K.L. Homan and M. Padberg. Solving airline crew scheduling problems by branch-and-cut. Management Science, 39(6):657-682, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and adequate control and flexibility to solve problems quickly and efficiently even if additional side constraints have to be met or additional subproblems have to be solved. In our experiments we use a subset of SPPs taken from Hoffman Padberg s problem suite of air crew scheduling problems [9]. To our knowledge, the only work on solving SPPs with constraint programming without using an ILP solver has been done by Carmen Gervet. She proposed an SPP solver using set constraints and employing a demanding formal apparatus. Her solver operates on sets of sets which complicates the ....

....But this turns out to be not powerful enough to solve larger instances of SPPs. To improve the situation, we explored two directions: Reducing the problem size of an SPP by performing preprocessing before solving it. We evaluated in detail different preprocessing steps from the OR literature [9] and propose a (to our knowledge) new preprocessing technique. This new technique allows for the considered problems a significantly improved problem size reduction with comparable computational effort with respect to standard preprocessing techniques (see Section 2) Improvement of the ....

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Karla L. Hoffman and Manfred Padberg. Solving airline crew scheduling problems by branchand -cut. Management Science, 39(6):657 -- 682, 1993.

....to be satis ed and the huge search space that has to be explored [6] The problem is often tackled by breaking it down into the crew pairing and the crew assignment subproblems, which are still hard problems. The crew pairing subproblem has been studied extensively and tackled with OR techniques [10, 1, 20], genetic algorithms [14] neural networks [4, 12] constraint programming [15] etc. Much work has been done also for the crew assignment problem, where pure OR methods have been applied [2, 16, 5] or hybrid methods that combine OR and constraint programming [8, 3, 17, 21, 7] In this paper, we ....

Homan K. L., Padberg M. Solving Airline Crew Scheduling Problems by Branch and Cut. Management Science, 39:657-682, 1993.

.... 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 address, the barrier code ....

.... rail problems with some tuning (and more time) In order to further determine the solution quality of the new code we have also run the active set code for more problems in the literature, such as all set covering problems from Beasleys OR library [42] and the NW problems of Hoffman and Padberg [34]. Most of these are quite small, so the scalability benefit of our algorithm is not important, but we obtain optimal or best known solutions for most problems with the same standard parameter settings as used above. 4 Parallelizing the original algorithm In Section 4.1 we first investigate how ....

K. L. Hoffman and M. Padberg. Solving airline crew scheduling problems by branchand -cut. Management Science, 39(6):657--682, 1993.

....S 1 , S n of S, each with a given cost c j = c(S j ) We wish to select the minimum weight subfamily of C that forms a partition of S. This problem is well studied and describes many important applications, including airline crew scheduling, vehicle routing, and political districting (see [41, 11, 51, 19]) To describe an integer programming formulation of the SPP, we construct matrix A, whose rows correspond to the members of S and whose columns correspond to the members of C. Entry a ij is 1 if the i th element of S is included in subset S j ; otherwise, we set a ij to zero. Then the problem ....

Ho#man, K., and Padberg, M.: Solving Airline Crew Scheduling Problems by Branch-and-cut. Management Science 39, 657, 1993

....in our experimentation we found that using rays that start at the LP optimum and varying only their directions is not enough for finding good feasible solutions: the origin of the rays must also be varied. Thus we were led to develop OCTANE as a tool to be used within a branch and cut framework [9, 18, 20], by running it from different nodes of the enumeration tree. Depending on the problem at hand, the goal might be to solve the problem to optimality (and use the heuristic to hopefully find an optimum solution earlier) or to find a relatively good solution relatively quickly. In the former case, ....

Hoffman, K. and Padberg, M. "Solving Airline Crew Scheduling Problems by Branch-and-Cut." Management Science, 39, 1993, 657-682.

....some characteristics of the polyhedral structure of the ALBP and applies that knowledge to resolve the WSP. Specifically, branch and cut has resulted in successful applications including those by Padberg and Rinaldi [17] for symmetric traveling salesman problems (TSP) and Hoffman and Padberg [9] for airline crew scheduling problems. In a related paper [20] we introduced families of valid inequalities for the ALBP and showed conditions under which they define facets for a certain relaxation of the ALBP. This paper, a continuation, presents a separation algorithm for each family of ....

....7 is the last cutpoint of 9 but E7 = s = 3, we proceed to 5 which is the last cutpoint of 7 and find that E5 = 2 = E9 59 = 3 1 = 2. Then, we consider task 1, the last cutpoint of 5, and obtain 9, 5, 1 as a series of tasks or endpoints, stations 3, 2, 1 as a series of stations, and intervals: [9, 5], 5, 1] For each interval n denoted by [t1, t2] in which t1 and t2 are endpoints of interval n, we denote n(t) as an attribute of task t, the amount of time available from station 2 t E 49 to 1 t E 50, 21 tt EE 51, after t has been assigned. We start by calculating n(1) 12 tt c EE ....

K. L. Hoffman, and M. Padberg, 1993. Solving Airline Crew Scheduling Problems by Branch-and-Cut. Mgmt. Sci. 39, 657-82.

.... problem and are meant to turn the theory into an algorithmic tool for the solution of practical problems (see for instance [RW87] FMW93] Moreover, the last decade has brought a wide range of interesting applications such as production planing problems ( RW87] airline scheduling problems ([HP93]) vehicle routing problems (see for instance [Po93] certain clustering and graph partitioning problems ( FMSWW94] or subproblems that arise within the design of electronic circuits or the design of mainframe computers ( We92] FGKKMW93] F93] in which the 0 1 knapsack problem is involved ....

K. L. Hoffman, M. Padberg, "Solving Airline Crew-Scheduling Problems by Branch-and-Cut", Working Paper, (1992).

....the relevant additional constraints are satisfied, and that the total involved costs are minimal. The set of feasible duties may be generated a priori, or it may be generated on the fly during the solution process. Set covering models have been popular in the airline industry for many years [12, 16]. However, in the railway industry the sizes of the crew scheduling instances are, in general, a magnitude larger than in the airline industry, which prohibited the application of these models in the railway industry until recently. But due to the increase in the computational power of nowadays ....

K.L. Hoffmann, and M.W. Padberg, "Solving airline crew scheduling problems by branch-and-cut," Management Science, 39 (1993) 657-682.

....SPP. These two problems are de ned as SPP: min c x; s.t. Ax = 1 (15) SCP: min c x; s.t. Ax 1 (16) where A is a 0 1 matrix, x a 0 1 vector, and c is the cost vector for the columns. Both problems are well studied and good algorithms solving (15) 16) can be found in the literature (see e.g. [9, 19]) A legal roster r for a crew member c is translated into a column of matrix A by inserting 1 s in the row i corresponding to the crew member and into all rows i that correspond to activities used in r. Especially in the beginning, however, it is unlikely and hard to guarantee that the columns ....

K.L. Homan and M. Padberg. Solving airline crew scheduling problems by branch-and-cut. Management Science, 39(6):657-682, 1993.

....Once these cuts are identified, they are typically embedded in a branch and cut enumeration scheme. In this methodology, a search tree is created, but unlike traditional branch and bound, cuts are added to the model at each node of the tree. Issues related to implementation are discussed in [G14, H5, K7, P1]. To date, there have been no applications of cutting plane techniques to bilevel programs. Integer Programming 127 3.4 BENDERS DECOMPOSITION FOR MIXED INTEGER LINEAR PROGRAMMING Enumeration methods discussed in Section 3.2 can be extended directly to solve mixed integer linear programs of ....

K.L. Hoffman and M. Padberg, "Solving Airline Crew Scheduling Problems by Branch-and-Cut," Management Science, Vol. 39, No. 6, pp. 657--682 (1993).

....full polyhedral description of these problems is unlikely. Branch and cut works likes branch and bound but tightens the bounds in every node of the tree by adding cuts. For a complete description of how such cuts are embedded into a tree search structure along with other tricks of the trade, see Ho#man and Padberg (1993). Because even small instances of the CAP1 may involve a huge number of columns (bids) the techniques described above need to be augmented with another method known as column generation. Introduced by Gilmore and Gomory (1961) it works by generating a column when needed rather than all at once. ....

....(1961) it works by generating a column when needed rather than all at once. An overview of such methods can be found in Barnhart et al. 1994) Later in this paper we illustrate how this idea could be implemented in an auction. One sign of how successful exact approaches are can be found in Ho#man and Padberg (1993). They report being able to find an optimal solution to an instance of SPA with 1,053,137 variables and 145 constraints in under 25 minutes. In auction terms this corresponds to a problem with 145 items and 1,053,137 bids. A major impetus behind the desire to solve large instances of SPA (and SPC) ....

K. Ho#man and M. W. Padberg, "Solving Airline Crew Scheduling Problems by Branch and Cut", Management Science, 39, 657--682, 1993.

....partitioning problem (SPP) is the zero one integer programming problem of the form: Minimise n X j=1 c j x j (1) Subject to n X j=1 a ij x j = 1, i = 1; m (2) x j 2 f0; 1g, j = 1; n (3) where a ij = 0 or 1. The best known application of the SPP is airline crew scheduling [1, 2, 4, 9, 12, 15]. In this formulation each row (i = 1; m) represents a flight leg (a takeoff and landing) that must be flown. The columns (j = 1; n) represent feasible round trip rotations for a crew (i.e. a sequence of flight legs for a crew that begin and end at individual base locations and ....

....an exact algorithm based on a new method, called column subtraction (or row sum) method, which is capable of solving large sparse instances of set covering, packing and partitioning problems. The most successful optimal solution algorithm so far appears to be the work of Hoffman and Padberg [12]. They presented an exact algorithm based on branch and cut (which involves solving the LP relaxation of the problem and incorporating cuts derived from polyhedral considerations) and reported optimal solutions for a large set of real world set partitioning problems. There have been relatively ....

[Article contains additional citation context not shown here]

K. Hoffman and M. Padberg. Solving airline crew-scheduling problems by branch-and-cut. Management Science, 39(6):657--682, 1993.

....x j 2 f0; 1g (j = 1; p) M) where A 2 R m Thetan , b 2 R m ; c 2 R n , and p n. 2 Basic Features of the Algorithm 2. 1 Preprocessing Problem preprocessing has been shown to be a very effective way of improving integer programming formulations prior to and during branch and bound [7, 10, 15, 19]. Rather than writing our own preprocessor, we have simply employed the CPLEX 3.0 3 preprocessor, invoking it not only once, but repeatedly until no further reductions result. In addition to applying standard linear programming (LP) reductions, also valid for integer programs, CPLEX applies ....

K. L. Hoffman and M. Padberg, "Solving airline crew-scheduling problems by branch-and-cut," Management Science 39 (1993) 657--682.

....was integral and where it was not, an integer solution could be found very quickly. Graves et al. 50] describe an interactive system which solves a sequence of subproblems by mathematical or local search techniques and can produce optimal solutions for small problems. Hoffman and Padberg [51] have developed a system which uses a set partitioning method and can run independently of any company. This system also produced many integer LP solutions and for the remaining problems an integer solution could be formed after applying a cutting plane method. Some papers have been published ....

K. L. Hoffman and M. Padberg. Solving Airline Crew Scheduling Problems by Branchand -Cut. Management Science, 39:657--682, 1993.

....relaxation for general mixed integer problems. In recent years valid inequalities from vertex packing relaxations have been shown to be valuable in deriving cutting planes for 0 1 integer programming, see for example Atamturk et al. 3] Borndorfer and Weismantel [5] and Ho#man and M. W. Padberg [9]. In 0 1 programming, a vertex 3 packing relaxation is obtained by considering pairwise conflicts between binary variables. We generalize this concept to mixed 0 1 integer programming by considering pairwise conflicts between continuous variables and binary variables as well. As far as we know ....

K. Ho#man and M. W. Padberg. Solving airline crew-scheduling problems by branchand -cut. Management Science, 39:667--682, 1993.

....n boxes of the same physical size but different weights into a minimal number of crates such that each crate can hold up to 3 boxes, and the total weight of each crate does not exceed 30 kg. We consider the example with 8 boxes weighing 12, 11, 10, 9, 7, 4, 3, 1 kg. 4. Set Partitioning Problem [8]: The problem is to collect a set M of subsets of N , M 2 N , such that they are pairwise disjoint, their union is N , and the sum of the weights of the subsets is minimal. We consider the example with 197 subsets over 17 elements [8] 5. Pigeon Hole Problem [2] The problem is to show that ....

....12, 11, 10, 9, 7, 4, 3, 1 kg. 4. Set Partitioning Problem [8] The problem is to collect a set M of subsets of N , M 2 N , such that they are pairwise disjoint, their union is N , and the sum of the weights of the subsets is minimal. We consider the example with 197 subsets over 17 elements [8]. 5. Pigeon Hole Problem [2] The problem is to show that it is not possible to put n 1 pigeons into n pigeon holes such that at most one pigeon is in each pigeon hole. We consider the example with 10 pigeons. These benchmarks are chosen for a number of reasons: Smith et al. 16] have ....

K. L. Hoffman, M. Padberg. Solving Airline Crew-Scheduling Problems by Branch-and-Cut. In Technical Report, George Mason University and New York University, USA, 1992.

....d pq , p; q = 1; 2; 3 with P i2Vp ;j2Vq d ij , p; q = 1; 2; 3. 4 The Capacity Formulation The basic solution approach we use is branch and cut to obtain the lower bound and the heuristic described below to obtain the upper bound. For details on the branch and cut approach, see [6] [7] or [14] The cuts used in our cutting plane procedure are the partition inequalities, total capacity inequalities, and rounded metric inequalities. 4.1 Initial Formulation To begin with, we first formulate the problem with only degree inequalities. These inequalities are special cases of the ....

Hoffman, K. and Padberg, M., "Solving Airline Crew Scheduling Problems by Branchand -cut", Management Science, 39(6), pp.657-682, (1993).

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K.L. Homan and M. Padberg. Solving airline crew scheduling problems by branch-and-cut. Management Science, 39(6):657-682, 1993.

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Ho#man K and Padberg M, Solving airline crew scheduling problems by branch and cut, Management Science, 39, 6, pp 657-682, 1993.

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K.L. Homan and M. Padberg. Solving airline crew scheduling problems by branch-and-cut. Management Science, 39(6):657-682, 1993.

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K.L. Hoffman and M. Padberg, "Solving airline crew scheduling problems by branch-and-cut", Management Science, 39, 657-682, (1993).

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K. L. Hoffman and M. Padberg. Solving airline crew scheduling problems by branch-andcut. Management Science, 39(6):657--682, 1993.

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K.L.Hoffman, M.Padberg. Solving Airline Crew Scheduling Problems by Branchand -Cut. Management Science, 39(6):657-682, 1993.

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Karla Hoffman and Manfred W. Padberg, `Solving airline crew scheduling problems by branch-and-cut', Management Science, 39, 657--682 (1993).

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Hoffman KL, Padberg M. Solving Airline Crew-Scheduling Problems by Branchand -Cut. Management Science 1993;39:657-682.

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