Correction of radiation sources of photometric devices using the linear programming method

Abstract

Linear programming (LP) is a branch of mathematical programming that studies methods for solving extreme problems, which are characterized by a linear relationship between variables and a linear criterion.

Linear programming methods are applied to practical problems in which: it is necessary to choose the best solution (optimal plan) from the set of possible ones; a solution can be expressed as a set of values of some variables; restrictions imposed on admissible solutions by specific conditions of the problem are formulated in the form of linear equations or inequalities; the goal is expressed as a linear function of the main variables. The values of the objective function, allowing you to compare different solutions, serve as a criterion for the quality of the solution.

For the practical solution of the problem by mathematical methods, it should first of all be written using mathematical expressions: equations, inequalities, and the like. Based on the features of linear programming problems noted above, we can outline the following general model formation scheme: selection of a certain number of variable quantities, the assignment of the numerical values of which uniquely determines one of the possible states of the phenomenon under study;  expression of the relationships inherent in the phenomenon under study in the form of mathematical relationships (equations, inequalities); these ratios form a system of problem constraints;  quantitative expression of the selected criterion of optimality in the form of an objective function;  mathematical formulation of the problem as the problem of finding the extremum of the objective function, subject to the fulfillment of the constraints imposed on the variables.

This article shows that if the reasons leading to the instability of the spectral characteristics of the emitter are not correlated, the problem of optimizing the correction of the emitter functioning can be formulated and solved by the linear programming method. Shown is a specific conventional example of optimization of the mode of correction of the operation of the emitter by the method of linear programming.