中国科技论文在线 int. j. prod. res., 1998, vol. 36, no. 3, 683± 694

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1 int. j. prod. res., 1998, vol. 36, no. 3, 683± 694 Improving job-shop schedules through critical pairwise exchanges C. CHU², J.-M. PROTH² * and C. WANG² In this paper, we consider a job-shop scheduling problem. The criterion to be minimized is the makespan. To reach this goal, we propose a heuristic algorithm which gradually improves a given schedule by reversing the order in which some tasks are performed on machines. The job-shop scheduling problem being modelled as a disjunctive graph, reversing the order of two consecutive tasks which are performed on a given machine is equivalent to reversing the direction of a critical disjunctive arc. The important fact is that, due to the results proposed in this paper, we are able to choose the critical disjunctive arc to be reversed such that the makespan decreases at each iteration if the critical path is unique; otherwise, at least as many iterations as the number of critical paths are needed. This approach is simple and easy to implement. 1. Introduction This paper studies the typical job-shop scheduling problem where a set of jobs with distinct task routes have to be processed by a nite set of machines. A job is processed by performing a given sequence of tasks, each task being assigned to a given machine. We assume that set-up times are included in the manufacturing times, and that no preemptive tasks are allowed, i.e. an operation in process cannot be interrupted. Furthermore, a machine performs at most one operation at a time, and machine breakdowns are not considered in this paper. The criterion to be optimized is the makespan, i.e. the maximum job completion time which is also known as the total schedule time (Barker and McMahon 1985). Minimizing the makespan leads to increasing utilization of resources and productivity, which are crucial in reducing manufacturing costs. This problem is known as one of the hardest combinatorial optimization problems for which it is extremely di cult to nd optimal solutions. Due to its computational complexity, the job-shop scheduling problem is usually solved in two ways, namely implicit enumeration methods and heuristic methods. The branch-andbound approach, which is an e ective implicit enumeration method, has been widely used. However, all the branch-and-bound algorithms found in the literature can only solve small-size job-shop scheduling problems. A recent publication indicated that the job-shop problem proposed by Muth and Thompson (1963) can be solved in 19 minutes by a branch-and-bound algorithm (Brucker et al. 1994). Actually, it is extremely hard for branch-and-bound algorithms and mathematical Revision received August ² INRIA-Lorraine, Technopoà le Metz 2000, 4 rue Marconi, Metz, France. Universite de Technologie de Troyes, De partement GSI, BP 2060, Troyes Cedex, France. Institute for Systems Research, University of Maryland, College Park, MD 20742, USA. Shenyang Institute of Automation, Chinese Academy of Sciences, 114 Nanta Street, Shenyang , P. R. of China. * To whom correspondence should be addressed. 0020± 7543/98 $ Ñ 1998 Taylor & Francis Ltd. 转载

2 684 C. Chu et al. programming formulations to be implemented in a production environment. McCarthy and Liu (1993) discussed the gap between scheduling theories and practical manufacturing requirements, and concluded that the utilization of classical scheduling theory in most production environments is marginal. Algorithms for optimal schedules are only of academic signi cance. Moreover, due to the frequent changes in the availability of resources and demands, the schedule has to be frequently recomputed. This impels users to select algorithms which require heavy computation (Grabot and Geneste 1994). To be more application-oriented, heuristic methods have been developed to nd near-optimal or good feasible solutions. While most heuristic methods are based on some dispatching priority rules, some other ones are more sophisticated (Werner and Winkler 1995). Surveys of priority scheduling rules can be found in former publications. The heuristic methods usually nd local optimal solutions in quite a small amount of time (Day and Hottenstein 1970, Panwalker and Iskander 1977). It should be noticed that there exist close interactions between implicit enumeration methods and heuristic methods. Good heuristics often contribute to the implementation of e ective upper and/or lower bounds in branchand-bound algorithms. On the other hand, heuristic algorithms may incorporate branch-and-bound procedures to compute the solutions corresponding to the most promising part of the branch tree (McCarthy and Liu 1993). In production practices, simple dispatching priority rules are often adopted to generate manufacturing schedules which may be further improved in terms of performance criteria. A rescheduling process is often required after getting the initial schedules (Wu and Liu 1995). This paper aims at developing such a rescheduling algorithm or, more precisely, an algorithm which improves an existing schedule by reducing the makespan. It starts by modelling the schedule as a disjunctive graph. We show that it is possible to select a critical disjunctive arc which, if reversed, reduces the length of the critical path. Reversing the critical disjunctive arc corresponds to reversing the order of two consecutive tasks on a particular machine, and reducing the length of the critical path corresponds to reducing the makespan. In 2, we present the disjunctive graph which models a scheduling problem and the graph, derived from the previous one, which models a feasible schedule. The main results of this paper are included in 3. In 4, we propose an e cient algorithm which takes advantage of the previous results. Section 5 is devoted to numerical examples. Section 6 is the conclusion. 2. Scheduling modelling A job-shop scheduling problem is often represented by a disjunctive graph (Lageweg et al. 1977, Adams et al. 1988, Carlier and Pinson 1989). In a disjunctive arc, vertices, drawn as circles, represent tasks. Conjunctive arcs, which are directed lines, represent precedence constraints among the tasks of the same job. Disjunctive arcs, which are pairs of opposite directed lines, represent possible precedence constraints among tasks belonging to di erent jobs, these tasks being performed on the same machine. Formally, a disjunctive graph modelling a scheduling problem is a 4-tuple F = (O,U, V, W ) where: O is the set of vertices which represent the tasks to be performed, including the ctitious start and nish tasks,

3 Critical pairwise exchanges in job shop scheduling 685 U is the set of conjunctive arcs representing the order in which the tasks belonging to the same jobs should be performed. V is the set of disjunctive arcs, and more precisely the set of pairs of opposite directed lines (i.e. arcs) which represent the possible precedence constraints among tasks belonging to di erent jobs but performed on the same machine. W is the set of weights associated with conjunctive and disjunctive arcs; a weight represents the duration of the task represented by the vertex which is the origin of the arc. The model of a schedule is derived from the disjunctive graph by removing one arc in each pair of opposite directed lines. If N is the total number of tasks, we number the tasks consecutively from 0 to N + 1, vertex 0 representing the ctitious start task and vertex N + 1 the ctitious nish task. Let us consider an example of three jobs, each of them including three tasks. The manufacturing processes are the following: Job J1: (O 1,O 2,O 3 ) Job J2: (O 4,O 5,O 6 ) Job J3: (O 7,O 8,O 9 ) Tasks O 1,O 5,O 8 are performed on machine M 1, tasks O 2,O 4,O 9 are performed on machine M 2, tasks O 3,O 6,O 7 are performed on machine M 3. The scheduling problem model is given in Fig. 1. In this model, for instance, arc (O 1,O 2 ) is a conjunctive arc which re ects the fact that O 1 should be performed before O 2 (O 1 and O 2 belong to the same job J 1 ). Arcs (O 1,O 8 ) and (O 8,O 1 ) are, for instance, a pair of opposite directed lines (i.e. conjunctive arcs) which express the facts that: (1) tasks O 1 and O 8 are performed on the same machine, and (2) O 1 can be performed before O 8 or, on the contrary, O 8 can be performed before O 1. By removing one arc in each pair of opposite directed lines, we transform the disjunctive graph into an ordinary graph. To be the model of a feasible schedule, it is necessary that this graph be acyclic (Lageweg et al. 1977, Adams et al. 1988), which means that no directed cycle exists. Figure 2 provides the model of a feasible schedule derived from the disjunctive graph presented in Fig. 1. In the ordinary graph representing a schedule, the directed path with the largest weighted length joining O 0 to O i is called the critical path related to O i. If i = N + 1, Figure 1. The scheduling problem model.

4 686 C. Chu et al. Figure 2. Model of a feasible schedule. we call this path the critical path of the ordinary graph. The vertices belonging to the critical path of the ordinary graph are called critical vertices, and the other vertices are the non-critical vertices. We refer to the disjunctive arcs belonging to the critical paths as the critical disjunctive arcs, the other ones being the non-critical disjunctive arcs. It is known that the length of the critical path of such an ordinary graph is the makespan of the corresponding schedule. As a consequence, the goal is to derive from the disjunctive graph an ordinary graph having a critical path the length of which is minimal. The critical path of the graph given in Fig. 2 is O 0 -O 7 -O 8 -O 9 -O 2 -O 3 -O 6 -O 10 and its length is 29. Another schedule can be obtained by reversing the direction of some disjunctive arcs, providing that the resulting graph is acyclic. For instance, the ordinary graph represented in Fig. 3 is obtained by reversing the disjunctive arc (O 3,O 6 ). The critical path of this new ordinary graph is O 0 -O 7 -O 8 -O 5 -O 6 -O 3 -O 10 and its length is 25. Thus, the schedule represented by this ordinary graph is better than the previous one. This suggests that the length of the critical path can be reduced by appropriately reversing the direction of some disjunctive arcs. The results presented in the next section will allow us to select the disjunctive arc to be reversed at each iteration. It is important to point out that the approach presented in this paper guarantees that the critical path is improved at each iteration, except if the conjunctive arc which is Figure 3. The second feasible schedule.

5 Critical pairwise exchanges in job shop scheduling 687 reversed does not belong to all the critical paths. In this case, at least as many iterations as the number of critical paths are necessary. 3. Local optimality As mentioned above, reversing the direction of a disjunctive arc leads to the model of a new schedule which is feasible if and only if the resulting graph is acyclic. Note that reversing the direction of a non-critical disjunctive arc cannot reduce the makespan since, in this case, the critical path of the initial graph is still a path from vertex 0 to vertex N + 1 in the resulting graph. As a consequence, only the critical disjunctive arcs should be considered to improve a schedule. Theorem 1 presented in this section shows that reversing the direction of a critical disjunctive arc in an acyclic graph which models a schedule leads to a graph which is still acyclic, and thus models another schedule. In other words, Theorem 1 guarantees that we can reverse the direction of any critical disjunctive arc without checking the cyclicity, assuming that the initial ordinary graph is acyclic, i.e. models a feasible schedule. Theorem 2 provides a necessary and su cient condition to improve the makespan by reversing a critical disjunctive arc, assuming that this arc belongs to all the critical paths. The corollary derived from Theorem 2 is the basis of the algorithm presented in the next section Theorem 1 The ordinary graph obtained by reversing a critical disjunctive arc in an acyclic ordinary graph is still acyclic. Proof: Let G = (O,U, V, W ) be an acyclic ordinary graph. Let (O i,o j ) Î V be a critical disjunctive arc, i.e. a disjunctive arc belonging to the critical path. We denote by G = (O,U, V, W ) the ordinary graph derived from G by reversing (O i,o j ). Let us assume that G is cyclic, i.e. that there exists a directed cycle belonging to G. Obviously, this cycle contains (O j,o i ), otherwise G would have been cyclic. Furthermore, having such a cycle means that there exists a same directed path joining O i to O j in G and G. This directed path has two properties of interest, that is: (1) it is made with more than one arc, otherwise it would be restricted to (O i,o j ) in G, and (2) the length of the rst arc of this path, i.e. the arc the origin of which is O i, is equal to the length of (O i,o j ). As a consequence, the length of the directed path joining O i and O j is greater than the length of (O i,o j ), which implies that (O i,o j ) does not belong to the critical path: this contradicts the hypothesis. Thus G is acyclic. Q.E.D. Theorem 1 indicates that reversing a disjunctive arc still leads to a feasible schedule, i.e. a schedule whose makespan is bounded. The remaining problem consists of de ning the disjunctive arcs to be reversed in order to reduce the length of the critical path. This will lead to a local optimum. The following de nitions will be used in the remainder of the presentation: L (G,i,j) is the longest path joining vertices O i and O j in the ordinary graph G, CP(G,i) is the longest path joining vertices O 0 and O N+1 among the paths containing vertex O i in the ordinary graph G,

6 688 C. Chu et al. CP(G) is the length of the critical path in G. According to the previous de nitions, the following relations are straightforward for G = (O,U, V, W ): and CP(G,i) = L (G,0,i) + L (G,i,N + 1) (1) CP(G) = Max i /O i Î O CP(G,i), (2) for i = 1,...,N Let O i and O j be two critical vertices, i.e. two vertices belonging to the critical path, and let us assume that (O i,o j ) Î V. Let also w i Î W be the length of arc (O i,o j ); w i is the duration of the task represented by O i. With these assumptions, the following two relations hold: CP(G) = L (G,0,j) + L (G,i,N + 1) - w i (3.1) This relation is obvious if we consider that both the longest path O 0 to O j (whose length is L (G,0,j)) and the longest path joining O i to O N+1 (whose length is L (G,i,N + 1)) contain arc (O i,o j ); as a consequence, the length of (O i,o j ) should be subtracted once to obtain the length of the critical path. Furthermore CP(G) = L (G,0,i) + L (G,j,N + 1) + w i (3.2) This relation is derived from the following remark: the longest path joining O 0 to O N+1 and which, as we know, contains (O i,o j ), is composed of the longest path joining O 0 to O i (whose length is L(G,0,i)), the arc (O i,o j ) (whose length is w i ), and the longest path joining O j to O N+1 (whose length is L (G,j,N + 1)). Theorem 2, presented hereafter, allows us to select the disjunctive arc to be reversed in order to reduce the length of the critical path. From a practical point of view, Theorem 2 allows us to select two consecutive tasks whose manufacturing order on one of the machines should be reversed to reduce the makespan. It is important to notice that Theorem 2 provides a necessary and su cient condition to reduce the makespan Theorem 2 Let G = (O,U, V, W ) be the ordinary graph representing a given schedule, G the ordinary graph obtained by reversing (O i,o j ) Î V, and G the ordinary graph derived from G by removing arc (O i,o j ). If (O i,o j ) belongs to all the critical paths of G, then CP(G ) < CP(G) if and only if: Proof: L (G,0,j) - L (G,0,j) + L (G,i,N + 1) - L(G,i,N + 1) > w i + w j. (4) (1) We rst prove that (4) is a su cient condition for CP(G ) < CP(G). Relation (4) can be rewritten as: L(G,0,j) + L (G,i,N + 1) - w i - [L (G,0,j) + L(G,i,N + 1) + w j]> 0, or, according to equation (3.1):

7 Critical pairwise exchanges in job shop scheduling 689 CP(G) - [L (G,0,j) + L (G,i,N + 1) + w j]> 0. (5) Furthermore, the longest path joining O 0 in O j in G is the same as the longest path joining O 0 to O j in G, otherwise it would contain (O i,o j ) and since it ends with O j, it would contain a circuit, which is contradictory with Theorem 1. Thus L (G,0,j) = L (G,0,j). (6) A similar argument applies for the paths joining O i to O N+1 in G and G, and thus: L (G,i,N + 1) = L (G,i,N + 1) (7) Consider equations (6), (7) and (3.2) applied to G, we have: L (G,0,j) + L(G,i,N + 1) + w j = L(G,0,j) + L (G,i,N + 1) + w j Using equation (8), inequality (5) becomes: CP(G) - CP(G ) > 0, Or: = CP(G ) (8) CP(G ) < CP(G) (2) We now prove that inequality (4) is a necessary condition for inequality CP(G ) < CP(G). According to equation (8): According to (3.1): CP(G ) = L(G,0,j) + L (G,i,N + 1) + w j. CP(G) = L(G,0,j) + L (G,i,N + 1) = w i And, since CP(G ) < CP(G), we have: L (G,0,j) + L (G,i,N + 1) - w i > L (G,0,j) + L (G,i,N + 1) + w j. Thus inequality (4) holds. Q.E.D. It should be noticed that, if the critical disjunctive arc which is reversed does not belong to all the critical paths, then only the lengths of the critical paths which contain this arc are reduced. Thus we may have to perform more than one iteration to reduce the makespan in this very particular case. Corollary: We consider an ordinary graph G = (O,U, V, W ) and the graphs G and G derived from G as explained in Theorem 2. Let (O i,o j ) Î V be the disjunctive arc under consideration. Then the length of a critical path of G derived from G by reversing (O i,o j ) can be expressed as follows: CP(G ) = Max{L (G,0,i) + L(G,i,N + 1), L (G,0,j) + L (G,j,N + 1), L(G,0,j) + L(G,i,N + 1) + w j,cp(g )}. The proof of the corollary can be simply drawn out from the proof of Theorem 2. Any disjunctive arc satisfying relation (4) is referred to as a variable arc. The variable arcs are the arcs to be considered to improve the makespan.

8 690 C. Chu et al. According to the previous results, it is easy to understand that a local optimum can be reached iteratively by reversing a variable arc at each iteration. As outlined above, the value of the makespan may remain the same for more than one iteration if the variable arc which is reversed does not belong to all the critical paths. In the next section, we present a greedy algorithm derived from the previous results and which leads to a local optimum. 4. Heuristic algorithm Given a job-shop scheduling problem modelled by a disjunctive graph F, it is possible to generate an initial schedule either at random or by applying some priority rules at the entrance of each machine. This initial schedule can be represented by an ordinary graph G = (O,U, V, W ). The second step of the algorithm consists of computing the critical path of G. This critical path is obtained by applying the following forward dynamic programming formulae: { L ( G, 0, 0) = 0( initialization) L (G,0,i) = Max{L(G,0,i C ) + w ic, L (G,0,i D ) + w id }. O ic (resp. O id ) is the vertex representing the task which immediately precedes the task represented by O i on the same machine (resp. in the same job). Indeed, w ic and w id are the duration of the tasks represented by O ic and O id, respectively. The third step of the algorithm consists of selecting a critical disjunctive arc which is a variable arc, i.e. a disjunctive arc belonging to the critical path and which veri es inequality (4). Let us consider a critical disjunctive arc (O i,o j ) where neither O i nor O j are extreme vertices, i.e. are either O 0 or O N+1. To check if this arc veri es inequality (4), we have to compute the four terms of the left side of this inequality: (1) L(G,0,j) has been computed in the second step of the algorithm, i.e. when computing the critical path: (2) L(G,0,j) = Max{L (G,0,i D ) + w id, L (G,0,j C ) + w jc }. L(G,0,i D ) + w id represents the minimal time to start the task represented by O i which should be at most equal to the starting time of the task represented by O j, since (O i,o j ) does not exist in G ; L(G,0,j C ) + w jc is the completion time of the task which precedes the task represented by O i in the same job; (3) L(G,i,N + 1) is obtained by subtracting L (G,0,i) from the length of the critical path, i.e. CP(G); (4) the only additional e ort required to check inequality (4) is the computation of L (G,i,N + 1). This term can be computed using the following relation: L(G,i,N + 1) = Max{L (G,i CS,N + 1), L(G,j DS,N + 1)}+ w i where O ics (resp. O jds ) is the vertex representing the task which follows the task represented by O i (resp. O j ) in the same job (resp. on the same machine). The method used to compute L (G,0,i) and L (G,i,N + 1) is of crucial importance for the e ciency of the proposed algorithm. We use Algorithms 7 and 8 presented by Gondran and Minoux (1979) to compute these paths. The complexity of the algorithm is O( R), where R is the number of arcs.

9 Critical pairwise exchanges in job shop scheduling 691 Since a vertex (except the two ctious vertices) has at most two predecessors (successors) that are of signi cance in computing a critical path, the complexity of these algorithms is O(N), where N is the number of vertices. The algorithm which leads to a local optimum can nally be summarized as follows. Algorithm: (1) Generate an initial schedule and derive its ordinary graph model. (2) Compute a critical path in this ordinary graph and identify the critical disjunctive arcs. (3) For each critical disjunctive arc (O i,o j ) compute: C = L(G,0,j) - L (G,0,j) + L (G,i,N + 1) - L (G,i,N + 1) - (w i - w j ) (4) If C > 0 for at least one arc (O i,o j ), then reverse arc (O i,o j ) and go to 2, otherwise stop the computation. 5. Numerical results The previous algorithm has been coded in C and runs on a SUN workstation. It was rst tested on some numerical problems quoted in the OR-Library of Internet (Mattfeld and Vaessens 1995). We applied three priority rules to each set of data. These priority rules are FCFS, SPT and MRW (Most Remaining Work). Each rule led to an initial schedule to which the previous approaches were applied. A subset of the numerical computations is given in Table 1. Columns IM give the value of the makespan corresponding to the initial schedule. Columns IMPR and PER provide respectively the makespan after applying the pairwise exchanges algorithm and the improvement expressed in percentages rounded to one decimal place. TIME contains the computation time required to improve the makespan. The rst ten data sets were provided by Applegate and Cook (1991). They are labelled as problems orb01 and orb10 in the OR-Library. Each set concerns 10 jobs, each of them being composed of 10 tasks, each task being performed on one of the 10 available machines: these problems are referred to as problems. Data sets 11 to 15 were provided by Lawrence (1984) and are labelled as problems la31 to la35 in the OR-Library. These problems are problems, i.e. 30 jobs, 10 tasks per job and 10 machines. Data sets 16 to 20 were also provided by Lawrence (1984) and are labelled as 15 problems. The last problems la36 to la40 in the OR-Library. They are four sets of data from Yamada and Nakano (1992). They are labelled as yn1 to yn4 in the OR-Library and are problems. As the reader can see, the solutions obtained by applying the priority rules are signi cantly improved by applying the approach proposed in this paper in most of the cases. Furthermore, the computation time is very small. Another conclusion is that the best nal solutions are not always the ones derived from the best initial solution. Another set of tests was carried out using ve groups of data where the duration of the tasks (manufacturing times or processing times) were generated at random following a uniform distribution on [5, 50]. Each group of data was composed of 10 examples. The examples in the ve groups were respectively ( ), ( ), ( ), ( ) and ( ) problems. Four initial schedules were generated for each problem using respectively priority rules

10 692 C. Chu et al. Initial state generated using FCFS Initial state generated using SPT Initial state generated using MRW Set of data IM IMPR % (s) IM IMPR % (s) IM IMPR % (s) Table 1. Initial state generated using FCFS Group IMa IMPRa Results related to data extracted from the OR-Library. (s) Initial state generated using SPT IMa IMPRa (s) Initial state generated using MRW Ima IMPRa (s) Initial state generated using ABZ IMa IMPRa (s) Table 2. Average results of the ve groups of problems. FCFS, SPT and MRW, and the Adam s algorithm (Adams et al. 1988). This algorithm is referred to as ABZ in Table 2. The average results for each group are presented in Table 2. For the ve groups of problems: (1) The average improvement of the initial schedules generated using FCFS priority rules ranges from 4. 32% to 5. 98%.

11 Critical pairwise exchanges in job shop scheduling 693 (2) The average improvement of the initial schedules generated using SPT priority rules ranges from 2. 38% to 4. 78%. (3) The average improvement of the initial schedules generated using MRW priority rules ranges from 0. 90% to 3. 14% (4) The average improvement of the initial schedules generated using Adam s algorithm ranges from 1. 71% to 4. 31% We also noticed that the average computation time required by Adam s algorithm to provide an initial schedule is contained between and s, while the computation time required to improve the initial schedule is contained between and s. According to the computation results, MRW rules lead to better initial schedules than FCFS and SPT rules. We also noticed that the priority rules enriched by the approach contained in this paper are very simple to implement and required a very small amount of time. Most of the time, they also lead to better schedules in terms of makespan than Adam s algorithm. 6. Conclusion The motivation of this paper was to develop a practical and e cient approach for improving the initial schedules generated by priority rules and algorithms found in the literature. With a disjunctive graph model, scheduling a job shop is equivalent to selecting directions from the disjunctive arcs. The makespan of a schedule equals the critical path length of the corresponding graph. An existing job shop schedule can be modi ed by reversing the direction of some disjunctive arcs of the graph. In this paper, we show how to select the disjunctive arcs to be reversed in order to improve the makespan. We derive from this result a simple, however quite e cient, heuristic algorithm for gradual improvements to initial schedules generated by priority rules or other algorithms proposed in former publications. Several numerical examples have been performed. They showed that the heuristic algorithm considerably improves the initial schedules except in a very few cases. In addition, the heuristic algorithm is very e cient in terms of computation time. Therefore, it is promising for future industrial implementations. References Adams, J., Balas, E., and Zawack, D., 1988, The shifting bottleneck procedure for job shop scheduling. Management Science, 34 (3), 391± 401. Applegate, D., and Cook, W., 1991, A computational study of the job-shop scheduling instance. ORSA Journal on Computing, 3, 149± 156. Barker, J. R., and McMahon, G. B., 1985, Scheduling the general job-shop. Management Science, 31 (5), 594± 598. Brucker, P., Jurisch, B., and Sievers, B., 1994, A branch and bound algorithm for the jobshop scheduling problem. Discrete Applied Mathematics, 49, 107± 127. Calier, J., and Pinson, E., 1989, An algorithm for solving the job-shop problem. Management Science, 35 (2), 164± 176. Day, J. E., and Hottenstein, M. L., 1970, Review of sequencing research. Naval Research L ogistics Quarterly, 17, 11± 38. Gondran, M., and Minoux, M., 1979, Graphes et Algorithmes (Paris: Editions Eyrolles), pp. 41± 44. Grabot, B., and Geneste, L., 1994, Dispatching rules in scheduling: a fuzzy approach. International Journal of Production Research, 32 (4), pp. 903± 915. Lageweg, B. J., Lenstra, J. K., and Rinnoykan, A. H. G., 1977, Job shop scheduling by implicit enumeration. Management Science, 24 (4), pp. 441± 450.

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