Immediate schedule adjustment and semidefinite relaxation

Jinling Zhao; Wei Chen; Su Zhang

doi:10.3934/jimo.2018062

Article Contents

2019, Volume 15, Issue 2: 633-645. Doi: 10.3934/jimo.2018062

This issue Previous Article Optimum management of the network of city bus routes based on a stochastic dynamic model Next Article Sufficiency and duality in non-smooth interval valued programming problems

Immediate schedule adjustment and semidefinite relaxation

1.
School of Mathematics and Physics, University of Science and Technology Beijing, 30 Xueyuan Road, Haidian District, Beijing 100083, China
2.
Business School, Nankai University, 94 Weijin Road, Nankai District, Tianjin 300071, China

* Corresponding author: Su Zhang
* Corresponding author: Su Zhang

Received: November 2017

Revised: January 2018

Early access: April 2018

Published: January 2019

The first author is supported by National Natural Science Foundation of China No. 11101028,11271206, and the Fundamental Research Funds for the Central Universities. The third author is supported by National Natural Science Foundation of China No. 11401322.

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Abstract

This paper considers the problem of temporary shortage of some resources within a project execution period. Mathematical models for two different cases of this problem are established. Semidefinite relaxation technique is applied to get immediate solvent of these models. Relationship between the models and their semidefinite relaxations is studied, and some numerical experiments are implemented, which show that these mathematical models are reasonable and feasible for practice, and semidefinite relaxation can efficiently solve the problem.

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Mathematics Subject Classification: Primary: 90C90; Secondary: 90B99.

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Figure 1. Project Network Diagram in Example 1

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Figure 2. Project Network Diagram in Example 2

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Table 1. Basic Notations

Symbol	Definition
$Pred(i)$	set of direct predecessors of Activity $i$
$Succ(i)$	set of direct successors of Activity $i$
$d_i$	processing time (or duration) of Activity $i$
$s_i$	start time of Activity $i$ according to the existing schedule
$f_i$	completion time of Activity $i$ according to the existing schedule
$R_k$	amount of originally available units of renewable resource $k$ in unit time
$r_{ik}$	usage of Activity $i$ of renewable resource $k$ in unit time
$t_0$	start time of the temporary shortage of resources
$T$	lasting time of the resources shortage
$\Delta R_k$	amount of decrement of resource $k$ in unit time

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Table 2. Comparing the 1/3-Method with the Rounding Method

$\Delta R_k$	1/3-Method	Rounding Method	SDR Opt.Val $t^*$	Opt.Val $t^*$	Delay Time
$2$	15	infeasible	15.0000	15	1
$5$	15	15	15.0000	15	1
$8$	18	infeasible	15.4583	18	4
$11$	19	19	16.7083	19	5

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