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

Ji-Bo Wang; Mengqi Liu; Na Yin; Ping Ji

doi:10.3934/jimo.2016060

Article Contents

2017, Volume 13, Issue 2: 1025-1039. Doi: 10.3934/jimo.2016060

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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
* Corresponding author

Received: October 2015

Revised: June 2016

Published: August 2017

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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Abstract

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.

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

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Table 1. Data of Example 1

$J_{j}$	$J_{1}$	$J_{2}$	$J_{3}$	$J_{4}$	$J_{5}$	$J_{6}$
$p_{j}$	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

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Table 2. 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

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Table 3. 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

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Table 4. 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

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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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