Research On Some Optimization Problems In Education Management System

As the result of increasing progress of society, the economic development changes gradually from the exploitation on the natural and capital resources to the contest on the knowledge and qualified scientists and technicians.The state synthetic power and competitive ability become more and more dependent upon the quality and creative ability on talented people. To keep the sustainable progress of society and economic development, education business should be developed greatly, such that to guarantee the well educated talent resource for carrying out the grand target of constructing the middle-affluent society.As a developing country, Chinese people's living standard including material consumption and civilization is still not high. For a long time, due to the influence of the ability to understand the world and level of knowledge, the method or idea to use scientific management for the government managers is still weak. This causes the lower work efficiency, serious waste on talent resources usage, and mistaking decision, etc. As the increasingly progress of science and technology during the recent years, the idea of scientific management has been paid more attention.Based on the author's macro-management experience during the past decades on education business development, this Ph.D.thesis present some beneficial theoretical research and solution finding methods to solve those unscientific management phenomena. Following the principle of "typical topic selection, original models, and reliable solution methods", this paper combines scientific research closely with practice, such that the research achievements have higher applicable values in practice.This thesis is divided into six chapters totally. Chapter 1 is introduction,chapter 2-5 give some theoretical researches on four typical optimization problems in education management, and the last chapter enumerates several rather representative application examples with some further discussion on results given in former chapters .In Chapter 1, the necessity of scientific management on education business is first described, and then the current research results on determinate and indeterminate optimization problems in this field is reviewed. From the simplex methods for solving the ordinary linear programming problems to solution finding methods for complex system optimization problems, determinate optimization has become an independent course for more than half a century. The bilevel programming models for solving the complex system optimization problems have been paid great attention for their particular theoretical advantage and outstanding actual application ability.The indeterminate phenomena exists extensively in daily life. Comparing with the determinate theory, the indeterminate theory has its huge advantage, it can solve many problems which can't be solved by determinate theory.The indeterminate theory is divided into fuzzy sets theory, stochastic theory, gray system theory , rough sets theory " etc. Each theory has its special research objects and methods. This thesis studied some optimization problems of education management system based on methods of fuzzy sets theory.At the end of chapter 1, four main characteristics and innovations of this thesis is concluded.In Chapter 2, a higher educational optimal investment bilevel programming model is presented. Based on the actual cases of higher education investment, a higher educational optimal investment bilevel programming model under the means of no difference is presented and discussed. A polynomial algorithm is designed to find the optimal solution, and the complexity of the algorithm is also analyzed. Through the solution finding procedure, one can get the optimal investment decision schemes for both provincial management department and higher educational institutes (universities, colleges or higher professional schools) simultaneously. This model can be applied to real investment managements notonly to provincial supervisory department in managing higher educational institutes' investment, but also to other departments in managing their subordinate sections. The detailed content of this chapter is arranged as follows:Since the higher educational investment has bilevel Characteristic (The upper level can be considered to be provincial supervisory department, and the lower level to be higher educational institutes). Section §2.2 describes a higher educational optimal investment bilevel programming model (2 — 1) with no consideration the case of higher educational institutes' self-financing investment.nmax J] Z(xi)s.t. 0 (i = !,?■? ,n) where: Z(xi) = maxwhere: Y(xi) =m= !,??? ,mTo solve this model, some definitions are defined as follows:Definition 2.1 A set S is said to be a permissible set of the model (2-1),ifS= \{x,y)t=i i=i m ￡ oyyy = xt; xt > 0;yy > 0; i = 1, ? ? ? ,n; j =I,-- ,m .Definition 2.2 Y{xi) is called as feasible solution set of the i-th sub-programming under the model (2 — 1).Definition 2.3 Set {x,y) e S, y = (y\--- ,yn) (y* € Yfa)), if V {% = I,--- ,n) is ￡/ie optimal solution of the i-th sub-programming under the model (2-1), then (x, y) is said to be the feasible solution of model (2 - 1), denoted byF.Definition 2.4 P = \ (x,y) max￡)Z(a:i), (x,y) 6 F I is defined as theL (*>y) ?=i Jopftmai solution sei of the model (2 — 1).In section §2,3, the proof of the existence of optimal solution for model (2 - 1) is given:Theorem 2,1 There certainly exists an optimal solution for model (2-1).In section §2.4, model (2 - 1) is analyzed, and transferred into a linear programming (2 — 2). The equivalence of the optimal solutions of both models is proved by the following lemma and theorem:Lemma 2,1 // (x, y, z)is the optimal solution of model (2 - 2), then (x, y) is a feasible solution of model (2-1).Theorem 2.2 // (x, y, z) is the optimal solution of model (2-2), then (x, y) also is the optimal solution of model (2 — 1), and objective functions of model (2-1) and (2 — 2) have the same optimal values.In section §2.5 and §2.6, a polynomial algorithm for solving model (2 - 1) is given, and its complexity is proved to be O(n2),In section §2.7, under the consideration of self-financing investment in higher educational institutes, suppose that there happen two different cases: One is that the self-financing investment amount of higher educational institutes is known by their superior departments. Another is that unknown. Then two different solution finding schemes for both cases to determine the optimal investment are given.In Chapter 3, a higher educational optimal investment bilevel programming model is studied under the hypothesis of prior development strategy. Prior development strategy has been becoming an important way for modern management business. Different management departments from different professions usually have different levels of prior development preference in macro-management. In this Chapter, the levels and steps of optimal investment programming model for higher education under the preference of prior development principles are analyzed first, two different models are derived: One is to ensure the optimalinvestments for prior development institutes. Another is to make the optimal inves tments for all left schools using residual fund with no difference, and then solution finding methods for both upper and lower level optimal investment decision-making are presented. By extending the feasible solution sets, the bilevel programming models are transferred into linear programming models. At last, the equivalence of the model solutions both original and transferred models is proved, and one polynomial algorithm to find the optimal solution is given also. Thes method given in this Chapter can be applied widely to any kind of decision-making problems on the assumption that prior development concepts are considered. The detailed content of this chapter is arranged as follows:In section §3.2, the higher educational optimal investment bilevel programming model is discussed under the hypothesis of prior development strategy. Particularly, subsection 3.2.1 gives the problem's description, subsection 3.2.2 discusses the levels of higher educational investment decision-making, and subsection 3.2.3 presents the steps of higher educational investment decision-making under the hypothesis of prior development strategy. For actual case, the investment procedure is divided into two steps:The first step is to invest to p kinds of prior development institutes, and p bilevel programming models are presented as (1) ? ? ? (p), where there have ki sub-programmings (I = 1, ? ? ? ,p) under the l-th. model:The second step is to invest to all institutes using the residual fund with no difference principle. Takes the maximum capacity of total campus students as the objective goal and establishes the listed bilevel programming model(p+l), where there have K sub-programminp under it:max J2 Z(xi) i&KIn section ?.3, the solution finding method for optimal investment is determined. In order to solve the models, models (!)晻?(p) and (p+1) are transferred into linear programming models (1? ???(p? and ((p + l)?in section ?.4 ,In section ?.5, by listed lemmas and theorems, the transferred models (1? ???(p?, ((p + 1)? and models (1) ???(p), (p + 1) are proved to be equivalent, respectively.Lemma 3.1 // (x, y, z) is the optimal solution of model (1? (I = 1, ???,p), then (x,y) is the feasible solution of model (I),Theorem 3.1 // (x,y,z) is the optimal solution of model (1? (I = 1, ???,p), then (x,y) is the optimal solution of model (I), and their optimal values are equal.Lemma 3.2 // (x,y, z) is the optimal solution of model ((p + 1)?, then (x, y) is the feasible solution of model (p+1).Theorem 3.2 // {x, y, z) is the optimal solution of model ((p +1)?, then (x,y) is the optimal solution of model (p +1),and their optimal values are equal.Section ?.6 presents a polynomial algorithm for finding the optimal solu-tions and values by solving models (1) ? ? ? (p) and (p+1), and the complexity of this algorithm is proved to be O(K2)OChapter 4 gives a fuzzy consistent sorting method for institutes' synthetic power. In this Chapter, all possible cases of higher educational institutes assessment are considered first, then based on the objective indexes, an index system for measuring synthetic power of higher educational institutes is established. Using the basic ideas of analytic hierarchy process (AHP) and fuzzy theory, fuzzy consistent matrixes are defined not only to decide the significant values of all indexes between different levels,but also to decide the superior degrees between higher educational institutes, such that the inconsistency of assessment matrixes can be avoided effectively and the work of correcting matrixes on inconsistency is reduced normally.After computing the weights of unitary vectors for both hierarchy single sorting indexes and hierarchy total sorting indexes, the sorting weight vectors can be computed for institutes' synthetic power, thus implies the final sort of higher educational institutes. The method given in this Chapter is easy and applicable, it can be conveniently used in sorting projects whose superior degree can be determined by the objective indexes.The detailed content of this chapter is arranged as follows:Section §4.2 presents a single objective index system according to the principle that one lower hierarchy index only related to one upper hierarchy index, see table 4-1. Section §4.3 gives some definitions of basic corresponding concepts, properties, and theorems:nDefinition 4.5 If the vector (a>i, ? ? ? ,u;n) satifies X^w,- = 1, then the vector (uii, ? ? ? ,uin) is called as unitary vector.Definition 4.6 Given an index system according to the principle that one lower hierarchy index only related to one upper hierarchy index. If an index in this system does not exist the lower related indexes,then this index is called the lowest hierarchy index.Theorem 4.3 Suppose a fuzzy mutually complementary matrix A —let r< — 52 Oifc (t = 1, ? ? ? , n), and set mathematical transformation k=ithen R = (ry)n> 0, u> 0, 9 > 0, then h is non-negative triangle fuzzy number.Definition 5.2 // non-negative triangle fuzzy number h = (e, w, 6) satisfies e = uj = 0, then It has the characteristic of "bad evaluation".Definition 5.3 If non-negative triangle fuzzy number h = (e,u,6) satisfies u = 6 ~ \, then Ti has the characteristic of "good evaluation".Definition 5.4 For fuzzy vector {hi,--- ,hm)T, if the fuzzy component "hj (j = 1, -.. ,m) is triangle fuzzy number, then (hi, ? ? ? , hm)T is triangle fuzzy vector.Definition 5.5 Set fuzzy vector (hi,--- Jim)T> for any fuzzy component %i andlij, ifJii is fuzzy prior to Hj according to certain principle of fuzzy prior{ notation: hit-hj, the same after), then there has hj^hj, Thus (hi, ? ? ? ,hm) is called as consistent fuzzy vector.Definition 5.6 Set fuzzy vector (hi,--- ,hm)T> for any fuzzy component hi,hjand h*, if h;￡hj and hj>-hk, then there has hi>-hk. Thus (hi,--- ,hm)T is called as harmonic fuzzy vector.Based on the principle of fuzzy prior given by Bass and Kwakernaak, one new concept of "L-unitary method" is defined,"L- unitary method" : For any non-negative triangle vector (hi, ? ■ ? , hm)T [where h7 = {￡j,ujj, 9j); j ~ 1, ■ ? ? , m), suppose that there exists at least one fuzzy component which doesn't have the characteristic of bad evaluation. Let V wi = P> then P > 0, Based on the principle of fuzzy prior given byBass and Kwakernaak, the fuzzy component which reaches the maximum value p is defined as the optimal one. Let h'j = (￡^,cjj,^) = (￡j/p,Uj/p,l A (dj/p)) (this implies that h'j (j = I,--- ,m) is still a triangle fuzzy number), thus (h'lt ? ? ■ , h'm)T is called as the L-unitary fuzzy vector of (hi,--- , hm)T.The "L- unitary method" is proved to have good properties by the listed theorems:Theorem 5.1 L-unitary fuzzy vector (h[, ■ ? ? , h'm)T of non-negative triangle fuzzy vector (hi, ? ? ? , hm)T is a consistent fuzzy vector.Theorem 5.2 L-unitary fuzzy vector (h^, ? ? ? Jt'm)T of non-negative triangle fuzzy vector (hi, ? ? ■ s hm)T is a harmonic fuzzy vector.Theorem 5.3 If one fuzzy component of a non-negative triangle fuzzy vector has the characteristic of bad evaluation, then its corresponding fuzzy component in L- unitary fuzzy vector also has the same characteristic.Theorem 5,4 // the optimal fuzzy component of a non-negative trianglefuzzy vector has the characteristic of good evaluation, then its L- unitary fuzzy vector is itself.Theorem 5.5 "L- unitary method" can keep the fuzzy component's prior order invariable for non-negative triangle fuzzy vector.For real number case of objective evaluation factors, subsection 5.4.2 gives a "simple proportion transition unitary method", makes the factor which has the maximum value also has the maximum L- unitary value.Section §5.5 gives the fuzzy synthetic evaluation values for all institutes, and transfers them into real numbers by Yager's F\ method. According to the terms listed in the original problem, to get the overall optimal decision-making, take the maxinium total fuzzy synthetic evaluation real values of all new projects as the objective goal, an integer linear programming model(5 - 3) is established:mmaxZ = ^2,Xj^>jY, Xj = rs s = I,--- ,SXj€R,s.t. < ￡) Xj = ki I = 1,- ■? ,L Xj € {0,1} j = l,--- ,mUsing the mathematic software, such as LINGO or MATLAB ,the optimal solution of this model can be obtained.For further discussion of the theory and methods presented in chapter 2-5, three application examples are given in Chapter 6: the first is provincial supervisory department's optimal investment decision-making for higher education based on the prior development strategy, the second is five universities' synthetic power sorting method in Shandong Province, the third is the application of fuzzy multi-factor decision-making for higher educational institutes setting.