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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

Korean

그럼에도 불구하고, 선형 프로그래밍은 가능한 세트의 최상위에 최적을 달성하고, 몇 가지 방향으로 최적의 계획의 작은 편차는 한계를 위반합니다. 지금까지, 우리는 경제의 공급 수요 균형과 그 총 국내 제품 (GDP)의 주어진 수준의 최적 가치를 보장하는 생산 계획 세트를 찾는 데 거의주의를 기울였으며, 균형 균형은 빈 내부와 함께 세트를 정의하기 때문에 직접적으로 적용 할 수 없습니다. 이 논문에서, 우리는 경제의 공급 수요 균형과 제한된 GDP의 최적 가치보다 작은 GDP를 보장하는 $(\varepsilon,δ$) 솔루션의 세트를 소개합니다. 이 논문에서 우리는 공식적인 솔루션 문제를 설정함으로써

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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