Generalized Benders Decomposition Method Research Articles

Grid-friendly peer-to-peer transactive energy market for inhibiting players’ strategic behavior

With the development of distributed renewable energy resources (RERs) in smart cities, the peer-to-peer (P2P) transactive energy market has become an effective way for RER owners to consume the surplus produced power. Nevertheless, the influence of grid usage costs in P2P energy trading has not been considered in prosumers’ strategic behaviors. To solve this issue, we analyze the network usage of P2P energy transactions and model a Nash–Stackelberg game to simulate the trading process of market participants. Firstly, we assume the prosumers can join the P2P energy market and determine their roles freely. The market manager adjusts the grid usage prices to influence the prosumers’ exchanged energy amounts. We then establish a monolithic VCG mechanism to inhibit the prosumers’ strategic behaviors while reducing the computational difficulty of the VCG mechanism. Finally, we develop a best-response (BR)-based algorithm to solve one energy prosumer’s cost minimization subproblem at a time to seek market equilibrium. Since the energy prosumers and market manager interact on the trading amount, the prosumers’ responses are modeled as bi-level mixed-integer nonlinear programming (BLMINLP). We employ the generalized Benders decomposition method to reduce the solution difficulty. Case studies verify the theoretical properties and show the practicability of the proposed algorithms.

A joint model of location, inventory and third-party logistics provider in supply chain network design

Integrated supply chain network design plays an increasingly important role in improving the operational efficiency of a company. This paper studies a novel joint supply chain network model that integrates location, inventory and third-party logistics provider decisions for consideration in a three-level supply chain involving manufacturers, distribution centers (DCs), and retailers. Different from traditional location-inventory problems, the proposed model not only determines the locations of manufacturers and DCs, and how to serve DCs and retailers by manufacturers and DCs, but also chooses potential third-party logistics providers to serve DCs and retailers. The goal is to design an efficient supply chain system to reduce the overall cost including facility cost, inventory cost, and transportation cost to meet the demands of retailers. A nonlinear integer programming model based on multigraph is proposed. To deal with the complex nonlinear terms, we reformulate a mixed-integer nonlinear programming model. Moreover, to deal with the difficulties caused by multigraph, a method based on the generalized Benders decomposition (GBD) is developed. Computational results clearly show the effectiveness of the proposed model and the GBD method. Comparative analysis shows the potential benefits of integrating the third-party logistics provider decisions into the location-inventory model.

Energy trade-offs in coupled ICM and electricity market under dynamic carbon emission intensity

With the adoption of the cap-based carbon market (CCM), a growth-limiting dilemma from the industrial growth and emission cap reduction has emerged, making it unsuitable for developing countries under rapid industrialization. In contrast, the intensity-based carbon market (ICM) provides a promising option targeting carbon emission intensity (CEI) reduction. However, due to the complexity of correlations between the ICM and electricity market (EM), the resulting energy trade-offs have not been investigated clearly. Thus, this paper quantifies the impact of market correlations on energy trade-offs under the dynamic CEI. Firstly, the CEI dynamic is proved and modeled as an inversely proportional function to the power output. Then, a stochastic EM clearing model for energy trade-offs restricted by ICM obligations and renewable uncertainties is designed. Since the model is a mixed-integer nonlinear problem with multiple scenarios, it is then reformulated and solved by the Generalized Benders Decomposition method. Finally, the proposed method is demonstrated on the IEEE 14- and 118-bus systems. The results show that the ICM effectiveness is dependent on CEI benchmarks and carbon prices, and the ignorance of the CEI dynamic can lead to the miscalculation of carbon emissions and sector profits.

Joint Routing and Charging Problem of Electric Vehicles with Incentive-Aware Customers Considering Spatio-Temporal Charging Prices

This paper investigates the scheduling problem of a fleet of electric vehicles, providing mobility as a service to a set of time-specified customers, where the operator needs to solve the routing and charging problem jointly for each EV. Hereby we consider incentive-aware customers and propose that the operator offers monetary incentives to customers in exchange for time flexibility. In this way, the fleet operator can achieve a routing and charging schedule with lower costs, whilst the customers receive monetary compensation for their flexibility. Specifically, we first propose a bi-level optimization model whereby the fleet operator optimizes the routing and charging schedule accounting for the spatio-temporal varying charging price, jointly with a monetary incentive to reimburse the delivery time flexibility experienced by the customers. Concurrently the customers choose their own time flexibility by minimizing their own cost. Second, we cope with the computational burden coming from this nonlinear bi-level optimization model with an accurate reformulation approach consisting of the KKT optimality conditions, a Big-M-based linearization method, and the zero duality gap of convex optimization problems. This way, we convert the proposed problem into a single-level optimization problem, which can be solved by a strengthened generalized Benders decomposition method holding a faster convergence rate than the generalized Benders decomposition method. To evaluate the effectiveness of the proposed mathematical model, we carry out numerous simulation experiments by using the VRP-REP data of Belgium. The numerical results showcase that the proposed mathematical model can reduce the delivery fees for the customers together with the cost of operation incurred by the fleet operator.

Generalized Benders Decomposition Method to Solve Big Mixed-Integer Nonlinear Optimization Problems with Convex Objective and Constraints Functions

The paper presents the Generalized Benders Decomposition (GBD) method, which is now one of the basic approaches to solve big mixed-integer nonlinear optimization problems. It concentrates on the basic formulation with convex objectives and constraints functions. Apart from the classical projection and representation theorems, a unified formulation of the master problem with nonlinear and linear cuts will be given. For the latter case the most effective and, at the same time, easy to implement computational algorithms will be pointed out.

Aging and Delay Analysis Based on Lyapunov Optimization and Martingale Theory

The age of information (AoI) is a key performance metric for data freshness in real-time systems, measuring the time elapsed between status updates received at a remote destination. This paper addresses the analysis of both the AoI and the delay for a system demanding timely status updates, consisting of an internet of things (IoT) community, and a multi-core edge computing node (EN). The IoT devices perform channel contention according to a slotted aloha non-orthogonal multiple access scheme with two transmission power levels to access the EN. The objective of the paper is the dynamic selection of both the EN computation capabilities and the suitable access probability for the two power levels, aiming at maximizing the time-average system utility, while guaranteeing the queues stability. The device end-to-end delay is analyzed by resorting to the martingale theoretical approximation under the condition that the Lyapunov optimization framework is performed on a time slot basis, hence leading to a novel instantaneous mixed integer problem formulation. Furthermore, the generalized benders decomposition method is employed to provide a suboptimal solution for the formulated problem. Finally, the accuracy of the proposed framework is validated by comparing the obtained analytical predictions with simulation results.

Evaluating Resilience of Electricity Distribution Networks via a Modification of Generalized Benders Decomposition Method

This article presents a computational framework to evaluate the resilience of electricity distribution networks (DNs) to a class of cyber-physical failures. In our model, we consider an attacker who targets multiple DN components to maximize the loss of the DN operator. We consider two types of operator responses: i) coordinated emergency response and ii) uncoordinated autonomous disconnects, which may lead to cascading failures. To evaluate resilience under response i), we consider a bilevel mixed-integer second-order cone program. This problem is computationally challenging to solve due to mixed-integer variables in the inner problem and nonconvex constraints. Our solution approach exploits the properties of power flows on radial DNs operating in a postattack contingency state to refine the classical generalized Benders decomposition method. Specifically, we suggest a modification of the right-hand side of the generalized Benders cut to be a value determined by the sum of values of a subset of inner dual cost coefficients. We show how the size of this subset can be selected to achieve a reasonable tradeoff between solution accuracy and computational effort. Furthermore, we evaluate DN resilience under response ii) by sequentially computing autonomous component disconnects due to operating bound violations resulting from the initial attack and the potential cascading failures. Our approach is useful for estimating the gain in resilience under response i), relative to ii).

Generic modelling and optimal day-ahead dispatch of micro-energy system considering the price-based integrated demand response

As an extension of micro-grid, micro-energy system (MES) is one of the important carriers for energy utilization in the future, and using energy prices as a controllable resource is conducive to improving the optimization potential of MES. Firstly, based on the energy hub model, a generic method for modelling the steady-state energy balance equation of MES is proposed. Then, considering the multi-energy substitution effect in the background of multi-energy coupling, the concept of price-based integrated demand response (P-IDR) is introduced. Meanwhile, based on the price elasticity theory and the discrete choice theory, the energy timing transfer characteristics and energy substitution characteristics in P-IDR is modelled. Furthermore, after taking into account the P-IDR, an MINLP model for day-ahead dispatch of MES is built, and a generalized benders decomposition method is used for solution. Case studies are conducted on an MES to verify the effectiveness of the proposed modelling method. The result shows that it is beneficial to improve renewable energy accommodation and reduce the peak and off-peak difference of energy load when the P-IDR is deployed. In addition, consider the energy substitution characteristics can reduce the user’s cost on energy purchase and be more in line with the user’s rational consumption behaviour.

Energy‐efficient scheduling for sustainable manufacturing systems with renewable energy resources

AbstractEnvironmentally friendly energy resources open a new opportunity to tackle the problem of energy security and climate change arising from wide use of fossil fuels. This paper focuses on optimizing the allocation of the energy generated by the renewable energy system to minimize the total electricity cost for sustainable manufacturing systems under time‐of‐use tariff by clipping the peak demand. A rolling horizon approach is adopted to handle the uncertainty caused by the weather change. A nonlinear mathematical programming model is established for each decision epoch based on the predicted energy generation and the probability distribution of power demand in the manufacturing plant. The objective function of the model is shown to be convex, Lipchitz‐continuous, and subdifferentiable. A generalized benders decomposition method based on the primal‐dual subgradient descent algorithm is proposed to solve the model. A series of numerical experiments is conducted to show the effectiveness of the solution approach and the significant benefits of using the renewable energy resources.

Joint Resource Allocation for Latency-Sensitive Services Over Mobile Edge Computing Networks With Caching

Mobile edge computing (MEC) has risen as a promising paradigm to provide high quality of experience via relocating the cloud server in close proximity to smart mobile devices (SMDs). In MEC networks, the MEC server with computation capability and storage resource can jointly execute the latency-sensitive offloading tasks and cache the contents requested by SMDs. In order to minimize the total latency consumption of the computation tasks, we jointly consider computation offloading, content caching, and resource allocation as an integrated model, which is formulated as a mixed integer nonlinear programming (MINLP) problem. We design an asymmetric search tree and improve the branch and bound method to obtain a set of accurate decisions and resource allocation strategies. Furthermore, we introduce the auxiliary variables to reformulate the proposed model and apply the modified generalized benders decomposition method to solve the MINLP problem in polynomial computation complexity time. Simulation results demonstrate the superiority of the proposed schemes.

Combined Optimization of Clustering Using Artificial Bee Colony Algorithm and Cooperative Beamforming in Green Cognitive Radio Networks with Relays

In this paper, our objective is to reduce the interference in the direction of Primary Users (PUs) and to decrease the power cost of Secondary Users (SUs) in the Cognitive underlay system. Artificial Bee Colony (ABC) algorithm, one of the most recently introduced meta-heuristic optimization algorithms is used here as the clustering algorithm. Here relay stations are introduced in cooperative beamforming technique, which are equipped with directional antennas that can efficiently improve the spectral efficiency and can reduce the interference to primary users in a cognitive underlay systems. In the proposed research, flexible cooperation is considered as a general case while full cooperation and inter-cell cooperation are considered as special cases. We jointly optimize the beamforming and the clustering to reduce the overall power consumption of the secondary users and to satisfy the primary users interference limits. Inorder to find optimal solution, an iterative algorithm called generalized Benders decomposition method is applied. We have investigated the benefits of ABC algorithm and showed the advantages of the relays in cognitive wireless networks.

Clustering and Cooperative Beamforming in Green Cognitive Radio Networks with Relays

A relay assisted cooperative beamforming technique can efficiently improve the spectral efficiency and can reduce the interference to primary users in a cognitive underlay systems. Our objective is to reduce the total power cost of Secondary Users (SU) while limiting the interference in the direction of Primary Users (PU). Kmean algorithm which is a partition based clustering algorithm is used here as the clustering algorithm. While considering flexible cooperation as general scheme, we collectively optimize the clustering and the beamforming to lower the overall power consumption while fulfilling the secondary users quality of service and satisfying the primary users interference temperature limits. An iterative algorithm called generalized Benders decomposition method is used to find an optimal solution. The indirect transmission mode of relays is considered in this work. The relays are equipped with directional antennas which will reduce the interference to the PU and meet the power cost requirement of SU under the assumptions that the clusters are non-overlapping and fixed. Thus spectral efficiency can be increased. Through simulations, we analyzed the advantages of algorithms and showed the benefits of the relays in cognitive wireless networks.

Water Resources Planning and Management by Use of Generalized Benders Decomposition Method to Solve Large-scale MINLP Problems

Water resources systems (WRS) models involve a large number of continuous and integer quantities. Water quality management problems also require the consideration of uncertainties related to the variability of flow streams and temperatures. WRS dynamics are primarily nonlinear. These characteristic features suggest the use of (stochastic) mixed-integer programming models, as well the use of sensibility analysis and simulations. Moreover, high dimensional real-world models and combinatorial alternatives require adequate tools for large-scale optimization models. These techniques consist of decomposition methods such as the generalized Benders decomposition (GBD) and the branch-and-bound enumerative algorithm. This contribution introduces the subject of modeling WRSs by use of GBD and branch-and-bound algorithms with numerical applications.

Generalized Benders Decomposition for one Class of MINLPs with Vector Conic Constraint

In this paper, we mainly study one class of mixed-integer nonlinear programming problems (MINLPs) with vector conic constraint in Banach spaces. Duality theory of convex vector optimization problems applied to this class of MINLPs is deeply investigated. With the help of duality, we use the generalized Benders decomposition method to establish an algorithm for solving this MINLP. Several convergence theorems on the algorithm are also presented. The convergence theorems generalize and extend the existing results on MINLPs in finite dimensional spaces.

Bicriterion discrete equilibrium network design problem

The budget network design problem and fixed‐charge network design problem imply different economic pursuits on travel cost and construction cost and structure these two cost components in different ways. A more general version of these two classic formulations is the biobjective network design problem. This article discusses an exact solution strategy for the biobjective discrete network design problem with equilibrium constraints, which eliminates the inexactness and incompleteness deficiencies pertaining to heuristics or metaheuristics presented in previous research. In particular, we adapted and justified a dichotomic solution framework for the biobjective network design problem, in which the complete solution set of the problem can be exhausted by repeatedly solving a parameterized scalar problem and updating the parameter set. A generalized Benders decomposition method, a widely used solution strategy for nonlinear mixed integer programming problems, is further implemented in the solution framework, which offers an efficient algorithmic tool for solution of the scalar problem. Numerical results obtained from the example problems justify the solution optimality, completeness, and efficiency of the presented solution method. © 2014 Wiley Periodicals, Inc. NETWORKS, Vol. 63(4), 286–305 2014

Integrating operations and marketing decisions using delayed differentiation of products and guaranteed delivery time under stochastic demand

In this research, we integrate the issues related to operations and marketing strategy of firms characterized by large product variety, short lead times, and demand variability in an assemble-to-order environment. The operations decisions are the inventory level of components and semi-finished goods, and configuration of semi-finished goods. The marketing decisions are the products price and a lead time guarantee which is uniform for all products. We develop an integrated mathematical model that captures trade-offs related to inventory of semi-finished goods, inventory of components, outsourcing costs, and customer demand based on guaranteed lead time and price.The mathematical model is a two-stage, stochastic, integer, and non-linear programming problem. In the first stage, prior to demand realization, the operation and marketing decisions are determined. In the second stage, inventory is allocated to meet the demand. The objective is to maximize the expected profit per-unit time. The computational results on the test problems provide managerial insights for firms faced with the conflicting needs of offering: (i) low prices, (ii) guaranteed and short lead time, and (iii) a large product variety by leveraging operations decisions.We also develop a solution procedure to solve large instances of the problem based on an accelerated version of the Generalized Benders’ Decomposition (GBD) method. The accelerating mechanism involves search intensification and diversification around solutions which improve the upper bound. The suggested GBD method gives a better solution and a tighter lower bound in a given time period than the conventional GBD implementation and the non-linear branch-and-bound method.

Joint Optimization of Clustering and Cooperative Beamforming in Green Cognitive Wireless Networks

We investigate the trade-off between performance and energy efficiency in cooperative cognitive underlay systems. By cooperating, cognitive base stations mitigate the interference and serve their users more effectively at the cost of spending more power. With a flexible cooperation scheme, we thus jointly optimize the clustering and the beamforming to minimize the overall power consumption while satisfying the secondary users' quality of service and respecting the primary users' interference limits. We formulate this problem as a mixed-integer nonlinear program and decompose it into a master problem and a beamforming subproblem. Then, we derive an iterative algorithm based on the generalized Benders decomposition method to find an optimal solution. Moreover, we propose simple techniques to speed up its convergence. When the clusters are fixed and non-overlapping, we also propose a decentralized algorithm with limited signaling schemes using the alternating direction method of multipliers. In contrast to previous works, our distributed algorithm handles the total interference power constraints coupling the cognitive base stations. Through simulations, we analyze the effectiveness and convergence of the proposed algorithms and show the benefits of the cooperation in cognitive wireless networks.

Inexact solution of NLP subproblems in MINLP

In the context of convex mixed integer nonlinear programming (MINLP), we investigate how the outer approximation method and the generalized Benders decomposition method are affected when the respective nonlinear programming (NLP) subproblems are solved inexactly. We show that the cuts in the corresponding master problems can be changed to incorporate the inexact residuals, still rendering equivalence and finiteness in the limit case. Some numerical results will be presented to illustrate the behavior of the methods under NLP subproblem inexactness.

Projeto de rede com custos convexos e balanceamento de fluxos

O problema de projeto de redes arborescentes de fonte única com custos convexos é abordado neste trabalho. Trata-se de problema referencial no projeto de redes de transporte de matéria, energia ou informação. Resulta do esforço de modelagem um programa não-linear inteiro misto de grande escala e de difícil resolução. Para superar tais dificuldades, dois métodos distintos são aplicados ao problema: O primeiro deles é a técnica de Decomposição de Benders Generalizada; o segundo método combina a técnica de Aproximação Externa com as idéias de projeção subjacentes à Decomposição de Benders para derivar a técnica OA-Híbrido. O método OA-Híbrido se mostra muito eficiente na solução de instâncias com até 702 arcos, obtendo custos de computação bastante razoáveis, tornando promissora a sua aplicação a problemas ainda mais sofisticados.

MULTICOMMODITY NETWORK EXPANSION UNDER ELASTIC DEMANDS

We analyze some issues of network design and bandwidth allocation in telecommunication systems with congestible resources. The work is closely related to network monitoring and traffic measurement functions that must be carried out on line, in order to overcome congestion caused by an unfavorable traffic pattern or by a failure. In addition to the traditional use of routing controls, our approach achieves network efficiency with capacity assignment and also imposing variable prices for the consumers. A generalized Benders decomposition method is applied to a mixed integer nonlinear programming formulation of the integrated problem of network design and operation. The method exploits the nature of the continuous subproblem, that is a large-scale convex network flow problem with demands sensitive to commodity prices. Some numerical experience suggests that the method is usefull to address both questions of global optimality and competitive pricing in such systems.