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### On n-tardy sets

Peter A. Cholak, Peter M. Gerdes, Karen Lange
2012 Annals of Pure and Applied Logic
We show there is a 3-tardy set A that is not computed by any 2-tardy set B. We also show that there are nonempty E properties Q n (A) such that if Q n (A) then A is properly n-tardy.  ...  Our study of n-tardy sets takes off from where Harrington and Soare left off. We answer all the open questions asked by Harrington and Soare about n-tardy sets.  ...  Is there a properly n + 1-tardy set that is not computed by any n-tardy sets? B.  ...

### On n-Tardy Sets [article]

Peter A. Cholak, Peter M. Gerdes, Karen Lange
2010 arXiv   pre-print
Our study of n-tardy sets takes off from where Harrington and Soare left off. We answer all the open questions asked by Harrington and Soare about n-tardy sets.  ...  We show there is a 3-tardy set A that is not computed by any 2-tardy set B. We also show that there are nonempty E properties Q_n(A) such that if Q_n(A) then A is properly n-tardy.  ...  Very tardy and n-tardy sets A degree is tardy if it is not a prompt degree. A set is very tardy if it is not almost prompt.  ...

### Correlation among tardiness-based measures for scheduling using priority dispatching rules

2013 International Journal of Production Research
Based on the behavior of pdrs in regards with maximum tardiness, two sets of rules have been identified; FIFO, EDD, SLACK and CR from one side and SI, SPT, MOD and MF on the other side.  ...  Thus we have s 2 T = 1 n − 1 n j=1 (T 2 j + T 2 − 2 T T j ) = 1 n − 1 ( n j=1 T 2 j + n T 2 − 2 T n j=1 T j ) = 1 n − 1 ( n j=1 T 2 j + n T 2 − 2 T n T ) = 1 n − 1 (nT 2 rms − n T 2 ) = n n − 1 (T 2 rms  ...

### Page 32 of The Journal of the Operational Research Society Vol. 44, Issue 1 [page]

1993 The Journal of the Operational Research Society
For the odd value of n;, the non-tardy set will contain one more job than the tardy set.  ...  And for the odd value of n; the non-tardy set will contain one more job than the tardy set.  ...

### Page 836 of The Journal of the Operational Research Society Vol. 44, Issue 8 [page]

1993 The Journal of the Operational Research Society
The MA sequence consists of two consecutive subsequences: the first of these includes all the early jobs (i.e. the non-tardy ones) and the second one includes all the tardy jobs.  ...  Denote the set of early jobs in the MA sequence by E* and the set of late (tardy) jobs by L*. Denote also the cardinalities of sets E* and L* by N,+ and N;,., respectively. Clearly, E* U L* = {1,...  ...

### Single Machine Scheduling and Selection to Minimize Total Flow Time with Minimum Number Tardy

Jerzy Kyparisis, Christos Douligeris
1993 Journal of the Operational Research Society
Denote the set of early jobs in the MA sequence by E* and the set of late (tardy) jobs by L*. Denote also the cardinalities of sets E* and L* by N,+ and N;,., respectively. Clearly, E* U L* = {1,...  ...  In addition, one has to consider additional pairs of sets (E’, L’), where the number of early jobs E’ is smaller than N;- but exceeds M.  ...

### Scheduling to minimize maximum earliness and number of tardy jobs where machine idle time is allowed

M Azizoglu, M Koksalan, S K Koksalan
2003 Journal of the Operational Research Society
Let the resulting number of tardy jobs be n’;. Then, n’;+n7(c) is a lower bound on nz.  ...  One branch places the selected job in the early set and the other branch places it in the tardy set. Denote E(a) and T(c) as the early and the tardy sets of partial solution o respectively.  ...

### Federated scheduling for stochastic parallel real-time tasks

Jing Li, Kunal Agrawal, Christopher Gill, Chenyang Lu
2014 2014 IEEE 20th International Conference on Embedded and Real-Time Computing Systems and Applications
All of them guarantee bounded expected tardiness and provide the same capacity augmentation bound; In practice, however, we expect them to provide different performances, both in terms of the task sets  ...  they can schedule and the actual tardiness they guarantee.  ...  [Mills10] If a set of sequential tasks τ low is scheduled on n low cores using GEDF and n low > τi∈τlow E [U i ], then the expected tardiness of each task is bounded.  ...

### Some recent research directions in the computably enumerable sets [article]

Peter Cholak
2013 arXiv   pre-print
As suggested by the title, this paper is a survey of recent results and questions on the collection of computably enumerable sets under inclusion.  ...  This is not a broad survey but one focused on the author's and a few others' current research.  ...  Tardy Sets. As mentioned above, there are some recent results on n-tardy and very tardy sets (a set is very tardy iff it is not almost prompt).  ...

### Page 215 of The Journal of the Operational Research Society Vol. 48, Issue 2 [page]

1997 The Journal of the Operational Research Society
Algorithm C: largest processing time on the critical path Input: 1, a; b;, d; fori=1,...,n. Step 1. Let the set of tardy jobs, T = {@} and k = n. Enter step 2. Step 2. If Cy 1) < dy for alli = 1,2,...  ...  Let the set of tardy jobs, T= {} and k = n. Enter step 2. If Cy a) < dq for alli =1,2,..., k, go to step 4; otherwise let x( j) be the first tardy job.  ...

### Page 663 of The Journal of the Operational Research Society Vol. 54, Issue 6 [page]

2003 The Journal of the Operational Research Society
Let the resulting number of tardy jobs be n’;. Then, n’;+n7(c) is a lower bound on nz.  ...  One branch places the selected job in the early set and the other branch places it in the tardy set. Denote E(a) and T(c) as the early and the tardy sets of partial solution o respectively.  ...

### Multi-objective fuzzy job shop scheduling

Manjeet Singh, Gürsel A. Süer, Feyzan Arikan
2014 International Journal of Fuzzy Computation and Modelling
maximum tardiness, respectively.  ...  The paper focuses on multi-objective scheduling in a job shop environment. One of the useful methods in multi-objective environment is the use of fuzzy operators in modelling the system.  ...  N t ( = 2, 3 and 4) Set Find Find n T TT T max n T = 2 42 22 n T = 3 26 11 n T = 4 22 8 Table 9 Values for T max (setting n T and TT) Set Set Find n T TT T max 2 42  ...

### Two-machine flowshop scheduling to minimize the number of tardy jobs

J N D Gupta, A M A Hariri
1997 Journal of the Operational Research Society
Let the set of tardy jobs, T= {} k =n, and i = 1.  ...  Algorithm C: largest processing time on the critical path Input: 1, a; b;, d; fori=1,...,n. Step 1. Let the set of tardy jobs, T = {@} and k = n. Enter step 2. Step 2. If Cy 1) < dy for alli = 1,2,...  ...

### Two-machine flowshop scheduling to minimize the number of tardy jobs

J N D Gupta, A M A Hariri
1997 Journal of the Operational Research Society
Let the set of tardy jobs, T= {} k =n, and i = 1.  ...  Algorithm C: largest processing time on the critical path Input: 1, a; b;, d; fori=1,...,n. Step 1. Let the set of tardy jobs, T = {@} and k = n. Enter step 2. Step 2. If Cy 1) < dy for alli = 1,2,...  ...

### Minimizing maximum tardiness and number of tardy jobs on parallel machines subject to minimum flow-time

J N D Gupta, A J Ruiz-torres, S Webster
2003 Journal of the Operational Research Society
V/,,) be an ordered set of Minimizing tardiness and tardy jobs on parallel machines Since the inputs require simple sorts that can O(n log) time, the overall time complexity of O(n logn) In algorithm P  ...  Go to Step 6. 0 and otherwise | set Eis <-*- <b, If Minimizing tardiness and tardy jobs on parallel machines Step 4: If jEA,, set w=(\j), A;=(A;U/), and enter Step 5; otherwise, schedule job / on machine  ...
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