Branching strategies for mixed-integer programs containing logical constraints and decomposable structure

Decision-making optimisation problems can include discrete selections, e.g. selecting a route, arranging non-overlapping items or designing a network of items. Branch-and-bound (B&B), a widely applied divide-and-conquer framework, often solves such problems by considering a continuous approximation, e.g. replacing discrete variable domains by a continuous superset. Such approximations weaken the logical relations, e.g. for discrete variables corresponding to Boolean variables. Branching in B&B reintroduces logical relations by dividing the search space. This thesis studies designing B&B branching strategies, i.e. how to divide the search space, for optimisation problems that contain both a logical and a continuous structure. We begin our study with a large-scale, industrially-relevant optimisation problem where the objective consists of machine-learnt gradient-boosted trees (GBTs) and convex penalty functions. GBT functions contain if-then queries which introduces a logical structure to this problem. We propose decomposition-based rigorous bounding strategies and an iterative heuristic that can be embedded into a B&B algorithm. We approach branching with two strategies: a pseudocost initialisation and strong branching that target the structure of GBT and convex penalty aspects of the optimisation objective, respectively. Computational tests show that our B&B approach outperforms state-of-the-art solvers in deriving rigorous bounds on optimality. Our second project investigates how satisfiability modulo theories (SMT) derived unsatisfiable cores may be utilised in a B&B context. Unsatisfiable cores are subsets of constraints that explain an infeasible result. We study two-dimensional bin packing (2BP) and develop a B&B algorithm that branches on SMT unsatisfiable cores. We use the unsatisfiable cores to derive cuts that break 2BP symmetries. Computational results show that our B&B algorithm solves 20% more instances when compared with commercial solvers on the tested instances. Finally, we study convex generalized disjunctive programming (GDP), a framework that supports logical variables and operators. Convex GDP includes disjunctions of mathematical constraints, which motivate branching by partitioning the disjunctions. We investigate separation by branching, i.e. eliminating solutions that prevent rigorous bound improvement, and propose a greedy algorithm for building the branches. We propose three scoring methods for selecting the next branching disjunction. We also analyse how to leverage infeasibility to expedite the B&B search. Computational results show that our scoring methods can reduce the number of explored B&B nodes by an order of magnitude when compared with scoring methods proposed in literature. Our infeasibility analysis further reduces the number of explored nodes.Open Acces

Train scheduling with application to the UK rail network

Nowadays, transforming the railway industry for better performance and making the best usage of the current capacity are the key issues in many countries. Operational research methods and in particular scheduling techniques have a substantial potential to offer algorithmic solutions to improve railway operation and control. This thesis looks at train scheduling and rescheduling problems in a microscopic level with regard to the track topology. All of the timetable components are fixed and we aim to minimize delay by considering a tardiness objective function and only allowing changes to the order and to the starting times of trains on blocks. Various operational and safety constraints should be considered. We have achieved further developments in the field including generalizations to the existing models in order to obtain a generic model that includes important additional constraints. We make use of the analogy between the train scheduling problem and job shop scheduling problem. The model is customized to the UK railway network and signaling system. Introduced solution methods are inspired by the successful results of the shifting bottleneck to solve the job shop scheduling problems. Several solution methods such as mathematical programming and different variants of the shifting bottleneck are investigated. The proposed methods are implemented on a real-world case study based on London Bridge area in the South East of the UK. It is a dense network of interconnected lines and complicated with regard to stations and junctions structure. Computational experiments show the efficiency and limitations of the mathematical programming model and one variant of the proposed shifting bottleneck algorithms. This study also addresses train routing and rerouting problems in a mesoscopic level regarding relaxing some of the detailed constraints. The aim is to make the best usage of routing options in the network to minimize delay propagation. In addition to train routes, train entry times and orders on track segment are defined. Hence, the routing and scheduling decisions are combined in the solutions arising from this problem. Train routing and rerouting problems are formulated as modified job shop problems to include the main safety and operational constraints. Novel shifting bottleneck algorithms are provided to solve the problem. Computational results are reported on the same case study based on London Bridge area and the results show the efficiency of one variant of the developed shifting bottleneck algorithms in terms of solution quality and runtime

Train Timetable Design for Shared Railway Systems using a Linear Programming Approach to Approximate Dynamic Programming

In the last 15 years, the use of rail infrastructure by different train operating companies (shared railway system) has been proposed as a way to improve infrastructure utilization and to increase efficiency in the railway industry. Shared use requires coordination between the infrastructure manager and multiple train operators in a competitive framework, so that regulators must design appropriate capacity pricing and allocation mechanisms. However, the resulting capacity utilization from a given mechanism in the railway industry cannot be known in the absence of operations. Therefore assessment of capacity requires the determination of the train timetable, which eliminates any potential conflicts in bids from the operators. Although there is a broad literature that proposes train timetabling methods for railway systems with single operators, there are few models for shared competitive railway systems. This paper proposes a train timetabling model for shared railway systems that explicitly considers network effects and the existence of multiple operators requesting to operate several types of trains traveling along different routes in the network. The model is formulated and solved both as a mixed integer linear programming (MILP) problem (using a commercial solver) and as a dynamic programming (DP) problem. We solve the DP formulation with a novel algorithm based on a linear programming (LP) approach to approximate dynamic programming (ADP) that can solve much larger problems than are computationally intractable with commercial MILP solvers. The model simulates the optimal decisions by an infrastructure manager for a shared railway system with respect to a given objective function and safety constraints. This model can be used to evaluate alternative capacity pricing and allocation mechanism. We demonstrate the method for one possible capacity pricing and allocation mechanism, and show how the competing demands and the decisions of the infrastructure manager under this mechanism impact the operations on a shared railway system for all stakeholders

Chance Constrained Mixed Integer Program: Bilinear and Linear Formulations, and Benders Decomposition

In this paper, we study chance constrained mixed integer program with consideration of recourse decisions and their incurred cost, developed on a finite discrete scenario set. Through studying a non-traditional bilinear mixed integer formulation, we derive its linear counterparts and show that they could be stronger than existing linear formulations. We also develop a variant of Jensen's inequality that extends the one for stochastic program. To solve this challenging problem, we present a variant of Benders decomposition method in bilinear form, which actually provides an easy-to-use algorithm framework for further improvements, along with a few enhancement strategies based on structural properties or Jensen's inequality. Computational study shows that the presented Benders decomposition method, jointly with appropriate enhancement techniques, outperforms a commercial solver by an order of magnitude on solving chance constrained program or detecting its infeasibility

Modular Supply Network Optimization of Renewable Ammonia and Methanol Co-production

To reduce the use of fossil fuels and other carbonaceous fuels, renewable energy sources such as solar, wind, geothermal energy have been suggested to be promising alternative energy that guarantee sustainable and clean environment. However, the availability of renewable energy has been limited due to its dependence on weather and geographical location. This challenge is intended to be solved by the utilization of the renewable energy in the production of chemical energy carriers. Hydrogen has been proposed as a potential renewable energy carrier, however, its chemical instability and high liquefaction energy makes researchers seek for other alternative energy carriers. Ammonia and methanol can serve as promising alternative energy carriers due to their chemical stability at room temperature, low liquefaction energy, high energy value. The co-production of these high energy dense energy carriers offers economic and environmental advantages since their synthesis involve the direct utilization of CO2 and common unit operations. This problem report aims to review the optimization of the co-production of methanol and ammonia from renewable energy. Form this review, research challenges and opportunities are identified in the following areas: (i) optimization of methanol and ammonia co-production under renewable and demand uncertainty, (ii) impacts of the modular exponent on the feasibility of co-production of ammonia and methanol, and (iii) development of modern computational tools for systems-based analysis