A Study of Combinatorial Optimization Problems in Industrial Computer Systems

Abstract: A combinatorial optimization problem is an optimization problem where the number of possible solutions are finite and grow combinatorially with the problem size. Combinatorial problems exist everywhere in industrial systems. This thesis focuses on solving three such problems which arise within two different areas where industrial computer systems are often used. Within embedded systems and real-time systems, we investigate the problems of allocating stack memory for an system where a shared stacks may be used, and of estimating the highest response time of a task in a system of industrial complexity. We propose a number of different algorithms to compute safe upper bounds on run-time stack usage whenever the system supports stack sharing. The algorithms have in common that they can exploit commonly-available information regarding timing behaviour of the tasks in the system. Given upper bounds on the individual stack usage of the tasks, it is possible to estimate the worst-case stack behaviour by analysing the possible and impossible preemption patterns. Using relations on offset and precedences, we form a preemption graph, which is further analysed to find safe upper-bounds on the maximal preemptions chain in the system. For the special case where all tasks exist in a single static schedule and share a single stack, we propose a polynomial algorithm to solve the problem. For generalizations of this problem, we propose an exact branch-and-bound algorithm for smaller problems and a polynomial heuristic algorithm for cases where the branch-and-bound algorithm fails to find a solution in reasonable time. All algorithms are evaluated in comprehensive experimental studies. The polynomial algorithm is implemented and shipped in the developer tool set for a commercial real-time operating system, Rubus OS. The second problem we study in the thesis is how to estimate the highest response time of a specified task in a complex industrial real-time system. The response-time analysis is done using a best-effort approach, where a detailed model of the system is simulated on input constructed using a local search procedure. In an evaluation on three different systems we can see that the new algorithm were able to produce higher response times much faster than what has previously been possible. Since the analysis is based on simulation and measurement, the results are not safe in the sense that they are always higher or equal to the true response time of the system. The value of the method lies instead in that it makes it possible to analyse complex industrial systems which cannot be analysed accurately using existing safe approaches. The third problem is in the area of maintenance planning, and focus on how to dynamically plan maintenance for industrial systems. Within this area we have focused on industrial gas turbines and rail vehicles.  We have developed algorithms and a planning tool which can be used to plan maintenance for gas turbines and other stationary machinery. In such problems, it is often the case that performing several maintenance actions at the same time is beneficial, since many of these jobs can be done in parallel, which reduces the total downtime of the unit. The core of the problem is therefore how to (or how not to) group maintenance activities so that a composite cost due to spare parts, labor and loss of production due to downtime is minimized. We allow each machine to have individual schedules for each component in the system. For rail vehicles, we have evaluated the effect of replanning maintenance in the case where the component maintenance deadline is set to reflect a maximum risk of breakdown in a Gaussian failure distribution. In such a model, we show by simulation that replanning of maintenance can reduce the number of maintenance stops when the variance and expected value of the distribution are increased.  For the gas turbine maintenance planning problem, we have evaluated the planning software on a real-world scenario from the oil and gas industry and compared it to the solutions obtained from a commercial integer programming solver. It is estimated that the availability increase from using our planning software is between 0.5 to 1.0 %, which is substantial considering that availability is currently already at 97-98 %.