scholarly journals A Heuristic Based On Makespan Lower Bounds in Flow Shops with Multiple Processors

2017 ◽  
Vol 5 (1) ◽  
pp. 33-43
Author(s):  
Vikram Srinivasan ◽  
Daryl Santos
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Flow Shop ◽  
Hybrid Flow Shop ◽  
Prior Work ◽  
Processing Stage ◽  
Flow Shops ◽  
Multiple Processors ◽  
Np Complete ◽  
Better Than

Minimum makespan scheduling of Flow Shops with Multiple Processors (FSMPs), also known as the Hybrid Flow Shop (HFS), is classified as NP complete. Thus, the FSMP largely depends on strong heuristics to develop solutions to makespan scheduling instances. An FSMP consists of m stages wherein each stage has one or more processors through which n jobs are scheduled. This paper presents a heuristic based on the lower bound developed in a prior work in order to determine good makespan solutions in the FSMP environment. In the environment studied in this work, the multiple machines available at a particular processing stage are identical processors. In order to evaluate the proposed heuristic, its performance is compared to makespans obtained via the use of modified pure flow shop heuristics. Results show that the proposed heuristic is indeed a strong heuristic for the FSMP and it provides makespans that are better than those provided by some of the already existing pure flow shop heuristics that have been adapted for the FSMP environment.

scholarly journals Lower bounds to eigenvalues of the Schrödinger equation by solution of a 90-y challenge

2020 ◽  
Vol 117 (28) ◽  
pp. 16181-16186
Author(s):  
Rocco Martinazzo ◽  
Eli Pollak
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Spectral Features ◽  
Numerical Examples ◽  
Tunneling Splittings ◽  
Order Of Magnitude ◽  
Bound Theorem ◽  
Math Phys ◽  
Better Than

The Ritz upper bound to eigenvalues of Hermitian operators is essential for many applications in science. It is a staple of quantum chemistry and physics computations. The lower bound devised by Temple in 1928 [G. Temple,Proc. R. Soc. A Math. Phys. Eng. Sci.119, 276–293 (1928)] is not, since it converges too slowly. The need for a good lower-bound theorem and algorithm cannot be overstated, since an upper bound alone is not sufficient for determining differences between eigenvalues such as tunneling splittings and spectral features. In this paper, after 90 y, we derive a generalization and improvement of Temple’s lower bound. Numerical examples based on implementation of the Lanczos tridiagonalization are provided for nontrivial lattice model Hamiltonians, exemplifying convergence over a range of 13 orders of magnitude. This lower bound is typically at least one order of magnitude better than Temple’s result. Its rate of convergence is comparable to that of the Ritz upper bound. It is not limited to ground states. These results complement Ritz’s upper bound and may turn the computation of lower bounds into a staple of eigenvalue and spectral problems in physics and chemistry.

Global lower bounds for flow shops with multiple processors

1995 ◽  
Vol 80 (1) ◽  
pp. 112-120 ◽  
Author(s):  
D.L. Santos ◽  
J.L. Hunsucker ◽  
D.E. Deal
Keyword(s):  
Lower Bounds ◽  
Flow Shops ◽  
Multiple Processors

Lower bounds for the Perron root of a non-negative irreducible matrix

1982 ◽  
Vol 92 (1) ◽  
pp. 49-54 ◽  
Author(s):  
Emeric Deutsch
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Rayleigh Quotient ◽  
Irreducible Matrix ◽  
Perron Root ◽  
Better Than

We derive a family of lower bounds for the Perron root of a non-negative irreducible matrix. These lower bounds are better than certain lower bounds of the Rayleigh quotient type, also derived in this paper. For the particular case of a symmetric non-negative irreducible matrix, our lower bound is always better than a corresponding Rayleigh quotient and, as shown in example 4, it can be infinitely better.

Double vertex-edge domination

2017 ◽  
Vol 09 (04) ◽  
pp. 1750045 ◽  
Author(s):  
Balakrishna Krishnakumari ◽  
Mustapha Chellali ◽  
Yanamandram B. Venkatakrishnan
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Dominating Set ◽  
Domination Number ◽  
Edge Incident ◽  
Minimum Cardinality ◽  
Edge Dominating Set ◽  
Number Formula ◽  
Number Of Leaves ◽  
Np Complete

A vertex [Formula: see text] of a graph [Formula: see text] is said to [Formula: see text]-dominate every edge incident to [Formula: see text], as well as every edge adjacent to these incident edges. A set [Formula: see text] is a vertex-edge dominating set (double vertex-edge dominating set, respectively) if every edge of [Formula: see text] is [Formula: see text]-dominated by at least one vertex (at least two vertices) of [Formula: see text] The minimum cardinality of a vertex-edge dominating set (double vertex-edge dominating set, respectively) of [Formula: see text] is the vertex-edge domination number [Formula: see text] (the double vertex-edge domination number [Formula: see text], respectively). In this paper, we initiate the study of double vertex-edge domination. We first show that determining the number [Formula: see text] for bipartite graphs is NP-complete. We also prove that for every nontrivial connected graphs [Formula: see text] [Formula: see text] and we characterize the trees [Formula: see text] with [Formula: see text] or [Formula: see text] Finally, we provide two lower bounds on the double ve-domination number of trees and unicycle graphs in terms of the order [Formula: see text] the number of leaves and support vertices, and we characterize the trees attaining the lower bound.

The (Coarse) Fine-Grained Structure of NP-Hard SAT and CSP Problems

2022 ◽  
Vol 14 (1) ◽  
pp. 1-54
Author(s):  
Victor Lagerkvist ◽  
Magnus Wahlström
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Constraint Satisfaction Problems ◽  
Close Connection ◽  
Algebraic Characterization ◽  
Sat Problem ◽  
Fine Grained ◽  
Algebraic Framework ◽  
Fine Grained Structure ◽  
Np Complete

We study the fine-grained complexity of NP-complete satisfiability (SAT) problems and constraint satisfaction problems (CSPs) in the context of the strong exponential-time hypothesis (SETH) , showing non-trivial lower and upper bounds on the running time. Here, by a non-trivial lower bound for a problem SAT (Γ) (respectively CSP (Γ)) with constraint language Γ, we mean a value c 0 > 1 such that the problem cannot be solved in time O ( c n ) for any c < c 0 unless SETH is false, while a non-trivial upper bound is simply an algorithm for the problem running in time O ( c n ) for some c < 2. Such lower bounds have proven extremely elusive, and except for cases where c 0 =2 effectively no such previous bound was known. We achieve this by employing an algebraic framework, studying constraint languages Γ in terms of their algebraic properties. We uncover a powerful algebraic framework where a mild restriction on the allowed constraints offers a concise algebraic characterization. On the relational side we restrict ourselves to Boolean languages closed under variable negation and partial assignment, called sign-symmetric languages. On the algebraic side this results in a description via partial operations arising from system of identities, with a close connection to operations resulting in tractable CSPs, such as near unanimity operations and edge operations . Using this connection we construct improved algorithms for several interesting classes of sign-symmetric languages, and prove explicit lower bounds under SETH. Thus, we find the first example of an NP-complete SAT problem with a non-trivial algorithm which also admits a non-trivial lower bound under SETH. This suggests a dichotomy conjecture with a close connection to the CSP dichotomy theorem: an NP-complete SAT problem admits an improved algorithm if and only if it admits a non-trivial partial invariant of the above form.

scholarly journals Multi-stage appointment scheduling for Outpatient Chemotherapy Unit: a case study

2021 ◽  
Author(s):  
Asma BOURAS ◽  
Malek Masmoudi ◽  
Nour El Houda SAADANI ◽  
Zied BAHROUN ◽  
Mohamed Amine ABDELJAOUAD
Keyword(s):  
Lower Bound ◽  
Flow Shop ◽  
Treatment Time ◽  
Flow Shop Scheduling ◽  
Mixed Integer ◽  
Computational Time ◽  
Mixed Integer Program ◽  
Hybrid Flow Shop ◽  
Multi Stage ◽  
Hybrid Flow Shop Scheduling

This paper deals with a multi stage hybrid flow-shop problem (HFSP) that arises in a privately Chemotherapy clinic. It aims to optimize the makespan of the daily chemotherapy activity. Each patient must respect the cyclic nature of chemotherapy treatment plans made by his referent oncologist while taking into account the high variability in resource requirements (treatment time, nurse time, pharmacy time). The problem requires the assignment of chemotherapy patients to oncologists, pharmacists, chemotherapy beds or chairs and nurses over a 1-day period. We provided a Mixed Integer Program (MIP) to model this issue, which can be considered as a five-stage hybrid flow-shop scheduling problem with additional resources, dedicated machines, and no-wait constraints.  Since this problem is known to be NP-hard, we provided a lower bound expression and developed an approximated solving algorithm: a tabu search inspired metaheuristic based on a constructive heuristic that can quickly reach satisfying results. To assess the empirical performance of the proposed approach, we conducted experiments on randomly generated instances based on real-world data of a Tunisian private clinic: Clinique Ennasr. Computational experiments show the efficiency of the proposed procedures: The mathematical model provided optimal solutions in reasonable computational time only for small instances (up to 10 patients).   Meta-heuristic’s results demonstrate, also, that the proposed approach offers good results in terms of solution quality and computational times with an average relative gap to the MIP solution equal to 3.13% and to the lower bound equal to 5.37% for small instances (up to 15 patients). The same gap to the lower bound increases to 25% for medium and large size instances (20-50 patients).

LOWER BOUNDS ON THE SECOND ORDER NONLINEARITY OF BOOLEAN FUNCTIONS

2011 ◽  
Vol 22 (06) ◽  
pp. 1331-1349 ◽  
Author(s):  
XUELIAN LI ◽  
YUPU HU ◽  
JUNTAO GAO
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Lower Bounds ◽  
Boolean Functions ◽  
Second Order ◽  
Algebraic Degree ◽  
Positive Integers ◽  
Second Order Nonlinearity ◽  
Better Than ◽  
Monomial Functions

It is a difficult task to compute the r-th order nonlinearity of a given function with algebraic degree strictly greater than r > 1. Though lower bounds on the second order nonlinearity are known only for a few particular functions, the majority of which are cubic. We investigate lower bounds on the second order nonlinearity of cubic Boolean functions [Formula: see text], where [Formula: see text], dl = 2il + 2jl + 1, m, il and jl are positive integers, n > il > jl. Furthermore, for a class of Boolean functions [Formula: see text] we deduce a tighter lower bound on the second order nonlinearity of the functions, where [Formula: see text], dl = 2ilγ + 2jlγ + 1, il > jl and γ ≠ 1 is a positive integer such that gcd(n,γ) = 1. Lower bounds on the second order nonlinearity of cubic monomial Boolean functions, represented by fμ(x) = Tr(μx2i+2j+1), [Formula: see text], i and j are positive integers such that i > j, were obtained by Gode and Gangopadhvay in 2009. In this paper, we first extend the results of Gode and Gangopadhvay from monomial Boolean functions to Boolean functions with more trace terms. We further generalize and improve the results to a wider range of n. Our bounds are better than those of Gode and Gangopadhvay for monomial functions fμ(x). Especially, our lower bounds on the second order nonlinearity of some Boolean functions F(x) are better than the existing ones.

A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers

10.37236/1188 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Geoffrey Exoo
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Ramsey Numbers

For $k \geq 5$, we establish new lower bounds on the Schur numbers $S(k)$ and on the k-color Ramsey numbers of $K_3$.

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