DCA for solving the scheduling of lifting vehicle in an automated port container terminal

The container was introduced as a universal carrier for various goods in the 1960s and soon became a standard worldwide transportation. The competitiveness of a container seaport is marked by different success factors, particularly the time in port for ships. Operational problems of container terminals is divided into several problems, such as assignment of vessels, loading/unloading and storage of the containers, quay cranes scheduling cite, planning yard cranes cite and assignment of storage containers cite. In this work, the study will focus on piloting yard trucks. Two different types of vehicles can be used, namely automated guided vehicles (AGVs) and lifting vehicles (LVs). An AGV receives a container from a quay crane and transports containers over fixed path. LVs are capable of lifting a container from the ground by itself. The model that we consider is formulated as a mixed integer programming problem, and the difficulty arises when the number of binary variables increases. There are a lot of algorithms designed for mixed integer programming problem such as Branch and Bound method, cutting plane algorithm, . . . By using an exact penalty technique we treat this problem as a DC program in the context of continuous optimization. Further, we combine the DCA with the classical Branch and Bound method for finding global solutions.     

An adaptive guidance meta-heuristic for the vehicle routing problem with splits and clustered backhauls

This paper presents the case study of an Italian carrier, Grendi Trasporti Marittimi, which provides freight transportation services by trucks and containers. Its trucks deliver container loads from a port to import customers and collect container loads from export customers to the same port. In this case study, all import customers in a route must be serviced before all export customers, each customer can be visited more than once and containers are never unloaded or reloaded from the truck chassis along any route. We model the problem using an Integer Linear Programming formulation and propose an Adaptive Guidance metaheuristic. Our extensive computational experiments show that the adaptive guidance algorithm is capable of determining good-quality solutions in many instances of practical or potential interest for the carrier within 10?min of computing time, whereas the mathematical formulation often fails to provide the first feasible solution within 3?h of computing time.     

Yard crane scheduling in port container terminals

Yard cranes are the most popular container handling equipment for loading containers onto or unloading containers from trucks in container yards of land scarce port container terminals. However, such equipment is bulky, and very often generates bottlenecks in the container flow in a terminal because of their slow operations. Hence, it is essential to develop good yard crane work schedules to ensure a high terminal throughput. This paper studies the problem of scheduling a yard crane to perform a given set of loading/unloading jobs with different ready times. The objective is to minimize the sum of job waiting times. A branch and bound algorithm is proposed to solve the scheduling problem optimally. Efficient and effective algorithms are proposed to find lower bounds and upper bounds. The performance of the proposed branch and bound algorithm is evaluated by a set of test problems generated based on real life data. The results show that the algorithm can find the optimal sequence for most problems of realistic sizes.     

A branch-and-price approach to the feeder network design problem

In this paper we consider the problem of designing a container liner shipping feeder network. The designer has to choose which port to serve during many rotations that start and end at a central hub. Many operational characteristics are considered, such as variable leg-by-leg speeds and cargo transit times. Realistic instances are generated from the LinerLib benchmark suite. The problem is solved with a branch-and-price algorithm, which can solve most instances to optimality within one hour. The results also provide insights on the cost structure and desirable features of optimal routes. These insights were obtained by means of an analysis where scenarios are generated varying internal and external conditions, such as fuel costs and port demands.     

Multiple machine continuous setup lotsizing with sequence-dependent setups

We address the short-term production planning and scheduling problem coming from the glass container industry. A furnace melts the glass that is distributed to a set of parallel molding machines. Both furnace and machine idleness are not allowed. The resulting multi-machine multi-item continuous setup lotsizing problem with a common resource has sequence-dependent setup times and costs. Production losses are penalized in the objective function since we deal with a capital intensive industry. We present two mixed integer programming formulations for this problem, which are reduced to a network flow type problem. The two formulations are improved by adding valid inequalities that lead to good lower bounds. We rely on a Lagrangian decomposition based heuristic for generating good feasible solutions. We report computational experiments for randomly generated instances and for real-life data on the aforementioned problem, as well as on a discrete lotsizing and scheduling version.     

The multiple container loading cost minimization problem

In the shipping and transportation industry, there are several types of standard containers with different dimensions and different associated costs. In this paper, we examine the multiple container loading cost minimization problem (MCLCMP), where the objective is to load products of various types into containers of various sizes so as to minimize the total cost. We transform the MCLCMP into an extended set cover problem that is formulated using linear integer programming and solve it with a heuristic to generate columns. Experiments on standard bin-packing instances show our approach is superior to prior approaches. Additionally, since the optimal solutions for existing test data is unknown, we propose a technique to generate test data with known optimal solutions for MCLCMP.     

Minimizing fleet operating costs for a container transportation company

This paper focuses on a fleet management problem that arises in container trucking industry. From the container transportation company perspective, the present and future operating costs to minimize can be divided in three components: the routing costs, the resource (i.e., driver and truck) assignment costs and the container repositioning costs (i.e., the costs of restoring a given container fleet distribution over the serviced territory, as requested by the shippers that own the containers).This real-world problem has been modeled as an integer programming problem. The proposed solution approach is based on the decomposition of this problem in three simpler sub-problems associated to each of the costs considered above.Numerical experiments on randomly generated instances, as well as on a real-world data set of an Italian container trucking company, are presented.     

A branch-and-price algorithm to solve the integrated berth allocation and yard assignment problem in bulk ports

In this research, two crucial optimization problems of berth allocation and yard assignment in the context of bulk ports are studied. We discuss how these problems are interrelated and can be combined and solved as a single large scale optimization problem. More importantly we highlight the differences in operations between bulk ports and container terminals which highlights the need to devise specific solutions for bulk ports. The objective is to minimize the total service time of vessels berthing at the port. We propose an exact solution algorithm based on a branch and price framework to solve the integrated problem. In the proposed model, the master problem is formulated as a set-partitioning problem, and subproblems to identify columns with negative reduced costs are solved using mixed integer programming. To obtain sub-optimal solutions quickly, a metaheuristic approach based on critical-shaking neighborhood search is presented. The proposed algorithms are tested and validated through numerical experiments based on instances inspired from real bulk port data. The results indicate that the algorithms can be successfully used to solve instances containing up to 40 vessels within reasonable computational time.     

A mathematical model of inter-terminal transportation

We present a novel integer programming model for analyzing inter-terminal transportation (ITT) in new and expanding sea ports. ITT is the movement of containers between terminals (sea, rail or otherwise) within a port. ITT represents a significant source of delay for containers being transshipped, which costs ports money and affects a port’s reputation. Our model assists ports in analyzing the impact of new infrastructure, the placement of terminals, and ITT vehicle investments. We provide analysis of ITT at two ports, the port of Hamburg, Germany and the Maasvlakte 1 & 2 area of the port of Rotterdam, The Netherlands, in which we solve a vehicle flow combined with a multi-commodity container flow on a congestion based time–space graph to optimality. We introduce a two-step solution procedure that computes a relaxation of the overall ITT problem in order to find solutions faster. Our graph contains special structures to model the long term loading and unloading of vehicles, and our model is general enough to model a number of important real-world aspects of ITT, such as traffic congestion, penalized late container delivery, multiple ITT transportation modes, and port infrastructure modifications. We show that our model can scale to real-world sizes and provide ports with important information for their long term decision making.     

Scheduling vessels and container-yard operations with conflicting objectives

We consider the problem of coordinating the operations of two supply chain partners: a foreign shipping company and a domestic port. The two partners have conflicting business objectives, and the issue is to determine the optimal cycle time, by which the shipping company removes the empty containers from the domestic port, so that the joint profit of the two partners is maximized. The domestic port prefers a shorter cycle time to mitigate its empty container accumulation and land use problems, while the shipping company wishes a longer cycle time to save its expensive vessel capacities. We propose an iterative procedure to search for this optimal cycle time. In each iteration, a candidate cycle time is evaluated by solving a deterministic vessel scheduling problem and a stochastic container-yard capacity optimization problem. We prove the properties of the vessel scheduling problem, derive the optimality condition under which the vessel scheduling problem can be decomposed, and show that the profit function of the domestic port is convex and thus the optimal container-yard capacity can be determined efficiently. Empirical observations on the algorithm computational performance collected from over 300 randomly generated test cases under various problem settings are reported.     

An exact method for scheduling a yard crane

This paper studies an operational problem arising at a container terminal, consisting of scheduling a yard crane to carry out a set of container storage and retrieval requests in a single container block. The objective is to minimize the total travel time of the crane to carry out all requests. The block has multiple input and output (I/O) points located at both the seaside and the landside. The crane must move retrieval containers from the block to the I/O points, and must move storage containers from the I/O points to the block. The problem is modeled as a continuous time integer programming model and the complexity is proven. We use intrinsic properties of the problem to propose a two-phase solution method to optimally solve the problem. In the first phase, we develop a merging algorithm which tries to patch subtours of an optimal solution of an assignment problem relaxation of the problem and obtain a complete crane tour without adding extra travel time to the optimal objective value of the relaxed problem. The algorithm requires common I/O points to patch subtours. This is efficient and often results in obtaining an optimal solution of the problem. If an optimal solution has not been obtained, the solution of the first phase is embedded in the second phase where a branch-and-bound algorithm is used to find an optimal solution. The numerical results show that the proposed method can quickly obtain an optimal solution of the problem. Compared to the random and Nearest Neighbor heuristics, the total travel time is on average reduced by more than 30% and 14%, respectively. We also validate the solution method at a terminal.     

Discrete Allocation and Replenishment of Resources with Binary Effectiveness

The problem considered is that of the allocation and replenishment of several resources in integer quantities in such a way as to maximize the sum of the returns from activities with concave return functions. All the resources are of the same physical type and each resource has an effectiveness of 0 or 1 against each activity, depending on the geographical locations of the resources and the activities or on other constraints. Solutions with objective values arbitrarily close to the optimal value are generated by the application of resourcewise optimization to an associated problem in continuous variables, and the rounding of a continuous solution to an integer solution according to given rules. The application of other continuous methods is indicated. Some properties of optimal integer solutions are derived.     

Modeling and algorithmic development of a staff scheduling problem

The problem of scheduling workers at a hub of a trucking system for the stripping and loading of the trucks and the determination of the optimal number of workers is a difficult problem. The trucks arrive at the facility at different (but known) times and may have their own scheduled departure times. This problem is like a set partitioning problem but with a side constraint pertaining to the dynamic arrival of the trucks. We develop a procedure to solve this problem that is based on the column generation technique, and the solution of a set covering problem to obtain the integer solution. The performance of the procedure is demonstrated by applying it to the real-life data obtained from a trucking company. Its effectiveness is illustrated by comparing it with a lower bound and a well-known heuristic procedure.     

Truck dock assignment problem with operational time constraint within crossdocks

In this paper, we consider a truck dock assignment problem with an operational time constraint in crossdocks where the number of trucks exceeds the number of docks available. The problem feasibility is affected by three factors: the arrival and departure time window of each truck, the operational time for cargo shipment among the docks, and the total capacity available to the crossdock. The objective is to find an optimal assignment of trucks that minimizes the operational cost of the cargo shipments and the total number of unfulfilled shipments at the same time. We combine the above two objectives into one term: the total cost, a sum of the total dock operational cost and the penalty cost for all the unfulfilled shipments. The problem is then formulated as an integer programming (IP) model. We find that as the problem size grows, the IP model size quickly expands to an extent that the ILOG CPLEX Solver can hardly manage. Therefore, two meta-heuristic approaches, Tabu Search (TS) and genetic algorithm (GA), are proposed. Computational experiments are conducted, showing that meta-heuristics, especially the Tabu search, dominate the CPLEX Solver in nearly all test cases adapted from industrial applications.     

A conic quadratic formulation for a class of convex congestion functions in network flow problems

In this paper we consider a multicommodity network flow problem with flow routing and discrete capacity expansion decisions. The problem involves trading off congestion and capacity assignment (or expansion) costs. In particular, we consider congestion costs involving convex, increasing power functions of flows on the arcs. We first observe that under certain conditions the congestion cost can be formulated as a convex function of the capacity level and the flow. Then, we show that the problem can be efficiently formulated by using conic quadratic inequalities. As most of the research on this problem is devoted to heuristic approaches, this study differs in showing that the problem can be solved to optimum by branch-and-bound solvers implementing the second-order cone programming (SOCP) algorithms. Computational experiments on the test problems from the literature show that the continuous relaxation of the formulation gives a tight lower bound and leads to optimal or near optimal integer solutions within reasonable CPU times.     

Mixed integer programming models for dispatching vehicles at a container terminal

This paper presents scheduling models for dispatching vehicles to accomplish a sequence of container jobs at the container terminal, in which the starting times as well as the order of vehicles for carrying out these jobs need to be determined. To deal with this scheduling problem, three mixed 0–1 integer programming models, Model I, Model II and Model III are provided. We present interesting techniques to reformulate the two mixed integer programming models, Model I and Model II, as pure 0–1 integer programming problems with simple constraint sets and present a lower bound for the optimal value of Model I. Model III is a complicated mixed integer programming model because it involves a set of non-smooth constraints, but it can be proved that its solutions may be obtained by the so-called greedy algorithm. We present numerical results showing that Model III is the best among these three models and the greedy algorithm is capable of solving large scale problems.     

On double cycling for container port productivity improvement

How quay cranes (QC) are scheduled is vital to the productivity of seaside container port operations. Double cycling concept is an operation strategy of loading the containers into ships as they are unloaded, thus improving the efficiency of a QC as well as the container port. Goodchild and Daganzo (Transp Sci 40(4):473–483, 2006) first described QC double cycling problem and solved the problem after formulating it into a two machine flow shop problem. Song (Port Technol Int 36:50–52, 2007) studied the formula to determine the optimal starting sequence for double cycling while reflecting on the practical issue of QC working direction. The above studies focused on a single QC double cycling and their empirical trials showed the double cycling could improve the productivity of each QC approximately by between 10 and 20 %. In Zhang and Kim (Comput Ind Eng 56(3):979–992, 2009), a multiple QC double cycling model was first suggested by formulating a mixed integer programming model to maximise the number of double cycles between multiple QCs. In the present paper we point out a flaw with the existing multiple QC double cycling model that lets cycles that are not implementable. In addition, the paper discusses the need for imposing constraints arising from real world requirements to the formulations aiming at double cycling.     

When is rounding allowed in integer nonlinear optimization?

In this paper we consider nonlinear integer optimization problems. Nonlinear integer programming has mainly been studied for special classes, such as convex and concave objective functions and polyhedral constraints. In this paper we follow an other approach which is not based on convexity or concavity. Studying geometric properties of the level sets and the feasible region, we identify cases in which an integer minimizer of a nonlinear program can be found by rounding (up or down) the coordinates of a solution to its continuous relaxation. We call this property rounding property. If it is satisfied, it enables us (for fixed dimension) to solve an integer programming problem in the same time complexity as its continuous relaxation. We also investigate the strong rounding property which allows rounding a solution to the continuous relaxation to the next integer solution and in turn yields that the integer version can be solved in the same time complexity as its continuous relaxation for arbitrary dimensions.     

A heuristic for sea-freight container selection,cargo allocation and cargo orientation

A model is proposed to generate solutions for container selection, for the allocation of cargo to containers, and for cargo orientation within a container. The model is in the form of a mixed integer program with the objective of minimizing the total shipping cost. The practical requirements of loading priority and weight distribution along the main dimension of the container are incorporated into the model. A heuristic solution strategy is proposed and used to control the computation time by pre-setting the search increments. Three case examples are presented. The first and third examples show that the proposed model can produce a better solution than the manual schedulers. The second example is taken from the literature and is compared with the solution generated by the proposed model, demonstrating that the practical considerations incorporated into the model do not necessarily lead to increased shipping costs.