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The ’Three Axial Sums’ problem with capacity restrictions. (English) Zbl 0698.90056

Special topics on mathematical economics and optimization theory, Methods Oper. Res. 61, 9-20 (1990).
[For the entire collection see Zbl 0687.00023.]
The authors consider the capacitated three axial sums problem \[ \min imize\quad \sum_{I}\sum_{J}\sum_{K}c_{ijk}x_{ijk} \] subject to \[ \sum_{J}\sum_{K}x_{ijk}=a_ i,\quad i\in I;\quad \sum_{I}\sum_{K}x_{ijk}=b_ j,\quad j\in J; \]
\[ \sum_{I}\sum_{J}x_{ijk}=e_ k,\quad k\in K;\quad x_{ijk}\geq 0,\quad x_{ijk}\leq d_{ijk},\quad i\in I,\quad j\in J,\quad k\in K. \] They formulate an associated uncapacitated problem (where the constraints of type \(x_{ijk}\leq d_{ijk}\) do not appear) and show a correspondence between feasible (and thus also optimal) solutions of the two problems which makes it possible to solve the original problem as an uncapacitated one by known methods.
Reviewer: J.Rohn

MSC:

90C08 Special problems of linear programming (transportation, multi-index, data envelopment analysis, etc.)

Citations:

Zbl 0687.00023