From SSKE

SEM - Mathematical Modelling of Economic Processes

The lectures are aimed at familiarizing the students to a selection of classical topics concerning operation research and mathematical optimization, with focus on economic modelling: optimization problems - such as maximization of profits or minimization of costs -, simple cases of so-called “market-problems”, in which there are groups of actors with opposing interests - treated within the two-person matrix games theory - or planning problems, like optimally achieving some goal in a finite number of steps over a finite time interval – based on the principle of optimality or on dynamic programming.

This is necessary as an introduction to service systems understanding in two aspects:

Firstly, the students need a preliminary background concerning the concept of economic modelling from the classical maximal profit / minimal cost perspective and from the classical perspective of production planning over finite time horizon. These major classes of problems are not eliminated but included into the service dominant logic approach, as tools for a correct economic analysis over short time intervals, in which static or quasi-static models are well suited, with the advantage of existing dedicated software for numerical computation.

Secondly, the candidates to the SEM master programme are graduates of several distinct licence programs, and only the graduates of System Engineering license programs (Automatic Control) have a preliminary knowledge of static optimization under constraints and of operation research; hence it is necessary to bring all the students to a common higher level of knowledge in this area.

In the first part of the course, some basic aspects concerning linear programming problems and classes of models are presented, accompanied by intuitive examples, illustrating the basic concepts of duality and shadow prices, and followed by a detailed case study dedicated to a linear model of production.

The general Lagrange approach for nonlinear constrained optimization is introduced next in an intuitive way, based on simple examples with geometric significance, in order to emphasize the interpretation of Lagrange multipliers as shadow prices, with direct relation to economic modelling. The linear programming approach is then presented as a particular case of the general Lagrange approach.

A brief introduction in the topics of two person matrix games emphasizes a class of models of competition and the concept of payoff, on one side, and, for mixed strategies, the fact that two person matrix games are only a particular case of the linear programming approaches. Network models are illustrated by a relevant example of maximal flow through a network.

Finally, in the part dedicated to dynamic models, the optimality principle is applied to economic planning problems and the state structured case is introduced, within a simple economic example, solved using the Bellmann equation.