46 research outputs found
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An extension of set partitioning with application to scheduling problems
The well known problems of set covering, set partitioning and set packing are defined and their interrelationship is considered. A natural generalisation called the extended set partitioning model is presented and the three standard models are shown to be special cases of this generalisation. In addition, the extended model includes another type of set problem which can be of greater use in certain applications. The model forms the basis of a computer assisted bus crew scheduling system developed by the authors. The system is in regular use by Dublin City Services in the Republic of Ireland. Finally, the equivalence between a special case of the set partitioning problem and the shortest route problem is considered and it is shown that this equivalence also applies to the extended model
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Combinatorial optimization and metaheuristics
Today, combinatorial optimization is one of the youngest and most active areas of discrete mathematics. It is a branch of optimization in applied mathematics and computer science, related to operational research, algorithm theory and computational complexity theory. It sits at the intersection of several fields, including artificial intelligence, mathematics and software engineering. Its increasing interest arises for the fact that a large number of scientific and industrial problems can be formulated as abstract combinatorial optimization problems, through graphs and/or (integer) linear programs. Some of these problems have polynomial-time (“efficient”) algorithms, while most of them are NP-hard, i.e. it is not proved that they can be solved in polynomial-time. Mainly, it means that it is not possible to guarantee that an exact solution to the problem can be found and one has to settle for an approximate solution with known performance guarantees. Indeed, the goal of approximate methods is to find “quickly” (reasonable run-times), with “high” probability, provable “good” solutions (low error from the real optimal solution). In the last 20 years, a new kind of algorithm commonly called metaheuristics have emerged in this class, which basically try to combine heuristics in high level frameworks aimed at efficiently and effectively exploring the search space. This report briefly outlines the components, concepts, advantages and disadvantages of different metaheuristic approaches from a conceptual point of view, in order to analyze their similarities and differences. The two very significant forces of intensification and diversification, that mainly determine the behavior of a metaheuristic, will be pointed out. The report concludes by exploring the importance of hybridization and integration methods
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Developments in linear and integer programming
In this review we describe recent developments in linear and integer (linear) programming. For over 50 years Operational Research practitioners have made use of linear optimisation models to aid decision making and over this period the size of problems that can be solved has increased dramatically, the time required to solve problems has decreased substantially and the flexibility of modelling and solving systems has increased steadily. Large models are no longer confined to large computers, and the flexibility of optimisation systems embedded in other decision support tools has made on-line decision making using linear programming a reality (and using integer programming a possibility). The review focuses on recent developments in algorithms, software and applications and investigates some connections between linear optimisation and other technologies
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A co-operative parallel heuristic for integer linear programming: Combining simulated annealing with branch & bound
This paper considers the exact approach of branch and bound (B&B) and the metaheuristic known as simulated annealing (SA) for processing integer programs (IP). We extend an existing SA implementation (GPSIMAN) for pure zero–one integer programs (PZIP) to process a wider class of IP models, namely mixed zero–one integer programs (MZIP). The extensions are based on depth-first B&B searches at different points within the SA framework. We refer to the resultant SA implementation as MIPSA. Furthermore, we have exploited the use of parallel computers by designing a co-operative parallel heuristic whereby concurrent executions of B&B and MIPSA, linked through a parallel computer, exchange information in order to influence their searches. Results reported for a wide range of models taken from a library of MIP benchmarks demonstrate the effectiveness of using a parallel computing approach which combines B&B with SA
Cutting plane methods for general integer programming
Integer programming (IP) problems are difficult to solve due to the integer restrictions imposed on them. A technique for solving these problems is the cutting plane method. In this method, linear constraints are added to the associated linear programming (LP) problem until an integer optimal solution is found. These constraints cut off part of the LP solution space but do not eliminate any feasible integer solution. In this report algorithms for solving IP due to Gomory and to Dantzig are presented. Two other cutting plane approaches and two extensions to Gomory's algorithm are also discussed. Although these methods are mathematically elegant they are known to have slow convergence and an explosive storage requirement. As a result cutting planes are generally not computationally successful
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Portfolio optimisation models and properties of return distributions
Mean-risk models have been widely used in portfolio optimisation. However, such models may
produce portfolios that are dominated with respect to second order stochastic dominance and therefore not
optimal for rational and risk-averse investors. This paper considers the problem of constructing a portfolio
which is nondominated with respect to second order stochastic dominance and whose return distribution
has specified desirable properties. The problem is multi-objective and is transformed into a single
objective problem by using the reference point method, in which target levels, known as aspiration points,
are specified for the objective function values. A model is proposed in which the aspiration points relate to
ordered return outcomes of the portfolio return. The model is extended by additionally specifying
reservation points, which act pre-emptively in the optimisation. The theoretical properties of the models
are studied. The performance of the models on real data drawn from the Hang Seng index is also
investigated
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Computer assisted modelling of linear, integer and separable programming problems
For mathematical programming (MP) to have greater impact upon the decision making process, MP software systems must offer suitable support in terms of model communication and modelling techniques . In this paper modelling techniques that allow logical restrictions to be modelled in integer programming terms are described and their implications discussed. In
addition it is demonstrated that many classes of non-linearities which are not variable separable may be reformulated in piecewise linear form. It is shown that analysis of bounds is necessary in the following three important contexts: model reduction, formulation of logical restrictions as 0-1 mixed integer programs and reformulation of nonlinear programs as variable separable programs, It is observed that as well as incorporating an interface between the modeller and the optimiser there is a need to make available to the modeller software facilities which support the modelling techniques described here
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A two-stage stochastic programming with recourse model for determining robust planting plans in horticulture
A two-stage stochastic programming with recourse model for the problem of determining optimal planting plans for a vegetable crop is presented in this paper. Uncertainty caused by factors such as weather on yields is a major influence on many systems arising in horticulture. Traditional linear programming models are generally unsatisfactory in dealing with the uncertainty and produce solutions that are considered to involve an unacceptable level of risk. The first stage of the model relates to finding a planting plan which is common to all scenarios and the second stage is concerned with deriving a harvesting schedule for each scenario. Solutions are obtained for a range of risk aversion factors that not only result in greater expected profit compared to the corresponding deterministic model, but also are more robust
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Linear, integer separable and fuzzy programming problems: a united approach towards automatic reformulation
For mathematical programming (MP) to have greater impact as a
decision tool, MP software systems must offer suitable support in
terms of model communication and modelling techniques. In this
paper modelling techniques that allow logical restrictions to be
modelled in integer programming terms are described and their
implications discussed. In addition it is demonstrated that many
classes of non-linearities which are not variable separable may be
after suitable algebraic manipulation put in a variable separable
form. The methods of reformulating the fuzzy linear programming
problem as a Max-Min problem is also introduced. It is shown that
analysis of bounds plays a key role in the following four important
contexts: model reduction, reformulation of logical restrictions
as 0-1 mixed integer programs, reformulation of nonlinear programs
as variable separable programs and reformulation of fuzzy linear
programs. It is observed that as well as incorporating an
interface between the modeller and the optimiser there is a need to
make available to the modeller software facilities which support the
model reformulation techniques described here
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Variable neighbourhood search for the minimum labelling Steiner tree problem
We present a study on heuristic solution approaches to the minimum labelling Steiner tree problem, an NP-hard graph problem related to the minimum labelling spanning tree problem. Given an undirected labelled connected graph, the aim is to find a spanning tree covering a given subset of nodes of the graph, whose edges have the smallest number of distinct labels. Such a model may be used to represent many real world problems in telecommunications and multimodal transportation networks. Several metaheuristics are proposed and evaluated. The approaches are compared to the widely adopted Pilot Method and it is shown that the Variable Neighbourhood Search that we propose is the most effective metaheuristic for the problem, obtaining high quality solutions in short computational running time