S. Ceria, P. Nobili, and A. Sassano. A lagrangian-based heuristic for large-scale Set Covering problems. Mathematical Programming, 81:215--228, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....in a cutting plane exact algorithm [1] Beasley [2] adopted several reduction procedures, in addition to a lagrangian approach similar to the one we describe in Section 3. Fisher and Kedia [13] added dual heuristics to this framework, which has been recently improved by Ceria, Nobili and Sassano [7]. Caprara, Fischetti and Toth [6] used a lagrangian heuristic coupled with a dynamic pricing scheme, in order to determine a good relaxation in short time, even in large scale problems. 3. The proposed approach This section presents an integer formulation of the unate covering problem and ....

S. Ceria, P. Nobili, and A. Sassano. A lagrangian-based heuristic for large-scale Set Covering problems. Mathematical Programming, 81:215--228, 1998.

....as follows: SCP: minimize cost(x) N, a ij = 1 (if i S j ) 0 (otherwise) It is understood that variable x j equals 1 if subset S j is in the selected family and 0 otherwise. The SCP has many practical applications, including bus, railway and airline crew scheduling [1, 8, 19], location problems of facilities [20] and logical analysis of data [6] It is known to be NP hard in the strong sense, and a number of exact and heuristic algorithms have been proposed in the literature. Among exact branch and bound algorithms, Fisher and Kedia proposed an exact branch andbound ....

.... with up to 400 rows and 4,000 columns [4] Among approximate methods, Jacobs and Brusco developed an algorithm of simulated annealing [17] Beasley and Chu presented a genetic algorithm [5] and Ceria, Nobili and Sassano presented a Lagrangian based heuristic for solving very large scale instances [8]. Caprara, Fischetti and Toth combined a Lagrangian based heuristic and greedy algorithm, and obtained impressive results for various types of instances, especially for very large scale instances with up to 5,000 rows and 1,000,000 columns [7] All these algorithms can be thought of as based on ....

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S. Ceria , P. Nobili and A. Sassano, "A Lagrangian-Based Heuristic for Large-Scale Set Covering Problems," Mathematical Programming, 81 (1998) 215--228.

.... and it is unlikely that there are polynomial algorithms which find approximations within a factor ln n from an optimal solution [25] Lagrangian relaxation approaches for set covering [10] have been investigated for railway problems of similar size to the problems in our test set, see e.g. [14, 15, 17]. Hoffman and Padberg [34] apply branch and cut techniques to smaller set partitioning problems with base constraints. Marsten et al. 39] apply the interior point solver of the linear programming package CPLEX [35] for set partitioning and covering and experience that for the problems they ....

....have base constraints. Note, that although the number of base constraints is small compared to the total number of constraints, these are quite dense. The rail problems have been used in the FASTER competition organized by Ferrovie dello Stato and the Italian Operational Research Society, AIRO, [15, 17]. These problems have only three different cost coefficients. The problem set3000 is a small but difficult problem which CPLEX [36] does not solve well. All problems are available from the authors. Several problems are so large that a code like CPLEX is not able to find a solution even to the ....

S. Ceria, P. Nobili, and A. Sassano. A Lagrangian-based heuristic for large-scale set covering problems. Mathematical Programming. To appear.

....data analysis. Interested readers are referred to the survey by Balas [1] and to the annotated bibliography by Ceria, Nobili and Sassano [16] There is also an extensive literature on numerical methods for solving such problems, both exact ( 2] 4] 9] 27] and approximate ( 5] 8] 12] [15], 22] 23] 25] An experimental comparison of the main methods is reported in the survey paper by Caprara, Fischetti, and Toth [13] Our objective is to analyse a probabilistic set covering problem, in which the right hand side of (1) is replaced by a binary random vector . Clearly, ....

S. Ceria, P. Nobili, A. Sassano, A Lagrangian-Based Heuristic for Large-Scale Set Covering Problems. Mathematical Programming 81 (1998), 215-228.

.... in Balas and Ho [3] and Balas and Carrera [2] The most effective heuristic approaches to SCP are those based on Lagrangian relaxation following the seminal work by Balas and Ho [3] and then the improvements by Beasley [6] Fisher and Kedia [20] Balas and Carrera [2] Ceria, Nobili and Sassano [15], and Wedelin [31] Lorena and Lopes [27] propose an analogous approach based on surrogate relaxation. Recently, Beasley and Chu [7] and Jacobs and Brusco [25] proposed a genetic and a local search algorithm, respectively. We next outline a heuristic method recently proposed by Caprara, Fischetti ....

S. Ceria, P. Nobili and A. Sassano, "A Lagrangian-Based Heuristic for LargeScale Set Covering Problems", Technical Report R.406, IASI-CNR, Rome, 1995, to appear in Mathematical Programming.

....proposed in [9] see the next section. Moreover, Caprara, Fischetti and Toth [9] present improvements on the standard way of defining the step size and the ascent direction within the subgradient optimization procedure. A primal dual subgradient approach is proposed by Ceria, Nobili and Sassano [12]. In particular, one can restate the LP dual problem (5) by explicitly imposing the upper bound c i = maxfc j : j 2 J i g to each dual variable u i , i 2 M . If constraints P i2I j u i c j , j 2 N , are relaxed in a Lagrangian way through a multiplier vector x, one obtains a Lagrangian dual ....

....in the paper. The heuristic algorithm of Lorena and Lopes [21] uses the surrogate relaxation described in the previous section, and embeds within the subgradient optimization procedure a greedy heuristic identical to the one proposed by Beasley [4] The approach of Ceria, Nobili and Sassano [12] is mainly intended for large scale SCP instances. Initially, a subgradient optimization procedure is applied to the whole problem so as to define a good core problem. Differently from Caprara, Fischetti and Toth [9] where the initial selection of the core problem is not a critical issue since ....

[Article contains additional citation context not shown here]

S. Ceria, P. Nobili and A. Sassano, "A Lagrangian-Based Heuristic for Large-Scale Set Covering Problems", Technical Report R.406, IASI-CNR, Rome, 1995, to appear in Mathematical Programming.

....to X j2N a ij x j 1; 8i 2 M x j 2 f0; 1g; 8j 2 N; where a ij = 1 (if i 2 S j ) 0 (otherwise) It is understood that variable x j equals 1 if subset S j is in the selected family and 0 otherwise. The SCP has many practical applications, including bus, railway and airline crew scheduling [1, 8, 19], location problems of facilities [20] and logical analysis of data [6] It is known to be NP hard in the strong sense, and a number of exact and heuristic algorithms have been proposed in the literature. Among exact branch and bound algorithms, Fisher and Kedia proposed an exact branch andbound ....

.... with up to 400 rows and 4,000 columns [4] Among approximate methods, Jacobs and Brusco developed an algorithm of simulated annealing [17] Beasley and Chu presented a genetic algorithm [5] and Ceria, Nobili and Sassano presented a Lagrangian based heuristic for solving very large scale instances [8]. Caprara, Fischetti and Toth combined a Lagrangian based heuristic and greedy algorithm, and obtained impressive results for various types of instances, especially for very large scale instances with up to 5,000 rows and 1,000,000 columns [7] All these algorithms are based on local search, in ....

[Article contains additional citation context not shown here]

S. Ceria , P. Nobili and A. Sassano, "A Lagrangian-Based Heuristic for Large-Scale Set Covering Problems," Mathematical Programming, 81 (1998) 215--228.

....generation is one of the best established optimization approaches to duty scheduling. The methodology is based on techniques such as duty generation by constrained shortest path computations ( 32] 24] 27 29] large scale techniques such as sifting, core, and active set strategies ( 12] [16], 15] 51] LP and other acceleration techniques ( 33] 23] heuristics ( 5] 8] 36] 54] 16] 15] and other contributions (see [7] 30] 52] 21] for surveys) Likewise, column generation modules form the optimization cores of several commercial duty scheduling systems, ....

.... on techniques such as duty generation by constrained shortest path computations ( 32] 24] 27 29] large scale techniques such as sifting, core, and active set strategies ( 12] 16] 15] 51] LP and other acceleration techniques ( 33] 23] heuristics ( 5] 8] 36] 54] [16], 15] and other contributions (see [7] 30] 52] 21] for surveys) Likewise, column generation modules form the optimization cores of several commercial duty scheduling systems, among them (in alphabetical order) the GENCOL optimizer of the HASTUS CREW OPT system (see [25] 31] 22] ....

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Ceria S., Nobili P., Sassano A. (1995) A Lagrangian-Based Heuristic for LargeScale Set Covering Problems. Working Paper, University of Roma La Sapienza

....costs, the optimal solution to the relaxed problem and so on. Although there are many early references on such approaches (e.g. see the 26 survey [15] by Beasley) more improvement is still expected, as can be understood by the outstanding results reported recently for the set covering problem [22, 23]. This type of approaches are mainly based on one of the following three ideas. ffl Reduce the search space by xing some variables by using the information of relative costs. ffl Generate feasible solutions of the original problem by modifying optimal (or su ciently good) solutions to the ....

S. Ceria, P. Nobili and A. Sassano, \A Lagrangian-based heuristic for large-scale set covering problems," Mathematical Programming, vol.81, pp.215-228, 1998.

....19 are unicost SCP. The algorithm is coded in C and the computations are done on a 400MHz Pentium II PC. The details of the OR library problems are given in Table C1 and our results can be found in Tables C2, C3 and C4 in Appendix C. The optimal or currently best known values are taken from refs. [1, 2, 5, 8, 3, 12]. Table 2: Mean relative deviation from the optimal (or best known) solution and normalized CPU times for the mean field method (MF) and other approaches. The results for the genetic algorithm (GA) are taken from [2] R Gr and Alt Gr are greedy heuristics algorithms from [5] BB is a branch and ....

....CPU times for the mean field method (MF) and other approaches. The results for the genetic algorithm (GA) are taken from [2] R Gr and Alt Gr are greedy heuristics algorithms from [5] BB is a branch and bound algorithm and LH is a Lagrangian based heuristic, with results taken from [12] and [3], respectively. y) The CPU times reported for these problems was 0.0 and a comparison is therefore not possible. Problem Rel. deviation ( Normalized) CPU Time set MF GA R Gr Alt Gr BB LH MF GA R Gr Alt Gr BB LH 4 2.1 0 1 11 5 2.7 0.09 1 3.6 6 3.5 0 1 ....

[Article contains additional citation context not shown here]

S. Ceria, P. Nobili and A. Sassano, "A Lagrangian-Based Heuristics for Large Scale Set-Covering Problems", Mathematical Programming 81 (1998) 215-228.

....heuristics for large scale SCP s act on a subset of the columns, called core, which is selected before the execution of the algorithm. In the static approach the core remains the same during the execution (cf. 6, 11] while in the dynamic approach it is updated using an adaptive mechanism (e.g. [7, 10, 9]) In this paper we propose a novel heuristic algorithm for large scale SCPs arising from crew sheduling problems in airline companies. At each iteration a near optimal cover is constructed using the information provided by the previous iterations to guide the search. The nal solution is the ....

S. Ceria, P. Nobili, and A. Sassano. A Lagrangian-based heuristic for large-scale set covering problems. Mathematical Programming, 1995. to appear.

....branch bound, but can be viewed as a Lagrangian based heuristic, which can efficiently find solutions to most of these problems in a few hours to within a thousandth of the optimal solution. For other Lagrangian relaxation methods applied to pairing problems in railway industry see for instance [4, 6], which also address problems of similar size using specialized heuristics. Recent work in airline crew scheduling include [3, 8, 12 14, 17] Usually, the ILPs considered there are smaller than those considered in this paper. For our machine model we assume P processing elements (PEs) ....

S. Ceria, P. Nobili, and A. Sassano. A Lagrangian-based heuristic for large-scale set covering problems. Technical report, Dipartimento di Informatica e Sistemistica, Universit`a di Roma, La Sapienza, Italy, 1995.

.... in Balas and Ho [3] and Balas and Carrera [2] The most effective heuristic approaches to SCP are those based on Lagrangian relaxation following the seminal work by Balas and Ho [3] and then the improvements by Beasley [7] Fisher and Kedia [31] Balas and Carrera [2] Ceria, Nobili and Sassano [25], and Wedelin [53] We next outline a heuristic which has been designed to attack very large scale instances. The technique outperforms previously published methods: in 92 out of the 94 instances in the literature the method found, within short computing time, the optimal (or the best known) ....

S. Ceria, P. Nobili and A. Sassano, "A Lagrangian-Based Heuristic for Large-Scale Set Covering Problems", Technical Report R.406, IASI-CNR, Rome, 1995, to appear in Mathematical Programming.

....19 are unicost SCP. The algorithm is coded in C and the computations are done on a 400MHz Pentium II PC. The details of the OR library problems are given in Table C1 and our results can be found in Tables C2, C3 and C4 in Appendix C. The optimal or currently best known values are taken from refs. [1, 2, 4, 7, 3, 11]. For each of the test SCPs, 10 trials of our algorithm are performed. In Tables C2 C4, the best and the average costs are listed for each problem. For 10 of the problems our method found the optimal solution. The MF results typically are within a few percent of the optimal solutions, as can be ....

S. Ceria, P. Nobili and A. Sassano, "A Lagrangian-Based Heuristics for Large Scale Set-Covering Problems", Mathematical Programming B 81 (1998).

....[3] and Balas and Carrera [2] The most effective heuristic approaches to SCP are those based on Lagrangian relaxation. These approaches follow the seminal work by Balas and Ho [3] and then the improvements by Beasley [6] Fisher and Kedia [19] Balas and Carrera [2] Ceria, Nobili and Sassano [15], and Wedelin [30] Lorena and Lopes [26] propose an analogous approach based on surrogate relaxation. Recently, Beasley and Chu [7] and Jacobs and Brusco [24] proposed a genetic and a local search algorithm, respectively. We next briefly outline the heuristic method proposed by Caprara, Fischetti ....

S. Ceria, P. Nobili and A. Sassano, "A Lagrangian-Based Heuristic for Large-Scale Set Covering Problems", Technical Report R.406, IASI-CNR, Rome, 1995, to appear in Mathematical Programming.

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S. Ceria, P. Nobili, and A. Sassano, "A Lagrangian-Based Heuristic for Large-Scale Set Covering Problems", Technical Report R.406, 1995, IASI--CNR, Roma, to appear in Mathematical Programming.

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