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.
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.) |