Adaptive Genetic Algorithm Based on Fuzzy Reasoning for the Multilevel Capacitated LotSizing Problem with Energy Consumption in Synchronizer Production
Abstract
:1. Introduction
2. Literature Review—Existing Approaches for the MLCLSP
2.1. DecompositionBased Approaches
2.2. Traditional Heuristic Methods
2.3. Bionic Algorithms
2.4. A Brief Summary
3. The Proposed FuzzyGA for MLCLSP
3.1. Problem Statement
3.2. Solving Model of MLCLSP with the FuzzyGA
Algorithm1 Capacity constraint algorithm 
Input: x_{it}, ${Y}_{it}$; output: x_{it} 

{production lot size ${x}_{it}$ of period t is obtained without capacity constraints sort in ascending order $Xu{p}_{{k}_{1}t}$. Then, find the minimum lot size $Xu{p}_{t}\left(1\right)$. Calculate ${C}_{r{k}_{1}}=\leftQC\left({k}_{1},t\right)\right{a}_{{k}_{1}1t}Xu{p}_{{k}_{1}t}\left(1\right),i=1$, Else calculate ${C}_{r{k}_{j}}=\leftQC\left({k}_{j},t\right)\right\left(\sum _{m=1}^{i1}{a}_{{k}_{j}mt}Xu{p}_{t}\left(m\right)+{a}_{{k}_{j}it}{x}_{it}\right)$ if ${C}_{r{k}_{j}}\le 0$. End this cycle and jump to Step 5 for the next cycle. End End 

{the corresponding lotsizing of $Xu{p}_{t}\left(i\right)$ is transferred to the production lotsizing of $t1$. Set the corresponding binary variable ${Y}_{it}$ to zero. Cyclic variable $i=i+1$. Calculate ${C}_{r{k}_{j}}={C}_{r{k}_{j}}{a}_{{k}_{j}it}Xu{p}_{{k}_{j}t}\left(i\right)$}. 

End 

If $t>0$, jump to Step 4. Else, update $Q$. If $QC=QCa\le 0$, output updated lotsizing, setup variable as result, and end the whole procedure; Else, the solution cannot satisfy the capacity constraint and end the whole procedure. End End 
Algorithm 2 Algorithm of solving fitness value 
Input: production lotsizing x_it, inventory I_it, and setup variable Y_it; output: fitness value: f 

3.3. Encoder Based on Hierarchical Structure
Algorithm 3 Algorithm of a lotsizing solution according to the demand constraints 
Input: setup variable, product demand, and material quantity relationship ${Y}_{it}$, ${D}_{it}$, ${R}_{ij}$; output: production lotsizing ${x}_{it}$ 

Use demand balance relation ${x}_{it}\left(1,i,{T}_{i}\left(j\right)\right)=\sum _{t={T}_{i}\left(j\right)}^{{T}_{i}\left(j+1\right)1}{D}_{it}\left(i,t\right)$ and solve production lotsizing ${X}_{it}$ of product i at different planning periods. Else, use demand balance relation ${x}_{it}\left(1,i,{T}_{i}\left(j\right)\right)=\sum _{t={T}_{i}\left(j\right)}^{{T}_{i}\left(j+1\right)1}{D}_{it}\left(i,t\right)+\sum _{{l}_{i}=1}^{l1}{R}_{ij}{X}_{it}\left({l}_{i},i,{T}_{i}\left(j\right)\right)$ and solve production lotsizing ${x}_{it}$. End 

3.4. Adaptive Optimization Process of Parameters Based on Fuzzy Theory
4. Experimental Setup
4.1. Experimental Data and Design Hypothesis
4.1.1. Experiment 1 Validation of FuzzyGA in Solving MLCLSP
4.1.2. Experiment 2 Adaptive Parameters of the FuzzyGA
4.1.3. Experiment 3 Performance of FuzzyGA in Solving MLCLSP
4.2. Performance Evaluation Metrics
5. Results and Discussion
5.1. Results and Discussion for Experiment 1
5.2. Results and Discussion for Experiment 2
5.3. Results and Discussion for Experiment 3
6. Conclusions and Prospects
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
 Awad, M.I.; Hassan, N.M. Joint decisions of machining process parameters setting and lotsize determination with environmental and quality cost consideration. J. Manuf. Syst. 2018, 46, 79–92. [Google Scholar] [CrossRef]
 Jia, S.; Yuan, Q.; Cai, W.; Li, M.; Li, Z. Energy modeling method of machineoperator system for sustainable machining. Energy Convers. Manag. 2018, 172, 265–276. [Google Scholar] [CrossRef]
 Klement, N.; Abdeljaouad, M.A.; Porto, L.; Silva, C. LotSizing and Scheduling for the Plastic Injection Molding Industry—A Hybrid Optimization Approach. Appl. Sci. 2021, 11, 1202. [Google Scholar] [CrossRef]
 Boonmee, A.; Sethanan, K. A GLNPSO for multilevel capacitated lotsizing and scheduling problem in the poultry industry. Eur. J. Oper. Res. 2016, 250, 652–665. [Google Scholar] [CrossRef]
 Berretta, R.; Rodrigues, L.F. A memetic algorithm for a multistage capacitated lotsizing problem. Int. J. Prod. Econ. 2004, 87, 67–81. [Google Scholar] [CrossRef]
 Buschkühl, L.; Sahling, F.; Helber, S.; Tempelmeier, H. Dynamic capacitated lotsizing problems: A classification and review of solution approaches. Or Spectr. 2010, 32, 231–261. [Google Scholar] [CrossRef]
 Duda, J.; Stawowy, A. Optimization Methods for LotSizing Problem in an Automated Foundry. Arch. Metall. Mater. 2013, 58, 863–866. [Google Scholar] [CrossRef] [Green Version]
 Jia, S.; Yuan, Q.; Lv, J.; Liu, Y.; Ren, D.; Zhang, Z. Therbligembedded value stream mapping method for lean energy machining. Energy 2017, 138, 1081–1098. [Google Scholar] [CrossRef] [Green Version]
 Jinxing, X.; Jiefang, D. Heuristic genetic algorithms for general capacitated lotsizing problems. Comput. Math. Appl. 2002, 44, 263–276. [Google Scholar]
 Yijun, L.; David, L. Selfadaptive randomized constructive heuristics for the multiitem capacitated lotsizing problem. arXiv 2021, arXiv:2103.04199. [Google Scholar]
 Furlan, M.M.; Santos, M.O. BFO: A hybrid bees algorithm for the multilevel capacitated lotsizing problem. J. Intell. Manuf. 2017, 28, 929–944. [Google Scholar] [CrossRef]
 Kirschstein, T.; Meisel, F. A dynamic multicommodity lotsizing problem with supplier selection storage selection and discounts for the process industry. Eur. J. Oper. Res. 2019, 279, 393–406. [Google Scholar] [CrossRef]
 Cunha, J.O.; Kramer, H.H.; Melo, R.A. On the computational complexity of uncapacitated multiplant lotsizing problems. Optim. Lett. 2021, 15, 803–812. [Google Scholar] [CrossRef]
 Han, Y.; Kaku, I.; Tang, J.; Dellaert, N.; Cai, J.; Li, Y. A Comparison of Particle Swarm Optimizations for Uncapacitated Multilevel Lotsizing Problems(Theory and Methodology). J. Jpn. Ind. Manag. Assoc. 2010, 61, 203–213. [Google Scholar]
 Seeanner, F. MultiStage Simultaneous LotSizing and Scheduling—Planning of Flow Lines with Shifting Bottlenecks; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2013. [Google Scholar] [CrossRef]
 Jia, S.; Yuan, Q.; Cai, W.; Lv, J.; Hu, L. Establishing prediction models for feeding power and material drilling power to support sustainable machining. Int. J. Adv. Manuf. Technol. 2019, 100, 2243–2253. [Google Scholar] [CrossRef]
 Jia, S.; Cai, W.; Liu, C.; Zhang, Z.; Bai, S.; Wang, Q.; Li, S.; Hu, L. Energy modeling and visualization analysis method of drilling processes in the manufacturing industry. Energy 2021, 228, 120567. [Google Scholar] [CrossRef]
 Zohali, H.; Naderi, B.; Mohammadi, M. The economic lot scheduling problem in limitedbuffer flexible flow shops: Mathematical models and a discrete fruit fly algorithm. Appl. Soft Comput. 2019, 80, 904–919. [Google Scholar] [CrossRef]
 Farooq, B.; Bao, J.; Raza, H.; Sun, Y.; Ma, Q. Flowshop path planning for multiautomated guided vehicles in intelligent textile spinning cyberphysical production systems dynamic environment. J. Manuf. Syst. 2021, 59, 98–116. [Google Scholar] [CrossRef]
 Che, Z.H.; Chiang, T.A.; Lin, T.T. A multiobjective genetic algorithm for assembly planning and supplier selection with capacity constraints. Appl. Soft Comput. 2021, 101, 107030. [Google Scholar] [CrossRef]
 Hadi, A.S.; Mehdi, R.B. Solving a flexible job shop lot sizing problem with shared operations using a selfadaptive COA. Int. J. Prod. Res. 2021, 59, 483–515. [Google Scholar]
 Andrés, A.; Mingyuan, C. Multilevel lotsizing with rawmaterial perishability, deterioration, and lotsizing ordering: An application of production planning in advanced composite manufacturing. Comput. Ind. Eng. 2020, 145, 106484. [Google Scholar]
 Sikora, R. A genetic algorithm for integrating lotsizing and sequencing in scheduling a capacitated flow line. Comput. Ind. Eng. 1996, 30, 969–981. [Google Scholar] [CrossRef]
 Soltanali, H.; Rohani, A.; Tabasizadeh, M.; AbbaspourFard, M.H.; Parida, A. An improved fuzzy inference systembased risk analysis approach with application to automotive production line. Neural Comput. Appl. 2020, 32, 10573–10591. [Google Scholar] [CrossRef]
 Tempelmeier, H.; Helber, S. A heuristic for dynamic multiitem multilevel capacitated lotsizing for general product structures. Eur. J. Oper. Res. 1984, 75, 296–311. [Google Scholar] [CrossRef]
 Boctor, F.F.; Poulin, P. Heuristics for Nproduct, Mstage, economic lot sizing and scheduling problem with dynamic demand. Int. J. Prod. Res. 2005, 43, 2809–2828. [Google Scholar] [CrossRef]
 Miller, A.J.; Nemhauser, G.L.; Savelsbergh, M. Solving MultiItem Capacitated LotSizing Problems with Setup Times by BranchandCut; CORE Discussion Papers; Université Catholique de Louvain: OttigniesLouvainlaNeuve, Belgium, 2000. [Google Scholar]
 Zhang, C.; Zhang, D.; Wu, T. Datadriven Branching and Selection for Lotsizing and Scheduling Problems with Sequencedependent Setups and Setup Carryover. Comput. Oper. Res. 2021, 132, 105289. [Google Scholar] [CrossRef]
 Ramezanian, R.; SaidiMehrabad, M.; Fattahi, P. MIP formulation and heuristics for multistage capacitated lotsizing and scheduling problem with availability constraints. J. Manuf. Syst. 2013, 32, 392–401. [Google Scholar] [CrossRef]
 Verma, M.; Sharma, R. Lagrangian relaxation and bounded variable linear programs to solve a twolevel capacitated lot sizing problem. In Proceedings of the 2011 3rd International Conference on Electronics Computer Technology, Kanyakumari, India, 8–10 April 2011; Volume 6, pp. 188–192. [Google Scholar]
 Tempelmeier, H.; Buschkühl, L. A heuristic for the dynamic multilevel capacitated lotsizing problem with linked lotsizes for general product structures. Or Spectr. 2009, 31, 385–404. [Google Scholar] [CrossRef]
 Wu, T.; Shi, L.; Geunes, J.; Akartunalı, K. An optimization framework for solving capacitated multilevel lotsizing problems with backlogging. Eur. J. Oper. Res. 2011, 214, 428–441. [Google Scholar] [CrossRef] [Green Version]
 Zhao, Q.; Xie, C.; Xiao, Y. A variable neighborhood decomposition search algorithm for multilevel capacitated lotsizing problems. Electron. Notes Discret. Math. 2012, 39, 129–135. [Google Scholar] [CrossRef]
 Chen, H. Fixandoptimize and variable neighborhood search approaches for multilevel capacitated lot sizing problems. Omega 2015, 56, 25–36. [Google Scholar] [CrossRef]
 Mohammadi, M.; Fatemi Ghomi SM, T.; Karimi, B.; Torabi, S.A. Rollinghorizon and fixandrelax heuristics for the multiproduct multilevel capacitated lotsizing problem with sequencedependent setups. J. Intell. Manuf. 2010, 21, 501–510. [Google Scholar] [CrossRef]
 Pitakaso, R.; Almeder, C.; Doerner, K.F.; Hartl, R.F. Combining exact and populationbased methods for the constrained multilevel lotsizing problem. Int. J. Prod. Res. 2006, 44, 4755–4771. [Google Scholar] [CrossRef]
 Toledo, C.F.M.; De Oliveira, R.R.R.; França, P.M. A Hybrid MultiPopulation Genetic Algorithm Applied to Solve the MultiLevel Capacitated Lot Sizing Problem with Backlogging; Elsevier Science Ltd.: Amsterdam, The Netherlands, 2013; Volume 40, pp. 910–919. [Google Scholar]
 Mohammadi, M.; Ghomi, S.M.T.F. Genetic algorithmbased heuristic for capacitated lotsizing problem in flow shops with sequencedependent setups. Expert Syst. Appl. 2011, 38, 7201–7207. [Google Scholar] [CrossRef]
 Duda, J. A hybrid genetic algorithm and variable neighborhood search for multifamily capacitated lotsizing problem. Electron. Notes Discret. Math. 2017, 58, 103–110. [Google Scholar] [CrossRef]
 Quadt, D.; Kuhn, H. Capacitated lotsizing with extensions: A review. 4OR 2008, 6, 61–83. [Google Scholar] [CrossRef]
t = 1,2, …,T:  Planning period 
i = 1,2, …,n:  Index for product 
k = 1,2, …,M:  Equipment number 
E(i):  Set of nextlevel products that need product i in the production process 
c_{it}:  Production cost unit product 
s_{it}:  Setup cost per production of lotsizing 
h_{it}:  Inventory cost unit product 
D_{it}:  External demand of product i in the planning period t 
Ca_{kt}:  Production capacity of equipment resources k in the planning period t 
R_{ij}:  The quantity of product i directly needed to produce one unit of product j (gozinto factor) 
ct_{kit}:  The time cost of unit production of the product i in the equipment resource k during the planning period t 
st_{kit}:  Time cost of production setup of the product i in the equipment resource k during the planning period t 
x_{it}:  Output of product i during the planning period t 
I_{it}:  Inventory of product i during the planning period t 
${Y}_{it}$:  Whether the product i is produced in the planning period t; ${Y}_{it}=1$, if ${x}_{it}>0,{Y}_{it}=0$ otherwise. 
IF  $\mathbf{PS}\left(\mathit{o}\mathit{b}{\mathit{j}}_{\mathit{v}\mathit{a}\mathit{l}\mathit{u}\mathit{e}}^{\prime}\right)$  $\mathbf{PM}\left(\mathit{o}\mathit{b}{\mathit{j}}_{\mathit{v}\mathit{a}\mathit{l}\mathit{u}\mathit{e}}^{\prime}\right)$  $\mathbf{PB}\left(\mathit{o}\mathit{b}{\mathit{j}}_{\mathit{v}\mathit{a}\mathit{l}\mathit{u}\mathit{e}}^{\prime}\right)$ 

THEN  PB (P_{m})  PM (P_{m})  PS (P_{m}) 
THEN  PB (P_{c})  PM (P_{c})  PS (P_{c}) 
Product Number $\mathit{i}$\Planning Period $\mathit{t}$  1  2  3  4 

1  28  14  9  32 
2  6  1  1  17 
3  1  9  4  11 
4  17  13  15  17 
5  18  2  4  10 
Equipment Number k\Planning Period t  1  2  3  4 

1  5666  5483  5427  5539 
2  5332  5437  5213  5147 
3  5189  5088  5711  5225 
4  5891  5676  5367  5148 
5  5835  5274  5249  5982 
6  5442  5948  5618  5596 
7  5694  5944  5902  5961 
8  5498  5342  5072  5554 
9  5625  5717  5492  5336 
Level 2\Level 1 Product Number  1  2 

3  2  1 
4  0  2 
5  1  1 
Equipment Number k  Product Number i  Planning Period t  

1  2  3  4  
1  1  34\2  31\4  32\3  30\8 
2  32\9  33\7  32\6  35\9  
3  31\3  33\8  33\9  34\9  
4  31\4  33\5  31\6  32\8  
5  33\7  35\7  32\4  30\3  
2  1  35\8  31\8  31\4  30\2 
2  30\2  33\5  33\5  33\6  
3  32\2  31\5  35\5  33\6  
4  34\4  34\2  35\2  35\9  
5  32\3  31\7  35\4  33\7  
3  1  33\7  35\4  34\7  30\3 
2  34\3  30\6  30\2  32\4  
3  31\3  31\8  35\9  32\4  
4  30\7  34\4  31\5  31\2  
5  33\3  30\7  33\3  33\9  
4  1  32\5  32\6  32\7  33\7 
2  34\3  30\4  30\6  30\7  
3  33\9  31\3  32\7  33\3  
4  33\3  33\9  30\9  34\7  
5  35\6  33\8  33\8  31\8  
5  1  33\4  32\6  32\6  34\8 
2  33\8  31\2  32\2  33\5  
3  34\7  31\5  34\8  31\5  
4  31\4  33\4  34\5  31\9  
5  30\2  33\5  33\5  30\8  
6  1  31\6  32\8  33\8  34\7 
2  31\6  34\5  31\4  31\4  
3  31\9  33\8  33\3  34\8  
4  31\3  31\9  33\3  34\3  
5  35\3  34\8  32\4  31\8  
7  1  31\4  32\5  31\2  30\9 
2  34\5  33\3  32\3  32\5  
3  34\6  33\9  32\4  33\8  
4  32\3  32\3  35\4  34\3  
5  33\8  31\2  31\3  31\7  
8  1  34\5  34\6  31\3  31\9 
2  33\8  31\7  32\7  33\6  
3  30\7  33\7  35\8  32\6  
4  30\2  32\6  32\6  31\2  
5  30\7  33\3  32\7  33\6  
9  1  30\8  32\4  30\8  35\2 
2  30\5  34\3  32\4  35\7  
3  35\7  32\7  34\7  32\3  
4  33\7  34\5  31\3  32\5  
5  30\2  32\7  35\5  35\9 
Parameters\Production Number  1  2  3  4  5 

Production cost ${c}_{it}$  39  36  31  37  35 
Inventory cost ${h}_{it}$  59  48  32  53  40 
Setup cost ${s}_{it}$  123  110  117  104  127 
Initial inventory ${I}_{i0}$  5  18  18  1  22 
Energy costs  1.5  1.2  2  2.2  1.8 
Fuzzy Sets\Magnitude  ${\mathit{P}}_{\mathit{m}1}$  ${\mathit{P}}_{\mathit{m}2}$  ${\mathit{P}}_{\mathit{m}3}$ 

PS  0.01  0.1  0.001 
PM  0.02  0.2  0.002 
PB  0.03  0.3  0.003 
Fuzzy Sets\Magnitude  ${\mathit{P}}_{\mathit{c}1}$  ${\mathit{P}}_{\mathit{c}2}$  ${\mathit{P}}_{\mathit{c}3}$ 

PS  0.1  0.1  0.1 
PM  0.7  0.3  0.5 
PB  0.9  0.9  0.9 
Product Number i \Planning Period t  1  2  3  4 

1  28  14  9  32 
2  7  0  1  17 
3  64  37  23  92 
4  31  13  17  51 
5  53  16  14  59 
Product Type N  Planning Period T  Number of Devices M  Traditional GA  FuzzyGA  

Optimal Value  Average Optimal Value  Iterations  Running Time (s)  Optimal Value  Average Optimal Value  Iterations  Running Time (s)  
5  5  9  90,101  94,619  95  4.5174  87,629  90,839  104  4.3237 
10  5  9  256,618  263,514  111  5.7030  242,461  246,960  105  5.5393 
20  5  9  999,289  1,061,107  94  7.0897  908,033  944,982  97  7.6957 
20  7  9  1,406,511  1,457,795  107  9.8041  1,349,712  1,378,720  93  9.2876 
20  10  9  1,950,637  2,115,856  100  12.7491  2,005,225  2,044,097  90  11.3101 
20  10  15  1,985,502  2,020,672  95  13.3185  1,888,792  1,905,142  85  12.2024 
20  10  20  2,515,585  2,596,667  112  16.4103  2,410,076  2,440,758  97  15.5662 
Product Type N  Planning Period T  Number of Devices M  The Proposed Iteration Condition  100 Times  300 Times  500 Times  

Optimal Value  Running Time (s)  Optimal Value  Running Time (s)  Optimal Value  Running Time (s)  Optimal Value  Running Time (s)  
5  5  9  87,629  4.3237  88,748  3.3064  88,080  9.7639  86,276  16.6609 
10  5  9  242,461  5.5393  244,647  4.3638  247,769  13.0195  241,662  21.7165 
20  5  9  908,033  7.6957  975,467  6.4136  947,128  21.6976  918,993  33.2535 
20  7  9  1,349,712  9.2876  1,402,439  8.2405  1,345,295  24.7846  1,214,054  42.3357 
20  10  9  2,005,225  11.3101  2,081,146  10.7705  1,877,778  33.7112  2,074,800  54.1895 
20  10  15  1,888,792  12.2024  2,148,849  11.9693  1,984,062  37.0395  1,977,120  58.4784 
20  10  20  2,410,076  15.5662  2,663,989  12.7645  2,512,804  38.8569  2,330,056  66.2304 
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. 
© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Wang, S.; Hui, J.; Zhu, B.; Liu, Y. Adaptive Genetic Algorithm Based on Fuzzy Reasoning for the Multilevel Capacitated LotSizing Problem with Energy Consumption in Synchronizer Production. Sustainability 2022, 14, 5072. https://doi.org/10.3390/su14095072
Wang S, Hui J, Zhu B, Liu Y. Adaptive Genetic Algorithm Based on Fuzzy Reasoning for the Multilevel Capacitated LotSizing Problem with Energy Consumption in Synchronizer Production. Sustainability. 2022; 14(9):5072. https://doi.org/10.3390/su14095072
Chicago/Turabian StyleWang, Shuai, Jizhuang Hui, Bin Zhu, and Ying Liu. 2022. "Adaptive Genetic Algorithm Based on Fuzzy Reasoning for the Multilevel Capacitated LotSizing Problem with Energy Consumption in Synchronizer Production" Sustainability 14, no. 9: 5072. https://doi.org/10.3390/su14095072