Output analysis for approximated stochastic programs

Because of incomplete information and also for the sake of numerical tractability one mostly solves an approximated stochastic program instead of the underlying ''true'' decision problem. However, without an additional analysis, the obtained output (the optimal value and optimal solutions of the approximated stochastic program) should not be used to replace the sought solution of the ''true'' problem. Methods of output analysis have to be tailored to the structure of the problem and they should also reflect the source, character and precision of the input data. The scope of various approaches based on results of asymptotic and robust statistics, of the moment problem and on general results of parametric programming will be discussed from the point of view of their applicability and possible extensions

Supply chain network design under uncertainty and risk

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.We consider the research problem of quantitative support for decision making in supply chain network design (SCND). We first identify the requirements for a comprehensive SCND as (i) a methodology to select uncertainties, (ii) a stochastic optimisation model, and (iii) an appropriate solution algorithm. We propose a process to select a manageable number of uncertainties to be included in a stochastic program for SCND. We develop a comprehensive two-stage stochastic program for SCND that includes uncertainty in demand, currency exchange rates, labour costs, productivity, supplier costs, and transport costs. Also, we consider conditional value at risk (CV@R) to explore the trade-off between risk and return. We use a scenario generator based on moment matching to represent the multivariate uncertainty. The resulting stochastic integer program is computationally challenging and we propose a novel iterative solution algorithm called adaptive scenario refinement (ASR) to process the problem. We describe the rationale underlying ASR, validate it for a set of benchmark problems, and discuss the benefits of the algorithm applied to our SCND problem. Finally, we demonstrate the benefits of the proposed model in a case study and show that multiple sources of uncertainty and risk are important to consider in the SCND. Whereas in the literature most research is on demand uncertainty, our study suggests that exchange rate uncertainty is more important for the choice of optimal supply chain strategies in international production networks. The SCND model and the use of the coherent downside risk measure in the stochastic program are innovative and novel; these and the ASR solution algorithm taken together make contributions to knowledge

Incorporating statistical model error into the calculation of acceptability prices of contingent claims

The determination of acceptability prices of contingent claims requires the choice of a stochastic model for the underlying asset price dynamics. Given this model, optimal bid and ask prices can be found by stochastic optimization. However, the model for the underlying asset price process is typically based on data and found by a statistical estimation procedure. We define a confidence set of possible estimated models by a nonparametric neighborhood of a baseline model. This neighborhood serves as ambiguity set for a multi-stage stochastic optimization problem under model uncertainty. We obtain distributionally robust solutions of the acceptability pricing problem and derive the dual problem formulation. Moreover, we prove a general large deviations result for the nested distance, which allows to relate the bid and ask prices under model ambiguity to the quality of the observed data.Comment: 27 pages, 2 figure

Design and architecture of a stochastic programming modelling system

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Decision making under uncertainty is an important yet challenging task; a number of alternative paradigms which address this problem have been proposed. Stochastic Programming (SP) and Robust Optimization (RO) are two such modelling ap-proaches, which we consider; these are natural extensions of Mathematical Pro-gramming modelling. The process that goes from the conceptualization of an SP model to its solution and the use of the optimization results is complex in respect to its deterministic counterpart. Many factors contribute to this complexity: (i) the representation of the random behaviour of the model parameters, (ii) the interfac-ing of the decision model with the model of randomness, (iii) the difficulty in solving (very) large model instances, (iv) the requirements for result analysis and perfor-mance evaluation through simulation techniques. An overview of the software tools which support stochastic programming modelling is given, and a conceptual struc-ture and the architecture of such tools are presented. This conceptualization is pre-sented as various interacting modules, namely (i) scenario generators, (ii) model generators, (iii) solvers and (iv) performance evaluation. Reflecting this research, we have redesigned and extended an established modelling system to support modelling under uncertainty. The collective system which integrates these other-wise disparate set of model formulations within a common framework is innovative and makes the resulting system a powerful modelling tool. The introduction of sce-nario generation in the ex-ante decision model and the integration with simulation and evaluation for the purpose of ex-post analysis by the use of workflows is novel and makes a contribution to knowledge

Scenario reduction heuristics for a rolling stochastic programming simulation of bulk energy flows with uncertain fuel costs

Stochastic programming is employed regularly to solve energy planning problems with uncertainties in costs, demands and other parameters. We formulated a stochastic program to quantify the impact of uncertain fuel costs in an aggregated U.S. bulk energy transportation network model. A rolling two-stage approach with discrete scenarios is implemented to mimic the decision process as realizations of the uncertain elements become known and forecasts of their values in future periods are updated. Compared to the expected value solution from the deterministic model, the recourse solution found from the stochastic model has higher total cost, lower natural gas consumption and less subregional power trade but a fuel mix that is closer to what actually occurred. The worth of solving the stochastic program lies in its capacity of better simulating the actual energy flows. Strategies including decomposition, aggregation and scenario reduction are adopted for reducing computational burden of the large-scale program due to a huge number of scenarios. We devised two heuristic algorithms, aiming to improve the scenario reduction algorithms, which select a subset of scenarios from the original set in order to reduce the problem size. The accelerated forward selection (AFS) algorithm is a heuristic based on the existing forward selection (FS) method. AFS\u27s selection of scenarios is very close to FS\u27s selection, while AFS greatly outperforms FS in efficiency. We also proposed the TCFS method of forward selection within clusters of transferred scenarios. TCFS clusters scenarios into groups according to their distinct impact on the key first-stage decisions before selecting a representative scenario from each group. In contrast to the problem independent selection process of FS, by making use of the problem information, TCFS achieves excellent accuracy and at the same time greatly mitigates the huge computation burden