Systems Analysis

2020/2021

Учебный год

ENG

Обучение ведется на английском языке

5

Кредиты

Автор программы

Беленький Александр Соломонович

Кто читает:

Департамент математики

Статус:

Курс по выбору

Когда читается:

4-й курс, 2, 3 модуль

Преподаватель

Беленький Александр Соломонович

Course Syllabus

Abstract

The course is an introduction to the tools and techniques of analyzing mathematical and applied systems, which are studied using the examples of the systems having linear or convex structures. It includes the studying of both how to develop and analyze mathematical models of applied systems in economics, sociology, and engineering, and how to analyze mathematical features of these systems. The main focus of the course is on systems with nonlinear objective functions, including the maximum functions of a finite number of convex and quasiconvexquasiconcave (monotone) functions in finite-dimensional Euclidean spaces, and linear constraints. The course consists of two interrelated parts—theoretical and applied ones—studied in parallel. In the theoretical part, the students study approaches to formulating hypotheses about the properties of applied systems and methodologies of analyzing these properties. In the applied part of the course, the approaches and the methodologies studied in its theoretical part are used to analyze practical problems in management systems in economics, in industry, in transport, in trade, in agriculture, in advertising, in electoral systems, in personnel deployment, and in banking.

Learning Objectives

• to further develop analytical abilities and the systems thinking of the students, formed in them as a result of studying general and special mathematical courses
• to study ideas and schemes of the methods for solving large-scale applied optimization problems with linear constraints that arise in economic, social, and industrial systems
• to prepare the students to successfully pass working interviews and to work at both public and commercial enterprises, organizations, and companies by forming in them abilities to “work in a team,” to make presentations for the leaders at every level and employer representatives, to demonstrate to company’s customers applications of a systems approach to every assignment and its advantages in dealing with both employer and customer problems

Expected Learning Outcomes

• master the elements of system analysis used in other mathematical and applied disciplines, studying which is part of their curricula
• be able to apply methods of systems analysis in analyzing and mathematical modeling problems and in solving applied problems in economic, social, and industrial systems
• be able to understand mathematical reasoning in publications on mathematical modeling and systems analysis of applied problems in economic, social, and industrial systems

Course Contents

• The science and art of applied systems analysis: basic principles, methodology, objects, objectives, and expected results of the application.
  Basic concepts, definitions and the terminology of systems analysis. Systems thinking: features and the ways of developing. System analysis and applied mathematics as elements of a tool for analyzing applied problems in economic and social systems. The problem of uncertainty and a variety of objectives in managing economic and social systems and a systems approach to its analysis. The students are expected to get familiar with a) the substance and the basic features of systems analysis as a field of science, and b) the spectrum of applied problems where the application of systems analysis approaches and techniques looks the most promising. The students are expected to get familiar with a) the substance and the basic features of systems analysis as a field of science, and b) the spectrum of applied problems where the application of systems analysis approaches and techniques looks the most promising.
• Economic and mathematical systems as the objects of applied systems analysis studies
  Management issues in economic systems. Mathematical methods of solving economic problems formulated with the use of economic and mathematical models. Decision making, operations research, mathematical programming, and optimal control as the basic tools for applied systems analysis. Examples of the statements of systems analysis problems studied in economic systems. The students are expected to study a) basic principles of mathematical modeling in applied economics, b) fundamental ideas of operations research, mathematical programming, and optimal control as basic tools of applied systems analysis, and c) general basic techniques of mathematically formulating economic problems being a subject of a quantitative analysis.
• Linear inequalities as a subject of studies for applied systems
  The axiomatic definition of real numbers and linear spaces. Examples of linear spaces. An overview of basic assertions on vector spaces, matrices and systems of linear equations and inequalities. Three basic problems of linear programming and their economic interpretation. Canonical and standard linear programming problems. The concept of duality in linear programming. Duality in linear models and competition. The formulation of the basic duality theorem of linear programming. The geometry of the simplest linear programming problems in two- and three-dimensional spaces. Economic and geometric interpretations of dual variables and the sensitivity analysis in linear programming. Examples of analyzing and finding solutions in the simplest problems of linear programming based on the duality theorem and the equilibrium theorem. The canonical linear programming problem. The weak complementary slackness theorem and the equilibrium concept. The statement of the theorem on strong complementary slackness. Economic and geometric interpretations of the complementary slackness. The formulation of the fundamental theorem of linear programming and the duality equilibrium theorem. The students will be explained a) the geometry of finite linear spaces, b) the fundamentals and the geometry of linear programming, including the complementary slackness and the equilibrium concept, and those of the duality theory, and c) the basic principles of analyzing applied problems based on the linear programming ideas.
• The theory of linear inequalities
  The dual linear subspace and the theorem on the rank of the conjugate subspace. Theorem on the solvability of systems of linear equations and its geometric interpretation. Non-negative and semipositive solutions of systems of linear equations and inequalities. The Farkas lemma. Corollaries of the Farkas lemma and their geometric interpretation. Theorems on alternatives: solving linear inequalities, the non-negative solutions of linear inequalities, semi-positive solutions of systems of homogeneous equations and inequalities. The basic non-negative solutions of systems of linear equations. A proof of the main duality theorem of linear programming for the standard linear programming problem and the proof of a strong complementary slackness theorem. The students will learn a) the Farkas lemma as the fundamental assertion underlying those aspects of the theory of linear inequalities that are widely applied in quantitative economics, and b) the use of the theory of linear inequalities in substantiating the duality theorem of linear programming and the strong complementary slackness theorem.
• Linear inequalities and convex cones
  Convex sets and convex cones. Operations on convex cones and their features. The duality theorem for convex cones. Finite cones and a representation of a set of non-negative solutions to a system of homogeneous linear equations and inequalities as a finite cone. Basic theorems on finite cones. Extreme vectors of convex cones and the extreme solutions of systems of linear homogeneous inequalities. Positively independent vectors and pointed cones. Extreme solutions of systems of linear homogeneous inequalities. Necessary and sufficient conditions for an extreme solution of a system of linear homogeneous inequalities. The structure of the set of solutions of non-homogeneous systems of linear inequalities. The students will study the theory of convex cones, including a few basic theorems on finite convex cones and their applications in the applied systems analysis of economic systems.
• Economic management problems with criteria in the forms of linear-fractional functions of vector variables.
  Convex polyhedral sets. Theorems on the representation of a convex polyhedral set. Monotone functions of a vector argument on convex sets. Examples of non-linear and nonfractional-linear monotone functions on convex sets. Proof of the monotonicity of the difference of two mutually inverse linear-fractional functions. Necessary and sufficient conditions for the monotonicity of a continuous function on polyhedral sets in finite-dimensional arithmetic space. A method of minimizing a monotone function over a polyhedral set having vertices. The students will study the fractional programming and the theory of optimizing monotone functions on convex polyhedral sets in finite-dimensional spaces. They will study a technique of formalizing applied economic problems as those of finding optimal values of monotone functions on bounded polyhedral sets.
• Discrete minimax problems in economics
  Discrete minimax problem with monotone functions on polyhedral sets. Convex cones in a finite space and basic arithmetic operations on them. The dual cone of a convex cone. The recessive cones of convex polyhedral sets. A sufficient condition for the solvability of a discrete minimax problem with monotone functions on a polyhedral set. Necessary and sufficient conditions for a minimax of monotone functions in discrete minimax problems on polyhedral sets and their verifiability for problems on polyhedral sets with vertices. The final method for finding the minimax of two monotone functions on polyhedral sets and an iterative method for finding the minimax of finite number of monotone functions on polyhedral sets having vertices. The students will acquire a basic technique of formulating applied economic problems as discrete minimax and maximin ones on polyhedral sets. They will learn how to formulate and prove extremum conditions in particular nonlinear problems formulated as discrete minimax and maximin problems on polyhedral sets, in particular, those with monotone functions.
• Continuous minimax problems in systems economic studies
  Continuous minimax problem on polyhedral sets with goal functions of two vector arguments possessing the feature of monotonicity on each vector argument and sufficient conditions for the solvability of these problems. Two-person games on polyhedral sets as a generalization of matrix games with mixed strategies. Necessary and sufficient conditions for the solvability of a two- person game on polyhedral sets of disjoint player strategies with a payoff function being a sum of a bilinear function of two vector arguments and two linear functions of the same vector arguments. A finite method for finding saddle points in two-person games on polyhedral sets with bilinear payoff function of vector arguments. Examples of formulations of management problems in economic systems in the form of games. The concept of equilibrium in games on a set of connected player strategies. Optimal player strategies in two-person games on polyhedral sets of connected player strategies. Necessary and sufficient conditions of an equilibrium in a two-person game on a polyhedral set of connected player strategies with the payoff function being the sum of a bilinear function of two vector arguments and two linear functions of the same vector arguments. This section of the course will be presented as an introduction in game theory techniques and their applications in analyzing economic systems functioning under uncertainty conditions, to let the students acquire basic techniques of both formulating such problems as game ones and finding equilibrium points in the corresponding games. The students will be explained how a game problem on polyhedral sets of feasible strategies of the players with a nonlinear payoff function being a sum of a bilinear and two linear functions of vector arguments belonging to these polyhedral sets can be reduced to a linear programming problem of a special structure, and how the dual theory of linear programming helps develop a finite method of finding an equilibrium in the initial game.
• Exchange models in economic systems
  Simple exchange models. Exchange matrices in exchange models. Semi-positive price vectors and equlibiria in a simple exchange model. A theorem on the existence of an quilibrium in a simple exchange model. Irreducible exchange matrices. A sufficient condition for the existence of a positive equilibrium price vector in a simple exchange model. The students will learn how to develop exchange models in applied economic systems and how to use these models in analyzing the potential of the economic systems described by such mathematical models.
• Optimally allocating new transport hubs and redistributing traffic flows in a region
• Optimization of planning advertising campaigns for goods and services
• Analyzing the effectiveness of the regional electrical grid and directions of its future development
• Analyzing the possibility of forming public-private partnerships and the private sector interest in investing in large-scale projects
• Organization of training and retraining of the personnel in an enterprise under changing economic conditions
• Forming reserve prices in tenders for the execution of a set of public procurement projects
• Estimating the expected number of service requests in certain queuing systems with a finite number of users (finite-source multi-server queuing systems)
• Estimating the number of "undecided" voters who may prefer one of the candidates as a result of their election campaigns

Assessment Elements

• Otest
• Oproject1
• Oproject2
• Oexam
  The exam is conducted orally (a survey based on the course materials). The exam is conducted on the Skype platform (https://www.skype.com/). You must connect to the exam according to the response schedule sent by the teacher to the students ' corporate emails on the eve of the exam. The student's computer must meet the following requirements: a working camera and microphone, and Skype support. To participate in the exam, the student must: put his / her photo on the avatar, appear for the exam according to the exact schedule, and turn on the camera and microphone when answering. During the exam, students are not allowed to turn off the camera, use notes and hints. A short-term communication failure during the exam is considered to be a communication failure of less than a minute. A long-term communication violation during the exam is considered to be a violation of a minute or more. If there is a long-term communication failure, the student cannot continue to participate in the exam. The retake procedure involves the use of complicated tasks.

Interim Assessment

• Interim assessment (3 module)
  0.5 * Oexam + 0.1 * Oproject1 + 0.15 * Oproject2 + 0.25 * Otest

Bibliography

Recommended Core Bibliography

• Математические задачи системного анализа, Моисеев, Н. Н., 1981

Recommended Additional Bibliography

• Выпуклый анализ и экстремальные задачи, Пшеничный, Б. Н., 1980