A branch-and-price algorithm for unrelated parallel machine scheduling with machine usage costs

This paper considers unrelated parallel machine scheduling involving machine usage costs, in addition to classic job completion time-related costs. The usage cost of each machine is made up of a fixed usage cost and a variable usage cost proportional to the total processing time of the jobs assigned to it. These features model many practical situations where machine usage costs include, for example, rental fees when the machines are not owned but rented. To tackle this problem, four mathematical models based on the Shortest Weighted Processing Time (SWPT) rule are introduced. Additionally, the problem is formulated into a set-partitioning model, for which a branch-and-price algorithm is proposed with an appropriate branching strategy. This facilitates the development of an efficient pseudo-polynomial dynamic programming algorithm and a polynomial-time heuristic to solve the pricing problem. Extensive numerical experiments demonstrate the superior performance of the proposed branch-and-price algorithm over the four SWPT-based mathematical formulations and an existing branch-and-price algorithm designed for a special case. Notably, it can optimally solve instances involving up to 225 jobs and 15 machines within one hour. Moreover, statistical analyses reveal that the proposed polynomial-time heuristic significantly reduces the computation time, and the mathematical model based on the contribution of every job to the total weighted completion time exhibits the best overall performance.