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Scheduling jobs with controllable processing time, truncated job-dependent learning and deterioration effects

  • * Corresponding author

    * Corresponding author 
The work described in this paper was partially supported by the grant from The Hong Kong Polytechnic University (PolyU projects G-YBFE and 4-BCBJ) and the National Natural Science Foundation of China (Grant Nos. 71471120 and 71471057).
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  • In this paper, we consider single machine scheduling problems with controllable processing time (resource allocation), truncated job-dependent learning and deterioration effects. The goal is to find the optimal sequence of jobs and the optimal resource allocation separately for minimizing a cost function containing makespan (total completion time, total absolute differences in completion times) and/or total resource cost. For two different processing time functions, i.e., a linear and a convex function of the amount of a common continuously divisible resource allocated to the job, we solve them in polynomial time respectively.

    Mathematics Subject Classification: Primary: 90B35; Secondary: 90C26.

    Citation:

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  • Table 1.  Data of Example 1

    $J_{j}$$J_{1}$$J_{2}$$J_{3}$$J_{4}$$J_{5}$$J_{6}$
    $p_{j}$1081118916
    $\beta_{j}$213234
    $\bar{u}_{j}$323122
    $v_{j}$1081211149
    $a_{j}$-0.25-0.15-0.2-0.1-0.3-0.25
     | Show Table
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    Table 2.  Values of $\Lambda_{jr}$

    ${j\backslash r}$${1}$${2}$${3}$${4}$${5}$${6}$
    $1$57.207643.311032.749722.291514.35007.0000
    $2$54.415239.839629.242120.485012.88246.1146
    $3$49.603839.183135.268026.280616.34387.7000
    $4$119.830492.749069.510149.399431.414415.0473
    $5$48.405735.240027.899319.860712.91506.3000
    $6$72.415248.138535.918728.446522.960011.2000
     | Show Table
    DownLoad: CSV

    Table 3.  Data of Example 2

    $J_{j}$$J_{1}$$J_{2}$$J_{3}$$J_{4}$$J_{5}$$J_{6}$
    $p_{j}$108111891
    $v_{j}$1081211149
    $a_{j}$-0.25-0.15-0.2-0.1-0.3-0.25
     | Show Table
    DownLoad: CSV

    Table 4.  Values of $\Theta_{jr}$

    ${j\backslash r}$${1}$${2}$${3}$${4}$${5}$${6}$
    $1$77.145664.129255.175147.385240.777232.0996
    $2$57.292549.878344.089838.598232.702125.2778
    $3$92.831278.972068.870059.716550.219538.6263
    $4$121.6433108.376797.102985.827473.259656.9729
    $5$89.996473.103062.051654.907547.569837.4467
    $6$98.375481.777070.358860.425251.998740.9331
    The bold numbers are the optimal solution
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    Table 5.  Main results of this paper ($\rho\in\{C_{\max},\sum C_j, TADC\}$)

    $1|p_{jr}^A(t,u_j)=p_j\max\left\{r^{a_j},b\right\}+c t-\theta_{j} u_{j}|\delta_1 \rho+\delta_2 \sum_{j=1}^{n}v_{j}u_{j}$$O(n^3)$Theorem 3.3
    $1| p_{jr}^A(t,u_j)= \left(\frac{p_j\max\left\{r^{a_j},b\right\}}{u_j}\right)^l +ct|\delta_1\rho+\delta_2 \sum_{j=1}^{n}v_{j}u_{j}$$O(n^3)$Theorem 4.4
    $1| p_{jr}^A(t,u_j)= \left(\frac{p_j\max\left\{r^{a},b\right\}}{u_j}\right)^l +ct|\delta_1\rho+\delta_2 \sum_{j=1}^{n}v_{j}u_{j}$$O(n\log n)$Theorem 4.6
    $1| p_{jr}^A(t,u_j)= \left(\frac{p_j\max\left\{r^{a_j},b\right\}}{u_j}\right)^l +ct,\sum_{j=1}^{n}u_{j}\leq U |\rho$$O(n^3)$Theorem 4.9
    $1| p_{jr}^A(t,u_j)= \left(\frac{p_j\max\left\{r^{a},b\right\}}{u_j}\right)^l +ct,\sum_{j=1}^{n}u_{j}\leq U|\rho$$O(n\log n)$Theorem 4.10
    $1| p_{jr}^A(t,u_j)= \left(\frac{p_j\max\left\{r^{a_j},b\right\}}{u_j}\right)^l +ct,\rho\leq R|\sum_{j=1}^{n}u_{j}$$O(n^3)$Theorem 4.13
    $1| p_{jr}^A(t,u_j)= \left(\frac{p_j\max\left\{r^{a},b\right\}}{u_j}\right)^l +ct,\rho\leq R| \sum_{j=1}^{n}u_{j}$$O(n\log n)$Theorem 4.14
    $1|p_{j}^A= \left(\frac{p_j\max\left\{r^{a_j},b\right\}}{u_j}\right)^l +c t|(\rho,\sum_{j=1}^{n}u_{j}) $$O(n^3)$Theorem 4.15
    $1|p_{j}^A= \left(\frac{p_j\max\left\{r^{a},b\right\}}{u_j}\right)^l +c t|(\rho,\sum_{j=1}^{n}u_{j}) $$O(n\log n)$Theorem 4.16
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