Dantzig-Wolfe Reformulation Research Articles

A stabilised scenario decomposition algorithm applied to stochastic unit commitment problems

In recent years the expansion of energy supplies from volatile renewable sources has triggered an increased interest in stochastic optimisation models for hydro-thermal unit commitment. Several studies have modelled this as a two-stage or multi-stage stochastic mixed-integer optimisation problem. Solving such problems directly is computationally intractable for large instances, and alternative approaches are required. In this paper we use a Dantzig–Wolfe reformulation to decompose the stochastic problem by scenarios. We derive and implement a column generation method with dual stabilisation and novel primal and dual initialisation techniques. A fast, novel schedule combination heuristic is used to construct very good primal solutions, and numerical results show that knowing these from the start improves the convergence of the column generation method significantly. We test our method on a central scheduling model based on the British National Grid and illustrate that convergence to within 0.1% of optimality can be achieved quickly.

GUB Flow Cover Inequalities and an Exact Algorithm for Transportation Service Procurement Problem with Total Quantity Discounts

In transportation service procurement, total quantity discounts offered by a strategic carrier are incentives to a shipper. While the shipper reduces total cost by activating discounts, the carrier can win a valuable contract as considerable freight volume is assigned. Moreover, a strategic shipper would achieve even more cost reduction by purchasing more freight volume than demand and securing more discounts. The total quantity discounts are thus important to the transportation service procurement problem, which optimizes to select carriers and allocate them freight volume with the aim of minimizing the total cost for the shipper. The transportation service procurement problem with total quantity discounts (TP-TQD) can be formulated into a multiple choice model. We introduce a new formulation based on Dantzig-Wolfe reformulation for the problem, whose lower bound from LP relaxation is proved to be tighter than that of the multiple choice model. Two sets of generalized upper bound (GUB) flow cover valid inequalities are proposed to strengthen the models and applicable to any problems with structure consisting of the more-than-demand constraints, capacities, and GUB constraints. Effective separation methods are developed. A branch-and-price-and-cut algorithm is developed to solve the TP-TQD. Extensive experiments on a set of test instances generated from practical data are carried out to evaluate the methods. The results indicate that the new valid inequalities are effective and the exact algorithm is competent to solve problem instances of practical sizes.

A Combined column generation and heuristics for railway short-term rolling stock planning with regular inspection constraints

The aim of railway rolling stock planning problem is to find an optimal allocation of train-sets for a given set of trips in the train timetable in order to minimize the total cost. We propose a column generation and Lagrangian relaxation heuristics for short-term rolling stock planning problems with regular inspection constraints. The problem is formulated as a subtour traveling salesman problem to find a set of elementary shortest cycles that cover all trips in the timetable. In the proposed method, a tight lower bound is obtained from the continuous relaxation of Dantzig–Wolfe reformulation by column generation. The pricing problem can be formulated as an elementary shortest cycle problem with resource constraints. A labeling algorithm is applied to solve the pricing problem. In order to reduce the computational effort, we apply a general state space augmenting algorithm to solve the pricing problems. Computational results show that the proposed column generation and Lagrangian relaxation heuristics can find good lower and upper bounds for 300 trips within reasonable computing time.

Stabilized column generation for the temporal knapsack problem using dual-optimal inequalities

We present two new methods to stabilize column-generation algorithms for the temporal knapsack problem (TKP). Caprara et al. (INFORMS J Comp 25(3):560–571, 2013] were the first to suggest the use of branch-and-price algorithms for Dantzig–Wolfe reformulations of the TKP. Herein, the respective pricing problems are smaller-sized TKP that can be solved with a general-purpose MIP solver or by dynamic programming. Our stabilization methods are tailored to the TKP as they use (deep) dual-optimal inequalities, that is, inequalities known to be fulfilled by all (at least some) optimal dual solutions to the linear relaxation. Extensive computational tests reveal that both new stabilization techniques are helpful. Several previously unsolved instances are now solved to proven optimality.

Solving the Temporal Knapsack Problem via Recursive Dantzig–Wolfe Reformulation

The Temporal Knapsack Problem (TKP) is a generalization of the standard Knapsack Problem where a time horizon is considered, and each item consumes the knapsack capacity during a limited time interval only. In this paper we solve the TKP using what we call a Recursive Dantzig–Wolfe Reformulation (DWR) method. The generic idea of Recursive DWR is to solve a Mixed Integer Program (MIP) by recursively applying DWR, i.e., by using DWR not only for solving the original MIP but also for recursively solving the pricing sub-problems. In a binary case (like the TKP), the Recursive DWR method can be performed in such a way that the only two components needed during the optimization are a Linear Programming solver and an algorithm for solving Knapsack Problems. The Recursive DWR allows us to solve Temporal Knapsack Problem instances through computation of strong dual bounds, which could not be obtained by exploiting the best-known previous approach based on DWR.

Column Generation‐Based Approach for Solving Large‐Scale Ready Mixed Concrete Delivery Dispatching Problems

AbstractReady mix concrete (RMC) dispatching forms a critical component of the construction supply chain. However, optimization approaches within the RMC dispatching continue to evolve due to the specific size, constraints, and objectives required of the application domain. In this article, we develop a column generation algorithm for vehicle routing problems (VRPs) with time window constraints as applied to RMC dispatching problems and examine the performance of the approach for this specific application domain. The objective of the problem is to find the minimum cost routes for a fleet of capacitated vehicles serving concrete to customers with known demand from depots within the allowable time window. The VRP is specified to cover the concrete delivery problem by adding additional constraints that reflect real situations. The introduced model is amenable to the Dantzig–Wolfe reformulation for solving pricing problems using a two‐staged methodology as proposed in this article. Further, under the mild assumption of homogeneity of the vehicles, the pricing sub‐problem can be viewed as a minimum‐cost multi‐commodity flow problem and solved in polynomial time using efficient network simplex method implementations. A large‐scale field collect data set is used for evaluating the model and the proposed solution method, with and without time window constraints. In addition, the method is compared with the exact solution found via enumeration. The results show that on average the proposed methodology attains near optimal solutions for many of the large sized models but is 10 times faster than branch‐and‐cut.

The multi-vehicle traveling purchaser problem with pairwise incompatibility constraints and unitary demands: A branch-and-price approach

In this work, we study a supplier selection and routing problem where a fleet of homogeneous vehicles with a predefined capacity is available for procuring different products from different suppliers with the aim to satisfy demand at the minimum traveling and purchasing cost. Decisions are further complicated by the presence of pairwise incompatibility constraints among products, implying the impossibility of loading two incompatible products on the same vehicle. The problem is known as the Multi-Vehicle Traveling Purchaser Problem with Pairwise Incompatibility Constraints. We study the special case in which the demand for each product is unitary and propose a column generation approach based on a Dantzig–Wolfe reformulation of the problem, where each column represents a feasible vehicle route associated with a compatible purchasing plan. To solve the pricing problem we propose a hybrid strategy exploiting the advantages of two alternative exact methods, a labeling algorithm solving a Resource-Constrained Elementary Shortest Path Problem on an expanded graph, and a tailored branch-and-cut algorithm. Due to the integrality request on variables, we embed the column generation in a branch-and-bound framework and propose different branching rules. Extensive tests, carried out on a large set of instances, show that our branch-and-price method performs well, improving on average, both in quality and in computational time, solutions obtained by a state-of-art branch-and-cut approach applied to a three-index connectivity constraints based formulation.

Mathematical programming approaches to pricing problems

There are many real cases where a company needs to determine the price of its products so as to maximise its revenue or profit.To do so, the company must consider customers' reactions to these prices, as they may refuse to buy a given product or service if its price is too high. This is commonly known in literature as a pricing problem.This class of problems, which is typically bilevel, was first studied in the 1990s and is NP-hard, although polynomial algorithms do exist for some particular cases. Many questions are still open on this subject.The aim of this thesis is to investigate mathematical properties of pricing problems, in order to find structural properties, formulations and solution methods that are as efficient as possible. In particular, we focus our attention on pricing problems over a network. In this framework, an authority owns a subset of arcs and imposes tolls on them, in an attempt to maximise his/her revenue, while users travel on the network, seeking for their minimum cost path.First, we provide a detailed review of the state of the art on bilevel pricing problems. Then, we consider a particular case where the authority is using an unit toll scheme on his/her subset of arcs, imposing either the same toll on all of them, or a toll proportional to a given parameter particular to each arc (for instance a per kilometre toll). We show that if tolls are all equal then the complexity of the problem is polynomial, whereas in case of proportional tolls it is pseudo-polynomial.We then address a robust approach taking into account uncertainty on parameters. We solve some polynomial cases of the pricing problem where uncertainty is considered using an interval representation.Finally, we focus on another particular case where toll arcs are connected such that they constitute a path, as occurs on highways. We develop a Dantzig-Wolfe reformulation and present a Branch-and-Cut-and-Price algorithm to solve it. Several improvements are proposed, both for the column generation algorithm used to solve the linear relaxation and for the branching part used to find integer solutions. Numerical results are also presented to highlight the efficiency of the proposed strategies. This problem is proved to be APX-hard and a theoretical comparison between our model and another one from the literature is carried out.

A branch-cut-and-price algorithm for the piecewise linear transportation problem

In this paper we present an exact solution method for the transportation problem with piecewise linear costs. This problem is fundamental within supply chain management and is a straightforward extension of the fixed-charge transportation problem. We consider two Dantzig–Wolfe reformulations and investigate their relative strength with respect to the linear programming (LP) relaxation, both theoretical and practical, through tests on a number of instances. Based on one of the proposed formulations we derive an exact method by branching and adding generalized upper bound constraints from violated cover inequalities. The proposed solution method is tested on a set of randomly generated instances and compares favorably to solving the model using a standard formulation solved by a state-of-the-art commercial solver.

A branch-price-and-cut algorithm for multi-mode resource leveling

We propose a new mixed integer programming formulation and solution algorithm for a multi-mode resource-constrained project scheduling problem with availability constraints (calendars) and the objective to minimize the resource availability cost. Our model exploits the problem structure and has an exponential number of variables, which necessitates a column generation approach. The linear programming relaxation is strengthened by adding valid inequalities that need to be carefully separated in order to show the desired effect. Integer optimal solutions are obtained by an exact state-of-the-art branch-price-and-cut algorithm.Classical time-indexed mixed integer programming formulations for similar problems quite often fail already on instances with only 30 jobs, depending on the network complexity and the total freedom of arranging jobs. A reason is the typically very weak linear programming relaxation. Our model can be interpreted as a non-trivial Dantzig–Wolfe reformulation of a time-indexed formulation. In particular, for larger instances our reformulation gives significantly tighter dual bounds, enabling us to optimally solve instances with 50 multi-mode jobs. This outperforms the classical formulation by far.

Automatic Dantzig–Wolfe reformulation of mixed integer programs

Dantzig–Wolfe decomposition (or reformulation) is well-known to provide strong dual bounds for specially structured mixed integer programs (MIPs). However, the method is not implemented in any state-of-the-art MIP solver as it is considered to require structural problem knowledge and tailoring to this structure. We provide a computational proof-of-concept that the reformulation can be automated. That is, we perform a rigorous experimental study, which results in identifying a score to estimate the quality of a decomposition: after building a set of potentially good candidates, we exploit such a score to detect which decomposition might be useful for Dantzig–Wolfe reformulation of a MIP. We experiment with general instances from MIPLIB2003 and MIPLIB2010 for which a decomposition method would not be the first choice, and demonstrate that strong dual bounds can be obtained from the automatically reformulated model using column generation. Our findings support the idea that Dantzig–Wolfe reformulation may hold more promise as a general-purpose tool than previously acknowledged by the research community.

Uncommon Dantzig-Wolfe Reformulation for the Temporal Knapsack Problem

We study a natural generalization of the knapsack problem, in which each item exists only for a given time interval. One has to select a subset of the items (as in the classical case), guaranteeing that for each time instant, the set of existing selected items has total weight no larger than the knapsack capacity. We focus on the exact solution of the problem, noting that prior to our work, the best method was the straightforward application of a general-purpose solver to the natural integer linear programming formulation. Our results indicate that much better results can be obtained by using the same general-purpose solver to tackle a nonstandard Dantzig-Wolfe reformulation in which subproblems are associated with groups of constraints. This is also interesting because the more natural Dantzig-Wolfe reformulation of single constraints performs extremely poorly in practice.

Exact Algorithms for a Bandwidth Packing Problem with Queueing Delay Guarantees

The bandwidth packing problem (BWP) concerns the selection of calls from a given set and the assignment of one path to each selected call. The ultimate aim of the BWP is to maximize profit while the routings of the selected calls observe the capacity constraints of the links. Here, we additionally consider queueing delays in the network, which may cause a deterioration in the quality of service to users if they exceed the acceptable limits. The integer programming formulation for the BWP with the queueing delay restriction contains a nonlinear constraint that is intrinsic to the model. We apply the Dantzig-Wolfe decomposition to this nonlinear constraint, and since the Dantzig-Wolfe decomposition has exponentially many variables, we propose the branch-and-price procedure to find optimal solutions. We also propose a generalized Dantzig-Wolfe reformulation based on the aggregation of variables, which makes our branch-and-price algorithm more competitive. Computational results on cases of randomly generated networks and some real-life telecommunication networks demonstrate that our algorithm performs well for large networks.

Dantzig-Wolfe Reformulation for the Network Pricing Problem with Connected Toll Arcs

This paper considers a pricing problem on a network with connected toll arcs and proposes a Dantzig-Wolfe reformulation for it. The model is solved with column generation and the gap between the optimal integer value and the linear relaxation optimal value is shown to be at least as good as the one from the mixed-integer formulation proposed in the literature. Numerical results on different sets of instances are reported, showing that in many cases the proposed model performs strictly better.

Exactly solving a two‐level location problem with modular node capacities

AbstractIn many telecommunication networks, a given set of client nodes must be served by different sets of facilities—providing different services and having different capabilities—which must be located and dimensioned in the design phase. Network topology must be designed as well, by assigning clients to facilities and facilities to higher level entities, when necessary. We tackle a particular location problem, where two sets of facilities have to be located, and in which different devices can be installed at each site, providing different capacities at different costs. We optimize location and dimensioning of these facilities simultaneously. We introduce a compact formulation of that problem, we use discretization and Dantzig–Wolfe reformulation techniques to improve models, and we design an exact optimization algorithm. We test our approach on a set of instances derived from existing literature on facility location. © 2011 Wiley Periodicals, Inc. NETWORKS, 2012

Column Generation for Extended Formulations

Working in an extended variable space allows one to develop tight reformulations for mixed integer programs (MIP). However, the size of the extended formulation grows rapidly too large for a direct treatment by a MIP-solver. When the extended formulation stems from a decomposition principle, as typical in practice, a column generation procedure that mimics that for the Dantzig-Wolfe reformulation can be applied to the extended formulation. Pricing subproblem solutions are expressed in the variables of the extended formulation and added to the current restricted version of the extended formulation along with the subproblem constraints that are active for the subproblem solution. Such “column-and-row generation” procedure is reviewed with the goal to analyse the approachʼs potential benefits compared to a standard column generation approach. Numerical experiments highlight a key observation: lifting pricing problem solutions in the space of the extended formulation permits their recombination into new subproblem solutions and results in faster convergence.

Stronger column generation bounds for the Minimum Cost Hop-and-root Constrained Forest Problem

Abstract In this paper, we present a Dantzig-Wolfe reformulation for the Minimum Cost Hop-and-root Constrained Forest Problem and discuss a Column Generation (CG) method to evaluate its Linear Programming (LP) bounds. For solving one of two types of pricing problems that arise in CG, we compared two solution strategies: Dynamic Programming and a Branch-and-cut (BC) algorithm. In general, CG performed much better when BC was used. Not only the LP bounds implied by the proposed reformulation are much stronger than the multi-commodity flow bounds from the literature, but also could be evaluated with less computational time. A Column Generation Heuristic was discussed and implemented, providing upper bounds that are, on average, within 2.3% of optimality.

Solving Nonlinear Covering Problems Arising in WLAN Design

Wireless local area networks (WLANs) are widely used for cable replacement and wireless Internet access. Because the medium access control (MAC) scheme of WLANs has a strong influence on network performance, it should be accounted for in WLAN design. This paper presents AP location models that optimize a network performance measure specifically for the MAC scheme of WLANs that represents the efficiency in sharing the wireless medium. For these models, we propose a solution framework based on an effective integer-linear programming Dantzig–Wolfe reformulation. This framework is applicable to any nonlinear covering problem where the objective function is a sum of contributions over the groundset elements (users in WLANs). Extensive computational results show that our solution strategy quickly yields optimal or near-optimal solutions for WLAN design instances of realistic size.

Classification of Dantzig–Wolfe reformulations for binary mixed integer programming problems

In this note, we provide a classification of Dantzig–Wolfe reformulations for Binary Mixed Integer Programming Problems. We specifically focus on modeling the binary conditions in the convexification approach to the Dantzig–Wolfe decomposition. For a general Binary Mixed Integer Programming problem, an extreme point of the overall problem does not necessarily correspond to an extreme point of the subproblem. Therefore, the binary conditions cannot in general be imposed on the new master problem variables but must be imposed on the original binary variables. In some cases, however, it is possible to impose the binary restrictions directly on the new master problem variables. The issue of imposing binary conditions on the original variables versus the master problem variables has not been discussed systematically for MIP problems in general in the literature and most of the research has been focused on the pure binary case. The classification indicates in which cases you can, and cannot, impose binary conditions on the new master problem variables.

Branching in branch-and-price: a generic scheme

Developing a branching scheme that is compatible with the column generation procedure can be challenging. Application specific and generic schemes have been proposed in the literature, but they have their drawbacks. One generic scheme is to implement standard branching in the space of the compact formulation to which the Dantzig-Wolfe reformulation was applied. However, in the presence of multiple identical subsystems, the mapping to the original variable space typically induces symmetries. An alternative, in an application specific context, can be to expand the compact formulation to offer a wider choice of branching variables. Other existing generic schemes for use in branch-and-price imply modifications to the pricing problem. This is a concern because the pricing oracle on which the method relies might become obsolete beyond the root node. This paper presents a generic branching scheme in which the pricing oracle of the root node remains of use after branching (assuming that the pricing oracle can handle bounds on the subproblem variables). The scheme does not require the use of an extended formulation of the original problem. It proceeds by recursively partitioning the subproblem solution set. Branching constraints are enforced in the pricing problem instead of being dualized via Lagrangian relaxation, and the pricing problem is solved by a limited number of calls to the pricing oracle. This generic scheme builds on previously proposed approaches and unifies them. We illustrate its use on the cutting stock and bin packing problems. This is the first branch-and-price algorithm capable of solving such problems to integrality without modifying the subproblem or expanding its variable space.