Generalized 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 generalized Benders decomposition approach for the mean-standard deviation shortest path problem

ABSTRACT This paper concentrates on the mean-standard deviation shortest path problem, which is an important extension of traditional shortest path problem. Due to the standard deviation term, the general formulation of this problem is nonlinear and concave. We transform this formulation into a mixed-integer conic quadratic program and develop a generalized Benders decomposition approach. The Benders master problem is a continuous conic quadratic program about travel time mean and standard deviation. The subproblem is a least expected travel time path problem with the variance limit. At each iteration, the subproblem generates a generalized Benders optimality cut for the relaxed Benders master problem. The relaxed Benders master problem provides an ascending lower bound and the subproblem produces a feasible solution to update the upper bound. In the numerical experiments, all instances in four transportation networks are solved optimally. This paper provides a novel solving scheme for the mean-standard deviation shortest path problem.

A multicut generalized benders decomposition approach for the integration of process operations and dynamic optimization for continuous systems

The integration of process operations and dynamic optimization leads to large scale optimization problems whose monolithic solution is challenging. In this paper we propose a new formulation of the integrated planning, scheduling, and dynamic optimization problem for continuous single stage systems. We analyze the structure of the problem using Stochastic Blockmodeling and we show that the estimated structure can be used as the basis for a multicut Generalized Benders decomposition (GBD) algorithm, which can solve the problem in reduced computational time. Furthermore, we propose an accelerated hybrid multicut algorithm which can lead to further reduction in computational time. Through case studies, we analyze the computational performance of the proposed formulation and decomposition based solution algorithms.

Integration of scheduling and control for batch process based on generalized Benders decomposition approach with genetic algorithm

Batch process plays an important role in various fields of industrial production, such as chemical engineering and pharmacy, given its characteristics such as small batch size, flexible production, and additional value of the product. Efforts to integrate the scheduling and process control for improving the benefits of batch process are recent. The integration of scheduling and process control is described by state equipment network which is closely related to the processing variables due to the feature of the network structure of the batch process where material splitting and mixing are allowed. The integrated formulation invokes logical disjunctions and operational dynamics which represents a typical mixed-logic dynamic optimization (MLDO) problem. To solve such a MLDO problem, we transform it into a mixed-integer nonlinear program (MINLP) using the Big M reformulation and the simultaneous collocation method. Then, the MINLP problems are solved through a generalized Benders decomposition (GBD) approach and genetic algorithm. The decomposed master problem is a scheduling problem with variable processing times, processing costs, and the Benders cut. Accordingly, the genetic algorithm is implemented to increase benefit. The primal problem comprises a set of separable dynamic optimization problems in the processing units. By collaboratively optimizing the process scheduling and dynamics, the proposed method substantially improves the overall economic performance of the batch production. At last, the feasibility and superiority of the proposed integration model and optimization algorithm can be determined by dealing with specific production instances.

Endogenous price zones and investment incentives in electricity markets: An application of multilevel optimization with graph partitioning

In the course of the energy transition, load and supply centers are growing apart in electricity markets worldwide, rendering regional price signals even more important to provide adequate locational investment incentives. This paper focuses on electricity markets that operate under a zonal pricing market design. For a fixed number of zones, we endogenously derive the optimal configuration of price zones and available transfer capacities on a network in order to optimally govern investment and production decisions in the long run. In a multilevel mixed-integer nonlinear model that contains a graph partitioning problem on the first level, we determine welfare-maximizing price zones and available transfer capacities for a given electricity market and analyze their impact on market outcomes. Using a generalized Benders decomposition approach developed in Grimm et al. (2019) and a problem-tailored scenario clustering for reducing the input data size, we are able to solve the model to global optimality even for large instances. We apply the approach to the German electricity market as an example to examine the impact of optimal zoning on key performance indicators such as welfare, generation mix and locations, or electricity prices. It turns out that even for a small number of price zones, an optimal configuration of zones induces a welfare level that almost approaches the first best.

Supplier Selection for Spare Parts Supply Chain Networks

Spare parts management encompasses all the spare parts decisions; thus, it has a significant impact on the company results. This paper studies the integration of supply chain network design and inventory control decisions into the spare parts management context. The paper presents a modelling structure that integrates supplier selection, warehouse location, consumption point assignments, and inventory control decisions in order to optimize the total system cost while satisfying the company requirements. Additionally, the proposed structure incorporates sourcing strategy selection (i.e. warehousing or cross-docking strategies). A decomposition-based solution strategy is proposed based on the Generalized Benders Decomposition approach. The modelling structure is specified considering an inventory control policy widely used in industrial sites. The proposed integrated approach is focused on supporting the decision-making process yielding a spare parts supply chain system performance improvement.

Endogenous Price Zones and Investment Incentives in Electricity Markets: An Application of Multilevel Optimization With Graph Partitioning

In the course of the energy transition, load and supply centers are growing apart in electricity markets worldwide, rendering regional price signals even more important to provide adequate locational investment incentives. This paper focuses on electricity markets that operate under a zonal pricing market design. For a fixed number of zones, we endogenously derive the optimal configuration of price zones and available transfer capacities on a network in order to optimally govern investment and production decisions in the long run. In a multilevel mixed-integer nonlinear model that contains a graph partitioning problem on the first level, we determine welfare-maximizing price zones and available transfer capacities for a given electricity market and analyze their impact on market outcomes. Using a generalized Benders decomposition approach developed in Grimm et al. (2019) and a problem-tailored scenario clustering for reducing the input data size, we are able to solve the model to global optimality even for large instances. We apply the approach to the German electricity market as an example to examine the impact of optimal zoning on key performance indicators such as welfare, generation mix and locations, or electricity prices. It turns out that even for a small number of price zones, an optimal configuration of zones induces a welfare level that almost approaches the first best.

Design of Dividing‐Wall Columns by Mixed‐Integer Nonlinear Programming Optimization

AbstractDividing‐wall columns (DWCs) offer an interesting alternative to conventional distillation towers. Savings of up to 30 % in both energy and capital cost are reported. While these high savings are realistic in some applications, DWCs offer less or no advantage at all in other applications. In any case, it is important to make a fair comparison between the conventional system and the DWC based on an optimized process design. Mixed‐integer nonlinear programming (MINLP), especially the presented modified Generalized Benders Decomposition approach, could be the tool to make this comparison, as shown here.

Channel assignment, link scheduling, routing, and rate control for multi-channel wireless mesh networks with directional antennas

The wireless mesh network (WMN) has attracted significant interests as a broadband wireless network to provide ubiquitous wireless access for broadband services. Especially with incorporating multiple orthogonal channels and multiple directional antennas into the WMN, each node can communicate with its neighbor nodes simultaneously without interference between them. However, as we allow more freedom, we need a more sophisticated algorithm to fully utilize it and developing such an algorithm is not easy in general. In this paper, we study a joint channel assignment, link scheduling, routing, and rate control problem for the WMN with multiple orthogonal channels and multiple directional antennas. This problem is inherently hard to solve, since the problem is formulated as a mixed integer nonlinear problem (MINLP). However, despite of its inherent difficulty, we develop an algorithm to solve the problem by using the generalized Benders decomposition approach [2]. The simulation results show the proposed algorithm provides the optimal solution to maximize the network utility, which is defined as the sum of utilities of all sessions.

Residential Demand Response Scheduling With Multiclass Appliances in the Smart Grid

In this paper, we study an electricity load scheduling problem in a residence. Compared with previous works in which only limited sets of appliances are considered, we classify various appliances into five sets considering their different energy consumption and operation characteristics, and provide mathematical models for them. With these appliance models, we propose an electricity load scheduling algorithm that controls the operation time and energy consumption level of each appliance adapting to time-of-use pricing in order to maximize the overall net utility of the residence while satisfying its budget limit. The optimization problem is formulated as a mixed integer nonlinear programming (MINLP) problem, which is in general, difficult to solve. In order to solve the problem, we use the generalized Benders decomposition approach with which we can solve the MINLP problem easily with low computational complexity. By solving the problem, we provide an algorithm to obtain the optimal electricity load scheduling of various appliances with different energy consumption and operation characteristics in a unified way.

Optimal design of mixed AC–DC distribution systems for commercial buildings: A Nonconvex Generalized Benders Decomposition approach

• We develop an MINLP model which determines the optimal design of a mixed AC–DC building electrical distribution system. • We develop a tailored global optimization algorithm, based on Nonconvex Generalized Benders Decomposition. • We provide a case study, including a medium office building. Direct current (DC) electricity distribution systems have been proposed as an alternative to traditional, alternating current (AC) distribution systems for commercial buildings. Partial replacement of AC distribution with DC distribution can improve service to DC loads and overall building energy efficiency. This article develops (i) a mixed-integer, nonlinear, nonconvex mathematical programming problem to determine maximally energy efficient designs for mixed AC–DC electricity distribution systems in commercial buildings, and (ii) describes a tailored global optimization algorithm based on Nonconvex Generalized Benders Decomposition. The results of three case studies demonstrate the strength of the decomposition approach compared to state-of-the-art general-purpose global solvers.

Formation of a strategic manufacturing and distribution network with transfer prices

We propose a profit maximization model for the decision support system of a firm that wishes to establish or rationalize a multinational manufacturing and distribution network to produce and deliver finished goods from sources to consumers. The model simultaneously evaluates all traditional location factors in a manufacturing and distribution network design problem and sets intra-firm transfer prices that take account of tax and exchange rate differentials between countries. Utilizing the generalized Benders decomposition approach, we exploit the partition between the product flow and the cash allocation (i.e., the pricing and revenue assignment) decisions in the supply chain to find near optimal model solutions. Our proposed profit maximizing strategic planning model produces intuitive results. We offer computational experiments to illustrate the potential valuable guidance the model can provide to a firm's supply chain design strategic planning process.

Integrated Scheduling and Dynamic Optimization of Complex Batch Processes with General Network Structure Using a Generalized Benders Decomposition Approach

We address the integration of scheduling and dynamic optimization for batch chemical processes. The processes can have complex network structures, allowing material splitting and mixing. The integrated problem is formulated as a mixed-integer dynamic optimization problem where a continuous-time scheduling model is linked to the dynamic models via processing times, processing costs, and batch sizes. To reduce the computational complexity, we develop a tailored and efficient decomposition method based on the framework of generalized Benders decomposition by exploiting the special structure of the integrated problem. The decomposed master problem is a scheduling problem with variable processing times and processing costs, as well as the Benders cuts. The primal problem comprises a set of separable dynamic optimization problems for the processing units. By collaboratively optimizing the process scheduling and the process dynamics, the proposed method substantially improve the overall economic performance of the batch production compared with the conventional sequential method which solves the scheduling problem and the dynamic optimization problems separately. In comparison with the simultaneous method which solves the integrated problem by a general-purpose MINLP solver directly, the proposed method can reduce computational times by orders of magnitude.

Optimal allocation of stock levels and stochastic customer demands to a capacitated resource

This paper considers a new class of stochastic resource allocation problems that requires simultaneously determining the customers that a capacitated resource must serve and the stock levels of multiple items that may be used in meeting these customers’ demands. Our model considers a reward (revenue) for serving each assigned customer, a variable cost for allocating each item to the resource, and a shortage cost for each unit of unsatisfied customer demand in a single-period context. The model maximizes the expected profit resulting from the assignment of customers and items to the resource while obeying the resource capacity constraint. We provide an exact solution method for this mixed integer nonlinear optimization problem using a Generalized Benders Decomposition approach. This decomposition approach uses Lagrangian relaxation to solve a constrained multi-item newsvendor subproblem and uses CPLEX to solve a mixed-integer linear master problem. We generate Benders cuts for the master problem by obtaining a series of subgradients of the subproblem’s convex objective function. In addition, we present a family of heuristic solution approaches and compare our methods with several MINLP (Mixed-Integer Nonlinear Programming) commercial solvers in order to benchmark their efficiency and quality.

Tree network design avoiding congestion

We introduce in this paper an optimal method for tree network design avoiding congestion. We see this problem arising in telecommunication and transportation networks as a flow extension of the Steiner problem in directed graphs, thus including as a particular case any alternative approach based on the minimum spanning tree problem. Our multi-commodity formulation is able to cope with the design of centralized computer networks, modern multi-cast multi-party or hub-based transportation trees. The objective in such cases is the minimization of the sum of the fixed (structural) and variable (operational) costs of all the arcs composing an arborescence that links the origin node (switching center, server, station) to every demand node (multi-cast participants, users in general). The non-linear multi-commodity flow model is solved by a generalized Benders decomposition approach.

Distribution system planning through a generalized Benders decomposition approach

In this paper we introduce a model to specify the distribution system for electrical power industry. The objective function includes maintenance and investment costs of substations, feeders and voltage regulators along with cost associated with energy loss in the feeders. In addition to the standard constraints to specify the network design and the power flow, additional constraints related to power drops in the feeders are taken into account. The model has a nonlinear objective function and nonlinear constraints. We first specify an equivalent formulation where the constraints are linear. The generalized Benders decomposition is used to deal with this model. Simple procedures are introduced to solve the subproblems and generate the cuts. To deal with the nonlinearity of the objective function of the master problem, and equivalent mixed-integer linear problem is introduced and solved by branch and bound.