A moving block sequence-based evolutionary algorithm for resource investment project scheduling problems

Xiaoxiao Yuan; Jing Liu; Xingxing Hao

doi:10.3934/bdia.2017007

Article Contents

2017, Volume 2, Issue 1: 39-58. Doi: 10.3934/bdia.2017007

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A moving block sequence-based evolutionary algorithm for resource investment project scheduling problems

Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education, Xidian University, Xi'an 710071, China

* Corresponding author: Jing Liu
* Corresponding author: Jing Liu

Published: January 2017

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Abstract

Inspired by the representation designed for floorplanning problems, in this paper, we proposed a new representation, namely the moving block sequence (MBS), for resource investment project scheduling problems (RIPSPs). Since each activity of a project in RIPSPs has fixed duration and resource demand, we consider an activity as a rectangle block whose width is equal to the duration of the activity and height the resource needed by the activity. Four move modes are designed for activities, by using which the activity can move to the appropriate position. Therefore, the new representation of the project of RIPSPs consists of two parts: an activity list and a move mode list. By initializing the move modes randomly for each activity and moving it appropriately, the activity list can be decoded into valid solutions of RIPSPs. Since the decoding method of MBS guarantees that after moved, each activity is scheduled in the left-most and bottom-most position within a coordinate, which means that each activity in the corresponding project is arranged as early as possible when the precedence constraints and resource demands are satisfied. In addition, the multiagent evolutionary algorithm (MAEA) is employed to incorporate with the newly designed MBS representation in solving RIPSPs. With the intrinsic properties of MBS in mind, four behaviors, namely the crossover, mutation, competition, and self-learning operators are designed for agents in MAEA. To test the performance of our algorithm, 450 problem instances are used and the experimental results demonstrate the good performance of the proposed representation.

Keywords:

Resource investment project scheduling problems,
moving block sequence representation,
multiagent evolutionary algorithm.

Mathematics Subject Classification: 90C59.

Citation:

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Figure 1. An example of precedence graph

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Figure 2. An example of precedence graph

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Figure 3. The types of overlaps. (a) All kinds of top-overlaps, (b) all kinds of right-overlaps

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Figure 4. Relative positions of $CoverLeftX$ and $CoverRightX$. (a) and (b) are the cases without violating precedence constraints. (c) and (d) are the cases violating precedence constraints

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Figure 5. The decoding process

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Figure 6. The agent lattice of MBS$_{\rm {MAEA}}$-RIPSP

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Table 1. The Percentages of Finding Optimal Solutions for MBS$_{\rm MAEA}$-RIPSP on Möhring Instances with 1000 Evaluations

$C_1/C_2/C_3/C_4$	$\theta = 1.0$	$\theta = 1.1$	$\theta = 1.2$	$\theta = 1.3$	$\theta = 1.4$	$\theta = 1.5$
1/1/1/1	1.00	1.00	1.00	1.00	0.067	0.033
3/1/1/1	1.00	0.90	0.90	0.80	0.133	0.067
1/3/1/1	1.00	1.00	1.00	0.80	0.700	0.333
1/1/3/1	1.00	1.00	1.00	1.00	0.950	0.067
1/1/1/3	1.00	1.00	1.00	0.60	1.00	0.067
3/3/1/1	1.00	0.50	0.80	0.80	0.00	0.333
3/1/3/1	1.00	1.00	0.90	0.85	0.60	0.333
3/1/1/3	1.00	0.75	0.95	0.75	0.533	0.067
1/3/3/1	1.00	1.00	1.00	1.00	0.70	0.033
1/3/1/3	1.00	1.00	1.00	1.00	0.80	0.00
1/1/3/3	1.00	1.00	1.00	1.00	0.60	0.00
3/3/3/1	1.00	1.00	0.90	1.00	0.333	0.20
3/3/1/3	1.00	0.45	1.00	0.75	0.133	0.033
3/1/3/3	1.00	1.00	0.95	0.85	0.167	0.067
1/3/3/3	1.00	1.00	1.00	1.00	0.70	0.067

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Table 2. The Comparison of Numbers of Generation to Reach to the Optimal Solutions between MBS$_{\rm MAEA}$-RIPSP and GA for Möhring Test Sets

$C_1/C_2/C_3/C_4$	$\theta = 1.0$		$\theta = 1.1$		$\theta = 1.2$		$\theta = 1.3$		$\theta = 1.4$		$\theta = 1.5$
$C_1/C_2/C_3/C_4$	MBS	GA	MBS	GA	MBS	GA	MBS	GA	MBS	GA	MBS	GA
1/1/1/1	1	1	1	2	1	1	1	6	6	21	14	1
3/1/1/1	1	1	6	1	1	2	1	2	7	20	27	1
1/3/1/1	1	2	1	33	1	4	1	1	2	43	13	1
1/1/3/1	1	1	1	1	1	1	1	1	1	23	4	3
1/1/1/3	1	1	1	1	1	3	2	1	12	1	4	5
3/3/1/1	1	1	1	28	1	2	1	2	5	1	7	1
3/1/3/1	1	1	1	1	1	2	1	1	8	11	8	9
3/1/1/3	1	2	1	2	1	6	2	1	9	8	10	2
1/3/3/1	1	1	1	1	1	8	2	1	1	15	2	3
1/3/1/3	1	1	1	2	1	1	1	1	8	6	20	1
1/1/3/3	1	3	1	3	1	2	1	2	5	37	12	1
3/3/3/1	1	1	1	2	1	2	1	2	9	14	13	1
3/3/1/3	1	1	4	1	1	1	3	2	8	8	22	1
3/1/3/3	1	2	1	21	1	1	2	1	31	18	4	5
1/3/3/3	1	2	1	2	1	3	3	1	19	3	49	23

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Table 3. Parameter Settings for J10, J14 and J20 Test Sets

Test Set	#Agent	MaxGen	ExcuteNum	$SelfLTime$	$P_{cro}$	$P_{mut}$	$P_{com}$
J10	$20\times 20$	10	10	12	0.95	0.85	0.9
J14	$20\times 20$	10	8	12	0.95	0.85	1.0
J20	$20\times 19$	10	8	12	0.95	0.85	1.0

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Table 4. Experimental Results of MBS$_{\rm MAEA}$-RIPSP for J10, J14 and J20 Test Sets

$\theta$	J10			J14			J20
$\theta$	Opt.%	Dev.%	Eva.	Opt.%	Dev.%	Eva.	Opt.%	Dev.%	Eva.
1.0	40.0	6.8896	6988	76.5	0.7970	3719	32.5	4.9126	7135
1.1	41.0	3.4006	7225	63.0	1.0457	4981	35.0	5.4335	7327
1.2	55.0	2.5922	5821	43.125	2.5851	7683	33.0	4.9258	7273
1.3	51.5	2.4822	6170	49.375	2.2975	7369	43.0	2.6173	7188
1.4	56.0	2.8036	6065	47.5	1.9321	6771	43.0	2.0079	7279
1.5	68.5	1.8743	4496	55.0	1.8272	7465	43.0	1.9728	7154

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Table 5. Comparisons between MBS$_{\rm MAEA}$-RIPSP and GA for J10, J14 and J20 Test Sets

Test Set	Opt.%		Dev.%
Test Set	MBS	GA	MBS	GA
J10	52.00	48.20	3.3404	0.23
J14	55.75	40.00	1.7474	0.32
J20	38.25	33.33	3.6445	4.68

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