Set Partitioning Model Research Articles

A branch-cut-and-price approach for the two-echelon vehicle routing problem with drones

We present a new set-partitioning model for the two-echelon vehicle routing problem with drones (2E-VRP-D). In this model, partial routes corresponding to drone movements are enumerated by an efficient dynamic program. A state-of-the-art exact branch-cut-and-price algorithm solves the model using a labeling algorithm for the pricing problems. We further propose an adaptation of the well-known rounded capacity cuts, specifically tailored to address this problem, as well as pre-processing methods designed to enhance the algorithm’s efficiency. In addition, we propose an effective heuristic branch-cut-and-price based on the exact algorithm. Computational experiments demonstrate that the exact algorithm solves all instances from the 2E-VRP-D literature and significantly increases the size of optimally solved clustered instances. Finally, a sensitivity analysis is conducted to assess the impact of the proposed enhancements.

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.

A branch-and-price algorithm for unrelated parallel machine scheduling with machine usage costs

This paper considers unrelated parallel machine scheduling involving machine usage costs, in addition to classic job completion time-related costs. The usage cost of each machine is made up of a fixed usage cost and a variable usage cost proportional to the total processing time of the jobs assigned to it. These features model many practical situations where machine usage costs include, for example, rental fees when the machines are not owned but rented. To tackle this problem, four mathematical models based on the Shortest Weighted Processing Time (SWPT) rule are introduced. Additionally, the problem is formulated into a set-partitioning model, for which a branch-and-price algorithm is proposed with an appropriate branching strategy. This facilitates the development of an efficient pseudo-polynomial dynamic programming algorithm and a polynomial-time heuristic to solve the pricing problem. Extensive numerical experiments demonstrate the superior performance of the proposed branch-and-price algorithm over the four SWPT-based mathematical formulations and an existing branch-and-price algorithm designed for a special case. Notably, it can optimally solve instances involving up to 225 jobs and 15 machines within one hour. Moreover, statistical analyses reveal that the proposed polynomial-time heuristic significantly reduces the computation time, and the mathematical model based on the contribution of every job to the total weighted completion time exhibits the best overall performance.

Delay-resistant robust vehicle routing with heterogeneous time windows

We consider a robust variant of the vehicle routing problem with heterogeneous time windows (RVRP-HTW) with a focus on delay-resistant solutions. Here, customers have different availability time windows for every vehicle and must be provided with a preferably tight appointment window for the planned service. Different vehicles are a possibility to model different days on which one physical vehicle can serve a customer. This is the main reason why different time windows for different vehicles are of high practical relevance. To ensure that the appointment windows are adhered to as much as possible, we introduce a new objective function that penalizes delays. Our novel approach allows us to find solutions that are robust with respect to uncertainties in travel and service times limited by a budget polytope. We present a set-partitioning model, the solution of which is based on column generation and employs a labeling algorithm that integrates robustness into the calculations and is adapted to our problem-specific constraints. In a Monte-Carlo simulation on real-life data, we evaluate this method in terms of runtime and solution quality. Our solutions show very good performance, even if the data is more uncertain than assumed for the optimization, incurring only marginal extra travel time compared to a naive deterministic planning scheme.

On the role of time‐of‐use electricity price in charge scheduling for electric bus fleets

AbstractAs electric buses become increasingly popular, it is imperative to optimize the schedules of electric buses with explicit consideration of their charging requirements. Unfortunately, existing studies failed to properly model the impacts of essential operating factors, including the time‐of‐use (TOU) electricity price, partial charging, and limited chargers. Our paper proposes a mixed‐integer nonlinear program to minimize the total operating cost of an electric bus fleet for fulfilling a group of timetabled trips, considering the above realistic features simultaneously. To effectively solve the problem of global optimality, we convert this model to an equivalent set partitioning model and develop a specialized branch‐and‐price approach subsequently. Our approach's computational efficiency is verified via extensive numerical experiments. The model is applied to a real‐world case study in Nanjing, China. Results show that incorporating the TOU electricity price into the electric bus scheduling problem can produce a sizeable cost saving of up to 22%. Managerial insights unveiled by the numerical results are discussed. These insights inform the practitioners of how the operations of electric buses can be both cost‐effective and grid‐friendly.

A matheuristic for aircraft maintenance routing problem incorporating cruise speed control

Aircraft maintenance routing problem incorporating cruise speed control (AMRP-CSC) is an extension of the well-known aircraft maintenance routing problem (AMRP) where the flexible cruise times of flights are considered in route construction. Changing cruise times in AMRP can transform the infeasible flight connections into feasible ones, resulting in a larger solution space and further opportunities for efficiently routing. This may open up the possibility for a substantial improvement in aircraft utilization, but also increases resolution complexity. In this study, a new solution methodology, namely a matheuristic approach, is proposed for this problem. It is composed of three main components: an improved ant colony optimization (IACO) algorithm, a set partitioning (SP) procedure, and a neighborhoods search (NS) procedure. The IACO algorithm serves as a route generator, populating a pool of routes with promising feasible aircraft maintenance routes. Then, a SP model, which features the high-quality columns corresponding to the routes in the pool, is solved to produce a possible better solution. Finally, this solution is further improved by a NS procedure that iteratively solves the reduced AMRP-CSC instances to optimality. This matheuristic approach is analyzed and tested using the data extracting from the Bureau of Transportation Statistics (BTS), and then its accuracy and the efficiency have been demonstrated by experiment analyses.

A branch-and-price-and-cut algorithm for time-dependent pollution routing problem

The time-dependent pollution routing problem (TDPRP) extends the pollution routing problem (PRP) cause it captures traffic congestion at peak periods in urban transportation. It concerns planning a fleet of homogeneous vehicles to serve all customers, jointly deciding their speed on every arc and the departure time from each node such that the total cost of routes reaches a minimum. To our knowledge, all solution methods for the TDPRP are founded on (meta-)heuristics. In this study, we propose an exact branch-and-price-and-cut (BPC) algorithm for the TDPRP. The master problem is a route selection problem formulated as a set-partitioning model, and the pricing problem is a time-dependent elementary shortest path problem with resource constraints (TDESPPRC), in which the speed and start time at each customer need to be decided. We devise a tailored label-setting algorithm to handle the pricing problem, and the master problem is solved by column generation. Then, two valid inequalities are employed to tighten the lower bound, and several accelerating techniques are used to speed up the label-setting algorithm. Meanwhile, we present an analytical characterization applied to different scenarios. Finally, extensive computational experiments show our algorithm outperform a commercial MIP solver, and the optimal solutions are found for more benchmark instances using less time.

Optimizing aircrew recovery considering long connections: A column generation based approach

Severe weather and air traffic flow control can often cause flight delays and disrupt the crew schedules, resulting in a too-short or a too-long connection between flights. Short connections need to be adjusted to ensure the airline’s smooth operations, while the long connection whose duration is only a bit lower than the threshold enabling the crew to leave the flying zone for a rest at airport terminals can be quite inefficient, as the crew’s ground standby is viewed as working all the time without executing any flight which adds to the shortage of crew resources. Thus, we propose a method of delaying flights to the point enabling crews to leave for a rest, which helps save the available working hours. The difficulty lies in the trade-off between the additional swap or delay cost in creating long connections and the possible global cost decrease due to savings in available working hours brought by long connections, which calls for optimization. In this paper, we introduce the crew recovery problem for one day of operation taking long connections into account. The problem is mathematically formulated as an arc-based integer programming model and a set partitioning model in order to determine the optimal flight recovery schedule and the crew reassignment. A column generation based solution approach is developed and the sub-problems consisting of the resource-constrained shortest path problem are solved by a labeling algorithm. Computational experiments based on real-world data from a major Chinese airline confirm the effectiveness and efficiency of the solution approaches. It is also demonstrated the save of available working hours resulting from adaptively creating long connections would decrease the recovery cost by about 18.87%–74.88%.

A column generation-based algorithm for gate assignment problem with combinational gates

The gate assignment is a complicated real-world problem occurring in airports. The present paper studies the gate assignment problem with combinational gates. The problem is multi-objective by nature, concerning four objectives. Among those objectives, the number of passengers allocated to nearby gates is aimed to be maximized. The remaining objectives are targeted to be minimized, i.e., the total passenger walking distance, the amount of carbon dioxide emitted by the aircraft assigned to remote gates, and the total distance traveled by the shuttle bus. To solve this problem, we formulate an integer programming model. A column generation-based algorithm based on a set-partitioning model is proposed to solve the problem effectively. In addition, the Pareto Local Search algorithm is adopted to obtain non-dominated solutions. Computational experiments show that the effectiveness of our model is verified through tests with instances of different sizes. The column generation-based algorithm can obtain the optimal solution or the approximately optimal solution with a small gap in a reasonable time. The results of the Pareto Local Search algorithm show the interaction between multiple objectives.

A branch-and-price algorithm for the airport gate assignment problem considering the trade-off between robustness and efficiency

In this paper, a novel objective for the airport gate assignment problem which considers the trade-off between robustness and operational efficiency in a unified dimension is developed. A task-based two-phase model is established, where the branch-and-price framework is embedded in the first stage. A set partitioning model is developed, where a series of aircrafts assigned to the same gate of a certain type are defined as a gate plan. The pricing subproblem is also decomposed into three parallel subproblems according to the gate size to account for the heterogeneity of the gates. Each gate plan is allocated to a physical gate in order to minimize the utilization of remote gates in the second stage. The pulse algorithm with two acceleration strategies, that generates a lower bound matrix in reverse order and applies path combination, is used to solve the pricing subproblems. Finally, extensive computational experiments are conducted to demonstrate the high performance of the proposed model, especially for large-scale instances.

A matheuristic algorithm for the share-a-ride problem

This research studies the Share-a-Ride Problem (SARP) in which a set of taxis is used to serve a set of package and passenger requests at the same time. The goal is to maximize the profit from serving the requests without violating given constraints. We develop a new matheuristic algorithm that combines the simulated annealing with mutation strategy (SAMS) and the set partitioning (SP) approach to improve the reported solutions of SARP benchmark instances. The SAMS consists of a time-slack strategy, neighborhood moves, a mutation strategy that depends on the time-slack strategy, and a penalty mechanism for infeasible solutions. The first phase of the proposed matheuristic uses SAMS to generate a set of feasible candidate routes. A route accumulation mechanism is added to the SAMS to keep the candidate routes in the route pool. The second phase of the matheuristic focuses on solving the set partitioning model to find the best route combination as the final solution. The proposed matheuristic is tested on existing small and large SARP instances, and a set of newly generated large SARP instances and then compared with SAMS algorithm. The experimental results show that the proposed matheuristic obtains optimal solutions to all the small SARP benchmark instances. For large instances, the proposed matheuristic outperforms SAMS algorithm as it obtains several new best solutions to SARP. Lastly, our algorithm obtains high-quality solutions to the new large SARP instances generated in this study.

An exact algorithm for the multi-trip container drayage problem with truck platooning

Container Drayage Problem (CDP) refers to the optimization problem of routing and scheduling a set of container trucks around a container terminal. Conventionally, a driver is stuck to one container truck and allowed to perform multiple trips (multi-trip) to the terminal within their working time. The recent development of automation technologies enables semi-autonomous trucks to follow the leading human-driven truck as a platoon on the road; therefore, truck platooning can save human labor and reduce the fuel cost of following trucks through aerodynamic drag reduction. In this paper, we study a multi-trip container drayage problem with truck platooning (MT-CDP-TP), where multi-trip, truck platooning, and fuel cost reduction are simultaneously considered in a CDP. Despite the operational benefits brought by the MT-CDP-TP, the problem is challenging to solve due to its NP-hardness when formulated as a multi-trip pickup and delivery problem with load-dependent cost. We propose a Branch-and-Price-and-Cut (BPC) algorithm, with a route-based set partitioning model and tight linear relaxations, to yield the exact solutions. Valid inequalities are generated based on a graph structure, where each node represents a feasible route, and each arc stands for the conflict between two routes. Moreover, we design a tailored pulse propagation algorithm with novel pruning procedures based on the dual information from the master problem and valid inequalities to solve the pricing problem efficiently. Extensive numerical experiments are conducted for performance validation, and the computational results show that the proposed exact algorithm can solve instances with up to 100 task nodes (i.e., 50 customers) and facilitate reducing the labor cost and fuel consumption by a wide margin in container drayage operations.

A Branch-and-Price Algorithm for the Online Scheduling of Valet Drivers

In the present paper, the online valet driving problem (OVDP) is studied. In this problem, customers request a valet driving service through the platform, then the valets arrive on e-bikes at the designated pickup location and drive the vehicle to the destination. The key task is to assign the valets effectively for driving orders to minimize the overall cost. To serve that purpose, we first propose a new online scheduling strategy that divides the planning horizon into several rounds with fixed length of time, and each round consists of pooling time and scheduling time. By including the features of online scheduling and the power level of e-bikes, this OVDP becomes more practical but nevertheless challenging. To solve the OVDP, we formulate it into a set partitioning model and design a branch-and-price (B&P) algorithm. To improve the computation efficiency, a label setting algorithm is incorporated to address the pricing subproblem, which is accelerated via a heuristic pricing method. As an essential part of the algorithm design, an artificial column technique and a greedy-based constructive heuristic are implemented to obtain the initial solution. Based on the numerical analysis of various scaled instances, it is verified that the proposed B&P algorithm is not only effective in optimum seeking, but also shows a high level of efficiency in comparison with the off-the-shelf commercial solvers. Furthermore, we also explore the impact of pooling and scheduling time on the OVDP and discover a bowl-shaped trend of the objective value with respect to the two time lengths.

Optimizing first-mile ridesharing services to intercity transit hubs

Travel to intercity transportation hubs, such as railway stations and airports, can be the most troublesome and inefficient part of the entire air/railway travel journey, as travelers often carry large luggage and have stringent arrival time requirements. Taking public transportation, such as metro and bus services, is inconvenient to carry luggage and less reliable in arrival time while taking taxi services could be less economical. As a result, providing reliable and convenient yet economical on-demand first-mile services for travelers to intercity transportation hubs is essential. This paper proposes a ridesharing approach for the first-mile transport system for travelers heading towards the intercity transportation hub and develops a mixed-integer linear programming (MILP) model with the objective of minimizing the total operating costs for ridesharing service operators. The MILP model considers (1) large luggage that may occupy seats when the car trunk is not large enough to place them; (2) passengers’ requirements on arrival time and ride time; and (3) travel time uncertainty ensuring that riders’ arrival time and ride time can be satisfied. A tailored adaptive large neighborhood search algorithm with an acceleration strategy is developed for obtaining robust near-optimal solutions within a reasonable time. To assess the solution quality, the MILP model is reformulated as a set-partitioning model, and the column generation algorithm is leveraged to determine a tight lower bound; a greedy algorithm is introduced to obtain an upper bound. Computational experiments on Shanghai South Railway Station demonstrate that ridesharing is an effective strategy for reducing overall travel costs while meeting the first-mile travel demand. In addition, it is essential to consider luggage and travel time uncertainty for determining ridesharing schemes.

A column generation-based heuristic for brachytherapy patient scheduling with multiple treatment sessions considering radioactive source decay and time constraints

Imbalanced and inefficient schedules of brachytherapy treatment result in long waiting times and a large number of waiting patients, which may raise many subsequent problems. In this paper, we solve the brachytherapy patient scheduling problem with multiple treatment sessions, simultaneously considering the half-life decaying effect of radioactive sources, as well as the strict time constraints on the time interval between any two consecutive treatment sessions and the unavailable time of treatment. Patients on the waiting list are given different weights (priorities) according to the severity of their illness and waiting time. The studied problem aims to efficiently schedule patients in a rolling way to maximize the sum of the weights of the chosen patients from the current waiting list on the premise that the already arranged patients can complete their multiple treatment sessions in the future under strict time constraints. We formulate the problem as an integer programming model and develop a column generation-based heuristic approach on a set partitioning. We propose a pricing algorithm that can add many good patient plans at one iteration and a speed-up strategy to improve the performance of column generation. Computational studies are conducted on abundant test instances generated based on real-world data to demonstrate the efficiency and the high-quality solutions of the proposed approach by comparing with the integer programming model, the set partitioning model, the globally optimal solution, and the “Sessions on the Same Day Every Week” (SSEW) rule currently used in most real-world hospitals. Furthermore, we validate that the proposed approach can reduce the number of waiting patients compared with the SSEW rule. Sensitivity analysis is carried out on critical parameters, and further managerial insights are derived.

Territorial design for customers with demand frequency

Territorial design is an important long-term decision for urban delivery service companies, in contexts where customers are partitioned into districts. This study focuses on a territorial design problem given the demand frequency of each customer, i.e., the estimated percentage of days with demand, over the planning horizon. This study formulates a set partitioning model and designs a column generation based algorithm to solve the problem. The algorithm decomposes the original problem into a restricted master problem (RMP) and a series of pricing problems (PPs), each limited to one district. A dynamic programming based method is designed to solve the PPs efficiently. To further accelerate the solution processes of the PPs and of the RMP, some tailored strategies are also embedded within the algorithm. Numerical experiments are conducted to validate the contributions of the dynamic programming and of the acceleration strategies. Some tests based on real-world cases are also performed in order to derive some managerial insights to support the practitioners’ decisions on service territory design.

An exact algorithm for the two-echelon vehicle routing problem with drones

This paper studies a new variant of the vehicle routing problem with drones, i.e., the two-echelon vehicle routing problem with drones, where multiple vehicles and drones work collaboratively to serve customers. Drones can perform multiple back-and-forth trips when their paired vehicle stops at a customer node, forming a two-echelon network. Several practical constraints such as customers’ delivery deadlines and drones’ energy capacity are considered. Different from existing studies, we treat the number of drones taken by each vehicle as a decision variable instead of a given parameter, which provides more flexibility for planning vehicle and drone routes. We first formulate this problem as a mixed-integer linear programming model, which is solvable by off-the-shelf commercial solvers. To tackle instances more efficiently, we next construct a set-partitioning model. To solve it, an exact branch-and-price algorithm is proposed, where a bidirectional labeling algorithm is used to solve the pricing problem. To speed up the algorithm, a tabu search algorithm is first applied before the exact labeling algorithm for finding desired columns in each iteration of the column generation process. Extensive numerical tests show that our algorithm can solve most instances within 25 customers to optimality in a short time frame and some instances of 35 customers to optimality within a three-hour time limit. Results also demonstrate that the allocation decisions of drones can help save the duration of all routes by 3.44% on average for 25-customer instances, compared to the case of fixing the number of paired drones on each vehicle. In addition, sensitivity analyses show that multiple strategies, e.g., adopting batteries of a higher energy density and developing faster drones, can be applied to further improve the delivery efficiency of a truck–drone system.

Branch-And-Price Algorithm for the Tramp Ship Routing and Scheduling Problem Considering Ship Speed and Payload

In recent years, increasing fuel prices, depressed market conditions and air pollution issues have brought huge challenges to the tramp shipping industry. This work investigates the tramp ship routing and scheduling problem considering ship speeds and payloads, aiming at minimizing ship fuel consumption. A mixed integer non-linear programming model with a discretized speed variable and a set partitioning model for this problem is established, and a branch-and-price algorithm is proposed to solve the problem. Through the column generation approach, the problem at each branch-and-bound node is decomposed into a linear programming master problem and a pricing problem of the elementary shortest path with resource constraints. A labeling algorithm is adopted for solving the pricing problem. Multiple groups of instances are generated to test the effectiveness of the proposed model and algorithm and analyze the impacts of ship speed, payload, and speed discretization on the solution. Computational experiments are conducted, which verifies the proposed scheduling routing method for tramp ships and confirms that adopting the proposed model can effectively reduce fuel consumption of tramp ships which can not only deepen the theory of tramp routing and scheduling, but also provide theoretical guidance to tramp ship company. The branch-and-price algorithm can effectively solve large-scale tramp ship routing and scheduling problems. Reasonable number of speed discretization points can bring a desirable trade-off between solution accuracy and algorithm runtime.

Minimizing user inconvenience and operational costs in a dial-a-flight problem for flying safaris

We study a Dial-a-Flight Problem faced by one of the major safari airline companies in Tanzania. Given a set of daily passenger requests and a fleet of heterogeneous airplanes, the problem requires to determine the best set of itineraries to transport the passengers from their origins to the requested destinations within specific time windows, while satisfying a number of operational constraints. The aim is to minimize user inconvenience, measured by delays with respect to the pre-defined time windows and by the number of intermediate stops in the itineraries, and operational cost. The problem is complicated by the high number of daily requests in peak touristic periods, and by the fact that refueling is possible only at a limited number of airstrips. We solve the problem by means of an adaptive large neighborhood search, which we enrich with local search operators and a set partitioning model. Extensive computational tests on real-world instances prove the effectiveness of the proposed algorithm, which can improve the solutions found by the company both in terms of operational cost and user inconvenience, in reasonable computational time.

The Pickup and Delivery Problem with Time Windows and Incompatibility Constraints in Cold Chain Transportation

This study investigates a new variant of the pickup and delivery problem with time windows (PDPTW) applied in cold chain transportation, which quantifies the effect of time on the quality of perishable products. Multiple commodities with incompatibility constraints are considered, where some types of products cannot be transported in a vehicle simultaneously because of their different properties and requirements for storage temperatures. The aim is to determine vehicles’ pickup and delivery routes as well as their departure times from the depot such that the travel cost and refrigeration cost of vehicles and the quality decay cost of products are minimized. We formulate this problem as a set partitioning model, which is solved exactly by a tailored branch-and-price (B&P) algorithm. To tackle the asymmetry issue arising from the pricing problem of the B&P framework, we develop a novel asymmetric bidirectional labeling algorithm. Benchmark instance sets based on real-world statistical data and classic PDPTW instance sets are first generated for this problem. Numerical results show that our B&P algorithm can solve most instances to optimality in an acceptable time frame. Moreover, our results demonstrate that integrating the refrigeration and quality decay costs into the objective function can significantly lower the total cost of cold chain transportation activities, compared with the widely adopted objective function minimizing only the travel cost. Funding: This work was supported by the National Key R&D Program of China [Grant 2018YFB1700600] and the National Natural Science Foundation of China [Grants 72101049, 71971090, 71821001]. Supplemental Material: The online supplement is available at https://doi.org/10.1287/trsc.2022.1167 .