Hybrid column generation for large-size Covering Integer Programs: Application to transportation planning
Introduction
Covering Integer Programming (CIP) is an NP-hard minimization problem that models real-case applications like location problems [8]. It can also appear as the master problem of a Dantzig–Wolfe decomposition in other applications, e.g. transportation problems [1], [29] and cutting stock problems [12]. This paper is devoted to the second category of large-size covering integer programs. For solving the linear relaxation of these problems, the column generation method is an efficient approach when the pricing subproblem can be solved in reasonable time. However, as it only provides a lower bound of the optimal solution, it is often combined with other solving approaches to obtain integer solutions: exact methods (e.g. branch-and-bound and cuts) or approximation methods (e.g. heuristics, metaheuristics and Lagrangian methods) [22], [17], [30], [19], [3]. We will denote by CG+MIP the two-stage process that consists in, firstly, solving the linear relaxation of the master problem by Column Generation (CG), and secondly, running a Mixed-Integer Programming (MIP) solver on the last restricted master problem in order to get integer solutions. Various improvements of column generation have been proposed both for the general process (e.g. initialization, resolution of the master problem and subproblems, choice of inserted columns) as for the integer resolution scheme (e.g. branch-and-price, column generation combined to heuristics or metaheuristics) [2], [26], [27], [16], [5]. Since the goal of the paper is not to design the best-possible solving method for a specific problem but to show the added value of hybridization in column generation before branching, we keep a basic MIP branching scheme without exploring branch-and-price techniques.
The main contribution of this paper is an original and efficient hybridization of CG with (i) the greedy heuristic of Dobson [10], denoted by Gr, which achieves a logarithmic approximation ratio for CIP, and (ii) an extension of Gr to large-size CIP problems, denoted by Gr+ Although the principle of such an extension was already described in [6], no paper has ever analyzed the fractional subproblems associated with Gr+ as it is done in this paper, nor studied how to implement it.
Numerical experiments are conducted on real-case instances of a locomotive assignment problem. Both CG+MIP and Gr+ are first evaluated separately on the problem. Although the greedy heuristic does not generally have outstanding performance in terms of solution quality, it shows however an interesting potential for generating diversified columns with controlled quality and running time. This is a strong motivation to design strategies for an efficient hybridization of the two resolution approaches. This hybrid scheme has two objectives:
- 1.
Accelerate the column generation process that solves the linear relaxation of the CIP master problem,
- 2.
output an integer solution that is strictly better than both the CG+MIP solution and the Gr+ solution.
The paper is organized as follows. In Section 2, we briefly present the CIP formulation and describe the standard Column Generation and Gr principles. Section 3 presents the Gr+ extension of Gr to large-size CIP, and makes the link between the heuristic subproblem and fractional optimization. The generic hybridization scheme is described in Section 4. Section 5 describes the transportation planning application and analyses the greedy subproblem tractability. Section 6 provides and analyses computational results on the real-case problem for CG+MIP and Gr+ tested separately, then for hybridization. Different hybridization strategies are proposed and evaluated. Final conclusions are given in Section 7.
Section snippets
Existing solving approaches for CIP
We first introduce in this section the formulation of CIP programs. We recall then the column generation principles on CIP. We finish by a brief description of Gr.
Extension of Gr to large size CIP using fractional optimization
To get the column of minimum ratio (step 1. of algorithm Gr), one has to evaluate for every and then find . Gr+ is the extension of Gr to a large-size context where explicit enumeration of J is impossible. Crama and Van de Klundert [6] called it greedy column generation. However, we prefer to use the term Gr+ to avoid confusion with the standard CG. The paper of Crama and Van de Klundert [6] focused on getting approximation results for large-size CIP without
Hybrid column generation
The main idea of the hybridization is to make cooperate CG and Gr+ in a more efficient way than just initializing CG with the columns output by Gr+ Indeed, our combination takes advantage of the CG's iterative scheme guided by the cost objective and exploits the diversification potential of Gr+ Both approaches CG and Gr+ can be seen as column generation processes with two different criteria. On the one hand, the standard CG is based on the reduced cost criterion which allows convergence to
The locomotive assignment application: CIP formulation and subproblems
A generic locomotive assignment problem (LAP) consists in assigning locomotives to trains so that each train is pulled by a consist of locomotives with sufficient pulling power, while minimizing a cost function associated to the use of locomotives. Compared to the railway problem studied by Ziarati et al. [29], we make no distinction between passive and active locomotives. We consider a set K of types of locomotives so that for , is the operational pulling power of a locomotive of type k
Numerical results of hybrid column generation on the locomotive assignment application
Numerical results on real-world instances provided by a large railway company are presented, for comparing first CG+MIP and Gr+ before hybridization. Also, in order to evaluate the quality of integer solutions, we have solved compact formulations using the MIP solver of Cplex within a time limit of 3600 s. All the computations reported in this paper have been carried out on a DELL Personal Computer with Pentium M-1.73 GHz processor and 2 GB RAM. The code has been written in C++ and the solver is
Conclusion
We presented in this paper an original hybrid resolution scheme for large-size Covering Integer Programs which combines CG with an approximation heuristic Gr+. Gr+ extends Gr, the greedy heuristic of Dobson, to the case when explicit enumeration of columns is not feasible, using the Dinkelbach algorithm for solving fractional subproblems. The generic hybrid scheme proposed in this paper exploits the diversification power of Gr+ which selects columns according to the criterion ‘cost/coverage’,
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