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The problem of choosing the optimal thicknesses of the layers of refractory and heat-insulating materials that provide the minimum total cost of the lining and fuel used to compensate for heat losses is considered. The proposed technique (when specifying the required temperature at the boundary of the lining layers as an additional condition) reduces the problem of conditional two-dimensional optimization of the objective function, which does not have an unconditional minimum and has a break line, to the problem of finding the minimum of a unimodal one-dimensional objective function. The relevance of this work is due to the fact that during the overhaul of the furnace it is possible to replace its lining using new materials and optimize costs taking into account the updated state of the range of materials and their prices.

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