Minimal proper non-IRUP instances of the one-dimensional Cutting Stock Problem

We consider the well-known one dimensional cutting stock problem (1CSP). Based on the pattern structure of the classical ILP formulation of Gilmore and Gomory, we can decompose the infinite set of 1CSP instances, with a fixed demand n, into a finite number of equivalence classes. We show up a strong relation to weighted simple games. Studying the integer round-up property we computationally show that all 1CSP instances with $n\le 9$ are proper IRUP, while we give examples of a proper non-IRUP instances with $n=10$. A gap larger than 1 occurs for $n=11$. The worst known gap is raised from 1.003 to 1.0625. The used algorithmic approaches are based on exhaustive enumeration and integer linear programming. Additionally we give some theoretical bounds showing that all 1CSP instances with some specific parameters have the proper IRUP.Comment: 14 pages, 2 figures, 2 table

Improved flow-based formulations for the skiving stock problem

Thanks to the rapidly advancing development of (commercial) MILP software and hardware components, pseudo-polynomial formulations have been established as a powerful tool for solving cutting and packing problems in recent years. In this paper, we focus on the one-dimensional skiving stock problem (SSP), where a given inventory of small items has to be recomposed to obtain a maximum number of larger objects, each satisfying a minimum threshold length. In the literature, different modeling approaches for the SSP have been proposed, and the standard flow-based formulation has turned out to lead to the best trade-off between efficiency and solution time. However, especially for instances of practically meaningful sizes, the resulting models involve very large numbers of variables and constraints, so that appropriate reduction techniques are required to decrease the numerical efforts. For that reason, this paper introduces two improved flow-based formulations for the skiving stock problem that are able to cope with much larger problem sizes. By means of extensive experiments, these new models are shown to possess significantly fewer variables as well as an average better computational performance compared to the standard arcflow formulation

Incremental Packing Problems: Algorithms and Polyhedra

In this thesis, we propose and study discrete, multi-period extensions of classical packing problems, a fundamental class of models in combinatorial optimization. Those extensions fall under the general name of incremental packing problems. In such models, we are given an added time component and different capacity constraints for each time. Over time, capacities are weakly increasing as resources increase, allowing more items to be selected. Once an item is selected, it cannot be removed in future times. The goal is to maximize some (possibly also time-dependent) objective function under such packing constraints. In Chapter 2, we study the generalized incremental knapsack problem, a multi-period extension to the classical knapsack problem. We present a policy that reduces the generalized incremental knapsack problem to sequentially solving multiple classical knapsack problems, for which many efficient algorithms are known. We call such an algorithm a single-time algorithm. We prove that this algorithm gives a (0.17 - ⋲)-approximation for the generalized incremental knapsack problem. Moreover, we show that the algorithm is very efficient in practice. On randomly generated instances of the generalized incremental knapsack problem, it returns near optimal solutions and runs much faster compared to Gurobi solving the problem using the standard integer programming formulation. In Chapter 3, we present additional approximation algorithms for the generalized incremental knapsack problem. We first give a polynomial-time (½-⋲)-approximation, improving upon the approximation ratio given in Chapter 2. This result is based on a new reformulation of the generalized incremental knapsack problem as a single-machine sequencing problem, which is addressed by blending dynamic programming techniques and the classical Shmoys-Tardos algorithm for the generalized assignment problem. Using the same sequencing reformulation, combined with further enumeration-based self-reinforcing ideas and new structural properties of nearly-optimal solutions, we give a quasi-polynomial time approximation scheme for the problem, thus ruling out the possibility that the generalized incremental knapsack problem is APX-hard under widely-believed complexity assumptions. In Chapter 4, we first turn our attention to the submodular monotone all-or-nothing incremental knapsack problem (IK-AoN), a special case of the submodular monotone function subject to a knapsack constraint extended to a multi-period setting. We show that each instance of IK-AoN can be reduced to a linear version of the problem. In particular, using a known PTAS for the linear version from literature as a subroutine, this implies that IK-AoN admits a PTAS. Next, we study special cases of the generalized incremental knapsack problem and provide improved approximation schemes for these special cases. In Chapter 5, we give a polynomial-time (¼-⋲)-approximation in expectation for the incremental generalized assignment problem, a multi-period extension of the generalized assignment problem. To develop this result, similar to the reformulation from Chapter 3, we reformulate the incremental generalized assignment problem as a multi-machine sequencing problem. Following the reformulation, we show that the (½-⋲)-approximation for the generalized incremental knapsack problem, combined with further randomized rounding techniques, can be leveraged to give a constant factor approximation in expectation for the incremental generalized assignment problem. In Chapter 6, we turn our attention to the incremental knapsack polytope. First, we extend one direction of Balas's characterization of 0/1-facets of the knapsack polytope to the incremental knapsack polytope. Starting from extended cover inequalities valid for the knapsack polytope, we show how to strengthen them to define facets for the incremental knapsack polytope. In particular, we prove that under the same conditions for which these inequalities define facets for the knapsack polytope, following our strengthening procedure, the resulting inequalities define facets for the incremental knapsack polytope. Then, as there are up to exponentially many such inequalities, we give separation algorithms for this class of inequalities

Conflict-Free Routing of Mobile Robots

The recent advances in perception have enabled the development of more autonomous mobile robots in the sense that they can operate in a more dynamic environment where obstacles surrounding the robot emerge, disappear, and move. The increased perception of Autonomous Mobile Robots (AMRs) allows them to plan detailed on-line trajectories in order to avoid previously unforeseen obstacles, making AMRs useful in dynamic environments where humans, traditional fork-lifts, and also other mobile robots operate. These abilities contributed to increase automation in logistic applications. This thesis discusses how to efficiently operate a fleet of AMRs and make sure that all tasks are successfully completed.Assigning robots to specific delivery tasks and deciding the routes they have to travel can be modelled as a variant of the classical Vehicle Routing Problem (VRP), the combinatorial optimization problem of designing routes for vehicles. In related research it has been extended to scheduling routes for vehicles to serve customers according to predetermined specifications, such as arrival time at a customer, amount of goods to deliver, etc.In this thesis we consider to schedule a fleet of robots such that areas avoid being congested, delivery time-windows are met, the need for robots to recharge is considered, while at the same time the robots have freedom to use alternative paths to handle changes in the environment. This particular version of the VRP, called CF-EVRP (Conflict-free Electrical Vehicle Routing Problem) is motivated by an industrial need. In this work we consider using optimizing general purpose solvers, in particular, MILP and SMT solvers are investigated. We run extensive computational analysis over well-known combinatorial optimization problems, such as job shop scheduling and bin-packing problems, to evaluate modeling techniques and the relative performance of state-of-the-art MILP and SMT solvers.We propose a monolithic model for the CF-EVRP as well as a compositional approach that decomposes the problem into sub-problems and formulate them as either MILP or SMT problems depending on what fits each particular problem best. The performance of the two approaches is evaluated on a set of CF-EVRP benchmark problems, showing the feasibility of using a compositional approach for solving practical fleet scheduling problems

Combinatorial optimisation for sustainable cloud computing

Enabled by both software and hardware advances, cloud computing has emerged as an efficient way to leverage economies of scale for building large computational infrastructures over a global network. While the cost of computation has dropped significantly for end users, the infrastructure supporting cloud computing systems has considerable economic and ecological costs. A key challenge for sustainable cloud computing systems in the near future is to maintain control over these costs. Amid the complexity of cloud computing systems, a cost analysis reveals a complex relationship between the infrastructure supporting actual computation on a physical level and how these physical assets are utilised. The central question tackled in this dissertation is how to best utilise these assets through efficient workload management policies. In recent years, workload consolidation has emerged as an effective approach to increase the efficiency of cloud systems. We propose to address aspects of this challenge by leveraging techniques from the realm of mathematical modeling and combinatorial optimisation. We introduce a novel combinatorial optimisation problem suitable for modeling core consolidation problems arising in workload management in data centres. This problem extends on the well-known bin packing problem. We develop competing models and optimisation techniques to solve this offline packing problem with state-of-the-art solvers. We then cast this newly defined combinatorial optimisation problem in an semi-online setting for which we propose an efficient assignment policy that is able to produce solutions for the semi-online problem in a competitive computational time. Stochastic aspects, which are often faced by cloud providers, are introduced in a richer model. We then show how predictive methods can help decision makers dealing with uncertainty in such dynamic and heterogeneous systems. We explore a similar but relaxed problem falling within the scope of proactive consolidation. This is a relaxed consolidation problem in which one decides which, when and where workload should be migrated to retain minimum energy cost. Finally, we discuss ongoing efforts to model and characterise the combinatorial hardness of bin packing instances, which in turn will be useful to study the various packing problems found in cloud computing environments