Benders Decomposition Approach Research Articles

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.

A Benders decomposition approach for a new sustainable pharmaceutical supply chain network: a case study in France

A Benders decomposition approach for a new sustainable pharmaceutical supply chain network: a case study in France

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.

An accelerated Benders decomposition approach for virtual base station formation in stochastic Cloud-RANs

In Cloud-RANs, the baseband unit (BBU) is separated from the remote radio head (RRH) at the cell site and migrated to a server cloud to decrease deployment and maintenance costs. A base station (BS) can be formed as a network slice in which the virtual network functions (VNFs) of its BBU can be placed on any server by network virtualization (NFV) technology and logically connected by software-defined networking (SDN). However, the uncertain demands of users make it very difficult to determine a network slice with appropriate VNF placement and flow routing between VNFs. We aim to minimize the cost of providing BS services while meeting their time-varying demands. We consider the network slice scheme and fronthaul establishment jointly and formulate this as an integer linear programming (ILP) problem. We propose scenario classification Benders decomposition (CBD) based on Benders decomposition (BD) but with fewer Benders cuts, accelerating the solution convergence. Moreover, we propose an efficient heuristic approach to obtain good solutions for the ILP problem based on the solutions of its relaxed problems obtained by BD or CBD. The numerical experiments show that when combining our heuristic approach with CBD, the results are close to the optimal solutions of the ILP problem, while the computation time is significantly shorter than those of previous approaches.

Integrated multi-product reverse supply chain design and disassembly line balancing under uncertainty

End-of-life (EOL) product recycling has received increasing attention because of potential environmental, social and economic benefits. A well-designed reverse supply chain (RSC) can efficiently handle EOL products. As the critical activity in the RSC, the disassembly process decomposes collected EOL products into components to fulfill the demands of remanufacturing plants. Efficiently coordinating RSC design and disassembly line balancing decisions may improve the whole system performance significantly, especially when facing multiple EOL products and uncertainty. This paper investigates a novel integrated RSC design and disassembly line balancing problem to handle multiple EOL products where the supply of EOL products, the demand for components, and task times in disassembly are stochastic. This complex problem needs to jointly determine the number and locations of disassembly plants, disassembly equipment procurement, disassembly line balancing, and inventory levels of both EOL products collected and components dismantled. The objectives are to maximize the expected profit and minimize the carbon emissions simultaneously. For the problem, a bi-objective two-stage stochastic programming model is formulated and an exact ɛ-constrained method is proposed, transforming the bi-objective problem into a series of single-objective problems. Especially, an improved Benders decomposition approach is developed to solve each single-objective problem efficiently. Numerical experiments comprising an illustrative case and 200 random instances are conducted to evaluate the performance of proposed methods. Moreover, some managerial insights are drawn.

A Benders decomposition approach for planning home blood donations

AbstractBlood donations are crucial for the health system. We consider the problem of planning blood donation services, where the donors are reached at home. The scope is to minimize the penalty for the unserved donors, while guaranteeing that the available resources for implementing the service are not exceeded and that the appointment preferences of the donors are met. We present an offline model for this setting, where the produced solution must be robust with respect to the availability of the donors, which is not known in advance and is managed in a stochastic way using scenarios. A Benders decomposition approach to solve this model is developed. The proposed algorithm is tested on real-life instances coming from the Milan department of the Associazione Volontari Italiani Sangue (AVIS).

Integrated commercial and operations planning model for schedule design, aircraft rotation and crew scheduling in airlines

AbstractThe commercial and operations planning in airlines has traditionally been a hierarchical process starting with flight schedule design, followed by fleet assignment, aircraft rotation planning and finally the crew scheduling. The hierarchical planning approach has a drawback that the optimal solution for a planning phase higher in hierarchy may either be infeasible for the subsequent phase or may lead to a sub‐optimal overall solution. In this paper, we solve a profit‐maximizing integrated planning model for clean‐sheet “rotated” schedule design with flight re‐time option and crew scheduling for a low‐cost carrier (LCC) in an emerging market. While the aircraft rotation problem has been traditionally modeled in the literature as a daily routing of individual aircraft for maintenance requirement, in this work we address the requirement of planned aircraft rotations as part of schedule design for LCCs. The planned aircraft routing is important in our case to create as many via‐flights as possible due to the underserved nature of the emerging market. We solve this large‐scale integer‐programming problem using two approaches – Benders Decomposition and Lagrangian Relaxation. For Lagrangian Relaxation, we exploit the special structure of our problem and intuitive understanding behind the Lagrangian duals to develop a multiplier adjustment approach to find an improved lower bound of integrated model solution. The crew‐pairing sub‐problem is solved using column generation through multi‐label shortest path algorithm followed by branch‐and‐price for integer solution. We test our solution methodology on a flight universe of 378 unique flights for different problem sizes by varying the number of aircraft available for operations. Our computational results show that within a reasonable run time of few hours both the approaches, Benders Decomposition and Lagrangian Relaxation, are successful in finding lower bounds of the integrated model solution, which are higher than the solution of traditional hierarchical approach by 0.5%–2.5%. We find Lagrangian Relaxation methodology to usually attain an improved solution faster than the Benders Decomposition approach, particularly for large‐scale problems.

Optimal Establishments of Massive Testing Programs to Combat COVID-19: A Perspective of Parallel-Machine Scheduling-Location (ScheLoc) Problem

Massive testing to identify COVID-19-infected people is crucial in combating COVID-19. However, from the perspective of facility location problems, many current massive testing programs are not properly set, leading to unreasonable travelling distances, long makespan, unbalanced workload, and long queues. This article proposes a decision framework for developing massive testing programs. Specifically, a biobjective parallel-testing-site Scheduling-location (ScheLoc) model is formulated, simultaneously minimizing the makespan and total travelling distance. The former can help reduce the time length of potential virus spread, and the latter can help alleviate the risk of virus spread and traveler inconvenience. To solve the proposed biobjective ScheLoc problem, in addition to the standard <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\bm {\epsilon }$</tex-math></inline-formula> -constraint method, we further develop two novel strategies. The first one iteratively solves simpler approximate MIP models (IMIP). The second innovatively extends the classical logic-based Benders decomposition approach to solve biobjective problems (B-LBBD). A Hong Kong-based case study shows that the proposed decision framework can significantly reduce the makespan and travelling distance (with a mean of 13 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\%$</tex-math></inline-formula> and 5.1 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\%$</tex-math></inline-formula> , respectively) and enhance workload balancing. Besides, the developed solution methods, especially the B-LBBD, outperform the adapted <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\bm {\epsilon }$</tex-math></inline-formula> -constraint method in various aspects.

Research on dynamic robust planning method for active distribution network considering correlation

The universality of load subjects in distribution network brings challenges to the reliability of distribution network planning results. In this paper, a two-stage dynamic robust distribution network planning method considering correlation is proposed. The method evaluates the correlation between random variables using the Spearman rank correlation coefficient, and converts the correlated random variables into mutually independent random variables by Cholesky decomposition and independent transformation; expresses the source-load uncertainty by a bounded interval without distribution, and describes the active distribution network planning as a dynamic zero-sum game problem by combining with the two-phase dynamic robust planning; use the Benders decomposition approach to tackle the issue; mathematical simulation is used to confirm the accuracy and efficacy of the method. The results show that the dynamic robustness planning method of active distribution network taking into account the correlation can accurately simulate the operation of active distribution network with uncertain boundaries, which enhances the reliability and economy of the active distribution network planning results.

A data-driven mathematical model to design a responsive-sustainable pharmaceutical supply chain network: a Benders decomposition approach

A data-driven mathematical model to design a responsive-sustainable pharmaceutical supply chain network: a Benders decomposition approach

Accelerating Benders decomposition approach for shared parking spaces allocation considering parking unpunctuality and no-shows

By allocating idle private parking spaces to demanders, shared parking reuses idle resources and effectively alleviates parking problems. In practice, there may be parking unpunctuality behavior of demanders, as well as the potential no-show behavior of owners and demanders. These stochastic factors affect the allocation of shared parking spaces. In this paper, we study the shared parking spaces allocation problem considering parking unpunctuality and no-shows. First, the allocation problem considering parking unpunctuality is formulated as a stochastic programming model, with the objective to maximize the profit of the shared parking platform. Sample average approximation (SAA) combined with an accelerating Benders decomposition (ABD) algorithm is exploited to solve the model effectively, where initial optimality cuts are explored to speed up the convergence of the algorithm. Then the proposed approach is extended to incorporate no-shows. Numerical experiments show great performance of the proposed approach. The experiments also show that the fluctuations of the unpunctual time and the no-show probability have a significant impact on the shared parking system, and balanced supply and demand is helpful to increase the profit of the platform and the satisfaction of demanders.

Distributed risk-constrained strategy management of cloud energy storage for residential and electricity market services provision

Energy storage systems can improve the performance of home energy management systems and also provide ancillary services in electricity markets. However, the economic and technical insufficiency of individual home storage units, makes employing them inefficient for domestic storage and market applications, respectively. Therefore, central storage units with capacity sharing capability to small consumers as Cloud Energy Storages (CESs) are becoming a promising solution, and are also potentially able to contribute to ancillary service provision in the market. In this context, this paper aims to develop a novel stochastic day-ahead distributed management framework for CES to simultaneously adopt the optimal strategy of storage allocation for photovoltaic-integrated homes and participation in energy and ancillary service markets. The developed distributed framework is formulated within the Benders decomposition approach in which CES strategy and home energy management systems are optimized through master and subproblems based on linear mathematical models, respectively. Furthermore, the uncertainty of market prices is handled by the Conditional Value at Risk (CVaR) method. The developed model is assessed through real-world case studies of ERCOT. The results highlight the effectiveness of the collaborative service framework for both CES and households. Energy arbitrage through local storage services compensates for CES's challenges such as degradation and low power rating in the provision of energy services. These improvements benefit CES owners with higher net revenues and households with reduced cost of electricity supply.

Decision‐dependent distributionally robust integrated generation, transmission, and storage expansion planning: An enhanced Benders decomposition approach

AbstractIntegrated generation, transmission, and storage expansion planning (IGT&amp;SP) is the cornerstone to realize low‐carbon transition considering security constraints in the long run. A novel IGT&amp;SP planning scheme is proposed to balance the planning cost, i.e. renewable energy sources (RESs) and energy storage systems, based on the distributionally ambiguity sets. A novel decision‐dependent ambiguity set is proposed to capture the relation between the uncertainties of RES output and long‐term planning. A two‐stage risk‐averse distributionally robust optimization is formulated, where the RESs, energy storage systems, and transmission line expansion are optimized in the first stage and a unit commitment problem is proposed in the second‐stage optimization to assess the performance of the expanded system. This problem is reformulated into a two‐stage optimization problem with complete mixed‐integer recourse, where the state variable is binary. A novel enhanced Benders decomposition algorithm is proposed to solve the IGT&amp;SEP efficiently, where the cutting planes are generated by a primal‐dual relaxation of the recourse problem. Simulations are conducted on the modified IEEE‐30 test system and modified IEEE‐118 test system. Compared with adjustable robust optimization and L1‐norm Wasserstein distance‐based distributionally robust optimization, numerical results verify the effectiveness of the proposed IGT&amp;SP, together with the solution algorithm.

Efficient solution approaches for the bi-criteria [formula omitted]-hub median and dispersion problem

In this paper, we study the bi-criteria p-hub median and dispersion problem, that arises in the design of hub networks where the dispersion of hubs is desired to mitigate the risk of disruptions. The problem is formulated as a bi-objective mixed integer program, where the first objective is to minimize the total cost of routing the flows through p hubs and the second objective is to maximize the minimum distance (or dispersion) among the selected p hub locations themselves. We present two exact solution approaches that guaranteed to obtain the entire non-dominated Pareto frontier. The first is a cutting plane method in which a p-hub median problem with a particular dispersion distance is solved at each iteration. Three formulations of the problem, based on the different type of cuts and preprocessing, are presented. We study the dominance relationship among the three formulations. Through computational experiments, we show that the proposed cutting plane method is efficient in solving medium size instances of the problem and our strongest formulation is at least 40% computationally faster than the others. For solving large instances of the problem, we present a decomposition method where the p-hub median problem with dispersion distance is solved using an accelerated Benders decomposition approach. We present several problem specific enhancements to the algorithm such as starting with a better solution, efficient ways of solving decomposed subproblem and adding Pareto optimal Benders cuts to the master problem. The computational results on the Turkish network (TR81), US423, and Australian Post (AP) dataset show that the cutting plane algorithm with the proposed decomposition procedure is three to four times faster than the commercial solver.

A Stochastic Optimization Approach to Energy-Efficient Underground Timetabling Under Uncertain Dwell and Running Times

We consider a problem from the context of energy-efficient underground railway timetabling, in which an existing timetable draft is improved by slightly changing departure and running times. In practice, synchronization between accelerating and braking trains to utilize regenerative braking plays a major role for the energy efficiency of a timetable. Because deviations from a planned timetable may lead to unnecessarily high energy consumption during actual operation, we include operational uncertainties in our model to create a timetable that remains energy efficient, even if deviations from the nominal timetable occur. To solve the problem, we use a scenario expansion model in conjunction with a Benders decomposition approach. As an alternative to solving the Benders subproblems, we present a heuristic sparse cut that can be computed efficiently. The resulting sparse-cut heuristic produces high-quality solutions on a set of real-world instances stemming from the Nürnberg underground system, outperforming the integrated mixed-integer programming approach as well as the basic Benders approach. Additionally, we evaluate two static recovery strategies—shortening dwell times as well as shortening dwell and running times—to determine the cost and benefit of handling delays using a simple static rule. In our experiments, we are able to reduce the energy consumption by up to 9.4% and confirm that delay recovery via shortening dwell times is an energy-efficient and effective way to increase punctuality at low cost in terms of energy. Funding: This research was supported by the Bavarian Ministry of Economic Affairs, Regional Development and Energy through the Center for Analytics – Data – Applications (ADA-Center) within the framework of “BAYERN DIGITAL II” (20-3410-2-9-8).

Network equilibrium of battery electric vehicles considering drivers’ resting behavior

Driving fatigue cost is a major component of vehicle drivers’ travel costs in an intercity or regional network. The charging behavior of electric vehicle (EV) drivers, which is generally synchronized with drivers’ resting behavior, can contribute to the mitigation of drivers’ fatigue, especially after a prolonged driving period. Overlooking the impact of driving fatigue on the travel cost may overestimate the side-effects of the charging behavior for EV drivers, and result in biased flow and charging demand distribution. In this study, by considering the fatigue as part of the travel cost, we make the first attempt to investigate the impact of EV drivers’ charging and resting behaviors on their fatigue cost, and thus their travel plans and resultant flow distribution across the network. To this end, a novel network equilibrium modeling framework is first developed to capture the interaction among EV drivers’ travel plans, which specify the routing, recharging, and resting plans on a general intercity or regional network where charging stations and rest stops are deployed. When traveling between their origins and destinations, EV drivers are assumed to determine their travel plans to minimize their individual travel cost composed of driving time, rest time, charging cost, and fatigue cost, while preventing their batteries from being exhausted. The equilibrium model is then formulated as a variational inequality and transformed into a nonlinear optimization problem. An efficient solution algorithm integrating column generation and Benders decomposition approach is proposed to solve the established optimization problem. Numerical examples are presented to demonstrate the performance of the proposed models and solution algorithm. Numerical results validate that considering the impact of driving fatigue on the travel cost emphasizes the need for en-route charging for EV drivers with long-distance trips, and has an appreciable impact on the network flow and charging demand distribution. In addition, large-sized batteries and fast chargers may not necessarily reduce drivers’ travel costs for long-distance travel since charging behavior can be synchronized with drivers’ resting behavior and thus contribute to the mitigation of drivers’ fatigue.

Integrated optimization of rolling stock allocation and train timetables for urban rail transit networks: A benders decomposition approach

We investigate the integrated optimization of rolling stock allocation and train timetables (RATT) in an urban rail transit network with multiple connected lines and rolling stock depots. Different from most existing research on single-line cases, we consider that the rail manager plans to allocate a certain fleet of rolling stock to depots to implement the operational timetables of multiple lines, where each depot can serve more than one line but has limited capacity. By means of the construction of a space–time network, we formulate RATT into a bi-objective mixed-integer linear programming (MILP) model. The model can simultaneously generate the rolling stock allocation plan and train schedules for the considered lines, with the aim of optimizing both the investment cost of rolling stock and the service quality of passengers. Due to the computational complexity of large-scale real-world instances, we develop a Benders decomposition-based solution algorithm, which decomposes the MILP into a rolling stock assignment problem and a set of independent subproblems, i.e., a train scheduling subproblem for each line. As the subproblems also contain integral variables, we further prove that a large portion of the subproblems, after proper reformulation, has the integrality property and can be solved rapidly via linear relaxation. We test our integrated approach and solution algorithms on real-world instances from the Beijing rail transit network. The results show that our solution approach can obtain near optimal solutions, while the commercial solvers cannot even return feasible solutions on real-world instances. Compared to the existing plan in the Beijing metro, our RATT approach can improve the service quality by 17.6%, on average, using the same fleet size of rolling stock.

A logic-based Benders decomposition solution approach for two covering problems that consider the underlying transportation

We investigate two maximal covering location problems with capacity restrictions, minimum workload, and transportation. The problems are inspired by a waste collection problem in which large waste containers are scattered throughout the municipality, and the residents bring their waste to these containers. We take the residents’ preferences into account when allocating them to locations. When a container is full, a vehicle transports an empty container from the disposal facility (depot) to that location and replaces it. We propose a mixed-integer linear programming formulation for the problems in which vehicles can carry one or two containers, and apply a logic-based Benders decomposition approach for the latter. Here, the sub problem is a multi-period minimum weight perfect matching problem. We show that our logic-based Benders decomposition approach outperforms the direct formulation in terms of solution quality and speed. We further show that transportation of two containers at a time reduces the distance to be driven by 29.5% on average, without compromising the covering level. Furthermore, we analyze the effect of imposing a minimum workload as well as the effect of changing the focus between transportation and covering.

Parallel Batch Processing for the Coating Problem

We solve a challenging scheduling problem with parallel batch processing and two-dimensional shelf strip packing constraints that arises in the tool coating field. Tools are assembled on so-called planetaries (batches) before they are loaded into coating machines to get coated. The assembling is not trivial and must fulfil specific constraints, which we refer to as shelf strip packing constraints. Further, each tool is associated with a starting time window s.t. tools can only be put on the same planetary if their time window overlap. The objective is to minimise the makespan and the number of required planetaries. Since the problem naturally decomposes into scheduling and packing parts, we tackle the problem with a two-phase logic-based Benders decomposition approach. The master problem assigns items to batches. The first phase solves as subproblem the packing problem by checking if the assignment is feasible, whereas the second phase solves the scheduling subproblem. The approach is compared with a monolithic mixed integer linear programming approach as well as a monolithic constraint programming approach. Experimental evaluation shows that our proposed approach outperforms the state-of-the-art benchmarks by solving more instances to optimality in a shorter time.

A Hybrid Algorithm of Ant Colony and Benders Decomposition for Large-Scale Mixed-Integer Linear Programming

We propose a hybrid approach for solving problems of mixed-integer linear programming with two types of variables (MILPTTV). The hybrid approach includes three algorithms: (1) Benders decomposition; (2) the ant colony algorithm; (3) the feasible heuristic method. We decompose the programming problem into a master problem and a subproblem using Benders decomposition. The ant colony algorithm generates an initial solution for the master problem. A feasible heuristic is used to obtain feasible solutions as input to the subproblem, whereas the subproblem is solved to optimality by using a linear programming solver that is named Linprog function in MATLAB R2016a. Over successive iterations, the master problem is refined by adding cuts from the subproblem and the master problem is solved by a linear programming solver Cplexmilp function. We compare the performance of the proposed hybrid approach against a general Benders decomposition approach as well as against a mixed-integer programming solver CPLEX (version 12.6) that is invoked by MATLAB on MIPLIB test problem mas76, mas74 and some large-scale numerical examples.