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Software tools for the investigation of stochastic programming problems

Stochastic Programming (SP) has become synonymous with optimum decision making under uncertainty. However, modelling and solving stochastic programming problems are challenging tasks. Many factors affect the complexity of SP models with respect to their deterministic counterparts: (i) the need for capturing the random behaviour of the models parameters, (ii) the difficulties in solving the model instances (which are usually very large), and (iii) the requirements for advanced analysis of the model results through simulation techniques. This research focuses on the investigation of stochastic programming problems using software tools, and the outcomes of this work address some of the aforementioned difficulties. The well known classes of stochastic programming models and their properties are first reviewed, and a critical analysis of the use of algebraic modelling languages and their extensions for the formulation of SP models is presented. The insight gained into the interaction of models of randomness with optimisation models has led to the design of a novel approach to extending algebraic modelling languages for SP modelling. This approach has been successfully applied to the AMPL and MPL language to incorporate new SP language constructs. The resulting languages SAMPL and SMPL support the formulation of scenario-based SP problems in a natural and concise way. The integration of modelling systems with specialised solvers for SP has also been investigated. The use of decomposition methods for solving recourse problems has been analysed in terms of scale-up properties; alternative formulations of chance-constrained problems, which can be formulated as deterministic mixed integer programs, are also illustrated. Computational and conceptual issues related to the value of stochastic solutions (VSS) have been analysed and discussed. A Stochastic Programming Integrated Environment (SPInE), which supports the SMPL and SAMPL extended modelling languages and embodies a stochastic solver has been designed and developed in this research. The features of the system include the integration with scenario generators and the analysis of the SP model solutions and validation through the use of simulation. A case study is presented to illustrate how the SPInE system provides practitioners with a powerful platform for the analysis of real life problems using stochastic programming techniques.

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Introduction and background 1 Chapter 1 Introduction and background Στοξαστικόσ (Stochastikós): based on guesswork, based on conjecture. 1.1 Optimum decision making under uncertainty Optimum decision making concerns a class of problems where it is necessary to make decisions to optimise one or more given objectives, subject to certain restrictions. Mathematical Programming (MP) models have made considerable contribution to optimum decision making, as they enable the modeller to capture the structure of the problem and to quantify the effects of the decisions in terms of the objectives set by the decision maker. Mathematical Programming models express the objective as a function of one or more decision variables. This objective function needs to be minimised or maximised, subject to a set of constraints given in the form of equalities or inequalities, which represent the structure of the real world problem. MP models can be classified according to two main criteria: • The type of relationships and objective functions • The type of decision variables Linear Programming (LP) models are characterised by objective function and constraints which are linear combinations of the decision variables, while Quadratic Programming (QP) models are characterised by a quadratic objective function. Following the second criterion, one can classify the models as Integer Programming (IP) models if all decision variables are integer, or Mixed Integer Programming (MIP) models if some of the variables are integer while the remaining are continuous, and so on. Linear Programming models were first introduced in the 40’s, and their use in real world applications has since become increasingly significant. The rapid growth in computational power, as well the developments in software tools for modelling and solving LP problems, are the main foundation for the success of these models. Unfortunately, their success has also showed up their limitations in many situations where these models cannot be employed with any confidence. The world of LP and IP is highly deterministic and the underlying assumption is that the parameters which are used to define the models are known with certainty.

Tesi di Dottorato

Dipartimento: Department of Mathematical Sciences

Autore: Patrick Valente Contatta »

Composta da 161 pagine.

 

Questa tesi ha raggiunto 510 click dal 20/03/2004.

 

Consultata integralmente una volta.

Disponibile in PDF, la consultazione è esclusivamente in formato digitale.