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 ....

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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).

....i j is used to initialize a tree clique with respect to G(A T ) that is iteratively extended greedily in the order of the z sorting of the rows until z i j becomes zero and the growing procedure stops. There is also a second variant for z cliques that is an adaptation of a similar routine by Hoffman and Padberg [1993]. Here we call the greedy heuristic once for each column of A and initialize the clique with the support of this column. Having detected a violated clique inequality in one of these ways, we lift randomly determined additional rows with zero z value sequentially into the inequality. This is done ....

Hoffman, K. L. and Padberg, M. W. (1993). Solving airline crew-scheduling problems by branch-and-cut.

....a subproblem referred to as the pricing problem. For a survey see Barnhart et al. 1996] One of the major application areas for these techniques has been the airline crew pairing or scheduling problem (see, for example Anbil et al. 1991] Barnhart et al. 1994] Desrosiers et al. 1991] Hoffman and Padberg [1993], Lavoie et al. 1988] Marsten et al. 1994] Minoux [1984] and Vance et al. 1995] A survey of work on the crew pairing problem can be found in Vance et al. 1995] The objective of the problem is to find a minimum cost assignment of flight crews to a given flight schedule. In particular, ....

....general classes. In the first class are algorithms where column generation is performed off line . That is, a set of pairings are enumerated up front and the crew pairing integer program (IP) is solved to optimality over this subset of pairings. An example of this type of approach can be found in Hoffman and Padberg [1993]. Because even moderate sized problems can have billions of pairings, these approaches must work on a very small subset. The second class of approaches use dynamic column generation to solve the linear programming (LP) relaxation of the set partitioning problem to optimality or near optimality. ....

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

.... of LP relaxations in the subproblems, but they can also be used for other purposes, e.g. for the separation of lift and project cutting planes of zero one optimization problems [BCC93a, BCC93b] and within heuristics for the determination of good feasible solutions in mixed integer programming [HP93] Therefore, we provide two basic interfaces for a linear program. The first one is in a very general form for linear programs defined by a constraint matrix stored in some sparse format. The second one is designed for the solution of the LP relaxations in the subproblem. The main differences to ....

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

....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) interconnected by some network. Each PE is a high performance RISC processor with at least two levels of cache and its own local memory. It will turn ....

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

....(pairings) that an airline crew might fly. Associated with each assignment of a crew to a particular flight leg is a cost, c j . The matrix elements a ij are defined by a ij = 1 if flight leg i is on rotation j 0 otherwise. 4) Airline crew scheduling is an economically significant problem [1, 3, 11, 16] and often a difficult one to solve. One approximate approach (as well as the starting point for most exact approaches) is to solve the linear programming (LP) relaxation of the SPP. A number of authors [3, 11, 21] have noted that for small SPP problems the solution to the LP relaxation either ....

.... among other problems, set partitioning [9] Desrosiers et al. developed an algorithm that uses a combination of Dantzig Wolfe decomposition with restricted column generation [8] Hoffman and Padberg report optimal solutions when they use branch and cut for a large set of real world SPP problems [16]. Several motivations for applying genetic algorithms to the set partitioning problem exist. First, since a GA works directly with integer solutions, there is no need to solve the LP relaxation. Second, genetic algorithms can provide flexibility in handling variations of the model such as ....

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K. Hoffman and M. Padberg. Solving Airline Crew-Scheduling Problems by Branch-andCut. Management Science, 39(6):657--682, 1993.

....and Sun s SunOS and Solaris. Bixby, Cook, Cox and Lee Parallel Mixed Integer Programming 3 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.

....bookkeeping and a much larger amount of memory. Thus a marriage of classical cutting planes and tree search is out of the question as far as the solution of large scale combinatorial optimization problems is concerned [16] This point is reemphasized in a recent paper of Hoffman and Padberg [12]: To share the set of generated constraints across different branches of the search tree is mathematically simply not correct when traditional cutting planes of integer programming, such as Gomory cuts or intersection cuts, are used. The main result of this paper is that Gomory cuts can be ....

....of integer programming, such as Gomory cuts or intersection cuts, are used. The main result of this paper is that Gomory cuts can be shared across different branches of the search tree for a mixed 0 1 program. Therefore, in the context of mixed 0 1 programming, the objections raised in [16] and [12] are moot. In the second part of the paper, we report on our computational experience with this approach. 2 Lifting Gomory s Mixed Integer Cuts In this section we prove our main result, namely a lifting theorem for Gomory cuts in the case of mixed 0 1 programs. A similar result for general (pure ....

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

....(this is often referred to as column or matrix generation) 2) With such a set as a starting point, the problem is then reformulated as finding the best subset of rotations such that each flight is covered precisely once. This transforms the problem into a set partitioning problem (see e.g. [13] and references therein) Solutions to this standard problem are then found by approximate methods based on e.g. linear programming; more recently an exact branch and cut method has been used [13] Even for moderate problem sizes, feasible rotations exist in astronomical numbers, and the pool ....

....is covered precisely once. This transforms the problem into a set partitioning problem (see e.g. 13] and references therein) Solutions to this standard problem are then found by approximate methods based on e.g. linear programming; more recently an exact branch and cut method has been used [13]. Even for moderate problem sizes, feasible rotations exist in astronomical numbers, and the pool has to be incomplete; this approach is therefore non exhaustive. Feedback ANN methods could be used to attack the resulting set partitioning problem. In fact, ANN methods have been successfully ....

K.L. Hoffman and M. Padberg, "Solving Airline Crew Scheduling Problems by Branch-andCut ", Management Science 39, 657 (1993).

....be the work of Hoffman and Padberg. They present an exact approach based on the use of branch and cut a branch and bound like scheme with additional preprocessing and constraint generation at each node in the search tree. They report optimal solutions for a large set of real world SPP problems [16]. 3 The Sequential Genetic Algorithm In this section we describe the sequential GA we used as the basis for the parallel genetic algorithm. The choice of algorithm, the selection of parameter settings, and the development of a local search heuristic to use with the sequential GA were the result ....

....be relatively small) A run was terminated either when the optimal solution was found z or when all subpopulations had performed 100,000 iterations. 5. 2 Test Problems To test the parallel genetic algorithm, we selected a subset of forty problems from the test set used by Hoffman and Padberg [16]. The test problems are given in Table 1, where they have been sorted according to increasing numbers of columns. The columns in this table are the test problem name, the number of rows and columns in the problem, the number of nonzeros in the A matrix, the optimal objective function value for ....

[Article contains additional citation context not shown here]

K. Hoffman and M. Padberg. Solving Airline Crew-Scheduling Problems by Branch-andCut. Management Science, 39(6):657--682, 1993.

....integer program is very small. Small problems often have integer solutions but may have a gap of up to a few per cent. Larger problems rarely have integer solutions to the continuous relaxation, but the gap is always extremely small. Of the 11 problems with more than 100 constraints considered in [16] the largest gap was 0.7 . In a study of over one hundred large problems from several European airlines, the gap was almost always less than 0.5 and for the largest problems (more than one million non zeros in the coefficient matrix) the typical gap was 0.1 [12] Branching on single variables is ....

....variables is generally very inefficient. 2] reports some success using the follow on branching strategy proposed by [21] Either leg number j follows after leg number i for a pairing in the solution or it does not. Using this fact may eliminate a large number of variables in both branches. [16] proposes a Branch and Cut algorithm for the Set Partitioning problem. First the LP relaxation is solved. If the optimal solution is fractional a violated valid inequality is generated if possible. Several classes of strong valid inequalities are used. Solutions are reported for a large number of ....

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

....a i x = 1, guarantees that the ith element is in exactly one set. The set partitioning problem has been extensively investigated because of its special structure and its numerous practical applications. The best known application is airline crew scheduling, see e.g. the recent reference [HP93]. Other applications include: truck scheduling; bus scheduling; facility location; circuit design and 85 capital investment. See e.g Garfinkel and Nemhauser [GN69] Marsten [MAR74] Balas and Padberg [BP76] Balas [BAL77] Nemhauser and Weber [NW79] Fisher and Kedia [FK90] Chan and Yano [CY92] ....

K. L. HOFFMAN and M. PADBERG. Solving airline crew-scheduling problems by branch-and-cut. Management Science, 6:657--682, 1993.

....j 0 otherwise. 1:6) y Numbers in square brackets refer to the numbered entries in the references. Airline crew scheduling is a very visible and economically significant problem. The operations research (OR) literature contains numerous references to the airline crew scheduling problem [2, 3, 4, 7, 25, 36, 46, 47]. Estimates of over a billion dollars a year for pilot and flight attendant expenses have been reported [1, 7] Even a small improvement over existing solutions can have a large economic benefit. At one time solutions to the SPP were generated manually. However, airline crew scheduling problems ....

....is the work of Eckstein [20] who has developed a general purpose mixed integer programming system for use on the CM 5 parallel computer and applied it to, among other problems, set partitioning. At the time of this writing the most successful approach appears to be the work of Hoffman and Padberg [36]. They present an exact approach based on the use of branch and cut a branch and bound like scheme where, however, additional preprocessing and constraint generation take place at each node in the search tree. An important component of their system is a high quality linear programming package ....

[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.

....by the company, which includes instances with sizes ranging from 50,000 variables and 500 constraints to 1,000,000 variables and 5,000 constraints. 1 Introduction The mathematical programming literature on the crew scheduling problem is vast in the context of the airline industry (see for example [1, 4, 7, 11, 12, 14]) but contains very few examples of problems from the railways. In this paper we present a Lagrangian based heuristic to solve largescale set covering problems arising from crew scheduling at the Italian railways. The main objective of the crew scheduling problem is to assign crews so that all ....

K.L. HOFFMAN and M. PADBERG, Solving airline crew-scheduling problems by branchand -cut, Management Science 39 657--682 (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 [8, 13, 15], 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 the optimum solution earlier) or to find relatively quickly a relatively good solution. In the former ....

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

....the set becomes empty. Finally, it should also be observed that many of the preprocessing and probing techniques can be implemented to run much faster for constraints with 0 1 coefficients only. Furthermore, pure 0 1 rows allow additional preprocessing techniques, such as row domination, see e.g. [1, 5, 7]. 5 Cut generation Any feasible solution to S defines a vertex packing in the conflict graph. Therefore, the vertex packing polytope associated with the conflict graph contains the convex hull of feasible solutions to S. Hence, valid inequalities for the vertex packing polytope, such as clique ....

....as identifying violated cliques. This approach does not require storage of a clique table and, since the conflict graphs are smaller, the identification of violated cliques is faster. However, the disadvantage is that we construct a partial conflict graph over and over again. Hoffman and Padberg [5] have used this approach successfully for set partitioning problems. However, for general integer programming problems, building the conflict graph on the fly is not practical, since small conflict graphs may not suffice and finding implications between pairs of variables is more complex and ....

K. Hoffman and M. Padberg. Solving airline crew-scheduling problems by branch-andcut. Management Science, 39:667--682, 1993.

.... scheduling area, scheduling of trucks, tanker routing, switching circuit design, assembly line balancing, capital equipment decisions, location of off shore drilling platforms, scheduling of ships, scheduling of airline fleets and airline crew scheduling (see Gershkoff [19] Hoffman and Padberg [21], Marsten et al. [27] Marsten and Shepardson [28] McCloskey and Hanssman [29] Detailed surveys on the SPP, including its applications, can be found in Balas and Padberg [6, 7] Christofides, et al. [13] Garfinkel [17] and in Garfinkel and Nemhauser [18] A popular application of the SPP is the ....

....selected in the optimal solution or not. CSPs are normally very large problems. For example, in excess of a billion feasible pairings could easily result for problems involving just 800 flight legs. Hence, the search space can be enormous. More details on CSPs can be found in Hoffman and Padberg [21]. 4. Search Space Reduction Techniques For problems with enormous solution space, the SA guided search for a good solution could be time consuming, due to the random nature of SA. Moreover, if the constraints are tight (as in the SPP) the feasibility restoration procedure, if used, could also 7 ....

[Article contains additional citation context not shown here]

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

....cutting plane technology. Motivated by work of Dantzig, Johnson and Fulkerson in the 50 s [DFJ54] Padberg, Groetschel and others have shown how cutting plane techniques could be used to strengthen the linear programming relaxations of many pure 0 1 integer programming problems [GH91, PR91, HP92] The strengthening is effected by studying the facets of the underlying polytope generated by the convex hull of 0 1 solutions. Knowledge of these facets leads to subroutines for recognizing inequalities violated by the current fractional solution. These violated inequalities can then be added ....

K. L. Hoffman and M. Padberg. Solving airline crew-scheduling problems by branchand -cut. 1992. preprint.

....a tree clique with respect to G(A T ) that is iteratively extended greedily in the order of the z sorting of the rows until z i j becomes zero and the growing procedure stops. There is also a second variant for z cliques that is an adaptation of a similar routine by Hoffman and Padberg [16]. Here we call the greedy heuristic once for each column of A and initialize the clique with the support of this column. Having detected a violated clique inequality in one of these ways, we lift randomly determined additional rows with zero z value sequentially into the inequality. This is done ....

K. L. Hoffman and M. W. Padberg, Solving airline crew-scheduling problems by branchand -cut, Mgmt. Sci., 39 (1993), pp. 657--682.

.... Grotschel, Junger, Reinelt (1988) for the max cut problem with applications to ground states in spin glasses and via minimization in VLSI design; Grotschel, Junger and Reinelt (1984) for the linear ordering problem with applications to triangulation of input output matrices and ranking in sports; Hoffman and Padberg (1992) for the set partitioning problem with applications to airline crew scheduling; Grotschel and Wakabayashi (1989) for the clique partitioning problem with applications to clustering in biology and the social sciences; Grotschel, Monma and Stoer (1992) for certain connectivity problems with ....

K. L. Hoffman and M. Padberg (1992): Solving airline crew-scheduling problems by branchand -cut, preprint, George Mason University.

....Crew Scheduling A survey of older work on crew scheduling can be found in Arabeyre et al. 1969) Etschmaier and Mathaisel (1985) provide a more recent, but by no means up to date, survey. Anbil et al. 1991a) 1991b) 1993) Barutt and Hull (1990) Desrosiers at al. 1991) Gershkoff (1989) Hoffman and Padberg (1993), and Lavoie et al. 1988) present more recent algorithms and practice. Most current approaches to crew scheduling center around the set partitioning problem min X p2P c p y p X p:i2p y p = 1 i 2 F (3) y p 2 f0; 1g p 2 P where y p = 1 if pairing p is in the solution, and 0 otherwise. F is ....

....all nonbasic columns in order to prove optimality. Other approaches have used constrained shortest path methods on specially structured networks to price out attractive pairings. See for example Minoux (1984) Lavoie, Minoux, and Odier (1988) Desrosiers et al. 1991) and Barnhart et al. 1994) Hoffman and Padberg (1993) found optimal integer solutions to problems with a maximum of 300,000 pairings using a branch and cut algorithm. In their approach, crew base constraints were explicitly considered. Barutt and Hull (1990) used parallel computing to enumerate pairings efficiently. 2 A Duty Period Based ....

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

....partitioning problem The set partitioning problem is a zero one optimization problems with side constraints of the form Ax = 1, where all coefficients of the matrix A are 0 or 1 and the right hand side 1 is a vector of ones. This problem has applications in airline crew scheduling. Reference: Hoffman and Padberg (1993) Steiner tree problem Given a connected graph G = V; E) T V , c 2 R E , find F E with c(F ) minimum such that F induces a tree in G and T S e2F e. This is the network survivability problem with d v 2 f0; 1g. An application is the design of telephone networks. Reference: Chopra, ....

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

.... x 3 w 2 1 x 1 Gamma x 2 w 3 0 x i = 0 or 1; i = 1; 3: w i = 0 or 1; i = 1; 3 These added variables are referred to as the weighted variables. The other variables are the unweighted variables. 2 Description of Algorithm 2. 1 Overview The basic approach is branch and cut [10, 11, 21, 29]. At each node of the tree we solve the linear programming (LP) relaxation obtained by replacing the integrality requirements by the simple bounds: 0 x i 1; i = 1; n; 0 w i 1; i = 1; m If the solution to the LP is integral, we compare it to the best integral solution so far. ....

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

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

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

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