scholarly journals New Upper Bounds for the Size of Permutation Codes via Linear Programming

10.37236/407 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Mathieu Bogaerts
Keyword(s):  
Linear Programming ◽  
Permutation Group ◽  
Association Scheme ◽  
Optimization Problem ◽  
Hamming Distance ◽  
Upper Bounds ◽  
Linear Inequalities ◽  
Permutation Codes ◽  
Permutation Code ◽  
Irreducible Characters

An $(n,d)$-permutation code of size $s$ is a subset $C$ of $S_n$ with $s$ elements such that the Hamming distance $d_H$ between any two distinct elements of $C$ is at least equal to $d$. In this paper, we give new upper bounds for the maximal size $\mu(n,d)$ of an $(n,d)$-permutation code of degree $n$ with $11\le n\le 14$. In order to obtain these bounds, we use the structure of association scheme of the permutation group $S_n$ and the irreducible characters of $S_n$. The upper bounds for $\mu(n,d)$ are determined solving an optimization problem with linear inequalities.

scholarly journals Max-min solution approach for multi-objective matrix game with fuzzy goals

2016 ◽  
Vol 26 (1) ◽  
pp. 51-60 ◽  
Author(s):  
Sandeep Kumar
Keyword(s):  
Linear Programming ◽  
Linear Programming Problem ◽  
Optimization Problem ◽  
Matrix Game ◽  
Game Model ◽  
Solution Approach ◽  
Multi Objective ◽  
Fuzzy Goals ◽  
Fuzzy Goal ◽  
Zero Sum

In this paper, we consider a multi-objective two person zero-sum matrix game with fuzzy goals, assuming that each player has a fuzzy goal for each of the payoffs. The max-min solution is formulated for this multi-objective game model, in which the optimization problem for each player is a linear programming problem. Every developed model for each player is demonstrated through a numerical example.

scholarly journals A Bound on Permutation Codes

10.37236/2929 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Jürgen Bierbrauer ◽  
Klaus Metsch
Keyword(s):  
Projective Plane ◽  
Symmetric Group ◽  
Minimum Distance ◽  
Main Part ◽  
Latin Squares ◽  
Permutation Codes ◽  
Permutation Code ◽  
Mutually Orthogonal Latin Squares ◽  
Hamming Metric ◽  
Completion Theorem

Consider the symmetric group $S_n$ with the Hamming metric. A  permutation code on $n$ symbols is a subset $C\subseteq S_n.$ If $C$ has minimum distance $\geq n-1,$ then $\vert C\vert\leq n^2-n.$ Equality can be reached if and only if a projective plane of order $n$ exists. Call $C$ embeddable if it is contained in a permutation code of minimum distance $n-1$ and cardinality $n^2-n.$ Let $\delta =\delta (C)=n^2-n-\vert C\vert$ be the deficiency of the permutation code $C\subseteq S_n$ of minimum distance $\geq n-1.$We prove that $C$ is embeddable if either $\delta\leq 2$ or if $(\delta^2-1)(\delta +1)^2<27(n+2)/16.$ The main part of the proof is an adaptation of the method used to obtain the famous Bruck completion theorem for mutually orthogonal latin squares.

scholarly journals Verifying a Solver for Linear Mixed Integer Arithmetic in Isabelle/HOL

2020 ◽  
pp. 233-250
Author(s):  
Ralph Bottesch ◽  
Max W. Haslbeck ◽  
Alban Reynaud ◽  
René Thiemann
Keyword(s):  
Linear Programming ◽  
Branch And Bound ◽  
Decision Procedure ◽  
Simplex Algorithm ◽  
Upper Bounds ◽  
Mixed Integer ◽  
Small Solution ◽  
Code Generator ◽  
Integer Arithmetic ◽  
Termination Proofs

AbstractWe implement a decision procedure for linear mixed integer arithmetic and formally verify its soundness in Isabelle/HOL. We further integrate this procedure into one application, namely into , a formally verified certifier to check untrusted termination proofs. This checking involves assertions of unsatisfiability of linear integer inequalities; previously, only a sufficient criterion for such checks was supported. To verify the soundness of the decision procedure, we first formalize the proof that every satisfiable set of linear integer inequalities also has a small solution, and give explicit upper bounds. To this end we mechanize several important theorems on linear programming, including statements on integrality and bounds. The procedure itself is then implemented as a branch-and-bound algorithm, and is available in several languages via Isabelle’s code generator. It internally relies upon an adapted version of an existing verified incremental simplex algorithm.

Association Schemes for Ordered Orthogonal Arrays and (T, M, S)-Nets

1999 ◽  
Vol 51 (2) ◽  
pp. 326-346 ◽  
Author(s):  
W. J. Martin ◽  
D. R. Stinson
Keyword(s):  
Linear Programming ◽  
Association Scheme ◽  
Type Theorem ◽  
Orthogonal Arrays ◽  
Association Schemes ◽  
Theoretic Approach ◽  
Linear Programming Bound ◽  
Ordered Orthogonal Arrays

AbstractIn an earlier paper [10], we studied a generalized Rao bound for ordered orthogonal arrays and (T, M, S)-nets. In this paper, we extend this to a coding-theoretic approach to ordered orthogonal arrays. Using a certain association scheme, we prove a MacWilliams-type theorem for linear ordered orthogonal arrays and linear ordered codes as well as a linear programming bound for the general case. We include some tables which compare this bound against two previously known bounds for ordered orthogonal arrays. Finally we show that, for even strength, the LP bound is always at least as strong as the generalized Rao bound.

Chandpur Enterprises Limited, Steel Division

2017 ◽  
pp. 1-4
Author(s):  
Samuel E. Bodily ◽  
Akshay Mittal
Keyword(s):  
Linear Programming ◽  
Linear Model ◽  
Raw Materials ◽  
Nonlinear Models ◽  
Upper Bounds ◽  
Raw Material ◽  
Shadow Prices ◽  
Lower And Upper Bounds ◽  
Optimization Function ◽  
Introductory Class

The managing director of a steel plant faces the decision of how much of each raw material to order for the plant for the following month. Due to lower and upper bounds on the amounts of each raw material in a batch and varying amounts of electricity and time consumed for different raw materials, one can't simply use the cheapest raw material. A linear program and the solver optimization function of Excel will provide the optimal amounts that meet the constraints. Interestingly, the best mixture for a batch is not the best mixture for a monthly plan. Shadow prices indicate the value of relaxing constraints. The typical monthly model from a student will be nonlinear, although it can be written as a linear model. This case provides the basis for an introductory class on linear programming and linear versus nonlinear models.

scholarly journals A dual-inexact fuzzy stochastic model for water resources management and non-point source pollution mitigation under multiple uncertainties

2014 ◽  
Vol 18 (5) ◽  
pp. 1793-1803 ◽  
Author(s):  
C. Dong ◽  
Q. Tan ◽  
G.-H. Huang ◽  
Y.-P. Cai
Keyword(s):  
Linear Programming ◽  
Point Source ◽  
Upper Bounds ◽  
Fuzzy Linear Programming ◽  
Chance Constrained Programming ◽  
Lower And Upper Bounds ◽  
Point Source Pollution ◽  
Pollution Mitigation ◽  
Trade Offs ◽  
Non Point Source Pollution

Abstract. In this research, a dual-inexact fuzzy stochastic programming (DIFSP) method was developed for supporting the planning of water and farmland use management system considering the non-point source pollution mitigation under uncertainty. The random boundary interval (RBI) was incorporated into DIFSP through integrating fuzzy linear programming (FLP) and chance-constrained programming (CCP) approaches within an interval linear programming (ILP) framework. This developed method could effectively tackle the uncertainties expressed as intervals and fuzzy sets. Moreover, the lower and upper bounds of RBI are continuous random variables, and the correlation existing between the lower and upper bounds can be tackled in RBI through the joint probability distribution function. And thus the subjectivity of decision making is greatly reduced, enhancing the stability and robustness of obtained solutions. The proposed method was then applied to solve a water and farmland use planning model (WFUPM) with non-point source pollution mitigation. The generated results could provide decision makers with detailed water supply–demand schemes involving diversified water-related activities under preferred satisfaction degrees. These useful solutions could allow more in-depth analyses of the trade-offs between humans and environment, as well as those between system optimality and reliability. In addition, comparative analyses on the solutions obtained from ICCP (Interval chance-constraints programming) and DIFSP demonstrated the higher application of this developed approach for supporting the water and farmland use system planning.

Sign in / Sign up

Export Citation Format

Share Document