Accelerated Benders Decomposition Algorithm Research Articles

Integrated optimization of vessel dispatching and empty container repositioning considering turnover time uncertainty

The global trade disproportion results in the accumulation of containers in import-dominated ports and shortages in export-dominated ports, causing congestion and high freight costs, thus hindering maritime shipping economy development. To address these issues, this study develops a stochastic programming model considering uncertain container turnover times. The model integrates decisions for vessel deployment and empty container repositioning over multiple planning periods through a two-stage decision process, aiming to minimize the total cost, including vessel deployment, container leasing, and penalty costs for unfulfilled demand. By formulating the scenario selection problem as a p-median problem, we effectively reduce the model size. We propose an accelerated Benders decomposition algorithm which leverages the independence of sub-problems in the second stage to enable parallel computation. Numerical experiments show that our Benders decomposition algorithm improves solution speed by over 63% compared to the Gurobi optimization solver. Furthermore, our integrated optimization approach proves to be more cost-effective than the reactive method used by shipping lines, achieving an average cost savings of 0.72%. Additionally, our method of constructing turnover time scenarios to address uncertainty saves approximately 0.45% in costs compared to using the probability distribution of container turnover time.

An accelerated Benders decomposition algorithm for the integrated storage space assignment, berth allocation, and yard crane deployment problem

In this paper, storage space assignment problem, berth allocation problem, and yard crane deployment problem are integrated with due attention to traffic congestion in the passing lines of storage yards. The paper provides a mixed-integer programming model for the integrated problem with the objective of minimizing the movement and operating costs of yard cranes, the cost of moving container flow between the storage yard and berths, and the delay cost of vessels. The planning horizon is considered to be one day and the berth layout is assumed to be discrete. Since the integrated storage space assignment and yard crane deployment problem is NP-hard and so is berth allocation problem, the integration of these three problems will be at least equally complex. Thus, an Accelerated Benders Decomposition Algorithm is also developed for solving the integrated problem. This is done by introducing a combinatorial Benders cut to the classical Benders decomposition algorithm to improve its performance and also adding 8 valid inequalities derived from the problem properties and assumptions to the algorithm formulation to accelerate the solution process. The proposed mathematical model and solution approach are evaluated by generating instances of different sizes for the problem and assessing the solution results. The results show the good performance of the proposed accelerated Benders decomposition algorithm. Taking the integrated approach to the modeling of the three problems rather than the non-integrated approach improved the cost of terminal operations by 11.15% on average.

An accelerated benders decomposition algorithm for the solution of the multi-trip time-dependent vehicle routing problem with time windows

The logistics companies executing the last mile delivery of goods in urban areas deal every day with the problem of routing their vehicles, while taking into account multiple trips per vehicle, time-dependent travel time, customers’ time windows and loading time at the depot simultaneously. This paper addresses this problem, known as Multi-Trip Time-Dependent Vehicle Routing Problem with Time Windows, aiming at its exact solution, minimizing the total travelled distance of a company's fleet. Based on a literature model, a new reformulation with reduced size is suggested. This reformulation is decomposed by applying the Benders method in an effective way, resulting in a subproblem with no duality gap. By exploiting the special features of the problem and the particular structure of the decomposition made, several novel valid inequalities are introduced, in order to both tighten the non-decomposed formulations and warm start the relaxed master problem to achieve less infeasible solutions and higher lower bounds. For the solution of the problem, an innovative algorithm is proposed, including suboptimal master solutions and a multi-cut generation procedure, which is based on the careful observation of the values of the Benders dual subproblem variables. The impact of the valid inequalities as well as two variants of the suggested algorithm are tested on benchmark data and they are compared with the non-decomposed models and a heuristic introduced in the literature. The computational results indicate improved efficiency and stronger bounds for the proposed algorithm.

The design of a sustainable-resilient forward-reverse logistics network considering resource sharing and using an accelerated benders decomposition algorithm

The design of a sustainable-resilient forward-reverse logistics network considering resource sharing and using an accelerated benders decomposition algorithm

Prevention of post-pandemic crises: A green sustainable and reliable healthcare supply chain network design for emergency medical products

The COVID-19 pandemic happened exactly when no one expected it and many people died due to the lack of medical equipment. Although pulse oximeters and thermometers were only one of the virus detection equipment, the lack of that equipment could lead to people not knowing about the disease and then the death of those people. Given that these medical devices are very vital in the era of a pandemic, and much less in the later, it is required to provide the best conditions for their use in a closed-loop healthcare supply chain network in post-pandemic. In this paper, a scenario-based two-stage stochastic programming three-objective model for designing a green closed-loop supply chain network is presented. Cost minimization, reliability maximization, and critical response maximization are the objectives of the proposed model. The third objective function is considered for the critical response based on the choice of transportation method. The greenhouse gas emissions for all supply chain elements are considered uncertain and controlled in several possible scenarios. By using an accelerated Benders decomposition algorithm, the mathematical model was solved in different dimensions and the result was analyzed and evaluated in different scenarios. The results demonstrate that the acceleration of Benders bonds reduces the convergence speed of the algorithm by 41%. Our study provides managerial insights into sustainability and reliability, enhancing healthcare supply chain resilience during and after pandemics.

Designing a portfolio-based closed-loop supply chain network for dairy products with a financial approach: Accelerated Benders decomposition algorithm

Product portfolio design is one of the important and effective factors in the financial and physical flows of various supply chains, especially dairy products. Accordingly, the financial and physical flows of the portfolio should be taken into consideration during the supply chain design. In this study, a closed-loop supply chain (CLSC) network is designed for dairy products aiming at maximizing the net cash flow from assets and maximizing the amounts paid to shareholders simultaneously. To find the optimal policy, an accelerated Benders decomposition (ABD) algorithm is implemented to tackle the complexity of the model. Moreover, three multi-objective optimization solution approaches of the weighted sum method (WSM), augmented ε-constraint (AEC), and fuzzy multi-objective programming (FMOP) are implemented to tackle the bi-objectiveness of the model. Next, a real case study in Iran is investigated to reveal the applicability of the developed methodology. Numerical results reveal that the ABD method reduces CPU time by about 10.8%. Moreover, the results of the case study demonstrate that by integrating financial and physical flows, an improvement of 4.8478% in the net cash flow from assets and a 2.3% improvement in the amounts paid to shareholders compared to the current situation.

A chance-constraint optimization model for a multi-echelon multi-product closed-loop supply chain considering brand diversity: An accelerated Benders decomposition algorithm

The closed-loop supply chain (CLSC) network design has become one of the most critical issues due to the importance of resource optimization. Moreover, increasing competition in commercial markets leads to the diversity of a brand’s product portfolio to meet the customer’s demand. Hence, this paper develops a multi-period, multi-brand stochastic mixed-integer linear programming (MILP) model with direct and indirect distribution for the proposed CLSC network. Because of the uncertain nature of demand, the uncertainties for new and second-hand product demand are considered. Besides, a chance-constraint optimization (CCO) approach is applied to deal with uncertainty. Moreover, the accelerated Benders decomposition (BD) algorithm is designed to solve the proposed model. Several test problems are created and used to solve the accelerated BD compared with the conventional BD algorithm to investigate this model. Finally, the results are compared and described analytically, and some future research is suggested.

Design of survivable wireless backhaul networks with reliability considerations

The ever-increasing need for stable, seamless, high-speed connectivity places an enormous burden on the communications infrastructure, comprised of access, backhaul, and core networks. Furthermore, recent advances in access technologies shift the capacity bottleneck towards the newly popularized wireless backhaul network, which is also susceptible to different types of failures. In this paper, a wireless backhaul network design under reliability and survivability constraints is formulated as a mixed integer linear programming (MILP) model to determine the minimum cost bandwidth assignment of wireless links. We developed an accelerated Benders decomposition algorithm to solve this model. Extensive computational experiments show the accuracy of the model proposed to tackle the problem and the efficiency of the solving method, particularly in larger networks.

A New Scenario Reduction Method Based on Higher-Order Moments

Scenario reduction is an effective method to ease the computational burden of stochastic programming problems, which aims at choosing a subset of scenarios that can better represent a large number of possible scenarios. Higher-order moments are critical in the scenario reduction process, especially for stochastic programming problems that are greatly affected by the moments. From this idea, we construct a mixed integer linear programming model to improve the reduction accuracy of traditional methods by minimizing the moments’ information loss between the original and reduced scenarios. An improved Benders decomposition algorithm is then designed to find an optimal solution for the model. Finally, the resulting scenarios are examined on an international portfolio selection problem. Empirical and comparative studies are also carried out to reveal the superiority of our proposed scenario reduction method over other existing approaches or models, together with the superior performance of the algorithm. Summary of Contribution: To effectively solve stochastic programming problems, the scenario reduction method has become an active research area to strike a balance between the fine representation of random variables and computational complexity. Thus, how to design a reasonable optimal scenario reduction model and effectively solve this complex model is very important and meaningful. On the other hand, for some stochastic programming problems, especially the portfolio selection problems, statistical properties of risky assets returns may play a more important role in the scenario reduction process. However, the traditional scenario reduction methods have ignored this point. Thus, in this paper, we propose a mixed integer linear programming model to improve the reduction accuracy by minimizing the higher-order moments’ information loss between the original and reduced scenarios. Furthermore, an accelerated Benders decomposition algorithm is also designed to solve the proposed model. Hence, the aim of this paper is to extend the existing scenario reduction method in substantial and meaningful ways.

Ring hierarchical hub network design problem: Exact and heuristic solution methods

In this paper, we study the hierarchical hub network design problem with a ring-star-star structure. In this problem, the hubs are located in two layers. In the first layer, central hubs (main hubs) are located in a ring structure, and in the second layer, secondary hubs are located. Each secondary hub is allocated to a central hub. Other demand points can also be allocated to each of these hubs (central hubs or secondary hubs). The objective is to minimize transportation costs on the network. This problem applies to communication networks when establishing direct links between demand nodes is not cost-effective, and there are two levels of service for customers. Also, the ring structure is used to reduce costs associated with full communication between central hubs. We present a mixed-integer programming model for the problem and report the results of the problem solving for a numerical example on the CAB dataset. Also, we present three solution methods for the problem. First, by exploiting the decomposable structure of the proposed model, we introduce an accelerated Benders decomposition algorithm. Then, we present a hybrid genetic algorithm that uses the Dijkstra algorithm to evaluate solutions. Next, we present a hybrid variable neighborhood search algorithm that uses the Dijkstra algorithm to calculate the cost of each solution. Also, a relax-and-fix algorithm is proposed to find near-optimal feasible solutions for the problem. Computational results are presented on the USA423 dataset. The results show that the proposed relax-and-fix algorithm can solve instances with up to 100 nodes. Also, the proposed hybrid algorithms can solve instances with up to 423 nodes. The performance of the proposed hybrid variable neighborhood search is better than the proposed hybrid genetic algorithm in solving large-sized instances. • The paper formulates the hierarchical hub network using a ring structure for the central hubs. • Minimizing the total flow cost while satisfying the demands is considered. • The paper develops solution methodologies based on Accelerated Benders Decomposition, Hybrid Genetic Algorithm, and Hybrid Variable Neighborhood Search Algorithm. • The paper presents computational studies using CAB and USA423 datasets.

An accelerated Benders decomposition algorithm for stochastic power system expansion planning using sample average approximation

An accelerated Benders decomposition algorithm for stochastic power system expansion planning using sample average approximation

A robust optimization model for a dynamic closed-loop supply chain network redesign using accelerated Benders decomposition

The objective of this study is to model a dynamic redesigning closed-loop supply chain network with capacity planning in order to minimize the costs of the network. The structure of this model consists of existing facilities including manufacturing plants, distribution and reworking centers. Any such structure should change due to fluctuations in demand in order to meet customer demand. Establishing new facilities, closing the existing ones, and adding discrete capacity levels to facilities, are among the decisions which lead to necessary changes in network structure. To make the issue more realistic, it is assumed that demand and returned products are stochastic. To solve the problem, a two-stage stochastic mixed integer linear programming is modelled, followed by writing a robust counterpart of the MILP model program. An accelerated Benders decomposition algorithm is proposed to solve this model. To increase the convergence trend of this proposed algorithm, valid-inequalities and Pareto optimal cut are combined to the model. The expected performance improvement based on applying valid-inequalities and Pareto optimal cut is expressed through numerical results obtained from different samples.

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.

An Accelerated Benders Decomposition Algorithm for Solving a Double-Type Double-Standard Maximal Covering Location Problem

In this paper, a double-type double-standard model (DtDsM) for maximal covering location problem is proposed which has several applications in determining the location of public emergency facilities. DtDsM includes two types of facilities: normal and backup facilities. Although backup facilities have a greater coverage distance, they offer not a full service but only a primary service. In DtDsM, each demand point must lie within the coverage distance of a backup facility if it does not lie in the coverage distance of a normal facility ensuring it to receive minimal primary services within a predetermined time. Furthermore, an accelerated Benders decomposition algorithm is proposed to solve the model. The speed and accuracy of the algorithm are compared with the commercial solver CPLEX.

Green closed-loop supply chain network design with stochastic demand: a new accelerated benders decomposition method

Changing the structure of supply chains to move towards less polluting industries and better performance has attracted many researchers in recent studies. Design of such networks is a process associated with uncertainties and control of the uncertainties during decision-making is of particular importance. In this paper, a two-stage stochastic programming model was presented for the design of a green closed-loop supply chain network. In order to reach the environmental goals, an upper bound of emission capability that helps governments and industries to control greenhouse gas emissions was considered. During the reverse logistics of this supply chain, waste materials are returned to the forward flow by the disassembly centers. To control the uncertainty of strategic decisions, demand and the upper bound of emission capacity with three possible scenarios is considered. To solve the model, a new accelerated Benders decomposition algorithm along with Pareto-Optimal-Cut was used. The efficiency of the proposed algorithm was compared with the regular Benders algorithm. The effect of different numerical values of parameters and probabilities of scenarios on the total cost was also examined.

A three-stage stochastic model for emergency relief planning considering secondary disasters

ABSTRACT In the real world, secondary disasters occur frequently after primary disasters, and their diverse and uncertain nature along with the destruction may make the emergency relief operations more challenging. A scenario-based three-stage stochastic programming model is proposed considering the correlation between primary and secondary disasters under uncertain conditions. In order to enhance the computational tractability of the model, an accelerated Benders decomposition algorithm is formulated. In addition, to tackle large-scale cases, an approximation method employing the worst-case scenario in the third stage is established to improve the computational tractability. A computational study is performed to highlight the significance of the model and the efficiency of the proposed solution strategy. The results indicate that, by considering secondary disasters, demand satisfaction can be considerably improved compared with considering only primary disasters.

An integrated (1, T) inventory policy and vehicle routing problem under uncertainty: an accelerated Benders decomposition algorithm

ABSTRACT Inventory control and logistics are both important to the success of a supply chain. These problems become highly complicated when the demand of end-users is uncertain. The demand uncertainty could have a considerable destructive effect on a supply chain, e.g. bullwhip effect. Therefore, this paper proposes a two-stage approach to study an inventory-routing problem for a three-echelon single-item supply chain, consisting of suppliers, manufacturers, distributors, and wholesalers. The approach utilizes the ordering policy in the first stage to determine the optimal replenishment ordering quantity of wholesalers under uncertainty. This stage aims to minimize the inventory holding and lost-sales costs. To the best of our knowledge, this is the first application of the policy to an inventory-routing problem. Afterward, a mathematical formulation is proposed in the second stage to study a new vehicle routing problem. This model minimizes the transportation cost. Based on the literature, the vehicle routing problem is NP-hard; therefore, a new accelerated Benders decomposition algorithm is developed to solve large-scale instances of the problem. The algorithm incorporates a modified -optimality accelerator. The computational results show the superior performance of the modified accelerator compared to the conventional one. To evaluate the performance of the accelerated Bender decomposition algorithm, we compare it with the mathematical programming and a Genetic algorithm using twenty benchmark examples. We also introduce and assess a real-case study in the automotive parts industry. Finally, we analyze some key features of the case study to provide managerial implications.

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.

Optimizing inland waterway port management decisions considering water level fluctuations

This study presents a multi-commodity, multi-period mixed-integer linear programming model to efficiently manage the inland waterway port operations while minimizing the overall supply chain cost. An accelerated Benders decomposition algorithm is developed to solve this challenging NP-hard problem. Proposed algorithm can produce high-quality solutions consistently within a reasonable amount of time. Numerical results show that considering water level fluctuations the test region is required to use an additional 81.5% barges and 39.7% towboats. Further, managerial insights are drawn for different key input parameters that impact port operations.

An accelerated benders decomposition algorithm for a bi-objective green closed loop supply chain network design problem

Recently, social awareness, governmental legislations and competitive business environment have spurred researchers to pay much attention to closed-loop supply chain network design. In order to support the arising trend, this paper presents a comprehensive mathematical model for a multi-period, multi-product, multi-modal and bi-objective green closed-loop supply chain. The objective of the model is to minimize the total cost and environmental emissions through making the best decisions on facility location, transportation amounts and inventory balances. According to the inherent complexity of the problem and considering multi-product, multi-period and multi- modality assumptions makes it hard to handle, and as for the solution approach, an effective accelerated benders decomposition algorithm is implemented. Then, computational results for a set of numerical example are discussed. Besides, the model and solution approach are applied on a wire-and-cable industry. Then, a sensitivity analysis is implemented in an effort to validate the model. Results reveal applicability of the proposed mathematical model and presented solution approach. Following the obtained results, it can be validly concluded that the suggested solution approach leads to more than 13 percent reduction in total cost for the studied case, and can be even employed for larger and more complex real-world industrial applications.