Pareto-optimal Cuts Research Articles

Improved Benders decomposition for stack-based yard template generation in an automated container terminal

Automated container terminals have become popular solutions to the growing need for container handling. We propose a two-stage mathematical model to design an efficient yard template for automated container terminals, using stacks as the allocation unit. In the first stage, a bi-objective model is developed to optimize landside and seaside operations in the terminal. A Benders decomposition algorithm is designed to solve the model. The proposed algorithm is improved by several acceleration strategies, including valid inequalities, pareto-optimal cuts, and ε−optimal approach. Drawing upon the solution derived from the first stage, the second stage assigns containers to specific bays, which is partitioned into multiple subproblems by capitalizing on the structure of it to enhance solvability. Numerous numerical experiments are conducted to verify the efficiency of the proposed algorithm. In addition, scenario analysis demonstrate that the stack-based yard template is more efficient than a cluster-based yard template for automated container terminals with multiple targets.

Exact Method for Production Hub Location

This study proposes a production hub location (PHL) problem that integrates the classical multiplant lot-sizing and hub location problems. The PHL problem is to determine the location of production facilities, lot-sizing, inventory, hub location, and the distribution of multiple commodities from plants to customers, with an objective to minimize the total production, setup, inventory, hub operating, and transportation costs. The PHL problem applies to manufacturing companies that either have built a hub-and-spoke distribution network or are accessible to such a network through collaborations with other third-party logistics companies. Because the PHL problem is [Formula: see text]-hard, we propose an exact method that integrates dynamic programming and Benders decomposition (DPBD) for solving the problem. The DPBD method is enhanced by exploring several problem properties, such as a multicut reformulation, the generation of Pareto-optimal cuts, a two-stage hub elimination and restoration procedure, and the inclusion of a novel heuristic procedure. We compare the PHL model with several related models theoretically and computationally with newly created benchmark instances. The computational results indicate that the proposed model can reduce the total costs and facilitate better network designs and system decisions, highlighting the value of an integrated approach. We also provide managerial insights into the benefits of integration and show the efficiency of the DPBD method through an extensive number of computational tests. History: Accepted by Andrea Lodi, Area Editor for Design and Analysis of Algorithms—Discrete. Supplemental Material: The online supplement is available at https://doi.org/10.1287/ijoc.2023.0339 .

A benders decomposition algorithm for a bid generation problem in the procurement of Three-echelon transportation services

In this paper, we address a bid generation problem in a combinatorial auction for the procurement of transportation services with time windows in a three-echelon transportation network. By considering the interconnectivity of transportation operations in different echelons, a path-based mathematical programming model is proposed for this problem. To solve the model with complicated linking constraints, a Benders decomposition algorithm in a heuristic manner is designed. The algorithm iteratively and alternatively solves a master problem that is a selective routing problem with time windows for the second echelon and two subproblems that are routing problems with time windows for the first and third echelon respectively. The basic Benders decomposition algorithm is enhanced by applying the acceleration techniques of multiple cuts, Pareto-optimal cuts and multiple cuts combined with Pareto-optimal cuts. Extensive computational experiments were performed to evaluate the effectiveness of the acceleration techniques and the efficiency of the algorithm. The computational results indicate that the enhanced algorithm significantly outperforms CPLEX in terms of solution quality and CPU time. Moreover, our sensitivity analysis on some key parameters of the model show that the lower the proportion of reserved requests, the larger of the capacity of each vehicle, and the higher the percentage of profit increase of a carrier after bidding.

A Bid Generation Problem in Truckload Transportation Service Procurement Considering Multiple Periods and Uncertainty: Model and Benders Decomposition Approach

Transportation service procurement is often realized by an auction. With the rolling horizon planning concept adopted in logistics, carriers usually plan their transportation operations of several periods (days) in advance. This implies that carriers must consider multiple periods when participating in combinatorial auctions organized by shippers. Since transportation requests in future cannot be foreseen, carriers must consider request uncertainty in such auctions. In this paper, we consider a carrier’s bid generation problem appeared in a multi-round combinatorial auction for truckload transportation service procurement with the consideration of multiple periods and request uncertainty. The problem is to maximize the total expected net profit of the carrier in a planning horizon of multiple periods by optimally determining the transportation requests to bid, the period to serve each request, and the routes to serve all requests including the carrier’s reserved requests. This problem is hard to solve because of its stochastic nature. By adopting the scenario approach of stochastic optimization, a mixed integer linear programming model is formulated for the problem. A Benders decomposition approach is then proposed to solve the model, with Pareto-optimal cuts to accelerate its solution process. The performance of the approach is evaluated by numerical experiments on randomly generated instances. The computational results demonstrate that the Bender decomposition approach is much more efficient than CPLEX solver in solving large instances of the problem. In addition, the value of considering uncertain requests and multi-period in the bid generation is evaluated.

Benders decomposition for internal truck renewal decision in green ports

ABSTRACT To effectively decrease emissions and pollution emitted by equipment in container ports, many ports began to update equipment using diesel fuels to achieve the aim of green ports. This paper studies an internal truck renewal problem in container ports and proposes a two-stage stochastic programming model to optimally determine internal truck composition optimization adjustment including three renewal modes, i.e. purchasing, retrofitting, and chartering. An exact solution procedure based on Benders decomposition accelerating by Pareto-optimal cuts is developed for the proposed model. Given the problem background of Shanghai Yangshan Deep Water Port, comprehensive computational experiments are carried out to verify the effectiveness of the proposed mathematical model and the performance of the solution approach. According to the numerical experiments, some suggestions for the management of renewing internal trucks in green ports are summarized at the end of the paper.

Benders decomposition applied to profit maximizing hub location problem with incomplete hub network

This paper addresses a profit maximizing multiple allocation hub network design problem, which is a hub location problem with incomplete hub network, aiming to determine the quantity and location of the hubs, allocate demand nodes to these hubs, establish the network topology, and route the demand flows to satisfy the demands between the origin and destination pairs that were selected to be served in order to obtain the highest profit. This problem also considers the multiple allocation strategy, does not allow direct connections between non-hub nodes, does not impose capacity constraints, and assumes that there are fixed costs involved in installing hubs and installing hub arcs. A new formulation for the problem is presented and it is proved that this formulation is stronger than another one in the literature, which models the same problem. Several versions of the Benders decomposition method, combining the generation of Pareto-optimal cuts, multiple cuts strategies, and Benders branch-and-cut scheme, are proposed and analyzed through extensive computational experiments on benchmark instances. The best-proposed Benders decomposition versions were compared with the CPLEX solver, with and without the activation of its Benders decomposition algorithm. The results showed that these algorithms perform much better than the solver versions. The proposed Benders decomposition approach can solve benchmark instances with up to 150 nodes.

A two-stage robust hub location problem with accelerated Benders decomposition algorithm

ABSTRACT In this paper, a two-stage robust optimisation is presented for an uncapacitated hub location problem in which demand is uncertain and the level of conservatism is controlled by an uncertainty budget. In the first stage, locations for establishing hub facilities were determined, and allocation decisions were made in the second stage. An accelerated Benders decomposition algorithm was used to solve the problem. Computational experiments showed better results in terms of number of iterations and computation time for Benders decomposition with Pareto-optimal cuts in comparison with the classical Benders decomposition algorithm. According to numerical analysis, it was concluded that increasing the uncertainty budget also increased total costs for more established hubs. To determine the uncertainty budget in an appropriate manner, a new expected aggregate function was introduced. The numerical studies demonstrated the usefulness of the proposed method in defining the appropriate uncertainty budget in the presence of uncertainty.

Combining Benders decomposition and column generation for scheduled service network design problem in rail freight transportation

This paper addresses the scheduled service network design problem for rail freight transportation. The proposed model integrates service selection and scheduling, and routing of time-dependent customer shipments based on a connection network representation of the associated operations and decisions along with their relations and time dimensions. Benefiting from the construction method of the connection network, the transfer restrictions of shipments between services can be easily incorporated into our models. We propose two model formulations based on arc variables and path variables respectively. A bespoken solution methodology that combines the Benders decomposition and column generation is proposed to solve the path-based model. Compared with the results of using GUROBI to solve the arc-based model, experimental results show that the proposed approach is effective, yielding high-quality solutions for all test instances, and the acceleration techniques, i.e. trust-region and Pareto-optimal cuts, can significantly improve the convergence efficiency of Benders decomposition.

Benders Decomposition for the Profit Maximizing Capacitated Hub Location Problem with Multiple Demand Classes

This paper models the profit maximizing capacitated hub location problem with multiple demand classes to determine an optimal hub network structure that allocates available capacities of hubs to satisfy demand for commodities from different market segments. A strong deterministic formulation of the problem is presented, and a Benders reformulation is described to optimally solve large-size instances of the problem. A new two-phase methodology is developed to decompose the Benders subproblem, and two effective separation routines are derived to strengthen the Benders optimality cuts. The algorithm is enhanced by the integration of improved variable-fixing techniques. The deterministic model is further extended by considering uncertainty associated with the demand to develop a two-stage stochastic program. To solve the stochastic version, a Monte Carlo simulation–based algorithm is developed that integrates a sample average approximation scheme with the proposed Benders decomposition algorithm. Novel acceleration techniques are presented to improve the convergence of the algorithm proposed for the stochastic version. The efficiency and robustness of the algorithms are evaluated through extensive computational experiments. Computational results show that large-scale instances with up to 500 nodes and three demand classes can be solved to optimality, and that the proposed separation routines generate cuts that provide significant speedups compared with using Pareto-optimal cuts. The developed two-phase methodology for solving the Benders subproblem as well as the variable-fixing and acceleration techniques can be used to solve other discrete location and network design problems.

Developing a new model for a competitive facility location problem considering sustainability using Markov chains

In this paper, we consider a competitive facility location problem. The problem is defined for a new company planning to locate some new facilities (chain stores) in a geographical region where some other competitors have already located their facilities. We aim to propose a new model that the attractiveness function of each facility is developed based on the sustainable aspects. The attractiveness is based on the two aspects of sustainable development measurements called flexibility in supplying diverse products by a facility and the productivity of services in a facility. The objective is to maximize the profit in which the revenue is based on the amount of market share captured by the new facilities and the cost includes the cost of locating new facilities, the total tax paid to the government due to emissions of pollutants, and the cost of customers’ dissatisfaction. The last type of cost occurs once a customer has to wait too long to receive services. Therefore, for providing justice in servicing, a chance constraint is proposed to control this dissatisfaction according to the Jackson Markov network. For an illustrative case study, the new company can capture 61.49% of the market share and 38.51% of the market remains for the existing companies. This can show that the current methodology leads to capture more market share in a competitive environment. An accelerated Benders’ decomposition is used to solve the model, which is enhanced by disaggregating cuts, valid inequalities, and Pareto-optimal cuts. The proposed solution approach can improve the solution about 10% against the best previous method. Managerial implications and numerical examples illustrate the effectiveness and the applicability of the mathematical model and the solution approach.

A two-stage robust optimisation for terminal traffic flow problem

Airport congestion witnesses potential conflicts: insufficient terminal airspace and delay propagation within scrambled the competition in the terminal manoeuvring area. Re-scheduling of flights is needed in numerous situations, heavy traffic in air segments, holding patterns, runway schedules and airport surface operations. Robust optimisation for terminal traffic flow problem, providing a practical point of view in hedging uncertainty, can leverage the adverse effect of uncertainty and schedule intervention. To avoid delay propagation throughout the air traffic flow network and reduce the vulnerability to disruption, this research adopts a two-stage robust optimisation approach in terminal traffic flow. It further enhances the quality of Pareto-optimality Benders-dual cutting plane based on core point approximation in the second stage recourse decision. The efficiency of the cutting plane algorithm is evaluated by a set of medium sized real-life scenarios. The numerical results show that the proposed scheme outperforms the well-known Pareto-optimal cuts in Benders-dual method from the literature.

Enhancing Benders decomposition algorithm to solve a combat logistics problem

This paper proposes a multi-time period, two-stage stochastic programming model for the design and management of a typical combat logistics problem. The design shall minimize the total path setup cost, commodity preposition and processing costs, and expected transportation, storage, and shortage costs across all possible path failure scenarios. Due to the complexity associated with solving the model, we propose an accelerated Benders decomposition algorithm to solve the model in a realistic-size network problem within a reasonable amount of time. The Benders decomposition algorithm incorporates several algorithmic improvements such as pareto-optimal cuts, multi-cuts, knapsack inequalities, integer cuts, input ordering, mean-value cuts, and the rolling horizon heuristic. Computational experiments are performed to assess the efficiency of different enhancement techniques within the Benders decomposition algorithm.

An Exact Algorithm for Multilevel Uncapacitated Facility Location

We study a general class of multilevel uncapacitated p-location problems in which the selection of links between levels of facilities is part of the decision process. We propose an exact algorithm based on a Benders reformulation to solve large-scale instances of the general problem and some well-known particular cases. We exploit the network flow structure of the reformulation to efficiently generate Pareto-optimal cuts. We perform extensive computational experiments to assess the performance of several different variants of the Benders algorithm. Results obtained on benchmark instances with up to 3,000 customers, 250 potential facilities, and four levels confirm its efficiency.

A hybrid robust stochastic programming for a bi-objective blood collection facilities problem (Case study: Iranian blood transfusion network)

ABSTRACTIn this paper, a generalized network optimization model is developed for a complex blood supply chain including a regionalized blood bank system. This system consist of collection sites, testing and processing facilities, storage facilities, distribution centers, as well as points of demand (hospitals). To keep the network in contradiction of the uncertainty, a consolidated approach based on a recently developed stochastic robust approach is extended. An accelerated stochastic Benders decomposition algorithm is proposed to solve the problem modeled in this paper. To speed up the convergence of the solution algorithm, valid inequalities are introduced to get better quality lower bounds. Numerical illustrations are given to verify the mathematical formulation and also to show the benefits of using the stochastic robust approach. At the end, the performance improvements attained by the valid inequalities and the Pareto-optimal cuts are demonstrated in a real-world application.

Optimizing Train Formation Problem With Car Flow Routing and Train Routing by Benders-and-Price Approach

As the core of the rail freight flow organization process, train formation problem (TFP) has attracted much attention. In Chinese practice, car flow routing, TFP, and train routing are usually optimized sequentially to reduce the complexity of computation, which may result in a local optimum, and even no feasible solution. To address this issue, this paper studies the integrated optimization of the three sub problems with aims to minimize the total cost of transportation cost, accumulation cost, and classification cost. An integer linear arc-based model incorporating the unitary and in-tree rules of a shipment is first formulated and solved by the state-of-the-art solver GUROBI. Since GUROBI can't deal with the large test cases, a path-based model is built and solved by a bespoken two-phase algorithm. The first phase of the algorithm is Benders-and-Price approach that combines Benders decomposition and column generation, and the second phase is to solve the arc-based model with some variables fixed as the corresponding values fetched from the first phase. The results show that the proposed algorithm outperforms GUROBI, and the acceleration techniques, i.e., trust region and Pareto-optimal cuts, can improve the convergence efficiency of Benders decomposition significantly.

Subloop-based reversal of port rotation directions for container liner shipping network alteration

Container liner shipping network alteration is a practical manner of shipping network design, which aims to make minor modifications to ameliorate the existing network. In a generic liner shipping network with butterfly ports, each ship route is separated into a set of subloops on the basis of its structure and internal butterfly ports. Reversing the subloop directions has an impact on the network-wide cost including inventory cost, transshipment cost, and slot-purchasing cost. This paper proposes a new destination-based nonlinear model for the subloop-based reversal of port rotation directions with the objective of minimizing the overall network-wide cost. We prove that the addressed problem is NP-hard. Next, the model is transformed to an equivalent mixed-integer linear programming model. Based on the structure of the reformulated model, we develop a Benders decomposition (BD) algorithm and a metaheuristic method to solve practical-size instances. Three acceleration strategies are incorporated into the BD algorithm, which are adding Pareto-optimal cuts, updating big-M coefficients and generating combinatorial Benders cuts. Case studies based on three small examples and an Asia-Europe-Oceania liner shipping network with a total of 46 ports are conducted. Results show that the problem could be efficiently solved by the accelerated BD algorithm and the optimization of subloop directions is conducive to decreasing the network-wide cost especially the inventory cost.

A bi-objective robust optimization model for a blood collection and testing problem: an accelerated stochastic Benders decomposition

Blood transfusion services are vital components of healthcare systems all over the world. In this paper, a generalized network optimization model is developed for a complex blood supply chain in accordance with Iranian blood transfusion organization (IBTO) structure. This structure consist of four types facilities. Blood collection centers, blood collection and processing centers, mobile teams and blood transfusion center have various duties in IBTO structure. The major contribution is to develop a novel hybrid approach based on stochastic programming, e-constraint and robust optimization (HSERO) to simultaneously model two types of uncertainties by including stochastic scenarios for total blood donations and polyhedral uncertainty sets for demands. An accelerated stochastic Benders decomposition algorithm is proposed to solve the problem modeled in this paper. To speed up the convergence of the solution algorithm, valid inequalities are introduced to get better quality lower bounds. In addition, a Pareto-optimal cut generation scheme is used to strengthen the Benders optimality cuts. Numerical illustrations are given to verify the mathematical formulation and also to show the benefits of using the HSERO approach. At the end, the performance improvements achieved by the valid inequalities and the Pareto-optimal cuts are demonstrated in a real world application.

Large-scale reverse supply chain network design: An accelerated Benders decomposition algorithm

Abstract Reverse logistics (RL) is a term that captures any process that involves the movement of goods or services from their destination back to their source. RL is increasingly gaining recognition as an essential component in the design process of supply chains. In this paper, we formulate a mixed-integer linear programming (MILP) model that aims to determine the optimal locations and capacities of various nodes such as inspection centers and remanufacturing facilities and to help decision makers find optimal transportation decisions and to determine the number and type of transportation units required in the network. We develop an exact method to solve large-scale real-sized instances of this problem. We initially attempt to solve the problem using the traditional Benders Decomposition (BD) technique which fails to solve the problem in reasonable computational times. We improve the traditional BD technique by adding several accelerating methods such as trust-region, logistics constraints, Pareto-optimal cuts, restructuring of the problem, and continuous relaxation of the integer variables to increase the convergence rate and to reduce the total number of cuts required in the master problem. In almost all the instances, we were able to reach optimal solutions. For the remaining instances, we succeed in solving the largest problem with an optimality gap of 0.5% and within a reasonable running time. The paper highlights and evaluates the performance and effectiveness of the different acceleration techniques of our improved BD algorithm along with the computational results.

The generalized rollon-rolloff vehicle routing problem and savings-based algorithm

Taking the waste collection management as the practical background, the rollon-rolloff vehicle routing problem (RRVRP) involves tractors pulling large containers between customer locations and the disposal facility. Each used-tractor begins and ends its route at the depot. Customer demand includes emptying a full container, ordering an empty container or changing a container. The objective is to find tractor routes that feature the minimum amount of total nonproductive time. In the literature the RRVRP is formulated as the node routing problem, and the “trip” definition that is the complete transport service of a container is used. To relax some assumptions of the RRVRP to cater to practical desires, we present a variant called the generalized RRVRP (G-RRVRP). The G-RRVRP generalizes the practical background, the objective function and the demand flow, and considers specially container loading and unloading time constraints. The G-RRVRP classifies demand into loaded-container demand and cargo demand with time windows. On condition of respecting container loading/unloading time and customer time windows, the G-RRVRP can design tractor routes for the synchronous scheduling of loaded and empty containers so as to ensure the timeliness of transport service. The G-RRVRP aims to minimize the total running cost of used tractors, instead of the total nonproductive time of tractors adopted by the RRVRP. A mixed integer linear programming model for the G-RRVRP is proposed. The Benders decomposition algorithm involving Pareto-optimal cuts and Benders decomposition-callback implementation, and a two-stage heuristic involving the savings algorithm followed by a local search phase are provided. The mathematical formulation and the two-stage heuristic are tested by solving 40 small-scale instances and 20 benchmark instances. Small-scale instances can be solved directly by CPLEX through the Benders decomposition strategies to find exact solutions. The case study indicates the applicability of the G-RRVRP model and the two-stage heuristic to realistic-size problems abstracted from intercity linehaul systems. The computational experiments and case study indicate that the heuristic can solve various instances of the G-RRVRP such that the solution quality and the computation time are acceptable.

An accelerating Benders’ decomposition approach to the integrated supply chain network design with distributed generation

Distributed generation (DG) is one of the new trends in distribution networks under the effect of various policies, such as distribution loss reduction. It is important to find the position and determine the size of DGs in the existing distribution network. In this paper, we consider DGs for the integrated supply chain networks and obtain the optimal locations, sizing, and the service areas of DGs. First, a mixed integer programming model is proposed to formulate the problem. Then, an accelerating Benders’ decomposition approach with two valid cuts and Pareto-optimal cuts is described for this problem. Computational results show the efficiency of the our model and proposed algorithm.