Control of a complex objects, states of which are describing by the matrix rating mechanism

The control problem of a multi-criteria object is considered. Controlled object that has several criteria that are significant for a decision maker. Each criterion characterizes a control object in terms of a particular result of activity or an efficiency indicator. To evaluate the effectiveness of the functioning of the managed facility as a whole, the rating matrix mechanism is used, taking into account all the criteria in the complex. The optimal control problem is formulated as a search for the values of aggregated criteria that provide a given value of a complex indicator with minimal costs for providing values of particular criteria. The generalized cost function was reduced to an equation with one variable. The analytical equation of the level line of the indicator aggregated as a result of the convolution of two criteria is obtained. The line equation is found for an arbitrary binary convolution matrix, including the elements of which are given continuous values. It is shown that the objective function is reduced to a fourth-order polynomial, which can be analytically solved using the Ferrari or Descartes-Euler methods.

It is shown that the task of searching for the values of two particular criteria describing the state of the control object for which the complex indicator calculated using the additive-multiplicative approach to complex assessment is equal to the given value and the costs for their provision are minimal, has a solution in general form for arbitrary nondecreasing convolution matrix of two criteria.

Particular solutions to the control problem are found using costly functions, which are the inverse function of the Cobb-Douglas production function. It was shown that the cost function of the aggregate indicator has additional terms and is described by an algebraic equation with nonzero coefficients for variables and an additional constant. Based on what it was concluded that the cost functions, which are the inverse function e of the Cobb-Douglas production function, can be applied to control objects that have only two criteria.

A similar formulation of the control problem for an arbitrary non-decreasing convolution matrix of two criteria is considered when using the additive-multiplicative approach to aggregation and when using cost functions described by a second-order algebraic equation in general form. As a result of the study, it is shown that the form of the cost function for the aggregated indicator is preserved. Thus, using cost functions in the form of second-order equations, the control problem has a solution in the general form for any number of criteria.