Finding dual of a scheduling LP formulation

Question

Suppose I have an LP formulation as such:

$\min\ \ \sum\limits_{i,j,t}\ w_{ij}x_{ijt} (\frac{t-r_j}{p_{ij}}+0.5)$

$\sum\limits_{i,t}\frac{x_{ijt}}{p_{ij}}=1\,\forall\ j$

$\sum\limits_{j}x_{ijt}\leq 1\,\forall \ i,t$

$x_{ijt}\geq 0\ \forall\ i,j,t\geq r_j$

For understanding the context of its formulation, please refer here.

My question is how to proceed to compute its dual problem. Now for standard notation like

$\min\ p'x$

$Ax\geq b, x\geq 0$

The dual is simply

$\max\ b'u$

$A'u\leq p, u\geq 0$

But the primal problem of interest has inequalities of the less than ($\leq$) form as well as equality constraints. So how do we proceed with this kind of non-standard form?

Answer 1

Two alternative approaches:

1. Rewrite the primal problem in standard form by replacing equality with two inequalities and multiplying both sides of $\le$ inequalities by $-1$ to reverse the sense to $\ge$.
2. For equality constraints in the primal, use free variables in the dual, and for $\le$ constraints in the primal, use $\le 0$ variables. The second approach is simpler. See this paper.