Approaches for solving some scheduling and routing problems

Abstract

We study approaches for finding good solutions, and lower bounds, for three difficult combinatorial optimisation problems.
The supply ship travelling salesman problem is a simplification of a situation faced by a naval logistics coordinator who must direct a support vessel tasked with resupplying ships in a fleet. It is a generalisation of the travelling salesman problem in which the nodes are in motion, each following some predetermined route. We apply dynamic programming state-space relaxation techniques, producing lower bounds for the problem that are 73% to 84% of the best solution, on average. We also apply heuristics to find good solutions to this NP-hard problem, showing that restricted dynamic programming approaches outperform simple 2-opt and 3-opt local search procedures for instances with 20 nodes.
We introduce the supply ship scheduling problem, another roblem inspired by a support vessel environment. We wish to minimise the number of mobile machines required to process a set of jobs; each job is in a different stationary location and features a fixed start time. Jobs may be simultaneously processed by multiple machines, obtaining a speed-up in processing time. We represent the problem as a directed graph and use the minimum flow in a transformed network to determine the minimum number of machines. We present a neighbourhood structure based on the maximum cut, applying it within descent and tabu search procedures. We construct a restricted dynamic programming based approach, but this is outperformed by the tabu search algorithm.
The task allocation problem, arising in distributed computing, is to assign a set of tasks to a set of processors so that the overall cost is minimised. Costs are incurred from processor usage, interprocessor communication and task execution. We construct, and try to improve, semidefinite programming relaxations to find lower bounds for variants of this NP-hard problem. We develop a branch-and-bound approach to find optimal solutions, but this is only effective for small instances.

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