Modelling and solving environments for mathematical programming (MP): a status review and new directions
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Modelling and solving environments for mathematical programming (MP): a status review and new directions B Dominguez-Ballesteros1, G Mitra1*, C Lucas1 and N-S Koutsoukis1 1
Brunel University, Uxbridge, Middlesex, UK
Languages and computing environments that support Mathematical Programming (MP) modelling continue to develop and evolve. In this paper: (i) We address declarative and procedural features of modelling languages. (ii) We assess developments which couple the modelling and the solving processes, typically column generation, the branch-and-price approach to Integer Programming (IP), and the sampling of scenarios within Stochastic Programming (SP). (iii) We consider how data modelling and symbolic modelling naturally come together and are used within the information value chain. (iv) Finally, we investigate the features of new tools, which support prototyping of optimisation based Decision Support (DS) applications. Journal of the Operational Research Society (2002) 53, 1072–1092. doi:10.1057/palgrave.jors.2601361 Keywords: linear programming; planning; algebraic modelling languages; optimisation; decision support
Introduction and background There is a re-emergence of interest in Mathematical Programming (MP) models for optimum Decision Support (DS) applications. To construct such DS applications it is often necessary to undertake feasibility studies, and construct and investigate optimisation models. This paper focuses on the tools and software environments which support such investigations.
The MP modelling process For an understanding of how MP is applied to industry it is essential to recognise the different stages in the process of model formulation, solution and results investigations. The different stages of the overall process as set out in Figure 1 are explained in the following way. Conceptualisation. Initially the problem owner analyses the real world problem. The available information is collected and used to study the ‘real’ issues; these are to establish the content and the relevance of the given problem, and what needs to be optimised. At this stage it is not necessary to think about the mathematical formulation of the problem in the form of an algebraic model. Instead it is necessary to consider the physical problem and the essential aspects which must be taken into account for decision making. *Correspondence: G Mitra, Centre for the Analysis of Risk and Optimisation Modelling Applications (CARISMA), Department of Mathematical Sciences, Brunel University, Uxbridge, Middlesex UB8 3PH, UK. E-mail: [email protected]
Algebraic form. Subsequently the modeller develops a mathematical formulation of the problem. In order to model a given problem as a MP it is necessary to specify an objective function to be maximised (or minimised) as well as the algebraic constraints in a suitable algorithmic form. This leads to constrained optimisation problems. This mathematical representat
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