Fleet deployment and demand fulfillment for container shipping liners

This article collectively addresses the issues of optimizing the sailing speed of containerships and fleet deployment in intercontinental container liner shipping, taking cargo time values into consideration. A mixed-integer nonlinear programming model is developed to maximize the gross profits of a container shipping line. The sailing speed is divided into 0.1 knot intervals and a reciprocal-discretization method is proposed to transform the model into an integer-based linear programming model, which can be solved using optimization solvers such as CPLEX. A case study is conducted based on the Far East-West America liner route of a global container shipping line to assess the effectiveness of the proposed model and the reciprocal-discretization method. The results show that when the cargo time value is taken into consideration, the increased sailing speed of containerships on long legs and the decreased number of deployed containerships will generate greater operating profits for the container shipping line. The rise in bunker price will lead to a stepwise decline in sailing speed of containerships and a stepwise growth in the number of deployed containerships. Moreover, the downward trend of sailing speed due to the increase of bunker price will be somewhat delayed as the unit container cargo value increases. Several useful insights are drawn from analysis of the results.

Reformulation, linearization, and a hybrid iterated local search algorithm for economic lot-sizing and sequencing in hybrid flow shop problems

A new, R-graph based, superstructure and corresponding MINLP model for designing conventional distillation columns are presented. A GDP representation (GR) of the superstructure is first constructed, then it is transformed to MINLP representation to which, in turn, additional trivial improvements are added. The new model has been tested on binary mixture examples, and the obtained results are compared to the results of an MINLP model which developed according to the GDP model of Yeomans and Grossmann. 9 The new model yields shorter computation time and provides better local optima. Additionally, the new model has been used for optimizing a complex multicomponent separation system consisting of several distillation columns. In order to handle such a complex system with a huge number of nonlinear equations, the outer approximation algorithm is modified to provide good initial values for the NLP subproblems. Distillation is one of the most widespread processes applied for separating multicomponent liquid mixtures. It is used for working up large volume or stream, and it requires high investment and operation outlays. The significance of the design of economically optimal distillation processes is of no question. Enormous interest has also been addressed to the area of designing optimal heat integrated distillation columns and distillation sequences. Minimizing the cost of a distillation process implies finding the optimal configuration of each individual column, as well. In the present article we consider staged column models only. The number of stages, and the stage numbers of the feed and side-stream points, are discrete decision variables. The total cost of a column may be modeled as the sum of the fix (capital) cost, depending on the number of stages and on the column diameter, and the variable (operation) cost related to the utility consumption. The objective of the design procedure is to find the optimal configuration of the column, which has the minimum total (annualized) cost. In order to model these processes, discrete decisions are required for calculating the number of stages. Optimizing single columns is a well-known procedure; all the basic textbooks outline how to do it in an easy manner. 1 After approximately determining the minimum and the estimated optimal number of theoretical stages, optimizing over the continuous variables is performed at varied values of the discrete variables. This is a 2-dimension discrete array of continuous optimization tasks in the case of a single-feed, two-product column, because there are merely two column sections in this case. This task becomes much more difficult if several feeds and side-products are to be taken into account. The real problem, however, is synthesizing a distillation sequence, or a system of advanced distillation columns, or a complex flowsheet containing distillation units when the number of distillation columns and their connections are not known in advance. In order to solve the complex synthesis problem outlined above, a proper superstructure for a single column, together with a proper generalized disjunctive programming (GDP) model or a proper mixed-integer nonlinear programming (MINLP) model, is a minimum requirement. Once such a model works well for a single column, the problem of more complex flowsheets may also be addressed. Mixed-integer nonlinear programming (MINLP) and generalized disjunctive programming (GDP) are the two exact methodologies commonly applied for solving process engineering problems with discrete decisions. The former includes algebraic equations describing the process, and binary variables related to discrete decisions. The latter method uses logic variables and expressions to represent the problem. 2 Both formulations have been successfully applied to rigorous models of distillation columns. Both methods apply a fixed maximum number of stages, and the actually used stages are selected from this set. Mathematical formulations that represent rigorous models of distillation column configurations fall into two categories: (i) one task-one equipment (OTOE) representations and (ii) variable task-equipment (VTE) representations. 3

A master–slave solving method with adaptive model reformulation technique for water network synthesis using MINLP

Optimal Antibody Purification Strategies Using Data-Driven Models

Microgrids have become valuable assets because they improve the reliability of consumers while integrating renewables via distributed energy resources (DERs). Thus, making them cost-efficient is essential to secure their proliferation. This paper proposes a new method for the optimal design of microgrids. The proposed two-stage method optimizes the size and the location of the DERs, i.e., the renewable energy sources (RESs), distributed generation (DG) units, and battery energy storage systems (BESSs). Furthermore, the overall operation of the microgrid is optimized using a stochastic scenario-based approach, considering grid-connected and unintentional islanded modes. The proposed method also considers internal network reinforcements. Thus, the first stage is an energy-based approach, formulated as a mixed-integer linear programming (MILP) problem, and it is used to size the DERs, whereas the second stage uses an optimal AC power flow (AC-OPF) to formulate a mixed-integer nonlinear programming (MINLP) model that allocates the DERs and selects the best conductor for each circuit. The multi-objective nature of the problem is addressed via Pareto optimization to analyze the trade-off between operational and capital costs. The MINLP model is linearized through piece-wise approximations and solved using commercial solvers. Furthermore, the impact of battery degradation is analyzed through a simple adaptation of the Stage 1 model. Results were obtained with data from the real university campus microgrid CampusGrid, located at the State University of Campinas (UNICAMP), in São Paulo, Brazil.

Effective liner shipping is important for the global seaborne trade. The volume of cargoes transported by liner shipping has been increasing over the past decades. Liner shipping companies face three levels of decision problems, including strategic, tactical, and operational problems. The tactical-level decisions are commonly made every three to 6 months. These decisions include: (1) port service frequency determination; (2) fleet deployment; (3) sailing speed optimization; and (4) vessel schedule design. Most of the concurrent liner shipping studies have addressed the tactical-level decision problems separately. Even though a few studies have proposed joint planning models that capture multiple decision problems at the same time, none of the conducted studies has integrated all the four tactical-level decision problems. To address this gap in the state-of-the-art, this study presents a holistic optimization model that addresses all the tactical-level liner shipping decision problems, aiming to maximize the total profit obtained from liner shipping services. The key route service cost components, found in the liner shipping literature, are considered in this study, which include: (1) vessel operational cost; (2) vessel chartering cost; (3) port handling cost; (4) port late arrival cost; (5) fuel consumption cost; (6) container inventory costs in sea and at ports of call; and (7) emission costs in sea and at ports of call. An exact optimization approach is adopted for the developed mathematical model. The computational experiments, conducted for a set of Asia-North America liner shipping routes, showcase the efficiency of the proposed approach and offer some important managerial insights.

This paper presents a new mixed-integer nonlinear programming (MINLP) model for the optimal operation of unbalanced three-phase droop-based microgrids. The proposed MINLP model can be seen as an extension of an optimal power flow for microgrids operating in islanded mode, that aims to minimize the total amount of unsupplied demand and the total distributed generator (DG) generation cost. Since the slack bus concept is not longer valid, the proposed model considers the frequency and voltage magnitude reference as variables. In this case, DGs units operate with droop control to balance the system and provide a frequency and voltage magnitude reference. Additionally, a set of efficient linearizations are introduced in order to approximate the original MINLP problem into a mixed-integer linear programming (MILP) model that can be solved using commercial solvers. The proposed model has been tested in a 25-bus unbalanced three-phase microgrid and a large 124-node grid, considering different operational and time-coupling constraints for the DGs and the battery systems (BSs). Load curtailment and different modes of operation for the wind turbines have also been tested. Finally, an error assessment between the original MINLP and the approximated MILP model has been conducted.

In this paper, we develop a mathematical model of a technician routeing and scheduling problem with time windows (TRSPTW) and overnight shifts, which we title the ‘overnight TRSPTW’. This problem is motivated by a real application in the telecommunications industry in Saskatchewan, Canada. A mixed-integer nonlinear programming (MINLP) model is employed to achieve two key objectives: (1) minimising the total costs associated with technicians and subcontractors, including travel costs, accommodation costs and penalty costs caused by late starts and (2) minimising imbalanced workloads. Furthermore, the present work aims to determine the daily assignment of technicians to communities, depots and routes; find a high-quality schedule for technicians’ start times in communities and lunch breaks and determine the daily assignment of tasks to the subcontractor. The bi-objective MINLP model used to solve the overnight TRSPTW is a typical NP-hard problem in combinatorial optimisation. As such, we introduce a new hybrid category of TRSPTW that combines centroid-based clustering, which is an unsupervised machine learning (ML) approach, with a bi-objective zebra optimisation algorithm (BOZOA). The resultant algorithm blends the advantages of the ZOA and ML to strike a balance between the exploration and exploitation of the solution region. Finally, we compare our results with those obtained using an exact solver for small-, medium-, and large-sized instances. The performance evaluation and validation results revealed that the proposed ML-based BOZOA provides very good performance in solving TRSPTWs at a variety of scales with respect to the optimality criteria, including, number of taken iterations, infeasibility, optimality error and complementarity compared with both an exact solver and two inspired algorithms from ZOA. Highlights An ML-based bi-objective zebra optimisation algorithm to treat large-scale TRSPs Centroid-based clustering on the population of zebras to avoid bias towards a specific search space Making a trade-off between exploration and exploitation of the feasible region in the developed algorithm A new MINLP model of a weighted bi-objective TRSP with limited capacity depots Workload function, penalty function for lateness, subcontracts, time windows for tasks and breaks Experiments using real data to show the performance of the model and solution method

An exact MINLP model for optimal location and sizing of DGs in distribution networks: A general algebraic modeling system approach

This research studies the cellular manufacturing system (CMS) controlled by kanban mechanism which defective items are produced in any production run of each product and rework is carried out to transform them into serviceable items. We consider and compare two different policies for rework where in the first policy rework is completed within the same production cycle and in the second policy rework done after N production cycles. Recently Aghajani et al. (2012) explain policy 2 and proposed a mixed-integer nonlinear programming (MINLP) model for this policy. In order to minimize total cost, MINLP model was developed for policy 1 to find simultaneously the optimal number of kanban, batch size, and number of batches. The cost function includes the cost of setup, holding, and transportation. Due to the high combinatorial structure of the problem, particle swarm optimization (PSO), and simulated annealing (SA) algorithms as meta-heuristic methods are proposed to solve the problem and numerical experiments are conducted to demonstrate the efficiency of the proposed algorithms. It is shown that both PSO and SA result are in a near optimal solution but the PSO algorithm gives a better performance than the SA method. Also, sensitivity analysis is carried out to study the effect of defective rate, holding cost, and setup cost variations on the total system cost is discussed the performance of these policies in different conditions.

ABSTRACTIn this paper, a multiobjective mixed-integer piecewise nonlinear programming model (MOMIPNLP) is built to formulate the management problem of urban mining system, where the decision variables are associated with buy-back pricing, choices of sites, transportation planning, and adjustment of production capacity. Different from the existing approaches, the social negative effect, generated from structural optimization of the recycling system, is minimized in our model, as well as the total recycling profit and utility from environmental improvement are jointly maximized. For solving the problem, the MOMIPNLP model is first transformed into an ordinary mixed-integer nonlinear programming model by variable substitution such that the piecewise feature of the model is removed. Then, based on technique of orthogonal design, a hybrid heuristic algorithm is developed to find an approximate Pareto-optimal solution, where genetic algorithm is used to optimize the structure of search neighborhood, and both local branching algorithm and relaxation-induced neighborhood search algorithm are employed to cut the searching branches and reduce the number of variables in each branch. Numerical experiments indicate that this algorithm spends less CPU (central processing unit) time in solving large-scale regional urban mining management problems, especially in comparison with the similar ones available in literature. By case study and sensitivity analysis, a number of practical managerial implications are revealed from the model.Implications: Since the metal stocks in society are reliable overground mineral sources, urban mining has been paid great attention as emerging strategic resources in an era of resource shortage. By mathematical modeling and development of efficient algorithms, this paper provides decision makers with useful suggestions on the optimal design of recycling system in urban mining. For example, this paper can answer how to encourage enterprises to join the recycling activities by government’s support and subsidies, whether the existing recycling system can meet the developmental requirements or not, and what is a reasonable adjustment of production capacity.

• Evacuation network design with road capacity improvement and uncertainty is stated. • A non-convex mixed-integer nonlinear program model is established. • Second-order cone programming reformulations are developed. • Benders decomposition with various acceleration strategies is devised. • Efficiency of proposed methods and comparison of existing method are explored. This work first presents a stochastic shelter location and evacuation planning problem with considering road capacity improvement strategies, in which the fixed setup cost of shelters and the improvement cost of road capacity are subject to a budget limit. To explicitly capture the impact of traffic volumes and road capacity improvement decisions on evacuation time, the Bureau of Public Roads function is employed. The problem is formulated as a non-convex mixed-integer nonlinear program (MINLP) model that is difficult to solve directly since the objective function is a multivariable non-convex nonlinear function. To tackle the non-convex MINLP, second-order cone programming (SOCP) reformulations that can be directly solved by using the state-of-the-art solvers are developed. Furthermore, a Benders decomposition (BD) approach that utilizes duality results of SOCP and employs acceleration strategies associated with valid inequalities, multi-cut, strengthened Benders cuts, knapsack inequalities, and callback routine, is proposed to solve large-scale problems. Moreover, extensive numerical experiments and a real-world case study (a potential hurricane risk zone in Texas, U.S.) are conducted to verify the applicability and effectiveness of the proposed model and solution approaches. Computational results show that the derived reformulations are competitive in dealing with small- and medium-scale problems, whereas BD approach demonstrates the best computational performance in solving large-scale problems. The devised acceleration strategies are effective in improving the computational efficiency of the BD approach. In addition, exerting investment for those shelters and arcs that are close to evacuation regions is useful to reduce the expected total evacuation time.

An Economic Analysis of the Future U.S. Biofuel Industry, Facility Location, and Supply Chain Network

In major cities, with the rapid urbanization and population growth, sustainable development has become a pressing global concern, leading to increasing traffic congestion, energy consumption, and environmental pollution. The transportation sector, responsible for nearly a quarter of global energy-related emissions, plays a vital role in addressing these issues. In response, numerous international frameworks have emphasized the need for cleaner, more efficient mobility solutions. In this context, shared mobility has emerged as a promising solution, bridging private car ownership and public transit to reduce emissions and promote efficiency. This thesis investigates two representative forms of shared mobility: electric vehicle carsharing (EVCS) and collaborative passenger-parcel ridesharing (CPPR), with a focus on operational optimization to support sustainable and economically viable urban mobility. The first two works in this thesis examine EVCS. The first study addresses station-based EVCS in a competitive market. A Generalized Nash Equilibrium (GNE) framework is formulated, where companies compete for limited infrastructure and user demand while jointly optimizing facility deployment, vehicle allocation, relocation, and vehicle-trip assignment. In addition, the study incorporates the role of government by proposing a bi-level model: the upper level represents the government's objective to maximize social welfare through infrastructure planning, while the lower level captures operator competition via the GNE framework. A surrogate optimization approach is adopted to solve the bi-level model, with an ADMM-based algorithm embedded at the lower level for efficient GNE computation. To further enhance carsharing flexibility, the second study focuses on free-floating EV carsharing systems with battery swapping (FEVCS-BS). Unlike the station-based mode, free-floating allows users to pick up and return vehicles at any location within the service area, offering greater convenience. However, this flexibility introduces operational challenges. The absence of fixed returning charging stations and the limited driving range of EVs makes vehicle relocation unavoidable for both vehicle recharging and rebalancing. To address this, two relocation strategies are considered: user-based relocation, driven by pricing incentives, and staff-based relocation, carried out by operational personnel. A mixed-integer nonlinear programming (MINLP) model is developed to capture users’ price-incentivized behavior and to jointly optimize hybrid relocation, fleet size, vehicle allocation, and trip assignment for profit maximization. An outer-inner approximation method with range reduction is applied to solve the nonconvex model and obtain a global optimum. Building on this foundation, the next two studies shift focus to another key form of shared mobility—ridesharing, where individual travelers with similar routes share a vehicle and split travel costs. However, conventional ridesharing often overlooks unused trunk space, missing the opportunity to fully utilize vehicle capacity. To address this inefficiency, we explore an innovative mode: collaborative passenger-parcel ridesharing (CPPR), which enables the joint transportation of passengers and parcels in the same vehicle. The third study explores the routing problem in CPPR while incorporating compensation affected passenger behavior. In this innovative mode, some passengers may be hesitant to share rides with parcels due to perceived inconvenience. Therefore, an effective compensation mechanism is essential to encourage participation. This study embeds compensation-related behavior into a routing framework and formulates the problem as a Compensation Scheme Design in a Split Delivery Vehicle Routing Problem with Time Windows for CPPR (C-SDVRPTW-CPPR). To solve it efficiently, a Surrogate-based Adaptive Large Neighborhood Search (SOT-ALNS) algorithm is developed to generate near-optimal routing plans. The fourth study extends the CRRP problem into a dynamic demand setting, where travel and delivery requests fluctuate throughout the day. Such an environment requires real-time decision-making while also ensuring long-term profitability. To tackle this, we formulate the Dynamic Vehicle Routing Problem with Time Windows for CPPR (DVRPTW-CPPR) and propose a Deep Reinforcement Learning (DRL) framework with a dynamic encoder-decoder architecture to learn optimal routing policies. In conclusion, this thesis explores the different shared mobility modes, including carsharing and ridesharing, to address critical challenges in operations. Through the proposed models and solution methods, operators can effectively manage operations, thereby supporting sustainability goals, and ensuring profitability of shared mobility services.