Benders Decomposition Algorithm Research Articles

Integrated optimization of multi-machine scheduling and berth allocation in automated ore terminals

As water transportation undergoes a new technological revolution, automated terminal technology has become a key issue in reducing costs and improving efficiency. However, the operation theory limitation makes it problematic for machines to cooperate effectively and operate profitably, which is notably salient at ore terminals with their unique structure. This study focuses on the integrated scheduling of a new automated ore terminal. It comprises three main processes: (a) berth allocation, (b) splitter scheduling, and (c) feeder scheduling. We attempt to tackle this issue by developing a mixed-integer linear programming (MILP) model to minimize the vessels’ maximum departure time. Considering the complexity of the MILP model, we propose a logic-based Benders decomposition algorithm to solve the efficiency problem. The results indicate that the algorithm can solve realistically sized problem instances. The main factors affecting the computation time are the congestion level, total cargo types, and vessel service time. The integrated scheduling methods can effectively solve the actual challenges faced by terminals and provide suggestions and references for terminal expansion planning.

A Study on Disrupted Flight Recovery Based on Logic-Based Benders Decomposition Method

Aiming at the disrupted flight recovery problem, this paper established a mixed-integer programming model based on the resource assignment model to minimize the recovery cost. To deal with the large-scale examples, the Logic-Based Benders decomposition algorithm is designed to divide the problem into a master problem and sub-problems. The algorithm uses MIP in the master problem to determine flight cancellations and aircraft replacements. In the sub-problems, MIP or CP is used to determine the departure time of delayed flights. Later, incorporating sectional constraints into the main problem and iterating until an optimal solution is obtained. Furthermore, the added cutting plane constraint in the iterations of the Benders decomposition algorithm are strengthened to eliminate more inferior solutions. By comparing the results of CPLEX, the Logic-Based Benders decomposition algorithm, and the enhanced Benders decomposition algorithm, it is verified that the improved Benders decomposition algorithm can solve large-scale examples more efficiently with a faster time and fewer iterations.

Designing a sustainable supply chain for battery PVC cases: A multi-objective optimization approach

Nowadays, Polyvinyl Chloride (PVC) and its derivatives are being used in a wide range of products, including car battery cases, due to their corrosion resistance and exceptional durability. As PVC production has grown, the amount of non-hazardous waste generated from PVC products has dramatically increased. In this context, incorporating PVC waste management into the PVC supply chain can effectively address the PVC waste generation issue. This study investigates the design of the Polyvinyl Chloride Closed-Loop Supply Chain (PVCCLSC) based on a triple-objective model for Abadan Petrochemical Company. The first objective aims to minimize the total cost of PVCCLSC, the second objective focuses on minimizing environmental impact, and the third objective maximizes the improvement of Occupational Health and Safety (OHS). In this study, an integrated solution approach is developed encompassing four main phases: (i) Developing an accelerated K-means clustering algorithm, along with the Elbow Method (EM) and Average Silhouette Method (ASM), to estimate the quality of returned PVC products, (ii) Employing a robust optimization method to capture the uncertainty, (iii) Implementing an augmented ε-constraint method to solve the proposed multi-objective model, and (iv) Applying a Bender’s Decomposition Algorithm (BDA) to solve the model and determine optimal solutions. As a result of the proposed BDA, relative optimality gaps have been reduced from 4.56 % in the conventional solution to zeros, and runtime has been reduced from 55 min to 20 min. Furthermore, as a result of this study, more than 44 % of returned PVC products should be recycled using the tertiary method to fulfill environmental requirements.

Multistage Stochastic optimization for mid-term integrated generation and maintenance scheduling of cascaded hydroelectric system with renewable energy uncertainty

The uncertainties resulting from the escalating penetration of renewable energy resources pose severe challenges to the efficient operation of modern power systems. Hydroelectricity is characterized by its flexibility, controllability, and reliability, and thus becomes one of the most ideal energy resources to hedge against such uncertainties. This paper studies the mid-term integrated generation and maintenance scheduling of a cascaded hydroelectric system (CHS) consisting of multiple cascaded reservoirs and hydroelectric units. To precisely describe the mid-term water regulation policies, the hydraulic coupling relationship and water-energy nexus of CHS are incorporated into the proposed optimization model. The uncertainties of natural water inflow and the power outputs of wind/solar energy generation are taken into consideration and captured via a stochastic process modeled by a scenario tree. A multistage stochastic optimization (MSO) approach is developed to coordinate the complementary operations of multiple energy resources, by optimizing the mid-term water resource management, generation scheduling, and maintenance scheduling of CHS. The proposed MSO model is formulated as a large-scale mixed-integer linear program that presents significant computational intractability. To address this issue, a tailored Benders decomposition algorithm is developed. Two real-world case studies are conducted to demonstrate the capability and characteristics of the proposed model and algorithm. The computational results show that the proposed MSO model can exploit the flexibility of hydroelectricity to efficiently respond to variable wind and solar power, and reserve water resources for the generation in peak months to reduce the consumption of fossil fuel. The proposed solution approach also exhibits promising computational efficiency when handling large-scale models.

A generalized Benders decomposition approach for the optimal design of a local multi-energy system

A local multi-energy system (LMES) is a decentralized energy system producing energy under multiple forms to satisfy the energy demand of a set of buildings located in its neighborhood. We study the problem of optimally designing an LMES over a multi-phase horizon. This problem is formulated as a large-size mixed-integer linear program with a block-decomposable structure involving mixed-integer sub-problems. We propose a new way to adapt a recently published framework for generalized Benders decomposition to our problem. This is done by exploiting the fact that the constraint matrix appearing in front of the first-stage variables in the coupling constraints is non-negative. The obtained generalized Benders decomposition algorithm relies on the use of a new type of non-convex Benders cuts involving indicator functions. We first prove that, under the assumption that all first-stage decision variables are integer and bounded, the finite and optimal convergence of our algorithm is guaranteed in theory. We then investigate how to obtain a good numerical performance in practice. Finally, we report the results of a computational study carried out on a real-life case study. These results show that the proposed algorithm clearly outperforms both a mathematical programming solver directly solving the problem as a whole and a state-of-the art hierarchical decomposition algorithm.

Power generation–network–load–energy storage co-planning under uncertainty

With the aggregation of renewable energy in the power system, the uncertainty caused by the renewable energy affects the planning and operation of power systems. Meanwhile, the existing planning models fail to consider renewable energy uncertainty methods, specifically concerning renewable energy confidence and future possible scenarios; thus, a confidence-based scenario cluster method is presented. A novel generator, network, load, and energy storage (GNLS) co-planning model is proposed in the paper. First, a confidence-based scenario cluster is built, which can reflect uncertainties by clustering and analyzing wind, solar, and load. Second, the proposed model focuses on load and energy storage co-planning, and in addition, relevant flexible indices are used to assess the model. Finally, the GNLS co-planning model is built as a bi-level stochastic model on continuous time scales. The model is solved using the Benders decomposition algorithm. The method in this paper is validated using an IEEE RTS 24-bus and a real test system in China to demonstrate the reduction in renewable energy curtailment and optimization of economic factors in power system planning.

Benders decomposition algorithms for minimizing the spread of harmful contagions in networks

The COVID-19 pandemic has been a recent example for the spread of a harmful contagion in large populations. Moreover, the spread of harmful contagions is not only restricted to an infectious disease, but is also relevant to computer viruses and malware in computer networks. Furthermore, the spread of fake news and propaganda in online social networks is also of major concern. In this study, we introduce the measure-based spread minimization problem (MBSMP), which can help policy makers in minimizing the spread of harmful contagions in large networks. We develop exact solution methods based on branch-and-Benders-cut algorithms that make use of the application of Benders decomposition method to two different mixed-integer programming formulations of the MBSMP: an arc-based formulation and a path-based formulation. We show that for both formulations the Benders optimality cuts can be generated using a combinatorial procedure rather than solving the dual subproblems using linear programming. Additional improvements such as using scenario-dependent extended seed sets, initial cuts, and a starting heuristic are also incorporated into our branch-and-Benders-cut algorithms. We investigate the contribution of various components of the solution algorithms to the performance on the basis of computational results obtained on a set of instances derived from existing ones in the literature.

Decomposition algorithms for the robust unidirectional quay crane scheduling problems

Our study focuses on a two-stage robust optimization model for the unidirectional quay crane scheduling problem with uncertain handling times at container terminals. We first implement two classical algorithms adopted in Li and Zhang (2021), the Benders decomposition algorithm and the column-and-constraint generation algorithm, to solve the robust model. Based on analytical and numerical comparisons of them, we design an exact hybrid algorithm to leverage capabilities of both algorithms by alternatively adding cuts of both kinds. Extensive experiments validate the effectiveness of this mechanism. Numerical experiments also reveal that the benefit and the cost of robustness fade away as the uncertainty budget increases and highlight the advantage of our robust approach to deal with uncertainty over a two-stage stochastic program under extreme situations.

Computationally efficient solution of mixed integer model predictive control problems via machine learning aided Benders Decomposition

Mixed integer Model Predictive Control (MPC) problems arise in the operation of systems where discrete and continuous decisions must be taken simultaneously to compensate for disturbances. The efficient solution of mixed integer MPC problems requires the computationally efficient online solution of mixed integer optimization problems, which are generally difficult to solve. In this paper, we propose a machine learning-based branch and check Generalized Benders Decomposition algorithm for the efficient solution of such problems. We use machine learning to approximate the effect of the complicating variables on the subproblem by approximating the Benders cuts without solving the subproblem, therefore, alleviating the need to solve the subproblem multiple times. The proposed approach is applied to a mixed integer economic MPC case study on the operation of chemical processes. We show that the proposed algorithm always finds feasible solutions to the optimization problem, given that the mixed integer MPC problem is feasible, and leads to a significant reduction in solution time (up to 97% or 50×) while incurring small error (in the order of 1%) compared to the application of standard and accelerated Generalized Benders Decomposition.

Robust two-stage optimization consensus models with uncertain costs

In the consensus-reaching process (CRP), decision-makers (DMs) frequently encounter the dilemma of too much uncertain information, which can lead the actual decision to deviate from the optimal solution obtained by the currently used consensus models. To do this, we construct two robust two-stage optimization consensus models with uncertain costs and obtain their robust two-stage counterparts. We then apply a Benders decomposition algorithm to solve the resulting models. Finally, the experimental results show that the new models are better suited for uncertain contexts and could help DMs produce more reliable choices.

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.

A Benders decomposition algorithm for resource allocation with multi-resource operations

This paper addresses a real life scheduling problem characterised by multi-resource operations whose completion simultaneously requires more than one (renewable) resource of different types. Such problems arise in various companies not only in manufacturing but also in services. Solving such a problem needs to address two interconnected subproblems: a sequencing subproblem and a resource-allocation subproblem if more than one resource is available in some types. The resource-allocation subproblem consists of allocating to each operation the resources it requires while the sequencing subproblem consists of determining the order in which each resource performs the operations assigned to it. This paper focuses on the resource-allocation subproblem. It generalises the basic scheduling problems which consist of determining the operations' starting or completion times for a given processing sequence for every resource. We consider a cost function taking into account the makespan, the cost of resource utilisation, and the load imbalance among the resources of the same type. We first formulate the problem into a mixed-integer linear program (MILP). To efficiently solve it, even in practical-sized instances, an exact algorithm called BD (Benders decomposition with enhancing cuts) is developed where the master problem only considers integer variables. We prove that the slave problem can be transformed into finding the longest paths in a digraph and therefore can be solved with the Bellman–Ford algorithm. To enhance the efficiency of the method, equivalent solutions are limited in the master problem. The performance of the approach is evaluated by comparing it against CPLEX, a state-of-the-art commonplace MILP solver, used to directly solve the initial MILP. The computational results demonstrate that BD provides competitive solutions in all upper and lower bounds. In particular, it improves, compared with CPLEX, the upper and lower bounds by 5.07% and 4.63%, respectively, in solving practical-sized instances. The experiment also shows that considering load balancing can make more rational use of resources and avoid adverse effects caused by excessive workload of staff and imbalanced use of equipment, which is very important in real-world production.

A heuristic-based benders decomposition for the hybrid relief distribution network minimizing deprivation cost of vulnerable areas

PurposeThe study aims to optimize truck routes by minimizing social and economic costs. It introduces a strategy involving diverse drones and their potential for reusing at DNs based on flight range. In HTDRP-DC, trucks can select and transport various drones to LDs to reduce deprivation time. This study estimates the nonlinear deprivation cost function using a linear two-piece-wise function, leading to MILP formulations. A heuristic-based Benders Decomposition approach is implemented to address medium and large instances. Valid inequalities and a heuristic method enhance convergence boundaries, ensuring an efficient solution methodology.Design/methodology/approachResearch has yet to address critical factors in disaster logistics: minimizing the social and economic costs simultaneously and using drones in relief distribution; deprivation as a social cost measures the human suffering from a shortage of relief supplies. The proposed hybrid truck-drone routing problem minimizing deprivation cost (HTDRP-DC) involves distributing relief supplies to dispersed demand nodes with undamaged (LDs) or damaged (DNs) access roads, utilizing multiple trucks and diverse drones. A Benders Decomposition approach is enhanced by accelerating techniques.FindingsIncorporating deprivation and economic costs results in selecting optimal routes, effectively reducing the time required to assist affected areas. Additionally, employing various drone types and their reuse in damaged nodes reduces deprivation time and associated deprivation costs. The study employs valid inequalities and the heuristic method to solve the master problem, substantially reducing computational time and iterations compared to GAMS and classical Benders Decomposition Algorithm. The proposed heuristic-based Benders Decomposition approach is applied to a disaster in Tehran, demonstrating efficient solutions for the HTDRP-DC regarding computational time and convergence rate.Originality/valueCurrent research introduces an HTDRP-DC problem that addresses minimizing deprivation costs considering the vehicle’s arrival time as the deprivation time, offering a unique solution to optimize route selection in relief distribution. Furthermore, integrating heuristic methods and valid inequalities into the Benders Decomposition approach enhances its effectiveness in solving complex routing challenges in disaster scenarios.

The flexible airport bus and last-mile ride-sharing problem: Math-heuristic and metaheuristic approaches

Airport buses play a crucial role in addressing the last-mile problem of air travel, especially in cities and countries lacking inner-city rail transit systems. Nevertheless, airport buses are currently witnessing a decline in ridership due to drawbacks such as long departure intervals, inflexible stops, and considerable distances between stops. Consequently, delivering high-quality airport bus services has become a pressing concern for public transport operators. Motivated by new flexible buses and ride-sharing services, this paper explores a flexible airport bus service that integrates ride-sharing services for passengers traveling from bus stops to their destinations. This problem entails integrated decisions involving bus stop selection, passenger assignment to drop-off bus stops, as well as bus and ride-sharing routing. Accordingly, this problem presents more challenges in decision-making than traditional flexible bus or ride-sharing routing problems. We first develop an arc-based mixed-integer linear programming model. Subsequently, we design a double decomposition math-heuristic algorithm that builds upon logic-based Benders decomposition and column generation algorithms to obtain a near-optimal solution within practical computation time limits for practical-scale instances. Additionally, we implement an adaptive large neighborhood search algorithm to evaluate the solution quality of this math-heuristic algorithm and to solve large-scale instances. To validate the effectiveness of both the model and the algorithms, we conduct numerical experiments using instances derived from Shenzhen airport bus lines. The experimental results demonstrate that the flexible service mode offers significant advantages in reducing both passenger ride time and vehicle mileage over traditional airport bus or taxi modes.

A Mutual Catastrophe Insurance Framework for Horizontal Collaboration in Prepositioning Strategic Reserves

Improving Disaster Preparedness Through Mutual Catastrophe Insurance In “A Mutual Catastrophe Insurance Framework for Horizontal Collaboration in Prepositioning Strategic Reserves,” H. Zbib, B. Balcik, M.-È. Rancourt, and G. Laporte present an innovative approach to collaborative disaster preparedness. The novel framework considers a risk-averse mutual insurer offering multiyear insurance contracts with coverage deductibles and limits to a portfolio of risk-averse policyholders. It is designed to foster horizontal collaboration among policyholders for joint disaster preparedness by effectively integrating operational and financial functions. The problem is modeled as a large-scale nonlinear multistage stochastic program and solved by using an effective Benders decomposition algorithm. The framework is validated with real data from 18 Caribbean countries focusing on hurricane preparedness. Given the predicted impacts of climate change, the proposed multiyear mutual catastrophe insurance framework promises to reshape global disaster preparedness and make a profound societal impact by providing a transparent disaster financing plan to protect vulnerable regions. The study’s findings stress the importance of long-term cooperation, prenegotiation of indemnification policies, and strategic setting of deductibles and limits by taking into account the correlation between policyholders.

Multi-objective mixed integer programming modelling for closed-loop supply chain network design: an enhanced Benders decomposition algorithm

Considering a closed-loop supply chain (CLSC) network with uncertain demand and recycling rates, this article innovatively designs a multi-objective mixed-integer programming model that incorporates corporate social responsibility (CSR), a facility retrofit strategy (FRS), a flexible supply strategy (FSS) and a vehicle selection strategy (VSS). Then, robust optimization methods are applied to construct robust models under three situations of uncertainty. For the computational complexity of large-scale problems, an enhanced Benders decomposition algorithm (EBDA) is designed. A numerical case analysis is conducted using five different scale instances. First, compared to other algorithms, EBDA accelerates the solution efficiency while ensuring convergence. Secondly, the sensitivity of the objective weights and the trade-offs of multiple objectives are analysed. Finally, the impact is analysed of the uncertain environment, CSR, an FRS, an FSS and a VSS on the CLSC network. Decision makers need to balance three objectives to manage CLSC and use these strategies appropriately to address the negative impact of an uncertain environment.

Global multi-sourcing network design with inventory planning under uncertainty

ABSTRACT In the current era of turbulence, demand uncertainty and significant supply delays have become recurrent challenges for every global supply chain, which are accentuated by geopolitical issues, regional conflicts, natural disasters, etc. Companies are keen to decide when and where nearshoring and offshoring their supply locations and to adapt their inventory control policies to cope with significant perturbations. To deal with these challenges, we investigate the integration of inventory planning at the strategic level through a novel setting of the location-inventory problem under uncertainty. This problem is characterised by inbound multi-sourcing, order splitting among multiple suppliers, and optimised inventory decisions from sourcing to customers under uncertain demand and lead time. A two-stage stochastic optimisation model is proposed and solved using a Benders decomposition algorithm, supported by a sample average approximation approach. The results show that both lead time and demand uncertainties impact strategic and tactical decisions of the global supply network. Specifically, multi-sourcing and inbound order splitting policies have proven beneficial when the network faces a high lead time uncertainty. Based on an illustrative case of a global automobile spare parts supply network under different supply-side scenarios, managerial insights are gained on the benefit of onshore, nearshore, and offshore decisions. When facing a high lead time uncertainty, as in the pandemic case or any geopolitical episode, our findings suggest the necessity to rely on nearshoring for the supply side.

Two-stage recoverable robust optimization for an integrated location–allocation and evacuation planning problem

We consider an integrated location–allocation and evacuation planning problem in a disaster context, where the effects of a disaster, including the uncertain capacities of relief facilities (rescue centers and distribution centers), uncertain demands for relief supplies and casualty treatment services, and uncertain availability of transportation links are characterized by a discrete scenario set. Instead of complete failures, we allow the disrupted relief facilities only lose part of their capacity. To deal with the uncertainties, we propose a two-stage recoverable robust optimization model, where the location decision of relief facilities, the allocation decision of delivering relief supplies from relief facilities to affected areas, the transfer decision of transporting casualties from affected areas to rescue centers etc are defined in two stages where the first-stage solution should be robust against the possible effects of a disaster that are revealed in the second stage, and the second-stage solution involves some recovery actions, which we term as multi-mitigation strategies: re-opening and re-operation, re-allocation, and relief supply sharing, to mitigate the effects. To solve the model to optimality, we develop a nested two-stage decomposition algorithm that iterates between a master problem considering only a subset of disaster scenarios solved by a Benders decomposition algorithm that incorporates some non-trivial acceleration strategies, and an adversarial separation problem that identifies disaster scenarios to enhance the worst-case recovery cost of the master problem. We introduce some warm-start techniques to accelerate the convergence of the solution algorithm. We conduct numerical studies on simulation instances to assess the performance of the solution algorithm, and analyze the robustness and recoverability of the model. We also conduct extensive numerical studies on realistic instances from Ya’an and Ganzi to demonstrate the benefits of accounting for recoverable robustness over a stochastic policy and a robust policy without recovery actions, the benefits of considering integrated optimization over sequential optimization, and the benefits of considering partial capacity loss and multi-mitigation strategies.

A Competitive Bilevel Programming Model for Green, CLSCs in Light of Government Incentives

The growth of world population has fueled environmental, legal, and social concerns, making governments and companies attempt to mitigate the environmental and social implications stemming from supply chain operations. The state-run Environmental Protection Agency has initially offered financial incentives (subsidies) meant to encourage supply chain managers to use cleaner technologies in order to minimize pollution. In today’s competitive markets, using green technologies remains vital. In the present project, we have examined a class of closed-loop supply chain competitive facility location-routing problems. According to the framework of the competition, one of the players, called the Leader, opens its facilities first. The second player, called the Follower, makes its decision when Leader’s location is known. Afterwards, each customer chooses an open facility based on some preference huff rules before returning the benefits to one of the two companies. The follower, under the influence of the leader’s decisions, performs the best reaction in order to obtain the maximum capture of the market. So, a bilevel mixed-integer linear programming model is formulated. The objective function at both levels includes market capture profit, fixed and operating costs, and financial incentives. A metaheuristic quantum binary particle swarm optimization (PSO) is developed via Benders decomposition algorithm to solve the proposed model. To evaluate the convergence rate and solution quality, the method is applied to some random test instances generated in the literature. The computational results indicate that the proposed method is capable of efficiently solving the model.

Supplier selection under disruption risk with hybrid procurement

In today’s global landscape, supply chains face a plethora of risks due to the increasingly dynamic and turbulent business environment. In this study, we introduce a supplier selection model with hybrid procurement under disruption risk. Specifically, we consider the supplier selection problem of a firm with multiple units (or business segments) demanding several products, raw materials, and services. The firm has a centralized supplier selection strategy at the corporate level. However, instead of allocating orders to selected suppliers, the corporate management allows each unit to use its most preferred suppliers from the selected ones in a decentralized manner. To mitigate the impact of disruptions, less preferred suppliers of each unit serve as a backup. We present a mathematical model for this problem and study the structural properties of our formulation. We develop an exact Benders decomposition algorithm, and also present a method to generate upper and lower bounds. We compare the performance of the proposed solution algorithms on large-scale instances. Finally, we present a numerical example to demonstrate the impact of decentralized procurement with different objectives. Our results show that the expected procurement cost can increase significantly if suppliers are selected without considering the decentralized decisions of units.