Benders Cuts Research Articles

A branch‐and‐cut algorithm for hub network design problems with profits

AbstractHub network design problems with profits (HNDPPs) extend the classical hub location problems (HLPs), with the profit maximization objective. HNDPPs aim to design a service network with hub nodes and hub links, select the commodities to be served, and determine the transportation routes for the selected commodities. In this article, we allow the transportation route of each commodity to contain more than one hub link, which generalizes existing HNDPPs and is known to be more profitable. A new path‐based formulation is proposed, which is then reformulated in a Benders decomposition fashion, and an exact branch‐and‐cut algorithm is developed to solve the Benders master problem. At the nodes of the enumeration tree, Benders decomposition algorithm is used to solve a linear relaxation of the Benders master problem. When implementing the Benders decomposition algorithm, the special structure of the subproblem is explored, based on which an efficient method is proposed to generate Benders cuts. In addition, refinements and a heuristic strategy are used to speed up the branch‐and‐cut algorithm. Extensive numerical experiments on benchmark instances have been carried out to verify the effectiveness and efficiency of the proposed solution method.

Refined cut selection for benders decomposition: applied to network capacity expansion problems

In this paper, we present a new perspective on cut generation in the context of Benders decomposition. The approach, which is based on the relation between the alternative polyhedron and the reverse polar set, helps us to improve established cut selection procedures for Benders cuts, like the one suggested by Fischetti et al. (Math Program Ser B 124(1–2):175–182, 2010). Our modified version of that criterion produces cuts which are always supporting and, unless in rare special cases, facet-defining. We discuss our approach in relation to the state of the art in cut generation for Benders decomposition. In particular, we refer to Pareto-optimality and facet-defining cuts and observe that each of these criteria can be matched to a particular subset of parametrizations for our cut generation framework. As a consequence, our framework covers the method to generate facet-defining cuts proposed by Conforti and Wolsey (Math Program Ser A 178:1–20, 2018) as a special case. We conclude the paper with a computational evaluation of the proposed cut selection method. For this, we use different instances of a capacity expansion problem for the european power system.

Exact Branch-Price-and-Cut for a Hospital Therapist Scheduling Problem with Flexible Service Locations and Time-Dependent Location Capacity

We study a new variant of the vehicle routing problem, which arises in hospital-wide scheduling of physical therapists. Multiple service locations exist for patients, and resource synchronization for the location capacities is required as only a limited number of patients can be treated at one location at a time. Additionally, operations synchronization between treatments is required as precedence relations exist. We develop an innovative exact branch-price-and-cut algorithm including two approaches targeting the synchronization constraints (1) based on branching on time windows and (2) based on adding combinatorial Benders cuts. We optimally solve realistic hospital instances with up to 120 treatments and find that branching on time windows performs better than adding cutting planes. Summary of Contribution: We present an exact branch-price-and-cut (BPC) algorithm for the therapist scheduling and routing problem (ThSRP), a daily planning problem arising at almost every hospital. The difficulty of this problem stems from its inherent structure that features routing and scheduling while considering multiple possible service locations with time-dependent location capacities. We model the ThSRP as a vehicle routing problem with time windows and flexible delivery locations and synchronization constraints, which are properties relevant to other vehicle routing problem variants as well. In our computational study, we show that the proposed exact BPC algorithm is capable of solving realistic hospital instances and can, thus, be used by hospital planners to derive better schedules with less manual work. Moreover, we show that time window branching can be a valid alternative to cutting planes when addressing synchronization constraints in a BPC algorithm.

Exact and heuristic algorithms for the maximum weighted submatrix coverage problem

The maximum weighted submatrix coverage problem is a recently introduced problem with applications in data mining. It is concerned with selecting K submatrices of a given numerical matrix such that the sum of the matrix-entries, which occur in at least one of the selected submatrices, is maximized. In the paper introducing the problem, a problem-specific constraint programming approach was developed and embedded in a large neighborhood-search to obtain a heuristic. A compact integer linear programming formulation was also presented, but deemed inefficient due to its size.In this paper, we introduce new integer linear programming formulations for the problem, one of them is based on Benders decomposition. The obtained Benders decomposition-based formulation has a nice combinatorial structure, i.e., there is no need to solve linear programs to separate Benders cuts. We present preprocessing procedures and valid inequalities for all formulations. We also develop a greedy randomized adaptive search procedure for the problem, which is enhanced with a local search.A computational study using the instances from literature is done to evaluate the effectiveness of our new approaches. Our algorithms manage to find improved primal solutions for ten out of 17 real-world instances, and optimality is proven for two real-world instances. Moreover, for over 700 of 1617 large-scale synthetic instances, our algorithms find improved primal solutions compared to the heuristics from the literature.

Reactive Power Management for Networked Microgrid Resilience in Extreme Conditions

The inadequate level of reactive power generation by distributed energy resources (DERs) in extreme conditions could render networked microgrids (NMGs) more susceptible to major contingencies. In extreme conditions when the NMGs islanding from the main grid might occur, the inadequacy of reactive power generation could make it more difficult for NMGs operators to maintain a reasonable margin for the resilient NMGs operation. This paper studies the resilient reactive power scheduling of NMGs in extreme conditions. A two-stage model is proposed which applies Here-And-Now (HAN) and Wait-And-See (WAS) decisions at the first and the second stages, respectively. The HAN and WAS stages are linked through a stochastic programming approach where the entire problem is formulated in mixed-integer linear programming. To address scalability and privacy concerns in NMGs operations, the proposed two-stage model is decentralized and decomposed into a master problem and three subproblems and linked by Benders cuts. The master problem identifies the least-cost operation plan for NMGs considering normal and extreme conditions at Stages 1 and 2. Subproblems I, II, and III satisfy the feasibility, security, and optimality of individual microgrid operations, respectively. The DER limitations in supplying reactive loads are more evident at Stage 2 during the islanding simulation, which is managed by the proposed extreme condition reactive power procurement approach. The effectiveness of proposed two-stage model in enhancing the resilience of the NMGs is assured by comprehensive simulation studies.

Combinatorial Benders decomposition for mixed-model two-sided assembly line balancing problem

In this work, we consider a mixed-model two-sided assembly line balancing problem for a given cycle time (MTALBP-I), in which the primary objective is to minimise the number of mated-stations (i.e. the length of the two-sided assembly line), while the number of stations, which describes the total number of operators, is also concerned. To solve this problem, we propose a combinatorial Benders decomposition-based exact algorithm, and develop a sequence-based enumerative search method to calculate effective combinatorial Benders cuts. To evaluate the performance of our proposed solution approach, we conduct extensive computational experiments on a set of benchmark instances, and the results demonstrate its efficiency of finding exact solutions even for large-sized instances.

Benders decomposition for a period-aggregated resource leveling problem with variable job duration

We consider a special case of resource leveling problem with variable job duration and period-aggregated resource consumption. The objective function is to minimize the total extra resource cost required to complete all jobs before the hard project deadline. All jobs must be implemented without any preemption. The planning horizon is divided into periods, each period is characterized by an available level of resources of each type. For each job, resource consumption varies from one period to another. It is possible to control the speed of the job execution: if more resources are allocated to a job, then the job is executed faster if fewer resources are used, the job takes more time. A problem solution defines the start and end times for each job, but also its resource usage per period. On contrary to former models of this problem, our formulation allows an independent resource usage for each resource type. This approach can reach more flexible solutions with lower cost objective function value. We develop a Benders decomposition algorithm to solve this new formulation. Several enhancements are also implemented, such as the construction of effective bounds and multi-cut generation with the single search tree (Branch&Benders cuts). Numerical experiments on several sets of instances demonstrate the advantage of the algorithm in comparison with the basic model and a CPLEX built-in Benders decomposition.

Logic-Based Benders Decomposition and Binary Decision Diagram Based Approaches for Stochastic Distributed Operating Room Scheduling

The distributed operating room (OR) scheduling problem aims to find an assignment of surgeries to ORs across collaborating hospitals that share their waiting lists and ORs. We propose a stochastic extension of this problem where surgery durations are considered to be uncertain. In order to obtain solutions for the challenging stochastic model, we use sample average approximation and develop two enhanced decomposition frameworks that use logic-based Benders (LBBD) optimality cuts and binary decision diagram based Benders cuts. Specifically, to the best of our knowledge, deriving LBBD optimality cuts in a stochastic programming context is new to the literature. Our computational experiments on a hospital data set illustrate that the stochastic formulation generates robust schedules and that our algorithms improve the computational efficiency. Summary of Contribution: We propose a new model for an important problem in healthcare scheduling, namely, stochastic distributed operating room scheduling, which is inspired by a current practice in Toronto, Ontario, Canada. We develop two decomposition methods that are computationally faster than solving the model directly via a state-of-the-art solver. We present both some theoretical results for our algorithms and numerical results for the evaluation of the model and algorithms. Compared with its deterministic counterpart in the literature, our model shows improvement in relevant evaluation metrics for the underlying scheduling problem. In addition, our algorithms exploit the structure of the model and improve its solvability. Those algorithms also have the potential to be used to tackle other planning and scheduling problems with a similar structure.

A Stochastic Programming Approach for Locating and Dispatching Two Types of Ambulances

Emergency Medical Service systems aim to respond to emergency calls in a timely manner and provide prehospital care to patients. This paper addresses the problem of locating multiple types of emergency vehicles to stations while taking into account that vehicles are dispatched to prioritized patients with different health needs. We propose a two-stage stochastic-programming model that determines how to locate two types of ambulances in the first stage and dispatch them to prioritized emergency patients in the second stage after call-arrival scenarios are disclosed. We demonstrate how the base model can be adapted to include nontransport vehicles. A model formulation generalizes the base model to consider probabilistic travel times and general utilities for dispatching ambulances to prioritized patients. We evaluate the benefit of the model using two case studies, a value of the stochastic solution approach, and a simulation analysis. The case study is extended to study how to locate vehicles in the model extension with nontransport vehicles. Stochastic-programming models are computationally challenging for large-scale problem instances, and, therefore, we propose a solution technique based on Benders cuts.

Transfer Learning for Constrained Stochastic Control Using Adjustable Benders Cuts

Stochastic optimal control problems are generally difficult to solve to optimality, and it is desirable to be able to transfer a high-quality solution learned for one such problem to a new, related problem at low computational cost. This is useful in cases such as a known change in dynamics, e.g., when a vehicle's payload increases. We describe a way of doing this for discrete time, infinite horizon problems on continuous spaces. The solutions take the form of Q-function approximations, consisting of “Benders cuts” that lower-bound the optimal Q-function. This form induces an easy to evaluate control policy. We describe how to adjust a set of existing cuts at low computational cost while preserving the lower bounding property for the new problem. This transfers prior training effort to the new problem far more cheaply than starting from scratch. We use random problem instances to demonstrate that our transfer procedure yields near-optimal results on the new model, outperforming a clipped linear-quadratic regulator (LQR) on average. The Benders approach, while model-based, is strongly related to model-free reinforcement learning, in that it uses sampled experience evaluating the so-called Bellman error to learn the value function.

Nutmeg: a MIP and CP Hybrid Solver Using Branch-and-Check

This paper describes the implementation of Nutmeg, a solver that hybridizes mixed integer linear programming and constraint programming using the branch-and-cut style of logic-based Benders decomposition known as branch-and-check. Given a high-level constraint programming model, Nutmeg automatically derives a mixed integer programming master problem that omits global constraints with weak linear relaxations, and a constraint programming subproblem identical to the original model. At every node in the branch-and-bound search tree, the linear relaxation computes dual bounds and proposes solutions, which are checked for feasibility of the omitted constraints in the constraint programming subproblem. In the case of infeasibility, conflict analysis generates Benders cuts, which are appended to the linear relaxation to cut off the candidate solution. Experimental results show that Nutmeg’s automatic decomposition outperforms pure constraint programming and pure mixed integer programming on problems known to have successful implementations of logic-based Benders decomposition, but performs poorly on general problems, which lack specific decomposable structure. Nonetheless, Nutmeg outperforms the standalone approaches on one problem with no known decomposable structure, providing preliminary indications that a hand-tailored decomposition for this problem could be worthwhile. On the whole, Nutmeg serves as a valuable tool for novice modelers to try hybrid solving and for expert modelers to quickly compare different logic-based Benders decompositions of their problems.

Probabilistic decomposition‐based security constrained transmission expansion planning incorporating distributed series reactor

This study presents a probabilistic transmission expansion planning model incorporating distributed series reactors, which are aimed at improving network flexibility. Although the whole problem is a mixed-integer non-linear programming problem, this study proposes an approximation method to linearise it in the structure of the Benders decomposition (BD) algorithm. In the first stage of the BD algorithm, optimal number of new transmission lines and distributed series reactors are determined. In the second stage, the developed optimal power flow problem, as a linear sub-problem, is performed for different scenarios of uncertainties and a set of probable contingencies. The Benders cuts are iteratively added to the first stage problem to decrease the optimality gap below a given threshold. The proposed model utilises the Monte Carlo simulation method to take into account uncertainty of wind generations and demands. Several case studies on three test systems are presented to validate the efficacy of the proposed approach.

Large-scale influence maximization via maximal covering location

Influence maximization aims at identifying a limited set of key individuals in a (social) network which spreads information based on some propagation model and maximizes the number of individuals reached. We show that influence maximization based on the probabilistic independent cascade model can be modeled as a stochastic maximal covering location problem. A reformulation based on Benders decomposition is proposed and a relation between obtained Benders optimality cuts and submodular cuts for correspondingly defined subsets is established. We introduce preprocessing tests, which allow us to remove variables from the model and develop efficient algorithms for the separation of Benders cuts. Both aspects are shown to be crucial ingredients of the developed branch-and-cut algorithm since real-life social network instances may be very large. In a computational study, the considered variants of this branch-and-cut algorithm outperform the state-of-the-art approach for influence maximization by orders of magnitude.

A heuristic benders-decomposition-based algorithm for transient stability constrained optimal power flow

Security Constrained Optimal Power Flow (SCOPF) is traditionally performed to obtain the optimal generation levels to achieve the minimum generation cost under operational constraints in normal and contingency conditions. In this paper the SCOPF problem with DC load flow equations is extended to consider the transient stability margin of the generating units as a heuristic decomposition algorithm. However, the proposed algorithm is compatible with any SCOPF formulation. Due to computational complexities of security constraints and transient stability assessment, Benders decomposition algorithm is utilized to handle the transient stability margin of generating units as separate sub-problems. A transient energy function is developed to assess the transient stability of generating units under credible faults using the cut-set energy function criterion. In case of violation of the required transient stability margin, Benders cuts are constructed and added to the master problem. The proposed decomposed Transient Stability Constrained OPF (TSCOPF) is converged in few iterations. The developed TSCOPF algorithm is implemented on the IEEE 39-bus dynamic test system to validate its accuracy and efficiency.

Co-Optimization of Generation Self-Scheduling and Coal Supply for Coal-Fired Power Plants

Generation self-scheduling and coal supply in coal-fired power plants are closely related but typically optimized separately. To enhance the optimal operation of power plants, we propose a coordinated optimal operation strategy of generation and coal management in this paper. Uncertainties in electricity prices and demands, coal prices, and coal inventory holding costs are captured and modeled by discrete scenarios. Emission constraints are introduced to control generation emissions. The heat loss caused by the weathering of coal during the storage is taken into account, which distinguishes the considered co-optimization problem from the previous ones. The proposed strategy is built on a mixed-integer linear programming-typed two-stage stochastic programming model, in which whether to purchase coal is determined in the first stage and the quantity of coal purchase, the coal inventory, and the economic generation dispatch are determined in the second stage. The objective is to maximize the expected profits. An improved Benders decomposition algorithm is developed to solve the problem where multiple Benders cuts are added in each iteration and valid inequalities are introduced to speed up the convergence of the algorithm. Numerical experiments demonstrate the effectiveness of the proposed strategy and algorithm.

Using the Thermal Inertia of Transmission Lines for Coping with Post-Contingency Overflows

For the corrective security-constrained optimal power flow (OPF) model, there exists a post-contingency stage due to the time delay of corrective measures. Line overflows in this stage may cause cascading failures. This paper proposes that the thermal inertia of transmission lines can be used to cope with post-contingency overflows. An enhanced security-constrained OPF model is established and line dynamic thermal behaviors are quantified. The post-contingency stage is divided into a response substage and a ramping substage and the highest temperatures are limited by thermal rating constraints. A solving strategy based on Benders decomposition is proposed to solve the established model. The original problem is decomposed into a master problem for preventive control and two subproblems for corrective control feasibility check and line thermal rating check. In each iteration, Benders cuts are generated for infeasible contingencies and returned into the master problem for adjusting the generation plan. Because the highest temperature function is implicit, an equivalent time method is presented to calculate its partial derivative in Benders cuts. The proposed model and approaches are validated on three test systems. Results show that the operation security is improved with a slight increase in total generation cost.

A Branch-And-Benders-Cut approach for the fault tolerant regenerator location problem

We study the fault tolerant regenerator location problem (FTRLP) in optical networks, where any single network link (optical fiber) might fail, and an optical signal can only travel a maximum distance (called the optical reach) before its quality deteriorates, needing regenerations by installing regenerators at network nodes. The FTRLP is to find a minimal number of nodes for regenerator deployment such that each pair of nodes can communicate with each other for each single-link fault (failure) scenario. To solve the FTRLP to optimality, we propose a branch-and-Benders-cut (BBC) approach. During the exploration of the branch-and-bound search tree, the algorithm adds combinatorial Benders cuts. Computational results are provided and demonstrate effectiveness of the BBC approach.

A Logic-Based Benders Decomposition Approach for the VNF Assignment Problem

Middleboxes have gained popularity due to the significant value-added services these network elements provide to traffic flows, in terms of enhanced performance and security. Policy-aware traffic flows usually need to traverse multiple middleboxes in a predefined order to satisfy their associated policy, also known as Service Function Chaining . Typically, Middleboxes run on specialized hardware, which make them highly inflexible to handle the unpredictable and fluctuating-nature of traffic, and contribute to significant capital and operational expenditures (Cap-ex and Op-ex) to provision, accommodate, and maintain them. Network Function Virtualization is a promising technology with the potential to tackle the aforementioned limitations of hardware middleboxes. Yet, NFV is still in its infancy, and there exists several technical challenges that need to be addressed, among which, the Virtual Network Function assignment problem tops the list. The VNF assignment problem stems from the newly gained flexibility in instantiating VNFs (on-demand) anywhere in the network. Subsequently, network providers must decide on the optimal placement of VNF instances which maximizes the number of admitted policy-aware traffic flows across their network. Existing work consists of Integer Linear Program (ILP) models, which are fairly unscalable, or heuristic-based approaches with no guarantee on the quality of the obtained solutions. This work proposes a novel Logic-Based Benders Decomposition (LBBD) based approach to solve the VNF assignment problem. It consists of decomposing the problem into two subproblems: a master and a subproblem; and at every iteration constructive Benders cuts are introduced to the master to tighten its search space. We compared the LBBD approach against the ILP and a heuristic method, and we show that our approach achieves the optimal solution (as opposed to heuristic-based methods) 700 times faster than the ILP.

A comparison of separation routines for benders optimality cuts for two-level facility location problems

This paper studies two-level uncapacitated facility location problems, a class of discrete location problems that consider different hierarchies of facilities and their interactions. Benders reformulations for both single and multiple assignment variants and while several separation procedures for three classes of Benders cuts are presented: standard optimality cuts, lifted optimality cuts, and non-dominated optimality cuts. Extensive computational experiments are performed on difficult and large-scale benchmark instances to assess the performance of the considered separation routines.

Resource adequacy in interdependent electricity markets undergoing heterogeneous expansion in renewable energy

In this study, the impact of the heterogeneous expansion of renewable energies on resource adequacy within interdependent electricity markets is investigated. The effect of increasing markets’ dependency on RA is also studied. A dynamic generation expansion model based on Benders decomposition is used to model the markets and coordinating entity to decompose the problem (keeping the markets independent in the master problem, and whole network constraints in the sub-problem). By using this method, the benders cuts are applied to the two different objective functions simultaneously. In a case study, the authors find that increasing markets’ dependency can postpone new unit installation time or decreases the need for new units in the market with more renewable resource but meanwhile the expected reserve margin stays at an acceptable level. According to the authors’ case study, the power plant's net present value increases by increasing markets’ dependency in the market with more renewable resources. The flexibility improves in both markets by increasing markets’ dependency.