# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021175
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## A revised imperialist competition algorithm for cellular manufacturing optimization based on product line design

 1 School of Management, Hangzhou Dianzi University, Hangzhou 310018, China 2 Department of Mathematics, China Jiliang University, Hangzhou 310018, China

* Corresponding author: Jufeng Wang

Received  February 2021 Revised  August 2021 Early access October 2021

Due to the fierce market competition, enterprises try to satisfy customers' requirements for personalized products in order to maximize profit or market share of their products. This not only needs to determine the product variants through product line design, but also needs to pay attention to resource allocation in the manufacturing process. This paper proposes a cellular manufacturing optimization model that considers the market and production. If the company excessively pursues the satisfaction of customers' personalized needs, the manufacturing time and cost may increase accordingly. Of course, with the restriction of production capacity in manufacturing cells and the expectation of reducing cost, managers cannot design attributes' levels of a product line casually, which may result in its unstable marketing share and profit. Therefore, the product demand influenced by customers' preferences could be a key factor to link market and production. The objective of propose model is to maximize product profit which consists of revenue and miscellaneous costs (material, processing, transportation, final assembly and fixed costs). A revised imperialist competitive algorithm (RICA) is developed to optimize the discrete problem. Extensive numerical experiments and t-test are carried out to verify the effect of this method. The results demonstrate the proficiency of RICA over another imperialist competitive algorithm based method and genetic algorithm in terms of solution quality.

Citation: Chunfeng Liu, Yuanyuan Liu, Jufeng Wang. A revised imperialist competition algorithm for cellular manufacturing optimization based on product line design. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021175
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##### References:
The encode scheme for an example solution
Flowchart of the proposed RICA
Crossover of country
Mutation of country
The diagram of the Taguchi experiment
Convergence diagram of a typical example
Literature review of research scenario and solution method
 Authors Cell formation Consumer preference Product line design Solution method Gupta et al. [14] $\surd$ - - GAs Kashan et al. [19] $\surd$ - - GBPSO Kamalakannan and Pandian [17] $\surd$ - - TS, MGE Bagheri and Bashiri [6] $\surd$ - - GICA Zohrevand et al. [50] $\surd$ - - TS-GA Jouzdani et al. [16] $\surd$ - - SA Li et al. [24] $\surd$ - - HHS Liu et al. [25] $\surd$ - - DBFA Zhao [49] $\surd$ - - Memetic Algorithm Mehdizadeh et al. [30] $\surd$ - - MOVDO Bootaki et al. [8] $\surd$ - - GA-AUGMECON Niakan et al. [36] $\surd$ - - NSGA-II Liu et al. [26] $\surd$ - - DICAP Howard and Sheth [15] - $\surd$ - Buyer Behavior Theory Norton [37] - $\surd$ - Coase Theorem Cao et al. [9] - $\surd$ - Ontology-based Ng and Law [35] - $\surd$ $\surd$ Fuzzy-ER Achabou et al. [2] - $\surd$ - Conjoint Analysis, Cluster Analysis Agnew and Dargusch [3] - $\surd$ $\surd$ BWS, DCE Grasso and Asioli[13] - $\surd$ $\surd$ DCMs Michalek et al. [32] - $\surd$ $\surd$ Conjoint Analysis Tookanlou and Wong [42] - $\surd$ $\surd$ Empirical Studies Tsafarakis et al. [43,44] - $\surd$ $\surd$ Hybrid PSO, FSTDE Kuzmanovic et al. [23] - $\surd$ $\surd$ Conjoint Analysis This paper $\surd$ $\surd$ $\surd$ RICA
 Authors Cell formation Consumer preference Product line design Solution method Gupta et al. [14] $\surd$ - - GAs Kashan et al. [19] $\surd$ - - GBPSO Kamalakannan and Pandian [17] $\surd$ - - TS, MGE Bagheri and Bashiri [6] $\surd$ - - GICA Zohrevand et al. [50] $\surd$ - - TS-GA Jouzdani et al. [16] $\surd$ - - SA Li et al. [24] $\surd$ - - HHS Liu et al. [25] $\surd$ - - DBFA Zhao [49] $\surd$ - - Memetic Algorithm Mehdizadeh et al. [30] $\surd$ - - MOVDO Bootaki et al. [8] $\surd$ - - GA-AUGMECON Niakan et al. [36] $\surd$ - - NSGA-II Liu et al. [26] $\surd$ - - DICAP Howard and Sheth [15] - $\surd$ - Buyer Behavior Theory Norton [37] - $\surd$ - Coase Theorem Cao et al. [9] - $\surd$ - Ontology-based Ng and Law [35] - $\surd$ $\surd$ Fuzzy-ER Achabou et al. [2] - $\surd$ - Conjoint Analysis, Cluster Analysis Agnew and Dargusch [3] - $\surd$ $\surd$ BWS, DCE Grasso and Asioli[13] - $\surd$ $\surd$ DCMs Michalek et al. [32] - $\surd$ $\surd$ Conjoint Analysis Tookanlou and Wong [42] - $\surd$ $\surd$ Empirical Studies Tsafarakis et al. [43,44] - $\surd$ $\surd$ Hybrid PSO, FSTDE Kuzmanovic et al. [23] - $\surd$ $\surd$ Conjoint Analysis This paper $\surd$ $\surd$ $\surd$ RICA
Attributes and levels of vacuum glass
 Component Attribute Level Component 1 Thickness Thin, thick, superthick The shape of the support Cylindrical, spherical, oval Glass shape Square, circular, rhombic Component 2 Color Blue, gray, green Light transmission Transparent, translucent, opaque Thermal insulation General, great, excellent Edge banding material Metallic, plastic, rubber Component 3 Decoration Retro, fashion, chinoiserie Welding of metallic layer Metal brazing, gastight welding, laser welding
 Component Attribute Level Component 1 Thickness Thin, thick, superthick The shape of the support Cylindrical, spherical, oval Glass shape Square, circular, rhombic Component 2 Color Blue, gray, green Light transmission Transparent, translucent, opaque Thermal insulation General, great, excellent Edge banding material Metallic, plastic, rubber Component 3 Decoration Retro, fashion, chinoiserie Welding of metallic layer Metal brazing, gastight welding, laser welding
Preference scores of each attribute
 Attribute1 (A1) Attribute 2 (A2) Attribute 3(A3) Level11(L11) Level12(L12) Level13(L13) Level21(L21) Level22(L22) Level23(L23) Level31(L31) Level32(L32) Level33(L33) Individual1 2(0.20) 3(0.30) 5(0.50) 1(0.13) 4(0.50) 3(0.38) 2(0.18) 5(0.45) 4(0.36) Individual2 2(0.33) 3(0.50) 1(0.17) 3(0.33) 4(0.44) 2(0.22) 3(0.30) 4(0.40) 3(0.30) Individual3 4(0.36) 2(0.18) 5(0.45) 4(0.44) 3(0.33) 2(0.22) 5(0.56) 1(0.11) 3(0.33)
 Attribute1 (A1) Attribute 2 (A2) Attribute 3(A3) Level11(L11) Level12(L12) Level13(L13) Level21(L21) Level22(L22) Level23(L23) Level31(L31) Level32(L32) Level33(L33) Individual1 2(0.20) 3(0.30) 5(0.50) 1(0.13) 4(0.50) 3(0.38) 2(0.18) 5(0.45) 4(0.36) Individual2 2(0.33) 3(0.50) 1(0.17) 3(0.33) 4(0.44) 2(0.22) 3(0.30) 4(0.40) 3(0.30) Individual3 4(0.36) 2(0.18) 5(0.45) 4(0.44) 3(0.33) 2(0.22) 5(0.56) 1(0.11) 3(0.33)
Utility of different products
 A1 A2 A3 Individual 1 Individual 2 Individual 3 Product 1 L13 L23 L31 1.06 0.69 1.23 Competitive product 1 L12 L22 L33 1.16 1.24 0.84 Competitive product 2 L11 L23 L32 1.03 0.95 0.69 Competitive product 3 L13 L22 L31 1.18 0.91 1.34 Probability PROBi1 0.24 0.19 0.30 Demand D1 (S=300) 73
 A1 A2 A3 Individual 1 Individual 2 Individual 3 Product 1 L13 L23 L31 1.06 0.69 1.23 Competitive product 1 L12 L22 L33 1.16 1.24 0.84 Competitive product 2 L11 L23 L32 1.03 0.95 0.69 Competitive product 3 L13 L22 L31 1.18 0.91 1.34 Probability PROBi1 0.24 0.19 0.30 Demand D1 (S=300) 73
Control parameters of RICA
 Control parameters Levels $N_{pop}$ (Npop): Number of countries 40, 50, 60 $N_{imp}$ (Nimp): Number of imperialists 8, 10, 12 $\xi$ (Xi): Weight coefficient for colony profit 0.1, 0.2, 0.3 $\rho$ (CORA): Cooling rate 0.6, 0.7, 0.8 $U$ (UN): The number of times for which the optimal profit remains unchanged 10, 12, 15 $\varepsilon$ (epsilon): A pre-given threshold 0.0001, 0.0005, 0,001
 Control parameters Levels $N_{pop}$ (Npop): Number of countries 40, 50, 60 $N_{imp}$ (Nimp): Number of imperialists 8, 10, 12 $\xi$ (Xi): Weight coefficient for colony profit 0.1, 0.2, 0.3 $\rho$ (CORA): Cooling rate 0.6, 0.7, 0.8 $U$ (UN): The number of times for which the optimal profit remains unchanged 10, 12, 15 $\varepsilon$ (epsilon): A pre-given threshold 0.0001, 0.0005, 0,001
Parameters of the proposed problem
 Parameter Value Min Max $M$ : Number of components of each product type (or number of machines in each cell) 3 $C$ : Number of cells 3 $A$ : Number of attributes of each product type 9 $L$ : Number of levels of each attribute 3 $J$ : Number of company's product types 8 $J $$^{'} : Number of competitive product types for each company's product type 5 I : Number of sample consumers 100 S : The size of represented population in the market 1000 \phi : Fixed cost per unit time 3 p$$ _{jsa}$ : Price of an attribute's level 10 20 $h $$_{jsa} : Material cost of an attribute's level 1 3 k$$ _{jsa}$ : Processing cost of an attribute's level 1 2 ${\theta _j}$ : Material handling cost each time 25 40 ${\Omega _j}$ : Unit assembly cost of each product type 2 5 ${\tau _{jsa}}$ : Processing time of an attribute's level 3 8 $\varphi$ : Preference score for an attribute's level 1 5
 Parameter Value Min Max $M$ : Number of components of each product type (or number of machines in each cell) 3 $C$ : Number of cells 3 $A$ : Number of attributes of each product type 9 $L$ : Number of levels of each attribute 3 $J$ : Number of company's product types 8 $J $$^{'} : Number of competitive product types for each company's product type 5 I : Number of sample consumers 100 S : The size of represented population in the market 1000 \phi : Fixed cost per unit time 3 p$$ _{jsa}$ : Price of an attribute's level 10 20 $h $$_{jsa} : Material cost of an attribute's level 1 3 k$$ _{jsa}$ : Processing cost of an attribute's level 1 2 ${\theta _j}$ : Material handling cost each time 25 40 ${\Omega _j}$ : Unit assembly cost of each product type 2 5 ${\tau _{jsa}}$ : Processing time of an attribute's level 3 8 $\varphi$ : Preference score for an attribute's level 1 5
Performance comparison between RICA, ICASA and GA for impact parameters
 $C=6, A=5, L=5$ $\mathop {\bar V}\nolimits_{RICA}$ $\mathop {\bar V}\nolimits_{ICASA}$ $\mathop {\bar V}\nolimits_{GA}$ $\mathop {\Delta \bar V}\nolimits_{ICASA}^{RICA} (\%)$ $\mathop {\Delta \bar V}\nolimits_{GA}^{RICA} (\%)$ $CPU$ (s) $M$ 6 398720 335469 322124 19 24 130 12 607478 517620 489546 17 24 203 18 797091 673714 657481 18 21 353 24 1010046 860332 819408 17 23 519 $M=14, C=5, L=5$ $\mathop {\bar V}\nolimits_{RICA}$ $\mathop {\bar V}\nolimits_{ICASA}$ $\mathop {\bar V}\nolimits_{GA}$ $\mathop {\Delta \bar V}\nolimits_{ICASA}^{RICA} (\%)$ $\mathop {\Delta \bar V}\nolimits_{GA}^{RICA} (\%)$ $CPU$ (s) $A$ 3 1371713 1201124 1156277 14 19 308 6 1000975 863354 829463 16 21 344 9 963418 837029 796480 15 21 433 12 978119 831539 791116 18 24 593 $M=12, C=5, A=10$ $\mathop {\bar V}\nolimits_{RICA}$ $\mathop {\bar V}\nolimits_{ICASA}$ $\mathop {\bar V}\nolimits_{GA}$ $\mathop {\Delta \bar V}\nolimits_{ICASA}^{RICA} (\%)$ $\mathop {\Delta \bar V}\nolimits_{GA}^{RICA} (\%)$ $CPU$ (s) $L$ 3 459945 402027 388192 14 18 176 10 916115 788887 741580 16 24 283 17 1343335 1154694 1104538 16 22 418 24 1601637 1344834 1285988 19 25 819 $M=17, A=7, L=3$ $\mathop {\bar V}\nolimits_{RICA}$ $\mathop {\bar V}\nolimits_{ICASA}$ $\mathop {\bar V}\nolimits_{GA}$ $\mathop {\Delta \bar V}\nolimits_{ICASA}^{RICA} (\%)$ $\mathop {\Delta \bar V}\nolimits_{GA}^{RICA} (\%)$ $CPU$ (s) $C$ 3 1304929 1156539 1111787 13 17 294 18 1427445 1198392 1155033 19 24 541 13 1403110 1178416 1137835 19 23 813 18 1421519 1222470 1173408 16 21 801
 $C=6, A=5, L=5$ $\mathop {\bar V}\nolimits_{RICA}$ $\mathop {\bar V}\nolimits_{ICASA}$ $\mathop {\bar V}\nolimits_{GA}$ $\mathop {\Delta \bar V}\nolimits_{ICASA}^{RICA} (\%)$ $\mathop {\Delta \bar V}\nolimits_{GA}^{RICA} (\%)$ $CPU$ (s) $M$ 6 398720 335469 322124 19 24 130 12 607478 517620 489546 17 24 203 18 797091 673714 657481 18 21 353 24 1010046 860332 819408 17 23 519 $M=14, C=5, L=5$ $\mathop {\bar V}\nolimits_{RICA}$ $\mathop {\bar V}\nolimits_{ICASA}$ $\mathop {\bar V}\nolimits_{GA}$ $\mathop {\Delta \bar V}\nolimits_{ICASA}^{RICA} (\%)$ $\mathop {\Delta \bar V}\nolimits_{GA}^{RICA} (\%)$ $CPU$ (s) $A$ 3 1371713 1201124 1156277 14 19 308 6 1000975 863354 829463 16 21 344 9 963418 837029 796480 15 21 433 12 978119 831539 791116 18 24 593 $M=12, C=5, A=10$ $\mathop {\bar V}\nolimits_{RICA}$ $\mathop {\bar V}\nolimits_{ICASA}$ $\mathop {\bar V}\nolimits_{GA}$ $\mathop {\Delta \bar V}\nolimits_{ICASA}^{RICA} (\%)$ $\mathop {\Delta \bar V}\nolimits_{GA}^{RICA} (\%)$ $CPU$ (s) $L$ 3 459945 402027 388192 14 18 176 10 916115 788887 741580 16 24 283 17 1343335 1154694 1104538 16 22 418 24 1601637 1344834 1285988 19 25 819 $M=17, A=7, L=3$ $\mathop {\bar V}\nolimits_{RICA}$ $\mathop {\bar V}\nolimits_{ICASA}$ $\mathop {\bar V}\nolimits_{GA}$ $\mathop {\Delta \bar V}\nolimits_{ICASA}^{RICA} (\%)$ $\mathop {\Delta \bar V}\nolimits_{GA}^{RICA} (\%)$ $CPU$ (s) $C$ 3 1304929 1156539 1111787 13 17 294 18 1427445 1198392 1155033 19 24 541 13 1403110 1178416 1137835 19 23 813 18 1421519 1222470 1173408 16 21 801
Statistical t-test results from SPSS for samples of entries 1-16
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