D. Applegate and W. Cook. A computational study of the job-shop scheduling problem, ORSA, Journal on Computing, 3:149156, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that the resulting algorithm is optimal in the sense that choosing to search a symbolic cost state with non minimal minimum cost can never reduce the number of symbolic cost states explored. The third contribution of this paper is a number of techniques inspired by branch and bound algorithms [AC91] that have been adopted in making the algorithm even more useful. These techniques are particularly useful for limiting the search space and for quickly nding solutions near to the minimum cost of reaching a goal state. To support this claim, we have implemented the algorithm in an experimental ....

....limited capacity, also limited to one in most cases. The purpose is to allocate starting times to the operations, such that the overall duration of the schedule, the makespan, is minimal. Many solution methods such as local search algorithms like simulated annealing [AvLLU94] shifting bottleneck [AC91] branch andbound [AC91] or even hybrid methods have been proposed [JM99] We apply Uppaal to 25 of the smaller Lawrence Job Shop problems. Our models are based on the timed automata models in [Feh99a] In order to estimate the lower bound on the remaining cost, we calculate for each job and ....

[Article contains additional citation context not shown here]

D. Applegate and W. Cook, A Computational Study of the Job-Shop Scheduling Problem, OSRA Journal on Computing 3 (1991), 149-156.

....di erent solution methods. But there also exist many developments that favor one of the methods and just use the other one to overcome certain weaknesses. Constrained local search, for example, uses LS as the predominant approach, with CP used to nd neighbors in a sparse and or large neighborhood [1, 10]. Focussing on constraint programming instead has yielded hybrids that use local search to adapt the variable and or value ordering in the search tree. For a more complete overview on the eld we refer to the recent tutorial by Focacci et al. 6] As with other methods, constraint programming ....

D. Applegate and W. Cook. A computational study of the job-shop scheduling problem, ORSA, Journal on Computing, 3:149156, 1991.

....) d(S) Hence, T must be last and the start time can be narrowed correspondingly, i.e. T s(S ) d(S ) Analogous rules hold for the detection that a task must be first. Several approaches differ in the amount of further propagation and the selection of the task sets S to consider [1, 2, 3, 6, 8]. We integrated a kind of edge finding in a propagator, which bases on an algorithm suggested in [8] for proving lower bounds of job shop problems. The algorithm (quadratic complexity) computes so called ascending sets of tasks. For tasks not in these sets, edge finding rules are applied. By ....

.... form of edge finding) in combination with a propagator comprising only the propagation of the reified approach above, was sufficient to solve hard scheduling benchmarks (see Section 4) To find the optimal solution quickly, we used repair and shuffle techniques for local optimization (see e.g. [1]) while the edge finding propagators are also stated for propagation. For this it was very convenient and inevitable to use programmable search in Oz and high level abstractions. Furthermore, we implemented a propagator for multi capacitated scheduling, where the resources may have a capacity ....

[Article contains additional citation context not shown here]

D. Applegate and W. Cook. A computational study of the job-shop scheduling problem. Operations Research Society of America, Journal on Computing, 3(2):149--156, 1991.

....approach (see [39, 7] allows the user to program some new constraints. But it has no support to apply more sophisticated algorithmic techniques to implement new constraints (see also Section 8 and [28] as, for example, required for global scheduling constraints employing edge finding (refer to [2, 1]) On the other hand, combinatorial problems can be tackled in a language like C together with a dedicated library for constraint solving (see for example, Ilog [17] Although many programming abstractions are provided through C classes, it is hard for a C library to provide an adequate ....

....application of method impose( keeps the propagator suspending on its parameters and introduces the propagator to the emulator. OZ C proc begin(lesseq, 2) OZ Expect pe; OZ EXPECT(pe, 0, expectIntVarBounds) OZ EXPECT(pe, 1, expectIntVarBounds) return pe.impose(new LessEq(OZ args[0] OZ args[1]) OZ C proc end 11 5 Additional expressiveness of the CPI This section explains the extended expressiveness of the CPI, which is desired to implement advanced propagators for demanding applications. All the discussed extensions are supported by adequate CPI abstractions that fit smoothly ....

[Article contains additional citation context not shown here]

D. Applegate and W. Cook. A computational study of the job-shop scheduling problem. Operations Research Society of America, Journal on Computing, 3(2):149--156, 1991.

....with the 1 and 2 parameter. The application of method impose( makes the propagator suspending on its parameters and introduces the propagator to the emulator. OZ C proc begin(lesseq, 2) OZ Expect pe; OZ EXPECT(pe, 0, expectIntVarBounds) return pe.impose(new LessEq(OZ args[0] OZ args[1]) OZ C proc end 5 Additional Expressiveness of the CPI This section explains the extended expressiveness of the CPI which is desired to implement advanced propagators for demanding applications. All the discussed extensions are supported by adequate CPI abstractions which fit smoothly in ....

....algorithms used for the supplied library constraints of ILOG SOLVER resp. Oz is beyond the scope of this paper. Benchmarking job shop problems. The following benchmarks compare Oz 2.0.3 with ILOG SCHEDULER 2. 2 for classical 10x10 job shop scheduling benchmarks for the proof of optimality [1]. In both systems we used the best strategy available in the corresponding libraries. In Table 2, the entry Fails denotes the number of failure nodes in the search tree needed for proving optimality. The entry CPU denotes the run time needed for proving optimality. The last two columns compare ....

D. Applegate and W. Cook. A computational study of the job-shop scheduling problem. Operations Research Society of America, Journal on Computing, 3(2):149--156, 1991.

....possible to create new propagators efficiently. This serialization propagator does not employ any domainspecific information other than that of a linear time scale. Global constraints, however, may also employ more elaborate propagation techniques for domain specific purposes. Applegate and Cook [1] proposed such a technique, called edge finding, see also [3] Variants of this technique enjoy a run time which essentially grows quadratically in the number of items to be serialized. In computational terms, even these more efficient variants are yet more expensive than the global constraint ....

APPLEGATE, D., AND COOK, W. A computational study of the job-shop scheduling problem. Operations Research Society of America, Journal on Computing 3, 2 (1991), 149-- 156.

....s(S T cannot be scheduled after all tasks in S . Hence, T must be first and corresponding propagators can be imposed, narrowing the start times. Analogously, if ) d(S) c(S ) Gamma s(T ) d(S) T must be last. For this kind of reasoning, the term edgefinding was coined in [2]. There are several variations of this idea in [5, 2, 6, 16] for the OR community and in [19, 7] for the constraint community; they differ in the amount of propagation and which sets S are considered for edgefinding (in principle, there are exponentially many) The resulting propagators do a lot ....

....in S . Hence, T must be first and corresponding propagators can be imposed, narrowing the start times. Analogously, if ) d(S) c(S ) Gamma s(T ) d(S) T must be last. For this kind of reasoning, the term edgefinding was coined in [2] There are several variations of this idea in [5, 2, 6, 16] for the OR community and in [19, 7] for the constraint community; they differ in the amount of propagation and which sets S are considered for edgefinding (in principle, there are exponentially many) The resulting propagators do a lot of propagation, but are also more expensive than e.g. reified ....

[Article contains additional citation context not shown here]

D. Applegate and W. Cook. A computational study of the job-shop scheduling problem. Operations Research Society of America, Journal on Computing, 3(2):149--156, 1991.

....a local search can be done at some choice points, in order to find rapidly good solutions in the corresponding sub space, and to quickly improve the upper bound (in case of a minimization task) As a special case of this technique, a local search at the root node is often used. Shuffling [Applegate Cook 91] in the field of disjunctive scheduling, is an often cited technique: a starting complete schedule is partially broken, then rebuilt by mean of a complete algorithm (known as edge finding) and the process is iterated. Pesant Gendreau 99] exploit the same schema in the context of Constraint ....

D. Applegate, W. Cook. A computational study of the job-shop scheduling problem. ORSA Journal On Computing, 3(2):149--156, 1991.

....has activated this field. They can be regarded as problem independent approaches in the sense that they are not for specific optimization problems; but rather, they give general frameworks to design optimization problems [15] Other heuristics include GRASP [2] the shifting bottleneck approach [11,24], and local search [18 19,25] A comprehensive survey of job shop scheduling techniques has been done by Jain and Meeran in [26] In this paper, we develop a genetic algorithm approach that employs a local search technique to treat the job shop scheduling problem. In previous work [9] we ....

D. Applegate and W.Cook," A Computational study of job shop scheduling problem", RSA Journal of Computing, 3:149-156, 1991.

....Kan [14] proved this problem to be NP Hard. One classic 10x10 job shop problem formulated by Muth and Thompson in 1963 was not solved until 1989 by Carlier and Pinson [6] Recently however, heuristics have been very successful in solving many of the early benchmark problems. Several authors [1,2,7,8,11,18,19,22] established additional benchmark problems as the heuristic approaches improved. A listing of 159 of these benchmark problems is located on J. Beasley s OR Library website (http: mscmga.ms.ic.ac.uk info.html) Authors commonly use a subset of these 159 benchmark problems for testing their job ....

Applegate D., W. Cook, (1991). "A computational study of the job shop scheduling problem," ORSA Journal on Computing, 3, 149-156.

....optimization problems they can reach a far better quality in a given time frame. But, local search algorithms cannot guarantee that they find a solution, and may be unable to find one. And thus, they are not the panacea. Several works have studied cooperation between local and systematic search [2,13,17, 36,43,44,48,54].Those hybrid approaches have led to good results on large scale problems. Three categories of hybrid approaches can be found in the literature: performing a local search before or after a systematic search; performing a systematic search improved with a local search at some point of the ....

D. Applegate, W. Cook, A computational study of the job-shop scheduling problem, ORSA J. Comput. 3 (2) (1991) 149--156.

....needs approximately 500,000 binary variables and 50,000 constraints. With the horizon of problems considered, it was not possible to solve problems having more than 6 jobs and 6 resources with this approach with CPLEX 7.2 MIP optimizer. The second formulation is a disjunctive formulation [1]. For each activity a ji , we introduce a continuous variable S ji representing its start time and we have the constraint: s min (a ji ) S ji s max (a ji ) A precedence constraint a ji a jk is obviously modeled by the constraint: S ji pt(a ji ) S jk A resource constraint over a set of ....

....instances. Using the job shop problem generator of Watson et al. 20] we generated 10 job shop problems in each of three sizes: 10 10, 15 10, and 20 10. Each problem instance has a work ow structure and has the duration of each activity drawn with uniform probability from the interval [1, 99]. The work ow structure means that the resources in the problem are partitioned into two sets of approximately equal size. The activities in each job, then, must use each of the resources in the rst partition before using any of the resources in the second. A randomly generated set of work ow job ....

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D. Applegate and W. Cook. A computational study of the job-shop scheduling problem. ORSA Journal on Computing, 3:149-156, 1991.

....feasible schedules [36] Since, each set of feasible permutations has a corresponding schedule, the objective of the JSP is to find, among the feasible permutations, the one with the smallest makespan. The JSP is NP hard [26] and has also proven to be computationally challenging. Exact methods [4, 7, 9, 10, 19] have been successful in solving small instances, including the notorious 10 10 instance of Fisher and Thompson [16] proposed in 1963 and only solved twenty years later. Problems of dimension 15 15 are still considered to be beyond the reach of today s exact methods. For such problems there ....

....of dimension 15 15 are still considered to be beyond the reach of today s exact methods. For such problems there is a need for good heuristics. Surveys of heuristic methods for the JSP are given in [30, 37] These include dispatching rules reviewed in [18] the shifting bottleneck approach [1, 4], local search [27, 28, 37] simulated annealing [27, 38] tabu search [28,29,36] and genetic algorithms [12] Recently, Binato et al. 6] described a greedy randomized adaptive search procedure (GRASP) for the JSP. A comprehensive survey of job shop scheduling techniques can be found in Jain and ....

D. Applegate and W.Cook. A computational study of the job-shop scheduling problem. ORSA Journal on Computing, 3:149--156, 1991.

.... (It is interesting to note that the instance of Muth and Thompson was one of the easier instances to solve using their technique) Slightly larger instances, however, are still currently intractable; they report instances of size 10 x 15, 15 x 20, 15 x 15 and 10 x 20 that they were unable to solve [4]. Formal Definition and Previous Results We formally define the job shop problem as follows. We are given a set l = m, m2, of machines, a set J = J, Jr of jobs, and a set 0 = Oili = 1, i,j = 1, n of operations, where i indexes the machine on which operation O i runs. Thus m ....

D. Applegate and B. Cook. A computational study of the job-shop scheduling problem. ORSA Journal of Computing, 3:149 156, 1991.

....a strategy for choosing between several states with the same cost. Our experiments show that the number of states searched can change drastically, depending on the order in which states with the same cost are searched. We also present a number of methods inspired by branch and bound algorithms [21] for reducing the part of the state space which needs to be searched. Since the order in which states are searched can drastically influence the number of states searched, we have added the possibility for the user to guide the search. This can be done by adding priorities to the model which are ....

....in the sense that choosing to search a symbolic cost state with non minimal minimum cost can never reduce the number of symbolic cost states explored. Formally C # C i# #u. C # (u) The third contribution of this paper is a number of techniques inspired by branch and bound algorithms [21] that have been adopted in making the algorithm even more useful. These techniques are particularly useful for limiting the search space and for quickly finding solutions near to the minimum cost of reaching a goal state. To support this claim, we have implemented the algorithm in an experimental ....

[Article contains additional citation context not shown here]

D. Applegate and W. Cook. A Computational Study of the Job-Shop Scheduling Problem. OSRA Journal on Computing 3, pages 149--156, 1991.

....instance to be acyclic if no job has two or more operations that need to run on any given machine. A single instance of acyclic job shop scheduling with 10 jobs, 10 machines and 100 operations resisted attempts at exact solution for 22 years until its resolution [17] See also Applegate Cook [6]. We will show here that good approximation algorithms do exist for job shop scheduling. There are two natural lower bounds on the makespan of any job shop instance: P max , the maximum total processing time needed for any job, and Pi max , the maximum total amount of time for which any machine ....

D. Applegate and W. Cook. A computational study of the job-shop scheduling problem. ORSA Journal of Computing, 3:149--156, 1991.

....of the search effort. Such is not the case for other problems. For example, the use of the edge finding mechanism enables the resolution of hard job shop scheduling problem instances such as those used by Applegate and Cook in their computational study of the job shop scheduling problem (cf. Applegate and Cook 91] The following table shows the results obtained on ten hard instances of the job shop scheduling problem. CPU times are expressed in seconds on a RS6000 workstation. The algorithm used to solve the problem is described in [Bapstiste et al. 95] The total number of backtracks over the ten ....

....It is, on the other hand, more effective in pruning the search space. Its integration in SCHEDULE allows users to enjoy the flexibility inherent to constraint programming, with performance in the same range of efficiency as specific operations research algorithms, such as the one reported in [Applegate and Cook 91] SCHEDULE is based on SOLVER, an extensible tool for constraint programming. This is quite important from an industrial point of view: users can extend the scheduling model and define solution search strategies with respect to the requirements of a specific application. The fact that SOLVER and ....

David Applegate and William Cook. A Computational Study of the Job-Shop Scheduling Problem. ORSA Journal on Computing, 3(2):149-156, 1991.

....Most of the successful exact approaches to minimize makespan within the disjunctive scheduling class rely on an Ph. Baptiste, C. Le Pape, W. Nuijten Adjustments for Cumulative Scheduling Problems 2 extensive use of satisfiability tests and of time bound adjustments for the One Machine Problem ([2, 3, 6, 9, 12, 24, 25, 29]) The satisfiability tests often consist in verifying that there exists a preemptive schedule. The time bound adjustments allow to tighten the release dates and the deadlines of the activities. The aim of this paper is to present counterparts of such techniques for the cumulative class. We ....

....activities A i of W PE (A i , t 1 , t 2 ) Note that for t 1 = t 2 , W PE (t 1 , t 2 ) is defined and, under the assumption r i p i d i , is equal to 0. Figure 2: The required energy consumption of an activity (release date 0, deadline 10, processing time 7 and resource requirement 2) over [2 7]. The activity must execute during at least 2 time units in [2 7] which corresponds to W PE (A, 2, 7) 2 (7 (2 0) 10 7) 4 Proposition 3. There is a feasible partially elastic schedule of I if and only if for any non empty interval [t 1 t 2 ] W PE (t 1 , t 2 ) C (t 2 t 1 ....

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D. Applegate and W. Cook, A Computational Study of the Job-Shop Scheduling Problem, ORSA Journal on Computing 3 (1991) 149-156.

....with style and technical points and no attempt was made to update the references in the body of the text. Instead, a retrospective section is included after the conclusion to assess the impact of the paper and to relate to some current and future research. 4 sizeMaster(112) sizeSquares([50,42,37,35,33,29,27,25,24,19,18,17,16,15,11,9,8,7,6,4,2]) FIGURE 2.1. The Data for the Perfect Square Problem 2. A Motivating Example To illustrate several features of cc(FD) we present a program to solve the so called perfect square problem. The purpose of the program is to build a square, called the master square, out of a number of given ....

....of using constraint languages: it is easier to come up with a better design for many problems. Of course, implementing this design in C will lead to better performance. Table 5. 3 compares cc(FD) with a specialized scheduling algorithm [5] The algorithm is not state of the art (see for instance [6, 2]) but the comparison is still signi cant because, on the one hand, the techniques in [2, 6] are very speci c and do not scale easily to other scheduling problems and, on the other hand, cc(FD) has not been designed with scheduling applications in mind at this stage. The applications are very ....

[Article contains additional citation context not shown here]

D. Applegate and W. Cook. A Computational Study of the Job Shop Scheduling Problem. ORSA J. of Comp., 3(2):149-156, 1991.

....reachability, viz. as the (im)possibility to reach a state that improves on a given optimality criterion. Such criteria distinguish scheduling algorithms from classical, full state space exploration model checking algorithms. They are used together with, for example, branch and bound techniques [AC91] to prune parts of the search tree that are guaranteed not to contain optimal solutions. This observation motivates research into the extension of model checking algorithms with optimality criteria. They provide a basis for the (cost ) guided exploration of state spaces, and improve the potential ....

D. Applegate and W. Cook. A Computational Study of the Job-Shop Scheduling Problem. OSRA Journal on Computing 3, pages 149-156, 1991.

....that the resulting algorithm is optimal in the sense that choosing to search a symbolic cost state with non minimal minimum cost can never reduce the number of symbolic cost states explored. The third contribution of this paper is a number of techniques inspired by branch and bound algorithms [AC91] that have been adopted in making the 2 Formally C 0 v C i 8u: C 0 (u) C(u) algorithm even more useful. These techniques are particularly useful for limiting the search space and for quickly nding solutions near to the minimum cost of reaching a goal state. To support this claim, we ....

....problems (la16 to la25) we did not nd optimal solutions though in some cases we were very close to the optimal solution. Since branch and bound algorithms generally do not scale too well when the number of machines and jobs increase, this is not surprising. The branch and bound algorithm for [AC91] who solves about 10 out the 15 problems in the same setting, faces the same problem. Note that the results of this algorithm depend sensitively on the choice of an upper bound. Also the algorithm used in [BJS95] who combines a good heuristic with an ecient branch and bound algorithm and thus ....

D. Applegate and W. Cook. A Computational Study of the Job-Shop Scheduling Problem. OSRA Journal on Computing 3, pages 149-156, 1991.

....machine scheduling problems, which require sequencing a set of jobs on a single machine with release times and 10 deadlines. Balas [8] introduced inequalities for this class of problems, that can be incorporated in the standard ATSP TW models in which node variables are used. Applegate and Cook [2] performed computational experiments with several of these classes for the job shop scheduling problem in which however a different objective function (minimization of the makespan) is considered. We refer to Queyranne and Schulz [31] for a comprehensive survey on polyhedral approaches to machine ....

D. Applegate and W. Cook. A computational study of the job--shop scheduling problem. ORSA Journal on Computing, 3:149--156, 1991.

....reachability, viz. as the (im)possibility to reach a state that improves on a given optimality criterion. Such criteria distinguish scheduling algorithms from classical, full state space exploration model checking algorithms. They are used together with, for example, branch and bound techniques [AC91] to prune parts of the search tree that are guaranteed not to contain optimal solutions. This observation motivates research into the extension of model checking algorithms with optimality criteria. They provide a basis for the guided exploration of state spaces, and improve the potential of ....

D. Applegate and W. Cook. A Computational Study of the Job-Shop Scheduling Problem. OSRA Journal on Computing 3, pages 149-156, 1991.

....indicate the order in which operations are processed on each machine. Once these variables are set, the time at which each operation starts processing can be easily computed. Algorithms of this type have been proposed by Carlier Pinson [7] Brucker, Jurisch Sievers [6] and Applegate Cook [2]. Although solving job shop scheduling problems to optimality is difficult, recently there has been progress developing heuristics that find good schedules. Adams, Balas Zawack [1] proposed the shifting bottleneck procedure, which uses a primitive form of iterated local search to produce ....

....(or tail) to the earliest time that the operation can start such that it processes continuously and such that there is a preemptive schedule for the remaining operations on that machine. Fisher, Lageweg, Lenstra Rinnooy Kan [11] examine surrogate duality while Balas [3] and Applegate Cook [2] attempt to find bounds by examining the polyhedral structure of the problem. Neither approach has proven to be practical so far. The strongest computational results of Applegate Cook [2] use only combinatorial bounds in their branch and bound algorithm. In this paper we present new lower ....

[Article contains additional citation context not shown here]

D. Applegate and W. Cook. A computational study of the job-shop scheduling problem. ORSA J. Comput., 3:149--156, 1991.

....that the resulting algorithm is optimal in the sense that choosing to search a symbolic cost state with non minimal minimum cost can never reduce the number of symbolic cost states explored. The third contribution of this paper is a number of techniques inspired by branch and bound algorithms [AC91] that have been adopted in making the algorithm even more useful. These techniques are particularly useful for limiting the search space and for quickly nding solutions near to the minimum cost of reaching a goal state. To support this claim, we have implemented the algorithm 2 Formally C 0 v ....

....limited capacity, also limited to one in most cases. The purpose is to allocate starting times to the operations, such that the overall duration of the schedule, the makespan, is minimal. Many solution methods such as local search algorithms like simulated annealing [AvLLU94] shifting bottleneck [AC91] branch andbound [AC91] or even hybrid methods have been proposed [JM99] In this section, we apply Uppaal to 25 of the smaller Lawrence Job Shop problems. 6 Our models are based on the timed automata models in [Feh99a] In order to estimate the lower bound on the remaining cost, we calculate ....

[Article contains additional citation context not shown here]

D. Applegate and W. Cook. A Computational Study of the Job-Shop Scheduling Problem. OSRA Journal on Computing 3, pages 149-156, 1991.

....solving the moderate sized classical problem with 10 jobs to be processed on 10 machines (c.f. Muth and Thompson, 1963] optimally took over 25 years in development (c.f. Carlier and Pinson, 1989] Due to the fact that even the best known exact algorithms (c.f. Brucker et al. 1994] or ([Applegate and Cook, 1991]) are not able to solve problems with 15 jobs and 15 machines in acceptable time, a long list of heuristics were developed in the last 25 years. In 1988 ( Balas et al. 1988] published a very promising concept with the shifting bottleneck procedure. The best currently known algorithms are the ....

Applegate, D. and Cook, W. (1991). A computational study of the job--shop scheduling problem. ORSA Journal on Computing, 3:149--156.

....the last 30 years. The most famous instance is a 10 10 problem of Fisher Thompson [MT63] that was left unsolved until 1989 when it was solved by Carlier Pinson [CP89] Classical benchmarks include problems randomly generated by Adams, Balas Zawak in 1988 [ABZ88] Applegate Cook in 1990 [AC91] and by Lawrence in 1984 [La84] Of the 40 problems published by Lawrence, one is still unsolved (a 20 10 referred to as LA29) The typical size of these benchmarks ranges from 10 5 to 30 10. 2.2 The branch and bound scheme with time windows Branch and bound algorithms have, however, ....

....tree (which corresponds to getting rid of a disjunction in the constraint formulation) There are many variations depending on which pair to pick, how to exploit the disjunctive constraint before the pair is actually ordered, etc. but the general strategy is almost always to order pairs of tasks [AC91] The domain associated with time(t i ) is represented as an interval : to each task t i , a window t i , t i d(t i ) is associated, where t i is the minimal starting date and t i is the maximal completion date (thus the starting date time(t i ) must be between t i and 5 t i d(t i ) ....

[Article contains additional citation context not shown here]

D. Applegate & B. Cook. A Computational Study of the Job Shop Scheduling Problem. Operations Research Society of America vol 3, no 2, 1991

....user can specify the problem and generate a situation specific scheduler. One of the benefits of a transformational approach to scheduling is the synthesis of specialized constraint management code. Previous systems for performing scheduling in AI (e.g. 13, 12, 48, 47] and Operations Research [2, 24] use constraint representations and operations that are geared for a broad class of problems, such as constraint satisfaction problems or linear programs. In contrast, transformational techniques can derive specialized representations for constraints and related data, and also derive efficient ....

Applegate, D., and Cook, W. A computational study of the job-shop scheduling problem. ORSA Journal on Computing 3, 2 (Spring 1991 1991), 149--156.

....constraints are often crucial for solving dicult problems. If such constraints have many parameters, they are often called global constraints. Resource constraints in scheduling are typically modeled using global constraints. For solving hard scheduling problems, a technique called edge nding [AC91] is used, which has been integrated in the constraint programming framework in several variants [CL94a, CL96a, CL97, CL94b, BPN95, W ur96] The basic idea of edge nding is to check whether a certain task t is to be placed before or after a set of other tasks T . In case this can be determined, ....

D. Applegate and W. Cook. A computational study of the job-shop scheduling problem. ORSA Journal on Computing, 3(2):149-156, 1991.

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D. Applegate and W. Cook. A computational study of the job-shop scheduling problem, ORSA, Journal on Computing, 3:149156, 1991.

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Applegate, D., & Cook, W. (1991). A computational study of the job-shop scheduling problem. ORSA Journal on Computing, 3 (2), 149--156.

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Applegate, D., and Cook., W. 1991. A computational study of the job-shop scheduling problem. ORSA Journal on Computing 3(2):149--156.

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D. Applegate and W. Cook. A computational study of the job shop scheduling problem. ORSA Journal on Computing, 3:149--156, 1991.

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Applegate D., Cook W., 1991. A computational study of the job-shop scheduling problem. ORSA Journal on Computing, 3(2), 149-156.

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D. Applegate, and W. Cook, A Computational Study of the Job-Shop Scheduling Problem, ORSA 149-156, Journal on Computing, Spring, 3(2), 1991.

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David Applegate and William Cook. A computational study of the job-shop scheduling problem. OSRA Journal on Computing, 3:149--156, 1991.

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D. Applegate and W. Cook. A Computational Study of the Job Shop Scheduling Problem. ORSA J. of Comp., 3(2):149-156, 1991.

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Applegate, D. and Cook, W. (1991). A Computational Study of the Job-Shop Scheduling Problem, ORSA Journal on Computing, 3(2), 149-156.

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Applegate D. and W. Cook (1991), "A computational study of the job-shop scheduling problem", ORSA Journal on Computing 3, 149--156.

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D. Applegate, B. Cook. A computational study of the job shop scheduling problem. Operations Research Society of America, 3(2), 1991.

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D. Applegate and W. Cook, "A computational study of the job-shop scheduling problem," ORSA J. Comput., vol. 3, no. 2, pp. 149--156, 1991.

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D. Applegate, and W. Cook, A Computational Study of the Job-Shop Scheduling Problem, ORSA 149-156, Journal on Computing, Spring, 3(2), 1991.

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D. Applegate and W. Cook, A computational study of the job-shop scheduling problem, ORSA Journal of Computing, 3 (1991), pp. 149-156.

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Applegate, D. and W. Cook. 1991. A computational study of the job shop scheduling problem. Orsa Journal on Computing. 3, 2 149 156.

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#401. Applegate, D., and W. Cook #1991#, A computational study of the job-shop scheduling problem, ORSA Journal on Computing

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D. Applegate and W. Cook, #A Computational Study of the Job-Shop Scheduling Problem," ORSA Journal on Computing 3 #1991# 149-156.

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Applegate, D., and Cook, W. 1991. A computational study of the job-shop scheduling problem. ORSA Journal on Computing 3:149--156.

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Applegate, D., Cook, W. "A Computational Study of the Job-Shop Scheduling Problem". In ORSA Journal of computing, 3(2): 149-156, 1991.

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David Applegate and William Cook [1991]. A Computational Study of the Job-Shop Scheduling Problem. ORSA Journal on Computing, 3(2):149-156, 1991.

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Applegate, D., and Cook, W. (1991) A computational study of the job-shop scheduling problem, ORSA Journal on Computing 3, pp. 149-156.

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