Distributionally robust scheduling algorithms for total flow time minimization on parallel machines using norm regularizations

In this paper, we study a distributionally robust parallel machines scheduling problem, minimizing the total flow time criterion. The distribution of uncertain processing times is subject to ambiguity belonging to a set of distributions with constrained mean and covariance. We show that the problem can be cast as a deterministic optimization problem, with the objective function composed of an expectation and a regularization term given as an ℓp norm. The main question we ask and answer is whether the particular choice of the used ℓp norm affects the computational complexity of the problem and the robustness of its solution. We prove that if durations of the jobs are independent, the solution in terms of any ℓp norm can be solved in a pseudopolynomial time, by the reduction to a non-linear bipartite matching problem. We also show an efficient, polynomial-time algorithm for ℓ1 case. Furthermore, for instances with dependent durations of the jobs, we propose computationally efficient formulation and an algorithm that uses ℓ1 norm. Moreover, we identify a class of covariance matrices admitting a faster, polynomial-time algorithm. The computational experiments show that the proposed algorithms provide solutions with a similar quality to the existing algorithms while having significantly better computational complexities.