Informatica

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, 2013). Metaheuristics are stochastic algorithms and often draw their inspiration from nature. Any metaheuristic algorithm is an amalgamation of two basic functions, i.e. exploration and exploitation (Blum and Roli, 2003). Exploration and exploitation are also sometimes referred to as diversification and intensification. The objective of diversification or exploration is to navigate through the search space to find out ‘potential regions or zones’ with ‘good solutions’. On the other hand, intensification or exploitation is related to thoroughly searching out the ‘potential region or zone’ to locate the best solution. All metaheuristic algorithms attempt to strike an optimal balance between diversification and intensification. This balance has a direct bearing on the convergence rate of the considered algorithm as well as its ability to find out diverse solutions. Many metaheuristic algorithms have been proposed so far in quest of the optimal balance between diversification and intensification. Yet, many more algorithms continue to be developed. Nevertheless, in recent times, researchers have established the suitability, applicability and often superiority of metaheuristics over the traditional approaches, which are deterministic and exact. For large-scale and complex problems, like chemical process optimization, structural optimization, etc., metaheuristics provide a good trade-off between solution quality and computational time. Some of the most popular metaheuristics are genetic algorithm (GA) (Holland, 1992), simulated annealing (Kirkpatrick et al., 1983), particle swarm optimization (PSO) (Kennedy and Eberhart, 1995), grey wolf optimizer (Mirjalili et al., 2014) etc.

In this decade, research on metaphor-based metaheuristics has received a tremendous impetus. A plethora of nature-inspired metaheuristics, like flower pollination algorithm (Yang, 2012), swallow swarm optimization algorithm (Neshat et al., 2013), grey wolf optimizer (Mirjalili et al., 2014), moth-flame optimization algorithm (Mirjalili, 2015), dragonfly algorithm (Mirjalili, 2016), grasshopper optimization algorithm (Saremi et al., 2017), artificial flora optimization algorithm (Cheng et al., 2018), seagull optimization algorithm (Dhiman and Kumar, 2019), marine predators algorithm (Faramarzi et al., 2020), arithmetic optimization algorithm (Abualigah et al., 2021), rat swarm optimization (Mzili et al., 2022), etc., has been developed in the last ten years. Besides proposing new metaheuristics, tons of work have also been carried out in the area of hybridization of metaheuristics, wherein existing algorithms are either merged with other algorithms or new features are introduced in the existing algorithms.

The DA, a population-based nature-inspired metaheuristic, was propounded by Mirjalili (2016), Mafarja et al. (2018). Just like PSO, DA is also guided by swarm intelligence. It mimics the static and dynamic swarming behaviours of dragonflies. Since its inception in 2015, DA has been applied to solve various classes of optimization problems ranging from continuous (Abedi and Gharehchopogh, 2020) to discrete (Jawad et al., 2021) and from unconstrained (Can and Alatas, 2017) to constrained (Khalilpourazari and Khalilpourazary, 2020). It has been successfully employed in both single-objective (Reddy, 2016) and multi-objective roles (Joshi et al., 2021). The hybridization of DA has also received a lot of attention lately. Debnath et al. (2021) developed a hybrid memory-based dragonfly algorithm with differential evolution (DADE), whereas, Shirani and Safi-Esfahani (2020) proposed a biogeography-based Mexican hat wavelet dragonfly algorithm (BMDA). Xu and Yan (2019) fused the classical DA with the Nelder-Mead algorithm to develop a hybrid Nelder-Mead algorithm and dragonfly algorithm (INMDA) to improve the local capacity for exploration. Ghanem and Jantan (2018) proposed a hybridization of dragonfly algorithm and artificial bee colony (HDA) to improve the convergence rate. Sree Ranjini and Murugan (2017) combined the exploration capability of DA with the exploitation capacity of PSO to develop a memory-based hybrid dragonfly algorithm (MHDA). Yu et al. (2020) proposed the quantum-behaved and Gaussian mutational dragonfly algorithm (QGDA) and (Sayed et al., 2019) developed the chaotic dragonfly algorithm (CDA) by seamlessly integrating chaos theory with classical DA.

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, 1989), optimal operation of alkylation unit (Sauer et al., 1964), reactor network design (Ryoo and Sahinidis, 1995) and Haverly’s pooling problem (Floudas and Pardalos, 1990). Hence, the algorithms considered in this paper are classical DA (Mirjalili, 2016), DADE (Debnath et al., 2021), QGDA (Yu et al., 2020), MHDA (Sree Ranjini and Murugan, 2017), CDA (Sayed et al., 2019), BMDA (Shirani and Safi-Esfahani, 2020), INMDA (Xu and Yan, 2019) and HAD (Ghanem and Jantan, 2018). The algorithms are comprehensively tested based on the optimal solution obtained, computational time and convergence rate. The derived optimal solutions are further validated from the viewpoint of the best solution, mean best solution and dispersion (standard deviation) of the solutions on repeated trials. The Friedman’s test rank is computed for each algorithm based on three criteria (best, mean and standard deviation) used for optimal solution analysis. Further, the opinions of five experts are aggregated using a fuzzy scale and a multi-criteria decision making (MCDM) tool in the form of fuzzy-measurement alternatives and ranking according to compromise solution (MARCOS) is adopted to identify the best algorithm based on the comprehensive analysis of the optimal solution, computational burden and convergence rate. The basic methodology followed in this paper can be represented in the form of a flowchart, as shown in Fig. 1.

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 $({f_{\text{mean}}})$, standard deviation $({f_{\text{std}}})$, CPU (run time) (in sec) and Friedman ranking. While comparing the performance of optimizers, the one having the lowest ${f_{\text{mean}}}$ value is always preferable (for minimization problem). If the ${f_{\text{mean}}}$ values of two optimizers become equal, their ${f_{\text{std}}}$ values can then be compared. In such cases, the optimizer having smaller ${f_{\text{std}}}$ value is more stable.

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 5. It can be interestingly noticed that for the four industrial chemical process problems, there are some discrepancies in the optimization performance of the eight DA variants in respect of computational time, Friedman’s rank based on the optimal solutions and convergence rate. For example, in case of the HEND problem, DADE ranks best in terms of computational time and convergence rate, but is inferior to QGDA and classical DA with respect to the optimal solution obtained. Thus, there is an ardent need to holistically analyse these algorithms and their performance on multiple case studies based on different evaluation criteria. Moreover, if the obtained rankings are directly aggregated, it would mean that equal importance is assigned to each of the criteria which would be an oversimplification of the problem. Thus, a fuzzy scale for assigning relative importance to each criterion is considered, as provided in Table 6.

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 7 shows the assigned importance for each criterion by each expert and the corresponding triangular fuzzy number. It can be observed that all the experts deem information related to the optimal solution (FNRT criterion) as relatively the most important one. Based on the aggregation of the triangular fuzzy numbers for each of the criteria for all the experts, the corresponding fuzzy criteria weights are obtained as $(3.8,4.6,5.8)$, $(5.8,7,7.8)$ and $(3.8,5,5.8)$ for T, F and C respectively.

It should be noted that only the aggregated values of T, F and C in Table 5 constitute the decision matrix. Thus, the initial decision matrix has an $8\times 3$ format (Table 8). It is normalized using equations (5) and (6). The normalized decision matrix is presented in Table 8. Sample calculations of normalization are shown below:

The fuzzy weighted normalized decision matrix is presented in Table 9 along with the computed values of ${\tilde{S}_{i}}$ parameter. Sample calculations for fuzzy weighted normalized decision matrix using equation (7) are shown below:

Similarly, sample calculations for ${\tilde{S}_{i}}$ parameter using equation (10) are shown below:

The corresponding fuzzy values of utility degree and utility function are subsequently calculated for each of the alternatives, as exhibited in Table 10. Sample calculations for ${\tilde{K}_{i}^{-}}$ using equation (8) are shown below: Similarly, sample calculations for ${\tilde{K}_{i}^{+}}$ using equation (9) are shown below: The ${\tilde{T}_{i}}$ and $d{f_{\text{crisp}}}$ are calculated using equations (11) and (14) respectively: The $f({\tilde{K}_{i}^{+}})$ and $f({\tilde{K}_{i}^{-}})$ are computed using equation (12) and (13), as shown below:

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 11. The ${K_{i}^{-}}$ and ${K_{i}^{+}}$ are computed as follows:

The $f({K_{i}^{-}})$ and $f({K_{i}^{+}})$ are computed as follows:

Similarly, the $f({K_{i}})$ is computed using equation (16):

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.