Metaheuristics are commonly employed as a means of solving many distinct kinds of optimization problems. Several natural-process-inspired metaheuristic optimizers have been introduced in the recent years. The convergence, computational burden and statistical relevance of metaheuristics should be studied and compared for their potential use in future algorithm design and implementation. In this paper, eight different variants of dragonfly algorithm, i.e. classical dragonfly algorithm (DA), hybrid memory-based dragonfly algorithm with differential evolution (DADE), quantum-behaved and Gaussian mutational dragonfly algorithm (QGDA), memory-based hybrid dragonfly algorithm (MHDA), chaotic dragonfly algorithm (CDA), biogeography-based Mexican hat wavelet dragonfly algorithm (BMDA), hybrid Nelder-Mead algorithm and dragonfly algorithm (INMDA), and hybridization of dragonfly algorithm and artificial bee colony (HDA) are applied to solve four industrial chemical process optimization problems. A fuzzy multi-criteria decision making tool in the form of fuzzy-measurement alternatives and ranking according to compromise solution (MARCOS) is adopted to ascertain the relative rankings of the DA variants with respect to computational time, Friedman’s rank based on optimal solutions and convergence rate. Based on the comprehensive testing of the algorithms, it is revealed that DADE, QGDA and classical DA are the top three DA variants in solving the industrial chemical process optimization problems under consideration.

The increasing population coupled with rapid urbanization has led to unprecedented demands for natural and man-made resources. The chemical industry which acts as the raw material provider to several critical sectors, like pharmaceuticals, construction, etc., must keep up with the pace of these ever-increasing demands. Setting up new production facilities to serve the increasing demand requires significant resources and is also highly capital-intensive. Improving yield and efficiency of the existing plants on the other hand just needs deployment of better managerial practices, and sound knowledge of the processes and their optimization.

Due to large number of input parameters involved in any of the typical industrial chemical processes, optimizing them using classical approaches, like one-factor-at-a-time (OFAT), Taguchi methodology, etc., may not be always feasible. Of late, metaheuristics, which are essentially general-purpose heuristic approaches, have become quite popular among the researchers working in the area of process optimization. Perhaps this popularity is mainly due to high-level problem-independent algorithmic framework of the metaheuristics (Sörensen and Glover,

In this decade, research on metaphor-based metaheuristics has received a tremendous impetus. A plethora of nature-inspired metaheuristics, like flower pollination algorithm (Yang,

The DA, a population-based nature-inspired metaheuristic, was propounded by Mirjalili (

In this paper, the performance of eight popular DA variants is compared based on four industrial chemical process problems (i.e. heat exchanger network design (Floudas and Ciric,

Flowchart of the adopted methodology.

The classical DA (Mirjalili,

Differential evolution (DE), in general, has high computational ability and a fast convergence rate. Akin to GA, DE explores the search space based on crossover and mutation. At the end of each cycle, DADE (Debnath

By implementing the concept of a quantum rotation gate, Yu

The lack of internal memory in the classical DA can cause premature convergence to local optima. To overcome this problem, Sree Ranjini and Murugan (

Sayed

To address the issue of premature convergence under heavy loads, Shirani and Safi-Esfahani (

Xu and Yan (

Ghanem and Jantan (

The MARCOS is an innovative MCDM approach that can be employed in many contexts (Chakraborty

The application steps of MARCOS in fuzzy environment are summarized as shown below:

From the defuzzified values of utility degrees and utility functions, the corresponding utility function of each of the alternatives with respect to anti-ideal and ideal solutions is computed using the following equation:

Based on the descending values of the utility function, the alternatives are finally sorted from the best to the worst, the best alternative having the maximum utility function value.

To assess and compare the relative performance of eight different DA variants, four different industrial chemical process optimization problems are considered in this paper as the test problems. All these four problems are constrained optimization problems. The numerical experiments are carried out on a Dell Inspiron 15-3567 series Windows System with Intel(R) CoreTM i7-7500U CPU @2.70 GHz, Clock Speed 2.9 Ghz, L2 Cache Size 512 and 8 GB RAM. To avoid any bias in the results, 30 independent trials are conducted for each of the DA algorithms on each test problem. The initial population size and maximum number of cycles for each DA variant are kept as 60 and 500 respectively. Thus, during each trial, 30000 function evaluations are carried out. The weight parameter in DA variants is assumed to be linearly decreasing from 0.9 to 0.4 as the number of cycles increases from 0 to 500. Similarly, the separation/alignment/cohesion weights in the considered DA variants are randomly varied between 0–0.1 for cycles less than 250. At 250 or more than 250 cycles, the separation/alignment/cohesion weight becomes 0.

The DA variants are subsequently ranked by comparing the algorithm’s mean

The main objective of the HEND problem (Floudas and Ciric,

The optimal values of the control variables obtained, and objective function values (i.e. minimum

Simulation results of the HEND problem.

DADE | QGDA | MHDA | BMDA | INMDA | HDA | CDA | DA | |

0.052351 | 0.011093 | |||||||

15.97275 | 16.66409 | 16.66889 | 16.66681 | 16.66669 | 16.66667 | 16.80441 | 16.66939 | |

87.17488 | 66.77105 | 0.83742 | 0.01812 | 0.003442 | 57.77835 | 47.67126 | ||

33.51656 | 99.85326 | 143.3811 | 197.9965 | 198.9125 | 123.661 | 124.0067 | 23.45766 | |

1971712 | 1999763 | 1999999 | 2000000 | 2000000 | 2000000 | 1958385 | 1999885 | |

595.4896 | 599.993 | 599.9197 | 599.9949 | 599.999 | 600 | 585.1924 | 599.8195 | |

101.3036 | 100.0403 | 100.0001 | 100 | 100 | 100 | 102.4928 | 100.0042 | |

599.2642 | 599.9864 | 599.9999 | 600 | 600 | 600 | 599.251 | 599.9945 | |

701.4332 | 700.0313 | 700.0001 | 700 | 700 | 700 | 704.8981 | 699.9442 | |

190.5047 | 189.305 | 189.3934 | 189.3181 | 189.3138 | 189.3116 | 192.599 | 189.3521 | |

191.4536 | 190.2539 | 190.3423 | 190.267 | 190.2627 | 190.2605 | 193.5479 | 190.301 | |

0.789 | 0.082 | 0.915 | 0.864 | 0.525 | 0.727 | 0.940 | 0.836 | |

Run time | 3.2125 | 4.60625 | 7.570313 | 7.254688 | 7.303125 | 5.41875 | 7.290625 | 5.2125 |

FNRT |
4.7 | 3.2 | 4.7 | 4.8 | 4.8 | 5.1 | 4.7 | 4 |

Based on the simulation results for this problem, the

Convergence curve for the HEND problem.

The basic objective of the OOAU problem (Sauer

Simulation results of the OOAU problem.

DADE | QGDA | MHDA | BMDA | INMDA | HDA | CDA | DA | |

1362.7004 | 1364.9895 | 1365.0069 | 1365.0087 | 1364.4943 | 1364.8813 | 1365.009 | 1365.0091 | |

99.957173 | 99.99925 | 99.999969 | 99.999997 | 99.997169 | 99.994546 | 100 | 99.999999 | |

2000.1839 | 2000.0086 | 2000.0046 | 2000.0001 | 2000.328 | 2000.0167 | 2000.0009 | 2000 | |

90.745206 | 90.740691 | 90.740725 | 90.740741 | 90.741674 | 90.740325 | 90.740738 | 90.740741 | |

91.03223 | 91.015261 | 91.015162 | 91.015122 | 91.018349 | 91.015422 | 91.015123 | 91.01512 | |

3.307297 | 3.2787429 | 3.2786118 | 3.2785546 | 3.2857938 | 3.280122 | 3.2785563 | 3.2785504 | |

141.48571 | 141.46021 | 141.46006 | 141.45996 | 141.46966 | 141.46005 | 141.45996 | 141.45996 | |

−136.97331 | −142.65288 | −142.70488 | −142.71839 | −141.13781 | −142.43987 | −142.71733 | −142.71923 | |

−136.18861 | −141.86818 | −141.92018 | −141.93369 | −140.35311 | −141.65517 | −141.93263 | −141.93453 | |

0.005292 | 0.0008229 | 0.049782 | 0.0451094 | 0.0024717 | 0.0104315 | 0.002072 | 0.0328824 | |

Run time | 4.0140625 | 6.9 | 9.18125 | 9.4265625 | 9.39375 | 5.9734375 | 9.515625 | 5.596875 |

FNRT |
3.4 | 3.2 | 5.1 | 5.2 | 5.8 | 5.2 | 3.9 | 3.4 |

Convergence diagram for the OOAU problem.

When this OOAU problem is solved using the eight DA variants, the corresponding values of the optimal control variables, and objective functions with respect to

The RND problem (Ryoo and Sahinidis,

Simulation results of the RND problem.

DADE | QGDA | MHDA | BMDA | INMDA | HDA | CDA | DA | |

0.9999763 | 0.9999369 | 0.3944072 | 0.3919993 | 0.9945312 | 0.9999841 | 0.3940459 | 0.9976073 | |

0.4354706 | 0.4665822 | 0.3943078 | 0.3918984 | 0.4360605 | 0.398256 | 0.3939394 | 0.4420572 | |

0.3746106 | 0.3746492 | 0.0054414 | 0.0001139 | 0.374615 | 0.0024281 | |||

0.3832139 | 0.3763675 | 0.3748098 | 0.3748497 | 0.384213 | 0.3879295 | 0.3748192 | 0.3825421 | |

0.0025721 | 0.0006354 | 15.739914 | 15.899662 | 0.1285526 | 0.002037 | 15.764036 | 0.0540139 | |

13.419795 | 11.83318 | 13.26011 | 15.640824 | 0.0001726 | 13.009488 | |||

−0.383214 | −0.376367 | −0.374810 | −0.374850 | −0.384213 | −0.387929 | −0.374819 | −0.382542 | |

−0.2162199 | −0.1976946 | −0.1846461 | −0.0677247 | −0.0037783 | −0.1555713 | −0.1157014 | −0.292938 | |

0.1684727 | 0.1727503 | 0.1832099 | 0.1193487 | 0.0110534 | 0.1777097 | 0.1727922 | 0.1403655 | |

Run time | 2.68125 | 4.9203125 | 8.4046875 | 8.440625 | 4.9046875 | 6.065625 | 8.0796875 | 8.4 |

FNRT |
4 | 3.7 | 4.3 | 4.5 | 7.4 | 4.8 | 4.8 | 2.5 |

The simulation-based results for this problem determine the optimal values of the control variables, and objective functions (i.e.

Convergence curve for the RND problem.

This HPP problem (Floudas and Pardalos,

Simulation results of the HPP problem.

DADE | QGDA | MHDA | BMDA | INMDA | HDA | CDA | DA | |

0.0001505 | 0.0001034 | 0.9561836 | 0.0181292 | 0.0001372 | 1.9123671 | |||

199.99996 | 199.99997 | 199.99999 | 199.99927 | 199.19222 | 199.93562 | 199.99961 | 198.38445 | |

0.0001411 | 0.0001513 | 0 | 0 | 0 | ||||

99.999692 | 99.999637 | 99.999648 | 99.999939 | 99.609244 | 99.999977 | 99.999819 | 99.218489 | |

0.9438356 | 0.0176439 | 0.000137 | 1.8876711 | |||||

99.999991 | 99.999994 | 99.99999 | 99.999345 | 99.595327 | 99.936127 | 99.999653 | 99.190654 | |

0.012348 | 0.0004854 | 0.0246959 | ||||||

99.999875 | 99.999876 | 99.999899 | 99.999928 | 99.596896 | 99.999491 | 99.999921 | 99.193792 | |

1.0000004 | 1.0000009 | 1.000001 | 1.0000002 | 0.9999999 | 1 | 0.9999992 | 0.9999999 | |

−400.00475 | −400.00546 | −400.00555 | −399.99661 | −397.34948 | −399.66011 | −400.00041 | −394.69895 | |

−399.04995 | −399.05066 | −399.05075 | −399.04181 | −396.39468 | −398.70531 | −399.04561 | −393.74415 | |

0.3171725 | 0.3181686 | 0.2615975 | 0.2643702 | 0.3889547 | 0.552458 | 0.3975508 | 0.2672359 | |

Run time | 2.0260417 | 4.109375 | 7.1916667 | 4.1302083 | 7.1614583 | 4.98125 | 7.1458333 | 7.1791667 |

FNRT |
4.10 | 3.77 | 5.00 | 4.77 | 4.93 | 5.27 | 4.33 | 3.83 |

Convergence curve for the HPP problem.

A summarized version of the rankings of the eight DA variants with respect to computational time (T), Friedman’s rank based on the derived optimal solutions (F) and convergence rate (C) for the four case studies under consideration is provided in Table

Summary of performance of the DA variants on the four case studies.

Problem | HEND | OOAU | RND | HPP | Aggregated | ||||||||||

Criteria | T | F | C | T | F | C | T | F | C | T | F | C | T | F | C |

DADE | 1 | 3 | 1 | 1 | 2 | 2 | 1 | 3 | 2 | 1 | 3 | 1 | 1.00 | 2.75 | 1.50 |

QGDA | 2 | 1 | 7 | 4 | 1 | 3 | 3 | 2 | 6 | 2 | 1 | 4 | 2.75 | 1.25 | 5.00 |

MHDA | 8 | 3 | 6 | 5 | 5 | 5 | 7 | 4 | 3 | 8 | 7 | 2 | 7.00 | 4.75 | 4.00 |

BMDA | 5 | 6 | 3 | 7 | 6 | 7 | 8 | 5 | 7 | 3 | 5 | 8 | 5.75 | 5.50 | 6.25 |

INMDA | 7 | 6 | 8 | 6 | 8 | 8 | 2 | 8 | 4 | 6 | 6 | 7 | 5.25 | 7.00 | 6.75 |

HDA | 4 | 8 | 2 | 3 | 6 | 4 | 4 | 6 | 1 | 4 | 8 | 5 | 3.75 | 7.00 | 3.00 |

CDA | 6 | 3 | 5 | 8 | 4 | 6 | 5 | 6 | 5 | 5 | 4 | 6 | 6.00 | 4.25 | 5.50 |

DA | 3 | 2 | 4 | 2 | 2 | 1 | 6 | 1 | 8 | 7 | 2 | 3 | 4.50 | 1.75 | 4.00 |

Fuzzy scale considered in this paper.

Linguistic term for criteria importance | Symbol | Triangular fuzzy number |

Extremely Poor | EP | |

Very Poor | VP | |

Poor | P | |

Medium Poor | MP | |

Medium | M | |

Medium Good | MG | |

Good | G | |

Very Good | VG | |

Extremely Good | EG |

Five experts (decision makers) are subsequently asked to provide their opinions on the importance on the three evaluation criteria using the fuzzy linguistic scale. Table

Importance assigned to each criterion by the experts.

Decision maker | Linguistic term | Triangular fuzzy number | ||||

T | F | C | T | F | C | |

Expert 1 | VG | EG | G | |||

Expert 2 | MG | G | VP | |||

Expert 3 | P | MG | M | |||

Expert 4 | MP | G | MG | |||

Expert 5 | M | VG | G |

It should be noted that only the aggregated values of T, F and C in Table

Decision matrix and its normalization.

Problem | Decision matrix | Normalized decision matrix | ||||

Criteria | T | F | C | T | F | C |

DADE | 1.00 | 2.75 | 1.50 | 1.0000 | 0.4545 | 1.0000 |

QGDA | 2.75 | 1.25 | 5.00 | 0.3636 | 1.0000 | 0.3000 |

MHDA | 7.00 | 4.75 | 4.00 | 0.1429 | 0.2632 | 0.3750 |

BMDA | 5.75 | 5.50 | 6.25 | 0.1739 | 0.2273 | 0.2400 |

INMDA | 5.25 | 7.00 | 6.75 | 0.1905 | 0.1786 | 0.2222 |

HDA | 3.75 | 7.00 | 3.00 | 0.2667 | 0.1786 | 0.5000 |

CDA | 6.00 | 4.25 | 5.50 | 0.1667 | 0.2941 | 0.2727 |

DA | 4.50 | 1.75 | 4.00 | 0.2222 | 0.7143 | 0.3750 |

Anti-ideal (AI) solution | 1.00 | 1.25 | 1.50 | 1.00 | 1.00 | 1.00 |

Ideal (ID) solutions | 7.00 | 7.00 | 6.75 | 0.1429 | 0.1786 | 0.2222 |

The fuzzy weighted normalized decision matrix is presented in Table

Similarly, sample calculations for

Fuzzy-weighted normalized decision matrix.

Alternative | T | F | C | |

AI | ||||

DADE | ||||

QGDA | ||||

MHDA | ||||

BMDA | ||||

INMDA | ||||

HDA | ||||

CDA | ||||

DA | ||||

ID |

Utility degree and utility function of the DA variants.

Alternative | Utility degree | Utility function | ||

DADE | ||||

QGDA | ||||

MHDA | ||||

BMDA | ||||

INMDA | ||||

HDA | ||||

CDA | ||||

DA |

The corresponding fuzzy values of utility degree and utility function are subsequently calculated for each of the alternatives, as exhibited in Table

Defuzzified utility degree, utility function and ranks of the DA variants.

Alternative | Rank | |||||||

DADE | 4.3510 | 0.7896 | 0.1536 | 0.8464 | 5.5101 | 0.1815 | 0.7682 | 1 |

QGDA | 3.4433 | 0.6249 | 0.1216 | 0.6698 | 7.2260 | 0.4929 | 0.4666 | 2 |

MHDA | 1.4799 | 0.2686 | 0.0522 | 0.2879 | 18.1402 | 2.4737 | 0.0809 | 5 |

BMDA | 1.2175 | 0.2210 | 0.0430 | 0.2368 | 22.2653 | 3.2224 | 0.0543 | 7 |

INMDA | 1.0991 | 0.1995 | 0.0388 | 0.2138 | 24.7705 | 3.6770 | 0.0441 | 8 |

HDA | 1.6884 | 0.3064 | 0.0596 | 0.3284 | 15.7763 | 2.0447 | 0.1060 | 4 |

CDA | 1.4187 | 0.2575 | 0.0501 | 0.2760 | 18.9661 | 2.6236 | 0.0742 | 6 |

DA | 2.6703 | 0.4846 | 0.0943 | 0.5194 | 9.6075 | 0.9251 | 0.2736 | 3 |

After deriving the utility degree and utility function for each alternative, their values are finally defuzzified. These defuzzified values of utility degree and utility function along with the final rankings of the alternatives are provided in Table

The

Similarly, the

Thus, based on the considered industrial chemical process optimization problems, the application of fuzzy MARCOS method leads to relative ranking of the eight DA variants as DADE > QGDA > DA > HDA > MHDA > CDA > BMDA > INMDA.

In the last decade, a plethora of optimization algorithms has been developed by the researchers to solve a variety of complex problems. Among various application fields, industrial process optimization is a realistic application area where an optimized solution can directly lead to real-world benefits. In this paper, the performance of eight different variants of DA is comprehensively studied based on four complex industrial chemical process optimization case studies. Evaluation of the considered DA variants is carried out from the standpoint of convergence criterion, time intensiveness and quality of the solution obtained. The quality of the solution is assessed while measuring the best solution derived, mean best solution obtained and dispersion of the derived solutions on 30 repeated trials. To amalgamate all this information on the solution quality, Friedman’s test ranks are also computed. Finally, employing a group decision-making approach under fuzzy environment, the information derived from convergence criterion, time intensiveness and solution quality is translated into a relative ranking of the eight DA variants as DADE > QGDA > DA > HDA > MHDA > CDA > BMDA > INMDA. It can be interestingly noted that despite its simplicity, DA outperforms many of its better endowed variants. The derived observations would thus help the future researchers in identifying the most promising DA variants. Moreover, the comprehensive methodology followed to evaluate the optimization techniques can also be replicated by the researchers for analysis of other algorithms as well.

However, despite the comprehensiveness of the study, these findings also come with caveats. The scalability of the tested algorithms and the computational resources required are potential limitations, as is the transferability of the current results to other, perhaps larger-scale industrial contexts. These factors may influence the broader applicability of the conclusions and are critical considerations for future research endeavours.

In terms of future scope, an expansion of this research to include a broader array of DA subtypes, such as those enhanced through hybridization with grey wolf optimization, genetic algorithms, and binary dragonfly improved particle swarm optimization can be undertaken. Furthermore, the potential of multi-objective DA variations remains an enticing prospect for further investigations. In light of this study’s scope and its constraints, particularly the length of this paper, a comprehensive discussion on every existing DA variant was not feasible. Yet, this constraint opens the door for future work that can explore these additional variants, ideally leading to the development of more refined, context-specific optimization tools. It is hoped the methodological rigour and the analytical framework presented herein will not only inform but also inspire subsequent research in this domain.

^{4}C hybrid composite using Taguchi-MARCOS method