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ArticlesA Hybrid Fuzzy-Crow Optimizer for Unconstrained and Constrained Engineering Design Problems
• Mona Gamal Gafar1,2, Ragab A. El-Sehiemy3, and Shahenda Sarhan4,*

Human-centric Computing and Information Sciences volume 12, Article number: 14 (2022)
https://doi.org/10.22967/HCIS.2022.12.014

Abstract

Constrained search space is considered a problematic field for engineers. It requires finding solutions for problems satisfying a number of predefined constraints while solving uncertain and ambiguous situations that realistic problems exhibit. As all feasible solutions should have degrees of truth to accommodate real design problems. The fuzzy set theory is able to handle uncertainty issues in real-time problems. In this paper, we introduce a hybrid fuzzy-crow framework for providing an optimal design of constrained and unconstrained engineering problems. The fuzzy-crow framework works on an initial population of fuzzy numbers for problem solutions. It benefits Zadeh extension principle for calculating problem fitness functions and constraints in addition to their membership degrees. The main features of the proposed framework are merging the merits of fuzzy logic and crow search optimization. The fuzzified objective and constraints are incorporated to obtain a fine-tuned solution at fast convergence of the non-dominated solutions. The proposed framework was evaluated based on statistical and convergence analysis using 10 benchmark test functions and five constrained engineering problems against some of the state of art. The results indicated the superiority of the proposed framework over the state of art in finding fine-tuned non-dominated optimized solutions in fuzzy search space.

Keywords

Crow Search Optimization Algorithm, Fuzzy System, Engineering Design Problem, Constrained and Unconstrained Optimization Problems

Introduction

Optimization of design problems is a methodology for optimizing the mathematical formulation of a design problem to support finding the optimal design based on problem variables, objectives, constraints and feasibility. In practical design problems, there are many variables and constraints that significantly influence the objective functions, while the solving algorithm could be trapped in local optimal [1, 2]. Design optimization problems could be classified based on constraints on the problem variables into unconstrained and constrained.
In constrained optimization problems, the objective function is maximized or minimized based on some constraints on problem variables. Some of the well-known constrained optimization problems are the tension/compression spring [3, 4], welded beam [5, 6], gear train [7, 8], and pressure vessel [9, 10]. On the other hand, in unconstrained optimization problems, the objective function is minimized based on real variables with no restrictions on their values or based on the feasibility. Minimization of functions of one variable and finding the maximum or minimum of a differentiable function of several variables are examples of unconstrained optimization.
In the last decade, the field of optimization rapidly and continuously developed where various theoretical, algorithmic, and computational proposals were integrated to handle different engineering and management problems. Optimization algorithms can be mainly classified into deterministic and stochastic. Deterministic optimization algorithms include linear programming, nonlinear programming, & mixed-integer nonlinear programming, and have several merits of the analytical properties that permit to produce a series of accepted solutions which lead to converging on a global optimal solution. Deterministic methods introduce general tools for optimization problems solving to reach a global or nearly global optimum solution [11, 12].
Stochastic algorithms are classified into Heuristic and meta-heuristic algorithms which represent the most used optimization algorithms. Heuristic algorithms designate a mathematical procedure for optimally determining a solution by iteratively trying to ameliorate an elect solution based on a given measure of quality, namely being problem-dependent. Meta-heuristic algorithms are problem-independent techniques that can be applied to multi-objective, multiple solution and nonlinear formulations real-world problems seeking to find high-quality solutions. Therefore, we have selected crow as one of the promising meta-heuristic stochastic algorithms to consider and modify to test its performance in some constrained and unconstrained design problems.
Different optimization algorithms were introduced to handle design optimization problems and for various engineering problems, as particle swarm optimization (PSO) [7], cuckoo search (CS) [1315], firefly algorithm [16, 17], ant colony optimization (ACO) [1820], Grey Wolf optimizer (GWO) [9, 21, 22], hybrid differential evolution (DE) algorithms [23], real and binary coded immunological algorithms [24], Jaya optimization algorithm[25], Elitist teaching–learning-based optimization algorithm (Elitist TLBO) [26], derivative-free [27], crow search algorithm (CSA) [2830], hunger games search (HGS)[31], slime mould algorithm (SMA)[32], monarch butterfly optimization (MBO)[33], Harris hawks optimization (HHO) [32], Grey Wolf optimization(GWO) [34], equilibrium optimizer (EO)[35], opposition-based moth swarm algorithm (OBMSA)[36], improved sine cosine algorithm with crossover scheme for global optimization (ISCA) [37], a better exploration strategy in Grey Wolf optimizer (IGWO)[38], Runge-Kutta method (RUN) [39], and colony predation algorithm (CPA) [40].
On the other hand, the traditional design problems depend on an absolute truth in the solution. The situation is either true or false which is not applicable to real world problems. All feasible solutions should have degrees of truth to accommodate real design problems. The fuzzy set theory is able to handle the uncertainty issues in real-time problems [41, 42]. The fuzzy variables are defined by a membership function that measures the degrees of truth of the variable values.
Fuzzy constrained design problems are defined by fuzzy variables and subjected to certain restrictions that can be also fuzzified. Then it is possible to reformulate the design problems to reflect fuzzy and ambiguous concepts, adding realistic dimension to the solution optimization process. Hence, real decision variables hold values with a membership degree. Under this approach, the pair composed by a value and membership degree defines the decision attribute. Furthermore, fuzzy objective function is calculated according to fuzzy methodologies, and the output is the fuzzified solution and membership degrees. Unfortunately, the end user is not interested in understanding fuzzy concepts. Therefore, the defuzzification process produces the scalar value of the objective function or problem solution. Fuzzy applications to different engineering sciences are reported in [4346].
In this paper, the authors propose fuzzified crow optimization algorithm (FCOA), an alternative optimization approach able to overcome the complexities presented in engineering problems, for solving engineering design problems in an uncertain search space. FCOA is inspired by CSA [28, 29]. Crow initial algorithm is improved by the Zadeh extension principle [47, 48] to calculate the fitness function on fuzzy control variables. The extension principle is a methodology for mapping fuzzy variables, fuzzy constraints, and the fuzzy objective function on the search universe and calculating the corresponding membership degrees.
FCOA provides many advantages over the state of art as the enhancement of accuracy and capabilities to handle the constraints presented in real problems. These engineering multi-constraints design problems proved the FCOA ability to find optimal solutions for the objective functions. We can summarize the contribution of this paper as follows.

Proposing an alternative optimization algorithm that merge the ability of fuzzy logic and crow search optimization algorithm able to overcome the complexities presented in engineering problems.

Enhancing the accuracy of handling the constraints in real problems.

Comparing FCOA with recent optimization search algorithms using 10 unconstrained test benchmarking and five engineering design problems.

The rest of the paper is organized as follows. The related work, theories and background are covered in Section 2. Section 3 focuses on the FCOA flowchart, pseudocode, and working mechanism. The fuzzy engineering design problems of experimental results and comparisons with the state of art are explained in Section 4. Then, the conclusion and findings of the research are mentioned in Section 5.

Related Work

Meta-Heuristic Swarm Intelligence Algorithms
Recently, meta-heuristic swarm intelligence algorithms have witnessed a huge continuous development. Many studies have proposed solutions to design optimization problems based on meta-heuristic swarm intelligence algorithms. In [49], the authors introduced a PSO based algorithm called particle swarm optimization mixed variables (PSOMV) for solving mixed-variable optimization problems through a mixed-variable encoding scheme that can simultaneously handle continuous and discrete decision variables. In [15], the authors believe that using CS with the bat algorithm amended with the Sugeno fuzzy inertia weight for updating velocity equation and boosting flexibility and diversity of the bat population will improve the bat algorithm capability to solve large scale constrained engineering design optimization problems and results stability.
In [50], the authors introduced a solution to the vehicle bracket shape optimization design problem based on a new meta-heuristic algorithm called seagull optimization algorithm (SOA) with finite element analysis to evaluate the function evaluations and Kriging model to estimate it. Literature [16] was shown a hybrid firefly algorithm with new grouping attraction inspired by the PSO position update methodology for solving IEEE CEC 2017 constrained optimization problems. In this hybrid, the fittest firefly contributes to deciding which fireflies join the group based on their fitness value. The authors indicated that the new grouping attraction had effectively reduced the number of attractions and fitness comparisons in the firefly groups.
In [18], the authors introduced two variants of ant colony algorithm improved ant colony optimization (IACO) and local IACO (LIACO). IACO tries to balance exploitation and exploration though the selection between the Brownian motion and Lévy flights where the LIACO enhanced the IACO performance by using local search in the colony. The proposed algorithms were evaluated using 22 real-world engineering optimization problems of the IEEE CEC 2011 proving superiority over that state of art.
The IEEE CEC 2014 and CEC 2017 were used in [37] to evaluate an enhanced version of sine cosine algorithm (SCA) called improved sine cosine algorithm (ISCA). This enhancement was proposed to decrease the diversity of SCA search equations by amalgamating self-learning and global search mechanisms with the crossover exploitation skills. The results indicated the superiority of the proposed algorithm over the state of art.
In [51], the authors tried to balance exploitation and exploration, dynamically rectify the convergence speed, handle the loss of the population diversity, and enhance the diversity of solutions of the grey wolf optimizer by proposing a multi-objective, multiple search strategies grey wolf optimizer (MMOGWO). To achieve these goals the proposed algorithm used adaptive chaotic mutation, boundary mutation, and elitism search strategies for finding a solution of multi-objective optimization problems.
Finally, opposition-based learning (OBL) has played a major role in enhancing swarm intelligence algorithms performance and solving the low convergence rate problem while solving engineering problems. In [38], the OBL was used to balance the GWO exploitation and exploration capabilities, and solve its low convergence rate problem. The proposed algorithm was evaluated using 23 standard benchmark problems, proving efficiency over GWO. Also, in [36] OBL was used to allow Moth swarm algorithm (MSA) to avoid being stuck in sub-optimal solutions that minimize its convergence speed. The OBMSA was evaluated using three constrained engineering problems proving superiority over MSA.

Crow Search Algorithm
Search space contains many feasible solutions to the problems under consideration. The optimal solution solves the objective function of the problem respecting the supplied constraints. CSA [28, 29] is
a search algorithm inspired by the crows as intelligent birds in looking for and hiding their food. The CSA is a meta-heuristic intelligent algorithm which searches for optimal solutions in a d-dimensional workspace. The algorithm initializes a number (N) of crows which lives in flocks and memorizes their food hiding places. Crow inew position is determined by their flight length according to an awareness probability (AP). Crow $i$ follow crow $j$ for theft by an expected AP greater than AP, as follows:

$X_{i,iter+1}=X_{i,iter}+ FL*F_{rand}*(mem_{j,iter}- X_{i,iter })$(1)

where $X_{i,iter}$ is the crow i current position, FL is the flight length, $F_{rand}$ is a random constant of uniform distribution in the interval [0,1], and $mem_{j,iter}$ is the j crow’s memory of hiding position.
Hence crows are very suspicious and protect their food hiding places. If the awareness probability is less than AP, then the crow’s next position is determined by Equation (2) as follows:

$X_{i,iter+1}=l-(l-u)*F_{rand}$(2)

where $l$ and $u$ are the lower and the upper bounds of the variable X,respectively, $F_{rand}$ is a random constant of uniform distribution in the interval [0,1].
The main objective of CSA is to maximize or minimize (based on the problem specification) the objective function. The fitness function of each crow is the objective function and the corresponding individual values in the memory location are the control variables. The convergence of the algorithm is related to finding the best solution, which is determined after completing a predetermined number of iterations. The number of iterations, flight length, and awareness probability are changed based on the problem context. Since the algorithm is based on random regulations, it could produce different solutions at every run. This problem is solved by running the algorithm several times and selecting the most-fit solution based on constraints supplied.
The CSA was previously used in different studies to handle optimization problems as in [52], and three search strategies—neighborhood based local search (NLS), non-neighborhood based global search (NGS) and wandering around based search (WAS)—were introduced to enhance the CSA search strategy. The proposed conscious neighborhood-based crow search algorithm (CCSA) using these search strategies tried to achieve a balance between local and global search to enhance crows’ movements in different search spaces. In [29], the authors introduced an improved spiral crow search algorithm (ISCSA) to balance exploitation and exploration of the CSA by updating the position equation using an optimal guidance position. Whereas, adding the Gaussian variation and random perturbation protects the proposed algorithm from being tapped in local optimization. Then ISCSA was tested using 23 benchmarks and four engineering design problems.
In [53], the CSA search strategy was modified by the space transform search (STS) to balance CSA exploration with less complexity while adjusting the convergence speed. STS-CSA was tested using IEEE CEC 2017 functions and tree engineering optimization problems. Finally, in [54] the CSA exploitation was enhanced by updating search equation based on the global best solution found inspired through the artificial bee colony algorithm search strategy. With all these updates, CSA still has shortcomings of being trapped in local optima, less explorative capabilities, and unstable convergence rate especially when handling complexities and constraints of engineering design problems. Therefore, in the next sub-section, we will proceed with clarifying the Zadeh extension principle and its role in enhancing the CSA fitness function.

Fuzzy Arithmetic’s & Zadeh Extension Principle
Fuzzy system [41, 42, 47] handles uncertainty and membership degree of truth in ambiguous workspace. Fuzzy objective functions depend on fuzzy variables in their representation constraints and intended solutions. The solution should be represented by a degree of truth or membership function. Zadeh extension principle [55, 56] introduces a mechanism for solving fuzzy equations. The steps of the Zadeh extension algorithm are as follows:
• 1.Set control fuzzy variables membership function (shown in Fig. 1).
• 2.Declare the Cartesian product of the control variables and corresponding membership values.
• 3.Compute the fuzzy objective function Zi based on the predefined Cartesian product.
• 4.Compute the minimal membership degree of Zi for the control fuzzy variables.
• 5.Defuzzify the fuzzy objective function (shown in Fig. 2) using the centroid function to get the scalar solution Z^* of the problem using Equation (3) as follows

$Z^*=\frac{∫μ_c (z)∙zdz}{∫μ_c(z) dz}$(3)

Fig. 1. Membership function for the control variable.

Fig. 2. Fitness defuzzification.

Proposed Fuzzy Based Crow Search Framework

The FCOA is an integrated hybrid framework. FCOA integrates the CSA with the Zadeh extension principle [55, 56] to find optimal solutions in a fuzzy search space. CSA is an evolutionary search mechanism proven to solve global optimization problems and utilizes a fitness function to reach the optimal solution. Unfortunately, CSA concentrates on regular and traditional search spaces. Due to vagueness and constraints of the fuzzy universe, finding feasible solution is a highly tricky problem. Therefore, the Zadeh extension principle is used to indicate the non-dominated solutions for MOPs in a fuzzy universe. FCOA profits the possibility and uncertainty of fuzzy numbers, and the universe of search space is defined by a set of fuzzy decision numbers which indicates the solution of the constrained objective function, thereby finding the suitable combination of fuzzy numbers that solve the problem. The extension principle states that fuzzy problem universe consisted of the Cartesian product of the fuzzy decision variables’ universes. Therefore, the objective function f is determined by the power sets of this Cartesian fuzzy input space. The resulting objective function is a fuzzy set of the solution defined as shown in Equation (4).

$f = P(X_1 × X_2×… ×X_n)  P(Y)$(4)

The membership degree of the fuzzy set of solutions is determined by Equation (5).

$μ(y) =\gcd_{y=f(x_1,x_2,…x_n}^{Max} {min⁡[μ(x_1),μ(x_2),…,μ(x_n )]}$(5)

where $X1, X2, …, Xn$ are the fuzzy decision variables.
Pseudocode 1. FCOA
Input: n crows N, flight lengthFLdecision variables d, awareness probability AP and MaxIt;
Output: Return the best fitness and membership degree.
1: Start
2: Crows=init (d, N)
3: Memories → Initialize_Memories(d,N)
4: [F, M]→Evaluate_fitness(Crows)
5: For (Iteration : Max-Iteration) do
6: If (R≥AP)
$X_{i,iter+1}=X_{i,iter}+ FL*F_{rand}*(mem_{j,iter}- X_{i,iter})$   $Else$   $X_{i,iter+1}=l-(l-u)*F_{rand}$
7: Check variables feasibility within boundaries
8: [F, M]→Evaluate_fitness($Crows$)
9: Update crows' memories

10: Iteration = Iteration+1.
11: End for
12: Sol→Defuzz ([F, M])
13: Select Best_Sol
14: Return best_Sol, membership and best crow position
End

Functions
Function Evaluate_fitness(Crows)
Inputs: Crows
Output: f, μ
1: $f = P(X_1 × X_2 ×… ×X_n) → P(Y)$
2: $μ(y) =_{y=f(x_1,x_2,…x_n}^{Max} {min⁡[μ(x_1 ),μ(x_2 ),…,μ(x_n )]}$
3: Return $f, μ$
End function

Function Defuzz ([f, M])
Input $f$ = fitmess, $M$ = membership
Output Sol
1: $Sol=\frac {∫μ_c(z)∙zdz}{∫μ_c(z) dz}$
2: Return Sol
End function
The main differences between the FCOA and the initial CSA are the steps of crow initializations and the fitness function evaluation.
• 1.The CSA initializes the crows by scalar value, whereas the FCOA utilizes fuzzy numbers in the crows for the problem solutions.
• 2.The CSA calculates regular fitness function and does not handle constraints’ evaluations in the algorithm.
• 3.The FCOA benefits the Zadeh extension principle described in Section 2.2 to calculate the fuzzified fitness functions and constraints using the crows’ fuzzy numbers. The fit crow is picked upon the best fitness, constraints satisfaction, and membership degrees.
A scrutiny observation will clarify the benefits of the FCOA as follows:
• FCOA can efficiently overcome the stagnation of the traditional COA by fine tuning control variables within their operating range.
• It can solve effectively design engineering optimization problems.
• It emphasizes non-dominated solutions by introducing fuzziness variables which enhance the obtained solutions.
• Fig. 3 and Pseudocode1 present the steps of the FCOA. The flow chart begins by initializing the FCOA parameters (flight length and the awareness probability) and the initial fuzzy-crow population. The number of fuzzy decision variables d and the crows number N formulate the problem search space as shown in Equation (6). Each fuzzy number is represented by a combination of value x_i and membership degreeμ_i.

(6)

Each crow has a fuzzy memory M with membership degree of μ which initially set to its initial position (Equation 7) and updated according to the crow with the best fuzzy fitness through the algorithm.

(7)

Fig. 3. Flowchart of FCOA.

From the previous example, we can calculate the space complexity of the proposed FCOA in terms of two factors which are the number of crows $N$ and number of fuzzy decision variables $d$ as $O(N+d)$, whereas the time complexity is $O(Nd)$ which is considered slightly high. However, we accepted it as our first aim is to provide optimal solution rather than a suboptimal solution with a lower complexity.
The crows’ memories are updated by Equation (8).

(8)

The Zadeh extension principle is used to calculate the crows’ fuzzy fitness function using Equations (4) and (5) according to each problem definition.

(9)

At the end of the iteration, the best solution and membership degree for the problem is selected. The overall solution is the best in the intersection of all solution sets.

Experimental Results

In this section, the validity of the proposed framework has been proved for its capability by using optimization problems. These problems are diverse in terms of decision variables and constraints. Therefore, they can efficiently evaluate new searching algorithms’ performance. These functions were tested after implementing the algorithm in the MATLAB environment. The algorithm is coded in MATLAB R2019b, which has an Intel Core I7 (1.8GHz) processor and 16 GB RAM memory.
Redefining the existing design problems using fuzzy decision variables would be feasible to test the FCOA algorithm. Famous constrained design problems are reformulated to be a fuzzy representation. The fuzzy formulation of the problems is designed through replacing traditional variables with fuzzy ones along with the corresponding membership function. Thex ̃ notation represents the corresponding fuzzy variable of x.
In experiments, the FCOA stands for the same number of crows’ N and max iteration number like CSA [28]. Therefore, the time and space consumption is close for both FCOA and CSA algorithms. The FCOA parameters like N, iterations, flight length FL and AP are represented in Table 1. The main comparison issues will be the worst, best, average, and standard deviation of all objective function values produced by the FCOA and other algorithms over 50 runs.

Table 1. FCOA parameters

Design problem N Iterations FL AP
Unconstrained test functions 50 500 2 0.1
Constrained engineering problems
Pressure vessel 50 5000 2 0.1
Tension/compression spring 50 1000 2 0.1
Welded beam 20 2000 2 0.1
Three-bar truss 50 500 2 0.1
Gear train 20 500 2 0.1
Data from [28].

Unconstrained Benchmark Functions
The proposed FCOA effectiveness and performance are evaluated using on 10 benchmark test functions for 2, 10, and 30 control variables. These data of unconstrained benchmark functions are represented in Table 2. The control variables scheduling and their related statistical results of the proposed FCOA for 50 trials on one unimodal and two multimodal functions are shown in Table 3. The results on the benchmark functions demonstrate that the proposed FCOA is close to global and near global minima with acceptable solution quality. In addition, Table 4 gives comparisons of FCOA fitness average and standard deviation on 50 trails for dimensions of 30 control variables and 10,000 iterations with recent literature algorithms such as HGS[31], SMA[32], OBMSA [36], opposition-based learning Grey WolfOptimizer(OBLGWO)[57], monarch butterfly optimization (MBO)[33], HHO [32], GWO [34], modified CSA (m-SCA)[58], and EO [35].

Table 2. Defination of benchmark test functions
Function Optimum Dim Formulation Range
Uni-modal F1 0 2, 10 $f(x) = \displaystyle\sum_{i=1}^n x_i^2$ [-100, 100]
F2 0 30 $f(x) = \displaystyle\sum_{i=1}^n |x_i| + \prod_{i=1}^n |x_i|$ [-10, 10]
F3 0 30 $f(x) = \displaystyle\sum_{i=1}^n (\displaystyle\sum_{j-1}^i x_j)^2$ [-100, 100]
F4 0 30 $max_i {|x_i |,1≤i≤n}$ -
F5 0 30 $f(x)= \displaystyle\sum_{i=1}^{n-1}\lfloor 100(x_{i+1}-x_i^2 )^2+(x_i-1)^2 \rfloor$ [-30, 30]
F6 0 30 $\displaystyle\sum_{i=1}^n([x_i+0.5])^2$ [-100, 100]
F7 0 30 $\displaystyle\sum_{i-1}^n ix_i^4 +random[0,1)$ [-128, 128]
Multi-modal F8 0 30 $f(x)= \displaystyle\sum_{i=1}^n[x_i^2-10 cos⁡〖(2πx_i )+10〗 ]$ [5.12, 5.12]
F9 0 2, 10 $f(x) = -20exp \left(-0.2\sqrt{\frac{1}{n}\displaystyle\sum_{i=1}^n x_i^2}\right) -exp \left(\frac{1}{n}\displaystyle\sum_{i=1}^n cos 2πx_i\right) + 20 + e$ [-32, 32]n
F10 0 2, 10 $f(x) = \frac{1}{4000}\displaystyle\sum_{i=1}^n x_i^2 -$\displaystyle\prod_{i=1}^n cos($\frac{x_i}{\sqrt{i}}+1)$ [-600, 600]n

Table 3. Scheduling of decision variables and statistical analysis for the three unconstrained tested benchmarking
Dimension Benchmarking test functions F1 F9 F10
n=2 x1 2.60E-12 1.31E-16 9.60E-09
x2 -4.46E-13 -3.21E-16 -4.64E-09
fitness function 6.95E-24 0 0
Best value 6.95E-24 0 0
Mean value 2.07E-21 2.18E-14 0.004044135
Worst value 1.57E-20 6.71E-14 0.190074367
STD 3.47E-21 1.74E-14 0.027725196
Mean number of evaluation 2000 2000 6000
n=10 x1 -1.17E-07 -7.02E-15 -9.29E-09
x2 -6.57E-09 3.51E-15 1.29E-08
x3 -1.85E-08 4.89E-16 1.19E-08
x4 3.28E-10 1.68E-15 1.44E-08
x5 -1.10E-07 1.62E-16 6.78E-09
x6 5.02E-08 -3.19E-15 -1.06E-08
x7 4.32E-09 -2.06E-15 -1.48E-08
x8 -4.55E-08 1.28E-15 -1.04E-09
x9 3.53E-08 -2.37E-15 5.88E-09
x10 -1.06E-07 -1.44E-14 -2.56E-08
fitness function 4.31E-14 2.04E-14 0
Best value 4.31E-14 2.04E-14 0
Mean value 1.37E-12 0.041729038 0.013028839
Worst value 6.38E-12 0.271359008 0.110783258
STD 1.42E-12 0.072123277 0.03562264
Mean number of evaluations 5500 6000 6000
Table 4. Comparisons of average and standard deviation of fitness functions on 50 trials for FCOA and literature algorithms
 Function Metric m-SCA GWO EO HGS SMA MBO HHO OBMSA OBLGWO FCOA F2 Avg. 9.11E-04 7.18E-17 6.23E-24 6.00E-168 2.51E-140 1.42E+09 1.83E-31 5.22E-194 5.29E-48 7.11E-15 SD 1.90E-03 0.029014 8.69E-24 0.00E+00 1.40E-139 5.38E+09 4.29E-31 0.00E+00 1.17E-47 5.79E-15 F3 Avg. 8.48E+02 3.29E-06 9.60E-09 1.20E-167 0.00E+00 4.80E+04 7.78E-49 5.02E-276 4.41E-21 4.18E-39 SD 5.49E+02 79.14958 4.21E-08 0.00E+00 0.00E+00 3.94E+04 4.33E-48 0.00E+00 2.32E-20 3.31E-37 F4 Avg. 7.07E-01 5.61E-07 2.19E-10 2.50E-137 6.14E-159 3.10E+01 1.05E-31 1.14E-166 7.92E-22 7.65E-35 SD 5.37E-01 1.315088 2.83E-10 9.80E-137 3.14E-158 2.62E+01 2.62E-31 0.00E+00 1.22E-21 4.32E-35 F5 Avg. 29.5658 26.81258 2.54E+01 1.92E+01 6.38E+00 6.96E+07 1.79E-02 2.77E+01 2.63E+01 1.41E-03 SD 2.432 69.90499 2.65E-01 9.77E+01 11.55532 9.20E+07 0.036307 5.70E-01 5.59E-01 0.019449 F6 Avg. 1.24E+00 0.816579 6.72E-06 7.78E-07 0.006317 2.71E+04 0.000152 1.83E-02 4.85E-05 1.00E-32 SD 5.12E-01 0.000126 5.88E-06 1.17E-06 0.003635 2.63E+04 0.000273 9.04E-03 2.03E-06 1.03E-30 F7 Avg. 1.95E-02 0.002213 5.95E-73 3.43E-04 0.000196 1.28E+02 0.000212 1.90E-04 2.97E-04 1.24E-03 SD 6.87E-03 0.100286 2.09E-72 4.66E-04 0.000189 1.07E+02 2.86E-04 1.40E-04 2.70E-04 1.30E-03 F8 Avg. 7.81E+01 0.310521 3.32E-02 0.00E+00 0.00E+00 2.06E+02 0.00E+00 0.00E+00 0.00E+00 0.00E+00 SD 5.21E+01 47.35612 1.82E-01 0.00E+00 0.00E+00 1.60E+02 0.00E+00 0.00E+00 0.00E+00 0.00E+00

Constrained Benchmark Functions
Pressure vessel design problem
The schematic of the pressure vessel design problem is clarified in Fig. 4 and the corresponding fuzzy mathematical representation is formulated as follows:

$Min.f(\tilde x)=0.6224 \tilde x_1 \tilde x_3 \tilde x_4 + 1.7781 \tilde x_2 + 3.1661 \tilde x_1^2 \tilde x_4 + 19.84 \tilde x_1^2 \tilde x_3$(10)

$g_1(\tilde x) = -(\tilde x_1) + 0.0193 \tilde x_3 ≤ 0$(11)

$g_2(\tilde x) = -(\tilde x_2) + 0.00954 \tilde x_3 ≤ 0$(12)

$g_3(\tilde x) = - π \tilde x_3^2 \tilde x_4 - 4/3 π \tilde x_3^3 + 1,296,000 ≤ 0$(13)

$g_4(\tilde x) = (\tilde x_4) - 240 ≤ 0$
$0 ≤ \tilde x_i ≤ 100 \mkern18mu \mkern18mu i=1,2$
$10 ≤ \tilde x_i ≤ 200 \mkern18mu ,\mkern18mu i=3,4$(14)

Fig. 4. Schematic of the pressure vessel design problem.

Table 5. FCOA best solution parameters values for pressure vessel problem
Parameter Value Parameter Value
$\tilde x_1$ 0.786 G1 0.037207
$\tilde x_2$ 0.5 G2 -0.09309
$\tilde x_3$ 42.6532 G3 -20.1206
$\tilde x_4$ 170 G4 -70
$f$ 6020.0258 Time 88.25578

Fig. 5. FCOA Convergence rate through finding pressure vessel problem best solution.

Table 5 shows the best solution found by the FCOA for the pressure vessel design problem and Fig. 5 represents the FCOA convergence rate from which we can figure that FCOA quickly reaches its best solution. Then, Fig. 6 clarifies a comparison between the FCOA, CSA [28, 29] and other algorithms in the field of optimizing design problems. These algorithms are GA3 [59], PSO-DE [60], elephant herding optimization (EHO) [61], DE[62], Gaussian quantum-behaved particle swarm optimization (G-QPSO)[63], QPSO [63], Moth-flame optimization (MFO) [64], WOA[65], grasshopper optimization algorithm (GOA) [66], HHO [67], artificial bee colony algorithm (ABC) [68], and Lévy flight distribution (LFD) [69]. The objective function is competing with the other algorithms. Therefore, the FCOA algorithm is optimal.

Fig. 6. Statistical indices for pressure vessel design problem (50 run) using different optimization algorithms

2Tension/compression spring design problem
Fig. 7 shows the tension/compression spring design problem schematic. The objective function and constraints of the fuzzy representation of the tension/compression spring problem is formulated as follows:

$Min.f(\tilde x)=((\tilde x_3) + 2) \tilde x_2 \tilde x_1^2$(15)

$S.t.$

$g_1 (\tilde x) = 1 - \frac{\tilde x_2^3 \tilde x_3}{71,785 \tilde x_1^4} ≤0$(16)

$g_2 (\tilde x) = \frac{4 \tilde x_2^2 - \tilde x_1 \tilde x_2}{12,566(\tilde x_2 \tilde x_1^3 - \tilde x_1^4} + \frac{1}{5108 \tilde x_1^2} ≤ 0$(17)

$g_3 (\tilde x) = 1-\frac{140.45 \tilde x_1}{\tilde x_2^2 \tilde x_3} ≤0$
$0.05 ≤ \tilde x_1 ≤ 2 \mkern18mu 0.25 ≤ \tilde x_2 ≤ 1.3 \mkern18mu 2 ≤ \tilde x_3 ≤ 15$ (18)

Fig. 7. Schematic diagram of tension/compression spring design problem.

Table 6 represents the optimal objective function of the tension/compression spring design problem. All constraints are satisfied, and the control variables are provided. The convergence rate in Fig. 8 proves the profession of the FCOA as it quickly reaches the best solution, and Fig. 9 represents the experiments’ comparisons. The algorithms used in comparisons are EHO [61], DE[62], MFO [64], WOA[65], GOA [66], HHO [67], LFD [69], and UPSO [70] methods, and the FCOA competes with these algorithms accordingly.

Table 6. FCOA best solution for tension/compression spring design problem
Parameter Value Parameter Value
$\tilde x_1$ 0.050001 G1 -0.00016
$\tilde x_2$ 0.31717 G2 -0.00071
$\tilde x_3$ 14.06501 G3 -3.96336
$f$ 0.012739 Time 14.27717

Fig. 8. FCOA Convergence rate through finding tension/compression spring design problem best solution.

Fig. 9. Statistical indices for tension/compression spring design problem (50 run) using different optimization algorithms.

Welded beam design problem Fig. 10 shows the schematic of welded beam design problem, and the objective function and constraints of the fuzzy representation of welded beam problem is formulated as follows:

$Min.f(\tilde x) = 1.10471\tilde ω^2 \tilde L + (0.04811\tilde d \tilde h(14 + \tilde L)$(19)

$S.t.$

(20)

where

(21)

Fig. 10. Welded beam design problem Schematic diagram.

Table 7. FCOA best solution for Welded beam design problem
Parameter Value Parameter Value
$\tilde x_1$ 0.205384 G1 -67.4507
$\tilde x_2$ 3.463224 G2 -572.226
$\tilde x_3$ 9.13174 G3 0
$\tilde x_4$ 0.205384 G4 -3.41986
$f$ 1.737105 G5 -0.08038
Time 48.5 G6 -0.23596
- - G7 -10.8717

Fig. 11. FCOA Convergence rate through finding Welded beam design problem best solution.

Table 7 demonstrates the minimal objective function along with the control variables values and the constraints values of the welded beam design problem resulting after 50 runs of the FCOA. The time column represents the average time of the algorithm producing the problem solutions. Then Fig. 11 clarifies the FCOA convergence rate for finding the best solution of the Welded beam design problem. The proposed FCOA competed with other optimization algorithms as shown in Fig. 12, such as EHO [61], DE [62], MFO [64], WOA [65], GOA [66], HHO [67], LFD [69], and CSA [28] methods. Therefore, the FCOA gives feasible optimal solution satisfying these constraints.

Fig. 12. Statistical indices for tension/compression spring design problem after 50 run using different optimization algorithms.

The three-bar truss design problem
Fig. 13 clarifies the schematic of the three-bar truss design problem, and the objective function and constraints of the three-bar truss problem fuzzy representation is formulated as follows:

$Min.f(\tilde x) = (2\sqrt 2 \tilde x_1 + \tilde x_2) × l$(22)

$S.t.$

$g_1 (\tilde x) = \frac{\sqrt 2 \tilde x_1 + \tilde x_2}{\sqrt 2 \tilde x_1^2 + 2 \tilde x_1 \tilde x_2} P-σ ≤ 0$(23)

$g_2 (\tilde x) = \frac{x_2}{\sqrt 2 \tilde x_1^2 + 2(\tilde x_1 \tilde x_2} P-σ ≤ 0$(24)

$g_3 (\tilde x) = \frac{1}{\sqrt 2 \tilde x_2 + \tilde x_1} P-σ ≤ 0$
$0 ≤ \tilde x_i ≤ 1 , i=1,2$
$l=100 cm,P=2 kN⁄cm^2 , σ=2 kN⁄cm^2$ (25)

Fig. 13. Three-bar truss design problem diagram.

Table 8 represents the objective function value of the three-bar problem along with the control variables values and the problem constraints, whereas Fig. 14 shows the convergence rate of the objective function using the FCOA from which we can see that the FCOA has converged as expected. Fig. 15 clarifies the statistical indices for three bar problem after 50 runs using different optimizations, in which the FCOA competes with CSA [28], PSO-DE [60], CS[71], MBA [59], and IGWO[72]. The comparison is based on the mean, standard deviation, and best objective functions. These experiments show that the FCOA produces the minimal objective function, and all constraints are satisfied accordingly.

Table 8. The best solution by FCOA for three bar problem
Parameter Value Parameter Value
$\tilde x_1$ 0.707839 G1 -0.00207
$\tilde x_2$ 0.220441 G2 -1.56804
$f$ 222.2514 G3 -0.03843
Time 1.214395 - -

Fig. 14. Convergence rate of FCOA for three bar problem.

Fig. 15. Statistical indices for three-bar problem after 50 run using different optimization algorithms.

Gear train design problem
Fig. 16 shows the gear train design problem, and the objective function and constraints of the fuzzy representation of gear train problem is formulated as follow:

$Min.f(\tilde x)=\left[\left(\frac{1}{6.931}\right)-\left(\frac{\tilde x_3 \tilde x_2}{\tilde x_1 \tilde x_4}\right)\right]^2$(26)

$S.t.$

$12≤(\tilde x_i) ≤ 60$

Fig. 16. Schematic of gear train problem.

Table 9 represents the gear train problem optimal solution, control variables, and time consumption, whereas Fig. 17 clarifies the convergence rate of the gear train solution by the FCOA, in which FCOA finds the minimal solution in an uncertain search space. While Fig. 18 introduces the statistical indices for gear train design problem after 50 runs using different optimization algorithms such as CSA[28], MBA [59], CS [71], ABC [68], IAPSO [73], PSO-GA [74], GA[75], and DE [75].

Table 9. The best solution by FCOA for three bar problem
Parameter Value Parameter Value
$\tilde x_1$ 48.70103 $\tilde x_4$ 54.74296
$\tilde x_2$ 20.15216 $f$ 1.83E-15
$\tilde x_3$ 19.0875 Time 8.68098

Fig. 17. Convergence rate of FCOA for finding best solution of gear train problem.

Fig. 18. Statistical indices for gear train design problem after 50 run using different optimization algorithms.

Discussion

This section presents a deep discussion on the obtained results in this paper. The validity of the proposed FCOA has been presented in Section 4 for five constrained and 10 unconstrained test functions. The considered problems are diverse in terms of decision variables and constraints. The statistical analyses that were provided prove the high robustness in terms of four indices specified as worst, mean, standard deviation, and best. For all these indices, the FCOA has the lowest level compared with other competitive optimization algorithms as represented in Figs. 6, 9, 12, 15, and 18. At first records, Fig. 6 solves the problem that was referred to as the pressure vessel design problem for 50 runs. The FCOA improves the solution this problem by reducing the solution from 6,020 in the worst fitness function achieved by 16,700 obtained with GOA. The improvement is approximately 36%.
In Fig. 9, the FCOA enhances the solution of the tension compression design problem by reducing the solution from 0.015 in the worst fitness function achieved by EHO. The improvement is approximately 16%, and the statistical indices are shown in comparison with UPSO, DE, MEO, WOA, HHO, and LED. In Fig. 12, the FCOA is applied to solve the welded beam design problem. Its fitness function equals 1.737105 which is the lowest values compared with EHO, DE, MFO, WOA, GOA, HHO, LFD, and CSA. The EHO has the worst fitness function that is increased more than the proposed FCOA by 37.87%. In Fig. 15, the FCOA reaches the best objective function of 222.2514 for the three-bar truss deign problem compared with the PSO-DE, CS, MBA, IGWA, and CSA which gives 264, achieving an improvement of 16%. Finally in Fig. 18, the FCOA achieves the best objective function of 1.82931E-15 for the gear train design problem compared with the FCOA with ABC, MBA, CS, CSA, IAPSO, PSO-GA, GA, and DE. Fast convergence rates are noticed for all studied cases as shown in Figs. 5, 8, 11, 14, and 17.
Based on the obtained simulation results, the FCOA has more robust solutions for all tested design problems compared with CSA as shown in the comparison Figs. 6, 9, 12, 15, and 18. With the FCOA, the solution is fine-tuned to avoid being trapped in local optima and achieves high explorative capabilities and fast convergence rates for the tested engineering design problems.

Conclusion

This paper introduces a novel crow search optimization framework incorporated with fuzzy logic for solving uncertain and ambiguous situations that real engineering design optimization problems exhibit. In this proposed algorithm, the Zadeh extension principle is used for computing fitness functions and constraints membership degrees for the CSA. Ten unconstrained benchmarking test functions and five constrained engineering design problems (pressure vessel, tension/compression spring, welded beam, three-bar truss, and gear train) were used to evaluate FCOA performance and effectiveness by finding global optimal solutions for these problems.
Statistical and convergence analysis results indicated highly promising & competitive performance and effectiveness of the FCOA compared to the state of art, as well as its ability to fine-tune control variables while satisfying all constraints under consideration deviating from violation boundaries and its closeness to global and near global minima with acceptable solution quality. Although for that we still need to perform some enhancement on the FCOA to especially improve its convergence rate in constrained design problems and to consider the related time and space complexity.
The main merits of the proposed algorithm could be summarized as follows:

During this initialization, the dominated crows is replaced by the nearest non-dominated one.

The carried-out results verified the validity and advantages of the proposed FCOA.

An improvement in the fitness function is achieved by the FCOA in the range of 16%–36% for the tested engineering design problem.

Acknowledgements

Not applicable.

Author’s Contributions

Conceptualization, SS, MGG. Investigation and methodology, SS,MGG, RAE. Project administration, RAE. Resources, MGG. Supervision, SS. Writing of the review and editing, SS, MGG. Software, MGG, SS. Validation, RAE. All the authors have proofread the final version.

Funding

None.

Competing Interests

The authors declare that they have no competing interests.

Author Biography

Name : Dr. Mona Gamal Gaffer
Affiliation :
Department of Computer Science, College of Science and Humanities in Al-Sulail, Prince Sattam bin Abdulaziz University, Kharj, Saudi Arabia.
Machine Learning and Information Retrieval Department, Artificial Intelligence, Kafrelsheikh University, Egypt.
Biography : MONA G. GAFAR received the B.Sc., M.Sc., and Ph.D. degrees from the Faculty of Computers and Information, Mansoura University, in 2006, 2010, and 2014, respectively. She is currently an Assistant Professor with the Department of Computer Science, College of Science and Humanities in Al-Sulail, Prince Sattam bin Abdulaziz University, Saudi Arabia. In addition, she is an associate professor with the Department of Machine Learning and Information Retrieval Department, Faculty of Artificial Intelligence, Kafrelsheikh University, Egypt. Her research interests involve artificial intelligence, data mining, software engineering and applications of modern optimization techniques

Name : Prof. Ragab A. El-Sehiemy
Affiliation : Electrical Engineering Department-Faculty of Engineering, Kafrelsheikh University, Egypt
Biography : Dr. Ragab A. El-Sehiemy received B.Sc. degree in electrical engineering, and M.Sc. and Ph.D. degrees in Electric Power Systems Engineering from Menoufia University, Egypt, in 1996, 2005, and 2008, respectively. He is currently a Professor with the Electrical Engineering Department, Faculty of Engineering, Kafrelsheikh University, Egypt. Dr. El Sehiemy has been Awarded Prof. Mohamed Mahmoud Khalifa Award in Electrical Power Engineering in 2016 from Academy of Research and Technology, Egypt. Dr. El Sehiemy research interests include power system optimization, operation and planning; smart grid and its applications; Renewable Energy, He is an Editor in the International Journal of Engineering Research in Africa.

Name : Dr. Shahenda Sarhan
Affiliation :
Faculty of Computers and Information Sciences, Mansoura University, Mansoura, Egypt Faculty of Computing and Information Technology, King Abdulaziz University, Jeddah, KSA
Biography : SHAHENDA SARHAN received the B.Sc., M.Sc., and Ph.D. degrees in computer sciences from Mansoura University, Egypt. Since 2012, she has been an Associate Professor in computer science with the Faculty of Computers and Information, Mansoura University. She is currently an Associate Professor with the Faculty of Computing and Information Technology, King Abdulaziz University. Her research interests include artificial intelligence and computer networks

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Mona Gamal Gafar1,2, Ragab A. El-Sehiemy3, and Shahenda Sarhan4,*, A Hybrid Fuzzy-Crow Optimizer for Unconstrained and Constrained Engineering Design Problems, Article number: 12:14 (2022) Cite this article 1 Accesses