A Hybrid Genetic Algorithm for the Simple Assembly Line Balancing Problem with a Fixed Number of Workstations
Abstract
:1. Introduction
The success of reformulation methods “gives a certain foundation for the concentration on finding effective procedures for SALBP1 in the past 50 years though SALBP2 is a problem which may even have the greater practical relevance because it arises whenever an existing line has to be rebalanced, while SALBP1 is relevant mainly in case of the first installation of a line.”
1.1. Problem Description
1.2. Review on the Resolution Methods
1.3. Outline of the Proposed Algorithm
2. Line Balancing with Incompatibilities between Tasks
2.1. Problem Description
 The problem may feature both type1 and type2 objectives, and the latter can be solved using a reformulation method;
 The SALBP is a simplified version of the IBTALBP and, as such, any lower bound for the SALBP is a valid lower bound for the modified problem;
 The problem is reversible (see Section 1.1).
2.2. Bounds and Reduction Rules
2.2.1. SALBP1 Lower Bounds
2.2.2. New Lower Bounds for the IBTALBP
2.2.3. Preprocessing Rules
2.3. Resolution by Means of Bounded Dynamic Programming
Algorithm 1 Outline of the BDP procedure. 
Input: Parameters w, t, c, and m, and instance definition

3. The Genetic Algorithm
3.1. Representation Scheme and Initialisation
3.2. Reformulation Method and Fitness Evaluation
3.3. Genetic Operations and Parallelisation Scheme
3.4. Overall Structure of the Genetic Algorithm
Algorithm 2 Outline of the complete procedure. 
Input: Parameters m, w, t, m, population Size, $mp$, time limit, l, and instance definition

4. Computational Experiments
 A random keyencoded hybrid GA version of the algorithm whose encoding of individuals is similar to the proposal found in [50]. The procedure is also a reformulation method that uses the same approach proposed in Section 3. The GA encodes each individual using a random valued weight for each task of the instance. In [50], these weights were decoded using a stationoriented constructive procedure. In our proposal, the weights substitute the criterion used to reorder tasks in the BDP (a higher weight indicates that the task should appear first in the ordering), which is then used during the fitness evaluation. The modified GA uses the initialisation, crossover, and mutation operators proposed in [50] to handle the weight encoding. In addition to the weight information, each individual also encodes a binary value to determine the direction of the precedence graph given to the BDP. The precedence graph part of each individual uses the operators defined in Section 3.3.
 A random sampling method. The reformulation approach proposed in Section 3.2 is used. The second phase—the upperbound search phase—is not conducted using a GA but rather by iteratively creating new IBTALBP instances using the procedure proposed in Section 3.1 and solving them using the BDP.
5. Discussion and Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A. Improved Solutions
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Population Size = 100  w = 500  l = 20 
Time limit = 3600 s  $mp$ = 0.03  t = 50 
Procedure  # Best  # Improves 

Literature  294  
BSBDP  293  1 
RKGA  296  6 
RSBDP  296  5 
IBTGA  301  8 
BS  RKGA (3600 s)  RSBDP (3600 s)  IBTGA (3600 s)  

Graph(m)  Lit.  BDP  Min.  Avg.  Min.  Avg.  Min.  Avg. 
Arcus2(19)  7922  7922  7921  7921.5  7921  7921.5  7921  7921.3 
Arcus2(20)  7524  7523  7523  7523.2  7523  7523  7523  7523.1 
Arcus2(21)  7187  7186  7184  7185.4  7184  7185.5  7184  7185.1 
Arcus2(22)  6856  6856  6859  6859  6858  6858  6850  6854.1 
Arcus2(23)  6560  6560  6559  6564  6560  6562.7  6559  6561.4 
Arcus2(24)  6282  6290  6286  6295  6286  6292.2  6280  6280.6 
Arcus2(25)  6101  6118  6108  6110.1  6105  6109.6  6096  6096.5 
Arcus2(26)  5855  5860  5864  5867.4  5854  5857.4  5851  5851.5 
Mukherje(20)  220  220  221  221  221  221  221  221 
Barthol2(50)  85  86  85  85  85  85.7  85  85.5 
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ÁlvarezMiranda, E.; Pereira, J.; TorrezMeruvia, H.; Vilà, M. A Hybrid Genetic Algorithm for the Simple Assembly Line Balancing Problem with a Fixed Number of Workstations. Mathematics 2021, 9, 2157. https://doi.org/10.3390/math9172157
ÁlvarezMiranda E, Pereira J, TorrezMeruvia H, Vilà M. A Hybrid Genetic Algorithm for the Simple Assembly Line Balancing Problem with a Fixed Number of Workstations. Mathematics. 2021; 9(17):2157. https://doi.org/10.3390/math9172157
Chicago/Turabian StyleÁlvarezMiranda, Eduardo, Jordi Pereira, Harold TorrezMeruvia, and Mariona Vilà. 2021. "A Hybrid Genetic Algorithm for the Simple Assembly Line Balancing Problem with a Fixed Number of Workstations" Mathematics 9, no. 17: 2157. https://doi.org/10.3390/math9172157