Introduction

INFORMATICA

Informatica

1822-88440868-4952

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Vilnius University

INFOR538

10.15388/23-INFOR538

Research Article

A Fuzzy MARCOS-Based Analysis of Dragonfly Algorithm Variants in Industrial Optimization Problems

Kalita

Kanak

K. Kalita is an associate professor at the Mechanical Engineering Department at Vel Tech University, India. With a dedicated focus on research, K. Kalita specializes in optimizing composite laminated structures. This specialization is complemented by a strong background in computational mechanics and soft computing techniques, contributing significantly to the field through innovative research and academic excellence.

Ganesh

Narayanan

N. Ganesh brings a wealth of experience to his role as a senior associate professor at the School of Computer Science and Engineering, Vellore Institute of Technology, Chennai Campus. With a career spanning nearly two decades in teaching, training and research, he has established himself as an authority in this field. His research interests are diverse and forward-thinking, encompassing software engineering, agile software development, prediction and optimization techniques, deep learning, image processing and data analytics.

Shankar

Rajendran

R. Shankar serves as an associate professor in the Department of Computer Science and Engineering at the Koneru Lakshmaiah Education Foundation in Vaddeswaram, India. His research interests are software development optimization, deep learning and data analytics.

Chakraborty

Shankar

s_chakraborty00@yahoo.co.in5∗

S. Chakraborty is a distinguished member of the faculty at Jadavpur University’s Department of Production Engineering in India. Renowned for contributions to academia and research, professor Chakraborty is a regular reviewer for several journals of international repute. The research interests of professor Chakraborty encompass a broad range of topics including operations research, multi-criteria decision making, statistical quality control and soft computing.

1Department of Mechanical Engineering, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Avadi, India 2University Centre for Research & Development, Chandigarh University, Mohali, 140413, India 3School of Computer Science and Engineering, Vellore Institute of Technology, Chennai, India 4Department of Computer Science and Engineering, Koneru Lakshmaiah Education Foundation, Vaddeswaram, India 5Department of Production Engineering, Jadavpur University, Kolkata, India.

∗Corresponding author.

2024

20112023

351155178122022112023

2024

Open access article under the CC BY license.

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.

Key words Dragonfly algorithm Process optimization MCDM MARCOS

1 Introduction

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.

Fig. 1

Flowchart of the adopted methodology.

2 Methods 2.1 Dragonfly Algorithm

The classical DA (Mirjalili, 2016) is a simple yet powerful metaheuristic algorithm mimicking the swarm behaviour of dragonflies. The social behaviour of dragonflies exhibited during searching and gathering of food as well as during foe avoidance forms the basis of the equation-based rules used to simulate and actuate DA. The DA is realized using the five parameters, e.g. separation, alignment, cohesion, attraction and distraction. Collision avoidance with neighbouring dragonflies is governed by separation. Velocity matching to the neighbouring individuals is carried out by alignment. Attractions towards the centre of mass of the neighbourhood and towards a food source are respectively governed by cohesion and attraction. Movement away from the enemy is controlled by distraction.

2.2 Hybrid Dragonfly Algorithm with Differential Evolution

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 et al., 2021) stores the best solution in its memory and continues the search with DE which promotes population diversity by employing mutation.

2.3 Quantum-Behaved and Gaussian Mutational Dragonfly Algorithm

By implementing the concept of a quantum rotation gate, Yu et al. (2020) endeavoured to strike a better balance between exploration and exploitation traits of DA. The Gaussian mutation is also incorporated into this algorithm to help generate diverse solutions.

2.4 Memory-Based Hybrid Dragonfly Algorithm

The lack of internal memory in the classical DA can cause premature convergence to local optima. To overcome this problem, Sree Ranjini and Murugan (2017) introduced certain features of PSO into DA and called the hybrid algorithm as MHDA. By endowing DA with internal memory, MHDA allows each dragonfly to keep track of its DA-pbest solution, i.e. coordinates of the best solution obtained by it so far. The MHDA has also access to DA-gbest, i.e. coordinates of the overall best solution obtained by the algorithm so far. After initial exploration of the search space by DA, exploitation of the promising search space zones is initialized by PSO considering DA-pbest and DA-gbest solutions.

2.5 Chaotic Dragonfly Algorithm

Sayed et al. (2019) employed ten chaotic maps to fine-tune the weights involved in the separation, alignment, cohesion, attraction and distraction parameters of the classical DA. The authors argued that as compared to DA, CDA would have an improved convergence rate, with the algorithmic complexity being at par with DA. The overall complexity of CDA is O(dM + MC), where d, M and C are the dimensions of the problem, number of dragonflies and objective function complexity respectively.

2.6 Biogeography-Based Mexican Hat Wavelet Dragonfly Algorithm

To address the issue of premature convergence under heavy loads, Shirani and Safi-Esfahani (2020) proposed a variant of DA called BMDA (biogeography-based algorithm, Mexican hat wavelet and dragonfly algorithm) that combines the migration process of the biogeography-based optimization (BBO) technique with the transformation process of DA’s Mexican hat wavelet.

2.7 Hybrid Nelder-Mead Algorithm and Dragonfly Algorithm

Xu and Yan (2019) argued that too many social interactions in DA would be responsible for reduced solution accuracy and premature convergence to local optima. These may be caused due to improper balance between diversification and intensification. Xu and Yan (2019) thus suggested hybridizing DA with an improved Nelder-Mead algorithm to improve its local search capacity.

2.8 Hybridization of Dragonfly Algorithm and Artificial Bee Colony

Ghanem and Jantan (2018) highlighted that the presence of Levy flight in the position update phase of DA would make it unable to effectively carry out a local search. To rectify this problem, Ghanem and Jantan (2018) suggested hybridization of DA and artificial bee colony (ABC) algorithm to make use of the exploitation and exploration abilities of DA along with the exploration ability of ABC.

2.9 Fuzzy MARCOS

The MARCOS is an innovative MCDM approach that can be employed in many contexts (Chakraborty et al., 2020; Stanković et al., 2020; Stević et al., 2020; Deveci et al., 2021; Biswal et al., 2023). Its computational strategy has been developed taking into account both the ideal and anti-ideal solutions (Bakır and Atalık, 2021). The utility degrees of the candidate alternatives are quantified, which are subsequently considered to evaluate the relative performance and rank each of the alternatives (Bakır et al., 2021; Badi et al., 2022). In this paper, MARCOS is integrated with fuzzy set theory to deal with the individual opinions of five experts with respect to computational time, Friedman’s rank based on optimal solutions and convergence rate leading to the relative ranking of the eight DA variants.

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

Step 1: Formulate the initial decision matrix ( X ), consisting of m possible choices (alternatives) and n evaluation criteria. (1) X = x 11 x 12 … x 1 j … x 1 n x 21 x 22 … x 2 j … x 2 n … … … … … … x i 1 x i 2 … x i j … x i n … … … … … … x m 1 x m 2 … x m j … x m n , where x i j is the performance of ith alternative against jth criterion.

Step 2: Develop the corresponding extended decision matrix ( X ′ ) while considering the anti-ideal (AI) and ideal (ID) solutions. (2) A I = min i x i j if j ∈ B and max i x i j if j ∈ C , (3) A I = max i x i j if j ∈ B and min i x i j if j ∈ C , where B is the set of beneficial criteria and C is the set of non-beneficial criteria. (4) X ′ = x a i 1 x a i 2 … x a i j … x a i n x 11 x 12 … x 1 j … x 1 n … … … … … … x i 1 x i 2 … x i j … x i n x m 1 x m 2 … x m j … x m n x i d 1 x i d 2 … x i d j … x i d n .

Step 3: Normalize the extended decision matrix using Eqs. (5) and (6) depending on the type of the criterion under consideration. (5) n i j = x i d x i j , if j ∈ C , (6) n i j = x i j x i d , if j ∈ B .

Step 4: Develop the weighted normalized fuzzy decision matrix. (7) v ˜ i j = ( v i j l , v i j m , v i j u ) = n i j ⊗ w ˜ j = ( n i j × w j l , n i j × w j m , n i j × w j u ) , where w ˜ j is the fuzzy weight assigned to jth criterion.

Step 5: Determine the utility degrees of each alternative using the following expressions: (8) K ˜ i − = S ˜ i S ˜ a i = ( s i l s a i u , s i m s a i m s i u s a i l ) , (9) K ˜ i + = S ˜ i s ˜ i d = ( s i l s i d u , s i m s i d m s i u s i d l ) , where S ˜ i ( s i l , s i m , s i u ) is the sum of elements of the weighted normalized fuzzy decision matrix and can be estimated using Eq. (10): (10) S ˜ i = ∑ j = 1 n v ˜ i j .

Step 6: Formulate the fuzzy matrix T ˜ i applying the following expression: (11) T ˜ i = t ˜ i = ( t i l , t i m , t i u ) = K ˜ i − ⊕ K ˜ i + = ( k i − l + k i + l , k i − m + k i + m , k i − u + k i + u ) .

Step 7: Evaluate the utility functions for both the ideal and anti-ideal solutions. (12) f ( K ˜ i + ) = K ˜ i − d f crisp = ( k i − l d f crisp , k i − m d f crisp , k i − u d f crisp ) , (13) f ( K ˜ i − ) = K ˜ i + d f crisp = ( k i + l d f crisp , k i + m d f crisp , k i + u d f crisp ) , where (14) d f crisp = l + 4 m + u 6 . The value of d f crisp is obtained from a fuzzy number D ˜, which can be estimated using Eq. (15): (15) D ˜ = ( d l , d m , d u ) = max i t ˜ i j .

Step 8: Determine the utility functions of all the alternatives.

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: (16) f ( K i ) = K i + + K i − 1 + 1 − f ( K i + ) f ( K i + M ) + 1 − f ( K i − ) f ( K i − ) .

Step 9: Rank the alternatives.

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.

3 Problem Description

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

4 Numerical Results on Chemical Process Optimization 4.1 Case Study 1: Heat Exchanger Network Design (HEND)

The main objective of the HEND problem (Floudas and Ciric, 1989) is to minimize the comprehensive area of HEND. This problem contains nine control variables and eight equality constraints. Mathematically, this minimization type HEND problem is defined as follows: (17) f ( x ¯ ) = 35 x 1 0.6 + 35 x 2 0.6 , h 1 ( x ¯ ) = 200 x 1 x 4 − x 3 = 0 , h 2 ( x ¯ ) = 200 x 2 x 6 − x 5 = 0 , h 3 ( x ¯ ) = x 3 − 10000 ( x 7 − 100 ) = 0 , h 4 ( x ¯ ) = x 5 − 10000 ( 300 − x 7 ) = 0 , h 5 ( x ¯ ) = x 3 − 10000 ( 600 − x 8 ) = 0 , h 6 ( x ¯ ) = x 5 − 10000 ( 900 − x 9 ) = 0 , h 7 ( x ¯ ) = x 4 ln ( x 8 − 100 ) − x 4 ln ( 600 − x 7 ) − x 8 + x 7 + 500 = 0 , h 8 ( x ¯ ) = x 6 ln ( x 9 − x 7 ) − x 6 ln ( 600 ) − x 9 + x 7 + 600 = 0 , 0 ⩽ x 1 ⩽ 10 , 0 ⩽ x 2 ⩽ 200 , 0 ⩽ x 3 ⩽ 100 , 0 ⩽ x 4 ⩽ 200 , 1000 ⩽ x 5 ⩽ 2000000 , 0 ⩽ x 6 ⩽ 600 , 100 ⩽ x 7 ⩽ 600 , 100 ⩽ x 8 ⩽ 600 , 100 ⩽ x 9 ⩽ 900.

The optimal values of the control variables obtained, and objective function values (i.e. minimum ( f min ), mean ( f mean ) and standard deviation ( f std ) of the eight DA variants (DADE, QGDA, MHDA, BMDA, INMDA, HDA, CDA and DA) are provided in Table 1.

Table 1

Simulation results of the HEND problem.

	DADE	QGDA	MHDA	BMDA	INMDA	HDA	CDA	DA
x 1	0.052351	1.44 E − 06	2.92 E − 05	4.58 E − 07	8.65 E − 08	4.61 E − 13	0.011093	4.58 E − 06
x 2	15.97275	16.66409	16.66889	16.66681	16.66669	16.66667	16.80441	16.66939
x 3	87.17488	66.77105	0.83742	0.01812	0.003442	1.58 E − 05	57.77835	47.67126
x 4	33.51656	99.85326	143.3811	197.9965	198.9125	123.661	124.0067	23.45766
x 5	1971712	1999763	1999999	2000000	2000000	2000000	1958385	1999885
x 6	595.4896	599.993	599.9197	599.9949	599.999	600	585.1924	599.8195
x 7	101.3036	100.0403	100.0001	100	100	100	102.4928	100.0042
x 8	599.2642	599.9864	599.9999	600	600	600	599.251	599.9945
x 9	701.4332	700.0313	700.0001	700	700	700	704.8981	699.9442
f min	190.5047	189.305	189.3934	189.3181	189.3138	189.3116	192.599	189.3521
f mean	191.4536	190.2539	190.3423	190.267	190.2627	190.2605	193.5479	190.301
f std	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 T Rank	4.7	3.2	4.7	4.8	4.8	5.1	4.7	4

Based on the simulation results for this problem, the f min values of DADE, QGDA, MHDA, BMDA, INMDA, HDA, CDA and DA are respectively observed as 190.5047, 189.305, 189.3934, 189.3181, 189.3138, 189.3116, 192.599 and 189.3521. In other words, the QGDA result is respectively 0.63 %, 0.047 %, 0.007 %, 0.005 %, 0.003 %, 1.710 % and 0.025 % lower (better) than the simulation-based results obtained using DADE, MHDA, BMDA, INMDA, HDA, CDA and DA. Table 1 also shows that DADE, QGDA, MHDA, BMDA, INMDA, HDA, CDA and DA provide the corresponding f mean values as 191.4536, 190.2539, 190.3423, 190.267, 190.2627, 190.2605, 193.5479 and 190.301 respectively without violating any of constraints. Thus, the resulting benefit for QGDA is 0.627 %, 0.046 %, 0.007 %, 0.005 %, 0.003 %, 1.702 % and 0.025 % as compared to that obtained from DADE, MHDA, BMDA, INMDA, HDA, CDA and DA respectively. On the other hand, the f std value for QGDA is noticed to be 0.082, which is lower by 89.607%, 91.038%, 90.509%, 84.381%, 88.721%, 91.277% and 90.191% than the simulation-based results derived from DADE, MHDA, BMDA, INMDA, HDA, CDA and DA respectively. Table 1 also shows the Friedman’s ranks for DADE, QGDA, MHDA, BMDA, INMDA, HDA, CDA and DA as 4.7, 3.2, 4.7, 4.8, 4.8, 5.1, 4.7 and 4 respectively. Thus, based on the Friedman’s rank test (FNRT) at 95% significance level, the ranking of the eight DA variants can be derived as QGDA > DA > DADE > MHDA > CDA > BMDA > INMDA > HDA. It is also interesting to note that according to the average run time, DADE is 30.258 %, 57.565 %, 55.718%, 56.012%, 40.715%, 55.937% and 38.369% faster than QGDA, MHDA, BMDA, INMDA, HDA, CDA and DA respectively. Although DADE is superior to QGDA, MHDA, BMDA, INMDA, HDA, CDA, and DA with respect to computational burden, QGDA is the second best in computational time. Figure 2 depicts the convergence curves of all the DA variants for this problem. Based on Fig. 2, it can be unveiled that DADE has a convergence advantage, which can find out a better solution with faster speed as compared to other DA variants. With respect to convergence rate, the considered DA variants can be ranked as DADE > HDA > BMDA > DA > CDA > MHDA > QGDA > INMDA.

Fig. 2

Convergence curve for the HEND problem.

4.2 Case Study 2: Optimal Operation of Alkylation Unit (OOAU)

The basic objective of the OOAU problem (Sauer et al., 1964) (containing seven variables and 14 inequality constraints) is to maximize the octane number of olefin feed in the presence of acid. The minimization type of the OOAU problem can be mathematically stated as shown below: (18) f ( x ¯ ) = 0.035 x 1 x 6 + 1.715 x 1 + 10.0 x 2 + 4.0565 x 3 − 0.063 x 3 x 5 , g 1 ( x ¯ ) = 0.0059553571 x 6 2 x 1 + 0.88392857 x 3 − 0.1175625 x 6 x 1 − x 1 ⩽ 0 , g 2 ( x ¯ ) = 1.1088 x 1 + 0.1303533 x 1 x 6 − 0.0066033 x 1 x 6 2 − x 3 ⩽ 0 , g 3 ( x ¯ ) = 6.66173269 x 6 2 − 56.596669 x 4 + 172.39878 x 5 − 10000 − 191.20592 x 6 ⩽ 0 , g 4 ( x ¯ ) = 1.08702 x 6 − 0.3762 x 6 2 + 0.32175 x 4 + 56.85075 − x 5 ⩽ 0 , g 5 ( x ¯ ) = 0.006198 x 7 x 4 x 3 + 2462.3121 x 2 − 25.125634 x 2 x 4 − x 3 x 4 ⩽ 0 , g 6 ( x ¯ ) = 161.18996 x 3 x 4 + 5000.0 x 2 x 4 − 489510.0 x 2 − x 3 x 4 x 7 ⩽ 0 , g 7 ( x ¯ ) = 0.33 x 7 x 4 + 44.333333 ⩽ 0 , g 8 ( x ¯ ) = 0.022556 x 5 − 1.0 x 2 − 0.007595 x 7 ⩽ 0 , g 9 ( x ¯ ) = 0.00061 x 3 − 1.0 − 0.0005 x 1 ⩽ 0 , g 10 ( x ¯ ) = 0.819672 x 1 − x 3 + 0.819672 ⩽ 0 , g 11 ( x ¯ ) = 24500.0 x 2 − 250.0.0 x 2 x 4 − x 3 x 4 ⩽ 0 , g 12 ( x ¯ ) = 1020.4082 x 2 x 4 + 1.2244898 x 3 x 4 − 100000 x 2 ⩽ 0 , g 13 ( x ¯ ) = 6.25 x 1 x 6 + 6.25 x 1 − 7.625 x 3 − 100000 ⩽ 0 , g 14 ( x ¯ ) = 1.22 x 3 − x 1 x 6 − x 1 ⩽ 0 , 10000 ⩽ x 1 ⩽ 2000 , 0 ⩽ x 2 ⩽ 100 , 2000 ⩽ x 3 ⩽ 4000 , 0 ⩽ x 4 ⩽ 100 , 0 ⩽ x 5 ⩽ 100 , 0 ⩽ x 6 ⩽ 20 , 0 ⩽ x 7 ⩽ 200.

Table 2

Simulation results of the OOAU problem.

	DADE	QGDA	MHDA	BMDA	INMDA	HDA	CDA	DA
x 1	1362.7004	1364.9895	1365.0069	1365.0087	1364.4943	1364.8813	1365.009	1365.0091
x 2	99.957173	99.99925	99.999969	99.999997	99.997169	99.994546	100	99.999999
x 3	2000.1839	2000.0086	2000.0046	2000.0001	2000.328	2000.0167	2000.0009	2000
x 4	90.745206	90.740691	90.740725	90.740741	90.741674	90.740325	90.740738	90.740741
x 5	91.03223	91.015261	91.015162	91.015122	91.018349	91.015422	91.015123	91.01512
x 6	3.307297	3.2787429	3.2786118	3.2785546	3.2857938	3.280122	3.2785563	3.2785504
x 7	141.48571	141.46021	141.46006	141.45996	141.46966	141.46005	141.45996	141.45996
f min	−136.97331	−142.65288	−142.70488	−142.71839	−141.13781	−142.43987	−142.71733	−142.71923
f mean	−136.18861	−141.86818	−141.92018	−141.93369	−140.35311	−141.65517	−141.93263	−141.93453
f std	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 Rank	3.4	3.2	5.1	5.2	5.8	5.2	3.9	3.4

Fig. 3

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 f min , f mean and f std are derived in Table 2. Using the simulation-based results, it is noticed that the f min values for DADE, QGDA, MHDA, BMDA, INMDA, HDA, CDA and DA are −136.97331, −142.65288, −142.70488, −142.71839, −141.13781, 142.43987, −142.71733 and −142.71923 respectively. It is revealed that the DA result is − 4.195 %, − 0.047 %, − 0.010 %, − 0.001 %, − 1.120 %, − 0.196 % and − 0.001 % lower (better) than that derived using DADE, QGDA, MHDA, BMDA, INMDA, HDA and CDA respectively. It can also be observed from Table 2 that the applications of DADE, QGDA, MHDA, BMDA, INMDA, HDA, CDA and DA result in the corresponding f mean values as − 136.18861, − 141.86818, − 141.92018, − 141.93369, − 140.35311, 141.65517, −141.93263 and −141.93453 respectively, without violating any of the constraints. Thus, the benefit achieved for DA is − 4.219 %, − 0.047 %, − 0.010 %, − 0.001 %, − 1.127 %, − 0.197 % and − 0.001 % as compared to that obtained from DADE, QGDA, MHDA, BMDA, INMDA, HDA and CDA respectively. On the other hand, the f std value for QGDA is estimated as 0.000823, which is lower by 84.45 %, 98.347 %, 98.176 %, 66.707 %, 92.111 %, 60.285 % and 97.497 % than that derived for DADE, QGDA, MHDA, BMDA, INMDA, CDA and DA respectively. For this example, the Friedman’s ranks for DADE, QGDA, MHDA, BMDA, INMDA, HDA, CDA and DA are noticed to be 3.4, 3.2, 5.1, 5.2, 5.8, 5.2, 3.9 and 3.4 respectively. Using the Friedman’s rank test at 95 % level of significance, the DA variants under consideration can be sorted as QGDA > DA > DADE > CDA > MHDA > BMDA > HDA > INMDA. It can be revealed that the average run time for DADE is 41.825%, 56.280%, 57.418%, 57.269%, 32.801%, 57.816% and 28.280% faster than QGDA, MHDA, BMDA, INMDA, HDA, CDA and DA respectively. Thus, with respect to computational burden, the eight DA variants can be ranked as DADE > DA > HDA > QGDA > MHDA > INMDA > BMDA > CDA. In Fig. 3, the corresponding convergence curves of the considered DA variants are depicted, which reveal that DA ranks first, followed by DADE, QGDA, HDA, MMDA, CDA, BMDA and INMDA, with respect to rate of convergence.

4.3 Case Study 3: Reactor Network Design (RND)

The RND problem (Ryoo and Sahinidis, 1995) deals with maximization of the concentration of a certain product. It consists of six variables, one inequality and four equality constraints. Mathematically, this minimization type RND problem can be stated as shown below: (19) f ( x ¯ ) = x 4 , h 1 ( x ¯ ) = k 1 x 2 x 5 + x 1 − 1 = 0 , h 2 ( x ¯ ) = k 3 x 3 x 5 + x 1 + x 3 − 1 = 0 , h 3 ( x ¯ ) = k 2 x 2 x 6 − x 1 − x 2 = 0 , h 4 ( x ¯ ) = k 4 x 4 x 6 + x 2 − x 1 + x 4 − x 3 = 0 , g 1 ( x ¯ ) = x 5 0.5 + x 6 0.5 ⩽ 4 , 0 ⩽ x 1 , x 2 , x 3 , x 4 ⩽ 1 , 0.00001 ⩽ x 5 , x 6 ⩽ 16 , where k 3 = 0.0391908, k 4 = 0.9 k 3 , k 1 = 0.09755988 and k 2 = 0.99 k 1 .

Table 3

Simulation results of the RND problem.

	DADE	QGDA	MHDA	BMDA	INMDA	HDA	CDA	DA
x 1	0.9999763	0.9999369	0.3944072	0.3919993	0.9945312	0.9999841	0.3940459	0.9976073
x 2	0.4354706	0.4665822	0.3943078	0.3918984	0.4360605	0.398256	0.3939394	0.4420572
x 3	2.495 E − 09	8.385 E − 09	0.3746106	0.3746492	0.0054414	0.0001139	0.374615	0.0024281
x 4	0.3832139	0.3763675	0.3748098	0.3748497	0.384213	0.3879295	0.3748192	0.3825421
x 5	0.0025721	0.0006354	15.739914	15.899662	0.1285526	0.002037	15.764036	0.0540139
x 6	13.419795	11.83318	1.037 E − 05	2.24 E − 05	13.26011	15.640824	0.0001726	13.009488
f min	−0.383214	−0.376367	−0.374810	−0.374850	−0.384213	−0.387929	−0.374819	−0.382542
f mean	−0.2162199	−0.1976946	−0.1846461	−0.0677247	−0.0037783	−0.1555713	−0.1157014	−0.292938
f std	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 Rank	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. f min , f mean and f std ) of the DA variants under consideration, as provided in Table 3. It can be unveiled from this table that the f min value of HDA is respectively 1.231 %, − 3.072 %, − 3.500 %, − 3.489 %, − 0.967 %, 0.000 %, − 3.498 % and − 1.408 % lower (better) than that obtained using DADE, QGDA, MHDA, BMDA, INMDA, CDA and DA respectively. Similarly, with respect to f mean values, the resulting benefit for DA is 0.0767, 0.0952, 0.1083, 0.2252, 0.2892, 0.1374 and 0.1772 as compared to that derived from DADE, QGDA, MHDA, BMDA, INMDA, HDA and CDA respectively. On the other hand, the f std value for INMDA is estimated as 0.011053, which is 93.439%, 93.602%, 93.967%, 90.739%, 93.78%, 93.603% and 92.1252% lower than that obtained from DADE, QGDA, MHDA, BMDA, HDA, CDA and DA respectively. The corresponding Friedman’s ranks for DADE, QGDA, MHDA, BMDA, INMDA, HDA, CDA and DA are 4, 3.7, 4.3, 4.5, 7.4, 4.8, 4.8 and 2.5 respectively. Thus, based on the Friedman’s rank test at 95 % significance level, the considered DA variants can be sorted as DA > QGDA > DADE > MHDA > BMDA > HDA > CDA > INMDA. However, in terms of computational time, these eight algorithms can be ranked as DADE > INMDA > QGDA > HDA > CDA > DA > MHDA > BMDA. Figure 4 exhibits the convergence curves of the DA variants for this problem. Thus, based on this figure, it can be concluded that HDA has a superior convergence advantage, helping in searching out a better solution with a faster speed, followed by DADE, MHDA, INMDA, CDA, QGDA, BMDA and DA.

Fig. 4

Convergence curve for the RND problem.

4.4 Case Study 4: Haverly’s Pooling Problem (HPP)

This HPP problem (Floudas and Pardalos, 1990) is of maximization type, containing nine variables, two inequality and four equality constraints. Mathematically, the HPP problem can be defined as below: (20) f ( x ¯ ) = 9 x 1 + 15 x 2 − 6 x 3 − 16 x 4 − 10 ( x 5 + x 6 ) , h 1 ( x ¯ ) = x 7 + x 8 − x 4 − x 3 = 0 , h 2 ( x ¯ ) = x 1 − x 5 − x 7 = 0 , h 3 ( x ¯ ) = x 2 − x 6 − x 8 = 0 , h 4 ( x ¯ ) = x 7 x 9 + x 8 x 9 − 3 x 3 − x 4 = 0 , g 1 ( x ¯ ) = x 7 x 9 + 2 x 5 − 2.5 x 1 ⩽ 0 , g 2 ( x ¯ ) = x 8 x 9 + 2 x 6 − 1.5 x 2 ⩽ 0 , 0 ⩽ x 1 , x 3 , x 4 , x 5 , x 6 , x 8 ⩽ 100 , 0 ⩽ x 2 , x 7 , x 9 ⩽ 200. This maximization problem (converted first to minimization type) is now solved using the eight DA variants along with determination of the optimal values of the control variables, and objective functions (with respect to f min , f mean and f std ), as shown in Table 4. Based on the derived results, it can be noticed that with respect to f min value, the MHDA result is − 0.0002 %, − 0.00002 %, − 0.0022 %, − 0.6684 %, − 0.0864 %, − 0.0013 % and − 1.3445 % lower than that obtained with DADE, QGDA, BMDA, INMDA, HDA, CDA and DA respectively. Similarly, with respect to f mean value, the resulting benefit for MHDA is − 0.00020 %, − 0.00002 %, − 0.00224 %, − 0.67006 %, − 0.08664 %, − 0.00129 % and − 1.34773 % as compared to that obtained using DADE, QGDA, BMDA, INMDA, HDA, CDA and DA respectively. Based on the simulation-based results, the f std value for MHDA is estimated as 0.261598, which is 17.52 %, 17.78%, 1.05%, 32.74%, 52.65%, 34.2% and 2.11% lower than that derived using DADE, QGDA, BMDA, INMDA, HDA, CDA and DA respectively. Table 4 also shows the results of the Friedman’s rank test at 95% significance level, which lead to the ranking of the considered DA variants as QGDA > DA > DADE > CDA > BMDA > INMDA > MHDA > HDA. However, with respect to computational burden, these algorithms can be sorted as DADE > QGDA > BMDA > HDA > CDA > INMDA > DA > MHDA. Figure 5 shows the corresponding convergence diagram of all the eight DA variants for this problem. Based on this figure, the ranking of the algorithms is noted as DADE > MHDA > DA > QGDA > HDA > CDA > INMDA > BMDA.

Table 4

Simulation results of the HPP problem.

	DADE	QGDA	MHDA	BMDA	INMDA	HDA	CDA	DA
x 1	0.0001505	9.983 E − 05	0.0001034	1.127 E − 05	0.9561836	0.0181292	0.0001372	1.9123671
x 2	199.99996	199.99997	199.99999	199.99927	199.19222	199.93562	199.99961	198.38445
x 3	9.63 E − 05	0.0001411	0.0001513	9.324 E − 06	0	0	2.284 E − 06	0
x 4	99.999692	99.999637	99.999648	99.999939	99.609244	99.999977	99.999819	99.218489
x 5	5.243 E − 05	1.227 E − 06	3.414 E − 06	2.773 E − 06	0.9438356	0.0176439	0.000137	1.8876711
x 6	99.999991	99.999994	99.99999	99.999345	99.595327	99.936127	99.999653	99.190654
x 7	8.313 E − 06	1.504 E − 06	5.335 E − 07	1.794 E − 05	0.012348	0.0004854	5.582 E − 07	0.0246959
x 8	99.999875	99.999876	99.999899	99.999928	99.596896	99.999491	99.999921	99.193792
x 9	1.0000004	1.0000009	1.000001	1.0000002	0.9999999	1	0.9999992	0.9999999
f min	−400.00475	−400.00546	−400.00555	−399.99661	−397.34948	−399.66011	−400.00041	−394.69895
f mean	−399.04995	−399.05066	−399.05075	−399.04181	−396.39468	−398.70531	−399.04561	−393.74415
f std	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 Rank	4.10	3.77	5.00	4.77	4.93	5.27	4.33	3.83

Fig. 5

Convergence curve for the HPP problem.

5 Fuzzy MARCOS-Based Ranking of the DA Variants

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.

Table 5

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

Table 6

Fuzzy scale considered in this paper.

Linguistic term for criteria importance	Symbol	Triangular fuzzy number
Extremely Poor	EP	( 1 , 1 , 1 )
Very Poor	VP	( 1 , 1 , 3 )
Poor	P	( 1 , 3 , 3 )
Medium Poor	MP	( 3 , 3 , 5 )
Medium	M	( 3 , 5 , 5 )
Medium Good	MG	( 5 , 5 , 7 )
Good	G	( 5 , 7 , 7 )
Very Good	VG	( 7 , 7 , 9 )
Extremely Good	EG	( 7 , 9 , 9 )

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.

Table 7

Importance assigned to each criterion by the experts.

Decision maker	Linguistic term			Triangular fuzzy number
Decision maker	T	F	C	T	F	C
Expert 1	VG	EG	G	( 7 , 7 , 9 )	( 7 , 9 , 9 )	( 5 , 7 , 7 )
Expert 2	MG	G	VP	( 5 , 5 , 7 )	( 5 , 5 , 7 )	( 1 , 1 , 3 )
Expert 3	P	MG	M	( 1 , 3 , 3 )	( 5 , 5 , 7 )	( 3 , 5 , 5 )
Expert 4	MP	G	MG	( 3 , 3 , 5 )	( 5 , 7 , 7 )	( 5 , 7 , 7 )
Expert 5	M	VG	G	( 3 , 5 , 5 )	( 7 , 7 , 9 )	( 5 , 7 , 7 )

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 × 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: n DADE , T = x i d x i j = 1 1 = 1 ; n DADE , F = 1.25 2.75 = 0.45 ; n DADE , C = 1.5 1.5 = 1 ; n QGDA , T = 1 2.75 = 0.36 ; n QGDA , F = 1.25 1.25 = 1 ; n QGDA , C = 1.5 5.0 = 0.3.

Table 8

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 9 along with the computed values of S ˜ i parameter. Sample calculations for fuzzy weighted normalized decision matrix using equation (7) are shown below: v ˜ D A D E , T = ( n i j × w j l , n i j × w j m , n i j × w j u ) = ( 1 × 3.8 , 1 × 4.6 , 1 × 5.8 ) v ˜ D A D E , T = ( 3.8 , 4.6 , 5.8 ) , v ˜ D A D E , F = ( 0.4545 × 5.8 , 0.4545 × 7 , 0.4545 × 7.8 ) = ( 2.636 , 3.182 , 3.545 ) , v ˜ D A D E , C = ( 1 × 3.8 , 1 × 5 , 1 × 5.8 ) = ( 3.8 , 5 , 5.8 ) , v ˜ Q G D A , T = ( 0.3636 × 3.8 , 0.3636 × 4.6 , 0.3636 × 5.8 ) = ( 1.382 , 1.673 , 2.109 ) , v ˜ Q G D A , F = ( 1 × 5.8 , 1 × 7 , 1 × 7.8 ) = ( 5.8 , 7 , 7.8 ) , v ˜ Q G D A , C = ( 0.3 × 3.8 , 0.3 × 5 , 0.3 × 5.8 ) = ( 1.14 , 1.5 , 1.74 ) .

Similarly, sample calculations for S ˜ i parameter using equation (10) are shown below: S ˜ D A D E = ∑ j = 1 n v ˜ i j = ( 3.8 + 2.636 + 3.8 , 4.6 + 3.182 + 5 , 5.8 + 3.545 + 5.8 ) = ( 10.236 , 12.782 , 15.145 ) , S ˜ Q G D A = ( 1.382 + 5.8 + 1.14 , 1.673 + 7 + 1.5 , 2.109 + 7.8 + 1.74 ) = ( 8.322 , 10.173 , 11.649 ) .

Table 9

Fuzzy-weighted normalized decision matrix.

Alternative	T	F	C	S ˜ i
AI	( 0.543 , 0.657 , 0.827 )	( 1.036 , 1.25 , 1.393 )	( 0.844 , 1.111 , 1.289 )	( 2.423 , 3.018 , 3.510 )
DADE	( 3.800 , 4.600 , 5.800 )	( 2.636 , 3.182 , 3.545 )	( 3.800 , 5.000 , 5.800 )	( 10.236 , 12.782 , 15.145 )
QGDA	( 1.382 , 1.673 , 2.109 )	( 5.800 , 7.000 , 7.800 )	( 1.140 , 1.500 , 1.740 )	( 8.322 , 10.173 , 11.649 )
MHDA	( 0.543 , 0.657 , 0.829 )	( 1.526 , 1.842 , 2.053 )	( 1.425 , 1.875 , 2.175 )	( 3.494 , 4.374 , 5.056 )
BMDA	( 0.661 , 0.800 , 1.009 )	( 1.318 , 1.591 , 1.773 )	( 0.912 , 1.200 , 1.392 )	( 2.891 , 3.591 , 4.173 )
INMDA	( 0.724 , 0.876 , 1.105 )	( 1.036 , 1.250 , 1.393 )	( 0.844 , 1.111 , 1.289 )	( 2.604 , 3.237 , 3.786 )
HDA	( 1.013 , 1.227 , 1.547 )	( 1.036 , 1.250 , 1.393 )	( 1.900 , 2.500 , 2.900 )	( 3.949 , 4.977 , 5.839 )
CDA	( 0.633 , 0.767 , 0.967 )	( 1.706 , 2.059 , 2.294 )	( 1.036 , 1.364 , 1.582 )	( 3.376 , 4.189 , 4.843 )
DA	( 0.844 , 1.022 , 1.289 )	( 4.143 , 5.000 , 5.571 )	( 1.425 , 1.875 , 2.175 )	( 6.412 , 7.897 , 9.035 )
ID	( 3.800 , 4.600 , 5.800 )	( 5.800 , 7.000 , 7.800 )	( 3.800 , 5.000 , 5.800 )	( 13.400 , 16.600 , 19.400 )

Table 10

Utility degree and utility function of the DA variants.

Alternative	Utility degree		Utility function
Alternative	K ˜ i −	K ˜ i +	f ( K ˜ i − )	f ( K ˜ i + )
DADE	( 2.916 , 4.235 , 6.251 )	( 0.528 , 0.770 , 1.130 )	( 0.103 , 0.150 , 0.220 )	( 0.567 , 0.824 , 1.216 )
QGDA	( 2.371 , 3.370 , 4.808 )	( 0.429 , 0.613 , 0.869 )	( 0.083 , 0.119 , 0.169 )	( 0.461 , 0.656 , 0.935 )
MHDA	( 0.995 , 1.449 , 2.087 )	( 0.180 , 0.263 , 0.377 )	( 0.035 , 0.051 , 0.073 )	( 0.194 , 0.282 , 0.406 )
BMDA	( 0.824 , 1.190 , 1.722 )	( 0.149 , 0.216 , 0.311 )	( 0.029 , 0.042 , 0.067 )	( 0.160 , 0.231 , 0.335 )
INMDA	( 0.742 , 1.073 , 1.563 )	( 0.134 , 0.195 , 0.283 )	( 0.026 , 0.038 , 0.055 )	( 0.144 , 0.209 , 0.304 )
HDA	( 1.125 , 1.649 , 2.410 )	( 0.204 , 0.300 , 0.436 )	( 0.040 , 0.058 , 0.085 )	( 0.219 , 0.321 , 0.469 )
CDA	( 0.962 , 1.388 , 1.999 )	( 0.174 , 0.252 , 0.361 )	( 0.034 , 0.049 , 0.070 )	( 0.187 , 0.270 , 0.389 )
DA	( 1.827 , 2.616 , 3.729 )	( 0.330 , 0.476 , 0.674 )	( 0.064 , 0.092 , 0.131 )	( 0.355 , 0.509 , 0.725 )

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 K ˜ i − using equation (8) are shown below: K ˜ DADE − = S ˜ i S ˜ a i = ( s i l s a i u , s i m s a i m s i u s a i l ) = ( 10.236 3.510 , 12.782 3.018 , 15.145 2.423 ) K ˜ DADE − = ( 2.916 , 4.235 , 6.251 ) , K ˜ QGDA − = ( 8.322 3.510 , 10.173 3.018 , 11.649 2.423 ) = ( 2.371 , 3.370 , 4.808 ) . Similarly, sample calculations for K ˜ i + using equation (9) are shown below: K ˜ DADE + = S ˜ i S ˜ i d = ( s i l s i d u , s i m s i d m s i u s i d l ) = ( 10.236 19.4 , 12.782 16.6 , 15.145 13.4 ) K ˜ DADE + = ( 0.528 , 0.770 , 1.130 ) , K ˜ QGDA + = ( 8.322 19.4 , 10.173 16.6 , 11.649 13.4 ) = ( 0.429 , 0.613 , 0.869 ) . The T ˜ i and d f crisp are calculated using equations (11) and (14) respectively: t ˜ DADE = ( t i l , t i m , t i u ) = K ˜ i − ⊕ K ˜ i + = ( k i − l + k i + l , k i − m + k i + m , k i − u + k i + u ) = ( 2.916 + 0.528 , 4.235 + 0.77 , 6.251 + 1.13 ) = ( 3.444 , 5.005 , 7.381 ) , t ˜ QGDA = ( t i l , t i m , t i u ) = K ˜ i − ⊕ K ˜ i + = ( k i − l + k i + l , k i − m + k i + m , k i − u + k i + u ) = ( 2.371 + 0.429 , 3.37 + 0.613 , 4.808 + 0.869 ) = ( 2.799 , 3.983 , 5.677 ) , d f crisp = l + 4 m + u 6 = 3.444 + ( 4 × 5.005 ) + 7.381 6 = 5.14. The f ( K ˜ i + ) and f ( K ˜ i − ) are computed using equation (12) and (13), as shown below: f ( K ˜ DADE + ) = K ˜ i − d f crisp = ( k i − l d f crisp , k i − m d f crisp , k i − u d f crisp ) = ( 2.916 5.14 , 4.235 5.14 , 6.251 5.14 ) f ( K ˜ DADE + ) = K ˜ i − d f crisp = ( k i − l d f crisp , k i − m d f crisp , k i − u d f crisp ) = ( 0.567 , 0.824 , 1.216 ) , f ( K ˜ QGDA + ) = ( 2.371 5.14 , 3.37 5.14 , 4.808 5.14 ) = ( 0.461 , 0.656 , 0.935 ) , f ( K ˜ DADE − ) = K ˜ i + d f crisp = ( k i + l d f crisp , k i + m d f crisp , k i + u d f crisp ) = ( 0.528 5.14 , 0.77 5.14 , 1.13 5.14 ) f ( K ˜ DADE − ) = K ˜ i + d f crisp = ( k i + l d f crisp , k i + m d f crisp , k i + u d f crisp ) = ( 0.103 , 0.150 , 0.220 ) , f ( K ˜ QGDA − ) = ( 0.429 5.14 , 0.613 5.14 , 0.869 5.14 ) = ( 0.083 , 0.119 , 0.169 ) .

Table 11

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

Alternative	K i −	K i +	f ( K i − )	f ( K i + )	( 1 − f ( K i − ) ) f ( K i − )	( 1 − f ( K i + ) ) f ( K i + )	f ( K i )	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 11. The K i − and K i + are computed as follows: K D A D E − = 2.916 + ( 4 × 4.235 ) + 6.251 6 = 4.351 , K Q G D A − = 2.371 + ( 4 × 3.37 ) + 4.808 6 = 3.443 , K D A D E + = 0.528 + ( 4 × 0.77 ) + 1.13 6 = 0.7896 , K Q G D A + = 0.429 + ( 4 × 0.613 ) + 0.869 6 = 0.6249.

The f ( K i − ) and f ( K i + ) are computed as follows: f ( K D A D E − ) = 0.103 + ( 4 × 0.15 ) + 0.22 6 = 0.1536 , f ( K Q G D A − ) = 0.083 + ( 4 × 0.119 ) + 0.169 6 = 0.1216 , f ( K D A D E + ) = 0.567 + ( 4 × 0.824 ) + 1.216 6 = 0.8464 , f ( K D A D E + ) = 0.461 + ( 4 × 0.656 ) + 0.935 6 = 0.6698.

Similarly, the f ( K i ) is computed using equation (16): f ( K D A D E ) = K i + + K i − 1 + 1 − f ( K i + ) f ( K i + ) + 1 − f ( K i − ) f ( K i − ) = 0.7896 + 4.3510 1 + ( 1 − 0.8464 ) 0.8464 + ( 1 − 0.1536 ) 0.1536 = 0.7682 , f ( K Q G D A ) = 0.6249 + 3.4433 1 + ( 1 − 0.6698 ) 0.6698 + ( 1 − 0.1216 ) 0.1216 = 0.4666.

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.

6 Conclusions

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.

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