Robust Optimization for a Maritime Inventory Routing Problem

A single‐product maritime inventory routing problem (MIRP) is studied in which the production and consumption rates vary over the planning horizon. The problem involves a heterogeneous fleet and multiple production and consumption ports with limited storage capacity. Two discrete time formulations are developed: an original model and a reformulated model that is a pure fixed charge network flow (FCNF) model with side constraints. Mixed integer sets arising from the decomposition of the formulations are identified. In particular, several lot‐sizing relaxations are derived for the formulations and used to establish valid inequalities to strengthen the proposed formulations. Until now, the derivation of models and valid inequalities for MIRPs has mainly been inspired by the developments in the routing community. Here, we have developed a new model leading to new valid inequalities for MIRPs obtained by generalizing valid inequalities from the recent lot‐sizing literature. Considering a set of instances based on real data, a computational study is conducted to test the formulations and the effectiveness of the valid inequalities. The FCNF formulation is generally much stronger than the original formulation. The developed valid inequalities reduce the integrality gap significantly for both formulations. By using a branch‐and‐bound scheme based on the strengthened FCNF formulation, most of our test instances are solved to optimality. © 2013 Wiley Periodicals, Inc. NETWORKS, Vol. 62(4), 297–314 2013

Collaborative robots are progressively employed to support human workers in assembly tasks or independently assigned to one station to complete assembly tasks in U-shaped assembly lines. This study develops two exact methods, based on constraint programming (CP) model and Combinatorial Benders’ decomposition algorithm (CBDA), to tackle the U-shaped assembly line balancing problem with collaborative robots. The CBDA is based on distinct decomposition strategies that address the master problem and the subproblem, formulated by a mixed integer linear programming (MILP) model and a CP model, which is designed to obtain a real objective value and generate cuts that refine the master problem. Additionally, MILP-based and CP-based local search techniques are proposed to improve the algorithm's performance. The computational study demonstrates that both the proposed methods, namely CP and CBDA, outperform the current state-of-the-art methods (the MILP model and simulated annealing algorithm). Furthermore, the CP outperforms the CBDA in terms of the number of optimal solutions, achieving the optimal solutions for 574 out of 1500 test instances, and might be regarded as the new state-of-the-art methodology.

: In many situations it is advantageous to solve the dual of a linear program rather than the primal. For example, an approach to solve two stage stochastic linear programs with recourse employs Bender's decomposition (Dantzig and Glynn 1990). Within this algorithm, cuts (necessary conditions for the second stage costs) are added sequentially to the master problem. In the primal version of the master problem, cuts appear as new rows. In each iteration of Bender's decomposition algorithm, when a new row is added, the pervious solution of the master problem gets infeasible. However, when solving the dual of the master problem, a cut appears as a column. When adding a new column, the previous solution of the master problem remains feasible and the new optimal solution can be obtained within a few simplex iterations. This report describes the changes made by the author in the MINOS file MI35INPT, so MINOS could transform the given primal linear program, specified in an MPS file, to its dual, solve the dual and in addition, write another MPS file which contains the specification of the dual problem.

We present two decomposition algorithms for single product deep-sea maritime inventory routing problems (MIRPs) that possess a core substructure common in many real-world applications. The problem involves routing vessels, each belonging to a particular vessel class, between loading and discharging ports, each belonging to a particular region. Our algorithms iteratively solve a MIRP by zooming out and then zooming in on the problem. Specifically, in the “zoomed out” phase, we solve a first-stage master problem in which aggregate information about regions and vessel classes is used to route vessels between regions, while only implicitly considering inventory and capacity requirements, berth limits, and other side constraints. In the “zoomed in” phase, we solve a series of second-stage subproblems, one for each region, in which individual vessels are routed through each region and load and discharge quantities are determined. Computational experience shows that an integrated approach that combines these two algorithms is vastly superior to solving the problem directly with a commercial mixed-integer programming solver.

We study two different decomposition algorithms for the general (nonconvex) partially separable nonlinear program (PSP): bilevel decomposition algorithms (BDAs) and Schur interior-point methods (SIPMs). BDAs solve the problem by breaking it into a master problem and a set of independent subproblems, forming a type of bilevel program. SIPMs, on the other hand, apply an interior-point technique to solve the problem in its original (integrated) form, but then use a Schur complement approach to solve the Newton system in a decentralized manner. Our first contribution is to establish a theoretical relationship between these two types of decomposition algorithms. This is a first step toward closing the gap between the incipient local convergence theory of BDAs and the mature local convergence theory of interior-point methods. Our second contribution is to show how SIPMs can be modified to solve problems for which the Schur complement matrix is not invertible in general. The importance of this contribution is that it substantially enlarges the class of problems that can be addressed with SIPMs.

Multidisciplinary design optimization (MDO) problems are engineering design problems that require the consideration of the interaction between several design disciplines. Due to the organizational aspects of MDO problems, decomposition algorithms are often the only feasible solution approach. Decomposition algorithms reformulate the MDO problem as a set of independent subproblems, one per discipline, and a coordinating master problem. A popular approach to MDO problems is bilevel decomposition algorithms. These algorithms use nonlinear optimization techniques to solve both the master problem and the subproblems. In this paper, we propose two new bilevel decomposition algorithms and analyze their properties. In particular, we show that the proposed problem formulations are mathematically equivalent to the original problem and that the proposed algorithms converge locally at a superlinear rate. Our computational experiments illustrate the numerical performance of the algorithms.

Two-stage recoverable robust optimization for an integrated location–allocation and evacuation planning problem

The economic and low‐carbon operation strategy of multi‐energy microgrids (MEM) has become an important research topic in smart grids. The operation of MEM is affected by uncertain factors from renewable energy and internal load. To handle uncertainties from both source and load sides, the authors propose a novel worst‐case‐and‐probability uncertainty sets and a novel worst‐expectation min‐max‐max‐min two‐stage four‐level robust optimisation (RO) model considering stepped carbon trading for MEM. This four‐level RO model is a more generalised model compared with the traditional two‐stage RO and distributionally robust optimisation based models. First, the unsolvable original four‐level model is decoupled into a master problem and a sub‐problem (SP). Second, Karush–Kuhn–Tucker condition is applied to convert SP into a solvable problem. Third, column and constraint generation (C&CG) algorithm is employed to solve the reformulated four‐level RO. Finally, the effectiveness of the proposed model and solution method is verified by case studies. The results indicate that the convergence behaviour of this solution method is excellent. The MEM operator can select appropriate robustness factors and comprehensive norms to control its conservatism level. Besides, MEM could reduce carbon emissions by participating in carbon trading and installing carbon capture devices.

We consider an inventory routing problem (IRP) in the liquefied natural gas (LNG) supply chain, called the LNG-IRP. Here, an actor is responsible for the LNG production and inventory management at the liquefaction plants, the routing and scheduling of a heterogeneous fleet of LNG ships, as well as the inventories and sales at the regasification terminals. Furthermore, all ports have a limited number of berths available for loading and unloading. The LNG-IRP is more complicated than many other maritime inventory routing problems because a constant rate of the cargo evaporates in the tanks each day and is used as fuel during transportation. In addition, a variable number of tanks are unloaded at the regasification terminals. We introduce a new path flow formulation for this problem arising from a novel decomposition scheme based on parts of a ship schedule, called duties. A ship schedule for the entire planning horizon can be divided into duties consisting of a visit to a liquefaction plant, then one or two visits to a regasification terminal before ending in a liquefaction plant. The solution method suggested is based on a priori generation of duties, and the formulation is strengthened by valid inequalities. The same problem was previously solved by a branch-price-and-cut algorithm for a schedule-based formulation. Computational results show that the new formulation provides tighter bounds than the previous schedule-based formulation. Furthermore, on a set of 27 benchmark instances, the proposed algorithm clearly outperforms the previous branch-price-and-cut algorithm both with regard to computational time and the number of problems solved within a 10-h time limit.

The choice of a suitable time representation plays a major role in designing a computationally efficient planning/scheduling framework. For the time representation to be effective, it needs to capture the nuances of the actual process, while ensuring that the resultant model is tractable and computationally efficient. In this paper, a novel time representation scheme, titled hybrid time discretization, which combines both continuous- and discrete-time representations in a single framework, is proposed and used to solve a complex multiproduct maritime inventory routing problem (MIRP) with undedicated compartments. Several continuous- and discrete-time-based frameworks have been proposed to solve the MIRP over the years. While single-/multigrid discretized frameworks do efficiently capture the time-sensitive nuances of the problem (such as active/inactive jetty hours and hourly/daily variations in the demands), they struggle to solve larger problem instances to the required optimality gap in a reasonable time. This is primarily due to the large disparities among the timings of different task components in the model. On the other hand, continuous-time representation can significantly reduce the problem size and computational effort; however, modeling the entire problem using the continuous-time-based framework could have problems in representing some time-sensitive intricacies in the demands and jetty/port activities. To allay the above-mentioned drawbacks, in this paper, a novel hybrid time representation is proposed, which combines a continuous-time representation with a multigrid discrete-time framework. The proposed novel framework is cast in a rolling horizon strategy and tested on various scenarios generated for the above-mentioned maritime transportation problem, which are shown to be significantly more complex than those presented in the literature. The proposed MILP framework is able to solve most of the test cases to the required optimality in a reasonable time without the need for heuristic or meta-heuristic assists. The proposed hybrid time representation could find significant utility in other lateral applications such as batch scheduling and inventory routing problems.

Adaptive and robust radiation therapy optimization for lung cancer

Maritime container supply chains (MCSCs) is exposed to various risks arising from both internal operations and the external environment, and the increasing complexity of the modern global logistics system makes the situation even worse, thus causing a significant challenge to the effective risk management of MCSCs. However, systematic studies on this topic are relatively few. In view of this, this study aims to explore and analyse various MCSC risks, develop suitable risk assessment methods, and evaluate the overall performance of MCSCs from a systematic perspective, so as to ensure the safety, reliability, and resilience of MCSCs. This research starts with the identification and classification of all possible risk factors that may be involved in an MCSC based on a comprehensive literature review, and the research results are further validated through a Delphi expert survey. The identified risk factors are then analysed, screened, and assessed in detail. The novelty of this study lies not only on the risk assessment of MCSCs under an uncertain environment from a supply chain level but also on the consideration of the impact of risk condition of each individual MCSC on the overall performance of the entire container supply network. The research results will provide useful insights and valuable information for both researchers and practitioners on the risk analysis and assessment of MCSCs, which is beneficial to different types of stakeholders involved in the maritime shipping industry. The work is also able to provide a theoretical foundation for risk-based decision making and shipping route optimisation in further work. Although the risk assessment methods are presented on the basis of the specific context in MCSCs, it is believed that, with domain-specific knowledge and data, they can also be tailored for a wide range of applications to evaluate the reliability and performance of other supply chain systems, especially where a high level of uncertainty is involved.

Almost Robust Discrete Optimization

Polygeneration, a multi-input multioutput energy conversion process which typically involves the coproduction of electricity and liquid synthetic fuels, is a promising technology which offers real potential toward the reduction of excessive energy consumption and consequent greenhouse gas emissions. The optimal design of such a complex and nonlinear process system under inevitable and unpredictable future uncertainty poses great challenges in terms of both modeling and corresponding solution strategies. In this paper, we propose a stochastic programming framework for the optimal design under uncertainty of polygeneration energy systems. On the basis of a detailed mixed-integer nonlinear programming (MINLP) model, proposed in our previous work, a two-stage stochastic programming problem is formulated, which is then converted into a large-scale multiperiod MINLP problem by employing cubature based integration and sampling techniques. A decomposition algorithm is utilized for the efficient solution of the multiperiod problem, which involves iterations between a set of nonlinear subproblems of much smaller size and a master mixed-integer linear programming problem. A case study is then presented, where detailed computational results and comparisons between optimal designs obtained for both the stochastic and deterministic cases are shown.

With the change of energy structure and the development of energy internet technology, the penetration rate of distributed generation is gradually increasing, and the location of distributed generation has an important impact on the distribution network line losses. In this paper, a mixed integer quadratic programming model with separability is considered to optimize the location of distributed generation and to reduce line losses in distribution networks, which measures line losses in distribution networks in terms of the quadratic term of current. In order to solve the problem of large-scale distributed power facility location problems with separability, a Benders decomposition algorithm incorporating matheuristics is proposed, which first decomposes the original problem into a mixed-integer linear programming master problem and a convex quadratic subproblem with separability, and then separates the generalized Benders cuts form solving the subproblem at the root node relaxed solution of the master problem. A simple in-out Stabilization technique is applied in order to efficiently raise the lower bound of the master problem at the root node. In order to avoid the use of branch-and-cut algorithms to solve the master problem to obtain optimal feasible solutions, two matheuristic repair strategies are proposed for the root node relaxation solution to obtain high-quality feasible solutions. Experimental results on 90 large-scale power facility location problems show that matheuristic repair strategies for the root node relaxation solution can find the global optimal solution.