests transmitted to B in order to reach the goals specified in A; a set [V.sup.B] whose elements will describe possible reactions or actions undertaken within set B in order to fulfill various demands resulted from A ; a class X of functions x: T [right arrow] X which represent different successions of events that can take place within the interaction between A and B, or different evolving states of the relations between them; a class [S.sup.A] of functions [s.sup.A] : T [right arrow] [V.sup.A] describing the social coordinators' strategies to achieve the goals mentioned in A ; a class [S.sup.B] of functions [s.sup.B] : T [right arrow] [V.sup.B] describing the strategies of actions to be taken by those decision factors that coordinate the activity within the educational network B; a function

[phi]:T x T x X x [S.sup.A] [right arrow] X,

whose values represent the states

x(t) = [phi](t,[t.sub.0],[x.sub.0],[s.sup.A](t)),

resulted from various initial states (events) ([t.sub.0], [x.sub.0]) [member of] T x X under the action of certain given (normative) requests [s.sup.A] [member of] [S.sup.A] at different moments in time t [member of] T; a function [psi] : T x X [right arrow] [V.sup.B], whose values [psi](t,x) characterize the actions undertaken by B at different moments in time t [member of] T and under different states x [member of] X of the rapport between the social demand and the educational system offer.

The ensemble of all the elements previously presented,

[SIGMA] = (T, X, [V.sup.A], [V.sup.B], X, [S.sup.A], [S.sup.B], [phi], [psi]),

builds a dynamic system, namely a mathematical category as it is known within the specialized literature. It is useful to mention that in the terminology used within the theory of dynamic systems, the elements of set T express different moments in time in the evolution of system [SIGMA]; the elements of set X designate the possible states of system [SIGMA] ; the elements of set [V.sup.A] designate the inputs or the signals that can be sent to system [SIGMA]; the elements of set [V.sup.B] designate the outputs or the signals that system [SIGMA] can send (transmit); the elements of set [S.sup.A] represent the input functions or the commands that can be transmitted to system [SIGMA] ; the elements of set [S.sup.B] represent the output functions or the signals that system [SIGMA] can transmit, the application

[phi] : T x T x X x [S.sup.A] [right arrow] X,

represents the transition function of system [SIGMA]'s states, while the application

[psi] : T x X [right arrow] [V.sup.B],

indicates the output signals transmitted by system [SIGMA] according to the moment chosen and the state of the system specific to that moment.

In the specific situations that we know, the consequences of various management strategies or policies can be evaluated by the achievements or damages caused. The evaluation possibility that we refer at, actually implies the existence of a modality to measure the efficiency or inefficiency of the actions undertaken. In the theory of optimal control, the notion of "measuring the efficiency or inefficiency of the commands transmitted to a dynamic system [SIGMA]" is mathematically modeled by functionals of the form

J : T x X x T x X x [S.sup.A] [right arrow] [??],

whose values J([t.sub.0],[x.sub.0],[t.sub.1],[x.sub.1], [s.sup.A]) are meant to express the cost of the commands (strategies) [s.sup.A] = [s.sup.A](t) that transit system [SIGMA] from the initial state ([t.sub.0], [x.sub.0]), [t.sub.0] [member of] T, [x.sub.0] [member of] X to the final state, ([t.sub.1], [x.sub.1]), [t.sub.1] [member of] T, [x.sub.1] [member of] X, [t.sub.1] > [t.sub.0], namely of those commands which satisfy the following relation

[x.sub.1] = [phi]([t.sub.1], [t.sub.0], [x.sub.0], [s.sup.A] ([t.sub.1])).

In the special case when set T is a subinterval of the real axis [??], functional J is usually represented under the integral form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

[t.sub.0], [t.sub.1] [member of] T, [t.sub.0] < [t.sub.1], [x.sub.0] [member of] X, [x.sub.1] = [phi]([t.sub.1], [t.sub.0], [x.sub.0], [s.sup.A]([t.sub.1])), x(t) = [phi](t, [t.sub.0], [x.sub.0], [s.sup.A](t)),

and I is a real function defined on the Cartesian product T x X x [V.sup.A] with the property that for any x [member of] X and [s.sup.A] [member of] [S.sup.A], the application

T [contains as member] t [right arrow] I (t, x(t), [s.sup.A](t)) [member of] [??],

is integrable on any closed and finite subinterval from T.

The problem of determining those strategies that ensure the achievement of the set goals at the lowest costs possible (or with higher benefits) is associated to some management activities mathematically modeled by means of dynamic systems [SIGMA] endowed with a functional J specialized in measuring the efficiency or the cost of different management decisions. Within the theory of dynamic systems, such a problem is known as the "optimal control problem".

By using the terminology applied within the theory of dynamic systems, the exact mathematical enunciation of an optimal control problem is the following:

Let [T.sub.0] and [T.sub.1] be two subsets of set T with the property that

sup [T.sub.0] < inf [T.sub.1],

and [X.sub.0], [X.sub.1] are two disjoint subsets of set X. For the problem we aim to define, the elements of set [T.sub.0] x [X.sub.0] represent those states of system [SIGMA] from which the commands are to begin, while the elements of set [T.sub.1] x [X.sub.1] represent the states which we want the system to reach, or the target states of the optimal control problem. Let [S.sup.A.sub.01] be the set of all the command functions from [S.sup.A] which evolves from a state ([t.sub.0], [x.sub.0]) from [T.sub.0] x [X.sub.0] into a state ([t.sub.1], [x.sub.1]) from [T.sub.1] x [X.sub.1], in other words, [s.sup.A] [member of] [S.sup.A.sub.01] if and only if there exist ([t.sub.0], [x.sub.0]) [member of] [T.sub.0] x [X.sub.0] and [t.sub.1] [member of] [T.sub.1] so that ([t.sub.1], [x.sub.1]) with

[x.sub.1] = [phi]([t.sub.1], [t.sub.0], [x.sub.0], [s.sup.A]([t.sub.1]),

belongs to the target set [T.sub.1] x [X.sub.1].

Given the fact that

J([t.sub.0],[x.sub.0],[t.sub.1],[x.sub.1],[s.sup.A]),

represents the cost (efficiency) of the command [s.sup.A] = [s.sup.A](t), which transits the system from state ([t.sub.0], [x.sub.0]) into state ([t.sub.1], [x.sub.1]), the optimal control problem that we refer to, relies on determining those commands [s.sup.A] = [s.sup.A](t) from [S.sup.A.sub.01] that minimize (or after case maximize) functional J, when [S.sub.01.sup.A] [not equal to] [empty set].

In order to solve optimal control problems, the theory in the field offers both general methods, as well as particular methods for specific situations. As documentation source we recommend a reference paper (Kalman et al., 1969), as well as the paper (Sandru et al., 2008) which contains considerable simplifications of the classical procedures of solving optimization problems.

3. COMMENTS

The theoretical results presented in this paper provide decision factors in institutions with a basis to carry out a thorough and reliable analysis of various situations which the field of education can be dealing with, in order to enable a sound judgment and the adoption of the best solutions by means of an effective set of mathematical instruments.

In the end, to those interested in applying issues related to education we recommend the paper (Sandru, 2008) which presents solving algorithms specialized on various concrete categories.

4. REFERENCES

Arrow, K., Hurwicz, L. & Uzawa, H. (1958). Studies in Linear and Nonlinear Programming, Stanford University Press, ISBN 0804705623, Stanford, California

Kalman, R. E.; Falb, P. L. & Arbib, M. A. (1969). Topics in Mathematical System Theory, McGraw-Hill, ISBN 0754321069, New York

Sandru, I. M. D. (2008). The optimal design of the quality management concepts using mathematical modeling techniques, In: Proceedings of the 10 th WSEAS International Conference on Mathematical and Computational Methods in Science and Engineering, Iliescu, M. et al., pp. 334-339, ISBN 978-960-474-019-2, Politehnica Univ., Nov. 2008, WSEAS Press, Bucharest.

Sandru, O. I., Vladareanu, L. & Sandru, A. (2008). A new method of approaching the problems of optimal control, In: Proceedings of the 10 th WSEAS International Conference on Mathematical and Computational Methods in Science and Engineering, Iliescu, M. et al., pp. 390-393, ISBN 978-960-474-019-2, Nov. 2008, WSEAS Press, Bucharest

Sandru, O. I. & Sandru, I. M. D. (2009). Regarding marketing problems as dynamic system theory problems, In: Proceedings of the WSEAS International Conference, Hashemi, S. & Vobach, C., pp. 183-187, ISBN 978-960-474-073-4, Univ. Houston, May 2009, WSEAS, Houston