Pricing Subproblem Research Articles

Optimization of demand-oriented train timetables under overtaking operations: A surrogate-dual-variable column generation for eliminating indivisibility

This paper aims to optimize demand-oriented train timetables under overtaking operations for a high-speed rail corridor. With the application of the constructed space-time network representation, the timetabling and skip-stopping decisions in response to passenger demand for heterogeneous train traffic are formulated into an integer linear programming model. Specifically, we carefully assign the hour-dependent origin-to-destination demand to the skip-stop-flexible timetable by using a group of tailored constraints that bind the origin station and destination station together. While solving the proposed model under the column-generation-based framework, the biggest barrier is that the pricing subproblem cannot be successfully solved through the standard dynamic programming algorithm, because the dual price from the demand constraint is dependent upon two coupled stations. To dynamically eliminate the indivisibility attached to the demand-oriented timetabling problem, we propose a novel approach to replace the two-station-dependent dual variable with its single-station-dependent surrogate counterpart. A branch-and-price-and-cut procedure is also conducted to achieve the corresponding integer solutions, where specific families of valid inequalities are selected to narrow the feasible solutions in the restricted master problem. Finally, numerical experiments are implemented to demonstrate the efficiency and effectiveness of the proposed method.

Dantzig-Wolfe decomposition for the facility location and production planning problem

There are several mathematical models proposed for the facility location and production planning problem in the literature. However, some of these models disregard what products each customer has ordered and neglect critical production-related constraints and setup decisions while some others do not well define the connection cost between customers and facilities. In this study, we propose two mathematical models to overcome the disadvantages aforementioned, along with their reformulations by item decomposition to improve lower bounds. We demonstrate that the pricing subproblems of the item decomposition are related to uncapacitated lot-sizing problems with the Wagner-Whitin property. This property is employed to enhance the performance of column generation for the item decomposition. Our computational results show that this item decomposition method can improve lower bounds over other classical lower bounding techniques, such as linear programming relaxation and model reformulation. Additionally, we implement the proposed item decomposition method to other benchmark problems in the literature and observe that our proposed method can improve the benchmark solutions with a statistical significance.

The Migratory Beekeeping Routing Problem: Model and an Exact Algorithm

Apiculture has gained worldwide interest because of its contributions to economic incomes and environmental conservation. In view of these, migratory beekeeping, as a high-yielding technique, is extensively adopted. However, because of the lack of an overall routing plan, beekeepers who follow the experiential migratory routes frequently encounter unexpected detours and suffer losses when faced with problems such as those related to nectar source capacities and the production of bee products. The migratory beekeeping routing problem (MBRP) is proposed based on the practical background of the commercial apiculture industry to optimize the global revenue for beekeepers by comprehensively considering nectar source allocation, migration, production and sales of bee products, and corresponding time decisions. The MBRP is a new variant of the vehicle routing problem but with significantly different production time decisions at the vertices (i.e., nectar sources). That is, only the overlaps between residence durations and flowering periods generate production benefits. Different sales visits cause different gains from the same products; in turn, these lead to different production time decisions at previously visited nectar source locations and even change the visits for production. To overcome the difficulty resulting from the complicated time decisions, we utilize the Dantzig–Wolfe decomposition method and propose a revised labeling algorithm for the pricing subproblems. The tests, performed on instances and a real-world case, demonstrate that the column generation method with the revised labeling algorithm is efficient for solving the MBRP. Compared with traditional routes, a more efficient overall routing schedule for migratory beekeepers is proposed. Summary of Contribution. Based on the practical background of commercial apiculture industry, this paper proposes a new type of routing problem named the migratory beekeeping routing problem (MBRP), which incorporates the selection of productive nodes and sales nodes as well as the production time decision at the productive nodes on a migratory beekeeping network. To overcome the difficulty resulting from the complicated time decisions, we utilize the Dantzig–Wolfe decomposition method and propose a revised labeling algorithm for the pricing subproblems. The tests, performed on instances and a real-world case, demonstrate that the column generation method with the revised labeling algorithm is efficient for solving the MBRP. Compared with traditional routes, a more efficient overall routing schedule for migratory beekeepers is proposed. Therefore, this paper is congruent with, and contributes to, the scope and mission of INFORMS Journal on Computing, especially the area of Network Optimization: Algorithms & Applications.

Shift Planning Under Delay Uncertainty at Air France: A Vehicle-Scheduling Problem with Outsourcing

Airlines must operate many jobs in airports, such as passenger check-in or runway tasks. In airlines’ hubs, airlines generally choose to perform these jobs with their own agents. Shift planning aims at building the sequences of jobs operated by the airline agents and has been widely studied given its impact on operating costs. The impact of delayed flights is generally not taken into account despite the propagation of flight delays along these sequences: If a flight is late, then the agents doing the corresponding jobs are delayed, and may arrive late to their next jobs and delay the corresponding flights. Since delay costs are much higher than the costs of outsourcing jobs, if the agent who is supposed to operate a job is still working elsewhere when the job begins, then airlines tend to outsource the job to their own dedicated team or to a third party. We introduce a stochastic version of the shift-planning problem that takes into account outsourcing costs due to delay. It can be seen as a natural stochastic generalization of the vehicle-scheduling problem in which delayed jobs are outsourced. We propose a column-generation approach to solve it, whose key element is the pricing subproblem algorithm, modeled as a stochastic resource-constrained shortest-path problem. Numerical results on Air France industrial instances show the benefits of using our stochastic version of the shift-planning problem and the efficiency of the solution method. Moving to the stochastic version enables Air France to reduce total operating costs by 3.5%–4.8% on instances with more than 200 jobs, and our algorithm can solve to near optimality instances with up to 400 jobs.

Scheduling jobs with release dates on identical parallel machines by minimizing the total weighted completion time

This paper addresses the problem of scheduling a set of jobs that are released over the time on a set of identical parallel machines, aiming at the minimization of the total weighted completion time. This problem, referred to as P|rj|∑wjCj, is of great importance in practice, because it models a variety of real-life applications. Despite its importance, the P|rj|∑wjCj has not received much attention in the recent literature. In this work, we fill this gap by proposing mixed integer linear programs and a tailored branch-and-price algorithm. Our branch-and-price relies on the decomposition of an arc-flow formulation and on the use of efficient exact and heuristic methods for solving the pricing subproblem. Computational experiments carried out on a set of randomly generated instances prove that the proposed methods can solve to the proven optimality instances with up to 200 jobs and 10 machines, and provide very low gaps for larger instances.

An exact algorithm for the multi-period inspector scheduling problem

In this paper, we study the multi-period inspector scheduling problem (MPISP). This problem aims to determine a set of routes for a team of inspectors performing inspection jobs in different locations across multiple days, with the objective of maximizing the total workloads that the inspectors undertake. Since an inspector can only perform inspections or travel during working periods and rest at other times, a route for an inspector is divided into several segments. This characteristic, on the one hand, differentiates the MPISP from many routing problems in the literature; on the other hand, however, makes the routing decisions more complicated and challenging. To solve the MPISP, we first formulate it into a set-packing model and then propose an exact branch-and-price algorithm. In particular, we design a tailored label-setting algorithm for the pricing subproblem, which is a variant of the elementary shortest path problem with resource constraints. Moreover, we implement some acceleration techniques, such as bidirectional search, label pruning, decremental search space relaxation, and heuristic column generator. Extensive computational experiments were conducted on a set of benchmark instances, and the results have demonstrated the effectiveness of the proposed algorithm.

On the exact solution of vehicle routing problems with backhauls

In this paper, we are interested in the exact solution of the vehicle routing problem with backhauls (VRPB), a classical vehicle routing variant with two types of customers: linehaul (delivery) and backhaul (pickup) ones. We propose two branch-cut-and-price (BCP) algorithms for the VRPB. The first one follows the traditional approach with one pricing subproblem, whereas the second one exploits the linehaul/backhaul customer partitioning and defines two pricing subproblems. The methods incorporate elements of state-of-the-art BCP algorithms, such as rounded capacity cuts, limited-memory rank-1 cuts, strong branching, route enumeration, arc elimination using reduced costs and dual stabilization. Computational experiments show that the proposed algorithms are capable of obtaining optimal solutions for all existing instances with up to 200 customers, many of them for the first time. The approach involving two pricing subproblems appears to be more efficient than the traditional one. We introduce new large instances and find tight bounds for them. We finally evaluate the performance of the algorithms on benchmark instances of the heterogeneous fixed fleet VRPB and the VRPB with time windows.

An exact algorithm for inland container transportation network design

In this paper, we investigate the inland depot location problem of the inland transportation system. In inland container transportation, empty containers are transported between depots and consignees/shippers, and empty containers should be repositioned after/before inbound/outbound full containers. We build a robust mathematical model that focuses on determining when and where consignees/shippers are assigned to. In addition, inter-depots empty container repositioning is implemented considering demand uncertainty. This paper proposes a branch-and-price algorithm that is based on Lagrangian relaxation and column generation. We show the optimality condition of the pricing subproblem and construct a simpler formulation in this paper. Computational experiments are performed with test instances that mimic real life. Our results also show that considering time compatibility of full and empty container routes is closer to reality and increases the utilization of empty containers in depots. The proposed algorithm yields promising solutions compared with CPLEX.

Exact algorithms to minimize makespan on single and parallel batch processing machines

Batch-processing machine scheduling problem is one of the challenging problems in the machine scheduling literature where machines are capable of processing a batch of jobs simultaneously. In this paper, we tackle single and parallel batch processing machine scheduling problems with the objective of minimizing makespan. We propose a reformulation for parallel batch processing machine scheduling, which is based on decomposition in two levels, and an exact algorithm for its solution. To the best of our knowledge, there is no exact algorithm to solve this problem in the literature, except for its formulations solved by off-the-shelf solvers. In the first part of the proposed algorithm, we solve the single batch processing machine problem by a column-and-cut generation algorithm that provides a lower bound for the parallel machine problem. The second part of our proposed algorithm employs a search mechanism to find the minimum makespan for the parallel machine problem, which entails the solution of the reformulation of this problem by column generation at every iteration. The novel aspect of this column generation algorithm is the integration of batch generation and machine schedule generation in a single pricing subproblem. We test the performance of the proposed algorithms on randomly generated instances and show that, on average, they outperform the off-the-shelf solver. The major findings are that the single machine problem provides tight lower and upper bounds for the parallel machine problem, and the proposed algorithm for the parallel machine problem solves more instances to optimality than that for the single machine problem.

A column generation and a post optimization VNS heuristic for the vehicle routing problem with multiple time windows

The Vehicle Routing Problem with Multiple Time Windows (VRPMTW) is a generalization of the Vehicle Routing Problem (VRP), where the customers have one or more time windows in which they can be visited. In this paper, we propose a Column Generation (CG) algorithm and a post optimization heuristic based on a Variable Neighborhood Search (VNS) to provide both lower and upper bounds for the cost of optimal solutions to VRPMTW. As in CG algorithms for VRP, the master problem is based on a Weighted Set Covering formulation. However, due to the multiple time windows, the pricing subproblem is an Elementary Shortest Path Problem with Multiple Time Windows and Capacity Constraints, which is more difficult to solve than the classical Elementary Shortest Path Problem with a Single Time Window and Capacity Constraints. Computational experiments were performed on 594 instances generated from classical Solomon instances with up to 17 customers. They showed that CG was able to produce lower bounds, within one hour of running time, for 66.7% of the instances. Besides, the post optimization heuristic was able to improve the solution provided by the VNS heuristic in 28.9%, finding integer optimal solutions for 39.9% of the instances. Moreover, for the instances where lower bounds are known, the average optimality gap was 6.0% on average.

Three-Dimensional Bin Packing and Mixed-Case Palletization

Despite its wide range of applications, the three-dimensional bin-packing problem is still one of the most difficult optimization problems to solve. Currently, medium- to large-size instances are only solved heuristically and remain out of reach of exact methods. This is particularly true for its practical variant, the mixed-case palletization problem, where item support is needed. This and the lack of a realistic benchmark data set are identified as major research gaps by a recent survey. In this work, we propose a novel formulation and a column-generation solution approach, where the pricing subproblem is a two-dimensional layer-generation problem. Layers are highly desirable in practical packings as they are easily packable and can accommodate important practical constraints such as item support, family groupings, isle friendliness, and load bearing. Being key to the success of the column-generation approach, the pricing subproblem is solved optimally as well as heuristically and is enhanced by using item grouping, item replacement, layer reorganization, and layer spacing. We conduct extensive computational experiments and compare against existing approaches. We also use industrial data to train and propose a realistic data set. The proposed approach outperforms the best-performing algorithm in the literature on most instances and succeeds to solve practical size instances in very reasonable computational times.

Cover Inequalities for a Vehicle Routing Problem with Time Windows and Shifts

This paper introduces the vehicle routing problem with time windows and shifts (VRPTWS). At the depot, several shifts with nonoverlapping operating periods are available to load the planned trucks. Each shift has a limited loading capacity. We solve the VRPTWS exactly by a branch-and-cut-and-price algorithm. The master problem is a set partitioning with an additional constraint for every shift. Each constraint requires the total quantity loaded in a shift to be less than its loading capacity. For every shift, a pricing subproblem is solved by a label-setting algorithm. Shift capacity constraints define knapsack inequalities; hence we use valid inequalities inspired from knapsack inequalities to strengthen the linear programming relaxation of the master problem when solved by column generation. In particular, we use a family of tailored robust cover inequalities and a family of new nonrobust cover inequalities. Numerical results show that nonrobust cover inequalities significantly improve the algorithm.

A Branch-and-Price Algorithm for the Integrated Berth Allocation and Quay Crane Assignment Problem

This paper integrates, from a tactical perspective, berth allocation and quay-crane assignment, two important, closely related decisions in container terminal operations in a single model. To obtain optimal solutions, a branch-and-price algorithm is sought in this paper under the framework of Dantzig–Wolfe decomposition. The algorithm decomposes the original problem to a master problem that links all vessels competing for the shared resources of berths and quay cranes and multiple per-vessel pricing subproblems that can be solved efficiently in polynomial time. Specifically, in the stage of generating promising initial feasible columns, three heuristics are adopted; during the column-updating stage, the subgradient-based Lagrangian relaxation is introduced to tackle the possibly encountered degeneracy phenomenon; the branching strategy is implemented in the pricing subproblem rather than in the master problem as reported in the literature with the benefit of avoiding incurring new dual prices, simplifying the branching process; and both breadth- and depth-first searching policies are tested to select the next node to explore. With real-life data, extensive numerical experiments are conducted to select the best choice for each stage with the strengths and drawbacks of each choice provided. And then the superiority of the branch-and-price algorithm configured with the selected combination of strategies is verified by comparing it with both CPLEX, a general-purpose solver, and a set-partitioning model, a dedicated algorithm reported in the literature. In addition, the decomposition by vessels adopted in this paper is also verified numerically by comparing it with the decomposition by berths reported in the literature, and the performance of the algorithm for other problem settings has also been tested. In conclusion, the numerical experiments show that our method outperforms the commercial solver and state-of-the-art solution methods reported in the literature in terms of both solution quality and computational time.

Aircraft routing and crew pairing: Updated algorithms at Air France

Aircraft routing and crew pairing problems aim at building the sequences of flight legs operated respectively by airplanes and by crews of an airline. Given their impact on airlines operating costs, both have been extensively studied for decades. Our goal is to provide reliable and easy to maintain frameworks for both problems at Air France. We propose simple approaches to deal with Air France current setting. For routing, we introduce an exact compact IP formulation that can be solved to optimality by current MIP solvers in at most a few minutes even on Air France largest instances. Regarding crew pairing, we provide a methodology to model the column generation pricing subproblem within a new resource constrained shortest path framework recently introduced by the first author. This new framework, which can be used as a black-box, leverages on bounds to discard partial solutions and speed-up the resolution. The resulting approach enables to solve to optimality Air France largest instances. Recent literature has focused on integrating aircraft routing and crew pairing problems. As a side result, we are able to solve to near optimality large industrial instances of the integrated problem by combining the aforementioned algorithms within a simple cut generating method.

REPR: Rule-Enhanced Penalized Regression

This article describes a new rule-enhanced penalized regression procedure for the generalized regression problem of predicting scalar responses from observation vectors in the absence of a preferred functional form. It enhances standard L1-penalized regression by adding dynamically generated rules, that is, new 0-1 covariates, corresponding to multidimensional “box” sets. In contrast to prior approaches to this class of problems, we draw heavily on standard (but non-polynomial-time) mathematical programming techniques, enhanced by parallel computing. We identify and incorporate new rules using a form of classical column generation and solve the resulting pricing subproblem, which is NP-hard, either exactly by a specialized parallel branch-and-bound method or by a greedy heuristic based on Kadane’s algorithm. The resulting rule-enhanced regression method can be computation intensive when we solve the subproblems exactly, but our computational tests suggest that it outperforms prior methods at making accurate and stable predictions from relatively small data samples. Through selective use of our greedy heuristic, we can make our method’s run time generally competitive with some established methods, without sacrificing prediction performance. We call our method’s pricing subproblem rectangular maximum agreement.

An empirical analysis of exact algorithms for the unbounded knapsack problem

This work presents an empirical analysis of exact algorithms for the unbounded knapsack problem, which includes seven algorithms from the literature, two commercial solvers, and more than ten thousand instances. The terminating step-off, a dynamic programming algorithm from 1966, presented the lowest mean time to solve the most recent benchmark from the literature. The threshold and collective dominances are properties of the unbounded knapsack problem first discussed in 1998, and exploited by the current state-of-the-art algorithms. The terminating step-off algorithm did not exploit such dominances, but has an alternative mechanism for dealing with dominances which does not explicitly exploits collective and threshold dominances. Also, the pricing subproblems found when solving hard cutting stock problems with column generation can cause branch-and-bound algorithms to display worst-case times. The authors present a new class of instances which favors the branch-and-bound approach over the dynamic programming approach but do not have high amounts of simple, multiple and collective dominated items. This behaviour illustrates how the definition of hard instances changes among algorithm approaches. The codes used for solving the unbounded knapsack problem and for instance generation are all available online.

Buffering locations in retail deliveries

This research helps a European retailer with its daily goods delivery operations. The challenge is the limited capacity of the transportation network in the Amsterdam area. This leads to very strict requirements for the delivery routing plans. In order to respond to this challenge both an analytical model and a numerical method are proposed. This method is capable of generating routing plans given all the necessary constraints including the cost structure. The model takes into account all incurred costs such as mileage, unsatisfied demand and waiting times; they all are weighted according to their relative importance. Furthermore, in order to reduce waiting times, buffering locations within the Amsterdam area are considered where a truck can park and wait until the loading zone at a store becomes available. The numerical method is approximate in nature and is based on the Column Generation technique. This technique allows iterative explorations of the search space by adding new promising one-truck routes (columns). The Regret construction heuristic is applied to generate an initial solution. New promising columns are generated by means of solving the Pricing Sub-problem which takes into account the duals of the Master problem relaxation. The analysis demonstrates that the buffers help to reduce the waiting times incurred by early arrivals without any drop in the total solution costs. Furthermore, a method is proposed to verify the usefulness of different buffering locations in the model.

Two-agent scheduling on unrelated parallel machines with total completion time and weighted number of tardy jobs criteria

This paper considers a two-agent scheduling problem in which each agent has a set of jobs that competes with that of another agent for the use of m unrelated parallel machines. Each agent aims to minimize a certain scheduling criterion related to the completion times of its jobs. The overall objective is to minimize the total completion time of the jobs of one agent while keeping the weighted number of tardy jobs of another agent within a given limit. We introduce a novel column generation scheme, called in-out column generation, to maintain the stability of dual variables and then embed this scheme into a branch-and-price framework. A greedy heuristic is used to obtain a set of initial columns to start the in-out column generation. The pricing subproblem in the column generation scheme is formulated as a single-machine scheduling problem that can be solved using dynamic programming techniques. An efficient branching strategy that is compatible with pricing subproblems is also proposed. The extensive computational results that are obtained by using randomly generated data demonstrate that our branch-and-price algorithm is singularly efficient and promising.

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 branch-and-price algorithm for the multi-trip multi-repairman problem with time windows

This study introduces the multi-trip multi-repairman problem with time windows where an integrated traveling cost involving distance-dependent and time-dependent costs needs to be minimized. The problem is formulated as two mixed integer programming models. A branch-and-price algorithm is proposed, in which two route-generating approaches are devised to handle the pricing sub-problem. A large number of instances are randomly generated based on an actual service network of China. The proposed algorithm is validated based on these instances and compared to the direct solving method using Cplex. The comparison to the single-trip mode indicates the advantages of the multi-trip mode.