Set-covering Formulation Research Articles

Reliable reserve-crew scheduling for airlines

We study the practical setting in which regular- and reserve-crew schedules are dynamically maintained up to the day of executing the schedule. At each day preceding the execution of the schedule, disruptions occur due to sudden unavailability of personnel, making the planned regular and reserve-crew schedules infeasible for its execution day. This paper studies the fundamental question how to repair the schedules’ infeasibility in the days preceding the execution, taking into account labor regulations. We propose a robust repair strategy that maintains flexibility in order to cope with additional future disruptions. The flexibility in reserve-crew usage is explicitly considered through evaluating the expected shortfall of the reserve-crew schedule based on a Markov chain formulation. The core of our approach relies on iteratively solving a set-covering formulation, which we call the Robust Crew Recovery Problem, which encapsulates this flexibility notion for reserve crew usage. A tailored branch-and-price algorithm is developed for solving the Robust Crew Recovery Problem to optimality. The corresponding pricing problem is efficiently solved by a newly developed pulse algorithm. Based on actual data from a medium-sized hub-and-spoke airline, we show that embracing our approach leads to fewer flight cancellations and fewer last-minute alterations, compared to repairing disrupted schedules without considering our robust measure.

A Numerically Exact Algorithm for the Bin-Packing Problem

We propose a numerically exact algorithm for solving the Bin-Packing Problem (BPP) based on a branch-price-and-cut framework combined with a pattern-enumeration method. Key to the algorithm is a novel technique for the computation of numerically safe dual bounds for the widely adopted set covering reformulation of the BPP (tightened with additional valid inequalities) with a precision that is higher than the one of general-purpose floating-point solvers. Our branch-price-and-cut algorithm also relies on an exact integer (fixed-point) label setting algorithm for solving the pricing problem associated with the tightened set-covering formulation. To the best of our knowledge, ours is the first algorithm for the BPP that is numerically exact and practical for solving large-scale instances. Extensive computational results on instances affected by notorious numerical difficulties (those of the Augmented Non-IRUP class) show that our exact algorithm outperforms all of the not numerically exact state-of-the-art algorithms based on branch-and-cut-and-price techniques that rely on a set-covering formulation of the BPP. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms − Discrete.

Interval scheduling with economies of scale

Motivated by applications in cloud computing, we study interval scheduling problems exhibiting economies of scale. An instance is given by a set of jobs, each with start time, end time, and a function representing the cost of scheduling a subset of jobs on the same machine. Specifically, we focus on the max-weight function and non-negative, non-decreasing concave functions of total schedule weight. The goal is a partition of the jobs that minimizes the total schedule cost, where overlapping jobs cannot be processed on the same machine. We propose a set cover formulation and a column generation algorithm to solve its linear relaxation. For the max-weight function, which is already NP-hard, we give a polynomial-time pricing algorithm; for the more general case of a function of the total weight, we have a pseudo-polynomial algorithm. To obtain integer solutions, we extend the column generation approach using branch-and-price. We computationally evaluate our methods on two different functions, using both random instances and instances derived from cloud computing data; our algorithm significantly outperforms known integer programming formulations (when these are available) and is able to provably optimize instances with hundreds of jobs.

Smooth and Flexible Dual Optimal Inequalities

We address the problem of accelerating column generation (CG) for set-covering formulations via dual optimal inequalities (DOIs). We study two novel classes of DOIs, which are referred to as Flexible DOIs (F-DOIs) and Smooth-DOIs (S-DOIs), respectively (and jointly as SF-DOIs). F-DOIs provide rebates for covering items more than necessary. S-DOIs describe the payment of a penalty to permit the undercoverage of items in exchange for the overinclusion of other items. Unlike other classes of DOIs from the literature, the S-DOIs and F-DOIs rely on very little problem-specific knowledge and, as such, have the potential to be applied to a vast number of problem domains. In particular, we discuss the application of the new DOIs to three relevant problems: the single-source capacitated facility location problem, the capacitated p-median problem, and the capacitated vehicle-routing problem. We provide computational evidence of the strength of the new inequalities by embedding them within a column-generation solver for these problems. Substantial speedups can be observed as when compared with a nonstabilized variant of the same CG procedure to achieve the linear-relaxation lower bound on problems with dense columns and structured assignment costs.

Solving robust bin-packing problems with a branch-and-price approach

One-dimensional bin-packing is a well-known combinatorial optimization problem which is strongly NP-hard. It consists of allocating a given set of items of different sizes into bins of the same capacity to minimize the number of bins used. The capacity of each bin cannot be exceeded. This paper deals with some variants of this problem to take into account the cases when there are items with uncertain sizes. The goal is to obtain robust solutions taking into account possible variations of item sizes around their nominal values. First, two robust approaches are considered which are based on a stability radius calculation, to ensure that the stability radius, measured either with the Manhattan or Chebyshev norm, is not below a given threshold. Then, a complementary robust approach is applied which is based on a relative resiliency calculation. To solve to optimality these robust variants of the bin-packing problem, a compact 0-1 linear programming formulation, which is also valid for the standard bin-packing problem, is introduced. Then, a Dantzig-Wolfe decomposition is suggested in order to provide a set-cover reformulation with a stronger linear relaxation, but an exponential number of columns. Finally, to obtain integer optimal solutions, a branch-and-price algorithm is developed, whose linear relaxation of the set-cover formulation is solved by a dynamic column generation. Numerical experiments are conducted on adapted benchmark sets from the literature. The performance of the branch-and-price algorithm allows us to investigate what protection against uncertainty is offered by each approach, and at which cost of robustness.

An Exact Method for the Minimum Feedback Arc Set Problem

A feedback arc set of a directed graph G is a subset of its arcs containing at least one arc of every cycle in G . Finding a feedback arc set of minimum cardinality is an NP-hard problem called the minimum feedback arc set problem . Numerically, the minimum set cover formulation of the minimum feedback arc set problem is appropriate as long as all simple cycles in G can be enumerated. Unfortunately, even those sparse graphs that are important for practical applications often have Ω (2 n ) simple cycles. Here we address precisely such situations: An exact method is proposed for sparse graphs that enumerates simple cycles in a lazy fashion and iteratively extends an incomplete cycle matrix. In all cases encountered so far, only a tractable number of cycles has to be enumerated until a minimum feedback arc set is found. The practical limits of the new method are evaluated on a test set containing computationally challenging sparse graphs, relevant for industrial applications. The 4,468 test graphs are of varying size and density and suitable for testing the scalability of exact algorithms over a wide range.

An enhanced branch-and-price algorithm for the integrated production and transportation scheduling problem

ABSTRACT Industry 4.0 technologies, such as artificial intelligence, the internet of things and 3D printing are aiding the manufacturers by complementing their skilled workforce and transforming the way factories are run. This paper studies an integrated production and transportation scheduling problem in the context of the spare parts supply chain by integrating 3D printing with JIT delivery systems. This work aims to find a synchronised production and distribution schedule that minimises the weighted sum of delivery times and transportation costs. Based on the characteristics of the problem, we propose a new set-covering formulation. An enhanced branch-and-price algorithm is designed to solve the problem instances to optimality. To expedite the column generation process, two acceleration strategies are also used. The computational results are in favour of the proposed algorithm and the acceleration strategies. Further, the results indicate that integrating the production and transportation scheduling decisions leads to an average savings of about 16.27% of the total costs.

Exact solutions for the 2d-strip packing problem using the positions-and-covering methodology.

We use the Positions and Covering methodology to obtain exact solutions for the two-dimensional, non-guillotine restricted, strip packing problem. In this classical NP-hard problem, a given set of rectangular items has to be packed into a strip of fixed weight and infinite height. The objective consists in determining the minimum height of the strip. The Positions and Covering methodology is based on a two-stage procedure. First, it is generated, in a pseudo-polynomial way, a set of valid positions in which an item can be packed into the strip. Then, by using a set-covering formulation, the best configuration of items into the strip is selected. Based on the literature benchmark, experimental results validate the quality of the solutions and method’s effectiveness for small and medium-size instances. To the best of our knowledge, this is the first approach that generates optimal solutions for some literature instances for which the optimal solution was unknown before this study.

Positions and covering: A two-stage methodology to obtain optimal solutions for the 2d-bin packing problem.

We present a two-stage methodology called Positions and Covering (P&C) to solve the two-dimensional bin packing problem (2D-BPP). The objective of this classical combinatorial NP-hard problem is to pack a set of items (small rectangles) in the minimum number of bins (larger rectangles). The first stage is the key-point of the Positions and Covering, where for each item, it is generated in a pseudo-polynomial way a set of valid positions that indicate the possible ways of packing the item into the bin. In the second stage, a new set-covering formulation, strengthen with three sets of valid inequalities, is used to select the optimal non-overlapping configuration of items for each bin. Experimental results for the P&C method are presented and compared with some of the best algorithms in the literature for small and medium size instances. Furthermore, we are considering both cases of the 2D-BPP, with and without rotations of the items by 90°. To the best of our knowledge, this is one of the first exact approaches to obtain optimal solutions for the rotation case.

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.

A column generation-based approach for proportionate flexible two-stage no-wait job shop scheduling

Job shop scheduling, as one of the classical scheduling problems, has been widely studied in literatures, and proved to be mostly NP-hard. Although it is extremely difficult to solve job shop scheduling with no-wait constraint to optimality, the two-machine no-wait job shop scheduling to minimise makespan could be solvable in polynomial time when each job has exactly two equal length operations (proportionate job shop). In the present paper, an extension is attempted by considering a proportionate flexible two-stage no-wait job shop scheduling problem with minimum makespan, and a set-covering formulation is put forward which contains a master problem and a pricing problem. To solve this problem, a column generation (CG)-based approach is implemented. In comparison, a mixed integer programming model is constructed and optimised by Cplex. A series of randomly generated numerical instances are calculated. And the testing result shows that the mixed integer model handled by Cplex can only solve small scale cases, while the proposed CG-based method can conquer larger size problems in acceptable time.

Pilot Study on the Application of Employee Scheduling for the Problem of Safety Instrumented System Design and Maintenance Planning for Remotely Located Oil and Gas Facilities

Abstract The technology of production, transportation, and processing of oil and gas involves various hazardous processes. To mitigate the risk that these processes pose, the technological solutions work closely with the automated control and safety systems. The design and organisation of maintenance for the automated safety instrumented systems (SIS) have a significant bearing on the overall safety of operations in this industry. Over the past few decades, many hydrocarbon resources have been discovered in unconventional environments, such as remote, offshore, and arctic locations. Transportation of engineering personnel to these remote locations and back, and thereby, the organisation of the shift work poses additional challenges for the petroleum sector. Under such circumstances, the workforce-related costs play a considerable role in the overall cost of the technological solution and thereby the decisions regarding the workforce organisation should be addressed in the framework of evaluating and choosing the appropriate safety measures. That is why the research presented in this paper aims to address the lifecycle of the technological solution integrating the problems of SIS design, maintenance planning, and employee scheduling into a single decision-making framework to optimise the set of technical and organisational safety measures inherent in the SIS. The performance and maintenance of the SIS are described with a Markov model of device failures, repairs and technological incidents occurrence. The employee scheduling part of the mathematical model utilises the set-covering formulation of maintenance crews taking particular trips. A black-box optimisation algorithm is used to find reasonable solutions to the integrated problem of engineering design and workforce planning. The decisions include the choices of the components and structures for the safety system, the facility overhaul frequencies, the maintenance personnel size, as well as the schedules of trips and shifts for the crews.

Service system design with immobile servers, stochastic demand and concave-cost capacity selection

The service system design problem is a location-allocation problem with service quality considerations that is often modeled as a network of M/M/1 queues to minimize facility setup, customer access, and waiting costs. Traditionally, capacity decisions are either ignored or modeled as a selection among discrete capacity levels. In this work, we study the general continuous capacity case and account for economies-of-scale in its cost through an increasing concave function. We focus on the special square-root case that has been shown to model capacity in terms of the number of servers needed under Poisson arrivals and exponential service times. The problem is formulated as a mixed-integer nonlinear program with concave and convex terms in the objective function. Two novel resolution approaches are proposed: In the first, the problem is reformulated as a mixed-integer quadratic program with fourth-degree polynomial equality constraints. These constraints and the quadratic objective function are approximated using piecewise-linear segments. In the second, we use Lagrangian relaxation to decompose the problem and reformulate the subproblems as second-order cone programs that are solved at multiple utilization levels. The Lagrangian multipliers are updated using a cutting-plane method and a feasible solution is obtained by solving the corresponding set-covering formulation. The solution approaches are tested and compared. The linearization approach provides high quality solutions within short computational times for small instances and lower accuracy; whereas the Lagrangian approach scales well as size increases.

A set-covering formulation for a drayage problem with single and double container loads

This paper addresses a drayage problem, which is motivated by the case study of a real carrier. Its trucks carry one or two containers from a port to importers and from exporters to the port. Since up to four customers can be served in each route, we propose a set-covering formulation for this problem where all possible routes are enumerated. This model can be efficiently solved to optimality by a commercial solver, significantly outperforming a previously proposed node-arc formulation. Moreover, the model can be effectively used to evaluate a new distribution policy, which results in an enlarged set of feasible routes and can increase savings w.r.t. the policy currently employed by the carrier.

LP-based working subsets for personnel scheduling: evaluation and augmentation

This paper evaluates the efficacy of LP-based working subsets for generalised set-covering formulations of personnel scheduling problems and presents a nearest-neighbour augmentation procedure for improving performance. Three experimental studies were completed in the evaluation process. In the first study, the LP-based working subset was sufficient to yield an optimal shift scheduling solution for 85% of the 24,300 test problems and the nearest-neighbour augmentation improved the percentage of optimal solutions to over 98%. The second study focused on more complex cyclic shift scheduling environments that permitted shift length, meal break and relief break flexibility. The adequacy of LP-based working subsets was supported by the provision of optimal shift scheduling solutions for 185 of 189 (98%) of the test problems. The third study examined a challenging tour scheduling environment for which globally-optimal benchmarks are not available. The superiority of the augmented LP-based working subset procedure was nevertheless evident, as it yielded better results than the (non-augmented) LP-based working subset for 92% of the test problems despite being constrained to only 40% of the allowed computation time for the non-augmented subsets. [Received 24 April 2017; Revised 25 August 2017; Revised 17 September 2017; Accepted 6 December 2017]

A heuristic for cumulative vehicle routing using column generation

Cumulative vehicle routing problems are a simplified model of fuel consumption in vehicle routing problems. Here we computationally study, an inexact approach for constructing solutions to cumulative vehicle routing problems based on rounding solutions to a linear program. The linear program is based on the set cover formulation and is solved using column generation. The pricing subproblem is solved heuristically using dynamic programming. Simulation results show that a simple scalable strategy gives solutions with cost close to the lower bound given by the linear programming relaxation. We also give theoretical bounds on the integrality gap of the set cover formulation.

Coupled segmentation and similarity detection for architectural models

Recent shape retrieval and interactive modeling algorithms enable the re-use of existing models in many applications. However, most of those techniques require a pre-labeled model with some semantic information. We introduce a fully automatic approach to simultaneously segment and detect similarities within an existing 3D architectural model. Our framework approaches the segmentation problem as a weighted minimum set cover over an input triangle soup, and maximizes the repetition of similar segments to find a best set of unique component types and instances. The solution for this set-cover formulation starts with a search space reduction to eliminate unlikely combinations of triangles, and continues with a combinatorial optimization within each disjoint subspace that outputs the components and their types. We show the discovered components of a variety of architectural models obtained from public databases. We demonstrate experiments testing the robustness of our algorithm, in terms of threshold sensitivity, vertex displacement, and triangulation variations of the original model. In addition, we compare our components with those of competing approaches and evaluate our results against user-based segmentations. We have processed a database of 50 buildings, with various structures and over 200K polygons per building, with a segmentation time averaging up to 4 minutes.

A goal-driven prototype column generation strategy for the multiple container loading cost minimization problem

In the multiple container loading cost minimization problem (MCLCMP), rectangular boxes of various dimensions are loaded into rectangular containers of various sizes so as to minimize the total shipping cost. The MCLCMP can be naturally modeled as a set cover problem. We generalize the set cover formulation by introducing a new parameter to model the gross volume utilization of containers in a solution. The state-of-the-art algorithm tackles the MCLCMP using the prototype column generation (PCG) technique. PCG is an effective technique for speeding up the column generation technique for extremely hard optimization problems where their corresponding pricing subproblems are NP-hard. We propose a new approach to the MCLCMP that combines the PCG technique with a goal-driven search. Our goal-driven prototype column generation (GD-PCG) algorithm improves the original PCG approach in three respects. Computational experiments suggest that all three enhancements are effective. Our GD-PCG algorithm produces significantly better solutions for the 350 existing benchmark instances than all other approaches in the literature using less computation time. We also generate two new set instances based on industrial data and the classical single container loading instances.

Modeling profit sharing in combinatorial exchanges by network flows

In this paper we study the possibilities of sharing profit in combinatorial procurement auctions and exchanges. Bundles of heterogeneous items are offered by the sellers, and the buyers can then place bundle bids on sets of these items. That way, both sellers and buyers can express synergies between items and avoid the well-known risk of exposure (see, e.g., Cramton, Shoham, & Steinberg, Combinatorial Auctions, 2006). The reassignment of items to participants is known as the Winner Determination Problem (WDP). We propose solving the WDP by using a Set Covering formulation, because profits are potentially higher than with the usual Set Partitioning formulation, and subsidies are unnecessary. The achieved benefit is then to be distributed amongst the participants of the auction, a process which is known as profit sharing. The literature on profit sharing provides various desirable criteria. We focus on three main properties we would like to guarantee: Budget balance, meaning that no more money is distributed than profit was generated, no negative transfers, which guarantees to each player that participation does not lead to a loss, and the core property, which provides every subcoalition with enough money to keep them from separating. We characterize all profit distributions that satisfy these three conditions by a monetary flow network and establish a connection to the famous VCG payment scheme (Clarke in Public Choice 18:19ff, 1971; Groves in Econometrica 41:617–631, 1973; Vickrey in J. Finance 16:8–37, 1961) and the Shapley Value (Shapley in Kuhn & Tucker (Eds.), Contributions to the Theory of Games, vol. II, pp. 307–317, 1953). Finally, we introduce a novel profit sharing scheme based on scaling down VCG discounts using max-min fairness principles to achieve budget balance. It approximates the Shapley Value linearly in the number of bundles and can be computed efficiently.

The School Bus Problem on Trees

The School Bus Problem is an NP-hard vehicle routing problem in which the goal is to route buses that transport children to a school such that for each child, the distance travelled on the bus does not exceed the shortest distance from the child’s home to the school by more than a given regret threshold. Subject to this constraint and bus capacity limit, the goal is to minimize the number of buses required. In this paper, we give a polynomial time 4-approximation algorithm when the children and school are located at vertices of a fixed tree. As a byproduct of our analysis, we show that the integrality gap of the natural set-cover formulation for this problem is also bounded by 4. We also present a constant factor approximation for the variant where we have a fixed number of buses to use, and the goal is to minimize the maximum regret.