Some formulas for Bell numbers

Komatsu, Takao; Pita-Ruiz, Claudio

doi:10.2298/fil1811881k

Cited by 4 publications

(1 citation statement)

References 8 publications

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“…Considering the problem definition, the calculation has been solved. Mathematically, the number of ways a set of n elements can be partitioned into nonempty subsets is called a Bell number [58] and is denoted

B(n)

. The n ‐th Bell number is given by.…”

Section: Problem Definitionmentioning

confidence: 99%

Heuristic approach for the multiobjective optimization of the customized bus scheduling problem

Liu

Zhang

2021

IET Intelligent Trans Sys

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Customized bus is a typical demand‐responsive transit (DRT) service that allows passengers to customize their bus route on demand and be picked up and dropped off at their desired location. A set of ride requests sent at various times by geographically dispersed passengers are operated in real time by the controller and then assigned to the shared bus with the assurance of maintaining the minimum vehicle and driver costs, which introduces a new challenge to scheduling. In this paper, a mixed integer linear programming (MILP) model is presented to solve the customized bus scheduling problem. To quickly generate satisfactory scheduling solutions in the case of single‐ and multi‐depots, a heuristic multi‐objective optimization algorithm based on preemptive scheduling is proposed, which can converge to the optimal solution among a set of vehicle blocks (consecutive trips by one bus). A real‐world customized bus scheduling problem from the public transport company of Chengdu, China that contains 2,000‐trip instances operated on a day is used to demonstrate that the proposed approach reduces total operating costs by 30.51% and 30.71%, compared with the simulated annealing algorithm and genetic algorithm, respectively.

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B(n)

. The n ‐th Bell number is given by.…”

Section: Problem Definitionmentioning

confidence: 99%

Heuristic approach for the multiobjective optimization of the customized bus scheduling problem

Liu

Zhang

2021

IET Intelligent Trans Sys

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Shifting powers in Spivey’s Bell number formula

Mansour

Rastegar

Roitershtein

et al. 2020

Quaestiones Mathematicae

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K-regular decomposable incidence structure of maximum degree

Stošović,

Katić,

Galić

2023

Mathematica Moravica

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This paper discusses incidence structures and their rank. The aim of this paper is to prove that there exists a regular decomposable incidence structure J = (P, B) of maximum degree depending on the size of the set and a predetermined rank. Furthermore, an algorithm for construction of this structures is given. In the proof of the main result, the points of the set P are shown by Euler's formula of complex number. Two examples of construction the described incidence structures of maximum degree 6 and maximum degree 30 are given.

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Some formulas for Bell numbers

Cited by 4 publications

References 8 publications

Heuristic approach for the multiobjective optimization of the customized bus scheduling problem

Heuristic approach for the multiobjective optimization of the customized bus scheduling problem

Shifting powers in Spivey’s Bell number formula

K-regular decomposable incidence structure of maximum degree

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