How to pack trapezoids: exact and evolutionary algorithms

The purposes of this paper are twofold. In the first, we describe an exact polynomial-time algorithm for the pair sequencing problem and show how this method can be used to pack fixed-height trapezoids into a single bin such that interitem wastage is minimised. We then go on to examine how this algorithm can be combined with bespoke evolutionary and local search methods for tackling the multiple-bin version of this problem—one that is closely related to one-dimensional bin packing. In the course of doing this, a number of ideas surrounding recombination, diversity, and genetic repair are also introduced and analysed

Modifying colourings between time-steps to tackle changes in dynamic random graphs

Many real world operational research problems can be formulated as graph colouring problems. Algorithms for this problem usually operate under the assumption that the size and constraints of a problem are fixed, allowing us to model the problem using a static graph. For many problems however, this is not the case and it would be more appropriate to model such problems using dynamic graphs. In this paper we will explore whether feasible colourings for one graph at time-step t can be modified into a colouring for a similar graph at time-step t+1 in some beneficial manner

Investigating edge-reordering procedures in a tabu search algorithm for the capacitated arc routing problem

This paper presents two ideas to guide a tabu search algorithm for the Capacitated Arc Routing Problem to a promising region of the solution space. Both ideas involve edge-reordering, although they work in different ways. One of them aims to directly tackle deadheading cycles, and the other tries to reorder edges with the aim of extending a scope of solutions that can be reached from a given solution. Experiments were performed on 134 benchmark instances of various sizes, and the two ideas were shown to have an ability to guide the search to good solutions. Possible issues that may arise when implementing these ideas are also discussed

Tackling the edge dynamic graph colouring problem with and without future adjacency information

Many real world operational research problems, such as frequency assignment and exam timetabling, can be reformulated as graph colouring problems (GCPs). Most algorithms for the GCP operate under the assumption that its constraints are fixed, allowing us to model the problem using a static graph. However, in many real-world cases this does not hold and it is more appropriate to model problems with constraints that change over time using an edge dynamic graph. Although exploring methods for colouring dynamic graphs has been identified as an area of interest with many real-world applications, to date, very little literature exists regarding such methods. In this paper we present several heuristic methods for modifying a feasible colouring at time-step t into an initial, but not necessarily feasible, colouring for a “similar” graph at time-step t+1t+1 . We will discuss two cases; (1) where changes occur at random, and (2) where probabilistic information about future changes is provided. Experimental results are also presented and the benefits of applying these particular modification methods are investigated

Towards objective measures of algorithm performance across instance space

This paper tackles the difficult but important task of objective algorithm performance assessment for optimization. Rather than reporting average performance of algorithms across a set of chosen instances, which may bias conclusions, we propose a methodology to enable the strengths and weaknesses of different optimization algorithms to be compared across a broader instance space. The results reported in a recent Computers and Operations Research paper comparing the performance of graph coloring heuristics are revisited with this new methodology to demonstrate (i) how pockets of the instance space can be found where algorithm performance varies significantly from the average performance of an algorithm; (ii) how the properties of the instances can be used to predict algorithm performance on previously unseen instances with high accuracy; and (iii) how the relative strengths and weaknesses of each algorithm can be visualized and measured objectively

Metaheuristics can solve sudoku puzzles

In this paper we present, to our knowledge, the first application of a metaheuristic technique to the very popular and NP-complete puzzle known as ‘sudoku’. We see that this stochastic search-based algorithm, which uses simulated annealing, is able to complete logic-solvable puzzle-instances that feature daily in many of the UK’s national newspapers. We also introduce a new method for producing sudoku problem instances (that are not necessarily logic-solvable) and use this together with the proposed SA algorithm to try and discover for what types of instances this algorithm is best suited. Consequently we notice the presence of an ‘easy-hard-easy’ style phase-transition similar to other problems encountered in operational research