A best possible algorithm for an online scheduling problem with position-based learning effect

Ran Ma; Lu Zhang; Yuzhong Zhang

doi:10.3934/jimo.2021144

Article Contents

2022, Volume 18, Issue 6: 3989-4010. Doi: 10.3934/jimo.2021144

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A best possible algorithm for an online scheduling problem with position-based learning effect

1.
School of Management Engineering, , Qingdao University of Technology, Qingdao 266525, China, University Research Center for Smart City Construction and Management of Shandong Province, Qingdao 266525, China
2.
Institute of Operations Research, School of Management, Qufu Normal University, Rizhao 276826, China

^*Corresponding author: Ran Ma
^*Corresponding author: Ran Ma

Received: December 31, 2020

Revised: March 31, 2021

Early access: September 2021

Published: September 2022

This research is supported by the National Natural Science Foundation of China (Grant Nos. 11501171, 11771251) and the Province Natural Science Foundation of Shandong (Grant No. ZR2020MA028)

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Abstract

In this paper, we focus on an online scheduling problem with position-based learning effect on a single machine, where the jobs are released online over time and preemption is not allowed. The information about each job $ J_j $, including the basic processing time $ p_j $ and the release time $ r_j $, is only available when it arrives. The actual processing time $ p_j' $ of each job $ J_j $ is defined as a function related to its position $ r $, i.e., $ p_j' = p_j(\alpha-r\beta) $, where $ \alpha $ and $ \beta $ are both nonnegative learning index. Our goal is to minimize the sum of completion time of all jobs. For this problem, we design a deterministic polynomial time online algorithm Delayed Shortest Basic Processing Time (DSBPT). In order to facilitate the understanding of the online algorithm, we present a relatively common and simple example to describe the execution process of the algorithm, and then by competitive analysis, we show that online algorithm DSBPT is a best possible online algorithm with a competitive ratio of 2.

Keywords:

Mathematics Subject Classification: Primary: 90B35.

Citation:

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Figure 1. The scheduling of DSBPT and an off-line optimal scheduling

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Figure 2. Block and subblock

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Figure 3. Reverse pair

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Table 1. Notations-1

Notation	Meaning
$ J_j $	the job of index $ j $, where $ j=1,2,\ldots,n $
$ I $	a job instance, $ I=\left\{J_1,J_2,\ldots,J_n\right\} $
$ r_j(I) $	the release time of job $ J_j $ in $ I $
$ p_j(I) $	the basic processing time of job $ J_j $ in $ I $
$ p'_j(I) $	the actual processing time of job $ J_j $ in $ I $
$ X(I) $	a feasible schedule of $ I $
$ S_j(X(I)) $	the starting time of job $ J_j $ in $ X(I) $
$ C_j(X(I)) $	the completion time of job $ J_j $ in $ X(I) $,
	$ C_j(X(I))=S_j(X(I))+p_j(I) $
$ Z_j(X(I)) $	the contribution of job $ J_j $ to the total objective value of $ I $ in $ X(I) $
$ Z(X(I)) $	the total objective value of $ I $ in $ X(I) $,
	$ Z(X(I))=\sum Z_j(X(I)) $
$ \pi(I) $	an off-line optimal schedule of $ I $
$ {\rm{OPT}}( I )$	the total objective value of $ I $ in $ \pi(I) $, $ {\rm{OPT}}(I)=\sum Z_j(\pi(I)) $

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Table 2. Notations-2

Notation	Meaning
$ \|T\| $	the number of jobs in $ T $
$ T(X(I)) $	the set of jobs of $ T $ in $ X(I) $
$ Z_T(X(I)) $	the total contribution of jobs of $ T $ to $ Z(X(I)) $,
	$ Z_T(X(I))=\sum_{J_j\in T} Z_j(X(I)) $
$ {\rm{OPT}}(T) $	the total objective value of jobs of $ T $ in an off-line optimal schedule, $ {\rm{OPT}}(T)=\sum Z_j(\pi(T)) $

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Algorithm 1 DSBPT

At each decision time

$ t $

, if the machine is idle and there exist valid jobs, the valid job with the shortest basic processing time is selected. If there are many such valid jobs, the job

$ J_j $

with the earliest release time is selected, presume its processing position is

$ k $

$ p_j(I)(\alpha-k\beta)\leq t $

, and process the job; otherwise the machine will remain idle and wait until the next decision time

$ t $

to make a decision.

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Algorithm 1: Online Algorithm DSBPT.

1: Input: job instance

$ I $

(

$ |I|\geq 1 $

) and learning index

$ \alpha $

and

$ \beta $

$ S $

is an empty sequence,

$ k\leftarrow 1 $

$ C\leftarrow 0 $

$ A $

is an available jobs set,

$ f(S)\leftarrow 0 $

$ dt\leftarrow 0 $

$ Ct\leftarrow 0 $

$ t\leftarrow \min\{r_j|J_j\in I\} $

3: While True do
4: update available jobs set A,

$ dt\leftarrow \max\{t,Ct\} $

$ D \leftarrow \{J_j\in A|\ the \ job \ has \ the \ shortest \ processing \ time \ first $

$ \qquad \qquad \qquad \qquad \ and \ then \ releases \ the \ earliest\} $

6: arbitrarily select a job

$ J_a $

from

$ D $

7: If

$ dt \geq p_a(\alpha-k\beta) $

then
8:

$ S\leftarrow S + \{J_a\} $

$ C\leftarrow dt+p_a(\alpha-k\beta) $

10:

$ f(S)\leftarrow f(S)+C $

11:

$ k\leftarrow k+1 $

12: end if
13: If

$ C>dt $

then
14:

$ Ct\leftarrow C $

15: end if
16: end while
17: return

$ S $

and

$ f(S) $

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Table 3. Information of instance $ I $

	$ J_1 $	$ J_2 $	$ J_3 $	$ J_4 $	$ J_5 $	$ J_6 $
release time	0	8	17	23	28	30
basic processing time	8	6	12	10	4	8

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Table 4. The selection process of jobs under $ \sigma(I) $

decision time	valid job set	selected job	sequence of finished jobs	current objective function value
0	$ J_1 $	None	None	0
8	$ J_1 $ $ J_2 $	$ J_2 $	None	0
14	$ J_1 $	$ J_1 $	$ J_2 $	14
17	$ J_3 $	None	$ J_2 $	14
20.8	$ J_3 $	$ J_3 $	$ J_2 $-$ J_1 $	34.8
23	$ J_4 $	None	$ J_2 $-$ J_1 $	34.8
28	$ J_4 $ $ J_5 $	None	$ J_2 $-$ J_1 $	34.8
29.2	$ J_4 $ $ J_5 $	$ J_5 $	$ J_2 $-$ J_1 $-$ J_3 $	64
30	$ J_4 $ $ J_6 $	None	$ J_2 $-$ J_1 $-$ J_3 $	64
31.4	$ J_4 $ $ J_6 $	$ J_6 $	$ J_2 $-$ J_1 $-$ J_3 $-$ J_5 $	95.4
34.6	$ J_4 $	$ J_4 $	$ J_2 $-$ J_1 $-$ J_3 $-$ J_5 $-$ J_6 $	130
37.1	None	None	$ J_2 $-$ J_1 $-$ J_3 $-$ J_5 $-$ J_6 $-$ J_4 $	167.1

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Table 5. The selection process of jobs under $ \pi(I) $

time	valid job set	selected job	sequence of finished jobs	current objective function value
0	$ J_1 $	$ J_1 $	None	0
8	$ J_2 $	$ J_2 $	$ J_1 $	8
13.1	None	None	$ J_1 $-$ J_2 $	21.1
17	$ J_3 $	$ J_3 $	$ J_1 $-$ J_2 $	21.1
23	$ J_4 $	None	$ J_1 $-$ J_2 $	21.1
25.4	$ J_4 $	$ J_4 $	$ J_1 $-$ J_2 $-$ J_3 $	46.5
28	$ J_5 $	None	$ J_1 $-$ J_2 $-$ J_3 $	46.5
30	$ J_5 $ $ J_6 $	None	$ J_1 $-$ J_2 $-$ J_3 $	46.5
30.9	$ J_5 $ $ J_6 $	$ J_5 $	$ J_1 $-$ J_2 $-$ J_3 $-$ J_4 $	77.4
32.5	$ J_6 $	$ J_6 $	$ J_1 $-$ J_2 $-$ J_3 $-$ J_4 $-$ J_5 $	109.9
34.5	None	None	$ J_1 $-$ J_2 $-$ J_3 $-$ J_4 $-$ J_5 $-$ J_6 $	144.4

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Table 6. The comparison between $ \sigma(I) $ and $ \pi(I) $

	job processing scheduling	objective function value
$ \sigma(I) $	$ J_2-J_1-J_3-J_5-J_6-J_4 $	167.1
$ \pi(I) $	$ J_1-J_2-J_3-J_4-J_5-J_6 $	144.4

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Algorithm 2 FDSBPT

At arbitrary time

$ t $

, if the machine is idle and there exist valid jobs, the valid job with the shortest basic processing time is selected. If there are multiple jobs with the shortest processing time, choose arbitrary job

$ J_j $

flexibly, presume its processing position is k, if

$ p_j(I)(\alpha-k\beta)\leq t $

, we arrange job

$ J_j $

on the idle machine at time

$ t $

; otherwise, we do nothing until the next time.

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Algorithm 2: Online Algorithm DSBPT.

1: Input: job instance

$ I $

(

$ |I|\geq 1 $

) and learning index

$ \alpha $

and

$ \beta $

$ S $

is an empty sequence,

$ k\leftarrow 1 $

$ C\leftarrow 0 $

$ A $

is an available jobs set,

$ f(S)\leftarrow 0 $

$ t\leftarrow 0 $

3: While True do
4: update available jobs set A
5:

$ D \leftarrow \{ {J_j} \in A|\;the\;job\;has\;the\;shortest\;processing\;time{\rm{\} }} $

6: arbitrarily select a job

$ J_a $

from

$ D $

7: If

$ t \geq p_a(\alpha-k\beta) $

then
8:

$ S\leftarrow S + \{J_a\} $

$ C\leftarrow t+p_a(\alpha-k\beta) $

10:

$ f(S)\leftarrow f(S)+C $

11:

$ k\leftarrow k+1 $

12: end if
13:

$ t\leftarrow t+1$

14: end while
15: return

$ S $

and

$ f(S) $

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Table 7. Information of instance $ I_c $

	$ J_1 $	$ J_2 $	$ J_3 $	$ J_4 $	$ J_5 $	$ J_6 $	$ J_7 $	$ J_8 $	$ J_9 $	$ J_{10} $
release time	0	8	17	23	28	30	30	33	35	39
basic processing time	8	6	12	10	4	8	12	9	10	11
	$ J_{11} $	$ J_{12} $	$ J_{13} $	$ J_{14} $	$ J_{15} $	$ J_{16} $	$ J_{17} $	$ J_{18} $	$ J_{19} $	$ J_{20} $
release time	41	42	46	48	48	48	48	50	52	52
basic processing time	13	6	7	9	8	8	9	5	13	8

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Table 8. The selection process of jobs under $\sigma(I_c)$

decision time	valid job set	selected job	current objective function value
0	$J_1$	None	0
8	$J_1$ $J_2$	$J_2$	0
14	$J_1$	$J_1$	14
17	$J_3$	None	14
21.84	$J_3$	$J_3$	35.84
23	$J_4$	None	35.84
28	$J_4$ $J_5$	None	35.84
30	$J_4$ $J_5$ $J_6$ $J_7$	None	35.84
33	$J_4$ $J_5$ $J_6$ $J_7$ $J_8$	None	35.84
33.36	$J_4$ $J_5$ $J_6$ $J_7$ $J_8$	$J_5$	69.2
35	$J_4$ $J_6$ $J_7$ $J_8$ $J_9$	None	69.2
37.12	$J_4$ $J_6$ $J_7$ $J_8$ $J_9$	$J_6$	106.32
39	$J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$	None	106.32
41	$J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$	None	106.32
42	$J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{12}$	None	106.32
44.48	$J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{12}$	$J_{12}$	150.8
46	$J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{13}$	None	150.8
48	$J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{13}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$	None	150.8
49.88	$J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{13}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$	$J_{13}$	200.68
50	$J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ $J_{18}$	None	200.68
52	$J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ $J_{18}$ $J_{19}$ $J_{20}$	None	200.68
56.04	$J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ $J_{18}$ $J_{19}$ $J_{20}$	$J_{18}$	256.72
60.34	$J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ $J_{19}$ $J_{20}$	$J_{15}$	317.06
67.06	$J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{16}$ $J_{17}$ $J_{19}$ $J_{20}$	$J_{16}$	384.12
73.62	$J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{17}$ $J_{19}$ $J_{20}$	$J_{20}$	457.74
80.02	$J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{17}$ $J_{19}$	$J_8$	537.76
87.04	$J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{17}$ $J_{19}$	$J_{14}$	624.8
93.88	$J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{17}$ $J_{19}$	$J_{17}$	718.68
100.54	$J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{19}$	$J_4$	819.22
107.74	$J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{19}$	$J_9$	926.96
114.74	$J_7$ $J_{10}$ $J_{11}$ $J_{19}$	$J_{10}$	1041.7
122.22	$J_7$ $J_{11}$ $J_{19}$	$J_7$	1163.92
130.14	$J_{11}$ $J_{19}$	$J_{11}$	1294.06
138.46	$J_{19}$	$J_{19}$	1432.52
146.52	None	None	1579.04

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Table 9. The selection process of jobs under π(I_c)

time	valid job set	selected job	current objective function value
0	$J_1$	$J_1$	0
8	$J_2$	$J_2$	8
13.88	None	None	21.88
17	$J_3$	$J_3$	21.88
28.52	$J_4$ $J_5$	$J_5$	50.4
32.28	$J_4$ $J_6$ $J_7$	$J_6$	82.68
39.64	$J_4$ $J_7$ $J_8$ $J_9$ $J_{10}$	$J_8$	122.32
47.74	$J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{12}$ $J_{13}$	$J_{12}$	170.06
53.02	$J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{13}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ $J_{18}$ $J_{19}$ $J_{20}$	$J_{18}$	223.08
57.32	$J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{13}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ $J_{19}$ $J_{20}$	$J_{13}$	280.4
63.2	$J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{15}$ $J_{16}$ $J_{17}$ $J_{19}$ $J_{20}$	$J_{15}$	343.6
69.76	$J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{16}$ $J_{17}$ $J_{19}$ $J_{20}$	$J_{16}$	413.36
76.16	$J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{17}$ $J_{19}$ $J_{20}$	$J_{20}$	489.52
82.4	$J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{14}$ $J_{17}$ $J_{19}$	$J_{14}$	571.92
89.24	$J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{17}$ $J_{19}$	$J_{17}$	661.16
95.9	$J_4$ $J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{19}$	$J_4$	757.06
103.1	$J_7$ $J_9$ $J_{10}$ $J_{11}$ $J_{19}$	$J_9$	860.16
110.1	$J_7$ $J_{10}$ $J_{11}$ $J_{19}$	$J_{10}$	970.26
117.58	$J_7$ $J_{11}$ $J_{19}$	$J_7$	1087.84
125.5	$J_{11}$ $J_{19}$	$J_{11}$	1213.34
133.82	$J_{19}$	$J_{19}$	1347.16
141.88	None	None	1489.04

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Table 10. The comparison between $ \sigma(I_c) $ and $ \pi(I_c) $

	job processing scheduling	objective function value
$ \sigma(I_c) $	$ J_2-J_1-J_3-J_5-J_6-J_{12}-J{13}-J_{18}-J_{15}-J_{16}-J_{20}-J_8-J_{14}-J_{17}-J_4-J_9-J_{10}-J_7-J_{11}-J_{19} $	1579.04
$ \pi(I_c) $	$ J_1-J_2-J_3-J_5-J_6-J_8-J_{12}-J_{18}-J{13}-J_{15}-J_{16}-J_{20}-J_{14}-J_{17}-J_4-J_9-J_{10}-J_7-J_{11}-J_{19} $	1489.04

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