Ketevan Kutkhashvili

Planning of projects in the long term by the private company depends on available resources (especially financial).The optimal distribution and management of resources is a complex task,and stability of firm or any large organization and proper decision making are of main importance for economic development .The paper will discuss one such problem and its mathematical model . 

In the work for creation of large system, requiring different kinds of material and financial resources with  limited amounts, mathematical model with two criteria is constructed.

Based on scheduling theory, the original model of scheduling problems has been constructed.It can perform its sub-tasks with a single-stage multiprocessor system.It is assumed,that the processors of this system are partially interchangeable and the sets of additional resources and partial order are empty.The times of tasks, entering the system, are not fixed and have random values.

The implementation of each project requires some financial cost.The income of the company at the beginning of each month is known.The number of projects is so large,that the total cost of implementing of all projects exceed the company's total revenue of the year.We need:1)select the maximum number of projects,that the company can implement during the year;2)create a continuous plan for the implementation of selected projects, according their financial costs.To solve the economic problem, we have set,we can build a mathematical model, making the best schedule for the projects implementation.

Some observations will be made about teaching mathematical subjects in the faculties of informatics,for example, solving applied mathematical problems, using computer.Programming of mathematical problems will greatly increase the motivation.Using computer programming in teaching of fundamental mathematical concepts,such as function,limit,derivative,etc. ,simplifies their learning and makes it more interesting.

The effective algorithm to order the sequence of tasks is constructed.The schedule length and maximal price of tasks implementation are considered as the measure of the algorithm effectiveness.The algorithm makes it possible to construct the schedule, giving optimal solution for both criteria.

The effective algorithm to order the sequence of tasks is constructed.The schedule length and maximal price of tasks implementation are considered as the measure of the algorithm effectiveness.The algorithm makes it possible to construct the schedule, giving optimal solution for both criteria.

The work is devoted to the construction of polynomial algorithm and test of its effectiveness for one problem, when the tasks are implemented by single- stage processors , the system of processors is partially interchangeable and neither partially ordered set, nor the set of additional resources are empty.Implementation of each task in conditions of different additional resources needs different price and the executing time of each task on any processor is constant.The measures of optimality are the length of schedule in terms of time and maximal cost of the system.The effectiveness of the algorithm is established.

In the process of teaching mathematics the use of computer programs not only diversifies traditional methods, but raises the motivation of students,provides visual instruction and differentiation of assignments for them, facilitates their examination.The use of electronic devices makes learning of mathematics more interesting,emotional,effective and attractive.Math-Bridge is the first All-European program of electronic education.Math-Bridge is easy- to- use and gives plenty of possibilities to students and teachers.

Renewal rate of engineering and technical knowledge increases fast.Mathematical subjects make the basis for constructing a number of different specialities. The purpose of the project is the perfection of mathematics teaching in engineering specialities and directions in the institutes of higher education.Math-Bridge is All-European web-oriented intellectual system for creation of on-line courses on mathematics and teaching mathematics on-line.In the report we briefly describe main possibilities of Math-Bridge electronic system implantation to educational process,effectiveness of new pictorial and information technologies.

For country technological and economical development it is important to improve the quality of mathematical education in such specialties,as science,technology,engineering and mathematics (STEM). Modern competence demands corresponding specialists training in general engineering and,particularly,in computer sciences,means implantation of new technologies of teaching.In the work there are offered the possibilities of computer technologies use in mathematical subjects in STEM specialties,the main structure of mathematics teaching and the results, obtained in the project.

In the work scheduling theory specific problem research is given,particularly,construction of automated system of educational schedules for secondary schools of Georgia.The purpose of the project is to construct such a flexible and easy-to-use package,which considers the specificity of the schools of different types and, as much as possible, will be adapted to both large in number and small schools and also to the schools with office system of teaching and without it.   

One of the areas of discrete optimization,in particular,the scheduling theory, is considered.As is well known,the problems of scheduling theory are of NP complexity and only in particular cases it is possible to construct the polinomial complexity algorithms.For the problem, considered in the work, the sets of additional resourses and the partially ordered set,defined as the task of system,are empty.The schedule length and maximal price of tasks execution are considered as an effectiveness measure.It is constructed an algorithm,using obtained optimal values,which allows to find Pareto-optimal solutions of the problem.

Given system of tasks with certain set of resourсes or devices has to be executed.In conditions of given properties of tasks and resources systems and certain restrictions on them, effective algorithm of tasks implementation shedule has to be constructed.As the measures of effectiveness are considered the price and running time.

On the basis of general theory methods mathematical model is constructed for scheduling theory problems, where tasks implementation is possible by single-step multiprocessing system,the processors are half interchangeable and the sets of additional resources and partial order are empty.The new multicriteria optimization algorithm (from the viewpoint of speed of acting and other certain criteria) is constructed.

The man in his activity is constantly faced with the situations, in which he has to make a choice.This paper discusses the problems,related to the choice of decisions in the presence of the number of criteria for some discrete tasks.

For any programming language the main principles of algorithmization are the basis for creation of good software,as the knowledge of the basis of algorithmization helps specialists to take the problem more systemic and to formulate it technically correctly.At the same time,the great importance is attached to data presentation and consideration of their whole processing.The teaching of algorithmization means acquiring new skills, which will help the programmer to formulate his ideas better and to improve his logic mentality.

 The sets of vector values and additional resources are empty,an algorithm of the polynomial complexity is constructed.Taking into account the obtained solutions,the algorithm proposes to the decision making person a system of possible compromise settlements,deviating from the optimal solution with a desirable error, in the conversational mode. 

The work is devoted to the research on one scheduling theory specific problem.New optimization algorithm is constructed and its effectiveness is established.The methods of combinatorics and graph theory are used.For multicriteria solving special interactive method is worked out.

The mathematical model, describing diffusion of contaminating ingredients in the air is constructed and  applied to forecasting of air contamination.The model is partial differential equation with initial and boundary conditions.The characteristic diffusion coefficients of the model are calculated in terms of the statistical analysis of data, taken from measuring stations.In the model  the temperature of the air,cloudiness,wind speed and its direction are considered.

A standard problem of discrete optimization from scheduling problem theory is considered.There are n tasks to be executed on m processors at some restrictions  and conditions.The purpose of the problem is to find an optimal execution order for the tasks,so that the given conditions were carried out simultaneously,if the construction of such schedule is basically possible .

Doctoral Thesis Referee

Master Theses Supervisor

Doctoral Thesis Supervisor/Co-supervisor

Scientific editor of monographs in foreign languages

Scientific editor of a monograph in Georgian

Editor-in-Chief of a peer-reviewed or professional journal / proceedings

Review of a scientific professional journal / proceedings

Member of the editorial board of a peer-reviewed scientific or professional journal / proceedings

Participation in a project / grant funded by an international organization

Participation in a project / grant funded from the state budget

Patent authorship

Membership of the Georgian National Academy of Science or Georgian Academy of Agricultural Sciences

Membership of an international professional organization

Membership of the Conference Organizing / Program Committee

National Award / Sectoral Award, Order, Medal, etc.

Honorary title

Monograph

Handbook

Research articles in high impact factor and local Scientific Journals

Publication in Scientific Conference Proceedings Indexed in Web of Science and Scopus