Original scientific paper

Interval method for interval linear program

R. Viher

Abstract

In the problem of interval linear programming (\ref{ijedan})
\begin{equation}\label{ijedan}
\begin{array}{c}
\max c^{T}x\\
a\leq Ax\leq b
\end{array}
\end{equation}
$x,c\in\mathbf{R}^{n}$; $a,b\in\mathbf{R}^{m}$; $A\in\mathbf{R}_{m}^{m\times n}$ ($A$ is of full row
rank) we introduce the new variable $t=c_{1}x_{1}+\cdots+c_{k}x_{k}+\cdots+c_{n}x_{n}$ and eliminate
the old variable $(c_{k}\neq 0)$
\[x_{k}=\frac{1}{c_{k}}t-\frac{c_{1}}{c_{k}}x_{1}-\cdots-\frac{c_{k-1}}{c_{k}}x_{k-1}-
\frac{c_{k+1}}{c_{k}}x_{k+1}-\cdots-\frac{c_{n}}{c_{k}}x_{n}.\]
So we come to the second form
\begin{equation}\label{idva}
\begin{array}{c}
\max t\\
a+\th\leq Bx'\leq b+\th
\end{array}
\end{equation}
$x'\in\mathbf{R}^{n-1}$; $a,b,h\in\mathbf{R}^{m}$; $B\in\mathbf{R}_{m-1}^{m\times (n-1)}$. It is known
when $c\in\mathcal{R}(A^{T})$ that problem (\ref{ijedan}) has an explicit solution. In this article we formulate
the analogous theorem for the second form (\ref{idva}), and then show the application of those results on the
problem of sensitivity analyses.

Keywords

interval method; interval linear program; first form of interval linear program; second form of interval linear program; explicit solution; sensitivity analyses

Hrčak ID:

739

URI

https://hrcak.srce.hr/739

Publication date:

20.6.2003.

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