Column Generation Algorithm Research Articles

A Sequential Follower Refinement Algorithm for Robust Surgery Scheduling

An algorithm for the two-stage robust optimization surgery-to-operating room allocation problem is presented. The second-stage problem is an integer linear program whose convex hull is approximated using three types of specialized valid inequalities and Chvátal-Gomory cuts. The resulting linear relaxation of the second-stage problem is then dualized and integrated into the first-stage problem. The resulting mixed integer linear program, which is an approximation of the original problem, is then solved using a commercial solver. If the solution of this model is not optimal for the second-stage problem, valid inequalities for the second-stage problem are generated, yielding a type of column-generation based approach that we refer to as the sequential follower refinement (SFR) algorithm. Data from an academic medical center are used to compare the computational performance of SFR with the constraint and column generation (C&CG) algorithm, which is the only exact approach that has been specifically applied for this problem in the literature. An extensive numerical study of SFR and its computational characteristics is presented that shows that SFR yields better-quality solutions compared with C&CG, even as the termination criterion of SFR is met much sooner, especially for problems involving higher number of surgeries. History: Accepted by Paul Brooks, Area Editor for Applications in Biology, Medicine, & Healthcare. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0191 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0191 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

A distributionally robust optimization approach for optimal load dispatch of energy hub considering multiple energy storage units and demand response programs

The coupling relationship among various energy sources has been consistently reinforced by the advancement of the energy Internet. In the context of modeling multi-energy systems, the energy hub (EH) represents a significant and valuable approach. To attain optimal operation of the EH, this study proposes an optimal load dispatch model for a system that incorporates a combined heat and power (CHP) unit, gas boiler (GB), electric chiller (EC), absorption chiller (AC), heat energy storage (HES) unit, and electrical energy storage (EES) unit. A two-stage distributionally robust optimization (DRO) method that is driven by data is employed to address the uncertainties of electricity prices. Additionally, the proposed two-stage DRO model is effectively solved through the utilization of the column and constraint generation (C&CG) algorithm. Three cases are discussed in the paper with various energy storage units and demand response (DR) settings. The simulation results show that by deploying energy storage units and participating in DR projects, the EH system can reduce the total costs by 2.56 % and 10 %, respectively. Meanwhile, simulation results reveal that the proposed data-driven DRO model can achieve a compromise between the economy and robustness of the scheduling model compared with stochastic optimization (SO) and robust optimization (RO) methods. The simulation results illustrate the effectiveness of the proposed model for ensuring the economical and efficient operation of the system in the face of electricity price uncertainties.

Multi-Timescale Trading Strategy for Renewable Power to Ammonia Virtual Power Plant in the Electricity, Hydrogen, and Ammonia Markets

Renewable power to ammonia (RePtA) is a prominent zero-carbon pathway for decarbonization. Due to the imbalance between renewables and production energy demand, the RePtA system relies on electricity exchange with the power grid. Participating in the electricity market as a virtual power plant (VPP) may help to reduce energy costs. However, the power profile of local photovoltaics and wind turbines is similar to those in the market, resulting in rising energy costs under the conventional strategy. Hence, we develop a multi-timescale trading strategy for the RePtA VPP in the electricity, hydrogen, and ammonia markets. By utilizing the hydrogen and ammonia buffer systems, the RePtA VPP can optimally coordinate production planning. Moreover, we find it possible to describe the trading of electricity, ammonia, and hydrogen in a unified timeframe. The two-stage robust optimization model of the electricity market is extended to multiple markets and solved by the column and constraint generation (CC&G) algorithm. The case is derived from an actual project in the Inner Mongolia Autonomous Region. Sensitivity analysis demonstrates the economic advantages of the RePtA VPP joining multiple markets over the conventional strategy and reveals the necessity of the hydrogen and ammonia buffer and reactor flexibility.

Two-stage Robust Optimization Scheduling Considering Wind Uncertainty and Multi-area Frequency Dynamic Safety

The large-scale wind power grid connection exacerbates the problem of uneven distribution of inertia between different regions, posing challenges to the frequency security defense of multi-area systems. This article proposes a two-stage robust optimization scheduling model that considers both the uncertainty of wind power output and the dynamic safety of multi-area frequencies. Firstly, according to the conditional value at risk theory, the operating cost and operating risk of the multi-region system are optimized, and the acceptable safety boundary of wind power is obtained. Secondly, multi-zone frequency dynamic safety verification is carried out in harsh wind power scenarios. Decouple the reserve capacity into non-accident reserve and accident reserve, and nest them separately in a two-stage model to achieve clear solution ideas. Finally, the two-stage optimization problem is solved through the column and constraint generation algorithm. The simulation results of numerical examples show that the model and method can effectively improve the frequency response capability of the multi-region system, so as to ensure the safe and economical operation of the system.

Coordination of preventive and emergency dispatch in renewable energy integrated power systems under extreme weather

AbstractIn recent years, frequent occurrence of extreme weather events has caused great losses to the power system. Meanwhile, the high penetration of renewable energy has brought new challenges to the safe operation of the system. Coordination of various measures against different stages of disaster can effectively reduce the overall loss and enhance system's resilience. This paper proposed a methodology to coordinate preventive and emergency dispatch stages based on the defender–attacker–defender model, in which various dispatching measures are modelled as defenders, whereas the extreme weather event and wind power's uncertainty are modelled as attackers. Specifically, a two‐stage three‐layer robust optimal dispatching model is established, in which conventional generator's output, wind farm's grid‐connected planning, and load shedding are taken as the main dispatching measures. To solve this model, the column and constraint generation algorithm is used to transform the three‐layer optimization model into a mixed integer linear problem . The effectiveness of the proposed model is verified in the modified IEEE RTS‐79 test system, and the results show that the coordination of preventive and emergency dispatch can effectively reduce the load outage cost under extreme weather, thus enhancing the ability of power system to cope with extreme scenarios.

Multi-trucks-and-drones cooperative pickup and delivery problem

This study aims to propose a decision methodology on scheduling trucks and drones for truck-and-drone cooperative delivery and pickup system. A fleet contains multiple truck groups; each truck group is a truck with carrying multiple drones. The fleet serves a set of dispersed customers who have the requirements of pickup and delivery services as well as their due time for service. A mixed-integer linear programming (MILP) model is formulated in this study for routing the trucks and drones in the fleet so that each customer’s pickup or delivery requirements could be served by either a truck or a drone before their required due time. For solving the MILP model efficiently, this study designs a novel hybrid algorithm by combining the column generation and the logic-based Benders decomposition. Based on the main frame of column generation algorithm, the hybrid algorithm uses logic-based Benders decomposition to solve the pricing problem, and dynamic programming to solve subproblems of logic-based Benders decomposition for the purpose of accelerating the whole algorithm’s solving process. Numerical experiments are also conducted on the context of the Hangzhou city so as to validate the efficiency of the proposed hybrid algorithm. Some managerial implications are also derived on the basis of some sensitivity analysis. The proposed methodology, i.e., the MILP model and the novel hybrid algorithm, is potentially useful for platform operators who run the truck-and-drone based urban delivery systems.

A tri-level programming-based frequency regulation market equilibrium under cyber attacks

Owing to their flexibility and rapid response, grid-connected distributed energy resources (DERs) are wielding growing influence in frequency regulation markets (FRMs). Nevertheless, compared with conventional large-scale generators, small-scale DERs are usually weakly shielded by cyber security measures. This offers attackers the opportunity of disrupting the clearing and settlement of FRMs by manipulating the bid information of DERs. In this paper, the frequency regulation market equilibrium is studied considering the dynamic gaming between attackers and defenders, both of which need the knowledge of FRMs for their interventions. Specifically, a tri-level programming model characterizing the attacker–defender–operator (ADO) interdiction problem in FRMs is developed and then analyzed using a column and constraint generation algorithm, thereby achieving market equilibrium representing the defender's best response to the attacker. The defense strategy in the market equilibrium can attenuate the negative influence of cyber attacks upon the FRMs to the maximum extent. Finally, based on the operating rules of the California Independent System Operator, the FRM process considering the ADO interdiction is simulated and the numerical equilibrium results are presented.

Vulnerability analysis of an electric vehicle fleet for car-sharing service under cyber attacks

The booming development of the car-sharing industry and the continuous increase in the market share of electric vehicles demand effective dispatching strategies that consider collaborative optimization of route selection and charging management. Furthermore, in recent years, network attacks have posed significant threats to traffic management centers. Therefore, this paper presents a pioneering work aimed at investigating the vulnerability of electric vehicle sharing services in traffic management systems under network attacks. Firstly, we develop a bilevel optimization model to capture the game between network attackers and electric vehicle fleet operators. The upper-level problem represents the behavior of network attackers, who aim to maximize the total cost by manipulating traffic control signals and sensor data. The lower-level problem determines the system operator's decisions on optimizing route selection and charging scheduling to minimize costs. Secondly, a nested column generation algorithm is employed to solve the bilevel optimization model. Finally, several sensitivity studies are conducted using real-world traffic networks to examine the effectiveness of the proposed model and the influence of various parameters on system performance.

Integrated multi-plant collaborative production, inventory, and hub–spoke delivery of make-to-order products

Motivated by make-to-order applications with committed delivery dates in a variety of industries, we investigate the integrated multi-plant collaborative production, inventory, and hub–spoke delivery problem in a complex production–distribution network. This network includes multi-location heterogeneous plants, distribution centers, and customers, for producing customized and splittable orders with one or more general-size multi-type jobs. Completed jobs are transported from plants to distribution centers, and then the orders whose all constituent jobs have arrived are delivered from distribution centers to customer sites. The objective is to make integrated scheduling decisions for production, inventory, and delivery, for minimizing total cost composed of production, transportation, tardiness, and inventory. We first formulate this problem as a mixed-integer programming model, and analyze its intractability by proving that the problem is NP-hard and no approximation algorithms exist with a constant worst-case ratio. We then reformulate this problem as a binary integer linear programming model to select a feasible schedule for each job, and propose a combined column generation and two-layer column enumeration algorithm to solve it. Through extensive numerical experiments, we demonstrate that our proposed algorithm is capable of generating optimal or near-optimal solutions expeditiously and outperforms four benchmark approaches, and gain valuable managerial insights for practitioners.

Capacity Value Assessment for a Combined Power Plant System of New Energy and Energy Storage Based on Robust Scheduling Rules

With the rapid increase in new energy penetration, the uncertainty of the power system increases sharply. We can smooth out fluctuations and promote the more grid-friendly integration of new energy by combining it with energy storage. This paper proposes an evaluation method for assessing the value of a combined power plant system of new energy and energy storage using robust scheduling rules. Firstly, the k-means clustering algorithm is improved by using the elbow method in order to generate typical scenarios that can be used for the operation optimization of the combined power plant system of new energy and energy storage. Then, a two-stage robust optimization model of the combined power plant system of new energy and energy storage with a min–max–min structure is constructed according to the uncertainty of new energy. In this model, the operation constraints and coordinated control of wind–solar–thermal–storage units are considered. By constructing the uncertainty set of the new energy output, the overall operating cost of the system is minimized and uncertainty adjustment parameters are introduced to flexibly adjust the conservatism of the scheduling rules. Furthermore, based on the column and constraint generation algorithm and strong duality theory, the original problem can be decomposed into a master problem and subproblems with mixed integer linear characteristics for an alternating solution, so as to obtain the optimal solution of the original problem, and finally obtain the robust scheduling rule with the lowest operating cost under the worst scenario. Finally, based on the wind and solar power output curves and the output of each unit under the robust scheduling rules, combined with the value estimation method of the combined power plant system of new energy and energy storage, the value of the combined power plant system of new energy and energy storage is evaluated. Through the establishment of models and example analysis, it is proven that raising the quantity of the grid-connected power generated with new energy will cause an increase in the volatility of the power system; it will also bring considerable benefits to new energy plants, and the energy storage can improve the stability of the system. The above can provide references for the subsequent energy storage configuration in the planning of a combined power plant system of new energy and energy storage.

Robust optimization method of power system multi resource reserve allocation considering wind power frequency regulation potential

With the large-scale grid integration of wind turbines, the traditional power system active standby capacity can no longer meet the frequency safe operation requirements. In order to further improve the rationality of power system active standby configuration, this paper proposes a power system multi-resource standby robust optimisation strategy considering the WP FR potential. Firstly, a frequency response model is derived for the WP integrated power system. Subsequently, a two-stage multi-resource reserve robust configuration model is constructed while acknowledging frequency constraints. Secondly, an uncertainty modeling method considering the temporal error correlation is proposed to accurately characterize the uncertainty of WP, and the original problem is transformed into a master-subproblem iteration solution by column and constraint generation algorithm. Finally, the efficiency of the proposed model is supported through the adjusted IEEE RTS-79 test system. The results indicate that the proposed model efficiently capitalizes on FR potential of WP and shows the flexibility of multi-resource control while guaranteeing system frequency stability and encouraging WP participation in reserve configuration. Additionally, the proposed uncertainty modeling method can enhance operational efficiency while maintaining robustness of resource allocation.

Robust optimization for the integrated berth allocation and quay crane assignment problem

AbstractThis paper studies the berth allocation and quay crane assignment problem (denoted by BACAP) under uncertainty. We assume that the ships' arrival and operation time is uncertain in this problem. We merge the proactive and reactive strategies to address the two‐stage robust optimization (denoted by RO) model for the BACAP to obtain a complete schedule with robustness. We obtain the berth allocation and quay crane assignment with a proactive strategy in the first stage. In the second stage, we formulate a rescheduling model with a reactive strategy considering the sensitivity towards the change in the complete schedule. The second stage model is based on the prospect theory, a quantitative way to describe the stakeholders' perception, including the port managers and shipowners, of the deviation from the baseline plan. The two stages are iterated until a favorable schedule with high robustness is found. To illustrate the superiority of the two‐stage robust optimization model with the prospect theory for the complete schedule, we give an intuitive example to compare the performance among the related models. The two‐stage RO model with the prospect theory for the complete schedule can generate a lower cost and higher robustness schedule. As for the solution methods, the column and constraint generation (denoted by C&CG) algorithm is applied to obtain the exact solution for the two‐stage RO model. Moreover, we propose the scenario‐constrained C&CG (denoted by SC) algorithm, which can reduce constraints and variables for the master problem to accelerate the solving process of the two‐stage RO model. In addition, the optimality of the SC algorithm is verified by analyzing the pattern of the occurrence of the worst‐case scenarios. Besides, to tackle the large‐scale instances, we propose the schedule‐fixed (denoted by SF) algorithm, in which the results of the previous iterations are treated as fixed. The SF algorithm can increase computing efficiency with a small gap compared to the optimal solution value. Furthermore, extensive numerical experiments are conducted on both real‐life instances and randomly generated instances to verify the superiority and generality of our model and algorithms.

Column generation algorithm based on recursive matrix for optimal development strategy of physical education teaching in colleges and universities

Abstract In this paper, in the process of solving the mathematical model of sports facility optimization, the integer linear programming issue of choosing a sports facility is solved using a branch delimitation approach paired with a recursive matrix-based column-generating method. As part of the process of creating a scientific evaluation system for physical education, a hierarchical gray correlation-based evaluation model for the quality of physical education in colleges and universities is built by combining the hierarchical analysis weight-solving method and the solution method of the gray correlation degree. After the analysis, 81.1% of the teachers and students were satisfied after physical education facilities were optimized with the physical education teaching facilities. The gray correlation coefficient of physical education hardware facilities and software is 0.7, and the gray correlation coefficient of physical education effect is 0.18.

Distributionally robust production and replenishment problem for hydrogen supply chains

This study investigates a two-stage production and replenishment problem for the hydrogen supply chain (HSC), where the uncertainty of demand and the risk of pipeline disruptions are simultaneously considered. We model this problem by adopting a distributionally robust optimization (DRO) approach. Before realizing the demands, the decision-maker decides the hydrogen production, storage, and truck-transported replenishment policies. Then, after demands and pipeline working status are realized, the pipeline-transported replenishment policy is determined in response to the first-stage decisions. The replenishment policies are determined under differentiated service level requirements. In addition, we characterize the uncertainty and correlation of demands through a factor-based model approach to accommodate the uncertainty of the distribution of demand factors. To solve the proposed model, we propose a column and constraint generation (CCG) algorithm, which can terminate in finite iterations. Through theoretical analysis and numerical experiments, we verify that the proposed model outperforms classical benchmark models in terms of total costs and are more robust. Finally, we examine the impact of the replenishment structure on the performance of the HSC and the computational efficiency of the proposed CCG algorithm.

Adaptive robust scheduling optimization of a smart commercial building considering joint energy and reserve markets

With the increase in renewable energy penetration and the widespread use of electric vehicles, the uncertainty of controllable resources adversely affects the economic scheduling of smart buildings. Meanwhile, the smart building can take advantage of its flexibility through multiple markets mechanism. Therefore, this paper proposes an optimal scheduling model for the smart commercial building, including electric vehicle charging, which partakes in the day-ahead energy reserve markets. Firstly, the behavior of electric vehicles is obtained through Monte Carlo simulation. This paper categorizes electric vehicles according to their behavioral characteristics and proposes corresponding scheduling strategies. Then, the joint energy and reserve markets mechanism is introduced, and the robust adaptive optimization (ARO) method is employed to deal with uncertainties. The proposed model is a three-level and two-stage model, which can be solved by the column and constraints generation algorithm. Finally, the economy and feasibility of the proposed model are validated through a case study. Results show that the ARO method saves 3%–4% of expenses more than the pure robust optimization method, and the proposed joint markets mechanism can increase profits by 1%–4%.

Robust mixed electron-photon radiation therapy planning for soft tissue sarcoma.

Mixed electron-photon beam radiation therapy (MBRT) is an emerging technique in which external electron and photon beams are simultaneously optimized into a single treatment plan. MBRT exploits the steep dose falloff and high surface dose of electrons while maintaining target conformity by leveraging the sharp penumbra ofphotons. This study investigates the dosimetric benefits of MBRT for soft tissue sarcoma (STS)patients. A retrospective cohort of 22 STS of the lower extremity treated with conventional photon-based Volumetric Modulated Arc Therapy (VMAT) were replanned with MBRT. Both VMAT and MBRT treatments were planned on the Varian TrueBeam linac using the Millenium multi-leaf collimator. No electron applicator, cutout or additional collimating devices were used for electron beams of MBRT plans. MBRT plans were optimized to use a combination of 6 MV photons and five electron energies (6, 9, 12, 16, 20 MeV) by a robust column generation algorithm. Electron beams in this study were planned at standard 100 cm source-axis distance (SAD). The dose to the clinical target volume (CTV), bone, normal tissue strip and other organs-at-risk (OARs) were compared using a Wilcoxon signed-ranktest. As part of the original VMAT treatment, tissue-equivalent bolus was required in 10 of the 22 patients. MBRT plans did not require bolus by virtue of the higher electron entrance dose. CTV coverage by the prescription dose was found to be clinically equivalent between plans of either modality: (MBRT) = 97.9 ± 0.2% versus (VMAT) = 98.1 ± 0.6% (p=0.34). Evaluating the absolute paired difference between doses to OARs in MBRT and VMAT plans, we observed lower to normal tissue in MBRT plans by 14.9 ± 3.2% ( ). Similarly, to bone was found to be decreased by 8.2 ± 4.0% ( ) of the bonevolume. For STS with subcutaneous involvement, MBRT offers statistically significant sparing of OARs without sacrificing target coverage when compared to VMAT. MBRT plans are deliverable on conventional linacs without the use of electron applicators, shortened source-to-surface distance (SSD) or bolus. This study shows that MBRT is a logistically feasible technique with clear dosimetricbenefits.

Distributionally Robust Programming of Berth-Allocation-with-Crane-Allocation Problem with Uncertain Quay-Crane-Handling Efficiency

In order to promote the efficient and intelligent construction of container ports, we focus on the optimization of berth-and-quay-crane (QC) allocation in tidal terminal operations. This paper investigates the quay-crane-profile-(QC-profile)-based assignment problem, and considers the uncertainty in QC profiles regarding QC efficiency for the first time. A mixed-integer programming (MIP) model is established for a discrete berth allocation with a crane-assignment problem (BACAP), considering the tide time window. We aim to minimize the total time loss caused by anchorage and the delay of vessels. Leveraging the theory of uncertainty optimization, the proposed deterministic model is extended into a stochastic programming (SP) model and a distributionally robust optimization (DRO) model, via the consideration of the random QC efficiency. To solve the proposed models, a column generation (CG) algorithm is employed, utilizing the mathematical method and subproblem-solving approach. The numerical experiments with different instances demonstrate that the DRO model yields a smaller variation in the objective function values, and the effectiveness of the CG method. The experimental results verify the robustness of the constructed models, and the efficiency of the proposed algorithm.

A branch-and-price method for a two-echelon location routing problem with recommended satellites

In city logistics, the two-echelon location routing problem is crucial in coordinating resources and radiating nodes, enabling the efficient planning of relay stations and vehicle distribution routes. However, with the expansion of distribution scopes and changes in urban development, fixed relay stations may not be able to adapt to the evolving network demand structure, leading to reduced efficiency. This paper proposes a two-echelon location routing problem with recommended satellites (2E-LRPRS) to address this issue. These satellites are chosen from the customer set and can be reoptimized to enhance the network's adaptability. A mixed-integer linear program (MILP) is formulated to minimize the distribution cost of the two-echelon network. To solve this problem, a column-generation algorithm (CG) is proposed, which decomposes the original problem into a linear relaxation main problem and a series of two-stage shortest-path subproblems with resource constraints. The branch-and-pricing algorithm (B&P) is then designed to obtain the integer solution. The experimental results demonstrate that the proposed B&P is significantly more efficient and effective than the MILP solver - Cplex. Furthermore, by selecting relay stations from customers, the recommended satellites can be flexibly adjusted based on the locations and demands of distribution customers, leading to an average operating cost savings of 8.43%.

Adaptive robust optimization for lot-sizing under yield uncertainty

In manufacturing environments, uncertain production yield directly impacts the quality and feasibility of the production planning decisions. This paper investigates the use of adaptive robust optimization to hedge against uncertain yield when determining a production plan, and to react properly when updated information unfolds. We first derive a myopic adaptive robust policy for the inventory management problem, a special case of the lot-sizing problem where the setup and the production costs are omitted. We show that the policy is optimal under mild assumptions. Second, we address a multi-period single-item lot-sizing problem with a backorder and uncertain yield via adaptive robust optimization. We formulate an adaptive robust model based on the budgeted uncertainty set, where we exploit a linear approximation to transform the quadratic constraints into a mixed-integer linear program. We also propose a column and constraint generation algorithm to solve the adaptive model exactly. Finally, we demonstrate the performances of the proposed approaches and the value of the adaptive robust solutions through extensive numerical experiments.

Joint location–inventory problem with economies and diseconomies of scale: A general model and an efficient column generation algorithm

This paper studies the location–inventory problem, which jointly optimizes the location, inventory, and allocation of distribution centers (DCs) to serve uncertain demand of multiple retailers. Problems of this kind have been extensively studied to carefully tradeoff various costs, including the fixed facility location cost and the variable facility operating cost Most existing works assumed the facility operating cost to be either a linear or a concave function of the volume of demand allocated. However, practical evidence suggests that when the volume of allocated demand exceeds a certain level, the marginal cost may increase due to overtime charge or facility congestion, leading to a complicated cost function. In this paper, we capture the facility operating cost as a general inverse S-shaped function of the volume of allocated demand, which is first concave due to economies of scale and then convex due to diseconomies of scale. We formulate this problem as a set-covering model and propose a column generation (CG) algorithm to solve its linear relaxation. The corresponding pricing problem has nice structural properties and can be solved efficiently by a branch-and-bound (B&B) method. The efficiency of the proposed algorithm is validated through extensive numerical studies, where two types of inverse S-shaped functions are considered to capture facility operating cost under various situations. Sensitivity analysis is also carried out to reveal managerial insights and analyze the delicate tradeoffs between several cost components.