Column Generation Research Articles

An exact algorithm for the multi-trip vehicle routing problem with time windows and multi-skilled manpower

Motivated by the challenges of non-emergency patient transportation services in the healthcare industry, this study investigated a multi-trip vehicle routing problem incorporating multi-skilled manpower with downgrading. We aimed to find an optimal plan for vehicle routing and multi-skilled manpower scheduling in tandem with the objective of minimizing the total cost, including travel and staff costs, without violating time windows and lunch break constraints. To address this, two mathematical models were formulated: an arc-flow model and a trip-based set-covering model. In addition, a branch-and-price-and-cut algorithm, based on the set-covering model, was proposed to solve practical-scale instances. To determine the feasibility of the integer solutions, we introduce a feasibility check model. To address the multi-trip characteristics of the proposed problem, a novel two-phase column generation algorithm was introduced to solve the subproblem. This approach differs from traditional one-phase labeling algorithms and involves a tailored labeling algorithm for obtaining non-dominated labels in the first phase and a strategy to identify the trip with the minimum reduced cost for each label in the second phase. Furthermore, novel and efficient staff-based inequalities were developed by improving the k-path inequalities. Extensive numerical experiments were conducted to demonstrate the solution performance of the proposed algorithm and reveal managerial insights for non-emergency ambulance operations. The results demonstrate that our algorithm can successfully solve instances with up to 50 patients to optimality within two hours. Moreover, we demonstrated the value of jointly optimizing vehicle routing and staff planning, which can result in significant cost savings of up to 19.4%.

Two-stage robust operation of electricity-gas-heat integrated multi-energy microgrids considering heterogeneous uncertainties

With the widespread adoption of combined heat and power and power-to-heat technologies, multi-energy microgrids (MEMGs) have been garnering significant research attention from both industry and academia. However, dealing with uncertainties from renewable energy and load and coordinating multiple energy carriers are the main challenges for MEMG operation. In this regard, a two-stage robust operation method of electricity-gas-heat integrated MEMGs considering heterogeneous uncertainties is proposed in this paper. First, network models for an electricity-gas-heat-based distribution-level MEMG are formulated considering the dynamic characteristics of gas and heat networks. Then, the power-to‑hydrogen-and-heat unit and ladder-type carbon trading mechanism are introduced to reduce the curtailment of wind power and carbon emissions. Further, a two-stage robust optimization (TSRO) method is applied to tackle uncertainties of wind power and load under extreme scenarios in the MEMG operation by iteratively solving the operation problem with the column and constraint generation (C&CG) algorithm. Finally, case studies are conducted to verify our proposed method, demonstrating that it can reduce the multi-energy supply cost while the stepped carbon trading mechanism can also significantly reduce carbon emissions.

Bi-level robust planning of hydrogen energy system for integrated electricity–heat–hydrogen energy system considering multimode utilization of hydrogen

This paper proposes a bi-level robust planning model to address the rational configuration of a hydrogen energy system, accounting for the impact of wind power uncertainty in an integrated electricity–heat–hydrogen energy system (IEHHES) with the increasing wind power penetration. The upper-level problem aims to determine the capacity of the hydrogen energy system in this system, whereas the lower-level problem is formulated as a two-stage robust optimization model to simulate the typical daily operational scheme of the system, considering uncertain wind turbine output and the on/off switching of devices. This model considers multiple hydrogen utilization modes and the physical characteristics of various hydrogen storage tanks. The complex bi-level robust model is intractable directly. Therefore, this bi-level robust planning model is initially decomposed into planning and operation subproblems. These subproblems are then can be solved in parallel using the alternating direction method of multipliers (ADMM) algorithm; the planning subproblem, a quadratic programming model, is solved using a commercial solver; the operating subproblem, a two-stage robust optimization model, is solved using the column and constraint generation algorithm. Finally, case studies validate the advantages and effectiveness of the developed model and algorithm.

A machine learning approach to two-stage adaptive robust optimization

We propose an approach based on machine learning to solve two-stage linear adaptive robust optimization (ARO) problems with binary here-and-now variables and polyhedral uncertainty sets. We encode the optimal here-and-now decisions, the worst-case scenarios associated with the optimal here-and-now decisions, and the optimal wait-and-see decisions into what we denote as the strategy. We solve multiple similar ARO instances in advance using the column and constraint generation algorithm and extract the optimal strategies to generate a training set. We train machine learning models that predict high-quality strategies for the here-and-now decisions, the worst-case scenarios associated with the optimal here-and-now decisions, and the wait-and-see decisions. The models can be applied to problems with varying dimensions. We also introduce novel methods to expedite training data generation and reduce the number of different target classes the machine learning algorithm needs to be trained on. We apply the proposed approach to the facility location, the multi-item inventory control and the unit commitment problems. Our approach solves ARO problems drastically faster than the state-of-the-art algorithms with high accuracy.

A data-driven robust optimization method based on scenario clustering for PVC production scheduling under uncertainty

In this paper, a data-driven robust optimization method based on scenario clustering is proposed for addressing energy consumption uncertainty characterized by correlation and multi-peaked distribution within the PVC production process. Firstly, principal component analysis (PCA) and kernel density estimation (KDE) methods are used to capture the correlation and distribution information effectively across multidimensional uncertain parameters; then a modified K-means clustering method based on density peaks is applied to cluster energy consumption scenarios and the flexible uncertainty subsets is established. A two-stage robust optimization model for the vinyl chloride production section is then proposed, and the column and constraint generation algorithm is applied to solved. Finally, the effectiveness of the proposed method is validated through a PVC production case study. Comparative results demonstrate that the proposed model reduces energy consumption under uncertainty and improves the robustness of PVC production scheduling.

Approximate linear programming for a queueing control problem

Admission decisions for loss systems accessed by multiple customer classes are a classical queueing control problem with a wide variety of applications. When a server is available, the decision is whether to admit an arriving customer and collect a lump-sum revenue. The system can be modeled as a continuous-time infinite-horizon dynamic program, but suffers from the curse of dimensionality when different customer classes have different service rates. We use approximate linear programming to solve the problem under three approximation architectures: affine, separable piecewise linear and finite affine. The finite affine approximation is a recently proposed generalization of the affine approximation, which allows for non-stationary parameters. For both affine and finite affine approximations, we derive equivalent, but more compact, formulations that can be efficiently solved. We propose a column generation algorithm for the separable piecewise linear approximation. Our numerical results show that the finite affine approximation can obtain the tightest bounds for 75% of the instances among the three approximations. Especially, when the number of servers is large and/or the load on the system is high, the finite affine approximation always achieves the tightest bounds. Regarding policy performance, the finite affine approximation has the best performance on average compared to the other two approximations and the achievable performance region method (Bertsimaset al., 1994, Kumar and Kumar, 1994). Furthermore, the finite affine approximation is 4 to 5 orders of magnitude faster than the achievable performance region method and the separable piecewise linear approximation for large-scale instances. Therefore, considering bounds, policy performance, and computational efficiency, the finite affine approximation emerges as a competitive approximation architecture for the class of problems studied here.

An exact method for vehicle routing problem with backhaul discounts in urban express delivery network

The surge in e-commerce has led to an increased demand for urban express services, requiring the strategic development of delivery networks that are both efficient and cost-effective. This study addresses a practical vehicle routing problem (VRP) in an urban express delivery network to minimize transportation costs. Specifically, it considers the implementation of backhaul discounts, a factor disregarded in the existing literature. This VRP is further complicated by various realistic constraints, including pickup and delivery, time windows, multiple trips, heterogeneous fleets, and docking capacity limitations, which make most general VRP solvers inapplicable. This study proposes a trip-based formulation to overcome this challenge and develop a tailored branch-and-price algorithm. Feasible trips are classified into four types to simplify the computation of backhaul discounts, thereby enhancing solution efficiency. Validation with real-world data from SF Express substantiates the efficacy of our method and yields insights for sustainable city logistics management. Moreover, our simplified column generation algorithm exhibits competitive performance, achieving optimal solutions expeditiously for the tested instances.

A robust optimization framework for two‐echelon vehicle and UAV routing for post‐disaster humanitarian logistics operations

AbstractProviding first aid and other supplies (e.g., epi‐pens, medical supplies, dry food, water) during and after a disaster is always challenging. The complexity of these operations increases when the transportation, power, and communications networks fail, leaving people stranded and unable to communicate their locations and needs. The advent of emerging technologies like uncrewed autonomous vehicles can help humanitarian logistics providers reach otherwise stranded populations after transportation network failures. However, due to the failures in telecommunication infrastructure, demand for emergency aid can become uncertain. To address the challenges of delivering emergency aid to trapped populations with failing infrastructure networks, we propose a novel robust computational framework for a two‐echelon vehicle routing problem that uses uncrewed autonomous vehicles (UAVs), or drones, for the deliveries. We formulate the problem as a two‐stage robust optimization model to handle demand uncertainty. Then, we propose a column‐and‐constraint generation approach for worst‐case demand scenario generation for a given set of truck and UAV routes. Moreover, we develop a decomposition scheme inspired by the column generation approach to generate UAV routes for a set of demand scenarios heuristically. Finally, we combine the decomposition scheme within the column‐and‐constraint generation approach to determine robust routes for both trucks (first echelon vehicles) and UAVs (second echelon vehicles), the time that affected communities are served, and the quantities of aid materials delivered. To validate our proposed algorithms, we use a simulated dataset that aims to recreate emergency aid requests in different areas of Puerto Rico after Hurricane Maria in 2017.

Investigating Large Neighbourhood Search for Bus Driver Scheduling

The Bus Driver Scheduling Problem (BDSP) is a combinatorial optimisation problem with high practical relevance. The aim is to assign bus drivers to predetermined routes while minimising a specified objective function that considers operating costs as well as employee satisfaction. Since we must satisfy several rules from a collective agreement and European regulations, the BDSP is highly constrained. Hence, using exact methods to solve large real-life-based instances is computationally too expensive, while heuristic methods still have a considerable gap to the optimum. Our paper presents a Large Neighbourhood Search (LNS) approach to solve the BDSP. We propose several novel destroy operators and an approach using column generation to repair the sub-problem. We analyse the impact of the destroy and repair operators and investigate various possibilities to select them, including adaptivity. The proposed approach improves all the upper bounds for larger instances that exact methods cannot solve, as well as for some mid-sized instances, and outperforms existing heuristic approaches for this problem on all benchmark instances.

Bibliometric Study of Metaheuristics Application for Solving Inventory Routing Problem

Bibliometric Study of Metaheuristics Application for Solving Inventory Routing Problem

Generalized Riskiness Index in Vehicle Routing Under Uncertain Travel Times: Formulations, Properties, and Exact Solution Framework

We consider a vehicle routing problem with time windows under uncertain travel times where the goal is to determine routes for a fleet of homogeneous vehicles to arrive at the locations of customers within their stipulated time windows to the maximum extent while ensuring that the total travel cost does not exceed a prescribed budget. Specifically, a novel performance measure that accounts for the riskiness associated with late arrivals at the customers, called the generalized riskiness index (GRI), is optimized. The GRI covers several existing riskiness indices as special cases and generates new ones. We demonstrate its salient managerial and computational properties to motivate it better. We propose alternative set partitioning-based models of the problem. To obtain the optimal solution, we develop an exact solution framework combining route enumeration and branch-price-and-cut algorithms, in which the GRI is dealt with in route enumeration and column generation subproblems. We mainly reduce the solution space by exploiting the GRI and budget constraints’ properties without losing optimality. The proposed method is tested on a collection of instances derived from the literature. The results show that a new instance of the GRI outperforms several existing riskiness indices in mitigating lateness. The exact method can solve instances with up to 100 nodes to optimality. It can consistently solve instances involving up to 50 nodes, outperforming state-of-the-art methods by more than doubling the manageable instance size. Funding: This work was supported by the National Natural Science Foundation of China [Grants 72101187, 72371204, 72021002, and 71901180], the Qatar National Research Fund [Grant ARG01-0430-230029], Natural Science Foundation of Sichuan Province [24NSFSC6232], and Guanghua Talent Project of the Southwestern University of Finance and Economics. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2023.0345 .

Risk-averse electrolyser sizing in industrial parks: An efficient stochastic-robust approach

Hydrogen is called to be one of the most important energy vectors in future energy systems. Nowadays, its use in the industry sector is prominent, finding multiple applications in fertilizer production or oil refining. In this sense, some industry sectors demand a considerable amount of hydrogen for their processes. In many cases, hydrogen must be purchased externally, which supposes a challenge due to hydrogen transportation is costly and few efficient. In this context, local hydrogen production through mature electrolysis technology may suppose an attractive alternative in industrial parks. This paper focuses on this topic, in particular, a risk-averse electrolyser sizing methodology is developed. The new approach accounts for uncertainties in electricity prices as well as local renewable generation and demand through an original hybrid stochastic-robust model. The developed uncertainty modelling is integrated into a novel four-level optimization framework, whose main result is the optimal electrolyser rated power. To efficiently attain the solution, an original hybridization of the Benders' decomposition and the Column and Constraint Generation Algorithm is proposed. The developed methodology results efficient in an illustrative three-industry park, showing that installing local hydrogen generation may reduce the amount of hydrogen purchased externally (by 38%) and project costs by 2.5%. Furthermore, increasing the robustness level leads to increase the project cost by 8%, while assuming unfavourable realization of uncertainties. Moreover, the developed tool is further validated in larger parks, involving an increasing number of industries, showing that the proposed methodology scales well with the size of the park.

Crew Scheduling and Routing Problem in Road Restoration via Branch-and-Price Algorithms

This paper addresses the single crew scheduling and routing problem in the context of road network repair and restoration, which is critical in assisting complex postdisaster decisions in humanitarian logistics settings. We present three novel formulations for this problem, which are the first suitable for column generation and branch-and-price (BP) algorithms. Specifically, our first formulation is based on enumerating crew schedules and routes while explicitly defining the relief paths. The second formulation relies on enumerating the schedules, routes, and relief paths. Finally, the third formulation builds upon the second one by including additional constraints and variables related to relief path decisions. Considering each formulation, we propose BP algorithms that rely on several enhancements, including a new dynamic programming labeling algorithm to efficiently solve the subproblems. Extensive computational results based on 648 benchmark instances reveal that our BP algorithms significantly outperform existing exact approaches, solving 450 instances to optimality, and remarkably 118 instances for the first time. Our framework is also very effective in improving the lower bounds, upper bounds, and optimality gaps that have been reported in the literature. Funding: This work was supported by the Fundação de Amparo à Pesquisa do Estado de São Paulo [Grants 15/26453-7, 16/01860-1, 16/15966-6, and 19/23596-2], the Conselho Nacional de Desenvolvimento Científico e Tecnológico [Grant 313220/2020-4], and the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior. Supplemental Material: The online appendices are available at https://doi.org/10.1287/trsc.2023.0227 .

An enhanced exact algorithm for the multi-trip vehicle routing problem with time windows and capacitated unloading station

This paper introduces a vehicle routing problem that simultaneously considers multiple trips, time windows, and a capacitated unloading station. This problem is a generalization of the multi-trip vehicle routing problem with time windows, which determines a set of least-cost vehicle routes to fulfill all customer demands while respecting the constraints of vehicle capacity and time windows. Due to restricted resources (e.g., equipment and labor force) at the depot, vehicles may need to wait in a queue for being unloaded when they arrive. This unloading capacity constraint significantly complicates the problem, as it causes a trip to involve three stages—traveling, waiting, and unloading. We formulate this problem as an arc flow model and a trip-based set partitioning model, where the latter is solved by a branch-price-and-cut (BPC) algorithm. To improve the computational aspect of the BPC framework, a two-phase column generation (CG) algorithm is designed. First, a bidirectional labeling algorithm is tailored to solve the pricing problem, where two accelerating strategies are employed to speed up the resolution process. Meanwhile, k-path inequalities and limited-memory subset row inequalities are utilized to tighten the linear relaxation of the master problem. Computational results based on the instances adapted from the well-known Solomon’s benchmark show that the developed BPC algorithm can solve most instances within 50 customers to optimality in a short time frame and some instances of 100 customers to optimality within a 3-hour time limit. Moreover, our BPC algorithm performs better than exact algorithms in the literature for similar problem variants in both solution quality and computing time.

Optimal autonomous truck platooning with detours, nonlinear costs, and a platoon size constraint

Autonomous trucks offer a promising avenue for enhancing the efficiency and reducing the environmental impact of road freight transportation. This paper examines a transitional phase towards a fully unmanned truck fleet, focusing on a platooning approach with a lead driver. We develop a novel optimization model for autonomous truck platooning that simultaneously considers platoon formation, scheduling, and routing to minimize costs related to labor and fuel. The model incorporates the possibility of detours, nonlinear fuel savings due to air-drag reduction, and the practical platoon size limit. We present an enhanced column generation method, termed the platoon-generation-and-updating approach, which demonstrates high effectiveness in reducing computational time and complexity. Our numerical analysis, based on the Hong Kong highway network, demonstrates the substantial cost advantages of autonomous truck platooning. It also investigates how platooning efficiency is influenced by various operating factors, including truck fleet size, platoon size restrictions, labor-to-fuel cost ratio, and the strictness of delivery time windows, with practical implications interwoven into the discussion.

Railway crew planning with fairness over time

Passenger railway operators typically employ large numbers of drivers and guards, and are interested in providing them with fair and attractive working conditions. At Netherlands Railways (NS), the largest passenger railway operator in The Netherlands, this challenge is addressed through the use of Sharing-Sweet-and-Sour rules, which specify a fair allocation of sweet (attractive) and sour (unattractive) work over the different crew bases. While these rules are currently implemented at the crew base level and in the tactical planning phase, NS is considering formulating these rules at the individual level, in the operational planning phase, and with respect to a given planning period. This gives rise to a new problem, which we call the railway crew planning problem with fairness over time. We propose a rolling horizon approach with a penalty-based feedback mechanism and a column generation heuristic to solve this problem. On several real-life instances from NS, including up to 572 unique guards, this method is able to satisfy the individual rules for on average 95.2% of the employees.

Column generation based solution for bi-objective gate assignment problems

Column generation based solution for bi-objective gate assignment problems

Subnetwork prediction approach for aircraft schedule recovery

The combinatorial optimization techniques for the aircraft schedule recovery problem are too expensive to compute. However, we require a reasonable solution within 1–2 min for real-time application. The computational time of traditional optimization algorithms will drastically reduce if we can accurately predict the subnetwork affected by the disruption. In this study, we advocate the use of machine learning (ML) as a promising tool to predict the subnetwork. The optimal recovery schedules for a large number of disruptions are used for offline training, and subsequently, the trained ML model is employed to reduce the search space for new disruption cases substantially. To achieve this, we employ a unique feature space specifically designed to capture the essential characteristics of the problem. A notable contribution is constructing a novel feature space based on revised aircraft schedules designed to capture the essential characteristics of the problem. The recursive feature elimination technique is employed for optimal feature selection. Five machine learning classifiers: Logistic Regression, Random Forest, Extreme gradient boosting, Light gradient boosting, and Categorical Boosting are compared. The performance is evaluated on real data obtained from an airline company. Our study demonstrates that, with the subnetwork of aircraft predicted by the classifier, the computational time of the column generation approach is remarkably reduced without deteriorating the solution quality in all the disruption instances tested, with more than 98% of the instances being solved within 60 s. The results highlight the remarkable advantage of integrating ML in the pre-optimization phase.

Multi-cell content caching: Optimization for cost and information freshness

In multi-access edge computing (MEC) systems, there are multiple local cache servers caching contents to satisfy the users’ requests, instead of letting the users download via the remote cloud server. In this paper, a multi-cell content scheduling problem (MCSP) in MEC systems is considered. Taking into account jointly the freshness of the cached contents and the traffic data costs, we study how to schedule content updates along time in a multi-cell setting. Different from single-cell scenarios, a user may have multiple candidate local cache servers, and thus the caching decisions in all cells must be jointly optimized. We first prove that MCSP is NP-hard, then we formulate MCSP using integer linear programming, by which the optimal scheduling can be obtained for small-scale instances. For problem solving of large scenarios, via a mathematical reformulation, we derive a scalable optimization algorithm based on repeated column generation. Our performance evaluation shows the effectiveness of the proposed algorithm in comparison to an off-the-shelf commercial solver and a popularity-based caching.

Optimal Scheduling of Integrated Energy System Considering Electric Vehicle Battery Swapping Station and Multiple Uncertainties

In recent years, there has been rapid advancement in new energy technologies aimed at mitigating greenhouse gas emissions stemming from fossil fuels. Nonetheless, uncertainties persist in both the power output of new energy sources and load. To effectively harness the economic and operational potential of an Integrated Energy System (IES), this paper introduces an enhanced uncertainty set. This set incorporates N-1 contingency considerations and the nuances of source–load distribution. This framework is applied to a robust optimization model for an Electric Vehicle Integrated Energy System (EV-IES), which includes Electric Vehicle Battery Swapping Station (EVBSS). Firstly, this paper establishes an IES model of the EVBSS, and then proceeds to classifies and schedules the large-scale battery groups within these stations. Secondly, this paper proposes an enhanced uncertainty set to account for the operational status of multiple units in the system. It also considers the output characteristics of both new energy sources and loads. Additionally, it takes into consideration the N-1 contingency state and multi-interval distribution characteristics. Subsequently, a multi-time-scale optimal scheduling model is established with the objective of minimizing the total cost of the IES. The day-ahead robust optimization fully considers the multivariate uncertainty of the IES. The solution employs the Nested Column and Constraint Generation (C&CG) algorithm, based on the distribution characteristics of multiple discrete variables in the model. The intraday optimal scheduling reallocates the power of each unit based on the robust optimization results from the day-ahead scheduling. Finally, the simulation results demonstrate that the proposed method effectively reduces the conservatism of the uncertainty set, ensuring economic and stable operation of the EV-IES while meeting the demands of electric vehicle users.