Dynamic model of the software development market based on the assignment problem on pain points

Abstract

The authors propose the formulation of a discrete dynamic model of the software developmentmarket (SM) based on the assignment problem (AP) on pain points (PP), which can also be obtained according to the scheme used in (Vasin, Grigorieva, Lesik, 2018), if we abandon the integer number of elements of the assignment matrix. However, there are also features: equilibrium prices can be calculated directly, and therefore a variational formulation of the internal problem of determining equilibrium prices based on Debreu's theorem (Debreu, 1954) is not required. The functions of changing the phase coordinates can be taken convex, for example, the norm of the difference in the square, and do not take into account the constant costs for each control switching. Such a statement is also given in this paper. If we have a dynamic expansion of the AP on PP, it is possible to determine the additional profit of the transport system through the use of futures. Formulas for the components of the gradient of the indicator are obtained. This allows us to organize a gradient method for solving a dynamic AP on PP. The authors also demonstrate an approximate algorithm and a model example of its use for solving the dynamic expansion of the AP on PP, based on solving the current static problem with an increment of those elements of the efficiency matrix that coincide with the corresponding elements of the optimal assignment matrix, if we abandon the integer nature of the assignment matrix. This is equivalent to randomization of the assignment problem, when the corresponding assignments are implemented with certain probabilities, which are used to determine the error of the approximate algorithm by comparing it with the exact solution obtained with the gradient method for sufficiently large values of penalty constants.