  We describe the torus-equivariant cohomology of weighted partial flag
orbifolds ${\mathrm{w}}\Sigma$ of type $A$. We establish counterparts of
several results known for the partial flag variety that collectively constitute
what we refer to as ``Schubert Calculus on ${\mathrm{w}}\Sigma$''. For the
weighed Schubert classes in ${\mathrm{w}}\Sigma$, we give the Chevalley's
formula. In addition, we define the weighted analogue of double Schubert
polynomials and give the corresponding Chevalley--Monk's formula.
