## 1 Introduction

*et al.*, 1983), particle swarm optimization (PSO) (Kennedy and Eberhart, 1995), grey wolf optimizer (Mirjalili

*et al.*, 2014) etc.

*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.

*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.

*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.

## 2 Methods

### 2.1 Dragonfly Algorithm

### 2.2 Hybrid Dragonfly Algorithm with Differential Evolution

*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

*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

### 2.5 Chaotic Dragonfly Algorithm

*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

### 2.7 Hybrid Nelder-Mead Algorithm and Dragonfly Algorithm

### 2.8 Hybridization of Dragonfly Algorithm and Artificial Bee Colony

### 2.9 Fuzzy MARCOS

*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.

**Step 1:**Formulate the initial decision matrix $(X)$, consisting of

*m*possible choices (alternatives) and

*n*evaluation criteria.

##### (1)

\[ X=\left[\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}{x_{11}}\hspace{1em}& {x_{12}}\hspace{1em}& \dots \hspace{1em}& {x_{1j}}\hspace{1em}& \dots \hspace{1em}& {x_{1n}}\\ {} {x_{21}}\hspace{1em}& {x_{22}}\hspace{1em}& \dots \hspace{1em}& {x_{2j}}\hspace{1em}& \dots \hspace{1em}& {x_{2n}}\\ {} \dots \hspace{1em}& \dots \hspace{1em}& \dots \hspace{1em}& \dots \hspace{1em}& \dots \hspace{1em}& \dots \\ {} {x_{i1}}\hspace{1em}& {x_{i2}}\hspace{1em}& \dots \hspace{1em}& {x_{ij}}\hspace{1em}& \dots \hspace{1em}& {x_{in}}\\ {} \dots \hspace{1em}& \dots \hspace{1em}& \dots \hspace{1em}& \dots \hspace{1em}& \dots \hspace{1em}& \dots \\ {} {x_{m1}}\hspace{1em}& {x_{m2}}\hspace{1em}& \dots \hspace{1em}& {x_{mj}}\hspace{1em}& \dots \hspace{1em}& {x_{mn}}\end{array}\right],\]*i*th alternative against

*j*th criterion.

**Step 2:**Develop the corresponding extended decision matrix $({X^{\prime }})$ while considering the anti-ideal (AI) and ideal (ID) solutions.

*B*is the set of beneficial criteria and

*C*is the set of non-beneficial criteria.

##### (4)

\[ {X^{\prime }}=\left[\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}{x_{ai1}}\hspace{1em}& {x_{ai2}}\hspace{1em}& \dots \hspace{1em}& {x_{aij}}\hspace{1em}& \dots \hspace{1em}& {x_{ain}}\\ {} {x_{11}}\hspace{1em}& {x_{12}}\hspace{1em}& \dots \hspace{1em}& {x_{1j}}\hspace{1em}& \dots \hspace{1em}& {x_{1n}}\\ {} \dots \hspace{1em}& \dots \hspace{1em}& \dots \hspace{1em}& \dots \hspace{1em}& \dots \hspace{1em}& \dots \\ {} {x_{i1}}\hspace{1em}& {x_{i2}}\hspace{1em}& \dots \hspace{1em}& {x_{ij}}\hspace{1em}& \dots \hspace{1em}& {x_{in}}\\ {} {x_{m1}}\hspace{1em}& {x_{m2}}\hspace{1em}& \dots \hspace{1em}& {x_{mj}}\hspace{1em}& \dots \hspace{1em}& {x_{mn}}\\ {} {x_{id1}}\hspace{1em}& {x_{id2}}\hspace{1em}& \dots \hspace{1em}& {x_{idj}}\hspace{1em}& \dots \hspace{1em}& {x_{idn}}\end{array}\right].\]**Step 3:**Normalize the extended decision matrix using Eqs. (5) and (6) depending on the type of the criterion under consideration.

**Step 4:**Develop the weighted normalized fuzzy decision matrix.

##### (7)

\[ {\tilde{v}_{ij}}=\big({v_{ij}^{l}},{v_{ij}^{m}},{v_{ij}^{u}}\big)={n_{ij}}\otimes {\widetilde{w}_{j}}=\big({n_{ij}}\times {w_{j}^{l}},{n_{ij}}\times {w_{j}^{m}},{n_{ij}}\times {w_{j}^{u}}\big),\]*j*th criterion.

**Step 5:**Determine the utility degrees of each alternative using the following expressions:

**Step 6:**Formulate the fuzzy matrix ${\tilde{T}_{i}}$ applying the following expression:

**Step 7:**Evaluate the utility functions for both the ideal and anti-ideal solutions.

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

**Step 9:**Rank the alternatives.

## 3 Problem Description

## 4 Numerical Results on Chemical Process Optimization

### 4.1 Case Study 1: Heat Exchanger Network Design (HEND)

##### (17)

\[\begin{aligned}{}& f(\bar{x})=35{x_{1}^{0.6}}+35{x_{2}^{0.6}},\\ {} & {h_{1}}(\bar{x})=200{x_{1}}{x_{4}}-{x_{3}}=0,\\ {} & {h_{2}}(\bar{x})=200{x_{2}}{x_{6}}-{x_{5}}=0,\\ {} & {h_{3}}(\bar{x})={x_{3}}-10000({x_{7}}-100)=0,\\ {} & {h_{4}}(\bar{x})={x_{5}}-10000(300-{x_{7}})=0,\\ {} & {h_{5}}(\bar{x})={x_{3}}-10000(600-{x_{8}})=0,\\ {} & {h_{6}}(\bar{x})={x_{5}}-10000(900-{x_{9}})=0,\\ {} & {h_{7}}(\bar{x})={x_{4}}\ln ({x_{8}}-100)-{x_{4}}\ln (600-{x_{7}})-{x_{8}}+{x_{7}}+500=0,\\ {} & {h_{8}}(\bar{x})={x_{6}}\ln ({x_{9}}-{x_{7}})-{x_{6}}\ln (600)-{x_{9}}+{x_{7}}+600=0,\\ {} & 0\leqslant {x_{1}}\leqslant 10,\hspace{1em}0\leqslant {x_{2}}\leqslant 200,\hspace{1em}0\leqslant {x_{3}}\leqslant 100,\hspace{1em}0\leqslant {x_{4}}\leqslant 200,\\ {} & 1000\leqslant {x_{5}}\leqslant 2000000,\hspace{1em}0\leqslant {x_{6}}\leqslant 600,\hspace{1em}100\leqslant {x_{7}}\leqslant 600,\hspace{1em}100\leqslant {x_{8}}\leqslant 600,\\ {} & 100\leqslant {x_{9}}\leqslant 900.\end{aligned}\]##### Table 1

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

${x_{1}}$ | 0.052351 | $1.44\mathrm{E}-06$ | $2.92\mathrm{E}-05$ | $4.58\mathrm{E}-07$ | $8.65\mathrm{E}-08$ | $4.61\mathrm{E}-13$ | 0.011093 | $4.58\mathrm{E}-06$ |

${x_{2}}$ | 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 | $1.58\mathrm{E}-05$ | 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 |

${f_{\text{min}}}$ | 190.5047 | 189.305 | 189.3934 | 189.3181 | 189.3138 | 189.3116 | 192.599 | 189.3521 |

${f_{\text{mean}}}$ | 191.4536 | 190.2539 | 190.3423 | 190.267 | 190.2627 | 190.2605 | 193.5479 | 190.301 |

${f_{\text{std}}}$ | 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 ${T_{\text{Rank}}}$ | 4.7 | 3.2 | 4.7 | 4.8 | 4.8 | 5.1 | 4.7 | 4 |

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

*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)

\[\begin{aligned}{}& f(\bar{x})=0.035{x_{1}}{x_{6}}+1.715{x_{1}}+10.0{x_{2}}+4.0565{x_{3}}-0.063{x_{3}}{x_{5}},\\ {} & {g_{1}}(\bar{x})=0.0059553571{x_{6}^{2}}{x_{1}}+0.88392857{x_{3}}-0.1175625{x_{6}}{x_{1}}-{x_{1}}\leqslant 0,\\ {} & {g_{2}}(\bar{x})=1.1088{x_{1}}+0.1303533{x_{1}}{x_{6}}-0.0066033{x_{1}}{x_{6}^{2}}-{x_{3}}\leqslant 0,\\ {} & {g_{3}}(\bar{x})=6.66173269{x_{6}^{2}}-56.596669{x_{4}}+172.39878{x_{5}}-10000-191.20592{x_{6}}\leqslant 0,\\ {} & {g_{4}}(\bar{x})=1.08702{x_{6}}-0.3762{x_{6}^{2}}+0.32175{x_{4}}+56.85075-{x_{5}}\leqslant 0,\\ {} & {g_{5}}(\bar{x})=0.006198{x_{7}}{x_{4}}{x_{3}}+2462.3121{x_{2}}-25.125634{x_{2}}{x_{4}}-{x_{3}}{x_{4}}\leqslant 0,\\ {} & {g_{6}}(\bar{x})=161.18996{x_{3}}{x_{4}}+5000.0{x_{2}}{x_{4}}-489510.0{x_{2}}-{x_{3}}{x_{4}}{x_{7}}\leqslant 0,\\ {} & {g_{7}}(\bar{x})=0.33{x_{7}}{x_{4}}+44.333333\leqslant 0,\\ {} & {g_{8}}(\bar{x})=0.022556{x_{5}}-1.0{x_{2}}-0.007595{x_{7}}\leqslant 0,\\ {} & {g_{9}}(\bar{x})=0.00061{x_{3}}-1.0-0.0005{x_{1}}\leqslant 0,\\ {} & {g_{10}}(\bar{x})=0.819672{x_{1}}-{x_{3}}+0.819672\leqslant 0,\\ {} & {g_{11}}(\bar{x})=24500.0{x_{2}}-250.0.0{x_{2}}{x_{4}}-{x_{3}}{x_{4}}\leqslant 0,\\ {} & {g_{12}}(\bar{x})=1020.4082{x_{2}}{x_{4}}+1.2244898{x_{3}}{x_{4}}-100000{x_{2}}\leqslant 0,\\ {} & {g_{13}}(\bar{x})=6.25{x_{1}}{x_{6}}+6.25{x_{1}}-7.625{x_{3}}-100000\leqslant 0,\\ {} & {g_{14}}(\bar{x})=1.22{x_{3}}-{x_{1}}{x_{6}}-{x_{1}}\leqslant 0,\\ {} & 10000\leqslant {x_{1}}\leqslant 2000,\hspace{1em}0\leqslant {x_{2}}\leqslant 100,\hspace{1em}2000\leqslant {x_{3}}\leqslant 4000,\hspace{1em}0\leqslant {x_{4}}\leqslant 100,\\ {} & 0\leqslant {x_{5}}\leqslant 100,\hspace{1em}0\leqslant {x_{6}}\leqslant 20,\hspace{1em}0\leqslant {x_{7}}\leqslant 200.\end{aligned}\]##### Table 2

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 |

${f_{\text{min}}}$ | −136.97331 | −142.65288 | −142.70488 | −142.71839 | −141.13781 | −142.43987 | −142.71733 | −142.71923 |

${f_{\text{mean}}}$ | −136.18861 | −141.86818 | −141.92018 | −141.93369 | −140.35311 | −141.65517 | −141.93263 | −141.93453 |

${f_{\text{std}}}$ | 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 ${_{\text{Rank}}}$ | 3.4 | 3.2 | 5.1 | 5.2 | 5.8 | 5.2 | 3.9 | 3.4 |

### 4.3 Case Study 3: Reactor Network Design (RND)

##### (19)

\[\begin{aligned}{}& f(\bar{x})={x_{4}},\\ {} & {h_{1}}(\bar{x})={k_{1}}{x_{2}}{x_{5}}+{x_{1}}-1=0,\\ {} & {h_{2}}(\bar{x})={k_{3}}{x_{3}}{x_{5}}+{x_{1}}+{x_{3}}-1=0,\\ {} & {h_{3}}(\bar{x})={k_{2}}{x_{2}}{x_{6}}-{x_{1}}-{x_{2}}=0,\\ {} & {h_{4}}(\bar{x})={k_{4}}{x_{4}}{x_{6}}+{x_{2}}-{x_{1}}+{x_{4}}-{x_{3}}=0,\\ {} & {g_{1}}(\bar{x})={x_{5}^{0.5}}+{x_{6}^{0.5}}\leqslant 4,\\ {} & 0\leqslant {x_{1}},{x_{2}},{x_{3}},{x_{4}}\leqslant 1,\hspace{1em}0.00001\leqslant {x_{5}},{x_{6}}\leqslant 16,\end{aligned}\]##### Table 3

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 |

${x_{3}}$ | $2.495\mathrm{E}-09$ | $8.385\mathrm{E}-09$ | 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 | $1.037\mathrm{E}-05$ | $2.24\mathrm{E}-05$ | 13.26011 | 15.640824 | 0.0001726 | 13.009488 |

${f_{\text{min}}}$ | −0.383214 | −0.376367 | −0.374810 | −0.374850 | −0.384213 | −0.387929 | −0.374819 | −0.382542 |

${f_{\text{mean}}}$ | −0.2162199 | −0.1976946 | −0.1846461 | −0.0677247 | −0.0037783 | −0.1555713 | −0.1157014 | −0.292938 |

${f_{\text{std}}}$ | 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 ${_{\text{Rank}}}$ | 4 | 3.7 | 4.3 | 4.5 | 7.4 | 4.8 | 4.8 | 2.5 |

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

##### (20)

\[\begin{aligned}{}& f(\bar{x})=9{x_{1}}+15{x_{2}}-6{x_{3}}-16{x_{4}}-10({x_{5}}+{x_{6}}),\\ {} & {h_{1}}(\bar{x})={x_{7}}+{x_{8}}-{x_{4}}-{x_{3}}=0,\\ {} & {h_{2}}(\bar{x})={x_{1}}-{x_{5}}-{x_{7}}=0,\\ {} & {h_{3}}(\bar{x})={x_{2}}-{x_{6}}-{x_{8}}=0,\\ {} & {h_{4}}(\bar{x})={x_{7}}{x_{9}}+{x_{8}}{x_{9}}-3{x_{3}}-{x_{4}}=0,\\ {} & {g_{1}}(\bar{x})={x_{7}}{x_{9}}+2{x_{5}}-2.5{x_{1}}\leqslant 0,\\ {} & {g_{2}}(\bar{x})={x_{8}}{x_{9}}+2{x_{6}}-1.5{x_{2}}\leqslant 0,\\ {} & 0\leqslant {x_{1}},{x_{3}},{x_{4}},{x_{5}},{x_{6}},{x_{8}}\leqslant 100,\hspace{1em}0\leqslant {x_{2}},{x_{7}},{x_{9}}\leqslant 200.\end{aligned}\]##### Table 4

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

${x_{1}}$ | 0.0001505 | $9.983\mathrm{E}-05$ | 0.0001034 | $1.127\mathrm{E}-05$ | 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 |

${x_{3}}$ | $9.63\mathrm{E}-05$ | 0.0001411 | 0.0001513 | $9.324\mathrm{E}-06$ | 0 | 0 | $2.284\mathrm{E}-06$ | 0 |

${x_{4}}$ | 99.999692 | 99.999637 | 99.999648 | 99.999939 | 99.609244 | 99.999977 | 99.999819 | 99.218489 |

${x_{5}}$ | $5.243\mathrm{E}-05$ | $1.227\mathrm{E}-06$ | $3.414\mathrm{E}-06$ | $2.773\mathrm{E}-06$ | 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 |

${x_{7}}$ | $8.313\mathrm{E}-06$ | $1.504\mathrm{E}-06$ | $5.335\mathrm{E}-07$ | $1.794\mathrm{E}-05$ | 0.012348 | 0.0004854 | $5.582\mathrm{E}-07$ | 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 |

${f_{\text{min}}}$ | −400.00475 | −400.00546 | −400.00555 | −399.99661 | −397.34948 | −399.66011 | −400.00041 | −394.69895 |

${f_{\text{mean}}}$ | −399.04995 | −399.05066 | −399.05075 | −399.04181 | −396.39468 | −398.70531 | −399.04561 | −393.74415 |

${f_{\text{std}}}$ | 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 ${_{\text{Rank}}}$ | 4.10 | 3.77 | 5.00 | 4.77 | 4.93 | 5.27 | 4.33 | 3.83 |

## 5 Fuzzy MARCOS-Based Ranking of the DA Variants

##### Table 5

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

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 | $(7,9,9)$ |

##### Table 7

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

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 | $(3,5,5)$ | $(7,7,9)$ | $(5,7,7)$ |

##### Table 8

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 |

##### Table 9

Alternative | T | F | C | ${\tilde{S}_{i}}$ |

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

Alternative | Utility degree | Utility function | ||

${\widetilde{K}_{i}^{-}}$ | ${\widetilde{K}_{i}^{+}}$ | $f({\widetilde{K}_{i}^{-}})$ | $f({\widetilde{K}_{i}^{+}})$ | |

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 | $(1.827,2.616,3.729)$ | $(0.330,0.476,0.674)$ | $(0.064,0.092,0.131)$ | $(0.355,0.509,0.725)$ |

##### Table 11

Alternative | ${K_{i}^{-}}$ | ${K_{i}^{+}}$ | $f({K_{i}^{-}})$ | $f({K_{i}^{+}})$ | $\frac{(1-f({K_{i}^{-}}))}{f({K_{i}^{-}})}$ | $\frac{(1-f({K_{i}^{+}}))}{f({K_{i}^{+}})}$ | $f({K_{i}})$ | 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 |