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April  2017, 13(2): 1025-1039. doi: 10.3934/jimo.2016060

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

 1 School of Science, Shenyang Aerospace University, Shenyang 110136, China 2 Business School, Hunan University, Changsha 410082, Hunan, China 3 Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

* Corresponding author

Received  October 2015 Revised  June 2016 Published  October 2016

Fund Project: 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).

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.

Citation: Ji-Bo Wang, Mengqi Liu, Na Yin, Ping Ji. Scheduling jobs with controllable processing time, truncated job-dependent learning and deterioration effects. Journal of Industrial & Management Optimization, 2017, 13 (2) : 1025-1039. doi: 10.3934/jimo.2016060
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##### References:
Data of Example 1
 $J_{j}$ $J_{1}$ $J_{2}$ $J_{3}$ $J_{4}$ $J_{5}$ $J_{6}$ $p_{j}$ 10 8 11 18 9 16 $\beta_{j}$ 2 1 3 2 3 4 $\bar{u}_{j}$ 3 2 3 1 2 2 $v_{j}$ 10 8 12 11 14 9 $a_{j}$ -0.25 -0.15 -0.2 -0.1 -0.3 -0.25
 $J_{j}$ $J_{1}$ $J_{2}$ $J_{3}$ $J_{4}$ $J_{5}$ $J_{6}$ $p_{j}$ 10 8 11 18 9 16 $\beta_{j}$ 2 1 3 2 3 4 $\bar{u}_{j}$ 3 2 3 1 2 2 $v_{j}$ 10 8 12 11 14 9 $a_{j}$ -0.25 -0.15 -0.2 -0.1 -0.3 -0.25
Values of $\Lambda_{jr}$
 ${j\backslash r}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ $1$ 57.2076 43.3110 32.7497 22.2915 14.3500 7.0000 $2$ 54.4152 39.8396 29.2421 20.4850 12.8824 6.1146 $3$ 49.6038 39.1831 35.2680 26.2806 16.3438 7.7000 $4$ 119.8304 92.7490 69.5101 49.3994 31.4144 15.0473 $5$ 48.4057 35.2400 27.8993 19.8607 12.9150 6.3000 $6$ 72.4152 48.1385 35.9187 28.4465 22.9600 11.2000
 ${j\backslash r}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ $1$ 57.2076 43.3110 32.7497 22.2915 14.3500 7.0000 $2$ 54.4152 39.8396 29.2421 20.4850 12.8824 6.1146 $3$ 49.6038 39.1831 35.2680 26.2806 16.3438 7.7000 $4$ 119.8304 92.7490 69.5101 49.3994 31.4144 15.0473 $5$ 48.4057 35.2400 27.8993 19.8607 12.9150 6.3000 $6$ 72.4152 48.1385 35.9187 28.4465 22.9600 11.2000
Data of Example 2
 $J_{j}$ $J_{1}$ $J_{2}$ $J_{3}$ $J_{4}$ $J_{5}$ $J_{6}$ $p_{j}$ 10 8 11 18 9 1 $v_{j}$ 10 8 12 11 14 9 $a_{j}$ -0.25 -0.15 -0.2 -0.1 -0.3 -0.25
 $J_{j}$ $J_{1}$ $J_{2}$ $J_{3}$ $J_{4}$ $J_{5}$ $J_{6}$ $p_{j}$ 10 8 11 18 9 1 $v_{j}$ 10 8 12 11 14 9 $a_{j}$ -0.25 -0.15 -0.2 -0.1 -0.3 -0.25
Values of $\Theta_{jr}$
 ${j\backslash r}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ $1$ 77.1456 64.1292 55.1751 47.3852 40.7772 32.0996 $2$ 57.2925 49.8783 44.0898 38.5982 32.7021 25.2778 $3$ 92.8312 78.9720 68.8700 59.7165 50.2195 38.6263 $4$ 121.6433 108.3767 97.1029 85.8274 73.2596 56.9729 $5$ 89.9964 73.1030 62.0516 54.9075 47.5698 37.4467 $6$ 98.3754 81.7770 70.3588 60.4252 51.9987 40.9331 The bold numbers are the optimal solution
 ${j\backslash r}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ $1$ 77.1456 64.1292 55.1751 47.3852 40.7772 32.0996 $2$ 57.2925 49.8783 44.0898 38.5982 32.7021 25.2778 $3$ 92.8312 78.9720 68.8700 59.7165 50.2195 38.6263 $4$ 121.6433 108.3767 97.1029 85.8274 73.2596 56.9729 $5$ 89.9964 73.1030 62.0516 54.9075 47.5698 37.4467 $6$ 98.3754 81.7770 70.3588 60.4252 51.9987 40.9331 The bold numbers are the optimal solution
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
 $1|p_{jr}^A(t,u_j)=p_j\max\left\{r^{a_j},b\right\}+c t-\theta_{j} u_{j}|\delta_1 \rho+\delta_2 \sum_{j=1}^{n}v_{j}u_{j}$ $O(n^3)$ 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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