A column generation approach for the split delivery vehicle routing problem

A column generation approach is presented for the split delivery vehicle routing problem with large demand. Columns include route and delivery amount information. Pricing sub-problems are solved by a limited-search-with-bound algorithm. Feasible solutions are obtained iteratively by fixing one route once. Numerical experiments show better solutions than in the literature.     

A specialized column generation approach for a vehicle routing problem with demand allocation

Motivated by logistical operations for a food bank, this paper addresses a class of vehicle routing problems with demand allocation considerations over a network of partner agencies locations and candidate delivery sites. Any delivery tour starts at a central depot operated by the food bank and selected delivery sites are sequentially visited in order to supply goods to a set of partner agencies who travel from their respective locations to their assigned delivery sites. The problem is modelled as a mixed-integer programme with the objective of minimizing a weighted average of the distances travelled by delivery vehicles and partner agencies, and is tackled via two heuristics. First, a relax-and-fix heuristic is presented for the proposed model and is computationally enhanced using two symmetry-defeating strategies. Second, the problem is reformulated as a set partitioning model with side packing constraints that prompts a specialized column generation approach. Computational experience is provided using realistic data instances to demonstrate the usefulness of the proposed heuristics and the importance of integrated solution techniques for this class of problems.     

A column generation algorithm for the vehicle routing problem with soft time windows

The Vehicle Routing Problem with Time Windows consists of computing a minimum cost set of routes for a fleet of vehicles of limited capacity visiting a given set of customers with known demand, with the additional constraint that each customer must be visited in a specified time window. We consider the case in which time window constraints are relaxed into “soft” constraints, that is penalty terms are added to the solution cost whenever a vehicle serves a customer outside of his time window. We present a branch-and-price algorithm which is the first exact optimization algorithm for this problem.     

A tutorial on column generation and branch-and-price for vehicle routing problems

This paper provides a tutorial on column generation and branch-and-price for vehicle routing problems. The main principles and the basic theory of the methods are first outlined. Some additional issues, including reinforcement of the relaxation or stabilization, complete the paper. For the sake of simplicity, this material is illustrated with the case of the vehicle routing problem with time windows.     

A column generation approach for the rail crew re-scheduling problem

When tracks are out of service for maintenance during a certain period, trains cannot be operated on those tracks. This leads to a modified timetable, and results in infeasible rolling stock and crew schedules. Therefore, these schedules need to be repaired. The topic of this paper is the re-scheduling of crew.     

A self-tuning heuristic for a multi-objective vehicle routing problem

In this study, a heuristic free from parameter tuning is introduced to solve the vehicle routing problem (VRP) with two conflicting objectives. The problem which has been presented is the designing of optimal routes: minimizing both the number of vehicles and the maximum route length. This problem, even in the case of its single objective form, is NP-hard. The proposed self-tuning heuristic (STH) is based on local search and has two parameters which are updated dynamically throughout the search process. The most important advantage of the algorithm is the application convenience for the end-users. STH is tested on the instances of a multi-objective problem in school bus routing and classical vehicle routing. Computational experiments, when compared with the prior approaches proposed for the multi-objective routing of school buses problem, confirm the effectiveness of STH. STH also finds high-quality solutions for multi-objective VRPs.     

A brain storm optimization approach for the cumulative capacitated vehicle routing problem

The cumulative capacitated vehicle routing problem (CCVRP) is a combinatorial optimization problem which aims to minimize the sum of arrival times at customers. This paper presents a brain storm optimization algorithm to solve the CCVRP. Based on the characteristics of the CCVRP, we design new convergent and divergent operations. The convergent operation picks up and perturbs the best-so-far solution. It decomposes the resulting solution into a set of independent partial solutions and then determines a set of subproblems which are smaller CCVRPs. Instead of directly generating solutions for the original problem, the divergent operation selects one of three operators to generate new solutions for subproblems and then assembles a solution to the original problem by using those new solutions to the subproblems. The proposed algorithm was tested on benchmark instances, some of which have more than 560 nodes. The results show that our algorithm is very effective in contrast to the existing algorithms. Most notably, the proposed algorithm can find new best solutions for 8 medium instances and 7 large instances within short time.     

A column generation approach to the discrete barycenter problem

The discrete Wasserstein barycenter problem is a minimum-cost mass transport problem for a set of discrete probability measures. Although an exact barycenter is computable through linear programming, the underlying linear program can be extremely large. For worst-case input, a best known linear programming formulation is exponential in the number of variables, but has a low number of constraints, making it an interesting candidate for column generation.In this paper, we devise and study two column generation strategies: a natural one based on a simplified computation of reduced costs, and one through a Dantzig–Wolfe decomposition. For the latter, we produce efficiently solvable subproblems, namely, a pricing problem in the form of a classical transportation problem. The two strategies begin with an efficient computation of an initial feasible solution. While the structure of the constraints leads to the computation of the reduced costs of all remaining variables for setup, both approaches may outperform a computation using the full program in speed, and dramatically so in memory requirement. In our computational experiments, we exhibit that, depending on the input, either strategy can become a best choice.     

A column generation approach for solving the patient admission scheduling problem

This paper addresses the Patient Admission Scheduling (PAS) problem. The PAS problem entails assigning elective patients to beds, while satisfying a number of hard constraints and as many soft constraints as is possible, and arises at all planning levels for hospital management. There exist a few, different variants of this problem. In this paper we consider one such variant and propose an optimization-based heuristic building on branch-and-bound, column generation, and dynamic constraint aggregation to solve it. We achieve tighter lower bounds than previously reported in the literature and, in addition, we are able to produce new best known solutions for five out of twelve instances from a publicly available repository.     

A column generation approach for the unconstrained binary quadratic programming problem

This paper proposes a column generation approach based on the Lagrangean relaxation with clusters to solve the unconstrained binary quadratic programming problem that consists of maximizing a quadratic objective function by the choice of suitable values for binary decision variables. The proposed method treats a mixed binary linear model for the quadratic problem with constraints represented by a graph. This graph is partitioned in clusters of vertices forming sub-problems whose solutions use the dual variables obtained by a coordinator problem. The column generation process presents alternative ways to find upper and lower bounds for the quadratic problem. Computational experiments were performed using hard instances and the proposed method was compared against other methods presenting improved results for most of these instances.     

Approximation algorithms for a vehicle routing problem

In this paper we investigate a vehicle routing problem motivated by a real-world application in cooperation with the German Automobile Association (ADAC). The general task is to assign service requests to service units and to plan tours for the units such as to minimize the overall cost. The characteristics of this large-scale problem due to the data volume involve strict real-time requirements. We show that the problem of finding a feasible dispatch for service units starting at their current position and serving at most k requests is NP-complete for each fixed k ≥ 2. We also present a polynomial time (2k − 1)-approximation algorithm, where again k denotes the maximal number of requests served by a single service unit. For the boundary case when k equals the total number |E| of requests (and thus there are no limitations on the tour length), we provide a -approximation. Finally, we extend our approximation results to include linear and quadratic lateness costs.     

A column generation approach for an employee scheduling problem with multiple shifts and work locations

This paper is concerned with the problem of assigning employees to a number of work centres taking into account employees' expressed preferences for specific shifts, off-days, and work centres. This particular problem is faced by the Kuwait National Petroleum Corporation that hires a firm to prepare schedules for assigning employees to about 86 stations distributed all over Kuwait. The number of variables in a mixed-integer programming model formulated for this problem is overwhelming, and hence, a direct solution to even the continuous relaxation of this model for relatively large-scale instances is inconceivable. However, we demonstrate that a column generation method, which exploits the special structures of the model, can readily solve the continuous relaxation of the model. Based on this column generation construct, we develop an effective heuristic to solve the problem. Computational results indicate that the proposed approach can facilitate the generation of good-quality schedules for even large-scale problem instances in a reasonable time.     

Model-hierarchical column generation and heuristic for the routing and wavelength assignment problem

The routing and wavelength assignment (RWA) problem typically occurs in wavelength division multiplexing optical networks. Given a number of available wavelengths, we consider here the problem of maximising the number of accepted connections with respect to the clash and continuity constraints. We first propose a new strategy which combines two existing models. This leads to an improved column generation scheme. We also present two heuristics to compute feasible solutions: a hybrid heuristic and the integer solution at the root node of the column generation. Our approaches are compared with the best existing results on a set of classic RWA instances.     

A column generation approach for the timetabling problem of Greek high schools

A column generation (CG) approach for the solution of timetabling problems is presented. This methodology could be used for various instances of the timetabling problem, although in this paper the solution of the high-school situation in Greece is presented. The results obtained show clearly that the CG approach that has been extremely successful in recent years in the solution of airline crew scheduling problems could also be very efficient and robust for the solution of timetabling problems. Several large timetabling problems corresponding to real problems have been successfully solved, with the solutions obtained feasible and of very high quality in accordance with the problem definition. In addition, none of the solutions contained any idle hour for any of the teachers, which was one of the main goals of this optimization effort.     

On a periodic vehicle routing problem

This paper deals with a study on a variant of the Periodic Vehicle Routing Problem (PVRP). As in the traditional Vehicle Routing Problem, customer locations each with a certain daily demand are given, as well as a set of capacitated vehicles. In addition, the PVRP has a horizon, say T days, and there is a frequency for each customer stating how often within this T-day period this customer must be visited. A solution to the PVRP consists of T sets of routes that jointly satisfy the demand constraints and the frequency constraints. The objective is to minimize the sum of the costs of all routes over the planning horizon. We develop different algorithms solving the instances of the case studied. Using these algorithms we are able to realize considerable cost reductions compared to the current situation.     

A branch-and-price algorithm for a vehicle routing with demand allocation problem

We investigate the vehicle routing with demand allocation problem where the decision-maker jointly optimizes the location of delivery sites, the assignment of customers to (preferably convenient) delivery sites, and the routing of vehicles operated from a central depot to serve customers at their designated sites. We propose an effective branch-and-price (B&P) algorithm that is demonstrated to greatly outperform the use of commercial branch-and-bound/cut solvers such as CPLEX. Central to the efficacy of the proposed B&P algorithm is the development of a specialized dynamic programming procedure that extends works on elementary shortest path problems with resource constraints in order to solve the more complex column generation pricing subproblem. Our computational study demonstrates the efficacy of the proposed approach using a set of 60 problem instances. Moreover, the proposed methodology has the merit of providing optimal solutions in run times that are significantly shorter than those reported for decomposition-based heuristics in the literature.     

A multi-objective approach for unmanned aerial vehicle routing problem with soft time windows constraints

Aerial robotics can be very useful to perform complex tasks in a distributed and cooperative fashion, such as localization of targets and search of point of interests (PoIs). In this work, we propose a distributed system of autonomous Unmanned Aerial Vehicles (UAVs), able to self-coordinate and cooperate in order to ensure both spatial and temporal coverage of specific time and spatial varying PoIs. In particular, we consider an UAVs system able to solve distributed dynamic scheduling problems, since each device is required to move towards a certain position in a certain time. We give a mathematical formulation of the problem as a multi-criteria optimization model, in which the total distances traveled by the UAVs (to be minimized), the customer satisfaction (to be maximized) and the number of used UAVs (to be minimized) are considered simultaneously. A dynamic variant of the basic optimization model, defined by considering the rolling horizon concept, is shown. We introduce a case study as an application scenario, where sport actions of a football match are filmed through a distributed UAVs system. The customer satisfaction and the traveled distance are used as performance parameters to evaluate the proposed approaches on the considered scenario.     

A path relinking algorithm for a multi-depot periodic vehicle routing problem

In this paper, we consider a multi-depot periodic vehicle routing problem which is characterized by the presence of a homogeneous fleet of vehicles, multiple depots, multiple periods, and two types of constraints that are often found in reality, i.e., vehicle capacity and route duration constraints. The objective is to minimize total travel costs. To tackle the problem, we propose an efficient path relinking algorithm whose exploration and exploitation strategies enable the algorithm to address the problem in two different settings: (1) As a stand-alone algorithm, and (2) As a part of a co-operative search algorithm called integrative co-operative search. The performance of the proposed path relinking algorithm is evaluated, in each of the above ways, based on standard benchmark instances. The computational results show that the developed PRA performs well, in both solution quality and computational efficiency.     

The single vehicle routing problem with toll-by-weight scheme: A branch-and-bound approach

Expressways in China make use of the toll-by-weight scheme, in which expressway tolls are collected based on the weight and traveling distance of the vehicle. Most vehicle routing models assume that the cost of traversing each edge is equivalent to edge length or some constant; as a result, such models cannot be practically applied to the Chinese expressway transportation system. This study addresses a new single vehicle routing problem that takes the vehicle’s (laden and unladen) weight into account. To solve this problem exactly, we provide a branch-and-bound algorithm with a provably valid lower bound measure, along with five dominance checkers for additional pruning. We analyze our algorithm using instances generated from standard TSP test cases, as well as two new sets of test cases based on real expressway information from the Gansu and Jiangxi provinces in China. The algorithm can be applied to any toll scheme in which the toll per unit distance monotonically increases with weight, even if the toll function is non-linear.