# Finding dual of a scheduling LP formulation

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?

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.