J.M. Moore. An n-job, one machine sequencing algorithm for minimizing the number of late jobs. Management Science, 15:102-109, 1968.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....2 Previous work. Work on (special cases of) JISP dates back to the 1950s. Jackson [17] proved that the earliest due date (EDD) greedy rule is an optimal algorithm for 1jjL max . This implies that if all jobs can be scheduled in an instance of 1jj P U j , then EDD nds such a schedule. Moore [24] gave a greedy O(n log n) time optimal algorithm for 1jj P U j . On the other hand, the weighted version 1jj P w j U j is NP hard (knapsack is a special case when all deadlines are equal) Sahni [26] presented a fully polynomial time approximation scheme for this problem. When release dates ....

J.M. Moore. An n-job, one machine sequencing algorithm for minimizing the number of late jobs. Management Science, 15:102-109, 1968.

....Previous work. Work on (special cases of) JISP dates back to the 1950s. Jackson [17] proved that the earliest due date (EDD) greedy rule is an optimal algorithm for 1jjL max . This implies that if all jobs can be scheduled in an instance of 1jj P U j , then EDD finds such a schedule. Moore [24] gave a greedy O(n log n) time optimal algorithm for 1jj P U j . On the other hand, the weighted version 1jj P w j U j is NP hard (KNAPSACK is a special case when all deadlines are equal) Sahni [26] presented a fully polynomial time approximation scheme for this problem. When release dates ....

J.M. Moore. An n-job, one machine sequencing algorithm for minimizing the number of late jobs. Management Science, 15:102--109, 1968.

....nondecreasing order of d i Gamma p i . This rule is known to minimize maximum earliness subject to no machine idle time. The Moore Hodgson algorithm (MH) minimizes the number of tardy jobs for a single machine in O(n log n) time. As an output of MH, we also obtain . Moore Hodgson Algorithm [Moo68]. Schedule job k last, where: 1. Job j is the first tardy job in EDD 2. J = f1; 2; jg 3. p i = max i2J p i (break ties with larger due date ) Lawler s O(n 2 ) algorithm minimizes the maximum cost f max subject to precedence constraints on a single machine. We use the notation F(B A) to ....

J. M. Moore. An n job, one machine sequencing algorithm for minimizing the number of late jobs. Management Science, 15:102--109, 1968.

....note that LCL is optimal also for 1jprecjf max . In the implementation of LCL for instances with precedence constraints, the only candidates for the last position are jobs that no other jobs depend on them. 3. 2 An Alternative Algorithm for 1jjf max The following algorithm, presented by Moore [8], is an alternative algorithm for 1jjf max . The algorithm proceeds in iterations. In each iteration we check if there exists a schedule in which f max B for some value, B. Thus, using binary search we can reach the optimal value efficiently. Initial upper and lower bounds for B are max(min) ....

J.M. Moore. An n-job, one machine sequencing algorithm for minimizing the number of late jobs. Management Science, 15:102-109, 1968. 7

....for it. It is informative to notice that if release times are added then both problems become strongly NP hard. 1jj P j w j U j remains strongly NP hard even for the unweighted version with release times [2] Yet, the unweighted problem without release times turns out to be polynomial solvable [7]. Thus, it seems that adding release times makes the problem much harder to solve. It suffices to know which jobs complete before their deadline because then we can find the schedule of these jobs by solving 1jjL max , which can be solved in polynomial time using the EDD rule [6] We now show a ....

....the optimal entry (this can be done by storing the decision made at each table entry in a separate Xw;j table) Comment: We immediately obtain an O i n 2 j algorithm for 1jj P j U j by setting w j = 1. The unweighted version is also solvable by a O i n log n j greedy algorithm algorithm [7] and a similar O i n P j p j j dynamic programming algorithm . Theorem 3.1 There exists a O( 1 ffl n 4 ) time (1 ffl) approximation algorithm for 1jj P j w j U j The above pseudo polynomial algorithm can be a basis for a FPTAS as follows. Since the algorithm runs in polynomial ....

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J.M. Moore, An n-job, one machine sequencing algorithm for minimizing the number of late jobs, Management Science 15 (1968), pp. 102-109.

....Scheduling to maximize completion count has been rather less studied. From a result of Lawler [5] it follows that off line scheduling to maximize CC is somewhat easier than with respect to the EPU metric, and can be done in polynomial time (O(n 5 ) where n is the number of tasks) Moore [7] presented a more efficient algorithm for the special case when all the tasks have equal request times (T:a = 0 for all T ) In this paper, we study the on line scheduling problem with respect to the completion count metric. Despite the fact that off line scheduling for CC is easier than for EPU, ....

....information the request times, execution requirements, and deadlines of all tasks in is known a priori,scheduling such a task set is not really an on line problem. Theorem 4 There are optimal (1 competitive) algorithms for scheduling equal request time task systems. Proof: Moore [7], presented an optimal algorithm for non preemptive scheduling to maximize task completions in equal request time task systems. However, when all tasks have the same request times, the problems of preemptivescheduling and non preemptivescheduling on a uniprocessor are equivalent, in the sense that ....

J. M. Moore. An n job, one machine sequencing algorithm for minimizing the number of late jobs. Management Science, 15:102--109, 1968.

....in which J i must execute if it is on time. With arbitrary release dates, the problem is NP Hard in the strong sense [LRKB77] However, many special cases and or relaxations of the problem are polynomially solvable. With equal release dates, the problem 1jj P U i is polynomially solvable [MOO68]. Lawler [LAW90] has proposed an O(n 5 ) dynamic programming algorithm for the preemptive case 1jr i ; pmtnj P U i . Note that an enhanced O(n 4 ) version of this algorithm is described in [BAP99a] Kise, Mine and Ibaraki [KIM78] have shown that when release dates and due dates are ....

J.M. MOORE. An n job, one machine sequencing algorithm for minimizing the number of late jobs. Management Science, 15(1):102--109, 1968.

....end EDD Figure 2: The EDD Algorithm When all tasks have the same request times, the problems of preemptive scheduling and nonpreemptive scheduling on a uniprocessor are identical, in the sense that every set of tasks that can be scheduled preemptively can also be scheduled without preemption. In [10], an algorithm is presented for non preemptive scheduling to maximize task completions for bulk request systems. This algorithm turns out to be essentially equivalent to EDD; we therefore refer the interested reader to [10] for a proof of the correctness of EDD. 3.2 Equal Execution Times (EET) ....

....can be scheduled preemptively can also be scheduled without preemption. In [10] an algorithm is presented for non preemptive scheduling to maximize task completions for bulk request systems. This algorithm turns out to be essentially equivalent to EDD; we therefore refer the interested reader to [10] for a proof of the correctness of EDD. 3.2 Equal Execution Times (EET) We now move on to considering the case where all tasks have equal execution times. A scheduling algorithm is said to use no inserted idle time if the processor is never idle while there are active non degenerate tasks that ....

J. Moore. An n job, one machine sequencing algorithm for minimizing the number of late jobs. In Management Science, 15(1), 1968.

....task systems. Proof Sketch: When all tasks have the same request times, the problems of preemptive scheduling and nonpreemptive scheduling on a uniprocessor are identical, in the sense that every set of tasks that can be scheduled preemptively can also be scheduled without preemption. In [6], an algorithm is presented for the nonpreemptive scheduling to maximize task completions for bulk request systems. This algorithm turns out to be essentially equivalent to EDD; we therefore refer the interested reader to [6] for a proof of the correctness of EDD. The basic idea of the EDD ....

....be scheduled preemptively can also be scheduled without preemption. In [6] an algorithm is presented for the nonpreemptive scheduling to maximize task completions for bulk request systems. This algorithm turns out to be essentially equivalent to EDD; we therefore refer the interested reader to [6] for a proof of the correctness of EDD. The basic idea of the EDD algorithm is to create a deadline ordered sequence of the entire task set and then to iteratively keep removing the largest execution time task from the sequence until the remaining set of tasks becomes feasible with an Earliest ....

J. Moore. An n job, one machine sequencing algorithm for minimizing the number of late jobs. In Management Science, 15(1), 1968. 9

....know ahead of time in what pattern the tasks will arrive. This problem corresponds to a standard job sequencing problem where each job has both a deadline and a ready time (the arrival time, before which the job cannot be scheduled) Kise, Ibaraki, and Mine 1978) Lawler 1964) Lawler 1976) (Moore 1968), Villareal and Bulfin 1983) We have found no previous work 7 combining deadlines, ready times, and the weighted task optimality criterion we give a polynomial time algorithm addressing all these factors. The problem is known NP hard without our assumption that each task has the same fixed ....

Moore 1968 Moore, J. M. An n-job, one-machine sequencing algorithm for minimizing the number of late jobs. Management Science 15:102-- 109. 1968.

....in polynomial time. This special case is sufficiently general to contain the problem 1 j j P U j without batch setup times (in 1 j j P U j , every jobs forms its own family and all family setup times are zero) Hence, our result generalizes the well known polynomial time algorithm of Moore [7]. Our solution approach to 1 j s f j P U j is as follows: We formulate 1 j s f j P U j with uniform family due dates as a dual resource allocation problem with tree structured constraints (cf. Section 3) Since this dual resource allocation problem can be solved in polynomial time by dynamic ....

J. Moore, An n job, one machine sequencing algorithm for minimizing the number of late jobs, Management Science 15, 1968, 102--109.

....with context switching. In other words, while completing a task the overhead due to context switching is minimum. However this scheme has a poor performance in terms of the class of problems that can be feasibly scheduled. Some systems with less time critical tasks could adopt this strategy. Moore [32] showed that the earliest deadline first policy was optimal for independent, nonpreemptable tasks executing upon a uniprocessor, making the assumption that the set of tasks had the same ready time. Bratley, Florian and Robillard [13] developed an enumeration algorithm which implicitly determines ....

Moore, J. M., An n job, one machine sequencing algorithm for minimizing the number of late jobs., Management Science, 15(1)., 1986.

....of late jobs ( P U j ) It is important to understand that this objective is different from minimizing the total tardiness. If a job J j is late, it can be scheduled arbitrarily after the jobs on time. The objective is then equivalent to sequencing the maximum number of jobs on time. Moore [10] proposed a well known O(n log n) algorithm to solve the 1j j P U j problem. Although this problem has drawn a lot of attention in the literature when additional constraints are incorporated or when weights on the jobs are considered (1j j P w j U j ) little work has been done when jobs have ....

Moore, J.M. (1968). A n job, one machine sequencing algorithm for minimizing the number of late jobs. Management Science 15(1), pp 102-109.

....of late jobs ( P U j ) It is important to understand that this objective is different from minimizing the total tardiness. If a job J j is late, it can be scheduled arbitrarily after the jobs on time. The objective is then equivalent to sequencing the maximum number of jobs on time. Moore [7] proposed an O(n log n) algorithm to solve the 1j j P U j problem. Although this problem has drawn a lot of attention in the literature when additional constraints are incorporated or when weights on the jobs are considered (1j j P w j U j ) little work has been done when jobs have release ....

Moore, J.M. (1968). A n job, one machine sequencing algorithm for minimizing the number of late jobs. Management Science 15(1), 102109.

....has been done on the general one machine scheduling problem, but the results presented here only concern the minimization of the number of late jobs. In the special case where processing times are equal to 1, Monma [Mon82] gives an O(n log n) algorithm for the 1jp j = 1j P U j problem. Moore [Moo68] develops an O(n log n) algorithm to solve the 1j j P U j problem. Unfortunately, the 1jr j j P U j problem, studied in this paper has been proved NP hard [Len77] But Kise, Ibaraki and Mine [Kis78] give an O(n 2 ) algorithm for the case where release dates and due dates are similarly ....

....for (11) to be satisfied. Therefore, c 1 strongly depends on the value chosen for M , and decreases as M increases. Thus, this lower bound will often be very poor. As previously mentioned, an important remark is that there is always an optimal sequence in which all late jobs are at the end (see [Moo68]) Hence, after solving POS1, one knows that there is an optimal solution in which there are at least f = dc 1e late jobs at the end. This can be fixed in POS1 by adding the constraint Uk = 1 k = n Gamma f 1; n where POS1f denotes the resulting model. Because the formulation POS1f is ....

Moore, J.M. (1968). A n job, one machine sequencing algorithm for minimizing the number of late jobs. Management Science 15(1), 102-109.

....that maximizes the number of OR tasks with essential predecessors that meet their deadline. To produce such a schedule, note that an OR task together with one predecessor subtree consisting of k i AND tasks may be thought of as one large task with processing time k i 1. Then the algorithm of [Moore68], which minimizes unit penalty on a single processor, may be used to schedule tasks with processing time (k i 1) to maximize the number of OR tasks that meet their deadline. In summary, the complexity of the skipped problem is always at least as high as the complexity of the unskipped problem. ....

....problems. a) Scheduling to meet deadlines with identical processing times on 1 processor. Deadline Location General Graph In Tree Simple In forest 1 deadline O(n) deadlines On all tasks NP C (Theorem 3.1) NP C (Theorem 3.6) NP C (Theorem 3.7) On OR tasks only NP C (Theorem 3.1) NP C (Theorem 3. 6) [Moore68] Algorithm (b) Scheduling to minimize completion time on m processors. Task Processing Time General Graph In Tree Identical NP C [Lawler89] NP C (Theorem 3.8) Arbitrary No Algorithm Path Balancing Algorithm 3.2.2. Scheduling to Minimize Completion Time Table 3.2 also gives the complexity of ....

Moore, J. M. An n Job, One Machine Sequencing Algorithm for Minimizing the Number of Late Jobs. Management Science (1968) vol. 15, pp. 102-109. 104

....(a) Scheduling to meet deadlines with identical processing times on 1 processor. Deadlines Location General Graph In Tree Simple In Forest 1 Deadline O(n) Deadlines On All Tasks NP C (Theorem 3.1) NP C (Theorem 4.1) NP C (Theorem 4.2) ON OR Tasks Only NP C (Theorem 3.1) NP C (Theorem 4. 1) [17] Algorithm (b) Scheduling to minimize completion time on m processors. Task Processing Time General Graph In Tree Identical NP C [15] 3=2 OPT ) NP C (Theorem 4.3) Arbitrary No Algorithm Path Balancing Heuristic 4. Skipped Problems. In an AND OR skipped scheduling problem, the inessential ....

....number of OR tasks that simultaneously meet their deadlines and have essential predecessors. To produce such a schedule, we note that an OR task together with one predecessor subtree consisting of k i AND tasks may be thought of as one large task with processing time k i 1. Then the algorithm of [17], which minimizes unit penalty on a single processor, may be used to schedule tasks with processing time (k i 1) to maximize the number of OR tasks that meet their deadline. In summary, we find that the complexity of the skipped problem is always at least as high as the complexity of the ....

J. M. Moore, An n job, one machine sequencing algorithm for minimizing the number of late jobs, Management Sci., 15 (1968), pp. 102-109.

....a moot point. One job needs to finish last, and it immediately follows that we can do no better than executing all of that job last. Thus, our greedy algorithm continues to apply. Theorem 3.1 ( Law73] Least Cost Last is an optimal algorithm for 1jprecjf max . 3.1. 2 An alternative approach Moore [Moo68] gave a different approach to 1jjf max that may be faster in some cases. His scheme is based on a reduction to the maximum lateness problem and its solution by the EDD rule. To see how an algorithm for L max can be applied to 1jjf max , suppose we want to know whether there is a schedule with f ....

....j ) time algorithm for exactly solving 1jj P w j U j . We remark that a similar dynamic program can be used to solve the problem in time O(n P p j ) which is effective when the processing times are polynomially bounded integers. We also note that a quite simple greedy algorithm due to Moore [Moo68] can solve the unweighted 1jj P U j problem in O(n log n) time. 3.3 Another dynamic program: PjjC max For a second example of the applicability of dynamic programming, we return to the NP hard problem PjjC max , and focus on a special case that is solvable in polynomial time the case in ....

J. M. Moore. An n-job, one machine sequencing algorithm for minimizing the number of late jobs. Management Science, 15:102--109, 1968.

....prec j L max Lawler [1973] 1 j p j = 1; intree (l) j L max Bruno et al. 1980] 1 j p j = 1; prec (l = 1) r j j L max Bruno et al. 1980] 1 j p j = 1; outtree (l) r j j P C j Section 5.1 1 j p j = 1; prec (l = 1) j P C j Section 5. 2 1 j sp Gamma graph j P w j C j Lawler [1978] 1 jj P U j Moore [1968] 1 j p j = 1; r j j P w j U j Assignment problem 1 j p j = 1; r j j P w j T j Assignment problem NP hard: 1 j chains (l) j C max Wikum et al. 1994] 1 j p j = 1; chains (l ij ) j C max Yu [1996] 1 j p j = 1; prec (l) j C max Timkovsky [1998A] 1 j p j = 1; intree (l) r j j C max Section ....

Moore, J.M. [1968] An n job, one machine sequencing algorithm for minimizing the number of late jobs, Management Sci. 15, 102-109.

....noted that, once again, the fact that our algorithm is greedy makes preemption a moot point. One job needs to finish last, and it immediately follows that we can do no better than executing all of that job last. Thus, our greedy algorithm continues to apply. 3.1. 2 An alternative approach Moore [Moo68] gave a different approach to 1jjf max that may be faster in some cases. His scheme is based on a reduction to the maximum lateness problem and its solution by the EDD rule. To see how an algorithm for L max can be applied to 1jjf max , suppose we want to know whether there is a schedule with f ....

....) time algorithm for exactly solving 1jj P w j U j . We remark that a similar dynamic program can be used to solve the problem in time O(n P p j ) which is effective when the processing times are polynomially bounded integers. We also note that a quite simple greedy algorithm due to Moore [Moo68] can solve the unweighted 1jj P U j problem in O(n log n) time. 3.3 Dynamic Programming for PjjC max For a second example of the applicability of dynamic programming, we return to the NP hard problem PjjC max , and focus on a special case that is solvable in polynomial time the case in which ....

J. M. Moore. An n-job, one machine sequencing algorithm for minimizing the number of late jobs. Management Science, 15:102--109, 1968.

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J.M. Moore. An n-job, one machine sequencing algorithm for minimizing the number of late jobs. Management Science, 15:102-109, 1968.

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MOORE, J. M. A n jobs, one machine sequencing algorithm for minimizing the number of later jobs. Management Science 15 (1968), 102--109.

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J. M. Moore, An n job, one machine sequencing algorithm for minimizing the number of late jobs, Management Sci. 15 (1968), 102-109.

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J. Michael Moore [1968]. An n Job, One Machine Sequencing Algorithm for Minimizing the Number of Late Jobs. Management Science, 15(1):102-109, 1968.

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J. M. Moore, An n job, one machine sequencing algorithm for minimizing the number of late jobs, Management Sci. 15 (1968), 102--109.

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