The linear problem of making solution with consider the risks and regrets

This paper deals with a linear programming problem with interval function coefficients. A new approach for a problem under uncertainty is proposed. It is based on two concepts: the guaranteed result and the minimax regret criterion. We consider the following linear programming problem under uncertainty
\[
\max_{x{\epsilon}X} f(x,y),
\]
where $f(x, y) = \sum_{i=1}^n{x_i}{y_i}$ is utility function, the feasible set $X$ is given by
\[
X = \{x = (x_1, x_2, ..., x_n) \in \mathbb{R}^n | Ax \leq b\}
\]
Here $A$ is an $m \times n$ matrix, $x$ and $b$ are $n$- and $m$-column vectors, respectively. The set of uncertainties is defined by
\[
Y = \{y = (y_1, y_2, ..., y_n) | a_i \leq y_i \leq b_i, i \in \{1, 2, ..., n\} \}
\]
Further, we decide the standard linear programming problem
\[
f(x, y) = \sum_{i=1}^n{a_i}{x_i}
\]
\[
\begin{cases}
x = (x_1, x_2, ..., x_n) \in X, \\
x_i \geq 0, i \in \{1, 2, ..., n\},
\end{cases}
\]
where $X = \{x = (x_1, x_2, ..., x_n) \in \mathbb{R}^n | Ax \leq b\}$ and $x^*_V \in X$ is solution.
The two-criteria problem
\[
\Gamma = \langle X, \{f_1(x) = R_V(x), f_2(x) = R_S(x)\} \rangle,
\]
is formalized. Here
\[
R_V(x) = f_V[x^*_V] - \sum_{i=1}^n{a_i}{x_i}
\]
is risk function,
\[
R_S(x) = \max_{y \in Y}\phi(x,y) - \min_{x \in X}\max_{y \in Y}\phi(x, y)
\]
is regret function, where $\phi = \max_{x \in X}f(x, y) - f(x, y)$.
Then we look for vector $x_P \in X$, which minimizes the function
\[
F(x) = R_V^2(x) + R_S^2(x), x \in X.
\]
Solution $x_P \in X$ is Pareto optimal for problem $\Gamma$. In this paper we construct the algorithm finding optimal solution $x_P \in X$. In order to illustrate the proposed solution method, a numerical example is given.

Keywords: linear programming problem, uncertainty, two-criteria problem, risk function, regret function, Pareto minimum.