Updating an Input–Output Matrix with Sign-preservation: Some Improved Objective Functions and their Solutions

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Abstract
This paper presents an easy-to-use method for updating input–output (IO) matrices with sign-preservation by combining Lagrangian multipliers and penalty functions. Biproportional methods such as the representative RAS are very simple and popular because a target matrix can be obtained simply by iterative computation. However, they cannot reasonably deal with matrices that include negative entries. Although a generalized version, GRAS, can do so, its objective function is questionable. In contrast, some non-biproportional methods such as those that take weighted or unweighted squared differences between the target and original matrix as objective functions can deal with negative entries, but it is difficult to guarantee the signs of entries. In this study, GRAS and some conventional objective functions were improved and their solutions for preserving the signs of entries are presented. Comparisons of applying these objective functions to a simple example show that both the Improved Normalized Squared Differences (INSD) function and the Improved GRAS (IGRAS) function yield a good target matrix and are close to each other; we suggest that INSD or IGRAS be used for updating IO transaction matrices in practice.