Fuzzy C-means-based scenario bundling for stochastic service network design

Stochastic service network designs with uncertain demand represented by a set of scenarios can be modelled as a large-scale two-stage stochastic mixed-integer program (SMIP). The progressive hedging algorithm (PHA) is a decomposition method for solving the resulting SMIP. The computational performance of the PHA can be greatly enhanced by decomposing according to scenario bundles instead of individual scenarios. At the heart of bundle-based decomposition is the method for grouping the scenarios into bundles. In this paper, we present a fuzzy c-means-based scenario bundling method to address this problem. Rather than full membership of a bundle, which is typically the case in existing scenario bundling strategies such as k-means, a scenario has partial membership in each of the bundles and can be assigned to more than one bundle in our method. Since the multiple bundle membership of a scenario induces overlap between the bundles, we empirically investigate whether and how the amount of overlap controlled by a fuzzy exponent would affect the performance of the PHA. Experimental results for a less-than-truckload transportation network optimization problem show that the number of iterations required by the PHA to achieve convergence reduces dramatically with large fuzzy exponents, whereas the computation time increases significantly. Experimental studies were conducted to find out a good fuzzy exponent to strike a trade-off between the solution quality and the computational time

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

On modelling planning under uncertainty in manufacturing

We present a modelling framework for two-stage and multi-stage mixed 0-1 problems under uncertainty for strategic Supply Chain Management, tactical production planning and operations assignment and scheduling. A scenario tree based scheme is used to represent the uncertainty. We present the Deterministic Equivalent Model of the stochastic mixed 0-1 programs with complete recourse that we study. The constraints are modelled by compact and splitting variable representations via scenarios

Integrated tactical planning in the lumber supply chain under demand and supply uncertainty

Lumber supply chain includes forests as suppliers, sawmills as production sites, distribution centers, and different types of customers. In this industry, the raw materials are logs that are shipped from forest contractors to sawmills. Logs are then sawn to green/finished lumbers in sawmills and are distributed to the lumber market through different channels. Unlike a traditional manufacturing industry, the lumber industry is characterized by a divergent product structure with the highly heterogeneous nature of its raw material (logs). Moreover, predicting the exact amount of the product demand and the availability of logs in the forest is impossible in this industry. Thus, considering random demand and supply in the lumber supply chain planning is essential. Integrated tactical planning in a supply chain incorporates the synchronized planning of procurement, production, distribution and sale activities in order to ensure that the customer demand is satisfied by the right product at the right time. Briefly, in this dissertation, we aim at developing integrated planning tools in lumber supply chains for making decisions in harvesting, material procurement, production, distribution, and sale activities in order to obtain a maximum robust profit and service level in the presence of uncertainty in the log supply and product demand. In order to gain the latter objectives, we can categorize this research into three phases. In the first phase, we investigate the integrated annual planning of harvesting, procurement, production, distribution, and sale activities in the lumber supply chain in a deterministic context. The problem is formulated as a mixed integer programming (MIP) model. The proposed model is applied on a real-size case study, which leads to a large-scale MIP model that cannot be solved by commercial solvers in a reasonable time. Consequently, we propose a Lagrangian Relaxation based heuristic algorithm in order to solve the latter MIP model. While improving significantly the convergence, the proposed algorithm also guarantees the feasibility of the converged solution. In the second phase, the uncertainty is incorporated in the lumber supply chain tactical planning problems. Thus, we propose a multi-stage stochastic mixed-integer programming (MS-MIP) model to address this problem. Due to the complexity of solving the latter MS-MIP model with commercial solvers or relevant solution methodologies in the literature, we develop a Hybrid Scenario Cluster Decomposition (HSCD) heuristic algorithm which is also amenable to parallelization. This algorithm decomposes the original scenario tree into a set of smaller sub-trees. Hence, the MS-MIP model is decomposed into smaller sub-models that are coordinated by Lagrangian terms in their objective functions. By embedding an ad-hoc heuristic and a Variable Fixing algorithm into the HSCD algorithm, we considerably improve its convergence and propose an implementable solution in a reasonable CPU time. Finally, due to the computational complexity of multi-stage stochastic programming approach, we confine our formulation to the robust optimization method. Hence, at the third phase of this research, we propose a robust planning model formulated based on cardinality-constrained method. The latter provides some insights into the adjustment of the level of robustness of the proposed plan over the planning horizon and protection against uncertainty. An extensive set of experiments based on Monte-Carlo simulation is also conducted in order to better validate the proposed robust optimization approach applied on the harvesting planning in lumber supply chains

Scheduling process operations under uncertainty and integration with long term planning

This thesis centers upon the application of mathematical modelling, optimization theory and uncertainty analysis to the problem of scheduling batch operations for large scale industries. Over the years, decision making strategies such as scheduling, that deals with allocation of plant resources, has been widely adopted by industries to efficiently carry out their operations and achieve the desired targets. In this thesis, the focus is on planning and scheduling under endogenous uncertainty in the context of multijob, multitasking batch plants. This class of scheduling problems are of practical importance, specially in the analytical services sector, where effective scheduling models could increase the efficiency in carrying out the plant operations and may lead to increased throughput, or reduced makespan, resulting in greater profits or customer satisfaction

Soft clustering-based scenario bundling for a progressive hedging heuristic in stochastic service network design

We present a method for bundling scenarios in a progressive hedging heuristic (PHH) applied to stochastic service network design, where the uncertain demand is represented by a finite number of scenarios. Given the number of scenario bundles, we first calculate a vector of probabilities for every scenario, which measures the association strength of a scenario to each bundle center. This membership score calculation is based on existing soft clustering algorithms such as Fuzzy C-Means (FCM) and Gaussian Mixture Models (GMM). After obtaining the probabilistic membership scores, we propose a strategy to determine the scenario-to-bundle assignment. By contrast, almost all existing scenario bundling methods such as K-Means (KM) assume before the scenario-to-bundle assignment that a scenario belongs to exactly one bundle, which is equivalent to requiring that the membership scores are Boolean values. The probabilistic membership scores bring many advantages over Boolean ones, such as the flexibility to create various degrees of overlap between scenario bundles and the capability to accommodate scenario bundles with different covariance structures. We empirically study the impacts of different degrees of overlap and covariance structures on PHH performance by comparing PHH based on FCM/GMM with that based on KM and the cover method, which represents the state-of-the-art scenario bundling algorithm for stochastic network design. The solution quality is measured against the lower bound provided by CPLEX. The experimental results show that, GMM-based PHH yields the best performance among all methods considered, achieving nearly equivalent solution quality in a fraction of the run-time of the other methods

Optimization Approaches for Electricity Generation Expansion Planning Under Uncertainty

In this dissertation, we study the long-term electricity infrastructure investment planning problems in the electrical power system. These long-term capacity expansion planning problems aim at making the most effective and efficient investment decisions on both thermal and wind power generation units. One of our research focuses are uncertainty modeling in these long-term decision-making problems in power systems, because power systems\u27 infrastructures require a large amount of investments, and need to stay in operation for a long time and accommodate many different scenarios in the future. The uncertainties we are addressing in this dissertation mainly include demands, electricity prices, investment and maintenance costs of power generation units. To address these future uncertainties in the decision-making process, this dissertation adopts two different optimization approaches: decision-dependent stochastic programming and adaptive robust optimization. In the decision-dependent stochastic programming approach, we consider the electricity prices and generation units\u27 investment and maintenance costs being endogenous uncertainties, and then design probability distribution functions of decision variables and input parameters based on well-established econometric theories, such as the discrete-choice theory and the economy-of-scale mechanism. In the adaptive robust optimization approach, we focus on finding the multistage adaptive robust solutions using affine policies while considering uncertain intervals of future demands. This dissertation mainly includes three research projects. The study of each project consists of two main parts, the formulation of its mathematical model and the development of solution algorithms for the model. This first problem concerns a large-scale investment problem on both thermal and wind power generation from an integrated angle without modeling all operational details. In this problem, we take a multistage decision-dependent stochastic programming approach while assuming uncertain electricity prices. We use a quasi-exact solution approach to solve this multistage stochastic nonlinear program. Numerical results show both computational efficient of the solutions approach and benefits of using our decision-dependent model over traditional stochastic programming models. The second problem concerns the long-term investment planning with detailed models of real-time operations. We also take a multistage decision-dependent stochastic programming approach to address endogenous uncertainties such as generation units\u27 investment and maintenance costs. However, the detailed modeling of operations makes the problem a bilevel optimization problem. We then transform it to a Mathematic Program with Equilibrium Constraints (MPEC) problem. We design an efficient algorithm based on Dantzig-Wolfe decomposition to solve this multistage stochastic MPEC problem. The last problem concerns a multistage adaptive investment planning problem while considering uncertain future demand at various locations. To solve this multi-level optimization problem, we take advantage of affine policies to transform it to a single-level optimization problem. Our numerical examples show the benefits of using this multistage adaptive robust planning model over both traditional stochastic programming and single-level robust optimization approaches. Based on numerical studies in the three projects, we conclude that our approaches provide effective and efficient modeling and computational tools for advanced power systems\u27 expansion planning