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doi: 10.3934/jimo.2021190
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## Bi-objective unrelated parallel machines scheduling problem with worker allocation and sequence dependent setup times considering machine eligibility and precedence constraints

 Department of Industrial Engineering and Management, Mazandaran University of Science and Technology, Babol, Iran

* Corresponding author: Javad Rezaeian

Received  January 2021 Revised  September 2021 Early access November 2021

In today's competitive world, scheduling problems are one of the most important and vital issues. In this study, a bi-objective unrelated parallel machine scheduling problem with worker allocation, sequence dependent setup times, precedence constraints, and machine eligibility is presented. The objective functions are to minimize the costs of tardiness and hiring workers. In order to formulate the proposed problem, a mixed-integer quadratic programming model is presented. A strategy called repair is also proposed to implement the precedence constraints. Because the problem is NP-hard, two metaheuristic algorithms, a multi-objective tabu search (MOTS) and a multi-objective simulated annealing (MOSA), are presented to tackle the problem. Furthermore, a hybrid metaheuristic algorithm is also developed. Finally, computational experiments are carried out to evaluate different test problems, and analysis of variance is done to compare the performance of the proposed algorithms. The results show that MOTS is doing better in terms of objective values and mean ideal distance (MID) metric, while the proposed hybrid algorithm outperforms in most cases, considering other employed comparison metrics.

Citation: Reza Alizadeh Foroutan, Javad Rezaeian, Milad Shafipour. Bi-objective unrelated parallel machines scheduling problem with worker allocation and sequence dependent setup times considering machine eligibility and precedence constraints. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021190
##### References:

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##### References:
Results of the test problem with $n = 8, m = 2$, and $w = 2$
The results of validation of solutions quality
]">Figure 3.  The flowchart of the MOTS algorithm [4]
Representation of a chromosome with $n = 10$, $m = 3$ and $k = 2$
An example of the precedence constraints
The implementation steps of the correcting algorithm
The swap operator
The reversion operator
The workers relocation operator
Pareto fronts of all three proposed metaheuristics, a medium test problem ($n = 20, m = 6, w = 5$)
Pareto fronts of all three proposed metaheuristics, a large test problem ($n = 30, m = 10, w = 6$)
Comparison of the hypervolume indicator for the problems of all sizes
Hypervolumes comparison, a medium test problem $(n = 20, m = 6, w = 5)$
Hypervolumes comparison, a large test problem $(n = 30, m = 10, w = 6)$
Pareto fronts of MOTS in previous iterations
Pareto fronts of MOSA in previous iterations
Pareto fronts of hybrid algorithm in previous iterations
The convergence curve of the proposed MOTS
The convergence curve of the proposed MOSA
The convergence curve of the proposed hybrid algorithm
The last significant change in the Pareto front of the proposed hybrid algorithm
A comparison of some most recent studies existing in the literature
 Constraints Reference Unrelated Parallel machines Worker flexibility/ allocation Machine eligibility Sequence dependent setup times Precedence constraints Objective function(s) Solution approach(es) Cota et al. [7] Yes No No Yes No Makespan; Total energy consumption ALNS, LA, MO-ALNS, MO-ALNS/D Zhu and Zhou [39] No No No No Yes Makespan; Total workload; Maximum machine workload EMOGWO Munoz-Villamizar et al. [26] No No No Yes No Makespan; Total earliness and tardiness; Cost minimization; Effectiveness optimization GAMS solver Arık and Toksarı [3] No No No No No Total tardiness penalty cost; Earliness penalty cost; Cost of setting due dates A fuzzy local search algorithm Gong et al. [17] No Yes No No No Makespan; Total worker cost; Green production indicator A hybrid evolutionary algorithm Zhang et al. [36] No No No No Yes Total penalties; Makespan; Total completion time Approximation algorithms Gong et al. [16] No Yes No No No Makespan A hybrid artificial bee colony Wang et al. [33] No No No No No Total energy consumption; Makespan An augmented $\varepsilon$-constraint; A heuristic method; NSGA-II Zhang et al. [38] Yes No No No No Total energy consumption; Makespan A heuristic evolutionary algorithm Lei et al. [24] Yes No No No No Makespan An imperialist competitive algorithm Khanh Van and Van Hop [21] Yes No No Yes No Total weighted earliness and tardiness; Makespan A hybrid algorithm based on GA and ISETP Kim et al. [22] No No No Yes Yes Total tardiness SA, GA This paper Yes Yes Yes Yes Yes Cost of tardiness; Cost of hiring workers MOTS, MOSA, A hybrid evolutionary algorithm
 Constraints Reference Unrelated Parallel machines Worker flexibility/ allocation Machine eligibility Sequence dependent setup times Precedence constraints Objective function(s) Solution approach(es) Cota et al. [7] Yes No No Yes No Makespan; Total energy consumption ALNS, LA, MO-ALNS, MO-ALNS/D Zhu and Zhou [39] No No No No Yes Makespan; Total workload; Maximum machine workload EMOGWO Munoz-Villamizar et al. [26] No No No Yes No Makespan; Total earliness and tardiness; Cost minimization; Effectiveness optimization GAMS solver Arık and Toksarı [3] No No No No No Total tardiness penalty cost; Earliness penalty cost; Cost of setting due dates A fuzzy local search algorithm Gong et al. [17] No Yes No No No Makespan; Total worker cost; Green production indicator A hybrid evolutionary algorithm Zhang et al. [36] No No No No Yes Total penalties; Makespan; Total completion time Approximation algorithms Gong et al. [16] No Yes No No No Makespan A hybrid artificial bee colony Wang et al. [33] No No No No No Total energy consumption; Makespan An augmented $\varepsilon$-constraint; A heuristic method; NSGA-II Zhang et al. [38] Yes No No No No Total energy consumption; Makespan A heuristic evolutionary algorithm Lei et al. [24] Yes No No No No Makespan An imperialist competitive algorithm Khanh Van and Van Hop [21] Yes No No Yes No Total weighted earliness and tardiness; Makespan A hybrid algorithm based on GA and ISETP Kim et al. [22] No No No Yes Yes Total tardiness SA, GA This paper Yes Yes Yes Yes Yes Cost of tardiness; Cost of hiring workers MOTS, MOSA, A hybrid evolutionary algorithm
Processing time ($T_{il}$) of the jobs
 $J_1$ $J_2$ $J_3$ $J_4$ $J_5$ $J_6$ $J_7$ $J_8$ $M_1$ $0$ $4$ $2$ $1$ $5$ $3$ $5$ $4$ $M_2$ $0$ $3$ $5$ $8$ $2$ $2$ $6$ $3$ $M_3$ $0$ $1$ $5$ $4$ $4$ $2$ $3$ $3$
 $J_1$ $J_2$ $J_3$ $J_4$ $J_5$ $J_6$ $J_7$ $J_8$ $M_1$ $0$ $4$ $2$ $1$ $5$ $3$ $5$ $4$ $M_2$ $0$ $3$ $5$ $8$ $2$ $2$ $6$ $3$ $M_3$ $0$ $1$ $5$ $4$ $4$ $2$ $3$ $3$
Delivery time and tardiness penalty of the jobs
 $J_1$ $J_2$ $J_3$ $J_4$ $J_5$ $J_6$ $J_7$ $J_8$ $D_j$ $10$ $13$ $8$ $15$ $6$ $16$ $14$ $18$ $\beta_j$ $4$ $3$ $7$ $5$ $9$ $8$ $6$ $2$
 $J_1$ $J_2$ $J_3$ $J_4$ $J_5$ $J_6$ $J_7$ $J_8$ $D_j$ $10$ $13$ $8$ $15$ $6$ $16$ $14$ $18$ $\beta_j$ $4$ $3$ $7$ $5$ $9$ $8$ $6$ $2$
Setup time for each machine
 $(j, l)\in A$ $J_{l=1}$ $J_2$ $J_3$ $J_4$ $J_5$ $J_6$ $J_7$ $J_8$ $J_{j=1}$ $3$ $3$ $1$ $3$ $3$ $1$ $3$ $1$ $J_2$ $3$ $2$ $3$ $1$ $3$ $1$ $3$ $2$ $J_3$ $3$ $1$ $1$ $2$ $3$ $3$ $2$ $1$ $J_4$ $3$ $2$ $2$ $3$ $3$ $3$ $1$ $3$ $M_1$ $J_5$ $2$ $2$ $1$ $2$ $3$ $1$ $1$ $3$ $J_6$ $1$ $3$ $1$ $3$ $1$ $1$ $2$ $2$ $J_7$ $2$ $1$ $2$ $3$ $2$ $1$ $1$ $1$ $J_8$ $2$ $2$ $1$ $3$ $2$ $1$ $2$ $1$ $(j, l)\in A$ $J_{l=1}$ $J_2$ $J_3$ $J_4$ $J_5$ $J_6$ $J_7$ $J_8$ $J_1$ $3$ $2$ $2$ $1$ $2$ $2$ $1$ $3$ $J_2$ $3$ $2$ $1$ $1$ $1$ $2$ $3$ $2$ $J_3$ $3$ $2$ $2$ $2$ $1$ $3$ $1$ $3$ $J_4$ $1$ $1$ $2$ $2$ $1$ $2$ $1$ $3$ $M_2$ $J_5$ $2$ $1$ $2$ $1$ $3$ $1$ $3$ $1$ $J_6$ $3$ $3$ $2$ $2$ $1$ $3$ $1$ $2$ $J_7$ $1$ $1$ $1$ $2$ $3$ $3$ $2$ $3$ $J_8$ $1$ $3$ $1$ $2$ $1$ $3$ $2$ $3$ $(j, l)\in A$ $J_{l=1}$ $J_2$ $J_3$ $J_4$ $J_5$ $J_6$ $J_7$ $J_8$ $J_1$ $1$ $1$ $3$ $3$ $1$ $3$ $2$ $3$ $J_2$ $3$ $1$ $1$ $3$ $3$ $2$ $1$ $2$ $J_3$ $1$ $1$ $3$ $1$ $2$ $3$ $1$ $1$ $J_4$ $3$ $1$ $2$ $2$ $3$ $2$ $2$ $1$ $M_3$ $J_5$ $1$ $3$ $1$ $2$ $2$ $3$ $2$ $2$ $J_6$ $1$ $3$ $1$ $3$ $1$ $2$ $3$ $2$ $J_7$ $1$ $1$ $3$ $2$ $2$ $2$ $1$ $3$ $J_8$ $1$ $1$ $2$ $2$ $3$ $1$ $3$ $3$
 $(j, l)\in A$ $J_{l=1}$ $J_2$ $J_3$ $J_4$ $J_5$ $J_6$ $J_7$ $J_8$ $J_{j=1}$ $3$ $3$ $1$ $3$ $3$ $1$ $3$ $1$ $J_2$ $3$ $2$ $3$ $1$ $3$ $1$ $3$ $2$ $J_3$ $3$ $1$ $1$ $2$ $3$ $3$ $2$ $1$ $J_4$ $3$ $2$ $2$ $3$ $3$ $3$ $1$ $3$ $M_1$ $J_5$ $2$ $2$ $1$ $2$ $3$ $1$ $1$ $3$ $J_6$ $1$ $3$ $1$ $3$ $1$ $1$ $2$ $2$ $J_7$ $2$ $1$ $2$ $3$ $2$ $1$ $1$ $1$ $J_8$ $2$ $2$ $1$ $3$ $2$ $1$ $2$ $1$ $(j, l)\in A$ $J_{l=1}$ $J_2$ $J_3$ $J_4$ $J_5$ $J_6$ $J_7$ $J_8$ $J_1$ $3$ $2$ $2$ $1$ $2$ $2$ $1$ $3$ $J_2$ $3$ $2$ $1$ $1$ $1$ $2$ $3$ $2$ $J_3$ $3$ $2$ $2$ $2$ $1$ $3$ $1$ $3$ $J_4$ $1$ $1$ $2$ $2$ $1$ $2$ $1$ $3$ $M_2$ $J_5$ $2$ $1$ $2$ $1$ $3$ $1$ $3$ $1$ $J_6$ $3$ $3$ $2$ $2$ $1$ $3$ $1$ $2$ $J_7$ $1$ $1$ $1$ $2$ $3$ $3$ $2$ $3$ $J_8$ $1$ $3$ $1$ $2$ $1$ $3$ $2$ $3$ $(j, l)\in A$ $J_{l=1}$ $J_2$ $J_3$ $J_4$ $J_5$ $J_6$ $J_7$ $J_8$ $J_1$ $1$ $1$ $3$ $3$ $1$ $3$ $2$ $3$ $J_2$ $3$ $1$ $1$ $3$ $3$ $2$ $1$ $2$ $J_3$ $1$ $1$ $3$ $1$ $2$ $3$ $1$ $1$ $J_4$ $3$ $1$ $2$ $2$ $3$ $2$ $2$ $1$ $M_3$ $J_5$ $1$ $3$ $1$ $2$ $2$ $3$ $2$ $2$ $J_6$ $1$ $3$ $1$ $3$ $1$ $2$ $3$ $2$ $J_7$ $1$ $1$ $3$ $2$ $2$ $2$ $1$ $3$ $J_8$ $1$ $1$ $2$ $2$ $3$ $1$ $3$ $3$
The availability of $Y_{il}$s
 $J_1$ $J_2$ $J_3$ $J_4$ $J_5$ $J_6$ $J_7$ $J_8$ $M_1$ $\star$ $\star$ $\star$ $\star$ $\star$ $\star$ $M_2$ $\star$ $\star$ $\star$ $\star$ $\star$ $M_3$ $\star$ $\star$ $\star$ $\star$
 $J_1$ $J_2$ $J_3$ $J_4$ $J_5$ $J_6$ $J_7$ $J_8$ $M_1$ $\star$ $\star$ $\star$ $\star$ $\star$ $\star$ $M_2$ $\star$ $\star$ $\star$ $\star$ $\star$ $M_3$ $\star$ $\star$ $\star$ $\star$
Results of the first test problem
 Job $S_{ikjl}$ $CS_l$ $C_l$ $T_{il}$ $T_l$ $X_{ikjl}=1$ $1$ $0$ $0$ $0$ $0$ $0$ $X_{1213}$ $2$ $1.2$ $10$ $14$ $4$ $1$ $X_{1228}$ $3$ $1.2$ $6$ $8$ $2$ $0$ $X_{1232}$ $4$ $2.4$ $17$ $25$ $8$ $10$ $X_{2217}$ $5$ $1.2$ $2$ $6$ $4$ $0$ $X_{2274}$ $6$ $3.6$ $14$ $16$ $2$ $0$ $X_{3215}$ $7$ $1.2$ $4$ $14$ $6$ $0$ $X_{3256}$ $8$ $2.4$ $20$ $24$ $4$ $6$ $F=0.6\star f_1+0.4\star f_2=39+448=487$
 Job $S_{ikjl}$ $CS_l$ $C_l$ $T_{il}$ $T_l$ $X_{ikjl}=1$ $1$ $0$ $0$ $0$ $0$ $0$ $X_{1213}$ $2$ $1.2$ $10$ $14$ $4$ $1$ $X_{1228}$ $3$ $1.2$ $6$ $8$ $2$ $0$ $X_{1232}$ $4$ $2.4$ $17$ $25$ $8$ $10$ $X_{2217}$ $5$ $1.2$ $2$ $6$ $4$ $0$ $X_{2274}$ $6$ $3.6$ $14$ $16$ $2$ $0$ $X_{3215}$ $7$ $1.2$ $4$ $14$ $6$ $0$ $X_{3256}$ $8$ $2.4$ $20$ $24$ $4$ $6$ $F=0.6\star f_1+0.4\star f_2=39+448=487$
Test problems used to validate the quality of solutions
 Test Problem $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$ $(n, m, w)$ $(4, 2, 2)$ $(5, 3, 2)$ $(5, 2, 2)$ $(6, 3, 2)$ $(6, 2, 2)$ $(6, 2, 2)$ $(7, 2, 2)$ $(7, 2, 3)$ $(7, 2, 3)$ $(8, 2, 2)$ with constraints √ √ √ √ √ √ without constraints √ √ √ √
 Test Problem $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$ $(n, m, w)$ $(4, 2, 2)$ $(5, 3, 2)$ $(5, 2, 2)$ $(6, 3, 2)$ $(6, 2, 2)$ $(6, 2, 2)$ $(7, 2, 2)$ $(7, 2, 3)$ $(7, 2, 3)$ $(8, 2, 2)$ with constraints √ √ √ √ √ √ without constraints √ √ √ √
Input data for test problems
 Parameter Generating function Jobs processing time $T_{il}\sim U[5\;\;15]$ Jobs delivery time $D_j\sim [20\;\;40]$ Setup times $S_{ijl}\sim [1\;\;7]$ Jobs tardiness penalties $\beta_j\sim [1\;\;10]$ Worker's skill $\pi_k\sim rand (0.5, 1.5)$
 Parameter Generating function Jobs processing time $T_{il}\sim U[5\;\;15]$ Jobs delivery time $D_j\sim [20\;\;40]$ Setup times $S_{ijl}\sim [1\;\;7]$ Jobs tardiness penalties $\beta_j\sim [1\;\;10]$ Worker's skill $\pi_k\sim rand (0.5, 1.5)$
Levels of the controller parameters
 Algorithm Parameter Levels MOTS $Tabu_{list}$ $2, \;4, \;6$ $N_{new}$ $10, \;20, \;40$ $max_{it}$ $30, \;50, \;70$ MOSA $max_{it}$ $100, \;200, \;300$ $Sub_{it}$ $20, \;25, \;30$ $T_0$ $60, \;100, \;150$ $\alpha$ $0.93, \;0.95, \;0.99$ Hybrid $Tabu_{list}$ $2, \;4, \;5$ $max_{it}$ $70, \;150, \;200$ $Sub_{it}$ $20, \;25, \;30$ $T_0$ $80, \;100, \;150$ $\alpha$ $0.93, \;0.95, \;0.99$
 Algorithm Parameter Levels MOTS $Tabu_{list}$ $2, \;4, \;6$ $N_{new}$ $10, \;20, \;40$ $max_{it}$ $30, \;50, \;70$ MOSA $max_{it}$ $100, \;200, \;300$ $Sub_{it}$ $20, \;25, \;30$ $T_0$ $60, \;100, \;150$ $\alpha$ $0.93, \;0.95, \;0.99$ Hybrid $Tabu_{list}$ $2, \;4, \;5$ $max_{it}$ $70, \;150, \;200$ $Sub_{it}$ $20, \;25, \;30$ $T_0$ $80, \;100, \;150$ $\alpha$ $0.93, \;0.95, \;0.99$
Ranking of factors (based on $SN$ ratios and mean of responses) and their best values for each algorithm
 Algorithm Parameter Rank ($S/N$ ratios) Rank (Means) Best value MOTS $Tabu_{list}$ $3$ $3$ $4$ $N_{new}$ $2$ $2$ $40$ $max_{it}$ $1$ $1$ $70$ MOSA $max_{it}$ $1$ $1$ $200$ $Sub_{it}$ $2$ $2$ $25$ $T_0$ $4$ $4$ $100$ $\alpha$ $3$ $3$ $0.93$ Hybrid $Tabu_{list}$ $1$ $2$ $5$ $max_{it}$ $3$ $1$ $70$ $Sub_{it}$ $2$ $3$ $25$ $T_0$ $4$ $4$ $150$ $\alpha$ $5$ $5$ $0.93$
 Algorithm Parameter Rank ($S/N$ ratios) Rank (Means) Best value MOTS $Tabu_{list}$ $3$ $3$ $4$ $N_{new}$ $2$ $2$ $40$ $max_{it}$ $1$ $1$ $70$ MOSA $max_{it}$ $1$ $1$ $200$ $Sub_{it}$ $2$ $2$ $25$ $T_0$ $4$ $4$ $100$ $\alpha$ $3$ $3$ $0.93$ Hybrid $Tabu_{list}$ $1$ $2$ $5$ $max_{it}$ $3$ $1$ $70$ $Sub_{it}$ $2$ $3$ $25$ $T_0$ $4$ $4$ $150$ $\alpha$ $5$ $5$ $0.93$
Computational results for the problems of all sizes: MOTS & MOSA
 Problem $(n, m, w)$ Obj. functions MOTS MOSA P1* P2 P3 P4 P1 P2 P3 $(8, 2, 2)$ f1 710 826 1661 1968 f2 4853 4769 4769 4769 $(8, 3, 2)$ f1 856 898 1550 f2 3656 4418 4418 $(10, 3, 2)$ f1 1063 1433 f2 3662 4880 $(10, 3, 3)$ f1 731 4762 f2 3065 3438 $(10, 4, 2)$ f1 983 1252 6937 6937 f2 5719 5314 2055 2229 $(12, 3, 2)$ f1 1460 3990 f2 6532 5869 $(12, 3, 3)$ f1 681 815 3837 f2 6358 5880 5557 $(12, 4, 2)$ f1 1207 1233 5487 f2 6445 6184 5399 $(15, 4, 3)$ f1 480 527 924 4346 4560 f2 10760 10542 10092 9807 9807 $(15, 6, 3)$ f1 2156 2176 2196 2579 5527 f2 9484 9224 8963 8703 9475 $(20, 4, 3)$ f1 6317 13816 f2 6943 9658 $(20, 6, 5)$ f1 2894 3466 3719 11392 12469 f2 8410 7044 6702 9826 9826 $(25, 4, 3)$ f1 12610 20565 f2 7749 12725 $(25, 6, 5)$ f1 3715 16032 16037 f2 8823 11685 11685 $(30, 8, 4)$ f1 4133 4560 19549 21010 22219 f2 14096 13584 15081 15081 15081 $(30, 10, 6)$ f1 3190 3444 3961 23057 25436 f2 13499 12942 12632 14132 14132 $(40, 10, 6)$ f1 8993 9944 38190 40236 f2 17455 17182 21838 21838 $(40, 12, 8)$ f1 6334 6421 7055 44881 31799 f2 18491 18383 18187 19731 19731 $(50, 12, 8)$ f1 13801 14655 73855 78444 80185 f2 19918 19450 19905 19905 19905 $(50, 15, 10)$ f1 11484 13550 47375 51987 59331 f2 22310 21220 26906 26906 26906 $P_z, (z = 1, 2, …, Z)$ denotes the $z$th pareto solution.
 Problem $(n, m, w)$ Obj. functions MOTS MOSA P1* P2 P3 P4 P1 P2 P3 $(8, 2, 2)$ f1 710 826 1661 1968 f2 4853 4769 4769 4769 $(8, 3, 2)$ f1 856 898 1550 f2 3656 4418 4418 $(10, 3, 2)$ f1 1063 1433 f2 3662 4880 $(10, 3, 3)$ f1 731 4762 f2 3065 3438 $(10, 4, 2)$ f1 983 1252 6937 6937 f2 5719 5314 2055 2229 $(12, 3, 2)$ f1 1460 3990 f2 6532 5869 $(12, 3, 3)$ f1 681 815 3837 f2 6358 5880 5557 $(12, 4, 2)$ f1 1207 1233 5487 f2 6445 6184 5399 $(15, 4, 3)$ f1 480 527 924 4346 4560 f2 10760 10542 10092 9807 9807 $(15, 6, 3)$ f1 2156 2176 2196 2579 5527 f2 9484 9224 8963 8703 9475 $(20, 4, 3)$ f1 6317 13816 f2 6943 9658 $(20, 6, 5)$ f1 2894 3466 3719 11392 12469 f2 8410 7044 6702 9826 9826 $(25, 4, 3)$ f1 12610 20565 f2 7749 12725 $(25, 6, 5)$ f1 3715 16032 16037 f2 8823 11685 11685 $(30, 8, 4)$ f1 4133 4560 19549 21010 22219 f2 14096 13584 15081 15081 15081 $(30, 10, 6)$ f1 3190 3444 3961 23057 25436 f2 13499 12942 12632 14132 14132 $(40, 10, 6)$ f1 8993 9944 38190 40236 f2 17455 17182 21838 21838 $(40, 12, 8)$ f1 6334 6421 7055 44881 31799 f2 18491 18383 18187 19731 19731 $(50, 12, 8)$ f1 13801 14655 73855 78444 80185 f2 19918 19450 19905 19905 19905 $(50, 15, 10)$ f1 11484 13550 47375 51987 59331 f2 22310 21220 26906 26906 26906 $P_z, (z = 1, 2, …, Z)$ denotes the $z$th pareto solution.
Computational results for the problems of all sizes: The proposed hybrid
 Problem $(n, m, w)$ Obj. functions Hybrid P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 $(8, 2, 2)$ f1 1524 1553 1617 f2 5191 5107 5022 $(8, 3, 2)$ f1 1216 1400 1624 1961 f2 4266 4266 4113 3961 $(10, 3, 2)$ f1 989 1405 1677 1711 1791 1867 1870 1896 f2 6097 5793 5793 5488 5184 5184 4880 4575 $(10, 3, 3)$ f1 2443 2586 2671 2695 2702 2704 f2 3295 3295 3208 3152 3152 3065 $(10, 4, 2)$ f1 858 1047 1196 1330 1452 1601 f2 6531 6531 6328 6125 5922 5719 $(12, 3, 2)$ f1 2106 2400 2491 2841 2946 f2 7526 7195 6863 6532 6201 $(12, 3, 3)$ f1 1428 2331 2669 2690 3088 3109 3258 3279 3334 3438 f2 8268 7790 7313 7046 6835 6568 6358 6091 5880 5613 $(12, 4, 2)$ f1 3812 3829 f2 6707 6184 $(15, 4, 3)$ f1 2871 3015 3589 3610 3617 3618 3619 f2 1080 1080 1078 1077 1074 1054 1052 $(15, 6, 3)$ f1 2100 2313 2576 2600 f2 9739 9739 9484 9483 $(20, 4, 3)$ f1 6887 6979 7519 f2 11292 10078 9755 $(20, 6, 5)$ f1 5712 7395 7406 7445 9537 9551 9718 9740 10162 10176 10365 f2 9718 9718 9587 9518 9518 9386 9309 9253 9044 8782 8779 $(25, 4, 3)$ f1 11769 11826 12126 13470 17244 17665 f2 10769 10769 10598 10598 10598 10237 $(25, 6, 5)$ f1 5851 6530 6679 6684 8501 8534 8715 9107 9122 f2 10561 10561 10389 10266 10240 10117 10114 10114 9991 $(30, 8, 4)$ f1 13183 13805 13942 14068 15660 16033 17741 17785 f2 16567 16411 16391 16211 16191 16010 15498 15142 $(30, 10, 6)$ f1 10899 10951 10968 13811 17298 18530 18552 f2 15173 15031 14762 14762 14566 14084 13899 $(40, 10, 6)$ f1 21130 22146 23254 f2 25930 25229 24604 $(40, 12, 8)$ f1 18544 18551 20567 20619 20629 21045 f2 21022 20914 20914 20910 20866 20866 $(50, 12, 8)$ f1 47230 47348 60202 62672 f2 23776 23729 23456 21764 $(50, 15, 10)$ f1 18791 23480 26830 28710 28254 29460 f2 28995 28995 28637 28376 28302 28202
 Problem $(n, m, w)$ Obj. functions Hybrid P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 $(8, 2, 2)$ f1 1524 1553 1617 f2 5191 5107 5022 $(8, 3, 2)$ f1 1216 1400 1624 1961 f2 4266 4266 4113 3961 $(10, 3, 2)$ f1 989 1405 1677 1711 1791 1867 1870 1896 f2 6097 5793 5793 5488 5184 5184 4880 4575 $(10, 3, 3)$ f1 2443 2586 2671 2695 2702 2704 f2 3295 3295 3208 3152 3152 3065 $(10, 4, 2)$ f1 858 1047 1196 1330 1452 1601 f2 6531 6531 6328 6125 5922 5719 $(12, 3, 2)$ f1 2106 2400 2491 2841 2946 f2 7526 7195 6863 6532 6201 $(12, 3, 3)$ f1 1428 2331 2669 2690 3088 3109 3258 3279 3334 3438 f2 8268 7790 7313 7046 6835 6568 6358 6091 5880 5613 $(12, 4, 2)$ f1 3812 3829 f2 6707 6184 $(15, 4, 3)$ f1 2871 3015 3589 3610 3617 3618 3619 f2 1080 1080 1078 1077 1074 1054 1052 $(15, 6, 3)$ f1 2100 2313 2576 2600 f2 9739 9739 9484 9483 $(20, 4, 3)$ f1 6887 6979 7519 f2 11292 10078 9755 $(20, 6, 5)$ f1 5712 7395 7406 7445 9537 9551 9718 9740 10162 10176 10365 f2 9718 9718 9587 9518 9518 9386 9309 9253 9044 8782 8779 $(25, 4, 3)$ f1 11769 11826 12126 13470 17244 17665 f2 10769 10769 10598 10598 10598 10237 $(25, 6, 5)$ f1 5851 6530 6679 6684 8501 8534 8715 9107 9122 f2 10561 10561 10389 10266 10240 10117 10114 10114 9991 $(30, 8, 4)$ f1 13183 13805 13942 14068 15660 16033 17741 17785 f2 16567 16411 16391 16211 16191 16010 15498 15142 $(30, 10, 6)$ f1 10899 10951 10968 13811 17298 18530 18552 f2 15173 15031 14762 14762 14566 14084 13899 $(40, 10, 6)$ f1 21130 22146 23254 f2 25930 25229 24604 $(40, 12, 8)$ f1 18544 18551 20567 20619 20629 21045 f2 21022 20914 20914 20910 20866 20866 $(50, 12, 8)$ f1 47230 47348 60202 62672 f2 23776 23729 23456 21764 $(50, 15, 10)$ f1 18791 23480 26830 28710 28254 29460 f2 28995 28995 28637 28376 28302 28202
Hypervolume indicator for the problems of all sizes
Comparisons results for the problems of all sizes
One-way ANOVA for all test problems (between groups)
 Metric Problem size Sum of Squares df Mean Square F. Sig. S-metric Small 8758922.903 2 4379461.452 5.519 0.012 Medium 8.541E7 2 4.271E7 3.688 0.050 Large 1.348E9 2 6.739E8 6.716 0.008 NP Small 130.083 2 65.042 24.500 0.000 Medium 81.333 2 40.667 7.562 0.005 Large 42.333 2 21.167 15.744 0.000 MID Small 6748717.583 2 3374358.792 2.312 0.124 Medium 1.932E8 2 9.662E7 4.538 0.029 Large 2.856E9 2 1.428E9 11.064 0.001 DM Small 8430960.250 2 4215480.125 10.568 0.001 Medium 7.217E7 2 3.608E7 7.737 0.005 Large 1.954E8 2 9.768E7 5.840 0.013 SNS Small 122196.583 2 61098.292 5.399 0.013 Medium 1.564E7 2 7817526.167 2.349 0.130 Large 1.236E7 2 6179340.222 0.094 0.911 Time Small 20371.750 2 10185.875 21.221 0.000 Medium 43694.111 2 21847.056 6.425 0.010 Large 177652.111 2 88826.056 10.736 0.001
 Metric Problem size Sum of Squares df Mean Square F. Sig. S-metric Small 8758922.903 2 4379461.452 5.519 0.012 Medium 8.541E7 2 4.271E7 3.688 0.050 Large 1.348E9 2 6.739E8 6.716 0.008 NP Small 130.083 2 65.042 24.500 0.000 Medium 81.333 2 40.667 7.562 0.005 Large 42.333 2 21.167 15.744 0.000 MID Small 6748717.583 2 3374358.792 2.312 0.124 Medium 1.932E8 2 9.662E7 4.538 0.029 Large 2.856E9 2 1.428E9 11.064 0.001 DM Small 8430960.250 2 4215480.125 10.568 0.001 Medium 7.217E7 2 3.608E7 7.737 0.005 Large 1.954E8 2 9.768E7 5.840 0.013 SNS Small 122196.583 2 61098.292 5.399 0.013 Medium 1.564E7 2 7817526.167 2.349 0.130 Large 1.236E7 2 6179340.222 0.094 0.911 Time Small 20371.750 2 10185.875 21.221 0.000 Medium 43694.111 2 21847.056 6.425 0.010 Large 177652.111 2 88826.056 10.736 0.001
Results of post-hoc comparisons
 Small Medium Large Metrics Order of best performances Significant difference Order of best performances Significant difference Order of best performances Significant difference S-metric MOTS-Hybrid-MOSA MOTS-MOSA MOTS-Hybrid-MOSA MOTS-MOSA MOTS-Hybrid-MOSA MOTS-MOSA NPS Hybrid-MOTS-MOSA Hybrid-MOSA, Hybrid-MOTS Hybrid-MOTS-MOSA - Hybrid-MOSA-MOTS Hybrid-MOSA, Hybrid-MOTS MID Hybrid-MOTS-MOSA - MOTS-Hybrid-MOSA MOTS-MOSA MOTS-Hybrid-MOSA MOTS-MOSA DM Hybrid-MOTS-MOSA Hybrid-MOSA, Hybrid-MOTS Hybrid-MOTS-MOSA - Hybrid-MOSA-MOTS Hybrid-MOTS SNS Hybrid-MOTS-MOSA Hybrid-MOSA Hybrid-MOTS-MOSA - Hybrid-MOSA-MOTS - Time Hybrid-MOSA-MOTS Hybrid-MOSA, Hybrid-MOTS Hybrid-MOSA-MOTS Hybrid-MOTS Hybrid-MOSA-MOTS Hybrid-MOSA, Hybrid-MOTS
 Small Medium Large Metrics Order of best performances Significant difference Order of best performances Significant difference Order of best performances Significant difference S-metric MOTS-Hybrid-MOSA MOTS-MOSA MOTS-Hybrid-MOSA MOTS-MOSA MOTS-Hybrid-MOSA MOTS-MOSA NPS Hybrid-MOTS-MOSA Hybrid-MOSA, Hybrid-MOTS Hybrid-MOTS-MOSA - Hybrid-MOSA-MOTS Hybrid-MOSA, Hybrid-MOTS MID Hybrid-MOTS-MOSA - MOTS-Hybrid-MOSA MOTS-MOSA MOTS-Hybrid-MOSA MOTS-MOSA DM Hybrid-MOTS-MOSA Hybrid-MOSA, Hybrid-MOTS Hybrid-MOTS-MOSA - Hybrid-MOSA-MOTS Hybrid-MOTS SNS Hybrid-MOTS-MOSA Hybrid-MOSA Hybrid-MOTS-MOSA - Hybrid-MOSA-MOTS - Time Hybrid-MOSA-MOTS Hybrid-MOSA, Hybrid-MOTS Hybrid-MOSA-MOTS Hybrid-MOTS Hybrid-MOSA-MOTS Hybrid-MOSA, Hybrid-MOTS
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