Robust and stochastic approaches to network capacity design under demand uncertainty

This thesis considers the network capacity design problem with demand uncertainty using the stochastic, robust and distributionally robust stochastic optimization approaches (DRSO). Network modeling in itself has found wide areas of application in most fields of human endeavor. The network would normally consist of source (origin) and sink (destination) nodes connected by arcs that allow for flows of an entity from the origin to the destination nodes. In this thesis, a special type of the minimum cost flow problem is addressed, the multi-commodity network flow problem. Commodities are the flow types that are transported on a shared network. Offered demands are, for the most part, unknown or uncertain, hence a model that immune against this uncertainty becomes the focus as well as the practicability of such models in the industry. This problem falls under the two-stage optimization framework where a decision is delayed in time to adjust for the first decision earlier made. The first stage decision is called the "here and now", while the second stage traffic re-adjustment is the "wait and see" decision. In the literature, the decision-maker is often believed to know the shape of the uncertainty, hence we address this by considering a data-driven uncertainty set. The research also addressed the non-linearity of cost function despite the abundance of literature assuming linearity and models proposed for this. This thesis consist of four main chapters excluding the "Introduction" chapter and the "Approaches to Optimization under Uncertainty" chapter where the methodologies are reviewed. The first of these four, Chapter 3, proposes the two models for the Robust Network Capacity Expansion Problem (RNCEP) with cost non-linearity. These two are the RNCEP with fixed-charge cost and RNCEP with piecewise-linear cost. The next chapter, Chapter 4, compares the RNCEP models under two types of uncertainties in order to address the issue of usefulness in a real world setting. The resulting two robust models are also comapared with the stochastic optimization model with distribution mean. Chapter 5 re-examines the earlier problem using machine learning approaches to generate the two uncertainty sets while the last of these chapters, Chapter 6, investigates DRSO model to network capacity planning and proposes an efficient solution technique

Multi-stage stochastic and robust optimization for closed-loop supply chain design

This dissertation focuses on formulating and solving multi-stage decision problems in uncertain environments using stochastic programming and robust optimization approaches. These approaches are applied to the design of closed-loop supply chain (CLSC) networks, which integrate both traditional flow and the reverse flow of products. The uncertainties associated with this application include forward demands, the quantity and quality of used products to be collected, and the carbon tax rate. The design decisions include long-term facility configurations as well as short-term contracts for transportation capacities by various modes that differ according to their variable costs, fixed costs, and emission rates. This dissertation consists of three papers. The first paper develops a multi-stage stochastic program for a CLSC network design problem with demands and quality of return uncertainties. The second paper focuses on robust optimization; particularly, the question of whether an adjustable robust counterpart (ARC) produces less conservative solutions than the robust counterpart (RC). Using the results of the second paper, a three-stage hybrid robust/stochastic program is proposed in the third paper, in which an ARC is formulated for a mixed integer linear programming model of the CLSC network design problem. In the first paper, a multi-stage stochastic program is proposed for the CLSC network design problem where facility locations are decided in the first stage and in subsequent stages, the capacities of transportation of different modes are contracted under uncertainty about the amounts of new and return products to transport among facilities. We explore the impact of the uncertain quality of returned products as well as uncertain demands with dependencies between periods. We investigate the stability of the solution obtained from scenario trees of varying granularity using a moment matching method for demands and distribution approximation for the quality of returns. Multi-stage solutions are evaluated in out-of-sample tests using simulated historical data and also compared with two-stage model. We observe an instance of overfitting, in which a scenario tree including more outcomes at each stage produces a dramatically different solution that has slightly higher average cost, compared to the solution from a less granular tree, when evaluated against the underlying simulated historical data. We also show that when the scenarios include demand dependencies, the solution performs better in out-of-sample simulation. In the second paper, the ARC of an uncertain linear program extends the RC by allowing some decision variables to adjust to the realizations of some uncertain parameters. The ARC may produce a less conservative solution than the RC does but cases are known in which it does not. While the literature documents some examples of cost savings provided by adjustability (particularly affine adjustability), it is not straightforward to determine in advance whether they will materialize. We establish conditions under which adjustability may lower the optimal cost with a numerical condition that can be checked in small representative instances. The provided conditions include the presence of at least two binding constraints at optimality of the RC formulation, and an adjustable variable that appears in both constraints with implicit bounds from above and below provided by different extreme values in the uncertainty set. The third paper concerns a CLSC network that is subject to uncertainty in demands for both new and returned products. The model structure also accommodates uncertainty in the carbon tax rate. The proposed model combines probabilistic scenarios for the demands and return quantities with an uncertainty set for the carbon tax rate. We constructed a three-stage hybrid robust/stochastic program in which the first stage decisions are long-term facility configurations, the second stage concerns the plan for distributing new and collecting returned products after realization of demands and returns but before realization of the carbon tax rate, and the numbers of transportation units of various modes, as the third stage decisions, are adjustable to the realization of the carbon tax level. For computational tractability, we restrict the transportation capacities to be affine functions of the carbon tax rates. By utilizing our findings in the second paper, we found conditions under which the ARC produces a less conservative solution. To solve the affinely adjustable version, Benders cuts are generated using recent duality developments for robust linear programs. Computational results show that the ability to adjust transportation mode capacities can substitute for building additional facilities as a way to respond to carbon tax uncertainty. The number of opened facilities in ARC solutions are decreased under uncertainty in demands and returns. The results confirm the reduction of total expected cost in the worst case of the carbon tax rate by increasing utilization of transportation modes with higher capacity per unit and lower emission rate

Water Distribution Networks Optimization Considering Uncertainties in the Demand Nodes

The fluctuation in the consumption of treated water is a situation that distribution networks gradually face. In times of greater demand, this consumption tends to suffer unnecessary impacts due to the lack of water. The uncertainty that occurs in water consumption can be mathematically modeled by a finite set of scenarios generated by a normal distribution and attributed to the network design. This study presents an optimization model to minimize network installation and operation costs under uncertainties in water demands. A Mixed Integer Nonlinear Programming model is proposed, considering the water flow directions in the pipes as unknown. A deterministic approach is used to solve the problem in three steps: First, the problem is solved with a nominal value for each uncertain parameter. In the second stage, the problem is solved for all scenarios, with the independent variables of the scenario being fixed and obtained from the solution reached in the first stage, known as the deterministic solution. Finally, all scenarios are solved without fixing any variable values, in a stochastic approach. Two case studies were used to test the applicability of the model and global optimization techniques were used to solve the problem. The results show that the stochastic solution can lead to optimal solutions for robust and flexible water distribution networks, capable of working under different conditions, considering the uncertainties of node demand and variable pipe directions.The authors gratefully acknowledge the financial support from the National Council for Scientific and Technological Development (Brazil), process 309026/2022-9

On robust network coding subgraph construction under uncertainty

We consider the problem of network coding subgraph construction in networks where there is uncertainty about link loss rates. For a given set of scenarios specified by an uncertainty set of link loss rates, we provide a robust optimization-based formulation to construct a single subgraph that would work relatively well across all scenarios. We show that this problem is coNP-hard in general for both objectives: minimizing cost of subgraph construction and maximizing throughput given a cost constraint. To solve the problem tractably, we approximate the problem by introducing path constraints, which results in polynomial time-solvable solution in terms of the problem size. The simulation results show that the robust optimization solution is better and more stable than the deterministic solution in terms of worst-case performance. From these results, we compare the tractability of robust network design problems with different uncertain network components and different problem formulations

Online Predictive Optimization Framework for Stochastic Demand-Responsive Transit Services

This study develops an online predictive optimization framework for dynamically operating a transit service in an area of crowd movements. The proposed framework integrates demand prediction and supply optimization to periodically redesign the service routes based on recently observed demand. To predict demand for the service, we use Quantile Regression to estimate the marginal distribution of movement counts between each pair of serviced locations. The framework then combines these marginals into a joint demand distribution by constructing a Gaussian copula, which captures the structure of correlation between the marginals. For supply optimization, we devise a linear programming model, which simultaneously determines the route structure and the service frequency according to the predicted demand. Importantly, our framework both preserves the uncertainty structure of future demand and leverages this for robust route optimization, while keeping both components decoupled. We evaluate our framework using a real-world case study of autonomous mobility in a university campus in Denmark. The results show that our framework often obtains the ground truth optimal solution, and can outperform conventional methods for route optimization, which do not leverage full predictive distributions.Comment: 34 pages, 12 figures, 5 table

Theory and Applications of Robust Optimization

In this paper we survey the primary research, both theoretical and applied, in the area of Robust Optimization (RO). Our focus is on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying prominent theoretical results of RO, we also present some recent results linking RO to adaptable models for multi-stage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering.Comment: 50 page