### Application of duality theory to solve two-criteria problem of linear programming for ecological-economic system

#### Abstract

In the paper, we investigate two-criterion optimization problem: maximization of one target function and minimization of another target function. To solve the offered two-criterion problem, the method of the main criterion is applied. We consider the problem of production activity of the ecological-economic system with the maximization of the value of the final product as the first target function and the minimization of emissions of polluters into the environment as the second target function. We constructed of two production functions (economic and ecological). To construct the economic production function, we select maximal producing of the final products in a costing form as the most essential (main) criterion. Also, there is introduced the appropriate data of the criterion level total volume of emissions of polluters into the environment. After this two-criteria problem is reduced to one-criteria problem. For the construction of ecological production function, the main criterion in the problem of the minimal general volume of emissions of polluters into the environment is defined. We use a parameter of the criterion level of the second criterion and obtained one-criterion problem. Therefore, investigation of the appropriate dual problems explicitly provides economic and ecological production functions to the deduced one-criterion problems. These functions in input two-criterion problem give way to optimal manage of ecological-economic system.

#### Keywords

optimal management, two-criterion problem, dual problem, target function, main criterion method, ecological-economic equilibrium, production function

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