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Immediate schedule adjustment and semidefinite relaxation

  • * Corresponding author: Su Zhang

    * Corresponding author: Su Zhang
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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  • 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.

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


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  • Figure 1.  Project Network Diagram in Example 1

    Figure 2.  Project Network Diagram in Example 2

    Table 1.  Basic Notations

    $Pred(i)$ 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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