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**Resource allocation among competing activities: A lexicographic minimax approach.**
*(English)*
Zbl 0627.90068

The minmax linear programming problem
\[
\min_{x}\max_{j}a_ j[d_ j-x_ j)/d_ j]
\]
subject to
\[
\sum^{n}_{j=1}a_{ij}x_ j\leq b_ i,\quad i=1,2,...,m,\quad 0\leq x_ j\leq d_ j
\]
is examined and a fast non-simplex algorithm developed which requires at most \(2n(m+1)\) divisions and multiplications. Then the idea of a lexicographic minmax algorithm for solving the original problem with the objective function
\[
lex\min_{x}\{\max_{j}a_ j[(d_ j-x_ j)/d_ j]
\]
is proposed that improves the result from the point of view of the vector of weighted deviations. The proposed algorithms promise to be effective for a given class of large scale resource allocation and production planning problems.

Reviewer: F.Turnovec

### Keywords:

minmax linear programming; non-simplex algorithm; lexicographic minmax algorithm; large scale resource allocation; production planning
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\textit{H. Luss} and \textit{D. R. Smith}, Oper. Res. Lett. 5, 227--231 (1986; Zbl 0627.90068)

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