scholarly journals Alternate Formulations of the Reducibility Problem of Open Shop Sequences Minimizing the Makespan

1970 ◽  
Vol 8 (1-2) ◽  
pp. 243-254
Author(s):  
Tanka Nath Dhamala
Keyword(s):  
Conflict Resolution ◽  
Decision Problem ◽  
Completion Time ◽  
Open Shop ◽  
Comparability Graph ◽  
Open Problems ◽  
Special Cases ◽  
Sequence Graph ◽  
Reducibility Problem ◽  
Maximum Completion Time

The decision problem whether a given open shop sequence, minimizing the maximum completion time, is irreducible has been considered in the last 20 years. The problem has diversified applications in industries and communications. By now, a number of algorithms based on the specific properties of the corresponding sequence graph are proposed. Thus the problem is solved only partially and only in some special cases, but not in general yet. A number of open problems and conjectures carried out in this research have been posed, so far. In this paper, we present a brief sketch of these ideas with different formulations of the reducibility of open shop sequences and expose how important are the roles of conflict resolution reaching a conclusion to its end. Paths on the so-called H-comparability graphs with respect to the implication classes play vital roles in it.Key words: Scheduling; Sequencing; Open shop; Comparability graph; reducibility; ComplexityDOI: http://dx.doi.org/10.3126/jie.v8i1-2.5117Journal of the Institute of Engineering Vol. 8, No. 1&2, 2010/2011Page: 243-254Uploaded Date: 20 July, 2011

scholarly journals Some Discrete Optimization Problems With Hamming and H-Comparability Graphs

2010 ◽  
Vol 27 (1-2) ◽  
pp. 167-176
Author(s):  
Tanka Nath Dhamala
Keyword(s):  
Discrete Optimization ◽  
Optimization Problems ◽  
Open Shop ◽  
Comparability Graph ◽  
Hamming Graph ◽  
Acyclic Orientation ◽  
Open Shop Problem ◽  
Sequence Graph ◽  
Hamming Graphs ◽  
Discrete Optimization Problems

Any H-comparability graph contains a Hamming graph as spanningsubgraph. An acyclic orientation of an H-comparability graph contains an acyclic orientation of the spanning Hamming graph, called sequence graph in the classical open-shop scheduling problem. We formulate different discrete optimization problems on the Hamming graphs and on H-comparability graphs and consider their complexity and relationship. Moreover, we explore the structures of these graphs in the class of irreducible sequences for the open shop problem in this paper.

Special Cases of the Decision Problem

1951 ◽  
Vol 49 (22) ◽  
pp. 203-221 ◽  
Author(s):  
Alonzo Church
Keyword(s):  
Decision Problem ◽  
Special Cases

Minimizing the total completion time in a unit-time open shop with release times

1997 ◽  
Vol 20 (5) ◽  
pp. 207-212 ◽  
Author(s):  
Thomas Tautenhahn ◽  
Gerhard J. Woeginger
Keyword(s):  
Completion Time ◽  
Open Shop ◽  
Total Completion Time ◽  
Release Times

scholarly journals On Reay's Relaxed Tverberg Conjecture and Generalizations of Conway's Thrackle Conjecture

10.37236/6516 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Megumi Asada ◽  
Ryan Chen ◽  
Florian Frick ◽  
Frederick Huang ◽  
Maxwell Polevy ◽  
...  
Keyword(s):  
Geometric Property ◽  
Convex Sets ◽  
Convex Hulls ◽  
Open Problems ◽  
Higher Dimensional ◽  
Special Cases ◽  
Minimum Number ◽  
Point Set ◽  
Colored Version ◽  
Special Case

Reay's relaxed Tverberg conjecture and Conway's thrackle conjecture are open problems about the geometry of pairwise intersections. Reay asked for the minimum number of points in Euclidean $d$-space that guarantees any such point set admits a partition into $r$ parts, any $k$ of whose convex hulls intersect. Here we give new and improved lower bounds for this number, which Reay conjectured to be independent of $k$. We prove a colored version of Reay's conjecture for $k$ sufficiently large, but nevertheless $k$ independent of dimension $d$. Pairwise intersecting convex hulls have severely restricted combinatorics. This is a higher-dimensional analogue of Conway's thrackle conjecture or its linear special case. We thus study convex-geometric and higher-dimensional analogues of the thrackle conjecture alongside Reay's problem and conjecture (and prove in two special cases) that the number of convex sets in the plane is bounded by the total number of vertices they involve whenever there exists a transversal set for their pairwise intersections. We thus isolate a geometric property that leads to bounds as in the thrackle conjecture. We also establish tight bounds for the number of facets of higher-dimensional analogues of linear thrackles and conjecture their continuous generalizations.

scholarly journals Branch and Bound Method to Solve Multi Objectives Function

2016 ◽  
Vol 12 (3) ◽  
pp. 5964-5974
Author(s):  
Tahani Jabbar Kahribt ◽  
Mohammed Kadhim Al- Zuwaini
Keyword(s):  
Lower Bounds ◽  
Branch And Bound ◽  
Single Machine ◽  
Upper Bound ◽  
Completion Time ◽  
Heuristic Method ◽  
Total Weighted Completion Time ◽  
Branch And Bound Method ◽  
Special Cases ◽  
Weighted Completion Time

This paper  presents  a  branch  and  bound  algorithm  for  sequencing  a  set  of  n independent  jobs  on  a single  machine  to  minimize sum of the discounted total weighted completion time and maximum lateness,  this problems is NP-hard. Two lower bounds were proposed and heuristic method to get an upper bound. Some special cases were  proved and some dominance rules were suggested and proved, the problem solved with up to 50 jobs.

On the reduction of the decision problem. First paper. Ackermann prefix, a single binary predicate

1939 ◽  
Vol 4 (1) ◽  
pp. 1-9 ◽  
Author(s):  
László Kalmár
Keyword(s):  
Decision Problem ◽  
Reduction Process ◽  
The Other ◽  
First Order ◽  
Other Hand ◽  
Special Cases ◽  
Binary Predicate ◽  
Ternary Variables ◽  
Order Predicate Calculus ◽  
Predicate Variable

1. Although the decision problem of the first order predicate calculus has been proved by Church to be unsolvable by any (general) recursive process, perhaps it is not superfluous to investigate the possible reductions of the general problem to simple special cases of it. Indeed, the situation after Church's discovery seems to be analogous to that in algebra after the Ruffini-Abel theorem; and investigations on the reduction of the decision problem might prepare the way for a theory in logic, analogous to that of Galois.It has been proved by Ackermann that any first order formula is equivalent to another having a prefix of the form(1) (Ex1)(x2)(Ex3)(x4)…(xm).On the other hand, I have proved that any first order formula is equivalent to some first order formula containing a single, binary, predicate variable. In the present paper, I shall show that both results can be combined; more explicitly, I shall prove theTheorem. To any given first order formula it is possible to construct an equivalent one with a prefix of the form (1) and a matrix containing no other predicate variable than a single binary one.2. Of course, this theorem cannot be proved by a mere application of the Ackermann reduction method and mine, one after the other. Indeed, Ackermann's method requires the introduction of three auxiliary predicate variables, two of them being ternary variables; on the other hand, my reduction process leads to a more complicated prefix, viz.,(2) (Ex1)…(Exm)(xm+1)(xm+2)(Exm+3)(Exm+4).

scholarly journals Minimizing Total Completion Time For Preemptive Scheduling With Release Dates And Deadline Constraints

2014 ◽  
Vol 39 (1) ◽  
pp. 17-26 ◽  
Author(s):  
Cheng He ◽  
Hao Lin ◽  
Yixun Lin ◽  
Junmei Dou
Keyword(s):  
Completion Time ◽  
Total Completion Time ◽  
Release Dates ◽  
Preemptive Scheduling ◽  
Release Date ◽  
Enumeration Algorithm ◽  
Processing Times ◽  
Special Cases ◽  
Deadline Constraints ◽  
Special Case

Abstract It is known that the single machine preemptive scheduling problem of minimizing total completion time with release date and deadline constraints is NP- hard. Du and Leung solved some special cases by the generalized Baker's algorithm and the generalized Smith's algorithm in O(n2) time. In this paper we give an O(n2) algorithm for the special case where the processing times and deadlines are agreeable. Moreover, for the case where the processing times and deadlines are disagreeable, we present two properties which could enable us to reduce the range of the enumeration algorithm

scholarly journals Polynomial Algorithm for Constructing Pareto-Optimal Schedules for Problem 1∣rj∣Lmax,Cmax

2020 ◽  
Author(s):  
Alexander A. Lazarev ◽  
Nikolay Pravdivets
Keyword(s):  
Completion Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Single Machine Scheduling ◽  
Maximum Lateness ◽  
Pareto Optimal ◽  
Due Dates ◽  
Processing Times ◽  
Pareto Optimal Set ◽  
Maximum Completion Time

In this chapter, we consider the single machine scheduling problem with given release dates, processing times, and due dates with two objective functions. The first one is to minimize the maximum lateness, that is, maximum difference between each job due date and its actual completion time. The second one is to minimize the maximum completion time, that is, to complete all the jobs as soon as possible. The problem is NP-hard in the strong sense. We provide a polynomial time algorithm for constructing a Pareto-optimal set of schedules on criteria of maximum lateness and maximum completion time, that is, problem 1 ∣ r j ∣ L max , C max , for the subcase of the problem: d 1 ≤ d 2 ≤ … ≤ d n ; d 1 − r 1 − p 1 ≥ d 2 − r 2 − p 2 ≥ … ≥ d n − r n − p n .

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