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Resource allocation flowshop scheduling with learning effect and slack due window assignment

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    *Corresponding author

This work was supported by the Liaoning Province Universities and Colleges Basic Scientific Research Project of Youth Project, Education Department of Liaoning (China) (Grant no. LQN2017ST04)

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  • We study flowshop scheduling problems with respect to slack due window assignments, which are operations in which jobs are assigned an individual due window. We combine learning effect and controllable processing times, in which the flowshop has a two-machine no-wait setup. The goal is to determine job sequence, slack due window based on common flow allowance, due window size, and resource allocation. We provide a bicriteria analysis for the scheduling and resource consumption costs. We show that the two costs can be solved in polynomial time utilizing three different combinations.

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

    Citation:

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  • Figure 1.  A two-machine no-waiting flowshop scheduling

    Table 1.  List of notations

    notations definitions
    $ n $ the total number of jobs
    $ J_{j} $ $ (j = 1,2,\ldots,n) $ index of job
    $ \overline{p}_{j} $ normal processing time of job $ J_{j} $
    $ {p}_{j} $ actual processing time of job $ J_{j} $
    $ a_{j} $ learning index of job $ J_{j} $
    $ b_{j} $ compression rate of job $ J_{j} $
    $ u_{j} $ amount of resource that can be allocated to job $ J_{j} $
    $ v_{j} $ per time unit cost associated with resource allocated
    for job $ J_{j} $
    $ \bar{d}_j $ due date for job $ J_j $
    $ a $ learning index
    $ \eta $ positive constant
    $ M_{i} $ $ (i = 1,2) $ index of machine
    $ O_{j,i} $ operation of job $ J_j $ processed on machine $ M_i $
    $ \overline{p}_{j,i} $ normal processing time of operation $ O_{j,i} $
    $ {p}_{j,i} $ actual processing time of operation $ O_{j,i} $
    $ a_{j,i} $ learning index of operation $ O_{j,i} $
    $ u_{j,i} $ amount of resource that can be allocated to
    operation $ O_{j,i} $
    $ v_{j,i} $ per time unit cost associated with resource allocated
    for operation $ O_{j,i} $
    $ C_{j,i} $ completion time of operation $ O_{j,i} $
    $ [d_j, d'_j] $ due window of job $ J_j $
    $ d_j $ starting time of due window for job $ J_j $
    $ d'_j $ finishing time of due window for job $ J_j $
    $ D' $ due window size
    $ C_j = C_{j,2} $ completion time of job $ J_{j} $
    $ W_j $ waiting time of job $ J_{j} $
    $ E_j $ earliness of job $ J_{j} $
    $ T_j $ tardiness of job $ J_{j} $
    $ C_{\max} $ makespan
    $ \sum_{j = 1}^n C_j $ total completion time
    $ \sum_{j = 1}^n W_j $ waiting completion time
    $ \sum_{j = 1}^n\sum_{h = j}^n|C_j-C_h| $ total absolute differences in completion times
    $ \sum_{j = 1}^n\sum_{h = j}^n|W_j-W_h| $ total absolute differences in waiting times
     | Show Table
    DownLoad: CSV

    Table 2.  Summary of Results, $Z$ is the special condition $v_{j, 1} \overline{p}_{j, 1} = v_{j, 2} \overline{p}_{j, 2} = P'_{j}$

    problem complexity reference
    $F2\left|NW, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}\right|A+ \sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ $O(n^3)$ Liu and Feng [12]
    $F2|NW, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, \sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}\leq X|A$ $O(n^3)$ Tian et al. [21]
    $F2\left|NW, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, A\leq Y\right|\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ $O(n^3)$ Tian et al. [21]
    $F2|NW, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}} \right)^{\eta}, Z, \sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}\leq X|A$ $O(n \log n)$ Tian et al. [21]
    $F2\left|NW, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, Z, A\leq Y\right|\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ $O(n \log n)$ Tian et al. [21]
    $F2\left|NW, CON, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta} \right|$ $\sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma d)+ \theta\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ $O(n^3)$ Gao et al. [4]
    $F2\left|NW, CON, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, Z\right|$ $\sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma d)+ \theta\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ $O(n\log n)$ Gao et al. [4]
    $F2\left|NW, CON, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, \sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}\leq X\right|$ $\sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma d)$ $O(n^3)$ Geng et al. [5]
    $F2|NW, CON, $ $p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, \sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma d)\leq Y|\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ $O(n^3)$ Geng et al. [5]
    $F2\left|NW, CON, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, Z, \sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}\leq X\right|$ $\sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma d)$ $O(n \log n)$ Geng et al. [5]
    $F2|NW, CON, $ $p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, Z, \sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma d)\leq Y|\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ $O(n\log n)$ Geng et al. [5]
    $F2\left|NW, SLK, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}\right|$ $\sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma q)+ \theta\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ $O(n^3)$ Sun et al. [20]
    $F2|NW, SLK, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, $ $ \sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}\leq X|\sum_{j = 1}^n(\alpha E_j+\beta T_j+ q)$ $O(n^3)$ Sun et al. [20]
    $F2|NW, SLK, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, $ $ \sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma q)\leq Y|\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ $O(n^3)$ Sun et al. [20]
    $F2\left|NW, SLK, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, Z\right|$ $\sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma q)+ \theta\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ $O(n \log n)$ Sun et al. [20]
    $F2|NW, SLK, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, Z, \sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}\leq X|\sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma q)$ $O(n \log n)$ Sun et al. [20]
    $F2|NW, SLK, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, Z, \sum_{j = 1}^n(\alpha E_j+\beta T_j+\gamma q)\leq Y|\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ $O(n\log n)$ Sun et al. [20]
    $F2\left|NW, CON-DW, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a_{j, i}}}{u_{j, i}} \right)^{\eta}\right|$ $\sum_{j = 1}^n(\alpha E_j+\beta T_j+\delta d +\gamma D)+ \theta\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ $O(n^3)$ Shi and Wang [19]
    $F2\left|NW, CON-DW, p_{j, i} = \left(\frac{\overline{p}_{{j, i}}r^{a}}{u_{j, i}}\right)^{\eta}, Z\right|$ $\sum_{j = 1}^n(\alpha E_j+\beta T_j+\delta d +\gamma D)+ \theta\sum_{i = 1}^2\sum_{j = 1}^n v_{j, i}u_{j, i}$ $O(n\log n)$ Shi and Wang [19]
    P1, P2, P3 $O(n^3)$ Theorem 1
    P1($Z$), P2($Z$), P3($Z$) $O(n\log n)$ Theorem 2
     | Show Table
    DownLoad: CSV

    Table 3.  Data for Example 1

    $ J_j $ $ J_1 $ $ J_2 $ $ J_3 $ $ J_4 $ $ J_5 $ $ J_6 $ $ J_7 $
    $ \overline{p}_{j,1} $ 4 6 8 9 10 5 18
    $ \overline{p}_{j,2} $ 1 3 5 6 4 17 3
    $ v_{j,1} $ 2 5 3 1 6 7 4
    $ v_{j,2} $ 3 4 2 4 5 6 8
    $ a_{j,1} $ -0.1 -0.2 -0.25 -0.12 -0.3 -0.15 -0.24
    $ a_{j,2} $ -0.3 -0.1 -0.35 -0.25 -0.22 -0.13 -0.32
     | Show Table
    DownLoad: CSV

    Table 4.  Weights for Example 1

    $ r $ 1 2 3 4 5 6 7
    $ {W'_r} $ 50 58 63 48 32 16 0
    $ {V'_r} $ -2 1 21 22 22 22 6
    $ {{\eta '}_r} $ 7.0711 7.4833 8.0000 8.3066 7.3485 6.1644 4.6904
    $ {{\vartheta '}_r} $ 7.4833 8.0000 8.3066 7.3485 6.1644 4.6904 2.4495
     | Show Table
    DownLoad: CSV

    Table 5.  Values $\lambda_{j, r}$ for Example 1

    ${j\backslash r}$ 1 2 3 4 5 6 7
    $1$ 79.0185 71.0153 66.8296 64.0572 62.0114 60.4040 59.0879
    $2$ 320.4994 285.0093 266.2270 253.7116 244.4413 237.1383 231.1477
    $3$ 275.0662 226.6246 202.4384 186.8978 175.6899 167.0453 160.0771
    $4$ 251.1229 217.0961 199.5329 188.0100 179.5710 172.9821 167.6170
    $5$ 564.1964 463.9798 413.9291 381.7710 358.5816 340.6988 326.2867
    $6$ 694.1769 631.6463 597.7243 574.7719 557.5800 543.9173 532.6284
    $7$ 396.4977 333.0474 300.8013 279.8553 264.6294 252.8128 243.2393
     | Show Table
    DownLoad: CSV

    Table 6.  Data for Example 2

    $ J_j $ $ J_1 $ $ J_2 $ $ J_3 $ $ J_4 $ $ J_5 $ $ J_6 $ $ J_7 $
    $ \overline{p}_{j,1} $ 4 6 5 7 8 7 6
    $ \overline{p}_{j,2} $ 2 8 10 14 10 7 5
    $ v_{j,1} $ 2 4 4 4 5 5 5
    $ v_{j,2} $ 4 3 2 2 4 5 6
    $ P_{j} $ 8 24 20 28 40 35 30
     | Show Table
    DownLoad: CSV

    Table 7.  Weights for Example 2

    $ r $ 1 2 3 4 5 6 7
    $ {W'_r} $ 50 58 63 48 32 16 0
    $ {V'_r} $ -2 1 21 22 22 22 6
    $ {{\eta '}_r} $ 7.0711 7.4833 8.0000 8.3066 7.3485 6.1644 4.6904
    $ {{\vartheta '}_r} $ 7.4833 8.0000 8.3066 7.3485 6.1644 4.6904 2.4495
    $ {\Psi'} _r $ 14.5544 13.4790 13.0900 11.8643 9.7939 7.5856 4.8381
     | Show Table
    DownLoad: CSV

    Table 8.  Computation time of Algorithm 1 in ms

    jobs ($ n $) Min Mean Max
    40 916 992.10 1,060
    60 1,368 1,491.60 1,991
    80 2,352 2,671.20 2,956
    100 3,972 4,232.70 4,618
    120 5,182 5,979.85 6,223
    140 7,775 7,868.50 8,171
    160 8,229 9,142.60 9,915
    180 10,892 11,514.70 12,325
    200 12,884 13,715.60 14,191
    220 14,134 15,125.50 16,119
    240 16,185 17,815.60 18,181
    260 18,347 19,752.40 20,282
    280 20,981 21,156.70 22,941
    300 23,823 24,715.90 25,671
     | Show Table
    DownLoad: CSV
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