91Ó°ÊÓ

Let I be the set of constraints that are not forced equal, so that Ie I it there is some feasible solution.

we show that $x^I=\left(X_1^I\mathrm{...}{X}_n^I\right)$defined by

$x^I=\left(X_1^I\mathrm{...}{X}_n^I\right)$such $I$that is satisfied without equality.

$X_1={∑}_{1∈I}^1{\mathrm{χ}}_i^I$characteristic solution indeed for any fired $I≤j≤m$, the constraint is satisfied by ‘$x$’ since summing the left – hand side and the right – hand side of the inequality over all ${\mathrm{χ}}^i$gives us

${∑}_i{a}_{j,i}({∑}_{(J∈x)}{x}_i^1≤|I|b_j$

That implies,
${∑}_i{a}_{j,i}({∑}_{(J∈x)}{x}_i^1{≤}_i∑a_{(j,i)}{x}_i≤b_j$

Furthermore, if 1th constraint is not forced – equal. So that

${∑}_i{a}_{j,i}{x}_i^I<b_j$

${∑}_i{a}_{j,i}({∑}_{J∈x}{x}_i^1≤|I|b_j$

So,
${∑}_i{a}_{j,i}{x}_i^I<b_j$and the constraint is satisfied without equality.

Thus, ‘$x$’ is a feasible solution where every
$J∈I$is satisfied without equality.

By the algorithm maintains a set of I constraints, initialized to be the empty set. And iterates.Through each constraint $j=1,2,\mathrm{....}m$.

Consider the$j$ iteration of the algorithm and let I denote the j constraint. We define a linear program, obtained from the original set of constraints as follows.

We introduct a new variable ${\mathrm{\AE ›}}_i$ and replace the ‘$j$’ constraint with the following constraint :

${\mathrm{\AE ›}}_j+{∑}_i{a}_{j,i}({∑}_{J∈x}{x}_i^1≤{∑}_i{a}_{j,i}{x}_i≤b_j$

We also add an objective function : max ${\mathrm{λ}}_i$. we then find a optimal solution to the resulting LP.