On integer linear programming formulations of a patrol boat scheduling problem with complete coverage requirements

2019 ◽  
pp. 154851291988395 ◽  
Author(s):  
Paul A Chircop ◽  
Timothy J Surendonk
Keyword(s):  
Linear Programming ◽  
Objective Function ◽  
Column Generation ◽  
Integer Linear Programming ◽  
Minimum Size ◽  
Scheduling Problem ◽  
Complete Coverage ◽  
Continuous Coverage ◽  
Alternative Objective Function ◽  
Theoretical Results

The Patrol Boat Scheduling Problem with Complete Coverage (PBSPCC) is concerned with finding a minimum size patrol boat fleet to provide continuous coverage at a set of maritime patrol regions, ensuring that there is at least one vessel on station in each patrol region at any given time. This requirement is complicated by the necessity for patrol vessels to be replenished on a regular basis in order to carry out patrol operations indefinitely. In this paper, we establish a number of important theoretical results for the PBSPCC. In particular, we establish a set of conditions under which an alternative objective function (minimize the total time not spent on patrol) can be used to derive a minimum size fleet. Preliminary results suggest that the new theoretical insights can be used as part of an acceleration strategy to improve the column generation runtime performance.

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.

scholarly journals A Hybrid IP/GA Approach to the Parallel Production Lines Scheduling Problem

2016 ◽  
Vol 2016 ◽  
pp. 1-11
Author(s):  
Huizhi Ren ◽  
Shenshen Sun
Keyword(s):  
Linear Programming ◽  
Integer Linear Programming ◽  
Mixed Integer Linear Programming ◽  
Time Window ◽  
Valid Inequalities ◽  
Mixed Integer ◽  
Scheduling Problem ◽  
Production Lines ◽  
Solution Approach ◽  
Cp Decomposition

A special parallel production lines scheduling problem is studied in this paper. Considering the time window and technical constraints, a mixed integer linear programming (MILP) model is formulated for the problem. A few valid inequalities are deduced and a hybrid mixed integer linear programming/constraint programming (MILP/CP) decomposition strategy is introduced. Based on them, a hybrid integer programming/genetic algorithm (IP/GA) approach is proposed to solve the problem. At last, the numerical experiments demonstrate that the proposed solution approach is effective and efficient.

Metode Pemrograman Linier Biner untuk Penjadwalan Kelas Tambahan

2020 ◽  
Vol 25 (2) ◽  
pp. 39-44
Author(s):  
Rena Melawati ◽  
◽  
Sri Pudjaprasetya ◽  
Novry Erwina ◽  
◽  
...  
Keyword(s):  
Mathematical Modeling ◽  
Linear Programming ◽  
Student Learning ◽  
Integer Linear Programming ◽  
Scheduling Problem ◽  
Binary Matrix ◽  
Binary Linear Programming ◽  
Teaching Load ◽  
Programming Tools ◽  
Mathematical Equations

This project discusses the methods of binary linear programming with an application of scheduling problem for students and teachers in an addi-tional class program. This topic was inspired by the the problems faced by several schools, which in preparing students for exams, often need to or-ganize additional class programs. Such a program is certainly efficient, because it can save teaching load, but is quite complicated in terms of scheduling. The desired schedule must fit the student learning times and teacher availability. Using mathematical modeling, conditions and regula-tions are expressed in the form of mathematical equations and or inequali-ties, which act as constraints. Next, formulating the problem in the stand-ard form, allow us to implement the integer linear programming tools available in Matlab. The output we obtained, were in the form of a binary matrix, directly representing the student learning schedule, as well as the teacher's teaching schedule.

scholarly journals Column Generation in Integer Linear Programming

2013 ◽  
pp. 235-259
Author(s):  
Irène Loiseau ◽  
Alberto Ceselli ◽  
Nelson Maculan ◽  
Matteo Salani
Keyword(s):  
Linear Programming ◽  
Column Generation ◽  
Integer Linear Programming

scholarly journals Transportation Optimization Models for Intermodal Networks with Fuzzy Node Capacity, Detour Factor, and Vehicle Utilization Constraints

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2109
Author(s):  
Chia-Nan Wang ◽  
Thanh-Tuan Dang ◽  
Tran Quynh Le ◽  
Panitan Kewcharoenwong
Keyword(s):  
Linear Programming ◽  
Objective Function ◽  
Integer Linear Programming ◽  
Mixed Integer Linear Programming ◽  
Deterministic Model ◽  
Mixed Integer ◽  
Model Parameters ◽  
Total Cost ◽  
The Impact ◽  
Node Capacity

This paper develops a mathematical model for intermodal freight transportation. It focuses on determining the flow of goods, the number of vehicles, and the transferred volume of goods transported from origin points to destination points. The model of this article is to minimize the total cost, which consists of fixed costs, transportation costs, intermodal transfer costs, and CO2 emission costs. It presents a mixed integer linear programming (MILP) model that minimizes total costs, and a fuzzy mixed integer linear programming (FMILP) model that minimizes imprecise total costs under conditions of uncertain data. In the models, node capacity, detour, and vehicle utilization are incorporated to estimate the performance impact. Additionally, a computational experiment is carried out to evaluate the impact of each constraint and to analyze the characteristics of the models under different scenarios. Developed models are tested using real data from a case study in Southern Vietnam in order to demonstrate their effectiveness. The results indicate that, although the objective function (total cost) increased by 20%, the problem became more realistic to address when the model was utilized to solve the constraints of node capacity, detour, and vehicle utilization. In addition, on the basis of the FMILP model, fuzziness is considered in order to investigate the impact of uncertainty in important model parameters. The optimal robust solution shows that the total cost of the FMILP model is enhanced by 4% compared with the total cost of the deterministic model. Another key measurement related to the achievement of global sustainable development goals is considered, reducing the additional intermodal transfer cost and the cost of CO2 emissions in the objective function.

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