Ant colony optimisation and local search for bin-packing and cutting stock problems

The Bin Packing Problem and the Cutting Stock Problem are two related classes of NP-hard combinatorial optimization problems. Exact solution methods can only be used for very small instances, so for real-world problems, we have to rely on heuristic methods. In recent years, researchers have started to apply evolutionary approaches to these problems, including Genetic Algorithms and Evolutionary Programming. In the work presented here, we used an ant colony optimization (ACO) approach to solve both Bin Packing and Cutting Stock Problems. We present a pure ACO approach, as well as an ACO approach augmented with a simple but very effective local search algorithm. It is shown that the pure ACO approach can compete with existing evolutionary methods, whereas the hybrid approach can outperform the best-known hybrid evolutionary solution methods for certain problem classes. The hybrid ACO approach is also shown to require different parameter values from the pure ACO approach and to give a more robust performance across different problems with a single set of parameter values. The local search algorithm is also run with random restarts and shown to perform significantly worse than when combined with ACO

A Majorization-Minimization Approach to Design of Power Transmission Networks

We propose an optimization approach to design cost-effective electrical power transmission networks. That is, we aim to select both the network structure and the line conductances (line sizes) so as to optimize the trade-off between network efficiency (low power dissipation within the transmission network) and the cost to build the network. We begin with a convex optimization method based on the paper ``Minimizing Effective Resistance of a Graph'' [Ghosh, Boyd \& Saberi]. We show that this (DC) resistive network method can be adapted to the context of AC power flow. However, that does not address the combinatorial aspect of selecting network structure. We approach this problem as selecting a subgraph within an over-complete network, posed as minimizing the (convex) network power dissipation plus a non-convex cost on line conductances that encourages sparse networks where many line conductances are set to zero. We develop a heuristic approach to solve this non-convex optimization problem using: (1) a continuation method to interpolate from the smooth, convex problem to the (non-smooth, non-convex) combinatorial problem, (2) the majorization-minimization algorithm to perform the necessary intermediate smooth but non-convex optimization steps. Ultimately, this involves solving a sequence of convex optimization problems in which we iteratively reweight a linear cost on line conductances to fit the actual non-convex cost. Several examples are presented which suggest that the overall method is a good heuristic for network design. We also consider how to obtain sparse networks that are still robust against failures of lines and/or generators.Comment: 8 pages, 3 figures. To appear in Proc. 49th IEEE Conference on Decision and Control (CDC '10

Exploring search space trees using an adapted version of Monte Carlo tree search for combinatorial optimization problems

In this article, a novel approach to solve combinatorial optimization problems is proposed. This approach makes use of a heuristic algorithm to explore the search space tree of a problem instance. The algorithm is based on Monte Carlo tree search, a popular algorithm in game playing that is used to explore game trees. By leveraging the combinatorial structure of a problem, several enhancements to the algorithm are proposed. These enhancements aim to efficiently explore the search space tree by pruning subtrees, using a heuristic simulation policy, reducing the domains of variables by eliminating dominated value assignments and using a beam width. They are demonstrated for two specific combinatorial optimization problems: the quay crane scheduling problem with non-crossing constraints and the 0-1 knapsack problem. Computational results show that the algorithm achieves promising results for both problems and eight new best solutions for a benchmark set of instances are found for the former problem. These results indicate that the algorithm is competitive with the state-of-the-art. Apart from this, the results also show evidence that the algorithm is able to learn to correct the incorrect choices made by constructive heuristics

Deterministic Annealing and Nonlinear Assignment

For combinatorial optimization problems that can be formulated as Ising or Potts spin systems, the Mean Field (MF) approximation yields a versatile and simple ANN heuristic, Deterministic Annealing. For assignment problems the situation is more complex -- the natural analog of the MF approximation lacks the simplicity present in the Potts and Ising cases. In this article the difficulties associated with this issue are investigated, and the options for solving them discussed. Improvements to existing Potts-based MF-inspired heuristics are suggested, and the possibilities for defining a proper variational approach are scrutinized.Comment: 15 pages, 3 figure

Exact and Heuristic Hybrid Approaches for Scheduling and Clustering Problems

This thesis deals with the design of exact and heuristic algorithms for scheduling and clustering combinatorial optimization problems. All the works are linked by the fact that all the presented methods arebasically hybrid algorithms, that mix techniques used in the world of combinatorial optimization. The algorithms are all efficient in practice, but the one presented in Chapter 4, that has mostly theoretical interest. Chapter 2 presents practical solution algorithms based on an ILP model for an energy scheduling combinatorial problem that arises in a smart building context. Chapter 3 presents a new cutting stock problem and introduce a mathematical formulation and a heuristic solution approach based on a heuristic column generation scheme. Chapter 4 provides an exact exponential algorithm, whose importance is only theoretical so far, for a classical scheduling problem: the Single Machine Total Tardiness Problem. The relevant aspect is that the designed algorithm has the best worst case complexity for the problem, that has been studied for several decades. Furthermore, such result is based on a new technique, called Branch and Merge, that avoids the solution of several equivalent sub-problems in a branching algorithm that requires polynomial space. As a consequence, such technique embeds in a branching algorithm ideas coming from other traditional computer science techniques such as dynamic programming and memorization, but keeping the space requirement polynomial. Chapter 5 provides an exact approach based on semidefinite programming and a matheuristic approach based on a quadratic solver for a fractional clustering combinatorial optimization problem, called Max-Mean Dispersion Problem. The matheuristic approach has the peculiarity of using a non-linear MIP solver. The proposed exact approach uses a general semidefinite programming relaxation and it is likely to be extended to other combinatorial problems with a fractional formulation. Chapter 6 proposes practical solution methods for a real world clustering problem arising in a smart city context. The solution algorithm is based on the solution of a Set Cover model via a commercial ILP solver. As a conclusion, the main contribution of this thesis is given by several approaches of practical or theoretical interest, for two classes of important combinatorial problems: clustering and scheduling. All the practical methods presented in the thesis are validated by extensive computational experiments, that compare the proposed methods with the ones available in the state of the art

Application of Quantum Annealing to Nurse Scheduling Problem

Quantum annealing is a promising heuristic method to solve combinatorial optimization problems, and efforts to quantify performance on real-world problems provide insights into how this approach may be best used in practice. We investigate the empirical performance of quantum annealing to solve the Nurse Scheduling Problem (NSP) with hard constraints using the D-Wave 2000Q quantum annealing device. NSP seeks the optimal assignment for a set of nurses to shifts under an accompanying set of constraints on schedule and personnel. After reducing NSP to a novel Ising-type Hamiltonian, we evaluate the solution quality obtained from the D-Wave 2000Q against the constraint requirements as well as the diversity of solutions. For the test problems explored here, our results indicate that quantum annealing recovers satisfying solutions for NSP and suggests the heuristic method is sufficient for practical use. Moreover, we observe that solution quality can be greatly improved through the use of reverse annealing, in which it is possible to refine a returned results by using the annealing process a second time. We compare the performance NSP using both forward and reverse annealing methods and describe how these approach might be used in practice.Comment: 20 pages, 13 figure