Question

In a satisfiable system of linear inequalities

$\begin{array}{l}\begin{array}{l}a11x1+···+a1nxn\le b1\\ :\end{array}\\ am1x1+···+amnxn\le bm\end{array}$

we describe the inequality as forced-equal if it is satisfied with equality by every solution x = $(x1,...,xn)$of the system. Equivalently,$Piajixi\le bj$ is not forced-equal if there exists an x that satisfies the whole system and such that $Piajixi\le bj$.

For example, in

$\begin{array}{l}\begin{array}{l}x1+x2\le 2\\ -x1-x2\le -2\end{array}\\ x1\le 1\\ -x2\le 0\end{array}$

Short answer

Describe the $j^{th}$inequality is forced equal solution of $x=\left(x\mathrm{1...}xn\right)$and get correct answer base on different formula to countthat satisfies the whole system and such that ${ƛ}_j+{\sum }_i{a}_{j,i}({\sum }_{(J\in x)}{x}_i^1\le {\sum }_i{a}_{(j,i)}{x}_i\sum b_j$

Step-by-step solution

Maximum Flow Diagram of Following Network

a) The claim hold trivially of all constants are forced-equal, so assume at least one constraint is not forced-equal.

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.

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

Formula base Calculation

${\sum }_i{a}_{j,i}({\sum }_{(J\in x)}{x}_i^1\le |I|b_j$

That implies,
${\sum }_i{a}_{j,i}({\sum }_{(J\in x)}{x}_i^1{\le }_i\sum a_{(j,i)}{x}_i\le b_j$

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

${\sum }_i{a}_{j,i}{x}_i^I<b_j$

${\sum }_i{a}_{j,i}({\sum }_{J\in x}{x}_i^1\le |I|b_j$

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

Thus, ‘$x$’ is a feasible solution where every
$J\in I$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.

Conclusion

We introduct a new variable ${ƛ}_i$ and replace the ‘$j$’ constraint with the following constraint :

${ƛ}_j+{\sum }_i{a}_{j,i}({\sum }_{J\in x}{x}_i^1\le {\sum }_i{a}_{j,i}{x}_i\le b_j$

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