This paper generalizes previous studies on genome rearrangement under biological constraints, using double cut and join (DCJ). We propose a model for weighted DCJ, along with a family of optimization problems called ϕ-MCPS (MiniMuM CoSt ParSiMoniouS SCenario), that are based on labeled graphs. We show how to compute solutions to general instances of ϕ-MCPS, given an algorithm to compute ϕ-MCPS on a circular genome with exactly one occurrence of each gene. These general instances can have an arbitrary number of circular and linear chromosomes, and arbitrary gene content. The practicality of the framework is displayed by presenting polynomial-time algorithms that generalize the results of Bulteau, Fertin, and Tannier on the Sorting by wDCJS anD inDelS in intergeneS problem, and that generalize previous results on the MiniMuM loCal ParSiMoniouS SCenario problem.