# American Institute of Mathematical Sciences

January  2017, 2(1): 39-58. doi: 10.3934/bdia.2017007

## 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

Published  September 2017

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.

Citation: Xiaoxiao Yuan, Jing Liu, Xingxing Hao. A moving block sequence-based evolutionary algorithm for resource investment project scheduling problems. Big Data & Information Analytics, 2017, 2 (1) : 39-58. doi: 10.3934/bdia.2017007
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An example of precedence graph
An example of precedence graph
The types of overlaps. (a) All kinds of top-overlaps, (b) all kinds of right-overlaps
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
The decoding process
The agent lattice of MBS$_{\rm {MAEA}}$-RIPSP
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
 $C_1/C_2/C_3/C_4$ $\theta = 1.0$ $\theta = 1.1$ $\theta = 1.2$ $\theta = 1.3$ $\theta = 1.4$ $\theta = 1.5$ 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
The Comparison of Numbers of Generation to Reach to the Optimal Solutions between MBS$_{\rm MAEA}$-RIPSP and GA for Möhring Test Sets
 $C_1/C_2/C_3/C_4$ $\theta = 1.0$ $\theta = 1.1$ $\theta = 1.2$ $\theta = 1.3$ $\theta = 1.4$ $\theta = 1.5$ 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
 $C_1/C_2/C_3/C_4$ $\theta = 1.0$ $\theta = 1.1$ $\theta = 1.2$ $\theta = 1.3$ $\theta = 1.4$ $\theta = 1.5$ 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
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
 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
Experimental Results of MBS$_{\rm MAEA}$-RIPSP for J10, J14 and J20 Test Sets
 $\theta$ J10 J14 J20 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
 $\theta$ J10 J14 J20 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
Comparisons between MBS$_{\rm MAEA}$-RIPSP and GA for J10, J14 and J20 Test Sets
 Test Set Opt.% Dev.% 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
 Test Set Opt.% Dev.% 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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