A mixed 0-1 linear programming formulation for the exact solution of the minimum linear arrangement problem

2009 ◽  
Vol 3 (4) ◽  
pp. 513-520 ◽  
Author(s):  
André R. S. Amaral
Keyword(s):  
Exact Solution ◽  
Linear Programming ◽  
Programming Formulation ◽  
Linear Arrangement ◽  
Arrangement Problem ◽  
Linear Programming Formulation ◽  
Linear Arrangement Problem

scholarly journals Generating lower bounds for the linear arrangement problem

1995 ◽  
Vol 59 (2) ◽  
pp. 137-151 ◽  
Author(s):  
Weiguo Liu ◽  
Anthony Vannelli
Keyword(s):  
Lower Bounds ◽  
Linear Arrangement ◽  
Arrangement Problem ◽  
Linear Arrangement Problem

scholarly journals Tournaments as Feedback Arc Sets

10.37236/1214 ◽  
1995 ◽  
Vol 2 (1) ◽  
Author(s):  
Garth Isaak
Keyword(s):  
Linear Programming ◽  
Lower Bounds ◽  
Integer Linear Programming ◽  
Minimum Size ◽  
Programming Formulation ◽  
Binary Expansion ◽  
Feedback Arc Set ◽  
Disjoint Cycles ◽  
Linear Programming Formulation ◽  
Integer Linear Programming Formulation

We examine the size $s(n)$ of a smallest tournament having the arcs of an acyclic tournament on $n$ vertices as a minimum feedback arc set. Using an integer linear programming formulation we obtain lower bounds $s(n) \geq 3n - 2 - \lfloor \log_2 n \rfloor$ or $s(n) \geq 3n - 1 - \lfloor \log_2 n \rfloor$, depending on the binary expansion of $n$. When $n = 2^k - 2^t$ we show that the bounds are tight with $s(n) = 3n - 2 - \lfloor \log_2 n \rfloor$. One view of this problem is that if the 'teams' in a tournament are ranked to minimize inconsistencies there is some tournament with $s(n)$ 'teams' in which $n$ are ranked wrong. We will also pose some questions about conditions on feedback arc sets, motivated by our proofs, which ensure equality between the maximum number of arc disjoint cycles and the minimum size of a feedback arc set in a tournament.

Preemptive scheduling with transportation delays between machines

2020 ◽  
Vol 15 (3) ◽  
pp. 829-847
Author(s):  
Ryma Zineb Badaoui ◽  
Mourad Boudhar ◽  
Mohammed Dahane
Keyword(s):  
Linear Programming ◽  
Neighborhood Search ◽  
Preemptive Scheduling ◽  
Scheduling Problem ◽  
Programming Formulation ◽  
Content Type ◽  
Decision Variables ◽  
Linear Programming Formulation ◽  
Local Search Strategy ◽  
Two Stages

Purpose This paper studies the preemptive scheduling problem of independent jobs on identical machines. The purpose of this paper is to minimize the makespan under the imposed constraints, namely, the ones that relate the transportation delays which are required to transport a preempted job from one machine to another. This study considers the case when the transportation delays are variable. Design/methodology/approach The contribution is twofold. First, this study proposes a new linear programming formulation in real and binary decision variables. Then, this study proposes and implements a solution strategy, which consists of two stages. The goal of the first stage is to obtain the best machines order using a local search strategy. For the second stage, the objective is to determine the best possible sequence of jobs. To solve the preemptive scheduling problem with transportation delays, this study proposes a heuristic and two metaheuristics (simulated annealing and variable neighborhood search), each with two modes of evaluation. Findings Computational experiments are presented and discussed on randomly generated instances. Practical implications The study has implications in various industrial environments when the preemption of jobs is allowed. Originality/value This study proposes a new linear programming formulation for the problem with variable transportation delays as well as a corresponding heuristic and metaheuristics.

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