91Ó°ÊÓ

Linear program is used for optimization tasks that has constraints and the optimization criterion as linear functions. A linear program has the set of variables that needs to be assign with the real values to satisfy the linear inequalities and to minimize or maximize a given linear objective function.

Let us assume a variable$\mathrm{z}$as the maximum absolute error such that:

$\mathrm{k}=\underset{1≤\mathrm{i}≤7}{\max }|{\mathrm{ax}}_{\mathrm{i}}+{\mathrm{by}}_{\mathrm{i}}−\mathrm{c}|$

To minimize the maximum absolute error $\left(\mathrm{k}\right)$, so we consider,

$\mathrm{k}â‰\yen \underset{1≤\mathrm{i}≤7}{\max }|{\mathrm{ax}}_{\mathrm{i}}+{\mathrm{by}}_{\mathrm{i}}−\mathrm{c}|$ for $0<i≤7$

To get absolute value we will remove modulus sign and split expression into two such that:

$kâ‰\yen a{x}_i+b{y}_i−c$ and $kâ‰\yen −a{x}_i−b{y}_i+c$

Since variable$k$is greater than some absolute value, add a constrain $kâ‰\yen 0$

Thus, the linear program is as follows,

Maximize: $-z$or Minimize:$z$

$\begin{array}{l}\mathrm{k}â‰\yen {\mathrm{ax}}_{\mathrm{i}}+{\mathrm{by}}_{\mathrm{i}}−\mathrm{c}\\ \mathrm{k}â‰\yen −{\mathrm{ax}}_{\mathrm{i}}−{\mathrm{by}}_{\mathrm{i}}+\mathrm{c}\\ \mathrm{k}â‰\yen 0\end{array}$

Therefore, the linear program that finds the maximum absolute error has been obtained.