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Input-output analysis with a sphere packing approach

Munkhbayar Byambadash, Tumendelger Lkhagvasuren, Энхбат Ренцен · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.23391 · cited by 0 works (OpenAlex) · open-access PDF

Japanese

しかし、線形プログラミングは、実行可能なセットの頂点で最適な範囲を達成し、いくつかの方向で最適なプランの小さな偏差は制約を破る。これまで、我々は、経済の供給と需要のバランスを保証する生産プランのセットを見つけることにほとんど注意を払われている。線形プログラミングは、実行可能なセットの最上位に最適なレベルを達成し、ボールパッキングアプローチは直接適用されません、バランスの方程式は、空の内部で強力なセットを定義します。この論文では、我々は、GDPが最大限の$\varepsilon、δ$の最適な値よりも低い可能性のあるGDPのバランスを保証する $(\vare

Machine translation (Meta M2M100); the English original follows.

Abstract

Input-output analysis with linear programming is widely used in the planning of national economies. However, linear programming attains its optimum at a vertex of the feasible set, and a small deviation of the optimal plan in some direction violates a constraint. So far, little attention has been paid to finding sets of production plans which guarantee both the supply-demand balance of an economy and a given level of its gross domestic product (GDP), and the sphere packing approach cannot be applied directly, since the balance equations define a set with an empty interior. In this paper, we introduce the set of $(\varepsilon,δ)$-solutions, in which GDP may be smaller than its optimal value by at most $\varepsilon$ and supply may differ from demand by at most $δ$, and we find a sphere with the maximum radius inscribed into this set by solving a linear programming problem. We prove that the optimal radius is positive for positive tolerances, that it is non-decreasing, concave and piecewise linear with respect to the tolerances, and that it is bounded by the narrowest balance constraint, which is determined by a sales-concentration index of sectors. The approach is applied to the official input-output tables of Mongolia for 2010-2019 and 2024. The classical optimal plan has zero robustness, while the robust plan keeps 99.73% of the maximal GDP and, in a Monte Carlo simulation, tolerates independent deviations of all outputs and final demands of up to $\pm$0.30%. The margin of robust plans is limited by sectors with concentrated sales, mainly education and, increasingly, export-oriented mining, which indicates that a diversification of the users of mining output widens the set of robust plans.

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Topics: Linear programming, Mathematics, Piecewise linear function, Mathematical optimization, Bounded function, Monte Carlo method

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