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

Authors:

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.

UDC:
519.87