Model and algorithms for multi-period sea cargo mix problem

In this paper, we consider the sea cargo mix problem in international ocean container shipping industry. We describe the characteristics of the cargo mix problem for the carrier in a multi-period planning horizon, and formulate it as a multi-dimensional multiple knapsack problem (MDMKP). In particular, the MDMKP is an optimization model that maximizes the total profit generated by all freight bookings accepted in a multi-period planning horizon subject to the limited shipping capacities. We propose two heuristic algorithms that can solve large scale problems with tens of thousands of decision variables in a short time. Finally, numerical experiments on a wide range of randomly generated problem instances are conducted to demonstrate the efficiency of the algorithms.     

Supply chain scheduling with receiving deadlines and non-linear penalty

We study the operations scheduling problem with delivery deadlines in a three-stage supply chain process consisting of (1) heterogeneous suppliers, (2) capacitated processing centres (PCs), and (3) a network of business customers. The suppliers make and ship semi-finished products to the PCs where products are finalized and packaged before they are shipped to customers. Each business customer has an order quantity to fulfil and a specified delivery date, and the customer network has a required service level so that if the total quantity delivered to the network falls below a given targeted fill rate, a non-linear penalty will apply. Since the PCs are capacitated and both shipping and production operations are non-instantaneous, not all the customer orders may be fulfilled on time. The optimization problem is therefore to select a subset of customers whose orders can be fulfilled on time and a subset of suppliers to ensure the supplies to minimize the total cost, which includes processing cost, shipping cost, cost of unfilled orders (if any), and a non-linear penalty if the target service level is not met. The general version of this problem is difficult because of its combinatorial nature. In this paper, we solve a special case of this problem when the number of PCs equals one, and develop a dynamic programming-based algorithm that identifies the optimal subset of customer orders to be fulfilled under each given utilization level of the PC capacity. We then construct a cost function of a recursive form, and prove that the resulting search algorithm always converges to the optimal solution within pseudo-polynomial time. Two numerical examples are presented to test the computational performance of the proposed algorithm.     

Impact analysis of maritime cabotage legislations on liner hub-and-spoke shipping network design

Maritime cabotage is a legislation published by a particular coastal country, which is used to conduct the cargo transportation between its two domestic ports. This paper proposes a two-phase mathematical programming model to formulate the liner hub-and-spoke shipping network design problem subject to the maritime cabotage legislations, i.e., the hub location and feeder allocation problem for phase I and the ship route design with ship fleet deployment problem for phase II. The problem in phase I is formulated as a mixed-integer linear programming model. By developing a hub port expanding technique, the problem in phase II is formulated as a vehicle routing problem with pickup and delivery. A Lagrangian relaxation based solution method is proposed to solve it. Numerical implementations based on the Asia–Europe–Oceania shipping services are carried out to account for the impact analysis of the maritime cabotage legislations on liner hub-and-spoke shipping network design problem.     

Routing and scheduling in project shipping

Tramp shipping companies are committed to transport a set of contracted cargoes and try to derive additional revenue from carrying optional spot cargoes. Here, we present a real life ship routing and scheduling problem for a shipping company operating in project shipping, a special segment of tramp shipping. This segment differs from more traditional tramp segments, as the cargoes are usually transported on a one-time basis. Because of the special nature of the cargoes, complicating requirements regarding stowage onboard the ships and cargo coupling must be considered while determining routes and schedules for the ships in the fleet. A mathematical model is presented and a tabu search heuristic is proposed to solve the problem. Computational results show that the tabu search heuristic provides optimal or near-optimal solutions in a reasonable amount of time, and that it can give significant improvements to manual planning for the shipping company.     

Designing robust liner shipping schedules: Optimizing recovery actions and buffer times

It is important for liner shipping companies to maintain cost efficient and robust liner shipping networks. Regularly, they set up pro-forma schedules, yet it is difficult to stay on time. We consider the problem of managing the delays. Therefore, we need to determine an optimal recovery policy and buffer time allocation to the ship route in order to minimize the total costs associated with delays and recovery actions, such as increasing sailing speed. We introduce a general framework consisting of a mixed integer programming formulation to solve discrete stochastic decision problems with short and long term decisions and apply this framework to the above described problem. Furthermore, we propose and test four heuristics for this problem. We compared the results of our method with an existing liner shipping route schedule and found a cost decrease of 28.9% after optimizing the buffer time distribution compared to the cost of sailing the current route schedule at constant speed.     

Berth scheduling for container terminals by using a sub-gradient optimization technique

A method for establishing a berth schedule consisting of berthing times and berthing positions of containerships in port container terminals is addressed in this paper. Each vessel requires a specific amount of space on the berth during a predetermined length of time for unloading and loading containers. The berth schedule must be constructed in a way to satisfy requests from carriers on berthing times and minimize handling efforts during ship operation. A mixed integer program is formulated for the berth-scheduling problem, which can be solved using a commercial package. In order to overcome the computational load of the mixed integer program, the formulation is converted into another integer linear program in which the solution space of the berth and the time is discretized. A Lagrangean relaxation model of the discretized model is solved using a sub-gradient optimization technique. Results of a numerical experiment are provided and discussed.     

Time-dependent optimization of a multi-item uncertain supply chain network: A hybrid approximation algorithm

We consider the uncertain least cost shipping problem. The input is a multi-item supply chain network with time-evolving uncertain costs and capacities. Exploiting the operational law of uncertainty theory, a mathematical model of the problem is established and the indeterminacy factors are tackled. We use the scaling idea together with transformation approach and uncertainty programming to develop a hybrid algorithm to optimize and obtain the uncertainty distribution of the total shipping cost. We analyze the practical performance of the algorithm and present an illustrative example.     

Matheuristic for the decentralized factories scheduling problem

In this study, a novel mixed integer linear programming (MILP) model is developed for the decentralized factories scheduling problem, where a set of transporters is used for shipping goods among parallel factories to minimize the production costs over all of the factories. Due to its typical features, such as multiple heterogeneous factories and transportation times, this problem is extremely difficult to solve, especially for large-scale problems. In order to address this difficulty, the main aim of this study is to develop a new solution algorithm based on the interoperation of metaheuristics and mathematical programming techniques to minimize the production costs for jobs. The algorithm comprises an electromagnetism-like algorithm and variable neighborhood search. In this hybridization based on MILP relaxation, the guiding principle involves the ordering of neighborhood structures. The results obtained by the proposed algorithm and MILP are analyzed and compared for various test problems.     

Distribution-free vessel deployment for liner shipping

One important problem faced by the liner shipping industry is the fleet deployment problem. In this problem, the number and type of vessels to be assigned to the various shipping routes need to be determined, in such a way that profit is maximized, while at the same time ensuring that (most of the time) sufficient vessel capacity exists to meet shipping demand. Thus far, the standard assumption has been that complete probability distributions can be readily specified to model the uncertainty in shipping demand. In this paper, it is argued that such distributions are hard, if not impossible, to obtain in practice. To relax this oftentimes restrictive assumption, a new distribution-free optimization model is proposed that only requires the specification of the mean, standard deviation and an upper bound on the shipping demand. The proposed model possesses a number of attractive properties: (1) It can be seen as a generalization of an existing variation of the liner fleet deployment model. (2) It remains a mixed integer linear program and (3) The model has a very intuitive interpretation. A numerical case study is provided to illustrate the model.     

The freight allocation problem with lane cost balancing constraint

We consider a problem faced by a buying office for one of the largest retail distributors in the world. The buying office plans the distribution of goods from Asia to various destinations across Europe. The goods are transported along shipping lanes by shipping companies, many of which have collaborated to form strategic alliances; each lane must be serviced by a minimum number of companies belonging to a minimum number of alliances. The task involves purchasing freight capacity from shipping companies for each lane based on projected demand, and subject to minimum quantity requirements for each selected shipping company, such that the total transportation cost is minimized. In addition, the allocation must not assign an overly high proportion of freight to the more expensive shipping companies servicing any particular lane, which we call the lane cost balancing constraint.This study is the first to consider the lane cost balancing constraint in the context of freight allocation. We formulate the freight allocation problem with this lane cost balancing constraint as a mixed integer programming model, and show that even finding a feasible solution to this problem is computationally intractable. Hence, in order to produce high-quality solutions in practice, we devised a meta-heuristic approach based on tabu search. Experiments show that our approach significantly outperforms the branch-and-cut approach of CPLEX 11.0 when the problem increases to practical size and the lane cost balancing constraint is tight. Our approach was developed into an application that is currently employed by decision-makers at the buying office in question.     

Hub-and-spoke network design and fleet deployment for string planning of liner shipping

All shipping liner companies divide their service regions into several rotations (strings) in order to operate their container vessels. A string is the ordered set of ports at which a container vessel will call. Each port is usually called at no more than twice along one string, although a single port may be called at several times on different strings. The size of string dictates the number of vessels required to offer a given frequency of service. In order to better use their shipping capacity, groups of Liner Service Providers sometimes make a short term agreement to merge some of their service routes (in a certain region) into one main ocean going rotation and p feeder rotations. In order to minimize the weighted sum of transit time, and fixed deployment costs, this paper proposes a mixed integer linear programming model of the network design, and an allocation of proper capacity size and frequency setting for every rotation. Given that none of the existing general-purpose MIP solvers is able to solve even very small problem instances in a reasonable time, we propose a Lagrangian decomposition approach which uses a heuristic procedure and is capable of obtaining practical and high quality solutions in reasonable times. The model will be applied on a real example, and we shall present some of the results obtained by our model which show how it facilitates a better use of assets and a significant reduction in the use of fuel, therefore allowing a more environmentally friendly service.     

Demurrage costs optimization in ports using binary mathematical programming models

Summary SODA is a system by which the schedule of ships in a port can be found in order to minimize the demurrage and operating costs associated with shipping operations. SODA constructs and resolves binary programming models using shipping data (arrival dates, demurrage costs, berths to be used, etc.) and available berths in a planning horizon. The model is of an adjustable precision: major accuracy implies greater calculation time. SODA provides an optimal result to a complex problem in a resolution time which is generally better than that needed for a manual solution.     

The bidding selection and assignment problem with minimum quantity commitment

This paper studies the bidding selection and assignment problem with a novel constraint, namely minimum quantity commitment (MQC), motivated by the Royal Philips Electronics Company. Responding to the stipulations by the US Federal Maritime Commission, any shipping agent transporting to the US must satisfy a minimum quantity of containers. To insure this MQC for shipping agents, the Royal Philips Electronics Company, with a large number of shipping needs, has to assign enough containers to each selected shipping agent to transport cargos to the US. This restriction creates difficulties for Philips as the company seeks to satisfy its shipping needs with minimum total costs. To solve this problem, we first formulate it by a mixed-integer programming model. In order for both linear programming relaxation and Lagrangian relaxation to provide good lower bounds, we then strengthen the model by a few valid inequalities. Furthermore, a Lagrangian-based heuristic and a branch and cut solver are applied to solve the problem. Extensive experiments show the effectiveness of all the models and methods.     

Modelling of containerized air cargo forwarding problems under uncertainty

This paper examines air container renting and cargo loading problems experienced by freight forwarding companies. Containers have to be booked in advance, in order to obtain discounted rental rates from airlines; renting or returning containers on the day of shipping will incur a heavy penalty. We first propose a mixed-integer model for the certain problem, in which shipment information is known with certainty, when booking. We then present a two-stage recourse model to handle the uncertainty problem, in which accurate shipment information cannot be obtained when booking, and all cargoes have to be shipped without delay. The first-stage decision is made at the booking stage, to book specific numbers of different types of containers. The second-stage decision is made on the day of shipping, depending on the extent to which the uncertainty has been realized. The decisions include number of additional containers of different types that are required to be rented, or the number of containers to be returned, under the scenario that might occur on the day of shipping. We then extend the recourse model into a robust model for dealing with the situation in which cargoes are allowed to be shipped later. The robust model provides a quantitative method to measure the trade-off between risk and cost. A series of experiments demonstrate the effectiveness of the robust model in dealing with risk and uncertainty.     

基于预判发货的网络零售商双模式批量配送研究

基于预判发货的背景,考虑订单处理中心和配送站之间存在第三方物流和自营物流两种配送模式,研究了B2C网络零售商的动态批量配送问题。首先利用混合整数规划构建了一个三级供应链系统下的动态批量配送模型,接着采用网络流规划的技术重新建模,并在其基础上对最优解的性质进行了分析,进而设计了计算时间复杂度为O(T²)的精确动态规划求解算法。最后用算例实验验证了该算法的有效性和适用性。     

基于可变航速的支线集装箱船舶调度优化模型与算法

随着航运市场的竞争不断加剧和集装箱船舶大型化的发展,越来越多的航运企业选择轴-辐式航运网络模式。支线船舶调度问题作为轴-辐式航运网络的重要组成部分受到研究者的高度关注。本文研究了可变航速和经济航速两种情境下的支线船舶调度问题,同时考虑枢纽港和喂给港的取送箱时间窗限制,以航运企业运营成本最小化为目标函数建立非线性混合整数规划模型。首先使用专业的规划求解器进行小规模算例的求解,验证了模型的准确性。同时运用改进的遗传算法对大规模支线船舶优化调度模型进行求解。为了提高求解效果,进一步设计了多智能体进化算法进行求解。数值结果表明,可变航速的运营成本低于经济航速的运营成本;在算法效率方面,改进遗传算法收敛速度较快,多智能体进化算法则可以提高求解精度。     

Scheduling real-time tasks for dependability

Real-time systems are increasingly used in applications whose failure may result in large economic and human costs. Since many such systems operate in environments that are non-deterministic, and possibly hazardous, it is extremely important that the systems be dependable, namely the deadlines of tasks must be met even in the presence of particular failures. In order to enhance the dependability of a real-time system, we study the problem of scheduling a set of real-time tasks to meet their deadlines in the presence of processor failures. We first prove that the problem of scheduling a set of non-preemptive tasks on more than two processors to tolerate one arbitrary processor failure is NP-complete even when the tasks share a common deadline. Heuristic algorithms are then proposed to solve this problem. The schedules generated by the heuristic algorithms can tolerate one arbitrary processor failure in the worst case. The analysis and experimental data show that the performance of the algorithms is near-optimal.     

A combined terminal location and ship routing problem

In this paper, we consider a combined terminal location and ship routing problem at Södra Cell AB. The purpose is to supply the customers' annual demand for pulp products while minimizing the distribution costs. Customers are supplied with various pulp products from pulp mills in Scandinavia by ships, trains, or lorries. The ship routes go from the pulp mills to terminals in Europe. From each terminal, the products are transported to customers by lorry, train, or barge. Some customers can be supplied directly from the pulp mills by trains or lorries. We have developed a mathematical model to select which terminals to use and, at the same time, determine the shipping routes. The mixed integer programming model was solved directly using a commercial solver. When the number of routes generated is large, the time required to obtain an optimal solution is too long. Hence, we have developed heuristics in order to obtain an acceptable solution in reasonable time. In addition to the basic case, five different scenarios were tested. Our heuristics provide solutions that are within 0.12% of the optimal ones.     

Optimising cargo loading and ship scheduling in tidal areas

This paper describes a framework that combines decision theory and stochastic optimisation techniques to address tide routing (i.e. optimisation of cargo loading and ship scheduling decisions in tidal ports and shallow seas). Unlike weather routing, tidal routing has been little investigated so far, especially from the perspective of risk analysis. Considering the journey of a bulk carrier between N ports, a shipping decision model is designed to compute cargo loading and scheduling decisions, given the time series of the sea level point forecasts in these ports. Two procedures based on particle swarm optimisation and Monte Carlo simulations are used to solve the shipping net benefit constrained optimisation problem. The outputs of probabilistic risk minimisation are compared with those of net benefit maximisation, the latter including the possibility of a ‘rule-of-the-thumb’ safety margin. Distributional robustness is discussed as well, with respect to the modelling of sea level residuals. Our technique is assessed on two realistic case studies in British ports. Results show that the decision taking into account the stochastic dimension of sea levels is not only robust in real port and weather conditions, but also closer to optimality than standard practices using a fixed safety margin. Furthermore, it is shown that the proposed technique remains more interesting when sea level variations are artificially increased beyond the extremes of the current residual models.     

Stability of the Optimal Solution to a Problem of Variational Assimilation with Error Covariance Matrices of Observational Data for a Sea Thermodynamics Model

A mathematical model of sea thermodynamics developed at the Institute of Numerical Mathematics of the Russian Academy of Sciences is considered. A problem of variational assimilation of daily-averaged sea surface temperature (SST) data with observational error covariance matrices is formulated and investigated. On the basis of variational assimilation of satellite observation data, an inverse problem of restoring the heat flux on the sea surface is solved. The stability of the optimal solution to the problem of variational data assimilation is studied, and the results of numerical experiments with the model for Baltic Sea dynamics are presented.