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

• *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)

• 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:

• 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

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

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

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 8.3066 7.3485 6.1644 4.6904 ${{\vartheta '}_r}$ 7.4833 8 8.3066 7.3485 6.1644 4.6904 2.4495

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 60.404 59.0879 $2$ 320.499 285.009 266.227 253.712 244.441 237.138 231.148 $3$ 275.066 226.625 202.438 186.898 175.69 167.045 160.077 $4$ 251.123 217.096 199.533 188.01 179.571 172.982 167.617 $5$ 564.196 463.98 413.929 381.771 358.582 340.699 326.287 $6$ 694.177 631.646 597.724 574.772 557.58 543.917 532.628 $7$ 396.498 333.047 300.801 279.855 264.629 252.813 243.239

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

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 8.3066 7.3485 6.1644 4.6904 ${{\vartheta '}_r}$ 7.4833 8 8.3066 7.3485 6.1644 4.6904 2.4495 ${\Psi'} _r$ 14.5544 13.479 13.09 11.8643 9.7939 7.5856 4.8381

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
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